Thèse De

Doctorat De

Sorbonne Université

Johannes Braathen

M Eli

Ben-Haim Examinateur

M Fawzi

Boudjema Rapporteur

M José

Ramon Espinosa

Rapporteur M Mark

Margarete Mühlleitner

Examinateur M Pietro Slavich

Terje Onkel

Nygaard

Luc Darmé

Frédéric Dreyer

Georgy Feher

Hugo Ricateau

Matthieu Sarkis

Oscar De Felice

Charles Cosnier-Horeau

Thomas Dupic

Chrysoula Markou

Alessandro Tartaglia

Constantin Babenko

Yifan Chen

Sophie Williamson

Kang Yang

Ruben Oncala Mesa

Yoan Gautier

Gaetan Laorgue

Enrico Russo

Giovanni Stagnitto

Pauline, Rémy, Loic Manon Louis

Cédric Hugo A Biljana

Le Bon

Automating Higgs precision calculations

I must thank members of LPTHE, and in particular of the Particle Physics and Cosmology group, for

support, and contributions throughout past years up until today.

First and foremost, I must thank my Ph.D. supervisors, Mark Goodsell and Pietro Slavich. They have always been patient and helpful in answering my numerous questions and correcting my many mistakes, and have guided me towards becoming a true scientist. I thank Mark for being, ever since I was his Master 2 intern, a positive force driving me forward in tackling new notions and calculations, and for encouraging and supporting me in all the projects I undertook during my thesis. I thank Pietro for his door always being open, for the time he took to answer in great detail all my questions both about Physics and administrative issues and for all his input and advice on my work. I have learned enormously from them, and under their guidance I have found avenues of research that I would like to pursue further in the future. They both helped me greatly in nding my post-doctoral position, and I am also deeply grateful to them for that. It has been an extraordinary opportunity as well as a true honour to be their student.

I am very grateful to the reviewers of my thesis, Fawzi Boudjema and José Ramon Espinosa, for the time they spent reading my manuscript in great detail and their very helpful comments on how to improve it. I would also like to thank members of the jury, Eli Ben-Haim and Margarete Mühlleitner, for participating in my thesis defense. I am deeply thankful towards Adam Falkowski who took the time to be my scientic tutor and helped me in my search for a post-doctoral position, and Soan Teber for his great kindness both as a teacher when I was a Master student and as my godfather in LPTHE.

It has been a great pleasure to collaborate during my thesis with Manuel Krauss, Toby Opferkuch, and Florian Staub, and I would like to thank them in particular for the time they took to answer my questions, and for the fruitful results of our work together. I would also like to thank Shinya Kanemura and Kodai Sakurai for our ongoing collaboration, which I very much look forward to continuing and developing in coming years.

I have had the pleasure and chance to meet many researchers in the course of my Ph.D. studies and I would like to thank especially Tomohiro Abe, Ben Allanach, Peter Athron, Emanuele Bagnaschi, Henning Bahl, Ulrich Ellwanger, Martin Gabelman, Thomas Hahn, Katsuya Hashino, Sven Heinemeyer, Yann Mambrini, Kentarou Mawatari, Kinya Oda, Werner Porod, Dominik Stöckinger, Alexander Voigt, and Georg Weiglein for many interesting and enriching discussions. I am also very grateful to Professors Hamaguchi, Hikasa, Hisano, Kanemura, Kobayashi, Murayama, and Shindou for the time they gave me when I toured Japan and visited their respective Universities and Institutes to give seminars in early summer 2017.

iii I was a young student; as well as Matthieu, Hugo, Constantin, Thomas, Yoan, and Sophie for all the time they spent answering my (often stupid) questions about both IT and Physics, and above all for the warm and friendly exchanges and interactions at and after work; and nally Luc, Yifan, Sophie, and Ruben for good memories from travelling to conferences. I very gratefully acknowledge nancial support from Fondation CFM pour la Recherche and Institut Lagrange de Paris, which allowed me to carry out my research in excellent conditions and enabled me to participate in numerous conferences and workshops. I would also like to thank Nathalie Bilimo for all the help she provided during my thesis, and in particular to establish my work contract.

It has been a great pleasure for me to teach for three years in the course 2P021: Électromagnétisme et électrocinétique, and I would like to thank Philippe Thomen, Laurent Coolen, and Michel Fioc who were in charge of the course during that time, as well as Laurent Teixeira for all his help with lab classes, and my friends Cédric Enesa and Andrea Mogini for our close collaboration on tutorials, initiated and organised at our many apéro evenings. I also thank my colleagues from the course 3P024: Projet en autonomie in which I have taught this last year.

Studying at Université Pierre et Marie Curie, now Sorbonne Université, for almost eight years more than one third of my life has been an extraordinary opportunity, and I have had the chance to meet many exceptionally talented and inspiring professors there. I would like to express all my gratitude to Eli Ben-Haim, Philippe Depondt, Thierry Hocquet, Édouard Kierlik, Émily Lamour, Vincent Minerbe, Jean-Michel Raymond, A. Marco Saitta, Jean-Paul Tavernet, Soan Teber, Jérome Tignion, Philippe Thomen, Matthieu Tissier, Maxime Wolfe, and Jean-Bernard Zuber.

I must also say how indebted I am to Danièle Guiron, Marie Levard, and Alexandre Guesnon for all they did for me during my earlier studies, and to Nicole Boret who rst taught me scientic rigor and gave me the love of Physics. Merci inniment, je vous serais toujours profondément reconnaissant.

I have had the chance of having many wonderful friends, who have supported through all my high-school and University studies. Most sincere thanks go to all my friends, Automating Higgs precision calculations (2HDMs) and the Georgi-Machacek model. In particular, we investigate the dangers involved with the common habit of trading Lagrangian parameters for masses using tree-level relations in non-supersymmetric models, and we show that loop corrections grow out of control well before the naive perturbativity bounds on couplings are reached.

Finally, in a last chapter, we consider the high-scale behaviour of (non-supersymmetric) models with extended Higgs sectors. More specically, we point out that the order at which couplings are extracted from the physical spectrum before being run with renormalisation group equations has strong eects on the values found at high energy scales, and thus on the high energy properties of the models. We illustrate this statement with both analytical and numerical results where the impact of the matching order is sizeable, in the context of three minimal extensions of the Standard Model: a singlet extension, a model with vector-like quarks, and nally a 2HDM. viii Automating

Higgs precision calculations

Résumé L'étude des propriétés du boson de Higgs, découvert au Grand Collisionneur de Hadrons du CERN, représente une excellente opportunité pour la recherche de Nouvelle Physique. En particulier, la masse de ce boson est maintenant mesurée expérimentalement avec une précision impressionnante, de l'ordre de 0.1%, tandis qu'elle est également prédite par certains modèles au-delà du Modèle Standard, notamment les modèles supersymétriques. L'objectif de cette thèse est de faire avancer le calcul des corrections radiatives aux masses des bosons de Higgs dans les modèles au-delà du Modèle Standard, ainsi que l'automatisation de ces calculs, an d'établir ou d'améliorer les limites sur les couplages entre la Nouvelle Physique et le boson de Higgs. Un premier chapitre est consacré au calcul des corrections dominantes à deux boucles, de la forme O(α s α t ), aux masses des scalaires neutres dans les modèles supersymétriques à jauginos de Dirac. Nous considérons à la fois des termes de masses de Dirac et de Majorana pour les jauginos (plus précisement les gluinos), et nous établissons des résultats pour des paramètres du secteur des tops/stops à la fois dans les schémas de renormalisations DR et on-shell (sur couche de masse). Les résultats analytiques que nous obtenons sont inclus dans des routines numériques disponibles publiquement, et constituent les premiers calculs explicites de corrections aux masses des bosons de Higgs à deux boucles dans des modèles supersymétriques au-delà du NMSSM.

Le sujet principal de cette thèse est l'étude de la Catastrophe des Bosons de Goldstone, un cas de divergences infrarouges dues aux bosons de Goldstones de masses nulles qui aecte les calculs de potentiels eectifs, d'équations de minimisation du potentiel (dites équations tadpoles) et d'énergies propres, ainsi que de comment surmonter ce problème. Nous étendons tout d'abord la procédure de resommation, développée récement pour le Modèle Standard et appliquée au MSSM, pour obtenir des équations tadpoles nies pour des théories de champs renormalisables générales. Nous démontrons ensuite qu'en adoptant un schéma de renormalisation sur couche de masse pour les masses des bosons de Goldstone, l'obtention de ces résultats aurait été facilitée, et qu'il est aussi possible d'éliminer toutes les divergences infrarouges dans les calculs de masses de scalaires. Nous présentons une approximation du potentiel eectif généralisée permettant de trouver des expressions de corrections de masses libres de toute divergence infrarouge, ainsi qu'un développement perturbatif des paramètres de masses permettant de résoudre directement (plutôt qu'itérativement) les équations tadpoles.

Ensuite, nous illustrons la mise en ÷uvre numérique de notre solution à la Catastrophe ix Automating Higgs precision calculations des Bosons de Goldstone dans le programme SARAH. Nous comparons les résultats que nous obtenons pour le Modèle Standard aux résultats existants, et trouvons un excellent accord. Nous étudions aussi de nouvelles corrections aux masses des bosons de Higgs en Split SUSY, dans le NMSSM, dans des modèles à deux doublets de Higgs, et dans le modèle de Georgi-Machacek. Nous nous intéressons notamment aux dangers encourus du fait du choix, habituel pour des modèles non-supersymétriques, d'échanger les paramètres du Lagrangien contre des masses en utilisant des relations à l'ordre des arbres, et nous montrons comment les corrections quantiques croissent hors de tout contrôle bien avant que les limites perturbatives habituelles (naïves) ne soit atteintes.

Finalement, nous considérons dans un dernier chapitre le comportement aux hautes énergies de modèles non-supersymétriques avec des secteurs scalaires étendus. Plus spéciquement, nous montrons que l'ordre (en théorie des perturbations) auquel les couplages sont extraits du spectre de masse physique, avant d'être évolués avec les équations du groupe de renormalisation, a des eets importants sur les valeurs des couplages obtenues aux hautes énergies et par conséquent sur le comportement à ces énergies des modèles considérés. Nous illustrons cette position avec à la fois des ré- The discovery of a 125-GeV Higgs-boson-like particle at the Large Hadron Collider at CERN in 2012 has been a ground-breaking event for Particle Physics. Not only did it complete the eld content of the Standard Model (SM), but it conrmed the role of Electroweak Symmetry Breaking (EWSB) as the origin of the masses of fermions and gauge bosons. However, the quest to understand the microscopic behaviour of Nature does not end here; indeed the Standard Model does not provide a quantum description of gravity, so at best it can only be valid up to the Planck scale M Pl ∼ 10 19 GeV (at which eects from quantum gravity must be taken into account). Furthermore, a number of signs seem to point towards the existence of yet-unknown phenomena that would also be at play in electroweak (and strong) interactions. These include theoretical arguments such as the hierarchy problem, the question of the stability of the SM electroweak vacuum, the strong-CP problem, the possibility of gauge coupling unication, and the hierarchy between the Yukawa couplings. Additionally there also exist experimental results that are incompatible or conicting with SM predictions: for example the existence of neutrino masses; the cosmological observations indicating the need for ination, for dark energy, and for some mechanism to explain baryogenesis; the astrophysical evidence for dark matter; and tensions (some of them long-standing) coming from the measurement of precision observables, such as the muon anomalous moment, or more recently anomalies in B-physics. It is therefore very likely that the SM is only the manifestation at currently accessible energy scales of a more fundamental underlying theory or in other words that the SM is an eective theory valid until some intermediate scale (i.e. between the Planck and electroweak scales) characterising the New Physics.

A tremendous number of Beyond-the-Standard-Model (BSM) theories have been devised to address some (or even all) of the deciencies of the SM and must now be confronted with experimental results. Indeed, the search for BSM Physics has become one of the major endeavours of modern High-Energy Physics and, among the dierent possible avenues, one can distinguish between direct searches i.e. trying to detect new particles or phenomena directly at experiments or indirect searches, in other words investigating possible eects of unknown Physics at higher scales on the properties (often masses or couplings) of presently observed particles. So far, potential new particles have eluded observation. However, the nding of the Higgs boson has opened a new area for indirect searches through the study both experimental and theoretical of its properties. In particular, the Higgs boson mass m h is now already measured to an exceptional level of precision of the order of one per mille while it is also predicted in certain BSM models.

A special class of BSM theories that exhibit a new type of symmetry relating bosonic and fermionic degrees of freedom called Supersymmetry (SUSY) stands out because of several theoretical and phenomenological advantages they oer: indeed, at the price of greatly extending the eld content of the theory (of which the enriched Introduction Higgs sector is then of particular interest), they can solve the hierarchy problem, allow gauge unication, provide candidates for dark matter particles, etc to cite only a few of the positive features. Another feature of supersymmetric models, of central importance here, is the fact that because of the relations between parameters of the theory imposed by Supersymmetry, the quartic coupling of the Higgs boson, which controls its tree-level mass, is actually determined in terms of the other couplings of the theory (including the electroweak gauge couplings), unlike what is the case for instance in the SM. The physical mass of the Higgs boson can then be evaluated by adding, to the known tree-level value, quantum (or radiative) corrections that are computed in the regime of perturbation theory, and it is therefore a prediction of these models.

As the radiative corrections to the Higgs mass involve, in principle, all particles that couple to the Higgs, the comparison between the experimentally measured value and the theoretical prediction for m h can constrain a large part of (if not the whole of ) the physical spectrum of the theory. This situation, combined with the realisation in the early 1990's that the predicted Higgs mass could be compatible with early experimental bounds, has triggered an active and long-lasting interest in the calculation of corrections to the mass of the Higgs boson, in the context of minimal supersymmetric extensions of the SM i.e. the Minimal Supersymmetric Standard Model (MSSM), and the Next-to-Minimal Supersymmetric Standard Model (NMSSM). Similar computations have also been performed in the SM itself, especially in recent years, because they allow a precise extraction of the Higgs quartic coupling from its measured mass in order to investigate the fate of the electroweak vacuum.

However, the eld of Higgs mass calculations in BSM (SUSY) models is currently confronted with a number of challenges. Among them, one can rst mention the fact that the accuracy to which the lightest Higgs mass is computed is quite far from the precision with which it is measured experimentally: the theoretical uncertainty is at best of the order of a few GeV in the MSSM, which has been studied extensively, but is much larger for other models where fewer corrections are known. Another issue is that the minimal SUSY models that have been studied most (especially the MSSM) are becoming increasingly unnatural and therefore less attractive, thus increasing the need to improve predictions in numerous other extensions of the SM. The aim of this thesis is to participate in addressing these problems, and to increase the number of models in which precise results for Higgs masses are available. We have done so in two dierent but complementary approaches, working both on explicit calculations of new corrections in particular (non-minimal supersymmetric) models and on computations for general renormalisable theories.

We begin this thesis with two introductory chapters that aim to provide a (pedagogical) overview of the physical and technical setting in which the work presented here has been carried out. In the rst of these two chapters, we start by recalling basic notions of the Standard Model and its Higgs sector, before turning to the eect of quantum corrections on the SM Higgs sector. This allows us to introduce the Goldstone Boson Catastrophe, which will be one of the central topics in the thesis, as well as to illustrate the need for some BSM Physics that would couple to the Higgs. Afterwards, we present the dierent BSM models relevant for this thesis: on the one hand, the framework of Supersymmetry and phenomenological SUSY models; and on the other hand, minimal extensions of the SM with enlarged Higgs sectors. The second introduction chapter is dedicated to a review of precision calculations of Higgs boson masses. After discussing technical points, such as renormalisation schemes and the dierent possible approaches to computing mass corrections, we review existing results in BSM models in particular in supersymmetric ones and we end the chapter by presenting the formalism and the Introduction tools that we will employ for generic calculations.

The third chapter is devoted to a particular class of supersymmetric theories, namely models with Dirac gauginos. These models have been receiving growing interest in recent years because of their rich phenomenology, of which some of the most favourable features are: (i) the possibility to evade both current limits on the detection of squarks (via the suppression of the production rates of coloured particles) and constraints from avour physics; and (ii) the increased naturalness of these models due to an enhancement of the lightest Higgs mass at tree level and to the milder dependence of corrections to scalar masses on gaugino masses. We perform the rst explicit calculation of two-loop corrections to Higgs masses in theories beyond the NMSSM by deriving the leading O(α t α s ) contributions in general Dirac gaugino models, using the eective potential technique. We then apply our results to the special cases of the MDGSSM and the MRSSM, two of the most popular Dirac gaugino models, and we also provide approximate expressions in several phenomenologically relevant limits. Moreover, we obtain corrections in terms of top/stop sector parameters renormalised either in an on-shell (OS) scheme and in the DR scheme, and we observe how the OS scheme avoids the appearance of large logarithmic contributions leading to a loss of accuracy, or even of perturbativity, when gluinos become much heavier than stops. Finally, by varying the scheme of the top/stop sector inputs, and by changing the determination of the strong gauge coupling, we are able to estimate the theoretical uncertainty of our results.

In the following chapters, we turn to generic calculations i.e. for general renormalisable models and investigate more specically the Goldstone Boson Catastrophe and its solution. This catastrophe is a case of infra-red divergence that plagues calculations of eective potentials, tadpole equations, and even self-energies in the SM and most BSM models. It arises because of the masses of the Goldstone bosons, which may run across zero under renormalisation group ow. This problem is especially severe and visible in the Landau gauge, where Goldstones have a vanishing tree-level mass, but this is also the gauge in which many two-loop results for generic theories have been derived (in MS/DR renormalisation scheme), as it allows simpler calculations. Therefore, we wish to address the problem in the Landau gauge so as to obtain infra-red-safe versions of these existing expressions. This is the subject of the fourth chapter, where at rst we generalise a resummation procedure that had been recently devised and employed for the SM and the MSSM, before nding a new solution by using an OS renormalisation for the Goldstone boson masses in practice, we nd modied expressions for some of the loop functions that appear in tadpoles. Furthermore, for the self-energy diagrams, we combine our on-shell Goldstone prescription with a generalised eective potential approximation (which we also devise) to obtain easily calculable divergence-free expressions for the relevant loop functions. Lastly, we also provide a way to solve loop-corrected tadpole equations directly instead of iteratively.

In the fth chapter, we implement our modied loop functions in the public code SARAH, opening the way for two-loop evaluations of neutral scalar (Higgs) masses in non-supersymmetric models including the SM with SARAH generated SPheno spec-Introduction in existing calculations in SUSY. Afterwards, we consider more BSM models both supersymmetric (Split SUSY), and non-supersymmetric (Two-Higgs-Doublet Models and the Georgi-Machacek model) and investigate previously unknown two-loop corrections to scalar masses. In particular, we point out the dangers associated with the common habit of using only tree-level relations to extract Lagrangian couplings from the mass spectrum in non-SUSY BSM models.

Finally, in the last chapter, we continue to study the behaviour of non-supersymmetric models, focusing more precisely on how the order in perturbation theory at which the scalar quartic couplings are extracted at a low scale from the mass spectrum aects the high-scale behaviour of the models i.e. the possibility of an instability of the electroweak vacuum vacuum, or of a loss of perturbativity or unitarity. We consider three dierent non-SUSY extensions of the SM namely a singlet extension, an extension with vector-like quarks and a Two-Higgs-Doublet Model. We show that the threshold corrections to the quartic couplings at low scales can be signicant, and moreover can have important eects after running the couplings to high scales. We therefore advocate employing a loop-corrected matching together with two-loop RGEs, which is possible using the corrections available in SARAH/SPheno. For the reader's convenience, a list of abbreviations is provided on page xi in the preface, and the denitions of all the loop functions that are used can be found in appendix B.1.1. Let us mention here also that we shall, throughout this thesis, follow the Einstein convention on summation of repeated indices (unless otherwise mentioned), and that we will sometimes refer to mass-squared parameters simply as masses (thereby committing a slight, but common, abuse) when there is no risk of ambiguity.

This thesis is based on the following publications:

• [START_REF] Braathen | Leading two-loop corrections to the Higgs boson masses in SUSY models with Dirac gauginos[END_REF] Johannes Braathen, Mark corresponding to chapter 6.

Chapter 1

The Higgs boson and Physics beyond the Standard Model

This chapter aims at providing a brief overview of Physics in the Higgs sector, in the Standard Model as well as in the models of new Physics that will be studied in the course of this thesis.

The Standard Model and the Higgs sector

The Standard Model (SM) of Particle Physics is currently the most successful theory of the fundamental interactions of Nature, unifying the descriptions of the strong, electromagnetic, and weak forces and of matter, as an SU (3) C ×SU (2) L ×U (1) Y gauge theory. Matter elds are divided between quarks, which are charged under colour i.e. under SU (3) C , and leptons, which are not. The study of the dynamics of quarks and gluons, the gauge bosons of SU (3) C , is an immensely rich eld by itself Quantum Chromodynamics (QCD) that will not be considered in this thesis. Note, however, that coloured particles in particular top quarks (and in supersymmetric models their superpartners, the stops) will still have eects on Higgs Physics. The remaining SU (2) L × U (1) Y part constitutes the electroweak (EW) sector of the Standard Model, also referred to as the Glashow-Weinberg-Salam (GWS) model.

Mass terms of gauge bosons and fermions

The rst step in the theoretical construction of the Standard Model is to nd a way to accommodate in the framework of a gauge and therefore renormalisable theory the weak interaction, observed from the 1930's and described rst by the nonrenormalisable Fermi theory. Some weak interaction processes, such as the muon decay to an electron µ -→ e -ν µ νe involve avour changing charged currents and must therefore be mediated by charged gauge bosons W ± . A natural choice of gauge group from which charged gauge bosons could be obtained is SU [START_REF] Braathen | Avoiding the Goldstone Boson Catastrophe in general renormalisable eld theories at two loops[END_REF], but it cannot be considered by itself because it can be shown that the neutral gauge boson associated with the third generator of SU (2) is not the photon. An additional U (1) group, with the weak hypercharge Y as its associated quantum number, thus needs to be introduced so that one can recover the photon among the gauge bosons, to ensure that QED is contained within the SM. However, means that only fermions of left-handed helicities couple to the SU (2) gauge bosons and for this reason the SU (2) gauge group is denoted with a lower index L. In turn, this means fermions are chiral, i.e. transform dierently under SU (2) L × U (1) Y gauge transformations depending on their left-or right-handed helicity. However then, any mass term of a fermion ψ (lepton or quark), that could be written -m ψ ψψ = -m ψ ψL ψ R + ψR ψ L

(1. 1.3) would also be forbidden because it would break gauge invariance. It is also sometimes said that chiral symmetry protects fermion masses as it is the dierent behaviour of left-and right-handed helicities that makes mass terms non-gauge-invariant.

Electroweak Symmetry Breaking and the Brout-Englert-Higgs mechanism

The solution to these two problems relies on the mechanism invented in 1964 by Brout and Englert [START_REF] Englert | Broken Symmetry and the Mass of Gauge Vector Mesons[END_REF] and by Higgs [START_REF] Higgs | Broken symmetries, massless particles and gauge elds[END_REF][START_REF] Higgs | Broken Symmetries and the Masses of Gauge Bosons[END_REF], for which a complex scalar Φ, transforming in the (1, 2, 1 2 ) representation of the SM gauge group, is added to the theory together with a potential that allows the spontaneous breaking of electroweak gauge group

SU (2) L × U (1) Y SSB ---→ U (1) QED .
Let us now illustrate how spontaneous symmetry breaking occurs in the electroweak sector and how this yields masses for gauge bosons and fermions. First, the Higgs doublet Φ can be written

Φ = G + 1 √ 2 v + h + iG 0 (1.1.4)
1.1 The Standard Model and the Higgs sector with v the Higgs vacuum expectation value (VEV), h the physical Higgs state, and G ± and G 0 respectively the charged and neutral would-be Goldstone bosons. 1 For convenience, we use some of the freedom that we are given by the gauge symmetry to rotate away any phase of the Higgs VEV, thereby ensuring that v is real and nonnegative. The Higgs potential V (0) is taken 2 to be

V (0) (Φ) = µ 2 |Φ| 2 + λ|Φ| 4 , (1.1.5)
where µ 2 is the Higgs mass term and λ the Higgs quartic coupling. As the potential need to be bounded from below, the Higgs quartic coupling must be non-negative (actually, it must even be positive if µ 2 is negative). One can then easily nd the minimisation condition of the scalar potential, also called the tadpole equation, ∂V (0) ∂h φ=0 = 0 = (µ 2 + λv 2 )v .

(1. 1.6) where we use φ = 0 as a shorthand notation for {h = 0, G 0 = 0, G ± = 0}. If µ 2 (and λ) are positive, this equation admits only one solution v = 0 , and the electroweak gauge symmetry is not broken (this is the unbroken, or symmetric, phase of the SM). However, if µ 2 is negative, eq. (1.1.6) also admits the solution

µ 2 + λv 2 = 0 ⇒ v = -µ 2 λ > 0 . (1.1.7)
Furthermore, it is then straightforward to verify that the solution with v > 0 is then the true minimum of the potential, while the solution v = 0 corresponds to an unstable local maximum of the potential. The non-zero value of the Higgs VEV breaks the SU (2) L × U (1) Y : this is the broken phase of the SM. Because the VEV does not carry electric charge, a residual U (1) symmetry is preserved in the minimum of the potential, corresponding to the gauge symmetry of QED. The spontaneous breaking of the EW gauge symmetry SU (2) L × U (1) Y down to the gauge symmetry of QED U (1) QED , realised by the Higgs VEV, is called the electroweak symmetry breaking (EWSB). Finally, the Higgs VEV can be related to the measured Fermi constant by matching the electroweak sector of the SM onto the Fermi theory at low energies, and

one nds v = 1 √ 2G F 246.22 GeV .
(1.1.8)

When considering the gauge bosons W a µ and B µ , respectively of the groups SU (2) L and U (1) Y , we obtain for the covariant derivative of Φ

(D µ Φ) i = ∂ µ Φ i -igW a µ σ a ij 2 Φ j -iY Φ g B µ Φ i ,
(1.1.9)

1 As we will see in what follows, Goldstones do not remain part of the physical spectrum af- ter the electroweak symmetry breaking, therefore they should in principle always be referred to as would-be Goldstone bosons. However, there is almost never any confusion possible with real Goldstone bosons from a spontaneously broken global symmetry, so we will very often to them as Goldstone bosons, in a slightly abusive way. 2 The (0) upper index denotes a tree-level quantity this will become clearer in section 1.1.4.

The Higgs boson and Physics beyond the Standard Model where g (g) is the gauge coupling for U (1) Y (SU (2) L ), Y Φ = 1 2 is the Higgs doublet hypercharge, i, j are SU (2) L indices and σ a are the Pauli matrices

σ 1 = 0 1 1 0 , σ 2 = 0 -i i 0 , σ 3 = 1 0 0 -1 .
(1.1.10)

If we now expand the gauge-invariant kinematic term for the Higgs doublet

|D µ Φ| 2 ⊃ ∂ µ G + ∂ µ G -+ 1 2 (∂ µ h) 2 + 1 2 ∂ µ G 0 2 + 1 4 g 2 v 2 W 1 µ -iW 2 µ √ 2
W 1µ + iW 2µ √ 2 + 1 8 (g 2 + g 2 )v 2 gW 3 µ -g B µ g 2 + g 2 2 + L φφV + L φV V + L φφV V + • • • (1.1.11)
L φφV , L φV V , and L φφV V contain interaction terms between the scalars and the gauge bosons that we will not expand here. With the simple eld redenition

W ± µ ≡ W 1 µ ∓ iW 2 µ √ 2 , Z µ ≡ gW 3 µ -g B µ g 2 + g 2 ,
(1.1.12)

we can rewrite the second line of equation (1.1.11) as

|D µ Φ| 2 ⊃ 1 4 g 2 v 2 W + µ W -µ + 1 8 (g 2 + g 2 )v 2 Z µ Z µ .
(1. 1.13) Through the EWSB the would-be Goldstone bosons G ± and G 0 become longitudinal components of the gauge bosons W ± and Z, which then acquire masses m W = 1 2 gv and m Z = 1 2 g 2 + g 2 v respectively. 3 This is the Brout-Englert-Higgs mechanism [START_REF] Coleman | All Possible Symmetries of the S Matrix[END_REF]. One last vector boson state, orthogonal to Z µ in eld space A µ ≡ g W 3 µ + gB µ g 2 + g 2 ,

(1. 1.14) remains massless, and is the photon i.e. the gauge boson of U (1) QED . The transformation from {W 3 µ , B µ } to {Z µ , A µ } can be seen as a rotation of angle θ W , the Weinberg (or weak mixing) angle, dened by tan θ W = g g .

(1. 1.15) The Weinberg angle also relates the couplings g , g to the electric charge e of the electron e = g cos θ W = g sin θ W = gg g 2 + g 2 .

(1. 1.16) Fermions also obtain masses from the scalar sector, via Yukawa interactions that are added by hand in the SM. Quarks and leptons come in both left-handed and righthanded helicities, and it is known from experiments that W bosons only couple to left-handed fermions. Therefore, fermions of dierent helicities come in dierent representations of SU (2) L singlet for right-handed fermions and doublet for left-handed ones as shown in table 1.1.

1.1 The Standard Model and the Higgs sector

Q f L = (u f L , d f L ) (3, 2, 1 6 ) Quarks u f R (3, 1, 2 3 ) d f R (3, 1, - 1 3 )
Leptons

L f L = (ν f L , e f L ) (1, 2, - 1
2 ) e f R

(1, 1, -1)

Table 1.1 Fermions and their representations under the SM gauge group SU (3) C × SU (2) L × U (1) Y . f = 1, 2, 3 is the family index. Indices L and R denote respectively left-and right-handed states.

With these assignments, gauge-invariant Yukawa terms can be written for the fermions as

L ⊃ -Y f g e ( LfL ) i Φ i e gR -Y f g d ( QfL ) i Φ i d gR -Y f g u ij ( QfL ) i Φ † j u gR + h.c.
, (1.1.17) where ij is the antisymmetric tensor. Note that, by construction, the SM does not accommodate neutrino masses and that there is no right-handed neutrino. The observation of neutrino oscillations by the Super-Kamiokande collaboration is a clear experimental evidence that neutrinos (at least from two of the three families) are massive and therefore that there must exist some Physics beyond the SM in the EW sector.

However, it is possible to extend the SM in relatively minimal ways to generate small neutrino masses (via e.g. the seesaw mechanism), but this lies beyond the scope both of this introduction and of this thesis, and will not be discussed more here.

Electroweak gauge xing

Something we have not considered yet is how to quantise the SU (2) L × U (1) Y theory.

At this point, the choice of gauge needs to be taken into consideration because the functional integration of the path integral must be done while still obeying the gaugexing condition. This is possible when following the quantisation procedure of Faddeev and Popov [START_REF] Faddeev | Feynman Diagrams for the Yang-Mills Field[END_REF], which will introduce gauge-dependent terms in the Lagrangian.

As mentioned above, a rst possible choice of gauge the unitary gauge has the advantage of eliminating the Goldstones altogether from the Lagrangian. However, it causes at the same time great complications to calculations in perturbation theory, therefore we will use instead R ξ gauges, which will prove to be signicantly more convenient. Following the Faddeev-Popov procedure [START_REF] Faddeev | Feynman Diagrams for the Yang-Mills Field[END_REF], the gauge-xing terms for SU (2) L and U (1) Y are found to be

G ± SU (2) L = 1 √ ξ ∂ µ W ± µ ∓ iξ g 2 vG ± G 3 SU (2) L = 1 √ ξ ∂ µ W 3 µ -ξ g 2 vG 0 G U (1) Y = 1 √ ξ ∂ µ B µ + ξ g 2 vG 0 (1.1.18)
The Higgs boson and Physics beyond the Standard Model and then we add to the Lagrangian additional terms

L ⊃ -G + SU (2) L G - SU (2) L - 1 2 (G 3 SU (2) L ) 2 + (G U (1) Y ) 2 = - 1 ξ ∂ µ W + µ ∂ ν W - ν + 1 2 (∂ µ W 3 µ ) 2 + 1 2 (∂ µ B µ ) 2 - i 2 gv ∂ µ W + µ G --∂ µ W - µ G + + 1 2 vG 0 g∂ µ W 3 µ -g ∂ µ B µ -ξ g 2 4 v 2 G + G --ξ g 2 + g 2 8 v 2 (G 0 ) 2 (1.1.19)
The gauge xing (1.1.18) was actually chosen in such a way that the resulting terms in the third line of eq. (1.1. [START_REF] Martin | Two loop eective potential for a general renormalizable theory and softly broken supersymmetry[END_REF]) would exactly cancel terms like v∂ µ G + W - µ or v∂ µ G 0 Z µ that appear in the expansion of |D µ Φ| 2 . The terms in the fourth line of eq. (1.1.19) give gauge-dependent contributions to the Goldstone masses, and will be considered as part of the scalar potential.

The Faddeev-Popov procedure also requires the introduction of ghosts ω a , ωb and one can show that they obtain a Lagrangian

L ghosts = ωa -(∂ µ D µ ) ab -ξ g 2 4 σ a ij Φ j σ b ik Φ k -iξ g 2 8 vG 0 δ ab ω b (1.1.20)
We see from the second and third terms in the above equation that the ghosts couple to the scalar sector. It is then convenient to x ξ = 0 to avoid couplings between scalars and ghosts: this is the Landau gauge. Note that this does not prevent purely gauge couplings between ghosts and gauge bosons, coming from the (∂ µ D µ ) ab operator in the ghost Lagrangian. In the Landau gauge, the gauge boson propagator becomes [START_REF] Peskin | An Introduction to quantum eld theory[END_REF] ∆ µν ξ ab

= -i k 2 -m 2 g µν - k µ k ν k 2 -ξm 2 (1 -ξ) ab → ∆ µν ξ=0 ab = -i k 2 -m 2 ab g µν - k µ k ν k 2 .
(1. 1.21) Another popular gauge choice is the Feynman gauge ξ = 1, which makes the gauge boson propagator even simpler

∆ µν ξ=1 ab = -ig µν k 2 -m 2 ab . (1.1.22)
The Landau and Feynman gauges are probably the two most common choices in the context of perturbative calculations and throughout this thesis we will use these gauges exclusively. Unless otherwise specied, analytic expressions will be given in the Landau gauge although, on the other hand, numerical calculations in the SARAH/SPheno framework that we will present in chapter 2 are performed in Feynman gauge.

Finally, note that in the limit ξ → ∞, the Goldstone bosons decouple entirely from the theory and can be eliminated from the Lagrangian: this is the quantised realisation of the unitary gauge [START_REF] Peskin | An Introduction to quantum eld theory[END_REF].

1.1 The Standard Model and the Higgs sector 1.1. [START_REF] Braathen | N -loop running should be combined with N -loop matching[END_REF] The Higgs sector at tree-level and beyond

The tree-level scalar masses are dened as the second derivatives of the potential with respect to the scalar elds:

(m 2 h ) tree = ∂ 2 V (0) ∂h 2 φ=0 = µ 2 + 3λv 2 (m 2 G ) tree = ∂ 2 V (0) ∂(G 0 ) 2 φ=0 = µ 2 + λv 2 + ξm 2 Z = µ 2 + λv 2 (1.1.23) (m 2 G ± ) tree = ∂ 2 V (0) ∂G + ∂G - φ=0 = µ 2 + λv 2 + ξm 2 W = µ 2 + λv 2
where the ξ-dependent terms come from the part of the gauge-xing terms equation (1.1.19) that we have absorbed in the scalar potential V (0) (see the discussion below eq. (1.1.19)). The last equalities for (m 2 G ) tree and (m 2 G ± ) tree follow from our choice to work in the Landau gauge ξ = 0.4 One can notice that the Goldstone boson masses are are gauge dependent, however, this is not a problem because the Goldstones are not physical degrees of freedom, having been absorbed by the gauge bosons. Moreover, one can obtain the tree-level masses in the minimum of the potential by imposing the minimum condition (1.1.7) and these read (m 2 h ) tree = 2λv 2 , and (m 2 G ) tree = (m 2 G ± ) tree = 0 .

(1. 1.24) Until now our discussion has remained only at tree-level, i.e. we have not been taking into account quantum corrections. Now that we have quantised the theory, we do not expect physical observables to be equal to results of tree-level calculations, but of loop-corrected ones. The works presented in this thesis are done in the regime of perturbation theory, i.e. when there is a small parameter (a coupling) with which one can perform a series expansion of the quantum corrections, and physical quantities or observables can then be written as 1.25) where Θ (n) denotes the n-loop correction to Θ.

Θ obs = Θ (0) + 1 16π 2 Θ (1) + 1 (16π 2 ) 2 Θ (2) + • • • = ∞ n=0 1 (16π 2 ) n Θ (n) (1.
First, we can consider the quantum corrections to the scalar potential, which are contained in the eective potential, dened as follows (see e.g. [START_REF] Jackiw | Functional evaluation of the eective potential[END_REF]):

(i) for a theory with scalars {φ i }, one starts with the generating functional of the theory (analogous to a partition function), with a set of external sources J i Z({J i (x)}) = i Dφ i (x) exp i d 4 x L({φ i (x)}) + J i (x)φ i (x)

(1.1.26)

(ii) from Z, one can then dene the generating functional of connected graphs W W ({J i (x)}) = -i log Z({J i (x)})

(1.1.27)

The Higgs boson and Physics beyond the Standard Model (iii) The next step is to perform a Legendre transform in order to obtain the eective action Γ, i.e. the generating functional of one-particle-irreducible (1PI) graphs Γ({ φi (x)}) ≡ W ({J i (x)})d 4 x φi (x)J i (x) , where φi (x) ≡ δW ({J i }) δJ i (x) .

(1.1.28)

(iv) It is common to further impose translational invariance of the φi , and then one nds Γ({ φi }) = -V e ({ φi }) d 4 x .

(1. 1.29) This last new object V e is the eective potential. Note that because the elds φi have been required to be invariant under space-time translations, they are often referred to as background elds.

One can show, following for example [START_REF] Jackiw | Functional evaluation of the eective potential[END_REF], that the one-loop contribution to V e is given by the supertrace formula

V (1) = 1 4
STr M 4 log M 2δ , (1.1.30) with the supertrace dened as

STr[M 2 ] ≡ s (-1) 2s (2s + 1) Tr[M 2 s ] , (1.1.31)
where s = 0, 1 2 , 1 are the possible values of the spin of the particle scalar, fermion or gauge boson that gives a contribution to V e , and with M 2 s the mass-squared matrix of the corresponding particle. The function log x denotes (see also eq. (B.1.2))

log x ≡ log x Q 2 ,
(1. 1.32) where Q is the renormalisation scale. Both the regularisation and the renormalisation of loop integrals will be discussed in section 2.2.1, and we will also show there how the renormalisation scale is introduced. Finally, δ is dened as

δ 0 = δ 1 2 = 3 2 ,
and

δ 1 =        3 2 in DR scheme, 5 6
in MS scheme.

(1. 1.33) The dierence between renormalisation schemes for δ 1 arises from the fact that gauge bosons have d = 4-2 component when using dimensional regularisation together with the modied minimal subtraction (MS) scheme, while they still have 4 components in the modied dimensional reduction (DR ) scheme. The DR renormalisation scheme is devised to avoid breaking Supersymmetry as is the case in MS, by working in d = 4-2

for momenta and momentum integration, while keeping 4 components for the vector elds. Renormalisation schemes will also be presented in more detail in section 2.2.1.

It was observed already in [START_REF] Jackiw | Functional evaluation of the eective potential[END_REF] that the eective potential is not gauge-invariant, however, it can serve to derive physically meaningful i.e. gauge invariant quantities, a very simple example of which is the value of the potential at its minimum. In particular, it was shown in [START_REF] Nielsen | On the Gauge Dependence of Spontaneous Symmetry Breaking in Gauge Theories[END_REF] that the existence of a minimum of the eective potential spontaneously breaking the gauge symmetry does not depend on the gauge in other

1. [START_REF] Braathen | Leading two-loop corrections to the Higgs boson masses in SUSY models with Dirac gauginos[END_REF] The Standard Model and the Higgs sector words, for the SM, the occurrence of the EWSB is gauge independent. The gauge (in)dependence of observables computed from the eective potential has been investigated extensively in the literature, but we will not discuss it further here the reader may refer e.g. to [1315] for recent developments, as well as to references therein.

In the case of the SM, one nds for the one-loop eective potential in the Landau gauge and MS scheme [START_REF] Ford | The Standard model eective potential at two loops[END_REF] V (1)

(h) = 1 4 m 4 h (h) log m 2 h (h) - 3 2 + 3 4 m 4 G (h) log m 2 G (h) - 3 2 -3 q=u,d,c,s,t,b m 4 q (h) log m 2 q (h) - 3 2 + 3 2 m 4 W (h) log m 2 W (h) - 5 6 + 3 4 m 4 Z (h) log m 2 Z (h) - 5 6
(1. 1.34) where

m 2 h (h) = µ 2 + 3λ(v + h) 2 , m 2 G (h) = µ 2 + λ(v + h) 2 , m q (h) = y q √ 2 (v + h) , m 2 W (h) = 1 4 g 2 (v + h) 2 , m 2 Z (h) = 1 4 (g 2 + g 2 )(v + h) 2 , (1.1.35)
are the (Higgs-)eld-dependent tree-level masses of the Higgs, Goldstones, quarks, and W and Z bosons respectively (and the y q are the diagonal Yukawa couplings of each quark). Generally, in the literature as well as in the remainder of this thesis, the eld dependence is not kept explicit, but is rather assumed implicitly. Moreover, in most cases, only the top quark contribution is kept because the top Yukawa coupling y t is much larger than those of the other quarks. Finally, it is common to express the one-loop contributions to the potential with the loop function f (x) = x 2 /4(log x-3/2) (also dened in eq. (B.1.3)), which will often be used throughout this thesis.

Higher-order contributions of two-loop order and more to V e are found by computing 1PI vacuum bubble diagrams. The two-loop results for the Standard Model were rst computed by Ford, Jack, and Jones, in [START_REF] Ford | The Standard model eective potential at two loops[END_REF], and are written out in equations (5.2)-(5.6) therein. More recently, leading results beyond two loops were calculated by Stephen Martin, with the three-loop leading contributions involving the strong and Yukawa couplings obtained in [START_REF] Martin | Three-loop Standard Model eective potential at leading order in strong and top Yukawa couplings[END_REF], and the four-loop part at leading order in QCD in [START_REF] Martin | Four-loop Standard Model eective potential at leading order in QCD[END_REF].

One important point to consider is that when working with the eective potential, the minimum condition is modied by the radiative corrections and becomes

∂V e ∂h φ=0 = ∂V (0) ∂h φ=0 + 1 16π 2
∂V (1) ∂h φ=0 + 1 (16π 2 ) 2 ∂V (2) ∂h φ=0

+ • • • = (µ 2 + λv 2 )v + 1 16π 2
∂V (1) ∂h φ=0 + 1 (16π 2 ) 2 ∂V (2) ∂h φ=0

+ • • • , (1.1.36)
which, in turn, means the solution of the tadpole equation is changed. Actually, there are dierent possible choices of the parameter(s) for which to solve the tadpole equation(s): the typical choices are either some mass parameter(s) of the Lagrangian (the Higgs mass term µ 2 in the SM), or some VEV(s) (v in the SM). Throughout this thesis, we will choose the former option and we will consider the values of the VEV, obtained by an independent calculation, as an input for Higgs mass computations. In

The Higgs boson and Physics beyond the Standard Model the case of the SM, we nd for example

µ 2 = -λv 2 - 1 v ∂∆V ∂h φ=0 (1.1.37)
with ∆V = V e -V (0) . Moreover, the tree-level masses in the minimum of the loopcorrected potential are also aected and are now

(m 2 h ) tree = µ 2 + 3λv 2 = 2λv 2 - 1 v ∂∆V ∂h φ=0 (m 2 G ) tree = µ 2 + λv 2 = - 1 v ∂∆V ∂h φ=0 (1.1.38)
In later chapters, we will be interested in calculations for generic models (both analytical and in relation to SARAH) and we will want to make use of existing results by Martin in particular [START_REF] Martin | Two loop eective potential for a general renormalizable theory and softly broken supersymmetry[END_REF][START_REF] Martin | Two loop scalar self energies in a general renormalizable theory at leading order in gauge couplings[END_REF] and by Goodsell, Nickel, and Staub [START_REF] Goodsell | Generic two-loop Higgs mass calculation from a diagrammatic approach[END_REF]. In refs. [1921], the masses that are used in the expressions of the potential at its minimum, as well as in the tadpole equation(s) and in the mass calculations, are the tree-level masses in the minimum of the (loop-corrected) eective potential, and we will follow the same choice throughout large parts of this thesis.

Finally, the scalar masses themselves receive radiative corrections that must be taken into account. Indeed the physical mass of a particle is dened by the pole of its full i.e. loop-corrected propagator (it is therefore also called the pole mass or on-shell (OS) mass) and diers from its tree-level mass. The radiative corrections to the masses are given by the self-energy diagrams, and the physical Higgs and Goldstone masses are found by solving the equations

(m 2 h ) OS = (m 2 h ) tree + Π hh p 2 = (m 2 h ) OS , (m 2 G 0 ) OS = (m 2 G ) tree + Π G 0 G 0 p 2 = (m 2 G ) OS , (1.1.39) (m 2 G ± ) OS = (m 2 G ) tree + Π G + G -p 2 = (m 2 G ) OS ,
which is typically done by numerical iterations.

In the Standard Model, the Higgs mass cannot be predicted theoretically as it depends on the Higgs quartic coupling λ, which has not yet been measured experimentally.

But, the discovery of the Higgs boson in 2012 and the precise measurement of its mass around [START_REF] Espinosa | Strong Electroweak Phase Transitions in the Standard Model with a Singlet[END_REF] GeV means now that the value of the Higgs quartic can be extracted from the knowledge of m h , at a given order in perturbation theory. In turn, the value of the Higgs quartic extracted from m h at the scale at which m h is computed, typically m t can be run to high scales in order to study the stability of the SM electroweak vacuum (we will come back to this in a short moment).

The expression of the one-loop Higgs self-energy is fairly short and simple to write, and reads in the Landau gauge (see e.g. [START_REF] Martin | Higgs boson mass in the Standard Model at two-loop order and beyond[END_REF])

Π (1) hh (p 2 ) = 3y 2 t (4m 2 t -p 2 )B(m 2 t , m 2 t ) -18λ 2 v 2 B(m 2 h , m 2 h ) + 3λ p 4 -m 4 h m 2 h B(m 2 G , m 2 G ) + 1 2 (g 2 + g 2 ) p 2 -3m 2 Z - p 4 4m 2 Z B(m 2 Z , m 2 Z ) -p 2 A(m 2 Z ) 2m 2 Z + 2m 2 Z + g 2 p 2 -3m 2 W - p 4 4m 2 W B(m 2 W , m 2 W ) -p 2 A(m 2 W ) 2m 2 W + 2m 2 W (1.1.40)
1.1 The Standard Model and the Higgs sector where the (one-)loop functions A(x) and B(x, y) = B(p 2 , x, y) are dened in eqs. (B. 1.6) and (B.1.7) (these functions are sometimes called Passarino-Veltman functions [START_REF] Passarino | One Loop Corrections for e+ e-Annihilation Into mu+ mu-in the Weinberg Model[END_REF]).

An equivalent equation in the Feynman gauge and gaugeless limit (i.e. g = g = 0) is also given in eq. (6.2.5).

The three most complete calculations of Higgs mass corrections available currently are by Buttazzo et al. 5 [25], by Martin and Roberston for the public code SMH [START_REF] Martin | Higgs boson mass in the Standard Model at two-loop order and beyond[END_REF], and by Kniehl, Pikelner, and Veretin [START_REF] Kniehl | Two-loop electroweak threshold corrections in the Standard Model[END_REF][START_REF] Kniehl | mr: a C++ library for the matching and running of the Standard Model parameters[END_REF].

1. The rst [START_REF] Buttazzo | Investigating the near-criticality of the Higgs boson[END_REF] is a full two-loop computation in a mixed MS-OS scheme and in the Feynman gauge, that serves as the rst step of a study of the (meta)stability of the electroweak vacuum. More precisely, the input parameters are G F , m W , m Z , m H , m t , taken as OS inputs, and the strong coupling constant α s that is taken in the MS scheme and at renormalisation scale equal to m Z (i.e. α s (m Z )). These are used to compute the MS values of λ, µ 2 , y t , g, g and g s at scale Q = m t , which are then run up to the Planck scale. The result for the Higgs quartic is given as an interpolating function of λ MS (Q = m t ) as a function of the Higgs and top quark masses.

2. The second [START_REF] Martin | Higgs boson mass in the Standard Model at two-loop order and beyond[END_REF] contains all two-loop order together with leading three-loop corrections to the Higgs mass, in pure MS scheme and Landau gauge. The analytical results are implemented in the numerical code SMH, which computes to the desired order in perturbation theory the Higgs mass as a function of the Higgs quartic λ (and of the renormalisation scale Q, the gauge couplings g, g , g 3 , the Higgs VEV and the top Yukawa) or inversely the Higgs quartic as a function of the Higgs mass. [START_REF] Kniehl | Two-loop electroweak threshold corrections in the Standard Model[END_REF] provides a complete two-loop extraction of the MS running parameters of the SM (in particular of the running mass of the Higgs) from the OS inputs of G F , m W , m Z , m H , m t , m b and the MS input α s (M Z ), and in a general R ξ gauge. These analytical results are available numerically in the public code mr [START_REF] Kniehl | mr: a C++ library for the matching and running of the Standard Model parameters[END_REF].

The last

The rst and second of the computations described above are discussed in more detail in section 5.2, where they are also compared to the result obtained with the latest version of SARAH. We will also review the state-of-the-art of Higgs mass calculations in BSM models in the next chapter.

The Goldstone Boson Catastrophe in the Standard Model

Before ending our review of Higgs physics in the Standard Model, we take the opportunity of discussing the Goldstone Boson Catastrophe in the context of the SM and the rst solutions that were found to address it in [START_REF] Elias-Miro | Taming Infrared Divergences in the Eective Potential[END_REF][START_REF] Martin | Taming the Goldstone contributions to the eective potential[END_REF] (see also [1315] for recent related work). First observed 6 (and named) for the Standard Model in [START_REF] Martin | Three-loop Standard Model eective potential at leading order in strong and top Yukawa couplings[END_REF], the Goldstone Boson Catastrophe is a case of infra-red divergence in the eective potential and its derivatives, as well as in scalar self-energies, due to the Goldstone bosons.

More precisely, if we look back at eq. (1.1.38), we see that the mass m 2 G that appears 5 Actually, this paper continues the work started in [START_REF] Degrassi | Higgs mass and vacuum stability in the Standard Model at NNLO[END_REF].

6 For the sake of completeness, let us mention that similar infra-red divergences had been encoun- tered from other states than Goldstones in [START_REF] Ford | The Eective potential and the renormalization group[END_REF][START_REF] Einhorn | The Eective potential, the renormalisation group and vacuum stability[END_REF], and from Goldstones in a two-loop calculation of the Higgs masses in the MSSM in the eective potential approximation [START_REF] Martin | Complete two loop eective potential approximation to the lightest Higgs scalar boson mass in supersymmetry[END_REF].

The Higgs boson and Physics beyond the Standard Model for the Goldstones in calculations the tree-level mass in the minimum of the loopcorrected potential is not zero and depends on the renormalisation scale Q. This comes from the renormalisation scale dependence of the corrections to the potential ∆V and means that under renormalisation group ow the Goldstone mass m 2 G can vanish or run to negative values. The origin of the problem lies in the fact that the -loop contributions to the SM eective potential contain terms that can be expanded

in the limit m 2 G → 0 in the form (m 2 G ) n log m m 2
G (where n ≥ 3and m ≤ ) and are thus infra-red divergent. Let us consider the severity of such terms depending on the order of n:

• rst, terms of the form log m m 2 G or log m m 2 G /(m 2
G ) p that appear from three-loop and four-loop orders respectively diverge in the limit m 2 G → 0, causing in turn an unphysical divergence of the potential itself;

• then, terms like m 2

G log m m 2
G present at two-loop and higher orders are not diver- gent i.e. are IR safe but if m 2 G becomes negative, the logarithms will acquire unphysical imaginary parts (these imaginary parts are deemed unphysical because they are not associated to any instability of the potential). Furthermore, the derivatives of such terms appear in the tadpole equations, and are divergent because

∂ ∂h m 2 G log m m 2 G min. = ∂m 2 G ∂h d dm 2 G m 2 G log m m 2 G min. = 2λv log m m 2 G + m log m-1 m 2 G , (1.1.41)
where we have used the derivative of the eld-dependent Goldstone mass from eq. (1.1.35).

• nally, m 4

G log m m 2
G terms, present already from the one-loop order, are also IR safe even if they can have imaginary parts, and it is only their second derivatives that will have divergences. Second derivatives of the eective potential are used to compute approximate scalar masses sometimes called eective potential masses corresponding to the vanishing external momentum limit of the full calculation with self-energies. To illustrate this, one can consider e.g. the one-loop Higgs self-energy given above in eq. (1.1.40) and take p 2 = 0 in that expression. The Goldstone term in Π

(1)

hh (p 2 = 0) is then -3λm 2 h B(0, m 2 G , m 2 G ) = +3λm 2 h log m 2 G (1.1.42)
which indeed corresponds to a second derivative of m 4 G log m m 2 G and diverges in the limit m 2 G → 0. The fact that such terms diverge from one-loop order and beyond was not deemed a serious problem because, until recently [START_REF] Braathen | Avoiding the Goldstone Boson Catastrophe in general renormalisable eld theories at two loops[END_REF], it was believed that the inclusion of external momentum in the mass calculations would cure all IR problems [START_REF] Martin | Complete two loop eective potential approximation to the lightest Higgs scalar boson mass in supersymmetry[END_REF]. As will be shown in chapter 4 (section 4.3), this statement holds at one-loop level (e.g. with the B(0, m 2 G , m 2 G ) term), but at two loops the inclusion of momentum is actually not by itself sucient to cure the IR divergences because some two-loop self-energy diagrams still diverge in the limit m 2 G → 0 even with non-zero external momentum.

It is important to further remark that a simple change of gauge is not sucient to avoid the Goldstone Boson Catastrophe. Indeed, the tree-level masses of the neutral or charged Goldstones in the minimum of the loop-corrected potential are in a general

R ξ gauge m 2 G 0,± = ξm 2 Z,W - 1 v ∂∆V ∂h φ=0 , (1.1.43)
and can therefore still run to negative values under renormalisation group ow (or if the corrections in the derivative term are large). It is only the scale at which the catastrophe appears that depends on the (R ξ ) gauge, not its appearance. One could also think of the unitary gauge, in which the Goldstones would disappear, but as was mentioned earlier loop calculations in this gauge are complicated to such a level that it is preferable to nd a solution to avoid the IR divergences in the Landau (or general R ξ ) gauge rather than to attempt two-loop calculations in the unitary gauge.

It should also be emphasised that the Goldstone Boson Catastrophe does not arise from a genuine physical problem, but is rather a technical issue of the perturbative calculation with (unphysical) Goldstones. Indeed, Goldstone bosons are not part of the physical spectrum of the SM (or of any spontaneously broken gauge theory) and therefore they cannot cause a physical divergence in calculations. Furthermore, the fact that the appearance of the divergences depends on the choice of gauge is also an indication of their lack of physical meaning. But at the same time, it also means that solutions must exist to cure the IR divergences in the calculations.

Initial methods to cure the Goldstone Boson Catastrophe in the eective potential and its rst derivative for the case of the Standard Model were found in references [START_REF] Elias-Miro | Taming Infrared Divergences in the Eective Potential[END_REF][START_REF] Martin | Taming the Goldstone contributions to the eective potential[END_REF] (and later applied to the MSSM in [START_REF] Kumar | Resummation of Goldstone boson contributions to the MSSM eective potential[END_REF]). In both [START_REF] Elias-Miro | Taming Infrared Divergences in the Eective Potential[END_REF] and [START_REF] Martin | Taming the Goldstone contributions to the eective potential[END_REF] the infra-red divergences are eliminated through a resummation of contributions from the Goldstone bosons to the eective potential, although the two methods dier slightly in the precise way the resummation is dened. The starting point of both methods is to observe that the most divergent Goldstone contribution in V e at given order in the pertubative series is the bubble diagram formed of a ring of -1 Goldstone propagators with -1 insertions of 1PI subdiagrams Γ(p 2 ) involving mass scales much larger than the Goldstone masses, as shown in gure 1.1. Moreover, to nd the leading divergence in the limit of vanishing m 2 G , it is sucient to consider the zero-momentum limit of Γ(p 2 )

and then we can write the (divergent) -loop Goldstone contribution to the eective potential as

V ( ) G = - 3i 2 C d d k(iΓ(0)) -1 i k 2 -m 2 G -1 = - 3i 2 (Γ(0)) -1 ( -1)! d dm 2 G -1 C d d k log(k 2 -m 2 G )
The Higgs boson and Physics beyond the Standard Model

= (Γ(0)) -1 ( -1)! d dm 2 G -1 3m 4 G 4 log m 2 G - 3 2 ,
(1. 1.44) where the factor 3 comes from the Goldstone multiplicity and the factor C is dened in eq. (B.1.1). We nd that we can sum these terms to all orders and obtain a corrected Goldstone contribution in eq. (1.1.34)

V

(1)

G = 3 4 m 2 G + Γ(0) 2 log m 2 G + Γ(0) - 3 2 ,
(1. 1.45) and a resummed eective potential Ve at -loop order

Ve = V e + 3 16π 2 1 4 m 2 G + Γ(0) 2 log m 2 G + Γ(0) - 3 2 - -1 n=0 (Γ(0)) n n! d dm 2 G n m 4 G 4 log m 2 G - 3 2 .
(1.1.46)

The term in the second line of eq. (1.1.46) avoids double counting of the Goldstone diagrams and ensure that the original and resummed eective potentials are equal up to terms of order higher than loops. With a well-chosen Γ(0) the above resummed potential and its rst derivative are both free of unphysical IR divergences and imaginary parts (mass calculations were not studied in [START_REF] Elias-Miro | Taming Infrared Divergences in the Eective Potential[END_REF][START_REF] Martin | Taming the Goldstone contributions to the eective potential[END_REF][START_REF] Kumar | Resummation of Goldstone boson contributions to the MSSM eective potential[END_REF]), and it is then the denition of Γ(0) that constitutes the dierence between the procedures in [START_REF] Elias-Miro | Taming Infrared Divergences in the Eective Potential[END_REF] and [START_REF] Martin | Taming the Goldstone contributions to the eective potential[END_REF], as we will see now.

On the one hand, in the approach of [START_REF] Martin | Taming the Goldstone contributions to the eective potential[END_REF][START_REF] Kumar | Resummation of Goldstone boson contributions to the MSSM eective potential[END_REF] the insertion Γ ≡ ∆ is obtained by an expansion of the eective potential for small m 2 G . In particular, if we loop-expand the insertion as

∆ = 1 16π 2 ∆ 1 + 1 (16π 2 ) 2 ∆ 2 + • • • , (1.1.47)
the leading term ∆ 1 can be found (together with another term denoted Ω) when expanding the two-loop potential for small m 2 G in the following way

V (2) (m 2 G ) = V (2) (m 2 G = 0) + 3 2 ∆ 1 A(G) + 3 2 Ωm 2 G + O(m 4 G ) , (1.1.48)
where A(x) is dened in eq. (B.1.6) and the factor 3 comes from the Goldstone multiplicity. Considering only ∆ 1 is actually sucient because the three-loop terms in the Standard Model potential that would correspond to ∆ 2 were not computed in [START_REF] Martin | Three-loop Standard Model eective potential at leading order in strong and top Yukawa couplings[END_REF].

In the end, the resummed potential at two-loop order can be written in a simple expression

Ve = V (0) + 1 16π 2 V (1) (m 2 G = 0) + 3f m 2 G + 1 16π 2 ∆ 1 + 1 (16π 2 ) 2 V (2) (m 2 G = 0) + 3 2 Ωm 2 G (1.1.49)
up to two-loop terms of order m 4

G that cause no divergence in the two-loop tadpole equation, and where f (x) is dened in eq. (B.1.3).

On the other hand, [START_REF] Elias-Miro | Taming Infrared Divergences in the Eective Potential[END_REF] shows that the resummation of the Goldstone diagrams contributing to the eective potential can be seen as similar to the construction of an Eective Field Theory describing soft Goldstones only, by integrating out heavy elds G respec- tively. The divergent diagrams involving Goldstones then correspond in the theory of soft Goldstones to a limited number of diagrams with soft Goldstones, and where masses and couplings have received threshold corrections. In particular, the masses of the soft Goldstones receive a threshold correction denoted Π g , from the integration of the other (heavy) elds, that is equal to the Goldstone self-energy evaluated at vanishing external momentum, minus the soft Goldstones part. The contributions of the Goldstone bosons to the eective potential of the SM can then be found in terms of diagrams in the low-energy theory of soft Goldstones, and notably, to the two-loop level the only such diagram is the one-loop vacuum bubble that reads

V (1) G = - 3i 2 C d d k log k 2 -m 2 G -Π g = 3 4 m 2 G + Π g 2 log m 2 G + Π g - 3 2 ,
(1. 1.50) which corresponds to eq. (1.1.45) with Π g playing the role of Γ(0). The separation between soft and hard modes of the Goldstone bosons may, at rst, seem non-trivial and somewhat ambiguous, however, it was suggested in [START_REF] Elias-Miro | Taming Infrared Divergences in the Eective Potential[END_REF] that the splitting of the momentum integrals of Goldstone propagators can performed consistently with the method of regions [START_REF] Beneke | Asymptotic expansion of Feynman integrals near threshold[END_REF][START_REF] Jantzen | Foundation and generalization of the expansion by regions[END_REF]. More recently, it was demonstrated in [START_REF] Espinosa | Resummation of Goldstone Infrared Divergences: A Proof to All Orders[END_REF] that the separation with the method of regions works to all orders in perturbation theory, thus (formally) extending this resummation prescription to any desired order. For more details on how the method of regions works and how to split momentum integrals with it, the reader may refer to the appendices of [START_REF] Espinosa | Resummation of Goldstone Infrared Divergences: A Proof to All Orders[END_REF].

To solve the Goldstone Boson Catastrophe in the SM to two-loop order, only the oneloop expressions of ∆ and Π g are needed, and both approaches to the resummation method yield the same results, with

∆ 1 = Π (1) g = -6y 2 t A(m 2 t ) + 3λA(m 2 h ) + 3 2 g 2 A(m 2 W ) + 2 3 m 2 W + 3 4 (g 2 + g 2 ) A(m 2 Z ) + 2 3 m 2 Z .
(1.1.51)

While providing accurate results for the two-loop eective potential and tadpole equation in the SM, these two methods are dicult to extend to other models and to automate, to a large part because of mixing among the Goldstones. Furthermore, a resummation of the eective potential does not by itself suce to cure IR divergences appearing in the calculation of mass corrections, because some diagrams diverge in the p 2 → 0 limit. This provides motivation to search for a solution to the Goldstone Boson Catastrophe for general models and that can be applied to mass diagrams, which will be the subject of chapter 4.

It is also worth mentioning that for the case of the SM an alternative solution to the Goldstone Boson Catastrophe was found, relying on the use of symmetry-improved two-particle-irreducible (2PI) eective action developed by Pilaftsis and Teresi [3739].

The 2PI eective action was rst introduced in [START_REF] Cornwall | Eective Action for Composite Operators[END_REF], essentially by a generalisation of the derivation of the 1PI eective action in the previous section by introducing bilocal sources in the standard and connected generating functionals Z and W (c.f.

7 Gluons and photons are also included in the low-energy part of this construction. and is therefore free of IR divergences caused by massless particles in the loops by construction. A further renement was introduced in [START_REF] Pilaftsis | Symmetry Improved CJT Eective Action[END_REF] with the symmetry-improved 2PI eective action that is built so as to respect (global) symmetries at any order in perturbation theory to which it is computed, by imposing on the dressed propagators the Ward identities associated with the symmetries. In [START_REF] Pilaftsis | Symmetry Improved 2PI Eective Action and the Infrared Divergences of the Standard Model[END_REF][START_REF] Pilaftsis | Symmetry-Improved 2PI Approach to the Goldstone-Boson IR Problem of the SM Eective Potential[END_REF], the use of this improved eective action for the SM was shown to avoid the IR divergences from the Goldstone bosons and the numerical results agree with those of [START_REF] Elias-Miro | Taming Infrared Divergences in the Eective Potential[END_REF][START_REF] Martin | Taming the Goldstone contributions to the eective potential[END_REF]. Conceptually, the 2PI eective action approach relates to the method we devise in chapter 4 because it amounts to a resummation of the propagators of all particles, while our method corresponds to a resummation of Goldstone propagators only. However, as this approach does not allow simple analytical calculations (instead the action and derived quantities must be computed numerically) and is dicult to apply to generic models beyond the SM and thus to automate, we will not discuss it further here.

Going beyond the Standard Model

The Standard Model suers from a number of issues: on the one hand, theoretical deciencies (some of which have been apparent since its conception) and, on the other hand, experimental observations that are conicting with the SM or that cannot be accommodated by it. Among these problems, one could mention, for example, on the theoretical side: the lack of explanations of ination, or of the mechanism of baryogenesis; the strong CP problem; or the absence of a description of quantum gravity. On the experimental side, signs that the SM is not a complete theory of Nature include the observational signs of the existence of dark matter for which the SM provides no candidate particle; the discovery of neutrino masses; or tensions that have appeared between SM predictions and experimental results e.g. for the muon anomalous moment or, more recently, in B Physics.

Complete lists of the problems of the SM can be found in the literature (see e.g. [START_REF] Quevedo | Cambridge Lectures on Supersymmetry and Extra Dimensions[END_REF][START_REF] Goodsell | Lectures on Supersymmetry[END_REF]), and therefore it makes little sense to attempt to provide an exhaustive presentation here. Instead, we will focus here on the two points that relate the most to the Higgs sector: the hierarchy problem and the fate of the electroweak vacuum.

The hierarchy problem

The hierarchy problem has several formulations, of which the rst and most straightforward one is to ask why the energy scales appearing in quantum theories of gravitational and electroweak interactions, respectively, are so dierent. More precisely, the typical scale at which quantum gravity eects appear, the Planck scale M Pl found as

M Pl ≡ c G 1.22 × 10 19 GeV (1.2.1)
is seventeen orders of magnitude larger than the scale of electroweak interactions, usually taken to be the Higgs VEV v = 246 GeV. As a matter of comparison, the typical energy scale of strong interactions (the QCD scale) is only two to three orders of magnitude smaller than the electroweak scale. This may seem as a simple matter of aesthetics, but it comes with other more clearly worrisome consequences.

Going beyond the Standard Model

One of them relates directly to the Higgs boson and to electroweak symmetry breaking:

to nd a value of the Higgs VEV v compatible with the W and Z bosons masses (or equivalently with the value derived from the measurement of the Fermi constant G F ) the Higgs mass term µ 2 must be of order ∼ -(100 GeV) 2 . However, scalar masses are not protected by either chiral or gauge symmetries as are fermion and gauge boson masses respectively (see section 1.1.1) and therefore the corrections to the Higgs mass term µ 2 can be huge actually they are proportional to the mass of the heaviest particle coupling to the Higgs. Obtaining the required value of the loop-corrected µ 2 would imply that some exceptionally precise cancellation occurs between its tree-level value and the radiative corrections, which seems very strange.

To illustrate this, let us consider a toy model in which a heavy Dirac fermion ψ, with a mass around the Planck scale M ψ M Pl , couples to a (light) real scalar S corresponding to the SM Higgs with a Lagrangian

L = ψ(i / ∂ -M ψ )ψ + 1 2 (∂ µ S) 2 - 1 2 m 2 S S 2 -y ψ ψψS (1.2.2)
Such a fermion can be thought of as a heavy particle appearing in a quantum theory of gravity, of which the SM is (supposed to be) an Eective Field Theory below M Pl . The one-loop threshold correction to m 2 S when integrating out ψ can be found in dimensional regularisation to be

δ (1) m 2 S (p 2 ) = -(-iy ψ ) 2 C d d ktr i(/ k + M ψ ) k 2 -M 2 ψ i((/ k -/ p) + M ψ ) (p -k) 2 -M 2 ψ = y 2 ψ 4π 2 A(M 2 ψ ) -2M 2 ψ - p 2 2 B(p 2 , M 2 ψ , M 2 ψ ) . (1.2.3)
after performing an MS renormalisation, and with the loop factor C dened in eq. (B.1.1). The natural value of the momentum owing in the loop is m S , i.e. a mass scale much smaller than the fermion mass M ψ , and furthermore, it is also common to perform the matching at scale equal to (or of the same order as) M ψ . Therefore, we can set p 2 ≈ m 2

S

M 2 ψ and Q = M ψ , which greatly simplies the above expression, and we nd

δ (1) m 2 S (0) = - y 2 ψ 4π 2 M 2 ψ .
(1.2.4)

When considering heavy fermions interacting with the SM scalar sector, corrections of the above form referred to as quadratic divergences 8 also occur for the Higgs mass parameter µ 2 , and can be much larger than the required value of ∼ -(100 GeV) 2 for µ 2 . The consequent problem of the necessary ne-tuning of (µ 2 ) tree the technical hierarchy problem has, for a long time, been a strong motivation for the search of mechanisms beyond the SM to protect scalar mass terms from (divergent) radiative corrections. Among the models that have been shown to solve this problem (a large number of which have been devised especially for the task) one can mention models with new symmetries, such as Supersymmetry [START_REF] Dimopoulos | Softly Broken Supersymmetry and SU(5)[END_REF] or composite Higgs models [START_REF] Dimopoulos | Massless Composites With Massive Constituents[END_REF][START_REF] Kaplan | Composite Higgs Scalars[END_REF]; models with extended space-time [START_REF] Randall | A Large mass hierarchy from a small extra dimension[END_REF][START_REF] Arkani-Hamed | The Hierarchy problem and new dimensions at a millimeter[END_REF]; or as was suggested more recently, new mechanisms like cosmological relaxation [START_REF] Graham | Cosmological Relaxation of the Electroweak Scale[END_REF].

An alternative point of view is not to regard the (technical) hierarchy problem as a problem of the theory, and require for example some anthropic principle to explain the value of µ 2 .

8 Original corrections to scalar masses from fermions were computed using regularisation with a cut-o scale ΛUV and were shown to behave like ∝ y 2 Λ 2 UV (both for SM and new heavy fermions). The scalar masses then diverge quadratically when ΛUV → ∞, hence the name of quadratic divergences.

V e (h) h v λ e (h) 4 h 4 , (1.2.5)
where the eective Higgs quartic λ e is related to the tree-level quartic λ by the inclusion of (Higgs-)eld-strength renormalisation eects and of higher-order corrections to the potential [START_REF] Degrassi | Higgs mass and vacuum stability in the Standard Model at NNLO[END_REF][START_REF] Buttazzo | Investigating the near-criticality of the Higgs boson[END_REF]. However, the dierence between λ e and λ becomes negligible for large eld values [START_REF] Degrassi | Higgs mass and vacuum stability in the Standard Model at NNLO[END_REF] and consequently the study of the behaviour of the potential for large Higgs elds amounts to studying the running of the Higgs (tree-level) quartic λ at high scales. To be more specic, one must remember that λ is not predicted in the SM but can, however, be extracted from the measurement of the Higgs mass m h [4952] using the known threshold corrections at the scale at which m h is measured typically taken to be Q = m t . Once λ(Q) has been extracted at some low scale, its value at any large scale can be found through renormalisation group running, and the renormalisation group equation (RGE) of λ is presently known to full three-loop order [START_REF] Bednyakov | Higgs self-coupling beta-function in the Standard Model at three loops[END_REF][START_REF] Chetyrkin | β-function for the Higgs self-interaction in the Standard Model at three-loop level[END_REF].

One of the points of interest in a study of the running of λ is its sign 9 at high scales. In particular, for the Higgs quartic to become negative is a worrisome sign for the theory, because it at least signals the existence of states of energy lower than that of the EW vacuum, and moreover, if λ remains negative at high scales, the scalar potential is not bounded from below. The EW vacuum in which we live is then not absolutely stable: it will be metastable or unstable depending on whether its lifetime (i.e. the typical time until the Higgs tunnels to the lower minimum of the potential) is larger or smaller than the age of the Universe.

An in-depth study of the stability of the electroweak vacuum using the above-mentioned state-of-the-art results was begun in [START_REF] Degrassi | Higgs mass and vacuum stability in the Standard Model at NNLO[END_REF] and continued in [START_REF] Buttazzo | Investigating the near-criticality of the Higgs boson[END_REF], and showed that if one extrapolates the SM up to the Planck scale, the measured values of the Higgs and top quark masses favour the vacuum being only metastable, as can be seen in gure 3 of [START_REF] Buttazzo | Investigating the near-criticality of the Higgs boson[END_REF]. Interestingly, the Higgs quartic becomes negative at a scale Q ∼ 10 10 GeV and remains negative at higher scales, with a value at the Planck scale very close to 0 λ(M Pl ) -0.014, c.f. equation (61e) of [START_REF] Buttazzo | Investigating the near-criticality of the Higgs boson[END_REF].

There are several possible interpretations of this result of metastability: at rst, one may argue that a metastable vacuum is not sucient to require the presence of some BSM Physics below the Planck scale, and that we may well live in a vacuum that is not the absolute minimum of the Higgs potential, because the characteristic tunnelling time to the absolute minimum is much larger than the age of the Universe. In turn, 9 As was noted in [START_REF] Buttazzo | Investigating the near-criticality of the Higgs boson[END_REF], λ is the only SM coupling whose β-function is not proportional to the coupling itself and therefore only λ can change sign. In terms of symmetries, this comes from the fact that the case λ = 0 does not have an enhanced symmetry with respect to λ = 0.

Supersymmetry

dierent options are available to explain the value of λ(M Pl ) found close to zero, and in particular one of them is that the small negative λ(M Pl ) only arises from threshold corrections to a matching condition between a high energy model and the SM, at some high scale, possibly as high as the Planck scale. For other possibilities, and further discussion of interpretations of the metastability of the electroweak vacuum for Particle Physics and Cosmology, the reader may consult [START_REF] Degrassi | Higgs mass and vacuum stability in the Standard Model at NNLO[END_REF][START_REF] Buttazzo | Investigating the near-criticality of the Higgs boson[END_REF] and references therein.

Another possibility is to consider the metastability of the electroweak vacuum as an indicator of the existence of New Physics, at some scale below or around where λ becomes negative, that is able somehow stabilise the potential and prevent λ from vanishing. In this context, the SM is viewed as an eective theory below the New Physics scale, or in other words, the scale at which λ runs to negative values (in the SM) can be seen as a cut-o scale up to which the SM can be considered valid.

Building models beyond the Standard Model

Before turning to the presentation of some BSM models, let us rst make a short comment on the two possible approaches to model building.

On the one hand, one can try to devise a complete theory valid up to high energies, possibly even up to the Planck scale that would solve some (or all) of the deciencies of the SM, and then study the consequences of such a model at energies around the electroweak scale. One of the best known examples of this approach called the top-down approach is the class of supersymmetric theories, in which fermionic and bosonic states are related by a new symmetry valid at high scales, but spontaneously broken at lower energies. In coming section 1.3 we will consider dierent Supersymmetry models studied in the course of this thesis, namely the MSSM, the NMSSM, and Dirac gaugino models.

On the other hand, another approach to the construction of models of New Physics is also possible: the bottom-up approach. The idea is to extend the Standard Model by new elds and/or new terms in the Lagrangian, at energy scales around or slightly above the electroweak scale, in order to address some of the issues of the Standard Model, or sometimes simply to study the phenomenological consequences of such an extension. We will be more specically interested in models with extended Higgs sectors, which we will review in section 1.4 (extensions of the SM by a singlet, by a doublet, or by two triplets).

Supersymmetry

This section aims at introducing, from a phenomenological point of view, some key notions about Supersymmetry and some supersymmetric models in particular Dirac gaugino models that will appear throughout this thesis. The reader may refer e.g.

to [START_REF] Quevedo | Cambridge Lectures on Supersymmetry and Extra Dimensions[END_REF][START_REF] Goodsell | Lectures on Supersymmetry[END_REF][START_REF] Martin | A Supersymmetry primer[END_REF][START_REF] Wess | Supersymmetry and supergravity[END_REF] for reviews of SUSY.

Supersymmetry (SUSY) extends the Poincaré group of space-time invariance by a new symmetry that relates fermions and bosons. We will now recall elements of the SUSY formalism that will be required in this thesis, before showing how in principle SUSY helps solve the hierarchy problem one great success of Supersymmetry and nally discussing the supersymmetric extensions of the SM that will be studied in this thesis.

The Higgs boson and Physics beyond the Standard Model

Note that in what follows we will only consider Supersymmetry to be a global symmetry, 10 and we will not be considering gravity.

1.3.1 Some basics of SUSY

ψ = ξ α χ α (1.3.1)
where ξ α and χ α are the complex and anti-commuting (Grassmanian) components of Weyl spinors ξ and χ of left-and right-handed helicities respectively. Dotted and undotted indices correspond to the two possible helicities of the spinors and take values 1 and 2. Non-barred and barred Weyl spinors correspond to two dierent representations of the Lorentz group SO(3, 1) ∼ = SU (2) L ⊕ SU (2) R , more precisely to the ( 1 2 , 0) and (0, 1 2 ) representations. These two representations can be related by complex conjugation, i.e.

χ α ≡ (χ α ) † , (1.3.2)
hence the bar on right-handed spinors.

One can change from upper to lower indices using the two-dimensional antisymmetric tensors αβ or α β with the relations

ξ α = αβ ξ β , ξ α = αβ ξ β , ξ α = α β ξ β , ξ α = α β ξ β , (1.3.3)
having chosen to take 12 = -12 = 1. Spinor indices will also often be contracted, with the following conventions

ξχ = ξ α χ α = -ξ α χ α , ξ χ = ξ α χ α = -ξ α χ α . (1.3.4)
The Dirac conjugate of the four-component spinor ψ can be written

ψ ≡ ψ † γ 0 = (χ α , ξ α) (1.3.5)
and with this, one can easily introduce the left-and right-handed projectors, using the identity

1 4 = P L + P R = P 2 L + P 2
R , in order to rewrite a Dirac mass term as

m ψ ψψ = m ψ ξχ + ξ χ . (1.3.6)
Furthermore the charge conjugate of the Dirac spinor ψ is

ψ C ≡ iγ 0 γ 2 ψ T = αβ 0 0 α β χ α ξ α = χ α ξ α (1.3.7)
so one nds that the Majorana condition

ψ C M = ψ M (1.3.8)
10 Local Supersymmetry is subject of research on its own, namely Supergravity.

Supersymmetry

implies for the Weyl spinor components that ξ = χ. Then one can write a Majorana spinor and a Majorana mass term in two-component language as

ψ M = ξ α ξ α ,
and

1 2 M ψ ψ M ψ M = 1 2 m ψ ξξ + ξ ξ .
(1.3.9)

1.3.1.2 Supersymmetry, the SUSY algebra and its representations

Symmetries play an important role in the study of Quantum Field Theories, and nogo theorems most famously the Coleman-Mandula theorem [START_REF] Coleman | All Possible Symmetries of the S Matrix[END_REF] were established, greatly limiting the possible space-time symmetries of physical theories. Actually, Coleman and Mandula showed in [START_REF] Coleman | All Possible Symmetries of the S Matrix[END_REF] that if one considers a physical theory in four dimensions, with only local interactions, and further supposes the states are in nite number and that some of these are massive, then the largest possible symmetry group when allowing only bosonic symmetry generators is the Poincaré group. However, Haag, opusza«ski, and Sohnius [START_REF] Haag | All Possible Generators of Supersymmetries of the s Matrix[END_REF] showed that it is possible to extend Poincaré symmetry by additional fermionic symmetry generators, i.e. generators that carry spin-1 2 , and therefore transform as spinors under transformations of the rest of the Poincaré group and have spinor indices (undotted or dotted). These generators will be denoted Q A α where index A labels the possible multiple generators and the Supersymmetry transformations they generate turn bosons into fermions and vice versa. Furthermore, because the fermions are complex states, the complex conjugates of the generators QA α will also be generators of dierent supersymmetric transformations.

From the above considerations, and with the additional requirement of having a closed algebra, one can derive the SUSY algebra see e.g. section 3.1 of [START_REF] Martin | A Supersymmetry primer[END_REF] for a derivation in the N = 1 case. In the presence of N supercharges labelled A, B,

• • • ∈ [1, • • • , N ],
the complete SUSY algebra is

{Q A α , QB α } = 2σ µ α αP µ δ AB {Q A α , Q B β } = αβ Z AB { QA α , QB β } = α β (Z AB ) * [P µ , Q A α ] = [P µ , QA α ] = 0 (1.3.10)
where σ µ = (1 2 , σ i ), P µ is the generator of translations, and Z AB are anti-symmetric tensors called the central charges. Note that we will also use σ µ = (1 2 , -σ i ).

While theories with several supersymmetries are studied, e.g. in the context of String Theory, most phenomenologically interesting SUSY models only have a single Super- symmetry (N = 1 SUSY). One important exception is the case of Dirac gaugino models which we will consider in depth in section 1.3.5 and in chapter 3 in which Supersymmetry in the gauge sector is extended to N = 2. However, even these models can be studied in the language of N = 1 SUSY, 11 and for this reason we will now only discuss representations of the SUSY algebra in the N = 1 case. 11 The states in the gauge sector that would live in a hypermultiplet in N = 2 SUSY namely the gauge boson, the gauginos and the (complex) adjoint gauge scalar can also be seen as one gauge supermultiplet, containing the gauge boson and one gaugino, plus a chiral supermultiplet in the adjoint representation of the gauge group, that will contain the other gaugino and the adjoint scalar.

The Higgs boson and Physics beyond the Standard Model One-particle states in a SUSY theory live in irreducible representations of the SUSY algebra, called supermultiplets, containing both fermionic and bosonic states that will be referred to as superpartners. Because gauge symmetries are internal and not spacetime symmetries, their generators must commute with SUSY generators and hence all particles of a given supermultiplet belong to the same representation of the gauge group. Moreover, it can be shown [START_REF] Quevedo | Cambridge Lectures on Supersymmetry and Extra Dimensions[END_REF][START_REF] Martin | A Supersymmetry primer[END_REF][START_REF] Wess | Supersymmetry and supergravity[END_REF] that the numbers of fermionic and bosonic states in a given supermultiplet must be the same, which will x the structure of representations. Indeed there will be two possible ways to construct supermultiplets, as summarised in table 1.2: either a Weyl spinor (two fermionic degrees of freedom when on-shell) together with a complex scalar (two bosonic degrees of freedom) forming a chiral supermultiplet; or a massless gauge boson (with two helicities hence two bosonic degrees of freedom when on-shell) together with a Weyl spinor forming a gauge (or vector ) supermultiplet. An important remark should be made here about the above counting of bosonic and fermionic degrees of freedom. The number of degrees of freedom that we considered both for fermions and gauge bosons is only valid on mass-shell, i.e. when the equations of motion of the given eld applies. O-shell this counting has to be modied, with four degrees of freedom for fermions and three for gauge bosons, which appears to contradict the requirement of having the same number of fermion and boson states in each supermultiplet. We can solve this problem by introducing new bosonic auxiliary elds to match the number of fermionic degrees of freedom in both chiral and gauge supermultiplets. For the chiral supermultiplet, we add a complex scalar F that vanishes on-shell and therefore has a non-propagating Lagrangian L aux. = F * F .

(1.3.11)

Similarly, for the gauge supermultiplet we add a real scalar D a a being the gauge group index with a Lagrangian

L aux. = 1 2 D a D a .
(1.3.12)

Another argument requiring the introduction of these elds is that the SUSY algebra would not close o-shell without them. The auxiliary elds are then introduced with SUSY transformation rules ensuring that the complete Lagrangian remains invariant under SUSY transformations.

Superspace formalism and superelds

The superspace formalism is a powerful tool that simplies the construction of supermultiplets and of Lagrangians in a SUSY-invariant way. Superspace is an eightdimensional manifold formed of normal space-time with its four bosonic coordinates x µ to which are added four fermionic coordinates as two two-component anticommuting spinors θ α and θ α (having mass dimension -1/2).

Supersymmetry

We can then build new elds on the superspace, superelds, that will contain all the components of supermultiplets, and to obtain their explicit expressions we can perform a power series expansion in terms of θ, θ, which must end at a nite order since the components of θ, θ are Grassmanian variables. A general supereld can be written [START_REF] Martin | A Supersymmetry primer[END_REF] S(x, θ, θ) = a + θξ + θ χ + θθb + θ θc + θσ µ θv µ + θ θθη + θθ θ ζ + θθ θ θd .

(1. 3.13) In this expansion, a, b, c, d are scalar components, ξ, η ( χ, ζ) are left-handed (righthanded) fermionic components and v µ is a vector component. This number of components is obviously much larger than the number of degrees of freedom in a supermultiplet so there must exist criteria to constrain or relate the components.

Before considering the dierent types of superelds, let us however rst mention briey the formulation of SUSY transformations in the superspace formalism. The SUSY generators can be written as dierential operators acting on superelds

Qα = i ∂ ∂θ α -σ µ θ α ∂ µ , Qα = -i ∂ ∂θ α + θσ µ α ∂ µ , Q α = i ∂ ∂ θ α -(σ µ θ) α∂ µ , Q α = -i ∂ ∂ θ α + (θσ µ ) α∂ µ . (1.3.14)
Then the transformation of a supereld S under a SUSY transformation of parameters and

¯ is √ 2δ S = S(x µ + i σ µ θ + i¯ σ µ θ, θ + , θ + ¯ ) -S(x µ , θ, θ) = -i Q + ¯ Q S(x µ , θ, θ) = α ∂ ∂θ α + ¯ α ∂ ∂ θ α + i( σ µ θ + ¯ σ µ θ)∂ µ S(x µ , θ, θ) . (1.3.15)
With the above equation, it is possible to derive the transformations of the component elds, either for a generic supereld or for a particular type of supereld chiral or real as we will see now.

To dene superelds for chiral supermultiplets, we need to introduce a last type of object: chiral (SUSY-)covariant derivatives. These are necessary to constrain chiral superelds and are dened as

D α = ∂ ∂θ α -i σ µ θ α ∂ µ , D α = - ∂ ∂θ α + i θσ µ α ∂ µ , D α = ∂ ∂ θ α -i(σ µ θ) α∂ µ , D α = - ∂ ∂ θ α + i(θσ µ ) α∂ µ , (1.3.16)
where the derivatives with respect to an anti-commuting variable are themselves dened by

∂θ β ∂θ α = δ β α , ∂ θ β ∂θ α = 0 , ∂ θ β ∂ θ α = δ α β , ∂θ β ∂ θ α = 0 .
(1.3.17)

A chiral supereld Φ is then dened by the condition

D αΦ = 0 , (1.3.18)
that strongly constraints the components of Φ. One can show that this gives

Φ = φ + √ 2θψ + θθF + i θσ µ θ∂ µ φ - i √ 2 θθ θσ µ ∂ µ ψ + 1 4 θθ θ θ∂ µ ∂ µ φ , (1.3.19)
The Higgs boson and Physics beyond the Standard Model with ψ a left-handed fermion, φ its complex scalar superpartner, and F the auxiliary eld. Taking the complex conjugate of Φ we can also obtain an antichiral supereld.

In turn, gauge supermultiplets are described by real superelds, i.e. superelds V that verify the condition V = V * .

(

L chiral = d 2 θd 2 θΦ * i Φ i + d 2 θ W ({Φ i }) θ=0 + h.c. , (1.3.22)
where W ({Φ i }) is a holomorphic function of the chiral superelds Φ i called the superpotential. This object contains supersymmetric mass terms and self-interactions of the scalars as well as the interactions of fermions with scalars. The most general renormalisable superpotential that can be written for a generic theory is

W ({Φ i }) = L i Φ i + 1 2 M ij Φ i Φ j + 1 6 y ijk Φ i Φ j Φ k . (1.3.23)
The requirement of gauge invariance means that L i terms are only allowed when Φ i is a gauge singlet (we will see that this is possible in the NMSSM and in Dirac gaugino models, but not in the MSSM).

If one introduces vector superelds V a associated with some general gauge group (which we will again index with a), the rst term in eq. (1.3.22) must be modied to remain supergauge invariant. Furthermore, another purely gauge term must be added containing kinetic terms of gauge bosons and gauginos. For this we need to introduce a eld-strength chiral supereld, dened as

W α ≡ - 1 4 D D e -V D α e V , (1.3.24)
and its components in the adjoint representation of the gauge group (and Wess-Zumino

gauge) W a α W α ≡ 2g a T a W a α (1.3.25) W a α = λ a α + θ α D a - i 2 (σ µ σ ν θ) α F a µν + iθθ(σ µ D µ λa ) α .
(1.3.26)

12 In eq. (1.3.21), we have adopted the Wess-Zumino (super)gauge which enables us to eliminate additional bosonic and fermionic auxiliary elds that would be present in the expression of V a in a general gauge. This choice being for the supergauge, the ordinary gauge is not xed. The reader can refer to e.g. section 4.5 of [START_REF] Martin | A Supersymmetry primer[END_REF] for more details.

Supersymmetry

Here T a are the generators of the gauge group in the adjoint representation, F a µν the usual eld-strength tensor and D µ the (gauge-)covariant derivative. In the end, one nds that, in all generality, the renormalisable terms that one can write are [START_REF] Goodsell | Lectures on Supersymmetry[END_REF][START_REF] Martin | A Supersymmetry primer[END_REF]

L chiral+gauge = d 2 θd 2 θ Φ * i (exp(2g a T a V a )) ij Φ j + d 2 θ W ({Φ i }) θ=0 + h.c. + d 2 θ W aα W a α θ=0 + h.c. . (1.3.27)
Note that for an Abelian gauge group, an additional type of term a Fayet-Iliopoulos term of the form

L FI = -2κ d 2 θd 2 θ V (1.3.28)
is also allowed.

Recalling that integration over a Grassmanian variable amounts to taking a derivative, we can nd

d 2 θd 2 θ Φ * i (exp(2g a T a V a )) ij Φ j = F * i F i + |D µ φ i | 2 + i ψi σ µ D µ ψ i - √ 2g a φ * i T a ij ψ α j λ a α - √ 2g a λa α ψ α i T a ij φ j + g a φ * i T a ij φ j D a (1.3.29) d 2 θ W ({Φ i }) θ=0 + h.c. = - 1 2 ∂ 2 W ∂Φ i ∂Φ j Φ i =φ i ψ i ψ j + ∂W ∂Φ i Φ i =φ i F i + h.c. = - 1 2 (M ij ψ i ψ j + h.c.) - 1 2 y ijk ψ i ψ j φ k + h.c. + ∂W ∂Φ i Φ i =φ i F i + h.c.
(1.3.30)

d 2 θ W aα W a α θ=0 + h.c. = 1 2 D a D a + i λa σ µ D µ λ a - 1 4 F aµν F a µν (1.3.31) d 2 θd 2 θ V = 1 2 D (1.3.32)
We can immediately recognise the usual kinetic terms for scalars, Weyl fermions (including gauginos) and gauge bosons, as well as SUSY masses for fermions and Yukawa interactions between scalars and fermions. New SUSY interactions also appear in the second line of eq. (1.3.29) between scalar and fermions. The auxiliary elds can then be eliminated using their equations of motion that read in the non-Abelian case

F * i = - ∂W ∂Φ i Φ i →φ i , D a = -g a φ * i T a ij φ j . (1.3.33)
Using the above equations of motion, terms in the Lagrangian containing the auxiliary elds can be rewritten, and one nds the (tree-level) scalar potential to be

V (0) ({φ i , φ * i }) = ∂W ∂Φ i Φ i →φ i 2 + 1 2 a g 2 a φ * i T a ij φ j 2 , (1.3.34)
where the two terms will be referred to as the F -term(s) and the D-term(s) Q φ i being the charges of the elds φ i under the U (1) group. With respect to the non-Abelian case, this yields an additional contribution to the scalar masses due to a term

L FI ⊃ gκQ φ i φ * i φ i , (1.3.36)
plus some constant terms. In the models considered in this thesis however there will be no Fayet-Iliopoulos terms in the low-energy part of the spectrum. 13

If we furthermore compute explicitly the F -terms for a general theory, with the superpotential as dened in eq. (1.3.23), we nd

V (0) F -terms = |L i | 2 + M * ij M ik φ * j φ k + 1 4 (y ijk ) * y ilm φ * j φ * k φ l φ m + L * i M ij φ j + 1 2 L * i y ijk φ j φ k + 1 2 M * ij y ikl φ * j φ k φ l + h.c. . (1.3.37)
Once we have dened the gauge group we are considering and the representations in which the scalars (or equivalently the chiral supermultiplets) are, we can compute the D-terms explicitly as well (see sections 1.3.4 and 1.3.5). It is worth noticing that from the structure of terms in 1.3.34 and the requirement of gauge invariance of W and L we know that the two scalars in the term (φ * i T a ij φ j ) 2 must be in the same representation, while φ j and φ k in the (y ijk ) * y ilm φ * j φ * k φ l φ m term can live in dierent representations of the gauge group.

In the end, we see that for a theory in which SUSY is unbroken it is sucient to give the (super)eld content (and the gauge group) together with the superpotential to x entirely the Lagrangian. However, as we will now see, Supersymmetry cannot be an exact symmetry of Nature and new SUSY-breaking terms will appear in the Lagrangian.

R-symmetry

The above description of supersymmetric Lagrangians can still, in some cases, accommodate an additional global U (1) R symmetry called R-symmetry under which the (chiral) superelds can be charged and the superspace coordinates θ,

θ transform as θ R -→ e iα θ, θ R -→ e -iα θ , (1.3.38)
where α is the parameter of the transformation (in other words θ and θ respectively carry R-charges +1 and -1). From this it can be shown (e.g. using their representation as dierential operators acting on superspace) that the SUSY generators have respective charges -1 and +1 under R transformations, and hence

[R, Q] = -Q, [R, Q] = Q . (1.3.39)
13 Fayet-Iliopoulos terms may appear in some mechanisms of spontaneous SUSY breaking at high- scales as we will briey discuss later.

Supersymmetry

Superelds can be assigned additive charges under the R-symmetry called R-charges and then transform as S(x, θ, θ)

R

-→ e ir S α S(x, e -iα θ, e iα θ) .

(1.3.40)

The supereld S * has R-charge -r S , and therefore real (i.e. gauge) superelds can not carry an R-charge. In the case of chiral superelds, as both the supereld itself and the superspace coordinates can carry an R-charge, the dierent components of the supereld will not have the same charge assignments: if the chiral supereld has an R-charge r Φ , then the scalar, fermion and auxiliary components have charges r Φ , r Φ -1, and r Φ -2 respectively. Moreover, one can nd from the charge assignments of θ and θ and the denitions in eq. (1.3.16) that the SUSY-covariant derivatives D α and D α have R-charges -1 and +1 and in turn one can deduce from this that eld-strength superelds have r W = 1.

As for any symmetry, the Lagrangian cannot carry any R-charge, which constrains the allowed terms and couplings in an R-symmetric SUSY model (depending on the charge assignments of the superelds in the model). The rst and third terms in the Lagrangian of a general SUSY model eq. (1.3.27) (as well as the Fayet-Iliopoulos term in eq. (1.3.28)) are always R-symmetric however, the second term will only be so provided that the superpotential W has an R-charge equal to 2.

We will encounter R-symmetry in the MRSSM a Dirac gaugino model devised to preserve a continuous R-symmetry as well as the particular subgroup Z 2 of U (1) R under which the MSSM and the NMSSM are built to be invariant. In both cases (U (1) R R-symmetry or R-parity), we will see that the new symmetry strongly constrains the superpotential and soft SUSY-breaking Lagrangian of the models.

SUSY breaking

An important observation one can make from the last equation in (1.3.10) is that as the momentum operator P µ commutes with the SUSY generators, the squared-mass operator P 2 will commute as well and therefore particles in a same supermultiplet must have the same mass. This would mean that if Supersymmetry was not broken, a Standard Model particle and its superpartner would have the identical masses; but this would imply for example selectrons (scalar superpartners of electrons) with a mass of 511 keV! As these would necessarily have been observed a long time ago, Supersymmetry has to be broken, at least at low energies.

The study of SUSY breaking and of the possible mechanisms at work is a vast subject, which partly lies beyond the scope of the phenomenological approach to Supersymmetry. It is well accepted that Supersymmetry is broken spontaneously, or in order words that the complete Lagrangian of the theory is SUSY invariant but that the vacuum of the theory is not. In fact, it can be shown that SUSY is spontaneously broken if any of the (scalar) auxiliary elds F i or D a have non-zero VEVs. The proof of this statement is simple and is as follows [START_REF] Martin | A Supersymmetry primer[END_REF] • rst if one takes the rst line of eq. (1.3.10) for (α, α) = (1, 1) or (2, 2) one can recover the Hamiltonian in terms of SUSY generators

{Q 1 , Q1 } + {Q 2 , Q2 } = 2σ µ 1 1P µ + 2σ µ 2 2P µ = 2 tr(σ µ )P µ = 4P 0 = 4H (1.3.41)
where the trace is in spinor space, and H ≡ P 0 is the Hamiltonian. • if we further consider that there are no space-time dependent eects (and no fermion condensates) then

H = d 3 x V (0) = d 3 x F * i F i + 1 2 (D a ) 2 (1.3.43)
and therefore • F -term (or O'Raifeartaigh) breaking, in which it is the auxiliary eld F of a gauge singlet chiral supereld that acquires a VEV.

vac|H|vac = d 3 x F * i F i + 1 2 D a
In neither case is it possible to accommodate the superelds required for SUSY breaking with the low-energy spectrum of a physically realistic SUSY theory. The SUSY breaking must then occur in some hidden sector and later be propagated to the observed sector by some new mechanism, such as for example 14 gauge mediation or Planck-scale suppressed mediation, typically at a very high-scale.

To study SUSY theories at phenomenologically accessible scales (MSSM, NMSSM, etc.), it is more convenient to work in an eective eld theory approach and consider an eective low-energy Lagrangian for the model in which all the hidden and mediation sectors are integrated out. The information about the precise mechanism of SUSY breaking is then contained in new non-renormalisable operators that arise from the matching with the complete SUSY theory and break Supersymmetry explicitly. A subset of these terms have the additional property of not spoiling the solution that SUSY provides to the technical hierarchy problem by reintroducing quadratic divergences in scalar masses [START_REF] Girardello | Soft Breaking of Supersymmetry[END_REF], and are hence called soft terms (because they only reintroduce soft, i.e logarithmic, divergences). More precisely, soft terms can be Majorana mass terms for gauginos, mass terms for scalars, bilinear and trilinear couplings between scalars, and tadpole terms, i.e.

L soft = - 1 2 M a λ a λ a + h.c. -m 2 ij φ * i φ j - 1 6 a ijk φ i φ j φ k + 1 2 b ij φ i φ j + t i φ i + h.c. . (1.3.45)
14 One could also mention mediation through anomalies or extra-dimension. See e.g. section 7 of [START_REF] Martin | A Supersymmetry primer[END_REF] for further discussion of how SUSY breaking can be mediated.

Supersymmetry

A further type of terms, of the form

L soft ⊃ - 1 2 c ijk φ * i φ j φ k , (1.3.46)
can be soft provided that there is no gauge singlet in the theory (the reason for this condition is that the above terms may generate divergences in singlet tadpoles). The possible soft terms in a given model are of course restricted because of gauge invariance, but it is interesting to note that Majorana masses for gauginos, as well as diagonal scalar masses (i.e. taking i = j in the second term of eq. (1.3.45)) are always allowed. Soft terms are yet further constrained in models that also respect an R-symmetry (as discussed in the previous section), however without specifying the R-charge assignments of the superelds, one can only observe that on the one hand, any diagonal soft scalar mass term is R-symmetric while, on the other hand, a Majorana mass term always breaks the R-symmetry (because the eld-strength superelds, and thus the gauginos, have R-charge +1).

As soft mass terms and couplings are all produced through the same SUSY breaking process, they are expected to lie around a common mass (or mass-squared) scale, usually called the (soft) SUSY breaking scale, which is also the energy scale around which SUSY particles should be found if they exist. As we will see in more detail in the next chapter when reviewing Higgs mass calculations, the SUSY scale was rst believed to be around, or a little above, the electroweak scale, however, as there is not yet any evidence of SUSY in experiments, it is currently expected to be of the order of a few TeV, or possibly much higher. As a very brief example, let us now examine the case of an operator coupling a eldstrength supereld W a of the observed sector to a chiral supereld X of the hidden sector

L ⊃ -d 2 θ c a X M W aα W a α θ=0 (1.3.47)
with c a some coupling and M a (heavy) mass scale, required to keep the coupling dimensionless this scale can be for example the Planck scale M Pl in models with Planck-scale-mediated SUSY breaking. If we now suppose that the scalar and fermion components of X vanish and that only the auxiliary eld F X remains and obtains a VEV F X , then the above operator gives a contribution to the Lagrangian

L ⊃ -d 2 θ c a θθ F X M W aα W a α θ=0 + h.c. = - 1 2 M a λ aα λ a α + h.c. . (1.3.48)
Hence, through this toy model of F -term breaking, we have generated in the low energy theory a Majorana mass term M a ≡ 2c a F X /M for the gauginos.

Similarly, we can obtain a soft mass term for the scalar component of a chiral supereld

Φ through an operator L ⊃ -d 2 θd 2 θ f X * X M 2 Φ * Φ = -m 2 φ φ * φ (1.3.49)
where

m 2 φ = f F X * F X /M

The hierarchy problem and Supersymmetry

In this section, we will show how Supersymmetry solves the hierarchy problem, presented in section 1.2.1. It is important to note, however, that Supersymmetry was not invented for the purpose of solving the hierarchy problem and that the fact that it does is only a fortuitous by-product of the extended symmetry. More precisely it is a consequence of one of the non-renormalisation theorems that have been proven for SUSY.

In particular it has been shown that the superpotential is not renormalised [START_REF] Grisaru | Improved Methods for Supergraphs[END_REF][START_REF] Seiberg | Naturalness versus supersymmetric nonrenormalization theorems[END_REF].

Let us investigate the cancellation of scalar mass corrections a bit more explicitly by considering a very simple toy model of a massless chiral supereld L interacting with another heavy chiral supereld H. In order to avoid mixing among the two superelds, we require the model, and hence the superpotential, to be invariant under the Z 2 transformation H → -H. The superpotential is then

W (Φ) = 1 2 M H 2 + 1 2 Y LH 2 .
(1.3.50)

Using the results of the previous sections, and assuming M and Y to be real, we nd the Lagrangian of this model to be

L = |∂ µ L| 2 + |∂ µ H| 2 + i ψL σ µ ∂ µ ψ L + i ψH σ µ ∂ µ ψ H -M 2 |H| 2 - 1 2 M ψ H ψ H + h.c. - 1 2 Y ψ H ψ H L + 2ψ L ψ H H + h.c. -Y M L|H| 2 + h.c. -Y 2 |L| 2 |H| 2 - 1 4 Y 2 |H| 4 (1.3.51)
There are three diagrams that will contribute to the radiative corrections to the light scalar mass, as shown in gure 1.2. We can compute each contribution and we nd 15

Π (A) LL * (p 2 ) = -(-iY 2 ) d d k i(2π) d i k 2 -M 2 = Y 2 16π 2 A(M 2 ) , (1.3.52) Π (B) LL * (p 2 ) = -(-iY M ) 2 d d k i(2π) d i k 2 -M 2 i (p + k) 2 -M 2 = - Y 2 M 2 16π 2 B(p 2 , M 2 , M 2 ) , (1.3.53)
15 Note that the coecient of the M 2 × B term in Π (C) LL * diers from the result in section 1.2.1 because a loop of Weyl fermions with mass insertions is not allowed here, as there is no ψH ψH L term.

1.3 Supersymmetry

Π (C) LL * (p 2 ) = + (-iY ) 2 2 d d k i(2π) d iσ µ α αk µ k 2 -M 2 i(σ ν ) αα (k ν + p ν ) (p + k) 2 -M 2 = Y 2 d d k i(2π) d k • (k + p) (k 2 -M 2 )((p + k) 2 -M 2 ) = - Y 2 16π 2 A(M 2 ) -M 2 - p 2 2 B(p 2 , M 2 , M 2 ) (1.3.54)
where we used σ µ α α(σ ν ) αα = tr σ µ σ ν = 2η µν (the reader may refer to [START_REF] Dreiner | Two-component spinor techniques and Feynman rules for quantum eld theory and supersymmetry[END_REF] for details on calculations with two-component spinors). The typical momentum at which these diagrams are is the tree-level light scalar mass, or in other words p 2 = 0, and we have then

Π (A) LL * (0) + Π (B) LL * (0) + Π (C) LL * (0) = 0 (1.3.55)
showing that the light scalar mass does not receive any correction, neither a quadratically divergent correction like existed in the SM, nor actually any nite correction.

The above discussion was in the case of unbroken Supersymmetry, but we will now show that the cancellation of quadratic divergences in scalar masses is not spoilt by soft SUSY breaking. For this purpose, let us suppose that the scalar component H of the supereld H in our toy model receives a soft mass, that we denote m 2 soft . H then has mass m 2 = M 2 + m 2 soft and consequently the expressions of the (A) and (B) parts of the L self-energy must be modied with the replacement of the masses in the loop functions M 2 → m 2 (note that the factor M 2 in front of the B function in eq. (1.3.53) is not changed because it comes from the L|H| 2 coupling in the Lagrangian). In turn, the neat cancellation of the quadratic divergences of each term in eq. (1.3.55) is also modied, however, because we are interested in the behaviour of the light scalar mass corrections for large M and because we expect the soft SUSY-breaking scale m soft to be much lower, we can obtain

A(m 2 ) -A(M 2 ) = m 2 log m 2 -m 2 -M 2 log M 2 + M 2 = (M 2 + m 2 soft ) log M 2 + log 1 + m 2 soft M 2 -M 2 log M 2 -m 2 soft = m 2 soft log M 2 , (1.3.56)
to leading order in m 2 soft /M 2 . Similarly, we have in the limit p 2 = 0

M 2 B 0 (M 2 , M 2 ) -B 0 (m 2 , m 2 ) = -m 2 soft + • • • . (1.3.57)
The one-loop corrections to the light scalar masses are in the end

Π (A) LL * (0) + Π (B) LL * (0) + Π (C) LL * (0) = Y 2 16π 2 m 2 soft log M 2 -m 2 soft . (1.3.58)
This equation does indeed show new terms with respect to the unbroken-SUSY case of equation (1.3.55), however, the dependence on the heavy mass M is logarithmic,

and not quadratic as we might have feared and as is the case in the SM. Such a logarithmic divergence in the scalar mass is far less severe than a quadratic one, and it is therefore referred to as a soft divergence (which also explain the name of soft SUSY breaking terms). A complete discussion, for a realistic model and taking into account the dierent types of soft terms that are allowed, was done in [START_REF] Girardello | Soft Breaking of Supersymmetry[END_REF], showing that quadratic divergences are not reintroduced because of soft SUSY breaking.

We have here shown that Supersymmetry is able to solve the technical hierarchy problem, and for this it was not necessary to suppose that the SUSY breaking scale is

The Higgs boson and Physics beyond the Standard Model around the electroweak scale. We did assume m soft M , which served in particular to simplify the term M 2 log(1+m 2 soft /M 2 ), but if we imagine once more that M ∼ M Pl , there is ample space for m soft to be small with respect to M but large with respect to the EW scale v. However, as the SUSY scale is being driven higher up by experiments comes the problem of explaining the hierarchy between the electroweak and SUSY scales: currently the ratio of these scales is expected to be of the order of or larger than 10. This is called the little hierarchy problem and currently poses a challenge to Supersymmetry, at least in its minimal versions (as the MSSM).

Minimal models

There are numerous ways of extending the Standard Model into Supersymmetric theories, and SUSY model building is an extremely active domain nowadays. In this section, we will start the discussion of the Supersymmetric models considered in this thesis by minimal extensions of the SM, the MSSM and the NMSSM.

The Minimal Supersymmetric Standard Model

The rst and most natural way to extend the SM in the context of a (low-energy) SUSY model is to nd the simplest way to assign SM elds in chiral and gauge supermultiplets, while adding only the smallest possible number of additional states to ensure the theoretical consistency of the model. This can actually be done quite simply: rst, the SM fermions quarks and leptons are part of chiral supermultiplets and their superpartners called squarks and sleptons belong to the same representations of the SM gauge group (c.f. table 1.1). Then the gauge bosons are put in gauge supermultiplets and have fermionic superpartners respectively gluinos, Winos and the Bino that transform in the adjoint representation of the corresponding component of the gauge group. Finally, the Higgs boson is also part of a chiral supermultiplet but here the situation requires a little more care, because if there were only one fermionic partner of the Higgs a higgsino the electroweak gauge symmetry would be anomalous. Furthermore, with only one Higgs chiral supereld, there would be no way to write a holomorphic superpotential leading to Yukawa interactions for both up-and down-type quarks, nor any possibility to obtain a (supersymmetric) mass term for the higgsino. To avoid these three problems, two Higgs doublet superelds with opposite weak hypercharges are needed and we will call them H u and H d , the subscripts u and d indicating which type of quarks (up-type or down-type) that each doublet couples to. Moreover, to forbid interactions that violate 16 B -L, the model can be required to be invariant under a discrete symmetry, called R-parity, dened by a new quantum number R P ≡ (-1) 3(B-L)+2s , (1.3.59) that takes values +1 for SM particles including the Higgs bosons and -1 for all the new superpartners. A very interesting consequence of this R-parity is that the lightest particle with R P = -1, i.e. the lightest superpartner (LSP), must be stable, which can obviously have strong phenomenological implications such as the LSP being a possible candidate of dark matter particle.

The model one nally obtains is the Minimal Supersymmetric Standard Model (MSSM), and its complete (super)eld content is summarised in table 1.3 (on page 49) together 16 B and L are respectively the baryon and lepton numbers.

W MSSM = µH u • H d + y f g u Q f • H u u c g -y f g d Q f • H d d c g -y f g e L f • H d e c g (1.3.60)
where we denote all superelds in bold to avoid confusion with their component elds, and we use a dot

• to indicate contracted SU (2) L indices as H u • H d = ij H ui H dj ,
and wrote family indices as in section 1.1. The MSSM Lagrangian also contains several soft SUSY breaking terms, that can be written in all generality as

L soft MSSM = -m 2 Hu H * u H u -m 2 H d H * d H d -(B µ H u • H d + h.c.) -(m 2 Q ) f g Q * f L QgL -(m 2 ū) f g ũc f R (ũ c gR ) * -(m 2 d) f g dc f R ( dc gR ) * -(m 2 L ) f g L * f L LgL -(m 2 ē ) f g ẽc f R (ẽ c gR ) * (1.3.61) - 1 2 M 3 ga ga + 1 2 M 2 W a W a + 1 2 M 1 B B + h.c. -(a u ) f g ũc f R QgL • H u + (a d ) f g dc f R QgL • H d + (a e ) f g ẽc f R LgL • H d + h.c. .
In the above soft Lagrangian, a u , a d , and a e are 3×3 complex matrices in family space, while m 2 Q , m 2 ū, m 2 d m 2 L , and m 2 ē are 3 × 3 Hermitian matrices in family space. We will follow the commonly choice of expressing soft trilinear couplings as the corresponding (dimensionless) superpotential coupling times a (mass-dimensional) term, which means here that the trilinears a u , a d , a e are related to the Yukawas by matrices A u , A d , A e such that a u = y u A u , etc. Moreover, the bilinear coupling B µ can always be chosen to be real and positive, through a redenition of the phase of one of the Higgs doublets.

Mostly because of the SUSY breaking terms, the number of additional free parameters (masses, mixing angles, and phases) in the MSSM with respect to the SM is huge it found in [START_REF] Dimopoulos | The Supersymmetric avor problem[END_REF] to be larger than a hundred. Fortunately, many of these parameters are severely constrained, especially by experimental data from avour physics (one could mention bounds on the µ → eγ process or on mixing between neutral kaons for example). Also, simplifying assumptions are often made from a theoretical point of view: for example when studying radiative corrections, it is common (as in chapter 3) to neglect 17 the eects of all leptons and of lighter quarks (i.e. quarks other than the top), and therefore assume that

y u =   0 0 0 0 0 0 0 0 y t   , a u =   0 0 0 0 0 0 0 0 y t A t   , y d = 0 , y e = 0 , a d = 0 , a e = 0 . (1.3.62)
In some cases, such as phenomenological studies of models where parameters are scanned over to determine the regions of parameter space that are still allowed by experimental results, stronger assumptions are made to reduce the number of free parameters to one that allows numerical scans. One strategy is to impose unication conditions on parameters at some Grand Unied Theory (GUT) scale (M GUT ), together with some particular mechanism of SUSY breaking at high scales. One example of a variant of the MSSM in this setting is the Constrained MSSM, or CMSSM 17 However, as we will see at the end of the next subsection, neglecting the bottom and tau Yukawa couplings is only possible when the ratio of Higgs VEVs tan β is not too large, because of the tan β enhancement both Yukawas receive (see eq. (1.3.84))

The Higgs boson and Physics beyond the Standard Model (often called mSUGRA because the SUSY breaking is Planck-scale mediated in this model) in which there are only ve free parameters: common masses of all sfermions m 0 (Q = M GUT ) and of gauginos m 1/2 (Q = M GUT ); a common soft trilinear coupling for all sfermions A 0 (Q = M GUT ) (i.e. A u = A d = A e = diag(A 0 , A 0 , A 0 )); the ratio of the Higgs doublet VEVs tan β (we will dene this ratio properly in what follows) and nally the sign of the SUSY Higgs mass µ. Similar models also exist assuming anomaly-mediated or gauge-mediated SUSY breaking, respectively the mAMSB and the mGMSB. Another possible strategy is to use phenomenological assumptions to limit the possible terms in the low-energy Lagrangian, an example of this being the phenomenological MSSM (or pMSSM) relying on three assumptions: that there are no new sources of CP-violation with respect to the SM, that there are no avour changing neutral currents and that there is rst and second generation universality. With these assumptions (motivated by experimental data), the pMSSM has 19 free parameters (see e.g. [START_REF] Djouadi | The Minimal supersymmetric standard model: Group summary report[END_REF] for more details). Additional constraints can be considered, limiting the number of parameters and to distinguish the variants of the pMSSM it is common to refer to the variant with the number of parameters (e.g. pMSSM11, pMSSM17, etc.). Such versions of the MSSM, with constraints from the GUT scale or from phenomenological assumptions, are under intense scrutiny from theorists and experimentalists and several studies with likelihood analyses and global ts have been performed in recent years, mostly by the GAMBIT [START_REF] Athron | Global ts of GUT-scale SUSY models with GAMBIT[END_REF][START_REF] Athron | A global t of the MSSM with GAMBIT[END_REF] and MasterCode [6769] groups.

The Higgs sector of the MSSM

Let us now investigate further the Higgs sector and how the electroweak symmetry is broken in the MSSM. First, each of the scalar components of the two doublet chiral superelds can be decomposed into their charged and neutral components

H u = (H + u , H 0 u ) , H d = (H 0 d , H - d )
.

( We therefore do not consider squark and slepton terms in the scalar potential and nd

V (0) = ∂W ∂H u Hu→Hu 2 + ∂W ∂H d H d →H d 2 + 1 2 g 2 Y Hu H * u H u + Y H d H * d H d 2 + 1 8 g 2 H - u (H 0 u ) * 1 1 -i 1 + i -1 H + u H 0 u + (H 0 d ) * H + d 1 1 -i 1 + i -1 H 0 d H - d 2 -L soft MSSM Hu,H d (1.3.64)
18 Note that spontaneous R-parity violation is actually excluded by experiments for the standard version of the MSSM [START_REF] Barbier | R-parity violating supersymmetry[END_REF].

Supersymmetry

= |µ| 2 |H 0 u | 2 + |H + u | 2 + |H 0 d | 2 + |H - d | 2 + m 2 Hu |H 0 u | 2 + |H + u | 2 + m 2 H d |H 0 d | 2 + |H - d | 2 + B µ (H + u H - d -H 0 u H 0 d ) + h.c. + 1 8 (g 2 + g 2 ) |H 0 u | 2 + |H + u | 2 -|H 0 d | 2 -|H - d | 2 2 + 1 2 g 2 H + u (H 0 d ) * + H 0 u H + d 2 .
In principle, we need to verify that we can break the electroweak gauge symmetry without also breaking U (1) QED with this potential. Actually this is quite straightforward [START_REF] Martin | A Supersymmetry primer[END_REF]: rst with an SU (2) L gauge transformation it is possible to rotate away the VEV from one of the components of one of the doublets, say H + u . Once that is done, one can take the derivative

∂V (0) ∂H + u min., with H + u =0 = B µ H - d + g 2 H 0 d * H 0 u * H - d = 0 (1.3.65)
If we suppose for a moment that H - d = 0, then the above equation implies that

H 0 d * H 0 u * = -B µ /g 2 ∈ R , (1.3
v 2 u + v 2 d = v 2 = 4m 2 Z g 2 + g 2 .
(1.3.67)

Another very useful quantity is the ratio of the Higgs VEVs

tan β ≡ v u v d , (1.3.68)
which will appear on many occasions.

In order to study the electroweak symmetry breaking, it is then sucient to consider only the terms in the potential involving H 0 u and H 0 d , i.e. use the potential

V (0) = |µ| 2 + m 2 Hu |H 0 u | 2 + |µ| 2 + m 2 H d |H 0 d | 2 -B µ H 0 u H 0 d + h.c. (1.3.69) + 1 8 (g 2 + g 2 ) |H 0 u | 2 -|H 0 d | 2 2 .
We notice at rst that along the direction H 0 u = H 0 d (called the D-at direction) the potential may not be bounded from below, unless [START_REF] Martin | A Supersymmetry primer[END_REF] 2B µ < 2|µ| 2 + m 2 Hu + m 2 H d .

(1.3.70)

Furthermore, as there are two states that can acquire VEVs, there remains two (nontrivial) minimum conditions for the potential given by its derivatives with respect to H 0 u and H 0 d . One can then ensure that v u = v d = 0 is an unstable maximum of V (0) by requiring that the Hessian matrix of V (0) at v u = v d = 0 must have at least one negative eigenvalue, which gives the additional condition on B µ [55]

B 2 µ > (|µ| 2 + m 2 Hu )(|µ| 2 + m 2 H d ) (1.3.71)
The Higgs boson and Physics beyond the Standard Model Under these assumptions, the tadpole equations at tree-level read

√ 2 v u ∂V (0) ∂H 0 u min. = m 2 Hu + |µ| 2 -B µ v d v u - 1 8 (g 2 + g 2 )(v 2 u -v 2 d ) = 0 = m 2 Hu + |µ| 2 -B µ cot β - m 2 Z 2 cos 2β = 0 , √ 2 v d ∂V (0) ∂H 0 d min. = m 2 H d + |µ| 2 -B µ v u v d + 1 8 (g 2 + g 2 )(v 2 u -v 2 d ) = 0 = m 2 Hu + |µ| 2 -B µ tan β + m 2 Z 2 cos 2β = 0 . (1.3.72)
The two most common choices for these tadpole equations are to solve for the soft

H + u H + d = s β ± c β ± -c β ± s β ± G + H + (1.3.73)
where we use the short-hand notations c x ≡ cos x and s x ≡ sin x. Turning now to H 0 u and H 0 d , we know that they must contain the neutral Goldstone G 0 as well as another CP-odd (pseudoscalar) Higgs A and two CP-even Higgses h and H. Conventionally, the mass eigenstates are dened by

H 0 u H 0 d = 1 √ 2 v sin β v cos β + 1 √ 2 c α s α -s α c α h H + 1 √ 2
s β 0 c β 0 -c β 0 s β 0 G 0 A . (1.3.74)
In general the lightest CP-even Higgs h is assumed to correspond to the Higgs boson found at the LHC, although scenarios exist in which it is the heavy Higgs H that corresponds to the observed state, and the lighter h escapes detection in a very narrow range of masses not probed by the LEP or the LHC (such scenarios are becoming increasingly restricted as experiments further probe the possible parameter space).

At tree-level, a straightforward calculation shows that β 0 = β ± = β and that α conventionally taken in the interval -π 2 , 0 (for

m 2 A > m 2 Z ) is related to β by the relation sin 2α sin 2β = - m 2 H + m 2 h m 2 H -m 2 h , (1.3.75) or equivalently tan 2α tan 2β = m 2 A + m 2 Z m 2 A -m 2 Z . (1.3.76)
The tree-level scalar masses are (at the minimum of the tree-level potential)

m 2 A = 2B µ sin 2β , m 2 h,H = 1 2 m 2 A + m 2 Z ∓ (m 2 A -m 2 Z ) 2 + 4m 2 Z m 2 A sin 2 2β 1/2 , m 2 H ± = m 2 A + m 2 W , (1.3.77)
1.3 Supersymmetry while the Goldstone masses are zero as expected. The m 2 Z terms in the CP-even Higgs masses arise from the fact that, as opposed to the SM (and many of its non-SUSY extensions) the Higgs quartic couplings are not free parameters in Supersymmetry, but are related to the electroweak gauge couplings by the D-terms. A major consequence of this is that there is an upper limit in the MSSM on the lightest Higgs mass at tree-level [START_REF] Inoue | Low-Energy Parameters and Particle Masses in a Supersymmetric Grand Unied Model[END_REF][START_REF] Flores | Higgs Masses in the Standard, Multi-Higgs and Supersymmetric Models[END_REF], namely

m 2 h ≤ m 2 Z cos 2 2β , (1.3
β = π 2 -α , (1.3.79)
and this can occur even without decoupling (e.g. because of some symmetry) one then talks of alignment without decoupling. Alignment, whether through decoupling or without it, is a powerful tool to relax experimental constraints, as signs of new Physics in the Higgs sector are hidden. If furthermore there is decoupling, then an extended Higgs sector cannot be distinguished from the SM one at currently accessible energy scales.

As was the case in the SM, the previous discussion of the Higgs sector is modied when one takes into account radiative corrections to the potential. First of all the tadpole equations (1.3.72) are modied by the inclusion of the derivatives of the radiative corrections to the potential ∆V = V e -V (0) with respect to H 0 u and H 0 d respectively.

This in turn changes the parameters for which the tadpole equations are solved (either

m 2 Hu , m 2 H d
or |µ| 2 , B µ ) and therefore the Goldstone tree-level masses as well. Unlike the SM, however, the tree-level masses in the minimum of the loop-corrected potential are not the same for the neutral and charged Goldstone bosons. These were shown in [START_REF] Kumar | Resummation of Goldstone boson contributions to the MSSM eective potential[END_REF] to be (up to a dierent choice of convention for the VEVs)

m 2 G 0 = -δ u s 2 β -δ d c 2 β - (δ u -δ d ) 2 8B µ s 3 2β + • • • m 2 G ± = -δ u s 2 β -δ d c 2 β - (δ u -δ d ) 2 8(B µ + g 2 v u v d /4) s 3 2β + • • • (1.3.80)
with the shorthand notations

δ u ≡ 1 √ 2v u ∂∆V ∂H 0 u min. , δ d ≡ 1 √ 2v d ∂∆V ∂H 0 d min.
.

(1.3.81)

In section 4.1.2, we will see how to extend the procedure for deriving the Goldstone tree-level masses in the minimum of the full potential in a general theory and we will

The Higgs boson and Physics beyond the Standard Model give explicitly the expressions of the leading (i.e. one-loop) terms see eq. (4.1.26). As we had found for the SM in eq. (1.1.38), the Goldstone masses are formally of one-loop order, but can vanish or become negative leading to a Goldstone Boson Catastrophe in the MSSM as well. This was rst addressed in Ref. [START_REF] Kumar | Resummation of Goldstone boson contributions to the MSSM eective potential[END_REF] where the resummation method of [START_REF] Martin | Taming the Goldstone contributions to the eective potential[END_REF] was applied to the MSSM to obtain an eective potential and tadpole diagrams free of infrared divergences. It is worth mentioning here that many calculations of Higgs masses in the MSSM have been performed in a so-called gaugeless limit, in which the electroweak gauge couplings g, g are set to zero, and where the Goldstone Boson

Catastrophe does not appear. The reason for this is that, as we saw earlier, the Higgs quartic coupling in the MSSM is related to the electroweak gauge couplings, therefore in the gaugeless limit the Higgs quartic coupling also vanishes and the Goldstones decouple from the Higgs bosons.

Finally, let us consider the interactions of the Higgs bosons with fermions and sfermions, and in particular the generation of fermion and sfermion masses. We saw in eq. (1.3.30) that the Yukawa interactions, between fermions and the Higgs doublets, are found via the second derivatives of the superpotential, which in this case gives

L ⊃ -y f g u u f L u c gR H 0 u -d f L u c gR H + u + y f g d u f L d c gR H - d -d f L d c gR H 0 d + y f g e ν f L e c gR H - d -e f L e c gR H 0 d . (1.3.82)
If we only consider the third generation fermions, we nd their tree-level masses to be

m t = y t √ 2 v u = y t √ 2 v sin β , m b = y b √ 2 v d = y b √ 2 v cos β , m τ = y τ √ 2 v d = y τ √ 2 v cos β , (1.3.83)
where y t , y b , and y τ the top, bottom and tau Yukawas are the largest eigenvalues of the three Yukawa coupling matrices respectively. With respect to the SM case, the top (or bottom and tau) Yukawa coupling is modied by a new factor sin β (cos β). This is especially noticeable when tan β 1 where (at tree-level)

y MSSM b y MSSM t = m b m t tan β = y SM b y SM t tan β (1.3.84) while y MSSM t = y SM t / sin β ≈ y SM t .
This means that in the large tan β limit of the MSSM, the bottom Yukawa (and similarly the tau one) can be much larger than their SM counterparts, so that even if it is still possible to neglect m b , m τ m t it is not possible to neglect consistently y b (and y τ ) before y t .

If we then turn to couplings between the Higgses and the sfermions, these can arise from dierent sources. The rst one is via the squared rst derivatives of the same Yukawa terms in the superpotential, giving quartic interactions of the form |sfermion-Higgs| 2 . To this are also added cross terms between the derivatives of the Yukawa and SUSY Higgs mass terms of W MSSM and soft trilinear couplings, both giving sfermionsfermion-Higgs interactions. Finally, a last type of terms come from the D-term part of the scalar Lagrangian yielding terms of the form |sfermion| 2 |Higgs| 2 . To study the masses of sfermions it actually suces to compute couplings between sfermions and the neutral Higgses only. Furthermore, in order to simplify the discussion, we will not consider rst and second generation sfermions that have very small (and hence usually 1.3 Supersymmetry neglected) couplings to the Higgs sector. We obtain in this case

L ⊃ -y 2 t ( tL t * L + tR t * R )|H 0 u | 2 -y 2 b ( bL b * L + bR b * R )|H 0 d | 2 -y 2 τ (τ L τ * L + τR τ * R )|H 0 d | 2 + µ * y t tL t * R (H 0 d ) * + y b bL b * R (H 0 u ) * + y τ τL τ * R (H 0 u ) * + h.c. -y t A t tL t * R H 0 u + y b A b bL b * R H 0 d + y τ A τ τL τ * R H 0 d + h.c. + 1 2 (|H 0 u | 2 -|H 0 d | 2 ) 1 2 g 2 - 1 6 g 2 tL t * L + 2 3 g 2 tR t * R + - 1 2 g 2 - 1 6 g 2 bL b * L - 1 3 g 2 bR b * R + - 1 2 g 2 + 1 2 g 2 τL τ * L -g 2 τR τ * R , (1.3.85)
where we use (a

u ) 33 = y t A t , (a d ) 33 = y b A b , (a e ) 33 = y τ A τ .
Here we included the contribution from the D-terms for completeness, however these are sometimes neglected as will be the case in chapter 3 (moreover they vanish in the gaugeless limit).

The eld-dependent stop mass matrix M 2 stops is dened as

L ⊃ -( t * L t * R )M 2 stops tL tR . (1.3.86)
In the notations of chapter 3 (themselves taken from [START_REF] Degrassi | On the neutral Higgs boson masses in the MSSM for arbitrary stop mixing[END_REF][START_REF] Degrassi | On the radiative corrections to the neutral Higgs boson masses in the NMSSM[END_REF]), and neglecting the D-term contributions, the stop mass matrix in the MSSM are found to be

M 2 stops = m 2 Q + |X| 2 X * X m 2 U + |X| 2 , (1.3.87)
where

m 2 Q = (m 2 Q ) 33 and m 2 U = (m 2
ū) 33 see eq. (1.3.61) and

|X| 2 = y 2 t |H 0 u | 2 , (1.3.88) X = y t A t H 0 u -µ * (H 0 d ) * . (1.3.89)
This matrix can be diagonalised to give two eld-dependent mass eigenvalues

m 2 t1,2 = 1 2 (m 2 Q + m 2 U + 2|X| 2 ) ± (m 2 Q -m 2 U ) 2 + 4| X| 2 , (1.3.90)
and similarly mass eigenvalues can be derived for sbottoms or staus. We will return to and rene the discussion of the stop masses in chapter 3 in the context of Dirac gaugino models. Before we end this section there is a last quantity that we need to dene, for the discussion of radiative corrections to Higgs masses, namely the stop mixing, that we dene at the minimum of the potential as

X t = X m t = A t -µ * cot β . (1.3.91)
Finally, note that in many cases (e.g. the real MSSM, or in our study of Dirac gaugino models in chapter 3) the supersymmetric Higgs mass parameter µ is taken to be real, which also has consequences for X and X t . For a squark in general, the mixing parameter can be dened as

X q = A q -µ * F(β) , (1.3.92)
where F(β) = cot β for q = u, c, t, and F(β) = tan β for q = d, s, b. Therefore, we will in what follows not consider the electroweak gauginos, but only the gluinos ga , that do not mix with fermions from the Higgs sector (because they are the only fermions to transform as octets under SU (3) C ). They couple to squarks with the strong gauge coupling, and for this reason their contributions to the neutral scalar masses from two-loop and beyond can be large and must be taken into account.

Shortcomings of the MSSM

One serious theoretical issue of the MSSM is the so-called µ-problem, i.e. the problem of generating the supersymmetric Higgs mass term µ that must be of the order of the SUSY breaking scale M SUSY (the scale of the SUSY breaking terms). The reason why µ must be of the order of M SUSY can be explained as follows [START_REF] Ellwanger | The Next-to-Minimal Supersymmetric Standard Model[END_REF]: on the one hand, µ cannot be zero, because (among other reasons) there is a lower bound on the possible value of |µ| coming from the experimental lower bound on chargino masses. On the other hand, if µ is too large then the extremum of the potential at H u = H d = 0 becomes a stable minimum and EWSB cannot take place (e.g. relation (1.3.71) cannot be veried). Therefore, the only scale that µ can be related to is the SUSY breaking scale M SUSY , which is also apparent from eq. (1.3.71) but is contradictory for a SUSY preserving parameter. As we will see in what follows, this problem can be solved in both the NMSSM and Dirac gaugino models [START_REF] Benakli | Generating mu and Bmu in models with Dirac Gauginos[END_REF], by generating the µ term through the VEV of a dynamical eld having the same quantum numbers as µ, i.e. a singlet.

Another issue of the MSSM is to nd a way to have suciently large radiative corrections to the Higgs mass to obtain a Higgs mass of 125 GeV even if the tree-level mass has a low upper-bound (1.3.78). As will be shown in the next chapter where we will discuss Higgs mass calculations, for the radiative corrections to be suciently large in the MSSM either of the following conditions needs to be fullled: large stop masses, or large stop mixing. Again the NMSSM and Dirac gaugino models improve the situation on this matter, by relaxing the upper-bound on m h (more precisely, the equivalents of eq. (1.3.78) contain additional terms, due to the singlet or the adjoint superelds respectively).

1.3 Supersymmetry Experimental searches have so far not yielded any sign of supersymmetric particles and thus raise the exclusion limits on possible masses for the superpartners e.g. gure 1.3

shows for a summary of SUSY searches from the ATLAS collaboration. In particular, the stops were thought to be among the rst superpartners to be observed at the LHC being coloured particles, with strong couplings to other particles due to the large g S and y t , and being lighter than rst and second family squarks in some models of SUSY breaking 19 but Runs 1 and 2 have only been able to set a lower limit on their mass. Figure 1.4 provides an example of the limits on m t1 obtained in the case of stop pair production. More generally, the latest public results for lower bounds on stop masses are at 95% condence level 1150 GeV from ATLAS [START_REF] Aaboud | Search for squarks and gluinos in nal states with jets and missing transverse momentum using 36 fb -1 of √ s=13 TeV pp collision data with the ATLAS detector[END_REF], and 1000 GeV from CMS [START_REF] Sirunyan | Search for natural and split supersymmetry in proton-proton collisions at √ s = 13 TeV in nal states with jets and missing transverse momentum[END_REF] in the case of compressed spectra the bounds are weaker, namely 430

GeV for ATLAS [START_REF] Aaboud | Search for dark matter and other new phenomena in events with an energetic jet and large missing transverse momentum using the ATLAS detector[END_REF] and 510 GeV for CMS [START_REF] Sirunyan | Search for the pair production of third-generation squarks with two-body decays to a bottom or charm quark and a neutralino in protonproton collisions at √ s = 13 TeV[END_REF].

As mentioned earlier, this then poses questions as to whether Supersymmetry, if it exists, provides a satisfactory answer to the hierarchy problem, because even if it can protect scalar masses from quadratic divergences, it appears to be reintroducing a little hierarchy problem between the electroweak and the SUSY breaking scales.

The Higgs boson and Physics beyond the Standard Model

[GeV]

χ ∼ W b → 1 t ~ / 1 0 χ ∼ t → 1 t ~1 0 χ ∼ b f f' → 1 t ~ / 1 0 χ ∼ W b → 1 t ~ / 1 0 χ ∼ t → 1 t ~1 0 χ ∼ b f f' → 1 t ~ / 1 0 χ ∼ W b → 1 t ~ / 1 0 χ ∼ t → 1 t ~1 0 χ ∼ b f f' → 1 t ~ / 1 0 χ ∼ c → 1 t ~-1 =8 TeV, 20 fb s t ) < m 1 0 χ ∼ , 1 t ~ m ( ∆ W + m b ) < m 1 0 χ ∼ , 1 t ~ m ( ∆ ) < 0 1 0 χ ∼ , 1 t ~ m ( ∆ 1 0 χ ∼ t → 1 t ~ / 1 0 χ ∼ W b → 1 t ~ / 1 0 χ ∼ c → 1 t ~ / 1 0 χ ∼ b f f' → 1 t ~ production, 1 t 1 t ~Status: Dec 2017 ATLAS Preliminary 1 0 χ ∼ W b 1 0 χ ∼ c 1 0 χ ∼ b f f'

Observed limits

Expected limits All limits at 95% CL In this pessimistic context, one of the very interesting phenomenological features of models with Dirac gauginos is that stop production is suppressed (as we will discuss in section 1.3.5.3) therefore allowing such models to evade some of the current collider bounds.

The Next-to-Minimal Supersymmetric Standard Model

As we have just seen, the MSSM, while solving some of the issues of the SM, still has a number of shortcomings. To address some of these and also because nothing

W NMSSM = (µ + λ S S)H u • H d + T S S + 1 2 µ S S 2 + 1 3 κS 3 + y f g u Q f • H u u c g -y f g d Q f • H d d c g -y f g e L f • H d e c g , (1.3.93)
while for the soft terms

L soft NMSSM = L soft MSSM + a λ S SH u • H d + 1 3 a κ S 3 + 1 2 B S S 2 + t S S + h.c. -m 2 S |S| 2 .
(1.3.94)

We choose to denote the superpotential coupling between the new singlet and the two Higgs doublets as λ S instead of λ (as is often written in the literature) to avoid risks of confusion with the SM Higgs quartic coupling λ. Note also that linear (tadpole) terms, both in the superpotential and in the soft SUSY-breaking Lagrangian are allowed for the new state because it is a gauge singlet. The new trilinear couplings are dened such that a λ S = λ S A λ S and a κ = κA κ . While the above equations describe the general NMSSM, one often prefers to consider a restricted version of the NMSSM with a scale-invariant superpotential (i.e. without any mass-dimensionful parameters)

obtained by imposing an additional, global, Z 3 symmetry to the theory under which all chiral superelds transform with a phase e i 2π 3 . This signicantly reduces the number of allowed parameters of the theory and the superpotential and soft Lagrangian are then

W NMSSM = λ S SH u • H d + 1 3 κS 3 + y f g u Q f • H u u c g -y f g d Q f • H d d c g -y f g e L f • H d e c g , L soft NMSSM = -(m 2 Q ) f g Q * f L QgL -(m 2 ū) f g ũc f R (ũ c gR ) * -(m 2 d) f g dc f R ( dc gR ) * -(m 2 L ) f g L * f L LgL -(m 2 ē ) f g ẽc f R (ẽ c gR ) * -m 2 Hu H * u H u -m 2 H d H * d H d (1.3.95) - 1 2 M 3 ga ga + 1 2 M 2 W a W a + 1 2 M 1 B B + h.c. -(a u ) f g ũc f R QgL • H u + (a d ) f g dc f R QgL • H d + (a e ) f g ẽc f R LgL • H d + h.c. + a λ S SH u • H d + 1 3 a κ S 3 + h.c. -m 2 S |S| 2 . (1.3.96)
The absence of any term in W NMSSM with a non-zero mass dimension means that there will be no situation analogous to the µ-problem of the MSSM, and moreover we see that, if the singlet scalar acquires a VEV S = v S / √ 2, eective µ and B µ terms can be generated as

µ e = 1 √ 2 λ S v S , (1.3.97) B e µ = 1 √ 2 λ S A λ S v S + 1 2 κ * λ S v 2 S = µ e A λ S + κ * v S √ 2 . (1.3.98)
As in this scenario the only parameters in the scalar potential with a mass dimension are soft SUSY-breaking terms, the singlet VEV and in turn the eective µ e and B e µ must also be of the order of the soft terms, therefore eliminating the possibility of an MSSM-like µ-problem.

The Higgs boson and Physics beyond the Standard Model

The Higgs sector of the NMSSM

The NMSSM has a vast and rich phenomenology, which we will not try to describe here, but instead we will now discuss some aspects of the Higgs sector of the NMSSM that will be useful for later chapters. First, the Higgs potential in the (Z 3 -invariant)

NMSSM is

V (0) = |µ + λ S S| 2 |H 0 u | 2 + |H + u | 2 + |H 0 d | 2 + |H - d | 2 + m 2 S |S| 2 + m 2 Hu |H 0 u | 2 + |H + u | 2 + m 2 H d |H 0 d | 2 + |H - d | 2 + λ S (H + u H - d -H 0 u H 0 d ) + κS 2 2 + 1 8 (g 2 + g 2 ) |H 0 u | 2 + |H + u | 2 -|H 0 d | 2 -|H - d | 2 2 + 1 2 g 2 H + u (H 0 d ) * + H 0 u H + d 2 + λ S A λ S S(H + u H - d -H 0 u H 0 d ) + 1 3 κA κ S 3 + h.c. , (1.3.99)
having used the same decomposition of the Higgs doublets as in the MSSM. The precise determination of the vacuum state of the NMSSM is a long and dicult task which we will not consider here (the reader may refer to [START_REF] Ellwanger | The Next-to-Minimal Supersymmetric Standard Model[END_REF][START_REF] Ellwanger | Phenomenology of supersymmetric models with a singlet[END_REF] for more details) but it can be shown that vacua can exist with non-zero VEVs for both doublets and the singlet. 20 There are then three non-trivial tadpole equations that are typically used to

eliminate m 2 Hu , m 2 H d
, and m 2 S . The free parameters left to describe the Higgs sector of the NMSSM at tree-level are then λ S , κ, A λ S , A κ , tan β, v S (or equivalently µ e ) .

(

It is worth mentioning nally that the condition A 2 κ 8m 2 S (9m 2 S ) ensures the existence of an (absolute) minimum with v S = 0.

m 2 h ≤ m 2 Z cos 2 2β + 1 2 λ 2 S v 2 sin 2 2β , (1.3.101)
where the second term shows an enhancement with respect to the MSSM, thanks to the presence of the additional singlet.

Another important consequence of the additional singlet and of its coupling λ S is that the NMSSM is the simplest SUSY models in which the Goldstone Boson Catastrophe is apparent even in the gaugeless limit. Indeed, as can be seen in eq. (1.3.99), λ S contributes to the Higgs triple and quartic couplings which then do not depend solely on g, g . The NMSSM hence provides an excellent setting to compare the numerical workarounds rst developed to circumvent the Goldstone Boson Catastrophe in SARAH/SPheno with the general solution presented in chapter 4.

Dirac gaugino models

Many options are available to go beyond the minimal supersymmetric extensions of the Standard Model MSSM and NMSSM and among them one of the most promising 20 Actually, depending on the parameters, the Z3-invariant NMSSM can allow several stable vacua which leads to a problem of domain walls in Cosmology, but once more this is far beyond the scope of this short review.

Supersymmetry

Supermultiplet Spin-0 Spin-

Q f QfL = (ũ f L , dfL ) Q f L = (u f L , d f L ) - (3, 2, 1 6 ) u c f ũc f R u c f R - ( 3, 1, -2 3 ) d c f dc f R d c f R - ( 3, 1, 1 3 ) (S)leptons L f LfL = (ν f L , ẽfL ) L f L = (ν f L , e f L ) - (1, 2, -1 2 ) e c f ẽc f R e c f R - (1, 1, 1)
Higgs/

H u H u = (H + u , H 0 u ) ( h+ u , h0 u ) - (1, 2, 1 2 )
Higgsinos

H d H d = (H 0 d , H - d ) ( h0 d , h- d ) - (1, 2, - 1 2 )
NMSSM singlet superfield

Singlet/singlino S S s - (1, 1, 0) MRSSM Higgs-like superfields Higgs-like R u (R + u , R 0 u ) (r + u , r0 u ) - (1, 2, -1 2 ) states R d (R 0 d , R - d ) (r 0 d , r- d ) - (1, 2, 1 2 )
Gauge sector

MSSM gauge superfields Gluons/Gluinos G - ga g a (8, 1, 0) W /Winos W - W a W a (1, 3, 0) B/Bino B - B B (1, 1, 0) MDGSSM/MRSSM adjoint chiral superfields Adjoint octet O O a χ a O - (8, 1, 0) Adjoint triplet T T a χ a T - (1, 3, 0) Adjoint singlet S S χ S - (1, 1, 0)
Table 1.3 Field contents and representation assignments of the MSSM, the NMSSM, the MDGSSM and the MRSSM. For the models beyond the MSSM, the eld content is that of the MSSM plus the relevant additional states. In the MDGSSM, additional Higgs-like lepton and fake electron superelds are sometimes added to ensure the unication of gauge couplings (see e.g. [START_REF] Benakli | The Di-Photon Excess in a Perturbative SUSY Model[END_REF]), but we will not consider these states in the context of the calculation of leading corrections to neutral scalar masses in chapter 3 so we do not mention them in this table. Finally, R-charges of the dierent states in the MRSSM are given separately in table 1.4.

is the class of models that allow Dirac masses for the gauginos [8691], in particular instead of but possibly in addition to Majorana ones. Such models have drawn attention from the community in recent years because of various advantages that they provide with respect to the MSSM or NMSSM, from both theoretical and phenomenological points of view. On the theory side, Dirac gaugino models have the attractive feature that they increase the naturalness of the model, because Supersymmetry is broken by supersoft operators [START_REF] Fox | Dirac gaugino masses and supersoft supersymmetry breaking[END_REF] and the SM-like Higgs boson mass is enhanced at tree level [START_REF] Benakli | Generating mu and Bmu in models with Dirac Gauginos[END_REF][START_REF] Belanger | Dark Matter with Dirac and Majorana Gaugino Masses[END_REF]. Moreover, from a phenomenological perspective, Dirac gaugino masses are also interesting as they relax constraints on squark masses (through suppressing

The We devote this section to an overview of Dirac gaugino models, for the needs of the study of two-loop corrections to neutral scalar masses in this context in chapter 3.

Extended Supersymmetry and supersoft SUSY breaking

The starting point for the construction of models with Dirac gauginos is to ask how much Supersymmetry (i.e. how many supercharges) can be allowed without encountering a theoretical inconsistency. First of all, the chiral nature of the SM fermions is a strong indication that, if it exists, Supersymmetry in the matter sector must have N = 1 because in N = 2 SUSY, hypermultiplets cannot be chiral. Furthermore, the number of matter states in a theory with N = 2 SUSY is so large that it would lead to Landau poles at low energies (well before the GUT or Planck scales), for example for the strong gauge coupling. However, in the gauge sector the situation is dierent and N = 2 Supersymmetry can be considered without encountering problematic Landau poles [START_REF] Fox | Dirac gaugino masses and supersoft supersymmetry breaking[END_REF] (although they would occur for N = 4). The extension of Supersymmetry for gauge states can be realised in an N = 1 language by adding chiral superelds Σ a transforming in the adjoint representation of the gauge group (called more shortly adjoint chiral superelds).

The enriched structure of the gauge sector allows for new SUSY breaking terms, from which the eponymous Dirac masses of the gauginos result. More precisely, in the context of soft SUSY breaking 21 , the new D-term breaking operators can appear with the form [START_REF] Fox | Dirac gaugino masses and supersoft supersymmetry breaking[END_REF] L

supersoft = d 2 θ √ 2 W α W a α Σ a M + h.c. (1.3.102)
where W = λ + θD + . . . is the eld-strength supereld associated with some spurious vector supereld of the hidden sector, whose auxiliary D component acquires a VEV D , and W a = λ a + θD a + . . . is the regular eld-strength supereld of the gauge group. Upon spontaneous breaking of Supersymmetry, the above operator can be rewritten in terms of the auxiliary eld VEV as

L supersoft = d 2 θ √ 2 D θ α W a α Σ a M + h.c. = d 2 θ √ 2m D θ α W a α Σ a + h.c. ⊃ -m D λ a χ a Σ + √ 2m D Σ a D a + h.c. , (1.3.103)
21 It has also been suggested, e.g. in ref. [START_REF] Martin | Nonstandard supersymmetry breaking and Dirac gaugino masses without supersoftness[END_REF], that Dirac masses could arise through other operators, but we do not consider them as they potentially correspond to a hard breaking of SUSY.

Supersymmetry

where we dened m D = D /M and expanded the adjoint chiral supereld as Σ a = Σ a + √ 2 θχ a Σ + . . . . We recall that to derive the last line in the above equation, we made use of the following spinor algebra relation [START_REF] Martin | A Supersymmetry primer[END_REF] (θξ)(θχ) = -1 2 (θθ)(ξχ) . the second term will generate, for the scalar components of the adjoint chiral superelds (the adjoint scalars), mass terms as well as trilinear interactions with Higgses and sfermions, which we collectively denote as φ. Indeed, integrating out the auxiliary eld D a gives for the adjoint scalars

L ⊃ -(m D Σ a + m * D Σ a * ) 2 - √ 2 g (m D Σ a + m * D Σ a * ) φ * t a φ , (1.3.105)
where t a are the generators of the gauge group in the representation appropriate to φ, and a sum over the gauge indices of φ is understood.

The reason why these new SUSY-breaking operators are called supersoft is actually one of the major features of models with Dirac gaugino, which is that these new operators do not reintroduce any logarithmic (i.e. soft) divergence in scalar masses. The only possible corrections are then nite ones, hence the name supersoft. A simple argument to prove this statement is provided in [START_REF] Fox | Dirac gaugino masses and supersoft supersymmetry breaking[END_REF]: suppose rst that the gaugino, having acquired a mass below the scale M , gives a logarithmically divergent contribution to the mass of a given scalar φ.

d 2 θd 2 θ (W α W α ) * W β W β M 6 Φ * Φ W → θ D -------→ d 2 θd 2 θ θ 2 θ2 m 4 D M 2 Φ * Φ = M ∼Λ m 4 D Λ 2 φ * φ , (1.3.106)
where Φ is the chiral supereld containing the scalar φ. There is however no way that this operator can cancel a logarithmic log Λ 2 divergence, and moreover it vanishes in the limit Λ → ∞. Therefore, we can conclude that our original assumption must have been wrong and that corrections to scalar masses from a Dirac gaugino must be nite. In terms of diagrams contributing to the mass of the scalar φ, this corresponds to a cancellation of the divergence of the fermion loop involving the fermionic partner of the scalar and a gaugino (left side of gure 1.5) against that of the new scalar diagram involving the scalar itself and the adjoint scalar (right side gure 1.5).

A brief overview of Dirac gaugino models

There is more than one way to construct a Dirac-gaugino extension of the MSSM.

The minimal choice, which we will denote as the Minimal Dirac Gaugino Supersym- in recent works [START_REF] Benakli | The Di-Photon Excess in a Perturbative SUSY Model[END_REF][START_REF] Benakli | Constrained minimal Dirac gaugino supersymmetric standard model[END_REF][START_REF] Goodsell | Dark matter scenarios in a constrained model with Dirac gauginos[END_REF] the term MDGSSM has also been used to describe a unied scenario where extra lepton-like states are added to ensure natural gauge-coupling unication, but the distinction will be irrelevant for the discussion in chapter 3.

Models with Dirac gauginos also provide a possible setting for building low-energy supersymmetric models that preserve a U (1) R symmetry or R-symmetry and not only an R-parity. First, one can realise that in an R-symmetry preserving extension of the MSSM, the usual Higgs doublets must have R-charge 0 otherwise their VEVs would break the R-symmetry and in turn from the structure of the Yukawa terms in the superpotential the quark and lepton superelds can be found to have R-charges +1 (because all terms in the superpotential must have total charge +2, see section 1.3.1.5).

If we then wish to construct a theory without large Majorana masses for the gauginos and to be able to benet from simpler SUSY-breaking scenarios [START_REF] Nelson | R symmetry breaking versus supersymmetry breaking[END_REF] we should avoid R-symmetry breaking in the soft terms, which also means removing the MSSM-like Aterms. Indeed, both Majorana mass terms for gauginos and A-terms are forbidden by R-symmetry because they correspond to Lagrangian terms with total R-charge +2, but it is important to note that the supersoft operator in eq. (1.3.102) is R-symmetric. Such a low-energy eective SUSY model with R-symmetry then naturally comes with Dirac gaugino masses and furthermore embeds into gauge-mediated scenarios [START_REF] Antoniadis | A New gauge mediation theory[END_REF]103108].

Moreover, the supersymmetric mass term µ is also forbidden by R-symmetry, but we may retain a B µ term (allowed by the symmetry) since it is required for EWSB and will not generate Majorana masses through renormalisation group evolution.

An early variant of an R-symmetric SUSY model, with an approximate R-symmetry, is the / µSSM or µ-less MSSM [START_REF] Nelson | The Minimal supersymmetric model without a mu term[END_REF], however, it should be noted that, as studied in ref. [START_REF] Benakli | Dirac Gauginos and the 125 GeV Higgs[END_REF], the / µSSM is currently challenged by electroweak precision measurements. On the other hand, if we choose to retain the R-symmetry as exact (possibly broken only by gravitational eects) then one popular construction is the Minimal R-symmetric Supersymmetric Standard Model, or MRSSM [START_REF] Kribs | Flavor in supersymmetry with an extended R-symmetry[END_REF]: two additional Higgs-like superelds are included, which have R-charge +2 and couple in the superpotential to the regular Higgs doublets but obtain no expectation value. They allow the Higgs elds H u and H d to both have non-zero supersymmetric mass terms (see e.g. eq. (1.3.124) in the following) and contribute to the EWSB without violating the R-symmetry [START_REF] Kribs | Flavor in supersymmetry with an extended R-symmetry[END_REF] (the R-charge assignments of all superelds of the MRSSM are given in table 1.4). An even more minimal realisation is the MMRSSM [START_REF] Frugiuele | Making the Sneutrino a Higgs with a U (1) R Lepton Number[END_REF][START_REF] Bertuzzo | Dirac gauginos, R symmetry and the 125 GeV Higgs[END_REF], where the down-type Higgs

Supersymmetry

H d and its R-partner are missing, a sneutrino then playing the role of H d . Another option to preserve R-symmetry is the supersymmetric one-Higgs-doublet model [START_REF] Davies | A Supersymmetric One Higgs Doublet Model[END_REF]: starting from the eld content of the MDGSSM, the singlet adjoint supereld is missing and the down-type Higgs does not develop an expectation value, therefore the bino is massless up to anomaly-mediation contributions.

Some aspects of the phenomenology of Dirac gaugino models

Dirac gaugino models also have several phenomenological properties that are especially attractive with respect to the MSSM (and NMSSM). The rst one can mention is that these models can naturally accommodate a relatively large hierarchy between gaugino (in particular gluino) and squark masses. The rationale for this is that, as we saw at the end of section 1.3.5.1, gaugino contributions to scalar mass parameters are smaller in models with Dirac gauginos than with Majorana gauginos (like in the MSSM) [START_REF] Kribs | Supersoft Supersymmetry is Super-Safe[END_REF].

Therefore the Dirac masses for gauginos can be taken to much larger values than in the case of Majorana masses without spoiling the naturalness of the model i.e. without increasing the necessary tuning, between tree-level values and radiative corrections, for scalar and in particular Higgs masses.

A second advantage is the suppression of the production rates of coloured SUSY particles [9395,[START_REF] Diessner | Squark production in R-symmetric SUSY with Dirac gluinos: NLO corrections[END_REF], which helps evade bounds from experimental searches. On the one hand, as Dirac gluinos can be heavy, both the gluino pair and the associated gluinosquark productions at colliders (e.g. the LHC) are typically suppressed by kinematics.

On the other hand, the total production of squark pairs either a squark plus an antisquark or two squarks is also suppressed because (i) processes with two squarks of the same handedness are absent when the gluino has a Dirac mass (these processes require a chirality ip, 22 which is only possible with a Majorana fermion), and furthermore (ii) several other channels involving the exchange of a gluino in a t-channel such as for example pp → qL qR , q * L q * R are more suppressed than in the MSSM (because the gluino is a Dirac instead of a Majorana fermion). The remaining contributions then give a much lower squark pair production cross-section compared to the MSSM, and therefore the constraints obtained for squark masses in the MSSM are relaxed.

Finally, Dirac gaugino models also have favourable properties with respect to avour constraints [START_REF] Kribs | Flavor in supersymmetry with an extended R-symmetry[END_REF][START_REF] Fok | µ to e in R-symmetric Supersymmetry[END_REF][START_REF] Hisano | Electric Dipole Moments in PseudoDirac Gauginos[END_REF][START_REF] Kumar | Neutrino Mass, Sneutrino Dark Matter and Signals of Lepton Flavor Violation in the MRSSM[END_REF], in particular when R-symmetry is preserved. More precisely, supersymmetric models are strongly constrained by experimental results on avour violating observables, among which the most constraining is kaon-antikaon (K -K) oscillations. This is sometimes referred to as the supersymmetric avour problem and in general SUSY models it is assumed to imply that the mechanism that mediates SUSY breaking between the hidden and observed sectors is avourblind. However, the possibility of constructing R-symmetric models of SUSY at low energies [START_REF] Kribs | Flavor in supersymmetry with an extended R-symmetry[END_REF] lifts this requirement, and allows parameter points with large avour violation to avoid current experimental results. A rst reason for this is that all left-right mixing terms (i.e. A-type trilinear couplings), and therefore in particular left-right avour violating terms, are forbidden by the R-symmetry. Moreover, if we take the example of kaon oscillations, it is known that new Physics contributions must be very 22 If one considers the expansion of the supersoft operator in eq. (1.3.103) and integrates out the auxiliary elds, one can nd that the fermion component of the adjoint supereld χ a Σ does not couple to Higgses and sfermions, while the MSSM-like gaugino state λ a does. For a (purely) Dirac gaugino this means that only one of its helicity states couples to Higgses and sfermions and hence chirality ip diagrams cannot exist. On the other hand, if the gaugino is a Majorana fermion, the two helicity states are equal see eq. (1.3.9) and the chirality ip is possible.

The Higgs boson and Physics beyond the Standard Model small, but in minimal SUSY models (e.g. the MSSM) with Majorana gauginos, potentially large and dangerous contributions arise from box diagrams involving eective operators between quarks and squarks like [START_REF] Kribs | Flavor in supersymmetry with an extended R-symmetry[END_REF] 1

m g d * R s * L dR s L (1.3.107)
Such a dimension-5 operator comes from integrating out the Majorana gauginos, but is forbidden in R-symmetry preserving models where the gaugino must be purely Dirac fermions. The leading operator obtained from integrating out a Dirac gaugino is a dimension-6 operator of the form

1 m 2 g dL ∂ µ s * L dL γ µ s L (1.3.108)
which is clearly more suppressed than the dimension-5 operator. Furthermore, as Dirac gauginos can naturally be signicantly heavier than Majorana ones, the potential SUSY contributions to kaon mixing become strongly suppressed, and much larger amounts of avour violation become acceptable. Other contributions to avour violation observables are also suppressed in models with Dirac gauginos: e.g the µ → eγ process requires a chirality ip [START_REF] Kribs | Flavor in supersymmetry with an extended R-symmetry[END_REF][START_REF] Fok | µ to e in R-symmetric Supersymmetry[END_REF] either in a gaugino or in a lepton line and, as we have discussed already, chirality ips are not possible with Dirac gauginos. Similarly, bounds on CP violation (for example from electric dipole moments) are weakened as well with Dirac gauginos [START_REF] Kribs | Flavor in supersymmetry with an extended R-symmetry[END_REF][START_REF] Hisano | Electric Dipole Moments in PseudoDirac Gauginos[END_REF]. We must emphasise however that these attractive results rely heavily on the gauginos being purely Dirac, and it was shown in [START_REF] Dudas | Flavour models with Dirac and fake gluinos[END_REF] that if gauginos have both Majorana and Dirac masses (as is the case in the MDGSSM for example) the suppression of dangerous contributions to avour or CP violating processes is reduced in most of the parameter space.

Let us now consider more closely the setting of our calculation in chapter 3, by discussing the properties of the adjoint scalars and of the gluinos in general Dirac gaugino models, before turning to the two models that will be studied, namely the MDGSSM and the MRSSM, and their respective Higgs sectors.

Supermultiplet

R-charge

(S)quark multiplets Q f , u c f , d c f 1 (S)lepton multiplets L f , e c f 1 Gauge multiplets G, W, B 0 Adjoint chiral multiplets O, T, S 0 MSSM-like Higgs multiplets H u , H d 0 Extra Higgs multiplets R u , R d 2
Table 1.4 Supermultiplets of the MRSSM and their charges under the U (1) R symmetry.

Properties of the adjoint scalars

We have already seen that Dirac masses for gauginos as well as masses and couplings for the adjoint scalars are generated by the supersoft operator see eqs.

L ⊃ -(m 2 Σ + 2 |m D | 2 ) Σ a * Σ a - 1 2 (B Σ + 2 m 2 D ) Σ a Σ a - 1 2 (B * Σ + 2 m * 2 D ) Σ a * Σ a * , (1.3.109)
where m 2 Σ includes in general contributions from both the superpotential and the soft SUSY-breaking Lagrangian, and B Σ is a soft SUSY-breaking bilinear term. In addition, mixing with the MSSM-like Higgs scalars may be induced, upon EWSB, by the D-term interactions in eq. (1.3.105), as well as by superpotential interactions.

We shall denote the adjoint multiplet for U (1) Y as a singlet S = S + √ 2 θχ S + . . . , the one for SU (2) L as a triplet T a = T a + √ 2 θχ a T + . . . , and the one for SU (3) as an octet O a = O a + √ 2 θχ a O + . . . . In chapter 3 we shall be interested only in the two-loop corrections to the Higgs masses involving the strong gauge coupling g s , thus the relevant trilinear couplings in eq. (1.3.105) will be the ones involving the octet scalar (also called sgluon) and the squarks.

We shall make the additional restriction that the octet scalar only interacts via the strong gauge coupling and the above trilinear terms, equivalent to the assumption that it has no superpotential couplings or soft trilinear couplings other than with itself. This shall simplify the computations, and it is true for almost all variants of Dirac gaugino models studied so far. To have renormalisable Yukawa couplings between the octet and the MSSM elds we would need to add new coloured states (such as a vector-like top).

However, in the most general version of the MDGSSM there could also be terms that violate the above assumption which have only recently attracted attention [START_REF] Benakli | The Di-Photon Excess in a Perturbative SUSY Model[END_REF][START_REF] Cohen | GeV Diphotons from Supersymmetry with Dirac Gauginos[END_REF] namely couplings between the singlet and the octet of the form

W ⊃ 1 2 λ SO S O a O a , L ⊃ - 1 2 T SO S O a O a + h.c. . (1.3.110)
The coupling λ SO is typically neglected because it violates R-symmetry and leads to Majorana gaugino masses: for example, in the restricted version of the MDGSSM or the / µSSM the R-symmetry violation is assumed to only occur in the Higgs sector and possibly only via gravitational eects. On the other hand, there is no symmetry preventing the generation of T SO , but it is typically dicult for it to obtain a phenomenologically signicant magnitude, hence it has been neglected see [START_REF] Benakli | The Di-Photon Excess in a Perturbative SUSY Model[END_REF] for a full discussion (and for cases when it could be large). Furthermore, T SO is irrelevant in the decoupling limit (when the singlet S is heavy) that we shall employ later in our simplied formulae.

With the above assumptions, we can make a rotation of the supereld O a such that we can take m D to be real without loss of generality, but we cannot simultaneously require that the soft SUSY-breaking bilinear B O be real without additionally imposing CP invariance. The octet mass terms are then

L ⊃ -m 2 O O a * O a - 1 2 B O O a O a - 1 2 B * O O a * O a * -m 2 D (O a + O a * ) 2 . (1.3.111)
If B O is not real, the real and imaginary parts of the octet scalar mix with each other. Their mass matrix can be diagonalised with a rotation by an angle φ O ,

O a = e iφ O √ 2 (O a 1 + i O a 2 ) , φ O = - 1 2 Arg B O + 2 m 2 D , (1.3.112)
to obtain the two mass eigenvalues

m 2 O 1,2 = m 2 O + 2 m 2 D ± |2 m 2 D + B O | . (1.3.113)
The Higgs boson and Physics beyond the Standard Model

Then the trilinear couplings of the octet mass eigenstates O a 1,2 to squarks qL and qR

read L ⊃ -2 g s m D (cos φ O O a 1 -sin φ O O a 2 ) (q * L t a qL -q * R t a qR ) , (1.3.114)
where t a are the generators of the fundamental representation of SU (3). where the latter is much larger than the soft SUSY-breaking mass terms. In particular,

cos φ O ≈ 1 + O m -4 D , sin φ O ≈ - Im(B O ) 4 m 2 D + O m -4 D . (1.3.115)
For the remainder of this thesis, we shall restrict our attention to the CP-conserving case. This is motivated by clarity and simplicity in the calculations, and also physically in that there are strong constraints upon CP violation, even in the Higgs sector [117 120]. However, we shall make an exception in allowing a non-zero angle φ O , because it is particularly simple to do so, and its eects are only felt at an order beyond that considered here: it generates CP-violating phases in the stop mass matrix at two loops, and in the Higgs mass at three. This is because the couplings in eq. (1.3.114) are real, and phases only appear in the octet scalar-gluino-gluino vertex.

Gluino masses and couplings

In the case of Dirac gauginos, there is mixing between the Weyl fermion of the gauge multiplet λ a and its Dirac partner χ a Σ . We shall allow in general both Majorana and Dirac masses which, in two-component notation, we write as

L ⊃ - 1 2 M λ λ a λ a - 1 2 M Σ χ a Σ χ a Σ -m D λ a χ a Σ + h.c. . (1.3.116)
As mentioned in the previous section, we can dene m D to be real. In general we cannot remove the phases from both M λ and M Σ ; however, as also mentioned above, we shall not consider CP violation in the gluino sector, and thus take all three masses to be real. We then rotate λ a and χ a Σ to mass eigenstates λ a 1 and λ a 2 via a mixing matrix R ij , so that

λ a = R 11 λ a 1 + R 12 λ a 2 , χ a Σ = R 21 λ a 1 + R 22 λ a 2 .
(

Ψ a D = λ a χa Σ . (1.3.118)
We recall that in the models of interest here there are no Yukawa couplings of the additional octet supereld, therefore the two gluino mass-eigenstates only couple to quarks and squarks via their gaugino component λ a . In particular, the couplings 1.3 Supersymmetry of each (four-component) gluino ga i are simply related to the couplings of the usual (N)MSSM gluino by an insertion of the mixing matrix:

L ⊃ - √ 2 g s R 1i q * L t a (g a i P L q) -(q P L ga i ) t a qR + h.c. , (1.3.119)
where a sum over the SU (3) indices of quarks and squarks is again understood. Consequently, as we shall see below, the gluino contribution to the two-loop eective potential in Dirac-gaugino models can be trivially recovered from the known results valid in the MSSM and in the NMSSM.

The MDGSSM and the MRSSM

We now consider the Higgs sector of the theory. Dirac gaugino models extend the (N)MSSM, so we shall assume that we have at least the usual two Higgs doublets H u and H d . To these we must add the adjoint scalars S and T a mentioned above, which mix with the Higgs elds. The couplings of the adjoint scalars, as well as the presence of any additional elds in the Higgs sector, will, however, depend on the model under consideration. In the following we shall focus on the minimal Dirac-gaugino extension of the MSSM, the MDGSSM, and on the minimal R-symmetric extension, the MRSSM.

In the MDGSSM there are no additional superelds apart from the adjoint ones, and the superpotential reads In the case of the MRSSM, we must add two superelds R u and R d with the same gauge quantum numbers as H d and H u , respectively, but with dierent charges under a conserved R-symmetry. The superpotential reads

W = W Yukawa + W MDGSSM , (1.3.120)
W Yukawa = Y f g u u c f Q g • H u -Y f g d d c f Q g • H d -Y f g e e c f L g • H d , (1.3.121)
W MDGSSM = (µ + λ S S) H u • H d + λ T H d • T a σ a H u + W Σ , (1.3
W = W Yukawa + W MRSSM , (1.3.123)
W MRSSM = (µ d + λ S d S) R d • H d + λ T d H d • T a σ a R d + (µ u + λ Su S) H u • R u + λ Tu R u • T a σ a H u , (1.3
m 2 h ≤ m 2 Z cos 2 2β + 1 2 (λ 2 S + λ 2 T )v 2 sin 2 2β . (1.3.125)
This shows a possible enhancement of the tree-level Higgs mass with respect to the MSSM and NMSSM, once more demonstrating the increase naturalness of models with Dirac gauginos.

Non-supersymmetric extensions of the Standard Model

We will now devote the last section of this chapter to a brief presentation of the nonsupersymmetric models that will be considered in this thesis. More specically, we will only consider here theories with extended Higgs sectors respectively extensions with a singlet scalar, with a second Higgs doublet, and with two triplets (one real and one complex). Such minimal extensions of the Standard Model (SM) are invaluable tools in the pursuit of physics beyond the SM (BSM). They oer the possibility of studying dierent eects at energy scales testable by the Large Hadron Collider (LHC) in a comparably clean environment i.e. the models typically contain the minimal numbers of new elds to exhibit novel phenomenology.

Singlet extensions of the Standard Model

The simplest and most straightforward way to extend the SM is to add a gauge singlet scalar, which can be either complex or real. Phenomenological motivations for the study of models with singlets include the fact that they can provide candidates for dark matter [121124], that they can also accommodate a strong rst order phase transition [START_REF] Espinosa | Strong Electroweak Phase Transitions in the Standard Model with a Singlet[END_REF] (one of the necessary conditions for electroweak baryogenesis), or that they could help stabilise the EW vacuum [126128]. It is worth noting nally that models with gauge singlet scalars could arise also as low energy limits of theories at higher energies, such as e.g. the NMSSM from which all superpartners (sfermions, gauginos, higgsinos) and one Higgs doublet would have been integrated out.

Non-supersymmetric extensions of the Standard Model

In chapter 6, we will consider a real singlet extension, which we will refer to as the Singlet-extended SM (SSM), with the following scalar potential

V (0) = µ 2 |Φ| 2 + 1 2 M 2 S S 2 + κ 1 |Φ| 2 S + 1 3 κ 2 S 3 + 1 2 λ|Φ| 4 + 1 2 λ SH S 2 |Φ| 2 + 1 2 λ S S 4 .
(1.4.1)

A linear (tadpole) term for the singlet should in principle be present, however, we have the freedom to redene the parameters of the potential by shifting the singlet by a constant (see e.g. Ref. [START_REF] Espinosa | Strong Electroweak Phase Transitions in the Standard Model with a Singlet[END_REF]) and we use this to remove the linear term. Furthermore, one should also note that the conventions for the SM-like Higgs quartic λ, and the other quartics λ S , λ SH , are chosen so as to match with the model le used in SARAH (having in mind the study of the SSM in chapter 6). In particular, λ is dened with a factor 2 with respect to the SM (eq. (1.1.5)). The Higgs doublet can be decomposed in the same way as in the SM see eq. (1.1.4) as a real CP-even Higgs scalar that here we denote φ, together with neutral and charged Goldstones G 0 , G ± , while the real scalar could in principle also be split into a VEV v S plus a real CP-even scalar s in practice, however, we will almost never be required to use such a decomposition, so we will mostly work with S. There are then two tadpole equations in the SSM, reading at tree-level

∂V (0) ∂φ min. = 0 = µ 2 v + κ 1 vv S + 1 2 λv 3 + 1 2 λ SH vv 2 S (1.4.2) ∂V (0) ∂s min. = 0 = M 2 S v S + 1 2 κ 1 v 2 + κ 2 v 2 S + 1 2 λ SH v 2 v S + 2λ S v 3 S (1.4.3)
One immediate change with respect to the SM is that in order for the neutral component of the Higgs doublet to obtain a non-zero VEV, it does not suce any more that µ 2 < 0, but instead the condition for EWSB is that

µ 2 + κ 1 v S + 1 2 λ SH v 2 S < 0 (1.4.4)
The singlet also obtains a VEV from the above potential interestingly eq. (1.4.3)

does not even admit v S = 0 as a solution when κ 1 = 0. The usual choice, which will be followed here, is then to solve the tadpole equations for µ 2 and M 2 S , which leaves as free parameters of the SSM λ, λ SH , λ S , κ 1 , κ 2 , v S .

(1.4.5)

The neutral CP-even component φ of the Higgs doublet mixes together with the singlet s to form two mass eigenstates, which we denote h and H. The scalar mixing angle α is dened by the relation

φ s = c α -s α s α c α h H (1.4.6)
and can be found with the relation

cot 2α = m 2 φφ -m 2 ss 2m 2 sφ = κ 1 v 2 + 2λv 2 v S -2κ 2 v 2 S -8λ S v 3 S 4κ 1 vv S + 4λ SH vv 2 S (1.4.7)
When studying the SSM (as is also the case for other non-SUSY BSM models), it is common to take the two mass eigenvalues m 2 h , m

λ = m 2 h + m 2 H t 2 α v 2 (1 + t 2 α ) , (1.4.8) λ S = κ 1 v 2 8v 3 S - κ 2 4v S + (m 2 H + m 2 h t 2 α ) 4(1 + t 2 α )v 2 S , (1.4.9
)

λ SH = - m 2 H t α -m 2 h t α + κ 1 v + κ 1 t 2 α v vv S (1 + t 2 α )
.

(1.4.10)

Moreover, in chapter 6, we will show how to extract at one-or two-loop order λ, λ S , and λ SH from an on-shell scalar spectrum, and the impact on the high-scale behaviour of the model of the order at which this matching is performed.

It is also possible to restrict the allowed couplings of the model by imposing a Z 2 symmetry under which the singlet is charged, or in other words transforms as S → -S. This new symmetry then forbids the dimensionful couplings κ 1 and κ 2 , reducing the scalar potential to

V (0) = µ 2 |Φ| 2 + 1 2 M 2 S S 2 + 1 2 λ|Φ| 4 + 1 2 λ SH S 2 |Φ| 2 + 1 2 λ S S 4 (1.4.11)
When M 2 S > 0 (as we will ensure), the Z 2 symmetry cannot be broken spontaneously and the singlet does not acquire a VEV (as can be seen from eq. (1.4.3)). We will use the acronym Z 2 SSM for this restricted version of the SSM (in reference to the additional Z 2 symmetry and to distinguish it from the general SSM). The fact that v S and κ 1 vanish means that there is only one useful tadpole equation, eq. (1.4.3) becoming trivial, and furthermore that there is no longer any mixing among the scalars which greatly simplies the derivation of an analytical formula for the two-loop extraction of the Higgs quartic λ (c.f. section 6.2.2.1).

Two-Higgs-Doublet Models

Another popular way of extending the SM Higgs sector is to add a second Higgs doublet, once again without modifying the remainder of the SM eld content. The resulting models are called Two-Higgs-Doublet Models (2HDMs, or sometimes THDMs), and come in dierent variants (or types) depending on how the two Higgs doublets couple to fermions, as will be discussed below. The reader may refer to [START_REF] Gunion | The Higgs Hunter's Guide[END_REF] or [START_REF] Branco | Theory and phenomenology of two-Higgs-doublet models[END_REF] for complete reviews of 2HDMs.

The most general renormalisable scalar potential that one can write for a model with two doublets dened with equal hypercharges takes the form

V (0) = m 2 11 Φ † 1 Φ 1 + m 2 22 Φ † 2 Φ 2 + m 2 12 Φ † 1 Φ 2 + Φ † 2 Φ 1 + λ 1 Φ † 1 Φ 1 2 + λ 2 Φ † 2 Φ 2 2 + λ 3 Φ † 1 Φ 1 Φ † 2 Φ 2 + λ 4 Φ † 1 Φ 2 Φ † 2 Φ 1 + 1 2 λ 5 Φ † 1 Φ 2 2 + Φ † 2 Φ 1 2 + λ 6 Φ † 1 Φ 1 Φ † 1 Φ 2 + Φ † 2 Φ 1 + λ 7 Φ † 2 Φ 2 Φ † 1 Φ 2 + Φ † 2 Φ 1 .
(1.4.12)

Note that our sign convention for m 2 12 diers from most denitions in the literature.

When including fermions and their Yukawa interactions with the Higgs sector, we want 1.4 Non-supersymmetric extensions of the Standard Model to avoid tree-level Higgs-mediated avour-changing neutral currents (FCNCs), which would be conicting with experimental results. These arise because if a certain type of fermions (leptons, or up-or down-type quarks) couples to both Higgs doublets the avour eigenstates do not in general correspond to the interaction eigenstates. We therefore choose to impose a Z 2 symmetry under which the scalar doublets transform as

Φ 1 → +Φ 1 , Φ 2 → -Φ 2 .
(1.4.13)

This forbids the couplings λ 6 and λ 7 , and in principle m 2 12 as well. The mass coupling m 2 12 is however left in the potential as a soft Z 2 -symmetry breaking term in the sense that a non-zero value of m 2 12 does not by itself reintroduce phenomenologically dangerous FCNCs.

This {Φ 1 , Φ 2 } basis is sometimes called the Z 2 basis and the scalar potential then reads

V (0) = m 2 11 Φ † 1 Φ 1 + m 2 22 Φ † 2 Φ 2 + m 2 12 Φ † 1 Φ 2 + Φ † 2 Φ 1 + λ 1 Φ † 1 Φ 1 2 + λ 2 Φ † 2 Φ 2 2 + λ 3 Φ † 1 Φ 1 Φ † 2 Φ 2 + λ 4 Φ † 1 Φ 2 Φ † 2 Φ 1 + 1 2 λ 5 Φ † 1 Φ 2 2 + Φ † 2 Φ 1 2
.

(1.4.14)

One or both doublet(s) Φ 1 and Φ 2 may acquire VEVs if m 2 ij has one or two negative eigenvalues, and we write the doublets and their VEVs as

Φ i = Φ + i Φ 0 i and Φ i = 1 √ 2 0 v i , for i = 1, 2.
(

Through a eld redenition, it is always possible to make m 2 12 and λ 5 real, but then the doublet VEVs may have complex phases which would lead to CP violation (see e.g [START_REF] Basler | The CP-Violating 2HDM in Light of a Strong First Order Electroweak Phase Transition and Implications for Higgs Pair Production[END_REF]). However, in the following we will consider only scenarios in which CP is not violated in the Higgs sector i.e CP-conserving 2HDMs and therefore we assume v i ∈ R, for i = 1, 2, here. We then dene the angle β through the usual relation

tan β = v 2 v 1 ⇔ v 1 = v cos β v 2 = v sin β (1.4.16)
where v is dened by

v 2 = v 2 1 + v 2 2 .
The mass parameters m 2 11 and m 2 22 can be determined using the tadpole equations, which read at tree-level

√ 2 v 1 ∂V (0) ∂Φ 0 1 min = 0 = m 2 11 + m 2 12 tan β + λ 1 cos 2 β + 1 2 (λ 3 + λ 4 + λ 5 ) sin 2 β v 2 , √ 2 v 2 ∂V (0) ∂Φ 0 2 min = 0 = m 2 22 + m 2 12 cot β + λ 2 sin 2 β + 1 2 (λ 3 + λ 4 + λ 5 ) cos 2 β v 2 ,
(1.4.17)

and the 2HDM scalar sector is thus left with seven free parameters which are λ i (for i ∈ {1, 2, 3, 4, 5}); m 2 12 ; tan β .

(1.4.18)

As in the case of the (real) MSSM, the two Higgs doublets of the (CP-conserving)

2HDM can be decomposed in terms of mass eigenstates, namely two CP-even Higgses, h and H, a CP-odd pseudoscalar Higgs A, and a charged Higgs H ± . A common practice, similar to what we saw in the previ is then to take as inputs the four masses

The Higgs boson and Physics beyond the Standard Model m h , m H , m A , m H ± as well as the mixing angle of the CP-even mass matrix α (or t α ≡ tan α), instead of the quartic couplings λ i (i ∈ {1, 2, 3, 4, 5}). The latter can in turn be computed at tree-level using the following relations

λ 1 = 1 + t 2 β 2(1 + t 2 α )v 2 m 2 h t 2 α + m 2 H + m 2 12 t β (1 + t 2 α ) , (1.4.19) λ 2 = m 2 12 (1 + t 2 β ) 2t 3 β v 2 + (1 + t 2 β ) m 2 h + m 2 H t 2 α 2t 2 β (1 + t 2 α )v 2 , (1.4.20) λ 3 = 1 (1 + t 2 α )t β v 2 m 2 H -m 2 h t α (1 + t 2 β ) + 2m 2 H ± (1 + t 2 α )t β + m 2 12 (1 + t 2 α )(1 + t 2 β ) , (1.4.21) λ 4 = 1 t β v 2 -m 2 12 (1 + t 2 β ) + m 2 A t β -2m 2 H ± t β , (1.4.22) λ 5 = 1 t β v 2 -m 2 12 (1 + t 2 β ) -m 2 A t β , (1.4.23)
using the shorthand notation t β = tan β.

Finally, assuming that one still wants to avoid tree-level FCNCs, there are several possible ways to assign transformation rules of the fermion under the Z 2 symmetry (see e.g. table 1 in [START_REF] Bernon | Scrutinizing the alignment limit in two-Higgs-doublet models: mh=125 GeV[END_REF]), and these correspond to dierent ways of coupling the Higgs doublets to fermions, i.e. to dierent types of 2HDMs. The two main types of 2HDMs that one then nds and that will be considered in chapter 5 are

• type I, in which only one of the doublets taken to be Φ 2 couples to the fermions, with a similar structure of couplings as the SM doublet;

• type II, in which the top quarks acquire masses from the doublet Φ 2 while the down quarks and leptons obtain theirs from the other doublet Φ 1 . This actually corresponds exactly to the Higgs-fermion couplings of the MSSM, and therefore one of the many motivation for the study of such 2HDMs is that they may appear as low energy limits of SUSY models, once all superpartners have been integrated

out.

Other variants of 2HDMs exist with dierent Z 2 charge assignments for the fermions (one can mention lepton-specic or ipped 2HDMs for example) and, furthermore, if one abandons the requirement of suppressing FCNCs at tree-level then one obtains a 2HDM where all (gauge invariant) Yukawa interactions between fermions and the two Higgs doublets are allowed, called type III but we will not consider any of these possibilities further in this thesis.

Before concluding this section, let us mention that it has been shown that a low energy Two-Higgs Doublet model with alignment in the Higgs sector can originate naturally from Dirac gaugino models at higher energies this, and the fact that including loop corrections does not spoil the alignment, was discussed recently in [START_REF] Benakli | Higgs alignment from extended supersymmetry[END_REF]. As mentioned earlier in the context of the MSSM, alignment can further help avoid the experimental detection of an extended Higgs sector, by making one of the CP-even Higgses Standard

Model-like and we will return to alignment in the Higgs sector of 2HDMs in chapter 5.

The Georgi-Machacek model

The Georgi-Machacek Model [START_REF] Georgi | [END_REF] extends the SM by one real scalar SU (2) L -triplet η with Y = 0 and one complex scalar SU (2) L -triplet χ with Y = 1, in a way that The two additional triplet scalars η and χ can be written as

η = 1 √ 2 η 0 - √ 2(η -) * - √ 2η - -η 0 , χ = 1 √ 2 χ - √ 2(χ 0 ) * - √ 2χ -- -χ - , (1.4.26)
however, a very compact form to write the Lagrangian in a SU (2) L ×SU (2) R invariant form is instead to express the SM doublet and the triplets as a bidoublet Φ and a bitriplet ∆

Φ = φ 0 * φ + φ -φ 0 , ∆ =   χ 0 * η + χ ++ χ -η 0 χ + χ --η -χ 0   .
(1.4.27)

Here, φ + and φ 0 are the charged and neutral components of the SM doublet (dened in eq. (1.1.4)) and φ -= φ + * . Using this notation, the scalar potential of the Georgi-Machacek model reads

V (0) (Φ, ∆) = µ 2 2 2 TrΦ † Φ + µ 2 3 2 Tr∆ † ∆ + λ 1 TrΦ † Φ 2 + λ 2 TrΦ † Φ Tr∆ † ∆ + λ 3 Tr∆ † ∆∆ † ∆ + λ 4 Tr∆ † ∆ 2 -λ 5 Tr Φ † σ a Φσ b Tr ∆ † t a ∆t b -M 1 Tr Φ † τ a Φτ b (U ∆U † ) ab -M 2 Tr ∆ † t a ∆t b (U ∆U † ) ab ,
τ a and t a are the SU (2) generators for the doublet and triplet representations respectively, while U is given for instance in [START_REF] Hartling | The decoupling limit in the Georgi-Machacek model[END_REF] as

U =    -1 √ 2 0 1 √ 2 -i √ 2 0 i √ 2 0 1 0    .
(1.4.28)

The doublet and the triplets obtain VEVs as

Φ = 1 √ 2 v φ 0 0 v φ , η = 1 √ 2 v η 0 0 -v η , χ = 0 v χ 0 0 , (1.4.29)
where the custodial symmetry enforces v η = v χ ≡ v T , and there are no tree-level contributions to the ρ parameter. They further full v 2 φ + 8v 2 T = v 2 , which allows us to dene

s H = sin Θ H = 2 √ 2v T v , c H = cos Θ H = v φ v .
(1.4.30)

The free parameters of the model are then

λ 1 . . . λ 5 , M 1 , M 2 , s H (1.4.31) since µ 2 2 , µ 2
3 can be eliminated by the tadpole equations. The physical eigenstates can be organised into representations of the custodial symmetry as

• a veplet consisting of a doubly charged, a singly charged, and a neutral CP-even scalar;

• a triplet consisting of a singly charged and a CP-odd neutral scalar; and,

• two CP-even singlets typically denoted h and H where the Standard Model Higgs-like boson is the lighter of the two.

Tree-level expressions for the triplet mass m 3 , veplet mass m 5 and singlet masses are given, for example, in [START_REF] De Florian | Handbook of LHC Higgs Cross Sections: 4. Deciphering the Nature of the Higgs Sector[END_REF] and read

m 2 5 = M 1 v 2 φ 4v T + 12M 2 v T + 3 2 λ 5 v 2 φ + 8λ 3 v 2 T , m 2 3 = M 1 4v T + 1 2 λ 5 v 2 , m 2 h,H = 1 2 m 2 11 + m 2 22 ∓ (m 2 11 -m 2 22 ) 2 + 4m 4 12 , (1.4.32)
with

m 2 11 = 8λ 1 v 2 φ , m 2 12 = - √ 3 2 v φ [M 1 -4(2λ 2 -λ 5 )v T ] , m 2 22 = M 1 v 2 φ 4v T -6M 2 v T + 8(λ 3 + 3λ 4 )v 2 T . (1.4.33)
Chapter 2

Precision calculations of the Higgs boson mass

The Higgs boson and its properties provide an excellent setting for indirect searches of Physics beyond the Standard Model. Indeed, as we have seen in the previous chapter, many BSM models exhibit an extended Higgs sector, or contain additional states that have noticeable eects on it (e.g. stops in SUSY).

In particular, the mass of the observed (SM-like) Higgs boson is sensitive to radiative corrections from heavy particles and is actually a prediction in numerous extensions of the SM, such as Supersymmetry, while it is now measured to an impressive precision of the order of 0.1% making it the ideal tool for BSM searches.

After recalling the available measurements of the Higgs mass, we will in this chapter review some of the main results from the past three decades of activity on Higgs mass calculations in BSM models.

Measurements of the Higgs mass

The SM-like Higgs boson was discovered by the ATLAS and CMS collaborations at the CERN Large Hadron Collider (LHC) [START_REF] Aad | Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC[END_REF][START_REF] Chatrchyan | Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC[END_REF]

Scalar mass calculations

In the Standard Model, the Higgs quartic coupling λ is a yet-unmeasured free parameter, and thus the tree-level Higgs mass cannot be predicted from a theoretical Precision calculations of the Higgs boson mass

A(x) = (16π 2 ) d 4 k (2π) 4 1 k 2 + x = 2 ∞ k=0 k 3 k 2 + x dk Cut-o Λ ------→ 2 Λ k=0 k 3 k 2 + x dk = Λ 2 + x log x Λ 2 + x .
(2.2.1)

Scalar mass calculations

However, this type of regularisation has several disadvantages, chief among which the fact that it explicitly breaks Lorentz invariance. It is therefore preferable to employ dimension regularisation (DREG) [START_REF] Hooft | Regularization and Renormalization of Gauge Fields[END_REF], because it preserves both Lorentz and gauge invariance as well as unitarity, and is of simple use. [START_REF] Braathen | Leading two-loop corrections to the Higgs boson masses in SUSY models with Dirac gauginos[END_REF] The idea is to consider the dimension of spacetime as a complex variable d that is expanded in the form d = 4-2 . The momentum integration is then not performed in 4 but in d dimensions, with the replacement (for Euclidean momentum)

(16π 2 ) d 4 k (2π) 4 DREG ----→ (16π 2 )µ 2 d d k (2π) d . (2.2.2)
The new parameter µ is an arbitrary (and unphysical) mass scale, sometimes called the regularisation scale, introduced to preserve the mass dimension of the integral and to be able to keep the usual dimensions of couplings. The UV divergence of a loop integral (coming from the large k region of the integration) is made apparent in DREG as poles of the form 1/ n with 1 ≤ n ≤ , being the loop order. If we return to our example of the A function, its expression becomes in DREG

A(x) = (16π 2 )µ 2 d d k (2π) d 1 k 2 + x = (16π 2 ) π d/2 µ 2 Γ 1 -d 2 (2π) d x d/2-1 = x - 1 + γ E -log 4π -log µ 2 + log x -1 (2.2.3)
where the 1/ pole is the translation of the UV divergence in 4 dimensions (as it divergences when → 0, i.e. d → 4), and γ E is the Euler-Mascheroni constant.

Still, new problems will appear when considering supersymmetric theories because in DREG the number of components of a vector boson is also changed to d and thus SUSY is broken explicitly by the regularisation, as a gauge supereld would not contain the same number of bosonic and fermionic components (4 -2 and 4 respectively).

A solution was found in [START_REF] Siegel | Supersymmetric Dimensional Regularization via Dimensional Reduction[END_REF][START_REF] Capper | Regularization by Dimensional Reduction of Supersymmetric and Nonsupersymmetric Gauge Theories[END_REF] with regularisation through dimensional reduction (DRED), of which the principle is to reduce the number of spacetime dimensions to d = 4-2 , similarly to DREG, but at the same time to maintain the number of components of vectors (and of tensors in general) to 4. This ensures that SUSY is preserved, in particular meaning that supersymmetric relations between couplings and SUSY Ward identities still hold, and a dierence between DREG and DRED only appears in diagrams involving vector bosons. In the language of DREG, DRED corresponds to the splitting of 4-component vectors into d-component vectors plus new (unphysical) -scalars that correspond to the remaining 2 components, and the mass and couplings of the -scalars are related to those of the vector bosons by Supersymmetry.

Once the divergent integral has been regularised, a choice remains on what condition to impose to eliminate the divergence or in other words on the choice of renormalisation scheme in order to derive the required counter-term(s). As an illustration, let us consider that the parameter that we compute is a mass, with the example2 of a scalar X whose tree-level mass is equal to its Lagrangian mass parameter. For this calculation, we will only need wave function and mass renormalisations and it therefore suces for us to write here the kinetic and mass terms of the Lagrangian describing the 1 It does however lead to complications when considering the fully antisymmetric tensor µνρσ , in particular when dening the matrix γ5, but we will not encounter this problem in what follows and therefore we will not say more about it.

Precision calculations of the Higgs boson mass behaviour of the scalar X. We start by writing this Lagrangian in terms of bare i.e.

non-renormalised quantities, as

L bare ⊃ 1 2 ∂ µ X 0 2 - 1 2 m 2 0,X (X 0 ) 2 .
(2.2.4)

Then, eld-strength renormalisation is introduced by relating the bare eld X 0 to the renormalised eld X with the rescaling

X 0 = Z 1/2 X X , (2.2.5)
after which the bare Lagrangian terms in equation (2.2.4) can be expressed as

L bare ⊃ 1 2 Z X (∂ µ X) 2 - 1 2 m 2 0,X Z X X 2 .
(2.2.6)

The eld-strength renormalisation counter-term δ CT Z X can now be dened as

δ CT Z X = Z X -1 .
(2.2.7)

In turn, we can introduce mass renormalisation, replacing the bare mass parameter m 2 0,X by the renormalised mass parameter m 2 X and the mass renormalisation counter-

term δ CT m 2 X , as m 2 0,X = m 2 X + δ CT m 2 X . (2.2.8)
While we only present here eld-strength and mass renormalisations, the same can be done for other couplings, and in the end it is possible to rewrite the bare Lagrangian as a renormalised Lagrangian L plus a counter-term Lagrangian δ CT L

L bare = L + δ CT L , (2.2.9)
with here,

L ⊃ 1 2 (∂ µ X) 2 - 1 2 m 2 X X 2 , (2.2.10) δ CT L ⊃ 1 2 δ CT Z X (∂ µ X) 2 - 1 2 δ CT m 2 X X 2 - 1 2 δ CT Z X m 2 X X 2 .
(2.2.11)

Two comments should be made here about the above equation for δ CT L. First, it should be noted that the above expression is the one-loop result of the counter-term Lagrangian (as we neglected a term δ CT Z X δ CT m 2 X that appears from two-loop order), but for the discussion in this section a one-loop discussion is sucient. Moreover, it would in principle have been possible to reabsorb the third term of δ CT L into a redenition of δ CT m 2 X .

The experimentally measurable pole mass M 2 X of X is then found in terms of the renormalised mass parameter m 2 X and of the renormalised self-energy Π X as

M 2 X = m 2 X + Π X (p 2 = M 2 X ) ,

Scalar mass calculations

A rst natural choice of scheme is to nd the counter-term by relating the quantity (mass, coupling, mixing angle, etc.) that is being computed to a physically meaningful observable. This is usually referred to as an on-shell (OS) renormalisation scheme, as it often corresponds to computing n-point functions or vertex functions with external legs set on-shell. Note however that for a generic quantity (e.g. a mixing angle) there may be more than one way to dene OS renormalisation conditions. The on-shell renormalisation scheme is dened here by the two following conditions

Π OS X (p 2 = M 2 X ) = 0 , (2.2.14) lim p 2 →m 2 X Π OS X (p 2 ) p 2 -m 2 X = 0 . (2.2.15)
This xes the two counter-terms to be

δ CT OS m 2 X = -Π X (p 2 = M 2 X ) , (2.2.16) δ CT OS Z X = d dp 2 Π X (p 2 ) p 2 =M 2 X .
(2.2.17)

and the Lagrangian mass parameter is in this OS scheme simply equal to the pole mass

(m 2 X ) OS = M 2 X .
( where Π reg. X is the nite part of the self-energy. This condition must hold for all p 2 , and renormalisation schemes based on minimal subtraction are said to be mass-independent schemes. One can nd of the counter-terms in these schemes to be

δ CT min.sub. m 2 X = -Π div X (p 2 = m 2 X ) , (2.2.20) δ CT min.sub. Z X = d dp 2 Π div X (p 2 ) p 2 =m 2 X . (2.2.21)
Finally, equation (2.2.12) becomes in minimal subtraction schemes

M 2 X = (m 2 X ) min.sub. + Π reg X (p 2 = M 2 X ) , (2.2
(m 2 X ) min.sub. = (m 2 X ) OS -Π reg X (p 2 = (m 2 X ) OS ) .
(2.2.23)

Both the results for the renormalised self-energy and the mass parameter retain a dependence on an unphysical scale Q, the renormalisation scale. We choose to dene it dierently from the regularisation scale µ in order to eliminate the constant piece -γ E + log 4π always associated with a term 1/ . In this modied minimal subtraction, Q is found following [START_REF] Bardeen | Deep Inelastic Scattering Beyond the Leading Order in Asymptotically Free Gauge Theories[END_REF][START_REF] Martin | Evaluation of two loop selfenergy basis integrals using dierential equations[END_REF] as

Q 2 = 4πe -γ E µ 2 .
(2.2.24)

Precision calculations of the Higgs boson mass Depending on whether DREG or DRED is used, the renormalisation schemes obtained with modied minimal subtraction are referred to as MS and DR respectively the bar dierentiating them from the MS and DR schemes obtained with the normal minimal subtraction.

A comment on the distinction between minimal subtraction and modied minimal subtraction is at hand here. In many QFT textbooks, the MS (DR) scheme is dened by starting from MS (DR) and requiring that the counter-term that cures the 1/ pole also cancel the constant -γ E + log 4π see e.g. equation (11.77) of [START_REF] Peskin | An Introduction to quantum eld theory[END_REF] while the renormalisation scale Q is always equal to the regularisation scale µ. However the prescription that we use, coming from the original paper on modied minimal subtraction [START_REF] Bardeen | Deep Inelastic Scattering Beyond the Leading Order in Asymptotically Free Gauge Theories[END_REF] (see in particular eq. ( 5.4) therein), is slightly dierent in that the counter-term is not modied (it only cancels the pole), but instead the distinction between MS (DR) and MS (DR) comes from the denition of Q, that is equal to µ for minimal subtraction but given by eq. (2.2.24) for modied minimal subtraction, i.e.

Q 2 MS (DR) = µ 2 and Q 2 MS (DR) = 4πe -γ E µ 2 (2.2.25)
Note that, while they may seem dierent, these two denitions of modied minimal subtraction are actually equivalent and do not lead to any dierence in the results that one obtains.

The parameters computed in either one of these schemes are said to be running because of their renormalisation scale dependence, and the relations between parameters in MS or DR schemes, which dier already from one-loop order, can be found in [START_REF] Martin | Regularization dependence of running couplings in softly broken supersymmetry[END_REF] these are obtained by relating parameters in both schemes to their physically meaningful OS counterpart, as we did with our toy example in eq. (2.2.23). Although running parameters are not directly measurable quantities, minimal subtraction schemes are widely employed because they signicantly simplify calculations with respect to the OS scheme.

A last subtlety pertaining to the DR scheme needs to be pointed out before we end this discussion. As mentioned already, DRED implies the addition of -scalars to ensure the equality of the numbers of fermionic and bosonic degrees of freedom in a gauge supermultiplet while working in d dimensions, and these will appear in calculations.

When Supersymmetry is unbroken, it relates their properties to those of the physical gauge bosons and no artefacts are left in the nal result of the computation, but once one considers soft SUSY breaking terms, the situation is altered and independentscalar soft mass terms m2 arise. In particular, the two-loop β-function of soft scalar masses and the one-loop correction to scalar masses exhibit an unwanted dependence on the unphysical m2 parameters [144147]. Fortunately, it has been shown that a simple redenition of the renormalisation scheme, more precisely a one-loop shift to the soft scalar masses as dened in equation ( 9) of [START_REF] Jack | Decoupling of the epsilon scalar mass in softly broken supersymmetry[END_REF], is sucient to eliminate all dependence on the -scalar masses. The resulting renormalisation scheme is denoted DR so as to distinguish it from the regular DR scheme [START_REF] Jack | Decoupling of the epsilon scalar mass in softly broken supersymmetry[END_REF].

Throughout this thesis, we will employ dimensional regularisation or reduction for calculations in non-supersymmetric or supersymmetric theories respectively, and we will provide results for both when considering generic theories. As for renormalisation schemes, we will employ OS, MS and DR schemes.

2.2 Scalar mass calculations

Calculations beyond leading order and choice of inputs

In the previous section, we have discussed how we regularise and renormalise a given calculation, but without considering its type or its context, and without discussing complications that arise when computing higher-order corrections (i.e. beyond one loop). Indeed, when considering a two-loop diagram we actually encounter two types of UV divergences:

• genuine two-loop divergences, which are cancelled by purely two-loop counterterms (e.g. by the two-loop contributions to δ CT m 2 X and δ CT Z X in our example of the previous section); and,

• divergences that come from subdiagrams, and that must be cancelled by a oneloop diagram with insertion of a one-loop counter-term. This can be interpreted as the need to renormalise the parameters that enter the one-loop corrections when performing a two-loop calculation.

The situation is similar, although even more complicated, when going to yet higher orders. In this thesis, we are interested in Higgs mass calculations, in the sense that we want to relate the Higgs mass(es) to the fundamental parameters of the theory we work in. When calculating corrections at two loops, we will obtain the genuine two-loop counter-terms using minimal subtraction, however, we remain free to choose dierent ways of renormalising the parameters in the contributions at lower orders, or in other words we have the liberty to choose the renormalisation scheme in which these parameters are derived, either an OS scheme or minimal subtraction (MS/DR ).

If one for example computes (as we do in chapter 3) the two-loop O(α t α s ) SUSY-QCD corrections 3 to Higgs masses, then one must specify in what scheme the parameters that appear in the corresponding lower-order corrections, here the one-loop O(α t ) terms, are computed. For SUSY-QCD corrections, these involve only the quark/squark sector and, for the more particular case of the O(α t α s ) corrections, these are the top and stop masses m t , m t1,2 , the stop mixing angle θ t and the stop trilinear coupling A t . As an on-shell renormalisation relates fundamental parameters to experimentally measured quantities, it is a natural choice for the mass parameters of observed particles, such as the top quark mass m t . However, for properties of particles that have not (yet) been detected, the choice is more open: on the one hand, one can relate them to quantities that could in principle be obtained in an experiment; or, on the other hand, they can be left as DR parameters, which simplies the expressions of counter-terms and is furthermore more convenient if said parameters are obtained through RGE running from a high-scale boundary condition.

The most convenient strategy (which we also follow) is to rst perform the two-loop calculation in the DR scheme, by which we mean that we compute (modied-)minimalsubtraction renormalised one-and two-loop corrections in terms of DR fundamental parameters. Then, once a DR result has been obtained, it can be converted to an OS scheme i.e. expressed in terms of input parameters that are renormalised on-shell by adding to the parameters entering at lower orders a shift obtained by the dierence of the counter-terms in OS and DR schemes see e.g. eq. (2.2.23) for the conversion of a generic mass parameter, and section 3.1.5 for details on the conversion of top/stop sector parameters. Note that for parameters that only appear in the last order of a given calculation, the choice of renormalisation scheme is irrelevant, as the conversion Precision calculations of the Higgs boson mass between schemes would correspond to a correction of higher order than the order that is considered for example in the case of the two-loop SUSY-QCD corrections to Higgs masses, one does not need to specify the scheme in which the gluino masses 4 m gi are computed because they only enter the calculation at two-loop order, and therefore shifting m gi between schemes would only give a correction at three-loop order.

Dierent types of mass calculations

After our discussion of renormalisation and of input choices in the previous two sections, we can now consider scalar (or in particular Higgs) mass calculations. We will rst introduce the dierent possible approaches to these calculations xed order (diagrammatic or with the eective potential) or EFT in a simple setting before turning, in the following section, to some of the results obtained for SUSY theories.

Fixed-order calculations

The standard procedure to derive the radiative corrections to masses of particles is to rely on the denition of the physical (or pole) masses as the poles of the renormalised propagator matrix or equivalently as the zeroes of the renormalised two-point function matrix. In the diagrammatic approach, the corrections to the two-point functions can be computed order by order in perturbation theory (hence the name of xed-order calculations) as one-particle-irreducible self-energy diagrams (i.e. two-point function diagrams).

Together with the self-energies, one must also take into account the tadpole diagrams.

Two dierent, although equivalent, points of view exist as to how tadpoles appear in mass calculations, and can be explained as follows:

• Calculations must be performed at the minimum of the loop-corrected scalar potential, as that is where (in eld space) masses are dened. In turn, this imposes relations between Lagrangian parameters i.e. for each tadpole equation, one parameter can be eliminated;

• In the process of renormalising the scalar sector of a given theory, tadpole counter-terms appear, and the best way to x these is to impose that the total loop-corrected tadpoles (i.e. the renormalised one-point function) vanish.

Throughout this thesis, we take the rst of these points of view, and we choose to perform calculations at the minimum of the scalar potential V e , computed to the same order as we work to for the mass calculations.

In the situation where one considers a scalar φ that does not mix with any other state, neither at tree level nor when including loop corrections, the pole mass of φ is found as the solution of the equation

M 2 φ = m 2 φ + Π φ (p 2 = M 2 φ ) , (2.2.26)
where m 2 φ is the Lagrangian mass parameter of φ, Π φ its renormalised self-energy.

Scalar mass calculations

However, in the presence of mixing among scalar states φ i , the situation becomes technically more involved and one needs to use a loop-corrected scalar mass matrix, which is obtained at n-loop order as

M 2 loop ij (p 2 ) = M 2 tree ij + ∆M 2 ij (p 2 ) = m 2 0,ij + Π ij (p 2 ) , (2.2.27)
where m 2 0,ij are tree-level mass parameters, Π ij are renormalised self-energies. The pole masses of the scalars are then found by solving for p 2 in the equation

det p 2 δ ij -(M) loop ij (p 2 ) = 0 , (2.2.28)
which is the generalisation of equation (2.2.26).

We show, as an example, in gures 2.2 and 2.3 the topologies allowed respectively for one-and two-loop self-energy diagrams, that contribute to mass corrections. Dierent loop functions are obtained depending on the type of particle (scalar, fermion, vector boson, or ghost) that runs in each internal line of every topology, nevertheless it is possible, using tensor reduction (and tools such as TARCER [START_REF] Mertig | TARCER: A Mathematica program for the reduction of two loop propagator integrals[END_REF]), to express all the functions corresponding to scalar self-energy diagrams in terms of scalar integrals

i.e integrals that involve only scalar propagators which can be computed in DRED or DREG. There are two scalar functions at one loop, A and B dened respectively in eqs. (B.1.6) and (B.1.7), and analytic expressions for both have been known for a long time (see e.g. [START_REF] Passarino | One Loop Corrections for e+ e-Annihilation Into mu+ mu-in the Weinberg Model[END_REF]). At two-loop order, there are three scalar functions S, U , and M (dened in eqs. (B.1.18) to (B.1.20)) that appear, together with A and B, in the expressions of all self-energy diagrams. However, their expressions are in general not known analytically for non-zero external momentum and for dierent masses of particles in the loops, but can be computed either by numerical integration (for example with the program SecDec [START_REF] Borowka | SecDec-3.0: numerical evaluation of multi-scale integrals beyond one loop[END_REF]), or by solving systems of dierential equations that relate the loop functions such dierential equations were derived in [START_REF] Martin | Evaluation of two loop selfenergy basis integrals using dierential equations[END_REF] and are implemented in the C++ program TSIL [START_REF] Martin | TSIL: A Program for the calculation of two-loop self-energy integrals[END_REF]. Precision calculations of the Higgs boson mass However, numerical evaluations of loop integrals with tools like SecDec or TSIL are costly (in terms of time, or equivalently computing power) and the eect of momentum at two-loop order is generally expected to be small. Therefore, a simpler approach is often taken by working in the limit of vanishing external momentum, which is actually equivalent to a calculation using second derivatives of the eective potential. This last statement can be proven as follows: we start from the eective action Γ dened in eq. (1.1.28), and that we can expand as

A B

Γ{φ i (x)} = d 4 x -V e + 1 2 Z ij ∂ µ φ i ∂ µ φ j + • • • . (2.2.29)
We furthermore know that by construction, Γ is the generating functional of 1PI graphs: therefore the second derivative of Γ with respect to two elds φ i (x), φ j (y) corresponds to the inverse propagator in position space ∆ -1 (xy), and its Fourier transform is then the inverse propagator in momentum space i.e. i∆ -1 (p

2 ) ≡ p 2 δ ij - m 2 0,ij -Π ij (p 2
∂ 2 V e ∂φ i ∂φ j min = -i∆ -1 (0) = m 2 0,ij + Π ij (0) , (2.2.30)
or equivalently (with V e = V (0) + ∆V )

∂ 2 ∆V ∂φ i ∂φ j min = Π ij (0) . (2.2.31)
The eective potential approach, or in other words using the second derivatives of the eective potential, greatly simplies the computation of mass corrections, and in particular, solving eq. (2.2.28) then amounts to nding the eigenvalues of the matrix

M 2 e ij = m 2 0,ij + ∂ 2 ∆V ∂φ i ∂φ j min .
(2.2.32)

The eective eld theory approach

The Higgs mass computed at renormalisation scale Q from the xed-order methods described above can schematically be written as

m 2 h = 1 (16π 2 ) n=0      {n(i)} i n(i)=n c n ({n(i)}) i log n(i) m 2 i Q 2      , (2.2.33)
where is the loop order, the index i labels the dierent particles that contribute to the Higgs mass (m 2 i are then the corresponding masses, etc.), and the factors c n ({n(i)}) can in general be found numerically.

When the hierachies between (some of ) the mass scales in the loop corrections become important, large logarithmic contributions may appear, causing the perturbative expansion done for xed-order calculations to become inadequate. Indeed, it is natural to compute the Higgs mass at a relatively low scale Q = m (most usually one takes the top mass, m = m t ) and if one or several particles have large masses, denoted M , then the -loop corrections will contain terms of the form log M/m, which are greatly enhanced when M m.

State-of-the-art of Higgs mass calculations

The solution to this problem is to use an eective eld theory (EFT) approach to the calculation, which allows a resummation of the large log M/m logarithms. The heavy particles, i.e. those of mass higher than, or of the order of, M , are integrated out at a high scale Q typically one takes Q = M and below the scale Q one works with an eective eld theory, i.e. a low-energy theory containing only light states. The usual way to integrate out heavy states is to perform a matching of n-point functions, computed (diagrammatically) respectively in the EFT and in the full theory at the scale Q and at the desired loop order (say -loop order), which reads

Γ -loop (n) (Q) = Γ -loop (n) (Q) , (2.2.34)
where Γ -loop

(n)
and Γ -loop

(n)
are functions of the parameters respectively in the low-energy and complete theories. Note that one could equivalently have performed a matching of the eective actions of the EFT and the complete theory to obtain the relations for the couplings. The matching then denes the masses and couplings that will appear in the EFT and the eects of heavy states at low energies come from two sources:

• via the matching conditions for the couplings that already existed in the full theory and receive threshold corrections in the EFT;

• via new operators that arise in the EFT from the matching, and in particular higher-dimensional (in other words non-renormalisable) operators, which are suppressed by powers of M (also obtained by the matching of n-point functions between high-and low-energy theories).

Properties of the light particles are then computed using the parameters and eld content of the low-energy theory, and with the additional higher-dimensional operators the impact of the latter however decreases as the ratio m/M tends to zero (as these operators are suppressed by powers of M ). For example, one nds that the (running) mass m2

φ of a scalar φ (not considering any mixing here) in the EFT is related at the matching scale Q to the mass parameter in the full theory by threshold corrections ∆m 2

φ as m2 φ (Q) = m 2 φ (Q) + ∆m 2 φ (Q) . (2.2.35)
The running parameter m2 φ (Q) can then be run to the scale at which one wants to compute the pole mass of φ, and then, following equation (2.2.26), one nds

M 2 φ = m 2 φ + Πφ (p 2 = M 2 φ ) , (2.2.36)
where Πφ is calculated in the low energy theory.

In this EFT approach, radiative corrections are resummed to N LL (i.e. (next-to-) leading logarithmic) order when performing the matching of couplings and the calculation of corrections at -loop order, together with + 1-loop running of the renormalisation group equations. 5

State-of-the-art of Higgs mass calculations

In this section, we will attempt to give a brief review of Higgs mass calculations in SUSY models. Providing an exhaustive description of all results in this eld during Precision calculations of the Higgs boson mass the past three decades would be a near-impossible task, and would go well beyond the scope of this thesis; hence we will instead only mention some of the main contributions, relevant for the lightest Higgs boson mass.

Real and complex MSSMs

As we discussed in the previous chapter, in the MSSM the Higgs quartic couplings are related to the electroweak gauge couplings and the tree-level mass of the lightest CP-even Higgs boson is

(m 2 h ) tree = 1 2 m 2 A + m 2 Z -(m 2 A -m 2 Z ) 2 + 4m 2 Z m 2 A sin 2 2β 1/2 (m 2 h ) tree ≤ m 2 Z cos 2 2β . (2.3.1)
The existence of an upper bound on (m 2 h ) tree is by itself a very important feature of SUSY models. However, from the above equation it may seem that the MSSM is excluded in view of the observation of a 125 GeV Higgs boson (and that it would have been already from the early experimental bounds from LEP). Fortunately, the tree-level Higgs mass is dramatically altered by radiative corrections.

The initial spark of Higgs mass calculations in SUSY was the study in refs.

[7375] of the impact of the one-loop corrections from tops and stops on m h and its theoretical upper bound, in the CP-conserving (i.e. real) MSSM. In these three references, the dominant O(α t ) (with α t ≡ y 2 t /4π) corrections 6 to the CP-even Higgs masses are computed in the eective potential approximation for [START_REF] Ellis | Radiative corrections to the masses of supersymmetric Higgs bosons[END_REF][START_REF] Okada | Upper bound of the lightest Higgs boson mass in the minimal supersymmetric standard model[END_REF], and a diagrammatic approach for [START_REF] Haber | Can the mass of the lightest Higgs boson of the minimal supersymmetric model be larger than m(Z)?[END_REF] and it is shown that for a large top mass 7 (and thus a large top Yukawa) the mass of lightest Higgs can become signicantly larger than m Z (the tree-level bound) due to the radiative corrections. The reason for this is that, while the top quark contributes negatively to scalar masses (because of the minus sign of the fermion loop), the contributions from the heavier stops are positive and larger in absolute value than those from the top. In fact, the one-loop corrections to the lightest Higgs mass from stops and tops can be as large as the tree-level value. While this may seem to contradict perturbativity, such huge corrections are actually not a problem here as they are driven by the large top Yukawa y t that only enters the computation at one-loop order. A major consequence of these papers is that not only does Supersymmetry predict a value for the Higgs mass, but that this value can be modied by large radiative corrections, which should therefore be computed to higher accuracy. This led to intensive work on the MSSM with real parameters in the 1990's, and later on its complex counterpart, with numerous xed-order and EFT calculations.

Chronologically, both approaches were considered from an early point, but for clarity we will discuss them separately here.

Fixed-order results

In the early 1990's, numerous papers studied Higgs masses in the xed-order approach, either in the eective potential approximation [START_REF] Ellis | Radiative corrections to the masses of supersymmetric Higgs bosons[END_REF][START_REF] Okada | Upper bound of the lightest Higgs boson mass in the minimal supersymmetric standard model[END_REF]151154], or with diagrammatic 6 Let us specify here that the corrections we are referring to are actually of the form m 4 t /v 2 ∝ αtm 2 t .

We also note that some other notational conventions for radiative corrections may be slightly abused in the following e.g. O(αtαs) as is commonly done in the literature.

7 One must remember that when these three papers where written, the top quark had not yet been detected and its mass was therefore not known. The value of the top mass that has been obtained from Tevatron and LHC measurements around 173 GeV is indeed large for the discussion here.

H 0 d = 1 √ 2 (v d + S 1 + iP 1 )
, and

H 0 u = 1 √ 2 (v u + S 2 + iP 2 ) . (2.3.2)
In the notations of section 1.3.4.2 we nd . We recall that these are of the form

det p 2 • 1 2 -M 2 S loop (p 2 ) = 0 , (2.3.3) with M 2 S loop 11 (p 2 ) = m 2 Z c 2 β + m 2 A s 2 β - 1 v d ∂∆V ∂S 1 min. + Π S 1 S 1 (p 2 ) , M 2 S loop 12 (p 2 ) = - 1 2 (m 2 Z + m 2 A )s 2β + Π S 1 S 2 (p 2 ) , M 2 S loop 22 (p 2 ) = m 2 Z s 2 β + m 2 A c 2 β - 1 v u ∂∆V ∂S 2 min. + Π S 2 S 2 (p 2 ) , (2.3
1 v d ∂V e ∂S 1 min. = 1 v d ∂V (0) ∂S 1 min. + 1 v d ∂∆V ∂S 1 min. = 0 , 1 v u ∂V e ∂S 2 min. = 1 v u ∂V (0) ∂S 2 min. + 1 v u ∂∆V ∂S 2 min. = 0 .
(2.3.5)

Furthermore, as S 1 and S 2 are the real components of H 0 u and H 0

d , it is clear that 1 v d ∂V (0) ∂S 1 min. = √ 2 v d ∂V (0) ∂H 0 d min. , 1 v d ∂V (0) ∂S 2 min. = √ 2 v u ∂V (0) ∂H 0 u min. , (2.3.6)
and the right-hand sides of these two relations are given in eq. (1.3.72). When applying the minimum conditions to the diagonal terms in eq. (2.3.4), we have split them between tree-and loop-level terms, as shown in (2.3.5), and we have simplied the expressions of the tree-level masses using the corresponding tree-level tadpole terms, i.e. equation (2.3.6).

One advantage of this approach is that the expressions (2.3.4) can be used for any desired order in perturbation theory, and that it also allows the denition of an eective mixing angle α at any order with the relation (see section 1.3.4.2 for its tree-level value)

M 2 P loop (p 2 ) 11 = m 2 A s 2 β - 1 v d ∂∆V ∂S 1 min. + Π P 1 P 1 (p 2 ) , M 2 P loop 12 (p 2 ) = 1 2 m 2 A s 2β + Π P 1 P 2 (p 2 ) , M 2 P loop 22 (p 2 ) = m 2 A c 2 β - 1 v u ∂∆V ∂S 2 min. + Π P 2 P 2 (p 2 ) , (2.3.8)
where m 2

A is again the running pseudoscalar mass.

The leading corrections to m 2 h at one-loop order are the O(α t ) and O(α b ) contributions where α b ≡ y 2 b /4π. In the decoupling limit m 2 A m 2

Z and for large degenerate soft masses for the two stops

m 2 Q = m 2 U = M 2 S m 2
t , the one-loop corrections to m 2 h can be written in the very simple form

∆ (1) m 2 h = 3m 4 t 2π 2 v 2 log M 2 S m 2 t + X2 t - X4 t 12 - y 4 b µ 4 tan 4 βv 2 32π 2 M 4 S + • • • , (2.3.9)
where we recall that X t is the stop mixing X t = A tµ cot β (here for the real MSSM in which µ ∈ R) and Xt = X t /M S . Complete results for the one-loop Higgs masses were obtained in [START_REF] Chankowski | Complete on-shell renormalization scheme for the minimal supersymmetric Higgs sector[END_REF][START_REF] Dabelstein | The One loop renormalization of the MSSM Higgs sector and its application to the neutral scalar Higgs masses[END_REF][START_REF] Pierce | Precision corrections in the minimal supersymmetric standard model[END_REF].

However, even with a full one-loop calculation, theoretical uncertainties remain large, and higher-order corrections are needed to obtain precise predictions. The rst twoloop results were derived for the approximation of zero external momentum [START_REF] Martin | Complete two loop eective potential approximation to the lightest Higgs scalar boson mass in supersymmetry[END_REF][START_REF] Degrassi | On the neutral Higgs boson masses in the MSSM for arbitrary stop mixing[END_REF]164175]. While the earliest of these papers have studied the O(α s α t ) and O(α 2 t ) contributions, ref. [START_REF] Martin | Complete two loop eective potential approximation to the lightest Higgs scalar boson mass in supersymmetry[END_REF] gave the complete two-loop corrections to the lightest Higgs mass in the eective potential approach, but encountered for the rst time the Goldstone Boson Catastrophe (because computations were performed beyond the gaugeless limit). More recently, the dominant momentum-dependent two-loop [176178] and the leading three-loop corrections [START_REF] Harlander | Higgs boson mass in supersymmetry to three loops[END_REF][START_REF] Kant | Light MSSM Higgs boson mass to three-loop accuracy[END_REF] have also been calculated.

When considering the complex MSSM, the distinction between CP-even and CP-odd

Higgses cannot be made anymore, and one must therefore work with a 4 × 4 mass matrix for all neutral states in the Higgs sector of the form SUSY-QCD corrections were given with full momentum dependence in [START_REF] Borowka | Complete two-loop QCD contributions to the lightest Higgs-boson mass in the MSSM with complex parameters[END_REF].

M 2 n loop (p 2 ) = M 2 S loop (p 2 ) M 2 SP loop (p 2 ) M 2 SP loop (p 2 ) * M 2 P loop (p 2 ) , ( 2
Since the late 1990's and early 2000's, results for xed-order Higgs mass calculations in the MSSM have been implemented in a number of public spectrum generators, i.e.

codes which determine the complete loop-corrected mass spectrum of the theory (and often use these masses to derive other physical observables). Currently, several such programs provide full one-loop and leading two-loop corrections to Higgs masses, and among the most widely-used of them, one can mention FeynHiggs [START_REF] Heinemeyer | FeynHiggs: A Program for the calculation of the masses of the neutral CP even Higgs bosons in the MSSM[END_REF], SoftSUSY [START_REF] Allanach | SOFTSUSY: a program for calculating supersymmetric spectra[END_REF][START_REF] Allanach | Higher order corrections and unication in the minimal supersymmetric standard model: SOFTSUSY3.5[END_REF], SuSpect [START_REF] Djouadi | SuSpect: A Fortran code for the supersymmetric and Higgs particle spectrum in the MSSM[END_REF], and SPheno [START_REF] Porod | SPheno, a program for calculating supersymmetric spectra, SUSY particle decays and SUSY particle production at e+ e-colliders[END_REF][START_REF] Porod | SPheno 3.1: Extensions including avour, CP-phases and models beyond the MSSM[END_REF]. FeynHiggs diers from the three other mentioned codes because of the corrections it implements and of the choice of renormalisation scheme it employs for the two-loop corrections: indeed it uses results in the on-shell scheme from [START_REF] Degrassi | On the neutral Higgs boson masses in the MSSM for arbitrary stop mixing[END_REF][START_REF] Brignole | On the O(alpha(t)**2) two loop corrections to the neutral Higgs boson masses in the MSSM[END_REF][START_REF] Brignole | On the two loop sbottom corrections to the neutral Higgs boson masses in the MSSM[END_REF][START_REF] Dedes | On the two loop Yukawa corrections to the MSSM Higgs boson masses at large tan beta[END_REF][START_REF] Borowka | Momentum-dependent two-loop QCD corrections to the neutral Higgs-boson masses in the MSSM[END_REF]184187], while SoftSUSY, SuSpect, and SPheno use results in the DR scheme from [START_REF] Degrassi | On the neutral Higgs boson masses in the MSSM for arbitrary stop mixing[END_REF][START_REF] Pierce | Precision corrections in the minimal supersymmetric standard model[END_REF][START_REF] Brignole | On the O(alpha(t)**2) two loop corrections to the neutral Higgs boson masses in the MSSM[END_REF][START_REF] Brignole | On the two loop sbottom corrections to the neutral Higgs boson masses in the MSSM[END_REF][START_REF] Dedes | On the two loop Yukawa corrections to the MSSM Higgs boson masses at large tan beta[END_REF][START_REF] Dedes | Two loop corrections to radiative electroweak symmetry breaking in the MSSM[END_REF]. We will return in more detail to SPheno, and especially to its extension to SUSY models beyond the MSSM and to non-SUSY models, in section 2.4.3.2.

A brief remark on the evaluation of the theoretical uncertainties on the determination of m h should be made here: in order to compare the result of a given calculation with the measured value of m h in a meaningful way, it is necessary to provide an estimate of the uncertainty in the theoretical computation. While crude estimates can be found for example by varying the renormalisation scale at which a computation is done, or by changing how parameters that enter the two-loop corrections are determined (these two methods help give an idea of the size of missing three-loop terms), obtaining a precise number is a dicult task, and furthermore depends on the parameter point for which the masses are computed. We will therefore only keep in mind the estimate of about 3 GeV, obtained in [START_REF] Degrassi | Towards high precision predictions for the MSSM Higgs sector[END_REF][START_REF] Allanach | Precise determination of the neutral Higgs boson masses in the MSSM[END_REF] for regions of the MSSM parameters favoured by naturalness.

The observed value of the Higgs mass, at 125 GeV, and the tree-level bound on the lightest Higgs mass m h in the MSSM, imply that the radiative corrections ∆m 2 h must be of the same order as the tree-level mass-squared m 2 h tree . If we assume that we are in the most favourable case in which the tree-level bound is saturated (this happens when tan β and m A are both large), the observed Higgs boson mass requires the stop masses to be of the order of 1 TeV in a scenario with maximal stop mixing i.e. X t /M S ∼ 2,

although the precise number is somewhat dependent on the renormalisation scheme in which X t is computed or stop masses larger than 10 TeV in the absence of mixing (X t = 0). In this regime, logarithms of the form log M S /m t become large and cause a (signicant) loss of accuracy in xed-order calculations, which has led to a renewal of the interest for the EFT approach.

EFT and hybrid results

Eective Field Theory techniques have also been applied to Higgs mass calculations, since the beginning of the 1990's, by matching the MSSM onto the SM or a (variant of ) 2HDM early examples of these two options include e.g. [199201] and [START_REF] Sasaki | Renormalization group analysis of the Higgs sector in the minimal supersymmetric standard model[END_REF][START_REF] Haber | The Renormalization group improved Higgs sector of the minimal supersymmetric model[END_REF] respectively. When integrating out SUSY partners, higher-dimensional operators are generated, which are suppressed by powers of v 2 /M 2 S where the Higgs VEV v characterises the EW scale and M S denotes the mass scale of the SUSY particles. However, SUSY particles were at rst expected to be found at scales not much higher than the electroweak scale, and in that case, xed-order calculations would not be suering from the presence of large logarithms while EFT calculations would require Precision calculations of the Higgs boson mass taking into account a large number of higher-dimensional operators to give a precise prediction for m h . For this reason, the EFT approach was until recently seldom used to compute complete radiative corrections to Higgs masses let us mention however ref. [START_REF] Espinosa | Radiative corrections to the Higgs boson mass for a hierarchical stop spectrum[END_REF] where an EFT served to study of the leading two-loop contributions to the Higgs mass in a scenario with a large hierarchy between the stops (the heavy stop being integrated out), and where the eect of higher-dimensional (dimension-6) operators is taken into account. Nevertheless, EFT techniques have been employed to extract the logarithmic contributions to m 2 h , with results obtained at one loop [199, 200, 202, 203], two loops [START_REF] Espinosa | Two loop radiative corrections to the mass of the lightest Higgs boson in supersymmetric standard models[END_REF]205209], and also three loops [START_REF] Degrassi | Towards high precision predictions for the MSSM Higgs sector[END_REF][START_REF] Martin | Three-loop corrections to the lightest Higgs scalar boson mass in supersymmetry[END_REF].

In later years, the discovery of the Higgs boson with a relatively heavy mass and the absence of signs of SUSY partners at the LHC have led the interest for EFTs to rise again in part because, for larger SUSY-breaking scale, xed-order calculations exhibit a loss of accuracy and at the same time the eects of higher-dimensional operators become smaller. A rst possible, purely EFT, strategy is to match the MSSM with the SM, and take advantage of the current knowledge of three-loop RGEs and full twoloop Higgs mass calculations in the SM see [START_REF] Buttazzo | Investigating the near-criticality of the Higgs boson[END_REF][START_REF] Bednyakov | Higgs self-coupling beta-function in the Standard Model at three loops[END_REF][START_REF] Chetyrkin | β-function for the Higgs self-interaction in the Standard Model at three-loop level[END_REF]. More precisely, the idea is to match at a scale Q the SM Higgs quartic coupling λ to its MSSM value (related to the EW gauge couplings), with threshold corrections that can be expanded in perturbation theory as

λ(Q) = 1 4 g 2 (Q) + g 2 (Q) cos 2 2β + n=1 1 (16π 2 ) n δ (n) λ , (2.3.11)
where is the loop order to which the matching is performed. The threshold corrections have been obtained to full one-loop order in [211213], and leading two-loop (as well as higher-order) contributions have been studied [START_REF] Espinosa | Complete two loop dominant corrections to the mass of the lightest CP even Higgs boson in the minimal supersymmetric standard model[END_REF]213216]. Combined with the above cited RGEs and corrections for the SM, this makes possible a full next-to-leading and partial next-to-next-to-leading logarithmic order resummation of the corrections to m h . Some of these results are implemented in public codes, such as SusyHD [START_REF] Vega | SusyHD: Higgs mass Determination in Supersymmetry[END_REF], MhEFT (based on [START_REF] Draper | Precise estimates of the Higgs mass in heavy supersymmetry[END_REF][START_REF] Lee | Higgs bosons in heavy supersymmetry with an intermediate m A[END_REF], and able to accomodate a 2HDM instead of the SM as low energy model), or HSSUSY [START_REF] Athron | Precise Higgs mass calculations in (non-)minimal supersymmetry at both high and low scales[END_REF] (itself a module of FlexibleSUSY [START_REF] Athron | FlexibleSUSYA spectrum generator generator for supersymmetric models[END_REF][START_REF] Athron | FlexibleSUSY 2.0: Extensions to investigate the phenomenology of SUSY and non-SUSY models[END_REF]).

Dierent approaches, which we will refer to as hybrid approaches, have also been developed recently in order to benet simultaneously from the advantages of xedorder and EFT calculations. One of these methods, devised in FlexibleEFTHiggs [START_REF] Athron | Precise Higgs mass calculations in (non-)minimal supersymmetry at both high and low scales[END_REF][START_REF] Athron | FlexibleSUSY 2.0: Extensions to investigate the phenomenology of SUSY and non-SUSY models[END_REF], constitutes a variation of the purely-EFT calculation described above in which the matching condition of the quartic constant as in equation (2.3.11) is replaced by a matching of Higgs pole masses between the SM and the MSSM, or any general BSM model. Indeed, it should be noted that with the framework of FlexibleSUSY/FlexibleEFTHiggs computations can be performed for a wide range of BSM models as is the case with the SARAH/SPheno framework, to which we will return in section 2.4.3. As in the SM the tree-level mass of the Higgs boson and in turn its pole mass, see eqs. (1.1.38) and (1.1.39) depends on the quartic coupling λ, we can extract it at a scale Q from the pole mass matching and we nd

λ(Q) = 1 2v 2 (M 2 h ) MSSM + 1 v ∂(∆V ) SM ∂h min.
-Π SM hh p 2 = (m 2 h ) MSSM .

(2.3.12)

where, quantities on the right-hand side of this equation are also computed at the matching scale Q. More specically, (M 2 h ) MSSM is the pole mass of the Higgs computed in the MSSM (or another BSM model), (∆V ) SM denotes the loop corrections to the SM eective potential and Π SM hh is the Higgs self-energy computed in the SM. The

State-of-the-art of Higgs mass calculations

FlexibleEFTHiggs procedure now allows the combination of a full one-loop calculation of m h , together with the resummation of large logarithms up to next-to-leading logarithmic order [START_REF] Athron | FlexibleSUSY 2.0: Extensions to investigate the phenomenology of SUSY and non-SUSY models[END_REF]. A similar calculation has also been made available (for generic models as well) in SARAH/SPheno in [START_REF] Staub | Improved predictions for intermediate and heavy Supersymmetry in the MSSM and beyond[END_REF].

For completeness, we should also mention that another hybrid approach has been taken in [222224]. The idea is to add to the Higgs mass computed in the Feynmandiagrammatic way an additional term equal to the corrections to m 2 h in the EFT approach, but from which two types of terms have been subtracted to avoid doublecounting of corrections: (i) the one-and two-loop logarithmic terms in the xed-order calculation (i.e. the terms in eq. (2.2.33) for = 1, 2); and (ii) the non-logarithmic parts of the EFT corrections. This approach then incorporates both the corrections obtained in the xed-order approach (and in particular takes into account terms of proportional to powers of v 2 /M 2 S ) and the higher-order logarithms resummed to leading and next- to-leading logarithmic orders in the EFT calculation this method proves especially useful for intermediate SUSY scales (where the theoretical uncertainties in the two standard approaches, xed-order and EFT, are large).

Recently, eorts on the determination of Higgs masses in the MSSM have been concentrated on three main areas: (i) completing the computation of the full two-loop momentum dependence in the diagrammatic approach (in the complex MSSM); (ii) improving EFT and hybrid calculations (extend pole mass matching to two loops; integrate out SUSY states gradually, with a tower of EFTs between the MSSM and the SM, etc.); (iii) obtaining the three-loop corrections to the scalar masses and making these new results available in the existing automated tools.

Supersymmetric models beyond the MSSM

Eorts to derive corrections to neutral scalar masses in the NMSSM have, until recent years, been somewhat more moderate and therefore the available results in these models are less advanced than in the MSSM. However, one-loop corrections have been studied in the CP-conserving NMSSM already since the 1990's, with corrections from tops and stops as well as from bottoms and sbottoms obtained, at zero external momentum, in [225228], while leading logarithmic contributions from charginos, neutralinos and scalars where given in [START_REF] Ellwanger | Yukawa induced radiative corrections to the lightest Higgs boson mass in the NMSSM[END_REF]. Complete one-loop diagrammatic calculations with full momentum dependence where carried out in [START_REF] Degrassi | On the radiative corrections to the neutral Higgs boson masses in the NMSSM[END_REF][START_REF] Staub | The Electroweak sector of the NMSSM at the one-loop level[END_REF] (in a pure DR scheme), in [START_REF] Ender | Analysis of the NMSSM Higgs Boson Masses at One-Loop Level[END_REF] (both in a mixed DR -OS scheme and a pure OS scheme) and in [START_REF] Drechsel | Precise Predictions for the Higgs-Boson Masses in the NMSSM[END_REF] (in a mixed DR -OS scheme as well) for the CP-conserving NMSSM. In the presence of CP violation, rst investigations of scalar masses at one loop were also performed in the eective potential limit [233237] and complete results were found in [START_REF] Graf | Higgs Boson Masses in the Complex NMSSM at One-Loop Level[END_REF] with a diagrammatic approach (previously one-loop and leading two-loop results had been obtained with EFT techniques in [START_REF] Cheung | The Higgs Boson Sector of the Next-to-MSSM with CP Violation[END_REF]). Ref. [START_REF] Degrassi | On the radiative corrections to the neutral Higgs boson masses in the NMSSM[END_REF] provided the rst explicit two-loop results, by deriving the O(α t α s +α b α s ) corrections to Higgs masses in the CP-conserving NMSSM using the eective potential (and a pure DR renormalisation scheme). These results were extended in [START_REF] Mühlleitner | Two-loop contributions of the order O(α t α s ) to the masses of the Higgs bosons in the CP-violating NMSSM[END_REF] to the case with CP violation, and with a mixed DR -OS scheme.

As for the MSSM, the analytic expressions that we have cited here have been implemented in a number of public spectrum generators written specically for the NMSSM, namely SoftSUSY [START_REF] Allanach | SOFTSUSY: a program for calculating supersymmetric spectra[END_REF][START_REF] Allanach | Next-to-Minimal SOFTSUSY[END_REF], NMSSMTools [242244], and NMSSMCALC [START_REF] Baglio | NMSSMCALC: A Program Package for the Calculation of Loop-Corrected Higgs Boson Masses and Decay Widths in the (Complex) NMSSM[END_REF][START_REF] King | Exploring the CP-violating NMSSM: EDM Constraints and Phenomenology[END_REF] (for detailed discussions and comparisons of these codes, the reader may also refer to [START_REF] Staub | Higgs mass predictions of public NMSSM spectrum generators[END_REF][START_REF] Drechsel | Higgs-Boson Masses and Mixing Matrices in the NMSSM: Analysis of On-Shell Calculations[END_REF]). It should be noted that SoftSUSY and NMSSMTools also include some Precision calculations of the Higgs boson mass partial contributions that depend only on the third family Yukawa couplings, adapted from the MSSM calculations i.e. adding corrections only to the 2×2 part of the NMSSM CP-even mass matrix 8 that corresponds to the states present in the MSSM. Moreover, NMSSMCALC takes into account terms that arise from the extraction of the DR running VEV at two-loop order, and give eects on the scalar masses formally of the form O(α t α s ) (these contributions are not available in the other codes). New corrections to Higgs masses in the NMSSM have also been obtained by employing results for generic theories implemented in SARAH/SPheno and FlexibleSUSY/FlexibleEFTHiggs instead of performing explicit calculations. In particular, ref. [START_REF] Goodsell | Two-loop corrections to the Higgs masses in the NMSSM[END_REF] investigated contributions beyond O(α t α s + α b α s ), more precisely the eects of the complete corrections of the form O(α 2 t + α 2 b + α 2 τ ) (and not only the corresponding MSSM contributions), and of terms involving the superpotential singlet coupling λ S .

For supersymmetric models beyond the NMSSM, no explicit expressions for contributions to Higgs masses at more than one-loop order had been obtained until the calculation of the leading two-loop O(α t α s ) corrections in models with Dirac gauginos that we will present in chapter 3. It should be noted, however, that SARAH/SPheno has allowed studies of Higgs masses at two loops since version 4.4.0 (see ref. [START_REF] Goodsell | Two-Loop Higgs mass calculations in supersymmetric models beyond the MSSM with SARAH and SPheno[END_REF]), and has indeed been used e.g. for the MDGSSM [START_REF] Goodsell | Dark matter scenarios in a constrained model with Dirac gauginos[END_REF] or the MRSSM [START_REF] Diessner | Two-loop correction to the Higgs boson mass in the MRSSM[END_REF][START_REF] Diessner | Exploring the Higgs sector of the MRSSM with a light scalar[END_REF].

Non-supersymmetric models

In most non-supersymmetric models, the scalar quartics are free parameters this is for example the case in BSM models, such as 2HDMs or the Georgi-Machacek model, but also in the Standard Model. The tree-level Higgs masses are then also free parameters, and it is often assumed that radiative corrections to scalar masses and mixing angles can be reabsorbed into a shift of the Lagrangian parameters (in particular of the quartics). There have therefore been very few studies of corrections to Higgs masses in non-SUSY extensions of the SM, and instead eorts have been concentrated on the calculation of corrections to Higgs couplings or observables (decay widths, cross sections, etc.), in terms of scalar (pole) masses see for example [253257].

There are nevertheless instances where it is useful to compute masses in such models.

On the one hand, in the context of EFT calculations it has become common to integrate out part of the spectrum of a SUSY model, and consider at low energies an extension of the SM with additional elds, and in particular with extended Higgs sectors e.g. consider a 2HDM as a low energy limit of the MSSM. The quartic couplings in the low energy model are then no longer free but are instead xed by the matching condition with the SUSY model (in which the quartics are related to the EW gauge couplings), and this allows predictive results. On the other hand, one is sometimes interested in deriving the values of the Lagrangian parameters of the scalar sector of a given model from the input of the mass spectrum, and as we will discuss in chapter 6, it is in general crucial to include the eects of quantum corrections in this extraction.

The simplest way to proceed is then to determine (typically by numerical iterations) the Lagrangian parameters that correspond to the desired loop-corrected spectrum through a calculation of the Higgs masses (see e.g. ref. [START_REF] Costa | Two-loop stability of a complex singlet extended Standard Model[END_REF] for an early example in the case of a singlet extension of the SM, and chapter 6 for a more general discussion).

8 We recall that in the CP-conserving NMSSM, there are three CP-even states: the real components S1, S2 of H 0 u and H 0 d as in the MSSM, plus the real component S3 of the singlet scalar. Their masses are described by a 3 × 3 (loop-corrected) mass matrix. What SoftSUSY and NMSSMTools are doing is to include the additional corrections only for the upper-left 2 × 2 part of that matrix.

Calculations in generic theories

Calculations in generic theories

In the previous sections, we have briey reviewed some of the results obtained since the early 1990's on the calculation of Higgs boson masses in the minimal supersymmetric extensions of the SM mostly the MSSM but also to a smaller extent the NMSSM.

While considerable eorts have been devoted to this task, achieving a remarkable level of accuracy, there is no guarantee that the MSSM or the NMSSM are realised by Nature, and a tremendous number of other phenomenologically viable models exist and receive interest, such as non-minimal supersymmetric models (Dirac gaugino models would for instance belong to this category) or non-supersymmetric theories. One may wish to study scalar masses in a large variety of such models with a similar level of precision as what has been obtained in the MSSM and NMSSM, and one then arrives to the question of how to proceed to avoid long and painful calculations for each model under consideration. The answer is automation, i.e. to perform the dicult computation of radiative corrections to an observable e.g masses once for a general model, and afterwards apply these results to any number of particular models as desired. The second step in this scheme involves translating the properties (eld content, Lagrangian parameters, etc.) of the given model(s) into the general setting in which one has performed the calculation of the radiative corrections. This is actually not a dicult task, provided that suitable conventions are employed for the generic models, and it can even be automated with computer programs e.g. with the Mathematica package SARAH, see section 2.4.3. This section will serve to introduce rst the necessary notations and the existing results for calculations of Higgs or more generally neutral scalar masses in general renormalisable eld theories, before presenting the framework provided by the public tools SARAH and SPheno for the automated and precise study of a wide range of BSM models. While we have already mentioned the fact that these two programs have made possible studies of non-minimal supersymmetric models (more specically Dirac gaugino models), it is important to note that they have also allowed the study of new two-loop contributions to Higgs masses in the MSSM and NMSSM not derived in explicit calculations. Such new corrections include the eects at two loops of non-minimal avour violation [START_REF] Goodsell | The Higgs Mass in the MSSM at two-loop order beyond minimal avour violation[END_REF] or of trilinear R-parity-violating operators [START_REF] Dreiner | On the two-loop corrections to the Higgs mass in trilinear R-parity violation[END_REF] in the MSSM, as well as the additional results in the NMSSM [START_REF] Goodsell | Two-loop corrections to the Higgs masses in the NMSSM[END_REF] that we mentioned previously.

Notations for general eld theories

We begin by presenting our notations for the purely scalar sector of a generic theory, following the procedure in [START_REF] Goodsell | Generic two-loop Higgs mass calculation from a diagrammatic approach[END_REF] (itself in the setting established in [START_REF] Martin | Two loop eective potential for a general renormalizable theory and softly broken supersymmetry[END_REF][START_REF] Martin | Two loop scalar self energies in a general renormalizable theory at leading order in gauge couplings[END_REF]). We introduce rst the scalar Lagrangian and potential in terms of real scalar elds ϕ 0 i , for which the Lagrangian mass parameters are not necessarily diagonal 9

L S = 1 2 (∂ µ ϕ 0 i ) 2 -V (0) ({ϕ 0 i }) , V (0) ({ϕ 0 i }) = ti ϕ 0 i + 1 2 m2 0,ij ϕ 0 i ϕ 0 j + 1 6 λijk 0 ϕ 0 i ϕ 0 j ϕ 0 k + 1 24 λijkl 0 ϕ 0 i ϕ 0 j ϕ 0 k ϕ 0 l . (2.4.1)
9 This is not uncommon when considering some particular theory: the Lagrangian is typically not written in terms of states with well-dened (i.e. diagonal) masses but in terms of interaction eigenstates (i.e. gauge and/or avour eigenstates). One then computes a mass matrix for the interaction eigenstates that can be diagonalised to obtain the mass eigenstates and eigenvalues see e.g. our discussion for stops in the MSSM and equations (1.3.86) to (1.3.90).

Precision calculations of the Higgs boson mass

Note that the generic scalar index i can correspond in a given model to a collection of avour (or family) indices, gauge indices, etc., and that we will also imply a summation on repeated indices. Also, unless otherwise specied, we will in the following assume all scalar couplings to be fully symmetric under the exchange of scalar indices. Furthermore, we have expanded the covariant derivatives in the kinetic terms and we will give the terms other than (∂ µ ϕ 0 i ) 2 separately as scalargauge-boson couplings in the following (we will proceed similarly for the fermion and gauge-boson kinetic terms).

We can then expand the scalars as uctuations φ 0 i around vacuum expectation values v i such that ϕ 0 i ≡ v i + φ 0

i and rewrite the above potential as

V (0) ({ϕ 0 i }) = V (0) (v i ) + ti 0 φ 0 i + 1 2 m2 0,ij φ 0 i φ 0 j + 1 6 λijk 0 φ 0 i φ 0 j φ 0 k + 1 24 λijkl 0 φ 0 i φ 0 j φ 0 k φ 0 l . (2.4.2)
with the redenitions

ti 0 = ti + m2 0,ij v j + 1 2 λijk 0 v j v k + 1 6 λijkl 0 v j v k v l , m2 0,ij = m2 0,ij + λijk 0 v k + 1 2 λijkl 0 v k v l , λijk 0 = λijk 0 + λijkl 0 v l , λijkl 0 = λijkl 0 . (2.4.3)
The tree-level tadpole equations here simply read ti 0 = 0, and with the rst equation in (2.4.3) we nd that these impose relations among the parameters in the original basis e.g. one can eliminate one mass parameter m2 0,ij per tadpole equation and we use these relations to simplify the expression of m2 0,ij . From this we can dene the eld-dependent masses and couplings,

m2 ij (φ 0 ) ≡ ∂ 2 V (0) ∂φ 0 i ∂φ 0 j = m2 0,ij + λijk 0 φ 0 k + 1 2 λijkl 0 φ 0 k φ 0 l , (2.4.4) λijk (φ 0 ) ≡ ∂ 3 V (0) ∂φ 0 i ∂φ 0 j ∂φ 0 k = λijk 0 + λijkl 0 φ 0 l , (2.4.5) λijkl (φ 0 ) ≡ ∂ 4 V (0) ∂φ 0 i ∂φ 0 j ∂φ 0 k ∂φ 0 l = λijkl 0 .
(2.4.6)

We then introduce a new basis { φi } and an orthogonal matrix R to diagonalise the tree-level mass matrix as

φ 0 i = Rij φj , (2.4.7)
and obtain the new masses and couplings (2.4.11)

m2 i δ ij = m2 kl Rki Rlj

Calculations in generic theories

As we mentioned in the case of the SM in the previous chapter, we shall take the expectation values v i to be xed inputs for the scalar mass calculations (i.e. we consider that they are computed at the required order in perturbation theory through a separate calculation), and instead we solve the tadpole equations for the mass-squared parameters in the Lagrangian. This means correcting the mass parameters, passing from m2 0,ij (which satisfy the tree-level tadpole equations) to new quantities m 2 ij (which obey the loop-corrected tadpole equations). Using the minimisation conditions, the relationship between them is

m 2 ij v j = m2 0,ij v j - ∂∆V ({m 2 ij }) ∂φ 0 i φ 0 i =0
.

(2.4.12)

In order to be able to use the results for two-loop radiative corrections of [1921], we follow the choice of these papers and employ the parameters m 2 ij as masses of scalars in all loop calculations. Diagonalising these requires the introduction of a new basis via φ 0 i = R ij φ j , having squared masses m 2 i and couplings λ ijk , λ ijkl .

To illustrate what we mean by the above denitions, let us relate these to the case of the Higgs boson in the SM, which we had discussed in section 1.1.4. First, the mass term in the original Lagrangian is m2 0,hh = µ 2 , and then if we expand the SM Lagrangian around the Higgs VEV (or equivalently if we compute the second derivative of V (0) with respect to h) we obtain m2 0,hh = µ 2 +3λv 2 . Now, as we explained above, the expressions of the m2 0,ij are modied because of the relations among parameters imposed by the tree-level tadpole equations. Solving the tadpole equation th 0 = µ 2 v + λv 3 = 0 for µ 2 (i.e. m2 0,hh ), we have m2 0,hh = 2λv 2 .

(

m 2 hh = µ 2 + 3λv 2 = 2λv 2 - 1 v ∂∆V ∂h min. = m2 0,hh - 1 v ∂∆V ∂h min. . (2.4.15)
This is the equivalent of equation (2.4.12) for the SM (which we already found in eq. (1.1.38)).

We can then consider fermions and gauge bosons, using the notation for a general renormalisable eld theory used in [START_REF] Martin | Two loop eective potential for a general renormalizable theory and softly broken supersymmetry[END_REF][START_REF] Martin | Two loop scalar self energies in a general renormalizable theory at leading order in gauge couplings[END_REF]. The fermions ψ I are taken in Weyl notation and their kinetic and mass Lagrangian terms are written as

L F = i ψI σ µ ∂ µ ψ I - 1 2 M IJ ψ I ψ J + M * IJ ψI ψJ .
(2.4.16)

Here we should point out that we have followed the choice of [START_REF] Goodsell | Generic two-loop Higgs mass calculation from a diagrammatic approach[END_REF] to denote the complex conjugation explicitly on M * IJ while in [19, 20] complex conjugation is implied solely by the lowering and raising of fermion indices. We assume the fermions to be dened in a basis where all their squared masses are diagonal, i.e.

M IJ M * JK = m 2 I δ I K .

(2.4.17)

Precision calculations of the Higgs boson mass

The Yukawa couplings are denoted collectively y IJk and are symmetric under the exchange I ↔ J, and thus the scalarfermion interaction terms read

L SF = - 1 2 y IJk ψ I ψ J φ k + h.c. . (2.4.18)
Finally, massive vector (gauge) bosons A a µ can be introduced with kinetic and (diago- nal) mass terms

L V = - 1 4 F µνa F a µν - 1 2 m 2 a A a µ A µa , (2.4.19)
where we have expanded the standard (non-Abelian) eld-strength tensor as

F a µν = F a µν -g abc A b µ A c ν , F a µν = ∂ µ A a ν -∂ ν A a µ .
(2.4.20)

We have expanded all kinetic terms and we can now give the interaction terms of gauge bosons with scalars, with fermions, or with themselves (and ghosts), as found in [START_REF] Martin | Two loop eective potential for a general renormalizable theory and softly broken supersymmetry[END_REF] L SV = -

1 2 g abi A a µ A µb φ i - 1 4 g abij A a µ A µb φ i φ j -g aij A a µ φ i ∂ µ φ j , L F V = g aJ I ψI σ µ ψ J A a µ , L gauge = g abc A a µ A b ν ∂ µ A νc -g abe g cde A µa A νb A c µ A d ν + g abc A a µ ω b ∂ µ ωc (2.4.21)
In the above interaction terms, the couplings g abi and g abij are symmetric under the interchange either of two vector indices or of two scalar indices. Moreover, g aij is antisymmetric under the exchange i ↔ j and g abc is fully antisymmetric. Details on how to diagonalise the fermion and gauge-boson squared masses and how to relate the couplings involving gauge bosons to the structure constants and generators of the gauge group can be found in [START_REF] Martin | Two loop eective potential for a general renormalizable theory and softly broken supersymmetry[END_REF].

Two-loop neutral scalar masses in generic theories

To calculate the Higgs boson masses in general eld theories we require the tadpole diagrams and self-energies. Two-loop order expressions for the former were given in [START_REF] Goodsell | Generic two-loop Higgs mass calculation from a diagrammatic approach[END_REF], which were derived from the general expression for the eective potential at two loops in the Landau gauge, given in [START_REF] Martin | Two loop eective potential for a general renormalizable theory and softly broken supersymmetry[END_REF]. Hence we must also use the selfenergies in the Landau gauge; these were given in [START_REF] Martin | Two loop scalar self energies in a general renormalizable theory at leading order in gauge couplings[END_REF], in terms of a basis of two-loop functions dened in [START_REF] Martin | Evaluation of two loop selfenergy basis integrals using dierential equations[END_REF], up to order g 2 in the gauge couplings. Afterwards, in [START_REF] Martin | Two-loop scalar self-energies and pole masses in a general renormalizable theory with massless gauge bosons[END_REF], these expressions were extended to order g 4 for unbroken gauge groups (i.e. when the gauge bosons remain massless). We therefore restrict ourselves to the gaugeless limit, where we ignore the contributions of broken gauge groups for two-loop corrections for the SM gauge group this means setting g = g = 0. As the electroweak gauge couplings g and g are smaller than the strong gauge coupling g s and the top Yukawa coupling (and in some models and scenarios 10 , also the remaining third generation Yukawa couplings), we expect the electroweak contributions that we neglect in this limit to be subdominant. The gaugeless limit has a number of advantages, chiey simplicity and speed of the calculation; but also the fact that we can compute the one-loop corrections in any gauge desired. Once we have dropped the electroweak contributions, it is also tempting to disregard the momentum-dependence of the loop 10 For example, in models with two Higgs doublets (2HDM, MSSM, etc.) when tan β is large.

Calculations in generic theories

functions, which is typically estimated to contribute at the same order (and indeed is so for the MSSM [START_REF] Borowka | Momentum-dependent two-loop QCD corrections to the neutral Higgs-boson masses in the MSSM[END_REF][START_REF] Degrassi | Two-loop QCD corrections to the MSSM Higgs masses beyond the eective-potential approximation[END_REF]) hence the popularity of calculations in the eective potential approach.

Note, however, that when addressing the Goldstone Boson Catastrophe in the context of mass calculations for generic models in chapter 4, we will nd it necessary to go beyond the (standard) eective potential approximation because the inclusion of external momentum is necessary to regulate the divergences of some of the two-loop self-energy diagrams when m 2 G → 0. We will therefore devise a generalised eective potential approximation to cure the divergent behaviour of the diagrams, while still beneting from the simplicity of calculations without the full momentum dependence see section 4.3.

The SARAH/SPheno framework

As discussed earlier, once expressions for the contributions to the eective potential, tadpole equations and self-energies have been obtained for a generic model, there remains to translate them for the model one is interested in. This is usually a simple, although tedious, task and can be automatised with programs that handle formal calculations. One and actually the rst such program is the Mathematica package SARAH [261266], and in this section we will give a brief presentation of the framework it provides, and in particular of how its interfacing with the Fortran spectrum generator SPheno [START_REF] Porod | SPheno, a program for calculating supersymmetric spectra, SUSY particle decays and SUSY particle production at e+ e-colliders[END_REF][START_REF] Porod | SPheno 3.1: Extensions including avour, CP-phases and models beyond the MSSM[END_REF] allows the study of a wide range of BSM theories taking into account higher-order corrections to physical quantities, including two-loop corrections to neutral scalar masses. We refer the reader to standard references such as [START_REF] Staub | Exploring new models in all detail with SARAH[END_REF][START_REF] Vicente | Computer tools in particle physics[END_REF] for more detailed explanations of how to use SARAH, and to [START_REF] Porod | SPheno, a program for calculating supersymmetric spectra, SUSY particle decays and SUSY particle production at e+ e-colliders[END_REF][START_REF] Porod | SPheno 3.1: Extensions including avour, CP-phases and models beyond the MSSM[END_REF] for SPheno. For completeness, we should also mention the existence of other codes that can compute the vertices (i.e. the Feynman rules) of a given model, namely FeynRules [START_REF] Christensen | FeynRules -Feynman rules made easy[END_REF] and LanHEP [START_REF] Semenov | LanHEP: A Package for automatic generation of Feynman rules in gauge models[END_REF]. It should be noted that FeynRules accomodates non-renormalisable operators which is currently not the case in SARAH, but at the same time it does not link to a spectrum generator code a major advantage of SARAH. Also, a spectrum generator for a generic BSM model can also be obtained alternatively by using FlexibleSUSY [START_REF] Athron | FlexibleSUSYA spectrum generator generator for supersymmetric models[END_REF][START_REF] Athron | FlexibleSUSY 2.0: Extensions to investigate the phenomenology of SUSY and non-SUSY models[END_REF] (based in part on SARAH to create a C++ spectrum generator).

However we will not consider these options any further in this thesis.

Analytic calculations with SARAH

SARAH was written originally for supersymmetric models to derive the tree-level mass matrices and vertices from the input of the superpotential (and soft SUSY-breaking Lagrangian if necessary) of the model and to provide these results as inputs for other codes calculating amplitudes (e.g. CalcHEP). However, it is now able to handle a wide range of models, both supersymmetric (and not only dened at high scales) and nonsupersymmetric. Also, calculations of radiative corrections to physical observables have been gradually included, and the current version of the code (4.12.3) provides:

• full two-loop RGEs, using results derived for general QFTs in [270273];

• one-and two-loop corrections to the eective potential, implementing results from [START_REF] Martin | Two loop eective potential for a general renormalizable theory and softly broken supersymmetry[END_REF];

• one-loop tadpole equations, and one-loop self-energies for all particles;

• two-loop tadpoles and neutral scalar self-energies. These were rst found by taking numerical derivatives of the two-loop eective potential [START_REF] Goodsell | Two-Loop Higgs mass calculations in supersymmetric models beyond the MSSM with SARAH and SPheno[END_REF] (after interfacing SARAH with SPheno), but, since version 4.5.0, analytic expressions are computed directly in SARAH with results obtained for the tadpoles in [START_REF] Goodsell | Generic two-loop Higgs mass calculation from a diagrammatic approach[END_REF], and implemented from refs. [START_REF] Martin | Two loop scalar self energies in a general renormalizable theory at leading order in gauge couplings[END_REF][START_REF] Martin | Evaluation of two loop selfenergy basis integrals using dierential equations[END_REF][START_REF] Martin | Two-loop scalar self-energies and pole masses in a general renormalizable theory with massless gauge bosons[END_REF] for the self-energies. Furthermore, since version 4.12.0, the solution to the Goldstone Boson Catastrophe that we present in chapter 4 is also implemented in SARAH, as will be discussed in chapter 5;

• one-loop corrections to two-body decays, using the analytic results from [START_REF] Goodsell | Generic calculation of two-body partial decay widths at the full one-loop level[END_REF].

The procedure to investigate a given BSM model is illustrated in gure 2.4 and can be explained as follows. First, the model to study is dened : i.e. the user species in a Mathematica package le (.m) the global and gauge symmetries of the model, the eld content (or supereld content for a SUSY model), the Lagrangian (and superpotential if relevant), as well as the VEVs of the states after spontaneous symmetry breaking (EWSB usually) and the way in which states mix after SSB. Then, the SARAH package can be called from a Mathematica notebook to compute and write out expressions for tree-level properties of the model (mass matrices, vertices, potential, etc.) as well as for the above-listed radiative corrections. All these expressions can be manipulated directly as formal objects by Mathematica, which allows for example to check the theoretical consistency of the model or to evolve with its RGE any Lagrangian parameter.

Interfaces with SPheno and other HEP codes

Furthermore, since [START_REF] Porod | SPheno 3.1: Extensions including avour, CP-phases and models beyond the MSSM[END_REF][START_REF] Staub | A Tool Box for Implementing Supersymmetric Models[END_REF], SARAH can be interfaced with the Fortran code SPheno to create a complete spectrum generator for the model under consideration. The program SPheno is a spectrum generator written for the MSSM, rst for variants dened at a high scale (e.g. mSUGRA) [START_REF] Porod | SPheno, a program for calculating supersymmetric spectra, SUSY particle decays and SUSY particle production at e+ e-colliders[END_REF], and later extended to various other scenarios. It computes the loop-corrected masses and mixings of particles in the spectrum, and in turn uses those to calculate other properties of the particles, such as their decay widths (for two-and three-body decays), as well as low energy precision observables, such as lepton anomalous moments or lepton dipole moments. To extend this program to more general BSM models (including non-SUSY ones), SARAH writes new Fortran modules that are then compiled together with the SPheno routines in particular routines to compute loop integrals. In the following, we will refer to the resulting program as a SARAH/SPheno, or simply SPheno, spectrum generator. A further extension in SARAH, called FlavorKit [START_REF] Porod | A Flavor Kit for BSM models[END_REF], makes possible calculations of numerous avour observables in the spectrum generator, for a wide range of models.

However, the computation of particle spectra and physical observables is not the nal step in the study of BSM theories, and there are numerous further avenues to investigate, both from the point of view of theoretical consistency and of phenomenology.

For this reason the numerical framework of SARAH/SPheno can be linked to more High-Energy Physics codes: on the one hand, Mathematica routines in SARAH can write input les for a range of codes; on the other hand, SARAH/SPheno based spectrum generators employ the standard convention of the Supersymmetry Les Houches Accord (SLHA) [START_REF] Skands | SUSY Les Houches accord: Interfacing SUSY spectrum calculators, decay packages, and event generators[END_REF][START_REF] Allanach | SUSY Les Houches Accord 2[END_REF] for their input and output formats. A rst possible direction is to determine whether the considered model can accommodate a vacuum state compatible with electroweak symmetry breaking, and moreover to verify the stability of this vacuum if it exists. Finding the global minimum of the scalar potential for a model with an extended Higgs sector is a priori a very complicated task, but it can be performed

Calculations in generic theories

to one-loop order by the program Vevacious [START_REF] Camargo-Molina | Vevacious: A Tool For Finding The Global Minima Of One-Loop Eective Potentials With Many Scalars[END_REF], which uses as inputs a model le created by SARAH, and spectrum les from SPheno.

Moreover, comparing the theoretical predictions of a BSM model to experimental result provides additional criteria to rule in or out parameter points of the model. A great wealth of experimental results is available from a wide range of contexts such as, for example, collider phenomenology in particular measurements related to Higgs physics or avour physics or dark matter. To exploit these results for general models, SARAH can be linked to a number of other codes made either: (i) to compute amplitudes or cross-sections; (ii) to generate simulated collider events; or (ii)

to verify whether the properties of a model are compatible with data. Among programs that belong to the rst and/or second of these categories, SARAH can create input les for CalcHEP [START_REF] Belyaev | CalcHEP 3.4 for collider physics within and beyond the Standard Model[END_REF], FeynCalc [START_REF] Hahn | Generating Feynman diagrams and amplitudes with FeynArts 3[END_REF][START_REF] Hahn | FormCalc 9 and Extensions[END_REF] or WHIZARD/O'Mega [START_REF] Kilian | WHIZARD: Simulating Multi-Particle Processes at LHC and ILC[END_REF][START_REF] Moretti | An Optimizing matrix element generator[END_REF], as well as create models les in the standardised UFO (for Universal FeynRules Output) format [START_REF] Degrande | UFO -The Universal FeynRules Output[END_REF], used by MadGraph5_aMC@NLO [START_REF] Alwall | The automated computation of tree-level and next-to-leading order dierential cross sections, and their matching to parton shower simulations[END_REF], GoSam [START_REF] Cullen | GOSAM -2.0: a tool for automated one-loop calculations within the Standard Model and beyond[END_REF], Herwig++ [START_REF] Bellm | Herwig++ 2.7 Release Note[END_REF], or Sherpa [START_REF] Höche | Beyond Standard Model calculations with Sherpa[END_REF]. Furthermore, SARAH/SPheno spectrum generators can provide the Higgs sector predictions (e.g. masses, decays, cross-sections, etc.) required as inputs for the programs HiggsBounds [290292] and HiggsSignals [START_REF] Bechtle | HiggsSignals: Confronting arbitrary Higgs sectors with measurements at the Tevatron and the LHC[END_REF][START_REF] Bechtle | Probing the Standard Model with Higgs signal rates from the Tevatron, the LHC and a future ILC[END_REF]. These two codes examine whether a given parameter point is allowed by current data or whether it is excluded at 95% condence level: the former by verifying whether the parameter point is excluded by searches of additional Higgs bosons; and the latter by determining if the Higgs properties obtained for the considered parameter point are compatible with currently available measurements for the observed Higgs. Lastly, the theoretical prediction for the dark matter properties for a generic model (i.e. relic density, direct and indirect detection rates, etc.) can be computed by using micrOMEGAs [295298] together with SARAH/SPheno. Indeed, implementing a new model in micrOMEGAs requires a CalcHEP model le, which can be generated automatically by SARAH together with a main le for micrOMEGAs. Once the executable micrOMEGAs program has been compiled, the dark matter calculations can be performed, using as inputs spectrum les from SARAH/SPheno.

Finally, we must also mention the Mathematica package SSP [START_REF] Staub | A Tool Box for Implementing Supersymmetric Models[END_REF] SSP stands for SARAH Scan and Plot that makes possible parameter scans with SARAH/SPheno and linked programs some of the gures in chapter 5 have been made with SSP.

Numerical set-up of the spectrum calculation

We end our presentation of the framework of SARAH and SPheno for Higgs mass calculations by describing how to calculate, in a xed-order approach, 11 the mass spectrum with a SARAH/SPheno spectrum generator. First, it is necessary to distinguish two types of inputs that will appear in the computations, namely SM parameters and additional parameters of the BSM model, which are treated dierently. The former are given in a particular block of the SLHA input le, called SMINPUTS, and consist of the Fermi constant G F and the pole masses of the Z boson, the top quark and tau lepton; together with three MS values: α MS s (m Z ), α MS (m Z ) and m MS b (m b ). From these inputs, the Higgs VEV as well as the gauge and third family Yukawa couplings, i.e. v, g s , g, g , y t , y b , y τ , (2.4.22) are extracted as MS/DR parameters and run to the scale at which the calculation of radiative corrections are performed. There are, in principle, two ways to do this 11 Since [START_REF] Staub | Improved predictions for intermediate and heavy Supersymmetry in the MSSM and beyond[END_REF], SARAH/SPheno also accomodates an EFT calculation, but we will not discuss it here.

Precision calculations of the Higgs boson mass in SPheno, which can be explained as follows (see also appendix A of [START_REF] Staub | Improved predictions for intermediate and heavy Supersymmetry in the MSSM and beyond[END_REF] for more details):

• the rst option follows the method from [START_REF] Pierce | Precision corrections in the minimal supersymmetric standard model[END_REF] and is referred to as one-scale matching. The idea is to extract at the scale m Z the values of the couplings and Higgs VEV in the BSM model under consideration, and in an MS or DR scheme (depending on whether the model is supersymmetric or not). These parameters can then be evolved using two-loop RGEs in the BSM model. However, this method encounters problems if some of the new states are heavy, because large logarithmic contributions appear in the threshold corrections that are applied at m Z ;

• another strategy, more suited for BSM models involving heavy particles, is the so-called two-scale matching. The parameters in (2.4.22) are extracted at m Z but this time as Standard Model parameters in the MS scheme. Afterwards, these are evolved with the known three-loop SM RGEs to the scale at which computations are performed, and are then converted to the BSM model (and appropriate scheme) by applying one-loop threshold corrections.

In the numerical studies that follow, we shall employ the latter approach.

The new parameters of the BSM theories are given in one or several dierent block(s) in the SLHA input, often exclusively the MINPAR block, and are taken as MS or DR parameters at some input scale, which can be varied freely e.g. it can be a high input scale for constrained SUSY models like mSUGRA, or simply be taken equal to the renormalisation scale employed in the rest of the code (we will often choose this when studying non-supersymmetric models). In the following we will call these extra parameters Θ(Q) for a generic model.

The dierent steps performed to compute the spectrum are then:

(i) The running couplings Θ(Q) are (if necessary) run the scale at which the masses and tadpoles are computed using two-loop RGEs generated by SARAH while the SM parameters are evolved to this same scale including all known SM corrections, i.e. three-loop running and two-loop matching for strong coupling g s and top Yukawa y t (following the second of the two methods discussed above).

(ii) The tree-level tadpoles T i are solved to x the remaining free parameters, which in what follows are typically the mass parameters µ 2 i |φ| 2 .

(iii) The tree-level mass are calculated by diagonalising the tree-level mass matrices (iv) The n-loop corrections to the tadpoles δ (n) t i are calculated. The imposed minimisation conditions are:

T i + n j δ (j) t i = 0 , (2.4.23)
which cause shifts in µ 2 i :

µ 2 i → µ 2 i + n j δ (j) µ 2 i .
(2.4.24)

Calculations in generic theories

(v) The one-and two-loop self-energies for real scalars are calculated for external gauge eigenstates. The pole masses are the eigenvalues of the loop-corrected mass matrix calculated as

M (n) φ (p 2 ) = M (2L) φ + n j=1 Π (j)
φ (p 2 ) .

(2.4.25)

Here,

M (2L) φ
is the tree-level mass matrix including the shifts eq. (2.4.24) computed to two-loop order.

The calculation of the one-loop self-energies is done iteratively for each eigenvalue i until the on-shell condition eigM

(n) φ (p 2 = m 2 φ i ) i ≡ m 2 φ i (2.4.26)
is fullled. The renormalised rotation matrix is taken to be the one calculated

for p 2 = m 2 φ 1
.

Precision calculations of the Higgs boson mass

User-dened model Recently, SARAH has made it possible to analyse at the two-loop level the Higgs sector of several non-minimal extensions of the MSSM, see [START_REF] Goodsell | Two-loop corrections to the Higgs masses in the NMSSM[END_REF][START_REF] Goodsell | The Higgs Mass in the MSSM at two-loop order beyond minimal avour violation[END_REF][START_REF] Dreiner | On the two-loop corrections to the Higgs mass in trilinear R-parity violation[END_REF][START_REF] Nickel | Precise determination of the Higgs mass in supersymmetric models with vectorlike tops and the impact on naturalness in minimal GMSB[END_REF][START_REF] Goodsell | The Higgs mass in the CP violating MSSM, NMSSM, and beyond[END_REF]. Of particular relevance for this work, it has allowed for Dirac-gaugino masses since version 3.2 [START_REF] Staub | SARAH 3.2: Dirac Gauginos, UFO output, and more[END_REF], incorporating also the results of ref. [START_REF] Goodsell | Two-loop RGEs with Dirac gaugino masses[END_REF]. Indeed, SARAH has been used for detailed phenomenological analyses of the MDGSSM at one loop in ref. [START_REF] Benakli | Dirac Gauginos and the 125 GeV Higgs[END_REF] and at two loops in refs. [START_REF] Benakli | Constrained minimal Dirac gaugino supersymmetric standard model[END_REF][START_REF] Goodsell | Dark matter scenarios in a constrained model with Dirac gauginos[END_REF]; and also for the MRSSM at one loop in ref. [START_REF] Dieÿner | Higgs boson mass and electroweak observables in the MRSSM[END_REF] and two loops in refs. [START_REF] Diessner | Two-loop correction to the Higgs boson mass in the MRSSM[END_REF][START_REF] Diessner | Exploring the Higgs sector of the MRSSM with a light scalar[END_REF]. However, while such a numerical tool for generic models fulls a signicant need of the community, it is also important to have explicit results for specic models, and not just as a cross-check.

Here, we shall compute the leading O(α t α s ) corrections to the neutral Higgs boson masses in both the MDGSSM and MRSSM, relying on the eective-potential techniques developed in ref. [START_REF] Degrassi | On the neutral Higgs boson masses in the MSSM for arbitrary stop mixing[END_REF] for the MSSM and in ref. [START_REF] Degrassi | On the radiative corrections to the neutral Higgs boson masses in the NMSSM[END_REF] for the NMSSM. This has the following advantages:

• We compute the O(α t α s ) corrections in both the DR and on-shell (OS) renormalisation schemes. The latter turns out to be particularly useful in scenarios with heavy gluinos a feature of many Dirac-gaugino models in the literature (as was also discussed in section 1.3.5.3) where the use of DR formulae for the two-loop Higgs-mass corrections can lead to large theoretical uncertainties.

Leading two-loop corrections to the Higgs boson masses in SUSY models with Dirac gauginos

• We have written a simple and fast stand-alone code implementing our results, which we make available upon request (indeed, a version of the code is already included in SARAH).

• We use our results to derive simple approximate expressions for the most important two-loop corrections, applicable in any Dirac gaugino model.

The outline of this chapter is as follows. We rst present in section 3.1 our results for the case of a general Dirac-gaugino model, and for the particular cases of the MDGSSM and the MRSSM, then show how we compute the shift to the OS scheme, and give simplied formulae for the SM-like Higgs boson mass either for a common SUSYbreaking scale or for a heavy Dirac gluino. In section 3.2 we give numerical examples of our results, illustrating the advantages of our approach and also discussing the inherent theoretical uncertainties. We conclude in section 3.3. Explicit expressions for the derivatives of the eective potential are given in appendix A.

Two-loop corrections in the eective potential approach

In this section we adapt to the calculation of two-loop corrections to the neutral Higgs masses in Dirac-gaugino models the eective-potential techniques developed in ref. [START_REF] Degrassi | On the neutral Higgs boson masses in the MSSM for arbitrary stop mixing[END_REF] for the MSSM and in ref. [START_REF] Degrassi | On the radiative corrections to the neutral Higgs boson masses in the NMSSM[END_REF] for the NMSSM. We start by deriving general results valid for all variants of Dirac-gaugino extensions of the MSSM, then we provide explicit formulae for the MDGSSM and MRSSM models discussed in section 1.3.5.6.

General results

The eective potential for the neutral Higgs sector can be decomposed as V eff = V (0) + ∆V , where ∆V incorporates the radiative corrections. We denote collectively as Φ 0 i the complex neutral scalars whose masses we want to calculate, and split them into vacuum expectation values v i , real scalars S i and pseudoscalars P i as

Φ 0 i ≡ v i + 1 √ 2 (S i + i P i ) . (3.1.1)
Then the mass matrices for the scalar and pseudoscalar elds can be decomposed as

M 2 S eff ij = M 2 S tree ij + ∆M 2 S ij , M 2 P eff ij = M 2 P tree ij + ∆M 2 P ij , (3.1.2)
and the radiative corrections to the mass matrices are

∆M 2 S ij = - 1 √ 2 δ ij v i ∂∆V ∂S i min + ∂ 2 ∆V ∂S i ∂S j min , (3.1.3) ∆M 2 P ij = - 1 √ 2 δ ij v i ∂∆V ∂S i min + ∂ 2 ∆V ∂P i ∂P j min , (3.1.4)
where v i , which we assume to be real, denote the VEVs of the full radiatively-corrected potential V eff , and the derivatives are in turn evaluated at the minimum of the potential. The single-derivative terms in eqs. 3.1 Two-loop corrections in the eective potential approach are used to remove the soft SUSY-breaking mass for a given eld Φ 0 i from the tree- level parts of the mass matrices. It is understood that those terms should be omitted for elds that do not develop a VEV (such as, e.g., the elds R u,d in the MRSSM).

Note that when using the minimum conditions in eqs. of the parameters appearing in ∆V . We restrict for simplicity our calculation to the so-called gaugeless limit, i.e. we neglect all corrections controlled by the electroweak gauge couplings g and g . At the two-loop level, we focus on the contributions to ∆V from top/stop loops that involve the strong interactions. In Dirac-gaugino models, this results in corrections to mass matrices and minimum conditions that are proportional to α s times various combinations of the top Yukawa coupling y t with the superpotential couplings of the singlet and triplet elds. It is therefore with a slight abuse of notation that we maintain the MSSM-inspired habit of denoting collectively those corrections as O(α t α s ).

As detailed in refs. [START_REF] Degrassi | On the neutral Higgs boson masses in the MSSM for arbitrary stop mixing[END_REF][START_REF] Degrassi | On the radiative corrections to the neutral Higgs boson masses in the NMSSM[END_REF], if we neglect the electroweak contributions to the stop mass matrix the parameters in the top/stop sector depend on the neutral Higgs elds only through two combinations, which we denote as X ≡ |X| e iϕ and X ≡ | X| e i φ. They enter the stop mass matrix as

M 2 stop = m 2 Q + |X| 2 X * X m 2 U + |X| 2 , (3.1.6)
where m 2 Q and m 2 U are the soft SUSY-breaking mass terms for the stops. While X = y t H 0 u both in the (N)MSSM and in Dirac-gaugino models, the precise form of X depends on the model under consideration and will be discussed later. For the time being, we only assume that X is real at the minimum of the potential, to prevent CPviolating contributions to the Higgs mass matrices. The top/stop O(α s ) contribution to ∆V can then be expressed in terms of ve eld-dependent parameters, which can be chosen as follows. The squared top and stop masses

m 2 t = |X| 2 , m 2 t1,2 = 1 2 (m 2 Q + m 2 U + 2 |X| 2 ) ± (m 2 Q -m 2 U ) 2 + 4 | X| 2 , (3.1.7)
a mixing angle θt , with 0 ≤ θt ≤ π/2, which diagonalises the stop mass matrix after the stop elds have been redened to make it real and symmetric

sin 2 θt = 2 | X| m 2 t1 -m 2 t2 , (3.1.8)
and a combination of the phases of X and X that we can choose as cos (ϕφ) = Re( X) Re(X) + Im( X) Im(X)

| X| |X| .

(3.1.9)

Finally, the gluino masses m gi and the octet masses m 2 O i do not depend on the Higgs background, since we neglect the singlet-octet couplings λ SO and T SO . In the following

Leading two-loop corrections to the Higgs boson masses in SUSY models with Dirac gauginos we will also refer to θ t , with -π/2 < θ t < π/2, i.e. the usual eld-independent mixing angle that diagonalises the stop mass matrix at the minimum of the scalar potential.

We nd general expressions for the top/stop contributions to the minimum conditions of the eective potential and to the corrections to the scalar and pseudoscalar mass matrices:

∂∆V ∂S i min = s 2θt ∂ X ∂S i F + √ 2 y t m t δ i2 G , (3.1.10) ∆M 2 S ij = s 2θt ∂ 2 X ∂S i ∂S j + 2 m 2 t1 -m 2 t2 ∂ X ∂S i ∂ X ∂S j - s 2θt √ 2 δ ij v i ∂ X ∂S j F + 2 y 2 t m 2 t δ i2 δ j2 F 1 + √ 2 m t y t s 2θt δ i2 ∂ X ∂S j + δ j2 ∂ X ∂S i F 2 + s 2 2θt ∂ X ∂S i ∂ X ∂S j F 3 , (3.1.11) ∆M 2 P ij = 1 m 2 t1 -m 2 t2 ∂ 2 | X| 2 ∂P i ∂P j - s 2θt √ 2 δ ij v i ∂ X ∂S j F + δ i2 v 2 X + √ 2 i ∂ X ∂P i δ j2 v 2 X + √ 2 i ∂ X ∂P j tan β F ϕ , (3.1.12)
where all quantities are understood as evaluated at the minimum of the potential, no summation is implied over repeated indices, the elds are ordered as (Φ 0 1 , Φ 0 2 , ...) = (H 0 d , H 0 u , ...) , and again the terms involving δ ij /v i should be omitted if Φ 0 i does not develop a VEV. The angle β is dened as in the MSSM by tan β = v 2 /v 1 . Here and thereafter we also adopt the shortcuts c φ ≡ cos φ and s φ ≡ sin φ for a generic angle φ.

The functions F, G, F 1 , F 2 , F 3 and F ϕ entering eqs. (3.1.10)(3.1.12) are combinations of the derivatives of ∆V . Explicit expressions for most of those functions can be found e.g. in ref. [START_REF] Degrassi | On the radiative corrections to the neutral Higgs boson masses in the NMSSM[END_REF], but we display all of them here for completeness: -

F = ∂∆V ∂m 2 t1 - ∂∆V ∂m 2 t2 - 4 c 2 2θt m 2 t1 -m 2 t2 ∂∆V ∂c 2 2 θt , (3.1.13) G = ∂∆V ∂m 2 t + ∂∆V ∂m 2 t1 + ∂∆V ∂m 2 t2 , (3.1.14)
F 1 = ∂ 2 ∆V (∂m 2 t ) 2 + ∂ 2 ∆V (∂m 2 t1 ) 2 + ∂ 2 ∆V (∂m 2 t2 ) 2 + 2 ∂ 2 ∆V ∂m 2 t ∂m 2 t1 + 2 ∂ 2 ∆V ∂m 2 t ∂m 2 t2 + 2 ∂ 2 ∆V ∂m 2 t1 ∂m 2 t2 , (3.1.15) F 2 = ∂ 2 ∆V (∂m 2 t1 ) 2 - ∂ 2 ∆V (∂m 2 t2 ) 2 + ∂ 2 ∆V ∂m 2 t ∂m 2 t1 - ∂ 2 ∆V ∂m 2 t ∂m 2 t2 - 4 c 2 2θt m 2 t1 -m 2 t2 ∂ 2 ∆V ∂c 2 2 θt ∂m 2 t + ∂ 2 ∆V ∂c 2 2 θt ∂m 2 t1 + ∂ 2 ∆V ∂c 2 2 θt
F 3 = ∂ 2 ∆V (∂m 2 t1 ) 2 + ∂ 2 ∆V (∂m 2 t2 ) 2 -2 ∂ 2 ∆V ∂m 2 t1 ∂m 2 t2 - 2 m 2 t1 -m 2 t2 ∂∆V ∂m 2 t1 - ∂∆V ∂m 2 t2 + 16 c 2 2θt (m 2 t1 -m 2 t2 ) 2 c 2 2θt ∂ 2 ∆V (∂c 2 2 θt ) 2 + 2 ∂∆V ∂c 2 2 θt
8 c 2 2θt m 2 t1 -m 2 t2 ∂ 2 ∆V ∂c 2 2 θt ∂m 2 t1 - ∂ 2 ∆V ∂c 2 2 θt ∂m 2 t2 , (3.1.17) F ϕ = - 2 z t cot β s 2 2θt (m 2 t1 -m 2 t2 ) 2 ∂∆V ∂c ϕt-φt , (3.1.18)
where we dened z t ≡ sign( X| min ).

Two-loop top/stop contributions to the eective potential

For the computation of the two-loop O(α t α s ) corrections to the Higgs mass matrices in models with Dirac gauginos we need the explicit expression for the top/stop O(α s ) contribution to ∆V , expressed in terms of the eld-dependent parameters dened in the previous section. In addition to the contributions of diagrams involving gluons, gluinos or the D-term-induced quartic stop couplings, which are in common with the (N)MSSM and can be found in ref. [START_REF] Degrassi | On the neutral Higgs boson masses in the MSSM for arbitrary stop mixing[END_REF], ∆V receives a contribution from the diagram shown in gure 3.1, involving stops and octet scalars.

We assume that the gaugino masses are real so that the diagonalising matrix R ij is real and R 2 1i is positive, but allow m gi to be negative. Since R 2 11 + R 2 12 = 1, we can simply write the top/stop O(α s ) contribution to the two-loop eective potential (in units of α s C F N c /(4π) 3 , where C F = 4/3 and N c = 3 are colour factors) as

∆V αs = 2 i=1 R 2 1i ∆V αs MSSM + ∆V αs octet , (3.1.19)
where ∆V αs MSSM is the analogous contribution in the (N)MSSM,

∆V αs MSSM = 2 J(m 2 t , m 2 t ) -4 m 2 t I(m 2 t , m 2 t , 0) + 2 m 2 t1 I(m 2 t1 , m 2 t1 , 0) + 2 L(m 2 t1 , m 2 gi , m 2 t ) -4 m t m gi s 2 θ c ϕ-φ I(m 2 t1 , m 2 gi , m 2 t ) + 1 2 (1 + c 2 2 θ) J(m 2 t1 , m 2 t1 ) + s 2 2 θ 2 J(m 2 t1 , m 2 t2 ) + m t1 ↔ m t2 , s 2 θ → -s 2 θ , ( 3
∆M 2 S 12 = -y 2 t m t μ s 2θt F 2 - 1 2 y 2 t A t μ s 2 2θt F 3 - y 2 t A t μ m 2 t1 -m 2 t2 F , (3.1.29) ∆M 2 S 22 = 2 y 2 t m 2 t F 1 + 2 y 2 t m t A t s 2θt F 2 + 1 2 y 2 t A 2 t s 2 2θt F 3 + y 2 t A t μ cot β m 2 t1 -m 2 t2 F , (3.1.30) ∆M 2 S 13 = 1 2 y t λ S m t μ cot β s 2 2θt F 3 - y t λ S m t A t -2 μ cot β m 2 t1 -m 2 t2 F , (3.1.31)
∆M 2

S 23 =-y t λ S m 2 t cot β s 2θt F 2 - 1 2 y t λ S A t m t cot β s 2 2θt F 3 - y t λ S m t A t cot β m 2 t1 -m 2 t2 F, (3.1.32)
∆M 2 where we dened μu ≡ µ u + λ Su v 3 + λ Tu v 4 .

S 33 = 1 2 λ 2 S m 2 t cot 2 β s 2 2θt F 3 + λ S m 2 t cot β A t + (λ S v 3 -μ) cot β v 3 (m 2 t1 -m 2 t2 ) F , (3.1.33) ∆M 2 S 14 = 1 2 y t λ T m t μ cot β s 2 2θt F 3 - y t λ T m t A t -2 μ cot β m 2 t1 -m 2 t2 F , (3.1.34) ∆M 2 S 24 =-y t λ T m 2 t cot β s 2θt F 2 - 1 2 y t λ T A t m t cot β s 2 2θt F 3 - y t λ T m t A t cot β m 2 t1 -m 2 t2 F, (3.1.35) ∆M 2 S 34 = 1 2 λ S λ T m 2 t cot 2 β s 2 2θt F 3 + λ S λ T m 2 t cot 2 β m 2 t1 -m 2 t2 F , (3.1.36) ∆M 2 S 44 = 1 2 λ 2 T m 2 t cot 2 β s 2 2θt F 3 + λ T m 2 t cot β A t + (λ T v 4 -μ) cot β v 4 (m 2 t1 -m 2 t2 ) F , ( 3
∆M 2 P 33 = λ S m 2 t cot β A t + (λ S v 3 -μ) cot β v 3 (m 2 t1 -m 2 t2 ) F + λ 2 S m 2 t cot β F ϕ , (3.1.43) ∆M 2 P 14 = y t λ T m t A t m 2 t1 -m 2 t2 F + y t λ T m t μ F ϕ , (3.1.44) ∆M 2 P 24 = y t λ T m t A t cot β m 2 t1 -m 2 t2 F + y t λ T m t μ cot β F ϕ , (3.1.45) ∆M 2 P 34 = λ S λ T m 2 t cot 2 β m 2 t1 -m 2 t2 F + λ S λ T m 2 t cot β F ϕ , (3.1.46) ∆M 2 P 44 = λ T m 2 t cot β A t + (λ T v 4 -μ) cot β v 4 (m 2 t1 -m 2 t2 ) F + λ 2 T m 2 t cot β F ϕ . ( 3
Φ 0 i = (H 0 d , H 0 u , S, T 0 , R 0 d , R 0
3.1 Two-loop corrections in the eective potential approach 3.1.5 On-shell parameters in the top/stop sector

The results presented so far for the two-loop corrections to the neutral Higgs masses in models with Dirac gauginos were obtained under the assumption that the parameters entering the tree-level and one-loop parts of the mass matrices are renormalised in the DR scheme. While this choice allows for a straightforward implementation of our results in automated calculations such as the one of SARAH, it is well known that, in the DR scheme, the Higgs-mass calculation can be plagued by unphysically large contributions if there is a hierarchy between the masses of the particles running in the loops [START_REF] Degrassi | On the neutral Higgs boson masses in the MSSM for arbitrary stop mixing[END_REF]. In particular, the contributions of two-loop diagrams involving stops and gluinos include terms proportional to m 2 gi /m ) sin 2θ t = 2 X| min (in general, the stop mixing X| min contains other terms in addition to m t A t , but they are exempt from O(α s ) corrections). Finally, since the VEVs v i are not renormalised at O(α s ), the top Yukawa coupling y t receives the same relative correction as the top mass. Dening x DR k = x OS k + δx k for each parameter x k ≡ (m t , m 2 t1 , m 2 t2 , θ t , A t ), the DR OS shifts of top and stop masses and mixing are given in terms of the nite parts (here denoted by a hat) of the top and stop self-energies

δm t = Σt (m t ) , δm 2 ti = Πii (m 2 ti ) (i = 1, 2), δθ t = 1 2 Π12 (m 2 t1 ) + Π12 (m 2 t2 ) m 2 t1 -m 2 t2 , (3.1.52)
and the shift for the trilinear coupling reads

δA t = δm 2 t1 -δm 2 t2 m 2 t1 -m 2 t2 - δm t m t + 2 cot 2θ t δθ t X| min . (3.1.53)
As in the case of the two-loop eective potential in eq. (3.1.19), the DR OS shifts δx k can be cast as

δx k = 2 i=1 R 2 1i (δx MSSM k ) i + δx octet k , ( 3
Π11 (m 2 t1 ) octet = g 2 s m 2 D 4π 2 C F c 2 φ O c 2 2θt B0 (m 2 t1 , m 2 t1 , m 2 O 1 ) + s 2 2θt B0 (m 2 t1 , m 2 t2 , m 2 O 1 )
Leading two-loop corrections to the Higgs boson masses in SUSY models with Dirac gauginos

+ (c φ O → s φ O , m O 1 → m O 2 ) , (3.1.55) Π22 (m 2 t2 ) octet = g 2 s m 2 D 4π 2 C F c 2 φ O c 2 2θt B0 (m 2 t2 , m 2 t2 , m 2 O 1 ) + s 2 2θt B0 (m 2 t2 , m 2 t1 , m 2 O 1 ) + (c φ O → s φ O , m O 1 → m O 2 ) , (3.1.56) Π12 (p 2 ) octet = - g 2 s m 2 D 4π 2 C F c 2 φ O c 2θt s 2θt B0 (p 2 , m 2 t1 , m 2 O 1 ) -B0 (p 2 , m 2 t2 , m 2 O 1 ) + (c φ O → s φ O , m O 1 → m O 2 ) , ( 3
F 1 = m 2 t1 ln m 2 t1 Q 2 -1 -m 2 t2 ln m 2 t2 Q 2 -1 , G 1 = m 2 t1 ln m 2 t1 Q 2 -1 + m 2 t2 ln m 2 t2 Q 2 -1 -2 m 2 t ln m 2 t Q 2 -1 , F 1 1 = ln m 2 t1 m 2 t2 m 4 t , F 1 2 = ln m 2 t1 m 2 t2 , F 1 3 = 2 - m 2 t1 + m 2 t2 m 2 t1 -m 2 t2 ln m 2 t1 m 2 t2 , F 1 ϕ = 0 , (3.1.59)
where Q is the renormalisation scale at which the parameters entering the tree-level and one-loop parts of the mass matrices are expressed. As mentioned above, the DR OS shifts derived in eq. (3.1.58) cancel the power-like dependence of the two-loop corrections on the gluino masses. (with k > 2) and tan β ↔ cot β must be performed in the formulae of sections 3.1.3 and 3.1.4. In the case of the bottom/sbottom corrections, however, passing from the DR scheme to the OS scheme would involve additional complications, as explained in ref. [START_REF] Brignole | On the two loop sbottom corrections to the neutral Higgs boson masses in the MSSM[END_REF].

3.1 Two-loop corrections in the eective potential approach

Simplied formulae

Having computed the general expressions for the two-loop corrections to the neutral Higgs masses in models with Dirac gauginos, it is now interesting to provide some approximate results for the dominant corrections to the mass of a SM-like Higgs. We focus on the case of a purely-Dirac mass term for the gluinos, which as mentioned earlier implies that we can set R 2 11 = R 2 12 = 1/2 and m g1 = -m g2 = m g , with m g ≡ m D . We also restrict ourselves to the decoupling limit in which all neutral states except a combination of H 0 d and H 0 u are heavy, so that (3.1.61)

H 0 d ≈ v + h √ 2 cos β + ... , H 0 u ≈ v + h √ 2 sin β + ... , (3.1
Finally, we assume that the superpotential couplings of the adjoint elds (e.g., the couplings λ S and λ T in the MDGSSM) are subdominant with respect to the top Yukawa coupling, so that we can focus on the two-loop corrections proportional to

α s m 4 t /v 2 .
With these restrictions, we shall give useful formulae valid for a phenomenologically interesting subspace of all extant Dirac gaugino models; while in the following we refer to simplied MDGSSM and MRSSM scenarios, this merely reects whether stop mixing is allowed.

Common SUSY-breaking scale

We rst consider a simplied MDGSSM scenario in which the soft SUSY-breaking masses for the two stops and the Dirac mass of the gluinos are large and degenerate,

i.e. m Q = m U = m g = M S with M S m t . Expanding our result 2 for the top/stop contributions to ∆m 2 h at the leading order in m t /M S , we can decompose it as

∆m 2 h ≈ 3 m 4 t 4 π 2 v 2 ln M 2 S m 2 t + X2 t - X4 t 12 + ∆m 2 h "MSSM" 2 + c 2 φ O ∆m 2 h O 1 2 + s 2 φ O ∆m 2 h O 2 2 , (3.1.62)
where Xt ≡ X t /M S , in which X t = A tμ cot β is the left-right mixing term in the stop mass matrix with μ dened as in section 3.1.3. The rst term in ∆m 2 h is the dominant 1-loop contribution from diagrams with top quarks or stop squarks, which is the same as in the MSSM. The second term is the O(α t α s ) contribution from twoloop, MSSM-like diagrams involving gluons, gluinos or a four-stop interaction. Under the assumption that the parameters m t , M S and A t entering the one-loop part of the correction are renormalised in the DR scheme at the scale Q, it reads

∆m 2 h "MSSM" 2 = α s m 4 t 2 π 3 v 2 ln 2 M 2 S m 2 t -2 ln 2 M 2 S Q 2 + 2 ln 2 m 2 t Q 2 + ln M 2 S m 2 t - 1
2 We have veried that, for MS = 1 TeV and for | Xt| up to the maximal mixing value of √ 6, the predictions for m h obtained with the simplied formulae of this section agree at the per-mil level with the unexpanded result. For larger MS the accuracy of our approximation improves, and for | Xt| > √ 6

it degrades.

Leading two-loop corrections to the Higgs boson masses in SUSY models with Dirac gauginos

+ X2 t 1 -2 ln M 2 S Q 2 - X4 t 12 .
(3.1.63)

We remark that this correction diers from the usual one in the MSSM, see e.g. eq. ( 21) of ref. [START_REF] Espinosa | MSSM lightest CP even Higgs boson mass to O(alpha(s) alpha(t)): The Eective potential approach[END_REF], due to the absence of terms involving odd powers of Xt . Indeed, those terms are actually proportional to the gluino masses, and in the considered scenario they cancel out of the sum over the gluino mass eigenstates, because m g1 = -m g2 .

If the parameters m t , M S and A t are renormalised in the OS scheme as described in section 3.1.5, the correction reads instead

∆m 2 h "MSSM" 2 = - 3 α s m 4 t 2 π 3 v 2 ln 2 M 2 S m 2 t + 2 + X2 t ln M 2 S m 2 t + X4 t 4 .
(3.1.64)

Note that the explicit dependence on the renormalisation scale Q drops out. Again, this correction diers from the usual one in the MSSM, see e.g. the rst line in eq. ( 20) of ref. [START_REF] Espinosa | Complete two loop dominant corrections to the mass of the lightest CP even Higgs boson in the minimal supersymmetric standard model[END_REF], due to the absence of a term linear in Xt .

Finally, the last two terms on the right-hand side of eq. (3.1.62) represent the O(α t α s ) contributions of two-loop diagrams with stops and octet scalars, which are specic to models with Dirac gluinos. In the DR scheme they read

∆m 2 h O i 2 = - α s m 4 t π 3 v 2 1 -ln M 2 S Q 2 + f (x i ) -X2 t 1 -ln m 2 O i Q 2 + 2 x i f (x i ) + X4 t 6 1 + 3 x i (1 + ln x i ) -ln m 2 O i Q 2 + 6 x 2 i f (x i ) , (3.1.65)
where

x i ≡ M 2 S /m 2 O i
, and the function f (x) is dened as

f (x) = 1 1 -4x ln x + x φ 1 4x , (3.1.66)
φ(z) being the function dened in eq. ( 45) of ref. [START_REF] Brignole | On the O(alpha(t)**2) two loop corrections to the neutral Higgs boson masses in the MSSM[END_REF]. Special limits of the function in eq. (3.1.66) above are f (1/4) = -2 (1 + ln 4)/3 and f (1) ≈ -0.781302. In the OS scheme the octet-scalar contributions receive at the leading order in m t /M S the shift

δ ∆m 2 h O i 2 = α s m 4 t π 3 v 2 B i - X2 t - X4 t 6 3 B i + 2 ln m 2 O i Q 2 -2 , (3.1.67)
where

B i ≡ B0 (M 2 S , M 2 S , m 2 O i ) = -ln(m 2 O i /Q 2 ) + g(M 2 S /m 2 O i ) , with the function g(x) dened as g(x) = 2 -1 -1 2x ln x -1 x √ 4 x -1 arctan √ 4 x -1 (x > 1/4) 2 -1 -1 2x ln x + 1 x √ 1 -4 x arctanh √ 1 -4 x (x < 1/4) . ( 3

MRSSM with heavy Dirac gluino

The second simplied scenario we consider is the R-symmetric model of section 3.1.4, in the limit of heavy Dirac gluino, i.e. m g m ti . This is a phenomenologically inter- esting limit because Dirac gaugino masses are supersoft, i.e. they can be substantially larger than the squark masses without spoiling the naturalness of the model [START_REF] Fox | Dirac gaugino masses and supersoft supersymmetry breaking[END_REF].

Two-loop corrections in the eective potential approach

In the MRSSM the left and right stops do not mix, hence we set θ t = 0 in our formulae, but we allow for the possibility of dierent stop masses m t1 and m t2 . In the decoupling limit of the Higgs sector, where we neglect the mixing with the heavy neutral states, the correction to the SM-like Higgs mass reduces to ∆m 2 h ≈ sin 2 β ∆M 2 S 22

. In analogy to eq. (3.1.62), the correction can in turn be decomposed in a dominant one-loop part, a two-loop, MSSM-like O(α t α s ) contribution and two-loop octet-scalar contributions:

∆m 2 h ≈ 3 m 4 t 8 π 2 v 2 ln m 2 t1 m 2 t2 m 4 t + ∆m 2 h "MSSM" 2 + c 2 φ O ∆m 2 h O 1 2 + s 2 φ O ∆m 2 h O 2 2 . (3.1.69)
Assuming that the top and stop masses in the one-loop part of the correction are DRrenormalised parameters at the scale Q, and expanding our results in inverse powers of m 2 g, the contribution of two-loop, MSSM-like diagrams involving gluons, gluinos or a four-stop coupling reads where the last term in square brackets represents the addition of terms obtained from the previous ones by swapping m 2 t1 and m 2 t2

∆m 2 h "MSSM" 2 = α s m 4 t 4 π 3 v 2 2 m 2 g m 2 t1 1 -ln m 2 g Q 2 + 2π 2 3 -2 -6 ln m 2 g m 2 t1 + 2 ln m 2 t Q 2 + 2m 2 t m 2 t1 1 -ln m 2 g Q 2 + ln 2 m 2 g m 2 t + ln 2 m 2 g m 2 t1 + 2 ln 2 m 2 t Q 2 -2 ln 2 m 2 t1 Q 2 + 2 m 2 t m 2 g 2π 2 3 2 + m 2 t1 m 2 t -2 -8 + m 2 t m 2 t1 ln m 2 g m 2 t -4 ln m 2 g m 2 t1 - m 2 t1 m 2 t 2 + 6 ln m 2 g m 2 t1 + ln m 2 g m 2 t + 2 2 + m 2 t1 m 2 t ln m 2 g m 2 t1 ln m 2 g m 2 t + O m -4 g + m 2 t1 -→ m 2 t2 , ( 3
2 = α s m 4 t 4 π 3 v 2 2π 2 3 -1 -6 ln m 2 g m 2 t -3 ln 2 m 2 t1 m 2 t + 2 ln 2 m 2 g m 2 t1 + m 2 t m 2 g 4π 2 3 2 + m 2 t1 m 2 t - 20
. By taking the limit m t1 = m t2 = m t in the equation above we recover eq. ( 42) of ref. [START_REF] Degrassi | On the neutral Higgs boson masses in the MSSM for arbitrary stop mixing[END_REF].

In the MRSSM, the contributions to ∆m 2 h arising from two-loop diagrams with stops and octet scalars allow for fairly compact expressions. If the stop masses in the oneloop part of the correction are renormalised in the DR scheme, those contributions read

∆m 2 h O i 2 = - α s m 4 t 2 π 3 v 2 m 2 g m 2 t1 1 -ln m 2 t1 Q 2 + f m 2 t1 m 2 O i + m 2 t1 -→ m 2 t2 , (3.1.72)
where f (x) is the function dened in eq. (3.1.66). For OS stop masses, the octet-scalar contributions to ∆m 2 we nd, in the DR scheme, ) . On the other hand, in the OS scheme we nd

h read instead ∆m 2 h O i 2 = - α s m 4 t 2 π 3 v 2 m 2 g m 2 t1 1 -ln m 2 t1 m 2 O i + f m 2 t1 m 2 O i -g m 2 t1 m 2 O i + m 2 t1 -→ m 2 t2 , ( 3
∆m 2 h O 1 2 = - α s m 4 t 4 π 3 v 2 m 2 g m 2 O 1 2 m 2 O 1 m 2 t1 1 -ln m 2 O 1 Q 2 + 2π 2 3 + 8 ln m 2 t1 m 2 O 1 + 2 ln 2 m 2 t1 m 2 O 1 + 4 m 2 t1 m 2 O 1 π 2 -2 + 10 ln m 2 t1 m 2 O 1 + 3 ln 2 m 2 t1 m 2 O 1 + O m -4 O 1 + m 2 t1 -→ m 2 t2 , ( 3
∆m 2 h O 1 2 = - α s m 4 t 4 π 3 v 2 m 2 g m 2 O 1 2π 2 3 -1 + 6 ln m 2 t1 m 2 O 1 + 2 ln 2 m 2 t1 m 2 O 1 + 4 m 2 t1 m 2 O 1 π 2 - 17 6 + 9 ln m 2 t1 m 2 O 1 + 3 ln 2 m 2 t1 m 2 O 1 + O m -4 O 1 + m 2 t1 -→ m 2 t2 . ( 3
m -4
g in the heavy-gluino limit. In summary, we nd that, in the MRSSM with heavy Dirac gluino, neither of the octet scalars can induce unphysically large contributions to ∆m 2 h , as long as the stop masses in the one-loop part of the correction are renormalised in the OS scheme.

Numerical examples

In this section we discuss the numerical impact of the two-loop O(α t α s ) corrections to the Higgs boson masses whose computation was described in the previous section. As we did for the simplied formulae of section 3.1.7, we focus on decoupling scenarios in which the lightest neutral scalar is SM-like and the superpotential couplings λ S,T are subdominant with respect to the top Yukawa coupling. Our purpose here is to elucidate the dependence of the corrections to the SM-like Higgs boson mass m h on relevant parameters such as the stop masses and mixing and the gluino masses, rather than provide accurate predictions for all Higgs boson masses in realistic scenarios. We therefore approximate the one-loop part of the corrections with the dominant top/stop contributions at vanishing external momentum, obtained by combining the formulae for the Higgs mass matrices given for MDGSSM and MRSSM in sections 3.1.3 and 3.1.4, respectively, with the one-loop functions given in eq. (3.1.59). We recall that a computation of the Higgs boson masses in models with Dirac gauginos could also be obtained in an automated way by means of the package SARAH [261266]. That would include the full one-loop corrections [START_REF] Staub | SARAH 3.2: Dirac Gauginos, UFO output, and more[END_REF] and the two-loop corrections computed in the gaugeless limit at vanishing external momentum [START_REF] Goodsell | Generic two-loop Higgs mass calculation from a diagrammatic approach[END_REF][START_REF] Goodsell | Two-Loop Higgs mass calculations in supersymmetric models beyond the MSSM with SARAH and SPheno[END_REF]. However, the computation implemented in SARAH employs the DR renormalisation scheme, and does not easily lend itself to an adaptation to the OS scheme which, as discussed in section 3.1.7.2, can be more appropriate in scenarios with heavy gluinos.

The SM parameters entering our computation of the Higgs boson masses, which we take from ref. [START_REF] Olive | Review of Particle Physics[END_REF], are the Z boson mass m Z = 91.1876 GeV, the Fermi constant G F = 1.16637 × 10 -5 GeV -2 (from which we extract v = (2 √ 2 G F ) -1/2 ≈ 174 GeV), the pole top-quark mass m t = 173.21 GeV and the strong gauge coupling of the Leading two-loop corrections to the Higgs boson masses in SUSY models with Dirac gauginos SM in the MS renormalisation scheme, α s (m Z ) = 0.1185. Concerning the SUSY parameters entering the scalar mass matrix at tree-level, we set λ S = λ T = 0 and push the parameters that determine the heavy-scalar masses to multi-TeV values, so that (m 2 h ) tree ≈ m 2 Z cos 2 2β. We also set tan β = 10, so that the tree-level mass of the SM-like Higgs boson is almost maximal but the corrections from diagrams involving sbottom squarks, which we neglect, are not particularly enhanced. For the parameters in the stop mass matrices we take degenerate soft SUSY-breaking masses m Q = m U = M S , we neglect D-term-induced electroweak contributions and we treat the whole leftright mixing term X t = A tµ cot β as a single input. Finally, for what concerns the parameters that determine the gluino and octet-scalar masses we focus again on the case of purely-Dirac gluinos, with m g1 = -m g2 = m g and R 2 11 = R 2 12 = 1/2 . We also take a vanishing soft SUSY-breaking bilinear B O , so that φ O = 0 and only the CP-even octet scalar O 1 , with mass m 2

O 1 = m 2 O + 4 m 2 g , participates in the O(α t α s )
corrections to the Higgs masses.

An example in the MDGSSM

In gure 3.2 we illustrate some dierences between the O(α t α s ) corrections to the SMlike Higgs boson mass in the MDGSSM and in the MSSM. We plot m h as a function of the ratio X t /M S , setting M S = 1.5 TeV and m g = m O = 2 TeV and adopting the OS renormalisation scheme for the parameters m t , M S and X t . We employ the renormalisation-group equations of the SM to evolve the coupling α s from the input scale m Z to the scale M S , then we convert it to the DR -renormalised coupling of the considered SUSY model, which we denote as αs (M S ), by including the appropriate threshold corrections (in this step, we assume that all soft SUSY-breaking squark masses are equal to M S ). The solid (black) and dashed (red) curves in gure 3.2 represent the SM-like Higgs boson mass in the MDGSSM and in the MSSM, respectively.

The comparison between the two curves highlights the fact that, in contrast with the case of the MSSM, in the MDGSSM with purely-Dirac gluinos the O(α t α s ) corrections to m h are symmetric with respect to a change of sign in X t . As mentioned in section 3.1.7.1, this stems from cancellations between terms proportional to odd powers of the gluino masses. In the points where m h is maximal, which in the OS calculation happens for |X t /M S | ≈ 2, the dierence between the MDGSSM and MSSM predictions for m h is about 1 or 2 GeV, depending on the sign of X t . Finally, the dotted (blue) curve in gure 3.2 represents the prediction for m h obtained in the MDGSSM by omitting the contributions of two-loop diagrams involving the octet scalars. The comparison between the solid and dotted curves shows that, in the considered point of the parameter space, the eect on m h of the octet-scalar contributions is positive but rather small, of the order of a few hundred MeV. Varying the parameters M S , m g and m O by factors of order two around the values used in gure 3.2, we nd that this is a typical size for the octet-scalar contributions to m h in the OS scheme.

A discussion of the theoretical uncertainty of our calculation is now in order. In our numerical examples we are not implementing the full one-loop corrections to the Higgs boson masses, nor the two-loop corrections beyond O(α t α s ) that are available in SARAH, in order to focus purely on the O(α t α s ) corrections. Therefore the only sources of uncertainty that we can meaningfully estimate are the uncomputed eects of O(α t α 2 s ), i.e. those arising from genuine three-loop diagrams with four strong-interaction vertices and from SUSY-QCD renormalisation eects of the parameters entering the one-and two-loop corrections. A common procedure for estimating those eects consists in comparing the results of the O(α t α s ) calculation of m h in the OS scheme with the  DR with α ˆs(M S )

OS with α ˆs(m t ) OS with -α s (m t ) The solid curve represents the original OS calculation; the dotted curve represents the DR calculation; the dashed and dot-dashed curves were obtained using αs (m t ) and α s (m t ), respectively, in the OS calculation instead of αs (M S ).

Leading two-loop corrections to the Higgs boson masses in SUSY models with Dirac gauginos results obtained by (i) converting the OS input parameters i.e., the top mass and the stop masses and mixing to the DR scheme by means of O(α s ) shifts, and (ii) computing m h using these DR parameters in both the one-loop and two-loop corrections, with the appropriate DR formulae for the O(α t α s ) corrections. The two sources of O(α t α 2 s ) discrepancies in such a comparison are the omission of terms quadratic in δx k in the expansion of the one-loop part of the corrections, eq. (3.1.58), and the dierent denition of the top and stop parameters entering the two-loop part of the corrections. In gure 3.3 we illustrate the renormalisation-scheme dependence of the O(α t α s ) determination of m h , in the same MDGSSM scenario as in gure 3.2.

The solid (black) curve represents the results of the original OS calculation, whereas the dotted (blue) curve represents the results of the DR calculation described above (note that both curves are plotted as functions of the ratio of OS parameters X t /M S ).

The comparison between the solid and dotted curves would suggest a rather small impact of the uncomputed O(α t α 2 s ) corrections, of the order of one GeV or even less (at least for the considered scenario).

Besides the top mass and the stop masses and mixing, there are a few more parameters entering the O(α t α s ) corrections to the Higgs boson masses whose O(α s ) denition amounts to a three-loop O(α t α 2 s ) eect, namely the gluino and octet-scalar masses and the strong gauge coupling itself. Concerning the masses, in an OS calculation it seems natural to interpret them as pole ones. For α s , on the other hand, there is no obvious on-shell denition available, and dierent choices of scheme, scale and even underlying theory while all formally equivalent at O(α t α s ) for the Higgs-mass calculation can lead to signicant variations in the numerical results. As mentioned earlier, the solid curve in gure 3.3 was obtained with top/stop parameters in the OS scheme, but with α s dened as the DR -renormalised coupling of the MDGSSM at the stop-mass scale, i.e. αs (M S ). However, since both stop squarks and top quarks enter the relevant two-loop diagrams, it would not seem unreasonable to evaluate the strong gauge coupling at the top-mass scale either. The dashed (red) and dot-dashed (green) curves in gure 3.3 represent the predictions for m h obtained with top/stop parameters still in the OS scheme, but with α s dened as the DR -renormalised coupling of the MDGSSM at the top-mass scale, αs (m t ), and as the MS-renormalised coupling of the SM at the same scale, α s (m t ), respectively. The comparison of these two curves with the solid curve shows that a variation in the denition of the coupling α s entering the two-loop corrections provides a less-optimistic estimate of the uncertainty associated to the O(α t α 2 s ) corrections compared with the scheme variation of the top/stop parameters. In particular, for the considered scenario the use of α s (m t ) would induce a negative variation with respect to the results obtained with αs (M S ) of about 4 GeV for X t ≈ 0 and about 7 GeV for |X t /M S | ≈ 2. In contrast, the use of αs (m t ) would induce a positive variation of about 1 GeV for X t ≈ 0 and about 2 GeV for |X t /M S | ≈ 2, i.e. more modest than the previous one but still larger than the one induced by a scheme change in the top/stop parameters. While remaining agnostic about the true size (and sign) of the three-loop O(α t α 2 s ) corrections, we take this as a cautionary tale against putting too much stock in any single estimate of the theoretical uncertainty of a xed-order calculation of m h in scenarios with TeV-scale superparticles.

An example in the MRSSM

In our second numerical example we consider the MRSSM, and illustrate the dependence of the SM-like Higgs boson mass on the gluino mass. In ref. [START_REF] Diessner | Two-loop correction to the Higgs boson mass in the MRSSM[END_REF] it was pointed out that, for multi-TeV values of m g, the contribution of two-loop diagrams involving octet scalars can increase the prediction for m h by more than 10 GeV. We will show that such large eects are related to the non-decoupling behaviour of the DR calculation of m h that we discussed in section 3.1.7.2, and that the octet-scalar contributions are much more modest in an OS calculation.

The upper (blue) and lower (red) solid curves in gure 3.4 represent the SM-like Higgs boson mass obtained from the DR calculation as a function of m g, with and without the octet-scalar contributions, respectively. We set m O = 2 TeV and M S = 1 TeV. The latter is interpreted as a DR -renormalised soft SUSY-breaking parameter evaluated at a scale equal to M S itself, which means that each point in the solid curves corresponds to a dierent value of the physical stop masses. Both curves show a marked dependence on m g, and the comparison between them shows that, for the highest value of m g considered in the plot, the eect on m h of the two-loop octet-scalar contributions does indeed grow to about 9 GeV. However, as can be seen in the explicit formulae for the two-loop corrections in the DR scheme of eqs. The dashed curve was obtained by converting the top and stop masses to the OS scheme and using the corresponding formulae for the O(α t α s ) corrections.

1072 GeV and 1392 GeV for the values of m g shown in the plot. If the octet-scalar contributions to the O(α s ) stop self-energies are omitted, the stop masses range instead between 1049 GeV and 346 GeV, i.e. they become smaller for increasing m g (indeed, in this case m g cannot be pushed to values much larger than those shown in the plot without rendering the stop masses tachyonic). The comparison between the solid and dashed curves shows that the scheme dependence of the O(α t α s ) calculation of m h becomes increasingly worse at large values of m g, especially in the lower curves where the octet-scalar contributions are omitted. Finally, the (black) dotted and dot-dashed curves in gure 3.4 represent the predictions for m h obtained directly from the OS calculation with and without octet-scalar contributions, respectively. In this case the input M S = 1 TeV is interpreted as an OS-renormalised parameter, meaning that the physical stop masses correspond to (M 2 S + m 2 t ) 1/2 ≈ 1015 GeV for all points in the curves. We stress that direct comparisons between these two curves and the solid (and dashed) ones would not be appropriate, because they refer to dierent points of the MRSSM parameter space. However, the dotted and dot-dashed curves show that, when the physical stop masses are taken as input, the prediction for m h in the MRSSM depends only mildly on the value of m g, and the eect of the octet-scalar contributions is below one GeV. This is explained by the fact that, as discussed in section 3.1.7.2, in the OS scheme there are no terms enhanced by m 2 g/M 2 S in either the gluino or the octet-scalar contributions to the O(α t α s ) corrections.

Before concluding, we note that there are extreme situations in which a DR calculation of m h is not workable at all, and a conversion to the OS scheme such as the one represented by the dashed lines in gure 3.4 is necessary. In the so-called supersoft scenario,

Conclusions

all soft SUSY-breaking masses vanish, and sizeable sfermion masses proportional to the Dirac-gaugino masses are induced only by radiative corrections. Such a scenario can be realised e.g. in the MRSSM by setting m O = 0 and M S = 0, where the latter is interpreted as a DR -renormalised parameter. At the scale where this condition is imposed, the DR stop masses coincide with the top mass, with the result that, in the DR calculation, the one-loop correction in the rst term of eq. had previously been applied to the MSSM [START_REF] Degrassi | On the neutral Higgs boson masses in the MSSM for arbitrary stop mixing[END_REF] and to the NMSSM [START_REF] Degrassi | On the radiative corrections to the neutral Higgs boson masses in the NMSSM[END_REF]. We obtained analytic formulae for the O(α t α s ) corrections to the scalar and pseudoscalar Higgs mass matrices valid for arbitrary choices of parameters in the squark and gaugino sectors, both in the DR and in the OS renormalisation schemes, which we make available upon request as a Fortran code. We also presented compact approximate formulae for the dominant corrections to the mass of the SM-like Higgs boson, valid under a number of simplifying assumptions for the SUSY parameters. Finally, we studied the numerical impact of the newly-computed corrections on the predictions for the SM-like Higgs boson mass in some representative scenarios. In particular, we elucidated the dierences between the predictions for m h in the MSSM and those in its Dirac-gaugino extensions; we discussed the theoretical uncertainty of our predictions stemming from uncomputed higher-order corrections; we stressed that a judicious choice of renormalisation scheme is required to obtain reliable predictions in scenarios where the gluinos are much heavier than the squarks, which can occur naturally in Dirac-gaugino models.

Chapter 4

Avoiding the Goldstone Boson Catastrophe in general renormalisable eld theories at two loops

We now return to the Goldstone Boson Catastrophe, which we already encountered for the SM and the MSSM in sections 1.1.5 and 1.3.4.2 respectively, and in this chapter we will demonstrate how to address it for general renormalisable eld theories.

This problem of infra-red divergences appears in generic models as in the SM and MSSM: for reasons of calculational simplicity, the eective potential beyond one loop has been calculated only in the Landau gauge (as this allows to decouple the ghosts), which means that the Goldstone bosons are treated as actual massless Goldstone bosons. Once loop corrections are taken into account, the mass-squared Lagrangian parameters of the Goldstone bosons determined from the tadpole equations 1 are small and can even be negative (as opposed to the pole mass, which is always zero) and this causes the loop integrals for the tadpoles to diverge or be complex. While this problem can in principle be circumvented by dropping the complex parts and changing the renormalisation scale to attempt to nd non-negligible positive squared masses, this is not easy to implement consistently in the context of automated calculations.

We presented in section 1.1.5 a solution for the tadpoles that was rst proposed in [START_REF] Elias-Miro | Taming Infrared Divergences in the Eective Potential[END_REF][START_REF] Martin | Taming the Goldstone contributions to the eective potential[END_REF] for the Standard Model and later applied to the MSSM in [START_REF] Kumar | Resummation of Goldstone boson contributions to the MSSM eective potential[END_REF]: (a subset of ) the terms involving the Goldstone boson should be resummed to all orders, roughly speaking replacing its mass-squared parameter (which appears in the loop functions)

with the equivalent mass parameter derived from the loop-corrected eective potential (i.e. zero, since it is a Goldstone boson). In section 4.2 we show how this can be extended to general renormalisable theories.

In [START_REF] Elias-Miro | Taming Infrared Divergences in the Eective Potential[END_REF][START_REF] Martin | Taming the Goldstone contributions to the eective potential[END_REF] it was noted that the Goldstone resummation would not regulate divergences in the second derivatives of the eective potential, and so to have a divergence-free calculation of the neutral scalar (i.e. Higgs) masses it would be necessary to include the external momentum in the self energies rather than using an eective potential approximation. This is particularly important because the zero momentum approximation is widely used to calculate the Higgs mass [1, 21, 32, 76, 77, 164175, 179, 180, Avoiding the Goldstone Boson Catastrophe in general renormalisable eld theories at two loops [START_REF] Martin | Three-loop corrections to the lightest Higgs scalar boson mass in supersymmetry[END_REF][START_REF] Mühlleitner | Two-loop contributions of the order O(α t α s ) to the masses of the Higgs bosons in the CP-violating NMSSM[END_REF][START_REF] Staub | Higgs mass predictions of public NMSSM spectrum generators[END_REF]250252,[START_REF] Goodsell | The Higgs Mass in the MSSM at two-loop order beyond minimal avour violation[END_REF][START_REF] Dreiner | On the two-loop corrections to the Higgs mass in trilinear R-parity violation[END_REF][START_REF] Nickel | Precise determination of the Higgs mass in supersymmetric models with vectorlike tops and the impact on naturalness in minimal GMSB[END_REF][START_REF] Goodsell | The Higgs mass in the CP violating MSSM, NMSSM, and beyond[END_REF] indeed there are few publicly available implementations of diagrammatic calculations of the Higgs mass beyond one loop in BSM theories which do not use it (some momentum-dependent diagrammatic calculations are available for the MSSM [176178,[START_REF] Borowka | Complete two-loop QCD contributions to the lightest Higgs-boson mass in the MSSM with complex parameters[END_REF]). While the Goldsone Boson Catastrophe can be avoided in the MSSM in the gaugeless limit (where the Goldstone boson does not couple to the Higgs, and so generates no divergences) it is of pressing concern for more general theories. Since the two-loop computation has become publicly available through SARAH [START_REF] Goodsell | Generic two-loop Higgs mass calculation from a diagrammatic approach[END_REF][START_REF] Goodsell | Two-Loop Higgs mass calculations in supersymmetric models beyond the MSSM with SARAH and SPheno[END_REF]261266], the Goldstone Boson Catastrophe as it aects that implementation has been discussed in [START_REF] Goodsell | Two-loop corrections to the Higgs masses in the NMSSM[END_REF][START_REF] Goodsell | The Higgs mass in the CP violating MSSM, NMSSM, and beyond[END_REF], and has recently manifested itself in [START_REF] Benakli | The Di-Photon Excess in a Perturbative SUSY Model[END_REF][START_REF] Goodsell | Dark matter scenarios in a constrained model with Dirac gauginos[END_REF][START_REF] Athron | Precise Higgs mass calculations in (non-)minimal supersymmetry at both high and low scales[END_REF]. Indeed, while the numerical impact of the problem in the Standard Model seems to be small (at least away from the divergent points, simply neglecting the imaginary part of the potential seems to give results close to those of the full solutions), in more complicated theories it can cause divergent contributions to the masses for many regions of the parameter space; for example, in [START_REF] Benakli | The Di-Photon Excess in a Perturbative SUSY Model[END_REF][START_REF] Goodsell | Dark matter scenarios in a constrained model with Dirac gauginos[END_REF] it was necessary to restrict to only the two-loop corrections proportional to the strong gauge coupling for those regions in performing parameter scans.

In section 4.3, we shall show that the inclusion of external momentum in the scalar self-energies does not by itself avoid all divergences. In fact, it is necessary to resum the Goldstone boson contributions in the mass diagrams too to cancel the divergences in a class of diagrams which do not depend on momentum. We will also show that the resummation can be implemented most easily to two-loop order by using an on-shell scheme for the Goldstone bosons. With these modications to cure the remaining divergences, the diagrammatic implementation in [START_REF] Goodsell | Generic two-loop Higgs mass calculation from a diagrammatic approach[END_REF] could in principle be extended to include the external momentum by changing the loop functions to those implemented in TSIL [START_REF] Martin | Evaluation of two loop selfenergy basis integrals using dierential equations[END_REF][START_REF] Martin | TSIL: A Program for the calculation of two-loop self-energy integrals[END_REF]. However, analytic expressions for general loop functions with momenta are not known: they are in general obtained by solving dierential equations, which is numerically expensive. Therefore, in appendix B.2 we give a complete set of analytic expressions for expansions of the necessary functions including all divergent and constant terms in an expansion of the four-momentum-squared s around zero (neglecting those of O(s)). This allows fast evaluation of a generalised eective potential approximation for the neutral scalar masses although for this part we shall be restricted to the gaugeless limit (setting the couplings of all broken gauge groups to zero) since the mass diagrams are known only up to second order in the gauge couplings.

A comment is at hand also about the gauge dependence of the results obtained in this chapter: our calculations are performed in the Landau gauge, and we could in principle expect results to dier if we used another gauge in particular the expressions we obtain with the on-shell Goldstone method only apply in the Landau gauge where the Goldstone pole mass is zero. However it should be noted that we use the gaugeless limit for the mass calculations, which means that there is no longer any gauge, and therefore we will not consider issues of gauge dependence further here.

Once the Goldstone Boson Catastrophe has been solved, using similar techniques it was shown in [START_REF] Elias-Miro | Taming Infrared Divergences in the Eective Potential[END_REF][START_REF] Martin | Taming the Goldstone contributions to the eective potential[END_REF][START_REF] Kumar | Resummation of Goldstone boson contributions to the MSSM eective potential[END_REF] that it is also possible to improve the solution of the tadpole equations for the other mass-squared parameters (not just the one corresponding to the tree-level Goldstone boson mass). In general, the same mass-squared parameters m 2 appear both as solutions of the tadpole equations, and in the loop functions, in the schematic form

m 2 = m 2 0 - 1 v ∂∆V (m 2 ) ∂v
where m 2 0 is the tree-level solution of the tadpole equation, v is some expectation value and ∆V are the loop corrections to the eective potential. Although resummation 4.1 The Goldstone Boson Catastrophe and resummation is not required for them (except perhaps for the Higgs boson, where the quantum corrections are so large that they force its tree-level mass to become negative we shall not discuss such a case here), these other mass-squared parameters can be expanded perturbatively in the loop functions so that the equations can be solved directly rather than iteratively. In other words, we nd only the tree-level values of the parameters on the right-hand side of the equation, and the loop-corrected solution on the left (as opposed to the loop-corrected value on both sides):

m 2 = m 2 0 - 1 v ∂∆V (m 2 0 ) ∂v -δ 1 v ∂∆V (m 2 0 ) ∂v ;
we shall refer to these throughout as self-consistent equations. In section 4.4, we will show how to carry out this procedure in general, showing that the formulae can be given in simpler form than in, e.g., [START_REF] Kumar | Resummation of Goldstone boson contributions to the MSSM eective potential[END_REF] for the MSSM case. We shall also go further and show how this shifts the mass diagrams.

Finally, a set of denitions of all the loop functions used throughout this chapter is provided in appendix B.1.

The Goldstone Boson Catastrophe and resummation

V (0) = v 2 4 (λv 2 + 2µ 2 ) + hv(v 2 λ + µ 2 ) + 1 2 (3v 2 λ + µ 2 )h 2 + 1 2 (µ 2 + λv 2 )G 2 + vλ(h 3 + hG 2 ) + λ 4 (h 4 + 2G 2 h 2 + G 4 ). (4.1.2) Dening m 2 G ≡ µ 2 + λv 2 , M 2 h ≡ µ 2 + 3λv
V e (v) ≡ V (0) h,G=0 + 1 16π 2 V (1) + 1 (16π 2 ) 2 V (2) + ... = V (0) h,G=0 + 1 16π 2 (f (m 2 G ) + f (M 2 h )) + λ (16π 2 ) 2 3 4 A(m 2 G ) 2 + 1 2 A(m 2 G )A(M 2 h ) + 3 4 A(M 2 h ) 2 - λ 2 v 2 (16π 2 ) 2 3I(M 2 h , M 2 h , M 2 h ) + I(M 2 h , m 2 G , m 2 G ) + ...
(M 2 h , m 2 G , m 2 G ) around m 2 G = 0 we nd 2 0 = ∂V e ∂v = m 2 G v + 2λv 16π 2 1 2 A(m 2 G ) + 3 2 A(M 2 h ) + 2λ 2 v (16π 2 ) 2 log m 2 G 3 2 A(m 2 G ) + 1 2 A(M 2 h ) + 2λv 2 M 2 h A(M 2 h )
+ other nonsingular terms. The log m 2 G terms on the second line are the manifestation of the Goldstone Boson Catastrophe: we cannot insert the tree-level solution m 2 G = 0 into them, and will have a complex potential if we nd m 2 G < 0. The solution proposed in [START_REF] Elias-Miro | Taming Infrared Divergences in the Eective Potential[END_REF][START_REF] Martin | Taming the Goldstone contributions to the eective potential[END_REF] is to resum the Goldstone boson propagators in the one-loop eective potential we make the substitution

V (1) e ⊃ - i 2 C d d k log(-k 2 + m 2 G ) → - i 2 C d d k log(-k 2 + m 2 G + Π GG (k 2 )) → - i 2 C d d k log(-k 2 + m 2 G + Π GG (0)) + ... (4.1.5)
where C is a constant dened in equation (B.1.1), and Π GG (k 2 ) is the Goldstone boson self energy, given here at one loop by The term involving only the Goldstone mass-squared will not have a well-dened derivative, and this also leads to divergences when we resum the eective potential at three loops and above. The prescription of [START_REF] Elias-Miro | Taming Infrared Divergences in the Eective Potential[END_REF] is to drop it in favour of

Π GG (k 2 ) = 1 16π 2 3λA(m 2 G ) + λA(M 2 h ) -4λ 2 v 2 B(k 2 , m 2 G , M 2 h ) .
Π g = 1 16π 2 Π (1)
g + ... where

Π (1) g (0) = 3λA(M 2 h ) = λA(M 2 h ) -4λ 2 v 2 B(0, 0, M 2 h ). (4.1.8)
Note that this does not correspond to dropping one particular class of diagrams (at one loop it is a combination of the one-and two-propagator diagrams) but instead must be dened in terms of dropping contributions from soft Goldstone bosons. Dening

∆ ≡ Π g (0) ≡ 1 16π 2 ∆ 1 + 1 (16π 2 ) 2 ∆ 2 + • • • , (4.1.9)
we then should use instead the resummed potential

Ve ≡ V e + 1 16π 2 f (m 2 G + ∆) - l-1 n=0 ∆ n n! ∂ ∂m 2 G n f (m 2 G ) (4.1.10)
2 Let us briey comment here on the choice of argument with respect to which we compute deriva- tives. Until now we have been taking derivatives with respect to a real scalar uctuation (or eld) in this case it would be h before setting all uctuations to zero (here h and G). However, this is exactly equivalent to setting rst the uctuations to zero and then taking a derivative with respect to the VEV (here v), because the real part of the uctuation and the VEV always come with the same prefactor in the present example, v and h always appear together in terms of the form (v + h).

where l is the loop order to which V e has been calculated; the terms in square brackets simply ensure that the potentials are identical up to l loops and only dier at higher orders. Performing this procedure for the potential above we nd

Ve =V (0) + 1 16π 2 (f (m 2 G + ∆) + f (M 2 h )) + λ (16π 2 ) 2 3 4 A(m 2 G ) 2 + 3 4 A(M 2 h ) 2 - λ 2 v 2 (16π 2 ) 2 3I(M 2 h , M 2 h , M 2 h ) + I(M 2 h , m 2 G , m 2 G ) + 1 λv 2 A(M 2 h )A(m 2 G ) . (4.1.11)
With the above procedure, we have resummed the leading divergences at two loops,

i.e. the terms of order m 2 G log m 2 G for small m 2 G (we expect m 2 G it to be of order a one-loop quantity at the minimum). If we are interested in the rst derivative of the potential then this is sucient; to nd the minimum to two-loop order we can expand the potential to order m 2 G with the help of eq. (B.1.54):

Ve = V (0) + 1 16π 2 f (m 2 G + ∆) + f (M 2 h ) + λ (16π 2 ) 2 3 4 A(M 2 h ) 2 (4.1.12) - λ 2 v 2 (16π 2 ) 2 3I(M 2 h , M 2 h , M 2 h ) + I(M 2 h , 0, 0) -2R SS (0, M 2 h )m 2 G + O(m 4 G ),
making the regularity apparent, although note that the higher-order terms still contain a m 4 G log m 2 G term. The tadpole equation, neglecting terms of three-loop order, is then

0 = v m 2 G + 1 16π 2 ∆ 1 + λv + ∆ /2 16π 2 A(m 2 G + ∆) (4.1.13) + 1 (16π 2 ) 2 λ 9λvA(M 2 h ) log M 2 h -2λ 2 v 3I(M 2 h , M 2 h , M 2 h ) + I(M 2 h , 0, 0) + λ 2 v 2 6λv 9U 0 (M 2 h , M 2 h , M 2 h , M 2 h ) + U 0 (M 2 h , M 2 h , 0, 0) + 4λvR SS (0, M 2 h ) .
Noting that the solution to the one-loop equation is m 2 G + 1 16π 2 ∆ 1 = 0, we see that we can neglect the A(m 2 G + ∆) term as it gives a correction of order three loops. We ought then to nd that we can identify the term in curly brackets with ∆ 2 : for a Goldstone boson we should nd m 2 G + Π GG (0) = 0, so we expect that we should be able in general to identify

1 v ∂V ( ) ∂v = Π ( )
GG (0), and therefore for our modied potential we should expect

1 v ∂ V ( ) ∂v = Π ( ) g (0). (4.1.14)
This leads to the prescription in [START_REF] Martin | Taming the Goldstone contributions to the eective potential[END_REF][START_REF] Kumar | Resummation of Goldstone boson contributions to the MSSM eective potential[END_REF], which is somewhat simpler: we expand the potential V e as a series in m 2 G : We can then use this as the denition of ∆ 1 instead of equation (4.1.9). We then resum the eective potential as

V (2) = V (2) | m 2 G =0 + 1 2 ∆ 1 A(m 2 G ) + 1 2 Ωm 2 G + O(m 4 G ).
Ve = V (0) + 1 16π 2 V (1) | m 2 G =0 + f m 2 G + ∆ + 1 (16π 2 ) 2 V (2) | m 2 G =0 + 1 2 Ωm 2 G . (4.1.16)
Avoiding the Goldstone Boson Catastrophe in general renormalisable eld theories at two loops By doing this, we immediately nd the expression in (4.1.12), with

Ω ≡ 4λ 2 v 2 R SS (0, M 2 h ).
(4.1.17)

When we take the derivative and expand up to two-loop order then the minimum is We shall follow this second procedure to nd the minimum condition in general renormalisable eld theories at two loop order.

at m 2 G + ∆ = 0 with ∆ 2 = 1 v ∂ ∂v V (2) | m 2 G =0 + λΩ .
We shall also consider a hybrid approach, which is to adopt an on-shell condition for the Goldstone boson: we dene

(m 2 G ) dim. reg ≡ (m 2 G ) OS -Π GG ((m 2 G ) OS ) = -Π GG (0). (4.1.19)
This is particularly eective at two loops, where we only need Π

(1) GG ; furthermore, since (m 2 G ) OS = 0, at this loop order there is no dierence between Π GG and Π g . Making the above substitution in the potential we nd exactly the same result as our resummed version in equation (4.1.12). However, we also have the advantage that we can make this substitution directly in the tadpole equation:

0 = v(m 2 G ) dim. reg + λv 16π 2 A((m 2 G ) dim. reg ) + 3λv 16π 2 A(M 2 h ) + 1 (16π 2 ) 2
∂V (2) ∂v

(4.1.20) = v(m 2 G ) dim. reg + 3λv 16π 2 A(M 2 h ) + 1 (16π 2 ) 2 lim (m 2 G ) OS →0 ∂V (2) ∂v ((m 2 G ) OS ) -3λ 2 vA(M 2 h ) log((m 2 G ) OS ) ,
which gives exactly the expression that we found above in (4.1.13). We shall nd in the following that this simple approach is also exactly what we need for the mass diagrams. However, we must rst introduce some notation and formalism to handle the general case when (potentially several) Goldstone bosons and neutral scalars can mix.

Goldstone bosons in general eld theories

In the previous subsection we considered the simplest possible model where there were only two real scalars which cannot mix. Once we consider more general theories, there can be more Goldstone bosons and, even when they have been identied at tree level, they can in general mix with other scalars (only pseudoscalars in the case of CP conservation) once loop corrections are included. This problem does not arise in the Standard Model as treated in references [START_REF] Elias-Miro | Taming Infrared Divergences in the Eective Potential[END_REF][START_REF] Martin | Taming the Goldstone contributions to the eective potential[END_REF], because all of the pseudoscalars are would-be Goldstone bosons and the neutral and charged Goldstones cannot mix, so can be treated as two separate sectors. In the MSSM, there are additional scalars and pseudoscalars, but in the CP-conserving case considered in [START_REF] Kumar | Resummation of Goldstone boson contributions to the MSSM eective potential[END_REF] the mixing is at most among pairs of elds, and could be written in each case in terms of mixing angles and 2 × 2 matrices.

To deal with Goldstone boson mixing in general theories (using the conventions introduced in section 2.4.1), we will need some notation and simple results. We start 4.1 The Goldstone Boson Catastrophe and resummation from a theory with a global symmetry such that the scalars transform under a set of innitesimal shifts as

φ i → φ i + G α G i .
Then the standard result is to expand

V (φ i + G α G i ) = V (φ i
) and dierentiate the relation once: When we sit at the minimum of the potential ∂V ∂φ 0 i = 0 but for a spontaneously broken symmetry α G i is not zero for all i, and thus we have a null eigenvector of the scalar mass matrix i.e. the Goldstone boson. For more than one symmetry broken then there will be multiple null eigenvectors and these should be formed into an orthonormal set.

G α G i ∂V ∂φ 0 i = 0, ∂( G α G i ) ∂φ 0 j ∂V ∂φ 0 i + G α G i ∂ 2 V ∂φ 0 i ∂φ 0 j = 0 .
Let us write the symmetry shifts as linear coecients α G i = a G ij φ 0 j after this has been performed so that

i α G i α G i = δ GG and then G G = φ G = R jG φ 0 j , where R jG = α G j . (4.1.22)
We use the index G now to refer to the Goldstone boson(s) in the diagonal basis.

The rst identity that we need arises from taking a further derivative of the above equations to give

G α G i ∂ 3 V ∂φ 0 i ∂φ 0 j ∂φ 0 k + ∂ 2 ( G α G i ) ∂φ 0 j ∂φ 0 k ∂V ∂φ 0 i + ∂( G α G i ) ∂φ 0 j ∂ 2 V ∂φ 0 i ∂φ 0 k + ∂( G α G i ) ∂φ 0 k ∂ 2 V ∂φ 0 i ∂φ 0 j = 0 → α G i α G j α G k ∂ 3 V ∂φ 0 i ∂φ 0 j φ 0 k =0 , (4.1.23)
i.e. there are no three-Goldstone couplings.

If we were able to work at the true minimum of the potential and with self-consistent values of all the parameters then this would be sucient. However, we must use the minimum conditions to determine the parameters a subset of the mass-squared parameters, in our case and this means that the above equations will be violated by loop corrections. In particular, the mass-squared parameter in the diagonal basis for the would-be Goldstone boson is no longer zero. To see this, let us dene the loop tadpoles

δ i ≡ 1 v i ∂∆V ∂φ 0 i φ 0 i =0 (4.1.24)
so that we can solve (2.4.12) with the commonly-made choice of

m 2 ij = -δ i δ ij + m2 0,ij . (4.1.25)
Note that this is the value at the minimum of the potential so δ i is not regarded as a function of {φ 0 i } when we take derivatives below. Now

m 2 G =(R T m 2 R) GG = - i R2 iG δ i + O(2 loops), (4.1.26)
i.e. we can use the tree-level rotation matrices to obtain the Goldstone mass from the loop tadpoles up to corrections of two-loop order, which is all we shall require in the following. This generalises, for example, equations (2.26) and (2.27) of [START_REF] Kumar | Resummation of Goldstone boson contributions to the MSSM eective potential[END_REF].

Avoiding the Goldstone Boson Catastrophe in general renormalisable eld theories at two loops Following equation (4.1.23) above, we then see that

λGG G = 0, λ GG G = O(1 loop) (4.1.27)
in general. This is a crucial result in the following, even if in theories that preserve CP both couplings are zero to all orders. For theories breaking CP that could generate such a term at one or two loops, when we expand the potential as a series in m 2 G as in section 4.1.1 (justied by it being a one-loop quantity) we shall also implicitly expand the Goldstone self-coupling λ GG G for the same reason; implicitly because we shall not need the higher-order terms and this just corresponds to setting λ GG G = 0 everywhere. Note that this is automatic once we also employ re-expansion of the tadpoles and masses in terms of tree-level parameters to obtain consistent tadpole equations in section 4.4.

In practice when we are considering the broken gauge groups to be SU (2) × U (1) Y the unbroken U (1) QED allows the Goldstones to be separated into one neutral and one (complex) charged Goldstone that cannot mix. Hence in the following to simplify the notation we will restrict to a single neutral Goldstone boson and drop the lower index G, but the treatment of the charged Goldstone is identical. In this case we can also write G α G i → a ij φ 0 j and thus R jG =

a ij v j √ a ij a ik v j v k
(where we now allow the normalisation of a ij v j to be arbitrary) for the linearly realised symmetries considered here.

Small m 2

G expansion of the eective potential for general theories

To close this section we can now apply the notation and machinery from the previous subsections to resum the general eective potential at two loops, generalising the procedure of [START_REF] Elias-Miro | Taming Infrared Divergences in the Eective Potential[END_REF][START_REF] Martin | Taming the Goldstone contributions to the eective potential[END_REF].

The total potential up to two loops expands as

V e = V (0) + 1 16π 2 V (1) + 1 (16π 2 ) 2 V (2) . (4.1.28)
For use in the elimination of the infrared divergences in the derivatives of the eective potential, we expand V e for small m 2 G . More precisely, we want to write the two-loop part of V e as

V (2) = V (2) | m G =0 + 1 2 A(m 2 G )∆ 1 + 1 2 m 2 G Ω + O(m 4 G ), (4.1.29)
where the quantities ∆ 1 and Ω are to be determined.

The two-loop potential splits into contributions [START_REF] Martin | Two loop eective potential for a general renormalizable theory and softly broken supersymmetry[END_REF]:

V (2) = V (2) SSS + V (2) SS + V (2)
F F S + V (2)
F F S + V (2)
SSV + V

(2)

SV + V (2) V V S + V (2)
F F V + V (2)
F F V + V (2) gauge (4.1.30)
where the subscripts denote the propagators in the loops as scalar, fermion or vector (gauge sector). The terms in the brackets will not be resummed (since they contain no scalars) and so can be taken to be unchanged from the expressions in [START_REF] Martin | Two loop eective potential for a general renormalizable theory and softly broken supersymmetry[END_REF]. The loop functions appearing in the other terms are recalled in the MS and DR schemes and Landau gauge in appendix B.1.1.2.

The Goldstone Boson Catastrophe and resummation

First, the scalar contributions to the eective potential at two-loop order V

(2)

S ≡ V (2) SSS + V (2) SS read V (2) SSS ≡ 1 12 (λ ijk ) 2 f SSS (m 2 i , m 2 j , m 2 k ), (4.1.31)
V

SS ≡

1 8 λ iijj f SS (m 2 i , m 2 j ), (4.1.32)
and these functions can be expanded using formulae (3.7), (3.8) of [START_REF] Kumar | Resummation of Goldstone boson contributions to the MSSM eective potential[END_REF]. Separating terms with one or more Goldstone bosons from the terms without any, and using the fact that λ GGG vanishes at leading order see the discussion around equation (4.1.27) we nd the expansion of V

(2) S :

V

S = V (2)
S | no GB + j,k =G 1 4 (λ Gjk ) 2 f SSS (0, m 2 j , m 2 k ) + k =G 1 4 (λ GGk ) 2 f SSS (0, 0, m 2 k ) + A(m 2 G ) j,k =G 1 4 (λ Gjk ) 2 P SS (m 2 j , m 2 k ) + j =G 1 4 λ GGjj A(m 2 j ) (2)
+ k =G 1 2 (λ GGk ) 2 P SS (0, m 2 k ) + m 2 G j,k =G 1 4 (λ Gjk ) 2 R SS (m 2 j , m 2 k ) + k =G 1 2 (λ GGk ) 2 R SS (0, m 2 k ) + O(m 4 G ), (4.1.33)
from which we can identify the scalar part of ∆ 1 and Ω Next, the terms in V (2) involving fermions and scalars are V

(∆ 1 ) S = j,k =G 1 2 (λ Gjk ) 2 P SS (m 2 j , m 2 k ) + j =G 1 2 λ GGjj A(m 2 j ) + k =G (λ GGk ) 2 P SS (0, m 2 k ), Ω S = j,k =G 1 2 (λ Gjk ) 2 R SS (m 2 j , m 2 k ) + k =G (λ GGk ) 2 R SS (0, m 2 k ).
F F S ≡ 1 2 y IJk y IJk f F F S (m 2 I , m 2 J , m 2 k ), (2)
V

F F S ≡ 1 2 Re y IJk y I J k M * II M * JJ f F F S (m 2 I , m 2 J , m 2 k ). (2)
Here, there are only two cases to consider, either k = G or k = G, and for the latter case, we can use eqs. (3.9) and (3.10) from [START_REF] Kumar | Resummation of Goldstone boson contributions to the MSSM eective potential[END_REF] to expand the loop functions for small m 2 G . We then obtain for (∆ 1 ) F S and Ω F S

(∆ 1 ) F S = y IJG y IJG P F F (m 2 I , m 2 J ) + Re y IJG y I J G M * II M * JJ P F F (m 2 I , m 2 J ), (4.1.37
)

Ω F S = y IJG y IJG R F F (m 2 I , m 2 J ) + Re y IJG y I J G M * II M * JJ R F F (m 2 I , m 2 J ). (4.1.38)
Finally, the terms with scalars and gauge bosons read

V (2) SSV = 1 4 (g aij ) 2 f SSV (m 2 i , m 2 j , m 2 a ), (4.1.39)

V

(2)

V S = 1 4 g aaii f V S (m 2 a , m 2 i ), (4.1.40)

V

(2) Avoiding the Goldstone Boson Catastrophe in general renormalisable eld theories at two loops

V V S = 1 4 (g abi ) 2 f V V S (m 2 a , m 2 b , m 2 i ).
As previously, we can expand these terms and separate the contributions of the Goldstone boson, and we nd

(∆ 1 ) V S = 3 2 g aaGG A(m 2 a ) + 1 2 (g abG ) 2 P V V (m 2 a , m 2 b ), (4.1.42
) The expansion (4.1.29) of V (2) enables us to rewrite the two-loop eective potential after resummation of the leading Goldstone boson contributions as

Ω V S = (g aGj ) 2 R SV (m 2 j , m 2 a ) + (g aGG ) 2 R SV (0, m 2 a ) + 1 2 (g abG ) 2 R V V (m 2 a , m 2 b ).
Ve = V (0) + 1 16π 2 V (1) | m 2 G =0 + f (m 2 G + ∆ G ) + 1 (16π 2 ) 2 V (2) | m 2 G =0 + 1 2 Ω m 2 G , Ω = Ω S + Ω F S + Ω V S , (4.1.44)
∆ G ≡ i R 2 iG 1 v i ∂ Veff ∂φ 0 i = 1 16π 2 (∆ 1 ) S + (∆ 1 ) F S + (∆ 1 ) V S + O(2 loop).
The minimum of this potential will be found at m 2 G + ∆ G = 0 (along with the minimisation conditions for the additional scalars) and clearly contains no logarithmic divergences for small m 2 G .

The above expression could now be used for studies of general theories: the simplest would be for numerical studies where the potential is evaluated as a function of the expectation values and the derivatives taken numerically, as performed for the MSSM in [START_REF] Martin | Complete two loop eective potential approximation to the lightest Higgs scalar boson mass in supersymmetry[END_REF][START_REF] Martin | Two loop eective potential for the minimal supersymmetric standard model[END_REF] and implemented generally in [START_REF] Goodsell | Two-Loop Higgs mass calculations in supersymmetric models beyond the MSSM with SARAH and SPheno[END_REF]. However, there are potential numerical instabilities when the expectation values of additional scalars are small, and for complicated models many evaluations of the potential are required which can be slow: it is therefore useful to have explicit expressions for the tadpoles, as were derived at two loops in [START_REF] Goodsell | Generic two-loop Higgs mass calculation from a diagrammatic approach[END_REF]. In the next section we shall compute these for the resummed potential.

Removing infra-red divergences in the minimum condition

In the previous section we derived the resummed two-loop eective potential expanded in m 2

G that explicitly contains no infra-red divergences in its derivatives. In this section we shall present these derivatives. However, we shall also present a new approach to the problem which allows us to calculate the derivatives simply, and so we shall also give our derivations. For the scalar-only diagrams we do this by three methods:

(i) The rst method is to generalise the approach of [START_REF] Martin | Taming the Goldstone contributions to the eective potential[END_REF][START_REF] Kumar | Resummation of Goldstone boson contributions to the MSSM eective potential[END_REF], and simply take the derivatives of the resummed potential (4.1.44). However, this has the disadvantage of requiring us to compute the derivative of the rotation matrix elements ∂R ij ∂φ 0 r ϕ=v and proves to be cumbersome: there are dramatic simplications in the nal result.

(ii) To avoid the derivatives of rotation matrix elements, we instead take the derivatives of Ve before diagonalising the mass matrix and singling out the Goldstone boson and expanding the potential in m 2 G . This leads to a simpler derivation of the results.

4.2 Removing infra-red divergences in the minimum condition (iii) For our third method, we introduce a new approach: we set the Goldstone boson mass on-shell in the (non-resummed) eective potential. We shall show that this gives the same result as the other methods but (much) more simply, and does not suer from the problem of needing to exclude Goldstone self interactions by hand. Furthermore, in the next section we shall employ this approach to compute the mass digrams, which would be more complicated using the alternative methods.

All-scalar diagrams 4.2.1.1 Elimination of the divergences by method (i)

Generalising the approach of [START_REF] Kumar | Resummation of Goldstone boson contributions to the MSSM eective potential[END_REF] to extract the tadpoles we take the derivatives of equation (4.1.33). Starting with the one-loop potential, we note that, since m 2 G +∆ G = 0 at the minimum, the derivative of f (m 2 G + ∆ G ) will vanish. Hence we only require

∂ V (1)
S ∂φ 0 r = i =G 1 2 A(m 2 i )λ iik R rk . (4.2.1)
Note that throughout we shall adopt the Einstein convention for summing repeated indices when all indices are to be summed over; when there is an index that is summed over only a subset (i.e. excluding the Goldstone boson indices) we shall write an explicit sum symbol.

For the two-loop terms, recall the scalar part

V (2) S =V (2)
SSS | no GB + j,k =G 1 4 (λ Gjk ) 2 f SSS (0, m 2 j , m 2 k ) + k =G 1 4 (λ GGk ) 2 f SSS (0, 0, m 2 k ) + V (2) SS | m 2 G =0 + 1 2 Ω S m 2 G . (4.2.2)
Treating each of these pieces in turn we nd:

∂V (2) SSS | no GB ∂φ 0 r = i,j,k =G 1 4 λ iil R rl (λ ijk ) 2 f (1,0,0) SSS (m 2 i , m 2 i ; m 2 j , m 2 k ) + 1 2 λ ijk λ i jk (R T ∂ r R) i i f SSS (m 2 i , m 2 j , m 2 k ) + 1 6 λ ijk λ ii jk R ri f SSS (m 2 i , m 2 j , m 2 k ) = R rl i,j,k =G 1 4 λ ijk λ i jk λ ii l U 0 (m 2 i , m 2 i ; m 2 j , m 2 k ) - 1 6 λ ijk λ iljk I(m 2 i , m 2 j , m 2 k ) (4.2.3)
Avoiding the Goldstone Boson Catastrophe in general renormalisable eld theories at two loops and similarly we see

∂ ∂φ 0 r j,k =G 1 4 (λ Gjk ) 2 f SSS (0, m 2 j , m 2 k ) = R rl j,k =G - 1 2 λ Gjk λ Gljk I(0, m 2 j , m 2 k ) + 1 4 λ Gjk λ Gj k λ jj l U 0 (m 2 j , m 2 j ; 0, m 2 k ) , (4.2.4)
∂ ∂φ 0 r k =G 1 4 (λ GGk ) 2 f SSS (0, 0, m 2 k ) = R rl k =G - 1 2 λ GGk λ GGlk I(0, 0, m 2 k ) + 1 4 λ GGk λ GGk λ kk l U 0 (m 2 k , m 2 k ; 0, 0) . (4.2.5)
Putting this all together we see that they combine to give the compact expression

∂V (2) SSS | m 2 G =0 ∂φ 0 r = R rl i =G,j,k i 1 4 λ ijk λ i jk λ ii l U 0 (m 2 i , m 2 i ; m 2 j , m 2 k ) - 1 6 λ ijk λ iljk I(m 2 i , m 2 j , m 2 k ) m 2 G →0 . (4.2.6)
Next we turn to the SS terms:

∂V

SS | m 2 G =0 ∂φ 0 r = R rl i,j =G - 1 4 λ iijj λ iil B(0, m 2 i , m 2 i )A(m 2 j ) + 1 2 λ ii jj (R T ∂ r R) i i A(m 2 i )A(m 2 j ) = 1 4 R rl i,j =G λ ii jj λ ii l P SS (m 2 i , m 2 i )A(m 2 j ) m 2 G =0 , (2)
where the two terms again combine into a single compact expression. The nal piece is

1 2 Ω S ∂m 2 G ∂φ 0 r =λ GGl R Gl (j,k) =(G,G) 1 4 (λ Gjk ) 2 R SS (m 2 j , m 2 k ) m 2 G =0 , (4.2.8)
using the expression of Ω S from eq. (4.1.34). The total scalar tadpole is then the sum of equations (4.2.6), (4.2.7) and (4.2.8). Clearly the simplicity of the nal result compared to the intermediate expressions implies that there should be a simpler way of deriving it as indeed we shall show.

Elimination of the divergences by method (ii)

From inspection it is clear that the one-loop tadpole is not divergent when we send m 2 G → 0. However, at two loops we found that the process of isolating the divergences in the potential, expanding it in the Goldstone mass, and then taking the derivatives was rather cumbersome due to the derivatives of the mixing matrix elements R ij .

Instead we could consider taking the derivatives before having cancelled out the divergent parts, and then ensure the cancellations later. Hence we rewrite the resummed eective potential as

Ve = V e + 1 16π 2 f m 2 G + ∆ G -f (m 2 G ) - 1 16π 2 1 2 A(m 2 G )∆ G , (4.2.9)
4.2 Removing infra-red divergences in the minimum condition using formulae (4.1.29) and (4.1.44). We expect the terms from the derivative of -1 2 A(m 2 G )∆ G to cancel o the IR divergences in the derivatives of V e . To show this, we use the expression of ∆ G derived in eq. (4.1.34). The relevant contribution to the minimum condition at two-loop order is

16π 2 ∂ ∂φ 0 r - 1 2 A(m 2 G )∆ G ϕ=v ⊃ - 1 2 ∂m 2 G ∂φ 0 r ϕ=v log m 2 G (∆ 1 ) S (4.2.10) = - 1 2 R rp λ GGp log m 2 G   (j,k) =(G,G) 1 2 (λ Gjk ) 2 P SS (m 2 j , m 2 k ) + j =G 1 2 λ GGjj A(m 2 j )   .
The purely scalar contribution to the non-resummed tadpoles is, at one-loop order

∂V (1) S ∂φ 0 r ϕ=v = 1 2 R rk λ iik A(m 2 i ) (4.2.11)
and at two loops

∂V (2) S ∂φ 0 r ϕ=v = R rp T p SS + T p SSS + T p SSSS , (4.2.12) where [21]
T

p SS = 1 4 λ jkll λ jkp f (1,0) SS (m 2 j , m 2 k ; m 2 l ) = 1 4 λ jkll λ jkp P SS (m 2 j , m 2 k )A(m 2 l ), (4.2.13)
T p SSS = 1 6 λ pjkl λ jkl f SSS (m 2 j , m 2 k , m 2 l ) = - 1 6 λ pjkl λ jkl I(m 2 j , m 2 k , m 2 l ), (4.2.14)
T p SSSS = 1 4 λ pjj λ jkl λ j kl f (1,0,0) SSS (m 2 j , m 2 j ; m 2 k , m 2 l ) = 1 4 λ pjj λ jkl λ j kl U 0 (m 2 j , m 2 j ; m 2 k , m 2 l ), (4.2.15)
with the notation f

(1,0,0) α dened in eq. (B. 1.40).

In these formulae, we can then consider separately the Goldstone contributions and investigate the divergent terms. We nd two types of divergent terms in eq. (4.2.12) :

• The rst type of divergent term comes from T SS , for j = k = G, and 3 l = G, and reads

∂V (2) S ∂φ 0 r ϕ=v ⊃ - 1 4 R rp l =G λ GGll λ GGp B 0 (m 2 G , m 2 G )A(m 2 l ) = 1 4 R rp l =G λ GGll λ GGp log m 2 G A(m 2 l ) (4.2.16)
• The other divergent terms, coming from T p SSSS with j = j = G, are

∂V (2) S ∂φ 0 r ϕ=v ⊃ 1 4 R rp λ pGG λ Gkl λ Gkl log m 2 G P SS (m 2 k , m 2 l ) (4.2.17) 3 The term with l = G is proportional to log m 2 G A(m 2 G ), which tends to zero when m 2 G → 0.
Avoiding the Goldstone Boson Catastrophe in general renormalisable eld theories at two loops

• A potentially more dangerous element of those terms, for the particular case k = l = G is not present as λ GGG = 0 (at least up to terms of one-loop order).

All the other terms in

∂V

(2) S ∂φ 0 r ϕ=v are regular in the limit m 2 G → 0.

After relabelling of the indices in the sums, we observe that the log m 2 G divergences from the terms in eqs. (4.2.16) and (4.2.17) cancel out perfectly with the ones from eq. (4.2.10). We can then take the limit m 2 G → 0 in the one-loop and two-loop parts of the minimum condition: this limit is regular in the one-loop tadpole (4.2.11) so we recover eq. (4.2.1), while we nd

∂ V (2)
S ∂φ 0 r ϕ=v = 1 4 R rp    j,k,l =G λ jkll λ jkp P SS (m 2 j , m 2 k )A(m 2 l ) + 2 k,l =G λ Gkll λ Gkp P SS (0, m 2 k )A(m 2 l )    + 1 6 R rp λ pjkl λ jkl f SSS (m 2 j , m 2 k , m 2 l ) m 2 G →0 + 1 4 R rp    (j,j ) =(G,G) λ pjj λ jkl λ j kl U 0 (m 2 j , m 2 j , m 2 k , m 2 l ) m 2 G →0 + (k,l) =(G,G) λ pGG (λ Gkl ) 2 R SS (m 2 k , m 2 l ) m 2 G →0    , (4.2.18)
at two-loop order. It is important to notice that all three functions f SSS , U 0 and R SS are regular when one of their arguments goes to zero, hence the result we nd is indeed free of infrared divergences.

Elimination of the divergences by setting the Goldstone boson onshell

Here we shall introduce a new approach to the Golstone Boson Catastrophe: we shall treat the Goldstone boson mass as an on-shell parameter and enforce that it is identically zero. This means replacing the dimensionally regularised (DR or MS) Goldstone mass by the on-shell (or pole) mass in the following way

(m 2 G ) dim. reg ≡ (m 2 G ) OS -Π (1)
GG (m 2 G ) OS = -Π (1)
GG 0

(4.2.19)
where the pole mass is (m 2 G ) OS = 0. Note that we only need the one-loop relation here, so any mixing in the mass terms between the Goldstone boson and other (pseudo-)scalars is irrelevant it would be proportional to (Π

iG ) 2 and thus a two-loop eect. When we write the eective potential in terms of the on-shell Goldstone boson mass we should nd that it is free of divergences. To do this, we shall start from the dimensionally regularised potential and substitute the Goldstone boson mass in equation (4.2.19), expanding out to the appropriate loop order; this gives the result that we would obtain by performing the calculation using the on-shell mass with the appropriate counterterms. For our case, we only need to use the one-loop self-energy 4.2 Removing infra-red divergences in the minimum condition in the one-loop tadpole; the scalar contribution to the Goldstone boson self-energy at one-loop order is

Π (1),S GG p 2 = 1 2 λ GGjj A(m 2 j ) - 1 2 (λ Gjk ) 2 B(p 2 , m 2 j , m 2 k ) (4.2.20)
where we again require the result λ GGG = 0 to leading order although in this case we could (if desired) make it an on-shell condition. Applying the above relation to the tadpole in eq. (4.2.11) we obtain the following shift to the two-loop tadpole:

1 2 R rp λ GGp A(m 2 G ) = 1 2 R rp λ GGp A((m 2 G ) OS ) - 1 2 R rp λ GGp log(m 2 G ) OS Π (1)
GG ((m 2 G ) OS ) + O(3 -loop) (4.2.21) → ∂V (2) S ∂φ 0 r ((m 2 G ) OS ) = ∂V (2) S ∂φ 0 r m 2 G →(m 2 G ) OS - 1 4 R rp λ GGp log(m 2 G ) OS λ GGjj A(m 2 j ) -(λ Gjk ) 2 B(0, m 2 j , m 2 k ) . Since B(0, m 2 j , m 2 k ) = -P SS (m 2 j , m 2
k ), these shifts correspond exactly to the divergent terms we saw in equations (4.2.16) and (4.2.17) and so when we formally take the limit (m 2 G ) OS → 0 we nd exactly the same tadpole given explicitly in (4.2.18) that we found by the two other methods. This derivation is certainly much faster than the rst method, but note that the principle is dierent to the previous calculations: there is no ad-hoc resummation, nor are we required to expand the potential as a series in m 2

G . However, perhaps remarkably, we nd exactly the same result for the tadpole that remains, implying that, at least at two loops, the two approaches are equivalent.

This new approach will prove to be simpler than both previous methods when we turn our attention to mass diagrams; for now we shall simply complete the set of tadpole equations.

Before moving on to diagrams with fermions, we shall comment on the prescription to follow when there is more than one Goldstone boson. In that case, since the Goldstone bosons are all degenerate the mutual mixing between them becomes a leading-order eect and we must diagonalise the self-energies Π GG on the subspace of indices G, G which run over all Goldstones. However, we can also easily write this in the nondiagonalised basis as a generalisation of (4.2.19):

(m 2 GG ) dim. reg ≡ (m 2 G ) OS -Π (1)
GG (m 2 G ) OS = -Π (1)
GG 0 ,

where formally all Goldstone bosons have the same mass m 2 G which we set to zero.

Then we can rewrite the tadpole as

1 2 R rp λ GG p A(m 2 GG ) = G 1 2 R rp λ GGp A((m 2 G ) OS ) (4.2.23) - G,G 1 2 R rp λ GG p log(m 2 G ) OS Π (1) GG ((m 2 G ) OS ) + O(3 -loop).
If the gauge group of the model of interest is just that of the Standard Model, then clearly the charged and neutral Goldstone bosons cannot mix, so this becomes trivial hence in the following we shall restrict for clarity to the one-Goldstone case. However, we shall later write the full result in the general case.

Avoiding the Goldstone Boson Catastrophe in general renormalisable eld theories at two loops

Diagrams with scalars and fermions

The one-loop tadpoles involving fermions are

∂V (1) F ∂φ 0 r ϕ=v =R rp T p F = -R rp Re[y KLp M * KL ] A(m 2 K ) + A(m 2 L ) (4.2.24)
and these do not present any divergence in the limit of vanishing Goldstone boson mass. The two-loop contributions are [START_REF] Goodsell | Generic two-loop Higgs mass calculation from a diagrammatic approach[END_REF],

∂V

F S ∂φ 0 r ϕ=v = R rp T p SSF F + T p F F F S , (2)
where

T p SSF F = 1 2 y IJk y IJl λ klp f (0,0 ,1)
F F S (m 2 I , m 2 J ; m 2 k , m 2 l ) -Re y IJk y I J k M * II M * JJ λ klp U 0 (m 2 k , m 2 l , m 2 I , m 2 J ), (4.2.26)
T p F F F S =2Re[y IJp y IKn y KLn M * JL ]T F F F S (m 2 I , m 2 J , m 2 K , m 2 n ) + 2Re[y IJp y IKn y JLn M * KL ]T F F F S (m 2 I , m 2 J , m 2 K , m 2 n ) -2Re[y IJp y KLn y M P n M * IK M * JM M * LP ]T F F F S (m 2 I , m 2 J , m 2 L , m 2 n ), (4.2.27)
with the loop functions from eq. (II.38) of [START_REF] Goodsell | Generic two-loop Higgs mass calculation from a diagrammatic approach[END_REF].

The second term T F F F S is regular when m 2 G → 0, because the loop functions, B 0 , I, U 0 , that appear in its expression are all regular when only one of their argument goes to zero. However, the k = l = G terms in T SSF F are divergent:

T p SSF F ⊃ 1 2 y IJG y IJG λ GGp log m 2 G P F F (m 2 I , m 2 J ) + 1 2 Re y IJG y I J G M * II M * JJ λ GGp log m 2 G P F F (m 2 I , m 2 J ). (4.2.28)
After either resummation or setting the Goldstone boson on-shell we nd the total, nite, two-loop contribution T p SSF F in equation (4.2.41) and note that T p F F F S is not modied from eq. (4.2.27).

Diagrams with scalars and gauge bosons

The one-loop tadpole involving (massive) gauge bosons is

∂V (1) V ∂φ 0 r ϕ=v =R rp T p V = 1 2 R rp g aap A(m 2 a ), (4.2.29)
which contains no scalar propagators so has no divergences in the Goldstone boson mass.

However, the gauge boson contribution to the one-loop scalar self-energy in Landau gauge is [START_REF] Martin | Two loop scalar self energies in a general renormalizable theory at leading order in gauge couplings[END_REF]:

Π (1,V ) ij =g aik g ajk B SV (m 2 k , m 2 a ) + 1 2 g aaij A V (m 2 a ) + 1 2 g abi g abj B V V (m 2 a , m 2 b ), (4.2.30)
where the loop functions are given in [START_REF] Martin | Two loop scalar self energies in a general renormalizable theory at leading order in gauge couplings[END_REF][START_REF] Martin | Evaluation of two loop selfenergy basis integrals using dierential equations[END_REF] but simplify for zero momentum in Landau gauge to

B SV (x, y)| p 2 =0 =0, A V (x)| p 2 =0 =3A(x) + 2xδ MS , B V V (x, y)| p 2 =0 =3P SS (x, y) + 2δ MS . (4.2.31)
Recall that there are six scalar-gauge boson contributions to the two-loop tadpole [START_REF] Goodsell | Generic two-loop Higgs mass calculation from a diagrammatic approach[END_REF]:

T p SSV = 1 2 g aij g akj λ ikp f (1,0,0) SSV (m 2 i , m 2 k ; m 2 j , m 2 a ) + 1 4 g aij g bij g abp f (0,0,1) SSV (m 2 i , m 2 j ; m 2 a , m 2 b ) (4.2.32) T p V S = 1 4 g abii g abp f (1,0) V S (m 2 a , m 2 b ; m 2 i ) + 1 4 g aaik λ ikp f (0,1) V S (m 2 a ; m 2 i , m 2 k ) (4.2.33) T p V V S = 1 2 g abi g cbi g acp f (1,0,0) V V S (m 2 a , m 2 c ; m 2 b , m 2 i ) + 1 4 g abi g abj λ ijp f (0,0,1) V V S (m 2 a , m 2 b ; m 2 i , m 2 j ). (4.2.34)
Of these only three are potentially singular f

(1,0,0) SSV , f (0,1)
V S and f (0,0,1) V V S ; from shifting the tadpoles we obtain

∆T p SV = - 1 2 λ GGr log m 2 G g aGk g aGk B SV (m 2 k , m 2 a ) + 1 2 g aaGG A V (m 2 a ) + 1 2 g abG g abG B V V (m 2 a , m 2 b )
≡ λ GGr g aGk g aGk ∆f

(1,0,0) SSV (m 2 G , m 2 G ; m 2 k , m 2 a ) (4.2.35)
+ λ GGr g aaGG ∆f (0,1)

V S (m 2 a , m 2 G , m 2 G ) + λ GGr g abG g abG ∆f (0,0,1) V V S (m 2 a , m 2 b ; m 2 G , m 2 G )
i.e. they correspond exactly to the potentially singular terms. However, note that B SV term is zero and indeed we nd that f

(1,0,0)

SSV (m 2 G , m 2 G ; m 2 k , m 2 a ) is non-singular; we nd f (1,0,0) SSV (m 2 G , m 2 G ; x, y) = -R SV (x, y) + O(m 2 G ) f (0,1) V S (x, m 2 G , m 2 G ) =(3A(x) + 2xδ M S ) log m 2 G + O(m 2 G ) f (0,0,1) V V S (y, z; m 2 G , m 2 G ) = -3P SS (y, z) + δ MS log m 2 G -R V V (y, z) + O(m 2 G ). (4.2.36)
We give the nal nite tadpoles in equation (4.2.44).

Total tadpole

Here we gather the results of the previous subsections and rewrite them for the most general case, that of multiple Goldstone bosons. The total tadpole, after curing the Goldstone boson catastrophe and taking m 2 G → 0 everywhere, is

∂ V (2) ∂φ 0 r = R rp T p SS + T p SSS + T p SSSS + T p SSF F + T p F F F S + T p SSV + T p V S + T p V V S + T p F F V + T p F F V + T p gauge . (4.2.37)
Avoiding the Goldstone Boson Catastrophe in general renormalisable eld theories at two loops The all-scalar diagrams are

T p SS = 1 4 j,k,l =G λ jkll λ jkp P SS (m 2 j , m 2 k )A(m 2 l ) + 1 2 k,l =G λ Gkll λ Gkp P SS (0, m 2 k )A(m 2 l ), (4.2.38)
T p SSS = 1 6 λ pjkl λ jkl f SSS (m 2 j , m 2 k , m 2 l ) m 2 G →0 , (4.2.39)
T p SSSS = 1 4 (j,j ) =(G,G ) λ pjj λ jkl λ j kl U 0 (m 2 j , m 2 j , m 2 k , m 2 l ) + 1 4 (k,l) =(G,G ) λ pGG λ Gkl λ G kl R SS (m 2 k , m 2 l ), (4.2.40)
where by (j, j ) = (G, G ) we mean that j, j are not both Goldstone indices. The fermion-scalar diagrams are

T p SSF F = (k,l) =(G,G ) 1 2 y IJk y IJl λ klp f (0,0,1) F F S (m 2 I , m 2 J ; m 2 k , m 2 l ) -Re y IJk y I J k M * II M * JJ λ klp U 0 (m 2 k , m 2 l , m 2 I , m 2 J ) + 1 2 λ GG p y IJG y IJG -I(m 2 I , m 2 J , 0) -(m 2 I + m 2 J )R SS (m 2 I , m 2 J ) -λ GG p Re y IJG y I J G M * II M * JJ R SS (m 2 I , m 2 J ), (4.2.41)
T p F F F S =T p F F F S m 2 G →0 , (4.2.42)
while the gauge boson-scalar tadpoles are

T p SSV = T p SSV m 2 G →0 , T p V S = 1 4 g abii g abp f (1,0) V S (m 2 a , m 2 b ; m 2 i ) m 2 G →0 + (i,k) =(G,G ) 1 4 g aaik λ ikp f (0,1) V S (m 2 a ; m 2 i , m 2 k ), T p V V S = 1 2 g abi g cbi g acp f (1,0,0) V V S (m 2 a , m 2 c ; m 2 b , m 2 i ) m 2 G →0 (4.2.43)
+ (i,j) =(G,G ) 1 4 g abi g abj λ ijp f (0,0,1) V V S (m 2 a , m 2 b ; m 2 i , m 2 j ) - 1 4 g abG g abG λ GG p R V V (m 2 a , m 2 b ).
Finally the gauge boson-fermion and gauge diagrams are not aected by the Goldstone boson catastrophe, as scalar masses do not appear in them, and can be found in the

appendix C.2 of [21] T p F F V = 2g aJ I g K bJ Re[M KI y I Ip ]f (1,0,0) F F V (m 2 I , m 2 K ; m 2 J , m 2 a ) + 1 2 g aJ I g I bJ g abp f (0,0,1) F F V (m 2 I , m 2 J ; m 2 a , m 2 b ), (4.2.44)
T p F F V = g aJ I g aJ I Re[y II p M * JJ ] f F F V (m 2 I , m 2 J , m 2 a ) + M 2 I f (1,0,0) F F V (m 2 I , m 2 I ; m 2 J , m 2 a ) + g aJ I g aJ I Re[M IK M KI M * JJ y KK p ]f (1,0,0) F F V (m 2 I , m 2 I ; m 2 J , m 2 a ) + 1 2 g aJ I g bJ I g abp M II M * JJ f (0,0,1) F F V (m 2 I , m 2 J ; m 2 a , m 2 b ), (4.2.45)
T p gauge = 1 4 g abc g dbc g adp f (1,0,0) gauge (m 2 a , m 2 d ; m 2 b , m 2 c ). (4.2.46)
4.3 Mass diagrams in the gaugeless limit

Mass diagrams in the gaugeless limit

As discussed in the introduction, the scalar masses are among the most interesting electroweak precision observables, and their calculation also suers from the Goldstone Boson Catastrophe. Earlier literature pointed out that the calculation in the eective potential approximation contains more severe divergences that cannot be solved by resummation, and thus the inclusion of the external momentum is necessary. However, we shall nd that there are also divergences that are not regulated by external momentum and thus both setting the Goldstone boson on-shell and external momentum are required to obtain nite, accurate results.

On the other hand, the eective potential approximation is still useful and has advantages over a full momentum-dependent result, chief among these being simplicity and speed of calculation. In particular, the evaluation of the loop functions at arbitrary external momentum requires the numerical solution of dierential equations [START_REF] Martin | Evaluation of two loop selfenergy basis integrals using dierential equations[END_REF] which, although implemented in the fast package TSIL [START_REF] Martin | TSIL: A Program for the calculation of two-loop self-energy integrals[END_REF], is still much slower than the zero-momentum functions, and when the functions must be repeatedly called can lead to times orders of magnitude longer for complicated models. Hence we shall consider expanding the two-loop self-energies as a series in s ≡ -p 2 (for metric signature (-, +, +, +)) as Π

-1 l,ij + 1 s Π (2) ij (s) = log(-s) s Π (2)
-1,ij + Π (2) l 2 ,ij log 2 (-s) + Π (2)
l,ij log(-s) + Π

and we shall neglect terms of O(s), giving a generalised eective potential approximation: for loop functions where the singular terms Π

(2)

-1 l,ij , Π (2)
-1,ij , Π (2)
l 2 ,ij , Π (2)
l,ij vanish the result is identical to the second derivative of the eective potential. This approximation is particularly good when the mass of the scalars considered is smaller than the scale of other particles that they couple to; but even when they are similar we nd that typically the dierence is only a few percent. This should then be within other uncertainties in the calculation for most purposes.

We shall perform our calculations using our procedure of taking the Goldstone boson mass(es) on-shell as before, working in the general case now of allowing multiple Goldstone bosons throughout. We shall make heavy use of the existing expressions for two-loop scalar self energies from [START_REF] Martin | Two loop scalar self energies in a general renormalizable theory at leading order in gauge couplings[END_REF]; however, these are only available up to second order in the gauge coupling. Hence we shall be restricted to work in the very popular gaugeless limit where we neglect the gauge couplings of broken gauge groups (including electromagnetism, since hypercharge and weak SU (2) are both broken so their gauge couplings are neglected). The two-loop self-energy in this limit can be decomposed as follows:

Π (2) ij =Π S ij + Π SF (W ) ij + Π SF 4 (M ) ij + Π S 2 F 3 (M ) ij + Π S 3 F 2 (V ) ij + Π SF 4 (V ) ij + Π SV ij + Π F V ij . (4.3.2)
This consists of scalar-only propagators, diagrams with scalar and fermion propagators, diagrams with scalar and vector propagators, and fermions and vectors. We

nd that Π SF 4 (M ) ij and Π SF 4 (V ) ij
are nonsingular as m 2 G → 0 and s → 0, so the relevant formulae in that limit are equations (B.15) and (B.28) of [START_REF] Goodsell | Generic two-loop Higgs mass calculation from a diagrammatic approach[END_REF]. Furthermore, in the gaugeless limit the Goldstone bosons do not couple to the vectors, so Π SV ij Avoiding the Goldstone Boson Catastrophe in general renormalisable eld theories at two loops and Π F V ij are unchanged from (B.36) and (B.41) of [21]. However, the remaining dia- grams require regulation: our new expressions for Π S ij are presented in section 4.3.1;

Π SF (W ) ij , Π S 2 F 3 (M ) ij and Π S 3 F 2 (V ) ij
are derived in section 4.3.2.

All-scalar terms

The two-loop scalar self-energy contribution with only scalar propagators is given by [START_REF] Martin | Two loop scalar self energies in a general renormalizable theory at leading order in gauge couplings[END_REF]:

Π S ij = 1 4 λ ijkl λ kmn λ lmn W SSSS (m 2 k , m 2 l , m 2 m , m 2 n ) + 1 4 λ ijkl λ klmm X SSS (m 2 k , m 2 l , m 2 m ) + 1 2 λ ikl λ jkm λ lmnn Y SSSS (m 2 k , m 2 l , m 2 m , m 2 n ) + 1 4 λ ikl λ jmn λ klmn Z SSSS (m 2 k , m 2 l , m 2 m , m 2 n ) + 1 6 λ iklm λ jklm S SSS (m 2 k , m 2 l , m 2 m ) + 1 2 λ ikl λ jkmn + λ jkl λ ikmn λ lmn U SSSS (m 2 k , m 2 l , m 2 m , m 2 n ) + 1 2 λ ikl λ jkm λ lnp λ mnp V SSSSS (m 2 k , m 2 l , m 2 m , m 2 n , m 2 p ) + 1 2 λ ikm λ jln λ klp λ mnp M SSSSS (m 2 k , m 2 l , m 2 m , m 2 n , m 2 p ). (4.3.3)
The loop integral functions are recalled in (B. 1.52).

When at most one of the propagators is a Goldstone boson, we can set m 2 G → 0, s → 0 and use the simplied expressions below (B.2) of [START_REF] Goodsell | Generic two-loop Higgs mass calculation from a diagrammatic approach[END_REF]. However, for cases including more Goldstone bosons we must look for singularities since, in general, only the S SSS term is regular. Furthermore, we can divide the functions into those regulated by the momentum and those that are not. In particular, by inspection we see that for two or more Goldstone bosons W, X, Y, V can be divergent as m 2 G → 0, even for nite momentum; this means those terms must be regulated by resummation or, in our case, by shifts from the one-loop self energy by putting the Goldstone bosons on shell. On the other hand, the terms U, M and Z must be regulated by including nite momentum.

It should be noted that the divergences that are not regulated by momentum all involve a Goldstone boson self-energy as a subdiagram. It is then logical to consider how they relate to the divergent terms in the tadpole graphs. If we consider the eective potential approximation and take the derivatives of the tadpoles as in [START_REF] Goodsell | Generic two-loop Higgs mass calculation from a diagrammatic approach[END_REF], then we see that the topologies X, Y, Z descend from the T SS graphs; S, U arise from T SSS ; and M, V, W from T SSSS . Then it is clear that, since the T SSS graphs contain no divergences, resummation is irrelevant for S and U , while T SS and T SSSS are both divergent when there is part of a Goldstone boson self-energy as a subdiagram. We also see that W and X topologies arise from T SSSS and T SS respectively by replacing a three-point vertex with a four-point one, and likewise V and Y arise by adding a leg connected directly by a propagator to the other leg; we illustrate this whole discussion in gure 4.1. Hence we expect that these special divergences should follow the same pattern as the tadpoles, and be cured in the same way. However, we shall also nd below some subtleties remain in the V topology.

Mass diagrams in the gaugeless limit

Self-consistent solution of the tadpole equations

For topology V , the rst combination is proportional to the scalar case in equations (4.3.12) and (4.3.13):

V SSSF F (k, m 2 G , m 2 G , x, y)+∆V SSSF F (k, m 2 G , m 2 G , x, y) = -2V SSSSS (k, m 2 G , m 2 G , x, y) -2B (s, m 2 G , k)P SS (x, y) → -2 Ṽ (k, x, y), (4.3.21)
while the second also contains an additional U function:

V SSSF F (k, m 2 G , m 2 G , x, y) + ∆V SSSF F (k, m 2 G , m 2 G , x, y) → -(x + y) Ṽ (k, x, y) + U (k, m 2 G , x, y). (4.3.22)
For this case, when k = m 2 G it is non-singular as in the scalar case, and when k = m 2 G we require the expansions with nite s from equation (4.3.12) and for U (0, 0, x, y) from appendix B.2.

Self-consistent solution of the tadpole equations

We have shown how to avoid the Goldstone Boson Catastrophe in general renormalisable eld theories, and how this can be applied to calculating neutral scalar masses in the gaugeless limit in a generalised eective potential approximation. However, we still have two possible ways of calculating both tadpoles and self-energies, which dier in terms of how we solve the tadpole equations. The choice arises because the mass parameters m 2 ij appear on both the left-and right-hand sides of equation (2.4.12) or equivalently, because when considering equation (4.1.25) the m 2 ij appear in the δ i .

Therefore we can:

1. Numerically solve equation (2.4.12) to nd the m 2 ij exactly.

2. Perturbatively expand the m 2 ij around the m2 0,ij so that

m 2 ij = m2 0,ij + δ (1) m 2 ij + δ (2) m 2 ij + ... (4.4.1)
and solve for the desired loop order.

Since the eective potential ∆V will only be computed to a given loop order, the two approaches are formally equivalent.

For the rst approach, in practice, this means that we must iteratively solve the tadpole equations: at each iteration we put m 2 i = R ki R li m 2 kl for the tree-level mass parameters, computing a new R each time and therefore modifying the couplings, and then set the Goldstone boson mass to zero in the loop functions and compute the tadpole equations from the expressions in section 4.2.

We nd in this case that the couplings are no longer guaranteed to satisfy certain relationships imposed by the broken symmetries; only the full on-shell amplitudes will satisfy the appropriate Slavnov-Taylor identities. This is only a problem for the coupling λ GG G between three Goldstone bosons, which is zero at tree-level and onshell; because the parameter in the Lagrangian will in general obtain a small non-zero value (in theories with CP-violation) and yet leads to divergent Goldstone boson selfenergies we must impose that this is also on-shell (i.e. zero). Since this coupling does not appear at one-loop in the calculation of the Higgs boson mass, taking this coupling Avoiding the Goldstone Boson Catastrophe in general renormalisable eld theories at two loops to vanish causes no shift at two loops. On the other hand, if we want to calculate the Goldstone boson self-energy at two loops then we do nd a set of shifts when we take this coupling on-shell: we would need to include the vertex corrections and dene a set of shifted loop functions for those contributions (which, of course, only aect the self-energies). We shall return to this in future work.

Instead, we will take the second approach in the list above, and while we cannot do the same as we did for the Goldstone boson and put the other scalars on shell, we can follow [START_REF] Kumar | Resummation of Goldstone boson contributions to the MSSM eective potential[END_REF] and re-expand the masses m 2 ij as a series in the couplings to one-loop order in the one-loop tadpole. We then use the tree-level masses in the loop functions and solve the tadpole equations perturbatively instead of iteratively as we have just described. Let us dene a set of masses m2 = {m 2 G , m2 i =G } i.e. we use the on-shell mass for the Goldstone, and the tree-level masses for the other scalars. To single out the Goldstone boson we use the tree-level mixing matrix RkG which in any case should correspond to the all-loop expression, depending as it does only on the symmetries and VEVs. We can rst dene a perturbation to the non-diagonal tree-level mass matrix

as ∆ij ≡ m 2 ij -m2 0,ij , (4.4.2)
which can then be rotated to the diagonal basis using rotation matrices R so that we obtain a perturbation ∆ ij in the basis that diagonalise the tree-level mass matrix.

If we use equation (4.1.25), we nd that

∆ij = -δ i δ ij , ⇒ ∆ ij ≡ -Rki Rkj δ k . (4.4.3)
However, this equation fails for pseudoscalars, and we will instead allow ∆ ij to be an implicit function of the tadpole shifts. Indeed, we solve the tadpole equations for some

variables {x i } with x i = c 0,i + c ij × ∂∆V ∂φ 0 j , (4.4.4)
and then

∆ij = k,l ∂m 2 ij ∂x k c kl ∂∆V ∂φ 0 l . (4.4.5)
For example, let us consider again the Abelian Goldstone model of section 4.1.1, with a single complex scalar Φ = 1 √ 2 (v + h + iG), and with a potential given in eq. (4.1.1). We solve the tadpole equations for the parameter µ 2 so that

µ 2 + λv 2 + 1 v ∆V ∂h = 0 .
M 2 h = µ 2 + 3λv 2 , m 2 G = µ 2 + λv 2 . (4.4.7) So in our notation, x h → µ 2 , c 0,h → -λv 2 , c hh → -1 v and so ∆ hh = - 1 v ∆V ∂h = ∆ GG . (4.4.8)
We can expand ∆ ij as usual in perturbation theory using

∆ ij = 1 16π 2 ∆ (1) ij + 1 (16π 2 ) 2 ∆ (2)
ij + ... to nd that we should shift the tadpoles according to

∂ V (2) ∂φ 0 r (m 2 ) = ∂ V (2) ∂φ 0 r ( m2 ) + 1 2 (i,i ) =(G,G ) Rrl λii l ∆ (1)
ii P SS ( m2 i , m2 i ) .

(4.4.9)

4.4 Self-consistent solution of the tadpole equations By (i, i ) = (G, G ) we mean that the sum over (i, i ) excludes the cases where both i and i are Goldstone boson indices. This allows us to express the ∆ ii entirely in terms of the tree-level m2 parameters and obtain a perturbative expansion for m 2 note that we should also replace all of the couplings λ ijk , λ ijkl etc and rotation matrices R ij with their tree-level values λijk , λijkl , Rij (we already implicitly used this to disregard the λ GGG terms). The only subtlety occurs when m2 i = m2 j for some i, j which is not ensured by a symmetry so that ∆ ii = ∆ jj ; in that case as usual the R matrix must be modied to diagonalise ∆

(1) ij on those indices. However the expression above is still valid in that case. Note that the shift only occurs for scalar propagators in the one-loop diagrams, which is why there is no modication of the fermionic or vector tadpole diagrams.

We can apply the same procedure to use the tree-level masses in the mass diagrams: after some algebra we nd (in the gaugeless limit otherwise we will have some additional shifts from scalar-vector diagrams) that

Π (2) ij (s, m 2 ) = Π (2) ij (s, m2 ) + (k,k ) =(G,G ) 1 2 λijkk ∆ (1)
kk P SS ( m2 k , m2 k ) (4.4.10)
λikl λjk l ∆

(1)

kk C(s, s, 0, m2 k , m2 l , m2 k ) ,
where we used the usual C function dened in eq. (B.1.17). These together then allow us to determine the scalar masses to be the values of s that give solutions to:

0 = Det sδ ij -m 2 0,ij + δ i ( m2 )δ ij - Π (1) ij (s, m2 ) 16π 2 - Π (2) ij (s, m2 ) (16π 2 ) 2 (4.4.11)
-

δ ij 2 1 (16π 2 ) 2 (j,j ) =(G,G ) Ril λjj l ∆ (1)
jj P SS ( m2 j , m2 j ) - Rii Rjj (16π 2 ) 2 (k,k ) =(G,G ) 1 2 λi j kk ∆ (1)
kk P SS ( m2 k , m2 k ) -λi kl λj k l ∆ (1) kk C(s, s, 0, m2 k , m2 l , m2 k ) .
Typically in spectrum generators the two-loop corrections are computed at xed momentum and then the eigenvalues of the above matrix computed iteratively. Since we have given the expansion of all the loop functions relevant for the two-loop corrections up to terms of order O(s), this could be generalised to include our simple momentum dependence for the two-loop part as in equation (4.3.1) without signicant loss of speed since the computationally expensive parts of the two-loop functions would only need to be evaluated once. However, since all of the expansions are strictly valid only up to two-loop order, the equation above could be solved perturbatively itself with no signicant loss of accuracy. Now, let us end this discussion by a comment on the relation between the on-shell method and the self-consistent solution of the tadpole equations. Since the Goldstone boson is massless at tree-level, this means that we automatically have the Goldstone boson on-shell. This means that the on shell and consistent solution approaches are more closely related than rst appears: since the Goldstone boson mass must be zero on-shell and we can identify the Goldstone boson eigenstates using a matrix R kG derived just from the broken symmetries (see sections 2.4.1 and 4.1.2) then the on-shell

Avoiding the Goldstone Boson Catastrophe in general renormalisable eld theories at two loops condition becomes

det(p 2 -m 2 ij -Π ij (p 2 )) = 0 → R kG R lG m 2 kl + Π GG (0) = 0.
(4.4.12)

and since m2 0,GG = 0 we have

δm 2 G = R kG R lG m 2 kl = -Π GG (0) ∆ GG = δm 2 G + O(2 -loop) → ∆ GG = -Π GG (0) + O(2 -loop), (4.4.13)
i.e. the approach of adjusting the loop functions (as we do when setting the Goldstone boson on-shell) or dening a set of shifts to the tadpoles and self-energies involving ∆ ij should give the same result when we just consider the shifts to the Goldstone boson masses, even though the expressions look very dierent.

Conclusions

We have presented a solution to the Goldstone Boson Catastrophe in general renormalisable theories to two-loop order. We showed that the approach of Goldstone boson resummation is equivalent (at least at two-loop order) to an on-shell scheme for the Goldstone boson(s), the latter being much more convenient calculationally. We then showed how there are a set of self-energy diagrams that also exhibit the Goldstone Boson Catastrophe even when external momentum is included but that our solution naturally avoids those singularities. We were then able to give expressions for a generalised eective potential approximation for neutral scalar masses in the gaugeless limit, that are free of infra-red divergences and give a good approximation to the full momentum-dependent result. This also included the re-expansion of the masses in terms of the values obtained from the tree-level tadpole equations, allowing a selfconsistent solution of the tadpole equations (i.e. equations where no terms to be solved for appear on both left and right hand sides).

The expressions obtained in this chapter now allow simple infra-red safe calculations in a wide variety of theories. Most practically, it would be simple to implement them in a package such as SARAH, to enable automated calculations for any model and avoid the problems seen, for example, in [START_REF] Athron | Precise Higgs mass calculations in (non-)minimal supersymmetry at both high and low scales[END_REF][START_REF] Goodsell | Two-loop corrections to the Higgs masses in the NMSSM[END_REF][START_REF] Goodsell | The Higgs mass in the CP violating MSSM, NMSSM, and beyond[END_REF], with the existing implementation. This should also enable more reliable and accurate explorations of the parameter space of many models with the SARAH/SPheno numerical framework in particular for non-supersymmetric models (such as 2HDMs), where the existing solution 4 to the Goldstone Boson Catastrophe is not particularly successful, relying as it does on there being a gauge-coupling dependent part of the scalar potential (as is the case in supersymmetric theories). This will be the subject of the next chapter. 4 We will review the previous approaches in SARAH in section 5.1. See also [START_REF] Goodsell | Two-Loop Higgs mass calculations in supersymmetric models beyond the MSSM with SARAH and SPheno[END_REF], appendix 2b of [START_REF] Goodsell | Two-loop corrections to the Higgs masses in the NMSSM[END_REF] and especially section 2 of [START_REF] Goodsell | The Higgs mass in the CP violating MSSM, NMSSM, and beyond[END_REF].

Supersymmetric and non-supersymmetric models without catastrophic Goldstone bosons Momentum-independent renormalisation schemes are the most convenient choices for applying to a large variety of models, and so all mass calculations in SARAH are performed in the MS or DR scheme. In contrast, on-shell schemes might oer some model dependent advantages. This is for instance the case in supersymmetric models with Dirac gauginos, which we studied in chapter 3. In particular, we found that when there is a large mass splitting between the stops and the gluino, an on-shell scheme leads to an improved convergence of the perturbative series. It is also very useful often if a DR and on-shell calculation exists for the same supersymmetric model:

the dierence between the results can be used as estimate of the missing higher-order corrections; this can now be done for the MSSM and certain classes of NMSSM and Dirac gaugino contributions. On the other hand, there has been hardly any discussion in the literature about radiative corrections to Higgs masses in non-supersymmetric BSM models. One reason for this, besides the technical hurdles, is that the additional freedom in non-supersymmetric models introduces a large number of free parameters,

i.e. in some cases it might be possible to absorb any nite correction in the scalar sector into the counter-terms of these parameters. Thus, it is often implicitly assumed that the masses, but also the mixing angles, in the extended Higgs sector in BSM could be kept at their tree-level values. However, this is fraught with danger:

(i) not all non-supersymmetric models really have a suciently large number of free parameters to absorb all radiative corrections. This is for instance the case in the Georgi-Machacek model; (ii) if a low-energy model is combined with an explicit UV completion (such as a GUT theory), the freedom to adjust the couplings is usually lost; (iii) using masses instead of couplings as input hides the presence of huge or even non-perturbative quartic couplings; (iv) even if parameters are checked with respect to simple limits such as λ < 4π or tree-level unitarity bounds, this does not guarantee that the considered parameter point is perturbative or that strongly coupled eects do not appear at energy scales already explored at the LHC. Partly motivated by the growing interest in exploring quantum corrections to non-supersymmetric models, we explore here in sections 5.5 and 5.6 the corrections to the Two-Higgs-Doublet Model (2HDM) and Georgi-Machacek model (GM), drawing attention to the fact that the corrections pass out of control well before the naive perturbativity or unitarity bounds.

Finally, an MS calculation has the advantage that it can give an estimate of the size of the theoretical uncertainty by varying the renormalisation scale. Moreover, to obtain more reliable results for the vacuum stability by considering the renormalisation group equation (RGE) improved eective potential, a translation of masses into MS parameters is necessary. We show in this work how these aspects can be analysed in non-supersymmetric models with the new calculation available now in SARAH.

The Goldstone Boson Catastrophe and its solutions

The Goldstone Boson Catastrophe appears because the mass-squared parameter(s) of the Goldstone boson(s) in the Lagrangian is(are) zero at tree-level, but non-zero once we take into account the loop corrections to the potential. Then at two loops and higher we must calculate loop corrections with a small and/or negative mass-squared parameter, which leads to large logarithms and/or phases this can in fact be a complete obstacle to a precise calculation.. This problem was only noticed in the rst attempt to go beyond the gaugeless limit in the MSSM at more than one loop [START_REF] Martin | Complete two loop eective potential approximation to the lightest Higgs scalar boson mass in supersymmetry[END_REF]. Indeed, in the MSSM, the gaugeless limit turns o the Goldstone boson couplings to the Higgs, and the other (momentum-dependent)

5.1 The Goldstone Boson Catastrophe and its solutions calculations that have been performed beyond this limit only consider the sector of the theory without the Goldstones. However, as soon as one considers non-minimal supersymmetric models in which trilinear interactions of the Higgs superelds occur in the superpotential, the gaugeless limit no longer oers much protection against the problem, since the quartic coupling is not determined by the gauge couplings; and this is also a generic feature of non-supersymmetric models (such as the Standard Model).

In the following we shall describe the previous approaches to the problem and the implementation of our new results in SARAH.

Previous approaches in SARAH

Up until now, in SARAH the catastrophe appeared in an even more acute form because all of the one-and two-loop tadpoles and self-energies are computed using the treelevel masses in the loops, so without a solution to the problem, the Goldstone bosons are massless and cause several loop functions to diverge. However, for supersymmetric models the original workaround implemented in [START_REF] Goodsell | Generic two-loop Higgs mass calculation from a diagrammatic approach[END_REF][START_REF] Goodsell | Two-Loop Higgs mass calculations in supersymmetric models beyond the MSSM with SARAH and SPheno[END_REF] and explored in more detail in [START_REF] Goodsell | The Higgs mass in the CP violating MSSM, NMSSM, and beyond[END_REF] relies on the fact that that the electroweak gauge couplings appear in the D-term potential.

1 We therefore used the tree-level parameters that are solutions of the full tree-level tadpole equations including the electroweak couplings to calculate the tree-level masses (but set the electroweak gauge couplings to zero in the mass matrices) used in the two-loop routines' loop functions. In other words, the masses in the loop functions are not at the minimum of the potential, and are typically tachyonic 2 , with a size of order the electroweak scale. Since we are neglecting two-loop corrections proportional to these couplings, this error is acceptable. On the other hand, for models beyond the MSSM (in particular, the NMSSM) there are typically regions of the parameter space where the Higgs sector masses still pass near to zero and cause the loop functions to diverge; for example such problems were observed in [START_REF] Benakli | The Di-Photon Excess in a Perturbative SUSY Model[END_REF][START_REF] Goodsell | Dark matter scenarios in a constrained model with Dirac gauginos[END_REF][START_REF] Athron | Precise Higgs mass calculations in (non-)minimal supersymmetry at both high and low scales[END_REF].

A more recent approach was to introduce regulator masses. All scalar masses in the two-loop routines which are below a certain threshold are set in terms of the renormalisation scale Q and a constant R:

m 2 S,min = RQ 2 (5.1.1)
This approach was introduced in SARAH to stabilise cases in which the D-term approach fails. This could either be, as demonstrated in an example in sec. 5.3, if other scalars articially become very light, or if the supersymmetric scale is much higher than the electroweak scale. However, in contrast to the D-term solution, this approach violates the symmetries of the theory and can lead to non-zero masses for Goldstone bosons.

Furthermore, there is no a priori indication for the optimal size of R; too large and the Goldstone/Higgs contributions are suppressed (because logarithmic contributions including them are articially reduced), too small and the results become numerically unstable, and the user must use trial and error. Finally, it implicitly assumes that the corrections coming from the Higgs/Goldstone bosons to the Higgs mass are small (so that modifying them is benign). This is not a good approximation in many non-SUSY models, and for this reason the newly implemented solution described in the next subsection allows non-SUSY models to be studied accurately for the rst time.

Supersymmetric and non-supersymmetric models without catastrophic Goldstone bosons 5.1.2 On-shell Goldstone bosons, consistent tadpole solutions, and the implementation in SARAH

In the previous chapter, a genuine solution was presented for generic eld theories, which we will now implement in SARAH. We should treat the Goldstone boson mass as an on-shell parameter, and a set of modied expressions for tadpoles and self-energies were derived indeed, it was shown that there were a class of loop diagrams that were not made nite purely by including external momenta. In addition, expressions for the consistent solution of the tadpole equations were given.

For the evaluation of tadpoles and self-energies we proposed in section 4.3 a generalised eective potential limit, where the self-energies are expanded in s = -p 2 (= m 2 on shell) and all terms of order O(s) are neglected (but crucially retaining terms that diverge at s = 0). We therefore require the following basis of loop functions, where {x, y, z, u, v} = 0 are masses squared:

Momentum independent : J(x), P SS (x, y), P SS (0, y), I(x, y, z), I(0, y, z), I(0, 0, z), U 0 (x, y, z, u), U 0 (0, y, z, u), U 0 (x, y, 0, u), U 0 (0, y, 0, u), U 0 (x, y, 0, 0), U 0 (0, y, 0, 0), M 0 (x, y, z, u, v), M 0 (0, y, z, u, v), M 0 (0, 0, z, u, v), M 0 (0, 0, 0, u, v), Ṽ (x, y, z) .

Momentum dependent : B(0, 0), M (x, 0, 0, 0, 0), M (0, y, 0, u, v), M (0, 0, 0, u, v), M (0, 0, 0, 0, v) U (0, 0, x, y), U (0, 0, 0, y) Ṽ (0, y, z), Ṽ (0, 0, z) .

(5.1.2)

All of these functions are implicitly dependent on the renormalisation scale Q, typically containing factors of log x ≡ log(x/Q 2 ). Expressions for all of these functions expanded up to O(1) in the external momenta (or the reference for them) are given in appendices B.2 and B.3. Note in particular that the functions Ṽ (x, y, z) are given in terms of the regularised function V (u, 0, y, z) dened in [START_REF] Martin | Two loop scalar self energies in a general renormalizable theory at leading order in gauge couplings[END_REF] 3 ; in appendix B.3 we derive explicit compact expressions for this function rst with full momentum dependence, and then expanded up to O(1) in the external momenta.

In our practical implementation in SARAH we have extended the available routines for calculating two-loop integrals with the missing ingredients to address the Goldstone boson catastrophe. Moreover, there are three loop functions involving fermions and gauge bosons which needed modication for the MS scheme as used for non-supersymmetric models, as compared to the DR for supersymmetric models; the tadpole and self-

energies contain ∂ V (2) ∂φ 0 r ⊃R rp T p F V , T p F V = g 2 d(I)C(I)Re(M II y II r ) × 1 2 F F V (x) , Π (2)
ij ⊃ Π F V ij = g 2 d(K)C(K) Re(y iKL y jKL )G F F (m 2 K , m 2 L )
5.1 The Goldstone Boson Catastrophe and its solutions

+ Re(y iKL y jK L M KK M LL )G F F (m 2 K , m 2 L ) , (5.1.3)
where V (2) is the two-loop contribution to the eective potential, d(I), C(I) are the dimension and quadratic Casimirs of representation I of the gauge group having coupling g, and the loop functions are: (

1 2 F F V (x) =4x 6 -7 log x + 3 log 2 x + δ MS 2 log x -1 , G F F (x,
Here δ MS is one for MS masses and zero for DR .

We have also implemented in SARAH the consistent tadpole solution of section 4.4, under the assumption that the variables {x i }, for which the tadpole equations are solved, are dimensionful and there is no explicit dependence of the trilinear/quartic couplings on them (only implicitly through the mixing matrices R); and also we assume that the fermion mass matrices do not depend on these parameters. These assumptions are fullled e.g. for {m Note that the solution to the Goldstone boson catastrophe exists only for the diagrammatic calculation (ag 8 → 3), but not for the eective potential calculations using numerical derivatives to obtain the tadpoles and self-energies (ag 8 → 1,2). By default, the new calculation is used now, but could be turned o if demanded (ag 151 → 0). In this case, it is usually necessary to include a non-zero regulator mass via ag 410 for non-supersymmetric models. In principle, there should not be any reason to revert to the old calculation with regulator masses except for double-checking the result.

The consistent tadpole solution is turned o by default but can be turned on by setting ag 150 → 1. This is because, while strictly it is more accurate to include it, there is also the possibility of numerical instability if the shift in the tree-level mass parameters is large; for example, if the expectation values of some scalars are small (such as e.g.

the neutral scalar of an electroweak triplet which must have a small expectation value from electroweak precision constraints) then the shift in the mass parameter can be much larger than the tree-level value and the perturbative solution fails. In such cases, it would be better to use a recursive approach which is currently not possible for the reasons given in section 4.4.

Standard Model

A rst comparison of our results with existing calculations

Now that two-loop corrections to scalar masses are available in SARAH, free of the Goldstone boson catastrophe, it is important to compare the results we obtain to other computations available in the literature, as a verication of our results and as a way to estimate the impact of missing corrections. We consider in this section the Higgs mass calculations in the Standard Model, and we will compare the results obtained with SPheno with the computations performed at complete two-loop calculation in [START_REF] Buttazzo | Investigating the near-criticality of the Higgs boson[END_REF], and the full two-loop (plus leading three-loop) Higgs mass calculation implemented in the public code SMH [START_REF] Martin | Higgs boson mass in the Standard Model at two-loop order and beyond[END_REF]. These works take into account two-loop electroweak corrections, which are not available for generic theories and are not included in our code, hence we will quantify the size of these eects, together with eects from momentum, and investigate the discrepancy in masses coming from the dierent determination of the top Yukawa coupling.

It is interesting to examine the way that the two calculations avoid the Goldstone Boson Catastrophe. The calculation of Buttazzo et al. [START_REF] Buttazzo | Investigating the near-criticality of the Higgs boson[END_REF] was performed in Feynman gauge and using certain parameters on-shell, whereas the results implemented in SMH are in a pure MS scheme and Landau gauge, which is closer to our approach. In the latter paper, some resummation is performed by hand to eliminate the divergence in the mass calculation; it is perhaps surprising that the absence of the function V (0, 0, y, z) from the basis in TSIL was not problematic, but there the calculation was performed by computing the set of integrals explicitly using TARCER [START_REF] Mertig | TARCER: A Mathematica program for the reduction of two loop propagator integrals[END_REF] rather than starting from a set of generic expressions, so the result was found directly in terms of the other basis functions. In principle this should agree with our equation (B.3.10).

A rst approach for the comparison between SPheno and [START_REF] Buttazzo | Investigating the near-criticality of the Higgs boson[END_REF] (5.2.1)

They furthermore took the experimentally determined central value of the Higgs mass to be 125.15 GeV, which we shall take as a reference value rather than an input. The use of consistent solutions to the tadpole equations as derived in section 4.4 has also been implemented in the SPheno code and this comparison in the context of the SM is a good occasion to study the eect of this additional shift to the tadpoles and mass diagrams, thus we compute the Higgs mass in this rst method both with and without using the consistent tadpole solutions. A second approach to compute m h with SPheno, which could potentially improve the comparison, is to use as well the same values for the top-Yukawa y t and electroweak gauge couplings g 1 , g 2 as those

given for each order in table 3 of [START_REF] Buttazzo | Investigating the near-criticality of the Higgs boson[END_REF].

We obtain another result for m h with SMH [START_REF] Martin | Higgs boson mass in the Standard Model at two-loop order and beyond[END_REF], and although this code is made to perform Higgs mass calculations in the Standard Model to partial three-loop order, we use it here with the three-loop corrections always switched o, for the purpose of our comparison with SPheno. We use the routine calc_Mh that gives for a given loop order the value of m h from the inputs of the renormalisation scale Q, the quartic coupling λ, the top-Yukawa y t , the Higgs VEV v, and the gauge couplings g 3 , g, g , all given at scale Q. In order to improve the comparison, we take the same values for the inputs as used at each order in SPheno. We give in table 5.1 the values we nd for the Higgs mass when taking the same values of λ as found in [START_REF] Buttazzo | Investigating the near-criticality of the Higgs boson[END_REF], with the two methods described above for SPheno and with SMH. Table 5.1 Values of the Higgs mass at scale Q = m t for the values of the quartic couplings λ found in [START_REF] Buttazzo | Investigating the near-criticality of the Higgs boson[END_REF] at tree level, one loop and two loops, in the two approaches we used for SPheno, and with SMH. The rst approach was to change only the SM parameter inputs while letting SPheno determine the top-Yukawa and electroweak gauge couplings, and the Higgs mass is then computed both with and without the consistent tadpole solutions. The second method was to take the same values of y t , g 1 , g 2 in SPheno as in [START_REF] Buttazzo | Investigating the near-criticality of the Higgs boson[END_REF] (and switch o the consistent tadpole routines). For SMH, the values of the input parameters the top-Yukawa, the electroweak gauge couplings, the Higgs VEV and the strong gauge coupling were taken from the outputs of the SPheno scans. Computations are made with SARAH-4.12.0, SPheno-4.0.3 and SMH-1.0 [START_REF] Martin | Higgs boson mass in the Standard Model at two-loop order and beyond[END_REF].

At tree-level, all the values we nd with SPheno and SMH obviously match as the treelevel Higgs mass only depends on λ and v which have almost the same values here, and the divergence from the value of 125.15 GeV is solely explained by the Higgs VEV which is not the same as in [START_REF] Buttazzo | Investigating the near-criticality of the Higgs boson[END_REF] since they take it as an on-shell parameter, while we use the MS value as described in [START_REF] Staub | Improved predictions for intermediate and heavy Supersymmetry in the MSSM and beyond[END_REF]. More importantly, the loop corrected values in Supersymmetric and non-supersymmetric models without catastrophic Goldstone bosons the dierent methods also agree quite well, thanks to the improved determination of the top Yukawa coupling y t (including leading two-loop eects) recently implemented in SARAH [START_REF] Staub | Improved predictions for intermediate and heavy Supersymmetry in the MSSM and beyond[END_REF], and at each order in perturbation theory the Higgs masses we nd are less than a GeV away from 125.15 GeV. It is interesting to note that the values of m h found using the SPheno code generated by SARAH version 4.9.3 in which y t is only determined at one-loop order are approximately 2 -2.5 GeV below those shown in table 5.1, and hence illustrate the importance of the precise determination of the top Yukawa coupling for calculations of m h . The small size of the dierence between the values found with the couplings computed by SPheno or taken from [START_REF] Buttazzo | Investigating the near-criticality of the Higgs boson[END_REF] a few tens of MeV at two loops tend to indicate that the precision of the extraction of y t in SPheno is now comparable to that in [START_REF] Buttazzo | Investigating the near-criticality of the Higgs boson[END_REF]. Considering now the eect of the consistent tadpole solutions that appears only in the two-loop masses we observe a small shift of about 30 MeV to m h , indicating that the perturbative expansion we perform in the tadpole equation is valid for the SM. Finally, the reasons explaining the remaining deviation of our results with respect to 125.15 GeV are the following:

(i) the dierence in the calculation of the Higgs VEV;

(ii) the two-loop electroweak corrections that are not (yet) implemented in SARAH;

(iii) the momentum dependence currently missing at two loops in SARAH.

The dierent value of the Higgs VEV is also quite certainly the main reason for the discrepancies between the values we obtain using SMH and those from [START_REF] Buttazzo | Investigating the near-criticality of the Higgs boson[END_REF]. I.e. it is because we use the VEV computed in SPheno in SMH, which does not correspond to the same accuracy of parameter extraction as used in [START_REF] Buttazzo | Investigating the near-criticality of the Higgs boson[END_REF], which would be required for a fair comparison directly between the two prior approaches: here our aim was to compare our result separately with [START_REF] Buttazzo | Investigating the near-criticality of the Higgs boson[END_REF] and SMH.

A further way to compare our results to those of [START_REF] Martin | Higgs boson mass in the Standard Model at two-loop order and beyond[END_REF] and [START_REF] Buttazzo | Investigating the near-criticality of the Higgs boson[END_REF] We observe that the change of λ between each order of the perturbation expansion is approximately the same in all four methods. Moreover, the value we extract at two loops with SPheno is very close to the value found in [START_REF] Buttazzo | Investigating the near-criticality of the Higgs boson[END_REF], only diering by 0.1%.

A detailed comparative study of SPheno and SMH results

After this rst comparison, we may now investigate in more depth the eects of the three sources of dierences on the Higgs masses listed above, using SPheno and SMH.

To begin with, we should consider the Higgs VEV and its calculation: in SMH, calculations are performed in the Landau gauge, while SPheno is by default set to use the

Standard Model

Feynman gauge, and while the Higgs mass should in principle be gauge independent, its vacuum expectation value is not, hence there is an inconsistency coming from the use of a Feynman gauge VEV in SMH. The easiest way to correct this is to switch the SPheno calculation to the Landau gauge we set in the code the gauge parameter ξ to a very small nite value to approach the limit of the Landau gauge (the current implementation gives a numerical divergence when ξ = 0) and then to use the new value of the Landau gauge VEV in SMH.

m 2 h = 2λv 2 + ∆ (1) M 2 h (m 2 h ) + ∆ (2) M 2 h (m 2 h ) (5.2.2)
where

∆ ( ) M 2 h (s) ≡ - 1 v ∂∆V ( ) ∂h h=G=G + =0 + Π ( ) hh (s) ≡ div Π ( ) hh (s) + ∆ ( ) M 2 h (0) + O(s), (5.2.3)
where div f (s) denotes all terms in f (s) that diverge as s → 0. Our SPheno code computes the one-loop corrections in any R ξ gauge with full momentum dependence, but the two-loop corrections are performed in a generalised eective potential approach i.e. we keep only the divergent part of the momentum dependence (see section 4.3

for more details). The momentum in the two-loop routines is xed (for speed of calculation) whereas that in the one-loop routines is adjusted to solve the on-shell condition:

s = 2λv 2 + ∆ (1) M 2 h (s) + ∆ (2) M 2 h,SPheno (s), ∆ (2) M 2 h,SPheno (s) ≡ div Π (2)
hh,gaugeless (s) + ∆ (2) M 2 h,gaugeless (0).

(5.2.4)

This begs the question of how to compare our result with SMH: ideally, we would like to extract a result from SMH which is comparable to ours. However, this is confounded by several factors: 4 In practice, there is always an additional residual gauge dependence as the Higgs mass is computed to nite order in perturbation theory and as not all parameters used for to compute m h are determined to the same loop order.

Supersymmetric and non-supersymmetric models without catastrophic Goldstone bosons (i) It is impossible to remove or extract the electroweak contributions in SMH, because the individual contributions in the computation diverge as the electroweak gauge couplings become zero; and the total result is not nite at vanishing external momentum.

(ii) To avoid the Goldstone boson catastrophe and ensure cancellation between Goldstone boson and longitudinal gauge boson diagrams, in the two-loop corrections in SMH the external momentum s has been replaced by 2λv 2 wherever it appears in a pre-factor (but not in the arguments of the loop functions).

(iii) The term

∆ (1) M 2 h ⊃ 3λ 16π 2 (s 2 -4λ 2 v 4 ) B(0, 0) 2λv 2 , (5.2.5)
which is part of the one-loop correction coming from Goldstone bosons and longitudinal gauge bosons, is moved into the two-loop corrections, with the justication that on-shell s = 2λv 2 + ∆ (1) M 2 h so will give a contribution at two-loop order when solving for the on-shell mass.

If it were not for point (2) above, it would perhaps have been possible to extract the result for the generalised eective potential approximation for the electroweak corrections. Instead, we will simply compare the results as we vary the momentum in SMH; by modifying slightly the source code, we obtain a version of SMH without the momentum dependence at two loops (but retaining the dependence at one loop).

Interestingly, the result of SMH is nite even when s = 0 meaning that the divergence as s → 0 has been removed. It turns out that this is because of the term (5.2.5), which has the eect of cancelling the divergences as s → 0 (even though this cancellation is ctitious). If we write δ (2) (s) for the missing momentum dependence in SMH from setting the coecients of loop functions equal to 2λv 2 , then we have

∆ (2) M 2 h,SMH (s) = 6λ 16π 2 ∆ (1) M 2 h (s) B(0, 0) + ∆ (2) M 2 h,gaugeless (s) + ∆ (2) M 2 h,electroweak (s) + δ (2) (s) = 6λ 16π 2 - 6λ 2 v 2 16π 2 B(0, 0) + ∆ (1) M 2 h,SMH (0) B(0, 0) + div Π (2)
hh,gaugeless (s) + Π

hh,electroweak (s) + δ (2) (s)

+ ∆ (2) M 2 h,gaugeless (0) + ∆ (2) M 2 h,electroweak (0) + δ(2) (0) + O(s), = 12λ 16π 2 ∆ (1) M 2 h (0) + ∆ (2) M 2 h,gaugeless (0) + ∆ (2) M 2 h,electroweak (0)
+ δ(2) (0) + O(s).

The cancellations of the divergences imply that div Π

hh,gaugeless (s) + Π

hh,electroweak (s) + δ (2) (s)

? = 1 (16π 2 ) 2 36λ 2 v 2 log 2 (-s) -72λ 2 v 2 log(-s) + 6λ 16π 2 ∆ (1) M 2 h (0) log(-s), (5.2.6)
where

(16π 2 )∆ (1) M 2 h (0) ≡ (16π 2 )∆ (1) M 2 h,SMH (0) -12λ 2 v 2 = -12λ 2 v 2 + 18λ 2 v 2 log(m 2 h ) -12y 2 t m 2 t log(m 2 t ) + g 2 Y + g 2 2 2 m 2 Z 3 log m 2 Z + 2 + g 2 2 m 2 W 3 log m 2 W + 2 .
(5.2.7)

On the other hand, by evaluating the diagrams for the Standard Model in the gaugeless limit retaining only the top Yukawa coupling and the Higgs quartic λ we nd div Π

(2)

hh,gaugeless (s) = 6λv 2 (16π 2 )
hh,gaugeless (0). We can then determine

∆ (2) M 2 h,electroweak (0) + δ (2) (0) = ∆ (2) M 2 h,SMH (0) -∆ (2) M 2 h,SPheno (-Q 2 ) - 12λ 16π 2 ∆ (1) M 2 h (0) . (5.2.9)
We nd that this residual dierence is tiny; at Q = m t = 173.34 GeV with the Higgs quartic λ = 0.12604, the top Yukawa y t = 0.9345, the gauge couplings (g 3 , g 2 , g Y ) = (1.1654, 0.6442, 0.2782), and the VEV v = 247.07 GeV we have:

∆ (2) M 2 h,electroweak (0) + δ (2) (0) -0.03(GeV) 2 = -0.0002%m 2 h ! (5.2.10)
This corresponds to a tiny value of the electroweak corrections; a similar observation was made in [START_REF] Buttazzo | Investigating the near-criticality of the Higgs boson[END_REF].

Finally, we compare the more physically meaningful dierences between the codes when we take s = m 2 h | tree in our routines. The values of the Higgs mass computed with SPheno after turning o the light SM fermion contributions and with the modied version of SMH is given in table 5.3, and strikingly they only dier by 40 MeV when we include the momentum dependence in SMH in other words, for Q = 173.34

GeV, the momentum dependence and electroweak corrections amount to only 0.03% of m h . We further examine the importance of both the momentum dependence and EW corrections by varying now the renormalisation scale at which we compute the Higgs mass: for this purpose, gure 5.1 shows the dierence of the two-loop masses between the two codes more precisely (m 2 h ) SPheno -(m 2 h ) SMH with and without momentum, as a function of the renormalisation scale Q (where the MS parameters are extracted by SPheno at each value while keeping λ xed rather than evolving the parameters: the idea is to show the importance of the choice of scale rather than the stability of the computation). While for large scales the two-loop momentum eects may become large (1 GeV or more), the electroweak corrections represent at most 0.2 GeV and even vanish for a scale close to the MS top mass. with s = m 2 h | tree , while in SMH the full two-loop dependence is implemented and is used to nd m h iteratively.

Momentum dependence

Implementing the solution to the Goldstone boson catastrophe in SARAH has required the insertion of external momentum in infra-red divergent loop integrals, and thus we should also investigate the impact of the momentum s = -p 2 on the Higgs mass calculation in SPheno. In practise, we have set for the majority of scans the external momentum for the two-loop calculations to be equal to m 2 h | tree but we will now vary the momentum to study its impact on m h . Table 5.4 shows the shift to the two-loop Higgs mass with respect to the value computed with s = (125 GeV) 2 for external momentum in loops equal to s = α × (125 GeV) 2 , where α ranges from 10 -6 to 10 6 and for λ = 0.126 and λ = 0.130. For all values of the external momentum considered here, the variation of the Higgs mass remains small: at most they become of order ∼ 0.13 GeV for α = 10 -6 (i.e.

√ s = 0.125 GeV), and while this eect is noticeable, it is far from the divergences that could have been feared when approaching the limit of s → 0. All in all, although pole masses as we compute here are in principle found as the zero of the inverse propagator, that has to be found iteratively as the self-energy contains momentum dependence, we see from the minute eects of momentum in the range α ∈ [1/2, 100], relevant for scalar masses, that we will not require an iterative solution and that simply taking s = (125 GeV) 2 in the loop diagrams with pseudoscalars will be a satisfactory approximation. In particular, changing s between m 2 h | tree and 125 GeV causes a dierence in m 2 h of less than an MeV.

We emphasise however that the eect of momentum on Goldstone boson mass diagrams discussed here is only a subset of the general momentum dependence of the twoloop masses, which should in principle be taken into account, as seen in the previous subsections.

The NMSSM

The NMSSM

As a second check of our new solution, and demonstration of its importance, we shall compare the results for the three dierent options to solve the Goldstone Boson Catastrophe in the example of the Next-to-minimal supersymmetric standard model (NMSSM) see [START_REF] Ellwanger | The Next-to-Minimal Supersymmetric Standard Model[END_REF] and references therein for a detailed description of the model, or section 1.3.4.5 for a brief introduction. Indeed, the NMSSM is the rst supersymmetric model for which the problems at certain points in the parameter space were found in earlier versions of SARAH. Here we shall show that this is avoided, and have a preliminary look at the impact of the consistent tadpole solutions.

We start with a test point dened by the following input parameters:

λ S = 0.7, κ = 0.25, A λ S = 1350 GeV, A κ = -500 GeV, µ eff = 600 GeV, and all other soft-masses set to 2 TeV. Table 5.5 gives the results for the Higgs masses obtained with the following calculations:

M 1 = M 2 = 1000 GeV, M 3 = 2000 GeV, D R = 10 -5 R = 10 -4 R = 10 -3 R = 10 -2 R = 10
(i) D-terms turned o in mass matrices but retained in tadpole solutions (as in previous versions of SARAH), labelled D in the table.

(ii) Regulator masses with R = 10 -5 10 -1 .

(iii) Goldstones set on shell, with and without consistent tadpole solutions, labelled OS and OS+Tad respectively.

We see from this table that there is an agreement in the light Higgs mass of about 0.4 GeV between all the calculations if R is chosen to be about 10 -2 .

While the new on-shell solution of the Goldstone boson catastrophe is optimal, between introducing a regulator R and the previous approach with neglected D-terms in the scalar mass matrix, the latter is preferred because one does not need to check for a suitable choice of R to stabilise the results. However, we can now consider parameter points where the old method fails. This is shown for the point dened by and all scalar soft-masses set to 2 TeV. The lightest scalar tree-level mass with and without the D-terms as function of λ S is shown in gure 5.2. One can see that for λ S 0.5, 0.8, the lightest scalar becomes massless in the limit of vanishing D-terms.

Thus, for these values, divergences in the two-loop corrections can be expected which are this time not associated with the Goldstone but with the lightest CP even state. We show the lightest Higgs mass in gure 5.3 as function of λ S for dierent methods to regulate the two-loop corrections. Obviously, the approach of neglecting electroweak D-terms fails for values of λ S at which the masses entering the loop calculations become very light. However, for very large values of λ S which are away from the poles, the agreement with the other calculations is also rather poor. In contrast, over the entire range of λ S we see a good agreement between the methods using regulator masses, if R = 10 -2 or 10 -3 is chosen, and the method of treating the Goldstones on-shell. It is interesting that for these values of R the minimum mass is √ R × M SUSY 100 GeV, i.e. logarithmic contributions involving the light scalars are being excised.

We note that the corrections from the consistent tadpole solution are small until λ S becomes large, at which point we see signicant deviations. However, as λ S approaches 0.9 we see from gure 5.2 that the tree-level lightest Higgs mass approaches zero, so we expect our perturbative calculation of the consistent tadpole solution to break down and become unreliable.

Split SUSY

In Split SUSY scenarios [START_REF] Bernal | The MSSM with heavy scalars[END_REF][START_REF] Giudice | Probing High-Scale and Split Supersymmetry with Higgs Mass Measurements[END_REF]304307], the SUSY scalars are much heavier than the gauginos and Higgsinos. Consequently, these models should be studied in an eective approach where all SUSY scalars are integrated out at some matching scale. The literature before. They are expected to be small since they originate from electroweak interactions at the matching scale (and so, admittedly, one could argue that we should neglect them in the gaugeless limit). We shall not discuss the absolute value of the Higgs mass, for which we would need to include all higher-order corrections to the matching that have been calculated elsewhere, but only on the impact of the new twoloop corrections. The overall size of these corrections is rather insensitive to the exact matching conditions and we will be using the above tree-level relations; but as we noted earlier, these should be a particularly good approximation for larger matching scales.

We make in addition the simplifying assumption that at M M the SUSY fermions are degenerate, i.e.

µ(M

M ) = M 1 (M M ) = M 2 (M M ) = M 3 (M M ) ≡ M F (5.4.8)
and thus we are left with three free parameters:

M F , M M , tan β.

SARAH uses two-loop RGEs for the running between M M and the renormalisation scale Q that we set to M F , which as mentioned above is necessary to avoid large logarithmic (5.5.11)

Using HiggsBounds we have veried that this point in parameter space is not excluded by the current experimental constraints.

At rst we only take into account the running of SM parameters, and we then consider that the above input parameters (i.e. those in eq. (5.5.11)) are given to SPheno as the values of the couplings at the scale Q at which the Higgs mass is computed, which we vary in the range [100 GeV, 10 000 GeV]. We nd the results shown in gure 5.5 for the tree-level, one-loop and two-loop Higgs mass m h . Since phenomenological analyses typically supply the quartic couplings without reference to a higher-energy theory or the scale where they are determined, this plot shows the importance of the choice of that scale.

We have veried that the renormalisation scale dependence of m h | tree (that is computed in terms of the parameters of the 2HDM scalar sector, i.e. the parameters in eq. (5.5.11) and v) is entirely due to the scale dependence of the Higgs VEV v,5 as the running of the quartic couplings is for the moment not applied. The renormalisation scale Q is seen to have only a limited eect on the two-loop value of m h which varies of about 2

GeV on the range of scales considered here, while the one-loop result varies by about 15 GeV. Since the two-loop curve is so at, this shows that most of the variation in with the RGEs included in SPheno. The dierence between this gure and gure 5.5 is that here we do not consider the inputs given in equation (5.5.11) to be taken at the scale Q at which we compute m h but at the scale m t , and then we evolve them from m t to Q. Red curve: tree-level; Blue dot-dashed curve: one-loop order; Green dashed curve: two-loop order.

the calculation of Higgs mass for the chosen quartics must come from variation of the Standard Model parameters, and that a two-loop calculation (rather than one-loop) is necessary not just for precision but also to ensure scale stability.

Using the two-loop RGEs implemented in SARAH/SPheno, we can also include the evolution of the 2HDM parameters to obtain a more complete scale dependence of the masses, as shown in gure 5.6. To be more specic, here we do not consider the inputs in eq. (5.5.11) to be given at the scale Q at which we compute m h , but instead we run these values from m t to Q.

Once more, the two-loop value of Higgs mass depends less on the renormalisation scale than the tree-level or one-loop values. This smaller dependence of the two-loop Higgs mass on Q, compared with the one-loop mass, even for choices of parameters that give large loop corrections is a rst verication of the validity of our new two-loop routines.

In the following we will therefore work at a xed scale Q = m t , condent that the results will be for the most part independent of this choice.

Quantum corrections to the alignment limit

The relations dening the alignment limit, in section 5.5.1, are only valid at tree level and we expect them to receive corrections at one-and two-loop order, and in this section we will discuss the importance of these eects on the mixing angle of the neutral CP-even scalars α.

Scanning over the dierent free couplings of the model m 2 12 and λ i (i ∈ {3, 4, 5}) we compare the values of the CP-even Higgs mixing angle α at tree level, one-loop and two-loop order, as shown in gure 5.7, and as expected, loop corrections cause Supersymmetric and non-supersymmetric models without catastrophic Goldstone bosons we choose one parameter as the abscissa; the tree-level alignment condition λ 1 = λ 2 = 1/2λ 345 plus the requirement that the Higgs mass is 125.09 GeV xes three parameters, namely λ 1 , λ 2 and either λ 4 for the bottom right plot or λ 5 for the other three; the remaining parameters are held xed at values λ 3 = 0.5, λ 4 = 0.5, m 2 12 = -1000 GeV 2 (when they are not otherwise varying). All plots are for tan β = 50. Red curve: tree-level; Blue dot-dashed curve: one-loop order; Green dashed curve: two-loop order. deviations from the tree-level relation t α = -1/t β ⇔ c β-α = 0. The observations we can make from these plots are the following:

(i) in the ranges of parameters that we considered, the eect of loop corrections on the value of α is small, at most of the order of 1%;

(ii) the one-loop corrections to α show very little dependence on the quartic couplings λ i=3,4,5 ;

(iii) it appears that for most parameter points, the two-loop corrections to α are of similar magnitude than the one-loop ones although somewhat smaller when

|λ i | 1.
(iv) for some parameter points however, the two-loop corrections to α become significantly larger than the one-loop corrections, see the lower right plot in gure 5.7.

We have veried that this happens when one of the quartic couplings λ i becomes large (typically |λ i | 1) in the plot mentioned above of -1/t α as a function of λ 5 it is λ 4 that becomes smaller than -1 . We may suspect the large two-loop eects are due to a loss of perturbativity: this will be discussed in more detail in the next section.

5.5 Two-Higgs-Doublet Model

Perturbativity constraints

It is common in practice to use the physical scalar masses, the Z 2 breaking parameter m 12 as well as the the angles α, β as input for the 2HDM in numerical studies. However, this input often hides that it corresponds to huge quartic couplings which spoil unitarity and the perturbative behaviour of the theory. Therefore, the constraints that all quartic couplings must be smaller than 4π as well as the tree-level unitarity constraints [310312] are applied to sort such points out. However, it was already shown in the SM that the limit of λ < 4π might be too weak [START_REF] Nierste | Higgs sector renormalization group in the MS and OMS scheme: The Breakdown of perturbation theory for a heavy Higgs[END_REF].

We now have all the machinery at hand to impose another constraint on the 2HDM model namely that the radiative corrections to the Higgs mass converge. We show here in one example that this can be a much stronger constraint than tree-level unitarity, while a more detailed analysis of this constraint on the parameter space of 2HDM models is left for future work.

We consider here a point for typeII dened by 6

m H = 593.6 GeV, m A = 535.2 GeV, m H + = 573.2 GeV, m 2 12 = -165675 GeV 2
, tan α = -0.235 , tan β = 1.017 .

(5.5.12)

Since the masses are treated as pole-masses and only tree-level relations are used in the above work, no scale for the MS parameters is given. On the other side, it is usually checked that the translation of these masses into quartic couplings results in parameters which are allowed by tree-level unitarity. However, this treatment implicitly assumes that one can dene at each loop level suitable counter-terms to renormalise the Higgs sector in a way that the masses can be kept constant, and that this renormalisation converges. This is however not the case for the parameter point dened by eq. (5.5.12)

as one can see as follows. Using the tree-level relations (1. To check the perturbative behaviour, we show the scale dependence in gure 5.8. Here, we used the values of the quartic couplings in eq. (5.5.13) as inputs computed from eq. (5.5.12) at scale m t , and evolved to the scale Q at which we compute the Higgs mass and checked the scale dependence of the Higgs mass at dierent loop levels.

For the evaluation of couplings at the considered scale, we used the two-loop RGEs calculated by SARAH. One sees that the scale dependence increases with increasing loop-level. Of course, one might wonder if this is just an eect from our choice to dene the quartic couplings at Q = m t as input. Therefore, we show in gure 5.9 the size of the loop corrections for dierent choices of our input scale Q. We see that the size of the loop corrections rapidly increases for Q > m t and the spread between oneand two-loop becomes even larger. Also choosing Q 160 GeV where the mass at one-and two-loop level seem to be roughly identical does not solve the problem: this is just a numerical coincidence and the scale dependence at two loops is even larger than at one loop.

Supersymmetric and non-supersymmetric models without catastrophic Goldstone bosons

Georgi-Machacek Model

Finally, we turn to the Georgi-Machacek model, described in section 1.4.3, to investigate the impact of the choice of inputs at dierent orders in perturbation theory. The free Lagrangian parameters in the Higgs sector, left after solving the tadpole equations, are the quartic couplings λ i (i ∈ {1, 2, 3, 4, 5}), the mass parameters M 1 , M 2 and the ratio of VEVs s H , but one can always use dierent inputs.

In particular, m h , s H , m 5 seem to be a suitable and popular [START_REF] De Florian | Handbook of LHC Higgs Cross Sections: 4. Deciphering the Nature of the Higgs Sector[END_REF] choice for the input parameters and can be traded for λ 1 , λ 5 and v T . In the following we shall do this using tree-level relations derived from those presented in section 1.4.3 (or [START_REF] De Florian | Handbook of LHC Higgs Cross Sections: 4. Deciphering the Nature of the Higgs Sector[END_REF]).

However, the choice to use masses instead of couplings as input can have the danger that one enters a non-perturbative regime without recognising it, as we already have pointed out for the 2HDM. We will discuss the importance of higher-order corrections in general in this model in the following: in contrast for instance to the 2HDM, it is not possible to renormalise all mixing angles and masses on-shell in this model. One reason for this is that the masses of the ve-plet are only exactly degenerate at tree- The scalar masses at tree-and loop-level for the parameter point λ 2 = λ 3 = λ 4 = 0, m 5 = 1 TeV and s H = 0.75. The renormalisation scale was set to m 5 .

level but the custodial symmetry is not protected against loop eects [START_REF] Blasi | Eects of custodial symmetry breaking in the Georgi-Machacek model at high energies[END_REF]. Therefore, the number of mass parameters but also of rotation angles is extended at the loop level: one needs three instead of two angles to diagonalise the loop-corrected CP-even mass matrix, and also the CP-odd and charged Higgs mass matrix no longer share the same angle. Therefore, an MS renormalisation of the scalar sector is the natural option to check the impact of higher order corrections to the masses and angles. We give in Tab. 5.6 the loop corrected masses for all scalars for the parameter point λ 2 = λ 3 = λ 4 = 0, m 5 = 1 TeV, and s H = 0.75.

We see in these numbers that not only a mass splitting between the components of the veplet and triplet is induced at the one-loop level, but also that the loop corrections to the SM-like Higgs scalar can be huge. One can understand these large loop corrections for the chosen parameter point to some extent analytically: the one-loop corrections to the (1, 1)-element of the CP even mass matrix are given in the eective potential in the limit m 5 v by

∆m 2 h ∼ v 2 8m 4 5 s 4 H 9π 2 v 4 .
(5.6.1)

Thus, for large values of m 5 and/or s H one can expect huge corrections to the mass.

Note, there are additional loop corrections to the o-diagonal elements of the scalar mass matrix which can have a signicant impact on the masses. Therefore, one needs a full numerical calculation already at the one-loop level to obtain an accurate number for the SM-like Higgs mass.

Before we further investigate the loop corrections, we want to comment briey on the choice for the renormalisation scale Q. In the SM, but also in other models like 2HDMs, it is suitable to set Q = m t to give a good convergence and ensure that there are no large logarithmic contributions from top loops. However, in the GM model the dominant loop corrections involve often scalar elds with masses of the order of m 5 . Therefore, the overall size of the loop corrections is usually smaller for Q = m 5 as one can see in gure 5.10.

We check now the Higgs mass in the (m 5 , s H ) plane proposed in [START_REF] De Florian | Handbook of LHC Higgs Cross Sections: 4. Deciphering the Nature of the Higgs Sector[END_REF] always using Q = m 5 . The other parameters are xed in this plane to We see that in the upper right corner in the (s H , m 5 ) plane the two-loop corrections are much larger than the one-loop ones and the Higgs can even become tachyonic.

m tree h = 125 GeV , M 1 = √ 2 s H v (m 2 5 + v 2 ) , M 2 = 1 6 M 1 λ 3 = -0.1 , λ 2 = 0.
For m 5 = 1 TeV, this already happens at s H > 0.5, while for m 5 = 1.5 TeV the upper limit of s H is as low as 0.25. For large m 5 this limit is much stronger than the one from perturbative unitarity of V V → V V scattering amplitudes which gives s H < 667 GeV m 5 [START_REF] Logan | All the generalized Georgi-Machacek models[END_REF]. Thus, even if it might still be possible to obtain the correct Higgs mass at two-loop level by adjusting the other input parameters or by absorbing nite corrections into counter-terms, the results in this parameter region should be taken with a lot of care. Most likely, they are meaningless. However, also for the other parameter regions with a reasonable hierarchy of the one-and two-loop corrections, one would need large adjustments in the input parameters to compensate for these loop corrections. These changes would then reect in the couplings and some decay widths of the 125-GeV scalar will deviate for large s H and/or m 5 clearly from the tree-level expectation. Finally, one can also see in gure 5.12 that the loop corrections to the masses of other scalars are sizeable and can shift the masses easily by tens to hundreds of GeV.

Conclusions

In this chapter we have presented several varied results relating to the calculation of two-loop corrections to the Higgs mass in general models. Chief among these are: (ii) We extended the derivation of shifts to the tadpoles and Higgs mass from consistently solving the tadpole equations by allowing fermion masses to be directly dependent on the parameters (such as µ in the MSSM), with the expressions given in appendix C.

(iii) We compared our results with those available for the Standard Model. In particular, this allowed a comparison within the same code of calculations in two dierent gauges, and we also found that the electroweak corrections are negligible, while those from momentum dependence are very small.

(iv) We showed that our new computation does indeed remove the instabilities (sharp peaks in the Higgs mass for certain parameter choices) in the previous approach for supersymmetric models; however, the reader should be aware that there are still some limitations when scalar masses in the loops become small compared to the renormalisation scale.

(v) We explored the corrections to the mixing angle in the alignment limit in the Two Higgs Doublet Model using the MS couplings as inputs, and found that provided the quartic couplings are chosen to be small, the loop corrections are safely under control.

Supersymmetric and non-supersymmetric models without catastrophic Goldstone bosons plane for the second (rst row) and third (second row) CP-even scalar.

(vi) We explored the 2HDM and Georgi-Machacek models with masses as physical inputs and using tree-level relations to obtain MS couplings, as commonly done in the literature. We nd that in most regions of the parameter space these lead to large quartic couplings, which rapidly lead to loss of control of the loop corrections. Perhaps surprisingly, this often occurs well before the couplings reach naive perturbativity bounds. In the next chapter we will continue investigating the use of masses (and mixing angles) as inputs, but we will go beyond tree-level relations and perform the extraction of the Lagrangian parameters at one-and two-loop orders.

All of the shown results are available to the community with SARAH version 4.12.0, and can hopefully contribute to an ecient and more precise study of many extensions of the SM; this should open the avenue to much future work. It would be particularly interesting to explore more carefully the relationship between on-shell and MS calculations in non-supersymmetric models, to better understand how the divergent behaviour of the masses that we observe for the MS scheme translates into dierences in physical couplings or even possibly ruling out certain parameter regions of models as unphysical. Note that this programme was continued in [START_REF] Krauss | Perturbativity Constraints in BSM Models[END_REF], where the renormalisation of the Georgi-Machacek model and the possible perturbativity constraints in that model were studied.

Chapter 6

Matching and running

In this last chapter, we investigate the high-scale behaviour of Higgs sectors beyond the Standard Model and in particular the importance of performing a proper matching of couplings before applying the renormalisation group equations.

Additional states in models with extended Higgs sectors alter the high-scale behaviour of the theory compared to the SM expectations. For instance, it is known today that the SM becomes metastable if it is extrapolated to very high energies [START_REF] Degrassi | Higgs mass and vacuum stability in the Standard Model at NNLO[END_REF][START_REF] Buttazzo | Investigating the near-criticality of the Higgs boson[END_REF][START_REF] Andreassen | Scale Invariant Instantons and the Complete Lifetime of the Standard Model[END_REF][START_REF] Spencer-Smith | Higgs Vacuum Stability in a Mass-Dependent Renormalisation Scheme[END_REF]: at a scale of 10 Many of these eects have already been studied in the literature for plethora of dierent models such as singlet extensions [START_REF] Lerner | Gauge singlet scalar as inaton and thermal relic dark matter[END_REF]126128,[START_REF] Pruna | Higgs singlet extension parameter space in the light of the LHC discovery[END_REF], triplet extensions [START_REF] Hamada | Landau pole in the Standard Model with weakly interacting scalar elds[END_REF][START_REF] Khan | Exploring Hyperchargeless Higgs Triplet Model up to the Planck Scale[END_REF], Two-Higgs-Doublet-Models (2HDMs) [323331] or models with vector-like fermions [START_REF] Xiao | Stabilizing electroweak vacuum in a vectorlike fermion model[END_REF].

These studies utilise the one-and sometimes even two-loop renormalisation group equations (RGEs). However, less care was was taken in the determination of the parameters which enter the RGE running. Often, two-loop RGEs were combined with a tree-level matching.

A proper determination including higher-order corrections of the quartic coupling, which enters the RGE running, was so far only performed for the SM [START_REF] Buttazzo | Investigating the near-criticality of the Higgs boson[END_REF]. 1 It was shown that even the two-loop shifts to λ are important for determining the fate of the model. This is remarkable, because it is well known that the loop corrections to the Higgs mass are small if they are calculated at Q = m t : the corresponding shifts in λ are only 2.5%. While the corrections from top quarks are of a similar order in many BSM models, other corrections like the ones from Higgs self-interactions can be much larger, as we found for example in the previous chapter for 2HDMs and the Georgi-1 Loop corrections in the scalar sector were taken into account in [START_REF] Kanemura | Radiative corrections to the Higgs boson couplings in the model with an additional real singlet scalar eld[END_REF] for a singlet extension and in [START_REF] Basler | The CP-Violating 2HDM in Light of a Strong First Order Electroweak Phase Transition and Implications for Higgs Pair Production[END_REF][START_REF] Basler | Strong First Order Electroweak Phase Transition in the CP-Conserving 2HDM Revisited[END_REF] for a 2HDM. These studies did not however investigate the impact on the high-scale behaviour of the model. In ref. [START_REF] Kobakhidze | Neutrino Masses and Higgs Vacuum Stability[END_REF] in turn, a one-loop matching has been performed for that purpose in the context of a seesaw-II as well as a left-right symmetric model.

6.1 Matching and Running

Matching

The renormalised coupling constants Θ i in d = 4 -2 dimensions, which enter the running, are related to the corresponding bare couplings Θ 0 i by

Θ 0 i µ -C i = Θ i + n a (n) i n . (6.1.3)
Here, C i are constant factors depending on the character of Θ i . 2 The coecients a i are the result of a perturbative expansion. In general, two approaches are possible to determine the Lagrangian parameters as function of physical observables such as masses:

(i) In an on-shell calculation the physical observables are identical at each loop-level, but all nite and innite corrections are absorbed into the counter-terms of the Lagrangian parameters (δΘ OS i ).

(ii) In an MS calculation the counter-terms of the Lagrangian parameters (δΘ MS i )

contain only the divergences. Therefore, the calculated masses depend on the loop-level at which the calculation is performed.

The bare Lagrangian parameters are identical in both cases

Θ 0 i = Θ OS i -δΘ OS i = Θ i (µ) -δΘ MS i . (6.1.4)
In an on-shell calculation, however, there is no generic set of renormalisation group equations known, and therefore to explore a theory at high energies it is necessary to use MS equations i.e. to extract the underlying MS parameters of the theory, and then run them. On the other hand, in an MS calculation, the physical parameters are functions of the Lagrangian parameters: so if we are given the physical quantities, we must invert these functions to extract the MS ones. This is where complications appear, and why many studies in the literature resort to simply using tree-level matching.

For example, suppose that we want to extract the quartic coupling of the SM from the Higgs mass (see eq. (1.1.5) for our denition of the Higgs potential). The Higgs mass M h is, however, calculated in terms of the underlying Lagrangian parameters as a loop expansion via the on-shell condition

M 2 h = 2λv 2 + ∞ n=1 1 (16π 2 ) n ∆ (n) M 2 h (λ) . (6.1.5)
This is in general a highly non-linear equation in λ, but fortunately since the series is perturbative we can solve it through expanding which leads to λ S < 4π/3 for small λ and λ SH . Although a seemingly weak constraint at rst sight, this can become a severe constraint particularly in the case of small v S , cf. eq. (1.4.9).

λ = λ (0) + 1 16π 2 δ (1) λ + 1 (16π 2 ) 2 δ (2) λ + . . . , (6.1.6) to nd λ (0) = M 2 h 2v 2 , δ (1) λ = - 1 2v 2 ∆ (1) M 2 h | λ=λ (0) , (6.1.7) δ (2) λ = - 1 2v 2 ∆ (1) λ ∂ ∂λ ∆ (1) M 2 h + ∆ (2) M 2 h λ=λ (0)
(

n, m) λ λ S λ SH Λ 4π [GeV] Λ unit. 4π [GeV]
(T, 1) In the following, we compare dierent approaches for the matching of the quartic couplings. In Fig. 6.1, we show the values of the quartic couplings at the scale 10 6 GeV as a function of the scale Q where the matching is performed. This is done by rst running the SM RGEs to the scale Q where we then match the quartic couplings to the spectrum at tree-level, one-loop as well as two-loop. The nal step is the running of the singlet-extended SM RGEs (both at one-or two-loop order) up to 10 6 GeV. On the left-hand plots, we use one-loop RGEs for the cases of tree-level matching (dotted) and one-loop matching (dashed). Although there is no dependence of the quartics on the matching scale when using tree-level matching, the scale dependence induced by the RGEs is larger than for the case of one-loop matching and one-loop RGEs. On the right-hand planes, we show the quartics at 10 6 GeV, evaluated with two-loop RGEs, when using one-loop (dashed) and two-loop matching (solid lines). Once again, the scale dependence is decreased when the highest available order of matching is employed (in this case, at two loops).

Also visible in Fig. 6.1 are the large dierences between the eventual coupling values when using the traditional approach of tree-level matching with one-loop RGEs and the approach of including all available corrections to the matching and RGE running (here two loops for both). This also means that large dierences are expected when evaluating the cut-o scale of a theory, i.e. the scale at which the model becomes nonperturbative or violates unitarity. In Tab. 6.1, we show the cut-o scale of a particular parameter point when using n-loop matching in conjunction with m-loop RGEs. We

show the scale at which the quartics become non-perturbative (Λ 4π ) separately from the case where either perturbativity or unitarity is violated (Λ unit. 4π ). The corresponding running of the individual couplings is displayed in Fig. 6.2. Note that we only display the cases (n, m) = (T, 1), (1, 1), (1, 2), and (2, 2) in this gure as dotted, dot-dashed, dashed and full lines corresponding to the comparison of N versus N -1 matching.

The impact of the two-loop RGEs is a moderation of the one-loop RGEs: while the one-loop β-function of λ S is given by β

λ S = 1 16π 2 (36λ 2 S + λ 2 SH ), so that λ S tends to grow very rapidly, there is a moderating term from the two-loop RGEs which goes with 1 (16π 2 ) 2 (-816λ 3 S -20λ S λ 2 SH ). Therefore, using the one-loop RGEs only, λ S grows large very quickly whereas the unitarity limit is reached at a much later scale when using two-loop RGEs. Nevertheless, a complete stalling of the evolution is typically only reached at λ S values which already violate the unitarity limit according to Eq. (6.2.4), see the black dashed (full) line between 10 9 and 10

Q [GeV] 10 -1 10 0 10 1 |λ i | λ S λ SH λ (2, 2) (1, 2) (1, 1) (T, 1) Figure 6.2
The running of the quartic couplings for the point given in Tab. 6.1. The line-styles refer to the loop order of the matching and RGE running as described in Tab. 6.1, namely (n, m) refers to the matching at n-loop order with m-loop RGEs. The solid red line is the 4π perturbativity limit, while the dashed-red line is the unitarity constraint of 4π/3 obtained from eq. (6.2.4) in the limit λ S λ SH , λ.

moderation of the evolution of λ SH and λ is not as pronounced. For λ SH , the corresponding β-function grows with 1 16π 2 12λ S λ SH with only a small moderating eect from the two-loop RGEs. As a consequence, it becomes larger than 4 π before λ S and then drags the latter with it. To summarise, in particular because of the large two-loop contributions to β λ S , there can be several orders of magnitude between the eventual cut-o scales when using one-or two-loop RGEs. The eect of using a higher-order (two-loop) matching instead of a tree-matching, in turn, is a reduction of the quartic couplings. The reason is the positive mass corrections to m H , leading to smaller MS couplings when doing the proper loop-level matching.

As shown here, the impact can be large and we observe positive shifts in the eventual cut-o scale by several orders of magnitude when including the matching.

Finally in Fig. 6.3, we show in the m H -v S plane the dierences between using N -1loop and N -loop matching when applying N -loop RGE running. The cut-o scale here and in what follows is dened as the scale at which either one of the couplings grows larger than 4π or any of the conditions for perturbative unitarity are violated, each evaluated with the running MS quartic couplings. The grey contours in the lefthand pane of Fig. 6.3 display the ratio of the evaluated cut-o scales for N = 1. In particular for small values of v S , which lead to large quartic couplings, the eects are quite drastic as loop eects become very important. The dierences between one-and Matching and running two-loop matching (shown as blue coloured contours) are signicantly milder in this region, the maximum dierence is just a factor of three. For large v S , instead, the quartic couplings are comparably small, leading to large cut-o scales in general. This also means, however, that during the long RGE running, small shifts in couplings can lead to more drastic eects as is seen in the upper region in the plot with v S 350 GeV. However, the cut-o scale dierences stay below an order of magnitude for N = 2. We show the ratio of the obtained cut-o given matching at N versus N -1 order using the RGEs at N -loop order. The coloured(grey) contours use the two(one)-loop RGEs, therefore showing the ratio of the matching at two(one)-loop versus one-loop(tree-level), respectively. Right: Ratio of the calculation performed using both matching and RGEs at two-loop order versus the leading order (treelevel matching and one-loop RGEs). The grey contours correspond to the ratios of the quartic coupling λ S for these two scenarios. Here we have xed the physical parameters such that m h = 125 GeV, tan α = 0.2, while the remaining parameters are chosen as κ 1 = 0 GeV and κ 2 = 1000 GeV.

On the right-hand side of Fig. 6.3 we present the dierence in cut-o scales between the most extreme cases, tree-level matching using one-loop RGEs versus two-loop matching using two-loop RGEs. In particular for small values of the singlet VEV, the eventual cut-o scale can be many orders of magnitude larger than the cut-o scale evaluated with tree-level matching. In grey contours we show the ratios of the singlet quartic couplings at Q = m t between the two matching approaches, λ

S /λ (T ) S . Already at the matching scale, dierences of an order of magnitude between tree and two-loop matching can appear, emphasising the requirement for proper matching and running when analysing the high-scale behaviour of a given model.

Singlet Extension with an additional Z 2 symmetry 6.2.2.1 Analytical approximation

We may now make a further simplication to the singlet extension studied in Sec. 6.2.1, namely adding an additional Z 2 symmetry under which the singlet scalar is charged this is the Z 2 SSM described in section 1.4.1. This symmetry forbids non-zero values for the couplings κ 1 , κ 2 and for the singlet VEV v S , and furthermore eliminates mixing in the Higgs sector. Therefore, the derivation of analytic expressions for the radiative 6.2 Models and results corrections to the matching of the Higgs quartic coupling λ, and their comparison to numerical studies, are signicantly simpler, and follow the procedure outlined in section 6.1.2. Here we will be interested in the part of the corrections that come on top of the purely SM corrections due to the singlet scalar, and shall give expressions including two-loop contributions.

The one-and two-loop corrections to the Higgs mass in the SM are well-known and small; however, in our model there may be large corrections from the singlet scalar.

In order to extract the two-loop contributions via eq. (6.2.1) we require the two-loop mass correction, and also the derivatives of the one-loop part. However, our two-loop calculation is performed in the gaugeless limit in Feynman gauge, so we require the full one-loop Higgs mass correction in this limit [START_REF] Martin | Two loop scalar self energies in a general renormalizable theory at leading order in gauge couplings[END_REF][START_REF] Martin | Evaluation of two loop selfenergy basis integrals using dierential equations[END_REF]). This gives us

∆ (1) M 2 h (p 2 ) = 3y 2 t (4m 2 t -p 2 )B(m 2 t , m 2 t ) - 3 2 λ 2 v 2 B(0, 0) - 9 2 λ 2 v 2 B(m 2 h , m 2 h ) - 1 2 v 2 λ 2 SH B(m 2 S , m 2
λ (0) = M 2 h v 2 , δ (1
) λ = δ (1) SM λ + 1 2 λ 2 SH B(m 2 S , m 2 S ) , δ (2) λ = δ (2) SM λ - 1 2v 2 ∆ (2) Z 2 SSM M 2 h λ=λ (0) + 3 2 λ 2 SH λB(m 2 S , m 2 S ) 3B(m 2 h , m 2 h ) + 3m 2 h B((m 2 h ) , m 2 h ) + B(0, 0) . (6.2.6)
where we use the shorthand notation B(x , y) ≡ ∂ ∂x B(x, y). We note that the infrared divergent piece B(0, 0) will cancel against an equivalent piece from ∆

(2)

Z 2 SSM M 2 h ,
similarly to the eect noted in [START_REF] Martin | Higgs boson mass in the Standard Model at two-loop order and beyond[END_REF].

For the Z 2 SSM , we obtain δ (2) -

λ = δ (2) SM λ - 1 2 λ SH λ 3 SH v 2 M SSSSS (m 2 S , m 2 S , m 2 S , m 2 S , m 2 h ) + 6λλ 2 SH v 2 M SSSSS (m 2 h , m 2 S , m 2 h , m 2 S , m 2 S ) -6λλ SH U SSSS (m 2 h , m 2 h , m 2 S , m 2 S ) -4λ 2 SH U SSSS (m 2 S , m 2 S , m 2 h , m 2 S ) + 9λ 2 λ SH v 2 V SSSSS (m 2 h , m 2 h , m 2 h , m 2 S , m 2 S ) + 2λ 3 SH v 2 V SSSSS (m 2 S , m 2 S , m 2 S , m 2 h , m 2 S ) -λ 2 SH Y SSSS (m 2 S , m 2 S , m 2 S , m 2 h ) -9λ 2 Y SSSS (m 2 h , m 2 h , m 2 h , m 2 S ) -12λ SH λ S Y SSSS (m 2 S , m 2 S , m 2 S , m 2 S ) -6λ SH λ S Z SSSS (m 2 S , m 2 S , m 2 S , m 2 S ) -12λλ SH Z SSSS (m 2 h , m 2 h , m 2 S , m 2 S ) M S [GeV] λ SH λ tree λ 1 λ 2 200
1 2v 2 λ 2 SH S SSS (m 2 h , m 2 S , m 2 S ) -I(m 2 h , m 2 S , m 2 S ) + 9 2 λ 2 SH λm 2 h B(m 2 S , m 2 S )B((m 2 h ) , m 2 h ). (6.2.7)
This expression is valid for the gaugeless limit but with generic external momentum (so we can take the momentum in the loop integrals on-shell as the procedure demands, if we wish). However, if we take the generalised eective potential limit introduced in section 4.3 and employed in SARAH, then the penultimate line vanishes and the loop functions simplify considerably. We can then obtain a further simplied version of this expression by replacing m 2 h by its tree-level value λv 2 and by performing an expansion in powers of v 2 /m 2 S and keeping only the leading and sub-leading terms, giving

δ (2) λ δ (2) SM λ + 9 4v 2 λ SH λA(m 2 S ) + λ 3 SH 1 -2 log m 2 S + log 2 m 2 S + 1 4 λ 2 SH λ -18 -6 log 2 m 2 S + (36 log m 2 h -12) log m 2 S + 3λ 2 SH λ S -1 + log m 2 S + log 2 m 2 S .
(6.2.8)

Numerical study

Because the Z 2 symmetry forbids some couplings, the corrections to the matching conditions can be understood in terms of only three parameters added to the SM ones:

λ SH , M S , and (to a lesser extent) λ S . The eects of using loop-corrected matching and RGEs in the Z 2 SSM are similar to those observed in Sec. 6.2.1 for the SSM, although for most values of λ SH and M S the shift to the quartic coupling has only a very small eect on the value of the cut-o scale. We give in Tab. 6.2 our results for λ obtained for the three dierent orders of matching, for both small and large λ SH and for two choices of M S . For small λ SH the one-loop shift to λ is small, because of a cancellation between the purely-SM part dominated by the eect of the top quark and the singlet part of δ (1) λ. If one then considers larger values of λ SH , the term from the singlet becomes dominant over the SM one, and δ (1) λ is a large negative shift the evolution of λ, extracted at dierent orders, as a function of λ SH is also shown in Fig. 6.5, discussed below. At two loops however, there is no cancellation between SM and singlet contributions, and δ (2) λ is always a negative shift to the Higgs quartic, as was observed previously for the general SSM. On the other hand, it is always small, showing importantly that perturbativity of the model is preserved.

Having fewer parameters allows for a more detailed study of the dierent phases of the theory. Indeed, there are two transitions that occur respectively Note that because of the cancellation that occurs in the one-loop correction for small λ SH (discussed in the main text), the curves we would have obtained using one-loop matching would have been very similar to those with tree-level matching.

• between a metastable and a stable vacuum of the theory; for the physically relevant values of λ around 0.25-0.26, this happens for λ SH ∼ 0.3 and depends very little on M S or λ S .

• between a UV-complete model to a UV-incomplete one in other words the cut-o scale of the model becomes smaller than the Planck scale for suciently large couplings. GeV the curve with tree-level matching (dashed line) does not, because of the negative shift to the initial value of λ at scale Q = m t at two-loop order. Two-loop corrections to the matching of λ may exclude some parameter points that appear viable when only using a tree-level matching and are therefore important in the discussion of allowed regions of parameter space. Comparing Matching and running the dashed and dotted lines, we also observe the stabilising eect of the use of the twoloop RGEs, as discussed in section 6.2.1. presents the whole range of couplings that we considered, while the right pane shows an enlargement of the region in which the transition between stable and metastable phases occur.

We observe that the UV-complete phase of the model corresponds to smaller values of the inputs at scale Q = m t which can easily be understood as large values of the couplings at m t naturally lead to even larger values at higher scales. Furthermore, we can see that the phase of the model with stable vacua is associated with larger values of λ SH , and that when λ decreases, the value of λ SH needed to ensure a stable vacuum increases. While the SM part of the β-function of λ is negative and tends to drive it to negative values, the additional piece in β λ in the Z 2 SSM is positive and is of the form β

λ ⊃ 1 16π 2 λ 2 SH . When lowering λ(m t ) a higher value of λ SH is needed so that the β-function of λ changes sign earlier, and that λ does not run negative at some scale.

The blue lines in Fig. 6.5 give λ(m t ), obtained from the requirement that m h = 125.1 GeV, as a function of λ SH . The dierent curves correspond to the dierent orders at which the matching can be done: dotted for tree-level matching, dashed for one-loop and solid for two-loop order. The most important point to notice is that, as for the vacuum stability as we saw with Fig. 6.4, there is a value of λ SH here around 0.65 for which the UV-completeness in other words whether perturbativity 6.2 Models and results or unitarity are broken at some scale below M P l of a given parameter point depends greatly on the order at which λ(m t ) has been extracted from the Higgs mass.

Vector-like quarks and stability of the SM

From the SM, it is known that the quartic coupling λ runs negative at a scale Q 10 9 -10 11 GeV, leading to a metastable but long-lived vacuum [START_REF] Degrassi | Higgs mass and vacuum stability in the Standard Model at NNLO[END_REF][START_REF] Buttazzo | Investigating the near-criticality of the Higgs boson[END_REF]. While extensions with a heavy singlet similar to the previous subsections can have a stabilising eect on the potential [START_REF] Elias-Miro | Stabilization of the Electroweak Vacuum by a Scalar Threshold Eect[END_REF][START_REF] Lebedev | On Stability of the Electroweak Vacuum and the Higgs Portal[END_REF], fermionic extensions typically have the opposite eect through the negative impact of the vector-like (VL) fermions Yukawa coupling on the running of λ, see e.g. [START_REF] Elias-Miro | Higgs mass implications on the stability of the electroweak vacuum[END_REF][START_REF] Bambhaniya | Naturalness, Vacuum Stability and Leptogenesis in the Minimal Seesaw Model[END_REF]. A model where the latter is compensated by the eect of the former is discussed in ref. [START_REF] Xiao | Stabilizing electroweak vacuum in a vectorlike fermion model[END_REF].

Here, we shall extend the SM by one generation of a VL quark doublet Q as well as an up-type quark singlet t with their corresponding counterparts Q , t , with quantum numbers under the SM gauge group of t : ( 3, 1, -2

3 ), t : (3, 1, 2 3 ), Q : (3, 2, 1 6 ), Q : ( 3, 2, - 1 6 ). The Lagrangian of the model reads (in terms of two-component spinors)

L = L SM -Y t Q • Ht + Ỹ t Q • H t + m T t t + m Q Q Q + h.c. . (6.2.9)
For simplicity we take m Q = m T ≡ M Q ; we then nd that, with the normalisation L ⊃ - Let us rst consider the impact of the new vector-like states on the running quartic Higgs coupling. For simplicity, we consider here and in the following examples only one extra non-zero Yukawa interaction Y t and consequently set Ỹ t = 0 as it does not play a role in the following discussion. 4 Then for matching at µ = m t with M Q < 1 TeV, the shifts to λ are less than 10% for Y t 0.7, but grow rapidly to ∼ 50% for Y t ∼ 1. On the other hand, the direct impact of Y t on the running of λ at one-loop is given by

16π 2 β (1) λ ⊃ 12Y 2 t (λ -Y 2 t ) , (6.2.11)
which contributes signicantly to the negative slope of λ for large values but plays a negligible role when Y t is small. In the latter case, the impact of the new fermions on the running of the gauge couplings may outweigh their direct impact on λ. Consider the potential of eq. (6.2.9). Due to the additional coloured fermions, the running of g s changes at one-loop to 16π 6.6 Simplied comparison between the running of λ in the SM with and without vector-like states. Here, we used full one-loop (dashed lines) and two-loop RGEs (full lines) in both models and as starting point the SM best-t values from ref. [START_REF] Buttazzo | Investigating the near-criticality of the Higgs boson[END_REF]. For the purple (black) lines we use Y t = 0.3 (0.7).

with respect to the SM. In total, both eects increase the inuence of the strong force on the running of λ, adding positively to the slope. The impact on λ is shown in Fig. 6.6 where the running λ is computed using two-loop RGEs when assuming the pure SM (blue) and the VL extension (purple and black). No matching was applied yet here (i.e. the shift in α S was also neglected) the changes in the VL case therefore entirely stem from the altered running of the gauge couplings, most importantly eq. (6.2.12).

As starting value for λ we used the best-t value from ref. [START_REF] Buttazzo | Investigating the near-criticality of the Higgs boson[END_REF]. The increased g s throughout higher energy scales leads to a positive contribution to the slope of λ. One can observe that it can even lead to a stabilisation of the potential at high energies as long as the direct impact of Y t is kept under control by taking it small this is seen in the purple curve where we have chosen Y t = 0.3. For larger values, the known destabilising eect can overcome the stabilisation from g s . As a consequence, the scale of metastability would coincide with the SM for values of Y t ∼ 0.5 and decreases quickly with larger values. This is also shown in the gure for Y t = 0.7 (black line) where λ enters the metastable region already at energies of ∼ 10 5 GeV. We remark that the inclusion of the shift in α S according to eq. (6.2.13) would lead to an even milder running of λ. In fact, just using the one-loop RGEs for the case Y t = 0.3 the quartic coupling stays positive over the entire energy range. We will discuss the eects of the proper matching, including the shifts in λ, in what follows.

Solving equation (6.2.10) for the matched λ VLQ at µ = m t and keeping Ỹ t = 0, we see that the shifts are slightly negative for small Y t 0.45 and positive for larger and one-loop (dashed) or two-loop (full lines) matching, and for comparison when using the best-t value λ SM = 0.25208 at the top mass scale (dotted). Black lines correspond to Λ 0 = 10 18 GeV, blue lines to 10 11 GeV. The background shows the two-loop shift in λ(m t ) in percent, dened as (λ (2) λ (1) )/λ (1) .

Yukawa couplings. This has as a consequence that for low Y t where the VL quarks help increasing the scale of metastability, loop corrections have the opposite eect. However, the size of the shifts stays below 1 %. In Fig. 6.7, we present as contour lines the predictions for the scale of metastability as a function of Y t and M Q = m Q = m T using tree-level (dotted), one-loop (dashed) as well as two-loop matching (full lines) while applying two-loop RGEs. The colour code in the background quanties the relative two-loop shift in λ(m t ), (λ (2) λ (1) )/λ (1) , which stays below roughly half a percent. Nevertheless, the impact of these small shifts is non-negligible: the corresponding Y t values at which λ crosses zero at a given scale typically change by more than 10 % between tree-and two-loop matching. That is, the scale at which metastability occurs is very sensitive on the starting value of λ meaning that matching is absolutely crucial to make reliable statements. After including the shifts for λ at µ = m t , the picture nevertheless remains the same as that for small Y t ; the impact of the VL quarks on α S can be such that the scale of metastability is increased with respect to the SM, leading to the possibility of absolute stability all the way up to the Planck scale.

The situation is reversed if Y t is large. In that case, we enter the known scenario in which the additional impact of Y t on the RGEs of λ drives it negative faster when compared to the SM, further destabilising the vacuum. We show this in Fig. Y t and M Q , using two-loop matching, whereas the contour lines represent the relative changes with respect to using one-loop matching, Λ

(2,2) 0 /Λ

(1,2) 0

. As expected, Λ 0 is well below the pure SM prediction, and becomes smaller for larger Y t and smaller M Q . The dierences in Λ 0 between one-and two-loop matching are quite mild here only O(100 %) since the two-loop corrections to λ are small. We remark however, that going one order lower and comparing tree-level with one-loop matching using one-loop RGEs, we would see up to an order of magnitude dierences in the eventual scale Λ 0 . Summarising, we have shown that vector-like quarks can have both a destabilising but also a stabilising eect on the SM Higgs potential, and that the inclusion of threshold eects are crucial for obtaining precise predictions about the fate of the electroweak vacuum.

Two-Higgs Doublet model

Finally, as a last example, we study the impact of loop-level matching on Two-Higgs Doublet models. Here we will restrict ourselves to the CP-conserving version with a softly broken Z 2 symmetry as presented in section 1.4.2. The scalar potential we consider is given in eq. (1.4.14), and we recall that, after EWSB, we decompose the scalar elds see eq. (1.4.15) according to

Φ k = φ + k 1 √ 2 (v k + φ 0 k + i σ k )
, i = 1, 2 , (6.2.14)

Models and results

where v 2 1 + v 2 2 = v 2 and we dene t β = tan β = v 2 /v 1 . The charged (neutral CP-odd) elds mix to one physical charged Higgs H ± (pseudoscalar A) and the corresponding would-be Goldstone bosons. At LO, the angle β coincides with the mixing angle in the pseudoscalar and charged Higgs sector. In the CP-even sector, there are two elds which mix to one light and one heavy eigenstate, with masses m h and m H . In the same fashion as for the models considered above, we can relate the scalar masses and mixing angles to the quartic couplings λ i (i ∈ {1, 2, 3, 4, 5}), and corresponding tree-level expressions are given in equations (1.4.19) to (1.4.23).

Analogously to the case of the singlet extension of the SM in section 6.2.1, we dene the cut-o scale of a particular scenario as the scale at which either one of the λ i becomes larger than 4 π or the unitarity constraints using the running couplings are violated. The latter are too long to show here but can e.g. easily be computed using the SARAH implementation of the model in conjunction with appendix D of Ref. [START_REF] Krauss | Perturbativity Constraints in BSM Models[END_REF].

First we are going to look at the matching at the top mass scale. It has already been pointed out in chapter 5 that the loop corrections to the mass spectrum of 2HDMs can be signicant. In Fig. 6.9, we show on the left-hand side the size of the individual couplings λ i for the three matching orders as a function of the charged Higgs mass. The leading order λ i are simple linear functions of this mass according to eqs. (1.4.19) to (1.4.23), whereas the λ i evaluated with higher-order matching contain the shifts due to self-energy and tadpole corrections. We see that large dierences of O(100 %)

or even larger can appear between leading and next-to-leading order. The size of the relative shifts is displayed on the right-hand side each. As expected for a converging perturbative series, the dierences between one-and two-loop matching are much less pronounced, however they can still range around tens of percent. Obviously these large dierences necessarily have a signicant eect on the validity of the theory at higher scales. In the following we will therefore investigate the changes in cut-o scales between the dierent approaches.

As mentioned in the introduction, the two-loop RGEs are well-known but often neglected in the literature although it is known that large dierences can appear, see e.g. ref. [START_REF] Chowdhury | Global ts of the two-loop renormalized Two-Higgs-Doublet model with soft Z 2 breaking[END_REF]. Similar to the singlet-extended SM, the two-loop RGEs tend to moderate the one-loop running. As a result, Landau poles typically appear at much higher scales when including the two-loop eects. For instance, for both i = 1 and 2, 16π 2 β λ i ⊃ 24 λ 2 i (1 -13 16π 2 λ i ). The two-loop contribution thereby counteracts the large one-loop slope, stalling the evolution for λ i just below 4 π. In contrast, the twoloop RGEs to λ 3 for instance do have a mitigating eect on the evolution, however a complete stalling only occurs for values much larger than 4π. In Fig. 6.10 we show for a particular parameter point the running of the couplings λ 1 (black) and λ 3 (blue) for n-loop matching and m-loop RGEs. The inuence of the two-loop RGEs can be best gauged for the cases (n, m) = (1, 1) versus (1, 2), displayed in dot-dashed and dashed lines, respectively. It is clearly seen that, while both quartic couplings run into a Landau pole close to Q = 3 TeV when using one-loop RGEs, the inclusion of the two-loop terms leads to a signicant attening and therefore splitting between the two cases.

Let us look at the impact of the threshold corrections next. As shown in the example of Fig. 6.9, the threshold corrections for the λ i can be signicant. This can also be seen in the starting values of the couplings at m t in Fig. 6.10: the values for λ 1 using treelevel (one-loop) [two-loop] matching are 1.88 (1.45) [1.14] whereas for λ 3 , the values are 5.7 (4.5) [4.3]. The decrease in value at higher loop orders comes from the fact that in this particular scenario, the average loop corrections to the scalar masses are . The loop-level spectrum was evaluated taking the treelevel values m H ± = 1.14 TeV and t α = -0.95 as inputs. The ratio of the loopcorrected charged Higgs mass to its tree-level input is shown as grey contour lines. The quartic couplings for the case of tree-matching were obtained using the leading order relations Eqs. (1.4.19) to (1.4.23), taking the spectrum of the twoloop calculation as input. We further xed tan β = 1.14 and applied the Yukawa scheme of type I.

positive. As a result, one obtains a negative shift in the λ i at the matching scale when imposing a given physical spectrum. However, it need not be the case that the cut-o scale is raised by higher loop eects. Indeed, for large values of |m 12 | and therefore large heavy scalar masses, the mass corrections can be large and negative leading to the opposite eects, i.e. a decreased cut-o scale due to larger quartic couplings after the inclusion of the proper matching.

An example where this happens is presented in Fig. 6.11. For this gure we have evaluated the spectrum at the two-loop level while xing the tree-level input values of t α as well as m H ± which enter the spectrum calculation at the loop level. Therefore, the loop-corrected m H ± varies over this plane. The grey contours show the ratio of the loop-corrected charged Higgs mass over the tree-level input, m

H ± /m (T ) H ± . To obtain the LO couplings, i.e. the case of tree-level matching, we take the scalar spectrum and calculate λ i according to eqs. (1.4.19) to (1.4.23). Finally, we run the couplings up in scale using two-loop RGEs for the two-loop-and one-loop RGEs for the tree-levelmatched couplings in order to evaluate the cut-o scale. The coloured contours show the ratio of the cut-o scales, Λ (T,1) /Λ (2,2) , obtained with tree-level matching and one-6.3 Conclusions loop RGEs and with two-loop matching and two-loop RGEs, respectively. In particular in the region where all heavy scalar masses are approximately equal, we observe large dierences in cut-o scales. In fact, while the tree-level matching approach suggests a cut-o at O(10 7 GeV), the full two-loop matching procedure demands new physics restoring unitarity and perturbativity already at the TeV scale.

Concluding, the conventional approach of tree-level matching and one-loop RGEs can both over-but also underestimate the cut-o scale by many orders of magnitude. It is therefore of crucial importance to (i) take into account the known RGEs beyond one-loop and to (ii) perform a loop-corrected matching of the couplings before running.

Conclusions

In this chapter, we have investigated the impact that matching plays in the highscale validity of minimal extensions of the Standard Model (SM). While for most non-supersymmetric models studies beyond tree-level matching and one-loop RGEs are rare or even absent, we have analysed in dierent scenarios the impact of both loop-corrected matching as well as using two-loop RGEs, highlighting the dierences with respect to previous approaches. For simple models, we provide an analytical computation of the matching conditions. We point out how sensitive the cut-o scale of the real-singlet-extended SM is on the loop order of both matching and RGE running and show that the scale dependence decreases when all available corrections are included in the matching. Imposing an additional Z 2 symmetry to this model furthermore enables us to study the fate of the electroweak vacuum as well as the UV completion analytically. We highlight regions of parameter space where the model can in principle be valid up to the Planck scale a statement which crucially depends on the proper matching of the quartic couplings at the low scale.

In a scenario where the SM is extended by vector-like quarks, we show that the impact of the latter can actually increase the Higgs quartic interaction such that it does not become negative at higher scales an observation that we have not encountered before in the literature. The reason is that, despite the negative impact of the additional Yukawa coupling on the running of λ, the presence of additional coloured states modies the running strong coupling in such a way that it adds positively to the βfunction of λ. Also in this scenario, the matching of λ before the RGE evolution has a signicant impact on the predicted high-scale behaviour of the model.

As a nal example we show in a Two-Higgs Doublet model that the loop-level matching of the quartic couplings can lead to signicant changes in both the MS values of scalar quartic couplings and subsequently the cut-o scale of the theory.

To conclude, we observe that robust statements about the UV-behaviour of nonsupersymmetric, weakly-coupled BSM models can only be made when including, at the very least, loop-level matching. We stress that the required loop corrections, as well as two-loop RGEs, are readily accessible with the computer tool SARAH for any generic renormalisable eld theory. In light of our results, we strongly encourage its use when accurate high-scale predictions are required.

Conclusion

The 125-GeV Higgs boson, discovered at the CERN Large Hadron Collider in 2012, represents an invaluable probe of BSM Physics, via the study of its properties. One property that has received considerable interest is the Higgs boson mass m h . Indeed, m h is now measured to a precision of about 0.1%, while intensive eorts have been spent on its computation, in the Standard Model, and in the BSM models like Supersymmetry where a tree-level value of m h is predicted by the theory. Adding then radiative corrections (which involve all the particle spectrum) to the tree-level contribution yields a result for the Higgs pole mass and allows to constrain the parameter space of these models. However, corrections to Higgs masses have only been studied (explicitly) beyond one loop in a limited number of BSM models (mostly in the minimal supersymmetric extensions of the SM, namely the MSSM and NMSSM), which means that precise predictions are missing for most models. Furthermore, even in the models that have concentrated most computational eorts, the accuracy of theoretical results is still not comparable to the experimental error on the Higgs mass measurement. The work presented in this thesis has therefore been aimed towards making new radiative corrections to scalar masses available in BSM models. We have done so with two complementary approaches: (i) by deriving explicitly new corrections in SUSY models beyond the MSSM and the NMSSM, and (ii) by addressing technical hurdles in the calculation of Higgs mass corrections for generic models and in the automation of these computations.

First, we have calculated the leading two-loop O(α t α s ) corrections to neutral scalar and pseudoscalar masses in supersymmetric theories with Dirac gauginos, in the effective potential approximation. These models constitute a very attractive class of non-minimal SUSY extensions of the SM because of their rich phenomenology, and some advantages they oer in particular their ability to evade experimental constraints and their increased naturalness. Our results have allowed us to compare the predictions for the lightest (CP-even) Higgs mass m h between the MSSM and the minimal Dirac gaugino model (the MDGSSM), and in particular we could investigate the eects of the new coloured particles, the octet scalars (or sgluons), on m h . We have also considered scenarios with large mass hierarchies between the gluinos and the stops, which remain natural because of the lesser dependence of the scalar masses on the Dirac mass of the gluinos. We could there point out the loss of accuracy associated with the choice of a DR renormalisation scheme for the parameters of the top/stop sector, and the fact that an on-shell scheme is more suitable in such scenarios. In particular, the phenomenologically relevant supersoft limit (where all soft SUSY-breaking terms vanish) can only be studied when using an OS scheme, as perturbativity is lost with the DR scheme.

In the remainder of the thesis, a particular emphasis has been put on the Goldstone Boson Catastrophe and its solution for general renormalisable theories. By employing Conclusion an on-shell scheme for the Goldstone boson masses, we have obtained infra-red-safe expressions for the loop functions that appear in two-loop tadpole and self-energy diagrams in the gaugeless limit and a generalised eective potential approach for the self-energies. We have also shown that by an expansion of the mass parameters in the loop functions, we could solve the tadpole equations directly, instead of iteratively.

Nevertheless, several questions remain concerning this catastrophe, such as how to extend its solution to higher orders. Recently, ref. [START_REF] Espinosa | Resummation of Goldstone Infrared Divergences: A Proof to All Orders[END_REF] proved that the resummation prescription rst presented in [START_REF] Elias-Miro | Taming Infrared Divergences in the Eective Potential[END_REF] (see the description in section 1.1.5) can be extended to all orders in perturbation theory. It would then be interesting to nd out how in practice to apply this method to general renormalisable theories, and to study further the relation between this resummation and the on-shell solution presented in chapter 4 (especially beyond two loops).

Our analytic results now allow investigating scalar masses at two-loop order in both supersymmetric and non-supersymmetric theories (Standard Model or extensions of it), without the numerical instabilities due to the Goldstones. This can be done either by applying the results to the desired model by hand, or by using automated tools like SARAH/SPheno, in which the modied loop functions are implemented. An important result we nd is the very good agreement between the value of the Higgs mass in the SM now obtained by SARAH/SPheno and the existing complete two-loop results from dedicated calculations [START_REF] Martin | Higgs boson mass in the Standard Model at two-loop order and beyond[END_REF][START_REF] Buttazzo | Investigating the near-criticality of the Higgs boson[END_REF], indicating that at least at the scale Q = m t at which we performed the comparison the purely-electroweak corrections and the two-loop momentum eects are both small. We have also considered several BSM models, illustrating in the NMSSM how well our on-shell method cures all the numerical instabilities caused by the Goldstone bosons, and investigating previously unknown two-loop corrections in various other models. This has in particular enabled us to make several ndings for non-supersymmetric models, among which: (i) that loop corrections often grow out of control well before the Lagrangian parameters (couplings) reach the naive perturbativity bounds (in turn this signals the danger of trading Lagrangian parameters for masses using only tree-level masses as is often done for these models); and (ii) that the order at which the scalar quartic couplings are extracted from the physical spectrum can aect their high-scale evolution signicantly, which can then have consequences on whether the considered parameter point is deemed valid or not.

There are numerous further avenues of research from this point. First of all, our investigations in non-supersymmetric models have shown the need for a more precise determination of perturbativity and unitarity constraints (note that in the continuation of the work presented in chapter 5, ref. [START_REF] Krauss | Perturbativity Constraints in BSM Models[END_REF] studied perturbativity bounds in the Georgi-Machacek model). For Higgs mass calculations, one should of course continue to improve the precision of theoretical predictions. On the one hand, for explicit computations, this is done by considering corrections in new models, as well as by deriving additional corrections in already studied theories. On the other hand, in the context of generic calculations, work is still needed to reach the level of precision currently available for the MSSM. A natural next step (also needed for the MSSM) would be to complete the set of two-loop mass contributions, by obtaining the momentum dependence of the diagrams that are of quartic order in the couplings of broken gauge groups (e.g. the electroweak couplings for a model with the same gauge group as the SM), even if we showed in chapter 5 that these must be very small in the case of the Standard Model. Moreover, in the context of numerical calculations in generic theories (for example in SARAH/SPheno), it would be desirable to nd an applicable way of going beyond the generalised eective potential approximation that is currently used. Indeed, it would be important to work towards a fast implementation of the two-loop Conclusion momentum dependence of general scalar mass corrections, in order for example to allow parameter scans (which cannot be performed currently with full two-loop momentum dependence because of excessively long computing times). It is also worth mentioning that in the last two years results have become available at three loops for the eective potential of generic theories [START_REF] Martin | Evaluation of the general 3-loop vacuum Feynman integral[END_REF][START_REF] Martin | Eective potential at three loops[END_REF], which may at some point serve for the calculation of tadpoles and mass corrections at three-loop order. Furthermore, in this work we have only considered xed-order calculations in generic models. However, as the scale of New Physics seems to be driven to higher values by experimental searches, it appears necessary to make EFT calculations possible in general theories. In particular, obtaining analytic expressions for the matching of scalar quartic couplings between two generic models would be very important.

Finally, one should mention also that we have focussed here on the Higgs mass because it is now extremely well measured, as opposed to other Higgs properties, e.g. couplings, that are currently measured with a much worse accuracy or not even measured at all (this is for example the case for the Higgs trilinear coupling). However, it remains important to compute radiative corrections to these other properties, in expectation of future experimental results.

When, hopefully, conclusive experimental signs of some New Physics coupling to the Higgs sector are found, the analytic results and numerical tools developed to perform precision calculations of Higgs properties both in specic and in generic models to which this thesis has contributed will play a crucial role in understanding the nature of the newly observed phenomena.

Appendix A Derivatives of the two-loop eective potential in models with Dirac gauginos

We present here the derivatives of the two-loop eective potential used to calculate the Higgs masses in section 3.1. We recall that the eective potential and its derivatives are expressed in units of α s C F N c /(4π) 3 . The derivatives of the rst term in eq. (3.1.19) can be trivially obtained by multiplying the formulae in appendix C of ref. [START_REF] Degrassi | On the radiative corrections to the neutral Higgs boson masses in the NMSSM[END_REF] by R 2 1i and summing over the two gluino masses m gi , hence we do not repeat them here. The only exception is the single derivative of ∆V αs MSSM with respect to m 2 t , which was not needed in ref. [START_REF] Degrassi | On the radiative corrections to the neutral Higgs boson masses in the NMSSM[END_REF].

O i Q 2 -1 -4 ln m 2 t1 Q 2 -1 -Φ(m 2 t1 , m 2 t1 , m 2 O i ) + 4 s 2 2θt ∆ O i m 2 O i m 2 t1 (m 2 O i -m 2 t1 -m 2 t2 ) ln m 2 O i Q 2 -(m 2 O i -m 2 t1 + m 2 t2 ) ln m 2 t1 Q 2 - m 2
∆ O i m 2 O i ln m 4 O i m 2 t1 m 2 t2 -(m 2 t1 -m 2 t2 ) ln m 2 t1 m 2 t2 -(m 2 O i -m 2 t1 -m 2 t2 )Φ(m 2 t1 , m 2 t2 , m 2 O i ) , (A.8)
where we used the shortcut B.2 Diagrams regulated by momentum we give the expansions for small external momentum s ≡ -p 2 of the diagrams that diverge as s → 0, taken from expanding expressions in [START_REF] Martin | Evaluation of two loop selfenergy basis integrals using dierential equations[END_REF][START_REF] Scharf | Scalar two loop integrals for gauge boson selfenergy diagrams with a massless fermion loop[END_REF] or found by newly solving or expanding the integral equations in [START_REF] Martin | Evaluation of two loop selfenergy basis integrals using dierential equations[END_REF]. Hence we stress that (most of ) this section contains new results not found elsewhere.

∆ O i ≡ (m 2 O i -m 2 t1 -m 2
B.2.1 Limits of the Z and U functions First, for Z, we only need the fact that

B(p 2 , m 2 G , m 2 G ) -→ m G →0
2log(-s).

(B.2.1)

Then, for the U function, taking one argument to zero does not cause any divergence, and we nd, looking at the integral denition (B.1.19) of U , that U (x, y, 0, 0), U (0, x, y, 0), U (x, 0, y, 0), U (x, 0, 0, 0) and U (0, y, 0, 0) are all regular so we can substitute them for U 0 + O(s). The only divergent function is U (0, 0, x, y) that has the form U (0, 0, x, y) = A U (x, y) log(-s) + B U (x, y) + O(s) which matches an expansion of the full momentum-dependence expression in equation (6.24) of [START_REF] Martin | Evaluation of two loop selfenergy basis integrals using dierential equations[END_REF].

Finally, U (0, 0, 0, 0) = 1 2 log(-s) -3 2 + 1

is not required, as it always appears with λ GGG as a factor, which is zero up to higher order corrections.

B.3 Additional expressions for Ṽ (x, 0, z, u)

The approximate formulae for U (0, 0, x, y), M (0, y, 0, u, v) and M (0, 0, 0, 0, v) have been checked against the numerical results from TSIL [START_REF] Martin | TSIL: A Program for the calculation of two-loop self-energy integrals[END_REF] and show excellent agreement until s becomes of the order of the arguments in the functions even when s is of the order of the mass parameters, the dierence between the approximate result and the numerical from TSIL is about 10%.

B.3 Additional expressions for Ṽ (x, 0, z, u)

One of the key functions of the basis set is V (x, y, z, u). This is dened as V (x, y, z, u) ≡ -∂ ∂y U (x, y, z, u).

(B.3.1)
It is singular as y → 0, so we dene the regularised version (dened with one fewer explicit index to [START_REF] Martin | Two loop scalar self energies in a general renormalizable theory at leading order in gauge couplings[END_REF]):

V (x, y, z) ≡ lim On the other hand, for x → 0 we cannot take the above limit. In principle we could start again from the expressions for V (x, y, z) given in the appendix of [START_REF] Martin | Two loop scalar self energies in a general renormalizable theory at leading order in gauge couplings[END_REF] and take the smooth limit x → 0. Instead, here we provide two direct derivations of a compact expression for Ṽ (0, z, u), with both general external momentum and then for our generalised eective potential limit. The starting point for the rst derivation is the set of dierential equations given in [START_REF] Martin | Evaluation of two loop selfenergy basis integrals using dierential equations[END_REF], in this case ≡ k U U U (x, y, z, u) + ∆,

where the coecients of the loop functions are themselves functions of s, x, y, z, u.

However, here we encounter the problem that several of these coecients are actually This in turn will lead to a shift in the neutralino and chargino masses, which lead to a shift to the two-loop tadpoles.

  Abbreviations

Figure 1 . 1

 11 Figure 1.1 Examples of the most severely divergent contributions of the Goldstone bosons to the eective potential at two, three, four and ve loops. The black blobs represent the insertions of the 1PI subdiagrams Γ(p 2 ).

1. 3

 3 Supersymmetrywith the assignments of each supereld in representations of the SU (3) C × SU (2) L × U (1) Y SM gauge group.The superpotential of the MSSM is

Figure 1 . 3

 13 Figure 1.3 Summary of lower limits on masses of SUSY particles, obtained by the dierent SUSY searches of the ATLAS collaboration by December 2017. This plot, along with other summary from ATLAS can be found on the web page: https:// atlas.web.cern.ch/Atlas/GROUPS/PHYSICS/CombinedSummaryPlots/SUSY/. A similar plot for CMS, updated for the Moriond 2017 conference, can be found at https://twiki.cern.ch/twiki/pub/CMSPublic/PhysicsResultsSUS/ Moriond2017_BarPlot.pdf.

Figure 1 . 4

 14 Figure 1.4 Exclusion limits, at 95% condence level, on stop and neutralino masses, resulting from the stop pair production search of the ATLAS collaboration. Dierent decay channels of the stops are considered, with quantities of data comprised between 3.2 and 36 fb -1 from Run 2 of the LHC at √ s = 13 TeV. A comparison with results from Run 1 (light blue) is also provided. This plot can be found on the web page: https://atlas.web.cern.ch/Atlas/GROUPS/PHYSICS/ CombinedSummaryPlots/SUSY/ .

  prevents the Higgs sector from being extended, it is common to study extensions of the MSSM, and the simplest and most popular among them is the Next-to-Minimal Supersymmetric Standard Model or NMSSM, which provides a solution to the µproblem of the MSSM and enhances the tree-level mass of the lightest Higgs boson with respect to the MSSM (therefore reducing the size of the radiative corrections 1.3 Supersymmetry required to have a 125 GeV Higgs). The NMSSM extends the MSSM by a gaugesinglet chiral supereld S, and comes with a superpotential that reads in all generality

  The three complex neutral scalars appearing in the above potential can be decomposed into mass eigenstates as three real scalars (CP-even) and three pseudoscalars (CP-odd) one of them being the neutral Goldstone while the two charged components of the Higgs doublets give a charged Higgs and a charged Goldstone boson. The upper bound on the lightest Higgs mass becomes in the NMSSM

  Higgs boson and Physics beyond the Standard Model production) [9395] and avour constraints [9698].Dirac gaugino masses require the addition of two fermionic degrees of freedom (i.e., an extra Weyl spinor) for each gaugino. We can then write a mass term that respects a global chiral symmetry, which in SUSY models is promoted to a global U (1) Rsymmetry. In turn, Supersymmetry requires the same number of extra scalar degrees of freedom as fermionic ones; this implies that after electroweak symmetry breaking (EWSB) we have four new neutral scalar degrees of freedom compared to the MSSM, which may mix with the neutral scalars of the Higgs sector. The new states are packaged in an adjoint chiral multiplet for each gauge group, which should also have couplings to the Higgs scalars, possibly enhancing the SM-like Higgs boson mass at both tree and loop level.

( 1 . 3 . 104 )

 13104 The rst term in the above equation(1.3.103) is a Dirac mass term for λ a and χ a Σ having the same form as equation (1.3.6) for a Dirac spinor made of λ a and χa Σ while

  If we use a cut-o regularisation (see footnote 8) this correction takes the form m 2 D log Λ 2 and the natural cut-o scale is Λ ∼ M .Then there must exist a counter-term operator that regulates this divergence in cuto regularisation, and it can be shown that the only possible supersymmetric and gauge-invariant operator associated with D-term SUSY breaking is of the form[START_REF] Fox | Dirac gaugino masses and supersoft supersymmetry breaking[END_REF]

Figure 1 . 5

 15 Figure 1.5 Diagrams contributing to the mass of a scalar φ: on the left hand side with φ, i.e. the fermionic superpartner of φ, and a Dirac gaugino (λ and χ Σ in terms of Weyl spinors); and on the right hand side with the scalar φ and the adjoint scalar Σ. The logarithmic divergences from both diagrams cancel out when the gauginos only have Dirac masses.

  (1.3.103) 1.3 Supersymmetry and(1.3.105). Considering now all sources of mass terms for the adjoint scalars Σ a , we obtain

2 D 2 D

 22 These couplings lead to new (compared to MSSM and NMSSM) contributions to the two-loop eective potential involving the octet scalars which will aect the Higgs masses. We remark that, since in eq.(1.3.111) the superpotential mass term m aects only the real part of the octet scalar, the mixing angle φ O is suppressed by m in the limit

.3. 117 )

 117 In four-component notation, this leads in general to two Majorana gauginos with dierent masses. In the case of a pure Dirac mass, however, we obtain two Majorana gauginos with degenerate masses |m λ 1 | = |m λ 2 | = |m D | , which can also be combined in a single Dirac gaugino, written in terms of Weyl spinors as

  .122) where σ a are Pauli matrices, and the dot-product denotes the antisymmetric contraction of the SU (2) L indices. In addition to the terms explicitly shown in eqs.(1.3.121) and(1.3.122), the most general renormalisable superpotential contains terms involving only the adjoint superelds namely, mass terms for each of them, all trilinear terms allowed by the gauge symmetries, and a linear term for the singlet which we denote collectively as W Σ . The most general soft SUSY-breaking Lagrangian for the MDGSSM contains non-holomorphic mass terms for all of the scalars, as well as Majorana mass terms for the gauginos, plus A-type (i.e., trilinear), B-type (i.e., bilinear) and tadpole (i.e., linear) holomorphic terms for the scalars with the same structure as the terms in the superpotential. With the assumption, discussed in section 1.3.5.4, that we neglect the couplings λ SO and T SO dened in eq. (1.3.110), the superpotential W Σ and the soft SUSY-breaking terms that involve only the adjoint elds are not relevant to the calculation of the two-loop O(α t α s ) corrections to the Higgs masses presented in chapter 3, apart from contributing to the masses and mixing of the adjoint elds as discussed in sections 1.3.5.4 and 1.3.5.5 above.

  .124) The Higgs boson and Physics beyond the Standard Model while all terms involving only the MSSM-like Higgs superelds and/or the adjoint superelds, such as those in eq.(1.3.122), are forbidden by the R-symmetry. The most general soft SUSY-breaking Lagrangian for the MRSSM contains non-holomorphic mass terms for all of the scalars, plus all of the holomorphic terms involving only the MSSM-like Higgs scalars and/or the adjoint scalars (which, as mentioned above, have no equivalent in the superpotential). In contrast, the R-symmetry forbids Majorana mass terms for the gauginos, and holomorphic terms for the scalars with the same structure as the terms in the MRSSM superpotential. The requirement that the Rsymmetry is conserved also means that the scalar doublets R u and R d do not develop a vacuum expectation value (VEV), and do not mix with either the MSSM-like Higgs scalars or the adjoint scalars.Due to the larger number of additional neutral scalar states, both in the MDGSSM and in the MRSSM, a presentation of the Higgs sector would become quite lengthy, and we refer instead the reader to[START_REF] Benakli | Generating mu and Bmu in models with Dirac Gauginos[END_REF][START_REF] Benakli | The Di-Photon Excess in a Perturbative SUSY Model[END_REF][START_REF] Belanger | Dark Matter with Dirac and Majorana Gaugino Masses[END_REF][START_REF] Benakli | Dirac Gauginos and the 125 GeV Higgs[END_REF] for discussions of the Higgs sector in the MDGSSM, and for calculations of the Higgs potential at tree and loop level. We however recall here one important result, namely the modied upper bound on the lightest Higgs mass, which reads in the MDGSSM

  as an excess in the distribution of reconstructed Higgs decay products for the h → γγ and h → ZZ * → 4 channels as a function of their invariant mass. Already from the moment of its rst discovery the Higgs boson mass was known to a precision of the order of one GeV, which have since been improved to the remarkable level of about 0.2 GeV. At the time of writing, the most accurate experimental determination of m h comes from the combined result of ATLAS and CMS using 5 fb -1 of data at √ s = 7 TeV and 20 fb -1 at √ s = 8 TeV, obtained during Run 1 of the LHC and reads [51]m exp.

h = 125 .

 125 09 ± 0.21 (stat.) ± 0.11 (syst.) .

( 2 . 1 of

 21 1.1)A more recent measurement was obtained by the ATLAS collaboration with 36.1 fb -Run 2 data at √ s = 13 TeV[START_REF] Collaboration | Measurement of the Higgs boson mass in the H → ZZ * → 4 and H → γγ channels with √ s=13TeV pp collisions using the ATLAS detector[END_REF], see gure 2.1.

Figure 2 . 1

 21 Figure 2.1 ATLAS results for the SM-like Higgs boson mass, obtained from LHC Run2 data with an integrated luminosity of 36.1 fb -1 at √ s = 13 TeV, in the h → γγ and h → ZZ * → 4 channels, as well as the combined result and comparison with the combined ATLAS and CMS measurement from Run 1. This plot is taken from [52].

Figure 2 . 2 Figure 2 . 3

 2223 Figure 2.2 Possible topologies of one-loop self-energy diagrams.

. 4 ) and where m 2 Z , m 2 A

 422 , and tan β are running parameters. Note that in the above expression, the minimum conditions have already been used to eliminate the soft mass parameters of the two Higgs doublets m 2 Hu and m 2 H d

  .3.10) where M 2 S,P loop are the 2 × 2 scalar and pseudoscalar mass matrices, obtained as in the case of the real MSSM, while M 2 SP loop are new, CP-violating, corrections. Results for calculations of neutral scalar masses in the complex MSSM were obtained at one loop in [181184], then the leading contributions at two loops with vanishing 2.3 State-of-the-art of Higgs mass calculations external momentum were found in [185188], and more recently the complete two-loop

( 2 . 4 . 10 )

 2410 Next we need to dene what happens when we introduce the loop corrections to the eective potential ∆V . Instead of ti 0 = 0, the loop-corrected tadpole equations are then

.4. 13 ) 14 )

 1314 When introducing loop corrections to the scalar potential in the SM, the tadpole equation is modied and becomes th 0 + ∂∆V ∂h min. = 0 , ⇒ µ 2 = -λv 2 -With this relation, the mass parameter m 2 ij for the Higgs boson (i.e. m 2 hh ) is nally

Figure 2 . 4

 24 Figure 2.4 Illustration of the study of a BSM model using SARAH and SPheno, and links to other High-Energy Physics codes.

  (3.1.3) and (3.1.4) arise when the minimum conditions of the potential, ∂V eff ∂S i min = 0 , (3.1.5)

  (3.1.2), (3.1.3) and (3.1.4), we have proceeded as for the MSSM in section 2.3.1, i.e. we have split the tadpole terms between tree-and loop-level, and the tree-level part has already been used to simplify the tree-level mass matrices. With a straightforward application of the chain rule for the derivatives of the eective potential, the mass-matrix corrections in eqs. (3.1.3) and (3.1.4) and the minimum conditions in eq. (3.1.5) can be computed by exploiting the Higgs-eld dependence

Figure 3 . 1

 31 Figure 3.1 Novel two-loop contribution to the eective potential involving stops and octet scalars.

.1. 47 ) 3 . 1 . 4

 47314 Mass corrections in the MRSSMThe MRSSM is dened to be R-symmetric, and has elds R u , R d which pair with the Higgs elds without themselves developing VEVs. In this model the gluino mass terms are purely Dirac, therefore, in our conventions, R 2 11 = R 2 12 = 1/2 and m g1 = -m g2 = m D . The trilinear Higgs-stop coupling A t is forbidden, and the term X dened in eq. (3.1.6) reads X = -y t µ u + λ Su S * + λ Tu T 0 * R 0 * u , (3.1.48) and vanishes at the minimum of the scalar potential, hence the stops do not mix. Moreover, the term proportional to c ϕ-φ in the second line of eq. (3.1.20) cancels out in the sum over the gluino masses. As a consequence, the radiative corrections induced by top/stop loops are remarkably simple. Ordering the neutral components of the elds as

  (3.1.23) its counterterm can be derived from those of the other four parameters via the relation

  .1.54) where (δx MSSM k ) i are obtained, with the trivial replacement m g → m gi , from the MSSM shifts given in appendix B of ref.[START_REF] Degrassi | On the neutral Higgs boson masses in the MSSM for arbitrary stop mixing[END_REF], whereas δx octet k are novel contributions involving the octet scalar. In particular, δm octet t = 0, and the remaining shifts can be obtained by combining as in eqs. (3.1.52) and (3.1.53) the octet contributions to the nite parts of the stop self-energies:

2 )

 2 is the nite part of the Passarino-Veltman function. The change in renormalisation scheme for the top/stop parameters entering the oneloop (1 ) part of the corrections to the Higgs mass matrices induces a shift in the two-loop (2 ) part of the corrections: Analogous expressions hold for the shifts in the two-loop part of the minimum conditions of the eective potential. The one-loop corrections entering the equation above can be obtained by inserting in eqs. (3.1.10)(3.1.12) the one-loop expressions for the functions F , G, F 1,2,3 and F ϕ . In units of N c /(16π 2 ), these read:

.1. 68 )

 68 Again, it can be easily checked that the explicit dependence on Q cancels out in the sum of eqs. (3.1.65) and (3.1.67).

2 t1 with m 2 t2. 2 g 2 h

 2222 .1.70) where the last term in square brackets represents the addition of terms obtained from the previous ones by replacing m From eq. (3.1.70) above it is clear that, in the DR scheme, the two-loop top-stop-gluino contributions to the SM-like Higgs mass can become unphysically large when m g m ti , due to the presence of terms enhanced by m 2 g/m 2 ti . This non-decoupling behaviour of the corrections to the Higgs mass in the DR scheme has already been discussed in the context of the MSSM in ref. [76]. Indeed, the correction in eq. (3.1.70) corresponds to the one obtained by setting µ = A t = 0 in the MSSM result. The terms enhanced by m the top and stop masses in the one-loop part of the correction as OS parameters. After including the resulting shifts in the two-loop correction, we nd ∆m

2 O 1 does 2 O 1 ≈ 4 m 2 D when m 2 D

 212122 .1.73) where g(x) is the function dened in eq.(3.1.68). It would appear from eqs.(3.1.72) and(3.1.73) above that, independently of the renormalisation scheme adopted for the stop masses, the octet-scalar contributions to ∆m 2 h are enhanced by a factor m 2 g. This is due to the fact that the trilinear squark-octet interaction, see eq.(1.3.114), is proportional to the Dirac mass term m D i.e., to m g . However, as discussed in section 1.3.5.4, one of the mass eigenvalues for the octet scalars to x the notation, let us assume it is m in turn grow with the gluino mass, namely m becomes much larger than the soft SUSY-breaking mass terms for the octet scalars. Expanding the corresponding contribution to ∆m 2 h in inverse powers of m 2 O 1

m 2 g

 2 .1.74) which does indeed contain potentially large terms enhanced by the ratio m 2 g/m 2 ti . Note that those terms cancel only partially the corresponding terms in the MSSM-like 3.2 Numerical examples contribution see the rst term in the curly brackets of eq. (3.1.70) leaving residues proportional to

Figure 3 . 2

 32 Figure 3.2 Mass of the SM-like Higgs boson as a function of (X t /M S ) OS , for tan β = 10, M S = 1.5 TeV and m g = m O = 2 TeV. The dashed curve represents the MSSM result, whereas the solid (dotted) curve represents the MDGSSM result with (without) the octet-scalar contributions.

Figure 3 . 3

 33 Figure 3.3 Dierent determinations of the SM-like Higgs boson mass in the MDGSSM as a function of (X t /M S ) OS , for the same choices of parameters as in gure 3.2.

Figure 3 . 4

 34 Figure 3.4 Mass of the SM-like Higgs boson as a function of m g in the MRSSM, for tan β = 10, M S = 1 TeV and m O = 2 TeV. The meaning of the dierent curves is explained in the text.

2 g/M 2 SFigure 3 . 5

 2235 Figure 3.5 Mass of the SM-like Higgs boson as a function of m g in the supersoft limit of the MRSSM, for tan β = 10. The solid curve represents the results of the DR calculation, in which the two-loop O(α t α s ) corrections become unphysically large.

2 g/m 2 t

 22 (3.1.69) vanishes, while the two-loop corrections in eqs.(3.1.70) and(3.1.74) contain terms enhanced by m (concerning the octet-scalar contributions, we recall that m O 1 = 2 m g in this scenario). Since the Dirac-gluino mass needs to be in the multi-TeV range to generate realistic values for the physical stop masses, the non-decoupling terms in the two-loop corrections can become unphysically large. This is illustrated by the solid (red) curve in gure 3.5, which represents the SM-like Higgs boson mass obtained with the DR calculation as a function of the gluino mass (here we x the renormalisation scale as Q = m t and use α s (m t ) in the two-loop corrections). It appears that the DR prediction for m h becomes essentially proportional to m g, and quickly grows to nonsensical values as the latter increases. In contrast, the dashed (blue) curve is obtained with the same procedure as the dashed curves in gure 3.4, i.e. by computing the physical stop masses at O(α s ) as a function of m g and using them in conjunction with the appropriate OS formulae for the O(α t α s ) corrections to m h . In our example the stop masses range between 302 GeV and 1272 GeV, while the SM-like Higgs boson mass shows only a mild dependence on m g and remains conned to values well below the observed one.

3. 3

 3 ConclusionsSupersymmetric models with Dirac gaugino masses have attracted considerable attention in the past few years, because they are subject to looser experimental constraints and require less ne-tuning than the MSSM. Besides the extended gaugino sector, such models feature additional colourless scalars which mix with the usual Higgs doublets of the MSSM, as well as additional coloured scalars in the octet representation of SU (3) which contribute to the Higgs boson masses at the two-loop level. In this chapter we presented a computation of the dominant two-loop corrections to the Higgs boson masses in Dirac-gaugino models, relying on eective-potential techniques that

4. 1 . 1 4 ( 4 . 1 . 1 ) 1 √ 2 (

 11441112 Abelian Goldstone modelLet us begin by recalling the problem of the Goldstone Boson Catastrophe. For simplicity we shall take the simplest Abelian Goldstone model dened by a complex scalar eld Φ (and no gauge group) with potential V = µ 2 |Φ| 2 + λ|Φ| and expand around an expectation value v as Φ = v + h + iG) to obtain

( 4 .

 4 1.3) where the one-loop functions f (x), A(x) and the two-loop function I(x, y, z) are dened in the appendix, equations (B.1.3), (B.1.6), and (B.1.24). The potential is regular as m G → 0 but does contain terms of order m 2 G log m 2 G (where log x is also dened in the Avoiding the Goldstone Boson Catastrophe in general renormalisable eld theories at two loops appendix eq. (B.1.2)) so that when we derive the tadpole equation and expand the derivative of I

( 4 . 1 . 6 )

 416 With zero external momentum, this becomesΠ GG (0) = 1 16π 2 λA(m 2 G ) + 3λA(M 2 h ) .

( 4 .

 4 1.41)

Figure 4 . 1

 41 Figure 4.1 Divergent scalar-only diagrams that require regulation (by resummation or using our on-shell scheme), even in the presence of external momentum. The light blue dashed lines marked with a small red G denote Goldstone boson propagators. The dark blobs in the diagrams on the right-hand side represent full one-loop one-particle-irreducible corrections inserted on the line. On the top line we show the tadpoles (with their clear relation to the sunset and gure-eight diagrams in the potential); on the lower two we show the corrections to the self-energies, which clearly follow the same pattern.

( 4 . 4 . 6 )

 446 However, both the mass of the Goldstone boson and the Higgs are controlled by the µ 2 parameter;

  y) =G DR F F (x, y) + 2δ M S x + y + 2J(x) + 2J(y)-(x + y) 2B(x, y) + xB(y, x ) + yB(x, y ) → s→0 2(x + y)[3U 0 (x, y, x, 0) + 3U 0 (x, y, y, 0) + 5P SS (x, y)] -6I(x, x, 0) -6I(y, y, 0) + 10J(x) + 10J(y) -16(x + y) + 4δ M S x + y + J(x) + J(y) + (x + y)P SS (x, y) , G F F (x, y) =G DR F F (x, y) -4δ M S 2B(x, y) + yB(x, y ) + xB(y, x ) → s→04 3U 0 (x, y, x, 0) + 3U 0 (x, y, y, 0) + 5P SS (x, y) -4 + 4δ M S 2P SS (x, y) + 1 .

  is to compute the Higgs mass with the quartic coupling λ ranging in the interval [0.125, 0.130], and only setting 5.2 Standard Model the SM inputs to the same values as in [25], which we recall here G F = 1.16638 × 10 -5 GeV -2 , α s (M Z ) = 0.1184, M Z = 91.1876 GeV, m t = 173.34 GeV.

Figure 5 . 1

 51 Figure 5.1 Dierence between the two-loop Higgs mass computed by SMH and SPheno (m 2 h ) SPheno -(m 2 h ) SMH as a function of the renormalisation scale Q, with (blue curve) and without (orange dashed curve) the momentum dependence at two loops in SMH. The Higgs quartic coupling is here λ = 0.12604. In SPheno the contributions of the light SM fermions are turned o and the external momentum in the two-loop routines is set to s = m 2 h | tree .

δm 2 hTable 5 . 4

 254 [GeV] λ α = 10 -6 α = 10 -4 α = 10 -2 α = 1/2 α = 1 Shift in GeV of the two-loop Higgs mass in the Standard Model computed with SPheno and with respect to the value obtained for p = 125 GeV for dierent values of the quartic coupling λ, and of the incoming momentum s = α × (125 GeV) 2 in the two loop routines.

(m Q ) 33 =

 33 1500 GeV, (m ū) 33 = 1000 GeV,(5.3.1)

1 [GeV 2 ]Figure 5 . 2

 1252 Figure 5.2 The lightest scalar mass squared for the parameter point dened by eq. (5.3.2) when calculating with and without D-term contributions.

Figure 5 . 3

 53 Figure 5.3 The lightest Higgs mass at the two-loop level for the parameter point dened by eq. (5.3.2) for dierent methods to regulate the two-loop corrections.

Figure 5 . 4

 54 Figure 5.4 The change in the Higgs mass in GeV due to the two-loop corrections involving the new Yukawa-like interactions g(1,2)(u,d) . On the top, we used tan β = 1 at M M and on the bottom tan β = 10. The left plots are with the consistent tadpole solutions, the right ones without.

  contributions from the gluino. The size of the two-loop corrections proportional to the g(1,2)(u,d) couplings in the (M M , M F ) plane is shown in gure 5.4 for tan β = 1 and 10 for a calculation with and without the consistent tadpole solutions explained in Supersymmetric and non-supersymmetric models without catastrophic Goldstone bosons

Figure 5 . 5

 55 Figure 5.5 Lightest Higgs mass m 2 h as a function of the renormalisation scale Q, considering only the running of SM parameters. In other words, for this gure,the values of the BSM parameters λ i , m 2 12 , and tan β given in eq. (5.5.11) are considered to be input values at the scale Q at which the Higgs mass is computed, instead of being evolved from 160 GeV to Q. Red curve: tree-level; Blue dot-dashed curve: one-loop order; Green dashed curve: two-loop order.

λ

  i couplings to ensure a two-loop Higgs mass of 125.09 GeV, at scale Q = 160 GeV, together with tree-level alignment and nd the following values λ 1 = λ 2 = 0.0911, λ 3 = 0.3322, λ 4 = 0.8000, λ 5 = -0.9500 m 2 12 = -50 000 GeV 2 , tan β = 50.

Figure 5 . 6

 56 Figure 5.6 Lightest Higgs mass m 2 h as a function of the renormalisation scale Q, taking into account the running of all parameters both SM and BSM ones

Figure 5 . 7

 57 Figure 5.7 -1/t α as a function of the o-diagonal mass term m 2 12 (upper left), and of quartic couplings λ 3 (upper right), λ 4 (lower left) and λ 5 (lower right) at each order in perturbation theory. For each plot we vary the parameters as follows:

  4.19) to(1.4.23), the input given in eq. (5.5.12) translates into the following 7 set of quartic couplings: λ 1 = 2.831 , λ 2 = -2.134 , λ 3 = 7.974 , λ 4 = -0.660 , λ 5 = 0.753 , (5.5.13) which full the tree-level unitarity constraints [310312].

Figure 5 . 8 Figure 5 . 9

 5859 Figure 5.8 The dependence of the lightest scalar mass on the renormalisation scaleQ, considering that the quartic couplings of eq. (5.5.13) are used as input at the scale Q = m t . Left: m h (Q) at tree-, one-loop, and two-loop levels; Right: ∆m h ≡ m h (m t )m h (Q) at tree-, one-loop, and two-loop levels

4 m 5 1000Figure 5 .

 55 Figure 5.10 The SM-like Higgs mass at tree-, one-and two-loop level for m 5 = 1 TeV and as a function of s H . The results are shown for two dierent choices for the renormalisation Q scale, namely Q = m t and Q = m 5 .

Figure 5 .

 5 Figure 5.11 First row: absolute size of the SM-like Higgs mass in the Georgi-Machacek model as a function of s H and m 5 and including one-(left) and two-loop (right) corrections. Second row: the size of the one-(left) and two-loop (right) corrections.

Figure 5 .

 5 Figure 5.12 The size of the one-(left) and two-loop (right) corrections in the (s H , m 5 )

,Figure 6 . 1

 61 Figure 6.1 Values of the running quartic couplings at the scale Q = 10 6 GeV using one-loop (left) and two-loop RGEs (right) as function of the matching scale atwhich the quartic couplings were calculated. The labels (n, m) refer to n-loop level matching of the quartics and m-loop RGEs. We choose the parameters of the singlet extended SM at the matching scale to be m h = 125 GeV, m H = 400 GeV, tan α = 0.3 and v S = 300 GeV. Cubic terms were set to zero to ensure a scale invariant input. Note that the y-axes ranges are dierent on each sub-gure.

8 ΛFigure 6 . 3

 863 Figure 6.3 Dierence in the predicted cut-o scale depending on the matching performed as a function of the singlet VEV v S and the heavy CP-even Higgs mass m H . Left: We show the ratio of the obtained cut-o given matching at N versus N -1 order using the RGEs at N -loop order. The coloured(grey) contours use the

Figure 6 . 4

 64 Figure 6.4 Running of the Higgs quartic coupling as a function of the renormalisation scale Q, having taken λ SH = 0.28, λ S = 0.1 and m S = 500 GeV. The value of λ(m t ) is obtained by requiring m h = 125.15 GeV, with dierent orders of matchings depending on the curve. The solid line corresponds to the use of two-loop matching and two-loop RGE running, the dashed line to tree-level matching and two-loop RGEs, and the dotted line to tree-level matching and one-loop RGEs.

Fig. 6 .

 6 Fig.6.4 shows an example in which the order of the matching performed to extract λ causes dierences in the stability of the vacuum of the theory. Indeed, while the curve with two-loop matching and two-loop RGE running (solid line) crosses to negative values of λ for 10 10 GeV Q 10 16 GeV the curve with tree-level matching (dashed line) does not, because of the negative shift to the initial value of λ at scale Q = m t at two-loop order. Two-loop corrections to the matching of λ may exclude some

Figure 6 . 5

 65 Figure 6.5 Dierent phases in the Z 2 SSM shown in the λ SHλ plane, where the couplings are taken at scale Q = m t . The orange shaded region of parameter space corresponds to UV-complete theories, i.e. none of the three quartic couplings (λ, λ SH , λ S ) become non-perturbative and the constraints from unitarity are not violated before the Planck scale; the black shaded region corresponds to theories with stable vacua. The thin blue lines give λ as a function of λ SH when imposing m h = 125.15 GeV with a matching condition at respectively tree-level (dotted curve), one-loop order (dashed curve) and two-loop order (solid line). The other parameters of the scalar sector are λ S = 0.1, and M S = 500 GeV.

Fig. 6 .

 6 Fig. 6.5 shows how both types of transitions occur in this model. The dierent domains in this gure were obtained as follows: we start with values of the couplings, at scale Q = m t , in the range λ ∈ [0, 0.35] and λ SH ∈ [0, 4], and take λ S = 0.1 and M S = 500 GeV. We then use two-loop RGEs to run the couplings up to the Planck scale, and we verify whether the Higgs quartic λ becomes negative at any point, and whether perturbativity or unitarity are lost below the Planck scale. The left pane of Fig. 6.5

Figure

  Figure 6.6 Simplied comparison between the running of λ in the SM with and

Figure 6 . 7

 67 Figure 6.7 Contours of the scale Λ 0 at which λ runs negative with full two-loop RGEs

  6.8. The background shading indicates the scale Λ 0 at which λ crosses zero as a function of 0.75 0.80 0.85 0.90 0.95 1.00 Y t

Figure 6 . 8

 68 Figure 6.8 The stale of metastability Λ 0 in the case of large Y t using two-loop running with two-loop matching for λ. The black contours show the size of Λ 0 with respect to using one-loop matching.

Figure 6 . 9

 69 Figure 6.9 The size of the quartic couplings at m t at LO (blue dotted), NLO (grey dashed) and NNLO (blue solid) as a function of m H ± . Here, we set the other physical masses and mixing angle to m h = 125 GeV, m H = 750 GeV, m A = 730 GeV and tan α = -0.71. The other model parameters are tan β = 1.4 and m 2 12 = -500 2 GeV 2 . On the right, we show the relative dierences δλ LO i ≡ |(λ (T ) iλ (1) i )/λ (T ) i | and δλ NLO i

Figure 6 .

 6 Figure 6.10 RGE running of the individual quartic couplings λ 1 (black) and λ 3 (blue) for the parameter point dened by m H = 511 GeV, m A= 607 GeV, m H ± = 605 GeV, t β = 1.45, t α = -0.82, m 2 12 = -(250GeV) 2 , using n-loop level matching and m-loop RGEs. The dotted lines stand for (T, 1), dot-dashed for (1,1), dashed for (1,2) and full lines for[START_REF] Braathen | Avoiding the Goldstone Boson Catastrophe in general renormalisable eld theories at two loops[END_REF][START_REF] Braathen | Avoiding the Goldstone Boson Catastrophe in general renormalisable eld theories at two loops[END_REF]. Coupling values of of ±4 π are indicated by a red solid line. In addition, we display the largest eigenvalue of the scalar 2 → 2 scattering amplitude in purple. The upper bound of 8π is indicated by a dashed red line. We used the Yukawa scheme of type II.

4 Λ

 4 (T,1) /Λ[START_REF] Braathen | Avoiding the Goldstone Boson Catastrophe in general renormalisable eld theories at two loops[END_REF][START_REF] Braathen | Avoiding the Goldstone Boson Catastrophe in general renormalisable eld theories at two loops[END_REF]

Figure 6 .

 6 Figure 6.11 Comparison of the cut-o scales between using tree matching and oneloop RGEs and both two-loop matching and RGEs for a 2HDM region with large m 2 12 = -(750 GeV) 2 . The loop-level spectrum was evaluated taking the treelevel values m H ± = 1.14 TeV and t α = -0.95 as inputs. The ratio of the loop-

t2 ) 2 - 9 ) 2 t1 ↔ m 2 t2,

 2922 The derivatives of ∆V O i that involve m 2 t2 can be trivially obtained from the ones in eqs. (A.5)(A.7) by means of the replacement m while the derivatives with respect to all other combinations of eld-dependent parameters vanish.

A

  U (x, y) = -1 + x log xy log y xy = P SS (x, y) → A U (x, x) = log x, x) -(x + y) log 2 y + 4x log x -4y log y + 2x log x log y -2(x + y)Li 2 1 -x) 8(x log xy log y) + (x + y) log 2 xlog 2 y-2(yx) log x log y -(x + y) Li 2 1written the B U coecient in two ways, one for computational simplicity, and the other to explicitly show the symmetry in x ↔ y. The limit as x → 0 can be smoothly taken to give U (0, 0, 0, u) = (log u -1) log(-s) -

u→0V

  (x, u, y, z) -1 sx ∂ ∂u I(u, y, z) .

(B. 3 . 2 )

 32 On the other hand, we require a slightly dierent regularised function:Ṽ (x, y, z) ≡ lim u→0 -V (x, u, y, z) + P SS (y, z)B(u, x ) .

(B. 3 . 3 )

 33 For the case x = 0, we can simply extract the result at vanishing external momentum:lim s→0 V (x, z, u) = lim y→0 -U 0 (x, y , z, u) -1 x P SS (z, u) log y = I(x, z, u) -I(0, z, u) x 2 = -1x U 0 (x, 0, z, u).

(B. 3 . 4 )

 34 Then constructing Ṽ gives lim s→0 Ṽ (x, z, u) =lim s→0 V (x, z, u) -1 x R SS (z, u) + P SS (z, u)(log x -1) = 1x U 0 (x, 0, z, u) + R SS (z, u) + P SS (z, u)(log x -1) .(B.3.5)

  y, z, u) =k U U U (x, y, z, u) + k U T 1 T (x, z, u) + k U T 2 T (u, x, z) + k U T 2 T (z, x, u) + k U S S(x, z, u) -1 2 (A(x) + A(z) + A(u) + I(y, z, u)) + k U B B(x, y) + k U (B.3.6)

  Goodsell and Pietro Slavich, Leading twoloop corrections to the Higgs boson masses in SUSY models with Dirac gauginos. Johannes Braathen, Mark Goodsell and Florian Staub, Supersymmetric and non-supersymmetric models without catastrophic Goldstone bosons. Eur. Johannes Braathen, Mark Goodsell, Manuel Krauss, Toby Opferkuch and Florian Staub, N -loop running should be combined with N -loop matching.

	Phys. J. C 11 (2017) 77: 757, arXiv:1706.05372 [hep-ph],
	that constitute chapters 4 and 5; and nally
	• [4] Phys. Rev. D 97, 015011, arXiv:1711.08460 [hep-ph],

JHEP 09 (2016) 045, arXiv:1606.09213 [hep-ph], that corresponds in most part to chapter 3; • [2] Johannes Braathen and Mark Goodsell, Avoiding the Goldstone Boson Catastrophe in general renormalisable eld theories at two loops. JHEP 12 (2016) 056, arXiv:1609.06977 [hep-ph], and • [3]

  1.1 The Standard Model and the Higgs sector and hard modes of the Goldstone bosons.7 Note that by soft and hard Goldstones we mean Goldstones of small momentum k 2 ∼ m 2 G or large momentum k 2 m 2

  The Higgs boson and Physics beyond the Standard Model eqs.(1.1.26) and (1.1.27)) and then by performing a double Legendre transform of the connected generating functional (i.e. modifying eq. (1.1.28)). It is found to be expressed in terms of background elds and dressed (i.e. renormalised) propagators,

  The Higgs boson and Physics beyond the Standard Model1.2.2 The stability of the electroweak vacuumAnother puzzle of the Standard Model that relates to the Higgs sector is the question of the stability of the electroweak vacuum. In the previous section, we have discussed the existence of minima of the Higgs potential that break the electroweak gauge symmetry,

	at tree-level (see eqs. (1.1.5) and (1.1.7)) or once quantum corrections have been taken
	into account (see eq. (1.1.36)). Nevertheless, one could also ask if other minima may
	exist, in particular for large Higgs eld values, and whether such minima could be more
	stable than the minimum associated with v	246 GeV. For large eld values i.e.
	h	v it is necessary to use a renormalisation-group-improved eective potential,
	which can be approximated to be

  1.3.1.1 Fermions in two-component notationIn the context of SUSY, the use of two-component Weyl spinors will prove greatly useful, instead of four-component Dirac spinors. Any Dirac spinor ψ can be written as

Table 1 .

 1

	Supermultiplet	Spin-0	Spin-	1 2	Spin-1
	Chiral supermultiplet	Complex scalar	Weyl fermion	-
	Gauge supermultiplet	-	Weyl fermion	Massless
			(gaugino)	gauge boson

2 Possible representations of the SUSY algebra (not including gravity).

  The Higgs boson and Physics beyond the Standard Model • from the above equation, it follows that if a vacuum state |vac is not invariant under some SUSY transformation, i.e. if there exists an α or α such that Q α |vac = 0 and/or Q α|vac = 0 then the energy of the vacuum state is non-zero

	vac|H|vac > 0	(1.3.42)

  2 . Such soft masses will appear for squarks, sleptons and Higgses (see e.g. eq.(1.3.61) in the case of the MSSM).

	The Higgs boson and Physics beyond the Standard Model
	(A)	(B)	(C)
	Figure 1.2 One-loop diagrams contributing to the light scalar self-energy. Thick lines
	represent heavy states (scalars and fermions) while thin lines are for light states.
	Finally, other types of SUSY breaking terms can exist in models of extended Super-
	symmetry, for example supersoft terms in Dirac gaugino models, which we will study
	in section 1.3.5.

  First of all, as the electroweak symmetry breaking is still SU (2) L × U (1) Y → U (1) QED the number of Goldstone bosons is the same as in the SM, i.e. two charged Goldstones G ± and a neutral (pseudoscalar) one G 0 . G + must then be expressed in terms of H +

	Higgs doublet masses m 2 Hu and m 2 H d	, or to solve for |µ| 2 and B µ .
	Now that we know what the vacuum expectation values of the doublets are, we can con-
	sider how to relate the gauge eigenstates H + u , H 0 u , H -d , and H 0 d to the mass eigenstates.
	and (H -d ) * , together with another charged state H + , a charged Higgs boson	u

  ± become very heavy and decouple from the model. At lower energies, only the light state h is then left, and it becomes Standard Model-like i.e. its interaction to SM particles become the same as in the SM. Such a situation, where one of the CP-even Higgs mass eigenstates is aligned in eld-space with the (complete) Higgs VEV v therefore becoming SM-like is called alignment. In terms of the above dened mixing angles, alignment implies that

	.78)
	while the masses of the other Higgs bosons can grow in principle to arbitrarily large
	values with B µ . Note that this bound can be derived easily by taking the derivative of m 2 h from eq. (1.3.77) with respect to m 2 A and observing that it is always positive and only tends to zero in the limit m 2 A m 2 Z . At rst sight, equation (1.3.78) seems to be
	incompatible with the observed Higgs mass of 125 GeV, but it was shown in the early
	1990's [7375] that radiative corrections from top and stop loops can be huge and
	pull the lightest Higgs mass up by a signicant amount (we will discuss this in more
	detail in the next chapter). The bound (1.3.78) on m 2 h is saturated in the so-called decoupling limit in which m 2 A m 2 Z so that the states H, A, H

  The Higgs boson and Physics beyond the Standard Model1.3.4.3 Gauginos in the MSSMAs we will shortly turn our focus to models with Dirac masses for gauginos, it is useful to rst shortly review gauginos and their masses in the context of the MSSM. Every gaugino in the MSSM comes as a Weyl spinor i.e. two complex degrees of freedom in the adjoint representation of the dierent components of the SM gauge group as the gauge bosons. Unlike SM fermions, fermionic superpartners of SM gauge bosons obtain masses from soft SUSY breaking terms such terms being allowed because gauge superelds, and hence the gauginos, are not chiral. More specically, these are Majorana mass terms see the fourth line of eq. (1.3.61) that can be generated by high-scale operators such as those shown in eqs.(1.3.47) and(1.3.48).After the electroweak symmetry breaking, the neutral electroweak gauginos B, W 3 mix with the neutral higgsinos h0 u , h0 d to form four neutralino mass eigenstates, while the charged electroweak gauginos W ± = ( W 1 ∓ i W 2 )/ √ 2 form, together with the charged higgsinos h+

u , hd , two chargino mass eigenstates. The neutralino and chargino sector can have a rich phenomenology: most importantly, the lightest neutralino is often the LSP and consequently a possible candidate of Dark Matter however its couplings to the Higgs scalars are always related to the electroweak gauge couplings and are hence small and negligible for our concern.

  2 H and the mixing angle α as inputs,The Higgs boson and Physics beyond the Standard Model and use them to compute the quartic couplings λ, λ SH , λ S . At tree-level, one can nd the quartic couplings to be

  1.4 Non-supersymmetric extensions of the Standard Model preserves custodial symmetry. Custodial symmetry is an accidental symmetry that appears in the SM, noticed because the Veltman ρ parameter, dened as While the above result provides a strong constraint for BSM models, it also tends to indicate that some additional (approximate) symmetry exists to protect ρ from receiving large quantum corrections. In fact, it turns out that the SM Higgs kinetic term (1.1.11) and potential (1.1.5) exhibit an approximate symmetry under SU (2) L × SU (2) R transformations which becomes exact in the limit g → 0. Upon electroweak symmetry breaking, it is broken down to an approximate SU (2) symmetry, called the custodial symmetry. The Georgi-Machacek model is built to maintain this symmetry in presence of an extended Higgs sector and in particular of doubly charged Higgs bosons that may have rich phenomenological properties.

	ρ ≡	m 2 W Z cos 2 θ W m 2	(1.4.24)
	is exactly equal to 1 at tree-level. Although this equality does not hold when radiative
	corrections are included, the ρ parameter has been measured to be extremely close to
	1, indeed
	ρ exp. = 1.0048 ± 0.0022 .	(1.4.25)

  ). But from eq. (2.2.29) it follows that

  2.3 State-of-the-art of Higgs mass calculations computations[START_REF] Haber | Can the mass of the lightest Higgs boson of the minimal supersymmetric model be larger than m(Z)?[END_REF] 155163]. Once radiative corrections are introduced in the real MSSM, mixing appears between the two CP-even Higgses h and H (dened in section 1.3.4.2). The easiest way to proceed, for a diagrammatic calculation, is then to compute a loop-corrected scalar mass matrix in the gauge-eigenstate basis {S 1 , S 2 } i.e. the basis formed by the two real components of H 0 d and H 0

	u , decomposed respec-
	tively as

  Precision calculations of the Higgs boson massWhen dening a loop-corrected mixing angle as in the above equation, it is necessary to specify which mass matrix M 2 S loop (p 2 ) the angle diagonalises i.e. to what value of p 2 it corresponds. We will compute it for p 2 = m 2 h , m 2 h being the lightest loop-corrected mass eigenvalue (corresponding to the lightest loop-corrected mass eigenstate that we denote h). , one can compute the corrections to the CP-odd states of the Higgs sector the pseudoscalar Higgs and the neutral Goldstone boson. As for the CP-even Higgses, we can obtain a loop-corrected pseudoscalar mass matrix M 2 P loop (p 2 ) in the gauge eigenstate basis {P 1 , P 2 }, formed of the two imaginary components of H 0 u and H 0

	Similarlyd , as
	cot 2α loop ≡	M 2 S	loop 11 -M 2 S S 12 2 M 2 loop	loop 22	.	(2.3.7)

  u ), we nd that the only non-vanishing contributions to the minimum conditions of the potential and to the Higgs mass matrices are

	∂∆V ∂S 2 min	=	√ 2 y t m t G ,			(3.1.49)
	∆M 2 S 22	= 2 y 2 t m 2 t F 1 ,			(3.1.50)
	∆M 2 S 66	=	∆M 2 P 66	=	y 2 t m 2 t1 -m 2 μ2 u t2	F ,

(3.1.51)

  3.1.6 Obtaining the O(α b α s ) corrections Our DR computation of the O(α t α s ) corrections allows us to obtain also the two-loop O(α b α s ) corrections induced by the bottom/sbottom sector, which can be relevant for large values of tan β. To this purpose, the substitutions t → b, u → d, ∂∆V /∂S 1 ↔ ∂∆V /∂S 2 , ∆M 2

	S,P 11	↔ ∆M 2 S,P 22	, ∆M 2 S,P 1k	↔ ∆M 2 S,P 2k

  .[START_REF] Grisaru | Improved Methods for Supergraphs[END_REF] where v ≈ 174 GeV, and all other elds have negligible mixing with the lightest scalar h, which is SM-like. We can then approximate the correction to the squared mass m 2

	h
	as
	∆m 2 h ≈ cos 2 β ∆M 2 S 11 + sin 2 β ∆M 2 S 22 + sin 2β ∆M 2 S 12 .

  .1.75) Thus, we see that in the OS scheme the contribution to ∆m 2 In contrast, for the contribution of the lightest octet scalar O 2 , whose squared mass does not grow with m 2 g, the unexpanded formulae in eqs. (3.1.72) and (3.1.73) should always be used. However, in the total correction to m 2 h see eq. (3.1.69) the m 2 g enhancement of ∆m 2

	h from two-loop diagrams involving the heaviest octet scalar O 1 does not grow unphysically large when m 2 g in-creases, because the ratio m 2 g/m 2 O 1 tends to 1/4. h O 2 is compensated 2 for by the factor s 2 φ

O

, which, as discussed in section 1.3.5.4, is in fact suppressed by

2

  Hu , m 2 H d } in the MSSM, but not {µ, B µ } chosen as parameters to solve the tadpole equations. On the other hand, we give expressions for the shifts to the tadpoles and self-energies when fermion masses depend on the {x i } in appendix C, and plan to implement these in future.The routines to calculate the consistent tadpole solution are generated during the output of SPheno code; this is fully automatised beginning with SARAH version 4.12.0 and the user can obtain a SPheno version for non-supersymmetric models as before with the dierence that two-loop mass corrections are now included. The only requirements are recent versions of SARAH and SPheno which are available at www.hepforge.org.

	Supersymmetric and non-supersymmetric models without catastrophic Goldstone
	bosons
	7 410 0	# Regulator mass
	The new features can now be adjusted in the Block SPHENOINPUT in the Les Houches
	input le:
	Block SPhenoInput #
	...
	7 0	# Skip two loop masses : True / False
	8 3	# Choose two -loop method
	150 1	# Use consistent tadpole solution : True / False
	151 1 True / False # Generalised effective potential calculations : →

Table 5 .

 5 is to nd for each order what value of the quartic Higgs coupling we need to obtain m h = 125.15 GeV, and our results are given in table 5.2. 2 Values of the Higgs quartic coupling λ extracted from m h = 125.15 GeV, at tree level, one loop and two loops. The methods we used are explained in the caption of table 5.1. Values found using SARAH-4.12.0 and SMH-1.0.

			λ in 1 st approach	λ in 1 st approach
	Loop order	λ found in [25] without consistent	with consistent	λ in 2 nd	λ with SMH
			tadpole solutions	tadpole solutions	approach
	Tree level	0.12917	0.12786	0.12786	0.12786	0.12786
	One loop	0.12774	0.12647	0.12647	0.12669	0.12580
	Two loops	0.12604	0.12613	0.12619	0.12614	0.12541

  The values we nd for m 2 h with the two codes for the two dierent choices of gauge parameter and xed values of Q and λ are given in table 5.3. The rst observation that can be made from these results is that the Higgs mass shows residual dependence on the gauge m 2 h varies by about 50 MeV between ξ = 1 and ξ = 0.01. This is explained mainly 4 by the dierence in the calculation of the MS value for the electroweak VEV in SPheno between the Feynman gauge and other gauges: in the case of Feynman gauge one-loop corrections from δ V B as well as two-loop corrections from δ r are included which are not available in a general R ξ gauge (see the appendix A of[START_REF] Staub | Improved predictions for intermediate and heavy Supersymmetry in the MSSM and beyond[END_REF] for details of the matching in Feynman gauge). On the other hand, in an R ξ gauge, the VEV is calculated from M 2,MS

	Z of the Z-boson at one-loop. What is more interesting is that the agreement between = 1/4(g 2 1 + g 2 2 )v 2 = M 2,pole Z ZZ where Π T ZZ is the transversal self-energy -Π T
	the two codes improves greatly once we use the Landau gauge in SPheno; indeed the
	dierence in the Higgs mass results is reduced from approximately 0.4 GeV to less than
	0.05 GeV.
	A second point we can study is the eect of the two-loop momentum dependence and
	two-loop electroweak corrections. Let us introduce the notation for calculating the
	pole mass via

  2 log(-s) λ 2 -14 + 18 log(m 2 h ) + 3 log(-s) (s) . But if we set s 2 fixed = -Q 2 in our routines we should cancel the divergent part exactly, and leave us only with Π

		-2y 2 t λ + (y 2 t -λ) log(m 2 t ) ,	(5.2.8)
	so we can see there are several remaining pieces that must be cancelled by
	div Π	(2) hh,electroweak (s) + δ (2)

Table 5 .

 5 [START_REF] Braathen | Supersymmetric and non-supersymmetric models without catastrophic Goldstone bosons[END_REF] Comparison of two-loop Higgs masses calculated with the codes SPheno and SMH, for dierent choices of gauge in SPheno and switching on and o the two-loop momentum dependence in SMH. The renormalisation scale is xed to Q = 173.34 GeV, and the Higgs quartic coupling is λ = 0.12604 and is not varied (the idea being to illustrate the importance of the choice of scale, rather than the stability of the result). All other inputs for SMH are taken to the same values as in SPheno. In SPheno the only two-loop momentum dependence is from pseudoscalar diagrams and only a generalised eective potential approach (see main text)

	Supersymmetric and non-supersymmetric models without catastrophic Goldstone
	bosons
			SPheno		SMH
	ξ	1	0.01	0.01	0
	v [GeV]	247.494		246.914
	y t	0.939	0.939	0.940
	(g 3 , g 2 , g Y )		(1.1654, 0.6452, 0.2780)
	2 momentum dependence Light SM fermions	partial s = m 2 h | tree s = m 2 partial h | tree s = m 2 partial h | tree iterative full yes yes no no	none s = 0 no
	m 2 h [GeV]	125.083	125.134	125.133	125.176	125.121

Table 5 .

 5 [START_REF] Englert | Broken Symmetry and the Mass of Gauge Vector Mesons[END_REF] The Higgs masses in the NMSSM (in GeV) for the parameter point dened by eq. (5.3.1) for dierent choices for the two-loop corrections.

	-1	OS	OS+Tad

Table 5 . 6

 56

				5.6 Georgi-Machacek Model
		tree	one-loop	two-loop
	m h 1 [GeV]	125.00	210.45	< 0
	m h 2 [GeV]	1000.00	950.56	916.96
	m h 3 [GeV]	1054.67	975.20	954.03
	m A 1 [GeV]	1049.31	998.41	896.13
	m H + 1 m H + 2 m H ++ [GeV] 1000.00 [GeV] 1000.00 [GeV] 1049.31	950.80 998.21 951.55	---

Table 6 .

 6 [START_REF] Braathen | Leading two-loop corrections to the Higgs boson masses in SUSY models with Dirac gauginos[END_REF] Values of the quartic couplings and the cut-o for dierent combinations of parameters, λ (n) , and RGEs, β (m) at the loop-orders n and m respectively. Λ 4π is the scale at which the quartic coupling rst exceed 4π while Λ unit. 4π is the naive 4π cut-o augmented by the unitarity constraint of Eq. 6.2.4. The observables are xed at the n-loop order to be m h = 125 GeV, m H = 700 GeV, tan α = 0.1 while the remaining input parameters are chosen as κ 1 = 0 GeV, κ 2 = 2000 GeV, v S = 175 GeV.

	(T, 2) (1, 1) (1, 2) (2, 1) (2, 2)	0.34 0.33 0.32	1.1 0.24 0.18	-1.1 -0.97 -0.94	6.4 • 10 3 8.0 • 10 6 6.4 • 10 8 1.3 • 10 12 5.1 • 10 10 1.0 • 10 14	3.2 • 10 3 1.3 • 10 4 3.2 • 10 8 2.5 • 10 9 2.5 • 10 10 2.0 • 10 11

  13 GeV (10 12 and 10 15 GeV). The

				6.2 Models and results
	10 5	10 8	10 11	10 14	10 17

  SH v 2 , m 2 h ≡ λv 2 which are the tree-level squared masses of the singlet and Higgs respectively, while a complete list of the denitions for our loop functions can be found in appendix B.1 (based upon the basis dened in

	(6.2.5)
	Here we have dened m 2 S ≡ M 2 S + 1 2 λ

S ) .

Table 6 .

 6 [START_REF] Braathen | Avoiding the Goldstone Boson Catastrophe in general renormalisable eld theories at two loops[END_REF] Values of the Higgs quartic λ obtained from matchings at tree-level, oneloop and two-loop orders, for dierent choices of M S and λ SH . The singlet quartic coupling λ S is set to be 0.1.

		0.25	0.2610	0.2627	0.2552
		3	0.2610	0.2166	0.2099
	500	0.25 3	0.2610 0.2610	0.2623 0.1885	0.2551 0.1794
	1000	0.25 3	0.2610 0.2610	0.2651 0.1548	0.2546 0.1385

  1 2 λ|H| 4 , the one-loop matching of the Higgs quartic coupling at scale µ givesλ SM = λ VLQ -1 16π 2 (Y t + Ỹ

		t ) 2 (5Y 2 t -2Y t	Ỹ t + 5 Ỹ 2 t ) + 6(Y 4 t + Ỹ 4 t ) log	M 2 Q µ 2
	+	2λ VLQ 16π 2 (Y t -Ỹ t ) 2 + 3(Y 2 t + Ỹ 2 t ) log	M 2 Q µ 2 .

(6.2.10)

  .e. it decreases more slowly when increasing the scale compared to the SM. In addition, we also obtain a shift in α MS S (m t ) of

	2 β (1) gs = -7 +	4 3	n T +	2 3	n Q g 3 s → -5g 3 s ,	(6.2.12)
	α MS S →	1 -		α MS S

iα MS S π log(M Q /m t ) (6.2.13)

4 Although this leads to a stable lightest VL quark, there could for instance be couplings to a hidden sector, leading to a relaxation of the direct collider constraints.

  Adapted to the Dirac-gaugino case, it reads Derivatives of the two-loop eective potential in models with Dirac gauginos where Q is the renormalisation scale, the function Φ(x, y, z) is dened in appendix D of ref.[START_REF] Degrassi | On the radiative corrections to the neutral Higgs boson masses in the NMSSM[END_REF], and we used the shortcut∆ gi ≡ (m 2 gim 2 tm 2 t1 ) 2 -4 m 2 t m 2 t1 .The derivatives of the octet-scalar contribution ∆V O i , computed at the minimum of

										(A.3)
	the potential, are
	∂∆V O i ∂c 2 2 θt	= -2 I(m 2 t1 , m 2 t1 , m 2 O i ) + I(m 2 t2 , m 2 t2 , m 2 O i ) -2I(m 2 t1 , m 2 t2 , m 2 O i ) , (A.4)
	∂∆V O i ∂m 2 t1	= 2 ln	m 2 t1 Q 2 -1	2	+ 2 s 2 2θt ln	m 2 O i m 2 t1	ln	m 2 t1 m 2 t2
		-2 c 2 2θt Φ(m 2 t1 , m 2 t1 , m 2 O i ) + s 2 2θt	m 2 O i -m 2 t1 m 2 O i	+ m 2 t2	Φ(m 2 t1 , m 2 t2 , m 2 O i ) ,
										(A.5)
	∂ 2 ∆V O i (∂m 2 t1 ) 2 = -	4 s 2 2θt m 2 t1	+	4 c 2 2θt O i -4m 2 m 2 t1	m 2 O i t1 m 2	ln	m 2
										∂∆V αs ∂m 2 t	=	2 i=1	R 2 1i	∂∆V gi t ∂m 2	,	(A.1)
	with
	∂∆V gi ∂m 2 t	= 2 m 2 t 3 -4 ln	m 2 t Q 2 + 3 ln 2 m 2 t Q 2
				+ 2 m 2 t ln	m 2 gi m 2 t	+ m 2 t1 2 -ln	m 2 t Q 2 -ln	m 2 gi Q 2	ln	m 2 t1 Q 2
				+ 2 (m 2 t1 -m 2 t ) -	m gi m 2 t1 m t	s 2θt	ln	m 2 t Q 2 ln	m 2 gi Q 2
				-2 m 2 gi 3 -2 ln	m 2 gi Q 2 -m 2 t1 4 -	5 m gi s 2θt m t
				-	m gi s 2θt m t		3 m 2 t -m 2 gi ln	m 2 gi m 2 t	+ m 2 t1 4 -ln	m 2 t Q 2 -ln	m 2 gi Q 2	ln	m 2 t1 Q 2
				+ 2	m 2 gi m 2 t	(m 2 gi -m 2 t -m 2 t1 ) -	∆ gi m 2 t
							+	m gi s 2θt m t	m 2 t -m 2 gi -m 2 t1 +	∆ gi 2 m 2 t	Φ(m 2 t1 , m 2 gi , m 2 t )
				+ m 2 t1 → m 2 t2 , s 2θt → -s 2θt ,	(A.2)

  Consistent solution of the tadpole equations with shifts to fermion masses while the shift to the scalar self-energy isδΠ (2),F ij = -2Re[y KLi y K Lj ] M * KJ δM K J + δM * KLi y K L j M * IK M * LL ] M * KJ δM IJ + δM * KJ M IJ C(s, s, 0, m 2 K , m 2 L , m 2 I ).To illustrate this, consider the MSSM, where the tadpole equations read(|µ| 2 + m 2 Hu )v u -B µ v d + (|µ| 2 + m 2 H d )v d -B µ v u -

											KJ	M K J
		× P SS (m 2 K , m 2 K ) -B(m 2 K , m 2 L )
				-m 2 K + m 2 K -s C(s, s, 0, m 2 K , m 2 L , m 2 K )
	+ 4Re[y KLi y K L j δM * KK M * LL ]B(m 2 K , m 2 L )
	+ 4Re[y (C.0.5)
						1 8	(g 2 Y + g 2 2 )(v 2 u -v 2 d )v u = -	∂∆V ∂v u
						1 8	(g 2 Y + g 2 2 )(v 2 u -v 2 d )v d = -	∂∆V ∂v d	.	(C.0.6)
	Solving for |µ| 2 we have
	|µ| 2 = -	M 2 Z 2	+	1 c 2β	m 2 Hu s 2 β -m 2 H d c 2 β +	1 v	s β	∂∆V ∂v u	-	1 v	c β	∂∆V ∂v d	,	(C.0.7)
	so we have
				δµ =	1 2µ * vc 2β	s β	∂∆V ∂v u	-c β	∂∆V ∂v d	.

(C.0.8)

This would be especially clear in the unitary gauge, in which the Goldstone bosons would not appear any more after the EWSB. However, the unitary gauge is a very inconvenient choice for calculations, and we will use instead R ξ gauges, as will be discussed in what follows.

Note also that when we introduce quantum corrections, the value of the gauge parameter ξ in the Landau gauge is not renormalised and remains 0 see e.g.[START_REF] Collins | Renormalization[END_REF].

In the CMSSM, for example, the soft squark masses unify at the GUT scale, and under RGEs running the soft mass of the stop decreases slower than for the other squarks because the contribution from yt.

A more detailed discussion of renormalisation (and in particular on-shell renormalisation), taking into account mixing between several scalars can be found e.g. in[START_REF] Pilaftsis | Resonant CP violation induced by particle mixing in transition amplitudes[END_REF].

(2.2.12) with,Π X (p 2 ) = Π X (p 2 ) + δ CT m 2 X -(p 2m 2 X )δ CT Z X , (2.2.13)and where Π X is the non-renormalised self-energy. It is important to note that, in the above equations, both m 2 X and Π X are scheme-dependent.

We recall that αt ≡ y 2 t /4π and αs ≡ g 2 s /4π.

We use here the plural form, having in mind Dirac gaugino models where there are two gluinos with masses that can a priori be dierent. In the (N)MSSM there is only one gluino and the sentence should then be changed to singular form.

Note that this does not however capture non-logarithmic corrections as we will return to in chapter

Note that we take the expectation values to be xed and loop-correct the mass-squared terms rather than vice versa as explained in more detail below eq.(1.1.36).

Indeed, the gaugeless limit (turning o the electroweak gauge couplings) completely cures the problem in the MSSM by eliminating all of the Goldstone boson couplings to the Higgs.

Since the mass was tachyonic and generally not small, we then neglected the imaginary part of the self energies/tadpoles.

Note that our Ṽ uses one fewer variable than the denition of V in[START_REF] Martin | Two loop scalar self energies in a general renormalizable theory at leading order in gauge couplings[END_REF] because in our case it always appears in the form V (u, 0, y, z).

We recall that SARAH/SPheno employs here a running VEV, extracted from the Z boson pole mass at one-loop order. The reader may refer to appendix A of[START_REF] Staub | Improved predictions for intermediate and heavy Supersymmetry in the MSSM and beyond[END_REF] for more details.

We used HiggsBounds[START_REF] Bechtle | HiggsBounds: Confronting Arbitrary Higgs Sectors with Exclusion Bounds from LEP and the Tevatron[END_REF][START_REF] Bechtle | HiggsBounds -4: Improved Tests of Extended Higgs Sectors against Exclusion Bounds from LEP, the Tevatron and the LHC[END_REF] to check that this point passes all current collider limits.

Note, negative λ2 is usually taken to be forbidden because the potential is unbounded from below.However, this only holds for the tree-level potential. If RGE eects are included, λ2 becomes positive after a few hundred GeV of running[START_REF] Staub | Reopen parameter regions in Two-Higgs Doublet Models[END_REF].

Gauge and Yukawa couplings have Ci = 1, quartic couplings Ci = 2.

Acknowledgements

The path to complete the work presented here has been long and at times dicult, and there are many people that I would now like to thank for their invaluable help, Acknowledgements Acknowledgements

Leading two-loop corrections to the Higgs boson masses in SUSY models with Dirac gauginos with

The two-loop integrals J(x, y), I(x, y, z) and L(x, y, z) entering eqs. (3.1.20) and (3.1.22) are dened, e.g., in eqs. (D1)(D3) of ref. [START_REF] Degrassi | On the radiative corrections to the neutral Higgs boson masses in the NMSSM[END_REF], and were rst introduced in ref. [START_REF] Ford | The Standard model eective potential at two loops[END_REF]. Explicit expressions for the derivatives of ∆V αs , valid for all Dirac-gaugino models considered in this chapter, are provided in appendix A.

We remark that, by using the minimally subtracted two-loop integrals of ref. [START_REF] Ford | The Standard model eective potential at two loops[END_REF], we are implicitly assuming a DR renormalisation for the parameters entering the tree-level and one-loop parts of the eective potential. Consequently, our results for the two-loop top/stop contributions to mass matrices and minimum conditions also assume that the corresponding tree-level and one-loop parts are expressed in terms of DR -renormalised parameters. We will describe in section 3.1.5 how our two-loop formulae should be modied if the top/stop parameters entering the one-loop part of the corrections are expressed in a dierent renormalisation scheme. For what concerns the parameters entering the tree-level mass matrices for scalars and pseudoscalars whose specic form depends on the Dirac-gaugino model under consideration they can be taken directly as DR -renormalised inputs at some reference scale Q, at least in the absence of any experimental information on an extended Higgs sector. Exceptions are given by the electroweak gauge couplings and by the combination of doublet VEVs v ≡ (v 2 1 +v 2 2 ) 1/2 , which in general should be extracted from experimentally known observables such as, e.g., the muon decay constant and the gauge-boson masses. As was pointed out for the NMSSM in ref. [START_REF] Mühlleitner | Two-loop contributions of the order O(α t α s ) to the masses of the Higgs bosons in the CP-violating NMSSM[END_REF], the extraction of the DR parameter v(Q) involves two-loop corrections whose eects on the scalar and pseudoscalar mass matrices are formally of the same order as some of the O(α t α s ) corrections computed here. 1 However, a twoloop determination of v(Q) goes beyond the scope of our calculation, as it requires two-loop contributions to the gauge-boson self-energies which cannot be obtained with eective-potential methods. Besides, ref. [START_REF] Mühlleitner | Two-loop contributions of the order O(α t α s ) to the masses of the Higgs bosons in the CP-violating NMSSM[END_REF] showed that, at least in the NMSSM scenarios considered in that paper, the O(α t α s ) eects on the scalar masses arising from the two-loop corrections to v are quite small, typically of the order of a hundred MeV.

Mass corrections in the MDGSSM

The MDGSSM contains a singlet S and an SU (2) triplet T a which mix with the usual Higgs elds H d and H u . In this model, the stop mixing term X dened in eq. (3.1.6) reads

.23)

where A t is the soft SUSY-breaking trilinear interaction term for Higgs and stops. We order the neutral components of the elds as Φ 0 i = (H 0 d , H 0 u , S, T 0 ) and expand them as in eq. (3.1.1). For the minimum conditions of the eective potential, eq. 1 These additional O(αtαs) eects arise from terms in the tree-level mass matrices in which v appears in combination with the singlet or triplet superpotential couplings. In contrast, in the MSSM all occurrences of v in the tree-level mass matrices are multiplied by the electroweak gauge couplings, thus they are not relevant in the gaugeless limit.

Avoiding the Goldstone Boson Catastrophe in general renormalisable eld theories at two loops

Goldstone shifts

To determine the eect on the mass diagrams, let us make the shifts using the method of an on-shell Goldstone boson. Recall that the contribution to the one-loop self-energy is Π

(1),S ij

(1),S GG (0) + λ iGl λ jG l B (s, m 2 G , m 2 l )Π

(1),S GG (0)

where B is dened in eq. (B.1.13), and These exactly cancel the divergent parts in the mass diagrams. In the case of the X and Y diagrams, they go further and leave no nite parts; for the W diagrams, what remains is

We have no further divergences in W (in particular, U 0 (x, y, 0, 0) is non-singular).

In the V diagrams there is also a nite piece that remains, since

4.3 Mass diagrams in the gaugeless limit we nd that

Since we take λ GGG = 0, we never have a divergence from n = p = G. On the other hand, when k = G we do have a divergence that needs regulating by the momentum; recalling B(s, 0, 0) = -log(-s) + 2 we can write

For the other cases we can set s = 0 and write

(4.3.13)

Momentum-regulated diagrams

There are other V SSSSS diagrams that are not regulated by the Goldstone boson shifts. While V SSSSS (x, y, z, 0, 0), V SSSSS (0, x, y, 0, z), V SSSSS (0, x, y, 0, 0), V SSSSS (x, 0, y, 0, 0) are all regular, the diagrams V SSSSS (0, 0, x, y, z) and V SSSSS (0, 0, x, 0, y) are divergent, and their expression may be found simply by using those for U (0, 0, x, y) and U (0, 0, 0, x) given in appendix B.2:

V SSSSS (0, 0, x, y, z) = 1 x U 0 (0, x, y, z) -U (0, 0, y, z) . All other V SSSSS diagrams are either regular or vanish due to the prefactor λ GGG .

The remining functions U SSSS , M SSSSS and Z SSSS require regulation by momentum:

we give expressions for the expansion of these in appendix B.2.

Avoiding the Goldstone Boson Catastrophe in general renormalisable eld theories at two loops

Fermion-scalar diagrams

The potentially singular mass diagrams are Π

, but among these there are only a subset once more that are regulated by the Goldstone boson shifts; indeed, as in the purely scalar case we nd that the topology M is purely regulated by momentum for which all of the limits of the loop functions are provided in appendix B.2. For the other two, there are exactly four diagrams to regulate, which will match exactly. They have the form [START_REF] Martin | Two loop scalar self energies in a general renormalizable theory at leading order in gauge couplings[END_REF]:

and the loop functions are dened in section B.1.1.2.

As in the scalar case, we look at the shift in the one-loop scalar mass contribution involving Goldstone bosons:

where

and compare to the relevant expressions for the loop functions:

We should deal with each of these in turn. Firstly for the W topology:

Supersymmetric and non-supersymmetric models without catastrophic Goldstone bosons

In the previous chapter, we have presented a general procedure to cure the Goldstone Boson Catastrophe in two-loop Higgs mass calculations for general renormalisable theories. Our method is based on setting the Goldstone boson propagators on-shell, and we have provided a complete set of modied loop functions for the tadpoles and self-energies that were nite. Thus, combining these results with those of [START_REF] Martin | Two loop scalar self energies in a general renormalizable theory at leading order in gauge couplings[END_REF][START_REF] Goodsell | Generic two-loop Higgs mass calculation from a diagrammatic approach[END_REF][START_REF] Martin | Evaluation of two loop selfenergy basis integrals using dierential equations[END_REF][START_REF] Martin | Two-loop scalar self-energies and pole masses in a general renormalizable theory with massless gauge bosons[END_REF] which provide fully generic expressions for the two-loop corrections to real scalar masses in supersymmetric and non-supersymmetric models, all ingredients are present to calculate Higgs masses up to two-loop order in any renormalisable model.

The generic expressions of [START_REF] Martin | Two loop scalar self energies in a general renormalizable theory at leading order in gauge couplings[END_REF][START_REF] Goodsell | Generic two-loop Higgs mass calculation from a diagrammatic approach[END_REF][START_REF] Martin | Evaluation of two loop selfenergy basis integrals using dierential equations[END_REF][START_REF] Martin | Two-loop scalar self-energies and pole masses in a general renormalizable theory with massless gauge bosons[END_REF] are already used by the Mathematica package SARAH [261266] to calculate in combination with SPheno [START_REF] Porod | SPheno, a program for calculating supersymmetric spectra, SUSY particle decays and SUSY particle production at e+ e-colliders[END_REF][START_REF] Porod | SPheno 3.1: Extensions including avour, CP-phases and models beyond the MSSM[END_REF] the Higgs masses in supersymmetric models at the two-loop level [START_REF] Goodsell | Generic two-loop Higgs mass calculation from a diagrammatic approach[END_REF][START_REF] Goodsell | Two-Loop Higgs mass calculations in supersymmetric models beyond the MSSM with SARAH and SPheno[END_REF][START_REF] Goodsell | The Higgs mass in the CP violating MSSM, NMSSM, and beyond[END_REF] for more details see section 2.4.3. Before version 4.12.0 of SARAH, the workaround for the Goldstone boson catastrophe in this setup was to introduce nite masses for the electroweak Goldstones by dropping the D-terms in the mass matrices. However, there were many regions of parameter space where the divergences reappeared (see e.g. [START_REF] Benakli | The Di-Photon Excess in a Perturbative SUSY Model[END_REF][START_REF] Goodsell | Dark matter scenarios in a constrained model with Dirac gauginos[END_REF][START_REF] Athron | Precise Higgs mass calculations in (non-)minimal supersymmetry at both high and low scales[END_REF]) and this does not work at all for non-supersymmetric models, which have no D-term potential. For this reason, we here implement in SARAH the results of chapter 4, and ll some additional technical gaps that we describe in section 5.1; in particular, we complete the basis of required loop functions. New versions of SARAH since 4.12.0 therefore now oer the possibility to calculate two-loop masses for neutral scalars in non-supersymmetric models, as well as substantially improving the calculation in supersymmetric ones. As the only non-supersymmetric model for which comparable results exist is the Standard Model, in section 5.2 we compare our new calculation against the public code SMH [START_REF] Martin | Higgs boson mass in the Standard Model at two-loop order and beyond[END_REF] and the results of Buttazzo et al. [START_REF] Buttazzo | Investigating the near-criticality of the Higgs boson[END_REF], nding excellent agreement (even if our results do not include all of the contributions included in those references). In section 5.3, we show how our new approach improves our previous calculation for supersymmetric models through the example of the NMSSM, for which our results should now be considered state of the art. We then illustrate our new routines by computing some new results in Split SUSY in section 5.4.

Supersymmetric and non-supersymmetric models without catastrophic Goldstone bosons

Lagrangian below this scale is given by

(5.4.7)

Here, g 1 and g 2 are the running gauge couplings of U (1) Y and SU (2) L and β is dened as the mixing angle of the two Higgs doublets in the MSSM (in contrast to the denition in the MSSM as a ratio of expectation values). There are important higherorder corrections to the matching conditions which are necessary to have a precise prediction for the Higgs mass at the low scale. In particular λ has been calculated including the two-loop SUSY corrections [START_REF] Bagnaschi | Higgs Mass and Unnatural Supersymmetry[END_REF][START_REF] Vega | SusyHD: Higgs mass Determination in Supersymmetry[END_REF][START_REF] Bagnaschi | Improved determination of the Higgs mass in the MSSM with heavy superpartners[END_REF]. The numerical value of these corrections depends on the many SUSY parameters at the matching scale; however, a commonly taken useful approximation is to give the scalars a common mass M mess , in which case the corrections can be given in terms of just this scale and the squark mixing. Moreover, in strict split SUSY where the fermion masses are protected by an R-symmetry (or another symmetry in Fake Split SUSY [START_REF] Benakli | A Fake Split Supersymmetry Model for the 126 GeV Higgs[END_REF][START_REF] Benakli | O)Mega Split[END_REF]) near the electroweak or TeV scale and well below M mess , the squark mixing must by very small. In which case, the leading corrections to the Higgs quartic coupling are purely electroweak at one loop, and at two loops contain no logarithmic terms meaning that they are very small (in particular since the strong gauge and top Yukawa couplings run to small values at higher scales), so using the tree-level relationship above can be good enough.

Below the scale M mess , we must run to the scale of the fermion masses, before also integrating them out, and then running to the electroweak scale in the Standard Model.

In some previous approaches, the running was performed all the way down to the electroweak scale, before calculating the Higgs mass in the full Lagrangian (5.4.2); however, it was found in [START_REF] Benakli | A Fake Split Supersymmetry Model for the 126 GeV Higgs[END_REF] that in this approach it is necessary to include the threeloop leading logarithm involving the gluino mass to obtain good agreement between the two results this is automatically resummed by the renormalisation group running in the former approach. In either case, the full contribution of the gauginos and Higgsinos to the matching conditions is only known in the literature to one-loop order [START_REF] Bagnaschi | Higgs Mass and Unnatural Supersymmetry[END_REF].

Hence in this section we are interested in the eect of the two-loop corrections to the Higgs mass stemming from the g(1,2)(u,d) couplings which have not been studied in the Supersymmetric and non-supersymmetric models without catastrophic Goldstone bosons section 4.4. We show here results for M F up to 5 TeV. In order not to increase the theoretical uncertainty in the presence of new fermions in the multi-TeV range, we made use of the functionality in SARAH to perform the Higgs mass calculation in the eective SM [START_REF] Staub | Improved predictions for intermediate and heavy Supersymmetry in the MSSM and beyond[END_REF]. For this purpose, a second matching is performed to extract λ at the renormalisation scale Q. The imposed matching condition is

i.e. we perform a matching of the Higgs pole masses as suggested in [START_REF] Athron | Precise Higgs mass calculations in (non-)minimal supersymmetry at both high and low scales[END_REF], from which an eective λ is derived. λ is then evolved to m t using three-loop RGEs of the SM. At m t the Higgs mass is calculated within the SM at the two-loop level. The additional loop-corrections discussed here enter the calculation of m Split h (M F ), and in turn that of λ SM (M F ).

We see that the additional corrections for SUSY fermions are always well below 1 GeV once the consistent solution to the tadpole equations are included. However, if those are not used, the misleading impression of sizeable corrections of a few GeV is given; it would be interesting to investigate this phenomenon further.

Two-Higgs-Doublet Model

In this section we will be interested in the eect of two-loop scalar mass corrections in the CP-conserving 2HDM. For most scans and gures presented below, we worked in the type-I 2HDM if not indicated otherwise. However as the dierence with type-II comes from the couplings of the scalars to the down-type quarks and to the leptons which are light and give much smaller contributions to the lightest Higgs mass than the top quark, we do not expect large eects on our results (even for large tan β, since the contributions typically involve the quark masses rather than just the couplings).

The alignment in Two-Higgs-Doublet Models

Instead of the Z 2 basis dened in eq. (1.4.14), it is often more convenient to work in another basis the so-called Higgs basis {H 1 , H 2 } where the neutral component of the doublet H 1 is aligned in eld space with the total VEV v, with a rotation of angle

We choose to write these two new doublets as

In this new basis, following the notation of [START_REF] Bernon | Scrutinizing the alignment limit in two-Higgs-doublet models: mh=125 GeV[END_REF], the potential can be written as

(5.5.3)

Two-Higgs-Doublet Model

The CP-even physical states are eigenstates of the mass matrix

where

which is diagonalised with an angle α, and are given by

.

(5.5.6)

The alignment limit is dened as the limit in which the neutral components of the Higgs-basis doublets are also mass eigenstates, or in other words, the limit in which one of the CP-even neutral scalar mass eigenstates is aligned with the VEV v. From eq. ( 5.5.6) we see that this can be realised in two ways:

(i) s β-α = 0 in which H carries the VEV and is identied with the SM-like Higgs.

(ii) c β-α = 0 which means that h is the SM-like Higgs.

We do not make any assumption on the size of the masses of the dierent scalars i.e. we do not suppose that we are in the decoupling limit as well. Consequently, at tree-level we only require Z 6 v 2 → 0, and hence with the expression of Z 6 derived in [START_REF] Bernon | Scrutinizing the alignment limit in two-Higgs-doublet models: mh=125 GeV[END_REF], we have

where λ 345 ≡ λ 3 + λ 4 + λ 5 . The simplest, and tan β-independent, way to full this condition is to have

which we will use in the following to constrain tree-level alignment. Also, we will require that the SM-like Higgs be the lightest mass eigenstate h (case (ii) above), by ensuring that

(5.5.9)

This implies that c β-α = 0, and thus, with the conventional choice that β ∈ [0, π 2 ] and |α| ≤ π 2 , we have that

(5.5.10)

The constrain for tree-level alignment given in eq. ( 5.5.8) reduces the number of free parameters of the model from seven to ve, as two of the quartic couplings (eg. λ 2 and λ 3 ) can be found as a function of the three other ones.

Renormalisation scale dependence of the Higgs mass computed with SPheno

The masses computed by SPheno are pole masses, which should in principle not depend on the renormalisation scale at which they are computed. Evaluating the variation of the masses with the scale Q hence provides a consistency check of our results and an estimate of the theoretical uncertainty as the variation of the two-loop masses with Q corresponds roughly to the three-loop corrections. For this purpose, we have tuned the

Matching and running

Machacek model where the two-loop corrections to the Higgs masses were calculated for the rst time.

We show in this chapter that higher-order corrections can be very important for the study of the UV behaviour of a theory leading to four main conclusions:

(i) The threshold corrections at low energies can lead to substantial shifts in the running parameters of a model.

(ii) The change from one-loop to two-loop running can atten the running at large values of the coupling, preventing the onset of a Landau pole at high energies leading to a form of asymptotic safety.

(iii) Alternatively, in the case where the running drives some quartic coupling negative, higher-order corrections can lead to signicant changes to the predicted scale of metastability.

(iv) As a by-product of the above, we nd that new fermionic elds at low energies can stabilise the SM potential.

We illustrate the above with a detailed examination of three examples: a singlet extension, the SM extended by vector-like quarks and a Two-Higgs-Doublet Model.

This chapter is organised at follows: in section 6.1 we give a step-by-step prescription for the general matching procedure including loop eects, as well as details into the procedure used to obtain higher order-corrections to the quartic couplings in the different models considered. Afterwards, we discuss in section 6.2 the numerical results, providing insights including approximate formulae.

Matching and Running

To extrapolate a theory from the electroweak scale to high energies, we require two ingredients:

(i) The value of the couplings at the low scale where the running starts;

(ii) The RGE running of all parameters.

Renormalisation Group Equations

We shall always work in the MS scheme. In this scheme, the β-functions, which describe the energy dependence of the parameters Θ, are dened as

Here, µ is an arbitrary mass scale. β i can be expanded in a perturbative series: Matching and running which is simple enough for the Standard Model and extensions without scalar mixing so we shall give analytical expressions in sections 6.2.2.1 and 6.2.3. We recall also that the Higgs VEV v used in the above equations and throughout this chapter is the running VEV (extracted from the Z boson pole mass in a separate calculation).

On the other hand, for more complicated models, we need to solve the equivalent of eq. (6.1.5) through iteration, and we shall adopt this approach in general for the numerical studies. More precisely, we will compute with the SARAH/SPheno spectrum generator created for the model at hand the mass spectrum using a set of MS parameters that we call in general Θ(Q) (for details on this, see the description of the SARAH/SPheno setup in section 2.4.3). If a chosen set of input parameters Θ(Q) results in the desired physical masses and mixing angles when using a N -loop calculation, we refer to them as N -loop couplings. Thus, with tree-level relations we have leading order (LO) parameters, while the one-and two-loop mass corrections result respectively in the NLO and NNLO couplings. Finding the correct set of MS couplings corresponding to the desired physical parameters at loop-level is non-trivial. In the numerical studies presented in section 6.2, we use a simple tting algorithm which varies the input parameters until the desired masses and mixing angles are obtained.

Models and results

Singlet Extension

We start by studying the singlet-extended SM (SSM), which has been described in section 1.4.1. For clarity, we recall that the scalar potential in the SSM reads Asides from the typical requirement that the quartic couplings remain perturbative, the constraints from perturbative unitarity need to be taken into account. For that, we can evaluate the scalar 2 → 2 scattering amplitude in the limit of high energies and demand that the eigenvalues stay below 8 π.

We also recall the following shorthand notations

where Q 2 = 4πe -γ E µ 2 is the renormalisation scale squared.

B.1.1 Denition of loop functions B.1.1.1 One-loop functions

In the expression of the one-loop eective potential, we make use of the function f dened as

Two important one-loop functions that will appear in the expression of the eective potential, of its derivatives and in the self-energies are the nite parts of

Denitions and expansions of loop functions namely A(x) ≡ lim

where

and

In two-loop order expressions, the function J is sometimes used, although it is equal to A J(x) = A(x).

(B.1.10)

A limit of particular interest of B is the limit of vanishing external momentum, that we denote B 0 , and is related to the P SS function we have used

xy .

(B. 1.11) and furthermore, we have that

The derivative of the B function with respect to one of the mass arguments is also used, with the notation

For the fermion and gauge boson contributions to the scalar self-energy we also use the functions P F F , P F F and P V V related to A and P SS as

In the context of the reexpansion of the mass diagrams, we also make use of the oneloop three-point function C(p 2 1 , p 2 2 , (p 1 + p 2 ) 2 , x, y, z), which is the nite part of the following integral

.

Two-loop functions

We recall the denition of the following two-loop integrals S(x, y, z)

of which we take the nite parts

We also require the related functions (where V diers slightly from [START_REF] Martin | Two loop scalar self energies in a general renormalizable theory at leading order in gauge couplings[END_REF]) The integral I is symmetric on all three indices, and thus U 0 is symmetric on x ↔ y and z ↔ u separately etc; the I integral is fundamental for the two-loop eective potential, all other functions being obtained from it and A(x). It can be written explicitly although the expression is rather involved; it can be found in equations (D1) to (D3) of [START_REF] Degrassi | On the radiative corrections to the neutral Higgs boson masses in the NMSSM[END_REF] although it was rst derived in [START_REF] Ford | The Standard model eective potential at two loops[END_REF]. Here we note the useful limiting cases

where c xxx ≈ 2.3439 is a constant.

Denitions and expansions of loop functions

The two-loop functions appearing in the eective potential were dened in [START_REF] Martin | Two loop eective potential for a general renormalizable theory and softly broken supersymmetry[END_REF] and read f SSS (x, y, z) = -I(x, y, z), f SSV (x, y, z) = 1 z (-x 2y 2z 2 + 2xy + 2xz + 2yz)I(x, y, z) + (xy) 2 I(0, x, y)

where J(x, y) ≡ J(x)J(y) = A(x)A(y). To these functions we must also add the scheme dependent functions f V S , f V V S , f F F V , f F F V and f gauge that we give for the DR and MS schemes (slightly modifying the notation of [START_REF] Martin | Two loop eective potential for a general renormalizable theory and softly broken supersymmetry[END_REF])

+ (xz) 2 I(0, x, z) + (yz) 2 I(0, y, z)z 2 I(0, 0, z) + (zxy)J(x, y) + yJ(x, z) + xJ(y, z)

(-x 4 -8x 3 y -8x 3 z + 32x 2 yz + 18y 2 z 2 )I(x, y, z) Taking derivatives of these functions with respect to one argument is required for the two-loop tadpoles, and we use the notations

For the mass diagrams, we require the following loop integral functions:

X SSS (x, y, z) = J(z)P SS (x, y), U SSSS (x, y, z, u) = U (x, y, z, u),

+(x ↔ y),

For completeness, we recall equations (3.7)-(3.10) from [START_REF] Kumar | Resummation of Goldstone boson contributions to the MSSM eective potential[END_REF] for the expansion of the loop functions appearing in the two-loop eective potential

Denitions and expansions of loop functions

where the R functions are dened in [START_REF] Kumar | Resummation of Goldstone boson contributions to the MSSM eective potential[END_REF] as

R SV (x, y) = 1 y 3(x + y)I(0, x, y) -3xI(0, 0, x) + 3A(x)A(y) + 2xy + y 2 , (B.1.65) R V V (x, y) = 1 4xy(xy) 2 3A(x)A(y) x 2 + y 2 + 6xy -24xy xA(x) + yA(y) + 14xy(x 2 + y 2 ) + 20x 2 y 2 + 3(x + y) 3 I(0, x, y) -3(xy) 2 xI(0, 0, x) + yI(0, 0, y) .

For the derivatives of the two-loop f functions, we use the following expansions f (0,0,1)

B.2 Diagrams regulated by momentum

When studying the mass terms, we encountered some diagrams for which the resummation of the Goldstone contributions provide no shift to regulate an infrared divergence and hence these diagrams must be regulated by momentum. More precisely, this is the case for the functions U , M , Z and for some of the V diagrams. In this section, Denitions and expansions of loop functions

B.2.2 Limits of the M function

Turning now to the M function, there are more cases to consider. In the case of only one argument vanishing, we see from the integral expression (B.1.20) that the function is regular. From eqs.(6.28) and (6.31) in [START_REF] Martin | Evaluation of two loop selfenergy basis integrals using dierential equations[END_REF], we also nd that M (x, y, z, 0, 0), M (x, y, 0, 0, v) and M (x, y, 0, 0, 0) are nite. Then we have

where

A M is symmetric on all three indices, and as M (0, 0, 0, u, 0) or M (0, y, 0, 0, 0) have prefactor λ GGG , we only need to consider the following cases

The only additional case is M (0, 0, 0, 0, v) that we have to consider separately and that we nd to be

Finally, the expression of M (x, 0, 0, 0, 0) with full momentum dependence can be found in equation (6.31) of [START_REF] Martin | Evaluation of two loop selfenergy basis integrals using dierential equations[END_REF], and becomes when expanding to leading order for small s M (x, 0, 0, 0, 0) = 1 6x 18 + π 2 -12 log(-s/x) + 3 log 2 (-s/x) .

(B.2.12)

Denitions and expansions of loop functions singular as y → 0 so we cannot simply evaluate the right-hand side of the equation to determine V (x, 0, z, u)! However, we can obtain such a closed-form expression by using the ansatz U (x, y, z, u) = f 0 (s; x, z, u) + f (s; x, z, u)A(y) + f 1 (s; x, z, u)y + O(y 2 )

and substituting it into the above dierential equation, to nd f and f 1 :

The form of f must correspond to the singularity; indeed we have

However, f 1 is more work; we eventually obtain in the limit x → 0 that we are interested in

and

The coecients dened in the above are

(B.3.12)

B.3 Additional expressions for Ṽ (x, 0, z, u)

If we then make our generalised eective potential expansion, we nd

We do not need the limit when u = z = 0 because in that case we have couplings λ GGG . However, for z = 0 or u = 0 we do see that there is a smooth limit of the above.

Let us dene

We can then write

We have

If we now substitute in the standard expressions for I(z, u, 0) then we can simplify the above to

We can also it in a shorter but less symmetric form

We can also take the limits:

Denitions and expansions of loop functions

B.3.1 Integral representation

Our expression for Ṽ actually lends itself to an interesting nite integral representation.

We start with the denition Ṽ (0, z, u) ≡ lim y→0 -V (y, y, z, u) + B(s, y, y )P SS (z, u) .

.

We can then integrate this expression. For the case x → 0 we can simplify a little:

This integral is nite; we have checked that explicitly performing the integral using TARCER [START_REF] Mertig | TARCER: A Mathematica program for the reduction of two loop propagator integrals[END_REF] exactly yields expression (B.3.10).

Appendix C

Consistent solution of the tadpole equations with shifts to fermion masses

Here we give the two-loop shifts to the tadpoles and self-energies due to shifts in fermion masses when we solve the tadpole equations consistently.

We denote the undiagonalised fermion mass matrix as m IJ . The mass-squared matrix is dened [START_REF] Martin | Two loop eective potential for a general renormalizable theory and softly broken supersymmetry[END_REF] as

and is diagonalised by a unitary matrix N dened such that

Then if the tree-level matrices depend on some parameters {x i } for which we solve the tadpole equations as in equation (4.4.4) we have

Then the shift to the fermion contribution to the tadpole is •

Summary of lower limits on masses of SUSY particles, obtained by the different SUSY searches of the ATLAS collaboration by December 2017. This plot, along with other summary from ATLAS can be found on the web page:

https://atlas.web.cern.ch/Atlas/GROUPS/PHYSICS/CombinedSummaryPlots/ SUSY/. A similar plot for CMS, updated for the Moriond 2017 conference, can be found at https://twiki.cern.ch/twiki/pub/CMSPublic/ PhysicsResultsSUS/Moriond2017_BarPlot.pdf.

Exclusion limits, at 95% condence level, on stop and neutralino masses, resulting from the stop pair production search of the ATLAS collaboration.

Dierent decay channels of the stops are considered, with quantities of data comprised between 3.2 and 36 fb -1 from Run 2 of the LHC at √ s = 13 TeV.

A comparison with results from Run 1 (light blue) is also provided. This plot can be found on the web page: https://atlas.web.cern.ch/Atlas/ GROUPS/PHYSICS/CombinedSummaryPlots/SUSY/ .

Diagrams contributing to the mass of a scalar φ: on the left hand side with φ, i.e. the fermionic superpartner of φ, and a Dirac gaugino (λ and χ Σ in terms of Weyl spinors); and on the right hand side with the scalar φ and the adjoint scalar Σ. The logarithmic divergences from both diagrams cancel out when the gauginos only have Dirac masses. 1. This plot is taken from [START_REF] Collaboration | Measurement of the Higgs boson mass in the H → ZZ * → 4 and H → γγ channels with √ s=13TeV pp collisions using the ATLAS detector[END_REF].

.

Possible topologies of one-loop self-energy diagrams.

•

Possible topologies of two-loop self-energy diagrams.

User-dened model Illustration of the study of a BSM model using SARAH and SPheno, and links to other High-Energy Physics codes.

Novel two-loop contribution to the eective potential involving stops and octet scalars. •

Dierent determinations of the SM-like Higgs boson mass in the MDGSSM as a function of (X t /M S ) OS , for the same choices of parameters as in gure 3.2. The solid curve represents the original OS calculation; the dotted curve represents the DR calculation; the dashed and dot-dashed curves were obtained using αs (m t ) and α s (m t ), respectively, in the OS calculation instead of αs (M S ).

Mass of the SM-like Higgs boson as a function of m g in the supersoft limit of the MRSSM, for tan β = 10. The solid curve represents the results of the DR calculation, in which the two-loop O(α t α s ) corrections become unphysically large. The dashed curve was obtained by converting the top and stop masses to the OS scheme and using the corresponding formulae The lightest scalar mass squared for the parameter point dened by eq. (5.3.2)

when calculating with and without D-term contributions. The lightest Higgs mass at the two-loop level for the parameter point dened by eq. (5.3.2) for dierent methods to regulate the two-loop corrections.

•

The change in the Higgs mass in GeV due to the two-loop corrections involving the new Yukawa-like interactions g(1,2)(u,d) . On the top, we used tan β = 1 at M M and on the bottom tan β = 10. The left plots are with the consistent tadpole solutions, the right ones without.

Lightest Higgs mass m 2 h as a function of the renormalisation scale Q, considering only the running of SM parameters. In other words, for this gure, the values of the BSM parameters λ i , m 2 12 , and tan β given in eq. (5.5.11) are considered to be input values at the scale Q at which the Higgs mass is computed, instead of being evolved from 160 GeV to Q. Red curve: treelevel; Blue dot-dashed curve: one-loop order; Green dashed curve: two-loop order.

Lightest Higgs mass m 2 h as a function of the renormalisation scale Q, taking into account the running of all parameters both SM and BSM ones with the RGEs included in SPheno. The dierence between this gure and gure 5.5 is that here we do not consider the inputs given in equation (5.5.11) to be taken at the scale Q at which we compute m h but at the scale m t , and then we evolve them from m t to Q. Red curve: tree-level; Blue dot-dashed curve: one-loop order; Green dashed curve: two-loop order. -1/t α as a function of the o-diagonal mass term m 2 12 (upper left), and of quartic couplings λ 3 (upper right), λ 4 (lower left) and λ 5 (lower right) at each order in perturbation theory. For each plot we vary the parameters as follows: we choose one parameter as the abscissa; the tree-level alignment condition λ 1 = λ 2 = 1/2λ 345 plus the requirement that the Higgs mass is 125.09 GeV xes three parameters, namely λ 1 , λ 2 and either λ 4 for the bottom right plot or λ 5 for the other three; the remaining parameters are held xed at values λ 3 = 0.5, λ 4 = 0.5, m 2 12 = -1000 GeV 2 (when they are not otherwise varying). All plots are for tan β = 50. Red curve: tree-level; Blue dot-dashed curve: one-loop order; Green dashed curve: two-loop order.

5.8

The dependence of the lightest scalar mass on the renormalisation scale Q, considering that the quartic couplings of eq. (5.5.13) are used as input at the scale Q = m t . Left: m h (Q) at tree-, one-loop, and two-loop levels; Right: ∆m h ≡ m h (m t )m h (Q) at tree-, one-loop, and two-loop levels • • 5.9

The size of the one-and two-loop corrections of the lightest scalar mass as function of the scale Q in at which the input masses of eq. (5.5.12) are translated into quartic couplings (or in other words, the scale at which the quartic couplings in eq. (5.5.13) are dened as inputs).

• •

The size of the one-(left) and two-loop (right) corrections in the (s H , m 5 ) plane for the second (rst row) and third (second row) CP-even scalar. • •

The running of the quartic couplings for the point given in Tab. 6.1. The line-styles refer to the loop order of the matching and RGE running as described in Tab. 6.1, namely (n, m) refers to the matching at n-loop order with m-loop RGEs. The solid red line is the 4π perturbativity limit, while the dashed-red line is the unitarity constraint of 4π/3 obtained from eq. (6.2.4) in the limit λ S λ SH , λ.

Dierence in the predicted cut-o scale depending on the matching performed as a function of the singlet VEV v S and the heavy CP-even Higgs mass m H . Left: We show the ratio of the obtained cut-o given matching at N versus N -1 order using the RGEs at N -loop order. The coloured(grey) contours use the two(one)-loop RGEs, therefore showing the ratio of the matching at two(one)-loop versus one-loop(tree-level), respectively. Right:

Ratio of the calculation performed using both matching and RGEs at two-loop order versus the leading order (tree-level matching and one-loop RGEs). The grey contours correspond to the ratios of the quartic coupling λ S for these two scenarios. Here we have xed the physical parameters such that m h = 125 GeV, tan α = 0.2, while the remaining parameters are chosen as κ 1 = 0 GeV and κ 2 = 1000 GeV.

Simplied comparison between the running of λ in the SM with and without vector-like states. Here, we used full one-loop (dashed lines) and two-loop RGEs (full lines) in both models and as starting point the SM best-t values from ref. [START_REF] Buttazzo | Investigating the near-criticality of the Higgs boson[END_REF]. For the purple (black) lines we use Y t = 0.3 (0.7). • 184 6.7

Contours of the scale Λ 0 at which λ runs negative with full two-loop RGEs and one-loop (dashed) or two-loop (full lines) matching, and for comparison when using the best-t value λ SM = 0.25208 at the top mass scale (dotted). Black lines correspond to Λ 0 = 10 18 GeV, blue lines to 10 11 GeV.

The background shows the two-loop shift in λ(m t ) in percent, dened as (λ (2) λ (1) )/λ (1) .

The stale of metastability Λ 0 in the case of large Y t using two-loop running with two-loop matching for λ. The black contours show the size of Λ 0 with respect to using one-loop matching.

List of Figures

11 Comparison of the cut-o scales between using tree matching and one-loop RGEs and both two-loop matching and RGEs for a 2HDM region with large m 2 12 = -(750 GeV) 2 . The loop-level spectrum was evaluated taking the tree-level values m H ± = 1.14 TeV and t α = -0.95 as inputs. The ratio of the loop-corrected charged Higgs mass to its tree-level input is shown as grey contour lines. The quartic couplings for the case of tree-matching were obtained using the leading order relations Eqs. (1.4.19) to (1.4.23), taking the spectrum of the two-loop calculation as input. We further xed tan β = 1.14 and applied the Yukawa scheme of type I.