These De

M Xiulei Ren

-S Zou

Professeur M J Meng

Professeur -S Geng

Professeur Mme C Alexandrou

Professeur M M Schindler

Professeur -H Zhu

M J Carbonell

NNT : 2015SACLS156

Keywords: effective field theory, chiral Lagrangians, lattice QCD, baryon masses, sigma terms

and Roper to the baryon masses are taken into account for the first time. A first analysis of lattice Roper masses is presented.

Abstract

Mass is one of the most fundamental properties of matter. Understanding its origin has long been a central topic in physics. According to modern particle and nuclear physics, the key to this issue is to understand the origin of nucleon (lowest-lying baryon) masses from the nonperturbative strong interaction. With the development of computing technologies, lattice Quantum Chromodynamics simulations provide great opportunities to understand the origin of mass from first principles. However, due to the limit of computational resources, lattice baryon masses have to be extrapolated to the physical point. Chiral perturbation theory, as an effective field theory of low-energy QCD, provides a model independent method to understand nonperturbative strong interactions and to guide the lattice multiple extrapolations. Therefore, we present the interplay between lattice QCD and chiral perturbation theory to systematically study the baryon masses.

In the SU(3) sector, we study the lowest-lying octet baryon masses in covariant baryon chiral perturbation theory with the extended-on-mass-shell scheme up to next-to-next-to-nextto-leading order. In order to consider lattice artifacts from finite lattice box sizes, finite-volume corrections to lattice baryon masses are estimated. By constructing chiral perturbation theory for Wilson fermions, we also obtain the discretization effects of finite lattice spacings. We perform a systematic study of all the latest n f = 2 + 1 lattice data with chiral extrapolation (m q → m phys. q ), finite-volume corrections (V → ∞), and continuum extrapolation (a → 0). We find that finite-volume corrections are important to describe the present lattice baryon masses. On the other hand, the discretization effects of lattice simulations up to O(a 2 ) are of the order 1% when a ∼ 0.1 fm and can be safely ignored. Furthermore, we find that the lattice data from different collaborations are consistent with each other, though their setups are quite different.

Using the chiral formulas of octet baryon masses, we accurately predict the octet baryon sigma terms via the Feynman-Hellmann theorem by analyzing the latest high-statistics lattice QCD data. Three key factors -lattice scale setting effects, chiral expansion truncations and strong-interaction isospin-breaking effects -are taken into account for the first time. In particular, the predicted pion-and strangeness-nucleon sigma terms, σ πN = 55(1)(4) MeV and σ sN = 27 [START_REF] Bietenholz | Tuning the strange quark mass in lattice simulations[END_REF] (4) MeV, are consistent with the most latest lattice results of nucleon sigma terms.

With the success in the study of octet baryon masses, we also present a systematic analysis of the lowest-lying decuplet baryon masses in covariant baryon chiral perturbation theory by simultaneously fitting n f = 2 + 1 lattice data. A good description for both the lattice and the experimental decuplet baryon masses is achieved. The convergence of covariant baryon chiral perturbation theory in the SU(3) sector is discussed. Furthermore, the pion-and strangenesssigma terms for decuplet baryons are predicted by the Feynman-Hellmann theorem.

In addition, understanding the excitation spectrum of hadrons is still a challenge, especially the first positive-parity nucleon resonance, the Roper(1440). The baryon spectrum shows a very unusual pattern that the Roper state is lower than the negative-parity state N (1535). Most lattice studies suggest that the Roper mass exhibits much larger chiral-log effects than that of the nucleon. Therefore, we calculate the Roper mass in chiral perturbation theory by explicitly including the nucleon/Delta contributions. The mixed contributions between nucleon

Résumé

La masse est une des propriétés les plus fondamentales de la matière. Comprendre son origine a longtemps été un sujet central en physique. D'après la physique nucléaire et la physique des particules modernes, la clef de ce problème réside dans la compréhension de l' origine de la masse du nucléon (baryon de masse la plus petite) à partir de l' interaction forte. Avec le développement des technologies informatiques, la chromodynamique quantique sur réseau offre la possibilité de comprendre l' origine de la masse à partir des premiers principes. Cependant, dû aux ressources de calcul limitées, les masses obtenues à partir des calculs sur réseau doivent être extrapolées jusqu'au point physique. La théorie chirale des perturbations en tant que théorie effective des champs de QCD à basse énergie est une méthode indépendante de modèle permettant de comprendre l' interaction forte dans la région non perturbative et de guider les diverses extrapolations nécessaires pour passer du résultat lattice au résultat physique. Le but de cette thèse est donc d'utiliser la complémentarité entre QCD sur réseau et théorie chirale des perturbations afin d'étudier de façon systématique les masses des baryons.

Nous étudions les masses de l'octet baryonique le plus léger dans le cadre de la théorie chirale covariante des perturbations pour les baryons. Nous utilisons la méthode "extended on mass shell" jusqu'à l'ordre trois fois sous dominant. Afin d'étudier les artefacts des calculs sur réseau dûs à la taille finie de la boîte nous calculons les effets de volume fini.

Adaptant la théorie chirale des perturbations à des fermions de Wilson nous obtenons aussi les effets de discrétisation dû au pas a fini du réseau. Nous étudions de façon systématique toutes les données réseau en tenant compte à la fois de l' extrapolation au continu (a → 0), des corrections de volume finie (V → ∞) et de l'extrapolation chirale (m π → m phys π où m π est la masse du pion). Nous démontrons l'importance des corrections de volume fini dans la description des masses des baryons sur réseau. Par contre les effets de discrétisation sont de l'ordre de 1% jusqu'à l'ordre a 2 et peuvent donc être ignorés sans problème. De plus nous trouvons que toutes les données sur réseau prises en compte sont consistentes entre elles malgré des différences notables dans les procédures adoptées.

Utilisant les formules chirales des masses des baryons nous prédisons de façon précise leurs termes sigma via le théorème de Feynman-Hellmann en analysant les données sur réseau les plus récentes dont la statistique est élevée. Les effets dûs au pas du réseau, à la troncation de la série de perturbation chirale et à la violation d'isospin de l'interaction forte sont pris en compte pour la première fois. En particulier le terme sigma pion nucléon with the following notations: t1 = 2t 1 + t 2 , t2 = 2t 3 + t 4 , and t3 = 4t 5 + t 6 . . . . 79 7.3 Coefficients of the NLO contributions to octet baryon masses [Eq. (7.16)]. . . . 81 7.4 Values of LECs from the best fits to the lattice and experimental data with different fitting strategies at O(p 2 ), O(p 3 ), and O(p 4 ), respectively. The estimator for the fits with and without the experimental decuplet masses, χ 2 /d.o.f. and χ 2 /d.o.f.

(Color online)

. Pion mass dependences of the lowest-lying decuplet baryon masses. Filled (open) symbols denote the lattice data points included in (excluded from) the fits, which are projected to have the physical strange-quark mass. The dot-dashed, the dashed, and the solid lines are the best NLO, NNLO and N 3 LO fits to the lattice data, respectively. In obtaining the BChPT results, the strange quark mass has been set to its physical value. The lattice points in the shaded region are not included in the fits. . . . . . . . . . . . . . . . . . . . 87 7.3 The QCDSF-UKQCD lattice data [START_REF] Bietenholz | Tuning the strange quark mass in lattice simulations[END_REF] in comparison with the N 3 LO BChPT.

The lattice data denoted by the blue filled squares are included in the fit; those by the green opened circles (with N s = 32) and the red diamonds (with N s = 24) are not. FVCs of the lattice results have been subtracted. The two flavour singlet quantities, X π and X ∆ , are defined as: X π = (M 2 π + 2M 2 K )/3, X ∆ = (2m ∆ + m Ω -)/3, respectively [START_REF] Bietenholz | Tuning the strange quark mass in lattice simulations[END_REF]. . . . . . . . . . . . . . . . . . . . . . . . . . 88 7.4 Comparison between the N 3 LO BChPT and the LHPC data [START_REF] Walker-Loud | Light hadron spectroscopy using domain wall valence quarks on an Asqtad sea[END_REF]. . . . . . . . . 89

Introduction

Mass, as an essential property of matter, was firstly introduced by Issac Newton in his book The Mathematical Principles of Natural Philosophy in 1687 [1]. According to the different methods to measure the mass, there are two definitions of mass: inertial and gravitational mass. Inertial mass, which can be measured by the use of Newton's second law F = ma, denotes the inertia of an object. Gravitational mass, which can be measured by Newton's law of gravity F = GM m/r 2 (F is the force between the masses M and m, G is the gravitational constant, and r is the distance between the centers of the masses), represents the response of one object to the gravitational force. With the requirement of Einstein's equivalence principle [2], gravitational mass is identical to inertial mass. Up to now, all experimental researches, e.g. the famous Eötvös experiment [3], also show that inertial mass and gravitational mass are equivalent.

Mass has been introduced in many physical theories as a fundamental quantity. However, understanding the origin of mass is still a hot topic in the exploration of our ordinary matter world 1 . In the 1960s, with the famous Higgs boson proposed, the masses of fundamental particles (such as quarks, leptons, gauge bosons) can be produced by the Brout-Englert-Higgs mechanism [4][5][6]. This mechanism was confirmed in 2012, when the Higgs particle was observed by the ATLAS [7] and the CMS [8] experiments at CERN's Large Hadron Collider. Following the discovery, Peter Higgs and François Englert were awarded the Nobel Prize in Physics in 2013 for their extraordinary work in theory. However, the Brout-Englert-Higgs mechanism does not tell the whole story. When one explores the production mechanism of mass of ordinary matter, one finds that the Higgs mechanism can only provide about 1%. Where is the majority of contributions coming from?

Origin of mass

In order to understand the origin of mass, let us first examine the hierarchy of ordinary matter. Ordinary matter is made up of atoms and molecules (an electrically neutral group of two or more atoms). Because the binding energies of atoms/molecules introduced by van der Waals' interaction and chemical bond are much weaker compared to atomic masses, the study of the origin of mass can be attributed to the understanding of atomic mass. As illustrated in Fig. 1.1, an atom is composed of one or more electrons [9] and a nucleus [10], which is a bound system of protons or/and neutrons [11]. According to the Standard Model of particle physics, the proton and the neutron are comprised by u, u, d and u, d, d three quarks [12,13], respectively. These fundamental particles within the atom interact with each other via four fundamental interactions: gravitational interaction, electromagnetic interaction, strong interaction, and weak interaction.

However, on the scale of atomic matter, the gravitational force can be neglected due to the weak interaction strength. The weak interaction can also be ignored, since it is responsible for the nuclear decay processes and irrelevant to the formation of bound systems. As a long-range force, the electromagnetic force plays an important role in binding electrons and nucleus to form an atom. And, the strong interaction, the strongest force of the four, is responsible for binding u, d quarks to form nucleons and binding nucleons to form a nucleus.

All in all, the mass of ordinary matter can be the sum of the masses of particles within the atom (shown in Fig. 1.1) and the energy of electromagnetic and strong interactions (according to Einstein's mass-energy relation E = mc 2 [START_REF] Einstein | Ist die trägheit eines körpers von seinem energieinhalt abhängig?[END_REF], with the light speed c = 1 in natural units).

At the hadronic level, the picture is quite intuitive that the mass of an atom equals to the masses of electrons and nucleus minus the binding energies between electrons and nucleus. The electron mass is m e = 511 keV, which is about 1/1836 of nucleon mass (M N = 939 MeV), and the mass loss induced by binding is extremely small (at electron volt (eV) level). Therefore, the mass of an atom is mainly from the nucleus mass. Because protons and neutrons are bound into a nucleus with the residual strong interaction, the mass of nucleus equals to the sum of masses of nucleons minus the binding energy. The average nuclear binding energy is about 8 MeV per nucleon, which only accounts for less than 1% of nucleon mass, therefore, the mass loss is negligible comparing to the total nuclear mass. Consequently, we find that the masses of nucleons provide the vast majority of atomic mass. In order to make this point clearly, we take iron-56 ( 56 Fe), one of the most tightly bound nuclei (binding energy is about 8.8 MeV per nucleon), for example. It contains 26 electrons, 26 protons, and 30 neutrons. The atomic mass of 56 Fe is 55.935 u (∼ 52103.1 MeV) in unified atomic mass units. The masses of electrons, 26 × 0.511 = 13.286 MeV, only account about 0.025% of total mass of 56 Fe. The major contributions are the nucleus mass of 56 the binding energy accounts about 0.95%. Therefore, the masses of nucleons provide 99% of the atomic mass of iron-56.

As mentioned above, the nucleon is made up of three quarks, which are bound together by strong interactions. But the physical picture becomes more complicated than the nucleus, since the sum of quark masses is about 10 MeV (m u = 2.3 +0. 7 -0.5 MeV, m d = 4.8 +0.5 -0.3 MeV at a renormalization scale µ ≈ 2 GeV [START_REF] Olive | Review of Particle Physics[END_REF]), which accounts for about 1% of nucleon mass. The other 99% contributions are the mass-energy from strong interaction. In order to clearly understand this phenomena, in Fig. 1.2, we present a more realistic picture of the composition of proton. We can see that the proton not only contains three valence quarks, but also includes tremendous sea quarks (quark and anti-quark pairs) and gluons, which provide most of proton mass. Therefore, almost 98% of the mass of atomic matter could be from the strong interaction. It would be of fundamental interest to understand the strong interaction to produce the nucleon masses.

Baryon masses from QCD

Quantum Chromodynamics (QCD), as an important component of the Standard Model of particle physics, is the part of the theory describing the strong interaction. The basic degrees of freedom are quarks and gluons. In principle, QCD could provide a reasonable explanation of the 98% of atomic mass. However, being a non-Abelian gauge theory, QCD possesses two peculiar properties: asymptotic freedom and confinement. As illustrated in Fig. 1.3, quarks The figure is taken from Ref. [START_REF] Bethke | Experimental tests of asymptotic freedom[END_REF]. and gluons interact very weakly at high energy, allowing perturbative expansion in powers of the running coupling constant α s . On the other hand, due to confinement, α s takes large value at low energies, and QCD becomes nonperturbative. In order to tackle this problem, lattice quantum chromodynamics (LQCD), effective field theories, and phenomenological quark models are developed. Because the former two methods are usually deemed as the theory of QCD, we would like to focus on their studies of nucleon (baryon) masses to understand the origin of atomic mass.

Lattice quantum chromodynamics simulation is a numerical method for solving QCD in the non perturbative regime with the QCD action directly discretized in a finite space-time hypercube. The corresponding lattice QCD gauge theory was firstly formulated by Wilson in 1974 [START_REF] Wilson | Confinement of Quarks[END_REF]. The quark fields are located on lattice sites and the gluon fields are defined as the links to connect adjacent two sites. With the development of computational ability and advances of numerical algorithms, lattice QCD has undergone three developing stages: the "quenched" approximation [START_REF] Hamber | Numerical Estimates of Hadronic Masses in a Pure SU(3) Gauge Theory[END_REF][START_REF] Weingarten | Monte Carlo Evaluation of Hadron Masses in Lattice Gauge Theories with Fermions[END_REF] in 1990s, the "partial-quenched" approximation [START_REF] Khan | Light hadron spectroscopy with two flavors of dynamical quarks on the lattice[END_REF][START_REF] Prelovsek | Scalar meson in dynamical and partially quenched two-flavor QCD: Lattice results and chiral loops[END_REF] in 2000s, and the fully dynamical simulations nowadays (around 2010). Therefore, lattice simulations provide a powerful framework for a quantitative analysis of the baryon masses from the nonperturbative strong interaction.

Recently, several LQCD collaborations have performed studies of lowest-lying octet and decuplet masses [START_REF] Dürr | Ab-Initio Determination of Light Hadron Masses[END_REF][START_REF] Aoki | 2+1 Flavor Lattice QCD toward the Physical Point[END_REF][START_REF] Aoki | Physical Point Simulation in 2+1 Flavor Lattice QCD[END_REF][START_REF] Walker-Loud | Light hadron spectroscopy using domain wall valence quarks on an Asqtad sea[END_REF][START_REF] Lin | First results from 2+1 dynamical quark flavors on an anisotropic lattice: Light-hadron spectroscopy and setting the strange-quark mass[END_REF][START_REF] Bietenholz | Tuning the strange quark mass in lattice simulations[END_REF][START_REF] Bietenholz | Flavour blindness and patterns of flavour symmetry breaking in lattice simulations of up, down and strange quarks[END_REF][START_REF] Beane | High Statistics Analysis using Anisotropic Clover Lattices: (IV) Volume Dependence of Light Hadron Masses[END_REF], especially the BMW Collaboration [START_REF] Dürr | Ab-Initio Determination of Light Hadron Masses[END_REF], which presented a significant explanation of the origin of mass from first principles. However, due to the limited computing resources, most lattice simulations have to employ larger than physical light-quark masses2 , finite lattice volumes and lattice spacings. In order to extract the observables at the physical point, one has to perform multiple extrapolations of lattice baryon masses to the physical point with physical quark masses (m q → m phys.

q

), to infinite space-time (L → ∞), and to the continuum (a → 0). For many observables, these extrapolations have led to uncertainties comparable to or even larger than the inherent statistical uncertainties.

On the theoretical side, chiral perturbation theory (ChPT), proposed by Weinberg [START_REF] Weinberg | Phenomenological Lagrangians[END_REF] in 1978, is an effective field theory for low-energy QCD. The quintessence of ChPT is the effective Lagrangian technique with an expansion in powers of external momenta and light quark masses, which is constrained by chiral symmetry and its breaking pattern. After the further generalization by Gasser and Leutwyler [START_REF] Gasser | Chiral Perturbation Theory to One Loop[END_REF][START_REF] Gasser | Chiral Perturbation Theory: Expansions in the Mass of the Strange Quark[END_REF], ChPT became a very popular method in studies of nonperturbative strong interaction physics [START_REF] Weinberg | Phenomenological Lagrangians[END_REF][START_REF] Gasser | Chiral Perturbation Theory to One Loop[END_REF][START_REF] Gasser | Chiral Perturbation Theory: Expansions in the Mass of the Strange Quark[END_REF][START_REF] Gasser | Nucleons with Chiral Loops[END_REF][START_REF] Meissner | Recent developments in chiral perturbation theory[END_REF][START_REF] Leutwyler | Principles of chiral perturbation theory[END_REF][START_REF] Bernard | Chiral dynamics in nucleons and nuclei[END_REF][START_REF] Pich | Chiral perturbation theory[END_REF][START_REF] Ecker | Chiral perturbation theory[END_REF][START_REF] Pich | Effective field theory: Course[END_REF][START_REF] Scherer | Introduction to chiral perturbation theory[END_REF][START_REF] Bernard | Chiral perturbation theory[END_REF][START_REF] Bijnens | Chiral perturbation theory beyond one loop[END_REF][START_REF] Bernard | Chiral Perturbation Theory and Baryon Properties[END_REF][START_REF] Scherer | A Primer for Chiral Perturbation Theory[END_REF]. At present, it has been successfully applied to the meson sector (including static properties of mesons, the meson-meson interactions, semileptonic decays, etc.) and the baryon sector (including static properties of baryons, baryon-baryon interactions and other related processes). In ChPT, there exist unknown low-energy constants (LECs) in the effective Lagrangians. But, once the values of LECs are determined, ChPT can provide a model independent tool to explore the low-energy QCD.

Therefore, the combination between lattice QCD and chiral perturbation theory is a powerful strategy to study the nonperturbative strong interaction, and it can provide a trustworthy interpretation of the majority of baryon masses from the strong interaction to further answer the origin of mass.

Gratifyingly, chiral perturbative theory provides a model independent framework to study the light-quark mass dependence, lattice volume effects and discretization effects of LQCD results. In the last decades, baryon chiral perturbation theory (BChPT) has been applied to study the ground state octet and decuplet baryon masses with different renormalization methods. Because the baryon masses do not vanish in the chiral limit, Weinberg powercounting is violated in BChPT if one is not careful. In order to restore chiral power counting, there are mainly three renormalization schemes, the so-called Heavy-Baryon (HB) ChPT [START_REF] Jenkins | Baryon chiral perturbation theory using a heavy fermion Lagrangian[END_REF], the infrared (IR) BChPT [START_REF] Becher | Baryon chiral perturbation theory in manifestly Lorentz invariant form[END_REF] and the extended-on-mass-shell (EOMS) [START_REF] Gegelia | Matching heavy particle approach to relativistic theory[END_REF][START_REF] Fuchs | Renormalization of relativistic baryon chiral perturbation theory and power counting[END_REF] BChPT. In addition to these dimensional renormalization schemes (MS and its derivatives), to speed up the convergence of BChPT, other renormalization/regularization schemes are also proposed, e.g., the cutoff scheme [START_REF] Bernard | Cutoff schemes in chiral perturbation theory and the quark mass expansion of the nucleon mass[END_REF], the finite range regulator (FRR) method [START_REF] Young | Octet baryon masses and sigma terms from an SU(3) chiral extrapolation[END_REF][START_REF] Young | Convergence of chiral effective field theory[END_REF][START_REF] Leinweber | Physical nucleon properties from lattice QCD[END_REF], and the partial summation approach [START_REF] Semke | Baryon self energies in the chiral loop expansion[END_REF]. Although the baryon masses have been systematically studied, most calculations are up to next-to-next-to-leading order (NNLO). If one wants to perform a higher order (e.g. next-to-next-to-next-to-leading order, N 3 LO) calculation, the number of LECs is too large to be determined by the experimental results of baryon masses. Fortunately, as above mentioned lattice baryon masses, such a study of N 3 LO expansion baryon masses becomes available by fitting the lattice data. This study also provides a good opportunity to study the convergence properties of SU(3) BChPT. Up to NNLO, it is found that HBChPT converges not well [START_REF] Ishikawa | SU(2) and SU(3) chiral perturbation theory analyses on baryon masses in 2+1 flavor lattice QCD[END_REF]. And, the situation is much better in FRR and EOMS BChPT in the analysis of PACS-CS and LHPC lattice data [START_REF] Martin Camalich | The lowest-lying baryon masses in covariant SU(3)-flavor chiral perturbation theory[END_REF][START_REF] Young | Octet baryon masses and sigma terms from an SU(3) chiral extrapolation[END_REF]. Furthermore, as a covariant method, the EOMS approach not only satisfies all fundamental symmetries and analyticity constraints, but also converges relatively fast. Therefore, in this work we would like to apply EOMS BChPT to perform the chiral extrapolation of baryon masses up to next-to-next-to-next-to-leading order. In order to apply this chiral formulas of baryon masses to analyze lattice data, one has to consider the artifacts from finite volumes and lattice spacings.

Because the commonly employed lattice box size L is about 3 ∼ 5 fm, finite-volume effects cannot be neglected or included in statistical errors. In Refs. [START_REF] Gasser | Light Quarks at Low Temperatures[END_REF][START_REF] Gasser | Spontaneously Broken Symmetries: Effective Lagrangians at Finite Volume[END_REF], Gasser and Leutwyler suggested that one could use ChPT to evaluate finite-volume corrections 3 . In the past decades, most studies have employed either HB ChPT or IR BChPT and focused on the two flavor sector (see, e.g., Refs. [START_REF] Khan | The Nucleon mass in N f = 2 lattice QCD: Finite size effects from chiral perturbation theory[END_REF][START_REF] Procura | Nucleon mass: From lattice QCD to the chiral limit[END_REF]). Recently, two studies in the three-flavor sector have been performed with HB ChPT [START_REF] Beane | High Statistics Analysis using Anisotropic Clover Lattices: (IV) Volume Dependence of Light Hadron Masses[END_REF] and EOMS BChPT [START_REF] Geng | Finite-volume effects on octetbaryon masses in covariant baryon chiral perturbation theory[END_REF] up to NNLO. It was pointed out that both HB ChPT and EOMS BChPT can describe the NPLQCD data well with different values of baryon octet and decuplet couplings [START_REF] Geng | Finite-volume effects on octetbaryon masses in covariant baryon chiral perturbation theory[END_REF]. However, it should be noted that the EOMS BChPT extrapolation to the physical point is in better agreement with data than HB's with the LECs determined from LQCD data.

In order to apply ChPT to study LQCD results, in principle, one should first take the continuum limit, since ChPT describes continuum QCD and is not valid for nonzero lattice spacing. In the last decades, studies of the discretization effects of LQCD have made great progress with Symanzik's effective field theory [START_REF] Symanzik | Continuum Limit and Improved Action in Lattice Theories. 1. Principles and φ 4 Theory[END_REF][START_REF] Symanzik | Continuum Limit and Improved Action in Lattice Theories. 2. O(N ) Nonlinear Sigma Model in Perturbation Theory[END_REF][START_REF] Sheikholeslami | Improved Continuum Limit Lattice Action for QCD with Wilson Fermions[END_REF][START_REF] Luscher | Chiral symmetry and O(a) improvement in lattice QCD[END_REF]. At present, there are three versions of discretized BChPT for Wilson fermions (WChPT), for staggered fermions (SChPT), and for twisted mass fermions (tmChPT), respectively. The discretization effects on the ground-state meson/baryon properties, such as masses, decay constants, electromagnetic form factors, etc., have been extensively studied in Wilson HBChPT [START_REF] Rupak | Chiral perturbation theory for the Wilson lattice action[END_REF][START_REF] Bar | Simulations with different lattice Dirac operators for valence and sea quarks[END_REF][START_REF] Aoki | Chiral perturbation theory with Wilson type fermions including a 2 effects: N f = 2 degenerate case[END_REF][START_REF] Bar | Chiral perturbation theory at O(a 2 ) for lattice QCD[END_REF][START_REF] Beane | Nucleon properties at finite lattice spacing in chiral perturbation theory[END_REF][START_REF] Tiburzi | Baryon masses at O(a 2 ) in chiral perturbation theory[END_REF][START_REF] Grigoryan | Vector meson mass corrections at O(a 2 ) in PQChPT with Wilson and Ginsparg-Wilson quarks[END_REF][START_REF] Arndt | Hadronic electromagnetic properties at finite lattice spacing[END_REF][START_REF] Bedaque | Effective Field Theory for the Anisotropic Wilson Lattice Action[END_REF]. Furthermore, similar studies have also been performed in SChPT [START_REF] Aubin | Pion and kaon masses in staggered chiral perturbation theory[END_REF][START_REF] Aubin | Pseudoscalar decay constants in staggered chiral perturbation theory[END_REF][START_REF] Sharpe | Staggered chiral perturbation theory at next-toleading order[END_REF][START_REF] Tiburzi | Baryons with Ginsparg-Wilson quarks in a staggered sea[END_REF][START_REF] Aubin | Staggered chiral perturbation theory with heavy-light mesons[END_REF][START_REF] Aubin | Staggered chiral perturbation theory for heavy-light mesons[END_REF][START_REF] Aubin | Heavy-light semileptonic decays in staggered chiral perturbation theory[END_REF][START_REF] Bernard | Chiral Perturbation Theory for All-Staggered Heavy-Light Mesons[END_REF] and tmChPT [START_REF] Scorzato | Pion mass splitting and phase structure in twisted mass QCD[END_REF][START_REF] Sharpe | The Phase diagram of twisted mass lattice QCD[END_REF][START_REF] Sharpe | Twisted mass chiral perturbation theory at next-to-leading order[END_REF][START_REF] Walker-Loud | Nucleon and Delta masses in twisted mass chiral perturbation theory[END_REF][START_REF] Sharpe | Observations on discretization errors in twisted-mass lattice QCD[END_REF][START_REF] Buchoff | pi-pi Scattering in Twisted Mass Chiral Perturbation Theory[END_REF][START_REF] Munster | Twisted mass chiral perturbation theory for 2+1+1 quark flavours[END_REF]. However, the corresponding studies are still missing in the framework of EOMS BChPT. It is interesting to note that recently several attempts have been made to determine the unknown LECs of WChPT [START_REF] Damgaard | Microscopic Spectrum of the Wilson Dirac Operator[END_REF][START_REF] Akemann | Spectrum of the Wilson Dirac Operator at Finite Lattice Spacings[END_REF][START_REF] Hansen | Constraint on the Low Energy Constants of Wilson Chiral Perturbation Theory[END_REF][START_REF] Hansen | Determining low-energy constants in partially quenched Wilson chiral perturbation theory[END_REF][START_REF] Herdoiza | Determination of Low-Energy Constants of Wilson Chiral Perturbation Theory[END_REF].

Furthermore, in SU(3) BChPT one should also be careful about the contributions of the decuplet resonances since the average mass gap between the baryon octet and the baryon decuplet δ = m D -m 0 ∼ 0.3 GeV is similar to the pion mass and well bellow those of the kaon and eta mesons, M K , M η . In Ref. [START_REF] Jenkins | Baryon masses in chiral perturbation theory[END_REF], HBChPT was enlarged to include the decuplet and applied to calculate the octet baryon masses up to O(p 3 ). It was shown in Ref. [START_REF] Bernard | Critical analysis of baryon masses and sigma terms in heavy baryon chiral perturbation theory[END_REF] that the effects of the virtual decuplet on the octet baryon masses start out at O(p 4 ) in the same framework. For the spin-dependent quantities, the virtual decuplet contributions are found to be important in HBChPT, such as magnetic moments [START_REF] Puglia | Octet baryon charge radii, chiral symmetry and decuplet intermediate states[END_REF] and axial vector form factors [START_REF] Jenkins | Chiral corrections to the baryon axial currents[END_REF][START_REF] Luty | Decouplet contributions to hyperon axial vector formfactors[END_REF]. In EOMS BChPT, the effects of the virtual decuplet are found to be negligible for the magnetic moments of the octet baryons if the "consistent" coupling scheme for the octetdecuplet-pseudoscalar coupling is adopted [START_REF] Geng | Leading-order decuplet contributions to the baryon magnetic moments in Chiral Perturbation Theory[END_REF]. On the other hand, up to NNLO, the virtual decuplet contributions seem to play an important role in describing the NPLQCD volumedependent data [START_REF] Geng | Finite-volume effects on octetbaryon masses in covariant baryon chiral perturbation theory[END_REF] and in the determination of the baryon sigma terms [START_REF] Alarcon | The strangeness content of the nucleon from effective field theory and phenomenology[END_REF]. Therefore, it is necessary to study the effects of the virtual decuplet on the light-quark mass and on the volume dependence of the LQCD data at N 3 LO.

To sum up, in this work, we would like to use EOMS BChPT to study octet and decuplet baryon masses up to next-to-next-to-next-to-leading order. By constructing the chiral effective field theory to take the lattice artifacts from finite-lattice volumes and lattice spacings into account self-consistently, we could perform a trustworthy analysis of the lattice baryon masses. With the obtained LECs, we could explore the convergent property of EOMS BChPT at different chiral orders and predict the baryon sigma terms to interpret the composition of baryon masses. We also can apply this strategy to study the baryon masses of excitation states by including the excitation resonances in chiral perturbation theory.

Outline of the dissertation

In Chapter 2, the basic ideas and fundamental knowledge of lattice QCD are briefly introduced. We take the lowest-lying octet baryons for example and review the recent lattice baryon masses in detail. As the theoretical basis, in Chapter 3, chiral perturbation theory is elaborated by including the essence of effective field theory, chiral symmetry of low-energy QCD and its spontaneous and explicit breaking pattern. Applying Weinberg's theorem, the general form of chiral effective Lagrangians is constructed for the meson and baryon sectors. We also discuss the three renormalization schemes to deal with the power-counting breaking terms appearing in the calculation of baryon chiral perturbation theory. Next, we present the application of ChPT to the study of lattice baryon masses in chapters 4-7. Firstly, in Chapter 4, we perform a systematic study of lowest-lying octet baryon masses in covariant baryon chiral perturbation theory with the extended-on-mass-shell scheme up to next-to-next-to-next-to-leading order. The corresponding meson-meson and meson-baryon chiral Lagrangians are given. After calculating Feynman diagrams, we obtain the chiral corrections to octet baryon self-energies and the finite-volume corrections at O(p 4 ). By constructing Wilson BChPT in the SU(3) sector, we obtain the discretization effects for finite lattice spacing. In Chapter 5, we utilize the theoretical expressions of octet baryon masses in a discretized finite-volume hypercube and perform a systematic study of all the published lattice QCD data by including chiral extrapolation of octet baryon masses, finite-volume effects, and finite-lattice spacing discretization effects. In addition, the virtual decuplet effects on the chiral extrapolation and finite-volume effects are systematically explored. In Chapter 6, we predict the octet baryon sigma terms, σ πB and σ sB , via the Feynman-Hellmann theorem by analyzing the latest high-statistics lattice QCD data. Three key factors in determining sigma terms are clarified for the first time, including effects of lattice scale setting, systematic uncertainties originating from chiral expansion truncations, and constraint of strong-interaction isospin breaking effects. As a natural extension, in Chapter 7, we present an analysis of lowest-lying decuplet baryon masses up to next-to-nextto-next-to-leading order in EOMS BChPT. In order to determine the unknown low-energy constants, we perform a simultaneous fit of lattice data from the PACS-CS, QCDSF-UKQCD, and HSC collaborations, with finite-volume corrections included self-consistently. The pionand strangeness-sigma terms of the decuplet baryons are also predicted. With the development of computer techniques, the baryon masses of excitation states are becoming available in lattice simulations, especially for the Roper. In Chapter 8, the nucleon, Delta and Roper masses and widths are systematically studied in an extension of chiral perturbation theory that includes the Delta-nucleon and Roper-nucleon mass differences as low-energy scales. The contributions due to the mixing between nucleon and Roper are taken into account explicitly. The virtual Roper effects on the nucleon and Delta masses are evaluated up to next-to-next-to-leading order, as well as the effects of the nucleon and Delta in the Roper mass and width. Finally, a brief summary and perspectives are given in Chapter 9. Furthermore, in Appendix A, we tabulate the latest n f = 2 + 1 lattice results of lowest-lying octet baryon masses. In Appendices B and C, the utilized Feynman rules and the method of loop integrals are presented, respectively. In Appendix D, the strong isospin-breaking effects on octet baryon masses are calculated.

Chapter 2

Lattice Quantum Chromodynamics

Lattice Quantum Chromodynamics, as a numerical method to deal with nonperturbative lowenergy QCD from first principles, was proposed by Wilson [START_REF] Wilson | Confinement of Quarks[END_REF] in the 1980s. LQCD has the same input parameters as QCD: the running coupling constant α s and the quark masses. There are three steps to carry out a nonperturbative lattice simulation. First, in order to deal with the renormalization problem of QCD, one should introduce an ultraviolet regulator into the discretized theory in terms of the finite space-time grid. Second, using the path integral and the Monte Carlo numerical method, the Green's functions of lattice QCD can be evaluated. Third, one should perform multiple extrapolations (chiral extrapolation, finite-volume corrections and continuum extrapolation) to obtain the observables of interest in continuum space-time. In this chapter, we present the basic ideas and fundamental knowledge of lattice QCD. Taking lattice octet baryon masses as an example, we briefly introduce the extraction method, the adopted lattice actions, volumes and spacings. And, recent LQCD studies of Roper mass are summarized. For more details and the recent developments of LQCD, we refer to the recent excellent reviews [START_REF] Montvay | Quantum fields on a lattice[END_REF][START_REF] Gupta | Introduction to lattice QCD: Course[END_REF][START_REF] Pierro | From Monte Carlo integration to lattice quantum chromodynamics: An Introduction[END_REF][START_REF] Smit | Introduction to quantum fields on a lattice: A robust mate[END_REF][START_REF] Rothe | Lattice gauge theories: An Introduction[END_REF][START_REF] Degrand | Lattice Methods for Quantum Chromodynamics[END_REF][START_REF] Hagler | Hadron structure from lattice quantum chromodynamics[END_REF][START_REF] Gattringer | Quantum Chromodynamics on the Lattice -An Introductory Presentation[END_REF][START_REF] Fodor | Light Hadron Masses from Lattice QCD[END_REF][START_REF] Aoki | Review of lattice results concerning low-energy particle physics[END_REF].

Fundamentals of lattice QCD

In this section, we overview the basic and key techniques in lattice QCD simulation. Because the transition matrix elements can be evaluated by using the Feynman path integral approach, we first discuss the technique of path integrals in LQCD.

In continuum QCD, the expectation value of one chronological operator Ô[ψ, ψ, A µ ] can be expressed in terms of a path integral in Euclidean space and time

1 O = 1 Z DψD ψ DA µ Ô[ψ, ψ, A µ ]e -( ψM ψ+S G ) , (2.1)
where ψ and ψ are quark and anti-quark fields, and A µ is gauge field. The M is M = / D + m with the gauge covariant derivative D µ = ∂ µ + igA µ and quark mass m. The gluon gauge action is

S G = 1 4 G µν G µν with the gluon field strength G µν = ∂ µ A ν -∂ ν A µ -ig[A µ , A ν ],
where g is the SU(3) structure constant. A µ = A a µ T a with the eight gluon fields A a µ and T a being the SU(3) generators in the 3 representation. The partition function Z in Euclidean space-time is [START_REF] Dirac | The Lagrangian in quantum mechanics[END_REF][START_REF] Feynman | Quantum Mechanics and Path Integrals[END_REF]

Z = DψD ψ DA µ e -( ψM ψ+S G ) .
(2.2)

After performing the integration over fermion fields analytically and using its Gaussian structure,

DψD ψe -ψM ψ ∝ det(M ), (2.3)
1 We want to mention that the expectation values Ô in the path integral approach correspond to timeordered correlation functions. we can express the expectation value as

O = 1 Z DA µ det(M ) Ô[M -1 , A µ ]e d 4 x(-1 4 Gµν G µν ) , (2.4)
with

Z = DA µ det(M )e d 4 x(-1 4 Gµν G µν ) . (2.5)
Therefore, the expectation value of Ô[ψ, ψ, A µ ] and the partition function are integrals over only the background gluon gauge configurations with the quark contribution encoded in the non-local term det(M ).

In order to evaluate the path integral [Eq. (2.1)] in lattice QCD, one can discretize a finite Euclidean space-time with length L into a number of unit cells with the lattice spacing a, as illustrated in Fig. 2.1 for three dimensions. Here we only focus on the common isotropic case in which the lattice spacings in all directions are equal. The corresponding temporal/spatial coordinates of lattice sites are

x µ = n µ a, with n µ = 0, • • • , N µ -1, (2.6)
where the a and N µ are the lattice spacing and the number of lattice sites in the direction µ, respectively. The lattice size in the direction µ is L µ = aN µ . In the studies of the properties of hadronic ground states, the number of lattices in temporal dimension (N t ) is always 3 ∼ 5 times larger than the one in the spacial dimension (N s ).

In the discretized space-time, the quark field (Ψ(x)) is located in the lattice site,

ψ → Ψ(x) = (u(x), d(x), s(x), . . . ) T , (2.7)
with three light quark fields u(x), d(x), s(x), and the gluon gauge field (U µ (x, y)) is defined as the link connecting the two adjacent sites,

A µ → U µ (x, y) = P exp(- y x A µ (x )dx ). (2.8)
In terms of discretized fields, the expectation value of operator O can be written as

O = 1 Z DΨD Ψ DU µ O[Ψ, Ψ, U µ ] e -( ΨM (U )Ψ+S G [U ]) , (2.9)
in the discretized space-time, with the quark operator M (U ) and the gluon gauge action S G (U ), which depend on the gauge field U . In principle, we can perform the integration over the quark fields, and obtain

O(q, q, U ) = 1 Z DU µ (x)detM (U )O[M -1 (U ), U µ ] e -S G (U ) .
(2.10)

Because there are 4 gauge links for each lattice site, and every gauge link is described by an SU(3) matrix with 8 parameters, the number of integration variables in Eq. (2.10) is huge:

N 3 s × N t × 4 × 8.
Therefore, direct numerical integration is impractical and one has to use Monte-Carlo techniques to generate all the sets of U on all links. The cost of lattice QCD is roughly expressed as [START_REF] Lepage | Strong interactions at low and intermediate energies[END_REF] 2

Cost ∝ L a 4 1 a 1 M 2 π a , (2.11)
where the first factor is the number of lattice sites (N 3 s N t ), and the second and third factors are due to "critical-slowing-down" of the algorithms used for the simulation. Here, we want to mention that, at the annual conference of lattice QCD 2001, a practical formula was given by Ukawa [START_REF] Ukawa | Computational cost of full QCD simulations experienced by CP-PACS and JLQCD Collaborations[END_REF] with the standard Hybrid-Monte-Carlo (HMC) algorithm employed the CP-PACS and JLQCD collaborations. In 2006, an updated formula of lattice cost was presented using the domain decomposition ideas (DD) with HMC algorithm [START_REF] Del Debbio | QCD with light Wilson quarks on fine lattices (I): First experiences and physics results[END_REF]. The number of N op of floating-point can be expressed as

N op = k 20 MeV m L 3 fm 5 0.1 fm a 6 Tflops × year, (2.12)
where m is the running sea-quark mass in the MS scheme at renormalization scale µ = 2 GeV and k 0.05 when one employs the O(a)-improved action. In order to guarantee precision and statistical quality, the lattice spacing a is usually taken about 0.1 fm, and the lattice box size is L = 3 ∼ 5 fm.

In the last century, under the limitation of computer resources, the "quenched" approximation was employed in the lattice simulations [START_REF] Hamber | Numerical Estimates of Hadronic Masses in a Pure SU(3) Gauge Theory[END_REF][START_REF] Weingarten | Monte Carlo Evaluation of Hadron Masses in Lattice Gauge Theories with Fermions[END_REF], wherein quark fields are treated as non-dynamical "frozen" variables. It corresponds to taking det(M ) = constant and neglecting quark loop effects. After that, the so called "partially quenched" approximation [START_REF] Khan | Light hadron spectroscopy with two flavors of dynamical quarks on the lattice[END_REF][START_REF] Prelovsek | Scalar meson in dynamical and partially quenched two-flavor QCD: Lattice results and chiral loops[END_REF], where the determinant det(M ) = det( / D(U ) + m sea ) and the quark propagator = ( / D + m val ) -1 with m sea = m val , was proposed. In this case, the quark loop contributions are partially taken into account. Nowadays, with the development of computing power and numerical algorithms, full-dynamical lattice simulations have become feasible. The international activity has begun to share the lattice QCD configurations, such as the International Lattice Data Grid (ILDG) [http://plone.jldg.org/]. From this website, one can find several large collections of ensembles of configurations.

When one discretizes the equation of motion for fermions, the fermion doubling problem appears, which is manifested through the existence of extra poles in the Dirac propagator on the lattice. These poles do not disappear even in the continuum limit a → 0. In order to explicitly show this problem, we write out the Dirac propagator in momentum space

D(p) = i(1/a) µ γ µ sin(ap µ ) + m [(1/a) µ sin(ap µ )] 2 + m 2 .
(2.13)

In addition to the physical pole at p 2 = -m 2 , there are also 15 additional poles located at the edges of the Brillouin zone. This is because of the chiral anomaly [START_REF] Witten | Global Aspects of Current Algebra[END_REF] 3 , which exists in the continuum theory, while it is absent in lattice QCD [START_REF] Kaplan | Chiral Symmetry and Lattice Fermions[END_REF].

In practical lattice calculations, one has to fix the fermion doubling problem by employing different types of fermion actions, such as staggered fermions [START_REF] Kogut | Hamiltonian Formulation of Wilson's Lattice Gauge Theories[END_REF][START_REF] Banks | Strong Coupling Calculations of Lattice Gauge Theories: (1+1)-Dimensional Exercises[END_REF][START_REF] Susskind | Lattice Fermions[END_REF], Wilson fermions [START_REF] Wilson | Proceedings of the First Half of the 1975 International School of Subnuclear Physics[END_REF], twisted mass fermions [START_REF] Frezzotti | A Local formulation of lattice QCD without unphysical fermion zero modes[END_REF][START_REF] Frezzotti | Lattice QCD with a chirally twisted mass term[END_REF], and chiral symmetric fermions [START_REF] Ginsparg | A Remnant of Chiral Symmetry on the Lattice[END_REF][START_REF] Narayanan | A Construction of lattice chiral gauge theories[END_REF][START_REF] Hasenfratz | The Index theorem in QCD with a finite cutoff[END_REF][START_REF] Luscher | Exact chiral symmetry on the lattice and the Ginsparg-Wilson relation[END_REF] (e.g. domain wall fermions, overlap fermions).

Extracting baryon masses

In this section, we present the techniques to extract physical observables, especially baryon masses. Hadron matrix elements can be calculated using two-or three-point correlation functions. In order to extract the baryon masses, we take the two-point correlation function for example:

0|O f (x, t)O † i (0, 0)|0 , (2.14)
with the "source" operator O i and the "sink" operator O f , representing the amplitude for creating a state with the quantum numbers of O i at space-time point (0, 0), annihilated by O f at (x, t). For the specific octet baryon state |B , the two operators should have non-zero overlaps with |B ( 0|O i,f |B = 0). One can perform Fourier-transforms in the spatial directions for Eq. (2.14)

C if (p, t) = x 0|O f (x, t)O † i (0, 0)|0 e -ip•x . (2.15)
Inserting a complete set of energy eigenstates, the correlation function becomes

C if (p, t) = n 1 2E n (p) 0|O f |n n|O † i |0 e -En(p)t , (2.16)
where the energy E n (p) is the nth-eigenvalue of H, and the factor 1/2E n (p) is due to the relativistic normalization. If the octet baryon state |B happens to be the lowest energy state, the correlation function can become

C if (t) t→∞ ---→ 1 2E 0 (p) 0|O f |B B|O † i |0 e -E 0 (p)t , (2.17)
since all other states die at the large enough t. Therefore, the octet baryon mass m B = E 0 can be extracted with the product of matrix elements 0|O f |B B|O † i |0 . In principle, the expression of the correlator C if (p, t) can also be used to obtain the energies of the excited hadron states, which have the same quantum numbers as the operators O i,f , by fitting the correlation function to a sum of exponentials. However, it is complicated to extract excited state masses from these exponents as the correlation functions decay quickly. The standard method is the variational method to determine the excited state hadron spectrum [START_REF] Michael | Adjoint Sources in Lattice Gauge Theory[END_REF][START_REF] Luscher | How to Calculate the Elastic Scattering Matrix in Twodimensional Quantum Field Theories by Numerical Simulation[END_REF]. This method has been widely applied in the study of the hadron spectrum. Recently, a new method called Athens Model Independent Analysis Scheme (AMIAS) was proposed in Ref. [START_REF] Papanicolas | A novel method of data analysis for hadronic physics[END_REF] and applied to study the excited state of nucleon [START_REF] Alexandrou | Novel analysis method for excited states in lattice QCD: The nucleon case[END_REF].

The physical quantities computed from lattice QCD always depend on the lattice unit (unknown lattice spacing a). Therefore, one has to use the scale-setting method, which is to choose a well-known quantity at the physical point to determine the lattice spacing a, to obtain the lattice results with the physical unit. We want to mention that a suitable physical quantity demands the following conditions: it is easily calculated in lattice QCD with small statistical errors, and it should have a weak dependence on the quark masses (for a recent review, see Ref. [START_REF] Sommer | Scale setting in lattice QCD[END_REF]). Commonly, there are two main scale setting methods: the massindependent (phenomenological) scales and the mass-dependent (theoretical) scales. For the former, one assumes that the lattice spacing is independent of the bare quark masses at a fixed bare coupling. Then, the lattice spacing a, which is the same for all lattice ensembles, can be determined by using the physical quantities, such as the Ω mass or the pseudoscalar meson decay constants f π , f K at the physical point. For the mass-dependent scales, the lattice spacing varies with different bare quark masses. The lattice spacing a can be determined by using a physical quantity, which is assumed to be independent with the quark masses, such as the Sommer scales r 0 , r 1 from the static quark potential V (r) [START_REF] Sommer | A New way to set the energy scale in lattice gauge theories and its applications to the static force and alpha-s in SU(2) Yang-Mills theory[END_REF],

r 2 dV (r)
dr r=r 0 = 1.65, r 2 dV (r)

dr r=r 1 = 1.0, (2.18)
and the scales t 0 , w 0 from the gradient current [START_REF] Lüscher | Properties and uses of the Wilson flow in lattice QCD[END_REF][START_REF] Borsanyi | High-precision scale setting in lattice QCD[END_REF],

E (t)| t=t 0 = 0.3, t dE (t) dt t=w 2 0 = 0.3, (2.19)
with

E (t) = t 2 E(t)
, where E(t) being the expectation value of the continuum-like action density G µν (t)G µν (t)/4.

Lattice data of baryon masses

In the past decades, with the increase of computing power and the continuous improvement of numerical algorithms, lattice QCD has been shown to be extremely successful in studying the nonperturbative regime of QCD. There are many applications to low-energy hadronic physics, e.g. the hadronic spectrum (especially the ground state hadron masses) [START_REF] Fodor | Light Hadron Masses from Lattice QCD[END_REF][START_REF] Morningstar | Progress with excited states in lattice QCD[END_REF], the structure of hadrons [START_REF] Abdel-Rehim | Nucleon and pion structure with lattice QCD simulations at physical value of the pion mass[END_REF], the mechanism of quark confinement and chiral symmetry breaking [START_REF] Doi | Lattice QCD analysis for relation between quark confinement and chiral symmetry breaking[END_REF], the equilibrium properties of QCD at finite temperature and finite chemical potential [START_REF] Aarts | QCD at nonzero chemical potential: recent progress on the lattice[END_REF], etc. Besides that, lattice simulations can vary the input parameters, especially the quark masses, to predict the dependence of observables on the running coupling constant α s [START_REF] Blossier | High statistics determination of the strong coupling constant in Taylor scheme and its OPE Wilson coefficient from lattice QCD with a dynamical charm[END_REF]. Here we would like to present the recent studies of baryon masses from lattice simulations. The latest developments of lattice QCD in the study of low-energy particle physics are reviewed in Ref. [START_REF] Aoki | Review of lattice results concerning low-energy particle physics[END_REF].

Recently, the lowest-lying baryon masses, composed of up, down and strange quarks, has been simulated by various LQCD collaborations [START_REF] Dürr | Ab-Initio Determination of Light Hadron Masses[END_REF][START_REF] Aoki | 2+1 Flavor Lattice QCD toward the Physical Point[END_REF][START_REF] Aoki | Physical Point Simulation in 2+1 Flavor Lattice QCD[END_REF][START_REF] Walker-Loud | Light hadron spectroscopy using domain wall valence quarks on an Asqtad sea[END_REF][START_REF] Lin | First results from 2+1 dynamical quark flavors on an anisotropic lattice: Light-hadron spectroscopy and setting the strange-quark mass[END_REF][START_REF] Bietenholz | Tuning the strange quark mass in lattice simulations[END_REF][START_REF] Bietenholz | Flavour blindness and patterns of flavour symmetry breaking in lattice simulations of up, down and strange quarks[END_REF][START_REF] Beane | High Statistics Analysis using Anisotropic Clover Lattices: (IV) Volume Dependence of Light Hadron Masses[END_REF][START_REF] Alexandrou | Baryon spectrum with N f = 2 + 1 + 1 twisted mass fermions[END_REF]. Here we want to mention that, as the first systematic and accurate study of proton and nucleon masses and other light hadrons (including the ground state of pseudoscalar octet meson, vector nonet meson, octet baryon and decuplet baryon masses), the BMW Collaboration provided an accurate interpretation of the origin of mass from first principles [START_REF] Dürr | Ab-Initio Determination of Light Hadron Masses[END_REF]. In the BMW calculations, the lightest pion mass is M π ≈ 190 MeV, very close to its physical value M phys. π = 139 MeV, which guarantees a valid chiral extrapolation to the physical region. Three different values of lattice spacing a = 0.125, 0.085, and 0.065 fm are employed to take into account lattice discretization effects and to perform the continuum extrapolation of lattice data. Several lattice box size L are considered, to include finite-volume corrections. The effects of the heavier c, b, and t quarks are included in the coupling constant and light quark masses. With the above considerations, the extrapolation results of light hadron spectrum are very close to the corresponding experimental data (Fig. 2.2). Furthermore, in Fig. 2.3, they present the chiral extrapolation of the nucleon lattice data with three different lattice spacings. We can see that the extrapolation result for the nucleon mass is 953(29) [START_REF] Weingarten | Monte Carlo Evaluation of Hadron Masses in Lattice Gauge Theories with Fermions[END_REF] MeV, which is almost the same as the experimental value m phys. N = 939(1) MeV [START_REF] Olive | Review of Particle Physics[END_REF]. More recently, the ETMC Collaboration have studied the lowest-lying baryon masses with n f = 2 + 1 + 1 twisted mass fermions [START_REF] Alexandrou | Baryon spectrum with N f = 2 + 1 + 1 twisted mass fermions[END_REF], where a strange and a charm quark masses fixed to approximately their physical values. The pseudoscalar masses in the range of 210 MeV to 430 MeV with three different lattice spacings a = 0.0936(13), 0.0823 (10), 0.0646 (7) fm.

During the same period, several lattice QCD collaborations have also performed n f = 2 + 1 simulations of light hadron spectrum [START_REF] Aoki | 2+1 Flavor Lattice QCD toward the Physical Point[END_REF][START_REF] Aoki | Physical Point Simulation in 2+1 Flavor Lattice QCD[END_REF][START_REF] Walker-Loud | Light hadron spectroscopy using domain wall valence quarks on an Asqtad sea[END_REF][START_REF] Lin | First results from 2+1 dynamical quark flavors on an anisotropic lattice: Light-hadron spectroscopy and setting the strange-quark mass[END_REF][START_REF] Bietenholz | Tuning the strange quark mass in lattice simulations[END_REF][START_REF] Bietenholz | Flavour blindness and patterns of flavour symmetry breaking in lattice simulations of up, down and strange quarks[END_REF][START_REF] Beane | High Statistics Analysis using Anisotropic Clover Lattices: (IV) Volume Dependence of Light Hadron Masses[END_REF]. In Table 2.1, we tabulate the quark/gluon gauge actions and lattice spacings which are utilized in the BMW [START_REF] Dürr | Ab-Initio Determination of Light Hadron Masses[END_REF], PACS-CS [START_REF] Aoki | 2+1 Flavor Lattice QCD toward the Physical Point[END_REF], LHPC [START_REF] Walker-Loud | Light hadron spectroscopy using domain wall valence quarks on an Asqtad sea[END_REF], HSC [START_REF] Lin | First results from 2+1 dynamical quark flavors on an anisotropic lattice: Light-hadron spectroscopy and setting the strange-quark mass[END_REF], QCDSF-UKQCD [START_REF] Bietenholz | Flavour blindness and patterns of flavour symmetry breaking in lattice simulations of up, down and strange quarks[END_REF], NPLQCD [START_REF] Beane | High Statistics Analysis using Anisotropic Clover Lattices: (IV) Volume Dependence of Light Hadron Masses[END_REF], and ETMC [START_REF] Alexandrou | Baryon spectrum with N f = 2 + 1 + 1 twisted mass fermions[END_REF] simulations. From Table 2.1, we can see that most lattice simulations employ a lattice spacing a ≈ 0.1 fm and the O(a)improved action. On the other hand, although all these five collaborations adopt different fermion and gauge actions, all of them are believed to lead to the same continuum theory -QCD. Therefore, it is crucial to clarify whether all these simulation results of octet baryon masses are consistent with each other [START_REF] Beringer | Review of Particle Physics (RPP)[END_REF]. It should be mentioned that, in this work, we would like to limit our studies in the SU(3) sector to analyze the lattice baryon masses from n f = 2 + 1 simulations. Therefore, the ETMC results and the unpublicable BMW results are not studied.

As has been stated in introduction, lattice calculations are performed with larger than physical light quark masses and finite volumes. In Fig. pion masses provides the opportunity to explore the applicability of ChPT for extrapolation of baryon masses. Although the light u/d quark masses adopted are always larger than their physical counterpart, the strange quark masses vary from collaboration to collaboration: those of the PACS-CS and LHPC collaborations are larger than the physical one; those of the HSC and NPLQCD groups are a bit smaller, whereas those of the QCDSF-UKQCD collaboration are all lighter than the physical one. In the L-M 2 π plane, it is seen that the PACS-CS and LHPC simulations adopt a single value of lattice volume; the HSC and QCDSF-UKQCD simulations use two different lattice volumes and the NPLQCD simulations are performed with four different lattice volumes in order to study the finite-volume effects on the octet baryon masses. Many of the simulations are still performed with M φ L from 3 to 5 and with M φ larger than 300 MeV. As a result, finite-volume corrections (FVCs) may not be negligible (see, e.g., Ref. [START_REF] Geng | Finite-volume effects on octetbaryon masses in covariant baryon chiral perturbation theory[END_REF]).

In Appendix A, we tabulate the octet baryon masses of the PACS-CS, the LHPC, the HSC, the QCDSF-UKQCD and the NPLQCD collaborations. The numbers are given in physical units using either the lattice scale specified in the original publications [START_REF] Aoki | 2+1 Flavor Lattice QCD toward the Physical Point[END_REF][START_REF] Walker-Loud | Light hadron spectroscopy using domain wall valence quarks on an Asqtad sea[END_REF][START_REF] Lin | First results from 2+1 dynamical quark flavors on an anisotropic lattice: Light-hadron spectroscopy and setting the strange-quark mass[END_REF][START_REF] Beane | High Statistics Analysis using Anisotropic Clover Lattices: (IV) Volume Dependence of Light Hadron Masses[END_REF] or the method of ratios such as QCDSF-UKQCD [START_REF] Bietenholz | Flavour blindness and patterns of flavour symmetry breaking in lattice simulations of up, down and strange quarks[END_REF].

Nowadays, the lowest-lying hadron spectrum is now well understood with controllable lattice artifacts [START_REF] Fodor | Light Hadron Masses from Lattice QCD[END_REF]. But, for the light quark hadrons, there are many excited states whose physical properties are poorly understood. In particular, the first even-parity excited state of the nucleon, J P = (1/2) + Roper resonance N (1440) or P 11 (1440), has been very interesting since its discovery [START_REF] Roper | Evidence for a P-11 Pion-Nucleon Resonance at 556 MeV[END_REF]. In the quark model, the Roper, as an S-wave excitation, is lighter than the first odd-parity excited state N (1530) or S 11 (1530), a P -wave excitation. This abnormal phenomenology stimulates a large number of studies to explain its structure with a possible exotic nature. Recently, several LQCD collaborations [START_REF] Alexandrou | Novel analysis method for excited states in lattice QCD: The nucleon case[END_REF][START_REF] Mahbub | Roper Resonance in 2+1 Flavor QCD[END_REF][START_REF] Edwards | Excited state baryon spectroscopy from lattice QCD[END_REF][START_REF] Engel | QCD with Two Light Dynamical Chirally Improved Quarks: Baryons[END_REF][START_REF] Alexandrou | Nucleon Excited States in N f =2 lattice QCD[END_REF][START_REF] Liu | The Roper Puzzle[END_REF] mass of Roper resonance. In Fig. 2.5, lattice Roper masses from the CSSM [START_REF] Mahbub | Roper Resonance in 2+1 Flavor QCD[END_REF], JLab [START_REF] Edwards | Excited state baryon spectroscopy from lattice QCD[END_REF], BGR [START_REF] Engel | QCD with Two Light Dynamical Chirally Improved Quarks: Baryons[END_REF], Cyprus [START_REF] Alexandrou | Novel analysis method for excited states in lattice QCD: The nucleon case[END_REF][START_REF] Alexandrou | Nucleon Excited States in N f =2 lattice QCD[END_REF], and χQCD [START_REF] Liu | The Roper Puzzle[END_REF] collaborations are illustrated. We can see that lattice Roper masses exhibit much larger chiral-log effects than that of the nucleon, except the χQCD simulation adopting the Sequential Empirical Bayesian method with the overlap fermion [START_REF] Liu | The Roper Puzzle[END_REF]. The Roper mass from χQCD shows almost the same pion mass dependence as the nucleon. Furthermore, most lattice data of Roper mass correspond to an unstable Roper particle for M π < 500 MeV. Therefore, LQCD calculation of Roper mass should need more efforts to clarify this large discrepancy.

Chapter 3

Chiral Perturbation Theory

In this chapter, we discuss the theoretical framework of the present dissertation. Chiral perturbation theory (ChPT), as an effective field theory of low-energy QCD, was firstly proposed by Weinberg [START_REF] Weinberg | Phenomenological Lagrangians[END_REF] in 1979. Since its proposal, it has been widely applied in the study of low-energy hadronic physics. Firstly, we give a general introduction about the basic ideas of effective field theory. Following that, we turn to the low-energy region of QCD and discuss chiral symmetry and its breaking pattern. We introduce the construction principles of chiral effective Lagrangians in the mesonic sector and extend chiral perturbation theory to the one-baryon sector with three different renormalization schemes. In this chapter, we cannot cover all the details of ChPT and we suggest the interested readers to refer to several excellent reviews [START_REF] Meissner | Recent developments in chiral perturbation theory[END_REF][START_REF] Leutwyler | Principles of chiral perturbation theory[END_REF][START_REF] Bernard | Chiral dynamics in nucleons and nuclei[END_REF][START_REF] Pich | Chiral perturbation theory[END_REF][START_REF] Ecker | Chiral perturbation theory[END_REF][START_REF] Pich | Effective field theory: Course[END_REF][START_REF] Scherer | Introduction to chiral perturbation theory[END_REF][START_REF] Bernard | Chiral perturbation theory[END_REF][START_REF] Bijnens | Chiral perturbation theory beyond one loop[END_REF][START_REF] Bernard | Chiral Perturbation Theory and Baryon Properties[END_REF][START_REF] Scherer | A Primer for Chiral Perturbation Theory[END_REF].

Effective field theory

Effective Field Theory (EFT) is a powerful theoretical tool to describe low-energy physics. Nowadays, the EFT technique has become popular in particle and nuclear physics. The basic idea of EFT is that the low energy dynamics does not depend on the details of the high energy dynamics, which can be integrated out. In order to construct an EFT to describe low-energy physics, one has to ensure a characteristic heavy scale Λ (called "hard energy scale") to separate low and high energy regions, and make an expansion in powers of Q/Λ with Q Λ, where Q is the soft momentum scale of the energy region of interest. When the relevant degrees of freedom (light fields) are determined, the general possible effective Lagrangian can be written down according to a theorem, proposed in Weinberg's original work [START_REF] Weinberg | Phenomenological Lagrangians[END_REF]: "If one writes down the most general possible Lagrangian, including all terms consistent with assumed symmetry principles, and then calculates matrix elements with this Lagrangian to any given order of perturbation theory, the result will simply be the most general possible S-matrix consistent with analyticity, perturbative unitary, cluster decomposition, and the assumed symmetry principles."

The general form of effective Lagrangians can be expressed as

L eff. = i c i O i , (3.1)
where the operators O i are constructed with the light fields, and the couplings c i are called lowenergy constants (LECs) and contain the information about heavy degrees of freedom which are integrated out in the EFT. In principle, the values of c i can be computed from a more fundamental high energy theory. If the fundamental theory is weakly coupled (e.g. QED), one can explicitly obtain the values of c i , but if the theory is strongly coupled (e.g. QCD), one usually treats the c i as free parameters, which can be fixed by the relevant experimental and the lattice data. Usually, we can organize the operators by using naïve dimensional analysis (NDA) [START_REF] Manohar | Chiral Quarks and the Nonrelativistic Quark Model[END_REF]. First, the canonical dimensions of the light fields can be determined by using the kinetic terms in the free Lagrangian. Thus, one can easily obtain the dimension of the operators O i , denoted as d i . In the four dimensional space-time, the estimated magnitudes of all the corresponding LECs should be

c i ∼ (4π) N -2 /Λ d i -4 .
According to the dimensions of operators, we can distinguish three types of operators with different behaviours:

• Relevant operators with d < 4: These operators become more relevant at low energy with large values of the coefficients c i ∼ Λ 4-d i . The number of possible relevant operators is small, such as unit operator (d = 0), boson mass terms (d = 2) and fermion mass terms (d = 3).

• Marginal operators with d = 4: These operators, lying between relevancy and irrelevancy, are equally important at all energy scales. There are some well-known examples: φ 4 term in scalar field theory, a Yukawa interaction ψψφ, and QED/QCD interactions.

• Irrelevant operators with d > 4: Their contributions are small at low energies with the values of couplings ∼ 1/Λ d-4 , which can suppress the contributions from these operators. But they cannot be ignored because they usually contain information about the high energy physics.

The operators mentioned above can be systematically organized based on their contributions in an EFT. This principle of organization is the so called "power counting". As mentioned in Ref. [START_REF] Ecker | Chiral perturbation theory[END_REF], for the weakly-coupled cases, the power counting is just a dimensional expansion, as we discussed above. This kind of EFT is the decoupling effective field theory. We can write the corresponding effective Lagrangian in a general form

L eff. = L ≤4 + L 5 + L 6 + • • • , (3.2)
where L ≤4 contains all terms with dimension d ≤ 4, L 5 contains the terms with dimension 5, and so on. Generally speaking, the first term L ≤4 , which only contains the relevant and marginal operators, can be renormalizable. If the irrelevant operators are included (such as L 5 , L 6 , • • • ), the EFT becomes renormalizable order by order. The corrections from the d > 4 parts are suppressed by (E/Λ) d-4 . There are several decoupling EFTs: QED, the Fermi theory of weak interaction, and the Standard Model. But for the un-decoupling EFT [START_REF] Ecker | Chiral perturbation theory[END_REF], the situation will be very different when one has to tackle the strongly-coupled theories with spontaneous symmetry breaking down. The fundamental degrees of freedom are replaced by the light pseudo-Goldstone bosons at low energies. There are two examples of non-decoupling EFTs, the Standard Model without Higgs bosons and chiral perturbation theory. In the next section, we will give the details of chiral perturbation theory. Generally, there are infinite terms in the effective Lagrangian. However, at a specific expansion order, EFT is described by a number of finite terms in the Lagrangian with several coupling constants. Once the effective Lagrangian is written out at a given order, the EFT can provide a powerful prediction ability for many different processes. Besides, there must exist a range of applicability for an EFT, which usually depends on the characteristic energy scale Λ.

At some high energy region, one has to find another high energy theory, which could lead to the corresponding EFT. In this section, we only give a brief overview of EFT, the interested readers can find more details in the reviews [START_REF] Georgi | Effective field theory[END_REF][START_REF] Kaplan | Effective field theories[END_REF][START_REF] Manohar | Effective field theories[END_REF][START_REF] Bedaque | Effective field theory for few nucleon systems[END_REF][173][START_REF] Brambilla | Effective field theories for heavy quarkonium[END_REF][START_REF] Ecker | Effective field theories[END_REF][START_REF] Burgess | Introduction to Effective Field Theory[END_REF].

Chiral perturbation theory

In this section, we begin to introduce a powerful EFT for the study of nonperturbative strong interactions. As mentioned above, this EFT should have the same symmetries as QCD and in particular chiral symmetry. That is why this EFT is named as Chiral Perturbation Theory. We discuss how to construct chiral perturbation theory from low-energy QCD, mainly focusing on the meson sector. Chiral perturbation theory in its modern form was introduced by Weinberg [START_REF] Weinberg | Phenomenological Lagrangians[END_REF], and Gasser and Leutwyler [START_REF] Gasser | Chiral Perturbation Theory to One Loop[END_REF]. The basic idea of ChPT is an two-fold expansion in small momentum and small quark masses.

Chiral symmetry and its breaking

In the Standard Model, there are six different quark flavors with three "light" quarks (u, d, s) and three "heavy" quarks (c, b, t), compared with the splitting energy scale Λ χ ∼ 1 GeV1 (In the following, this energy scale is denoted as the chiral symmetry breaking scale). Because we mainly talk about the low-energy region in the following, we will focus on the three-flavor QCD. The corresponding Lagrangian can be expressed as

L QCD = f =u,d,s (q f i / Dq f -m f qf q f ) - 1 4 G µν,a G µν a . (3.3)
In order to better present the approximate chiral symmetry, we would like to rewrite the QCD Lagrangian as

L QCD = L 0 QCD + L M , (3.4)
with

L 0 QCD = f =u,d,s qf i / Dq f - 1 4 G µν,a G µν a , L M = qMq, (3.5)
where the three light quark masses are collected in a diagonal matrix M = diag(m u , m d , m s ). Firstly, we introduce two projection operators P L,R with the chirality operator γ 5 ,

P L = 1 2 (1 -γ 5 ), P R = 1 2 (1 + γ 5 ). (3.6)
Thus, the quark spinor field q f can be decomposed into left-and right-handed components, q f,L and q f,R , respectively,

q f = 1 2 (1 -γ 5 )q f + 1 2 (1 + γ 5 )q f = P L q f + P R q f ≡ q f,L + q f,R . (3.7)
In terms of this decomposition, one can re-express the Lagrangian L 0 QCD as

L 0 QCD = f =u,d,s qf,L i / Dq f,L + qf,R i / Dq f,R - 1 4 G µν,a G µν a , (3.8)
with the total separation between left-and right-handed quark fields. The above Lagrangian has a classical global U (3) L × U (3) R symmetry. According to group theory, the U (3) L × U (3) R can be expressed as

U (3) L × U (3) R ≡ SU (3) L × SU (3) R × U (1) V × U (1) A , (3.9)
where the vectorial symmetry U (1) V is the conservation of baryon number, but the axial symmetry U (1) A is broken by instanton effects (U (1) A anomaly) [START_REF] Hooft | A Planar Diagram Theory for Strong Interactions[END_REF]. Because the baryon number is invariant in the strong interaction, the Lagrangian L 0 QCD is actually invariant under the remaining SU (3

) L × SU (3) R transformation, q f,L → e i λ a 2 α a L q f,L , q f,R → e i λ a 2 α a R q f,R , (3.10)
with the Gell-Mann SU (3) matrices λ a (a = 1, . . . , 8). By using Nöther's theorem [START_REF] Hill | Hamilton's principle and the conservation theorems of mathematical physics[END_REF][START_REF] Noether | Invariant Variation Problems[END_REF][START_REF] Gell-Mann | The axial vector current in beta decay[END_REF], one can establish the connections between continuous symmetries and conserved quantities. For the chiral symmetry group SU (3) L × SU (3) R , there are 2(n2 f -1) = 16 (with n f = 3) Nöther currents divided into left-and right-handed conserved currents,

J µ,a L = qL γ µ λ a 2 q L , J µ,a R = qR γ µ λ a 2 q R , (3.11)
with ∂ µ J µ,a L/R = 0. Usually, one can use linear combinations of the above chiral currents to obtain the vector and axial-vector currents

V µ,a = J µ,a R + J µ,a L = qγ µ λ a 2 q,
A µ,a = J µ,a R -J µ,a L = qγ µ γ 5 λ a 2 q.

(3.12)

The corresponding conserved Nöther charges are

Q a V = d 3 xq † (x) λ a 2 q(x), with dQ a V dt = 0, Q a A = d 3 xq † (x)γ 5 λ a 2 q(x), with dQ a A dt = 0. (3.13)
Before we go on to discuss chiral symmetry breaking, we want to mention that there are two realizations of chiral symmetry 2 . One is the Wigner-Weyl mode, which is the linear representation, with the trivial (empty

) vacuum (Q a V |0 = Q a A |0 = 0).
Another is the Nambu-Goldstone mode, which is the non-linear realization, with a non-trivial vacuum (Q a

V |0 = 0, Q a A |0 = 0).
Up to now, the discussion about chiral symmetry is based on the chiral limit with three zero light-quark masses. But, in fact, chiral symmetry is not quite exact in QCD. Recalling the full QCD Lagrangian at low-energies [Eq. (3.4)], the mass part is

L M = -qMq = -(q R Mq L + qL Mq R ). (3.14)
We can see that left-and right-handed quark fields are mixed by the quark-mass matrix. Therefore, chiral symmetry is explicitly broken by the non-zero quark masses.

Due to the small masses of u, d, and s quarks, QCD possesses an approximate chiral symmetry and this symmetry should be approximately presented in the hadronic spectrum. That means there should exist (almost) degenerate parity doublets3 under Wigner's realization mode of chiral symmetry. In experiment, one can find two observed bands with different parity, but they are not degenerate. Furthermore, the masses of lowest-lying pseudoscalar mesons (π, K, η) are very small in comparison with other hadronic states. Therefore, in order to explain these two observations, we can deduce that chiral symmetry is conserved in the QCD Lagrangian but it is spontaneously broken by the vacuum state. In the Nambu-Goldstone realization mode, the global chiral symmetry SU (3

) L × SU (3) R is spontaneously broken to the vectorial subgroup SU (3) V = SU (3) L+R , SU (3) L × SU (3) R SSB --→ SU (3) V , (3.15)
where SU (3) V is the symmetry group of vacuum. According to Goldstone's theorem [START_REF] Goldstone | Field Theories with Superconductor Solutions[END_REF], in this process, an octet of pseudoscalar massless Goldstone bosons should be produced due to the eight broken generators of chiral symmetry group. If we denote G = SU (3) L × SU (3) R and H = SU (3) V , the Goldstone boson manifold is the coset space G/H which is isomorphic to SU (3). Furthermore, when considering that the light quark masses are not zero but small, we can easily find the candidates for the eight Goldstone bosons in the hadronic spectrum. There are three pions, π 0,± , four kaons, K ±,0 , K0 , and the eta, η. Therefore, we can explain the above two experimental phenomena by using Goldstone's theorem.

Lowest order chiral effective Lagrangian

In the preceding section, we identified the relevant degrees of freedom, the pseudoscalar mesons π, K, η, at low-energy region. Therefore, the chiral perturbation theory can be constructed by applying Weinberg's theorem. The corresponding effective Lagrangian should demand all the symmetries of QCD, such as Lorentz invariant, SU (3) C gauge invariance, charge conjugation (C), parity (P), and time reversal (T) symmetries, and chiral symmetry. Because chiral symmetry is spontaneously broken, one cannot directly write down the effective Lagrangian from the nonperturbative QCD Lagrangian due to the replacement of quarks and gluons with Goldstone bosons. In the 1970s, the standard method to construct effective field theory for a spontaneously broken symmetry was formulated by Callan, Coleman, Wess, and Zumino [START_REF] Coleman | Structure of phenomenological Lagrangians. 1[END_REF][START_REF] Callan | Structure of phenomenological Lagrangians. 2[END_REF]. Usually, the Goldstone boson fields (π, K, η) are collected in a unitary matrix field U (φ)

U (φ) = exp i φ F φ , (3.16
)

φ = 8 a=1 φ a λ a = √ 2    1 √ 2 π 0 + 1 √ 6 η π + K + π - -1 √ 2 π 0 + 1 √ 6 η K 0 K - K0 -2 √ 6 η    , (3.17)
with the meson decay constant F φ and the law of transformation for U (φ)

U (φ) → g R U (φ)g † L , U † (φ) → g L U † (φ)g † R , (3.18)
where g L and g R are SU (3) L and SU (3) R transformations, respectively. On the other hand, a more basic matrix field u(φ) is the square root of U (φ) with the chiral transformation

u(φ) → g R u(φ)h(g, φ) -1 , (3.19)
with the unbroken transformation h(g, φ) ∈ SU (3) V .

In the following, we could construct the general effective Lagrangians in terms of U (φ) or u(φ). As a first step, we want to discuss how to obtain the lowest order chiral Lagrangian corresponding to the QCD Lagrangian L 0 QCD in the chiral limit. Since the Goldstone bosons are pseudoscalar, in order to guarantee parity invariance, the effective Lagrangian should contain an even number of meson fields. More precisely, one U should accompany U † due to the parity transformation φ parity ---→ -φ/U parity ---→ U † . The most general invariant term can be the product of terms with the form,

U U † • U U † • ... , (3.20)
where ... denotes the trace of flavor space. At lowest order, due to U U † = I, in order to obtain a non-trivial interaction, the derivative of U is required. With the limitation of Lorentz invariance, one has to introduce even numbers of derivative on the meson fields. The only possible term of lowest order is ∂ µ U ∂ µ U † . We will investigate whether this term is invariant or not under the chiral transition. The elements of ∂ µ U ∂ µ U † have the following transformation properties,

∂ µ U → g R ∂ µ U g † L , ∂ µ U † → g L ∂ µ U † g † R . (3.21)
It is easy to see that

∂ µ U ∂ µ U † → g R ∂ µ U g † L g L ∂ µ U † g † R = g † R g R ∂ µ U ∂ µ U † = ∂ µ U ∂ µ U † . (3.22)
Finally, we obtain the lowest order chiral effective Lagrangian

L (2) φ = F 2 φ 4 ∂ µ U ∂ µ U † , (3.23)
where the superscript 2 denotes two derivatives. The coefficient F 2 φ /4 is to ensure the kinetic terms having the traditional form with 1 2 ∂ µ φ a ∂ µ φ a . This can be clearly shown when one performs the expansion of

U U = 1 + iφ F φ + • • • , U † = 1 - iφ F φ + • • • , (3.24)
∂ µ U = i F φ ∂ µ φ + • • • , ∂ µ U † = - i F φ ∂ µ φ + • • • , (3.25)
as the effective Lagrangian becomes

L (2) φ = 1 2 ∂ µ φ∂ µ φ + 1 12F 2 φ (φ∂ µ φ)(φ∂ µ φ) + O φ 6 F 4 φ . (3.26)
Here we want to mention that the Lagrangian L

φ trivially satisfies U (1) V invariance with baryon number zero.

As mentioned in the preceding section, chiral symmetry is explicitly broken by the nonzero quark masses. When constructing the effective Lagrangian, one has to incorporate the consequences of mixing left-and right-handed fields. According to the discussion in Ref. [START_REF] Georgi | Weak Interactions and Modern Particle Theory[END_REF], if the quark mass matrix transformed as

M → g R Mg † L , (3.27)
the mass term L M would be invariant under the chiral transformation. We can construct the lowest order term in the effective Lagrangian with M using the same arguments as before.

There is one term MU † + U M which is Lorentz invariant, and parity and chiral symmetric. Therefore, the lowest order Lagrangian can be written as

L SB = F 2 φ B 2 0 2 MU † + U M , (3.28)
where the subscript (SB) denotes the chiral symmetry explicitly breaking in QCD. The constant B 0 = -0|qq|0 /F 2 φ relates to the quark condensate and cannot fixed by symmetry requirements alone.

After expanding U and U † in powers of φ, we obtain

L SB = -B 0 Mφ 2 + B 0 6F 2 φ Mφ 4 + O φ 6 F 4 φ . (3.29)
The first term provides the relationships between the masses of Goldstone bosons and the quark masses with exact isospin symmetry

4 (m u = m d ≡ m l ), M 2 π = 2B 0 m l , M 2 K = B 0 (m l + m s ), M 2 η = 2 3 B 0 (m l + 2m s ), (3.30)
and gives the Gell-Mann-Okubo relation [185-187]

4M 2 K = M 2 π + 3M 2 η , (3.31)
which is not dependent on B 0 . Furthermore, the second term of Eq. (3.29) gives higher order corrections to the masses of pseudoscalar mesons. In Appendix D, we also calculate the strong isospin-splitting effects on the meson masses. Combining Eq. (3.23) and Eq. (3.28), the lowest order chiral Lagrangian is

L (2) φ = F 2 φ 4 ∂ µ U ∂ µ U † + 2B 2 0 MU † + U M . (3.32)

General chiral effective Lagrangians

The general form of the chiral Lagrangian for the mesonic sector is

L eff φ = n L (2n) φ [U, U † , ∂ µ U, ∂ µ U † , M], n = 1, 2, • • • , (3.33)
with the even order only. The superscripts 2n refer to the power of the momentum and quark mass expansion. As discussed in the above subsection, the index 2 denotes two derivatives terms or a single quark mass term. The higher order Lagrangians, e.g. L

φ , L

φ , contains more complicated terms. The expression of L (4) φ is given in Eq. (4.2) of Chapter 4. The chiral Lagrangians are organized in a momentum expansion based on the chiral counting rules

U, U † ∼ O(p 0 ), ∂ µ U, ∂ µ U † ∼ O(p 1 ), M ∼ O(p 2 ), (3.34)
with the small momentum p. This can be easy to understand with the derivative generating four momentum, ∂ µ ∼ p µ , and the on-shell condition M ∝ M 2 φ = p 2 . The above discussion is at the ordinary QCD level. When one wants to study the interactions with external fields at the low energies, a more general QCD Lagrangian is

L External QCD = L 0 QCD + f =u,d,s qf γ µ (v µ + γ 5 a µ )q f - f =u,d,s qf (s -iγ 5 ψ p )q f , (3.35)
where the external vector v µ , axial-vector a µ , scalar s, and pseudoscalar ψ p fields are Hermitian 3×3 matrices in flavor space. In this situation, the effective field theory becomes more powerful and can be used to study the electromagnetic interactions and semileptonic weak interactions. Applying Weinberg's theorem, the extended QCD Lagrangian, L External QCD , can be mapped into the most general effective Lagrangian

L External QCD [q f , q f , G µν , v µ , a µ , s, ψ p ] → L eff φ [U, ∂ µ U, v µ , a µ , s, ψ p ], (3.36)
which has been given in Ref. [START_REF] Gasser | Chiral Perturbation Theory: Expansions in the Mass of the Strange Quark[END_REF]. The chiral counting rules for the new induced terms are

v µ , a µ ∼ O(p 1 ), s, ψ p ∼ O(p 2 ). (3.37)
Additionally, let us re-examine the lowest order of chiral Lagrangian which naturally includes the external field interactions,

L (2) φ = F φ 4 D µ U † D µ U + U † χ + χ † U , (3.38)
where v µ , a µ can only appear through the covariant derivatives

D µ U = ∂ µ U -ir µ U + iU l µ , (3.39)
with l µ = v µ -a µ , r µ = v µ + a µ . The scalar field s and the pseudoscalar field

ψ p are encoded in χ = 2B 0 (s + iψ p ). (3.40)
If taking the external field l µ and r µ as

l µ = r µ = -eQA µ , (3.41)
with the external electromagnetic field A µ and Q = diag(2/3, -1/3, -1/3), or,

l µ = - g √ 2 (W + µ T + + H.c.), r µ = 0, (3.42)
with the gauge coupling g, the massive charged weak bosons

W ± µ = (W 1µ ∓ iW 2µ )/ √ 2,
and

T + =    0 V ud V us 0 0 0 0 0 0    , (3.43)
one can study the electromagnetic interaction of Goldstone bosons or the semileptonic decay.

Weinberg's power counting

In principle, the chiral effective Lagrangian contains an infinite number of terms with an infinite number of parameters. One has to find a systematic method to collect the important contributions at a specific chiral order. This concept is called "power counting". Under the rule of power counting, we also can test the convergence of the chiral expansion by increasing chiral order. It was shown that, in the meson sector, there is one-to-one correspondence between chiral orders and loop diagrams [START_REF] Gasser | Chiral Perturbation Theory to One Loop[END_REF]. The general form of an amplitude A for a given Feynman diagram is

A ∝ (d 4 p) L 1 (p 2 ) N M d p dN d ≡ p n ChPT , (3.44)
where L is the number of meson loops, N M is the number of internal lines (meson propagators), N d is the number of vertices with dimension d obtained from chiral Lagrangian L

φ . In a massindependent subtraction scheme, the only dimensional parameters are the momentum p and the pion mass. The chiral dimension n ChPT can be written as

n ChPT = 4L -2N M + d dN d .
(3.45)

The chiral Lagrangian starts at leading order d ≥ 2, therefore, the chiral dimension should be n ChPT ≥ 2.

Chiral perturbation theory in baryon sector

Meson-baryon effective Lagrangians

As an effective field theory of low-energy QCD, chiral perturbation theory can also be applied to study baryon systems. The difference between the meson sector and the baryon sector is that the baryon mass is not zero at the chiral limit. We take the lowest-lying octet baryon as an example to construct the general effective Lagrangian. The octet baryon fields can be collected in a traceless 3

× 3 matrix B, B = 8 a=1 B a λ a √ 2 =    1 √ 2 Σ 0 + 1 √ 6 Λ Σ + p Σ - -1 √ 2 Σ 0 + 1 √ 6 Λ n Ξ - Ξ 0 -2 √ 6 Λ    . (3.46)
Under the chiral symmetry SU (3) L × SU (3) R , the baryon field B transforms as any matter field,

B → B = KBK † , (3.47)
with K(U, g L , g R ) the compensator field [START_REF] Ecker | Chiral perturbation theory[END_REF] representing an element of the conserved subgroup SU (3) V . For convenience, we introduce two definitions, the so-called chiral connection Γ µ ,

Γ µ = 1 2 u † (∂ µ -ir µ )u + u(∂ µ -il µ )u † , (3.48)
and the axial current u µ ,

u µ = i u † (∂ µ -ir µ )u -u(∂ µ -il µ )u † . (3.49)
They transform as

Γ µ → KΓ µ K † -(∂ µ K)K † , u µ → Ku µ K † . (3.50)
The covariant derivative of the baryon field is defined as

D µ B = ∂ µ B + [Γ µ , B], (3.51)
with the transformation law

D µ B → K(D µ B)K † . (3.52)
As given in Ref. [START_REF] Krause | Baryon Matrix Elements of the Vector Current in Chiral Perturbation Theory[END_REF], the chiral counting scheme of the building blocks is

B, B = O(p 0 ), D µ B = O(p 0 ), (i / D -m 0 )B = O(p 1 ), 1, γ µ , γ 5 γ µ , σ µν = O(p 0 ), γ 5 = O(p). (3.53)
This counting rule can be understood from bilinears BΓB with the plane waves solutions of the free Dirac equation. The details can be seen in the page 154 of Ref. [START_REF] Scherer | A Primer for Chiral Perturbation Theory[END_REF]. With the contraction between Dirac matrices and derivatives of baryon field, the most general Lagrangians can be expressed as

L eff φB = L (1) φB + L (2) φB + L (3) φB + L (4) φB + • • • , (3.54)
with the gradual increase of chiral order. The lowest order of meson-baryon effective Lagrangian L

φB can be expressed as

L (1) φB = B(i / D -m 0 )B + D 2 Bγ µ γ 5 {u µ , B} + F 2 Bγ µ γ 5 [u µ , B] , (3.55)
where m 0 denotes the baryon mass in the chiral limit, and the constants D and F are the axialvector coupling constants, which can be determined from the baryon semi-leptonic decays.

After including baryons in the ChPT, the chiral order of a specific diagram with L loops is calculated as

n ChPT = 4L -2N M -N B + d dV d , (3.56)
where N B is the number of internal baryon propagators. However, because the baryon mass does not vanish in the chiral limit and its value is close to the chiral symmetry breaking scale Λ χ , Weinberg's power counting is naively violated in the baryon chiral perturbation theory 5 . This problem was first pointed out by Gasser, Sainio, and Svarc [START_REF] Gasser | Nucleons with Chiral Loops[END_REF] in the 1990s. In order to illustrate the power-counting breaking (PCB) terms that appear in the loop calculations, we take a scalar loop integral with one pion and one nucleon propagators for example,

H(p 2 , d) = -i d d q (2π) d 1 q 2 -m 2 π + iε 1 (p -q) 2 -m 2 0 + iε . (3.57)
By using the dimensional regularization scheme and limiting d → 4, one obtains

H = - 1 16π 2 -2 + log m 2 0 µ 2 + m 2 π m 2 0 log m π m 0 + 2m π m 0 1 - m 2 π 4m 2 0 arccos m π 2m 0 . (3.58)
According to the chiral order of Eq. (3.56), this Feynman diagram with one scalar loop is O(p 1 ), therefore, the first two terms

- 1 16π 2 -2 + log m 2 0 µ 2 , (3.59)
are the power-counting breaking terms and should be removed from the loop contributions (equivalent to be absorbed by counter terms).

In order to systematically restore the PC, the so-called Heavy-Baryon (HB) ChPT was first proposed by Jenkins and Manohar [START_REF] Jenkins | Baryon chiral perturbation theory using a heavy fermion Lagrangian[END_REF], considering baryons as heavy static sources. Later, covariant BChPT implementing a consistent PC with different renormalization methods have been developed, such as the infrared (IR) [START_REF] Becher | Baryon chiral perturbation theory in manifestly Lorentz invariant form[END_REF] and the extended-on-mass-shell (EOMS) [START_REF] Gegelia | Matching heavy particle approach to relativistic theory[END_REF][START_REF] Fuchs | Renormalization of relativistic baryon chiral perturbation theory and power counting[END_REF] renormalization schemes. In the following subsections, we will briefly summarize the essential features of HB ChPT, IR BChPT and EOMS BChPT.

In addition to the afore-mentioned dimensional renormalization schemes (MS and its derivatives), other renormalization/regularization schemes have been proposed, such as the cutoff scheme [START_REF] Bernard | Cutoff schemes in chiral perturbation theory and the quark mass expansion of the nucleon mass[END_REF], the finite range regulator (FRR) method [START_REF] Young | Convergence of chiral effective field theory[END_REF][START_REF] Leinweber | Physical nucleon properties from lattice QCD[END_REF], and the partial summation approach [START_REF] Semke | Baryon self energies in the chiral loop expansion[END_REF] etc., to speed up the convergence of BChPT.

Heavy baryon approach

Drawing on the experience of heavy quark physics and of QED applied to atomic and molecular physics, the basic idea of heavy baryon scheme [START_REF] Jenkins | Baryon chiral perturbation theory using a heavy fermion Lagrangian[END_REF] is to take the baryon as heavy (nearly onshell) and divide the baryon momentum into a large piece and a small residual piece,

p µ = m 0 v µ + k µ , (3.60)
where v µ is the baryon velocity with v 2 = 1 and v 0 ≥ 1, and k µ denotes the soft residual momentum with k µ • v m 0 , Λ χ (at rest frame with v µ = (1, 0, 0, 0)). Therefore, in the heavy baryon scheme, one can perform a two-fold expansion in terms of

k Λ χ , k m 0 . (3.61)
According to the decomposition of baryon momentum, the baryon field can also be separated into light and heavy components

B(x) = e im 0 v•x [B v (x) + H v (x)] , (3.62)
with light B v and heavy H v fields

B v (x) ≡ e im 0 v•x P v+ B(x), H v (x) ≡ e im 0 v•x P v-B(x). (3.63)
Here, the projection operators P v± are defined as

P v± ≡ 1 ± / v 2 , (3.64)
with the properties of

P v+ + P v-= 1, P 2 v± = P v± , P v± P v∓ = 0. (3.65)
It should be mentioned that the light field B v (x) is usually called heavy baryon field and ∂ µ B v (x) produces a small residual momentum k µ . From the leading order covariant effective Lagrangian [Eq. (3.55)], one can perform the path integral and integrate out the heavy component H v (x) to obtain the corresponding heavy baryon effective Lagrangians in terms of heavy baryon fields B v (x) with definite velocity v µ ,

L(1) φB = Bv iv • DB v -D Bv S µ {u µ , B v } -F Bv S µ [u µ , B v ] , (3.66)
where ˆindicates the heavy-baryon formalism. The derivative on the heavy baryon field is defined as the same in Eq. (3.51) with the replacement of B → B v . The covariant spin-operator

S µ = i 2 γ 5 σ µν v ν is the spin matrix with S • v = 0, S 2 = 1 -d 4 , {S µ , S ν } = 1 2 (v µ v ν -g µν ), [S µ , S ν ] = iε µναβ v α S β , (3.67)
in d space-time dimensions.

It should be noted that the leading order Lagrangian LφB [Eq. (3.66)] does not have the baryon mass m 0 terms. The baryon mass is shuffled from the propagators to the vertices. Therefore, heavy baryon ChPT has the same power counting as Weinberg's power counting for mesons with the chiral counting order of building blocks

B v , Bv ∼ O(p 0 ), v µ , S µ , ∼ O(p 0 ) D µ B v , u µ ∼ O(p 1 ). (3.68)
The propagator of heavy baryon, derived from L(1) φB , can be written as

G v (k) = P v+ v • k + iε . (3.69)
In order to better understand the HB method, let us reconsider the integral of a scalar loop diagram with one meson and one baryon propagator, which can be expressed as

H HB = -i d d q (2π) d 1 (q 2 -m 2 π + iε) 1 v • (k -q) + iε = - 1 16π 2 πm π 2m 0 , (3.70)
in the limit d → 4. We can see that the HB result is clearly of order O(p 1 ) as required from the counting rules.

HBChPT has been successfully applied to study low-energy physical phenomena, particularly in two-flavor sector [START_REF] Bernard | Chiral dynamics in nucleons and nuclei[END_REF]. However, in order to ensure HBChPT is reparameterization invariant or Lorentz invariant at a specific order, one has to take into account the 1/m 0 correction Lagrangians [START_REF] Luke | Reparametrization invariance constraints on heavy particle effective field theories[END_REF]. Therefore, the heavy baryon effective Lagrangians have more terms than the covariant counterparts at the same chiral order [START_REF] Ecker | Low-energy expansion of the pion -nucleon Lagrangian[END_REF]. One also found that the convergence of HBChPT is not fast enough in the calculation of some physical quantities, such as electroweak form factors [START_REF] Bernard | Nucleon electroweak form-factors: Analysis of their spectral functions[END_REF] and electromagnetic form factors [START_REF] Kaiser | Spectral functions of isoscalar scalar and isovector electromagnetic formfactors of the nucleon at two loop order[END_REF] of nucleon.

Infrared regularization

In order to avoid the shortcomings of HBChPT, one has to find a regularization method to combine the proper power counting of the HB approach and proper analyticity. As a first attempt, Tang and Ellis [START_REF] Tang | A New approach to chiral perturbation theory for matter fields[END_REF][START_REF] Ellis | Pion nucleon scattering in a new approach to chiral perturbation theory[END_REF] pointed out that the loop integral can be divided into soft-and hard-momentum parts. The hard-momentum terms are always polynomials and can be totally absorbed into the LECs. However, this division relies on specific loop integrals which cannot be always valid. Becher and Leutwyler [START_REF] Becher | Baryon chiral perturbation theory in manifestly Lorentz invariant form[END_REF] proposed a systematic method to extract the soft-and hard-momentum parts, which is known as Infrared Regularization (IR) of baryon chiral perturbation theory.

To illustrate the main idea of IR BChPT, we would like to also take one scalar loop integral [Eq. (3.57)] for example. It is clear that the infrared singularity appears in the region of integration of m π → 0. Because the nucleon momentum is close to the mass shell (m 2 0 -p 2 ) ∼ O(p), the infrared singularity is of the order O(p d-3 ). On the other hand, the so called hardmomentum parts with large momenta are just the ordinary Taylor expansion of momentum and can be totally absorbed into the LECs.

Following the discussion of Ref. [START_REF] Becher | Baryon chiral perturbation theory in manifestly Lorentz invariant form[END_REF], in order to isolate the infrared parts, one can introduce two dimensionless variables,

α = m π m 0 , Ω = p 2 -m 2 0 -m 2 π 2m π m 0 , (3.71)
which count as O(p) and O(p 0 ), respectively. After using the standard Feynman parametrization and performing the momentum shift k → k + pz, the scalar loop becomes

H(p 2 , d) = -i 1 0 dz d d q (2π) d dz [q 2 -M(z)] 2 , (3.72)
with

M(z) = m 0 [z 2 -2αΩz(1 -z) + α 2 (1 -z) 2 ]. (3.73)
Performing the integration over k, we can get

H(p 2 , d) = κ(d) 1 0 dz[M(z) -iε] d 2 -2 = κ(d) 1 0 dz 1 M(z) -iε 4-d 2 , (3.74) with κ(d) = Γ(2 -d/2)/(4π) d/2
. Therefore, the infrared singularity originates where M(z) goes to zero as m π → 0 at the small values of z. In order to isolate the divergent part, we perform a change of variable z = αx, then

H(p 2 , d) = κ(d) 1 α 0 dxα[M(αx) -iε] d 2 -2 . (3.75)
With m π → 0, the upper limit of integral x = 1/α → ∞, therefore, the integral I, which contains the same infrared singularity, can be expressed as

I ≡ κ(d)α d-3 ∞ 0 [D(x) -iε] d 2 -2 dx, (3.76)
where

D(x) = 1 -2Ωx + x 2 + 2αx(Ωx -1) + α 2 x 2 . (3.77)
In order to satisfy the relation of H = I -R, the regular part of H is defined as

R ≡ κ(d) ∞ 1 dz[M(z) -iε] d 2 -2 . (3.78)
For arbitrary values of d, the explicit expressions for H, I, and R involve hypergeometric functions. The chiral expansion of I has the form

I = O(p d-3 ) + O(p d-2 ) + O(p d-1 ) + • • • , (3.79)
while for any value of d the corresponding expansion of R is

R = O(p 0 ) + O(p 1 ) + O(p 2 ) + • • • . (3.80)
This expansion form can be easily understood at the threshold region, p 2 = (m π + m 0 ) 2 . The infrared part reads

I thr. = κ(d)α d-3 ∞ 0 dx [(1 + α)x -1] 2 -iε d 2 -2 = κ(d)α d-3 1 (d -3)(1 + α) 1 (n -3)(1 + α) = Γ(2 -d 2 ) (4π) d 2 (d -3) m d-3 π m 0 + m π , (3.81)
while the regular part is

R thr. = Γ(2 -d 2 ) (4π) d 2 (d -3) m d-3 0 m 0 + m π . (3.82)
Furthermore, according to the definitions of I [Eq. (3.76)]and R [Eq. (3.78)], we can explicitly compute the loop diagrams taking the limit of d → 4,

I = - 1 16π 2 - m π 2m 0 + m π m 0 log m π m 0 2m π m 0 + 1 - m 2 π 4m 2 0 arccos - m π 2m 0 , (3.83)
R = 1 16π 2 -2 + log m 2 0 µ 2 + m 2 π m 2 0 - 2m π m 0 1 - m 2 π 4m 2 0 arccos 1 - m 2 π 2m 2 0 , = 1 16π 2 -2 + log m 2 0 µ 2 - m 2 π m 2 0 + m 4 π 6m 4 0 + • • • . (3.84)
From the general expansion form of I and R, one can conclude that the infrared part I contains non-integer powers of momenta and quark masses, while the regular part R is the ordinary Taylor expansion in momenta and quark masses. Therefore, the regular part R can be totally absorbed by the LECs appearing in the chiral effective Lagrangian, and the power counting scheme is valid with the replacement of the general integral H by the corresponding infrared part I. Besides, the infrared part contains the full 1/m 0 correction terms, which represents the infinite sum of the kinetic energy corrections to the baryon propagator. Therefore, one has to carefully choose a proper regularization scale µ to absorb the divergences from 1/m 0 higher order corrections. This problem has been stressed in Ref. [START_REF] Becher | Baryon chiral perturbation theory in manifestly Lorentz invariant form[END_REF].

Comparison with Eq. (3.57) and Eq. (3.83), the function arccos(m π /2m 0 ) appearing in the MS calculation becomes arccos(-m π /2m 0 ) in IR scheme. This transformation has large effects on the light-quark mass evolution of some physical observables, e.g. the nucleon/baryon magnetic moments [START_REF] Pascalutsa | A Derivative of the Gerasimov-Drell-Hearn sum rule[END_REF][START_REF] Geng | Leading SU(3)breaking corrections to the baryon magnetic moments in Chiral Perturbation Theory[END_REF]. Furthermore, the analyticity of the infrared method is also broken at a scale of twice the baryon mass, although this large scale should not effect the lowenergy expansions. At present, there are several studies to reformulate the IRBChPT [START_REF] Schindler | Infrared regularization of baryon chiral perturbation theory reformulated[END_REF][START_REF] Dorati | Chiral perturbation theory and the first moments of the generalized parton distributions in a nucleon[END_REF] to overcome these drawbacks.

Extended-on-mass-shell scheme

The extended-on-mass-shell (EOMS) scheme, which also deals with the power-counting breaking problem, was originally proposed by Gegelia et al. [START_REF] Gegelia | Matching heavy particle approach to relativistic theory[END_REF] and has been applied to study the nucleon mass in detail in Ref. [START_REF] Fuchs | Renormalization of relativistic baryon chiral perturbation theory and power counting[END_REF]. As mentioned in the above subsection, the regular part R is considered as the power-counting breaking terms and totally removed in infrared chiral perturbation theory. However, one also found that R can contain power counting allowed terms, which are not necessarily absorbed by the counter terms. Therefore, Gegelia et al. [START_REF] Gegelia | Matching heavy particle approach to relativistic theory[END_REF] tried to separate the exact power-counting breaking terms from the loop integral and proposed the extended-on-mass-shell (EOMS) scheme 6 . More precisely, the basic idea of the EOMS method is to absorb the PCB terms by performing finite subtractions from the results of loop integration with the MS regularization scheme. Results must satisfy the power-counting rules.

The key technical point for the EOMS scheme is to extract the power-counting breaking terms from the results of loop integration. In order to illustrate this point clearly, as presented in Ref. [START_REF] Scherer | Introduction to chiral perturbation theory[END_REF], we would like to consider the scalar loop H in the chiral limit,

H(p 2 , d, m π = 0) = -i d d q (2π) d 1 q 2 + iε 1 (p -q) 2 -m 2 0 + iε . (3.85)
After introducing a small dimensionless quantity

∆ = p 2 -m 2 0 m 2 0 , (3.86)
which counts as O(q) due to the baryon mass closing to the mass shell, the integral H can be written as

H(p 2 , d, m π = 0) = κ(d)m d-4 0 1 0 dz[C(z, ∆)] d 2 -2 , (3.87)
with C(z, ∆) = z 2 -∆z(1 -z) -iε. Using the identity

1 0 dz[C(z, ∆)] d 2 -2 = (-∆) d 2 -2 1 0 dz z d 2 -2 1 - 1 + ∆ ∆ z d 2 -2 , (3.88)
and the hypergeometric function

F F (a, b; c; z) = Γ(c) Γ(b)Γ(c -b) 1 0 dt t b-1 (1 -t) c-b-1 (1 -tz) -a , Re(c) > Re(b) > 0, (3.89)
one can obtain the loop function

H(p 2 , d, m π = 0) = κ(d)m d-4 0 Γ d 2 -1 Γ d 2 (-∆) d 2 -2 F 2 - d 2 , d 2 ; d 2 ; 1 + ∆ ∆ , (3.90
)

with a = 2 -d/2, b = d/2 -1, c = d/2, and z = (1 + ∆)/∆. Applying the properties of F , F (a, b; c; z) = (1 -z) -a F a, c -b; c; z z -1 , (3.91)
F (a, b; c; z) = F (b, a; c; z), (3.92)
6 Since the substraction point is p 2 = m 2 0 , the renormalization condition is denoted "extended on-mass-shell" scheme in analogy with the on-mass-shell renormalization scheme in renormalizable theories.

the loop function becomes as

H(p 2 , d, m π = 0) = κ(d)m d-4 0 Γ d 2 -1 Γ d 2 F 1, 2 - d 2 ; d 2 ; p 2 m 2 0 = m d-4 0 (4π) d 2 Γ 2 -d 2 d -3 F 1, 2 - d 2 ; 4 -d; -∆ + (-∆) d-3 Γ d 2 -1 Γ(3 -d)F d 2 -1, d -2; n -2; -∆ ≡ F (d, ∆) + ∆ d-3 G(d, ∆), (3.93)
where F (d, ∆) and G(d, ∆) are proportional to hypergeometric functions. We can expand the hypergeometric function F for |z| < 1,

F (a, b; c; z) = 1 + ab c z + a(a + 1)b(b + 1) c(c + 1) z 2 2 + • • • , (3.94)
and find

F (d, ∆) = O(p 0 ) + O(p 1 ) + O(p 2 ) + • • • , (3.95) G(d, ∆) = O(p d-3 ) + O(p d-2 ) + O(p d-1 ) + • • • . (3.96)
Therefore, the PCB terms are contained in the first term of loop function and should be subtracted. Generally, there are two approaches to systematically obtain the PCB terms. First, one can calculate a Feynman diagram using the MS scheme to obtain analytical results, and pick out the subtracting PCB terms by using the chiral counting order for the building blocks. This method is obvious when the loop integral has a simple Lorentz structure and can easily generate explicit expressions for the analytic terms. For example, one can calculate the one-loop chiral corrections to the nucleon self-energy. But, in most cases the loop integrals are rather difficult to perform explicitly, therefore, one has to find another method to determine the subtraction terms for these cases.

In the second alternative, one can first expand the integrand in terms of the small quantities, e.g. m π , p, or 1/m 0 to identify the relevant terms of the final PCB terms with the naive chiral counting analysis, and then perform the integration to obtain the corresponding PCB terms. Here, we want to mention that, since the PCB terms are always finite and analytic, this method should always work.

After using the EOMS scheme, the one scalar loop integral Eq. (3.57) becomes

H EOMS = - 1 16π 2 m 2 π m 2 0 log m π m 0 + 2m π m 0 1 - m 2 π 4m 2 0 arccos m π 2m 0 . (3.97)
We can see that H EOMS is the same as Eq. (3.57) except for the power-counting breaking terms. Finally, we would like to summarize the differences among the MS, HB, IR and EOMS renormalization schemes. As an important feature of ChPT, the HB, IR, and EOMS methods satisfy the power-counting rule. More precisely, comparing with the Eqs. (3.70), (3.83), and (3.97), we can find that the results of HB scheme only contain terms of the specified chiral order, while the other two methods also have a series of higher order terms (recoil corrections). In addition, one can obtain the HB results by expanding the IR results in powers of 1/m 0 Black and blue circles indicate the strict PC terms and the higher order corrections, respectively. Half-and full-filled circles denote the analytical terms which have the different analyticity (The details can be seen in text). The figure is taken from Ref. [START_REF] Geng | Recent developments in SU(3) covariant baryon chiral perturbation theory[END_REF].

up to specific orders. Both HBChPT and IR BChPT spoil the analytical structure of loop amplitudes, the EOMS scheme has the same analyticity as the MS. These two features are clearly illustrated in Fig. 3.1 for the case of the pion-nucleon scattering.

Chapter 4

Octet Baryon Masses in ChPT

In this chapter, we perform the chiral expansion of octet baryon masses in covariant baryon chiral perturbation theory with the extend-on-mass-shell scheme up to next-to-next-to-nextto-leading order. The virtual decuplet baryon contributions are explicitly included. In order to take into account lattice QCD artifacts, finite-volume corrections are carefully examined in the same framework. Furthermore, finite lattice spacing discretization effects are calculated by constructing covariant baryon chiral perturbation theory for Wilson fermions.

Chiral expansion of octet baryon self-energies

In the past decades, the ground-state octet baryon masses have been studied extensively [START_REF] Jenkins | Baryon masses in chiral perturbation theory[END_REF][START_REF] Bernard | Critical analysis of baryon masses and sigma terms in heavy baryon chiral perturbation theory[END_REF][START_REF] Banerjee | Baryon mass splittings in chiral perturbation theory[END_REF][START_REF] Borasoy | Chiral Expansion of Baryon Masses and σ-Terms[END_REF][START_REF] Walker-Loud | Octet baryon masses in partially quenched chiral perturbation theory[END_REF][START_REF] Ellis | Baryon masses in chiral perturbation theory with infrared regularization[END_REF][START_REF] Frink | Chiral extrapolations of baryon masses for unquenched three flavor lattice simulations[END_REF][START_REF] Frink | Baryon masses, chiral extrapolations, and all that[END_REF][START_REF] Lehnhart | Baryon masses and nucleon sigma terms in manifestly Lorentz-invariant baryon chiral perturbation theory[END_REF][START_REF] Martin Camalich | The lowest-lying baryon masses in covariant SU(3)-flavor chiral perturbation theory[END_REF][START_REF] Young | Octet baryon masses and sigma terms from an SU(3) chiral extrapolation[END_REF][START_REF] Semke | Baryon self energies in the chiral loop expansion[END_REF][START_REF] Semke | On the possibility of a discontinuous quark-mass dependence of baryon octet and decuplet masses[END_REF][START_REF] Semke | On the quark-mass dependence of the baryon ground-state masses[END_REF][START_REF] Bruns | Chiral extrapolation of baryon mass ratios[END_REF][START_REF] Lutz | On the consistency of recent QCD lattice data of the baryon ground-state masses[END_REF]. It is found that SU(3) HBChPT converges rather slowly [START_REF] Ishikawa | SU(2) and SU(3) chiral perturbation theory analyses on baryon masses in 2+1 flavor lattice QCD[END_REF] in several versions of BChPT. Most calculations are performed only up to NNLO because of the many unknown LECs at N 3 LO except those of Refs. [START_REF] Borasoy | Chiral Expansion of Baryon Masses and σ-Terms[END_REF][START_REF] Walker-Loud | Octet baryon masses in partially quenched chiral perturbation theory[END_REF][START_REF] Frink | Chiral extrapolations of baryon masses for unquenched three flavor lattice simulations[END_REF][START_REF] Semke | On the quark-mass dependence of the baryon ground-state masses[END_REF][START_REF] Bruns | Chiral extrapolation of baryon mass ratios[END_REF][START_REF] Lutz | On the consistency of recent QCD lattice data of the baryon ground-state masses[END_REF]. Regarding chiral extrapolations, Young and Thomas [START_REF] Young | Octet baryon masses and sigma terms from an SU(3) chiral extrapolation[END_REF] obtained very good results using the FRR scheme, which induced a form factor in HBChPT to decrease the loop contributions, up to NNLO by fitting the LHPC [START_REF] Walker-Loud | Light hadron spectroscopy using domain wall valence quarks on an Asqtad sea[END_REF] and PACS-CS [START_REF] Aoki | 2+1 Flavor Lattice QCD toward the Physical Point[END_REF] lattice data. In Ref. [START_REF] Martin Camalich | The lowest-lying baryon masses in covariant SU(3)-flavor chiral perturbation theory[END_REF], we applied the NNLO EOMS-BChPT to analyze the same lattice data and found that the EOMS-BChPT can provide a better description of lattice data and is more suitable for chiral extrapolation purposes than HBChPT and NLO BChPT.

Recently, using a partial summation scheme up to N 3 LO, Semke and Lutz [START_REF] Semke | On the quark-mass dependence of the baryon ground-state masses[END_REF][START_REF] Lutz | On the consistency of recent QCD lattice data of the baryon ground-state masses[END_REF][START_REF] Semke | Strangeness in the baryon ground states[END_REF][START_REF] Lutz | On finite volume effects in the chiral extrapolation of baryon masses[END_REF] found that the BMW [START_REF] Dürr | Ab-Initio Determination of Light Hadron Masses[END_REF], HSC [START_REF] Lin | First results from 2+1 dynamical quark flavors on an anisotropic lattice: Light-hadron spectroscopy and setting the strange-quark mass[END_REF], PACS-CS [START_REF] Aoki | 2+1 Flavor Lattice QCD toward the Physical Point[END_REF], LHPC [START_REF] Walker-Loud | Light hadron spectroscopy using domain wall valence quarks on an Asqtad sea[END_REF], QCDSF-UKQCD [START_REF] Bietenholz | Flavour blindness and patterns of flavour symmetry breaking in lattice simulations of up, down and strange quarks[END_REF], and ETMC [START_REF] Alexandrou | Baryon spectrum with N f = 2 + 1 + 1 twisted mass fermions[END_REF] lattice results can be well described.

On the other hand, up to now, a simultaneous description of all the n f = 2 + 1 lattice data with finite-volume effects taken into account self-consistently is still missing.1 Such a study is necessary for exploring the convergence properties of the SU(3) BChPT by numerically evaluating the contributions of different orders. And, it is also helpful to test the consistency between different lattice simulations. Furthermore, it also provides a good opportunity to determine/constrain the many unknown LECs of BChPT at N 3 LO.

Chiral effective Lagrangians

In this subsection, we collect the relevant chiral Lagrangians for the calculation of octet baryon masses in the u, d, s three-flavor sector up to N 3 LO. Firstly, we would like to present the mesonbaryon effective Lagrangians without including decuplet baryon interactions. After that, the related decuplet baryon Lagrangians are discussed.

Chiral effective Lagrangians without decuplet

The Lagrangians can be written as the sum of a mesonic part and a meson-baryon part:

L eff = L (2) φ + L (4) φ + L (1) φB + L (2) φB + L (3) φB + L (4) φB , (4.1)
where superscript (i) denotes the corresponding chiral order O(p i ), φ = (π, K, η) represents the pseudoscalar Nambu-Goldstone boson fields, and B = (N, Λ, Σ, Ξ) lowest-lying octet baryons.

Pseudoscalar meson Lagrangians

The lowest order meson Lagrangian has been given in Eq. (3.38) of Chapter 3, and the corresponding notations can be found in there.

The most general meson Lagrangian at O(q 4 ) allowed by symmetries has the following form [START_REF] Gasser | Chiral Perturbation Theory: Expansions in the Mass of the Strange Quark[END_REF]:

L (4) φ = L 1 [ D µ U (D µ U ) † ] 2 + L 2 D µ U (D ν U ) † D µ U (D ν U ) † +L 3 D µ U (D µ U ) † D ν D(D ν U ) † + L 4 D µ U (D µ U ) † χU † + U χ † +L 5 D µ U (D µ U ) † (χU † + U χ † ) + L 6 [ χU † + U χ † ] 2 +L 7 [ χU † -U χ † ] 2 + L 8 U χ † U χ † + χU † χU † -iL 9 f R µν D µ U (D ν U ) † + f L µν (D µ U ) † D ν U + L 10 U f L µν U † f µν R +H 1 f R µν f µν R + f L µν f µν L + H 2 χχ † , (4.2)
where the field-strength tensors are defined as

f µν R = ∂ µ r ν -∂ ν r µ -i[r µ , r ν ] and f µν L = ∂ µ l ν - ∂ ν l µ -i[l µ , l ν ] with r µ = v µ + a µ , l µ = v µ -a µ with v µ
and a µ the external vector and axial currents. The LECs L 1,••• ,10 and H 1,2 are scale-dependent and absorb the infinities generated by the one-loop graphs.

Pseudoscalar meson-octet baryon Lagrangians

The effective pseudoscalar meson-octet baryon Lagrangians contain terms of odd and even chiral orders,

L eff φB = L (1) φB + L (2) φB + L (3) φB + L (4) φB . (4.3)
The lowest order meson-baryon Lagrangian has been presented in Eq. (3.55) of Chapter 3. The meson-baryon Lagrangian at order O(p 2 ) can be written as

L (2) φB = L (2, sb) φB + L (2) φB . (4.4)
This separation is motivated by the fact that the first part appears in tree and loop graphs, whereas the latter only contributes to masses via loops. The explicit chiral symmetry breaking part reads as

L (2,sb) φB = b 0 χ + BB + b D B{χ + , B} + b F B[χ + , B] , (4.5)
where b 0 , b D , and b F are LECs, and χ + = u † χu † + uχ † u. For the latter part, we take the same form as in Ref. [START_REF] Oller | Meson-baryon effective chiral lagrangians to O(q 3 )[END_REF]:

L (2) φB = b 1 B[u µ , [u µ , B]] + b 2 B{u µ , {u µ , B}} +b 3 B{u µ , [u µ , B]} + b 4 BB u µ u µ +ib 5 B[u µ , [u ν , γ µ D ν B]] - B← - D ν [u ν , [u µ , γ µ B]] +ib 6 B[u µ , {u ν , γ µ D ν B}] - B← - D ν {u ν , [u µ , γ µ B]} +ib 7 B{u µ , {u ν , γ µ D ν B}} - B← - D ν {u ν , {u µ , γ µ B}} +ib 8 Bγ µ D ν B - B← - D ν γ µ B u µ u ν + • • • , (4.6)
where b 1,••• ,4 have dimension mass -1 and b 5,••• ,8 have dimension mass -2 . If one works for a set of fixed quark masses (e.g., Ref. [START_REF] Frink | Baryon masses, chiral extrapolations, and all that[END_REF]), all terms with one or two covariant derivatives can be absorbed in the structures proportional to b 1,••• ,4 . However, for our purposes, we need to retain all the terms because they lead to different quark mass dependencies.

The contributions from the third chiral order Lagrangian are at least O(p 5 ), which beyond the accuracy of our studies. The fourth-order effective Lagrangian relevant to our study is [START_REF] Borasoy | Chiral Expansion of Baryon Masses and σ-Terms[END_REF]:

L (4) φB = d 1 B[χ + , [χ + , B]] + d 2 B[χ + , {χ + , B}] +d 3 B{χ + , {χ + , B}} + d 4 Bχ + χ + B +d 5 B[χ + , B] χ + + d 7 BB χ + 2 +d 8 BB χ 2 + . (4.7)

Chiral effective Lagrangians with decuplet

The baryon decuplet consists of a SU(3)-flavor multiplet of spin-3/2 resonances. Usually, the spin-3/2 particle can be described by the Rarita-Schwinger (RS) field [START_REF] Rarita | On a theory of particles with half integral spin[END_REF], ψ µ . The corresponding free Lagrangian is

L free RS = ψµ (iγ µνα ∂ α -M γ µν )ψ ν , (4.8)
with the mass M and the totally antisymmetric gamma matrix products

γ µν = 1 2 [γ µ , γ ν ], γ µνα = 1 2 {γ µν , γ α } = -iε µναβ γ β γ 5 (using the convention ε 0123 = -ε 0123 = +1
). After, one can obtain the Euler-Lagrange field equations

(i / ∂ -M )ψ µ = 0, (4.9)
∂ µ ψ µ = 0, (4.10)
γ µ ψ µ = 0. (4.11)
Due to the last two constraints, it is clearly that the 16 components of vector-spinor field are reduced to 4 independent components which is the physical number of spin degrees of freedom of the massive spin-3/2 particle. For the interacting case, because the physical and unphysical components may couple in a nontrivial way, the situation becomes more complicated. The most general form of an interaction among a spin-3/2 particle, a nucleon (Ψ) and a (pseudo-)scalar boson (φ) is given by [START_REF] Nath | Uniqueness of the interaction involving spin 3/2 particles[END_REF],

L Int = g ψµ g µν -(z + 1 2 )γ µ γ ν Ψ∂ ν φ + H.c., (4.12)
where g denotes the coupling constant and z is the off-shell parameter. This is the so-called inconsistent coupling and may often involve the unphysical spin-1/2 components. On the other hand, the "consistent" spin-3/2 couplings, proposed by Pascalutsa [START_REF] Pascalutsa | Quantization of an interacting spin-3/2 field and the Delta isobar[END_REF][START_REF] Pascalutsa | Field theory of nucleon to higher spin baryon transitions[END_REF], satisfy the gauge invariant (ψ µ → φ µ + ∂ µ ε, with a spinor field ε) and remove the spin-1/2 background,

L GI Int = gε µναβ (∂ µ ψν )γ 5 γ α Ψ∂ β φ + H.c.. (4.13)
Following, we would like to employ this method to include the decuplet baryon in the chiral effective Lagrangians.

For the decuplet baryon, T abc µ , the physical fields are assigned to the tensor as

T 111 = ∆ ++ , T 112 = ∆ + / √ 3, T 122 = ∆ 0 / √ 3, T 222 = ∆ -, T 113 = Σ * + / √ 3, T 123 = Σ * 0 / √ 6, T 223 = Σ * -/ √ 3, T 133 = Ξ * 0 / √ 3, T 233 = Ξ * -/ √
3, and T 333 = Ω -. The covariant free Lagrangian for decuplet baryons is

L D = T abc µ (iγ µνα D α -m D γ µν ) T abc ν , (4.14)
where m D is the decuplet-baryon mass in the chiral limit and the derivative of the decuplet baryon fields is defined as

D ν T abc µ = ∂ ν T abc µ + (Γ ν , T µ ) abc , (4.15)
with the definition (X,

T µ ) abc ≡ (X) a d T dbc µ +(X) b d T adc µ +(X) c d T abd µ .
In the previous and following Lagrangians, we always apply the Einstein notation to sum over any repeated SU(3)-index, and (X) a b denotes the element of row a and column b of the matrix representation of X. The O(p 2 ) Lagrangian for decuplet baryons is:

L (2) φD = t 0 2 T abc µ g µν T abc ν χ + + t D 2 T abc µ g µν (χ + , T ν ) abc , (4.16)
where the parameters t 0 , t D are two unknown LECs. At O(p 3 ), the chiral effective Lagrangian, describing the interaction of octet and decuplet baryons with pseudoscalar mesons, can be written as [START_REF] Geng | Leading-order decuplet contributions to the baryon magnetic moments in Chiral Perturbation Theory[END_REF] L

(1) φBD = iC m D F φ ε abc (∂ α T ade µ )γ αµν B e c ∂ ν φ d b + H.c., (4.17)
where the coefficient C denotes the φBT "consistent" coupling.

S 0 (x) = -i 0|T [ψ B (x) ψB (0)]|0 = 1 / p -m 0 -Σ( / p) , (4.18)
where Σ( / p) corresponds to the baryon self-energy,

m B -m 0 -Σ( / p = m B ) = 0, ⇒ m B = m 0 + Σ( / p = m B ). (4.19)
The leading contribution to the self-energy, Σ a = m

B = Σ c + Σ d + Σ e .
We remark that due to parity conservation, there are no first order contributions. The baryon mass up to fourth order in chiral expansion can be expressed as

m B = m 0 + m (2) B + m (3) B + m (4) B . (4.20)
The tree diagrams [Fig.

G b = i d 4 k (2π) 4 / kγ 5 1 / p -/ k -m 0 + iε / kγ 5 1 k 2 -M 2 φ + iε , (4.21)
G d = i d 4 k (2π) 4 {1, k µ k µ , k µ k ν p µ γ ν } 1 k 2 -M 2 φ + iε , (4.22
)

G e = i m (2) B d 4 k (2π) 4 / kγ 5 1 / p -/ k -m 0 + iε 2 / kγ 5 1 k 2 -M 2 φ + iε , (4.23)
where M φ represents the mass of a Nambu-Goldstone boson. These loop integrations can be calculated in the standard way as given in Appendix C. The above loop functions contain PCB terms and therefore additional steps need to be taken to conserve a proper chiral powercounting scheme. As we have demonstrated before, the EOMS scheme has shown a relatively better convergence property than HB or IR approaches. Therefore, we would like to employ the EOMS scheme to subtract the PCB terms. After calculating all the Feynman diagrams shown in Fig. 4.1, we obtain the chiral expansion of octet baryon masses up to N 3 LO. At O(p 2 ), the tree level contribution provides the leading order (LO) SU(3)-breaking corrections to the chiral limit octet baryon mass

m (2) B = φ=π, K ξ (a) B,φ M 2 φ , (4.24)
where the coefficients ξ

-(2b 0 + 4b F ) -2 3 (3b 0 -2b D ) -(2b 0 + 4b D ) -(2b 0 -4b F ) ξ (a) B,K -(4b 0 + 4b D -4b F ) - 2
3 (6b 0 + 8b D ) -4b 0 -(4b 0 + 4b D + 4b F )
N Λ Σ Ξ ξ (b) B,π 3
2 (D + F ) 2 2D 2 2 3 (D 2 + 6F 2 ) 3 2 (D -F ) 2 ξ (b) B,K 1 3 (5D 2 -6DF + 9F 2 ) 2 3 (D 2 + 9F 2 ) 2(D 2 + F 2 ) 1 3 (5D 2 + 6DF + 9F 2 ) ξ (b) B,η 1 6 (D -3F ) 2
B = 1 (4πF φ ) 2 φ=π, K, η ξ (b) B,φ H (b) B (M φ ), (4.25)
where the coefficients ξ

(b)
B,φ are given in Table 4.2, and the corresponding loop functions

H (b) B (M φ ) is H (b) B (M φ ) = - 2M 3 φ m 0   4m 2 0 -M 2 φ arctan   M φ 4m 2 0 -M 2 φ   + 4m 2 0 -M 2 φ arctan   2m 2 0 -M 2 φ M φ 4m 2 0 -M 2 φ   + M φ log M φ m 0   . (4.26)
The NNLO SU(3)-breaking corrections to the octet baryon masses are

m (4) B = ξ (c) B,π M 4 π + ξ (c) B,K M 4 K + ξ (c) B,πK M 2 π M 2 K + 1 (4πF φ ) 2 φ=π, K, η ξ (d,1) B,φ H (d,1) B (M φ ) + ξ (d,2) B,φ H (d,2) B (M φ ) + ξ (d,3) B,φ H (d,3) B (M φ ) + 1 (4πF φ ) 2 φ=π, K, η B =N, Λ, Σ, Ξ ξ (e) BB ,φ • H (e) B,B (M φ ). (4.27)
The first three terms of Eq. (4.27) are the tree contributions of diagram Fig. 4.1(c), and the corresponding coefficients ξ

(c) B,π , ξ (c) B,K , ξ (c)
B,πK can be found in Table 4. BB ,φ can be found in Table 4.5. After using the EOMS scheme to remove the PCB terms, the loop functions are written as:

(b) (c) (a)
H (d,1) B (M φ ) = M 2 φ 1 + ln µ 2 M 2 φ , (4.28)
H (d,2) B (M φ ) = M 4 φ 1 + ln µ 2 M 2 φ , (4.29)
H (d,3) B (M φ ) = m 0 M 4 φ 4 1 + ln µ 2 M 2 φ + 1 8 M 4 φ , (4.30)
H (e) B,B (M φ ) = 2M 3 φ m 2 0 4m 2 0 -M 2 φ 6m 2 0 (m (2)
B -m (2)
B ) -M 2 φ (2m (2)
B -m (2)
B ) arccos

M φ 2m 0 -M 2 φ 3(m (2)
B -m

(2)

B ) + 3m 2 0 (m (2)
B -m

(2)

B ) -M 2 φ (2m (2)
B -m

(2) B ) m 2 0 ln M 2 φ m 2 0 + (m (2)
B + m (2)
B ) ln

m 2 0 µ 2 , (4.31)
where m

B and m

B are the corresponding LO SU(3) corrections to octet baryon masses given in Eq. (4.24).

Virtual decuplet contributions

Up to N 3 LO, the octet baryon masses with the virtual decuplet contributions can be written as

m B = m 0 + m (2) B + m (3) B + m (4) B + m (D) B , (4.32)
N Λ Σ Ξ ξ (d,1) B,π -3(2b 0 + b D + b F )m 2 π -2(3b 0 + b D )m 2 π -6(b 0 + b D )m 2 π -3(2b 0 + b D -b F )m 2 π ξ (d,1) B,K -2(4b 0 + 3b D -b F )m 2 K -4 3 (6b 0 + 5b D )m 2 K -4(2b 0 + b D )m 2 K -2(4b 0 + 3b D + b F )m 2 K ξ (d,1) B,η -1 3 8(b 0 + b D -b F )m 2 K -2 9 4(3b 0 + 4b D )m 2 K -2 3 4b 0 m 2 K -1 3 8(b 0 + b D + b F )m 2 K -(2b 0 + 3b D -5b F )m 2 π -(3b 0 + 7b D )m 2 π +(b D -b 0 )m 2 π -(2b 0 + 3b D + 5b F )m 2 π ξ (d,2) B,π 3(b 1 + b 2 + b 3 + 2b 4 ) 2(2b 2 + 3b 4 ) 2(4b 1 + 2b 2 + 3b 4 ) 3(b 1 + b 2 -b 3 + 2b 4 ) ξ (d,2) B,K 2(3b 1 + 3b 2 -b 3 + 4b 4 ) 4 3 (9b 1 + b 2 + 6b 4 ) 4(b 1 + b 2 + 2b 4 ) 2(3b 1 + 3b 2 + b 3 + 4b 4 ) ξ (d,2) B,η 1 3 (9b 1 + b 2 -3b 3 + 6b 4 ) 2(2b 2 + b 4 ) 2 3 (2b 2 + 3b 4 ) 1 3 (9b 1 + b 2 + 3b 3 + 6b 4 ) ξ (d,3) B,π 6(b 5 + b 6 + b 7 + 2b 8 ) 4(2b 7 + 3b 8 ) 4(4b 5 + 2b 7 + 3b 8 ) 6(b 5 -b 6 + b 7 + 2b 8 ) ξ (d,3) B,K 4(3b 5 -b 6 + 3b 7 + 4b 8 ) 8 3 (9b 5 + b 7 + 6b 8 ) 8(b 5 + b 7 + 2b 8 ) 4(3b 5 + b 6 + 3b 7 + 4b 8 ) ξ (d,3) B,η 2 3 (9b 5 -3b 6 + b 7 + 6b 8 ) 4(2b 7 + b 8 ) 4 3 (2b 7 + 3b 8 ) 2 3 (9b 5 + 3b 6 + b 7 + 6b 8 )
N Λ Σ Ξ ξ (e) N N π = 3 4 (D + F ) 2 ξ (e) ΛN K = 1 6 (D + 3F ) 2 ξ (e) ΣN K = 1 2 (D -F ) 2 ξ (e) ΞΛK = 1 12 (D -3F ) 2 ξ (e) N N η = 1 12 (D -3F ) 2 ξ (e) ΛΛη = 1 3 D 2 ξ (e) ΣΛπ = 1 3 D 2 ξ (e) ΞΣK = 3 4 (D + F ) 2 ξ (e) N ΛK = 1 12 (D + 3F ) 2 ξ (e) ΛΣπ = D 2 ξ (e) ΣΣπ = 2F 2 ξ (e) ΞΞπ = 3 4 (D -F ) 2 ξ (e) N ΣK = 3 4 (D -F ) 2 ξ (e) ΛΞK = 1 6 (D -3F ) 2 ξ (e) ΣΣη = 1 3 D 2 ξ ΞΞη = 1 12 (D + 3F ) 2 ξ (e) ΣΞK = 1 2 (D + F ) 2
where m

(D) B
denotes the contributions of virtual decuplet resonances up to N 3 LO. After calculating the Feynman diagrams shown in Fig. 4.2 and subtracting the PCB terms with the EOMS scheme, the virtual decuplet contributions to the octet baryon masses can be expressed as

m (D) B = 1 (4πF φ ) 2 φ=π, K, η ξ (a) BD,φ H (a) B,D (M φ ) + 1 (4πF φ ) 2 φ=π, K, η D =∆, Σ * , Ξ * , Ω - ξ (b/c) BD ,φ H (b/c) B,D (M φ ). (4.33)
The first term of Eq. BD,φ are listed in Table 4.6 and the loop function

H (a) B,D (M φ ) is H (a) B,D (M φ ) = 1 24m 0 m 2 D M 2 φ 2m 4 0 + 4m 3 0 m D -7m 2 0 M 2 φ -4m 0 m D (m 2 D + M 2 φ ) -2(m 4 D + 3m 2 D M 2 φ -M 4 φ ) - 1 12m 3 0 m 2 D M 4 φ ln M φ m D 6m 4 0 + 6m 3 0 m D + m 2 0 (6m 2 D -4M 2 φ ) +m 0 (6m 3 D -2m D M 2 φ ) + 6m 4 D -4m 2 D M 2 φ + M 4 φ + 1 12m 3 0 m 2 D (m 0 -m D )(m 0 + m D ) 3 ln m D M φ m 2 D -m 2 0 × m 4 0 -2m 2 0 (m 2 D + 2M 2 φ ) + 2m 0 m D M 2 φ + m 4 D -4m 2 D M 2 φ - 1 12m 3 0 m 2 D √ W (m 2 0 -2m 0 m D + m 2 D -M 2 φ ) 2 (m 2 0 + 2m 0 m D + m 2 D -M 2 φ ) 3 × arctan m 2 0 + m 2 D -M 2 φ W -arctan m 2 0 -m 2 D + M 2 φ W , (4.34)
where

W = -m 4 0 -(m 2 D -M 2 φ ) 2 + 2m 2 0 (m 2 D + M 2 φ ). The next
H (b/c) B,D (M φ ) = 1 24m 2 0 m 2 D m (2) B M 2 φ 2(11m 2 0 + 8m 0 m D + 9m 2 D ) ×(m 2 0 -m 2 D ) + (5m 2 0 + 8m 0 m D + 18m 2 D )M 2 φ -6M 4 φ + 1 36m 0 m 3 D m (2) D M 2 φ -34m 4 0 -24m 3 0 m D + 30m 4 D -6m 2 D M 2 φ -6M 4 φ + 6m 0 m D (4m 2 D + M 2 φ ) + 3m 2 0 (4m 2 D + 7M 2 φ ) - 1 12m 4 0 m 2 D m (2)
B M 2 φ ln M φ m D 12m 5 D (m 0 + m D ) +6(m 4 0 -m 2 0 m 2 D -2m 0 m 3 D -3m 4 D )M 2 φ + 4(m 2 0 + m 0 m D + 3m 2 D )M 4 φ -3M 6 φ + 1 6m 3 0 m 3 D m (2) D M 2 φ ln M φ m D m 5 D (9m 0 + 8m D ) +3(m 2 0 -m 2 D )(2m 2 0 + m 0 m D + 2m 2 D )M 2 φ -m 0 (4m 0 + m D )M 4 φ + M 6 φ - 1 12m 4 0 m 3 D M 2 φ (m 0 -m D ) 2 (m 0 + m D ) 4 ln m D M φ m 2 D -m 2 0 × m D -5m 2 0 + 2m 0 m D -3m 2 D m (2)
B + 2m 0 m 2 0 -m 0 m D + 3m 2 D m (2) D - 1 6m 3 D m 0 M 4 φ ln m D M φ µ 2 6m D (m 0 + m D )m (2)
B -m 0 (4m 0 + 3m D )m (2) D + 1 12m 4 0 m 3 D √ W (m 2 0 -2m 0 m D + m 2 D -M 2 φ )(m 2 0 + 2m 0 m D + m 2 D -M 2 φ ) 2 × 3m D (m 2 D -M 2 φ ) 2 m (2) B -2m 0 (m 2 D -M 2 φ ) (3m 2 D + M 2 φ )m (2) D + m 2 D m (2)
N Λ Σ Ξ ξ (a) BD,π 16
3 C 2 4C 2 8 9 C 2 4 3 C 2 ξ (a) BD,K 4 3 C 2 8 3 C 2 40 9 C 2 4C 2 ξ (a) BD,η 0 0 4 3 C 2 4 3 C 2
N Λ Σ Ξ ξ (b/c) N ∆,π = 4C 2 ξ (b/c) ΛΣ * ,π = 3C 2 ξ (b/c) Σ∆,K = 8 3 C 2 ξ (b/c) ΞΣ * ,K = C 2 ξ (b/c) N Σ * ,K = C 2 ξ (b/c) ΛΞ * ,K = 2C 2 ξ (b/c) ΣΣ * ,π = 2 3 C 2 ξ (b/c) ΞΞ * ,π = C 2 ξ (b/c) ΣΣ * ,η = C 2 ξ (b/c) ΞΞ * ,η = C 2 ξ (b/c) ΣΞ * ,K = 2 3 C 2 ξ (b/c) Ξ∆ -,K = 2C 2
∆ Σ * Ξ * Ω - ξ D,π t 0 + 3t D t 0 + t D t 0 -t D t 0 -3t D ξ D,K 2t 0 2t 0 + 2t D 2t 0 + 4t D 2t 0 + 6t D +2m 2 0 m 2 D (m 2 D + M 2 φ )(m (2)
B + m

(2)

D ) + 2m 3 0 2(m 2 D -M 2 φ )m (2)
D + m 2 D m (2) B -m 4 0 m D 5m (2)
B + 2m

(2) D + 2m 5 0 m (2) D × arctan m 2 0 + m 2 D -M 2 φ W + arctan m 2 0 -m 2 D + M 2 φ W , (4.35)
where the NLO octet baryon mass, m

B , is given in Eq. (4.32) and the NLO decuplet baryon mass is m

(2)

D = - φ=π,K ξ (2)
D,φ M 2 φ . (4.36)
The corresponding coefficients ξ

D,φ are listed in Table 4.8. It should be noted that in order to obtain the results of Eq. (4.33), the decuplet-octet mass difference, δ = m D -m 0 , is considered up to all orders [START_REF] Martin Camalich | The lowest-lying baryon masses in covariant SU(3)-flavor chiral perturbation theory[END_REF]. Here we want to mention that there are two diagrams (presented in Fig. 4.3) contributed to the octet baryon masses up to N 3 LO. In Ref. [207], it was pointed that their contributions are small and should be started at O(p 5 ) in EOMS BChPT. Therefore, we do not include them in this work.

At N 3 LO, a replacement of meson masses by their O(p 4 ) counterparts in m

B generates N 3 LO contributions to m can be found in Ref. [START_REF] Gasser | Chiral Perturbation Theory: Expansions in the Mass of the Strange Quark[END_REF], which read as

M 2 π,4 = M 2 π,2 1 + M 2 π,2 32π 2 F 2 φ ln M 2 π,2 µ 2 - M 2 η,2 96π 2 F 2 φ ln M 2 η,2 µ 2 + 16 F 2 φ M 2 π,2 2 + M 2 K,2 (2L r 6 -L r 4 ) + M 2 π,2 2 (2L r 8 -L r 5 ) , (4.37)
M 2 K,4 = M 2 K,2 1 + M 2 η,2 48π 2 F 2 φ ln M 2 η,2 µ 2 + 16 F 2 φ M 2 π,2 2 + M 2 K,2 (2L r 6 -L r 4 ) + M 2 K,2 2 (2L r 8 -L r 5 ) , (4.38)
M 2 η,4 = M 2 π,2 M 2 η,2 96π 2 F 2 φ ln M 2 η,2 µ 2 - M 2 π,2 32π 2 F 2 φ ln M 2 π,2 µ 2 + M 2 K,2 48π 2 F 2 φ ln M 2 K,2 µ 2 +M 2 η,2 1 + M 2 K,2 16π 2 F 2 φ ln M 2 K,2 µ 2 - M η,2 24π 2 F 2 φ ln M η,2 µ 2 + 16 F 2 φ M 2 π,2 2 + M 2 K,2 (2L r 6 -L r 4 ) + 8 M 2 η,2 F 2 φ (2L r 8 -L r 5 ) + 128 9 (M 2 K,2 -M 2 π,2 ) 2 F 2 φ (3L r 7 + L r 8 ). (4.39)
The empirical values of 2L r 6 -L r 4 = -0.17 × 10 -3 and 2L r 8 -L r 5 = -0.22 × 10 -3 , and 3L r 7 + L r 8 = -0.15 × 10 -3 are taken from the latest global fit [START_REF] Bijnens | A new global fit of the L r i at next-to-next-to-leading order in Chiral Perturbation Theory[END_REF], which are evaluated at the renormalization scale µ = 0.77 GeV. 2 To be consistent with our renormalization scale used for the one-baryon sector, we have re-evaluated the L r i 's at µ = 1 GeV.

Finite-volume corrections

Because lattice QCD simulations are performed in a finite hypercube, the momenta of virtual particles are discretized. As a result, the simulated results are different from those of infinite space-time. The difference is termed as finite-volume corrections. In cases where M φ L 1, the so-called p-regime, ChPT provides a model-independent framework to study FVCs [START_REF] Gasser | Light Quarks at Low Temperatures[END_REF][START_REF] Gasser | Spontaneously Broken Symmetries: Effective Lagrangians at Finite Volume[END_REF].

Physically, finite-volume corrections can be easily understood. Because of the existence of space-time boundaries, the allowed momenta of virtual particles become discretized, i.e, one has to replace a momentum integral by a finite sum of discretized momenta,

∞ -∞ dk → N n=-N +1 2π L n, (4.40)
with N = L/(2a) (assuming periodical boundary conditions). In LQCD simulations of zerotemperature physics, the temporal extent is generally larger than the spatial extent such that the integral in temporal dimension can be treated as if it extends from -∞ and ∞. As a result, only the integral in spatial dimensions should be replaced by an infinite sum. Furthermore, it is obvious that only loop diagrams are affected by the existence of space-time boundaries.

In the following, we take the one-loop diagram of Fig. 4.1(c) as an example to present the details of FVCs calculation. The definition of FVCs is the following difference:

δG (c) B = G (c) B (L) -G (c) B (∞), (4.41)
where

G (c) B (L) and G (c)
B (∞) denote the integrals calculated in a finite hypercube and in an infinite space-time, respectively. These quantities have several features that make calculations more feasible than a direct computation of G

G (c) B (∞) = i 1 0 dx d 4 k (2π) 4 / k(/ k -/ p + m 0 )/ k ((k -px) 2 -M 2 B ) 2 = i 1 0 dx d 4 k (2π) 4 / k(k 2 -2k • p) + 2k 2 M 0 ((k -px) 2 -M 2 B ) 2 , (4.42)
where

M 2 B = x 2 m 2 0 + (1 -x)M 2 φ -iε.
Calculating the integral [Eq. (4.42)] in a finite hypercube requires treating the temporal and spatial dimensions differently. We choose to work in the baryon rest frame, i.e., p µ = (m 0 , 0). In this frame,

G (c) B = i 1 0 dx dk 0 2π d k (2π) 3 (γ 0 k 0 -γ • k)(k 2 0 -k 2 -2k 0 m 0 ) + 2(k 2 0 -k 2 )m 0 (k 0 -xm 0 ) 2 -k 2 -M 2 B 2 . ( 4.43)
This can be easily calculated by performing a shift in k 0 (k 0 → k 0 + xm 0 ), Wick rotating k 0 (k 0 → ik 0 ), and then performing the integration over k 0 . The result is

G (c) B = 1 0 dx d k (2π) 3   1 2 m 0 (2x + 1) 1
k 2 + M 2 B 1/2 - 1 4 m 0 (m 2 0 x 3 + M 2 B (x + 2)) 1 k 2 + M 2 B 3/2   . (4.44)
After utilizing the master formula [START_REF] Beane | Nucleon masses and magnetic moments in a finite volume[END_REF], one can easily obtain

δG (c) B = 1 0 dx 1 2 m 0 (2x + 1)δ 1/2 (M 2 B ) - 1 4 m 0 (m 2 0 x 3 + M 2 B (x + 2))δ 3/2 (M 2 B ) , (4.45)
where

δ r (M 2 ) = 2 -1/2-r ( √ M 2 ) 3-2r π 3/2 Γ(r) n =0 (L √ M 2 | n|) -3/2+r K 3/2-r (L √ M 2 | n|), (4.46)
where K n (z) is the modified Bessel function of the second kind, and

n =0 ≡ ∞ nx=-∞ ∞ ny=-∞ ∞ nz=-∞ (1 -δ(| n|, 0)) with n = (n x , n y , n z ).
In a similar way, one can calculate FVCs of Fig. 4. 1(d,e)

δG (d,1
)

B (M φ ) = 1 2 δ 1/2 M 2 φ , (4.47)
δG (d,2) B (M φ ) = M 2 φ 2 δ 1/2 M 2 φ , (4.48)
δG

(d,3) B (M φ ) = m 0 2 δ -1/2 M 2 φ , (4.49)
δG

(e) B,B = 1 0 dx - 1 2 (2xm (2)
B + m (2) B ) δ 1/2 (M 2 B ) - 1 4 M 2 φ (x -1) m (2)
B (1 + x) + m (2) B -2xm 2 0 m (2)
B (5x 2 -3) + m

(2)

B (4x + 3) δ 3/2 (M 2 B ) - 1 4 6m 4 0 (x + 1)x 3 m (2) B (x -1) + m (2) B -3m 2 0 M 2 φ × x(x -1)(x + 2) m (2)
B (x -1) + m

1 0 dx m 2 0 (m 0 (1 -x) + m D ) 6m 2 D δ 1/2 M 2 D - m 2 0 (m 0 (1 -x) + m D ) M 2 D 6m 2 D δ 3/2 M 2 D , (4.51)
and

δG (b/c) B,D = m 0 6m 3 D 1 0 dx 2m 2 0 (x -1)m (2) D -m 0 m D m (2)
D + 3m

(2)

B (x -1) + 2m 2 D m (2) B δ 1/2 M 2 D + 3m 3 0 m D x(x -1) 2 m (2) B -2m 2 0 M 2 D (x -1)m (2) D +3m 2 0 m 2 D x(x -1) m (2)
D -m (2) B + m 0 m D M 2 D × 3m (2)
B (x -1) + m (2) D -3m 0 m 3 D m (2) D x -2m 2 D M 2 D m (2) B δ 3/2 M 2 D +3m 0 m D M 2 D x -m 2 0 (x -1) 2 m (2) B +m 0 m D (x -1) m (2) B -m (2) D + m 2 D m (2) D δ 5/2 M 2 D , (4.52)
respectively, with

M 2 D = x 2 m 2 0 -x(m 2 0 -m 2 D ) + (1 -x)M 2 φ -iε.

Finite lattice spacings discretization effects

To apply ChPT to the study of lattice simulations, in principle, one should first take the continuum limit of LQCD data, since ChPT describes continuum QCD and is not valid for nonzero lattice spacing. However, nowadays it is a common practice to assume that lattice spacing artifacts for current LQCD setups of a ≈ 0.1 fm are small and can be treated as systematic uncertainties.

In order to study discretization effects on LQCD simulations, one can first write down Symanzik's effective field theory [START_REF] Symanzik | Continuum Limit and Improved Action in Lattice Theories. 1. Principles and φ 4 Theory[END_REF][START_REF] Symanzik | Continuum Limit and Improved Action in Lattice Theories. 2. O(N ) Nonlinear Sigma Model in Perturbation Theory[END_REF][START_REF] Sheikholeslami | Improved Continuum Limit Lattice Action for QCD with Wilson Fermions[END_REF][START_REF] Luscher | Chiral symmetry and O(a) improvement in lattice QCD[END_REF], a continuum effective field theory which describes the lattice field theory close to the continuum limit, and then one can extend ChPT to be consistent with this EFT with additional symmetry breaking parameters. In this way, the chiral expansion results can naturally encode lattice spacing effects (see, e.g. Ref. [START_REF] Bar | Chiral perturbation theory at non-zero lattice spacing[END_REF]). Sharpe and Singleton [START_REF] Sharpe | Spontaneous flavor and parity breaking with Wilson fermions[END_REF] and Lee and Sharpe [START_REF] Lee | Partial flavor symmetry restoration for chiral staggered fermions[END_REF] first extended ChPT to include finite lattice spacing effects up to O(a) for Wilson fermions [START_REF] Wilson | Confinement of Quarks[END_REF] (WChPT) and staggered fermions [START_REF] Kogut | Hamiltonian Formulation of Wilson's Lattice Gauge Theories[END_REF][START_REF] Susskind | Lattice Fermions[END_REF] (SChPT), respectively. Later, Munster and Schmidt [START_REF] Munster | Chiral perturbation theory for lattice QCD with a twisted mass term[END_REF] applied WChPT to the study of discretization artifacts of twisted mass fermions (tmChPT) [START_REF] Frezzotti | A Local formulation of lattice QCD without unphysical fermion zero modes[END_REF][START_REF] Frezzotti | Lattice QCD with a chirally twisted mass term[END_REF].

In the past decade, discretization effects on the ground-state meson/baryon properties, such as masses, decay constants, electromagnetic form factors, etc., have been extensively studied in WChPT. In the mesonic sector, the masses and decay constants of the Nambu-Glodstone mesons were first studied up to O(m 2 q ) and O(a) for the Wilson action [START_REF] Rupak | Chiral perturbation theory for the Wilson lattice action[END_REF] and for the mixed action [START_REF] Bar | Simulations with different lattice Dirac operators for valence and sea quarks[END_REF], where Wilson sea quarks and Ginsparg-Wilson valence quarks are employed. These studies were subsequently extended to next-to-leading order (up to O(a 2 )) [START_REF] Aoki | Chiral perturbation theory with Wilson type fermions including a 2 effects: N f = 2 degenerate case[END_REF][START_REF] Bar | Chiral perturbation theory at O(a 2 ) for lattice QCD[END_REF]. In the onebaryon sector, a systematic study of the nucleon properties up to O(a) was first performed by Beane and Savage for both the mixed and the unmixed action [START_REF] Beane | Nucleon properties at finite lattice spacing in chiral perturbation theory[END_REF]. The electromagnetic properties of the octet mesons as well as of the octet and decuplet baryons were also studied up to O(a) for both the mixed and the unmixed action [START_REF] Arndt | Hadronic electromagnetic properties at finite lattice spacing[END_REF]. Discretization effects on the nucleon and ∆ masses [START_REF] Tiburzi | Baryon masses at O(a 2 ) in chiral perturbation theory[END_REF] as well as on the vector meson masses [START_REF] Grigoryan | Vector meson mass corrections at O(a 2 ) in PQChPT with Wilson and Ginsparg-Wilson quarks[END_REF] were also studied up to O(a 2 ). The EFT for the anisotropic Wilson lattice action has been formulated up to O(a 2 ) [START_REF] Bedaque | Effective Field Theory for the Anisotropic Wilson Lattice Action[END_REF] as well.

In this subsection, we aim to study the discretization effects of the LQCD simulations of the ground-state octet baryon masses up to O(a 2 ) in covariant BChPT with the EOMS renormalization scheme. Firstly, we briefly review the continuum effective action up to and including O(a 2 ). After, we follow closely the procedure and notations of Ref. [START_REF] Tiburzi | Baryon masses at O(a 2 ) in chiral perturbation theory[END_REF] and construct for the first time the chiral Lagrangians incorporating a finite lattice spacing for the Wilson action in the u, d, and s three-flavor one-baryon sector.

Continuum effective action

Close to the continuum limit, LQCD can be described by an effective action, the 'Symanzik action' [START_REF] Symanzik | Continuum Limit and Improved Action in Lattice Theories. 1. Principles and φ 4 Theory[END_REF][START_REF] Symanzik | Continuum Limit and Improved Action in Lattice Theories. 2. O(N ) Nonlinear Sigma Model in Perturbation Theory[END_REF], which is expanded in powers of the lattice spacing a as

S eff = S 0 + aS 1 + a 2 S 2 + • • • = d 4 x(L (4) + aL (5) + a 2 L (6) + • • • ), (4.53)
where the index i = 4, 5, ... denotes the dimension mass. The L (4) is the normal (continuum) QCD Lagrangian and the two new terms L (5) and L (6) are introduced to include the discretization effects of LQCD. The Lagrangian L (5) contains chiral breaking terms only, while L (6) contains both chiral invariant and breaking terms. In the u, d, and s three-flavor sector, QCD Lagrangian is

L (4) = ψq (i / D -M)ψ q , (4.54)
with the quark field ψ q and the quark mass matrix M = diag(m l , m l , m s ) in the isospin limit (m u = m d ≡ m l ), and / D = D µ γ µ with D µ the covariant derivative. At O(a), there is only the Pauli term left by using equations of motion to redefine the effective fields [START_REF] Luscher | Chiral symmetry and O(a) improvement in lattice QCD[END_REF] aL (5) = ac SW ψσ µν G µν ω q ψ, (

where G µν = [D µ , D ν ] and c SW is the Sheikholeslami-Wohlert (SW) [START_REF] Sheikholeslami | Improved Continuum Limit Lattice Action for QCD with Wilson Fermions[END_REF] coefficient that must be determined numerically. The ω q (q = u, d, s) is a constant which is determined by the kind of lattice fermions employed in LQCD simulations: ω q = 1 for Wilson fermions [START_REF] Wilson | Confinement of Quarks[END_REF] and ω q = 0 for Ginsparg-Wilson (GW) fermions [START_REF] Ginsparg | A Remnant of Chiral Symmetry on the Lattice[END_REF]. Similar to the quark masses, the ω q 's are usually collected in the Wilson matrix,

W = diag(ω l , ω l , ω s ), (4.56)
with conserved isospin symmetry (ω u = ω d ≡ ω l ). This term breaks chiral symmetry in precisely the same way as the quark mass term. It should be noted that the Pauli term can be canceled by adding the clover term to the lattice action [START_REF] Bar | Chiral perturbation theory at O(a 2 ) for lattice QCD[END_REF], resulting in the O(a)-improved Wilson fermion action [START_REF] Sheikholeslami | Improved Continuum Limit Lattice Action for QCD with Wilson Fermions[END_REF][START_REF] Luscher | Chiral symmetry and O(a) improvement in lattice QCD[END_REF][START_REF] Luscher | Nonperturbative O(a) improvement of lattice QCD[END_REF][START_REF] Frezzotti | Chirally improving Wilson fermions-1: O(a) improvement[END_REF]. Up to O(a 2 ), the Symanzik action for Wilson fermions has been extensively studied in Refs. [START_REF] Sheikholeslami | Improved Continuum Limit Lattice Action for QCD with Wilson Fermions[END_REF][START_REF] Aoki | Chiral perturbation theory with Wilson type fermions including a 2 effects: N f = 2 degenerate case[END_REF][START_REF] Bar | Chiral perturbation theory at O(a 2 ) for lattice QCD[END_REF]. In total, there are 18 operators appearing in L (6) . They can be classified into operators of the following five types according to whether or not they break chiral symmetry and the O(4) rotation symmetry [START_REF] Tiburzi | Baryon masses at O(a 2 ) in chiral perturbation theory[END_REF]:

• L (6) 1 : quark bilinear operators that conserve chiral symmetry, ψ / D 3 ψ, ψ(D µ D µ / D + / DD µ D µ )ψ, ψD µ / DD µ ψ. (4.57) • L (6)
2 : quark bilinear operators that break chiral symmetry, ψm q D µ D µ ψ, m q ψD µ D µ ψ, ψm q iσ µν G µν ψ, m q ψiσ µν G µν ψ. (4.58)

• L

3 : four-quark operators that conserve chiral symmetry,

( ψγ µ ψ) 2 , ( ψγ µ γ 5 ψ) 2 , ( ψt a γ µ ψ) 2 , ( ψt a γ µ γ 5 ψ) 2 , (4.59)
where t a are the SU(3) generators, a = 1, • • • , 8.

• L

4 : four-quark operators that break chiral symmetry, ( ψψ) 2 , ( ψγ 5 ψ) 2 , ( ψσ µν ψ) 2 , ( ψt a ψ) 2 , ( ψt a γ 5 ψ) 2 , ( ψt a σ µν ψ) 2 . (4.60)

• L

5 : quark bilinear operators that break the O(4) rotation symmetry,

ψγ µ D µ D µ D µ ψ. (4.61)
It should be noted that fermionic operators that conserve chiral symmetry first appear at O(a 2 ).

Wilson chiral Lagrangians

In order to construct the chiral Lagrangians of WChPT, one has to write down the most general Lagrangians that are invariant under the symmetries of the continuum EFT. This can be done by following the standard procedure of spurion analysis [START_REF] Aoki | Chiral perturbation theory with Wilson type fermions including a 2 effects: N f = 2 degenerate case[END_REF][START_REF] Bar | Chiral perturbation theory at O(a 2 ) for lattice QCD[END_REF]. In practice, in order to obtain the corresponding a-dependent chiral Lagrangians, one only needs to know which symmetries are broken and how [START_REF] Tiburzi | Baryon masses at O(a 2 ) in chiral perturbation theory[END_REF]. Before writing down the chiral Lagrangians up to O(a 2 ), one has to first specify a chiral power-counting scheme, which should be enlarged to include the lattice spacing a. In LQCD simulations, the following hierarchy of energy scales is satisfied:

m q Λ QCD 1 a . (4.62)
If one assumes that the size of chiral symmetry breaking due to light-quark masses and discretization effects are of comparable size, as done in Refs. [START_REF] Bar | Chiral perturbation theory at O(a 2 ) for lattice QCD[END_REF][START_REF] Beane | Nucleon properties at finite lattice spacing in chiral perturbation theory[END_REF][START_REF] Tiburzi | Baryon masses at O(a 2 ) in chiral perturbation theory[END_REF], one has the following expansion parameters:

ε 2 ∼ m q Λ QCD ∼ aΛ QCD , (4.63)
where ε denotes a generic small quantity and Λ QCD ≈ 300 MeV denotes the typical low energy scale of QCD. Up to O(a 2 ), the a-dependent chiral Lagrangians contain terms of O (a, am q , a 2 ) and can be written as

L eff a = L O(a) + L O(amq) +L O(a 2 ) 1 + L O(a 2 ) 2 + L O(a 2 ) 3 + L O(a 2 ) 4 + L O(a 2 ) 5 , (4.64)
and L O(a 2 ) i (i = 1, . . . , 5) are the five classes of chiral Lagrangians corresponding to the previous five types of operators appearing in the Symanzik action at O(a 2 ).

The chiral Lagrangian at O(a) can be written as

L O(a) = b0 BB ρ + + bD B[ρ + , B] -+ bF B[ρ + , B] + , (4.65)
where b0 , bD , and bF are the unknown LECs of dimension mass -1 . The new operator ρ + = u † ρu † + uρ † u and ρ + transforms under chiral rotation (R), parity transformation (P), charge conjugation transformation (C) and hermitic conjugation transformation in the following way:

ρ + R -→ hρ + h † with h ∈ SU(3) V , ρ + P -→ ρ + , ρ + C -→ ρ T + , and ρ + h.c.
--→ ρ + . The matrix ρ is related to the Wilson matrix [Eq. (4.56)] via [START_REF] Rupak | Chiral perturbation theory for the Wilson lattice action[END_REF] which introduces explicit chiral symmetry breaking because of the finite lattice spacing a. The constant W 0 = -0|qσ µν G µν q|0 /F 2 φ is an unknown dimensional quantity that is related to the scale Λ χ .

ρ = 2ac SW W 0 W, (4.66)
The O(am q ) Lagrangian has the following form:

L O(amq) = b1 Bχ + ρ + B + b2 Bχ + Bρ + + b3 Bρ + Bχ + + b4 BBχ + ρ + + b5 Bχ + ρ + B + b6 Bρ + χ + B + b7 B[χ + , B] ρ + + b8 B{χ + , B} ρ + + b9 B[ρ + , B] χ + + b10 B{ρ + , B} χ + + b11 BB χ + ρ + + b12 BB χ + ρ + , (4.67)
where b1,...,12 are unknown LECs of dimension mass -3 . One can eliminate the b3 term by use of the following identity valid for any 3 × 3 matrix A derived from the Cayley-Hamilton identity [START_REF] Zhang | Trace Formulae of Characteristic Polynomial and Cayley-Hamilton's Theorem, and Applications to Chiral Perturbation Theory and General Relativity[END_REF]:

perm=6 A 1 A 2 A 3 A 4 - perm=8 A 1 A 2 A 3 A 4 - perm=3 A 1 A 2 A 3 A 4 + perm=6 A 1 A 2 A 3 A 4 -A 1 A 2 A 3 A 4 = 0, (4.68)
where 'perm' stands for permutation number. In the end, there are 11 independent terms left.

At O(a 2 ), the previous five operators in the Symanzik action can be mapped into the EFT with five classes of chiral Lagrangians L O(a 2 ) i (i = 1, . . . , 5). Following the notation of Ref. [START_REF] Tiburzi | Baryon masses at O(a 2 ) in chiral perturbation theory[END_REF], the first class of chiral Lagrangians can be written as

L O(a 2 ) 1 = a 2 c 2 SW W 2 0 c1 BB + c2 O + BB + c3 B[O + , B] + + c4 B[O + , B] -, (4.69)
where the operator O + is defined as

O + = 2 u † (W -W)u + u(W -W)u † , (4.70)
with

W = 1 -W = diag(1 -ω l , 1 -ω l , 1 -ω s )
, and c1,...,4 are the unknown LECs of dimension mass -3 . Because the second type of operators have an insertion of the quark mass m q , the chiral order of the corresponding chiral Lagrangians is at least O(p 6 ), which is beyond the present work and will not be shown.

There are seven independent terms in the third class of chiral Lagrangians

L O(a 2 ) 3 = a 2 c 2 SW W 2 0 ē1 B[O + , [O + , B]] + ē2 B[O + , {O + , B}] +ē 3 B{O + , {O + , B}} + ē4 BO + O + B +ē 5 B[O + , B] O + + ē6 B{O + , B} O + +ē 7 BB O + 2 + ē8 BB O 2 + , (4.71)
where the ēi are the unknown LECs of dimension mass -3 . Furthermore, we can eliminate the ē6 term by use of the Cayley-Hamilton identity [START_REF] Zhang | Trace Formulae of Characteristic Polynomial and Cayley-Hamilton's Theorem, and Applications to Chiral Perturbation Theory and General Relativity[END_REF]:

B{X 2 , B} + BXBX - 1 2 BB X 2 -BX BX = 0, (4.72)
with X = O + -1 3 O + being a 3 × 3 traceless matrix. Four-quark operators that break chiral symmetry can be mapped into the following chiral Lagrangian:

L O(a 2 ) 4 = d1 B[ρ + , [ρ + , B]] + d2 B[ρ + , {ρ + , B}] + d3 B{ρ + , {ρ + , B}} + d4 Bρ + ρ + B + d5 B[ρ + , B] ρ + + d7 BB ρ + 2 + d8 BB ρ 2 + , (4.73)
with the seven unknown LECs di of dimension mass -3 . Because the chiral transformation properties of ρ + and χ + are the same, the chiral Lagrangian has the same form as the corresponding fourth-order chiral Lagrangian of ChPT.

For the O(4) breaking operators, the mapped chiral Lagrangian can be written as

L O(a 2 ) 5 = a 2 c 2 SW W 2 0 f1 BD µ D µ D µ D µ B + f2 O + BD µ D µ D µ D µ B + f3 BD µ D µ D µ D µ [O + , B] + + f4 BD µ D µ D µ D µ [O + , B] -, (4.74)
where the fi are the unknown LECs of dimension mass -3 . Their contributions to the octet baryon masses can be absorbed by the terms of class one, i.e., Eq. (4.69).

Discretization effects up to O(a 2 )

The octet baryon masses up to N 3 LO and with finite lattice spacing a contributions up to O(a 2 ) can be expressed as

m B = m 0 + m (2)
B + m

(3) B + m (4) B + m (a) B , (4.75)
where m (a)

B denotes the discretization effects up to O(a 2 ). In our power-counting scheme, it contains the following three contributions:

m (a) B = m O(a) B + m O(amq) B + m O(a 2 ) B . (4.76)
Here, we need to mention that virtual decuplet contributions are not explicitly included, since their effects on the chiral extrapolation and the finite-volume corrections are relatively small [START_REF] Ren | Virtual decuplet effects on octet baryon masses in covariant baryon chiral perturbation theory[END_REF].

In (4.77) Table 4.9: Coefficients of the O(am q ) contributions to octet baryon masses (Eq. 4.78).

ξ l ξ s N B1 + 2 B3 B2 + B3 Λ 1 3 ( B1 + B2 + 6 B3 ) 1 3 (2 B1 + 2 B2 + 3 B3 ) Σ B1 + B2 + 2 B3 B3 Ξ B2 + 2 B3 B1 + B3
The O(am q ) contributions can be written as

m O(amq) B = -16ac SW W 0 B 0 (ξ l m l + ξ s m s ) = -8ac SW W 0 ξ l M 2 π + ξ s (2M 2 K -M 2 π ) , (4.78)
and the coefficients ξ l and ξ s are tabulated in Table 4.9. We have introduced the following combinations of LECs: b1 + b2 + 3 b7 + 3 b8 = B1 , b4 -3 b7 + 3 b8 = B2 , and 2 b10 + 3 b11 + b12 = B3 . Hence, there are 3 independent combinations. In obtaining the above results, the light-quark masses have been replaced by the leading-order pseudoscalar meson masses:

m l = 1 2B 0 M 2 π and m s = 1 2B 0 (2M 2 K -M 2 π ).
The O(a 2 ) contributions are not only from the fourth-order tree-level diagram Fig. 4.4(b), but also from the one-loop diagrams of Fig. 4.4(c,d) Chapter 5

m O(a 2 ) B = -a 2 c 2 SW W 2 0 C + 16 D + 16 Ē - 1 (4πF φ ) 2 ac SW W 0 π, K, η ξ (c) B,φ H (c) B (M φ ) + 1 (4πF φ ) 2 π, K, η ξ (d) BB ,φ H (d) B,B (M φ ), (4.79
BB (M φ ) read H (c) B (M φ ) = M 2 φ 1 + ln µ 2 M 2 φ , (4.80)
H (d) BB (M φ ) = m O(a) B   2M 5 φ m 2 0 4m 2 0 -M 2 φ arccos M φ 2m 0 + M 4 φ m 2 0 ln M 2 φ m 2 0 +2M 2 φ ln m 2 0 µ 2 , ( 4

Systematic study of lattice octet baryon masses

In this chapter, we discuss a systematic study of the lattice octet baryon masses by performing the chiral extrapolation of LQCD data, self-consistently including finite-volume corrections and finite lattice spacing discretization effects of LQCD. In order to fix all the unknown LECs and test the consistency of current lattice simulations, we perform a simultaneous fit of all the publicly available n f = 2 + 1 LQCD data from the PACS-CS, LHPC, HSC, QCDSF-UKQCD and NPLQCD collaborations. The contributions of virtual decuplet baryons are also explicitly taken into account to study their effects on the chiral extrapolation of octet baryon masses and finite-volume corrections.

Chiral extrapolation of octet baryon masses

In this section, we study light quark mass dependence of lowest-lying octet baryon masses using the N 3 LO EOMS-BChPT mass formulas [Eq. (4.20)] by fitting n f = 2 + 1 LQCD simulation results. As mentioned above, there are 19 unknown LECs, which cannot be fully determined by the lattice data of a single LQCD simulation. Therefore, we decided to fit all the published lattice results of ground-state octet baryon masses obtained by different collaborations, including the PACS-CS [START_REF] Aoki | 2+1 Flavor Lattice QCD toward the Physical Point[END_REF], LHPC [START_REF] Walker-Loud | Light hadron spectroscopy using domain wall valence quarks on an Asqtad sea[END_REF], HSC [START_REF] Lin | First results from 2+1 dynamical quark flavors on an anisotropic lattice: Light-hadron spectroscopy and setting the strange-quark mass[END_REF], QCDSF-UKQCD [START_REF] Bietenholz | Flavour blindness and patterns of flavour symmetry breaking in lattice simulations of up, down and strange quarks[END_REF] and NLPQCD [START_REF] Beane | High Statistics Analysis using Anisotropic Clover Lattices: (IV) Volume Dependence of Light Hadron Masses[END_REF] collaborations. In performing such a study, we can test the consistency of all these lattice simulations, which used totally different setups.

In the fit process, we use the meson decay constant F φ = 0.0871 GeV [START_REF] Amoros | QCD isospin breaking in meson masses, decay constants and quark mass ratios[END_REF]. In principle at N 3 LO one can use either the chiral limit value F φ = 0.0871 GeV obtained from a two-loop ChPT calculation [START_REF] Amoros | QCD isospin breaking in meson masses, decay constants and quark mass ratios[END_REF], or the SU(3) averaged value, F φ = 1.17F π with F π = 0.0924 GeV as in Ref. [START_REF] Martin Camalich | The lowest-lying baryon masses in covariant SU(3)-flavor chiral perturbation theory[END_REF]. The difference is of higher chiral order. In practice, we found that at N 3 LO the results are not sensitive to these two options while at NNLO the SU(3) averaged value is preferred by the LQCD data. For the baryon axial coupling constants D and F , we use the standard value D = 0.8 and F = 0.46 with D + F = 1.26 as determined from nuclear beta decay. We have allowed D and F to vary in the fits and found that the optimal values determined by the lattice data are consistent with the phenomenological values. The renormalization scale µ is set at 1 GeV, as in Ref. [START_REF] Martin Camalich | The lowest-lying baryon masses in covariant SU(3)-flavor chiral perturbation theory[END_REF].

As mentioned in Chapter 2, the fitted lattice data should fulfil M π < 500 MeV and M φ L > 4 to ensure that the N 3 LO BChPT is valid for these pion (light-quark) masses and lattice volumes. After selection, there are 11 data sets from the PACS-CS, LHPC, HSC, QCDSF-UKQCD and NPLQCD collaborations, respectively. The results of different collaborations are not correlated with each other, but the data from the same collaboration are partially correlated by the uncertainties propagated from the determination of the lattice spacing. Therefore, in order to correctly calculate the χ 2 we incorporate the inverse of the resulting correlation matrix C ij = σ i σ j δ ij + ∆a i ∆a j for each lattice ensemble (see Ref. [START_REF] Martin Camalich | The lowest-lying baryon masses in covariant SU(3)-flavor chiral perturbation theory[END_REF]) , where the σ i are the lattice statistical errors and the ∆a i are the fully-correlated errors propagated from the determination of lattice spacing a i . The FVCs to the baryon masses are consistently calculated in the EOMS-BChPT framework as explained in Section 4.3.

We perform a χ 2 fit to the lattice data and the physical octet baryon masses by varying the 19 LECs. The so-obtained values of the LECs from the best fits are listed in Table 5.1. We remark that in the following the experimental octet baryon masses are always included in the fit unless otherwise stated. This way, the fitted values of LECs are better constrained.

For the sake of comparison, we have fitted lattice data using the NLO and NNLO EOMS-BChPT. It should be noted that since at NLO, ChPT does not generate any FVCs, we have shifted LQCD data by subtracting FVCs calculated by N 3 LO EOMS-BChPT with the LECs determined from the corresponding best fit. The values of the LECs b 0 , b D , b F , and m 0 are tabulated in Table 5.1. An order-by-order improvement is clearly seen, with decreasing χ 2 /d.o.f. at each increasing order. Apparently, only using the O(p 3 ) chiral expansion, we cannot give a good description of the lattice data from the five collaborations. The corresponding χ 2 /d.o.f. is about 8.6. On the other hand, in the N 3 LO fit of lattice data and experimental octet baryon Table 5.2: Finite-volume corrections (in units of MeV) to lattice octet baryon masses in covariant BChPT up to N 3 LO. masses, χ 2 /d.o.f. = 1.0. In addition, the values of the fitted LECs all look very natural. 1 It should be noted that the baryon mass in the chiral limit, m 0 = 880 MeV, seems to be consistent with the SU(2)-BChPT value [START_REF] Procura | Nucleon mass: From lattice QCD to the chiral limit[END_REF][START_REF] Procura | Nucleon mass, sigma term and lattice QCD[END_REF]. Finally, it is important to point out that including the FVCs is important to understand the LQCD results in ChPT at N 3 LO. In Table 5.2, we tabulate the values of FVCs calculated in N 3 LO BChPT for each lattice ensemble in our fit. We can see that, in the SU(3) sector, FVCs are sizable even M φ L > 4. Without FVCs taken into account, the best fit to lattice data yields χ 2 /d.o.f. ∼ 1.9.

M π δm N δm Λ δm Σ δm Ξ M π L M K L M η L PACS-CS 296
In Fig. 5.1, setting the strange-quark mass to its physical value, we plot the light-quark mass evolution of N , Λ, Σ and Ξ as functions of M 2 π using the LECs from Table 5.1. We can see that the NNLO fitting results are more curved and do not describe well lattice data. On the contrary the N 3 LO fit can give both a good description of lattice data and experimental results. The rather linear dependence of the lattice data on M 2 π at large light quark masses, which are exhibited both by the lattice data [START_REF] Walker-Loud | Light hadron spectroscopy using domain wall valence quarks on an Asqtad sea[END_REF] and reported by other groups, is clearly seen.

Virtual decuplet contributions

In this section, we perform a systematic study of virtual decuplet effects on the chiral extrapolation of octet baryon masses and the corresponding finite-volume corrections of lattice by explicitly including their contributions and comparing the fitting results with the octet-only ones.

Light-quark mass dependence of the octet baryon masses

M π 2 [GeV 2 ] Octet Baryon Masses [GeV] Ν Λ Σ Ξ M π 2 [GeV 2 ] Octet Baryon Masses [GeV] Ν Λ Σ Ξ 1 1.2 1.4 1.6 0 0.1 0.2 0.3 0.4 M π 2 [GeV 2 ] Octet Baryon Masses [GeV] Ν Λ Σ Ξ 0 0.1 0.2 0.3 0.4 0.5 M π 2 [GeV 2 ] Octet Baryon Masses [GeV] Ν Λ Σ Ξ Fit NLO Fit NNLO Fit N 3 LO Expt.
PACS-CS LHPC QCDSF-UKQCD HSC

D to the physical decuplet baryon masses. Because t 0 and m D cannot be disentangled at the physical point, one only obtains a combination of m D and t 0 with

m eff D = m D -t 0 (2M 2 K + M 2 π ) = 1.
215 GeV and t D = -0.326 GeV -1 . In the following, the octet-decuplet mass splitting δ = m D -m 0 is fixed to be 0.231 GeV -the average mass gap between the octet and decuplet baryons. Therefore, one can fix the four LECs in the following way: m D = m 0 + 0.231 GeV, t 0 = (m 0 -0.984)/0.507 GeV -1 , t D = -0.326 GeV -1 and C = 0.85. As a result, the same 19 LECs as those in the octet-only BChPT need to be determined. The other fixed coupling constants are the same as previous section: the meson decay constant F φ = 0.0871 GeV, the baryon axial coupling constants D = 0.8, F = 0.46 and the renormalization scale µ = 1 GeV.

We utilize the formulas of Eq. (4.32) to fit the lattice octet baryon masses and the experimental values [START_REF] Beringer | Review of Particle Physics (RPP)[END_REF]. The LQCD data to be studied are taken from the PACS-CS [START_REF] Aoki | 2+1 Flavor Lattice QCD toward the Physical Point[END_REF], LHPC [START_REF] Walker-Loud | Light hadron spectroscopy using domain wall valence quarks on an Asqtad sea[END_REF], HSC [START_REF] Lin | First results from 2+1 dynamical quark flavors on an anisotropic lattice: Light-hadron spectroscopy and setting the strange-quark mass[END_REF], QCDSF-UKQCD [START_REF] Bietenholz | Flavour blindness and patterns of flavour symmetry breaking in lattice simulations of up, down and strange quarks[END_REF] and NPLQCD [START_REF] Beane | High Statistics Analysis using Anisotropic Clover Lattices: (IV) Volume Dependence of Light Hadron Masses[END_REF] data satisfying M π < 500 MeV and M φ L > 4, the same data set as previous section. The so-obtained values of the LECs from the best fit and the corresponding χ 2 /d.o.f. are tabulated in Table 5.1. For χ 2 /d.o.f., from the comparison with the octet-only best fit results, one can conclude that the inclusion of virtual decuplet baryons does not change the description of LQCD data. On the other hand, the values of LECs have changed a lot, as can be clearly seen from Table 5.1 3 . This confirms the assumption that using only octet baryon mass data, one cannot disentangle virtual decuplet contributions from those of virtual octet baryons and tree-level diagrams [START_REF] Ren | Octet baryon masses in next-to-next-to-next-to-leading order covariant baryon chiral perturbation theory[END_REF]. In other words, for the static properties of octet baryons, most contributions of the virtual decuplet are hidden in the relevant LECs, as one naively expects. Below, we will see that their inclusion, however, does improve the description of the volume-dependence of LQCD data, as also noted in Ref. [START_REF] Geng | Finite-volume effects on octetbaryon masses in covariant baryon chiral perturbation theory[END_REF].

In Fig. 5.2, setting the strange-quark mass to its physical value, we show the pion mass dependence of octet baryon masses in the N 3 LO EOMS BChPT with and without virtual decuplet baryon contributions. It is clear that the two N 3 LO fits give the same description of lattice data, as can be inferred from the same χ 2 /d.o.f. shown in Table 5.3.

Finite-volume corrections to the octet baryon masses

In Ref. [START_REF] Geng | Finite-volume effects on octetbaryon masses in covariant baryon chiral perturbation theory[END_REF], we have studied the FVCs to lowest-lying octet baryon masses using the EOMS BChPT up to NNLO, and found that finite-volume effects are very important and cannot be neglected. Therefore, in this work FVCs are self-consistently included in Eq. (4.32) to analyze the lattice data. The NPLQCD [START_REF] Beane | High Statistics Analysis using Anisotropic Clover Lattices: (IV) Volume Dependence of Light Hadron Masses[END_REF] simulation is performed with the same pion mass of M π 390 MeV and at four different lattice sizes L ∼ 2.0, 2.5, 3.0 and 3.9 fm. Therefore, it provides a good opportunity to study FVCs to the octet-baryon masses.

In Fig. 5.3, we contrast the NPLQCD data with the N 3 LO EOMS BChPT using the LECs from Table 5.1. As stated in Ref. [START_REF] Ren | Octet baryon masses in next-to-next-to-next-to-leading order covariant baryon chiral perturbation theory[END_REF], three sets of the NPLQCD data with M φ L > 4 are included in lattice data and denoted by solid points in Fig. 5.3. Another set with M π L = 3.86 (hollow points) is not included. Both the octet-only and the octet plus decuplet (O+D) BChPT can give a reasonable description of the FVCs. In the e -mπL /(m π L) ≤ 0.2 region, these two fits give essentially the same results. With the increase of e -mπL /(m π L) (the decrease of lattice size L), the O+D BChPT results are in better agreement the NPLQCD data, especially for the nucleon mass. It seems that the virtual decuplet baryons can help to improve the description of the FVCs, although the BChPT results are still a bit larger than the LQCD data at small M φ L. It is also indicated that including M φ L = 3.86 could give a better description of FVCs. Therefore, in the following, we would like to slightly loose the constraint of M φ L with M φ L > 3.8.

Continuum extrapolation of lattice data

In this section, we employ the octet baryon masses obtained in Wilson covariant BChPT up to N 3 LO [Eq. (4.75)] to estimate discretization effects of the current LQCD simulations by performing a simultaneous fit of the latest n f = 2 + 1 LQCD data, which are obtained with the O(a)-improved Wilson action.

At present, most LQCD simulations employ a single lattice spacing a and take discretization effects as systematic uncertainties. A similar strategy has been adopted by theoretical studies. On the other hand, one may combine the LQCD simulations from different collaborations and perform a quantitative study of the discretization effects. Among the latest LQCD simulations, several collaborations employed the O(a)-improved or 'clover' Wilson action, e.g. PACS-CS (with a = 0.0907 [START_REF] Einstein | Ist die trägheit eines körpers von seinem energieinhalt abhängig?[END_REF] fm and c SW = 1.715), QCDSF-UKQCD (with a = 0.0795(3) fm and c SW = 2.65), HSC and NPLQCD (with a s = 0.1227(8) fm, a t = 0.03506(23) fm, c s SW = 2.6, and c t SW = 1.8) collaborations. These simulations are performed at three different values of lattice spacing a and with different light-quark masses and, therefore, in principle allow for a quantitative study of the discretization effects on the octet baryon masses.

0.8 1 1.2 1.4 1.6 M π 2 [GeV 2 ] Octet Baryon Masses [GeV] Ν Λ Σ Ξ M π 2 [GeV 2 ] Octet Baryon Masses [GeV] Ν Λ Σ Ξ 1 1.2 1.4 1.6 0 0.1 0.2 0.3 0.4 M π 2 [GeV 2 ] Octet Baryon Masses [GeV] Ν Λ Σ Ξ 0 0.1 0.2 0.3 0.4 0.5 M π 2 [GeV 2 ]
Octet Baryon Masses It should be noted that both the HSC [START_REF] Lin | First results from 2+1 dynamical quark flavors on an anisotropic lattice: Light-hadron spectroscopy and setting the strange-quark mass[END_REF] and the NPLQCD [START_REF] Beane | High Statistics Analysis using Anisotropic Clover Lattices: (IV) Volume Dependence of Light Hadron Masses[END_REF] simulations employed the anisotropic clover fermion action [START_REF] Chen | Heavy quarks on anisotropic lattices: The Charmonium spectrum[END_REF]. In this action, the temporal lattice spacing is chosen to be much smaller than the spatial lattice spacing. The EFT for such a LQCD setup has been worked out in Ref. [START_REF] Bedaque | Effective Field Theory for the Anisotropic Wilson Lattice Action[END_REF]. It is in principle more appropriate for the study of lattice data of HSC and NPLQCD collaborations. On the other hand, this EFT contains more LECs to discriminate the temporal and spatial lattice spacing effects. As we will see, present limited LQCD data do not allow us to perform such a study. Therefore, in our study we assume that these simulations are performed with a single lattice spacing, a s , and we treat the difference between a s and a t as higher-order effects.

M Ξ [GeV] Exp[-m π L]/m π L
As in Refs. [START_REF] Ren | Virtual decuplet effects on octet baryon masses in covariant baryon chiral perturbation theory[END_REF][START_REF] Ren | Decuplet baryon masses in covariant baryon chiral perturbation theory[END_REF], we focus on LQCD data from the above four collaborations with M π < 500 MeV and M φ L > 3.8 to ensure the applicability of SU(3) covariant BChPT. In total, there are 12 sets of LQCD data from the PACS-CS (3 sets), QCDSF-UKQCD (2 sets), HSC (3 sets), and NPLQCD (4 sets). In order to better ascertain the values of LECs, the experimental octet baryon masses are also included in the fits.

In the O(a)-improved Wilson action, the Pauli term aL (5) is eliminated. As a result, discretization effects originate only from the O(am q ) and O(a 2 ) terms. Therefore, only the fourth-order tree-level diagrams contribute, while the leading order tree-level diagram and the

B = m O(amq) B + m O(a 2 ) B = -8ac SW W 0 ξ l M 2 π + ξ s (2M 2 K -M 2 π ) -16a 2 c 2 SW W 2 0 X, (a)
only contain 4 new independent combinations of LECs, i.e., B1 , B2 , B3 , and X. Together with the 19 unknown LECs appearing in the octet baryon masses in the continuum, there are in total 23 free LECs that need to be fixed 4 . The other parameters are fixed as the same as before: F φ = 0.0871 GeV, D = 0.8, F = 0.46, and µ = 1 GeV.

In order to study the discretization effects on the octet baryon masses, we perform two fits. First, we use the continuum octet baryon mass formulas to fit the LQCD and experimental data. Second, the mass formulas of Eq. (4.75) with discretization effects taken into account are employed to fit the same data. In both fits, the finite-volume corrections to the LQCD simulations are always taken into account self-consistently [START_REF] Ren | Octet baryon masses in next-to-next-to-next-to-leading order covariant baryon chiral perturbation theory[END_REF]. The LECs, together with the χ 2 /d.o.f., obtained from the two best fits are tabulated in Table 5.3. It is clear that the 19 LECs remain similar whether or not discretization effects are taken into account. The total χ 2 changes from 30 for the first fit to 28 for the second fit, indicating that the data can be described slightly better. On the other hand, the χ 2 /d.o.f. slightly increases from 0.91 to 0.97, implying that discretization effects do not play an important role in describing the present LQCD data. 5 This justifies their treatment as systematic uncertainties without being taken into account explicitly in the fitting, as done in most previous theoretical and LQCD studies. It should be noted that the one-sigma uncertainties of the LECs B1 , B2 , B3 , and, particularly, X are rather large. This shows clearly the need to perform LQCD simulations at multiple lattice spacings in order to pin down more precisely discretization effects, which has long been recognized [START_REF] Kronfeld | Uses of effective field theory in lattice QCD: Chapter 39[END_REF].

In the above fits we have included the experimental data to better constrain the large number of LECs appearing at N 3 LO. We can of course drop the experimental data, redo the fit, and calculate the octet baryon masses at the physical point. Such a procedure should be taken with caution, however, for the following reasons. First, we have a large number of unknown LECs (about 20). Second, the lightest LQCD data point has a M π about 300 MeV, and it is still a bit away from the physical point. Third, all the χ 2 /d.o.f. are close to 1. These factors can make the extrapolations unstable with respect to moderate changes of the LECs. In Table 5.4, we tabulate the extrapolated octet baryon masses with two sets of LECs, determined from the fits in which finite lattice spacing effects are either taken into account or neglected. It is clear that the extrapolated masses agree within uncertainties, and so do the corresponding LECs (not shown). Nevertheless, the extrapolated nucleon mass still deviates about 60-80 MeV from its physical value, calling for LQCD simulations with smaller light-quark masses (than studied in the present work).

In Fig. 5.4, we show the evolution of discretization effects as a function of the lattice spacing for three different pion masses with the relevant LECs determined from the second fit. It is seen that the discretization effects increase almost linearly with increasing lattice spacing a for fixed pion mass. For fixed a, they increase with increasing pion mass as well. Furthermore, essentially no curvature is observed. It is clear that in our present work the O(am q ) terms dominate over the O(a 2 ) terms6 . It should be stressed that the LEC X is consistent with zero and a fit without the O(a 2 ) contributions would have yielded very similar results as shown in Table 5.3 and Fig. 5.4. For a lattice spacing up to a = 0.15 fm, the finite lattice spacing effects on the baryon masses are less than 2%, consistent with the LQCD study of Ref. [START_REF] Durr | Scaling study of dynamical smeared-link clover fermions[END_REF].

The above results can be naively understood in the following way. Recall that m q /Λ QCD ∼ aΛ QCD in our power-counting scheme. If we take m s = 100 MeV, Λ QCD = 300 MeV, and a = 0.1 fm, we obtain m q /Λ QCD ≈ 0.3 and aΛ QCD ≈ 0.15. If we further assume that all the LECs are of natural size, i.e., ∼ 1, we then expect O(m 2 q ) : O(am q ) : O(a 2 ) = 4 : 2 : 1. Remember that the quark masses are larger than their physical values while the lattice spacing is fixed to be around 0.1 fm in the LQCD simulations, our actual numerical results seem to support this naive argument. Furthermore, we would like to point out that the a-dependent LECs B1 , B2 , B3 , and X are of natural size. The values in Table 5.3 appear to be small the LQCD simulations of the nucleon mass by considering only O(a 2 ) terms, and they showed that finite-volume corrections and finite lattice spacing effects are of similar size. In our present work we find that they are indeed of similar size, but the O(amq) contributions are larger than the O(a 2 ) ones.

- B /m B , as functions of lattice spacing a for M π = 300, 400, and 500 MeV, respectively. The SW coefficient is set at c SW = 1.715, the value of the PACS-CS Collaboration. The strange quark mass is fixed at its physical value dictated by the leading order ChPT. because we have set the dimensional quantity W 0 to be 1 GeV 3 . Its more 'proper' value can be estimated by noting the following relations W 0 a ∼ B 0 m q and M 2 π ∝ 2B 0 m q (in the leading-order ChPT), which yields W 0 ≈ 0.02 GeV 3 . With this value, the LECs turn out to be B1 = -0.0605 GeV -3 , B2 = -0.234 GeV -3 , B3 = 0.172 GeV -3 , and X = 0.152 GeV -3 , which are of natural size as expected.

Chapter 6

Octet baryon sigma terms

In Chapter 4, we have obtained the chiral expansion of octet baryon masses. As an application, we would like to utilize the Feynman-Hellmann theorem to predict the octet baryon sigma terms by analyzing the high-statistics n f = 2 + 1 lattice QCD data. In order to perform an accurate prediction of sigma terms, several key factors are systematically taken into account and clarified for the first time, including the effects of lattice scale setting, systematic uncertainties originating from chiral expansion truncations, and constraint of strong-interaction isospin breaking effects.

Introduction

Understanding the sea-quark structure of nucleon has long been a central topic in nuclear physics [START_REF] Chang | Flavor Structure of the Nucleon Sea[END_REF]. Of particular interest are the contributions of the ss component since nucleon contains no valence strange quarks, e.g., the strangeness contribution to proton spin [START_REF] Bali | Strangeness Contribution to the Proton Spin from Lattice QCD[END_REF] and to electric and magnetic form factors [START_REF] Ahmed | New Precision Limit on the Strange Vector Form Factors of the Proton[END_REF]. In this context, the strangeness-nucleon sigma term, σ sN = m s N |ss|N , plays an important role as it relates to the scalar strangeness content of nucleon, the composition of nucleon mass, KN scatterings, counting rates in Higgs boson searches [START_REF] Cheng | Chiral Symmetry and the Higgs Nucleon Coupling[END_REF], and the precise measurement of the Standard Model parameters in pp collisions at LHC [START_REF] Aad | Expected Performance of the ATLAS Experiment -Detector, Trigger and Physics[END_REF]. Furthermore, the uncertainty in σ sN is the principal source in predicting the cross section of certain candidate dark matter particles interacting with nucleons [START_REF] Bottino | Implications for relic neutralinos of the theoretical uncertainties in the neutralino nucleon cross-section[END_REF].

Although the pion-nucleon sigma term σ πN = m l N |ūu + dd|N can be determined from pion-nucleon scattering [START_REF] Gasser | Sigma term update[END_REF][START_REF] Alarcon | The chiral representation of the πN scattering amplitude and the pion-nucleon sigma term[END_REF], historically the strangeness-nucleon sigma term has been determined indirectly via the nonsinglet matrix element σ 0 = m l N |ūu+ dd-2ss|N , which yields a value ranging from 0 to 300 MeV [START_REF] Alarcon | The strangeness content of the nucleon from effective field theory and phenomenology[END_REF]. In principle, LQCD provides a model independent way in the determination of baryon sigma terms by either computing three-point [START_REF] Takeda | Nucleon strange quark content from two-flavor lattice QCD with exact chiral symmetry[END_REF][START_REF] Babich | Exploring strange nucleon form factors on the lattice[END_REF][START_REF] Bali | The strange and light quark contributions to the nucleon mass from Lattice QCD[END_REF][START_REF] Dinter | Sigma terms and strangeness content of the nucleon with N f = 2 + 1 + 1 twisted mass fermions[END_REF][START_REF] Engelhardt | Strange quark contributions to nucleon mass and spin from lattice QCD[END_REF][START_REF] Ohki | Nucleon strange quark content from N f = 2 + 1 lattice QCD with exact chiral symmetry[END_REF][START_REF] Gong | Strangeness and charmness content of nucleon from overlap fermions on 2+1-flavor domain-wall fermion configurations[END_REF][START_REF] Alexandrou | Strangeness of the nucleon from lattice QCD[END_REF] or two-point correlation functions (the so-called spectrum method) [START_REF] Young | Octet baryon masses and sigma terms from an SU(3) chiral extrapolation[END_REF][START_REF] Ren | Octet baryon masses in next-to-next-to-next-to-leading order covariant baryon chiral perturbation theory[END_REF][START_REF] Semke | Strangeness in the baryon ground states[END_REF][START_REF] Ohki | Nucleon strange quark content from N f = 2 + 1 lattice QCD with exact chiral symmetry[END_REF][START_REF] Durr | Sigma term and strangeness content of octet baryons[END_REF][START_REF] Horsley | Hyperon sigma terms for 2+1 quark flavours[END_REF][START_REF] Shanahan | Sigma terms from an SU(3) chiral extrapolation[END_REF][START_REF] Jung | Nucleon mass and strange content from (2+1)-flavor Domain Wall Fermion[END_REF][START_REF] Junnarkar | The Scalar Strange Content of the Nucleon from Lattice QCD[END_REF]. Although tremendous efforts have been made in this endeavor, due to the many systematic and statistical uncertainties inherent in these studies, no consensus has been reached on the value of the scalar strangeness content of nucleon.

The most important sources of systematic uncertainties originate from the so-called chiral extrapolations. In the u, d, and s flavor sector, a proper formulation of baryon chiral perturbation theory that satisfies all symmetry and analyticity constraints is known to be essential to properly describe the nonperturbative regime of QCD. In this sense, the extended-on-massshell formulation [START_REF] Gegelia | Matching heavy particle approach to relativistic theory[END_REF] has shown a number of both formal and practical advantages, whose applications have solved a number of long-existing puzzles in the one-baryon sector [START_REF] Geng | Recent developments in SU(3) covariant baryon chiral perturbation theory[END_REF]. Its applications in the studies of the LQCD octet baryon masses turn out to be very successful as well [START_REF] Ren | Octet baryon masses in next-to-next-to-next-to-leading order covariant baryon chiral perturbation theory[END_REF][START_REF] Ren | Virtual decuplet effects on octet baryon masses in covariant baryon chiral perturbation theory[END_REF][START_REF] Ren | Decuplet baryon masses in covariant baryon chiral perturbation theory[END_REF]. Furthermore, as demonstrated recently, mass dependent and mass independent lattice scale setting methods can result in a σ sN different by a factor of three [START_REF] Shanahan | Sigma terms from an SU(3) chiral extrapolation[END_REF][START_REF] Shanahan | Scale setting, sigma terms and the Feynman-Hellman theorem[END_REF].

Therefore, it casts doubts on the determination of σ sN from a single data set with a particular scale setting method. To say the least, systematic uncertainties might be well underestimated.

In this chapter, utilizing the latest and high-statistics n f = 2 + 1 LQCD simulations of the octet baryon masses from the PACS-CS [START_REF] Aoki | 2+1 Flavor Lattice QCD toward the Physical Point[END_REF], LHPC [START_REF] Walker-Loud | Light hadron spectroscopy using domain wall valence quarks on an Asqtad sea[END_REF], and QCDSF-UKQCD [START_REF] Bietenholz | Flavour blindness and patterns of flavour symmetry breaking in lattice simulations of up, down and strange quarks[END_REF] collaborations and the Feynman-Hellmann theorem, paying special attention to the lattice scale setting, we report a determination of the baryon sigma terms, particularly the strangenessnucleon sigma term, in covariant BChPT up to N 3 LO.

The Feynman-Hellmann theorem [START_REF] Feynman | Forces in Molecules[END_REF] dictates that in the isospin limit the baryon sigma terms can be calculated from the quark mass dependence of the octet baryon masses, m B , in the following way:

σ πB = m l B|ūu + dd|B ≡ m l ∂m B ∂m l , (6.1
)

σ sB = m s B|ss|B ≡ m s ∂m B ∂m s . (6.2)
The chiral expansion of octet baryon masses up to N 3 LO, m B , is already given in Eq. (4.20) of Chapter 4. The virtual decuplet contributions are not explicitly included, since their effects on the chiral extrapolation and FVCs are shown to be relatively small [START_REF] Ren | Virtual decuplet effects on octet baryon masses in covariant baryon chiral perturbation theory[END_REF]. For convenience in the following discussion, we would like to mention that there are 19 unknown LECs:

m 0 , b 0 , b D , b F , b 1-8 , d 1-5,7,8
to be determined by fitting the LQCD data [START_REF] Ren | Octet baryon masses in next-to-next-to-next-to-leading order covariant baryon chiral perturbation theory[END_REF], and the others are fixed at the following values: D = 0.8, F = 0.46 [START_REF] Cabibbo | Semileptonic hyperon decays[END_REF], F φ = 0.0871 GeV [START_REF] Amoros | QCD isospin breaking in meson masses, decay constants and quark mass ratios[END_REF], and µ = 1 GeV.

Three key factors for accurate determination

In order to obtain an accurate determination of baryon sigma terms, a careful examination of LQCD data is essential, since not all of them are of the same quality though they are largely consistent with each other as shown in Refs. [START_REF] Ren | Octet baryon masses in next-to-next-to-next-to-leading order covariant baryon chiral perturbation theory[END_REF][START_REF] Lutz | On finite volume effects in the chiral extrapolation of baryon masses[END_REF]. For instance, the statistics of the HSC simulations needs to be improved [START_REF] Lin | First results from 2+1 dynamical quark flavors on an anisotropic lattice: Light-hadron spectroscopy and setting the strange-quark mass[END_REF] while the NPLQCD simulations are performed at one single combination of light-quark and strange-quark masses [START_REF] Beane | High Statistics Analysis using Anisotropic Clover Lattices: (IV) Volume Dependence of Light Hadron Masses[END_REF], which offers little constraint on the quark mass dependence of baryon masses. The BMW simulations [START_REF] Dürr | Ab-Initio Determination of Light Hadron Masses[END_REF], though of high quality, are not publicly available. This leaves the PACS-CS [START_REF] Aoki | 2+1 Flavor Lattice QCD toward the Physical Point[END_REF], LHPC [START_REF] Walker-Loud | Light hadron spectroscopy using domain wall valence quarks on an Asqtad sea[END_REF], and QCDSF-UKQCD [START_REF] Bietenholz | Flavour blindness and patterns of flavour symmetry breaking in lattice simulations of up, down and strange quarks[END_REF] data for our study. It is important to note that most LQCD simulations fix the strange-quark mass close to its physical value and vary the light-quark masses. As a result, they are suitable to study the light-quark mass dependence but not the strange-quark mass dependence. In this respect, the QCDSF-UKQCD simulations are of particular importance because they provide a dependence of baryon masses on the strange-quark mass in a region not accessible in other simulations. In order to stay within the application region of BChPT, we only choose LQCD data satisfying the following two criteria: M π < 500 MeV 1 and M φ L > 3.8, as in Refs. [START_REF] Ren | Virtual decuplet effects on octet baryon masses in covariant baryon chiral perturbation theory[END_REF][START_REF] Ren | Baryon chiral perturbation theory with Wilson fermions up to O(a 2 ) and discretization effects of latest n f = 2 + 1 LQCD octet baryon masses[END_REF]. It should be noted that the later criterium is relaxed for the QCDSF-UKQCD data (the smallest M φ L taken into account is 2.932) since their FVCs are small because of the use of ratio method [START_REF] Bietenholz | Flavour blindness and patterns of flavour symmetry breaking in lattice simulations of up, down and strange quarks[END_REF]. 1 We have checked that reducing the cut on Mπ down to Mπ = 400 MeV or Mπ = 360 MeV has little effect on our numerical results, but since our χ 2 /d.o.f. (see Table 6.1) is already about 1, there is no need to further decrease the cut on Mπ. A second issue relates to the scale setting of LQCD simulations as mentioned in Chapter 2. For the spectrum determination of baryon sigma terms, it was pointed out in Ref. [START_REF] Shanahan | Sigma terms from an SU(3) chiral extrapolation[END_REF][START_REF] Shanahan | Scale setting, sigma terms and the Feynman-Hellman theorem[END_REF] that using the Sommer scale r 0 [START_REF] Sommer | A New way to set the energy scale in lattice gauge theories and its applications to the static force and alpha-s in SU(2) Yang-Mills theory[END_REF] to fix the lattice spacing of the PACS-CS data can change the prediction of the strangeness-nucleon sigma term by a factor of two to three in the FRR BChPT up to NNLO. However, in our following studies, we found that the scale setting effects on sigma terms are small. It was claimed that using Sommer scale r 1 for the purpose of lattice scale setting is preferred in the LHPC simulations [START_REF] Walker-Loud | Evidence for non-analytic light quark mass dependence in the baryon spectrum[END_REF] as well. As a result, it is necessary to understand how different scale setting methods for the same simulations affect the prediction of the baryon sigma terms. Unfortunately, such a systematic study is still missing. In addition, instead of relying on the scale determined by the LQCD collaborations themselves, one can fix the lattice scale self-consistently in the BChPT study of the LQCD dimensionless data, as recently done in Ref. [START_REF] Lutz | On finite volume effects in the chiral extrapolation of baryon masses[END_REF]. In the present work, all the three alternative ways of lattice scale setting will be studied and their effects on the predicted sigma terms examined and quantified.

At N 3 LO, the large number of unknown LECs should be better constrained in order to give a reliable prediction of the baryon sigma terms. In this work, we employ the latest LQCD results on the strong isospin-splitting effects on octet baryon masses to further constrain the LECs. The following values are used: δm N = -2.50 [START_REF] Borasoy | Chiral Expansion of Baryon Masses and σ-Terms[END_REF] MeV, δm Σ = -7.67 [START_REF] Luscher | Volume Dependence of the Energy Spectrum in Massive Quantum Field Theories. 1. Stable Particle States[END_REF] [START_REF] Sharpe | Twisted mass chiral perturbation theory at next-to-leading order[END_REF] MeV and δm Ξ = -5.87 [START_REF] Ishikawa | SU(2) and SU(3) chiral perturbation theory analyses on baryon masses in 2+1 flavor lattice QCD[END_REF] [START_REF] Bernard | Chiral perturbation theory[END_REF] MeV at the physical point. The δm N is chosen such as to cover all the recent results [START_REF] Beane | Strong-isospin violation in the neutron-proton mass difference from fully-dynamical lattice QCD and PQQCD[END_REF][START_REF] Borsanyi | Isospin splittings in the light baryon octet from lattice QCD and QED[END_REF], while the δm Σ and δm Ξ are taken from Ref. [START_REF] Borsanyi | Isospin splittings in the light baryon octet from lattice QCD and QED[END_REF]. It should be pointed out that no new unknown LECs need to be introduced to calculate isospin-breaking corrections up to the order at which we work. For more details about the strong isospin-splitting effects on octet baryon masses, please refer to Appendix D.

Predicted baryon sigma terms

In the mass formulas of octet baryon, the LQCD and experimental meson masses are described by NLO ChPT [START_REF] Gasser | Chiral Perturbation Theory: Expansions in the Mass of the Strange Quark[END_REF] with the LECs of Ref. [START_REF] Bijnens | The Second DAPhNE Physics Handbook[END_REF]. FVCs [START_REF] Hasenfratz | Goldstone Boson Related Finite Size Effects in Field Theory and Critical Phenomena With O(N ) Symmetry[END_REF] are taken into account but found to play a negligible role. In Table 6.1, we tabulate the values of LECs and the corresponding χ2 /d.o.f. from three best fits to lattice and experimental octet baryon masses. In the first fit, we use the lattice spacings a determined by LQCD collaborations themselves to obtain the hadron masses in physical units as done in Ref. [START_REF] Ren | Octet baryon masses in next-to-next-to-next-to-leading order covariant baryon chiral perturbation theory[END_REF]. In the second fit, we determine the lattice spacings a self-consistently. Interestingly, we find that the so-determined lattice spacings a are close to the ones determined by LQCD collaborations themselves. Specifically, the PACS-CS deviation is 2.5%, the LHPC deviation is 4.1%, and the QCDSF-UKQCD deviation is 2.1%. The corresponding χ 2 /d.o.f. also look similar. While in the third fit, we adopt the so-called mass dependent scale setting (MDS), either from r 0 for the PACS-CS data with r 0 (phys.) = 0.465 (12) fm [START_REF] Aubin | Light hadrons with improved staggered quarks: Approaching the continuum limit[END_REF], r 1 for the LHPC data with r 1 (phys.) = 0.31174 [START_REF] Khan | Light hadron spectroscopy with two flavors of dynamical quarks on the lattice[END_REF]

fm [242] 2 , or X π = (M 2 π + 2M 2 K )/3
for the QCDSF-UKQCD data with X π (phys.) = 0.4109 GeV [START_REF] Bietenholz | Flavour blindness and patterns of flavour symmetry breaking in lattice simulations of up, down and strange quarks[END_REF]. The third fit yields a smaller χ 2 /d.o.f. and different LECs compared to the other two fits.

In Fig. 6.1, we show octet baryon masses as functions of M 2 π and 2M 2 K -M 2 π using the LECs from Table 6.1 with the physical light-(right panel) and strange-quark (left panel) masses. In Table 6.2: Predicted pion-and strangeness-sigma terms of the octet baryons (in units of MeV) by the N 3 LO BChPT with the LECs of Table 6 (10) order to cross-check the validity of our N 3 LO BChPT fit, the BMW collaboration data [START_REF] Durr | Sigma term and strangeness content of octet baryons[END_REF] are shown as well. It is clear that our three fits yield similar results and are all consistent with the high-quality BMW data, which are not included in our fits.

Using the best fit LECs, we predict the sigma terms of the octet baryons and tabulate the results in Table 6.2. Our predictions given by LECs of Table 6.1 are consistent with each other within uncertainties, and the scale setting effects on the sigma terms seem to be small. Therefore, we take the central values from the fit to the mass independent a fixed LQCD simulations as our final results, and treat the difference between three lattice scale settings as systematic uncertainties, which are given in the second parenthesis of the second column of Table 6.2. It is clear that for σ πN , uncertainties due to scale setting are dominant, while for σ sN statistics errors are much larger, calling for improved LQCD simulations. It should be noted that we have studied the effects of virtual decuplet baryons and variation of the LECs D, F , F φ , and found that the induced uncertainties are negligible compared to those shown in Table 6.2. We want to mention that, as shown in Ref. [START_REF] Ren | Baryon chiral perturbation theory with Wilson fermions up to O(a 2 ) and discretization effects of latest n f = 2 + 1 LQCD octet baryon masses[END_REF], continuum extrapolations have no visible effects on the predicted sigma terms. Furthermore, there are also other related quantities, which often appear in the literature, including the strangeness content (y B ) and the so-called "dimensionless sigma terms" (f πB , f sB ):

y B = 2 B|ss|B B|ūu + dd|B = m l m s 2σ sB σ πB , (6.3)
f πB = m l B|ūu + dd|B M B = σ πB M B , (6.4
)

f sB = m s B|ss|B M B = σ sB M B , (6.5)
are also calculated and tabulated in Table 6.3. The pion-nucleon sigma term, σ πN = 55(1)(4) MeV, is in reasonable agreement with the latest πN scattering study, σ πN = 59 (7) MeV [START_REF] Alarcon | The chiral representation of the πN scattering amplitude and the pion-nucleon sigma term[END_REF], and also the systematic study of n f = 2+1 LQCD simulations on the nucleon mass, σ πN = 52(3)(8) MeV [207], but larger than that of Ref. [START_REF] Lutz | On finite volume effects in the chiral extrapolation of baryon masses[END_REF], σ πN = 39 +2 -1 MeV. Our predicted σ sN is compared with those of earlier studies in Fig. 6.2, classified into three groups according to the methods by which they are determined. The first group is the results reported by the n f = 2 + 1 LQCD simulations, while the second and third groups are predicted by the NNLO and N 3 LO BChPT, respectively. Our results are consistent with the latest LQCD determinations and those of NNLO BChPT studies. However, the prediction of the only other N 3 LO study in the partial summation approach [START_REF] Lutz | On finite volume effects in the chiral extrapolation of baryon masses[END_REF]

m B [GeV] 2M K 2 -M π 2 [GeV 2 ]
Figure 6.1: Octet baryon masses as a function of M 2 π and 2M 2 K -M 2 π vs the BMW LQCD data (data points are taken from Fig. [2] of Ref. [START_REF] Durr | Sigma term and strangeness content of octet baryons[END_REF]). The solid, dashed, and dot-dashed lines are obtained with the LECs from the three fits of Table 6.1. On the left and right panels, the strange-quark mass and the light-quark mass are fixed at their respective physical values. consistent with our result and most LQCD results. It should be noted that, in last two months, the BMW and χQCD collaborations have updated their nucleon sigma terms. More surprisingly, the latest nucleon strangeness sigma term from the BMW Collaboration changes from σ sN = 34(15)(25) [START_REF] Durr | Sigma term and strangeness content of octet baryons[END_REF] to σ sN = 105(41)(37) MeV [START_REF] Durr | Lattice computation of the nucleon scalar quark contents at the physical point[END_REF], while the χQCD result only changes a little bit (σ sN = 32.3(4.7)(4.9) MeV [START_REF] Yang | πN and strangeness sigma terms at the physical point with chiral fermions[END_REF]). Therefore, one has to put more efforts to clarify this large distinction of σ sN .

A note of caution is in order. Clearly, using the spectrum method to determine baryon sigma terms depends critically on the details of LQCD simulations. Lattice scale setting is just one of the sources for potentially large systematic errors. We have studied three common alternative strategies and found that the resulting predictions remain almost the same. Nevertheless, our studies do not exclude the possibility that predictions can change in more rare scenarios. In addition, other LQCD artifacts (such as lattice spacing discretization effects) not addressed in the present work that affect little the baryon masses may have an impact on the predicted baryon sigma terms, which is, however, beyond the scope of present work.

Given the fact that BChPT plays an important role in predicting baryon sigma terms, it is of particular importance to assess the uncertainties of truncating chiral expansions. It becomes even more important in the u, d, and s three-flavor sector, where convergence is governed by the relative large ratio of m K /Λ ChPT ≈ 0.5. Previous studies either stayed at NNLO or N 3 LO and, therefore, were unable to perform such an analysis except those of Refs. [START_REF] Ren | Octet baryon masses in next-to-next-to-next-to-leading order covariant baryon chiral perturbation theory[END_REF][START_REF] Ren | Virtual decuplet effects on octet baryon masses in covariant baryon chiral perturbation theory[END_REF], which, however, focused on a global study of baryon masses and did not include all the QCDSF-UKQCD data that provide further constraints on the strange-quark mass dependence of the octet baryon masses.

To understand quantitatively the convergence issue, we have studied at NNLO the octet baryon masses of the PACS-CS, LHPC and QCDSF-UKQCD data obtained with the lattice spacings a given by the LQCD collaborations themselves. We have allowed the LEC F φ to vary to get an estimation of the induced variation. All the obtained χ 2 /d.o.f. is larger than 1, indicating that higher-order chiral contributions need to be taken into account. In addition, we have employed the FRR method, which is known to converge relatively faster, to study the same data and found no qualitative difference with the EOMS approach. We noted that if one allows the F φ to deviate from the chiral limit value to take into account SU(3) breaking effects, the so-obtained F φ is close to its SU(3) average 1.17f π with f π = 92.1 MeV [START_REF] Beringer | Review of Particle Physics (RPP)[END_REF].

The predicted strangeness-nucleon sigma term is shown in Fig. 6.2. It is clear that the NNLO result has a much smaller uncertainty compared to the N 3 LO one mainly because LECs are over constrained by LQCD simulations. It should be mentioned that in the Feynman-Hellmann method the large m s multiplying the derivative enhances the uncertainty in the determination of strangeness-baryon sigma term, which seems to dominate the uncertainty and therefore puts an upper limit in the precision one can achieve.

To summarize, we have determined the octet baryon sigma terms using covariant baryon chiral perturbation theory up to next-to-next-to-next-to-leading order. We found σ πN = 55(1)(4) MeV and σ sN = 27(27)(4) MeV. Special attention was paid to uncertainties induced by the lattice scale setting method, which, however, were found to be small, in contrast with previous studies. Other uncertainties, such as those induced by truncating chiral expansions and variations of LECs were also studied in detail. In addition, we have used the strong-interaction isospin-splitting effects from the LQCD simulations to further constrain the relevant LECs. Our results indicate a small scalar strangeness content in nucleon, consistent with the strangeness contribution to proton spin and to the electromagnetic form factors of nucleon.

Chapter 7

Decuplet baryon masses in ChPT

As a natural extension, in this chapter, we present an analysis of lowest-lying decuplet baryon masses in covariant baryon chiral perturbation theory with the extended-on-mass-shell scheme up to next-to-next-to-next-to-leading order. In order to determine the unknown low-energy constants, we perform a simultaneous fit of n f = 2+1 lattice data from the PACS-CS, QCDSF-UKQCD, and HSC collaborations, self-consistently including finite-volume corrections. We also tentatively explore the convergence of the SU(3) BChPT. In addition, the pion-and strangeness-sigma terms of decuplet baryons are predicted using the Feynman-Hellmann theorem.

Introduction

In the past few years several studies of lowest-lying baryon decuplet masses have been performed on the lattice [START_REF] Dürr | Ab-Initio Determination of Light Hadron Masses[END_REF][START_REF] Aoki | 2+1 Flavor Lattice QCD toward the Physical Point[END_REF][START_REF] Aoki | Physical Point Simulation in 2+1 Flavor Lattice QCD[END_REF][START_REF] Walker-Loud | Light hadron spectroscopy using domain wall valence quarks on an Asqtad sea[END_REF][START_REF] Lin | First results from 2+1 dynamical quark flavors on an anisotropic lattice: Light-hadron spectroscopy and setting the strange-quark mass[END_REF][START_REF] Bietenholz | Tuning the strange quark mass in lattice simulations[END_REF][START_REF] Bietenholz | Flavour blindness and patterns of flavour symmetry breaking in lattice simulations of up, down and strange quarks[END_REF][START_REF] Beane | High Statistics Analysis using Anisotropic Clover Lattices: (IV) Volume Dependence of Light Hadron Masses[END_REF]. Just like the baryon octet case, under the limitation of computing resources, decuplet baryon masses from LQCD should be extrapolated to the physical point with the help of chiral perturbation theory. At present, the ground-state octet baryon masses have been studied rather extensively in baryon chiral perturbation theory up to N 3 LO [56-58, 61-65, 200]. In contrast, there are only a few studies of n f = 2 + 1 LQCD decuplet baryon masses [START_REF] Walker-Loud | Light hadron spectroscopy using domain wall valence quarks on an Asqtad sea[END_REF][START_REF] Semke | On the quark-mass dependence of the baryon ground-state masses[END_REF][START_REF] Lutz | On the consistency of recent QCD lattice data of the baryon ground-state masses[END_REF][START_REF] Ishikawa | SU(2) and SU(3) chiral perturbation theory analyses on baryon masses in 2+1 flavor lattice QCD[END_REF][START_REF] Semke | Strangeness in the baryon ground states[END_REF], despite many studies limited either to the SU(2) sector or to the quenched LQCD data [START_REF] Jenkins | Baryon masses in chiral perturbation theory[END_REF][START_REF] Banerjee | Baryon mass splittings in chiral perturbation theory[END_REF][START_REF] Martin Camalich | The lowest-lying baryon masses in covariant SU(3)-flavor chiral perturbation theory[END_REF][START_REF] Semke | On the possibility of a discontinuous quark-mass dependence of baryon octet and decuplet masses[END_REF][START_REF] Ishikawa | SU(2) and SU(3) chiral perturbation theory analyses on baryon masses in 2+1 flavor lattice QCD[END_REF][START_REF] Walker-Loud | Nucleon and Delta masses in twisted mass chiral perturbation theory[END_REF][START_REF] Leinweber | Baryon masses from lattice QCD: Beyond the perturbative chiral regime[END_REF][START_REF] Bernard | Chiral extrapolations and the covariant small scale expansion[END_REF][START_REF] Hacker | Including the Delta(1232) resonance in baryon chiral perturbation theory[END_REF][START_REF] Pascalutsa | The Nucleon and delta-resonance masses in relativistic chiral effective-field theory[END_REF][START_REF] Tiburzi | Strong isospin breaking in the nucleon and Delta masses[END_REF][START_REF] Lebed | Baryon masses at second order in chiral perturbation theory[END_REF][START_REF] Banerjee | The Decuplet revisited in χPT[END_REF][START_REF] Tiburzi | Hyperons in Two Flavor Chiral Perturbation Theory[END_REF][START_REF] Tiburzi | Decuplet baryon masses in partially quenched chiral perturbation theory[END_REF][START_REF] Tiburzi | Baryon masses in partially quenched heavy hadron chiral perturbation theory[END_REF][START_REF] Bedaque | Baryon masses at second order in large N chiral perturbation theory[END_REF][START_REF] Cordon | Baryon Masses and Axial Couplings in the Combined 1/N c and Chiral Expansions[END_REF]. In Refs. [START_REF] Walker-Loud | Light hadron spectroscopy using domain wall valence quarks on an Asqtad sea[END_REF][START_REF] Ishikawa | SU(2) and SU(3) chiral perturbation theory analyses on baryon masses in 2+1 flavor lattice QCD[END_REF], it was shown that HB ChPT at NNLO cannot describe the LHPC and PACS-CS decuplet baryon masses. In Ref. [START_REF] Martin Camalich | The lowest-lying baryon masses in covariant SU(3)-flavor chiral perturbation theory[END_REF], the PACS-CS and LHPC decuplet baryon data were also studied by using EOMS BChPT up to NNLO and a reasonable description of the LQCD data was achieved, contrary to the HBChPT studies of Refs. [START_REF] Walker-Loud | Light hadron spectroscopy using domain wall valence quarks on an Asqtad sea[END_REF][START_REF] Ishikawa | SU(2) and SU(3) chiral perturbation theory analyses on baryon masses in 2+1 flavor lattice QCD[END_REF]. In Refs. [START_REF] Semke | On the quark-mass dependence of the baryon ground-state masses[END_REF], Semeke and Lutz studied the BMW [START_REF] Dürr | Ab-Initio Determination of Light Hadron Masses[END_REF] lattice data for octet and decuplet baryon masses up to N 3 LO in BChPT with the partial summation scheme. It was shown that the light-quark mass dependence of decuplet baryon masses can be well described. However, FVCs to lattice data are not taken into account self-consistently. Whereas, in our previous studies [START_REF] Geng | Finite-volume effects on octetbaryon masses in covariant baryon chiral perturbation theory[END_REF][START_REF] Ren | Octet baryon masses in next-to-next-to-next-to-leading order covariant baryon chiral perturbation theory[END_REF][START_REF] Ren | Virtual decuplet effects on octet baryon masses in covariant baryon chiral perturbation theory[END_REF], it has been shown that FVCs need to be taken into account self-consistently in order to achieve a χ 2 /d.o.f about 1 in the description of current n f = 2 + 1 LQCD octet baryon masses.

Given the fact that a simultaneous description of n f = 2+1 LQCD decuplet baryon masses with FVCs taken into account self-consistently is still missing and that the EOMS BChPT can describe lattice octet baryon masses rather well [START_REF] Martin Camalich | The lowest-lying baryon masses in covariant SU(3)-flavor chiral perturbation theory[END_REF][START_REF] Geng | Finite-volume effects on octetbaryon masses in covariant baryon chiral perturbation theory[END_REF][START_REF] Ren | Octet baryon masses in next-to-next-to-next-to-leading order covariant baryon chiral perturbation theory[END_REF][START_REF] Ren | Virtual decuplet effects on octet baryon masses in covariant baryon chiral perturbation theory[END_REF], it is timely to perform a thorough study of lowest-lying decuplet baryon masses in EOMS BChPT up to N 3 LO.

Chiral correction to decuplet baryon masses

In this section, we collect the relevant chiral effective Lagrangians. For convenience of the following discussion, we would like to keep the relevant notations, which are already given in Chapter 4. After that, we calculate the chiral corrections to decuplet baryon masses and the corresponding FVCs in covariant BChPT up to N 3 LO.

Chiral effective Lagrangians

The chiral effective Lagrangians relevant to the present study can be written as the sum of a mesonic part and a meson-baryon part:

L eff = L (2) φ + L (4) φ + L (1) φD + L (2) φD + L (4)
φD .

(

The Lagrangians L

φ and L

φ of the mesonic sector are given in Eq. (3.38) of Chapter 3 and Eq. (4.2) of Chapter 4, respectively. For the decuplet baryon fields in the ChPT, the construction principles can be seen in Chapter 4. The leading-order meson-baryon Lagrangian is

L (1) φD = L D + L (1) φBD + L (1) φDD , (7.2)
where L D denotes the covariant free Lagrangian, and L

φBD and L

φDD describe the interaction of the octet-and decuplet-baryons with the pseudoscalar mesons. The expressions of L D and L

(1) φBD are given in Eq. (4.14) and Eq. (4.17), respectively. And, L

φDD has the following form:

L (1) φDD = iH m D F φ T abc µ γ µνρσ γ 5 ∂ ρ T abd ν ∂ σ φ c d , (7.3)
where we have used the so-called "consistent" coupling scheme for the meson-octet-decuplet vertices [START_REF] Pascalutsa | Quantization of an interacting spin-3/2 field and the Delta isobar[END_REF][START_REF] Pascalutsa | Field theory of nucleon to higher spin baryon transitions[END_REF]. The meson-baryon Lagrangian at order O(p 2 ) can be written as

L (2) φD = L (2, sb) φB + L (2, sb) φD + L (2) φD . (7.4)
The first and second terms, L

(2, sb) φD

, denote the explicit chiral symmetry breaking part. Their expressions are given in Eq. (4.5) and Eq. (4.16) of Chapter 4, respectively. For the chiral symmetry conserving part, L

φD , one has nine terms, following the conventions of Refs. [START_REF] Semke | On the quark-mass dependence of the baryon ground-state masses[END_REF][START_REF] Lutz | Large-N c operator analysis of 2-body meson-baryon counterterms in the chiral Lagrangian[END_REF], Wave function renormalization diagrams are not explicitly shown but included in the calculation.

L (2) φD = 1 F 2 φ t 1 T abc µ g µν (∂ σ φ∂ σ φ) d c T abd ν + t 2 T abc µ (∂ µ φ∂ ν φ) d c + (∂ ν φ∂ µ φ) d c T ν,abd +t 3 Tµ,abc ∂ ν φ a d ε bde T µ,f gc ∂ ν φ h f ε ghe +t 4 Tµ,abc ∂ µ φ a d ε bde ∂ ν φ h f ε ghe + ∂ ν φ a d ε bde ∂ µ φ h f ε ghe T ν,f gc +t 5 T abc µ g µν T abc ν ∂ σ φ∂ σ φ + t 6 T abc µ T ν,abc ∂ µ φ∂ ν φ +t 7 T abc α (∂ µ φ∂ ν φ) d c iγ µ ∂ ν T α,abd + T abc α (∂ ν φ∂ µ φ) d c iγ µ ∂ ν T α,abd + H.c. +t 8 Tα,abc ∂ µ φ a d ε bde iγ µ ∂ ν T α,f gc ∂ ν φ h f ε ghe
+ Tα,abc ∂ ν φ a d ε bde iγ µ ∂ ν T α,f gc ∂ µ φ h f ε ghe + H.c. +t 9 T abc α iγ µ ∂ ν T α,abc ∂ µ φ∂ ν φ + H.c. , (7.5)
where t 1,...,6 have dimension mass -1 and t 7,...,9 have dimension mass -2 . The fourth order chiral effective Lagrangians contain five LECs (see also Refs. [START_REF] Semke | On the quark-mass dependence of the baryon ground-state masses[END_REF][START_REF] Tiburzi | Decuplet baryon masses in partially quenched chiral perturbation theory[END_REF]):

L (4) φD = e 1 T abc µ g µν χ 2 + c d T abd ν + e 2 T abc µ (χ + ) d c g µν (χ + ) b e T aed ν +e 3 T abc µ g µν (χ + ) c d T abd ν χ + + e 4 T abc µ g µν T abc ν χ + 2 +e 5 T abc µ g µν T abc ν χ 2 + , (7.6)
where e 1-5 are the unknown LECs.

Decuplet baryon self-energies

In this subsection, decuplet baryon masses are calculated in the limit of exact isospin symmetry. Formally, up to O(p 4 ) baryon masses can be written as

M D = m D + m (2)
D + m (3)
D + m (4) D , (7.7)
where m

(2)

D , m (3)
D , and m

D are the LO, NLO, and NNLO SU(3)-breaking corrections to decuplet baryon masses, respectively. The corresponding Feynman diagrams are shown in Fig. 7.1, and the explicit expression of decuplet baryon masses is

M D = m D + ξ (a) D,π M 2 π + ξ (a) D,K M 2 K + 1 (4πF φ ) 2 φ=π,K,η ξ (b) D,φ H (b) D (M φ ) + ξ (c) D,φ H (c) D (M φ ) +ξ (d) D,π M 4 π + ξ (d) D,K M 4 K + ξ (d) D,πK M 2 π M 2 K + 1 (4πF φ ) 2 φ=π, K, η ξ (e,1) D,φ H (e,1)
D (M φ ) + ξ (e,2) D,φ H (e,2)
D (M φ ) + ξ (e,3) D,φ H (e,3) D (M φ ) - 1 (4πF φ ) 2 φ=π, K, η B=N, Λ, Σ, Ξ ξ (f ) DB,φ H (f ) D,B (M φ ) - 1 (4πF φ ) 2 φ=π, K, η D =∆, Σ * , Ξ * , Ω - ξ (g) DD ,φ H (g) D,D (M φ ), (7.8)
where ξ (i) 's and H (i) 's are the corresponding coefficients and loop functions with the subscript i denoting the corresponding diagrams shown in Fig. 7.1. The ξ (i) 's are tabulated in Tables 7.1 and 7.2. In Eq. (7.8), the loop functions

∆ Σ * Ξ * Ω - ξ (a) D,π t 0 + 3t D t 0 + t D t 0 -t D t 0 -3t D ξ (a) D,K 2t 0 2t 0 + 2t D 2t 0 + 4t D 2t 0 + 6t D ξ (b) D,π 4 3 C 2 10 9 C 2 2 3 C 2 0 ξ (b) D,K 4 3 C 2 8 9 C 2 4 3 C 2 8 3 C 2 ξ (b) D,η 0 2 3 C 2 2 3 C 2 0 ξ (c) D,π 50 27 H 2 80 81 H 2 10 27 H 2 0 ξ
(c) D,K 20 27 H 2 160 81 H 2 20 9 H 2
H (b) D , H (c)
D , H

D , H

D,B and H (g) D,D are obtained by using the MS renormalization scheme to remove the divergent pieces and the EOMS renormalization scheme to remove the PCB terms [START_REF] Gegelia | Matching heavy particle approach to relativistic theory[END_REF][START_REF] Fuchs | Renormalization of relativistic baryon chiral perturbation theory and power counting[END_REF][START_REF] Geng | Recent developments in SU(3) covariant baryon chiral perturbation theory[END_REF]. The explicit expressions of

H (b) D , H (c) D , H (e) D , H (f ) D,B , H (g)
D,D are given in the following:

H (b) D (M φ ) = 1 48m 3 D M 2 φ M 2 φ (6m 2 0 + 4m 0 m D + 7m 2 D ) +2(m 0 -m D )(m 0 + m D ) 3 -2M 4 φ + 1 24m 5 D M 4 φ log M φ m 0 -2M 2 φ (2m 2 0 + m 0 m D + 2m 2 D ) +6(m 4 0 + m 3 0 m D + m 2 0 m 2 D + m 0 m 3 D + m 4 D ) + M 4 φ + 1 24m 5 D (m 0 -m D )(m 0 + m D ) 3 log m 0 M φ m 2 0 -m 2 D × 2M 2 φ (-2m 2 0 + m 0 m D -2m 2 D ) + (m 2 0 -m 2 D ) 2 - 1 24m 5 D √ W (m 2 0 -2m 0 m D + m 2 D -M 2 φ ) 2 (m 2 0 + 2m 0 m D + m 2 D -M 2 φ ) 3 × arctan m 2 0 -m 2 D -M 2 φ √ W -arctan m 2 0 + m 2 D -M 2 φ √ W , (7.9)
D,π 3
2 (2t 0 + 3t D )M 2 π 3(t 0 + t D )M 2 π 3 2 (2t 0 + t D )M 2 π 3t 0 M 2 π ξ (e,1) D,K (4t 0 + 3t D )M 2 K 4(t 0 + t D )M 2 K (4t 0 + 5t D )M 2 K 2(2t 0 + 3t D )M 2 K ξ (e,1) D,η 1 6 8t 0 M 2 K -(2t 0 -3t D )M 2 π 1 3 (t 0 + t D )(4M 2 K -M 2 π ) 1 6 8(t 0 + 2t D )M 2 K -(2t 0 + 7t D )M 2 π 1 3 4(t 0 + 3t D )M 2 K -(t 0 + 6t D )M 2 π ξ (e,2) D,π -1 2 (3 t1 + 2 t2 + 3 t3 ) - 1 6 (6 t1
+ 5 t2

+ 9 t3 ) - 1 2 ( t1
+ t2 + 3 t3 ) -3 2 t3 ξ (e,2) D,K -( t1 + t2 + 2 t3 ) - 2 3 (2 t1
+ t2 + 3 t3 ) - 1 3 (5 t1
+ 3 t2 + 6 t3 ) -2( t1 + t2 + t3 ) ξ (e,2) D,η -1 6 ( t1 + 3 t3
)

-1 6 (2 t1 + 3 t2 + 3 t3 ) - 1 2 ( t1 + t2 + t3
)

-1 6 (4 t1 + 3 t3 ) ξ (e,
-4 3 (4t 7 + 3t 9 ) ξ (f ) DN,{π,K,η} {2C 2 , 0, 0} {0, 2 3 C 2 , 0} {0, 0, 0} {0, 0, 0} ξ (f ) DΛ,{π,K,η} {0, 0, 0} {C 2 , 0, 0} {0, C 2 , 0} {0, 0, 0} ξ (f ) DΣ,{π,K,η} {0, 2C 2 , 0} { 2 3 C 2 , 0, C 2 } {0, C 2 , 0} {0, 0, 0} ξ (f ) DΞ,{π,K,η} {0, 0, 0} {0, 2 3 C 2 , 0} {C 2 , 0, C 2 } {0, 0, 4C 2 } ξ (g) D∆,{π,K,η} { 5 3 H 2 , 0, 1 3 H 2 } {0, 8 9 H 2 , 0} {0, 0, 0} {0, 0, 0} ξ (g) DΣ * ,{π,K,η} {0, 2 3 H 2 , 0} { 8 9 H 2 , 0, 0} {0, 4 3 H 2 , 0} {0, 0, 0} ξ (g) DΞ * ,{π,K,η} {0, 0, 0} {0, 8 9 H 2 , 0} { 1 3 H 2 , 0, 1 3 H 2 } {0, 4 3 H 2 , 0} ξ (g) DΩ - ,{π,K,η} {0, 0, 0} {0, 0, 0} {0, 2 3 H 2 , 0} {0, 0, 4 3 H 2 } H (c) D (M φ ) = 5M 3 φ 72m 5 D (4m 2 D -M 2 φ ) 5/2 ×   arctan   M φ 4m 2 D -M 2 φ   + arctan   2m 2 D -M 2 φ M φ 4m 2 D M 2 φ -M 4 φ     + 5M 4 φ 144m 5 D 17m 4 D -2m 2 D M 2 φ + 2(30m 4 D -10m 2 D M 2 φ + M 4 φ ) log M φ m 0 , (7.10)
H (e,1)

D (M φ ) = M 2 φ 1 + ln µ 2 M 2 φ , (7.11)
H (e,2)

D (M φ ) = M 4 φ 1 + ln µ 2 M 2 φ , (7.12)
H (e,3)

D (M φ ) = m D M 4 φ 4 1 + ln µ 2 M 2 φ + 1 8 M 4 φ , (7.13)
H (f ) D,B (M φ ) = 1 144m 4 D m (2) D M 2 φ 90m 4 0 + 96m 3 0 m D + 36m 2 0 m 2 D + 48m 0 m 3 D -22m 4 D -3 30m 2 0 + 16m 0 m D + 19m 2 D M 2 φ + 30M 4 φ - 1 12m 3 D m (2) B M 2 φ 8m 3 0 + 8m 2 0 m D + 4m 0 m 2 D + 7m 3 D -(4m 0 + m D )M 2 φ + 1 24m 6 D m (2) D M 2 φ -6(5m 4 0 + 4m 3 0 m D + 3m 2 0 m 2 D + 2m 0 m 3 D + m 4 D )M 2 φ +4m 5 0 (5m 0 + 6m D ) + 4(5m 2 0 + 2m 0 m D + 3m 2 D )M 4 φ -5M 6 φ log M φ m 0 - 1 12m 5 D m (2) B M 2 φ 3m 4 0 (4m 0 + 5m D ) -3(4m 3 0 + 3m 2 0 m D + 2m 0 m 2 D + m 3 D )M 2 φ + (4m 0 + m D )M 4 φ log M φ m 0 + 1 24m 6 D (m 0 -m D ) 2 (m 0 + m D ) 4 log(m 0 M φ ) -log(m 2 0 -m 2 D ) -iπ × 2m D m (2)
B (4m 0 -m D ) -m (2)
D (5m 2 0 -2m 0 m D + 3m 2 D ) + 1 12 M 2 φ 3m (2)
B + 2m

(2) D log m 0 M φ µ 2 + 1 24m 6 D √ W (m 2 0 -2m 0 m D + m 2 D -M 2 φ )(m 2 0 + 2m 0 m D + m 2 D -M 2 φ ) 2 × 5m 4 0 m (2) D -2m 2 0 5M 2 φ m (2)
D + m 2 D m (2)
D -m (2) B -2m 3 0 m D m (2)
D + 4m (2) B + 2m 0 m D (m 2 D + M 2 φ ) m (2)
D + 4m (2) B -(m 2 D -M 2 φ ) 5M 2 φ m (2)
D + m 2 D 3m (2)
D + 2m (2) B × arctan m 2 0 -m 2 D -M 2 φ √ W -arctan m 2 0 + m 2 D -M 2 φ √ W , (7.14)
H (g) D,D (M φ ) = M 2 φ 432m 4 D 4m 4 D 132m (2)
D -97m (2) D +30M 4 φ 3m (2)
D + 2m (2) D -15m 2 D M 2 φ 31m (2)
D + 14m (2) D + 5M 2 φ 72m 6 D log M φ m D -30m 4 D M 2 φ m (2)
D + 48m 6 D m (2)
D -m (2) D +10m 2 D M 4 φ 2m (2)
D + m (2) D -M 6 φ 3m (2)
D + 2m (2) D + 5M 2 φ 36 12m (2)
D -7m (2) D log m 2 D µ 2 - 5 72m 6 D M 3 φ (4m 2 D -M 2 φ ) 3/2 × 2m 2 D m (2)
D -m (2) D -M 2 φ 3m (2)
D + 2m (2) D ×   arctan M φ 4m 2 D -M 2 φ + arctan 2m 2 D -M 2 φ M φ 4m 2 D -M 2 φ   . (7.15)
In Eqs. (7.10,7.14,7.15

), W = -m 4 0 -(m 2 D -M 2 φ ) 2 + 2m 2 0 (m 2 D + M 2 φ ), m (2)
D and m

B are the NLO decuplet and octet baryon masses, where m

with the corresponding coefficients ξ

B,φ listed in Table 7.3. It should be noted that in the evaluation of diagrams Fig. 7.1(f,g), we have only kept terms linear in M i (i = 1, . . . , 10) are taken from the latest global fit [START_REF] Bijnens | A new global fit of the L r i at next-to-next-to-leading order in Chiral Perturbation Theory[END_REF]. In order to be consistent with our renormalization scale used for the baryon sector, we have re-evaluated the LECs at µ = 1 GeV. The details can be found in Ref. [START_REF] Ren | Octet baryon masses in next-to-next-to-next-to-leading order covariant baryon chiral perturbation theory[END_REF].

Finite-volume corrections

As emphasized in Refs. [START_REF] Geng | Finite-volume effects on octetbaryon masses in covariant baryon chiral perturbation theory[END_REF][START_REF] Ren | Octet baryon masses in next-to-next-to-next-to-leading order covariant baryon chiral perturbation theory[END_REF][START_REF] Ren | Virtual decuplet effects on octet baryon masses in covariant baryon chiral perturbation theory[END_REF], finite-volume corrections have to be taken into account in studying the current LQCD data. In the case of decuplet baryon masses, they have been studied up to NNLO in EOMS BChPT [START_REF] Martin Camalich | The lowest-lying baryon masses in covariant SU(3)-flavor chiral perturbation theory[END_REF] and in HB ChPT [START_REF] Ishikawa | SU(2) and SU(3) chiral perturbation theory analyses on baryon masses in 2+1 flavor lattice QCD[END_REF]. In the following, we extend the study up to N 3 LO in EOMS BChPT.

The FVCs can be easily evaluated following the standard technique. One chooses the baryon rest frame, i.e., p µ = (m D , 0), performs a momentum shift and Wick rotation, integrates over Table 7.3: Coefficients of the NLO contributions to octet baryon masses [Eq. (7.16)].

N Λ Σ Ξ ξ (a) B,π -(2b 0 + 4b F ) - 2
3 (3b 0 -2b D ) -(2b 0 + 4b D ) -(2b 0 -4b F ) ξ (a) B,K -(4b 0 + 4b D -4b F ) - 2
3 (6b 0 + 8b D ) -4b 0 -(4b 0 + 4b D + 4b F )
the temporal dimension, and obtains the results expressed in terms of the master formulas given in Ref. [START_REF] Beane | Nucleon masses and magnetic moments in a finite volume[END_REF]. The details can be seen in Section 4.2 of Chapter 4.

To proceed with the above procedure, one should note that since Lorentz invariance is lost in finite volume, the mass term in the loop functions is identified as the term having the structure of δ ij . This can be easily seen by noticing that in the rest frame the zero component of decuplet baryon field vanishes because of the on-shell condition p µ T µ = 0. For instance, the loop function of diagram Fig. 7.1(b), after Feynman parameterisation, becomes:

G (b) D ∝ d 4 k (2π) 4 (m D (x -1) -m B )k α k β k 2 -M (b) D 2 2 , (7.17)
where

M (b) D 2 = (x 2 -x)m 2 D + xm 2 0 + (1 -x)M 2 φ -iε.
To evaluate its contribution to the decuplet baryon mass, one simply replaces k α k β with δ ij k 2 /3 in the numerator. Following the procedure specified above, one can then easily obtain the FVCs to the loop function of diagram Fig. 7.1(b),

δG (b) D (M φ ) ≡ G (b) D (L) -G (b) D (∞) = -1 12
1 0 dx [m 0 -m D (x -1)] δ 1/2 (M (b) D 2 ) -M (b) D 2 δ 3/2 (M (b) D 2
) , (

where the "master" formula δ r (M 2 ) is defined as

δ r (M 2 ) = 2 -1/2-r ( √ M 2 ) 3-2r π 3/2 Γ(r) n =0 (L √ M 2 | n|) -3/2+r K 3/2-r (L √ M 2 | n|), (7.19)
where K n (z) is the modified Bessel function of the second kind, and

n =0 ≡ ∞ nx=-∞ ∞ ny=-∞ ∞ nz=-∞ (1 -δ(| n|, 0)) with n = (n x , n y , n z ).
One also can obtain the FVCs of other loop diagrams in Fig.

D (M φ ) = 5 36 1 0 dx m D (x -2) δ 1/2 (M (c) D 2 ) -M (c) D 2 δ 3/2 (M (c) D 2 ) , (7.20)
with

M (c) D 2 = x 2 m 2 D + (1 -x)M 2 φ -iε.
Taking the limit of m D → ∞, Eq. (7.18) and Eq. (7.20) reduce to

δG (b) D (M φ ) HB = - 1 8 ∞ 0 dx δ 1/2 (β 2 ∆ ) -β 2 ∆ δ 3/2 (β 2 ∆ ) , (7.21) δG (c)
D (M φ ) HB = 1 2 ∞ 0 dx δ 1/2 (β 2 ) -β 2 δ 3/2 (β 2 ) , (7.22)
where

β ∆ = x 2 -2xδ + M 2 φ and β = x 2 + M 2 φ .
They agree with the HB ChPT results of Ref. [START_REF] Ishikawa | SU(2) and SU(3) chiral perturbation theory analyses on baryon masses in 2+1 flavor lattice QCD[END_REF].

FVCs to the N 3 LO one-loop diagrams Fig. 7.1 (e,f,g) have the following form:

δG (e,1)

D (M φ ) = 1 2 δ 1/2 (M 2 φ ), (7.23) δG (e,2)
D (M φ ) = 1 2 M 2 φ δ 1/2 (M 2 φ ), (7.24) δG (e,3)
D (M φ ) = 1 2 m D δ -1/2 (M 2 φ ), (7.25) δG (f )
D,B (M φ ) = 1 12 1 0 dx m (2)
D (x -1) -m (2) B • δ 1/2 (M (b) D 2 ) + (1 -x)m (2) D 2M (b) D 2 + M 2 φ (x -1) -(m 0 + m D )(m 0 + 2m D )x + 2m 2 D x 2 + M (b) D 2 + 3m 0 x(m 0 + m D (1 -x)) m (2) B • δ 3/2 (M (b) D 2 ) +M (b) D 2 (x -1)m (2) D M (b) D 2 + M 2 φ (x -1) -(m 0 + m D )(m 0 + 2m D )x + 2m 2 D x 2 -3m 0 x(m 0 + m D (1 -x))m (2) B • δ 5/2 (M (b) D 2 ) , (7.26) δG (g)
D,D (M φ ) = 5 36 1 0 dx (3x -5)m (2)
D + (3 -2x)m (2)
D • δ 1/2 (M (c) D 2 ) + M (c) D 2 (10 -6x) + 2m 2 D x(2x -3) + M 2 φ (-3x 2 + 8x -5) m (2) D + M (c) D 2 (2x -3) -3m 2 D (x 2 -2x) m (2) D • δ 3/2 (M (c) D 2 ) + M (c) D 2 m 2 D (6x -4x 2 ) + M (c) D 2 (3x -5) + M 2 φ (x -1)(3x -5) m (2) D + 3M (c) D 2 m 2 D (x 2 -2x)m (2) D • δ 5/2 (M (c) D 2
) .

There is a technical problem in computing δG , appearing in Eq. (7.18) and Eq. (7.26), can become negative for m D > m 0 +M φ in x ∈ [0, 1], i.e., when baryon can decay into the intermediate channel. In order to deal with this problem, we adopt the strategy proposed in Ref. [START_REF] Bernard | The Delta-resonance in a finite volume[END_REF] and replace the original δ r (M 2 ) with three parts by introducing a new scale µ satisfying µ < m 0 + M φ , i.e.,

δ r (M 2 ) = g r 1 -g r 2 + g r 3 , (7.28)
where the g r 1,2,3 are

g r 1 = 1 L 3 k      1 4π 2 n 2 L 2 + M 2 (m 2 D ) r - 1 4π 2 n 2 L 2 + M 2 (µ 2 ) r + r(x 2 -x)(m 2 D -µ 2 ) 4π 2 n 2 L 2 + M 2 (µ 2 ) r+1      , g r 2 = +∞ 0 k 2 dk 2π 2      1 k 2 + M 2 (m 2 D ) r - 1 k 2 + M 2 (µ 2 ) r + r(x 2 -x)(m 2 D -µ 2 ) k 2 + M 2 (µ 2 ) r+1      , g r 3 = δ r M 2 (µ 2 ) -r(x 2 -x)(m 2 D -µ 2 )δ r+1 M 2 (µ 2 ) , (7.29)
and

M 2 (m 2 D ) = (x 2 -x)m 2 D + xm 2 0 + (1 -x)M 2 φ -iε, (7.30)
M 2 (µ 2 ) = (x 2 -x)µ 2 + xm 2 0 + (1 -x)M 2 φ -iε. (7.31)
To take into account of FVCs in the study of the LQCD data, one simply replaces the loop functions H of Eq. (7.8) by H = H + δG with the δGs calculated above.

Systematic study of lattice decuplet baryon masses

In this section, we perform a simultaneous fit of the n f = 2 + 1 LQCD data from the PACS-CS [START_REF] Aoki | 2+1 Flavor Lattice QCD toward the Physical Point[END_REF], QCDSF-UKQCD [START_REF] Bietenholz | Flavour blindness and patterns of flavour symmetry breaking in lattice simulations of up, down and strange quarks[END_REF], and HSC [START_REF] Lin | First results from 2+1 dynamical quark flavors on an anisotropic lattice: Light-hadron spectroscopy and setting the strange-quark mass[END_REF] Collaborations and the experimental data [START_REF] Beringer | Review of Particle Physics (RPP)[END_REF] to determine the 17 unknown LECs, m D , t D , t 0•••9 , and e 1•••5 . Since t 1 , t 2 , t 3 , t 4 , t 5 , and t 6 appear in combinations, effectively we have only 14 independent LECs. The pion or light-quark mass dependence of the decuplet baryon masses is studied in the NLO, NNLO, and N 3 LO EOMS BChPT. Using the so-obtained LECs, we also carry out a detailed study on the QCDSF-UKQCD and LHPC data to test the applicability of the N 3 LO BChPT and the consistency between different LQCD simulations. Furthermore, the pion-and strangeness-baryon sigma terms are predicted by the use of the Feynman-Hellmann theorem.

Lattice data of decuplet baryon masses

Up to now, five collaborations have reported n f = 2 + 1 simulations of the decuplet baryon masses, i.e., the BMW [START_REF] Dürr | Ab-Initio Determination of Light Hadron Masses[END_REF], PACS-CS [START_REF] Aoki | 2+1 Flavor Lattice QCD toward the Physical Point[END_REF], LHPC [START_REF] Walker-Loud | Light hadron spectroscopy using domain wall valence quarks on an Asqtad sea[END_REF], HSC [START_REF] Lin | First results from 2+1 dynamical quark flavors on an anisotropic lattice: Light-hadron spectroscopy and setting the strange-quark mass[END_REF], and QCDSF-UKQCD [START_REF] Bietenholz | Flavour blindness and patterns of flavour symmetry breaking in lattice simulations of up, down and strange quarks[END_REF] collaborations. Because the BMW data are not publicly available and the data of the LHPC Collaboration seem to suffer some systematic errors, as shown in their chiral extrapolation result on the ∆(1232) mass, which is much higher than its physical value [START_REF] Walker-Loud | Light hadron spectroscopy using domain wall valence quarks on an Asqtad sea[END_REF], we will concentrate on the data of the PACS-CS, QCDSF-UKQCD, and HSC collaborations. Following the criteria used in our previous studies [START_REF] Ren | Virtual decuplet effects on octet baryon masses in covariant baryon chiral perturbation theory[END_REF], we only select the LQCD data that satisfy M π < 0.5 GeV and M φ L > 3.8. As a result, there are eight sets of data from the PACS-CS (3 sets), QCDSF-UKQCD (2 sets), and HSC (3 sets) Collaborations. Among the eight LQCD data sets studied, only in the ensemble with M π = 296 MeV from the PACS-CS Collaboration, can the decay ∆ → N + π happen. It should be noted that the PACS-CS Collaboration measured the lowest energy levels of the vector meson and decuplet baryon channels, which are different from the true resonance masses. The resulting difference for the ρ meson is estimated to be 5 percent using Lüscher's formula [START_REF] Aoki | 2+1 Flavor Lattice QCD toward the Physical Point[END_REF]. We will comment on this later.

It should be mentioned that the O(a)-improved Wilson action was used by all the above collaborations except the LHPC Collaboration, which employed a mixed action. The O(a)improved action has the favorable property that the leading order corrections from the finite lattice spacing are eliminated. The finite lattice spacing corrections of the mixed action of the LHPC Collaboration were also shown to be small [START_REF] Walker-Loud | Light hadron spectroscopy using domain wall valence quarks on an Asqtad sea[END_REF]. Therefore, in the present work we assume that the discretization artifacts of the present LQCD simulations are small and can be ignored, and will leave a detailed study on finite lattice spacing artifacts to a future study (for a recent study of the discretization effects on the octet baryon masses, see Ref. [START_REF] Ren | Baryon chiral perturbation theory with Wilson fermions up to O(a 2 ) and discretization effects of latest n f = 2 + 1 LQCD octet baryon masses[END_REF]).

Before we perform a simultaneous fit of LQCD data, we specify our strategy to fix some of the LECs in the N 3 LO BChPT mass formulas Eq. (7.8). For the meson decay constant, we use F φ = 0.0871 GeV. The φBD coupling is fixed to the SU(3)-average value among the different decuplet-to-octet pionic decay channels, C = 0.85 [START_REF] Alarcon | The strangeness content of the nucleon from effective field theory and phenomenology[END_REF]. The φDD coupling H is barely known, and we fix it using the large N c relation H A = (9/5)g A , where g A and H A are the nucleon and ∆ axial charges. With g A = 1.26, this yields the φDD coupling H = H A /2 = 1.13. In the loop function Eq. (7.27), the LO corrections to virtual octet masses are included, therefore, there are 4 more LECs m 0 , b 0 , b D , and b F related to octet baryon masses up to O(p 2 ). Similar to the determination of decuplet baryon masses at O(p 2 ) in Chapter 5, their values can be obtained by fitting the physical octet baryon masses with the NLO octet mass formula

M B = m 0 -m (2)
B . Because at the same pion masses, the m 0 and b 0 cannot be disentangled, we only obtain In the fitting process, we incorporate the inverse of the correlation matrix C ij = σ i σ j δ ij + ∆a i ∆a j for each lattice ensemble to calculate the χ 2 , where σ i are the lattice statistical errors and ∆a i are the fully-correlated errors propagated from the determination of a i . This is because the data from different collaborations are not correlated with each other, but the data form the same collaboration are partially correlated by the uncertainties propagated from the determination of lattice spacing a.

m eff 0 = m 0 -b 0 (4M 2 K + 2M 2 π ), b D = 0.06 GeV -1 ,

Chiral extrapolation

In this subsection, we proceed to study the eight sets of lattice data for decuplet baryon masses by using the N 3 LO BChPT mass formulas [Eq. (7.8)]. In order to constrain better the values of LECs, we include the corresponding experimental data in the fits. The obtained 14 LECs from the best fits are tabulated in Table 7.4. For the sake of comparison, we also perform fits at NLO 1 and NNLO. Up to NNLO, there are only three LECs, i.e., m D , t 0 , and t D .

It is clear that the NLO fit (without loop contributions) already describe the LQCD simulations very well. The description becomes a bit worse at NNLO. 2 The χ 2 /d.o.f. for the N 3 LO is as low as 0.20. Therefore we confirm that the PACS-CS, QCDSF-UKQCD, and HSC data are consistent with each other, although their setups are different. Furthermore, it seems that lattice decuplet baryon masses are almost linear in M 2 π , as demonstrated by the good fit obtained at NLO, χ 2 /d.o.f. * = 0.44.

The values of 14 LECs seem very natural, except that the LECS t1 , t2 , t3 , and t 7 might be slightly large. If we had constrained their values to lie between -1 to 1 in the fitting process, we would have obtained a χ 2 /d.o.f. = 0.26, instead of 0.20. It seems that the present LQCD simulations are not precise enough or are too limited to put a stringent constraint on the values of all the LECs appearing up to N 3 LO, because the NLO fit already yields a χ 2 /d.o.f. smaller than 1. This is further confirmed by the relatively large correlation observed between some of the LECs, e.g., between t1 and t2 , among t 7 , t 8 , and t 9 , and among e 1 , e 3 , and e 5 . We found 1 Because at O(p 2 ), BChPT does not generate any FVCs, we have adjusted lattice data by subtracting the FVCs calculated by the N 3 LO EOMS BChPT with virtual octet contributions taken into account.

2 Without the contributions of the virtual octet baryons, the NNLO description would be much better, with a χ 2 /d.o.f. ≈ 2. As mentioned earlier, the lightest lattice point with m π = 296 MeV of the PACS-CS Collaboration suffers from potentially large systematic errors. If we had performed the fit without this point, we would have obtained a χ 2 /d.o.f. = 0.24, slightly larger than the χ 2 /d.o.f. = 0.20 of Table 7.4. In addition, the values of the corresponding LECs would change moderately. On the other hand, the extrapolations with the LECs determined from the fit excluding physical masses became much worse. This seems to suggest that the inclusion of the lightest PACS-CS point is reasonable, keeping in mind the caveat that they suffer from potentially large systematic errors. This is also the strategy adopted by the PACS-CS collaboration [START_REF] Ishikawa | SU(2) and SU(3) chiral perturbation theory analyses on baryon masses in 2+1 flavor lattice QCD[END_REF] and other similar studies [START_REF] Semke | Strangeness in the baryon ground states[END_REF].

In Fig. 7.2, we show the ∆, Σ * , Ξ * , and Ω -masses as functions of M 2 π , where the strange quark mass is set to its physical value. It is clear that lattice data are rather linear in M 2 π . The O(p3 ) BChPT results show strong curvature and cannot describe LQCD data. A good description can only be achieved up to N 3 LO. 3 In Fig. 7.2, we also show those data of the PACS-CS and HSC collaborations, which are excluded from the fit. The O(p 4 ) BChPT can describe reasonably well those data as well.

It should be emphasised that the setups of the QCDSF-UKQCD simulations are rather [START_REF] Bietenholz | Tuning the strange quark mass in lattice simulations[END_REF]. In this way, the corresponding kaon and eta masses can be smaller than the pion mass. On the other hand, the FVCs from the kaon and eta loops can become comparable or even larger than that induced by the pion loop, because the M φ L can simultaneously become smaller than 4. Therefore, the QCDSF-UKQCD data provide us an opportunity to test BChPT in the world of small strange quark masses and small lattice volumes.

1 1.2 1.4 1.6 1.8 M π 2 [GeV 2 ] Decuplet Baryon Masses [GeV] ∆ Σ * Ξ * Ω - M π 2 [GeV 2 ] Decuplet Baryon Masses [GeV] ∆ Σ * Ξ * Ω -
In Fig. 7.3, the QCDSF-UKQCD lattice data are compared with the N 3 LO BChPT. The lattice points included in the fit are denoted by solid points and those excluded from the fit by hollow points. All lattice points are shifted by FVCs and the kaon mass is fixed using the function M 2 K = a + bM 2 π for the lattice ensemble with a and b determined in Appendix II of Ref. [START_REF] Ren | Octet baryon masses in next-to-next-to-next-to-leading order covariant baryon chiral perturbation theory[END_REF]. It is clear that the N 3 LO BChPT can describe reasonably well the QCDSF-UKQCD 0.8 0.9

M D /X ∆ M π 2 /X π 2 N 3 LO-II Fitted N s =32 N s =24
Expt.

Figure 7.3:

The QCDSF-UKQCD lattice data [START_REF] Bietenholz | Tuning the strange quark mass in lattice simulations[END_REF] in comparison with the N 3 LO BChPT. The lattice data denoted by the blue filled squares are included in the fit; those by the green opened circles (with N s = 32) and the red diamonds (with N s = 24) are not. FVCs of the lattice results have been subtracted. The two flavour singlet quantities, X π and X ∆ , are defined as:

X π = (M 2 π + 2M 2 K )/3, X ∆ = (2m ∆ + m Ω -)/3
, respectively [START_REF] Bietenholz | Tuning the strange quark mass in lattice simulations[END_REF].

data obtained in both large (N s = 32) and small (N s = 24) volumes with both heavy and light pion masses. However, it should be pointed out that the ratio method eliminates to a large extent the FVCs. In other words, to plot/study the data this way one can neglect FVCs, as noticed in Ref. [START_REF] Bietenholz | Flavour blindness and patterns of flavour symmetry breaking in lattice simulations of up, down and strange quarks[END_REF].

We would like to point out that in the above fits we have not included the LHPC data, while in Refs. [START_REF] Ren | Octet baryon masses in next-to-next-to-next-to-leading order covariant baryon chiral perturbation theory[END_REF][START_REF] Ren | Virtual decuplet effects on octet baryon masses in covariant baryon chiral perturbation theory[END_REF] we have studied their data for octet baryon masses. The reason is that the LHPC decuplet baryon data do not seem to be consistent with those of the PACS-CS, QCDSF-UKQCD, and HSC collaborations. This is clearly demonstrated in Fig. 7.4, where the LHPC data are contrasted with the N 3 LO BChPT with the LECs of N 3 LO-II tabulated in Table I, and the corresponding kaon mass is fixed using M 2 K = a+bM 2 π with a and b determined in Ref. [START_REF] Ren | Octet baryon masses in next-to-next-to-next-to-leading order covariant baryon chiral perturbation theory[END_REF]. It is clear that the dependence of lattice data on M 2 π seems to be flatter than suggested by N 3 LO BChPT. In Ref. [START_REF] Walker-Loud | Light hadron spectroscopy using domain wall valence quarks on an Asqtad sea[END_REF], it was noticed that it is difficult to extrapolate lattice data to the physical ∆(1232) mass. Our study seems to confirm their finding. If we had included the LHPC data4 (three sets of them satisfying our selection criteria), we would have obtained a χ 2 /d.o.f. = 2.4.

Finite-volume corrections play an important role in describing LQCD data as pointed out in the present context in Refs. [START_REF] Geng | Finite-volume effects on octetbaryon masses in covariant baryon chiral perturbation theory[END_REF][START_REF] Ren | Octet baryon masses in next-to-next-to-next-to-leading order covariant baryon chiral perturbation theory[END_REF]. In Table 7.5, we show the finite-volume corrections calculated in N 3 LO BChPT with the LECs determined above. Not surprisingly, the finitevolume corrections to the QCDSF-UKQCD data are the largest, which can be easily understood from the arguments given above.

Furthermore, in order to quantify the effects of loop contributions involving virtual octet the phenomenological values we used above, which can be seen as evidence for the existence of non-analytical chiral contributions following the argument given in Ref. [START_REF] Walker-Loud | Evidence for non-analytic light quark mass dependence in the baryon spectrum[END_REF]. One should note that because of the small difference between the χ 2 /d.o.f. obtained here and the χ 2 /d.o.f. obtained by putting C and H to zero, this evidence is rather weak in the present case.

Convergence of SU(3) EOMS BChPT

Convergence of BChPT in the u, d, and s three-flavor sector has been under debate for many years. See, e.g., Refs. [START_REF] Bernard | Cutoff schemes in chiral perturbation theory and the quark mass expansion of the nucleon mass[END_REF][START_REF] Young | Convergence of chiral effective field theory[END_REF][START_REF] Geng | Recent developments in SU(3) covariant baryon chiral perturbation theory[END_REF][START_REF] Donoghue | SU(3) baryon chiral perturbation theory and long distance regularization[END_REF] and references cited therein. 5 One prominent example is the magnetic moments of octet baryons. In Ref. [START_REF] Geng | Leading SU(3)breaking corrections to the baryon magnetic moments in Chiral Perturbation Theory[END_REF], it has been shown that compared to HB ChPT and IR BChPT, EOMS BChPT converges relatively faster. The same has been found for octet baryon masses [START_REF] Martin Camalich | The lowest-lying baryon masses in covariant SU(3)-flavor chiral perturbation theory[END_REF]. Nevertheless, even in EOMS BChPT, convergence is relatively slow because of the large expansion parameter, M K /Λ ChPT . Naively, each higher order contribution is only suppressed by about one half at the physical point, which can even be further reduced for LQCD simulations with larger light-quark masses. In the following, we would like to examine the contributions of different chiral orders. In Table 7. 4 should expect a smaller χ 2 /d.o.f. at NNLO and N 3 LO. This is indeed the case. Since the NLO results already describe LQCD data very well, it is not surprising that up to NNLO and N 3 LO, there should be some reshuffling of contributions of different order. This can be clearly seen from Table 7.6, where contributions of different chiral order to decuplet baryon masses at the physical point are tabulated. Once loop diagrams are included, a naive comparison of p 0 (m D ), p 2 , p 3 , and p 4 contributions turns out to be a disaster. At NNLO, the p 2 contributions can be a factor of 2 larger than m D , while at N 3 LO, the p 3 and p 4 contributions are opposite and become comparable to or even larger than the p 2 contributions, particularly for the decuplet baryons containing strangeness.

On the other hand, up to one-loop level, it might be more proper to judge convergence by comparing tree and loop contributions. In Figs. π . At NNLO, the p 3 contributions can reach about 50% of the tree contributions, while at N 3 LO the loop contributions become about 10% ∼ 20% of the tree contributions. These results suggest that the chiral expansions are convergent as expected.

Pion-and strangeness-decuplet baryon sigma terms

The baryon sigma terms are important quantities in understanding the chiral condensate and the composition of the baryons. At present, there are no direct LQCD simulation of these quantities for decuplet baryons (In Ref. [START_REF] Alexandrou | Sigma-terms and axial charges for hyperons and charmed baryons[END_REF], the very preliminary studies of the decuplet baryon sigma terms was reported.). On the other hand, one can calculate the decuplet baryon sigma terms σ πD and σ sD using BChPT, once the relevant LECs are fixed, via the Feynman-Hellmann theorem. See, e.g., Refs. [START_REF] Ren | Octet baryon masses in next-to-next-to-next-to-leading order covariant baryon chiral perturbation theory[END_REF] for relevant formulas.

Using the LECs given in Table 7.4, we calculate the sigma terms of the baryon decuplet at the physical point, and the results are listed in Table 7.7. For comparison, we also tabulate the results of Refs. [START_REF] Martin Camalich | The lowest-lying baryon masses in covariant SU(3)-flavor chiral perturbation theory[END_REF][START_REF] Semke | Strangeness in the baryon ground states[END_REF]. The difference between our O(p 3 ) predictions with those of Ref. [START_REF] Martin Camalich | The lowest-lying baryon masses in covariant SU(3)-flavor chiral perturbation theory[END_REF] reflects the influence of the LQCD data and the fitting strategy. However, our N 3 LO results are consistent with those of Ref. [START_REF] Semke | Strangeness in the baryon ground states[END_REF] within uncertainties. sum rule, which can be understood if they act as chiral partners in a reducible representation of the chiral symmetry group SU (2) L × SU (2) R [START_REF] Beane | The Role of the Roper in QCD[END_REF].

Much work has been carried out in ChPT including explicitly pions, nucleons and Deltas -for a comprehensive review, see for example Ref. [START_REF] Bernard | Chiral Perturbation Theory and Baryon Properties[END_REF]. In contrast, the next nucleon excited state, the Roper, has received considerably less attention. The Roper-nucleon splitting is still somewhat smaller than Λ χ at the physical pion mass, and could decrease towards the chiral limit. Therefore, it can be included as an explicit degree of freedom in ChPT. Various quantities and processes have been studied from this point of view, including: Roper octet contributions to decuplet masses and magnetic moments [START_REF] Banerjee | The Decuplet revisited in χPT[END_REF]; virtual Roper contributions to πN → ππN threshold amplitudes [START_REF] Bernard | The Reaction πN → ππN at threshold in chiral perturbation theory[END_REF] and to elastic πN scattering [START_REF] Long | The Role of the Roper in Chiral Perturbation Theory[END_REF][START_REF] Datta | Pion -nucleon phase shifts in heavy baryon chiral perturbation theory[END_REF]; chiral corrections to the Roper self-energy [START_REF] Borasoy | Chiral corrections to the Roper mass[END_REF][START_REF] Djukanovic | Chiral structure of the Roper resonance using complex-mass scheme[END_REF]; and Roper magnetic moment [START_REF] Bauer | Magnetic moment of the Roper resonance[END_REF]. The Roper has also been included indirectly via resonance saturation of LECs in the calculations of baryon self-energies [START_REF] Borasoy | Chiral Expansion of Baryon Masses and σ-Terms[END_REF] and of nonleptonic hyperon decays [START_REF] Borasoy | Resonances in radiative hyperon decays[END_REF], and explicitly in the chiral unitarity approach to study meson-baryon dynamics [START_REF] Meissner | Chiral unitary meson baryon dynamics in the presence of resonances: Elastic pion nucleon scattering[END_REF].

Perhaps the most noticeable effect of the Roper in ChPT is, not surprisingly, in the reaction that led to its discovery [START_REF] Roper | Evidence for a P-11 Pion-Nucleon Resonance at 556 MeV[END_REF], elastic πN scattering. In ChPT, whether without [START_REF] Mojzis | Elastic pi N scattering to O(p 3 ) in heavy baryon chiral perturbation theory[END_REF][START_REF] Fettes | Pion -nucleon scattering in chiral perturbation theory. 1. Isospin symmetric case[END_REF][START_REF] Fettes | Pion nucleon scattering in chiral perturbation theory. 2.: Fourth order calculation[END_REF][START_REF] Fettes | Complete analysis of pion nucleon scattering in chiral perturbation theory to third order[END_REF][START_REF] Becher | Low energy analysis of πN → πN[END_REF][START_REF] Alarcon | πN scattering in relativistic baryon chiral perturbation theory revisited[END_REF] or with [START_REF] Ellis | Pion nucleon scattering in a new approach to chiral perturbation theory[END_REF][START_REF] Datta | Pion -nucleon phase shifts in heavy baryon chiral perturbation theory[END_REF][START_REF] Fettes | Pion -nucleon scattering in an effective chiral field theory with explicit spin 3/2 fields[END_REF][START_REF] Torikoshi | Low-energy pion nucleon scattering in the heavy baryon and infrared schemes[END_REF][START_REF] Long | pi N Scattering in the Delta(1232) Region in an Effective Field Theory[END_REF] an explicit Delta, the empirical near-threshold behavior of the P 11 phase shift is reproduced in lowest orders, with a monotonically decreasing function of energy. Yet the empirical phase shift turns up and crosses zero in the Delta region, which can only be achieved if a nominally higher-order effect provides an opposite contribution to cancels those of lowest orders. This lack of convergence is not a problem when the Delta region is considered beyond the range of EFT, but needs to be addressed as we extend this range [START_REF] Long | pi N Scattering in the Delta(1232) Region in an Effective Field Theory[END_REF]. Once the Roper is included explicitly as a low-energy degree of freedom, Roper pole diagrams, suppressed only by the relatively small energy difference δ ∆N -δ RN , appear already at the lowest non-trivial order and naturally produce the observed phase-shift behavior [START_REF] Long | The Role of the Roper in Chiral Perturbation Theory[END_REF].

The goal of the present work is to initiate a systematic study of nucleon, Delta and Roper properties within ChPT. As LQCD continues to pick up momentum, we can anticipate increasingly precise data for a variety of baryon properties at pion masses from Λ χ down to the physical value, and perhaps below. Over such a broad range, the simplest approach is to treat the baryon mass differences in the same footing as the pion mass, and consider momenta Q ∼ m π ∼ δ ∆N ∼ δ RN . We express observables as expansions in Q/Λ χ . In this way, we generalize the simpler countings of ChPT without [START_REF] Jenkins | Baryon chiral perturbation theory using a heavy fermion Lagrangian[END_REF] and with [START_REF] Jenkins | Chiral corrections to the baryon axial currents[END_REF] Deltas, where δ ∆N and/or δ RN are considered high scales and the effects of Deltas and/or Ropers are relegated to LECs.

As a first step, we reconsider the Roper self-energy [START_REF] Borasoy | Chiral corrections to the Roper mass[END_REF][START_REF] Djukanovic | Chiral structure of the Roper resonance using complex-mass scheme[END_REF] to next-to-next-to-leading order including explicitly the pion-mass dependence of the mixing between nucleon and Roper. We also calculate the virtual Roper effects on the nucleon and Delta self-energies to the same order. Thus, we obtain the full one-loop corrections to the nucleon mass, and Delta and Roper masses and widths. Many LQCD studies exist already of nucleon and Delta masses, indicating a relatively good convergence of ChPT, see for example Ref. [START_REF] Alexandrou | Light baryon masses with dynamical twisted mass fermions[END_REF]. We hope our results will prove useful in similar analysis of upcoming data.

Chiral effective Lagrangians in SU(2) sector

In this section, we present the effective Lagrangian relevant for the chiral corrections to the nucleon, Delta, and Roper self-energies up to NNLO. Further background and references can be found in Ref. [START_REF] Bernard | Chiral Perturbation Theory and Baryon Properties[END_REF].

The nonrelativistic nature of nucleons in the regime of validity of the EFT means that we can integrate out antinucleons and limit ourselves to interactions with at most one baryon. We can split the effective Lagrangian as

L = L π + L πN + L πN ∆ + L πN R + L π∆R , (8.1)
which describe, respectively, interactions among pions, between pions and nucleons, of Deltas with pions and nucleons, of Ropers with pions and nucleons, and of Ropers with pions and Deltas. We do not consider electromagnetic effects nor isospin violation stemming from the quark mass difference, but extension is straightforward.

The effective Lagrangian can be ordered,

L = n ChPT L (n ChPT ) , (8.2)
according to the (integer) chiral index [START_REF] Weinberg | Phenomenological Lagrangians[END_REF][START_REF] Weinberg | Effective chiral Lagrangians for nucleon -pion interactions and nuclear forces[END_REF]

n ChPT = d + f /2 -2 ≥ 0, (8.3)
where d and f are, respectively, the number of derivatives and low-energy scales, and the number of fermion fields entering each interaction. In the mesonic and one-baryon sectors 1 the chiral index tracks the number of powers of Λ -1 χ expected from the naive dimensional analysis (NDA) [START_REF] Manohar | Chiral Quarks and the Nonrelativistic Quark Model[END_REF] that underlies the EFT expansion. It exhibits the lowest order where the corresponding interaction appears. The lower bound in Eq. (8.3) stems from the pattern of chiral-symmetry breaking in QCD, and guarantees that there is a leading order from which the expansion can be constructed.

In the next subsections we describe each term in Eq. (8.1), making specific choices of fields. At a given order, results can only depend on these choices up to higher-order terms, that is, results for different choices must fall within the theoretical error bars. The choice of fields is guided by the symmetries we want to implement, namely chiral symmetry and Lorentz invariance in the nonrelativistic regime.

Each interaction with symmetry transformation properties (i.e., invariant or transforming like the quark mass terms in the QCD Lagrangian) introduces a LEC. The LECs can be determined by fitting LQCD data at different values of m π , and particular combinations can be obtained from experimental data at the physical point. We assume that throughout the low-mass region we have two small expansion parameters δ N ∆ /Λ χ and δ N R /Λ χ in addition to m π /Λ χ . Thus all LECs, which are defined in the chiral limit (m π = 0), can be written as series in these parameters. The coefficients of these series cannot be determined from experiment or LQCD, except perhaps if QCD is deformed with the introduction of further independent parameters such as the number of colors. For simplicity we leave these series implicit.

Pion Lagrangian

We use an exponential realization of the pion fields (π 0 , π ± ),

u = exp i Φ 2f π , Φ = τ a π a = π 0 √ 2π + √ 2π --π 0 , (8.4)
where τ a is the isospin Pauli matrix and f π = 92.1 MeV [START_REF] Olive | Review of Particle Physics[END_REF] is the pion decay constant in the chiral limit. The chiral lagrangian is built from covariant combinations of pion fields such as the pion vector and axial currents

Γ µ = 1 2 u † ∂ µ u + u∂ µ u † = 1 2 τ a τ a Γ µ , u µ = i u † ∂ µ u -u∂ µ u † = 1 2 τ a τ a u µ , (8.5)
and

χ ± = u † χu † ± uχ † u, (8.6)
where • • • denotes the trace in flavor space, and χ = 2B 0 M, M = diag(m u , m d ), represents the explicit breaking of chiral symmetry by the up-and down quark masses m u and m d . Here B 0 = -qq /f 2 π is related to the quark condensate qq 0 in the chiral limit. From the pion Lagrangian, we need only the n ChPT = 0 part, which can be written as

L (0) π = f 2 π 4 [ u µ u µ + χ + ] . (8.7)
The first term includes the pion kinetic energy while the second gives rise to the pion mass,

m 2 π = 1 4 (χ + χ † ) . (8.8)
In the following only the pion propagator obtained from Eq. (8.7) appears.

Nucleon Chiral Lagrangian

Since the nucleon mass in the chiral limit M N 0 is of the order of the breakdown scale of the EFT, we use a heavy nucleon field

N = p n (8.9)
which does not have a trivial evolution factor exp(-iM N 0 v •x), where v µ is the nucleon velocity (v µ = (1, 0) in the rest frame). This amounts to measuring nucleon energies with respect to the nucleon rest energy in the chiral limit, M N 0 . With this choice of field, all spin physics enters through the covariant spin-operator S µ = iγ 5 σ µν v ν /2, with the basic properties

S • v = 0, {S µ , S ν } = 1 2 (v µ v ν -g µν ), [S µ , S ν ] = iε µναβ v α S β , (8.10)
in the convention ε 0123 = 1. Lorentz invariance at a given order is ensured by 1/M N 0 corrections introduced via reparametrization invariance [START_REF] Luke | Reparametrization invariance constraints on heavy particle effective field theories[END_REF]. Interactions with pions are written with the chiral covariant derivative

D µ N = ∂ µ N + Γ µ N. (8.11)
The pion-nucleon effective Lagrangian with lowest chiral index can be written as

L (0) πN = N (iv • D + g A S • u)N, (8.12)
where g A = O(1) is the axial vector coupling at the chiral limit. This Lagragian gives rise to a static nucleon propagator and the usual derivative pion-nucleon coupling. At the next order we have recoil corrections, the sigma term, and new interactions, L

πN = 1 M N 0 N (v • D) 2 -D 2 -ig A {S • D, v • u} N +c N 1 χ + N N -c N 2 (v • u) 2 N N + c N 3 2 u • u N N + • • • , ( (1)
) N N + • • • = e N 1 m 4 π N N + • • • , (8.15)
where e N 1 = (16e 38 + 2e 115 + 2e 116 ) is a combination of the three LECs e 38 , e 115 , and e 116 of Ref. [START_REF] Fettes | The Chiral effective pion nucleon Lagrangian of order p 4[END_REF]. We expect e N 1 = O(1/Λ 3 χ ).

Delta Chiral Lagrangian

The Delta is a spin-3/2, isospin-3/2 resonance and, therefore, it is described by a field

∆ µ =      ∆ ++ µ ∆ + µ ∆ 0 µ ∆ - µ      (8.16)
where each charge state has four spin components. In order to construct isospin conserving interactions we need the isospin 3/2 analogs of the Pauli matrices,

T 1 = 2 3      0 √ 3/2 0 0 √ 3/2 0 1 0 0 1 0 √ 3/2 0 0 √ 3/2 0      , (8.17
)

T 2 = 2i 3      0 - √ 3/2 0 0 √ 3/2 0 -1 0 0 1 0 - √ 3/2 0 0 √ 3/2 0      , T 3 =      1 0 0 0 0 1/3 0 0 0 0 -1/3 0 0 0 0 -1      , (8.18)
2 Sometimes further terms, such as

L (2) πN = N (B32δ 3 ∆N + B23δ∆N χ+ )N, (8.14)
are written explicitly [START_REF] Bernard | Chiral extrapolations and the covariant small scale expansion[END_REF]. Here they are absorbed in the LECs MN0 and cN1 in the ciral limit.

with T a T a = 5 3 , as well as the isospin 1/2-to-3/2 transition matrices,

t 1 = 1 √ 6 - √ 3 0 1 0 0 -1 0 √ 3 , t 2 = -i √ 6 √ 3 0 1 0 0 1 0 √ 3 , t 3 = 2 3 0 0 1 0 0 0 1 0 , (8.19) with t a t b † = δ ab -1
3 τ a τ b . Since Delta excitation requires only δ ∆N in energy, we choose the phase of the field so that the same factor M N 0 v • x is absent. As a consequence, Delta energies are also measured with respect to the nucleon's. The Delta is then also a static field, but with a residual mass.

The chiral Lagrangian for the Delta and its transition to the nucleon has a parallel form to the Lagrangian of the previous subsection. The chiral covariant derivative on Delta field is written as

D ν ∆ µ = ∂ ν ∆ µ + T a τ a Γ ν ∆ µ . (8.20)
At lowest index,

L (0) πN ∆ = -∆µ (iv • D -δ N ∆ )∆ µ -H A ∆µ S ν T a τ a u ν ∆ µ - 1 4 h A ∆µ t a τ a u µ N + H.c. . (8.21)
The first term provides the Delta propagator

S µν (k) = - i v • k -δ N ∆ + iε v µ v ν -g µν - 4 3 S µ S ν , (8.22)
where δ N ∆ = M ∆0 -M N 0 is the mass splitting between Delta and nucleon in the chiral limit. The LECs H A , h A = O(1) are axial vector couplings of the Delta and of the Delta-nucleon transition, respectively. At next order we again need recoil corrections and seagull vertices, L

πN ∆ = - 1 M N 0 ∆µ (v • D) 2 -D 2 ∆ µ + iH A M N 0 ∆µ {S • D, v ν T a τ a u ν }∆ µ + h A 4M N 0 i ∆µ ← - D µ v ν t a τ a u ν N + H.c. -∆µ c ∆1 χ + -c ∆2 (v • u) 2 + c ∆3 2 u 2 ∆ µ + • • • , (1)
where c ∆1 , c ∆ 2 , c ∆3 = O(1/Λ χ ) are LECs 3 . As in the nucleon sector, of higher orders we need only the m 4 π correction to the baryon mass, L

πN ∆ = -e ∆1 m 4 π ∆µ ∆ µ + • • • , (3)
with e ∆1 = O(1/Λ 3 χ ) yet another LEC.

3 Note that in covariant ChPT there are two more terms [START_REF] Geng | Nucleon-to-Delta axial transition form factors in relativistic baryon chiral perturbation theory[END_REF] in L

πN ∆ which are relevant in the study of baryon masses up to NNLO [207,[START_REF] Bernard | Chiral extrapolations and the covariant small scale expansion[END_REF],

L (1) πN ∆ = -ib3 N T a ω a,µν γµ∆ν - b6 M∆ N T a (∂µ∆ν )ω a,µν + H.c., (8.24)
with ω a µν = 1 2 τ a [Dµ, uν ] . However these terms are redundant in the heavy baryon expansion [START_REF] Long | Heavy-particle formalism with Foldy-Wouthuysen representation[END_REF][START_REF] Siemens | The reaction πN → ππN in chiral effective field theory with explicit ∆(1232) degrees of freedom[END_REF].

Roper Chiral Lagrangian

The Roper has the same spin/isospin numbers as the nucleon, just a higher mass. The machinery we employ to build the interactions of its field

R = R + R 0 (8.26)
is the same as for the nucleon. The only differences are that i) it has a residual mass in the chiral limit, and ii) it can mix with the nucleon with a mixing angle that depends on the quark masses.

We define the nucleon and Roper fields such that they are mass eigenstates in the chiral limit. Then we can simply add to the LO nucleon Lagrangian Eq. (8.12) the lowest-order Roper terms

L (0) πN R = R(iv • D -δ RN + g R S • u)R + g N R N S • uR + h.c. , (8.27)
where δ RN = M R0 -M N 0 denotes the mass splitting between nucleon and Roper in the chiral limit, g R = O(1) is the axial-vector coupling of the Roper, and g N R = O(1) the axial-vector transition coupling between Roper and nucleon. Terms at next order have a form familiar from Eq. (8.13), L

R = 1 M N 0 R (v • D) 2 -D 2 -ig R {S • D, v • u} R +c R1 χ + RR -c R2 (v • u) 2 RR + c R3 2 u • u RR (8.28) +c N R χ + N R + h.c. - ig N R M N 0 R{S • D, v • u}N + h.c. + • • • , (1) πN
where c R1 , c R 2 , c R3 = O(1/Λ χ ) are LECs to be fixed. Noteworthy is the Roper-nucleon mixing linear in the quark masses with LEC c N R = O(1/Λ χ ), which we kep here explicit in contrast to what is done in Ref. [START_REF] Borasoy | Chiral corrections to the Roper mass[END_REF]. When one is interested in Roper effects at a fixed pion mass, for example at its physical value (e.g. Refs. [START_REF] Long | The Role of the Roper in Chiral Perturbation Theory[END_REF][START_REF] Bernard | The Reaction πN → ππN at threshold in chiral perturbation theory[END_REF][START_REF] Datta | Pion -nucleon phase shifts in heavy baryon chiral perturbation theory[END_REF]), we can perform a new rotation to re-diagonalize the fields at that mass and eliminate c N R . Here we keep track of its effects as the pion mass is varied. As before, the only higher-order term we need is

L (3) πN R = e R1 m 4 π RR + . . . . (8.30)
with e R1 = O(1/Λ 3 χ ) another LEC. In addition we need the Roper-Delta transitions at the two lowest orders, .31) and

L (0) π∆R = - 1 4 g ∆R ∆µ t a τ a u µ R + H.c. ( 8
L (1) π∆R = ig ∆R 4M N 0 ∆µ ← - D µ v ν t a τ a u ν R + H.c. , (8.32)
where g ∆R = O(1) is the axial-vector transition coupling.

Baryon mass and width matrices

Perturbatively calculation of the baryon self-energies

Firstly, let us consider the two-point function of the baryon field B(x)

S 0 (x) = -i 0|T [B(x) B(0)]|0 = 1 p 0 -δ -Σ(p 0 ) , (8.33)
where the momentum matrix p 0 , the mass splitting matrix δ and the self-energy matrix Σ(p 0 ), respectively, can be written as

p 0 =    p 0 N N 0 p 0 N R 0 p 0 ∆∆ 0 p 0 RN 0 p 0 RR    , δ =    0 0 0 0 δ N ∆ 0 0 0 δ N R    , (8.34)
Σ(p 0 ) =    Σ N N (p 0 N N ) 0 Σ N R (p 0 N R ) 0 Σ ∆∆ (p 0 ∆∆ ) 0 Σ RN (p 0 RN ) 0 Σ RR (p 0 RR )    , (8.35)
where the elements of self-energy matrix, Σ BB (p 0 BB ) with B, B = N, ∆, R, can be expanded up to fourth order

Σ BB (p 0 BB ) = Σ (2)
BB + Σ

BB (p 0 BB ) + Σ

BB (p 0 BB ) + • • • . (4)
The baryon masses are defined as M B = M N 0 + Re[p 0 ] at the pole with p 0 = δ + Σ(p 0 ). And the corresponding widths are defined as Γ B = -2Im[Σ(p 0 )]. In order to obtain the baryon masses and widths, one has to calculate the matrix equation perturbatively in terms of the expansion of p 0 , p 0 = p 0 (1) + p 0 (2) + p 0 (3) + p 0 (4) + • • • , (8.37)

with p 0 (1) = δ =    0 0 0 0 δ N ∆ 0 0 0 δ N R    , p 0 (2) =    Σ (2) N N 0 Σ (2) N R 0 Σ (2) ∆∆ 0 Σ (2) RN 0 Σ (2) RR    , (8.38)
p 0 (3) =    Σ (3) N N (0) 0 Σ (3) N R (0) 0 Σ (3) ∆∆ (δ N ∆ ) 0 Σ (3) RN (0) 0 Σ (3) RR (δ N R )    , (8.39)
and

p 0 (4) =     Σ (2) N N Σ (3) N N (0) 0 Σ (2) N R Σ (3) N R (0) 0 Σ (2) ∆∆ Σ (3) ∆∆ (δ N ∆ ) 0 Σ (2)
RN Σ

(3)

RN (0) 0 Σ (2)
RR Σ

(3)

RR (δ N R )     +    Σ (4) N N (0) 0 Σ (4) N R (0) 0 Σ (4) ∆∆ (δ N ∆ ) 0 Σ (4) RN (0) 0 Σ (4) RR (δ N R )   

Diagonalization of the baryon mass and width matrices

In order to obtain the masses and widths of nucleon, Delta, and Roper, one should diagonalize the above two matrices [Eq. (8.41) and Eq. (8.45)] simultaneously. We encode the baryon mass and width matrices into a single matrix

√ W -i Γ RR 2 , (8.52)
with

W = (M N N -M RR ) 2 + 4M N R M RN + i(M N N -M RR )Γ RR - Γ 2 RR 4 . (8.53)
The corresponding three eigenvectors after normalization are

V N =       -|M RN | M RN √ W-M N N +M RR -i Γ RR 2 4M 2 RN +( √ W-M N N +M RR -i Γ RR 2 ) 2 0 2|M RN | 4M 2 RN +( √ W-M N N +M RR -i Γ RR 2 ) 2       , V ∆ =    0 1 0    , (8.54)
V R =       |M RN | M RN √ W+M N N -M RR +i Γ RR 2 4M 2 RN +( √ W+M N N -M RR +i Γ RR 2 ) 2 0 2|M RN | 4M 2 RN +( √ W+M N N -M RR +i Γ RR 2 ) 2       . (8.55) F(x) = 1 4 x 2 + 1 2 (1 - 3 2 x 2 ) log x 2 4 +        (1 -x 2 ) 3/2 log 1 x + 1 x 2 -1 , 0 ≤ x < 1, (x 2 -1) 3/2 arccos 1 x , |x| ≥ 1, -(1 -x 2 ) 3/2 iπ -log -1 x + 1 x 2 -1 , -1 < x < 0. (8.62)
The evolution of F(x) is illustrated in Fig. 8.2. There are six different values of F(x) with x = ±m π /δ (δ = δ N ∆ , δ = δ ∆R and δ = δ N R ) at the physical point are presented. We also investigate the limit behavior of F(x): when x → 0, the loop function F(x) → 0 and when x → ∞, the loop function F(x) → π 2 x 3 . Up to NNLO, the needed one loop diagrams are collected in Fig. 8.3, and the corresponding chiral correction to the nucleon self-energy can be expressed as

Σ(4) N N (0) = -e N 1 m 4 π + 3 64π 2 f 2 π (8c N 1 -c N 2 -4c N 3 )m 4 π log m 2 π µ 2 + 3c N 2 128π 2 f 2 π m 4 π - 3 4(4πf π ) 2 g 2 A M N 0 log m 2 π µ 2 + 1 m 4 π + h 2 A 4π 2 f 2 π (c N 1 -c ∆1 ) m 2 π δ 2 N ∆ J m π δ N ∆ , δ N ∆ µ + m 4 π - 5 8(4πf π ) 2 h 2 A M N 0 log m 2 π µ 2 + 9 10 m 4 π - h 2 A 24π 2 f 2 π δ 4 N ∆ M N 0 F m π δ N ∆ -4 -2 0 2 4 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 F(x) x Re[F(x)] Im[F(x)]
+ 9g 2 N R (4πf π ) 2 (c N 1 -c R1 ) m 2 π δ 2 N R J m π δ N R , δ N R µ + 2 3 m 4 π - 3 4(4πf π ) 2 g 2 N R M N 0 log m 2 π µ 2 + 1 m 4 π - 9g 2 N R 2(4πf π ) 2 δ 4 N R M N 0 F m π δ N R - 3 4π 2 f 2 π c N R g A g N R m 2 π δ 2 N R H m π δ N R , δ N R µ + m 4 π , (8.63)
where the function, J (x, y), from loop diagrams Fig. 8.3(a,b,c), is defined as

J (x, y) = x 2 log(x 2 y 2 ) + 2 log 4 x 2 -        4 √ 1 -x 2 log 1 x + 1 x 2 -1 , 0 < x < 1, (-4) √ x 2 -1 arccos 1 x , |x| ≥ 1, (-4) √ 1 -x 2 iπ -log -1 x + 1 x 2 -1 , -1 < x < 0. (8.64)
The asymptotic behavior of J (x, y) is: J (x, y) → -x 2 +x 2 log(4y 2 ) when x → 0 and J (x, y) → x 2 log(x 2 y 2 ) when x → ∞. Due to the mixing between nucleon and Roper fields, two new loop diagrams [Fig. 8.3(d,e)] should be taken into account. The corresponding three-point function H(x, y) is defined as

H(x, y) = 3 2 x 2 log(x 2 y 2 ) + πx 3 + log 4 x 2 -        2(1 -x 2 ) 3/2 log 1 x + 1 x 2 -1 , 0 < x < 1, 2(x 2 -1) 3/2 arccos 1 x , |x| ≥ 1, -2(1 -x 2 ) 3/2 iπ -log -1 x + 1 x 2 -1 , -1 < x < 0.
(8.65)

Furthermore, the limit behavior is: H(x, y) → -1 2 x 2 + 3 2 x 2 log(4y 2 ) when x → 0 and H(x, y) →

∆∆ . Other notations are the same as in Fig. 8.1.

Σ (2) ∆∆ = -4c ∆1 m 2 π , (8.66)
Σ (3) ∆∆ (δ N ∆ ) = - 25H 2 A 864πf 2 π m 3 π + h 2 A 3(4πf π ) 2 δ 3 N ∆ F(- m π δ N ∆ ) - g 2 ∆R 3(4πf π ) 2 δ 3 ∆R F( m π δ ∆R ), (8.67)
Σ( 4)

∆∆ (δ N ∆ ) = -e ∆1 m 4 π + 3 64π 2 f 2 π (8c ∆1 -c ∆2 -4c ∆3 )m 4 π log m 2 π µ 2 + 3c ∆2 128π 2 f 2 π m 4 π - 25 96(4πf π ) 2 H 2 A M N 0 log m 2 π µ 2 + 19 10 m 4 π + h 2 A (4πf π ) 2 (c ∆1 -c N 1 )m 2 π δ 2 N ∆ J (- m π δ N ∆ , - δ N ∆ µ ) - 5 32(4πf π ) 2 h 2 A M N 0 log m 2 π µ 2 - 1 10 m 4 π - 5h 2 A 6(4πf π ) 2 δ 4 N ∆ M N 0 F(- m π δ N ∆ ) + g 2 ∆R (4πf π ) 2 (c ∆1 -c R1 )m 2 π δ 2 ∆R J ( m π δ ∆R , δ ∆R µ ) - 5 32(4πf π ) 2 g 2 ∆R M N 0 log m 2 π µ 2 - 1 10 m 4 π - 5g 2 ∆R 6(4πf π ) 2 δ 4 ∆R M N 0 F( m π δ ∆R ) - 2 (4πf π ) 2 c N R h A g ∆R G(m π , δ N ∆ , δ N R , µ) - 4 3 m 4 π , (8.68)
G(m π , δ N ∆ , δ ∆R , µ) = m 4 π log m 2 π µ 2 + 2 3 δ 3 N ∆ m 2 π δ N ∆ + δ ∆R log 4δ 2 N ∆ m 2 π + 2 3 m 2 π δ 3 ∆R δ N ∆ + δ ∆R log 4δ 2 ∆R m 2 π + 4 3 m 2 π δ N ∆ + δ ∆R                -(δ 2 N ∆ -m 2 π ) 3/2 log δ N ∆ mπ + δ 2 N ∆ m 2 π -1 , δ N ∆ > m π , (m 2 π -δ N ∆ ) 3/2 arccos -δ N ∆ mπ , |δ N ∆ | < m π , -(δ 2 N ∆ -m 2 π ) 3/2 iπ -log -δ N ∆ mπ + δ 2 N ∆ m 2 π -1 , δ N ∆ < -m π , - 4
m 2 π δ N ∆ + δ ∆R                -(δ 2 ∆R -m 2 π ) 3/2 log -δ ∆R mπ + δ 2 ∆R m 2 π -1 , δ ∆R > m π , (m 2 π -δ 2 ∆R ) 3/2 arccos δ ∆R mπ , |δ ∆R | < m π , -(δ 2 ∆R -m 2 π ) 3/2 iπ -log δ ∆R mπ + δ 2 ∆R m 2 π -1 , δ ∆R < -m π . ( 3

Diagonal terms of the Roper self-energy

Just the same as the case of the nucleon, all the loop diagrams that contribute to the selfenergy of Roper up to and including NNLO are presented in Figs. (8.6-8.7). The different order contributions to the Roper self-energy are

Σ (2) RR = -4c R1 m 2 π , (8.70)
Σ (3) RR (δ N R ) = - 3g 2 R 32πf 2 π m 3 π + 3g 2 N R (4πf π ) 2 δ 3 N R F(- m π δ N R ) + g 2 ∆R 12π 2 f 2 π δ 3 ∆R F(- m π δ ∆R ), (8.71)
Σ( 4) .72) Mixed terms between nucleon and Roper Because Roper, as the first excitation state of nucleon, has the same quantum numbers as nucleon, there exists mixing between nucleon and Roper which is allowed by chiral symmetry. According to the mass formulas of nucleon [Eq. (8.57)] and Roper [Eq. (8.59)], we only need the mixed contributions up to NLO, including Σ

RR (δ N R ) = -e R1 m 4 π + 3 64π 2 f 2 π (8c R1 -c R2 -4c R3 )m 4 π log m 2 π µ 2 + 3c R2 128π 2 f 2 π m 4 π - 3 4(4πf π ) 2 g 2 R M N 0 log m 2 π µ 2 + 1 m 4 π + 9g 2 N R (4πf π ) 2 (c R1 -c N 1 ) m 2 π δ 2 N R J (- m π δ N R , - δ N R µ ) + 2 3 m 4 π - 3 4(4πf π ) 2 g 2 N R M N 0 log m 2 π µ 2 + 1 m 4 π - 9g 2 N R 2(4πf π ) 2 δ 4 N R M N 0 F(- m π δ N R ) + g 2 ∆R 4π 2 f 2 π (c R1 -c ∆1 ) m 2 π δ 2 ∆R J (- m π δ ∆R , - δ ∆R µ ) + m 4 π - 5 8(4πf π ) 2 g 2 ∆R M N 0 log m 2 π µ 2 + 9 10 m 4 π - g 2 ∆R 24π 2 f 2 π δ 4 ∆R M N 0 F(- m π δ ∆R ) - 3 4π 2 f 2 π c N R g R g N R m 2 π δ 2 N R H(- m π δ N R , - δ N R µ ) + m 4 π . ( 8
(2) N R , Σ (2) RN , Σ (3)
N R , and Σ

RN . As illustrated in Fig. (8.8), the mixed term up to leading order is

= - 3g A g N R 32πf 2 π m 3 π - h A g ∆R 12π 2 f 2 π δ 3 N ∆ F( m π δ N ∆ ) - 3g N R g R (4πf π ) 2 δ 3 N R F( m π δ N R
).

(8.74)

Results and Discussion

Up to now, we have obtained the chiral expansions of the self-energies of nucleon, Delta, and Roper up to NNLO. Now, we can give explicit expressions of the baryon masses and widths. As a tentative application, we would like to simultaneously analyze the lattice data of nucleon, Delta, and Roper masses to explore the quark mass dependence of Roper mass and the nucleon-Roper mixing effects.

Nucleon, Delta and Roper masses

In terms of Eqs. (8.57-8.59), we could explicitly write out the expressions of nucleon, Delta and Roper masses at O(ε 4 ).

• The nucleon mass M N is

M N = M N 0 -4c N 1 m 2 π - 3g 2 A 32πf 2 π m 3 π -e N 1 - 16c 2 N R δ N R m 4 π + 1 128π 2 f 2 π 3c N 2 + 32(c N 1 -c ∆1 )h 2 A + 48(c N 1 -c R1 )g 2 N R -96c N R g A g N R m 4 π - 3 256π 2 f 2 π M N 0 (4g 2 A + 3h 2 A + 4g 2 N R )m 4 π - 3c N R g A g N R 4πf 2 π δ N R m 5 π - 64c 2 N R (c N 1 -c R1 ) δ 2 N R m 6 π + 1 128π 2 f 2 π 6(8c N 1 -c N 2 -4c N 3 ) - 1 M N 0 (6g 2 A + 5h 2 A + 6g 2 N R ) m 4 π log m 2 π µ 2 - 1 24π 2 f 2 π δ 3 N ∆ 2h 2 A + h 2 A δ N ∆ M N 0 + 16c N R h A g ∆R m 2 π δ N R F( m π δ N ∆ ) - 3 32π 2 f 2 π δ 3 N R 2g 2 N R + 3g 2 N R δ N R M N 0 + 16c N R g N R g R m 2 π δ N R F( m π δ N R ) + h 2 A 4π 2 f 2 π (c N 1 -c ∆1 )m 2 π δ 2 N ∆ J m π δ N ∆ , δ N ∆ µ + 9g 2 N R (4πf π ) 2 (c N 1 -c R1 )m 2 π δ 2 N R J m π δ N R , δ N R µ - 3g A g N R 4π 2 f 2 π c N R m 2 π δ 2 N R H m π δ N R , δ N R µ . (8.75)
• The Delta mass M ∆ is

M ∆ = M N 0 + δ N ∆ -4c ∆1 m 2 π - 25H 2 A 864πf 2 π m 3 π -e ∆1 m 4 π + 1 384π 2 f 2 π (9c ∆2 + 64c N R h A g ∆R )m 4 π - 1 3072π 2 f 2 π M N 0 95H 2 A -3(h 2 A + g 2 ∆R ) m 4 π + 1 1536π 2 f 2 π 72(8c ∆1 -c ∆2 -4c ∆3 ) - 5 M N 0 (5H 2 A + 3h 2 A + 3g 2 ∆R ) m 4 π log m 2 π µ 2 + h 2 A 6(4πf π ) 2 2δ 3 N ∆ - 5δ 4 N ∆ M N 0 F(- m π δ N ∆ ) - g 2 ∆R 6(4πf π ) 2 2δ 3 ∆R + 5δ 4 ∆R M N 0 F( m π δ ∆R ) + h 2 A (4πf π ) 2 (c ∆1 -c N 1 )m 2 π δ 2 N ∆ J (- m π δ N ∆ , - δ N ∆ µ ) + g 2 ∆R (4πf π ) 2 (c ∆1 -c R1 )m 2 π δ 2 ∆R J ( m π δ ∆R , δ ∆R µ ) - 2h A g ∆R (4πf π ) 2 c N R G(m π , δ N ∆ , δ N R , µ). (8.76) • The Roper mass M R is M R = M N 0 + δ N R -4c R1 m 2 π - 3g 2 R 32πf 2 π m 3 π -e R1 + 16c 2 N R δ N R m 4 π + 1 128π 2 f 2 π 3c R2 + 32(c R1 -c ∆1 )g 2 ∆R + 48(c R1 -c N 1 )g 2 N R -96c N R g R g N R m 4 π - 3 256π 2 f 2 π M N 0 (4g 2 R + 4g 2 N R + 3g 2 ∆R )m 4 π + 3c N R g A g N R 4πf 2 π δ N R m 5 π - 64c 2 N R (c R1 -c N 1 ) δ 2 N R m 6 π + 1 128π 2 f 2 π 6(8c R1 -c R2 -4c R3 ) - 1 M N 0 (6g 2 R + 6g 2 N R + 5g 2 ∆R ) m 4 π log m 2 π µ 2 + 3 32π 2 f 2 π δ 3 N R g 2 N R 2 - 3δ N R M N 0 F(- m π δ N R ) + 16c N R g R g N R m 2 π δ N R F( m π δ N R ) + 1 24π 2 f 2 π δ 3 ∆R g 2 ∆R 2 - δ ∆R M N 0 F(- m π δ ∆R ) + 16c N R h A g ∆R m 2 π δ N R F( m π δ N ∆ ) + 9g 2 N R (4πf π ) 2 (c R1 -c N 1 )m 2 π δ 2 N R J - m π δ N R , - δ N R µ + g 2 ∆R 4π 2 f 2 π (c R1 -c ∆1 )m 2 π δ 2 ∆R J - m π δ ∆R , - δ ∆R µ - 3g R g N R 4π 2 f 2 π c N R m 2 π δ 2 N R H - m π δ N R , - δ N R µ . (8.77)
We want to mention that, the chiral expansions of nucleon, Delta and Roper masses have been carried out in Refs. [START_REF] Bernard | Cutoff schemes in chiral perturbation theory and the quark mass expansion of the nucleon mass[END_REF][START_REF] Procura | Nucleon mass: From lattice QCD to the chiral limit[END_REF][START_REF] Bernard | Chiral extrapolations and the covariant small scale expansion[END_REF][START_REF] Tiburzi | Strong isospin breaking in the nucleon and Delta masses[END_REF][START_REF] Borasoy | Chiral corrections to the Roper mass[END_REF][START_REF] Djukanovic | Chiral structure of the Roper resonance using complex-mass scheme[END_REF][START_REF] Bernard | The Mass of the Delta resonance in a finite volume: fourth-order calculation[END_REF]. Our nucleon mass, without the virtual Delta and Roper contributions, is the same as Eq.( 11) of Ref. [START_REF] Bernard | Cutoff schemes in chiral perturbation theory and the quark mass expansion of the nucleon mass[END_REF] and Eq.(A.1) of Ref. [START_REF] Bernard | The Mass of the Delta resonance in a finite volume: fourth-order calculation[END_REF]. But, for the nucleon and Delta masses, even without the virtual Roper contributions, our results are different with Ref. [START_REF] Tiburzi | Strong isospin breaking in the nucleon and Delta masses[END_REF], which did not perform the expansion in δ ∆N and δ RN .

Delta and Roper widths

The width of resonance can be obtained by the imaginary of self-energy with Γ B = -2Im[Σ B ]. By using the Eq. (8.58) and Eq. (8.59), we obtain:

• The width of Delta Γ ∆ Γ ∆ = π 3 h 2 A (4πf π ) 2 (δ 2 N ∆ -m 2 π ) 3/2 2 - 5δ N ∆ M N 0 -8π h 2 A (4πf π ) 2 (c ∆1 -c N 1 )δ N ∆ m 2 π δ 2 N ∆ -m 2 π + 16π 3 h A g ∆R (4πf π ) 2 c N R m 2 π δ N R ((δ N ∆ -δ N R ) 2 -m 2 π ) 3/2 . (8.78) • The width of Roper Γ R Γ R = 3πg N R (4πf π ) 2 2g N R - 3g N R δ N R M N 0 + 16g R c N R m 2 π δ N R (δ 2 N R -m 2 π ) 3/2 + g 2 ∆R 12πf 2 π 2 - δ ∆R M N 0 (δ 2 ∆R -m 2 π ) 3/2 -24π 3g 2 N R (4πf π ) 2 (c R1 -c N 1 )m 2 π δ N R δ 2 N R -m 2 π -32π g 2 ∆R (4πf π ) 2 (c R1 -c ∆1 )m 2 π δ ∆R δ 2 ∆R -m 2 π . (8.79)
In the next subsection, we will employ the widths of Delta and Roper to determine/constrain the values of the related LECs.

Analysis of lattice data of nucleon, Delta and Roper masses

Due to the unstable Roper state with m π ≤ 500 MeV in LQCD, in principle, the lattice Roper mass is not suitable to be studied in ChPT. As illustration of our calculations, we would like to perform the chiral extrapolation of Roper mass by analyzing the lattice data.

However, because there is a large number of LECs [START_REF] Aoki | Physical Point Simulation in 2+1 Flavor Lattice QCD[END_REF] appearing in the nucleon, Delta, and Roper masses up to NNLO, we do not expect to determine all the LECs by just fitting the lattice baryon masses. In order to tentatively study the chiral extrapolation of Roper mass, we would like to utilize the NLO baryon masses, which contains 10 unknown LECs (m 0 , δ ∆N , ∆ RN , c N 1 , c ∆1 , c R1 , c N R , g N R , g ∆R , and g R ), to simultaneously study the lattice data of nucleon, Delta, and Roper masses and their experimental results. At present, there are many lattice simulations on the nucleon mass, such as the n f = 2 simulations from the ETM [START_REF] Alexandrou | Low-lying baryon spectrum with two dynamical twisted mass fermions[END_REF], Mainz [START_REF] Capitani | The nucleon axial charge from lattice QCD with controlled errors[END_REF], QCDSF [START_REF] Bali | Nucleon mass and sigma term from lattice QCD with two light fermion flavors[END_REF], and BGR [START_REF] Engel | QCD with Two Light Dynamical Chirally Improved Quarks: Baryons[END_REF] collaborations, and the n f = 2 + 1 simulations from the PACS-CS [START_REF] Aoki | 2+1 Flavor Lattice QCD toward the Physical Point[END_REF], LHPC [START_REF] Walker-Loud | Light hadron spectroscopy using domain wall valence quarks on an Asqtad sea[END_REF], HSC [START_REF] Lin | First results from 2+1 dynamical quark flavors on an anisotropic lattice: Light-hadron spectroscopy and setting the strange-quark mass[END_REF], RBC-UKQCD [START_REF] Jung | Nucleon mass and strange content from (2+1)-flavor Domain Wall Fermion[END_REF] and BMW [START_REF] Dürr | Ab-Initio Determination of Light Hadron Masses[END_REF] collaborations. For the lattice Delta masses, several collaborations have reported their results, the ETM [START_REF] Alexandrou | Low-lying baryon spectrum with two dynamical twisted mass fermions[END_REF] and BGR [START_REF] Engel | QCD with Two Light Dynamical Chirally Improved Quarks: Baryons[END_REF] collaborations with n f = 2, and the PACS-CS [START_REF] Aoki | 2+1 Flavor Lattice QCD toward the Physical Point[END_REF], HSC [START_REF] Engel | QCD with Two Light Dynamical Chirally Improved Quarks: Baryons[END_REF] collaborations with n f = 2 + 1. But the situation for the lattice Roper masses is unsatisfactory. As mentioned in Chapter 2, the lattice data of Roper masses from the CSSM [START_REF] Mahbub | Roper Resonance in 2+1 Flavor QCD[END_REF], JLab [START_REF] Edwards | Excited state baryon spectroscopy from lattice QCD[END_REF], BGR [START_REF] Engel | QCD with Two Light Dynamical Chirally Improved Quarks: Baryons[END_REF], Cyprus [START_REF] Alexandrou | Novel analysis method for excited states in lattice QCD: The nucleon case[END_REF] collaborations is large different with the results of χQCD [START_REF] Liu | The Roper Puzzle[END_REF] simulation. In the sense of the tentative study of lattice Roper masses, we could divide the lattice results of Roper masses into two groups: one group is from the CSSM, JLab, BGR, and Cyprus collaborations, another from χQCD Collaboration.

In the following, we perform a simultaneous fit of the nucleon, Delta, and Roper masses of the LQCD by using the chiral expansions of baryon masses at O(ε 3 ). The lattice data should fulfill m 2 π < 0.15 GeV 2 to ensure the validity of NLO ChPT. After selection, the number of lattice points is 25 for the nucleon, 6 for the Delta, and 6(4) from group one (two) for the Roper. We denote the lattice data points of nucleon, Delta, and lattice data of Roper masses from group one as Set-I. And, Set-II contains the same nucleon and Delta lattice data as Set-I and the lattice Roper masses from group two. The pion decay constant f π is taken its latest experimental result f π = 92.1 MeV [START_REF] Olive | Review of Particle Physics[END_REF]. The nucleon-Delta coupling h A is fixed as h A = 2.09 by using the Delta width. And, the nucleon-Roper and Delta-Roper couplings g N R and g ∆R are constrained by the Breit-Wigner width of Roper Γ R = 350 ± 100 MeV.

We perform a χ 2 fit to the lattice data and the physical baryon masses. The obtained LECs and χ 2 /d.o.f. from the best fits are tabulated in Table 8.1. Besides, the fit-χ 2 for nucleon, Delta, and Roper are also given. The two fit strategies, named as Fit-I and Fit-II, give essentially the same description of nucleon and Delta masses of LQCD. This is also illustrated in Fig. 8.10 and Fig. 8.11 of the chiral extrapolation of lattice data. Due to the large difference of the fitted Roper masses in Set-I and Set-II, we can see that the values of LECs related to Roper masses, e.g. δ RN , c R1 , are very different. In Fig. 8.12, we present the quark mass dependence of Roper masses from Set-I (left panel) and Set-II (right panel). As we expected, the lattice data from the CSSM, JLab, BGR, Cyprus collaborations present a large chiral-log effect, whereas the χQCD results give a mild quark mass dependence of Roper mass. In Fit-I, as anticipated, the Roper-nucleon mass difference in the chiral limit is relatively small, and comparable to the Delta-nucleon splitting. The value of c N R ∼ 0 denotes that the mixed effects of nucleon and Roper are very small at O(p 3 ). But, on the other hand, the sizable error of c N R also indicates the mixing effects should be carefully investigated when better lattice data becomes available. And, it would be interest to study its effects up to NNLO. Chapter 9

Conclusion

The origin of mass of ordinary matter is linked to the origin of nucleon masses. The components of a nucleon mass can be divided into two parts: one is the three valence quark masses from the Brout-Englert-Higgs mechanism, which can only provide 1% contributions, and the other is from the nonperturbative strong interaction, which accounts for ∼99% of the nucleon mass.

In this dissertation, we present an integration of chiral perturbation theory and lattice QCD to study lattice baryon masses and to understand the mechanism of the origin of mass.

In the SU(3) sector, we performed a systematic study of the lattice results of lowest-lying octet and decuplet baryon masses in covariant baryon chiral perturbation theory with the extended-on-mass-shell renormalization scheme.

• For lowest-lying octet baryons, we calculated the baryon self-energies up to next-to-nextto-next-to-leading order in the chiral expansion by using EOMS BChPT. The corresponding finite-volume corrections to the lattice data were evaluated up to O(p 4 ). By constructing the discretized BChPT for Wilson fermions, we obtained the finite lattice spacing discretization effects of lattice QCD. Then, we performed a systematic study of lattice baryon masses from the PACS-CS, LHPC, QCDSF-UKQCD, HSC, and NPLQCD collaborations by exploring the chiral extrapolation of baryon masses, and evaluating finite-volume and finite lattice spacing discretization effects of LQCD.

We confirmed that covariant BChPT in the three-flavor sector converges as expected, relatively slowly as dictated by M K /Λ ChPT but with clear improvement order by order, at least concerning the octet baryon masses. A successful simultaneous fit of all the latest 2+1 flavors LQCD simulations indicates that the lattice octet baryon masses are consistent with each other, though their setups are quite different. We also found the finite-volume corrections of LQCD are important. And, for the lattice discretization effects, our studies showed that the treatment of discretization effects as systematic uncertainties in the previous studies of the LQCD octet baryon masses seems to be justified. It was shown that the discretization effects on the octet baryon masses are less than 2% for lattice spacings up to 0.15 fm, in agreement with other LQCD studies. Furthermore, it is also shown that the contributions of virtual decuplet baryons affect little the light-quark mass dependence of the octet baryon masses, indicating that their effects can be embodied in the LECs of the octet only version. On the other hand, a slightly better description of FVCs can be achieved once virtual decuplet baryons are taken into account.

• Applying the obtained chiral formulas for the octet baryon masses at O(p 4 ), we have performed an accurate prediction of the octet baryon sigma terms by systematically analyzing the high statistic lattice baryon masses from PACS-CS, LHPC, QCDSF-UKQCD collaborations. We found σ πN = 55(1)(4) MeV and σ sN = 27(27)(4) MeV. Special attention was paid to uncertainties induced by the lattice scale setting method, which, however, were found to be small, in contrast with previous studies. Other uncertainties, such as those induced by truncating chiral expansions and variations of LECs were also studied in detail. In addition, we have used the strong-interaction isospin-splitting effects from the LQCD simulations to further constrain the relevant LECs. Our results indicate a small scalar strangeness content in the nucleon, consistent with the strangeness contribution to the proton spin and to the electromagnetic form factors of the nucleon.

• As a natural extension, we have also studied the lowest-lying decuplet baryon masses in covariant baryon chiral perturbation theory with the extended-on-mass-shell scheme up to next-to-next-to-next-to-leading order. Through a simultaneous fit of the n f = 2+1 lattice data from the PACS-CS, QCDSF-UKQCD, and HSC collaborations, the 14 unknown lowenergy constants are determined. In fitting the lattice data, finite-volume corrections are taken into account self-consistently. A χ 2 /d.o.f. = 0.23 is achieved for the eight sets of lattice data satisfying M 2 π < 0.25 GeV 2 and M φ L > 3.8. Our studies show that the chiral expansions are convergent as expected and the results of the PACS-CS, QCDSF-UKQCD, and HSC collaborations seem to be consistent with each other, but not those of the LHPC Collaboration which employs the hybrid action. We have predicted the sigma terms of the decuplet baryons by use of the Feynman-Hellmann theorem, which should be compared to the lattice data in the future. It should be noted that our present study suffers from the limited range of the LQCD data (in terms of the input parameters) and the rather large number of unknown low-energy constants. Future refined LQCD simulations with various light-quark and strange quark masses, lattice volume and lattice spacing will be extremely welcome to put covariant baryon chiral perturbation theory to a more stringent test than did in the present work.

From the above studies, we present a nice interplay of lattice QCD and chiral perturbation theory to study the lowest-lying baryon masses. By performing the chiral extrapolation of lattice baryon masses to the physical point, we could reproduce the baryon masses. This could confirm that the nonperturbative strong interaction can provide the 99% of nucleon (baryon) masses.

With the studies of excited states becoming popular in lattice simulations, we also explored the chiral corrections to the self-energies of Roper, the first excitation state of nucleon. We performed a systematic study of the nucleon, Delta and Roper masses and widths in an extension of chiral perturbation theory that includes the Delta-nucleon and Roper-nucleon mass differences as expansion degrees of freedom. The contributions due to the mixing between nucleon and Roper induced by explicit chiral symmetry breaking were taken into account explicitly. The virtual Roper effects on the nucleon and Delta masses were evaluated up to next-to-next-to-leading order, as well as the effects of the nucleon and Delta in the Roper mass and width.

Lattice QCD has been successful to calculate many physical properties of lowest-lying states. The ground-state hadron spectrum is well understood [START_REF] Fodor | Light Hadron Masses from Lattice QCD[END_REF], and the corresponding numerical artifacts from continuum extrapolation and finite-volume corrections are well controlled. Therefore, computing the spectrum of excited mesons and baryons in lattice QCD becomes interesting. At present, there are several lattice collaborations which performed the study of excited hadron spectrum, such as the CSSM Collaboration [START_REF] Mahbub | Isolating Excited States of the Nucleon in Lattice QCD[END_REF][START_REF] Mahbub | Ordering of Spin-1/2 Excitations of the Nucleon in Lattice QCD[END_REF][START_REF] Menadue | Isolating the Λ(1405) in Lattice QCD[END_REF], the BGR Collaboration [START_REF] Engel | QCD with Two Light Dynamical Chirally Improved Quarks: Baryons[END_REF][START_REF] Engel | Low-lying Λ baryons from the lattice[END_REF], the HSC Collaboration [START_REF] Bulava | Nucleon, ∆ and Ω excited states in N f = 2 + 1 lattice QCD[END_REF][START_REF] Liu | Excited and exotic charmonium spectroscopy from lattice QCD[END_REF], the Cyprus Collaboration [START_REF] Alexandrou | Novel analysis method for excited states in lattice QCD: The nucleon case[END_REF][START_REF] Alexandrou | Nucleon Excited States in N f =2 lattice QCD[END_REF]. But, such calculations are still very challenging. Although, in this dissertation, we have studied Roper mass in chiral perturbation theory, the corresponding lattice Roper mass is not very clear and not suitable to perform the chiral extrapolation. We believe that, in the future, when the lattice QCD studies of Roper mass are mature, we can then use the same ideas to perform a systematic study of Roper mass by including the finite-volume corrections and finite lattice spacing discretization effects. Furthermore, we also want to study the nucleon axial charge g A including the Roper effects and the Roper axial charge g R in chiral perturbation theory.

In this dissertation, the isospin symmetry between u and d quarks is always taken exactly with m u = m d ≡ m l in the lattice baryon masses and the chiral corrections to baryon selfenergies 1 . Although the difference between u, d quarks is very small, it plays important roles in the study of low-energy physics. If one understands the baryon mass splitting from isospin symmetry breaking, especially for the proton and neutron masses splitting, it would be very helpful to the study of β decay and to explore new physics [START_REF] González-Alonso | Isospin breaking in the nucleon mass and the sensitivity of β decays to new physics[END_REF]. More recently, with the advance of computer techniques and computational resources, the effects of isospin symmetry breaking are starting to be included in lattice simulations. There are several studies of nucleon and baryon masses combining QCD and QED [START_REF] Beane | Strong-isospin violation in the neutron-proton mass difference from fully-dynamical lattice QCD and PQQCD[END_REF][START_REF] Borsanyi | Isospin splittings in the light baryon octet from lattice QCD and QED[END_REF][START_REF] Blum | Electromagnetic mass splittings of the low lying hadrons and quark masses from 2+1 flavor lattice QCD+QED[END_REF][START_REF] Horsley | Isospin breaking in octet baryon mass splittings[END_REF][START_REF] Shanahan | Strong contribution to octet baryon mass splittings[END_REF][START_REF] De Divitiis | Leading isospin breaking effects on the lattice[END_REF][START_REF] Borsanyi | Ab initio calculation of the neutron-proton mass difference[END_REF][START_REF] Thomas | Electromagnetic Contribution to the Proton-Neutron Mass Splitting[END_REF][START_REF] Erben | Dispersive estimate of the electromagnetic charge symmetry violation in the octet baryon masses[END_REF]. In Ref.

[323], Davoudi and Savage have studied the electromagnetic corrections to the masses of meson, baryon and nuclei in a finite volume. In a near future, we would like to construct chiral effective Lagrangians including virtual photons, and study the electromagnetic corrections to the octet baryon masses in chiral perturbation theory.

Furthermore, studies of hadron resonance structure become a hot topic with the discovery of the exotic (X, Y , and Z) hadron states by experiments. Although chiral perturbation theory is a powerful theoretical tool to study the low-energy physics, its application energy region is around 1 GeV. Therefore, unitary chiral perturbation theory (UChPT) [START_REF] Kaiser | Chiral dynamics and the low-energy kaon -nucleon interaction[END_REF][START_REF] Oller | Chiral unitary approach to meson meson and mesonbaryon interactions and nuclear applications[END_REF], which employs unitary techniques to conventional chiral perturbation theory, enables the study of higherenergy regions and describes low-lying resonances. With my collaborators, we have studied five different subjects: (1) We studied for the first time the antineutrino-proton reactions to produce the Λ(1405) by using the UChPT, which generates two Λ(1405) poles [START_REF] Ren | Antineutrino induced Λ(1405) production off the proton[END_REF]; (2) We presented a new reaction η c (η c (2S)) → φK * K * to generate the h 1 (1830) by using UChPT, which can provide a suggestion for experimentalists to confirm the existence of h 1 (1830) state [START_REF] Ren | Test of the h 1 (1830) made of K * K * with the η c → φK * K * decay[END_REF]; (3) We calculated the interaction kernel up to O(p 2 ) to generate the lowest-lying axial-vector mesons in the UChPT to explore the effects of higher order contributions [START_REF] Zhou | Pseudoscalar meson and vector meson interactions and dynamically generated axial-vector mesons[END_REF]; (4) We calculated the energy levels of the KK * system in the f 1 (1285) channel in finite volume by utilizing the UChPT [START_REF] Geng | S-wave KK * interactions in a finite volume and the f 1 (1285)[END_REF]; (5) With the three-body resonances drawing much attention, we also studied the ρ, D * , D * three-body system by using the fixed-center approximation to the Faddeev equation in Ref. [START_REF] Bayar | States of ρD * D * with J = 3 within the fixed center approximation to the Faddeev equations[END_REF]. Recently, With the discovery of pentaquark states P c (4380) + and P c (4450) + by LHCb [START_REF] Aaij | Observation of J/ψp Resonances Consistent with Pentaquark States in Λ 0 b → J/ψKp Decays[END_REF], it would be interesting to explore the nature and the properties of pentaquark states with those methods. lattices. The resulting pion masses range from 229 MeV to 449 MeV. In the simulations, they kept the singlet quark mass fixed and tuned the quark masses to ensure that the kaon always has a mass less than the physical one. It should be noted that in Table A.4 we did not tabulate the 16 3 × 32 and the 32 3 × 64 three-flavor simulation results, which have meson masses out of the range specified above. √ 3D cos(ε) -3F sin(ε)

2 Σ 0 4F 2 2 3 D 2 D 2 + F 2 D 2 + F 2 2 3 D 2 Σ -

σ πN = 55 ( 1 )( 4 ) 16 4. 1 56 5. 1 65 6. 1 70 6. 3 71 7. 1

 5514161561651703711 MeV et le "strangeness sigma term" σ sN = 27(27)(4) MeV sont en accord avec les résultats réseau les plus récents. Au vue des succès rencontrés lors de l'étude des masses de l'octet baryonique nous avons aussi fait une analyse systématique des masses du décuplet baryonique le plus léger dans la théorie chirale covariante des perturbations pour les baryons en fittant de façon simultanée les données réseau n f = 2 + 1. Une bonne description à la fois des données réseau et des masses expérimentales est obtenue. La convergence de la série de perturbations dans le secteur SU(3) est discutée. De plus les termes sigma sont prédits à l'aide du théorème de Feynman-Hellmann. Enfin comprendre le spectre d'excitation des hadrons est encore un challenge. En particulier le spectre des baryons a une structure très inhabituelle, la résonance Roper (1440) de parité positive étant plus légère que l'état de parité négative N(1535). La plupart des études sur réseau suggère que les effets des log chiraux sont plus importants pour la masse de la résonance Roper que pour celle des nucléons. Nous avons donc calculé la masse de cette résonance en théorie chirale des perturbations en tenant en compte de façon explicite des contributions du nucléon et du ∆. Les contributions venant du mélange entre le nucléon et la résonance Roper sont étudiées pour la première fois. Une première analyse de la masse de cette particule est présentée. Mots cls: thorie effective des champs, Lagrangiens chiraux, QCD sur rseau, masses des baryons, termes sigma List of Tables 2.1 The quark/gluon gauge action, and lattice spacings which are employed in the BMW, PACS-CS, LHPC, HSC, QCDSF-UKQCD, NPLQCD, and ETMC simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Coefficients of the NLO contribution to the self-energy of octet baryons [Eq. (4.24)]. 42 4.2 Coefficients of the NNLO contribution to the self-energy of octet baryons [Eq. (4.25)]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.3 Coefficients of the N 3 LO tree contribution to the self-energy of octet baryons [Eq. (4.27)]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.4 Coefficients of the tadpole contribution to the self-energy of octet baryons [Eq. (4.27)]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.5 Coefficients of the loop contributions (Fig. 4.1e) to the self-energy of octet baryons [Eq. (4.27)]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.6 Coefficients of the NNLO virtual decuplet contribution to the self-energy of octet baryons [Eq. (4.33)]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.7 Coefficients of loop diagrams (Fig. 4.2(b/c)) to the self-energy of octet baryons [Eq. (4.33)]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.8 Coefficients of LO contribution to the self-energy of decuplet baryons [Eq. (4.36)]. 46 4.9 Coefficients of the O(am q ) contributions to octet baryon masses (Eq. 4.78). . . 55 4.10 Coefficients of the tadpole diagram contributions to octet baryon masses (Eq. (4.79)). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Values of the LECs and fit-χ 2 /d.o.f. from the best fits. We have performed fits to the LQCD and experimental data at O(p 2 ), O(p 3 ), and O(p 4 ) , without ( / D) and with (D) the explicit contributions of the virtual decuplet baryons. . . . . . 58 5.2 Finite-volume corrections (in units of MeV) to lattice octet baryon masses in covariant BChPT up to N 3 LO. . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 5.3 Values of the LECs from the best fit to the LQCD data and the experimental data at O(p 4 ) with and without discretization effects. . . . . . . . . . . . . . . . 64 5.4 Extrapolated octet baryon masses (in units of MeV) to physical point with the LECs determined by fitting to LQCD data alone. . . . . . . . . . . . . . . . . . Values of the LECs from the best fits to LQCD and experimental octet baryon masses up to N 3 LO. The lattice scale in each simulation is determined using both the mass independent scale setting (MIS) and the MDS methods. In the MIS, both the original lattice spacings determined by the LQCD collaborations themselves "a fixed" and the self-consistently determined lattice spacings "a free" are used (see text for details). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 6.2 Predicted pion-and strangeness-sigma terms of the octet baryons (in units of MeV) by the N 3 LO BChPT with the LECs of Table 6.1. . . . . . . . . . . . . . The strangeness content and the "dimensionless sigma terms" of the octet baryons at the physical point. The first error is statistical and the second one is systematic, estimated by taking half the difference between the N 3 LO result and the NNLO result. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Coefficients of the NLO and NNLO contributions to decuplet baryon masses [Eq. (7.8)]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 7.2 Coefficients of the N 3 LO contributions to decuplet baryon masses [Eq. (7.8)],

7. 5 90 7. 6 91 8. 1

 5906911 Ratio of one-loop and tree contributions to the decuplet baryon masses, p 3 /(p 2 + m D ) , as a function of pion mass. The strange quark mass is set at its physical value. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ratio of one-loop and tree contributions to the decuplet baryon masses, (p 3 + p 4 )/(p 2 + m D ) , as a function of pion mass. The strange quark mass is set at its physical value. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . One loop self-energy diagrams contributing to M (3) N N . The dashed, solid, doublesolid, and triple-solid lines correspond to the pion, nucleon, ∆, and Roper, respectively. Black dots indicate an insertion from the dimension one chiral Lagrangians. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 The evolution of F(x). Solid (dash) line corresponds to the real (imaginary) part of F(x). Red circles denote F(x) with x = ±m phys. π /δ phys. N ∆ , blue squares and green diamonds are the results with x = ±m phys. π /δ phys. ∆R (δ phys. ∆R = m phys. ∆ -m phys. R ) and x = ±m phys. π /δ phys. N R . The full and hollow symbols are the real and imaginary results of F(x), respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 One loop self-energy diagrams contributing to M (4) N N . The dashed, solid, doublesolid, and triple-solid lines correspond to the pion, nucleon, ∆, and Roper, respectively. The filled circles denote an insertion from the dimension one chiral Lagrangians, the black boxes indicate O(p 2 ) mass insertions, and the empty circles are the couplings from the 1/M B correction Lagrangains. Wave function renormalization diagrams are not explicitly shown but included in the calculation.105 8.4 One loop self-energy diagrams contributing to M (3) ∆∆ . Other notations are the same as in Fig. 8.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 One loop self-energy diagrams contributing to M (4) ∆∆ . Other notations are the same as in Fig. 8.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 One loop self-energy diagrams contributing to M

( 3 )

 3 RR . Other notations are the same as in Fig. 8.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 One loop diagrams self-energy contributing to M (4) RR . Other notations are the same as in Fig. 8.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 1

Figure 1 . 1 :

 11 Figure 1.1: Structure within the atom.

Figure 1 . 2 :

 12 Figure 1.2: The composition of proton.

Figure 1 . 3 :

 13 Figure 1.3: Running coupling constant α s (Q) as a function of the respective energy scale Q.The figure is taken from Ref.[START_REF] Bethke | Experimental tests of asymptotic freedom[END_REF].

Figure 2 . 1 :

 21 Figure 2.1: The fundamental ingredients of lattice QCD simulations in three dimensions: lattice site, gauge link, plaquette (the smallest loop).

Figure 2 . 2 :

 22 Figure 2.2: Extrapolation results for the light hadron spectrum from the BMW Collaboration. Horizontal lines and bands are experimental values and the LQCD results are denoted as solid circles. The figure is taken from Ref. [22].

Figure 2 . 3 :

 23 Figure 2.3: The extrapolation of the nucleon and omega masses as function of the pion mass M 2 π for all three values of the lattice spacing. The figure is taken from Ref. [22].

  2.4, we show the lattice simulation points of the PACS-CS, LHPC, HSC, QCDSF-UKQCD and NPLQCD collaborations in the (2M 2 K -M 2 π ) -M 2 π plane and in the L-M 2 π plane. There is only one point for the NPLQCD simulation at M π = 389 MeV on the left panel, because the main purpose of the NPLQCD collaboration was to study finite-volume effects on the baryon masses. The large range of light

Figure 2 . 5 :

 25 Figure 2.5: Roper mass from the CSSM, JLab, BGR, Cyprus, and χQCD collaborations.

Figure 3 . 1 :

 31 Figure3.1: Power counting scheme of BChPT with MS, HB, IR, and EOMS renormalization schemes. Pink-filled points denote the contact terms. Red-filled circle are the PCB terms. Black and blue circles indicate the strict PC terms and the higher order corrections, respectively. Half-and full-filled circles denote the analytical terms which have the different analyticity (The details can be seen in text). The figure is taken from Ref.[START_REF] Geng | Recent developments in SU(3) covariant baryon chiral perturbation theory[END_REF].

4. 1 . 2

 12 Octet baryon self-energies up to N 3 LO Baryon self-energy up to N 3 LO The physical baryon mass is defined at the pole, / p = m B , in the two-point function of the baryon field ψ B (x)

( 2 )

 2 B , is of order O(p 2 ) [Fig. 4.1(a)]. The self-energy Σ b = m (3) B of the one-loop diagram [Fig. 4.1(b)] is of order O(p 3 ). One tree diagram contribution from L (4) φB [Fig. 4.1(c)] and two loop diagrams [Figs. 4.1(d,e)] are of order O(p 4 ),

Figure 4 . 1 :

 41 Figure 4.1: Feynman diagrams contributing to the octet-baryon masses up to O(p 4 ) in the EOMS-BChPT. The solid lines correspond to octet-baryons and dashed lines refer to Goldstone bosons. The black boxes (diamonds) indicate second (fourth) order couplings. The solid dot indicates an insertion from the dimension one meson-baryon Lagrangians. Wave function renormalization diagrams are not explicitly shown but included in the calculation.

  m

  4.1(a,c)] can be calculated straightforwardly. The corresponding results are shown below. The three one-loop diagrams [Figs. 4.1(b,d,e)] yield, generically,

  (a)B,φ are listed in Table4.1.

  + 3F ) 2 At O(p 3 ) diagram Fig. 4.1(b) gives the NLO SU(3)-breaking corrections to octet baryon masses m (3)

3 .

 3 The next term is the contribution from the tadpole diagram Fig. 4.1(d) and the Clebsch-Gordan coefficients are listed in Table 4.4. The last term is from the one-loop diagram of Fig. 4.1(e), together with the wave function renormalization diagrams not shown, and ξ (e)

Figure 4 . 2 :

 42 Figure 4.2: Feynman diagrams contributing to the octet baryon masses with the intermediate decuplet resonances. The solid lines correspond to octet baryons, the double lines to decuplet baryons, and the dashed lines denote pseudoscalar mesons. Black dots indicate an insertion from the dimension one chiral Lagrangian [Eq. (4.17)], and black boxes (diamonds) indicate O(p 2 ) mass insertions.

  (4.33) is the NNLO contributions of Feynman diagram Fig. 4.2(a). The corresponding coefficients ξ (a)

  term is the virtual decuplet contribution at O(p 4 ) from the one-loop diagram of Fig. 4.2(b) and the wave function renormalization diagrams of Fig. 4.2(c). The Clebsch-Gordan coefficients ξ (b/c) BD ,φ are tabulated in Table 4.7, and the loop function H (b/c) B,D (M φ ) has the following form:

( 4 )Figure 4 . 3 :

 443 Figure 4.3: Two feynman diagrams contributing to the octet baryon masses with the intermediate decuplet resonances. The other notations are the same as Fig. 4.2.

  have the same ultraviolet behavior, δG (c) B are finite and can therefore be calculated in four dimensions. Second, the unwelcome PCB terms appearing in a covariant baryon ChPT calculation are absent because they emerge from short-distance physics while such short-distance properties are the same in G (c) B (L) and G (c) B (∞). As a result, PCB terms vanish in the differences δG (c) B and no power-counting-restoration schemes, such as EOMS or IR, are needed to calculate δG (c) B . Using the Feynman parameterization, one obtains

50 )

 50 In addition, when the virtual decuplet baryon contributions are taken into account, the FVCs to the loop results of Fig.4.2(a,b,c) are δG

Figure 4 . 4 :

 44 Figure 4.4: Feynman diagrams contributing to the a-dependence of octet baryon masses up to O(a 2 ). The solid lines represent octet baryons and the dashed lines denote pseudoscalar mesons. The boxes (diamonds) indicate the O(a) (O(a 2 )) vertices. The circle-cross is an insertion from the L O(a) . The wave function renormalization diagrams are not explicitly shown but included in the calculation.

  the case of unmixed Wilson action, where the u, d, and s quarks are all Wilson fermions, the Wilson matrix can be written as W = diag(1, 1, 1). One can easily compute the O(a) contributions of the diagram Fig. 4.4(a) to the octet baryon masses, m O(a) B = -4ac SW W 0 (3 b0 + 2 bD ).

  )where C = c1 + 4(3c 2 + 2c 3 ), D = 4 d3 + 9 d7 + 3 d8 , and Ē = 4ē 3 + 9ē 7 + 3ē 8 . We introduce C + 16 D + 16 Ē = 16 X as one free LEC in the fitting process. The second line of Eq. (4.79) is for the contributions from the tadpole diagram of Fig. 4.4c, and the corresponding coefficients ξ (c) B,φ are listed in Table 4.10. The last term is for the contributions from the one-loop diagram of Fig. 4.4d, and the coefficients ξ(d) BB ,φ are already given in Table 4.5. The loop diagrams H (c) B (M φ ) and H (d)

  .81) where the m O(a) B is for the leading-order discretization effects of Eq. (4.77).Table 4.10: Coefficients of the tadpole diagram contributions to octet baryon masses (Eq. (4.79)).

Up to N 3

 3 LO, there are 19 unknown LECs (m 0 , b 0 , b D , b F , b 1-8 , and d 1-5, 7, 8 ) in the octetonly EOMS BChPT [Eq. (4.20)]. To take into account the contributions of the decuplet 1 We have checked that removing from lattice data the two lattice points of LHPC and HSC with Mπ > 400 MeV and MK > 580 MeV does not change qualitatively our results.

Figure 5 . 1 :

 51 Figure 5.1: The lowest-lying baryon octet masses as functions of the pion mass. The dotdashed lines and the dotted lines are the best NLO and NNLO fits to lattice data. The bands correspond to the best N 3 LO fits to lattice results. In obtaining the ChPT results, the strangeness quark mass has been fixed at its physical value. The lattice data points are obtained from the original ones by setting the strange quark mass at its physical value.

Figure 5 . 2 :

 52 Figure 5.2: Pion mass dependence of the LQCD data in comparison with the best fits of the EOMS BChPT up to N 3 LO with (solid lines) and without (dashed lines) the virtual decuplet contributions. The lattice data have been extrapolated to the physical strange-quark mass and infinite space-time.

Figure 5 . 3 :

 53 Figure 5.3: Lattice volume dependence of the NPLQCD data in comparison with the EOMS BChPT up to N 3 LO with (solid lines) and without (dashed lines) the virtual decuplet contributions. The full black points with M φ L > 4 are included in the fit data sets, while the hollow point with M φ L = 3.86 are not.

Figure 5 . 4 :

 54 Figure 5.4: (color online). Finite lattice spacing effects on the octet baryon masses, R B = m (a)

Figure 6 . 2 :

 62 Figure 6.2: Strangeness-nucleon sigma term determined from different studies. The purple and pink bands are our NNLO and N 3 LO results, respectively. Data points are taken from the following references: BMW [238], QCDSF-UKQCD [239], MILC(2013) [254], JLQCD [235], Engelhardt [234], Junnarkar & Walker-Loud [242], χQCD [236], Martin-Camalich et al. [56], Shanahan et al. [240], Lutz et al. [201].

Figure 7 . 1 :

 71 Figure 7.1: Feynman diagrams contributing to the decuplet-baryon masses up to O(p 4 ) in EOMS BChPT. The single lines correspond to octet-baryons, double lines to decuplet baryons and dashed lines to mesons. The black boxes (diamond) indicate second (fourth) order couplings. The solid dot indicates an insertion from the dimension one meson-baryon Lagrangians. Wave function renormalization diagrams are not explicitly shown but included in the calculation.

( 2 )

 2 D is given in Eq. (7.8), and m (2) B has the following form: m

B

  , in accordance with our power-counting. At N 3 LO, the pesudoscalar meson masses, appearing in m (2) D , should be replaced by their O(p 4 ) counterparts. The explicit expressions of the meson masses up to O(p 4 ) are given in Eqs. (4.37-4.39) of Chapter 4. The empirical values of the LECs L r

7 . 1 .

 71 For the NNLO one-loop diagram of Fig. (7.1c), one obtains δG (c)

  (M φ ) with the master formula [Eq. (7.19)], because M (b) D 2

  and b F = -0.231 GeV -1 . The octet-decuplet mass splitting δ = 0.231 GeV is taken as the average physical masses gap. As a result, m 0 and b 0 can be expressed as m 0 = m D -0.231 GeV and b 0 = (m D -1.423)/1.014 GeV -1 .

Figure 7 . 2 :

 72 Figure 7.2: (Color online). Pion mass dependences of the lowest-lying decuplet baryon masses.Filled (open) symbols denote the lattice data points included in (excluded from) the fits, which are projected to have the physical strange-quark mass. The dot-dashed, the dashed, and the solid lines are the best NLO, NNLO and N 3 LO fits to the lattice data, respectively. In obtaining the BChPT results, the strange quark mass has been set to its physical value. The lattice points in the shaded region are not included in the fits.

Figure 7 . 4 :

 74 Figure 7.4: Comparison between the N 3 LO BChPT and the LHPC data[START_REF] Walker-Loud | Light hadron spectroscopy using domain wall valence quarks on an Asqtad sea[END_REF].

Figure 7 . 5 :

 75 Figure 7.5: Ratio of one-loop and tree contributions to the decuplet baryon masses, p 3 /(p 2 + m D ) , as a function of pion mass. The strange quark mass is set at its physical value.

Figure 7 . 6 :

 76 Figure 7.6: Ratio of one-loop and tree contributions to the decuplet baryon masses, (p 3 + p 4 )/(p 2 + m D ) , as a function of pion mass. The strange quark mass is set at its physical value.

  7.5 and 7.6 p 3 /(p 2 + m D ) and (p 3 + p 4 )/(p 2 + m D ) are shown as a function of M 2

. 13 )

 13 with c N 1 , c N 2 , c N 3 unknown LECs expected to be of O(1/Λ χ ). The term with LEC c N 1 gives an m 2 π correction to the nucleon mass, while the c N 1 , c N 2 , c N 3 terms provide seagull vertices. For our purposes we also require the m 4 π correction to the mass contained in the third-order Lagrangian 2 , L (3) πN = e 38 χ + 2 N N + e 115 χχ † N N + e 116 (detχ + detχ †

Figure 8 . 2 :

 82 Figure 8.2: The evolution of F(x). Solid (dash) line corresponds to the real (imaginary) part of F(x). Red circles denote F(x) with x = ±m phys. π /δ phys. N ∆ , blue squares and green diamonds are the results with x = ±m phys. π /δ phys. ∆R (δ phys. ∆R = m phys. ∆ -m phys.R) and x = ±m phys. π /δ phys. N R . The full and hollow symbols are the real and imaginary results of F(x), respectively.

Figure 8 . 3 :

 83 Figure 8.3: One loop self-energy diagrams contributing to M (4) N N . The dashed, solid, doublesolid, and triple-solid lines correspond to the pion, nucleon, ∆, and Roper, respectively. The filled circles denote an insertion from the dimension one chiral Lagrangians, the black boxes indicate O(p 2 ) mass insertions, and the empty circles are the couplings from the 1/M B correction Lagrangains. Wave function renormalization diagrams are not explicitly shown but included in the calculation.

3 2 x 2 log(x 2 y 2 )Figure 8 . 4 :

 32284 Figure 8.4: One loop self-energy diagrams contributing to M

Figure 8 . 5 :

 85 Figure 8.5: One loop self-energy diagrams contributing to M (4) ∆∆ . Other notations are the same as in Fig. 8.3.

. 69 )Figure 8 . 6 : 3 )

 69863 Figure 8.6: One loop self-energy diagrams contributing to M (3) RR . Other notations are the same as in Fig. 8.1.

Figure 8 . 7 : 4 )

 874 Figure 8.7: One loop diagrams self-energy contributing to M (4) RR . Other notations are the same as in Fig. 8.3.

Figure 8 . 8 :

 88 Figure 8.8: Leading order diagrams of mixed terms between nucleon and Roper. Other notations are the same as in Fig. 8.3.

Figure 8 . 9 :

 89 Figure 8.9: Next-to-leading order diagrams of mixed terms between nucleon and Roper. Other notations are the same as in Fig. 8.1.

Figure 8 . 10 :

 810 Figure 8.10: Chiral extrapolation of nucleon mass. Left panel and right panel are the results from Fit-I and Fit-II, respectively.

Figure 8 . 11 :

 811 Figure 8.11: Chiral extrapolation of Delta mass. Left panel and right panel are the results from Fit-I and Fit-II, respectively.

Figure 8 . 12 :

 812 Figure 8.12: Chiral extrapolation of Roper mass. Left panel and right panel are the results from Fit-I and Fit-II, respectively.

8 9 2 9 ( 3 (

 823 cos 2 (ε) (3L 4 + L 5 -6L 6 -6L 7 -6L 8 + L 5 cos(2ε)) ξ η,ud 4L 4 + L 5 -8L 6 + 4L 7 ) -2 cos(2ε)(3L 4 + 2L 5 -6L 6 + 12L 7 ) + 2 cos(4ε)L 5 ) ξη,us

2 9 [ 3 ( 2 9 [ 3 ( 5 ] 4 3 4 3 4 3B

 23235444 4L 4 + L 5 -8L 6 + 4L 7 ) -√ 3 sin(2ε)(3L 4 + 2L 5 -6L 6 + 12L 7 + 2L 5 cos(2ε))] + 2 9 [cos(2ε)(3L 4 + 2L 5 -6L 6 + 12L 7 ) -cos(4ε)L 5 ] ξ η,ds 4L 4 + L 5 -8L 6 + 4L 7 ) + √ 3 sin(2ε)(3L 4 + 2L 5 -6L 6 + 12L 7 + 2L 5 cos(2ε))] + 2 9 [cos(2ε)(3L 4 + 2L 5 -6L 6 + 12L 7 ) -cos(4ε)LStrong isospin-breaking effects on octet baryon massesThe effective pseudoscalar meson-octet baryon Lagrangians have been given in Eq. (4.3) of Chatper 4. Due to the isospin breaking of u and d quarks, the quark mass matrix M == diag(m l , m l , m s ) (with m l ≡ m u = m d ) appeared in Eq. (4.3) should be replaced by M = diag(m u , m d , m s ). After calculating the Feynman diagrams given in Fig. 4.1, we can obtain the strong isospin-breaking effects on octet baryon masses up to N 3 LO,m B = m 0 + m 0 + b D + b F ) 4b 0 4(b 0 + b D -b F ) n 4b 0 4(b 0 + b D + b F ) 4(b 0 + b D -b F ) Λ (3b 0 + b D ) (3b 0 + b D ) (3b 0 + 4b D ) Σ + 4(b 0 + b D + b F ) 4(b 0 + b D -b F ) 4b 0 Σ 0 4(b 0 + b D ) 4(b 0 + b D ) 4b 0 Σ - 4(b 0 + b D -b F ) 4(b 0 + b D + b F ) 4b 0 Ξ 0 4b 0 4(b 0 + b D -b F ) 4(b 0 + b D + b F ) Ξ - 4(b 0 + b D -b F ) 4b 0 4(b 0 + b D + b F )At O(p 3 ), the chiral corrections to the baryon self-energies can be expressed as m (M φ ) is already given in Eq. (4.26), and the coefficients ξ (b) B,φ given in Table D.8. Table D.8: The coefficients ξ (b) B,φ in Eq. (D.12).

√ 3 ( 2 (D -F ) 2 2 3 (D 2 + 3F 2 ) 1 18 √ 3 ( 2 1 18 √ 3 ( 2 3 (D 2 + F 2 ) (D -F ) 2 1 18 √ 3 (

 3223218321832222183 D -3F ) sin(ε) -3(D + F ) cos(ε) D -3F ) cos(ε) + 3(D + F ) sin(ε) n (D + F ) D -3F ) sin(ε) + 3(D + F ) cos(ε) 2 D -3F ) cos(ε) -3(D + F ) sin(ε)

2 3 (D 2 + 3F 2 ) 2 9 √ 2 (D + F ) 2 (D -F ) 2 2 9 √√ 3 ( 2 (D + F ) 2 2 3 (D 2 + 3F 2 ) 1 18 √ 3 ( 4 3 2 3 2 3 4 (b + b 3 + b 4 ) 2 (b 1 -b 2 + b 3 + 2b 4 ) 2 (b 1 + b 2 + b 3 + 2b 4 ) Σ 0 4 (2b + b 4 ) 2 (b 1 + b 3 + 2b 4 ) 2 (b 1 + b 3 + 2b 4 ) Σ - 4 (b + b 3 + b 4 ) 2 (b 1 + b 2 + b 3 + 2b 4 ) 2 (b 1 -b 2 + b 3 + 2b 4 ) Ξ 0 2 (b -b 2 + b 3 + 2b 4 ) 4 (b 1 + b 3 + b 4 ) 2 (b 1 + b 2 + b 3 + 2b 4 ) Ξ - 2 (b -b 2 + b 3 + 2b 4 ) 2 (b 1 + b 2 + b 3 + 2b 4 ) 4 (b 1 + b 3 + b 4 ) 2 3 3 4 3 2 3 3 (b 5 - 3 (3b 5 - 2 3 3 (b 5 -√ 3 (3b 5 - 2 3 2 3 4 3 2 √ 4 3 4 3 2 √ 3b 6 2 3 2 3 5 + b 6 + b 7 + 2b 8 ) 4 (b 5 -b 6 + b 7 + 2b 8 ) 8 (b 5 + b 7 + b 8 ) n 4 (b 5 + b 6 + b 7 + 2b 8 ) 8 (b 5 + b 7 + b 8 ) 4 (b 5 -b 6 + b 7 + 2b 8 ) Λ 8 3 (2b 7 + 3b 8 ) 4 3 4 3

 32922293223218342243421234212344213421343421234212322344134212322342123441342342353523535224244262256784567885745678857845678744 3D sin(ε) -3F cos(ε) 3D cos(ε) + 3F sin(ε) D + 3F ) sin(ε) -3(D -F ) cos(ε) D + 3F ) cos(ε) + 3(D -F ) sin(ε) Table D.11: The coefficients ξ (d,2)B,φ in Eq. (D.13).π 0 p 2(3b 1 + b 3 + 3b 4 ) + (-3b 1 + 3b 2 + b 3 ) cos(2ε) + √ 3(3b 1 + b 2 -b 3 ) sin(2ε) n 2(3b 1 + b 3 + 3b 4 ) + (-3b 1 + 3b 2 + b 3 ) cos(2ε) -√ 3(3b 1 + b 2 -b 3 ) sin(2ε) Λ (4b 3 + 3b 4 -2b 3 cos(2ε)) Σ + 3b 1 + b 3 + 3b 4 + (3b 1 -b 3 ) cos(2ε) + √ 3b 2 sin(2ε) Σ 0 (4b 3 + 3b 4 + 2b 3 cos(2ε)) Σ - 3b 1 + b 3 + 3b 4 + (3b 1 -b 3 ) cos(2ε) -√ 3b 2 sin(2ε) Ξ 0 2(3b 1 + b 3 + 3b 4 ) -(3b 1 + 3b 2 -b 3 ) cos(2ε) -√ 3(3b 1 -b 2 -b 3 ) sin(2ε) Ξ - 2(3b 1 + b 3 + 3b 4 ) -(3b 1 + 3b 2 -b 3 ) cos(2ε) + √ 3(3b 1 -b 2 -b 3 ) sin(2ε) η p 2(3b 1 + b 3 + 3b 4 ) -(-3b 1 + 3b 2 + b 3 ) cos(2ε) -√ 3(3b 1 + b 2 -b 3 ) sin(2ε) n 2(3b 1 + b 3 + 3b 4 ) -(-3b 1 + 3b 2 + b 3 ) cos(2ε) + √ 3(3b 1 + b 2 -b 3 ) sin(2ε) Λ (4b 3 + 3b 4 + 2b 3 cos(2ε)) Σ + 3b 1 + b 3 + 3b 4 -(3b 1 -b 3 ) cos(2ε) -√ 3b 2 sin(2ε) Σ 0 (4b 3 + 3b 4 -2b 3 cos(2ε)) Σ - 3b 1 + b 3 + 3b 4 -(3b 1 -b 3 ) cos(2ε) + √ 3b 2 sin(2ε) Ξ 0 2(3b 1 + b 3 + 3b 4 ) + (3b 1 + 3b 2 -b 3 ) cos(2ε) + √ 3(3b 1 -b 2 -b 3 ) sin(2ε) Ξ - 2(3b 1 + b 3 + 3b 4 ) + (3b 1 + 3b 2 -b 3 ) cos(2ε) -√ 3(3b 1 -b 2 -b 3 ) sin(2ε) π ± K 0 K ± p 2(b + b 2 + b 3 + 2b 4 ) 2(b 1 -b 2 + b 3 + 2b 4 ) 4(b 1 + b 3 + b 4 ) n 2(b + b 2 + b 3 + 2b 4 ) 4(b 1 + b 3 + b 4 ) 2(b 1 -b 2 + b 3 + 2b 4 ) Λ (2b 3 + 3b 4 ) (9b 1 + b 3 + 6b 4 ) (9b 1 + b 3 + 6b 4 ) Σ + b 5 + b 6 + b 7 + 2b 8 ) cos(ε) + √ 3(3b 5 + 3b 6 -b 7 ) sin(ε) n (b 5 + b 6 + b 7 + 2b 8 ) cos(ε) -√ 3(3b 5 + 3b 6 -b 7 ) sin(ε) Λ (2b 7 + 3b 8 ) cos(ε) Σ + 4(2b 5 + b 8 ) cos(ε) Σ 0 4(2b 7 + b 8 ) cos(ε) Σ - 4(2b 5 + b 8 ) cos(ε) Ξ 0 b 6 + b 7 + 2b 8 ) cos(ε) -√ 3b 6 -b 7 ) sin(ε) Ξ - b 6 + b 7 + 2b 8 ) cos(ε) + 3b 6 -b 7 ) sin(ε) η p (9b 5 -3b 6 + b 7 + 6b 8 ) cos(ε) -√ 3(3b 5 -b 6 -b 7 ) sin(ε) n (9b 5 -3b 6 + b 7 + 6b 8 ) cos(ε) + √ 3(3b 5 -b 6 -b 7 ) sin(ε) Λ 4(2b 7 + b 8 ) cos(ε) Σ + (2b 7 + 3b 8 ) cos(ε) -3b 6 sin(ε) Σ 0 (2b 7 + 3b 8 ) cos(ε) Σ - (2b 7 + 3b 8 ) cos(ε) + sin(ε) Ξ 0 (9b 5 + 3b 6 + b 7 + 6b 8 ) cos(ε) + √ 3(3b 5 + b 6 -b 7 ) sin(ε) Ξ - (9b 5 + 3b 6 + b 7 + 6b 8 ) cos(ε) -√ 3(3b 5 + b 6 -b 7 ) sin(ε) (9b 5 + b 7 + 6b 8 ) (9b 5 + b 7 + 6b 8 ) Σ + 8(b 5 + b 7 + b 8 ) 4(b 5 -b 6 + b 7 + 2b 8 ) 4(b 5 + b 6 + b 7 + 2b 8 ) Σ 0 8(2b 5 + b 8 ) 4(b 5 + b 7 + 2b 8 ) 4(b 5 + b 7 + 2b 8 ) Σ - 8(b 5 + b 7 + b 8 ) 4(b 5 + b 6 + b 7 + 2b 8 ) 4(b 5 -b 6 + b 7 + 2b 8 ) Ξ 0 4(b 5 -b 6 + b 7 + 2b 8 ) 8(b 5 + b 7 + b 8 ) 4(b 5 + b 6 + b 7 + 2b 8 ) Ξ - 4(b 5 -b 6 + b 7 + 2b 8 ) 4(b 5 + b 6 + b 7 + 2b 8 ) 8(b 5 + b 7 + b 8 ) Table D.13: The coefficients ξ (e) pB ,φ in Eq. (D.13).

Table 2 .

 2 1: The quark/gluon gauge action, and lattice spacings which are employed in the BMW, PACS-CS, LHPC, HSC, QCDSF-UKQCD, NPLQCD, and ETMC simulations.

	lattice coll.	quark action	gluon action	lattice spacing a [fm]
				∼ 0.125, ∼ 0.085
	BMW	clover-improved Wilson Symanzik-improved	∼ 0.065
	PACS-CS	O(a)-improved Wilson	Iwasaki	0.0907(13)
	LHPC	asqtad sea/ domain wall valence	Symanzik-improved 0.12406(248)
	HSC	anisotropic clover	Symanzik-improved	a t = 0.03506(23) a s = 0.1227(8)
	QCDSF-UKQCD O(a)-improved Wilson	Symanzik-improved 0.0795(3)
	NPLQCD	anisotropic clover	Symanzik-improved	a t = 0.03506(23) a s = 0.1227(8)
				0.0936(13), 0.0823(10)
	ETMC	twisted mass Wilson	Iwasaki improved	0.0646(7)

Table 4 .

 4 1: Coefficients of the NLO contribution to the self-energy of octet baryons [Eq. (4.24)].

		N	Λ	Σ	Ξ
	ξ	(a) B,π

Table 4 .

 4 2: Coefficients of the NNLO contribution to the self-energy of octet baryons [Eq. (4.25)].

Table 4 .

 4 3: Coefficients of the N 3 LO tree contribution to the self-energy of octet baryons [Eq. (4.27)].

	N	Λ	Σ	Ξ
	(c) B,π ξ

-4(4d 1 + 2d 5 + d 7 + 3d 8 ) -4(4d 3 + 8 3 d 4 + d 7 + 3d 8 ) -4(4d 3 + d 7 + 3d 8 ) -4(4d 1 -2d 5 + d 7 + 3d 8 ) ξ (c) B,K -16 (d 1 -d 2 + d 3 -16( 8 3 d 3 + 2 3 d 4 + d 7 + d 8 ) -16(d 7 + d 8 ) -16 (d 1 + d 2 + d 3 -d 5 + d 7 + d 8 ) +d 5 + d 7 + d 8 ) ξ (c) B,πK 8 (4d 1 -2d 2 -d 5 16( 8 3 d 3 + 4 3 d 4 -d 7 + d 8 ) -16(d 7 -d 8 ) 8 (4d 1 + 2d 2 + d 5 -2d 7 + 2d 8 ) -2d 7 + 2d 8 )

Table 4 .

 4 4: Coefficients of the tadpole contribution to the self-energy of octet baryons [Eq. (4.27)].

Table 4 . 5

 45

: Coefficients of the loop contributions (Fig. 4.1e) to the self-energy of octet baryons [Eq. (4.27)].

Table 4 .

 4 B

	6: Coefficients of the NNLO virtual decuplet contribution to the self-energy of octet
	baryons [Eq. (4.33)].

Table 4 .

 4 7: Coefficients of loop diagrams (Fig. 4.2(b/c)) to the self-energy of octet baryons [Eq. (4.33)].

Table 4 .

 4 8: Coefficients of LO contribution to the self-energy of decuplet baryons [Eq. (4.36)].

Table 5 . 1 :

 51 Values of the LECs and fit-χ 2 /d.o.f. from the best fits. We have performed fits to the LQCD and experimental data at O(p 2 ), O(p 3 ), and O(p 4 ) , without ( / D) and with (D) the explicit contributions of the virtual decuplet baryons.

		/ D-O(p 2 )	/ D-O(p 3 )	/ D-O(p 4 )	D-O(p 4 )
	m 0 [MeV]	900(6)	767(6)	880(22)	908(24)
	b 0 [GeV -1 ]	-0.273(6)	-0.886(5)	-0.609(19)	-0.744(16)
	b D [GeV -1 ]	0.0506(17)	0.0482(17)	0.225(34)	0.355(20)
	b F [GeV -1 ]	-0.179(1)	-0.514(1)	-0.404(27)	-0.552(28)
	b 1 [GeV -1 ]	-	-	0.550(44)	1.08(6)
	b 2 [GeV -1 ]	-	-	-0.706(99)	0.431(93)
	b 3 [GeV -1 ]	-	-	-0.674(115)	-1.83(15)
	b 4 [GeV -1 ]	-	-	-0.843(81)	-1.57(4)
	b 5 [GeV -2 ]	-	-	-0.555(144)	-0.355(74)
	b 6 [GeV -2 ]	-	-	0.160(95)	-0.423(117)
	b 7 [GeV -2 ]	-	-	1.98(18)	2.79(15)
	b 8 [GeV -2 ]	-	-	0.473(65)	-1.73(6)
	d 1 [GeV -3 ]	-	-	0.0340(143)	0.0157(130)
	d 2 [GeV -3 ]	-	-	0.296(53)	0.445(57)
	d 3 [GeV -3 ]	-	-	0.0431(304)	0.328(18)
	d 4 [GeV -3 ]	-	-	0.234(67)	-0.117(59)
	d 5 [GeV -3 ]	-	-	-0.328(60)	-0.853(77)
	d 7 [GeV -3 ]	-	-	-0.0358(269)	-0.425(39)
	d 8 [GeV -3 ]	-	-	-0.107(32)	-0.557(56)
	χ 2 /d.o.f.	11.8	8.6	1.0	1.0

Table 5 . 3 :

 53 Values of the LECs from the best fit to the LQCD data and the experimental data at O(p 4 ) with and without discretization effects.

		BChPT	WBChPT		BChPT	WBChPT
	m 0 [MeV]	910(20)	915(20)	d 1 [GeV -3 ]	0.0295(124)	-0.0196(121)
	b 0 [GeV -1 ]	-0.579(56)	-0.557(50)	d 2 [GeV -3 ]	0.342(65)	0.230(58)
	b D [GeV -1 ]	0.211(56)	0.201(48)	d 3 [GeV -3 ]	-0.0314(63)	-0.0557(56)
	b F [GeV -1 ]	-0.434(43)	-0.359(41)	d 4 [GeV -3 ]	0.372(114)	0.304(1008)
	b 1 [GeV -1 ]	0.730(10)	0.810(8)	d 5 [GeV -3 ]	-0.401(110)	-0.237(88)
	b 2 [GeV -1 ]	-1.21(18)	-0.819(26)	d 7 [GeV -3 ]	-0.0913(58)	-0.104(48)
	b 3 [GeV -1 ]	-0.340(153)	-0.357(12)	d 8 [GeV -3 ]	-0.132(79)	-0.0417(67)
	b 4 [GeV -1 ]	-0.776(16)	-0.780(15)	B1 [GeV -3 ]×10 -2	-	-0.121(103)
	b 5 [GeV -2 ]	-1.15(287)	-1.34(23)	B2 [GeV -3 ]×10 -2	-	-0.467(109)
	b 6 [GeV -2 ]	0.778(390)	0.889(199)	B3 [GeV -3 ]×10 -2	-	0.344(267)
	b 7 [GeV -2 ]	0.899(26)	0.787(14)	X [GeV -3 ]×10 -4	-	0.606(5723)
	b 8 [GeV -2 ]	0.627(37)	0.817(28)
	χ 2	30.0	28.0	χ 2 /d.o.f.	0.91	0.97

tadpole/one-loop diagrams do not contribute. In the end, the discretization effects refer to the Eq. (4.78) and Eq. (4.79), m

Table 5 .

 5 4: Extrapolated octet baryon masses (in units of MeV) to physical point with the LECs determined by fitting to LQCD data alone.

		BChPT	WBChPT	Exp. [161]
	χ 2 /d.o.f.	0.89	1.0	-
	m N	889(21)	865(39)	940(2)
	m Λ	1113(17)	1087(41)	1116(1)
	m Σ	1163(19)	1139(42)	1193(5)
	m Ξ	1333(16)	1309(41)	1318(4)

Table 6 .

 6 1: Values of the LECs from the best fits to LQCD and experimental octet baryon masses up to N 3 LO. The lattice scale in each simulation is determined using both the mass independent scale setting (MIS) and the MDS methods. In the MIS, both the original lattice spacings determined by the LQCD collaborations themselves "a fixed" and the self-consistently determined lattice spacings "a free" are used (see text for details).

		MIS		MDS
		a fixed	a free
	m 0 [MeV]	884(11)	877(10)	887(10)
	b 0 [GeV -1 ]	-0.998(2)	-0.967(6)	-0.911(10)
	b D [GeV -1 ]	0.179(5)	0.188(7)	0.039(15)
	b F [GeV -1 ]	-0.390(17)	-0.367(21)	-0.343(37)
	b 1 [GeV -1 ]	0.351(9)	0.348(4)	-0.070(23)
	b 2 [GeV -1 ]	0.582(55)	0.486(11)	0.567(75)
	b 3 [GeV -1 ]	-0.827(107)	-0.699(169)	-0.553(214)
	b 4 [GeV -1 ]	-0.732(27)	-0.966(8)	-1.30(4)
	b 5 [GeV -2 ]	-0.476(30)	-0.347(17)	-0.513(89)
	b 6 [GeV -2 ]	0.165(158)	0.166(173)	-0.0397(1574)
	b 7 [GeV -2 ]	-1.10(11)	-0.915(26)	-1.27(8)
	b 8 [GeV -2 ]	-1.84(4)	-1.13(7)	0.192(30)
	d 1 [GeV -3 ]	0.0327(79)	0.0314(72)	0.0623(116)
	d 2 [GeV -3 ]	0.313(26)	0.269(42)	0.325(54)
	d 3 [GeV -3 ]	-0.0346(87)	-0.0199(81)	-0.0879(136)
	d 4 [GeV -3 ]	0.271(30)	0.230(24)	0.365(23)
	d 5 [GeV -3 ]	-0.350(28)	-0.302(50)	-0.326(66)
	d 7 [GeV -3 ]	-0.435(10)	-0.352(8)	-0.322(7)
	d 8 [GeV -3 ]	-0.566(24)	-0.456(30)	-0.459(33)
	χ 2 /d.o.f.	0.87	0.88	0.53

Table 6 .

 6 3: The strangeness content and the "dimensionless sigma terms" of the octet baryons at the physical point. The first error is statistical and the second one is systematic, estimated by taking half the difference between the N 3 LO result and the NNLO result.

			.1.
		MIS		MDS
		a fixed	a free
	σ πN	55(1)(4)	54(1)	51(2)
	σ πΛ	32(1)(2)	32(1)	30(2)
	σ πΣ	34(1)(3)	33(1)	37(2)
	σ πΞ	16(1)(2)	18(2)	15(3)
	σ sN	27(27)(4)	23(19)	26(21)
	σ sΛ	185(24)(17)	192(15)	168(14)
	σ sΣ	210(26)(42)	216(16)	252(15)
	σ sΞ	333(25)(13)	346(15)	340(13)

  is not

		1.5						1.5
		1.4						1.4
		1.3						1.3
	m B [GeV]	1.1 1.2						1.1 1.2
		0.8 0.9 1			MIS: a fixed MIS: a free MDS Exp.	Ξ Σ Λ Ν	0.8 0.9 1
		0	0.04	0.08	0.12	0.16	0.2	0.45	0.5	0.55	0.6	0.65	0.7
				M π 2 [GeV 2 ]

Table 7 .

 7 1: Coefficients of the NLO and NNLO contributions to decuplet baryon masses [Eq. (7.8)].

Table 7 .

 7

	2: Coefficients of the N 3 = 2t 1 +t 2 , t2 = 2t 4 , 3 +t LO contributions to decuplet baryon masses [Eq. (7.8)], with the following notations: t1	and t3 6 . 5 + t = 4t	Σ * Ξ * Ω -∆	5 ) 4 3 (3e 1 -e 2 + e 3 + 3e 4 + 9e 5 ) 4 3 (3e 1 -e 2 -e 3 + 3e 4 + 9e 5 ) 4(e 1 + e 2 -e 3 + e 4 + 3e 5 ) 2 + e 3 + e 4 + 3e 1 + e 4(e ξ (d) D,π (d) 16 3 (e 1 + e 3 + 3e 4 + 3e 5 ) 16 3 (2e 1 + e 2 + 2e 3 + 3e 4 + 3e 5 ) 16(e 1 + e 2 + e 5 ) 4 + e 3 + e 16(e 5 ) 4 + e ξ D,K (d) 5 ) 4 + e -16 3 (e 1 -e 2 -e 3 -3e 4 + 3e 5 ) -8 3 (4e 1 -e 3 -6e 4 + 6e 5 ) 2 -e 1 + e -16(e 5 ) 4 -2e 8(e 3 + 2e ξ D,πK	ξ (e,1)

Table 7 .

 7 4: Values of LECs from the best fits to the lattice and experimental data with different fitting strategies at O(p 2 ), O(p 3 ), and O(p 4 ), respectively. The estimator for the fits with and without the experimental decuplet masses, χ 2 /d.o.f. and χ 2 /d.o.f. of them to zero only slightly increases the χ 2 /d.o.f.. In short, the values of the N 3 LO LECs should be viewed in the present context and used with care.

	, are given in the last two

*

Table 7 .

 7 5: Finite-volume corrections (in units of MeV) to LQCD decuplet baryon masses in covariant BChPT up to N 3 LO.

		M π	δm ∆	δm Σ *	δm Ξ *	δm Ω -	M π L	M K L	M η L
	PACS-CS	296	14	5	0	-3	4.3	8.7	9.8
		384	5	2	1	1	5.7	8.6	9.3
		411	4	2	0	1	6.0	9.3	10.2
	QCDSF-UKQCD	320	20	13	8	4	4.1	5.8	6.2
		411	50	50	50	50	3.95	3.95	3.95
	HSC	383	4	2	1	0	5.7	8.1	8.8
		389	42	27	14	3	3.9	5.4	5.9
		449	28	19	11	4	4.5	5.8	6.2
		1.9
	M D [GeV]	1.5 1.7
		1.3					N 3 LO-II
							LHPC
		1.1	0	0.1	0.2	0.3	0.4	0.5
					M π 2 [GeV 2 ]

Table 7 .

 7 and Fig.7.2, one notices that the NLO BChPT can already describe LQCD data very well, but experimental data are missed a little bit. If the chiral expansions work, one 6: Contributions of different chiral order to the decuplet baryon masses at the physical point (in units of GeV).

			∆			Σ *			Ξ *			Ω -
	mD	p 2	p 3	p 4	p 2	p 3	p 4	p 2	p 3	p 4	p 2	p 3	p 4
	NLO 1.135 0.104 -	-	0.248 -	-	0.392 -	-	0.537 -	-
	NNLO 0.870 0.737 -0.383 -	1.089 -0.582 -	1.441 -0.785 -	1.793 -0.991 -
	N 3 LO 1.152 0.046 -0.429 0.463 0.158 -0.652 0.728 0.270 -0.878 0.988 0.382 -1.106 1.244

Table 7 .

 7 7: Pion-and strangeness-sigma terms of the decuplet baryons at the physical point. The first error is statistical and the second is systematic, estimated by taking half the difference between the N 3 LO result and the NNLO result.

			O(p 3 )			O(p 4 )
		w/o octet	w/ octet	Ref. [56]	w/o octet	w/ octet	Ref. [200]
	σ π∆	50(1)	64(1)	55(4)(18)	42(1)(4)	28(1)(8)	34(3)
	σ πΣ *	32(1)	44(1)	39(3)(13)	24(1)(4)	22(2)(9)	28(2)
	σ πΞ *	17(1)	26(1)	22(3)(7)	8(2)(5)	11(2)(6)	18(4)
	σ πΩ -	6(1)	8(1)	5(2)(1)	-5(2)(6)	5(2)(2)	10(4)
	σ s∆	70(13)	93(12)	56(24)(1)	99(24)(15)	88(22)(3)	41(41)
	σ sΣ *	198(13)	181(13)	160(28)(7)	233(23)(18)	243(24)(31)	211(44)
	σ sΞ *	298(13)	258(14)	274(32)(9)	375(24)(39)	391(24)(67)	373(53)
	σ sΩ -	370(14)	326(15)	360(34)(26)	507(25)(69)	528(26)(101)	510(50)

1

  Note that Lagrangians in these sectors are frequently labeled not by n ChPT but by d, for example

	L	(n ChPT +2) π	and L (n ChPT +1) πN	.

Table 8 . 1 :

 81 Values of the LECs from the best fit to the nucleon, Delta, and Roper lattice data and experimental results.

	Fit-I	Fit-II

Table A .

 A 4: Masses of the pseudoscalar mesons and the lowest-lying baryons (in units of MeV) obtained by the QCDSF-UKQCD Collaboration (TABLE XX, TABLE XXII and TABLE XXIII of Ref.[START_REF] Bietenholz | Flavour blindness and patterns of flavour symmetry breaking in lattice simulations of up, down and strange quarks[END_REF]). The uncertainties have the same origin as those in Table A.1. Table D.4: The coefficients ξ φ,φ in Eq. (D.8). 2m d -5ms + 6(mu + ms) cos(2ε) -(m d + ms -2mu) cos(4ε) + √ 3 ((ms -m d ) sin(4ε) -6(mu + ms) sin(2ε)) ξ φ,η -5mu -2m d -5ms -6(mu + ms) cos(2ε) -(m d + ms -2mu) cos(4ε) + √ 3 ((ms -m d ) sin(4ε) + 6(mu + ms) sin(2ε)) Table D.5: The coefficients ξ φ,φ in Eq. (D.8). 5m d -5ms + 6(m d + ms) cos(2ε) -(mu + ms -2m d ) cos(4ε) + √ 3 ((mu -ms) sin(4ε) + 6(m d + ms) sin(2ε))ξ φ,η -2mu -5m d -5ms -6(m d + ms) cos(2ε) -(mu + ms -2m d ) cos(4ε) + √ 3 ((mu -ms) sin(4ε) -6(m d + ms) sin(2ε)) ξ φ,K ± 0 ξ φ,K 0 0 Table D.6: The coefficients ξ φ,qq in Eq. (D.8). + L 5 -4L 6 -2L 8 2L 4 + L 5 -4L 6 -2L 8 0 ξ φ,dd 2L 4 + L 5 -4L 6 -2L 8 0 2L 4 + L 5 -4L 6 -2L 8 ξ φ,ss 0 2L 4 + L 5 -4L 6 -2L 8 2L 4 + L 5 -4L 6 -2L 8 ξ φ,ud 4L 4 + 2L 5 -8L 6 -4L 8 2L 4 -4L 6 2L 4 -4L 6 ξ φ,us 2L 4 -4L 6 4L 4 + 2L 5 -8L 6 -4L 8 2L 4 -4L 6 ξ φ,ds 2L 4 -4L 6 2L 4 -4L 6 4L 4 + 2L 5 -8L 6 -4L 8 2L 4 + L 5 -4L 6 -4L 7 -4L 8 ) + 2 √ 3 sin(2ε)(3L 4 + 2L 5 -6L 6 -6L 7 -6L 8 + L 5 cos(2ε))] + 1 9 [2 cos(2ε)(3L 4 + 2L 5 -6L 6 -6L 7 -6L 8 ) -cos(4ε)L 5 ] ξ π 0 ,dd 1 9 [6(2L 4 + L 5 -4L 6 -4L 7 -4L 8 ) -2 √ 3 sin(2ε)(3L 4 + 2L 5 -6L 6 -6L 7 -6L 8 + L 5 cos(2ε))] + 1 9 [2 cos(2ε)(3L 4 + 2L 5 -6L 6 -6L 7 -6L 8 ) -cos(4ε)L 5 ] ξ π 0 ,ss 8 9 sin 2 (2ε) (3L 4 + L 5 -6L 6 -6L 7 -6L 8 -L 5 cos(2ε)) ξ π 0 ,ud 2 9 (3(4L 4 + L 5 -8L 6 + 4L 7 ) + 2 cos(2ε)(3L 4 + 2L 5 -6L 6 + 12L 7 ) + 2 cos(4ε)L 5 ) ξ π 0 ,us 2 9 [3(4L 4 + L 5 -8L 6 + 4L 7 ) + √ 3 sin(2ε)(3L 4 + 2L 5 -6L 6 + 12L 7 -2L 5 cos(2ε))] -2 9 [cos(2ε)(3L 4 + 2L 5 -6L 6 + 12L 7 ) + cos(4ε)L 5 ] ξ π 0 ,ds 2 9 [3(4L 4 + L 5 -8L 6 + 4L 7 ) -√ 3 sin(2ε)(3L 4 + 2L 5 -6L 6 + 12L 7 -2L 5 cos(2ε))] -2 9 [cos(2ε)(3L 4 + 2L 5 -6L 6 + 12L 7 ) + cos(4ε)L 5 ] η 2L 4 + L 5 -4L 6 -4L 7 -4L 8 ) -2 √ 3 sin(2ε)(3L 4 + 2L 5 -6L 6 -6L 7 -6L 8 -L 5 cos(2ε))] -19 [2 cos(2ε)(3L 4 + 2L 5 -6L 6 -6L 7 -6L 8 ) + cos(4ε)L 5 ] ξ η,dd 1 9 [6(2L 4 + L 5 -4L 6 -4L 7 -4L 8 ) + 2 √ 3 sin(2ε)(3L 4 + 2L 5 -6L 6 -6L 7 -6L 8 -L 5 cos(2ε))] -1 9 [2 cos(2ε)(3L 4 + 2L 5 -6L 6 -6L 7 -6L 8 ) + cos(4ε)L 5 ] ξη,ss

			K ±
	ξ φ,π ±		0
	ξ φ,π 0 -5mu -ξ φ,K ±	0
	ξ φ,K 0		0
			K 0
	ξ φ,π ±		0
	ξ φ,π 0	-2mu -π ±	K ±	K 0
	ξ φ,uu	2L 4 π 0
	ξ π 0 ,uu 9 [6(ξη,uu 1 1 9 [6(

The ordinary matter is always called baryonic matter, which is not only the everything you see (e.g. the sun, rivers, mountains, mobile devices, food etc.), but also the nonvisible matter composed of baryons.

We also notice that few lattice simulations with physical light-quark masses have become available (see, e.g., Refs.[30,

31]) lately, which (will) largely reduce the systematic uncertainties related to chiral extrapolations to the physical light-quark masses.

The finite-volume corrections of scattering process can be studied by using the Lüscher formula[START_REF] Luscher | Volume Dependence of the Energy Spectrum in Massive Quantum Field Theories. 1. Stable Particle States[END_REF] and its resummed version.

Frequently, a symmetry of a classical theory is also a symmetry of the quantum theory based on the same Lagrangian. When it is not, the symmetry is said to be anomalous.

Λχ is associated with the mass of rho meson, mρ = 770 MeV, and the chiral symmetry breaking scale 4πfπ ∼ 1170 MeV.

Generally, a classical symmetry is realized in quantum field theory in two different ways depending on how the vacuum responds to a symmetry transformation.

For a given spin and parity hadronic state, there would be another state with the same spin but the opposite parity.

The up and down quarks have very small masses (mu ∼ m d ∼ few MeV) compared to ΛQCD.

It is easy to see from the derivative of baryon field, i∂ µ B = p µ B B = [(mB, 0) + (EB -mB, pB)]B, where the first term is the baryon mass in the chiral limit and cannot be thought as "small quantity".

In Ref.[START_REF] Lutz | On the consistency of recent QCD lattice data of the baryon ground-state masses[END_REF], Semke and Lutz showed that their partial summation approach can reproduce the results of the HSC and QCDSF-UKQCD collaborations by fitting the BMW, PACS-CS and LHPC data.

It should be noted that the uncertainties of the L r i are quite large. Because the effects of their contributions are found to be small, we do not take into account the uncertainties of these LECs in our fit of LQCD mass data.

In Refs.[START_REF] Martin Camalich | The lowest-lying baryon masses in covariant SU(3)-flavor chiral perturbation theory[END_REF][START_REF] Geng | Leading-order decuplet contributions to the baryon magnetic moments in Chiral Perturbation Theory[END_REF] the value of C is fixed from the ∆(1232) → πN decay rate, which yields C = 1.0. But in our previous study of the NPLQCD data, this coupling turned out to be somewhat smaller[START_REF] Geng | Finite-volume effects on octetbaryon masses in covariant baryon chiral perturbation theory[END_REF].

The same phenomenon has been observed in the studies of octet baryon magnetic moments[START_REF] Geng | Leading-order decuplet contributions to the baryon magnetic moments in Chiral Perturbation Theory[END_REF] and octet baryon masses up to NNLO[START_REF] Alarcon | The strangeness content of the nucleon from effective field theory and phenomenology[END_REF].

In our fits, we set W0 at 1 GeV 3 . Later a more proper value will be used to check the naturalness of the resulting LECs, B1, B2, B3, and X.

This is in contrast with the finite-volume effects. In Ref.[START_REF] Ren | Octet baryon masses in next-to-next-to-next-to-leading order covariant baryon chiral perturbation theory[END_REF], it is shown that a self-consistent treatment of finite-volume effects is essential to obtain a χ 2 /d.o.f. about 1.

In Ref. [207], Alvarez-Ruso et al. performed a phenomenological study of the continuum extrapolation of

Technically, this scale setting should be classified as a mass independent scale setting. Here, we slightly misuse the terminology to distinguish it from the one used in the LHPC original publication[START_REF] Walker-Loud | Light hadron spectroscopy using domain wall valence quarks on an Asqtad sea[END_REF].

In principle, at NNLO, we can use for the meson decay constant its SU(3) average, F φ = 1.17fπ with fπ = 92.4 MeV. This improves a lot the NNLO fit.

It needs to be mentioned that in Ref.[START_REF] Walker-Loud | Evidence for non-analytic light quark mass dependence in the baryon spectrum[END_REF], a different way of setting the lattice scale has been used to obtain the decuplet baryon masses of the LHPC Collaboration[START_REF] Walker-Loud | Light hadron spectroscopy using domain wall valence quarks on an Asqtad sea[END_REF] in physical units.

For related discussions in the mesonic sector, see, e.g., Refs.[START_REF] Descotes-Genon | Resumming QCD vacuum fluctuations in three flavor chiral perturbation theory[END_REF][START_REF] Bernard | Chiral dynamics with strange quarks in the light of recent lattice simulations[END_REF], where the so-called resummed chiral perturbation theory has been shown to exhibit better convergence than conventional chiral perturbation theory. To our knowledge, no similar studies exist in the one-baryon sector.

We want to mention that, in Appendix D, the strong isospin-breaking effects are calculated.

Under CPT symmetry, M π + = M π -, M K + = M K -, and M K 0 = M K0 .

Acknowledgments

After five years Ph.D. studies, I am becoming more professional, confident, courageous, and wise. These changes cannot be separated from the people who have ever given me guidance, assistance, and support.

First of all, I would like to express my heartfelt indebtedness to my supervisor Prof. Jie Meng. As a talented physicist, his enthusiasm, dedication, self-discipline, art of managing research group, way of thinking physics are truly inspiring me. He taught the basic principles and specifications of research when I was a novice. He always pushed and encouraged me to keep this work going well. Besides the physics, more important, I benefit a lot from his unique way of looking at the society and world.

I sincerely appreciate to my vice-supervisor Prof. Li-Sheng Geng, not only for teaching me the chiral perturbation theory and related numerical skills, but for the development of my views on these and related subjects. His efficient working style and good living habits have a great influence on me. He is a wonderful mentor, collaborator, and friend. I really treasure the time to get along with him.

With the help of Prof. Meng and Prof. Geng and the financial support from the Chinese Scholarship Council, I had the opportunity to visit the Institut de Physique Nucléaire Orsay, Université Paris-Sud, to be a joint-Ph.D. student under the supervision of Prof. Ubirajara van Kolck. I deeply indebted him for teaching me an amazing viewpoint of the physical world with effective field theories. It is enlightening and a rare pleasure to discuss with him. Under his encouragements, I began to learn the heavy baryon chiral perturbation theory. He is very patient and meticulous to modify my dissertation and prepare all matters of my defense from French side.

I am truly grateful to Prof. Véronique Bernard, who greatly helped me both in physics and in life when I was in France. With the help of her, I learnt the infrared baryon chiral perturbation theory, and several key problems appeared in this work were solved. She was deeply concerned about my dissertation and suggested many valuable improvements. I appreciate Dr. Manuel Pavon Valderrama for taking care of my routine affairs in IPNO. Thanks the members of theoretical physics group of IPNO, Prof. Elias Khan, Prof. Michael Urban, Prof. Marcella Grasso, Prof. Denis Lacroix, Prof. Nguyen van Giai, Prof. Bachir Moussallam, Prof. Shuck Peter, Dr. Jerry Yang, Mario Scanchez, Dr. Yusuke Tanimura, Noel Martin, Peng Yin, for hospitality and valuable discussions. I sincerely cherished the memory with my dear friends, Zi-Zhao Zong, Chao Meng, Bin Wang, Meng Wang, Ya-Qiong Wang, Yong-Hao Chen, Ying Wang, Wen-Long Zhang, Yan-Zhou Wang, Dan-Qiu Lin, in Paris.

With the financial support of Beihang University, I had the opportunity to visit IFIC, Valencia University for three months. I sincerely appreciate my collaborator Prof. Eulogio Oset, and his clear physical intuition and high efficiency are really impressive. Fruitful collaborations with Prof. Oset, Prof. Luis Alvarez-Ruso, Prof. Manuel Jose Vicente Vacas, Dr. Melahat Bayar, enlarged my research interests to the hadron structure. Thanks Prof. Juan Miguel Nieves Pamplona, Prof. Wei-Hong Liang, Dr. Tim Ledwig, Dr. Chu-Wen Xiao, Dr. En Wang, Dr. Toshikaka Uchino, Astrid Hiller Blin, Jorgivan Morais Dias, for their helpful discussions. I also thank my friends, Dr. En Wang, Dr. Chu-Wen Xiao, Dr. Jie Lu, Dan Zhou, Ying-Yi Lou, for warmhearted help in my daily life when I was in Spain.

Thanks my undergraduate advisors, Prof. Shou-Yu Wang and Prof. Bin Qi. The initial scientific research triggered my interest in nuclear physics and laid a solid foundation for my Ph.D. studies. Thanks Dr. Hao-Zhao Liang and Dr. Peng-Wei Zhao for the valuable suggestions. Every time I felt inspired after chatting with them. Thanks Prof. Toki Hiroshi, Prof. Emiko Hiayam for the useful discussions when they visited Beihang University. Thanks Dr. Jorge Martin Camalich for nice collaboration.

For four years studying in Beijing, I would like to thank the members of the nuclear theory group of Beihang University, Dr. Hua-Xing Chen, Dr. Chun-Yan Song, Xiao-Dong Xu, Guang-Zhi Xu, Kai-Wen Li, Dan Zhou, Zhi Shi, Xue-Wei Xia, Jun-Xu Lu, Er-Liang Cui, Zhu-Li, Zhao-Xi Li, Bin-Zhao, Xiao-Ying Qu, and the members of JCNP group of Peking University, Prof. Shuang-Quan Zhang, Dr. Qi-Bo Chen, Ling-Shuang Song, Shi-Hang Shen, Zheng-Xue Ren, and the former members, Dr. Wei Zhang, Dr. Lu-Lu Li, Dr. Jin-Niu Hu, Dr. Ying Zhang, Dr. Lang Liu, Dr. Zhen-Hua Zhang, Dr. Jian Li, Dr. Zhong-Ming Niu, Dr. Yi-Fei Niu, Dr. Ting-Ting Sun, Dr. Ying Chen, Ling-Fei Yu, Ji-Xuan Wei, Yu Zhou, Tian-Tian Wang, Yu-Zhi Zhou, Ming Li, for their helpful discussions and kind help in my daily life.

I am grateful to my thesis committee members Prof. Bing-Song Zou, Prof. Constantia Alexandrou, Prof. Matthias Schindler, Prof. Jaume Carbonell, Prof. Li-Hua Zhu, and Prof. Bing-Wei Long, for their assistance and helpful discussions. Thanks Ling-Shuang Song, Kai-Wen Li, Dan Zhou, Jun-Xu Lu, Rong An, for their valuable suggestions to the dissertation. Thanks the financial support from the Innovation Foundation of Beihang University for Ph.D. Graduates.

Thanks my dear friends, Lin Yu who gives me many advices in life and witnesses my growth and changes in this five years, Fu-He Liang, Chang-Hai Shan, Guang Li, Kai Zhang, Yan-Lei Zhang, and Ting-Wei Niu for their helpful encouragements and hospitality in Jinan and Jiaxiang, especially for Yan-Lei Zhang who designed several wonderful figures in the dissertation; De-Jing Yu, Yong-Qiang Pan, Ji-Long Wu, and Mu Cao who, interesting guys, provided a warm dormitory in undergraduate and cherish our friendship after graduation. Thanks my roommates, Dian Zhang and Guo Nan for their tolerance to my shortcomings in daily life. Thanks Jing Lu and Ge Zhang who accompanied me some periods in my life and left many happiness memories. Wish you have a bright future.

Finally, I must express my deepest gratitude to my family and my parents for their love! Without their permanent support and encouragement, I could't go so far! Xiu-Lei Beijing, China 2015. 10.22

Chapter 8

Roper mass and width in ChPT

With the development of computer techniques, the baryon masses of excited states are becoming available in lattice QCD, especially for the Roper. In this chapter, the nucleon, Delta, and Roper masses and widths are calculated in chiral perturbation theory up to next-to-next-toleading order. The effects of the mixing between the nucleon and the Roper are taken into account for the first time. We also tentatively perform an analysis of the lattice Roper masses to explore the quark mass dependence. Here, we want to mention that this work is still in progress, and the obtained results should be thought as preliminary.

Introduction

The light-quark spectrum provides a great opportunity to study QCD in its nonperturbative regime. In particular, the first even-parity excited state of the nucleon, the J P = (1/2) + Roper resonance N (1440) or P 11 (1440), has been a puzzle since its discovery [START_REF] Roper | Evidence for a P-11 Pion-Nucleon Resonance at 556 MeV[END_REF]. In the constituent quark model, the Roper, as an S-wave excitation, tends to be above the first oddparity excited state N (1530) or S 11 (1530), a P -wave excitation, a problem that is ameliorated if Goldstone-boson exchange is introduced. Several exotic interpretations have been proposed, such as a hybrid baryon [START_REF] Barnes | Where Are Hermaphrodite Baryons?[END_REF][START_REF] Li | Photoproduction signatures of hybrid baryons: An Application of the quark model with gluonic degrees of freedom[END_REF], a pentaquark state [START_REF] Krehl | What is the structure of the Roper resonance?[END_REF], and a breathing mode of the ground state [START_REF] Guichon | A Nonstatic Bag Model for the Roper Resonances[END_REF]. This puzzle has made the Roper a focus of the N * program at JLab, and in Bonn, Mainz, and Japan [START_REF] Aznauryan | Studies of Nucleon Resonance Structure in Exclusive Meson Electroproduction[END_REF][START_REF] Crede | Progress towards understanding baryon resonances[END_REF][START_REF] Olive | Review of Particle Physics[END_REF].

The recent outburst of interest in lattice QCD has not resolved this puzzle. The lowlying hadron spectrum is now well understood within controllable lattice artifacts [START_REF] Fodor | Light Hadron Masses from Lattice QCD[END_REF], but lattice simulations of excited states have not yet reached the same level of maturity. Several LQCD collaborations have published results for the Roper mass at various values of the pion mass m π [START_REF] Alexandrou | Novel analysis method for excited states in lattice QCD: The nucleon case[END_REF][START_REF] Mahbub | Roper Resonance in 2+1 Flavor QCD[END_REF][START_REF] Edwards | Excited state baryon spectroscopy from lattice QCD[END_REF][START_REF] Alexandrou | Nucleon Excited States in N f =2 lattice QCD[END_REF][START_REF] Liu | The Roper Puzzle[END_REF][START_REF] Engel | Meson and baryon spectrum for QCD with two light dynamical quarks[END_REF]. Except for the χQCD collaboration [START_REF] Liu | The Roper Puzzle[END_REF] which finds relatively small masses with a chiral behavior similar to that of nucleons, most of the results are in broad agreement. For m π ≤ 500 MeV, the proton is light enough for the Roper to be unstable, yet, it is treated as a stable particle. Taken at face value, however, these results imply that there is a strong decrease in the Roper mass as m π approaches the physical point. This, in turn, suggests that the Roper could be considerably closer to the nucleon in the chiral limit than in the real world.

There are other clues that indicate an important role for the Roper with respect to chiral symmetry. The importance of the Delta isobar ∆(1232) in low-energy phenomena stems from the relatively small Delta-nucleon mass difference δ ∆N = m ∆ -m N 290 MeV, at least compared to most mesons. The Roper-nucleon mass difference δ RN = m R -m N 490 MeV is less than twice as large, and the ratio of Roper-to-Delta widths is roughly of the order of (δ RN /δ ∆N ) 3 , as one might expect from widths that come primarily from a one-pion-loop selfenergy [START_REF] Long | The Role of the Roper in Chiral Perturbation Theory[END_REF]. Moreover, the nucleon, Delta and Roper nearly saturate the Adler-Weisberger

Here we would like to mention that the mixed terms Σ N R and Σ RN are the same. The baryon mass matrix, M B = M N 0 + Re[p 0 ], can be expressed as

with the diagonal terms M N N , M ∆∆ , and

N N + M

(3)

and the mixed terms

N R + M

where

BB are the real parts of baryon self-energies: ) ) . (8.44)

Similarly, we also can define the matrix of baryon widths

with the diagonal terms Γ N N , Γ ∆∆ , and

and the mixed terms

Because the nucleon is the lightest baryon state, the imaginary part of self-energies Σ N N (0) and Σ N R (0) = Σ RN (0) should be zero, therefore,

the expressions of baryon masses and widths are

N N + M

(3)

RN M

(3) N R

N N . The dashed, solid, doublesolid, and triple-solid lines correspond to the pion, nucleon, ∆, and Roper, respectively. Black dots indicate an insertion from the dimension one chiral Lagrangians.

∆∆ -i

N R M

(3)

Therefore, in order to obtain the explicit expressions of baryon masses and widths, one needs the diagonal terms of baryon self-energies up to NNLO and the mixed terms between nucleon and Roper up to NLO.

Baryon self-energies up to NNLO

Diagonal terms of the nucleon self-energy

The leading order contribution to the nucleon self-energy is

At O(ε 3 ), the contributing loop diagrams are collected in Fig. 8.1. After calculating the diagrams using the HBChPT, we can obtain the chiral corrections to the nucleon self-energy

where the loop function F(x) is defined as

At O(ε 3 ), the one-loop diagrams contributing to the mixed terms Σ

N R and Σ

RN are shown in Fig. 8.9. The mixed terms Σ N N , and can be expressed as

simulation results

In this section, we briefly summarize some key ingredients of the LQCD simulations of the PACS-CS [START_REF] Aoki | 2+1 Flavor Lattice QCD toward the Physical Point[END_REF], LHPC [START_REF] Walker-Loud | Light hadron spectroscopy using domain wall valence quarks on an Asqtad sea[END_REF], HSC [START_REF] Lin | First results from 2+1 dynamical quark flavors on an anisotropic lattice: Light-hadron spectroscopy and setting the strange-quark mass[END_REF], QCDSF-UKQCD [START_REF] Bietenholz | Flavour blindness and patterns of flavour symmetry breaking in lattice simulations of up, down and strange quarks[END_REF] and NPLQCD [START_REF] Beane | High Statistics Analysis using Anisotropic Clover Lattices: (IV) Volume Dependence of Light Hadron Masses[END_REF] collaborations, which are relevant to our study. In addition, we tabulate the simulated octet baryon masses in physical units, which satisfy M 2 π < 0.5 GeV 2 , M 2 K < 0.7 GeV 2 , and the pseudoscalar meson mass times lattice box size M φ L > 3.

PACS-CS [23]

The PACS-CS Collaboration employs the nonperturbatively O(a)-improved Wilson quark action and the Iwasaki gauge action. Numerical simulations are carried out at the lattice spacing of a = 0.0907 [START_REF] Einstein | Ist die trägheit eines körpers von seinem energieinhalt abhängig?[END_REF] fm, on a 32 3 × 64 lattice with the use of the domain-decomposed HMC algorithm to reduce the up-down quark mass, which is about 3 MeV. For the strange quark part they improve the PHMC algorithm with the UV-filtering procedure. Their simulation points cover from 701 MeV to 156 MeV, but the lightest point has a M π L ≈ 2.9, which might induce large finite volume corrections.

Table A.1: Masses of the pseudoscalar mesons and the octet baryons (in units of MeV) obtained by the PACS-CS Collaboration (TABLE III of Ref. [START_REF] Aoki | 2+1 Flavor Lattice QCD toward the Physical Point[END_REF].) The first number in the parentheses is the statistical uncertainty and second is that from the uncertainty of the lattice spacing. The lattice data sets with stars demand M π < 500 MeV and M φ L > 4. LHPC [START_REF] Walker-Loud | Light hadron spectroscopy using domain wall valence quarks on an Asqtad sea[END_REF] The LHPC Collaboration calculates the light hadron spectrum in full QCD using a mixed action that exploits the lattice chiral symmetry provided by domain wall valence quarks (the DWF valence quark action) and ensembles of computationally economical improved staggered sea quark configurations (the so-called asqtad action). The lattice spacing is determined to be a = 0.12406 [START_REF] Borsanyi | Isospin splittings in the light baryon octet from lattice QCD and QED[END_REF] HSC [START_REF] Lin | First results from 2+1 dynamical quark flavors on an anisotropic lattice: Light-hadron spectroscopy and setting the strange-quark mass[END_REF] The HSC Collaboration uses a Symanzik-improved action with tree-level tadpole-improved coefficients for the gauge sector and the anisotropic clover fermion action for the fermion sector. The lattice spacings are a s = 0.1227( 8) fm and a t = 0.003506 [START_REF] Aoki | 2+1 Flavor Lattice QCD toward the Physical Point[END_REF] [START_REF] Lin | First results from 2+1 dynamical quark flavors on an anisotropic lattice: Light-hadron spectroscopy and setting the strange-quark mass[END_REF]). The uncertainties have the same origin as those in Table A. 1.

QCDSF-UKQCD [28]

The QCDSF-UKQCD Collaboration employs the particular clover action, which has a single iterated mild stout smearing, and the (tree-level) Symanzik improved gluon action, which contains the gluon action and the three-flavor Wilson-Dirac fermion action. The simulations are carried out at the lattice spacing of a = 0.075 -0.078 fm, on 16 3 × 32, 24 3 × 48 and 32 3 × 64

NPLQCD [29]

The NPLQCD Collaboration mainly studied finite-volume effects on the octet baryon masses. Simulations are performed with n f = 2 + 1 anisotropic clover Wilson action in four lattice volumes with spatial extent L ∼ 2.0, 2.5, 3.0 and 3.9 fm. The anisotropic lattice spacing in the spatial direction is b s ∼ 0.123 fm and b t = b s /3.5 in the time direction. The pion mass is fixed at M π ∼ 390 MeV.

Feynman Rules

Here we would like to collect the relevant Feynman rules in the calculation of octet and decuplet baryon masses.

Propagators

Pseudoscalar meson propagator:

Octet baryon propagator:

Decuplet baryon propagator:

3)

The covariant spin-3/2 projection operator is defined as

Vertices

For the calculation of octet baryon masses

From L

(1) φB :

From L

(2)

From L

(1)

For the calculation of decuplet baryon masses

From L

(1)

φD :

From L

(1)

Loop Integrals

Feynman parameterization

The Feynman parameterization is a way to fraction with a product in the denominator:

invented by Richard Feynman to calculate loop integrals.

In our studies, we use the following Feynman parameterization:

Loop functions

In the calculation of the loop diagrams, we have used the d-dimension integrals in Minkowski space:

• is a combination symmetric with respect to the permutation of any pair of indices. After expansion in powers of ε up to O(ε), we can easily obtain the following integral identities:

(g ab g cd + g ac g bd + g ad g bc ).

(C.13)

Appendix D

Strong isospin-breaking effects on octet baryon masses

In Chapter 5, the chiral expansions of octet baryon masses are obtained under the assumption that the isospin symmetry of u, d quarks is always conservation. Although the isospin breaking corrections to the octet baryon masses are small in comparison with octet baryon masses, as pointed in Ref. [332], their effects are of fundamental importance to describe the structure of ordinary matter. There are two different sources of isospin symmetry breaking. One is from the small difference between the masses of u and d quarks, which are contributed by strong interaction. Another one is from the electric charge differences between u and d quarks. In this section, we would like to focus on the former one and calculate the strong isospin-breaking effects on the masses of pseudoscalar meson and octet baryon at O(p 4 ).

Isospin breaking effects to pseudoscalar meson masses

Expanding the lowest order meson Lagrangian [Eq. (3.38)] in powers of meson fields, one can obtain

Because the masses difference between u and d quarks, mixing of π 0 and η would be appear. In order to remove the mixing to obtain the meson masses, one should rotate the meson field with

where the lowest order mixing angel ε is

Therefore, the meson masses can be expressed at leading order 1 as

Up to next-to-leading order, the contributions to the meson masses by including the effects of isospin breaking read as

where M 2,2 φ and M 2,4 φ denote the M 2 φ at the leading order and the next-to-leading order, respectively. The corresponding coefficients ξ φ,φ and ξ φ,qq are listed in Tables (D.1-D.5) and Table D.6. The loop function H φ is

Table D.1: The coefficients ξ φ,φ in Eq. (D.8).

The coefficients ξ φ,φ in Eq. (D.8). Up to N 3 LO, the contributions to octet baryon masses are Table D.9: The coefficients ξ

B,ud , ξ

Table D.10: The coefficients ξ

B,φ in Eq. (D.13).

nB ,φ in Eq. (D.13).

ΛB ,φ in Eq. (D.13).

Σ + B ,φ in Eq. (D.13).

Table D.17: The coefficients ξ (e) Σ 0 B ,φ in Eq. (D.13).

Σ -B ,φ in Eq. (D.13).

Ξ 0 B ,φ in Eq. (D.13).

Ξ -B ,φ in Eq. (D.13). Nous présentons également une analyse systématique des masses les plus basses décuplet des baryons dans baryon covariante théorie de la perturbation chirale en ajustant simultanément les données nf = 2 + 1 en treillis.

Enfin, nous étudions également la masse Roper en théorie de perturbation chirale en incluant explicitement le nucléon / contributions Delta. Les contributions mixtes entre nucléons et Roper aux masses des baryons sont pris en compte pour la première fois. Une première analyse des treillis masses Roper est présenté.

Title : Effective Field Theory for Baryon Masses

Keywords : effective field theory, chiral Lagrangians, lattice QCD, baryon masses, sigma terms Abstract : Understanding nonperturbative QCD to produce the light baryon masses is of fundamental interest. Nowadays, interplay between lattice QCD and chiral perturbation theory is becoming a powerful tool to study light baryon masses. Utilizing recent lattice QCD simulations and chiral perturbation theory, we present a detailed study of the masses of the lowest-lying octet and decuplet baryon masses as well as the mass and width of the Roper resonance.

We study the lowest-lying octet baryon masses in covariant baryon chiral perturbation theory with the extended-on-mass-shell scheme up to next-to-next-to-next-to-leading order. In order to consider lattice artifacts, finite-volume corrections and finite lattice spacing discretization effects to lattice baryon masses are estimated. We perform a systematic study of all the latest nf = 2 + 1 lattice data with chiral extrapolation, finite-volume corrections, and continuum extrapolation. The lattice artifacts are well understood and discussed. We also find that the lattice data from different collaborations are consistent with each other. Using the chiral formulas of octet baryon masses, we accurately predict the octet baryon sigma terms via the Feynman-Hellmann theorem. The value determined for the nucleon strangeness sigma term is in agreement with lattice determinations.

We also present a systematic analysis of the lowest-lying decuplet baryon masses in covariant baryon chiral perturbation theory by simultaneously fitting nf = 2+1 lattice data.

Finally, we also study the Roper mass in chiral perturbation theory by explicitly including the nucleon/Delta contributions. The mixed contributions between nucleon and Roper to the baryon masses are taken into account for the first time. A first analysis of lattice Roper masses is presented.