Jure Bericic

Meriem Benali

Du Bist Ein

Keywords: electromagnetic probe, generalized polarizabilities, virtual Compton scattering, Hadron physics, proton structure, low energy expansion, dispersion relations. Zusammenfassung supplementary virtual photon . . . . . . . . . .

This long manuscrit, and the hidden work behind it, is not the result of only one person's eort. It would simply never be possible without the precious help of the people who deserve more than those few thanks.

Thanks to Hélène Fonvieille for sharing your wide knowledge with me (it was needed so many times in the production of this work) but also for sharing your doubts, too often hidden by teachers they are nevertheless important for understanding science. Then, thank you for your science of mentoring, this rare ability to know how to balance between a direct help and autonomous work. Thank you also for being someone of true kindness, who is really concerned about the future of students, from them and me you deserve a mountain of thanks.

Thanks to Achim Denig for having been there every time to control and check my work despite the distance and the travels. Thank you for the constructive discussions and

Résumé Introduction

Un siècle après sa découverte le proton est encore aujourd'hui un objet complexe source de questionnements. Nous savons que sa structure interne est gouvernée par l'interaction forte dont la constante de couplage varie avec l'énergie. Cette variation induit une dichotomie dans notre interprétation des expériences. A haute énergie, lorsque la constante de couplage diminue, nous pouvons interpréter les résultats par l'approche perturbative de la chromodynamique quantique. An de décrire le comportement du proton à basse énergie, alors que la constante de couplage augmente, nous devons utiliser les degrés de liberté hadroniques tels qu'ils sont présentés, par exemple, dans la théorie des perturbations chirales. A échelle d'énergie intermédiaire, dans la région des résonances nucléoniques, nous ne disposons pas d'une théorie descriptive consensuelle.

Le travail présenté dans cette thèse s'inscrit dans le cadre de ces énergies intermédiaires. En eet les polarisabilités généralisées (GPs) sont des quantités fondamentales paramétrant la réaction du proton à ces énergies. Les GPs entrent en jeu lors du processus de diusion Compton virtuelle γ * p → γp (VCS) où elles dépendent du carré du quadri-moment porté par le photon virtuel : Q 2 . Particulièrement, α E (Q 2 ) et β M (Q 2 ) les polarisabilités généralisées électrique et magnétique du proton décrivent la déformation locale de celui-ci lorsqu'il subit un champ électromagnétique.

Ces deux quantités furent mesurées par le passé, à diérentes valeurs de Q 2 , par trois expériences VCS à JLab, Bates et MAMI. Les résultats obtenus ont montré un comportement des GPs en fonction de Q 2 en désaccord avec les prédictions faites par diérents modèles théoriques. En eet, la forme dipolaire prédite pour α E (Q 2 ) est contestée dans la région entourant les résultats de MAMI autour de Q 2 = 0.33 GeV 2 .

Il fut donc décidé à MAMI de réaliser des nouvelles mesures de α E (Q 2 ) et β M (Q 2 ) à trois valeurs inédites de Q 2 : 0.1, 0.2 et 0.5 GeV 2 . Le travail présenté décrit les mesures réalisées à Q 2 = 0.2 GeV 2 . Il est divisé en cinq chapitres ; le premier décrit le cadre théorique des GPs ; le second présente les diérent modèles théoriques et résume les résultats obtenus par les expériences précédentes ; le troisième chapitre détaille l'appareillage expérimental ; la calibration de celui-ci est expliquée dans le quatrième chapitre ; le cinquième chapitre est consacré à la description des ts de l'eet des GPs.

Chapitre 1

Le premier chapitre est dédié à la présentation des polarisabilités généralisées, leur description phénoménologique, leur formalisme et la description cinématique du processus VCS.

Facteurs de forme du proton

Le chapitre commence par la description d'autre grandeurs fondamentales caractérisant la structure interne du proton : les facteurs de forme électrique et magnétique G p E (Q 2 ) et G p M (Q 2 ). Ces deux quantités décrivent la répartition des charges et des moments magnétiques à l'intérieur du proton. Les facteurs de forme paramètrent la diusion élastique d'un électron sur le proton. Lors de ce processus le photon virtuel échangé sonde la structure interne du proton à une échelle dépendant du quadri-moment transféré Q 2 . Cette dépendance d'échelle est utilisée pour obtenir la répartition interne des charges et des moments magnétiques via une transformée de Fourier. Ainsi, les facteurs de forme du proton permettent par analogie de comprendre ses polarisabilités généralisées.

Puisque les facteurs de forme interviennent dans tous les processus d'électrodynamique quantique (QED) ils contribuent aux résultats mesurés dans cette expérience. Or, leur comportement en fonction de Q 2 fut mesuré par plusieurs expériences et de nombreuses paramétrisations de ce comportement existent. L'incertitude engendrée par cette diversité est un des points clé de l'analyse, il est détaillé dans le chapitre cinq.

Les polarisabilités du proton

Les secondes quantités fondamentales du proton présentées dans le premier chapitre sont les polarisabilités. Ce sont des grandeurs scalaires dénies dans le cadre de la diusion Compton réelle (RCS), lorsqu'un photon réel interagit avec le proton.

Ces quantités décrivent le proton dans sa globalité, comme peuvent le faire sa charge ou son moment magnétique, elles sont alors à diérencier des facteurs de forme et des polarisabilités généralisées qui décrivent le proton d'un point de vue local. α E la polarisabilité électrique de RCS quantie la déformation du proton dans un champ électrique, β M la polarisabilité magnétique de RCS quantie la déformation du proton dans un champ magnétique.

Les polarisabilités, mesurées depuis les années 1960, ont prouvé l'extrême rigidité du proton. Celui-ci, placé dans un champ électromagnétique, aura une déformation de l'ordre de quelques centièmes de pourcent de son volume.

Les polarisabilités généralisées du proton

La dernière partie du premier chapitre est consacrée aux polarisabilités généralisées qui gouvernent le processus de diusion Compton Virtuelle. Celui-ci est accessible lors de l'électro-production de photon ep → epγ dont la cinématique est tout d'abord introduite. L'amplitude totale du processus d'électro-production de photon est une somme de trois amplitudes : Bethe-Heitler, VCS Born et VCS non-Born. Leur description théorique, par le théorème d'énergie de Low, permet d'écrire la section ecace totale comme un développement en puissance de q cm l'impulsion du photon réel produit, en dessous du seuil de production d'un pion.

Le processus est alors décrit comme une somme de deux sections ecaces. La première, dite de Bethe-Heitler+Born et notée σ BH+B , correspond aux amplitudes de Bethe-Heitler, de VCS Born et à leur interférence. Elle est parfaitement calculable grâce à l'électrodynamique quantique et sa valeur dépend seulement du choix xant la paramétrisation des facteurs de forme. La seconde correspond aux interférences entre les amplitudes précédentes et celle de VCS non-Born, elle n'est déterminée qu'à l'ordre premier du développement en puissance de q cm . A cet ordre l'eet des GPs entre en jeu via l'intermédiaire des fonctions de structure :

P LL -P T T / et P LT .

Ainsi en mesurant les sections ecaces expérimentales il est possible, grâce au théorème d'énergie de Low, de mesurer l'eet des GPs via les fonctions de structure. Pour ce faire il sut de comparer la section ecace théorique σ BH+B à celle mesurée, la diérence correspond alors à l'eet des GPs lorsque l'on néglige les ordres supérieurs en q cm . Chapitre 2

Le chapitre deux est divisé en trois sections ; la première est exclusivement dédiée à la présentation du modèle des relations de dispersion ; dans la seconde les diérents modèles prédictifs sont présentés ; la dernière section décrit et regroupe les résultats des expériences précédentes.

Le modèle des relations de dispersion

Le modèle des relations de dispersion (modèle DR) a la particularité d'utiliser α E (Q 2 ) et β M (Q 2 ) comme paramètres libres. Ainsi le modèle permet de calculer des sections ecaces d'électro-production de photon pour n'importe quelle valeur des GPs. Il peut alors être utilisé pour mesurer les GPs en comparant les sections ecaces obtenues dans notre expérience à celles calculées par le modèle. Cette méthode est l'une des deux approches que nous avons utilisées pour obtenir α E (Q 2 ) et β M (Q 2 ).

Plus précisément le modèle DR utilise la théorie des relations de dispersion et des principes fondamentaux de la physique pour construire douze amplitudes décrivant le processus VCS. La partie réelle de chaque amplitude est reliée à sa partie imaginaire via des relations, dites de dispersion, semblables aux relations de Kramers-Kronig. Ensuite la partie imaginaire de ces amplitudes est donnée par le calcule des processus de photo-absorption. Pour ces calculs seul le diagramme dominant est considéré, γ * N → N π, et son estimation est basée sur le modèle MAID. Ce modèle est valable sur un grand domaine de Q 2 et aussi au-dessus du seuil de production de pion, jusque dans la région de la résonance ∆.

Modèles prédictifs

Dans la seconde section de ce chapitre des modèles prédictifs sont présentés. Ils ne sont pas utilisés pour mesurer les GPs mais peuvent être comparés aux résultats obtenus. Les modèles sont : le modèle des perturbations chirales, le modèle de Lagrangien eectif, le modèle Sigma linéaire et le modèle des quarks non-relativistes.

Pour les polarisabilités issues de RCS ces modèles fournissent des valeurs très diérentes, illustrant ainsi la complexité des phénomènes d'énergie intermédiaire.

Les expériences précédentes

La dernière section est un résumé des conditions cinématiques et des résultats des expériences précédentes. Les deux expériences ayant eu lieu à MAMI ainsi que celle réalisée à JLab sont présentées. Aussi, est souligné dans cette section, le rôle particulier de l'expérience Bates qui a permis de révéler l'impact non négligeable des ordres supérieurs lors d'une analyse basée sur le théorème d'énergie de Low.

Grâce aux résultats de l'expérience Bates les mesures présentées ici sont réalisées en tenant compte de ces eets.

Dans cette section est aussi présentée une expérience parallèle à la nôtre visant à mesurer la polarisabilité généralisée électrique pour une valeur identique de Q 2 = 0.2GeV 2 mais par une méthode diérente basée sur la résonance ∆ et le modèle DR.

Enn le chapitre est conclu par la présentation graphique de l'ensemble des résultats obtenus par les expériences VCS lors d'analyses basées sur le théorème de Low ou basées sur le modèle DR.

Chapitre 3

Le troisième chapitre décrit l'appareillage utilisé dans la collaboration A1 utilisant le microtron de Mayence (MAMI). Il est divisé en deux sections, les deux premières introduisent le dispositif utilisé, la troisième insiste sur un point essentiel de l'analyse : le choix de l'espace des phases exploré.

Dispositif expérimental

Les deux premières sections présentent une description de l'expérience. Les caractéristiques techniques du microtron de Mayence sont détaillées dans la première section. Dans la seconde le fonctionnement des deux spectromètres du hall A1 est montré.

Dans l'expérience l'électron initial est fourni par MAMI, la particularité du microtron est son faisceau dit à haut cycle utile particulièrement intéressant pour les expériences de détection en coïncidence, comme le sont les expériences VCS. Le proton initial est quant à lui fourni par la cible d'hydrogène liquide. Le contrôle permanent de la cible lors des expériences réalisées dans le hall A1 permet de maitriser la luminosité expérimentale.

Les spectromètres, A et B, sont constitué chacun de plusieurs sous-détecteurs : deux chambres à ls, deux plans de scintillateurs et un détecteur Cherenkov.

Ils disposent en outre d'un système d'aimants permettant de dévier les particules détectées en coïncidence : l'électron et le proton.

permet en outre de vérier l'optique des spectromètres, le calcul de luminosité et la sélection d'évènements. Dans une seconde approche les quantités reconstruites sont comparées avec celles obtenues par la simulation, la qualité de la calibration est alors vériée pour la totalité des observables de l'expérience.

Ce contrôle des calibrations aura permis de trouver et corriger les inconsistances expérimentales, entre autre le mauvais alignement vertical des détecteurs.

Finalement, les rares inconsistances restantes seront traitées dans le calcul nal d'erreurs systématiques. Par exemple une erreur de calcul de luminosité, due à des eets locaux d'ébullition de la cible, fut estimée et corrigée. La variation dans l'application de cette correction engendre une erreur systématique.

La simulation

La simulation d'évènements VCS et d'acceptance des détecteurs est un outil obligatoire à l'obtention d'une mesure de section ecace précise. L'outil VCSSIM, développé à l'université de Gent, est divisé en trois étapes. Dans la première des évènements d'électro-production de photon sont générés selon la section ecace théorique. Les eets de perte d'énergie et de diusions multiples sont aussi simulés. Dans la seconde étape, les signaux produits sont dégradés par des eets de résolution des détecteurs. La troisième étape consiste en la reconstruction de ces évènements dans le but de reproduire les distributions expérimentales. Elles sont ensuite utilisées pour le contrôle des calibrations.

En plus de ces étapes VCSSIM produit un calcul de luminosité simulée dépendant des caractéristiques de la cible d'hydrogène liquide, cette quantité intervient elle aussi dans la mesure des sections ecaces VCS.

Conclusion

Pour la mesure présentée le travail de calibration est crucial, en eet l'impact des GPs ne représente que quelques pourcents de la section ecace mesurée, celle-ci doit donc être obtenue avec une excellente précision. La conclusion insiste sur la qualité de la calibration de ces données.

Chapitre 5

Le chapitre cinq est divisé en huit sections. La première section présente certains outils à disposition pour l'analyse : les codes de calcul des sections ecaces théoriques. La seconde section est dédiée aux calculs de corrections radiatives.

La troisième section décrit la mesure des sections ecaces d'electroproduction de photon. La quatrième section détaille la renormalisation des sections ecaces et les dernières sections sont consacrées aux ts des GPs selon les approches basées sur le théorème d'énergie de Low ou le modèle DR.

Les corrections radiatives

Certains processus sont détectés dans l'expérience et identiés malgré les sélections comme étant des évènements ep → epγ. Par exemple les diagrammes d'ordres supérieurs contenant des boucles de photon virtuel ou représentant l'émission de photon réel supplémentaire ne peuvent être dissociés du signal mesuré. Il faut donc appliquer des corrections sous la forme d'une renormalisation des sections ecaces.

Ces corrections sont calculées par une approche théorique ; certains diagrammes sont parfaitement déterminés grâce à la QED, d'autres, où la structure du proton intervient, requièrent des estimations. L'ensemble des diagrammes est tout d'abord présenté ainsi que les diérentes approximations eectuées. Ainsi un facteur de renormalisation F rad est calculé, il est fonction de l'espace de phase et des choix théoriques.

Les diérentes applications du terme F rad sont alors testées dans l'ensemble de l'espace des phases de l'expérience. Dans cette partie il est démontré que, malgré ces variations, une valeur moyenne du terme correctif peut être appliquée identiquement à toutes les sections ecaces mesurées. Les incertitudes ainsi générées sont estimées et considérées dans les erreurs systématiques.

Mesure des sections ecaces

La mesure des sections ecaces expérimentales est ici décrite. Puisqu'elles dépendent de quantités simulées et de sections ecaces théoriques, diérentes approches sont possibles (par exemple un eet des GPs peut être considéré dans les sections ecaces théoriques, il sera estimé à partir d'une première itération). Ces diérentes approches sont considérées dans le bilan nal d'erreurs systématiques.

Bien que la section ecace théorique dépende du choix de paramétrisation des facteurs de forme, il est montré dans cette section que la mesure de la section ecace expérimentale, elle, n'en dépend pas. Cette section décrit aussi le calcul d'erreurs statistiques et présente graphiquement certaines des 7784 mesures eectuées.

La renormalisation

Les sections ecaces mesurées dépendent du calcul des luminosités expérimentale et simulée, de l'ecacité du détecteur et de la qualité de la simulation. Dans ce type d'expérience ces facteurs seront toujours déterminés avec un biais intrinsèque.

C'est pourquoi il est nécessaire de renormaliser les sections ecaces. Dans cette partie le processus de renormalisation est détaillé, il est eectué grâce à des congurations dites LOW, a bas q cm , dans lesquels l'eet des GPs est faible. Ainsi en comparant les sections ecaces expérimentales aux sections ecaces théoriques σ BH+B il est possible de calculer, par minimisation de χ 2 un facteur de renormalisation F norm .

Ce facteur ainsi déterminé sera alors dépendant du choix de paramétrisation des facteurs de forme utilisée dans les calculs de QED. Néanmoins il est démontré dans cette section qu'il est important de conserver cette dépendance an d'obtenir une mesure des eets de GPs indépendante du choix de facteurs de forme. En eet la dépendance induite permet de compenser celle contenue dans les sections ecaces théoriques lors des extractions des GPs.

Une autre dépendance importante est induite dans le calcul de F norm par le choix d'utiliser un eet de GP théorique, même si celui-ci est minime. Cette dépendance doit être soustraite. Ceci est assuré par un processus itératif ; il consiste à tter un eet de GPs pour une normalisation donnée, celui-ci sera ensuite réintroduit dans les sections ecaces théoriques pour la détermination d'une nouvelle renormalisation. Il est ici montré qu'après quelques itérations une convergence s'établit indépendamment du choix initial de l'eet des GPs.

Les taux de comptage ayant été corrigés des petites inecacités de détecteur le facteur de renormalisation peut ensuite être considéré comme constant sur l'espace des phases mesuré. Les sections ecaces sont donc renormalisées globalement et l'incertitude sur F norm est considérée dans le bilan nal d'erreurs systématiques.

Extraction des GPs par le théorème d'énergie de Low (t

LEX)

Dans cette partie est présentée l'extraction des fonctions de structure basée sur le théorème de Low. Les étapes permettant la mesures sont détaillées et le t nal est présenté.

Selon le théorème d'énergie de Low, en négligeant les ordres supérieurs en q cm , la section ecace d'electroproduction de photon s'écrit comme la somme de la section ecace σ BH+B et d'un terme dépendant de l'espace des phases et des fonctions de structure. La mesure de celles-ci est alors réalisée lors d'un t des sections ecaces expérimentales par minimisation d'un χ 2 où les fonctions de structure sont les deux paramètres libres.

La force de cette approche est qu'elle est indépendante de modèle, sa faiblesse réside dans l'absence de traitement des ordres supérieurs en q cm , dont l'eet n'est pourtant pas toujours négligeable. Ils sont considérés ici par une sélection d'analyse inédite. Elle repose sur la détermination de l'eet des ordres supérieurs par le modèle DR. En eet ce modèle permet de calculer des sections ecaces théoriques complètes, c-à-d contenant aussi l'eet des ordres supérieurs en q cm . Donc, pour un choix arbitraire des GPs, les sections ecaces théoriques obtenues à partir du modèle DR sont comparées aux sections ecaces théoriques basées sur le théorème de Low et établies pour des valeurs identiques des GPs. La comparaison permet d'estimer l'eet des ordres supérieurs en chaque point de l'espace des phases, un critère sélectif est ensuite appliqué an de limiter leur impact sur la mesure.

La sélection introduit alors deux degrés de liberté : les valeurs théoriques des fonctions de structure et la valeur du critère sélectif. Dans cette section il est montré comment ces degrés de liberté sont traités. Un processus itératif permet de se défaire du choix arbitraire initial des GPs et la stabilité du résultat des ts permet de xer la valeur optimale du critère sélectif.

Finalement, les critères libres étant xés, un t dit nominal des fonctions de structure est déni. Son χ 2 réduit, de 1.21, déterminé à partir de 944 points traduit la bonne qualité du t. Les erreurs statistiques sont elles aussi déduites du t. Le modèle DR est ensuite utilisé pour trouver les valeurs de α E (Q 2 = 0.2 GeV 2 ) et β M (Q 2 = 0.2 GeV 2 ) correspondantes aux fonctions de structure ainsi obtenues.

Extraction des GPs par le modèle DR (t DR)

Dans cette section le modèle DR est utilisé pour obtenir la mesure des GPs et des fonctions de structure à Q 2 = 0.2 GeV 2 . Contrairement au t LEX, l'approche DR permet d'utiliser tous les points de section ecace, y compris ceux pour lesquels q cm < 126 M eV /c. Pour cela il faut comparer, lors d'une minimisation de χ 2 , les sections ecaces expérimentales mesurées à celles calculées par le modèle en chaque point de l'espace des phases et pour diérentes valeurs de α E (Q

2 = 0.2 GeV 2 ) et β M (Q 2 = 0.2 GeV 2
). Ainsi les valeurs des GPs donnant la meilleure correspondance entre les sections ecaces théoriques et expérimentales, sur l'ensemble de l'espace des phases, seront déterminées.

La première étape du t consiste donc à déterminer un χ 2 correspondant à un choix de valeurs des GPs discrétisées sur une grille en α E (Q 2 ) , β M (Q 2 ). L'ensemble des χ 2 ainsi déterminé est utilisé pour identier la position du minimum sur les n÷uds de la grille. La seconde étape utilise un t paraboloïde pour trouver cette position avec plus de précision.

La diculté technique de la méthode réside dans le calcul d'une très grande quantité de sections ecaces théoriques. Elles sont obtenues pour chaque point de l'espace des phases et pour chaque valeur des GPs, représentant un total de 3 024 000 points. Le temps de calcul demandé limite alors la possibilité d'explorer plusieurs variantes du t.

Les GPs sont obtenues avec un χ 2 minimum tté assez bon, valant 1.43 pour 4429 points de mesure. Les erreurs statistiques sont déduites des iso-contours du χ 2 . Les valeurs des fonctions de structure correspondantes sont aussi obtenues.

Bilan d'erreurs systématiques

Les erreurs systématiques sur les observables : GPs et fonctions de structure, sont déterminées par deux approches indépendantes.

La première consiste à modier la renormalisation des sections ecaces dans le but de représenter l'eet de l'ensemble des biais systématiques. Les diérents biais sont identiés et estimés, chacun en terme d'eets sur le facteur de renormalisation.

Les estimations reposent sur les observations expérimentales.

La seconde approche est, elle, basée sur plusieurs mesures indépendantes des fonctions de structure par le théorème d'énergie de Low. Chaque paramètre variable de l'analyse, comme la sévérité des sélections ou les osets expérimentaux, est alors modié et son eet sur les fonctions de structure quantié. La somme quadratique des eets permet de déduire l'erreur systématique nale.

Ces deux approches indépendantes montrent des résultats en parfait accord, la première méthode est ensuite choisie pour sa simplicité d'utilisation.

Conclusion

Dans le travail présenté les fonctions de structure ont été déterminées à Q 2 = 0.2 GeV 2 par deux approches indépendantes, la première basée sur le théorème d'énergie de Low et la seconde reposant sur le modèle des relations de dispersion. Soit, pour les fonctions de structure : q cm dans l'approche basée sur le théorème de Low.

P LL -P T T / (LEX) = 18.
Les mesures des polarisabilités généralisées électrique et magnétique sont elles aussi réalisées selon les deux approches :

α E (Q 2 ) (LEX) = 3.90 ± 0.14 stat ± 0.92 syst (10 -4 .f m 3 ), β M (Q 2 ) (LEX) = 1.14 ± 0.15 stat ± 0.78 syst (10 -4 .f m 3 ). α E (Q 2 ) (DR) = 4.05 ± 0.10 stat ± 0.98 syst (10 -4 .f m 3 ), β M (Q 2 ) (DR) = 1
.08 ± 0.12 stat ± 0.90 syst (10 -4 .f m 3 ).

Les valeurs sont en accord. Les mesures ne sont toutefois pas indépendantes car l'obtention des résultats par le théorème d'énergie de Low repose en partie sur le modèle DR, qui a fourni la valeur des GPs de spin.

Tous ces résultats sont déterminés à l'aide d'un grand nombre d'évènements, réduisant ainsi les erreurs statistiques. L'incertitude est donc dominée par les erreurs systématiques.

Finalement, les autres mesures de l'expérience, obtenues à Q 2 = 0.1 GeV 2 et Q 2 = 0.45 GeV 2 peuvent être regroupées avec celles des expériences précédentes et celles présentées dans cette thèse pour avoir la vue d'ensemble la plus complète sur le comportement des GPs selon Q 2 . La rapide variation de α E (Q 2 ) autour de 0.33 GeV 2 , brisant la forme dipolaire prédite par plusieurs modèles, est maintenant accentuée. Elle constitue une interrogation qui n'est pas comprise à l'heure actuelle.

Quant au comportement de β M (Q 2 ) selon Q 2 il est ici établi que les erreurs systématiques, quoiqu'optimisées dans notre analyse, restent dominantes et limitent la capacité des mesures à discriminer parmi les prédictions des modèles.

5.4.1

The iterative procedure in the LEX analysis . . . . . . . . .

5.4.2

The agreement between settings and between q cm -bins . . .

Introduction

In 1911 Ernest Rutherford discovered that the atoms have a heavy nucleus [START_REF] Rutherford | The scattering of alpha and beta particles by matter and the structure of the atom[END_REF].

Ten years later he was able to perform the rst articial transmutation and by this way proved that these nuclei were composed of elementary bricks: the protons. Thereafter he even anticipated the existence of the neutron which was later discovered by Chadwick in 1932 [START_REF] Chadwick | Possible Existence of a Neutron[END_REF].

One century after its discovery the proton is still under investigation, and nowadays it is the proton internal structure which is a source of questioning. We know that its structure is governed by the strong interaction but we have to face a dichotomy: the coupling constant of the strong interaction decreases at high energy and allows for an interpretation of the experiments in terms of perturbative Quantum ChromoDynamics (QCD), while at low energy, the coupling constant increases and asks for a description in terms of the hadronic degrees of freedom, such as Chiral Perturbation Theory.

In between, at excitation energies from ≈ 100 to ≈ 2000 MeV, both perturbation schemes are powerless to fully describe the proton interactions. This is the region of nucleon resonances which are still not described by a consensual theory.

In order to improve our knowledge, it is therefore essential to precisely extract hadronic structure information. In this aim the Compton scattering experiments are precious tools.

The strength of the Compton scattering experiments is that a part of the reaction relies on the electromagnetic interaction which is described by the extremely precise theory of Quantum ElectroDynamics (QED) [START_REF] Fermi | Quantum theory of radiation[END_REF]. Thus, thanks to its excellent precision, the electromagnetic probe will provide observables which are well understood and have an unambiguous physical interpretation.

The proton Generalized Polarizabilities (GPs) are one of those observables accessible via Compton scattering, more precisely via Virtual Compton Scattering (VCS) when a virtual photon scatters on a proton and a real photon is emitted.

The GPs depend on the virtuality Q 2 of the initial photon. In this context the GPs are dynamical properties which describe the collective behavior of the proton components when it is submitted to an applied electromagnetic eld. The emitted photon, of small energy, can then be seen as the quasi-static electromagnetic eld while the virtual photon will be the probe of the deformed proton.

The GPs were rst introduced for nuclei by Arenhövel and Drechsel [START_REF] Arenhövel | Generalized nuclear polarizabilities in (e, e' γ) coincidence experiments[END_REF] as a concept analogous to the classical description of the light scattering on atoms and molecules. Later, in 1995, P.A.M Guichon et al. studied the case of the nucleon, and derived a Low Energy Theorem for VCS on the proton [START_REF] Pierre | Virtual Compton scattering and generalized polarizabilities of the proton[END_REF], in a formalism that involves six independent GPs [START_REF] Drechsel | Generalized polarizabilities and the spin averaged amplitude in virtual Compton scattering o the nucleon[END_REF] at lowest-order. This triggered a series of dedicated VCS experiments throughout the world.

Among the six proton GPs the electric and magnetic GPs are the main topic of this work. They are scalar quantities which provide information on the proton local distribution of induced electric and magnetic dipoles and will help to understand its internal structure. For instance polarizabilities play a role in the proton charge radius puzzle, through the two-photon exchange process. More specically, the GPs are probes which help us to understand the contributions from the proton mesonic cloud and its quark core.

The VCS process γ * p → γp is accessible via the reaction of exclusive photon electroproduction ep → epγ. Studies of such a low cross-section process became possible with high duty-factor electron accelerators, and high-resolution spectrometers. The electric and magnetic GPs were measured for the rst time at MAMI [START_REF] Roche | The First determination of generalized polarizabilities of the proton by a virtual Compton scattering experiment[END_REF], later, Bates and JLab experiments were able to provide new values at dierent observation scales. However the current world data have shown an unexpected behavior with respect to the observation scale in the region around Q 2 = 0.33 GeV 2 .

The present work is a part of the experiment done at MAMI from 2011 to 2015, with the aim to understand better the Q 2 -behavior of the electric and magnetic GPs of the proton [START_REF] Merkel | A study of the Q 2 -dependence of the structure functions and the generalized polarizabilities in VCS at MAMI[END_REF]. Measurements were performed at three new values of Q 2 : 0.1, 0.2 and 0.45 GeV 2 and the present work covers the analysis of the data at

Q 2 = 0.2 GeV 2 .
The thesis content is as follows. First the theoretical context and physics motivations are introduced. Then the MAMI accelerator, the high-resolution spectrometers and the data-taking conditions are explained. Finally the data analysis is presented in detail and divided in two parts. The rst-level analysis concerns the treatment of experimental data: calibrations and event selection. The second-level analysis consists in determining the ve-fold dierential cross section d 5 σ(ep → epγ) and in extracting the VCS observables: the structure functions P LL -P T T / and P LT , and the electric and magnetic GPs α

E (Q 2 ) and β M (Q 2 ) at Q 2 = 0.2 GeV 2 .
Chapter 1

Introduction to Real and Virtual Compton Scattering

In this chapter the GPs will be presented from their phenomenological description to the theoretical tool which made possible their extraction from experiments.

In section 1.1 more basic nucleon observables are rst introduced, i.e. the electromagnetic form factors. This is because they directly intervene in the GP extraction and their phenomenological description simplify the physical understanding of GPs.

Then in section 1.2 the Real Compton scattering polarizabilities are described in order to introduce the GP formalism.

Finally, in the last section 1.3 the GPs are detailed together with the Virtual Compton Scattering process which allows to access them.

Proton form factors

The electric form factor G p E (q 2 ) and the magnetic form factor G p M (q 2 ) are the most basic quantities which describe the proton's internal structure. They naturally intervene in quantum electrodynamics (QED) processes when a charged particle interacts with the proton. The virtual photon is the vector boson of such electromagnetic interactions and carries a four-momentum squared q 2 (q 2 = q 0 2 -q 2 being negative for a space-like photon). The form factors will then depend on q 2 which denes the scale of the interaction.

The form factors express the spatial properties of the proton. More precisely, they contain the information about how the charges and the currents are distributed in the proton volume. This spatial information is encoded in the q 2 dependency. To understand how, it is common to express the momentum carried by the virtual photon in terms of its wavelength. According to de Broglie, the relation between the wavelength and the momentum is :

λ = h | q| , (1.1)
with h the Planck's constant. The de Broglie equation denes the link between the momentum exchanged and the interaction scale. The larger the momentum is the smaller the wavelength becomes, and so, the more localized the interaction 1.1. PROTON FORM FACTORS is. If the virtual photon's wavelength is large compared to the proton size, the interacting charged particle will be only sensitive to its global properties, the total proton charge and magnetic moment. If the photon's wavelength is small enough, the internal structure of the proton can be probed. The same reasoning can be applied to the four-momentum q 2 , yielding: λ = h/ -q 2 . Then the q 2 -dependency of form factors can be translated in terms of spatial distribution via a Fourier transform 1 . The Fourier transform of the electric or magnetic form factors will lead to the radial distribution of, respectively, the charge or the current :

G p E (q 2 ) F T → ρ electric ( r), G p M (q 2 ) F T → ρ magnetization ( r). (1.2)
In equation 1.2, ρ electric ( r) is the charge density along r. However this nonrelativistic representation of the proton charge density is criticized [START_REF] Miller | Charge Density of the Neutron[END_REF].

Indeed at relativistic energy, like in most hadron physics experiments nowadays, the scattered proton moves at high velocity. As the proton size and shape are not relativistically invariant quantities, its structure looks dierent in dierent frames. Also the relativistic process of pair creation further compromises a nonrelativistic density interpretation. In his paper, Miller [START_REF] Miller | Charge Density of the Neutron[END_REF] presents another density representation which is valid in a particular dynamics, the light-front dynamics.

Its use in association with a particular frame, called the Drell-Yan frame, will allow a relativistic interpretation of the proton charge density: ρ(b) with b the transverse distance in the frame. Unlike the density from equation 1.2, this new density is two-dimensional and so is the Fourier transform dened in this formalism.

This interpretation has also the advantage to use directly the more fundamental form factors F 1 (q 2 ) and F 2 (q 2 ) called the Dirac and Pauli form factors. Indeed G p E (q 2 ) and G p M (q 2 ) are not as fundamental, they are linear combinations of F 1 (q 2 ) and F 2 (q 2 ) built to avoid the interferences due to the F 1 × F 2 product in the (ep → ep) elastic cross-section. The physical meaning and the preferential use of G p E (q 2 ) and G p M (q 2 ) is then emphasized by the non-relativistic interpretation of equation 1.2. The linear combinations are, for a space-like virtual photon:

G p E (q 2 ) = F 1 (q 2 ) + q 2 4m 2 p F 2 (q 2 ), (1.3a) G p M (q 2 ) = F 1 (q 2 ) + F 2 (q 2 ), (1.3b)
where m p is the proton mass. So the density interpretation of form factors is still discussed nowadays. More generally the proton structure can be represented by two dierent pictures: a non-relativistic one, described by eective eld theory, and a relativistic one which uses the quantum theory of quarks and gluons (QCD).

Anyway, none of these descriptions are fully correct [START_REF] Vanderhaeghen | Long Range Structure of the Nucleon[END_REF]. As will be explained in section 1.3 the two interpretations also impact our representation of generalized polarizabilities.

1 this property of the Fourier transform is shown in a particular frame: the Breit frame. It is the frame in which the proton momentum is reversed in the scattering process. Elastic electron-proton scattering is the main process to measure the proton form factors, for two reasons. First the electron is a pointlike particle, so the structure eects in the cross section will only be due to the proton. The second reason is the simplicity of the process, i.e. both the electron and the proton stay in their ground state and no other particles are created. So in the one-photon exchange approximation we can perfectly describe the interaction using QED.

In this framework, elastic electron scattering is illustrated in gure 1.1. The process contains an incident electron with a four-momentum k = (E, k) which exchanges a virtual photon γ * with the proton target. The outgoing electron has a four-momentum k' = (E , k ). The protons are dened by the four-momenta p and p'. The four-momentum transferred by the virtual photon is q = kk' = p' -p. It is common practice to dene the quantity Q 2 , as the opposite of q 2 (when Q 2 is smaller the scale of the interaction is also larger). We recall where the form factors G p E (Q 2 ) and G p M (Q 2 ) intervene in the elastic cross section.

The general case of elastic electron scattering o a target is characterized by the following cross section:

dσ dΩ lab = dσ dΩ M ott E E |F (Q 2 )| 2 , (1.4)
where dσ dΩ M ott is the Mott cross section [START_REF] Hofstadter | Electron scattering and nuclear structure[END_REF] corresponding to the particular case of a spinless-particle target in the relativistic case. It takes the following form:

dσ dΩ M ott = Z 2 α 2 QED cos 2 θe 2 4E 2 sin 4 θe 2 , (1.5)
with Z the charge of the target in units of the elementary charge, α QED the ne structure constant 2 , and θ e the scattering angle of the electron in the laboratory frame.

In the general case of equation 1.4 the term E E is the target recoil correction term:

E E = 1 1+2 E M target
sin 2 θe 2 which tends to 1 when M target >> E.

2 α QED = e 2 /4π 0 c, with 0 the vacuum permittivity and = h/2π.

PROTON FORM FACTORS

The last term of equation 1.4, |F (Q 2 )| 2 , depends on the target internal structure. F (Q 2 ) is the target form factor that we can express specically for the proton. According to [START_REF] Rosenbluth | High Energy Elastic Scattering of Electrons on Protons[END_REF], in the case of an elastic electron scattering o the proton, equation 1.4 becomes the Rosenbluth formula:

dσ dΩ = dσ dΩ M ott E E G p E (Q 2 ) 2 +τ G p M (Q 2 ) 2 1 + τ + 2τ G p M (Q 2 ) 2 tan 2 ( θe 2 ) , (1.6)
with τ = Q 2 /4m 2 p .

In the limit Q 2 = 0, the form factors are normalized to the proton total charge in units of the electron charge, and to the proton total magnetic moment in units of the nuclear magneton µ N :

G p E (0) = 1 , G p M (0) = 2.793.
1.1.2 Form factor measurements and particular values at

Q 2 = 0.2 GeV 2

Experimental procedure

Elastic electron scattering o the proton was measured since the fties [START_REF] Hofstadter | Nuclear and nucleon scattering of high-energy electrons[END_REF] and continues nowadays, at Mainz [START_REF] Bernauer | Electric and magnetic form factors of the proton[END_REF] and other laboratories in the world (see [START_REF] Punjabi | The Structure of the Nucleon: Elastic Electromagnetic Form Factors[END_REF] for a review).

There are two kinds of experiments that can access form factors. The rst ones measure directly the electron scattering cross section and use equation 1.6 to extract form factors separately. The second ones require a polarized electron beam and the knowledge of outgoing-proton polarization to measure the asymmetry and then deduce the ratio G E /G M .

For the experiments dedicated to cross section measurement the classical way to extract form factors from cross section is called the Rosenbluth separation (equation 1.6). This method uses cross sections comparison at the same Q 2 but dierent angles θ e to obtain linear combinations of G p E (Q 2 ) and G p M (Q 2 ) which allow their extraction. The Rosenbluth separation has the advantage to provide modelindependent results. But one can also do a direct t of form factors based on a model shape as it has been done at MAMI [START_REF] Bernauer | Electric and magnetic form factors of the proton[END_REF] recently.

The polarization experiments consist in measuring the longitudinal and transverse polarization components of the recoil proton. The two components are obtained simultaneously by the single measurement of the azimuthal angular distribution of the scattered proton in a focal-plane polarimeter. The ratio G E /G M is then directly deduced with reduced systematic uncertainties. This constitutes the major advantage of this method, compared to cross-section measurement.

There is an inconsistency between the results from unpolarized cross-section and polarization experiments. It is believed to be a consequence of the one-photonexchange approximation. Because of the diverging results in the two-photon exchange correction, it is so far an unresolved puzzle [START_REF] Bernauer | Electric and magnetic form factors of the proton[END_REF].

There is actually a third method to measure the proton form factors which has been explored at MAMI recently. It is based on (e, e ) cross-section measurements in the radiative tail and aimed at determining the form factors at very low Q 2 [START_REF] Mihovilovic | Initial state radiation experiment at MAMI[END_REF].

Values at

Q 2 = 0.2 GeV 2
Many experiments led to dierent G p E (Q 2 ) and G p M (Q 2 ) results over time, thus there are many dierent parametrizations of the form factors versus Q 2 . However in order to extract the proton GPs at Q 2 = 0.2 GeV 2 we need to x a form-factor parametrization. In section 5.4.3 we will see how the choice of proton form factors impacts our VCS analysis.

At small Q 2 , the form-factor behavior follows more or less the standard dipole parametrization dened as :

G D = (1 + Q 2 0.71 ) -2 , (1.7)
with Q 2 expressed in GeV 2 . Most parametrizations exhibit a few percent deviation from the standard dipole. The agreement between various reliable parametrizations is not better than 1-2%.

So, for our GP extraction, it was decided to use three dierent parametrizations of form factors at Q 2 = 0.2 GeV 2 . The Friedrich-Walcher parametrization [START_REF] Friedrich | A Coherent interpretation of the form-factors of the nucleon in terms of a pion cloud and constituent quarks[END_REF] gives central values of form factors. The Arrington et al. [START_REF] Arrington | Global analysis of proton elastic form factor data with two-photon exchange corrections[END_REF] and Bernauer et al. [START_REF] Bernauer | High-precision determination of the electric and magnetic form factors of the proton[END_REF] parametrizations were chosen for their very dierent values of The polarizabilities are dened in the real Compton scattering (RCS) context, illustrated in gure 1.3, when an incident real photon with a four-momentum q = (q 0 , q) interacts with the proton by the reaction γp → γp. There is an innite number of polarizabilities and so far only the six rst ones 3 were extracted.

G p E (Q 2 ) and G p M (Q 2 ) at Q 2 = 0.2 GeV 2 . The Q 2 -
The electric and magnetic polarizabilities α E and β M describe the response of, respectively, the proton charge and magnetic moment distributions to the static external eld. The four other polarizabilities: γ E1 , γ M 1 , γ E2 and γ M 2 are quantities of higher dimension that describe the proton behavior when a ip of its spin is involved. Recently, the spin polarizabilities were measured for the rst time separately at MAMI [START_REF] Martel | Measurements of Double-Polarized Compton Scattering Asymmetries and Extraction of the Proton Spin Polarizabilities[END_REF]. The physical meaning of some other polarizabilities is detailed in [START_REF] Babusci | Lowenergy Compton scattering of polarized photons on polarized nucleons[END_REF].

To better understand the role of the electric and magnetic polarizabilities α E and β M it is convenient to use the pion cloud model. In this model, neither proved to be valid or not [START_REF] Vanderhaeghen | Long Range Structure of the Nucleon[END_REF], the proton is represented by three valence quarks surrounded by a pion cloud. The pions are spinless and considered as neutral or charged (positively or negatively). The valence quarks are, of course, charged and do have an intrinsic magnetic moment due to their spin.

One considers that initially the positive and negative charges are distributed in such a way that they share the same center of distribution. When an electric eld E is applied to the proton the positive and negative charges are pushed in opposite directions. This generates an induced electric dipole moment d which is proportional to E :

d = α E E, (1.8)
and the factor of proportionality α E is the electric polarizability. α E is expressed in units of a volume (f m 3 ). Its value is xed by the proton stiness, the stronger the proton's constituents are bound the smaller α E is. Now if one applies a magnetic eld H to the proton the valence-quarks magnetic moment tends to become aligned with H and creates a paramagnetic moment µ, whereas H will induce currents in the pion cloud which will oppose a diamagnetic moment to H and reduce µ. One can express the induced magnetic dipole as:

µ = β M H = (β dia M + β para M ) H, (1.9)
and due to these opposite eects the magnetic polarizability β M will be smaller than α E . 3 First to appear in the q development, detailed in section 1.2.1. The RCS process is the tool to access polarizabilities. Its kinematics is dened by the incoming real photon four-momentum q = (q 0 , q), the one of the outgoing real photon q' and by the initial and nal protons four-momenta p and p'. The blob represented in gure 1.3 illustrates the production of "proton excited states". Although there is not enough energy to produce such states at low energies (typically below the pion threshold) they intervene in the process as virtual intermediate states. So the magnitude of the proton deformation, expressed by the polarizabilities, depends on the proton excited states and the expression of a "deformed" proton adopted in this thesis refers to the virtual states eect. This is expressed by the general relation (in a non-relativistic approach at the rst-order perturbation), e.g. for the electric polarizability:

α E = 2 N * =N | < N * |d z |N > | 2 E N * -E N , (1.10)
where N * indicates a nucleon resonance and d z the electric dipole-moment operator [START_REF] Huillier | Diusion Compton Virtuelle a basse energie[END_REF].

At very low incident photon energy q 0 → 0 the RCS amplitude is perfectly described by the Thomson amplitude [START_REF] Olmos De Leon | Low-energy Compton scattering and the polarizabilities of the proton[END_REF]. It depends only on proton's static properties such as the mass, the charge and the anomalous magnetic moment κ.

When the photon energy increases, but is still below the pion production threshold (q 0 π = 144.7 M eV /c in the lab frame where the proton is at rest), the amplitude can be expressed as an expansion of terms ordered in powers of q 0 and q 0 . This expansion is based on the Low Energy Theorem which rst appears in the article of F. E. Low [START_REF] Low | Scattering of light of very low frequency by systems of spin 1/2[END_REF]. The complete amplitude was then developed by Petrun'kin [25] up to the order q 0 q 0 , in the lab frame. The amplitude is therefore:

f P et = -e 2 mp • +i(q 0 + q 0 ) e 2 4m 2 p (1 + 2κ) σ • ( × ) -i(q 0 + q 0 ) e 2 4m 2 p (1 + κ) 2 σ • [( n × ) × ( n × )] +i e 2 2m 2 p (1 + κ)[q 0 ( n • ) σ • ( n × ) -q 0 ( n • ) σ • ( n × )] +q 0 q 0 e 2 4m 3 p (2κ + κ 2 ) • -q 0 q 0 e 2 4m 3 p (1 + κ) 2 ( n × ) • ( n × )( n • n) +q 0 q 0 e 2 4m 3 p ( n × ) • ( n × ) +q 0 q 0 [ᾱ E • + βM ( n × ) • ( n × )]
ᾱE = α E + e 2 3mp < r 2 >, (1.12)
with < r 2 > the mean squared radius of the charge distribution. The Petrun'kin expansion is valid for q 0 80 M eV ; at higher energy other terms arise, especially the next-order terms are described by the spin polarizabilities γ E1 , γ M 1 , γ E2 and γ M 2 . Keeping every higher-order term in the squaring of the Petrun'kin amplitude will lead to the Real Compton Scattering cross section [START_REF] Olmos De Leon | Low-energy Compton scattering and the polarizabilities of the proton[END_REF]:

dσ dΩ Lab = dσ dΩ P owell -q 0 q 0 ( q 0 q 0 ) 2 e 2 mp ᾱE + βM 2

(1 + z) 2 + ᾱE -βM 2 (1 -z) 2 + o(q 3 ) (1.13)
where z = cos(θ γ ). The term o(q 3 ) contains other polarizabilities (among which the spin polarizabilities), it is neglected in the low-energy regime. The Powell cross section that appears in equation 1.13 expresses the scattering of the photon on a pointlike proton :

dσ dΩ P owell = 1 2 ( q 0 q 0 ) 2 e 2 mp 2 1 + z 2 + q 0 q 0 m 2 p ((1 -z) 2 + a 0 + a 1 z + a 2 z 2 )
, (1.14) with the coecients :

a 0 = 2κ + 9 2 κ 2 + 3κ 3 + 3 4 κ 4 , (1.15a
)

a 1 = -4κ -5κ 2 -2κ 3 , (1.15b)
a 2 = 2κ + 1 2 κ 2 -κ 3 - 1 4 κ 4 .
(1.15c)

Measuring the RCS cross section at dierent kinematics (dierent angles θ γ and dierent q 0 ) will then allow to extract the polarizabilities thanks to equation 1.13.

Proton polarizabilities measurement

The proton polarizabilities in RCS are extracted since 1960, with the rst pioneering experiment [26], and nowadays they are still under investigation. In order to improve measurements an additional constraint may be added to the global extraction procedure. It requires the measurement of the total photo-absorption 4 The retardation correction is related to the propagation of the electromagnetic wave, of non-zero frequency, through the nite size of the nucleon [START_REF] Huillier | Diusion Compton Virtuelle a basse energie[END_REF]. 1.3. GENERALIZED POLARIZABILITIES OF THE PROTON cross section σ γ (q 0 ). Indeed the uncertainty on polarizabilities may be reduced by using the Baldin sum rule [27] for the proton:

ᾱE + βM = 1 2π ∞ q 0 π σγ (q 0 ) q 0 2 dq 0 = (13.8 ± 0.4) • 10 -4 f m 3 , (1.16)
where the sum rule value is taken from [START_REF] Olmos De Leon | Low-energy Compton scattering and the polarizabilities of the proton[END_REF].

The 2014 Particle Data Group [START_REF] Olive | Particle data group[END_REF] gives the following value for the electric and magnetic polarizabilities of the proton: ᾱE = (11.5 ± 0.4) • 10 -4 f m 3 , βM = (2.5 ± 0.4) • 10 -4 f m 3 .

These results demonstrate the extreme proton stiness. To better understand this, it is convenient to compare the proton polarizability to its volume. α E is of the order of 10 -4 times its volume, therefore the proton's internal structure is barely deformed by quasi-static electric elds. For an atom the electric polarizability is of the order of its volume, therefore the proton components are considerably more linked by the strong interaction than the atom components are by the electromagnetic interaction.

Generalized polarizabilities of the proton

Just like for the form factors, the polarizabilities can be described as a function of a four-momentum transfer q 2 . Taking into account this dependency will lead to Generalized Polarizabilities (GPs). They are fundamental quantities which describe the proton deformation in an applied electromagnetic eld, but unlike the RCS polarizabilities, which express the global deformation, the generalized polarizabilities contain the information about the local deformation. The mechanism of how this local information is encoded in the q 2 -dependency is similar to the one described in section 1.1 for form factors.

The generalization of the electric polarizability will then be α E (q 2 ) and will contain information about the local electric deformation (for simplication the generalized polarizabilities are written without their bar). Thereafter, in the same way it was shown for form factors, applying a Fourier transform to α E (q 2 ) will allow us to draw a representation of this electric deformation along r. Similar considerations with an applied magnetic eld will lead to pictures of the local magnetic polarization, which is described through the q 2 -dependency of the magnetic polarizability, the so-called generalized magnetic polarizability β M (q 2 ).

The other RCS polarizabilities, such as spin polarizabilities, can also be extended to their q 2 -dependency.

As mentioned in section 1.1 there are two possible representations of the proton: one in the standard dynamics and the other in the light-front dynamics. is obtained with the standard dynamics. A recent interpretation of the GPs, which was performed in the light-front dynamics by Gorchtein et al. [START_REF] Gorchtein | Light-front interpretation of Proton Generalized Polarizabilities[END_REF], is presented in gure 1.5. The gure shows the induced polarization in a proton submitted to an electric eld along the x-axis, which is globally a dipole pattern. In ref [START_REF] Gorchtein | Light-front interpretation of Proton Generalized Polarizabilities[END_REF] it is shown how dierent parametrizations of the electric GP modify the local polarization distribution. These two representations of the proton allow us to visualize the deformation of the charge density in an external eld and map out the transverse position space dependence of the induced polarization. But, once again, gures 1.4 and 1.5 cannot be compared since they are obtained in completely dierent formalisms.

Virtual Compton scattering o a proton

The tool we used to extract GPs is the Virtual Compton scattering process (VCS) γ * + p → p + γ, represented in gure 1.6 (b) and (c). Indeed, just like for form factors, it is the virtuality of the photon that allows to explore the local information inside the nucleon. So the experimentally studied process is e + p → e + p + γ, the so-called photon electroproduction which is fully detailed in gure 1.6. This reaction is the combination of three sub-processes experimentally indiscernible:

-The Bethe-Heitler (BH) process corresponding to electron bremsstrahlung and represented in gure 1.6-a.

-The Born part of VCS that is bremsstrahlung from a "non-deformed" proton, gure 1.6-b.

-The non-Born part of VCS where virtual excited states intervene, gure 1.6c. In this process the real-photon emission is done by the "deformed" proton and depends on the GPs. A theoretical convention denes a last process as a part of non-Born VCS, it is the π 0 exchange in the t-channel represented in gure 1.7 (see also gure 2.5).

Therefore this particular process also contributes to the GPs and more exactly to the spin-ip GPs [START_REF] Metz | Generalized polarizabilities of the nucleon studied in the linear sigma model. 2[END_REF].

VCS Kinematics

In this section we will describe the VCS kinematics and dene its main variables.

In a VCS event the incident electron with a four-momentum k = (E, k) exchanges a virtual photon γ * with the proton target. The outgoing electron has a fourmomentum k' = (E , k ). The protons are again dened by the four-momenta p and p'. The four-momentum transferred by the virtual photon is again q = (q 0 , q) with q 0 = E -E and q = k -k . All those notations are summarized in table 1.3.1.

One has to note that for the VCS process the photon virtuality is equal to: q 2 = (k -k') 2 whereas for the BH process the photon virtuality is equal to: (p' -p) 2 .

Particle e e'

γ * p p' γ Four-momentum k k' q = k-k' p p' q' Mass squared m 2 e m 2 e q 2 = -Q 2 < 0 m 2 p m 2 p 0 Momentum vector k k q = k -k p p q
Modulus of momentum vector k k q p p q

Helicity and Spin projection h h λ σ σ λ In this thesis the kinematics will be described in two frames, the virtualphoton/target-proton center-of-mass (CM) and the laboratory where the initial proton is at rest. Variables of the CM will be written with the cm index. Lab variables will be written with no index.

For VCS two planes, presented in gure 1.8 can be distinguished:

-the scattering plane dened by both electron vectors k and k .

-the reaction plane dened by the recoiling-proton vector p and the real photon vector q .

Both planes contain the virtual-photon vector q, and so the two planes rotate around the q-axis dening the angle ϕ, being the same in the two frames: ϕ cm = ϕ. The angle between the real-photon vector q cm and the virtual-photon vector q cm is θ cm . The angle between the electron vectors is θ e,cm (θ e in the lab frame) .

Because of the Lorentz boost from the CM to the lab system, the recoilingprotons are contained in a narrow cone around the direction of the virtual photon.

For instance, at Q 2 = 0.2 GeV 2 the cone angle varies from θ cone = ±5 • to θ cone = ±14 • . So having a detector with a relatively small acceptance will be enough to cover a large range in θ cm .

To completely determine the kinematics of the photon-electroproduction reaction ve independent variables are necessary. Two sets of variables may be used:

(k, k , θ e , θ cm , ϕ cm ) and (q cm , q cm , , θ cm , ϕ cm ), where is the linear polarization parameter of the virtual photon.

The following relations (1.17 and 1.18) are used to switch from a set to the other : boosted in a cone around the virtual photon vector q as dened in the lab frame.

Q 2 = 4kk sin 2 θ e 2 , (1.17a)
s = -Q 2 + m 2 p + 2m p q 0 , (1.17b)
= [1 + 2 q 2 Q 2 tan 2 θ e 2 ] -1 , (1.17c)
with √ s the total energy in the center-of-mass.

q 0 cm = s-Q 2 -m 2 p 2 √ s , q cm = Q 2 + q 0 cm 2 , q 0 cm = s-m 2 p 2 √ s , q cm = q 0 cm , p 0 cm = s+Q 2 +m 2 p 2 √ s , p cm = q cm , p 0 cm = s+m 2 p 2 √ s , p cm = q cm .
(1.18)

For our experiment we choose to use the (q cm , q cm , , θ cm , ϕ cm ) set. One also has to consider the Q2 variable, introduced in [START_REF] Pierre | Virtual Compton scattering and generalized polarizabilities of the proton[END_REF], which is Q 2 at the limit of q cm → 0 at a xed q cm :

Q2 = Q 2 | q cm →0 = -2m p q0 cm , q0 cm = q 0 cm | q cm →0 = m p -m 2 p + q 2 cm . (1.19)
As a consequence, GPs can be considered as function of q cm or Q2 equivalently.

In our experiment q cm is considered xed at 458 M eV /c, which corresponds to Q2 ≈ 0.199 GeV 2 wgile our experimental < Q 2 > is 0.2 GeV 2 . The two quantities are close, for now on Q2 will be labeled as Q 2 .

The three contributions to the photon electroproduction amplitude

Now that the main variables are dened we will detail the three contributions to the photon-electroproduction amplitude and then write the cross section. The photon-electroproduction contributions, represented in gure 1.6, lead to the total amplitude T ee γ :

T ee γ = T BH + T B + T N B , (1.20)
where T BH , T B and T N B are respectively the amplitude of the Bethe-Heitler, Born and non-Born processes of gure 1.6.

The BH process is divided in two sub-processes: the real photon emission by the electron which is well described by QED and the virtual photon absorption by the proton which depends on form factors. So to fully calculate the Bethe-Heitler amplitude T BH one needs to choose a form-factor parametrization.

The Born process also depends only on the knowledge of form factors and T B can be calculated via QED.

The non-Born amplitude (NB) is the unknown term of equation 1.20 that we are interested in. The complete NB amplitude is not known analytically but a rst-order calculation was performed by Guichon et al. [START_REF] Pierre | Virtual Compton scattering and generalized polarizabilities of the proton[END_REF]; in section 1.3.3 we will remind how the GPs arise from the amplitude.

The cross section is of course the object experimentally accessible. For the work presented in this thesis we measured the unpolarized photon electroproduction cross-section dened as [START_REF] Pierre | Virtual Compton scattering o the nucleon[END_REF]:

d 5 σ exp dk dΩ k dΩ p cm = (2π) -5 64mp k k s-m 2 p s M (1.21)
with dΩ k the nal-electron solid angle in the lab, and Ω p cm the nal-proton solid angle in the center-of-mass frame. M is the Lorentz invariant dened by :

M = 1 4 σσ hλ |T ee γ | 2 . (1.22)
For an unpolarized beam and unknown nal-particles polarization the photonelectroproduction amplitude squared has to be summed over all initial and nal spin states. For experimentalist's convenience the cross section of equation 1.21 is then written as :

d 5 σ exp = Φq cm • M (1.23)
where Φq cm is the phase-space factor given by :

Φq cm = (2π) -5 64mp k k 2 √ s • q cm . (1.24)
The amplitude squared is :

|T ee γ | 2 = |T BH | 2 + |T B | 2 + |T N B | 2 +(T BH * T B + T BH T B * ) + (T BH * T N B + T BH T N B * ) +(T B * T N B + T B T N B * ).
(1.25)

In the photon electroproduction total cross section the most occurring process depends on the region of the phase space. T BH is the dominant term in the socalled Bethe-Heitler peaks when the outgoing photon is emitted in the direction of one of the electrons. In this particular region it is impossible to extract anything else than the Bethe-Heitler cross section. In other regions of the phase space the Born process will sometimes dominate over BH, but the non-Born process is always small in comparison. Actually only the interference between the non-Born and the sum of the two other processes allows a GP extraction. So equation 1.25 can be written as:

|T ee γ | 2 = |T BH+B | 2 + |T N B | 2 + (T BH+B T N B * + T BH+B * T N B ) (1.26)
with |T BH+B | 2 that accounts for Born process, Bethe-Heitler process and their interference.

Figure 1.9 illustrates the importance of each contribution for a given kinematics. The BH peaks are located in-plane, near θ cm = 70 • and θ cm = 110 • . The GP eect, theoretically calculated and magnied on the gure, will only modify the complete photon electroproduction cross section by a few percent. It is this small eect that we want to measure.

Low Energy Theorem

The Low Energy Theorem (LET or LEX for Low energy EXpansion) oers a model-independent way to express the photon electroproduction cross-section. It is based on perturbation theory and contains only few hypotheses (for instance the one-photon exchange approximation). The LET for VCS, derived by Guichon et al. [START_REF] Pierre | Virtual Compton scattering and generalized polarizabilities of the proton[END_REF] was inspired by the work of F.E. Low [START_REF] Low | Bremsstrahlung of very low-energy quanta in elementary particle collisions[END_REF]. Thanks to the theorem the three amplitudes of the photon electroproduction process can be expanded in powers of q cm , in the vicinity of q cm = 0, i.e. for the Bethe-Heitler amplitude :

T BH = a BH -1 q cm + a BH 0 + q cm a BH 1 + O BH (q cm 2 ), (1.27)
with the coecients a BH i independent of q cm and O BH (q cm 2 ) the higher-order terms. The Born amplitude will follow the same kind of expansion and is also known to start at order q cm -1 .

For the non-Born process, an expansion in powers of q cm was also proved to be possible [START_REF] Pierre | Virtual Compton scattering and generalized polarizabilities of the proton[END_REF] and can be summarized as :

T N B = q cm a N B 1 + O N B (q cm 2 ), (1.28)
ing to a BH -1 × a N B 1 , a B -1 × a N B 1 and a N B 1 × a N B
1 , this last one being too small to be considered in the rst order. As the Born and BH amplitudes are perfectly determined by QED the two amplitudes and their interaction are gathered in a completely known cross-section: σ BH+B .

Then from equation 1.21 one can write a LEX cross-section :

σ LEX = σ BH+B + Φq cm [M 0 -M BH+B 0 ] + O(q cm 2 ), (1.29
)

with [M 0 -M BH+B 0
] the term corresponding to the interference between the BH+B amplitude and the non-Born amplitude at lowest order. O(q cm 2 ) will then contain every term of higher order.

So the Low Energy Theorem provides a model-independent expression of the photon electroproduction cross section σ LEX which is valid below the pion production threshold: q cm ≤ 126 M eV /c. In the following part we will detail the expression of [M 0 -M BH+B 0 ].

Physical Observables

The way that the GPs arise from T N B is well explained in [START_REF] Pierre | Virtual Compton scattering and generalized polarizabilities of the proton[END_REF] and [START_REF] Pierre | Virtual Compton scattering o the nucleon[END_REF], in this section we will briey summarize the results. The non-Born amplitude is:

T N B (λ , λ) = * µ (λ )H µν N B ν (λ), (1.30)
with ν the polarization four-vector in the Lorentz gauge as dened in [5]. H µν N B is the hadronic tensor, it is the unknown object parametrized by the GPs.

In order to get an adequate parametrization of H µν N B a multipole expansion is used in the CM frame. The multipole expansion is done in a spherical harmonic basis with its coecients characterized by the multipoles : H (ρ L ,ρL)S N B (q cm , q cm ).

These unknown quantities will describe the electromagnetic transitions undergone by the proton. L (L ) is the angular momentum of the initial (nal) electromagnetic transition. S stands for the spin-nature of the transition: S = 0 (or S = 1) for a non-spin-ip transition (or a spin-ip transition). ρ (and ρ ) characterizes the nature of the initial (nal) multipole : ρ = 0 (charge), ρ = 1 (magnetic), ρ = 2 (electric) and ρ = 3 (longitudinal). But gauge invariance relates the charge and longitudinal multipoles, restricting ρ to (0,1,2).

An important step (detailed in [START_REF] Pierre | Virtual Compton scattering o the nucleon[END_REF]) uses parity symmetry, conservation of angular momentum and kinematical properties to express the expansion with only ten independent multipoles: (11,22)1 N B , H (11,11)0 N B , H (11,11)1 N B , H (21,01)

H (11,00)1 N B , H (11,02)1 N B , H
0 N B , H (21,01)1 N B , H (21,21)0 N B , H (21,21)1 N B , H (21,12)1 N B , (1.31)
all being functions of q cm and q cm . The low energy behavior of those multipoles is parametrized by introducing functions, which are the ten GPs:

P (ρ L ,ρL)S (q cm ) = Lim q cm →0 1 q cm 1 q L cm H (ρ L ,ρL)S N B (q cm , q cm ) (1.32)
with ρ , L', ρ L and S corresponding to the upper indices in equation 1.31.

Drechsel et al. [START_REF] Drechsel | Structure analysis of the virtual Compton scattering amplitude at low-energies[END_REF] have shown that some of these multipoles can be expressed as a function of the same GPs, reducing the number of independent GPs [START_REF] Pierre | Virtual Compton scattering o the nucleon[END_REF]:

H (11,00)1 N B (q cm , q cm ) = q cm √ 3 qcm 2 q 0 cm P (01,01)1 (q cm ) -1 √ 2 q cm 2 P (11,02)1 (q cm ) + O(q cm 2 ), H (11,02)1 N B (q cm , q cm ) = q cm q cm 2 P (11,02)1 (q cm ) + O(q cm 2 ),
H (11,11)S N B (q cm , q cm ) = q cm q cm P (11,11)S (q cm ) + O(q cm 2 ), H (21,01)S N B (q cm , q cm ) = -q cm q cm √ 2P (01,01)S (q cm ) + O(q cm 2 ), H (21,12)1 N B (q cm , q cm ) = -q cm q cm 2 √ 2P (01,12)1 (q cm ) + O(q cm 2 ), H (11,22)1 N B (q cm , q cm ) = q cm q cm P (11,11)1 (q cm ) + O(q cm 2 ), H (21,21)0 N B (q cm , q cm ) = -q cm q 0 cm P (11,11)0 (q cm ) + O(q cm 2 ), H (21,21)1 N B (q cm , q cm ) = -q cm 2 qcm 2 q 0 cm P (11,11)1 (q cm ) -√ 2q cm 2 P (01,12)1 (q cm ) + O(q cm 2 ).

(1.33)

Finally only 6 GPs are necessary to parametrized the hadronic tensor at rst order in q cm , They are : P (01,01)S=0,1 (q cm ), P (11,11)S=0,1 (q cm ), P (11,02)1 (q cm ), P (01,12)1 (q cm ).

(1.34) S = 0 and S = 1 correspond respectively to scalar and spin-ip (or spin) GPs.

This particular denition of GPs allows to nd the connection with the polarizabilities from RCS. The electric polarizability is given by:

P (01,01)0 (q cm = 0) = -4π e 2 2 3 ᾱE . (1.35)
The magnetic polarizability is given by: P (11,11)

0 (q cm = 0) = -4π e 2 8 3 βM . (1.36)
Those relations can also be written at Q 2 = 0, e.g. P (11,11)

0 (Q 2 ) = -4π e 2 8 3 β M (Q 2 ).
The other four GPs are related to the spin-ip polarizabilities of RCS, the detail of this link can be found in [START_REF] Vanderhaeghen | Real and virtual Compton scattering o the nucleon[END_REF].

Structure functions

Now that the amplitude is parametrized by the six GPs we are able to detail the term [M 0 -M BH+B 0 ] from equation 1.29. To do this it is convenient to dene new observables: the structure functions. They are linear combinations of GPs expressed as :

P LL (q cm ) = -2 √ 6m p G E P (01,01)0 (q cm ), (1.37a)
P T T (q cm ) = -3G M q cm 2 q 0 cm (P (11,11)1 (q cm ) -√ 2 q 0 cm P (01,12)1 (q cm )),

(1.37b) (11,11)0 (q cm ) + 3 2

P LT (q cm ) = 3 2 m p q cm Q G E P
Qq cm q 0 cm G M P (01,01)1 (q cm ).

(1.37c)

The important features to remind from these equations are the following:

-P LL is proportional to the electric GP α E (Q 2 ).

-P T T is a combination of spin GPs.

-P LT has a term proportional to the magnetic GP β M (Q 2 ) and a term proportional to a spin GP.

-the electric form factor G E multiplies the scalar GPs.

-the magnetic form factor G M multiplies the spin GPs.

The nal expression of the photon electroproduction cross-section that we need to extract the GPs is :

σ LEX = σ BH+B +Φq cm [v LL (P LL (q cm ) - P T T (q cm ) ) + v LT P LT (q cm )] + O(q cm 2 ), (1.38)
where v LL and v LT are kinematical coecients, detailed in appendix A, which do not depend on q cm but on (q cm , , θ cm , ϕ). Here again, σ is a short notation for d 5 σ exp /dk dΩ k dΩ p cm .

Chapter 2 Theoretical models and motivations for a VCS experiment

In this chapter the theoretical models and their GP predictions are introduced, then the results obtained by the previous VCS experiments are presented. Section 2.1 details the dispersion relation model, which has to be separated from the others since it is the only one which allows a direct extraction of GPs from an unpolarized experiment. In contrast, the LEX approach described in chapter 1 only allows to extract structure functions, i.e. linear combinations of GPs.

Section 2.2 introduces the other theoretical models and presents their GP predictions.

Section 2.3 reviews the previous experiments and their results.

In the last section 2.4 the context of our GP extraction is presented and the motivations for performing this new experiment are explained.

The Dispersion Relation model

The dispersion relation (DR) model was developed by B. Pasquini et al. [37] [38] for RCS and VCS. One of its features is that it cannot predict the scalar GPs behavior versus Q 2 . Indeed, in the model, these GPs contain parameters that are free. The positive consequence is that we can use the DR model to realize a GP extraction by comparing its calculation to our experimental data (of course the extraction will then be model-dependent). In this section only the basics of the DR model are introduced, the details of our DR analysis will be discussed later in section 5.6.

Formalism of the model

The polarizabilities are elementary structure constants of every composite system, from protons to macroscopic media. For each system the dipole polarizabilities describe how it reacts to an external electromagnetic eld. For a macroscopic medium this phenomenological description is done thanks to the dispersion theory and leads to the Kramers-Kronig dispersion relations. So the idea is to apply the dispersion theory to the proton in order to describe the VCS process and access the GPs.

The twelve DR model amplitudes

The construction of the model for RCS and VCS is presented in [START_REF] Drechsel | Dispersion relations in real and virtual Compton scattering[END_REF], and its particular application to VCS is further detailed in [START_REF] Pasquini | Dispersion relation formalism for virtual Compton scattering o the proton[END_REF]. This part presents only the three important steps which are related to the three physics principles that intervene in the model construction: relativistic covariance, causality and unitarity.

The goal is to calculate the particular VCS amplitude:

T V CS = T B + T N B , (2.1)
and more precisely to express the VCS tensor :

M µν = M µν B + M µν N B . (2.2)
We start with the denition of variables used in the DR formalism, the Mandelstam variables:

s = (q + p) 2 , t = (q -q') 2 , u = (q -p') 2 (2.3)
and an additional variable, ν, dened as :

ν = s-u 4mp .
(2.4)

Then the rst step is to construct a complete set of twelve amplitudes in accordance with relativity principles [START_REF] Drechsel | Lowenergy and low momentum representation of the virtual Compton scattering amplitude[END_REF]. An important property for setting up a DR formalism is for those twelve amplitudes to not contain kinematical singularities, thereafter the VCS tensor becomes:

M µν = 12 i=1 F i (Q 2 , ν, t)ρ µν i , (2.5)
with ρ µν i being twelve independent tensors and F i (Q 2 , ν, t) the amplitudes.

The second step consists in the application of causality that requires certain analytic properties of the amplitudes. This leads to relations connecting the real and imaginary parts of the amplitudes, just like the Kramers-Kronig relation, the so-called unsubtracted dispersion relation:

Re[F N B i (Q 2 , ν, t)] = 2 π P +∞ ν thr dν ν Ims[F i (Q 2 ,ν ,t)] ν 2 -ν 2 , (2.6) with Re[F N B i (Q 2 , ν, t)] the real part of the non-Born amplitudes at xed Q 2 and t. Im s [F i (Q 2 , ν , t)]
is the imaginary part of the VCS amplitudes above the pion production threshold, dened Finally, to express GPs from this formalism one has to consider that GPs are dened at the limit q cm → 0. In the DR model, this corresponds to the limit: ν → 0 and t → -Q 2 at nite Q 2 . Therefore, the GPs are expressed as linear combinations of the invariant amplitudes

1 by: ν ≥ ν thr = m π + (m 2 π + t/2 + Q 2 /2)/(2m p ). 1 m π is the pion mass.
F N B i (Q 2 , ν = 0, t = -Q 2 ).
The particular case of the F 1 , F 5 and F 2 amplitudes To satisfy the dispersion relation 2.6 the imaginary amplitudes have to drop fast enough at high energy, which is not the case for two of the twelve amplitudes, F 1 (Q 2 , ν, t) and F 5 (Q 2 , ν, t). Therefore it was decided to add an asymptotic behavior to the dispersion relation 2.6 which becomes:

ReF N B 1,5 (Q 2 , ν, t) = 2 π P νmax ν thr dν ν ImsF 1,5 (Q 2 ,ν ,t) ν 2 -ν 2 + F as 1,5 (Q 2 , ν, t), (2.7)
with ν max xed in the model at 1.5 GeV and F as 1,5 (Q 2 , ν, t) the asymptotic contribution. The asymptotic contributions are treated dierently for F 5 and F 1 .

The F 5 (Q 2 , ν, t) amplitude intervenes in the determination of the spin-ip GPs and its asymptotic contribution is xed in the model. Therefore the DR model cannot be used to extract the spin-ip GPs from experimental data.

The F 1 (Q 2 , ν, t) amplitude is related to the magnetic GP β M (Q 2 ) and its asymptotic contribution is not xed in the model. So the F 1 (Q 2 , ν = 0, t = -Q 2 ) asymptotic contribution depends on a free parameter Λ β which appears in the expression of β M (Q 2 ) :

β DR M (Q 2 ) = β πN M (Q 2 ) + [β M (Q 2 )-β πN M (Q 2 )] Q 2 =0 (1+Q 2 /Λ 2 β ) 2 . (2.8)
Λ β is a free parameter that has to be tted on experimental data to extract β M (Q 2 ). The dipole form in equation 2.8 is arbitrary. It provides nevertheless an easy parametrization of the Q

2 -dependence of β DR M (Q 2 ) asymptotic part. F 2 (Q 2 , ν, t) is the next amplitude discussed here, it is related to the sum β M (Q 2 ) + α E (Q 2 ).
Even if this amplitude satises the dispersion relation 2.6 one has still to consider contributions beyond πN in the calculation of its imaginary part. Indeed the use of DR to the RCS case has shown that approximating the imaginary amplitude to πN contributions only, failed to predict β M (Q 2 = 0) precisely enough. One has to consider higher orders that are not calculated yet.

To this aim, an asymptotic contribution is also considered for F 2 (Q 2 , ν, t). That nally leads to the following DR expression of α E (Q 2 ) :

α DR E (Q 2 ) = α πN E (Q 2 ) + [α E (Q 2 )-α πN E (Q 2 )] Q 2 =0 (1+Q 2 /Λ 2 α ) 2 , (2.9)
where Λ α is the second free parameter of the model.

Results of calculation in the DR model

One may wonder which contribution is more relevant in the DR model for the scalar GPs: the πN one or the asymptotic one. Figure 2.2 displays the asymptotic and πN contributions for the two scalar GPs (these results are obtained with an arbitrary choice : Λ α = Λ β = 0.7 GeV ). As it appears the electric GP is dominated by its asymptotic contribution while the magnetic GP is a sensitive interplay between the two contributions. Although the model focuses on the VCS process, it has the merit to provide also the calculation of the full photon electroproduction process, i.e. including the Bethe-Heitler part, and making it directly comparable to experiment.

It is important to emphasize the role of the DR model in our GP extraction from the LEX analysis. Since the relation between structure functions and scalar GPs depends on spin-ip GPs, which remain experimentally unknown, the LEX approach based on equation 1.38 is powerless to obtain GPs directly. We will need a prediction of the spin-ip GPs in order to xe these quantities in equations 1.37 and for this task we chose the DR model prediction.

Other models

The other models are predictive, in the sense that they predict all the GPs with no free parameters. While they cannot be used for a GP extraction from experiment, it is still interesting to compare their calculations.

Chiral perturbation theories

The chiral perturbation theory (χPT) is an eective eld theory that successfully calculates meson properties. In χPT the chiral symmetry and its breaking are implemented to a low-energy Lagrangian. Then the theory of perturbation is used to expand the calculation along a chosen quantity of the considered interaction; for instance the proton four-momentum and the order n at which the calculation stops are written as O(p n ). That way, chiral perturbation theories predict fundamental quantities, among which the RCS polarizabilities, the scalar GPs and the spin GPs. The rst description of χPT was done by Weinberg [START_REF] Weinberg | Phenomenological Lagrangians[END_REF], more information about such theories applied to baryons can be found in the review paper from V.

Bernard [START_REF] Bernard | Chiral Perturbation Theory and Baryon Properties[END_REF].

For VCS two dierent chiral perturbation theories give a prediction for the GPs: the heavy baryon chiral perturbation theory (HBχPT) [START_REF] Bernard | Chiral dynamics in nucleons and nuclei[END_REF] and the baryon chiral perturbation theory (BχPT) [START_REF] Bernard | Chiral expansion of the nucleon's electromagnetic polarizabilities[END_REF]. For both, all the low-energy constants that appear in the constructed Lagrangian are known. So these highly constrained models will predict GP values to compare with experimental data.

The BχPT just extends the theory to include a matter eld. So a new eective meson-baryon Lagrangian is constructed. The theory only considers the three low-mass quarks (u,d,s), the SU(3) avor symmetry and the eight pseudo-scalar mesons that result from its breaking (the pions, the kaons and the η). The results presented in table 2.2.1 were obtained by V. Bernard et al. [START_REF] Bernard | Chiral expansion of the nucleon's electromagnetic polarizabilities[END_REF] at order O(p 3 ).

In the HBχPT framework the nucleon is considered as extremely heavy, and so non-relativistic. This idea was introduced by Jenkins and Manohar [START_REF] Jenkins | Baryon chiral perturbation theory using a heavy fermion Lagrangian[END_REF] and requires to use the SU(2) symmetry instead of SU(3). As a result only three pseudo-scalars are predicted by the breaking of the symmetry, they are the pions.

Thus the model introduces the nucleon's pion cloud and its interaction with the VCS photon. Figure 2.4 shows the diagrams typically calculated and the HBχPT calculation of GPs from [START_REF] Hemmert | Generalized polarizabilities of the nucleon in chiral eective theories[END_REF] is drawn in gure 2.6. The ∆(1232)-resonance is ignored by theses theories, but it has been shown that including the ∆(1232)-resonance as an explicit degree of freedom impacts signicantly the predictions. Furthermore, in their paper [START_REF] Lensky | Predictions of chiral perturbation theory for Compton scattering o protons[END_REF] The predictions are presented for BχPT and HBχPT; the calculations are done up to the order O(p 3 ) and even O(p 4 ) for NEW (but the ∆(1232)-resonance is not considered at the O(p 4 ) order) . The compilation of RCS measurements from [START_REF] Olive | Particle data group[END_REF] is shown as a comparison.

BχPT

BχPT (NEW) HBχPT HBχPT (NEW) PDG [START_REF] Olive | Particle data group[END_REF] ᾱE (10 -4 f m 3 ) The electromagnetic couplings of these resonances are adjusted on photo-decay amplitude ts to experimental data. The X symbol of gure 2.5 stands for the π 0 or the σ (ππ) meson and the coupling constants are deduced from other models (more informations are given in [START_REF] Vanderhaeghen | Virtual Compton scattering study below pion production threshold[END_REF]).

The main limitation to the model comes from the evaluation of the πγγ coupling constant. Indeed the experimental photo-decay amplitude only xes the on-shell eects. This may explain the discrepancy between the experimental RCS polarizabilities and the ones predicted by the model.

A new determination of polarizabilities was performed in an eective Lagrangian model framework [START_REF] Yu | Nucleon polarizabilities in virtual Compton scattering[END_REF]. In this work a more general treatment of the ∆(1232) contribution was implemented by adding an o-shell degree of freedom. This calculation is drawn in gure 2.6. The prediction of α E (Q 2 = 0) is still in bad agreement with the experimental values, as for the work of reference [START_REF] Vanderhaeghen | Virtual Compton scattering study below pion production threshold[END_REF].

Linear sigma model

In the linear sigma model (LSM), originally developed in reference [START_REF] Gell-Mann | The axial vector current in beta decay[END_REF], the proton structure is described by a pointlike nucleon that interacts with pions and sigmas.

The LSM was developed for VCS by A. Metz and D. Drechsel relation gives rise to the Goldberger-Treiman relation [START_REF] Goldberger | Decay of the pi meson[END_REF] that is used to determine the pion-nucleon coupling constant g πN :

g πN = g A m N Fπ = 13.4, (2.10)
with m N the nucleon mass, g A the axial coupling constant and F π the pion decay constant. g πN is used to calculate all the one-loop diagrams (gure 2.4) contributing to the GPs.

The LSM considers an innite sigma mass to avoid all the Feynman diagrams containing the sigma propagator. The calculation [START_REF] Metz | Generalized polarizabilities of the nucleon studied in the linear sigma model[END_REF][32] predicts the following RCS polarizabilities:

ᾱE = 7.5 • 10 -4 f m 3 , βM = -2 • 10 -4 f m 3 .
Since the LSM ignores the ∆(1232)-resonance the magnetic polarizability is low. The Q 2 -behavior of scalar and spin GPs according to LSM is represented in gure 2.6.

The non-relativistic quark model nRQM

The nRQM is a general model for nucleon structure which was introduced by O.

W. Greenberg and Resniko [START_REF] Greenberg | Symmetric Quark Model of Baryon Resonances[END_REF]; it was later improved by Isgur and Karl [START_REF] Isgur | P Wave Baryons in the Quark Model[END_REF].

The strength of the model comes from its simplicity; indeed, in the approximation of equal masses for quarks only three parameters dene the model:

-The quark mass which is xed thanks to the proton magnetic moment. -The chromatic ne structure constant α s which explains the mass degeneracy between the proton and the ∆(1232).

The model was then adapted to Compton scattering and polarizabilities by G. Q. Liu et al. [START_REF] Liu | Virtual Compton scattering from the proton and the properties of nucleon excited states[END_REF]. It considers that the electric polarizability is ruled by the electromagnetic transition E1 between the proton and its resonances. The magnetic polarizability is deduced from the M1 transition P→ ∆(1232). The predicted behavior of GPs is represented in gure 2.6. As mentioned in [START_REF] Pasquini | Generalized polarizabilities of the proton in a constituent quark model revisited[END_REF] there is room to improve the model since it does not take into account important considerations such as relativity, gauge invariance and chiral symmetry.

However the predictions provide an order-of-magnitude estimation for the nucleon resonance contributions.

As a conclusion, gure 2.6 from [START_REF] Pasquini | Generalized polarizabilities of the proton in a constituent quark model revisited[END_REF] shows the predictions of all the presented models. Models agree on a smooth fall-o shape for the electric GP, they also agree to predict a more complicated shape for the magnetic GP. The spin-ip GP predictions are rarely in agreement from a model to another.

As a more general conclusion, the polarizabilities in RCS and furthermore in VCS are good observables to test the validity of various models of nucleon structure. Experimental data on polarizabilities are therefore welcome, and the present thesis is part of this experimental eort.

Previous and current VCS Experiments

In this section we will remind the previous scalar GPs measurements by presenting the explored phase space, the results and the specicities of each analysis. Then we will briey introduce a recent measurement at MAMI that uses the ∆ resonance to extract the electric GP at Q 2 = 0.2 GeV 2 .

The rst MAMI experiment

The very rst dedicated VCS experiment has been performed in 1996 at MAMI by the A1 collaboration [START_REF] Roche | The First determination of generalized polarizabilities of the proton by a virtual Compton scattering experiment[END_REF]. This experiment is quite similar to the one presented in this thesis. The absolute photon electroproduction dierential cross section has been measured at a virtuality Q 2 = 0.33 GeV 2 and = 0.62.

The beam energy was about 855 MeV. The scattered electron and the recoiling proton were detected in coincidence with the same two high-resolution magnetic full thin lines: dispersion-relation calculation [START_REF] Metz | Dispersion relation formalism for virtual Compton scattering and the generalized polarizabilities of the nucleon[END_REF]. Figure taken from [START_REF] Pasquini | Generalized Polarizabilities in a Constituent Quark Model[END_REF]. spectrometers that we used: A and B. The same liquid hydrogen target of 49.5 mm long was also used.

The explored phase space was dened by: -a range in q cm going from low q cm (33.6 MeV/c) for the renormalization, up to close to the pion production threshold (q cm (max) = 111.5 M eV /c).

-an in-plane measurement, mostly at ϕ = 180 • (which is the hemisphere opposite to the BH peaks).

-basically all polar angles of the nal photon, with θ cm from -141 • to 6 • .

The LEX approach was used to extract the structure functions with the following results [START_REF] Roche | The First determination of generalized polarizabilities of the proton by a virtual Compton scattering experiment[END_REF]:

P LL -P T T / = (23.7 ± 2.2 ± 0.6 ± 4.3) GeV -2 , P LT = (-5.0 ± 0.8 ± 1.1 ± 1.4) GeV -2 .
The rst error is statistical only. The two others are systematic: the rst one corresponds to the normalization of the angular distributions, the second one stems from the distortion of the distributions.

Later, a DR analysis was performed and conrmed the results [START_REF] Hose | Virtual Compton scattering at MAMI[END_REF] (table 4, method 4):

P LL -P T T / (DR) = (23.2 ± 3.0) GeV -2 , P LT (DR) = (-3.2 ± 2.0) GeV -2 ,
where the errors are purely statistical.

JLab VCS experiment (E93-050)

The JLab measurements were performed in 1998 [63] [64]. An electron beam of 4.030 GeV energy was directed onto a 15 cm liquid hydrogen target. The two high-resolution spectrometers of Hall A were used to detect the scattered electron and the outgoing proton in coincidence.

Extractions of GPs were done at two setups (I and II) corresponding to two values of Q 2 : 0.92 and 1.76 GeV 2 for an of respectively 0.95 and 0.88. The setup-I at Q 2 = 0.92 GeV 2 was split in two settings : I-a that contained events under pion production threshold and I-b that was exploring the ∆(1232) resonance.

Measurements were done with in-plane and out-of-plane data.

The cross sections were calculated using the proton form-factor parametrization from Brash et al. [START_REF] Brash | New empirical ts to the proton electromagnetic form-factors[END_REF]. Both LEX and DR analysis have been performed (since the I-b setting was above the pion production threshold no LEX approach was used for it). Table 2.3.2 presents the results for the structure functions.

analysis

Q 2 (GeV 2 ) P LL -P T T / (GeV -2 ) P LT (GeV -2 ) I-a LEX 0.92
1.77 ± 0.24 ± 0.70 -0.56 ± 0.12 ± 0.17

MIT-Bates experiment

The MIT-Bates experiment was performed in 2000-2001 [66][67]. In this experiment the GPs were extracted at a very low Q 2 of 0.057 GeV 2 and = 0.9. The target was 1.6 cm of liquid hydrogen and the experiment used Out-of-Plane Spectrometers for proton detection.

The Bates VCS experiment was designed to make an azimuthal separation of P LL -P T T / and P LT by taking data simultaneously at ϕ angles of 90 • , 180 • and 270

• , at xed θ cm = 90 • . Data were taken at photon energies q cm ranging from 43

MeV up to 115 MeV.

Both DR and LEX analyses have been performed with very dierent results

for P LT :

P LL -P T T / (DR) = (46.7 ± 4.9 stat ± 3.4 syst ) GeV -2 , P LT (DR) = (-8.9 ± 4.2 stat ± 1.4 syst ) GeV -2 . P LL -P T T / (LEX) = (54.5 ± 4.8 stat ± 3.4 syst ) GeV -2 , P LT (LEX) = (-20.4 ± 2.9 stat ± 1.4 syst ) GeV -2 .
The Bates experiment was the rst one to explicitly show that one has to carefully use the LEX. Indeed the LEX extraction is only valid in kinematics where O(q cm 2 ) in equation 1.38 is small enough to be neglected. The Bates experiment proved that the higher orders had a strong impact on its in-plane kinematics (where ϕ = 180 • ) and were responsible of the large dierence between the LEX and DR P LT values.

From the RCS electric polarizability and this new measurement one was able to determine the polarization radius of the proton: < r α >. This radius, which is about twice larger than the proton charge radius, is an evidence of the mesonic eects in the electric polarizability.

The Doubly polarized VCS experiment at MAMI

In 2005-2006 a new experiment was done at MAMI [68][69]. This experiment has been performed at = 0.645 and at the same Q 2 as the rst one, 0.33 GeV 2 , with the aim of gaining insight into the proton spin-ip GPs for the rst time.

This requires to measure doubly polarized VCS. The studied reaction was then e p → e p γ, obtained thanks to the MAMI capacity to produce a polarized electron beam; the outgoing proton polarization was measured by a focal-plane polarimeter.

The P ⊥ LT structure function was extracted for the rst time. Since this experi- ment is kinematically close to the rst MAMI experiment the scalar GPs were also extracted, by an unpolarized analysis, in order to conrm the previous result.

Only a LEX analysis was performed, with several dierent parametrizations for the proton form factors. The P LL -P T T / results are in agreement from a parametrization to an other but important variations were observed for P LT . With the Friedrich-Walcher parametrization of form factors [START_REF] Friedrich | A Coherent interpretation of the form-factors of the nucleon in terms of a pion cloud and constituent quarks[END_REF] the following structure functions were extracted:

P LL -P T T / = (27.1 ± 1.9 stat ± 3.0 syst ) GeV -2 , P LT = (-8.0 ± 0.7 stat ± 2.2 syst ) GeV -2 .
An iterative approach was used to determine the nal values, i.e. the obtained structure functions were reintroduced in the LEX analysis at each step. The comparison with the rst MAMI experiment tends to conrm the non-trivial Q 2 behavior observed in this region.

The VCS ∆ experiment at MAMI

In parallel to our experiment, another VCS experiment was performed at MAMI in 2013 at Q 2 = 0.2 GeV 2 and is currently in the very end of the analysis process [START_REF] Sparveris | A1 Proposal, Proposed experiment to study the nucleon structure by measurement of Virtual Compton Scattering at the Delta resonance at MAMI[END_REF].

It also uses the standard A1 setup, with the two spectrometers A and B detecting the outgoing electron and proton and using the 49.5 mm long liquid hydrogen target. But the measured phase space is dierent since this experiment explores the ∆(1232) resonance region, i.e. above the pion production threshold.

So this experiment did not use a LEX approach but measured the in-plane azimuthal asymmetry of the photon electroproduction cross section:

A (ϕ=0) = [σ ϕ=0 -σ ϕ=180 ]/[σ ϕ=0 + σ ϕ=180 ], (2.11)
which is sensitive to both the electric GP and the Coulomb quadrupole amplitude (C2)

2 . The experiment aims to extract these two quantities. The sensitivity to each one varies in a dierent way versus θ cm . A large range in θ cm is then explored in order to extract those two quantities from the same reaction channel.

Above the pion production threshold the only way to access α E (Q 2 ) is thanks to the DR model, so the experiment will provide a model-dependent extraction of the electric GP, that will be compared with our own results at the same value of

Q 2 = 0.2 GeV 2 .

Motivation for a new experiment: vcsq2

This section describes the motivations for our experiment [START_REF] Merkel | A study of the Q 2 -dependence of the structure functions and the generalized polarizabilities in VCS at MAMI[END_REF] which is designed to extract the electric and magnetic GP at three new values of Q 2 = 0.1; 0.

Q 2 α E (Q 2 ) β M (Q 2 ) (GeV 2 ) (10 -4 f m 3 ) (10 -4 f m 3 )
LEX analysis MIT-Bates* 0.057 0.9 The measurements are done at dierent , the exact values are specied in table 2.4.

Some points are shifted in Q 2 for visibility.

This situation was the physical motivation to explore in more depth the intermediate Q 2 range with a new VCS MAMI-experiment, at Q 2 values that will surround the previous one: The world results for the second structure function P LT are represented with respect to Q 2 in gure 2.8. For P LT , and so for the magnetic GP, some models predict a Q 2 -shape that contains an extremum in the low-Q 2 region, due to the interplay between the paramagnetic and the diamagnetic proton responses. But, since the experimental uncertainties are large relative to the P LT values, it is dicult to conrm the predictions. However the three new measurements, performed with a large statistics, may help to clear the picture. Investigating the GPs behavior will allow to gain a deeper understanding of the structure and the dynamics of the proton.

Q 2 = 0.1 GeV 2 , Q 2 = 0.2 GeV

Experimental conguration

The following chapter is dedicated to the experimental conguration, from the presentation of the facility and the spectrometers to the presentation of our experimental choices. The great advantage of MAMI is its capacity to provide a high intensity beam with a 100% duty cycle. Indeed the frequency of accelerating cavities is 2.45 GHz, this generates a beam with a neat temporal structure, i.e. each bunch of electrons is separated in time by only 0.4 ns. Thus the beam temporal structure is invisible for the detectors, for instance of A1, which have a time resolution of the same order of magnitude. The beam is then called a continuous wave beam. Typically in our experiment, with particles detected in coincidence by two spectrometers, the high duty cycle of MAMI is essential to increase the ratio of true coincidences over accidental ones.

Operating principles of MAMI

This section is a brief description of the MAMI design which is more detailed in [START_REF] Jankowiak | The Mainz Microtron MAMI: Past and future[END_REF]. Like in all microtrons the electron acceleration is done through linear sections. MAMI was designed to realize this acceleration while saving space. A RTM, gure 3.2, is used to avoid the building of a too long accelerator.

It contains a LINAC combined with two magnets that will bend the electron trajectory in order to drive it back to the linear section, where the electron will be accelerated again. So electrons will make several turns and at each turn their energy increases. The electron trajectory therefore increases in radius at each turn, giving to the RTM this particular design which inspired its name.

After 18 turns in RTM1 the electrons reach an energy of 14 MeV and are injected in the second and larger RTM. In RTM2 they again undergo the same acceleration process and after 51 turns their energy is about 180 MeV. At this A HDSM is composed of four 90

• -bending magnets and two linear accelerators.

The frequency of each LINAC is adjusted in order to prevent the beam from instabilities. After 43 turns in this last stage the electrons can reach an energy of 1508 MeV before getting directed to experimental halls.

A1 experimental hall

The physics program of the A1 collaboration is focused on electroproduction reactions on nuclear targets that probe hadron structure, such as: elastic scattering processes, electroproduction of mesons or photons, etc. The A1 experimental hall is easily identiable in gure 3.1 by its three-spectrometer set-up [START_REF] Blomqvist | The three-spectrometer facility at the Mainz microtron MAMI[END_REF]. These large and heavy spectrometers can be independently rotated around the central axis where the target is placed. Each spectrometer can be operated in a single, double or triple coincidence mode depending on the studied physical process. It is also possible to raise the spectrometerB in an out-of plane position and thus reach specic regions of the angular phase space.

The liquid hydrogen target

The vacuum scattering chamber, at the pivot point of the three spectrometers, holds several targets on a ladder. The ladder moves to select which target will be hit by the beam depending on the need of the experiment. For our experiment the carbon target was used for calibration, the aluminum oxide screen was needed for beam position monitoring and the liquid hydrogen target was the one necessary to realize the actual data taking.

In order to perform our GP extraction we need to control and measure the luminosity because we measure an absolute cross section. But, the luminosity The target average density and temperature are 0.069 g.cm -3 and 22 • K, its pressure is approximately 2 bar.

The high-resolution spectrometers

In our experiment the particles are detected in coincidence, the electron by spectrometer B and the proton by spectrometer A. The deecting power of the spectrometers allows to reconstruct four particle properties at the target level in the considered spectrometer frame: θ 0 and φ 0 the particle vertical and horizontal angles, y 0 the coordinate of the particle track along the transverse horizontal axis, Thereafter the informations from the beam position and both spectrometers are gathered in order to reconstruct, in the lab frame, the vertex position and the missing mass.

The following sub-section summarizes the spectrometer reference frames and optics. In the next subsection the spectrometer A and B characteristics are detailed.

Spectrometer frames, Focal plane frame and optics

Two frames are important to dene: the focal plane reference frame where the track reconstruction is performed; and the spectrometer frame which is at the target side. The spectrometer frame diers from the laboratory frame only by a rotation corresponding to the spectrometer position in the hall. The focal plane is a virtual plane located at the exit of the magnets, which is dened by an x axis perpendicular to the particle track and an upwards z-axis in the direction of the particle trajectory (the coordinates are represented in gure 3.6).

Figure 3.6 is a schematic representation of a particle track in spectrometer A.

The dashed blue line is the reference track, i.e. the trajectory of a particle of The coecients have been determined by calibration measurements on specic targets with specic collimators [START_REF] Blomqvist | The three-spectrometer facility at the Mainz microtron MAMI[END_REF].

δ = 5 ijkl=0 D ijkl (x f p -x ref ) i (θ f p -θ ref ) j (y f p -y ref ) k (φ f p -φ ref ) l , θ 0 = 5 ijkl=0 T ijkl (x f p -x ref ) i (θ f p -θ ref ) j (y f p -y ref ) k (φ f p -φ ref ) l , y 0 = 5 ijkl=0 Y ijkl (x f p -x ref ) i (θ f p -θ ref ) j (y f p -y ref ) k (φ f p -φ ref ) l , φ 0 = 5 ijkl=0 P ijkl (x f p -x ref ) i (θ f p -θ ref ) j (y f p -y ref ) k (φ f p -φ ref ) l , (3.1

Spectrometer A and B global characteristics

The spectrometers A and B share the same principle but dier by their size, capacity and conception. The spectrometer C, not detailed here, is a scaled-down version of spectrometer A.

The spectrometer A holds four magnets, a quadrupole, a sextupole and two dipoles, in a so-called QSDD conguration. The two dipoles generate the particles dispersion while the quadrupole applies a focus to particles in the transverse direction. The sextupole performs corrections of aberrations in the dispersive plane.

By consequence of this particular quadrupole-sextupole design, the spectrometer's angular acceptance is large but the reconstruction is parallel-to-point in the transverse direction 1 , which means that a segment in target coordinates corresponds to a point in the focal-plane coordinates, and so the reconstruction of y A 0 suers from a poor resolution. However this allows the precise rst-order determination of φ A 0 through the measurement of y f p [START_REF] Blomqvist | The three-spectrometer facility at the Mainz microtron MAMI[END_REF].

In the dispersive direction the reconstruction is point-to-point 2 , so the dis- persive angle θ A 0 and the relative momentum δ A are reconstructed with a good 1 Implying < y 0 |y f p >≈ 0, while point-to-point would imply < ϕ 0 |y f p >≈ 0. 2 Implying < θ 0 |x f p >= 0.

resolution.

The spectrometer B compensates the spectrometer A's poor resolution in transverse direction by a more simple design. Indeed it contains a single dipole with a eld gradient that adds quadrupole and sextupole contributions to ensure the focus. As a result, a point-to-point optics is obtained both in the dispersive and non-dispersive planes, and the momentum and position resolutions of spectrometer B are high. The counterpart is its small solid-angle and momentum acceptance.

The other important characteristic of spectrometer B is its capacity to be tilted above the horizontal plane up to 10

• in the so called out-of-plane position. This will allows us to measure particles from the photon electroproduction process when ϕ cm = 0 and access specic and relevant regions of the phase space.

The main properties of the two spectrometers are summarized in table 3

Focal-plane detectors

For each spectrometer the standard detector package is composed of two pairs of vertical drift chambers (VDC), two scintillator planes and a Cerenkov detector (the conguration can be changed to use specic detectors like a polarimeter). In this section, based on [START_REF] Blomqvist | The three-spectrometer facility at the Mainz microtron MAMI[END_REF], the detectors are introduced in an order that follows the particle trajectory in the spectrometer.

Vertical drift chambers

The vertical drift chambers (VDC) are the rst stage of detection. They aim to determine the focal-plane coordinates (x, θ, y, φ) f p .

Their operating principle is represented in gure 3.7. A VDC has a planar structure with a plane of wires which is surrounded by two planes of high voltage cathodes. Two kinds of wires compose the wire plane: the wider wires which are called potential wires; and the thinner signal wires. The chamber is lled with a mixture of equal parts of argon and isobutane, plus 1.5% of ethanol.

When a charged particle passes through the VDC it ionizes the gases generating secondary electrons. The voltage dierence between cathodes and wires will make the electrons drift. Then when they approach a signal wire, the secondary electrons at their turn start to ionize the gases and generate an electron avalanche.

The potential wires serve to optimize the eld lines repartition. The start of the avalanche is detected in the signal wires. The signal of wires close to the particle trajectory starts earlier than the signal from more distant wires. • with respect to the reference trajectory. In this way the number of wires hit by a particle is increased.

To measure the four coordinates in the focal plane each VDC holds two planes of wires. The X 1 and X 2 planes, composed of wires parallel to ( y f p ), determine the particle momentum and vertical angle. The S 1 and S 2 planes, with wires rotated from 40

• with respect to the X plane, measure the two other coordinates.

Thanks to this design the overall tracking eciency is better than 99.9% (99% for a single plane) and the following resolutions are achieved [77] :

δx f p = 100 µm δy f p = 200 µm δθ f p = 0.3 mrad δφ f p = 1 mrad.

Scintillators

The next detectors encountered by a particle are two scintillator planes; one which is 3 mm thick, called "dE" for energy loss, and the second which is 10 mm thick and called "ToF" (for Time of Flight). Each plane of spectrometer A (B) has 15 The third use of scintillators is for particle identication. Indeed the PMT signal is also integrated by an ADC and this information is proportional to the energy lost by the particle in the scintillator paddle. A proton of a given momentum will deliver more energy than a pion of the same momentum in our low-energy regime.

Cherenkov detector

The last sub-detector is a threshold gas Cerenkov counter, which is used for particle identication between pions and electrons. Pions of a given momentum cross the detector at a lower speed than electrons of the same momentum, the Cerenkov detector is adjusted in a way that only the electrons generate Cerenkov light (i.e. Aqua++ combines them in the event builder. The trigger process is further detailed in [START_REF] Blomqvist | The three-spectrometer facility at the Mainz microtron MAMI[END_REF].

After the detection of a particle the recorded informations in the runle are:

-the drift time of each activated wire -the TDC and ADC information in each scintillator -the PMT signal of Cerenkov detectors -the target conditions -the wobbler information -the content of scalers for luminosity calculation.

Those informations are sucient to reconstruct the particle momentum and trajectory, and also the vertex position in the target.

The A1 software stack

The A1 collaboration uses an in-house built software package [START_REF] Weis | Data acquisition and analysis for the 3-spectrometer-setup at MAMI[END_REF] that allows to read out the detector electronics, do the data acquisition and perform both online and oine analysis. The four softwares involved are: Mezzo for the slow control, Aqua++ for the data acquisition, Lumi++ for luminosity calculation, and Cola++ for the analyses.

Mezzo is the slow control software, i.e. it records numerous parameters describing the status of the apparatus: spectrometers, beam and target. Its readout is sent directly to the central process via fast Ethernet at a typical frequency of 1

Hz, and thus can be seen by the user to check the experiment conditions online.

Aqua++ merges all data from spectrometers into events and saves them on a disk. The duration of data collection is limited, a single data-taking is a so-called run, and the choice of run's length is left to the user.

Coincidence events are recorded, but the single-arm events are also kept for the use of calibration. For this purpose a counter module is a part of the experimental trigger, it allows the prescale of single-arm events from each spectrometer separately.

Figure 3.9: Schematic view of the Aqua++ event builder threads, from [START_REF] Weis | Data acquisition and analysis for the 3-spectrometer-setup at MAMI[END_REF].

Figure 3.9 is a schematic view of the Aqua++ event builder threads [START_REF] Weis | Data acquisition and analysis for the 3-spectrometer-setup at MAMI[END_REF], it illustrates how the event builder merges informations from each spectrometer's front-end, from a user interface (SQL run database) and from the Mezzo slow control. In this way, the slow control data are also available for oine analysis.

Lumi++ is the software used for online and oine luminosity calculations. The integrated luminosity of a run is the product of incident electrons number during the run, by the number of target nuclei per unit area:

L = Q tot e • ρx tg N A A , (3.2)
with e = 1.6 • 10 -19 C the elementary charge, N A = 6.022 • 10 23 mol -1 the Avogadro constant, A = 1.008 g • mol -1 the hydrogen molar mass and ρ the target density. xtg is the averaged target length 3 . The total charge Q tot is obtained by the 3 The particular shape of the target with its round endcaps is not considered by Lumi++. integration of the beam current (measured by a Foerster probe) in time intervals (measured by scalers). All these informations are gathered over time and used by Lumi++. Furthermore the software also uses informations from other scalers which measure the dead time.

Cola++ is the data analysis software that is used both in online and oine analyses. It reads run les to build histograms readable by the user and also provides tools for: the manipulation of four-vectors, the customization of data cuts, and the design of events weighting. To function properly Cola++ needs three input les:

-The run.db le (for run database le) contains parameters, ordered in time, that describe the experimental conditions and have to be calibrated, see chapter 4.

-The .col le contains the list of histograms to be produced by Cola++ and the analysis conditions: the events selections, the denition of variables elaborated from raw variables (from run les), the events weight, etc.

-The .tma les (for transfer matrix les) detail the optics parameters dened in section 3.2.2.

Choice of kinematical settings

The VCS phase space is dened by the ve variables (q cm , q cm , , θ cm , ϕ cm ). In the lab frame particles from the ep → epγ process are emitted in a large range of directions, but with our spectrometers acceptance we only have access to a limited part of the VCS phase space. Moreover all the phase-space regions are not of equal interest. For instance in the Bethe-Heitler peaks the electron bremsstrahlung is too dominant to perform a GP extraction. So we need to carefully decide where the extraction will be done and so where the A1 spectrometers are positioned.

The leptonic-side variables

Among the ve important variables, three are xed by the leptonic part of VCS (q cm , q cm , ). These variables are xed by the electron lab-frame variables: (k, k , θ e ), and so they are xed by the electron spectrometer and the beam. Their value is deduced from three constraints.

First we want to perform our measurement around Q 2 = 0.2 GeV 2 , this xes q cm . Secondly, since in the LEX equation 1.38 the GP eect depends on q cm , we need to increase q cm to magnify the eect. But in order to perform our LEX analysis q cm has to remain under the pion production threshold.

The third and last constraint is that we want to have as high as possible, i.e.

close to 1. Indeed in equation 1.38 is in factor of the structure functions through the v LL and v LT coecients.

Another criterion could be considered: it is essential to have a large range in q cm to increase the lever arm in our LEX analysis. Fortunately this is ensured by the large momentum acceptance of the spectrometers.

The center-of-mass angles of the Compton Scattering process

We have more freedom in our choice of center-of-mass angles θ cm and ϕ cm (cf. gure 1.8) and it seems a priori that any value out of the Bethe-Heitler peaks would be relevant for a GP extraction. However it appears that it is not that simple.

The explored region in θ cm and ϕ cm depends on the spectrometers acceptance. It also depends on Q 2 , the higher Q 2 is the narrower will be the cone drawn by the outgoing proton and the larger will be the phase space probed in our experiment.

But at Q 2 = 0.2 GeV 2 it is impossible to explore the whole (θ cm ,ϕ cm ) space with the acceptance of our spectrometers with only one setting. So we have to decide which settings are relevant and three criteria motivated our choice.

The rst criterion is the importance of the GP term compared to the cross section σ BH+B . At q cm = 112.5 M eV /c, gure 3.10 shows the relative theoretical GP eect according to the LET: relative GP ef f ect = Φq cm [v LL (P LL (q cm ) -P T T (qcm) ) + v LT P LT (q cm )] σ BH+B . (3.3) Two relevant regions of the (θ cm ,ϕ cm )-plane appear to have an important GP eect:

-The rst region for ϕ cm ≈ 0 and cos(θ cm ) ≈ -1 with a GP eect of approximately +20%.

-The second region when ϕ cm ≈ ±100 • and cos(θ cm ) ≈ 0 with a GP eect of approximately -10%.

We will now consider these two regions and we will check if they also satisfy the two remaining criteria. If so, our spectrometers will be xed in order that their acceptance roughly corresponds to the considered region.

The second criterion is the weight of each structure function independently in the GP eect of equation 3.3. We wanted to realize two settings, one which will be more sensitive to the electric GP and another that will be more sensitive to the magnetic GP. This selective sensitivity is determined by the values of the coecients v LL and v LT , according to equation 1.38. Figures 3.11 and 3.12 show the theoretical behavior of the coecients v LL and v LT in the (cos(θ cm ),ϕ cm )plane at xed q cm = 458 M eV /c and = 0.85. The previously selected region for ϕ cm ≈ 0 and cos(θ cm ) ≈ -1 is still interesting since it corresponds to a domain where v LL ≈ 0. The second selected region where ϕ cm ≈ ±100 • and cos(θ cm ) ≈ 0 contains points where v LT = 0 and therefore should be highly sensitive to the P LL -P T T / structure function. So the selected regions also satisfy this second criterion.

The third and last criterion is to have a large range in v LL and v LT coecients.

Indeed to realize the nal structure function t one has to dispose of a large lever arm in the coecients in order to increase the global sensitivity of our t. Since the two selected regions stand for respectively opposite value of the coecients, as it appears on gures 3.11 and 3.12, they will oer the necessary lever arm.

A last criterion: the LEX validity

From the BATES experiment [START_REF] Bourgeois | Measurements of the generalized electric and magnetic polarizabilities of the proton at low Q**2 using the VCS reaction[END_REF][67] we know that ignoring the higher-order terms in the LEX analysis is not always safe. Indeed the higher-order terms ( O(q cm 2 )

in equation 1.38) depend on the (θ cm ,ϕ cm ) angles and become in some cases too strong to be neglected. So we wanted to identify the regions where higher-order eects are small enough in order to realize a LEX extraction as meaningful as possible.

To do that we build a theoretically-based criterion which provides an evaluation of the higher-order eect. For a coherent and xed choice of structure functions we calculate the theoretical photon electroproduction cross-section from LEX (σ LEX ), i.e. truncated at rst order. We also calculate the theoretical cross section predicted by the DR model (σ DR ) where we keep the exact same choice of structure functions.

Even if the higher-order terms are not specically expressed in the DR cross section they are included as a whole in the calculation. Then the comparison of the DR cross section to the LEX one will provide an estimation of the higherorder terms. So O DR (q cm 2 ) is the criterion representing the relative importance of higher-order terms in the LEX cross section, according to the DR model. It is written as:

O DR (q cm 2 ) = |σ LEX -σ DR | * 100 σ BH+B .
(3.4) Figure 3.13 shows the value of this theoretical criterion in the (cos(θ cm ),ϕ cm )-plane and also a two-dimensional representation where only the bins where O DR (q cm 2 ) < 2% remain (in blue). This second representation helps to identify the considered acceptance areas which are represented by red perimeters, the dashed black area stands for the BH peaks. Later in section 5.5.2 we will see how applying a cut on O DR (q cm 2 ) impacts our analysis.

The rst selected region (for ϕ cm ≈ 0 and cos(θ cm ) ≈ -1) appears to be of interest since it contains kinematical points where the higher-order eect is negligible according to the DR model. But the region also contains kinematical points where the higher orders seem to be important and where a LEX analysis may be adventurous. Later in section 5.5.2 we will show how we can get rid of these kinematical points. The higher orders appear to be smoother in the second selected region (ϕ cm ≈ ±100 • and cos(θ cm ) ≈ 0). 2 ) in the (cos(θ cm ),ϕ cm )-plane at q cm = 87.5 M eV /c and (right) two-dimensional representation of the angular region where O DR (q cm 2 ) < 2% (blue), the red squares correspond to the two areas of interest to perform the GP extraction.

To conclude, all the presented criteria are fullled by the regions indicated in red, which were at the basis of the denition of our spectrometer settings. The contour drawn by the spectrometers acceptance is actually larger and varies with respect to q cm .

3.3.4

The Q 2 = 0.2 GeV 2 settings Two main settings (corresponding to dierent spectrometers position) were designed in order to probe the regions of interest. A so-called in-plane setting containing the (ϕ cm ≈ 0 and cos(θ cm ) ≈ -1) region which will be more sensitive to P LT and β M (Q 2 ), and a so-called out-of-plane setting which used the out-ofplane capability of spectrometer B to probe the second region more sensitive to P LL -P T T / and α E (Q 2 ). These two settings are centered on values of q cm as high as possible but still below the pion threshold.

An important step of the analysis is the renormalization procedure which will be presented in section 5.4. To perform it we also need data where the GP eect is very small. This is realized with supplementary settings centered at low values of q cm (≈ 25 -50 MeV/c). They are called the low-q cm settings.

The data taking at Q 2 = 0.2 GeV 2 took place in 2011 but a problem was encountered that required us to redo the out-of-plane setting in 2015. We then decided to add another setting in a third region of interest, the forward out-ofplane setting. Table 3.3.4 gathers the spectrometers settings analyzed at Q 2 = 0.2GeV 2 .

setting reconstructed by both spectrometers A and B. A wrong calibration will then impact directly its resolution, so by minimizing the resolution of M 2 X one can identify the optimal oset values.

E beam P B θ B OOP B P A θ A name (MeV) MeV/c ( o ) ( o ) MeV/c ( o ) Q 2 = 0.
For the ep → epγ reaction, the missing-mass squared of the real photon is centered on 0 M eV 2 with a radiative tail at positive M 2 X . The next channel is the pion production ep → epπ 0 which corresponds to a missing-mass squared M 2 X = 18225 M eV 2 . This reaction constitutes a second source of true coincidences that is unwanted in our VCS analysis. Therefore, the missing-mass squared is used as a selection tool. The total distribution of gure 4.1 is a convolution of several ones: the peaks from γ and π 0 events, but also broader distributions that we want to remove from our analysis. One is from random coincidences (left of gure 4.2) and the other is generated by the target walls (right of gure 4.2).

The random coincidences distribution also presents a tiny peak of ep → epγ events which is a consequence of the spectrometers time resolution. The target walls distribution is comprised of random coincidences but also of true coincidences from ep → epγ and ep → epπ 0 reactions on a proton in a nucleus. These events occur on protons which are not at rest, and so, they are not to analyse. This spectrum shows also a small peak on hydrogen, due to an imperfect cut on the vertex position.

The goal of the calibration presented in the following is to have a good control of the data selections that keep the events from the ep → epγ reaction. X for random coincidences (left) and events from the target walls (right).

Calibration of VDCs

Disabling wires

It happens that VDC wires are dead or inecient. Such wires are easy to identify since their counting rate will be unexpectedly smaller than the one of neighbouring wires. In the same way the so-called hot wires with a too high counting rate have to be disabled since they trigger even when their is no particle passing them. The procedure is described in [START_REF] Bericic | Measurement of Generalized Polarizabilities of the Proton by Virtual Compton Scattering[END_REF].

TDC osets of drift time

The drift time in VDCs is used for track reconstruction. For each of the four VDC planes the measured drift time will depend on an oset that has to be calibrated.

In order to realize the calibration of the four VDCs we use the track-reconstruction errors of the four focal-plane coordinates provided by Cola++.

So for each VDC we test a range of oset values, then we calculate, for a run, the mean value of the error on reconstructed tracks. Figure 4.3 shows how the mean value of the error on the spectrometer B s2-plane varies with the oset (in units of TDC channel). In this way the oset that minimizes the reconstruction errors is identied.

Drift velocity

The drift velocity is also a parameter to calibrate; it may vary with time, for instance when an isobutane bottle change occurs. The calibration is done thanks to the error on track reconstruction in the same way as for TDC osets.

Once the calibration is done one has to ensure that the optimal value is stable over time, i.e. for each data run of a setting. If it is not, a new calibration is done and applied to the problematic runs.

Drift time criterion

The timing dierences between signals in neighbouring wires can be used to dierentiate between good events from the ionization of the measured particle and the One has to set three parameters in order to lter out wrong events by this criterion. The procedure is described in [START_REF] Bericic | Measurement of Generalized Polarizabilities of the Proton by Virtual Compton Scattering[END_REF].

Tracking quality and eciency

The quality of the VDC tracking can be checked over time by plotting the evolution of the resolution on the four focal plane coordinates in a complete setting (variables labeled as ∆x, ∆y, ∆θ, ∆φ in Cola++). This should remain stable and is a good indication if the calibration is incomplete. This was checked for our ve settings and all the instabilities were corrected.

The eciency per wire is at least superior to 90%. Such an eciency is enough to get close to a 100% eciency for the tracking. Therefore we do not apply any correction for tracking ineciency.

Calibration of Cerenkov detectors

In our experiment we use the information from the Cerenkov detector in spectrometer B which is dedicated to the separation of electrons from pions. Nevertheless we did not make use of it since the other analysis cuts were sucient to supress the pions (this is presented in section 4.3). So the only calibration we did consists in shifting the PMT pedestal values to zero in order to align them.

Scintillators

Timing calibrations

The standard calibration of scintillators timing is made in three steps that are not independent. So one has to do several times the three calibrations in an iterative process.

The rst step consists in plotting the coincidence time T AB between the two spectrometers for the complete setting. Then the global timing oset is modied in order to center the T AB distribution on zero, as presented in gure 4.5.

Secondly, the same type of correction has to be applied per scintillator paddle.

Indeed the signals from some paddles could start sooner and be compensated by signals from other paddles which start later; this gives a global timing spectrum which is well centered on zero but wider than it should be. So the per-paddle signal is homogenized.

The last step is the correction of the time walk phenomenon. It occurs when the detection of a signal triggers thanks to a threshold discriminator. Two pulses (having more or less a Gaussian shape) that start at the same time will trigger at dierent times if their maximum height is dierent. In Cola++ the true signal start t 0 is approximated by:

t 0 = t -b • 1 √ Q T , (4.2)
with t the measured start of the signal, Q T the total charge collected (in units of ADC) and b a free parameter to calibrate. To do so, one has to check the distribution of scintillator timing versus the signal integral measured by the ADC. If a correlation is observed is has to be removed by changing the free parameter value. This will modify the global timing calibration.

ADC

Scintillators provide timing information but also measure the energy loss of passing particles (in units of ADC channel). In our experiment this information can be used for particle identication between π + and protons in spectrometer A.

In each paddle the ADC spectrum is composed of two peaks, one corresponding to pions and the other (at higher energy) to protons. The calibration of the ADC consists in rescaling the spectrum in such a way that pions and protons peaks are aligned for each paddle. Thus, if one wants to perform a particle identication selection the same cut can be applied to every paddle. It has to be done for both scintillator planes (dE and ToF) of the spectrometer A. The calibration description is detailed in [START_REF] Bericic | Measurement of Generalized Polarizabilities of the Proton by Virtual Compton Scattering[END_REF]. In section 4.3.4 we show that the PID selection based on scintillator is actually unnecessary.

Eciency

Since the trigger is done by the scintillators, some events may be lost if the scintillators eciency is not 100%. Indeed there are small ineciencies which change from a paddle to another. So one has to correct the event loss in order to precisely measure the ep → epγ cross section.

For this purpose we use eciency runs taken with a specic target, the carbon target. Since one cannot measure the eciency of the triggering plane, two runs are required, one with the dE plane as a trigger and the other with the ToF plane as a trigger.

The eciency of a scintillator plane is determined as the ratio:

ef f = N track T DC N track , (4.3)
where N track is the number of (single-arm) events having a good VDC track, and N track T DC is the number of events having in addition a TDC information in the plane under study [START_REF] Schlimme | Scintillator eciency computer code[END_REF]. The eciency is determined in small two-dimensional bins in the scintillator coordinates (X scint , Y scint ).

The correction is applied to our data as a weight function:

f (X scint , Y scint ) = 1 ef f (X scint ,Y scint ) , (4.4)
for each triggering plane [START_REF] Fonvieille | A study of scintillator eciencies[END_REF]. Usually the eciency is superior to 98%, it never gets lower than 90% for the worst paddles.

Optics calibration

The spectrometer optics is an essential part of the particle reconstruction. We used as a starting point the transfer matrix les of the A1 distribution; then, adjustments were necessary for our data analysis, due mainly to the high values of central momenta for spectrometer B, and also partially for spectrometer A (see table 3.3.4).

The calibration of spectrometers optics is a broad topic, and sometimes in our case it gives several solutions due to the fact that some adjustments are based only on the M 2 X variable. Indeed it happens that two dierent oset choices lead to comparable results in terms of resolution on missing-mass and vertex reconstruction. We then have to stick with a reasonable choice and see how using other optics parameters would aect the extraction of physics observables. This will be considered in the systematic errors presented in section 5.7.

Matrix elements

The rst step of optics calibration is the adjustment of polynomial coecients presented section 3.2.2 and gathered in a tma le. This was done in [START_REF] Fonvieille | New optic les for spectrometer B at high momentum[END_REF] for spectrometer B thanks to carbon target runs and sieveslit collimators which select only particles emitted at well-known angles (θ 0 , φ 0 ). The coecients P ijkl and T ijkl are adjusted until the reconstructed track reproduces the sieve-slit holes conguration.

A second tool is the vertex reconstruction for cryo-target runs.

General osets

Beside the calibration of polynomial coecients we had to adjust supplementary coecients that correspond to global osets of the four target coordinates (δ, θ 0 , y 0 , φ 0 ). For instance a 2 mm shift of the target position along the beam axis is taken into account by a supplementary coecient, called ∆Y 0 (B).

The angular osets (∆θ 0 (A), ∆φ 0 (A), ∆θ 0 (B), ∆φ 0 (B)) were adjusted thanks to symmetry properties of our cross section. For instance the geometry of our experiment constrains the (θ 0 )-spectrum to have symmetrical edges for the inplane settings. If it is not the case, global osets ∆θ 0 (A) or ∆θ 0 (B) are applied to correct the distribution.

The ∆Y 0 (B) coecient was corrected thanks to carbon runs. We assumed that the carbon target is well-centered in the A1 hall and used this information to modify ∆Y 0 (B) until we obtained a reconstructed vertex centered on zcarbonfoil = 0.

This simple procedure is complicated by transverse beam position eects, as presented later in section 4.2.1. The consequence is that one has to carefully select a carbon run with a beam well-centered transversally (x lab = 0) in order to calibrate ∆Y 0 (B). The diculty to select such runs is explained 4.2.1 for the cryo-target and is the same for the carbon target.

Our calibration of ∆Y 0 (B) lead to consider an o-centered cryo-target: Z lab cryo-target = -2.2 ± 0.5 (mm) . The same order of magnitude of o-centered values (-2 to -3 mm) was found for other data sets of our experiment and also conrmed independently by other experiments [START_REF] Mihovilovic | Alignment measurement[END_REF].

Magnets saturation in spectrometer A

The spectrometer-A optics we used is valid in a wide momentum range. However, magnet saturation eects occur above a certain eld value. Typically this will aect the track reconstruction of particles having a momentum larger than ≈ 580 MeV/c. This is the case for the in-plane setting.

A specic calibration led to make small changes in the coecients P 0010 , P 1010 and T 0100 of spectrometer A [START_REF] Fonvieille | Spec A optic versus magnetic eld[END_REF] for this setting.

Beside the coecients correction one also has to change the reference momentum in spectrometer A, indeed the saturation of the magnetic eld changes its value. To consider this, an analytic parametrization was determined by J. Roche and is described in her thesis [START_REF] Roche | Diusion Compton virtuelle à MAMI[END_REF]. For our in-plane setting, we applied the following

Specic raw data calibration

After our rst-pass analysis we observed important inconsistencies in our crosssection measurements. They were caused by unexpected features. In this section we present how we successfully solved these issues in our nal analysis.

Beam position

During the analysis of our settings an issue concerning the beam position was found. The beam position is not as stable as we thought. The beam conditions are checked continuously, but its exact position on the cryo-target (or carbon-target) cannot be observed by MAMI operators.

At the beginning of data taking the beam position is xed by eye thanks to an aluminum screen. However, there is a way to establish its position over time in the x lab direction. This position, which is averaged over the wobbler, is noted X beam,lab . Indeed the vertex reconstruction by spectrometer B depends directly on X beam,lab . The target centering along z lab can be measured run per run, using the V ertex Z spectrum as shown in gure 4.6. We make the simple and reasonable assumption that, in reality, the cryo-target does not move along z lab throughout our data taking.

If we observe some instabilities in the reconstructed Z-position of the target with time, as shown by the upper plot of gure 4.9, very likely it was the beam position which was unstable instead. Then we can adjust the value of X beam,lab (t) in software, in order to stabilize V ertex Z. In practice this is done by adjusting the parameter Beam.of f set.x in the run.db le, for groups of runs. The result is shown in the lower plot of gure 4.9. The spread of the reconstruction has changed from ±0.5 mm to ±0.2 mm.

Snow thickness and spectrometer B momentum

The spectrometer B reference-momentum P ref

B

is measured by a Hall probe with a precision not better than a few percent 1 . If the value considered by Cola++ is dierent than the real one this will make the missing mass resolution wider.

1 P ref A
is measured with a much better precision by an NMR probe.

The cryo-target cell is cooled to around 22

• K, which is cold enough to adsorb any residual gas molecule in the vacuum chamber that comes into contact with its outer surface. This creates a layer of so-called target snow. The snow quantity mainly depends on vacuum conditions.

The scattered particles will lose energy in this supplementary snow layer, proportionally to the product of snow density by its thickness. This is a parameter, considered by Cola++ and the simulation, which aects the missing mass reconstruction, so it has to be correctly estimated. We chose to work at constant density and adjust only the snow thickness.

The snow thickness and P ref B are two independent parameters that need to be calibrated using the missing-mass squared resolution.

The calibration, described in [START_REF] Bericic | Measurement of Generalized Polarizabilities of the Proton by Virtual Compton Scattering[END_REF] and [START_REF] Correa | Calibration of VDC and snow thickness in view of VCS analyses[END_REF], uses two constraints on the reconstructed M 2 X : the peak width has to be minimized while its position has to be centered on the simulated one. The calibration consists then to test a range of values for one of the two parameters and select the optimal oset that minimizes the missing-mass squared width while keeping it well-centered. This will aect the second parameter optimal position and will require it to be calibrated in the same way. Several iterations are needed.

The P ref

B

parameter is stable in time but the snow thickness is run-dependent.

Indeed the vacuum conditions are not always stable for a complete setting. So, once the two parameters are xed for the rst runs of a setting, the calibration of the snow thickness has to be continued for the following runs.

Vertical misalignment

Another complication arose from our three in-plane settings, i.e. INP, LOW and LOW-bis. Because of the ϕ cm -symmetry of the ep → epγ cross-section (see gure 4.13) the in-plane data should result in a symmetrical ϕ cm -distribution of events. Figure 4.11 shows that for the in-plane setting the negative ϕ cm region contains more events. We found no combination of spectrometer optics that could give at the same time a symmetrical ϕ cm -distribution and a minimized missing-mass resolution.

Therefore we concluded that our experiment eectively suers from a vertical misalignment. The calculation of the ϕ cm variable involves both spectrometers A and B, therefore the origin of this vertical misalignment cannot be identied unambiguously. We identied four possible causes:

-A vertical beam position oset This solution is actually unrealistic. To obtain this kind of ϕ cm -asymmetry the oset should be too big (>2mm) and would have been seen during the data taking.

-A non-zero out-of-plane angle for spec B It is possible to imagine that the spectrometer B was not exactly in the in-plane position during the 2011 data taking. The study has shown that a 2.7 mrad oset in spectrometer B vertical angle is needed to correct the ϕ cm -asymmetry. This appears unlikely since it implies a very large spatial displacement at the entrance of the spectrometer (i.e. 2.7mr * 3.368m ≈ 9mm).

-A vertical oset of spectrometer collimators Although this seems unrealistic for spectrometer B (it would also imply a collimator oset of 9 mm), it can be considered for spectrometer A. In this spectrometer, a large oset in vertical angle translates into a small spatial displacement, due to the small target-to-collimator distance (0.565m).

-A small oor sagging in A1

The A1 oor sagging was proven by survey measurements [START_REF] Mihovilovic | Alignment measurement[END_REF] and is probably the most realistic cause of our vertical misalignment. For our three in-plane settings as our spectrometers are approximately in the same angular position (w.r.t. the beam, see table 3.3.4) the oor sagging should lead to the same kind of osets.

Our assumption A combined eect of those causes is also possible. It is then hard to identify the real osets, especially because the oor sagging may aect osets in a complicated way. However we assume that for such a small misalignment any vertical oset among the four listed is equivalent, and we can use any of them as a representation of the reality. This non-trivial assumption is supported by our tests of dierent vertical osets, of various nature. They all lead to the same optimized M 2 X , the same ϕ cmdistribution and similar cross-section measurement.

We then decided to use the spec-A-collimator oset to correct the vertical misalignment, i.e. to be the representation of our vertical misalignment. We did a missing mass minimization for each of the three settings and obtained three oset values, presented in gure 4.12. This gure 4.12 shows that the three independent results are close (as would be the case for the oor sagging hypothesis). The average obtained oset, of -3.35 mrad, corresponds to a reasonably small vertical displacement of 0.565m * 3.35mr = 1.89 mm at the level of the collimator entrance.

The simulation (see section 4.5) is therefore run with the same spec-A collimator oset : -3.35 mrad. So the simulated ϕ cm -distribution will also be nonsymmetrical but should match the experimental one.

This correction had a strong an positive impact on our measured cross section as illustrated in gure 4.13. This cross section is now symmetric w.r.t. ϕ cm = 0, as it should be in theory.

Luminosity corrections

Another unexpected issue arose from the experimental luminosity calculation. The luminosity is given for each run by Lumi++ as presented in section 3.2.4. This luminosity-per-run needs to be corrected from the dead time DT run , i.e. the time after each event during which the detectors are not able to record another event.

It is calculated by Lumi++ in % and is used to correct the luminosity for run i as:

L corrected i = L raw i * (1 -DT i /100), (4.6)
with the dead time which depends on the kinematics, for instance it is equal to 6% for the settings LOW and OOP, for the settings INP it varies from 5 to 8%. The problem was that Lumi++ miscalculated the dead time for some runs.

We then built a correction procedure that uses reliable information to recalculate the dead time [START_REF] Correa | Method to correct the coincidence dead time from lumi++ ouput les[END_REF].

First we assume that the dead time calculation is stable from a run to another when the beam conditions are stable. This is the case for a given setting. We observed that, independently of beam current variations, the dead time can be expressed as:

DT i = K • N i t i , (4.7)
with N i the number of events recorded, t i the run duration and K a constant linked to the setting that we have to determine.

The second step was then the determination of K. K was adjusted by averaging it on the runs having an initially reliable dead time value. The third and last step used equation 4.7 to recalculate the dead time for non-reliable runs.

The errors generated by this method were estimated from the reliable runs.

The error is only up to 1% of the dead time, i.e. negligible.

Later, a modied version of Lumi++ provided by H. Merkel [START_REF] Merkel | Luminosity computer code[END_REF] conrmed our result, the corrected luminosities-per-run found by the two methods dier by only 0.09%.

Choice of 3D Binning and Analysis cuts

Now that our detectors are calibrated and that the distributions are corrected we can apply selections in order to only consider ep → epγ events for the cross-section measurement. In this section the three mandatory selections are detailed rst and the necessity of optional selections is then discussed. Afterwards the binning of the three-dimensional phase-space is presented. Spectrometer-momentum acceptance selections are always used in addition to the discussed selections. One requires |δ B | < 7.5% and -6% < δ A < 15% .

Cut on coincidence time

The time selection is the rst mandatory cut. The coincidence time, detailed in section 3.2.3 and provided by scintillators, contains accidental coincidences under the peak of true coincidences that have to be removed.

So we estimate this amount by counting the number of events in the at part of the distribution at each side of the peak, as presented in gure 4.14. Then we apply a weighted function that subtracts in the selected region the estimated number of accidental coincidences.

The random coincidences represent 6% of the events in the peak region in gure 4.14.

Vertex selection

The vertex selection is the second mandatory cut. As stated before we need protons at rest in order to precisely measure the photon electroproduction cross-section.

So we want to remove the population of gure 4.2 (right plot) from gure 4.1.

The target walls are identied in gure 4.6 as peaks, the selection simply consist in requiring M IN < V ertex Z < M AX with M IN and M AX chosen suciently far from the peaks to include the resolution eects.

Missing-mass squared selection

The M 2 X selection is the third mandatory cut. As explained in section 4.1.1 the missing-mass squared is a signature of the produced real-photon, it is therefore essential to put a cut on M 2 X around 0.

The peak has a very long radiative tail for positive M 2 X which corresponds to events which are reconstructed with a poor resolution due to energy loss of a second radiated photon. The selection will remove them. The cut we applied depends on the width of the M 2 X peak, and so it varies from a setting to another.

Setting

PID selection based on energy deposited

At this point it is still possible a priori for unwanted events to pass the previously presented cuts, so we tested if a supplementary selection is required, namely the particle identication (PID) selection based on scintillator ADC signal in spectrometer A. The PID selection would consist in applying a two-dimensional cut on gure 4.15, but it is worthless. Indeed after applying the three mandatory selections the events identied as pions on this gure are strongly suppressed. As an example, for the setting LOW, these events represent only 0.1% of the total.

This result was similar for all our settings and we decided to not apply this selection.

PID selection based on Cerenkov detector

The Cerenkov detector of spectrometer B is used to dierentiate π -froms electrons. Figure 4.16 shows the measured signal for pions, which appears as a narrow peak on the left (it is the electronic pedestal), and for electrons, corresponding to the broad bump. The selection would consist in applying a cut on this gure.

Control of the optics quality

The missing-mass squared peak position and the target position in Z are two quantities which should not depend on the reconstructed target coordinates (x 0 , θ 0 , y 0 , φ 0 ) or the focal-plane coordinates (x, θ, y, φ) f p . By plotting the reconstructed coordinate with respect to the missing-mass squared or to the V ertex Z we should then see no correlation. Correlations would be the sign of a residual default in the optics (matrix elements or osets). For each spectrometer variable and each setting we checked the absence of such correlations. Figure 4.17 shows two examples of missing-mass squared control plots. The left one is obtained for the spectrometer A of the setting LOW-bis, no correlation appears between θ f p (A) and the missing-mass. The right plot shows a small correlation for the spectrometer B of the 2015 OOP setting. This is nevertheless the best we could obtain and we consider that such a small correlation has no consequence on our measurement. Figure 4.18 illustrates a well-reconstructed V ertex Z. The target shape clearly appears and does not depend on the reconstructed variable θ f p (B).

The optics we used satises our correlation criteria, since only small correlations remain. It is impossible to obtain perfect optics. We chose one optics as a reference for the nominal analysis, and other solutions for optics have been considered for the purpose of systematic studies.

Event rate stability

Another control allows to check in the same time the consistency of our data selection and the quality of our luminosity calculation. It consists in calculating for each run the number of ep → epγ events divided by the dead-time corrected luminosity of the run.

For each run i, the number N ep→epγ of ep → epγ events can be expressed as: where the integral (presented in more detail in section 5.3) is essentially runindependent. So the ratio N ep→epγ i /L corrected i must be run-independent (for a given setting).

N ep→epγ i = L corrected i • d 5 σ dΩ 5 dΩ 5 , (4.8)
Figure 4.19 shows this ratio with respect to the run label for the 2015 OOP setting. Each run has its own length but the ratio is constant over time. This is not true for one run which will be removed of the analysis (this run was interrupted during data taking because of detector issues).

The ratio for the INP setting shows a particular shape, gure 4.20. Indeed after the run 70 the beam current was increased from 10 µA to 15 µA. This seems correlated with a ≈2% decrease of the plotted ratio. We believe that the current was too high for the second part of the data taking and that it was responsible of a slight local target boiling. This phenomenon changes the target density in a way that is not always detected by the sensors.

From those data we calculated that the luminosity for the second part of the setting is overestimated by 2% and this corresponds to a 1% overestimation of the total INP setting luminosity. So we apply this 1% in our nominal analysis and consider its eect in our calculation of the systematic error (presented in section 5.7).

The settings LOW and LOW-bis presented a constant ratio. Thanks to this control plot we know that we can trust the luminosity calculation and the data

The Monte Carlo Simulation: VCSSIM

The simulation is a mandatory tool for VCS analysis. It is necessary for the determination of the absolute cross section and also provides essential informations to understand possible experimental problems.

The Monte Carlo simulation VCSSIM was rst developed at Gent University by L. Van Hoorebeke for the needs of the rst VCS experiment at MAMI. It was then adapted to the JLab hall A spectrometers, and later, Peter Janssens implemented supplementary useful features [START_REF] Janssens | Double-polarized virtual Compton scattering as a probe of the proton structure[END_REF]. This last version is the one we used for our GP extraction, up to small modications.

VCSSIM is composed of three separate Fortran codes corresponding to three steps:

-The rst step generates photon-electroproduction events at the target level according to a given cross section. Then energy loss from ionization and multiple scattering are considered along the path from the target to the spectrometers entrance in order to reject events out of the experimental acceptance.

-The second step applies spectrometer resolution to the generated events.

The deterioration of target variables in the simulation allows to reproduce the experimental behavior caused by the VDC resolution.

-The third and last step is the reconstruction of events from the previous step ouput. The variables thus obtained can be analysed with event selection cuts and be compared to the experimental spectra. If the whole procedure is correctly applied the distributions from Cola++ and VCSSIM should match.

In addition to the event generation, VCSSIM also calculates a simulated luminosity L sim which has to be used for the cross-section measurement (see section 5.3.1).

The VCSSIM program is fully detailed in [START_REF] Janssens | Monte Carlo simulation of virtual Compton scattering below pion threshold[END_REF]. In this section its main features are summarized.

Event generation

Before the event generation a simulation phase-space is determined, i.e. a range of values for each of the ve variables (k, k , θ e , θ cm , ϕ cm ) is xed. Two constraints determine the size of the simulated phase-space. Firstly, it has to be large enough to generate every particle that could reach the spectrometers acceptance. Since energy loss and multiple scattering aect the direction and momenta of particles, the simulated phase space has to be larger than the angular and momentum acceptances of the spectrometers. Secondly, it has to be not too large in order to save calculation time.

Then the event generation starts and an interaction point is sampled. The generation takes into account the beam position (including the wobbler information) and the beam energy loss in the target. The full photon-electroproduction kinematics is generated, by sampling uniformly in (k , θ e , ϕ e , ϕ cm , cos(θ cm )), with ϕ e the angle between the scattering plane and the horizontal plane containing the axis of the spectrometers. Instead of the acceptance-rejection method used in [START_REF] Janssens | Monte Carlo simulation of virtual Compton scattering below pion threshold[END_REF],

here we use all samples that go through the acceptance, and the σ BH+B cross section is used as a weight to the event. This procedure increases the eciency of the Monte Carlo. The price to pay is a special treatment of the statistical uctuations induced by the simulation (see section 5.3.1). The part of the radiative corrections that is acceptance-dependent (i.e. internal Bremsstralhung) is also included at this level.

Finally energy losses and multiple scattering are applied to the outgoing particles which are tracked up to the entrance of the spectrometers. Here, the simulation rejects events that do not pass through the collimators, or are not in the momentum bandwidth of the magnets.

Resolution eects and event reconstruction

To properly reproduce the experimental spectra resolution eects have to be added to the simulation. This is done by the second step of VCSSIM. It reads the rst step output, which are the particles coordinates at the target level, and computes their focal-plane coordinates using a transfer matrix.

Then, in the focal plane, two types of errors are sampled: the multiple scattering in the detector materials and the resolution of the wire chambers. Those errors are propagated back to the target [START_REF] Janssens | Double-polarized virtual Compton scattering as a probe of the proton structure[END_REF], where an additional error is also considered in order to reproduce eects such as: magnetic elds imperfections, beam position and energy instabilities, etc.

Finally the simulation's second step provides a le that contains the target variables of the two detected particles, degraded by resolution eects.

The last step of VCSSIM uses the second step output to reconstruct the kinematics of each event in the same way that it is done in the experimental analysis, i.e. by taking into account the energy loss of the particles in the target. By coupling informations from the two spectrometers, one reconstructs all important variables, such as M 2 X , or the variables for the three-dimensional binning: q cm , ϕ cm and cos(θ cm ). Vertex Z is obtained by coupling informations from spectrometer B and the beam.

Luminosity calculation

The simulated luminosity L sim is mandatory to measure the photon electroproduction cross-section as it is presented in section 5.3. L sim is calculated at the end of a simulated run. First a luminosity phase-space (L.P.S) is dened, it is smaller than the acceptance phase-space in the three dimensions (k e , θ e , ϕ e ). In the L.P.S the cross section is integrated and the number of samples N is counted. Then the simulated luminosity is given by:

L sim = N L.P.S d 5 σ
dk dΩ e dΩγγ,cm dk dΩ e dΩ γγ,cm

Conclusion

To conclude, VCSSIM is required for two reasons. Firstly, it calculates an 'eective solid angle', i.e. that contains the detector geometry plus the various resolution eects. This quantity is necessary in order to determine a cross section accurately.

Secondly VCSSIM gives insights in order to check the experimental data.

The parallel analysis at Q 2 = 0.1 GeV 2 , described in [START_REF] Bericic | Measurement of Generalized Polarizabilities of the Proton by Virtual Compton Scattering[END_REF], was performed thanks to another simulation: Simul++. We beneted from having two simulations to perform agreement tests and the results were in accord at the two percent level.

Simulated and experimental data comparison

To ensure that both our experimental and simulated data are correctly analyzed we can compare their distribution of observables. In the following gures the normalization of the simulation is xed in order to correspond to the same luminosity as in experiment, the simulated distributions are obtained by using σ BH+B +GP (LEX)

as a weight function. We dispose of a large number of observables, reconstructed coordinates, vertex, missing-mass squared, particles momentum, etc... We found a satisfying matching for each setting. Only a small disagreement remains. The φ 0 angle reconstructed by spectrometer A presents a dierence for one specic q cm bin as presented in gure 4.22.

Other choices of optics and osets can x this issue but deteriorate other distributions matching. The choice to keep this small disagreement has been considered in systematic error studies.

Finally, gure 4.23 presents the simulated and experimental distributions of the missing-mass squared for the four settings. The only perfect matching appears for the setting INP (bottom left). This is because we optimized the simulated resolution on this one, after this optimization the distributions for the 2015-OOP setting (bottom right) are in less good agreement. This is not an issue since the missing mass cuts are very far from the main peak. For the low energy settings (top right and left), the experimental resolution is slightly better than the simulated one. Globally, the agreement of simulated and experimental distributions for M 2 X is quite satisfactory.

Conclusion

The calibrations for VCS analysis have been done extensively. The principal unexpected issues: large snow thickness, vertical misalignment, local target boiling, magnet saturation eects, have been xed. As an example, the left plot of gure 4.24 shows that the distributions of ϕ cm from the experiment and from VC-SSIM both present an asymmetry w.r.t zero, which is due to the vertical osets we implemented in the experiment and the simulation. The matching is satisfactory.

The right plot of gure 4.24 is obtained when implementing a GP eect in the weighted function used by VCSSIM, i.e. the weight is calculated as σBH + B + GP (LEX). The rather poor agreement observed between experiment and simulation is due to the omission of the higher-order terms in q cm in the weighted function. It illustrates the limits of the LEX extraction which is discussed in the following chapter. Chapter 5

Photon electroproduction cross-section and extraction of VCS observables

This chapter describes the second-level analysis: the cross-section determination and the LEX and DR ts yielding the VCS observables: structure functions and GPs.

Section 5.1 presents the theoretical tools we used in the analysis. In section 5.2 we explain the calculation of radiative corrections at Q 2 = 0.2 GeV 2 . Section 5.3 is dedicated to cross-section measurement: method and results. Section 5.4 introduces an important step of the analysis which is the renormalization procedure.

Finally sections 5.5 to 5.8 present the LEX and DR ts and the results in terms of structure functions and GPs.

Theoretical tools

In this section are presented the two analysis tools essential to the LEX and DR extraction of GPs: the two computer codes which provide respectively the theoretical LEX and DR cross-sections.

In addition to the presented tools we also make use of the A1 software previously presented, especially Cola++ 1 .

The BHB code

The BHB code is a C++ program developed by M. Vanderhaeghen [START_REF] Vanderhaeghen | VCS-BHB computer code[END_REF] which calculates the photon electroproduction cross-section in a chosen point of the phase space d 5 σ th (P ).

The parameters required by the code are the parametrizations of the proton form factors (G E ,G M ) and the ve variables (k, k , θ e , θ cm , ϕ cm ) to dene the phasespace point. Then, the user chooses which cross section is calculated by the code among: BH, Born or BH+B. The user can also enter values of structure functions 1 Some graphics contents from Cola++ are drawn thank to ROOT from CERN [START_REF]Computer code[END_REF].

(P LL -P T T / , P LT ) to calculate the full photon electroproduction cross-section according to the LEX equation 1.38.

The DR code

The second theoretical code, used in our analysis, provides the photon electroproduction cross-section according to the DR model developed by B.Pasquini [START_REF] Pasquini | VCS-DR computer code, version maid[END_REF].

Here again the code calculates the theoretical cross section in a point of the phase space d 5 σ DR (P ).

In addition to the kinematical parameters from the (q cm , q cm , , θ cm , ϕ cm )-set the code requires values for the DR parameters Λ α and Λ β (presented in section 2.1).

Just like for the BHB code the user also has to specify a choice of proton formfactor parametrization among the available ones.

To calculate the full DR cross-section the code also needs values of polarizabilities from RCS; they are xed at the following values from [START_REF] Olmos De Leon | Low-energy Compton scattering and the polarizabilities of the proton[END_REF]:

ᾱE = 12.1 • 10 -4 f m 3 , βM = 1.6 • 10 -4 f m 3 .

Radiative corrections

Radiative corrections are known to be an important correction to electron scattering processes in general, due to the fact that the electron is a light particle that radiates photons easily.

To study the photon electroproduction reaction ep → epγ we only detect the scattered electron and proton, the real photon is identied by the missing-mass method. So we do not control all the processes that occur and which impact the cross-section measurement. For instance an event including a supplementary real or virtual photon could be counted as a rst-order photon electroproduction event, while it is not. Such processes, called radiative processes, must be estimated and corrected in order to remove their contribution to the cross section, i.e. the cross section corresponding to the rst-order graphs only (see gure 1.6). This is done by applying a correction factor, F rad , to the measured raw crosssection:

σ exp = σ raw * F rad . (5.1)
Radiative corrections to the ep → epγ reaction have been studied extensively in [START_REF] Vanderhaeghen | QED radiative corrections to virtual Compton scattering[END_REF] which served as a basis for the analysis of all VCS experiments. We will here summarize how we put these calculations into practice, more details can be found in [START_REF] Correa | VCS Experiment -Radiative corrections[END_REF]. Generally F rad is written as a sum of contributions which are noted as δ-terms, each one corresponding to a specic process. We consider the following sum:

1 F rad = 1 + δ vac + δ vertex + δ r + δ 1 + δ (0) 2 , (5.2)
where the δ-terms are explained in section 5.2.1 which also details the specic choices that have been done for our F rad estimation. Section 5.2.2 summarizes the results from [START_REF] Correa | VCS Experiment -Radiative corrections[END_REF]: value of the correction and uncertainty.

Physical processes and calculation choices

As can be seen from g 1.6, the amplitude of the bare diagrams is of order (e 3 ), leading to a cross-section of order (e 6 ). Radiative corrections have been calculated [START_REF] Vanderhaeghen | QED radiative corrections to virtual Compton scattering[END_REF] to the next order, i.e. including the following contributions:

-the emission of one supplementary real photon (amplitude of order (e 4 ) and cross-section of order (e 8 ))

-one supplementary virtual photon (amplitude of order (e 5 ) interfering with the bare amplitude, (e 3 ), which leads to a cross-section of order (e 8 )).

The supplementary photons can be emitted from the proton side or from the electron side. The calculation of all the δ-terms requires theoretical approximations (like neglecting the higher-order corrections) and the use of model (for the protonside correction). As a consequence we have to consider a 1% incompressibletheoretical error [START_REF] Vanderhaeghen | QED radiative corrections to virtual Compton scattering[END_REF] in our F rad determination.

Some of the diagrams of physical processes are available in appendix C and so are the mathematical equations used to estimate the correction terms. Only the specicities of our determination are presented here. Such processes are taken into account through the δ r term expressed as a sum of two terms: δ r = δ r1 +δ r2 . δ r1 gathers correction terms which depend on ∆E s and δ r2 contains correction terms which do not. ∆E s is the maximum energy that the soft photon can reach, it depends on the upper cut on the missing-mass squared used in the event analysis:

∆E 2 s = cut 4 , (5.3)
where cut is dened in section 4.3 and depends on the settings as presented in table 4.3.3. Typically, an upper cut value of M 2 X = 2000 M eV 2 corresponds to ∆E 2 s = 22 M eV . δ r1 and δ r2 are an important part of the radiative corrections since they can both reach separately 20% in some analysis bins.

δ r1 is already implemented in VCSSIM: applying a missing-mass squared cut to the simulation impacts the nal measured cross section in the same way it would be done by adding δ r1 to F rad . It is then important to separate δ r in those two terms in order to not consider δ r1 twice in our analysis. So, the corrections due to supplementary real photon considered in our F rad are only due to the δ r2 contribution which is detailed in appendix C.

δ vac and δ ver : supplementary virtual photon A supplementary virtual photon is produced by three kinds of processes, the vacuum polarization process (gure 5.2): the electron self-energy process (gure C.2) and processes with an additional loop between lines (gure 5.3). The virtual-photon contribution that we used has the same expression as the virtual-photon radiative correction to the elastic reaction ep → e p . Using the elastic case correction allowed us to avoid the complicated integral calculation involved in the photon-electroproduction case and due to the grahs 1 to 6 of gure 5.3. This choice, supported by the work of D. Marchand [START_REF] Marchand | Calcul des corrections radiatives à la diusion Compton virtuelle / Mesure absolue de l'énergie du faisceau d'électrons de Jeerson Lab[END_REF], leads to a ≈ 0.8% systematic error on the total virtual-photon correction.

The electron self-energy processes, gure C.2, are not considered in our calculation but are a part of the 0.8% error already considered. δ 1 and δ (0) 2 : proton-side corrections A supplementary photon can also be produced from the proton side, by rare proton bremsstrahlung and vertex corrections. Unlike corrections from the electron side, which can be calculated model-independently from QED, the proton-side corrections will depend on a model for the proton structure. This model dependence will become important if one wants a experimental precision at the 1% level [START_REF] Vanderhaeghen | QED radiative corrections to virtual Compton scattering[END_REF].

(1) 2 , which also take into account the two-photon exchange process (direct and crossed). The terms considered are from the elastic case but can be applied for photon electroproduction with a negligible error [START_REF] Maximon | Radiative corrections to electron proton scattering[END_REF].

To determine the proton-side corrections we decided to not consider the δ (1) 2 term; this results in a systematic error on our nal F rad which is estimated to be ±0.002 (this is detailed in appendix C).

Higher-order correction terms

Now, one may consider more supplementary virtual or real photons, the so-called higher-order correction terms. The work presented in [START_REF] Yennie | The infrared divergence phenomena and high-energy processes[END_REF] proposes to estimate the eect of higher orders by considering a exponentiated form for the correction factor:

1 F rad = e δver+δr (1 - δvac+δ 1 +δ (0) 2 2 ) 2 (5.4)
By comparing the exponentiated form to equation 5.2, we checked that the higher-order contribution is eectively smaller than the 1% theoretical error which already accounts for it. We nally decided to calculate the radiative corrections by the expression of equation 5.2.

Value of the global radiative correction factor

An important question was to determine if one can apply F rad as a global correction to the measured cross section or if it varies bin-per-bin in (q cm , cos(θ cm ), ϕ cm ). This was answered in [START_REF] Correa | VCS Experiment -Radiative corrections[END_REF], where the calculation of every correction term in the whole analysis phase-space showed only very small variations and leads to the conclusion that we can apply a global correction:

1 F rad = 1.057 .

Systematic error on F rad

The rst error is generated by the phase-space averaging. Figure 5.4 shows the F rad behavior at q cm = 112.5 M eV /c. For those bins one gets 1/F rad = 1.058±0.002 RM S (the maximum dierence between the averaged F rad and one bin value is ±0.006). One also has to check the F rad behavior for the bins at q cm = 37.5 M eV /c at which the renormalization is computed (see section 5.4). Figure 5.5 shows that 1/F rad = 1.056 ± RM S 0.001 (the maximum dierence calculated is ±0.003) which means that we can consider the radiative correction as a constant in our whole phase space.

A second error is induced by the ∆E s quantity. Indeed this one varies from a setting to another whereas we used it as a constant in the F rad determination.

However we observed that changing it to its maximal variation only induces a ±0.001 change on 1/F rad .

The quadratic sum of these two errors leads us to consider a total error induced by our phase-space averaging 2 of ±0.003 and the maximum variation of F rad is ±0.007.

The complete error budget on F rad is: 2 The averaging is done with an acceptance cut (the gradient selection), without it the error would have been larger. so the measurement requires enough statistics, and systematic errors reduced to the 1-2% level. To achieve that we used a method based on a simulation (VCSSIM) which is described in [START_REF] Janssens | Monte Carlo simulation of virtual Compton scattering below pion threshold[END_REF]. We summarize it in the next sections.

The simulation-based method

First, the classical formula that expresses the photon electroproduction crosssection in one bin is:

d 5 σ exp dk dΩ k dΩ p cm = d 5 σ dΩ 5 exp = N exp L exp • ∆Ω 5 , (5.5)
with N exp the number of events in the bin, L exp the experimental luminosity and ∆Ω 5 the solid angle of the ve-dimensional bin. ∆Ω 5 is dicult to determine in a simple way because the acceptance is not constant in a single bin. So ∆Ω 5 is reproduced thanks to VCSSIM in order to obtain the cross section by the following method.

From equation 5.5 we have in one experimental bin (i):

N exp i = L exp i σ exp (Ω)A(Ω)dΩ 5 , (5.6)
where A(Ω) is the complicated acceptance function which varies between 0 and 1 in the bin and σ exp (Ω) stands for d 5 σ/dΩ 5 . We introduce a point P 0 of the dierential phase-space dΩ 5 which is, in our case, the center of the bin. The dierential cross section in any point P of the considered bin can be expressed as: σ exp (P ) = σ exp (P 0 ) 1 + σ exp (P ) -σ exp (P 0 ) σ exp (P 0 ) .

(5.7)

Applying this to equation 5.6 leads to:

N exp i L exp = σ exp (P 0 ) 1 + σ exp (Ω) -σ exp (P 0 ) σ exp (P 0 ) A(Ω)dΩ 5 .
(5.8)

Now, if one makes use of a simulation faithful to the experiment, equation 5.8 can be written for the simulated events as well:

W sim i L sim = σ th (P 0 ) 1 + σ th (Ω) -σ th (P 0 ) σ th (P 0 ) A(Ω)dΩ 5 , (5.9)
with W sim i the number of events in the simulated bin (in our case it is a weighted sum, the weight being the theoretical BH+B cross section). L sim is the luminosity calculated by VCSSIM and σ th (P 0 ) can be either the BH+B cross section or the BH+B cross section increased by a theoretical GP eect. In our case we did not implement a GP eect in σ th (P 0 ) but we considered this possibility in the systematic studies of section 5.7.2.

The integrals of equations 5.8 and 5.9 can be considered as very close, since σ th reproduces well the relative variations of the true cross section in the bins.

Finally, using equations 5.8 and 5.9 leads to the important cross-section determination formula:

σ exp (P 0 ) = N exp i W sim i • L sim L exp • σ BH+B (P 0 ).
(5.10)

To calculate σ BH+B (P 0 ) we can choose dierent form-factor parametrizations as discussed in section 1.1.2. Here, one just needs to use the same parametrization for the theoretical cross section (σ BH+B (P 0 )) and for the BH+B cross section used as a weight event per event, in W sim i . Indeed, a change of form-factor parametrization roughly acts as a rescaling of the cross section (see section 5.4.3), which is canceled in the ratio σ BH+B (P 0 )/W sim i . In this way the measured cross section will not depend on the form-factor parametrization choice, in rst approximation.

Statistical error

Once the experimental cross section from equation 5.10 is measured we have to determine its statistical error. We considered statistical uctuations coming from the experimental and simulated number of counts:

∆σ exp σ exp = ∆N exp i N exp i 2 + ∆W sim i W sim i 2 .
(5.11)

The statistical error on L sim is negligible, by construction of VCSSIM. The error on L exp is of systematic nature and will be addressed in section 5.7.

N exp i depends on the coincidence selection described in section 4.3. Its statistical error is then calculated by considering the uctuations from event counts both in the coincidence peak and in the accidental plateau.

As explained in section 4.5 we implemented a weight-procedure in VCSSIM:

W sim i in bin i is equal to the sum N sim i j
w j , with w j = σ BH+B (j). For such quantities the statistical uctuations give [START_REF] Fonvieille | Treatment of the statistical error bar related to the VCSSIM simulation[END_REF]:

(∆W sim i ) 2 = N sim i N sim i -1 ( N sim i j w 2 j ) -(< W sim i >) 2 , (5.12)
with < W sim i > the mean of the weight distribution. In our case it can be simplied to the following expression [START_REF] Fonvieille | Treatment of the statistical error bar related to the VCSSIM simulation[END_REF]:

(∆W sim i ) 2 = N sim i j w 2 j .
(5.13)

The simulation is run on a larger statistics than the experiment, so that the term ∆W sim i /W sim i is not dominant in equation 5.11.

Cross-section results

We measured 7784 cross-section points distributed in the ve settings, some of them are presented in the next gures. They are obtained without renormalization factor, but they are corrected from the radiative processes by means of the factor F rad (i.e σ exp = σ raw * F rad ).

In the analysis some bins have a low number of events, typically bins at the acceptance edges. These bins are removed at the cross-section level by a cut on

N exp i : N exp i > 20
.

No cut was applied on the number of simulated events which are more numerous per bin in our case. be dierentiated at such a small q cm . The symmetry of the cross section in ϕ cm is reproduced, which proves that the vertical misalignment is successfully corrected.

The cross sections for the three other q cm bins (q cm = 12.5 / 62.5 / 87.5 M eV /c)

are not presented but are also in agreement with the theory, and so are the crosssection points from the setting LOW-bis. This kind of observations motivated us to carefully handle the higher orders in our LEX extraction (see section 5.5.2).

When |ϕ cm | is far from 0 it is hard to distinguish which curve among (BH+B) and (BH+B + GPs) is more faithful to the data. The GPs, especially β M (Q 2 ) for this setting, have to be extracted from the small dierences observed, this illustrates the diculty of the analysis. Finally gures 5.9 and 5.10 gather the cross section data from the OOP setting.

These points are more sensitive to α E (Q 2 ) but again the GP eect is hard to

Renormalization procedure

The cross sections presented in the previous section may suer from a global normalization factor. It could be caused by dierent sources, for example:

-a global loss of events due to detectors ineciencies not fully corrected.

-a miscalculation of the experimental luminosity. Errors may come from approximation on the target length (due to its shape), target density or residual biases in the deadtime correction.

-a bad estimation of radiative corrections; as stated in section 5.2, they suer from a lack of theoretical knowledge and the calculation has an uncertainty of 1-2%.

So we need to apply a correction factor, F norm , in order to compensate for all possible biases that act as a global factor: where σ measured = σ raw * F rad .

σ exp i = σ measured i * F norm ,
The F norm factor can be estimated thanks to the low-energy settings (LOW and LOW-bis). Indeed at low q cm , i.e. q cm < 50M eV /c the measured cross section depends only weakly on the GP eect, and so, we can normalize it to the theoretical cross section without including a too important unknown parameter.

An important point is that the renormalization procedure is possible only at low q cm (<50 MeV/c). At higher q cm the eects of normalization and polarizabilities are mixed together and cannot be disentangled.

Another important point is that, when we allow σ exp to be renormalized, the LEX and DR ts are sensitive only to the shape of the cross section versus (q cm , cos(θ cm ), ϕ cm ) without being disturbed by the absolute normalization.

Using the low-energy settings we identify F norm thanks to a χ 2 -minimization:

χ 2 norm = n i=1 σ measured i * F norm -σ th i ∆σ measured i (stat) * F norm 2 , (5.15)
where F norm is the free parameter of the minimization, (i = 1 → n) refers to bins at low q cm . Several features arise from the calculation. Firstly, the theoretical cross-section σ th i has to be as close as possible to the reality, and so, it must include a theoretical GP eect even if it is small at small q cm . This eect will be calculated thanks to the LEX, following equation 1.38. It will depend on input values of the structure functions P LL -P T T / and P LT .

Typically the GP eect is of the order of 0.5% and 1.5% of the total cross-section at respectively q cm = 12.5 M eV /c and q cm = 37.5 M eV /c. In section 5.4.1 we will present an iterative process that takes care of this dependency.

The second feature concerns the consistency between settings. The data at Q 2 = 0.2 GeV 2 were all collected in the same conditions and the settings were calibrated by following the same principles. So the need of a renormalization should be identical for all settings. This is checked further for our settings LOW and LOW-bis in section 5.4.2.

The last feature is less straightforward. The theoretical cross section depends on a chosen parametrization of form factors. As stated in section 1.1.2, signicant dierences exist from a parametrization to another. Actually, the renormalization factor determined by formula 5.15 will depend on the parametrization chosen in σ th i . We consider three renormalization factors corresponding to three specic form-factor parametrizations:

-F norm F W for the parametrization from J.Friedrich and T.Walcher [START_REF] Friedrich | A Coherent interpretation of the form-factors of the nucleon in terms of a pion cloud and constituent quarks[END_REF].

-F norm B for the parametrization from J. C. Bernauer et al. [START_REF] Bernauer | High-precision determination of the electric and magnetic form factors of the proton[END_REF].

-F norm A for the parametrization from J. Arrington et al. [START_REF] Arrington | Global analysis of proton elastic form factor data with two-photon exchange corrections[END_REF].

Consequently the renormalized cross section is also form-factor dependent (for instance σ exp F W = σ measured * F norm F W ). When no index is specied the quantities refer to our nominal choice, i.e. the parametrization from J.Friedrich and T.Walcher. In section 5.4.3 we will see how the form-factor dependency is exploited positively in our analysis.

The iterative procedure in the LEX analysis

The iterative procedure is the way to take into account the GP-eect dependency from σ th in the estimation of F norm . It consists in extracting the structure functions by the LEX analysis (presented section 5.5) and using the result to compute a new theoretical cross section. The initial choice of the theoretical cross section has no impact, the results fully converge after a few iterations. As an illustration gure 5.12 represents the procedure behavior for three dierent initial starting points: the red points show the steps if one considers no GP eect in the rst place, the blue points are for a so-called absurd choice of GP eect with P LL -P T T / = 40 GeV -2 and P LT = 2 GeV -2 (which is far from our nal result). The black cross is obtained with P LL -P T T / = 19.5 GeV -2 and P LT = -4.0 GeV -2 which are values close to our nal result, a so-called coherent choice. The renormalization factor F norm at each step is also indicated in this gure. As a check, we tried very dierent LEX extractions of structure functions (with and without the bins selection described in section 5.5) and we found that it weakly impacts the convergence point of the interations in terms of F norm : ∆F norm = ±0.002. So the iterative procedure is necessary to get the best determination of F norm .

However, some VCS experiments did not apply any renormalization, for example because of lack of low-q cm data [68].

5.4.2

The agreement between settings and between q cm -bins

In this section we present the consequence on the F norm determination if one considers dierent groups of data (various settings and bins) in the minimization of χ 2 norm . The results presented here are obtained with the Friedrich-Walcher parametrization of form factors. An iterative procedure has been performed and only the nal values of F norm are shown. To test the dierent data groups we can choose between two low-energy settings (LOW and LOW-bis) and each one of them contains two sets of ≈ 700 low-q cm bins: one set at q cm = 12.5 M eV /c and the other at q cm = 37.5 M eV /c. F norm results after iterative process for several combinations of selected bins and settings; 0 stands for the bin at q cm = 12.5 M eV /c, 1 is for q cm = 37.5 M eV /c and 0+1 means that the two bins are considered. The statistical error on F norm is ±0.002.

The data from settings LOW and LOW-bis were calibrated independently. The observed good agreement comforts us in the quality of our calibration principles.

The three last lines of the table illustrate the consequence of selecting bins at the edge of spectrometers acceptance. Indeed the calibration is optimized for the bins at the center of the spectrometers acceptance, i.e. at q cm = 37.5 M eV /c for setting LOW and LOW-bis and at q cm = 87.5 M eV /c and q cm = 112.5 M eV /c for the others. Only data from q cm = 37.5 M eV /c-bins were considered for the determination of our nal renormalization factor (the starred line in table 5.4.2).

Finally we estimated that the renormalization factor is determined to ±0.006, F norm results after dierent iterative processes, all using q cm = 37.5 M eV /c-data but considering dierent form-factor parametrizations.

Consequently the renormalized cross section will dier from a parametrization to another and so we have three sets of experimental cross section:

σ exp F W / σ exp A / σ exp B .
A priori, the form-factor dependency of the renormalized experimental crosssection appears to be a problem, but, it is actually a strength of the LEX extraction of structure functions.

Indeed the LEX extraction (presented in section 5.5) consists in computing the dierence between the renormalized cross section and a theoretical one, and the two are form-factor dependent (σ exp F orm-F actors and σ th F orm-F actors ). As will be shown below, changing the form-factor parametrization in the (BH+B) cross section roughly rescales it by a global factor. Therefore the LEX t can adapt itself to such a change if we allow also the experimental cross section to be changed by a global factor. This is the guiding principle of our renormalization procedure.

Inuence of form-factor parametrization on the (BH+B) cross section

We computed the following ratio of theoretical BH+B cross-sections in each analysis bin (i):

R A/B i = σ BH+B A σ BH+B B . (5.16)
The index 'A' still refers to the form-factor parametrization from Arrington et al. [START_REF] Arrington | Global analysis of proton elastic form factor data with two-photon exchange corrections[END_REF], and 'B' to the one from Bernauer et al. [START_REF] Bernauer | High-precision determination of the electric and magnetic form factors of the proton[END_REF]. Figures 5.13 and 5.14 show the ratio for the bins at respectively q cm = 37.5 M eV /c and q cm = 112.5 M eV /c. The observed variations are small, i.e. 0.974 < R A/B < 0.982. So changing the form-factor paramerization does act like a global rescaling of the theoretical cross section, to better than 1% at q cm = 112.5 M eV /c, and to better than 0.4% at q cm = 37.5 M eV /c. 5.16: Bins (in blue) that satisfy the mask selection in the (ϕ cm , cos(θ cm ))plane when K thr = 2.5%, P LL -P T T / = 17.9 (GeV -2 ) and P LT = -3.7 (GeV -2 ) at q cm = 87.5 M eV /c (left) and q cm = 112.5 M eV /c (right). Figure 5.17: Bins, in blue, that satisfy the mask selection in the (ϕ cm , cos(θ cm ))plane at q cm = 112.5 M eV /c. The bins in red contain data that are rejected by this selection.

The K thr dependency

We tested for a given GP eect (P LL -P T T / = 17.9 (GeV -2 ) and P LT = -3.7 (GeV -2 ) the consequences on the LEX extraction if one changes the value of the K thr criterion. This is represented in gure 5.18. Two regions arise, one which is unstable for a restrictive K thr < 2% (especially for P LT ) and the second, more stable, for K thr in the range [2; 5]%.

The red points are obtained when no mask selection is applied, i.e. K thr = ∞ . It appears that removing this selection has no eect on P LT but has an important eect on P LL -P T T / . We observed the same behavior with other structure-Table 5.5.3: Data used for the nal LEX t. As a reminder, the q cm values have to be below the pion production threshold.

We found a satisfying reduced χ 2 , i.e. 1.26 for n.d.f=944 and the extracted structure functions are 3 : P LL -P T T / = 18.10 ± 0.56 stat (GeV -2 ) P LT = -3.59 ± 0.31 stat (GeV -2 ).

(5.23) to nd the electric and magnetic GPs. 3 These results dier slightly from the ones of Table 5.4.3-c (for the Friedrich-Walcher parametrization), because of dierent inputs to build the mask.

For this purpose we used the DR model. The method simply consists in using the DR code to nd the Λ parameters corresponding to our measured values of the LEX structure functions at Q 2 = 0.2 GeV 2 , q cm = 458 M eV /c and = 0.85. Then the code is used a second time to nd the GPs at Q 2 = 0.2 GeV 2 . In this way the spin GPs are calculated by the DR model and deducted from the structure functions.

The procedure will induce a form-factor dependency, but it is small and impacts the results very weakly.

We found:

α E (Q 2 = 0.2 GeV 2 ) = 3.90 ± 0.14 stat (10 -4 f m 3 ) β M (Q 2 = 0.2 GeV 2 ) = 1.14 ± 0.15 stat (10 -4 f m 3 ) .

The DR analysis

The second extraction of GPs that we have done is directly based on the DR model. It requires again the experimental cross sections, renormalized by the factor F norm F W = 0.98 which corresponds to using the Friedrich-Walcher parametrization of proton form factors.

We did not perform a DR t for three dierent form-factor parametrizations, as we did for the LEX t. It would have required a too long computational time.

However, we expect that the DR extraction of GPs will be essentially independent of the form-factor choice, since the principles of the tting method are the same as for the LEX analysis.

Fit description

Contrary to the LEX t, the DR t is not analytical, because the theoretical crosssection σ DR depends on the polarizabilities in a very complex manner. We still compare our experimental cross sections to the DR calculation by a χ 2 method, but the minimum has to be found purely numerically.

We rst need to compute a large amount of DR cross sections. Indeed, we need the cross sections to be calculated in the 10080 bins of our analysis (10080/2 actually because the theoretical cross section is perfectly ϕ cm -symmetric) and we also need them to be calculated for dierent input values of the GPs.

So for the rst step, we built a regular grid of 25 values of α E (Q 2 = 0.2 GeV 2 ) times 24 values of β M (Q 2 = 0.2 GeV 2 ) in the range allowed by the model, i.e. 600 grid nodes, and we computed the 10080/2 cross sections for each of the 600 grid points. We calculated a total of 3, 024, 000 theoretical cross sections, which represents a long computational time.

The χ 2 is then written as:

χ 2 DR (α E (Q 2 ), β M (Q 2 )) = N i=1 σ exp F W (i) -σ DR F W (i, α E (Q 2 ), β M (Q 2 )) ∆σ exp F W (i)(stat) 2
. (5.24) The 600 χ 2 -values show a clear and unique minimum as represented in gure 5.20, and near the minimum, the χ 2 follows a paraboloid shape. ). E will be the minimum χ 2 , B its position along the x-axis (corresponding to the extracted electric GP) and D its position along the y-axis (corresponding to the extracted magnetic GP). The tted function also allows to extract the statistical errors on α E (Q 2 ) and β M (Q 2 ) by drawing the contour at χ 2 min + 1 [START_REF] James | Function minimization and error analysis[END_REF].

In contrary to the LEX t that requires supplementary selections, none of them are necessary for the DR t, which is conceptually much simpler 4 . Each of the 600 χ 2

DR was then calculated thanks to 4431 cross-section points 5 distributed in our dierent settings, as presented in table 5.6.1. Another dierence with the LEX t is that the DR extraction is valid even above the pion production threshold, allowing us to use two more q cm bins, i.e. bins at q cm = 137.5 M eV /c and q cm = 162.5 M eV /c. 4 the gradient selection was already not mandatory for the LEX t, for the DR t its impact is also small in terms of extracted GPs, so we decided to not use it. 5 we measured 7784 cross sections but some are measured several times by dierent settings, here the measurements are averaged over settings. α E (Q 2 ) = 4.05 ± 0.10 stat (10 -4 .f m 3 ) β M (Q 2 ) = 1.08 ± 0.12 stat (10 -4 .f m 3 ) .

The corresponding Λ parameters are: Λ α = 0.631 ± 0.011 stat (GeV ) Λ β = 0.745 ± 0.021 stat (GeV ) , which lead to the following structure functions: P LL -P T T / = 18.70 ± 0.38 stat (GeV -2 ) P LT = -3.48 ± 0.25 stat (GeV -2 ) .

The DR t hides an undiscussed feature. The model depends on the RCS electric and magnetic polarizabilities used as input in the code. Indeed the RCS polarizabilities appear in equations 2.8 and 2.9. Moreover, the value of βM has recently changed ( βM = (2.5 ± 0.4) • 10 -4 f m 3 [28]) and may change again in the future. So, one can wonder if variations of RCS polarizabilities would impact our DR analysis.

It is likely that a change in the RCS values in equations 2.8 and 2.9 will be compensated in our DR t by a change of the tted Λ parameters, leaving nally the GPs and the structure functions unchanged [START_REF] Fonvieille | [END_REF]. However, this remains to be veried.

Systematic errors

In this section we explain the choice that has been made to estimate the systematic errors in our analyses. We describe the two estimation methods that we used for the LEX t, and we conclude more briey for the DR t.

First method: changing the renormalization of the experimental cross section

The rst method consists in using the renormalization factor F norm to estimate the systematic errors. In this section we assume that the global systematic error can be obtained by considering a simple change of overall renormalization.

As explained in section 5.4 we believe that the renormalization factor corrects our data from a global bias with a excellent accuracy. However it does not prevent our analysis from inhomogeneous issues which can be due for example to:

-angular distortions caused by bias in optics or local loss of events. By observations of variations between dierent analyses at dierent stages of our progress we can estimate this as a ±0.01 variation of F norm .

-non-homogenous radiative eects that are ignored by our averaging of F rad . This eect shows variations that are equivalent at maximum to a ±0.007 variation of F norm (cf. section 5.2.2).

-non-homogenous renormalization factor. In section 5.4.3 we have observed that the renormalization factor undergoes variations from a setting to another (cf. table 5.4.2). We consider that they induce a systematic error of ±0.006 of F norm .

-error induced by applying a renormalization factor determined by low-energy settings to other settings. By doing this we assumed that the main part of systematic errors have a common origin. But we observed a ±0.006 variation between the LOW and LOW-bis renormalization factors. We can then estimate the systematic error brought by this assumption to be about the same order.

-moreover we can add the systematic error induced by the use of dierent conditions for the LEX t in the iterative procedure which allows to determine F norm . It has been shown in section 5.4.3 that it is a ±0.002 eect on F norm .

Finally by summing quadratically those ve systematic-error sources we estimate our total systematic error as an uncertainty of ±0.015 on the renormalization factor, or, in other terms, a total systematic uncertainty of ±1.5% on each crosssection point. This eect is propagated to the LEX and DR analysis, by simply changing the global renormalization factor that enters in the ts. For the LEX analysis we nd that ∆F norm ± 0.015 induces the following changes in the physics observables:

∆P LL -P T T / = ∓3.64 syst (GeV -2 ) ∆P LT = ∓1.57 syst (GeV -2 ).

The errors are anti-correlated with respect to ∆F norm .

Second method : multiple LEX analyses

The second method we used to estimate the systematic error is based on multiple analyses of the data. We had the chance to produce independent LEX analyses with dierent choices, i.e. dierent: spectrometers optics, target positioning, analysis cuts, renormalization factors, bin selection. So we can estimate the main systematic eects, which are the following:

-using the renormalization factor from another analysis to the nominal one induces structure-function variations of ∆P LL -P T T / = ±2.35 (GeV -2 ) and ∆P LT = ±1.02 (GeV -2 ).

-changing the cut on the reconstructed vertex by ±6 mm induces structurefunction variations of ∆P LL -P T T / = ±0.21 (GeV -2 ) and ∆P LT = ±0.30 (GeV -2 ).

-multiplying the upper cut on the missing-mass squared by 2 or 0.5 induces structure-function variations of ∆P LL -P T T / = ±1.67 (GeV -2 ) and ∆P LT = ±0.33 (GeV -2 ).

-using a theoretical cross-section in VCSSIM that contains a GP eect (based on our nal result) induces structure-function variations of ∆P LL -P T T / = ±0.15 (GeV -2 ) and ∆P LT = ±0.11 (GeV -2 ).

-not considering the local target boiling described in section 4.4.2 induces structure-function variations of ∆P LL -P T T / = ±0.1 (GeV -2 ) and ∆P LT = ±0.8 (GeV -2 ).

-considering a supplementary analysis cut in order to remove the unconsistency of gure 4.22 for the setting INP induces structure-function variations of ∆P LL -P T T / = ±1.02 (GeV -2 ) and ∆P LT = ±0.92 (GeV -2 ).

-considering dierent form-factor parametrizations induces ∆P LL -P T T / = ±0.2 (GeV -2 ) and ∆P LT = ±0.2 (GeV -2 ) (cf. table 5.4. 3-c).

-changing the K thr value (dened section 5.5.2) inside the stability plateau of gure 5.18 induces ∆P LL -P T T / = ±0.5 (GeV -2 ) and ∆P LT = ±0.1 (GeV -2 ).

-not using the gradient selection induces structure-functions variations of ∆P LL -P T T / = ±0.4 (GeV -2 ) and ∆P LT = ±0.01 (GeV -2 ).

If we do the quadratic sum we nd: ∆P LL -P T T / = ±3.14 (GeV -2 ) ∆P LT = ±1.66 (GeV -2 ) , which agrees well with the previous determination. The correlations of signs cannot be determined by this method of estimation of systematic errors.

Conclusion on systematic errors

For the LEX t we have been able to estimate the systematic errors on the tted observables by two methods, and they are in good agreement. We decided to use the rst method because of its simplicity of application. The systematic errors for the structure functions, presented in section 5.8, are obtained this way. They are then propagated to the LEX GP determination of section 5.5.4.

For the DR t we have not done such an extensive study, therefore, thanks to its simplicity, we will adopt the rst method, i.e. we apply a change ∆F norm = ±0.015 in order to nd the systematic error on the observables extracted by the DR t.

We nd The correlations of sign with ∆F norm observed here are due to the use of the rst method and will not be kept in the following.

Final results

The rst t, based on the LET, is in principle model-independent, technically easy to implement but requires to be done carefully because of higher-orders in q cm . While our LEX result for P LT is almost not impacted by the mask selection, our LEX result for P LL -P T T / is very sensitive to it: the tted value changes from 15 GeV -2 (without mask) to 18 GeV -2 (with mask selection), i.e. it changes by 3 GeV -2 , which is almost equal to one systematic standard deviation. This can be seen as having introduced some degree of DR-model-dependency in our LEX t. The LEX GPs extracted in section 5.5.4 depend even more obviously on the DR model.

The second extraction, based entirely on the DR model, is more straightforward but is technically harder to implement because of the computational time.

To perform both extractions we used cross sections corrected from radiative eects, and renormalized. We have shown the importance of the renormalization process which minimizes the systematic eects in our analyses and makes the extractions independent of form-factor parametrizations.

The extracted structure functions are summarized in section 5.8.1 and the extracted GPs are summarized in section 5.8.2.

Thanks to the strength of the renormalization process we were able to quantify the systematic errors for both extractions, LEX and DR. And thanks to the exibility of the LEX t we were able to check the systematic errors by doing several independent analyses.

Structure functions

The nal extracted values for the LEX t are: These two sets of results are in excellent agreement. The agreement is even more meaningful and satisfying for P LT which, for the LEX extraction, does not depend on the mask selection and so is entirely independent of the DR model. Figure 5.22 presents a non-simple Q 2 behavior of P LL -P T T / which demands to be explored further in the region around Q 2 = 0.33 GeV 2 . Our tted value of P LT is rather small and also not in smooth agreement with the results at Q 2 = 0.33 GeV 2 . Also, the picture at Q 2 = 0 GeV 2 may change in the future due to the new measurements of the RCS polarizabilities.

GPs

The nal extracted GPs by means of the LEX t plus DR-subtraction of the spin-ip GPs are:

α E (Q 2 ) (LEX) = 3.90 ± 0.14 stat ± 0.92 syst (10 -4 .f m 3 ) β M (Q 2 ) (LEX) = 1.14 ± 0.15 stat ± 0.78 syst (10 -4 .f m 3 ).

From the DR t we obtained: α E (Q 2 ) (DR) = 4.05 ± 0.10 stat ± 0.98 syst (10 -4 .f m 3 ) β M (Q 2 ) (DR) = 1.08 ± 0.12 stat ± 0.90 syst (10 -4 .f m 3 ).

The two set of results are in excellent agreement. Figures 5.24

Conclusion

In this thesis we analysed the data of a photon electroproduction (ep → epγ) experiment at MAMI, at Q 2 = 0.2 GeV 2 . We extracted a number of observables characterizing the VCS process at low energy: the structure functions P LL -P T T / and P LT , and the electric and magnetic GPs α E (Q 2 ), β M (Q 2 ).

This study was motivated by the previous VCS experiments. Their results have raised questions, namely about the Q 2 -behavior of the electric generalized polarizability which demanded to be explored. This work is a part of a more complete experiment that also covered two other Q 2 -values: 0.1 and 0.45 GeV 2 , in order to obtain a more complete picture of the situation in the region around

Q 2 = 0.33 GeV 2 .
The rst diculty of such experiments is the smallness of the ep → epγ cross section, requiring to accumulate enough statistics. Our experiment is a highstatistics one, with ≈ 10 6 events identied as ep → epγ when summed over our ve settings at Q 2 = 0.2 GeV 2 .

The second and main diculty to extract GPs comes from their small contribution (0-15%) to the photon electroproduction cross-section, requiring to measure this cross section with small systematic uncertainties, of the order of a few percent at maximum. In order to obtain the systematic precision of ±1.5% our experiment had a few strong points. First, we beneted of the MAMI facility and the A1 spectrometers. The high-duty cycle of the accelerator and the high resolution of the spectrometers are mandatory for such experiments. The calibration work was also a determinant part of our analysis. Its relevance was conrmed by the use of a Monte Carlo simulation. The last strong aspect is the renormalization procedure. Indeed our renormalization work highly diminished the impact of the possible experimental inconsistencies and of the theoretical uncertainties, e.g. the blur brought by the limited form-factor knowledge.

We were able to extract the structure functions and the GPs by two dierent approaches. The rst one is based on the LET and was widely explored in this thesis. The second extraction, based on the dispersion relation model, was more straightforward but explored to a lesser extent. Those two dierent types of analysis were already done by previous experiments, but our experiment also contains two specicities.

The rst one resides in the choice of the analysed phase-space. The measured cross-section points were distributed in dierent angular regions of interest, thus increasing the sensitivity of our analyses to the structure functions and GPs.

The second original feature is the use of the DR model to select the analysed data in the LEX analysis. The previous observations (from the Bates experiment)

and the ones we did during our analysis are evidences that the higher-order terms in the nal photon energy q cm impact our results. The mask selection that we built, based on the estimation of the higher-order terms by the DR model, is then an essential feature of our LEX analysis.

The two dierent extractions, LEX and DR, are in very good agreement, within their statistical error which is quite small. Actually, the results are dominated by the systematics which are dicult to reduce in VCS experiments.

The parallel analyses at Q 2 = 0.45 GeV The generalized magnetic polarizability of the proton is smaller and by consequence its extraction is more dicult due to the systematic errors. This complicates the observation of an extremun in the β M (Q 2 ) (or P LT ) Q 2 -behavior. However one observes also a non-smooth behavior with respect to the previous MAMI results. Because of the recent changes in the RCS magnetic polarizability value, we can anticipate that gure 5.25 will also change at Q 2 = 0 in the future.

To conclude, the study of the GP behavior requires more investigations and will be hopefully pushed further [START_REF] Sparveris | Jeerson Lab Proposal PR12-15-001[END_REF]. In a near future, our nal electric GP result will be confronted to the result of another VCS experiment done at MAMI, in the ∆-resonance region, also at Q 2 = 0. )).

(C.3)

The vertex process from gure 5.3 leads to δ ver : A third term δ

δ vertex = α QED π ( 3 2 ln( Q 2 m 2 e ) -2 -
ρ 2 = Q 2 + 4M 2 N x = (Q+ρ) 2
(1) 2 also describes the proton-side processes. It is presented in [92] as being about the same order of magnitude than δ (0) 2 , i.e. ≈ 0.02% of F rad in our kinematics. δ (1) 2 was not determined in our analysis but its omission is considered in the error budget on F rad .

1. 1 .

 1 PROTON FORM FACTORS 1.1.1 Elastic electron-proton scattering

Figure 1 . 1 :

 11 Figure 1.1: Diagram of elastic electron scattering o the proton. The blob indicates that the proton's internal structure is probed.

  behavior of those three parametrizations is presented in gure 1.2.

Figure 1 . 2 :

 12 Figure 1.2: Behavior of electric (left) and magnetic (right) proton form factors, divided by the dipole, as a function of Q 2 . The parametrizations are from Bernauer et al. [19] (dashed green line), Arrington et al. [18] (red dotted line) and Friedrich-Walcher [17] (blue full line). The vertical line shows the value at Q 2 = 0.2 GeV 2 .

Figure 1 . 3 :

 13 Figure 1.3: Diagram of Real Compton Scattering.

  of the incoming and scattered photons, n and n their unit momentum vector and σ the Pauli spin matrices. In the Petrun'kin amplitude of equation 1.11 the rst term is the Thomson scattering amplitude, the six following lines are due to the proton magnetic moment. The polarizabilities arise in the last line through ᾱE and βM . ᾱE and βM are the polarizabilities that are experimentally accessible, also called dynamical. They dier from the static polarizabilitites α E and β M of the previous subsection by a corrective term called the retardation correction 4 . For instance one can write:

Figure 1 .

 1 Figure 1.4 from [29][30] is one of these representations. It is a theoretical picture obtained in the framework of the heavy-baryon chiral perturbation theory (HBχPT O(p 3 )). The gure shows the local induced electric dipole moments as an electric eld is applied along the x axis; as we can see the local deformation also possesses a component transverse to the electric eld.

Figure 1 . 4

 14

Figure 1 . 4 :

 14 Figure 1.4: Nucleon polarization induced by an electric eld along the x axis, calculated in the framework of HBχPT O(p 3 ).Figure taken from [29].

  Figure taken from [29].

Figure 1 . 5 :

 15 Figure 1.5: Induced polarization in a proton of denite light-cone helicity when submitted to an electromagnetic eld. The light (dark) regions correspond to the largest (smallest) values. Figure taken from [31].

Figure 1 . 6 :

 16 Figure 1.6: Diagrams of the photon electroproduction process: (a) bremsstrahlung from the electrons side, (b) Born part of VCS, (c) non-Born part of VCS.

Figure 1 . 7 :

 17 Figure 1.7: π 0 exchange in the t-channel.

Figure 1 . 8 :

 18 Figure 1.8: VCS represented in the CM and laboratory frames. (Left) VCS variables as they are dened in the CM. (Right) representation of the nal proton

Figure 1 . 9 :

 19 Figure 1.9: In-plane cross section of the photon electroproduction process with respect to θ cm : (green dashed line) BH cross section; (light-blue dashed dotted line) Born cross section; (black full line) cross section of (BH+Born) processes including their interference; (red dotted line) photon electroproduction cross-section considering a theoretical GP eect, multiplied by 3 for visibility. The kinematics are dened by q cm = 458 M eV /c, q cm = 100 M eV /c, = 0.85 and ϕ = 180 • (negative θ cm ) or ϕ = 0 • (positive θ cm ), leading to Q 2 = 0.199 GeV 2 .

  The third and last step is based on the use of unitarity to replace the imaginary part of equation 2.6 by the absorption amplitudes. The absorption amplitudes represented in gure 2.1 represent all possible intermediate states X other than one single nucleon. But being at low energy, only the dominant term of an intermediate pion-nucleon state (πN ) is considered in the DR model. The amplitudes for (γ * p → πN ) and (πN → γp ) are obtained thanks to the MAID model from [40] including both resonant and non-resonant contributions.

Figure 2 . 1 :

 21 Figure 2.1: The absorptive part of the γ * p → γp reaction.

Figure 2 . 2 :

 22 Figure 2.2: Scalar GPs from the DR model for xed parameters Λ α = Λ β = 0.7 GeV . The red dashed line is the asymptotic contribution, the blue dotted line is the πN contribution and the black full line is the sum of the two.

Figure 2 . 3 :

 23 Figure 2.3: Scalar GPs from the DR model for dierent values of the Λ parameters. Red full line Λ = 0.3 GeV , green dashed line Λ = 0.5 GeV , light blue light dashed line Λ = 0.7 GeV , dark blue dotted line Λ = 0.9 GeV and black line Λ = 1.1 GeV .

Figure 2 . 4 :

 24 Figure 2.4: Typical pion loop diagram for Compton scattering.

  Figure 2.5: VCS Non-Born diagrams calculated in the eective Lagrangian model [49]. The N * and X symbols are specied in the text.

2. 3 .

 3 PREVIOUS AND CURRENT VCS EXPERIMENTS -The harmonic oscillator energy which is xed to reproduce the mass dierence between the baryon resonances.

  The model was later reconsidered by B. Pasquini, S. Scherer and D. Drechsel [59] for a new prediction of GPs. The following resonances were used: D 13 (1520), S 11 (1535), S 31 (1620), S 11 (1650), S 13 (1700), D 33 (1700) and recoil eects were implemented in the model. The following values were predicted for the scalar RCS polarizabilities: ᾱE = 5.5 • 10 -4 f m 3 , βM = 4.7 • 10 -4 f m 3 .

Figure 2 . 6 :

 26 Figure 2.6: Results for the six GPs in dierent model calculations as a function of Q2 . Full line: nRQM (CQM*100) [59], dashed line: linear sigma model [52], dotted line: eective Lagrangian model [50], dashed-dotted lines: HBχPT [47],

Figure 2 .

 2 [START_REF] Roche | The First determination of generalized polarizabilities of the proton by a virtual Compton scattering experiment[END_REF] represents the world measurements for the structure function P LL -P T T / . In the gure the DR prediction, if one xes Λ α = 0.7 GeV for all Q 2 , is represented as the dashed curve. This clearly shows the tension between the three-times conrmed Q 2 = 0.33 GeV 2 measurement and the smooth fall-o described by the curve and the other data points.

Figure 2 . 7 :

 27 Figure 2.7: World measurements of P LL -P T T / performed at JLab, Bates and MAMI by LEX analysis (empty red crosses) and DR analysis (full blue crosses) with statistical (inner) and total (quadratically summed, outer) error bars. The DR model prediction assuming a dipole behavior for the asymptotic part of α E (Q 2 ) with Λ α = 0.7 GeV, and a xed = 0.85 is also plotted (light blue dashed line).

  2 and Q 2 = 0.45 GeV 2 . I was in charge of analysing the Q 2 = 0.2 GeV 2 data set and two PHD students were in charge of the other data sets[START_REF] Bericic | Measurement of Generalized Polarizabilities of the Proton by Virtual Compton Scattering[END_REF][START_REF] Benali | Structure interne du proton à haute et basse énergie par la diusion Compton virtuelle[END_REF].

Figure 2 . 8 :

 28 Figure 2.8: World measurements of P LT performed at JLab, Bates and MAMI by LEX analysis (empty red crosses) and DR analysis (full blue crosses) with statistical (inner) and total (outer) error bars. The DR model prediction assuming a dipole behavior for the asymptotic part of β M (Q 2 ) with Λ α = 0.7 GeV is also plotted (light blue dashed line). Some points are shifted in Q 2 for visibility.

Sections 3 .

 3 1 and 3.2 are focused on the description of the Mainz Microton and the three-spectrometer facility of the A1 collaboration. Section 3.3 explains which experimental congurations are more relevant to perform a GP extraction leading to our choice of kinematical settings. 3.1 MAMI 3.1.1 The Mainz facility The Mainz Microtron (MAMI), located at the Institut für Kernphysik (KPH), is an accelerator for electron beams. The MAMI operation principles were developed by Prof. Dr. Helmut Herminghaus in cooperation with Dr. Karl-Heinz Kaiser [73]. The accelerator, operational since 1979, is regularly improved. The latest stage of development is MAMI-C, nalized in 2006, it provides an unpolarized (polarized) electron beam up to 100 µA (20 µA) and can reach an energy of 1.5 GeV.

Figure 3 .Figure 3 . 1 :

 331 Figure 3.1 is a layout of the facility. In each hall a collaboration is in charge of an experimental setup and follows a specic physics program. The A2 collaboration uses tagged real-photons to study processes such as: total photo-absorption, real Compton scattering and meson photo-production. The A4 collaboration was focused on parity violation in electron scattering. The physics motivation of the X1 collaboration is centered on the use of X-rays to determine material or biological properties.

  Figure 3.1 shows the electron path through the Mainz Microtron. After the choice of beam polarization (two dierent sources are available for polarized and unpolarized electrons) the electrons are sent through the injector. It is a linear accelerator (LINAC) that will increase the electron energy up to 3.5 MeV before entering the race-track microtrons (RTM).

Figure 3 . 2 :

 32 Figure 3.2: Schematic of a racetrack microtron (RTM) [74].

Figure 3 . 3 :

 33 Figure 3.3: Schematic of a harmonic double-sided microtron (HDSM) [74].

  of electron scattering experiments depends on target density (and indirectly its temperature). Furthermore the electrons will deposit energy in the target, warming it and changing its density. It is thus important to stabilize the target temperature by a cooling system. The liquid hydrogen target was built on the concept of closedcircuit loop. The circulation of liquid hydrogen is powered by a ventilator that ensures a constant ow in the target. Then the heat deposited by the electron beam dissipates during the passage of liquid hydrogen in the heat exchanger. The heat exchanger is cooled by a Philips cryo-generator. The whole scattering chamber and the Philips-Machine are represented in gure 3.4.A second important point is to avoid the local density variation inside the cryo-target. A too localized and intense beam can even create local boiling. The "wobbler" mechanism, that displaces the beam transversely by a few millimeters along a determined pattern at a frequency of several kHz, will prevent the target from local boiling. The target dimensions are represented in gure 3.4, its length is 49.5 mm for a width of 11.5 mm. The cell walls are made of a HAVAR foil of 10µm thickness.

Figure 3 . 4 :

 34 Figure 3.4: Schematic of the A1 vacuum scattering chamber and the liquid hydrogen target (bottom left).

Figure 3 . 5 :

 35 Figure 3.5: Schematic of the A1 three-spectrometers setup [75].

  ) with (x ref , θ ref , y ref , φ ref ) the focal-plane coordinates of the reference track.

Figure 3 . 6 :

 36 Figure 3.6: Schematic of a particle track in spectrometer A and representation of frame coordinates. ( x, z) is the dispersive plane, ( y, z) is the non-dispersive plane [76].

  Figure 3.7 is a schematic representation of the drift time as measured by a time-to-digital converter (TDC), the common stop being given by the scintillators.

Figure 3 . 7 :

 37 Figure 3.7: Schematic of the electric eld lines in a VDC when a particle crosses the detectors and representation of the corresponding recorded signal in TDC channels, from [78]. The longer signals correspond to closer wires.

( 14 )

 14 paddles linked via a light-guide to two photomultipliers (PMT). The PMT signals are sent to time-to-digital converters (TDCs) and analog-to-digital converters (ADCs) providing informations that have a threefold use: Firstly, since their response signals are fast they serve as trigger for the data acquisition.Secondly, they are used for timing measurements. As mentioned before, scintillators give the stop signal for VDC channels but they also allow a reconstruction of the coincidence time. When a particle hits the scintillator in, for instance, spectrometer A, it opens a ≈ 80 ns window. Then the acquisition waits for a second signal from the other spectrometer scintillators. The measured time between these two TDC signals (after ne corrections) is presented in gure 3.8 (without any selection of events). We distinguish two kinds of coincidences: the accidental coincidences which are randomly distributed in time; and the true coincidences which are in a peak centered on 0. Figure3.8 illustrates the very good time resolution (σ ≈ 0.40 ns) and the advantage of a high duty cycle as MAMI (the ratio of true coincidences over random ones is ≈ 15).

Figure 3 . 8 :

 38 Figure 3.8: Coincidence time measurements by scintillators (after software corrections), before (left) and after (right) zooming the y axis. The accidental coincidences appear as a at distribution, the true coincidences are distributed in a narrow peak.

10

  MeV electrons are faster than light in the medium which has a refraction index n = 1.0014). The Cerenkov process occurs in a decauorobutane (C 4 F 10 ) gas at atmospheric pressure [77]. The emitted light is reected by mirrors to photomultipliers; there are twelve mirrors in spectrometer A and ve in spectrometer B. 3.2.4 Trigger and data acquisition: the A1 software Trigger and data acquisition The trigger is independent in each spectrometer and the trigger conditions can be set dierently in each spectrometer. The minimal trigger condition is a single signal in only one scintillator PMT but, to minimize the background, a coincidence between the left and right PMTs of the scintillator paddle is required. Then one can ask for the particle to be detected in one of the two planes (dE or ToF) or in both of them. After being analyzed by the programmable logic unit (PLU) the signals are sent to a fast programmable gate array (FPGA) that gather signals from the three spectrometers. Then if the FPGA accepts an event a signal is sent to the frontend electronics of spectrometers to start the read-out process of digitized detector signals. Finally the data are sent to a workstation where the acquisition program

Figure 3 . 10 :

 310 Figure 3.10: Theoretical relative GP eect according to the LET and equation 3.3 for P LL -P T T / = 20 GeV -2 and P LT = -4 GeV -2 at q cm = 112.5 M eV /c. The blank region at the center of the plot is removed for clarity reasons and corresponds to the Bethe-Heitler peaks.

Figure 3 . 11 :

 311 Figure 3.11: Theoretical behavior of v LL in the (cos(θ cm ),ϕ cm )-plane at xed q cm = 458 M eV /c and = 0.85.

Figure 3 . 12 :

 312 Figure 3.12: Theoretical behavior of v LT in the (cos(θ cm ),ϕ cm )-plane at xed q cm = 458 M eV /c and = 0.85.

Figure 3 .

 3 Figure 3.13: (Left) behavior of O DR (q cm2 ) in the (cos(θ cm ),ϕ cm )-plane at q cm = 87.5 M eV /c and (right) two-dimensional representation of the angular region where O DR (q cm 2 ) < 2% (blue), the red squares correspond to the two areas of interest to

P

  A ) are the central momenta and (θ B , θ A ) are the spectrometer angles w.r.t. the beam. The scattered electron and proton are respectively detected in spectrometer B and A.

Figure 4 .

 4 1 shows an example of the missing-mass squared as reconstructed by Cola++.

Figure 4 . 1 :

 41 Figure 4.1: The missing-mass squared as reconstructed by Cola++ for all coincidences of the setting INP. The left peak corresponds to ep → epγ events, the small peak at the right to the ep → epπ 0 reaction.

Figure 4 . 2 :

 42 Figure 4.2: Setting INP: reconstructed M 2

Figure 4 . 3 :

 43 Figure 4.3: Mean value of the error on reconstruction per track with respect to a range of TDC osets, for the spectrometer B S2 layer; the variable is labeled as B.s2.chi_sqr in Cola++. The error bars come from the determination of the mean by ROOT.

Figure 4 .

 4 4 is a schematic representation of those events in a VDC plane.

Figure 4 . 4 :

 44 Figure 4.4: A hot wire, a good event and a δ-electron event in a schematic representation of a VDC plane from [71]. The vertical lines represent the time it took for the signal to arise after the passage of a particle through the VDC.

Figure 4 . 5 :

 45 Figure 4.5: Setting INP: coincidence time spectrum between spectrometers A and B before oset correction. The FWHM of the peak is equal to 0.9 ns.

Figure 4 . 6 shows

 46 the typical shape of the target along the beam axis as seen by spectrometer B; the peaks at the edges are caused by the walls of the target which generate more particle at our kinematics. The calibration of spectrometer B optics coecients Y ijkl is optimal when the peaks are sharper.

Figure 4 . 6 :

 46 Figure 4.6: Vertex reconstruction for the setting INP. V ertex Z is reconstructed with the standard optics from A1 (black) and with our optimized optics (red), for which the reconstruction has improved.

  954 -0.278(M eV /c), 579.954 being the value measured by the NMR probe.

Figure 4 .

 4 [START_REF] Roche | The First determination of generalized polarizabilities of the proton by a virtual Compton scattering experiment[END_REF] shows a snapshot of the aluminum screen hit by the beam. Then the beam is turned o, the aluminum screen is replaced by the cryo-target, and the beam is turned on again. At this point there is no possibility to check its exact position versus time.

Figure 4 . 7 :

 47 Figure 4.7: Snapshot of the aluminum screen hit by the beam. The composition of the screen makes it glow where the electrons strike it. The bright rectangle shows the wobbler size in horizontal (x) and vertical (y); here (±2 mm)(±1 mm).

Figure 4 . 8 :

 48 Figure 4.8: (Top view) Schematic representation of the error ∆Z on the reconstructed vertex position induced by an error on the beam position ∆X beam,lab .

Figure 4 .

 4 Figure 4.8 is a schematic representation of how the beam position acts on the vertex reconstruction. A change of beam position along x lab , from the black line

Figure 4 . 9 :

 49 Figure 4.9: Mean position of the reconstructed V ertex Z by spectrometer B for all the runs of the setting INP chronologically ordered. The top gure is for Beam.of f set.x = 0 and Y specB = 0, the bottom gure is after corrections: Beam.of f set.x adjusted per groups of runs, plus a global oset ∆Y 0 (B). The red dots stand for runs taken just after a beam positioning using the aluminum screen.

Figure 4 . 10 illustrates 2 X

 4102 for the setting INP how the increasing snow thickness aects the M peak position over time and how we corrected for it.

Figure 4 . 10 : 2 X

 4102 Figure 4.10: Position of the M 2 X peak with respect to the run label, before (black) and after (red) snow and P ref B corrections for the setting INP. The error bars are given by the t of the mean position. The large gap at run #6 corresponds to a large snow deposit after a long beam interruption. During this data taking the snow thickness slowly grows and the peak position slowly increases which result in a small slope on the present gure (black points).

Figure 4 . 11 :

 411 Figure 4.11: Experimental ϕ cm -distribution for VCS events, setting INP.

Figure 4 .

 4 Figure 4.12: Spec-A collimator vertical oset (red line) obtained by missing-mass resolution minimization for three in-plane settings. The error bars (green area) correspond to the t error on the determination of the minimum. The combined error (dashed blue area) is obtained by the quadratic sum of errors.

Figure 4 . 13 :

 413 Figure 4.13: Measured cross section (black crosses) at q cm = 87.5M eV /c and cos(θ cm ) = -0.675 for the setting INP before (top) and after (bottom) vertical misalignment correction. The curves are the theoretical cross section without (red) and with (blue) a GP eect.

Figure 4 .

 4 Figure 4.14: Setting INP coincidence time measurement. The red areas are used to estimate the number of accidental coincidences under the peak (blue area). The coincidence-time cut requires |T AB | < 4 ns.

Figure 4 .

 4 Figure 4.15 shows the energy deposited by protons and pions in spectrometer A scintillators after their ADC calibration and without the three mandatory selections. The protons deposit more energy and appear as a clear population at the top right of the gure.

Figure 4 . 15 :

 415 Figure 4.15: Energy deposited by protons and pions in spectrometer A scintillators (in units of ADC channels) after ADC calibration, without any analysis cut, setting INP.

Figure 4 . 17 :

 417 Figure 4.17: Example of optics control plots. (Left) reconstructed θ f p angle by the spectrometer A of the setting LOW-bis versus M 2 X . (Right) reconstructed θ f p angle by the spectrometer B of the setting OOP 2015 versus M 2 X .

Figure 4 . 18 :

 418 Figure 4.18: Reconstructed angle θ f p by the spectrometer B of the setting LOWbis with respect to the reconstructed vertex position. The darker areas correspond to the target walls. As expected, the shape of the target does not depend on the θ f p (B) angle.

Figure 4 . 19 :

 419 Figure 4.19: Ratio of ep → epγ events divided by the corrected luminosity for each run of the setting OOP-2015 with its statistical error. The ratio is expected to be constant.

Figure 4 .

 4 Figure 4.20: (Top) ratio of ep → epγ events divided by the corrected luminosity for each run of the setting INP: The ratio is slightly decreased after the run # 70. (Bottom) beam current for each run of the setting INP.

(4. 9 )

 9 times a correction factor related to the full distribution of incoming momentum k e[START_REF] Janssens | Monte Carlo simulation of virtual Compton scattering below pion threshold[END_REF].4.6. SIMULATED AND EXPERIMENTAL DATA COMPARISON

Figure 4 .

 4 21 is the reconstructed target angle φ 0 by the spectrometer A of the setting LOW-bis from VC-SSIM (blue) and from experiment (red); the gure shows a perfect matching. Such agreements are obtained for almost all variables.

Figure 4 . 21 :

 421 Figure 4.21: Reconstructed target angle by the spectrometer A of the setting LOWbis from VCSSIM (blue) and from experiment (red). The integral of simulated events is multiplied by the ratio L exp /L sim .

Figure 4 .

 4 Figure 4.22: φ 0 angle reconstructed by spectrometer A at q cm = 87.5 M eV /c for the INP setting. A small disagreement is found on the left edge of the distributions.

Figure 4 .

 4 Figure 4.23: Missing-mass squared distributions from VCSSIM (blue) and from the experiment (red) with the xed normalizations for the four analyzed settings.

Figure 4 .

 4 Figure 4.24: (Left) setting LOW ϕ cm distributions from VCSSIM (blue) and from the experiment (red) with xed normalizations at q cm = 37.5 (M eV /c). (Right) setting INP ϕ cm distributions, at q cm = 112.5 (M eV /c), when one considers a GP eect in VCSSIM (blue).

  δ r : Electron-side supplementary real photon A large part of the radiative corrections is due to supplementary electron bremsstrahlung of a so-called soft photon. This second real photon may be emitted before or after the hadronic part of the interaction. Figure 5.1 is an example of one of the considered diagrams, where the Bethe-Heitler process comes with one supplementary real photon in red.

Figure 5 . 1 :

 51 Figure 5.1: One possibility of Bethe-Heitler diagram with one supplementary real photon (red).

Figure 5 . 2 :

 52 Figure 5.2: Vacuum polarization diagrams. The particles are represented by the line symbols used before.

Figure 5 . 3 :

 53 Figure 5.3: Vertex process diagrams.

Figure 5 . 4 :

 54 Figure 5.4: 1/F rad behavior in the (cos(θ cm ), ϕ cm )-plane at q cm = 112.5 M eV /c. The central part (white) is removed by the gradient-selection (cf. section 5.5.1).

Figure 5 . 6 :Figure 5 . 7 :

 5657 Figure 5.6: Setting LOW: measured ep → epγ cross section at xed q cm = 37.5 M eV /c with respect to ϕ cm for the rst-half of cos(θ cm )-bins. The error bars are statistical. The blue full line is the theoretical cross section σ BH+B , the red dotted line is σ BH+B plus a polarizability eect calculated by the LEX with P LL -P T T / = 18.1 GeV -2 and P LT = -3.6 GeV -2 .

Figure 5 .

 5 Figure 5.8 presents the measured cross section from the setting INP at xed q cm = 112.5 M eV /c. When |ϕ cm | → 0 the measured points tend to lie between the theoretical curves with and without a GP eect. This may already indicate the importance of higher orders in the theoretical curve (BH+B+GPs) which is built thanks to the LEX formalism (a similar observation was done in gure 4.24).

Figure 5 . 8 :

 58 Figure 5.8: Setting INP: measured ep → epγ cross section at xed q cm = 112.5 M eV /c with respect to ϕ cm for all the cos(θ cm )-bins. The curves follow the convention of gure 5.6.

( 5 . 14 )Figure 5 . 9 :

 51459 Figure 5.9: Setting OOP: measured ep → epγ cross section at xed q cm = 112.5 M eV /c with respect to ϕ cm for the rst-half of cos(θ cm )-bins. The curves follow the convention of gure 5.6.

Figure 5 .

 5 Figure 5.10: Setting OOP: measured ep → epγ cross section at xed q cm = 112.5 M eV /c with respect to ϕ cm for the second-half of cos(θ cm )-bins. The curves follow the convention of gure 5.6.

Figure 5 .

 5 Figure 5.11 is a schematic representation of the iterative procedure. It starts by the χ 2 minimization of F norm with any initial choice of theoretical cross section. The obtained F norm 0 is then used to renormalize the measured cross section (equation 5.14). Afterwards these experimental cross sections are used in a LEX t to extract the structure functions which will be used to compute a new theoretical cross section. This last one is then used for a second determination of the renormalization factor F norm 1 , and the procedure continues.The iterative process stops when the dierence F norm n-1 -F norm n = ±0.002, which is the statistical precision on F norm (deduced from the points corresponding to χ 2 norm + 1). Obviously the iterative process has to be done keeping the form-factor parametrization xed, and the bins selection xed.

Figure 5 . 11 :

 511 Figure 5.11: Schematic representation of the iterative procedure. The t of F norm is described by equation 5.15, the LEX t itself is described in section 5.5 and the theoretical cross section is calculated thanks to equation 1.38.

Figure 5 . 12 :

 512 Figure 5.12: Illustration of three iterative processes. Each point is the structurefunction extraction (with statistical errors) obtained from the cross section renormalized by the indicated value of F norm . The colors stand for dierent starting points of the iterative process as explained in the text. Here all procedures converge to the point P LL -P T T / = 19 GeV -2 , P LT = -3.7 GeV -2 and F norm = 0.980.

  4.3-c: LEX extraction of structure functions before renormalization (F norm = 1) and after (F norm = F norm f orm-f actor ) for the three selected parametrizations of form factors. For every extraction the statistical errors are ∆P LL -P T T / = ±0.64 (GeV -2 ) and ∆P LT = ±0.34 (GeV -2 ).

Figure 5 . 13 :

 513 Figure 5.13:The ratio R A/B of two theoretical BH+B cross-sections in the (cos(θ cm ), ϕ cm )-plane at q cm = 37.5 M eV /c.

Figure 5 . 14 :

 514 Figure 5.14: The ratio R A/B of two theoretical BH+B cross-sections in the (cos(θ cm ), ϕ cm )-plane at q cm = 112.5 M eV /c.

Figure

  Figure5.16: Bins (in blue) that satisfy the mask selection in the (ϕ cm , cos(θ cm ))plane when K thr = 2.5%, P LL -P T T / = 17.9 (GeV -2 ) and P LT = -3.7 (GeV -2 ) at q cm = 87.5 M eV /c (left) and q cm = 112.5 M eV /c (right).

Figure 5 .

 5 Figure 5.19 shows the χ 2 LEX variations in the (P LL -P T T / , P LT )-plane. The elliptic shape is due to the linearity of the problem. A small correlation is observed between the two structure-function errors. The inner red ellipse expresses the standard deviation of each structure function separately, these are the errors mentioned in equation 5.23. The outer red ellipse expresses the standard deviation of both structure functions simultaneously [98].

Figure 5 .

 5 Figure 5.19: χ 2 LEX variations in the (P LL -P T T / , P LT )-plane. The red ellipses correspond to contours at χ 2 min + 1 and χ 2 min + 2.41 (where χ 2 min is not reduced).

5. 5 . 4

 54 From structure functions to generalized polarizabilities Once the structure functions are extracted, the LEX analysis is powerless to give individual GP values. So the next step requires the direct intervention of a model

Figure 5 .

 5 Figure 5.20: χ 2 DR of the DR analysis in the 600 (α E (Q 2 ), β M (Q 2 ))-bins.

6 :

 6 ∆α E (Q 2 ) = ∓0.98 syst (10 -4 .f m 3 ) ∆β M (Q 2 ) = ±0.90 syst (10 -4 .f m 3 ) , ∆Λ α = ∓0.121 syst (GeV ) ∆Λ β = ∓0.188 syst (GeV ) , ∆P LL -P T T / = ∓3.93 syst (GeV -2 )∆P LT = ∓1.86 syst (GeV -2 ) .

P

  LL -P T T / (LEX) = 18.10 ± 0.56 stat ± 3.64 syst (GeV -2 ) P LT (LEX) = -3.59 ± 0.31 stat ± 1.57 syst (GeV -2 ), and the DR t gives: P LL -P T T / (DR) = 18.70 ± 0.38 stat ± 3.93 syst (GeV -2 ) P LT (DR) = -3.48 ± 0.25 stat ± 1.86 syst (GeV -2 ).

Figures 5 .

 5 Figures 5.22 and 5.23 show those measurements together with the world data.

Figure 5 . 22 :

 522 Figure 5.22: World data for P LL -P T T / including the present extractions at Q 2 = 0.2 GeV 2 (in black). The data and curve follow the conventions of Figure 2.7.The green crosses are the preliminary results from[START_REF] Benali | Structure interne du proton à haute et basse énergie par la diusion Compton virtuelle[END_REF] (Q 2 = 0.45 GeV 2 ) and[START_REF] Bericic | Measurement of Generalized Polarizabilities of the Proton by Virtual Compton Scattering[END_REF]

  and 5.25 allow to compare the present measurements with the world data, and the same comments can be made as for the structure functions.

Figure 5 . 23 :

 523 Figure 5.23: World data for P LT including the present extractions at Q 2 = 0.2 GeV 2 (in black). The data and curve follow the conventions of Figure 2.8.The green crosses are the preliminary results from[START_REF] Benali | Structure interne du proton à haute et basse énergie par la diusion Compton virtuelle[END_REF] (Q 2 = 0.45 GeV 2 ) and[START_REF] Bericic | Measurement of Generalized Polarizabilities of the Proton by Virtual Compton Scattering[END_REF]

Figure 5 . 24 :

 524 Figure 5.24: World data for α E (Q 2 ) including the present extractions at Q 2 = 0.2 GeV 2 (in black). The data and curve follow the conventions of Figure 2.7.The green crosses are the preliminary results from[START_REF] Benali | Structure interne du proton à haute et basse énergie par la diusion Compton virtuelle[END_REF] (Q 2 = 0.45 GeV 2 ) and[START_REF] Bericic | Measurement of Generalized Polarizabilities of the Proton by Virtual Compton Scattering[END_REF]

Figure 5 . 25 :

 525 Figure 5.25: World data for β M (Q 2 ) including the present extractions at Q 2 = 0.2 GeV 2 (in black). The data and curve follow the conventions of Figure 2.8.The green crosses are the preliminary results from[START_REF] Benali | Structure interne du proton à haute et basse énergie par la diusion Compton virtuelle[END_REF] (Q 2 = 0.45 GeV 2 ) and[START_REF] Bericic | Measurement of Generalized Polarizabilities of the Proton by Virtual Compton Scattering[END_REF]

  2 [72] and Q 2 = 0.1 GeV 2[START_REF] Bericic | Measurement of Generalized Polarizabilities of the Proton by Virtual Compton Scattering[END_REF], associated with this one, have established a preliminary behavior of the electric GP. This behavior is not in smooth agreement with the previous MAMI results. Exploring a GP extraction at Q 2 = 0.33 GeV 2 in a wider angular phase space (namely outof-plane, as in our experiment) would certainly give more insight in the present puzzle.

(C. 2 )

 2 with ∆E s dened in equation 5.3. C.2 δ vac and δ ver : supplementary virtual photon

Figure C. 2 :

 2 Figure C.2: Electron self-energy diagrams.

1 2 ln 2 ( Q 2 m 2 e) + π 2 6

 226 ).

(C. 4 )C. 3 δ| 2 M 2 N

 4322 vac and δ ver : proton-side correction This correction is described through two terms which also take into account the two-photon exchange process (direct and crossed) :δ 1 = 2αem π (ln( 4 * (∆Es) 2 Q 2 x )ln(η) + Sp(1 -η x ) -Sp(1 -1 ηx ) P N | ln(x) -1) + 1 + E N | P N | (-1 2 ln 2 (x) -ln(x)ln( ρ ) + lnx -Sp(1 -1 x 2 ) + 2Sp(-1 x ) + π 2 6 )) (C.6)With M N the mass of the baryon (here proton), E N (| P N |) the lab energy (momentum) of the recoiling nucleon and:

4M 2 N

 2 η = E e /E el = 1 + E e (1 -cos(θ e )/M N where E e , E el and θ e are respectively in the lab frame: the energy of the incoming electron, the energy of the elastic outgoing electron and the angle between the outgoing and incoming electrons.

  (DR) = 18.70 ± 0.38 stat ± 3.93 syst (GeV -2 ), P LT (DR) = -3.48 ± 0.25 stat ± 1.86 syst (GeV -2 ).

	Les résultats des deux méthodes indépendantes sont en parfait accord pour
	la première fonction de structure. Pour la seconde les résultats obtenus sont en
	accord, à condition d'appliquer le critère sélectif basé sur le modèle DR lors du t
	LEX. L'indépendance des deux méthodes est alors diminuée. Cependant il a été
	montré qu'il est indispensable de tenir compte de l'eet des ordres supérieurs en

10 ± 0.56 stat ± 3.64 syst (GeV -2 ), P LT (LEX) = -3.59 ± 0.31 stat ± 1.57 syst (GeV -2 ), P LL -P T T /

Contents 1

 1 Introduction to Real and Virtual Compton Scattering 1.1 Proton form factors . . . . . . . . . . . . . . . . . . . . . . . . . . . GeV 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . Choice of kinematical settings . . . . . . . . . . . . . . . . . . . . . 3.3.1 The leptonic-side variables . . . . . . . . . . . . . . . . . . . The Q 2 = 0.2 GeV 2 settings . . . . . . . . . . . . . . . . . . Specic raw data calibration . . . . . . . . . . . . . . . . . . . . . .

	1.1.1 1.1.2 0.2 3 Experimental conguration Elastic electron-proton scattering . . . . . . . . . . . . . . . Form factor measurements and particular values at Q 2 = 3.1 MAMI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 The Mainz facility . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Operating principles of MAMI . . . . . . . . . . . . . . . . . 3.2 Trigger and data acquisition: the A1 software . . . . . . . . 3.3 3.3.2 The center-of-mass angles of the Compton Scattering process 3.3.3 A last criterion: the LEX validity . . . . . . . . . . . . . . . 3.3.4 4 Experimental Analysis 4.1 Standard raw data calibration . . . . . . . . . . . . . . . . . . . . . 4.1.1 The photon missing-mass squared . . . . . . . . . . . . . . . 4.1.2 Calibration of VDCs . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Calibration of Cerenkov detectors . . . . . . . . . . . . . . . 4.1.4 Scintillators . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.5 Optics calibration . . . . . . . . . . . . . . . . . . . . . . . . 4.2 5.2.1 Physical processes and calculation choices . . . . . . . . . . 5.2.2 Value of the global radiative correction factor . . . . . . . . 5.2.3 Systematic error on F rad . . . . . . . . . . . . . . . . . . . . 5.3 Measurement of cross sections . . . . . . . . . . . . . . . . . . . . . 5.3.1 The simulation-based method . . . . . . . . . . . . . . . . . A1 experimental hall . 3.2.4 5.3.2 Statistical error .

1.2

Polarizabilities of the proton: Real Compton scattering . . . . . . . 1.2.1 RCS amplitude and cross-section . . . . . . . . . . . . . . . 1.2.2 Proton polarizabilities measurement . . . . . . . . . . . . . . 1.3 Generalized polarizabilities of the proton . . . . . . . . . . . . . . . 1.3.1 Virtual Compton scattering o a proton . . . . . . . . . . . 1.3.2 Low Energy Theorem . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Physical Observables . . . . . . . . . . . . . . . . . . . . . . 2 Theoretical models and motivations for a VCS experiment 2.1 The Dispersion Relation model . . . . . . . . . . . . . . . . . . . . 2.1.1 Formalism of the model . . . . . . . . . . . . . . . . . . . . 2.1.2 Results of calculation in the DR model . . . . . . . . . . . . 2.2 Other models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Chiral perturbation theories . . . . . . . . . . . . . . . . . . 2.2.2 Eective Lagrangian model . . . . . . . . . . . . . . . . . . 2.2.3 Linear sigma model . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 The non-relativistic quark model nRQM . . . . . . . . . . . 2.3 Previous and current VCS Experiments . . . . . . . . . . . . . . . . 2.3.1 The rst MAMI experiment . . . . . . . . . . . . . . . . . . 2.3.2 JLab VCS experiment (E93-050) . . . . . . . . . . . . . . . 2.3.3 MIT-Bates experiment . . . . . . . . . . . . . . . . . . . . . 2.3.4 The Doubly polarized VCS experiment at MAMI . . . . . . 2.3.5 The VCS ∆ experiment at MAMI . . . . . . . . . . . . . . 2.4 Motivation for a new experiment: vcsq2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 The liquid hydrogen target . . . . . . . . . . . . . . . . . . . 3.2.2 The high-resolution spectrometers . . . . . . . . . . . . . . . 3.2.3 Focal-plane detectors . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Beam position . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Snow thickness and spectrometer B momentum . . . . . . . 4.2.3 Vertical misalignment . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Luminosity corrections . . . . . . . . . . . . . . . . . . . . . 4.3 Choice of 3D Binning and Analysis cuts . . . . . . . . . . . . . . . 4.3.1 Cut on coincidence time . . . . . . . . . . . . . . . . . . . . 4.3.2 Vertex selection . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Missing-mass squared selection . . . . . . . . . . . . . . . .

4.3.4 PID selection based on energy deposited . . . . . . . . . . . 4.3.5 PID selection based on Cerenkov detector . . . . . . . . . . 4.3.6 Three-dimensional binning of the phase space . . . . . . . . 4.4 Calibration checks . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Control of the optics quality . . . . . . . . . . . . . . . . . . 4.4.2 Event rate stability . . . . . . . . . . . . . . . . . . . . . . . 4.5 The Monte Carlo Simulation: VCSSIM . . . . . . . . . . . . . . . . 4.6 Simulated and experimental data comparison . . . . . . . . . . . . . 4.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Photon electroproduction cross-section and extraction of VCS observables 5.1 Theoretical tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 The BHB code . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 The DR code . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Radiative corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Cross-section results . . . . . . . . . . . . . . . . . . . . . . 5.4 Renormalization procedure . . . . . . . . . . . . . . . . . . . . . . .

  The LEX t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . Systematic errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . Final results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

		5.4.3	Canceling the form-factor dependency in the LEX extraction
	5.5
		5.5.1	Bin selection: gradient cut	. . . . . . . . . . . . . . . . .
		5.5.2	Bin selection: mask selection	. . . . . . . . . . . . . . . .
		5.5.3
		5.5.4	From structure functions to generalized polarizabilities . . .
	5.6	The DR analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . .
		5.6.1	Fit description	. . . . . . . . . . . . . . . . . . . . . . . . .
		5.6.2	The results of the DR t . . . . . . . . . . . . . . . . . . . .
	5.7
		5.7.1	First method: changing the renormalization of the experi-
			mental cross section . . . . . . . . . . . . . . . . . . . . . . .
		5.7.2	Second method : multiple LEX analyses . . . . . . . . . . .
	5.8

5.8.1 Structure functions . . . . . . . . . . . . . . . . . . . . . . . 5.8.2 GPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A VCS coecients in the LEX expansion B Spin GPs contribution to the structure functions according to the DR model C Radiative corrections to ep → epγ

Table 1 .

 1

3.1 : Main variables of the VCS kinematics.

Table 2 .

 2 Lensky et al. performed a new Lorentz-covariant calculation of the RCS electric and magnetic polarizabilities. Considering both the ∆(1232)-resonance and the new Lorentz-covariant calculation deeply impacts the predictions; results obtained by this approach are labeled as NEW in table 2.2.1. 2.1 from[START_REF] Lensky | Predictions of chiral perturbation theory for Compton scattering o protons[END_REF] illustrates a variety of results provided by χPT theories.

Table 2 .

 2 2.1 : RCS electric and magnetic polarizability predictions from dierent BχPT and HBχPT calculations as presented in[START_REF] Lensky | Predictions of chiral perturbation theory for Compton scattering o protons[END_REF].As it appears from table 2.2.1, there is a large sensitivity to the calculation choice in χPT predictions. Also, BχPT and HBχPT are in poor agreement.

		6.8	10.8	12.2	20.8	11.5±0.4
	βM (10 -4 f m 3 )	-1.8	4.0	1.2	14.7	2.5±0.4

2.2.2 Eective Lagrangian model

The eective Lagrangian model was elaborated by M.Vanderhaeghen

[START_REF] Vanderhaeghen | Virtual Compton scattering study below pion production threshold[END_REF]

, it consists in a phenomenological approach to calculate the polarizabilities. The idea is to evaluate the Non-Born contribution to photon electroproduction by summing the Feynman diagrams of dominant intermediate states. This relativistic model calculates the contributions of the non-Born diagrams represented in gure 2.5, with N * representing the following intermediates states : ∆(1232), P 11 (1440), D 13 (1520), S 11 (1535), S 31 (1620), S 11 (1650) and D 33 (1700).

Table 2 .

 2 3.2 : JLab structure function measurement for all settings and both types of analysis. The rst error is statistical, the second is systematic.

	I-a DR	0.92	1.70 ± 0.21 ± 0.89	-0.36 ± 0.10 ± 0.27
	I-b DR	0.92	1.50 ± 0.18 ± 0.19	-0.71 ± 0.07 ± 0.05
	II LEX	1.76	0.54 ± 0.09 ± 0.20	-0.04 ± 0.05 ± 0.06
	II DR	1.76	0.40 ± 0.05 ± 0.16	-0.09 ± 0.02 ± 0.03

Table 2 .

 2 2; 0.45 GeV 2 . Since 1996 a series of VCS experiments performed at MAMI, JLab and Bates have provided experimental results for the electric and magnetic GPs at dierent Q 2 values from 0.06 GeV 2 to 1.76 GeV 2 . Bates and JLab measurements were at respectively very low and very high Q 2 while MAMI was exploring the intermediate range. The results are gathered in table 2.4. 2 C2 gives us information about the sphericity of the proton. ± 2.2 ± 0.6 ± 4.3 -5.0 ± 0.8 ± 1.1 ± 1.4 4 : Structure functions world data. The rst error is statistical, the second is systematic.

	Experiment	Q 2 (GeV 2 )		P LL -P T T / (GeV -2 )	P LT (GeV -2 )
				LEX analysis
	MIT-Bates [67]	0.057	0.90	54.5 ± 4.8 ± 3.4	-20.4 ± 2.9 ± 1.4
	MAMI rst [62] 23.7 polar-MAMI [69] 0.33 0.62 0.33 0.645 27.1 ± 1.9 ± 3.0	-8.0 ± 0.7 ± 2.2
	JLab (I-a) [63]	0.92	0.95	1.77 ± 0.24 ± 0.70	-0.56 ± 0.12 ± 0.17
	JLab (II) [63]	1.76	0.88	0.54 ± 0.09 ± 0.20	-0.04 ± 0.05 ± 0.06
				DR analysis
	MIT-Bates [67]	0.057	0.90	46.7 ± 4.9 ± 3.4	-8.9 ± 4.2 ± 1.4
	MAMI rst [62]	0.33	0.62	23.2 ± 3.0 stat	-3.2 ± 2.0 stat
	JLab (I-a) [63]	0.92	0.95	1.70 ± 0.21 ± 0.89	-0.36 ± 0.10 ± 0.27
	JLab (I-b) [63]	0.92	0.95	1.50 ± 0.18 ± 0.19	-0.71 ± 0.07 ± 0.05
	JLab (II) [63]	1.76	0.88	0.40 ± 0.05 ± 0.16	-0.09 ± 0.02 ± 0.03

Table 2 .

 2 [START_REF] Arenhövel | Generalized nuclear polarizabilities in (e, e' γ) coincidence experiments[END_REF]-bis contains the world data for α E (Q 2 ) and β M (Q 2 ). The presented LEX analyses show values obtained by a DR-subtraction of the spin-ip GPs.

	Experiment

Table 2 .

 2 4-bis: Generalized polarizabilities world data. The rst error is statistical, the second is systematic. The star stands for values that we calculated ourselves with the DR code (described section 5.1.2) from the published LEX structure functions of table 2.4, the X means that we were not able to compute a DR value from the structure function.Although all models (except the DR model) predict a single-dipole behavior for α E (Q 2 ) the two MAMI experiments at Q 2 = 0.33 GeV 2 seem to contradict this prediction. The α

	MAMI rst*	0.33	0.62	6.90 ± 0.72 ± 1.41 2.52 ± 0.50 ± 1.11
	polar-MAMI*	0.33	0.645	8.05 ± 0.62 ± 0.98	X
	JLab (I-a)*	0.92	0.95	1.04 ± 0.20 ± 0.57 0.40 ± 0.18 ± 0.25
	JLab (II)*	1.76	0.88	0.85 ± 0.19 ± 0.42 -0.10 ± 0.17 ± 0.21
			DR analysis
	MIT-Bates [67]	0.057	0.90	7.85 ± 0.87 ± 0.60 2.69 ± 1.48 ± 0.49
	MAMI rst*	0.33	0.62	6.75 ± 0.98 stat	1.39 ± 1.26 stat
	JLab (I-a) [63]	0.92	0.95	1.02 ± 0.18 ± 0.77 0.13 ± 0.15 ± 0.42
	JLab (I-b) [63]	0.92	0.95	0.85 ± 0.15 ± 0.16 0.66 ± 0.11 ± 0.07
	JLab (II) [63]	1.76	0.88	0.52 ± 0.12 ± 0.35 0.10 ± 0.07 ± 0.12

9.22 ± 0.85 ± 0.60 6.77 ± 1.04 ± 0.48 E (Q 2 ) single-dipole behavior is translated in a smooth fall-o for the P LL -P T T / structure function (the predictions of the DR model are presented in appendix B).

Table 3 .

 3 2.2: Parameters of the spectrometers A and B. Values are taken from [77].

	.2.2.

2

  GeV 2

	904.9 904.9 1002.4 1002.4 OOP-forward 1002.4 770 723 LOW 715 LOW-bis 766 INP 766 OOP	32.5 32.5 30.4 29.2 29.2	0.0 0.0 0.0 8.5 8.5	462 442 580 486 411	52.2 52.2 51.5 51.0 42.5

Table 3 .

 3

3.4: The analyzed kinematical settings for spectrometer A and B. (P B ,

Table 4 .

 4

	Setting	INP	LOW	LOW-bis	2011 OOP	2015 OOP	OOP-forward
	Snow
	thickness	0.781	0.808	1.28	>4	0	0
	(mm)

2.2 contains the estimated snow quantities (averaged over the runs) for the Q 2 = 0.2 GeV 2 settings. The vacuum conditions in 2011 for the OOP-setting were so bad that the snow quantity hampers the VCS analysis, the OOP data were collected again in 2015 with perfect vacuum conditions.

Table 4 .

 4 2.2: Average snow thickness found for each setting with a snow density xed at 1.g.cm -3 .

Table 4 .

 4 3.3: M 2 X and V ertex Z cuts for every analyzed setting.

Table 5 .

 5 4.2 gathers the results. The six rst lines show an excellent agreement, giving an average value of F norm = 0.979.

	Bin(s)	Setting(s)	F norm χ 2 norm
	0+1	LOW+LOW-bis	0.983	1.28
	0+1	LOW	0.983	1.17
	0+1	LOW-bis	0.985	1.20
	*1 LOW+LOW-bis 0.980 1.36
	1	LOW	0.985	1.26
	1	LOW-bis	0.973	1.14
	0	LOW+LOW-bis	0.997	1.16
	0	LOW	0.95	1.00
	0	LOW-bis	1.06	1.13

Table 5 . 4

 54

.2:

Table 5 .

 5

	corresponding to the maximal spread among the six rst lines of table 5.4.2.
	5.4.3 Canceling the form-factor dependency in the LEX ex-
	traction
	Let's now consider several iterative procedures performed with dierent form-factor
	parametrizations of σ th i in equation 5.15. They will inevitably result in dierent
	values of F norm .
	Table 5.4.3 presents the dierent F norm obtained for three choices of form-factor
	parametrizations.
	Form-factor	Friedrich-	Arrington	Bernauer
	parametrization	Walcher [17]	et al. [18]	et al. [19]
	F norm	F norm F W = 0.980 F norm A	= 0.969 F norm B	= 0.993

4.3:

2

  GeV 2 [70]. C.2. δ V AC AND δ V ER : SUPPLEMENTARY VIRTUAL PHOTON Sp() the Spence function. E e , E e and θ e are respectively the energies of incoming and outgoing electrons and the polar angle of the outgoing electron dened in a particular frame. The considered frame depends on the experiment, i.e. elastic experiment, (ep → epγ) with photon-detection experiment or (ep → epγ)with proton-detection experiment. In our case:E e = mp M m2 (E e -Q 2 2mp -k.p' mp ) E e = mp M m2 (E e + Q 2 2mp -k'.p' mp ) sin 2 ( θ e /2) = EeE eEe E e sin 2 (θ e /2) M m2 = 2 • ∆E s .

	δ r2 =	α QED π (-1 2 ln 2 ( Ee E e	) + 1 2 ln 2 ( Q 2 m 2 e	) -π 2 3 + Sp(cos 2 ( θe 2 ))),	(C.1)

with

The systematic errors present a small asymmetry in positive and negative, the upper limit is kept to express the error symmetrically.

Acknowledgements

Chapter 4 Experimental Analysis

In this chapter we present the rst-level analysis work. The detectors need to be calibrated, the experimental data have to be checked and the VCS events must be selected. Sections 4.1 and 4.2 contain the description of detectors calibration. The rst section is a brief introduction of standard A1 calibrations while the second section describes with more detail the unusual issues we encountered. Section 4.3 presents the data selections and binning. Sections 4.4 and 4.6 gather the control gures used to ensure our analysis correctness, while section 4.5 is devoted to the simulation of the experiment.

Standard raw data calibration

The calibration of focal-plane detectors is essential for using clean data-selection and for extracting the GPs with minimized systematic errors. The procedures we followed are summarized here and described with more details in [START_REF] Bericic | Measurement of Generalized Polarizabilities of the Proton by Virtual Compton Scattering[END_REF].

The calibrations presented in this section are made thanks to the raw informations from detectors but also some specic variables dened in Cola++ (missing mass, errors estimated on track reconstruction, etc...). The settings are calibrated independently by following the same procedure. For some calibrations a data selection is required.

The photon missing-mass squared

The missing-mass squared, M 2 X is a relativistic invariant, dened as the squared mass of the undetected particles (the real photon in our experiment) by:

with E X = (E p +E e )-(E p +E e ) the missing energy and p x = ( k + p)-( k + p )

the missing momentum, both being dened thanks to conservation laws.

Beside the reaction identication, the missing-mass squared is a powerful tool to calibrate our detectors. Indeed M 2 X is deduced from the four-momentum vectors However after applying the three mandatory selections the events identied as pions on this gure represent only 0.3% of the total, therefore we did not apply any Cerenkov cut. Actually, the three mandatory selections are ecient and sucient for all our settings.

Three-dimensional binning of the phase space

The phase-space point depends on the ve variables of the VCS kinematics (q cm , q cm , , θ cm , ϕ cm ). In our settings, two of these variables cover a narrow range, q cm ≈ 458 M eV /c and ≈ 0.85 , therefore they are not binned. The three others are used to build a three-dimensional phase space as presented in table 4.3.6.

q cm (MeV/c) ϕ cm ( the resolution on (q cm , θ cm , ϕ cm ) which we estimated thanks to the simulation.

Calibration checks

In this section control plots are gathered in order to conclude on the experimental calibration.

Figure 5.5: 1/F rad behavior in the (cos(θ cm ), ϕ cm )-plane at q cm = 37.5 M eV /c. P SA, the maximum uncertainty brought by our phase-space averaging. theo, the incompressible theoretical error. EC, the uncertainty brought by our choice to use the elastic-case determination of δ vertex and δ vac .

P SO, the uncertainty brought by the proton-side omission of δ

(1) 2 . As a nal result, we get:

which is used to correct all our cross-section data, using equation 5.1. This factor is also considered in the normalization of experimental distributions when the reconstructed quantities are compared to the simulated ones (section 4.6).

The ±0.014 systematic uncertainty will largely disappear in the renormalization procedure described in section 5.4, and a small remaining part will contribute to our systematic errors.

Measurement of cross sections

Measuring the cross section of equation 1.38 is the rst step of the second-level analysis. We want to extract a GP eect which is a few percent of the cross section, for these three dierent choices, obtained before and after the renormalization.

With F norm = 1, there is no iterative process to do, and the structure functions obtained by the LEX t are clearly not in agreement within their statistical error bar. After having done the iterative process, leading to the F norm given in the table, the extracted structure functions come to very close agreement. This denitively proves the form-factor independency of our analysis and the need of a renormalization process.

Finally, the extraction presented in the following sections is done with crosssections renormalized by: F norm F W = 0.980 ± 0.002 stat ± 0.006 syst ± 0.002 syst , with the rst systematic error which is due to acceptance features and the second one which is a consequence of the use of dierent conditions for the LEX t in the iterative procedure which allows to determine F norm .

The LEX t

The LEX extraction of structure functions is based on equation 1.38 and consists rst to determine (in each analysis bin) the quantity ∆M related to the dierence between the experimental cross section and the BH+B one: ∆M = σ exp -σ BH+B Φq cm = [v LL (P LL (q cm ) -P T T (qcm) ) + v LT P LT (q cm )] + O(q cm 2 ),

(5.17

where σ exp is renormalized with the same form-factor parametrization that is used to compute σ BH+B . One denes Ψ 0 as the term independent of q cm : Ψ 0 = ∆M (q cm → 0) = [v LL (P LL -P T T ) + v LT P LT ].

(5.18)

If O(q cm 2 ) is small enough, ∆M should be independent of q cm . The q cm - independency of ∆M can be checked for xed values of θ cm and ϕ cm by plotting the experimental ∆M term as a function of q cm at xed angles (θ cm , ϕ cm ). In appendix D, gure D.1 shows this behavior for selected angular bins. In the ideal case of negligible higher orders our ∆M measurements at xed (cos(θ cm ), ϕ cm ) but dierent q cm are just several measurements of the same Ψ exp 0 . Ψ exp 0 is then deduced by averaging the ∆M terms in a (cos(θ cm ), ϕ cm )-bin over the q cm values.

Thereafter, equation 5.18 can be rewritten as:

and so the structure functions are simply the slope and intercept of a linear function that can be determined by a χ 2 -minimization: One has to note that changing the cross section global normalization (F norm = 0.98) has few impact on χ 2 LEX which is more sensitive to phase space depending factor. It is also possible to extract the structure functions by a more direct t of the renormalized experimental cross section:

with σ th the cross-section calculated according to equation 1.38 (with O(q cm 2 ) = 0) and the structure functions as free parameters of the t. To minimize χ 2 LEX or χ 2 LEX,2 leads strictly to the same structure-functions result since both ts are based on the same assumption.

In the following sections we will discuss the bin selections and their impact on the LEX t.

Bin selection: gradient cut

The gradient selection is designed in order to remove the Bethe-Heitler peaks from the analysis. Even if our data are out of this region it is still interesting to build the cut for two reasons. First, this selection was also used to identify where the radiative corrections can be averaged. Secondly the bins removed by this cut are at the acceptance edges, i.e. where experimental uncertainties are larger.

The selection is detailed in appendix E. It consists in a threshold cut on the cross-section variation from a bin to its neighbour. It is essentially designed that way in order to be close to our acceptance edges as presented in gure E.1. The use of the gradient selection is not essential to the LEX t and its omission is accounted in the systematic error budget on structure functions (see section 5.7).

The impact of this cut is positive in that it improves the minimum χ 2 of the LEX t. Also, although this cut removes 20% of the analysis bins, it has only a small impact on the nal statistical errors. It increases the errors by 3% and 6%

for P LT and P LL -P T T / respectively.

Bin selection: mask selection

The LEX extraction requires that the higher orders O(q cm 2 ) are negligible, but this is not true in the whole phase space. The rst warning of the importance of higher orders was given by the Bates LEX analysis as discussed in section 2.3.3.

In order to identify where the higher-order terms are small enough to be neglected, we designed a cut based on the DR model, the so-called mask selection.

The selection uses the estimation of higher orders discussed in section 3.3.3.

As a reminder, we have dened O DR (q cm 2 ) as the criterion representing the relative importance of higher-order terms in the LEX expansion according to the DR model. Now we consider the following cut:

with the cross sections σ LEX and σ DR being purely theoretical and calculated with the same input values of the structure functions. In rst approximation K thr may vary from 1-2% to 5-6%. The lower limit is xed by the magnitude of the systematic error on σ exp . The upper limit is given by the maximum value that O DR (q cm 2 ) can reach, presented in gure 3.13.

Figure 5.16 shows the angular bins that satisfy the selection with the following input value of structure functions in equation 5.22 P LL -P T T / = 17.9 (GeV -2 ) and P LT = -3.7 (GeV -2 ) and with K thr = 2.5%. Since at low q cm the GP eect is small equation 5.22 simply comes to comparing the BH+B cross section with itself and removes no bins. At q cm = 112.5 M eV /c the selection becomes very restrictive but we believe this is a necessity if one wants to get rid of higher-orders eect.

Figure 5.17 shows the eect of this selection on the analysis for the q cm = 112.5 M eV /c-bins. Cross sections are measured in every bin (red and blue) on this plot but only the blue ones are kept for the LEX analysis. Three regions corresponding to the three high-q cm settings are also shown. This mask selection induces two dependencies that need to be controlled. Firstly, we have to see the eect induced by K thr variations. Secondly the theoretical cross-sections σ DR and σ LEX depend on input values of structure functions and we have to study the impact of changing them within a reasonable range. function choices to build the mask criterion. We nally choose K thr = 2.5% as a nominal cut.

Theoretical structure functions dependency

We tested the impact of changing the theoretical structure functions when building the mask selection. We started from a given value and made the structure functions vary independently over a large range. Applying a mask selection has an important impact on the extracted structure functions, at least for P LL -P T T / . Globally, any of the mask selections of table 5.5.2 gives results in agreement within ±1 GeV -2 for P LL -P T T / and ±0.5 GeV -2 for P LT .

To remove this last uncertainty we used another iterative procedure. The structure functions extracted by the LEX are used to build a new mask selection, which is used to redo the LEX t, and so on. The iterations are performed at constant K thr .

Similar to the iterative procedure relative to the renormalization process, we tested two starting points. Firstly, we used as input the structure functions obtained when no mask selection is applied (line 1 of table 5.5.2). Secondly, we tried to iterate starting from the second line of the table.

The results converge to the following input values of structure functions for the mask selection:

which also coincide with the outputs of the LEX t.

Conclusion

Once one trusts the data calibration and the cross-section renormalization, the main feature of the LEX analysis concerns the handling of higher orders in q cm .

To this aim, we decided to use a mask selection based on the DR model, even if by doing so we lose a small part of the model-independency of the LEX analysis.

In our experiment we also have a direct hint of higher-order terms at an early level: we observed during the data analysis that some experimental distributions do not follow the simulated ones when we include a GP eect in the weight of simulated events; this is case namely for the ϕ cm -distribution (cf. gure 4.24).

This discrepancy was due to the theoretical GP eect which is badly estimated by the LET and so our data needed to be carefully handled.

LEX extractions can be done by considering dierent subgroups of settings and bins, and will provide various results. We believe that the best t is the one which gathers all data in order to increase the sensitivity to the two structure functions, and to increases the lever arm in the v LL and v LT coecients. In appendix D.2 the consequences of considering subgroups of data are shown.

Nominal LEX results

Finally we performed the nominal analysis thanks to 946 Ψ 0 data points as presented in table 5.5. q cm -bins used q cm -range q cm -bins used q cm -range Once the GPs are extracted we need to use again the DR code to nd the corresponding structure functions. By doing that we reintroduce a form-factor dependency, but as in section 5.5.4 it is a very small one.

The results of the DR t

Finally, we found a reduced χ 2 of 1.43 for n.d.f = 4429 and the extracted GPs are:

Appendix A

VCS coecients in the LEX expansion

In the LET expression, v LL and v LT are angular dependent functions. We recall their expression here, following the notations of [START_REF] Pierre | Virtual Compton scattering o the nucleon[END_REF]:

with q cm and ε xed in our experiment, and:

Appendix B Spin GPs contribution to the structure functions according to the DR model

From equations 1.37a and 1.37b one observes that P LL -P T T / is directly proportional to α E (Q 2 ) if one neglects its spin-GP contribution, which is entirely contained in P T T . Figure B.1 shows the P T T value with respect to Q 2 according to the DR model, it reaches a maximum of ≈ -2 GeV -2 which is, according to our extractions, a small part of P LL -P T T / . This explains the similar behaviors of P LL -P T T / and α E (Q 2 ).

According to the DR model, the spin-GP contribution to P LT is more important. In gure B.1 the value of K spin • P (01,01)1 (q cm ) is plotted versus Q 2 (K spin being dened as the coecient multiplying the spin GP in P LT in equation 1.37c, i.e. K spin = 3G M Qq cm /2 q 0 cm ). Appendix D Supplementary studies for the LEX t D.1 The ∆M stability with respect to q cm ∆M , as dened equation 5.17, is a tool to control the LEX t. Figure D.1 shows some typical ∆M behaviors versus q cm , it allows to check if ∆M is independent of q cm as required by the LEX t, and if the points measured by dierent settings are in agreement.

A Ψ 0 value is obtained by averaging these measurements, if and only if the bins pass the mask selection (described in section 5.5.2).

For the rst two plots at the top of gure D.1, none of the measurements satisfy the mask selection and no Ψ 0 is calculated. This signies that, according to the DR model, the higher orders are not negligible in these bins, and so, one should observe a non-at q cm behavior of ∆M .

For the two plots in the middle of gure D.1, the Ψ 0 value is not calculated from all the data, the high q cm are rejected by the mask selection; the red band corre- sponds to the averaging of the three dierent measurements at q cm = 62.5 M eV /c.

For the two last plots at the bottom of gure D.1 every measurement passes the mask selection, this signies that according to the DR model one can neglect the higher orders in those particular bins, and that the dierent ∆M points should not depend on q cm . Globally on these six plots, the points measured by dierent settings are in good agreement. The absence of q cm -dependence of ∆M is hard to judge by eye, and for this we rely on the theoretical criterion of our mask selection.

D.2 LEX ts with various subgroups of data

In section 3.3 we explained the importance of choosing the analyzed phase space and we detailed the reasons that motivated our choices. In this section we observe the consequences of ignoring these recommendations, it is illustrated in tables D.2a and D.2-b. Table D.2-a presents the results obtained by considering each setting separately, the renormalization factor and the analysis selections are applied. One observes that the setting INP, which is more sensitive to P LT gives a result close to the Figure D.1: Measured ∆M with respect to q cm for six selected bins; cos(θ cm ) = -0.675 and 55 • ≤ ϕ cm ≤105 • . The points are from dierent settings: INP (black), OOP(red) and LOW+LOW-bis (green). The light-red area corresponds to the quantity Ψ 0 ± ∆Ψ 0 tted in the (cos(θ cm ), ϕ cm )-bin . Some points are shifted in q cm for visibility. measured one for this structure function. On the contrary, the setting OOP is more sensitive to P LL -P T T / and governs the extraction of this structure function. The OOP-forward setting presents data collected afterwards, with a low statistics and gives results with high statistical errors. The two settings at low q cm are clearly less sensitive to the GP eect and are individually not relevant for a structure-function q cm -range is the one covered by each setting. The errors on the structure functions are statistical.

Table D.2-b presents the results obtained by considering each q cm -bin separately, the renormalization factor and the analysis selections are applied. It illus-trates the irrelevance of the low q cm for a LEX t and the importance of gathering the high-q cm data; by taking each q cm individually the results disagree. LEX ts of structure functions for the ve q cm bins separately. The value of q cm is the one at the center of the bin. The errors are statistical.

We believe that the best t is the one which gathers all data for the reasons explained in section 3.3.

One can do two side remarks about the numbers in table D.2-b:

-for each bin in q cm , the number of cross-section points in (cos(θ cm ), ϕ cm ) is given by n.d.f + 2. These cross-section points are the ones passing the mask criterion, and they are setting-averaged, as indicated in the table .   -the sum of the ve numbers (n.d.f. + 2) is equal to 2675, which is therefore our total number of setting-averaged cross section points. Figure D.1 allows to visualize how several individual data points, at dierent q cm , are gathered together in order to yield only one Ψ 0 data point. This explains why our nominal LEX t (section 5.5.3) uses 946 data points, in terms of Ψ 0 .

Appendix E Gradient selection

The gradient selection is essentially designed to follow the acceptance edges in (cos(θ cm ), ϕ cm ), which themselves depend on q cm . To test if a bin (i) is kept we compare its BH+B cross-section with the one of the four direct neighbourbins (j = 1 → 4) in the (ϕ cm , cos(θ cm ))-plane and with the one of the four bins (k = 1 → 4) in the diagonal directions. The eight conditions to satisfy are:

This design is the best if one wants to respect our specic acceptance; applying a cut on relative cross-section dierences would generate complicated and unwanted shapes.

Abstract

This work presents the measurement of the generalized electric α E (Q 2 ) and magnetic β M (Q 2 ) polarisabilities (GPs) of the proton. The GPs are dened in the Virtual Compton Scattering (VCS) context, i.e. the reaction γ * p → γp, where Q 2 is the four-momentum transfer of the virtual photon. The GPs are a generalization of polarizabilities measured in real Compton scattering (RCS) by taking into account the Q 2 -dependency. They are dynamical properties of the proton when it is deformed by an applied electromagnetic eld. α E (Q 2 ) (or β M (Q 2 )) gives access to the local polarization (or magnetization) density of the deformed proton.

The studied VCS process is accessible by the photon-electroproduction reaction (ep → epγ). The GP eect is a 1 -15% contribution to the the ep → epγ cross section, requiring a high-precision measurement. The present work is a part of an experiment conducted by the A1 collaboration at MAMI at three new Q 2 values: 0.1,0.2 and 0.45 GeV 2 . This thesis details the extraction at 0.2 GeV 2 . The experiment uses the 1 GeV electron beam, the 5 cm liquid hydrogen target and spectrometers A (B) to detect the nal electron (proton). The rst-level analysis includes a detailed calibration of experimental data, and the use of a simulation of the experiment. The measurement of the unpolarized ep → epγ cross section is described with two of its important features: the correction of the radiative eects and the renormalization. Two dierent frameworks are used to extract the GPs:

Résumé

Ce travail présente la mesure des polarisabilités généralisées (GPs) électrique α E (Q 2 ) et magnétique β M (Q 2 ) du proton à Q 2 = 0.2 GeV 2 . Les GPs sont dénies dans le contexte de la diusion Compton virtuelle (VCS), γ * p → γp, où Q 2 est le quadri-moment de transfert du photon virtuel. Les GPs sont la généralisation des polarisabilités mesurées en diusion Compton réelle (RCS) en considérant leur dépendance en Q 2 . Ce sont des propriétés dynamiques du proton dénissant sa déformation lorsqu'un champ électromagnétique lui est appliqué. α E (Q 2 ) (ou β M (Q 2 )) donne accès à la densité de polarisation (magnétisation) locale du proton déformé. L'eet des GPs ne contribuant qu'à 1 -15% de la section ecace ep → epγ une mesure de haute précision est requise. Ce travail s'inscrit dans le