Renormalisation de la charge

self-énergies : Re ˆΣ^T_W(M_W² ) = 0, Re ˆΣ^T_W^′(M_W² ) = 0, Re ˆΣ^T_ZZ(M_Z²) = 0, Re ˆΣ^T_ZZ^′ (M_Z²) = 0, ˆ Σ^T_AA(0) = 0, Σ^ˆT_AA^′ (0) = 0, ˆ Σ^T_AZ(0) = 0, Re ˆΣ^T_AZ(M_Z²) = 0. (6.21) On obtient finalement les contretermes de masse,

δM_W² = ReΣ^T_W(M_W² ), δM_Z² = ReΣ^T_ZZ(M_Z²), (6.22) ainsi que les constantes de renormalisation de fonctions d’onde suivantes :

δZ_ZZ = −ReΣ^TZZ^′ (M_Z²), δZZA = ² M2 Z Σ^T_AZ(0), δZ_AZ = − ² M2 Z ReΣ^T_AZ(M_Z²), δZAA = −Σ^TAA^′ (0), δZW = −ReΣ^TW^′(M_W² ), 0 = Σ^T_AA(0). (6.23)

La dernière relation est due à la symétrie électromagnétique qui garantit la nullité de la masse du photon à tout ordre. Il est également utile de renormaliser l’angle de Weinberg. Dans le cadre d’un schéma sur couche de masse, il est naturel d’utiliser sa définition à partir des masses des bosons W et Z :

s²_W = 1 − ^M 2 W M2 Z . (6.24)

En imposant que cette relation reste valable à tous les ordres, on obtient alors les contretermes suivants : δsW sW = −¹ 2 c2 W s2 W  δM2 W M2 W − ^δM 2 Z M2 Z  , (6.25) δcW cW = −^s 2 W c2 W δsW sW . (6.26)

6.4 Renormalisation de la charge

Pour renormaliser la charge électrique, on peut par exemple la mesurer à une certaine énergie et imposer la condition que le couplage électromagnétique e soit égal à cette valeur à tout ordre à l’énergie considérée. On obtient alors immédiatement les contretermes. Dans SloopS, on choisit de se placer dans la limite de Thomson, correspondant à une impulsion de transfert

Figure 6.1: Correction à une boucle et de contreterme au vertex eeγ.

nulle k2 = 0. À l’ordre des arbres, le vertex électromagnétique de l’électron, eeγ, correspond à l’amplitude suivante :

Γ^eeγ_µ = −ieγµ. (6.27)

Les corrections de boucle et de contretermes à ce vertex sont représentées sur la figure 6.1. Ils modifient la relation précédente pour donner une contribution totale à une boucle, dans la limite de Thomson,

ˆ

Γ^γee_µ (0, 0) = Γ^γee_µ (0, 0) + δΓ^γee_µ (0, 0), (6.28) où δΓγee

µ va dépendre des contretermes de la charge électrique et des constantes de renormali-sation des fonctions d’onde du photon et des électrons. Or, une identité de Ward [103] permet de relier ces dernières à Γγee

µ (0, 0). L’amplitude devient alors : ˆ Γ^γee_µ (0, 0) = Γ^eeγ_µ  1 + ^δe e ⁺ 1 2^δZAA− ¹₂^s_c^W W δZZA  . (6.29)

Cette relation ne dépend plus des constantes de renormalisation de la fonction d’onde de l’électron, elle en fait universelle et valable pour tous les fermions chargés (à un facteur de charge près).

On souhaite que la contribution virtuelle totale ne modifie pas le couplage, ce qui revient à imposer la condition :

ˆ

Γ^γee_µ (0, 0) = −ieγµ. (6.30)

Le contreterme de la charge électrique s’exprime alors : δe

e = −¹₂δZAA+ ¹ 2

sW

c_W^δZZA. (6.31)

En utilisant l’expression de la masse du boson W , équation (1.29) on peut désormais extraire le contreterme de la vev du boson de Higgs :

δv v ⁼ δMW MW − ^δe_e + ^δsW sW . (6.32)

6.5 Secteur de Higgs

Bien que le secteur de Higgs du Modèle Standard soit différent de celui du NMSSM, on présente sa renormalisation en guise d’introduction.

6.5. SECTEUR DE HIGGS Pour calculer la vev du Higgs, on cherche le minimum du potentiel de Higgs, équation (1.21), qui s’obtient à l’ordre des arbres en annulant sa partie linéaire :

V_Higgs^lin = T_h⁽⁰⁾0 h⁰. (6.33) À une boucle, cette partie linéaire est modifiée par l’apparition de termes dénommés tadpoles, ainsi que du contreterme associé, représentés par les diagrammes de Feynman suivants : Afin

+

h

⊗

h

de garder le minimum du potentiel inchangé, on impose que la somme de ces contributions soit nulle,

T_h⁽¹⁾0 + δTh0 = 0, (6.34)

donnant alors immédiatement l’expression du contreterme.

Il est aussi nécessaire de renormaliser le terme de masse du boson de Higgs. Dans le but de supprimer les corrections aux pattes externes des diagrammes, on renormalise également la fonction d’onde du boson de Higgs :

h⁰ →  1 + ¹ 2^δZh0  h⁰, m2 h0 → m2 h0 + δm2 h0. (6.35)

La self-énergie renormalisée du champ de Higgs s’écrit alors : ˆ

Σh0(p²) = Σh0(p²) − δmh²0 + δZh0(p²− m²h0). (6.36) Comme auparavant, dans l’esprit d’un schéma de renormalisation sur couche de masse, on impose que le propagateur ait un pôle à la masse m2

h0 et que son résidu soit égal à 1. Cela se traduit par les conditions,

Re ˆΣ_h0(m²_h0) = 0,

Re ˆΣ^′_h0(m²_h0) = 0. (6.37) conduisant aux relations suivantes pour les contretermes :

δm²_h0 = ReΣh0(m²_h0),

Chapter 7

Renormalisation du NMSSM dans SloopS:

secteurs des neutralinos, charginos et

sfermions.

Contents

5.1 Introduction . . . 80 5.2 CP-even Higgs sector of the NMSSM . . . 82 5.3 The H → γγ and H → γZ decay widths . . . 83 5.4 Numerical evaluation of Γ_γγ and Γ_γγ with SloopS . . . 84 5.5 Numerical investigation . . . 86 5.6 Conclusion . . . 89

Ce chapitre est consacré à la renormalisation des secteurs des neutralinos, des charginos et des sfermions du NMSSM dans SloopS, dans le but de calculer des observables à l’ordre d’une boucle. Plusieurs schémas de renormalisation ont été implémentés, afin de comparer les résultats obtenus pour les prédictions de différentes observables physiques. Nous nous sommes en particulier intéressés aux masses corrigés des neutralinos, ainsi qu’aux largeurs de désinté-gration des neutralinos, des charginos et des squarks. Les corrections QED et QCD ont été prises en compte pour les processus faisant intervenir des particules chargées. Une analyse des corrections a également été effectuée.

NEUTRALINOS, CHARGINOS ET SFERMIONS.

One-loop renormalisation of the NMSSM in SloopS : the

neutralino-chargino and sfermion sectors

Geneviève Bélanger1, Vincent Bizouard1, F.Boudjema1 and Guillaume Chalons2 1

LAPTh, Université de Savoie, CNRS, 9 Chemin de Bellevue, B.P. 110, F-74941 Annecy-le-Vieux, France

2

LPSC, Université Grenoble-Alpes, CNRS/IN2P3, Grenoble INP, 53 rue des Martyrs,F-38026 Grenoble, France

Abstract

We present the renormalisation of the neutralino, chargino and sfermion sectors of the NMSSM and its imple-mentation within SloopS, a code for the automatic computation of one-loop corrections initially developed for the standard model and the MSSM. Different renormalisation schemes are implemented and the importance of the choice of the renormalisation scheme is emphasized. The one-loop electroweak corrections to masses and to partial widths of neutralinos and charginos into final states with one gauge boson are computed. One-loop electroweak and QCD corrections to the partial widths of third generation sfermions into a fermion and a gauge boson are also computed.

7.1 Introduction

Supersymmetry has long been considered as the most natural extension of the standard model that can address the hierarchy problem while providing a dark matter candidate. The discovery of a Higgs boson with a mass of 125 GeV whose properties are compatible with those of the Standard Model is a great achievement of the first Run of the LHC [4, 110] and in some sense supports supersymmetry. Indeed a Higgs with a mass below 130 GeV is a prediction of the minimal supersymmetric standard model (MSSM). The fact that the observed mass is so close to the upper limit set in the MSSM however raises the issue of naturalness [61, 111]. Moreover, in the MSSM there is no relation between the higgsino mass parameter, µ, and the soft supersymmetry breaking Lagrangian. Indeed the term µH1H2 that induces a mixing between the two Higgs doublets is present in the superpotential before the symmetry breaking. Both these problems are solved in the singlet extension of the MSSM, the Next-to-minimal supersymmetric standard model (NMSSM) where the µ parameter is generated through the vacuum expectation value of the scalar component of the additional singlet superfield, hence is naturally at the electroweak scale. Moreover new terms in the superpotential give a contribution to quartic couplings that lead to an increase to the tree-level mass value of the lightest Higgs, thus more easily explaining the observed value of the Higgs mass [58,112] without relying on very large corrections from the stop/top sector. Although fine-tuning issues remain [59, 67, 77, 113] they are not as severe as in the MSSM.

The Higgs discovery thus led to a renewed interest in the NMSSM both at the theoretical and experimental level with new studies of specific signatures of the NMSSM Higgs sector [76, 114] and/or of the neutralino and sfermion sectors [115–117] being pursued at the LHC. With the exciting possibility of discovering new particles at the second Run of the LHC, it becomes even more important for a correct interpretation of a future new particle signal to

7.1. INTRODUCTION know precisely the particle spectrum as well as to make precise predictions for the relevant production and decay processes. The importance of loop corrections to the Higgs mass in supersymmetry can not be stressed enough, the large radiative corrections from the top and stop sector are necessary to raise the Higgs mass beyond the bounds imposed by LEP and to bring it in the range compatible with the LHC. Higher-order corrections are also of relevance for supersymmetric particles, higher-order SUSY-QCD and electroweak corrections to the full SUSY spectrum have been computed for some time in the MSSM and are incorporated in several public codes [118–121]. More recently higher-order corrections to Higgs and sparticle masses have been extended to the NMSSM [94,122] and several public codes incorporate these corrections, NMSSMTools [97, 98], SPheno [123, 124] , SoftSUSY [125], NMSSMCalc [95] or FlexibleSUSY [126]. Moreover higher-order corrections to decays have also been computed with some of these codes [127–129].

Higher order corrections to processes involving dark matter annihilation are also an essential ingredient in ascertaining whether a putative supersymmetric dark matter candidate observed at the LHC would be compatible with cosmological observations [130], in particular with the value of the relic density. The relic density is now known with a very high precision (at the percent level) within the standard cosmological model. Hence is much smaller than typical one-loop corrections in supersymmetry. Within the MSSM it was shown that QCD corrections to neutralino annihilation processes could well exceed the percent level [131] especially when the process features Higgs exchange, and similarly for QCD corrections to coannihilation pro-cesses for propro-cesses involving squarks or quarks [132–134]. Moreover electroweak corrections to neutralino annihilation processes were shown, in some scenarios, to easily exceed the per-cent level [84, 86, 135, 136] and so were electroweak corrections to some gaugino coannihilation processes [85].

The code SloopS was developed for the MSSM with the objective of computing one-loop corrections for collider and dark matter observables in supersymmetry. The complete renormal-isation of the model was performed in [33,34] and several renormalrenormal-isation schemes were imple-mented. This code relies on an improved version of LanHEP [28,137,138] for the generation of Feynman rules and counter terms. The model file generated is then interfaced to FeynArts [30], FormCalC[32] and LoopTools for the automatic computation of one-loop processes [139]. One-loop corrections to masses, two-body decays and production cross sections at colliders were realized together with one-loop corrections for various dark matter annihilation processes [85]. The aim of this paper is to perform the renormalisation of the neutralino/chargino sector as well as the sfermion sector of the NMSSM and to implement it within SloopS, thus providing full one-loop corrections to the sparticle masses and enabling the one-loop computation of any decay or scattering process involving these particles and/or SM ones. This is a natural ex-tension of the work performed in [33, 34] for the MSSM. The complete renormalisation of the NMSSM, including the very important Higgs sector is left for a separate publication. For the neutralino/chargino sector, different schemes are defined, these include either mixed DR-on-shell schemes where t_β is DR or pure on-shell schemes where input parameters are taken from the neutralino/chargino sector and from the Higgs sector. The dependence of the results on the renormalisation scheme can thus be studied. The sfermion sector is very similar to that of the MSSM and the renormalisation procedure proposed in [34] is implemented. As an application of the code, we provide numerical results for two-body decays of neutralinos, charginos and

NEUTRALINOS, CHARGINOS ET SFERMIONS. sfermions.

The paper is organized as follows. Section 2 contains a brief description of the NMSSM. The renormalisation of the neutralino and chargino sector is described in section 3 while the renormalisation of the sfermion sector is done in section 4. Section 5 and 6 contain numerical results. The different renormalisation schemes are compared and the electroweak corrections to some decays of neutralinos and charginos into 2-body final states involving a gauge boson are computed for several benchmarks. The decays of third generation sfermions into a fermion and a neutralino or chargino are presented in section 6. Section 7 contains our conclusions.