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

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Building models beyond the Standard Model

Before turning to the presentation of some BSM models, let us rst make a short comment on the two possible approaches to model building.
On the one hand, one can try to devise a complete theory valid up to high energies, possibly even up to the Planck scale that would solve some (or all) of the deciencies of the SM, and then study the consequences of such a model at energies around the electroweak scale. One of the best known examples of this approach called the top-down approach is the class of supersymmetric theories, in which fermionic and bosonic states are related by a new symmetry valid at high scales, but spontaneously broken at lower energies. In coming section 1.3 we will consider dierent Supersymmetry models studied in the course of this thesis, namely the MSSM, the NMSSM, and Dirac gaugino models. On the other hand, another approach to the construction of models of New Physics is also possible: the bottom-up approach. The idea is to extend the Standard Model by new elds and/or new terms in the Lagrangian, at energy scales around or slightly above the electroweak scale, in order to address some of the issues of the Standard Model, or sometimes simply to study the phenomenological consequences of such an extension. We will be more specically interested in models with extended Higgs sectors, which we will review in section 1.4 (extensions of the SM by a singlet, by a doublet, or by two triplets).

Supersymmetry, the SUSY algebra and its representations

Symmetries play an important role in the study of Quantum Field Theories, and nogo theorems most famously the Coleman-Mandula theorem [57] were established, greatly limiting the possible space-time symmetries of physical theories. Actually, Coleman and Mandula showed in [57] that if one considers a physical theory in four dimensions, with only local interactions, and further supposes the states are in nite number and that some of these are massive, then the largest possible symmetry group when allowing only bosonic symmetry generators is the Poincaré group. However, Haag, Šopusza«ski, and Sohnius [58] showed that it is possible to extend Poincaré symmetry by additional fermionic symmetry generators, i:e: generators that carry spin- 1 2 , and therefore transform as spinors under transformations of the rest of the Poincaré group and have spinor indices (undotted or dotted). These generators will be denoted QA where index A labels the possible multiple generators and the Supersymmetry transformations they generate turn bosons into fermions and vice versa. Furthermore, because the fermions are complex states, the complex conjugates of the generators Q A_ will also be generators of dierent supersymmetric transformations.
From the above considerations, and with the additional requirement of having a closed algebra, one can derive the SUSY algebra see e:g: section 3.1 of [55] for a derivation in the N = 1 case. In the presence of N supercharges labelled A; B; 2 [1; ;N], the complete SUSY algebra .

The hierarchy problem and Supersymmetry

In this section, we will show how Supersymmetry solves the hierarchy problem, presented in section 1.2.1. It is important to note, however, that Supersymmetry was not invented for the purpose of solving the hierarchy problem and that the fact that it does is only a fortuitous by-product of the extended symmetry. More precisely it is a consequence of one of the non-renormalisation theorems that have been proven for SUSY. In particular it has been shown that the superpotential is not renormalised [60, 61].
Let us investigate the cancellation of scalar mass corrections a bit more explicitly by considering a very simple toy model of a massless chiral supereld L interacting with another heavy chiral supereld H. In order to avoid mixing among the two superelds, we require the model, and hence the superpotential, to be invariant under the Z2 transformation H ! 􀀀H. The superpotential is then W() = 1 2 MH2 + 1 2 Y LH2 .

The Minimal Supersymmetric Standard Model

The rst and most natural way to extend the SM in the context of a (low-energy) SUSY model is to nd the simplest way to assign SM elds in chiral and gauge supermultiplets, while adding only the smallest possible number of additional states to ensure the theoretical consistency of the model. This can actually be done quite simply: rst, the SM fermions quarks and leptons are part of chiral supermultiplets and their superpartners called squarks and sleptons belong to the same representations of  the SM gauge group (c:f: table 1.1). Then the gauge bosons are put in gauge supermultiplets and have fermionic superpartners respectively gluinos, Winos and the Bino that transform in the adjoint representation of the corresponding component of the gauge group. Finally, the Higgs boson is also part of a chiral supermultiplet but here the situation requires a little more care, because if there were only one fermionic partner of the Higgs a higgsino the electroweak gauge symmetry would be anomalous.
Furthermore, with only one Higgs chiral supereld, there would be no way to write a holomorphic superpotential leading to Yukawa interactions for both up- and down-type quarks, nor any possibility to obtain a (supersymmetric) mass term for the higgsino. To avoid these three problems, two Higgs doublet superelds with opposite weak hypercharges are needed and we will call them Hu and Hd, the subscripts u and d indicating which type of quarks (up-type or down-type) that each doublet couples to. Moreover, to forbid interactions that violate16 B 􀀀 L, the model can be required to be invariant under a discrete symmetay, called R-parity, dened by a new quantum number RP (􀀀1)3(B􀀀L)+2s .

Gauginos in the MSSM

As we will shortly turn our focus to models with Dirac masses for gauginos, it is useful to rst shortly review gauginos and their masses in the context of the MSSM. Every gaugino in the MSSM comes as a Weyl spinor i:e: two complex degrees of freedom in the adjoint representation of the dierent components of the SM gauge group as the gauge bosons. Unlike SM fermions, fermionic superpartners of SM gauge bosons obtain masses from soft SUSY breaking terms such terms being allowed because gauge superelds, and hence the gauginos, are not chiral. More specically, these are Majorana mass terms see the fourth line of eq. (1.3.61) that can be generated by high-scale operators such as those shown in eqs. (1.3.47) and (1.3.48). After the electroweak symmetry breaking, the neutral electroweak gauginos ~B; ~W 3 mix with the neutral higgsinos ~h0 u; ~h0 d to form four neutralino mass eigenstates, while the charged electroweak gauginos ~W = ( ~W 1 i ~W 2)=p2 form, together with the charged higgsinos ~h +u ; ~h 􀀀d , two chargino mass eigenstates. The neutralino and chargino sector can have a rich phenomenology: most importantly, the lightest neutralino is often the LSP and consequently a possible candidate of Dark Matter however its couplings to the Higgs scalars are always related to the electroweak gauge couplings and are hence small and negligible for our concern.
Therefore, we will in what follows not consider the electroweak gauginos, but only the gluinos ~ga, that do not mix with fermions from the Higgs sector (because they are the only fermions to transform as octets under SU(3)C). They couple to squarks with the strong gauge coupling, and for this reason their contributions to the neutral scalar masses from two-loop and beyond can be large and must be taken into account.

Shortcomings of the MSSM

One serious theoretical issue of the MSSM is the so-called -problem, i:e: the problem of generating the supersymmetric Higgs mass term that must be of the order of the SUSY breaking scale MSUSY (the scale of the SUSY breaking terms). The reason why must be of the order of MSUSY can be explained as follows [78]: on the one hand, cannot be zero, because (among other reasons) there is a lower bound on the possible value of jj coming from the experimental lower bound on chargino masses. On the other hand, if is too large then the extremum of the potential at Hu = Hd = 0 becomes a stable minimum and EWSB cannot take place (e:g: relation (1.3.71) cannot be veried). Therefore, the only scale that can be related to is the SUSY breaking scale MSUSY, which is also apparent from eq. (1.3.71) but is contradictory for a SUSY preserving parameter. As we will see in what follows, this problem can be solved in both the NMSSM and Dirac gaugino models [79], by generating the term through the VEV of a dynamical eld having the same quantum numbers as , i:e: a singlet.
Another issue of the MSSM is to nd a way to have suciently large radiative corrections to the Higgs mass to obtain a Higgs mass of 125 GeV even if the tree-level mass has a low upper-bound (1.3.78). As will be shown in the next chapter where we will discuss Higgs mass calculations, for the radiative corrections to be suciently large in the MSSM either of the following conditions needs to be fullled: large stop masses, or large stop mixing. Again the NMSSM and Dirac gaugino models improve the situation on this matter, by relaxing the upper-bound on mh (more precisely, the equivalents of eq. (1.3.78) contain additional terms, due to the singlet or the adjoint superelds respectively).

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Extended Supersymmetry and supersoft SUSY breaking

The starting point for the construction of models with Dirac gauginos is to ask how much Supersymmetry (i:e: how many supercharges) can be allowed without encountering a theoretical inconsistency. First of all, the chiral nature of the SM fermions is a strong indication that, if it exists, Supersymmetry in the matter sector must have N = 1 because in N = 2 SUSY, hypermultiplets cannot be chiral. Furthermore, the number of matter states in a theory with N = 2 SUSY is so large that it would lead to Landau poles at low energies (well before the GUT or Planck scales), for example for the strong gauge coupling. However, in the gauge sector the situation is dierent and N = 2 Supersymmetry can be considered without encountering problematic Landau poles [89] (although they would occur for N = 4). The extension of Supersymmetry for gauge states can be realised in an N = 1 language by adding chiral superelds a transforming in the adjoint representation of the gauge group (called more shortly adjoint chiral superelds).

Table of contents :

Introduction
1 The Higgs boson and Physics beyond the Standard Model 
1.1 The Standard Model and the Higgs sector
1.1.1 Mass terms of gauge bosons and fermions
1.1.2 Electroweak Symmetry Breaking and the Brout-Englert-Higgs mechanism
1.1.3 Electroweak gauge xing
1.1.4 The Higgs sector at tree-level and beyond
1.1.5 The Goldstone Boson Catastrophe in the Standard Model
1.2 Going beyond the Standard Model
1.2.1 The hierarchy problem
1.2.2 The stability of the electroweak vacuum
1.2.3 Building models beyond the Standard Model
1.3 Supersymmetry
1.3.1 Some basics of SUSY
1.3.1.1 Fermions in two-component notation
1.3.1.2 Supersymmetry, the SUSY algebra and its representations
1.3.1.3 Superspace formalism and superelds
1.3.1.4 Superpotential and supersymmetric Lagrangians
1.3.1.5 R-symmetry
1.3.2 SUSY breaking
1.3.3 The hierarchy problem and Supersymmetry
1.3.4 Minimal models
1.3.4.1 The Minimal Supersymmetric Standard Model
1.3.4.2 The Higgs sector of the MSSM
1.3.4.3 Gauginos in the MSSM
1.3.4.4 Shortcomings of the MSSM
1.3.4.5 The Next-to-Minimal Supersymmetric Standard Model
1.3.4.6 The Higgs sector of the NMSSM
1.3.5 Dirac gaugino models
1.3.5.1 Extended Supersymmetry and supersoft SUSY breaking
1.3.5.2 A brief overview of Dirac gaugino models
1.3.5.3 Some aspects of the phenomenology of Dirac gaugino models
1.3.5.4 Properties of the adjoint scalars
1.3.5.5 Gluino masses and couplings
1.3.5.6 The MDGSSM and the MRSSM
1.4 Non-supersymmetric extensions of the Standard Model
1.4.1 Singlet extensions of the Standard Model
1.4.2 Two-Higgs-Doublet Models
1.4.3 The Georgi-Machacek model
2 Precision calculations of the Higgs boson mass 
2.1 Measurements of the Higgs mass
2.2 Scalar mass calculations
2.2.1 Regularisation and renormalisation schemes
2.2.2 Calculations beyond leading order and choice of inputs
2.2.3 Dierent types of mass calculations
2.2.3.1 Fixed-order calculations
2.2.3.2 The eective eld theory approach
2.3 State-of-the-art of Higgs mass calculations
2.3.1 Real and complex MSSMs
2.3.1.1 Fixed-order results
2.3.1.2 EFT and hybrid results
2.3.2 Supersymmetric models beyond the MSSM
2.3.3 Non-supersymmetric models
2.4 Calculations in generic theories
2.4.1 Notations for general eld theories
2.4.2 Two-loop neutral scalar masses in generic theories
2.4.3 The SARAH/SPheno framework
2.4.3.1 Analytic calculations with SARAH 87
2.4.3.2 Interfaces with SPheno and other HEP codes
2.4.3.3 Numerical set-up of the spectrum calculation
3 Leading two-loop corrections to the Higgs boson masses in SUSY models with Dirac gauginos 
3.1 Two-loop corrections in the eective potential approach
3.1.1 General results
3.1.2 Two-loop top/stop contributions to the eective potential
3.1.3 Mass corrections in the MDGSSM
3.1.4 Mass corrections in the MRSSM
3.1.5 On-shell parameters in the top/stop sector
3.1.6 Obtaining the O(bs) corrections
3.1.7 Simplied formulae
3.1.7.1 Common SUSY-breaking scale
3.1.7.2 MRSSM with heavy Dirac gluino
3.2 Numerical examples
3.2.1 An example in the MDGSSM
3.2.2 An example in the MRSSM
3.3 Conclusions
4 Avoiding the Goldstone Boson Catastrophe in general renormalisable eld theories at two loops 
4.1 The Goldstone Boson Catastrophe and resummation
4.1.1 Abelian Goldstone model
4.1.2 Goldstone bosons in general eld theories
4.1.3 Small m2 G expansion of the eective potential for general theories
4.2 Removing infra-red divergences in the minimum condition
4.2.1 All-scalar diagrams
4.2.1.1 Elimination of the divergences by method (i)
4.2.1.2 Elimination of the divergences by method (ii)
4.2.1.3 Elimination of the divergences by setting the Goldstone boson on-shell
4.2.2 Diagrams with scalars and fermions
4.2.3 Diagrams with scalars and gauge bosons
4.2.4 Total tadpole
4.3 Mass diagrams in the gaugeless limit
4.3.1 All-scalar terms
4.3.1.1 Goldstone shifts
4.3.1.2 Momentum-regulated diagrams
4.3.2 Fermion-scalar diagrams
4.4 Self-consistent solution of the tadpole equations
4.5 Conclusions
5 Supersymmetric and non-supersymmetric models without catastrophic Goldstone bosons 
5.1 The Goldstone Boson Catastrophe and its solutions
5.1.1 Previous approaches in SARAH
5.1.2 On-shell Goldstone bosons, consistent tadpole solutions, and the implementation in SARAH
5.2 Standard Model
5.2.1 A rst comparison of our results with existing calculations
5.2.2 A detailed comparative study of SPheno and SMH results
5.2.3 Momentum dependence
5.3 The NMSSM
5.4 Split SUSY
5.5 Two-Higgs-Doublet Model
5.5.1 The alignment in Two-Higgs-Doublet Models
5.5.2 Renormalisation scale dependence of the Higgs mass computed with SPheno
5.5.3 Quantum corrections to the alignment limit
5.5.4 Perturbativity constraints
5.6 Georgi-Machacek Model
5.7 Conclusions
6 Matching and running 
6.1 Matching and Running
6.1.1 Renormalisation Group Equations
6.1.2 Matching
6.2 Models and results
6.2.1 Singlet Extension
6.2.2 Singlet Extension with an additional Z2 symmetry
6.2.2.1 Analytical approximation
6.2.2.2 Numerical study
6.2.3 Vector-like quarks and stability of the SM
6.2.4 Two-Higgs Doublet model
6.3 Conclusions
Conclusion
A Derivatives of the two-loop eective potential in models with Dirac gauginos 
B Denitions and expansions of loop functions 
B.1 Loop functions
B.1.1 Denition of loop functions
B.1.1.1 One-loop functions
B.1.1.2 Two-loop functions
B.1.2 Small m2 G expansion
B.2 Diagrams regulated by momentum
B.2.1 Limits of the Z and U functions
B.2.2 Limits of the M function
B.3 Additional expressions for ~ V (x; 0; z; u)
B.3.1 Integral representation
C Consistent solution of the tadpole equations with shifts to fermion masses 
Bibliography 

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