Lepton flavor violating Higgs decay in the ελ minimal supersymmetric standard model

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(1)PONTIFICIA UNIVERSIDAD CATÓLICA DE CHILE FACULTAD DE FÍSICA INSTITUTO DE FÍSICA. Lepton flavor violating Higgs decay in the λ-Minimal Supersymmetric Standard Model BY Pablo Candia da Silva. Report presented before the Facultad de Fı́sica of the Pontificia Universidad Católica de Chile in partial fulfillment of the requirements for the degree of Magı́ster en Fı́sica.. Supervisor. :. Dr. Marco Aurelio Dı́az Gutiérrez (PUC Chile). Evaluation committee. :. Dr. Jorge Luis Alfaro Solı́s (PUC Chile) Dr. Máximo Bañados Lira (PUC Chile). Santiago, Chile.

(2) c 2018, Pablo Candia da Silva Se autoriza la reproducción total o parcial, con fines académicos, por cualquier medio o procedimiento, incluyendo la cita bibliográfica del documento.. 2.

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(5) Abstract Besides the fact that the SM has to be extended in order to account for neutrino masses, an interesting implication of this discovery is the violation of flavour symmetry in the leptonic sector. This feature is known as lepton flavour violation (LFV), and in principle, it could allow processes such as `i → `j γ to occur when i 6= j. This work focuses on the calculation of Br(h → eµ) in the Minimal Supersymmetric Standard Model (MSSM) with broken R-parity, which is a symmetry that implies both lepton and baryon number. If the latter is conserved to prevent proton decay, this theory is very well suited to describe tiny (eV) neutrino masses, and furthermore, LFV processes are enhanced via R-parity violating (RpV) parameters, which sets a good scenario for probing physics beyond the SM. Our preliminary results show that for two working scenarios where ML = 2 TeV and 10 TeV, Br(h → eµ) cannot exceed the value of 10−10 (respectively 10−7 ) if neutrino constraints are satisfied..

(6) Contents 1 Introduction. 1. 2 The Standard Model and its limitations. 3. 2.1. 2.2. Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3. 2.1.1. Poincaré group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3. 2.1.2. Gauge group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 5. 2.1.3. Particle content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 7. 2.1.4. Spontaneous symmetry breaking in the SM . . . . . . . . . . . . . .. 8. 2.1.5. Fermion sector of the SM . . . . . . . . . . . . . . . . . . . . . . . .. 11. Going beyond the Standard Model . . . . . . . . . . . . . . . . . . . . . . .. 12. 2.2.1. Hierarchy problem . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 13. 2.2.2. Neutrino masses and oscillations . . . . . . . . . . . . . . . . . . . .. 15. 2.2.3. Dark Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 19. 2.2.4. Lepton Flavour Violation . . . . . . . . . . . . . . . . . . . . . . . .. 21. 3 The Minimal Supersymmetric Standard Model 3.1. 3.2. 25. Supersymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 25. 3.1.1. Super Poincaré Group . . . . . . . . . . . . . . . . . . . . . . . . . .. 25. 3.1.2. Hierarchy problem revisited . . . . . . . . . . . . . . . . . . . . . . .. 27. Superspace Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 28. 3.2.1. General superfield . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 28. 3.2.2. Chiral covariant derivatives . . . . . . . . . . . . . . . . . . . . . . .. 30. 3.2.3. Right and left chiral superfields . . . . . . . . . . . . . . . . . . . . .. 31. ii.

(7) CONTENTS. 3.2.4. Vector superfields . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 33. 3.2.5. Formalism for Abelian gauge theories . . . . . . . . . . . . . . . . .. 36. 3.2.6. Non-Abelian case . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 37. The Minimal Supersymmetric Standard Model . . . . . . . . . . . . . . . .. 39. 3.3.1. Particle content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 39. 3.3.2. Kinetic Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 41. 3.3.3. Gauge Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 42. 3.3.4. Superpotential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 43. 3.4. Soft SUSY breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 44. 3.5. Spontaneous Symmetry Breaking . . . . . . . . . . . . . . . . . . . . . . . .. 45. 3.6. Decoupling limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 46. 3.7. R-parity breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 48. 3.7.1. R-parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 48. 3.7.2. R-parity violating superpotential . . . . . . . . . . . . . . . . . . . .. 49. 3.7.3. Neutrino masses in the RpV MSSM . . . . . . . . . . . . . . . . . .. 50. 3.7.4. One-loop neutrino masses . . . . . . . . . . . . . . . . . . . . . . . .. 52. 3.7.5. Neutrino masses and mixing angles . . . . . . . . . . . . . . . . . . .. 53. 3.3. 4 λ-MSSM. 56. 4.1. Motivation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 56. 4.2. Superpotential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 57. 4.3. Radiative neutrino masses . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 57. 4.4. R-parity violation and the decoupling limit . . . . . . . . . . . . . . . . . .. 60. 5 LFV decay h → eµ. 62. 5.1. Background of LFV Higgs decays . . . . . . . . . . . . . . . . . . . . . . . .. 62. 5.2. Higgs production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 63. 5.3. Decay width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 64. 5.4. Parameter space and numerical results . . . . . . . . . . . . . . . . . . . . .. 66. 5.4.1. Working scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 66. 5.4.2. Agreement with neutrino data . . . . . . . . . . . . . . . . . . . . .. 69. iii.

(8) CONTENTS. 6 Summary and future work. 73. A Spinors and conventions. 75. A.1 Spinor identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 76. B Mass matrices of the RpV MSSM. 79. C Couplings. 83. iv.

(9) List of Tables 2.1. Particle content of the SM . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 8. 2.2. Neutrino parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 19. 2.3. Bounds for some LFV processes . . . . . . . . . . . . . . . . . . . . . . . . .. 23. 3.1. Chiral superfield content of the MSSM . . . . . . . . . . . . . . . . . . . . .. 40. 3.2. Vector superfield content of the MSSM . . . . . . . . . . . . . . . . . . . . .. 40. 5.1. Parameter space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 67. 5.2. RpV parameters and neutrino observables . . . . . . . . . . . . . . . . . . .. 70. v.

(10) List of Figures 2.1. Composition of the universe . . . . . . . . . . . . . . . . . . . . . . . . . . .. 19. 4.1. Optional caption for list of figures . . . . . . . . . . . . . . . . . . . . . . .. 58. 4.2. Decoupling limit masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 60. 4.3. Decoupling limit couplings . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 60. 5.1. Generic diagrams for the principal Higgs production mechanisms at de LHC. 64. 5.2. P. mν vs. Br(h → eµ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 68. 5.3. P. mν vs. Br(h → eµ) in another benchmark . . . . . . . . . . . . . . . . .. 69. 5.4. χ2 dependence with TRpV couplings . . . . . . . . . . . . . . . . . . . . . .. 70. 5.5. χ2 dependence with Br(h → eµ) . . . . . . . . . . . . . . . . . . . . . . . .. 71. 5.6. Neutrino masses vs. TRpV parameters at P1 . . . . . . . . . . . . . . . . .. 72.

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(12) Chapter 1. Introduction Our current understanding of the elementary particles and their interactions relies on the Standard Model of particle physics (SM), which is a theory build upon the principles of quantum mechanics, Lorentz invariance and gauge symmetry. This gauge theory is based on the theoretical concepts that were developed during the XX century, with special emphasis in the 1950-1970 period, and its predictive power has not been surpassed up to this date. We know, however, that physics beyond the SM exists. Among other things, the SM cannot account for neutrino flavour oscillations, which in turn imply the massive nature of these particles. In addition, there is compelling evidence of the existence of dark matter, an invisible agent that exerts a gravitational force on celestial bodies that cannot be explained by visible matter alone, and for which the SM doesn’t provide any explanation. From the theoretical standpoint there are also motivations for extending the SM, such as explanations for the existence of three lepton and quark families or the pursue of a quantum theory of gravity, just to name a few. One of the most popular extensions is a supersymmetric version of the SM. Supersymmetry, which is by the way one of the main motivations behind the construction of the Large Hadron Collider, is the symmetry under certain class of transformations that exchange fermions and bosons. The Minimal Supersymmetric Standard Model (MSSM), the smallest supersymmetric extension of the SM, roughly doubles the number of particles that have been discovered until now, and 1.

(13) CHAPTER 1. INTRODUCTION. showcases many features that make it an interesting option for physics beyond the SM. For instance, in this theory the electro-weak and strong forces that are described by the SM converge into a hypothetical unique force at a sufficiently high energy scale, known as the Grand Unification scale. Furthermore, this theory offers a solution to the fine tuning problem that affects the masses of fundamental scalars, such as the Higgs boson. In this research, we studied a MSSM version that violates lepton number, and which is motivated by neutrino physics and dark matter. The study focuses in the calculation of lepton flavour violating effects in the Higgs sector, which come as a by-product of the mechanism that generates neutrino masses. In particular, we calculated the branching ratio of the decay h → eµ in order to see how close it is to the reach of collider experiments. In chapter 2, the SM is briefly presented, and some of its main shortcomings are explained. In chapter 3, we introduce the superfield formalism, which leads to the MSSM Lagrangian when the SM fields are replaced by superfields and SUSY is softly broken. Some characteristics of the MSSM scalar sector are discussed, as well as R-parity and its implications. Chapter 4 is dedicated to motivate and explain the particular supersymmetric model that was used for the calculation. Chapter 5 shows the results that were found for Br(h → eµ), and neutrino masses are discussed within the context of this particular model. Lastly, chapter 6 summarizes the content of this thesis, discusses future work, and explains the conclusions that were drawn from this research.. 2.

(14) Chapter 2. The Standard Model and its limitations 2.1. Standard Model. The Standard Model relies on three basic principles: invariance under the Poincaré group, gauge symmetry and renormalizability. We begin with a brief discussion of the Poincaré group, which will be later generalized to the super-Poincaré group and will serve as a starting point for introducing SUSY theories.. 2.1.1. Poincaré group. Relativistic quantum field theories, as we know, rest upon the principles of quantum mechanics and special relativity. Several approaches exist when it comes to explain the latter (we won’t speak about the former), but for our purposes, it is convenient to say that special relativity consists of the invariance of physical laws under transformations of the Poincaré group. This continuous group includes translations in time and space, rotations in space, and boosts, or in other words, Lorentz transformations and space-time translations. As any 3.

(15) CHAPTER 2. THE STANDARD MODEL AND ITS LIMITATIONS. 10-parameter group, it has 10 generators, and its algebra is given by. [Pµ , Pν ] = 0,. (2.1a). [Mµν , Pρ ] = i(ηµρ Pν − ηνρ Pµ ),. (2.1b). [Mµν , Mρσ ] = i(ηµρ Mνσ − ηµσ Mνρ − ηνρ Mµσ + ηνσ Mµρ ).. (2.1c). where Mµν are the six generators of Lorentz transformations (antisymmetric) and Pµ are the generators of translations. It can be shown that this algebra has two Casimir invariants, W 2 and P 2 , where Wµ is the Pauli-Lubański vector 1 Wµ = − µρσλ M ρσ P λ . 2 Hence, it is said that one-particle states fall into irreducible representations of the Poincaré group, and are classified by the mass and spin quantum numbers (m2 , s) associated to (P 2 , W 2 ). Quantum field theories consist of realizations of this algebra. Regarding the construction of Lagrangians, three kinds of fields are our building blocks: • Scalar fields, that remain invariant under a Lorentz transformation Λ, • spin-1/2 fields, which transform as i. µν. ψ(x) → ψ 0 (x) = e 2 ωµν M ψ(Λ−1 x), where M µν =. i µ ν [γ , γ ] 4. can be shown to satisfy (2.1c), and • spin 1 fields, which undergo V ν (x) → V 0µ (x) = Λµν V ν (Λ−1 x) when a Lorentz transformation is made. Lagrangians of relativistic field theories must be invariant under such transformations. 4.

(16) 2.1. STANDARD MODEL. 2.1.2. Gauge group. The gauge group of the SM is SU (3)C ×SU (2)L ×U (1)Y , where L stands for the left-handed fermions, C for colour charge, and Y for hypercharge. As the SU (2) transformations act only upon left-handed particles, they are arranged in doublets that transform under some representation of SU (2). SU (3) transformations, on the other hand, act only upon particles that have color charge, or quarks. Lastly, the hypercharge assignation dictates how fermions transform under U (1). Therefore, a general gauge transformation for a particle with hypercharge Y is. ψ → e−igs γ. A (x)T A. e−igw β. a (x)τ a. e−igy α(x)Y /2 ψ ,. (2.2). where gs , gw , and gy are the gauge couplings of each group. The first exponential function corresponds to a finite SU (3) transformation, and the matrices T A , A = 1, ..., 8 are the generators of the Lie algebra of SU (3). Particles with no colour charge transform in the trivial representation of the group, which means that all the generators are 0. It is said, then, that the particle transforms as a singlet under SU (3). Otherwise, the particle can transform in the fundamental representation of SU (3), where the generators are T A = 12 λA , and the eight 3 × 3 traceless matrices λA are the Gell-Mann matrices. Any set of generators T A must fulfill the SU (3) algebra [T A , T B ] = if ABC T C , where f ABC are the structure constants of SU (3). The second exponential corresponds to a finite SU (2) transformation, and in the fundamental representation, τ a = 12 σ a , where σ a are the three Pauli matrices. The Lie algebra of the group is [τ a , τ b ] = itabc τ c , where tabc = abc , and abc is the Levi-Civita symbol, as one could guess from the conmutation relation of the Pauli matrices. Finally, the third exponential corresponds to a U (1) abelian transformation generated by Y , which is simply a number that corresponds to the hypercharge that is assigned to each particle. Someone may wonder why the parameters that determine the gauge transformations α(x), β(x) and γ(x) are spacetime dependent. This is the main feature of gauge transformations: they are local. A theory that remains invariant under such a transformation is a gauge 5.

(17) CHAPTER 2. THE STANDARD MODEL AND ITS LIMITATIONS. theory. In the case of the Dirac Lagrangian, for instance, Lorentz symmetry is enough to ensure invariance under global continuous transformations, but if the transformation is local, the derivative in the kinetic term spoils this cancellation of phases. This is the reason for introducing the covariant derivative, which renders these terms gauge invariant by means of the minimal coupling, a method which consists of replacing the partial derivative for the covariant one. This derivative is constructed by introducing one gauge field for every generator of the group, and it depends of the representation under which the field transforms. We write down the general form of the covariant derivative according to this prescription as follows. A a a Dµ = ∂µ − igs GA µ T − igw Wµ τ − igy Bµ. Y . 2. (2.3). Gauge invariance can be shown provided that the gauge fields transform as. A A ABC B GA γ (x)GC µ → Gµ − ∂µ γ + gs f µ. (2.4). Wµa → Wµa − ∂µ β a + gw tabc β b (x)Wµc. (2.5). Bµ → Bµ − ∂µ α.. (2.6). As we introduced the gauge fields, we need to account for their dynamics. This is accomplished by adding spin-1 field kinetic terms, which can be made from the field strength tensors. We define the field strength tensors as. A A ABC B C GA Gµ G ν µν = ∂µ Gν − ∂ν Gµ + gs f. (2.7). a Wµν = ∂µ Wνa − ∂ν Wµa + gw tabc Wµb Wνc. (2.8). Bµν = ∂µ Bν − ∂ν Bµ .. (2.9). In order to make a gauge invariant term from these, it suffices to consider, for instance, a W µν , as can be shown with the help of the transformations (2.6). The gauge LaWµν a. grangian of the SM is given by 1 1 a µν 1 A µν Lgauge = − Bµν B µν − Wµν Wa − Gµν GA . 4 4 4 6. (2.10).

(18) 2.1. STANDARD MODEL. 2.1.3. Particle content. At this point, we can already discuss the particle content of the SM. When it comes to fermions, we have two possible chiralities: they can be right-handed or left-handed, except for the neutrinos since no right-handed neutrino has been ever detected. Irrespectively of their chiralities, fundamental fermions can also be separed in leptons and quarks. As explained above, both left-handed leptons and quarks transform as doublets under SU (2), with the difference that leptons are singlets under SU (3) while quarks transform as triplets under SU (3). On the other hand, right-handed quarks transform as SU (2) singlets and SU (3) triplets, whereas right-handed leptons are singlets under both SU (2) and SU (3) gauge groups. The representation space of SU (3) receives the name of color space, and it is related to strong interactions, hence, we say that quarks come in red, blue and green colours. The SM has three generations of particles that are treated on an equal foot, each one with bigger masses than the previous. The first family is composed by the electron (e), the electron neutrino (νe ), the up quark (u) and the down quark (d); the second family has the muon (µ), the muon neutrino (νµ ), the charm quark (c) and the strange quark (s); and the third family comes with the tau (τ ), the tau neutrino (ντ ), the top quark (t) and the bottom quark (b). All of them come in right-handed and left-handed copies, except for the neutrinos. Henceforth, electrons, muons, and taus will be denoted by eiL,R , with i = 1, 2, 3 respectively. The same applies to quarks, where uiL,R , i = 1, 2, 3 correspond to up, charm and top, and diL,R , i = 1, 2, 3 refers to down, strange and bottom. In addition to these fermions, there is a fundamental and recently discovered ingredient of the SM. It corresponds to the Higgs boson, which is a scalar field that is the key of the mechanism that gives mass to all fundamental particles of the SM (Higgs mechanism). By the moment, let’s consider that the Higgs boson is contained in a SU (2) doublet of complex scalar fields. The particle content is summarized in table 2.1. Hypercharges may seem a bit awkward, but their meaning will be clearer in the following section. The eight GA µ are gluons, which mediate the strong interaction, and the meaning of the 3 Wµa ’s together with Bµ will be clearer later. 7.

(19) CHAPTER 2. THE STANDARD MODEL AND ITS LIMITATIONS. SU (3). SU (2). Y. 1. 2. -1. 1. 1. -2. 3. 2. 1/3. uiR. 3. 1. 4/3. diR. 3. 1. -2/3. 1. 2. 1. Bµ. 1. 1. 0. Wµa. 1. 3. 0. GA µ. 8. 1. 0. Particles Li = (νiL. eiL )T. eiR Qi = (uiL. φ = (φ1. diL )T. φ2 )T. Table 2.1: Particle content of the SM. Leptons are in the first row, quarks appear in the second, the third row contains the Higgs boson, and the last contains the gauge bosons. The SU (2) and SU (3) labels are given by the dimension of each representation, and Y is the hypercharge of each field. The i index denotes the family, and the colour index is omited. The index a goes from 1 to 3, while A goes from 1 to 8. Note that 1 denotes the trivial representation of the group (singlet under the gauge group).. 2.1.4. Spontaneous symmetry breaking in the SM. For practical purposes, it is convenient to begin the analysis with the Higgs part of the Lagrangian:. LHiggs = (Dµ φ)† (Dµ φ) − V (φ) ,. (2.11). where V (φ) is a potential that spontaneously breaks the SU (2)L × U (1)Y symmetry, and is given by. V (φ) = µ2 φ† φ + λ(φ† φ)2 , with µ2 < 0 , 8. (2.12).

(20) 2.1. STANDARD MODEL which has a minimum when φ21 + φ22 + φ23 + φ24 = −µ2 /(2λ). We choose a vacuum such that  . 1 0 hφi0 = √   , 2 v. (2.13). in which we defined the vacuum expectation value (vev) of the Higgs as v =. p. −µ2 /λ. It. is clear that although the Higgs Lagrangian is invariant under SU (2) × U (1), this vacuum solution is not. This is known as spontaneous symmetry breaking (SSB). We can prove that the vacuum is invariant under the transformations generated by Q = τ 3 + Y /2, and is affected by the other 3 independent generators. By virtue of the Goldstone theorem [1], we can foresee that there will be 3 massless Goldstone bosons, which yields 3 massive gauge bosons and a massless one [2, 3, 4]. We can use the unitary gauge in order to get rid of the Goldstone bosons and write the scalar doublet as follows: . . 1  0  φ= √   . 2 v+h. (2.14). Now, we expand the kinetic term of the Higgs boson for finding the gauge boson masses. Only quadratic terms are relevant, so we have. † µ Lkin Higgs = (Dµ φ) (D φ). .  . 2. 1  gy Bµ + gw Wµ3 gw (Wµ1 − iW 2 ) 0 =     + ... 8 g (W 1 + iW 2 ) g B − g W 3 v w y µ w µ µ =. i v 2 h 2 1 1µ 2 gw Wµ W + Wµ2 W 2µ + gy2 Bµ B µ − 2gw gy Wµ3 B µ + gw Wµ3 W 3µ + ... 8 . =. v2 8. . . 2 gw Wµ1 W 1µ + Wµ2 W 2µ +. We can define W ± =. √1 2. v2. h. i. gy2. Bµ , Wµ3  8 −gw gy . . −gy gw   2 gw. . Bµ. W 3µ.    + ... .. W 1 ∓ iW 2 , and its easy to check that the mass matrix above is . rendered diagonal by means of the following rotation of the gauge fields: 9.

(21) CHAPTER 2. THE STANDARD MODEL AND ITS LIMITATIONS. Zµ = q Aµ = q. 1. . gw Wµ3 − gy Bµ. . gy Wµ3 + gw Bµ ,. 2 gy2 + gw. 1 2 gy2 + gw. . (2.15). . (2.16). so we obtain. Lkin Higgs ⊃. v 2 2 + −µ v 2 2 2 gw Wµ W + gy + gw Zµ Z µ + 0 · Aµ Aµ . 4 8. (2.17). The masses of the gauge bosons can be read as. MW =. v gw , 2. MZ =. vq 2 2, gy + gw 2. MA = 0,. (2.18). which means three massive and a massless gauge boson, as anticipated. The Weinberg angle is defined by the relation tan θw = gy /gw . Furthermore, defining τ ± = τ 1 ± iτ 2 , the SU (2)L × U (1)Y covariant derivative can be recast as. gw Y Dµ = ∂µ − i √ τ + Wµ+ + τ − Wµ− − iZµ gw cw τ 3 − gy sw 2 2 . . . − igw sw Aµ. Y τ + 2 3. . .. (2.19) We can see how the coupling to the photon is given by the electric charge operator Q = τ 3 + Y /2 that was mentioned earlier. After spontaneous symmetry breaking, the gauge Lagrangian reads. 1 1 + µν 1 A µν Lgauge = − Fµν F µν − Wµν W− − Gµν GA , 4 4 4. (2.20). ± is the one of the gauge where Fµν is the field strength tensor of the photon Aµ and Wµν. bosons Wµ± . 10.

(22) 2.1. STANDARD MODEL. 2.1.5. Fermion sector of the SM. We already showed the Higgs Lagrangian and the gauge boson kinetic terms. Its time to speak about fermions. All fermions in the SM model are spin-1/2 particles, and their free propagation is described by means of the Dirac equation. However, in order to construct the SM, it is convenient to arrange the fermions in SU (2) doublets as shown in table 2.1, and promote partial to covariant derivatives. The kinetic term is. Lkin = i. 3 X. . Qi σ µ Dµ Qi + Li σ µ Dµ Li + uiR σ µ Dµ uiR + eiR σ µ Dµ eiR + diR σ µ Dµ diR ,. i=1. (2.21) where the index i denotes the family to which each doublet/singlet belongs. The covariant derivative is given by (2.19), and the representation of the generators that act on a given fermion is shown in table 2.1. The couplings of the fermions to the gauge bosons can be found in a straightforward manner by expanding the covariant derivative in the kinetic term. It is a fact that every fundamental fermion is massive. In the SM, fermion masses are generated by the following Yukawa type interactions:. −LY uk = Yij` Li φ ejR + Yijd Qi φ djR + Yiju Qi φ̃ ujR + h.c. .. (2.22). The matrices Y ` , Y d and Y u are named lepton, down and up type Yukawa couplings respectively, and they are dimensionless. Furthermore, we defined φ̃ = iσ2 φ∗ . It’s easy to check that these terms are gauge invariant by using the transformation rules for each field. After SSB, this yields the following mass terms for the fermions v ` d u √ −Lmass = Y e e + Y d d + Y u u Y uk ij iL jR ij iL jR ij iL jR + h.c. . 2. (2.23). If we consider the up and down quarks in (2.23) to be related to their corresponding mass eigenstates by. . uiL = UL†. . . uiR = UR†. (m). ij. ujL , 11. . (m). ij. ujR. (2.24).

(23) CHAPTER 2. THE STANDARD MODEL AND ITS LIMITATIONS. and. . † diL = DL. . . † diR = DR. (m). ij. djL ,. . (m). ij. djR. (2.25). where DL,R and UL,R are unitary matrices and the superscript (m) refers to mass eigenstates, then, these matrices will have a family mixing effect in the coupling of the quarks to the W boson, as can be easily seen by writing down the coupling in the mass eigenstate basis:.  . LqqW = gw uiL.  0 diL σ µ  Wµ− . . . Wµ+  uiL  0. . diL. . gw = √ uiL σ µ Wµ+ diL + h.c 2 gw (m) † = √ ujL (m) (UL )ji σ µ Wµ+ (DL )ik dkL + h.c. 2 gw (m) = √ ujL (m) (VCKM )jk σ µ Wµ+ dkL + h.c. . 2. (2.26) (2.27) (2.28) (2.29). † Here we defined the Cabbibo-Kobayashi-Maskawa (CKM) matrix as VCKM = UL DL , and. it can be shown that this matrix can be parametrized with 3 angles and one phase [5]. What happens with this family mixing effect in the lepton sector? This will be the subject of subsection 2.2.4, and it is related to one of the main limitations of the SM.. 2.2. Going beyond the Standard Model. When it comes to the description of fundamental particles, the SM is certainly outstanding. However, there are shortcomings of the theory that make it unable to account for some natural phenomena, as well as some points that don’t seem to be fully understood. Among these issues we can mention the hierarchy problem, neutrino oscillations and mixings, dark matter and the matter-antimatter asymmetry. In the following, we make a brief exposition of the first three, as the hierarchy problem is one of the main motivations for supersymmetry and R-parity violation is an appealing possibility for explaining neutrino masses and dark matter. 12.

(24) 2.2. GOING BEYOND THE STANDARD MODEL. 2.2.1. Hierarchy problem. If we take a look at eq. (2.23), we can see that the mass of the fermions are given by v mf = √ Y f , 2. (2.30). where f is the fermion and Y f is the correspondent Yukawa entry in the basis where it is diagonal. The masses of the gauge bosons are given by eq. (2.18), and expanding the scalar potential (2.12), we find. m2h = 2λv 2. (2.31). for the Higgs tree level mass. However, we know that all these masses receive quantum corrections, so we could ask: how much fine-tuning is needed if we want to have the right physical mass for the particles when those corrections are considered? In the case of fermions, the massless theory exhibits chiral symmetry, which corresponds to the rotation ψL → eiθL ψL ,. ψR → ψR. or ψL → ψL ,. ψR → eiθR ψR .. So, this symmetry should be preserved when we take mf → 0. In consequence, loop corrections for the fermions have the form. δmf ∼ mf ln. ΛN P mf. ! , + # ΛN P. (2.32). where ΛN P corresponds to the scale at which new physics appear, and the linear term is forbidden because we want to preserve chiral symmetry when mf → 0. For the gauge bosons, gauge invariance has to be recovered in the massless limit, and the corrections have the form. δMA2. ∼. MA2 ln. ΛN P mf 13. ! , + # Λ2N P. (2.33).

(25) CHAPTER 2. THE STANDARD MODEL AND ITS LIMITATIONS. where the quadratic term on ΛN P cannot appear because it would spoil gauge symmetry. In the case of the Higgs boson, there is no symmetry that forbids the appeareance of quadratic terms in ΛN P , so the correction will have the form. δm2h ∼ ... + #Λ2N P + ... ,. (2.34). which shows that the quadratic term pushes the Higgs mass towards ΛN P . By means of this argument, we can see that the Higgs mass grows quadratically with any mass scale that represents new physics, and a high degree of fine-tuning is needed if we want it to be of a few hundred GeV, contrary to what happens to fermions or gauge bosons. The problem posed by this gap between the electoweak scale (∼ 100 GeV) and the scale of new physics is known as the hierarchy problem. In many textbooks, a cutoff scale is used in place of ΛN P , something that would not apply if we choose dimensional regularization for loop integrals. However, despite the fact that dimensional regularization gets rid of the quadratic divergence provided by the cutoff regulator, this quadratic divergence of the Higgs mass could enter, for instance, by means of the mass of a new particle that couples directly or indirectly to the Higgs [6]. Some proposals that could help to solve the hierarchy problem are [6]. • Supersymmetry, whose solution will be sketched in subsection 3.1.2. • Technicolor, which prescinds of the Higgs boson and postulates that the chiral symmetries of the fermions are broken by a new gauge interaction. • Composite Higgs, a model in which the Higgs is not elementary, and it dissolves at some energy scale. • Extra dimensions in which the (4 + d)-dimensional Planck scale is reduced to the weak scale under some assumptions.. The first is the only one that will be adressed in this work. 14.

(26) 2.2. GOING BEYOND THE STANDARD MODEL. 2.2.2. Neutrino masses and oscillations. Back in the time when the SM was being developed, the neutrino was thought to be a massless particle. It was hypothesized by Pauli in 1930 to preserve energy and angular momentum in beta decay, and finally detected in 1956. However, it was not until 1998 that neutrino oscillation experiments detected flavour change in atmospheric neutrinos [7], which implies that they have a non-zero mass. We denote the weak eigenstate neutrinos as να , where α = e, µ, τ , whilst the mass eigenstates are labeled as νi , with i = 1, 3, 4. W boson decays will always result in a charged lepton plus a neutrino νi , so that in the process W + → `+ α + ν, the neutrino corresponds to. |νLα i =. X. ∗ Uαi |νLi i.. (2.35). i. In other words, an α flavoured neutrino is defined as the neutrino produced together with the charged lepton α. The matrix U is known as the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix. As the mass eigenstates have a well-defined mass, the time evolution of the state is given by. |νLα (t)i =. X. ∗ −iEi t Uαi e |νLi i,. (2.36). i. so the probability of finding a neutrino of flavor β at time t is. P (να → νβ ) =. X. ∗ ∗ −i(Ei −Ej )t Uαi Uβi Uαj Uβj e .. (2.37). i,j. Now, we use the well-known dispersion relation of a relativistic particle. E=. q. p2 + m2 ,. (2.38). from which it follows that. Ei t − Ej t ≈ L 15. m2i − m2j 2E. (2.39).

(27) CHAPTER 2. THE STANDARD MODEL AND ITS LIMITATIONS. under the (fairly criticized) approximation pi = pj = E and t = L, where L is the distance traveled by the neutrino. With this result and by manipulating (2.37), we find. P (να → νβ ) = δαβ − 4. X. ∗ ∗ <(Uαi Uβi Uαj Uβj ). sin. ∆m2ij L 4E. 2. i>j. +2. X. ∗ ∗ =(Uαi Uβi Uαj Uβj ). sin. i>j. ∆m2ij L 2E. !. !. (2.40). for the probability to find a β flavor neutrino. This shows that neutrino oscillations depend fundamentally of the squared mass difference of the neutrinos, the distance that they travel, and their energy. From this formula, it can be seen that a non-zero probability for neutrino oscillations implies that they must have mass. The PMNS matrix is usually parametrized as [8] . c12 c13. s12 c13.   iδ U = −s12 c23 − c12 s23 s13 e . s12 c23 − c12 c23 s13 eiδ. c12 c23 − s12 s23 s13 eiδ −c12 s23 − s12 c23 s13 eiδ. . s13 e−iδ  . s23 c13  , c23 c13. (2.41). . where cij ≡ cos θij , sij ≡ sin θij and δ is a CP-violating phase. The mixing angles can be defined via. |Ue2 |2 , 1 − |Ue3 |2 |Uµ3 |2 = , 1 − |Ue3 |2. sin2 θ12 =. (2.42a). sin2 θ23. (2.42b). sin2 θ13 = |Ue3 |2 .. (2.42c). There are three kind of neutrinos: solar, atmospheric and reactor. However, we will refer in more detail to two kinds of major interest. As the name suggests, solar neutrinos come from the sun. They are generated in nuclear reactions that lead to. 4p →4 He + 2e+ + 2νe . 16. (2.43).

(28) 2.2. GOING BEYOND THE STANDARD MODEL. The flux of these neutrinos was first calculated by J. Bahcall in 1964 [9]. However, detectors measured roughly a third of the theoretical flux [10]. This was one of the first hints of neutrino oscillations, and it is known as the solar neutrino problem. In 2001, for the first time, the SNO experiment [11] measured data that required neutrino oscillations among three generations, and the results for the flux of active neutrinos was in close agreement with solar models. On the other hand, atmospheric neutrinos come from the interaction of cosmic rays and the earth atmosphere, and they are mostly produced in the following process:. π + −→ µ+ + νµ µ+ −→ e+ + νe + ν µ .. (2.44). From this, we should expect the following ratio for neutrinos that hit the detector on the ground. N (νµ + ν µ ) ∼ 2, N (νe + ν e ). (2.45). which is, indeed, what more detailed calculations indicate [12]. However, when the flux was experimentally measured, it turned out to be nearer to 1/2. Fortunately, the earth provides a good environment for the study of atmospheric neutrinos, as the distance L that neutrinos travel can be controlled by studying particles that arrive from different points of the earth. Reactor neutrinos refer to the ones that are generated in nuclear reactions that occur in nuclear power plants. They are the main source of human-generated neutrinos. But, how are these facts related to the famous solar and atmospheric angles and mass differences? One way of seeing this is considering the following inequality:. |∆m212 | (∼ 10−4 eV 2 ) < |∆m213 | (∼ 10−3 eV 2 ).. (2.46). In the case when E/L ∼ |∆m213 |, which is the case of atmospheric and reactor neutrinos, the contributions of ∆m212 and ∆m223 can be neglected in eq. (2.40), which results in the 17.

(29) CHAPTER 2. THE STANDARD MODEL AND ITS LIMITATIONS. formula [13] !. 2. P (νe → νe ) ≈ sin 2θ13 sin. 2. ∆m213 L , 4E. (2.47). in such a way that the value of θ13 can be found with the amplitude of the oscillation. This angle turns out to be small, and under the approximation of small θ13 , we have !. 2. P (νµ → ντ ) ≈ sin 2θ23 sin. 2. ∆m213 L , 4E. (2.48). which can be used to measure θ23 . Therefore, atmospheric and reactor neutrino experiments are used to determine ∆m2atm ≡ ∆m213 , θatm ≡ θ23 , and θreac ≡ θ13 , the reactor angle. In the case of E/L ∼ |∆m212 |, we have ". P (νe → νe ) ≈ c413 1 − sin2 2θ12 sin2. ∆m212 L 4E. !#. + s413 ,. (2.49). which allows to find θ12 with the amplitude of the oscillation provided that θ13 is close to zero. Consequently, solar neutrino experiments allow to measure ∆m2solar ≡ ∆m212 and θsolar ≡ θ12 . Extensions of the SM that describe neutrino masses and mixings are numerous. One of the most compelling options nowadays relies on the seesaw mechanism. If neutrinos are Majorana particles, the smallness of their mass could hint that they arise from physics of a new scale that supresses their values. Within this logic, it would be promising to study BSM physics that conduces to the Weinberg operator [14]. Depending on how this operator is generated at higher energies, three cases are distinguished [13]: • Type-I seesaw, which consists of the SM and heavy singlet fermions, • Type-II seesaw, or SM plus a heavy scalar triplet, • Type-III seesaw, or SM plus a heavy fermion triplets. Supersymmetry is much more non-minimal than these extensions. So, we won’t discuss them in this work, where the neutrino mass mechanism arises in the SUSY context. In 18.

(30) 2.2. GOING BEYOND THE STANDARD MODEL. Parameter. 3σ range. ∆m221 [10−5 eV2 ]. 7.05 – 8.14. |∆m231 | [10−3 eV2 ]. 2.43 – 2.67. sin2 θ12 /10−1. 2.73 – 3.79. sin2 θ23 /10−1. 3.84 – 6.35. sin2 θ13 /10−2. 1.89 – 2.39. Table 2.2: Neutrino parameters that will comprise the phenomenological constraints for the results of this work. The values were taken from [15].. any case, further explanations can be found in [13], [17]. Some of the latest values for neutrino parameters appear in table 2.2, and the model that is discussed in chapter 4 has to satisfy these constraints. Note that the values in table 2.2 assume normal ordering (NO) of neutrino masses, which means m3 > m2 > m1 .. 2.2.3. Dark Matter. During the last decades, cosmological observations have given rise to some puzzles that cast doubt over our knowledge in particle physics and gravitation. Here, we briefly discuss the case of dark matter. We follow the references [8, 16].. Figure 2.1: Composition of the universe.. The widely accepted ΛCDM or standard model of Big Bang cosmology hypothesizes that the matter that we are used to (baryonic matter) constitutes only a 5% of the universe. The remaining 95% is separated in dark matter (DM) and dark energy (26% and 69% 19.

(31) CHAPTER 2. THE STANDARD MODEL AND ITS LIMITATIONS. respectively). The Λ is associated to the latter, while CDM stands for cold dark matter. Dark matter is presumably made of unknown particles that are invisible to light (hence the name) and move slowly compared to the speed of light (cold), and it has never been detected. However, there is consensus that some effects in astronomical observations are indisputable proofs of its existence. The first evidence for dark matter was found in the 1930’s, when it was observed that various celestial bodies were moving faster than predicted by the laws of gravity applied to the surrounding visible objects. The 1970’s brought the confirmation that this effect is exhibited by every galaxy, which suggests that some sort of invisible (or dark) matter is imparting the additional gravitational pull needed to explain the observations. The study of gravitational lensing in the Milky Way and other galaxies has also shown that visible matter is insufficient to explain the deflection of the light that comes from distant sources, and suggests that galaxies have to be more massive than they seem. The cosmic microwave background (CMB) also provides a strong evidence for the existence of DM, and in fact, effects related to its anisotropy can be properly fitted with ΛCDM if the percentages mentioned above are assumed for each constituent. There are many possible explanations for these observations, and hypothetical particles refered to as dark matter candidates have been postulated since the emergence of this mystery. Until some decades ago, massive compact halo objects (MACHOs) such as neutron stars, white dwarfs, black holes, among other faint objects, were the natural answer to the DM enigma. However, in recent years, estimations have discarded the possibility that these objects represent a major fraction of the dark matter in the universe. On the other hand, particle physics has a wide variety of non-baryonic DM candidates, such as the axion and sterile neutrinos, among many others. Weakly interacting massive particles (WIMPs) are a broad class of DM candidates that appear in extensions of the SM, and they solve the problem under certain assumptions about their mass. Examples are the lightest supersymmetric particle (LSP) in the MSSM, or additional Higgs boson states in two Higgs doublet models. There are many options for the LSP, including the gravitino and the lightest neutralino, the latter being the most likely option. When R-parity is broken (section 3.7), the gravitino becomes a good candidate for cold. 20.

(32) 2.2. GOING BEYOND THE STANDARD MODEL. dark matter. Even if its stability is not protected by R-parity, the corresponding decay rates are supressed by the Planck scale and the small R-parity violating couplings in such a way that it could be stable over cosmological times [18]. Furthermore, gravitino decays produce γ-ray lines that have been contrasted with Fermi-LAT data in order to find bound for the gravitino mass [19]. One of the motivations for the model that will be explained in chapter 4 is related to gravitino dark matter and cosmological observations.. 2.2.4. Lepton Flavour Violation. Unexplained phenomena such as the ones that we have just reviewed are always perplexing. But it is also interesting to discuss what could be observed in the case that some theoretical scenario turns out to be true. Hence, we make a brief exposition about LFV and its relation with neutrino masses, and why is it important to look for signals of LFV in order to probe new physics, closely following a review on the subject [20]. Leptons come in three generations, distinguished by the flavour quantum number. This quantum number is assigned as. Leptons =.     (e−    . −. (µ       (τ −. νe ). Le = +1. νµ ). Lµ = +1. ντ ). Lτ = +1. ,. and the same with opposite sign for antiparticles. Clearly, the electronic number of a muon is zero and so on for the other families. A reasonable question arises: is flavour a good quantum number for leptons? Or in other words, is flavour quantum number conserved for SM leptons? Recalling our disgression about the CKM matrix, we might guess that the coupling with the W boson could be a problem. 1 L``w = √ gw eαL σ µ Wµ− ναL . 2. (2.50). Let’s say that leptons and neutrino eigenstates are related to the weak basis by means of 21.

(33) CHAPTER 2. THE STANDARD MODEL AND ITS LIMITATIONS. unitary matrices V` and Vν :. ναL =. 3 X. (Vν )αi νiL ,. eαL =. i=1. 3 X. (V` )αi eiL .. (2.51). i=1. In the mass eigenstate basis, this is 1 L``w = √ gw (V`† Vν )ij eiL σ µ Wµ− νjL . 2. (2.52). In the SM, however, neutrinos are massless and we can rotate them in family space so that V` = Vν . As a consequence of this, we can define new flavour numbers for the eigenstates in such a way that this coupling will conserve lepton flavour number. In other words, there’s no LFV in the SM. However, neutrino oscillations experiments show that neutrinos are massive, and thus we know that the SM has to be extended for, at least, accounting for these masses. In theories with massive neutrinos, (V`† Vν ) ≡ UP M N S ,. (2.53). so the PMNS matrix is a responsible for LFV in SM extensions. Now, if we think about new processes in such theories, the following diagram comes to mind:. γ. W− µ− Uµ1. W− ν1. ∗ Ue1. e− , (2.54). which shows that any theory that describes neutrino masses necessarily has LFV at loop level. In other words, neutrino masses imply LFV, and that’s why LFV is important to probe physics beyond the SM. Today, we know that neutrinos are massive and hence some LFV processes are expected to occur. However, not every single extension of the SM will result in interesting LFV effects 22.

(34) 2.2. GOING BEYOND THE STANDARD MODEL. Process. Bound. Experiment. BR(µ → eγ). < 4.2 × 10−13. MEG at PSI (2016). BR(τ → eγ). < 3.3 × 10−8. BaBar (2010). BR(τ → µγ). < 4.4 × 10−8. BaBar (2010). BR(h → eµ). < 3.5 × 10−4. LHC CMS (2016). BR(h → eτ ). < 6.1 × 10−3. LHC CMS (2017). BR(h → µτ ). < 2.5 × 10−3. LHC CMS (2017). Table 2.3: Bounds for some LFV processes. In this work, we intend to study the process h → eµ. The values were taken from [20].. from a phenomenological point of view. Consider, for instance, a minimal extension of the SM that adds right handed neutrinos which interact with the Higgs via the following Yukawa coupling:. Lmass = −Yijν Li φ̃ νjR + h.c. (2.55). A detailed calculation shows that in this model, the branching ratio BR(µ → eγ) is given by [21] 3 ∗ m2 Uµk Uek 3α X νk BR(µ → eγ) = 2 32π k=1 MW. 2. ∼ 10−55 .. It would be rather euphemistic to say that this number is beyond the reach of our current detectors, and it shows that neutrino masses alone don’t necessarily imply observable LFV. As a consequence, LFV signals in particle accelerators would indicate that new physics are non-minimal extensions of the SM. Recent experimental bounds for some observables can be seen in table 2.3. Even if neutrino masses result in LFV through loop diagrams that involve the W boson, they are not the sole culprit of this effect. In fact, LFV in the Higgs sector may appear due to a misalignment between the coupling to the Higgs boson (or any other scalar, for that matter) and the charged lepton mass matrix. A general Lagrangian for describing this effect is [20] 23.

(35) CHAPTER 2. THE STANDARD MODEL AND ITS LIMITATIONS. 1 LHLF V = −MiD eiL eiR − √ Yije h eiL ejR + h.c. . 2 If we assume that M D is a diagonal matrix, it is evident that the second term will generate an unavoidable family mixing because the charged leptons that appear in the Lagrangian are mass eigenstates. A typical scenario for this misalignment is a model with two Higgs doublets, whose lepton Yukawa Lagrangian is given by. (1). (2). L = −Yij Li φ1 ejR − Yij Li φ2 ejR + h.c. .. (2.56). In this model, the charged lepton mass matrix is given by i 1 h Mij = √ Y (1) v1 + Y (2) v2 , ij 2. (2.57). while the couplings of charged leptons with Higgses are. (1). (2). L = −Yij h1 eiL ejR − Yij h2 eiL ejR + h.c. ,. (2.58). from which LFV follows due to the fact that diagonalizing the mass matrix generally leads to non-diagonal couplings in the mass eigenstate basis.. 24.

(36) Chapter 3. The Minimal Supersymmetric Standard Model 3.1. Supersymmetry. Supersymmetry is a hipothetical space-time symmetry that implies the existence of a boson for every fermion and viceversa. The smallest supersymmetric extension of the SM is known as the Minimal Supersymmetric Standard Model, which is the main subject of this section. Roughly speaking, this symmetry enforces the existence of a fermion for every SM boson and a boson for every SM fermion, and SUSY transformations exchange one in the other. These particles are said to be superpartners of each other, and new bosonic particles have the same names as their fermion superpartners, but with an s- at the beginning (slepton, squark), while -ino at the end is added for fermionic superpartners (gaugino, Higgsino). Let’s begin by discussing the algebra of this space-time symmetry which is, by the way, an exception to an important no-go theorem.. 3.1.1. Super Poincaré Group. The Poincaré group was discussed in subsection 2.1.1. Combining the symmetry under this group with an internal symmetry group, we managed to show how the SM is built. We could ask ourselves: are there more ways to combine space-time and internal symmetries? 25.

(37) CHAPTER 3. THE MINIMAL SUPERSYMMETRIC STANDARD MODEL. Sidney Coleman and Jeffrey Mandula showed that if G is the symmetry group of the S matrix, under reasonable assumptions,. ‘G is locally isomorphic to the direct product of the Poincaré group and an internal symmetry group’ [22], or as rephrased in [23], ‘space-time and internal symmetries cannot be combined in any but a trivial way’.. However, Haag, Lopuszański and Sohnius, among others, later proved that this theorem can be bypassed by allowing anticommuting symmetry generators [24], and this is how the SUSY algebra was born. As for this thesis we’ve just scratched the surface when it comes to group theory, let’s simply state the main points. The algebra of Supersymmetry is Poincaré (eqs. (2.1a) to (2.1c)) plus. [M µν , QA ] =. 1 µν B (σ )A QB 2. (3.1a). [QA , P µ ] = QȦ , P µ = 0. h. i. (3.1b). h. i. (3.1c). {QA , QḂ } = 2 σAḂ Pµ ,. (3.1d). [QA , QB ] = QȦ , QḂ = 0. where Q and Q are the SUSY generators, or supercharges, and σ µν is defined in appendix A. This is the super-Poincaré group. Note that we only introduced one supercharge and its complex conjugate, which is known as the N = 1 case, and is the case of interest for this work. It can be shown that, as [Q, P ] = 0, representations of the super-Poincaré group fall into multiplets of equal mass but different spin. These multiplets, which contain the same number of fermionic and bosonic degrees of freedom (d.o.f.), are known as supermultiplets. Furthermore, because the SUSY generators commute with gauge transformations, particles within the same supermultiplet transform under the same representation of the gauge groups. The massless representations of N = 1 SUSY are separated in a chiral supermultiplet, which contains a complex scalar and a Weyl spinor; and a vector supermultiplet, which 26.

(38) 3.1. SUPERSYMMETRY. contains a Weyl spinor (gaugino) plus a gauge field. Each one of these supermultiplets has 2 bosonic d.o.f. and 2 fermionic d.o.f., and for N = 1, they are the only representations that involve states with spin ≤ 1. In section 3.2, a prescription for constructing such theories will be explained. Note that if, for instance, the superpartner of the electron was a scalar with exactly the same mass as the electron (511 keV), it should have been discovered by now, which indicates that supersymmetry has to be broken. There are multiple ways to break SUSY, but just one will be mentioned when we discuss the MSSM.. 3.1.2. Hierarchy problem revisited. In the first chapter, we got a glimpse of the hierarchy problem and how quantum corrections can be a hassle when it comes to their contributions to the mass of scalars. This was explained by the fact that, somehow, fermion masses and vector boson masses are both ‘protected’ by symmetries, which isn’t the case of the Higgs mass, that is very sensitive to new physics scales. However, we have just talked about supersymmetry and some of its interesting features. Could this symmetry be a miraculous cure to the illness that seems to affect the Higgs boson mass? Indeed, the existence of a boson for every fermion guarantees that the bad behavior of loop contributions to scalar masses will be tamed by loop contributions coming from superpartners. This can be seen in the diagrams of (3.2).. f˜. f S. S (3.2) δm2S = −. |λf |2 2 Λ + ... 8π 2. δm2S =. λf˜ 16π 2. Λ2 + .... If Λ is any parameter of the theory with units of mass (in most discussions, a cutoff scale), in the limit of preserved SUSY, we have that |λf |2 = λf˜, and there is an exact cancellation of the quadratic divergence in the diagrams. However, we mentioned that if SUSY is 27.

(39) CHAPTER 3. THE MINIMAL SUPERSYMMETRIC STANDARD MODEL. realised in nature, it must be broken. In that case, we have λ δm2S = m2sof t ln 16π 2. Λ msof t. !. + ... ,. (3.3). where msof t is a mass parameter which explicitly violates SUSY. This renders the Higgs mass much more natural provided that msof t lies in the electroweak scale. The absence of superpartners in this energy scale has sparked some controversy around what is known as the little hierarchy problem [25]. These discussions, however, go far beyond the reach of what we intend to analyze in this work.. 3.2. Superspace Formalism. In the same way that we made an extension from the Poincaré group to super-Poincaré, it seems rather natural to extend spacetime, a capital concept in special relativity, to a superspace [26]. In order to do so, we add four extra spinoral variables, so that we are now . . able to define a superspace as the space spanned by the supercoordinates xµ , θA , θȦ . These new coordinates have to be treated with the rules and conventions that apply to Grassmann variables. Some important properties are listed section 4.1 of [27]. In the following sections, we will follow ref [27] very closely.. 3.2.1. General superfield. We define a superfield as a function Φ(z) ≡ Φ(xµ , θ, θ) that is defined over the superspace. It is not difficult to check that the definitions. . QA = − i ∂A + Ȧ. . Ḃ µ iσA θ ∂µ Ḃ. . ,. (3.4a). . Ȧ. Q = − i ∂ + iσ µȦB θB ∂µ ,. (3.4b). Pµ = i ∂µ. (3.4c). provide a realization of the SUSY algebra (eqs. (3.1a) to (3.1d)) acting on the superfields. If we consider an infinitesimal SUSY transformation acting on a superfield Φ(z) that is parametrized by the infinitesimal Grassmann variables and , we obtain 28.

(40) 3.2. SUPERSPACE FORMALISM. δΦ(xµ , θ, θ) = i(Q + Q)Φ(xµ , θ, θ) = Φ(xµ − iθσ µ + iσ µ θ, θ + , θ + ) − Φ(xµ , θ, θ).. (3.5). In the last equality, a first order Taylor expansion in and was made. We can see that a SUSY transformation acting on superfunctions correspond to a traslation in superspace.. Now, we are going to consider a general superfield. Our goal is to find a field theoretic realization of the super-Poincaré algebra, so this general superfield has to include all kind of fields that will appear in the SUSY invariant Lagrangian. With the help of Weyl spinor identities, we can prove that the most general superfield can be written as follows:. S(z) = s(x) +. √. 2θξ(x) +. √. 2 θχ(x) + θθM (x) + θθN (x) + θσ µ θAµ (x) + θθθλ(x). 1 + θθθζ(x) + θθθθD(x). 2. (3.6). The field content (or component fields) of the general superfield is given by four scalars s(x), M (x), N (x), D(x), two left handed Weyl spinors ξ(x), ζ(x), two right handed Weyl spinors χ(x), λ(x), and a single vector field Aµ (x). This makes a total of 16 bosonic and 16 fermionic real fields. Furthermore, it is clear that the product of superfields will be general superfields.. If we want to find the infinitesimal SUSY transformation for the component fields, we need to compute S + δS and rearrange the terms in order to attain the form of the general superfield of eq. (3.6). Using the equations (3.4b) and (3.4c), then. δS = i(Q + Q)S 29. (3.7).

(41) CHAPTER 3. THE MINIMAL SUPERSYMMETRIC STANDARD MODEL. will yield (after some algebra) the transformations. √ √ δs = 2ξ + 2χ. (3.8a). √. δ( 2ξA ) = 2A M + (σ µ )A (−i∂µ s + Aµ ) √ δ( 2χȦ ) = 2Ȧ N − (σ µ )Ȧ (i∂µ s + Aµ ). (3.8b) (3.8c). i δM = λ + √ ∂µ ξσ µ 2 i δN = ζ − √ σ µ ∂µ χ 2. (3.8d) (3.8e). √ √ i i δAµ = σµ λ + ζσµ − √ ∂µ ξ + √ ∂µ χ + 2σµν ∂ ν ξ − 2σ µν ∂ ν χ 2 2 i Ȧ δλ = Ȧ D − Ȧ ∂ µ Aµ − i(σ µ )Ȧ ∂µ M + (σ µν )Ȧ ∂µ Aν 2 i δζA = A D + A ∂ µ Aµ − i(σ µ )A ∂µ N + (σ µν )A ∂µ Aν 2 δD = i∂µ (ζσ µ + λσ µ ).. (3.8f) (3.8g) (3.8h) (3.8i). Eq. (3.8i) is specially important, as it shows that the coefficient of the θθθθ transforms as a total derivative. So, a D-term, as it is usually denoted, can be used to construct a SUSY invariant Lagrangian because it contributes nothing but a total derivative to the SUSY variation.. 3.2.2. Chiral covariant derivatives. As the spinorial derivatives ∂A and ∂Ȧ do not commute with the SUSY charges (3.4b) and (3.4c), we define a covariant derivative that commutes with these transformations as. Ḃ. µ DA ≡ ∂A − i σ A θ ∂µ , Ḃ Ȧ. Ȧ. D ≡ ∂ − i σ µȦB θB ∂µ ,. (3.9a) (3.9b). and it can be proved that. Ḃ. Ḃ. {DA , QB } = {DA , Q } = {DȦ , Q } = {DȦ , QB } = 0. 30. (3.10).

(42) 3.2. SUPERSPACE FORMALISM. Another important property is. DA DB DC = DȦ DḂ DĊ = 0 .. 3.2.3. (3.11). Right and left chiral superfields. As mentioned, representations of the SUSY algebra give rise to particle supermultiplets. We can study how a particular case of this property is realized in the superspace by imposing the following conditions. DȦ Φ = 0. (3.12). DA Φ† = 0.. (3.13). A superfield that satisfies eq. (3.12) (or (3.12)) receives the name of left (right) chiral superfield. In order to solve these conditions, it is useful to note that we can define y µ ≡ xµ − iθσ µ θ and make the shift. (xµ , θ, θ). −→. (y µ , θ, θ). −→. µ DA = ∂A − 2iσA θ ∂µ(y) Ḃ. −→. D. (3.14a). so that the covariant derivatives becomes. Ḃ. µ DA = ∂A − i σ A θ ∂µ Ḃ Ȧ. Ȧ. D = ∂ − i σ µȦB θB ∂µ. Ḃ. (y). Ȧ(y). Ȧ. =∂ .. (3.14b) (3.14c). Its straightforward to see that, in these new coordinates, any superfield Φ(y, θ) will satisfy (y). the condition (3.12). The same can be done for the condition (3.13) by defining y, DȦ. and DA(y) as the conjugates of the definitions above, so that a superfield Φ† (y, θ) will automatically satisfy it. The decompositions of these superfields in terms of the component 31.

(43) CHAPTER 3. THE MINIMAL SUPERSYMMETRIC STANDARD MODEL. fields are given by. Φ(y, θ) = φ(y) +. √. 2θξ(y) + θθF (y) , √ Φ† (y, θ) = φ∗ (y) + 2θξ(y) + θθF ∗ (y) .. (3.15a) (3.15b). We can return to the former set of coordinates by replacing y = x − θσθ (respectively y = x + θσθ) and making a Taylor expansion in the anticommuting variables. In (x, θ, θ) coordinates, the left and right chiral superfields are. √ 1 i Φ(x, θ, θ) = φ(x) − iθσ µ θ∂µ φ(x) − θθθθ∂ µ ∂µ φ(x) + 2θξ(x) + √ θθ∂µ ξ(x)σ µ θ 4 2 + θθF (x) ,. (3.16a). √ 1 i Φ† (x, θ, θ) = φ∗ (x) + iθσ µ θ∂µ φ∗ (x) − θθθθ∂ µ ∂µ φ∗ (x) + 2θξ(x) − √ θθθσ µ ∂µ ξ(x) 4 2 + θθF ∗ (x) .. (3.16b). Clearly, the second is the complex conjugate of the first. Furthermore, if we compare the superfield (3.16a) with the general superfield (3.6) it is easy to guess the SUSY transformations for the components of the chiral superfield by making the correct substitutions in the transformations (3.8a) to (3.8i). We get. √ δφ = 2ξ, √ √ δξA = 2A F − 2(iσ µ )A ∂µ φ, √ δF = i∂µ ( 2ξσ µ ).. (3.17a) (3.17b) (3.17c). We can note that, in accordance with the discussion about supermultiplets, this superfield has the same number of bosonic and fermionic degrees of freedom (φ(x) and F (x) make four bosonic d.o.f, while a Weyl spinor with complex components makes four fermionic d.o.f). Another important remark about the chiral superfield is that the component field that appears with the θθ term of the expansion (usually denoted as F -term) transforms as a total derivative, so it is a good candidate for a Lagrangian density. Lastly, it is 32.

(44) 3.2. SUPERSPACE FORMALISM. straightforward to see that the product of left (right) chiral superfields will also be a left (right) chiral superfield. This means that we can always construct a SUSY invariant Lagrangian density by taking the F -term of a polinomial of left chiral or right chiral superfields.. 3.2.4. Vector superfields. A vector superfield satisfies the constrain V = V † . By applying this condition to the general superfield (3.6), we obtain. V (x, θ, θ) = s(x) +. √. 2θξ(x) +. √. 2θξ(x) + θθM (x) + θθM ∗ (x) + θσ µ θAµ (x) + θθθλ(x). 1 + θθθλ(x) + θθθθD(x), 2. (3.18). where s(x), Aµ and D(x) are real fields. However, in order to define a supergauge transformation, it is convenient to recast this expression as. V (x, θ, θ) = s(x) +. √. 2θξ(x) +. √. 2θξ(x) + θθM (x) + θθM ∗ (x) + θσ µ θAµ (x). i i + θθθ λ(x) − √ σ µ ∂µ ξ(x) + θθθ λ(x) − √ σ µ ∂µ ξ(x) 2 2 1 1 + θθθθ D(x) − ∂ µ ∂µ s(x) , 2 2 . . . . (3.19). the supergauge transformation is defined as V → V + i(Λ − Λ† ), where Λ is a chiral superfield. We can write this in a general way as. iΛ − iΛ† = 2<φ(x) +. √. 2θχ(x) +. √. 2θχ(x) + θθF (x) + θθF ∗ (x) − 2θσ µ θ∂µ =φ(x). i i 1 − √ θθθσ µ ∂µ χ(x) − √ θθθσ µ ∂µ χ(x) − θθθθ∂ µ ∂µ <φ(x). 2 2 2 33. (3.20).

(45) CHAPTER 3. THE MINIMAL SUPERSYMMETRIC STANDARD MODEL. Making the supergauge transformation, we can see that the component field transform as. s → s + 2<φ. (3.21a). ξ →ξ+χ. (3.21b). M →M +F. (3.21c). Aµ → Aµ − 2∂µ =φ. (3.21d). λ→λ. (3.21e). D→D. (3.21f). 1 1 D − ∂ µ ∂µ s → D − ∂ µ ∂µ s. 2 2. (3.21g). The last transformation rule can be trivially derived from the previous ones, but it is important to stress the fact that the D-term of a vector superfield is invariant under supergauge transformations, as can be read from this rule. Now, the main observation is about to be made. As the name suggests, the vector superfield should describe the gauge boson with its superpartner (the gaugino), and the general idea is to find a gauge such that the vector superfield indeed consists only of a vector field, its superpartner, and an auxiliary field. This is called the Wess-Zumino gauge, and it can be fixed by choosing s = −2<φ, ξ = −χ, and F = −M . The imaginary part of φ remains free, and this consists of the everyday U (1) gauge freedom. In this gauge, the vector superfield can be written as. 1 VW Z = θσ µ θAµ (x) + θθθλ(x) + θθθλ(x) + θθθθD(x). 2. (3.22). This could be seen just as a particular choice of gauge with no use when it comes to build a SUSY invariant action, as the constraints imposed over the component fields of Λ are not SUSY invariant, but as the D-term of a vector superfield is both SUSY and supergauge invariant, it can be calculated in the WZ gauge and the result will be the same as if it was calculated in any other gauge. However, what is the big deal about calculating things in this particular gauge? Because of the Grassmann nature of the supercoordinates, any higher-than-three power of 34.

(46) 3.2. SUPERSPACE FORMALISM. VW Z will vanish. More explicitly, the expression 1 2 µ VW Z = θθθθA Aµ , 2. (3.23). confirms this claim. This property will be specially useful for calculating exponentials of VW Z in later sections, as the power series truncates at second order. For describing the dynamics of the vector fields, we need to define a chiral field-strength superfield. We define it as. 1 WA = − DDDA V 4 1 W Ȧ = − DDDȦ V, 4. (3.24) (3.25). which can be shown to be gauge invariant considering the fact that the Λ (or Λ† ) that defines the supergauge transformation is a left (or right) chiral superfield, and thus satisfies DȦ Λ = 0 (or, respectively, DA Λ† = 0). Furthermore, as the product of three chiral covariant derivatives is zero, WA and W Ȧ are respectively left and right chiral superfields. After some algebra, it can be shown that. 1 Ȧ W A WA + W Ȧ W 4 . F. 1 1 i i = D2 − Fµν F µν + λσ µ ∂µ λ − ∂µ λσ µ λ, 2 4 2 2. (3.26). which is the kinetic term for the gauge bosons and the gauginos, plus a term that involves the auxiliary fields D (which can be ”eliminated” by means of the equations of motion). The F subscript indicates that we are taking the F -term. We just mentioned how to construct a kinetic term for the gauge bosons and the superpartners. What about the kinetic terms of the fields that appear in the chiral supermultiplet? This question can be answered by analizing the vector superfield Φ†i Φj . Its D-term (denoted by the D subscript) is given by. h. Φ†i Φj. i D. 1 1 1 i i = Fi∗ Fj + ∂µ φ∗i ∂ µ φj − ∂µ ∂ µ φ∗i φj − φ∗i ∂µ ∂ µ φj + ξj σ µ ∂µ ξ i − ∂µ ξj σ µ ξ i . (3.27) 2 4 4 2 2 35.

(47) CHAPTER 3. THE MINIMAL SUPERSYMMETRIC STANDARD MODEL. Clearly, this term contains the kinetic terms of the particles that appear in the chiral supermultiplet. A closer look leads to the conclusion that, except for a total derivative, we have just found what is known as the Wess-Zumino Lagrangian density, which is the simplest field realization of SUSY.. 3.2.5. Formalism for Abelian gauge theories. We already defined a supergauge transformation acting on a vector superfield. Moreover, we also managed to construct a supergauge invariant field-strength superfield. Now, it is time to define a supergauge invariant kinetic term for the particles of the chiral supermultiplet. We define a U (1) supergauge transformation acting over left and right chiral superfields as. 0. Φi = e−2igti Λ Φi , 0. (3.28). †. Φi† = e2igti Λ Φ†i ,. (3.29). where Λ and Λ† are left and right chiral superfields respectively, g is the gauge coupling constant, and ti is the U (1) charge of the field. The gauge invariant kinetic term is [Φ†i e2gti V Φi ]D , where V is a vector superfield that transforms as. 0. V = V + i(Λ − Λ† ).. (3.30). The kinetic term is usually calculated in the Wess-Zumino gauge for simplicity. With all these considerations, the Lagrangian for the Abelian gauge theory is 1 Ȧ L= W A WA + W Ȧ W 4 . F. h. + Φ†i e2gti V Φi + ηV. i D. + [W(Φi ) + h.c.]F ,. (3.31). where W(Φi ) is the superpotential of the theory, which can be introduced by the fact that supersymmetry allows any polynomial of left or right chiral superfields to be added to the Lagrangian. Now, we will show the non-Abelian generalization of this construction with the general component form for the Lagrangian. 36.

(48) 3.2. SUPERSPACE FORMALISM. 3.2.6. Non-Abelian case. For this case, we consider that our superfields Φ are arranged in multiplets that transform under a representation R of a simple gauge group G with coupling constant g. Let’s consider a nonabelian gauge transformation acting on the superfields Φ and Φ† :. aT a. Φ0i = (e−2igΛ. ) ij Φ j. a†T a. 0. Φ†i = Φ†j (e2igΛ. )j i. where Λa (z) and Λa† (z) are left and right chiral superfunctions that determine the gauge transformations and the T a are the generators of the group’s algebra in the representation R. They satisfy. [T a .T b ] = if abc T c. ;. Tr[T a T b ] = κ δ ab. (3.32). where f abc are the structure constants of the algebra and κ is the representation constant of R. For simplicity, we define. Vi j = 2gT ai j V a. ;. Λij = 2gT ai j Λa .. (3.33). Where V a are vector superfields that transform in the adjoint representation of the group. Now, the generalized supergauge transformation is. 0. †. eV = e−iΛ eV eiΛ 0. e−V = e−iΛ eV eiΛ. †. (3.34) (3.35). This transformation rule implies that Φ†i (eV )ij Φj is gauge invariant. Furthermore, using the Baker-Campbell-Haussdorf formula and ignoring second order terms in Ω, Ω† , we have. 0. eV = eV +i(Λ−Λ. † )+ i [V,Λ+Λ† ]+ i [V,[V,Λ−Λ† ]]+... 2 12. 37. (3.36).

(49) CHAPTER 3. THE MINIMAL SUPERSYMMETRIC STANDARD MODEL. We can read the infinitesimal transformation for V :. i i δV ≡ V 0 − V = i(Λ − Λ† ) + [V, Λ + Λ† ] + [V, [V, Λ − Λ† ]] + ... 2 12. (3.37). or. i a V 0 − V a = i(Λa − Λa† ) + gf abc V b (Λc − Λc† ) − g 2 + f abc f cde V b V d (Λe − Λe† )... (3.38) 3 This means that we can work in the Wess-Zumino gauge given that we choose the right a. Λa − Λ† . As in the abelian case, we now define the field-strength chiral superfield:. 1 WA = − DDe−V DA eV , 4 1 Ȧ Ȧ W = − DDeV D e−V . 4. (3.39) (3.40). It’s possible to verify that this superfields transform as below:. WA 0 = e−iΛ WA eiΛ , W. Ȧ. 0. Ȧ. = eiΛ W e−iΛ .. (3.41) (3.42). We see that the trace of the product of two W ’s is a good candidate for a gauge invariant term. Let’s calculate it using the WZ gauge. We begin with the expansion. 1 1 e−V DA eV = DA V − [V, DA V ] + [V, [V, DA V ]] + ... 2 6. (3.43). Defining WA = 2gT a WAa , we have. 1 a abc b c WAa = − DD DA VW (DA VW Z + igf Z )VW Z . 4. We only keep terms up to second order in VW Z . The field strength superfield reads 38. (3.44).