Deviations of the leptonic branching ratios of the heavy Higgs in the MSSM with broken R parity

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(1)Department of Physics, Institute of Physics and Astronomy.. Deviations of the Leptonic Branching Ratios of the heavy Higgs in the MSSM with broken R-parity por ANDRÉS IGNACIO MÉNDEZ LEIVA. Thesis submitted to the Department of Physics of the Pontifica Universidad Católica in partial fulfillment of the requirements for the degree of MSc. in Physics.. Thesis Committee: Prof. Marco Aurelio Diaz Gutierrez (supervisor) Prof. Benjamin Koch Prof. Juan Pedro Ochoa Ricoux. November, 2018 Santiago, Chile. c 2018, Andrés Ignacio Méndez Leiva.

(2) c 2018, Andrés Ignacio Méndez Leiva. Se autoriza la reproducción total o parcial, con fines académicos, por cualquier medio o procedimiento, incluyendo la cita bibliográfica del documento..

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(4) ABSTRACT. We performed a phenomenological study of the leptonic decay modes of a heavy Higgs in a supersymmetric scenario which we called the λ-MSSM, in which we added -bilinear and λ-trilinear terms to break R-parity explicitly, constrained by the most recent neutrino experimental data. In particular, we explored the possibility of decreasing the coupling associated to the τ τ mode, in order to reduce the large number of events predicted by other typical R-parity conserving (RpC) models to justify the non-observation of any heavy scalar resonance, while opening regions of the parameter space which were previously excluded. We also study the possibility of enhancing the µµ mode due to its importance in future experimental searches. We found that even with a loose cosmological bound given by the sum of the neutrino masses, the deviations for the τ τ mode are smaller than a 1%, while for the µµ mode it can be up to 50%. However, once we constrained our parameters to reproduce the most recent neutrino observables associated with neutrino oscillation, we found that the deviations attained in each mode are extremely small, leading to deviations of ∼ 0.0001% for the µµ mode and ∼ 0.0000000001% for the τ τ model, making the λ-MSSM indistinguishable from the RpC-MSSM case in the leptonic channels.. ii.

(5) Contents 1 INTRODUCTION. 1. 2 THE STANDARD MODEL AND BEYOND. 4. 2.1. The Poincaré Group . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4. 2.2. The Standard Model Lagrangian . . . . . . . . . . . . . . . . . . . . . .. 7. 2.2.1. Gauge Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 8. 2.2.2. Matter Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. 2.2.3. Higgs Sector and Spontaneous Symmetry Breaking. 2.3. . . . . . . . . . 11. Beyond the Standard Model . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3.1. Neutrino Oscillations . . . . . . . . . . . . . . . . . . . . . . . . 14. . . . . . . . . . . . . . . . . . . . 18. 2.3.1.1. Solar Neutrino Problem. 2.3.1.2. Atmospheric Neutrino Problem. . . . . . . . . . . . . . . . 19. 2.3.2. The Hierarchy Problem . . . . . . . . . . . . . . . . . . . . . . . 20. 2.3.3. Dark Matter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25. 3 INTRODUCTION TO SUPERSYMMETRY. 27. 3.1. The Super Poincaré Algebra . . . . . . . . . . . . . . . . . . . . . . . . 28. 3.2. Superspace Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.2.1. Chiral Superfields . . . . . . . . . . . . . . . . . . . . . . . . . . 33. 3.2.2. Vector Superfield . . . . . . . . . . . . . . . . . . . . . . . . . . 36. 3.3. Invariant chiral interactions . . . . . . . . . . . . . . . . . . . . . . . . 38. 3.4. Non abelian Gauge Theory . . . . . . . . . . . . . . . . . . . . . . . . . 39. 4 THE MINIMAL SUPERSYMMETRIC STANDARD MODEL 4.1. 42. Particle Content of the MSSM . . . . . . . . . . . . . . . . . . . . . . . 42 iii.

(6) Contents. 4.1.1. Fermionic Sector . . . . . . . . . . . . . . . . . . . . . . . . . . 43. 4.1.2. Gauge Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44. 4.1.3. Higgs Sector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45. 4.2. MSSM Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46. 4.3. R-Parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49. 4.4. MSSM Superpotential . . . . . . . . . . . . . . . . . . . . . . . . . . . 50. 4.5. Soft Supersymmetry Breaking . . . . . . . . . . . . . . . . . . . . . . . 52 4.5.1. 4.6. Electro-Weak Symmetry Breaking . . . . . . . . . . . . . . . . . . 53. Scalar Sector and the Decoupling Limit . . . . . . . . . . . . . . . . . . 56. 5 R-PARITY VIOLATION IN THE MSSM. 59. 5.1. R-parity Violation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59. 5.2. R-Parity Violation and Neutrino Masses . . . . . . . . . . . . . . . . . . 60. 5.3. 5.2.1. Tree-Level Neutrino Masses . . . . . . . . . . . . . . . . . . . . . 61. 5.2.2. One-loop Neutrino Masses . . . . . . . . . . . . . . . . . . . . . . 63. A glimpse of RpV Gravitino Dark Matter . . . . . . . . . . . . . . . . . 65. 6 THE MODEL: MSSM WITH BRpV & λ-TRpV. 67. 6.1. The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67. 6.2. Superpotential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68. 6.3. Fermion Mass Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 68. 6.4. Neutrino Physics in the MSSM with BRpV & λ-TRpV . . . . . . . . . . 73 6.4.1. 6.4.2. 7. iv. Radiative Corrections . . . . . . . . . . . . . . . . . . . . . . . . 74. . . . . . . . . . . . . . . . . . . . 74. 6.4.1.1. Bottom-Sbottom loop. 6.4.1.2. Charged Fermion-Charged Scalar loop. . . . . . . . . . . . . 75. Neutrino Observables . . . . . . . . . . . . . . . . . . . . . . . . 76. LEPTONIC DECAYS OF THE HEAVY HIGGS. 77. 7.1. Searches of an MSSM Heavy Higgs . . . . . . . . . . . . . . . . . . . . 77. 7.2. RpV Modified Leptonic Couplings . . . . . . . . . . . . . . . . . . . . . 80. 7.3. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 7.3.1. Parameter Space . . . . . . . . . . . . . . . . . . . . . . . . . . 83.

(7) v. Contents. 7.3.2. Leptonic decay modes in RpV scenario . . . . . . . . . . . . . . . . 87. 7.3.3. Experimental constrains of the parameter space . . . . . . . . . . . 89. 7.3.4. Fitting the most recent Neutrino Data . . . . . . . . . . . . . . . . 89. 7.3.5. H → µ− µ+ and H → τ − τ + revisted . . . . . . . . . . . . . . . . 97. 8 SUMMARY AND FUTURE WORK. 100. A Mass Matrices. 109. A.1 CP-even Neutral Scalars Mass Matrix: . . . . . . . . . . . . . . . . . . . 109 A.2 Charged Scalars Mass Matrix: . . . . . . . . . . . . . . . . . . . . . . . 110 A.3 Bottom Squarks Mass Matrix: . . . . . . . . . . . . . . . . . . . . . . . 111. B Relevant Couplings. 112. B.1 Neutral CP-even scalar- Chargino - Chargino . . . . . . . . . . . . . . . 112 B.2 Bottom-Neutralino-Sbottom . . . . . . . . . . . . . . . . . . . . . . . . 112 B.3 Chargino-Neutralino-Charged Scalar . . . . . . . . . . . . . . . . . . . . 113.

(8) List of Tables 2.1. Particle content of the Gauge Sector in the SM with the associated gauge groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2.2. 9. Particle content of the Matter Sector of the SM with their respective representation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. 2.3. Latest results on neutrino observables in the 3σ range according to [1]. 20. 4.1. Chiral Supermultiplet content of the MSSM with their respective particles and superpartners. The index i is a family index running from 1 to 3, such that ei = {e, µ, τ } for leptons and ui = {u, c, t} for up type quarks. The same applies for the rest of the scalar and fermionic particles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44. 4.2. Gauge Supermultiplet content of the MSSM. Vectorial notation is used for Winos and W Bosons, such that each one have 3 different components. For Gluons and Gluinos, the index a = 1, ..., 8 as usual.. 45. 4.3. Chiral Supermultiplet content of the scalar sector of the MSSM. . . . 45. 7.1. SUSY parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84. 7.2. RpV parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84. 7.3. Experimental values of neutrino observables and the predicted ones in three different benchmarks with their corresponding χ2 . . . . . . . 91. 7.4. RpV parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93. vi.

(9) List of Figures 7.1. Most relevant MSSM heavy Higgs branching ratios (left) and number of events predicted for these modes (right).. 7.2. . . . . . . . . . . . . . . 79. Expected number of events registered by ATLAS at the end of Run 2 (left) and expected number of events at the end of the HL-LH (right). 80. 7.3. Plot showing that as MA grows, the heavy CP-even Higgs H and the CP-odd scalar A become mass degenerate while the tree-level mass of the SM-like Higgs attains a maximum value MZ in the decoupling limit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85. 7.4. Plot showing that in the decoupling limit the heavy CP-even Higgs H decouples from the gauge bosons V = {W ± , Z, γ} while for the light Higgs the same couplings approach the SM values. . . . . . . . . 86. 7.5. SM-like Higgs mass dependency of Xt /Ms with the observed value of ∼125 GeV highlighted as a red line. . . . . . . . . . . . . . . . . . . . 87. 7.6. Ratios between the branching ratios in the RpC and RpV case BRRpV /BRRpC against the mass of the heavy CP-scalar mH for the µµ mode (up) and the τ τ mode (down) in the unconstrained parameter space specified in table 7.2.. 7.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88. Dependency of the deviations on the branching ratios showed as the fraction between BRRpV /BRRpC for the µµ mode (up) and the τ τ mode (down) against the tree-level mass of the heaviest neutrino mtree ν3 varying the parameters according to table 7.2. . . . . . . . . . . . . . 90. 7.8. χ2 dependence of λ121 and λ122 when the TRpV are varied independently in the range [−0.002, 0.002]. . . . . . . . . . . . . . . . . . . . 92 vii.

(10) List of Figures. 7.9. viii. Surface plot showing the χ2 dependence of λ121 and λ122 (left) and the χ2 dependence of 2 and 3 (right) while keeping the other parameters fixed around BM3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94. 7.10 Scatter plot showing the 2d projection of the minimum of figure 7.9 in the χ2 -λ121 and χ2 -λ122 planes. . . . . . . . . . . . . . . . . . . . . 94 7.11 Squared neutrino masses m2νi versus TRpV couplings λ121 (left) and λ122 (right) around BM3.. . . . . . . . . . . . . . . . . . . . . . . . . 95. 7.12 Relation between χ2 and BRRpV /BRRpC for the µµ (up) and the τ τ (down) modes moving λ121 and λ122 in the range [−0.002, 0.002] around BM3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99.

(11) CHAPTER. 1. INTRODUCTION. From time immemorial, mankind has tried to find an answer for the ultimate underlying substance or principle by which all things exist. The first half of the 20th century was key to this endeavor, as the discovery of quantum mechanics leaded scientists to an unprecedented level of understanding on the fundamental building blocks of matter. Since then, many efforts have been made trying to classify and understand the properties of these fundamental blocks, which has leaded to notorious discoveries over the years and to the development of the Standard Model of particle physics, the most successful and most accurate physical theory ever made, which describes with stunning accuracy the underlying workings of reality. Our current understanding therefore is almost entirely encapsulated by the Standard Model. It correctly describes the strong nuclear force, the weak force and the electromagnetism, and classifies all the elementary particle known so far.. However, despite its widely accepted success, there are phenomena that cannot be explained within the context of the SM. For instance, it does not incorporate gravity as a fundamental interaction, its particle content is not enough to explain Dark. 1.

(12) 2. matter nor its cosmological effects, and it does not incorporate any mechanism of neutrino mass generation and oscillations, suggesting the existence of a more fundamental theory that could explain these deficiencies. In this regard, supersymmetry (SUSY), a theory which extends the symmetries of space and time, has been one of the most appealing options for decades as it introduces new particles that can account for the current limitations of the SM and also contains interesting features from the theoretical point of view, such as the unification of the gauge couplings and it also offers a natural solution to the high sensitivity of scalar masses to loop correction, problem known as the hierarchy problem.. Motivated by the recent exciting discoveries of a scalar resonance compatible with the SM Higgs Boson in 2012 and the confirmation of Neutrino Oscillations in 2014, this work focuses mainly in the study of the scalar sector of the Minimal Supersymmetric Standard Model (MSSM) with specific lepton number violating (LNV) terms. The main goal is to study te possibility of predict the latest 2017 neutrino data within this model in order to constrain the LNV couplings introduced and make a phenomenological analysis on how this could affect the leptonic branching ratios of a heavy version of the SM-like Higgs in order to give an explanation for the non-observation of any heavy resonance in such channels despite the large number of events predicted in most of the parameter space. In order to achieve this, we study the deviations of the leptonic couplings of the Heavy CP-even scalar, H, due to the LNV couplings in a region compatible with neutrino observables, and see if its possible to find deviations large enough to differentiate this specific model from the other more typical lepton number conserving supersymmetric scenarios.. This work is organized as follows: in Chapter 2 a brief introduction to the Standard Mode is given together with its limitations that motivate the many extensions of it, in particular Supersymmetry, which is introduced later on in Chapter 3. Here, the construction of SUSY invariant lagrangians is explained making used of the superspace formalism. In Chapter 4, the superspace formalism is applied to the SM by replacing the usual SM fields for superfields in order to obtain the MSSM.

(13) 3. Chapter 1. INTRODUCTION. lagrangian. Important features that will be used in this work such as soft SUSY breaking, R-partiy and the Decoupling Limit are also discussed here. Chapter 5 is entirely devoted to explain the breaking of R-parity in the MSSM and the mechanism of neutrino mass generation as one of the most relevant implications of it. In Chapter 6 we present the superpotential that defines the particular model used for this thesis and the corrections taken into account to predict neutrino observables. In Chapter 7 we explain our numerical calculations and the parameter space we worked with to compute the leptonic branching ratios of the Heavy CP-even scalar. We also discussed different benchmark scenarios and the predicted neutrino observables in this model. Finally, Chapter 8 summarizes the whole content of this thesis together with the results obtained and we discussed some conclusions that can be made based on this research..

(14) CHAPTER. 2. THE STANDARD MODEL AND BEYOND. In this chapter we review the general aspects of the Standard Model as well as its limitations. We start with the mathematical description of the Poincaré Group in Section 2.1 which will serve as starting point to introduce Supersymmetry later on. In Section 2.2 we specify the particle content of the SM and we write the Lagrangian explicitly. Lastly, in Section 2.3 we give a brief description of some phenomena that cannot be explained by the SM, focusing on Neutrino Oscillations and the Hierarchy Problem, which are the most relevant ones regarding this work.. 2.1. The Poincaré Group. The most fundamental components of any quantum field theory (QFT) are the principles of quantum mechanics (QM) and special relativity (SR). These, of course, were the starting point for the development of the Standard Model. According to SR, the laws of physics are coordinate independent, which means that they should remain. 4.

(15) 5. Chapter 2. THE STANDARD MODEL AND BEYOND. unchanged if the coordinate system is rotated, boosted or translated. These transformations form the Poincaré Group, defining the symmetries of the space-time.. The Poincaré group, also called the inhomogeneous Lorentz group ISO(3, 1), is the semi-direct product of the Lorentz group and space-time translations. It has 10 generators in total, which correspond to 4 translation generators P µ and 6 Lorentz generators M µν . For an element of this group T (Λ, a), the action on the 4-dimensional Minkowski space-time is defined as, 0. x µ = T (Λ, a)xµ = Λµν xν + aµ. (2.1). Its Lie algebra in terms of the 10 generators P µ and M µν can be written as, h i µν ρσ µρ νσ µσ νρ µσ µρ M ,M = −i η M − η M − ηνρM + ηνσM h i P ρ , M µν = i η ρµ P ν − η ρν P µ h i P µ, P ν = 0. (2.2) (2.3) (2.4). We can also define the generators of spacial rotations J i and the generators of Lorentz boosts K i according to, J i = ijk M jk. K i = M 0i. (2.5). Consequently, in terms of these generators, the Lorentz algebra becomes, [Ji , Jj ] = iijk Jk. [Ji , Kj ] = iijk Kk. [Ki , Kj ] = −iijk Jk. (2.6). We can classify the irreducible unitary representations of the Poincaré group using the Casimir invariants: generators that commute with all the other ones and allow us to label the different states on which the Poincaré generators act. The Poincaré group has two Casimir invariants:.

(16) 2.1. The Poincaré Group. P 2 = P µ Pµ = m2. W 2 = W µ Wµ. 6. (2.7). where Wµ is the Pauli-Lubanski pseudovector: 1 Wµ = µνρσ M νρ P σ 2. (2.8). The eigenvalues of these two invariants serve to label the representations of the group. The first is associated with the mass and the second with the spin (or helicity). Therefore, the physically relevant representations may thus be classified as massive or massless representations, each one fundamentally different from each other. This is also known as Wigner’s Classification [2].. The Poincaré group has non-unitary finite-dimensional irreducible representations too, which are also useful as they act on fields. To find such representations we notice that we can define two independent operators, 1 Ai = (Ji + iKi ) 2. 1 Bi = (Ji − iKi ) 2. (2.9). where each one satisfies the SU (2) algebra, [Ai , Aj ] = iijk Ak. [Bi , Bj ] = iijk Bk. [Ai , Bj ] = 0. (2.10). Thus, the generators of the Lorentz group decomposes into two commuting sets of SU (2) generators, and so the finite-dimensional representations of the group are 0. specified by two integers or half integers spin quantum numbers (n, n ). The most important representations caracterized by these numbers are: • (0, 0) → Scalar representation • ( 12 , 0) → Left-handed spinorial representation • (0, 12 ) → Left-handed spinorial representation • ( 12 , 12 ) → Left-handed spinorial representation.

(17) 7. Chapter 2. THE STANDARD MODEL AND BEYOND. Each one of these representations will have an associated field transforming accordingly, and so the different particles in nature can be represented by scalar fields, left- or right-handed spinor fields and vector fields. As we will see in the next section, these are the building blocks of any theory of elementary particles such as the Standard Model.. 2.2. The Standard Model Lagrangian. The internal symmetries of the Standard Model are defined by the group GSM = SU (3)c × SU (2)L × U (1)Y which describe the strong, weak and electromagnetic interactions. The different subscripts indicate the charges associated with each group: color, weak-isospin and hypercharge. All the particles in the SM transform under a certain representation of these groups, which defines the type of interactions in which they participate. For instance, only particles transforming under SU (3)c carry a color charge and take part in strong interactions, while particles transforming under SU (2)L × U (1)Y participate in electro-weak interactions.. In general, a field ψ(x) would transform under a certain representation of the SM group GSM as, G. SM ψ(x) −−− −→ ψ 0 (x) = e−igs γ. a (x)T a 3. ~. ~. Y. e−igw β(x)·T2 e−igy α(x) 2 ψ(x). (2.11). where α(x), β(x) and γ(x) are transformation parameters which define the specific Gauge, T3a (a = 1...8), T~2 and Y /2 are the generators of SU (3)c × SU (2)L × U (1)Y respectively and gs , gW and gY are the gauge couplings of each group. Because (2.11) defines a local transformation, we need to introduce the covariant derivative to ensure gauge invariance: ∂µ −→ Dµ = ∂µ − igy Y Bµ − igw Wµi Twi − igs Giµ Tsi. (2.12).

(18) 2.2. The Standard Model Lagrangian. 8. Such that, G. SM Dµ ψ −−− −→ Dµ ψ. 0. ~. ~. ~. = eigs~γ (x)·Ts eigw β(x)·Tw eigy α(x)Y Dµ ψ. (2.13). The definition given in (2.12) depends on the field upon which the derivative is acting. For instance, if a field transforms as a singlet under one of the GSM groups, it means the generators associated to such group should be zero when they act on the field. We are going to classify the particle content of the SM with their respective Lagrangians in three different sectors: the Gauge sector, Matter sector and the Scalar sector.. 2.2.1. Gauge Sector. The Gauge sector describes the force carriers, particles that give rise to forces between other particles, mediating the different interactions. These carriers are represented by Gauge Bosons, spin 1 particles which arise from the group structure of the theory.. The first of these groups SU (3)c , is the responsible for the color charge of quarks and it encodes the underlying symmetry of strong interactions. It is defined by eight generators Tsa , each one with a gauge boson associated to it transforming in the adjoint representation. These eight field are called are gluons gµa and they are the mediators of the strong force. The second relevant group is the weak isospin group SU (2)L . It describe the transformation of left handed fermion as isospin doublets and it is defined by three generators T~w with 3 gauge bosons associated to it: the W-bosons, denoted by Wµa , responsible for the weak interactions. Lastly, the hypercharge group U (1)Y has only one generator and one gauge boson associated to it, the B-boson Bµ , which acts partialy as a force carrier in electromagnetic inter-.

(19) 9. Chapter 2. THE STANDARD MODEL AND BEYOND. actions . To summarize, all the previously mentioned particles with their respective groups are listed in Table 2.1 .. The corresponding Lagrangian for the Gauge fields is given by, 1 1 a µν 1 a µν Wa − Gµν Ga LGauge = − Bµν B µν − Wµν 4 4 4. (2.14). where Bµν , Waµν and Gaµν are the field strength tensors defined by, Bµν =∂µ Bν − ∂ν Bµ. (2.15). a Wµν ≡Wµν Twa = ∂µ Wν − ∂ν Wµ + igw [Wµ , Wν ]. (2.16). Gµν ≡Gaµν Tsa = ∂µ Gν − ∂ν Gµ + igs [Gµ , Gν ]. (2.17). which transform in the adjoint representation of their respective symmetry groups so the Lagrangian (2.14) is Gauge invariant. 0 Bµν −→ Bµν = Bµν. (2.18). † 0 Wµν −→ Wµν = UW (x)Wµν UW (x). (2.19). Gµν −→ G0µν = US (x)Gµν US† (x). (2.20). Gauge Bosons Gauge Group B-boson Bµ U (1)Y. Coupling gY. SU (3)c × SU (2)L × U (1)Y (1, 1, 0). W-bosons Wµa. SU (2)L. gw. (1, 3 , 0). Gluons gµa. SU (3)c. gs. (8 , 1, 0). Table 2.1: Particle content of the Gauge Sector in the SM with the associated gauge groups..

(20) 2.2. The Standard Model Lagrangian. 2.2.2. 10. Matter Sector. The matter sector contains all elementary fermions which can be separated into quarks and leptons. This particles are represented by two-component Weyl spinors classified by their chirality. Thus, all fermions have a right-handed (RH) and a left-handed (LH) component, except for the neutrino, which it’s only left-handed, and the way in which they couple is determinate by their transformation properties under GSM .. When it comes to quarks, there are six in total which can be up-type quarks (up, charm, top) or down-type quarks (down, strange, bottom) and they are classified in three different families. Both LH and RH components transform as triplets under SU (3)c , thus carrying a color charge c = {red, blue, green} and taking part in strong interactions . Respect to SU (2)L , only LH quarks transform as doublets under such group while RH quarks transform as singlets. The different quarks are grouped as:.      tαL cαL       , = , α α α sL bL dL . qLαi. uαL. (2.21). uαRi = uαR , cαR , tαR. (2.22). dαRi = dαR , sαR , bαR. (2.23). Where α is a color index (which is often omitted), L and R represent chirality and i = 1, 2, 3 is an index that indicates one of the three families of quarks.. On the other hand, leptons don’t play any role concerning the strong interactions, thus being singlets under SU (3)c . Left-handed leptons transform as doublets and right-handed leptons as singlets under SU (2)L . Just like quarks, there are six leptons.

(21) 11. Chapter 2. THE STANDARD MODEL AND BEYOND. present in nature grouped in three families. Explicitly,  `Li =  eRi =. νeL eL. . .  , . eR. ,. νµL µL. µR. . .  ,  ,. ντ L τL.  . τR. (2.24) (2.25). The kinetic Lagrangian for quarks and leptons is given by:. Lmatter. 3 n o X µ µ µ µ µ ¯ ¯ = i`Li σ̄ Dµ `Li + iēRi σ̄ Dµ eRi + iq̄Li σ̄ Dµ qLi + iūRi σ̄ Dµ uRi + idRi σ̄ Dµ dRi i=1. (2.26) where i is a family index and the covariant derivative is given by (2.12) depending on each field. The different particles and their transformation properties under the Gauge groups is shown in Table 2.2 below. Fermions LH lepton. Field Rep lLi. SU (3)c × SU (2)L × U (1)Y 4 (1, 2,-1/2). RH lepton. eRi. (1, 1, -1). LH quark. qLi. (3, 2, 1/6). RH up quark. uRi. (3, 1, 2/3). RH down quark. dR i. (3, 1, 1/3). Table 2.2: Particle content of the Matter Sector of the SM with their respective representation.. 2.2.3. Higgs Sector and Spontaneous Symmetry Breaking. It is a well known experimental fact that excepting for the photon and the neutrinos, all the other Gauge bosons and fermions are massive, and so the original Gauge invariance of the SM should be spontaneously broken, leaving SU (3)c × U (1)em as.

(22) 2.2. The Standard Model Lagrangian. 12. a remaining symmetry. For this, we introduce the Higgs field, an SU (2)L doublet scalar field with hypercharge Y = 1 :  Φ=. φ+ φ0.  (2.27). . The corresponding Lagrangian for this field is given by LHiggs = (Dµ Φ)† (Dµ Φ) − V (Φ† Φ). (2.28). where V (Φ† Φ) is the symmetry breaking potential: V (Φ† Φ) =. µ2 † λ Φ Φ + (Φ† Φ)2 2 4. (2.29). with λ > 0 and µ2 < 0. In this way, the minimum of the potentail is degenerate. Since the system has set of degenerate vacuum states and a non-zero groundstate, the Higgs fields acquire a vacuum expectation value VEV:   0 1 hΦi = √   2 v. r with v =. −. µ2 λ. (2.30). After choosing the groundstate, we can write the physical Higgs field in the Unitary Gauge in terms of the VEV as: . 0. . 1  Φ =√  2 v+h. (2.31). Expanding the Lagrangian in terms of (2.31), one finds that the Gauge bosons mix among themselves to give rise to the physical mass eigenstates:.

(23) 13. Chapter 2. THE STANDARD MODEL AND BEYOND. 1 Wµ± = (Wµ1 ∓ iWµ2 ) 2 1 Aµ = p 2 (gy Wµ3 + gw Bµ ) gw + gy2 1 Zµ = p 2 (gw Wµ3 − gy Bµ ) 2 gw + gy. (2.32) (2.33) (2.34). After SSB, only the photon Aµ remains massless, while the masses of the other two Gauge fields can be expressed in terms of the VEV as, gw v Mw = 2. ,. p 2 gw + gy2 Mz = 2 v. (2.35). On the other hand, the fermion masses result from Yukawa-type interactions with the Higgs fields. The corresponding Lagrangian is:. −LY uk. 3 n o X Yeij `¯Li Φ eRj + Ydij q̄Li Φ dRj + Yuij Q̄Li iσ2 Φ∗ uRj =. (2.36). i,j=1. where both i,j are family indices and Yeij , Ydij , Yuij are the Yukawa couplings which generically are 3 × 3 complex matrices. As a result of SSB, all fermions except for the neutrinos get a mass proportional to their respective Yukawa couplings.. 2.3. Beyond the Standard Model. There are a number of reasons to believe that the SM cannot be the ultimate theory to describe fundamental particles, and it has to be regarded as an effective theory only valid up to some energy scale, usually the electro-weak scale. We can categorized these reasons as physical phenomena in nature that cannot be described by the SM, such as Gravity, and aesthetic flaws of the model itself, such as the naturalness.

(24) 2.3. Beyond the Standard Model. 14. of different free parameters of the model.. Although physics beyond the SM (BSM) is a vast subject and there are many ways of approaching it, we will only focus on giving a brief explanation of Neutrino Oscillations, the Hierarchy Problem and Dark Matter, which are the main limitations of the SM concerning this work.. 2.3.1. Neutrino Oscillations. Neutrinos are fundamental particles first postulated in 1930 by W. Pauli to explain the apparent missing energy in β-decay processes associated to nuclear reactions. Over the years, different experiments established the existence of such particle as an electrically neutral spin-1/2 fermion. However, since neutrinos only interact with matter via the weak force making them extremely difficult to detect directly, it took many years to understand the main properties of these elusive particles, and it wasn’t until 1970 that the weak interactions of neutrinos were correctly described by the Standard Model by considering neutrinos as massless particles.. Nowadays, it is a well established fact that neutrinos come in three different types or flavors, one for each of the three charged lepton families: the electronic neutrino νe , the muon neutrino νµ and the tau neutrino ντ . Furthermore, we know from the existing data that because weak interactions violate parity, only left-handed neutrinos are allowed to participate in them. Consequently, in the Standard Model neutrinos are described as left-handed massless fermions, with an individual lepton family number associated to them which is exactly conserved in electro-weak interactions. However, if neutrinos were massive, then such conservation law would be only approximate and transitions between the different flavor neutrinos would be allowed. This phenomenon is known as Neutrino Oscillation..

(25) 15. Chapter 2. THE STANDARD MODEL AND BEYOND. Neutrino oscillations as a flavour changing mechanism due to possibility of neutrinos having a small but non-zero mass, were first suggested by Pontecorvo in 1957 and developed by Maki, Nakagawa and Sakata in 1962. Different experiments along the last two decades involving solar, atmospheric and reactor neutrinos have provided compelling evidence for the existence of such phenomenon. Among these experiments, the ones leaded by T. Kajita of the Super-Kamiokande Collaboration [3] and by Arthur B. McDonald of the Sudbury Neutrino Observatory (SNO) Collaboration [4], validated the actual existence of neutrino oscillation and therefore, the fact that neutrinos are indeed massive particles.. This discovery constitute one of the major breakthrough for particle physics of the last decade, as it directly presents experimental evidence for the incompleteness of the Standard Model, opening a window to the discovery of new physics beyond our current understanding of nature.. In view of the importance of neutrino oscillations, it is convenient to have a clear understanding of the main aspects of the theory. The full description of the theory of neutrino oscillations together with the experimental results and their interpretations have been discussed and reviewed in many papers [1] [5] [6]. In what follows, we present a brief and general explanation based in a simple but useful quantum mechanical description of the phenomenon.. Consider a neutrino state with a well defined flavour |να i with α = e, µ, τ that is related to the mass eigenstates |νi i by a unitary mixing matrix U:. |να i =. 3 X. Uαi |νi i. (2.37). i=1. If the mass eigenstates |νi i have a definite energy Ei and their time evolution is governed by the Schrodinger equation, these eigenstates should evolve as plane waves: i. ∂ |νi (t)i = Ei |νi i ∂t. −→. |νi (t)i = e−iEi t |νi i. (2.38).

(26) 2.3. Beyond the Standard Model. 16. In the ultrarelativistic limit, the energy of a mass eigenstate neutrino with momentum p~ is given by, Ei =. q. p~2 + m2i ' E +. m2i 2E. (2.39). where E = |~p|2 . Consequently, using (2.37) and (2.38) we can find the time evolution of the flavour neutrino state: |να (t)i =. 3 X. X. Uαi e−iEi t |νi i =. i=1. ( 3 X. ) ∗ Uαi e−iEi t Uβi. |νβ i. (2.40). i=1. β=e,µ,τ. We can see from the previous equation that even though we started in t = 0 with a state with a definite flavour |να i , for t > 0 it evolves s a superposition of different falvours when the mixing matrix U is different from the identity. Therefore, the probability of a neutrino transitioning from a flavour state α to another flavour state β at a time t is:. 2. Pνα →νβ (t) = | hνβ | να (t)i | =. 3 X. 2 −iEi t. Uαi e. ∗ Uβi. =. 3 X. ∗ ∗ Uαi Uβi Uαj Uβj e−i(Ei −Ej )t. i,j=1. i=1. (2.41) If we assume that all massive neutrinos carry the same momentum p~, the energy difference can be written in terms of the mass difference: Ei − Ej =. ∆m2ij 2E. (2.42). where ∆m2ij ≡ m2i − m2j . However, time is not measure in neutrino oscillation experiments, so in order to have a measurable flavour transition probability we can express (2.41) in terms of the approximate distance L between the detector and the source from where neutrinos are emitted.. Pνα →νβ (L) =. 3 X. ∗ ∗ Uαj Uβj e−i Uαi Uβi. ∆m2 ij L 2E. (2.43). i,j=1. Since the mixing matrix U is unitary, if the neutrinos were massless or if the three.

(27) 17. Chapter 2. THE STANDARD MODEL AND BEYOND. masses were equal, then ∆m2 = 0 and there wouldn’t be any transition between flavour states. Therefore we need at least two massive neutrinos to explain this phenomenon.. The mixing matrix U, also called the Pontecorvo-Maki-Nakagawa-Sakata matrix UP M N S , may in general be complex. Considering the case of three generations of massive neutrinos, the UP M N S matrix can be parametrized in terms of three mixing angles and three phase parameters. It is often convenient to write it in a decomposed form since the experimental data can be approximately analyzed in terms of oscillations between just two flavour neutrino states. Then, the UP M N S mixing matrix can be written as:. UP M N S.  1 0   = 0 cos θ23  0 − sin θ23  cos θ12   × − sin θ12 . . . −iδ. cos θ13 0 sin θ13 e     0 1 0 sin θ23  ×    − sin θ13 e−iδ 0 cos θ13 cos θ23    sin θ12 0 1 0 0       iα /2 1 × cos θ12 0 0 e 0     iα2 /2 0 1 0 0 e 0.     . (2.44). where θij are de three different mixing angles and δ is a CP-violating Dirac phase which is the only physical phase if the neutrinos are Dirac fermions. On the contrary, if neutrinos are Majorana fermions, two additional CP-violating phases α1 ,α2 are necessary. These phases are important in processes such as neutrinoless double beta decay [7], but for our purposes the most relevant parameters to characterize neutrino oscillations are the mass difference ∆m2 and the mixing angles, so we can assume CP conservation and set all the phases to zero. This way, the parametrization of the UP M N S is given by, . c12 c13. s12 c13. s13. .     UP M N S = −s12 c23 − c12 s23 s13 c12 c23 − s12 s23 s13 s23 c13    s12 s23 − c12 c23 s13 −c12 s23 − s12 c23 s13 c23 c13. (2.45).

(28) 2.3. Beyond the Standard Model. 18. where cij = cos θij and sij = sin θij . It is possible to measure some of the relevant parameters by counting the disappearance or the appearance of neutrinos given an initial flux of a certain flavor state. Neutrinos coming from the sun and from atmospheric processes were the principal source for experiments of such kinds, which gave rise to the so-called solar and atmospheric neutrino problems.. 2.3.1.1. Solar Neutrino Problem. One of the main sources of neutrinos is the Sun. Due to the thermonuclear reactions in the solar core, over 60 billion neutrinos per square centimeter reach the Earth every second. These reactions only produce electron neutrinos, which makes it very convenient for the study of oscillation processes. Since 1960, scientist had calculated the approximate number of neutrinos produced in the sun, but independently of the model they used, the total amount of neutrinos measured on Earth seems to be less of what it was theoretically predicted. The observation that, compared to the theoretical predictions, the flux of neutrinos from the Sun measured on Earth appears anomalously low is usually called the Solar Neutrino Problem.. This problem persisted for decades, and many people thought that the theoretical predictions were initially wrong. However, with the improvement of solar models along the years, neutrino oscillations became the most accepted proposal to solve the solar problem, as it would indicate that electron neutrinos must have change in flavour on their way from the Sun, therefore explaining the deficit of the captured electron-neutrinos on Earth.. Conclusive evidence to support the idea of oscillating electron neutrinos first came from the Sudbury Neutrino Observatory (SNO), which started observations in 1999. It measure only a third of the expected number of electron neutrinos that should have been caught in their detectors [8]. This was confirmed later by the KamLND [9], which showed clear evidence for disappearance of electron anti-neutrinos in reactor.

(29) 19. Chapter 2. THE STANDARD MODEL AND BEYOND. experiments, consistent with the expectation from the solar neutrino results. Their results contributed mainly to the measurements of ∆m212 ≡ ∆m2sol and θ12 ≡ θsol .. 2.3.1.2. Atmospheric Neutrino Problem. Another source for detectable neutrinos are the atmospheric processes induced by the incoming cosmic rays. These are mainly composed of high energetic particles such as protons, heavy nuclei and electrons. When cosmic rays hit the different elements on the atmosphere they produce a cascade of hadron which decay during flight producing the atmospheric neutrinos. The principal source of neutrinos comes from pion decays: π + → µ+ νµ. ,. µ+ → e+ νe ν̄µ. (2.46). π − → µ− ν̄µ. ,. µ+ → e− ν̄e νµ. (2.47). Then, the atmospheric neutrinos measured on the surface are mainly electron neutrinos and muon neutrinos. At moderate energies, one would expect the ratio R between the number of these two kinds of neutrinos to be []: R=. Nµ νµ + ν̄µ = '2 Ne νe + ν̄e. (2.48). However, many experiments such as Super-Kamiokande [10], Soudan2 [11] and IMB [12], have measure an smaller ratio, indicating that either there was less muon neutrinos in the data than in the prediction or there was more electron neutrinos, or both. This discrepancy is what is known as the Atmospheric Neutrino Problem. Different results also suggest that the dominant oscillation mode for atmospheric neutrinos is from muon neutrinos to tau neutrinos, as the number of measure electron neutrinos is not enhanced. Super-Kamiokande experiment also measured the direction of the incoming neutrinos, as they can reach the detector from all angles. The results showed that at high energy, nearly half of the muon neutrinos coming up from below the detector were missing..

(30) 2.3. Beyond the Standard Model. 20. All these results can also be explained within the context of changing flavor processes, as the different neutrinos produced in the atmosphere will oscillate in their way to the detector. Neutrinos arriving at different angles also travel different distances, which can explain the disappearance of muon neutrinos coming from below the detector since they have to travel a longer distance than the ones coming from above. Most of these experiments measures effects on ∆m213 ≡ ∆m2atm and θ23 ≡ θatm .. In this work we show a predictive scheme for neutrino mixing angles and masses which can account for the observed atmospheric and solar neutrino problems. Since neutrino data will be the main phenomenological constrain for our parameter space, we show the current best-fit values and 3σ allowed ranges for the relevant parameters in Table 2.3 below, Parameter. Best fit. 3σ. sin2 θ12. 0.297. 0.250 − 0.354. sin2 θ23. 0.437. 0.379 − 0.616. sin2 θ12. 0.0214. 0.0185 − 0.0246. ∆m212 [10−5 eV2 ]. 3.37. 6.93 − 7.97. |∆m231 |[10−3 eV2 ]. 2.50. 2.37 − 2.63. Table 2.3: Latest results on neutrino observables in the 3σ range according to [1].. 2.3.2. The Hierarchy Problem. Over the years, the Hierarchy Problem has been considered for many theorists as an undesirable feature and a serious imperfection of the Standard Model that implies a lack of understanding of its fundamental properties. Although it doesn’t address.

(31) 21. Chapter 2. THE STANDARD MODEL AND BEYOND. any problem of inconsistency with the current experimental data nor with unexplained physical phenomena as the previously mentioned extensions, the hierarchy problem deals with the more general question about the natural sizes of parameters in a quantum field theory and why the EW symmetry breaking scale in the SM is so small compared to the natural scale of unification or gravitational phenomena.. Even though there are various levels to the hierarchy problem, it is usually formulated as a fine-tunning problem that arises once we consider the SM as an effective field theory coming from a much larger theory which includes unknown new physics at a higher energy scale Λ. At the very least we expect that such new energy scale should be of the order of the Planck scale MP ∼ 1019 GeV, when quantum gravity effects become important. An unpleasant fine-tunning is then required to stabilize the large gap in between energy scales, ranging from the electroweak scale 102 GeV to the high energy scale Λ ∼ MP , against quantum corrections that push the tree-level masses of scalar particles to extremely unnatural high values [13].. In the SM the only physical scalar field is the Higgs field, and its vacuum expectation value v ' 246 GeV characterize the electroweak sector and sets the scale of all tree-level masses in the theory. Because the SM is renormalizable, the masses of the different particles receive finite higher-order corrections which depend on the energy scale of new physics, being the Higgs mass mH ' 125 GeV the most sensible to such corrections as it is approximately sixteen orders of magnitude smaller than the apparent energy cutoff Λ.. We can infer the sensitivity of the Higgs mass calculating loop corrections up to a certain cut-off momentum set by the new energy scale. At one loop level, the Higgs mass receives corrections from fermion loops (with the top-quark loop being.

(32) 2.3. Beyond the Standard Model. 22. the most relevant one), gauge boson loops and self interactions.. φ. φ, = φ. φ + φ λ. yt. yt. φ + φ. g. g. φ. These loops are quadratically divergent and go like, δm2H. Z ∼. Λ. d4 k. 1 ∼ Λ2 k2. (2.49). Doing the explicit calculation, we obtain that these corrections are given by, δm2H(1l). Λ2 = 16π 2. 2 9 2 3 2 2 6λ − gW + gY − 6yt ∼ 1019 GeV 4 4. (2.50). After renormalization, the physical Higgs mass have to be written in terms of the bare mass parameter m2H(bare) and the radiative corrections previously computed, thus m2H(phys) = m2H(bare) + δm2H(1l) | {z } | {z }. ∼(102 GeV)2. (2.51). ∼(1019 GeV)2. We can now see that the quadratic divergence of the loop integrals forces us to start with an equally large but negative value for the bare parameter in the Lagrangian if we want to agree with the experimental mass of the Higgs boson. This large cancellation is precisely what makes the Standard Model unnatural, as it requires an extreme fine-tunning of its parameters. Moreover, such specific adjustment of the unrenormalized parameters is not valid to all orders in perturbation theory as new quantum corrections would spoil it, and so the fine-tunning process has to be done order by order.. This dependency on the new energy scale not only affects the mass of the Higgs boson, but also all masses in the SM. However, unlike scalars, the radiative corrections to the masses of fermions and gauge bosons are proportional to the masses.

(33) 23. Chapter 2. THE STANDARD MODEL AND BEYOND. of the particles themselves and no quadratic divergence is present. For the case of ferminons this happens because there is a chiral symmetry in the limit where the fermion masses go to zero: 0. ψ → ψ = eiαγ5 ψ. (2.52). This symmetry protects the masses of fermions from large quantum corrections as it ensure that such corrections will vanish in the massless limit. Since mf is the only parameter which breaks this symmetry, all radiative corrections must be proportional to the mass itself and the dependency on the energy scale can only be logarithmic. δmf ∼ mf ln Λ. (2.53). Hence, the loop corrections are suppressed by the smallness of the tree-level parameter mf and no unpleasant fine-tunning is necessary. It is then said that small fermion masses are technically natural [14]. Similarly, the masses of the gauge bosons are protected by gauge symmetry, which forbids any explicit mass term at all orders in perturbation theory.. Although the problem of the UV quadratic divergences is manifest if one uses a momentum cut-off regulator, the hierarchy problem is independent of the the renormalization scheme. If for example we use dimensional regularization to calculate the radiatives corrections, there would be no quadratic divergence nor explicit dependency on the UV scale. However, it is important to understand that the hierarchy problem is not about the divergence itself. The main problem is that the scalar particles such as the Higgs boson are sensitive to any high energy scale.. Considering the dimensional regularization approach, if an additional particle S couples directly or indirectly to the Higgs field at a new energy scale set by its mass mS , the one loop contribution to the Higgs mass from this particle would be of the form:.

(34) 2.3. Beyond the Standard Model. δm2H(1l). ∼. λS m2S. ln. Λ mS. 24. + finite. (2.54). We can see that the quadratic divergences no longer appear in the radiative corrections, and only a logarithmic divergence appears, which comes from the poles of the Gamma function. Nevertheless, the logarithmic piece and the finite part of (2.54) are proportional to the mass scale mS of the new physics, and so the quadratic dependence on the new energy scale is reintroduced by this parameter.. The hierarchy problem of the SM has driven the search for more fundamental theories in which fine-tunned parameters are no longer necessary to describe it. Many interesting theories have come from such searches, being Supersymmetry (SUSY) the most remarkable of all [15]. Although yet to be discovered, SUSY gives an elegant solution to the hierarchy problem by enlarging the space-time symmetries introducing transformation which relate bosons and fermions. This new symmetry protects the scalar masses from large quantum corrections just as the chiral or gauge symmetry protects the masses of the rest of the particles, as it introduces new bosonic or fermionic particles, the so-called superpartners, for each of the original SM particles, and relates the different couplings in such a way that the radiative corrections cancel among each other preventing the quadratic sensitivity to Λ.. There are of course many other possible solutions to this problem such as technicolor, composite Higgs models, extra dimensions or models based in other global symmetries. However, we will only focus on SUSY which will be formally introduced in the next chapters..

(35) 25. Chapter 2. THE STANDARD MODEL AND BEYOND. 2.3.3. Dark Matter. One of the most puzzling limitations in our current understanding of the Cosmos is that baryonic matter, that is, the ordinary matter that forms planets, stars, galaxies, and the gas in between, made up of protons and neutrons, is not all there is out there. In fact, from cosmological observations we now know that it is actually the less dominant form of matter, representing only a small fraction of all the energy present in the universe.. The first hypothesis for the existence of another but invisible kind of matter came from astronomer Fritz Zwichy in 1933, who measuring the radial velocities of galaxies in the Coma Cluster, found anomalous velocity dispersions, indicating that the density in the cluster was much higher than the one produced by luminous matter alone. He then conclude that for that to be true, a non-luminous form of matter or Dark Matter (DM) should exist in a much higher density than ordinary radiating matter [16]. Later in the 70’s, measurements of rotation curves of spiral galaxies also found evidence for a big amount of mass missing which was required to explain the observations. In the following decades, the Dark Matter hypothesis grew strong, and many precise observations on galactic and cluster scales, mass determinations using gravitational lensing, and measurements of the distribution of hot X-ray emitting gas in galaxies accounted for its existence.. Nowadays, the Cosmic Microwave Background (CMB) is the most precise way to map the density of matter in the early Universe. By analyzing the CMB and its small fluctuations cosmologists were able to determine that our Universe is composed of around a 5% baryonic matter, 27% cold dark matter and 68% dark energy [17]. However, despite all the evidence from astronomical and cosmological measurements, Dark Matter can only be observed indirectly through its gravitational influence on ordinary matter, and so the nature of it remains a mystery. The most natural and simplest assumption is to consider DM as a new kind of elementary particle yet to.

(36) 2.3. Beyond the Standard Model. 26. be discovered which couples to gravity but doesn’t interact with the electromagnetic force and at most it could interact weakly with the rest of the SM particles. Within this scope, commonly referred DM candidates are Weakly Interacting Particles (WIMPs), Axions and Sterile Neutrinos. Nevertheless, there is no particle in the SM meeting the requirements to be such particles, and so its description would necessarily require an extension of the SM.. One of the most appealing extensions to the Standard Model is Supersymmetry. Among the SUSY particles that can be regarded as viable DM candidates are the lightest supersymmetric particle (LSP), which for the MSSM is the neutralino, and the gravitino, which is the superpartner of the graviton which would come from a quantum gravity theory in case R-Parity is not conserved. These particles are both electrically neutral and thus are ideal WIMP-like candidates for DM. However, for our purposes, we will only consider the gravitino as a suitable DM candidate as the breaking of R-Parity is essential in our model. The consequences of this will be explained further on.. Nevertheless, whatever candidate might be considered, no direct evidence have been observed yet and the hunt for dark matter continues up to this date in different experiments, imposing more strict bounds on the possible mass of these hypothetical particles. These searches also presents the opportunity to understand the nature of DM, maybe opening a window to new symmetries, new types of fundamental particles and new forces, while at the same time opening up a new field in astronomy and cosmology..

(37) CHAPTER. 3. INTRODUCTION TO SUPERSYMMETRY. The concept of Supersymmetry (SUSY) as a way of relating bosons and fermions was first introduced in the context of hadronic physics in 1966 by Hironari Miyazawa [18]. Although it didn’t gain much acceptance at that time, a new boson-fermion symmetry was discovered in 1971 by Ramond, Neveu and Schwarz in the context of strong interactions and string theory [19], which leaded to the rediscovery of supersymmetry in quantum field theories as an extended space-time symmetry relating elementary bosonic and fermionic particles, which first realization in four dimensional space-time was formulated by Wess and Zumino in 1974 [20]. Since then, the study of supersymmetric field theories has become one of the major endeavors in physics and its mathematical structure has subsequently been applied to many other areas of physics.. Within the context of quantum field theories, SUSY is commonly presented as a way of avoiding the Coleman-Mandula Theorem proposed in 1967 [21], which demonstrates that under some physically reasonable assumptions, the most general symmetry group of the S-matrix can only be a direct product of the Poincaré group and. 27.

(38) 3.1. The Super Poincaré Algebra. 28. an internal symmetry group. This means that is no possible to combine space-time symmetries with internal symmetries in any but a trivial way. Nevertheless, it is possible to evade this by allowing fermionic generators. This leads to the notion of SUSY as an extension of the Poincaré algebra by adding fermionic generators Q, Q̄, which turn bosons into fermions and vice versa: Q|bosoni ' |fermioni. Q|fermioni ' |bosoni. (3.1). In fact, it was in 1975 when Haag, Lopuszański and Sohnius (HLS) generalized the Coleman-Mandula theorem by proving that the most general continuous symmetry of the S-matrix is that pertaining to a Z2 -graded Lie algebra where the odd generators belong to the spinorial representation of the homogeneous Lorentz group and the even generators are a direct sum of the Poincaré and other internal symmetry generators [22]. In other words, the HLS theorem states that supersymmetry is indeed the only possible non-trivial extension of the symmetries of the S-matrix when the notion of a Lie algebra is generalized to a superalgebra (or graded Lie algebra), which contains anticommutation relations and fermionic generators.. The aim of this chapter is to present the supersymmetric algebra associated to the Super Poincaré group and to discuss their corresponding particle representations known as supermultiplets . To have a field representation of such supermultiplets, the superspace formalism is introduced in Section 3.2 which will be essential to construct supergauge invariant actions later on.. 3.1. The Super Poincaré Algebra. The simplest supersymmetric extension of the Poincaré algebra allowed by the HLS theorem is called the Super Poincaré Algebra. This superalegra has an odd sector with N fermionic generators, also known as supercharges QIα , QI† α̇ (I = 1... N ),.

(39) 29. Chapter 3. INTRODUCTION TO SUPERSYMMETRY. responsible for the transformation between fermions and bosons which satisfy anticommutation relations, and an even sector corresponding to the usual Lie algebra of the Poincaré group together with a set of scalar generators associated to an internal symmetry group.. We will focus on minimal supersymmetry in four dimensions also known as N = 1 SUSY, in which only one supercharge Qα , Q†α̇ , is added. In two-component Weyl spinor notation, the Super Poincaré algebra ca be written as. h i β µν Qα , M = σ µν α Qβ h i β̇ Q†α̇ , M µν = σ̄ µν α̇ Q†β̇ h i h i Qα , P µ = Q†α̇ , P µ = 0 n o † Qα , Qβ̇ = 2 σ µ αβ̇ Pµ o n n o Qα , Qβ = Q†α̇ , Q†β̇ = 0. (3.2) (3.3) (3.4) (3.5) (3.6). The irreducible representations of this superalgebra as single particle states are called supermultiplet representations, which are composed of bosonic and fermionic individual states. Since P 2 commutes with all the SUSY generators, all particles within the same irreducible supermultiplet should be degenerate in mass, and since the same holds for the scalar generators of the internal symmetry group, all the states whithin such supermultiplet should have the same internal quantum numbers too. Furthermore, it can be proved that the number of bosonic and fermionic degrees of freedom (d.o.f.) within a supermultiplet has to be the same [23].. For N = 1 SUSY, we can classify supermultiplets dependeing on their particle content as Chiral Supermultiplets or Vector Supermultiplets. The first ones are composed by a complex scalar state with spin s = 0 together with a two-component Weyl fermion state s = 1/2, while the vector supermultiplet contains a Weyl fermion state s = 1/2 and a real vector state s = 1, which will be interpreted as a gauge boson later on..

(40) 3.2. Superspace Formalism. 30. The fact that SUSY representations are given in terms of supermultiplets, which contain particles with opposite statistics, provides the solution to the hierarchy problem discussed in Section 2.3.2, as the superpartners will introduce new quantum corrections to the different masses of the scalar particles which will cancel out the contributions of their SM counterparts.. Although a field representation for the individual states of a supermultiplet can be constructed, it is more convenient to introduce the superspace formalism and find the superfield representations which will help us to construct Lagrangians which are manifestly SUSY invariant. This will be addressed in the next section.. 3.2. Superspace Formalism. Quantum field theories are commonly described in Minkowski space-time because in such formulation it is possible to construct Lagrangians where Poincare Symmetry is manifest. However, this is not the case for supersymmetric field theories. In order to have a framework where supersymmetry is inherently manifest, it is necessary to define an extension of Minkowski space-time, known as Superspace, by which the extra space-time symmetries associated to the SUSY generators are taken into account by adding a set of 2+2 spinorial coordinates θα , θ̄α̇ which represent four independent constant anticommuting Grassman numbers grouped as two-component Weyl spinors, {θα }α=1,2. and. {θ̄α̇ }α̇=1̇,2̇. (3.7). Therefore, the set of supercoordinates which spans the superspace is given by {xµ , θα , θ̄α̇ }, where the new coordinates spanning the fermionic subspace satisfy the usual anti-.

(41) 31. Chapter 3. INTRODUCTION TO SUPERSYMMETRY. commutation relations for Grassmann variables {θα , θβ } = 0. {θ̄α̇ , θ̄β̇ } = 0. {θα , θ̄β̇ } = 0. (3.8). The building blocks of any supersymmetric theory defined over superspace are functions of the superspace coordinates. These functions are known as Superfields. Because of the anticommutative properties of the new supercoordinates, Superfields can always be expanded in Taylor series and its expansion will be finite. Hence, the most general scalar Superfield S = S(x, θ, θ̄) can be written as,. S(x, θ, θ̄) =f (x) +. √. 2θξ(x) +. √ 2θ̄χ̄(x) + θθM (x) + θ̄θ̄N (x) + θσ µ θ̄Aµ (x). 1 + θθθ̄λ̄(x) + θ̄θ̄θζ(x) + θθθ̄θ̄D(x) 2 (3.9). In this representation, the expansion coefficients are known as component fields and they represent the usual fields in 4 dimensional Minkowski space. These can be bosonic fields, such as f (x), M (x), N (x), Aµ (x), D(x) or fermionic fields, such as ξ(x), χ(x), λ(x), ζ(x). Furthermore, a supersymmetry transformation is defined as the translation in superspace of a superfield S(x, θ, θ̄): S(x + δx, θ + δθ, θ̄ + δ θ̄) = S(x, θ, θ̄) + δS(x, θ, θ̄). (3.10). where δxµ = iθσ µ ¯ − iσ µ. ,. δθ = . ,. δ θ̄ = ¯. (3.11). Defining the SUSY generatos Qα and Q̄α̇ as differential operators acting on superfields satisfying the SUSY algebra given by Eq. (3.2)-(3.6), Qα = −i∂α − (σ µ )αβ̇ θ̄β̇ ∂µ. (3.12). Q̄α̇ = i∂¯α̇ + θβ (σ µ )β α̇ ∂µ. (3.13).

(42) 3.2. Superspace Formalism. 32. we can write the SUSY variation of a superfield as, δ S(x, θ, θ̄) = (iQ + +i¯Q̄)S(x, θ, θ̄). (3.14). We can also express this transformation in terms of infinitesimal variations of the different component fields as follows: √ 2¯ξ¯. (3.15). 2δξα = 2α M + (σ µ ¯)α (−i∂µ f + Aµ ). (3.16). 2δ χ̄α̇ = 2¯α̇ N − (σ̄ µ )α̇ (i∂µ f + Aµ ). (3.17). δf = √ √. √. 2ξ +. i δM = ¯λ̄ + √ ∂µ ξσ µ ξ¯ 2 i δN = ζ − √ σ µ ∂µ χ̄ 2. (3.18) (3.19). √ √ i i δAµ = σµ λ̄ + ζσµ ¯ − √ ∂µ ξ + √ ∂µ χ̄¯ 2σµν ∂ ν ξ − + 2¯σ̄µν ∂ ν χ̄ 2 2 i δ λ̄α̇ = ¯α̇ D − ¯α̇ ∂ µ Aµ − i(σ̄ µ )α̇ ∂µ M + (σ µν ¯)α̇ ∂µ Aν 2 i δζα = α D + √ α ∂ µ Aµ − i(σ µ ¯)α ∂ν N − (σµν)α ∂µ Aν 2 δD = i∂µ (ζσ µ ¯ + λ̄σ̄ µ ). (3.20) (3.21) (3.22) (3.23). Due to the transformations given above, it is easy to see that the integral of any superfield in the superspace coordinates is SUSY invariant: Z δ I[S] =. 4. dx. Z. 4. Z. d θ δ S(x, θ, θ̄) =. d4 x δD(x). (3.24). Which is a total derivative according to (3.23) and thus δ I[S] = 0. Moreover, it can be proved that (3.9) is a reducible representation and so it is possible to reduce the number of components fields by imposing some SUSY invariant constraints in order to have suitable supermultiplet representations.. To do so, we define the chiral (anti-chiral) covariant derivatives Dα and D̄α̇ , that anticommute with the SUSY generators Q and Q̄, which can be used to con-.

(43) 33. Chapter 3. INTRODUCTION TO SUPERSYMMETRY. struct supersymmetric Lagrangians: Dα = ∂α − i(σ µ )αβ̇ θ̄β̇ ∂µ. D̄α̇ = ∂¯α̇ − i(σ̄ µ )α̇β θβ ∂µ. (3.25). and they satisfy the following anticommutation relations {Dα , Qβ } = {Dα , Q̄β̇ } = 0. (3.26). {D̄α̇ , Q̄β̇ } = {Dα , Q̄β̇ } = 0. (3.27). such that δ(Dα S) = Dα (δS). Given these definitions it is now possible to find the field representations of Chiral and Vector supermultiplets.. 3.2.1. Chiral Superfields. A chiral superfield corresponds to the field realization in superspace of the chiral supermultiplet and it is defined as a superfield that satisfies the following condition: D̄α̇ Φ(x, θ, θ̄) = 0. (3.28). The same can be done with the hermitian congugate of Φ, defining the anti-chiral superfields as those superfields satisfying: Dα Φ† (x, θ, θ̄) = 0. (3.29). To find the decomposition of a chiral superfield in terms of its component fields it is useful to define a set of new variables given by, y µ ≡ xµ + iθσ µ θ̄. ȳ µ ≡ xµ − iθσ µ θ̄. (3.30). θ0 ≡ θ. θ0 ≡ θ. (3.31). θ̄0 ≡ θ̄. θ̄0 ≡ θ̄. (3.32).

(44) 3.2. Superspace Formalism. 34. such that the new coordinates satisfy: ∂ β ν D̄α̇ y = −∂α̇ − iθ σβ α̇ ν (xµ + iθσ µ θ̄) = 0 ∂x µ ν β̇ ∂ Dα ȳ = ∂α + iσαβ̇ θ (xµ − iθσ µ θ̄) = 0 ∂xν µ. (3.33) (3.34). This means that any superfield Φ1 that depends on (y µ , θ) explicitly but not on θ̄ satisfies the constraint (3.28), and a field Φ2 that depends on (ȳ µ , θ̄) explicitly but not on θ satisfies (3.29), as it would be the case for Φ2 = Φ†1 . Therefore, D̄α̇ Φ(y, θ) = 0. (3.35). Dα Φ† (ȳ µ , θ̄) = 0. (3.36). and the decomposition in terms of its component fields is given by, Φ(y, θ) = φ(y) +. √. 2θξ(y) + θθF (y) √ ¯ + θ̄θ̄F ∗ (ȳ) Φ† (ȳ, θ̄) = φ∗ (ȳ) + 2θ̄ξ(y). (3.37) (3.38). Using (3.30)-(3.32), we can express the superfields in the original supercoordinates: √ 1 i Φ(y, θ) = φ(x) − iθσ µ θ̄∂µ φ(x) − θθθ̄θ̄∂ µ ∂µ φ(x) + 2θξ(x) + √ θθ∂µ ξσ µ θ̄ + θθF (x) 4 2 (3.39) √ 1 ¯ − √i θ̄θ̄θσ µ ∂µ ξ(x) ¯ + θ̄θ̄F ∗ (x) Φ† (ȳ, θ̄) = φ∗ (x) + iθσ µ θ̄∂µ φ∗ (x) − θθθ̄θ̄∂ µ ∂µ φ∗ (x) + 2θ̄ξ(x) 4 2 (3.40) Now it can be seen that it’s possible to map a chiral superfield directly from the.

(45) 35. Chapter 3. INTRODUCTION TO SUPERSYMMETRY. generic superfield defined in (3.9) by using the following substitution, f →φ. ξ→ξ. (3.41). M →F. χ→0. (3.42). N →0. ζ→0. (3.43). i λ → √ ∂µ ξσ µ 2 i λ̄ → − √ σ̄ µ ∂µ ξ 2. Aµ → − i∂µ φ 1 D → − ∂ µ ∂µ φ 2. (3.44) (3.45). and so the SUSY transformation of the component fields of a chiral superfield can be deduced directly from (3.15)-(3.23): √ 2ξ √ √ δξα = 2α − 2i(σ µ ¯)α ∂µ φ √ δF = i∂µ ( 2ξσ µ ¯) δφ =. (3.46) (3.47) (3.48). Since the variation of the θθ-component of a chiral superfield is always a total spacetime derivative, it can be used to construct a SUSY invariant Lagrangian density. These invariant terms are called F-terms and they can be obtained by integrating appropriately in the supercoordinates: Z [Φ]F =. 2. 2. †. 2. d θ d θ̄ δ (θ̄) Φ. [Φ ]F =. Z. d2 θ d2 θ̄ δ 2 (θ) Φ†. (3.49). However, the product of a chiral and an anti-chiral superfield results in a real superfield, and in such cases the F-term won’t yield a supersymmetric action. Instead, the θθθ̄θ̄-component should be used. These are called D-terms, and they can also be obtained under a suitable integration in the supercoordinates, L=. [Φ†i Φj ]D. Z =. d2 θ d2 θ̄ Φ†j Φi. (3.50). When dealing with a single chiral superfield Φi , Eq.(3.50) results in the the kinetic.

(46) 3.2. Superspace Formalism. 36. terms of a complex scalar field and of a Weyl fermion: 1 1 1 L = Fi∗ Fi + ∂µ ∂ µ φ∗i φi − ∂µ φ∗i ∂ µ φi − φ∗i ∂µ ∂ µ φi + iξi σ µ ∂µ ξ¯i 4 2 4. (3.51). Which is the Lagrangian density of the free chiral supermultiplet, and it describes the simplest realization of a SUSY invariant field theory in 4-dimensional Minkowski space, known as the Wess-Zumino model [20].. 3.2.2. Vector Superfield. A Vector Superfield gives a superfield representation of the vector supermultiplet, and it is defined by a reality condition: V (x, θ, θ̄) = V † (x, θ, θ̄). (3.52). If V has the form of given in (3.9), then the reality condition (3.52) implies that the component fields have to satisfy the following relations: f = f∗ ≡ C. χ̄ = ξ¯. (3.53). N = M∗. ζ=λ. (3.54). D = D∗. Aµ = A∗µ. (3.55). This means that a vector superfield contains three real component fields C(x), Aµ (x) and D(x), a complex field M (x) and two complex Weyl spinors ξ(x) and λ(x). The most general expansion of V can be expressed as: √ √ i µ ∗ µ ¯ V (x, θ, θ̄) =C(x) + 2θξ + 2θ̄ξ + θθM + θ̄θ̄M + θσ θ̄Aµ + θθθ̄ λ̄ − √ σ̄ ∂µ ξ 2 i 1 1 + θ̄θ̄θ λ − √ σ µ ∂µ ξ¯ + θθθ̄θ̄ D(x) − ∂ µ ∂µ C (3.56) 2 2 2.

(47) 37. Chapter 3. INTRODUCTION TO SUPERSYMMETRY. However, the vector superfield contains too many degrees of freedom to be a representation of a vector supermultiplet. This issue can be resolved by defining a generalize abelian supergauge transformation as: V (x, θ, θ̄) −→ V (x, θ, θ̄) + iΛ − iΛ†. (3.57). Where Λ is a chiral superfield. The components fields transform under (3.57) according to, C → C + 2 Reφ M →M +F D→D. ξ →ξ+χ Aµ → Aµ − 2∂µ Imφ λ→λ. (3.58) (3.59) (3.60). Using this extra freedom, it is possible to gauge away the fields C, M , N , and ξ under an appropriate choose of Λ, leaving exactly the fields we want to have a representation of the vector supermultiplet. We also recover the usual gauge transformation for the field Aµ . This particular gauge it’s called the Wess-Zumino Gauge. In this gauge, a vector superfield has the following form, 1 VW Z = θσ µ θ̄Aµ (x) + θθθ̄λ̄(x) + θ̄θ̄θλ(x) + θθθ̄θ̄D(x) 2. (3.61). To write down kinetic termms for vector superfields we define the chiral and antichiral spinor superfield: 1 Wα = − D̄D̄Dα V 4 1 W̄ᾱ = − DDD̄ᾱ V 4. (3.62) (3.63). Because analytic functions of chiral superfields are chiral superfields themselves, we can construct SUSY invariants Lagrangian densities by taking the F-component of.

(48) 3.3. Invariant chiral interactions. 38. a product of spinor superfields, L = [W α Wα ]F + [W̄α̇ W̄ α̇ ]F. (3.64). Using (3.49), we obtain gauge kinetic terms which allow us to write the SUSY invariant Lagrangian for U (1) abelian theory describing the dynamics of a free vector supermultiplet: 1 1 L = D2 + iλσ µ ∂µ λ̄ − Fµν F µν 2 4. 3.3. (3.65). Invariant chiral interactions. In Section 3.2.1 and Section 3.2.2 we discussed how to construct invariant Lagrangian densities describing free chiral and vector supermultiplets. Here we show how to add interaction terms of chiral superfields through a Superpotential.. The superpotential is a SUSY invariant and renormalizable function of chiral (antichiral) superfields. The most general superpotential that can be written is an holomorphic cubic polynomial of chiral superfields: 1 1 W = hi Φi + mij Φi Φj + Φi Φj Φk 2 3!. (3.66). The corresponding interacting Lagrangian will be given by the F-term of (3.66): L = [W(Φ)]F + h.c.. (3.67). In terms of the component fields, Eq. (3.67) can be expressed as, L=. 1 1 hk + mik φi + fijk φi φj Fk − ξi ξj (mij + fijk φk ) + h.c. 2 2. (3.68).