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The Higgs As a Pseudo-Goldstone Boson

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Universidad Autónoma de Madrid Facultad de Ciencias Departamento de Física Teórica The Higgs as a pseudo-Goldstone boson Memoria de Tesis Doctoral realizada por Sara Saa Espina y presentada ante el Departamento de Física Teórica de la Universidad Autónoma de Madrid para la obtención del Título de Doctora en Física Teórica. Tesis Doctoral dirigida por M. Belén Gavela Legazpi, Catedrática del Departamento de Física Teórica de la Universidad Autónoma de Madrid. Madrid, 27 de junio de 2017 Contents Purpose and motivation 1 Objetivo y motivación 4 I Foundations 9 1 The Standard Model (SM) 11 1.1 Gauge fields and fermions 11 1.2 Renormalizability and unitarity 14 2 Symmetry and spontaneous breaking 17 2.1 Symmetry types and breakings 17 2.2 Spontaneous breaking of a global symmetry 18 2.2.1 Goldstone theorem 20 2.2.2 Spontaneous breaking of the chiral symmetry in QCD 21 2.3 Spontaneous breaking of a gauge symmetry 26 3 The Higgs of the SM 29 3.1 The EWSB mechanism 29 3.2 SM Higgs boson phenomenology at the LHC: production and decay 32 3.3 The Higgs boson from experiment 34 3.4 Triviality and Stability 38 4 Having a light Higgs 41 4.1 Naturalness and the “Hierarchy Problem” 41 4.2 Higgsless EWSB 44 4.3 The Higgs as a pseudo-GB 45 II Higgs Effective Field Theory (HEFT) 49 5 Effective Lagrangians 51 5.1 Generalities of EFTs 51 5.2 The Chiral Effective Lagrangian 52 5.3 Including a light Higgs 54 5.4 Linear vs. Non-linear approach 56 6 Renormalization of the scalar sector 59 6.1 Renormalization in EFTs 59 iii Contents 6.2 The scalar Lagrangian 60 6.2.1 Parametrization of the GB matrix 62 6.2.2 Counterterm Lagrangian 64 6.3 Renormalization of the off-shell Green functions 66 6.3.1 1-point functions 67 6.3.2 2-point functions 67 6.3.3 3-point functions 69 6.3.4 4-point functions 72 6.3.5 Comparison with previous works and SM limit 75 6.3.6 Dealing with the apparent non-invariant divergences (NIDs) 76 6.4 Renormalization Group Equations 78 7 Complete renormalization 81 7.1 A gauge and Yukawa theory for a manifold of scalars 81 7.1.1 One-loop renormalization 83 7.2 One-loop HEFT 85 7.2.1 Renormalization of the Leading Lagrangian 88 7.2.2 Renormalization of the Sub-leading Lagrangian 90 III Linear sigma model for a pseudo-Goldstone Higgs 95 8 Presentation of the model 97 8.1 The SO(5)/SO(4) scalar sector 98 8.1.1 The scalar potential 99 8.1.2 Scalar-gauge boson couplings 102 8.1.3 Renormalization and scalar tree-level decays 102 8.1.4 Scalar parameter space 103 8.2 Fermionic sector 106 8.2.1 The fermionic Lagrangian 108 8.3 Coleman–Weinberg potential 110 9 Phenomenology 113 9.1 Bound from Higgs measurements 113 9.2 Precision electroweak constraints 114 b 9.2.1 S, T and gL 114 9.2.2 Scalar contributions in the linear SO(5) model: h and σ 115 9.2.3 Fermionic contributions 117 9.3 Higgs and σ coupling to gluons 122 9.3.1 h ↔ gg transitions 123 9.3.2 σ ↔ gg transitions 124 9.4 Higgs and σ decay into γγ 126 9.4.1 h ↔ γγ transitions 126 9.4.2 σ ↔ γγ transitions 128 9.5 The σ resonance at the LHC 129 10 Connection with EFTs 133 10.1 Model independent analysis 134 iv Contents 10.1.1 Impact on Higgs observables 140 10.2 Explicit fermion sector 143 10.2.1 Integrating out only the heavy fermions 144 10.2.2 Integrating out only the heavy scalar 146 10.2.3 Combining the two limits 147 IV HEFT and Baryon Number Violation 149 11 Baryon non-invariant couplings in Higgs effective field theory 151 11.1 The BNV HEFT Lagrangian 152 11.2 Comparison with the SMEFT 153 11.3 Flavor contraction counting 155 Summary and conclusions 157 Resumen y conclusiones 161 Bibliography 167 v List of Figures 1.1 Particle content of the SM 12 2.1 Spontaneous symmetry breaking in a ferromagnet 18 2.2 Paraboloid versus Mexican hat potentials 19 3.1 Diagrams contributing to WW scattering 32 3.2 Local p-value as a function of the Higgs mass observed in the CMS and ATLAS experiments 32 3.3 Computed SM Higgs production cross sections and branching ratios 33 3.4 Higgs production channels at LHC 33 3.5 Higgs decay channel into photons 34 3.6 ATLAS and CMS best fits for the signal strengths 36 3.7 ATLAS and CMS combined fit of κg vs κγ. 37 3.8 ATLAS and CMS combined measurements of coupling modifiers. 38 3.9 Fit for Higgs couplings as a function of the particle mass. 39 3.10 Regions for the stability of the SM vacuum in terms of Higgs and top pole masses. 40 4.1 Relaxion potential 43 6.1 Diagram contributing to the Higgs 1-point function. 67 6.2 Diagrams contributing to the π self-energy. 68 6.3 Diagrams contributing to the Higgs self-energy. 69 6.4 Diagrams contributing to the hh → h amplitude, not including dia- grams obtained by crossing. 70 6.5 Diagrams contributing to the ππ → h scattering amplitude, not in- cluding diagrams obtained by crossing. 71 6.6 Diagrams contributing to the hh → hh amplitude, not including dia- grams obtained by crossing. 73 6.7 Diagrams contributing to the ππ → hh amplitude, not including di- agrams obtained by crossing. 73 6.8 Diagrams contributing to the ππ → ππ amplitude, not including diagrams obtained by crossing. 75 8.1 Schematics of the SO(5) → SO(4) model 97 2 8.2 Parameter space for mσ and sin γ. 105 8.3 Schematics of light fermion mass generation 107 9.1 Uncombined contributions of the scalar sector (black curve) and the exotic fermionic sector to the parameters S and T . 116 vi List of Figures 9.2 Combined contributions to S and T from the scalar sector and the exotic fermionic sector. 117 9.3 Scalar and fermionic impact on the T parameter and on the Z-bL-bL L coupling gb . 120 9.4 Examples of correlations between fermion mixings and EW precision measurements. 122 9.5 LHC constraints on sin2 γ versus σ mass parameter space neglecting exotic heavy fermion contributions. 130 9.6 LHC constraints on sin2 γ versus σ mass parameter space considering non-negligible exotic heavy fermion contributions. 131 10.1 Schematic fermion mass operator at low scales with arbitrary inser- tions of the scalar fields. 135 vii List of Tables 1.1 Transformation properties of the SM fermions 13 6.1 Operators in the L4 Lagrangian for the Higgs and GBs. 63 6.2 Counterterms from the non-linear Lagrangian (only h and GBs) con- tributing to Green functions 67 8.1 Heavy fermion charges assignments under the different groups. 108 10.1 Yukawa effective interactions for different fermion embeddings in SO(5)135 10.2 Effective operators and first corrections in 1/λ 139 10.3 Table with the definitions for the renormalization factors. 144 10.4 Leading order effective operators with heavy fermions integrated out. 145 10.5 Effective operators up to order f/Mi and 1/λ 148 viii Purpose and motivation An extremely compact and elegant formulation of the dynamics of the known funda- mental particles is given by the so-called Standard Model (SM) of particle physics, which has withstood all experimental tests since its conception around forty years ago. Aside from strong interactions, the Lagrangian of the SM describes both the electromagnetic and weak interactions by an SU(2)L × U(1)Y gauge-invariant the- ory. In 2012 LHC data showed the existence of a spin zero resonance of mass around 125 GeV [1, 2]. This particle has been identified with the boson of the Higgs mech- anism [3–6], responsible for the mass of the SM particles. Nevertheless, despite its general success, there remain observations without expla- nation within the SM. Only around 5% of the Universe energy content is explained by ordinary matter. Then, around 27% of the energy balance seems to require an explanation in terms of particles, the so-called “dark matter”, not accounted for in the SM, while the rest is known as the “dark energy”, which encodes the accelerated expansion of the Universe. Besides, the SM does not explain massive neutrinos and it contains no mechanism to describe the matter/antimatter asymmetry present in the Universe either. Moreover, the SM has three unsatisfactory internal issues: it provides no hint about the hierarchy in particle flavors observed in nature (also known as the flavor puzzle), the absence of CP violation in strong interactions requires a stringent fine-tuning and the Higgs mass seems unnaturally light if there is New Physics (NP) to which it couples. The latter is often referred to as the so-called electroweak (EW) hierarchy problem. This might not be a real problem in the sense that the SM is mathe- matically consistent; but taking naturalness seriously could be a way to provide a good intuition on NP. Furthermore, Higgs physics has an impact on most of the SM problems mentioned above. As a consequence, a different nature of the Higgs particle than the SM one can lead to a change in the overall current picture. Despite having a lot of information on the interactions of the Goldstone bosons (GBs), which are absorbed to become the longitudinal components of the EW gauge bosons, little is known about the underlying dynamics of the electroweak symmetry breaking (EWSB).
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