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Dilaton As the Higgs Particle

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Dilaton as the Higgs Particle by Hassan Saadi A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENT FOR THE DEGREE OF Master of Science in THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES (Mathematics) The University of British Columbia (Vancouver) December 2016 c Hassan Saadi,2016 Abstract This essay focuses on exploring dilatons as an alternate model to the Higgs mechanism. An introductory analysis to the Higgs mechanism, effective potential method, and dilatons is provided. Then, three different models are explored on how to obtain a light dilaton that emerges as a pseudo Goldstone boson because of the spontaneously broken approximate scale invariance. This light dilaton is shown to have properties that are, in general, similar to the Higgs boson with minor differences that can differentiate between the two models in collider experiments. ii Preface This dissertation is original, unpublished, independent work by the author, Hassan Saadi. iii Table of Contents Abstract......................................... ii Preface......................................... iii Acknowledgements................................... vi 1 Introduction......................................1 1.1 The Higgs Mechanism..............................2 1.2 The Effective Potential Description.......................5 1.2.1 Formalism.................................6 1.2.2 Renormalization Group.........................8 1.2.3 Application to φ4 Theory........................ 10 2 Dilatons........................................ 13 2.1 Introduction.................................... 13 2.2 Model 1: Distinguishing the Higgs Boson from the Dilaton at the Large Hadron Collider.................................. 16 2.2.1 Electroweak Sector............................ 17 2.2.2 Dilaton Self-coupling........................... 18 2.2.3 Couplings to Massless Gauge Bosons.................. 19 2.3 Model 2: Effective Theory of a Light Dilaton................. 22 2.3.1 Effective Theory of a Dilaton...................... 22 2.3.2 Dilaton Interactions in a Conformal SM................ 29 iv 2.4 Model 3: A Very Light Dilaton......................... 33 2.4.1 The Model................................ 34 2.4.2 Dilaton.................................. 39 2.4.3 Phase Structure............................. 41 3 Conclusion....................................... 43 Bibliography...................................... 43 Appendices....................................... 46 A Dirac and Gell-Mann Matrices......................... 46 B Integral with a Cutoff.............................. 49 C Beta Function for the Gauge Coupling..................... 51 D The Renormalized Mass in Fermions...................... 52 v Acknowledgments I would like to thank my supervisor, Professor Gordon Semenoff, in the Physics Depart- ment for introducing me to the field and helping me in my thesis. I am really glad that I got the opportunity to work on this interesting project. I would also like to thank my su- pervisor Professor Nassif Ghoussoub in the Mathematics Department for his support and help. Moreover, I express my sincere appreciation to all the helpful staff of the Institute of Applied Mathematics, the Mathematics Department, and the Physics Department. And finally, I would like to thank my parents for supporting me in pursuing my dreams. vi Chapter 1 Introduction The Standard Model (SM) in particle physics consists of three generations of quarks and 1 leptons, four bosons, and the Higgs boson. The quarks are spin- 2 particles with 6 flavors: up, down, charm, strange, top, and bottom 0 1 0 1 0 1 u c t @ A ; @ A ; @ A (1.1) d s b L L L And the leptons also with 6 flavors: electron, electron neutrino, muon, muon neutrino, tau, and tau neutrino 0 1 0 1 0 1 νe νµ ντ @ A ; @ A ; @ A (1.2) e− µ− τ − L L L where the subscript L denotes left-handed components. The SM has been consistent with experiment. However, the SM does not explain some phenomena such as dark matter, dark energy, gravity, neutrino oscillations, and the hierarchy problem (why the Higgs' mass is small and doesn't receive radiative corrections). In this chapter, how the Higgs mechanism gives mass to the particles of SM is demonstrated. Then, the effective potential description is derived. 1 1.1 The Higgs Mechanism Let's define 0 1 νe Le = @ A (1.3) e− L 0 1 u Lq = @ A (1.4) d L In the following equations, one generation of quarks and leptons is considered and the neutrino is massless. The generalization to any number of generations is straightforward. The Lagrangian of the SM consists of four parts: gauge, fermion, Yukawa, and Higgs (or scalar) LSM = Lgauge + Lfermion + LY + LHiggs (1.5) 1 1 1 L = − B Bµν − (W a )2 − T r(G Gµν) (1.6) gauge 4 µν 4 µν 2 µν i i L = ¯ iγµ @ − g0B + gτ W a +e ¯ iγµ @ − ig0B e fermion Le µ 2 µ 2 a µ Le R µ µ R i i + ¯ iγ @ + g0B + τ W a + ig λ Ga Lq µ 6 µ 2 a µ s a µ Lq 2i +u ¯ iγµ @ + g0B + ig λ Gµa u R µ 3 µ s a R i + d¯ iγµ @ − g0B + ig λ Ga (1.7) R µ 3 µ s a µ h y ¯ i h ¯ y ¯ i LY = −ζe e¯Rφ Le + Le φeR − ζd dRφ Lq + Lq φdR h ¯y ¯ ¯ i − ζu u¯Rφ Lq + Lq φuR (1.8) y 2 y y 2 LHiggs = (Dµφ) (Dµφ) − µ φ φ − λ(φ φ) (1.9) 2 where Bµν = @µBν − @νBµ =) U(1)Y gauge field (1.10) a a a b c Wµν = @µWν − @νWµ + gabcWµWν =) SU(2)L gauge fields (1.11) a a a b c Gµν = @µGν − @νGµ + gsabcGµGν =) SU(3)c gauge fields (1.12) 0 p 1 3 + a Wµ 2Wµ τaWµ = @p A (1.13) − 3 2Wµ −Wµ i i D φ = @ + g0B + gτ W a φ (1.14) µ µ 2 µ 2 a µ 0 1 0 1 φ+ φ0∗ ¯ ∗ φ ≡ @ A ! φ = iτ2φ = @ A (1.15) φ0 −φ+∗ µ and ζ is Yukawa's coupling, abc is an antisymmetric three-index symbol, γ are Dirac matrices, τa are Pauli matrices, and λa are Gell-mann matrices listed in appendix A. Now let's modify the potential by assuming that µ2 < 0 so that we have a Mexican- hat potential. Spontaneous symmetry breaking occurs when the vacuum picks up an expectation value. Let's choose the vacuum expectation value of the scalar field to be 0 1 0 hφi0 = @ A (1.16) pν 2 where ν = p−µ2/λ. When the vacuum picks up an expectation value it breaks the symmetries of SU(2)L and U(1)Y , but it preserves U(1)EM symmetry. φ can be expressed in a new gauge φ0 without a phase as 0 1 a π τa 0 φ(x) = exp i @ A (1.17) 2ν νp+ρ 2 0 1 a 0 π τa 0 φ (x) = exp − i φ = @ A (1.18) 2ν νp+ρ 2 where ρ(x) and π(x) are associated with the radial and angular axises respectively in the Mexican hat potential of the Higgs, τa are pauli matrices, and πa are real fields (would-be 3 Goldstone). Expressing (1.14) after φ picking up an expectation value 0 p 1 + ρ+ν 1 −i 2gWµ 2 Dµφ = p @ A (1.19) 2 0 3 ρ+ν @µρ − i(g Bµ − gWµ ) 2 Expanding the Higgs Lagrangian (1.9) about the minimum of the Higgs potential by using (1.13) and (1.19) yields g2 1 L = W µW +(ρ + ν)2 + (g0Bµ − gW µ)(g0B − gW 3)(ρ + ν)2 Higgs 4 − µ 8 3 µ µ 1 1 1 + @µρ∂ ρ + µ2(ν + ρ)2 − λ(ν + ρ)4 (1.20) 2 µ 2 4 3 We rotate Bµ and Wµ so that the photon is massless 0 1 0 1 0 1 0 Zµ sin(θW ) − cos(θW ) Bµ @ A = @ A @ A (1.21) 3 Aµ cos(θW ) sin(θW ) Wµ where g0 sin(θW ) = (1.22) pg2 + g02 g cos(θW ) = (1.23) pg2 + g02 and the masses of the particle can be read off the Lagrangian (1.20)(Quigg, 2013) ν p M = g2 + g02 ' 91:2 GeV (1.24) Z 2 1 M ± = gν ' 80:4 GeV (1.25) W 2 MW ± ) = cos(θW ) (1.26) MZ p p −1=2 0 −1=2 and ν = (GF 2) ' 246 GeV , so φ 0 = (GF 8) ' 174 GeV where GF = −5 −2 1:663787(6) × 10 GeV is Fermi constant and sin(θW ) = 0:2122 is the weak mixing parameter. The Yukawa Lagrangian (1.8) becomes ρ + ν h i ρ + ν h ¯ ¯ i LY = −ζe p e¯ReL +e ¯LeR − ζd p dRdL + dLdR 2 2 ρ + ν h i − ζu p u¯RuL +u ¯LuR (1.27) 2 4 and the masses of the electron, up and down quarks can be read off ζeν me = p (1.28) 2 ζdν md = p (1.29) 2 ζuν mu = p (1.30) 2 It has been shown above how gauge bosons, leptons, and quarks acquire mass when the symmetry of the Higgs potential is broken when it picks up a vacuum expectation value. This mechanism is called Glashow-Salam-Weinberg model (GSW) when applied to the electroweak theory. Hence, the GSW model describes how the group SU(2)L ⊗ U(1)Y breaks to the electromagnetic gauge group U(1)EM Higgs SU(3)c ⊗ SU(2)L ⊗ U(1)Y −−−! SU(3)c ⊗ U(1)EM (1.31) | {z } GSW U(1)EM The original four generators (τa;Y ) are broken, but the electric charge operator Q = 1 2 (τ3 + Y ) is not. Hence, by Goldstone theorem the photon and the gluons are massless. One of the problems with the Higgs model is that it's not a dynamical model, that's to say, it does not explain why the symmetry was broken in the first place. The Higgs model suffers from the hierarchy problem that implies that the square mass of the Higgs should be finely tuned (to cancel radiative quantum corrections) with the correct sign in order to produce the scale ν.
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