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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 . 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 because of the spontaneously broken approximate . This light dilaton is shown to have properties that are, in general, similar to the 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This dissertation is original, unpublished, independent work by the author, Hassan Saadi.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 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 ...... 16 2.2.1 Electroweak Sector...... 17 2.2.2 Dilaton Self-coupling...... 18 2.2.3 Couplings to Massless Gauge ...... 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 for the Gauge Coupling...... 51 D The Renormalized in ...... 52

v Acknowledgments

I would like to thank my supervisor, Professor Gordon Semenoff, in the 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 (SM) in consists of three generations of and 1 , four bosons, and the Higgs boson. The quarks are - 2 particles with 6 flavors: up, down, charm, strange, top, and bottom       u c t   ,   ,   (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       νe νµ ντ   ,   ,   (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 energy, gravity, neutrino oscillations, and the (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   νe ψLe =   (1.3) e− L   u ψLq =   (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, , 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 = ψ¯ iγµ ∂ − g0B + gτ W a ψ +e ¯ iγµ ∂ − ig0B e fermion Le µ 2 µ 2 a µ Le R µ µ R  i i  + ψ¯ iγ ∂ + g0B + τ W a + ig λ Ga ψ 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 † ¯ i h ¯ † ¯ i LY = −ζe e¯Rφ ψLe + ψLe φeR − ζd dRφ ψLq + ψLq φdR h ¯† ¯ ¯ i − ζu u¯Rφ ψLq + ψLq φuR (1.8)

† 2 † † 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)  √  3 + a Wµ 2Wµ τaWµ = √  (1.13) − 3 2Wµ −Wµ  i i  D φ = ∂ + g0B + gτ W a φ (1.14) µ µ 2 µ 2 a µ     φ+ φ0∗ ¯ ∗ φ ≡   → φ = iτ2φ =   (1.15) φ0 −φ+∗

µ and ζ is Yukawa’s coupling, abc is an antisymmetric three-index symbol, γ are Dirac matrices, τa are , 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 breaking occurs when the vacuum picks up an expectation value. Let’s choose the of the scalar field to be   0 hφi0 =   (1.16) √ν 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   a  π τa  0 φ(x) = exp i   (1.17) 2ν ν√+ρ 2   a 0  π τa  0 φ (x) = exp − i φ =   (1.18) 2ν ν√+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  √  + ρ+ν 1 −i 2gWµ 2 Dµφ = √   (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 is massless       0 Zµ sin(θW ) − cos(θW ) Bµ   =     (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 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 √ √ −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 √ e¯ReL +e ¯LeR − ζd √ dRdL + dLdR 2 2 ρ + ν h i − ζu √ u¯RuL +u ¯LuR (1.27) 2

4 and the masses of the electron, up and down quarks can be read off

ζeν me = √ (1.28) 2 ζdν md = √ (1.29) 2 ζuν mu = √ (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 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. Moreover, the SM does not specify the value of the Higgs boson mass (MH ' 126 GeV ) exactly, it only narrows it to a certain interval. Hence, the Higgs mass is a free parameter.

1.2 The Effective Potential Description

The method of semi-classical approximation is used in the effective potential description where we expand around the classical solution and see how quantum fluctuations affect the model. In this section, the effective potential method is derived. Then, we are going to apply this method to a φ4 theory.

5 1.2.1 Formalism

Consider a scalar field theory

Z R 4 Z = ei W (J) = Dφei[S(φ)+ d xJ(x)φ(x)] (1.32)

Expanding W in a Taylor series

X 1 Z W = d4x ...d4x Gn(x ...x )J(x )...J(x ) (1.33) n! 1 n 1 n 1 n n Green functions can be obtained from W by differentiating W with respect to J(x); that’s to say δW 1 Z R 4 φ (x) ≡ = Dφei[S(φ)+ d xJ(x)φ(x)]φ(x) (1.34) c δJ(x) Z

By performing a Legendre transformation, a functional effective action Γ(φc) can be ob- tained from W as Z 4 Γ(φc) = W (J) − d xJ(x)φc(x) (1.35)

By substituting (1.34) in (1.35) to eliminate φc we get Z h1 i Γ(φ ) = d4x (∂φ )2Z(φ ) − V (φ ) + ... (1.36) c 2 c c eff c where (...) means higher powers of derivatives. Note that Z Z δΓ(φc) 4 δJ(x) δW (J) 4 δJ(x) = d x − d x φc(x) − J(y) = −J(y) (1.37) δφc(y) δφc(y) δJ(x) δφc(y) This is similar to the relation between free energy and energy in thermodynamics F = E − TS. J and φ are conjugates such as T and S. We can deduce from (1.36) and (1.37) that dV (φ ) eff c = J (1.38) dφc and if there is no external source dV (φ ) eff c = 0 (1.39) dφc This means that the expectation value of φ is determined by minimizing the effective potential in the absence of an external source. Hence, the effective potential method is

6 useful in studying of the vacuum of a theory. In order to obtain the first order in quantum fluctuations we vary the generating functional

Z in (1.32) with respect to φ evaluated at φc h i δ S(φ) + R d4yJ(y)φ(y) δZ 2 0 = = −∂ φc(x) − V (φc(x)) + J(x) = 0 (1.40) δφ φc δφ(x) φc

Let’s write φ = φc + ϕ in (1.32) and expand to second order in ϕ

Z R 4 Z = e(i/~) W (J) = Dφ e(i/~)[S(φ)+ d xJ(x)φ(x)] Z (i/ )[S(φ )+R d4xJ(x)φ (x)] (i/ ) R d4x 1 [(∂ϕ)2−V 00(φ )ϕ2)] ' e ~ c c Dϕ e ~ 2 c Z (i/ )[S(φ )+R d4xJ(x)φ (x)] (−i/ ) R d4x 1 ϕ[(∂)2+V 00(φ )]ϕ = e ~ c c Dϕ e ~ 2 c

R 4 1 = e(i/~)[S(φc)+ d xJ(x)φc(x)] p 2 00 det(∂ + V (φc))   R 4 1 2 00 (i/~)[S(φc)+ d xJ(x)φc(x)]− T r log[∂ +V (φc)] = e 2 (1.41)

Hence, we obtain

Z i   W (J) = S(φ ) + d4xJ(x)φ (x) + ~T r log[∂2 + V 00(φ )] + O( 2) (1.42) c c 2 c ~

substituting this in (1.35) (the c subscript is dropped but implied for φ)

i   Γ(φ) = S(φ) + ~T r log[∂2 + V 00(φ)] + O( 2) (1.43) 2 ~

We are interested in cases when the vacuum expectation value is transitionally invariant. Therefore, φ will be independent of x, and the trace expression in momentum space becomes

  Z T r log[∂2 + V 00(φ)] = d4x hx| log[∂2 + V 00(φ)] |xi

Z Z d4k = d4x hx|ki hk| log[∂2 + V 00(φ)] |ki hk|xi (2π)4 Z Z d4k = d4x log[−k2 + V 00(φ)] (1.44) (2π)4

7 The effective potential according to equation (1.36)

i Z d4k hk2 − V 00(φ)i V (φ) = V (φ) − ~ log + O( 2) (1.45) eff 2 (2π)4 k2 ~ where the 1/k2 is added to make the logarithm argument dimensionless because we need to add a term independent of φ. Note that the same expression holds for a fermion field but with the opposite sign for the integral. Moreover, expression (1.45) is quadratically divergent. So, three counterterms are introduced in the Lagrangian to make the model finite, two of them in the effective potential as (after converting to Euclidean space) Z 4 2 00 ~ d kE hkE + V (φ)i 1 2 1 4 2 Veff (φ) = V (φ) + 4 log 2 + Bφ + Cφ + O(~ ) (1.46) 2 (2π) kE 2 4! 2 2 Integrating this expression up to a cutoff kE = Λ yields (the integral is done in Appendix B)

Λ2 (V 00(φ))2  Λ2 1 1 1 V (φ) = V (φ) + V 00(φ) − log( ) − + Bφ2 + Cφ4 (1.47) eff 32π2 64π2 V 00(φ) 2 2 4!

1.2.2

The relation between the bare and renormalized Green’s functions of scalar fields is

n (n) 2 (n) GB (p, mB, µB, gB) = Zφ G (p, m, µ, g) (1.48)

The renormalized Green’s function is finite. Because we are working in the continuum approach of renormalization group, observables are independent of the scales that we define our renormalization quantities. This implies

n d (n) 2  ∂ n µ dZφ ∂g ∂ ∂m ∂  (n) µ GB = 0 = Zφ µ + + µ + µ G (1.49) dµ ∂µ 2 Zφ dµ ∂µ ∂g ∂µ ∂m Define

µ dZφ γφ ≡ (1.50) Zφ dµ ∂g β ≡ µ (1.51) g ∂µ µ ∂m γ ≡ (1.52) m m ∂µ

8 and then we obtain  ∂ n ∂ ∂  µ + γ + β + γ m G(n) = 0 (1.53) ∂µ 2 φ ∂g m ∂m which is the Callan-Symanzik equation. The effective action can be written as ∞ X i Z Z Γ[φ ] = d4x ... d4x Γ(n)(x , ..., x )φ (x )...φ (x ) (1.54) cl n! 1 n 1 n cl 1 cl n n=2 And the relation between the renormalized and bare proper vertex functions is

n (n) 2 (n) Γ (p, m, µ, g) = Zφ ΓB (p, mB, gB) (1.55)

and the Callan-Symanzik equation becomes  ∂ ∂ ∂  µ + β + γ m − nγ Γ(n) = 0 (1.56) ∂µ ∂g m ∂m φ By using equation (1.54), equation (1.56) becomes Z  ∂ ∂ ∂ 4 δ  µ + β + γmm − γφ d x φcl(x) Γ[φcl, µ, g, m] = 0 (1.57) ∂µ ∂g ∂m δφcl(x) Now we apply (1.57) to (1.36) to get  ∂ ∂ ∂ δ  µ + β + γmm − γφ φcl(x) V = 0 (1.58) ∂µ ∂g ∂m δφcl(x)  ∂ ∂ ∂ δ  µ + β + γmm − γφ φcl(x) + 2γφ Z = 0 (1.59) ∂µ ∂g ∂m δφcl(x) Define φ  t = ln cl (1.60) µ β β¯ = (1.61) 1 − γφ γ γ¯ = φ (1.62) 1 − γφ Hence, equations (1.58) and (1.59) for a φ4 massless theory become  ∂ ∂  − + β¯ + 4¯γ V (4)(t, g) = 0 (1.63) ∂t ∂g  ∂ ∂  − + β¯ + 2¯γ Z(t, g) = 0 (1.64) ∂t ∂g

9 (4) ∂4V where V = 4 and renormalization conditions ∂φc

V (4)(0, g) = g , Z(0, g) = 1 (1.65)

Combining (1.63) and (1.64), we get

1 ∂ γ¯ = Z(0, g) (1.66) 2 ∂t ∂ β¯ = V (4)(0, g) − 4¯γg (1.67) ∂t

Equations (1.63) and (1.64) are special cases of

 ∂ ∂  − + β¯ + nγ¯ F (t, g) = 0 (1.68) ∂t ∂g

Now g0 is define as the solution of the ordinary differential equation

dg0 = β¯(g0) (1.69) dt with the boundary condition g0(0, g) = g (1.70)

The general solution to (1.68) is

h Z t i F (t, g) = f(g0(t, g)) exp n dt γ¯(g0(t, g)) (1.71) 0

where f is an arbitrary function.

1.2.3 Application to φ4 Theory

If we have a potential with λ V (φ) = φ4 (1.72) 4! then, taking derivatives of the potential and then substituting in (1.47)

 Λ2 1   λ λ2 λφ2 1 1  V (φ) = λ + B φ2 + + (log − ) + C φ4 (1.73) eff 64π2 2 4! (16π)2 2Λ2 2 4!

10 To cancel the first infinite term in (1.73) that comes from quantum fluctuations, we can impose that the renormalized mass should vanish. This implies

d2V Λ2 eff = 0 =⇒ B = −λ (1.74) 2 2 dφc 0 32π

4 d Veff However, we cannot do the same thing with 4 = 0 because the logarithm function dφc is not define at zero. In order to avoid this problem, φ is going to be evaluated at an arbitrary mass M d4V eff = λ(M) (1.75) 4 dφc M where the λ(M) (or λ(µ)) depends on the choice of M. Imposing (1.75) we obtain 3λ2  λM 2 11 C = − log( ) + (1.76) 32π2 2Λ2 3 Substituting B and C in (1.73) we get

λ λ2φ4  φ2 25 V (φ) = φ4 + log( ) − (1.77) eff 4! 256π2 M 2 6 where φ = φc (Coleman and Weinberg, 1973). Differentiating the potential in (1.77)

3λ2t V (4) = λ + (1.78) 16π2

At one loop order, there is no correction to the wave function renormalization, so Z = 1, and from (1.66) we obtainγ ¯ = 0, and from (1.67) we obtain

3λ2 β¯ = (1.79) 16π2

At one loop order, the differential equation (1.69) is

dλ0 3λ02 = (1.80) dt 16π2

The solution to this differential equation is

0 λ λ = 3λt (1.81) 1 − 16π2

11 and the effective potential becomes

(4) λ V = 3λt (1.82) 1 − 16π2 Note that to first order, equation (1.82) agrees with (1.78).

12 Chapter 2

Dilatons

In this chapter, an introduction to dilatons is provided with an application. Then, three models of dilatons are provided to explain the existence of a light Higgs boson in the SM.

2.1 Introduction

Scale invariance is a symmetry of spacetime when the coupling constant is dimensionless. Hence, a theory possessing this symmetry should be invariant when the energy or length scales change. Dilatations or scale transformations are defined as

λ : xµ → eλxµ (2.1) where x is a spacetime point and λ is a real number. This transformation can be realized linearly on fields λ : φ(x) → eλdφ(eλx) (2.2) where d is a scaling dimension. The infinitesimal transformations are

δxµ = xµ δλ (2.3)

µ δφ = (d + x ∂µ)φ δλ (2.4)

According to Noether’s theorem, if there is a continuous global symmetry, then there µ µ exists a conserved current S associated with this symmetry such that ∂µS = 0. We can

13 define a physical energy-momentum tensor δS Θµν = −2 M (2.5) δgµν

(where SM is the action for matter fields) that is symmetric and gauge invariant but it is different from the canonical energy-momentum tensor define as

µν δL ν i µν T = i ∂ φ − g L (2.6) δ(∂µφ ) where φi here are the fields in the theory. It is different because the second tensor is not symmetric, gauge invariant, and traceless. However, it can be improved by adding some terms. That is why sometimes the physical energy-momentum tensor is called the improved energy-momentum tensor when dilatations are considered (Callan et al., 1970). The dilatation current is linked to Θµν such that under scale transformation the trace of the physical energy-momentum is zero

µ µν µ µ S = Θ xν −→ ∂µS = Θµ = 0 (2.7)

To generalize the equations above, consider a Lagrangian in a basis of anomalous dimension eigen-operators X L = gi(µ)Oi(x) (2.8) i where [Oi] = di and transforms as

λdi λ Oi(x) → e Oi(e x) (2.9)

and µ is a renormalization scale that transforms under scale transformations as

µ → e−λµ (2.10)

This yields X X ∂L δL = g (µ)(d + xµ∂ )O (x) + β (g) (2.11) i i µ i i ∂g i i i

∂gi(µ) where βi(g) = µ ∂µ . From the definition of the scale current we obtain X X ∂L ∂ Sµ = θµ = g (µ)(d − 4)O (x) + β (g) (2.12) µ µ i i i i ∂g i i i

14 Therefore, if di = 4 and βi = 0, then there is scale invariance. Consider a spin-half nucleon and a pseudoscalar meson interacting with the nucleon through Yukawa’s coupling. We are going to divide the Lagrangian into a scale-symmetric part and a scale-breaking part

L = Ls + LB (2.13) 1 λ L = iψγ¯ ∂µψ + ∂ φ∂µφ + g ψγ¯ ψφ − 0 φ4 (2.14) s µ 2 µ 0 5 4! 1 L = −m ψψ¯ − µ2φ2 (2.15) B 0 2 0 where the zero subscript means bare masses and coupling constants. According to (2.1) and (2.2) transformations we get 3 δψ = ( + x ∂µ)ψ (2.16) 2 µ µ δφ = (1 + xµ∂ )φ (2.17)

µ δLs = (4 + xµ∂ )Ls (2.18)

Hence, according to (2.12), di = 4 and βi = 0 for Ls, so it is scale invariant. However, 1 δL = −(3 + x ∂µ)m ψψ¯ − (2 + x ∂µ)µ2φ2 (2.19) B µ 0 2 µ 0 After integration by parts, we obtain

¯ 2 2 µ ∆ = m0ψψ + µ0φ −→ ∂µS = ∆ (2.20)

In order to make the Lagrangian scale invariant, we can introduce a new field χ(x) in order to capture this symmetry non-linearly. By (2.2), χ should transform as

λ : χ(x) → eλχ(eλx) (2.21) and χ(x) = feσ(x)/f (2.22) where σ is a new field, and f = hχi is the order parameter where the scale symmetry breaks. σ transforms non-linearly as σ(x) σ(eλx) → + λ (2.23) f f

15 In order to make a theory scale invariant now, we multiply the scale-breaking parts by appropriate powers of eσ/f and add a free σ Lagrangian. For instance, the meson-nucleon Lagrangian (2.13) becomes (Coleman, 1985) µ2 f 2 L = L − m ψψe¯ σ/f − 0 φ2e2σ/f + ∂ eσ/f ∂µeσ/f s 0 2 2 µ χ µ2 χ 1 = L − m ψψ¯ − 0 φ2( )2 + ∂ χ∂µχ (2.24) s 0 f 2 f 2 µ In order to generalize this example to any Lagrangian, we modify

 χχ4−di g (µ) → g µ (2.25) i i f f Parameterizing χ in terms of the fluctuations around the vacuum expectation value as χ¯(x) = χ(x) − f yields 1 χ¯ L = ∂ χ∂¯ µχ¯ + Θµ + ... (2.26) χ 2 µ f µ

2.2 Model 1: Distinguishing the Higgs Boson from the Dila- ton at the Large Hadron Collider

In this section a model proposed by (Goldberger et al., 2008) is explored. The symmetry breaking of scale invariance is related to the breaking of gauge symmetry in the SM because the interactions in the SM are nearly conformal up to the QCD scale. A dilaton is a pseudo- that emerges because scale invariance is broken. The Higgs and dilaton have similar properties because they are related to . In this model, the electroweak symmetry breaking (EWSB) sector is strongly coupled; the conformal breaking scale f might not be equal to ν. So, the scale of spontaneous symmetry breaking ΛCFT ∼ 4πf causes EWSB at a scale ΛEW ∼ 4πν ≤ ΛCFT . Notice that if the dilaton is realized non-linearly, then the dilaton and the Higgs couple to the SM at the tree level in the same way with the difference that in dilaton we replace ν with f. Hence, if ν and f are close, then it would be hard to differentiate between a pseudo-dilaton and the Higgs. However, adding a term to the Lagrangian that explicitly breaks conformal symmetry would allow us to distinguish the dilaton from the Higgs.

16 2.2.1 Electroweak Sector

The electroweak chiral Lagrangian must respect two symmetries. The first symmetry is

SU(2)L ⊗ U(1) gauge symmetry. The second symmetry comes from experimental con- straints that the Higgs sector respect approximately SU(2)L ⊗ SU(2)C symmetry where

SU(2)C is custodial symmetry (Datta et al.(2009)). Accommodating SU(2)L ⊗ SU(2)C symmetry as (2, 2) requires a non-linear realization of the symmetry where U(x) is a dimensionless unimodular matrix field that is unitary (compare this to (1.17))

i P3 π τ a U(x) = e ν a=1 a (2.27) with a covariant derivative

τ 1 D U = ∂ U + ig B U 3 − ig W aτ U (2.28) µ µ 1 µ 2 2 2 µ a

The terms

† † T ≡ Uτ3U ,Vµ ≡ (DµU)U

Wµν ≡ ∂µWν − ∂νWµ + ig2[Wµ,Wν ] (2.29) respect SU(2)L covariance and U(1) invariance with T , Vµ , and Wµν having dimensions of zero, one, and two respectively. The chiral Lagrangian is

1 1 1   L = − (B )2 − T r(W 2 ) + ν2T r (D U)†(DµU) + ... (2.30) χEW 4 µν 2 µν 4 µ where Bµν as in (1.10). The first two terms have dimension four and are the kinetic terms. The third term has dimension two. The written terms in (2.30) are the only ones in the

Lagrangian when the limit of MH → ∞ of the linear theory is considered at the tree level. Yukawa’s Lagrangian become

LY = −Q¯LUmqqR − L¯LUmllR + h.c. (2.31)

where mq/ν and ml/ν are Yukawa’s matrices for quarks and leptons respectively. Note

that mq,l is a 2 × 2 diagonal matrix of 3 × 3 blocks, and in ml the lower block is zero.

17 Adding the fermion kinetic Lagrangian Lψ to the picture, we obtain the full electroweak chiral Lagrangian at low energies

L = LχEW + LY + Lψ (2.32)

In order to make LχSM scale invariance, we multiply the scale breaking part by appropriate powers of χ/f as in (2.24). LχSM couples to fermions and as in the SM with a minimal Higgs boson by replacing ν → νχ/f (expanding around hχi = f)

 χ¯ χ¯2 h 1 i χ¯ X L = 2 + M 2 W +W −µ + M 2 Z Zµ + M ψψ¯ (2.33) χSM f f 2 W µ 2 Z µ f ψ ψ

2.2.2 Dilaton Self-coupling

If there are no terms that break the symmetry explicitly, then the dilaton self-interactions become 1 c L = ∂ χ∂µχ + 4 (∂ χ∂µχ)2 + ... (2.34) χ 2 µ (4πχ)4 µ where the constant c4 ∼ O(1) is determined by the underlying Conformal Theory (CFT). However, if there are terms that break the symmetry explicitly, then this effect can be incorporated in the Lagrangian by adding an operator O(x) with scaling dimension

∆O 6= 4

LCFT → LCFT + λOO(x) (2.35)

To realize this symmetry breaking in the low-energy effective theory, a spurion field is introduced that constrains the non-derivative interactions of χ(x) as ∞ X χn(∆O−4) V (χ) = χ4 c (∆ ) (2.36) n O f n=0 n where cn ∼ λ0 is determined by the underlying CFT. We are interested in the case where the explicit breaking of the conformal symmetry is small. This can be accomplished either by an operator O that is nearly marginal (|∆O − 4|  1), or by a coefficient λO that is chosen to be small in units of f. For m and f fixed, the potential is 1 λ m2 V (χ) = m2χ2 + χ3 (2.37) 2 3! f

18 2 d2V Note that the potential of the dilaton is determined in terms of mχ = dχ2 > 0 and VEV hχi = f. To leading order, the cubic coupling is  (∆O + 1) + O(λO), when λO  1 λ = (2.38) 5 + O(|∆O − 4|) when |∆O − 4|  1

Conformal algebra and unitarity dictate λ > 2. For λO  1 and ∆O = 2, the Higgs coupling is recovered. Moreover, the nearly marginal case can be realized by Coleman- Weinberg effective potential up to corrections as 1 m2 h χ i V (χ) = χ4 4 ln( ) − 1 + O(|∆ − 4|2) (2.39) 16 f 2 f O The couplings in (2.33) can receive radiative corrections of order m2. For instance, because 2 2 2 mt Λ the top is heavy, it contributes δm ∼ 16π2f 2 where Λ is an ultraviolet (UV) cutoff. 3 mt Λ Moreover, the cubic coupling receives 16π2f 3 . These radiative corrections vanish under the assumption that in the limit of λO → 0, the theory is scale invariant. In order to incorporate this idea in the low-energy model, we make the UV cutoff dependent on χ χ Λ → Λ (2.40) f This means that χ serves as a conformal compensator as before.

2.2.3 Couplings to Massless Gauge Bosons

There are loop contributions Hγγ and Hgg because the Higgs couples to the massive gauge bosons and fermions. As mentioned above, dilatons couples to the SM like the Higgs; hence, there are dilaton couplings χγγ and χgg. These loop effects are dominated by heavy mass particles such as the . In the SM, the Higgs couples to gluons as (the Higgs background is considered to be spacetime independent) Z 1 ˆa 2 ˆaµν Γ[A, H] = Gµν(−q)Π(q ,H)G (q) + ... (2.41) 4 q where Π(q2,H) is the vacuum polarization function that includes the loop contributions from heavy particles. The one-loop contribution of fermions is 1 4 X Z 1 hx(1 − x) + 2m2H†H/ν2 i Π(q2,H) = − dx x(1 − x) × ln i (2.42) g2(µ) (4π)2 µ2 i 0

19 2 2 q is considered to be of order MH when dealing with Higgs processes. For light particles, 2 2 2 mi  q ≈ MH . Hence, the vacuum polarization function is independent of the Higgs field, and light particles have negligible effect to Hgg couplings. However, for heavy 2 particles such as the top quark, mi  MH , the q term is neglected and the Higgs couples to gluons through an operator

αs X h L = bi (Ga )2 (2.43) hGG 8π 0 ν µν i

H†H 1 h a where we have performed an expansion ln( ν2 ) = 2 + ν + ... and Gµν is the canonically i normalized field strength. We sum over heavy particles only and b0 is the contribu- i 3 b0g tion of each heavy particle to the one-loop QCD beta function βi(g) = 16π2 . For instance, 2 the top quark has b0 = 3 . Because the Higgs and dilatons couple to massless gauge bosons, we can make the replace- ment m2 χ2 2 i H†H → m2 (2.44) ν2 i f 2 in (2.42) and the Lagrangian becomes

hα α i χ L = EM c (F )2 + s c (G )2 ln( ) (2.45) χ 8π EM µν 8π G µν f where cEM and cG are the coefficients of the one-loop beta functions for the gauge couplings e and gs. As we have done before, we can split the sum to run over light and heavy particles by comparing them to the dilaton mass. Moreover, if we want to consider QCD to be conformal, then by conformal invariance

X X b0 + b0 = 0 (2.46) light heavy Hence, the effective coupling becomes

α χ¯ L = − s blight (Ga )2 (2.47) χgg 8π 0 f µν

light 2 where b0 = −11 + 3 nlight. Now, if the top quark is heavier than the dilaton mass, then nlight = 5, otherwise nlight = 6. Equation (2.47) shows a tenfold increase in the coupling

20 strength compared to the Higgs. Therefore, the coupling of dilaton to gluons is enhanced, while to the is suppressed. In the Higgs model, the Higgs might receive corrections from particles beyond the SM that are heavy with dimension-six operator as α L ⊃ s c H†H(Ga )2 (2.48) hGG 4π hg µν depending on the value of chg. On the other hand, in the dilaton model, corrections come from higher dimension operators as c L ⊃ g2 χg D Ga DαGµνα (2.49) χgg s (4πχ)2 α µν These operators are suppressed by powers of m2/f 2  1 in comparison with conformal terms. Hence, it is hard to distinguish between the Higgs and the dilaton because it is not clear whether corrections are coming from heavy particles beyond the SM in the Higgs model or enhancement of the coupling strength in the dilaton model. The low-energy Lagrangian of the dilaton below the scale 4πf is 1 1 λ m2 L = ∂ χ∂¯ µχ¯ − m2χ¯2 + χ¯3 χ 2 µ 2 3! f χ¯ X 2¯χ χ¯2 h 1 i + M ψψ¯ + ( + ) M 2 W +W −µ + M 2 Z Zµ f ψ f f 2 W µ 2 Z µ ψ α χ¯ α χ¯ + EM c (F )2 + s c (G )2 (2.50) 8π EM f µν 8π G f µν If strong and electromagnetic interactions are considered in the conformal sector at high

energies, then cEM and cG become   17 − 9 when MW < m < mt, cEM = (2.51)  11 − 3 when m > mt 2 c = 11 − n (2.52) G 3 light The branching ratios to WW ,ZZ, and fermions in dilatons are the same as in the Higgs.

The important parameters are f and m, and the Lagrangian has three couplings: λ, cG, and cEM .

21 2.3 Model 2: Effective Theory of a Light Dilaton

In this section a model proposed by (Chacko and Mishra, 2013) is demonstrated. Accord- ing to precision electroweak tests, the SM Higgs is the favored model. And it remains to solve the hierarchy problem. The simplest model of electroweak symmetry broken by strong dynamics is disfavored by experiment. A dilaton emerges in the low energy theory as a massless scalar when an exact conformal symmetry is spontaneously broken. We are interested in theories of electroweak symmetry breaking where there are operators that break conformal symmetry explicitly and increase in the infrared to become strong at the breaking scale. Hence, there does not exist an obvious reason for the existence of a light dilaton in the low energy theory.

2.3.1 Effective Theory of a Dilaton

A model for the dilaton is constructed based on a marginal operator that breaks conformal symmetry yielding a naturally light dilaton. In the beginning, we are going to work in the limit of exact conformal invariance, then we incorporate violating effects.

Effective Theory in the Limit of Exact Conformal Invariance

The theory is scale invariant in the absence of effects that break conformal symmetry. The Lagrangian has derivative terms

1 c Z∂ χ∂µχ + (∂ χ∂µχ)2 + ... (2.53) 2 µ χ4 µ and the potential is Z2κ V (χ) = 0 χ4 (2.54) 4! There is a preferred expectation value f = hχi when there is a non-derivative term in the Lagrangian even when there are no terms that break conformal symmetry explicitly. The effective potential is obtained and used to find the dilaton mass. A mass-independent

22 2 scheme (MS) is used. The coupling constants c, Z κ0, etc and Z are labeled by dimen-

sionless gi. We will begin with one loop analysis and then generalize to any number of loops. 0 At one loop, gi evolves to gi as dg χ g0 = g − i log (2.55) i i d log µ f

0 2 All gis are considered zero but Z and Z κ0. After this modification, the dilaton potential becomes h d(Z2κ ) χiχ4 V (χ) = Z2κ − 0 log (2.56) 0 d log µ f 4! Now the potential does not have scale invariance. Note that the wave function renormal- ization does not exist at one loop order

d log Z = −2γ = 0 (2.57) d log µ

From perturbation theory we get

dκ 3κ2 0 = 0 (2.58) d log µ 16π2

At this stage we can set Z to one. The Lagrangian can be made conformal by making the renormalization scale µ depends onχ ¯ as in (2.10). Therefore, making the coupling constants dimensionless, and the theory would look massless. The kinetic part becomes

1 Z∂¯ χ∂µχ (2.59) 2 µ

where Z¯ to one loop order is

dZ µ Z¯ = Z − log (2.60) d log µ f

Note that Z¯ is equal to Z since at one loop order there is no wave function renormalization. We can set Z¯ to one now. The potential becomes

h d(Z2κ ) χiχ4 V (χ) = κ¯ − 0 log (2.61) 0 d log µ µ 4!

23 whereκ ¯0 is d(Z2κ ) µ κ¯ = Z2κ − 0 log (2.62) 0 0 d log µ f We use Coleman-Weinberg equation in order to get the effective potential at one loop order 1 X  M 2  1 V = V ± M 4 log i − (2.63) eff 64π2 i µ2 2 i Hence, we obtain ! 3κ2 h  µ2  1i χ4 V (χ ) = κ − 0 log − cl (2.64) eff cl 0 2 1 2 32π 2 κ0f 2 4! Conformal invariance can be manifested by expressing 3¯κ2 h κ¯ 1i κˆ =κ ¯ + 0 log( 0 ) − (2.65) 0 0 32π2 2 2 so that the potential is κˆ V (χ ) = 0 χ4 (2.66) eff cl 4! cl

Forκ ˆ0 > 0, conformal symmetry is not broken because the breaking scale hχi = f is

driven to zero. Forκ ˆ0 < 0, conformal symmetry is broken. And forκ ˆ0 = 0, there exist a stable minimum corresponding to a massless dilaton. This result can be generalized to any number of loops by generalizing the coupling con- stants (2.55), potential (2.56), wave function renormalization (2.60), modified potential

(2.61), andκ ¯0 in (2.62) as

∞ n n X (−1) d gi h χin g0 = g + log (2.67) i i n! d log µn f n=1 ( ∞ n n 2 ) 4 X (−1) d (Z κ0)h χin χ V (χ) = log (2.68) n! d log µn f 4! n=0 ∞ X (−1)m dmZ h µim Z¯ = log (2.69) m! d log µm f m=0 ( ∞ ) X (−1)n h χin χ4 V (χ) = κ¯ log (2.70) n! 0,n f 4! n=0 ∞ m n+m 2 X (−1) d (Z κ0)h µim κ¯ = log (2.71) 0,n m! d log µn+m f m=0

24 0 respectively. The beta functions for all gis andκ ¯0,n are zero because we have conformal invariance. In order to generalize the effective potential, it is easier to work in the method of renormalization group than work with counter terms in the Lagrangian method in section (1.2). According to (1.58), the Callan-Symanzik equation becomes  ∂ ∂ ∂  µ + βi − γχcl Veff (χcl, g¯i, µ) = 0 (2.72) ∂µ ∂g¯i ∂χcl

Because we have conformal invariance, βi and γ of χ are zero. Hence, (2.72) becomes ∂ µ V (χ , g¯ , µ) = 0 (2.73) ∂µ eff cl i The effective potential is κˆ V (χ ) = 0 χ4 (2.74) eff cl 4! cl and it only has a stable minimum whenκ ˆ0 = 0.

Conformal Symmetry Violating Effects

Now let us consider effects that break conformal symmetry explicitly. Consider an operator with scaling dimension ∆ in the Lagrangian

L = LCFT + λOO(x) (2.75)

The dimensionless coupling constant becomes

ˆ ∆−4 λO = λO µ (2.76)

ˆ ˆ Let us assume λO  1 in order to be able to apply perturbation theory. λO in this case satisfies the renormalization group differential equation d log λˆ O = −(4 − ∆) (2.77) d log µ derived from (2.76). Under the transformation x → e−ωx, the operator transforms as ω∆ O(x) → e O(x). In order for the Lagrangian (2.75) to be scale invariant, λO must be promoted to a spurion as in (2.25) as

(4−∆)ω λO → e λO (2.78)

25 As before, we make the renormalization scale depends on the conformal compensator as

µχ = µχ¯. The dilaton potential now is n 2 ∞ 2− Z κ0 X Z 2 κn V (χ) = χ4 − λn χ(4−n) (2.79) 4! 4! O n=1 where  = 4 − ∆. At one loop order (2.79) becomes Z2κ Z∆/2κ V (χ) = 0 χ4 − 1 λ χ∆ (2.80) 4! 4! O where κ0 and κ1 are coupling constants. From (2.55), we know how the coupling constants evolve in the renormalization group. Hence, modifying the potential to be ( ) d(Z2κ ) χ χ4 V (χ) = Z2κ − 0 log 0 d log µ f 4! ( ) d(Z∆/2κ ) χ λ χ∆ − Z∆/2κ − 1 log O (2.81) 1 d log µ f 4!

0 2 ∆/2 All gis are considered zero but Z,Z κ0, and Z κ1. From perturbation theory we get dκ 3κ2 0 = 0 (2.82) d log µ 16π2 dκ ∆(∆ − 1)κ κ 1 = 1 0 (2.83) d log µ 32π2 In order to make coupling constants dimensionless, the coefficient of the dilaton kinetic term becomes dZ µ Z¯ = Z − log (2.84) d log µ f The potential is modified to ( ) d(Z2κ ) χ χ4 V (χ) = κ¯ − 0 log 0 d log µ µ 4! ( ) d(Z∆/2κ ) χ λ χ∆ − κ¯ − 1 log O (2.85) 1 d log µ µ 4! where d(Z2κ ) µ κ¯ = Z2κ − 0 log (2.86) 0 0 d log µ f d(Z∆/2κ ) µ κ¯ = Z∆/2κ − 1 log (2.87) 1 1 d log µ f

26 λO has a mass dimension of 4−∆ because it is considered as a spurion. The beta functions forκ ¯0 andκ ¯1 vanish because of conformal invariance by construction. We use (2.63) to find the effective potential at one loop order κ¯ κ¯ κ¯ h 3¯κ λ ∆(∆ − 1)¯κ ih 1i V (χ ) = 0 χ4 − 1 λ χ∆ + 0 0 χ4 − O 1 χ∆ Σ − (2.88) eff cl 4! cl 4! O cl 4! 32π2 cl 64π2 cl 2 where hκ¯ κ¯ i Σ = log 0 − 1 ∆(∆ − 1)λ χ∆−4 (2.89) 2 4! O cl In order to make the analysis easier, we are going to neglect the loop suppressed terms in (2.88) (including them does not change the conclusion). The minimum for the tree level potential is 4¯κ f (∆−4) = 0 (2.90) κ¯1λO∆ Hence, the dilaton mass is κ¯ κ¯ 00 2 1 ∆−2 0 2 Veff (χcl) = mσ = λO∆(4 − ∆)f = 4 (4 − ∆)f (2.91) χcl=f 4! 4! 2 ˆ by using (2.90). If we have weak coupling in our CFT, thenκ ¯0, κ¯1  (4π) , λO  1 ∆−4 implies λOf  1, and loop suppressed terms in (2.88) are negligible in this regime. 2 ˆ On the other hand, if we have strong coupling in our CFT, thenκ ¯0,¯κ1 ∼ O(4π) , and λO to be small at the scale f does not hold according to equation (2.90). The mass of the dilaton from equation (2.91) is not light anymore. Hence, if there is a CFT that is strongly coupled and it is broken explicitly by a relevant operator that grows in the infrared, then there is no obvious reason for a light dilaton to exist. Nevertheless, if (4 − ∆)  1 in (2.91) even in the strongly coupled scenario, then the dila- ton mass would become small in comparison with the strong coupling scale Λ. Therefore, if a CFT is broken by a marginal operator, then a light dilaton would exist with a mass √ that scales as mσ ∼ 4 − ∆.

This result can be generalized beyond small λO. Two things must be taken into consider- ation for this to happen. Firstly, equation (2.77) must be generalized to

d log λˆ O = −g(λˆ ) (2.92) d log µ O

27 ˆ ˆ where g(λO) is a polynomial in λO in general as ∞ ˆ X ˆn g(λO) = cnλO (2.93) n=0 ˆ Note that the lowest order in (2.93) matches (2.77) when λO  1

ˆ g(λO) = c0 = (4 − ∆) (2.94)

In a strongly coupled CFT, cn ∼ O(1) for n ≥ 1 generally. Secondly, higher order terms in (2.79) become important and must be considered. Integrating equation (2.92) ! Z dλˆ 1 G(λˆ ) = exp − O (2.95) O ˆ ˆ λO g(λO)

ˆ −1 where G(λO)µ is a renormalization group invariant. As in (2.78), we need to promote ˆ −1 G(λO)µ to make the Lagrangian (2.75) scale invariant as

ˆ ˆ −ω λO(µ) → λO(µe ) (2.96) ˆ −1 −ω ˆ −1 G(λO)µ → e G(λO)µ (2.97)

From (2.95), we can deduce that

¯ ˆ ˆ λO = λO[1 + g(λO) log µ] (2.98)

Furthermore, using (2.55) and (2.97), we can define

h χi Ω(¯ λˆ , χ/µ) = λˆ 1 − g(λˆ ) log (2.99) O O O µ such that it is invariant under infinitesimal scale transformations. Now that Ω¯ is a poly- ˆ nomial in λO. ˆ For µ ∼ f the scale of symmetry breaking and g(λO)  1, Ω¯ can be approximated as

ˆ χ−g(λO) Ω(¯ λˆ , χ/µ) = λˆ (2.100) O O µ

The potential becomes to leading order in Ω¯

χ4 V (χ) = (κ − κ Ω)¯ (2.101) 4! 0 1

28 and the dilaton mass is κ m2 = 4 0 g(λˆ )f 2 (2.102) σ 4! O There is similarity between the expressions of the dilaton mass in (2.91) and (2.102). The dilaton mass in (2.102) implies that the mass of the dilaton is determined by the scaling ˆ behavior of the operator O to be marginal at the breaking scale µ = f with g(λO)  1. ˆ ˆ However, if a CFT is strongly coupled, then λO and g(λO) are not small. Therefore, the existence of a light dilaton in such models has some tuning. Including higher order ˆ corrections in λO does not change this result. We can write

ˆ ˆ g(λO) =  + δg(λO) (2.103)

ˆ where δg(λO) captures higher order corrections. For further discussion on how to construct ˆ an effective dilaton theory that includes higher orders in λO check (Chacko and Mishra, 2013).

2.3.2 Dilaton Interactions in a Conformal SM

When there is exact conformal invariance, the dilaton interactions with the SM is deter- mined by realizing the symmetry non-linearly in the low energy effective theory. On the other hand, if there is an operator that breaks this symmetry explicitly, then we should expect deviations that correct the coupling of the dilaton to the SM. All gauge kinetic terms are expressed as 1 − F F µν (2.104) 4g2 µν In the limit of exact conformal symmetry, dilatons couple to the gauge bosons such as W as χ2 m2 W W +W µ− (2.105) f g2 µ σ From (2.22) we have χ = fe f . We expand χ to leading order in f −1 in terms of σ to obtain σ m2 2 W W +W µ− (2.106) f g2 µ

29 ˆ Now we include effects that break conformal symmetry. We will assume that  > |δg(λO)| ˆ at the breaking scale, and we will show that the same result holds when  < |δg(λO)|. The conformal compensator kinetic part becomes

∞ 1h X i 1 + α λ¯n χ(−n) ∂ χ∂µχ (2.107) 2 χ,n O µ n=1 where αχ,n ∼ O(1) depend on the operator O and the particular CFT in use. The dilaton kinetic term becomes ∞ 1h X i 1 + α λ¯n f (−n) ∂ σ∂µσ (2.108) 2 χ,n O µ n=1 To make the dilaton kinetic term canonical, σ is rescaled which changes the value of the breaking scale f by an order in (2.106) without changing the form of the interaction. Therefore, corrections to the dilaton kinetic term does not change the form of interactions σ between the dilaton and the SM to leading order in f . The gauge kinetic term (2.104) becomes

∞ 1 h X i − 1 + α λ¯n χ(−n) F F µν (2.109) 4ˆg2 W,n O µν n=1 where αW,n ∼ O(1) are dimensionless. The physical gauge coupling is modified to be

∞ 1 1 h X i = 1 + α λ¯n f (−n) (2.110) g2 4ˆg2 W,n O n=1 To leading order in σ (2.109) is c¯ σ  W F F µν (2.111) 4g2 f µν where P∞ nα λ¯n f (−n) c¯ = n=1 W,n O (2.112) W P∞ ¯n (−n) 1 + n=1 αW,nλOf ¯ − In a strongly coupled CFTc ¯W ∼ O(λOf ), making the coupling of the dilaton suppressed 2 ˆ mσ by λO, which is of order Λ2 . The gauge boson mass (2.105) becomes

∞ χ2h X imˆ 2 1 + β λ¯n χ(−n) W W +W µ− (2.113) f W,n O g2 µ n=1

30 −1 where βW,n ∼ O(1). Expanding (2.113) to leading order in f in terms of σ

σ m2 W [2 + c ]W +W µ− (2.114) f g2 W µ

The physical mass of the W boson is defined as " # 1 + P∞ β λ¯n f (−n) m2 =m ˆ 2 n=1 W,n O (2.115) W W P∞ ¯n (−n) 1 + n=1 αW,nλOf

where P∞ nβ λ¯n f (−n) c = − n=1 W,n O (2.116) W P∞ ¯n (−n) 1 + n=1 βW,nλOf 2 ˆ mσ Corrections to the dilaton coupling in this case is also of order λO ∼ Λ2 . ˆ ¯ − If  < |δg(λO)| at the scale of symmetry breaking, then we replace λOχ in (2.107), ¯ (2.109), and (2.113) by Ω(λO, χ), which in this regime becomes ¯ ¯ λO Ω(λO, χ) = ¯ (2.117) 1 + c1λO log χ

2 ¯ and corrections to the dilaton couplings are suppressed by c1Ω (λO, f) which is of order 2 mσ Λ2 again. The dilaton also couples to the massless gauge bosons, namely, photon and gluon. Note that conformal symmetry for massless bosons is not broken at the classical level like massive bosons W and Z, only at the quantum level. Therefore, under x → e−ωx trans- µν formation, FµνF becomes

b F F µν(x) → e4ω(1 + < g2ω)F F µν(x) (2.118) µν 8π2 µν

b< where 8π2 is determined as (2.51). If we want this term to be conformally invariant, then the dilaton coupling should be

b χ < log F F µν (2.119) 32π2 f µν which is similar to (2.45). Expanding this term to leading order in f −1 in terms of σ

b σ < F F µν (2.120) 32π2 f µν

31 Comparing (2.106) and (2.120), we can infer that the dilaton couples more strongly to massive gauge bosons than to massless gauge bosons. Now we include violating effects of conformal invariance would yield couplings between the conformal compensator χ and the kinetic term as

∞ 1 h X i − 1 + α λ¯n χ(−n) F F µν (2.121) 4ˆg2 A,n O µν n=1 The physical gauge coupling is modified to be

∞ 1 1 h X i = 1 + α λ¯n f (−n) (2.122) g2 gˆ2 A,n O n=1 Expand (2.121) in terms of σ, and adding the effect of (2.120), the dilaton couples to massless gauge bosons as σ h b c i < + A  F F µν (2.123) f 32π2 4g2 µν where cA is dimensionless P∞ nα λ¯n f (−n) c = n=1 A,n O (2.124) A P∞ ¯n (−n) 1 + n=1 αA,nλOf ¯ − ˆ In a strongly coupled CFT cA ∼ O(λOf ) ∼ λO at the breaking scale f. The corrections 2 mσ are suppressed by Λ2 as before. In the limit of exact conformal invariance, the dilaton couples to fermions as

χ m ψψ¯ (2.125) f ψ

Expanding (2.125) in terms of σ yields

m σ ψ ψψ¯ (2.126) f

Now including effects that break conformal symmetry would modify (2.125) as

∞ χh X i 1 + β λ¯n χ(−n) mˆ ψψ¯ (2.127) f ψ,n O ψ n=1 where βψ,n are dimensionless, andm ˆ ψ is the fermion mass in the unperturbed theory. Note that we assumed that the approximate U(3) flavor symmetry is not broken by the

32 operator O; hence, ensuring that the dilaton couples diagonally in the mass basis. The fermion kinetic term also receives corrections of the form

∞ h X ¯n (−n)i ¯ µ 1 + αψ,nλOχ ψγ ∂µψ (2.128) n=1 Combing (2.127) and (2.128), we obtain the correction to the dilaton coupling

m σ ψ [1 + c ]ψψ¯ (2.129) f ψ

2 ¯ − mσ In a strongly coupled CFT cψ ∼ O(λOf ); hence, the corrections are suppressed by Λ2 .

2.4 Model 3: A Very Light Dilaton

In this section a model proposed by (Grinstein and Uttayarat, 2011) is explored. As we have seen in section (1.2), quantum effects destroy scale invariance even when there is no explicit mass terms in the Lagrangian. The goal is to construct a model where the dilaton mass is small; that is to say, the mass can be arbitrarily small while the rest of the spectrum is kept constant and interacting. This can be achieved by starting with an interacting field theory that has a perturbative attractive infrared fixed point and scalars. By following a renormalization group trajectory that is approaching a fixed point, spontaneous symmetry breaking can be found. A specific model will be considered below where as Yang-Mills gauge coupling runs towards the Banks-Zaks IR-fixed point (Banks and Zaks, 1982), it drives Yukawa couplings and the scalar towards the values of non-trivial fixed point also. A non-trivial minimum for the scalar field may be developed from the relative values of the coupling constants. The parameters in the model can either cause spontaneous symmetry breaking of scale invariance or not. If the couplings are close to the boundary of these two regimes, then a light dilaton emerges in units of its decay constant. Note that the rest of the spectrum is interacting and robust under changes in the parameter space to keep the dilaton mass light.

33 2.4.1 The Model

In this model we consider a SU(N) that has two scalars and two ψi,

χk with nf = nχ + nψ = 2nχ flavors. Scalars are considered as singlets, and spinors as vector-like in the fundamental representation. Let us consider this Lagrangian

nχ nχ 1 X X 1 1 L = − T r(F F µν) + ψ¯jiDψ/ + χ¯kiDχ/ + (∂ φ )2 + (∂ φ )2 2 µν j k 2 µ 1 2 µ 2 j=1 k=1 1 1 1 − y (ψψ¯ +χχ ¯ )φ − y (ψχ¯ + h.c.)φ − λ φ4 − λ φ4 − λ φ2φ2 (2.130) 1 1 2 2 24 1 1 24 2 2 4 3 1 2 This Lagrangian has scale invariance at the classical level, discrete symmetry

φ1 → φ1, φ2 → −φ2, ψ → ψ, χ → −χ (2.131)

and global transformations ψ → Uψ, χ → Uχ. For nf small, the gauge coupling increases in the infrared similar to QCD. There are Goldstone bosons in the spectrum because the

chiral symmetry SU(nf ) ⊗ SU(nf ) is spontaneously broken. On the other hand, we are

interested in large nf where the gauge coupling decreases in the ultraviolet but does not increase in the infrared.

Fixed Point Structure

We consider large values of nf and N, two loop order for gauge coupling, and one loop order for Yukawa and scalar couplings. The β functions are (Machacek and Vaughn, 1983)(Machacek and Vaughn, 1984)(Machacek and Vaughn, 1985)(Caswell, 1974) ∂g δN 25N 2 g5 (16π2) = − g3 + ∂t 3 2 16π2 ∂y (16π2) 1 = 4y y2 + 11N 2y3 − 3Ng2y ∂t 1 2 1 1 ∂y (16π2) 2 = 3y2y + 11N 2y3 − 3Ng2y ∂t 1 2 2 2 ∂λ (16π2) 1 = 3λ2 + 3λ2 + 44N 2λ y2 − 264N 2y4 ∂t 1 3 1 1 1 ∂λ (16π2) 2 = 3λ2 + 3λ2 + 44N 2λ y2 − 264N 2y4 ∂t 2 3 2 2 2 ∂λ (16π2) 3 = λ λ + λ λ + 4λ2 + 22N 2λ y2 + 22N 2λ y2 − 264N 2y2y2 (2.132) ∂t 1 3 2 3 3 3 1 3 2 1 2

34 11N δ where nf = 2 (1 − 11 ) and terms of order O(δ) are neglected except in βg. Calculation for the beta function of the gauge coupling is done in Appendix C. We should show that there exist a non-trivial fixed point. In order for this to occur, the gauge coupling should start from a UV value that is smaller than the fixed point. Then, the last term in the beta functions of Yukawa couplings is negative and dominates until the positive non-linear terms start to dominate over the negative linear terms. The same process applies to the scalars. We find the zero of equations (2.132) to find the fixed point. Note that after finding the zero of the gauge coupling, it is used in the subsequent equations for Yukawa couplings, and then these are used for the scalar self-coupling. To leading order in N −1, the fixed point is at the zero of β functions

2 δ 3 g2 18 g2 g2 = 16π2 , y2 = y2 = ∗ , λ = λ = λ = ∗ (2.133) ∗ 75 N 1∗ 2∗ 11 N 1∗ 2∗ 3∗ 11 N

Theories with scalars and fermions in 4 dimensions do not have non-trivial infrared fixed point. However, this is not the case here because the gauge coupling is driving the other couplings toward the fixed point.

Vacuum Structure

The potential has a trivial minimum classically, hφ1i = hφ2i = 0. On the other hand, including quantum effects changes the vacuum as discussed in section (1.2). The effective potential at one loop in the MS scheme is

1 1 1 V = − λ φ4 − λ φ4 − λ φ2φ2 eff 24 1 1 24 2 2 4 3 1 2 11N 2M 4  M 2 3 11N 2M 4  M 2 3 − f+ ln f+ − − f− ln f− − (64π2) 2µ2 2 (64π2) 2µ2 2 M 4  M 2 3 M 4  M 2 3 + s+ ln s+ − + s− ln s− − (2.134) (64π2) µ2 2 (64π2) µ2 2

35 where

Mf± = y1φ1 ± y2φ2 (λ + λ )φ2 + (λ + λ )φ2 M 2 = 1 3 1 2 3 2 s± 4 p(λ − λ )2φ4 + (λ − λ )2φ4 − 2(λ λ − λ λ − λ λ − 7λ2)φ2φ2 ± 1 3 1 2 3 2 1 2 1 3 2 3 3 1 2(2.135) 4

As it was discussed in (Coleman and Weinberg, 1973), including masses for the spinors and scalars does not change the conclusion. It is hard to look for a minimum for this potential. However, we can use discrete symmetry (2.131) of the vacuum to find a local

minimum. Along φ2 = 0, the effective potential becomes

λ (λ φ2)2  λ φ2 3 (λ φ2)2  λ φ2 3 22N 2y4φ4  y2φ2 3 V = 1 φ4+ 1 1 ln( 1 1 )− + 3 1 ln( 3 1 )− − 1 1 ln( 1 1 )− eff 24 1 256π2 2µ2 2 256π2 2µ2 2 64π2 µ2 2 (2.136) with an extremum

∂ Veff (hφ1i) = 0 ∂φ1 λ λ2  λ hφ i2  λ2  λ hφ i2  =⇒ − 1 = 1 ln( 1 1 ) − 1 + 3 ln( 3 1 ) − 1 6 64π2 2µ2 64π2 2µ2 88N 2y4  y2 hφ i2  − 1 ln( 1 1 ) − 1 (2.137) 64π2 µ2

Note that we can have dimensional transmutation where a dimensionless parameter λ1

is replaced by a dimensional vacuum expectation value hφ1i. Large logarithms might spoil perturbative analysis in the effective potential. To avoid this problem, the condition 2 λ1 hφ1i 16π2 ln( µ2 )  1 must hold. Hence, λ1 can be expressed by using the last two terms in (2.137), and the condition becomes

λ2 − 88N 2y4 hφ i2 3 1 ln2 1  1 (2.138) (16π2)2 µ2

We need to show that the extremum that we found is a local minimum by checking that the eigenvalues of the mass matrix are both positive. Note that mixed derivatives terms

vanish at hφ1i because we have discrete symmetry and because we are working along

36 φ2 = 0. The two eigenvalues are 2 2 2 4 ∂ λ3 − 88N y1 2 2 Veff (hφ1i , 0) = 2 hφ1i (2.139) ∂φ1 32π 2 2 2 ∂ λ3 2 λ3(λ2 + 4λ3) hφ1i  λ3 hφ1i  2 Veff (hφ1i , 0) = hφ1i − 2 ln 2 + 1 ∂φ2 2 64π 2µ 264N 2y2y2 hφ i2  y2 hφ i2 1 − 1 2 1 ln 1 1 − (2.140) 64π2 µ2 3 The first eigenvalue is positive given

2 2 4 ε ≡ λ3 − 88N y1 > 0 (2.141)

And the second eigenvalue is positive if the one loop terms are small in comparison with the tree level term. Note that the value of the effective potential at hφ1i is negative λ2 − 88N 2y4 ε V (hφ i) = − 3 1 hφ i4 = − hφ i4 (2.142) eff 1 512π2 1 512π2 1

The importance of the second scalar field φ2 is that it provides us with a non-trivial minimum in the effective potential by perturbative analysis. To make ε small, we should

choose an expectation value such that it is smaller than a fixed renormalization scale µ0,

hφ1i  µ0. The coupling constant will run until the value of the scale µ starts to be

close to the heaviest particle mass in the model. At this point µ . hφ1i, the trajectory is modified. The coupling constant can run far enough to get arbitrarily close to the infrared

fixed point as long as we start with a small hφ1i. Moreover, if there are other minima

outside φ2 = 0 axis, then the same analysis applies to the global minimum. Note that scale invariance of the classical Lagrangian is broken by two effects. The first effect is the non-trivial minimum found at one loop order. The second effect is quantum fluctuations at one loop order too. If the first effect is dominant, then a pseudo Goldstone boson would appear because we have spontaneously broken approximate scale invariance. But if quantum fluctuations are dominant, then there would be no pseudo Goldstone boson.

Particle Spectrum

If hφ1i = hφ2i = 0, then all the particles are massless because the symmetry is not broken. Calculations are performed by considering one loop order so to examine the invariance of

37 physical quantities in the renormalization group flow. The fermion self-energy at large N is dominated by the gauge interaction. To one loop order, the fermion self-energy is

iΣ(p/) = i(Am + Bp/) (2.143) g2 N  y2ν2  A = − 3 ln 1 + 4 ,B = 1 (2.144) 16π2 2 µ2 in Landau gauge. Therefore, the masses of ψ and χ are

h g2 N  y2ν2 i M (µ) = M (µ) = y ν 1 − 3 ln 1 − 4 (2.145) ψ χ 1 16π2 2 µ2

Derivation of the renormalized mass in the MS scheme for fermions is explained in Ap- pendix D. The masses of scalar fields φ1 and φ2 are

λ ν2 3λ2ν2  λ ν2 5 2π  3λ2ν2  λ ν2 1 2λ  M 2 = 1 + 1 ln 1 − + √ + 3 ln 3 − − 1 φ1 2 2 2 2 2 64π 2µ 3 3 3 64π 2µ 3 3λ3 " 22N 2y2 λ ν2  λ ν2  y2ν2  + 1 y2ν2 − 1 − 3 y2ν2 − 1 ln 1 16π2 1 12 1 12 µ2 Z 1 #  2 2 x(1 − x) 2  λ1  − 3 dx y1ν − λ1ν ln 1 − x(1 − x) 2 0 2 2y1 " 3λ2ν2  2 2π  3λ2ν2 2 2λ  22N 2y2  λ ν2  1 √ 3 1 1 2 2 1 = 2 − + + 2 − + 2 − 2 y1ν − 64π 3 3 3 64π 3 3λ3 16π 12 Z 1 #  2 2 x(1 − x) 2  λ1  − 3 dx y1ν − λ1ν ln 1 − x(1 − x) 2 0 2 2y1 λ2 − 88N 2y4 ε ' 3 1 ν2 = ν2 (2.146) 32π2 32π2

38 λ ν2 λ λ ν2  λ ν2  λ λ ν2  λ ν2  M 2 = 3 + 1 3 ln 1 − 1 + 2 3 ln 3 − 1 φ2 2 64π2 2µ2 64π2 2µ2 2 2 2 Z 1 λ3ν  λ3ν  2 λ1  + 2 ln 2 + dx ln x + (1 − x) 16π 2µ 0 λ3 " 22N 2y2 λ ν2  λ ν2  y2ν2  + 2 y2ν2 − 3 − 3 y2ν2 − 3 ln 1 16π2 1 12 1 12 µ2 Z 1 #  2 2 x(1 − x) 2  λ3  − 3 dx y1ν − λ3ν ln 1 − x(1 − x) 2 0 2 2y1 λ ν2 λ λ ν2  λ ν2  λ2ν2  λ ν2  ' 3 + 2 3 ln 3 − 1 + 3 ln 3 − 2 2 64π2 2µ2 16π2 2µ2 " 22N 2y2 λ ν2  λ ν2  y2ν2  + 2 y2ν2 − 3 − 3 y2ν2 − 3 ln 1 16π2 1 12 1 12 µ2 Z 1 #  2 2 x(1 − x) 2  λ3  − 3 dx y1ν − λ3ν ln 1 − x(1 − x) 2 (2.147) 0 2 2y1

The first lines in (2.146) and (2.147) are the pole mass expressions at one loop order. In the second line in (2.146), equation (2.137) is used. And in the other lines, the approximation

that λ1 is small at the scale µ0 and the condition (2.138) are used. The vacuum expectation

value ν has the anomalous dimension of φ1

∂ν 11N 2y2 = γ ν = − 1 ν (2.148) ∂t φ1 16π2   where t = ln µ as it is defined in (1.60) with a minus sign difference in convention. µ0 Using β functions in (2.132) and the above expression we get

∂M ∂M ∂M ∂M ψ = χ = φ1 = φ2 = 0 (2.149) ∂t ∂t ∂t ∂t

Hence, the masses are RG-invariant at two loops order.

2.4.2 Dilaton

Dilatation Current

The dilatation current, Dµ defined in (2.7) corresponds to the improved energy momentum tensor defined in (2.5). As mentioned in section (2.1), the canonical energy momentum

39 tensor can be improved by adding terms. Therefore, the improved energy momentum tensor from Lagrangian (2.130) and equation (2.6) is

1 1 Θµν = −F aµλF aν + χi¯ (γµDν + γνDµ)χ + ψi¯ (γµDν + γνDµ)ψ λ 2 2 1 + ∂µφ ∂νφ − κ(∂µ∂ν − gµν∂2)φ2 − gµνL (2.150) i i 2 i

where the term proportional to κ is the improvement term (Coleman and Jackiw, 1971). 1 The integral of this term vanishes because it is a total derivative with κ = 3 . At the classical level, this current is conserved; hence, we have scale invariance. However, at the quantum level it is not, and we get a trace anomaly (Coleman and Jackiw, 1971)(Adler et al., 1977) φ4 Θµ = γ φ ∂2φ + (4γ λ − β ) 1 + ... (2.151) µ φ1 1 1 φ1 1 λ1 24

Dilaton

The dilaton state |σi can be created by acting on the vacuum with the spontaneously broken dilatation current. Similar to PCAC, we define a dilaton mass Mσ and a decay

constant fσ as µ µ 2 h0| ∂µD |σi = h0| Θµ |σix=0 = −fσMσ (2.152)

Let us consider the energy momentum tensor

f h0| Θµν(x) |σi = σ (pµpν − gµνp2)eip.x (2.153) 3

2 2 In (2.153), the momentum is defined on-shell p = Mσ .

The goal is to identify Mσ and fσ with a state in the spectrum that we have for our dilaton model in the previous section. As explained before, if we have approximate symmetry, we expect the dilaton to be light, and it couples to the stress energy tensor. Notice that Mφ1

fits this description because it is lighter than Mφ2 . In order to show that it couples to the stress energy tensor, as before, we chose to expand the fields at hφ1i = ν and hφ2i = 0, and we notice that φ1 is the only linear field term. Shifting the field φ1 in the Lagrangian

40 (2.150), the terms proportional to pµpν contribute as 1 Θµν = − ν∂µ∂νφ + ... (2.154) 3 1

Comparing (2.153) and (2.154), we identify fσ = ν and Mσ = Mφ1 . Moreover, the

anomaly in (2.151) also supports this identification as (after shifting φ1) ν3φ Θµ = γ ν∂2φ + (4γ λ − β ) 1 + ... (2.155) µ φ1 1 φ1 1 λ1 6 Hence, to lowest order we obtain λ2 + λ2 − 88N 2y4 h0| Θµ |σi = −γ νp2 − 1 3 1 ν3 + ... (2.156) µ x=0 φ1 32π2 This result matches the definition in (2.152) with the identifications ε f = ν , M 2 = M 2 = ν2 (2.157) σ σ φ1 32π2 2 3 2 which are RG-invariant and λ1ν and γφ1 νMσ terms were dropped for consistency.

2.4.3 Phase Structure

As mentioned before, a condition was set for λ1 to be small in order to have dimensional transmutation. Then, the model has a non-trivial minimum given that condition (2.141) is satisfied. The two conditions previously mentioned are not satisfied near the infrared fixed point. On the other hand, if we start at a point at large RG-time t and then follow the flow to the infrared, then there always exist points that are arbitrarily close to the infrared fixed point where spontaneous symmetry breaking occurs. So we choose at a renormalization scale µ0 coupling constants that provide a non-trivial minimum and satisfy hφ1i  µ0. In the mass independent scheme, the coupling constants will run towards the infrared fixed point. The smaller hφ1i, the closer we get to the infrared fixed point. Since we are close to the infrared fixed point, massive particles are integrated out, Yukawa and self-couplings are also not important. Hence, the beta function is governed by massless

Yang-Mills vectors, and increases as µ is decreased below hφ1i in the form 2 2 g∗ g (µ) ≈ 2 (2.158) g∗ 22N µ 1 + 2 ln 16π 3 hφ1i

41 The effective theory is similar to the theory of with mass

3 16π2 − 225 22N g2 − Mg ∼ hφ1i e ∗ = hφ1i e 44δ (2.159)

Hence, now we have nf massive fermions, two massive scalars, glueballs with mass Mg in (2.159), and the dilaton is identified with the lighter scalar with mass (2.157). However, when ε is not small, then perturbation theory fails, and the true vacuum exists at hφ1i =

hφ2i = 0 where no spontaneous symmetry breaking happens; and hence, all particles are massless. From this picture we can infer that there are two phases at a fixed renormalization scale

µ0. In the first region, ε is not small, hφ1i = 0, particles are massless, and RG trajectories

flow into the infrared fixed point. In the second region, ε is small, hφ1i = ν, particles have masses, and RG trajectories get close but do not end at the infrared fixed point. Adding relevant perturbations in the Lagrangian breaks scale invariance explicitly. Therefore, if these mass perturbations are smaller than the masses we obtained in this section, then qualitatively nothing changes, and quantitatively the corrections are small. On the other hand, if these mass perturbations are large enough, then a pseudo Goldstone boson will not exist.

42 Chapter 3

Conclusion

In this essay, an introduction to the Higgs mechanism, effective potential method, and dilatons is given. Various dilaton models are explored. In model 1, a pseudo-dilaton cou- ples to the SM energy momentum tensor with strength proportional to f −1. It was shown that the dilaton coupling to gluons is enhanced making it possible to differentiate it from the Higgs boson in collider physics experiments. In model 2, an effective theory of a light dilaton is constructed that includes explicit conformal symmetry breaking effects to any number of loops. Then, by considering a marginal operator that breaks the symmetry, it is shown that the existence of a light dilaton in a strongly coupled CFT is associated with some tuning. Finally, corrections to the dilaton interactions with a conformal SM is explained. In model 3, a SU(N) gauge theory with two spinors and scalars is considered. Then, a non-trivial minimum in the effective potential is found, and dimensional trans- mutation happens. Hence, scale invariance is broken. Provided that certain conditions are satisfied on some parameters, one of the scalars is identified with a light dilaton.

43 Bibliography

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44 A. Datta, D. Mukhopadhyaya, and A. Raychaudhuri. Physics at the Large Hadron Col- lider. Springer, 2009.

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45 Appendices

A Dirac and Gell-Mann Matrices

Dirac matrices satisfy anticommutation relations

{γµ, γν} ≡ γµγν + γνγµ = 2ηµν (A.1)

where   1 0 0 0     µν 0 −1 0 0  η =   (A.2)   0 0 −1 0    0 0 0 −1 Dirac matrices in Dirac representation are     0 I 0 j 0 σj γ =   , γ =   (A.3) 0 −I −σj 0

where I is the 2 × 2 identity matrix, and σj are Pauli 2 × 2 matrices       0 1 0 −i 1 0 σx =   , σy =   , σz =   (A.4) 1 0 i 0 0 −1

Pauli matrices satisfy commutation and anti commutation relations

ijk [σi, σj] = 2iε σk (A.5)

{σi, σj} = 2δijI (A.6)

46 where εijk is   +1, for even permutations of 123  ijk  ε = −1, for odd permutations of 123 (A.7)   0, otherwise

We can define γ5 as

5 0 1 2 3 γ = iγ γ γ γ = −iγ0γ1γ2γ3 = γ5 i = ε γµγνγργσ (A.8) 4! µνρσ

where εµνρσ is the Levi-Civita symbol defined as  +1, for even permutations of 0123   εµνρσ = −1, for odd permutations of 0123 (A.9)   0, otherwise

and

{γ5, γµ} = 0 (A.10)   0 I γ5 =   (A.11) I 0

Dirac matrices in the Weyl representation are the same for γj but different for γ0 and γ5     0 0 I 5 −I 0 γ =   , γ =   (A.12) I 0 0 I

Gell-Mann matrices form a Lie algebra that satisfy the commutation relations

[T a,T b] = if abcT c (A.13)

47 where f abc are the structure constants. For SU(3), Gell-Mann matrices in the basis a 1 a T = 2 λ are       0 1 0 0 −i 0 1 0 0       1   2   3   λ = 1 0 0 , λ = i 0 0 , λ = 0 −1 0 ,       0 0 0 0 0 0 0 0 0       0 0 1 0 0 −i 0 0 0       4   5   6   λ = 0 0 0 , λ = 0 0 0  , λ = 0 0 1 ,       1 0 0 i 0 0 0 1 0     0 0 0 1 0 0   1   7   8   λ = 0 0 −i , λ = √ 0 1 0  , (A.14)   3   0 i 0 0 0 −2

Note that there exist a Cartan subalgebra because λ3 and λ8 commute. This implies that 1 there are two quantum numbers that are observable simultaneously; isospin I3 = 2 λ3 and hypercharge Y = √1 λ . 3 8 Generally, equation (A.13) holds for any representation matrices T a for a gauge group G. The quadratic Casimir operator in an irreducible representation R is defined as

a a C2(R) ≡ T T (A.15)

where summation over a is implied and it is a multiple of the identity. One can choose to normalize the structure constants as

X acd bcd ab ab f f = Nδ = C2(G)δ (A.16) c,d

where C2(G) is the Casimir operator in the adjoint representation. The generators in the fundamental representation are normalized as 1 T r(T aT b) = δab (A.17) 2 The Dynkin index is defined as

a b ab T rR(T T ) ≡ S2(R)δ (A.18)

48 Setting a = b in equation (A.18) and summing over a we obtain

d(G)S2(R) = d(R)C2(R) (A.19)

where d(R) is the representation dimension (d(fundamental) = N and d(adjoint) = N 2 − 1), d(G) is the Lie group dimension. For instance, SU(N) has a dimension d(SU(N)) = 2 N 2−1 N − 1, C2(fundamental) = 2N , and C2(G) = C2(adjoint) = N. Note that in the adjoint representation d(R) = d(G); hence, Dynkin index and the quadratic Casimir operator are the same in this case S2(R) = C2(G) = N. In the fundamental representation 1 we have S2(R) = 2 from equation (A.17). Note that C2(F ) and C2(S) are the Casimir

operators for fermion and scalar representations, respectively, S2(F ) and S2(S) are the Dynkin indices for fermion and scalar representations, respectively.

B Integral with a Cutoff

In order to do the integral with a cutoff in (1.46), this identity should be used

Z Z Λ2 d4kG(k2) = π2 dk2k2G(k2) (B.1) 0 Now we have

Z 4 Z Λ2 ~ d kE h 2 00 2 i ~ 2 h 2 00 2 i 2 4 log(kE + V (φ)) − log(kE) = 5 2 dkE log(kE + V (φ)) − log(kE) kE 2 (2π) 2 π 0 (B.2) 2 Let kE = x and integrate by parts to get

2 Z Λ Λ4 1 x log(x + V 00(φ))dx = log(Λ2 − ) + V 00(φ)Λ2 0 2 2 (V 00(φ))2 (V 00(φ))2 1 − log(Λ2) + log(V 00(φ) − ) (B.3) 2 2 2 2 Z Λ Λ4 1 x log(x) = log(Λ2 − ) (B.4) 0 2 2 Hence, the integral in (B.2) becomes

h V 00(φ) 1i ~ V 00(φ)Λ2 + ~ (V 00(φ))2 log( ) − (B.5) 32π2 64π2 Λ2 2

49 Performing the same analysis in the MS scheme where equation (1.45) is

i   V (φ) = V (φ) − ~T r log[∂2 + V 00(φ)] (B.6) eff 2

Going to the momentum space and to the Euclidean space, we obtain

Z dDp Z dDp ln(−p2 + V 00(φ)) = i E ln(p2 + V 00(φ)) (2π)D (2π)D E ∂ Z dDp 1 = −i E D 2 00 α ∂α (2π) (pE + V (φ)) α=0 ∂  1 Γ(α − D ) 1  = −i 2 ∂α (4π)D/2 Γ(α) (V 00(φ))α−D/2 α=0 Γ(−D/2) = −i (V 00(φ))D/2 (B.7) (4π)D/2

Where D = 4−2 is the number of dimension. To obtain ordinary dimension, we introduce an arbitrary scaleµ ˜, and the effective potential becomes

1 Γ(−D/2) V (φ) = − µ˜4−D (V 00(φ))D/2 + V (φ) eff 2 (4π)D/2 1 V 00(φ)− = − (V 00(φ))2Γ(−2 + ) + V (φ) (B.8) 2(4π)2 2πµ˜2

1 1 3 In the limit of  → 0, Γ(−2 + ) = 2 (  − γ + 2 ). Hence, (B.8) becomes 1 1 3 4πµ˜2  V (φ) = − (V 00(φ))2 − γ + + log + V (φ) eff 64π2  2 V 00(φ) 1 3 Λ˜ 2  = − (V 00(φ))2 + log + V (φ) (B.9) 64π2 2 V 00(φ) where √ − γ + 1 Λ˜ = 4πe 2 2 µ˜ (B.10)

50 C Beta Function for the Gauge Coupling

According to (Machacek and Vaughn, 1983), the beta function at two loop order for the gauge field is

g3 g5 β(g) = − b − b (C.1) (4π)2 0 (4π)4 1 11 4 1 2κ b = C (G) − κS (F ) − S (S) + Y (F ) (C.2) 0 3 2 3 2 6 2 (4π)2 4 34 20 b = [C (G)]2 − κ[4C (F ) + C (G)]S (F ) 1 3 2 2 3 2 2 1 − [2C (S) + C (G)]S (S) (C.3) 2 3 2 2

From Appendix A, SU(N) has

N 2 − 1 C (G) = N,S (F ) = n ,C (F ) = ,S (S) = 0 ,C (S) = 0 2 2 f 2 2N 2 2 (C.4) 1 with κ = 2 , 1 for two-component fermions or four-component fermions respectively. Hence, b0 becomes 11 4 1 Nδ b = N − n = (C.5) 0 3 3 2 f 3

2nf with δ = 11(1 − 11N ). According to (Caswell, 1974), a gauge theory is not expected to flip the sign of the beta function and approach zero to be valid in perturbation theory. 11N Fixing the lowest order term b0 = 0 implies nf = 2 . Using this in b1 yields 34 1 N 2 − 1 20N b = N 2 − [4( ) + ]n 1 3 2 2N 3 f 34 1 10 11 = N 2 − [ (N 2 − 1) + N] N 3 N 3 2 34 11 11 55 25 = N 2 − N 2 + − N 2 ' − N 2 (C.6) 3 2 2 3 2

51 D The Renormalized Mass in Fermions

The form of the self-energy loop graph by using Feynman parameters is

Z 4 2 d k µ i(k/ + m) µ −i iΣ2(p/) = (ie) γ γ (2π)4 k2 − m2 + iε (k − p)2 + iε Z 4 Z 1 2 d k 2k/ − 4m = e 4 dx 2 2 2 2 (D.1) (2π) 0 [(k − m )(1 − x) + (p − k) x + iε]

Completing the square and shifting k → k + px yields

Z 1 Z 4 2 d k xp/ − 2m iΣ2(p/) = 2e dx 4 2 2 2 2 (D.2) 0 (2π) [k − (1 − x)(m − p x) + iε]

We regularize this integral by using dimensional in d = 4 − ε dimensions. The self-energy becomes

Z 1 Z d 2 4−d h i d k 1 Σ2(p/) = −ie µ (d − 2)xp/ − dm d 2 2 2 2 0 (2π) [k − (1 − x)(m − p x) + iε] α Z 1 h2 µ˜2 i = dx[(2 − ε)xp/ − (4 − ε)m] + ln 2 2 (D.3) 4π 0 ε (1 − x)(m − p x) whereµ ˜2 ≡ 4πe−γE µ2. The renormalized and bare Green’s functions are related as

1 1 i iGR(p/) = iGbare(p/) = 1 + δ2 1 + δ2 p/ − m0 + Σ2(p/) + ... i = (D.4) p/ − m0 + δ2p/ − m0δ2 + Σ2(p/) + ...

Using m0 = mR + mRδm it becomes

i iGR(p/) = (D.5) p/ − mR + ΣR(p/)

4 where ΣR(p/) = Σ2(p/) + δ2p/ − (δm + δ2)mR + O(e ). In the modified minimal subtraction 2 (MS), the γE and ln(4π) factors are removed. Expanding (D.3) inµ ˜ we obtain the counter terms

α 2 δ = − ( + ln(4πe−γE )) (D.6) 2 4π ε 3α 2 δ = − ( + ln(4πe−γE )) (D.7) m 4π ε

52 Hence, the self-energy becomes UV finite in the form ( ) α Z 1 h µ2 i ΣR(p/) = dx (xp/ − 2mR) ln 2 2 − xp/ + mR (D.8) 2π 0 (1 − x)(mR − p x)

We can relate the pole mass (which is related to the physical mass) and the renormalized mass by looking for a pole in the of equation (D.5) at p/ = mP . This condition gives

mP − mR + ΣR(mP ) = 0 (D.9)

To leading order, we take mP = mR and do the integral by parts in (D.8) to obtain

2 h α  µ  2 i mR = mP + ΣR(mp) = mp 1 − 4 + 3 ln 2 + O(α ) (D.10) 4π mP

Note that the renormalized mass in the MS scheme depends on the arbitrary scale µ.

53