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Dilaton as the Higgs particle Saadi, Hassan 2017

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Dilaton as the Higgs ParticlebyHassan SaadiA THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENT FOR THE DEGREE OFMaster of ScienceinTHE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES(Mathematics)The University of British Columbia(Vancouver)December 2016c©Hassan Saadi,2016AbstractThis essay focuses on exploring dilatons as an alternate model to the Higgs mechanism. Anintroductory analysis to the Higgs mechanism, effective potential method, and dilatons isprovided. Then, three different models are explored on how to obtain a light dilaton thatemerges as a pseudo Goldstone boson because of the spontaneously broken approximatescale invariance. This light dilaton is shown to have properties that are, in general, similarto the Higgs boson with minor differences that can differentiate between the two modelsin collider experiments.iiPrefaceThis dissertation is original, unpublished, independent work by the author, Hassan Saadi.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 The Higgs Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 The Effective Potential Description . . . . . . . . . . . . . . . . . . . . . . . 51.2.1 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2.2 Renormalization Group . . . . . . . . . . . . . . . . . . . . . . . . . 81.2.3 Application to φ4 Theory . . . . . . . . . . . . . . . . . . . . . . . . 102 Dilatons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Model 1: Distinguishing the Higgs Boson from the Dilaton at the LargeHadron Collider . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2.1 Electroweak Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2.2 Dilaton Self-coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2.3 Couplings to Massless Gauge Bosons . . . . . . . . . . . . . . . . . . 192.3 Model 2: Effective Theory of a Light Dilaton . . . . . . . . . . . . . . . . . 222.3.1 Effective Theory of a Dilaton . . . . . . . . . . . . . . . . . . . . . . 222.3.2 Dilaton Interactions in a Conformal SM . . . . . . . . . . . . . . . . 29iv2.4 Model 3: A Very Light Dilaton . . . . . . . . . . . . . . . . . . . . . . . . . 332.4.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.4.2 Dilaton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.4.3 Phase Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46A Dirac and Gell-Mann Matrices . . . . . . . . . . . . . . . . . . . . . . . . . 46B Integral with a Cutoff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49C Beta Function for the Gauge Coupling . . . . . . . . . . . . . . . . . . . . . 51D The Renormalized Mass in Fermions . . . . . . . . . . . . . . . . . . . . . . 52vAcknowledgmentsI 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 Igot 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 andhelp. Moreover, I express my sincere appreciation to all the helpful staff of the Institute ofApplied Mathematics, the Mathematics Department, and the Physics Department. Andfinally, I would like to thank my parents for supporting me in pursuing my dreams.viChapter 1IntroductionThe Standard Model (SM) in particle physics consists of three generations of quarks andleptons, four bosons, and the Higgs boson. The quarks are spin-12 particles with 6 flavors:up, down, charm, strange, top, and bottomudL,csL,tbL(1.1)And the leptons also with 6 flavors: electron, electron neutrino, muon, muon neutrino,tau, and tau neutrino νee−L,νµµ−L,νττ−L(1.2)where the subscript L denotes left-handed components. The SM has been consistent withexperiment. 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 Higgsmechanism gives mass to the particles of SM is demonstrated. Then, the effective potentialdescription is derived.11.1 The Higgs MechanismLet’s defineψLe =νee−L(1.3)ψLq =udL(1.4)In the following equations, one generation of quarks and leptons is considered and theneutrino 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 (orscalar)LSM = Lgauge + Lfermion + LY + LHiggs (1.5)Lgauge = −14BµνBµν − 14(W aµν)2 − 12Tr(GµνGµν) (1.6)Lfermion = ψ¯Leiγµ(∂µ − i2g′Bµ +i2gτaWaµ)ψLe + e¯Riγµ(∂µ − ig′Bµ)eR+ ψ¯Lq iγ(∂µ +i6g′Bµ +i2τaWaµ + igsλaGaµ)ψLq+ u¯Riγµ(∂µ +2i3g′Bµ + igsλaGµa)uR+ d¯Riγµ(∂µ − i3g′Bµ + igsλaGaµ)(1.7)LY = −ζe[e¯Rφ†ψLe + ψ¯LeφeR]− ζd[d¯Rφ†ψLq + ψ¯LqφdR]− ζu[u¯Rφ¯†ψLq + ψ¯Lq φ¯uR](1.8)LHiggs = (Dµφ)†(Dµφ)− µ2φ†φ− λ(φ†φ)2 (1.9)2whereBµν = ∂µBν − ∂νBµ =⇒ U(1)Y gauge field (1.10)W aµν = ∂µWaν − ∂νW aµ + gabcW bµW cν =⇒ SU(2)L gauge fields (1.11)Gaµν = ∂µGaν − ∂νGaµ + gsabcGbµGcν =⇒ SU(3)c gauge fields (1.12)τaWaµ = W 3µ √2W+µ√2W−µ −W 3µ (1.13)Dµφ =(∂µ +i2g′Bµ +i2gτaWaµ)φ (1.14)φ ≡φ+φ0→ φ¯ = iτ2φ∗ = φ0∗−φ+∗ (1.15)and ζ is Yukawa’s coupling, abc is an antisymmetric three-index symbol, γµ are Diracmatrices, 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 anexpectation value. Let’s choose the vacuum expectation value of the scalar field to be〈φ〉0 = 0ν√2 (1.16)where ν =√−µ2/λ. When the vacuum picks up an expectation value it breaks thesymmetries of SU(2)L and U(1)Y , but it preserves U(1)EM symmetry. φ can be expressedin a new gauge φ′ without a phase asφ(x) = exp(ipiaτa2ν) 0ν+ρ√2 (1.17)φ′(x) = exp(− ipiaτa2ν)φ = 0ν+ρ√2 (1.18)where ρ(x) and pi(x) are associated with the radial and angular axises respectively in theMexican hat potential of the Higgs, τa are pauli matrices, and pia are real fields (would-be3Goldstone). Expressing (1.14) after φ picking up an expectation valueDµφ =1√2 −i√2gW+µ ρ+ν2∂µρ− i(g′Bµ − gW 3µ)ρ+ν2 (1.19)Expanding the Higgs Lagrangian (1.9) about the minimum of the Higgs potential by using(1.13) and (1.19) yieldsLHiggs = g24Wµ−W+µ (ρ+ ν)2 +18(g′Bµ − gWµ3 )(g′Bµ − gW 3µ)(ρ+ ν)2+12∂µρ∂µρ+12µ2(ν + ρ)2 − 14λ(ν + ρ)4 (1.20)We rotate Bµ and W3µ so that the photon is masslessZ0µAµ =sin(θW ) − cos(θW )cos(θW ) sin(θW )BµW 3µ (1.21)wheresin(θW ) =g′√g2 + g′2(1.22)cos(θW ) =g√g2 + g′2(1.23)and the masses of the particle can be read off the Lagrangian (1.20) (Quigg, 2013)MZ =ν2√g2 + g′2 ' 91.2 GeV (1.24)MW± =12gν ' 80.4 GeV (1.25)⇒ MW±MZ= cos(θW ) (1.26)and ν = (GF√2)−1/2 ' 246 GeV , so 〈φ0〉0= (GF√8)−1/2 ' 174 GeV where GF =1.663787(6) × 10−5 GeV −2 is Fermi constant and sin(θW ) = 0.2122 is the weak mixingparameter. The Yukawa Lagrangian (1.8) becomesLY = −ζe ρ+ ν√2[e¯ReL + e¯LeR]− ζd ρ+ ν√2[d¯RdL + d¯LdR]− ζu ρ+ ν√2[u¯RuL + u¯LuR](1.27)4and the masses of the electron, up and down quarks can be read offme =ζeν√2(1.28)md =ζdν√2(1.29)mu =ζuν√2(1.30)It has been shown above how gauge bosons, leptons, and quarks acquire mass when thesymmetry 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 theelectroweak theory. Hence, the GSW model describes how the group SU(2)L ⊗ U(1)Ybreaks to the electromagnetic gauge group U(1)EMSU(3)c ⊗ SU(2)L ⊗ U(1)Y︸ ︷︷ ︸GSW U(1)EMHiggs−−−→ SU(3)c ⊗ U(1)EM (1.31)The original four generators (τa, Y ) are broken, but the electric charge operator Q =12(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 tosay, it does not explain why the symmetry was broken in the first place. The Higgs modelsuffers from the hierarchy problem that implies that the square mass of the Higgs shouldbe finely tuned (to cancel radiative quantum corrections) with the correct sign in orderto produce the scale ν. Moreover, the SM does not specify the value of the Higgs bosonmass (MH ' 126 GeV ) exactly, it only narrows it to a certain interval. Hence, the Higgsmass is a free parameter.1.2 The Effective Potential DescriptionThe method of semi-classical approximation is used in the effective potential descriptionwhere we expand around the classical solution and see how quantum fluctuations affectthe model. In this section, the effective potential method is derived. Then, we are goingto apply this method to a φ4 theory.51.2.1 FormalismConsider a scalar field theoryZ = eiW (J) =∫Dφei[S(φ)+∫d4xJ(x)φ(x)] (1.32)Expanding W in a Taylor seriesW =∑n1n!∫d4x1...d4xnGn(x1...xn)J(x1)...J(xn) (1.33)Green functions can be obtained from W by differentiating W with respect to J(x); that’sto sayφc(x) ≡ δWδJ(x)=1Z∫Dφei[S(φ)+∫d4xJ(x)φ(x)]φ(x) (1.34)By performing a Legendre transformation, a functional effective action Γ(φc) can be ob-tained from W asΓ(φc) = W (J)−∫d4xJ(x)φc(x) (1.35)By substituting (1.34) in (1.35) to eliminate φc we getΓ(φc) =∫d4x[12(∂φc)2Z(φc)− Veff (φc) + ...](1.36)where (...) means higher powers of derivatives. Note thatδΓ(φc)δφc(y)=∫d4xδJ(x)δφc(y)δW (J)δJ(x)−∫d4xδJ(x)δφc(y)φc(x)− J(y) = −J(y) (1.37)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)thatdVeff (φc)dφc= J (1.38)and if there is no external sourcedVeff (φc)dφc= 0 (1.39)This means that the expectation value of φ is determined by minimizing the effectivepotential in the absence of an external source. Hence, the effective potential method is6useful in studying symmetry breaking of the vacuum of a theory.In order to obtain the first order in quantum fluctuations we vary the generating functionalZ in (1.32) with respect to φ evaluated at φcδZδφ∣∣∣φc=δ[S(φ) +∫d4yJ(y)φ(y)]δφ(x)∣∣∣φc= −∂2φc(x)− V ′(φc(x)) + J(x) = 0 (1.40)Let’s write φ = φc + ϕ in (1.32) and expand to second order in ϕZ = e(i/~)W (J) =∫Dφe(i/~)[S(φ)+∫d4xJ(x)φ(x)]' e(i/~)[S(φc)+∫d4xJ(x)φc(x)]∫Dϕe(i/~)∫d4x 12[(∂ϕ)2−V ′′(φc)ϕ2)]= e(i/~)[S(φc)+∫d4xJ(x)φc(x)]∫Dϕe(−i/~)∫d4x 12ϕ[(∂)2+V ′′(φc)]ϕ= e(i/~)[S(φc)+∫d4xJ(x)φc(x)] 1√det(∂2 + V ′′(φc))= e(i/~)[S(φc)+∫d4xJ(x)φc(x)]− 12Tr(log[∂2+V ′′(φc)])(1.41)Hence, we obtainW (J) = S(φc) +∫d4xJ(x)φc(x) +i~2Tr(log[∂2 + V ′′(φc)])+O(~2) (1.42)substituting this in (1.35) (the c subscript is dropped but implied for φ)Γ(φ) = S(φ) +i~2Tr(log[∂2 + V ′′(φ)])+O(~2) (1.43)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 spacebecomesTr(log[∂2 + V ′′(φ)])=∫d4x 〈x| log[∂2 + V ′′(φ)] |x〉=∫d4x∫d4k(2pi)4〈x|k〉 〈k| log[∂2 + V ′′(φ)] |k〉 〈k|x〉=∫d4x∫d4k(2pi)4log[−k2 + V ′′(φ)] (1.44)7The effective potential according to equation (1.36)Veff (φ) = V (φ)− i~2∫d4k(2pi)4log[k2 − V ′′(φ)k2]+O(~2) (1.45)where the 1/k2 is added to make the logarithm argument dimensionless because we needto add a term independent of φ. Note that the same expression holds for a fermion fieldbut with the opposite sign for the integral. Moreover, expression (1.45) is quadraticallydivergent. So, three counterterms are introduced in the Lagrangian to make the modelfinite, two of them in the effective potential as (after converting to Euclidean space)Veff (φ) = V (φ) +~2∫d4kE(2pi)4log[k2E + V ′′(φ)k2E]+12Bφ2 +14!Cφ4 +O(~2) (1.46)Integrating this expression up to a cutoff k2E = Λ2 yields (the integral is done in AppendixB)Veff (φ) = V (φ) +Λ232pi2V ′′(φ)− (V′′(φ))264pi2(log(Λ2V ′′(φ))− 12)+12Bφ2 +14!Cφ4 (1.47)1.2.2 Renormalization GroupThe relation between the bare and renormalized Green’s functions of scalar fields isG(n)B (p,mB, µB, gB) = Zn2φ G(n)(p,m, µ, g) (1.48)The renormalized Green’s function is finite. Because we are working in the continuumapproach of renormalization group, observables are independent of the scales that wedefine our renormalization quantities. This impliesµddµG(n)B = 0 = Zn2φ(µ∂∂µ+n2µZφdZφdµ+ µ∂g∂µ∂∂g+ µ∂m∂µ∂∂m)G(n) (1.49)Defineγφ ≡ µZφdZφdµ(1.50)βg ≡ µ∂g∂µ(1.51)γm ≡ µm∂m∂µ(1.52)8and then we obtain (µ∂∂µ+n2γφ + β∂∂g+ γmm∂∂m)G(n) = 0 (1.53)which is the Callan-Symanzik equation.The effective action can be written asΓ[φcl] =∞∑n=2in!∫d4x1...∫d4xn Γ(n)(x1, ..., xn)φcl(x1)...φcl(xn) (1.54)And the relation between the renormalized and bare proper vertex functions isΓ(n)(p,m, µ, g) = Zn2φ Γ(n)B (p,mB, gB) (1.55)and the Callan-Symanzik equation becomes(µ∂∂µ+ β∂∂g+ γmm∂∂m− nγφ)Γ(n) = 0 (1.56)By using equation (1.54), equation (1.56) becomes(µ∂∂µ+ β∂∂g+ γmm∂∂m− γφ∫d4xφcl(x)δδφcl(x))Γ[φcl, µ, g,m] = 0 (1.57)Now we apply (1.57) to (1.36) to get(µ∂∂µ+ β∂∂g+ γmm∂∂m− γφ φcl(x) δδφcl(x))V = 0 (1.58)(µ∂∂µ+ β∂∂g+ γmm∂∂m− γφ φcl(x) δδφcl(x)+ 2γφ)Z = 0 (1.59)Definet = ln(φclµ)(1.60)β¯ =β1− γφ (1.61)γ¯ =γφ1− γφ (1.62)Hence, equations (1.58) and (1.59) for a φ4 massless theory become(− ∂∂t+ β¯∂∂g+ 4γ¯)V (4)(t, g) = 0 (1.63)(− ∂∂t+ β¯∂∂g+ 2γ¯)Z(t, g) = 0 (1.64)9where V (4) = ∂4V∂φ4cand renormalization conditionsV (4)(0, g) = g , Z(0, g) = 1 (1.65)Combining (1.63) and (1.64), we getγ¯ =12∂∂tZ(0, g) (1.66)β¯ =∂∂tV (4)(0, g)− 4γ¯g (1.67)Equations (1.63) and (1.64) are special cases of(− ∂∂t+ β¯∂∂g+ nγ¯)F (t, g) = 0 (1.68)Now g′ is define as the solution of the ordinary differential equationdg′dt= β¯(g′) (1.69)with the boundary conditiong′(0, g) = g (1.70)The general solution to (1.68) isF (t, g) = f(g′(t, g)) exp[n∫ t0dt γ¯(g′(t, g))](1.71)where f is an arbitrary function.1.2.3 Application to φ4 TheoryIf we have a potential withV (φ) =λ4!φ4 (1.72)then, taking derivatives of the potential and then substituting in (1.47)Veff (φ) =( Λ264pi2λ+12B)φ2 +( λ4!+λ2(16pi)2(logλφ22Λ2− 12) +14!C)φ4 (1.73)10To cancel the first infinite term in (1.73) that comes from quantum fluctuations, we canimpose that the renormalized mass should vanish. This impliesd2Veffdφ2c∣∣∣0= 0 =⇒ B = −λ Λ232pi2(1.74)However, we cannot do the same thing withd4Veffdφ4c= 0 because the logarithm functionis not define at zero. In order to avoid this problem, φ is going to be evaluated at anarbitrary mass Md4Veffdφ4c∣∣∣M= λ(M) (1.75)where the coupling constant λ(M) (or λ(µ)) depends on the choice of M . Imposing (1.75)we obtainC = − 3λ232pi2(log(λM22Λ2) +113)(1.76)Substituting B and C in (1.73) we getVeff (φ) =λ4!φ4 +λ2φ4256pi2(log(φ2M2)− 256)(1.77)where φ = φc (Coleman and Weinberg, 1973).Differentiating the potential in (1.77)V (4) = λ+3λ2t16pi2(1.78)At one loop order, there is no correction to the wave function renormalization, so Z = 1,and from (1.66) we obtain γ¯ = 0, and from (1.67) we obtainβ¯ =3λ216pi2(1.79)At one loop order, the differential equation (1.69) isdλ′dt=3λ′216pi2(1.80)The solution to this differential equation isλ′ =λ1− 3λt16pi2(1.81)11and the effective potential becomesV (4) =λ1− 3λt16pi2(1.82)Note that to first order, equation (1.82) agrees with (1.78).12Chapter 2DilatonsIn this chapter, an introduction to dilatons is provided with an application. Then, threemodels of dilatons are provided to explain the existence of a light Higgs boson in the SM.2.1 IntroductionScale 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 lengthscales change. Dilatations or scale transformations are defined asλ : xµ → eλxµ (2.1)where x is a spacetime point and λ is a real number. This transformation can be realizedlinearly on fieldsλ : φ(x)→ eλdφ(eλx) (2.2)where d is a scaling dimension. The infinitesimal transformations areδxµ = xµ δλ (2.3)δφ = (d+ xµ∂µ)φ δλ (2.4)According to Noether’s theorem, if there is a continuous global symmetry, then thereexists a conserved current Sµ associated with this symmetry such that ∂µSµ = 0. We can13define a physical energy-momentum tensorΘµν = −2δSMδgµν(2.5)(where SM is the action for matter fields) that is symmetric and gauge invariant but it isdifferent from the canonical energy-momentum tensor define asTµν =δLδ(∂µφi)∂νφi − gµνL (2.6)where φi here are the fields in the theory. It is different because the second tensor isnot symmetric, gauge invariant, and traceless. However, it can be improved by addingsome terms. That is why sometimes the physical energy-momentum tensor is called theimproved 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 ofthe physical energy-momentum is zeroSµ = Θµνxν −→ ∂µSµ = Θµµ = 0 (2.7)To generalize the equations above, consider a Lagrangian in a basis of anomalous dimensioneigen-operatorsL =∑igi(µ)Oi(x) (2.8)where [Oi] = di and transforms asOi(x)→ eλdiOi(eλx) (2.9)and µ is a renormalization scale that transforms under scale transformations asµ→ e−λµ (2.10)This yieldsδL =∑igi(µ)(di + xµ∂µ)Oi(x) +∑iβi(g)∂L∂gi(2.11)where βi(g) = µ∂gi(µ)∂µ . From the definition of the scale current we obtain∂µSµ = θµµ =∑igi(µ)(di − 4)Oi(x) +∑iβi(g)∂L∂gi(2.12)14Therefore, if di = 4 and βi = 0, then there is scale invariance.Consider a spin-half nucleon and a pseudoscalar meson interacting with the nucleonthrough Yukawa’s coupling. We are going to divide the Lagrangian into a scale-symmetricpart and a scale-breaking partL = Ls + LB (2.13)Ls = iψ¯γµ∂µψ + 12∂µφ∂µφ+ g0ψ¯γ5ψφ− λ04!φ4 (2.14)LB = −m0ψ¯ψ − 12µ20φ2 (2.15)where the zero subscript means bare masses and coupling constants. According to (2.1)and (2.2) transformations we getδψ = (32+ xµ∂µ)ψ (2.16)δφ = (1 + xµ∂µ)φ (2.17)δLs = (4 + xµ∂µ)Ls (2.18)Hence, according to (2.12), di = 4 and βi = 0 for Ls, so it is scale invariant. However,δLB = −(3 + xµ∂µ)m0ψ¯ψ − 12(2 + xµ∂µ)µ20φ2 (2.19)After integration by parts, we obtain∆ = m0ψ¯ψ + µ20φ2 −→ ∂µSµ = ∆ (2.20)In order to make the Lagrangian scale invariant, we can introduce a new field χ(x) inorder to capture this symmetry non-linearly. By (2.2), χ should transform asλ : χ(x)→ eλχ(eλx) (2.21)andχ(x) = feσ(x)/f (2.22)where σ is a new field, and f = 〈χ〉 is the order parameter where the scale symmetrybreaks. σ transforms non-linearly asσ(x)f→ σ(eλx)f+ λ (2.23)15In order to make a theory scale invariant now, we multiply the scale-breaking parts byappropriate powers of eσ/f and add a free σ Lagrangian. For instance, the meson-nucleonLagrangian (2.13) becomes (Coleman, 1985)L = Ls −m0ψ¯ψeσ/f − µ202φ2e2σ/f +f22∂µeσ/f∂µeσ/f= Ls −m0ψ¯ψχf− µ202φ2(χf)2 +12∂µχ∂µχ (2.24)In order to generalize this example to any Lagrangian, we modifygi(µ)→ gi(µχf)(χf)4−di(2.25)Parameterizing χ in terms of the fluctuations around the vacuum expectation value asχ¯(x) = χ(x)− f yieldsLχ = 12∂µχ¯∂µχ¯+χ¯fΘµµ + ... (2.26)2.2 Model 1: Distinguishing the Higgs Boson from the Dila-ton at the Large Hadron ColliderIn this section a model proposed by (Goldberger et al., 2008) is explored. The symmetrybreaking of scale invariance is related to the breaking of gauge symmetry in the SMbecause the interactions in the SM are nearly conformal up to the QCD scale. A dilatonis a pseudo-Goldstone boson that emerges because scale invariance is broken. The Higgsand dilaton have similar properties because they are related to conformal symmetry. Inthis model, the electroweak symmetry breaking (EWSB) sector is strongly coupled; theconformal breaking scale f might not be equal to ν. So, the scale of spontaneous symmetrybreaking ΛCFT ∼ 4pif causes EWSB at a scale ΛEW ∼ 4piν ≤ ΛCFT . Notice that if thedilaton is realized non-linearly, then the dilaton and the Higgs couple to the SM at thetree 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 andthe Higgs. However, adding a term to the Lagrangian that explicitly breaks conformalsymmetry would allow us to distinguish the dilaton from the Higgs.162.2.1 Electroweak SectorThe electroweak chiral Lagrangian must respect two symmetries. The first symmetry isSU(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 whereSU(2)C is custodial symmetry (Datta et al. (2009)). Accommodating SU(2)L ⊗ SU(2)Csymmetry as (2, 2) requires a non-linear realization of the symmetry where U(x) is adimensionless unimodular matrix field that is unitary (compare this to (1.17))U(x) = eiν∑3a=1 piaτa(2.27)with a covariant derivativeDµU = ∂µU + ig1BµUτ32− 12ig2WaµτaU (2.28)The termsT ≡ 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 dimensionsof zero, one, and two respectively. The chiral Lagrangian isLχEW = −14(Bµν)2 − 12Tr(W 2µν) +14ν2Tr((DµU)†(DµU))+ ... (2.30)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 theLagrangian when the limit of MH →∞ of the linear theory is considered at the tree level.Yukawa’s Lagrangian becomeLY = −Q¯LUmqqR − L¯LUmllR + h.c. (2.31)where mq/ν and ml/ν are Yukawa’s matrices for quarks and leptons respectively. Notethat mq,l is a 2 × 2 diagonal matrix of 3 × 3 blocks, and in ml the lower block is zero.17Adding the fermion kinetic Lagrangian Lψ to the picture, we obtain the full electroweakchiral Lagrangian at low energiesL = LχEW + LY + Lψ (2.32)In order to make LχSM scale invariance, we multiply the scale breaking part by appropriatepowers of χ/f as in (2.24). LχSM couples to fermions and gauge boson as in the SM witha minimal Higgs boson by replacing ν → νχ/f (expanding around 〈χ〉 = f)LχSM =(2χ¯f+χ¯2f2)[M2WW+µ W−µ +12M2ZZµZµ]+χ¯f∑ψMψψ¯ψ (2.33)2.2.2 Dilaton Self-couplingIf there are no terms that break the symmetry explicitly, then the dilaton self-interactionsbecomeLχ = 12∂µχ∂µχ+c4(4piχ)4(∂µχ∂µχ)2 + ... (2.34)where the constant c4 ∼ O(1) is determined by the underlying Conformal Field Theory(CFT). However, if there are terms that break the symmetry explicitly, then this effectcan be incorporated in the Lagrangian by adding an operator O(x) with scaling dimension∆O 6= 4LCFT → LCFT + λOO(x) (2.35)To realize this symmetry breaking in the low-energy effective theory, a spurion field isintroduced that constrains the non-derivative interactions of χ(x) asV (χ) = χ4∞∑n=0cn(∆O)(χf)n(∆O−4)(2.36)where cn ∼ λn0 is determined by the underlying CFT. We are interested in the case wherethe explicit breaking of the conformal symmetry is small. This can be accomplished eitherby an operator O that is nearly marginal (|∆O − 4|  1), or by a coefficient λO that ischosen to be small in units of f . For m and f fixed, the potential isV (χ) =12m2χ2 +λ3!m2fχ3 (2.37)18Note that the potential of the dilaton is determined in terms of m2χ =d2Vdχ2> 0 and VEV〈χ〉 = f . To leading order, the cubic coupling isλ =(∆O + 1) +O(λO), when λO  15 +O(|∆O − 4|) when |∆O − 4|  1(2.38)Conformal algebra and unitarity dictate λ > 2. For λO  1 and ∆O = 2, the Higgscoupling is recovered. Moreover, the nearly marginal case can be realized by Coleman-Weinberg effective potential up to corrections asV (χ) =116m2f2χ4[4 ln(χf)− 1]+O(|∆O − 4|2) (2.39)The couplings in (2.33) can receive radiative corrections of order m2. For instance, becausethe top quark is heavy, it contributes δm2 ∼ m2tΛ216pi2f2where Λ is an ultraviolet (UV) cutoff.Moreover, the cubic coupling receivesm3tΛ16pi2f3. These radiative corrections vanish underthe assumption that in the limit of λO → 0, the theory is scale invariant. In order toincorporate this idea in the low-energy model, we make the UV cutoff dependent on χΛ→ Λχf(2.40)This means that χ serves as a conformal compensator as before.2.2.3 Couplings to Massless Gauge BosonsThere are loop contributions Hγγ and Hgg because the Higgs couples to the massivegauge bosons and fermions. As mentioned above, dilatons couples to the SM like theHiggs; hence, there are dilaton couplings χγγ and χgg. These loop effects are dominatedby heavy mass particles such as the top quark. In the SM, the Higgs couples to gluons as(the Higgs background is considered to be spacetime independent)Γ[A,H] =14∫qGˆaµν(−q)Π(q2, H)Gˆaµν(q) + ... (2.41)where Π(q2, H) is the vacuum polarization function that includes the loop contributionsfrom heavy particles. The one-loop contribution of fermions isΠ(q2, H) =1g2(µ)− 4(4pi)2∑i∫ 10dxx(1− x)× ln[x(1− x) + 2m2iH†H/ν2µ2](2.42)19q2 is considered to be of order M2H when dealing with Higgs processes. For light particles,m2i  q2 ≈ M2H . Hence, the vacuum polarization function is independent of the Higgsfield, and light particles have negligible effect to Hgg couplings. However, for heavyparticles such as the top quark, mi MH , the q2 term is neglected and the Higgs couplesto gluons through an operatorLhGG = αs8pi∑ibi0hν(Gaµν)2 (2.43)where we have performed an expansion ln(H†Hν2) = 12 +hν + ... and Gaµν is the canonicallynormalized gluon field strength. We sum over heavy particles only and bi0 is the contribu-tion of each heavy particle to the one-loop QCD beta function βi(g) =bi0g316pi2. For instance,the top quark has b0 =23 .Because the Higgs and dilatons couple to massless gauge bosons, we can make the replace-ment2m2iν2H†H → m2iχ2f2(2.44)in (2.42) and the Lagrangian becomesLχ =[αEM8picEM (Fµν)2 +αs8picG(Gµν)2]ln(χf) (2.45)where cEM and cG are the coefficients of the one-loop beta functions for the gauge couplingse and gs. As we have done before, we can split the sum to run over light and heavy particlesby comparing them to the dilaton mass. Moreover, if we want to consider QCD to beconformal, then by conformal invariance∑lightb0 +∑heavyb0 = 0 (2.46)Hence, the effective coupling becomesLχgg = −αs8piblight0χ¯f(Gaµν)2 (2.47)where blight0 = −11 + 23nlight. Now, if the top quark is heavier than the dilaton mass, thennlight = 5, otherwise nlight = 6. Equation (2.47) shows a tenfold increase in the coupling20strength compared to the Higgs. Therefore, the coupling of dilaton to gluons is enhanced,while to the photons is suppressed.In the Higgs model, the Higgs might receive corrections from particles beyond the SMthat are heavy with dimension-six operator asLhGG ⊃ αs4pichgH†H(Gaµν)2 (2.48)depending on the value of chg. On the other hand, in the dilaton model, corrections comefrom higher dimension operators asLχgg ⊃ g2scχg(4piχ)2DαGaµνDαGµνα (2.49)These operators are suppressed by powers of m2/f2  1 in comparison with conformalanomaly terms. Hence, it is hard to distinguish between the Higgs and the dilaton becauseit is not clear whether corrections are coming from heavy particles beyond the SM in theHiggs model or enhancement of the coupling strength in the dilaton model.The low-energy Lagrangian of the dilaton below the scale 4pif isLχ = 12∂µχ¯∂µχ¯− 12m2χ¯2 +λ3!m2fχ¯3+χ¯f∑ψMψψ¯ψ + (2χ¯f+χ¯2f2)[M2WW+µ W−µ +12M2ZZµZµ]+αEM8picEMχ¯f(Fµν)2 +αs8picGχ¯f(Gµν)2 (2.50)If strong and electromagnetic interactions are considered in the conformal sector at highenergies, then cEM and cG becomecEM =−179 when MW < m < mt,−113 when m > mt(2.51)cG = 11− 23nlight (2.52)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 .212.3 Model 2: Effective Theory of a Light DilatonIn 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 remainsto solve the hierarchy problem. The simplest model of electroweak symmetry broken bystrong dynamics is disfavored by experiment. A dilaton emerges in the low energy theoryas a massless scalar when an exact conformal symmetry is spontaneously broken. We areinterested in theories of electroweak symmetry breaking where there are operators thatbreak conformal symmetry explicitly and increase in the infrared to become strong at thebreaking scale. Hence, there does not exist an obvious reason for the existence of a lightdilaton in the low energy theory.2.3.1 Effective Theory of a DilatonA model for the dilaton is constructed based on a marginal operator that breaks conformalsymmetry yielding a naturally light dilaton. In the beginning, we are going to work in thelimit of exact conformal invariance, then we incorporate violating effects.Effective Theory in the Limit of Exact Conformal InvarianceThe theory is scale invariant in the absence of effects that break conformal symmetry.The Lagrangian has derivative terms12Z∂µχ∂µχ+cχ4(∂µχ∂µχ)2 + ... (2.53)and the potential isV (χ) =Z2κ04!χ4 (2.54)There is a preferred expectation value f = 〈χ〉 when there is a non-derivative term inthe 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-independent22scheme (MS) is used. The coupling constants c, Z2κ0, etc and Z are labeled by dimen-sionless gi. We will begin with one loop analysis and then generalize to any number ofloops.At one loop, gi evolves to g′i asg′i = gi −dgid logµlog(χf)(2.55)All g′is are considered zero but Z and Z2κ0. After this modification, the dilaton potentialbecomesV (χ) =[Z2κ0 − d(Z2κ0)d logµlog(χf)]χ44!(2.56)Now the potential does not have scale invariance. Note that the wave function renormal-ization does not exist at one loop orderd logZd logµ= −2γ = 0 (2.57)From perturbation theory we getdκ0d logµ=3κ2016pi2(2.58)At this stage we can set Z to one.The Lagrangian can be made conformal by making the renormalization scale µ dependson χ¯ as in (2.10). Therefore, making the coupling constants dimensionless, and the theorywould look massless. The kinetic part becomes12Z¯∂µχ∂µχ (2.59)where Z¯ to one loop order isZ¯ = Z − dZd logµlog(µf)(2.60)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 becomesV (χ) =[κ¯0 − d(Z2κ0)d logµlog(χµ)]χ44!(2.61)23where κ¯0 isκ¯0 = Z2κ0 − d(Z2κ0)d logµlog(µf)(2.62)We use Coleman-Weinberg equation in order to get the effective potential at one looporderVeff = V ± 164pi2∑iM4i(log(M2iµ2)− 12)(2.63)Hence, we obtainVeff (χcl) =(κ0 − 3κ2032pi2[log( µ212κ0f2)− 12])χ4cl4!(2.64)Conformal invariance can be manifested by expressingκˆ0 = κ¯0 +3κ¯2032pi2[log(κ¯02)− 12](2.65)so that the potential isVeff (χcl) =κˆ04!χ4cl (2.66)For κˆ0 > 0, conformal symmetry is not broken because the breaking scale 〈χ〉 = f isdriven to zero. For κˆ0 < 0, conformal symmetry is broken. And for κˆ0 = 0, there exist astable 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), and κ¯0 in (2.62) asg′i = gi +∞∑n=1(−1)nn!dngid logµn[log(χf)]n(2.67)V (χ) ={ ∞∑n=0(−1)nn!dn(Z2κ0)d logµn[log(χf)]n}χ44!(2.68)Z¯ =∞∑m=0(−1)mm!dmZd logµm[log(µf)]m(2.69)V (χ) ={ ∞∑n=0(−1)nn!κ¯0,n[log(χf)]n}χ44!(2.70)κ¯0,n =∞∑m=0(−1)mm!dn+m(Z2κ0)d logµn+m[log(µf)]m(2.71)24respectively. The beta functions for all g′is and κ¯0,n are zero because we have conformalinvariance. In order to generalize the effective potential, it is easier to work in the methodof renormalization group than work with counter terms in the Lagrangian method insection (1.2). According to (1.58), the Callan-Symanzik equation becomes(µ∂∂µ+ βi∂∂g¯i− γχcl ∂∂χcl)Veff (χcl, g¯i, µ) = 0 (2.72)Because we have conformal invariance, βi and γ of χ are zero. Hence, (2.72) becomesµ∂∂µVeff (χcl, g¯i, µ) = 0 (2.73)The effective potential isVeff (χcl) =κˆ04!χ4cl (2.74)and it only has a stable minimum when κˆ0 = 0.Conformal Symmetry Violating EffectsNow let us consider effects that break conformal symmetry explicitly. Consider an operatorwith scaling dimension ∆ in the LagrangianL = LCFT + λOO(x) (2.75)The dimensionless coupling constant becomesλˆO = λO µ∆−4 (2.76)Let us assume λˆO  1 in order to be able to apply perturbation theory. λˆO in this casesatisfies the renormalization group differential equationd log λˆOd logµ= −(4−∆) (2.77)derived from (2.76). Under the transformation x → e−ωx, the operator transforms asO(x) → eω∆O(x). In order for the Lagrangian (2.75) to be scale invariant, λO must bepromoted to a spurion as in (2.25) asλO → e(4−∆)ωλO (2.78)25As before, we make the renormalization scale depends on the conformal compensator asµχ = µχ¯. The dilaton potential now isV (χ) =Z2κ04!χ4 −∞∑n=1Z2−n2 κn4!λnOχ(4−n) (2.79)where  = 4−∆.At one loop order (2.79) becomesV (χ) =Z2κ04!χ4 − Z∆/2κ14!λOχ∆ (2.80)where κ0 and κ1 are coupling constants. From (2.55), we know how the coupling constantsevolve in the renormalization group. Hence, modifying the potential to beV (χ) ={Z2κ0 − d(Z2κ0)d logµlog(χf)}χ44!−{Z∆/2κ1 − d(Z∆/2κ1)d logµlog(χf)}λOχ∆4!(2.81)All g′is are considered zero but Z,Z2κ0, and Z∆/2κ1. From perturbation theory we getdκ0d logµ=3κ2016pi2(2.82)dκ1d logµ=∆(∆− 1)κ1κ032pi2(2.83)In order to make coupling constants dimensionless, the coefficient of the dilaton kineticterm becomesZ¯ = Z − dZd logµlog(µf)(2.84)The potential is modified toV (χ) ={κ¯0 − d(Z2κ0)d logµlog(χµ)}χ44!−{κ¯1 − d(Z∆/2κ1)d logµlog(χµ)}λOχ∆4!(2.85)whereκ¯0 = Z2κ0 − d(Z2κ0)d logµlog(µf)(2.86)κ¯1 = Z∆/2κ1 − d(Z∆/2κ1)d logµlog(µf)(2.87)26λO has a mass dimension of 4−∆ because it is considered as a spurion. The beta functionsfor κ¯0 and κ¯1 vanish because of conformal invariance by construction. We use (2.63) tofind the effective potential at one loop orderVeff (χcl) =κ¯04!χ4cl −κ¯14!λOχ∆cl +κ¯04![ 3κ¯032pi2χ4cl −λO∆(∆− 1)κ¯164pi2χ∆cl][Σ− 12](2.88)whereΣ = log[ κ¯02− κ¯14!∆(∆− 1)λOχ∆−4cl](2.89)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 levelpotential isf (∆−4) =4κ¯0κ¯1λO∆(2.90)Hence, the dilaton mass isV ′′eff (χcl)∣∣∣χcl=f= m2σ =κ¯14!λO∆(4−∆)f∆−2 = 4 κ¯04!(4−∆)f2 (2.91)by using (2.90). If we have weak coupling in our CFT, then κ¯0, κ¯1  (4pi)2, λˆO  1implies λOf∆−4  1, and loop suppressed terms in (2.88) are negligible in this regime.On the other hand, if we have strong coupling in our CFT, then κ¯0,κ¯1 ∼ O(4pi)2, and λˆOto be small at the scale f does not hold according to equation (2.90). The mass of thedilaton from equation (2.91) is not light anymore. Hence, if there is a CFT that is stronglycoupled and it is broken explicitly by a relevant operator that grows in the infrared, thenthere 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 massthat 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 tod log λˆOd logµ= −g(λˆO) (2.92)27where g(λˆO) is a polynomial in λˆO in general asg(λˆO) =∞∑n=0cnλˆnO (2.93)Note that the lowest order in (2.93) matches (2.77) when λˆO  1g(λˆO) = c0 = (4−∆) (2.94)In a strongly coupled CFT, cn ∼ O(1) for n ≥ 1 generally. Secondly, higher order termsin (2.79) become important and must be considered. Integrating equation (2.92)G(λˆO) = exp(−∫dλˆOλˆO1g(λˆO))(2.95)where G(λˆO)µ−1 is a renormalization group invariant. As in (2.78), we need to promoteG(λˆO)µ−1 to make the Lagrangian (2.75) scale invariant asλˆO(µ) → λˆO(µe−ω) (2.96)G(λˆO)µ−1 → e−ωG(λˆO)µ−1 (2.97)From (2.95), we can deduce thatλ¯O = λˆO[1 + g(λˆO) logµ] (2.98)Furthermore, using (2.55) and (2.97), we can defineΩ¯(λˆO, χ/µ) = λˆO[1− g(λˆO) log(χµ)](2.99)such that it is invariant under infinitesimal scale transformations. Now that Ω¯ is a poly-nomial in λˆO.For µ ∼ f the scale of symmetry breaking and g(λˆO) 1, Ω¯ can be approximated asΩ¯(λˆO, χ/µ) = λˆO(χµ)−g(λˆO)(2.100)The potential becomes to leading order in Ω¯V (χ) =χ44!(κ0 − κ1Ω¯) (2.101)28and the dilaton mass ism2σ = 4κ04!g(λˆO)f2 (2.102)There is similarity between the expressions of the dilaton mass in (2.91) and (2.102). Thedilaton mass in (2.102) implies that the mass of the dilaton is determined by the scalingbehavior of the operator O to be marginal at the breaking scale µ = f with g(λˆO)  1.However, if a CFT is strongly coupled, then λˆO and g(λˆO) are not small. Therefore,the existence of a light dilaton in such models has some tuning. Including higher ordercorrections in λˆO does not change this result. We can writeg(λˆO) = + δg(λˆO) (2.103)where δg(λˆO) captures higher order corrections. For further discussion on how to constructan effective dilaton theory that includes higher orders in λˆO check (Chacko and Mishra,2013).2.3.2 Dilaton Interactions in a Conformal SMWhen 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 theother hand, if there is an operator that breaks this symmetry explicitly, then we shouldexpect deviations that correct the coupling of the dilaton to the SM. All gauge kineticterms are expressed as− 14g2FµνFµν (2.104)In the limit of exact conformal symmetry, dilatons couple to the gauge bosons such as Was (χf)2m2Wg2W+µ Wµ− (2.105)From (2.22) we have χ = feσf . We expand χ to leading order in f−1 in terms of σ toobtain2σfm2Wg2W+µ Wµ− (2.106)29Now we include effects that break conformal symmetry. We will assume that  > |δg(λˆO)|at the breaking scale, and we will show that the same result holds when  < |δg(λˆO)|.The conformal compensator kinetic part becomes12[1 +∞∑n=1αχ,nλ¯nOχ(−n)]∂µχ∂µχ (2.107)where αχ,n ∼ O(1) depend on the operator O and the particular CFT in use. The dilatonkinetic term becomes12[1 +∞∑n=1αχ,nλ¯nOf(−n)]∂µσ∂µσ (2.108)To make the dilaton kinetic term canonical, σ is rescaled which changes the value of thebreaking 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 interactionsbetween the dilaton and the SM to leading order in σf .The gauge kinetic term (2.104) becomes− 14gˆ2[1 +∞∑n=1αW,nλ¯nOχ(−n)]FµνFµν (2.109)where αW,n ∼ O(1) are dimensionless. The physical gauge coupling is modified to be1g2=14gˆ2[1 +∞∑n=1αW,nλ¯nOf(−n)](2.110)To leading order in σ (2.109) isc¯W4g2σfFµνFµν (2.111)wherec¯W =∑∞n=1 nαW,nλ¯nOf(−n)1 +∑∞n=1 αW,nλ¯nOf (−n)(2.112)In a strongly coupled CFT c¯W ∼ O(λ¯Of−), making the coupling of the dilaton suppressedby λˆO, which is of orderm2σΛ2. The gauge boson mass (2.105) becomes(χf)2[1 +∞∑n=1βW,nλ¯nOχ(−n)]mˆ2Wg2W+µ Wµ− (2.113)30where βW,n ∼ O(1). Expanding (2.113) to leading order in f−1in terms of σσfm2Wg2[2 + cW ]W+µ Wµ− (2.114)The physical mass of the W boson is defined asm2W = mˆ2W[1 +∑∞n=1 βW,nλ¯nOf(−n)1 +∑∞n=1 αW,nλ¯nOf (−n)](2.115)wherecW = −∑∞n=1 nβW,nλ¯nOf(−n)1 +∑∞n=1 βW,nλ¯nOf (−n)(2.116)Corrections to the dilaton coupling in this case is also of order λˆO ∼ m2σΛ2.If  < |δg(λˆO)| at the scale of symmetry breaking, then we replace λ¯Oχ− in (2.107),(2.109), and (2.113) by Ω(λ¯O, χ), which in this regime becomesΩ(λ¯O, χ) =λ¯O1 + c1λ¯O logχ(2.117)and corrections to the dilaton couplings are suppressed by c1Ω2(λ¯O, f) which is of orderm2σΛ2again.The dilaton also couples to the massless gauge bosons, namely, photon and gluon. Notethat conformal symmetry for massless bosons is not broken at the classical level likemassive bosons W and Z, only at the quantum level. Therefore, under x → e−ωx trans-formation, FµνFµν becomesFµνFµν(x)→ e4ω(1 + b<8pi2g2ω)FµνFµν(x) (2.118)where b<8pi2is determined as (2.51). If we want this term to be conformally invariant, thenthe dilaton coupling should beb<32pi2log(χf)FµνFµν (2.119)which is similar to (2.45). Expanding this term to leading order in f−1 in terms of σb<32pi2σfFµνFµν (2.120)31Comparing (2.106) and (2.120), we can infer that the dilaton couples more strongly tomassive gauge bosons than to massless gauge bosons.Now we include violating effects of conformal invariance would yield couplings betweenthe conformal compensator χ and the kinetic term as− 14gˆ2[1 +∞∑n=1αA,nλ¯nOχ(−n)]FµνFµν (2.121)The physical gauge coupling is modified to be1g2=1gˆ2[1 +∞∑n=1αA,nλ¯nOf(−n)](2.122)Expand (2.121) in terms of σ, and adding the effect of (2.120), the dilaton couples tomassless gauge bosons asσf[ b<32pi2+cA4g2]FµνFµν (2.123)where cA is dimensionlesscA =∑∞n=1 nαA,nλ¯nOf(−n)1 +∑∞n=1 αA,nλ¯nOf (−n)(2.124)In a strongly coupled CFT cA ∼ O(λ¯Of−) ∼ λˆO at the breaking scale f . The correctionsare suppressed by m2σΛ2as before.In the limit of exact conformal invariance, the dilaton couples to fermions asχfmψψ¯ψ (2.125)Expanding (2.125) in terms of σ yieldsσmψfψ¯ψ (2.126)Now including effects that break conformal symmetry would modify (2.125) asχf[1 +∞∑n=1βψ,nλ¯nOχ(−n)]mˆψψ¯ψ (2.127)where βψ,n are dimensionless, and mˆψ is the fermion mass in the unperturbed theory.Note that we assumed that the approximate U(3) flavor symmetry is not broken by the32operator O; hence, ensuring that the dilaton couples diagonally in the mass basis. Thefermion kinetic term also receives corrections of the form[1 +∞∑n=1αψ,nλ¯nOχ(−n)]ψ¯γµ∂µψ (2.128)Combing (2.127) and (2.128), we obtain the correction to the dilaton couplingσmψf[1 + cψ]ψ¯ψ (2.129)In a strongly coupled CFT cψ ∼ O(λ¯Of−); hence, the corrections are suppressed by m2σΛ2.2.4 Model 3: A Very Light DilatonIn this section a model proposed by (Grinstein and Uttayarat, 2011) is explored. As wehave seen in section (1.2), quantum effects destroy scale invariance even when there is noexplicit mass terms in the Lagrangian. The goal is to construct a model where the dilatonmass is small; that is to say, the mass can be arbitrarily small while the rest of the spectrumis kept constant and interacting. This can be achieved by starting with an interacting fieldtheory that has a perturbative attractive infrared fixed point and scalars. By following arenormalization group trajectory that is approaching a fixed point, spontaneous symmetrybreaking can be found. A specific model will be considered below where as Yang-Millsgauge coupling runs towards the Banks-Zaks IR-fixed point (Banks and Zaks, 1982), itdrives 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 thecoupling constants. The parameters in the model can either cause spontaneous symmetrybreaking of scale invariance or not. If the couplings are close to the boundary of these tworegimes, then a light dilaton emerges in units of its decay constant. Note that the rest ofthe spectrum is interacting and robust under changes in the parameter space to keep thedilaton mass light.332.4.1 The ModelIn this model we consider a SU(N) gauge theory that has two scalars and two spinors ψi,χk with nf = nχ + nψ = 2nχ flavors. Scalars are considered as singlets, and spinors asvector-like in the fundamental representation. Let us consider this LagrangianL = −12Tr(FµνFµν) +nχ∑j=1ψ¯ji /Dψj +nχ∑k=1χ¯ki /Dχk +12(∂µφ1)2 +12(∂µφ2)2− y1(ψ¯ψ + χ¯χ)φ1 − y2(ψ¯χ+ h.c.)φ2 − 124λ1φ41 −124λ2φ42 −14λ3φ21φ22 (2.130)This Lagrangian has scale invariance at the classical level, discrete symmetryφ1 → φ1, φ2 → −φ2, ψ → ψ, χ→ −χ (2.131)and global transformations ψ → Uψ, χ→ Uχ. For nf small, the gauge coupling increasesin the infrared similar to QCD. There are Goldstone bosons in the spectrum because thechiral symmetry SU(nf ) ⊗ SU(nf ) is spontaneously broken. On the other hand, we areinterested in large nf where the gauge coupling decreases in the ultraviolet but does notincrease in the infrared.Fixed Point StructureWe consider large values of nf and N , two loop order for gauge coupling, and one looporder for Yukawa and scalar couplings. The β functions are (Machacek and Vaughn,1983)(Machacek and Vaughn, 1984)(Machacek and Vaughn, 1985)(Caswell, 1974)(16pi2)∂g∂t= −δN3g3 +25N22g516pi2(16pi2)∂y1∂t= 4y1y22 + 11N2y31 − 3Ng2y1(16pi2)∂y2∂t= 3y21y2 + 11N2y32 − 3Ng2y2(16pi2)∂λ1∂t= 3λ21 + 3λ23 + 44N2λ1y21 − 264N2y41(16pi2)∂λ2∂t= 3λ22 + 3λ23 + 44N2λ2y22 − 264N2y42(16pi2)∂λ3∂t= λ1λ3 + λ2λ3 + 4λ23 + 22N2λ3y21 + 22N2λ3y22 − 264N2y21y22 (2.132)34where nf =11N2 (1− δ11) and terms of order O(δ) are neglected except in βg. Calculationfor 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, thegauge 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 untilthe positive non-linear terms start to dominate over the negative linear terms. The sameprocess applies to the scalars. We find the zero of equations (2.132) to find the fixedpoint. Note that after finding the zero of the gauge coupling, it is used in the subsequentequations for Yukawa couplings, and then these are used for the scalar self-coupling. Toleading order in N−1, the fixed point is at the zero of β functionsg2∗ = 16pi2 275δN, y21∗ = y22∗ =311g2∗N, λ1∗ = λ2∗ = λ3∗ =1811g2∗N(2.133)Theories with scalars and fermions in 4 dimensions do not have non-trivial infrared fixedpoint. However, this is not the case here because the gauge coupling is driving the othercouplings toward the fixed point.Vacuum StructureThe potential has a trivial minimum classically, 〈φ1〉 = 〈φ2〉 = 0. On the other hand,including quantum effects changes the vacuum as discussed in section (1.2). The effectivepotential at one loop in the MS scheme isVeff = − 124λ1φ41 −124λ2φ42 −14λ3φ21φ22− 11N2M4f+(64pi2)(lnM2f+2µ2− 32)− 11N2M4f−(64pi2)(lnM2f−2µ2− 32)+M4s+(64pi2)(lnM2s+µ2− 32)+M4s−(64pi2)(lnM2s−µ2− 32)(2.134)35whereMf± = y1φ1 ± y2φ2M2s± =(λ1 + λ3)φ21 + (λ2 + λ3)φ224±√(λ1 − λ3)2φ41 + (λ2 − λ3)2φ42 − 2(λ1λ2 − λ1λ3 − λ2λ3 − 7λ23)φ21φ224(2.135)As it was discussed in (Coleman and Weinberg, 1973), including masses for the spinorsand scalars does not change the conclusion. It is hard to look for a minimum for thispotential. However, we can use discrete symmetry (2.131) of the vacuum to find a localminimum. Along φ2 = 0, the effective potential becomesVeff =λ124φ41+(λ1φ21)2256pi2(ln(λ1φ212µ2)−32)+(λ3φ21)2256pi2(ln(λ3φ212µ2)−32)−22N2y41φ4164pi2(ln(y21φ21µ2)−32)(2.136)with an extremum∂∂φ1Veff (〈φ1〉) = 0=⇒ −λ16=λ2164pi2(ln(λ1 〈φ1〉22µ2)− 1)+λ2364pi2(ln(λ3 〈φ1〉22µ2)− 1)− 88N2y4164pi2(ln(y21 〈φ1〉2µ2)− 1)(2.137)Note that we can have dimensional transmutation where a dimensionless parameter λ1is replaced by a dimensional vacuum expectation value 〈φ1〉. Large logarithms mightspoil perturbative analysis in the effective potential. To avoid this problem, the conditionλ116pi2ln( 〈φ1〉2µ2)  1 must hold. Hence, λ1 can be expressed by using the last two terms in(2.137), and the condition becomesλ23 − 88N2y41(16pi2)2ln2〈φ1〉2µ2 1 (2.138)We need to show that the extremum that we found is a local minimum by checking thatthe eigenvalues of the mass matrix are both positive. Note that mixed derivatives termsvanish at 〈φ1〉 because we have discrete symmetry and because we are working along36φ2 = 0. The two eigenvalues are∂2∂φ21Veff (〈φ1〉 , 0) = λ23 − 88N2y4132pi2〈φ1〉2 (2.139)∂2∂φ22Veff (〈φ1〉 , 0) = λ32〈φ1〉2 − λ3(λ2 + 4λ3) 〈φ1〉264pi2(lnλ3 〈φ1〉22µ2+ 1)− 264N2y21y22 〈φ1〉264pi2(lny21 〈φ1〉2µ2− 13)(2.140)The first eigenvalue is positive givenε ≡ λ23 − 88N2y41 > 0 (2.141)And the second eigenvalue is positive if the one loop terms are small in comparison withthe tree level term. Note that the value of the effective potential at 〈φ1〉 is negativeVeff (〈φ1〉) = −λ23 − 88N2y41512pi2〈φ1〉4 = − ε512pi2〈φ1〉4 (2.142)The importance of the second scalar field φ2 is that it provides us with a non-trivialminimum in the effective potential by perturbative analysis. To make ε small, we shouldchoose an expectation value such that it is smaller than a fixed renormalization scale µ0,〈φ1〉  µ0. The coupling constant will run until the value of the scale µ starts to beclose to the heaviest particle mass in the model. At this point µ . 〈φ1〉, the trajectory ismodified. The coupling constant can run far enough to get arbitrarily close to the infraredfixed point as long as we start with a small 〈φ1〉. Moreover, if there are other minimaoutside φ2 = 0 axis, then the same analysis applies to the global minimum. Note thatscale invariance of the classical Lagrangian is broken by two effects. The first effect is thenon-trivial minimum found at one loop order. The second effect is quantum fluctuationsat one loop order too. If the first effect is dominant, then a pseudo Goldstone bosonwould appear because we have spontaneously broken approximate scale invariance. Butif quantum fluctuations are dominant, then there would be no pseudo Goldstone boson.Particle SpectrumIf 〈φ1〉 = 〈φ2〉 = 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 of37physical quantities in the renormalization group flow.The fermion self-energy at large N is dominated by the gauge interaction. To one looporder, the fermion self-energy isiΣ(/p) = i(Am+B/p) (2.143)A =g216pi2N2(− 3 ln y21ν2µ2+ 4), B = 1 (2.144)in Landau gauge. Therefore, the masses of ψ and χ areMψ(µ) = Mχ(µ) = y1ν[1− g216pi2N2(3 lny21ν2µ2− 4)](2.145)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 areM2φ1 =λ1ν22+3λ21ν264pi2(lnλ1ν22µ2− 53+2pi3√3)+3λ23ν264pi2(lnλ3ν22µ2− 13− 2λ13λ3)+22N2y2116pi2[y21ν2 − λ1ν212− 3(y21ν2 − λ1ν212)(lny21ν2µ2)− 3∫ 10dx(y21ν2 − x(1− x)2λ1ν2)ln(1− x(1− x) λ12y21)]=3λ21ν264pi2(− 23+2pi3√3)+3λ23ν264pi2(23− 2λ13λ3)+22N2y2116pi2[− 2(y21ν2 − λ1ν212)− 3∫ 10dx(y21ν2 − x(1− x)2λ1ν2)ln(1− x(1− x) λ12y21)]' λ23 − 88N2y4132pi2ν2 =ε32pi2ν2 (2.146)38M2φ2 =λ3ν22+λ1λ3ν264pi2(lnλ1ν22µ2− 1)+λ2λ3ν264pi2(lnλ3ν22µ2− 1)+λ23ν216pi2(lnλ3ν22µ2+∫ 10dx ln(x2 + (1− x)λ1λ3))+22N2y2216pi2[y21ν2 − λ3ν212− 3(y21ν2 − λ3ν212)(lny21ν2µ2)− 3∫ 10dx(y21ν2 − x(1− x)2λ3ν2)ln(1− x(1− x) λ32y21)]' λ3ν22+λ2λ3ν264pi2(lnλ3ν22µ2− 1)+λ23ν216pi2(lnλ3ν22µ2− 2)+22N2y2216pi2[y21ν2 − λ3ν212− 3(y21ν2 − λ3ν212)(lny21ν2µ2)− 3∫ 10dx(y21ν2 − x(1− x)2λ3ν2)ln(1− x(1− x) λ32y21)](2.147)The first lines in (2.146) and (2.147) are the pole mass expressions at one loop order. In thesecond line in (2.146), equation (2.137) is used. And in the other lines, the approximationthat λ1 is small at the scale µ0 and the condition (2.138) are used. The vacuum expectationvalue ν has the anomalous dimension of φ1∂ν∂t= γφ1ν = −11N2y2116pi2ν (2.148)where t = ln(µµ0)as it is defined in (1.60) with a minus sign difference in convention.Using β functions in (2.132) and the above expression we get∂Mψ∂t=∂Mχ∂t=∂Mφ1∂t=∂Mφ2∂t= 0 (2.149)Hence, the masses are RG-invariant at two loops order.2.4.2 DilatonDilatation CurrentThe dilatation current, Dµ defined in (2.7) corresponds to the improved energy momentumtensor defined in (2.5). As mentioned in section (2.1), the canonical energy momentum39tensor can be improved by adding terms. Therefore, the improved energy momentumtensor from Lagrangian (2.130) and equation (2.6) isΘµν = −F aµλF aνλ +12χ¯i(γµDν + γνDµ)χ+12ψ¯i(γµDν + γνDµ)ψ+ ∂µφi∂νφi − 12κ(∂µ∂ν − gµν∂2)φ2i − gµνL (2.150)where the term proportional to κ is the improvement term (Coleman and Jackiw, 1971).The integral of this term vanishes because it is a total derivative with κ = 13 . At theclassical level, this current is conserved; hence, we have scale invariance. However, at thequantum level it is not, and we get a trace anomaly (Coleman and Jackiw, 1971)(Adleret al., 1977)Θµµ = γφ1φ1∂2φ1 + (4γφ1λ1 − βλ1)φ4124+ ... (2.151)DilatonThe dilaton state |σ〉 can be created by acting on the vacuum with the spontaneouslybroken dilatation current. Similar to PCAC, we define a dilaton mass Mσ and a decayconstant fσ as〈0| ∂µDµ |σ〉 = 〈0|Θµµ |σ〉x=0 = −fσM2σ (2.152)Let us consider the energy momentum tensor〈0|Θµν(x) |σ〉 = fσ3(pµpν − gµνp2)eip.x (2.153)In (2.153), the momentum is defined on-shell p2 = M2σ .The goal is to identify Mσ and fσ with a state in the spectrum that we have for our dilatonmodel in the previous section. As explained before, if we have approximate symmetry, weexpect the dilaton to be light, and it couples to the stress energy tensor. Notice that Mφ1fits this description because it is lighter than Mφ2 . In order to show that it couples to thestress energy tensor, as before, we chose to expand the fields at 〈φ1〉 = ν and 〈φ2〉 = 0,and we notice that φ1 is the only linear field term. Shifting the field φ1 in the Lagrangian40(2.150), the terms proportional to pµpν contribute asΘµν = −13ν∂µ∂νφ1 + ... (2.154)Comparing (2.153) and (2.154), we identify fσ = ν and Mσ = Mφ1 . Moreover, theanomaly in (2.151) also supports this identification as (after shifting φ1)Θµµ = γφ1ν∂2φ1 + (4γφ1λ1 − βλ1)ν3φ16+ ... (2.155)Hence, to lowest order we obtain〈0|Θµµ |σ〉x=0 = −γφ1νp2 −λ21 + λ23 − 88N2y4132pi2ν3 + ... (2.156)This result matches the definition in (2.152) with the identificationsfσ = ν , M2σ = M2φ1 =ε32pi2ν2 (2.157)which are RG-invariant and λ21ν3 and γφ1νM2σ terms were dropped for consistency.2.4.3 Phase StructureAs mentioned before, a condition was set for λ1 to be small in order to have dimensionaltransmutation. Then, the model has a non-trivial minimum given that condition (2.141) issatisfied. The two conditions previously mentioned are not satisfied near the infrared fixedpoint. On the other hand, if we start at a point at large RG-time t and then follow the flowto the infrared, then there always exist points that are arbitrarily close to the infrared fixedpoint where spontaneous symmetry breaking occurs. So we choose at a renormalizationscale µ0 coupling constants that provide a non-trivial minimum and satisfy 〈φ1〉  µ0. Inthe mass independent scheme, the coupling constants will run towards the infrared fixedpoint. The smaller 〈φ1〉, the closer we get to the infrared fixed point.Since we are close to the infrared fixed point, massive particles are integrated out, Yukawaand self-couplings are also not important. Hence, the beta function is governed by masslessYang-Mills vectors, and increases as µ is decreased below 〈φ1〉 in the formg2(µ) ≈ g2∗1 + g2∗16pi222N3 lnµ〈φ1〉(2.158)41The effective theory is similar to the theory of glueballs with massMg ∼ 〈φ1〉 e−322N16pi2g2∗ = 〈φ1〉 e− 22544δ (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 〈φ1〉 =〈φ2〉 = 0 where no spontaneous symmetry breaking happens; and hence, all particles aremassless.From this picture we can infer that there are two phases at a fixed renormalization scaleµ0. In the first region, ε is not small, 〈φ1〉 = 0, particles are massless, and RG trajectoriesflow into the infrared fixed point. In the second region, ε is small, 〈φ1〉 = ν, particles havemasses, and RG trajectories get close but do not end at the infrared fixed point. Addingrelevant perturbations in the Lagrangian breaks scale invariance explicitly. Therefore, ifthese mass perturbations are smaller than the masses we obtained in this section, thenqualitatively nothing changes, and quantitatively the corrections are small. On the otherhand, if these mass perturbations are large enough, then a pseudo Goldstone boson willnot exist.42Chapter 3ConclusionIn this essay, an introduction to the Higgs mechanism, effective potential method, anddilatons 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 shownthat the dilaton coupling to gluons is enhanced making it possible to differentiate it fromthe Higgs boson in collider physics experiments. In model 2, an effective theory of a lightdilaton is constructed that includes explicit conformal symmetry breaking effects to anynumber 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 associatedwith some tuning. Finally, corrections to the dilaton interactions with a conformal SM isexplained. 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 conditionsare satisfied on some parameters, one of the scalars is identified with a light dilaton.43BibliographyS. L. Adler, J. C. Collins, and A. Duncan. Energy-momentum-tensor trace anomaly inspin-1/2 quantum electrodynamics. Phys. Rev. D, 15, 1977. doi: http://dx.doi.org/10.1103/PhysRevD.15.1712.T. Banks and A. Zaks. On the Phase Structure of Vector-Like Gauge Theories withMassless Fermions. Nucl. Phys. B, 196, 1982. doi: 10.1016/0550-3213(82)90035-9.C. Callan, S. Coleman, and R. Jackiw. A new improved energy momentum tensor. Ann.Phys., 59, 1970. doi: 10.1016/0003-4916(70)90394-5.W. E. Caswell. Asymptotic Behavior of Non-Abelian Gauge Theories to Two-Loop Order.Phys. Rev. Lett, 33, 1974. doi: http://dx.doi.org/10.1103/PhysRevLett.33.244.Z. Chacko and R. K. Mishra. Effective Theory of a Light Dilaton. Physical Review D, 87,2013. doi: http://dx.doi.org/10.1103/PhysRevD.87.115006.S. Coleman. Aspects of Symmetry. Cambridge University Press, 1985.S. Coleman and E. Weinberg. Radiative Corrections as the Origin of Spontaneous Sym-metry Breaking. Physical Review D, 7(6):1888–1910, 1973. doi: http://dx.doi.org/10.1103/PhysRevD.7.1888.S. R. Coleman and R. Jackiw. Why dilatation generators do not generate dilatations.Ann. Phys., 67, 1971. doi: 10.1016/0003-4916(71)90153-9.44A. Datta, D. Mukhopadhyaya, and A. Raychaudhuri. Physics at the Large Hadron Col-lider. Springer, 2009.W. D. Goldberger, B. Grinstein, and W. Skiba. Distinguishing the Higgs Boson fromthe Dilaton at the Large Hadron Collider. Physical Review Letters, 100, 2008. doi:http://dx.doi.org/10.1103/PhysRevLett.100.111802.B. Grinstein and P. Uttayarat. A Very Light Dilaton. Journal of High Energy Physics,1107, 2011. doi: 10.1007/JHEP07(2011)038.M. Machacek and M. 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Prince-ton University Press, 2013.45AppendicesA Dirac and Gell-Mann MatricesDirac matrices satisfy anticommutation relations{γµ, γν} ≡ γµγν + γνγµ = 2ηµν (A.1)whereηµν =1 0 0 00 −1 0 00 0 −1 00 0 0 −1 (A.2)Dirac matrices in Dirac representation areγ0 =I 00 −I , γj = 0 σj−σj 0 (A.3)where I is the 2× 2 identity matrix, and σj are Pauli 2× 2 matricesσx =0 11 0 , σy =0 −ii 0 , σz =1 00 −1 (A.4)Pauli matrices satisfy commutation and anti commutation relations[σi, σj ] = 2iεijkσk (A.5){σi, σj} = 2δijI (A.6)46where εijk isεijk =+1, for even permutations of 123−1, for odd permutations of 1230, otherwise(A.7)We can define γ5 asγ5 = iγ0γ1γ2γ3 = −iγ0γ1γ2γ3 = γ5=i4!εµνρσγµγνγργσ (A.8)where εµνρσ is the Levi-Civita symbol defined asεµνρσ =+1, for even permutations of 0123−1, for odd permutations of 01230, otherwise(A.9)and{γ5, γµ} = 0 (A.10)γ5 =0 II 0 (A.11)Dirac matrices in the Weyl representation are the same for γj but different for γ0 and γ5γ0 =0 II 0 , γ5 =−I 00 I (A.12)Gell-Mann matrices form a Lie algebra that satisfy the commutation relations[T a, T b] = ifabcT c (A.13)47where fabc are the structure constants. For SU(3), Gell-Mann matrices in the basisT a = 12λa areλ1 =0 1 01 0 00 0 0 , λ2 =0 −i 0i 0 00 0 0 , λ3 =1 0 00 −1 00 0 0 ,λ4 =0 0 10 0 01 0 0 , λ5 =0 0 −i0 0 0i 0 0 , λ6 =0 0 00 0 10 1 0 ,λ7 =0 0 00 0 −i0 i 0 , λ8 = 1√31 0 00 1 00 0 −2 , (A.14)Note that there exist a Cartan subalgebra because λ3 and λ8 commute. This implies thatthere are two quantum numbers that are observable simultaneously; isospin I3 =12λ3 andhypercharge Y = 1√3λ8.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 asC2(R) ≡ T aT a (A.15)where summation over a is implied and it is a multiple of the identity. One can choose tonormalize the structure constants as∑c,dfacdf bcd = Nδab = C2(G)δab (A.16)where C2(G) is the Casimir operator in the adjoint representation. The generators in thefundamental representation are normalized asTr(T aT b) =12δab (A.17)The Dynkin index is defined asTrR(TaT b) ≡ S2(R)δab (A.18)48Setting a = b in equation (A.18) and summing over a we obtaind(G)S2(R) = d(R)C2(R) (A.19)where d(R) is the representation dimension (d(fundamental) = N and d(adjoint) = N2−1), d(G) is the Lie group dimension. For instance, SU(N) has a dimension d(SU(N)) =N2 − 1, C2(fundamental) = N2−12N , and C2(G) = C2(adjoint) = N . Note that in theadjoint representation d(R) = d(G); hence, Dynkin index and the quadratic Casimiroperator are the same in this case S2(R) = C2(G) = N . In the fundamental representationwe have S2(R) =12 from equation (A.17). Note that C2(F ) and C2(S) are the Casimiroperators for fermion and scalar representations, respectively, S2(F ) and S2(S) are theDynkin indices for fermion and scalar representations, respectively.B Integral with a CutoffIn order to do the integral with a cutoff in (1.46), this identity should be used∫d4kG(k2) = pi2∫ Λ20dk2k2G(k2) (B.1)Now we have~2∫d4kE(2pi)4[log(k2E + V′′(φ))− log(k2E)]=~25pi2∫ Λ20dk2E[log(k2E + V′′(φ))− log(k2E)]k2E(B.2)Let k2E = x and integrate by parts to get∫ Λ20x log(x+ V ′′(φ))dx =Λ42log(Λ2 − 12) + V ′′(φ)Λ2− (V′′(φ))22log(Λ2) +(V ′′(φ))22log(V ′′(φ)− 12) (B.3)∫ Λ20x log(x) =Λ42log(Λ2 − 12) (B.4)Hence, the integral in (B.2) becomes~32pi2V ′′(φ)Λ2 +~64pi2(V ′′(φ))2[log(V ′′(φ)Λ2)− 12](B.5)49Performing the same analysis in the MS scheme where equation (1.45) isVeff (φ) = V (φ)− i~2Tr(log[∂2 + V ′′(φ)])(B.6)Going to the momentum space and to the Euclidean space, we obtain∫dDp(2pi)Dln(−p2 + V ′′(φ)) = i∫dDpE(2pi)Dln(p2E + V′′(φ))= −i ∂∂α∫dDpE(2pi)D1(p2E + V′′(φ))α∣∣∣α=0= −i ∂∂α( 1(4pi)D/2Γ(α− D2 )Γ(α)1(V ′′(φ))α−D/2)∣∣∣α=0= −iΓ(−D/2)(4pi)D/2(V ′′(φ))D/2 (B.7)Where D = 4−2 is the number of dimension. To obtain ordinary dimension, we introducean arbitrary scale µ˜, and the effective potential becomesVeff (φ) = −12µ˜4−DΓ(−D/2)(4pi)D/2(V ′′(φ))D/2 + V (φ)= − 12(4pi)2(V ′′(φ))2Γ(−2 + )(V ′′(φ)2piµ˜2)−+ V (φ) (B.8)In the limit of → 0, Γ(−2 + ) = 12(1 − γ + 32). Hence, (B.8) becomesVeff (φ) = − 164pi2(V ′′(φ))2(1− γ + 32+ log4piµ˜2V ′′(φ))+ V (φ)= − 164pi2(V ′′(φ))2(32+ logΛ˜2V ′′(φ))+ V (φ) (B.9)whereΛ˜ =√4pie−γ2+ 12 µ˜ (B.10)50C Beta Function for the Gauge CouplingAccording to (Machacek and Vaughn, 1983), the beta function at two loop order for thegauge field isβ(g) = − g3(4pi)2b0 − g5(4pi)4b1 (C.1)b0 =113C2(G)− 43κS2(F )− 16S2(S) +2κ(4pi)2Y4(F ) (C.2)b1 =343[C2(G)]2 − κ[4C2(F ) + 203C2(G)]S2(F )− [2C2(S) + 13C2(G)]S2(S) (C.3)From Appendix A, SU(N) hasC2(G) = N , S2(F ) = nf , C2(F ) =N2 − 12N, S2(S) = 0 , C2(S) = 0(C.4)with κ = 12 , 1 for two-component fermions or four-component fermions respectively. Hence,b0 becomesb0 =113N − 4312nf =Nδ3(C.5)with δ = 11(1 − 2nf11N ). According to (Caswell, 1974), a gauge theory is not expected toflip the sign of the beta function and approach zero to be valid in perturbation theory.Fixing the lowest order term b0 = 0 implies nf =11N2 . Using this in b1 yieldsb1 =343N2 − 12[4(N2 − 12N) +20N3]nf=343N2 − [ 1N(N2 − 1) + 103N ]112N=343N2 − 112N2 +112− 553N2 ' −252N2 (C.6)51D The Renormalized Mass in FermionsThe form of the self-energy loop graph by using Feynman parameters isiΣ2(/p) = (ie)2∫d4k(2pi)4γµi(/k +m)k2 −m2 + iεγµ −i(k − p)2 + iε= e2∫d4k(2pi)4∫ 10dx2/k − 4m[(k2 −m2)(1− x) + (p− k)2x+ iε]2 (D.1)Completing the square and shifting k → k + px yieldsiΣ2(/p) = 2e2∫ 10dx∫d4k(2pi)4x/p− 2m[k2 − (1− x)(m2 − p2x) + iε]2 (D.2)We regularize this integral by using dimensional regularization in d = 4 − ε dimensions.The self-energy becomesΣ2(/p) = −ie2µ4−d∫ 10[(d− 2)x/p− dm] ∫ ddk(2pi)d1[k2 − (1− x)(m2 − p2x) + iε]2=α4pi∫ 10dx[(2− ε)x/p− (4− ε)m][2ε+ lnµ˜2(1− x)(m2 − p2x)](D.3)where µ˜2 ≡ 4pie−γEµ2. The renormalized and bare Green’s functions are related asiGR(/p) =11 + δ2iGbare(/p) =11 + δ2i/p−m0 + Σ2(/p) + ...=i/p−m0 + δ2/p−m0δ2 + Σ2(/p) + ... (D.4)Using m0 = mR +mRδm it becomesiGR(/p) =i/p−mR + ΣR(/p) (D.5)where ΣR(/p) = Σ2(/p) + δ2/p− (δm + δ2)mR +O(e4). In the modified minimal subtraction(MS), the γE and ln(4pi) factors are removed. Expanding (D.3) in µ˜2 we obtain thecounter termsδ2 = − α4pi(2ε+ ln(4pie−γE )) (D.6)δm = −3α4pi(2ε+ ln(4pie−γE )) (D.7)52Hence, the self-energy becomes UV finite in the formΣR(/p) =α2pi∫ 10dx{(x/p− 2mR)[lnµ2(1− x)(m2R − p2x)]− x/p+mR}(D.8)We can relate the pole mass (which is related to the physical mass) and the renormalizedmass by looking for a pole in the propagator of equation (D.5) at /p = mP . This conditiongivesmP −mR + ΣR(mP ) = 0 (D.9)To leading order, we take mP = mR and do the integral by parts in (D.8) to obtainmR = mP + ΣR(mp) = mp[1− α4pi(4 + 3 lnµ2m2P)+O(α2)](D.10)Note that the renormalized mass in the MS scheme depends on the arbitrary scale µ.53

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