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The Higgs Boson and Electroweak Symmetry Breaking 2. Models of EWSB M. E. Peskin SLAC Summer Institute 2011 In the previous lecture, I discussed the simplest model of EWSB, the Minimal Standard Model. This model turned out to be a little too simple. It could describe EWSB, but it could not explain its physical origin. In this lecture, I would like to discuss four models that have been put forward to explain the physics of EWSB: • • • • Technicolor Supersymmetry Extra Dimensions Little Higgs I hope this will give you an idea of the variety of possiblities for the next scale in elementary particle physics. To begin, I would like to explain the logic of these choices and state the modest goal of this lecture. At the end of the previous lecture, I argued that the MSM Higgs theory was flawed because it did not allow any understanding of the origin of electroweak symmetry breaking. Because the Higgs mass term receives quadratically divergent radiative correction, which can be of either sign, it is impossible, within the MSM, to explain why the parameter µ2 is small and negative. In this lecture, I will present some models that predict electroweak symmetry breaking through straightforward mechanisms in quantum field theory. In each model, two things must happen: First, the quadratically divergent contributions to the Higgs field mass term must not appear. This will happen if there is a symmetry that forbids the Higgs field mass term. Please note that 2 2 δL = −µ |ϕ| is invariant to all of the usual symmetries of quantum field theory, so this step already entails severe restrictions. Second, the resulting model should show an instability to !ϕ" = # 0 as the result of a calculation. In this lecture, I will not construct complete models. I will only go far enough to demonstrate the physics that causes !ϕ" = # 0 Concentrate for the moment on the first goal -- removing the Higgs mass term by a symmetry. There are two possible approaches: 1. Assume that the Higgs field is composite. In particular, if we build the Higgs field out of fermions, the underlying theory has no quadratic divergences. 2. Assume that the Higgs field is elementary. Then some unusual symmetry is required to forbid the Higgs mass term. Here are three examples: a. δϕ = # b. δϕ = #µ Aµ combined with gauge symmetry c. δϕ = #ψ realized if ϕ is a Goldstone boson combined with chiral symmetry The models I will discuss realize these four possibilities. Technicolor: ϕ was introduced by Higgs in analogy to the theory of superconductivity. There, Landau and Ginzburg had introduced ϕ as a phenomenological charged quantum fluid. Their equations account for the Meissner effect, quantized flux tubes, critical fields and Type I-Type II transitions, ... Bardeen, Cooper, and Schrieffer showed that pairs of electrons near the Fermi surface can form bound states that condense into the macroscopic wavefunction ϕ at low temperatures. This suggests that we should build the Higgs field as a composite of some strongly interacting fermions that form bound states. Weinberg, Susskind: QCD has strong interactions, and also fermion pair condensation. For 2 flavors i i L = q iL γ · DqL + q iR γ · DqR has the global symmetry SU(2)xSU(2)xU(1). If quarks have strong interactions, scalar combinations of q and q should condense into a macroscopic wavefunction in the vacuum ! " state: i q jL qR = ∆δij != 0 Act with global symmetries. We find a manifold of vacuum ! " states q j q i = ∆V L R ij In any given state, SU(2)xSU(2)xU(1) is spontaneously broken to SU(2)xU(1). The degrees of freedom of V(x) are Goldstone bosons: iπ a σ a /fπ V =e Identify π with the 3 pi mesons. a 2 2 m ! m Nambu and Jona-Lasinio: this is why π ρ The pions would be massless-exact Goldstones, if mu = md = 0 fπ is the pion decay constant = 93 MeV ! " µ5a ! b !0| J π = ipµ fπ δ ab Now couple this system to SU(2)xU(1) gauge bosons 1 1 1 uR (0, Y + ) dR (0, Y − ) qL ( , Y ) 2 2 2 In the presence of the condensate, only the gauge symmetry iα(I 3 +Y ) qL → e qL iα(I 3 +Y ) qR → e qR is preserved. So, Q = I3+Y must correspond to a massless gauge boson. The 3 other generators must correspond to massive gauge bosons. The symmetry breaking occurs precisely because the W boson has a purely left-handed coupling. To compute the masses explicitly, write a σ 1 a ! ! Dµ V = ∂µ V − igW V − ig BY V + ig BV (Y ± ) 2 2 1 2 L = fπ tr Dµ V † Dµ V 4 for !V " = 1 !2 ! " ! ! g g g g 1 2 + + − − 3 3 3 L = fπ tr !! √ W σ + √ W σ + W σ − Bσ !! 4 2 2 2 2 " # 2 fπ 2 + − 1 g W W + (gW 3 − g " B)2 = 4 2 This is the same structure as the MSM, where we found m2W g2 2 = v 4 2 !2 g + g m2Z = v2 4 To obtain the correct masses, we need a scaled-up QCD “technicolor” - in which mT ρ v ∼ mρ · ∼ 2 TeV fπ The analogue of the Higgs boson in this theory is the σ or a0 0++ resonance of QCD. In technicolor, this appears as a peak (better, a shoulder) in S-wave WW and ZZ scattering at ~ 1600 GeV. A much better signal is the T ρ , which appears as a resonance in P-wave WW scattering and in e+ e− → W + W − 'R/MR This resonance is already strongly constrained by LEP 2 data. 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 ALEPH excluded 95% CL 0 200 400 600 800 1000 1200 1400 1600 1800 MR (GeV) There are more serious phenomenological problems with technicolor. Mixing of the T ρ with W, Z alters the precision electroweak predictions at the 2-3% level, increasing sin2 θw and mW above the MSM expectation. Corrections of order 3% are also expected in Γ(Z → bb) . Fermion masses are generated by higher-dimension operators 1 i j q Q q u L R L R 2 M There is no simple mechanism for flavor conservation, so we 0 0 expect large corrections to K , B mixing, b → sγ Fixes for these problems may exist if the strong-coupling theory has special properties (“walking technicolor”, “conformal fixed point”). Supersymmetry: Supersymmetry (SUSY) is a symmetry that relates bosons and fermions with the same SU(2)xU(1) quantum numbers. This is a very deep theoretical idea whose full discussion is beyond the scope of this lecture. The Standard Model contains many fermions fields, a separate one for each left- or right-handed quark or lepton. A supersymmetric extension of the Standard Model therefore contains a huge number of elementary scalar fields. The mass terms for all of these fields are forbidden by the combination of the SU(2)XU(1) symmetry and supersymmetry. We need to address: Why is there an instability that generates a Higgs field v.e.v.? And, why does no other scalar field obtain a v.e.v.? In this discussion, only a few aspects of SUSY will be important: • Every boson or fermion in the theory has a partner, with spin differing by 1/2 • Coupling constants in the renormalizable interactions of the partners are equal to the corresponding Standard Model parameters • SUSY must be spontaneously broken; this is parameterized by “soft” SU(2)xU(1)-invariant mass terms. • Quadratic divergences in boson mass terms cancel between loop diagrams with bosons and fermions h + ~h ~h The Higgs boson sector of SUSY is unexpectedly complex. 1 H1 , H2 with Y = ± 2 There must be two Higgs doublet fields including only one gives an anomalous gauge theory SU(2) SU(2) U(1) + ~h 1 U(1) ~h 2 SU(2) SU(2) quark-Higgs couplings are only of the form L= ij −λd dR H1α "αβ Qβ − λij u uR H2α "αβ Qβ so it is still true that all flavor violation can be moved into the CKM matrix. (But, soft SUSY-breaking terms may provide new sources of flavor violation.) H1 , H2 have 8 degrees of freedom ➤ ❨3 eaten Goldstones) + (CP even h0 , H 0) + (CP odd A0 ) + ( H + , H −) The Higgs mass terms come from soft SUSY breaking: 2 2 L = −MH1 |H1 |2 − MH2 |H2 |2 These parameters do not have huge additive corrections, but they do evolve - on a log scale - due to RG effects. For example, the coupling to SU(2) gauginos gives 3 d 2 MH2 = − αw m(w) ! 2 − ··· d log Q 2π ~ w ~h 2 A more important effect is the coupling to top quarks. t̃HLu ~t H2 M2 2 This sends MH to negative values as Q decreases. 2 H2 2 2 , M!tR These fields The same effect applies to M! tL compete with the Higgs field to go unstable. SUSY thus raises a new question about EWSB. SUSY rationalizes the elementary scalar field, but in the process it introduces many elementary scalars. Any one can obtain a vacuum expectation value. If !H2 " = # 0 , we break SU(2)xU(1). ! # If " tR != 0 , we break color SU(3) but preserve SU(2). Which behavior is predicted ? Assume that all soft scalar masses are equal at a very high mass scale; integrate the RG equations down: Here are the relevant RG equations. Notice that the Higgs boson is favored, both by color/SU(2) factors and by the influence of the gluino: d λ2t 2 2 2 2 2 M!tL = + · 1 · (M + M + M + A H2 t) − 2 ! t L ! t R d log Q (4π) 2 d λ 2 M!t2R = + t 2 · 2 · (M!t2L + M!t2R + MH2 + A2t ) − d log Q (4π) d λ2t 2 2 2 2 2 MH2 = + · 3 · (M + M + M + A H2 t) !tL !tR d log Q (4π)2 8 αs m(! g )2 − · · · 3π 8 αs m(! g )2 − · · · 3π If the Higgs coupling to the top quark is the largest coupling in the theory, this effect is likely to dominate and drive EWSB. Extra Dimensions: Another of our possibilities is to have a symmetry that connects the Higgs field to a gauge field. The combination of this symmetry with gauge invariance then forbids the Higgs mass term. A realization of this is to assume that the universe has an extra space dimension X M = (xµ , x5 ) Then gauge fields will have 5 components, all linked by gauge symmetry AM (X) = (Aµ (X), A5 (X)) 5 It is attractive to identify some components A with the Higgs scalar field. This idea is called “gauge-Higgs unification”. Fayet, Hall and Nomura, Hosotani and Shimizu A simple model that contains the right ingredients is an SU(3) gauge theory on M 4 × S 1 , a 5-dimensional universe with one dimension compactified as a (flat) circle of circumference R . 5 Momenta in the 5 direction are quantized: p = (2πn + δ)/R The parameter δ depends on the boundary conditions around the 5th dimension. H = SU(2)XU(1) appears as a subgroup of G= SU(3). Boundary conditions are provided such that, µ for components in H, only A (X) have δ = 0 modes for components in G/H, only A5 (X) have δ = 0 modes Fermions in 5 dimensions are 4-component spinors with no Weyl reduction. In terms of 4-dimensional spinors Ψ= ! ψL ψR " In models, Ψ obeys boundary conditions so that only one of (ψL , ψR ) will have δ = 0 modes. To zeroth order, we can have a model with one Ψ field in a 3 of SU(3), such that 3 1 2 have δ = 0 modes. , ψL ) and ψR (ψL This fields are coupled by the gauge field components (A513 , A523 ) These can be identified with (tL , bL ) tR (ϕ+ , ϕ0 ) In models of this type, A5 has an instability toward a nonzero v.e.v. I will demonstrate this instability in a simpler context. Assume that we have a single Ψ field coupled to a U(1) gauge field, in a 5-d space compactified on a circle. I will compute the potential for A5 from the functional integral ! " DΨ exp[− d5 x{Ψi " ∂ Ψ + gΨA5 γ 5 Ψ}] = exp[−(V ol)3 RT · V (A5 )] and show that it has a maximum at A5 = 0 . Then A5 will run down to its minimum, acquiring a v.e.v. This is the Hosotani-Toms mechanism. From the previous slide, dV (A ) =g 5 dA 5 ! " # 5 5 d x Ψ(x)γ Ψ(x) = −g In infinite flat 5-d space ! " Ψ(x)Ψ(y) = (i ! ∂ ) times the phase ! d x tr[γ 5 5 " # Ψ(x)Ψ(x) ] γ · (x − y) 1 = −3i 2 2 3 8π (x − y) 8π (x − y)5 exp{igA5 (x5 − y 5 )} In the compact geometry ! " 5 −gtr[γ Ψ(x)Ψ(x) ] = (n = 0) + Integrating up, ∞ # n=−∞,n#=0 3 nR inRA5 +ig 2 e 5 2π (nR) ∞ # 1 3 5 (− sin(nRA )) = 0+ 2 4 π (nR) n=1 ∞ ! 3 1 5 5 V (A ) = 2 cos(nRA )+C 5 π n=1 (nR) 5 A = 0 , signalling an Thus, V (A ) has a maximum at 5 instability to symmetry breaking. Hosotani explains this behavior in the following way: In statistical mechanics, the functional integral representation of the sum over states tr[e−βH ] is represented by antiperiodic boundary conditions for fermion fields. Thus, when we make A5 dynamical, it seeks the configuration in which Ψ obtains a phase (-1) going around the circle. This instability, in the more complex model of the top quark mass described earlier, drives electroweak symmetry breaking. Little Higgs: Return to the idea that the Higgs boson is a composite of strongly-interacting fermions. The problems we met with this idea can be ameliorated by raising the strong-interaction scale. Then we can implement a different strategy (KaplanGeorgi). Let the strong-interaction symmetry breaking preserve SU(2) xU(1). Let the multiplet of Goldstone bosons include the Higgs doublet ϕ . Then, by coupling to gauge fields or to new particles, break down the constraints that keep ϕ exactly massless. Here is a simple realization: Arkani-Hamed, Cohen, Katz, Nelson Consider a gauge theory with the symmetry SU(3)xSU(3)xU(1), broken by strong interactions to SU(3)xU(1). This gives an SU(3) octet of Goldstone bosons ! " 2iΠa ta /f V =e a a 2iΠ t = Φ −H † H φ in which the H is a doublet. We will want f ∼ 1 TeV , Mρ ∼ 10 TeV All fields in Π must be massless if the SU(3) symmetries V → ΛR V are respected. † ΛL Coupling this structure to the top quarks. We need to put top quarks into the representations u UR uR χL = b U L with an extra singlet quark. The Lagrangian is L = −λ1 f ( 0 0 uR ) V χL − λ2 f U R UL The first term has the symmetry The second term has the symmetry V → V Λ†L χ → ΛL χ V → ΛR V Either symmetry suffices to insure that H is exactly massless. Thus, to build a Higgs potential, we need to involve both interaction terms. Transform to the top quark mass eigenstates: λ2 uR − λ1 UR tL = u L tR = ! 2 λ1 + λ22 λ1 uR + λ2 UR TL = UL TR = ! 2 2 + λ λ 2 1 ! mT = λ21 + λ22 f then the H vertices are: tR tL H TR = −iλt λt = ! λ1 λ2 λ21 + λ22 tL H λT = ! = −iλT λ21 λ21 + λ22 Using this structure, compute corrections to the H mass: tL tR ! d4 k 1 (2π)4 k 2 = −6λ2T ! 1 d4 k (2π)4 k 2 − m2T λT = +6 f ! d4 k mT (2π)4 k 2 − m2T = −6λ2t H tL H TR T H By the relations on the previous page, the quadratic divergences of these diagrams cancel. What is left over is: m2H M2 M2 λ21 λ22 f 2 λ2T m2T log 2 = −3 log 2 = −3 2 2 8π mT 8π mT If the top quark is heavy, this set of contributions can dominate the H mass and produce EWSB. If the T has a mass below 2.5 TeV, it can be found at the LHC. Then the relation that implies the divergence cancellation: λ2t + λ2T mT = f λT can be tested experimentally. The quadratic divergences of H mass diagrams with W, Z are naturally cancelled by contributions from new W, Z bosons with mass of 1-2 TeV. If this model is correct, these bosons ought to appear soon in searches at the LHC. In this lecture, I have described 4 possible models of EWSB. We do not know whether the Higgs boson is elementary or composite; I have presented models of both types. In each model, the Higgs boson is part of a larger superstructure that will be revealed when we experiment at multi-100 GeV eneriges at the LHC and the Linear Collider. These superstructures can also include candidates for the dark matter particle, as Tim Tait will discuss. Each option leads to its own characteristic set of new particles to be discovered. Very soon, we are going to find out whether one of these models, or some different one, will solve the mystery of electroweak symmetry breaking.