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The Higgs Boson and Electroweak Symmetry Breaking

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The Higgs Boson and Electroweak Symmetry Breaking 2. Models of EWSB M. E. Peskin SLAC Summer Institute, 2004 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 cold not explain its physical origin. In this lecture, I would like to discuss three models that have been put forward to explain the physics of EWSB: • Technicolor • Supersymmetry • Little Higgs I hope this will give you an idea of the variety of possiblities for the next scale in elementary particle physics. 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 i i L = qLγ · DqL + qRγ · 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: j i = ∆ = 0 !qLqR" δij ! Act with global symmetries. We find a manifold of vacuum states j i = ∆ !qLqR" Vij 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: a a V = eiπ σ /fπ a Identify π with the 3 pi mesons. m2 m2 Nambu and Jona-Lasinio: this is why π ! ρ The pions would be massless-exact Goldstones, if mu = md = 0 f π is the pion decay constant = 93 MeV µ5a b µ ab !0| J !π " = ip fπδ ! Now couple this system to SU(2)xU(1) gauge bosons 1 1 1 qL ( , Y ) uR (0, Y + ) dR (0, Y − ) 2 2 2 Only the gauge symmetry 3 3 iα(I +Y ) iα(I +Y ) qL → e qL qR → e qR is conserved. 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 a σ ! ! 1 DµV = ∂µV − igW V − ig BY V + ig BV (Y ± ) 1 2 2 L = 2 tr † µ 4fπ DµV D V for !V " = 1 " 2 1 2 g + + g − − g 3 3 g 3 = fπ tr W σ + W σ + W σ Bσ L 4 !√ √ 2 − 2 ! ! 2 2 ! 2 fπ ! − 1 " ! = g2W! +W + (gW 3 g B)2 ! (1) 4 " 2 − # This is the same structure as the MSM, where we found g2 g2 + g!2 m2 = v2 m2 = v2 W 4 Z 4 To obtain the correct masses, we need a scaled-up QCD - “technicolor” - in which v mT ρ ∼ 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 − This resonance is already strongly constrained by LEP 2 data. 4.5 4 ALEPH 3.5 R 3 /M 2.5 R ' 2 1.5 1 excluded 0.5 95% CL 0 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 2 predictions at the 2-3% level, increasing s in θ w and m W 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 Q q q uj M 2 L R L R 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 which will be explained at the school next week. You will hear then about a number of phenomenological advantages of supersymmetry as a theory of the next energy scale. SUSY is particularly interesting for the problem of EWSB because it gives a rationale and a restrictive theoretical framework for introducing elementary scalar fields. 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 Y = ± There must be two Higgs doublet fields with 2 including only one gives an anomalous gauge theory SU(2) SU(2) U(1) + U(1) ~ ~h h1 2 SU(2) SU(2) quark-Higgs couplings are only of the form L = − ij − ij λd dRH1α"αβQβ λu uRH2α"αβQβ 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.) H 1 , H 2 have 8 degrees of freedom ➤ 0 0 ❨3 eaten Goldstones) + (CP even h , H ) − + (CP odd A 0 ) + ( H + , H ) The Higgs mass terms come from soft SUSY breaking: 2 2 2 2 L = −MH1|H1| − MH2|H2| 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 w~ d 2 3 2 MH2 = − αwm(w!) − · · · d log Q 2π ~ H h2 2 A more important effect is the coupling to top quarks. ~ tL H2 t 2 R This sends M H 2 to negative values as Q decreases. 2 2 The same effect applies to M !t L , M !t R These fields 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 ! H 2 " #= 0 , we break SU(2)xU(1). If ! t R # != 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: 2 d 2 λt 2 2 2 2 8 2 M = + · 1 · (M + M + MH2 + At ) − αsm(g!) − · · · d log Q !tL (4π)2 !tL !tR 3π 2 d 2 λt 2 2 2 2 8 2 M = + · 2 · (M + M + MH2 + At ) − αsm(g!) − · · · d log Q !tR (4π)2 !tL !tR 3π 2 d 2 λt 2 2 2 2 MH2 = + · 3 · (M + M + MH2 + At ) (1) d log Q (4π)2 !tL !tR If the Higgs coupling to the top quark is the largest coupling in the theory, this effect is likely to dominate and drive EWSB. 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 (Kaplan-Georgi). 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 a a a a Φ H V = e2iΠ t /f 2iΠ t = ! −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 → ΛRV ΛL are respected. Coupling this structure to the top quarks. We need to put top quarks into the representations u χL = b UR uR U L with an extra singlet quark. The Lagrangian is L = −λ1f ( 0 0 uR ) V χL − λ2fU RUL † The first term has the symmetry V → V ΛL χ → ΛLχ The second term has the symmetry V → ΛRV Either symmetry suffices to insure that H is exactly massless. Thus, to build a Higgs potential, we need to involve both interaction terms.
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