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Home , Little Higgs, Technicolor (physics)

The 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 . 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 • • Little Higgs I hope this will give you an idea of the variety of possiblities for the next scale in elementary . 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 that form bound states. Weinberg, Susskind:

QCD has strong interactions, and also 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 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 . 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 SU(2) SU(2) U(1) + U(1) ~ ~h h1 2 SU(2) SU(2) -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 .

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 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. Transform to the top quark mass eigenstates: λ u − λ U t = u t = 2 R 1 R L L R 2 2 !λ1 + λ2 λ u + λ U T = U T = 1 R 2 R L L R 2 2 !λ1 + λ2 2 2 mT = !λ1 + λ2f then the H vertices are:

tR TR H = −iλt H = −iλT tL tL λ λ λ2 λ = 1 2 λ = 1 t 2 2 T 2 2 !λ1 + λ2 !λ1 + λ2 Using this structure, compute corrections to the H mass: t d4k L − λ2 1 = 6 t 4 2 H ! (2π) k tR t L d4k − λ2 1 H = 6 T 4 2 2 ! (2π) k − mT TR

4 λT d k mT T = +6 4 2 2 f ! (2π) k − mT H By the relations on the previous page, the quadratic divergences of these diagrams cancel. What is left over is: 2 2 2 2 2 2 2 2 − λ1λ2f M − λT mT M mH = 3 2 log 2 = 3 2 log 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: 2 2 mT λ + λ = t T 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.

These bosons can be found and studied at the LHC and the Linear Collider. In this lecture, I have described 3 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.

Each option leads to its own characteristic set of new particles to be discovered.

Very soon, we are going to find out the answer ! And, as always, the answer will lead us to new and still deeper Great Questions.