The Higgs Bosonand

Electroweak Symmetry Breaking

2. Models of EWSB

M. E. PeskinSLAC 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:

I hope this will give you an idea of the variety of possiblities for the next scale in elementary particle physics.

• Technicolor• Supersymmetry• Little Higgs

Technicolor:

was introduced by Higgs in analogy to the theory of superconductivity.

There, Landau and Ginzburg had introduced 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 at low temperatures.

This suggests that we should build the Higgs field as a composite of some strongly interacting fermions that form bound states.

ϕ

ϕ

ϕ

Weinberg, Susskind:

QCD has strong interactions, and also fermion pair condensation. For 2 flavors

has the global symmetry SU(2)xSU(2)xU(1).

If quarks have strong interactions, scalar combinations of and should condense into a macroscopic wavefunction in the vacuum state:

Act with global symmetries. We find a manifold of vacuum states

In any given state, SU(2)xSU(2)xU(1) is spontaneously broken to SU(2)xU(1).

L = qiLγ · DqiL + q

iRγ · Dq

iR

q

q

〈qjLq

iR

〉= ∆δij != 0

〈q

jLq

iR

〉= ∆Vij

The degrees of freedom of V(x) are Goldstone bosons:

Identify with the 3 pi mesons.

Nambu and Jona-Lasinio: this is whyThe pions would be massless-exact Goldstones, if

is the pion decay constant = 93 MeV

V = eiπaσa/fπ

πa

m2

π ! m2

ρ

mu = md = 0

fπ

〈0|Jµ5a∣∣πb

〉= ipµfπδ

ab

Now couple this system to SU(2)xU(1) gauge bosons

Only the gauge symmetry

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.

qL (1

2, Y ) uR (0, Y +

1

2) dR (0, Y −

1

2)

qL → eiα(I3+Y )

qL qR → eiα(I3+Y )

qR

To compute the masses explicitly, write

for

This is the same structure as the MSM, where we found

DµV = ∂µV − igWaσa

2V − ig′BY V + ig′BV (Y ±

1

2)

L =1

4f2π tr DµV

†DµV

〈V 〉 = 1

L = 14f2

πtr

∣∣∣∣ g√2W+σ+ +g√2W−σ− +

g

2W 3σ3 − g

′

2Bσ3

∣∣∣∣2

=f2

π

4

[g2W+W− +

1

2(gW 3 − g′B)2

](1)

m2

W =g2

4v2

m2

Z =g2 + g′2

4v2

To obtain the correct masses, we need a scaled-up QCD - “technicolor” - in which

The analogue of the Higgs boson in this theory is the or resonance of QCD.

In technicolor, this appears as a peak (better, a shoulder) in S-wave WW and ZZ scattering at ~ 1600 GeV.

mTρ ∼ mρ ·v

fπ∼ 2 TeV

σ

!

a0

0++

A much better signal is the , which appears as a resonance in P-wave WW scattering and in

This resonance is already strongly constrained by LEP 2 data.

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 200 400 600 800 1000 1200 1400 1600 1800

M (GeV)

/M

excluded95% CL

ALEPH

e+e−

→ W+W

−

Tρ

There are more serious phenomenological problems with technicolor.

Mixing of the with W, Z alters the precision electroweak predictions at the 2-3% level, increasing and above the MSM expectation. Corrections of order 3% are also expected in

Fermion masses are generated by higher-dimension operators

There is no simple mechanism for flavor conservation, so we expect large corrections to , mixing,

Fixes for these problems may exist if the strong-coupling theory has special properties (“walking technicolor”, “conformal fixed point”).

sin2θw

1

M2Q

i

LqRqLujR

K0

B0 b → sγ

Γ(Z → bb)

mW

Tρ

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.

There must be two Higgs doublet fields with

including only one gives an anomalous gauge theory

quark-Higgs couplings are only of the form

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.)

have 8 degrees of freedom ➤

❨3 eaten Goldstones) + (CP even ) + (CP odd ) + ( )

L = −λijd dRH1α"αβQβ − λ

iju uRH2α"αβQβ

H1 , H2

h0 , H0

A0 H+ , H−

H1 , H2

SU(2)

SU(2)

U(1)

SU(2)

SU(2)

U(1)+h1 2

h~~

Y = ±1

2

The Higgs mass terms come from soft SUSY breaking:

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

A more important effect is the coupling to top quarks.

This sends to negative values as Q decreases.

L = −M2H1|H1|2 − M2H2|H2|

2

d

d log QM2H2 = −

3

2παwm(w̃)

2− · · ·

M2

H2

2h~

~

H2

w

~t

t

L

RH2

The same effect applies to 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 , we break SU(2)xU(1).

If , 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:

M2t̃L

, M2t̃R

〈H2〉 #= 0〈t̃R

〉!= 0

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

d log QM2

t̃L= +

λ2t(4π)2

· 1 · (M2t̃L

+ M2t̃R

+ M2H2 + A2

t ) −8

3παsm(g̃)

2− · · ·

d

d log QM2

t̃R= +

λ2t(4π)2

· 2 · (M2t̃L

+ M2t̃R

+ M2H2 + A2

t ) −8

3παsm(g̃)

2− · · ·

d

d log QM2H2 = +

λ2t(4π)2

· 3 · (M2t̃L

+ M2t̃R

+ M2H2 + A2

t ) (1)

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 .

Then, by coupling to gauge fields or to new particles, break down the constraints that keep exactly massless.

ϕ

ϕ

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

in which the H is a doublet.

We will want

All fields in must be massless if the SU(3) symmetries

are respected.

V = e2iΠata/f 2iΠata =

(Φ H

−H† φ

)

f ∼ 1 TeV , Mρ ∼ 10 TeV

Π

V → ΛRV Λ†L

Coupling this structure to the top quarks. We need to put top quarks into the representations

with an extra singlet quark. The Lagrangian is

The first term has the symmetry

The second term has the symmetry

Either symmetry suffices to insure that H is exactly massless. Thus, to build a Higgs potential, we need to involve both interaction terms.

V → V Λ†L

χ → ΛLχ

χL =

ubU

L

UR uR

L = −λ1f ( 0 0 uR ) V χL − λ2fURUL

V → ΛRV

Transform to the top quark mass eigenstates:

then the H vertices are:

tL = uL tR =λ2uR − λ1UR√

λ21

+ λ22

TL = UL TR =λ1uR + λ2UR√

λ21

+ λ22

mT =

√λ2

1+ λ2

2f

= −iλt = −iλT

λt =λ1λ2√λ2

1+ λ2

2

λT =λ21√

λ21

+ λ22

tR

tLH

TR

tLH

Using this structure, compute corrections to the H mass:

= −6λ2t

∫d4k

(2π)41

k2

= −6λ2T

∫d4k

(2π)41

k2 − m2T

= +6λTf

∫d4k

(2π)4mT

k2 − m2T

m2H = −3λ21λ

22f

2

8π2log

M2

m2T

= −3λ2

Tm2

T

8π2log

M2

m2T

t

t

L

RH

tL

RH

T

T

H

By the relations on the previous page, the quadratic divergences of these diagrams cancel. What is left over is:

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:

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.

mTf

=λ2t + λ

2

T

λT

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.