Download The Higgs Boson and Electroweak Symmetry Breaking

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.