Electroweak Symmetry Breaking

Electroweak Symmetry Breaking
Wolfgang Kilian
University of Siegen
Central European School in Particle Physics
Prague, Sept. 2007

PART I
The pedestrian way to the TeV scale

Effective Theory
Building Blocks: Quarks and Leptons
Q L =
L L =
(
UL
)
, Q
D R =
L
)
, L
E R =
L
(
NL
(
UR
)
,
D R
)
.
E R
(
NR
and the photon
A µ

Effective theory:
Generic Lagrangian consistent with the known or assumed
symmetries.
L = L(Q L , Q R , L L , L R , A)
Here: Electromagnetic gauge invariance

Organizing principle:
Dimensionality of the terms in the Lagrangian
◮ Dimension < 4: “relevant operators”
◮ Dimension = 4: “marginal operators”
◮ Dimension > 4: “irrelevant operators”

Start with the lowest dimension:
Dimension 2: Photon mass – forbidden by electromagnetic gauge
symmetry
Dimension 3: Fermion Masses:
L 3 = −( ¯Q L M Q Q R + ¯L L M L L R + h.c.)
− ( ¯N c L M N L
N L + ¯N c R M N R
N R ),
This contains both Dirac and Majorana mass terms

Gauge invariance relates kinetic energy and electromagnetic
interactions:
D µ = ∂ µ + ieqA µ ,
A µν = ∂ µ A ν − ∂ ν A µ ,
so the massless QED Lagrangian is
L 4 = ¯Q L i /DQ L + ¯Q R i /DQ R + ¯L L i /DL L + ¯L R i /DL R − 1 4 A µνA µν .
Dimension-4 Lagrangian: coefficients have zero mass dimension.
Numeric couplings:
q Q = y Q + τ 3
2
with constants y Q = 1 3 and y L = −1.
, q L = y L + τ 3
,
2

Weak interactions: Fermi model
L 6 = −4 √ 2 G F
(
2J
+
µ J µ− + ρ ∗ J 0 µJ µ0) ,
with charged and neutral currents
J µ ± = √ 1 [
¯QL τ ± γ µ V (†)
2
CKM Q L + ¯L
]
L τ ± γ µ L L ,
[
Jµ 0 = ¯Q L
(−q Q sw 2 + τ 3
2
(
+¯L L −q L sw 2 + τ 3
2
)
γ µ Q L + ¯Q R (−q Q s 2 w )γ µ Q R
)
γ µ L L + ¯L R (−q L s 2 w )γ µ L R
]
.
Dimension-6 Lagrangian: coefficients have negative mass
dimension.

Proposition:
This model has a hidden SU(2) × U(1) invariance.
. . . that is (almost) a gauge invariance, i.e., local symmetry

The current-current interaction allows for rewriting it as couplings
of currents to auxiliary vector fields W + µ , W − µ , Z µ :
where
L 6 = −g W (W +µ J + µ + W −µ J − µ ) − g Z (Z µ J 0 µ) + L 2(W ) ,
L 2(W ) = M 2 W W +µ W − µ + 1 2 M2 Z Z µ Z µ .
The vector fields are auxiliary; they can be eliminated by their
equations of motion:
W ± µ
= g W
MW
2 Jµ ∓ ,
Z µ = g Z
MZ
2 Jµ,
0

We have obtained vector-current couplings
gW 2 = 4M2 W
v 2 and gZ 2 = ρ 4MZ
2 ∗
v 2 .
and the characteristic scale of electroweak symmetry:
v ≡ ( √ 2 G F ) −1/2 = 246 GeV

To identify SU(2) L × U(1), we introduce a SU(2) L triplet of vector
fields W a µ and a U(1) vector field B µ :
and
W + = 1 √
2
(W 1 − iW 2 )
W − = 1 √
2
(W 1 + iW 2 )
Z = c w (1 + x Z )W 3 − s w (1 + y Z )B,
A = s w (1 + x A )W 3 + c w (1 + y A )B,
(Apart from A µ , all vector fields are still auxiliary)

The Lagrangian of a gauge-invariant theory should have the form
L = ¯Q L i /D L Q L + ¯Q R i /D R Q R + ¯L L i /D L L L + ¯L R i /D R L R
+ kinetic terms for gauge bosons
Covariant derivatives for SU(2) L × U(1) have the generic form
(
D Lµ = ∂ µ + i g 1q ′ − g 2
′
(
D Rµ = ∂ µ + i g 3q ′ − g 4
′
τ 3 )
B µ + igW a τ a
µ
2
2 ,
τ 3
2
)
B µ ,
The current-vector couplings of the weak interactions do have this
form, if we identify

Charged coupling:
g W = g.
No right-handed W couplings, and correct neutral-W
normalization:
U(1) anomaly free:
g Z = 1
c w
(1 − x Z )g
g = e
s w
(1 + x A )
g i ′ = g ′ , i = 1, 2, 3, 4,
y A + x Z = x A + y Z
g ′ = e (1 + y A )
c w

The symmetry is in the covariant derivatives, but not in the mass
terms.
To correct this, introduce a 2 × 2 matrix-valued field Σ with
Σ † Σ = ΣΣ † = 1
Such a field has three free parameters w 1 , w 2 , w 3 :
Σ(x) = exp
(− i )
v w(x) with w(x) = w a (x) τ a , a = 1, 2, 3,
We define the gauge transformation of Σ and thus the covariant
derivative of Σ:
D µ Σ = ∂ µ Σ + igW µ Σ − ig ′ ΣB µ ,

With this definition, we rewrite
and
L 3 = −( ¯Q L ΣM Q Q R + ¯L L ΣM L L R + h.c.)
− ¯L c 1 + τ 3
L Σ∗ M NL ΣL L −
2
¯L c R M 1 + τ 3
N R
L R ,
2
L 2(W ) = v 2
4 tr [
(D µ Σ)(D µ Σ) †] − β ′ v 2
where some abbreviations are useful:
V µ = Σ(D µ Σ) † and T = Στ 3 Σ † ,
8 tr [TV µ] tr [TV µ ]

Intermediate Result:
With the help of auxiliary fields, we have made explicit a hidden
symmetry of the weak interactions . . .
. . . except for the gauge-boson kinetic energy:
L em = − 1 4 A µνA µν
Therefore, if we argue that SU(2) × U(1) symmetry has physical
content, we predict the existence of physical gauge bosons, and
their associated kinetic terms:

Field-strength tensors:
W µν = ∂ µ W ν − ∂ ν W µ + ig[W µ , W ν ],
B µν = ∂ µ B ν − ∂ ν B µ ,
where
⇒ kinetic term
W µ = Wµ
a τ a
2
and B µ = B µ
τ 3
2 .
L 4(W ) = − 1 2 tr [W µνW µν ] − 1 2 tr [B µνB µν ] .

Experimental Consequences I
◮ Existence of two charged (W + , W − ) and one neutral (Z)
vector boson
◮ This was confirmed in the 80’s, and the properties of those
were measured in detail in the 90’s.
◮ Symmetry predicts the existence of those, but neither masses
nor couplings. (Note the appearance of x A , y A , x Z .) These are
free parameters.

Theoretical interpretation
We have just rewritten the electromagnetic + weak interactions as
a model with local SU(2) L × U(1) Y gauge invariance.
Due to gauge invariance, we can choose w a = 0 and thus Σ = 1 as
a gauge, so the fields w a are unphysical. Thus, the model has no
physical degrees of freedom in addition to the fermions and
massive vector bosons.
Turning this around, any model of fermions and massive vector
bosons can be written as a spontaneously broken gauge theory.

We can, however, keep w a ≠ 0 and derive Feynman rules for those.
(Including some gauge-fixing term), they become interacting scalar
fields. This implies expanding the Lagrangian around some ground
state, e.g., Σ = 1. This ground state is not gauge-invariant.
We therefore have a spontaneously broken local SU(2) L × U(1) Y
gauge invariance. The field Σ may be called a Higgs field, and the
generation of fermion masses by the Σ ground state value may be
called the Higgs mechanism.

Conclusion I
Calling the combination of electromagnetic and weak interactions a
spontaneously broken instead of simply a softly broken gauge
theory, without introducing further physical degrees of freedom, is
a statement without content.
Thus, so far this was an academic exercise.
New physical degrees of freedom require an experiment for their
detection.

But first, let us complete the effective theory.
So far, we have written all terms required by the low-energy
structure of weak and electromagnetic interactions, plus
gauge-kinetic terms for the additional gauge bosons. (All)
Lagrangian terms have mass dimension ≤ 4.
But we can add a lot of terms of dimension ≤ 4 that have no
direct impact on low-energy phenomenology.

Anomalous couplings
L 1 = α 1 gg ′ tr
[ΣB µν Σ † W µν]
[
]
L 2 = iα 2 g ′ tr ΣB µν Σ † [V µ , V ν ]
L 3 = iα 3 g tr [W µν [V µ , V ν ]]
L 4 = α 4 (tr [V µ V ν ]) 2
L 5 = α 5 (tr [V µ V µ ]) 2
L 6 = α 6 tr [V µ V ν ] tr [TV µ ] tr [TV ν ]
L 7 = α 7 tr [V µ V µ ] tr [TV ν ] tr [TV ν ]
L 8 = 1 4 α 8g 2 (tr [T W µν ]) 2
L 9 = i 2 α 9g tr [T W µν ] tr [T [V µ , V ν ]]
L 10 = 1 2 α 10(tr [TV µ ] tr [TV ν ]) 2
L 11 = α 11 gɛ µνρλ tr [TV µ ] tr [V ν W ρλ ]

The physical significance of those couplings:
◮ α 1 and α 8 replace two of the x, y mixing parameters
introduced earlier. A more well-known parameterization calls
them S, (T ), and U.
◮ The parameters α 2 , α 3 , α 9 , α 11 affect triple-gauge couplings.
Including them, the triple couplings of W , Z and photon
become arbitrary, consistent with e.m. gauge invariance.
◮ The remaining parameters introduce arbitrary quartic gauge
couplings.
Summary: All couplings are free, there is no prediction from
electroweak gauge invariance alone.
Any meaningful prediction draws from additional assumptions on
(extra) structure in the Higgs sector.

Intermediate Result: The effective theory
L = v 2
4 tr [
(D µ Σ)(D µ Σ) †]
− ( ¯Q L ΣM Q Q R + ¯L L ΣM L L R + h.c.)
− ¯L c 1 + τ 3
L Σ∗ M NL ΣL L −
2
¯L c R M 1 + τ 3
N R
L R
2
+ ¯Q L i /DQ L + ¯Q R i /DQ R + ¯L L i /DL L + ¯L R i /DL R
− 1 2 tr [W µνW µν ] − 1 2 tr [B µνB µν ]
− β ′ v 2
8 tr [TV ∑11
µ] tr [TV µ ] +
i=1
L i

Experimental Consequences II
◮ Unless there happen to be new degrees of freedom, the
parameters α i are the measurable quantities that complete
our picture of the effective theory of electroweak interactions.
◮ So far, all measurements that have been possible have
resulted in values consistent with zero (to be precise, at the
order of a few percent, as predicted by loop effects).

Theoretical Interpretation
While mathematically, local gauge invariance is an arbitrary
construct possible for any model of fermions and massive vector
bosons –
◮ due to the observed smallness of the anomalous couplings,
there appears to be physical meaning in the notion of
SU(2) L × U(1) Y as the spontaneously broken electroweak
symmetry.

PART II
Higgsology

Remark on the anomalous couplings:
They cannot be exactly zero; all operators appear as counterterms
to one-loop divergencies of the effective Lagrangian. For a precise
definition, one needs a renormalization scheme.
They cannot be exactly zero; scattering amplitudes of massive
vector bosons are unbounded at high energies and violate the
unitarity limit, hence corrections must be present such that the
unitarity limit is not violated.
This is true only for part of the couplings. If (as experimentally
confirmed),
β ′ ≈ 0, i.e. M W = c w M Z ,
half of the operators (those containing the T factor) do not appear
as one-loop counterterms.

L 1 = α 1 gg ′ tr
[ΣB µν Σ † W µν]
[
]
L 2 = iα 2 g ′ tr ΣB µν Σ † [V µ , V ν ]
L 3 = iα 3 g tr [W µν [V µ , V ν ]]
L 4 = α 4 (tr [V µ V ν ]) 2
L 5 = α 5 (tr [V µ V µ ]) 2
L 6 = α 6 tr [V µ V ν ] tr [TV µ ] tr [TV ν ]
L 7 = α 7 tr [V µ V µ ] tr [TV ν ] tr [TV ν ]
L 8 = 1 4 α 8g 2 (tr [T W µν ]) 2
L 9 = i 2 α 9g tr [T W µν ] tr [T [V µ , V ν ]]
L 10 = 1 2 α 10(tr [TV µ ] tr [TV ν ]) 2
L 11 = α 11 gɛ µνρλ tr [TV µ ] tr [V ν W ρλ ]

L 1 = α 1 gg ′ tr
[ΣB µν Σ † W µν]
[
]
L 2 = iα 2 g ′ tr ΣB µν Σ † [V µ , V ν ]
L 3 = iα 3 g tr [W µν [V µ , V ν ]]
L 4 = α 4 (tr [V µ V ν ]) 2
L 5 = α 5 (tr [V µ V µ ]) 2

Custodial SU(2) symmetry
This experimental fact (β ′ ≈ 0) can be formulated as an extra
(approximate) symmetry of the Lagrangian which, if exact, would
forbid the T terms:
Symmetry breaking pattern:
instead of
SU(2) L × SU(2) R → SU(2) C
SU(2) L × U(1) Y → U(1) em
The symmetry is not realized in the gauge sector (otherwise, we
must have right-handed W bosons), neither in the fermion sector
(otherwise, t- and b masses would be identical, and quark mixing
would be absent), but it appears to be an important property of
the Higgs sector that does not follow from the original symmetry
assumption.

More precise definition:
If we set
e → 0 and s w → 0 with g = e/s w fixed
and, in the fermion sector
m u = m d which implies V CKM = 1
the electroweak effective Lagrangian exhibits the
symmetry-breaking pattern
SU(2) L × SU(2) R → SU(2) C
Consequently, (loop) corrections to ρ = MW 2 /(c2 w MZ 2 ) = 1 are of
order
e 2
16π 2
and
(m t − m b ) 2
16π 2 v 2

Theoretical Interpretation
Any model that explains electroweak symmetry breaking should
have an explanation for those two observations:
◮ the anomalous couplings are small
◮ the Higgs sector has an approximate custodial symmetry

Looking at the field Σ:
Σ(x) = exp
(− i )
v w(x)
with w(x) = w a (x) τ a , a = 1, 2, 3,
So far, it was a mathematical construct, and the scalar fields w a
parameterizing Σ were unphysical. In fact, their scattering
amplitudes are related to the scattering amplitudes of
longitudinally polarized massive vector bosons.
Alternatively, there may be more content associated to Σ.
Simple possibility: relax the constraint on Σ:
with a real constant c.
Σ † Σ = c × 1 (1)

This allows for fluctuations of the modulus of the Higgs field:
(
Σ(x) = 1 + H )
exp
(− i )
v v w(x) with w(x) = w a (x) τ a ,
The new field H is physical.
This modification allows for a reparameterization
Σ = 1 v
(
(v + H ′ ) − iw ′)
and, recalling that this is a 2 × 2 matrix, we can write this matrix
also as
Σ = 1 ( )
( √ )
−i 2w
+
˜φ φ where φ =
v
v + H + iw 3
and ˜φ = −iτ 2 φ ∗ .

Standard Model Power Counting
Since we no longer have an infinite series in the expansion of Σ, we
may rather extract the factor 1/v and assign φ the mass dimension
1 (instead of zero). This turns all marginal operators with Σ
factors into irrelevant operators.
⇒ Anomalous couplings get a natural prefactor (m H /Λ) n , if the
Lagrangian term contains n factors of Σ.
⇒ The custodial-symmetry violating terms get extra factors of
(m H /Λ) due to the Σ factors in the T definition
⇒ Either m H is small, or Λ is large, or both. Incidentally, with
zero anomalous couplings the model (the Standard Model) is
renormalizable, so the anomalous couplings are not needed as
counterterms.

Theoretical interpretation
Models of electroweak symmetry breaking should
◮ if they contain no light Higgs, either provide a large cutoff
scale Λ or some other means that suppresses anomalous
couplings, and implement custodial symmetry.
◮ contain a light Higgs. However, the interactions of this Higgs
particle have to be further investigated.
◮ contain a more complicated Higgs sector, that, however, has
the same effect as a light Higgs on suppressing those terms,
and should respect custodial symmetry.

The Higgs-less case
Assume that no Higgs will be found at LHC and ILC, i.e., no light
Higgs state exists.
Can the cutoff scale Λ be large

Unitarity argument: Scattering amplitudes of massive vector
bosons violate unitarity at √ s ≤ 1 TeV. This should be accessible
at the LHC.
Loop argument: All (custodial-symmetric) anomalous couplings are
required as counterterms to loop diagrams. The values depend on
the renormalization scheme, but generically are of the order
1/16π 2 .
⇒ The cutoff cannot be larger than about
Λ ≈ 4πv ≈ 3 TeV
(The unitarity limit is even lower.)

With a cutoff of, e.g., 1.5 TeV, the typical size of anomalous
couplings is
|α| ∼ v 2
Λ 2 ∼ 3 %.
This is marginally consistent with data.
⇒ The Higgs-less scenario is not favored by data, but still possible.
In a model without Higgs, there should not be large enhancement
factors, rather some extra built-in suppression of anomalous
coefficients.

The Standard Model Case
Now, let us include a Higgs boson ⇒ the Standard Model.
This allows us to ignore anomalous couplings. (Their natural size
would be m 2 H /Λ2 .)
With zero anomalous couplings, the SM fits data well for a Higgs
of 115 GeV, at most some 200 GeV (the smaller, the better).
In the SM, custodial-symmetry violating terms are naturally
suppressed, sufficient for a good fit without them.

In the SM, the Higgs boson is a dynamical particle, and there are
loop corrections from its interactions with
◮ the top quark.
◮ the weak gauge bosons,
◮ with itself,
◮ and others.
This results in quantum corrections to the Higgs mass of the order
of
∆mH 2 ∼
m2 t M
(4πv) 2 Λ2 W
2 m
,
(4πv) 2 Λ2 H
2 ,
(4πv) 2 Λ2
where Λ is some cutoff. The top-loop correction violates custodial
symmetry.

If the mass correction is to be of the same order as the Higgs mass
itself, we get the order-of-magnitude estimate
Λ 4πv = 3 TeV.
This is the same estimate as for the Higgs-less case.
⇒ The introduction of the (light) Higgs boson does not really
improve the situation over the no-Higgs case.
In a model with a light Higgs, there should be some mechanism
that removes the cutoff dependence from the Higgs self-energy.

A Heavy Higgs
The constraints from top and W loops can be avoided if the Higgs
mass itself is rather heavy. (The limit is below 1 TeV.)
In this case, only Higgs self-couplings count. With increasing
energy, the Higgs self-coupling becomes non-perturbative.
The result is a Landau pole in the Higgs self-coupling at about
Λ ∼ m H exp 8π2 v 2
6m 2 H
(Similarly, for a too light Higgs the top-loop drives the
self-coupling negative at high energies.)
Only for m H ∼ 130 GeV, these cutoffs can be pushed to infinity.

Multiple Higgses
So far, we have put constraints on the Higgs matrix field Σ:
Σ † Σ = 1 or Σ † Σ = c × 1.
We could remove these constraints (keeping the transformation
properties) and make Σ an arbitrary complex 2 × 2 matrix.
Then, the two columns are no longer related:
Σ = 1 ( )
˜φ φ
v
where φ and ˜φ are two independent complex doublet vectors.
This is a (type-II) two-Higgs-doublet model.

The ground state 〈Σ〉 could take an arbitrary direction
while still
〈Σ〉 = (v 1 + v 2 ) 1 2 + (v 1 − v 2 ) τ 3
2
〈Σ † Σ〉 = (v 2 1 + v 2 2 ) × 1 = v 2 × 1
The ratio v 1 /v 2 = tan β is a new parameter, a measure of
custodial symmetry violation in the Higgs sector.
A number of new self-couplings in the Higgs sector is possible,
some of them custodial symmetry violating.

Phenomenology:
Two neutral Higgs bosons h and H;
A triplet of charged and “pseudoscalar” H + , A 0 , H − .
This model has a decoupling limit where H, H + , A 0 , H − become
heavy (and degenerate).

An infinite number of Multi-Higgs models is possible. There is a
distinction:
◮ Custodial symmetry is present in interactions and vacuum
expectation values (e.g., SM). Then, everything is consistent
with data.
◮ Custodial symmetry is present in interactions, not necessarily
in vacuum expectation values (tan β). Examples: Models with
only doublets and singlets. Then, there are some loop
corrections to the ρ parameter.
◮ Custodial symmetry is violated (e.g., lone Higgs triplets).
Then, there are tree-level corrections to the ρ parameter.
Such states should be rather heavy (TeV).
Furthermore, important constraints come from the absence of
FCNC.

Example for a nontrivial Higgs sector with custodial symmetry:
Higgs multiplet
⎛
φ ++ χ + ⎞
¯φ0
Φ = ⎝ φ + χ 0 ¯φ− ⎠
φ 0 χ − ¯φ−−

Experimental Consequences III
Establish or exclude the (light) Higgs boson.
◮ If not there, there must be effects in the TeV range that
account for Higgs-less electroweak symmetry breaking and cut
off the low-energy effective theory.
◮ If there is a light Higgs boson, there should be effects that cut
off the corrections to its self-coupling.
◮ The Higgs-sector structure may not be minimal; identify all
scalar states associated with it.

The LHC is well suited to study the Higgs-less case
and will be able to find the most relevant (neutral) Higgs state,
irrespective of its decay modes.
However, it is not clear how far other Higgs-sector states can be
covered. This will be the domain of the ILC, if at all.

PART III
Signal processes at colliders

Not Relying on the Higgs
While anomalous couplings have been measured (rather,
constrained) before, these measurements will be refined, and new,
previously unaccessible couplings will be measured.
First of all, there are the vector-boson mixing parameters. These
are determined from
◮ The W and Z masses
◮ The W and Z gauge couplings, extracted from their decay
widths and branching ratios
◮ The Fermi “constant”, anomalous contributions to which can
be identified by looking at, e.g., e + e − → µ + µ − at high energy

These measurements have been done at LEP, to a good precision.
The LHC will not significantly improve the current situation.
For any improvement in this area, a new experimental program of
Z and W measurements must be performed at an e + e − collider
(ILC). This is not likely to happen very soon.
⇒ The constraints on mixing parameters will essentially remain in
their current state.

Next, there are triple gauge couplings.
The LEP II limits are marginally sensitive to new physics. The
limits clearly need improvement.
At the LHC, qq → W + W − can be identified in four-fermion final
states up to multi-TeV energies. Thus, new physics effects
(resonances) are likely to be accessible.
However, at hadron colliders this measurement suffers from
unavoidable ambiguities; a meaningful determination of anomalous
couplings is difficult.
This is clearly the domain of the ILC.

Quartic gauge couplings have never been measured before.
There are two channels:
1. Triple vector-boson production
2. Quasi-elastic vector boson scattering
Both result in 6-fermion final states, with distinctive kinematics.

Here, sensitivity studies for the LHC are still not complete.
Main reason:
◮ Complexity of the processes: need viable Monte-Carlo
simulation programs
Comparable studies for the ILC have been done, with encouraging
results (α parameters can be determined to percent-level accuracy).
This analysis will be done at the LHC, even if a Higgs boson exists.

Higgs Measurements
The Higgs state gives mass to vector bosons. This has to be
checked.
The coupling to vector bosons is reflected in production channels
W + W − → H and ZZ → H
If the Higgs mass is sufficiently high ( 130 GeV), there are also
corresponding decay channels
H → W + W − and H → ZZ
These are all accessible at the LHC and ILC; furthermore, the same
coupling is probed in
(f ¯f →)Z ∗ → HZ and (f ¯f ′ →)W ∗ → HW

To establish the Higgs as the carrier of electroweak symmetry
breaking, the absolute value of the HWW and HZZ couplings (cf.
custodial symmetry!) must be determined experimentally.
At the ILC, this is possible using the inclusive channel
e + e − → Z + X
where X includes all decay modes of H.
At the LHC, this is not possible.
However, at the LHC (and the ILC) it can be checked that the
Higgs cancels the rise of WW scattering amplitudes, i.e., the key
experiment is
W + W − → W + W − etc.
at high energies, just as in the no-Higgs case.

The Higgs field gives mass to fermions. This also has to be
checked.
◮ The coupling to top quarks can be measured directly in t¯tH
production, or H → t¯t decays, if available
◮ The coupling to lighter fermions is more difficult at the LHC
(exception: τ), but straightforward at the ILC
◮ Indirectly, the main LHC production mechanism
gg → H(→ γγ) probes the top-quark coupling. However,
without a direct determination of the HWW and HZZ
couplings, this does not identify H as the Higgs boson.
(cf. Top-Higgs, pseudo-axions, techni-pions, . . . )

The Higgs field has a potential that contains a quadratic and a
quartic term:
V (H) = − µ2
2 (v + H)2 + λ (v + H)4
4
Knowing v and m H = √ 2λ v fixes λ, and thus the cubic and
quartic Higgs self-couplings.
Measuring the Higgs self-coupling establishes this theoretical idea
and distinguishes the Higgs from a scalar particle “accidentally”
coupled to W and Z.
Probes:
W + W − → HH,
Z ∗ → ZHH
⇒ impossible at LHC (small window at SLHC if Higgs not too
light)
⇒ possible at ILC, if Higgs not too heavy

σtot [fb]
1000
MH = 115 GeV
¯νeνeH
100
10
ZH
1
¯νeνeHH
0.1
ZHH
0.01
0.001
10 4 ZHHH
√ s [TeV]
¯νeνeHHH
1 2 5 10 20

Is the Higgs sector complete
. . . we may never have a final answer to this.
At least,
◮ At the LHC, one may detect Higgs-sector particles that decay
to leptons (τ) or top quarks, or have high decay-product
multiplicity
◮ At the ILC, one may detect Higgs-sector particles with any
decay mode that are not too heavy
Furthermore, an incompleteness of the Higgs sector manifests itself
in anomalous Higgs couplings.

Cutoff Physics
If there is a Higgs boson, there are probably new fields (particles)
coupled to it that cut off the Higgs self-energy.
They are coupled to the Higgs. Therefore,
◮ If they are lighter than the Higgs, they should appear as a
Higgs decay mode.
◮ If they can be produced by other means, Higgs bosons should
appear in their decays.
A measurement of those couplings would establish the cut-off
nature of these new degrees of freedom.

Another cutoff possibility would be Higgs compositeness.
◮ This is probably difficult to see at the LHC, it might require
much higher energy. However, the Higgs form-factors would
manifest themselves, to first order, as anomalous couplings.

Experimental Consequences IV
◮ If a Higgs is found, all couplings and properties must be
measured with the best possible accuracy.

PART IV
Gallery of Models

So far, we followed a purely bottom-up approach. We could learn
that
◮ A minimal spontaneously broken SU(2) L × U(1) Y gauge
theory is a good description of electroweak physics
◮ The Higgs “mechanism” may involve either no Higgs particle,
a single Higgs particle, or more. The Higgs case (SM) is
appealing because all precision data nicely fit, but
◮ In either case, there is a strong motivation for additional new
physics in the TeV range, if not lower.
Furthermore, the theory of electroweak symmetry breaking should
account for
1. Small anomalous couplings
2. Approximate custodial symmetry

Technicolor
You should have realized that electroweak symmetry breaking and
(QCD) chiral symmetry breaking are analogous to each other.
This analogy may be carried to the end.
Imagine that there is a pair of electroweak doublets (U, D) L,R of
new massless fields, coupled to a confining “technicolor” gauge
force, with characteristic scale few TeV (instead of 100 MeV).
⇒ Then, chiral symmetry breaking would occur at TeV energies
⇒ This would imply electroweak symmetry breaking at
TeV/few π.
⇒ Custodial symmetry would be realized
⇒ There would be WW -resonances in the TeV range, but
probably no significant scalar one, i.e., no Higgs.

Problem 1: Quarks and leptons do not acquire masses this way (no
chiral symmetry breaking).
◮ they must be mixed with the technifermions
⇒ Extended Technicolor
Problem 2: Extended Technicolor generates FCNC, but no
sufficient t/b masses (even c).
◮ Technicolor dynamics must be much different from QCD
⇒ Walking Technicolor
Problem 3: Technicolor gives a too large contribution to the S
parameter
⇒ Walking Technicolor
Experiment: WW scattering, light techni-mesons

Higgsless Extra dimensions
The rise of WW scattering amplitudes would be dampened by a
scalar resonance (the Higgs) in the s-channel.
It would equivalently be dampened by t-channel vector exchange.
But we would need an infinite number of those, with a coupling
sum-rule satisfied.
◮ Such a tower of W /Z resonances would appear if there was a
fifth dimension in which the electroweak gauge bosons could
propagate.
◮ This is embedded in a warped geometry (Randall-Sundrum),
where scale transformations up to the Planck scale are related
to translations in the extra dimension.

Problem 1: Custodial symmetry
◮ In the “bulk”, there is a SU(2) R symmetry.
Problem 2: S Parameter
◮ The calculation is questionable.
Problem 3: Flavor Physics
◮ Fermions are slightly delocalized in the extra dimension. This
would also explain the hierarchical CKM structure.
Experiment: WW scattering, gravitational effects
Note: The phenomenology is close to Walking Technicolor. It may
even be equivalent (AdS/CFT)

Topcolor
There is a Higgs, but it is a composite state, so the compositeness
scale is a cutoff.
The Higgs is composed of top quarks (maybe mixed with a heavy
top-quark partner χ).
◮ Top-quarks couple to a different QCD (Top-color). At TeV
energies, there is dynamical symmetry breaking
SU(3) 12
C
× SU(3)3 C → SU(3)QCD C
◮ Topcolor exchange is not confining, but strong enough for a
bound state H ∼ t¯t.

Problem 1: Electroweak symmetry breaking parameters
◮ Combine with technicolor. The weaknesses of both model
seem to cancel out.
Problem 2: Custodial symmetry
◮ t and b are distinguished by an extra U(1) quantum number
(tilting). This is no worse than the SM hypercharge coupling.
Experiment:
+ Top-pions, decaying to t and b
+ Top-quark “resonance” χ
+ Z ′ resonance with characteristic top-quark couplings
+ Top-gluons or colorons

Elementary Higgs
The previous models implemented “dynamical” symmetry
breaking, where no elementary Higgs bosons were involved.
Note that in any such model, one can define a formal composite
Higgs field out of the symmetry-breaking currents, and in some
models (e.g., topcolor), there is actually a corresponding physical,
but composite state. So there is always a Higgs mechanism.
Can there be a truly elementary Higgs boson
From the perspective of the fundamental theory (the Planck
scale), this would be an almost exactly massless scalar. In four
dimensions, there are two mechanisms that generate massless
scalars:
1. The Goldstone theorem: spontaneous breaking of a global
symmetry
2. Supersymmetry: a scalar superpartner of a chiral fermion

Radiative Symmetry Breaking
The electroweak Higgs boson is part of an electroweak doublet; it
has weak charge, hypercharge, and fermionic couplings.
For the Higgs potential, loop effects are important, maybe
dominant.
⇒ Coleman-Weinberg-Linde potential (1-loop)
V (φ) = aφ 2 + bφ 4
+ Λ2
32π 2 STr [ M 2 (φ) ]
+ 1
64π 2 STr [
M 4 (φ)
(
log M2 (φ)
Λ 2 − 1 )]
2

MSSM
In supersymmetric models, the second line vanishes.
In the MSSM, the tree-level quartic coupling is a combination of
gauge couplings (thus rather small), while the tree-level mass
squared is a combination of the µ-term with the
soft-supersymmetry breaking masses m Hu , m Hd . This combination
is typically positive.
Thus, the last term is responsible for electroweak symmetry
breaking.
As a result, in the MSSM, the electroweak scale v is proportional
to the soft-SUSY-breaking scale or the µ parameter µH u H d
(whichever dominates).
Explaining electroweak symmetry breaking is thus linked to
explaining supersymmetry breaking

Problem 1: Soft supersymmetry breaking is difficult to explain as
spontaneous symmetry breaking.
◮ . . . but not impossible. However, such models require a
“hidden sector”.
Problem 2: Soft supersymmetry breaking parameters carry flavor
quantum numbers and thus cause FCNC
◮ This puts strong constraints on the mechanism of SUSY
breaking
Problem 3: What relates the µ parameter with soft SUSY
breaking
◮ Either another constraint on soft SUSY breaking, or additional
dynamics: NMSSM

NMSSM
In the NMSSM, the µ parameter is the ground-state value of an
additional Higgs field S:
µ = 〈S〉
This completely relates EWSB to soft-SUSY breaking (typically
assuming a trilinear superpotential term S 3 and additional
soft-breaking terms).
Alternatively, singlet symmetry breaking may also be due to
radiative symmetry breaking. This requires additional superfields.

Little Higgs
In SUSY models, the second line of the CW potential vanishes to
all orders.
However, to keep the cutoff Λ high, it suffices if it vanishes to
one-loop order.
This is realized in Little-Higgs models.

How to make up a Little Higgs model:
1. Invent fields and interactions such that the model contains
two “sectors”.
2. The complete model has a gauge symmetry which is
spontaneosly broken down to the SM symmetry
SU(2) L × U(1) Y at a scale F (TeV range).
3. Consider now the models where either sector is removed. In
these limiting cases, there should be a spontaneously broken
global symmetry. The associated Goldstone multiplet should
contain the SM Higgs doublet.
⇒ Since to get rid of this Goldstone nature, the Higgs must
“feel” both sectors simultaneously, the quadratic term in the
potential appears first at two-loop order.

Template realizations:
◮ Littlest-Higgs type:
◮ Global symmetry group (SU(5)) which embeds two copies of
the SM symmetry (SU(2) 2 × U(1) 2 ). Breaking
SU(5) → SO(5) results in the SM.
◮ Irreducible scalar multiplet which contains the SM Higgs
◮ Simplest-Higgs type:
◮ Simple gauge group (SU(4)) which embeds the SM symmetry
◮ Multiple copies of the scalar multiplet which contains the SM
Higgs

In Little Higgs models, electroweak symmetry breaking is the result
of the Coleman-Weinberg mechanism
Similar to the MSSM, a partner of the top quark plays the
dominant role.
In general, the spectrum exhibits heavy vector bosons (W ′ , Z ′ ),
heavy quarks (T , . . .), and heavy scalars (Φ), often an enlarged
Higgs sector of light scalars.

Furthermore
◮ Many models of EWSB proposed today implement extra
dimensions, often associated with SUSY
◮ Dynamical extra-dimension models contain scalars (moduli,
dilatons, radions) associated with fluctuating scales (sizes) of
the extra dimensions. Since the Higgs boson in the
electroweak theory implements breaking of scale invariance,
there may be a close relation
◮ Alternatively, Higgs-gauge unification identifies the Higgs with
the fifth component of a five-dimensional gauge field, the
other four components resulting in an ordinary gauge field
◮ There is no clear distinction between strongly-interacting and
weakly-interacting models

Final Summary
◮ Electroweak interactions are described by a minimal effective
theory of spontaneously broken SU(2) × U(1) gauge
symmetry.
◮ The physical mechanism of electroweak symmetry breaking is
unknown. Economical (SM-like or non-SM-like) descriptions
appear incomplete and point to new physics at TeV energies.
◮ Therefore, this issue can be settled not by theoretical
arguments, but only by experimental data. However,
interpreting the LHC/ILC results and making up the theory of
electroweak symmetry breaking will require a major theoretical
effort.