The Physics of Electroweak Symmetry BreakingPhysik Master, Fruhjahrssemester 2009

Jose SantiagoITP ETH Zurich

June 27, 2009

Contents

1 Introduction 4

2 Symmetries in Physics and their Fate 52.1 Classical Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Continuous Symmetries and Conservation Laws . . . . . . . . . . . . . . . 62.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3.1 Scalars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3.2 Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4 Symmetries and Degeneracy . . . . . . . . . . . . . . . . . . . . . . . . . . 102.5 Spontaneous Breaking of a Global Symmetry . . . . . . . . . . . . . . . . . 11

2.5.1 Spontaneously Broken U(1) Symmetry . . . . . . . . . . . . . . . . 112.5.2 Goldstone’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.6 Spontaneous Breaking of a Local Symmetry . . . . . . . . . . . . . . . . . 162.6.1 An Abelian Example . . . . . . . . . . . . . . . . . . . . . . . . . . 172.6.2 Spontaneous Symmetry Breaking of non-Abelian Gauge Symmetries 202.6.3 Formal Aspects of the Higgs Mechanism . . . . . . . . . . . . . . . 21

2.7 The sigma model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.7.1 Different Representations of the Sigma Model . . . . . . . . . . . . 27

2.8 Non-linear Realization of a Symmetry . . . . . . . . . . . . . . . . . . . . . 32

3 Electroweak Symmetry in the Standard Model 363.1 Electroweak Symmetry and its Breaking . . . . . . . . . . . . . . . . . . . 363.2 The Need for an Electroweak Symmetry Breaking Sector . . . . . . . . . . 40

3.2.1 Custodial symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . 433.2.2 Unitarity Violation . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.3 The SM EWSB sector: unitarity restoration with one scalar . . . . . . . . 483.3.1 Higgs unitarization of longitudinal gauge boson scattering . . . . . 483.3.2 Constraints on the Higgs mass . . . . . . . . . . . . . . . . . . . . . 50

3.4 The hierarchy problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.5 The Standard Model as an Effective Theory . . . . . . . . . . . . . . . . . 55

3.5.1 Precision tests of the Standard Model . . . . . . . . . . . . . . . . . 563.5.2 Constraints on new physics . . . . . . . . . . . . . . . . . . . . . . 573.5.3 Contribution of dimension 6 operators . . . . . . . . . . . . . . . . 65

1

3.5.4 A Simple Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4 Electroweak Symmetry Breaking in Supersymmetric Models 714.1 Motivation for Supersymmetry . . . . . . . . . . . . . . . . . . . . . . . . . 714.2 Formalism of Supersymmetry: Building Supersymmetric Lagrangians . . . 72

4.2.1 The Simplest Supersymmetric Lagrangian: Non-interacting Wess-Zumino Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.2.2 Non-gauge Interactions of Chiral Multiplets . . . . . . . . . . . . . 724.2.3 Lagrangians for Gauge Multiplets . . . . . . . . . . . . . . . . . . . 734.2.4 The hierarchy problem in SUSY . . . . . . . . . . . . . . . . . . . . 734.2.5 Soft Supersymmetry Breaking Interactions . . . . . . . . . . . . . . 754.2.6 The minimal supersymmetric standard model . . . . . . . . . . . . 75

5 Unitarity Restoration by an Infinite Tower of Resonances: Technicolor 815.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.2 Low energy QCD and chiral symmetry breaking . . . . . . . . . . . . . . . 82

5.2.1 The pattern of chiral symmetry breaking and restoration of unitarity 825.2.2 Turning on weak interactions . . . . . . . . . . . . . . . . . . . . . 85

5.3 Technicolor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895.3.1 The simplest example: minimal technicolor model of Weinberg and

Susskind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 905.3.2 Phenomenological issues with technicolor and possible resolutions . 915.3.3 New Developments in Technicolor . . . . . . . . . . . . . . . . . . . 97

6 Composite Higgs Models 1036.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1036.2 Composite Higgs Models: General Structure . . . . . . . . . . . . . . . . . 104

6.2.1 The minimal composite Higgs model . . . . . . . . . . . . . . . . . 1046.2.2 The minimal (custodial) composite Higgs model . . . . . . . . . . . 1056.2.3 EWSB in the minimal composite Higgs model . . . . . . . . . . . . 110

6.3 Fermions in Composite Higgs Models . . . . . . . . . . . . . . . . . . . . . 1206.3.1 Fermion masses the wrong way . . . . . . . . . . . . . . . . . . . . 1216.3.2 Fermion masses the right way . . . . . . . . . . . . . . . . . . . . . 1226.3.3 Partial compositeness . . . . . . . . . . . . . . . . . . . . . . . . . . 1236.3.4 Fermion contribution to the Higgs potential . . . . . . . . . . . . . 125

7 Models with Extra Dimensions 1297.1 5D models compactified on a circle . . . . . . . . . . . . . . . . . . . . . . 129

7.1.1 Gauge theories in a circle . . . . . . . . . . . . . . . . . . . . . . . . 1297.1.2 Fermions in a circle . . . . . . . . . . . . . . . . . . . . . . . . . . . 1317.1.3 Couplings in extra dimensions . . . . . . . . . . . . . . . . . . . . . 132

7.2 5D models compactified on an interval . . . . . . . . . . . . . . . . . . . . 132

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7.2.1 Models in an interval . . . . . . . . . . . . . . . . . . . . . . . . . . 1337.2.2 Holographic Method . . . . . . . . . . . . . . . . . . . . . . . . . . 136

8 Conclusions 141

3

Chapter 1

Introduction

Our goal in this course in to discuss our current understanding of electroweak symmetrybreaking (EWSB), including the lack of experimental evindence for the precise mechanismthat triggers it. We will also investigate different alternative realizations, starting withthe Standard Model (SM) and moving on to different models beyond the SM (BSM), theirstrengths and weaknesses and the current constraints on them. The ultimate goal is todevelop the knowledge and tools to properly interpret the results that the LHC will provideus with in order to fully understand the process of EWSB.

These notes are meant as a guide to the different topics we will cover in the course butare not going to be comprehensive. We will try to always give references to the literatureand text books to complement the discussion here.

4

Chapter 2

Symmetries in Physics and their Fate

Literature: The topics in this Chapter are covered in many text-books. I have mainly followed Pokorski [1], Donoghue, Golowich andHolstein [2], Georgi [3] and Peskin and Schroeder [4]. Particular top-ics are also covered in lecture notes or review articles that will becited correspondingly. Finally, I have used extensively Contino’s notes(http://indico.phys.ucl.ac.be/conferenceDisplay.py?confId=148).

2.1 Classical Equations of Motion

Our main tool in this course will be quantum field theory and its building block, the action,

S[φ] =

∫d4xL(φ(x), ∂µφ(x)), (2.1)

where L is the Lagrangian density (from now on denoted simply as the Lagrangian), thatwe assume depends only on fields and their first derivatives (and we have not made explicitpossible extra space-time, internal or flavor indices in the fields). The classical equationsof motion for the fields are obtained from Hamilton’s principle of stationary action, forarbitrary variations of the fields (but with fixed “boundary conditions”). That is,

δS = 0, (2.2)

for arbitrary variations of the fields,

φ(x)→ φ(x) + δφ(x), (2.3)

with δφ(x) = 0 for the boundary of integration Ω (which can include spatial infinity).Computing explicitly the variation of the action we obtain

δS =

∫Ω

d4x

[δLδφδφ+

δLδ∂µφ

δ∂µφ

]=

∫Ω

d4x

[δLδφ− ∂µ

δLδ∂µφ

]δφ+ ∂µ

[δLδ∂µφ

δφ

],

(2.4)

5

where we have used that the coordinates x do not change in the variation,

δ∂µφ = ∂µδφ,

and integration by parts. The second term in square brackets results in a surface termthat vanishes because the variation is zero at the boundary. The action will be stationaryfor arbitrary variations if the classical equations of motion are satisfied

δLδφνi− ∂µ

δLδ∂µφνi

= 0. (2.5)

We have explicitly written the corresponding equations for an arbitrary number of flavors(denoted by i) of vector fields (with an explicit space-time index ν) as an example. Notethat any two Lagrangians that differ by a total detivative of an arbitrary function of thefields,

L′ = L+ ∂µVµ(φ),

give exactly the same equations of motion, due to de vanishing of the variation of the fieldsat the boundary.

2.2 Continuous Symmetries and Conservation Laws

A symmetry of the action is some (infinitesimal) change of the fields, δφ, such that theaction remains invariant,

S[φ+ δφ] = S[φ], (2.6)

or in terms of the Lagrangian,

L+ δL ≡ L(φ+ δφ, ∂µφ+ δ∂µφ) = L(φ, ∂µφ) + ∂µVµ(φ, ∂µφ, δφ). (2.7)

Here we don’t assume that the variation of the fields vanishes at the boundary but thatthe fields and their derivatives vanish fast enough at the boundaries to make the boundaryterm vanish.

Symmetries have important consequences. They imply conserved currents and definitestatements about the spectrum of the theory. The first consequence is given by Noether’stheorem, which states that for any invariance of the action under a continuous transfor-mation of the fields, there exists a classical charge Q which is time-independent (Q = 0)and is associated with a conserved current, ∂µJ

µ. The theorem includes both internal andspace-time symmetries. For a continuous internal symmetry, it is just a simple consequenceof the classical equations of motion. If we impose the equations of motion on (2.4), weobtain

δL =[EoM

]δφ+ ∂µ

[δLδ∂µφ

δφ

]= ∂µ

[δLδ∂µφ

δφ

]= ∂µV

µ. (2.8)

6

Thus, there is a current associated to the symmetry,

Jµ ≡ δLδ∂µφ

δφ− V µ, (2.9)

that is conserved,∂µJ

µ = 0, (2.10)

and an associated charge

Q(t) ≡∫d3x J0(t, x), (2.11)

which is a constant of motion,

Q =

∫d3x ∂0J0 = −

∫d3x ∂iJ

i = 0,

provided the currents fall off sufficiently rapidly at the space boundary Ω.Let us consider the case that the symmetry is not only a symmetry of the action but

also of the Lagrangia, V µ = 0. Furthermore, let’s assume the symmetry is a linear unitarytransformation of the fields,

δφ = iεaTaφ, (2.12)

where εa are a set of infinitesimal parameters (that for the moment will be consideredindependent of space-time, i.e. we will be considering a global symmetry) and T a,a = 1, . . . ,m are a set of N × N hermitian matrices acting on flavor space (φ is a vectorin such space), which satisfy the Lie algebra of the corresponding symmetry group,

[T a, T b] = ifabcTc, TrT aT b ∝ δab. (2.13)

Applying our general expression of Noether’s current to this particular case for V µ = 0 andusing that the εa are arbitrary parameters, we get the conserved currents (with standardnormalization)

Jµa = −i δLδ(∂µφ)

T aφ, a = 1, . . . ,m. (2.14)

Our derivation of Noether’s theorem also tells us what the divergence of the current is inthe case that the transformation is not a symmetry of the Lagrangian,

∂µJµ = δL. (2.15)

Before going into some examples of conserved currents in physical systems, it is importantto emphasize that the charges associated to these currents,

Qa(t) ≡∫d3x J0

a(t, x), (2.16)

even if they are not conserved (because the transformation is not a good symmetry of theaction or lagrangian), they are still the generators of the transformation (2.12). First, they

7

obviously satisfy the same commutation relations as the matrices T a. Second, they actuallygenerate the field transformations (2.12) through Poisson brackets. Let us introduce theconjugate momenta for the fields φ(x) (that we assume different from zero)

Π(t, x) =δL

δ(∂0φ(t, x)). (2.17)

The Hamiltonian of the system reads

H =

∫d3x [Π∂0φ− L], (2.18)

and the Poisson bracket of two functionals F1,2 of the fields Π and φ is defined as

F1(t, x), F2(t, x) =

∫d3z

[δF1(t, x)

δφ(t, z)

δF2(t, y)

δΠ(t, z)− δF1(t, x)

δΠ(t, z)

δF2(t, y)

δφ(t, z)

]. (2.19)

In particular we have

Π(t, x), φ(t, y) = −δ(x− y), (2.20)

Π(t, x),Π(t, y) = φ(t, x), φ(t, y) = 0. (2.21)

The Noether charge can be written as

Qa(t) =

∫d3xΠ(t, x)(−iT a)φ(t, x). (2.22)

Computing its Poisson bracket with φ we get

Qa(t), φ(t,x) = (iT a)φ(t,x), (2.23)

and thus, Qa generates the transformation on the fields (independently of whether it isconserved or not).

If the transformation does not only affect the fields but also the coordinates, Noether’stheorem has to be generalized. We will show here the result (see Pokorski for a fulldiscussion). Assuming

x′µ = xµ + δxµ = xµ + εµ(x), (2.24)

andφ′(x′) = exp[−iT (ε)]φ(x), (2.25)

so that for infinitesimal transformations we have

φ′(x′) = φ′(x) + δxµ∂µφ(x) + . . . , (2.26)

∂′µφ′(x′) = ∂µφ

′(x) + δxν∂µ∂νφ(x) + . . . . (2.27)

A sufficient condition for the transformation to be a symmetry of the action is

L(φ′(x′), ∂′φ′(x′))d4x′ = [L(φ(x), ∂φ(x))− ∂µV µ(φ(x))]d4x. (2.28)

It can then be seen that

∂µJµ ≡ ∂µ

[Lgµρ −

δLδ(∂µφ)

∂ρφ

]δxρ +

δLδ(∂µφ)

δφ+ V µ

= 0. (2.29)

8

Note: All we have said in this section is strictly true for Classi-cal Field Theory. What happens when one goes to QFT? The ef-fects of symmetries generalize quite straightforwardly, with somesubtleties. For one, the fields are no longer classical fields butquantum operators, and as such, transform as

φi(x)→ Uφi(x)U−1 = [exp(−iθaT a)]ijφj(x). (2.30)

Also, Poisson brackets go to standard (anti)commutators,

[Qa, φ] = −T aφ (2.31)

i.e. , → −i[, ]. One particle states in QFT correspondingto the quantum excitations of a particular field are constructedfrom the creation operators of the corresponding field acting onthe vacuum,

a†ψ(k)|0〉 = |kψ〉, (2.32)

and therefore how the spectrum behaves under the symmetrydepends on how the vacuum does. Finally, QFT involves, atthe quantum level, singularities that have to be regularized. Itis sometimes the case that no regularization is possible that itis compatible with the classical symmetry. The system is thensaid to be anomalous under the symmetry (it is symmetric at theclassical level but not at the quantum level). In the path integralformulation of QFT the system is anomalous when the integralmeasure is not invariant under the symmetry.

2.3 Examples

A set of nice examples are given by Georgi [3], involving both scalars and fermions. Weencourage the interested student to study them in detail. Here we just mention what theexamples are:

2.3.1 Scalars

Consider a set of N real scalar fields. The Kinetic Lagrangian reads,

LK =1

2∂µφ

TS∂µφ, (2.33)

with S a symmetric, positive definite matrix. We can then define a linear transformationon the fields φ → Lφ, with L satisfying LTSL = 1, (L is given by an orthogonal matrixthat diagonalizes S times a rescaling of the diagonal entries) so that the kinetic Lagrangian

9

now reads,

LK =1

2∂µφ

T∂µφ. (2.34)

This Lagrangian has an obvious SO(N) symmetry. Masses and other couplings can breakthis symmetry as discussed in Georgi’s book.

2.3.2 Fermions

Consider now a set of N massless, free, spin-1/2, Dirac fermions. The Lagrangian reads,

L = iψγµ∂µψ, (2.35)

which has a U(N) symmetry. This is not the largest symmetry of the kinetic Lagrangian,though. Writing the fermions in their chirality components we have

L = iψLγµ∂µψL + iψRγ

µ∂µψR, (2.36)

making a larger, chiral, U(N)L × U(N)R symmetry apparent. A Dirac mass term, mixingleft and right chiralities, will break this chiral symmetry. If a subgroup of n masses aredegenerate, the corresponding vector subgroup, SU(n)V of equal left and right rotationsin the subspace of degenerate fermions will survive.

2.4 Symmetries and Degeneracy

One important result in quantum mechanics is that states that are related by a symmetryof the system have to be degenerate. This is a simple consequence of the fact that thegenerator of the symmetry, is the conserved Noether’s charge, Qa. Being conserved, itcommutes with the Hamiltonian and therefore leads to degenerate states. Let us considertwo eigenstates of the Hamiltonian related by a symmetry, |ψ〉 = Qa|χ〉. Then we have,

Eψ|ψ〉 = H|ψ〉 = HQa|χ〉 = QaH|χ〉 = EχQa|χ〉 = Eχ|ψ〉, (2.37)

and therefore Eψ = Eχ. Physical states are grouped in representations of the symmetry, allwith the same energy (mass). This result -states related by a symmetry are degenerate- isequaly valid in quantum field theory. However, in QFT, the one-particle states associatedto two fields which are related by a symmetry do not need to be degenerate. The reason isthat in QFT, the one-particle states are created from the vacuum, and for the two physicalstates to be related, not only the fields have to be related by the symmetry but the vacuumhas to be invariant under the symmetry. Let’s see why in a bit more detail. Let’s considerthe two fields φ1(x) and φ2(x), corresponding to particles of type “1” and “2”, respectively,and that are related by the action of some symmetry,

φ1(x) = i[Q, φ2(x)], (2.38)

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with Q a hermitian operator generating the symmetry. The same is then true for thecorresponding creation and annihilation operators,

a1 = i[Q, a2]. (2.39)

The particle states are then related as follows,

|1〉 = a†1|0〉 = i[Q, a†2]|0〉 = iQa†2|0〉 − ia†2Q|0〉 = iQ|2〉 − ia†2Q|0〉. (2.40)

Thus, the particle states satisfy |1〉 = iQ|2〉 only if the vacuum is invariant Q|0〉 = 0.If the vacuum is invariant, then the physical particle states are related by the symmetrywhen the corresponding fields are and have therefore to be degenerate. If the vacuum isnot invariant (we then say that the symmetry is spontaneously broken), then it is not truethat particle states whose corresponding fields are related by the symmetry have to bedegenerate.

2.5 Spontaneous Breaking of a Global Symmetry

System with spontaneously broken symmetries, i.e. those for which the Lagrangian isinvariant under the symmetry but the vacuum is not, are interesting and relevant to thereal world. We have seen in the previous section that a spontaneously broken symmetry nolonger implies degeneracy of the physical states. Another important result in spontaneouslybroken symmetries is Goldstone’s Theorem, which states that for every spontaneouslybroken continuous global symmetry, there is a massless scalar boson with the same quantumnumbers of the broken generator. Before going into the details of the theorem, we willdiscuss a very simple example.

2.5.1 Spontaneously Broken U(1) Symmetry

Consider a complex scalar with Lagrangian,

L = ∂µφ∗∂µφ− V (|φ|) = ∂µφ

∗∂µφ− [µ2φ∗φ+ λ(φ∗φ)2]. (2.41)

This system has a U(1) global symmetry φ→ eiαφ. Assuming that a perturbative expan-sion exists, the vacuum of the system will be close to the classical vacuum, given by theconstant field configuration that minimizes the potential. If µ2 > 0, then the minimum ofthe potential is 〈φ〉 = 0 (recall λ has to be positive for the Hamiltonian to be boundedfrom below) and the vacuum is also invariant under the symmetry. If on the other handwe have µ2 < 0, then the minimum of the potential happens for any field configurationsuch that

|φ|2 = −µ2

2λ≡ v2 > 0. (2.42)

One important result in quantum field theory (in infinite space) is that the vacuum statefor a theory that satisfies the standard QFT properties (including the clustering principle)

11

can be only one point in the manifold of degenerate minima of the potential. In our case,that means that we have to choose one particular phase for the true vacuum of our system.In particular, we can choose the vacuum to be real,

〈φ〉 = v, (2.43)

and define a new field, φ′ = φ− v, which has zero vev (and whose quantum excitations aretherefore orthogonal to the vacuum state). In terms of this new field, we have, definingthe real and imaginary parts of φ′ = (φ1 + iφ2)/

√2, the following Lagrangian (up to an

irrelevant constant term)

L =1

2

[∂µφ1∂

µφ1 + ∂µφ2∂µφ2

]− 1

24λv2φ2

1 −√

2λvφ1(φ21 + φ2

2)− λ

4(φ2

1 + φ22)2, (2.44)

which is the Lagrangian of two real scalars, one with positive mass squared, m21 = 4λv2,

and one massless, with triple and quartic interactions among them. Alternatively, insteadof using the real and imaginary parts of φ′ as the relevant degrees of freedom, one can usethe modulus and the phase,

φ =

(v +

ρ√2

)eiη/(

√2v), (2.45)

with 〈ρ〉 = 〈η〉 = 0. We then have

φ∗φ = (v + ρ/√

2)2, (2.46)

and therefore the potential is clearly independent of η. The whole Lagrangian reads, forthese fields,

L =1

2∂µρ∂

µρ+1

2

(1 +

ρ√2v

)2

∂µη∂µη − 1

24λv2ρ2 −

√2λvρ3 − λ

4ρ4. (2.47)

In particular we now have that the massless scalar, η, has disappeared from the potentialand it only couples derivatively.

We will see in the next sections and examples that this particular choice of variablesgeneralizes to arbitrary symmetry breaking patterns and has several advantages.

Note: For a continuous symmetry, we have to have a manifold ofdegenerate minima of the potential, since given any minimum, themanifold in the field space obtained by continuously transformingwith the symmetry the original minimum gives the same value ofthe potential (since it is invariant under the symmetry). Thisalso ensures that the choice of one particular vacuum, enforcedby the properties of QFT in infinite space, is physically equivalentto any other choice, since they are related by a symmetry of theLagrangian.

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2.5.2 Goldstone’s Theorem

In the previous example we have seen how the spontaneous breaking of the global U(1)symmetry resulted in a massless scalar that, with the right choice of variables, coupledonly derivatively. This result generalizes to an arbitrary pattern of spontaneous globalsymmetry breaking as we discuss now. Let us discuss it in the context of a system of(fundamental or composite) scalar fields with the following Lagrangian,

L =1

2∂µφ∂

µφ− V (φ), (2.48)

where φ is some multiplet of spliness fields and V (φ) is invariant under some symmetrygroup action

δφ = iεaTaφ, (2.49)

with T a imaginary antisymmetry matrices (so that φ is hermitian). In perturbation theory,the vacuum, around we should perturb, is given by the minimum of the potential. Thusthe fields φ will have a vev that minimize the potential 〈φ〉 = λ, satisfying

Vj(λ) = 0, Vjk(λ) ≥ 0. (2.50)

We have used the notation

Vj1...jn =∂n

∂φj1 . . . φjnV (φ).

Expanding around the minimum, we have, for the physical perturbations φ′ = φ− λ,

L =1

2∂µφ

′∂µφ′ − V (λ)− 1

2Vij(λ)φ′iφ

′j + . . . . (2.51)

Thus, the term proportional to Vij(λ) is the mass squared term for the physical fields,which is definite positive (no tachyons).

Now, this vacuum, λ, does not need to be invariant under the symmetry. In particular,the generators of the symmetry group G can be separated in generators, Y i that annihilatethe vacuuum and generators, X a that do not:

(T a) = (Y i, X a), Y iλ = 0, X aλ 6= 0, (2.52)

where Y i form a subgroup H of G. We then say that the symmetry is spontaneouslybroken from G to H. Since the potential is invariant under the symmetry, we have

0 = V (φ+ δφ)− V (φ) = iVk(φ)εa(Ta)klφl. (2.53)

Differenciating this expression with respect to φj and using that εa is arbitrary, and eval-uating the resulting quantity at φ = λ we get

0 = Vjk(λ)(T a)klλl + Vk(λ)(T a)kj = Vjk(λ)(T a)klλl = M2T aλ, (2.54)

13

where we have used that λ is a minimum of the potential and that the second derivative atλ gives the mass squared matrix for the physical fields. This equality is trivially satisfiedfor the generators of the unbroken group, as they annihilate the vacuum. The condition ishowever non-trivial for the generators of the coset space (the broken generators) implyingthat X aλ is an eigenvector of M2 with zero eigenvalue. This corresponds to a masslessscalar field, given by

φTX aλ. (2.55)

Note that there is one massless scalar per broken generator. Also note that it has the samequantum numbers under the unbroken subgroup as the corresponding generator. Finally,note that we have chosen one set of λ that minimizes the potential. However, since thepotential is invariant under the symmetry G, it is obvious that any λ′ = gλ, with g anelement of G is also a minimum of the potential. Since G is a continuous group, we havea continuous manifold of degenerate minima. Physically, only one of these minima can beused (in an infinite space), since going from one to another would require an infinite changein all space. However, all the different minima are physically equivalent and any choice wemake can be converted in a different one by a simple relabeling of the generators, withoutaltering the physics.

Note: Assume [Y i, Y j] = icijkY k + icijaX a and apply it to thevacuum (note that the commutator annihilates the vacuum)

0 = icijkY kλ+ icijaX aλ = icijaX aλ, (2.56)

which implies that cija = 0 and H is a subgroup. This also implies(from the antisymmetry of the structure constants) that

[Y i, X a] = iciabX b, (2.57)

and therefore X a transform under some representation of the sub-group H. In general, the commutator of two broken generatorsdoes not live in H. However, if there is a parity operation Pwhich leaves the Lie algebra of the group G invariant and is suchthat

PY iP−1 = Y i, PX aP−1 = −X a, (2.58)

then we obviously have

[X a, X b] = icabiY i. (2.59)

Our previous proof of Goldstone’s theorem has been based on the analysis of the classicalpotential and therefore relies on perturbation theory. Goldstone’s theorem is however moregeneral and can be proven to all orders in perturbation theory [5]. The details can be foundin the original article or in Pokorski’s book. Here we will just sketch the proof following

14

Pokorski. The idea is to prove that the spontaneous breaking of a global continuoussymmetry implies the existence of poles at p2 = 0 in certain Green’s functions and thenthat this poles imply the presence of massless scalar particles in the physical spectrum.

Let us consider the following Green’s function

Gaµ,k(x− y) = 〈0|Tjaµ(x)φk(y)|0〉, (2.60)

where jaµ is the current corresponding to a generator Qa of the symmetry G and φk belongsto an irreducible multiplet of real scalar fields. This Green’s function satisfies a Wardidentity that can be obtained by differentiating it being careful with the derivative of theθ functions involved in the time ordering,

∂µ(x)Gaµ,k(x− y) = δ(x0 − y0)〈0|[ja0 (x), φk(y)]|0〉 = δ(x0 − y0)〈0| − T akjφj(y)δ(x− y)|0〉

= −δ(x− y)T akj〈0|φj(y)|0〉 = −δ(x− y)T akj〈0|φj(0)|0〉. (2.61)

We have used the transformation properties of the field as generated by the Noether’scurrent and translational invariance of the vacuum. This Ward identity, in the case ofconserved currents, is valid both for bare and renormalized fields. The Fourier transformof the Ward identity reads

ipµGaµ,k(p) = T akj〈0|φj(0)|0〉, (2.62)

where

Gaµ,k(x− y) =

∫d4p

(2π)4exp[−ip(x− y)]Ga

µ,k(p). (2.63)

Plugging the most general form of the (Fourier-transformed) Green’s function as allowedby Lorentz invariance, Ga

µ,k(p) = pµFak (p2), in the Ward’s identity we get

F ak (p2) = − i

p2T akj〈0|φj(0)|0〉, (2.64)

which implies that the Green’s function corresponding to a generator that does not anni-hilate the vacuum,

T akj〈0|φj(0)|0〉 6= 0, (2.65)

has a pole at p2 = 0.Let us now consider the following matrix element

〈0|jaµ(x)|πk(p)〉 = ifak pµe−ipx, (2.66)

where |πk(p)〉 describes a particle of mass mk which is a quantum of the field φk. Thereduction formula relates this matrix element to the Green’s function Gs

µ,k(p). Defining

Gkk′ =

∫d4q

(2π)4

−δkk′q2 −m2

k + iεe−iq(y−x), (2.67)∫

d4yG−1(x− y)G(y − z) = δ(x− z), (2.68)

15

we get

〈0|jaµ(x)|πk(p)〉 =

∫d4yd4ze−ipzGa

µ,k′(x− y)iG−1kk′(y − z)

= limp2→m2

k

e−ipxGaµ,k′(p)iG

−1kk′(p)

= − limp2→m2

k

e−ipxGaµ,k′(p)i(p

2 −m2k). (2.69)

Putting together (2.66) with this equation we get,

limp2→m2

k

Gaµ,k′(p)i(p

2 −m2k) = −fak pµ, (2.70)

which implies mk = 0, for those Green’s functions that have a massless pole (i.e. thosecorresponding to generators that don’t annihilate the vacuum), as we wanted to prove.This proof also gives us the value of fak ,

fak = iT akj〈0|φj(0)|0〉. (2.71)

Thus, there must be a massless boson, |Πa(p)〉 = i|πk(p)〉T akj〈0|φj(0)|0〉, corresponding toeach broken generator.

Note: Pokorski goes far beyond the discussion we have included here. In-teresting topics related to spontaneously broken global symmetries includesystems with spontaneous and explicit symmetry breaking and patterns ofsymmetry breaking.

2.6 Spontaneous Breaking of a Local Symmetry

In the previous section we discussed Goldstone’s theorem, which implies the existence of amassless scalar for each spontaneously broken generator of a continuous global symmetry.What happens if the symmetry that is sponatenously broken is local? In particular thecorresponding global symmetry given by keeping the gauge parameters constant is also asymmetry of the Lagrangian so Goldstone’s theorem should also apply, right? The answeris actually no. The reason is that in our non-perturbative proof of the theorem, Lorentzinvariance and a Hilbert space with positive norm were required. However, gauge theoriesdo not satisfy both requirements simultaneously. In a covariant gauge, the theory containsstates of negative norm whereas in a gauge in which the theory has only states of positivenorm, it is not manifestly covariant. This means that in gauge theories, Goldstone’stheorem does not hold and, as we will see, the Higgs mechanism operates.

16

2.6.1 An Abelian Example

Let us start with the simple example of a complex scalar with a local U(1) symmetry.

L = −1

4F µνFµν + (Dµφ)∗Dµφ− λ(φ∗φ− v2)2, (2.72)

where Dµφ = (∂µ + igAµ)φ. This system is invariant under a local U(1) transformation

φ(x)→ eiα(x)φ(x), Aµ(x)→ Aµ(x)− 1

g∂µα(x). (2.73)

If v2 > 0, the minimum of the potential occurs for |φ|2 = v2 and we have spontaneousbreaking of the local symmetry. Expanding the physical fields around the true vacuum,

φ(x) = v +1√2

(h(x) + iπ(x)), (2.74)

and using

Dµφ =1√2

[∂µh− gAµπ] +i√2

[∂µπ + gA(√

2v + h)], (2.75)

the Lagrangian reads,

L = −1

4FµνF

µν +1

2[∂µh− gAµπ]2 +

1

2[∂µπ + gA(

√2v + h)]2

−λ[√

2vh+h2 + π2

2

]2

= −1

4FµνF

µν +1

2(∂µh)2 +

1

2(∂µπ)2

+1

22g2v2AµA

µ − 1

24λv2h2 +

√2gvAµ∂

µπ

+gAµ(h∂µπ − π∂µh) + g2AµAµ(√

2vh+ h2/2 + π2/2)

−λ4

(h2 + π2)2 − λ√

2vh(h2 + π2). (2.76)

Note that the kinetic mixing between Aµ and π makes it difficult to interpret the degrees offreedom in this Lagrangian. Up to this kinetic mixing, the Lagrangian seems to representa massive gauge field, with mass mA =

√2gv, a massive scalar with mass mh = 2

√λv,

a massless scalar π and their interactions. Note that π, which is the field that with itskinetic mixing is creating the problem of interpretation, is what would correspond to theGoldstone boson if the symmetry was global. How can we get a proper identification of therelevant degrees of freedom? There are several things we can do but they all go throughfixing the gauge in a way that the relevant degrees of freedom are manifest. The simplestpossibility is to go to the so-called unitary gauge, in which the make a gauge transformationthat locally removes the would-be Goldstone boson. In our particular example, we justneed to do a gauge transformation that removes, locally, the phase of φ, so that φ remains

17

real. This is easier to see in the polar coordinates (2.45). It is then clear that we can reachthe unitary gauge by performing a gauge transformation with α(x) = −η(x)/(

√2v), so

thatφU(x) = v + ρ(x)/

√2, AUµ (x) = Aµ + (1/

√2vg)∂µη, (2.77)

and the Lagrangian then reads,

L = −1

4(FU

µν)2 +

1

2(∂µρ)2 +

1

22g2(v + ρ/

√2)2(AUµ )2 − 1

24λv2ρ2 −

√2λvρ3 − λ

4ρ4. (2.78)

The Lagrangian has simplified enormously. Also, the particle content is apparent in theunitary gauge, in which there is no kinetic mixing between different fields. We have amassive gauge boson, a massive scalar and their interactions. Note also that the would-beGoldstone boson has completely disappeared from the spectrum. The number of degrees offreedom is however unchanged. Originally we had two real scalar degrees of freedom plus amassless gauge boson (another two degrees of freedom, the two transverse polarizations),in the Unitary gauge we still have four real degrees of freedom, one in ρ and three inthe massive gauge boson (two transverse plus the longitudinal polarization). It is usuallysaid that the would-be Goldstone boson is combined with the gauge boson to providethe longitudinal component (it is eaten by the gauge boson to gain its mass). Althoughin the unitary gauge the particle content is manifest (all dof are physical), other gaugesare sometimes useful, in particular in explicit calculations (and also in the discussion ofrenormalizability of gauge theories). Let us go to a covariant gauge by means of thefollowing gauge fixing term,

LGF = −(∂µAµ)2/2a. (2.79)

The quadratic part of the gauge-fixed Lagrangian then reads

L+ LGF = −1

2Aµ

[−gµν∂2 +

(1− 1

a

)∂µ∂ν −m2

Agµν

]Aν

+1

2(∂µh)2 − 1

2m2hh

2 +1

2(∂µπ)2 +mAAµ∂

µπ + . . . . (2.80)

Let us now consider the free particles (which are not necessarily the physical states) andinclude the interactions as a perturbation. In the Landau gauge (a = 0) we have (we do notcare about h which does not affect the argument) a massless gauge boson with propagator

− iDµν ≡ −i

p2 + iε(gµν − pµpν/p2) ≡ − i

p2 + iεPµν , (2.81)

where in the second equality we have defined the transverse projector, and a massless wouldbe Goldstone with propagator,

i

p2 + iε. (2.82)

The full gauge boson propagator is then

∆µν(p) = −iDµν1

1− Π(p2), (2.83)

18

where Π(p2) is defined by the gauge-invariant 1PI vacuum polarization tensor (here, sincewe are considering some of the quadratic terms in the Lagrangian as a perturbation, by1PI we really mean 1 gauge boson irreducible)

1PI

µ ν≡ iΠµν(p) = iΠ(p2)(p2ηµν − pµpν) = iΠ(p2)p2Pµν .

The exact propagator is then given by the infinite series of diagrams shown

= + 1PI + 1PI 1PI + . . .

which results in the following form of the exact propagator (we do not write explicitly thefactors of iε in the intermediate calculations)

∆µν = − i

p2Pµν +

(− i

p2Pµρ

)iΠ(p2)p2Pρσ

(− i

p2P σν

)+ . . .

= − i

p2Pµν

[1 + Π(p2) + Π(p2)2 + . . .

]= −iDµν 1

1− Π(p2), (2.84)

where we have made use of the fact that PµνPνσ = Pµσ (it is a projector) and have explicitlysummed the geometric series. Now, to leading order in the interactions, we get a directcontribution to Πµν from the mass term,

iΠµν(p2)m = im2

Agµν , (2.85)

plus a contribution from the exchange of a φ field

Aµ

π

Aν= (−mApµ) i

p2 (mapν) = −im2Apµpνp2 .

Thus, the total contribution is transverse as is should and we get

Π(p2) = m2A/p

2, (2.86)

so that the total propagator reads,

∆µν(p) = − i

p2 −m2A

Pµν , (2.87)

which corresponds to the massive propagator in the Landau gauge. We see then how theexchange of the Goldstone boson provides the correct pole in the propagator as predictedby the gauge invariance of the Lagrangian.

19

Another set of gauges that are interesting are the so-called Rξ or renormalizable gauges.They are obtained by means of the following gauge fixing term

LGF = − 1

2ξ(∂µA

µ − ξmAπ)2. (2.88)

The quadratic part of the Lagrangian then reads

L+ LGF = −1

2Aµ

[−gµν∂2 +

(1− 1

ξ

)∂µ∂ν −m2

Agµν

]Aν

+1

2(∂µh)2 − 1

2m2hh

2 +1

2(∂µπ)2 − ξ

2m2Aπ

2 + . . . . (2.89)

In particular note that the gauge fixing term removed the kinetic mixing between the gaugeboson and the would-be Goldstone. In a general Rξ gauge, the propagators for the differentfields are

Aµ :−i

p2 −m2A + iε

[gµν − (1− ξ) pµpν

p2 − ξm2A

]= −igµν − pµpν/m

2A

p2 −m2A + iε

− i pµpν/m2A

p2 − ξm2A + iε

, (2.90)

π : i/(p2 − ξm2A + iε), (2.91)

h : i/(p2 −m2h + iε). (2.92)

The Landau gauge corresponds to ξ = 0 whereas the unitary gauge corresponds to ξ →∞.In the unitary gauge the corresponding Goldstone boson does not propagate and we recoveronly physical states. In a renormalizable gauge (ξ 6= ∞) we have written the gaugepropagator as the propagator of a standard massive gauge boson (in unitary gauge) plusa term correponding to a scalar negative norm boson coupled to the source of the vectorboson derivatively. Another gauge that is sometimes useful for calculations is the ’t Hooft-Feynman gauge, ξ = 1, in which the gauge boson has a very simple propagator ∝ gµν andthe Goldstone boson has mass mA.

2.6.2 Spontaneous Symmetry Breaking of non-Abelian GaugeSymmetries

Our previous analysis goes almost unchanged to the non-abelian case. In this section andthe next we follow section 20.1 of [4], to which we refer the reader for further details.

We will consider a system of real scalars invariant under a local non-abelian symmetrywith generators ta,

φ→ (1 + iαa(x)ta)φ = (1− αa(x)T a)φ, (2.93)

where in the second equality we have used the fact that the fields are real (so that the ta

are hermitian and pure imaginary) to write ta = iT a with T a real and antisymmetric. Thecovariant derivative reads,

Dµφ = (∂µ + gAaµTa)φ. (2.94)

20

The scalar kinetic term reads

1

2(Dµφi)

2 =1

2(∂µφi)

2 + gAaµ∂µφiT

aijφj +

1

2g2AaµA

µb (T aφ)i(T

bφ)i. (2.95)

Let us now assume that the scalar potential is such that some of the fields acquire a vacuumexpectation vacuum (vev) that spontaneously breaks the symmetry,

〈φi〉 = λi, (2.96)

and expand around the correct vacuum φ → λ + φ. The scalar kinetic term, expandedaround the vacuum λ gives us the mass matrix for the gauge bosons,

∆L =1

2m2abA

aµA

bµ =1

2g2(T aλ)i(T

bλ)iAaµA

bµ. (2.97)

Note that this mass matrix is positive definite, since in any basis, the diagonal entries arestrictly non-negative. Also note that, if for any a we have T aλ = 0, then the correspondinggauge boson is massless and the symmetry along that generator is not broken. Also in thekinetic term there is a kinetic mixing between the Goldstone bosons (recall the Goldstonebosons were πa = φTT aλ) and the gauge bosons corresponding to the broken generators,

∆L = gAaµ∂µφi(T

aλ)i. (2.98)

Now we can repeat the calculation that we did for the abelian symmetry of the 1PI vacuumpolarization tensor for the massless gauge bosons, including the mass and mixing withGoldstones as a perturbation, replacing (2.85) and (2.6.1) with the ones we have computedhere. That gives, for the vacuum polarization tensor

iΠµν(p) = ig2(T aλ)i(Tbλ)igµν +

(gpµ(T aλ)i

) ip2

(− gpν(T bλ)i

)= im2

ab

(gµν −

pµpνp2

), (2.99)

which again is transverse as required by gauge invariance but gives a mass to the gaugebosons associated to the broken generators. A similar discussion about the gauge fixing aswas done in the section of the abelian symmetry can be done here (note however that thequantization of non-abelian gauge symmetries, although also a resonably straight forwardgeneralization of the case of abelian symmetries, involves new features that are worthstudying, see Peskin and Schroeder’s book for instance).

2.6.3 Formal Aspects of the Higgs Mechanism

All we have said in the last two sections about the Higgs mechanism relied on the existanceof a set of scalars that acquired a vev. Spontaneous symmetry breaking of global and localsymmetries is however more general and does not require the presence of fundamental

21

scalars. In fact, no fundamental scalar degrees of freedom seem to play a role in knownexamples of global symmetry breaking, like superconductivity or chiral symmetry breakingin QCD. Of course, we know from Goldstone’s theorem that some scalar degrees of freedomwill be present in systems with spontaneous symmetry breaking, the Goldstone bosons, butthese do not need to be fundamental scalars and can instead be composite objects madeout of fermions or other species. In this section, we will discuss the Higgs mechanismin a more general set-up, in which we do not start with a set of fundamental scalarsdescribing the process of spontaneous symmetry breaking. The final result will however beessentially unchanged with respect to the examples we saw in the last two sections. Thepresence of massless scalar degrees of freedom associated to the broken symmetries (theGoldstone bosons) will play a crucial role in giving mass to the broken gauge bosons in agauge invariant way, i.e. keeping the vacuum polarization tensor properly transverse. Ourstarting point is a general system with a global symmetry G, represented by the LagrangianL0. The corresponding Noether current,

Jaµ = − δL0

δ∂µφT aφ, (2.100)

(note that φ here represents all the degrees of freedom in the system that, let us stress itagain, are not necessarily scalars) satisfies

δL0 = ∂µJaµ = 0, (2.101)

We can promote the global symmetry to a local one δφ = −αa(x)T aφ by coupling thesystem to a set of gauge bosons through the conserved Noether currents,

L = L0 + gAaµJaµ +O(A2), (2.102)

where we have not made explicit the terms of order higher than linear that are required tomake the Lagrangian gauge invariant. The important feature is that, at the linear level,the coupling of the gauge bosons to our system is fixed by gauge invariance and it goesthrough the Noether currents of the global symmetry. Thus, in any computation that in-volves vertices with only one gauge boson, we ca use the Lagrangian above. These Noethercurrents are closely related to the corresponding Goldstone bosons. At low energies, Gold-stone bosons are just infinitesimal rotations of the vacuum, Qa|0〉, where Qa is the chargeassociated to the Noether current. Jaµ has therefore the right quantum numbers to createor destroy a Goldstone boson from the vacuum. We can parametrize the matrix elementof such process as

〈0|Jaµ(x)|πk(p)〉 = −ipµF ak e−ipx, (2.103)

where p is the on-shell momentum of the Goldstone boson and F ak is a matrix of contants

that vanish for unbroken generators. From the conservation of the Noether current wehave

0 = ∂µ〈0|Jaµ(x)|πk(p)〉 = −p2F ak e−ipx, (2.104)

which agrees with the fact that the Goldstone bosons are massless.

22

We can now see how the Higgs mechanism operates in our general system. By nowwe know that, in order to do that, we have to study the vacuum polarization tensor ofthe corresponding gauge bosons. The Ward identities associated to the gauge symmetryrequire it to be transverse, of it has to have the form

iΠµν(p) = i

(gµν −

pµpνp2

)(m2

ab +O(p2)). (2.105)

Computing the non-singular term in our general theory is not easy but the singular partcomes exclusively from the exchange of our massless Goldstones, whose coupling to thegauge bosons can be deduced from (2.102) and (2.103) and is given by

µ, a j= gpµF a

j .

The Goldstone exchange contribution to the vacuum polarization tensor is therefore

Aaµ

πj

Abν=(−gpµF a

j

)ip2

(gpνF

bj

)= −ipµpν

p2 g2F aj F

bj .

Thus, we have the mass term for the gauge bosons,

m2ab = g2F a

j Fbj . (2.106)

Note that, given the generality of the system, we have not been able to compute the non-singular part of the vacuum polarization tensor. We have instead used gauge invarianceto deduce its form (transverse) and computed the singular contribution generated by theexchange of the massless Goldstone. This process is completely general, and can be appliedto any model with local spontaneous symmetry breaking.

2.7 The sigma model

Let us now discuss a simple model that will be useful for many aspects of the course. Theσ model was introduced for the first time by Gell-Mann and Levy [6]. In the preparation ofthese notes, I have mainly used [1] and [2]. The linear σ model contains two Dirac fermionsu, d that we will jointly denote by

ψ =

(ud

)and four real scalars σ′ and ~π = (π1, π2, π3). The lagrangian is given by

Llinσ =

1

2(∂µσ

′)2 +1

2(∂µπ

i)2 − µ2

2(σ′2 + ~π2)− γ

4(σ′2 + ~π2)2

+ ψi∂ψ + gψ(σ′ + i~σ · ~πγ5)ψ, (2.107)

23

where ~σ are the Pauli matrices. As we saw in the previous examples, the fermion kineticterm has an SU(2)L × SU(2)R global symmetry (we neglect for the moment the U(1)factors, one of which corresponds to baryon number and the other one is anomalous). Wewill want to insist that the full lagrangian is invariant under this chiral symmetry. Thisrequirement determines the transformation properties of the scalar fields. In particular,writing the fermions in their corresponding chiral components we get

Llinσ =

1

4Tr(∂µΣ†∂µΣ)− µ2

4Tr(Σ†Σ)− λ

16[Tr(Σ†Σ)]2

+ ψLi∂ψL + ψRi∂ψR + gψLΣψR + gψRΣ†ψL, (2.108)

where we have defined the combination of scalars Σ ≡ σ′ + i~σ · ~π, which satisfies

σ′2 + ~π2 =1

2Tr(Σ†Σ). (2.109)

In order for the Lagrangian to be invariant under the chiral symmetry, Σ has to transformas a bidoublet,

ψL,R → UL,RψL,R, Σ→ ULΣU †R, (2.110)

with UL,R arbitrary SU(2) matrices

UL,R = exp(−iαaL,Rσa/2). (2.111)

Using the following properties of the Pauli matrices, Trσi = 0 and σiσj = iεijkσk + δij, wehave

σ′ =1

2TrΣ, πk = − i

2Tr[σkΣ].

The corresponding infinitesimal transformations for the scalars are then

σ′ → σ′ +1

2(~αL − ~αR) · ~π, (2.112)

πk → πk − 1

2(αkL − αkR)σ′ − 1

2εklmπl(αmL + αmR ). (2.113)

In particular, under an isospin rotation (αL = αR) σ′ and ~π transform, respectively, as asinglet and a triplet, whereas under an axial rotation (αL = −αR), all four fields mix,

σ′ → σ′ + ~αL · ~π, (2.114)

~π → ~π − σ′~αL. (2.115)

Noether’s theorem tells us that the invariance of the Lagrangian implies the correspondingconserved currents. Using the general definition of Noether’s current and the explicitexpression for the variation of the different fields under a L and R rotation, we get

JkLµ = ψLγµσk

2ψL −

i

8Tr[σk(Σ∂µΣ† − ∂µΣΣ†)]

= ψLγµσk

2ψL −

1

2(σ′∂µπ

k − πk∂µσ′) +1

2εklmπl∂µπ

m, (2.116)

24

and

JkRµ = ψRγµσk

2ψR +

i

8Tr[σk(∂µΣ†Σ− Σ†∂µΣ)]

= ψRγµσk

2ψR +

1

2(σ′∂µπ

k − πk∂µσ′) +1

2εklmπl∂µπ

m. (2.117)

These currents can be combined into conserved vector currents,

V kµ = JkLµ + JkRµ = ψγµ

σk

2ψ + εklmπl∂µπ

m, (2.118)

and axial-vector currents

Akµ = JkLµ − JkRµ = ψγµγ5σk

2ψ + πk∂µσ

′ − σ′∂µπk. (2.119)

By now we know too well that the invariance of the Lagrangian is not necessarily the fullstory. Depending on the parameters of the scalar potential, the vacuum of the systemmight be non invariant under the chiral symmetry, which is then spontaneously broken.The scalar potential can be written, up to an irrelevant constant factor

V (σ′, πk) =λ

4

(σ′2 + ~π2 +

µ2

λ

)2

. (2.120)

If µ2 > 0, then the minimum of the potential is for σ′ = πk = 0, and the vacuum isalso invariant under the chiral symmetry. This means that our fields represent quantumexcitations around the true vacuum and they correspond to degenerate multiplets (all fourscalars have the same mass, similarly for the two fermions). If on the other hand we haveµ2 < 0, then the minimum of the potential (and therefore the vacuum of the system in theclassical approximation) satisfies

σ′2 + ~π2 = −µ2

λ≡ v2. (2.121)

The true vacuum of the system corresponds to a point in this manifold. Any choiceis physically equivalent, since they are all related by the symmetry of the Lagrangian(different choices just amount to a renaming of the broken and unbroken symmetries) butthe vacuum is one and only one of the states satisfying (2.121) and not a linear combinationof different ones (in infinite space-time). One choice that we can always make is

〈σ′〉 = v, 〈πk〉 = 0. (2.122)

With this choice, the vacuum is still invariant under the isospin (vector) symmetry but notunder the axial symmetry. Thus, this pattern of symmetry breaking (which in the currentmodel is the only possible one) corresponds to SU(2)L × SU(2)R being spontaneously

25

broken to SU(2)V . Let us see what implications this pattern of symmetry breaking has forthe spectrum.

In order to preserve orthogonality of the vacuum to one-particle states, we define thephysical field,

σ = σ′ − v, (2.123)

so that 〈σ〉 = 0. The Lagrangian now reads, up to an irrelevant constant,

L = ψi∂ψ + gvψψ +1

2[(∂µσ)2 − 2λv2σ2] +

1

2(∂µ~π)2

+ gψ[σ + i~σ · ~π]ψ − λ

4(σ2 + ~π2)2 − λvσ(σ2 + ~π2), (2.124)

which describes a nucleon of mass −gv, a σ meson of mass mσ =√

2λv and an isospintriplet of massless pions, ~π, and their interactions. We can also compute, in the treeapproximation,

〈0|Aaµ(x)|πb(q)〉 = 〈0|σ′|0〉〈0|∂µπa(x)|πb(q)〉 = −〈0|σ′|0〉iqµδabe−iqx = ifπqµδabe−iqx.

(2.125)Thus, we have fπ = −〈0|σ′|0〉 = −v. All other elements in Aaµ(x) will give zero whenapplied over the vacuum.

Note that, as predicted by Goldstone’s theorem, we have found three massless Gold-stones, which correspond to the three broken generators. (G = SU(2)L × SU(2)R has sixgenerators out of which the three in H = SU(2)V remain unbroken, leaving three brokengenerators.) Using the commutation relations of the generators

[T iL,R, TjL,R] = iεijkT kL,R, [T iL, T

jR] = 0, (2.126)

we obtain[T iV,A, T

jV,A] = iεijkT kV , [T iV , T

jA] = iεijkT kA. (2.127)

In particular, the last equation tells us that the broken generators transform as a tripletunder the unbroken isospin group. The Goldstone bosons transforming as an isospin triplettherefore agrees also with the theorem.

What would happen if we include an explicit symmetry breaking term? Let us considerwe add an isospin preserving but chiral breaking term like

L′ = −εσ′ = − ε2

TrΣ. (2.128)

This has several implications. The axial-vector current is no longer conserved,

∂µAaµ = δAaL′ = −εδAaσ′ = −επa, (2.129)

and the corresponding charge is time dependent,

dQaA

dt= −ε

∫d3x πa. (2.130)

26

Recall however that this charge still generates the corresponding transformations on thefields. Another implication is that the minimum of the potential is modified. We now have

∂σ′V = σ′[µ2 + γ(σ′ 2 + ~π2)

]+ ε, (2.131)

∂πiV = πi[µ2 + γ(σ′ 2 + ~π2)

], (2.132)

so that the degeneracy of the minimum in the potential is lifted and the vacuum is alignedwith the perturbation,

〈πi〉 = 0, 〈σ′〉 = v, (2.133)

with µ2v + γv3 + ε = 0. Defining σ = σ′ − v we get a mass for the pions

m2π = µ2 + λv2 = −ε/v = ε/fπ. (2.134)

This gives us the divergence of the axial-vector current as

∂µAaµ = −fπm2ππ

a(x). (2.135)

This equation is the basis of the PCAC (partially conserved axial vector current) approxi-mation.

2.7.1 Different Representations of the Sigma Model

In the previous section we saw that the σ model, for µ2 < 0, represents massive nucleonsand a scalar singlet plus a triplet of massless pions. If we are interested in energies muchsmaller than the scale of symmetry breaking, we should be able to integrate out the σ fieldand obtain an effective theory containing only the pions. However, it is not obvious how todo that. Taking the limit v →∞ does not seem to really decouple σ, since the coupling σπ2

grows with v. The reason is that if we integrate out σ we are explicitly breaking the chiralsymmetry. This is a typical situation in spontaneously broken continuous symmetries, inwhich the decoupling theorem does not apply. It turn out, however, that we can still writean effective Lagrangian only in terms of the pion fields that is invariant under the fullchiral symmetry. This is at the expense of having a much more complicated, non-linear,transformation rule for the Goldstone bosons. This non-linear realization of the symmetrytakes advantage of the fact that the Goldstone bosons are a parametrization of the manifoldof degenerate “vacua” and therefore can be represented by the local orientation with respectto the vacuum. In the next section we will describe the general theory behind non-linearrepresentations of spontaneously broken gauge symmetries. Here, we will just see it in theexample of the σ model. Let us consider the following representation of the σ model,

Σ = σ′ + i~σ · ~π = S ′U, U = exp(i~σ · ~ξ/v). (2.136)

The exponential can be computed using the properties of the Pauli matrices, which resultin,

U = cos

(ξ

v

)+ i

~σ · ~ξξ

sin

(ξ

v

), (2.137)

27

where ξ =√ξ2

1 + ξ22 + ξ2

3 . Equivalently, we have

σ′ = S ′ cosξ

v= S ′ + . . . , πi = S ′

ξi

ξsin

ξ

v= S ′

ξi

v+ . . . . (2.138)

In particular note that S ′ 2 = σ′ 2 + ~π2 = Tr(Σ†Σ)/2 is invariant under the full chiralsymmetry. Also we have, in the new coordinates, the following vacuum,

〈S〉 ≡ 〈S ′ − v〉 = 0, 〈ξi〉 = 0. (2.139)

Inserting this representation in the Lagrangian, we get

L = ψi∂ψ + g(v + S)(ψLUψR + ψRU

†ψL)

+1

2

[(∂µS)2 − 2λv2S2

]+

(v + S)2

4Tr(∂µU∂

uU †)− λvS3 − λ

4S4. (2.140)

Note that this is just a redefinition of the fields, so the physics is completely unchanged.Also note that the new massless Goldstones ξi no longer appear in the potential but haveonly derivative (bosonic) interactions. Finally, the matrix U transforms under the chiralgroup the same way Σ does (since S ′ does not transform),

U → LUR†. (2.141)

We have said that the new representation is just a redefinition of fields and thereforedoes not change the physics. This is in fact a powerful theorem in field theory on repre-sentation independence first proved by R. Haag [7]. It states that if two fields are relatednonlinearly, e.g. φ = χF (χ) with F (0) = 1, then the same experimental observables resultif one calculates with the field φ using L(φ) or instead with χ using L

(χF (χ)

). The proof

consists basically of demonstrating that (i) two S-matrices are equivalent if they have thesame single particle singularities, and (ii) since F (0) = 1, φ and χ have the same free fieldbehaviour and single particle singularities.

Let us see in action Haag’s theorem using the two different representations of the σmodel. Let us compute scattering amplitudes of Goldstone bosons. In particular, we willcompute the scattering π+π0 → π+π0. The relevant diagrams are shown below.

π+ π+

π0 π0(a)

π+ π+

π0 π0

σ(S)

(b)

1. Linear representation.

The relevant part of the Lagrangian reads

∆L = −λ4

(~π2)2 − λvσ~π2. (2.142)

28

With this Lagrangian, the two diagrams in the figure contribute to the scatteringamplitude

Mπ+π0→π+π0 = −2iλ+ (−2iλv)2 i

p2 −m2σ

= −2iλ

[1 +

2λv2

p2 − 2λv2

]=ip2

v2+ . . . .

(2.143)Note that, although the couplings are non-derivative, the constant term at low en-ergies cancels and we are left with an amplitude that vanishes in the low energylimit.

2. Non-linear representation.

In this case we have the Lagrangian

∆L =1

4(v + S)2Tr

(∂µU∂

µU †)

+ . . . . (2.144)

The contribution proportional to diagram (b) in the figure is of order p4 (two factorsat each vertex) and can be neglected at low energies. The contribution of diagram (a)is of order p2 as can be clearly seen by writing the relevant part of the Lagrangianin terms of the Goldstone fields ~ξ. Using the explicit form of U in terms of theGoldstone fields, Eq. (2.137), and ∂µξ = (~ξ · ∂µ~ξ)/ξ, we get

∆L =1

4(v + S)2 2

v2

(∂µξ)

2 +sin2(ξ/v)

(ξ/v)2

[(∂µ~ξ)

2 − (∂µξ)2]

=1

6v2

[(~ξ · ∂µ~ξ)2 − ~ξ2(∂µ~ξ · ∂µ~ξ)

]+S

v(∂µ~ξ)

2 + . . . , (2.145)

where we have also explicitly shown the coupling of the scalar to the Goldstone bosonsto show that it will contribute at order p4 in the amplitude and also for future use.

Note: The same result could have been obtained from the expansion of theexponential and the commutation relations of the Pauli matrices. Care hasto be taken when using this approach in including all relevant terms to thisorder (in particular there is a contribution to order ξ4 that comes from thelinear term in ∂µU times the triple Goldstone term in ∂µU † and viceversa).

The corresponding amplitude reads

Mπ+π0→π+π0 =i(p′+ − p+)2

v2+ . . . =

ip2

v2+ . . . . (2.146)

As expected, we see that the amplitude for the Goldstone boson scattering is identicalin both representations, although the final result can preceed through different diagrams indifferent representations. We have also seen that Goldstone boson scattering is proportional

29

to p2 and therefore vanishes at low energies. This is a completely general property ofGoldstone bosons and is obvious in the non-linear representation, since their only couplingsare derivative. Haag’s theorem guarantees that we can use either (or any other one)representation for the sigma model and all the physical results that we compute will bethe same. This is very useful, as it allows us to use whichever representation makes ourcalculations (or our understanding of the physics involved) easier. We have actually seen anexample in the scattering amplitude we have computed. Obtaining the correct contributionat low energies ∝ p2 required a cancelation between different contributions in the linearrealization whereas it was immediate in the non-linear representation. In this non-linearrepresentation, we were able to compute the Goldstone boson scattering at low energieswithout the contribution of the field S, i.e., we had a low energy effective Lagrangian forthe Goldstone bosons that did not involve the scalar singlet S. The reason is that we havereplaced the particle content in the linear representation, an isospin singlet σ plus a tripletπ transforming joinly (linearly) as a bidoublet under the full chiral symmetry, with the oneof the non-linear representation, a singlet (under the full chiral symmetry) S, plus three

scalars ~ξ which transform non-linearly under the chiral symmetry, as defined in (2.141)

(note that the quantity U transforms linearly under the chiral symmetry, but the fields ~ξtransform non-linearly). There is actually a subset of chiral transformations under which

the fields ~ξ transform linearly, the vector (isospin) transformations. Indeed, if we takeL = R = 1 + i~α · ~σ + . . . we have

LUR† = L

[∞∑n=0

1

n!(i~ξ · ~σ/v)n

]L† =

∞∑n=0

1

n!(iξi/v LσiL†)n

=∞∑n=0

1

n!(iξi/v Rijσ

j)n = exp(i~ξ′ · ~σ/v), (2.147)

where ξ′ j = Rijξi = RT

jiξi and R is the generator of the linear representation of the

broken generator. Thus, we have seen that the Goldstone bosons in the non-linear rep-resentation transform non-linearly under the chiral transformation, to compensate for thenon-transformation properties of S but they transform linearly under the unbroken group(recall that σ was already a singlet under the isospin symmetry so nothing to compensatehere). This is just one example of a very general result that we will discuss in the nextlecture.

Let us summarize here what we have learned from the σ model. We have seen how thesigma model allows for a very simple description of the spontaneous breakdown of the chiralSU(2)L×SU(2)R symmetry down to its vectorial, isospin, SU(2)V subgroup. In the processfermions can acquire a mass (which is forbidden by the chiral symmetry, now spontaneouslybroken), one of the scalar degrees of freedom, a singlet under the vectorial symmetry, alsoacquires a mass. Finally there are three scalar degrees of freedom that remain massless.These are the Goldstone bosons predicted by Goldstone’s theorem. They have the quantumnumbers of the broken generators arise and how they do not mix with the fourth scalardegree of freedom under isospin transformations. Chiral rotations, however mix the four

30

of them, making it difficult to obtain a low energy effective Lagrangian only in terms ofthe Goldstone fields without explicitly breaking the chiral symmetry. This difficulty wasovercome by changing the linear representation to the non-linear one, in which we replacethe four scalar degrees of freedom by a scalar that is a singlet under the full chiral symmetryplus a triplet of scalars that transform linearly (as a triplet) under the unbroken isospinsymmetry but transform non-linearly under chiral rotations (to compensate for the non-transformation of the singlet). In this situation, we can actually decouple the singlet in achirally invariant way. This can be done by simply freezing its value to the vev v. Thisway, we are left with the so-called non-linear sigma model, with Lagrangian,

Lnon−linσ = ψi∂ψ + gv

(ψLUψR + ψRU

†ψL)

+v2

4Tr(∂µU∂

uU †). (2.148)

Haag’s theorem ensures that the physical calculations performed with both representationsare the same. Of course, in the non-linear sigma model, we have decoupled one of theparticles and it can only be considered an effective theory that will eventually break down.One can consider the non-linear sigma model as the leading term in an expansion inmomentum of the most general effective Lagrangian with chiral symmetries. This is theapproach of chiral perturbation theory to low energy QCD.

Note: Note that we could have used the equivalence SU(2)L × SU(2)R ≡SO(4) to write the non-linear representation in the equivalent form

φ0 =

π1(x)π2(x)π3(x)σ(x)

= (v + S) exp(iξaX a/v)

0001

, (2.149)

whereX a are the generators of SO(4) corresponding to the broken axial-vector

transformations. Written this way, the S and ~ξ fields correspond exactly to theones we used in the non-linear representation. In particular we can decouplethe singlet by just fixing it to its vev to obtain the non-linear sigma model.This can be written in terms of the exponential of the pion fields,

exp(iξaX a/v)(0 0 0 1

)T, (2.150)

or equivalently, in terms of a set of real scalar fields transforming as the (4)representation of SO(4), φ(x), subject to the chirally invariant constraint

φ2(x) = v2. (2.151)

Again, both representations are completely equivalent, although one could bemore useful than the other in particular situations.

31

2.8 Non-linear Realization of a Symmetry

Literature: Here we will follow the discussion in Pokorski and also the nicereview by Feruglio [8]. I have also found useful the lectures on chiral sym-metry by Peskin [9] and the discussion in Weinberg’s second QFT book [10].Obviously, the original papers by CCWZ [11, 12] are also worth reading.

We have seen in the previous sections how theories with spontaneously broken sym-metries have less restrictions on the spectrum of the theory. Fields transforming underdifferent representations of the unbroken symmetry do not need to have the same masses.It is then tempting to write our theory in terms of multiplets of the unbroken symmetry,instead of multiplets under the full symmetry. The fact that the symmetry is spontaneouslybroken, however, (i.e. the Lagrangian is still invariant under the symmetry) imposes someconstraints on the couplings of the different fields. How can we then impose the rightconstraints from the symmetry but work with building blocks that have good representa-tion properties only under the unbroken symmetry? The answer, in the case of the sigmamodel, was to work with the non-linear representation. In that case, we could work withbuilding blocks that had good (linear) transformation properties under unbroken group,

the scalar singlet S and the triplet of Goldstones ~ξ. With these building blocks, we couldconstruct a theory that was invariant under the full chiral symmetry, at the expense thatthe Goldstone fields transform non-linearly under the (broken) chiral symmetry. This pro-cedure, can actually be generalized to an arbitrary symmetry breaking pattern. The trickthat we mentioned but didn’t fully exploit in the example of the sigma model is the factthat the Goldstone bosons can be considered a local rotation of the vacuum among thecontinuous set of degenerate vacua,

|α〉 = exp(iαaX a)|0〉. (2.152)

In the non-linear representation, a field is given by the singlet S times a set of angles ξa(x)specifying the vacuum orientation which sets the Goldstones locally to zero.

Let us see the general theory of non-linear realization of a broken symmetry. This wasfirst developed by Coleman, Wess and Zumino and Callan, Coleman, Wess and Zumino(CCWZ) in [11, 12].

One can introduce the non-linear realization of a group on a manifold at a very formallevel, which shows the generality of the procedure. Feruglio discusses these formal develop-ments in his review and we will discuss them in the lectures. Here we will just summarizethe main results, without further proof or discussion. The set up is a real manifold M(which in our field theoretic description will be a set of scalar fields) and a Lie group Gacting on it. The manifold has a special point, the origin (for us it will be the vacuum), suchthat a subgroup H of G leaves the origin invariant. The first important result is Haag’stheorem that we discussed in previous lectures. It describes “allowed” transformations asthose that have Jacobian one at the origin. They give raise to identical S-matrix elements.The second result is that it is always possible to choose coordinates on M (scalar field

32

representation) such that the action of H on M is linear. The corresponding coordinatesare said to be in standard form. A final theorem states that any non-linear representationof the group G on M can be always brought to the standard form by means of an allowedcoordinate transformation.

Let us now be a bit more explicit. We will consider a scalar theory and choose co-ordinates on the manifold of scalars that generate the non-linear realization of the groupin its standard form. Let us consider a theory with some symmetry group G which isspontaneously broken to a subgroup H. Let G be a compact, connectetd, semi-simple Liegroup. In some neighbourhood of the identity of G, we can write an element of G as,

g = exp(iξaX a) exp(iuiY i), (2.153)

where as in previous sections X a and Y i are, respectively, the broken and unbroken gen-erators.

Note: The scalar fields ξ(x) are coordinates of the manifold of left cosetsG/H at each point of space-time. The set of group elements l(ξ) = exp(iξaX a)parametrizes this manifold. The decomposition (2.153) means that once wehave a parametrization l(ξ), each group element g can be uniquely decom-posed into a product g = lh, where h ∈ H. l is the representative member ofthe coset to which g belongs and h connects l to g within the coset.

Let Φ0(x) be a field which transforms according to a linear representation of G. Let uswrite it as follows,

Φ0(x) = exp(−iξaX a)Φ(x) ≡ UΦ, (2.154)

where X a here denotes the matrix representation of the broken generators appropriate forΦ0(x). Let us now perform an arbitrary G rotation of the field Φ0,

Φ′0(x) = exp(−iαaGa)Φ0(x) = exp(−iαaGa) exp(−iξaX a)Φ(x)

= exp(−iα′aGa)Φ(x) = exp(−iξ′a(ξ, α)X a) exp(−iui(ξ, α)Y i)Φ(x). (2.155)

In terms of the group elements, we have just used the property that the product of anelement of the group g with another element of the group, in particular an element ofthe coset l(ξ) is inside the group and therefore can be decomposed as the product of anelement of the coset times an element of the unbroken subgroup, gl(ξ) = g′(g, ξ) = l(ξ′)h,where ξ′ = ξ′(g, ξ) and h = h(g, ξ). Thus, the field Φ0 can be represented by the fieldsξhata(x) and Φ(x) for which a general global transformation of the group G is realized asa non-linear transformation of the fields ξa and a local transformation belonging to theunbroken subgroup H on the multiplet Φ,

ξa(x) → ξ′a(x) = ξ′a(g, ξ(x)), (2.156)

Φ(x) → exp(iui(x)Y i)Φ(x) = h(g, ξ(x))Φ. (2.157)

The transformation properties are defined by (2.155). We say that the fields ξ,Φ form anon-linear realization of the group G. What happens if the transformation is under the

33

unbroken subgroup H? In that case the transformation of the Goldstone fields simplify,becoming linear,

Φ′0(x) = exp(−iαiY i) exp(−iξaX a)Φ(x) = exp(−iαiY i) exp(−iξaX a) exp(iαiY i) exp(−iαiY i)Φ(x)

= = exp(−iξaRabXb) exp(−iαiY i)Φ(x) = exp(−iξ′bX b) exp(−iαiY i)Φ(x), (2.158)

where Rab is the matrix representation of the linear transformation of the broken generatorsunder the unbroken group,

exp(−iαiY i)X a exp(iαiY i) = RabXb. (2.159)

We would like to use this non-linear realization of the group to construct G-invariantLagrangians. The problem is that, despite the fact that the symmetry is global, thenon-linear realization involves the Goldstone fields and therefore is space-time dependent.Thus, derivatives have to be transformed into covariant derivatives. Furthermore, the non-linearity of the transformation makes the transformation of ∂µξ complicated. Instead ofstarting with this object, we will consider one that has simpler transformation properties,

e−iξ·X∂µeiξ·X = iDa

µXa + iEi

µYi ≡ iDµ + iEµ, (2.160)

where we have used that the object we are considering is a generator of the group and cantherefore be expanded in generators. Let us consider now ∂µΦ0,

∂µΦ0 = ∂µ

[eiξ·XΦ

]= eiξ·Xe−iξ·X∂µ

[eiξ·XΦ

]= eiξ·X

[iDa

µXa + iEi

µYiΦ + ∂µΦ

]= eiξ·X

[iDa

µXaΦ + (∂µ + iEi

µYi)Φ]. (2.161)

But ∂µΦ0 transforms under a general global transformation g the same as Φ0 does, thus,the term in brackets transforms under g as Φ does[

iDaµX

aΦ + (∂µ + iEiµY

i)Φ]→ h(g, ξ(x))

[iDa

µXaΦ + (∂µ + iEi

µYi)Φ]. (2.162)

Since different irrepresentations of H do not mix in Φ under a G transformation, we canset to zero all elements in Φ except those belonging to a single Ri. But then the twoterms in brackets belong to different H representations in R. Thus, they must transformindependently. On one hand we have

DaµX

aΦ→ hDaµX

aΦ = D′aµXaΦ′ = hh−1D′aµX

ahΦ. (2.163)

We therefore have

D′µ = hDµh−1 = Db

µeiu·YXbe

−iu·Y = DbµRbaXa. (2.164)

34

Similarly,

(∂µ + iEµ · Y )Φ → h(∂µ + iEµ)Φ = (∂µ + iE ′)Φ′ = (∂µ + iE ′)hΦ

= h(∂µ + ih−1E ′h+ h−1∂µh)Φ. (2.165)

Thus,E ′µ = hEµh

−1 + i(∂µh)h−1 = hEµh−1 − ih∂µh−1. (2.166)

We therefore have three basic ingredients, Φ, Dµ and Eµ on which a general g transfor-mation acts as a local h(g, ξ(x)) ∈ H transformation belonging to the unbroken subgroup.Thus, a general H invariant Lagrangian built out of functions of these building blocks isautomatically G-invariant. Let us summarize here the transformations of these fields,

Φ → h(g, ξ(x))Φ, (2.167)

Dµ → h(g, ξ(x))Dµh−1(g, ξ(x)), Da

µ → RTabDbµ, (2.168)

Eµ → h(g, ξ(x))Eµh−1(g, ξ(x))− ih(g, ξ(x))∂µh

−1(g, ξ(x)), (2.169)

where R is the matrix representation of H under which the broken generators transform.In particular, the quantity Eµ ≡ ∂µ + iEµ acts as an H-covariant derivative,

EµΦ→ h(g, ξ(x))EµΦ. (2.170)

With these building blocks we can construct in a very simple way G-invariant La-grangians out of multiplets of the unbroken group. What happens if the symmetry is(partially) gauged? Essentially the same formalism can be used, with the replacement ofthe usual derivative with the gauge covariant derivative

∂µ → ∂µ + iAµ = ∂µ + iAaµGa.

We define now,

e−iξ·X(∂µ + iAµ)eiξ·X = iDaµX

a + iEiµY

i = iDµ + iEµ, (2.171)

where now Dµ = Dµ(ξ, A) , Eµ = Eµ(ξ, A).

(∂µ + iAµ)Φ0 = (∂µ + iAµ)eiξ·XΦ = eiξ·X(∂µ + e−iξ·X∂µeiξ·X + ie−iξ·XAµe

iξ·X)Φ

= eiξ·X∂µ + e−iξ·X

[∂µ + iAµ

]eiξ·X

Φ

= eiξ·XiDµ + (∂µ + iEµ)

Φ. (2.172)

But since under a local G transformation, (∂µ + iAµ)Φ0 transforms as Φ0 does, that meansthat [iDµ + (∂µ + iEµ)]Φ transforms the same way Φ does. Therefore, under local Gtransformations we have,

Φ → h(g(x), ξ(x))Φ, (2.173)

Dµ → h(g(x), ξ(x))Dµh−1(g(x), ξ(x)), (2.174)

EµΦ ≡ (∂µ + iEµ)Φ→ h(g(x), ξ(x))EµΦ. (2.175)

Thus, we can build an invariant Lagrangian under local G tranformations by constructing,out of Φ, Dµ and εmu a Lagrangian invariant under local H transformations.

35

Chapter 3

Electroweak Symmetry in theStandard Model

Literature: The Standard Model is covered in many reviews and textbooks.Here we are going to take a slightly different approach than just introducingthe SM but instead we will describe the experimentally confirmed part ofthe SM and argue that an EWSB sector is needed. We will then describethe EWSB sector in the SM, including its sucesses and problems. This willmotivate alternative EWSB sectors that we will consider in the rest of thecourse. In general, I have used Contino’s notes(http://indico.phys.ucl.ac.be/conferenceDisplay.py?confId=148)as a rough guide. Some other references I found particularly useful duringthe preparation of these notes are the books by Peskin and Schroeder [4];Pokorski [1] and Donoghue, Golowich and Holdstein [2]. Very interesting asan overview of electroweak symmetry breaking in and beyond the StandardModel are the lectures by Grojean [13]. Also very interesting are the lecturesby Veltman on the Higgs system [14].

3.1 Electroweak Symmetry and its Breaking

Most of the Standard Model (SM) of particle physics has been tested to extremely goodaccuracy up to energies of the order of the electroweak symmetry breaking (EWSB) scalev ∼ 174 GeV. It is based on a Yang-Mills theory with an SU(3)C ×SU(2)L×U(1)Y gaugegroup spontaneously broken to SU(3)C × U(1)Q and three families of fermions with thequantum numbers displayed in Table 3.1, in which we have also displayed the chiralityof each fermion (we use ψL,R = PL,Rψ with PL,R ≡ (1 ∓ γ5)/2 the chirality projectors).Neutrino masses also require the introduction of RH neutrinos, which are singlets underthe SM gauge symmetry. In these lectures we are interested in the mechanism of EWSBand will disregard neutrino masses altogether. The particular mechanism of EWSB hashowever escaped experimental confirmation so far. The SM solution for EWSB, despite its

36

Table 3.1: Quantum numbers for a family of SM fermions. The hypercharge assignment isarbitrary, fixed by convention.

qL =

(uLdL

)uR dR lL =

(νLeL

)eR

(3, 2)1/6 (3, 1)2/3 (3, 1)−1/3 (1, 2)−1/2 (1, 1)−1

simplicity, is not fully satisfactory. Assuming that naturalness is a good guiding principle,new physics is required at around the TeV scale either to replace the Higgs as the mediatorfor EWSB or to make it more natural, in a way that we will discuss in these lectures.We will then dedicate this lecture to study some of the features of the mechanism ofEWSB in the SM, its successes and weaknesses, as a motivation for the need of physicsbeyond the SM (BSM) and a guide for what makes a model BSM interesting, relevant, andphenomenologically viable.

We are therefore going to describe the Lagrangian of the SM, as it has been experi-mentally confirmed so far. We will start with the kinetic Lagrangian, which reads

Lkin = −1

4GaµνG

aµν − 1

4W iµνW

i µν − 1

4BµνB

µν + ψLiDψL + ψRiDψR, (3.1)

where ψL,R run over all the SM fermions of the appropriate chirality. Following standardnotation (see for instance [4]) we have defined the covariant derivative by

Dµ = ∂µ − igaT aAaµ = ∂µ − i[gstaGaµ + gT iLW

iµ + g′Y Bµ], (3.2)

where we have denoted T a and Aaµ the general list of generators and gauge bosons andta and T iL are the SU(3)C and SU(2)L generators in the corresponding representation ofthe field the covariant derivative is acting on, normalized to Tr[TαT β] = δαβ/2 for thefundamental representation (i.e. they are ta = λa/2, T iL = σi/2 for a color triplet andweak doublet, with λ and σ the Gell-mann and Pauli matrices, respectively). The gaugeboson field tensors are defined in general by

[Dµ, Dν ] = −igaF aµνT

a ⇒ F aµν = ∂µA

aν − ∂νAaµ + gaf

abcAbµAcν . (3.3)

In particular, for our choice of gauge groups, we have

Gaµν ≡ ∂µG

aν − ∂νGa

µ + gsfabcGb

µGcν , (3.4)

W iµν ≡ ∂µW

iν − ∂νGj

µ + gεijkW jµW

kν , (3.5)

Bµν ≡ ∂µBν − ∂νBµ. (3.6)

As advertised, this Lagrangian has a local SU(3)C×SU(2)L×U(1)Y symmetry. Defining

37

U(x) ≡ exp[iT aαa(x)], with T a running over all 12 (=8+3+1) generators, we have

ψ(x) → U(x)ψ(x) ≈ (1 + iT aαa(x) + . . .)ψ(x), (3.7)

Aµ(x) ≡ Aaµ(x)T a → U

(Aµ +

i

g∂µ

)U † ≈ Aµ + i[α,Aµ] +

1

g∂µα, (3.8)

Dµψ(x) → U(Dµψ). (3.9)

Apart from this local symmetry, we know from our examples in previous lectures thatthe kinetic term for the fermions has an extra U(3)5 global symmetry (in our previouslanguage this would be U(3)uL × U(3)uR × U(3)dL × U(3)dR for the quark system with therestriction that uL and dL are tied together by the SU(2)L symmetry and therefore thereis only one “Left” rotation in the quark sector. Similarly, for the lepton sector we have acommon left rotation for the three lL and a right rotation for the eR).

One important property of the SM gauge symmetry is that it is (its electroweak part)chiral, by which we mean that left-handed fermions transform in different representationsof the gauge group than right-handed fermions. In particular, a mass term for the fermions,which mixes left and right components, is not gauge invariant and is therefore forbiddenunless the symmetry is spontaneously broken. Obviously, the gauge invariance also forbidsa mass term for the gauge bosons which, unless the symmetry is spontanously broken,should all be massless.

Note: Chirality of the electroweak symmetry implies other constraints,namely, the possible presence of anomalies. This constraints the quantumnumbers of the chiral fields. The fermion content of the SM is anomaly free.

Experimentally, we observe however that fermions and some of the gauge bosons havefinite masses. This means that the gauge symmetry must be spontaneously broken andexperimental data indicate that it is broken down to QCD, SU(3)C , and electromagnetism,U(1)em. These two symmetries are vector-like, which means that all fermions have left andright components transforming under the same representation. Thus, Dirac masses areallowed by the unbroken gauge invariance. Similarly, the gauge bosons corresponding tothe broken generators, will get masses through the Higgs mechanism.

Traditionally, one would write here the gauge invariant Higgs Lagrangian, that in theSM is responsible for spontaneously breaking the electroweak symmetry. Instead, we willfollow our experimental guiding principle and write down the mass Lagrangian for thefermions and gauge bosons.

Lmass = Lgmass + Lfmass. (3.10)

For the electroweak gauge bosons, we have,

Lgmass =1

2m2W [(W 1

µ)2 + (W 2µ)2] +

1

2m2Z(cWW

3µ − sWBµ)2, (3.11)

where cW , sW are the cosine and sine of the Weinberg angle that define which combinationof W 3 and B becomes the massive Z. This Lagrangian gives mass to W 1,2

µ and a linear

38

combination of W 3µ and Bµ, leaving massless the orthogonal combination sWW

3µ + cWBµ.

Performing the following redefinition of fields,

W±µ =

W 1µ ∓ iW 2

µ√2

,

(ZµAµ

)=

(cW −sWsW cW

)(W 3µ

Bµ

)⇔(W 3µ

Bµ

)=

(cW sW−sW cW

)(ZµAµ

),

(3.12)where W±

µ have electric charged ±1, Zµ has charge 0 and Aµ is the massless photon, wehave

Lgmass = m2WW

+µ W

−µ +1

2m2Z(Zµ)2. (3.13)

This mass Lagrangian breaks (we assume the breaking is spontaneous) the electroweaksymmetry SU(2)L × U(1)Y → U(1)em

i(Dµ−∂µ) = gT 3LW

3µ+g′Y Bµ+. . . = (gsWT

3L+g′cWY )Aµ+(gcWT

3L−g′sWY )Zµ+. . . = eQAµ+. . . ,

(3.14)which implies,

eQ = gsW

(T 3L +

g′

g

cWsW

Y

)→ gsW (T 3

L + Y ), (3.15)

where we have use the fact that the normalization of the U(1)Y factor is not fixed to makeY (g′cW )/(gsW )→ Y , (this is actually the convention used in Table 3.1), or equivalently

g′ = gcW/sW ⇒sW = g′/

√g2 + g′ 2,

cW = g/√g2 + g′ 2,

(3.16)

Thus, we havee = gsW = g′cW , Q = T 3

L + Y. (3.17)

The mass lagrangian for the fermions reads (we neglect neutrino masses here),

Lfmass = −3∑

ij=1

uiLm

uiju

jR + diLm

dijd

jR + eiLm

eije

jR + h.c.

. (3.18)

Using the global symmetries of the kinetic Lagrangian, we can put the mass matrices inthe following form

muij = V †ijm

ui , md

ij = mdi δij, me

ij = meiδij, (3.19)

with Vij the unitary 3 × 3 CKM matrix (alternatively, one could put the up quark sectormass diagonal and the down quark sector one diagonal up to a left rotation given by theCKM matrix) and mu

i ,mdi ,m

ei the diagonal masses of the charge 2/3 quarks, charge −1/3

quarks and charged leptons, respectively. Note that the mass terms we have written forthe fermions are compatible with the unbroken electromagnetic symmetry (but not withthe original electroweak symmetry). Also, these mass terms break the global symmetriesdown to U(1)B and U(1)Li .

39

The total Lagrangian, after EWSB, reads (we do not display the QCD part from nowon, since it does not play any role in EWSB),

L = −1

2W+µνW

−µν − 1

4AµνA

µν − 1

4ZµνZ

µν +m2WW

+µ W

−µ +1

2m2Z(Zµ)2

+∑[

ψLiDψL + ψRiDψR

]−[uiLV

†ijm

uju

jR + diLm

di diR + eiLm

eieiR + h.c.

], (3.20)

where the (electroweak part of the) covariant derivative now reads

i(Dµ − ∂µ) =g√2

(T+W+µ + T−W−

µ ) +g

cW[T 3L − s2

WQ]Zµ + eQAµ, (3.21)

and we have defined T± = T 1 ± iT 2 = (σ1 ± iσ2)/2.

3.2 The Need for an Electroweak Symmetry Breaking

Sector

In the previous section we have described the Lagrangian of the spontaneously brokenSU(2)L × U(1)Y → U(1)em model that very successfully reproduces experimental dataup to the electroweak scale. From previous lectures we know that in order for a localsymmetry to be broken in a spontaneous way (so that the Lagrangian is still invariantunder the symmetry), the Higgs mechanism has to be in action. For that, we just need thewould-be Goldstone bosons that will combine with the broken gauge bosons to give thema mass in a gauge invariant way (i.e. resulting in Ward identity preserving, transversevacuum polarization tensors). The question now is, can we do with just these (three)would-be Goldstone bosons and write the Lagrangian using a non-linear representation?The answer is, of course, yes, we can, but not for long. It turns out that the theorywe just wrote does not preserve some of the fundamental features of realistic quantumfield theories at high energies. In particular, the theory as written, violates unitarity athigh energies. One can describe the physics at the EWSB scale by means of the so-calledelectroweak chiral Lagrangian, that uses the non-linear representation, keeping only theGoldstone bosons that are required to give masses to the W and the Z. We will do so byusing the formalism we developed in the previous lectures. We will start by parametrizingthe generators of the group in terms unbroken generators,

Q = T 3L + Y,

and generators of the coset, for which we choose,

T iL.

Let us now start with a singlet under the unbroken group,

Φ(x) =

(01

). (3.22)

40

In order for this to be a singlet under the unbroken subgroup, it has to have zero electriccharge and therefore hypercharge Y = Q − T 3

L = 1/2. This field can be promoted to afull representation of the electroweak group by multiplying it with the exponential of theGoldstone bosons,

Φ0 = exp[i~π · ~σ/(√

2v)]Φ0, (3.23)

transforms as a (1, 2)1/2 under the full gauge symmetry,

Φ0 → ei~αL·~σ/2eiαY /2Φ0. (3.24)

We prefer to work with the Goldstone fields in terms of a 2 × 2 matrix by constructinganother doublet out of the conjugate of Φ0,

Φ0 = iσ2Φ∗0 ∼ (1, 2)−1/2, (3.25)

and constructing the matrix

Σ = (Φ0 Φ0) = exp(i~π · ~σ/(√

2v)). (3.26)

The SU(2)L × U(1)Y symmetry can be seen acting on Σ as

Σ→ eiwiLσ

i/2Σe−iwY σ3/2, (3.27)

(with fermions we need to also gauge U(1)B−L in this minimal setup). Thus, we canwrite a fully gauge invariant Lagrangian involving only the massive gauge bosons (plus thephoton). The building blocks of this theory are

• The field strengths: Wµν ≡ W iµνσ

i/2, Bµν .

• The Lorentz vector Vµ ≡ (DµΣ)Σ†, involving the covariant derivative of Σ,

DµΣ ≡ ∂µΣ− i[g2W Iµσ

IΣ− g′

2Σσ3Bµ]. (3.28)

• The combination of Σ−fields: T ≡ Σσ3Σ†.

Except for Bµν , which is invariant under SU(2)L × U(1)Y , the transformation of all theother combinations of fields, denoted collectively by Φ, under the full SU(2)L × U(1)Ysymmetry is

Φ→ eiwILσ

I/2Φe−iwILσ

I/2. (3.29)

Doing an expansion in momenta and keeping up to second order we can write the ChiralEW Lagrangian

LEWCh = −1

2Tr(WµνW

µν)− 1

4BµνB

µν +v2

2Tr[(DµΣ)†DµΣ]

+ a0v2

4[Tr(TVµ)]2 + a1i

gg′

2BµνTr(TWµν) + a8

g2

4[Tr(TWµν)]

2 + . . . .(3.30)

41

The coefficients of the first three terms are chosen so that the kinetic terms of the gaugeand Goldstone bosons are canonically normalized. Let us expand the kinetic term of theGoldstones. First, we expand DµΣ in powers of the Goldstone fields,

DµΣ = i

[∂µπ√

2v− g

2Wµ +

g′

2σ3Bµ

]+ . . .

= i

[2∑b=1

(∂µπ

b

√2v− g

2W bµ

)σb +

(∂µπ

3

√2v− g

2W 3µ +

g′

2Bµ

)σ3

]+ . . . , (3.31)

where the dots denote terms with more than one field. The kinetic term for the Goldstonescan then be written

LΣ =v2

2Tr[(DµΣ)†DµΣ] = |∂µπ+ − g√

2vW+

µ |2 +1

2(∂µπ

3 − g√2cW

vZµ)2. (3.32)

In the last equality we have used the definition of the Z boson and cW that we introducedin the previous section. Also, we have defined π± = (π1 ∓ iπ2)/

√2. The masses of the

gauge bosons are aparent in the unitary gauge (Σ = 12×2, πi = 0)

L =1

2g2v2W+

µ W−µ +

1

4v2√g2 + g′ 2ZµZ

µ = m2WW

+µ W

−µ +1

2m2ZZµZ

µ. (3.33)

Thus we getmW = g v√

2

mZ =√g2 + g′ 2 v√

2

⇒ ρ ≡ m2

W

m2Zc

2W

= 1. (3.34)

This relation between the W and Z masses is not coincidence. There is a symmetry thatguarantees this result. The importance of this relation, as parametrized by ρ (or throughits close cousin the T parameter T ∼ ρ − 1), is that it has been measured to with a permille accuracy, finding very good agreement with the SM prediction.

Instead of the unitary gauge, one can fix the gauge to an Rξ gauge by adding thefollowing gauge-fixing term

LG.F. = − 1

2ξ

(∂µW i

µ + ξgv√

2πi)2

− 1

2ξ(∂µBµ − ξ

g′v√2π3)2. (3.35)

In this family of gauges, the gauge-Goldstone boson kinetic mixing is cancelled. The gaugeboson propagators read, for the W±,

−ik2 −m2

i

[gµν − (1− ξ) kµkν

k2 − ξm2i

], (3.36)

where mi = mW ,mZ ,mA for the W , Z and A, respectively. The Goldstones, on the otherhand, have propagators,

i

k2 − ξm2i

, (3.37)

where mi = mW ,mZ where corresponds.

42

3.2.1 Custodial symmetry

Let us discuss what the symmetry reason behind the relation ρ = 1 is. The pattern ofsymmetry breaking described by Σ has a larger global symmetry than SU(2)×U(1). TheU(1)Y symmetry acts on the right of Σ with the σ3 Pauli matrix. We can extend thisU(1) symmetry to a full SU(2)R (global) symmetry under which the kinetic term LΣ isstill invariant. The vacuum is given by Σ = 12×2, which is invariant under the vectorSU(2)L+R symmetry (equal left and right rotations). Now, note that the left rotationactually coincides with the gauge SU(2)L symmetry, under which the three gauge bosonsW Iµ transform as a triplet (the adjoing of SU(2)L). The remaining symmetry after EWSB

is the simultaneous rotation from the left and the right with the same matrix, but inparticular there is an arbitrary left rotation that remains unbroken. Thus, W I

µ transform asa triplet under the remaining custodial symmetry and therefore they must receive the samecontribution to their mass from EWSB. This global symmetry pattern SU(2)L×SU(2)R →SU(2)L+R has been dubbed custodial symmetry, as it protects the ρ (or T ) parameter.Indeed, the presence of a symmetry under which the three SU(2)L gauge bosons transformas a triplet, together with unbroken U(1)em gauge invariance, ensure ρ = 1.

Note: Let us show that custodial symmetry plus unbroken U(1)em implyρ = 1. First write the Z and the photon as arbitrary combinations of W 3 andB, (

ZA

)=

(c s−s c

)(W 3

B

), (3.38)

where c ≡ cos θ, s ≡ sin θ with θ an arbitrary angle, to be determined byQ = T 3 + Y . The coupling of the photon is then

gT 3W 3 + g′Y B = (sgT 3 + cg′Y )A+ . . . = eQA+ . . . , (3.39)

from which we obtain c = g/√g2 + g′ 2, s = g′/

√g2 + g′ 2. Now allowing for

an arbitrary mass matrix in the neutral sector we get

L0m =

v2

4

(W 3 B

)(g2 aa b

)(W 3

B

)(3.40)

=(Z A

)( c2g2 − 2sca+ s2b sc(g2 − b) + (c2 − s2)asc(g2 − b) + (c2 − s2)a s2g2 + 2sca+ s2b

)(ZA

).

Requiring the photon mass and mixing with the Z to vanish determines aand b and therefore the Z mass, which is given by

1

2m2Z =

v2

4(g2 + g′ 2) =

1

2

m2W

c2W

, (3.41)

and therefore ρ = 1.

43

Note however that the custodial symmetry is a good symmetry of LΣ but not of thefull electroweak chiral Lagrangian. In particular T = Σσ3Σ† is not invariant under the fullSU(2)R global symmetry and therefore operators involving it can give a direct contributionto the ρ parameter. Actually, the operator with coefficient a0 does induce a correction toit.

Which sectors of the SM break the custodial symmetry? One we have already seen,the hypercharge. That means that at loop level, there will be extra corrections to ρproportional to g′. Another sector that breaks the custodial symmetry is the fermionicsector, through the Yukawa couplings. These can be written as

L =(tL bL

)Σ

(λttRλbbR

). (3.42)

We have written the couplings in a way that makes it transparent the breaking of thecustodial symmetry. It is the difference between the top and bottom Yukawa couplingsthat break the global SU(2)R symmetry, and therefore the custodial symmetry. This meansthat, due to the large difference between the top and bottom masses, one can expect arelatively large contribution from this sector to ρ at one loop. This contribution is infact an interesting example of non-decoupling of heavy physics, in which the conditions ofthe Appelquist-Carazzone theorem are not satisfied. The reason is that in the heavy toplimit, the large mass comes from a dimensionless coupling that becomes large, thus hittingstrong coupling. Furthermore, the resulting low energy theory is non-renormalizable andtherefore non-vanishing effects in the large top mass limit can be expected.

We will compute the leading contribution to ρ in the heavy top limit as an exercise.An important lesson to learn from this discussion, and from the top contribution is

that tree level violations of the custodial symmetry by new physics will be in general verystrongly constrained and therefore custodial symmetry should be a natural ingredient inBSM model building. Also, non-decoupling physics that violates the custodial symmetry,can have non-negligible effects on ρ at the loop level and should be considered in modelsof physics BSM.

3.2.2 Unitarity Violation

The model as we have described it so far, can be used to describe the physics up tothe EW scale that has been experimentally tested. It is however not consistent whenextrapolated at higher energies (another question that we will address later is whether thedescription of experimental data in terms of this model is accurate or not). The reason isthat the scattering of longitudinal gauge bosons (or equivalently of the Goldstone bosonsthat are eaten by them to become massive) grows with the energy. However, cross sectionscannot be arbitrary large, as they are constrained by unitarity. The growth with energyof longitudinal gauge boson scattering can be easily understood by noting that a massivespin-1 particle with momentum kµ can be represented by

Aµ = εµ exp(ikνxν), (3.43)

44

where the polarization vector satisfies εµεµ = −1 and kµε

µ = 0. Thus, if we assume thevector is moving in the z direction (kµ = (E, 0, 0, k), with kµkµ = E2 − k2 = M2), thethree polarizations can be written as

εµ+ =1√2

(0, 1, i, 0), εµ− =1√2

(0, 1,−i, 0), (3.44)

for the LH and RH transverse polarizations and

εµL =

(k

M, 0, 0,

E

M

)=kµ

M+O

(M

E

), (3.45)

for the longitudinal one.

Exercise: Confirm the form of the longitudinal polarization. Hint: Start inthe rest frame and perform a boost to with momentum k along the z axis. Computethe boost of the polarization vector along that direction ε = (0, 0, 0, 1). Using theon-shell condition, check that at high energies, the longitudinal polarizationaligns with the momentum.

Thus, we see that at high energies, the longitudinal polarizations aligns with the momentumof the gauge boson. This causes a growth in the scattering amplitude that is incompatiblewith unitarity. Let us first try to understand the result and then compute it using a neattrick to simplify the calculation. Let us assume we are in the unitary gauge, in which Σ isequal to the identity, and let us consider the scattering of longitudinal W bosons,

W+LW

−L → W+

LW−L , (3.46)

which proceeds through the three diagrams shown in Fig. 3.1. The naive energy dependence

W +W +

W − W −

W +W +

W − W −

W +W +

W − W −

Figure 3.1: Diagrams contributing to longitudinal W scattering. In the second and thirddiagrams, either the Z or the γ are exchanged in the s and t channels, respectively.

of the amplitude can be easily estimated by recalling that εL ∼ E, the trilinear couplingsare proportional to momentum (and therefore can pick up a power of the energy), and thepropagators go like ∼ (1 +E2)/E2. Thus, the contact interaction diagram has potentially

45

O(E4) contributions whereas the two diagramas with exchange of Z and A are potentiallyO(E6). It turns out that the terms∼ E6 coming from the pµpν/m2 terms in the propagatorscancel out on each diagram, making each individual diagram of order∼ E4. If the couplingscome from an F 2

µν term, these leading order ∼ E4 cancels among the three diagrams. Thereis however a residual O(E2) term that does not cancel. This term will violate unitarity athigh energies. The result of the full scattering amplitude is in fact

M(W+LW

−L → W+

LW−L ) =

g2

4m2W

(s+ t). (3.47)

Instead of doing the calculation for the gauge bosons, we will make use of the Equiva-lence Theorem that states that amplitudes involving longitudinally polarized gauge bosonsare equal to amplitudes with the corresponding Goldstone bosons as external particles upto corrections of order m2/E2, with m the mass of the gauge boson and E the energyinvolved in the scattering. At leading order in m2

W/E2, we can therefore replace the ex-

ternal W± with the corresponding Goldstone bosons π±. We will furthermore work in thet’Hooft-Feynman gauge (ξ = 1), in which the gauge boson propagators have the simpleform,

−igµν

p2 −m2, (3.48)

and therefore they go like∼ 1/E2 at large energies. The counting now goes as follows, gaugecouplings of two Goldstone bosons are momentum dependent and therefore go like ∼ Ewhereas a Goldstone boson quartic coupling has two powers of momentum and thereforegoes like ∼ E2. Thus, the diagrams with an internal gauge boson propagator go like∼ E0 whereas the contact interaction grows with energy like ∼ E2. Thus, this is the onlydiagram we have to compute. The quartic interaction of the Goldstone bosons is given bythe corresponding term in LΣ. We have actually computed it in the study of the sigmamodel getting,

LΣ =1

12v2

[(πi∂µπ

i)2−πiπi(∂µπj∂µπj)]+ . . . =

g2

24m2W

(π−∂µπ+−π+∂µπ

−)2 + . . . . (3.49)

Using this expression, we obtain for the amplitude,

M(π+π− → π+π−) =g2

4m2W

(s+ t)[1 +O

(m2W

E2

)]. (3.50)

Let us now be more quantitative on the violation of unitarity. In order to do that, weexpand the amplitude in partial waves,

M = 16π∞∑l=0

(2l + 1)Pl(cos θ)al, (3.51)

where Pl are the Legendre polynomials, that satisfy the orthonormality condition∫ 1

−1

dxPl(x)Pl′(x) =2

(2l + 1)δll′ . (3.52)

46

The first few polynomials are

P0(x) = 1, (3.53)

P1(x) = x, (3.54)

P2(x) =1

2(3x2 − 1). (3.55)

They all satisfy Pl(1) = 1. Using the fact that for a 2 → 2 process, the cross section isgiven by dσ/dΩ = |M|2/(64π2s), with dΩ = 2πd cos θ, we get

σ =8π

s

∞∑l,l′=0

(2l + 1)(2l′ + 1)ala∗l′

∫ 1

−1

dxPl(x)Pl′(x)

=16π

s

∞∑l=0

(2l + 1)|al|2. (3.56)

The optical theorem relates the total cross section with the imaginary part of the forwardamplitude,

σ =1

sImM(θ = 0) =

16π

s

∞∑l=0

(2l + 1)Im(al) =16π

s

∞∑l=0

(2l + 1)|al|2. (3.57)

In practice, this has to be satisfied for each partial wave. Thus, we have

|al|2 = Im(al)⇒ Re(al)2 +

(Im(al)−

1

2

)2

=1

4, (3.58)

which is a circle in the complex plane of radius 1/2, centered at i/2. Therefore we have|Re(al)| < 1/2. 1 This has to be applied to the partial wave amplitudes, that read,

al =1

32π

∫ 1

−1

d cos θ Pl(cos θ)M =1

16π(s− 4m2W )

∫ 0

−s+4m2W

dt Pl

(1 +

2t

s− 4m2W

)M,

(3.59)where in the second equality we have used properties of a 2→ 2 scattering process of fourparticles with the same mass.

Note: Recall that, for a 2→ 2 process of four particles with the same mass wehave s = 4E2, t = −2p2(1− cos θ), where E, p and θ are the energy, modulusof trimomentum and scattering angle of the collision (thus m2 = E2− p2), wehave ∫ 1

−1

d cos θ =2

s− 4m2

∫ 0

−s+4m2

dt, (3.60)

and cos θ = 1 + 2t/(s− 4m2).

1This is an all order in perturbation theory statement, imaginary parts of the scattering amplitude aregenerated at the next order in perturbation theory.

47

Inserting the expression we got for the amplitude, we have, for the first few partial waves

a0 =g2

128π

s

m2W

(1 +Om

2W

s

),

a1 =g2

384π

s

m2W

(1 +Om

2W

s

). (3.61)

Requiring Re(a0) ≤ 1/2 we obtain that unitarity is violated at a scale

Λunit. =√smax =

8√π

gmW ≈ 22mW ≈ 1.7 TeV. (3.62)

A more detailed calculation, involving several coupled channels, gives a bound

Λunit. . 700 GeV. (3.63)

It should be noted however that these bounds are approximate bounds on the perturbativeviolation of unitarity. They are not strict, as non-perturbative effects will not turn onabruptly. Nevertheless they are a good indicator that something has to happen at energies. 1 TeV to unitarize longitudinal gauge boson scattering. We do not know yet whatthis is but it is clear that the quantum field theory that is represented by the electroweakchiral Lagrangian, necessarily breaks down and has to be replaced by something else atsuch energies. This is the main motivation for the LHC that will extensively explore themulti-TeV region.

Thus, we see that although current experimental data can be explained in terms ofjust the observed particles (including the longitudinal components of the gauge bosons or,equivalently, the would-be Goldstone bosons), unitarity requires an electroweak symmetrybreaking sector, that unitarizes longitudinal gauge boson scattering. In the rest of thecourse we will study in some detail different alternatives for this EWSB sector, the exper-imental constraints on them and their advantages and disadvantages. We start with theSM solution in the next section.

3.3 The SM EWSB sector: unitarity restoration with

one scalar

3.3.1 Higgs unitarization of longitudinal gauge boson scattering

The SM simply consists on linearizing the non-linear chiral electroweak Lagrangian. Weknow how to do that linearization from our analysis of the sigma model, we just need todefine the linearly transforming field,

φ = (v +h√2

)Σ

(01

)=

(iπ+

v + h−iπ3√

2

)≡(φ+

φ0

)∼ (1, 2)1/2, (3.64)

48

where in the last equality we have denoted the quantum numbers of the linear multiplet.Note that in the linear representation we have included a fourth scalar, the Higgs boson,that completes the linear multiplet. We can construct, out of the complex conjugateanother doublet with opposite hypercharge,

φ ≡ iσ2φ∗ =

(φ∗0−φ−

)∼ (1, 2)−1/2. (3.65)

The mass terms for the gauge bosons and fermions are derived in the SM from the gaugeinvariant scalar Lagrangian,

LSM = Lkin + |Dµφ|2−λ[(φ†φ)− v2

]2

−[qiLφV

†ijλ

uju

jR + qiLφλ

iddiR + liLφλ

ieeiR + h.c.

]. (3.66)

The Higgs does not really play any role in the mass generation for the gauge bosonsor fermions (as we discussed in the previous sections, the Goldstones are enough to dothat), however it does play an important role in making the model consistent at highenergies. In particular, the scalar kinetic term includes couplings between the Higgs field hand the gauge bosons and therefore it contributes to longitudinal gauge boson scattering.These contributions are represented in Fig. 3.2. The gauge couplings to the Higgs are not

W +W +

W − W −

H

W +W +

W − W −

H

Figure 3.2: Diagrams contributing to longitudinal W scattering from the exchange of theHiggs in the s and t channels, respectively.

momentum dependent but are proportional to the corresponding gauge boson mass. Thus,we get a contribution that goes lie ∼ E2 at large energies (∼ E4 from the longitudinalpolarizations times ∼ E−2 from the propagator). Assuming 2m2

W s, t and replacingεµL ≈ kµ/m, the full amplitude reads,

M =(igm2

W )2

4m4W

[s2

s−m2H

+t2

t−m2H

]= − g2

4m2W

[s2

s−m2H

+1

t−m2H

]. (3.67)

Adding this contribution to the one we obtained for the higgless scattering, we get

M(W+LW

−L → W+

LW−L ) =

g2

4m2W

[s+ t− s2

s−m2H

− t2

t−m2H

]= − g2

4m2W

m2H

[s

s−m2H

+t

t−m2H

]. (3.68)

49

In particular, the full amplitude no longer grows at high energies. The contribution oforder ∼ E2 cancels exactly between the gauge and Higgs terms, thus the Higgs unitarizesexactly longitudinal gauge boson scattering. Of course it can only unitarize it at energiesof the order of the Higgs mass itself. Unitarity therefore still puts an upper bound on theHiggs boson mass. In particular, inserting the amplitude we have obtained in the s-wavepartial amplitude and taking the large s limit, we get

a0 ∼g2

32π

m2H

m2W

. (3.69)

Requiring Re a0 ≤ 1/2 we obtain

mH ≤4√π

gmW ≈ 11mW ≈ 880 GeV. (3.70)

3.3.2 Constraints on the Higgs mass

As we have seen in the previous section, although the Higgs mass is a free parameter inthe SM,

m2H = 4λv2, (3.71)

it is constrained by unitarity. In fact there are other constraints on the Higgs mass thatwe will discuss in this section.

Strong coupling

A simple constraint on the Higgs mass is simply strong coupling. The heavier the Higgs is,the larger λ has to be and therefore the scalar self-interactions become stronger. This istypical of any spontaneous symmetry breaking sector. The closer to the unitarity cut-offthe scalar sector is, the more strongly coupled it has to be, since perturbative unitarity issaturated and beyond that point, unitarity has to be satisfied by higher order terms. Also,a heavier Higgs has larger and larger decay width,

Γ(h→ W+W−) ≈ 1

32π2

m3h

v2. (3.72)

For mh & 1 TeV, the width becomes of the same order as the mass itself and the verynotion of the Higgs as a particle does not make much sense.

Triviality

At the quantum level, the coefficients of the Higgs potential

V (h) = −1

2µ2h2 +

1

4λh4 (3.73)

50

change with energy. The RGE for the Higgs quartic reads, at one loop,

16π2 dλ

d lnQ= 24λ2 − (3g′ 2 + 9g2 − 12λ2

t )λ+3

8g′ 4 +

3

4g′ 2g2 +

9

8g4 − 6λ4

t + . . . . (3.74)

In the limit of large Higgs mass (large λ), the first term dominates, making the Higgs massa growing function of Q. The solution to the λ dominated RGE is

λ(Q) =m2H

4v2 − (3/2π2)m2H ln(Q/

√2v)

, (3.75)

which presents a Landau pole at

Q =√

2v exp8π2v2

3m2H

. (3.76)

Some form of new physics has to enter at energies smaller than this value in order toprevent the theory to explode. This gives a cut-off scale for the SM,

Λtriviality =√

2v exp8πv2

3m2H

. (3.77)

Equivalently, for a fixed value of the SM cut-off, this gives an upper bound on the Higgsmass. In particular, we cannot take the limit Λ → ∞, since that leads to λ = 0 (trivialtheory) for which no EWSB can occur.

Stability

Let us now consider the small Higgs mass limit. In that case it is not the quartic itself butthe top Yukawa that dominates the RGE for λ. The negative sign in that case makes λ adecreasing function of Q. The energy dependence of the top Yukawa coupling is given bythe RGE

16π2 dλtd lnQ

=9

2λ3t + . . . . (3.78)

The solution of the system of two RGEs gives,

λ2t (Q) =

λ2t,0

1− (9/16π2)λ2t,0 ln(Q/Q0)

, (3.79)

λ(Q) = λ0 −(3/8π2)λ4

t,0 ln(Q/Q0)

1− (9/16π2)λ2t,0 ln(Q/Q0)

. (3.80)

At large energies the top Yukawa drives λ to negative values. Again, some new physicsmust enter before λ = 0 to avoid the potential to be unbounded from below (in practice onecould allow the potential to be unbounded from below provided we live in a long enoughlived local minimum of the potential). This implies another cut-off for the SM,

Λstab. =√

2v exp4π2m2

H

6λ4tv

2. (3.81)

51

Now for a fixed value of the cut-off, the Higgs mass cannot be smaller than some value asgiven by the previous equation.

These bounds have been summarized in Fig. 3.3 (taken from [15])

100

200

300

400

500

600

1 10 102

Hig

gs m

ass

(GeV

)

Λ (TeV)

Vacuum Stability

Triviality

Electroweak

Standard Model

Figure 3.3: Bounds on the Higgs mass from triviality and stability. Figure taken from [15].

In the next few lectures, we will see that the presence of these cut-offs in the SM makesthe SM realization of EWSB quite unnatural. This unnaturalness problem of the SMwill motivate us to continue our quest in the search of alternative realizations of EWSB.Another motivation to study other realizations of EWSB beyond the SM is the fact thatthere is no dynamical mechanism for EWSB in the SM. The EW symmetry is brokenbecause we put, by hand, a negative mass squared in the Higgs potential. Before jumpinginto other models of EWSB we will however review the amazing consistency of the SM

52

with current experimental data and we will unveil the second crucial role the Higgs bosonplays: consistency with electroweak precision tests. Any of the models of new physics thatwe will study along the course, apart from improving on the unnaturalness problem of theSM and, if possible, provide a dynamical mechanism for EWSB, they should also fulfill theimportant roles that the Higgs plays in the SM, restoration of unitarity and compatibilitywith EWPT.

3.4 The hierarchy problem

The hierarchy problem is based on the observation that theories with light fundamentalscalars are not natural. It is usually stated as “Quantum corrections to the Higgs mass arequadratically divergent”, or in equations,

δm2H =

[1

4(9g2 + 2g′ 2)− 6y2

t + 6λ] Λ2

32π2, (3.82)

where yt is the top Yukawa coupling (we have neglected the contribution from lighterfermions) and we have assumed a common cut-off Λ to regulate the momentum integrals.Thus, the natural value of the Higgs mass is the cut-off of the theory. In order for theEWSB scale v to be much lower than the cut-off we need a delicate cancellation betweenthis quantum corrections and the bare parameters of the model. It is natural to assumethat at least the Planck mass, which is the scale at which gravity becomes strong andquantum gravity effects are relevant, is a cut-off of the SM as we know it. If there isnothing but the SM between the scale of EWSB and the Planck mass, the bare parametersof the Higgs potential have to be adjusted to cancel the quantum corrections to one partin ∼ 1015! This is definitely not a satisfactory feature of the mechanism of EWSB in theSM.

We would like to stress that the hierarchy problem is a real problem that can be statedwithout any mention to cut-off or regularization dependence. One could naively arguethat the quadratic divergence is an artificial result or the regularization used and thatone can simply avoid it (and therefore prove that it is a bogus problem) by going todimensional regularization, in no power divergences can appear. The precise statement ofthe hierarchy problem is that the mass of a fundamental scalar is quadratically sensitive tohigh energy thresholds and therefore, it is unnatural for a scalar to be much lighter thanany other scale in the model. This is a statement about renormalized quantities and hasnothing to do with the regularization method used. This feature is actually an example ofa more general property, first discussed by ’t Hooft, which says that a small parameter istechnically natural if the system acquires a larger symmetry when such parameter is setto zero. An example is a massive fermion

L = ψiDψ +mψψ = ψLiDψL + ψRiDψR +m(ψLψR + ψRψL). (3.83)

This theory has a global U(1) symmetry ψ → eiαψ. However, if we take the mass to zero,there is a larger, chiral symmetry ψ → eiγ

5αψ. Thus, the fermion mass can be naturally

53

much lighter than other scales in the theory. Technically what happens is that correctionsto the fermion mass, that violate the chiral symmetry, have to proceed through a couplingthat breaks the chiral symmetry. But the fermion mass itself is the only parameter thatbreaks the symmetry. Thus, quantum corrections to the fermion mass are proportional tothe fermion mass itself and remain small if we started with a small tree level value. Anotherexample is gauge boson masses, that have an associated gauge symmetry when the massis taken to zero. There is however in general no symmetry associated to a massless scalarwith non-derivative couplings and therefore light scalars are unnatural.

Let us study these features in a toy model. The model under consideration is a Yukawatype theory with two scalars and a fermion. One of the scalars is massless at tree level,while the other fields will have arbitrary masses. The Lagrangian is given by

L =1

2(∂µφ)2 +

1

2(∂µΦ)2 + ψi∂ψ−

1

2m2φφ

2− 1

2m2

ΦΦ2−mψψψ−1

4λφ2Φ2− yφφψψ− yΦΦψψ.

(3.84)In principle we could have written more terms in the Lagrangian but we just want to seehow a heavy threshold affects differently scalar and fermion masses. Doing the calculationin dimensional regularization and the MS renormalization scheme we obtain at one loop,for the scalar mass,

δm2φ =

y2φ

4π2m2ψ

[1− 2 ln

m2ψ

µ2+O(m2

φ/m2ψ)]− λ

32π2m2

Φ

[1− ln

m2Φ

µ2

]. (3.85)

The correction to the fermion self-energy gives a fermion mass

δmψ = mψ

[5

4− 3

2lnm2

Φ

µ2+O(m2

ψ/m2Φ)]

+ (Φ→ φ). (3.86)

Exercise: Compute the corrections in Eqs.(3.85) and (3.86). Hint: Computethe corresponding one loop self-energies in dimensional regularization; perform theFeynman parameter integrals; evaluate the self-energies at p2 = m2 and expandthe result in inverse powers of large masses. The scalar self-energy has two typesof contributions, one with the other scalar and one with the fermion running in theloop; the fermion has only one type of contribution.

We see that, if ψ or Φ are heavy, the φ mass is quadratically sensitive to these large scales,even if we set mφ = 0 at tree level. The fermion on the other hand, even if the scalarsare extremely heavy, is only logarithmically sensitive to the heavy scale. Furthermore, thecorrections to the fermion mass are proportional to the fermion mass itself (multiplicativerenormalization) and therefore if we start with a light fermion, it will remain light afterradiative corrections have been taken into account. Thus, fermion masses are stable againstradiative corrections whereas scalar masses are unstable, quadratically sensitive to higherenergy thresholds.

There is a related difference between scalars versus fermions or gauge bosons, whichhas to do with the number of degrees of freedom for massless versus massive particles with

54

or without spin. This is summarized in Table 3.2 in which we see that massive fermions orgauge bosons have more degrees of freedom that the corresponding massless fields. Theseextra degrees of freedom have to come from somewhere to give mass to the correspondingfield. This feature reflects the enhanced symmetry in the massless case. Scalars on theother hand have the same number of degrees of freedom independently of their mass. Thus,it is not much different to have massless or massive scalar, a footprint of their instabilityagainst radiative corrections.

Table 3.2: Number of degrees of freedom for massive versus massless particles of differentspins and associated symmetry for the massless case

spin massive massless symmetry

0 1 1 -12

4 2 chiral1 3 2 gauge

The hierarchy problem is a serious issue. One could try to argue that the SM being arenormalizable theory, it does not need to have a UV cut-off. It is not easy to reconcilethat with the existance of the Planck mass, at which gravity becomes strongly coupled buteven without resorting to gravity, we have seen that essentially for any given Higgs massthere is a UV cut-off due to triviality of stability. Thus, unless we are willing to assumea extremelly fine-tuned realization of EWSB, we should expect some new physics not farfrom the EW scale stabilizing the Higgs mass (or replacing the Higgs altogether). Beforeconsidering this possibility, it is useful to analyze the degree of compatibility of the SMwith current experimental data. This will make explicit the second crucial role that theHiggs boson plays in the SM, namely compatibility with EWPT, and also with show howmodels of new physics can be constrained by current experimental data.

3.5 The Standard Model as an Effective Theory

In the previous lecture we saw that the EWSB sector of the SM is not fully satisfactory atthe theoretical level and that one can confidently expect new physics to show up aroundthe TeV scale to stabilize the EWSB scale. However, it is also true that, as a description ofcurrent experimental data, the SM is extremelly successful. In this lecture we will discusshow successful it is in a quantitative way and, at the same time, we will discuss how to testmodels of new physics against current experimental data. This is of course not a courseon precision tests of the SM and therefore we will not go into the details of all observablesthat have been experimentally tested. We will be content with having a clear idea of whichare the most constrained observables and a simple way of estimating bounds on the scaleof new physics from electroweak precision tests (EWPT).

55

3.5.1 Precision tests of the Standard Model

The main experimental tests of the SM come from low energy (mainly leptons off nucleons)scattering data, precision CP, P or flavour violating experiments, e+e− scattering at andaround the Z pole (LEP and SLC), e+e− scattering above the Z pole, up to energies ∼ 200GeV (LEP2) and Tevatron data (pp collider at 2 TeV center of mass energy).

The gauge sector of the SM has three independent parameters, g, g′ and v. Includingthe scalar sector, we only have one further parameter, the quartic coupling λ, that canbe exchanged by the Higgs mass mH . Finally, adding the fermion sector, we have alarge number of new parameters, the Yukawa couplings. However, only the top Yukawacoupling is reasonably large, all the other ones (or the relevant combinations appearing inthe CKM matrix) are small as corresponds to their small masses and mixing angles. Thus,excluding flavour physics and very small parameters, the SM has five independent relevantparameters,

g, g′, v, yt, λ, (3.87)

in terms of which, we can obtain definite predictions in the SM for any observable. It is cus-tomary to exchange these parameters with the observables that have been experimentallymeasured with the best precision, usually,

MZ , GF , αem,mt,mH , (3.88)

which are respectively the Z mass, the Fermi Constant as measured in muon decay, theelectromagnetic coupling constant, the top mass and the Higgs mass. Of course the Higgsmass has not been measured yet, but the fact that we have not found it yet means thatit is heavy, which in turns implies that λ is large. Thus, we cannot neglect it as it cangive important corrections to EW observables at the quantum level (this is true thanks tothe extreme precision of the experimental data). We can then express, in the context ofthe SM, all other observables in terms of these five and compare these predictions withexperimental data. The SM predictions have been computed to at least one loop (and insome observables to more than two loops) and a comparison with experimental data resultsin an excellent fit (see for instance [16]). The result of the fit can be summarized in thefollowing points.

• The observables at the Z pole are measured typically with . 10−3 relative precision.Thus, the quantum structure of the SM (∼ 1/16π2 ∼ 10−2) is fully probed by theseexperimental data.

• Some of the observables that show some discrepancies are the total hadronic crosssection at the Z pole, σh (pull=+2), and the forward-backwards asymmetry ofe+e− → bb at the Z pole, AbFB (pull=-2.7). This latter observable is somewhatproblematic because the other observable that is also sensitive to the b couplingsto the Z, Rb ≡ Γ(bb)/Γ(had), is in very good agreement with the SM prediction(pull=+0.8).

56

• The dependence of some observables on mt is quadratic, therefore the top mass isstrongly constrained indirectly by EWPT and the result agrees very well with directmeasurements at Tevatron (although the latest measurements of mt and mW at theTevatron start to show some tension). The dependence on the Higgs mass is onlylogarithmic, thus the sensitivity much weaker. Nevertheless, experimental data is soprecise that a bound on the SM Higgs mass can be put

46 GeV ≤ mH ≤ 154 GeV, (from EWPT at 90% C.L. in the SM). (3.89)

It is important however to emphasize that this bound only applies to the SM Higgs.One could have a heavier Higgs whose effects on EWP observables are compensatedby some new physics.

Note: Alternatively this constraint can be seen as the second crucial rolethe Higgs plays in the SM, making it compatible with EWPT. The Higgsnot only mediates longitudinal gauge boson gauge boson scattering to renderit unitary but it also mediates quantum corrections to EW observables thatare compatible with experiment is the Higgs is light enough (and no otherphysics beyond the SM is present).

• LEP2 data is far less precise (∼ 10−2 relative precision) than the Z pole data butthis is compensated by the gain in energy,

√s ≤ 209 GeV and LEP2 data is very

important as a test of new physics.

Another important piece of information is the unsuccessful Higgs search at LEP, that hasput a limit on a SM Higgs mass [17]

mH ≥ 114.4 GeV (for a SM Higgs at 95% C.L. from direct searches). (3.90)

Non-SM Higgses can be lighter than that limit, see [18] for a recent review, although thereis a model-independent limit mH ≥ 82 GeV.

3.5.2 Constraints on new physics

Having seen how well the SM compares with experimental data, the next step is to learnhow to perform a similar comparison of models of new physics. Since the SM agrees so wellwith experimental data, it is natural to assume that the new physics has a high enoughtypical scale for its effects to be a small correction to SM physics. In principle, one cantake their favourite model of new physics, with their new parameters beyond the ones ofthe SM, and compute all precision observables in terms of these parameters. A fit to theEWP observables will then decide whether the particular model is excluded or compatiblewith current data. A more interesting way of doing this study, however, is to use EffectiveLagrangians (EL) as a tool to parametrize physics beyond the SM. In this way, we can do

57

the analysis for a completely general extension of the SM (with some mild assumptions)and perform the comparison with experiment. This will put bounds on the coefficients ofthe different operators appearing in the EL. We then only need to compute the values ofsuch coefficients in our particular model and will automatically know the constraints onour model.

Effective theories are based on the idea that, at some particular energy, at which we aredoing an observation (experiment), the details at much higher energies (shorter distances)are irrelevant for the description of the observation. Or said in a different way, when doingexperiment with a typical wave-length, the details of the universe at much shorter distancesare averaged out and can be simply parametrized by a number of unknown coefficients.

In our particular case, the idea is that, provided we pick up the right degrees of free-dom and symmetries relevant at the energies at which we are doing experiment, we canparametrize the unknown short-distance physics by an infinite expansion of operators builtwith the chosen degrees of freedom and preserving the relevant symmetries,

Leff =∑d

Ld =∑d

∑i

α(d)i O

(d)i , (3.91)

where Ld is the sum of all operators with mass dimension d, [O(d)i ] = d. In D space-time

dimensions the Lagrangian has mass dimension D. Thus, the coefficients of the expansionhave [α(d)] = D − d. In particular, for D = 4, we have that the coefficients have massdimension [α(d)] = 4− d, which is negative for all operators of dimension higher than four.

Note: The mass dimension of the Lagrangian in a D−dimensional spacetime is D, since the action S =

∫dDxL is dimensionless. In an field theory

in D dimensions, the mass dimension of the different fields can be foundby looking at their kinetic terms, taking into account that a derivative hasmass dimension 1 for arbitrary D. Thus, scalars and gauge bosons have massdimension [φ] = [AM ] = D/2 − 1 whereas fermions have mass dimension[ψ] = (D − 1)/2. In particular, taking D = 4 we recover the usual massdimension 1 for bosons and 3/2 for fermions. The mass dimension for the restof the parameters in the Lagrangian can be obtained as outlined above.

This EL will parametrize physics up to some scale Λ at which a new threshold appearsand our model has to be completed by a more fundamental theory that incorporates newstates. Although it looks hopeless to try to do any physics with such a Lagrangian, thathas an infinite number of terms, the reality is luckly quite the opposite. Recall that thecoefficient of dimension higher than D have negative mass dimension and are thereforesuppressed by the cut-off scale Λ. For instance in four dimensions we have

α(d>4) =a(d)

Λd−4, (3.92)

with a(d) a dimensionless constant which is expected to vary from ∼ 4π for strongly coupledtheories to ∼ 1 for weakly coupled theories that couple at tree level with the SM or even

58

∼ 1/16π2 if the interaction only proceeds through quantum effects. Thus operators ofhigher dimensions will be suppressed by increasing powers of E/Λ, where E is the energyat which we are doing experiment. The crucial point is that if we are interested in describingthe world around us, we are nevertheless limited by experimental precision and thereforewe do not need theoretical predictions with infinite precision. Thus, as long as we onlyneed some finite precision in our calculations, we can keep a finite number of operatorsin the EL, and throw away all operators with a dimension higher than the critical oneto obtain the required precision. This allows us to cut the sum in d in Eq. (3.91). Thenumber of possible operators of some dimension d, built with a finite number of fields isfinite, although it can be very large if the number of fields or the dimension are large.Another important property is the fact that operators that are related by the classicalequations of motion are redundant [19]. This means that, two operators that are relatedby the classical equation of motion give the same physics, even including quantum effects.Thus, we can reduce the sum over i in Eq. (3.91) by eliminating those operators that areredundant by the classical equations of motion.

Note that an EL is non-renormalizable as we have coefficients with negative mass di-mensions (precisely the virture of EL at low energies). This means that an infinite numberof counterterms will be required to renormalize the theory. This is obviously expectedfrom the fact that our theory breaks down at high energies and furthermore already in-cludes an infinite number of possible operators allowed by the symmetries (and thereforehas an infinite number of counterterms). Our saviour is again the finite precision that werequire on our predictions. Assuming that we only need a finite precision, we only needto keep a finite number of operators and higher order counterterms are irrelevant for theexperimental predictions.

Given the agreement between the SM and experimental data, a very reasonable choiceof relevant degrees of freedom and symmetries are those of the SM. The choice is whetherwe keep the Higgs boson or we work in the non-linear sigma model representation of theSM (known as the electroweak chiral Lagrangian). In both cases it is easy to computethe first few terms of the EL. For instance, assuming the SM with the Higgs boson in thelinear representation, there is only one allowed operator of dimension five, assuming leptonnumber violation [20, 21, 22]

L5 =a(5)

Λεijεkl l

ciRφ

jφllkL + h.c., (3.93)

which gives a Majorana mass to neutrinos. Given the smallness of such masses, it is naturalto assume approximate lepton and baryon number conservation as did W. Buchmuller andD. Wyler when they classified all possible non-redundant operators of dimension six [23].The total number of such operators, up to flavour indices, is 81. One can then takethese 81 operators and study their effects on EWPT. Such an exercise has been recentlyperformed by several groups [24, 25, 26] (in the latter reference they make special emphasisin determining what the minimal set of operators is constrained by EWPT).

In this lecture we will be a bit less ambitious and will consider a simplifying assumptionin order to have a smaller set of parameters. We follow [27] (although we will use a slightly

59

different normalization of the fields) in assuming that new physics is universal in thefollowing sense. We assume that there are some gauge boson fields W I

µ and Bµ to whichthe light fermions couple as they do to the SM gauge bosons,

L = −gW IµJ

µI − g

′BµJµY + . . . . (3.94)

We have defined the fermion currents as

JµI =∑f

ψfLσI

2γµψfL, (3.95)

JµY =∑f

ψfYfγµψf , (3.96)

where the sum runs over all light fermions in the SM. Note that these interpolating gaugefields are not necessarily the SM gauge bosons but can have a component of new physics.The important feature, that is the very definition of these interpolating fields (and of theuniversality of new physics) is that the only gauge interactions (apart from QCD of thelight fermions is the one given in Eq. (3.94). We will also assume that the threshold ofnew physics is far enough above the relevant energies that we can safely expand in powersof energy (or momentum). Finally we will just assume unbroken QED (and in particularelectric charge conservation). With these assumptions all effects of new physics relevantfor EWPT can be encoded in the self-energies of the interpolating fields. In the spirit ofeffective theories we can split these self-energies in two parts, one local tree level correctionfrom new physics and another that contains all loop corrections from the SM fields. Itis the first part that we are considering here (all effects of the SM, including quantumcorrections, will be taken into account in the second calculable part). Furthermore, we cankeep only the terms proportional to ηµν in the two point function, as the terms proportionalto the external momenta pµpν will vanish or be negligible when contracted with conservedcurrents or currents built with light fermions. Thus, we can parametrize the most general,U(1)Q gauge invariant, universal Lagrangian by (we do not explicitly write the vectorindices that are contracted with ηµν)

L = −W+Π+−(p2)W− − 1

2W 3Π33(p2)W 3 −W 3Π3B(p2)B − 1

2BΠBB(p2)B. (3.97)

Furthermore, the assumption that new physics occurs at a high scale allows us to expandthe two point functions in a momentum expansion,

Π(p2) = Π(0) + p2Π′(0) +1

2(p2)2Π′′(0) + . . . , (3.98)

where we have kept only operators of dimension six or lower (recall that the self-energieshave mass dimension 2). Thus, assuming we can keep only up to dimension six operators,we have 3 × 4 = 12 independent coefficients (the two point function and the first twoderivatives at p2 = 0 for each of the four combinations). Not all of those are however

60

independent. First, three of these coefficients can be removed by canonically normalizingthe fields. This corresponds to the determination of g, g′ and v in the SM,

Π′+−(0) = Π′BB(0) = 1,

Π+−(0) = −m2W = −(80.425 GeV)2. (3.99)

The remaining 9 parameters are not yet fully independent. The reason is that we have notyet required that U(1)Q is unbroken (other than electric charge conservation). Imposingconservation of the U(1) group generated by Q = T3 + Y we obtain the following twoconsistency conditions

g′ 2Π33 + g2ΠBB + 2gg′Π3B = 0,

gΠBB + g′Π30 = 0. (3.100)

Exercise: Prove the two consistency conditions in Eq. (3.100). Hint: Assumethe photon and the Z are an arbitrary unitary rotation of W 3 and B; determinethe rotation angle in terms of g and g′ by requiring that the photon couples to Q;then impose that the photon is massless and does not mix with the Z.

We are therefore left with 7 = 12 − 3 − 2 coefficients that parametrize any new universalphysics beyond the SM. A smart choice of these seven independent parameters is given inTable 3.3.

Table 3.3: Coefficients of the most general Lagrangian of new universal physics BSM. Theexpressions below are also valid when the normalization conditions, Eq. (3.99), have notbeen imposed.

Coefficients Dimension-6 operator SU(2)C SU(2)L

S = gg′

Π′3B(0)

Π′+−(0)(φ†σIφ)W I

µνBµν + −

T = Π33(0)−Π+−(0)−Π+−(0)

|φ†Dµφ|2 − −

U =Π′+−(0)−Π′33(0)

Π′+−(0)Dim. 8 − −

V = −Π+−(0)2

(Π′′33(0)−Π′′+−(0)

Π′+−(0)

)Dim. 10 − −

X = −Π+−(0)2Π′+−(0)

Π′′3B(0)√Π′+−(0)Π′BB(0)

Dim. 8 + −

Y = −Π+−(0)2Π′+−(0)

Π′′BB(0)

Π′BB(0)(∂ρBµν)

2 + +

W = −Π+−(0)2Π′+−(0)

Π′′33(0)

Π′+−(0)(DρW

Iµν)

2 + +

61

These coefficients are related to the Peskin-Takeuchi S, T, U paremeters [28] by

S = 4s2W S/αem, T = T /αem, U = −4s2

W U/αem. (3.101)

We have also shown in the table which SU(2)L × U(1)Y−invariant dimension 6 operator(assuming a fundamental Higgs) generates the corresponding operator and whether theypreserve or violate custodial symmetry and SU(2)L symmetry. From the dimension of thedifferent operators we see that there is a hierarchy between the different coefficients,

U ∼ m2W

Λ2T , V ∼ m4

W

Λ4T , X ∼ m2

W

Λ2S, (3.102)

where we have assumed that operators preserving/breaking the same groups of symmetriesare generated at a similar scale. Thus, in models with new universal physics, there are fouroblique parameters that fully parametrize corrections to EWPT,

S, T ,W, Y. (3.103)

In particular, we have that, for universal physics, T is related to the ρ parameter,

ρ = 1 + T , Universal physics. (3.104)

This could be expected from the fact that T is the only of the four parameters that violatescustodial symmetry. A fit to these four coefficients was performed in [27] with the resultshown in Table 3.4. The main conclusion is that these four coefficients are constrainedto be . 10−3. It is interesting to note that LEP data alone does not allow to constraintindependently the four parameters, but only three combinations of them. It is LEP2 (lessprecise but higher energies) data that allows for an independent determination of the fourparameters. Also note that the result of the fit depends on the Higgs mass. The reasonis the logarithmic dependence of the different coefficients (mainly S and T ) on the Higgsmass (it is only the SM part that depends on the Higgs mass, but upon comparison withexperiment, the constraint on new physics is modified accordingly). It is interesting to

Table 3.4: Fit to universal corrections to the SM

Fit 103S 103T 103W 103Y115 GeV Higgs 0.0± 1.3 0.1± 0.9 0.1± 1.2 −0.4± 0.8800 GeV Higgs −0.9± 1.3 2.1± 1.0 0.0± 1.2 −0.2± 0.8

compute the leading dependence on mt and mH of the relevant self-energies that affectour four parameters. The leading top mass dependence is quadratic, and represents a niceexample of non-applicability of the decoupling theorem. The dependence on the Higgsmass is only logarithmic as implied by the screening theorem [29],

S =GFm

2W

12√

2π2ln

(m2h

m2h ref

)+ . . . , T = −3GFm

2W

4√

2π2

g′ 2

g2ln

(m2h

m2h ref

)+ . . . . (3.105)

62

Let us start with the mt dependence of the ρ parameter. This correction has nothing todo with the gauge symmetry, but rather with the violation of custodial symmetry dueto the difference between the top and bottom Yukawa couplings. In particular, it is dueto the large value of the top Yukawa (and the small value of the bottom one). We cantherefore do the calculation in the limit of large top mass. In that limit, the top doesnot decouple, because we are taking a dimensionless coupling very large (and the resultingeffective theory is non-renormalizable). In this limit, it is the coupling to the Goldstonebosons that grows with the top Yukawa and therefore we can do the calculation in thegauge-less limit, given by the Lagrangian,

Lgauge−less = ψiDψ + |Dµφ|2 − V (φ)− (λtqLφtR + h.c.), (3.106)

where the gauge fields are not dynamical but just considered external classical sources.Recall Eq. (3.32) for the kinetic term of the scalar, including the Goldstone bosons,

LKin(πi) = Z(+)2 |∂µπ+ − g√

2vW+

µ |2 +1

2Z

(3)2 |∂µπ3 − g√

2cWvZµ|2, (3.107)

where we have now included arbitrary wave-function renormalization constants Z(+)2 and

Z(3)2 , for π+ and π3, respectively, which will be 1 at tree level but will receive corrections at

higher orders in perturbation theory, while preserving the same covariant structure of thekinetic terms. Expanding the squares in (3.107) we get an all order result, in the gauge-lesslimit for the ρ parameter,

ρ =m2W

m2Zc

2W

=Z

(+)2

Z(3)2

, (3.108)

in terms of the ratio of the wave-function renormalization of the two eaten Goldstones.Thus, we only need to compute the wave-function renormalization of the Goldstone bosonswhich, to leading order, is given by the p2 term in the expansion around p0 = 0 (weare neglecting masses of order mW against mt) of the diagrams shown in Fig. 3.4. The

t t

b t

π+ π3

Figure 3.4: Goldstone boson self energies

63

self-energy for the charged Goldstone bosons reads, in dimensional regularization,

M2+(p2) =

3

16π2

(λ2

t + λ2b)[− a(mb)− a(mt) + (p2 −m2

b −m2t )B0(p;mb,mt)

]+ 4λtλbmtmbB0(p;mb,mt)

(3.109)

=3

16π2v2m2t (p

2 −m2t )B0(p; 0,mt) + . . . =

3

16π2v2m2t

(∆ +

1

2− ln

m2t

µ2

)p2 + . . . ,

where we have used the corresponding Passarino-Veltman functions, which are describedat the end of this lecture. In the second equality we have neglected terms proportionalto mb whereas in the last one we have expanded in powers of p2 around p2 = 0 andretained the term proportional to p2, which is the one that will determine the wave-functionrenormalization. We have defined ∆ ≡ 1/ε − γE + ln 4π. The self-energy of the neutralGoldstone is

M23(p2) = − 3

16π2λ2t

[2a(mt)− p2B0(p;mt,mt)

]+ (mt, λt → mb, λb)

=3

16π2v2m2tp

2B0(p;mt,mt) + . . . =3

16π2v2m2tp

2

(∆− ln

m2t

µ2

)+ . . . .(3.110)

Again in the first equality we have neglected terms proportional to the bottom mass (andthose that do not contribute to the wave-fucntion renormalization of the Goldstone boson)and in the second one we have kept the term that renormalizes the wave function. Thus,we have for the ρ parameter

ρ =Z

(+)2

Z(3)2

≈1 + 3

16π2λ2t

(∆ + 1

2− ln

m2t

µ2

)1 + 3

16π2λ2t

(∆− ln

m2t

µ2

) ≈ 1 +3

32π2λ2t = 1 +

3

8

GF√2

m2t

π2. (3.111)

Using similar methods it can be shown that another observable receiving quadratic mt

corrections is the coupling of the LH bottom to the Z. If we parametrize this coupling by

− g

2cW[1− 2

3s2W + δgb]bLγ

µbLZµ, (3.112)

the leading correction to δgb is (see for instance Pokorski’s text book [1]),

δgb = −GFm2t

4π2√

2. (3.113)

Similarly, the dependence on the Higgs mass, can be obtained by computing the W 3 − Bself-energy, with a charged Goldstone running inside the loop for the S parameter andthe self-energy of the charged Goldstones with a hypercharge gauge boson (and a chargedGoldstone) running in the loop for the T parameter (note that we need a hyperchargegauge boson running in the loop as this is the only source of custodial violation in thegauge sector). For details of these calculations, see for instance [30].

64

3.5.3 Contribution of dimension 6 operators

Let us compute the contribution to the four relevant oblique parameters in the case ofuniversal physics from the corresponding dimension 6 operators.

• T

LT =αTΛ2|φ†Dµφ|2 =

αTΛ2

∣∣∣∣∣v2(0 1

)( g√2W+µ

−g2W 3µ + g′

2Bµ

)∣∣∣∣∣2

=αTΛ2

v4

4(gW 3

µ − g′Bµ)2 =αTΛ2

v4

4[g2W 3

µWµ3 + g′ 2BµB

µ − 2gg′W 3µB

µ].(3.114)

Thus, we have

T =Π33 − Π+−

m2W

= −αT2

v4g2

Λ2m2W

= −αTv2

Λ2. (3.115)

• S

LS =αSΛ2φ†σIφW I

µνBµν = −αS

v2

Λ2W 3µνB

µν

= −2αSv2

Λ2∂µW

3ν (∂µBν − ∂νBµ) = −2αS

v2

Λ2W 3ν p

2ηµνBµ + . . . (3.116)

Thus

S =g

g′Π′3B = 2

g

g′αS

v2

Λ2, (3.117)

• Y

LY =αYΛ2

(∂ρBµν)2 = −2

αYΛ2∂µBν∂

2Bµν

= 2αYΛ2

(p2)2BνηµνBµ + · · · = −1

2

(p2)2

2

(−8

αYΛ2

)BµB

µ + . . . (3.118)

Thus,

Y =m2W

2Π′′BB = −4αY

m2W

Λ2. (3.119)

• W (similar to Y )

LW =αWΛ2

(DρWIµν)

2 = −2αWΛ2

DµWIνD

2W I µν + . . .

= 2αWΛ2

(p2)2W Iν η

µνW Iµ + · · · = −1

2

(p2)2

2

(−8

αWΛ2

)W IµW

I µ + . . .(3.120)

Thus,

W =m2W

2Π′′33 = −4αW

m2W

Λ2. (3.121)

65

3.5.4 A Simple Example

As a simple example of universal physics let us consider the effect of a heavy copy of thehypercharge gauge boson,

L = LSM + LB′ , (3.122)

where the relevant parts of the two Lagrangians can be schematically written as

LSM = −1

2W µ

3 (p2 −m2W )W3µ − t0m2

WWµ3 Bµ

− 1

2Bµ(p2 − t20m2

W )Bµ + gJ3µL W3µ + g′JµYBµ, (3.123)

LB′ = −1

2B′µ(p2 −M2)B′µ + g′JµYB

′µ. (3.124)

we have defined t0 = s0/c0 = g′/g and the standard fermion currents JµY =∑

ψ ψY γµψ,

J i µL =∑

ψ ψTiLγ

µψ. Let us proceed now along the standard route by integrating out theheavy boson. This can be done at tree level by introducing the solution of the classicalequation of motion back in the Lagrangian and expanding in inverse powers of the heavymass. The classical equation of motion for B′ reads

(M2 − p2)B′µ + g′JYµ = 0⇒ B′ = −g′ JµY

M2+O

(p2

M2

). (3.125)

Inserting it back in the Lagrangian and retaining only up to dimension 6 operators we get

LB′ = −1

2g′ 2

JYµ (p2 −M2)JµYM4

− g′JµY g′ J

Yµ

M2+ . . . = −1

2g′ 2

JYµ JµY

M2+ . . . . (3.126)

Thus, integrating out B′ leads to four-fermion interactions. This four fermion interaction,however can be written in terms of purely oblique corrections. To see that, we can makeuse of the classical equations of motion for B,

g′JµY = (p2 − t20m2W )Bµ + t0m

2WW

µ3 . (3.127)

Using this, we can write the contact interaction as

− 1

2g′ 2

J2Y

M2= −1

2B

(p2 − t20m2W )2

M2B − 1

2W3

t20m4W

M2W3 − t0m2

WW3(p2 − t20m2

W )

M2B, (3.128)

Adding this term to the SM Lagrangian we get

L6 = −1

2W µ

3

[p2 −m2

W

(1− t20

m2W

M2

)]W3µ − t0m2

WWµ3

[ p2

M2+ 1− t20

m2W

M2

]Bµ

−1

2Bµ[p2(1− 2t20

m2W

M2

)− t20m2

W

(1− t20

m2W

M2

)+

p4

M2

]Bµ

+gJ3µL W3µ + g′JµYBµ +O(p6) +O(1/M4). (3.129)

66

The mass Lagrangian for the neutral gauge bosons reads,

Lmass6 =

(1− t20

m2W

M2

)LmassSM , (3.130)

which guarantees that it is diagonalized in exactly the same way as in the SM,

Z = c0W3 − s0B, A = s0W3 + c0B, (3.131)

with a massless photon and a modified Z mass,

m2Z =

(1− t20

m2W

M2

)m2Z SM =

(1− t20

m2W

M2

)m2W

c20

. (3.132)

Since the rotation is the same and the couplings to fermions are also the same, the photon(the massless combination) couples to Q = T 3

L + Y as it should. In particular the W massis not modified, so we have

ρ = 1 + αT =m2W

m2Zc

20

= 1 + t20m2W

M2+ . . . , (3.133)

which is a direct contribution to the T parameter.Note that with the use of the equation of motion for B we have been able to write the

original effective Lagrangian with four-fermion interactions in a purely oblique form, i.e.corrections only to the gauge boson self-energies. The reason is that the model of newphysics we have considered fall into the category of universal new physics. This is easy tosee by noting that the fermion gauge couplings can be written in the form,

∆L = gJ3µL W3µ + g′JµY Bµ, (3.134)

where we have defined the interpolating fields W3 ≡ W3, B ≡ B + B′. Following therecipe for models with universal new physics, we know that all the relevant correctionscome then from the self energies of these interpolating fields. Thus, we need to computethe effective Lagrangian for W3 and B, instead of the one for the original fields. In factthis is the Lagrangian we have computed (after the use of equation of motions for B totranform the four-fermion interactions into purely oblique corrections). A simpler way toobtain the same lagrangian, in the case of universal physics, is to integrate out not themassive field but the combination of the massive field and the SM one that doesn’t coupleto fermions (this is usually called the holographic approached, after its use in models withextra dimensions with a holographic motivation). Let’s do it that way. The full Lagrangianfor the neutral gauge bosons can be written, in obvious notation,

L = −1

2W3Π33W3 −W3Π3BB −

1

2BΠBBB −

1

2B′ΠB′B′B

′ + gJ3W3 + gJy(B +B′)

= −1

2W3Π33W3 + gJ3W3 + gJyB+ −

1

2W3Π3BB+ −

1

8B+( + )B+

−1

8B−( + )B− −

1

4B+( − )B− −

1

2W3Π3BB−, (3.135)

67

where we have defined B± ≡ B±B′ and ( ± ) ≡ ΠBB ±ΠB′B′ . The classical EoM for B−can be written as

B− = −( − )B+ + 2Π3BW3

( + ). (3.136)

Inserting it back in the Lagrangian we get

L = −1

2W3Π33W3 −W3Π3BB −

1

2BΠBBB −

1

2B′ΠB′B′B

′ + gJ3W3 + gJy(B +B′)

= −1

2W3Π33W3 + gJ3W3 + gJyB+ −

1

2W3Π3BB+ −

1

8B+( + )B+

1

8

[( − )B+ + 2Π3BW3]2

( + ). (3.137)

This effective Lagrangian, when expanded in powers of momenta, gives exactly Eq. (3.129).Such Lagrangian can be written in the standard form of universal new physics, Eq. (3.98)with

Π33 = −m2W

(1− t20

m2W

M2

)+ p2 + . . . , (3.138)

ΠBB = −m2W t

20

(1− t20

m2W

M2

)+ p2

(1− 2t20

m2W

M2

)+p4

2

2

M2+ . . . , (3.139)

Π3B = m2W t0

(1− t20

m2W

M2

)+ p2t0

m2W

M2+ . . . . (3.140)

We have not modified the charged current sector, so the two normalization conditionsin Eq. (3.99) involving the charged sector are automatically satisfied. The one involvingB is not, however and in order to get Π′

BB(0) = 1 we need to perform the following

renormalization of the B field,

B → NBB ≡(

1 + t20m2W

M2

)B. (3.141)

In particular this means that g′ → NBg′, ΠBB → N2

BΠBB and Π3B → NBΠ3B (but notethat in our expressions we still have t0 = g′/g). In order to avoid confusion, we will writeexplicitly the factors of NB and the two point functions refer to the ones we have written.With these new redefined two point functions, it is trivial to check that Eqs. (3.100), whichstates the preservation of QED, are satisfied. Also, we can now compute all the relevantoblique parameters. In particular we have,

T =Π33(0)− Π+−(0)

m2W

=Π33(0)

m2W

+ 1 = −(

1− t20m2W

M2

)+ 1 = t20

m2W

M2, (3.142)

S =g

NBg′NBΠ′3B(0) =

g

g′Π′3B(0) =

g

g′t0m2W

M2=m2W

M2, (3.143)

Y =m2W

2N2BΠ′′BB(0) =

m2W

2

(1 + 2t20

m2W

M2

)2

M2=m2W

M2, (3.144)

W =m2W

2Π′′33(0) = 0. (3.145)

68

Note that, in this particular case and at the order we are working, the canonical normaliza-tion of B didn’t actually have any effect. Also note recall T = αT so indeed we reproduceour previous calculation of the T parameter.

3.5.5 Summary

We have seen in this section that the SM is a very good description of the availableexperimental data. We have discussed how to use EWPT to constraint models of newphysics and seen that in general, very small deviations are allowed. This has lead to theso called little hierarchy problem. New physics is required by naturalness to be below∼ 1 TeV in order to stabilize the EWSB scale, however, EWPT push the scale of newphysics closer to the multi-TeV scaler. This means that, either the new physics has somespecial properties that somehow hides their effects at low energies while still being ableto stabilize the EWSB scale or new fine-tuning is reintroduced in the theory. Successfulmodels of new physics typically have such property of hiding large effects. Custodiallypreserving new physics is a very simple example. Theories with a discrete symmetry thatforbids tree-level interations between new physics and the SM particles (like R parity inSUSY or T parity in Little Higgs models) prevents large tree level corrections while stillallowing for a resolution of the hierarchy problem -which is occurs at the quantum level-.Nevertheless, most of models of new physics, still suffer from some amount of fine-tuning,which is typically of the order of ∼ few % (thus the name little hierarchy, as opposed tothe ∼ 10−15 fine-tuning in the SM).

In the following lectures we will overview some ideas on how to solve the hierarchyproblem and discuss their successes and weaknesses.

69

Note: Some properties of Passarino-Veltman functions. We followthe discussion in [31]. The one and two point PV functions can be defined as:

i

16π2A(m1) = µ2ε

∫dnk

(2π)n1

N1

, (3.146)

i

16π2[B0, B

µ, Bµν ](12) = µ2ε

∫dnk

(2π)n[1, kµ, kµkν ]

N1N2

, (3.147)

where n = 4− 2ε and

N1 = k2 −m21 + iε, (3.148)

N2 = (k + p1)2 −m22 + iε. (3.149)

The exact result for A and B0 read, using ∆ ≡ 1ε− γE + ln(4π),

A(m) = m2[∆ + 1− ln(m2/µ2)], (3.150)

and

B0(q2;m1,m2) = ∆− lnm1m2

µ2+ 2 (3.151)

+1

q2

[(m2

2 −m21) ln

m1

m2

+√λ(q2 + iε,m2

1,m22)ArcCosh[

m21 +m2

2 − q2 − iε2m1m2

]],

where we have used the definitions,

λ(a, b, c) ≡ a2 + b2 + c2 − 2ab− 2ac− 2bc, (3.152)

andArcCosh(z) = ln(z +

√z2 − 1). (3.153)

We can also write the higher order two point functions in terms of these ones.For instance

Bµ =pµ12p2

[A(m1)− A(m2) + (m2

2 −m21 − p2)B0(p2;m1,m2)

]. (3.154)

Some important expansions of B0 are

B0(m2,m,M) = 1− lnM2

µ2+

(1

2− ln

M2

m2

)m2

M2+O

(m4

M4

),(3.155)

B0(m2,M,M) = − lnM2

µ2+

m2

6M2+O

(m4

M4

). (3.156)

70

Chapter 4

Electroweak Symmetry Breaking inSupersymmetric Models

Note: Supersymmetric theories are very well covered in many textbooksand review articles. We will follow in the lectures mainly Martin’s Supersym-metry Primer [32]. In fact, we are not going to produce lecture notes for thesupersymmetry lectures. Instead, we will mention what is being explained ineach lecture and where it can be found in Martin’s review. (The referenceswill be to v5 of the review.)

4.1 Motivation for Supersymmetry

Where: Section 1 of the primer

Contrary to scalars with non-derivative interactions, fermion masses are protected bychiral symmetry so that light fermions are natural. One possible way of solving the hi-erarchy problem is to use a symmetry that relates scalars to fermions, so that the chiralprotection is extended to the Higgs boson. Such a possibility is actually highly non-trivialand the resulting symmetry, called supersymmetry, is very constrained in models with anon-trivial S-matrix. Once we have a symmetry that relates scalars to fermions, or moregenerally, bosons to fermions. It is easy to see how loop contributions to the Higgs massreceive contributions from both particles and their superpartners and the two contributionscan exactly cancel. This is a good motivation to investigate further the implications ofsupersymmetry.

71

4.2 Formalism of Supersymmetry: Building Super-

symmetric Lagrangians

After a review of two component Weyl spinor notation for fermions in four dimensions(section 2 of the primer) we go on to build supersymmetric Lagrangians.

4.2.1 The Simplest Supersymmetric Lagrangian: Non-interactingWess-Zumino Model

Where: Section 3.1 of the primer.

We start with a Weyl fermion and a massless complex scalar (2 degrees of freedomeach), assume the simplest possible supersymmetric transformation for the scalar

δφ = εψ, δφ∗ = ε†ψ†, (4.1)

with ε an infinitesimal symmetry parameter, which is a spinor with mass dimension −1/2,and guess what the supersymmetry transformation of the fermion should be so that theLagrangian is invariant up to a total derivative.

We then have to check that the supersymmetry algebra closes, which means that thecommutator of two succesive supersymmetric transformations is a symmetry of the La-grangian. It turn out that the commutator is proportional to the four-momentum up tothe equations of motion for the fermion. That means that the algebra closes on-shell but notoff-shell (which makes sense because off-shell, the fermion has four degrees of freedom andnot the two of an off-shell complex scalar). This is fixed by introducing a non-propagatingauxiliary (F) field (a complex scalar -2 dof- with mass dimension -2) whose supersymmet-ric transformation is proportional to the equations of motion of the fermion. This waythe supersymmetry algebra also closes off-shell. This forms the chiral supermultiplet, thatcontains a massless scalar and a massless fermion.

4.2.2 Non-gauge Interactions of Chiral Multiplets

Where: Section 3.2 of the primer.

The most general renormalizable Lagrangian involving chiral supermultiplets that isinvariant under the supersymmetry transformation (up to total derivatives) can be obtainedin terms of a holomorphic function of the scalar components (non-auxiliary) of the chiralsupermultiplets involved. By holomorphic we mean that it is a function of φi and not ofφ∗ i. This function is called the superpotential,

W = Liφi1

2M ijφiφj +

1

6yijkφiφjφk. (4.2)

72

The superpotential has mass dimension 3 and determines the interactions as follows (afterintegrating out the auxiliary fields)

Lint = −1

2

(W ijψiψj +W ∗

ijψ† iψ† j

)−W iW ∗

i , (4.3)

where

Wij...k ≡δW

δφiδφj . . . δφk. (4.4)

In particular, this determines (part of) the scalar potential.

4.2.3 Lagrangians for Gauge Multiplets

Where: Sections 3.3 and 3.4 of the primer.

A supersymmetric Lagrangian for gauge bosons contains a number of gauge bosonstransforming in the adjoint of some Lie group, a similar number of fermions also trans-forming in the adjoint (2 dof for each type of field per element of the adjoint) plus the samenumber of real scalar auxiliary fields to close the supersymmetry algebra off-shell. Oncewe introduce the auxiliary fields and write reasonable supersymmetric transformations fothe different fields (taking into account reality of the adjoint representation and that gaugeinvariance has to be preserved) it is easy to check the invariance of the kinetic Lagrangianfor the different fields (that involves covariant derivatives, both in the Lagrangian and inthe supersymmetric transformations and therefore includes gauge couplings) and that thesupersymmetry algebra closes.

In order to get gauge interactions for chiral supermultiplets, we transform normalderivatives into covariant derivatives (both in the Lagrangian and the supersymmetrytransformations). This is not enough however to obtain a supersymmetric Lagrangian.The reason is that this only couples the gauge boson members of the vector multipletwith the chiral multiplets but supersymmetry relates the gauge boson to the gauginos andthe Da auxiliary fields. Thus, we need to include direct couplings between those and thefermions and sfermions of the chiral supermultiplet. There are only three gauge invariantpossible couplings, whose coefficients are fixed by the requirement of a supersymmetryinvariant Lagrangian. In the process, an extra term in the supersymmetric transformationof the F auxiliary fields of the chiral supermultiplets involving the gauginos is required.

With that we have all renormalizable interactions compatible with gauge invariance andsupersymmetry. Integrating out the Da auxiliary fields we get a new D−term contributionto the scalar potential which is a quartic term with gauge couplings as coupling constant.

4.2.4 The hierarchy problem in SUSY

Let us see how supersymmetry solves the hierarchy problem. We consider as an example thetop mass contribution to the Higgs mass squared. Supersymmetry requires a superpartner

73

for each SM particle.qL ↔ qL = (tL, bL), tR ↔ tR. (4.5)

Let us consider the following interaction Lagrangian

Lint = −(λF tRφ†qL + h.c.) + λL|φ†qL|2 + λR|tRφ|2. (4.6)

To this Lagrangian we could add (and we will in the next section) other terms that aregauge invariant but supersymmetry breaking

Lsoft = λLR(tRqiLεijφj + h.c.) + q†LqLm

2L + t†RtRm

2R. (4.7)

Let us assume for simplicity that λLR = 0. The one loop correction to the Higgs mass isgiven by

δm2H = −|λF |

22Nc

16π2[Λ2 − 6m2

F logΛ

mF

+ 2m2F ]

−∑s=L,R

λsNc

16π2[−Λ2 + 2m2

s logΛ

ms

] +(λsv)2Nc

16π2[−1 + 2 log

Λ

ms

]

, (4.8)

where mF ≡ λFv/√

2. If we have λL = λR = |λF |2, as required by SUSY, we get an exactcancellation of quadratic divergencies,

δm2H = −2Nc|λF |2

16π2

(m2

L +m2R − 2m2

F ) logΛ

mL

− 6m2F log

mL

mR

+ (m2R + 2m2

F ) logmL

mR

. (4.9)

In fact, if SUSY was exactly unbroken (Lsoft = 0), then we have mL = mR = mF and thewhole contribution cancels exactly

δm2H = 0 (unbroken SUSY). (4.10)

However mL = mR = mF is not allowed phenomenologically but we have seen that we canadd SUSY breaking terms that do not spoil the cancellation of quadratic divergencies (softSUSY breaking terms). If mF mL ∼ mR and defining m2

t≡ (m2

L +m2R)/2, we get

δm2H ≈ −

2Nc

16π2|λt|2m2

t logΛ2

m2t

, (4.11)

so that we see that physically, the quadratic divergence is cut-off by the stop mass.

74

4.2.5 Soft Supersymmetry Breaking Interactions

Where: Section 4 of the primer

Supersymmetry requires degeneracy between members of a supermultiplet. However,we have not observed experimentally the superpartners of the SM particles. This meansthat supersymmetry must be spontaneously broken. We can proceed to study models ofspontaneous supersymmetry breaking or we can parametrize all possible supersymmetrybreaking terms that do not spoil the cancellation of quadratic contributions to the Higgspotential. These are the so-called soft supersymmetry breaking terms, and correspondto operators with mass dimension smaller than four (i.e. involving dimensionful couplingconstants). The soft supersymmetry breaking Lagrangian of a general theory has the form

Lsoft = −(

1

2Maλ

aλa +1

6aijkφiφjφk +

1

2bijφiφj + tiφj

)+ h.c.− (m2)ijφ

∗ jφi, (4.12)

which are enough to give masses to all superpartners. One operator that is formally softbut that might give rise to quadratic divergencies in the Higgs mass is

− 1

2cjki φ

∗ iφjφk + h.c., (4.13)

although in models of spontaneous supersymmetry breaking it is typically negligible small.

4.2.6 The minimal supersymmetric standard model

In the SM we have just one scalar doublet φ, that gives masses to both T 3 = ±1/2components. This is done by coupling one of the fields to φ and one to φ∗. In SUSY however,non-gauge couplings come from the superpotential, which is a holomorphic function. Thatmeans that we can couple the fields to either φ or φ∗, but not to both. We thus need twoscalar doublets in the minimal supersymmetric extension of the SM,

φu =

(H0u

H−u

)∼ (2)−1/2, φd =

(H+d

H0d

)∼ (2)+1/2. (4.14)

This constraint can be also understood from the fact that scalars in SUSY bring alongthe corresponding superpartners, which are Weyl fermions and therefore contribute toanomalies. Adding two doublets with opposite hypercharges provides the simplest way ofcancelling such anomalies. Apart from this subtlety, the MSSM corresponds to the obvioussupersymmetryzation of the SM.

The MSSM superpotential reads

WMSSM = −uyuQHu − dydQHd − eyeLHd + µHuHd. (4.15)

Note in particular that there is not HHf term allowed by gauge invariance and thereforenot quartic Higgs coupling is generated from the superpotential.

75

The soft supersymmetry breaking Lagrangian in the MSSM reads

LMSSMsoft = −1

2(M3gg +M2WW +M1BB + h.c.)

−(˜uauQHu + ˜dadQHd − ˜eaeQHd + h.c.)

−Q†m2QQ− L†m2

LL− ˜um2u˜u† − ˜dm2

d˜d† − ˜em2

e˜e†

−m2HuH

∗uHu −m2

HdH∗dHd − (bHuHd + h.c.). (4.16)

Let us look more in detail at the Higgs potential. We have two scalar (complex)doublets, 8 dof in total. We will break the SU(2)L × U(1)Y gauge symmetry to U(1)Q,eating three Goldstone bosons in the way, thus we are left with 5 real physical scalars inthe spectrum (from the Higgs sector). The Higgs potential can be written as

V = (|µ|2 +m2Hu)(|H0

u|2 + |H+u |2) + (|µ|2 +m2

Hd)(|H0

d |2 + |H−d |2)

+ [b(H+u H

−d −H

0uH

0d) + h.c.] +

1

8(g2 + g′ 2)(|H0

u|2 + |H+u |2 − |H0

d |2 − |H−d |2)2

+1

2g2|H+

u H0 ∗d +H0

uH−∗d |

2. (4.17)

Using SU(2)L rotations we can always fix 〈H+u 〉 = 0. We then have

0 =∂V

∂H+d

= (|µ|2 +M2Hd

)H−d +1

4(g2 + g′ 2)(H0

u|2 − |H0d |2 − |H−d |

2)(−H−d ) +g2

2|H0

u|2H−d

= H−d [(|µ|2 +M2Hd

)− 1

4(g2 + g′ 2)(H0

u|2 − |H0d |2 − |H−d |

2) +g2

2|H0

u|2], (4.18)

so that 〈H−d 〉 = 0 is a minimum of the potential for which QED remains unbroken. Usingthese two assumptions we are left with the following potential

V = (|µ|2 +m2Hu)|H0

u|2 + (|µ|2 +m2Hd

)|H0d |2− [bH0

uH0d + h.c.] +

1

8(g2 + g′ 2)(|H0

u|2− |H0d |2)2.

(4.19)The only complex parameter is b, but we can make it real by absorbing its phase in H0

u orH0d . Thus, we can take b real and positive. But then we have

0 =∂V

∂H0u

= [|µ|2 +m2Hu +

g2 + g′ 2

4(|H0

u|2 − |H0d |2)]H0 ∗

u − bH0d , (4.20)

which implies that the phases of H0u,d are opposite and can therefore be eliminated by a

U(1)Y transformation (recall the two Higgses have opposite hypercharges). Thus we cantake

〈H0u,d〉 = vu,d, (4.21)

real and positive. This means that the Higgs potential is CP conserving and therefore wecan classify the 5 physical scalars according to their CP properties.

76

The positive quartic coupling ensures stability except for the D-flat direction vu = vd.Positivity of the potential along that flat direction requires

2b < 2|µ|2 +m2Hu +m2

Hd. (4.22)

Similarly, requiring that a combination of H0u and H0

d gets a negative mass squared atH0u = H0

d = 0 impliesb2 > (|µ|2 +m2

Hu)(|µ|2 +m2Hd

). (4.23)

In particular, for mHu = mHd the two conditions are incompatible. A particular case isa supersymmetric theory (no SUSY breaking) for with both vanish. Thus, if SUSY isunbroken, EWSB does not occur.

Let us denotetan β ≡ vu

vd. (4.24)

We have

v2u + v2

d = v2 = (174 GeV)2 =2m2

Z

g2 + g′ 2. (4.25)

tan β is currently not fixed by experiment. The different parameters are related by thecondition of minimum of the potential

∂H0uV = 0 ⇒ m2

Hu + |µ|2 − b cot β − m2Z

2cos 2β = 0, (4.26)

∂H0dV = 0 ⇒ m2

Hd+ |µ|2 − b tan β +

m2Z

2cos 2β = 0. (4.27)

In particular we have

sin 2β =2b

m2Hu

+m2Hd

+ 2|µ|2, (4.28)

and

m2Z =

|m2Hd−m2

Hu|√

1− sin2 2β−m2

Hu −m2Hd− 2|µ|2, (4.29)

which exemplify the so-called µ problem, which stands for the fact that all these quantities,|µ|2,m2

Hu,d, b seem to have to be of the same order but the former is SUSY preserving

whereas the latter three are SUSY breaking and therefore they are expected to have verydifferent origins.

We can write the two scalar doublets in terms of mass and CP eigenstates(H0u

H0d

)=

(vuvd

)+

1√2

(cα sα−sα cα

)(h0

H0

)+

i√2

(sβ cβ−cβ sβ

)(G0

A0

), (4.30)

and (H+u

H−∗d

)=

(sβ cβ−cβ sβ

)(G+

H+

), (4.31)

77

where G±,0 are the eaten Goldstones, h0 and H0 are CP even neutral scalars, A0 is a CPodd neutral scalar and H+ is a CP even charge (charge +1) scalar. The resulting masseigenstates are

mG±,0 = 0, (4.32)

m2A0 =

2b

sin 2β= 2|µ|2 +m2

Hu +m2Hd, (4.33)

m2h0,H0 =

1

2(m2

A0 +m2Z ∓

√(m2

A0 −m2Z)2 + 4m2

Zm2A0 sin2 2β), (4.34)

m2H± = m2

A0 +m2W , (4.35)

and the rotation angles are related by

sin 2α

sin 2β= −

m2H0

+m2h0

m2H0−m2

h0

,tan 2α

tan 2β=m2A0 +m2

Z

m2A0 −m2

Z

. (4.36)

From these equations we see that mA0,H0,H± can be arbitrarily large, as they grow withb/ sin 2β. mh0 on the other hand is bounded from above

mh0 ≤ mZ | cos 2β|. (4.37)

This is a tree level result but already shows one of the problems with supersymmetrictheories (or at least minimal ones). The fact that LEP didn’t find the Higgs is, in mostor parameter space, incompatible with this inequality. loop contributions modify thisconstraint, making it still compatible with LEP bounds, if only at the prize of some amountof fine-tuning.

Let us consider for simplicity the decoupling limit mA0 mZ . In that case we alsohave mH0,H±ggmZ . In that limit the only light scalar is h0, which behaves like the SMHiggs. We thus obtain the following contribution from the top/stop loops

m2h0 ≈ m2

Z cos2 2β +Nc

4π

m4t

v2log

m2t

m2t

, (4.38)

where we have neglected for simplicity the mixing between tL,R and have taken mtL=

mtR= mt. In this expression we have assumed EWSB (so that the physical Higgs mass

squared is determined by v, tan β and the quartic coupling). In the best case scenario weget

mh2 . 135 GeV, (4.39)

assuming mt . TeV. Even extending the spectrum, as long as the new particles are belowaround one TeV and couplings remain perturbative all the way to the unification scale, itis difficult to get beyond ∼ 150 GeV. The reason we have required the stop mass to bebelow 1 TeV is that the improvement in the Higgs mass bound can only be obtained at theexpense of fine-tuning. We do not see it in the expression of the Higgs mass because in away we have cheated when writing it. In fact, from our discussion of the Hierarchy problem

78

we know that, at energies below the scale of the superpartners we just have the SM andwe know in the SM the Higgs mass squared is quadratically sensitive to the UV physics.Where has this quadratic sensitivity gone? The reason it didn’t show up explicitly is that weassumed EWSB, and therefore the quadratically divergent contribution to m2

h0 is fixed byv and tan β and the calculation we have shown is really due to the one loop contribution tothe Higgs quartic coupling (which is not quadratically but just logarithmically divergent).The fine-tuning problem can be understood by computing the Higgs mass squared beforeEWSB. At tree level we have (in the decoupling limit we have h0 =

√2(sβH

0u + cβH

0d)),

V = 2h20(|µ|2 +m2

Hu)s2β + (|µ|2 +m2

Hd)c2β − 2bsβcβ

= 2h20|µ|2(1− s2

2β) +m2Hus

2β(1− 2c2

β) +m2Hdc2β(1− 2s2

β)= 2h2

0|µ|2c22β −m2

Hus2βc2β +m2

Hdc2βc2β+ . . . , (4.40)

so thatm2h0 = 4|µ|2c2

2β − c2β(m2Hus

2β +m2

Hdc2β). (4.41)

The one loop contribution on the other hand is

δm2h0 ≈

Nc

4π|λt|2m2

t logΛ2

m2t

, (4.42)

which, as expected, is quadratically sensitive to the stop mass. Large physical mh0 requiresa largemt but a large stop mass gives a large loop correction to δm2

h0 that has to be cancelledby the tree level correction thus implying fine-tuning. As an example if we take

mt ≈ 700 GeV, Λ ≈ 1015 GeV, tan β ≈ 20− 30, (4.43)

so that mh0 ≈ 120 GeV, we get

δm2h0 ≈ 106 ⇒ F.T. ≈

∣∣∣∣δm2h0

m2h0

∣∣∣∣ ≈ 106

104≈ 102 ⇒∼ %F.T.. (4.44)

Which shows the fine-tuning problem in the MSSM. This fine-tuning problem can beimproved with a smaller cut-off Λ and also by the adition of new fields that can give extracontributions to the Higgs potential (for instance in the nMSSM).

The MSSM, up to some technical problems, for which solutions have been proposed, anda mild fine-tuning, is a well defined and successful theory of EWSB up to very high energies.Two other interesting aspects of SUSY is that, when evolved towards high energies, thethree gauge coupling constants seem to converge to much better precision than in the SM(gauge coupling unification), and the fact that EWSB can be generated radiatively, relatedto the large top Yukawa coupling.

One aparent drawback of the MSSM is that baryon and lepton numbers are no longeraccidental symmetries, as we can write terms in the superpotential that violate them

W = λuLQ˜D + λ2

˜U ˜D ˜D + . . . . (4.45)

79

This would induce fast proton decay, which is phenomenologically ruled out. Howeverwe can define a discrete parity, R parity, that is + on the SM particles and - on thesupersymmetric partners. This ensures that proton decay operators are forbidden by thesymmetry. It also ensures that there is no tree level coupling between SM particles and onesuperpartner. This is actually a great advantage, since it automatically cancels most of thetree level contributions of new particles to electroweak precision observables, thus allowingfor very light superpartners without contradicting current experimental observables (flavoris a completely separate issue that requires some protection). Note that couplings “inpairs”, which is what is needed for the one loop solution to the Hierarchy problem arestill allowed by R parity and therefore the nice features of supersymmetric models are notspoiled by it.

80

Chapter 5

Unitarity Restoration by an InfiniteTower of Resonances: Technicolor

Literature: In the preparation of these lectures I have found useful thereviews by Farhi and Susskind [33] and the lectures by Peskin [34], Lane [35]and R. Contino(http://indico.phys.ucl.ac.be/conferenceDisplay.py?confId=148)

5.1 Introduction

In previous lectures, we have seen that EWSB in the SM although quite minimal, is notcompletely satisfactory. There is no dynamical explanation for the origin of EWSB plusthe actual mechanism suffers from the hierarchy problem. It is actually amusing to noticethat, supporting the idea that the naturalness problem of the SM is a real problem, nofundamental scalars have so far been found in nature. In case you are wondering, this isnot because no symmetry breaking process has been observed in nature. On the contrary,we have examples of symmetry breaking in which nature has chosen an explicit realizationthat does not suffer from the same problems of the SM. We will start our lecture with adiscussion of low energy QCD and the spontaneous breaking of chiral symmetry. This spon-taneous symmetry breaking does not suffer from any hierarchy problem. The scatteringof Goldstone bosons is unitarized by an infinite tower of hadronic resonances (compositestates) and not surprisingly, there is no fundamental scalars in the model. These ideasmotivated a scaled up version of QCD as a natural explanation of EWSB. The idea be-hind technicolor is thus simple and beautiful, and it is reminiscent of a symmetry breakingmechanism that we have actually observed in nature. Nevertheless, particular realizationsof this idea are not necessarily so nice, as they encounters some phenomenological problemsthat force us to complicate models quite a bit. Before getting to the actual discussion oftechnicolor ideas we will review what happens in the known example of symmetry breakingby strong interactions: chiral symmetry breaking in low energy QCD.

81

5.2 Low energy QCD and chiral symmetry breaking

5.2.1 The pattern of chiral symmetry breaking and restorationof unitarity

We have taken this discussion from [4] Let us consider QCD with two massless flavors(given the smallness of the u and d masses, this is not a bad approximation of the realworld). The fermionic sector of the QCD Lagrangian reads,

L = qiDq = qLiDqL + qRiDqR, (5.1)

where we have defined the doublet of fields qL,R = (uL,R, dL,R)T and Dµ is the QCD covari-ant derivative. This Lagrangian is invariant under the following global transformations(

uLdL

)→ UL

(uLdL

),

(uRdR

)→ UR

(uRdR

), (5.2)

where UL,R are arbitrary 2 × 2 unitary matrices. Separating the U(1) and the SU(2)parts of the transformations, we get that the Lagrangian has, at the classical level anSU(2)L × SU(2)R × U(1)L × U(1)R symmetry with associated currents

jµL,R = qL,RγµqL,R, jµaL,R = qL,Rγ

µσa

2qL,R, (5.3)

where σa are the Pauli matrices. The sum of left and right handed currents give (threetimes) the baryon number and the isospin currents,

jµ = qγµq, jµa = qγµσa

2q, (5.4)

whereas the difference gives the axial currents

jµ 5 = qγµγ5q, jµ 5 a = qγµγ5σa

2q. (5.5)

The former of the two is anomalous and therefore not a real symmetry of the Lagrangian.jµ 5 a on the other hand is a real symmetry of the Lagrangian, that is however spontaneouslybroken by the QCD vacuum.

At low energies, QCD becomes strongly coupled and quark-anti quark pairs boundtogether and condense. The vacuum state with a quark anti-quark condensate is charac-terized by a non-zero vev for the scalar operator

〈0|qq|0〉 = 〈0|qLqR + qRqL|0〉 6= 0, (5.6)

which is then non-invariant under a chiral transformation UL 6= UR. Note however that thevectorial SU(2), characterized by UL = UR leaves the vacuum invariant. Thus, the axialsymmetry is spontaneously broken by the QCD vacuum.

82

Note 1: The statement above is just a hypothesis, as due to strong couplingwe cannot compute the true QCD vacuum. This pattern of chiral symmetrybreaking is however well motivated phenomenologically. The reason is thatthe generators of the left and right transformations are related by parity

PQaL,RP

−1 = QaR,L,

which means that for each isospin multiplet with a fixed parity, we shouldfind in nature a degenerate multiplet with opposite parity. The lack of suchobservational evidence strongly suggests spontaneous breaking of the chiralsymmetry.

Note 2: Recall that we have already seen this pattern of symmetry breakingin two different situations, one is the sigma model (that was invented preciselyas a model of chiral symmetry breaking in QCD -before QCD was invented-)and the other is the global custodial symmetry in the SM.

The corresponding Goldstone bosons associated to the broken symmetry are created fromthe vacuum by the axial currents,

〈0|jµ 5 a(x)|πb(p)〉 = −ipµfπδabe−ip·x, (5.7)

where πa are the Golstone bosons of the broken chiral symmetry. fπ is known as the piondecay constant and has dimension of mass. It is guaranteed to have the same value for thethree Goldstone bosons due to the unbroken vectorial symmetry (isospin), under whichthe three Goldstone bosons transform as a triplet. There is actually an isospin tripletof mesons, the pions, which are much lighter than the rest of the hadronic resonances,mπ± = 139 MeV, mπ0 = 135 MeV, mρ = 770 MeV, and are threfore naturally identifiedwith the Goldstone bosons of the broken axial symmetry. This allows to measure fπ inpion decays, with the result fπ ≈ 93 MeV. At energies much smaller than mhad we canintegrate out the hadrons and we are left with just the low energy effective Lagrangian ofthe Goldstone bosons, the chiral Lagrangian,

L = L(2) + L(4) + . . . , (5.8)

with

L(2) =f 2π

4Tr(∂µΣ†∂µΣ), (5.9)

andL(4) = α1[Tr(∂µΣ†∂µΣ)]2,+α2Tr(∂µΣ†∂νΣ)Tr(∂µΣ†∂νΣ), (5.10)

where we have defined the matrix of Goldstone bosons as we did in the SM

Σ = ei~π·~σ/(√

2fπ). (5.11)

83

This chiral Lagrangian, represented by the non-linear sigma model, is identical in essenceto the EW Chiral Lagrangian, that represents the SM without a Higgs. Thus, we know itwill have the same kind of sick behaviour in the scattering amplitudes for Goldstone bosonsat high energies. Indeed, the pion scattering amplitude grows with the energy as (E/mρ)

2,where mρ represents the cutoff of the non-linear sigma model, i.e. the mass of the firsthadronic resonance. However, QCD is a renormalizable well-defined theory so somethingmust restore unitarity in pions scattering. The SM solution is to unitarize this amplitudeby the exchange of a fundamental scalar, the Higgs boson, and provided the Higgs mass isnot too high, this occurs in a perturbative region. The SM represents a linear sigma modelUV completion of the chiral non-linear sigma model. In the case of low energy QCD thereis no fundamental scalar and the amplitude grows until it hits strong coupling. In thatcase higher order terms in the perturbative expansion become important and, eventually,one has to include all the terms or, equivalently, include explicitly the hadronic resonances.These hadronic resonances are the ones that, in QCD, restore unitarity of the Goldstoneboson scattering. In particular, it is an experimental fact that the first resonances areresponsible in the most part for such unitarization (vector meson dominance): it is theexchange of the ρ and the a1 that mainly unitarize pion scattering. The interesting featureis that, because the main source of unitarization is different in each case, a partial wavedecomposition of the amplitude gives a different answer for different channels, dependingon which way the non-linear sigma model is completed in the UV. For instance, in thelinear sigma model case, there will be a resonance in the partial wave with zero spinand isospin, whereas in the QCD case, the resonance appears in the spin 1, isospin 1channel, corresponding to the vector resonance exchanged, the ρ. In fact, experimentaldata confirms the latter picture as shown in Fig. 5.1, in which we show the experimentaldata on the vector, isospin 1 partial amplitude in pion-pion scattering, together with theLO chiral prediction (O(E2)) in dashed line and the NLO one (O(E4)) in solid line. We

Figure 5.1: Experimental data on the vector, isospion 1 partial amplitude of pion pionscattering together with the LO (dashed) and NLO (solid) chiral predictions.

84

see how the LO chiral prediction does not show any effect of the resonance, whereas theNLO starts showing the effect of the ρ (an imaginary part of the amplitude is generatedat one loop). Experimental data shows the actual resonance that would require all termsin the perturbative expansion to be reproduced. If the restoration of unitarity was due toa scalar (even if not composite, if the restoration of unitarity was done at weak couplingand there were no strongly coupled interactions) we would have found a similar behaviourin the T 0

0 partial amplitude instead.Thus, the picture of low energy QCD, for two massless quarks, is that the original

SU(2)L × SU(2)R × U(1)V is spontaneously broken to SU(2)V × U(1)V , leading to threeGoldstone bosons, the pions, associated to the spontaneous breaking of the axial symmetry.Goldstone boson scattering does not violate unitarity due to the exchange of resonancesof the strongly coupled theory that completes in the UV the non-linear sigma model thatrepresents the interactions of the pions. Note that the pions themselves are compositestates of the strongly interacting theory (they are qq bound states).

5.2.2 Turning on weak interactions

Let us now discuss what happens if one considers the EW gauge bosons coupled to thissystem (but assuming exact chiral symmetry, i.e. exactly massless u and d quarks). It turnsout that the SU(2)L×U(1) symmetry is actually a subgroup of the global symmetry of theoriginal Lagrangian, corresponding to the generators of the global SU(2)L and Y = TR3 +B

2,

with B the baryon number (1/3 for quarks), which are weakly gauged (by which we meanthat we assume the corresponding global symmetry is actually a local symmetry, withits associated gauge bosons, and small coupling constant). The QCD vacuum breaks theoriginal symmetry down to SU(2)V × U(1)B and therefore the surviving gauge group is aU(1) with generator Q = TL3 +TR3 +B/2 = TL3 + Y , which is, precisely, electromagnetism.

Note: The gauge sector has actually nothing to do with the U(1)B group -itis not charged under it-. Thus, in what gauge bosons regards, the EWSB isfixed by the symmetry breaking pattern SU(2)L×SU(2)R → SU(2)V , whichguarantees that the U(1) group generated by T 3

L + T 3R = Q (for the gauge

bosons) is unbroken.

This spontaneous breaking of the SU(2)L×U(1) symmetry means that the W± and the Zget a mass by eating the corresponding would-be Goldstone bosons, which are nothing butthe pions of the chiral symmetry breaking. This is a particular case of the discussion wehad at the beginning of the course when discussing the Higgs mechanism. In order to seethis more explicitly, let us consider the corrections to the EW gauge boson propagators,which can be written, at tree level in the Landau (ξ = 0) gauge as (recall that, at thislevel, the EW gauge bosons are still massless),

Gµν(p) = − i

p2

(ηµν −

pµpνp2

)≡ − i

p2Pµν , (5.12)

85

where in the second equality we have defined the transverse projector. Let us denote the1PI two point function by

1PI

µ ν≡ iΠµν(p) = iΠ(p2)(p2ηµν − pµpν) = iΠ(p2)p2Pµν .

The exact two point function is then given by the infinite series of diagrams shown

= + 1PI + 1PI 1PI + . . .

which results in the following form of the exact two-point function

Gµν = − i

p2Pµν +

(− i

p2Pµρ

)iΠ(p2)p2Pρσ

(− i

p2P σν

)+ . . .

= − i

p2Pµν

[1 + Π(p2) + Π(p2)2 + . . .

]= − i

p2Pµν 1

1− Π(p2), (5.13)

where we have made use of the fact that PµνPνσ = Pµσ (it is a projector) and have explicitlysummed the geometric series. It is then clear that the gauge boson can only acquire a massif Π(p2) has a pole at p2 = 0. This pole is actually present in the theory, and correspondsto the massless Goldstone boson. Thus, we expect the correction to the gauge boson massto come from the exchange of the massless Golstone bosons in the gauge boson two pointfunction. Let us compute such correction. The coupling of the EW gauge bosons to thecorresponding currents is,

L = −gW aµJ

µaL − g

′Bµ(Jµ 3R +

1

6Jµ). (5.14)

Using this coupling, together with the form of the pion matrix element, Eq. (5.7), we findthe following expression for the annihilation of a pion into a gauge boson,

W a πb= ig

(−1

2

)(−ifπpµδab) = −g

2fπp

µδab,

B πb= ig′

(12

)(−ifπpµδ3b) = g′

2fπp

µδab,

where the factors of ±1/2 come from the left and right chirality projectors in the decom-position of the gauge currents into the vector and axial vector currents. Using that thepropagator of the Goldstone boson is iδab/p2 we obtain, for the self-energies of the differentgauge bosons

86

W aµ W b

ν=(g2fπpµ

)ip2

(−g

2fπpν

)δab = −ig2

4f 2πpµpνp2 δab

W aµ Bν

=(−g′

2fπpµ

)ip2

(−g

2fπpν

)δa3 = igg

′

4f 2πpµpνp2 δa3

Bµ Bν=(−g′

2fπpµ

)ip2

(g′

2fπpν

)= −ig′ 2

4f 2πpµpνp2

From the general form of the 1PI two point function, Eq. (5.13), we obtain (note that wehave explicitly computed the term in iΠµν = iΠ(p2)(−pµpν + . . .) and therefore have totake into account the minus sign),

Πab(p2) =

g2

4

f 2π

p2δab + . . . , (5.15)

ΠaB(p2) = ΠBa(p2) = −gg

′

4

f 2π

p2δa3 + . . . , (5.16)

ΠBB(p2) =g′ 2

4

f 2π

p2+ . . . , (5.17)

where the dots represent other contributions that do not have poles at p2 = 0. From thiswe see that the EW gauge bosons acquire a mass matrix of the form

LM =1

2

f 2π

4

(W 1 W 2 W 3 B

)g2 0 0 00 g2 0 00 0 g2 −gg′0 0 −gg′ g′ 2

W 1

W 2

W 3

B

. (5.18)

This mass matrix results in the following masses for the gauge bosons,

m2W = m2

Zc2W = g2f

2π

4, m2

A = 0. (5.19)

Thus we see that the photon remains massless and the W and Z masses follow the by nowwell known relation ρ = 1. The reason for that is simply that QCD is custodially symmetric(the global symmetry pattern is the custodially preserving SU(2)L×SU(2)R → SU(2)L+R).Thus, we see that QCD by itself spontaneously breaks the EW symmetry, maintaining thecondition ρ = 1. Of course, this is not the full story for EWSB since, for instance, the Wmass turns out to be

mW (QCD) = g93 MeV

2≈ 46 MeV, (5.20)

which is far from the experimentally measured mW ≈ 80.4 GeV. Also, the pions have beenmeasured to be physical fields, and not the unphysical Goldstone bosons that are eaten bythe gauge bosons to become massive.

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Despite these drawbacks, QCD still represents, at the qualitative level, a successfulrealization of EWSB. The important question is, apart from the fact that this realizationof symmetry breaking and unitarization of Goldstone boson scattering by resonances of astrongly coupled theory as opposed to a fundamental scalar, has been observed in nature,is it any better than the realization of EWSB in the SM through the Higgs? The answer isdefinitely yes, because the breaking of chiral symmetry by QCD does not suffer from thehierarchy problem. The reason is that QCD is an asymptotically free theory that condensesin the IR, generating the QCD scale, Λ ∼ 300 MeV, by dimensional transmutation. Theone loop running of the strong coupling is given by

αs(Q2) =

α0s

1 + b0α0s

4πln(Q2/Q2

0), (5.21)

where b0 = 11− 2nf/3 and α0s ≡ αs(Q

20). It is conventional to define ΛQCD as the scale at

which the strong coupling constant blows up,

1 + b0α0s

4πln(Λ/Q2

0) = 0, (5.22)

in terms of which we can write the strong coupling constant as

αs(Q2) =

4π

b0 ln(Q2/Λ2). (5.23)

Let us assume now that there is a high energy threshold at which new particles are present.At these energies, the strong coupling constant is small and corrections to it remain in theperturbative regime. Thus, the threshold effects will have a small impact on the value ofthe strong coupling constant at such large energies Λ, g

(0)s . Now the question is, given that

threshold effect on g(0)s , do we expect the strong coupling scale ΛQCD to be sensitive to this

enormously high cutoff? The answer is no, we can actually naturally expect ΛQCD to beexponentially smaller than Λ as can be seen by simple solving for Λ in terms of g0

s in theabove equation. The result is

ΛQCD = Λ exp

[− 2π

α0sb0

]= Λ exp

[− 8π

g(0) 2s b0

]. (5.24)

Thus, we see that, as long as gs is not large in the UV, the QCD scale is preturbativelystable against UV thresholds. This is why it is natural to have ΛQCD ≈ 300 MeV despitethe fact that we have observed other thresholds in nature, for instance the EWSB scalev ∼ 174 GeV. For instance, a coupling constant of order g0 ∼ 0.3 at the Planck mass willgenerate a strong coupling scale ΛQCD ∼ 1 GeV. Note that this result does not mean thatΛQCD is predicted to be the value we measured. What we have seen is that, provided thetheory is asymptotically free, so that the coupling constant in the UV (where possible newthresholds will appear) is small, the scale of symmetry breaking generated by dimensionaltransmutation is naturally exponentially smaller than the cut-off of the theory, whetherthis is ∼ 1 GeV, ∼ 1 TeV or ∼ 10−5GeV depends on the actual value of the cut-off and thecoupling constant but the important feature is that a large hierarchy of scales is naturallypredicted.

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5.3 Technicolor

In this lecture we will discuss ideas motivated by the observed spontaneous breaking ofthe chiral symmetry by the QCD vacuum and its successful realization of EWSB at thequalitative level. The simplest idea is to consider a “scaled-up” version of QCD [36, 37] inwhich,

fπ = 93 MeV→ Fπ =√

2v = 246 GeV. (5.25)

Of course, we do not have to have precisely three colors in this new version of QCD.Let us therefore consider an SU(NTC) technicolor gauge group with a dynamically bro-ken SU(2)L × SU(2)R → SU(2)L+R global symmetry. The scale of dynamical symmetrybreaking will be assumed to be ΛTC ∼ TeV and the three technipions will be the Goldstonebosons eaten by the W and the Z to get their masses. In order to be able to scale-up theresults of QCD to technicolor with an arbitrary number of techni-colors (NTC), we needto know how different quantities scale with N . This is not known in general but we havesome very simple scaling properties in the large-N limit (see [38, 39]). Assuming SU(NTC)is a confining theory for large NTC , it is possible to write the n-point Green’s functions ofquark bilinears, in the large NTC limit, as and infinite sum over stable intermediate mesonstates created out of the vacuum. In particular, for instance, the two point functions canbe written as an infinite sum over stable resonances,

〈TJµV JνV 〉 = (q2gµν − qµqν)

∑n

f 2ρn

q2 −m2ρn

, (5.26)

〈TJµAJνA〉 = (q2gµν − qµqν)

∑n

[ f 2an

q2 −m2an

+1

q2f 2π

]. (5.27)

Note the massless pole corresponding to the Goldstone bosons in the axial currents. Usingthe scaling with large NTC of the n-point bilinear functions, we can obtain the followingscaling for different relevant masses and couplings,

mρ ∼ constant, (5.28)

fπ,ρ ∼√NTC

4π, (5.29)

gππρ ∼4π√NTC

. (5.30)

In particular, both mρ and fπgρ are independent of NTC in the large-N limit. This scalingproperties assume that ΛQCD is constant (which also implies mρ constant). However, wehave seen that the electroweak gauge boson masses are proportional to f 2

π rather thanΛ2QCD. If we keep fπ fixed as we vary the number of technicolors, then Λ ∼ mρ ∼ 1/

√N .

In particular we have, for instance,

mρTC ∼√

3

NTC

Fπfπmρ ≈

√3

NTC

246

0.0930.77 ≈ 1.8 TeV (for NTC = 4). (5.31)

89

Thus, smaller NTC implies stronger coupling (gρ ∼ 1/√N) and heavier ρTC (recall that

this apparent dependence of mρTC on N is due to the fact that we have fixed Fπ).Thus, the EW symmetry is broken by both the technicolor sector and QCD. The W

and the Z eat a combination of π and πTC to get a mass,

|χ〉 = sinα|π〉+ cosα|πTC〉, (5.32)

where v2 = f 2π + F 2

π and tanα = fπ/Fπ 1. The longitudinal components of the EWgauge bosons are therefore mostly the technipions whereas the physical pions are mostlythe QCD pions.

Note: This is easily seen by notizing that

〈0|Jµ 5 a|πb〉 = −ipµfπδab, (5.33)

〈0|Jµ 5 a|πbTC〉 = −ipµFπδab, (5.34)

and that〈0|Jµ 5 a|physical pion〉 = 0. (5.35)

If we have ND technifermions that are doublets under SU(2)L and color singlets, therelation between Fπ and mW gets modified to

mW =

√ND

2gFπ, (5.36)

and therefore Fπ = 246 GeV/√ND. Thus, in general, if we have a larger number of

technicolors or of technidoublets, we have to replace the technipion decay constant withan effective one

Fπ →√NDNTC/3Fπ, (5.37)

so that its effect is effectively larger and therefore the actual value of Fπ can be smaller.Note however that more technidoublets imply more Goldstone bosons. The new flavorsymmetry must therefore be broken by gauge interactions so that these Goldstone bosonsacquire masses.

5.3.1 The simplest example: minimal technicolor model of Wein-berg and Susskind

Before going into the challenges that technicolor models have to overcome it is instructiveto briefly introduce the simplest technicolor model introduced by Weinberg [36] and bySusskind [37]. The model has an SU(NT )×SU(3)×SU(2)L×U(1)Y gauge symmetry. Inaddition to the SM fields, we add one flavor doublet of color singlet technifermions, (T,B).

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They have quantum numbers,

QaL =

(TB

)aL

∼ (NT , 1, 2, 0), (5.38)

QaR =

(TB

)aR

∼(

(NT , 1, 1/2, 0)(NT , 1,−1/2, 0)

), (5.39)

where a = 1, . . . , NT and we have grouped the two RH techniquarks into a doublet tomake explicit the global SU(2)L × SU(2)R × [U(1)A] × U(1)B (the axial abelian group iswritten in square brackets to remind the reader that it is anomalous and therefore nota true symmetry). This spectrum is anomaly free provided NT is even. We assume thisnew technicolor group will become strongly coupled at a scale ΛT at which techniquarcondensates will form. Identically to what happens in low energy QCD, the fact that weget QQ condensates and that LH and RH techniquarks transform differently under theelectroweak group, will induce a sponatenous breaking of the electroweak symmetry. Withjust one set of doublets, we have exactly the same symmetry breaking pattern as in thechiral Lagrangian, three Goldstone bosons are generated, which are then eaten by theelectroweak gauge bosons. The masses of the different resonances in the spectrum can beestimated by scaling with NT (and ND, the number of SU(2)L doublets if we include morethan one) QCD results.

5.3.2 Phenomenological issues with technicolor and possible res-olutions

This minimal version of technicolor, despite successfully describing EWSB in a natural wayat the qualitative level has some phenomenological problems. We briefly summarize belowthe most important and the suggested solutions. An up to date discussion can be foundin [40].

EWPT

Scaling up properties of QCD give estimations of the S parameter that are incompatiblewith experiment. Being a strongly coupled theory, we cannot compute the actual value ofthe S parameter in technicolor models. Peskin and Takeuchi used detailed properties ofQCD, like vector meson dominance (saturation by first resonances) and large-N behaviourto get an estimation [28]

S = −4πd

dq2

(ΠV V (q2)− ΠAA(q2)

) ∣∣∣∣q2=0

≈ 4π

(1 +

m2ρT

m2a1T

)F 2π

m2ρT

≈ 0.25NDNTC

3, (5.40)

which, taking into account the experimental limit S . 0.3 (assuming a similar contributionto T , otherwise the limit is more strict), seems to indicate that if technicolor models are just

91

a scaled up version of QCD, they are most likely excluded experimentally. Alternatively,we can see the limit above as a bound on the technirho mass

mρT &√ND1.8 TeV, (5.41)

which is difficult to make compatible with Eq. (5.31), which was computed for ND = 1.Furthermore, even that estimation was difficult to make compatible with restoration ofunitarity in longitudinal gauge boson scattering if ρT is the lightest mode responsible forsuch restoration.

A solution to this problem has necessarily to do with technicolor models that are nota scaled-up version of QCD. Walking technicolor, that we will discuss latter, is such anexample.

Fermion masses and FCNC

So far we have not said anything about how the SM fermions should get a mass. In orderfor the techniquark condensates to give them a mass, we need fermions to couple to thetechnicolor currents. This is done in Extended Technicolor models, in which SU(3)C , allflavor symmetries, and the technicolor group are assumed to be subgroups of an extendedtechnicolor gauge group, which is broken at a scale ΛETC ,

SU(NETC)ΛETC−→ SU(3)C × SU(NTC). (5.42)

Actually, the different flavor symmetries should be broken at different scales in order to givedifferent masses to the SM fermions. The corresponding massive flavor gauge bosons willgenerate couplings between SM fermions and technifermions in the form of four-fermionoperators. After integrating out the extended technicolor gauge bosons, we end up withan effective Lagrangian of the form

g2ETC

(αab

(T γµtaT )(T γµtbT )

Λ2ETC

∣∣∣ETC

+ βab(T γµt

aq)(qγµtbT )

Λ2ETC

∣∣∣ETC

+ γab(qγµt

aq)(qγµtbq)

Λ2ETC

∣∣∣ETC

),

(5.43)where the subscript ETC indicates that the corresponding operators are generated at ascale ΛETC , ta,b denote de ETC generators and gETC is the ETC gauge coupling. The firsttype of terms will generate masses for the technipions (the goldstones that are not eatenby the electroweak gauge bosons), the second type will generate fermion masses and finallythe third type can generate dangerous FCNC interactions. Let us look at the terms theSM generating fermion masses in a bit more detail, Fierz rearranging the correspondingoperators, we get (

βg2ETC〈T T 〉|ETC

Λ2ETC

)qq. (5.44)

This gives a mass for the corresponding SM fermion, which runs logarithmically belowΛETC . In order to compute the SM fermion mass, we need to solve the Callan-Symanzik

92

equation for the techniquark condensate,

〈ΨΨ〉ETC = 〈ΨΨ〉TC exp

(∫ ΛETC

ΛTC

d log µ γm

), (5.45)

where the anomalous dimension for the operator ΨΨ is given in perturbation theory by

γm(µ) =3C2(R)

2παTC(µ) + . . . , (5.46)

with C2(R) the quadratic Casimir of the technifermions in theR representation of SU(NTC).For the fundamental representation, we have C2(NTC) = (N2

TC − 1)/2NTC . If technicolorbehaves like QCD in the sense that the coupling constant decreases very fast with theenergy, then the anomalous dimension can be taken essentially vanishing and we get theresult for the techniquark condensate

〈ΨΨ〉ETC ≈ 〈ΨΨ〉TC . (5.47)

Thus, the SM fermion mass, at the ETC scale, reads

mq(ETC) ∼ 1

Λ2ETC

〈ΨΨ〉ETC ∼1

Λ2ETC

〈ΨΨ〉TC = ΛTC

(ΛTC

ΛETC

)2

=4πF 3

π

Λ2ETC

. (5.48)

Where the last equality is an estimation based on the NJL model. We have also assumeda similar behaviour (that we will have to modify in more realistic set-ups) to QCD in thesense that the anomalous dimension becomes very small at higher energies. We thereforeobtain the relation between the scale at which the corresponding flavor symmetry has tobe broken and the generated fermion mass,

ΛETC(q) ≈

√√√√4π23/2v3

mqN32D

≈ 13 TeV

N3/4D

√1 GeV

mq

. (5.49)

We have taken into account that the condensation scale is lower for a higher number oftechnidoublets ND. We therefore see that fermion masses put an upper bound on the scaleof ETC breaking. Furthermore, this upper bound is of the order of (∼ 10 TeV for light GeV-ish quarks and seems hopelessly small (∼ TeV) for the top. This is already a problem, sinceit is not obvious how to generate the top mass, but even a bigger problem is the fact thatthis same scale is the one that suppresses the dangerous FCNC interactions. For a realistictheory of flavor, we must have different SM fermions to couple to the same technifermion(or to two technifermions that are themselves related by the flavor symmetries). Thus,flavour violating four fermion interactions among the SM fermions, i.e. non-zero familydependent γ couplings, are generically expected. Note that even if these four-fermioninteractions among SM fermions are flavor diagonal (but not proportional to the identity),they will induce FCNC when we rotate to the physical basis. For instance, one expects,

L =g2ETCV

2ds

Λ2ETC

(dΓµs) (dΓ′µs) + h.c., (5.50)

93

where Γµ,Γ′µ are either of γµ(1 ± γ5)/2, and we have written the generated operator in

terms of a dimensionless coefficient Vds which depends on the particular flavor structure ofthe theory but is not expected to be much smaller than the product of the correspondingCKM elements. Using the experimental values of ∆mK and εK , we obtain the followingapproximate bounds on the scale of ETC,

ΛETC &

1300 TeV × gETC

√Re(V 2

ds),

16000 TeV × gETC√

Im(V 2ds).

(5.51)

This very stringent lower bound on the ETC scale is incompatible with our previous esti-mate of the upper bound value of ΛETC to give the SM fermions the right masses. Notehowever that we have made an important assumption, namely that the dynamics of TCis identical to that of QCD and in particular that the TC coupling is very small at largeenergies. This is clearly not necessarily the case in a general strongly coupled theory andwe will see in the next section how departing from this assumption improves dramaticallythe behaviour of technicolor models.

Departing from QCD: Walking technicolor

All the main phenomenological problems with ETC came from the fact that we are consid-ering a scaled-up version of QCD. In particular, we are assuming that the theory goes toits asymptotically freedom regime very fast above ΛTC . This forced us to have quite lowΛETC scales in order to give the SM fermion the right masses. But these values generateFCNC that are in gross contradiction with observed experimental data. Also, the assump-tion of vector meson dominance and the relation between the first vector and axial-vectorresonances in the technicolor spectrum (features just taken as they are in QCD) led to atoo large contribution to the S parameter. What if ETC is actually not a scaled-up version

β

gTC

g*TC

Figure 5.2: Strongly coupled IR fixed point. The theory remains nearly conformal until itcondenses.

of QCD? What if, in particular, the technicolor coupling contant, instead of going to a very

94

small value at energies just above ΛTC , remains almost constant (it does not run but walks)all the way to ΛETC? Imagine that the theory has a non-perturbative IR fixed point athigh energies as displayed in Fig. 5.2. If the theory is at its conformal point at E ∼ ΛETC ,it will behave like an approximately conformal theory for ΛTC ≤ E ≤ ΛETC , spontaneouslybroken at ΛTC . In that case, any operator is characterized by its scaling dimension at thefixed point, d∗, which, due to the strong coupling regime, can be significantly different fromthe classical dimension.

In particular, the solution of the Callan-Symanzik equation for the techniquark con-densate, assuming a constant anomalous dimension, reads,

〈ΨΨ〉ETC = 〈ΨΨ〉TC(

ΛTC

ΛETC

)γm. (5.52)

The SM fermion mass at the ETC scale then reads,

mq(ETC) ∼ 〈ΨΨ〉Λ2ETC

∼ 〈ΨΨ〉TCΛ2ETC

(ΛTC

ΛETC

)γm∼ ΛTC

(ΛTC

ΛETC

)2+γm

. (5.53)

Thus, we see that a large negative anomalous dimension can notably change the scalingbehaviour, allowing for a higher value of ΛETC and therefore ameliorating the FCNC prob-lem. In the extreme walking limit, in which αTC(µ) is constant, it is possible to obtain anapproximate non-perturbative formula for the anomalous dimension,

γm(µ) = −1 +√

1− αTC(µ)/α∗TC , (5.54)

where α∗TC = π3C2(R)

. It is believed that γm = −1 is the signal of spontaneous chiral

symmetry breaking. In extended technicolor, αTC(µ) is assumed to be near its criticalvalue α∗TC for ΛTC . µ . ΛETC , thus γm & −1 and

mq(ETC) ∼ ΛTCΛTC

ΛETC

, (5.55)

and can do with a safer value of the ETC scale to generate the fermion masses,

ΛETC ∼270 TeV

N3/4D

√1 GeV

mq

. (5.56)

Note that, contrary to the terms that give masses to the fermions, which in the extremewalking limit are suppressed by just one power of ΛETC , the FCNC operators, that involvefour SM fermions, are still suppressed, even in walking technicolor models, by two powers ofΛETC . This means that compatibility between quark masses and FCNC and CP violationis much improved, although some tension is still present.

We have thus seen how conformal behavior at strong coupling, walking technicolor,allows to generate light fermion masses compatible with enough suppression of FCNC (orat least importantly improving the situation). There is also some evidence that nearconformal behavior can naturally reduce the contribution to the S parameter.The main argument is that near conformal dynamics is usually accompanied by a nearlyparity symmetric spectrum, i.e. degenerate vector and axial resonances. This degeneracyautomatically minimizes the contribution to the S parameter.

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Top quark mass and walking technicolor

One problem that not even near conformal ETC can solve is the generation of the topmass mt ≈ 175 GeV. The scale ΛETC needed to generate the top mass is of order ΛETC ∼TeV for simple ETC and ΛETC ∼ 20 TeV for walking technicolor. The former is obviouslytoo low, and even the assumption of four-fermion interactions is no longer valid. Thelatter is somewhat higher, although it still seems to be difficult to make it compatible withFCNC constraints. Some attempts try to separate the ETC scale for different generations(Tumbling technicolor theories), thus improving the FCNC. However, it is not easy togenerate the corresponding cascading of scales solely from gauge dynamics. Furthermore,even if we manage to separate the third generation from the other two, the bottom stillcomes together with the top and it is very difficult to get the hierarchy in masses betweenthe top and the bottom without inducing large corrections to the Zbb coupling, which arestrongly constrained experimentally.

We list now two of the suggested solutions to the top mass problem:

Topcolor-assisted technicolor One of the proposed solutions is called top color assistedtechnicolor. The basic idea is to assume two separated technicolor sectors, one thatcondenses at a scale ∼ TeV and couples preferentially to the top, inducing a topcondensate that creates a composite state with the quantum numbers of the top.The second technicolor sector is a regular ETC that gives small masses to the SMfermions. In particular, the top quark is an admixture of a top that gets a small massfrom the ETC sector and the composite with the quantum numbers of the top in thestronger technicolor sector. In this way the large top mass is naturally explained anddoes not introduce large corrections to the lighter sector, since it does not perturbstrongly the ETC sector that gives mass to the lighter fermions (and that, with a bitof walking can safely avoid FCNC).

An alternative is that top condensation induces all the required EWSB, i.e. it is notthe condensation of new technifermions that drives EWSB but a tt condensate. Theproblem is that, for the top to condense, one would need it to couple more stronglythan its mass indicates. It would have to be mt ∼ 600 GeV for its “Yukawa coupling”to be strong enough to produce the condensation. Top see-saw models propose thatthe EWSB mass of the top is actually of that size ∼ 600 GeV but it mixes withother heavier composites that make the physical mode lighter (in a small see-saw) togive the observed mass. From the low energy perspective, we can model this processthrough a 600 GeV top mass (from EWSB) that mixes with a ∼ TeV vector-likequark singlet. The lightest physical admixture of both quarks has a mass ∼ 175 andis the top we measure.

It is not easy to make minimal top condensation models (with or without top see-saw) compatible with experimental observations, in particular when corrections tothe Zbb coupling are considered.

Larger anomalous dimensions The limit γ . 1 for condensation is based on approxi-mate models and numerical simulations but there is no formal proof of such bound.

96

Unitarity imposes a firm bound given by γ ≤ 2. If one allows anomalous dimensionsof order γ ∼ 1.5, the top mass can be generated consistently with the bounds fromFCNC and EWPT.

5.3.3 New Developments in Technicolor

The problems we have outlined in the previous sections made technicolor fade againstits main competitor, SUSY, and in the last decade against new ideas in model building,some of which we will discuss in these lectures. The fact that SUSY was not found atLEP and the experimental bound on the Higgs mass starts to impose some amount offine-tuning on minimal supersymmetric models, gave some new momentum to technicolortheories. It was also realized that some new ideas in model building could actually be usedto understand better technicolor models. Thus, in the last few years, new more realistictechnicolor models have emerged and their phenomenological implications will be studiedat the LHC. Here we briefly describe two of these new routes in technicolor models.

Minimal walking models

The models we have described so far are based on SU(NTC) with fermions transforming inthe fundamental representation NTC . The reason is that in the end, our only experiencewith these kind of models comes from QCD, for which that is the case. Furthermore, wetypically need to use large-NTC results, which made models less compatible with experi-mental data (for instance the S parameter naively grows with the number of technicolors).In recent years, an effort is being carried out to analyze the conformal window as a functionof, not only the number of technicolor or techniflavors (number of weak doublets) but alsoas a function of the representation of the group. The results of these analyses seem topoint to very minimal models, with a very small number of technicolors and techniflavorsas being within the conformal window (thus leading to the required walking), resulting inthe minimal naive contribution to the S parameter (defined as the one loop contribution ofthe technifermions assuming momentum independent constituent masses), and even allow-ing for the presence of realistic dark matter candidates. An interesting example of thesenew technicolor models is the Ultra Minimal near Conformal Technicolor model [41]. Themodel consists of an SU(2) technicolor group with two Dirac flavors in the fundamentalrepresentation, the LH components transforming as a SU(2)L doublet with zero hyper-charge, the RH components as SU(2)L singlets with hypercharge ±1

2, respectively, plus

two Weyl fermions, which are singlets under the electroweak symmetry but transform inthe adjoint representation of the technicolor group. These new fermions transformingin a different representation of the group, bring the model into the conformal window,allowing for the model to walk. A detailed analysis of the symmetry breaking patternshows an unbroken global technibaryon number. In some circunstances, the lightest tech-nibaryon, which is stable due to technibaryon number conservation, is weakly interactingand electrically neutral and can therefore serve as a good dark matter candidate. It canbe even possible to generate the dark matter relic density through a technibaryon number

97

asymmetry, like it happens for the ordinary baryon density. In some models, the two canbe related, which might explain the similarity of baryon to dark matter density today.

Unfortunately, some of the features of these models, like the existence of a conformalwindow for such small number of technifermions, can only be estimated due to the strongcoupling. An important amount of resources is being currently used in studying thesefeatures in the lattice, which will provide a definite answer to some of these questions.

New tools for technicolor: extra dimensional models

As we said above, one of the main drawbacks of technicolor is the impossibility of preciselycompute many of the phenomenological implications of the theory. That means that it isdifficult to exclude these models with a high degree of confidence but it also means thatit is extremely difficult to make reliable predictions. The situation changed in some wayin the last decade when, due to the original work of Maldacena [42], it was realized thatsome strongly coupled gauge theories in d dimensions could be related to weakly coupled“dual” gravity theories in d + 1 dimensions. The original idea, and the one that has themost “experimental” support (here by experimental we do not really mean particle physicsexperiments that theoretical tests that the duality has passed), is stated in a 10-dimensionalstring theory, compactified on Ω × AdS5, where Ω is a 5-dimensional compact space andAdS5 is 5-dimensional anti-deSitter space and its dual, that is a conformal theory (forinstance N = 4 SYM for type II-B string theory with Ω = S5) defined on the boundary ofAdS5. The large-N limit of the conformal field theory corresponds to the weak couplinglimit (the supergravity limit of the string theory) in the string side of the duality.

The AdS/CFT correspondence is therefore a duality between strongly coupled confor-mal theories and weakly coupled gravity theories defined in one more dimension. Thissounds precisely like the kind of thing we need in order to study technicolor models. Wewant models that are near conformal and strongly coupled but we would like to be ableto perform calculations with them. In order to do that, we only need to go to the dualweakly coupled models and make the corresponding calculations there. In fact, at thattime, Randall and Sundrum (RS) had proposed a gravity model (which was soon extendedto incorporate the full SM) in AdS5 to solve the hierarchy problem in a purely geometricway [43]. It didn’t take long to put these two ideas together and to conjecture a dualitybetween RS models and strongly coupled conformal theories in four dimensions. This newconjecture rests on much less firm ground. It does not rely on string theory or highlysupersymmetric gauge theories and it might be very well that we cannot compute whichconformal theory our RS model is dual to. However, it is true that these 5D models sharea lot of features in common with strongly coupled conformal theories and a qualitativeAdS/CFT dictionary has been developed that seems to work pretty well. The main entriesin the dictionary relevant for model building, are compiled in table 5.1, taken from [44].Given these successes, even if we don’t know which 4D CFT we are really studying, it isquite likely that there is one CFT that is (at least approximately) dual to our 5D modelsin AdS5. We will in fact make more use of this duality in future lectures.

5D models of technicolor are defined on a slice of AdS5, which has a metric, in confor-

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Bulk of AdS ↔ CFT

Coordinate (z) along AdS ↔ Energy scale in CFT

Appearance of UV brane ↔ CFT has a cutoff

Appearance of IR brane ↔ conformal symmetry brokenspontaneously by CFT

KK modes localized on IR brane ↔ composites of CFT

Modes on the UV brane ↔ Elementary fields coupled to CFT

Gauge fields in bulk ↔ CFT has a global symmetry

Bulk gauge symmetry brokenon UV brane

↔ Global symmetry not gauged

Bulk gauge symmetry unbrokenon UV brane

↔ Global symmetry weakly gauged

Higgs on IR brane ↔ CFT becoming strong producescomposite Higgs

Bulk gauge symmetry brokenon IR brane by BC’s

↔ Strong dynamics that breaksCFT also breaks gauge symmetry

Table 5.1: Relevant rules for model building using the AdS/CFT correspondence

mally flat coordinates,

ds2 =L2

0

z2(ηµνdx

µdxν − dz2), (5.57)

with M−1Pl ≈ L0 ≤ z ≤ L1 ≈ TeV−1 the coordinate along the extra dimensions. This form

of the metric shows the scale invariance of AdS, as it is invariant under x, z → λx, λz, whichhints at the conformal invariance of the dual theory (in fact the isometry group of AdS5 isSO(2, 4), which corresponds to the conformal group in four dimensions). The conformalinvariance is broken explicitly at the so called UV brane (z = L0), which represents inthe dual theory the coupling of external sources (the SM particles) to the CFT. It is alsobroken, although this time spontaneously due to condensation (in the dual theory), atthe IR brane (z = L1). Having obtained a near conformal theory we now need the chiralsymmetry breaking SU(2)L × SU(2)R → SU(2)V . The dictionary tells us that globalsymmetries in the CFT correspond to bulk gauge symmetries in the 5D side. Thus, weinclude an SU(2)L×SU(2)R×U(1)X bulk gauge symmetry. This will furthermore ensurethe required global custodial symmetry that guarantees a small contribution to the Tparameter.

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Note: The original RS models, which did not have technicolor models as agoal, included a simpler SU(2)L×U(1)Y bulk gauge group. Not surprisingly,this lead to a very large contribution to the T parameter, which was subse-quently fixed by enlarging the bulk gauge group, after the realization of theimplications on the global symmetries in the dual picture.

The extra U(1)X group is introduced to obtain the correct hypercharge assignments. Aswe said, the IR brane represents the spontaneous breaking of conformal invariance due tocondensation, thus we would like to have, at the IR brane, the bulk SU(2)L × SU(2)Rsymmetry been spontaneously broken to SU(2)V . This can be done in the 5D model bymeans of a Higgs but that will actually leave a physical Higgs field in the spectrum, morealong the lines of composite Higgs models that we will describe in the future. Anotherpossibility, more in the spirit of technicolor models, is to break the symmetry by boundaryconditions. This is a spontaneous breaking that does not leave any scalar in the low energyspectrum.

Note: It is not obvious that breaking gauge symmetries by boundary condi-tions is a spontaneous breaking. However, it was shown in [45] that consistentboundary conditions can be obtained from localized scalars acquiring a vevand taking this vev (and thus the corresponding physical Higgses) to infinity.

The UV brane, on the other hand, represents the explicit conformal breaking due to thecoupling with the elementary, SM, fields. Thus, the bulk gauge symmetry, must be brokenby boundary conditions to SU(2)L×U(1)Y , where the hypercharge is a combination of thethird component of SU(2)R and the U(1)X . This model thus has all the right ingredientsof a realistic technicolor model. Furthermore, in the limit that the 5D gauge coupling issmall, we have a weakly coupled theory that will allow us to perform calculations. In thismodel, for instance, anomalous dimensions are related to the localization of the differentfields in the extra dimension and can be very easily computed. This gives us the scalingof mass terms and different couplings that successfully reproduce the SM fermion masses.Furthermore, the same mechanism that suppressed the masses of the light fermions alsosuppresses FCNC for them. Although models that have been constructed still need someamount of cancellations to suppress enough the most dangerous FCNC processes, they arenaturally close to current limits.

Apart from being able to compute the scaling of fermion masses and FCNC processes,we are also able to compute explicitly the restoration of unitarity in longitudinal gaugeboson scattering for these models. It was shown in [45] that, as long as the breaking byboundary conditions is consistent (with the variation of the action in the bulk and branes),unitarity is exactly restored (in the sense that terms growing with energy like ∼ E4 and∼ E2 vanish exactly) through the exchange of the whole tower of KK modes (resonancesin the 4D dual) in the elastic channel. Still, the constant term can violate unitarity if thelightest KK mode is too heavy (similar to what happens in the SM with a too heavy Higgs).This imposes a bound on the lightest KK mode of the order of ∼ 700 GeV. This bound is

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great for phenomenology, but it is clear that it might give rise to problems with EWPTand in particular with the S parameter. Before discussing that, it is worth mentioning thatthe fact that all KK modes contribute to the restoration of unitarity, means that inelasticchannels will open at higher energies. In fact, it was shown that when inelastic channelsare included, unitarity is violated at some high energy. This is due to the fact that 5Dtheories are intrinsically non-renormalizable and can only be considered effective theoriesbelow some cut-off. In fact, the unitarity bound from inelastic scattering approximatelyagrees with the naive cut-off of the 5D theory.

The fact that we need so light resonances to restore unitarity, means that more likely,we will have problems with EWPT. In fact, we also have that weak coupling in the 5Dpicture corresponds to large-N limit of the CFT, which goes in the direction of makingthe S parameter larger. The bound on the S parameter imposes an upper bound on N ,which in turn imposes a lower bound on the 5D coupling. In fact, in minimal models, thetwo bounds are contradictory, and the models is excluded. Model building in the 5D side,however, allows us, at the expense of ∼ few % fine-tuning, to compensate the contributionto the S parameter with a vertex correction so that the total contribution to the “effective”S parameter cancels.

The heaviness of the top still poses a problem in these models although again modelbuilding techniques in 5D allow for Higgsless models that pass all EWPT constraints attree level (with few per cent fine-tuning), have realistic fermion and gauge boson massesand preserve unitarity in longitudinal gauge boson scattering up to the cut-off of the theory.Loop corrections can be important in the case of the top but they can be argued to be of thesame size as higher dimensional operators arising at the cut-off of the 5D theory. We aretherefore in a situation in which we have gained some computational power, in the sensethat we can compute our model to be realistic in the leading color approximation (leadingcorrections in 1/N). Subleading color contributions (loop corrections on the 5D side)can be substantial in some cases but they are formally of the same order as uncalculablecorrections coming from higher dimensional operators. This is somewhat uncomfortablebut it is difficult to get further in calculability using 5D theories. Even more unsettling isthe fact that although FCNC are well under control within the 5D, the same uncalculablehigher dimensional operators, could contribute to light fermion FCNC well above currentexperimental data (this is forced upon in Higgsless models, whereas it can be avoidedin other RS models, because of the requirement of cancellation of the S parameter withfermion vertex corrections).

More freedom, including some control over quantum corrections, can be obtained bymeans of the so called deconstruction of extra-dimensional models. This can be viewed asa coarse discretization of the extra dimension. The resulting theory is a weakly coupled4D theory, that corresponds to the low energy effective theory of the extra dimensionalone. It can be realized by means of a non-linear sigma model, in which case we will haveto replace it with a UV completion at its cut-off, or by means of a linear sigma model, thathas the ugly inconvenient of reintroducing the hierarchy problem.

In a way, both the 5D and the deconstructed Higgsless models sacrifice the solution ofthe big hierarchy problem (they could be considered duals to strongly coupled quasi-CFTs

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that solve the hierarchy problem but there are features of these 4D CFTs that we cannotcompute in our intrinsically effective weakly coupled descriptions). This has become a bitof a trend in the last few years. I personally think that we should not forget the largehierarchy problem as the best motivation for physics beyond the SM but agree that, withthe onset of the LHC in front of us, it is useful to prioritize our efforts. And possiblesolutions of the little hierarchy problem that can have phenomenological implications atthe LHC seem like good places to start looking. A wealth of new models, including in a waythe ones we have mentioned in this section, have been proposed in the last few years thatsolve the little hierarchy problem, allowing for a natural Higgs mass -or scale of electroweaksymmetry breaking- with a cut-off that is compatible with current experimental constraintsand at the same time have observable consequences at the LHC. If any of these modelsis confirmed, we might have to think about what UV completion takes over at the newcut-off of the theory, which typically is ∼ 1− 10 TeV.

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Chapter 6

Composite Higgs Models

Literature: This section has been taken almost entirely from RobertoContino’s notes.

6.1 Introduction

Technicolor models are based on a beautiful, simple idea, which is actually realized innature for another example of spontaneous symmetry breaking. Unfortunately, particularrealizations of this simple idea clash with experimental observations and it is difficult toobtain realistic models that are not unnatural. Composite Higgs models [46, 47, 48] aroseas an intermediate case between a fundamental scalar Higgs and technicolor models. Inthese models, the Higgs boson exists but it is a composite state of a strongly coupled newinteraction. This allows these models to have some of the nice properties of technicolormodels while avoiding some of the most constraining phenomenological problems:

• Being a bounds state, the Higgs is not sensitive to UV effects above the composite-ness scale. This solves the hierarchy problem (instability of the Higgs mass againstradiative corrections).

• We furthermore assume the Higgs is light as compare with the rest of the resonancesbecause it is the Goldstone boson of a dynamically broken global symmetry of thestrong sector.

• A light Higgs can (partially) restore unitarity in longitudinal gauge boson scattering,thus allowing for heavier vector resonances in the strong sector, compatible withEWPT. (The flavor problem will be discussed below).

• Composite Higgs models interpolate between technicolor and the SM with a funda-mental Higgs. In principle, we can take the limit in which the vector resonances getheavier and heavier, but keeping the mass of the Goldstones at a few GeV scale. Inthat limit, the only effect of the strong sector is a scalar, with a natural cut-off, the

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composite scale, that is very high. This is just the SM limit. Of course, in that casekeeping the Higgs in the light spectrum requires fine-tuning. As we will see below,the Higgs mass is generated by radiative corrections but it is still proportional to thecomposite scale and keeping it much smaller than the compositeness scale requiresto fine-tune this one loop correction.

6.2 Composite Higgs Models: General Structure

The starting point for composite Higgs models is very similar to technicolor models. Weassume there is a strongly interacting sector with a global symmetry group G that isspontaneously broken, due to condensation, to a subgroup H. The SM group should be asubgroup of H. Furthermore, G/H should contain an SU(2)L doublet with hypercharge±1/2. In that case we have:

• In the absence of the external gauging of GSM , H is unbroken, and the Higgs is anexact Goldstone. This means that its potential vanishes at tree level (it can onlyhave derivative couplings).

• The explicit breaking due to the gauging of GSM will induce a potential for theHiggs (it is really a pseudo-Goldstone boson). The orientation of GSM compared toH in the true vacuum (the misalignment of the vacuum) is a new parameter (withrespect to technicolor theories), ε = v/Fπ. This parameter also parameterizes theinterpolation between the SM (ε 1) and pure technicolor (ε→ 1).

6.2.1 The minimal composite Higgs model

Let us consider the minimal composite Higgs model. We want H to include the electroweakgroup, so minimality requires it to be the electroweak group H = SU(2)L × U(1)Y . Wealso want to have at least four Goldstone bosons that can act as the Higgs. The minimalchoice would be a group with 8 = 4 + 4 generators. Let us try with G = SU(3). G hasindeed H as a subgroup and the right number of generators so that there are 4 Goldstonebosons in the coset space. The next requirement is that the four Goldstones transform asthe Higgs under H. We use the Gell-Mann matrices as the generators of SU(3),

T a =λa

2, a = 1, . . . 8. (6.1)

They satisfy the commutation relations, [T a, T b] = ifabcTc with,

f123 = 1, f458 = f678 =

√3

2,

f147 = f165 = f246 = f257 = f345 = f376 =1

2, (6.2)

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all the others (that are not related by total antisymmetry) vanish. In particular, we have(a, b, c = 1, 2, 3)

[T a, T b] = iεabcT c, [T a, T 8] = 0, (6.3)

so that T a generate an SU(2) subgroup and T 8 generates a U(1) subgroup. The cosetspace is spanned by T a with a = 4, 5, 6, 7. Defining T+ ≡ T 4 − iT 5 and T 0 = T 6 − iT 7,and grouping them in a two dimensional vector,

Tφ =

(T+

T 0

), (6.4)

and using the commutation relations, we get,

[T a, Tφ] = −σa

2Tφ, [T 8, Tφ] = −

√3

2Tφ. (6.5)

Thus, defining gSM = g and g′SM =√

3g, we have that Tφ (and therefore the Goldstonesassociated to them) transforms as a doublet under SU(2) with hypercharge 1/2, just likethe SM Higgs. However, this has fixed the hypercharge coupling constant in terms of theSU(3) coupling (as opposed to the one of a U(1) group that does not come from a unifiedgroup). This fixed value is in disagreement with observation, as the sine of the Weinbergangle becomes,

s2W =

g′ 2SMg′ 2SM + g2

SM

=3g2

SM

3g2SM + g2

SM

=3

46= s2

W exp ≈ 0.231. (6.6)

This problem is easily fixed by starting with an extra U(1) field under which the Higgs isnot charged. A linear combination of the U(1) that comes from the SU(3) and the newone will give rise to the hypercharge, now with a free coupling. This minimal model has aphenomenological problem. The symmetry breaking pattern is not custodially symmetricand hadronic resonances will in general induce a too large T parameter in conflict withexperimental data.

6.2.2 The minimal (custodial) composite Higgs model

Let us now construct a fully realistic model. The model will be minimal in the sense thatthe Goldstones in G/H still contain only the Higgs boson but now H is such that the globalsymmetry breaking pattern is custodial invariant. Custodial symmetry requires H to be atleast SU(2)L × SU(2) ∼ SO(4) and the Higgs to transform as a bidoublete (equivalentlyas a fundamental of SO(4)). In order to get only 4 Goldstones we need G to have 10generators (SO(4) has six generators). As we will see, SO(5) fulfills all the requirements.In order to generate a non-trivial Weinberg angle, we will add an extra U(1) group so thatin total we have G = SO(5)× U(1)X and H = SO(4)× U(1)X . We have

# Goldstones = dim (SO(5)/SO(4)) = 4, (6.7)

105

so that we have four real scalars as the only Golstones.We will use the following basis of SO(5) generators, in the fundamental representation,

(T aL)i,j = − i2

[1

2εabc(δbi δ

cj − δbjδci ) + (δai δ

4j − δaj δ4

i )],

(T aR)i,j = − i2

[1

2εabc(δbi δ

cj − δbjδci )− (δai δ

4j − δaj δ4

i )],

(T aC)ij = − i√2

(δai δ5j − δaj δ5

i ), (6.8)

with a = 1, 2, 3, run over the SU(2)L and SU(2)R unbroken groups, a = 1, 2, 3, 4 run overthe coset space SO(5)/SO(4) and i, j = 1, . . . , 5. They are normalized to TrTATB = δAB.The corresponding commutation relations can be worked out to be,

[T aL, TbL] = iεabcT cL, [T aR, T

bR] = iεabcT cR, [T aL, T

bR] = 0, (6.9)

[T aC , TbC ] =

i

2εabc(T cL + T cR), [T aC , T

4C ] =

i

2(T aL − T aR), (6.10)

[T aL,R, TbC ] =

i

2(εabcT cC ± δabT 4

C), [T aL,R, T4C ] = ∓ i

2T aC . (6.11)

In particular, it can be checked from these commutation relations that the generators inthe coset space (and therefore the corresponding Goldstone bosons) transform as a (2, 2)of SU(2)L×SU(2)R (or equivalently as a fundamental, (4) of SO(4)). Note that these areprecisely the quantum numbers of the SM Higgs. The Goldstone bosons can be describedin terms of a field Σ:

Σ = Σ0eΠ/Fπ , (6.12)

where Σ0 = (0, 0, 0, 0, 1), Π = −iT aCha√

2. Expanding the exponential we get,

Σ =sinh/Fπ

h(h1, h2, h3, h4, h coth/Fπ), h =

√h2a. (6.13)

The most general SO(5) × U(1)X invariant Lagrangian, build out of the Goldstonebosons and the external (elementary) gauge fields (for the moment we consider the gaugefields as classical sources), at quadratic order in the gauge fields, is,

Lχ =1

2(PT )µν

[ΠX

0 (p2)XµXν + Π0(p2)Tr(AµAν) + Π1(p2)ΣAµAνΣT]

+ . . . , (6.14)

where (PT )µν = ηµν − pµpν/p2 is the transverse projector so that terms with two gauge

fields are the (quadratic) transverse part of Fµν , which transforms as Fµν → ΩFµνΩ−1,

with Ω an SO(5) transformation. Also we have Σ → ΣΩ−1, we have treated Σ as anexternal classical background (it has no momentum) and Π0,1(p2) and ΠX

0 (p2) are formfactors that encode the dynamics of the strong sector (heavy resonances plus fluctuationsof the Goldstones around the vacuum Σ).

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We can get some information on the form factors by expanding around the SO(4)-preserving vacuum, Σ0,

L =1

2(PT )µν

[ΠX

0 (p2)XµXν + Π0(p2)AaµAaν + (Π0 +

1

2Π1)AaµA

aν

], (6.15)

where Aa here represents the unbroken sources, Aa the broken ones, we have used that(TΣT

0 )i = Ti5 and (T aL)i5 = (T aR)i5 = 0, (T aC)i5 = − i√2δai and (Σ0T )i = T5i with (T aL)5i =

(T aR)5i = 0, (T aC)5i = i√2δai . We thus get

Πa ≡ Π0, Πa = Π0 +1

2Π1. (6.16)

Now, remember that we can only have a massless pole (due to the exchange of the Goldstonebosons) in the two point function of the broken generators. In particular, in the large Nlimit we have

(PT )µνΠa(p2) = 〈JaµJaν 〉 = (p2ηµν − pµpν)

∑n

F 2ρ,n

p2 −m2ρ,n

= (PT )µν∑n

p2F 2ρ,n

p2 −m2ρ,n

,

(PT )µνΠa(p2) = 〈J aµJ aν 〉 = (p2ηµν − pµpν)

[ F 2π

2p2+∑n

F 2a,n

p2 −m2a,n

]= (PT )µν

[F 2π

2+∑n

p2F 2a,n

p2 −m2a,n

]. (6.17)

Comparing the above expressions we get

Π1(0) = F 2π , Π0(0) = 0, ΠX

0 (0) = 0. (6.18)

Now using Σ = (h1sh, h2sh, h3sh, h4sh, ch), with hi ≡ hi/h, sh ≡ sin(h/Fπ) and ch =cos(h/Fπ), and turning on only the SU(2)L × U(1)Y physical gauge fields, with Xµ =A3Rµ = Bµ (Y = TR3 +X), we get

L =1

2(PT )µν

[(ΠX

0 +Π0 +s2h

4Π1)BµBν +(Π0 +

s2h

4Π1)LaµL

aν +2s2

hΠ1H†TLa Y HL

aµBν

], (6.19)

where we have definedH ≡ (h1, h2, h3, h4). (6.20)

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Note: The SM is embedded in the SU(2)L × SU(2)R × U(1)X as follows.

L = −1

4

[1

g2L

L2µν +

1

g2R

R2µν +

1

g2X

X2µν

]= −1

4

[1

g2L

L2µν +

1

g2R

(Rbµν)

2 +1

g′ 2B2µν +

1

g2Z′Z ′ 2µν

],

(6.21)where we have defined

R3µ = Bµ +

g2R

g2R + g2

X

Z ′µ, Xµ = Bµ −g2X

g2R + g2

X

Z ′µ, (6.22)

and the couplings and charges

g′ =gRgX√g2R + g2

X

, gZ′ =√g2R + g2

X , (6.23)

and

Y = TR3 +QX , QZ′ =g2RT

R3 − g2

XQX

g2R + g2

X

. (6.24)

With these redefinitions, the covariant derivative reads, in both basis,

iDµ = LaTLa +RaTRa +QXX+. . . = LaTLa +R

bTRb +YB+QZ′Z′+. . . . (6.25)

Note that the relation between the two bases changes correpondingly if weuse canonical normalization.

Expanding at order O(p2) and setting H = (0, 0, 1, 0), we get (using H†TLa Y HLaµBν →

−14L3µBν)

L = (PT )µν

[1

2

F 2πs

2h

4(BµBν + L3

µL3ν − 2L3

µBν) +F 2πs

2h

4L+µL−ν

]+

1

2(PT )µνp

2[Π′0(0)LaµL

aν + (Π′0(0) + ΠX ′

0 (0))BµBν −s2h

2Π′1(0)L3

µBν

]+ . . . .(6.26)

Note that so far we have not used canonically normalized gauge fields. The couplingconstants are defined by canonically normalizing them,

Π′0(0) = − 1

g2,Π′0(0) + ΠX ′

0 (0) = − 1

g′ 2. (6.27)

With canonical fields, we have

m2W = g2F

2π sin2(〈h〉/F 2

π )

4=g2v2

4, (6.28)

where we have defined v = 246 GeV = Fπ sin 〈h〉Fπ

. Thus, we have,

ε =v

Fπ= sin

〈h〉Fπ

. (6.29)

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In particular, note that the vev of the composite Higgs is not v but related to it throughthe equation above. If we expand around its vev, we can compute the couplings of thephysical Higgs to the gauge bosons,

ha =

00

〈h〉+ h0

. (6.30)

Expading to linear order in h, we have

F 2πs

2h = F 2

π

[s2〈h〉+ 2s〈h〉c〈h〉

h

Fπ+ (c2

〈h〉− s2〈h〉)

h2

F 2π

. . .]

= v2 + 2v√

1− ε2h+ (1− 2ε2)h2 + . . . .

(6.31)Thus, the couplings of the gauge bosons to the Higgs, are reduced with respect to the SMones by

gV V h = gSMV V h√

1− ε2,

gV V hh = gSMV V hh (1− 2ε2)(V = W,Z). (6.32)

This reduction in the Higgs couplings is a generic feature of composite Higgs models. Theparticular dependence on ε depends on the group structure but the reduction is generic.This can be understood by noting that the Goldstones can be parametrized as a belongingto a full representation of the group G, as we did in our case with Σ. However, only thepart transforming under H will give masses to the gauge bosons (in our case Σ1,...,4). Thatpart mixes with the part along the coset (Σ5 in our case), which is a singlet under theSM. The fact that the Higgs lives partly in a singlet reduces its coupling to the SM fieldsaccordingly to that mixing.

This reduced couplings have two obvious effects in the two important roles that theHiggs played in the SM, restoration of unitarity in longitudinal gauge boson scattering andcontribution to EWPT to make the SM compatible with experimental data. Regarding theformer, the amplitude for longitudinal charged gauge boson scattering, taking into accountthe shift in the Higgs couplings, reads,

M(W+W− → W+W−) =g2

4m2W

[s− ss(1− ε

2)

s−m2h

+ t− tt(1− ε2)

t−m2h

]=

g2

4m2W

[sε2 −m2

h(1− ε2)s

s−m2h

+ (s→ t)]. (6.33)

In particular, when ε→ 0 we recover the SM result and the Higgs (which couples as the SMHiggs in that case) is fully responsible for the restoration of unitarity. The limit ε→ 1, onthe other hand, results in no contribution from the Higgs and restoration of unitarity hasto be done by the tower of hadronic resonances (technicolor limit). For intermediate values0 < ε < 1, there is still a term that grows with energy (sε2) and therefore longitudinal

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gauge boson scattering violates unitarity (unless restored by the hadronic resonances) ata scale

Λunitarity =ΛNo Higgs

unitarity

ε. (6.34)

In order for the theory to make sense above that scale, we need the hadronic resonancesto be lighter than that scale mρ ≤ Λunitarity.

The implications of the reduced couplings for EWPT are also easy to estimate. Theleading contribution to the S and T from diagrams including the Higgs, read, in the SMand in the heavy Higgs approximation,

S, T = aS,T logmh + bS,T , (6.35)

where aS,T and bS,T are constants. This contribution can be seen as the Higgs mass cutting-off the UV sensitivity of loops involving gauge bosons. In the SM, the Higgs couplings areprecisely the correct ones to exactly cancel off such UV dependence. In our case, thecouplings are modified and they will not cancel exactly the UV dependence. In the sameheavy Higgs limit, and calling Λ the UV cut-off of our effective description, we get

S, T = aS,T [(1− ε2) logmh + ε2 log Λ] + bS,T

= aS,T [logmh + log(Λ/mh)ε2 ] + bS,T = aS,T log

[mh

(Λ

mh

)ε2]+ bS,T , (6.36)

which simply amounts to using an effective, heavier, Higgs mass in EWPT

mEWPT,eff = mh

(Λ

mh

)ε2. (6.37)

This implies a positive contribution to S and a negative contribution to T ,

∆S =1

12πlog

(m2

EWPT,eff

m2h,ref

), ∆T = − 3

16πc2W

log

(m2

EWPT,eff

m2h,ref

). (6.38)

Of course, we get again the SM result mEWPT,eff = mh for ε→ 0 and the technicolor resultmEWPT,eff = Λ for ε→ 1.

6.2.3 EWSB in the minimal composite Higgs model

In this section we are going to study the one loop potential generated for the Higgs. This isgiven by the Coleman-Weinberg effective potential, which determines the correct vacuumin the presence of explicitly symmetry breaking terms. Before doing the calculation in thecase of the minimal composite Higgs model, we will review a very illuminating examplefrom QCD.

110

Electromagnetic contribution to the pion mass

The QCD example we would like to discuss as a useful motivation of the Higgs potentialin our minimal composite Higgs model is the electromagnetic contribution to the pionmass. Let us consider again 2 quark QCD in the chiral limit (massless quarks). As weknow well, this theory has a global SU(2)L × SU(2)R symmetry spontaneously broken bythe QCD vacuum to SU(2)V . The corresponding Goldstone bosons are the pions. In thechiral limit they are exactly massless. In fact, the have no potential in that limit as alltheir interactions have to be derivative. Gauging part of the global symmetry actuallybreaks it and a potential for the Goldstone bosons (and in particular a mass) is generatedat the quantum level. In order to study such effect, we will turn on the electromagneticinteraction (we gauge the U(1)Q subgroup of H). In practice this means that we includean elementary photon, Aµ, external to the strongly coupled theory (QCD), see Fig. 6.1.We have not introduced at the moment the W± and Z bosons. Furthermore, U(1)Q is not

QCDSU(2)L £ SU(2)R

SU(2)V

A¹

Figure 6.1: Elementary photon external to the strongly coupled theory (QCD). The pionsare the Goldstone Bosons of the spontaneous symmetry breaking of the global symmetry.

broken by the QCD vacuum (it is embedded in the unbroken group, with Q = TL3 +TR3 , asfar as the gauge bosons are concerned). This means that the photon does not eat any ofthe pions and they remain in the spectrum as massless scalars at tree level. The gauging ofthe electromagnetic group generates a potential for the (charged) pions at one loop. Ourgoal in this section is to compute such potential.

We first write the most general SU(2)L × SU(2)R invariant Lagrangian. We use thenon-linear sigma model description of the Goldstone bosons, treating them as constantclassical fields

Σ = eiπaσa/fπ , Σ→ ULΣU †R. (6.39)

To make the analysis easier, we consider gauge bosons for the full SU(2)L×SU(2)R group.They are taken as external sources (we will switch off all of them except for the photon

111

when time comes). At the quadratic level in those fields we have

L =1

2(PT )µν

[ΠL(p2)Tr

(LµLν

)+ ΠR(p2)Tr

(RµRν

)− ΠLR(p2)Tr

(Σ†LµΣRν

)], (6.40)

ΠL(p2), ΠR(p2) and ΠLR(p2) are form factors that encode the effects of the exchange of thehadronic resonances. We can again get information from the form factors by going to theQCD vacuum Σ = 12×2. It is useful to rewrite the action in terms of vector (unbroken)and axial (broken) bosons,

Vµ =Rµ + Lµ√

2, Aµ =

Rµ − Lµ√2

. (6.41)

We get

L =1

2(PT )µν

[1

2

(ΠL+ΠR+ΠLR

)Tr(VµVν)+

1

2

(ΠL+ΠR−ΠLR

)Tr(AµAν)+

1

2

(ΠR−ΠL

)Tr(VµAν+AµVν)

],

(6.42)where

ΠV V =ΠL + ΠR − ΠLR

2, (6.43)

ΠAA =ΠL + ΠR + ΠLR

2, (6.44)

ΠV A =ΠR − ΠL

2. (6.45)

Now we know that the only correlator that can have a massless pole (due to the exchangeof the Goldstones) is 〈AµAν〉,

〈AµAν〉 = (p2ηµν − pµpν)[f 2

π

p2+ . . .

]. (6.46)

This impliesΠV V (0) = ΠV A(0) = 0, ΠAA(0) = f 2

π , (6.47)

and therefore

ΠL(0) = ΠR(0) =1

2ΠLR(0) =

f 2π

2. (6.48)

One other interesting implication is the fact that the left-right form factor

ΠLR(p2) = ΠAA(p2)− ΠV V (p2), (6.49)

is equal to the difference between the axial and vector correlators. In that sense it is anorder parameter of chiral symmetry breaking, since it is zero or not depending on whetherthe chiral symmetry is (un)broken. Also, at energies much higher than ΛQCD at which thebreaking of the chiral symmetry is irrelevant, both correlators should be similar.

112

The effective Lagrangian that involves the photon and the pions is easy to obtain, bysetting Lµ = Rµ = Qvµ, where vµ represents the photon and

Q =

(23

00 −1

3

). (6.50)

We can also use the explicit form of Σ = cos(π/fπ) + iπ ·~σ sin(π/fπ), with π ≡√πaπa and

π ≡ ~π/π. The resulting effective Lagrangian reads,

L =1

2(PT )µνvµvν

[(ΠL(p2) + ΠR(p2))Tr(Q2)− ΠLR(p2)Tr(Σ†QΣQ)

]=

1

2(PT )µνvµvν

[(ΠL(p2) + ΠR(p2)− ΠLR(p2))Tr(Q2)

+ ΠLR(p2)sin2(π/fπ)

π2(π2

1 + π22)], (6.51)

where we have used that Tr [Σ†QΣQ] = Tr[Q2]− sin2(π/f)(π21 + π2

2)/π2. As we could haveexpected, we see that the photon does not induce any correction (as it doesn’t couple toit directly) to the neutral pion but only to the charged ones. We can therefore set in theeffective Lagrangian π3 = 0 and therefore π2

1 + π22 = π2 to get

L =1

2(PT )µνv

µvν[Πγ(p

2) + ΠLR(p2) sin2(π/fπ)], (6.52)

where we have defined Πγ(p2) ≡ 2Tr(Q2)ΠV V (p2).

We can now compute the one loop potential for the pions generated at one loop byphoton exchange. This is given by the Coleman-Weinberg potential, that resums all massinsertions with zero external momentum at one loop (with the photon circulating in theloop). Introducing a generic ξ gauge,

LGF = − 1

2g2ξ(∂µv

µ)2,

we have for the propagator

i[(PT )µν

1

Πγ(p2)− ξ g

2

p2(PL)µν

], (6.53)

and for the “mass terms”i sin2(π/fπ)ΠLR(p2)(PT )µν . (6.54)

The contribution of one loop diagrams with increasing number of mass insertions thengives

V (π) = 3i

∫d4p

(2π)4

∞∑n=1

1

2n

[i2

ΠLR(p2)

Πγ(p2)sin2(π/fπ)

]n=

3

32π2

∫ ∞0

dQ2Q2 log[1 +

ΠLR(Q2)

Πγ(Q2)sin2(π/fπ)

]. (6.55)

113

In the first line the 3 corresponds to the three polarization of the photon for a non-zeropion vev, the factor of 1

2ncomes from symmetry factors in the diagrams whereas in the

last equation we have resummed the series into a logarithm, Wick rotated to Euclideanmomentum Q2 = −p2 and performed the angular integration

∫dΩ = 2π2.

In order to continue with our calculation, we need to argue on the behaviour of theform factors at large Euclidean momenta and therefore on the convergence of the integral.Recall that ΠLR is an order parameter of chiral symmetry breaking and therefore we expectit go to go zero at large momenta. Thus, the integral is expected to be convergent anddominated by momenta smaller than the composite scale (this is what we mean by sayingthat pions are composite states and their masses are not UV sensitive). This statementcan be made more precise by means of the operator product expansion (OPE) of thecorresponding currents.

The time ordered product of two operators, O1,2(x), can be expressed as a sum of localoperators times c-number coefficients that depend on the separation,

TO1(x)O2(0) =∑n

cn12(x)On(0). (6.56)

This equality is at the level of the operators, which imply the equality of any Green functionmade with them. The sum extends over all operators with the same quantum numbers ofthe product O1O2.

The OPE of the product two conserved currents, has the form, in Euclidean momentumspace,

i

∫d4x eiq·xTJµ(x)Jν(0) = (q2ηµν − qµqν)

∑n

c(n)(q)On(0). (6.57)

Purely based on dimensional analysis, it is clear that operators with larger mass dimension,are suppressed by a higher power of q2 at large q,

c(n)(q2) ∼ q−[On]. (6.58)

In QCD, the gauge invariant (color singlets) operators with dimension 6 or lower, spin zero(otherwise they cannot contribute to the vev without breaking Lorentz invariance) are

1 (Unity Operator) (d = 0) (6.59)

OM = ψMψ (d = 4) (6.60)

OG = GaµνG

µνa (d = 4) (6.61)

Oσ = ψσµνT aMψGaµν (d = 6) (6.62)

OΓ = (ψΓ1ψ)(ψΓ2ψ) (d = 6) (6.63)

Of = fabcGaµνG

bνγG

cγµ (d = 6) (6.64)

where M and M are matrices in flavor space, whose elements are proportional to the quarkmasses and Γ1,2 stand for some matrices acting on color, flavor and Lorentz indices. Outof these operators,

114

• The operators OM and Oσ break explicitly the chiral symmetry and therefore vanishin the chiral limit (massless quarks).

• OΓ is the only chiral invariant operator whose vev spontaneously breaks the chiralsymmetry, distinguishing between the axial and vector currents.

As a consequence of the previous results, we have

ΠLR(q2) = q2COΓ(q2)〈OΓ〉+ . . . = q2

[δ

q6+O(1/q8)

]. (6.65)

The coefficient δ has been computed in perturbation theory by Shifman, Vainshtein andZakharov [49],

δ = +8π2(αs/π +O(α2s))(〈ψψ〉)2, (6.66)

using the fact that 〈OΓ〉 factorizes in (〈ψψ〉)2 in the large-N limit. The OPE then guaranteesthe convergence of the integral, since

Πγ(q2) = −q

2

e2+O(q0), (6.67)

for large q2. In fact, the integral can be reasonably well approximated by expanding thelogarithm and keeping only the lowest term in Πγ,

V (π) ≈ 3

8πα sin2

(π

fπ

)∫ ∞0

dQ2 ΠLR(Q2). (6.68)

The integral can be now performed in the large-N limit as follows. We know thecorrelators in the large-N limit,

ΠV V (q2) = q2∑n

f 2ρ,n

q2 −m2ρ,n

, (6.69)

ΠAA(q2) = q2

[f 2π

q2+∑n

f 2a,n

q2 −m2a,n

]. (6.70)

From the OPE we know the large q2 behaviour of the LR correlator, which can be translatedinto sum rules for the decay constants and masses of the hadronic resonances in the largeN limit

ΠLR(Q2) ∼ 1

Q4+O(Q−6), ⇒

limQ2→∞ΠLR(Q2) = 0limQ2→∞Q

2ΠLR(Q2) = 0(6.71)

If we insert in these equations the expression of the LR correlator in terms of the vector andaxial ones in the large N limit, ΠLR = ΠAA − ΠV V , we get the first and second Weinbergsum rules, ∑

n

[f 2ρ,n − f 2

a,n

]= f 2

π , (6.72)∑n

[f 2ρ,nm

2ρ,n − f 2

a,nm2a,n

]= 0, (6.73)

115

where we have used

ΠLR = ΠAA − ΠV V = f 2π +

∑n

[f 2a,n

(1 +

m2a,n

p2+ . . .

)− f 2

ρ,n

(1 +

m2ρ,n

p2+ . . .

)]. (6.74)

A further simplification comes from the assumption that the sum is dominated bythe first vector and axial resonances, ρ, a1. This so-called vector-meson dominance isexperimentally confirmed in the case of QCD. In that case the first two Weinberg sumrules read

f 2ρ − f 2

a1= f 2

π , f 2ρm

2ρ − f 2

a1m2a1

= 0, (6.75)

which give

f 2ρ = f 2

π

m2a1

m2a1−m2

ρ

, f 2a1

= f 2π

m2ρ

m2a1−m2

ρ

. (6.76)

Thus, the Weinberg sum rules, together with vector-meson dominance and the large Nlimit, give, for the large Q2 limit of the LR correlator

ΠLR(Q2) ≈ f 2π

[1 +

m2ρ

m2a1−m2

ρ

Q2

Q2 +m2a1

−m2a1

m2a1−m2

ρ

Q2

Q2 +m2ρ

+ . . .

]. (6.77)

Its integral can then be approximated by

V (π) ≈ 3α

8π2sin2

(π

fπ

)∫ ∞0

dQ2 ΠLR(Q2) ≈ 3α

8π2sin2

(π

fπ

)f 2π

m2a1m2ρ

m2a1−m2

ρ

log

(m2a1

m2ρ

).

(6.78)Since we observe experimentally ma1 > mρ, the integral is positive and the potentialgenerated for the pion is minimized for

sin

(〈π〉fπ

)= 0. (6.79)

Thus, the radiative corrections align the vacuum along the U(1)Q preserving direction.This is in fact a more general result than the approximation we have used here to deriveit. It is a consequence of the general property

ΠLR(Q2) ≥ 0, for 0 ≤ Q2 ≤ ∞, (6.80)

as proved by Witten in the case of QCD [50].Let us summarize the result we have obtained. Gauging the electromagnetic current

explicitly breaks the symmetry, inducing a potential for the pseudo-Goldstone bosons (thepions) at the quantum level. The neutral pion is not affected by this and remains anexact Goldstone in this approximation. The potential generated for the charged pionsinduces a mass squared for them, which is positive, and therefore the electromagneticgauge symmetry is not spontaneously broken in the process. All this was in the chiral

116

limit. If we turn on the u, d quark masses, the neutral pion (and also the charged ones)will get a mass from the quark masses, but the mass difference between charged and neutralpions is still dominated by the one loop electromagnetic contribution we just computed.Our calculation gives,

m2π± −m2

π0 ≈3α

4π

m2a1m2ρ

m2a1−m2

ρ

log

(m2a1

m2ρ

). (6.81)

This result was first obtained in [51] who also used ma1 ≈√

2mρ to get

∆m2γ ≡ m2

π± −m2π0 ≈

3α

2πm2ρ log 2 ≈ 1430 MeV2. (6.82)

This is the electromagnetic contribution to the charged pion mass. The explicit chiralsymmetry breaking from the finite quark masses gives a common mass to all three pionson top of which the electromagnetic contribution sits. Thus we have

mπ± =√m2π0 + ∆m2

γ ≈ mπ0 +∆m2γ

2mπ0

≈ mπ0 + 5.3 MeV, (6.83)

which is not far from the experimental result

mπ± −mπ0 = 4.6 MeV. (6.84)

Using the large N limit of the correlators we can compute other observables. Anexample in the QCD chiral Lagrangian that be relevant in composite Higgs models is thekinetic mixing between left and right fields,

Lχ =f 2π

4Tr[(DµΣ)†DµΣ

]+ . . .+ L10Tr

[Σ†LµνΣR

µν]

+ . . .

= −1

2(PT )µν

[f 2π − 4p2L10 + . . .

]Tr[Σ†LµΣRν

]+ . . . , (6.85)

so that the LR correlator, in terms of chiral coefficients reads

ΠLR(p2) = f 2π − p24L10 + . . . , (6.86)

or alternatively

L10 = −4[Π′AA(0)− Π′V V (0)

], (6.87)

where the prime denotes derivation with respect to p2. The chiral coefficient L10 is theQCD analogue of the S parameter for electroweak symmetry breaking. Inserting in thisexpression Weinberg sum rules, we get

L10 = −4

(f 2ρ

m2ρ

−f 2a1

m2a1

)= −4f 2

π

(1

m2ρ

+1

m2a1

)≤ 0. (6.88)

Note that it has a well defined sign. In fact, we will obtain a similar result in compositeHiggs models for the S parameter.

117

One loop potential in the minimal composite Higgs model

The previous discussion of the electromagnetic contribution to the pion mass can be trans-lated almost unchanged to the Higgs potential in composite Higgs models. For simplicity,we will consider the contribution of the SU(2)L gauge fields, neglecting the smaller (g′ g)contribution from the U(1)Y one. The effective Lagrangian reads in that case

LSU(2)L =1

2(PT )µν

(Π0 +

s2h

4Π1

)LaµL

aν . (6.89)

Introducing again a generic ξ gauge,

LGF = − 1

2g2ξ(∂µL

aµ)2,

we have for the propagator

i[(PT )µν

1

Π0(p2)− ξ g

2

p2(PL)µν

], (6.90)

and for the “mass terms”

isin2(h/Fπ)

4Π1(p2)(PT )µν . (6.91)

The contribution of one loop diagrams with increasing number of mass insertions thengives

V (π) =9

2

∫ ∞0

d4Q

(2π)4log[1 +

Π1(Q2)

Π0(Q2)

sin2(h/Fπ)

4

]. (6.92)

The factor of 9 stands for 3 polarizations times three SU(2)L gauge bosons. Once again,we have to argue on the convergence of this integral. Recall that Π0 is related to thecorrelator of two unbroken currents

〈Jµa Jνa 〉 = (PT )µνΠ0(p2), (6.93)

whereas Π1 is to the difference between broken and unbroken

〈Jµa Jνa 〉 − 〈Jµa J

νa 〉 = −1

2(PT )µνΠ1(p2). (6.94)

Note that much like ΠLR for QCD, Π1 here is an order parameter of the symmetry breaking,which is sensitive to EWSB at energies of order Fπ but goes to zero at momenta Q2 Fπ.It is in this sense that composite models solve the hierarchy problem, since the Higgspotential is insensitive to scales much higher than the compositeness scale. The correctway of making these statements more precise is by using the OPE, as we did in the QCDcase. The operator with the smallest dimension contributing to both broken and unbrokencorrelators is the identity operator,

〈Jµa,aJνa,a〉 = (q2ηµν − qµqν)

[C

(1)a,a(q

2) +∑n>1

C(n)a,a (Q2)〈On〉

]. (6.95)

118

For instance, if the strong sector is asymptotically free, we have, for the large Q2 limit

C(1)(Q2) = − N

24π2log

Q2

µ2+O(αs). (6.96)

Thus, if we want the integral that determines the Higgs potential to be convergent, weneed the first operator contributing to the difference of broken and unbroken generators tobe dimension 5 or greater,

〈Jµa Jνa 〉 − 〈Jµa J

νa 〉 = (q2ηµν − qµqν)

[C(5)(q2)〈O5〉+ . . .

], (6.97)

so that

Π1(Q2) ∼ 1

Qn−2=

1

(Q2)n/2−1, n ≥ 5 (for Q2 →∞). (6.98)

Let us therefore assume that Π1(Q2 → ∞) goes to zero fast enough for the integral toconverge, expanding the Logarithm we get,

V (h) =9

8

g2

16π2sin2

(h

Fπ

)∫ ∞0

dQ2 Π1(Q2). (6.99)

The analogy with QCD suggests that Π1(Q2) is positive, and therefore EWSB is nottriggered by this correction (gauge interactions tend to align the vacuum with the symmetrypreserving one). This was overcome in the original models by Georgi and Kaplan byintroducing an extra gauged U(1)A group in the elementary sector, in addition to the SMgauge group SU(2)L × U(1)Y . The crucial property to trigger EWSB was that the newgauge symmetry was not a subset of the unbroken H symmetry of the strong interaction(whereas the SM group is). This way the radiative corrections from U(1)A destabilized theEW-preserving vacuum and EWSB could be triggered. In that case, we would have

ε =v

Fπ= sin

〈h〉Fπ6= 0 (6.100)

at the minimum of the potential.The reason behind this extra complication of the model was that, at the time of the

original models, the top quark was thought to be much lighter than the electroweak gaugebosons and therefore its possible contribution (which as we will see comes with the oppositesign) to the Higgs potential would be negligible. The fact that the top is so heavy changesthe picture completely and it is in fact the top contribution the most important one. Wewill discuss such contribution in future lectures. Also, we will use five-dimensional modelsto compute the corresponding form factors and that way being able to fully compute theHiggs potential in a realistic composite Higgs model. For the moment, let us assume thereis some extra contribution to the Higgs potential that induces EWSB, ε 6= 0. As wementioned, the S parameter is, in the electroweak theory, the analogue of the L10 chiralcoefficient. From Eq.(6.26) we have

L = −(PT )µνp2 s

2h

4Π′1(0)L3

µBν + . . . , (6.101)

119

which implies an S parameter (note that we have to canonically normalize the correspond-ing gauge fields)

S = 16πΠ′3B(0) = 16πs2h

4Π′1(0)4πε2Π′1(0) = 8πε2

(f 2ρ

m2ρ

−f 2a1

m2a1

)= 4πε2F 2

π

(1

m2ρ

+1

m2a1

).

(6.102)If we now use ε2F 2

π = v2 and a relation similar to the QCD one between the hadronicmasses m2

a1∼√

2m2ρ, we get

S ≈ 6π

(v2

m2ρ

). (6.103)

This expression for S in terms of v and mρ is in fact very similar for composite Higgsmodels and for technicolor models. The difference is in the relation between these twoparameters. In technicolor we have

v = Fπ ∼√N

4πmρ,⇒ STC ≈

3N

8π, (6.104)

whereas in composite Higgs models we have

v = εFπ = ε

√N

4πmρ. (6.105)

Thus, ε controls the ratio v/mρ and therefore the contribution to the S parameter

SCH ∼ ε2STC . (6.106)

A bound on S then turns into an upper bound on ε

S ≤ 0.3⇒ ε2 ≤ 48π

60N≈ 1

4

(10

N

), (6.107)

where we have defined F 2π

m2ρ≡ N

16π2 .

This bound on ε is mild enough to allow for phenomenologically viable composite Higgsmodels without too much fine-tuning. We have however not discussed yet one of the otherbig problems in technicolor theories, the generation of fermion masses and the associatedpresence of dangerous flavor violating processes. In fact, composite Higgs models suffersome of the same problems (mildly improved by the presence of the ε parameter) as tech-nicolor models in regards of fermion masses, if fermion masses are generated the same wayas in technicolor.

6.3 Fermions in Composite Higgs Models

Fermions can get a mass in composite Higgs models in a similar way to technicolor theories,by coupling to some composite operator which can be the Higgs itself or can couple to theHiggs so that electroweak symmetry breaking is propagated to the fermions and a massfor them is generated.

120

6.3.1 Fermion masses the wrong way

Let’s assume, just as in technicolor, that the coupling to the composite operator is bilinear,

L = λqqO(x). (6.108)

The operatorO has the quantum numbers of the Higgs and therefore its square can generatea correction to the Higgs mass. In order for it not to be sensitive to UV physics (so thatthe hierarchy problem is not reintroduced), the anomalous dimensions has to be [O2] ≥ 4(the operator has to be irrelevant). At weak coupling or in the large N limit, this boundimplies

[O] ≥ 2. (6.109)

Then the fermion masses scale as

mq ∼ ΛTC

(ΛTC

Λ

)[O]−1

, ΛTC ∼ mρ. (6.110)

The bound on the anomalous dimension of the corresponding operator means that Λ cannotbe too large but that means that FCNC are not enough suppressed and therefore in conflictwith observation,

(qq)2

Λ2.

So far, we essentially have the same result as in technicolor theories. One difference withcomposite Higgs models is that Fπ and therefore mρ can be made arbitrarily large bymaking ε small,

mρ ≈v

ε

4π√N. (6.111)

Thus, FCNC can be evaded by taking ε 1. Unfortunately, this can be done only at theexpense of fine-tuning. This is the problem we have found already, we can technically takethe SM limit in composite Higgs models and all the problems associated to technicolorlike theories will disappear, but of course the Hierarchy problem will then reappear. Wesaw that with unitarity restoration in longitudinal gauge boson scattering, we saw it inthe contribution to the S parameter and we see it now in the contribution to FCNC fromthe generation of fermion masses. The former two are mild enough to make the modelcompatible with just a mild fine-tuning. The FCNC however, is so restrictive that fullyreintroduces the hierarchy problem. In fact, for ε 1, the following estimate holds,

µ2 ∼ g2

16π2m2ρ, λ ∼ g2

N, (6.112)

so that the natural scale for the Higgs vev is

v2 ∼ µ2/λ ∼ N

16π2m2ρ ∼ f 2

π . (6.113)

Thus in general we have several contributions that are naturally of order F 2π but their sum

has to be fine-tuned to give a result ε2F 2π , the naive fine-tuning is therefore ε2.

121

6.3.2 Fermion masses the right way

There is a way of solving the FCNC problem that we observed in the previous section.The solution is to couple the elementary fermions to the composite states linearly insteadof quadratically [52]

L = λ[qO + h.c.]. (6.114)

In that case, the quark masses scale like

mq ∼ εmρ

(mρ

Λ

)2γ

, (6.115)

where γ = [Oψ]− 4 = [O] + 32− 4 = [O]− 5

2. The bound on [O] from the requirement of

no UV instabilities is

[O∂O] ≥ 4⇒ [O] ≥ 3

2(large N). (6.116)

Thus we get2γ = 2[O]− 5 ≥ −2. (6.117)

Recall that the unitarity bound ([O] ≥ 3/2) is compatible with no reintroduction of UVinstabilities. In particular, we can now generate heavy enough masses with a large valueof Λ provided γ is close enough to zero. The FCNC four-fermion operators involvingelementary fields are then suppressed by the safely large scale Λ and therefore compatiblewith observations.

The linear coupling of elementary fermions to composite operators not only improvesthe constraints from FCNC, they also allow us to generate hierarchical fermion masses ina natural way. The reason is the following. Suppose that at some high scale Λ ΛEW ,the strong sector is in the vicinity of an IR fixed points, so that the theory behaves likea conformal field theory at energies below Λ. In that case, the fermion masses can beestimated to be

mq ∼ λL(µ)λR(µ)N

16π2εmρ, (6.118)

where µ ∼ ΛEW and λL,R are the corresponding Wilson coefficients for the couplings ofqL,R to the corresponding composite Operators. If the corresponding anomalous dimensionsγL,R are small and positive, we have

µd

dµλ = γλ⇒ λ(µ) ≈ λ0

(µΛ

)γ. (6.119)

Given that µ/Λ 1, we have that small changes in the anomalous dimensions can inducevery large changes in the low energy fermion masses

mq ∼(mρ

Λ

)γL+γR√N

4πv . (6.120)

Is this enough to generate the large top mass? The answer is yes, provided the anoma-lous dimensions are negative. In that case the corresponding operator is relevant and the

122

coupling λ grows in the IR. The growth of λ at lower energies forces us to include higherorders in the RGE for λ (due to wave-function renormalization),

µd

dµλ = γλ+ c

N

16π2λ3. (6.121)

If c is positive, λ continues to grow in the IR until it hits a new IR fixed point

λ∗ ≈√−γc

4π√N. (6.122)

Note that the IR fixed point is still perturbative (in λ) in the large N limit. We can thisway get a large enough mass to reproduce the top mass. For instance, if both OL,R arerelevant, we get

mt ∼4π√Nv√γLγR. (6.123)

Thus, the linear coupling of elementary fermions to composite operators allowed us tohave a large UV scale Λ to suppress FCNC while still generating large enough fermionmasses and even the top mass. This construction has even more interesting features thatwe will discuss in the next few sections.

Note that there are new possible contributions to FCNC induced by the exchange ofcomposite states at the scale mρ. However, the coupling of any SM fermion to the com-posite sector occurs through the Wilson coefficients λi and therefore FCNC four-fermioninteractions are proportional to the SM fermion masses,

qiqjqkqlΛ2

∼√λiλjλkλl

m2ρ

, (6.124)

this naturally predict small flavor violation for light SM fermions and large violationsin top couplings. The former, which are very stronly constrained experimentally, aresuppressed enough to be compatible with experiment whereas the latter are essentiallyuntested experimentally but will be subject to study at the LHC.

6.3.3 Partial compositeness

The linear coupling of elementary fermions to the composite operators induces anotherinteresting feature: partial compositeness. Partial compositeness stands for the followingproperty. Imagine the operator O to which the elementary fermion couples linearly hasthe right quantum numbers to excite a tower of massive fermionic composite states ξn,

〈0|O|ξn〉 = ∆n. (6.125)

The linear couplingL = ψO + h.c., (6.126)

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induces then at low energies a mass mixing between the composite fermionic states andthe elementary fermions

Lmix =∑n

∆n(ψξn + h.c.). (6.127)

Note: The same coupling is induced between elementary gauge bosons andvector resonances that are excited by the corresponding conserved current,

〈0|Jµ|ρn〉 = εµmρ,nfρ,n. (6.128)

We then haveLmix =

∑n

mρ,nfρ,nAµρµn, (6.129)

which is known in the literature as ρ photon mixing.

This mass mixing means that the physical states, before EWSB, will be an admixture ofelementary and composite states, thus the name partial compositeness. Take for simplicityonly one resonance, the Lagrangian in the elementary/composite basis reads,

L = ψLi∂ψL + χ(i∂ −m)ξ + ∆(ψLξR + h.c.). (6.130)

Due to the mixing term, ψL and χ are not mass eigenstates. They can be easily diagonalizedby taking (

ψlLψhL

)=

(cosφ sinφ− sinφ cosφ

)(ψLξL

), (6.131)

with tanφ = ∆/m. The resulting Lagrangian reads

L → ψlLi∂ψlL + ψh(i∂ −

m

cosφ)ψh. (6.132)

ψl,h represent the light (SM) and heavy physical fields, respectively. Note that the SMfermions are still massless (before EWSB), this is due to conservation of the fermionicindex. Charged fermions can only get Dirac masses that mix a LH field with a RH field.The fermion content we started with was two LH fields and one RH field, thus, afterrotation we will be left with a single LH field that has to be massless. Once EWS isbroken, it will marry another massless fermion with the opposite chirality to get a mass.The angle φ parametrizes the degree of compositeness of the SM fermions:

• The larger φ the more composite a SM particle is.

• In order to solve the hierarchy problem, the Higgs should be totally composite.

• Heavier particles (after EWSB) couple more to the Higgs and therefore are morecomposite, lighter particles are more elementary

λ = g∗ sinφL sinφR. (6.133)

124

The same considerations apply to the gauge sector

L = − 1

4g2e

F 2e,µν −

1

4g2∗F 2c,µν +

M2∗

2

(geg∗Aeµ − Acµ

)2

, (6.134)

which can be diagonalized by(Aµρµ

)=

(cos θ sin θ− sin θ cos θ

)(AeµAcµ

), (6.135)

to give

L = −1

2F 2µν +

1

2

(DµρνDνρµ −DµρνDµρν

)+

M2∗

2 cos2 θρ2µ + . . . . (6.136)

This simple model is very useful to parametrized general composite Higgs models and theirextra dimensional counterparts, see [53] for more details.

6.3.4 Fermion contribution to the Higgs potential

We saw in the previous section that the top gets its large mass as a consequence of it beingpartially composite, its couplings to the composite sector are large, λ(µ) ∼ 4π

√−γ/√N ,

and that induces a large coupling to the composite sector. But this large coupling suggeststhat the top sector could significantly contribute to the Higgs potential at one loop. Recallthat the top is an elementary field, external to the strong sector, that couples linearly (andstrongly) to some of the composite states. This coupling breaks the global symmetry andinduces a potential for the pseudo-Goldstone bosons (our composite Higgs) at one loop.Let us investigate such contribution.

First we have to choose a representation for the fermions. Each SM fermion will cou-ple linearly to an operator of the strongly coupled theory which has to transform as arepresentation of SO(5)× U(1)X

∆L = λ(ψO + h.c.). (6.137)

As we did with the gauge bosons, the simplest thing to do is to promote the SM to fullrepresentations of SO(5)×U(1)X by means of spurions that we will set to zero at the righttime. The smallest representations we can embed the SM fermions are the spinorial (4),fundamental (5) or adjoint (10) representations of SO(5). The advantage of the latter twois that they allow us to implement a L↔ R parity within the custodial symmetry that canbe used to protect the Zbb coupling. For simplicity we will consider just the top sector, asthe bottom and other light fermions are not expected to give a sizable contribution to theHiggs potential. We choose to incorporate the SM fermions into fundamentals of SO(5),

ΨqL =

(q′LqL)

u′L

, ΨtR =

(qtRq′ tR)

tR

, (6.138)

125

where both multiplets have QX = 2/3 and only qL and tR are elementary fields (theSM third generation quark doublet and the RH top), all other fields are the spurionsneeded to complete full SO(5) representations. The most general SO(5)×U(1)X invariantLagrangian, at the quadratic order and in momentum space, is

∆L =∑

r=qL,tR

Ψirp(δijΠr

0(p) + ΣiΣjΠr1(p)

)Ψjr

+ ΨiqL

(δijM0(p) + ΣiΣjM1(p)

)ΨjuR + h.c., (6.139)

where i = 1, . . . , 5 run over the SO(5) indices in the fundamental representation, ΠqL,uR0,1 (p)

and M0,1(p) are form factors that parametrize the effect of the composite sector and wehave

Σ = sh(h1, h2, h3, h4, ch/sh). (6.140)

Setting to zero the spution fields, we get the effective Lagrangian for the SM fields and theHiggs,

Leff = qLp

[Πq

0(p2) +s2h

2Πq

1(p2)HcHc†]qL + uRp

[Πu

0(p2) +s2h

2Πu

1(p2)

]uR

+shch√

2Mu

1 (p2)qLHcuR + h.c., (6.141)

wher we have defined

Hc =

(−h3 − ih4

h1 + ih2

), (6.142)

and we have

Πq0,1 = ΠqL

0,1, Πu0 = ΠuR

0 + ΠuR1 , Πu

1 = −1

2ΠuR

1 , Mu1 = M1. (6.143)

From this effective Lagrangian we can give a good estimation of the top mass by settingp2 = 0,

mt ≈shch√

2

Mu1 (0)√

ZuLZuR, (6.144)

where the wave function renormalization constants are

ZuL,R = Πq,u0 (0) +

s2h

2Πq,u

1 (0). (6.145)

The effective Lagrangian also allows us to compute the contribution to the Higgs effectivepotential, through the Coleman-Weinberg formula

V (h) = −2Nc

∫d4Q

(2π)4

log

(1 +

s2h

2

Πq1

Πq0

)+ log

(1 +

s2h

2

Πu1

Πu0

)+ log

(1− s2

hc2h

2

(Mu1 )2

(−Q2)[Πq0 + Πq

1s2h/2][Πu

0 + Πu1s

2h/2]

). (6.146)

126

The first two contributions come from diagrams that only involve tL or only involve tR andthey are similar to the ones we found for gauge bosons. The last contribution comes fromdiagrams that involve both tL and tR. In that case, the full term proportional to p givesus the propagator

i

p(Πq,u0 + Πq,u

1 s2h/2)

, (6.147)

and the term proportional to Mu1 gives us the vertex (note that in order to close the loop

we need two of those, thus the square in the contribution to the Higgs potential). Now if weassume as usual that the integrals are convergent and that we can expand the logarithms,we get a potential of the form

V (h) ≈ αs2h − βs2

hc2h = (α− β)s2

h + βs4h, (6.148)

where

α = −2Nc

∫d4Q

(2π)4

1

2

Πq1

Πq0

+1

2

Πu1

Πu0

, (6.149)

β = 2Nc

∫d4Q

(2π)4

1

2

(Mu1 )2

(Q2)Πq0Πu

0

. (6.150)

In this approximation, we have

FπV′ = ch[2(α− β)sh + 4βs3

h] = s2h[α− β + 2βs2h], (6.151)

F 2πV′′ = 2[α− β + 2βs2

h] + s2h[4βshch] = 2[α− β + 2βs2h] + 8βs2

hc2h, (6.152)

where a prime here denotes derivative with respect to h. In particular we have EWSB(V ′′(0) < 0) if α−β < 0. In that case, there is a minimum of the potential with 0 < sh < 1for β > 0 and β > |α|

〈s2h〉 = ε2 =

β − α2β

. (6.153)

Recall that if the minimum is at sh = 1 we go back to the technicolor limit. We have toadd this top contribution to the gauge boson one, which is in general smaller (althoughfor realistic values of the higgs vev becomes relevant). Whether one can actually get areasonable vev for the Higgs or not depends on the actual values of the integrals. Onething we can do (we’ll see it in future lectures) is to resort to five-dimensional modelsin warped extra dimensions, which are weakly coupled duals of these composite Higgsmodels. In the 5D picture, one can compute the form factors and the integrals and thefollowing results can be found. In general |αL,R| are parametrically larger than β, however,we naturally have that αL and αR have opposite signs and therefore a partial cancellationis possible, leading to β > |αL + αR| and therefore to a natural realistic pattern EWSB.

Our potential also gives us an estimate of the Higgs mass,

m2h = V ′′(hmin) =

8β

F 2π

s2hc

2h. (6.154)

127

Now we have

β =

∫d4Q

(2π)4

2Nc

Q2F (Q2), (6.155)

where F (Q2) is defined in Eq. (6.150). If we define some scale Λ at which the integral iseffectively cut-off

Λ2 ≡ 2

∫ ∞0

dQQF (Q2)

F (0), (6.156)

that we would expect to be of the order of the mass of the first fermionic resonances, andwe use the approximate expression for the top mass

m2t ≈

s2hc

2h

2F (0), (6.157)

we get for the Higgs mass

m2h ≈

Nc

π2

m2t

v2ε2Λ2. (6.158)

Numerically we get

mh ≈ 190 GeV( ε

0.5

)( Λ

1 TeV

), (6.159)

thus one expects a light Higgs in this scenario (in practice, 5D models typically give aneven lighter Higgs, mostly because of the presence of light fermionic composites that implya lower value of Λ).

128

Chapter 7

Models with Extra Dimensions

Literature: This section has been taken almost entirely from RobertoContino’s notes.

Models with more than four space-time dimensions offer new possibilities for a naturalrealization of EWSB. They can provide new mechanism to explain the stability of the EWscale against UV physics and also other observed hierarchies in nature, like fermion masses.We will be interested in them also as new realizations of strongly coupled theories and inparticular of composite Higgs models.

The simplest way to make models with extra dimensions compatible with observationis to compactify the extra dimensions with a size below current experimental reach. Letus discuss how one can analyze these models. We will take as an example models withone extra dimension. Although more dimensions can give rise to new phenomenology, thesimpler 5D models provide enough intuition and tools to study other models.

7.1 5D models compactified on a circle

7.1.1 Gauge theories in a circle

Let us consider a gauge theory in 5D with the fifth dimension compactified on a circle ofradius R. The action reads,

S =

∫d4xdy−1

4FMNF

MN =

∫d4xdy−1

4FµνF

µν +1

2Fµ 5F

µ5

=

∫d4xdy−1

4FµνF

µν − 1

2Aµ∂

25A

µ +1

2∂µA5∂

µA5 − ∂5Aµ∂µA5, (7.1)

where we have denoted with M,N all five dimensions, µ, ν the four extended dimensions(denoted with an x, whereas the extra dimension is denoted with a y) and used a mostlyminus metric ηMN = diag(1,−1,−1,−1,−1). We have also integrated by parts when

129

necessary (in the circle there are no remaining boundary terms). Let us now insert theFourier expansion of AM ,

AM(x, y) =∞∑n=0

1√πR

[1

1 + δn,0Ae(n)M (x) cos(ny/R) + A

o(n)M (x) sin(ny/R)

], (7.2)

in the action and integrate over the extra dimension y to get

S =

∫d4x

∑n

−1

4F e(n)µν F e(n)µν−1

4F o(n)µν F o(n)µν+

1

2(mnA

e(n)µ +∂µA

o(n)5 )2+

1

2(mnA

o(n)µ −∂µA

e(n)5 )2,

(7.3)where mn ≡ n/R and we have used the orthonormality of the Fourier basis. Note that,

except for n = 0, Ao,e(n)5 is the eaten Goldstone boson of A

e,o(n)µ to give it a mass. Thus, we

have described the 5D gauge boson as a massless 4D gauge boson, Ae(0)µ , a 4D massless scalar

Ae(0)5 , and two towers of massive gauge bosons with masses mn = n/R. We have considered

here a U(1) gauge group as an example. Since we are solving only for the quadratic termsso far, the result is identical for non-abelian gauge groups, although in that case, themassless scalar would transform in the adjoint representation of the corresponding gaugegroup. Also note that in the R → 0 limit, all massive modes decouple and we are onlyleft with 4D massless fields (the so called zero modes) which is the mathematical way ofsaying that we cannot see the extra dimension if it is small enough. This also means thatat energies below the inverse of the size of the extra dimension, we just see the world asfour-dimensional. Only when we probe energies high enough, we will start perceiving the5D nature (or equivalently, the tower of KK modes).

Gauges

Consider the case of a non compact extra dimension (R → ∞). We can then get to theA5 = 0 gauge by means of the gauge transformation

Ω = P exp[ig

∫ y

−∞dy′A5(x, y′)

], ∂yΩ = igA5Ω. (7.4)

Under this transformation, A5 transforms as,

A′5 =i

gΩ−1[∂5 − igA5]Ω =

i

gΩ−1[igA5 − igA5]Ω = 0. (7.5)

However, if R is finite, Ω is not periodic, due to Ae(0)5 (x) (the lower limit of the integral

would be in that case 0). We can still use Ω ≡ Ω exp[−igyAe(0)5 ], which is periodic. In that

case we haveA′5 = A

e(0)5 (x). (7.6)

Thus, we can eliminate the y dependent part of A5 but not the zero mode (which agreeswith the fact that it is not an eaten Goldstone boson). This corresponds then to the

130

unitary gauge, in which all would-be Goldstone bosons are eaten and only physical modesappear.

Anoter alternative is to go to an Rξ type gauge by adding the following gauge-fixingterm

LGF =

∫d4xdy

[− 1

2ξ(∂µA

µ − ξ∂5A5)2

], (7.7)

which eliminates the Aµ − A5 mixing

L+LGF =

∫d4xdy

−1

2Aν

[(1− 1

2ξ

)∂µ∂ν − gµν∂ρ∂ρ

]Aµ −

1

2Aµ∂

25A

µ − 1

2A5∂

ρ∂ρA5 +ξ

2A5∂

25A5

.

(7.8)

7.1.2 Fermions in a circle

Let us now consider fermions in our 5D space. One important feature of fermions in 5D isthat the smallest spinorial representation of the Lorentz group in 5D is a four-componentDirac fermion. The Gamma matrices in 5D are

ΓM = (γµ,−iγ5), (7.9)

so that γ5 is part of Lorentz transformation and ΨL and ΨR are related by them. Thisalso means that we can actually write a Dirac mass term. The action reads,

S =

∫d4xdyΨ(iΓM∂M−M)Ψ =

∫d4xdyΨLi∂ΨL+ΨRi∂ΨR−ΨL[∂5+M ]ΨR+ΨR[∂5−M ]ΨL.

(7.10)Inserting again the corresponding Fourier expansion, we get

S =

∫d4xdy

∑n

ψe(n)L i∂ψ

e(n)L + ψ

o(n)L i∂ψ

o(n)L + ψ

e(n)R i∂ψ

e(n)R + ψ

o(n)R i∂ψ

o(n)R (7.11)

−mn[ψe(n)L ψ

o(n)R − ψo(n)

L ψe(n)R + h.c.]−M [ψ

e(n)L ψ

e(n)R + ψ

o(n)L ψ

o(n)R + h.c.].

The quadratic terms are not diagonal in the Fourier basis, however, they only mix level bylevel with a mass matrix(

ψe(n)L ψ

o(n)L

)( M mn

−mn M

)(ψe(n)R

ψo(n)R

)+ h.c., (7.12)

which can be diagonalized with the rotation(ψe(n)L,R

ψo(n)L,R

)→ 1√

2

(i 1−i 1

)(ψe(n)L,R

ψo(n)L,R

), n 6= 0. (7.13)

131

The action reads, in the new basis,

S →∫d4xdy

∑n

ψe(n)L i∂ψ

e(n)L + ψ

o(n)L i∂ψ

o(n)L + ψ

e(n)R i∂ψ

e(n)R + ψ

o(n)R i∂ψ

o(n)R

−[(M + imn)ψe(n)L ψ

e(n)R + (M − imn)ψ

o(n)L ψ

o(n)R + h.c.], (7.14)

so that we end up with a vector-like “zero mode”, ψe(0)L,R , with mass M and two towers of

“massive KK modes” with masses,√M2 + n2

R2 . As with the gauge bosons, even if M = 0,

we have just zero modes below energies ∼ R−1 so that for small enough extra dimensionsthey are undetectable.

7.1.3 Couplings in extra dimensions

So far we have described the quadratic terms in the action of 5D fields in terms of infinitetowers of 4D KK modes. Let us consider now the interactions among different fields. Asan example, we take the coupling of fermions and gauge bosons. The relevant part of theaction reads

∆S =

∫d4xdyg5ΨγµΨAµ =

∑mnr

∫d4x

g5√πR

[ ∫dyfΨLm fΨL

n fArπR

]ψ

(m)L γµψ

(n)L A(r)

µ + L→ R,

(7.15)where we have used the general KK expansions

Aµ =∑n

fAn√πR

A(n)µ , ΨL,R =

∑n

fΨL,Rn√πR

ψ(n)L,R. (7.16)

The mass dimensions of different fields and couplings are not the same if 5D as in 4D.Taking into account that the action has to be dimensionless, we get the following massdimensions

[Aµ] =3

2, [Ψ] = 2, [g5] = −1

2. (7.17)

In particular, g5/√πR is dimensionless. The structure of the coupling above is generic, the

effective 4D coupling is given by the 5D coupling, properly normalized, times the overlapof the corresponding KK mode wave functions.

7.2 5D models compactified on an interval

The model we just outlined above is not realistic. For once, we have obtained vector-likefermions contrary to what we observe in the SM fermions, which are chiral. Second, wehave obtained a massless scalar, which is the remnant of a higher-dimensional gauge field(and a closer look at its properties shows that the field is massless because a protectionfrom the 5D gauge invariance), and therefore an excellent candidate for a naturally light

132

Higgs boson. However, it transforms in the adjoint representation of the correspondinggauge group, again different from what we observe in nature (the Higgs is a doublet, asopposed to a triplet of SU(2)L). These problems can be both solved if we compactify ona singular space, instead of a smooth one like the circle. The simplest possibility is toassume the extra dimension is compactified not in a circle but in an S1/Z2 orbifold, whichis just a circle with the points y → −y identified. This is a smooth manifold except at thetwo fixed points y = 0, πR. Fields can be classified into even or odd under the y → −y Z2

symmetry so that, out of the two towers (sines and cosines) of the Fourier expansion, onlyhalf of it, the one with the right parity properties, will survive. In fact, it is common toconsider more general orbifolds, like S1/(Z2 × Z ′2), in which independent parities at y = 0and y = πR are assigned to the different fields. We will in fact consider a compactificationthat is equivalent to the most general 5D orbifold, an interval 0 ≤ y ≤ L. We now have twoboundaries in our extra dimension, which can have their corresponding boundary operatorsand that will receive contributions from boundary terms whenever we integrate by parts.Our fields will have boundary conditions that have to be compatible with the variation ofthe boundary action.

7.2.1 Models in an interval

Let us start with the example of a fermion in an interval. Assuming for the moment noboundary actions, we have

S =

∫d4xdy

1

2(ΨiΓM∂MΨ− i(∂MΨ)†Γ0ΓMΨ)−MΨΨ

, (7.18)

where we have written the action in an explicitly hermitian way. The variation of theaction then reads

δS =

∫d4xdy

1

2(δΨiΓM∂MΨ + ΨiΓM∂MδΨ− i(∂MδΨ)†Γ0ΓMΨ− i(∂MΨ)†Γ0ΓMδΨ)

−MδΨΨ−MΨδΨ

=

∫d4xdy

δΨ(iΓM∂M −M)Ψ + [(i∂MΨ)ΓM −MΨ]δΨ

+

1

2

∫d4x[Ψγ5δΨ− δΨγ5Ψ

]L0, (7.19)

where the last term corresponds to the boundary contribution from integration by parts.The variation of the bulk part gives the bulk equations of motion

i∂ΨL,R + (±∂5 −M)ΨR,L = 0, (7.20)

whereas the boundary part of the variation provides the consistent boundary conditions.As an example, we can choose Dirichlet boundary conditions for either of the two chiralities,

133

but once the b.c. for one chirality is chosen, the b.c. for the other chirality is fixed by theequations of motion,

ΨR,L(yi) = 0⇒ (∂5 ±M)ΨL,R

∣∣∣yi

= 0. (7.21)

We denote these simple fermionic boundary conditions by [±,±], where the first sign refersto the b.c. at y = 0 and the second at y = L with a [+] denoting ΨR = 0 whereas a [−]denotes ΨL = 0. More general boundary conditions can be imposed if boundary terms arepresent in the action. Let us for the moment see where this simple choice takes us. Let usconsider a general KK expansion

ΨL,R(x, y) =∑n

fL,Rn (y)ψ(n)L,R(x), (7.22)

where we assume that the 4D fields satisfy the 4D Dirac equation

i∂ψ(n)L,R = mnψ

(n)R,L. (7.23)

Inserting this expansion in the equations of motion we obtain the equation for the fermionicprofiles

(±∂5 −M)fR,Ln = −mnfL,Rn , (7.24)

which is a system of two coupled first order differential equations. We can decouple themby iteration to obtain

[∂25 ± (∂5M)−M2 +m2

n]fL,Rn = 0. (7.25)

The solutions depend on the value of M and the choice of boundary conditions. Let usdiscuss the presence of possible massless zero modes. If m0 = 0 then the tow first orderequations are already decoupled and we can solve them immediately,

fL,R0 = AL,Re∓My, (7.26)

with AL,R the normalization constant to be determined by the normalization condition∫ L

0

dyfLmfLn =

∫ L

0

dyfRmfRn = δmn. (7.27)

Now, a Dirichlet b.c. at any boundary forces the corresponding normalization constant tovanish so that we will only have a zero mode for the following b.c.

[++] ⇒ fL0 (y) = Ae−My, (7.28)

[−−] ⇒ fR0 (y) = AeMy, (7.29)

[±,∓] ⇒ no zero mode. (7.30)

Thus, contrary to the circle, the interval allows for chiral zero modes.

134

A similar game can be played with gauge bosons. In that case, simple boundaryconditions compatible with the variation of the action are [+] ≡ A5 = 0 and [−] ≡ Aµ = 0.Again, independent boundary conditions can be imposed at each brane. The result is,

[++] ⇒ A(0)µ , (7.31)

[−−] ⇒ A(0)5 , (7.32)

[±,∓] ⇒ no zero mode. (7.33)

In particular we now have zero modes for Aµ and A5 along different gauge directions,which means that now the scalar zero mode does not need to transform under the adjointrepresentation of the gauge group with [++] b.c.. In fact, these choices of b.c. allow us tobreak the gauge group in a spontaneous way. The existence of Aµ zero modes correspondsto unbroken gauge symmetries at low energies. Twisted b.c. correspond to the symmetriesbeing broken locally at the corresponding boundary. Finally, symmetries broken at bothbranes result in zero modes for A5. Recall that this latter zero mode is physical and cannotbe gauged away.

Let us consider a general gauge group G broken to a subgroup H0 at the boundaryy = 0 and to a subgroup H1 at y = L. This is accomplished by imposing the followingboundary conditions

Aa [++] ⇒ T a ∈ AlgH0 ∩H1, (7.34)

Aa [+−] ⇒ T a ∈ AlgH0/(H0 ∩H1), (7.35)

Aa [−+] ⇒ T a ∈ AlgH1/(H0 ∩H1), (7.36)

Aa [−−] ⇒ T a ∈ AlgG/(H0 ∪H1). (7.37)

In particular we have that the low energy gauge symmetry group is H0 ∩H1 and A(0)5 lives

on the coset space G/(H0 ∪ H1). We could now use this structure to realize a patternof symmetry breaking that we are familiar with. Let us consider a bulk SO(5) × U(1)Xsymmetry, broken to SO(4)×U(1)X at y = L and to SU(2)L×U(1)Y (with Y = T 3

R +X)at y = 0. In that case we have

Aa [++] ∈ AlgSU(2)L × U(1)Y , (7.38)

Aa [−+] ∈ Alg(SO(4)× U(1)X)/(SU(2)L × U(1)Y ), (7.39)

Aa [−−] ∈ AlgSO(5)/SO(4). (7.40)

This theory has an SU(2)L × U(1)Y unbroken gauge invariance, a massless scalar Aa5transforming as a 4 of SO(4) (an SU(2)L × SU(2)R bidoublet), plus a tower of massivevectors forming split SO(5) × U(1)X adjoints. The fact that the boundary conditionsexplicitly break the SO(4) symmetry imply that a potential for A5 will be generated at theloop level. Locality and gauge invariance in the extra dimension however guarantees thatthis potential is finite, as the only gauge invariant contribution has to come from a non-local Wilson line. In order to make the connection with composite Higgs models, we will

135

compute the one loop contribution to the A5 potential from bulk fermions. We could do itby inserting all KK modes in the corresponding Coleman-Weinberg expression. However,there is another method that will give the result in a way that is completely paralel to theone we used for composite Higgs models.

7.2.2 Holographic Method

Our goal is to compute the effective Lagrangian for the low energy fields, including theA5, in a way that can be directly parametrized by the effective Lagrangian we used incomposite Higgs models. That way we can borrow our calculation of the Higgs potentialfrom last chapter and we will have directly our answer. The idea is to compute the effectiveLagrangian in two steps.

1. First, we integrate out the bulk degrees of freedom by using the 5D equations ofmotion (at the classical level) for fixed values of the fields on one boundary, sayy = 0.

Z =

∫dΦeiS[Φ] =

∫dφ0e

iS0[φ0]

∫φ0

ei(S[Φ]−S0[φ0]) =

∫dφ0e

iS0[φ0]+iSeff [φ0]. (7.41)

The resulting effective action Seff [φ0] is in general a 4D non-local action for theboundary fields φ0.

2. Using the 4D boundary action (S0 +Seff) we compute the A5 potential by performingloops of the boundary matter fields φ0. Since we aim at its potential, we can treatA

(0)5 as a classical external background (its legs have no momentum).

This procedure is equivalent to the KK calculation since we know that hte potential is anon-local effect, that requires any 5D loop to propagate from one brane to the other andtherefore all 5D loops receive a contribution from loops of φ0.

Another important observation is the following. First, we can always go to an almostaxial gauge in which A5 = A

(0)5 . Then we can always make a field redefinition of the form,

Φ→ Φ′ ≡ ΩΦ, AM → A′M ≡i

gΩ−1[∂M − igAM ]Ω, (7.42)

whereΩ = eiθ(x,y) = e−ig

R Ly dy′ A

(0)5 (x). (7.43)

This field redefinition is a gauge transformation except that it does not satisfy the cor-responding boundary condition at y = 0 (it does not vanish). At every other point ofthe extra dimension it coincides with the b.c. that makes A5 = 0. Thus, with this fieldredefinition, we eliminat A5 from everywhere in the bulk and at the y = L brane, leavingall the dependence of A

(0)5 at the y = 0 brane. A5 disappears from everywhere except at

the corresponding b.c., which is now

φ′0 ≡ Φ′(y = 0) = eiθ(y=0)φ0 ≡ eiθ0φ0, (7.44)

136

where θ0 ≡ −La0A(0)5 .

To summarize, the theory with A5 and boundary conditions φ0 is equivalent to a theorywith vanishing A5 and boundary conditions φ′0 = eiθ0φ0. Thanks to this observation, wecan perform the calculation of the potential by deriving the boundary action for the fieldsφ′0 (with A5 = 0), and at the end of the calculation reinterpret φ′0 = eiθ0φ0.

Let us apply this holographic method to bulk fermions and then we will use the resultsto compute the effective potential in a toy model of electroweak symmetry breaking. Thebulk fermionic action reads

S =1

g25

∫d4xdy

1

2(ΨiΓM∂MΨ− i(∂MΨ)†Γ0ΓMΨ)−MΨΨ

, (7.45)

where we have written the action in an explicitly hermitian way and we have factored outa global factor g−2

5 so that the 5D fermions have mass dimension 3/2 as in 4D. Recall thatthe variation of the action then reads

g25δS =

∫d4xdy

1

2(δΨiΓM∂MΨ + ΨiΓM∂MδΨ− i(∂MδΨ)†Γ0ΓMΨ− i(∂MΨ)†Γ0ΓMδΨ)

−MδΨΨ−MΨδΨ

=

∫d4xdy

δΨ(iΓM∂M −M)Ψ + [(i∂MΨ)ΓM −MΨ]δΨ

+

1

2

∫d4x[Ψγ5δΨ− δΨγ5Ψ

]L0. (7.46)

Now, instead of using Dirichlet b.c. for one of the two chiralities as we did when expandingin KK modes (this choice guaranteed the variation of the boundary action vanished), wewant to choose a holographic boundary condition. Again, we cannot choose independentb.c. for each chirality and we will choose,

ΨL(y = 0) = ψ0L, (7.47)

whereas ΨR is free to vary at y = 0. At y = L on the other hand we will choose standardDirichlet b.c. for one of the two chiralities. Note that with this choice of b.c., the boundaryaction no longer vanishes, since we now have, at y = 0, δΨL = 0 but ΨLδΨR 6= 0. Thus,we have to add an extra boundary action so that our boundary conditions are compatiblewith the variational principle of the action. The required boundary action is [54]

S0 =1

2

1

g25

∫d4xdy [ΨLΨR + ΨRΨL]

∣∣∣y=0

. (7.48)

Following the holographic approach, we have to solve the bulk equations of motion withb.c. as in Eq. (7.47) and insert the solution back in the action. The bulk equation of motionis the Dirac equation that when inserted back in the 5D bulk action makes it vanish, thus

137

the only remaining part of the action after inserting the EoM is the boundary term we hadto add

Seff =1

2g25

∫d4x[ψ0

Lψ0R + ψ0

Rψ0L], (7.49)

where we have defined ψ0R ≡ ΨR(y = 0), to be determined in terms of ψ0

L by the corre-sponding solution of the equations of motion.

The bulk EoM read,− pΨL,R + (±∂5 −M)ΨR,L = 0, (7.50)

where we have Fourier transformed the four extended dimensions into momentum space,denoted by pµ. We will solve these equations by using the ansatz

ΨL,R(p, y) = fL,R(p, y)ψL,R(p), (7.51)

with p ≡ √pµpµ. We can always require that the four dimensional fields satisfy the off-shell4D Dirac equation,

pψL,R(p) = pψR,L(p), (7.52)

so that we obtain EoM for the 5D profiles

− pfL,R + (±∂5 −M)fR,L = 0. (7.53)

These two coupled first order equations can be decoupled by iteration

(−∂25 +M2)fL,R = p2fL,R ⇒ (∂2

5 + ω2)fL,R = 0, (7.54)

where we have defined ω2 ≡ p2 −M2. The precise solution depends on the b.c. at y = L,we have two possibilities:

• Case L+, ΨR(L) = 0. The solution reads

fL(p, y) =1

p

ω cos[ω(y − L)]−M sin[ω(y − L)]

, (7.55)

fR(p, y) = sin[ω(y − L)]. (7.56)

• Case L−, ΨL(L) = 0. The solution reads

fL(p, y) = sin[ω(y − L)], (7.57)

fR(p, y) = −1

p

ω cos[ω(y − L)] +M sin[ω(y − L)]

. (7.58)

We can now write the contribution to the boundary action. Recall

ψ0R = ΨR(p, 0) = fR(p, 0)ψR(p) = fR(p, 0) p

pψL(p) =

fR(p, 0)

fL(p, 0)p

pψ0L, (7.59)

138

so that the boundary action reads

Seff =1

g25

∫d4x ψ0

L

fR(p, 0)

fL(p, 0)p

pψ0L ≡

L

g25

∫d4x ψ0

LΣ(p)pψ0L, (7.60)

where we have defined

Σ(p) ≡ 1

pL

fR(p, 0)

fL(p, 0). (7.61)

Let us now apply this formalism to a toy model of EWSB. Instead of a realisticSO(5)/SO(4) symmetry breaking pattern, we will consider a bulk SO(2) gauge theorybroken on both branes and an SO(2) bulk fermion doublet. The boundary conditions are

Aµ ∼ [−−],

(Ψ1 ∼ [++]Ψ2 ∼ [−−]

). (7.62)

The effective action for the boundary fields reads

Seff =L

g25

∫d4x

(ψ

0(1)L ψ

0(2)L

)p

(Σ(+)(p) 0

0 Σ(−)(p)

)(ψ

0(1)L

ψ0(2)L

), (7.63)

where we have defined the Σ functions for the L± branches,

Σ(+) = − 1

ML+ ωL cot(ωL), (7.64)

Σ(−) =ω cot(ωL)−M

p2L. (7.65)

Now we take into account that the fields we are using are really the rotated ones, ψ0(i)L →

(eiθ0)ijψ0(j)L , with

eiθ0 = e−Ah(x)/f =

(cos(h/f) − sin(h/f)sin(h/f) cos(h/f)

), (7.66)

where f ≡ 1/√Lg2

5, h is the properly normalized A(0)5 and

A ≡(

0 1−1 0

). (7.67)

The now Higgs dependent effective action reads

Seff =L

g25

∫d4x

(ψ

0(1)L ψ

0(2)L

)p

(Σ(+) − s2

h(Σ(+) − Σ(−)) −shch(Σ(+) − Σ(−))

−shch(Σ(+) − Σ(−)) Σ(−) + s2h(Σ

(+) − Σ(−))

)(ψ

0(1)L

ψ0(2)L

),

(7.68)where we have defined sh ≡ sinh/f and ch ≡ cosh/f . In order to impose the [−, .] b.c. for

Ψ2 we have to set ψ0(2)L to zero and keep only ψ

0(1)L as a dynamical variable. The resulting

action is

Seff =L

g25

∫d4x ψ

0(1)L p

[Σ(+) − sin2 h(x)

f(Σ(+) − Σ(−))

]ψ

0(1)L . (7.69)

139

This expression closely resembles the effective Lagrangian for fermions and the Higgs incomposite Higgs models. In both cases the Higgs enters as an “angular variable”, ascorresponds to a Higgs that is originally a Goldstone boson. Also the dynamics of thenew physics (composite states in one case, KK modes in the other) is encoded in the formfactors Σ(±)(p). The difference now is that we have explicit expressions for the form factorsand can therefore compute exactly the Higgs potential. In fact, we can directly use theresults we obtained for general form factors in the case of composite Higgs models to writedown the contribution to the Higgs potential in our extra dimensional toy model,

V (h) = −2

∫d4Q

(2π)4log

[1− sin2

(h

f

)Σ(+)(Q)− Σ(−)(Q)

Σ(+)(Q)

], (7.70)

where Q2 = −p2 is the Euclidean momentum. The convergence of the integral depens onthe convergence properties of Σ(+) − Σ(−), which as usual is an order parameter. In ourcase we have, in Euclidean momentum

Σ(+)(Q)− Σ(−)(Q)

Σ(+)(Q)= 1− 1

Q2

[ω2Ecoth2(ωEL)−M2

], (7.71)

so that at large momentum, this order parameter vanishes exponentially,

Σ(+)(Q)− Σ(−)(Q)

Σ(+)(Q)∼ −4

(1 +

M2

Q2

)e−2QL, (for QL,Q/M 1). (7.72)

This exponential convergence corresponds, from a 4D conformal field theory point of view,to the conformal symmetry being spontaneously broken by an operator of infinite scalingdimension. Since the convergence is so strong, we get a good approximation by expandingthe logarithm, getting

V (h) ≈ −2

8π2

∫ ∞0

dQQ(Q2 +M2)(coth2ωEL− 1) sin2 h/f = − 1

4π2

1

L4f(m) sin2 h

f, (7.73)

where we have defined

f(m) ≡∫ ∞m

dωEω3E(coth2ωE − 1), (7.74)

with, for instance f(0) = 32ζ(3) ≈ 1.8.

This calculation shows how the corresponding Higgs potential is finite and how fermionscan give a contribution to the Higgs potential with non-vanishing minimum (note the sign ofthe contribution). More realistic models include realistic symmetry structure (for instanceSO(5)/SO(4)) and warped extra dimensions instead of flat space. A minimal realisticmodel of natural electroweak symmetry breaking in models with warped extra dimensionswas proposed in [55].

140

Chapter 8

Conclusions

The goal of these lectures was to introduce the main implications of symmetries and symme-try breaking and their application to the exciting case of electroweak symmetry breaking.After a discussion of general topics related to symmetries, we have focused on EWSB inthe standard model and beyond. The hierarchy problem and natural realizations of EWSBhave been the main guiding principle. Of course, even a full semester course cannot beenough to cover the vast amount of literature and model building that has taken place inthe last few decades on the subject of EWSB. Thus, the choice of topics has been quitebiased. The main deciding factor has been the extremely good set of lecture notes by R.Contino, that we kindly allowed me to use as a draft script. Although there have beenquite a number of additions and some subtractions, the main line of argumentation andchoice of topics are mostly due to him. Of course I have tried to give it my personal pointas often as possible (and in doing so I might have introduced some mistakes in the notesthat are only my responsibility). In particular I hope that the emphasis in ideas ratherthan particular models will be enough to overcome the lack of discussion in some interest-ing developments in theoretical particle physics and EWSB that we did not have time tocover in the lectures.

141

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