the standard model · electroweak gauge theory, which describes the weak and electro-magnetic...

T H O M A S B E C H E R A N D M A R T I N H O F E R I C H T E R

T H E S TA N D A R D M O D E L

U N I V E R S I TÄT B E R N

courses for master students at the university of bern, spring semesters 2017, 2020

[email protected], [email protected]

This script is adapted from lecture notes obtained from Thomas Teubner, who inherited them fromSacha Davidson, who in turn had taken them from Adrian Signer. Large parts (in particular chapters 3

and 6) go back to an even earlier version by Douglas Ross, modified by Adrian Signer. Chapters 1, 2, 4,and 5 were rewritten by Sacha Davidson, following Guido Altarelli’s Standard Model course at the LesHouches summer school 1990. We thank Adrian Signer and Thomas Teubner for allowing us to use andmodify the LaTeX source of their lecture notes for our courses. Compared to the original version, wehave extended the Sections 3, 4, 7, and 8, and added Sections 9, 10, and 11. Despite the large number ofco-authors, we take full responsibility for typos and mistakes introduced or overlooked by us.

Bern, May 2017 and May 2020

Contents

1 Introduction 5

2 QED, an Abelian Gauge Theory 9

3 Non-Abelian Gauge Theories 17

4 Quantum Chromodynamics 31

5 Spontaneous Symmetry Breaking 41

6 The Standard Model with one Family 51

7 Additional Generations, the CKM Matrix, and CP Violation 69

8 Neutrinos 89

9 Anomalies and Anomaly Cancellation 105

10 Non-perturbative effects 117

11 Physics beyond the Standard Model 121

1Introduction

The Standard Model (SM) of particle physics describes all knownparticles and their interactions, except gravity. After more thanthirty years of experimental tests at ever increasing precision, thistheory it still consistent with all available data, with no compellingevidence for physics beyond.1 Over the years, experiments have 1 We discard some deviations in

electroweak precision measurementswhich are far from conclusive; thereis also a 3− 4 σ deviation betweenmeasurement and SM prediction ofg− 2 of the muon.

verified all the different elements of the theory: starting with thepresence of “neutral current” weak interactions2, to the existence of

2 Gargamelle experiment, 1974.

the Z and W bosons3, to the Cabibbo-Kobayashi-Maskawa (CKM)

3 UA1 and UA2 collaborations, 1983.

structure of quark mixing, and culminating with the recent discov-ery of a Higgs-like boson.4 The only observed deviation from the

4 ATLAS and CMS, July 2012.original version of the model are the neutrino masses, which are,however, easily accommodated.

Despite this tremendous success, most particle physicists haveambivalent feelings about the SM.5 The theory has some very at- 5 As, perhaps, evidenced by its horrible

name.tractive features, but also suffers from several problems, and in-volves many free parameters. Before turning to its deficiencies, letus first discuss its best feature: it provides a unified description, interms of “gauge theories”, of the strong, electromagnetic and weakinteractions. A gauge theory is one that possesses invariance undera set of “local transformations”, i.e. transformations whose param-eters are space-time dependent. Electromagnetism is a well-knownexample of a gauge theory. In this case the gauge transformationsare local complex phase transformations of the fields of chargedparticles, and gauge invariance necessitates the introduction of amassless vector (spin-1) particle, called the photon, whose exchangemediates the electromagnetic interactions.

In the 1950’s Yang and Mills considered (as a purely mathe-matical exercise) extending gauge invariance to include local non-abelian (i.e. non-commuting) transformations such as SU(2).6 In 6 Chen-Ning Yang and Robert L. Mills.

Conservation of Isotopic Spin andIsotopic Gauge Invariance. Phys.Rev.,96:191–195, 1954

this case one needs a set of massless vector fields (three in the caseof SU(2)), which were formally called “Yang-Mills” fields, but arenow known as “gauge fields”. In order to apply such a gauge the-ory to weak interactions, one considers particles which transforminto each other under the weak interaction, such as a u-quark and ad-quark, or an electron and a neutrino, to be arranged in doubletsof weak isospin. The three gauge bosons are interpreted as the W±

and Z bosons, that mediate weak interactions in the same way that

6 thomas becher, martin hoferichter

the photon mediates electromagnetic interactions.The difficulty in the case of weak interactions was that they are

known to be short range, mediated by very massive vector bosons,whereas Yang-Mills fields are required to be massless in order topreserve gauge invariance. The apparent paradox was solved by theapplication of the “Higgs mechanism”.1 This is a prescription for 1 Peter W. Higgs. Broken symmetries,

massless particles and gauge fields.Phys.Lett., 12:132–133, 1964; F. Englertand R. Brout. Broken Symmetry andthe Mass of Gauge Vector Mesons.Phys.Rev.Lett., 13:321–323, 1964; andPeter W. Higgs. Broken Symmetriesand the Masses of Gauge Bosons.Phys.Rev.Lett., 13:508–509, 1964

breaking the gauge symmetry spontaneously. In this scenario onestarts with a theory that possesses the required gauge invariance,but where the ground state of the theory is not invariant underthe gauge transformations. The breaking of the invariance arisesin the quantization of the theory, whereas the Lagrangian onlycontains terms which are invariant. One of the consequences of thisis that the gauge bosons acquire a mass and the theory can thus beapplied to weak interactions.

Spontaneous symmetry breaking and the Higgs mechanism haveanother extremely important consequence. It leads to a renormaliz-able theory with massive vector bosons.2 This means that one can 2 Gerard ’t Hooft and M.J.G. Veltman.

Regularization and Renormalization ofGauge Fields. Nucl.Phys., B44:189–213,1972

carry out a programme of renormalization in which the infinitiesthat arise in higher-order calculations can be reabsorbed into theparameters of the Lagrangian (as in the case of QED). Had one sim-ply broken the gauge invariance explicitly by adding mass termsfor the gauge bosons, the resulting theory would not have beenrenormalizable and therefore could not have been used to carry outperturbative calculations. A consequence of the Higgs mechanismis the existence of a scalar (spin-0) particle, the Higgs boson.

The remaining step was to apply the ideas of gauge theories tothe strong interaction.3 The gauge theory of the strong interaction 3 H. Fritzsch, Murray Gell-Mann, and

H. Leutwyler. Advantages of theColor Octet Gluon Picture. Phys.Lett.,B47:365–368, 1973

is called “Quantum Chromo Dynamics” (QCD). In this theory thequarks possess an internal property called “colour” and the gaugetransformations are local transformations between quarks of dif-ferent colours. The gauge bosons of QCD are called “gluons” andthey mediate the strong interaction. The union of QCD and theelectroweak gauge theory, which describes the weak and electro-magnetic interactions, is known as the Standard Model. It has avery simple structure and the different forces of nature are treatedin the same fashion, i.e. as gauge theories. This is quite remarkable,given how different the weak, electromagnetic and strong interac-tions are at low energies.

Even neglecting neutrino masses4, it has nineteen fundamental 4 Accounting for neutrino massesrequires at least seven additionalparameters.

parameters, most of which are associated with the masses of thegauge bosons, the quarks and leptons, and the Higgs. Neverthe-less these are not all independent and, for example, the ratio ofthe W and Z boson masses are (correctly) predicted by the model.Since the theory is renormalizable, perturbative calculations canbe performed at higher order that predict cross sections and de-cay rates for both strongly and weakly interacting processes. Thesepredictions, when confronted with experimental data, have beenconfirmed very successfully. As both predictions and data are be-coming more and more precise, the tests of the Standard Model are

the standard model 7

becoming increasingly stringent.In this course, we will discuss all the different elements of the

Standard Model in detail, and also discuss some of the associatedphysics. We then turn to some of the theoretical problems of thetheory (strong CP-problem, naturalness, triviality) and will brieflydiscuss a number of proposals for physics beyond the StandardModel, which were invented to address some of its shortcomings.

2QED, an Abelian Gauge Theory

The aim of this lecture is to start from a symmetry of the fermionLagrangian and show that “gauging” this symmetry (= making itwell behaved) implies classical electromagnetism with its gauge in-variance, the eeγ interaction, and that the photon must be massless.

2.1 Preliminaries

In the Field Theory lectures at this school, the quantum theory ofan interacting scalar field was introduced, and the voyage fromthe Lagrangian to the Feynman rules was made. Fermions can bequantised in a similar way, and the propagators one obtains arethe Green functions for the Dirac wave equation (the inverse ofthe Dirac operator) of the QED/QCD course. In this course, I willstart from the Lagrangian (as opposed to the wave equation) ofa free Dirac fermion, and add interactions, to construct the Stan-dard Model Lagrangian in classical field theory. That is, the fieldsare treated as functions, and I will not discuss creation and anni-hiliation operators. However, to extract Feynman rules from theLagrangian, I will implicitly rely on the rules developed for scalarfields in the Field Theory course.

2.2 Gauge Transformations

Consider the Lagrangian density for a free Dirac field ψ:

L = ψ(iγµ∂µ −m

)ψ (2.1)

This Lagrangian density is invariant under a phase transformationof the fermion field

ψ → eiωψ, ψ → e−iωψ, (2.2)

where ω is a real constant (i.e. independent of x) and ψ is the con-jugate field.

The set of all numbers eiω form a group1. This particular group 1 A group is a mathematical term for aset, where multiplication of elements isdefined and results in another elementof the set. Furthermore, there has tobe a 1 element (s.t. 1× a = a) andan inverse (s.t. a× a−1 = 1) for eachelement a of the set.

is “abelian” which is to say that any two elements of the groupcommute. This just means that

eiω1 eiω2 = eiω2 eiω1 . (2.3)


This particular group is called U(1) which means the group of allunitary 1× 1 matrices. A unitary matrix satisfies U+ = U−1 withU+ being the adjoint matrix.

We can now state the invariance of the Lagrangian eq. (2.1) un-der phase transformations in a more fancy way by saying that theLagrangian is invariant under global U(1) transformations. Byglobal we mean that ω does not depend on x.

For the purposes of these lectures it will usually be sufficient toconsider infinitesimal group transformations, i.e. we assume thatthe parameter ω is sufficiently small that we can expand in ω andneglect all but the linear term. Thus we write

eiω = 1 + i ω + O(ω2). (2.4)

Under such infinitesimal phase transformations the field ψ changesaccording to

ψ→ ψ + δψ = ψ + i ω ψ, (2.5)

and the conjugate field ψ by

ψ→ ψ + δψ = ψ− i ω ψ, (2.6)

such that the Lagrangian density remains unchanged (to order ω).At this point we should note that global transformations are not

very attractive from a theoretical point of view. The reason is thatmaking the same transformation at every space-time point requiresthat all these points ’know’ about the transformation. But if I wereto make a certain transformation at the top of Mont Blanc, how cana point somewhere in England know about it? It would take sometime for a signal to travel from the Alps to England.

Thus, we have two options at this point. Either, we simply notethe invariance of eq. (2.1) under global U(1) transformations andput this aside as a curiosity, or we insist that invariance undergauge transformations is a fundamental property of nature. If wetake the latter option we have to require invariance under localtransformations. Local means that the parameter of the transfor-mation, ω, now depends on the space-time point x. Such local (i.e.space-time dependent) transformations are called “gauge transfor-mations”.

If the parameter ω depends on the space-time point then thefield ψ transforms as follows under infinitesimal transformations

δψ(x) = i ω(x)ψ(x); δψ(x) = −i ω(x)ψ(x). (2.7)

Note that the Lagrangian density eq. (2.1) now is no longer invari-ant under these transformations, because of the partial derivativebetween ψ and ψ. This derivative will act on the space-time depen-dent parameter ω(x) such that the Lagrangian density changes byan amount δL, where

δL = −ψ(x) γµ[∂µω(x)

]ψ(x). (2.8)

The square brackets in [∂µω(x)] are introduced to indicate thatthe derivative ∂µ acts only inside the brackets. It turns out that we


can restore gauge invariance if we assume that the fermion fieldinteracts with a vector field Aµ, called a “gauge field”, with aninteraction term

− e ψ γµ Aµψ (2.9)

added to the Lagrangian density which now becomes

L = ψ(iγµ

(∂µ + i e Aµ

)−m

)ψ. (2.10)

In order for this to work we must also assume that apart from thefermion field transforming under a gauge transformation accordingto eq. (2.7) the gauge field, Aµ, also changes according to

−eAµ → −e(Aµ + δAµ(x)) = −e Aµ + ∂µ ω(x). (2.11)

So δAµ(x) = −∂µ ω(x)/e.

Exercise 1.1Using eqs. (2.7) and (2.11) show that under a gauge transformationδ(−e ψ γµ Aµψ) = ψ(x) γµ

[∂µω(x)

]ψ(x).

This change exactly cancels with eq. (2.8), so that once this inter-action term has been added the gauge invariance is restored. Werecognize eq. (2.10) as being the fermionic part of the Lagrangiandensity for QED, where e is the electric charge of the fermion andAµ is the photon field.

In order to have a proper quantum field theory, in which we canexpand the photon field Aµ in terms of creation and annihilationoperators for photons, we need a kinetic term for the photon, i.e.a term which is quadratic in the derivative of the field Aµ. With-out such a term the Euler-Lagrange equation for the gauge fieldwould be an algebraic equation and we could use it to eliminatethe gauge field altogether from the Lagrangian. We need to ensurethat in introducing a kinetic term we do not spoil the invarianceunder gauge transformations. This is achieved by defining the fieldstrength tensor, Fµν, as

Fµν ≡ ∂µ Aν − ∂ν Aµ, (2.12)

where the derivative is understood to act on the A-field only.1 It is 1 Strictly speaking we should thereforewrite Fµν = [∂µ Aν] − [∂ν Aµ]; youwill find that the brackets are oftenomitted.

easy to see that under the gauge transformation eq. (2.11) each ofthe two terms on the right hand side of eq. (2.12) change, but thechanges cancel out. Thus we may add to the Lagrangian any termwhich depends on Fµν (and which is Lorentz invariant, thus, withall Lorentz indices contracted). Such a term is aFµνFµν. This givesthe desired term which is quadratic in the derivative of the field Aµ.If we choose the constant a to be −1/4 then the Lagrange equationsof motion match exactly (the relativistic formulation of) Maxwell’sequations.2 2 The determination of this constant

a is the only place that a match toQED has been used. The rest of theLagrangian density is obtained purelyfrom the requirement of local U(1)invariance. A different constant wouldsimply mean a different normalizationof the photon field.

We have thus arrived at the Lagrangian density for QED, butfrom the viewpoint of demanding invariance under U(1) gauge


transformations rather than starting with Maxwell’s equations andformulating the equivalent quantum field theory.

The Lagrangian density for QED is:

L = −14

FµνFµν + ψ(iγµ

(∂µ + i e Aµ

)−m

)ψ. (2.13)

Exercise 1.2Starting with the Lagrangian density for QED write down theEuler-Lagrange equations for the gauge field Aµ and show that thisresults in Maxwell’s equations.

In the Field Theory lectures, we have seen that a term λφ4 inthe Lagrangian gave 4!λ as the coupling of four φs in perturbationtheory. Neglecting the combinatoric factors, it is plausible thateq. (2.13) gives the γee Feynman Rule used in the QED course,−ieγµ, for negatively charged particles.

Note that we are not allowed to add a mass term for the photon.A term such as M2 Aµ Aµ added to the Lagrangian density is notinvariant under gauge transformations as it would lead to

δL =2M2

eAµ(x)∂µω(x) 6= 0. (2.14)

Thus the masslessness of the photon can be understood in terms ofthe requirement that the Lagrangian be gauge invariant.

2.3 Covariant Derivatives

Before leaving the abelian case, we will introduce the concept ofa “covariant derivative”, which will be an invaluable tool whenwe extend these ideas to non-abelian gauge theories. We alreadynoted that the derivative of a field, which transforms under gaugetransformations does not transform as a field. Put more bluntly,the ordinary derivative of a gauge field is just not meaningful in agauge theory. The derivative along the direction of some unit vectornµ is obtained as a limit

nµ∂µψ(x) = limε→0

1ε[ψ(x + nε)− ψ(x)] (2.15)

but the two fields transform differently under the gauge transfor-mation because they live at different points. In order to be ableto compare the two fields, we need to introduce a quantity whichcompensates the phase difference at neighbouring points. This“link field” U(y, x) must transform as

U(y, x)→ eiω(y) U(y, x) e−iω(x) (2.16)

and should have the properties U(x, x) = 1, U†(x, y) = U(y, x)and |U(x, y)| = 1. In terms of the link field, we can then define thecovariant derivative

nµDµψ(x) = limε→0

1ε[ψ(x + nε)−U(x + nε, x)ψ(x)] (2.17)


If gauge theories are discretized, one will indeed work with the linkfield U(y, x).1 On the other hand, in the continuum limit, we can 1 The fermion fields will live on a

grid of space-time points, and thelink fields will connect neighbouringpoints.

parameterize the link field as

U(x + nε, x) = 1 + ie nµ Aµ(x)ε +O(ε2) . (2.18)

The gauge field Aµ(x) is also called the “connection”. In terms ofthe connection, we then have

Dµ ≡ ∂µ + i e Aµ. (2.19)

It has the property that given the transformations of the fermionfield eq. (2.7) and the gauge field eq. (2.11) the quantity Dµψ trans-forms in the same way under gauge transformations as ψ. Let usnote that the same problem with derivatives arises in curved space-time: simply taking a derivative with respect to coordinates is notmeaningful since a coordinate change affects different points differ-ently. Also in this case, one needs to introduce a covariant deriva-tive which involves the “affine” (or “Levi-Civita”) connection.

We may thus rewrite the QED Lagrangian density as

L = −14

FµνFµν + ψ(iγµDµ −m

)ψ. (2.20)

Furthermore the field strength Fµν can be expressed in terms of thecommutator of two covariant derivatives, i.e.

Fµν = − ie[Dµ, Dν

]

= − ie[∂µ, ∂ν] + [∂µ, Aν] + [Aµ, ∂ν] + i e [Aµ, Aν]

= ∂µ Aν − ∂ν Aµ, (2.21)

where in the last line we have adopted the conventional notationagain and left out the square brackets. Notice that when usingeq. (2.21) the derivatives act only on the A-field.

2.4 Gauge Fixing

The guiding principle of this chapter has been to hold on to theU(1) symmetry. This forced us to introduce a new massless fieldAµ which we could interpret as the photon. In this subsection wewill try to quantise the photon field (e.g. calculate its propagator)by naively following the prescription used for scalars and fermions,which will not work. This should not be surprising, because Aµ

has four real components, introduced to maintain gauge symme-try. However the physical photon has two polarisation states andthe presence of the unphysical polarizations in the formulation ofthe theory leads to technical difficulties which can be resolved by“fixing the gauge” (breaking our precious gauge symmetry) in theLagrangian in such a way as to maintain the gauge symmetry inobservables. In this section, we will introduce a gauge fixing termby hand, without showing that the corresponding prescription will


indeed leave physical observables invariant. A detailed discussionof gauge fixing as well as a derivation of the gauge-fixing terms inthe Lagrangian will be given in Chapter 3.4, for a general gaugetheory. Since this derivation is quite technical we for the momentsimply illustrate the problem and its solution.

In general, if the part of the action that is quadratic in some fieldφ(x) is given in terms of the Fourier transform φ(p) by

Sφ =∫ d4 p

(2π)4 φ(−p)O(p)φ(p), (2.22)

then the propagator for the field φ may be written as

iO−1(p). (2.23)

In the case of QED the part of the Lagrangian that is quadratic inthe photon field is given by −1/4 FµνFµν = −1/2 Aµ

(−gµν∂σ∂σ + ∂µ∂ν

)Aν,

where we have used partial integration to obtain the second expres-sion. In momentum space, the quadratic part of the action is thengiven by

SA =d4 p(2π)4

12

Aµ(−p)(−gµν p2 + pµ pν

)Aν(p). (2.24)

Unfortunately the operator(−gµν p2 + pµ pν

)does not have an in-

verse. This can be most easily seen by noting(−gµν p2 + pµ pν

)pν =

0. This means that the operator(−gµν p2 + pµ pν

)has an eigen-

vector (pν) with eigenvalue 0 and is therefore not invertible. Thecorresponding eigenvector corresponds to an unphysical longitudi-nal photon polarization. While these polarizations are not physical,they prevent us from inverting the matrix and it thus seems we arenot able to write down the propagator of the photon. We solve thisproblem by adding to the Lagrangian density a gauge fixing term

− 12(1− ξ)

(∂µ Aµ

)2 . (2.25)

This term breaks explicit gauge invariance and is motivated bythe Lorenz-gauge condition ∂µ Aµ = 0. However, the term doesnot enforce this condition, it simply gives different action to fieldconfigurations with different ∂µ Aµ.

With this term included (again in momentum space), SA be-comes

SA =∫ d4 p

(2π)412

Aµ(−p)(−gµν p2 − ξ

1− ξpµ pν

)Aν(p), (2.26)

and, noting the relation(

gµν p2 +ξ

1− ξpµ pν

)(gνρ − ξ

pν pρ

p2

)= p2g ρ

µ , (2.27)

we see that the propagator for the photon may now be written as

−i(

gµν − ξpµ pν

p2

)1p2 . (2.28)


The special choice ξ = 0 is known as the Feynman gauge. In thisgauge the propagator eq. (2.28) is particularly simple and we willuse it most of the time.

This procedure of gauge fixing seems strange: first we workedhard to get a gauge invariant Lagrangian, and then we spoil gaugeinvariance by introducing a gauge fixing term. The point is that wehave to fix the gauge in order to be able to perform a calculation,but the term we introduced is such that gauge invariant physicalquantities are unaffected by it. This last statement is nontrivial andwill be proven in Chapter 3.4 below.

Since the gauge term eq. (2.25) does not affect physical quanti-ties, the dependence on the gauge parameter ξ must cancel and it isirrelevant what value for ξ we choose. The choice ξ = 0 is simplya matter of convenience. A more careful procedure is to leave ξ ar-bitrary and check that all ξ-dependence in the final result cancels.This gives a strong check on calculations, however, at the price ofmaking the them much more tedious.

The procedure of fixing the gauge in order to be able to performa calculation, even though the final result does not depend on howwe have fixed the gauge, can be understood by the following anal-ogy. Assume we wanted to calculate some scalar quantity (say thetime it takes for a point mass to get from one point to another) inour ordinary 3-dimensional Euclidean space. To do so, we choose acoordinate system, perform the calculation and get our final result.Of course, the result does not depend on how we choose the coor-dinate system, but in order to be able to perform the calculation wehave to fix it somehow. Picking a coordinate system correspondsto fixing a gauge, and the independence of the result on the co-ordinate system chosen corresponds to the gauge invariance ofphysical quantities. To take this one step further we remark that notall quantities are independent of the coordinate system. For exam-ple, the x-coordinate of the position of the point mass at a certaintime depends on our choice. Similarly, there are important quan-tities that are gauge dependent. One example is the gauge bosonpropagator given in eq. (2.28). However, all measurable quantities(observables) are gauge invariant.

Finally we should mention that eq. (2.25) is by far not the onlyway to fix the gauge but it will be sufficient for these lectures toconsider gauges defined through eq. (2.25). These gauges are calledcovariant gauges.

2.5 Summary

• It is possible for the Lagrangian for a (complex) Dirac field to beinvariant under local U(1) transformations (phase rotations), inwhich the phase parameter depends on space-time. In order toaccomplish this we include an interaction with a vector gaugeboson which transforms under the local (gauge) transformationaccording to eq. (2.11).


• This interaction is encoded by replacing the derivative ∂µ by thecovariant derivative Dµ defined by eq. (2.19). Dµ ψ transformsunder gauge transformations as e−iω Dµ ψ.

• The kinetic term for the gauge boson is − 14 FµνFµν, where Fµν is

proportional to the commutator[Dµ, Dν

]and is invariant under

gauge transformations.

• The gauge boson must be massless, since a term proportional toAµ Aµ is not invariant under gauge transformations and hencenot included in the Lagrangian.

• The resulting Lagrangian is identical to that of QED.

• In order to define the propagator we have to specify a certaingauge; the resulting gauge dependence cancels in physical ob-servables.

3Non-Abelian Gauge Theories

In this lecture, the “gauge” concept will be constructed so that thegauge bosons have self-interactions — as are observed among thegluons of QCD, and the W±, Z and γ of the electroweak sector.However, the gauge bosons will still be massless. (We will see howto give the W± and Z their observed masses in the Higgs chapter.)

3.1 Global Non-Abelian Transformations

We apply the ideas of the previous lecture to the case where thetransformations do not commute with each other, i.e. the group is“non-abelian”.

Consider n free fermion fields ψi, arranged in a multiplet ψ:

ψ =

ψ1

ψ2

.

.ψn

(3.1)

for which the Lagrangian density is

L = ψ(iγµ∂µ −m

)ψ,

≡ ψi (iγµ∂µ −m

)ψi, (3.2)

where the index i is summed from 1 to n. Eq. (3.2) is therefore ashorthand for

L = ψ1 (iγµ∂µ −m

)ψ 1 + ψ

2 (iγµ∂µ −m)

ψ 2 + . . . . (3.3)

The Lagangian density (3.2) is invariant under (space-time inde-pendent) complex rotations in ψi space:

ψ→ Uψ, ψ→ ψU†, (3.4)

where U is an n× n matrix such that

UU† = 1, det[U] = 1. (3.5)

The transformation (3.4) is called an internal symmetry, whichrotates the fields (e.g. quarks of different colour) among themselves.


The group of matrices satisfying the conditions (3.5) is calledSU(n). This is the group of special, unitary n × n matrices. Spe-cial in this context means that the determinant is equal to 1. Inorder to specify an SU(n) matrix completely we need n2 − 1 realparameters. Indeed, we need 2n2 real parameters to determine anarbitrary complex n× n matrix. But there are n2 constraints due tothe unitary requirements and one additional constraint due to therequirement det = 1.

An arbitrary SU(n) matrix can be written as

U = eiωa Ta(3.6)

where we have adopted Einstein’s summation convention to sumover the repeated index, i.e.

ωa Ta ≡n2−1

∑a=1

ωa Ta . (3.7)

The ωa are real parameters which specify the group element, andthe matrices Ta are called the generators of the group.

Exercise 2.1Show that the unitarity of the SU(n) matrices entails hermiticity ofthe generators and that the requirement of det = 1 means that thegenerators have to be traceless.

In the case of U(1) there was just one trivial generator. Here wehave n2 − 1 generators Ta. There is still some freedom left of howto normalize the generators. We will adopt the usual normalizationconvention

tr(TaTb) =12

δab. (3.8)

The reason we can always enforce eq. (3.8) is that tr(TaTb) is a realmatrix symmetric in a ↔ b. Thus it can be diagonalized. If youhave problems getting on friendly terms with the concept of gener-ators, for the moment you can think of them as traceless, hermitiann× n matrices. (This is, however, not the complete picture.)

The crucial new feature of the group SU(n) is that two elementsof SU(n) generally do not commute, i.e.

eiωa1 Ta

eiωb2 Tb 6= eiωb

2 Tbeiωa

1 Ta(3.9)

(compare to eq. (2.3)). The product of matrix exponentials such as(3.6) can be obtained using the Baker-Campbell-Hausdorff formula

eX eY = eX+Y+ 12 [X,Y]+... , (3.10)

where the dots stand for higher-commutator terms. This formulamakes it clear that the basic object determining the multiplicationproperties of the group is the commutator of generators, which hasthe form

[Ta, Tb] ≡ i f abcTc 6= 0 , (3.11)


where we defined the structure constants of the group, f abc, andused the summation convention again. The structure constantsencode group multiplication and are totally antisymmetric. Thiscan be seen as follows: from eq. (3.11) it is obvious that f abc =

− f bac. To convince us of the antisymmetry in the other indices aswell, we note that multiplying eq. (3.11) by Td and taking the trace,using eq. (3.8), we get 1/2 i f abd = tr(TaTbTd) − tr(TbTaTd) =

tr(TaTbTd)− tr(TaTdTb).Recalling eq. (2.2) we see that the fermion field, ψi carries an

index i, which takes the value i ∈ 1 . . . n on which the matrix (3.6)acts. There are other possibilities one could consider. We could bemore general and only require that the generators satisfy eq. (3.11)for some set of m × m matrices Ta

R. A set of matrices fulfilling thecommutation relations is called a m-dimensional representationR of the group and the matrices Ta are called the fundamentalrepresentation of SU(n). A second representation, which is alson-dimensional is

Tan = −(Ta)T = −(Ta)∗ (3.12)

This is how the anti-fermion ψi is transforming (recall that (Ta)† =

Ta) and is called the conjugate representation. Interestingly, thestructure constants f abc provide a third example. Setting

(TaA)bc = −i f abc , (3.13)

where b and c are now viewed as the indices of the (n2 − 1)-dimensional matrix, one can easily verify that they commutationrelations eq. (3.11). This is called the adjoint representation of thegroup and one can consider fermions transforming under this rep-resentation. However, it turns out that all standard-model fermionstransform in the fundamental representation (or, in some cases,trivially under certain groups).

Exercise 2.2Show that the conjugate and adjoint representations are indeed rep-resentations of SU(n). What is the conjugate representation of theadjoint representation?

3.2 Non-Abelian Gauge Fields

Now suppose we allow the transformation U to depend on space-time. Then the Lagrangian density changes by δL under this “non-abelian gauge transformation”, where

δL = ψ γµ(∂µU

)ψ. (3.14)

The local gauge symmetry can be restored by introducing a covari-ant derivative Dµ, giving interactions with gauge bosons, such that

DµU(x)ψ(x) = U(x)Dµψ(x). (3.15)


This is like the electromagnetic case, except that Dµ is now a ma-trix,

iDµ = iI∂µ − gAµ (3.16)

where Aµ = Ta Aaµ. It contains n2 − 1 vector (spin one) gauge

bosons, Aaµ, one for each generator of SU(n). Under a gauge trans-

formation U, Aµ should transform as

Aµ → UAµU† +ig(∂µU

)U†. (3.17)

This ensures that the Lagrangian density

L = ψ(iγµDµ −m

)ψ (3.18)

is invariant under local SU(n) gauge transformations. It can bechecked that eq. (3.17) reduces to the gauge transformation of elec-tromagnetism in the abelian limit.

Exercise 2.3Perform an infinitesimal gauge transformation on ψ, ψ and D, using(3.6), and show that to linear order in the ωa, ψγµDµ ψ is invariant.

Exercise 2.4Show that in the SU(2) case, the covariant derivative is

iDµ =

(i∂µ − g

2 W3µ − g

2 (W1µ − iW2

µ)

− g2 (W

1µ + iW2

µ) i∂µ + g2 W3

µ

),

and find the usual charged current interactions for the lepton dou-blet

ψ =

(ν

e

)

by defining W± = (W1 ∓ iW2)/√

2.

Exercise 2.5Include the U(1) hypercharge interaction in the previous question;show that the covariant derivative acting on the lepton doublet (ofhypercharge Y = −1/2) is

iDµ =

(i∂µ − g

2 W3µ − g′YBµ − g

2 (W1µ − iW2

µ)

− g2 (W

1µ + iW2

µ) i∂µ + g2 W3

µ − g′YBµ

).

Define (Zµ

Aµ

)=

(cos θW − sin θW

sin θW cos θW

)(W3

µ

Bµ

)

and write the diagonal (neutral) interactions in terms of Zµ and Aµ.Extract sin θW in terms of g and g′. (Recall that the photon does notinteract with the neutrino.)

The kinetic term for the gauge bosons is again constructed fromthe field strengths Fa

µν which are defined from the commutator of


two covariant derivatives,

Fµν = − ig[Dµ, Dν

], (3.19)

where the matrix Fµν is given by

Fµν = TaFaµν, (3.20)

withFa

µν = ∂µ Aaν − ∂ν Aa

µ − g f abc Abµ Ac

ν. (3.21)

Notice that Fµν is gauge variant, unlike the U(1) case. We know thetransformation of D from (3.17), so

[Dµ, Dν

]→ U

[Dµ, Dν

]U†. (3.22)

The gauge invariant kinetic term for the gauge bosons is therefore

−12

Tr FµνFµν = −14

FaµνFa µν, (3.23)

where the trace is in SU(n) space, and summation over the index ais implied.

In sharp contrast with the abelian case, this term does not onlycontain terms which are quadratic in the derivatives of the gaugeboson fields, but also the terms

g f abc(∂µ Aaν)Ab

µ Acν −

14

g2 f abc f ade Abµ Ac

ν Adµ Ae

ν. (3.24)

This means that there is a very important difference betweenabelian and non-abelian gauge theories. For non-abelian gaugetheories the gauge bosons interact with each other via both three-point and four-point interaction terms. The three point interactionterm contains a derivative, which means that the Feynman rule forthe three-point vertex involves the momenta of the particles goinginto the vertex. We shall write down the Feynman rules in detaillater.

Once again, a mass term for the gauge bosons is forbidden,since a term proportional to Aa

µ Aa µ is not invariant under gaugetransformations.

3.3 Gauge Fixing

As in the case of QED, we need to add a gauge-fixing term in orderto be able to derive a propagator for the gauge bosons. There is onecomplication, which becomes relevant for the purpose of perform-ing higher loop calculations with non-abelian gauge theories: If onegoes through the formalism of gauge-fixing carefully, as we willdo in the next section, it turns out that at higher orders extra loopdiagrams emerge. These diagrams involve additional particles andare known as a “Faddeev-Popov ghosts”. For each gauge field thereis such a ghost field. These are not to be interpreted as physical


scalar particles which could in principle be observed experimen-tally, but merely as part of the gauge-fixing programme. For thisreason they are referred to as “ghosts”. Furthermore they have twopeculiarities:

1. They only occur inside loops. This is because they are not re-ally particles and cannot occur in initial or final states, but areintroduced to clean up a difficulty that arises in the gauge-fixingmechanism.

2. They behave like fermions even though they are scalars (spinzero). This means that we need to count a minus sign for eachloop of Faddeev-Popov ghosts in any Feynman diagram.

We shall display the Feynman rules for these ghosts later.Thus, for example, the Feynman diagrams which contribute to

the one-loop corrections to the gauge boson propagator are

+ - -

(a) (b) (c) (d)

Figure 1: One-loop corrections to thegauge-boson propagator.

Diagram (a) involves the three-point interaction between thegauge bosons, diagram (b) involves the four-point interaction be-tween the gauge bosons, diagram (c) involves a loop of fermions,and diagram (d) is the extra diagram involving the Faddeev-Popovghosts. Note that both diagrams (c) and (d) have a minus sign infront of them because both fermions and Faddeev-Popov ghostsobey Fermi statistics.

3.4 The Faddeev-Popov Lagrangian

We now derive the gauge fixing terms needed to quantize thegauge theory. By far the simplest method is to use the path inte-gral, which for a gauge theory naively takes the form

Z =∫DAµ exp

(iS[Aµ]

)with S[Aµ] = −

14

∫d4x Fa

µνFµνa . (3.25)

We have left out the fermions, since they do not play a role for thefollowing discussion. The problem with the above expression isthat many field configurations lead to exactly the same value of theaction. In particular, all pure gauge configurations

Aµ(x) = − ig

U(x)∂µU†(x) (3.26)

give a vanishing action, since they can be obtained from Aµ(x) = 0with a gauge transformation. For gauge invariant quantities, theintegration over physically equivalent gauge field configura-tions amounts to a trivial but infinite prefactor, so that (3.25) isill-defined.1 1 The problem does not arise in the

discretized version of gauge theories ifone works with the link fields U(y, x)instead of the gauge field Aµ(x). Thelink fields U(y, x) are elements ofthe compact gauge group so that theintegration over the symmetry groupis finite. In lattice simulations, one istherefore not forced to fix the gauge.


To obtain a meaningful expression, we would like to factor outthis integration. Faddeev and Popov 2 came up with a general 2 L. D. Faddeev and V. N. Popov.

Feynman Diagrams for the Yang-MillsField. Phys. Lett., 25B:29–30, 1967.[,325(1967); ,325(1967)]

method to do this.

A simple example

Before applying their method to the the path integral, let us go overthe necessary steps for an ordinary integral. Consider

I =∫

dx∫

dy f (x, y) (3.27)

where f (x, y) is a rotation invariant function. The rotation invari-ance corresponds to the gauge invariance of the path integral inte-grand. We want to bring this into the form

I =∫ 2π

0dθ∫ ∞

0dr F(r) , (3.28)

where the first factor is the trivial integration over the symmetrygroup. For this simple integral, the problem is solved by usingspherical coordinates, but for the gauge symmetry we do not knowhow to make a coordinate transformation in which the trivial partof the integral factors out. Faddeev and Popov provided a generalmethod to achieve this goal and we now illustrate it in our trivialexample, before applying exactly the same procedure to the pathintegral for gauge theory.

As a first step, we fix a direction (a gauge) by the condition

yθ = x sin θ + y cos θ!= 0 (3.29)

However, integrating over only a single direction might be danger-ous. To be sure to maintain rotation invariance, we also integrateover all directions in the form

∫ 2π

0dθδ(yθ)

∣∣∣∣∂yθ

∂θ

∣∣∣∣ = 2 . (3.30)

The factor 2 arises because the directions θ and θ + π are equiva-lent. The jacobian is

∣∣∣∣∂yθ

∂θ

∣∣∣∣yθ=0

= x cos θ − y sin θ|yθ=0 =√

x2 + y2 (3.31)

Now we insert (3.30) into the original integral

I =∫ 2π

0dθ∫

dx∫

dy δ(yθ)12

√x2 + y2 f (x, y) (3.32)

Now comes the crucial step: we rotate our coordinate system by anangle θ so that the new coordinates (x′, y′) are given by

y′ = yθ x′ = x cos θ − y sin θ (3.33)

After this transformation the integrand no longer depends on θ andwe have thus factored out the trivial integration over the symmetry


group:

I =∫ 2π

0dθ∫

dx′∫

dy′ δ(y′)12

√x′2 + y′2 f (x′, y′)

= (2π)∫

dx′x′

2f (x′, 0) (3.34)

Note that we have made use of rotation invariance of the integrandto replace f (x, y)→ f (x′, y′).

The real thing

We now apply the same technique to the path integral (3.25). Wewill consider a general linear gauge fixing functional G(A) andthen integrate over all gauges in the form1 1 The identity (3.35) only holds if the

solution to the δ-function conditionis unique, see (3.30). It isn’t, becauseof “Gribov copies” but these do notcontribute in perturbation theory.

1 =∫Dα δ(G(Aα))det

(δG(Aα)

δα

)(3.35)

with α = αaTa and where the gauge transformed field is defined as

Aµα = eiα

[Aµ − i

g∂µ

]e−iα (3.36)

The above two equations are completely analogous to (3.29) and(3.30) in our simple example. Note that the jacobian in (3.35) isindependent of α for a linear gauge fixing condition.

We now proceed in the same way as above, by first inserting(3.35) into the path integral (3.25) and then performing a variabletransformation Aµ → Aµ

α . The transformation is a shift, followed bya unitary rotation and leaves the measure invariant. Dropping theindex α on the gauge field, we arrive at

Z = (∫Dα)×

∫DAµ δ(G(A))det

(δG(A)

δα

)exp

(iS[Aµ]

). (3.37)

We have succeeded to factor out the integration over the gaugegroup, at the price of making the functional integral more com-plicated. The remaining task is to bring the integral into a formsuitable for perturbation theory.

To do so, we choose the gauge fixing condition to have the form

G(A) = ga(A)−ωa(x) (3.38)

for some function ω(x) . Popular choices for the remainder are

ga(A) = ∂µ Aaµ (“Lorenz gauge”) (3.39)

ga(A) = nµ Aaµ (“axial gauge”) (3.40)

The first one is most often adapted, since it does not require theintroduction an additional external vector, as is the case for axialgauge. To get rid of the δ-functional, we then integrate over all


functions ωa(x) with Gaussian weight:

Z =∫Dω exp

(−i∫

d4x(ωa(x))2

2(1− ξ)

)

×∫DAµ δ(ga(A)−ωa)det

(δga(A)

δα

)exp

(iS[Aµ]

)

=∫DAµ det

(δga(A)

δα

)exp

(iS[Aµ]− i

∫d4x

12(1− ξ)

(ga(A))2)

.

(3.41)

The extra term is precisely what was added to the QED Lagrangianto achieve gauge fixing. The only remaining problem is the pres-ence of the jacobian determinant. To compute the determinant inperturbation theory, we now represent it as an integral over auxil-iary Grassman “ghost” fields ηa and ηa:

det(

δga(A)

δα

)=∫DηDη exp

(−i∫

d4xd4yηa(x)δga(A(x))

δαb(y)ηb(y)

).

(3.42)Note that these fields do not carry a Dirac index and transform asscalars under Lorentz transformations. Since they have the wrongrelation between spin and statistics the Feynman-De Witt-Faddeev-Popov ghosts η and η cannot be interpreted as physical particles. Infact, their role is precisely to cancel the unphysical degrees of free-dom in the gauge field Aµ. However, perturbation theory for theseunphysical degrees of freedom works exactly as in the standardcase.

Let us now work out the explicit form of the ghost action forLorentz gauge. We have

Taga(A) = Ta∂µ Aαaµ = ∂µeiα

(Aµ −

ig

∂µ

)e−iα

= ∂µ

(Aµ −

1g

∂µα + i[α, Aµ

])+O(α2)

= Ta∂µ

(Aa

µ −1g

∂µαa − fabcαb Acµ

)+O(α2) (3.43)

The functional derivative thus yields

δga(A(x))δαb(y)

= − 1g

(δab + g fabc∂µ Ac

µ + g fabc Acµ∂µ)

δ(4)(x− y)

(3.44)When inserted into (3.42), the first term on the right gives a kineticterm for the ghost fields, while the second one gives an interactionbetween the gauge fields and the ghosts. Because of the prefactor1/g, the kinetic term is not properly normalised. One then rescalesthe ghost fields by a factor

√g to bring the action into canonical

form.In an Abelian gauge theory fabc = 0. In this case, the ghost fields

no longer interact with the gauge fields and can immediately beintegrated out so that only the gauge fixing term remains in theaction.


Exercise 2.4Derive the gauge-fixing Lagrangian in axial gauge ga(A) = nµ Aa

µ

and derive the associated gauge-boson propagator Gµν(k) . Showthat for ξ = 1 this propagator fulfils nµGµν(k) = nνGµν(k) = 0 andthat this implies that for this choice the ghost fields do not interact.The axial gauge has furthermore the property that

limk2→0

k2kµGµν(k) = 0 , (3.45)

so that the unphysical polarisation does not have an associatedpropagator pole. For this reason, these gauges are also referred toas “physical” gauges. Their disadvantage is the necessity for anexternal reference vector and the complicated form of the gauge-boson propagator.

3.5 The Lagrangian for a General Non-Abelian Gauge Theory

Let us summarize what we have found so far: Consider a gaugegroup G of “dimension” N (for SU(n) : N ≡ n2 − 1), whose Ngenerators, Ta, obey the commutation relations

[Ta, Tb

]= i fabcTc,

where fabc are called the “structure constants” of the group.The Lagrangian density for a gauge theory with this group, with

a fermion multiplet ψi, is given by

L = −14

FaµνFa µν + iψ

(γµDµ −mI

)ψ− 1

2(1− ξ)(∂µ Aa

µ)2 + LFP

(3.46)where

Faµν = ∂µ Aa

ν − ∂ν Aaµ − g f abc Ab

µ Acν, (3.47)

Dµ = ∂µI + i g Ta Aaµ (3.48)

and

LFP = −ηa∂µ∂µηa + g facb(∂µηa) Ac

µ ηb. (3.49)

Under an infinitesimal gauge transformation the N gauge bosonsAa

µ change by an amount that contains a term which is not linear inAa

µ:

δAaµ(x) = − f abc Ab

µ(x)ωc(x) +1g

∂µωa(x), (3.50)

whereas the field strengths Faµν transform by a change

δFaµν(x) = − f abc Fb

µν(x)ωc. (3.51)

In other words, they transform as the “adjoint” representation ofthe group (which has as many components as there are genera-tors). This means that the quantity Fa

µνFa µν (summation over a, µ, ν

implied) is invariant under gauge transformations.


3.6 Feynman Rules

The Feynman rules for such a gauge theory can be read off directlyfrom the Lagrangian. As mentioned previously, the propagatorsare obtained by taking all terms bilinear in the field and invertingthe corresponding operator (and multiplying by i). The rules forthe vertices are obtained by simply taking (i times) the factor whichmultiplies the corresponding term in the Lagrangian. The explicitrules are given in the following.

Vertices

(Note that all momenta are defined as flowing into the vertex!)

µ a

p1

ρ c

p3

ν b

p2−g fabc

(gµν (p1 − p2)ρ + gνρ (p2 − p3)µ + gρµ (p3 − p1)ν

)

−i g2feabfecd (gµρgνσ − gµσgνρ)

−i g2feacfebd (gµνgρσ − gµσgνρ)

−i g2feadfebc (gµνgρσ − gµρgνσ)

µa ν

b

σ d ρ c

µ a

j i

−i g γµ (T a)ij

µ a

c bq

g fabc qµ

Propagators

Gluon: −i δab gµν/p2p

aµ

bν

Fermion: i δij(γµpµ + m)/(p2 − m2)

pi j

Faddeev-Popov ghost: i δab/p2p

a b


3.7 An Example

As an example of the application of these Feynman rules, we con-sider the process of Compton scattering, but this time for the scat-tering of non-abelian gauge bosons and fermions, rather than pho-tons. We need to calculate the amplitude for a gauge boson of mo-mentum p2 and gauge label a to scatter off a fermion of momentump1 and gauge label i producing a fermion of momentum p3 andgauge label j and a gauge boson of momentum p4 and gauge labelb. Note that i, j ∈ 1 . . . n whereas a, b ∈ 1 . . . n2 − 1. In additionto the two Feynman diagrams one gets in the QED case there is athird diagram involving the self-interaction of the gauge bosons.

i k jp1 p3

(p1 + p2)

p2 p4

µ νa b

i k jp1 p3

p2 p4

µ ν

a b

i jp1 p3

(p4 − p2)

ρ c

σ

p2 p4a b

µ ν

(a) (b) (c)

Figure 2: Lowest order diagrams forvector-boson fermion scattering.We will assume that the fermions are massless (i.e. that we are at

sufficiently high energies so that we may neglect their masses), andwork in terms of the Mandelstam variables

s = (p1 + p2)2 = (p3 + p4)

2,

t = (p1 − p3)2 = (p2 − p4)

2,

u = (p1 − p4)2 = (p2 − p3)

2.

The polarizations are accounted for by contracting the ampli-tude obtained for the above diagrams with the polarization vectorsεµ(λ2) and εν(λ4). Each diagram consists of two vertices and apropagator and so their contributions can be read off from theFeynman rules.

For diagram (a) we get

εµ(λ2)εν(λ4)uj(p3)(−i g γν(Tb)k

j

)(i

γ · (p1 + p2)

s

)(−i g γµ(Ta)i

k

)ui(p1)

= −ig2

sεµ(λ2)εν(λ4)u (p3) (γ

νγ · (p1 + p2)γµ)(

TbTa)

u(p1).

For diagram (b) we get

εµ(λ2)εν(λ4)uj(p3)(−i g γµ(Ta)k

j

)(i

γ · (p1 − p4)

u

)(−i g γν(Tb)i

k

)ui(p1)

= −ig2

uεµ(λ2)εν(λ4)u (p3) (γ

νγ · (p1 − p4)γµ)(

TaTb)

u(p1).

Note that here the order of the T matrices is the other way aroundcompared to diagram (a).


Diagram (c) involves the three-point gauge-boson self-coupling.Since the Feynman rule for this vertex is given with incoming mo-menta, it is useful to replace the outgoing gauge-boson momentump4 by −p4 and understand this to be an incoming momentum.Note that the internal gauge-boson line carries momentum p4 − p2

coming into the vertex. The three incoming momenta that are tobe substituted into the Feynman rule for the vertex are thereforep2, −p4, p4 − p2. The vertex thus becomes

−g fabc(

gµν(p2 + p4)ρ + gρν(p2 − 2p4)µ + gµρ(p4 − 2p2)ν

),

and the diagram gives

εµ(λ2)εν(λ4)uj(p3)

(−i g γσ(Tc)i

j

)ui(p1)

(−i

gρσ

t

)

× (−g fabc)(

gµν(p2 + p4)ρ + gρν(p2 − 2p4)µ + gµρ(p4 − 2p2)ν

)

= −ig2

tεµ(λ2)ε

ν(λ4)u (p3)[Ta, Tb

]γρu(p1)

×(

gµν (p2 + p4)ρ − 2(p4)µgνρ − 2(p2)νgµρ

),

where in the last step we have used the commutation relationeq. (3.11) and the fact that the polarization vectors are transverseso that p2 · ε(λ2) = 0 and p4 · ε(λ4) = 0.

Exercise 2.5Draw all the Feynman diagrams for the tree level amplitude for twogauge bosons with momenta p1 and p2 to scatter into two gaugebosons with momenta q1 and q2. Label the momenta of the externalgauge boson lines.

3.8 Summary

• A non-abelian gauge theory is one in which the Lagrangian isinvariant under local transformations of a non-abelian group.

• This invariance is achieved by introducing a gauge boson foreach generator of the group. The partial derivative in the La-grangian for the fermion field is replaced by a covariant deriva-tive as defined in eq. (3.48).

• The gauge bosons transform under infinitesimal gauge transfor-mations in a non-linear way given by eq. (3.50).

• The field strengths, Faµν, are obtained from the commutator of

two covariant derivatives and are given by eq. (3.47). They trans-form as the adjoint representation under gauge transformationssuch that the quantity Fa

µνFa µν is invariant.

• FaµνFa µν contains terms which are cubic and quartic in the gauge

bosons, indicating that these gauge bosons interact with eachother.


• The gauge-fixing mechanism leads to the introduction of Faddeev-Popov ghosts which are scalar particles that occur only insideloops and obey Fermi statistics.

4Quantum Chromodynamics

Quantum Chromodynamics (QCD) is the theory of the strong in-teraction. It is nothing but a non-abelian gauge theory with thegroup SU(3). The fermions on which transform under the gaugegroup are called quarks and are described by a field ψi where iruns from 1 to 3. The quantum number associated with the label iis called colour. The standard model contains six flavors of quarks(up, down, charm, strange, top, bottom) and each flavor comes inthree colors. Their interactions with the Higgs boson are quite dif-ferent and as a result some of the quarks are very light (up, down,strange), while others are extremely heavy. In particular the topquark, which is as heavy as a Gold nucleus. The eight gauge bosonswhich have to be introduced in order to preserve local gauge invari-ance are the eight ‘gluons’. These are taken to be the carriers whichmediate the strong interaction in the same way that photons are thecarriers which mediate the electromagnetic interactions.

The Feynman rules for QCD are therefore simply the Feynmanrules listed in the previous lecture, with the gauge coupling con-stant, g, taken to be the strong coupling, gs, (more about this later),the generators Ta taken to be the eight generators of SU(3) in thetriplet representation, and f abc, a, b, c, = 1 . . . 8 are the structureconstants of SU(3) (you can look them up in a book but normallyyou will not need their explicit form).

The discovery that the strong interaction is described by a non-abelian gauge theory in the early 70’s ended a long period of de-spair in particle physics. Since the 50’s people discovered more andmore hadrons, i.e. strongly interacting particles. This was called the“particle zoo” and there wasn’t much hope for a theory describingall these particles and interactions. Freeman Dyson proclaimed in1958 that "the correct theory will not be found in a hundred years!".Furthermore, all the mathematical difficulties of quantum field the-ory had discouraged people so that Landau concluded that “theHamiltonian method for strong interactions is dead and must beburied, although of course with deserved honor”.

However, a combination of experimental and theoretical discov-eries over the next decade then lead to the emergence of QCD:

1. Quark model: Gell-Mann, Nishijima, Ne’eman and Zweig discov-


ered that the existing hadrons could be classified by assumingthat they were built from hypothetical “quark” constituents withspin 1/2 and electric charge +2/3 (up quark) and −1/3 (downand strange quarks). For example

|π+〉 ∼ |ud〉 , |π−〉 ∼ |ud〉 , |π0〉 ∼ 1√2

(|uu〉+ |dd〉

), (4.1)

and

|p〉 ∼ |uud〉 |n〉 ∼ |udd〉 . (4.2)

In addition, by introducing a third quark s one could also ex-plain the pattern of strange mesons and baryons. In addition toclassifying the known hadrons, the model predicted the existenceof an additional spin 3/2 baryon, the Ω−, which was discoveredin 1964.

It was later observed that the quarks should have an additionalquantum number, now called color. This was necessary becausethe Pauli principle forbids a spin 3/2 state made out of identicalquarks

|∆++〉 ∼ |u↑ u↑ u↑〉 . (4.3)

Distinguishing the quarks with an additional color quantumnumbers and then imposing that all physical states are colorsinglets can explain the existence of the ∆++ particle.

2. Deep inelastic scattering and the parton model: Experimental re-sults of e−p scattering at high energies could be explained byassuming that the electrons scattered off point-like constituents(“partons”), which carried part of the proton momentum. Feyn-man developed the parton model in which the incoming protonis views as a "box of partons sharing the proton momentum andpractically free". In 1968 Callan and Gross show a way of dis-tinguishing the spin of the partons. The experimental resultsstrongly favor spin 1/2.

3. Renormalization 1971 ’t Hooft and Veltman show that Yang-Millstheories are renormalisable.

4. In 1972 Gell-Mann and Fritzsch propose at a conference an SU(3)gauge theory of quarks and gluons for the strong interaction. Ina 1973 paper with Leutwyler they discuss the advantages of thismodel.

5. Asymptotic freedom: One of the problems for many quantum fieldtheories is that the coupling becomes stronger and stronger athigh energies so that the theory will eventually break down atvery high energies. Gross and Wilczek set out to prove that allquantum field theories have this property. However in 1973,together with Politzer they find an exception to this rule: non-abelian gauge theories are asymptotically free.


We will now discuss the concept of a running coupling andasymptotic freedom in more detail and will then discuss how per-turbative QCD computations are useful at high energies. Instead ofdeep-inelatic scattering, we will illustrate this with the scatteringprocess e+e− → “hadrons”.

4.1 Running Coupling

The coupling for the strong interaction is the QCD gauge coupling,gs. We usually work in terms of αs defined as

αs =g2

s4π

. (4.4)

Since the interactions are strong, we would expect αs to be toolarge to perform reliable calculations in perturbation theory. On theother hand the Feynman rules are only useful within the context ofperturbation theory.

This difficulty is resolved when we understand that ‘couplingconstants’ are not constant at all. The electromagnetic fine structureconstant, α, has the value 1/137 only at energies which are not largecompared to the electron mass. At higher energies it is larger thanthis. For example, at LEP energies it takes a value close to 1/129. Incontrast to QED, it turns out that in the non-abelian gauge theoriesof the Standard Model the weak and the strong coupling decrease asthe energy increases.

To see how this works within the context of QCD we note thatwhen we perform higher order perturbative calculations there areloop diagrams which have the effect of ‘dressing’ the couplings. Forexample, the one-loop diagrams which dress the coupling betweena quark and a gluon are:

where

= + - -

are the diagrams needed to calculate the one-loop corrections tothe gluon propagator.

These diagrams contain UV divergences and need to be renor-malized, e.g. by subtracting at some renormalization scale µ. Thisscale then appears inside a logarithm for the renormalized quan-tities. This means that if the squared momenta of all the externalparticles coming into the vertex are of order Q2, where Q µ,then the above diagrams give rise to a correction which contains a


logarithm of the ratio Q2/µ2:

−α2s β0 ln

(Q2/µ2

). (4.5)

This correction is interpreted as the correction to the effective QCDcoupling, αs(Q2), at momentum scale Q, i.e.

αs(Q2) = αs(µ2) − αs(µ

2)2 β0 ln(

Q2/µ2)

+ . . . . (4.6)

The coefficient β0 is calculated to be

β0 =11 Nc − 2 n f

12 π, (4.7)

where Nc is the number of colours (=3), n f is the number of activeflavours, i.e. the number of flavours whose mass threshold is be-low the momentum scale Q. Note that β0 is positive, which meansthat the coefficient in front of the logarithm in eq. (4.6) is negative,so that the effective coupling decreases as the momentum scale isincreased.

A more precise analysis shows that the effective coupling obeysthe differential equation

∂ αs(Q2)

∂ ln(Q2)= β

(αs(Q2)

), (4.8)

where β has the perturbative expansion

β(αs) = −β0 α2s − β1 α3

s +O(α4s ) + . . . . (4.9)

In order to solve this differential equation we need a boundaryvalue. Nowadays this is usually taken to be the measured value ofthe coupling at scale of the Z boson mass, MZ = 91.19 GeV, whichis measured to be

αs(M2Z) = 0.118± 0.002 . (4.10)

This is one of the free parameters of the Standard Model.1 1 Previously the solution to eq. (4.8)(to leading order) was written asαs(Q2) = 4π/β0 ln(Q2/Λ2

QCD) and thescale ΛQCD was used as the standardparameter which sets the scale for themagnitude of the strong coupling. Thisturns out to be rather inconvenientsince it needs to be adjusted everytime higher order corrections are takeninto consideration and the numberof active flavours has to be specified.The detour via ΛQCD also introducesadditional truncation errors and cancomplicate the error analysis.

The running of αs(Q2) is shown in figure 1. We can see that formomentum scales above about 2 GeV the coupling is less than 0.3so that one can hope to carry out reliable perturbative calculationsfor QCD processes with energy scales larger than this.

Gauge invariance requires that the gauge coupling for the inter-action between gluons must be exactly the same as the gauge cou-pling for the interaction between quarks and gluons. The β-functioncould therefore have been calculated from the higher order correc-tions to the three-gluon (or four-gluon) vertex and must yield thesame result, despite the fact that it is calculated from a completelydifferent set of diagrams.

Exercise 3.1Draw the Feynman diagrams needed for the calculation of theone-loop correction to the triple gluon coupling (don’t forget theFaddeev-Popov ghost loops).


0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

1 10 100

αs(Q2)

√Q2 (GeV )

Figure 1: The running of αs(Q2) withβ taken to two loops.

Exercise 3.2Solve equation (4.8) using β to leading order only, and calculate thevalue of αs at a momentum scale of 10 GeV. Use the value at MZ

given by eq. (4.10). Calculate also the error in αs at 10 GeV.

4.2 Quark (and Gluon) Confinement

This argument can be inverted to provide an answer to the ques-tion of why we have never seen quarks or gluons in a laboratory.Asymptotic Freedom tells us that the effective coupling betweenquarks becomes weaker at shorter distances (equivalent to higherenergies/momentum scales). Conversely it implies that the effec-tive coupling grows as we go to larger distances. Therefore, thecomplicated system of gluon exchanges which leads to the bind-ing of quarks (and antiquarks) inside hadrons leads to a strongerand stronger binding as we attempt to pull the quarks apart. Thismeans that we can never isolate a quark (or a gluon) at large dis-tances since we require more and more energy to overcome thebinding as the distance between the quarks grows. Instead, whenthe energy contained in the ‘string’ of bound gluons and quarks be-comes large enough, the colour-string breaks and more quarks arecreated, leaving more colourless hadrons, but no isolated, colouredquarks.

The upshot of this is that the only free particles which can be ob-served at macroscopic distances from each other are colour singlets.This mechanism is known as “quark confinement”. The details ofhow it works are not fully understood. Nevertheless the argumentpresented here is suggestive of such confinement and at the levelof non-perturbative field theory, lattice calculations have confirmedthat for non-abelian gauge theories the binding energy does indeedgrow as the distance between quarks increases.1 1 Lattice QCD simulations have also

succeeded in calculating the spectrumof many observed hadrons and alsohadronic matrix elements for certainprocesses from ‘first principles’, i.e.without using perturbative expansionsor phenomenological models.

Thus we have two different pictures of the world of strong in-teractions: On one hand, at sufficiently short distances, which canbe probed at sufficiently large energies, we can consider quarks


and gluons (partons) interacting with each other. In this regime wecan perform calculations of the scattering cross sections betweenquarks and gluons (called the “partonic hard cross section”) in per-turbation theory because the running coupling is sufficiently small.On the other hand, before we can make a direct comparison withwhat is observed in accelerator experiments, we need to take intoaccount the fact that the quarks and gluons bind (hadronize) intocolour singlet hadrons, and it is only these colour singlet statesthat are observed directly. The mechanism for this hadronizationis beyond the scope of perturbation theory and not understood indetail. Nevertheless Monte Carlo programs have been developedwhich simulate the hadronization in such a way that the results ofthe short-distance perturbative calculations at the level of quarksand gluons can be confronted with experiments measuring hadronsin a successful way.

Thus, for example, if we wish to calculate the cross section foran electron-positron annihilation into three jets (at high energies),we first calculate, in perturbation theory, the process for electronplus positron to annihilate into a virtual photon (or Z boson) whichthen decays into a quark and antiquark, and an emitted gluon. Atleading order the two Feynman diagrams for this process are:1 1 The contraction of the one loop

diagram (where a gluon connectsthe quark and antiquark) with thee+e− → qq amplitude is of the sameorder αs and has to be taken intoaccount to get an infra-red finite result.However, it does not lead to a three-jetevent (on the partonic level).

e+

e−

q

q

γ∗g e+

e−

q

q

γ∗

g

Exercise 3.3Draw the (tree level) Feynman diagrams for the process e+e− →4jets. Consider only one photon exchange plus the QCD contribu-tions (do not include Z boson exchange or WW production).

4.3 The R-ratio

Since the quarks and gluons are not observed directly, but onlyafter they form hadrons, we cannot immediately compare the re-sult with experimental data. To be able to compare, one either hasto choose an observable which is insensitive to hadronisation ef-fects, such as the R-ratio computed below, or one has to modelthese effects. In pratice, this modeling is often done using Monte-Carlo simulation programs, which compute the the radiation of thequarks and gluons and then use a model to form hadrons out ofthem.

Also, if we want to calculate scattering processes including initialstate hadrons we need to account for the probability of findinga particular quark or gluon inside an initial hadron with a givenfraction of the initial hadron’s momentum (these are called “partondistribution functions”).


A detailed discussion of these aspects is beyond the scope of thislecture, but we will now look at the simplest example of a high-energy process which is insensitive to hadronisation, namely thecross section ratio

R(Q) =σ(e+e− → “hadrons”)

σ(e+e− → µ+µ−). (4.11)

at collision energy Q. Experimentally, this can be obtained bycounting the number of collisions events with only a muon pairsin the final state and those with an arbitrary number of hadrons.The R ratio is obtained by taking the ratio of these two numbers,since the luminosity factor, which relates the cross section to thenumber of events drops out in the ratio. The numerator in the Rratio is an inclusive cross section, since we sum over all hadronicfinal states.

Let us now compute R in perturbation theory. In perturbationtheory, we do not have any hadrons, but we can hope to computethe inclusive cross section using quark and gluon final states. It isnot clear that this will work, but since we sum over the complete setof hadron states, we can hope that we get the same result summingover quarks and gluons, which corresponds to a different set ofQCD basis states. The lowest order amplitudes for the numeratorand denominator have the form

Mq =e+

e−

q

q

γ∗

, Mµ =e+

e−

µ−

µ+

γ∗

(4.12)

Contributions from final states with additional gluons (see Fig-ure 4.2) are suppressed by powers of the couplings constant. Aslong as the coupling constant is small at the relevant energy, thesecontributions should be suppressed. To obtain the R-ratio, we couldsquare the amplitudesMq andMµ, compute the cross section andthen take the ratio. However, the two diagrams are identical upto the values of the mass and the charge, and since we considerhigh-energy collisions, the effect of the quark and lepton massesshould be small and we neglect them. In this approximation, wehaveMq = eqMµ for a given quark. To get the value of R at agiven center of mass energy Q, we have to take into account all thequarks that are light enough to be produced and remember thatthere are Nc = 3 colors for each quark flavor. Squaring the ampli-tudes, we then obtain for the ratio

R(Q) = Nc ∑mq<Q

e2q =

2 for Q < 2mc103 for 2mc < Q < 2mb

113 for 2mb < Q MZ

, (4.13)

We impose the restriction Q MZ since one needs to includenot only photon but also Z-boson exchange if the energy becomeslarger. This can of course be done but our simple result does nottake the Z-exchange into account. A second limitation of the resultis that the massless approximation is not valid when Q ≈ 2 mq,


the threshold for producing the corresponding quark pair. Evenworse, near the quark threshold Q ≈ 2mq, the amount of kineticenergy which is available to the quarks is very small and the QCDcoupling becomes very large so that the expansion in the couplingwill break down.

6 46. Plots of cross sections and related quantities

σ and R in e+e− Collisions

10-8

10-7

10-6

10-5

10-4

10-3

10-2

1 10 102

σ[m

b]

ω

ρ

φ

ρ′

J/ψ

ψ(2S)Υ

Z

10-1

1

10

102

103

1 10 102

Rω

ρ

φ

ρ′

J/ψ ψ(2S)

Υ

Z

√s [GeV]

Figure 46.6: World data on the total cross section of e+e− → hadrons and the ratio R(s) = σ(e+e− → hadrons, s)/σ(e+e− → µ+µ−, s).σ(e+e− → hadrons, s) is the experimental cross section corrected for initial state radiation and electron-positron vertex loops, σ(e+e− →µ+µ−, s) = 4πα2(s)/3s. Data errors are total below 2 GeV and statistical above 2 GeV. The curves are an educative guide: the broken one(green) is a naive quark-parton model prediction, and the solid one (red) is 3-loop pQCD prediction (see “Quantum Chromodynamics” section ofthis Review, Eq. (9.7) or, for more details, K. G. Chetyrkin et al., Nucl. Phys. B586, 56 (2000) (Erratum ibid. B634, 413 (2002)). Breit-Wignerparameterizations of J/ψ, ψ(2S), and Υ(nS), n = 1, 2, 3, 4 are also shown. The full list of references to the original data and the details ofthe R ratio extraction from them can be found in [arXiv:hep-ph/0312114]. Corresponding computer-readable data files are available athttp://pdg.lbl.gov/current/xsect/. (Courtesy of the COMPAS (Protvino) and HEPDATA (Durham) Groups, May 2010.)

Figure 2: The R-ratio as a function ofthe center-of-mass energy Q =

√s.

The plot shows data points togetherwith the theoretical predictions whichare shown as lines. The leading-orderprediction is the dashed green line,while the next-to-next-to-leading orderresult is shown as a red line.

The plot in Figure 2, taken from the PDG1, nicely illustrates 1 C. Patrignani et al. Review of ParticlePhysics. Chin. Phys., C40(10):100001,2016

many of the features discussed so far. First of all, for large ener-gies, the perturbative computation describes the data very well onceZ-exchange is taken into account. (The presence of the Z-bosonmanifests itself as a peak near its mass.) The agreement is also verygood at lower energies, as long as Q is not near a quark threshold.Near the thresholds, the QCD interactions become strong and thequarks form bounds states which manifest themselves as peaksin the R-ratio. One also observes that the R-ratio increases eachtime we cross a heavy-quark threshold, as predicted in (4.13). Atvery low energies, on the other hand, the perturbative prediction isuseless since the interaction becomes strong and the cross sectionis dominated by states with just a few hadrons. The plot demon-strates the dual nature of QCD: at low energies, the theory givesrise to strong interactions which bind quarks and gluons into color-neutral bound states, while at large energies, we deal with a theoryof almost free quarks and gluons whose properties can be studiedperturbatively. Let us note that there is a theoretical framework,the Operator Product Expansion (OPE), which can be used to an-alyze the R-ratio systematically. In this framework one can showthat there are indeed no hadronisation corrections to the R-ratiofor Q → ∞ and one can obtain the corrections at finite Q in anexpansion in 1/Q2.


4.4 θ-Parameter of QCD

There is one more gauge invariant term that can be written down inthe QCD Lagrangian:

Lθ = θg2

s64π2 εµνρσFa

µνFaρσ. (4.14)

Here εµνρσ is the totally antisymmetric tensor (in four dimensions).Since we should work with the most general gauge invariant La-grangian there is no reason to omit this term. However, adding thisterm to the Lagrangian leads to a problem, called the “strong CPproblem”.

To understand the nature of the problem, we first convince our-selves that this term violates CP. In QED we would have

εµνρσFµνFρσ = −8E · B, (4.15)

and for QCD we have a similar expression except that Ea and Ba

carry a colour index — they are known as the chromoelectric andchromomagnetic fields. Under charge conjugation both the electricand magnetic field change sign. But under parity the electric field,which is a proper vector, changes sign, whereas the magnetic field,which is a polar vector, does not change sign. Thus we see that theterm E · B is odd under CP.

For this reason, the parameter θ in front of this term must beexceedingly small in order not to give rise to strong interactioncontributions to CP violating quantities such as the electric dipolemoment of the neutron. The current experimental limits on thisdipole moment tell us that θ < 10−10. Thus we are tempted to thinkthat θ is zero. Nevertheless, strictly speaking θ is a free parameterof QCD, and is sometimes considered to be the nineteenth freeparameter of the Standard Model.

Of course we simply could set θ to zero (or a very small number)and be happy with it.1 However, whenever a free parameter is zero 1 To be precise, setting θ → 0 in the

Lagrangian would not be enough, asθ 6= 0 can also be generated throughhigher order electroweak radiativecorrections, requiring a fine-tuningbeyond θ → 0.

or extremely small, we would like to understand the reason. Thefact that we do not know why this term is absent (or so small) is thestrong CP problem.

There are several possible solutions to the strong CP problemthat offer explanations as to why this term is absent (or small). Onepossible solution is through imposing an additional symmetry,leading to the postulation of a new, hypothetical, weakly interactingparticle, called the “(Peccei-Quinn) axion”. Unfortunately none ofthese solutions have been confirmed yet and the problem is stillunresolved.

Another question is why is this not a problem in QED? In facta term like eq. (4.14) can also be written down in QED. A thor-ough discussion of this point is beyond the scope of this lecture.Suffice to say that this term can be written (in QED and QCD) asa total divergence, so it seems that it can be eliminated from theLagrangian altogether. However, in QCD (but not in QED) there


are non-perturbative effects from the non-trivial topological struc-ture of the vacuum (somewhat related to so called “instantons” youprobably have heard about) which prevent us from neglecting theθ-term.

4.5 Summary

• Quarks transform as a triplet representation of colour SU(3)(each quark can have one of three colours).

• The eight gauge bosons of QCD are the gluons which are thecarriers that mediate the strong interaction.

• The coupling of quarks to gluons (and gluons to each other) de-creases as the energy scale increases. Therefore, at high energiesone can perform reliable perturbative calculations for stronglyinteracting processes.

• As the distance between quarks increases the binding increases,such that it is impossible to isolate individual quarks or gluons.The only observable particles are colour singlet hadrons. Pertur-bative calculations performed at the quark and gluon level mustbe supplemented by accounting for the recombination of finalstate quarks and gluons into observed hadrons as well as theprobability of finding these quarks and gluons inside the initialstate hadrons (if applicable).

• QCD admits a gauge invariant strong CP violating term witha coefficient θ. This parameter is known to be very small fromlimits on CP violating phenomena such as the electric dipolemoment of the neutron.

5Spontaneous Symmetry Breaking

We have seen that in an unbroken gauge theory the gauge bosonsmust be massless. This is exactly what we want for QED (masslessphoton) and QCD (massless gluons). However, if we wish to extendthe ideas of describing interactions by a gauge theory to the weakinteractions, the symmetry must somehow be broken since thecarriers of the weak interactions (W and Z bosons) are massive(weak interactions are very short range). We could simply break thesymmetry by hand by adding a mass term for the gauge bosons,which we know violates the gauge symmetry. However, this woulddestroy renormalizability of our theory.

Renormalizable theories are preferred because they are more pre-dictive. As discussed in the Field Theory and QED lectures, thereare divergent results (infinities) in QED and QCD, and these aresaid to be renormalizable theories. So what could be worse abouta non-renormalizable theory? The critical issue is the number ofdivergences: few in a renormalizable theory, and infinite in the non-renormalizable case. Associated to every divergence is a parameterthat must be extracted from data, so renormalizable theories canmake testable predictions once a few parameters are measured.For instance, in QCD, the coupling gs has a divergence. But onceαs is measured in one process, the theory can be tested in otherprocesses.1 1 It should be noted that effective

field theories, though formally notrenormalizable, can nevertheless bevery valuable as they often allow fora simplified description of a more‘complete’ or fundamental theoryin a resticted energy range. Popularexamples are Chiral PerturbationTheory, Heavy Quark Effective Theoryand Non-Relativistic QCD.

In this chapter we will discuss a way to give masses to the Wand Z, called “spontaneous symmetry breaking”, which maintainsthe renormalizability of the theory. In this scenario the Lagrangianmaintains its symmetry under a set of local gauge transformations.On the other hand, the lowest energy state, which we interpret asthe vacuum (or ground state), is not a singlet of the gauge sym-metry. There is an infinite number of states each with the sameground-state energy and nature chooses one of these states as the‘true’ vacuum.

5.1 Massive Gauge Bosons and Renormalizability

In this subsection we will convince ourselves that simply addingby hand a mass term for the gauge bosons will destroy the renor-malizability of the theory. It will not be a rigorous argument, but


will illustrate the difference between introducing mass terms forthe gauge bosons in a brute force way and introducing them viaspontaneous symmetry breaking.

Higher order (loop) corrections generate ultraviolet divergences.In a renormalizable theory, these divergences can be absorbed intothe parameters of the theory we started with, and in this way canbe ‘hidden’. As we go to higher orders we need to absorb moreand more terms into these parameters, but there are only as manydivergent quantities as there are parameters. So, for instance, inQED the Lagrangian we start with contains the fermion field, thegauge boson field, and interactions whose strength is controlled bye and m. Being a renormalizable theory, all divergences of diagramscan be absorbed into these quantities (irrespective of the number ofloops or legs), and once e and m are measured, all other observables(cross sections, g− 2, etc.) can be predicted.

In order to ensure that this programme can be carried out therehave to be restrictions on the allowed interaction terms. Further-more all the propagators have to decrease like 1/p2 as the mo-mentum p → ∞. Note that this is how the massless gauge-bosonpropagator eq. (2.28) behaves. If these conditions are not fulfilled,then the theory generates more and more divergent terms as onecalculates to higher orders, and it is not possible to absorb these di-vergences into the parameters of the theory. Such theories are saidto be “non-renormalizable”.

Now we can convince ourselves that simply adding a mass termM2 Aµ Aµ to the Lagrangian given in eq. (3.46) will lead to a non-renormalizable theory. To start with we note that such a term willmodify the propagator. Collecting all terms bilinear in the gaugefields in momentum space we get (in Feynman gauge)

12

Aµ

(−gµν(p2 −M2) + pµ pν

)Aν. (5.1)

We have to invert this operator to get the propagator which nowtakes the form

ip2 −M2

(−gµν +

pµ pν

M2

). (5.2)

Note that this propagator, eq. (5.2), has a much worse ultravioletbehavior in that it goes to a constant for p → ∞. Thus, it is clearthat the ultraviolet properties of a theory with a propagator asgiven in eq. (5.2) are worse than for a theory with a propagator asgiven in eq. (2.28). According to our discussion at the beginningof this subsection we conclude that without the explicit mass termM2 Aµ Aµ the theory is renormalizable, whereas with this termit is not. In fact, it is precisely the gauge symmetry that ensuresrenormalizability. Breaking this symmetry results in the loss ofrenormalizability.

The aim of spontaneous symmetry breaking is to break thegauge symmetry in a more subtle way, such that we can still givethe gauge bosons a mass but retain renormalizability.


5.2 Spontaneous Symmetry Breaking

Spontaneous symmetry breaking is a phenomenon that is by farnot restricted to gauge symmetries. It is a subtle way to break asymmetry by still requiring that the Lagrangian remains invariantunder the symmetry transformation. However, the ground state ofthe symmetry is not invariant, i.e. not a singlet under a symmetrytransformation.

In order to illustrate the idea of spontaneous symmetry break-ing, consider a pen that is completely symmetric with respect torotations around its axis. If we balance this pen on its tip on a table,and start to press on it with a force precisely along the axis we havea perfectly symmetric situation. This corresponds to a Lagrangianwhich is symmetric (under rotations around the axis of the pen inthis case). However, if we increase the force, at some point the penwill bend (and eventually break). The question then is in which di-rection will it bend. Of course we do not know, since all directionsare equal. But the pen will pick one and by doing so it will breakthe rotational symmetry. This is spontaneous symmetry breaking.

A better example can be given by looking at a point mass in apotential

V(~r) = µ2~r ·~r + λ(~r ·~r)2. (5.3)

This potential is symmetric under rotations and we assume λ > 0(otherwise there would be no stable ground state). For µ2 > 0 thepotential has a minimum at~r = 0, thus the point mass will simplyfall to this point. The situation is more interesting if µ2 < 0. Fortwo dimensions the potential is shown in Fig. 1. If the point masssits at~r = 0 the system is not in the ground state but the situa-tion is completely symmetric. In order to reach the ground state,the symmetry has to be broken, i.e. if the point mass wants to rolldown, it has to decide in which direction. Any direction is equallygood, but one has to be picked. This is exactly what spontaneoussymmetry breaking means. The Lagrangian (here the potential) issymmetric (here under rotations around the z-axis), but the groundstate (here the position of the point mass once it rolled down) isnot. Let us formulate this in a slightly more mathematical wayfor gauge symmetries. We denote the ground state by |0〉. A spon-taneously broken gauge theory is a theory whose Lagrangian isinvariant under gauge transformations, which is exactly what wehave done in chapters 1 and 2. The new feature in a spontaneouslybroken theory is that the ground state is not invariant under gaugetransformations. This means

e−iωaTa |0〉 6= |0〉 (5.4)

which entailsTa |0〉 6= 0 for some a. (5.5)

Eq. (5.5) follows from eq. (5.4) upon expansion in ωa. Thus, thetheory is spontaneously broken if there exists at least one generatorthat does not annihilate the vacuum.


y

V(r)

x

Figure 1: A potential that leads tospontaneous symmetry breaking.

In the next section we will explore the concept of spontaneoussymmetry breaking in the context of gauge symmetries in moredetail, and we will see that, indeed, this way of breaking the gaugesymmetry has all the desired features.

5.3 The Abelian Higgs Model

For simplicity, we will start by spontaneously breaking the U(1)gauge symmetry in a theory of one complex scalar field. In theStandard Model, it will be a non-abelian gauge theory that is spon-taneously broken, but all the important ideas can simply be trans-lated from the U(1) case considered here.

The Lagrangian density for a gauged complex scalar field, with amass term and a quartic self-interaction, may be written as

L = (DµΦ)∗ DµΦ− 14

FµνFµν −V(Φ), (5.6)

where the potential V(Φ), is given by

V(Φ) = µ2Φ∗Φ + λ |Φ∗Φ|2 , (5.7)

and the covariant derivative Dµ and the field-strength tensor Fµν

are given in eqs. (2.19) and (2.12) respectively. This Lagrangian isinvariant under U(1) gauge transformations

Φ → e−iω(x)Φ. (5.8)

Provided µ2 is positive this potential has a minimum at Φ = 0. Wecall the Φ = 0 state the vacuum and expand Φ in terms of creationand annihilation operators that populate the higher energy states.In terms of a quantum field theory, where Φ is an operator, the pre-cise statement is that the operator Φ has zero vacuum expectationvalue, i.e. 〈0|Φ|0〉 = 0.


Now suppose we reverse the sign of µ2, so that the potentialbecomes

V(Φ) = −µ2Φ∗Φ + λ |Φ∗Φ|2 , (5.9)

with µ2 > 0. We see that this potential no longer has a minimum atΦ = 0, but a (local) maximum. The minimum occurs at

Φ = eiθ

√µ2

2λ≡ eiθ v√

2, (5.10)

where θ can take any value from 0 to 2π. There is an infinite num-ber of states each with the same lowest energy, i.e. we have a de-generate vacuum. The symmetry breaking occurs in the choicemade for the value of θ which represents the true vacuum. For con-venience we shall choose θ = 0 to be our vacuum. Such a choiceconstitutes a spontaneous breaking of the U(1) invariance, since aU(1) transformation takes us to a different lowest energy state. Inother words the vacuum breaks U(1) invariance. In quantum fieldtheory we say that the field Φ has a non-zero vacuum expectationvalue

〈Φ〉 =v√2

. (5.11)

But this means that there are ‘excitations’ with zero energy, thattake us from the vacuum to one of the other states with the sameenergy. The only particles which can have zero energy are masslessparticles (with zero momentum). We therefore expect a masslessparticle in such a theory.

To see that we do indeed get a massless particle, let us expand Φaround its vacuum expectation value,

Φ =eiφ/v√

2

(µ√λ+ H

)' 1√

2

(µ√λ+ H + iφ

). (5.12)

The fields H and φ have zero vacuum expectation values and it isthese fields that are expanded in terms of creation and annihilationoperators of the particles that populate the excited states. Of course,it is the H-field that corresponds to the Higgs field.

We now want to write the Lagrangian in terms of the H and φ

fields. In order to get the potential we insert eq. (5.12) into eq. (5.9)and find

V = µ2H2 + µ√

λ(

H3 + φ2H)+

λ

4

(H4 + φ4 + 2H2 φ2

)+

µ4

4λ.

(5.13)Note that in eq. (5.13) there is a mass term for the H-field, µ2H2 ≡MH/2H2, where we have defined1 1 Note that for a real field φ represent-

ing a particle of mass m the mass termis 1

2 m2φ2, whereas for a complex fieldthe mass term is m2φ†φ.

MH =√

2µ. (5.14)

However, there is no mass term for the field φ. Thus φ is a field fora massless particle called the “Goldstone boson”. We will look atthis issue in a more general way in section 5.4. Next let us considerthe kinetic term. We plug eq. (5.12) into (DµΦ)∗DµΦ and get

(DµΦ)∗ DµΦ =12

∂µH∂µH +12

∂µφ∂µφ +12

g2v2 Aµ Aµ +12

g2 Aµ Aµ(H2 + φ2)

− gAµ

(φ∂µH − H∂µφ

)+ gvAµ∂µφ + g2vAµ AµH. (5.15)


There are several important features in eq. (5.15). Firstly, the gaugeboson has acquired a mass term 1/2g2v2 Aµ Aµ ≡ 1/2M2

A Aµ Aµ,where we have defined

MA = gv. (5.16)

Secondly, there is a coupling of the gauge field to the H-field,

g2vAµ AµH = gMA Aµ AµH. (5.17)

It is important to remember that this coupling is proportional tothe mass of the gauge boson. Finally, there is also the bilinear termg v Aµ∂µφ, which after integrating by parts (for the action S) may bewritten as −MA φ ∂µ Aµ. This mixes the Goldstone boson, φ, withthe longitudinal component of the gauge boson, with strength MA

(when the gauge-boson field Aµ is separated into its transverse andlongitudinal components, Aµ = AL

µ + ATµ , where ∂µ AT

µ = 0).Later on, we will use the gauge freedom to get rid of this mixingterm.

5.4 Goldstone Bosons

In the previous subsection we have seen that there is a masslessboson, called the Goldstone boson, associated with the flat directionin the potential. Goldstone’s theorem describes the appearance ofmassless bosons when a global (not gauge) symmetry is sponta-neously broken.

Suppose we have a theory whose Lagrangian is invariant under asymmetry group G with N generators Ta and the symmetry groupof the vacuum forms a subgroup H of G, with m generators. Thismeans that the vacuum state is still invariant under transformationsgenerated by the m generators of H, but not the remaining N − mgenerators of the original symmetry group G. Thus we have

Ta|0〉 = 0 a = 1 . . . m,

Ta|0〉 6= 0 a = m + 1 . . . N. (5.18)

Goldstone’s theorem states that there will be N −m massless parti-cles (one for each broken generator of the group). The case consid-ered in this section is special in that there is only one generator ofthe symmetry group (i.e. N = 1) which is broken by the vacuum.Thus, there is no generator that leaves the vacuum invariant (i.e.m = 0) and we get N −m = 1 Goldstone boson.

Like all good general theorems, Goldstone’s theorem has a loop-hole, which arises when one considers a gauge theory, i.e. whenone allows the original symmetry transformations to be local. In aspontaneously broken gauge theory, the choice of which vacuumis the true vacuum is equivalent to choosing a gauge, which is nec-essary in order to be able to quantize the theory. What this meansis that the Goldstone bosons, which can, in principle, transformthe vacuum into any of the states degenerate with the vacuum,now affect transitions into states which are not consistent with the


original gauge choice. This means that the Goldstone bosons are“unphysical” and are often called “Goldstone ghosts”.

On the other hand the quantum degrees of freedom associatedwith the Goldstone bosons are certainly there ab initio (before achoice of gauge is made). What happens to them? A massless vec-tor boson has only two degrees of freedom (the two directions ofpolarization of a photon), whereas a massive vector (spin-one) par-ticle has three possible values for the helicity of the particle. In aspontaneously broken gauge theory, the Goldstone boson asso-ciated with each broken generator provides the third degree offreedom for the gauge bosons. This means that the gauge bosonsbecome massive. The Goldstone boson is said to be “eaten” by thegauge boson. This is related to the mixing term between Aµ

L andφ of the previous subsection. Thus, in our abelian model, the twodegrees of freedom of the complex field Φ turn out to be the Higgsfield and the longitudinal component of the (now massive) gaugeboson. There is no physical, massless particle associated with thedegree of freedom φ present in Φ.

5.5 The Unitary Gauge

As mentioned above, we want to use the gauge freedom to choose agauge such that there are no mixing terms between the longitudinalcomponent of the gauge field and the Goldstone boson. Recall

Φ =1√2(v + H) eiφ/v =

1√2

(µ√λ+ H + iφ + . . .

), (5.19)

where the dots stand for nonlinear terms in φ. Next we make agauge transformation (see eq. (2.2))

Φ→ Φ′ = e−iφ/v Φ. (5.20)

In other words, we fix the gauge such that the imaginary part of Φvanishes. Under the gauge transformation eq. (5.20) the gauge fieldtransforms according to (see eq. (2.11))

Aµ → A′µ = Aµ +1

gv[∂µφ]. (5.21)

It is in fact the superposition of Aµ and φ which make up the phys-ical field. Note that the change from Aµ to A′µ made in eq. (5.21)affects only the longitudinal component. If we now express the La-grangian in terms of Φ′ and A′µ there will be no mixing term. Evenbetter, the φ field vanishes altogether! This can easily be seen bynoting that under a gauge transformation the covariant derivativeDµΦ transforms in the same way as Φ, thus

DµΦ→ (DµΦ)′ = e−iφ/vDµΦ = e−iφ/v 1√2

(∂µH + igA′µ(v + H)

),

(5.22)and (DµΦ)′∗(DµΦ)′ is independent of φ. Performing the algebra(and dropping the ′ for the A-field) we get the Lagrangian in the


unitary gauge

L =12

∂µH∂µH +M2

A2

Aµ Aµ − 14

FµνFµν − M2H

2H2

+ gMA Aµ AµH +g2

2Aµ AµH2 − λ

4H4 −

√λ

2MH H3, (5.23)

with MA and MH as defined in eqs. (5.16) and (5.14), respectively.All the terms quadratic in Aµ may be written (in momentum space)as

Aµ(−p)(−gµν p2 + pµ pν + gµν M2

A

)Aν(p). (5.24)

The gauge boson propagator is the inverse of the coefficient ofAµ(−p)Aν(p), which is

−i

(gµν −

pµ pν

M2A

)1

(p2 −M2A)

. (5.25)

This is the usual expression for the propagator of a massive spin-one particle, eq. (5.2). The only other remaining particle is thescalar, H, with mass mH =

√2 µ, which is the Higgs boson. This

is a physical particle, which interacts with the gauge boson and alsohas cubic and quartic self-interactions. The Lagrangian given ineq. (5.23) leads to the following vertices and Feynman rules:

µ

ν

µ

ν

2 i e2gµν

2 i eMAgµν

6 i λ

6 i mH

√2λ

The advantage of the unitary gauge is that no unphysical parti-cles appear, i.e. the φ-field has completely disappeared. The disad-vantage is that the propagator of the gauge field, eq. (5.25), behavesas p0 for p → ∞. As discussed in section 5.1 this seems to indicatethat the theory is non-renormalizable. It seems that we have notgained anything at all by breaking the theory spontaneously ratherthan by simply adding a mass term by hand. Fortunately this is nottrue. In order to see that the theory is still renormalizable, in spite


of eq. (5.25), it is very useful to consider a different type of gauges,namely the Rξ gauges discussed in the next subsection.

5.6 Rξ Gauges (Feynman Gauge)

The class of Rξ gauges is a more conventional way to fix the gauge.Recall that in QED we fixed the gauge by adding a term, eq. (2.25),in the Lagrangian. This is exactly what we do here. The gaugefixing term we are adding to the Lagrangian density eq. (5.6) is

LR ≡ − 12(1− ξ)

(∂µ Aµ − (1− ξ)MAφ

)2

= − 12(1− ξ)

∂µ Aµ∂ν Aν + MAφ∂µ Aµ − 1− ξ

2M2

Aφ2.(5.26)

Again, the special value ξ = 0 corresponds to the Feynman gauge.The second term in eq. (5.26) cancels precisely the mixing term ineq. (5.15). Thus, we have achieved our goal. Note however, that inthis case, contrary to the unitary gauge, the unphysical φ-field doesnot disappear. The first term in eq. (5.26) is bilinear in the gaugefield, thus it contributes to the gauge-boson propagator. The termsbilinear in the A-field are

−12

Aµ(−p)(−gµν(p2 −M2

a) + pµ pν −pµ pν

1− ξ

)Aν(p) (5.27)

which leads to the gauge boson propagator

−i(p2 −M2

A)

(gµν − ξ

pµ pν

p2 − (1− ξ)M2A

). (5.28)

In the Feynman gauge, the propagator becomes particularly simple.The crucial feature of eq. (5.28), however, is that this propagatorbehaves as p−2 for p → ∞. Thus, this class of gauges is manifestlyrenormalizable. There is, however, a price to pay: The Goldstoneboson is still present. It has acquired a mass, MA, from the gaugefixing term, and it has interactions with the gauge boson, withthe Higgs scalar and with itself. Furthermore, for the purposesof higher order corrections in non-Abelian theories, we need tointroduce Faddeev-Popov ghosts which interact with the gaugebosons, the Higgs scalar and the Goldstone bosons.

Let us stress that there is no contradiction at all between the ap-parent non-renormalizability of the theory in the unitary gaugeand the manifest renormalizability in the Rξ gauge. Since physicalquantities are gauge invariant, any physical quantity can be calcu-lated in a gauge where renormalizability is manifest. As mentionedabove, the price we pay for this is that there are more particlesand many more interactions, leading to a plethora of Feynmandiagrams. We therefore only work in such gauges if we want tocompute higher order corrections. For the rest of these lectures weshall confine ourselves to tree-level calculations and work solely inthe unitary gauge.


Nevertheless, one cannot over-stress the fact that it is only whenthe gauge bosons acquire masses through the Higgs mechanismthat we have a renormalizable theory. It is this mechanism thatmakes it possible to write down a consistent Quantum Field Theorywhich describes the weak interactions.

5.7 Summary

• In the case of a gauge theory the Goldstone bosons provide thelongitudinal component of the gauge bosons, which thereforeacquire a mass. The mass is proportional to the magnitude of thevacuum expectation value and the gauge coupling constant. TheGoldstone bosons themselves are unphysical.

• It is possible to work in the unitary gauge where the Goldstoneboson fields are set to zero.

• When gauge bosons acquire masses by this (Higgs) mechanism,renormalizability is maintained. This can be seen explicitly ifone works in a Rξ gauge, in which the gauge boson propagatordecreases like 1/p2 as p → ∞. This is a necessary condition forrenormalizability. If one does work in such a gauge, however,one needs to work with Goldstone boson fields, even though theGoldstone bosons are unphysical. The number of interactionsand the number of Feynman graphs required for the calculationof some processes is then greatly increased.

6The Standard Model with one Family

To write down the Lagrangian of a theory, one first needs to choosethe symmetries (gauge and global) and the particle content, andthen write down every allowed renormalizable interaction. In thissection we shall use this recipe to construct the Standard Modelwith one family. The Lagrangian should contain pieces

L(SM,1) = Lgauge bosons + Lfermion masses + LfermionKT + LHiggs. (6.1)

The terms are written out in eqns. (6.15), (6.29), (6.30) and (6.55).

6.1 Left- and Right- Handed Fermions

The weak interactions are known to violate parity. Parity non-invariant interactions for fermions can be constructed by givingdifferent interactions to the “left-handed” and “right-handed” com-ponents defined in eq. (6.4). Thus, in writing down the StandardModel, we will treat the left-handed and right-handed parts sepa-rately.

A Dirac field, ψ, representing a fermion, can be expressed as thesum of a left-handed part, ψL, and a right-handed part, ψR,

ψ = ψL + ψR, (6.2)

where

ψL = PL ψ with PL =(1− γ5)

2, (6.3)

ψR = PR ψ with PR =(1 + γ5)

2. (6.4)

PL and PR are projection operators, i.e.

PL PL = PL, PR PR = PR and PL PR = 0 = PR PL. (6.5)

They project out the left-handed (negative) and right-handed (pos-itive) chirality states of the fermion, respectively. This is the defi-nition of chirality, which is a property of fermion fields, but not aphysical observable.

The kinetic term of the Dirac Lagrangian and the interactionterm of a fermion with a vector field can also be written as a sum of


two terms, each involving only one chirality

ψ γµ∂µψ = ψL γµ∂µψL + ψR γµ∂µψR, (6.6)

ψ γµ Aµψ = ψL γµ AµψL + ψR γµ AµψR. (6.7)

On the other hand, a mass term mixes the two chiralities:

mψ ψ = mψL ψR + mψR ψL. (6.8)

Exercise 5.1Use (γ5)

2 = 1 to verify eq. (6.5) and ψ = ψ†γ0, γ5 † = γ5 as well asγ5γµ = −γµγ5 to verify eq. (6.7).

In the limit where the fermions are massless (or sufficiently rel-ativistic), chirality becomes helicity, which is the projection of thespin on the direction of motion and which is a physical observable.Thus, if the fermions are massless, we can treat the left-handed andright-handed chiralities as separate particles of conserved helicity.We can understand this physically from the following simple con-sideration. If a fermion is massive and is moving in the positive zdirection, along which its spin is having a positive component sothat the helicity is positive in this frame, one can always boost intoa frame in which the fermion is moving in the negative z direction,but with this spin component unchanged. In the new frame thehelicity will hence be negative. On the other hand, if the particle ismassless and travels with the speed of light, no such boost is possi-ble, and in that case helicity/chirality is a good quantum number.

Exercise 5.2For a massless spinor

u(p) =1√E

(E χ

~σ · ~p χ

),

where χ is a two-component spinor, show that

(1± γ5)u(p)

are eigenstates of~σ · ~p/E with eigenvalues ±1, respectively. Take

γ5 =

(0 11 0

),

and in 4× 4 matrix notation~σ · ~p means(

~σ · ~p 00 ~σ · ~p

).

6.2 Symmetries and Particle Content

We have made all the preparations to write down a gauge invariantLagrangian. We now only have to pick the gauge group and the


matter content of the theory. It should be noticed that there are notheoretical reasons to pick a certain group or certain matter content.To match experimental observations we pick the gauge group forthe Standard Model to be

U(1)Y × SU(2)× SU(3). (6.9)

To indicate that the abelian U(1) group is not the gauge groupof QED but of hypercharge a subscript Y has been added. Thecorresponding coupling and gauge boson is denoted by g′ and Bµ

respectively.The SU(2) group has three generators (Ta = σa/2), the cou-

pling is denoted by g and the three gauge bosons are denotedby W1

µ, W2µ, W3

µ. None of these gauge bosons (and neither Bµ) arephysical particles. As we will see, linear combinations of thesegauge bosons will make up the photon as well as the W± and the Zbosons.

Finally, the SU(3) is the group of the strong interaction. Thecorresponding eight gauge bosons are the gluons. In this section wewill concentrate on the other two groups, with one generation offermions. The strong interaction is dealt with in section 4, and extragenerations are introduced in the next chapter.

As matter content for the first family, we have

qL ≡(

uL

dL

); uR; dR; `L ≡

(νL

eL

); eR; νR!!. (6.10)

Note that a right-handed neutrino νR has appeared. It is a gaugesinglet (no strong interaction, no weak interactions, no electriccharge), so is unneccessary in a model with massless neutrinos.However, neutrinos are now known to have small masses, whichcan be described by adding the right-handed field νR. Neutrinomasses will be discussed further in chapter 8.

Note also that the left- and right-handed fermion componentshave been given different weak interactions. The Standard Modelis constructed this way, because the weak interactions are knownto violate parity. The left-handed components form doublets underSU(2) whereas the right-handed components are singlets. Thismeans that under SU(2) gauge transformations we have

eR → e′R = eR, (6.11)

`L → `′L = e−iωaTa`L. (6.12)

Thus, the SU(2) singlets eR, νR, uR and dR are invariant underSU(2) transformations and do not couple to the correspondinggauge bosons W1

µ, W2µ, W3

µ.Since this separation of the electron into its left- and right-

handed helicity only makes sense for a massless electron we alsoneed to assume that the electron is massless in the exact SU(2) limitand that the mass for the electron arises as a result of spontaneoussymmetry breaking in a similar way as the masses for the gaugebosons arise. We will come back to this later.


Under U(1)Y gauge transformations the matter fields transformas

ψ→ ψ′ = e−iωY(ψ)ψ (6.13)

where Y is the hypercharge of the particle under consideration.It is chosen to give the observed electric charge of the particles.The explicit values for the hypercharges of the particles listed ineq. (6.10) are as follows:

Y(`L) = −12

Y(eR) = −1 Y(νR) = 0,

Y(qL) =16

Y(uR) =23

, Y(dR) = −13

. (6.14)

Under SU(3) the lepton fields `L, eR, νR are singlets, i.e. they donot transform at all. This means that they do not couple to thegluons. The quarks on the other hand form triplets under SU(3).The strong interaction does not distinguish between left- and right-handed particles.

We have now listed all fermions that belong to the first family,together with their transformation properties under the variousgauge transformations. However, since we ultimately want massiveweak gauge bosons, we will have to break the U(1)Y × SU(2) gaugegroup spontaneously, by introducing some type of Higgs scalar.The transformation properties of this scalar will be deduced in thediscussion of fermion masses.

6.3 Kinetic Terms for the Gauge Bosons

The gauge kinetic terms for abelian and non-abelian theories werepresented in the first two lectures. From the general expression ofeq. (3.46), we extract for the SM gauge bosons:

L = −14

BµνBµν− 14

FaµνFa µν− 1

4FA

µνFA µν +Lgauge−fixing +LFP ghosts.(6.15)

Here Bµν = ∂µBν − ∂νBµ is the hypercharge field strength, the sec-ond term contains the SU(2) field strength, so a runs from one tothree (over the three vector bosons of SU(2)), and the third term isthe gluon kinetic term, so A = 1 . . . 8. To do an explicit perturba-tive calculation, additional gauge fixing terms, and Fadeev-Popovghosts, must be included. The form of these terms depends on thechoice of gauge.

6.4 Fermion Masses and Yukawa Couplings

We cannot have an explicit mass term for the quarks or electrons,since a mass term mixes left-handed and right-handed fermionsand we have assigned these to different multiplets of weak SU(2).However, if an SU(2) doublet Higgs is introduced, there is a gaugeinvariant interaction that will look like a mass when the Higgs gets


a vacuum expectation value (“vev”). Such an interaction is called a‘Yukawa interaction’ and is written as

LYukawa = −Ye `LiΦieR + h.c. , (6.16)

where h.c. means ‘hermitian conjugate’. Note that the Higgs dou-blet must have Y = 1/2 to ensure that this term has zero weakhypercharge.

Recalling eq. (6.19) we introduce a scalar “Higgs” field, which isa doublet under SU(2), singlet under SU(3) (no colour), and has ascalar potential as given in eq. (5.9), i.e.

V(Φ) = −µ2Φ∗Φ + λ |Φ∗Φ|2 . (6.17)

This potential has a minimum at Φ∗Φ = 12 µ2/λ, so some compo-

nent of the Higgs doublet should get a vev. In the unitary gauge,this vev can be written as

〈Φ〉 = 1√2

(0v

)(6.18)

with v = µ/√

λ.Recall from the previous chapter that Φ can be written as its

“radial” degree of freedom times an exponential containing thebroken generators of the gauge symmetry:

Φ =ei(ωaTa−ω3Y)√

2

(0

v + H

). (6.19)

The unitary gauge choice consists of absorbing this exponentialwith a gauge transformation, so that in the unitary gauge eq. (6.16)is

LYukawa = − Ye√2

(νL eL

)( 0v + H

)eR + h.c. . (6.20)

The part proportional to the vev is simply

− Ye v√2

(eL eR + eR eL) = −Ye v√2

e e , (6.21)

and we see that the electron has acquired a mass which is propor-tional to the vev of the scalar field. This immediately gives us arelation for the Yukawa coupling in terms of the electron mass, me,and the W mass, MW :

Ye = gme√2 MW

. (6.22)

Thus, as for the gauge bosons, the strength of the coupling of theHiggs to fermions is proportional to the mass of the fermions.

The quarks also acquire a mass through the spontaneous sym-metry breaking mechanism, via their Yukawa coupling with thescalars. The interaction term

−Yd qLi Φi dR + h.c. (6.23)


gives a mass to the d quark when we replace Φi by its vev. Thismass md is given by

md =Yd√

2v =

√2

Yd MWg

. (6.24)

Since the vev is in the lower component of the Higgs doublet, wemust do a little more work to obtain a mass for the upper element uof the quark doublet. In the case of SU(2) there is a second way inwhich we can construct an invariant for the Yukawa interaction:

−Yu εij qLi Φ∗ juR + h.c. (i, j = 1, 2), (6.25)

where εij is the two-dimensional antisymmetric tensor. Note that

Φc = εijΦj∗ =

(0 1−1 0

)(Φ∗+Φ∗0

)(6.26)

has Y = −1/2, as required by the U(1) symmetry. This term doesindeed give a mass mu to the u quark, where

mu =Yu√

2v =

√2

Yu MWg

. (6.27)

So the SM Higgs scalar couples to both the u and d quark, withinteraction terms

− gmu

2 MWu H u − g

md2 MW

d H d . (6.28)

The terms in the Lagrangian that give masses to the first generationquarks and charged leptons are

Lfermion masses = −Ye lLiΦieR − Yd qL

i Φi dR − Yu εij qLi Φ∗ juR + h.c. .

(6.29)We could also have included a Yukawa mass term for the neu-

trinos: −Yν εij `LiΦ∗ jνR+ h.c. However, neutrino masses do not

neccessarily arise from a Yukawa interaction (this will be discussedin chapter 8).

6.5 Kinetic Terms for Fermions

The fermionic kinetic terms should be familiar from chapter 3:

LfermionKT = i `LT

γµDµ `L + i eR γµDµ eR + i νR γµ∂µ νR

+ i qLT γµDµ qL + i dR γµDµ dR + i uR γµ∂µ uR (6.30)

where the covariant derivatives include the hypercharge, SU(2) andSU(3) gauge bosons as required. For instance:

Dµ = ∂µ + igTaWaµ + ig′Y(`L)Bµ for `L , (6.31)

Dµ = ∂µ + ig′Y(eR)Bµ for eR , (6.32)

Dµ = ∂µ + igsTsaGa

µ + ig′Y(dR)Bµ for dR , (6.33)


where the strong coupling (gs), the eight generators of SU(3) (Tsa)

and the corresponding gluon fields (Gaµ) have been introduced, and

Y( f ) is the hypercharge of fermion f .This gives the following interaction terms between the leptons

and the gauge bosons:

− g2

(νL

eL

)T

γµ

((W3

µ

√2 W+

µ√2 W−µ −W3

µ

)− tan θW Bµ

)(νL

eL

)

+ i g tan θW eR γµBµeR , (6.34)

where we have used W± = (W1 ∓ iW2)/√

2. The fields Bµ andW3

µ are replaced by the physical particles Zµ and Aµ through the‘rotation’

Zµ ≡ cos θW W3µ − sin θW Bµ , (6.35)

Aµ ≡ cos θW Bµ + sin θWW3µ . (6.36)

(In the exercises of chapter 3 these definitions followed from requir-ing that the photon does not interact with the neutrino. In section6.6 we will see that the photon is also massless.)

Writing out the projection operators for left- and right-handedfermions, eqs. (6.3) and (6.4), we obtain the following interactions:

1. A coupling of the charged vector bosons W± which mediatetransitions between neutrinos and electrons (or u and d quarks)with an interaction term

− g2√

2ν γµ

(1− γ5

)e W+

µ −g

2√

2u γµ

(1− γ5

)d W+

µ + h.c.

(6.37)(h.c. means ‘hermitian conjugate’ and gives the interaction in-volving an emitted W+

µ where the incoming particle is a neutrino(or u) and the outgoing particle is an electron (or d).)

2. The usual coupling of the photon with the charged fermions is(using, for instance, the relation eq. (6.54)):

g sin θW e γµ e Aµ −23

g sin θW u γµ u Aµ +13

g sin θW d γµ d Aµ .(6.38)

Note that the left- and right-handed fermions have exactly thesame coupling to the photon so that the electromagnetic cou-pling turns out to be purely vector (i.e. no γ5 term).

3. The coupling of neutrinos to the neutral weak gauge boson Zµ:

− g4 cos θW

ν γµ(

1− γ5)

ν Zµ . (6.39)

4. The coupling of both the left- and right-handed electron to theZ:

g4 cos θW

e(

γµ(

1− γ5)− 4 sin2 θWγµ

)e Zµ . (6.40)


5. The coupling of the quarks to the Z can be written in the generalform

− g2 cos θW

qi

(T3

i γµ(

1− γ5)− 2Qi sin2 θWγµ

)qi Zµ , (6.41)

where quark i has the third component of weak isospin T3i and

electric charge Qi.

From these terms in the Lagrangian we can directly read off theFeynman rules for the three-point vertices with two fermions andone weak gauge boson. Then we can use these vertices to calculateweak interactions of the quarks and leptons. This allows us, for ex-ample, to calculate the total decay width of the Z or W boson, bycalculating the decay width into all possible quarks and leptons.However, quarks are not free particles, so for exclusive processes,in which we trigger on known initial or final state hadrons, infor-mation is needed about the probability to find a quark with givenproperties inside an initial hadron or the probability that a quarkwith given properties will decay (“fragment”) into a final statehadron.

Exercise 5.3The decay rate for the Z into a fermion-antifermion pair, Z → f f , is

Γ =1

2MZ

∫dLIPS |M|2 =

164 π2 MZ

∫dΩ |M|2,

where dLIPS stands for the Lorentz invariant phase space measurefor the two final-state fermions, and

∫dΩ is the integral over the

solid angle (of one final-state particle).Write the general interaction term for the coupling of the Z bosonto a fermion as

− g2 cos θW

γµ(

v f − a f γ5)

.

Show that the squared matrix element, summed over the spins ofthe (outgoing) fermions and averaged over the spin of the (incom-ing) Z boson is

|M|2 = − 112

gµνg2

cos2 θW

((v f )

2 + (a f )2)

Tr (γµγ · k1γνγ · k2) ,

where k1 and k2 are the momenta of the outgoing fermions and thegauge polarization sum is

∑λ

ε(λ)∗µ ε

(λ)ν = −gµν +

qµ qν

M2Z

(q = k1 + k2 is the initial momentum of the Z boson). Hence showthat

Γ =1

48 π

g2

cos2 θW

((v f )

2 + (a f )2)

MZ.

Neglect the masses of the fermions in comparison to the Z mass.


Exercise 5.4The Z boson can decay leptonically into a pair of neutrinos orcharged leptons of all three generations and hadronically into uquarks, d quarks, c quarks, s quarks, or b quarks (c quarks couplelike u quarks, whereas s quarks and b quarks couple like d quarks).Deduce the values of v f and a f for each of these cases and con-sequently estimate the decay width of the Z boson. (The currentexperimental value is 2.4952± 0.0023 GeV.)[Take MZ = 91.19 GeV, sin2 θW = 0.23, and the fine-structure con-stant α = 1/129 (why this value?)].

6.6 The Higgs Part and Gauge Boson Masses

The Higgs doublet Lagrangian should contain a “spontaneoussymmetry breaking” potential which will give the Higgs a vev andself-interactions, and kinetic terms which will generate the gaugeboson masses and interactions between the Higgs and the gaugebosons. We first consider the potential:

V(Φ) = −µ2Φ∗i Φi + λ(

Φ∗i Φi)2

. (6.42)

This potential has a minimum at Φ+i Φi =

12 µ2/λ. Writing Φ in the

form of eq. (6.19) and replacing this in the potential eq. (6.42), wefind that we get a mass term for the real Higgs field H, with valuemH =

√2µ. As expected, the ωa do not appear in the potential. In

an ungauged theory, they would be the massless goldstone bosons.In a gauge theory like the Standard Model, they will reappear asthe longitudinal degrees of freedom of the massive gauge bosons.

The remaining term of the Φ Lagrangian is the kinetic term(DµΦ)†(DµΦ). Looking at this term more carefully will help us tounderstand where the “physical” gauge bosons (i.e. the W±, Z andphoton) come from, and how they are related to the W1

µ, W2µ, W3

µ, Bµ.To see the effect of the Higgs vev on the gauge boson masses, it ismost simple to work in the unitary gauge, that is, we absorb theexponential of eq. (6.19) with a gauge transformation. In this gauge,the covariant derivative acting on the Higgs doublet is

Dµ Φ =1√2

(∂µ + i

g2

(W3

µ

√2 W+

µ√2 W−µ −W3

µ

)+ i

g′

2Bµ

)(0

v + H

),

(6.43)so that

∣∣Dµ Φ∣∣2 =

12(∂µ H)2 +

g2v2

4W+ µW−µ +

v2

8

(g W3

µ − g′Bµ

)2

+ interaction terms , (6.44)

where the ‘interaction terms’ are terms involving three fields (twogauge fields and the H-field). Eq. (6.44) tells us that the W3

µ and Bµ

fields mix (as do W1µ and W2

µ) and the physical gauge bosons must


be superpositions of these fields, such that there are no mixingterms. Thus we define

Zµ ≡ cos θW W3µ − sin θW Bµ , (6.45)

Aµ ≡ cos θW Bµ + sin θWW3µ , (6.46)

with the weak mixing angle θW (“Weinberg angle”) defined by

tan θW ≡g′

g. (6.47)

With this eq. (6.44) is rewritten as

∣∣Dµ Φ∣∣2 =

12(∂µ H)2 +

g2v2

4W+

µ W− µ +v2g2

8 cos2 θWZµZµ + 0Aµ Aµ .

(6.48)Here we see how SU(2) and U(1) are unified (or at least ‘entan-gled’) in the sense that the neutral gauge boson that acquires amass through the Higgs mechanism is the linear superposition of agauge boson from the SU(2) and the U(1)Y gauge boson.

From eq. (6.48) we can read off the masses of the gauge bosons.The last term tells us that the linear combination eq. (6.46) remainsmassless. This field is identified with the photon. For the otherfields we have

MW =12

gv , MZ =12

gvcos θW

. (6.49)

The Z boson mediates the neutral current weak interactions.These were not observed until after the development of the theory.From the magnitude of amplitudes involving weak neutral currents(exchange of a Z boson), one can infer the (tree level) magnitude ofthe weak mixing angle, θW . The ratio of the masses of the Z and Wbosons is a prediction of the Standard Model. More precisely, wedefine a quantity known as the ρ-parameter by

M2W = ρ M2

Z cos2 θW . (6.50)

In the Standard Model ρ = 1 at tree level. In higher ordersthere is a small correction, which depends on the definition usedfor sin θW (that is, which loop corrections are included in sin θW).Note that the ρ-parameter would be very different from one ifthe symmetry breaking were due to a scalar multiplet which wasnot a doublet of weak isospin. Accurate measurements of the ρ-parameter and other so-called electro-weak precision observables,together with their prediction at higher order within the SM, serveas very powerful tests of the SM. The Higgs enters in virtual loops,allowing for an indirect determination of its mass through fits ofthe predictions to the data (see the homepage of the ElectroweakWorking Group, http://lepewwg.web.cern.ch/LEPEWWG for moreinformation).

The spontaneous symmetry breaking mechanism breaks SU(2)×U(1)Y down to U(1). It is this surviving U(1) that is identified as


the U(1) of electromagnetism. It is not the U(1)Y of the originalgauge group but a set of transformations generated by a particularlinear combination of the original U(1) and rotations about thethird axis of weak isospin. To see this we note that the explicitrepresentation of the generator Y as a 2× 2 matrix, which can becombined with the explicit representation of T1, T2 and T3, is givenby

Y =12

(1 00 1

). (6.51)

The factor 1/2 ensures the normalization1 condition eq. (3.8). Using 1 We warn the reader that in theliterature sometimes a different nor-malization is used such that eq. (6.52)reads Q = Y/2 + T3.

eq. (6.51) it can easily be seen that the symmetry associated withthe generator

Q ≡ Y + T3 =

(1 00 0

)(6.52)

is not broken, i.e. Q|0〉 = 0 (see eq. (5.5)). Thus, starting with fourgenerators, we get only three Goldstone bosons. These will becomethe longitudinal components of three gauge bosons, thereby givingthem a mass, whereas the fourth is left massless.

The coupling of any particle to the photon is always proportionalto

g sin θW (Y + T3) = g sin θW Q . (6.53)

Thus we can identify g sin θW with one unit of electric charge, andwe have the relationship between the weak coupling g and theelectron charge e,

e = g sin θW . (6.54)

We end this subsection by giving the remaining pieces of the SMLagrangian from eqs. (6.44) and (6.42),

LHiggs =∣∣Dµ Φ

∣∣2 − µ2Φ∗i Φi + λ(

Φ∗i Φi)2

=12(∂µ H)2 + µ2H2 +

g2v2

4W+ µW−µ +

v2g2

8 cos2 θWZµZµ

+ interaction terms . (6.55)

6.7 Classifying the Free Parameters

The free parameters in the Standard Model for one generation are:

• The two gauge couplings for the SU(2) and U(1) gauge groups,g and g′.

• The two parameters µ and λ in the scalar potential V(Φ).

• The Yukawa coupling constants Yu, Yd, Ye and Yν.

It is convenient to replace these parameters by others, whichare more directly measurable in experiments, namely e, sin θW , me

and mW , and mH , mu, md and mν. (Note that the gauge sector iswell measured, but the quark masses are not directly observable;although we see neutrino mass differences, measuring the absolute


mass scale is difficult — and the neutrino masses might not bedirectly proportional to Yukawa couplings anyway.) The relationbetween these physical parameters and the parameters of the initialLagrangian are

tan θW =g′

g, (6.56)

e = g sin θW , (6.57)

mH =√

2µ , (6.58)

MW =g µ

2√

λ, (6.59)

me = Yeµ√λ

. (6.60)

Note that when we add more generations of fermions, we willacquire more parameters: additional masses (or yukawa couplings,i.e. 4 parameters per generation), and also mixing angles, as we willsee in the next chapter.

In terms of these measured quantities, the Z mass, MZ, and theFermi-coupling, GF, are predictions of the SM (although historicallyGF was known for many years before the discovery of the W boson,and its value was used to predict the W mass).

6.8 Feynman Rules in the Unitary Gauge (for one Lepton Gen-eration)

For completeness, we now list the full set of Feynman rules for thecase of a single family of leptons.

Propagators

All propagators carry momentum p.


Wµ ν −i (gµν − pµ pν/M2

W )/(p2 − M2W )

Zµ ν −i (gµν − pµ pν/M2

Z)/(p2 − M2Z)

Aµ ν −i gµν/p2

ei (γ · p + me)/(p2 − m2

e)

νi γ · p/p2

Hi/(p2 − m2

H)

Three-point gauge-boson couplings

All momenta are defined as incoming.

Aρ

W−µ W+

ν

p3

p1 p2i g sin θW ((p1 − p2)ρ gµν + (p2 − p3)µ gνρ + (p3 − p1)ν gρµ)

Zρ

W−µ W+

ν

p3

p1 p2i g cos θW ((p1 − p2)ρ gµν + (p2 − p3)µ gνρ + (p3 − p1)ν gρµ)


Four-point gauge-boson couplings

W−ρ W+

σ

W−µ W+

ν

i g2 (2gµρ gνσ − gµν gρσ − gµσ gνρ)

Zρ Zσ

W−µ W+

ν

i g2 cos2 θW (2gµν gρσ − gµρ gνσ − gµσ gνρ)

Aρ Aσ

W−µ W+

ν

i g2 sin2 θW (2gµν gρσ − gµρ gνσ − gµσ gνρ)

Zρ Aσ

W−µ W+

ν

i g2 cos θW sin θW (2gµν gρσ − gµρ gνσ − gµσ gνρ)


Three-point couplings with Higgs scalars

H H

− 32 i g m2

H/MW

H

e e

− 12 i g me/MW

H

W−µ W+

ν

i g MW gµν

H

Zµ Zν

i(g/ cos2 θW

)MW gµν


Four-point couplings with Higgs scalars

H H

H H

− 34 i g2 (m2

H/M2W )

H H

W−µ W+

ν

12 i g2 gµν

H H

Zµ Zν

12 i

(g2/ cos2 θW

)gµν

Fermion interactions with gauge bosons

All momenta are defined as incoming.


W−µ

e ν

− i(g/2

√2)

γµ

(1 − γ5

)

Aµ

e e

i g sin θW γµ

Zµ

e e

14 i (g/ cos θW ) γµ

(1 − 4 sin2 θW − γ5

)

Zµ

ν ν

− 14 i (g/ cos θW ) γµ

(1 − γ5

)

6.9 Summary

• Weak interactions are mediated by the SU(2) gauge bosons,which act only on the left-handed components of fermions.

• The (left-handed) neutrino and left-handed component of theelectron form an SU(2) doublet, whereas the right-handed com-ponents of the electron and neutino are SU(2) singlets. Similarlyfor the quarks.

• There is also a weak hypercharge U(1)Y gauge symmetry. Bothleft- and right-handed quarks transform under this U(1)Y witha hypercharge which is related to the electric charge by the re-lation eq. (6.54). The left-handed leptons and the eR also carryhypercharge, but the νR has no SM gauge interactions.

• In the symmetry limit (before spontaneous symmetry breaking)the fermions with SU(2) gauge interactions are massless.1 The 1 This does not apply to νR, which can

have an explicit mass termspontaneous symmetry breaking mechanism which gives a vevto the scalar field also generates the fermion masses.


• The scalar multiplet that is responsible for the spontaneous sym-metry breaking also carries weak hypercharge. As a result, oneneutral gauge boson (the Z) acquires a mass, whereas its orthog-onal superposition is the massless photon. The magnitude of theelectron charge, e, is then given by e = g sin θW .

• The weak interactions proceed via the exchange of massivecharged or neutral gauge bosons. The old four-fermi weakHamiltonian is an effective Hamiltonian which is valid for lowenergy processes in which all momenta are small compared withthe W mass. The Fermi coupling is obtained in terms of e, MW

and sin θW by eq. (7.30).

7Additional Generations, the CKM Matrix, and CP Vi-olation

In the previous section, the Lagrangian of the Standard Model withone family was given. Here we include additional “families” (or“generations”) and discuss some of the phenomenological conse-quences in the quark sector. Family-changing processes among theleptons will be discussed in the neutrino chapter.

7.1 Adding Additional Generations

The additional generations to be added to the SM are labeled as

I II III(

νe

e

) (νµ

µ

) (ντ

τ

)

(ud

) (cs

) (tb

). (7.1)

The new generations II and III have the same quantum numbersas I, so that the only difference concerns their Yukawa interactions.Indeed, the masses in the different generations differ substantially 1 1 M. Tanabashi et al. Review of Particle

Physics. Phys. Rev., D98(3):030001, 2018

me = 0.5109989461(31)MeV, mµ = 105.6583745(24)MeV,

mτ = 1776.86(12)MeV,

mu = 2.16+0.49−0.26 MeV, mc = 1.27(2)GeV,

mt = 172.9(4)GeV,

md = 4.67+0.48−0.17 MeV, ms = 93+11

−5 MeV,

mb = 4.18+0.03−0.02 GeV, (7.2)

in the case of the quarks covering five orders of magnitude!2 2 The definition of masses also dependson the scale, in the same way as thedefinition of coupling “constants” asdiscussed in chapter 4 does. Especiallyfor the light quarks the resulting scaleand scheme dependence is significant,but does not change the overall orderof magnitude.

To obtain the most general Lagrangian for n generations it istherefore sufficient to promote the Yukawa couplings to matrices ingeneration space

LYukawa = −[Ye]ij`Li Φ eRj − [Yd]ijqLi Φ dRj − [Yu]ijqLi Φc uRj + h.c.,(7.3)


where eRi = (eR, µR, τR), uRi = (uR, cR, tR), etc. At first sight itappears that this introduces 6n2 real parameters, but most of themare unphysical and can be removed by unitary transformations ofthe fields, e.g.,

eLi → VeijeLj, eRi → Ue

ijeRj, (7.4)

does not change the physics for arbitrary unitary matrices Ue, Ve,i.e., we can replace the Yukawa matrix by

Ye → V†YeU. (7.5)

To count the number of physical parameters we first consider Y†e Ye,

which under the transformation (7.5) goes to U†Y†e YeU. Since Y†

e Ye

is Hermitian, it can be diagonalized by such a unitary transforma-tion with non-negative eigenvalues1 1~v†Y†

e Ye~v = |Ye~v|2 ≥ 0, ∀~v.

Y†e Ye → diag(c2

e , c2µ, c2

τ) (7.6)

Finally, we can choose another arbitrary unitary matrix W in such away that

Ye = Wdiag(ce, cµ, cτ), ci ≥ 0. (7.7)

Since the matrix V is still arbitrary up to this point, we can stillchoose V = W† and thereby define a basis in which Ye is diagonal

Ye = diag(ce, cµ, cτ). (7.8)

That is, the interaction terms in the Lagrangian and the mass termscan be simultaneously diagonalized.

The up-type Yukawa matrix Yu can be diagonalized in the sameway by choosing unitary transformations UuR and VqL , but after-wards the field redefinitions of qL have been exploited and there-fore Yd stays non-diagonal (alternatively, we could choose a basis inwhich Yd is diagonal, but not Yu). Explicitly, we have

LYukawa,d = −qL VqL W†d︸︷︷︸

VCKM

diag(cd, cs, cb)ΦdR + h.c.

→ −qLVCKMdiag(cd, cs, cb)V†CKMΦdR + h.c., (7.9)

where in the final step we have used the remaining freedom in thechoice of the dR fields. The unitary matrix VCKM is named afterCabibbo,2 Kobayashi, Maskawa.3 2 Nicola Cabibbo. Unitary Symmetry

and Leptonic Decays. Phys. Rev. Lett.,10:531–533, 1963

3 Makoto Kobayashi and ToshihideMaskawa. CP Violation in the Renor-malizable Theory of Weak Interaction.Prog. Theor. Phys., 49:652–657, 1973

After spontaneous symmetry breaking the Yukawa terms takethe form

LYukawa = − ¯dVCKMdiag(md, ms, mb)V†CKMd

− udiag(mu, mc, mt)u− ediag(me, mµ, mτ)e. (7.10)

Accordingly, the d-quark mass term is not diagonal, and the masseigenstates d (“mass basis”) differ from the original fields d (“inter-action basis”) by the unitary CKM transformation

dsb

= V†

CKM

dsb

. (7.11)


For the charged-current interactions this implies

Lcc = −g√2

uct

T

/W+PL

dsb

= − g√2

uct

T

/W+PLVCKM

dsb

. (7.12)

Unless the CKM matrix is diagonal, this term produces interactionsamong the three generations, only suppressed by the correspondingCKM element

VCKM =

Vud Vus Vub

Vcd Vcs Vcb

Vtd Vts Vtb

. (7.13)

Naively, this leaves 2n2 − n2 = n2 real parameters fro a generalunitary matrix, but we can further exploit phase redefinitions of theindividual quark fields to absorb unphysical parameters

q→ e−iφq q, q = u, d, s, c, b, t. (7.14)

Before tackling the general case, we first consider a fictitiousworld with two generations. In this case, these phase transforma-tions would lead to

VCKM =

(Vud Vus

Vcd Vcs

)→(

Vudei(φu−φd) Vusei(φu−φs)

Vcdei(φc−φd) Vcsei(φc−φs)

). (7.15)

Three of these phase differences can be chosen freely, but the fourthis then fixed

φc − φs = φc − φd − (φu − φd) + φu − φs. (7.16)

In particular, we can choose Vud > 0, Vus > 0, Vcs > 0. Unitarity forn = 2 implies

|Vud|2 + |Vus|2 = 1, |Vcd|2 + |Vcs|2 = 1,

V∗udVcd + V∗usVcs = 0. (7.17)

Setting Vud = cos θC, the unitarity relations (7.17) determine all theother entries, especially Vcd = − sin θC comes out negative. Theresult

VC(KM) =

(cos θC sin θC

− sin θC cos θC

)(7.18)

is simply a rotation in two dimensions parameterized in terms ofthe Cabibbo angle θC. In particular, all complex phases could beremoved.

For general n the counting is as follows: after removing the 2n−1 independent phase differences, we are left with n2 − (2n− 1) =

(n− 1)2 real parameters. In n dimensions there are(

n2

)=

n2(n− 1) (7.19)


rotations, which leaves

(n− 1)2 − n2(n− 1) =

12(n− 1)(n− 2) (7.20)

complex phases. The physical case n = 3 is therefore the firstin which a complex phase arises. With angles θ12, θ13, θ23, andcij = cos θij, sij = sin θij, the typical parameterization reads

VCKM =

1 0 00 c23 s23

0 −s23 c23

c13 0 s13e−iδ

0 1 0−s13eiδ 0 c13

c12 s12 0−s12 c12 0

0 0 1

=

c12c13 s12c13 s13e−iδ

−s12c23 − c12s23s13eiδ c12c23 − s12s23s13eiδ s23c13

s12s23 − c12c23s13eiδ −c12s23 − s12c23s13eiδ c23c13

.

(7.21)

The complex phase δ is the only source of CP violation in the Stan-dard Model (apart from the QCD θ-term). As we will see later,VCKM is “almost diagonal,” with angles

θ12 ≈ 13°, θ13 ≈ 0.2°, θ23 ≈ 2.4°. (7.22)

The complex phase itself is relative large, δ ≈ 69°, but its effect isalways accompanied by the small θ13. In fact, the size of CP viola-tion in the Standard Model is suppressed by the product of all threeangles, as reflected by the numerical value of the Jarlskog invariantJ ≈ 3× 10−5, see section 7.5.

7.2 Fermi Theory and Muon Decay

Fermi theory1 refers to Fermi’s description of neutron β decay 1 E. Fermi. An attempt of a theory ofbeta radiation. 1. Z. Phys., 88:161–177,1934

n→ pe−νe in terms of the 4-fermion interaction

L = − GF√2

eγµν pγµn, (7.23)

which phenomenologically works reasonably well. Later on, it wasrealized by Lee and Yang2 that the same approach would also work 2 T. D. Lee and Chen-Ning Yang.

Question of Parity Conservation inWeak Interactions. Phys. Rev., 104:254–258, 1956

if parity were violated, by adding terms with γ5 to the Lagrangian.Shortly thereafter, parity violation was indeed discovered in nuclearβ decays. 3 The principle of the experiment is illustrated in Fig. 1: 3 C. S. Wu, E. Ambler, R. W. Hayward,

D. D. Hoppes, and R. P. Hudson. Ex-perimental Test of Parity Conservationin β Decay. Phys. Rev., 105:1413–1414,1957

if parity were conserved, the configurations in which the electron(anti-neutrino) spins are opposite to the direction of motion shouldbe equally probable to the flipped one, but experiment shows aclear preference for the e− and νe being left-handed. This leads tothe V − A structure of weak interactions.

For muon decay one may write a similar 4-fermion interaction

L = −2√

2GFeLγµνeLνµLγµµL + h.c., (7.24)

which also gives a good description of muon decay µ− → e−νeνµ.However, such interactions are problematic because of their lack of


60Co → 60Ni

Jz = 5 Jz = 4

+P↔

e−

νe e−

νe

Figure 1: Parity violation in nuclear βdecay.

renormalizability. This follows already from the fact that their massdimension is 6, requiring a dimensionful coupling GF to restore thecorrect dimensions. Explicitly, the resulting problems arise whenconsidering the cross section for the related process µ−νµ → e−νe

for large center-of-mass energy s = (pµ + pνµ)2. The squared matrix

element derived from (7.24) will behave as

∑spins|M|2 ∼ G2

Fs2, (7.25)

where one power in√

s comes from each spin sum ∑s u(p, s)u(p, s) =

/p + m etc. Together with the flux factor the cross section still risesaccording to

σ =G2

Fs3π

(7.26)

as a function of s, which demonstrates that the theory breaks downat high energies. Formally, this argument is related to unitarity ofthe S-matrix, S†S = 1, which implies that the matrix element andthus the cross section cannot increase asymptotically. In fact thescale of the dimensionful coupling indicates at which scale thisbreakdown occurs, as we will now derive in more detail.1 1 A similar unitarity argument for WW

scattering showed, even before theHiggs discovery, that new degrees offreedom were required at the latestaround the TeV scale.

In the Standard Model muon decay is mediated by the exchangeof a W− boson:

µ

νµ

νe

eW−

Figure 2: Muon decay.

with matrix element

iM =

(− ig√

2

)2 14

u(pνµ)γµ(1− γ5)u(pµ)

× u(pe)γν(1− γ5)ν(pνe)

−igµν

q2 −M2W

, (7.27)


and momentum transfer q2 = (pνµ − pµ)2. Since the momentumtransfer is much smaller than MW , we can expand

1q2 −M2

W= − 1

M2W

(1 +

q2

M2W

+ · · ·)

, (7.28)

to obtain

iM = −ig2

8M2W

u(pνµ)γµ(1− γ5)u(pµ)u(pe)γµ(1− γ5)ν(pνe). (7.29)

Matching this expression onto (7.24), we have thus re-derived the4-fermion interaction as the low-energy limit of the Standard Modeland found the relation

GF√2=

g2

8M2W

=1

2v2 , (7.30)

implyingv = (

√2GF)

−1/2 = 246 GeV. (7.31)

In particular, had we written the coefficient in (7.24) simply as1/Λ2, we would have concluded1 1 Another popular convention absorbs

the√

2 into the vev v = 174 GeV.

Λ =v√2= 174 GeV, (7.32)

which differs from MW just by the numerical factor√

2/g.For µ−νµ → e−νe at asymptotically high energies we can neglect

the MW mass, and indeed the extra suppression by s2 from the Wpropagators makes the cross section (7.26) decrease according toσ ∼ 1/s. The Fermi theory is thus an example of a low-energyeffective theory, valid within a certain energy domain, but breakingdown at energies where degrees of freedom can be probed that arenot included in the theory. The ratio of the relevant scales, in thecase of muon decay mµ/MW , is the expansion parameter of theeffective theory. In the end, it is the presence of the factor 1/M2

Win (7.30) that makes the weak interactions “weak.”

Similar arguments also apply to neutral-current processes. Forexample, we can have elastic scattering of muon-type neutrinosagainst electrons via the exchange of the Z boson. The Feynmandiagram for such a process is:

νµ

νµ

ee

Z

Figure 3: Example of a neutral currentreaction: νµe− scattering.

Experimentally, one can thus have scattering events (for a beamof muon neutrinos impinging an a target) that do not have muons


in the final state (as would be the case for a charged-current reac-tion), and observing such events would be evidence for the exis-tence of a neutral mediator such as the Z-boson. This is how theGargamelle experiment demonstrated the existence of neutral cur-rents. 2 2 F. J. Hasert et al. Observation of

Neutrino Like Interactions WithoutMuon Or Electron in the GargamelleNeutrino Experiment. Phys. Lett.,46B:138–140, 1973

7.3 Flavor-Changing Neutral Currents (FCNCs)

The charged-current interactions can mediate flavor change evenbetween different families, see (7.12), simply suppressed by the cor-responding CKM matrix element. For the neutral currents, this isnot the case since the interaction term is diagonal both in interac-tion and mass basis

Lnc = −g

2 cos θW

¯d¯s¯b

T

/Z(T3(1− γ5)− 2Q sin2 θW

)

dsb

= − g2 cos θW

dsb

T

/Z(T3(1− γ5)− 2Q sin2 θW

)

dsb

. (7.33)

The absence of flavor-changing neutral currents (FCNCs) at treelevel is an important feature of the Standard Model.

At higher orders FCNCs do appear, e.g., b → sγ and b → sZtransitions can proceed by the following so-called “penguin” dia-grams

b s

u, c, t

W−

γ, Z

b s

W−

u, c, t

γ, Z

Figure 4: Examples of 1-loop FCNCs inthe Standard Model.

which obviously do not look anything like penguins (see 1 for 1 Mikhail A. Shifman. Forewordto ITEP lectures in particle physics.hep-ph/9510397, 1995

the origin of this ridiculous name). More importantly, such con-tributions are suppressed not only by CKM matrix elements, butfor several other reasons as well: for both diagrams, the amplitudetakes the generic form

M = ∑q=u,c,t

VqbV∗qs f (mq)g2e2

16π2 cos2 θW, (7.34)

where the explicit form of the loop function f (mq), dependingon the quark mass mq, differs for each diagram. The suppressionfactors are

1. Loop factor g2/(16π2 cos2 θW),

2. CKM suppression: |VqbV∗qs| 1,


3. Mass suppression: for mq → 0, the amplitude becomes

M→ g2e2

16π2 cos2 θWf (0) ∑

q=u,c,tVqbV∗qs = 0, (7.35)

because of unitarity of the CKM matrix.

The loop factor reflects the fact that the process itself is only al-lowed at higher orders in perturbation theory. One of the CKMfactors has to be off-diagonal and thus small (this one is similarto charged-current transitions between generations). The finalsuppression that arises due to unitarity of the CKM matrix is anexample of the so-called “GIM mechanism” first discovered in thecontext of KL → µ+µ− decays,1 before the discovery of the c quark. 1 S. L. Glashow, J. Iliopoulos, and

L. Maiani. Weak Interactions withLepton-Hadron Symmetry. Phys. Rev.,D2:1285–1292, 1970

Without the c quark, only the left diagram in

d

s

W−

W+

µ

µ

νu c+

d

s

W−

W+

µ

µ

ν

Figure 5: The decay K0 → µ+µ−

(the K0 consists of a d quark and ans antiquark). There are also penguincontributions as in Fig. 4, with aµ+µ− pair attached to γ, Z, as well asdiagrams involving the t quark.

contributes, which does provide a suppression by the loop factorand VudV∗us ∼ cos θC sin θC (in the two-flavor theory), but this wouldnot suffice to explain the experimentally observed rate. The ideabehind the GIM mechanism is that in the massless limit the addi-tional c-quark diagram is identical to the u-quark one apart from adifferent CKM factor VcdV∗cs ∼ − cos θC sin θC, and thus cancels themass-independent part with corrections that scale as O(m2

c /M2W).

In fact, this relation was used (successfully) to estimate the mass ofthe c quark prior to its discovery!

Another interesting probe is the analogous decay involving bquarks, Bs,d → µ+µ−, where Bs(d) consists of a b antiquark and an s(d) quark. There are again penguin and box contributions, in closeanalogy to the KL → µ+µ− decay. To actually predict the branchingfraction one also needs input on the decay constants of the Bs,d

mesons (which take into account the fact that the quarks are boundinto a meson). All the suppression factors (including yet another“chiral” suppression by m2

µ/m2b) lead one to predict tiny branching

fractions 2 2 Christoph Bobeth, Martin Gorbahn,Thomas Hermann, Mikolaj Misiak,Emmanuel Stamou, and MatthiasSteinhauser. Bs,d → l+ l− in theStandard Model with Reduced The-oretical Uncertainty. Phys. Rev. Lett.,112:101801, 2014

Br(Bs → µ+µ−) =Γ(Bs → µ+µ−)

Γtotal= 3.65(23)× 10−9,

Br(Bd → µ+µ−) =Γ(Bd → µ+µ−)

Γtotal= 1.06(9)× 10−10. (7.36)

These processes are therefore exremely sensitive to physics beyondthe Standard Model. Current experimental results 3 3 M. Tanabashi et al. Review of Particle

Physics. Phys. Rev., D98(3):030001, 2018

Br(Bs → µ+µ−) = 2.7+0.6−0.5 × 10−9,

Br(Bd → µ+µ−) = 1.6+1.6−1.4 × 10−10, (7.37)


do not yet point towards a significant tension with (7.36), but ac-tually favor branching fractions even smaller than in the StandardModel.

7.4 Determination of the CKM elements Vij

The main challenge in determining the CKM matrix elements fromexperiment is that quarks always appear bound in hadrons, andthese bound-state effects need to be taken into account somehow.To do so, we cannot rely on perturbation theory in αs because inthe relevant low-energy domain the strong coupling becomes toolarge to permit a perturbative expansion. For that reason, non-perturbative input is required already for the simplest processesfrom which one could try to extract the Vij.

As an example, let us consider π− → µ−νµ. The π− is composedof a d quark and a u antiquark, so at the quark level the processoccurs via the exchange of a W−

π−d

u

ν

µW−

Figure 6: Leptonic π− decay.

with vertex proportional to Vud. We can write for the amplitude

iM = 〈µ−(q)νµ(q′)|π−(p)〉

=

(− ig√

2

)2

Vud14

u(q)γµ(1− γ5)v(q′)−igµν

p2 −M2W

× i〈0|u(0)γν(1− γ5)d(0)|π−(p)〉, (7.38)

where the matrix element

〈0|u(0)γµ(1− γ5)d(0)|π−(p)〉 (7.39)

cannot be obtained from a perturbative calculation, but needs tobe evaluated in full QCD. Here, the notation d(0) etc. refers toquark fields evaluated at x = 0, while for the external states theirmomenta are indicated in brackets.

As a first step we note that parity is conserved by the stronginteractions. Recalling that a Dirac spinor ψ(t,~x) transforms as

Pψ(t,~x)P−1 = γ0ψ(t,−~x) (7.40)

under the parity transformation P, one finds that

PψγµψP−1 = η(µ)ψγµψ, (7.41)

where η(0) = 1, η(i) = −1, i.e., the vector current in (7.39) indeedtransforms as expected for a vector. In contrast, the axial-vector


current produces an overall sign under this transformation

Pψγµγ5ψP−1 = −η(µ)ψγµγ5ψ, (7.42)

it is thus said to be “odd” under parity. With parity a conservedquantity of QCD, but the pion a pseudoscalar (also odd underparity), this implies that the vector current cannot contribute. Forthe axial-vector current

jµA(x) = u(x)γµγ5d(x) (7.43)

it suffices to consider the matrix element at x = 0 because undertranslations P

〈0|jµA(x)|π−(p)〉 = 〈0|eiPx jµA(0)e

−iPx|π−(p)〉= e−ip·x〈0|jµA(0)|π−(p)〉, (7.44)

and the exponential is the standard plane wave assumed in thetransition to momentum space. The remaining matrix element hasto be proportional to pµ because this is the only Lorentz vectoravailable, but the phase and normalization is a matter of conven-tion. One possible choice is

〈0|jµA(x)|π−(p)〉 = i

√2Fπ pµ. (7.45)

The quantity Fπ ≈ 92.2 MeV is called the “pion decay constant”since it parameterizes the transition between the pion and the vac-uum. In terms of Fπ the amplitude then becomes

M = GFVudFπ u(q)(/q + /q′)(1− γ5)v(q′)

= GFVudFπmµu(q)(1− γ5)v(q′), (7.46)

and accordingly for the spin-averaged squared amplitude

∑spins|M|2 = G2

F|Vud|2F2πm2

µTr((/q + mµ)(1− γ5)/q′(1 + γ5))

= 8G2F|Vud|2F2

πm2µq · q′

= 4G2F|Vud|2F2

πm2µ(M2

π −m2µ), (7.47)

due to 2q · q′ = p2 − q2 − q′2 = M2π −m2

µ.In combination with the phase space integral

12Mπ

∫ d3q(2π)32E

∫ d3q′

(2π)32E′(2π)4δ(p− q− q′)

=1

2Mπ(2π)2 4π∫ ∞

0

dq q2

2E2qδ(Mπ − E− q)

∣∣E=√

M2π+q2

=1

8πMπ

qE

11 + q

E

∣∣∣∣q=

M2π−m2

µ2Mπ

=M2

π −m2µ

16πM3π

, (7.48)

the decay rate becomes

Γ(π− → µ−νµ) =G2

F|Vud|2F2π

4πm2

µ Mπ

(1−

m2µ

M2π

)2

. (7.49)


The life time of the pion would thus determine |Vud| if the decayconstant could be determined from elsewhere, and similarly onecould extract other CKM elements from leptonic decays (` = e, µ, τ

where kinematically allowed)

π− → `−ν` ↔ |Vud|, B− → `−ν` ↔ |Vub|,K− → `−ν` ↔ |Vus|, B−c → `−ν` ↔ |Vcb|,D− → `−ν` ↔ |Vcd|,D−s → `−ν` ↔ |Vcs|. (7.50)

In practice, this is not necessarily the most precise way to extractthe |Vij| because of the need for independent input on the de-cay constants, although those can nowadays be calculated by nu-merically solving QCD on a grid of space-time (“lattice QCD”).However, other more complicated observables may have smallerhadronic uncertainties, e.g., |Vub| can be extracted from inclusive Bdecays

Γ(B− → `−ν` + Xu) = Γ(b→ `−ν` + Xu) +O(

Λ2

m2b

), (7.51)

and similarly |Vcb| from

Γ(B− → `−ν` + Xc) = Γ(b→ `−ν` + Xc) +O(

Λ2

m2b

), (7.52)

where Xu(c) denote all hadronic or partonic states involving u orc quarks. The scale Λ that occurs in the correction terms denotesa typical QCD scale around 1 GeV. Further, in addition to latticeQCD one can also use light-quark symmetries (isospin u↔ d, flavorSU(3) u ↔ d ↔ s) or heavy-quark symmetries (b ↔ c) to constrainhadronic matrix elements.

Once the |Vij| are determined from different processes, an impor-tant check concerns unitarity of the CKM matrix, which demandsthat

|Vud|2 + |Vus|2 + |Vub|2 = 1,

|Vcd|2 + |Vcs|2 + |Vcb|2 = 1. (7.53)

By far the most stringent test is provided by the first row, below thelevel of 10−3. In fact, there are currently hints for a deficit around2–3 σ, depending on the hadronic input used in the various CKMdeterminations.

7.5 CP Violation

In this section we study CP violation in more detail. To this end, letus consider the down-type Yukawa Lagrangian

LYukawa,d = −[Yd]ijqLi Φ dRj − [Y∗d ]ijdR j Φ† qLi, (7.54)


where we have now written the Hermitian conjugate explicitly.Let us further summarize the transformation properties of Diracspinors ψ and scalars φ under C and P transformations

ψ(x) C→ i(ψ(x)γ0γ2)T , ψ(x) P→ γ0ψ(t,−~x),

φ(x) C→ φ∗(x), φ(x) P→ φ(t,−~x), (7.55)

where for a scalar charged under SU(2)L also the SU(2) indicesneed to be considered (e.g., εijΦ∗j for the Higgs doublet undercharge conjugation). In particular, we find for the Dirac bilinears1 1 To derive this, first write down

the transformation rules for ψ andremember that the fermion fieldsanticommute.

ψ1ψ2CP→ ψ2ψ1, ψ1γ5ψ2

CP→ −ψ2γ5ψ1,

ψ1Lψ2RCP→ ψ2Rψ1L. (7.56)

Accordingly,

LYukawa,dCP→ −[Yd]ijdR j Φ† qLi − [Y∗d ]ijqLi Φ dRj, (7.57)

and invariance under CP is only guaranteed if Yd = Y∗d . We alreadyknow that for the two-flavor case we can always choose Yd real andtherefore there cannot be a physical CP-violating effect.

A general measure of CP violation is the so-called Jarlskog in-variant J, which can be defined by

Im[VijVklV∗il V∗kj

]= J ∑

mnεikmεjln. (7.58)

Note that there is no sum over i, j, k, l. The advantage of this mea-sure of CP violation is that it is invariant under phase redefinitions,i.e., invariant under different choices how to parameterize the CKMmatrix. For our choice (7.21) one finds

J = c12c23c213s12s23s13 sin δ ∼ 3× 10−5, (7.59)

which indeed displays the suppression in all three CKM angles.Furthermore, it turns out that the complex phases can be removedif there are mass degeneracies, more precisely, if either two up-typeor two down-type quarks have the same mass. This means that inaddition to J > 0 we need that

Fu = (m2t −m2

u)(m2u −m2

c )(m2c −m2

t ) 6= 0,

Fd = (m2d −m2

s )(m2s −m2

b)(m2b −m2

d) 6= 0. (7.60)

VudV ∗ub VtdV ∗

tb

VcdV∗cb

α

βγ

Figure 7: The unitarity triangle.

The requirement of unitarity puts various constraints on theelements of the CKM matrix. For example we have

Vud V∗ub + Vcd V∗cb + Vtd V∗tb = 0. (7.61)


This can be represented as a triangle in the complex plane knownas the “unitarity triangle,” see Fig. 7. The angles of the triangle arerelated to ratios of elements of the CKM matrix

α = − arg

Vtd V∗tbVud V∗ub

, γ = − arg

Vud V∗ubVcd V∗cb

,

β = − arg

Vtd V∗tbVcd V∗cb

. (7.62)

A popular representation of the CKM matrix is the Wolfensteinparameterization, which uses the parameters A, assumed to be oforder unity, a complex number (ρ + i η), and a small number λ, thelatter is approximately equal to sin θC. In terms of these parametersthe CKM matrix is written as

VCKM =

1− λ2/2 λ A λ3 (ρ− i η)

−λ 1− λ2/2 A λ2

Aλ3 (1− ρ− i η) −A λ2 1

+O(λ4).

(7.63)We stress that this parameterization is only approximate, but itdisplays nicely the structure of the CKM matrix. E.g., transitionsbetween the first and third generations are suppressed by λ3, whichreflects the phenomenological finding that θ13 is very small. TheO(λ4) corrections are necessary to ensure exact unitarity. Finally,we note that if the sides of the unitarity triangle in Fig. 7 are nor-malized by |VcdV∗cb| (so that the lower side extends from (0, 0) to(1, 0) in the complex plane), the apex is given by (ρ, η) and the areaof the triangle equals J/2.

We already discussed unitarity tests involving only the modulusof the CKM elements, see (7.53). In addition, stringent tests of theflavor structure of the Standard Model can be obtained from theunitarity triangles such as (7.61), via measurements of their angles.To this end, we need to consider CP-violating processes, which willbe the subject of the remainder of this chapter.

7.6 Meson Mixing and CP Violation

Historically, kaon mixing was the process in which the violationof CP symmetry was first demonstrated experimentally. To seehow this works we first review some basic properties of the neutralkaons.1 The flavor eigenstates are K0 (composed of a d quark and 1 Andrzej J. Buras. Weak Hamiltonian,

CP violation and rare decays. hep-ph/9806471, 1998

an s anti-quark) and K0 (s quark and d anti-quark), but they canmix because of box diagrams with double W exchange:

s

d

W−

W+

d

s

u, c, tu, c, t

Figure 8: K0–K0 mixing.


As we will see, this implies that the flavor eigenstates are notidentical to the mass eigenstates. First, we note that under CPtransformation

CP|K0〉 = −|K0〉, CP|K0〉 = −|K0〉, (7.64)

because K0, K0 are each other’s anti-particles and both are pseu-doscalar mesons (and thus odd under parity). In the absence ofmixing effects, the time evolution of a state |K0(t)〉 would be givenby

|K0(t)〉 = |K0(0)〉e−iHt, H = M− iΓ2

, (7.65)

with mass M and width Γ. Including the mixing, we need to con-sider the system

idψ(t)

dt= Hψ(t), ψ(t) =

(|K0(t)〉|K0(t)〉

), (7.66)

where H = M − iΓ/2 is composed of two Hermitian matricesM and Γ.1 Using their Hermiticity as well as M11 = M22 ≡ M, 1 This Hamiltonian is not Hermitian

because we did not include the statesinto which the kaons decay.

Γ11 = Γ22 ≡ Γ (as follows from CPT invariance), the Hamiltonianbecomes

H =

(M− i Γ

2 M12 − i Γ122

M∗12 − i Γ∗122 M− i Γ

2

). (7.67)

The diagonalization gives the eigenstates2 2 The labels derive from L=long andS=short, in reference to the life time ofthe KL and KS mesons.

|KL,S〉 =1√2

[(1 + ε)|K0〉 ± (1− ε)|K0〉

], (7.68)

where we expanded in ε defined by

1− ε

1 + ε=

√√√√M∗12 − i Γ∗122

M12 − i Γ122

. (7.69)

The eigenvalues are

ML,S = M± Re Q, ΓL,S = Γ∓ 2Im Q, (7.70)

with

Q =

√(M12 − i

Γ12

2

)(M∗12 − i

Γ∗122

). (7.71)

The mass eigenstates KS and KL are almost identical to the CPeigenstates

|K1〉 =1√2

(|K0〉 − |K0〉

), |K2〉 =

1√2

(|K0〉+ |K0〉

), (7.72)

which fulfill CP|K1〉 = |K1〉 and CP|K2〉 = −|K2〉, but due tothe mixing there is a small contamination from the opposite CPeigenstate

|KS〉 = |K1〉+ ε|K2〉, |KL〉 = |K2〉+ ε|K1〉. (7.73)


For kaon mixing ε will turn out around O(10−3), so that we canneglect the imaginary parts of M12 and Γ12 to obtain

∆MK = ML −MS = 2Re M12, ∆ΓK = ΓL − ΓS = 2Re Γ12. (7.74)

Experimentally, these numbers are

∆MK = 3.483(7)× 10−15 GeV, ∆ΓK = −7.340(3)× 10−15 GeV.(7.75)

After reviewing this mixing formalism, we now return to theexperimental signatures of CP violation. This involves the decaychannels into 2π (CP even) and 3π (CP odd), and accordingly theKL decays preferably into 3π (via K2) and the KS into 2π (via K1).This explains the huge difference in life times between KL and KS.Now, due to the small admixture of the opposite CP states, thedecays KL → 2π and KS → 3π are still possible, but they aresuppressed by the CP-violating parameter ε. This violation of CPsymmetry is called “indirect,” because it does not actually proceedvia explicit breaking of CP symmetry in the decay, but solely viathe mixing effect. “Direct” CP violation has also been observedin kaon decays, but it is suppressed by another factor O(10−3). Ameasure for indirect CP violation is the ratio of decay amplitudes

ε =A(KL → (ππ)I=0)

A(KS → (ππ)I=0), (7.76)

where I = 0 refers to strong isospin. In general, one can write thesematrix elements in terms of isospin amplitudes AI for K → ππ

decays, I = 0, 2, from which the “strong” QCD phase δI has alreadybeen removed. This phase only has to do with QCD dynamics,but is not related at all to CP violation, which is all included inthe “weak” phases that remain in the AI . We will return later tothe role of strong and weak phases in constructing observablessensitive to CP violation, but for now we only quote the resultingexpression1 1 In practice the difference between ε

and ε amounts to a phase convention,i.e., one can simply choose Im A0 = 0.ε = ε + iξ, ξ =

Im A0

Re A0. (7.77)

A measure of direct CP violation, i.e., a direct transition to a state ofopposite CP, is given by the parameter

ε′ =1√2

Im(

A2

A0

)eiΦ, Φ =

π

2+ δ2 − δ0, (7.78)

which, in contrast to ε, is now indeed sensitive to the weak phasesin the AI . The experimental values are

|ε| = 2.228(11)× 10−3, Reε′

ε= 1.66(23)× 10−3. (7.79)

In addition, the phases of both ε and ε′ happen to be close to 45°,so that to good approximation ε′/ε is real.

This discussion largely translates to other neutral mesons as well,e.g., for neutral B mesons we have the mixing of B0

d–B0d and B0

s –

B0s , depending on whether the light quark besides the b is d or s.


b

d, s

W−

W+

d, s

b

u, c, tu, c, t

Figure 9: B0d,s–B0

d,s mixing.

The mixing diagram is the same as before, see Fig. 9, and also themixing formalism is essentially identical.

One can again define mass eigenstates

|BL〉 = p|B0〉+ q|B0〉, |BH〉 = p|B0〉 − q|B0〉, (7.80)

where |p|2 + |q|2 = 1, but 〈BL|BH〉 = |p|2 − |q|2 6= 0, and in thenotation analogous to (7.67)

(pq

)2

=M12 − iΓ12/2M∗12 − iΓ∗12/2

. (7.81)

Defining φ = arg(−M12/Γ12), we have

(pq

)2

=M12

M∗12

1 + i∣∣∣ Γ12

2M12

∣∣∣ eiφ

1 + i∣∣∣ Γ12

2M12

∣∣∣ e−iφ, (7.82)

so CP violation in the mixing occurs if φ 6= 0. For B mesons oneactually has Γ12 M12, so that one can neglect CP violation in themixing. Moreover, when calculating M12 from the loop diagram inFig. 9, the total contribution may be written as

M12 = ∑i=u,c,t

∑j=u,c,t

Vib V∗id V∗jb Vjd aij. (7.83)

Once again, if all the masses of the quarks were equal, then theamplitudes aij would all be equal, and the sum would vanish by theunitarity constraints imposed on the elements Vij. Since the quarksdo not all have the same mass, there is some residual contribution.Indeed, the above diagram is dominated by the term in which a tquark is exchanged on both sides, since the t quark is much moremassive than the rest. Restricting ourselves to the t-quark-exchangecontribution, we can thus read off the phase without calculating thediagram itself. It is given by the phase of the products of the CKMmatrix elements entering in the diagram, (V∗td Vtb)

2, which leads to

pq=

V∗td Vtb

Vtd V∗tb(7.84)

in the B system, a relation we will need below. For completeness,we summarize the relevant scaling in the K, D, and B systems

∆MK ∼ ∆ΓK ∼ ΓK,

∆MD ∼ ∆ΓD ΓD,

∆MB ∼ ΓB ∆ΓB. (7.85)


7.7 CP-violating Processes

To observe CP violation one needs to find a process for which

CP violation: P(A→ B) 6= P(A→ B), (7.86)

where A and B are the CP conjugate states of A and B. Comparingthis to

T violation: P(A→ B) 6= P(B→ A),

CPT violation: P(A→ B) 6= P(B→ A), (7.87)

we indeed see that by the CPT theorem the violation of CP and Tare equivalent. In practice, one considers CP asymmetries

ACP =P(A→ B)− P(A→ B)P(A→ B) + P(A→ B)

, (7.88)

where the probabilities are proportional to the squared matrixelements |M|2. To obtain a direct signal, one needs the interferenceof two contributions, e.g., writing

M = A1 +A2, Ai = |Ai|eiδweaki eiδstrong

i , (7.89)

where only δweaki changes sign under CP violation, one finds

ACP = − 2|A1A2| sin ∆δweak sin ∆δstrong

|A1|2 + |A2|2 + 2|A1A2| cos ∆δweak cos ∆δstrong , (7.90)

with ∆δweak = δweak1 − δweak

2 and ∆δstrong = δstrong1 − δ

strong2 .

Accordingly, ACP is only non-zero if both relative phases are. Anexample for these strong phases is provided by the δI encounteredin the K → ππ matrix elements above.

The second option to observe CP violation is through mixing,as in the case of KL → 2π decays, and, third, one can also havea combination of the two. That is, let us consider a CP eigenstatef = f as well as a B-meson decay B → f with amplitude A f . Thenthe same final state can be reached by first mixing B into B withsubsequent decay via A f . In this case,

ACP ∝ Im[

qpA f

A f

]. (7.91)

We will now consider this last option in more detail, which canbe realized in so-called tagged B-meson decays. The idea is asfollows: suppose we produce a bb pair in e+e− collisions, whichwill hadronize into a B and a B. Once we observed (“tagged”) oneof the B mesons at a time t0, we know whether it is the B or theanti-B, and thus know the identity also of the other B meson atthis time. However, the second B meson may decay at a later timet0 + ∆t. In case both mesons are detected in the B or the B state, weknow that an oscillation has to have happened in the time ∆t. Inpractice, it is not the time, but the distance traveled before decayingthat is measured in experiment.


Now suppose that at time t = 0 we prepare a state that is purelyB0. Accounting for the fact that the B0 meson has a decay rate Γ,the time evolution becomes1 1 This again neglects ∆Γ ∆M.

|B(t)〉 = e−iMt e−Γt/2[

cos(

∆M2

t)|B0〉+ i

qp

sin(

∆M2

t)|B0〉

].

(7.92)For the direct decay into f this implies

〈 f |H|B(t)〉 = e−iMt e−Γt/2[

cos(

∆M2

t)A f + i

qp

sin(

∆M2

t)A f

],

(7.93)where H indicates the weak Hamiltonian. Similarly, if we had pre-pared a B0 at t = 0, the amplitude to find the state | f 〉 would be

〈 f |H|B(t)〉 = e−iMt e−Γt/2[

cos(

∆M2

t)A f − i

pq

sin(

∆M2

t)A f

].

(7.94)For the CP asymmetry this implies

ACP =Γ(B(t)→ f )− Γ(B(t)→ f )Γ(B(t)→ f ) + Γ(B(t)→ f )

= − sin (∆M t) Im

(qpA f

A f

).

(7.95)For example, if the state | f 〉 is the CP-even two-pion state |π0π0〉,the Feynman diagram at the quark level for A2π is

W−b

d

u

u

dd

Figure 10: Quark-level process induc-ing the decay B→ ππ.

To fully calculate the decay amplitudes we would need to knowthe wave functions for the mesons in terms of the constituentquark-antiquark pairs, but for the ratio A2π/A2π we just needthe ratios of the CKM matrix elements occurring in this diagram,

A2π

A2π=

Vub V∗udV∗ub Vud

, (7.96)

so that

Im(

qpA2π

A2π

)= Im

(Vtd V∗tbV∗td Vtb

Vub V∗udV∗ub Vud

)= − sin(2α). (7.97)

Unfortunately, the same decay also receives contributions frompenguin diagrams, which impedes the extraction of α.1 A cleaner 1 Merge the u and u lines in Fig. 10,

attach a gluon or photon, and anotheruu pair for the final state.

probe (with less “penguin pollution”) can be obtained from | f 〉 =|J/ψ KS〉 (“golden channel”). In this case the quark-level Feynmandiagram is shown in Fig. 11.

The advantage of this process is that the part of the penguindiagrams that enters with a different phase is quite small, which


W−b

d

c

c

sd

Figure 11: Quark-level process induc-ing B0 → J/ψKS.

makes this process the cleanest and most precise probe of the uni-tarity triangle. However, we encounter an additional complicationin that the KS is not actually a CP eigenstate. Also, the phases inthe CKM matrix introduced in the ratio of pK/qK (the analog ofp/q), calculated from diagrams similar to the ones for the B0 sys-tem (but with the b quark replaced by an s quark) differ from p/q.In this case it is the diagram with an internal c-quark exchange thatdominates (although the mass of the c is much smaller than the t-quark mass, the CKM matrix elements are much larger for c-quarkexchange than for t-quark exchange and this effect dominates here),so we have a factor

pKqK

=V∗cd Vcs

Vcd V∗cs(7.98)

that enters in the ratio of the decay amplitudes, giving

AJ/ψ KS

AJ/ψ KS

= −Vcb V∗csV∗cb Vcs

V∗cd Vcs

Vcd V∗cs= −Vcb V∗cd

V∗cb Vcd, (7.99)

where the minus occurs because the J/ψ KS state is (approximately)CP odd. In the end, this leads to

Im

(qpAJ/ψ KS

AJ/ψ KS

)= −Im

(Vtd V∗tbV∗td Vtb

Vcb V∗cdV∗cb Vcd

)= sin(2β). (7.100)

This resulting constraint is indeed one of the most precise in cur-rent tests of the unitarity triangle, see Fig. 12.

7.8 Summary

• Additional generations may be added, with gauge interactionscopied from the first, but in this case one can have mass-mixingbetween quarks of different generations. In terms of the masseigenstates, the charged W bosons mediate transitions betweena T3 = + 1

2 quark (u, c, or t) and a superposition of T3 = − 12

quarks (d, s, and b). In two generations, this mechanism allowsweak interactions that violate strangeness conservation, and themixing matrix has only one independent parameter, the Cabibboangle. Including a third generation, the mixing matrix for theT3 = −1/2 quarks (d, s, and b) is the CKM matrix. This matrixhas four independent parameters (three angles plus one phase),so that some of the matrix elements may be complex.


Figure 12: Unitarity triangle accordingto CKMFitter 2019.

• The unitarity of the mixing matrix guarantees that there are noflavor-changing neutral processes at tree level in the StandardModel. Weak interactions involving the exchange of a Z bosondo not change flavor.

• The complex phase that appears in the CKM matrix gives riseto CP violation, apart from (potentially) the QCD θ term theonly source thereof in the Standard Model. However, the sizeof CP violation is suppressed by the small mixing angles, aparameterization-independent measure is the Jarlskog invari-ant J ∼ 3× 10−5.

• CP violation in the CKM matrix contributes to the mixing ofneutral K, D, and B mesons, with eigenstates of the mass matri-ces that are not CP eigenstates. This is called indirect CP viola-tion.

• The CKM matrix also introduces phases in the ratios of the decayamplitudes for B0 and B0 to a given CP eigenstate. Products ofthe phase of the mass mixing and the ratio of the decay ampli-tudes can be observed directly in tagged B-meson experiments,and the angles α and β of the unitarity triangle can be directlymeasured.

8Neutrinos

In its original formulation, the Standard Model had masslessneutrinos—neutrino masses were not measured at the time. Af-ter the observation of neutrino masses, see Sect. 8.2, we now knowthat neutrinos have very small masses and the Lagrangian has tobe modified to account for these. There are different possibilitiesfor neutrino mass terms and we do not yet know which option isrealized in nature. The mass options are

1. Dirac masses,

2. Majorana masses,

3. General case, neutrino see-saw.

We will first discuss the possible scenarios and their differences.Afterwards, we will consider in more detail neutrino oscillationsas well as other experiments dedicated to studying the remainingopen questions in the neutrino sector.

8.1 Neutrino masses

Dirac neutrinos

The simplest option to introduce neutrino masses proceeds viaright-handed neutrino fields νR, in complete analogy to the massterm for the quarks and charged leptons. For a single generation,the Yukawa term takes the form

LYukawa,ν = −Yν`LΦcνR + h.c. (8.1)

The left-handed leptons have Y = −1/2 as does the conjugateHiggs field Y = −1/2. As a consequence, the hypercharge of νR

must be zero. The right-handed neutrino therefore does not carryany charge at all, while all other fields in the SM are charged undersome gauge group. This special property of νR makes it possibleto also write down other mass terms, as will be discussed below.After spontaneous symmetry breaking, the conjugate Higgs takesthe form

Φc =1√2

(v + H

0

)(8.2)


in unitary gauge and inserting this into (8.1) we end up with thestandard mass term

LYukawa,ν = −Yν v√2(νLνR + νRνL) = −mννν (8.3)

for the neutrino. For simplicity, we have written down the massterm for a single generation but since the construction is completelyanalogous to the quark sector, it is clear what happens in the caseof three generations. As for the quark case, it will no longer bepossible to diagonalize both Yukawa matrices. After diagonalizingthe charged lepton Yukawa matrix, we will thus be left with a non-diagonal matrix for the neutrino Yukawa couplings. This matrix,called the Pontecorvo–Maki–Nagakawa–Sakata (PMNS) matrix, isthe exact analog of the CKM matrix. It is written in the form

UPMNS =

Ue1 Ue2 Ue3

Uµ1 Uµ2 Uµ3

Uτ1 Uτ2 Uτ3

(8.4)

and as in the quark case, it shows the composition of the masseigenstates ν1, ν2, and ν3 in terms of the interaction eigenstates νe,νµ, and ντ . For example, the neutrino νµ is the state produced whena W+ boson decays into a µ+ and a neutrino. This neutrino is asuperposition of the three mass eigenstates,

νµ = Uµ1ν1 + Uµ2ν2 + Uµ3ν3 . (8.5)

The unitary matrix UPMNS can be viewed as the basis change fromthe interaction to the mass eigenstate basis, as was the case for theCKM matrix in the quark sector. By the same arguments this matrixhas four parameters, three rotations and one complex, CP-violatingphase δ. Again, a popular parameterization reads

UPMNS =

1 0 00 c23 s23

0 −s23 c23

c13 0 s13e−iδ

0 1 0−s13eiδ 0 c13

c12 s12 0−s12 c12 0

0 0 1

=

c12c13 s12c13 s13e−iδ



,

(8.6)

in terms of angles θ12, θ13, θ23, and cij = cos θij, sij = sin θij. Akey difference to the CKM matrix is that the PMNS matrix is nolonger close to diagonal. We will return below to the question howto measure its entries.

Majorana neutrinos

To write down Dirac masses, we needed to introduce right-handedneutrinos into the SM. Surprisingly, it is also possible to write downmass terms without introducing additional degrees of freedom,using just the left-handed fields present in the massless case.1 To 1 Ettore Majorana. A symmetric theory

of electrons and positrons. Nuovo Cim.,14:171–184, 1937


do so we will need the charge conjugation matrix, which has theproperty

CγTµ C−1 = −γµ. (8.7)

In the Dirac and the chiral representation,1 the matrix C = iγ2γ01 “Chiral” because γ5 =

(−1 00 1

)is

diagonal. The only other difference tothe Dirac representation concerns γ0 =(

0 1

1 0

)instead of γ0 =

(1 00 −1

).

fulfills this equation. Note that it has the properties

C−1 = CT = C† = −C, Cγ0 = −γ0C. (8.8)

In the chiral representation of the Dirac algebra, the explicit form ofthe matrix is

C =

(iσ2 00 iσ2

)with iσ2 = ε =

(0 1−1 0

). (8.9)

The anti-particle of a left-handed field is right-handed and thecharge-conjugated field ψc

L(x) = C ψTL (x) transforms like ψR(x)

under Lorentz transformations. We can therefore hope to constructa mass term of the form

ψcL(x)ψL(x) + h.c. = ψT

L CψL − ψ†LCψ∗L. (8.10)

To obtain the right-hand side, we have rewritten

ψcL = (ψc

L)† γ0 = ψ∗L C†γ0 = ψT

L γ0 (−C) γ0 = ψTL C. (8.11)

Let us consider a free field theory with this mass term

L = ψLi/∂ψL −m2

(ψT

L CψL − ψ†LCψ∗L

). (8.12)

To check that this theory indeed describes a fermion of mass m, onecan derive the classical equations of motion, which are obtained byvarying ψL and ψ∗L (or, alternatively and equivalently, the real andimaginary part of the fields) and imposing that the variation of theaction vanish. From the second variation, we get

∂L∂ψ∗L

= γ0i/∂ψL + mCψ∗L = 0. (8.13)

Note that the mass term in L is quadratic in ψ∗L, the factor 2 whentaking the derivative cancels the 1/2 in the prefactor. To computethe variation with respect to ψL, it is easiest to first integrate by partso that the derivative acts on ψ∗L. The variation of ψL then yields

∂L∂ψL

= ψL(−i←−/∂ )−mψT

L C = 0. (8.14)

Transposing and using (8.7) to rewrite the transposed Dirac matri-ces in terms of the usual ones, this equation becomes

i/∂γ0Cψ∗L + mψL = 0. (8.15)

To verify that equations (8.13) and (8.15) imply that the componentsof the field fulfill the relativistic wave equation, we compute

0 = −i/∂γ0 × (8.13) = ∂2ψL −mi/∂γ0Cψ∗L(8.15)=

(∂2 + m2

)ψL, (8.16)


exactly as the Dirac Lagrangian (8.12) describes relativistic fermionsof mass m. There are, however, important differences. The Diracequation involves four spinor components describing a particle andan anti-particle, each with two spin states. In contrast, the field ψL

effectively only involves two components: the Majorana fermion isits own anti-particle. The Dirac Lagrangian is invariant under phaseredefinitions ψ → eiφ ψ, while the Majorana Lagrangian is not.This phase symmetry forms the basis of gauge invariance. Since itis absent, the Majorana neutrinos cannot carry gauge charges. ForDirac fermions, the global phase symmetry leads to a conservedquantity, fermion number, which is violated for Majorana fermions.

Since the left-handed lepton field `L carries hypercharge Y =

−1/2, a Majorana mass term for this field would violate gaugeinvariance. We can, however consider the combination

ψL ≡ εijΦi`Lj = (Φc)∗ `L, (8.17)

which is invariant under both SU(2)L and U(1)Y gauge transforma-tions and write a Majorana mass term for this field

LMajorana = − λ

MψT

L C ψL + h.c. = − λ

M(`T

L Φ†c )C (Φ∗c `L) + h.c.

(8.18)This operator has dimension d = 5 and is called the Weinbergterm. We have written the operator coefficient as a dimensionlesscoupling λ, divided by a mass scale M. Adding a dimension-fiveoperator to the Standard Model destroys its renormalizability, sothat the resulting theory should be viewed as a low-energy effectivetheory, just like the Fermi theory of the weak interactions. Also forthe Fermi theory, the coupling was proportional to an inverse massscale GF/

√2 = g2/(8M2

W) and the corresponding effective theorybreaks down when the energies become as large as MW (becausereal Ws start to get produced, which are not included in the low-energy effective theory). The same should be true for the StandardModel with a Weinberg term. Around the scale M, new particlesshould be present and the theory with (8.18) should be replacedby a renormalizable theory that includes the additional degrees offreedom. The simplest such theory will be discussed below.

Inserting the vacuum expectation value of the Higgs field (8.18)becomes

LMajorana = −λv2

2MνT

L CνL + . . . (8.19)

and we are left with a Majorana mass mν = λv2

M . Assuming λ ∼ 1and comparing to the known small values of mν . 1eV, we con-clude that the scale M is quite high, M & 1013 GeV.1 The Majo- 1 The other option would be a very

small value of λ and a lower scale ofM.

rana mass term explains why the neutrinos are much lighter thanthe other fermions: in this framework the reason would be thatthe masses are suppressed by the scale M compared to the otherfermions.

In the case of three generations, the prefactor of the Weinbergterm (8.18) becomes a matrix mij = λij v2/M, where i and j are gen-eration indices. This matrix can be chosen symmetric, because the


operator is symmetric under i ↔ j, but is in general complex. Tounderstand how many parameters are involved consider the matrixH = m m†. This is a symmetric, positive matrix that can be diago-nalized by a unitary matrix A. The same matrix also diagonalizesthe matrix m,

AmAT = diag(eiφ1 m1, eiφ2 m2, eiφ3 m2), (8.20)

but with complex eigenvalues. Note that we can always redefinethe phases of the fields to remove the phase factors in this matrix.The physical neutrino masses mi are the absolute values of theeigenvalues. Our discussion is quite similar to the one we had forthe derivation of the CKM matrix (or the PMNS matrix in the Diraccase), except for the phases in the mass matrix. If we allowed forarbitrary phases in the mass matrix, the number of parameters ofthe PMNS matrix for the Majorana case could be reduced to four,but the phases in the mass matrix would be additional parameters.The overall phase of the matrix does not lead to observable conse-quences, but the phase differences do. We can instead remove thephases in the mass matrix and use a PMNS matrix of the conven-tional form1 1 Frank T. Avignone, III, Steven R.

Elliott, and Jonathan Engel. DoubleBeta Decay, Majorana Neutrinos,and Neutrino Mass. Rev. Mod. Phys.,80:481–516, 2008

UPMNS =

c12c13 s12c13 s13e−iδ



×

eiα1/2 0 00 eiα2/2 00 0 1

. (8.21)

The first line is the standard Dirac parameterization used also forthe CKM matrix and the matrix in the second line contains theMajorana phase-differences α1 and α2. These phases are only visiblein CP-violating processes that also violate lepton number and arecurrently completely unknown.

General case, neutrino see-saw

Even if one insists on renormalizability, the Dirac neutrino massesdiscussed above were not the most general form of the mass term.Since the right-handed neutrinos νR do not carry any charge, it ispossible to write down a Majorana mass term for these fields, inaddition to the Dirac mass term (8.1). The Majorana mass term forthree generations of right-handed neutrinos takes the form

LMajorana = −12

MijνTRi C νRj + h.c. (8.22)

To understand the effect of this term, let us consider a single gener-ation and collect the left- and right-handed fermions into a spinor

ψ(x) =(

νcL

νR

)(8.23)


The full mass term can then be written in the form

LMajorana = −12

ψc

(0 mm M

)ψ + h.c. = −1

2νc

R M νR −m νL νR + h.c.,

(8.24)where the Dirac mass m arises after spontaneous symmetry break-ing. The upper left entry of the matrix is zero, because a Majoranamass term for `L is forbidden by gauge invariance. The eigenvaluesof the mass matrix are

λ± =12

(M±

√M2 + 4m2

). (8.25)

If the Majorana mass of the right-handed field is large M m, wehave λ+ ≈ M and λ− = −m2

M and the associated eigenvectors are

iψa ≈ ψL +mM

ψR, ψb ≈ ψR −mM

ψL. (8.26)

We thus end up with two Majorana fermions. The field ψa describesa light fermion with mass ma ≈ m2

M and the field ψb a heavy onewith mb ≈ M. Note that we have introduced a phase factor eiπ/2 forthe field ψa to make the mass parameter positive. These phase fac-tors and their significance were discussed at the end of the previoussection. At low energies, only the light particles will be present andthe theory will match onto the effective theory defined by (8.19).The heavy scale M present in the Weinberg operator (8.18) is simplythe mass of the heavy Majorana neutrino. This idea to produce alight neutrino at the expense of introducing a mixing with a right-handed field at a high scale M is called the sea-saw mechanism.1 1 Peter Minkowski. µ → eγ at a Rate

of One Out of 109 Muon Decays? Phys.Lett., 67B:421–428, 1977

One can make this even more explicit by considering the dia-gram involving a left-handed fermion scattering off a Higgs bosonand turning into its anti-particle.

Φ

νL

νR

p2≪M2

−→

Φ

νL

Figure 1: The Weinberg operator is ob-tained after expanding the propagatorof νR at low energies, analogously tothe case of Fermi theory.

This process, νLΦ → νcLΦ∗, is mediated by an intermediate

right-handed neutrino. At low energies, we can approximate theintermediate Majorana propagator (the Feynman rules for Majoranafermions are a bit tricky2) as 2 Howard E. Haber and Gordon L.

Kane. The Search for Supersymmetry:Probing Physics Beyond the StandardModel. Phys. Rept., 117:75–263, 1985;and Ansgar Denner, H. Eck, O. Hahn,and J. Kublbeck. Feynman rules forfermion number violating interactions.Nucl. Phys., B387:467–481, 1992

−iC [/p −M]−1 ≈ iC M−1. (8.27)

The approximated result is the same as what is obtained from theWeinberg operator (8.18) with λ = (Y∗ν )2. The same construction forthe case of three generations, leads to the Weinberg term

LMajorana = −(Y∗ν M−1Y†ν )ij(`

TLiΦ

†c )C (Φ∗c `Lj) + h.c. (8.28)


It is therefore quite natural to expect a Weinberg term generatingMajorana masses at low energies. Of course, heavy right-handedneutrinos is just the simplest scenario for an underlying renormal-izable theory at high energies. Other, more complicated scenar-ios have been studied in the literature and one can also entertainthe possibility that the right-handed neutrinos are light but veryweakly coupled. They could then provide a dark matter candidate.Also, since we only heave measured neutrino mass differences, itis in principle possible that one of the neutrinos is massless. Thisscenario could be realized by introducing only two right-handedneutrinos instead of three.

To distinguish Majorana from Dirac neutrinos, one needs a pro-cess sensitive to mass effects. One such process is neutrinolessdouble β-decay, denoted by 0νββ. This process, where a heavynucleus decays emitting two electrons but no neutrinos, violateslepton number by two units. The corresponding Feynman diagramis shown on the left of Fig. (2). The violation of lepton numbers isillustrated by the fact that the fermion arrows would clash on theneutrino propagator. Observing this decay would establish thatneutrinos are indeed Majorana fermions.

W

W

νi

d

d

u

e−

e−

u

Figure 3: Feynman diagram of the transitiondd → uue−e− which induces ββ0ν decay.

d

d u

u

e−

e−

W+

W+

νe

νe

black box

ββ0ν

Figure 4: νe → νe transition diagramthrough a ββ0ν black box [94].

Let us consider the neutrino propagator. Taking into account the Majorana condition inEq. (4.3), we have

⟨0|T (νeL(x1)νTeL(x2)|0⟩ = − 1 − γ5

2

!

i

U2ei⟨0|T (νi(x1)νi(x2))|0⟩

1 − γ5

2C

= − i

(2π)4

!

i

"d4q e−iq·(x1−x2)

U2eimi

q2 − m2i

1 − γ5

2C. (4.6)

Thus, the neutrino propagator is proportional to mi. It is obvious from Eq. (4.6) that this isconnected with the fact that only left-handed neutrino fields enter into the Hamiltonian ofweak interactions. In the case of massless neutrinos (mi = 0, with i = 1, 2, 3), in accordancewith the theorem on the equivalence of the theories with massless Majorana and Diracneutrinos (see Refs. 95, 96), the matrix element of neutrinoless double-β decay is equal tozero.

Let us consider the second term with p1 ! p2 of the matrix element in Eq. (4.5). Wehave

uL(p1)γα(1 − γ5)γβCuLT (p2) = uL(p2)C

T γTβ (1 − γT

5 )γTα uT

L(p1)

− uL(p2)γβ(1 − γ5)γαCuTL(p1). (4.7)

andT (Jβ(x2)J

α(x1)) = T (Jα(x1)Jβ(x2)). (4.8)

From Eqs. (4.7) and (4.8), it follows that the second term of the matrix element in Eq. (4.5)

20

23

Neutrinoless Double Beta Decay

The sum of the electron energies gives a spike at the endpoint of the “neutrino-full” double beta decay.

Figure 2: Left: Feynman diagram for0νββ. Right: energy spectrum of 2νββand 0νββ.

However, also with Dirac neutrinos the double beta decay intotwo electrons and two anti-neutrinos, 2νββ, would be allowed.In this decay, part of the energy is taken away by the neutrinos.The electron energy spectrum is peaked at intermediate energiesand goes to zero near the maximum allowed energy. In contrast,electrons in the 0νββ decay carry the maximum energy availablein the decay. The experimental signature for this process is thus apeak near the end point. The decay rate, i.e., the height of the peak,is proportional to the neutrino mass, Γ0νββ ∝ m2

ββ, more precisely tothe linear combination

mββ =

∣∣∣∣ ∑k=1,2,3

mkU2ek

∣∣∣∣

=∣∣∣m1|Ue1|2 + m2|Ue2|2ei(α2−α1) + m3|Ue3|2e−i(α1+2δ)

∣∣∣2. (8.29)

Examples for isotopes currently used for 0νββ searches include136Xe and 76Ge. Besides distinguishing between the Dirac and Ma-jorana scenarios, the observation of 0νββ would also provide aconstraint on the Majorana phases, as they appear in the expressionfor mββ. However, it is not guaranteed that, if observed, the processwould be mediated by the exchange of a light Majorana neutrino,as the exchange of heavy new degrees of freedom would lead to


the same signature. In either case, observing 0νββ decay woulddemonstrate the existence of new Majorana particles.

Since 0νββ decay has not been observed yet, we cannot draw anyconclusions on the PMNS and mass parameters from mββ exceptfor limits, which currently reach down to mββ < (0.061–0.165) eVat 90% CL,1 where the range reflects the uncertainty in the nu- 1 A. Gando et al. Search for Majorana

Neutrinos near the Inverted MassHierarchy Region with KamLAND-Zen. Phys. Rev. Lett., 117(8):082503,2016. [Addendum: Phys.Rev.Lett. 117,109903 (2016)]

clear matrix elements. Most of the phenomenological constraintsoriginate from neutrino oscillations, but before turning to theseprocesses we briefly mention two other sources of information onneutrino masses:

1. Tritium β decay spectrum: the endpoint of the decay spectrum(i.e., the maximal energy the electron can carry away) is sensitiveto the neutrino mass

m2β = ∑

k=1,2,3m2

k |Uek|2. (8.30)

The current best limit from the KATRIN experiment gives mβ <

1.1 eV at 90% CL,2 with a final projected sensitivity of 0.2 eV. 2 M. Aker et al. Improved Upper Limiton the Neutrino Mass from a DirectKinematic Method by KATRIN. Phys.Rev. Lett., 123(22):221802, 2019

Note that contrary to (8.29) all the phases drop out in mβ.

2. Cosmology: at some point finite neutrino masses leave imprintsin several cosmological observables. The current constraintsgive3 3 N. Aghanim et al. Planck 2018

results. VI. Cosmological parameters. 7

2018∑

k=1,2,3mk < 0.12 eV at 95% CL. (8.31)

8.2 Neutrino Oscillations

The physical consequences of mixing angles are quite differentbetween the lepton and quark sectors. This is because neutrinosare very light and have only weak interactions. In the quark sectorone can differentiate D → Kµν from D → πµν, because the π andK have strong and electromagnetic interactions, which allows usto track them in the detector, and they have sufficiently differentmasses that the tracks are distinguishable. This is not the case intrying to distinguish µ→ eν3ν2 from µ→ eν3ν1.

The small masses and weak interactions of neutrinos imply thatthe wave packets corresponding to different neutrino mass eigen-states remain superposed over long distances. The effects of flavormixing can therefore be seen via oscillations.

For simplicity we will consider the case of two generations first,which in the charged lepton sector we will take to be the electronand muon.4 We label the neutrino mass eigenstates as ν1 and ν2. 4 Of course, in the Standard Model we

have three families, but the importantconcepts can be understood in thesimpler case.

They are related by a rotation(

νe

νµ

)=

(cos θ sin θ

− sin θ cos θ

)(ν1

ν2

), (8.32)

where θ is the analog of the Cabibbo angle in the case of two-familyquark mixing.


Now we would like to compute the amplitude for an oscillationprocess. Suppose that we have an initial beam of muons that decaysto relativistic neutrinos of energy E and momentum k. The neutri-nos travel a distance L = τ to a detector where they produce an e ora µ by charged-current scattering. The amplitude will be

Aµα ∼∑j

Uµj × e−i(Ejτ−kj L) ×U∗αj, (8.33)

where the three pieces arise from production, propagation, anddetection.

First, suppose we could neglect the neutrino masses, so (Ej, k j) =

(En, kn) for any j, n. The propagation exponential can then be fac-tored out, and (8.33) is the unitarity condition for UPMNS,

UµjU∗αj = δµα. (8.34)

For quarks, with three generations, this is the same relation thatgives rise to the unitarity triangle.

Now we allow the neutrinos to have small masses, m E, k, sothat L ' τ remains. Then the exponent can be written as

−i(Ejτ − k jL) ' −i(Ej − k j)L = −iE2

j − k2j

E + kL ' −i

m2j

2EL, (8.35)

such that

Pµα = |Aµα|2 = |∑j

Uµje−im2

j L/(2E)U∗αj|2. (8.36)

Using the explicit form of U given in (8.32) one obtains the muonsurvival probability

Pµµ = 1− sin2(2θ) sin2 (m22 −m2

1)L4E

. (8.37)

In reality, there are three generations of leptons in the StandardModel, and there are three mass eigenstates in the sum of (8.36).The three mixing angles introduced in (8.6) can be measured asfollows:

1. “Solar” angle θ12: solar neutrinos are produced as νe in thesun through the pp chain and the CNO cycle. By the time theyarrive on Earth, some of them have oscillated to other flavorstates, observable as a reduction in the expected flux of electronneutrinos. Solar neutrino experiments are sensitive to the massdifference |∆m2

21| = |m22 −m2

1| ∼ 7.5× 10−5 eV2.

2. “Atmospheric” angle θ23: atmospheric neutrinos are producedby decays of pions and kaons, which in turn come from theinteraction of cosmic rays with nucleons in the atmosphere.Apart from the angle θ23, one can extract |∆m2

32| = |m23 − m2

2| ∼2.4× 10−3 eV2.

3. The measurement of the remaining angle θ13 requires reactorneutrinos.


Note that, strictly speaking, each of these processes are sensitiveto all PMNS parameters, so that a global analysis in a three familyscheme is required, leading to

θ12 ' 34°, θ23 ' 48°, θ13 ' 8.5°. (8.38)

Note that, unlike in the quark sector, some mixing angles are large,so that the PMNS mixing matrix is far from diagonal.

Although there are three generations, the oscillation probabilitieswe observe can be approximately described in terms of the two-neutrino formalism. To understand why this is the case, let usreturn to the oscillation amplitude Aαβ(L), and imagine it as thesum of three vectors in the complex plane. If α = β, the unitaritycondition at L = 0 says they should sum to a vector of length one.If α 6= β, then they should sum to zero and this is the unitaritytriangle. At non-zero L, two of the vectors rotate in the complexplane, with frequencies (m2

j − m2i )/(2E)—so neutrino oscillations

correspond, in some sense, to time-dependent non-unitarity.Next, let us consider the oscillation probabilities Pµα, mea-

sured for atmospheric neutrinos, on length scales correspondingto |∆m2

32|. The solar mass difference can be neglected, because|∆m2

21| |∆m232|, so there is only one relevant mass difference,

and the survival probability behaves as for two generations. Thiscan be visualized in the complex plane, where only the vectorUµ3U∗α3 rotates with L. The stationary sum Uµ2U∗α2 + Uµ1U∗α1 canbe treated as a single vector, so this looks like a two generation sys-tem. Therefore, “atmospheric” oscillations can be approximated astwo-neutrino oscillations because the atmospheric mass differenceis very large compared to the solar one.

In the case of the solar mass difference, the two neutrino ap-proximation is good because θ13 is small. The observed survivalprobability is Pee and since Ue3 Uej, j = 1, 2, the last term can bedropped in

Aee = ∑j

Ueje−im2

j L/(2E)U∗ej. (8.39)

For completeness, we also quote the general result

Pαβ = δαβ − 4 ∑i<j

Re[UαiU∗βiU

∗αjUβj

]sin2 Xij

+ 2 ∑i<j

Im[UαiU∗βiU

∗αjUβj

]sin(2Xij), (8.40)

where

Xij =(m2

i −m2j )L

4E= 1.267

∆m2ij

eV2L/E

m/MeV. (8.41)

So far, we did not comment on the sign of the mass differences.From matter effects in the sun (see next section), we know that∆m2

21 > 0, which leaves two mass orderings

1. Normal hierarchy: m1 < m2 < m3

2. Inverted hierarchy: m3 < m1 < m2


It is presently unknown which of these two possibilities is realizedin nature. However, we can still conclude that the lowest possiblemaximal neutrino mass is

√|∆m32| ' 0.05 eV, which is getting close

to the cosmological bound (8.31).One way to determine the neutrino hierarchy relies on long base-

line oscillation experiments, which are sensitive to the second termin (8.40). This term is CP violating, as can be understood from thefact that antineutrinos follow the same equation but with U → U∗,which changes the sign of the second term but not the first. In fact,we see the leptonic analog of the Jarlskog invariant emerge

Im[UαiU∗βiU

∗αjUβj

]= J ∑

γkεαβγεijk, (8.42)

again parameterizing the size of CP violation. Apart from measur-ing the hierarchy of neutrino masses, long base-line experimentscan therefore also measure the currently unknown CP phase δ, butof course not the Majorana phases α1, α2 given that lepton numberis conserved. Currently, there are some first hints for a normal massordering,1 and 3σ evidence of the CP phase being non-vanishing2 1 M. A. Acero et al. New constraints

on oscillation parameters from νeappearance and νµ disappearance inthe NOvA experiment. Phys. Rev.,D98:032012, 2018

2 K. Abe et al. Constraint on theMatter-Antimatter Symmetry-ViolatingPhase in Neutrino Oscillations. Nature,580(7803):339–344, 2020

δ = −108+40−33° (NH), δ = −79+28

−31° (IH), (8.43)

for normal (NH) and inverted (IH) hierarchy, respectively, indicat-ing that the CP phase may be close to maximal.

A second possibility to determine the mass hierarchy, assumingthat neutrinos are Majorana particles, goes back to 0νββ decay, seeFig. 3. The bands for normal and inverted hierarchy are defined bythe other PMNS mixing parameters. The next generation of theseexperiments will be able to probe the parameter space correspond-ing to the inverted hierarchy.

(eV)lightestm

4−103−

10 2−10 1−10

3−10

2−10

1−10

1

IH

NH

A

50 100 150

Ca

Ge

SeZr

Mo

CdTe

Te

Xe

Nd

(eV

)m

Figure 3: Parameter space of 0νββsearches, as a function of the lightestneutrino mass and mββ.


8.3 MSW effect

We first rederive two-generation neutrino oscillations using quan-tum mechanics, and then use the results to discuss neutrino oscil-lations in matter. In short, electron neutrinos acquire an effectivemass term from their interactions with dense matter—this is theMSW effect—which can have significant effects in the sun and insupernovae, and over long baselines in the Earth.

In the mass eigenbasis we have the Schrödinger equation

iddt

Ψ = HΨ, (8.44)

with a diagonal Hamiltonian

H =

(E1 00 E2

). (8.45)

This Schrödinger equation can easily be solved. Defining our initialstates at t = 0 as |1〉 ≡ |1(t = 0)〉, |2〉 ≡ |2(t = 0)〉 we get the timedependent states

|1(t)〉 = e−iE1t|1〉,|2(t)〉 = e−iE1t|2〉. (8.46)

Let us repeat the last few steps in the interaction eigenbasis. Multi-plying (8.44) by V (the rotation from (8.32) ) from the left we get thecorresponding Schrödinger equation as

iddt

Ψ = HΨ, (8.47)

with

H ≡ V · H ·V−1 =

(a + b c

c a− b

), (8.48)

where

a =12(E1 + E2), (8.49)

b =12(E1 − E2) cos(2θ), (8.50)

c = −12(E1 − E2) sin(2θ). (8.51)

The crucial feature of the new Hamiltonian is that it is no longerdiagonal. As a result, if we start at time t = 0 with an interactioneigenstate |α〉, then at a later time we get a superposition of |α〉and |β〉 interaction eigenstates. Indeed, using (8.32) for the time-dependent states, we get

|α(t)〉 = e−iE1t cos θ |1〉+ e−iE2t sin θ |2〉, (8.52)

|β(t)〉 = −e−iE1t sin θ |1〉+ e−iE2t cos θ |2〉. (8.53)

Let us now use this relation to compute the oscillation probabilityPα→β(t), defined as follows: assume that at t = 0 we know that


our state is a pure interaction eigenstate |α〉. To be concrete wecan assume this to be an electron neutrino νe created in the sun.Pα→β(t) then gives us the probability that at a later time t thisstate has evolved into an interaction eigenstate |β〉. Of course, thisprobability is simply the absolute value of the amplitude squared

Pα→β(t) = |〈β|α(t)〉|2

=∣∣∣sin θ cos θ

(e−iE1t − e−iE2t

)∣∣∣2

=12

sin2(2θ) (1− cos(E2 − E1)t)

= sin2(2θ) sin2(

E2 − E1

2t)

. (8.54)

In the first step we have used (8.52) and the orthogonality of themass eigenstates 〈i|j〉 = δij. The expression for Pα→β(t) can bebrought into a more useful form by noting that

Ei =√

p2 + m2i = p +

m2i

2p+ . . . (8.55)

and, therefore,

12(E2 − E1) '

m22 −m2

14E

≡ ∆m221

4E, (8.56)

where E is the energy of the beam.1 Furthermore, since the neutri- 1 This argument can be made morerigorously using wave packets.nos travel (almost) at the speed of light, we have L = vt = ct = t,

where L is the distance traveled by the neutrino. Thus, we arrive atthe final expression for the oscillation probability,

Pα→β(t) = sin2(2θ) sin2

(L

∆m221

4E

). (8.57)

Equation (8.57) has the expected properties in that the probabil-ity vanishes for L → 0, θ → 0, and most notably for ∆m2

21 → 0.This last limit tells us that there is no mixing if the two neutrinospecies have the same mass and, in particular, if they are massless.Moreover, we see again that this (CP-conserving) part of the oscil-lation probability is only sensitive to the absolute value of the massdifferences.

So far we have considered oscillations in vacuum, i.e., we haveassumed that the neutrinos were traveling through the vacuum.While this is true most of the time, the neutrinos produced in thesun first have to travel through the sun before they can reach us.The matter surrounding the neutrinos can have a crucial effect onthe oscillation probability for the neutrinos. This effect is called thematter effect or the Mikheyev–Smirnov–Wolfenstein (MSW) effect.2 2 L. Wolfenstein. Neutrino Oscillations

in Matter. Phys. Rev. D, 17:2369–2374,1978; and S.P. Mikheyev and A.Yu.Smirnov. Resonance Amplificationof Oscillations in Matter and Spec-troscopy of Solar Neutrinos. Sov. J.Nucl. Phys., 42:913–917, 1985

The crucial question is: how does the Hamiltonian H (8.48)change through interactions of the neutrinos with surroundingmatter? There are both neutral- and charged-current interactions.As we have learned, neutral-current interactions are mediated bythe exchange of a Z boson. Taking into account that the surround-ing matter is basically made of protons, neutrons, and electrons, atypical Feynman diagram is:


νx νx

p, n, e p, n, e

Z

Figure 4: Neutral-current interaction ofneutrinos with matter.

The important point is that these interactions are independent ofthe flavor x of the neutrino. Thus they affect the two diagonal en-tries of the Hamiltonian in the same way. This means they change ain (8.49), i.e., the Hamiltonian is modified by a → a. As we will seelater, this change is irrelevant.

The charged-current interactions are mediated by a W±. A typi-cal Feynman diagram is:

νe e−

e− νe

W

Figure 5: Charged-current interaction.In contrast to the neutral-currentreactions, only electron neutrinosparticipate.

These interactions take place only for electron neutrinos sincethere are no µs (or τs) in the surrounding matter. In our conven-tion where we identify the |α〉 state with an electron neutrino, thismeans that only the top-left entry of the Hamiltonian (8.48) is mod-ified. Thus, including the matter effects we arrive at the followingHamiltonian,

HMSW =

(a + b + w c

c a− b

), (8.58)

where w comes from the charged-current interactions. The explicitform of w is not important for us. What we want to know is howthe w-term modifies the mixing angle. To find the modified mixingangle θMSW, we have to diagonalize HMSW, i.e., we have to find

VMSW =

(cos θMSW sin θMSW

− sin θMSW cos θMSW

), (8.59)

such that

HMSW ≡ V−1MSW · HMSW ·VMSW (8.60)

is diagonal. If we plug the explicit forms for VMSW, (8.59), andHMSW, (8.58), into (8.60) we find the off-diagonal terms of HMSW tobe

c cos(2θMSW) +2b + w

2sin(2θMSW). (8.61)


This vanishes for

tan(2θMSW) = − 2c2b + w

=−∆m2

21 sin(2θ)

4Ew− ∆m221 cos(2θ)

. (8.62)

We note that θMSW does not depend on a, thus as mentionedabove, the change a → a induced by the neutral-current inter-actions does not matter at all. The important point is that for4Ew ∼ ∆m2

21 cos(2θ) there can be a dramatic effect and the oscil-lation probability can increase substantially. In fact, this effect isvery important in the explanation of experimental results. Equa-tion (8.62) also shows that strictly speaking the MSW effect is onlysensitive to the sign of ∆m2

21 cos(2θ12), but in the usual conventionfor the mixing angle θ12, solar neutrino oscillations imply ∆m2

21 > 0.

8.4 Summary

• While it is by now clear that neutrinos have masses, it is not yetclear whether they are Dirac or Majorana particles. This can, inprinciple, be distinguished from neutrinoless double-β decay.

• When neutrino masses are included in the Lagrangian, mixingangles appear at the charged current vertex, as in the quark case.

• The experimental signature of (small) neutrino masses is oscil-lations: a neutrino produced from one flavor of charged lepton,can be detected by the appearance of a different charged lepton.Thus, an electron neutrino produced in the sun can arrive as aneutrino of a different flavor on Earth.

• If the neutrinos travel through matter rather than the vacuum theoscillation pattern can change dramatically (MSW effect).

• The see-saw mechanism provides us with an explanation of whythe neutrino masses are so much smaller than the other leptonmasses.

• From neutrino oscillation experiments, we know two mass differ-ences, and thanks to the MSW effect one sign. Likewise, the mix-ing angles have been determined rather accurately from neutrinooscillations. Unknown are still (1) the mass ordering (normal orinverted), (2) the absolute mass scale, (3) the CP phase δ, and, incase of Majorana neutrinos, (4) the two Majorana phases.

9Anomalies and Anomaly Cancellation

Not all symmetries of a classical theory survive quantization. If aclassical symmetry is no longer present in the quantum theory, wecall it anomalous and the breaking of the symmetry is called ananomaly.

The technical reason for the appearance of anomalies in quan-tum field theory is the necessity of regularization. The regulatorsused to make sense of quantum field theory typically break somesymmetries of the classical theory. In many cases it is possible toregularize in such a way that a symmetry is preserved or to at leastrecover the symmetry when the regulator is switched off and thetheory is renormalized. Anomalies arise when this is not possibleso that the symmetry is destroyed by quantum effects.

We should distinguish anomalies in global and in local sym-metries. An anomalous global symmetry is not a problem, butwill have interesting physics consequences. An anomalous gaugesymmetry, on the other hand, is a disaster: It implies that the corre-sponding theory is ill-defined. Without gauge symmetry, one endsup with unphysical degrees of freedom and negative probabilities.

An example of an anomalous global symmetry of the StandardModel, which will be discussed in detail below, is baryon num-ber NB = 1

3 (nq − nq). All the interaction terms in the Lagrangianpreserve it, but it is violated by quantum effects. A physics conse-quence of the anomaly is that it is possible that the very early uni-verse had NB = 0 (i.e. that there were an equal number of quarksand anti-quarks) and that the baryon asymmetry we observe today(much more protons than anti-protons) developed later when theuniverse expanded.1 2 1 The other ingredients to generate

such an asymmetry are C and CPviolation as well as out-of-equilibriumevolution during part of the time.These conditions for baryogenesis arecalled the Sakharov conditions.2 A. D. Sakharov. Violation of CPInvariance, C Asymmetry, and BaryonAsymmetry of the Universe. Pisma Zh.Eksp. Teor. Fiz., 5:32–35, 1967

The most important anomaly in the context of the StandardModel is the axial anomaly (also called Adler–Bell–Jackiw (ABJ)anomaly or triangle anomaly). Due to this anomaly, chiral gaugesymmetries are in general anomalous. Since the Standard Modelis an example of a chiral gauge theory, it is important to verifythat its gauge symmetries are non-anomalous. We will see thatanomaly cancellation implies quite non-trivial relations among thehypercharges of the different particles.


9.1 The axial anomaly

In a massless gauge theory (let us consider massless QED for themoment), there is a classical symmetry

ψL → eiαL ψL, ψR → eiαR ψR. (9.1)

The (classically!) conserved currents can be derived using the in-variance of the Lagrangian

L = ψLi /DψL + ψR i /D ψR −14

Fµν Fµν (9.2)

or obtained directly using Noether’s theorem. This gives

0 = δL =∂L∂ψL

δψL +∂L

∂∂µψLδ∂µψL

= ∂µ∂L

∂∂µψLδψL +

∂L∂∂µψL

∂µδψL

= ∂µ

[∂L

∂∂µψLδψL

]= ∂µ

[ψLγµψL

], (9.3)

and similarly for ψR. Instead of the currents

jµL = ψLγµψL , jµR = ψRγµψR, (9.4)

we can also consider the vector and axial-vector currents

jµV = jµ

L + jµR = ψγµψ , jµA = jµ

R − jµL = ψγµγ5ψ . (9.5)

The vector current is associated with the symmetry transformationswhere α = αL = αR, which form the basis of gauge symmetry.The current is conserved, ∂µ jµ

V = 0, and the associated conservedquantity is the electric charge

Q = −e∫

d3x j0V(x) , (9.6)

and Q/(−e) is the fermion number, i.e. the number of fermionsminus the number of anti-fermions. In contrast, the axial current isanomalous. Its divergence is nonzero and given by

∂µ jµA = − e2

16π2 εµνρσFµνFρσ. (9.7)

This operator relation is quite remarkable. The term on the right-hand side is a quantum correction proportional to e2, not visible inthe classical theory. It is generated by a one-loop triangle diagram,which we will compute below. One would expect that the relationwill receive corrections of O(e4), but this is not the case, the relationis exact.1 One way to see how the anomaly arises is that a divergent 1 Stephen L. Adler and William A.

Bardeen. Absence of higher ordercorrections in the anomalous axialvector divergence equation. Phys. Rev.,182:1517–1536, 1969

integral multiplies an expression that naively vanishes. To workwith well-defined quantities, one needs to regularize the theory.The regularization induces a violation of the axial symmetry thatgoes away when the regularization is removed, but since the vio-lation gets multiplied by a divergent integral, the final result is theterm on the right-hand side of (9.7).


While the final result (9.7) is of course independent of deriva-tion and the regularization scheme, it is instructive to derive theanomaly in different regularizations. Here, we will consider explic-itly the one-loop integral, but note that the same result can also bederived in the path integral formalism, where the anomaly arisesbecause the path integral measure is not invariant under the axialtransformation.2 2 Kazuo Fujikawa. Path Integral

Measure for Gauge Invariant FermionTheories. Phys. Rev. Lett., 42:1195–1198,1979

In the following, we will verify that the relation (9.7) is correctat one-loop order by computing the matrix element of the axialcurrent for a final state with two photons

Fµ =∫

d4xe−iq·x〈γ(p)γ(k)| jµA(x) |0〉

= (2π)4δ(p + k− q)ε∗ν(p)ε∗λ(k)Mµνλ(p, k). (9.8)

We then show that qµFµ 6= 0 and verify that the violation is indeedgiven by the matrix element of the operator on the right-hand sideof (9.7).

The leading contribution to the quantity Fµ arises at one-looporder and is obtained by computing the two triangle loop diagramsshown below.

p, ν

k, λ

q, µl

l + p

l − k

p, ν

k, λ

q, µl

l − p

l + k

(A) (B)

Figure 1: Triangle loop diagramscontributing to 〈γ(p)γ(k)| jµ

A(x) |0〉.

Using the QED Feynman rules together with the Feynman rulefor the current jµ

A, which is iγµγ5, diagram A is given by1 1 Note the sign for the closed fermionloop. Since the anomaly arises fromthe UV part of the integral, all massescan be neglected.Mµνλ

A (p, k) = −ie2∫ ddl

(2π)d Tr[

γµγ5/l − /k(l − k)2 γλ /l

l2 γν /l + /p(l + p)2

].

(9.9)The second diagram is obtained by switching the two photons ofthe first one, i.e.Mµνλ

B (p, k) =MµλνA (k, p). We have regularized the

loop integral by working in d = 4− 2ε dimensions.To check current conservation, we contract the diagram with qµ

so that the trace starts with /qγ5. In four dimensions, we can write

/qγ5 = (/l + /p − /l + /k)γ5 = (/l + /p)γ5 + γ5(/l − /k). (9.10)

If we now consider qµMµνλA (p, k) and insert the right-hand side of

this relation into the numerator of the diagram, each of the termscancels a propagator

iqµMµνλA (p, k)

= e2∫ ddl

(2π)d Tr[

γ5γλ /ll2 γν /l + /p

(l + p)2 + γ5/l − /k(l − k)2 γλ /l

l2 γν

]

= e2∫ ddl

(2π)d Tr[

γ5γλ /ll2 γν /l + /p

(l + p)2 − γ5γν /ll2 γλ /l + /k

(l + k)2

], (9.11)


where the last line follows after the shift l → l + k in the secondterm and subsequent permutation of γν to the left due to cyclicityof the trace. This expression is then manifestly antisymmetric underν↔ λ and p↔ k so that qµ(Mµνλ

A (p, k) +MµνλB (p, k)) = 0!

There are several ways to see the mistake in this derivation, allof which are related to the regularization of divergent loop inte-grals, e.g. the shift l → l + k is only permitted once a particularregularization is adopted, otherwise the expression is ill-defined. Indimensional regularization the problem is then how to deal with γ5

in d dimension. Over the years, many different schemes have beendeveloped, a crucial test being that the anomaly triangle comesout correct. Here, we will present the calculation in the ’t Hooft–Veltman scheme.1 The idea is as follows: we split the loop mo- 1 Gerard ’t Hooft and M. J. G. Veltman.

Regularization and Renormalization ofGauge Fields. Nucl. Phys., B44:189–213,1972

mentum into a four-dimensional part l‖ and a (d− 4)-dimensionalremainder l⊥

l = l‖ + l⊥ . (9.12)

The four-dimensional γ-matrices anti-commute with γ5, while theadditional (d− 4)-dimensional ones are chosen to commute so thatwe have

/l γ5 = −γ5 /l ‖ + γ5 /l ⊥ = −γ5 /l + 2γ5 /l ⊥ . (9.13)

Using this result, we can derive a version of (9.10) valid in d dimen-sions

/qγ5 = (/l + /p− /l + /k)γ5 = (/l + /p)γ5 + γ5(/l − /k)− 2γ5 /l ⊥. (9.14)

We have already shown that the first two terms on the right-handside give a vanishing contribution after adding diagram B, so wejust need to analyze the final term

iqµMµνλA (p, k)

= e2∫ ddl

(2π)d Tr[−2γ5/l ⊥

/l − /k(l − k)2 γλ /l

l2 γν /l + /p(l + p)2

]. (9.15)

To proceed further, we introduce Feynman parameters

1ABC

=∫ 1

0dx∫ 1−x

0dy

2

(Ax + By + C(1− x− y))3 (9.16)

with A = (l − k)2, B = (l + p)2, and C = l2. We then shift l →l + kx− py to obtain

iqµMµνλA (p, k) = 2e2

∫ 1

0dx∫ x

0dy∫ ddl

(2π)dN

(l2 − ∆)3 , (9.17)

where∆ = −x(1− x)k2 − y(1− y)p2 − 2xy k · p. (9.18)

The numerator can be simplified to N = Tr[γ5γνγλ/p/k

](−2l2

⊥)after the shift. This form is obtained using the following properties:

1. Traces of γ5 with less than 4 γµ matrices vanish.


2. We need both independent vectors p and k, otherwise the con-traction with the ε tensor from the trace would vanish.

3. Odd powers of /l ⊥ or /l ‖ give a vanishing contribution since therest of the integral is even in these momenta after the shift.

4. /l ⊥ anti-commutes with the 4-dim γµ.

Exercise 9.1After Feynman parameterization we arrive at the following loopintegral:

∫ddl

1(l2 − ∆)3 Tr

[γ5 /l ⊥ (/l + (x− 1)/k − y/p) γν (/l + x/k − y/p)

× γλ(/l + x/k + (1− y)/p)]

.

The dependence of the numerator on the Feynman parameters xand y arose from the shift l → l + xk − yp. Using the argumentssketched above, show that the numerator simplifies to

l2⊥ Tr

[γ5γνγλ

/p/k]

.

To evaluate the remaining integral, we can use that due toLorentz symmetry of the integral, we can replace

l2⊥ = g⊥µνlµlν = g⊥µν

gµν

dl2 =

d− 4d

l2 (9.19)

under the integrand. The remaining integral is

d− 4d

∫ ddl(2π)d

l2

(l2 − ∆)3 =d− 4

di

(4π)d/2d4

Γ(2− d/2)(∆)d/2−2

= −12

i(4π)2 +O(d− 4). (9.20)

Using

Tr[γ5γνγλ

/p/k]= −4iενλρσ pρkσ (9.21)

we then obtain the final result

iqµMµνλA (p, k) =

e2

4π2 ενλρσ pρkσ, (9.22)

which is symmetric under the interchange of the two photons sothat the second diagram gives an identical contribution. In sum-mary we then have

〈γ(p)γ(k)| ∂µ jµA(0) |0〉 =−

e2

2π2 ε∗ν(p)ε∗λ(k)ενλρσ(ipρ)(ikσ) (9.23)

= − e2

16π2 〈γ(p)γ(k)| εµνρσFµνFρσ |0〉,

consistent with the ABJ anomaly (9.7). We will now explore theimplications of the axial anomaly in different contexts.


9.2 Chiral Anomaly in QCD

The previous calculation is readily generalized to QCD, which inthe massless limit exhibits a classical

U(n f )L ×U(n f )R = SU(n f )L × SU(n f )R ×U(1)L ×U(1)R (9.24)

symmetry, or equivalently in terms of vector and axial-vector com-binations. The relevant currents are then

jµA = qγµγ5q, jµa

A = qγµγ5taq, (9.25)

where the ta are generators in flavor space. In close analogy toQED, the divergence of the SU(n f ) axial-vector current becomes

∂µ jµaA = − g2

s16π2 εµνρσFb

µνFcρσTr(taTbTc), (9.26)

where the Ta are SU(Nc) generators. Accordingly, the trace factor-izes Tr(taTbTc) = Tr(ta)Tr(TbTc) in the two different spaces andvanishes because the generators are traceless. This is no longer truefor the singlet combination, for which ta is replaced by the unitmatrix, and the corresponding current becomes anomalous

∂µ jµA = −g2

s n f

32π2 εµνρσFaµνFa

ρσ. (9.27)

Note that the divergence of the singlet current takes a form remi-niscent of the θ term in (4.14), and indeed there is a relation to theaxial U(1) anomaly of QCD.

The SU(n f )L × SU(n f )R symmetry is broken spontaneously toits vector sub group SU(n f )V , leading to the appearance of n2

f − 1Goldstone bosons.1 If U(1)A were a real symmetry, one would 1 π±, π0 for n f = 2, plus K±, K0, K0, η

for n f = 3.expect n2f Goldstone bosons instead (or a parity symmetry in the

spectrum if the symmetry were not spontaneously broken), but thelightest candidate, the η′, is already much heavier than the other(pseudo) Goldstone bosons.2 Accordingly, the η′ indeed receives 2 “Pseudo” because the finite quark

masses break chiral symmetry explic-itly.

part of its mass from the anomaly (9.27).A second consequence concerns the decay of the π0 → γγ. In

QCD+QED3 the anomaly takes the form 3 Axial current from QCD, vectorcurrents from QED, and quarks in thetriangle diagram.

∂µ jµaA = − e2Nc

16π2 εµνρσFµνFρσTr(taQ2), (9.28)

where Q = diag(2/3,−1/3, . . .) is the matrix of the quark charges.In particular, the a = 3 current is anomalous

∂µ jµ3A = − e2

16π2 εµνρσFµνFρσ. (9.29)

Because the π0 has the same quantum numbers as the jµ3A current,

this anomaly mediates the decay π0 → γγ, even in massless QCD.To obtain the decay rate, we start from

F(x) = 〈γγ|∂µ jµ3A (x)|0〉 (9.30)


and consider the limit x0 → ∞. In this limit, the Fourier transformdevelops a pole at q2 = 0 (since the pion is massless in the chirallimit)

F(q) =∫

d4xF(x)e−iq·x =i

q2 〈π0(q)|∂µ jµ3

A |0〉〈γγ|π0(q)〉+ . . . (9.31)

Using the definition of the pion decay constant, which, up to thefactor

√2, is identical to the charged-pion case (7.45) due to isospin

symmetry, we have

〈0|∂µ jµ3A |π0(q)〉 = −q2Fπ , (9.32)

and therefore F(q) = −iFπ〈γγ|π0(q)〉. The decay amplitude canthus be derived by isolating the pion pole from the vacuum matrixelement in terms of the decay decay constant of the pion, leading to

M(π0 → 2γ) = 〈γγ|π0〉 = iFπ〈γγ|∂µ jµ3

A |0〉

= − iFπ

e2

32π2 〈γγ|εµνρσFµνFρσ|0〉

= −ie2

4π2Fπενλρσε∗ν(p)ε∗λ(k)pρkσ. (9.33)

Finally, we calculate the rate as

Γ(π0 → γγ) =1

2Mπ

18π

12 ∑

polarizations|M(π0 → 2γ)|2

=α2M3

π

64π3F2π= 7.749(15) eV, (9.34)

where the factor 1/2 in the first line is a symmetry factor that ac-counts for the two indistinguishable photons in the final-statephase space integration. Corrections to this equation scale asM2

π/(4πFπ)2 . 2%. The experimental value from the PrimEx exper-iment1 Γ(π0 → γγ) = 7.802(117) eV in fact agrees perfectly with 1 I. Larin et al. Precision measurement

of the neutral pion lifetime. Science,368:506–509, 2020

our prediction (9.34), with the central values less than 1% part!2 In2 The agreement of this leading-ordercalculation looks a bit better than itreally is because the physical piondecay constant already subsumes somecorrections O(M2

π).

addition to the π0 → γγ decay, with a branching fraction of almost99%, also the subleading decay channels (e+e−γ, e+e−, e+e−e+e−)are driven by the anomaly. This is a key difference to the charged-pion decay mediated by the weak charged current. In consequence,the lifetime of the neutral pion is shorter by more than 8 orders ofmagnitude.

Moreover, the good agreement of (9.34) with experiment is of-ten quoted as evidence that QCD has Nc = 3, since formally thedivergence (9.28) scales with Nc. This is misleading. To ensureanomaly cancellation in the Standard Model, we need to fulfill re-lations such as (9.39) below, which implies that it is not possible tosimply change Nc without also changing the hypercharge of thequarks. Working this out in detail, one finds that the decay rate(9.34) remains constant as Nc changes.3 Of course, there is a lot of 3 O. Bar and U. J. Wiese. Can one see

the number of colors? Nucl. Phys.,B609:225–246, 2001

other evidence for Nc = 3, for example the measurements of theR-ratio with the theoretical predictions for Nc = 3, as discussed inchapter 4.


9.3 Anomaly cancellation in the Standard Model

In contrast to the global symmetries discussed in the previous sec-tion, anomalies in the gauge symmetries of the Standard Modelwould be disastrous (leading to all sorts of inconsistencies). Toshow that the Standard Model is a well defined quantum field the-ory, we need to demonstrate that all such anomalies are absent.Since the SU(2)L and U(1)Y part of the gauge symmetries act dif-ferently on left- and right-handed fields, anomalies could arise. Toverify that the anomalous contributions cancel, we have to analyzeall triangle diagrams with different gauge bosons on the outside.

Bν , W jν , Ab

ν

Bλ, W kλ , Ac

λ

Bµ, W iµ

Figure 2: The set of triangle loop dia-grams that could induce anomalies inthe Standard Model gauge symmetries.

In these diagrams, the γ5 part of the coupling of the left bo-son, Bµ or Wi

µ, supplies the (potentially anomalous) axial current.Nonzero contributions can then arise when both bosons on theright couple with the vector part so that we have a loop with oneγ5, or both with their axial coupling so that we have three γ5 matri-ces in the loop.

Let us go over the different contributions, starting with the trian-gle involving three SU(2) bosons. In terms of its Dirac algebra andloop integration the computation is analogous to the QED triangle,but in the present case the vertices involve SU(2) matrices so thatthe triangle is proportional to

W jν

W kλ

W iµ

∝ Tr[ti

tj, tk]

,

where the anti-commutator arises because we have to add the sec-ond diagram where the gauge-bosons are exchanged. The anomalyvanishes for SU(2) because

Tr[ti

tj, tk]

= Tr[

ti 12

δjk]=

12

δjkTr[ti]= 0 , (9.35)

but would be nonzero for SU(n) with n > 2. Analogously, thetriangles with a single Wi

µ or Aaµ boson vanish. Let us consider the

remaining triangles in turn

Abν

Acλ

Bµ

∝ ∑q

Tr[(YqR −YqL)

Ta, Tb

]= δab ∑

q

(YqR −YqL

). (9.36)


The relative sign of the left- and right-handed hypercharges arisesbecause the axial part of the two currents has the opposite sign.With hypercharges YqR = Qq and YqR = Qq − T3,

∑q

(YqR −YqL

)= Nc ∑

qT3 = Ncng

(12− 1

2

)= 0 , (9.37)

where ng = 3 is the number of generations.

W jν

W kλ

Bµ q,

∝ ∑q

Tr[YqL

tj, tk

]= Nc δjk ∑

fYfL . (9.38)

Due to the presence of the Ws only left-handed fields are present,but the sum now runs over both quarks and leptons. Evaluating thesum over all fermions f , we find

∑f

YfL = ng

(Nc

6− 1

2

)= 0, (9.39)

implying that there is an interesting relation between the numberof colors and the hypercharges to make the Standard Model consis-tent. Finally,

Bν

Bλ

Bµ ∝ ∑f(Y3

fR−Y3

fL).

The relative sign arises again from the sign difference in the axialpart of the left- and right-handed currents. It is quite remarkablethat there is a constraint on the third powers of the hypercharges.And, as expected, the sum evaluates to zero after summing over theleft- and right handed quarks and leptons in a generation, i.e.,

−(− 1

2

)3

︸︷︷︸νL

+ 0︸︷︷︸νR

−(− 1

2

)3

︸︷︷︸eL

+(− 1)3

︸︷︷︸eR

+ Nc

[− 2

(16

)3

︸︷︷︸uL ,dL

+

(23

)3

︸︷︷︸uR

+

(− 1

3

)3

︸︷︷︸dR

]

= −34+

Nc

4= 0. (9.40)

Let us note in passing that one can derive a final constraint byconsidering the Standard Model in a weak gravitational field andevaluating the triangle diagram of a current from Bµ coupling totwo gravitons. The condition for this gravitational anomaly to can-cel is

∑f

(YfR −YfL

)= 0 , (9.41)


which has the same form as the gluonic anomaly (9.37) but dueto the nature of gravity, the sum runs over all fermions. Also thiscondition is fulfilled, because Y(νL) + Y(eL)− Y(eR) = 2(−1/2)−(−1) = 0. In total, we find that anomaly cancellation imposes thefollowing four conditions on the hypercharges

1. ∑q(YqR −YqL

)= 0,

2. ∑ f YfL = 0,

3. ∑ f (Y3fR−Y3

fL) = 0,

4. ∑ f(YfR −YfL

)= 0,

all of which are fulfilled in the Standard Model.

9.4 Baryon number

While the gauge symmetries are free of anomalies, there are globalsymmetries in the Standard Model that do become anomalous. Thecurrents associated with quark number and lepton number

Jµq =

2ng

∑k=1

qkγµqk , Jµ` =

2ng

∑k=1

¯kγµ`k (9.42)

are anomalous in the presence of SU(2)L. This looks surprisingat first sight since these are vector currents, but the γ5 matricesarise from the coupling the Ws. Specifically, the following trianglediagrams are nonzero

W 3ν

W 3λ

Jµq

∝ ngNc

[ (12

)2

︸︷︷︸uL ,cL ,tL

+

(−1

2

)2

︸︷︷︸dL ,sL ,bL

],

W 3ν

W 3λ

Jµ

∝ ng

[ (12

)2

︸︷︷︸νL

+

(−1

2

)2

︸︷︷︸eL

],

However, the combination JµB − Jµ

L is conserved, where JµB = 1

3 Jµq is

called the baryon number and JµL = Jµ

` lepton number. Similarly,electron, muon, and τ lepton number are conserved classically,broken by quantum effects, but the difference of either pair is againfree of anomalies. This set of symmetries (baryon number and thevarious lepton numbers) are also called “accidental symmetries” ofthe Standard Model, because they are not imposed but arise fromthe gauge-invariant terms allowed up to dimension-4 level.

At low energies or low temperatures, the violations of baryonnumber and lepton number are negligibly small, but they play arole at high temperatures in the early universe. In fact, there are nobaryon-number-violating terms in perturbation theory at all, for the


same reason that the θ-term in QCD requires non-trivial topologicalconfigurations (“instantons”): the divergence of the current canbe written as a total derivative and thus removed from the theoryas long as there are no non-negligible contributions from surfaceterms.

9.5 Summary

• It is not always possible to maintain classical symmetries whena system is quantized. The violations of classical symmetries byquantum effects are called anomalies.

• Anomalies in global symmetries simply imply the absence of thesymmetry under consideration. The special form of the violationleads to interesting physics such as the π0 → γγ decay.

• Anomalies in its gauge symmetries would invalidate the Stan-dard Model. The absence of anomalies requires a set of con-straints on the charges of the Standard Model fields, including arelation between the number of colors and the hypercharges.

• Baryon and lepton number separately are violated by non-perturbative effects in the Standard Model, but the combinationB− L is conserved.

10Non-perturbative effects

So far, the entire discussion of the Standard Model has proceededin the context of perturbation theory, but we have already encoun-tered two examples for which a perturbative description is notsufficient. This concerns the θ term in QCD, see chapter 4, andbaryon number violation through anomalies, as discussed in chap-ter 9. Both effects are related to non-perturbative solutions of thefield equations. In this chapter, we will take a closer look at suchnon-perturbative effects in the Standard Model.

10.1 QCD instantons and θ vacua

We start from the gauge transformation (3.17) applied to A = 0 and

U1(x) =x2 − d2

x2 + d2 +2idτ · xx2 + d2 , (10.1)

where d is an arbitrary parameter and τ refers to the vector of Paulimatrices, corresponding to any SU(2) subgroup of SU(3)c. Thecorresponding gauge-transformed potential

A(1)j (x) = − i

gs

(∂jU(x)

)U−1(x) (10.2)

= − 2dgs(x2 + d2)2

[τj(d2 − x2) + 2xjτ · x− 2d(x× τ)j

],

is an example of a gauge configuration with a non-trivial topologi-cal charge (also call winding number)

n =ig3

s24π2

∫d3x Tr

(Ai(x)Aj(x)Ak(x)

)εijk, (10.3)

it carries n = 1. Similarly, configurations with arbitrary n can beconstructed by repeated application of Un = (U1)

n, and gaugepotentials can be classified into distinct classes according to theirtopological charge n. One can actually construct explicit solutionsof the Yang–Mills theory in Euclidean space time, and such classicalsolutions with non-trivial n are called instanton solutions or justinstantons.

The existence of such solutions is the reason why, in general, theθ-term in QCD cannot be ignored, because even though it can be


written as a covariant derivative

Lθ =θg2

s32π2 Fa

µν Fµνa =

θg2s

16π2 ∂µKµ, (10.4)

with Fµνa = 1

2 εµνλσFaλσ and

Kµ = 2εµνρσTr[

Aν

(∂ρAσ +

2igs

3AρAσ

)], (10.5)

a finite contribution may arise from surface terms. E.g., startingfrom a potential A = 0 at t = −∞ and interpolating to A = A(1) att = +∞, one has

g2s

32π2

∫d4x Fa

µν Fµνa =

g2s

16π2

∫d4x ∂µKµ =

g2s

16π2

∫d3x K0

∣∣∣t=+∞

t=−∞

=ig3

s24π2

∫d3x Tr

(A(1)

i A(1)j A(1)

k)εijk = 1, (10.6)

where we have used that asymptotically Faµν = 0. In general, the θ

term measures the change in the winding number

g2s

32π2

∫d4x Fa

µν Fµνa = n+ − n− (10.7)

between the asymptotic configurations. Since there are gaugetransformations that take gauge configurations across topologicalclasses, it is clear that the QCD vacuum must contain contributionsfrom all of them. Ultimately, this implies that a term such as (10.4)has to be added to the QCD Lagrangian.

The question still remains if the associated parameter θ is ac-tually physical, i.e., if it can lead to any observable CP-violatingeffect. This question is related to the axial anomaly

∂µ jµA = −

g2s n f

16π2 Faµν Fµν

a = −g2

s n f

8π2 ∂µKµ. (10.8)

Since, formally, the right-hand side is a total derivative, one couldjust define a new conserved current

jµA = jµA +

g2s n f

8π2 Kµ. (10.9)

This current indeed defines a conserved charge

QA =∫

d3x j0A, (10.10)

but is not gauge invariant. That is, by applying the gauge transfor-mation U1, this charge will change according to

QA → QA − 2n f , (10.11)

in such a way that the different θ vacua are related by chiral U(1)A

transformations.To understand the physical consequences, one needs to include

the terms in ∂µ jµA that arise from the quark masses. When diago-

nalizing the quark mass matrix, we performed separately left- and


right-handed transformations, and thereby shifted all CP violationinto the CKM matrix. However, since this includes an axial U(1)transformation, one obtains a shift in the value of θ, so that actuallythe combination

θ = θ + arg det(M) (10.12)

enters as prefactor of the θ term, where θ is the original Lagrangianparameter and M the quark mass matrix. The strong CP problemthus amounts to the question why this combination θ is so small.If any of the quark masses were zero, one could actually performan axial phase transformation, and use this additional freedom toremove any dependence on θ altogether. However, this “solution”is not realized in nature,1 and since all quark masses are positive 1 Even if one quark were massless,

implying θ = 0, one would still haveto explain why, so in this sense theproblem of the smallness of θ wouldsimply be shifted to elsewhere.

there is an observable CP-violating effect from θ, unless additionalsymmetries are assumed to enforce θ = 0. Such a mechanismwas proposed by Peccei–Quinn,2 but so far the so-called “axion,”

2 R.D. Peccei and Helen R. Quinn.CP Conservation in the Presence ofInstantons. Phys. Rev. Lett., 38:1440–1443, 1977

which would arise from to the spontaneous breaking of that newsymmetry, has not been observed.

10.2 Electroweak instantons

In the electroweak sector of the Standard Model one could alsowrite a θ term

LEWθ =

g2θEW

32π2 WiµνWµν

i , (10.13)

so one may ask if θEW should be counted as another free parameter.In principle, this could happen because there are SU(2) instantonconfigurations (called “sphalerons”), in contrast to an Abelian U(1)theory, which does not have solutions with non-trivial topology.However, in this case the divergence of the B and L currents isproportional to the corresponding θ-term alone,

∂µ jµB = ∂µ jµL = −

g2n f

32π2 WiµνWµν

i , (10.14)

so that one can always find a transformation of quark and leptonfields

q→ e−iαB/3q, `→ e−iαL`, (10.15)

to make the transformed electroweak angle

θEW → θEW + n f (αB + αL) (10.16)

vanish. It is therefore not observable in the Standard Model. Infact, one would need physics beyond the Standard Model thatsimultaneously violates baryon number B and lepton number L,otherwise, the remaining rotation could still be used to enforceθEW = 0.1 1 Pavel Fileviez Pérez and Hiren H.

Patel. The Electroweak Vacuum Angle.Phys. Lett. B, 732:241–243, 2014

This does not mean, however, that there are no non-perturbativeeffects in the SU(2)L theory, the sphalerons solutions are still al-lowed, and they are the source of baryon number (or lepton num-ber) violation in the Standard Model. To estimate the size of the


effect, one needs to study the action at these field configurations,which gives a scaling

e−S = e−8nπ2/g2, (10.17)

which illustrates the non-perturbative character of these solutions:a Taylor expansion around g = 0 is not possible, since the functionand all derivatives vanish. In QCD these effects can thus be sizable,of order e−2πn/αs , but for the SU(2)L coupling the scaling goes as(n = 1)

e−8π2/g2= e−8π2 sin2 θW /e2

= e−2π sin2 θW /α ∼ e−188, (10.18)

for sin2 θW and α evaluated at the electroweak scale. The action fac-tor e−S can also be interpreted as a tunneling amplitude, from onetopological sector of the theory to another. At high temperatures,say above 1 TeV, one could therefore expect that thermal fluctu-ations can take the vacuum over the barrier, instead of having totunnel through it. This is how the sphaleron processes can becomerelevant in the early universe, e.g., they can be used to turn a viola-tion of lepton number into one of baryon number (“leptogenesis”).

10.3 Summary

• Non-Abelian gauge theories have vacua with a non-trivial topo-logical structure (“instanton solutions”). These effects cannot bedescribed in perturbation theory.

• In QCD, the instantons are related to the θ term. As long as allquark masses do not vanish, this leads to an observable CP-violating effect. The experimental observation that the coefficientis tiny is called the “strong CP problem.”

• Electroweak instantons (“sphalerons”) are the only source ofbaryon and lepton number violation in the Standard Model.Their amplitudes are negligibly small at low energies and tem-peratures, but can become relevant in the early universe.

11Physics beyond the Standard Model

The Standard Model has a number of shortcomings. The two mostglaring ones are that it does not describe all forces, since the theorydoes not include gravity, and that it does not account for all matter,since astronomical measurements indicate the presence of a largeamount of additional dark matter. Further, it is known that theamount of CP violation in the Standard Model is not sufficient toexplain the matter–antimatter asymmetry in the universe and finiteneutrino masses in some way or another point to physics beyondthe Standard Model (BSM). On top of this, there is a subtle theoret-ical issue, triviality, which indicates that the SM is incomplete andneeds to be embedded into a different theory at higher energies.This issue is related to the so-called naturalness of the Higgs mass.Without specifying a concrete BSM, its effects can be parameterizedin an effective field theory (SMEFT).

11.1 Triviality

From the existence of gravity, we know that the Standard Model isnot a complete theory of nature. However, even an analysis of theStandard Model itself reveals that it cannot be valid to arbitraryhigh energies. We encountered the problem when discussing run-ning couplings and asymptotic freedom. The Higgs sector essen-tially consists of a φ4-theory, for which one finds that the runningcoupling takes the form

λ(µ) =λ(µ0)

1− 34π3 λ(µ0) log µ2

µ20

. (11.1)

If we start at some low scale µ0 and study the coupling at higherscales, we observe that it grows. Formally, at some very large µ =

Λ, the logarithm will compensate the suppression by the couplingconstant and we will have

34π3 log

µ2

µ20

λ(µ0) = 1 (11.2)

at which point the coupling λ(µ = Λ) diverges. This is called theLaudau-pole in the running coupling. Of course, (11.1) is not ex-act since it was obtained from solving the renormalization group


equation of the running coupling at one-loop accuracy. Further-more, the higher-order corrections become larger and larger as oneapproaches the Landau pole so that perturbation theory can nolonger be trusted. However, it is certainly disconcerting that the φ4

coupling becomes stronger and stronger at high energies.In the Standard Model, we have MH =

√2λv = 125 GeV and

v2 = (√

2GF)−1 = 1.16× 10−5 GeV, which yields v = 246.7 GeV and

λ = 0.16. These are tree-level relations, but the appropriate scale inthe coupling is the scale at which we perform our measurements, soλ ≡ λ(MH) = 0.16. If we check what at what scale the Landau poleoccurs for this value of the coupling, we get

Λ = v exp

(4π3v2

3M2H

)= 8× 1071 GeV. (11.3)

This is a huge scale, much larger even than the Planck mass, soin practice the presence of the Landau pose is not a limitation forthe Standard Model. Note however, that the scale is very sensitiveto the Higgs mass. For a heavy Higgs MH = 1 TeV, we wouldobtain Λ = 3 TeV. Let us also note that our discussion is incom-plete, since we should compute the β-function in the full StandardModel. In particular, one should include the contribution of the topquarks, which couple strongly to the Higgs boson due to their largeYukawa couplings. However, qualitatively the full treatment leadsto the same conclusions.

Since the perturbative treatment is not appropriate near the Lan-dau pole, one should analyze the theory non-perturbatively, whichcan be done by discretizing the theory on a space-time lattice andevaluating the path integral numerically. While this is difficult todo for the Standard Model as a whole, φ4 theory has been stud-ied non-perturbatively in great detail. When putting the theoryon a lattice, the lattice spacing a acts as a UV cutoff for the theory,ΛUV ∼ 1/a. One can then compute the renormalized couplingλ(µ) at µ = v as a function of the value of the bare coupling usedas an input at a given cutoff ΛUV. Since the bare coupling corre-sponds to a coupling at a high scale µ ∼ ΛUV, one finds that therenormalized coupling is smaller than the input and to arrive ata given fixed value of λ(v) = λ0 one has to increase the value ofthe bare coupling more and more as one makes the lattice spacingsmaller. However, once the scale ΛUV is too large, or equivalentlya too small, it becomes impossible to reach the desired value of thephysical coupling λ(v), independently of how large the bare couplingis chosen. In other words, it is not possible to take the lattice spac-ing all the way to zero without ending up with a free theory! Thisproperty is called triviality. It implies that one can only make senseof interacting φ4 theory if there is a UV cutoff in place at some highscale, i.e. if it is viewed as a low-energy effective theory, such asFermi theory. The same appears to be true for the Standard Model.


11.2 Naturalness

The arguments above indicate that we should view the StandardModel as a low-energy effective theory to some more fundamentaltheory at higher energies. The parameters of the Standard Modelcould then be obtained by “integrating out” the physics at higherenergies in very much the same way as we did for Fermi theory.1 1 The term “integrating out” refers

to a path integral treatment, whereone would simply integrate over theheavy fields to obtain a theory withonly light fields. In our treatment,one instead writes down the mostgeneral theory of the light fields andthen determines the couplings of thevarious operators with a matchingcomputation, i.e. by computing thesame quantity in both theories andadjusting the couplings of the low-energy theory to reproduce the resultof the full theory.

For Fermi theory, we could compute the parameter GF = 14√

2g2

M2W

by matching to the Standard Model. Fermi theory breaks down atthe scale Λ ∼ MW when there is enough energy to produce theW-bosons, which are not included in Fermi theory. Similarly, weexpect the Standard Model to break down above some currentlyunknown scale Λ where new particles will arise.

Since we do not know the scale Λ at which it breaks down andsince we do not know what theory replaces it, we cannot reallycompute the parameters of the Standard Model, but we can discusshow they scale with Λ. Following Wilson, we can distinguish thefollowing type of operators in the effective theory:

Coefficients Operator dimension n UV sensitivityg, Y, λ 4 log Λ marginale.g. GF n > 4

1Λn−4 irrelevant

µ, MH n< 4 Λ4−n relevant

Table 11.1: Operator classification.

In the left column, we have indicated some examples of couplingconstants, multiplying operators of the corresponding dimension.These are also called Wilson coefficients. In the Fermi theory Λ ∼MW and the four-fermion operator is dimension n = 6. Accordingto the table its Wilson coefficient should therefore scale as 1/M2

W ,which is indeed the case. Of course, to get the exact value GF =

14√

2g2

M2W

it is necessary to perform the matching computation we

did earlier, but for a crude order of magnitude estimate the aboveclassification is sufficient.

Except for the mass term of the Higgs field, all operators in theStandard Model are n = 4. These marginal operators includethe gauge field couplings (with coefficients g, g′, and gs), Yukawaterms Y, and the self-interaction of the Higgs field governed by λ,and they are not very sensitive to contributions from heavy parti-cles. The dependence only arises in loop matching diagrams andis logarithmic. If there are additional heavy particles in the funda-mental theory, they will generate operators of any dimension oncethey are integrated out. One should therefore add to the StandardModel Lagrangian also irrelevant operators of dimension n > 4.These include four-fermion and many other operators at dimensionn = 6 (in total there are 59 operators1), but only a single opera- 1 B. Grzadkowski, M. Iskrzynski,

M. Misiak, and J. Rosiek. Dimension-Six Terms in the Standard ModelLagrangian. JHEP, 10:085, 2010

tor at n = 5, namely the Weinberg term that provides Majoranamasses to the neutrinos. The fact that we have not seen any signof the additional operators seems to indicate that Λ is high. Sincethe operators are irrelevant, their contributions are suppressed by


powers of Λ. The experiments we have performed probe scales upto

∼ 2 TeV LHC direct searches,∼ 5 TeV EW precision measurements ,∼ 100 TeV flavor physics, FCNC.

These number have to be taken with a grain of salt. It could forexample be that the BSM physics is at a lower scale but is such thatit does not induce large FCNCs. Also, the values of the scales Λfor flavor and precision measurements assume that the coefficientof the operator is a number of order one times the appropriatepower of Λ. In reality, there can be additional powers of couplingconstants, as is the case of GF. If it is weakly coupled, the BSMphysics can thus arise at a much lower scale than the values givenabove. Since the direct searches have so far not yielded any results,there is now a large effort to systematically include the effects ofn = 6 operators into Standard Model predictions and to then fit fortheir Wilson coefficients.

In principle, triviality implies an upper bound on the value of Λ,but as we have seen above, the Higgs-boson mass value is such thatthis bound is very weak. In principle, it is possible that the scaleΛ could be as large as the Planck mass mPlanck = 1019 GeV, butthis now brings us to the naturalness problem. The operator Φ†Φis a relevant operator and the same arguments that show that thecoefficients of the irrelevant operators are suppressed by powers ofΛ imply that the coefficient µ2 of Φ†Φ is enhanced

µ2 ∼ M2H = c Λ2 = (125 GeV)2. (11.4)

If Λ is very large, this implies that the dimensionless constant c isvery small (e.g. c = 10−34 for Λ = mPlanck). This is unnatural: if aphysical system is governed by a scale Λ, we expect dimensionfulquantities to be of order Λ to the appropriate power, times coeffi-cients of O(1). Turning this around, we see that the low value ofMH seems to indicate BSM physics at a low scale. Indeed, natural-ness was one of the main arguments that we should expect BSMphysics at the LHC. Furthermore the BSM physics at the low scaleshould be such that it shields the Higgs mass from contributionsassociated with BSM physics at even higher scales such as mPlanck.

Often, the naturalness problem is phrased in terms of quadraticdivergences in loop integrals. When one determines the renormal-ized Higgs mass at one loop, one has

MH(µ) = MbareH + Π(0) , (11.5)

where Π(q2) is the self-energy and MH(µ) is the renormalizedmass parameter (not the physical mass determined by the positionof the pole in the propagator). Considering, for example, the contri-bution of a fermion with a Yukawa coupling Yf and mass m f to the


self energy

Π f (0) = (−1)Y2f

∫ d4k(2π)4 tr

[(/k + m f )

2

(k2 −m2f )

2

]

= −4 Y2f

∫ d4k(2π)4

[1

(k2 −m2f )

+2m2

f

(k2 −m2f )

2

]. (11.6)

The first integral on the right-hand side is quadratically divergent.If one cuts off the integral at some high scale Λ, then Π f (0) ∝ Λ2.This divergent term then has to cancel against an equally large termin Mbare

H and this is considered unnatural. However, this way ofarguing is not necessarily the most convincing, even though it leadsto the same conclusion we reached above: what is unsatisfactoryabout this form of stating the problem is that neither the bare massnor the regularized loop integral are physical. One could for exam-ple regularize the loop integral in d = 4− 2ε dimensions and thequadratic divergence would then not be visible. Rather than focus-ing on the divergence, it is better to consider the finite remainderof the integral. This remainder is proportional to m2

f . Such a correc-tion is small for Standard Model fermions, but will be large if theBSM theory includes heavy fermions. In this case, the low-energyHiggs mass will receive a large matching correction proportionalto m2

f when the particle is integrated out and it would therefore beunnatural for the low-energy Higgs mass to be small. In this way ofstating the problem, the contribution should be viewed as a match-ing correction, i.e., a contribution to the low-energy mass parameterfrom integrating out a heavy particle. The above loop integral thensimply provides an explicit example of the general statement thatthe matching correction to the Higgs mass from heavy states areproportional to the mass of the heavy state squared.

There are many BSM scenarios in the literature that proposesolutions to this problem. Here are a few selected proposals:

1. The Higgs mass could be protected by supersymmetry. In asupersymmetric theory, boson and fermion loops cancel eachother in a given supermultiplet so that one naturally obtainssmall corrections to the Higgs mass when integrating out heavyparticles. In supersymmetric theories, the cancellation of bosonicand fermionic contributions within a supermultiplet eliminatesquadratic divergences. However, it does not eliminate the finitepieces which arise when supersymmetry is broken, i.e. whenthe masses of the superpartners are heavy. The stop t, the scalarsuperpartner of the top, for example gives a contribution

Πt(0) =3Y2

t8π2 m2

t logm2

tµ2 + . . . , (11.7)

where Yt is the top Yukawa. The coupling is the same as the onefor the top, thanks to supersymmetry. For m2

t = m2t this con-

tribution cancels against the contribution of the top quark. Animportant part of the cancellation is the fact that fermion loop


comes with an extra minus sign, compared to the bosonic loop.However, if the stop is much heavier than the top quark, we endup with the usual naturalness problem. SUSY is therefore onlya viable solution to the naturalness problem if the superpart-ners are not too heavy and the LHC measurements push thesuperpartner masses to higher and higher values, which creates atension with naturalness.

2. The Higgs boson could be a pseudo Goldstone-boson of a spon-taneously broken symmetry. It could be lighter than the otherBSM particles for the same reason that the pions are lighter thanthe proton. In such a scenario, the Higgs is typically a compos-ite particle, again similar to the pion. The difficulty is that suchscenarios involve a strongly coupled gauge theory beyond theStandard Model. It is technically difficult to analyze such theo-ries and it is presumably a bad sign that so far no signs of theadditional strong dynamics have been seen. In such a scenario,one would for example expect heavy resonances, similar to theρ-meson in QCD.

3. mPlanck only appears large,

m2Planck = m2+n

Pl Rn , (11.8)

because there are n large extra dimensions, while the fundamen-tal Planck scale in (4 + n) dimensions is small, mPl ∼ 1 TeV. Thisis an interesting scenario and versions of it enjoyed a lot of popu-larity about a decade ago. However, so far no signs of large extradimensions have been found. Furthermore, one should then ex-plain why the extra dimensions are so large (1/R ∼ mPl seemsnatural, but we need 1/R mPl). This second point is addressedby Randall–Sundrum models, a different type extra dimensionalscenario.1 1 Lisa Randall and Raman Sundrum.

A Large mass hierarchy from a smallextra dimension. Phys. Rev. Lett.,83:3370–3373, 1999

4. Gravity is special, so maybe it does not introduce contributionsto the Higgs mass at the high scale Λ = mPlanck. While there isno good reason for this scenario, it is also not excluded since wedo not have a good understanding of quantum gravity. If, forsome unknown reason, the Higgs mass is small at a high scaleµ ∼ mPlanck and there is no additional BSM physics betweenthe Planck and the electroweak scale, then MH will remain smalldown to low scales since its evolution in the Standard Model isgoverned by the RG equation

µdM2

Hdµ

=3M2

H8π2

(2λ + y2

t −3g′2

4− 3g2

20

), (11.9)

which induces only logartihmic changes. The problem with thisscenario is of course that it relies on postulated special propertiesof quantum gravity.

5. In so-called Grand Unified Theories (GUT), in which the differ-ent forces of the Standard Model are unified at higher energies,


one expects new degrees of freedom that couple to quarks andleptons simultaneously (“leptoquarks”). Low-energy versionsof such scenarios have become popular recently because theycould explain BSM hints that have been observed in semileptonicB-meson decays.

It is perhaps fair to say that, while there are interesting proposalsto solve the hierarchy problem, all of them suffer from difficultiesand it is often hard to fully flesh out a given scenario, without run-ning into trouble from the fact that most known observations agreenicely with the Standard Model without signs of BSM physics. TheBSM option that has been worked out in most detail and most thor-oughly tested is low-energy supersymmetry, but is now disfavoredby LHC data. However, to decide among the different scenariosexperimental guidance will be needed! It looks very hard to figureout what theory replaces the Standard Model without experimentalobservations of BSM physics; hopefully one of the small deviationswhich are currently observed will soon solidify into a clear observa-tion of BSM physics!

11.3 Summary

The Standard Model should be viewed as a low-energy effectivetheory for a more fundamental theory, in the same way as Fermitheory is the low-energy theory for the Standard Model. Argumentsin favor of this point of view are:

• The theory does not include (quantum-)gravity.

• There are strong indications for additional dark matter, beyondthe matter content of the Standard Model.

• The Standard Model cannot explain the matter–antimatter asym-metry in the universe, so new sources of CP violation are re-quired.

• Neutrino masses point to BSM degrees of freedom (although notstrictly necessary).

• The theory is likely a trivial quantum field theory, i.e. it cannotbe defined without an energy cutoff at a very high scale.

However, the effective field theory view point also leads to a prob-lem. If there is BSM physics at a large scale Λ, one would generi-cally expect a large Higgs mass MH ∼ Λ in the Standard Model.Naturalness thus suggests BSM physics at low energies and theBSM physics should remove the sensitivity of the Higgs mass to thelarger scale, an idea that now seems disfavored by LHC data.

the standard model · electroweak gauge theory, which describes the weak and electro-magnetic...

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