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ADVANCED QUANTUM FIELD THEORY

Syllabus

•Non-Abelian gauge theories

•Higher order perturbative corrections inφ3theory

•Renormalization

•Renormalization in QED

•The renormalization group -β-functions

•Infrared and collinear singularities

•Causality, unitarity and dispersion relations.

•Anomalies

1

1 Non-Abelian Gauge Theories1.1 QED as an Abelian Gauge TheoryGauge transformationsConsider the Lagrangian density for a free Dirac fieldψ:

L=

ψ(iγμ∂μ-m)ψ(1.1)

Now this Lagrangian density is invariant under a phase transformation of the fermion field

ψ→eiωψ,

since the conjugate field

ψtransforms as

ψ→e-iωψ.

The set of all such phase transformations is called the "groupU(1)" and it is said to be "Abelian" which means that any two elements of the group commute.This just means that e iω1eiω2=eiω2eiω1. For the purposes of these lectures it will usually be sufficient to consider infinitesimal group transformations, i.e. we assume that the parameterωis sufficiently small that we can expand inωand neglect all but the linear term. Thus we write e iω= 1 +iω+O(ω2). Under such infinitesimal phase transformations the fieldψchanges byδψ, where

δψ=iω ψ,

and the conjugate field

ψbyδψ, where

ψ=-iωψ,

such that the Lagrangian density remains unchanged (to orderω). Now suppose that we wish to allow the parameterωto depend on space-time. In that case, infinitesimal transformations we have

δψ(x) =iω(x)ψ(x),(1.2)

2

δψ(x) =-iω(x)ψ(x).(1.3)

Such local (i.e. space-time dependent) transformations are called"gauge transformations". Note now that the Lagrangian density (1.1) isno longerinvariant under these transforma- tions, because of the partial derivative that is interposed between

ψandψ, which will act

on the space-time dependent parameterω(x), such that the Lagrangian density changes by an amountδL, where

δL=-

It turns out that we can repair the damage if we assume that the fermion field interacts with a vector fieldAμ, called a "gauge field", with an interaction term -e

ψ γμAμψ

added to the Lagrangian density which now becomes L= In order for this to work we must also assume that apart the fermion field transforming under a gauge transformation according to (1.2, 1.3), the gauge field,Aμ, also changes by δA

μwhere

δA

μ(x) =-1

e∂μω(x).(1.6) This change exactly cancels with eq.(1.4), so that once this interaction term has been added the gauge invariance is restored. We recognize eq.(1.5) as being the fermionic part of the Lagrangian density for QED, where eis the electric charge of the fermion andAμis the photon field. In order to have a proper Quantum Field Theory, in which we can expand the photon field, A μ, in terms of creation and annihlation operators for photons, we need a kinetic term for the field,Aμ, i.e. a term which is quadratic in the derivative of the field. We need to ensure that in introducing such a term we do not spoil the invariance under gauge transformations. This is achieved by defining the field strength,Fμνas F It is easy to see that under the gauge transformation (1.6) each of the two terms on the R.H.S. of eq.(1.7) changes, but the changes cancel out. Thus we may add to the Lagrangian any term which depends onFμν(and which is Lorentz invariant - so we must contract all Lorentz indices). Such a term isaFμνFμν, which gives the desired term which is quadratic in the derivative of the fieldAμ, and furthermore if we choose the constantato be-1 4then the Lagrange equations of motion match exactly the (relativistic formulation) of Maxwell"s equations.

†The determination of this constantais theonlyplace that a match to QED has been used. The rest of

the Lagrangian density is obtained purely from the requirement of localU(1) invariance. 3 We have thus arrived at the Lagrangian density for QED, but from the viewpoint of de- manding invariance underU(1) gauge transformations rather than starting with Maxwell"s equations and formulating the equivalent Quantum Field Theory.

The Lagrangian density is:

L=-1 Note that we arenotallowed to add a mass term for the photon. A term such asM2AμAμ added to the Lagrangian density is not invariant under gauge transformations, but would give us a transformation

δL=-2M2

eAμ(x)∂μω(x). Thus the masslessness of the photon can be understood in terms of the requirement that the

Lagrangian be gauge invariant.

Covariant derivatives

It is useful to introduce the concept of a "covariant derivative".This is not essential for Abelian gauge theories, but will be an invaluable tool when we extend these ideas to non-

Abelian gauge theories.

The covariant derivativeDμis defined to be

D

μ=∂μ+ieAμ.(1.9)

This has the property that given the transformations of the fermion field (1.2) and the gauge field (1.6) the quantityDμψis tansforming the same way (covariantly) asψunder gauge transformation: δD

μψ=iω(x)Dμψ

We may thus rewrite the QED Lagrangian density as

L=-1 Furthermore the field strengthFμνcan be expressed in terms of the commmutator of two covariant derivatives, i.e. F

μν=-i

e[Dμ,Dν] =-ie[∂μ,∂ν]+[∂μ,Aν]+[Aμ,∂ν]+ie[Aμ,Aν] =∂μAν-∂νAμ(1.11)

4

1.2 Non-Abelian gauge transformationsWe now move on to apply the ideas of the previous lecture to the casewhere the transfor-

mations are "non-Abelian", i.e. different elements of the group do not commute with each other. As an example we take use isospin, although this can easily be extended to other Lie groups. The fermion field,ψi, now carries an indexi, which takes the value 1 if the fermion is a u-type quark and 2 of the fermion is ad-type quark. The conjugate field is written

ψi.

The Lagrangian density for a free isodoublet is

L=

ψi(iγμ∂μ-m)ψi,(1.12)

where the indexiis summed over 1 and 2. Eq.(1.12) is therefore shorthand forquotesdbs_dbs7.pdfusesText_5

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