Advanced Quantum Field Theory

Roberto Casalbuoni

Dipartimento di Fisica

Universita di Firenze

Lectures given at the Florence University during the academic year

Contents

Index

Notations and Conventions

Units Relativity and Tensors The Noethers theorem for relativistic elds Field Quantization

The Real Scalar Field The Charged Scalar Field The Dirac Field The Electromagnetic Field

Perturbation Theory The Scattering Matrix Wicks theorem Feynman diagrams in momentum space The cross section

Oneloop renormalization

Divergences of the Feynman integrals Higher order corrections The analysis by counterterms Dimensional regularization of the Feynman integrals Integration in arbitrary dimensions One loop regularization of QED One loop renormalization

Vacuum expectation values and the S matrix

In and Out states The S matrix The reduction formalism

Path integral formulation of quantum mechanics

Feynmans formulation of quantum mechanics Path integral in conguration space

The physical interpretation of the path integral The free particle The case of a quadratic action Functional formalism General properties of the path integral The generating functional of the Greens functions The Greens functions for the harmonic oscillator

The path integral in eld theory

The path integral for a free scalar eld The generating functional of the connected Greens functions The perturbative expansion for the theory The Feynmans rules in momentum space Power counting in Regularization in Renormalization in the theory

The renormalization group

The renormalization group equations Renormalization conditions Application to QED Properties of the renormalization group equations The coecients of the renormalization group equations and the renor

malization conditions

The path integral for Fermi elds

Fermionic oscillators and Grassmann algebras Integration over Grassmann variables The path integral for the fermionic theories

The quantization of the gauge elds

QED as a gauge theory Non abelian gauge theories Path integral quantization of the gauge theories Path integral quantization of QED Path integral quantization of the non abelian gauge theories The function in non abelian gauge theories

Spontaneous symmetry breaking

The linear model Spontaneous symmetry breaking The Goldstone theorem The Higgs mechanism Quantization of a spontaneously broken gauge theory in the R gauge

cancellation in perturbation theory

The Standard model of the electroweak interactions

The Standard Model of the electroweak interactions The Higgs sector in the Standard Model The electroweak interactions of quarks and the Kobayashi Maskawa

Cabibbo matrix The parameters of the SM

A

A Properties of the real antisymmetric matrices

Chapter

Notations and Conventions

Units

In quantum relativistic theories the two fundamental constants c e h the lightvelocity and the Planck constant respectively appear everywhere Therefore it isconvenient to choose a unit system where their numerical value is given by

c h

For the electromagnetism we will use the Heaviside Lorentz system where we takealso

From the relation c it follows

In these units the Coulomb force is given by

jF j ee

jx xj

and the Maxwell equations appear without any visible constant For instance thegauss law is

r E

dove is the charge density The dimensionless ne structure constant

e

hc

is given by

e

Any physical quantity can be expressed equivalently by using as fundamentalunit energy mass lenght or time in an equivalent fashion In fact from our choicethe following equivalence relations follow

ct time lenght

E mc energy mass

E pv energy momentum

Et h energy time lenght

In practice it is enough to notice that the product ch has dimensions E Therefore

ch mt sec J sec J mt

Recalling that eV e J

it follows

ch MeV mt MeV fermi

From which MeV fermi

Using this relation we can easily convert a quantity given in MeV the typical unitused in elementary particle physics in fermi For instance using the fact that alsothe elementary particle masses are usually given in MeV the wave lenght of anelectron is given by

eCompton

me

MeV MeV fermi

MeV fermi

Therefore the approximate relation to keep in mind is MeV fermi Fur thermore using

c fermi sec

we get fermi sec

and MeV sec

Also using barn cm

it follows from GeV mbarn

Relativity and Tensors

Our conventions are as follows the metric tensor g is diagonal with eigenvalues The position and momentum four vectors are given by

x t x p E p

where x and p are the three dimensional position and momentum The scalar prod uct between two four vectors is given by

a b abg ab ab ab a b

where the indices have been lowered by

a ga

and can be raised by using the inverse metric tensor g g The four gradientis dened as

x

t

x

t r

The four momentum operator in position space is

p i

x itir

The following relations will be useful

p pp

x

x

x p Et p x

The Noethers theorem for relativistic elds

We will now review the Noethers theorem This allows to relate symmetries of theaction with conserved quantities More precisely given a transformation involvingboth the elds and the coordinates if it happens that the action is invariant underthis transformation then a conservation law follows When the transformationsare limited to the elds one speaks about internal transformations When bothtypes of transformations are involved the total variation of a local quantity F xthat is a function of the space time point is given by

F x F x F x F x x F x

F x F x xF x

x

The total variation keeps into account both the variation of the reference frameand the form variation of F It is then convenient to dene a local variation F depending only on the form variation

F x F x F x

Then we get

F x F x xF x

x

Let us now start form a generic four dimensional action

S

ZV

dx Li x i N

and let us consider a generic variation of the elds and of the coordinates x x x

ix ix ix ix x

x

If the action is invariant under the transformation then

SV SV

This gives rise to the conservation equation

Lx L

ii x

Li

i

This is the general result expressing the local conservation of the quantity in paren thesis According to the choice one does for the variations x and i and ofthe corresponding symmetries of the action one gets dierent kind of conservedquantities

Let us start with an action invariant under space and time translations In thecase we take x a with a independent on x e i From the general resultin eq we get the following local conservation law

T

Li

i Lg T

T is called the energy momentum tensor of the system From its local conservationwe get four constant of motion

P

ZdxT

P is the four momentum of the system In the case of internal symmetries we takex The conserved current will be

J Li

i Li

i J

with an associated constant of motion given by

Q

ZdxJ

In general if the system has more that one internal symmetry we may have morethat one conserved charge Q that is we have a conserved charge for any

The last case we will consider is the invariance with respect to Lorentz transfor mations Let us recall that they are dened as the transformations leaving invariantthe norm of a four vector

x x

For an innitesimal transformation

x x x x

with

We see that the number of independent parameters characterizing a Lorentz trans formation is six As well known three of them correspond to spatial rotationswhereas the remaining three correspond to Lorentz boosts In general the relativis tic elds are chosen to belong to a representation of the Lorentz group for instancethe Klein Gordon eld belongs to the scalar representation This means that undera Lorentz transformation the components of the eld mix together as for instancea vector eld does under rotations Therefore the transformation law of the eldsi under an innitesimal Lorentz transformation can be written as

i

ij

j

where we have required that the transformation of the elds is of rst order in theLorentz parameters The coecients antisymmetric in the indices dene a matrix in the indices i j which can be shown to be the representative ofthe innitesimal generators of the Lorentz group in the eld representation Usingthis equation and the expression for x we get the local conservation law

Li

i Lgx

Li

ij

j

T x T

x

Li

ij

j

and dening watch at the change of sign

M xT

xT

Li

ij

j

it follows the existence of six locally conserved currents one for each Lorentz trans formation

M

and consequently six constants of motion notice that the lower indices are antisym metric

M

ZdxM

Three of these constants the ones with and assuming spatial values are nothingbut the components of the angular momentum of the eld

Field Quantization

The quantization procedure for a eld goes through the following steps

construction of the lagrangian density and determination of the canonical mo mentum density

x L

x

where the index describes both the spin and internal degrees of freedom

quantization through the requirement of the equal time canonical commutationrelations for a spin integer eld or through canonical anticommutation relationsfor half integer elds

x ty t ix y

x t y t x ty t

In the case of a free eld or for an interacting eld in the interaction picture andtherefore evolving with the free hamiltonian we can add the following steps

expansion of x in terms of a complete set of solutions of the Klein Gordonequation allowing the denition of creation and annihilation operators

construction of the Fock space through the creation and annihilation operators

In the following we will give the main results about the quantization of the scalarthe Dirac and the electromagnetic eld

The Real Scalar Field

The relativistic real scalar free eld x obeys the Klein Gordon equation

mx

which can be derived from the variation of the action

S

ZdxL

where L is the lagrangian density

L

m

giving rise to the canonical momentum

L

From this we get the equal time canonical commutation relations

x t y t ix y x t y t x t y t

By using the box normalization a complete set of solutions of the Klein Gordonequation is given by

fkx pV

pk

eikx f k x pV

pk

eikx

corresponding respectively to positive and negative energies with

k k

qjkj m

andk

Ln

withn ni ni ni ni ZZ

and L being the side of the quantization box V L For any two solutions f andg of the Klein Gordon equation the following quantity is a constant of motion anddenes a scalar product not positive denite

hf jgi i

Zdxf t g

wheref t g f g f g

The solutions of eq are orthogonal with respect to this scalar product forinstance

hfkjfki kk Yi

nini

We can go to the continuum normalization by the substitution

pV p

and sending the Kr onecker delta function into the Dirac delta function The expan sion of the eld in terms of these solutions is then in the continuum

x

Zdkfkxa

k f k xayk

or more explicitly

x

Zdk

pk

akeikx aykeikx

using the orthogonality properties of the solutions one can invert this expression

ak i

Zdxf k x

t x ayk i

Zdxx

t fkx

and obtain the commutation relations

ak ayk k k

ak ak ayk ayk

From these equations one constructs the Fock space starting from the vacuum statewhich is dened by

akji

for any k The statesayk ayknji

describe n Bose particles of equal mass m ki and of total four momentumk k kn This follows by recalling the expression for the energy momentum tensor that for the Klein Gordon eld is

T

mg

from which

P H

ZdxT

Zdx

jrj m

P i

ZdxT i

Zdx

xi P

Zdxr

and for the normal ordered operator P we get the normal ordering is dened byputting all the creation operators to the left of the annihilation operators

P

Zdk ka

ykak

implying P aykji ka

ykji

The propagator for the scalar eld is given by

hjT xyji iF x y

where

F x Z

dk

eikxF k

and

F k

k m i

The Charged Scalar Field

The charged scalar eld can be described in terms of two real scalar elds of equalmass

L

Xi

i

imi

and we can write immediately the canonical commutation relations

ix t jy t iijx y

ix t jy t ix t jy t

The charged eld is better understood in a complex basis

p i y

p i

The lagrangian density becomes

L ymy

The theory is invariant under the phase transformation ei and therefore itadmits the conserved current see eq

j i

y y

with the corresponding constant of motion

Q i

Zdx yt

The commutation relations in the new basis are given by

x t yy t ix y

and

x t y t yx t yy t

x t y t yx t yy t

Let us notice that these commutation relations could have also been obtained directlyfrom the lagrangian since

L

y y L y

Using the expansion for the real elds

ix

Zdk

hfkxai

k f k xayik

i

we get

x

Zdk

fkx

pak iak f k x

payk iayk

Introducing the combinations

ak pak iak bk

pak iak

it follows

x R

dkhfkxa

k f k xbyk

iyx

Rdk

hfkxb

k f k xayk

i

from which we can evaluate the commutation relations for the creation and annihi lation operators in the complex basis

ak ayk bk byk k k

ak bk ak byk

We get also

P

Zdkk

Xi

ayikaik Z

dkk

haykak bykbk

i

Therefore the operators ayk e byk both create particles states with momentumk as the original operators ayi For the normal ordered charge operator we get

Q

Zdk

haykak bykbk

i

showing explicitly that ay and by create particles of charge and respectivelyThe only non vanishing propagator for the scalar eld is given by

hjT xyyji iF x y

The Dirac Field

The Dirac eld is described by the action

S

ZV

dx !i" m

where we have used the following notations for four vectors contracted with the matrices

"v v

The canonical momenta result to be

L

iy y L y

The canonical momenta do not depend on the velocities In principle this createsa problem in going to the hamiltonian formalism In fact a rigorous treatmentrequires an extension of the classical hamiltonian approach which was performed byDirac himself In this particular case the result one gets is the same as proceedingin a naive way For this reason we will avoid to describe this extension and wewill proceed as in the standard case From the general expression for the energymomentum tensor see eq we get

T i ! g

!i" m

and using the Dirac equationT i !

The momentum of the eld is given by

P k

Zdx T k P i

Zdx yr

and the hamiltonian

H

Zdx T H i

Zdx yt

For a Lorentz transformation the eld variation is dened by

i

ij

j

In the Dirac case one has

x x x i

x

from which

i

Therefore the angular momentum density see eq is

M i !

x x i

i !

x x

By taking the spatial components we obtain

J MMM

Zdx y

ix r

where is the identity matrix in dimensions and we have dened

The expression of J shows the decomposition of the total angular momentum inthe orbital and in the spin part The theory has a further conserved quantity the

current ! resulting from the phase invariance eiThe decomposition of the Dirac eld in plane waves is obtained by using the

spinors upn e vpn solutions of the Dirac equation in momentum space forpositive and negative energies respectively We write

x Xn

Zd pp

rm

Ep

hbp nup neipx dyp nvp neipx

i

yx Xn

Zd pp

rm

Ep

hdp n!vp neipx byp n!up neipx

i

where Ep

pjpj m We will collect here the various properties of the spinorsThe four vector n is the one that identies the direction of the spin quantization inthe rest frame For instance by quantizing the spin along the third axis one takesn in the rest frame as nR Then n is obtained by boosting from therest frame to the frame where the particle has four momentum p

Dirac equation

"pmup n !up n"pm

"pmvp n !vp n"pm

Orthogonality

!up nup n !vp nvp n nn

uyp nup n vyp nvp n Ep

mnn

!vp nup n vyp nup n

where if p Ep p then p Epp Completeness

Xn

up n!up n "pm

mXn

vp n!vp n "pm

m

The Dirac eld is quantized by canonical anticommutation relations in order tosatisfy the Pauli principle

x t y t yx t yy t

andx t y t ix y

or x t yy t

x y

The normal ordered four momentum is given by

P Xn

Zdp pb

yp nbp n dyp ndp n

If we couple the Dirac eld to the electromagnetism through the minimal substitu tion we nd that the free action becomes

S

ZV

dx !i" e "Am

Therefore the electromagnetic eld is coupled to the conserved current

j e !

This forces us to say that the integral of the fourth component of the current shouldbe the charge operator In fact we nd

Q e

Zdx y

Xn

Zdp e

byp nbp n dyp ndp n

The expressions for the momentum angular momentum and charge operators showthat the operators byp and dyp create out of the vacuum particles of spin four momentum p and charge e and e respectively

The propagator for the Dirac eld is given by

hjT x !yji iSF x y

where

SF x i" mF x

Zdk

eikxSF k

and

SF k "k m

k m i

"k m i

The Electromagnetic Field

The lagrangian density for the free electromagnetic eld expressed in terms of thefour vector potential is

L

FF

whereF A A

and

E rA A

t B r A

from whichE F F F B F F F

The resulting equations of motion are

A A

The potentials are dened up to a gauge transformation

Ax Ax Ax x

In fact A and A satisfy the same equations of motion and give rise to the same

electromagnetic eld It is possible to use the gauge invariance to require some

particular condition on the eld A For instance we can perform a gauge transfor mation in such a way that the transformed eld satises

A

This is called the Lorentz gauge Or we can choose a gauge such that

A r A

This is called the Coulomb gauge Since in this gauge all the gauge freedom is com pletely xed in contrast with the Lorentz gauge it is easy to count the independentdegrees of freedom of the electromagnetic eld From the equations we seethat A has only two degrees of freedom Another way of showing that there areonly two independent degrees of freedom is through the equations of motion Letus consider the four dimensional Fourier transform of Ax

Ax

Zdk eikxAk

Substituting this expression in the equations of motion we get

kAk kkAk

Let us now decompose Ak in terms of four independent four vectors which can

be chosen as k Ek k Ek and two further four vectors ek orthogonal to k

ke

The decomposition of Ak reads

Ak ake bkk ckk

From the equations of motion we get

kae bk ck kbk ck k

The term in bk cancels therefore it is left undetermined by the equations of motionFor the other quantities we have

kak ck

The arbitrariness of bk is a consequence of the gauge invariance In fact if wegauge transform Ax

Ax Ax x

thenAk Ak ikk

where

x

Zdk eikxk

Since the gauge transformation amounts to a translation in bk by an arbitraryfunction of k we can always to choose it equal to zero Therefore we are left with thetwo degrees of freedom described by the amplitudes ak Furthermorethese amplitudes are dierent from zero only if the dispersion relation k issatised This shows that the corresponding quanta will have zero mass With thechoice bk the eld Ak becomes

Ak akek

showing that kAk Therefore the choice bk is equivalent to the choiceof the Lorentz gauge

Let us consider now the quantization of this theory Due to the lack of manifestcovariance of the Coulomb gauge we will discuss here the quantization in the Lorentzgauge where however we will encounter other problems If we want to maintain theexplicit covariance of the theory we have to require non trivial commutation relationsfor all the component of the eld That is

Ax ty t ig

x y

Ax t Ay t x ty t

with

L A

To evaluate the conjugated momenta is better to write the lagrangian density seeeq in the following form

L

A AA

A

AA

AA

ThereforeLA

A A F

implying

L A

F

It follows

L A

We see that it is impossible to satisfy the condition

Ax ty t ix y

We can try to nd a solution to this problem modifying the lagrangian density insuch a way that But doing so we will not recover the Maxwell equationHowever we can take advantage of the gauge symmetry modifying the lagrangiandensity in such a way to recover the equations of motion in a particular gauge Forinstance in the Lorentz gauge we have

Ax

and this equation can be obtained by the lagrangian density

L

AA

just think to the Klein Gordon case The minus sign is necessary to recover apositive hamiltonian density We now express this lagrangian density in terms ofthe gauge invariant one given in eq To this end we observe that thedierence between the two lagrangian densities is nothing but the second term ofeq

AA

AA

AA

AA

AA

A

Then up to a four divergence we can write the new lagrangian density in the form

L

FF

A

One can check that this form gives the correct equations of motion In fact from

LA

A A gALA

we get A A A A

the term

A

which is not gauge invariant is called the gauge xing term More generally wecould add to the original lagrangian density a term of the form

A

The corresponding equations of motion would be

A A

In the following we will use From eq we see that

L A

A

In the Lorentz gauge we nd again To avoid the corresponding problemwe can ask that A does not hold as an operator condition but rather as acondition upon the physical states

hphysjAjphysi

The price to pay to quantize the theory in a covariant way is to work in a Hilbertspace much bigger than the physical one The physical states span a subspace whichis dened by the previous relation A further bonus is that in this way one has todo with local commutation relations On the contrary in the Coulomb gauge oneneeds to introduce non local commutation relations for the canonical variables Wewill come back later to the condition

Since we dont have to worry any more about the operator condition wecan proceed with our program of canonical quantization The canonical momentumdensities are

L A

F gA

or explicitly

A A r Ai iA Ai Ai iA

Since the spatial gradient of the eld commutes with the eld itself at equal timethe canonical commutator gives rise to

Ax t Ay t igx y

To get the quanta of the eld we look for plane wave solutions of the wave equationWe need four independent four vectors in order to expand the solutions in themomentum space In a given frame let us consider the unit four vector whichdenes the time axis This must be a time like vector n and we will choosen For instance n Then we take two four vectors

in

the plane orthogonal to n and k Notice that now k since we are consideringsolutions of the wave equation Therefore

k n

The four vectors being orthogonal to n are space like then they will be chosen

orthogonal and normalized in the following way

Next we dene a unit space like four vector orthogonal to n and lying in the planek n

n

with

By construction is orthogonal to This four vector is completely xed bythe previous conditions and we get

k n kn

n k

As a last unit four vector we choose n

n

These four vectors are orthonormal

g

and linearly independent Then they satisfy the completeness relation

g g

In the frame where n and k k k we have

The plane wave expansion of A is

Ax

Zdkp

k

X

khake

ikx aykeikx

i

where we have included the hermiticity condition for Ax For any xed thisexpansion is the same as the one that we wrote for the Klein Gordon eld with thesubstitution

ak ak Then from eq

kak i

Zdx f k x

t Ax

with the functions fkx dened as in eq Using the orthogonality of the

s we nd

ak ig

Zdx

kf k xt Ax

and analogously

ayk ig

Zdx

kAxt fkx

From these expressions we can evaluate the commutator

ak ayk

Zdx dy

h f k x

fkyigg

kkgx y

f k xfky

iggkkgx y

i

Zdx f k xi

t fkxg

and using the orthogonality relations for the functions fk

ak ayk

gk k

Analogouslyak ak

ayk ayk

The commutation rules we have derived for the operators ak create some problemLet us consider a one particle state

j i Z

dk fkaykji

its norm is given by

h j i

Zdk dk f kfkhjakaykji

Zdk dk f kfkhjak aykji

gZ

dk jfkj

Therefore the states with have negative norm This problem does not comeout completely unexpected In fact our expectation is that only the transversestates are physical states For the moment being we have ignored thegauge xing condition hphysjAjphysi but its meaning is that only part ofthe total Hilbert space is physical Therefore the relevant thing is to show that thestates satisfying the Lorentz condition have positive norm To discuss the gaugexing condition let us notice that formulated in the way we did being bilinear inthe states it could destroy the linearity of the Hilbert space So we will try tomodify the condition in a linear one

Ajphysi

But this would be a too strong requirement Not even the vacuum state satises itHowever if we consider the positive and negative frequency parts of the eld

A x

Zdkp

k

X

kakeikx A

x Axy

it is possible to weaken the condition and require

A xjphysi

This allows us to satisfy automatically the original requirement

hphysjA A

jphysi

To make this condition more explicit let us evaluate the four divergence of A

iA x

Zdkp

keikx

X

k kak

Using eq we get

k n k k n k

from whichak akjphysi

Notice that

ak ak ayk

ayk k k k k

Let us denote by #kn n the state with n scalar photons that is with polarization

k and with n longitudinal photons that is with polarization

k Then

the following states satisfy the condition

#mk

m$ayk ayk

m#k

These states have vanishing norm

jj#mkjj

More generally we can make the following observation Let us consider the numberoperator for scalar and longitudinal photons

N

Zdk aykak aykak

Notice the minus sign that is a consequence of the commutation relations and itensures that N has positive eigenvalues For instance

Naykji Z

dk aykak a

ykji aykji

Let us consider a physical state with a total number n of scalar and longitudinalphotons Then

hnjN jni

since a and a act in the same way on a physical state see eq It follows

nhnjni

Therefore all the physical states with a total denite number of scalar and longitu dinal photons have zero norm except for the vacuum state n Then

hnjni n

A generic physical state with zero transverse photons is a linear superposition of theprevious states

ji cjiXi

cijii

This state has a positive denite norm

hji jcj

The proof that a physical state has a positive norm can be extended to the case inwhich also transverse photons are present Of course the coecients ci appearingin the expression of a physical state are completely arbitrary but this is not goingto modify the values of the observables For instance consider the hamiltonian wehave

H

Zdx A L

Zdx

F A A A

FF

A

One can easily show the hamiltonian is given by the sum of all the degrees of freedomappearing in A

H

Zdx

Xi

Ai rAi

A

rA

Zdk k

X

aykak aykak

Since on the physical states a and a act in the same way we get

hphysjHjphysi hphysjZ

dk k

X

aykakjphysi

The generic physical state is of the form jT i ji with ji dened as in eq Since only jT i contributes to the evaluation of an observable quantity we can always choose ji proportional to ji However this does not mean thatwe are always working in the restricted physical space because in a sum over theintermediate states we need to include all the degrees of freedom This is crucial forthe explicit covariance and locality of the theory

The arbitrariness in dening the state ji has in fact a very simple interpreta tion It corresponds to add to A a four gradient that is it corresponds to performa gauge transformation Consider the following matrix element

hjAxji Xnm

c ncmhnjAxjmi

Since A change the occupation number by one unit and all the states jni havezero norm except for the state with n the only non vanishing contributionscome from n m and n m

hjAxji c chjZ

dkpk

eikx kak kakji cc

In order to satisfy the gauge condition the state ji must be of the form

ji Z

dq fqayq ayqji

and therefore

hjAxji Z

dkpk

k kc ceikxfk cc

From eqs and we have

k

k n

from whichhjAxji x

with

x

Zdkp

k

n k ic ce

ikxfk cc

It is important to notice that this gauge transformation leaves A in the Lorentzgauge since

because the momentum k inside the integral satises k From the expansionin eq one gets immediately the expression for the photon propagator

hjT AxAyji igZ

dk

eikxk i

By dening

Dx

Zdk

eikxDk

and

Dk

k i

we gethjT AxAyji igDx y

Perturbation Theory

The Scattering Matrix

In order to describe a scattering process we will assign to the vector of state acondition at t

j#i j#ii

where the state #i will be specied by assigning the set of incoming free particlesin terms of eigenstates of momentum spin and so on For instance in QED we willhave to specify how many electrons positrons and photons are in the initial stateand we will have to specify their momenta the spin projection of fermions and thepolarization of the photons The equations of motion will tell us how this stateevolves with time and it will be possible to evaluate the state at t whereideally we will detect the nal states In practice the preparation and the detectionprocesses are made at some nite times It follows that our ideal description willbe correct only if these times are much bigger than the typical interaction time ofthe scattering process Once we know # we are interested to evaluate theprobability amplitude of detecting at t a given set of free particles speciedby a vector state #f The amplitude is

Sfi h#f j#i

We will dene the S matrix as the operator that give us j#i once we knowj#i

j#i Sj#i

The amplitude Sfi is thenSfi h#f jSj#ii

Therefore Sfi is the S matrix element between free states j#ii and h#f j are calledin and out states respectively In the interaction representation dened by

j#ti eiHStj#Sti Ot eiH

StOSte

iHSt

where the index S identies the Schr odinger representation and HS is the free hamil

tonian the state vectors satisfy the Schr odinger equation with the interaction hamil tonian evaluated in the interaction picture

i

tj#ti HI j#ti

whereas the operators evolve with the free hamiltonian To evaluate the S matrixwe rst transform the Schr odinger equation in the interaction representation in anintegral equation

j#ti j#i i

Z t

dt H

Itj#ti

One can verify that this is indeed a solution and that it satises explicitly theboundary condition at t The perturbative expansion consists in evaluatingj#ti by iterating this integral equation The result in terms of ordered T productsis

S Xn

inn$

Z

dt

Z

dtn T

HIt HItn

The T product of n terms means that the factors have to be written from left toright with decreasing times For instance if t t tn then

THIt HItn

HIt HItn

This result can be written in a more compact form by introducing the T orderedexponential

S TeiZ

dt HIt

This expression is a symbolic one and it is really dened by its series expansion Themotivation for introducing the T ordered exponential is that it satises the followingfactorization property

Te

Z t

t

Otdt T

e

Z t

t

OtdtTe

Z t

t

Otdt

If there are no derivative interactions we have

S TeiZ

dt HIt

Tei

Zdx Lint

It follows that if the theory is Lorentz invariant also the S matrix enjoys the sameproperty

Wicks theorem

The matrix elements of the S matrix between free particle states can be expressedas vacuum expectation values VEVs of T products These V EV s satisfy animportant theorem due to Wick that states that the T products of an arbitrarynumber of free elds the ones we have to do in the interaction representation canbe expressed as combinations of T products among two elds that is in terms ofFeynman propagators The Wicks theorem is summarized in the following equation

TeiZ

dx jxx

eiZ

dx jxx e

Zdx dy jxjyhjT xyji

where x is a free real scalar eld and jx an ordinary real function The previousformula can be easily extended to charged scalar fermionic and photon elds TheWicks theorem is then obtained by expanding both sides of this equation in powersof jx and taking the VEV of both sides Let us now expand both sides of eq in a series of jx and compare term by term We will use the simpliednotation i xi We get

T

T hjT ji

T X

ij k i hjT jkji

T X

ij k l

h ij hjT klji

hjT ijjihjT kljii

and so on By taking the VEV of these expression and recalling that the VEV ofa normal product is zero we get the Wicks theorem The T product of two eldoperators is sometimes called the contraction of the two operators Therefore toevaluate the VEV of a T product of an arbitrary number of free elds it is enoughto consider all the possible contractions of the elds appearing in the T productFor instance from the last of the previous relations we get

hjT ji X

ij k lhjT ijjihjT klji

An analogous theorem holds for the photon eld For the fermions one has toremember that the T product is dened in a slightly dierent way This gives aminus sign any time we have a permutation of the fermion elds which is odd withrespect to the original ordering As an illustration the previous formula becomes

hjT ji X

ij k lP hjT ijjihjT klji

where P is the sign of the permutation i j k l with respect to the funda mental one appearing on the right hand side More explicitly

hjT ji hjT jihjT ji hjT jihjT ji hjT jihjT ji

Feynman diagrams in momentum space

We will discuss here the Feynman rules for the scalar and spinor QED Let us recallthat the electromagnetic interaction is introduced via the minimal substitution

ieA

For instance for the charged scalar eld we get

Lfree ymy

FF

ieAy ieAmy

FF

The interaction part is given by

Lint iey

yA eAy

Therefore the electromagnetic eld is coupled to the current

j iey

y

but another interaction term appears the seagull term In the spinor case we havea simpler situation

Lfree !i" m

FF

!i" e "Am

FF

giving the interaction termLint e !A

In a typical experiment in particle physics we prepare beams of particles with def inite momentum polarization etc At the same time we measure momenta andpolarizations of the nal states For this reason it is convenient to work in a mo mentum representation The result for a generic matrix element of the S matrix canbe expressed in the following form

hf jSjii Xin

pin Xout

poutY

ext bosons

r

EV

Yext fermions

rm

V EM

Here f i and we are using the box normalization The Feynman amplitude Moften one denes a matrix T wich diers fromM by a factor i can be obtained bydrawing at a given perturbative order all the connected and topologically inequiv alent Feynmans diagrams The diagrams are obtained by combining together thegraphics elements given in Figures and The amplitudeM is given by thesum of the amplitudes associated to these diagrams after extraction of the factor

Pp in

Fermion line

Photon line

Scalar line

Fig Graphical representation for both internal and external particle lines

QED vertex

Scalar QED seagull

Fig Graphical representation for the vertices in scalar and spinor QED

At each diagram we associate an amplitude obtained by multiplying together thefollowing factors

for each ingoing and%or outgoing external photon line a factor or its com plex conjugate for outgoing photons if we consider complex polarizations asthe circular one

for each ingoing and%or outgoing external fermionic line a factor up and%or!up

for each ingoing and%or outgoing external antifermionic line a factor !vpand%or vp

for each internal scalar line a factor iF p propagator

for each internal photon line a factor igDp propagator

for each internal fermion line a factor iSF p propagator

for each vertex in scalar QED a factor iep pP

in pi P

out pfwhere p and p are the momenta of the scalar particle taking p ingoing and p

outgoing

For each seagull term in scalar QED a factor ieg

Pin pi

Pout pf

for each vertex in spinor QED a factor ieP

ingoing piP

outgoing pf

Integrate over all the internal momenta with measure dp This givesrise automatically to the conservation of the total four momentum

In a graph containing closed lines loops a factor for each fermionicloop see Fig

Fermion loop

Fig A fermion loop can appear inside any Feynman diagram attaching therest of the diagram by photonic lines

In the following we will use need also the rules for the interaction with a staticexternal em eld The correspending graphical element is given in Fig The

rule is to treat it as a normal photon except that one needs to substitute the wavefunction of the photon

EV

q

with the Fourier transform of the external eld

Aext q

Zdqeiq xAext

x

where q is the momentum in transfer at the vertex where the external eld line isattached to Also since the static external eld breaks translational invariance weloose the factor

Pin pin

Pout pout

Fig Graphical representation for an external electromagnetic eld Theupper end of the line can be attached to any fermion line

The crosssection

Let us consider a scattering process with a set of initial particles with four momentapi Ei pi which collide and produce a set of nal particles with four momentapf Ef pf From the rules of the previous Chapter we know that each externalphoton line contributes with a factor V E whereas each external fermionicline contributes with mV E Furthermore the conservation of the total fourmomentum gives a term

Pi pi

Pf pf If we exclude the uninteresting

case f i we can write

Sfi

Xi

pi Xf

pf

Yfermioni

m

V E

Ybosoni

V E

M

where M is the Feynman amplitude which can be evaluated by using the rules ofthe previous Section

Let us consider the typical case of a two particle collision giving rise to an Nparticles nal state Therefore the probability per unit time of the transition isgiven by w jSfij with

w V Pf PiYi

V Ei

Yf

V Ef

Yfermioni

mjMj

where for reasons of convenience we have written

m

V E m

V E

w gives the probability per unit time of a transition toward a state with well denedquantum numbers but we are rather interested to the nal states having momentabetween pf and pf dpf Since in the volume V the momentum is given by p nL the number of nal states is given by

L

dp

The cross section is dened as the probability per unit time divided by the &ux ofthe ingoing particles and has the dimensions of a length to the square

t t

In fact the &ux is dened as vrel vrelV since in our normalization we have aparticle in the volume V and vrel is the relative velocity of the ingoing particlesFor the bosons this follows from the normalization condition of the functions fkxFor the fermions recall that in the box normalization the wave function isr

m

EVupeipx

from which ZV

dxx

ZV

dxm

V Euypup

Then the cross section for getting the nal states with momenta between pf e pfdpfis given by

d wV

vreldNF w

V

vrel

Yf

V dpf

We obtain

d V

vrel

Yf

V dpf

V Pf Pi

V EE

Yf

V Ef

Yfermioni

mjMj

Pf Pi

EEvrel

Yfermioni

mYf

dpfEf

jMj

Notice that the dependence on the quantization volume V disappears as it shouldbe in the nal equation Furthermore the total cross section which is obtained byintegrating the previous expression over all the nal momenta is Lorentz invariant

In fact as it follows from the Feynman rulesM is invariant as the factors dpEFurthermore

vrel v v pE

pE

but in the frame where the particle is at rest laboratory frame we have p m and vrel v from which

EEjvrelj EmjpjE

mjpj m

qE m

qm

E m

m

qp p m

m

We see that also this factor is Lorentz invariant

Chapter

Oneloop renormalization

Divergences of the Feynman integrals

Let us consider the Coulomb scattering If we expand the S matrix up to thethird order in the electric charge and we assume that the external eld is weakenough such to take only the rst order we can easily see that the relevant Feynmandiagrams are the ones of Fig

p’−p

p’p=

p’−p

p’p

p’−p

p’

p

p−k

k

p’−p

p’

p’−k

k

p

p’−p

k k+q

p’p

p’−p

p’−k

kp’

p−κ

pa) b)

c) d)

Fig The Feynman diagram for the Coulomb scattering at the third orderin the electric charge and at the rst order in the external eld

The Coulomb scattering can be used in principle to dene the physical electriccharge of the electron This is done assuming that the amplitude is linear in ephysfrom which we get an expansion of the type

ephys e ae e ae

in terms of the parameter e which appears in the original lagrangian The rstproblem we encounter is that we would like to have the results of our calculation interms of measured quantities as ephys This could be done by inverting the previousexpansion but and here comes the second problem the coecient of the expan sion are divergent quantities To show this consider for instance the self energycontribution to one of the external photons as the one in Fig a We have

Ma !upieAext p p

i

"pm i

iep

up

where

iep

Zdk

ie ig

k i

i

"p "k m iie

eZ

dk

k i

"p "k m i

or

p i

Zdk

"p "k m

p k m i

k i

For large momentum k the integrand behaves as k and the integral divergeslinearly Analogously one can check that all the other third order contributionsdiverge Let us write explicitly the amplitudes for the other diagrams

Mb !upiepi

"p m iieAext

p pup

Mc !upie igq i

ieqAext p pup q p p

where

ieq Z

dk

Tr

i

"k "q m iie i

"k m iie

the minus sign originates from the fermion loop and therefore

q i

Zdk

Tr

"k "q m

"k m i

The last contribution is

Md !upieep pupAext p p

where

ep p Z

dk

ie i

"p "k m i

i

"p "k m iie ig

k i

or

p p iZ

dk

"p "k m i

"p "k m i

k i

The problem of the divergences is a serious one and in order to give some sense tothe theory we have to dene a way to dene our integrals This is what is called theregularization procedure of the Feynman integrals That is we give a prescription inorder to make the integrals nite This can be done in various ways as introducingan ultraviolet cut o or as we shall see later by the more convenient means ofdimensional regularization However we want that the theory does not depend onthe way in which we dene the integrals otherwise we would have to look for somephysical meaning of the regularization procedure we choose This bring us to theother problem the inversion of eq Since now the coecients are nite we canindeed perform the inversion and obtaining e as a function of ephys and obtain allthe observables in terms of the physical electric charge that is the one measured inthe Coulomb scattering By doing so a priori we will introduce in the observablesa dependence on the renormalization procedure We will say that the theory isrenormalizable when this dependence cancels out Thinking to the regularization interms of a cut o this means that considering the observable quantities in terms ofephys and removing the cut o that is by taking the limit for the cut o going tothe innity the result should be nite Of course this cancellation is not obviousat all and in fact in most of the theories this does not happen However there isa restrict class of renormalizable theories as for instance QED We will not discussthe renormalization at all order and neither we will prove which criteria a theoryshould satisfy in order to be renormalizable We will give these criteria withouta proof but we will try only to justify them On a physical basis As far QED isconcerned we will study in detail the renormalization at one loop

Higher order corrections

In this section we will illustrate in a simplied context the previous considerationsConsider again the Coulomb scattering At the lowest order the Feynman amplitudein presence of an external static electromagnetic eld is simply given by the followingexpression see Fig f i

Sfi Ef Eim

V EM

whereM !upieupAext

p p

and

Aext q

Zdx Aext

xeiq x

is the Fourier transform of the static em eld In the case of a point like sourceof charge Ze we have

Aext x Ze

jxj

For the following it is more convenient to re express the previous amplitude in termsof the external current generating the external em eld Since we have

A j

we can invert this equation in momentum space by getting

Aq gq i

jq

from which

M !upieup ig

q iijext p p

withjext x Zex

From these expressions one can get easily the dierential cross section dd' Thisquantity could be measured and then we could compare the result with the theoret ical result obtained from the previous expression In this way we get a measure ofthe electric charge However the result is tied to a theoretical equation evaluated atthe lowest order in e In order to get a better determination one has to evaluate thehigher order corrections and again extract the value of e from the data The lowestorder result is at the order e in the electric charge the next to leading contribu tions ar at the order e For instance the diagram in Fig c gives one of thesecontribution This is usually referred to as the vacuum polarization contributionand in this Section we will consider only this correction just to exemplify how therenormalization procedure works This contribution is given in eqs and The total result is

M Mc !upieup igq i

igq i

ieigq i

ijxt q

The sum gives a result very similar the to one obtained from M alone but withthe photon propagator modied in the following way

igq

igq

igq

ieqigq

igq

ie

q

As discussed in the previous Section is a divergent quantity In fact at highvalues of the momentum it behaves asZ

dp

p

and therefore it diverges in a quadratic way This follows by evaluating the integralby cutting the upper limit by a cut o As we shall see later on the real divergenceis weaker due to the gauge invariance We will see that the correct behaviour is givenby Z

dp

p

The corresponding divergence is only logarithmic but in any case the integral isdivergent By evaluating it with a cut o we get

q gqIq

The terms omitted are proportional to qq and since the photon is always attachedto a conserved current they do not contribute to the nal result The functions Iqcan be evaluated by various tricks we will evaluate it explicitly in the followingand the result is

Iq

log

m

Z

dz z z log

qz z

m

For small values of q q m one can expand the logarithm obtaining

Iq

log

m

q

m

Summarizing the propagator undergoes the following modication

igq

igq

eIq

This relation is shown in graphical terms in Fig

+

Fig The vacuum polarization correction to the propagator

The total result is then

M Mc !upieupiq

e

log

m e

q

m

iZe

in the limit q the result looks exactly as at the lowest order except that weneed to introduce a renormalized charge eR given by

eR e

e

log

m

In term of eR and taking into account that we are working at the order e we canwrite

M Mc !ufieRuiiq

eR

q

m

iZeR

Now we compare this result with the experiment where we measure dd' in thelimit q We can extract again from the data the value of the electric charge butthis time we will measure eR Notice that in this expression there are no more diver gent quantities since they have been eliminated by the denition of the renormalizedcharge It must be stressed that the quantity that is xed by the experiments isnot the (parameter( e appearing in the original lagrangian but rather eR Thismeans that the quantity e is not really dened unless we specify how to relate itto measured quantities This relation is usually referred to as the renormalizationcondition As we have noticed the nal expression we got does not contain anyexplicit divergent quantity Furthermore also eR which is obtained from the exper imental data is a nite quantity It follows that the parameter e must diverge for In other words the divergence which we have obtained in the amplitude iscompensated by the divergence implicit in e Therefore the procedure outlined herehas meaning only if we start with an innite charge Said that the renormalizationprocedure is a way of reorganizing the terms of the perturbative expansion in sucha way that the parameter controlling it is the renormalized charge eR If this ispossible the various terms in the expansion turn out to be nite and the theory issaid to be renormalizable As already stressed this is not automatically satisedIn fact renormalizability is not a generic feature of the relativistic quantum eldtheories Rather the class of renormalizable theories that is the theories for whichthe previous procedure is meaningful is quite restricted

We have seen that the measured electric charge is dierent from the quantityappearing in the expression of the vertex It will be convenient in the following to callthis quantity eB tha bare charge On the contrary the renormalized charge will becalled e rather than eR as done previously For the moment being we have includedonly the vacuum polarization corrections however if we restrict our problem to thecharge renormalization the divergent part coming from the self energy and vertexcorrections cancel out see later Let us consider now a process like e eNeglecting other corrections the total amplitude is given by Fig

We will dene the charge by choosing an arbitrary value of the square of themomentum in transfer q such that Q q In this way the renormalizedcharge depends on the scale In the previous example we were using q Thegraphs in Fig sum up to give omitting the contribution of the fermionic

+ +

e B

e B

e B

e B

e B

eB

e B

e B

e B

e B

e B

e B

µ−

e−e−

e−

µ−µ−

q +

Fig The vacuum polarization contributions to the e e scattering

lines

eB

higq

igq

ieBigq

igq

ieBigq

ieB igq

iQ

By considering only the g contribution to we get

eBigq

eBIq

eBIq

Q

This expression looks as the lowest order contribution to the propagator be deningthe renormalized electric charge as

e eB eBI eBI

Since the fermionic contribution is the same for all the diagrams by calling it F Qwe get the amplitude in the form

MeB eBF Q eBIQ eBI

Q

where eBF Q is the amplitude at the lowest order in eB As we know IQ is adivergent quantity and therefore the amplitude is not dened However followingthe discussion of the previous Section we can try to re express everything in termsof the renormalized charge We can easily invert by series the eq obtaining at the order e

eB e eI

By substituting this expression in Me we get

MeB e eI F Q eIQ eF Q eIQ I MRe

From the expression for IQ we get

IQ

log

m

Z

dz z z log

Qz z

m

Since the divergent part is momentum independent it cancels out in the expressionfor MRe Furthermore the renormalized charge e will be measured through thecross section and therefore by denition it will be nite Therefore the chargerenormalization makes the amplitude nite Evaluating IQ I per large Q

and we get

eIQ I

Z

dz z z log

m Qz z

m z z

Z

dz z z logQ

log

Q

and in this limit

MRe eF Q

log

Q

At rst sight MRe depends on but actually this is not the case In fact e isdened through the eq and therefore it depends also on What is goingon is that the implicit dependecnce of e on cancels out the explicit dependenceof MRe This follows from MRe MeB and on the observation that MeBdoes not depend on This can be formalized in a way that amounts to introducethe concept of renormalization group that we will discuss in much more detail lateron In fact let us be more explicit by writing

MeB MRe

By dierentiating this expression with respect to we get

dMeB

dMR

MR

e

e

This equation say formally that the amplitude expressed as a function of the renor malized charge is indeed independent on This equation is the prototype of therenormalization group equations and tells us how we have to change the couplingwhen we change the denition scale

Going back to the original expression for Me we see that this is given bythe tree amplitude eBF Q times a geometric series arising from the iteration of

the vacuum polarization diagram Therefore it comes natural to dene a couplingrunning with the momentum of the photon

eQ eB eBIQ eBI

Q eB eBIQ

Also this expression is cooked up in terms of the divergent quantities eB and IQbut again eQ is nite This can be shown explicitly by evaluating the renormal ized charge e in terms of the bare one

e eB

eBI

We can put the last two equations in the form

eQ

eB IQ

e

eB I

and subtracting them

eQ

e IQ I

we get

eQ e

eIQ I

By using this result we can write MRe as

MRe eQF Q

It follows that the true expansion parameter in the perturbative series is the runningcoupling eQ By using eq we obtain

Q

log

Q

We see that Q increases with Q therefore the perturbative approach loosesvalidity for those values Q of such that Q that is when

log

Q

or

Q

e

For we getQ

e

Actually Q is divergent for

Q

e

e

This value of Q is referred to as the Landau pole Therefore in the case of QED theperturbative approach maintains its validity up to gigantic values of the momentaThis discussion shows clearly that the behaviour of the running coupling constantis determined by IQ that is from the vaccum polarization contribution In par ticular Q increases with Q since in the expression eIQ I thecoecient of the logarithm is negative equal to We will see that inthe so called non abelian gauge theories the running is opposite that is the theorybecomes more and more perturbative by increasing the energy

The analysis by counterterms

The previous way of dening a renormalizable theory amounts to say that the origi nal parameters in the lagrangian as e should be innite and that their divergencesshould compensate the divergences of the Feynman diagrams Then one can try toseparate the innite from the nite part of the parameters this separation is ambigu ous see later The innite contributions are called counterterms and by denitionthey have the same operator structure of the original terms in the lagrangian Onthe other hand the procedure of regularization can be performed by adding to theoriginal lagrangian counterterms cooked in such a way that their contribution killsthe divergent part of the Feynman integrals This means that the coecients ofthese counterterms have to be innite However they can also be regularized in thesame way as the other integrals We see that the theory will be renormalizable ifthe counterterms we add to make the theory nite have the same structure of theoriginal terms in the lagrangian in fact if this is the case they can be absorbedin the original parameters which however are arbitrary because they have to bexed by the experiments renormalization conditions We will show now that atone loop the divergent contributions come only from the three diagrams discussedbefore In fact let us dene D as the supercial degree of divergence of a one loopdiagram the dierence between the number of integration over the mimentumand the number of powers of momentum coming from the propagators we assumehere that no powers of momenta are coming from the vertices which is not true inscalar QED We get

D IF IB

where IF and IB are the number of internal fermionic and bosonic lines Goingthrough the loop one has to have a number of vertices V equal to the number ofinternal lines

V IF IB

Also if at each vertex there are Ni lines of the particle of type i then we have

NiV Ii Ei

from whichV IF EF V IB EB

From these equations we can get V IF and IB in terms of EF and EB

IF

EF EB IB

EF V EF EB

from which

D

EF EB

It follows that the only one loop supercially divergent diagrams in spinor QED arethe ones in Fig In fact all the diagrams with an odd number of externalphotons vanish This is trivial at one loop since the trace of an odd number of matrices is zero In general Furrys theorem it follows from the conservation ofcharge conjugation since the photon is an eigenstate of C with eigenvalue equal to

Furthermore the eective degree of divergence can be less than the supercialdegree In fact gauge invariance often lowers the divergence For instance thediagram of Fig with four external photons light light scattering has D logarithmic divergence but gauge invariance makes it nite Also the vacuumpolarization diagram has an eective degree of divergence equal to Thereforethere are only three divergent one loop diagrams and all the divergences can bebrought back to the three functions p q e p p This does not meanthat an arbitrary diagram is not divergence but it can be made nite if the previousfunctions are such In such a case one has only to show that eliminating these threedivergences primitive divergences the theory is automatically nite In particularwe will show that the divergent part of p can be absorbed into the denitionof the mass of the electron and a redenition of the electron eld wave functionrenormalization The divergence in the photon self energy can be absorbedin the wave function renormalization of the photon the mass of the photon is notrenormalized due to the gauge invariance And nally the divergence of p

pgoes into the denition of the parameter e To realize this program we divide upthe lagrangian density in two parts one written in terms of the physical parametersthe other will contain the counterterms We will call also the original parametersand elds of the theory the bare parameters and the bare elds and we will use anindex B in order to distinguish them from the physical quantities Therefore the

E EF B D

0 2 2

0 4 0

2 0 1

2 1 0

Fig The supercially divergent oneloop diagrams in spinor QED

two pieces of the lagrangian should look like as follows the piece in terms of thephysical parametersLp

Lp !i" m e !A

FF

A

and the counter terms piece Lct

Lct iB ! " A ! C

F E

A

eD ! "A

and we have to require that la sum of these two contributions should coincide withthe original lagrangian written in terms of the bare quantities Adding together Lp

and Lct we get

L Bi ! " mA ! eD !A C

FF

gauge xing

where for sake of simplicity we have omitted the gauge xing term Dening therenormalization constant of the elds

Z D Z B Z C

we write the bare elds as

B Z A

B Z A

we obtain

L i !B "B m A

Z

!BB eZ

ZZ

!BBAB

FBF

B gauge xing

and putting

mB m A

Z eB

eZ

ZZ

we get

L i !B "B mB!BB eB !BBA

B

FBF

B gauge xing

So we have succeeded in the wanted separation Notice that the division of the pa rameters in physical and counter term part is well dened because the nite pieceis xed to be an observable quantity This requirement gives the renormalizationconditions The counter terms A B are determined recursively at each per turbative order in such a way to eliminate the divergent parts and to respect therenormalization conditions We will see later how this works in practice at one looplevel Another observation is that Z and Z have to do with the self energy of theelectron and as such they depend on the electron mass Therefore if we considerthe theory for a dierent particle as the muon which has the same interactions asthe electron and diers only for the value of the mass m me one wouldget a dierent bare electric charge for the two particles Or phrased in a dierentway one would have to tune the bare electric charge at dierent values in order toget the same physical charge This looks very unnatural but the gauge invarianceof the theory implies that at all the perturbative orders Z Z As a consequenceeB eZ

and since Z comes from the photon self energy the relation between

the bare and the physical electric charge is universal that is it does not depend onthe kind of charged particle under consideration

Summarizing one starts dividing the bare lagrangian in two pieces Then weregularize the theory giving some prescription to get nite Feynman integrals Thepart containing the counter terms is determined order by order by requiring thatthe divergences of the Feynman integrals which come about when removing theregularization are cancelled out by the counter term contributions Since the sepa ration of an innite quantity into an innite plus a nite term is not well dened weuse the renormalization conditions to x the nite part After evaluating a physicalquantity we remove the regularization Notice that although the counter terms aredivergent quantity when we remove the cut o we will order them according to thepower of the coupling in which we are doing the perturbative calculation That is wehave a double limit one in the coupling and the other in some parameter regulatorwhich denes the regularization The order of the limit is rst to work at some orderin the coupling at xed regulator and then remove the regularization

Before going into the calculations for QED we want to illustrate some generalresults about the renormalization If one considers only theory involving scalar

fermion and massless spin as the photon elds it is not dicult to constructan algorithm which allows to evaluate the ultraviolet that is for large momentadivergence of any Feynman diagram In the case of the electron self energy seeFigs a and b one has an integration over the four momentum p and abehaviour of the integrand coming from the propagators as p giving a lineardivergence it turns out that the divergence is only logarithmic From this countingone can see that only the lagrangian densities containing monomials in the eldswith mass dimension smaller or equal to the number of space time dimensions havea nite number of divergent diagrams It turns out also that these are renormalizabletheories a part some small technicalities The mass dimensions of the elds can beeasily evaluated from the observation that the action is dimensionless in our unitsh Therefore in n space time dimensions the lagrangian density dened asZ

dnx L

has a mass dimension n Looking at the kinetic terms of the bosonic elds twoderivatives and of the fermionic elds one derivative we see that

dim dimA n

dim

n

In particular in dimensions the bosonic elds have dimension and the fermionic Then we see that QED is renormalizable since all the terms in the lagrangiandensity have dimensions smaller or equal to

dim ! dim !A dimA

The condition on the dimensions of the operators appearing in the lagrangian canbe translated into a condition over the coupling constants In fact each monomialOi will appear multiplied by a coupling gi

L Xi

giOi

thereforedimgi dimOi

The renormalizability requiresdimOi

from whichdimgi

that is the couplings must have positive dimension in mass are to be dimensionlessIn QED the only couplings are the mass of the electron and the electric charge

which is dimensionless As a further example consider a single scalar eld Themost general renormalizable lagrangian density is characterized by two parameters

L

m

Here has dimension and is dimensionless We see that the linear models arerenormalizable theories

Giving these facts let us try to understand what makes renormalizable and nonrenormalizable theories dierent In the renormalizable case if we have writtenthe most general lagrangian the only divergent diagrams which appear are theones corresponding to the processes described by the operators appearing in thelagrangian Therefore adding to L the counter term

Lct Xi

giOi

we can choose the gi in such a way to cancel order by order the divergences Thetheory depends on a nite number of arbitrary parameters equal to the number ofparameters gi Therefore the theory is a predictive one In the non renormalizablecase the number of divergent diagrams increase with the perturbative order Ateach order we have to introduce new counter terms having an operator structuredierent from the original one At the end the theory will depend on an innitenumber of arbitrary parameters As an example consider a fermionic theory with aninteraction of the type ! Since this term has dimension the relative couplinghas dimension

Lint g !

At one loop the theory gives rise to the divergent diagrams of Fig Thedivergence of the rst diagram can be absorbed into a counter term of the originaltype

g !

The other two need counter terms of the type

g ! g !

These counter terms originate new one loop divergent diagrams as for instance theones in Fig The rst diagram modies the already introduced counter term ! but the second one needs a new counter term

g !

This process never endsThe renormalization requirement restricts in a fantastic way the possible eld

theories However one could think that this requirement is too technical and onecould imagine other ways of giving a meaning to lagrangians which do not satisfy this

a) b)

c)

Fig Divergent diagrams coming from the interaction !

condition But suppose we try to give a meaning to a non renormalizable lagrangiansimply requiring that it gives rise to a consistent theory at any energy We will showthat this does not happen Consider again a theory with a four fermion interactionSince dim g if we consider the scattering in the high energylimit where we can neglect all the masses on dimensional ground we get that thetotal cross section behaves like

gE

Analogously in any non renormalizable theory being there couplings with negativedimensions the cross section will increase with the energy But the cross section hasto do with the S matrix which is unitary Since a unitary matrix has eigenvalues of

a) b)

Fig Divergent diagrams coming from the interactions ! and !

modulus it follows that its matrix elements must be bounded Translating thisargument in the cross section one gets the bound

c

E

where c is some constant For the previous example we get

gE c

This implies a violation of the S matrix unitarity at energies such that

E r

c

g

It follows that we can give a meaning also to non renormalizable theories but onlyfor a limited range of values of the energy This range is xed by the value of thenon renormalizable coupling It is not diculty to realize that non renormalizabilityand bad behaviour of the amplitudes at high energies are strictly connected

Dimensional regularization of the Feynman

integrals

As we have discussed in the previous Section we need a procedure to give sense atthe otherwise divergent Feynman diagrams The simplest of these procedures is justto introduce a cut o and dene our integrals as

R lim

R

Of course

in the spirit of renormalization we have rst to perform the perturbative expansionand then take the limit over the cut o Although this procedure is very simple itresults to be inadequate in situations like gauge theories In fact one can show thatthe cut o regularization breaks the translational invariance and creates problemsin gauge theories One kind of regularization which nowadays is very much used isthe dimensional regularization This consists in considering the integration in anarbitrary number of space time dimensions and taking the limit of four dimensionsat the end This way of regularizing is very convenient because it respects all thesymmetries In fact except for very few cases the symmetries do not depend on thenumber of space time dimensions Let us what is dimensional regularization aboutWe want to evaluate integrals of the type

Ik

Zdp F p k

with F p k p or p The idea is that integrating on a lower number ofdimensions the integral improves the convergence properties in the ultraviolet For

instance if F p k p the integral is convergent in and in dimensionsTherefore we would like to introduce a quantity

I k

Zdp F p k

to be regarded as a function of the complex variable Then if we can denea complex function I k on the entire complex plane with denite singularityand as such that it coincides with I on some common domain then by analyticcontinuation I and I dene the same analytic function A simple example of thisprocedure is given by the Eulers ) This complex function is dened for Re z by the integral representation

)z

Z

dt ettz

If Re z the integral diverges as

dt

tjRe zj

in the limit t However it is easy to get a representation valid also for Re z Let us divide the integration region in two parts dened by a parameter

)z

Z

dt ettz Z

dt ettz

Expanding the exponential in the rst integral and integrating term by term we get

)z Xn

nn$

Z

dt tnz Z

dt ettz

Xn

nn$

nz

n z

Z

dt ettz

The second integral converges for any z since This expression coincides withthe representation for the ) function for Re z but it is dened also for Re z where it has simple poles located at z n Therefore it is a meaningful expressionon all the complex plane z Notice that in order to isolate the divergences we needto introduce an arbitrary parameter However the result does not depend on theparticular value of this parameter This the Weierstrass representation of the Euler)z From this example we see that we need the following three steps

Find a domain where I k is convergent Typically this will be for Re

Construct an analytic function identical to I k in the domain of conver gence but dened on a larger domain including the point

At the end of the calculation take the limit

Integration in arbitrary dimensions

Let us consider the integral

IN

ZdNpF p

with N an integer number and p a vector in an Euclidean N dimensional spaceSince the integrand is invariant under rotations of the N dimensional vector p wecan perform the angular integration by means of

dNp d'NpNdp

where d'N is the solid angle element in N dimensions ThereforeRd'N SN with

SN the surface of the unit sphere in N dimensions Then

IN SN

Z

pNF pdp

The value of the sphere surface can be evaluated by the following trick Consider

I

Z

exdx

p

By taking N of these factors we get

IN

Zdx dxNex

xN N

The same integral can be evaluated in polar coordinates

N SN

Z

Ned

By putting x we have

N

SN

Z

xNexdx

SN)

N

where we have used the representation of the Euler ) function given in the previousSection Therefore

SN N

)N

and

IN N

)N

Z

xNF xdx

with x p

The integrals we will be interested in are of the type

IMN

ZdNp

p a iA

with p a vector in a Ndimensional Minkowski space We can perform an anti clockwise rotation of Wicks rotation in the complex plane of p without hittingany singularity Then we do a change of variables p ip and we use the denitionp p PN

i pi obtaining

IMN i

ZdNp

p aA iAIN

where IN is an integral of the kind discussed at the beginning of this Section withF x given by

F x x aA

It follows

IN N

)N

Z

xN

x aAdx

By putting x ay we get

IN aNAN

)N

Z

yN yAdx

and recalling the integral representation for the Euler Bx y function valid forRex y

Bx y )x)y

)x y

Z

tx txydt

it follows

IN N)AN

)A

aAN

We have obtained this representation for N and ReA N But weknow how to extend the Euler gamma function to the entire complex plane andtherefore we can extend this formula to complex dimensions N

I )A

)A

aA

This shows that I has simple poles located at

AA

Therefore our integral will be perfectly dened at all such that AA At the end we will have to consider the limit The original integral inMinkowski space is thenZ

dp

p aA iA)A

)A

aA

For the following it will be useful to derive another formula Let us put in theprevious equation p p k and b a k thenZ

dp

p p k k aA iA)A

)A

aA

from whichZdp

p p k bA iA)A

)A

k bA

Dierentiating with respect to k we get various useful relations asZdp

pp p k bA

iA)A

)A

kk bA

and Zdp

ppp p k bA

iA

)Ak bA

)A kk

gk

b)A

Since at the end of our calculation we will have to take the limit it will beuseful to recall the expansion of the Gamma function around its poles

)

O

where

is the Euler Mascheroni constant and for n

)n nn$

n O

where

n

n

One loop regularization of QED

In this Section we will regularize by using dimensional regularization the relevantdivergent quantities in QED that is p q e

p p Furthermore in orderto dene the counter terms we will determine the expressions which become divergentin the limit of It will be also convenient to introduce a parameter suchthat these quantities have the same dimensions in the space with d as in d

The algebra of the Dirac matrices is also easily extended to arbitrary dimensions dFor instance starting from

g

we get d

and

d

Other relations can be obtained by starting from the algebraic properties of the matrices Let us start with the electron self energy which we will require to havedimension as in d From eq we have

p iZ

dk

"p "k m

p k m i

k i

In order to use the equations of the previous Section it is convenient to combinetogether the denominators of this expression into a single one This is done by usinga formula due to Feynman

ab

Z

dz

az b z

which is proven using

ab

b a

a

b

b a

Z b

a

dx

x

and doing the change of variables

x az b z

We get

p iZ

dz

Zdk

"p "k m

p kz mz k z

The denominator can be written in the following way

pz mz k p kz

and the term p k can be eliminated through the following change of variablesk k pz We nd

p mz k pz p k pzz k mz pz z

It follows we put k k

p iZ

dz

Zdk

"p z "k m

k mz pz z

The linear term in k has vanishing integral therefore

p iZ

dz

Zdk

"p z m

k mz pz z

and integrating over k

p iZ

dz

i

)

)

"p z m

mz pz z

By dening we get

p Z

dz

)

"p z m

mz pz z

Contracting the matrices

p

)

Z

dz "p z m

mz pz z

we obtain

p

)

Z

dz"p z m "p zm

mz pz z

Dening

A "p z m B "p z m C mz pz z

and expanding for

p

Z

dz

A

B A

h

logC

i

Z

dz

A

A logC B A

"p m

"p m "p m

Z

dz"p z m logmz pz z

"p m nite terms

Consider now the vacuum polarization see eq

q iZ

dk

Tr

"k "q m

"k m

iZ

dk

Tr"k m"k "q m

k mk q m

Using again the Feynman representation for the denominators

q iZ

dz

Zdk

Tr"k m"k "q m

k m z k q mz

We write the denominator as

k qz m k qz

through the change of variable k k qz we cancel the mixed term obtaining

k qz k qz qz qz m k qz zm

and therefore we put again k k

q iZ

dz

Zdk

Tr"k "qz m"k "q z m

k qz zm

Since the integral of the odd terms in k is zero it is enough to evaluate the contri bution of the even term to the trace

Treven Tr"k "qz m"k "q zpari mTr

Tr"k"k Tr"q "qz z mTr

If we dene the as matrices of dimension we can repeat the standardcalculation by obtaining a factor instead of Therefore

Treven kk gk qq gq

z z mg

kk z zqq gq gk

m qz z

and we nd

q iZ

dz

Zdk

h kkk qz zm

z zqq gq

k qz zm g

k qz zm

i

From the relations of the previous Section we getZdp

ppp a

ig)

a

whereas Zdp

p a i)

a

Therefore the rst and the third contribution to the vacuum polarization cancel outand we are left with

q iqqgqZ

dz zzZ

dk

k qz zm

Notice that the original integral was quadratically divergent but due to the previouscancellation the divergence is only logarithmic The reason is again gauge invari ance In fact it is possible to show that this implies qq Performing themomentum integration we have

q iqq gq

Z

dz z zi

)

m qz z

By putting again and expanding the previous expression

q qq gq

Z

dz z z

)

m qz z

qq gq

Z

dz z z)

m qz z

log qq gq

Z

dz z z

h

logC

i

where C is denite in eq Finally

q

qq gq

Zdz z z

log

h

logC

i

qq gq

Z

dz z z log

m qz z

or in abbreviated way

q

qq gq

nite terms

We have now to evaluate p p From eq we have

p p iZ

dk

"p "k m

p k m

"p "k m

p k m

k

the general formula to reduce n denominators to a single one is

nYi

ai n $

Z

nYi

diPn

i i

Pn

i iain

To show this equation notice that

nYi

ai

Z

nYi

die

nXi

iai

Introducing the identity

Z

dnXi

i

and changing variables i i

nYi

ai

Z

nYi

didnXi

ie

nXi

iai

Z

nYi

dind

e

nXi

iai

nXi

i

The integration over i can be restricted to the interval due to the deltafunction and furthermore Z

dne n $

n

In our case we get

p p iZ

dk

Z

dx

Z x

dy

"p "k m"p "k mk x y p kxmx p ky my

The denominator is

k mx y px py k px py

Changing variable k k px py

k px py mx y px py k px py px py

k mx y px x py y p pxy

Letting again k k

p p iZ

dx

Z x

dy

Zdk

"p y "px "k m"p x "py "k mk mx y px x py y p pxy

The odd term in k is zero after integration the term in k is logarithmically di vergent whereas the remaining part is convergent Separating the divergent piece from the convergent one

we get for the rst term

p p i

Z

dx

Z x

dy

Zdk

kk

k mx y px x py y p pxy

iZ

dx

Z x

dyi)

)

mx y px x py y p pxy

)

Z

dx

Z x

dy

mx y px x py y p pxy

Using the relation

d

we obtain

p p

)

Z

dx

Z x

dy

mx y px x py y p pxy

Z

dx

Z x

dy

mx y px x py y p pxy

Z

dx

Z x

dy

log

mx y px x py y p pxy

and nally

p p

nite terms

In the convergent part we can put directly

p p i

Z

dx

Z x

dyi)

"p y "pxm"p x "py mmx y px x py y p pxy

Z

dx

Z x

dy

"p y "pxm"p x "py mmx y px x py y p pxy

One loop renormalization

We summarize here the results of the previous Section

p

"p m f p

q qq gq

f q

qq gq

q

p p

f

p p

where the functions with the superscript f represent the nite contributions Letus start discussing the electron self energy As shown in eq the eect of pis to correct the electron propagator In fact we have see Fig

SF p i

"pm

i

"pmiep

i

"pm

Fig The loop expansion for the electron propagator

from which at the same order in the perturbative expansion

SF p i

"pm

ep

"pm

i

"pm ep

Therefore the eect of the divergent terms is to modify the coecients of "p and m

iSF p "pm ep "p

e

m

e

nite terms

This allows us to dene the counter terms to be added to the lagrangian expressedin terms of the physical parameters in such a way to cancel these divergences

Lct iB ! " A !

Lct modies the Feynman rules adding two particles interacting terms Thesecan be easily evaluated noticing that the expression of the propagator taking intoaccount Lct is

i

B"p m A i

"pm

B"p A

"pm

i

"pm

i

"pmiB"p iA

i

"pm

where consistently with our expansion we have taken only the rst order terms inA and B We can associate to these two terms the diagrams of Fig withcontributions iA to the mass term and iB"p to "p

-iA iBp/

Fig The counter terms for the selfenergy p in the gure should be read as"p

The propagator at the second order in the coupling constant is then given by addingthe diagrams of Fig At this order we get

SF p i

"pm

i

"pm

iep iB"p iA

i

"pm

and the correction to the free propagator is given by

ep B"p A

e

B

"p

e

m A

nite terms

We can now x the counter terms by choosing

B e

F

m

=

Fig The second order contributions at the electron propagator

A me

Fm

m

with F andFm nite for Notice also that these two functions are dimen sionless and arbitrary up to this moment However they can be determined by therenormalization conditions that is by xing the arbitrary constants appearing inthe lagrangian In fact given

iSF p )p "pm B"p A ep

we can require that at the physical pole "p m the propagator coincides with thefree propagator

SF p i

"pm for "p m

From here we get two conditions The rst one is

)"p m e

"pmef "p m e

F

"p

me

Fm

from which

ef"p m me

F

me

Fm

The second condition is)p

"p

pm

which gives

ef p

"p

pm

e

F

One should be careful because these particular conditions of renormalization givessome problems related to the zero mass of the photon In fact one nds some ill dened integral in the infrared region However these are harmless divergencesbecause giving a small to the photon and letting it to zero at the end of the calcula tions gives rise to nite results Notice that these conditions of renormalization havethe advantage of being expressed directly in terms of the measured parameters asthe electron mass However one could renormalize at an arbitrary mass scale M Inthis case the parameters comparing in Lp are not the directly measured parametersbut they can be correlated to the actual parameters by evaluating some observablequantity From this point of view one could avoid the problems mentioned above bychoosing a dierent point of renormalization

As far as the vacuum polarization is concerned gives rise to the followingcorrection of the photon propagator illustrated in Fig

Dq

igq

igq

ieqigq

Fig The loop expansion for the photon propagator

from which

Dq

igq

igq

ieqq gqq

igq

igq

eq

iqqq

eq

We see that the one loop propagator has a divergent part in g and also divergentand nite pieces in the term proportional to the momenta Therefore the propagatoris not any more in the Feynman gauge It follows that we should add to Lp

L

FF

A

AgA

the two counterterms

Lct CFF

E

A

C

FF

A

E C

A

As for the electron propagator we can look at these two contributions as pertur bations to the free lagrangian and evaluate the corresponding Feynman rules orevaluate the eect on the propagator and then expand in the counter terms Theeect of these two terms to the equation which denes the propagator in momentumspace is

Cqg C EqqDq ig

We solve this equation by putting

Dq qg qqq

Substituting in the previous equation we determine e The result is

i

q C i

qC E

C E

The free propagator including the corrections at the rst order in C and E is

Dq igq

C iqqq

C E

and the total propagator

Dq

igq

eq C iqqq

eq C E

We can now choose E and cancel the divergence by the choice

C e

F

m

In fact we are free to choose the nite term of the gauge xing since this choice doesnot change the physics This is because the terms proportional to qq as it followsfrom the gauge invariance and the conservation of the electromagnetic current Forinstance if we have a vertex with a virtual photon that is the vertex is connectedto an internal photon line and two external electrons the term proportional to qqis saturated with

!upup q p p

The result is zero bay taking into account the Dirac equation Let us see what thefull propagator says about the mass of the photon We have

Dq

igq

F ef

iqqq

F ef

igq

iqqq

where ef F

At the second order in the electric charge we can write the propagator in the form

Dq

iq q

g

qqq

q

and we see that the propagator has a pole at q since q is nite for q Therefore the photon remains massless after renormalization This is part of a rathergeneral aspect of renormalization which says that if the regularization procedurerespects the symmetries of the original lagrangian the symmetries are preserved atany perturbative order An exception is the case of the anomalous symmetries whichare symmetries at the classical level but are broken by the quantum correctionsAll the anomalous symmetries can be easily identied in a given theory

Also in this case we will require that at the physical pole q the propagatorhas the free form that is

from whichF ef

For small momenta we can write

q eqdfq

dq

q

since F is a constant Using the expression for f from the previous Section weget

f q

log

m

Z

dz z zq

m

and

q eq

m

from which

D ig

q

eq

m

gauge terms

The rst term q gives rise to the Coulomb potential er Therefore this ismodied by a constant term in momentum space or by a term proportional to thedelta function in the real space

eZ

dt

Zdq

eiqx x

q e

m

e

r e

mr

This modication of the Coulomb potential modies the energy levels of the hydro gen atom and it is one of the contributions to the Lamb shift which produces asplitting of the levels S and P The total Lamb shift is the sum of all theself energy and vertex corrections and turns out to be about MHz Thecontribution we have just calculated is only MHz but it is important sincethe agreement between experiment and theory is of the order of MHz

We have now to discuss the vertex corrections We have seen that the divergentcontribution is

p p and this is proportional to The counter term to add

to interacting part of Lp

Lintp e !A

isLct eD !A

The complete vertex is given by see Fig

ie e D

(counter-term)

Fig The one loop vertex corrections

We x the counter term by

D e

F

with F the same as in eq The reason is that with this choice we get theequality of the wave function renormalization factors Z Z This equality followsfrom the conservation of the current and therefore we can use the arbitrariness inthe nite part of the vertex counter term to guarantee it The one loop vertex isthen

ef

F

One can see that this choice of F is such that for spinors on shell the vertex is thefree one

!upupjpm !upup

We will evaluate now the radiative corrections to the g of the electron Here gis the gyromagnetic ratio which the Dirac equation predicts to be equal to be twoTo this end we rst need to prove the Gordon identity for the current of a Diracparticle

!upup !upp p

m

i

mq

up

con q p p For the proof we start from

"pup m pup

and"pup m

Subtracting these two expressions we obtain

up

pm i

mp

up

An analogous operation on the barred spinor leads to the result We observe alsothat the Gordon identity shows immediately that g because it implies that thecoupling with the electromagnetic eld is just

e

mF

q

To evaluate the correction to this term from the one loop diagrams it is enough toevaluate the matrix element

e!up p pup

in the limit p p and for on shell momenta In fact contributes only to

the terms in and in the previous limit they have to build up the free vertexas implied by the renormalization condition Therefore we will ignore all the termsproportional to and we will take the rst order in the momentum q For momenta

on shell the denominator of is given by

mx ymx xmy y mxy mx y

In order to evaluate the numerator let us dene

V !up "p y "pxm "p x "py m up

Using "p "pp and an analogous equation for "p we can bring "p to act onthe spinor at the right of the expression and "p on the spinor at the left obtaining

V !upmy yp "px

mx xp "py up

Making use of

g

"p"p "p"p

we get

V !uph mxy my xm p

myp mx ym p x ym

y ym mxp mx x xymiup

and for the piece which does not contain

V m!uppy xy x px xy y

up

Therefore the relevant part of the vertex contribution is

e!up p pup e

Z

dx

Z x

dy

m!up

hp

y xy x

x y

px xy y

x y

iup

By changing variable z x y we haveZ

dx

Z x

dyy xy x

x y

Z

dx

Z

x

dz

x

z x

z

Z

x log x x

x log x x

x

log x x

x

x

from which

e!up p pup e

m!upp pup

Using the Gordon in this expression and eliminating the further contribution in we obtain

e!up p pupjmom magn ie

m!upqup

Finally we have to add this correction to the vertex part taken at p p whichcoincides with the free vertex

!up ie

mq

up !upp pm

i

m

e

q

up

Therefore the correction is

e

m e

m

Recalling that g is the ratio between S B and em we get

g

O

This correction was evaluated by Schwinger in Actually we know the rstthree terms of the expansion

ath

g

to be compared with the experimental value

aexp

Chapter

Vacuum expectation values and

the S matrix

In and Outstates

For the future we will need a formulation of eld theory showing that any S matrixelement can be reduced to the evaluation of the vacuum expectation value of a T product of eld operators Such a formulation is called LSZ from the authors lLehaman Symanzik and Zimmermann This formalism is based on a series ofvery general properties as displacement and Lorentz invariance It is interestingto see how far one can go in determining the features of a eld theory by usingonly invariance arguments In the following we will work in the Heisenberg picturethat is the operators will be time dependent and evolving with the full hamiltonianFormally this representation can be obtained from eq by sending the freehamiltonian into the full hamiltonian It should be also kept in mind that sinceno exact solution of an interacting eld theory is known we are not sure that theassumptions we are going to make are consistent Said that our assumptions are

The eigenvalues of the four momentum lie within the forward light cone

p pp p

There exists a nondegenerate Lorentz invariant states of minimum energyThis is called the vacuum state

# ji

By convention we assume that the corresponding energy is zero

Pji

From eq we get also

Pji Pji

Since p is a Lorentz invariant condition it follows that # appears as thevacuum states in all the Lorentz frames

For each stable particle of mass m there exists a stable stae of single particle

# jpi

where # is a momentum eigenstate with eigenvalue p such that p m

Neglecting the complications from the zero mass states we can add a furtherrequirement the vacuum and the single particle states give a discrete spectrumin p For instance the case of the mesons is given in Fig

xxxxxxxxx

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

xxxx

xx

xxxx

E

p

Vacuum

m

2m

Continuum

singl

e par

ticle

state

Fig Energymomentum spectrum for particles of mass m

Remember that the pion is not a stable particle and decays into ! witha lifetime of the order of sec but this number is very large withrespect to the natural decay frequency m sec m MeV Therefore by neglecting the weak interactions responsible for the decay wecan assume that the pion is stable The same argument can be applied toquarks and leptons that will be the main actors in the following Howeverwe mentioned also the pions to say that the arguments presented here canbe applied in the appropriate range of energy also to composite particlesNotice that this assumption is very much in the spirit of perturbation theoryThat is we introduce a eld for each particle assuming that the interactionsdo not modify the spectrum in a violent way

The main tools of investigation in the eld of elementary particle physics arethe scattering experiments Ideally these experiments are realized by preparing at

t a system of free particles That is assuming that they do not interact eachother this requires that the particles are well separated in space These particleswill interact with a target at some nite time After that scattering process weimagine that at t the emerging particles will behave again as free particlesIn real life the times t correspond to laboratory time That is we preparethe beam at some nite time T the scattering process will occur around the timet assuming this as the time scale and then we will measure the outgoingparticles at a time T The ideal description considered before can be considered ascorrect if the interaction time t around t is much smaller that T For instanceimagine an electron prepared in an eigenstate of momentum p impinging over anatom Here there is a further simplication that we are doing since in a nite time ora nite volume we cannot prepare a state of denite momentum This is one of thereasons why analyzing a scattering process it is always convenient to quantize in abox of the dimension of the experiment itself that is about the distance between thesource of the beam and the detector The same consideration holds for the energythat is one has always to remember that the experiments is taking place between tonite times T and T Having prepared the electron in the initial state at the timeT the particle will travel since it will reach the target where it will be subject tothe interaction with the atoms of target itself The interaction time will be of theorder of the time necessary to cross the atomic dimensions for the case of a singlescattering or at maximum of the order of the thickness of the target for a multiplescattering After that we will let again the electron to move freely since it will reachthe detector where we will measure its properties as momentum and polarization Inconclusion we can think to describe the particles at t in terms of free eldsexcept that they will be subject to the self interaction which as we know cannotbe neglected The free elds that we are going to use will be denoted by inx fort and by outx for t The interacting eld x can be thought ofas constructed in terms of these free elds operators and such that at t itreduces to outx or inx In order that the in and out elds describe correctlythe incoming and outgoing free particles we have to require some properties Inparticular we will require

inx must transform as the corresponding x with respect to the symme tries of the theory In particular with respect to translation it must satisfy

P iinxx

equivalent to require

inx a eiP ainxeiP a

inx must satisfy the free Klein Gordon equation for a scalar eld or theDirac free equation for a spinor eld etc corresponding to the physical

mass mminx

Notice that this requirement implies that we have taken into account the self interactions of the eld which are responsible for the mass renormalization wewill mention the wave function renormalization later

The same properties are required for the out elds By using these two propertiesit follows at once that the in eld creates from the vacuum the physical one particlestate from the vacuum In fact let jpi an eigenstate ofP with eigenvalue p Wehave

i

xhpjinxji hpjP inxji phpjinxji

By applying ix once more we nd

hpjinxji phpjinxji

and using the Klein Gordon equation

p mhpjinxji

We see that the only state that inx can create out of the vacuum is the one withp such that p m From our starting hypotheses we have that this is the stateof single particle with mass m Since the in and out elds are free elds we canapply all the corresponding formalism In particular they can be Fourier expanded

inx

Zdk

hainkfkx ayinkf

kxi

with

fkx p

keikx k

qk m

Inverting this relation we nd

aink i

Zdx

f k xinx f

kxinx

from which

Pi aink

Zdx

f k x

xinx

xi

xf k x

inx

xi

Z

dx

f k x

xi

xinx

xf k x

xi

inx

kiaink

Since f k x and inx both satisfy the free Klein Gordon equation we see that theoperators dened in eq are time independent This follows immediately by

recalling that if x and x are both solutions of the Klein Gordon equationthen the four vector

j x

x

is a conserved current and therefore the charge

Q

Zdxj

is a constant of motion Using this it follows

P aink

Zdx

f k x

xinx

x

xf k x

inx

x

Z

dx

f k x

x

xinx

xf k x

x

inx

kaink

In conclusionP aink kaink

Also by hermitian conjugation we get

P ayink ka

yink

It follows that ayink creates states of four momentum k whereas aink annihilatesstates with momentum k For instance

Payinka

yinkji P a

yinka

yinkji k ka

yinka

yinkji

Painkayinkji P ainka

yinkji k kainka

yinkji

By using the condition on the spectrum of the momentum operator we get also

ainkji

since otherwise this state would have a negative eigenvalue of the energy We getalso

hp pM injk kN ini

unless M N and the set p pM is identical to the set k kNLet us now look for a relation between the in and the eld The eld x

will satisfy an equation of the type

mx jx

where jx the scalar current contains the self interactions of the elds and possibleinteractions with other elds in the theory We need to solve this equation with the

boundary conditions at t relating x with the in and out elds Notice thatin writing the previous equation we have included the eects of mass renormalizationinside the current jx but we know that at the same time we have also a wavefunction renormalization Therefore asymptotically the eld x cannot be thesame as the in and out elds but it will dier by a factor

pZ where

pZ is the

wave function renormalization Therefore the required asymptotic conditions are

limt

x limt

pZinx lim

tx lim

t

pZoutx

The solution for x is then

x pZinx

Zdyretx ymjy

We will see in a moment the meaning of Z in this context The function retxm

is the retarded Greens function dened by

mretxm x

such thatretxm

for x

The need of Z in this context is because we want the the in eld properly normalizedin order to produce single particle states In fact both hjinxji e hjxji havethe same x dependence they must be both proportional to expipx see eq but x may create out of the vacuum states other than the single particle stateand therefore the normalization of the two previous matrix elements cannot be thesame

It can be shown that the asymptotic condition cannot hold as an operatorstatement This is because it is not possible to isolate the operator x from thescalar current jx at t since this includes also the self interactions Howeverthe statement can be given in a weak form for the matrix elements The correctway of doing it is rst to smear out the eld over a space like region

ft i

Zdx f x tx t f

x tx t

with fx t an arbitrary normalizable solution of the Klein Gordon equation

mfx

Then the asymptotic condition is formulated as

limt

hjftji pZ lim

thjftinji

where ji and ji are any two normalizable states

Similar considerations can be made for the out eld In particular we can nd arelation between the in and out elds

x pZoutx

Zdyadvx ymjy

whereadvxm

for x

madvxm x

is the advanced Greens function By comparing this expression with eq wend p

Zinx pZoutx

Zdyx ymjy

withx ym advx ymretx ym

The construction of the out operators of creation and annihilation goes along thesame lines discussed for the in operators

The S matrix

We have shown in the previous Section that an initial state of n non interactingparticles can be described by

jp pn ini ayinp ayinpnj ini j ini

where stands for the all collection of the quantum numbers of the initial stateAfter the scattering process we are interested to evaluate the probability amplitudefor the nal state to be a state of non interacting particles described by the out operators

jp pn outi ayoutp ayoutpnj outi j outi

The probability amplitude is given by the S matrix elements

S h outj ini

We can also dene an operator S transforming the in states into the out states

h injS h outj

such thatS h injSj ini

From the denition we can derive some important properties for the operator S

Since the vacuum is stable and not degenerate it follows jSj and there fore

hinjS houtj eihinj

By choosing the phase equal to zero we get

j outi j ini ji

In an analogous way the stability of the one particle state requires

jp ini jp outi

The operator S maps the in to the out elds

inx SoutxS

In fact let us consider the following matrix element

h outjoutj ini h injSoutj ini

But h outjout is an out state and therefore

h outjout h injinS

Then

h outjoutxj ini h injSoutxj ini h injinxSj ini

which shows the relation

The S is unitary This follows from the very denition implying

Syj ini j outi

from whichh injSSyj ini h outj outi

showing thatSSy

From eq and the unitarity of S we get

j ini Syj outi Sj outi

S is Lorentz invariant as it follows from the fact that transform in into out elds having the same Lorentz properties For the same reason the S matrixis invariant under any symmetry of the theory

The reduction formalism

In this Section we will describe a systematic way of evaluating the S matrix elementsin terms of the vacuum expectation values of T products of eld operators Let usstart considering the following S matrix element

Sp h outj p ini

where j p ini is the state of a set of particles plus an additional incoming particleof momentum p Here we will consider the case of scalar particles but the formalismcan be easily extended to other cases We want to show that by using the asymptoticcondition it is possible to extract the particle of momentum p from the state byintroducing the corresponding eld operator Let us consider the following identities

h outj p ini h outjayinpj ini h outjayoutpj ini h outjayinp ayoutpj ini h p outj ini i

Zdx

hfpx

xh outjinx outxj ini

xfpx

h outjinx outxj ini

i

where we have used the adjoint of eq We have denoted j p outi the out state obtained by removing the particle with momentum p if present in the stateotherwise it must be taken as the null vector As we know the integral appearingin the previous expression is time independent since fpx inx and outx aresolutions of the Klein Gordon equation Therefore we are allowed to make thefollowing substitutions in the weak sense since we are taking matrix elements

limx

inx limx

pZx

limx

outx limx

pZx

obtaining

h outj p ini h p outj ini

ipZ

lim

x lim

x

Zdx

hfpx

xh outjxj ini

xfpx

h outjxj ini

i

By using the identityZ

dx

x

gx

xgx

xgx

gx

lim

x lim

x

Zdx

gx

xgx

xgx

gx

Z

dx gx gx gxgx

we can write

h outj p ini h p outj ini

ipZ

Zdxh outj

hfpx x fpxx

ij ini

By using the Klein Gordon equation for fpx we get

h outj p ini h p outj ini

ipZ

Zdxh outj

hfpx x

r mfpx

x

ij ini

Integrating by parts

h outj p ini h p outj ini

ipZ

Zdxfpxmh outjxj ini

We can iterate this procedure in order to extract all the in and out particles fromthe states Let us see also the case in which we want to extract from the previousmatrix element an out particle This exercise is interesting because it shows how theT product comes about Let us suppose that the out state is of the type pwith p the momentum of a single particle We want to extract this particle fromthe state

h outjxj ini h p outjxj ini h outjaoutpxj ini h outjaoutpx xainp

j ini h outjxainpj ini h outjxj p ini i

Zdyhfpy

yh outjoutyx xinyj ini

yf py

h outjoutyx xinyj ini

i

Using again the time independence of the integral we have

h outjoutyx xinyj ini

pZ

lim

y lim

y

h outjT yxj ini

and eq

h outjxj ini h outjxj p ini

ipZ

lim

y lim

y

Zdyhf py

yh outjT yxj ini

yf py

h outjT yxj ini

i h outjxj p ini

ipZ

Zdyh outj

hf py

yT yx f pyT yx

ij ini

h outjxj p ini

ipZ

Zdyf pymyh outjT yxj ini

Substituting inside the

h outj p ini h p outj ini

ipZ

Zdxfpx mxh outjxj p ini

ipZ

Zdxdy

fpxfpymx myh outjT yxj ini

More generally if we want to remove all the particles in q qm and in p pnwith the condition pi qj we get

hp pm outjq qn ini

ipZ

mn Z mnYij

dxidyjfqixif

pjyj mximyj

hjT y ynx xmji

If some of the qis are equal to some of the pjs we can reduce further the matrixelements of the type h outjp ini Therefore we have shown that it is possible toevaluate all the S matrix elements once one knows the vacuum expectation values ofthe T products called also the n point Greens function The result obtained herecan be easily extended to generic elds simply by substituting to the wave functionsfpx the corresponding solutions of the wave equation for instance for spinor eldsspinors times plane waves and to the Klein Gordon operator the corresponding waveoperators for the spinor case the Dirac operator i m In the Dirac case onehalso to change i

pZ to ipZ for fermions and to i

pZ for antifermions

where Z is the wave function renormalization for the spinor eld

We will show in the following that a simple technique to evaluate perturbativelythe vacuum expectation value of n elds a n point Greens function can be obtainedin the context of the path integral formulation of quantum mechanics

Chapter

Path integral formulation of

quantum mechanics

Feynmans formulation of quantum mechan

ics

In Feynman gave a formulation of quantum mechanics quite dierent from thestandard one This formulation did not make use of operators and of Schr odingerequation but rather it was giving an explicit expression for the quantum mechanicalamplitude although using a mathematical formalism which was far from being welldened which was obtained directly from the physical idea that the probabilityamplitude to go from one state to another can be obtained by summing togetherall the possible ways with an appropriate weight This was as apply the ordinarycomposition laws in probability theory to the probability amplitudes rememberthat the probability is the square of the amplitude The key observation to get theappropriate weight was taken by the Diracs book where it was shown that for onedegree of freedom the probability amplitude

hq ttjq ti

for t innitesimal is given by the eiS where S is the classical action to go from q atthe time t to q at the tome tt In fact we will show that the Feynmans formu lation can be derived from the usual formulation of quantum mechanics Althoughthe path integral formulation as the Feynmans approach is usually called is justa dierent way of formulating quantum mechanics at the beginning of the s itturned out to be a crucial tool for the quantization of gauge theories Also it is onethe few methods which can be used for studying situations outside the perturba tive approach In order to derive the Feynmans formulation we will start studyinga quantum system described by one degree of freedom and we will see later howthis can be extended also to innite degrees of freedom as needed in quantum eldtheory First of all we notice that the quantum mechanical problem is to evaluate

the time evolution which is given by the propagator This can be evaluated in anybasis for instance in conguration space its matrix elements are given by

hq tjq ti t t

It will be convenient to work in the Heisenberg representation that is using time dependent operators qt The states of the previous equation are dened as eigen states of the operator qt at the time t that is

qtjq ti jq tiq

It follows immediately that the relation between these states at dierent times isgiven by

jq ti eiHtjq i

for a time independent hamiltonian The matrix element can be evaluatedby slicing the time interval t t in innitesimal pieces and using the completenessrelation we get

hq tjq ti limn

Zdq dqnhq tjqn tnihqn tnjqn tni

hq tjq ti

wheretk t k t t tn t

Therefore

hq tjq ti limn

Z nYk

dqk

nYk

hqk tkjqk tki

We have still to evaluate the matrix element but now for an innitesimal timeinterval let us use again the completeness in momentum space with states takenat a time t in between tk e tk

hqk tk jqk tki Z

dphqk tk jp tihp tjqk tki

The mixed matrix elements can be written as

hqk tk jp ti hqk jeiHpqtk tjp i

remember %h making use of the canonical commutation relations for p andq the hamiltonian can be written in such a way that all the operators q are at theleft of all the operators p The result will be denoted by the symbol Hqp Bydoing so we get

hqk tk jp ti hqk jp i iHqk ptk t

hqk jp ieiHqk ptk t

The function Hq p is obtained by substituting in Hqp the operators q andp with their eigenvalues Notice that although Hqp and Hqp are the sameoperator Hq p e Hq p are in general dierent functions In this way we get

hqk tk jp ti p

eipqk Hqk ptk t

and

hp tjqk tki p

eipqk Hqk pt tk

where H is the eigenvalue of the operatorHpq dened by putting the operatorsq to the right of p by using the canonical commutation relations By choosingt tk tk and the previous two equations we get

hqk tk jqk ti Z

dp

eipqk qkHcqk qk p

where

Hcqk qk p

Hqk p Hqk p

Since in the limit we have

hqk tk jqk tki qk qk

it is reasonable to dene a variable velocity as

qk qk qk

and writeHcqk qk p Hqk p

In the same limit Hqk p is nothing but the hamiltonian of the system evaluatedin the phase space Therefore

hqk tk jqk tki Z

dp

eiLqk p

whereLqk p p qk Hqk p

is the lagrangian in phase space Coming back to and using we obtain

hq tjq ti limn

Z nYk

dqk

nYk

dpk

e

inXk

Lqk pk

In the limit the argument of the exponential gives the integral between t and t ofthe lagrangian in the phase space therefore we get the canonical action

S

Z t

t

dt Lq p

In the limit n the multiple integral in eq is called the functionalintegral and we will denote it symbolically as

hq tjq ti Z qt

qt

dqtdptei

Z t

t

dtLq p

where

dqt Y

ttt

dqt dpt Y

ttt

dpt

The asymmetry between the integration in qt and pt implies that the integral isnot invariant under a generic canonical transformation Notice that the integrationis made over all the functions qt such that qt q e qt q On the contrarythere are no limitations on pt as it required by the uncertainty principle

Path integral in conguration space

The path integral in eq can be easily written as a functional integral inconguration space when the hamiltonian of the system is of the type

Hqp p

m V q

In such a case

Hq p Hq p Hq p p

m V q

The momentum integration can be done naively

hq tjq ti limn

Z nYk

dqk

nYk

dpk

eipkqk qk pkm V qk V qk

limn

Z nYk

dqk

rm

i

n nYk

emiqk qk

eiV qk V qk

limn

Z rm

i

n nYk

dqk

nYk

eimqk qk V qk V qk

with the result

hq tjq ti limn

Z rm

i

n nYk

dqk

e

inXk

mqk qk V qk V qk

We have also

limn

nXk

m

qk qk

V qk V qk

Z t

t

dtLq q

where Lq q is the lagrangian in conguration space Therefore we will write

hq tjq ti Z qt

qt

Dqtei

Z t

t

dtLq q

with the functional measure given by

Dqt limn

rm

i

n nYk

dqk

It is the expression really dened by which was originally formulatedby Feynman In fact it has a simple physical But before discussing the interpre tation let us notice that for N degrees of freedom and with hamiltonians of thetype

Hqipi NXi

pim

V qi i N

the expression can be easily generalized However for arbitrary system onehas to start with the formulation in the phase space of the previous Section whichalso can be easily generalized to an arbitrary number of degrees of freedom

For discussing the interpretation of the expression let us recall the bound ary conditions

qt q qt q

q q'

t

t'

Fig The Feynman denition for the path integral

The expression in the exponent of is nothing but the action evaluated for acontinuous path starting from q at time t and ending at q at time t This path isapproximated in eq by many innitesimal straight paths as shown in Fig

Therefore the interpretation is that in order to evaluate the amplitude to go fromq t to q t is obtained by multiplying together the amplitudes corresponding tothe innitesimal straight paths Each of these amplitudes neglecting the normal ization factor

pmi is given by the exponential of i times the action evaluated

along the path Furthermore we need to sum or integrate functionally over all thepossible paths joining the two given end points In a more concise way one says thatthe amplitude to from one point to another is obtained by summing over all thepossible paths joining the two points with a weight proportional to the exponentialof iS where S is the classical action evaluated along the path

The physical interpretation of the path inte

gral

Feynman arrived to the path integral formulation in conguration space on a morephysical basis First of all he knew that the amplitude for innitesimal time dier ences is proportional to the exponential of iS with S the classical action evaluatedalong the innitesimal path From that one can derive directly the nal formula inthe following way Let us consider the classical double slit experiment as illustratedin Fig where S is the electron source The electrons come out of the slit Aand arrive to the screen C where they are detected by D going through one of thetwo silts A or B By counting the electrons with the detectpr D we can determineas a function of x how many electrons arrive on C and therefore the probabilitydistribution of the electrons on the screen C Under the conditions of Fig that is with both slits and open one gets the distribution given in Fig In order to explain this distribution it would seem reasonable to make the followingassumptions

S

D

x

A B C

2

1

Fig The double slit experiment

x

P

Fig The experimental probability distribution P with both slits open

Each electron must go through the slit or through the slit

As a consequence the probability for the electron to reach the screen at thepoint x must be given by the sum of the probability to reach x through withthe probability to go through

These hypotheses can be easily checked by closing one of the two slits and doingagain the experiment The results obtained in the two cases are given in Fig

x

P P2 1

Fig The experimental probability distributions P and P obtained keepingopen the sli or the slit respectively

We see that P P P therefore one of our hypotheses must be wrong Infact quantum mechanics tells us that we can say that an electron went through oneof the two slits with absolute certainty only if we detect its passage through theslit Only in this case one would get P PP In fact this is the result that oneobtains when doing the experiment with a further detector telling us which of theholes have been crossed by the electron This means that with this experimentalsetting the interference gets destroyed On the contrary the experiment tells us thatwhen we do not detect the hole crossed by the electron the total amplitude for theprocess is given by the sum of the amplitudes corresponding to the two dierentpossibilities that is

P jAj jAAj jAjjAjA AAA

jAjjAj PP

The two expressions dier for a term which is not positive denite and which is theresponsible for the interference pattern Therefore quantum mechanics tells us thatthe usual rules for combining together probabilities are correct but the result thatwe get follows from our ignorance about the path followed by the particle unlesswe measure it That is it does not make sense under such circumstances dene aconcept as the path followed by a particle The Feynnmans interpretation is a littlebit dierent although at the end the theories turn out to be equivalent He saysthat it is still possible to speak about trajectories of the electron if one gives up tothe usual rules of combining probabilities This point of view requires taking intoaccount dierent type of alternatives according to dierent experimental settingsIn particular one denes the following alternatives

Exclusive alternative We will speak about exclusive alternatives when weuse an apparatus to determine the alternative chosen by the system For in stance one we use a detector to determine the slit crossed by the electrons Inthis case the total probability is given by the sum of the probabilities corre sponding to the dierent alternatives

P Xi

Pi

Interfering alternatives This is the situation where the alternative chosenby the system is not detected by an experimental apparatus In this casewe associate to each alternative an amplitude Ai and assume that the totalamplitude is given by the sum of all the amplitudes Ai

A Xi

Ai

The total probability for the process is evaluated as

P jAj jXi

Aij

Therefore the probability rules are strictly related to the experimental setting Thereare situations in which one may have both type of alternatives For instance if wewant to know the total probability of having an electron in C within the intervala b having a detector in C and not looking at the slit crossed by the electrons Inthis case the total probability will be given by

P

Z b

a

dxjAx Axj

Since the alternatives of getting the electron at dierent points in C are exclusiveones

From this point of view it is well conceivable to speak about trajectories of theelectron but we have to associate to each possible path an amplitude and then sumall over the possible paths For instance suppose that we want to evaluate theamplitude to go from S to the screen C when the screen B is eliminated This canbe done by increasing the number of holes in the screen B Then if we denote byAx y the amplitude for the electron to reach x going through the hole placed ata distance y from the center of B the total amplitude will be given by

Ax

ZdyAx y

We can continue by using more and more screen of type B with more and moreholes That is we can construct a lattice of points between A and C and construct

all the possible paths from A to the point x on C joining the sites of the latticeas shown in Fig By denoting with C the generic path we get that the totalamplitude is given by

Ax XCAx C

S

D

x

A C

C

C

1

2

Fig The lattice for the evaluation of equation

From this point it is easy to get the path integral expression for the total am plitude since each path is obtained by joining together many innitesimal pathsand for each of these paths the amplitude is proportional to expiS where S is theaction for the innitesimal path Therefore for the single path C the amplitude willbe given by expiSC This is the nal result except for a proportionality factorThis is easy to get if one starts from the phase space formulation This is importantbecause in many cases the simple result outlined above is not the correct one Forinstance consider the following lagrangian

L m

qfq V q

where fq is an arbitrary function of q The canonical momentum is given by

p m qfq

and the corresponding hamiltonian is

H

mfqp V q

Notice that for qk qk we have f!qk fqk fqk where !qk qk qk Therefore form eq we get

hq tjq ti limn

Z nYk

dqk

nYk

dpk

eipkqk qk pkmf!qk V qk V qk

This expression is formally the same as the one in eq after sending m mf!qk Therefore we get

hq tjq ti limn

Z rmf!qk

i

n nYk

dqk

e

inXk

mf!qkqk qk V qk V qk

We can rewrite this expression in a way similar to the one used in eq

hq tjq ti Z qt

qt

Dqtei

Z t

t

dtLq q

with Lq q given in but the functional measure is now

Dqt limn

rmf!qk

i

n nYk

dqk

The formal expression for the path integral is always the same but the integrationmeasure must be determined by using the original phase space formulation In thecontinuum limit we have

limn

nYk

pf!qk lim

ne

nXk

log f!qk

limn

e

nXk

log f!qk

e

Z t

t

dt log fq

where we have used

limn

nXk

Z t

t

dt limn

hk

th tk

Therefore the result can be also interpreted by saying that the classical lagrangiangets modied by quantum eects as follows

L L i

log fq

The reason why we say that this is a quantum modication is because the secondterm must be proportional to %h as it follows from dimensional arguments

An interesting question is how one gets the classical limit from this formulationLet us recall that the amplitude for a single path is expiSC%h where we haveput back the factor %h To nd the total amplitude we must sum over all the possiblepaths C Since the amplitudes are phase factors the will cancel except for the pathswhich are very close to a stationary points of S Around this point the phaseswill add coherently whereas they will cancel going away from this point due to thefast variation of the phase Roughly we can expect that the coherence eect willtake place for those paths such that their phase diers from the stationary valueSCclass%h of the order of one For a macroscopical particle m gr and times ofabout sec the typical action is about %h and the amplitude gets contributiononly from the classical path For instance take a free particle starting from theorigin at t and arriving at x at the time t The classical path the one makingS stationary is

xclasst xt

t

The corresponding value for S is

Sxclass

m

Z t

q

mxt

Let us now consider another path joining the origin with x as for instance

xt xt

t

The corresponding action is

Sx

mxt

and we get

S Sx Sxclass

mxt

Let us take t sec and x cm We get S m For a macroscopic particlesay m gr and recalling that %h ergsec we get S %h and thisnon classical path can be safely ignored However for an electron m grand S %h It follows that in the microscopic world many paths may contributeto the amplitude

The free particle

We shall now consider the case of the one dimensional free motion In this case wecan easily make the integrations going back to the original denition eq The expression we wish to evaluate is

hq tjq ti limn

Z rm

i

n nYk

dqk

e

inXk

mqk qk

since the lagrangian is

L

m q

The integrations can be done immediately starting from the gaussian integralZ

dxeax

bx

raeba Re a

which allows us to evaluate the expressionZdxeax x bx x

r

a beabx x

a b

let us start by integrating qZdqe

imq q q q

ri

meimq q

i

m

m

i

eimq q

Multiplying by the exponential depending on q and integrating over this variablewe get

i

m

m

i

Z

dqeimq q

q q

i

m

m

i

i

m

eimq q

i

m

m

i

eimq q

By repeating this procedure after n steps we nd

Z nYk

dqk

e

inXk

mqk qk

i

m

n

m

in

eimqn qn

The rst factor in the right hand side of this equation cancels out an analogousfactor in the integration measure see eq

hq tjq ti

m

it t

eimq qt t

This expression can be generalized to a free system with N degrees of freedom

hqi tjqi ti

m

it t

N

e

imNXi

qi qit t

i N

Notice also thatlimtt

hq tjq ti q q

The classical path connecting q t to q t for the free particle is given by

qcl q t

t tq q

The corresponding classical action is

Scl

Z t

t

m

qcld

m

q q

t t

Z t

t

d m

q q

t t

Therefore our result can be written as

hq tjq ti

m

it t

eiScl

We shall see that for all the lagrangians quadratic in the coordinates an in themomenta this result that is that the amplitude is proportional to the exponentialof iS with S the action evaluated along the classical path is generally true

Of course one could formulate quantum mechanics directly in terms of the pathintegral It is not dicult to show that it is possible to recover all the results of theordinary formulation and prove that one has complete equivalence However we willnot do it here but we will limit ourselves to notice that the formulations agree inthe free case we take a xed initial point of propagation for instance q t Then we dene the wave function as

q t hq tj i Z qt

DqteiS

For the free particle we get

q t h m

it

i

eimq

t

This is indeed the wave function for a free particle in conguration space as it canbe checked by going in the momentum space

p t h m

it

i

Zdq eipqei

mq

t h m

it

i

it

m

ei p

mt eiEt

where E pm

The case of a quadratic action

In this Section we will consider the case of a quadratic lagrangian for a single degreeof freedom it can be easily extended to many degrees of freedom

L

m q b qq

cq d q eq f

To evaluate the integral let us start by the Euler Lagrange equations associ ated to eq

d

dm q bq d b q cq e

that ism q bq d cq e

Let us denote by qcl the solution of the classical equations of motion satisfyingthe boundary conditions

qclt q qclt q

We then perform the change of variables

q qcl

which in terms of our original denition means

k qk qclk k n

Since this is a translation the functional measure remains invariant The new vari able satises the boundary condition

t t

The lagrangian in terms of the new variables is given by

Lq Lqcl Lqcl

m b

c

d e m qcl b qcl b qcl cqcl

Inside the action we can integrate by parts the terms containing both qcl and Byusing the boundary conditions for we get

Lq Lqcl

m b

c d e

hm qcl bqcl cqcl

i

Recalling the equations of motion for qcl and performing a further integration byparts we nd

Lq Lqcl

m b

c d e

h d e

i

Lqcl

m b

c

Therefore we can write the path integral as follows

hq tjq ti eiSclq qZ t

t

Dei

Z t

t

m b

cd

Notice that here all the dependence on the variables q and q is in the classical actionIn fact the path integral which remains to be evaluated depends only on t and t

due to the boundary conditions satised by For the future purposes of extensionof this approach to eld theory this term turns out to be irrelevant However it isimportant for systems with a nite number of degrees of freedom and therefore wewill give here an idea how one can evaluate it in general First let us notice that theterm b is a trivial one since we can absorb it into a re denition of c throughan integration by part

b d

d

b

b

and deningc c b

In this way we get

i

Z t

t

m

cd lim

ninXk

m

k k

ck

k

limn

nXk

im

k kk

i

ck

k

limn

iT

where we have used the boundary conditions on n and we haveintroduced the vector

n

and the matrix

m

c c cn cn

Therefore

Gt t Z t

t

Dei

Z t

t

m b

cd

limn

m

i

n

Z nYk

dk

ei

T

Since is a symmetric matrix it can be diagonalized by an orthogonal transforma tion

UTDU

The eigenvectors of U are real and the Jacobian of the transformation isone since detjU j We getZ

nYk

dk

ei

T

Z nYk

dk

ei

TD nYk

r

ik np

deti

We get

Gt t limn

m

i

n n

deti

limn

m

i

imndeti

Let us dene

ft t limn

i

m

ndeti

then

Gt t r

m

ift t

In order to evaluate ft t let us dene for j n

Pj

m

c c c cj cj

andpj detjPjj

Clearly i

m

ndeti pn

For the rst values of j pj is given by

p

mc

p

mc

p

p

mc

p p

Therefore we nd the recurrency relation

pj

mcj

pj pj j n

where we dene p It is convenient to put this expression in the form

pj pj pj

cjpjm

It is also convenient to introduce the nite dierence operator

pj pj pj

a time t j and a function such that

lim

pj

in the limit in which we keep xed It follows

lim

pj d

d

The equation can be written as

pj cjpjm

therefore we get the following dierential equation for

d

d c

m

Furthermore satises the following boundary conditions

t lim

p

d

djt lim

p p

lim

mc

Recalling eq we have

ft t lim

pn t

Therefore for quadratic lagrangian we get the general result

hq tjq ti r

m

ift teiScl

with ft t t

Examples

The free particle

We need only to evaluate ft t The function satises

d

d t

d

djt

with solution t

from whichft t t t

in agreement with the direct calculation

The harmonic oscillator

In this case b and c m Therefore

d

d t

d

djt

from which

sin t

by putting T t t

ft t

sinT

To evaluate the classical action we need to solve the classical equations of motion

m q mq

with boundary conditionsqt q qt q

The general solution is given by

q A sin t B cos t

From the boundary conditions

A q q cosT

sinT B q

and substituting inside the lagrangian

L

m q

mq

m

A B cos t AB sin t

Recalling the following integrals

Z t

t

d cos t sin T

Z t

t

d sin t cos T

we get

Scl

m

sinT

q q cosT qq

and

hq tjq ti r

m

i sinTe

i

m

sinT

q q cosT qq

In the limit we get back correctly the free case

Functional formalism

The most natural mathematical setting to deal with the path integral is the theoryof functionals The concept of functional is nothing but an extension of the conceptof function Recall that a function is simply a mapping from a manifold M toR that a law which associates a number to each point of the manifold A realfunctional is dened as an application from a space of functions and therefore an dimensional space to R There are many spaces of functions For instancethe space of the square integrable real functions of a real variable Therefore a real

functional associate a real number to a real function If we denote the space offunctions by L a real functional is the mapping

F L R

We will make use of the notation

F F F L

to denote the functional of dened in eq Examples of functionals are

LL the space of the integrable functions in R with respect tothe measure wxdx

F

Z

dxwxx

LL space of the square integrable functions in R

F e

Zdxx

LC space of the continuous functions

F x

F is the functional that associates to each continuous function its value at x

The next step is to dene the derivative of a functional We need to understandhow a functional varies when we vary the function A simple way of understandthis point is to consider some path on the denition manifold of the functions Forinstance consider functions from R to R and take an interval t t of R madeof many innitesimal pieces as we did for dening the path integral Then we canapproximate a function t with a convenient limit of its discrete approximation n obtained by evaluating the functions at the discrete points in whichthe interval has been divided In this approximation we can think of the functionalF as an application from Rn R That is

F F n F i i n

Then we can consider the variation of F i when we vary the quantities i

F i i F i nXj

F i

jj

Let us put

Ki

F

i

where is the innitesimal interval where we evaluate the discrete apoproximationto the function Then for

F i i F i nXj

KjjZ

Kdx

that is

F

ZKxxdx

Since x is the variation of the function evaluated at x we will dene the functional derivative as

F

ZF

xxdx

In other words to evaluate the functional derivative of a functional we rst eval uate its innitesimal variation and then we can get the functional derivative bycomparison with the previous formula Looking at the eq we get

F

x lim

F i

i

i xi xi x i xn xn

i n

let us now evaluate the functional derivatives of the previous functionals

F

Zwxxdx

givingF

x wx

Then

F e

Zdxx

Z

xx

and thereforeF

x xF

In order to evaluate the derivative of F let us write

F

Zxx xdx

In this way F looks formally as a functional of type F and

F

x x x

Let us consider also

F e

ZdxdyKx yxy

then

F e

ZdxdyKx yxy

Z

dxdyKx yxy

where we have used the symmetry of the integral with respect to x y We obtain

F

x

ZdyKx yyF

A further example is the action functional In fact the action is a functional of thepath

Sq

Z t

t

Lq qdt

By varying it

Sq

Z t

t

L

qq

L

q q

dt

Z t

t

L

q d

dt

L

q

qdt

Since we consider variations with respect to functions with xed boundary condi tions we have qt qt We get

S

qt

L

qt d

dt

L

qt

Starting from the functional derivative one can generalize the Taylor expansionto the functional case This generalization is called the Volterra expansion of afunctional

F Xk

k$

Zdx dxk kF

x xk

x xk

All these denitions are easily generalized to the case of a function from Rm Rnor from a manifold M to a manifold N

General properties of the path integral

In this Section we would like to stress some important properties of the path integralnamely

Invariance of the functional measure under translations It followsimmediately from the denition that

q q t t

p p for any t t

Factorization of the path integral From the very denition of the path itfollows that in order to evaluate the amplitude for going from q at time t to q

at time t we can rst go to q( to a time t t( t sum up over all the pathsfromq a q( and from q( to q and eventually integrating over all the possibleintermediate points q( see Fig

q q"

t

t"

q'

t'

Fig The factorization of the path integral

In equations we get

hq tjq ti

Z qt

qt

dqdpei

Z t

t

Ld i

Z t

t

Ld

Zdq(

Z qt

qt

dqdpei

Z t

t

Ld

Z qt

qt

dqdpei

Z t

t

Ld

Zdq(hq tjq( t(ihq( t(jq ti

as it follows from the factorization properties of the measure

dq Y

tt

dq dq(Y

tt

dqY

tt

dq

dp Y

ttdp

Ytt

dpY

ttdp

It is clear that the factorization property is equivalent to the completeness of thestates

Using these properties it is possible to show that the wave function as denedin satises the Schr odinger equation

The next important property of the path integral is that it gives an easy way ofevaluating expectation values To this end we will consider expressions of the typewhich will be called the average value of F

Z qt

qt

dqdpF q peiS

where F is an arbitrary functional of q p and of their time derivatives Let us startwith Z qt

qt

dqdpeiSqt( t t( t

We begin by the observation that

Z qt

qt

dqdpeiSqt qZ qt

qt

dqdpeiS qhq tjq ti

since qt satises the boundary conditions qt q qt q By using thefactorization we get

Z qt

qt

dqdpeiSqt(

Zdq(

Z qt

qt

dqdpeiSqt(

Z qt

qt

dqdpeiS

Zdq(hq tjq( t(iq(hq( t(jq ti

Since jq ti is an eigenstate of qt it follows

Z qt

qt

dqdpeiSqt(

Zdq(hq tjqt(jq( t(ihq( t(jq ti

hq tjqt(jq ti

Therefore the average value of qt( evaluated with expiS coincides with the ex pectation value of the operator qt(

Now let us study the average value of qtqt con t t t t By proceedingas before we will get the expectation value of qtqt if t t or qtqt if

t t In terms of the T product we getZ qt

qt

dqdpeiSqtqt hq tjT qtqtjq ti

AnalogouslyZ qt

qt

dqdpeiSqt qtn hq tjT qt qtnjq ti

Let us now consider a functional of qtsuch to admit an expansion in a Volterrasseries Z qt

qt

dqdpF qeiS

Xk

k$

Zdt dtk kF q

qt qtkq

Z qt

qt

dqdpeiSqt qtk

Xk

k$

Zdt dtk kF q

qt qtkq

hq tjT qt qtkjq ti

hq tjT F qjq ti

where

T F q Xk

k$

Zdt dtk kF q

qt qtkq

T qt qtk

If the functional depends also on the time derivatives a certain caution is necessaryFor instance let us study the average value of qt qt t t t tZ qt

qt

dqdpeiSqt qt

lim

Z qt

qt

dqdpeiSqtqt qt

lim

hq tjT

qtqt

T

qtqt

jq ti

d

dthq tjT

qtqt

jq ti

Let us dene an ordered T product or covariant T product as

T qt qt

d

dtTqtqt

Then it is not dicult to show that for an arbitrary functional of qt e qt one has

Z qt

qt

dqdpF q qeiS hq tjT F q qjq ti

The proof can be done by expanding F q q in a Volterras series of q and q inte grating by parts and using At the end one has to integrate again by partsThe T product is obtained as in

The power of this formulation of quantum mechanics comes about when we wantto get the fundamental properties of quantum mechanics as the equations of motionfor the operators the commutation relations and so on so forth The fundamentalrelation follows from the following property of the path integralZ qt

qt

dqdp

q

hF q peiS

i

This is a simple consequence of the denition of the path integral and of the func tional derivative

limn

Z nYk

dqk

nYk

dpk

qk

F q qne

inXk

Lqk pk

of course we are assuming that the integrand goes to zero for large values of the qisTo this end one has to dene the integral in an appropriate way as it will be shownlater on On the other hand the previous equation follows from the invarianceunder translation of the functional measure In fact let us introduce the followingnotation

dqp dqdp

It follows for an innitesimal function such that t t

Z qt

qt

dqp Aq

Z qt

qt

dqp Aq

Z qt

qt

dqp Aq

Z qt

qt

dqp

Zt

Aq

qtdt

Proving the validity of eq By performing explicitly the functional derivativein eq we get a relation which is the basis of the Schwingers formulation ofquantum mechanics that is

ihq tjT

F q

S

q

jq ti hq tjT

F q

q

jq ti

The importance of this relation follows from the fact that we can use this relationto get all the properties of quantum mechanics by selecting the functional F in a

convenient way For instance by choosing F q we get

hq tjT

S

qt

jq ti

that is the quantum equations of motion With the choice F q q we get

hq tjT

qt

S

qt

jq ti ihq tjq tit t

and since this relation holds for arbitrary states

T

qt

S

qt

it t

For simplicity let us consider the following action

S

Z t

t

m q V q

dt

thenS

qt

m qt

V qt

qt

from which

T

qt

m qt

V q

qt

it t

By using the denition of the T product we get

T qt qt d

dtT qtqt

d

dtt tqtqt t tqtqt

d

dt

h t tqtqt t tqtqt

thetat tqt qt t t qtqti

t tqt qt t t qtqt T qt qt

t tqt qt T qt qt

Inserting this inside the eq

mt tqt qt T

qt

m qt

V

qt

it t

and using the equations of motion we get

qt m qt i

Therefore we get the canonical commutator

qtpt i

The generating functional of the Greens func

tions

In eld theory the relevant amplitudes are the ones evaluated between t andt We have seen that using the reduction formulas we are lead to evaluatethe vacuum expectation value of T products of local operators Therefore we willconsider matrix elements of the type limiting ourselves for the moment being toa single degree of freedom

hjT qt qtnji

Here ji is the ground state of system Notice that more generally one can considerthe matrix element

hq tjT qt qtnjq ti Z qt

qt

dqdpeiSqt qtn

These matrix elements can be derived in a more compact form by introducing thegenerating functional

hq tjq tiJ

Z qt

qt

dqdpeiSJ

where

SJ

Z t

t

d p q H Jq

and J is an arbitrary external source By dierentiating this expression withrespect to the external source and evaluating the derivatives at J we get

hq tjT qt qtnjq ti

i

nn

Jt Jtnhq tjq tiJ

J

Now we want to see how it is possible to bring back the evaluation of the matrixelement in to the knowledge of the generating functional Let us introducethe wave function of the ground state

#q t eiEthqji hqjeiHtji hq tji

then

hjT qt qtnji Z

dqdqhjq tihq tjT qt qtnjq tihq tji

Zdqdq#

q thq tjT qt qtnjq ti#q t

Next we dene the functional ZJ the generating functional of the vacuum ampli tudes that is

ZJ

Zdqdq#

q thq tjq tiJ#q t hjiJ

It follows

hjT qt qtnji

i

nn

Jt JtnZJ J

We want to show that it is possible evaluate ZJ as a path integral with arbitraryboundary conditions on q and q That is we want to prove the following relation

ZJ limti ti

eiEt t

# q#q

hq tjq tiJ

wit q and q arbitrary and with the ground state wave functions taken at t Tothis end let us consider an external source Jt vanishing outside the interval t twith t t t t Then we can write

hq tjq tiJ

Zdqdqhq tjq tihq tjq tiJhq tjq ti

with

hq tjq ti hqjeiHt tjqi Xn

#nq#

nqeiEnt

t

Since E is the lowest eigenvalue we get

limti

eiEthq tjq ti lim

ti

Xn

#nq#

nqeiEnt

eiEn Et

#q#

qeiEt

#

q t#q

Analogously

hq tjq ti Xn

#nq#

nqeiEnt

t

and

limti

eiEthq tjq ti limti

Xn

#nq#

nqeiEnt

eiEn Et

#q#

qeiEt

#q

t# q

And nally we get the result

imtiteiEt thq tjq tiJ

Zdqdq#

q#q#

q thq tjq tiJ#q

t

# q#q

ZJ

Let us notice again that the values q and q are completely arbitrary and furthermorethat the coecient in front of the matrix element is J independent Therefore it isphysically irrelevant In fact the relevant physical quantities are the ratios

hjT qt qtnjihji

i

n

Z

n

Jt JtnZJ J

where the J independent factor cancels out Then we can write

ZJ limtiti

N

Z qt

qt

Dqei

Z t

t

L Jqd

where N is a J independent normalization factor This expression suggests the de nition of an euclidean generating functional ZEJ This is obtained by introducingan euclidean time it Wicks rotation

ZEJ lim

N

Z q

q

Dqe

Z

Lq idq

d Jqd

Consider for instance the case

L

m

dq

dt V q

from which

Lq idq

d

m

dq

d

V q

It is then convenient to dene the euclidean lagrangian and euclidean action

LE

m

dq

d

V q

SE

Z

LEd

Notice that the euclidean lagrangian coincides formally with the hamiltonian Moregenerally

LE Lq i dqd

Therefore

ZEJ lim

N

Z q

q

DqeSE

Z

Jqd

Since eSE is a positive denite functional the path integral turns out to be welldened and convergent if SE is positive denitethe vacuum expectation values ofthe operators qt can be evaluated by using ZE and continuing the result for realtimes

Z

nZJ

Jt JtnJ

in

ZE

nZEJ

J JnJiiti

Since the values of q and q are arbitrary a standard choice is to set them at zero

The Greens functions for the harmonic oscil

lator

It is interesting to evaluate the generating functional for the harmonic oscillatorstarting from both the equations and Let us start from eq where we do not need to specify the boundary conditions

ZJ

ZDq#

q teiSJ#q tdqdq

with

#q t m

e

mq

e i

t

# q

t m

e

mq

e

i

t

and

SJ

Z t

t

m q

mq Jq

d

In this calculation we are not interested in the normalization factors but only onthe J dependence Then let us consider the argument of the exponential in eq and for the moment being let us neglect the time dependent terms from theoscillator wave functions

arg

mq q i

Z t

t

m q

mq Jq

d

let us perform the change of variable

q x x

where we will try to arrange x in such a way to extract all the dependence fromthe source J out of the integral We have

arg

mxt xt

mxt

xtmxtxt xtxt

i

Z t

t

m x

mx

d i

Z t

t

m x

mx Jx

d

i

Z t

t

m x x mxx Jx

d

Integrating by parts the last integral can be written as

imx xt

t i

Z t

t

m x mx J xd

and choosing x as a solution of the wave equation in presence of the source J

x x

mJ

we getimxt xt xt xt

Let us take the following boundary conditions on x

i xt xt

i xt xt

In this way the term cancels the mixed terms

arg

mxt xt

mxt xt

i

Z t

t

m x

mxd

i

Z t

t

m x

mx Jxd

Integrating by parts the last term we get

i

Z t

t

m x

mx Jxd

Z t

t

mx x x

mJ

Jxd

i

mx x

t

t

mxt

xt i

Z t

t

Jxd

Keeping in mind the boundary conditions the term coming from the inte gration by parts cancels the second term in eq and therefore

arg

mxt xt i

Z t

t

m x

mxd

i

Z t

t

Jxd

The J dependence is now in the last term and we can extract it from the integralThe rst two term give rise to the functional at J We obtain

ZJ e

i

Z t

t

JxdZ

with x solution of the equations of motion with boundary conditions For t t this solution can be written as

x i

Z t

t

sJsds

with d

d

s i

m s

andi t s t s i t s t s

These equations are easily solved fort t s and t s

Aei

Bei

Therefore we have

Aei Bei

The constants A e B are xed by the requirement that satises eq We have

AB iAei iBei

and AB iAB

from which

A B A

m

and nally

m

hei ei

i

The functional we were looking for is

ZJ e

Z t

t

dsdsJss sJsZ

We see that s s is the vacuum expectation value of the T product of theposition operators at the times s and s

s s

Z

ZJ

JsJs

J

hjT qsqsji

hji

The analogue calculation in the euclidean version is simpler since the path integralis of the type already considered quadratic action

ZEJ eSclE

Here SclE is th euclidean classical action We have already noticed that here the

boundary conditions are arbitrary and therefore we choose q for we have to solve the equations of motion in presence of the external source

q

SE

Z

Jqd

m q mq J

that is

q q

mJ

Let us now dene the euclidean Greens functiond

d

DE

m

with the boundary conditionlim

DE

then the solution of the eq is

q

Z

DE sJsds

We can evaluate DE starting from the Fourier transform

DE

ZdeiDE

Substituting inside we get

Z

dei DE

m

from which

DE

m

and

DE

m

Z

d

ei

To evaluate this integral we close the integral with the circle at the in the lower plane for or in the upper plane for The result is

DE

m

eii

me

By doing the same calculation for we nally get

DE

mej j

The classical action is given by

SE Z

Jqd

Z

m

q q

mJ

q

Jq

d

Z

Jqd

and therefore

ZEJ e

Z

Jqd

Z e

Z

JsDEs sJsdsds

Z

It follows

ZE

ZEJ

JsJs

J

DEs s

and using the eq

Z

ZJ

JtJt

J

ZE

ZEJ

JJ

Jiiti

we gett DEit

To compare these two expressions let us introduce the Fourier transform of tSince it satises eq we obtain

t

Zdeit

with

i

m

that is

i

m

Whereas eq does not present any problem in eq weneed to dene the singularities present in the denominator In fact this expressionhas two poles at and therefore we need to specify how the integral isdened In order to reproduce the euclidean result it is necessary to specify theintegration path as in Fig For instance for t by closing the integral inthe lower plane we get

i

mi

eit

meit

Im

Re

ν

ν− ω + ω

Fig The integration path for equation

We would get the same result by integrating along the real axis but translatingthe denominator of i in such a way to move the pole at in the lowerplane and the one at in the upper plane We get

t lim

i

m

Z

eit i

d

With this prescription we can safely rotate the integration path anti clockwise by since we do not encounter any singularity see Fig

t i

m

Z i

i

eit

d

We have omitted here the limit since the expression is now regular Performingthe change of variable i

t

m

Z

et

d DEit

Therefore we have shown how it works the analytic continuation implicit in eq By doing so we have also seen that the Feynman prescription the i termis equivalent to the euclidean formulation One could arrive to this connection also

Im

Re

− ω +

ε

ν

νi

εi

ω −

Fig The Wick rotation in the plane

by noticing that the Feynman prescription is equivalent to substitute in the pathintegral the term

e i

m

Zqtdt

with

e i

m i

Zqtdt

Therefore the Feynman prescription is there to make the integral convergent

e

m

Zqtdt

The same convergence is ensured by the euclidean formulation Therefore we haveshown that in the case of quadratic actions the path integral can be formulatedas an euclidean integral making it a well dened mathematical expression Thisfact is enough to guarantee that all the manipulations that we make with the pathintegral in the perturbative expansion are well dened since we will be expandingaround the quadratic case

Chapter

The path integral in eld theory

The path integral for a free scalar eld

We will extend now the previous approach to the case of a scalar eld theory Wewill show later how to extend the formulation to the case of Fermi elds Let us startwith a free scalar eld We have denoted the quantum operator by x Here wewill denote the classical eld by x For a neutral particle this is a real functionand its dynamics is described by the lagrangian

S

ZV

dx

m Z t

t

dt

Zdx

m

A free eld theory can be seen as a continuous collection of non interacting harmonicoscillators As it is well known this can be seen by looking for the normal modesTo this end let us consider the Fourier transform of the eld

x t

Zdkei

k xqk t

By substituting inside eq we get

S

ZV

dx

Zdk

dk

qk t qk t

k k mqk tqk teik x ik x

Z t

t

dt

Zdk

qk t qk t jkj mqk tqk t

Since x t is a real eld

qk t qk t

and

S

Z t

t

dt

Zdk

hj qk tj

kjqk tj

i

k jkj m

Therefore the system is equivalent to a continuous set of complex oscillators qk tor a pair of real oscillators The equations of motion are

qk t kqk t

We are now in the position of evaluating the generating functional ZJ by addingan external source term to the action

S

ZV

dx

m J

From the Fourier decomposition we get

S

Z t

t

dt

Zdk

hj qk tj

kjqk tj

i Jk tqk t

where we have also expanded Jx in terms of its Fourier components Let us

separate qk t and Jk t into their real and imaginary parts

qk t xk t iyk t

Jk t lk t imk t

It follows from the reality conditions that the real parts are even functions of kwhereas the imaginary parts are odd functions

xk t xk t lk t lk tyk t yk t mk t mk t

Now we can use the result found in the previous Section for a single oscillator it willbe enough to take the mass equal to one and sum over all the degrees of freedomIn particular we get for the exponent in eq

Z t

t

dsdsJss sJs

Z t

t

dsdsZ

dk

hlk ss skl

k s

mk ss skmk si

Z t

t

dsdsZ

dk

Jk ss skJ

k s

where we have made use of eq Going back to Jx t we get

ZdxdyJx

Zdk

s ske

ik x yJy

i

ZdxdyJxF x ymJy

where we have dened x x s y y s and

iF xm

Zdk

ske

ik x lim

Zdk

i

k m ieikx

Again we have put k k Therefore the generating functional is

ZJ e i

ZdxdyJxF x ymJy

Z

In particular we get

hjihjT xxji

Z

ZJ

JxJx

J

iF x xm

We could repeat the exercise in the euclidean formulation with the result

ZEJ e

ZdxdyJxGx ymJy

ZE

where

Gxm

Zdk

k meikx

We go from Minkowskian to the Euclidean description through the substitution

iF G

For the future it will be convenient to use the following notation for the N pointsGreens functions

GNx xN hjT x xNjihji

In the free case we can get evaluate easily the generic GN In fact GN the in

dex recalls that we are in the free case is obtained by developing the generatingfunctional

ZJ

Z

Xn

Zdx dxnJx Jxni

n

n$

hjT x xnjihji

Xn

Zdx dxnJx Jxni

n

n$Gn x xn

By comparison with the expansion of eq

ZJ

Z i

Zdxd

xJxJxF x x

Zdxd

xJxJxF x x

Z

dxdxJxJxF x x

Since ZJ is even in J we have

Gk x xk

For G we get back eq To evaluate G

we have rst to notice that

all the GN s are symmetric in their arguments therefore in order to extract the

coecients we need to symmetrize Therefore we writeZdx dxJx JxF x xF x x

Zdx dxJx Jx

hF x xF x x

F x xF x x F x xF x xi

It follows

G x x

hF x xF x x F x xF x x

F x xF x xi

If we represent G x y by a line as in Fig G

will be given by Fig

since it is obtained in terms of products of G Therefore Gn

is given bya combination of products of n two point functions We will call disconnected adiagram in which we can isolate two subdiagrams with no connecting lines Wesee that in the free case all the non vanishing Greens functions are disconnectedexcept for the two point function Therefore it is natural to introduce a functionalgenerating only the connected Greens functions In the free case we can put

x y

Fig The two point Green s function G x y

x x

x x

1 2

3 4

x1 x1x2x2

x3

x3x4

x4

++

Fig The expansion of the four point Green s function G x x x x

ZJ eiW J

with

W J

Zdxd

xJxJxF x x W

and we have

Z

ZJ

JxJx

J

iW J

JxJx

J

iF x x

Therefore the derivatives of W J give the connected diagrams with the propernormalizationthat is divided by Z In the next Section we will see that thisproperty generalizes to the interacting case That is W J is dened through eq and its Volterra expansion gives rise to the connected Greens functions

iW J Xn

in

n$

Zdx dxnJx JxnGn

c x xn

where

Gnc x xn

hjihjT x xnjiconn

The index (conn( denotes the connected Greens functions Notice that once we ndthe connected Greens functions the theory is completely solved since the generatingfunctional is recovered by exponentiation of W J

The generating functional of the connected

Greens functions

Let us start by dening recursively the connected Greens functions

Gx Gc x G

c

Gx x Gc x x G

c xGc x G

c Gc

Gx x x Gc x x x G

c xGc x x

Gc xG

c x x G

c xGc x x G

c xGc xG

c x

Gc G

c Gc G

c

On the right hand side we have used a symbolic notation by adding together thetopological equivalent congurations To dene G

nc x xn we decompose the

set x xn in all the possible subsets and then we write

Gnx xn X

Gnc xp xpn Gn

c xq xqn Gnkc xr xrnk

where we sum over all the indices ni such that n nk n and over all thepermutations from to n of the indices in the set p pn etc A graphical

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

= c

xxxxxxxxxxxxxxxxxxxxxxxxx

=c

c+

xxxxxxxxxxxxxxxxxxxxxxxxx

=

c

+cc

c3 +

c

c

c

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

= c +c

c3 + 4

c

c+

+ 6

c

c

c +cc

cc

Fig The connected Green s functions

description is given in Fig By using these denitions it is not dicult tocheck that W J is the generating functional of the connected Greens functions It

is enough to compare the expansion of ZJ in series of Gnc with the expansion from

the Gn This can be done by induction We will limit ourselves to an explicit checkup to the th order By dening G and using the same symbolic notation asbefore we get

ZJ

Z

Xn

in

n$JnGn eiW J W

exp

iG

c J i

$Gc J

i

$Gc J

i

$Gc J

iGc J

i

$Gc J

i

$Gc J

i

$Gc J

$

iG

c

J

i

$Gc

J iG

c Gc J

i

Gc G

c J

$

iG

c

J

iG

c

Gc J

$

hiG

c

Ji

iJGc

i

$

hGc G

c

i

i

$JhGc G

c Gc G

c

i

i

$JhGc G

c

G

c Gc G

c

Gc G

c

i

The perturbative expansion for the theory

Let us consider an interacting scalar eld The action will be of the type

S

ZV

dx

m V

where V is the interaction potential A typical example is the theory withthe potential given by

V

$

The derivation of the perturbative expansion is very simple in this formalism Wetake advantage of the following identity valid for any functional F

ZDxF e

iS i

ZdxJxx

F

i

J

ZDxe

iS i

ZdxJxx

We separate in eq the quadratic part of the action S from the interactingpart and then we use the previous identity

ZJ N

ZDxe

iS i

ZdxJxx

N

ZDxe

iZ

dxV eiS i

ZdxJxx

eiZ

dxV

i

J

ZJ

Notice that we have suppressed the temporal limits in the expression for the gen erating functional ZJ is the generating functional for the free case evaluated ineq Therefore we get

ZJ eiW J ZeiZ

dxV

i

J

e i

ZdxdyJxF x yJy

In this expression we have suppressed the mass in the argument of F For thefollowing calculation it will be useful to introduce the following shorthand notationZ

dx dxnF x xn hF x xni

Then eq can be rewritten in the following way

ZJ eiW J eihV

i

J

ie i

hJF Ji

Z

eihV

i

J

ieiWJ

This gives rise to the following expression for the generating functional of the con nected Greens functions

W J i log

eiWJ

BeihV

i

J

i

CA eiWJ

WJ i log

eiWJ

BeihV

i

J

i

CA eiWJ

This expression can be easily expanded in V In fact by dening

J eiWJ

BeihV

i

J

i

CA eiWJ

we see that we can expand W J in a series of Notice also that the expansion isin term of the interaction lagrangian At the second order in we get

W J WJ i

Let us now consider the case of the functional J can be in turn expanded ina series of the dimensionless coupling

It follows

W J WJ i i

Then from eq we get

i

$eiWJ h

i

J

ieiWJ

$eiWJ h

i

J

ih

i

J

ieiWJ

By using the following list of functional derivatives

JxeiWJ ix JeiWJ

JxeiWJ ix x x x JJ eiWJ

JxeiWJ

x xx J

ix x x JJJeiWJ

JxeiWJ x x ix xx x JJ

x x x x JJJJeiWJ

where the integration on the variables is understood we obtain

i

$

hhy y y y JJJJi

ihy yy y JJi hy yii

The quantity can be made in terms of by writing

$eiWJ h

i

J

i eiWJ eiWJ h

i

J

ieiWJ

i

$eiWJ h

i

J

ieiWJ

Then we have

h

i

J

ieiWJ

h

eiWJ

Jxi h

eiWJ

JxJx

i

heiWJ

JxJx

i heiWJ

Jx

Jx

i

eiWJ h Jx

i

and therefore

i

$eiWJ

hh

eiWJ

JxJx

i

heiWJ

JxJx

i heiWJ

Jx

Jx

i

eiWJ h Jx

ii

The gives a disconnected contribution since there are no propagators connectingthe two terms In fact the previous expression shows that contains a term whichcancels precisely the term in eq By using the expression for and eqs we get

i

$

i

$

h x xx J

ix x x JJJ

y xy y y JJJ iy yy xy J

ix x x x JJ

y xy y JJ

iy yy x

ix J

y xy J

y xi

By omitting the terms independent on the external source we have

i

hJ x

x y

x xx yy y

y Ji

i

hJ xx yy yx Ji

hJ xx xx yy y y JJJi

hJJ x xx yy y JJi

i

hJJJ x x xx yy

y y JJJi

Now by dierentiating functionally eq and putting J we can evaluatethe connected Greens functions For instance recalling that

iGc x x

W J

JxJx

J

we get

Gc x x iF x x

ZdxF x xF x xF x x

iZ

dxdyF x xh

F x y

F x yF x xF y y

iF y x

i

ZdxdyF x x

F x yF y yF x x

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

xxxx

xxxx

x x1 2= x

xxx

xxxx

x1 x2xx

xxxx

x1 x2+

xxxx

x +

+ xx

xxxx

x1 x2xxxx

xxxx

x y+ x

xxxxx

xxxx

x1 x2x +

xxxx y

xxxx

xxxx

x1 x2xx

x y

Fig The graphical expansion for the two point connected Green s functionGc x y

In Fig we give a graphical form to this expression as we see the resulthas an easy interpretation The diagrams of Fig are given in terms oftwo elements the propagator F x y corresponding to the lines and the vertex

corresponding to a point joining four lines In order to get Gnc we have to

draw all the possible connected diagrams containing a number of interaction verticescorresponding to the xed perturbative order The analytic expression is obtainedassociating a factor iF xy to each line connecting the point x with the point yand a factor i$ for each interaction vertex The numerical factors appearingin eq can be obtained by counting the possible ways to draw a given diagramFor instance let us consider the diagram of Fig This diagram contains thetwo external propagators the internal one and the vertex at x To get the numericalfactor we count the number of ways to attach the external propagators to the vertexafter that we have only a possibility to attach the vertex at the internal propagatorThere are four ways to attach the the vertex at the rst propagator and three forthe second Therefore

$

= xx

xxxx

xxxx

x1 xxxxx

xxxx

x1 x2x xxxx

xxxx

x x2

xxxx

x

Fig How to get the numerical factor for the rst order contributions toGc x x

where the factor $ comes from the vertex We proceed in the same way for thediagram of Fig

= xx

xx

x1

xxxxx

xxxx

x1 x2xxxx

xxxx

x yxxxx

xxxx

x2

yx y

Fig How to get the numerical factor for one of the second order contributions to G

c x x

= xxxx

xxxx

xx

xxxx

xxxx

x xx

xx1

2 3

4

xx

xxxx

xxxx

xx xxxx

x x

xx1

2 3

4

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

xxxx

xxxxxx

xxxx

xxxx

x xx

xx1

2 3

4

xx

xxxx

xxxx

xxxx

xxxx

x xx

xx1

2 3

4

xxxx

xx

xx

xx

xxxx

x x

xxx1

2 3

4

xxxx

xxxx

xxxx

xx xx

x x

xxx1

2 3

4xxxx

xx xxxx

xxxx

y

yy y+

+ +

+

+

+

xx xxxx

xxxx

xxxx

xxxx x

x1

x3x2

x4

x y

xxxx

xxxx

xxxx

xxxx xx

xx

xxxx

y

x

+

+ +

x1

x2

x4

x3

+ xxxx

xxxx

xx xx

xx xx

x y

x1

x2

x4

x3

Fig The graphical expansion up to the second order of Gc x x x x

There are four ways to attach the rst vertex to the rst propagator and four forattaching the second vertex to the second propagator Three ways for attaching thesecond leg of the rst vertex to the second vertex and two ways of attaching the thethird leg of the rst vertex to the second one Therefore we get

$

The four point function can be evaluated in analogous way and the result is givenin Fig Of course we can get it by functional dierentiation of W J Therst order contribution is

Gc x x x x i

ZdxF xxF xxF xxF xx

where the numerical coecient is one since there are of attaching the vertexto the four external lines

The Feynmans rules in momentum space

The rules for the perturbative expansion of the Greens functions in space time thatwe got in the previous Section can be easily extended to the momentum space Wewill consider again the two point function given in eq where we will take

separately the various contributions in that we will denote by Gic where i

stands for the power of let us also dene the Fourier transform of Gc x x as

Gc p p

Zdxd

xGc x xe

ipx ipx

Then we get

Gc p p p p

i

p m i

and

Gc p p

i

Zdxd

xdxeipx ipx

Zdp

eipx x i

p m i

Z

dq

i

q m i

Zdk

eikx x i

k m i

i

Zdp

dq

dk

p kp pp k

i

p m i

i

q m i

i

k m i

The product of the functions may be rewritten in such a way to pull out theoverall four momentum conservation

p pp pp k

By integrating over p and k we get

Gc p p i

pp

i

p m i

Zdq

i

q m i

Also in momentum space we can make use of a diagrammatic expansion with thefollowing rules

For each propagator draw a line with associated momentum p see Fig

:εp m2 2_ + i

___________i

Fig The propagator in momentum space

For each factor i$ draw a vertex with the convention that the momentum&ux is zero see Fig

λi4 = 0

p

p

p

p

p p

1

2 3

4

: _4!__ p p1 2 3 ++ +,

Fig The vertex in momentum space

To get Gnc draw all the topological inequivalent diagrams after having xed

the external legs The number of ways of drawing a given diagram is itstopological weight The contribution of such a diagram is multiplied by itstopological weight

After the requirement of the conservation of the four momentum at each vertexwe need to integrate over all the independent internal four momenta withweight Z

dq

A more systematic way is to associate a factor

i$

Xi

pi

to each vertex and integrate over all the momenta of the internal lines Thisgives automatically a factor

nXi

pi n numero linee esterne

corresponding to the conservation of the total four momentum of the diagram

By using these rules we can easily evaluate the nd order contribution of Fig

1

2

3q

q

q

p2

p1

Fig One of the second order contributions to Gc

We get

i$

i

p m i

Z Yi

dqi

i

qi m i

p q q qq q q p

i

p m i

p m ip p

Z

Yi

dqi

i

qi m i

p q q q

Power counting in

We start by going to the euclidean formulation Let us denote the time by tR Therelation with the euclidean time is tE itR Recalling from the Section thatfor the harmonic oscillator of unit mass one has we have dened xR tR xxE tE x

iF xR

Zdp

tR

ke

ip x

Zdp

DEtE

ke

ip x

ZdpE

eipExEpE m

where pE EE p and we have used eq for unit mass and with EEFrom this we get the following modications to Feynmans rules in the euclidean

version

Associate to each line a propagator

p m

To each vertex associate a factor

$

This rule follows because the weight in the euclidean path integral is given by

eSE instead of eiS

The other rules of the previous Section are unchanged The reason to use theeuclidean version is because in this way the divergences of the integrals becomemanifest by simply going in polar coordinates

We will now in the position to make a general analysis of the divergences Forsimplicity we will consider only scalar particles First of all consider the numberof independent momenta in a given diagram or the number of (loops( L Thisis given by denition by the number of momenta upon which we are performingthe integration since they are not xed by the four momentum conservation at thevertices Since one of this conservation gives rise to the overall four momentumconservation we see that the number of loops is given by

L I V I V

where I is the number of internal lines and V the number of vertices As in Section we will be interested in evaluating the supercial degree of divergence of adiagram To this end notice that

The L independent integrations give a factorQL

k ddpk where d is the number

of space time dimensions

Each internal line gives rise to a propagator containing two inverse powers inmomenta

IYi

pi m

Given that the supercial degree of divergence is dened as

D dL I

Notice that D has to do with a common scaling of all the integration variablestherefore it may well arise the situation where D corresponding to a convergentsituation but there are divergent sub integrations Therefore we will need a moreaccurate analysis that we will do at the end of this Section For the moment beinglet us stay with this denition We will try now to get a relation between D andthe number of external legs in a diagram By putting VN equal to the total numberof vertices with N lines we have see Section

NVN E I

with E the number of external legs and I the number of internal ones recall weare dealing with scalar particles only Eliminating I between this relation and theprevious one we get

I

NVN E

from which

D dIVNI dIdVN d

NVNEdVN

This relation can be rewritten as

D d

d E

N

dN

VN

In particular for d we have

D E N VN

For the theory this relation becomes particularly simple since then D dependsonly on the number of external legs

D E

Therefore the only diagrams supercially divergent D are the ones with E and E

As noticed before this does not prove that a diagram with E or D isconvergent Let us discuss more in detail this point Let us consider the diagram ofFig and let us suppose that the diagrams and have degrees of supercialdivergence D and D respectively Since they are connected by n lines we haven independent momenta and therefore

1 2

nD D1 2

Fig Two diagrams connected by nlines

D D D n n D D n

Therefore we may have D with D and D positive that is both divergentHowever for n we have

D D D

and therefore in order to have D with or being divergent diagrams we needD or D to be negative enough to overcome the other To exclude the case n means to consider only the one particle irreducible PI Feynmans diagrams InFig we give an example of a one particle reducible diagram

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

Fig Oneparticle reducible Feynman s diagram

The reason of excluding these diagrams from our considerations is that the mo mentum of the internal line connecting the two diagrams is xed and therefore itcannot give rise to divergences Furthermore the one particle reducible diagramscan be easily reconstructed from the PIs After that we look for the diagramshaving D but yet divergent in order to have a complete classication of thedivergences The idea is simply to look for the reducibility of the diagram We willnow stay with the theory Then we can consider the particle reduciblediagrams In fact for the nature of the vertex it is impossible to have a connected particle irreducible diagram We begin with the particle reducibility In thiscase we look for diagrams of the type of Fig

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

1

N

Fig Threeparticle reducible Feynman s diagram

In this case we have D and D N therefore

D D D N

In order the supercial degree of divergence is negative we must have N Inthis diagram we have a primitively divergent four point function and a diagram withD which can be again analyzed in the same way In analogous way we lookfor two particle reducible diagrams with D and we extract contributions of theforms given in Fig

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

1

N

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

1

N

Fig Twoparticle reducible Feynman s diagram

In the rst case we have D and D N therefore

D D D N

Again this requires N The second diagram has againD and we can iteratethe procedure In the second case we have D and D N therefore

D D D N

requiring N By proceeding in this way we see that truly convergent diagramsdo not contain hidden two and four point divergent functions What we have shownis that for the theory in dimensions a PI Feynman diagram is convergentif D and it cannot be split up into or particle reducible parts containingtwo and four point divergent functions This turns out to be a general theoremdue to Weinberg valid in any eld theory and it says that a Feynman diagram isconvergent if its supercial degree of divergence and that of all its subgraphs arenegative

In the case of the theory the Weinbergs theorem tells us that all the diver gences rely in the two and four point functions If it happens that all the divergentdiagrams in the primitive sense specied above correspond to terms which arepresent in the original lagrangian we say that the theory is renormalizable Ingeneral we can look at this question by studying the equation for the supercialdivergence that we got before

D d

d E

N

dN

VN

For d we recallD E N VN

We see that D increases with VN for N Therefore we have innite divergentdiagrams and the theory is not renormalizable We can have a renormalizable scalartheory only for N meaning that the dimension N of the vertex operator N must be less or equal to An interesting case is the one of the theory in dimensions For d we get

D E N VN

and for N D E

and the only divergent functions are the one two and three point ones In d we have dim and again in order to have renormalizability we must have thatthe dimensions of the vertex operator must be less or equal to

Regularization in

In this Section we will evaluate the one loop contributions to the two and four point Greens functions by using the dimensional regularization To this end let usconsider the action in dimensions d

S

Zdx

m

$

In our unit system we have %h c and therefore the action S should bedimensionless As discussed in Section this allows to evaluate the dimensions ofthe elds and we recall that for the scalar eld we have

dim d

From this it follows at once that

dimm dim

As we did in QED for the electric charge here too it is convenient to introduce adimensionful parameter in order to be able to dene a dimensionless coupling

new old

Then the action will be new

S

Zdx

m

$

The Feynmans rules are slightly modied

The scalar product among two four vectors becomes a sum over compo nents

In the loop integrals we will have a factorZdp

and the function in dimensions will be dened byZdp

Xi

pi

The coupling goes into

Also we will do all the calculation in the euclidean version Remember thatin Section we gave the relevant integrals both in the euclidean and in theminkowskian version

We start our calculation from the rst order contribution to the two point func tion corresponding to the diagram tadpole of Fig This is given by

p p

p mT

p m

p

p

2p1

Fig The oneloop contribution to the twopoint function in the theory

with

T

Zdp

p m

)

m

m

m

)

For we get

T

m

log

m

m

log

m

3

4p1

p2

p

p

p - k

k

Fig The oneloop contribution to the fourpoint function in the theory

This expression has a simple pole at and a nite part depending explicitly on

Let us consider now the one loop contribution to the four point function Thecorresponding diagram is given in Fig with an amplitude

p p p pY

k

pk mT

with

T

Zdk

k m

k p m p p p

We reduce the two denominator to a single one through the equation see Section

k m

k p m

Z

dx

xk m x k p m

Z

dx

xk m x k m k p p

Z

dx

k m xp k p

the denominator can be written as

k m xp k p k xp m p x p x

k xp m px x

Since our integral is a convergent one it is possible to translate k into k p xobtaining

T

Z

dx

Zdk

k m px x

By performing the momentum integration we get

T

Z

dx

)

m px x

In the limit

T

log

Z

dx

log

m px x

Z

dx logm px x

Also the x integration can be done by using the formulaZ

dx log

ax x

p a log

p a p a

a

The nal result is

T

h

log

m

s

m

plog

q m

p q

m

p

i

Renormalization in the theory

In order to renormalize the theory we proceed as in QED That is we split theoriginal lagrangian expressed in terms of bare couplings and bare elds in a part

Lp

m

$

depending on the renormalized parameters m and and eld and in the counter terms contribution

Lct

Z

m

$

The sum of these two expressions

L

Z

m m

$

gives back the original lagrangian if we redene couplings and elds as follows

B Z Z

mB

m m

Z m mZ

B

Z Z

In fact

L

B

B

m

BB

$B

The counter terms are then evaluated by the requirement of removing the di vergences arising in the limit except for a nite part which is xed by therenormalization conditions From the results of the previous Section by summing up

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

xx xx = xx xx xxxx xx+

xxxx

xxxx

xxxx

xxxx+ + xx

xxxxxx

xxxx

+

Fig The contributions to the teopoint connected Green s function

all the PI contributions we get the following expression for the two point connectedGreens function see Fig

Gc p p p p

p m

p mT

p m

p p

p m T

Therefore the eect of the rst order contribution is to produce a mass correctiondetermined by T The divergent part can be taken in care by the counterterm m

which produces an additional contribution to the one loop diagram represented inFig We choose the counterterm as

m

m

F

m

with and F an arbitrary dimensionless function Adding the contributionof the counterterm to the previous result we get

Gc p p p p

p m T

p mm

p m

xx x_ m 2δ=

Fig The Feynman rule for the counterterm m

at the order we have

Gc p p p p

p m m T

Of course the same result could have been obtained by noticing that the countertermmodies the propagator by shiftingm intomm Now the combination mTis nite and given by

m T

m

F log

m

The nite result at the order is then

Gc p p pp

p m

hF log m

i

This expression has a pole in Minkowski space and we can dene the renormalizationcondition by requiring that the inverse propagator at the physical mass is

p mphys

This choice determines uniquely the F term However we will discuss more in detailthe renormalization conditions in the next Chapter when we will speak about therenormalization group Notice that at one loop there is no wave function renormal ization but this comes about at two loops as we shall see later

Consider now the four point connected Greens function Its diagrammatic ex pansion at one loop is given in Fig again we have included only PI diagrams

From the results of the previous Section and introducing the Mandelstam invari ants

s p p t p p

u p p

we get

Gc p p p p p p p p

Yk

pk m

log

m

As t u

= xx

xxxx

xxxx

xxxx

xxxx

xxxx

xx

xxxx xx

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

+ xx x

xxxxxx

xxxx

xx

xxxx

xx

xxxx

xxxx

xxxx

xx

+

+ +xxxx

xxxx

xx

xxxx

xxxx

xxxx

Fig The oneloop expansion for the fourpoint Green s function

with

As t u X

zstu

r

m

zlog

q m

z q

m

z

The divergent part can be disposed o by the counterterm

$

$

G

m

with G an arbitrary dimensionless function This counterrtem produces an addi tional contribution with a Feynman rule represented in Fig

_ 22_

µω

δλ( )=

Fig The Feynman s rule for the couterterm

By adding the counterterm we get

Gc p p p p p p p p

Yk

pk m

G log

m

As t u

A possible renormalization condition is to x the scattering amplitude at some valueof the invariants for instance s m

phys t u to be equal to the lowestorder value

Gc p p p p p p p p

Yk

pk m

In any case G is completely xed by this conditionWithout entering into the calculation we want now show how the renormalization

program works at two loops We will examine only the two point function Itscomplete expansion up to two loops including also the counterterm contributions isgiven in Fig By omitting the external propagators and the delta function of

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

xxxx

xxxx = xx

xxxxxx

xxxx

xxxx

xxxx+ +

+ xx xxxx xx + xxxx

xxxx

xx

xxxx

xx xx+ +

xx xx+ xxx x+

a) b)

c) d)

Fig The twoloop expansion for the twopoint Green s function

conservation of the four momentum the various two loop contributions are noticethat a counterterm inside a loop gives a second order contribution as a pure two loops diagram

Contribution from the diagram a of Fig

p

m

m

log

m

Contribution from the diagram b of Fig

log

m

Contribution from the diagram c of Fig

m

G log

m

Contribution from the diagram d of Fig

m

F log

m

Adding together these terms we nd

p

m

m

F G

where we have used This expression is still divergent but at theorder Therefore we can make nite the diagram by modifying at the secondorder the m counterterm

m

m

with m given in eq and

m m

F G F

m

and F a new dimensionless function The further divergence in p can be eliminatedthrough the wave function renormalization counterterm that is

Z

with the choice

Z

H

m

with H dimensionless All the functions F F G H are nite in the limit Notice also that we are not introducing more and more freedom becauseonly particular combinations of these functions appear in the counterterms and areprecisely these combinations which are xed by the renormalization conditions Alsothe function H is xed by the renormalization condition on the inverse propagatorwhich are really two conditions a the position of the pole b the residuum at thepole

Fig Insertion of counterterms in multiloop diagrams

The previous procedure can be easily extended to higher and higher orders inthe expansion in the coupling constant What makes the theory renormalizableis the fact that the new divergences that one encounters can all be disposed oby modifying the counterterms m and This depends from the fact that thedivergent terms are constant or quadratic in the external momentum and thereforethey can be reproduced by operators as

and The non trivial part of

the renormalization program is just the last one In fact consider diagrams of thetype represented in Fig where we have taken out all the possible externallines These diagrams contain terms coming from the counterterms times log p

terms from the loop integration In the previous two loop calculation we had nosuch terms but we got their strict relatives logm If present the log p

terms would be a real disaster since they are non local in conguration space andas such they cannot be absorbed by local operators However if one makes all therequired subtractions that is one subtracts correctly also the diagrams of the typein Fig it is possible to show that the log p terms cancel out For instancein the expression for m

there are not such terms In fact they cancel in adding upthe four two loop contributions

Chapter

The renormalization group

The renormalization group equations

For the following considerations it will be convenient to introduce the one particleirreducible truncated n point functions )np p pn dened by removing thepropagators on the external legs and omitting the delta function expressing theconservation of the four momentum

nXi

pi)np p pn G

nPIp p pn

nYi

DF pi

where

DF p

ZdxeipxhjT xji

is the exact euclidean propagator We have two ways of evaluating these functionswe can start from the original lagrangian and evaluate them in terms of the barecouplings constants in the case of B and mB On the other hand we canexpress the bare parameters in terms of the renormalized ones and at the sametime renormalizing the operators through the wave function renormalization B

Z obtaining the relation

)nB p p pnB mB Zn

)np p pnm

The factor Zn originates from a factor Z

n coming from the renormalization

factors in Gnc and from n factors Z

from the inverse propagators Of course in a

renormalizable theory the )ns are nite by construction In the previous relationwe must regard the renormalized parameters as function of the bare ones In the case this means and m depending on B and mB However the right handside of the previous equation does not depend on the choice of the parameter This implies that also the functions )n do not depend on except for the eldrescaling factor Z

n These considerations can be condensed into a dierential

equation obtained by requiring that the total derivative with respect to of theright hand side of eq vanishes

m

m n

logZ

)np p pnm

The content of this relation is that a change of the scale can be compensated by aconvenient change of the couplings and of the eld normalization Notice that since)n is nite in the limit it follows that also the derivative of logZ is niteIt is convenient to dene the following dimensionless coecients

m

logZ

d

m

m

m

m

m

obtaining

mm

m nd

)np p pnm

We can try to eliminate from this equation the term To this end let us considerthe dimensions of )n In dimensions d the delta function has dimensions and dim therefore

Gnc p pn

Z nYi

dxieipx pnxnhjT x xnjiconn

has dimensions nn n The propagator DF p has dimensions and we get

dim)n

n n dim)n

n n

and putting

dim)n

n

n

Therefore )n must be a homogeneous function of degree n n in thedimensionful parameters pi m e Introducing a common scale s for the momentawe get from the Euler theorem

s

sm

m n

n

)nspim

Since everything is nite for we can take the limit and eliminate betweenthis equation and eq

s s

m m

m nd n

)nspim

This equation tells us how the )n scale with the momenta and allows us to evaluatethem at momenta spi once we know them at momenta pi To this end one needsto know the coecients m and d For this purpose one needs to specify therenormalization conditions since these coecients depend on the arbitrary functionsF G

In order to use the renormalization group equations we need to know more aboutthe structure of the functions d and m We have seen that the counterterms aresingular expressions in therefore it will be possible to express them as a Laurentseries in the renormalized parameters

B

a

Xk

akk

mB m

b

Xk

bkk

Z

c

Xk

ckk

with the coecients functions of the dimensionless parameters and m Noticethat a b and c may depend on but only in an analytical way Recalling fromthe previous Section the expressions for the counterterms at the second order in

G

m

m

F

m

F G F

Z

H

we get for the bare parameters

B

G

O

mB m

h

F

F G F

H

iO

Z

H

O

Therefore the coecients are given by

a

G O

a

O

b

F

F

H O

b

F G

O

b

O

c

H O

c

O

Notice that the dependence onm comes only from the arbitrary functions deningthe nite parts of the counterterms This is easily understood since the divergentterms originate from the large momentum dependence where the masses can beneglected This is an essential feature since it is very dicult to solve the grouprenormalization equations when the coecients m e d depend on both variables andm On the other hand since these coecients depend on the renormalizationconditions it is possible to dene a scheme where they do not depend on mAnother option would be to try to solve the equations in the large momentum limitwhere the mass dependence can be neglected

Callan and Szymanzik have considered a dierent version of the renormalizationgroup equations They have considered the dependence of )n on the physicalmass In that case one can show that and d depend only on Furthermorein their equation the terms m and do not appear In their place there is ainhomogeneous term in the large momentum limit

Renormalization conditions

Let us consider again the renormalization conditions The renormalization procedureintroduces in the theory a certain amount of arbitrariness in terms of the scale andof the arbitrary functions F F Recall that the lagrangian has been decomposedas

L Lp Lct

therefore the nite part of Lct can be absorbed into a redenition of the parametersappearing in L since both have the same operatorial structure This is another wayof saying that the nite terms in Lct are xed once we assign and m In turn theseare xed by relating them to some observable quantity as a scattering cross sectionFurthermore we have to x a scale at which we dene the parameters and mThe renormalization group equations then tell us how to change the couplings whenwe change in order to leave invariant the physical quantities As we have saidthe procedure for xing m and is largely arbitrary and we will exhibit variouspossibilities To do that it is convenient to write explicitly the expression for thetwo and four point truncated amplitudes

)p

H

p

m

F log

m

F

)p p p p

log

m

As t uG

In the two point function we have omitted the nite terms coming from the secondorder contributions for which we have given only the divergent part let us nowdiscuss various possibilities of renormalization conditions sometimes one speaks ofsubtraction conditions

A rst possibility is to renormalize at zero momenta that is requiring

)pp

p mA

)p p p ppi

A

where mA and A are quantities to be xed by comparing some theoretical

cross section with the experimental data Then comparing with eqs and and using As t u for pi we get

HA FA

logm

A

GA

logm

A

Another possibility would be to x the momenta at some physical value in theMinkowski space For instance we could require )p !m where !m isthe physical mass and then x ) with all the momenta on shell

To choose the subtraction point at zero momenta is generally dangerous formassless particles in this way one would introduce spurious infrared diver gences As a consequence a popular choice is to subtract at an arbitraryvalue of the momenta For instance one requires

)p p mB p M

)p p p p B s t u M

Notice that the point chosen for the four point function is a non physical oneIn this case we get

HB FB

log mB

GB

log mB

AMMM

This way of renormalize has however the problem of leaving an explicit depen dence on the mass in the coecients of the renormalization group equations

The last way of renormalizing we will consider is due to Weinberg and tHooft and the idea is simply to put equal to zero all the arbitrary functionsF F at each order in the expansion In this way the coecients ai biand ci appearing in the Laurent series dening the bare couplings turn out tobe mass independent

The Weinberg and t Hooft prescription allows to evaluate the renormalizationgroup equations coecients in a very simple way For instance let us start with thebare coupling

B

Xk

ak

k

Dierentiating with respect to and recalling that B does not depend on thisvariable we get

Xk

ak

k

Xk

akk

where

ak dak

d

Since is analytic for for small we can write A Bobtaining

Xk

ak

k

AB

Xk

akk

and identifying the coecients of the various powers in

B

a ABa

ak Aak Bak

from whichB

A

d

d

a

We then get

d

d

a

and for

d

d

a

and

d

d

ak

dakd

d

d

a

From the expression of the previous Section for a at the second order in we get

This equation can be integrated in order to know how we have to change when wechange the value of the scale The function is known as the running couplingconstant Its knowledge together with the knowledge of m allows us to integrateeasily the renormalization group equations In fact let us dene

t log s

and let us consider the following partial dierential equation we shall see that therenormalization group equations can be brought back to this form

tF xi t ixi

xiF xi t

The general solution to this equation is obtained by using the method of the (char acteristics( That is one considers the integral curves the characteristic curvesdened by the ordinary dierential equations

dxit

dt ixit

with given boundary conditions xi !xi Then the general solution is

F xi t xit

where!xi F !xi

is an arbitrary function to be determined by the condition at t required for F In fact we can check that satises the dierential equation

txit

dxit

dt

xi i

xi

In our case the equation contains a non derivative term

tGxi t ixi

xiGxi t xiGxi t

with xi a given function By putting

Gxi t e

Z t

dtxitF xi t

we see that F xi t satises eq and therefore the general solution will be

Gxi t e

Z t

dtxitxit

with xit satisfying eq For the )n we get t log s

)nspim )npims s snen

Z s

dsds

s

with

ss

s s

sms

s msms

and s ms mThis result tells us that when we perform a re scaling on the momenta the

amplitudes )n do not scale only with the trivial factor sn due to their dimensionsbut that they undergo also a non trivial scaling due to d which is necessaryto compensate the variations of and m with the scale

Application to QED

In this Section we will apply the previous results to QED In particular we want tocheck that the running coupling obtained via the renormalization group equationsis the same that we have dened in Section First of all let us start with theQED lagrangian in d dimensions neglecting the gauge xing terms

S

Zdx

!i" m e ! "A

FF

Since

dim ! "A

it followsdime

Therefore we deneenew eold

eold

As before we write the lagrangian as LpLct with we ignore here the gauge xingterm

Lp !i" m e ! "A

FF

and

Lct iZ! " m ! eZ

! "A Z

FF

giving

L i Z ! " m m ! e Z ! "A Z

FF

After re scaling of the elds one gets back the original lagrangian via in particularthe identication

eB e Z

Z Z e Z

where we have used Z Z From Section we recall that

Z C e

and therefore

eB e

e

We proceed as in the previous Section by dening

eB

e

Xk

akk

with

a e

Requiring that eB is independent and in the limit we get

e e

e

d

de

a

from which

e e

The dierential equation dening the running coupling constant is

e

e

e

Integrating this equation we obtain

e

log K

where K is an integration constant which can be eliminated by evaluating e atanother momentum scale Q We get

Q

log

Q

or

Q

log Q

which agrees with the result obtained in Section

Properties of the renormalization group equa

tions

Integrating eq

d

d

O

we get

log

s

s

where s is a reference scale and s s Therefore

s

s

log s

From this expression we see that starting from s increases with Supposethat we start with a small in such a way that the perturbative expansion is sensibleHowever increasing we will need to add more and more terms to the expansiondue to the increasing of Therefore the theory allows a perturbative expansiononly at small mass scales or equivalently at large distances In this theory we cangive a denite sense to the asymptotic states The increasing of is related to thesign of the function For the theory and for QED the function is positivehowever if it would be negative the theory would become perturbative at large massscales or at small distances In this case one could solve the renormalization groupequations at large momenta by using the perturbative expansion at a lower orderto evaluate the and the other coecients However in this case the coupling isincreasing at large distances and this would be a problem for our approach based onth in and out states As we shall see this is the situation in QCD here the eldsthat are useful to describe the dynamics at short distances the quark elds cannotdescribe the asymptotic states which can be rather described by bound states ofquarks as mesons and baryons

Going back to the eq we see that assuming the validity of the expressionfor the function up to large momenta the coupling constant becomes innite atthe value of given by

se

s

which is a very big scale if s is small The pole in eq is called the Landaupole see Section in the case of QED Of course there are no motivations whyeq should be valid for large values of

Let us now discuss several possible scenarios starting from a theory where for

If for any than the running coupling is always increasing and itwill become innite at some value of If this happens for a nite we saythat we have a Landau pole at that scale

Suppose that is positive at small and that it becomes negative goingthrough a zero at F see Fig

The point F is called a xed point since if we start with the initial condition F than remains xed at that value To study the behaviour of aroundthe xed point let us expand the around its zero

F F

It follows

d

d F

F

F

β(λ)

λλ

Sλ SλSλ

Fig The behaviour of giving rise to an ultraviolet xed point

from whichd

F F

d

Integrating this equation we get

Fs F

s

F

Notice that the sign of F plays here a very important role In the presentcase the sign is negative since for F and for F For large values of we have that F independently on the initial values For this reason F is referred to as a ultraviolet UV xed point seeFig The large scale or small distance behaviour of a eld theory of

S

Fλ

λ

µ µ

Fig The behaviour of close to a UV xed point

this type would be dictated by the value of F In particular if F ands F we would be always in the perturbative regime Otherwise startingfrom s F the theory would become perturbative at large scales see Fig About this point notice that is a xed point since However this is an infrared IR xed point since the coupling goes to zero for

independently on the initial value s We say also that the xed pointis repulsive since goes away from the xed point for increasing whereas aUV xed point is said to be attractive

Let us consider now the case of for small values of and decreasingmonotonically For instance suppose a with a Integratingthe equation for the running coupling we get

s

as logs

In this case is a decreasing function of and goes to zero for that is is a UV xed point This property is known as (asymptoyic freedom(and it holds for non abelian gauge theories that we will describe later on inthe course these are the only dimensional theories to share this propertyIn higher dimensions other theories enjoy this property For instance in d the theory is asymptotically free Notice that also this time the runningcoupling has a pole but at a smaller scale than s

s e

as

Therefore the coupling increases at large distances

Finally let us consider the case of for small values of becomes zeroat F and then positive Then F By expanding around F weget by the same analysis we did before

Fs F

s

F

and we see that F in a way independent on s whereas it goesaway from s for increasing Therefore F is an IR xed point see Fig

Let us now derive the perturbative expressions for m and d using the samemethod followed for *e possibile ricavare We dierentiate the eq withrespect to notice that in this renormalization scheme b

mm

Xk

bkk

m

Xk

bkk

from which

m

Xk

bkk

Xk

bkk

S

λ

µµ

Fλ

Fig The behaviour of close to an IR xed point

where we have made use of the denition m mm Using eq we nd

m dbd

mbk dbkd

d

d

a

dbkd

or

m

dbd

and

dbd

bk dbkd

d

d

a

dbkd

Using the eq for b we get for m

m

O

Let us now consider Z recall that although this quantity is innite in the limit d is nite By dierentiating eq we nd c

Zd

Z

d

d

Xk

ckk

Xk

ckk

or

Xk

ckk

d

AB

Xk

ckk

Using again the expression for see eq

d

dcd

dckd

ckdcd

dckd

d

d

a

At the lowest order using eq we get

d

O

To conclude this Section let us study the behaviour of )n when the theory hasa UV xed point at F Since for large scales F we may suppose that mand d go to mF e dF respectively Then integrating the following equationsee eq

sdm

ds mmF

we havems msmF

Analogously Z s

ds

sdF dF log s

Then for large values of s see eq

)nspim snndF )npimsmF F

Therefore if mF we can safely neglect the mass on the right hand sideof this equation From this equation we understand also the reason why d is calledthe anomalous dimension

In non abelian gauge theories there is a UV xed point at where m and therefore one can neglect the masses at large momenta

To justify why we have assumed that in the evaluation of m and d the onlyimportant contribution comes from F we notice that the integralZ s

ds

sms

can be rewritten using dss d

Z s

dm

Therefore barring the case in which m and have simultaneous zeroes the integralis dominated by the points where there is a zero of the function

The coecients of the renormalization group

equations and the renormalization conditions

We have already noticed that the coecients of the renormalization group equationsdepend on the subtraction procedure In particular using the rst two renormal ization conditions considered in Section a mass dependence of the coecients isintroduced To study better this problem let us consider for generic renormaliza tion conditions the expression for the bare coupling

B

a

Xk

akk

Dierentiating with respect to we nd

a

Xk

akk

a

Xk

akk

m

a

Xk

akk

where the dot stands for the derivative with respect to m Notice also that seeeq

m

m

m

With a familiar procedure we put AB obtaining

a Ba

a Aa Ba m

m a

At the second order in

B

G

G

G

and

Aa

G

m

m

G

Since the right hand side is at least of the second order in neglecting higher orderterms we get

A

m

G

and for

m

G

We see directly how the function depends on the subtraction procedure Similarexpression can be obtained for m e d Therefore when using subtraction procedure

such that the coecients of the renormalization group equations depend explicitlyon the masses it is convenient to look for solutions in a region of momenta wherethe masses can be neglected Another possibility is to choose the arbitrary scale such that m can be neglected

Of course changing the renormalization procedure will induce a nite renormal ization in the couplings For instance we get expressions of the type

f

m

O

Therefore we obtain

f

m

f f

m

m f

Working in a region where we can neglect the mass we have

f

and we see that a xed point at F implies a xed point in Thereforethe presence of a xed point does not depend on the scheme used at least in thisapproximation It is possible to show that also the sign of the function is scheme independent

Chapter

The path integral for Fermi elds

Fermionic oscillators and Grassmann algebras

The Feynmans formulation of the path integral can be extended to eld theoriesfor fermions However since the path integral is strictly related to the classicaldescription of the theory it could be nice to have such a (classical( description forfermionic systems In fact it turns out that it is possible to have a (classical( calledpseudoclassical description in a way such that its quantization leads naturally toanticommutation relations In this theory one introduces anticommuting variablesalready at the classical level This type of mathematical objects form the so calledGrassmann algebras Before proceeding further let us introduce some concepts aboutthese algebras which will turn out useful in the following

First of all we recall that an algebra V is nothing but a vector space where itis dened a bilinear mapping V V V the algebra product A Grassmannalgebra Gn is a vector space of dimensions n and it can be described in terms of ngenerators i i n satisfying the relations

ij ji i j i j n

Then the elements constructed by all the possible products among the generatorsform a basis of Gn

i ij ijk n

Notice that there are no other possible elements since

i

from eq The subset of elements of Gn generated by the product of an even

number of generators is called the even part of the algebra Gn whereas the subsetgenerated by the product of an odd number of generators is the odd part Gn Itfollows

Gn Gn Gn

A decomposition of this kind is called a grading of the vector space if it is possibleto assign a grade to each element of the subspaces In this case the grading is asfollows

If x Gn degx

If x Gn degx Notice that this grading is also consistent with the algebra product since if x y arehomogeneous elements of Gn that is to say even or odd then

degxy degxdegy

Since we have no time to enter into the details of the classical mechanics ofthe Grassmann variables we will formulate the path integral starting directly fromquantum mechanics Let us recall that in the Feynmans formulation one introducesstates which are eigenstates of a complete set of observables describing the congu ration space of the system In the case of eld theories one takes eigenstates of theeld operators in the Heisenberg representation

x tjx ti xjx ti

this is possible since the elds commute at equal times

x t y t

and as a consequence it is possible to diagonalize simultaneously at xed time theset of operators x t For Fermi elds such a possibility does not arise since theyanticommute at equal times Therefore it is not possible to dene simultaneouseigenstates of Fermi elds To gain a better understanding of this point let us stayfor the moment with free elds and let us recall that this is nothing but a set ofharmonic oscillators This is true also for Fermi elds except for the fact that theseoscillators are of Fermi type that is they are described by creation and annihilationoperators satisfying the anticommutation relations

ai aj hayi a

yj

i

hai a

yj

i ij

For greater simplicity take the case of a single Fermi oscillator and let us try todiagonalize the corresponding annihilation operator

aji ji

Then from a as it foillows from eqs we must have Thereforethe eigenvalue cannot be an ordinary number However it can be thought as anodd element of a Grassmann algebra If we have n Fermi oscillators and we require

aij ni ij ni

we getai ajj ni j aij ni jij ni

But aiaj ajai and therefore

i j

We see that the equations are meaningful if we identify the eigenvalues iwith the generators of a Grassmann algebra Gn In this case the equation canbe solved easily

ji eayji

where ji is the ground state of the oscillator In fact

aji a ayji aayji ayaji ji ayji ji

where we have used

eay ay

since a We have also

ayji ay ayji ayji

ayji

ji

Here we have used the left derivative on the Gn dened by

iii ik iiii ik iiii ik

kikiii ik

Suppose we want to dene a scalar product in the vector space of the eigenstatesof a notice that these are vector spaces with coecients in a Grassmann algebramathematically such an object is called a modulus over the algebra Then it isconvenient to extend the algebra G generated by to a comples Grassmann algebraG generated by and The operation ( ( is dened as the complex conjugationon the ordinary numbers and it is an involution automorphism of the algebra withsquare one with the property

ii ik ik ii

In the algebra G we can dene normalized eigenstates

ji eayjiN

where N G and the phase is xed by the convention

N N

The corresponding bra will be

hj N hjea

the normalization factor N is determined by the condition

hji N hjeaeayji N hj a ayji N hj aayji N N e

It followsN e

Let us recall some simple properties of a Fermi oscillator the occupation numberoperator has eigenvalues and In fact

aya ayaaya ay ayaa aya

from whichayaaya

Starting from the state with zero eigenvalue for aya we can build up only theeigenstate with eigenvalue ayji since ayji Therefore the space of theeigenstates of aya is a two dimensional space By dening the ground state as

ji

one sees easily that

a

ay

one writes down a as the most general matrix and then requires aji a and

a ay

The state with occupation number equal to is

ji

In this basis the state ji is given by

ji eea

y ji e ayji

e ji ji e

whereashj e

We see that in order to introduce eigenstates of the operator a has forced us togeneralize the ordinary notion of vector space on the real or on the complex numbersto a vector space on a Grassmann algebra a modulus In this space the linearcombinations of the vectors are taken with coecients in G or more generally inGn

Let us now come to the denition of the integral over a Grassmann algebraTo this end we recall that the crucial element in order to derive the path integralformalism from quantum mechanics is the completeness relation for the eigenstatesof the position operator Z

dqjqihqj

What we are trying to do here is to think to the operators a as generalized posi tion operators and therefore we would like its eigenstates to satisfy a completenessrelation of the previous type Therefore we will (dene( the integration over theGrassmann algebra G by requiring that the states dened in eqs and satisfy Z

ddjihj

where the identity on the right hand side must be the identity on the two dimensionalvector space Using the explicit representation for the statesZ

ddjihj

Zdde

Zdde

Zdd

Therefore the integration must satisfyZdd

Zdd

Zdd

Zdd

These relations determine the integration in a unique way since they x the valueof the integral over all the basis elements of G The integral over a generic elementof the algebra is then dened by linearity the previous integration rules can berewritten as integration rules over the algebra G requiring

d d d d d d

Then we can reproduce the rules and by requiringZd

Zd

For n Fermi oscillators we get

j ni e

nXi

ii

e

nXi

iayi

ji

with Z nYi

di di

j nih nj

and integration rules Zdij ij

Assuming alsodi j di dj

Notice that the measure d is translationally invariant In fact if f a b and

G Zdf

Zdf

since Zdb

Integration over Grassmann variables

Let us consider the Grassmann algebra Gn Its generic element can be expandedover the basis elements as

f n fXi

fiiXi

fiii Xiin

fni ini in

Of course the coecients of the expansion are antisymmetric If we integrate fover the algebra Gn only the last element of the expansion gives a non vanishingcontribution Noticing also that for the antisymmetry we haveX

iinfni ini in n$fn n n

it follows Zdn df n n$fn n

An interesting property of the integration rules for Grassmann variables is the inte gration by parts formula This can be seen either by the observation that a derivativedestroys a factor i in each element of the basis and thereforeZ

dn d

jf n

either by the property that the integral is invariant under translations see the pre vious Section and Section Let us now consider the transformation propertiesof the measure under a change of variable For simplicity consider a linear transfor mation

i Xk

aikk

The eq must be required to hold in each basis since it can be seen easilythat the previous transformation leaves invariant the multiplication rules In fact

ij cikcjhkh cikcjhhk cjhcikhk ji

where cij is the inverse of the matrix aij Therefore we requireZdij

Zdi

j ij

By putting

di Xk

dkbki

we obtain Zdij

Z Xhk

dkbkiajhh

Xhk

bkiajhhk abij ij

from whichbij aij

The transformation properties of the measure are

dn d detjajdn d

where we have used the anticommutation properties of di among themselves Wesee also that

detjaj det

and

dn d det

dn d

Therefore the rule for a linear change of variables isZdn dfi

Zdn ddet

fii

This rule should be compared with the one for commuting variablesZdx dxnfxi

Zdx dxndet

xx

fxixi

We see that the Grassmann measure transforms according to the inverse jacobianrather than with the jacobian itself As in the commuting case the more importantintegral to be evaluated for the path integral approach is the gaussian one

I

Zdn de

Xij

Aijij

Z

dn deTA

The matrix A is supposed to be real antisymmetric

AT A

As shown in the Appendix a real and antisymmetric matrix can be reduced to theform

As

by means of an orthogonal transformation S Then let us do the following changeof variables

i Xj

ST ijj

we getTA ST AST TSAST TAs

and using detjSj

I

Zdn de

TAs

The expression in the exponent is given by

TAs

n

n

n for n even

TAs

n

nn for n odd

From eq we get contribution to the integral only from the term in theexpansion of the exponential TAs

n for n even In the case of n odd we get

I since TAs does not contain n Therefore for n even

I

Zdn de

TAs

Zdn d

n$

TAs

n

Zdn d

n$n

$ n

n

npdetA

where we have used detjAsj detjAj ThenZdn de

TA pdetA

This result is valid also for n odd since in this case detjAj A similar resultholds for commuting variables In fact let us dene

J

Zdx dxnex

TAx

with A a real and symmetric matrix A can be diagonalized by an orthogonaltransformation with the result

J

Zdx dxnex

TAdx

n

pdetA

Again the result for fermionic variables is opposite to the result of the commutingcase Another useful integral is

IA

Zdn de

TA T

with i Grassmann variables To evaluate this integral let us put

A

We get

TA T

T TAA A T A

TA

TA

where we have used the antisymmetry of A and T T Since the measureis invariant under translations we nd

IA

Zdn de

TA TA eTApdetA

again to be compared with the result for commuting variables

IA J

Zdx dxnex

TAx JTx eJTAJ

np

detA

The previous equations can be generalized to complex variablesZdd

dndne

yA detjAj

and Zdd

dndne

yA y y eyAdetjAj

The second of these equations follows from the rst one as in the real case we willprove the rst one for the case n Starting fromZ

ddeTA A

and putting

p ip

we get

TA iA

and recalling that remember that here one has to use the inverse of the jacobian

dd idd

we get Zdde

TA i

ZddeiA

and Zdde

iA iA

The path integral for the fermionic theories

In this Section we will discuss the path integral formulation for a system describedby anticommuting variables Considering a Fermi oscillator one could start fromthe ordinary quantum mechanical description and then by using the completeness interms of Grassmann variable it would be possible to derive the corresponding pathintegral formulation We will give here only the result of this analysis It turns outthat the amplitude among eigenstates of the Fermi annihilation operator is given by

h tj ti Z tt

tt

D et t iS

whereD

Ydd

and

S

Z t

t

i

H

dt

where for the Fermi oscillator H The expression is very similar to theone obtained in the commuting case if we realize that the previous formula is theanalogue in the phase space using complex coordinates of the type q ip It shouldbe noticed the particular boundary conditions for the variables originating fromthe fact that the equations of motion are of the rst order Correspondingly wecan assign only the coordinates at a particular time In any case since at the endwe are interested in eld theory where only the generating functional is relevantand where the boundary conditions are arbitrary for instance we can take the eldsvanishing at t and in any case do not aect the dependence of the generatingfunctional on the external source The extension to the Fermi eld is done exactlyas we have done in the bosonic case and correspondingly the generating functionalfor the free Dirac theory will be

Z !

ZD !e

i

Z !i m ! !

where and ! are the external sources of Grassmann character Using the inte gration formula we get formally

Z ! Zei! i

i mi

Zei!

i m

where the determinant of the Dirac operator has been included in the term at zerosource In this expression we have to specify how to take the inverse operator Weknow that this is nothing but the Dirac propagator dened by

i mSF x y x y

It follows

Z ! ZeiZ

dxdy!xSF x yy

with

SF x lim

Zdk

%k m ieikx lim

Zdk

%k m

k m ieikx

The Greens functions are obtained by dierentiating the generating functional withrespect to the external sources Grassmann dierentiation In particular the twopoint function is given by

Z

Z

!xy

hjihjT x!yji iSF x y

The T product in this expression is the one for the Fermi elds as it follows fromthe derivation made in the bosonic case taking into account that the classical eldof this case are anticommuting

We will not insist here further on this discussion but all we have done in thebosonic case can be easily generalized for Fermi elds

Chapter

The quantization of the gauge

elds

QED as a gauge theory

Many eld theories possess global symmetries These are transformations leavinginvariant the action of the system and are characterized by a certain number ofparameters which are independent on the space time point As a prototype we canconsider the free Dirac lagrangian

L !xi mx

which is invariant under the global phase transformation

x x eiQx

If one has more than one eld Q is a diagonal matrix having as eigenvalues thecharges of the dierent elds measured in unit e For instance a term as !with a scalar eld is invariant by choosing Q Q and Q This is a so called abelian symmetry since

eiQeiQ ei Q eiQeiQ

It is also referred to as a U symmetry The physical meaning of this invariancelies in the possibility of assigning the phase to the elds in an arbitrary way withoutchanging the observable quantities This way of thinking is in some sort of contradic tion with causality since it requires to assign the phase of the elds simultaneouslyat all space time points It looks more physical to require the possibility of assigningthe phase in an arbitrary way at each space time point This invariance formulatedby Weyl in was called gauge invariance The free lagrangian cannot begauge invariant due to the derivative coming from the kinetic term The idea is

simply to generalize the derivative to a so called covariant derivative D havingthe property that D transforms as that is

Dx Dx eiQxDx

In this case the term!D

will be invariant as the mass term under the local phase transformation To con struct the covariant derivative we need to enlarge the eld content of the theoryby introducing a vector eld the gauge eld A in the following way

D ieQA

The transformation law of A is obtained from eq

ieQA ieQA

x

ieQAeiQx

eiQx ieQA

e

from which

A A

e

The lagrangian density

L !iD m !i ieQAm L e !QA

is then invariant under gauge transformations or under the local group U We seealso that by requiring local invariance we reproduce the electromagnetic interactionas obtained through the minimal substitution we discussed before

In order to determine the kinetic term for the vector eld A we notice that eq implies that under a gauge transformation the covariant derivative undergoesa unitary transformation

D D eiQxDe

iQx

Then also the commutator of two covariant derivatives

D D ieQA ieQA ieQF

withF A A

transforms in the same way

F eiQxFeiQx F

The last equality follows from the commutativity of F with the phase factor Thecomplete lagrangian density is then

L L LA !i ieQAm

FF

The gauge principle has automatically generated an interaction between the gaugeeld and the charged eld We notice also that gauge invariance prevents anymass term

MAA Therefore the photon eld is massless Also since the

local invariance implies the global ones by using the Noethers theorem we nd theconserved current as

j L

!Q

from which eliminating the innitesimal parameter

J !Q

Nonabelian gauge theories

The approach of the previous section can be easily extended to local non abeliansymmetries We will consider the case of N Dirac elds The free lagrangian

L NXa

!ai ma

is invariant under the global transformation

+x +x A+x

where A is a unitary N N matrix and we have denoted by + the column vectorwith components a In a more general situation the actual symmetry could be asubgroup of UN For instance when the masses are not all equal So we willconsider here the gauging of a subgroup G of UN The elds ax will belong ingeneral to some reducible representation of G Denoting by U the generic elementof G we will write the corresponding matrix Uab acting upon the elds a as

U eiATA U G

where TA denote the generators of the Lie algebra associated to G LieG that isthe vector space spanned by the innitesimal generators of the group in the fermionrepresentation The generators TA satisfy the algebra

TA TB ifABC TC

where fABC are the structure constants of LieG For instance if G SU andwe take the fermions in the fundamental representation

+

we have

TA A

A

where A are the Pauli matrices In the general case the TAs are N N hermitianmatrices that we will choose normalized in such a way that

TrTATB

AB

To make local the transformation means to promote the parameters A tospace time functions

A Ax

Notice that now the group does not need to be abelian and therefore in general

eiATAeiATA eiATA

eiATA

Let us now proceed to the case of the local symmetry by dening again the conceptof covariant derivative

D+x D+x UxDx

We will put againD igB

where B is a N N matrix acting upon +x In components

Dab ab

igBab

The eq implies

D+ igBUx+

Ux+ UxUxigBUx+ Ux+

Ux UxigBUx UxUx+

thereforeUxigB

Ux UxUx igB

and

Bx UxBxU

x i

gUxUx

For an innitesimal transformation

Ux iAxTA

we get

Bx iAxTA Bx

gAxT

A

Since Bx acquires a term proportional to TA the transformation law is consistentwith a B linear in the generators of the Lie algebra that is

Bab AAT

Aab

The transformation law for A becomes

AC fABC AA

B

gC

The dierence with respect to the abelian case is that the eld undergoes also ahomogeneous transformation

The kinetic term for the gauge elds is constructed as in the abelian case Infact the quantity

D D+ igF+

in virtue of the eq transforms as + under gauge transformations that is

D D+ igF +

igF Ux+

UxD D+ Ux igF+

This time the tensor F is not invariant but transforms homogeneously since itdoes not commute with the gauge transformation as in the abelian case

F UxFU

x

The invariant kinetic term will be assumed as

LA

TrFF

Let us now evaluate F

igF D D igB igB

igB B gB B

orF B B igB B

in componentsF F

C TC

withF C A

C AC gfABC A

AAB

The main feature of the non abelian gauge theories is the bilinear term in the pre vious expression Such a term comes because fABC expressing the fact that G isnot abelian The kinetic term for the gauge eld expressed in components is givenby

LA

XA

FAFA

Therefore whereas in the abelian case LA is a free lagrangian it contains onlyquadratic terms now it contains interaction terms cubic and quartic in the eldsThe physical motivation lies in the fact that the gauge elds couple to everythingwhich transforms in a non trivial way under the gauge group Therefore they couplealso to themselves remember the homogeneous piece of transformation

To derive the equations of motion for the gauge elds let us consider the totalaction Z

V

dx!+i m+ g !+B

+ SA

where

SA

ZV

dx TrFF

and the variation of SA

SA ZV

dxTrFF

Using the denition for the eld strength we get

F B igBB igBB

from which

SA ZV

dxTrF B igBB igBB

where we have taken into account the antisymmetry properties of F Integratingby parts we obtain

SA ZV

dxTrF B igBFB igF BB

ZV

dxTr F igB F

B

ZV

dx F igB F

A AA

where we used the cyclic property of the trace By taking into account also the freeterm for the Dirac elds and the interaction we nd the equations of motion

FA igB F

A g !+TA+

i m+ gB+

From the rst equation we see that the currents !+TA+ are not conserved In fact

the conserved currents turn out to be

JA !+TA+ iB F

A

The reason is that under a global transformation of the symmetry group the gaugeelds are not invariant said in dierent words they are charged elds with respectto the gauge elds In fact we can verify immediately that the previous currents areprecisely the Noethers currents Under a global variation we have

AC fABC AA

B + iATA+

and we get

j L+

+L

AC

AC

from whichj !+AT

A+ F C fABC AAB

In the case of simple compact Lie groups one can dene fABC fABC with theproperty fABC fBCA It follows

F C fABCABT

A iB F iB FATA

Thereforej !+AT

A+ iB FAA

After division by A we get the Noethers currents The contribution of thegauge elds to the currents is also crucial in order they are conserved quantities Infact the divergence of the fermionic contribution is given by

!+TA+ ig !+TAB+ ig !+B

TA+ ig !+TA B+

which vanishes for abelian gauge elds whereas it is compensated by the gaugeelds contribution in the non abelian case

Path integral quantization of the gauge theo

ries

As we have seen at the beginning of this course that the quantization of the elec tromagnetic eld is far from being trivial the diculties being related to the gauge

invariance of the theory In this Section we would like to discuss how these dicul ties arise also in the path integral quantization To this end let us recall the generalscheme of eld quantization in this approach First of all we will deal only with theperturbative approach meaning that the only thing that one needs to evaluate isthe generating functional for the free case The free case is dened by the quadraticpart of the action Therefore the evaluation of the path integral reduces to calculatea Gaussian integral of the type given in eqs and for the fermionic andbosonic case respectively For instance in the bosonic case one evaluates

IA J

Zdx dxnex

TAx JTx eJTAJ

np

detA

In our case the operator A appearing in the gaussian factor is nothing but the waveoperator The result of the integration is meaningful only if the operator A is non singular it needs to have an inverse otherwise the integral is not well dened Thisis precisely the point where we run into problems in the case of a gauge theoryRemember from eq that the wave operator for the electromagnetic eld inmomentum space is given by

kg kkAk

We see that the operator has a null eigenvector k since

kg kkk

and therefore it is a singular operator This is strictly related to the gauge invariancewhich in momentum space reads

Ak Ak kk

In fact we see that the equations of motion are invariant under a gauge transforma tion precisely because the wave operator has k as a null eigenvector Another wayof seeing this problem is the following in order to get physical results as to evaluateS matrix elements it is necessary to integrate over gauge invariant functionalsZ

DAF AeiS

with F A F A here A

is the gauge transformed of A But being the action

and the functional F gauge invariant it follows that the integrand is invariant alongthe gauge group orbits The gauge group orbits are dened as the set of points inthe eld space which can be reached by a given A via a gauge transformation Inother words given A its orbit is given by all the elds A

obtained by varying theparameters of the gauge group that is varying ' see Fig For instance inthe abelian case

A A '

A

A'

µ

µ

Ω

ΩA

A'

µ

µ

Fig The orbits of the gauge group in the eld space

Since the orbits dene equivalence classes or cosets we can imagine the eld spaceas the set of all the orbits Correspondingly we can divide our functional integralin a part parallel and in one perpendicular to the orbits Then the reason why ourfunctional integral is not well dened depends from the fact that the integrand alongthe space parallel to the orbits is innite the integrand being gauge invariant doesnot depend on the corresponding variables However this observation suggests asimple solution to our problem We could dene the integral by dividing it by theintegral along the part parallel to the orbit notice that is the same as the volumeof the gauge group In order to gain a clear understanding of this problem let usconsider the following very simple example in two dimensions

Z

Z

dxdye

ax y

This is as reducing the space time two two points with x and y the possible values ofthe eld in those points At the same time the exponent can be seen as an euclideanaction with the gauge symmetry

x x ' y y '

The integral is innite due to this invariance In fact this implies that theintegrand depends only on x y and therefore the integration over the remainingvariable gives rise to an innite result In order to proceed along the lines we havedescribed above let us partition the space of the eld that is the two dimensionalspace x y in the gauge group orbits Given a point x y the group orbit isgiven by the set of points x x ' y y ' corresponding to the linex y x y as shown in Fig We can perform the integral by integrating

x + y = c

f(x,y) = 0

(x , y )

(x , y )

0 0

Ω Ω

x

y

Fig The eld space x y partitioned in the orbits of the gauge group Weshow also the line x y c and the arbitrary line fx y dashed

along the perpendicular space the line x y c c is a constant and then alongthe parallel space It is this last integral which gives a divergent result since we areadding up the same contribution an innite number of times therefore the obviousway to do is to introduce as integration variables x y and x y integrating onlyon x y Or said in a dierent way perform both integrals and dividing up bythe integral over x y which cancels the divergence at the numerator This can bedone in a more systematic way by using the following identity

Z

d'

x y '

Then we multiply Z per by this factor obtaining

Z

Zd'

Z dxdy

x y '

e

ax y

The integral inside the parenthesis is what we are looking for since it is evaluatedalong the line xy constant Of course the result should not depend on the choiceof this constant it corresponds to translate the integration line in a way parallel tothe orbits In fact by doing the change of variables

x x ' y y '

we get

Z

Zd'

Z dxdy

x y

e

ax y

As promised the innite factor is now factorized out and we see also clearly thatthis nothing but the volume of the gauge group ' parameterize the gauge groupwhich in the present case is isomorphic to R Then we dene our integral as

Z Z

VG

Zdxdy

x y

e

ax y

with

VG

Zd'

In this way the integral is dened by integrating over the particular surface xy Of course we would get the same result integrating over any other line or surfacein the general case having the property of intersecting each orbit only once Forinstance let us choose the line see Fig

fx y

Then proceeding as in eq we consider the following identity which allowsus to dene a gauge invariant function f x y

f x y

Zd'

fx y

fx y

Zd' fx ' y '

In the previous case we had fx y x y giving rise to fx y x y' from which fx y Multiplying Z by the expression we get

Z

Zd'

Z dxdyfx y fx ' y ' e

ax y

Let us show that fx y is gauge invariant In fact by the transformation x x ' y y ' we obtain

fx ' y 'Z

d' fx ' ' y ' '

and changing the integration variable ' ' '

fx ' y 'Z

d' fx ' y '

Comparison with eq gives

fx y fx ' y '

The integral inside the parenthesis in eq does not depend on ' that is isgauge invariant This can be shown exactly as we did before in the case fx y x y Therefore we can again factorize the integration over ' being the sumof innite gauge copies

Z

Zd'

Z dxdyfx y fx y e

ax y

and dene

Z Z

VG

Zdxdyfx y fx y e

ax y

The function f x y called the Faddeev Popov determinant can be calculateddirectly from its denition To this end notice that in f x y appearsmultiplied by fx y As a consequence we need to know f only at points veryclose at the surface fx y But in this case the equation fx y hasthe only solution ' since by hypothesis the surface fx y crosses the gaugeorbits only once Therefore we can write

fx y

f

'

'

giving rise to

f x y f

'

where f fx y Clearly f represents the answer of the gauge xing f to an innitesimal gauge transformation As a last remark notice that our originalintegral can be written as

Z

Zdxe

xiAijxj

with

A a

a singular matrix As we have already noticed the singularity of A is related to thegauge invariance and the solution we have given here solves also the problem of thesingularity of A In fact integrating over x y only means to integrate only on theeigenvectors of A corresponding to non zero eigenvalues

Since the solution we have found to our problem is of geometrical type it is veryeasy to generalize it to the case of a gauge eld theory abelian or not We start bychoosing a surface in the gauge eld space given by

fBAB B n

where n is the number of parameters of LieG Also this surface should intersect thegauge orbits only once To write down the analogue of equation we need alsoa measure on the gauge group This can be obtained as follows We have seen thatwe are interested in the solutions of

fBA

around fBA therefore it is enough to know the measure around the identityof the gauge group In such a case the transformation ' can be written as

' iAxTA

Since at the rst order in the group parameters

'' iA ATA

we see that the invariant measure of the group around the identity is given by

d' YAx

dAx

It follows that for innitesimal transformations

d'' d'' d'

next we dene the Faddeev Popov determinant as in

f A

Zd'fAA

In this expression we have introduced a functional delta function dened by theidentity

Zd''

having the usual properties extended to the functional case We can see that f isa gauge invariant functional since

f A

Zd'fAA

and changing variables '' ' we get

f a

Zd''fAA

Zd'fAA

Consider now the naive path integral

Z

ZDAF Ae

i

ZLAd

x

where F is a gauge invariant functional Since in the following F will not play anyparticular role we will discuss only the case F but all the following consider ations are valid for an arbitrary gauge invariant F We multiply Z by the identity

Z

Zd'

Z DAe

i

ZLAd

xfAA

f A

The functional measure is invariant under gauge transformations

A 'A'

i

g''

since the homogeneous part has determinant one and the inhomogeneous part givesrise to a translation Then by performing the change of variables A A

andusing the gauge invariance of DA LA and f A we get

Z

Zd'

Z DAe

i

ZLAd

xfAAf A

Again we dene the functional integral as

Z Z

VG

ZRd'

ZDAe

i

ZLAd

xfAAf A

The evaluation of f goes as before Since in the previous equation it appearsmultiplied by fAA the only solution of fAA

in this neighborhood is

' or A therefore

fAA det

fAA

B

A

A

from which

f A det

fAA

B

A

Once again the Faddeev Popov determinant f A is the answer of the gauge xingto an innitesimal gauge transformation The nal expression that we get for Z is

Z

ZDAfAAdet

fAA

B

A

ei

ZLAd

x

This expression gives the right measure to be used in the path integral quantizationof the gauge elds In particular the vacuum expectation value vev of a gaugeeld functional will be dened as

hjT F Ajif ZDAf fe

iSF A

where the index f species the gauge in which we evaluate the vev of F Aboutthis point we can prove an important result that is if F A is a gauge invariantfunctional

F A F A

than its vev does not depend on the gauge xing fA We will prove thisresult by showing the relation between the vevs of F A in due dierent gaugesConsider the identity

hjT F Ajif ZDAffe

iSF A

Zd'f f

where we have used eq and dened f A fA By change of variable

A A we get

hjT F Ajif

Zd'

ZDAf

feiSF A

ff

Zd'

ZDAe

iSff

f fF A

where in the second line we have made a further change of variables ' ' andused the invariance of d' Then

hjT F Ajif Z

d'hjT f fF Ajif

This is the relation we were looking for If F is gauge invariant we get at once

hjT F Ajif hjT F Ajif

For the following we will dene a generating functional

Zf A

ZDAf fe

iSei

ZdxAA

A

which is not gauge invariant but in terms of which we can easily construct thevevs of gauge invariant quantities since it is built up in terms of the correctfunctional measure

For the purpose of building up the perturbative theory it is convenient to writethe integrand of eq in the form expiSe where Se is an eective action

taking into account the gauge xing f and the Faddeev Popov determinantWe will discuss later the exponentiation of f As far as f it is concerned wecan exponentiate it by using its Fourier representation or more conveniently bychoosing the gauge condition in the form fAA BAx with BAx a set ofarbitrary functions independent on the gauge elds Then

fBA det

fAABA

B

A

det

fAA

B

A

f A

and therefore

f A

Zd'fA BA

We can further multiply this identity by the following expression which does notdepend on the gauge elds

constant

ZDBe

i

Zdx

XA

BAx

We get

constant f A

Zd'DBe

i

Zdx

XA

fA x

fA BA

f A

Zd'e

i

Zdx

XA

fA x

Multiplying this expression by the functional we can show as before that thegauge volume factor factorizes allowing us to dene a class of generating functionals

ZA

ZDAf Ae

i

ZdxLA

XA

fA

ei

ZdxAA

A

Finally dening

Sf

Zdx

XA

fAx

we nd the analogue of eq is

hjT F Ajif Z

d'hjT feiS

fF Ajif

showing again that the vev a gauge invariant functional gives rise to a gaugeinvariant vev for the corresponding operator

Path integral quantization of QED

Let us now discuss the path integral quantization of QED in a covariant gauge Westart from the lagrangian

L ! i ieAm

FF

withF A A

invariant under

x x eiexxAx A

x Ax

ex

The Lorentz gauge is dened by

fA Ax

and

fA Ax

ex

withfA

y

ex

x y

Then the Faddeev Popov determinant turns out to be eld independent and it canbe absorbed into the normalization of the generating functional which is given by

Z ! N

ZD !DAe

i

ZdxL

Ae

i

Z! ! A

dx

We extract the interaction term in the usual way

Z ! eie

Zdx

i

i

i

!

Z !

with

Z ! N

ZD !DAe

i

ZdxL

Ae

i

Z! ! A

dx

Writing

SA

ZFF

dx

Zdx Ag

A

using A and integrating over the Fermi elds we get

Z ! eih!xSF x yyi

NZDAA

e

i

hAxA

xieihxAxi

Then we exponentiate the A through its Fourier transform

A

ZDCeihCxA

xi ZDCeihCxAxi

obtaining the following expression for Z

Z ! eih!xSF x yyi

NZDADCe

ihAxA

x x CxAxi

Doing the gaussian integral over A we get

Z ! eih!xSF x yyi

NZDCe

hix Cx

i

xy

iy Cyi

eih!xSF x yyiei

h i

ZDCe

ihC

i i

hC

Ci

After integrating by parts we can rewrite the integral over Cx as

ZDCe

ihC

i i

hCCi

and doing again this gaussian integral we nd

e

h i

i

i

i e

i

h

i

The nal result is

Z ! Zeih!xSF x yyie i

h

g

i

Introducing the function

G lim

Zdk

gk i

kk

k i

eikx

we write

Z ! Zeih!xSF x yye i

hxGx yyi

The photon propagator in this gauge is given by

hjihjT AxAxji iGx y

Notice that this propagator is transverse that is it satises the Lorentz condition

Gx

The Feynmans rules in this gauge are the same discussed in Section exceptfor th photon propagator which is

i

gk i

kkk i

Of course the additional term in the propagator proportional to kk does not aectthe physics since the photon is coupled to a conserved current

We can perform an analogous calculation but using the gauge xing in the formf B discussed at the end of the previous Section that is using the generatingfunctional

Z ! N

ZD !DAe

i

Zdx

L

A

e

Zdx

! ! A

In this case the only eect of the gauge xing is to change the lagrangian L toL L A

Of course the addition of this term removes the problemof the singular wave operator In fact we can writeZ

dxL

ZdxA

g

A

Therefore we have to evaluate the functional integral

Z ! Neih!xSF x yyiZDAe

i

Zdx

AKA

A

with

K g

obtaining

Z ! Zeih!xSF x yyiei

hxGx yyi

whereKG

x x

We can see that K is non singular for Going to momentum space eq givesZ

dk

gk

kk

Gkeikx

Zdk

eikx

or gk

kk

Gk

This can be solved by writing the most general second rank symmetric tensor func tion of k

Gk Ag Bkk

with the coecients such that

Ak

A

Bk

Solving these equations

Gk gk

kkk

The singularities are dened by the usual i term and we get

Gx lim

Zdk

gk i

kk

k i

eikx

The discussion made at the beginning of this Section corresponds to the socalled Landau gauge In fact for

lim

e i

ZdxA

A

The choice corresponding to the quantization made in Section is Inthis case

Gx gF xm

This is called the Feynmans gauge As we see the parameter selects dierentcovariant gauges As already stressed the physics does not depend on since thephoton couples to the fermionic current ! which is conserved in the abelian caseand as a consequence the dependent part of the propagator being proportionalto kk does not contribute The situation is quite dierent in the non abelian casewhere the fermionic current is not conserved since the gauge elds are not neutralTherefore we need further contributions in order to cancel the dependent termsSuch contributions arise from the Faddeev Popov determinant

Path integral quantization of the nonabelian

gauge theories

In the case of a non abelian gauge theory there are several dierences with respect toQED The pure gauge lagrangian is not quadratic it contains three linear and four linear interaction terms and furthermore the Faddeev Popov determinant typicallydepends on the gauge elds In this Section we will consider the pure gauge casesince the fermionic coupling is completely analogue to the abelian case Separatingthe lagrangian in the quadratic part plus the rest we get

LA

XC

FCFC

AC AC gfABC AAAB

A

C AC gfDEC A

DAE

L

A gfABC AAABA

C

gfABC fDEC A

AABADAE

LA LI

A

where LA and LI

A are the quadratic and the interacting part respectively Theinteraction terms can be extracted by the functional integration by the usual trick

Z eiS

IA

i

A

Z

A

with SIA

RdxLI

A and

Z N

ZDAe

i

Zdx

LA

XA

AA

ei

ZdxAA

Af A

Consider now f A In the Lorentz gauge

fAA AA

Under an innitesimal gauge transformation for later convenience we have changedthe denition of the gauge parameters sending A gA

AC gfABC AA

B C

we havefC A

A

C gfABC AA

B C

Then

fC A x

By

A

BCx

x y gfBAC x yA

A MCBx y

We see that f A depends explicitly on the gauge elds In order to get an ex pression for the generating functional which is convenient r deriving the Feynmanrules it is useful to give to it an exponential form This can be done recalling fromeq that a determinant can be written as a gaussian integral over Grassmannvariables

f A det

fAA x

By

A

N

ZDc ce

i

ZcAxMABx ycByd

xdy

The Grassmann elds cAx are called the ghost elds It should be noticed thatthe ghost carry zero spin and therefore in their particle interpretation lead to aviolation of the spin statistics theorem However this fact has no consequence sinceour generating functional do not generate amplitudes with external ghosts Theaction for the ghost elds can be read from the previous equation and we get

SFPG

ZdxcAx

AB gfACB A

C

cBx

The ghosts are zero mass particles interacting with the gauge elds through theinteraction term

SIFPG g

ZdxcAf

ACB A

CcB

In order to deal with this interaction it turns out convenient to dene a new gener ating functional where we introduce also external sources for the ghost elds

Z eiS

IA

i

eiSI

FPG

i

i

Z

with

Z

ZDADc c

ei

Zdx

LA

XA

AA

AAA

ei

Zdx

cAcA cA

A AcA

The integration over AA like in the abelian case For the ghost eld we use eq

with the result

ZDc ceihcAcA cA

A AcAi e

hiA i

iAi

eihA

Ai

Dening

ABx AB lim

Zdk

eikxk i

we get

Z e

i

hA xG

ABx yByieihA

xABx yByi

withGAB ABG

x y

It is then easy to read the Feynmans rules For the propagators we get

µ, ν,A B

k

Fig Gauge eld propagator

iABg

kkk i

k i

A B

k

Fig Ghost eld propagator

iAB

k i

The Feynmans rules for the vertices are obtained by taking the Fourier transform

of the interaction parts here we have written fABC fABC

µ,

ν,

A

B

C

k

k

k

1

2

3

λ,

Fig Threelinear gluon vertex

gfABCk k k gk k gk k gk k

µ,

ν,

A

B C

k

k

k

1

2

3

λ,

ρ, D

k4

Fig Fourlinear gluon vertex

igXi

kihfABEfCDEgg gg

fACEfBDEgg gg fADEfCBEgg ggi

µ, A

B

k

C

p q

Fig Ghostgluon vertex

gfABCk p qp

When we have fermions coupled to the gauge elds we need to add the rulefor the vertex the propagator is the usual one By a simple comparison with thecoupling in QED we see that in this case the rule for the vertex is

µ, A

k

p p'

b cβ γ, ,

Fig Fermiongluon vertex

igTAbc

p p k

The function in nonabelian gauge theories

We will skip a careful treatment of the renormalization in non abelian gauge the ories However since all the interaction terms have dimensions the theory isrenormalizable by power counting On the other hand we will evaluate the betafunction since in these theories there is the possibility of having giving riseto asymptotic freedom As it is known QCD the theory of strong interactions is infact a gauge theory based on the gauge group SU and it is precisely ths featureof being an asymptotically free theory that has made it very attractive phenomeno logically In non abelian gauge theories there are several ways to dene the couplingin fact we can look at the fermionic coupling at the trilinear or at the four linearcouplings If the theory is gauge invariant and the renormalization should preservethis feature all these ways should give the same result This is in fact ensured bya set of identities among the renormalization constants Ward identities which arethe analogue of the equality Z Z in QED Therefore we will study the deni tion coming from the fermionic coupling In this case the relation between the barecoupling and the renormalized one is the same as in QED see Section

gB gZ

ZZ

where Z is the vertex renormalization constant and Z and Z are the wave func tion renormalization constants for the fermion and the gauge particle respectively

As we shall see one of the dierences with QED is that now Z Z As a conse quence we will need to evaluate all the three renormalization constants Let us startwith the fermion self energy The one loop contribution is given in Fig

k

ppp k_

Fig Oneloop fermion selfenergy

We will work in the Feynman t Hooft gauge corresponding to the choice forthe gauge eld propagator see eq The Feynman rules for this diagramare the same as in QED except for the group factors In order to avoid group theorycomplications we will consider the case of a gauge group SU with fermions in thespinor representation where all the group factors are easily evaluated At the sametime we will exhibit also the general results for a theory SUN with fermions inthe fundamental representation that is the one of dimensions N Then recallingthe expression for the electron self energy in QED see eq we get

p XA

TATAQEDp XA

TATA

"p m f

QEDp

For SU we have

XA

TATAab XA

A

A

ab

ab

In the case of a group SUN and fermions in the fundamental representation it ispossible to show that X

A

TATAab CF ab

with

CF N

N

This result is justied by the observation that the operatorP

A TATA called the

Casimir operator commutes with all the generators and therefore it must be propor tional to the identity matrix From the previous expression for we nd immediatelysee Section

Z g

CF

= + +

+ + + +

+ +

a)

b) c) d)

e) f)

Fig Oneloop expansion of the vacuum polarization diagram in nonabeliangauge theories

The next task is the evaluation of Z To this end we need the one loop expansionof the gauge particle propagator the vacuum polarization diagram This is givenin Fig Since we are using dimensional regularization we can see immediately that the lastthree diagrams diagrams d e and f of Fig are zero this is not true inother regularizations In fact all these diagrams involve the integralZ

dk

k

which in dimensional regularization gives including a mass termZdk

k a i) a

By taking the limit a we see that for the integral vanishesLet us begin our calculation from the gauge particle contributions to the vacuum

polarization The kinematics is specied in Fig We will evaluate the trun cated Greens function which diers by a factor ig from the vacuum polarizationvacuum as dened in Section We have

igaAB

gfADCfDBC

Zdk

E

ip k

ik

The factor is a symmetry factor associated to the diagram and the tensor E

comes from the trilinear vertices It is given by

E gp k gk p gp k

gp k gp k gk p

gp k p k

k pk p

p kk p k pp k p kp k

p p

p + k

k

A, B,

C,

D,

µ νρ

σ

Fig Oneloop gauge particle contribution to the vacuum polarization

The group factor is easily evaluated for SU since in this case fABC ABC Then we get

ADCDBC AB

The general result for SUN is taking into account that fABC are completelyantisymmetric

fADCfDBC CGAB

withCG N

This result can be understood by noticing that TABC ifABC gives a represen tation of the Lie algebra of the group the adjoint representation and thereforeCG is nothing but the Casimir of the adjoint By using the standard trick wereduce the two denominators in the integral to a single one

k

k p

Z

dz

k z zp

withk k pz

Operating this substitution into the tensor E taking into account that the inte gration over the odd terms in k gives zero and that for symmetry we can make thesubstitution

kk

g

k

we get by calling k with k

E gk

gp

z z

pp

z

z z z

ak

b

Performing the integral over k with the formulas of Section and putting every thing together we get

igaAB

gCGAB

i

Z

dzz zp) a ) b

By taking only the term in the expansion and integrating over zwe get

igaAB i

g

CGAB

gp

pp

As we see this contribution is not transverse that is it is not proportional topp gp

as it should be in fact we will see that adding to this diagramthe contribution of the ghost loop we recover the transverse factor The ghost con tribution is the part b of the expansion of Fig The relevant kinematics isgiven in Fig

p p

p + k

k

A, B,µ ν

C

D

Fig Oneloop ghost contribution to the vacuum polarization

We have

igbAB gfACDfBDC

Zdk

i

p ki

kp kk

By the standard procedure we get k k pz

igbAB gCGAB

Z

dz

Zdk

kk

ppz z

k z zp

where we have eliminated the terms linear in k Then we perform the integral overk

igbAB ig

CGAB

Z

dz z z

p) pp)

We perform now the integral in z Then taking the limit and keeping onlythe leading term we get

igbAB i

g

CGAB

pg

pp

Putting together this contribution with the former one we get

igaAB

bAB i

g

CGAB

pp gp

The fermionic loop contribution is given in Fig and except for the groupfactor is the same evaluated in Section

p p

p + k

k

A, B,µ ν

Fig Oneloop fermion contribution to the vacuum polarization

The leading contribution is

igcAB igTrTATBQED

ig

nF AB

pp gp

where we have used the conventional normalization for the group generators in thefermionic representation

TrTATB

nF AB

where nF is the number of fundamental representations For instance in the caseof QCD we have three dierent kind of quarks corresponding to dierent &avorsThen the one loop leading term contribution to the vacuum polarization is givenby

igAB igaAB

bAB

cAB

i g

CG

nF

pp gp

By comparison with Section we get

Z g

CG

nF

We are now left with the vertex corrections The fermion contribution is given bythe diagram of Fig

k pp'

p + kp' + k

A

BB

Fig Fermion oneloop vertex correction

Again this is the same as in QED except for the group factors the leading contri bution is

igAf i g

TBTATB

The group factor can be elaborated as follows

TBTATB TBTBTA ifABCTC CF TA ifABCTBTC

CF TA

fABCfBCDTD

CF

CG

TA

Therefore we obtain

igAf i g

CF

CG

TA

The nal diagram to be evaluated is the vertex correction from the trilinear gaugecoupling given in Fig We get q p p

igAg igTB

Zdk

i"k

ki

p k

gfABC gq p k gp p k gp k q

ip k

igTC

k

pp'

B

p - kp' - k

C

A,µ

Fig Oneloop vertex correction from the trilinear gauge coupling

Since we are interested only in the divergent piece the calculation can be simplieda lot by the observation that the denominator goes as k whereas the numeratorgoes at most as k Therefore the divergent piece is obtained by neglecting all theexternal momenta Therefore the expression simplies to

igAg igfABCTBTC

Zdk

"kg

k gk gkk

The group factor gives

fABCTBTC i

fABCfBCDTD

i

CGTA

whereas the numerator gives simply k Then we have

igAg g

CGTA

Zdk

k i g

CGTA

Adding up the two vertex contributions we obtain

igA igAf A

g i g

CF CGTA

Again comparison with Section gives

Z g

CF CG

Using eq we have

gB g Z Z

Z

g

a

and collecting everything together we get

a g

CG

nF

The evaluation of g goes as in the QED case see Section that is

g g

g

d

dg

a

a a a

that is

g g

CG

nF

In the case of QCD where the gauge group is SU we have

CG

and therefore

QCDg g

nF

We see that QCD is asymptotically free g for

nF

Chapter

Spontaneous symmetry breaking

The linear model

In this Section and in the following we will study from a classical point of viewsome eld theory with particular symmetry properties We will start examining thelinear model This is a model for N scalar elds with a symmetry ON Thelagrangian is given by

L

NXi

ii

NXi

ii

NXi

ii

This lagrangian is invariant under linear transformations acting upon the vector N and leaving invariant its norm

jj NXi

ii

Consider an innitesimal transformation from now on we will omit the index ofsum over the indices which are repeated

i ijj

The condition for letting the norm invariant gives

j j jj

from which

or in componentsiijj

This is satised byij ji

showing that the rotations in N dimensions depend on NN parameters Fora nite transformation we have

jj jj

withi Sijj

implyingSST

In fact by exponentiating the innitesimal transformation one gets

S e

withT

implying that S is an orthogonal transformation The matrices S form the rotationgroup in N dimensions ON

The Noethers theorem implies a conserved current for any symmetry of thetheory In this case we will get NN conserved quantities It is useful forfurther generalizations to write the innitesimal transformation in the form

i ijj i

ABT

ABij j i j N AB N

which is similar to what we did in Section when we discussed the Lorentz trans formations By comparison we see that the matrices TAB are given by

TABij iAi

Bj Aj

Bi

It is not dicult to show that these matrices satisfy the algebra

TAB TCD iACTBD iADTBC iBDTAC iBCTAD

This is nothing but the Lie algebra of the group ON and the TAB are the in nitesimal generators of the group Applying now the Noethers theorem we ndthe conserved current

j L

ii i

iABT

ABij j

since the NN parameters AB are linearly independent we nd the NN conserved currents

JAB iiTABij j

In the case of N the symmetry is the same as for the charged eld in a realbasis and the only conserved current is given by

J J

One can easily check that the charges associated to the conserved currents closethe same Lie algebra as the generators TAB More generally if we have conservedcurrents given by

jA iiTAij j

withTA TB ifABCTC

then using the canonical commutation relations we get

QA QB ifABCQC

with

QA

Zdx jA x i

Zdx iT

Aij j

A particular example is the case N We parameterize our elds in the form

These elds can be arranged into a matrix

M i

where are the Pauli matrices Noticing that is pure imaginary and realand that anticommutes with and we get

M M

Furthermore we have the relation

jj jj

TrM yM

Using this it is easy to write the lagrangian for the model in the form

L

TrM

yM

TrM yM

TrM yM

This lagrangian is invariant under the following transformation of the matrix M

M LMRy

where L and R are two special that is with determinant equal to unitary matricesthat is LR SU The reason to restrict these matrices to be special is thatonly in this way the transformed matrix satisfy the condition In fact if A isa matrix with detA then

AT A

Therefore for M LMRy we get

M L

MRT LMRT

and from

L L

yT Ly

L

RT R Ry

since the L and R are independent transformations the invariance group in thisbasis is SUL SUR In fact this group and O are related by the followingobservation the transformation M LMRy is a linear transformation on thematrix elements of M but from the relation we see that M LMRy leavesthe norm of the vector invariant and therefore the same must be truefor the linear transformation acting upon the matrix elements of M that is on and Therefore this transformation must belongs to O This shows that thetwo groups SU SU and O are homomorphic actually there is a to relationship since L and R dene the same S as L and R

We can evaluate the eect of an innitesimal transformation To this end wewill consider separately left and right transformations We parameterize the trans formations as follows

L i

L R i

R

then we get

LM i

L M i

L i

L i

L L

where we have usedij ij iijkk

SinceLM L iL

we get

L

L L

L L

Analogously we obtain

R

R R

R R

The combined transformation is given by

L R

L R L R

and we can check immediately that

as it must be for a transformation leaving the form jj invariant Of particular

interest are the transformations with L R In this case we have L R and

M LMLy

These transformations span a subgroup SU of SULSUR called the diag onal subgroup In this case we have

We see that the transformations corresponding to the diagonal SU are the rota tions in the dimensional space spanned by These rotations dene a subgroupO of the original symmetry group O From the Noethers theorem we get theconserved currents

jL

L

L L

and dividing by LJL

and analogouslyJR

Using the canonical commutation relations one can verify that the correspondingcharges satisfy the Lie algebra of SUL SUR

QLi Q

Lj iijkQ

Lk QR

i QRj iijkQ

Rk QL

i QRj

By taking the following combinations of the currents

JV

JL JR JA

JL JR

one hasJV

andJA

The corresponding algebra of charges is

QVi Q

Vj iijkQ

Vk QV

i QAj iijkQ

Ak QA

i QAj iijkQ

Vk

These equations show that QVi are the innitesimal generators of a subgroup SU

of SUL SUR which is the diagonal subgroup as it follows from

QVi j iijkk QV

i

In the following we will be interested in treating the interacting eld theoriesby using the perturbation theory As in the quantum mechanical case this is welldened only when we are considering the theory close to a minimum the energy ofthe system In fact if we are going to expand around a maximum the oscillation ofthe system can become very large leading us outside of the domain of perturbationtheory In the case of the linear model the energy is given by

H

ZdxH

Zdx

NXi

L i

i L

Zdx

NXi

i jrij V jj

Since in the last member of this relation the rst two terms are positive deniteit follows that the absolute minimum is obtained for constant eld congurationssuch that

V jji

Let us call by vi the generic solution to this equation in general it could happen thatthe absolute minimum is degenerate Then the condition for getting a minimum isthat the eigenvalues of the matrix of the second derivatives of the potential at thestationary point are denite positive In this case we dene new elds by shiftingthe original elds by

i i i vi

The lagrangian density becomes

L

i

i V j vj

Expanding V in series of i we get

V V jvj

V

ij

v

ij

This equation shows that the particle masses are given by the eigenvalues of thesecond derivative of the potential at the minimum In the case of the linear modelwe have

V

jj

jj

ThereforeV

i i ijj

In order to have a solution to the stationary condition we must have i or

jj

This equation has real solutions only if However in order to have apotential bounded from below one has to require therefore we may have nonzero solutions to the minimum condition only if But notice that in thiscase cannot be identied with a physical mass these are given by the eigenvaluesof the matrix of the second derivatives of the potential at the minimum and theyare positive denite by denition We will study this case in the following SectionsIn the case of the minimum is given by i and one can study thetheory by taking the term jj as a small perturbation that is requiring thatboth and the values of i the &uctuations are small The free theory is givenby the quadratic terms in the lagrangian density and they describe N particles ofcommon mass m Furthermore both the free and the interacting theories are ONsymmetric

Spontaneous symmetry breaking

In this Section we will see that the linear model with is just but an exampleof a general phenomenon which goes under the name of spontaneous symmetrybreaking of symmetry This phenomenon lies at the basis of the modern descriptionof phase transitions and it has acquired a capital relevance in the last years in alleld of physics The idea is very simple and consists in the observation that atheory with hamiltonian invariant under a symmetry group may not show explicitlythe symmetry at the level of the solutions As we shall see this may happen whenthe following conditions are realized

The theory is invariant under a symmetry group G

The fundamental state of the theory is degenerate and transforms in a nontrivial way under the symmetry group

Just as an example consider a scalar eld described by a lagrangian invariant underparity

P

The lagrangian density will be of the type

L

V

If the vacuum state is non degenerate barring a phase factor we must have

P ji ji

since P commutes with the hamiltonian It follows

hjji hjPPPP ji hjPPji hjji

from whichhjji

Things change if the fundamental state is degenerate This would be the case in theexample if

V

with In fact this potential has two minima located at

v v

r

By denoting with jRi e jLi the two states corresponding to the classical congura tions v we have

P jRi jLi jRi

ThereforehRjjRi hRjPPPP jRi hLjjLi

which now does not imply that the expectation value of the eld vanishes In thefollowing we will be rather interested in the case of continuous symmetries So letus consider two scalar elds and a lagrangian density with symmetry O

L

where

For there is a unique fundamental state minimum of the potential whereas for there are innite degenerate states given by

jj v

with v dened as in By denoting with R the operator rotating the eldsin the plane in the non degenerate case we have

Rji ji

andhjji hjRRRRji hjji

since In the case degenerate case we have

Rji ji

where ji is one of the innitely many degenerate fundamental states lying on the

circle jj v Then

hjiji hjRRiRRji hji ji

withi RiR

i

Again the expectation value of the eld contrarily to the non degenerate statedoes not need to vanish The situation can be described qualitatively saying that theexistence of a degenerate fundamental state forces the system to choose one of theseequivalent states and consequently to break the symmetry But the breaking is onlyat the level of the solutions the lagrangian and the equations of motion preservethe symmetry One can easily construct classical systems exhibiting spontaneoussymmetry breaking For instance a classical particle in a double well potentialThis system has parity invariance x x where x is the particle position Theequilibrium positions are around the minima positions x If we put the particleclose to x it will perform oscillations around that point and the original symmetryis lost A further example is given by a ferromagnet which has an hamiltonianinvariant under rotations but below the Curie temperature exhibits spontaneousmagnetization breaking in this way the symmetry These situations are typical forthe so called second order phase transitions One can describe them through theLandau free energy which depends on two dierent kind of parameters

Control parameters as for the scalar eld and the temperature for theferromagnet

Order parameters as the expectation value of the scalar eld or as themagnetization

The system goes from one phase to another varying the control parameters and thephase transition is characterized by the order parameters which assume dierentvalues in dierent phases In the previous examples the order parameters were zeroin the symmetric phase and dierent from zero in the broken phase

The situation is slightly more involved at the quantum level since spontaneoussymmetry breaking cannot happen in nite systems This follows from the existenceof the tunnel eect Let us consider again a particle in a double well potential andrecall that we have dened the fundamental states through the correspondence withthe classical minima

x x jRix x jLi

But the tunnel eect gives rise to a transition between these two states and as aconsequence it removes the degeneracy In fact due to the transition the hamiltonianacquires a non zero matrix element between the states jRi and jLi By denotingwith H the matrix of the hamiltonian between these two states we get

H

The eigenvalues of H are

We have no more degeneracy and the eigenstates are

jSi pjRi jLi

with eigenvalue ES and

jAi pjRi jLi

with eigenvalue EA One can show that and therefore the funda mental state is the symmetric one jSi This situation gives rise to the well knowneect of quantum oscillations We can express the states jRi and jLi in terms ofthe energy eigenstates

jRi pjSi jAi

jLi pjSi jAi

Let us now prepare a state at t by putting the particle in the right minimumThis is not an energy eigenstate and its time evolution is given by

jR ti p

eiEStjSi eiEAtjAi

peiESt

jSi eitE jAi

with E EA ES Therefore for t E the state jRi transforms into thestate jLi The state oscillates with a period given by

T

E

In nature there are nite systems as sugar molecules which seem to exhibit sponta neous symmetry breaking In fact one observes right handed and left handed sugarmolecules The explanation is simply that the energy dierence E is so small thatthe oscillation period is of the order of years

The splitting of the fundamental states decreases with the height of the potentialbetween two minima therefore for innite systems the previous mechanism doesnot work and we may have spontaneous symmetry breaking In fact coming backto the scalar eld example its expectation value on the vacuum must be a constantas it follows from the translational invariance of the vacuum

hjxji hjeiPxeiPxji hjji v

and the energy dierence between the maximum at and the minimum at v becomes innite in the limit of innite volume

H H v ZV

dx

v

v

ZV

dx

V

The Goldstone theorem

From our point of view the most interesting consequence of spontaneous symmetrybreaking is the Goldstone theorem This theorem says that for any continuoussymmetry spontaneously broken there exists a massless particle the Goldstoneboson The theorem holds rigorously in a local eld theory under the followinghypotheses

The spontaneous broken symmetry must be a continuous one

The theory must be manifestly covariant

The Hilbert space of the theory must have a denite positive norm

We will limit ourselves to analyze the theorem in the case of a classical scalar eldtheory Let us start considering the lagrangian for the linear model with invarianceON

L

The conditions that V must satisfy in order to have a minimum are

V

l l ljj

with solutions

l jj v v

r

The character of the stationary points can be studied by evaluating the secondderivatives

V

lm lm

jj lm

We have two possibilities

we have only one real solution given by which is a minimumsince

V

lm lm

there are innite solutions among which is a maximum Thepoints of the sphere jj v are degenerate minima In fact by choosingl vlN as a representative point we get

V

lm vlNmN

Expanding the potential around this minimum we get

V Vminimum

V

lm

minimum

l vlNm vmN

If we are going to make a perturbative expansion the right elds to be used arel vlN and their mass is just given by the coecient of the quadratic term

Mlm

V

lm

minimo

lNmN

Therefore the masses of the elds a a N and N v are givenby

ma m

By deningm

we can write the potential as a function of the new elds

V m

m

rm

NXa

a

NXa

a

In this form the original symmetry ON is broken However a residual symmetryON is left In fact V depends only on the combination

PNa a and it is

invariant under rotations around the axis we have chosen as representative for thefundamental state v It must be stressed that this is not the most generalpotential invariant under ON In fact the most general potential up to thefourth order in the elds describing N scalar elds with a symmetry ON would depend on coupling constants whereas the one we got depends only on thetwo parameters m and Therefore spontaneous symmetry breaking puts heavyconstraints on the dynamics of the system We have also seen that we have N massless scalars Clearly the rotations along the rst N directions leave thepotential invariant whereas the N rotations on the planes a N move awayfrom the surface of the minima This can be seen also in terms of generators Sincethe eld we have chosen as representative of the ground state is ijmin viN wehave

T abij jjmin iai

bj bi

aj vjN

since a b N and

T aNij jjmin iai

Nj Ni

aj vjN ivai

Therefore we have N broken symmetries and N massless scalars Thegenerators of ON divide up naturally in the generators of the vacuum symmetry

here ON and in the so called broken generators each of them correspondingto a massless Goldstone boson In general if the original symmetry group G of thetheory is spontaneously broken down to a subgroup H which is the symmetry ofthe vacuum the Goldstone bosons correspond to the generators of G which are leftafter subtracting the generators of H Intuitively one can understand the origin ofthe massless particles noticing that the broken generators allow transitions from apossible vacuum to another Since these states are degenerate the operation doesnot cost any energy From the relativistic dispersion relation this implies that wemust have massless particles One can say that Goldstone bosons correspond to &atdirections in the potential

The Higgs mechanism

We have seen that the spontaneous symmetry breaking mechanism in the case ofcontinuous symmetry leads to massless scalar particles the Goldstone bosons Alsogauge theories lead to massless vector bosons in fact as in the electromagnetic casegauge invariance forbids the presence in the lagrangian of terms quadratic in theelds Unfortunately in nature the only massless particles we know are the photonand perhaps the neutrinos which however are fermions But once one couplesspontaneous symmetry breaking to a gauge symmetry things change In fact if welook back at the hypotheses underlying a gauge theory it turns out that Goldstonetheorem does not hold in this context The reason is that it is impossible to quantizea gauge theory in a way which is at the same time manifestly covariant and has aHilbert space with positive denite metric This is well known already for theelectromagnetic eld where one has to choose the gauge before quantization Whathappens is that if one chooses a physical gauge as the Coulomb gauge in orderto have a Hilbert space spanned by only the physical states than the theory loosesthe manifest covariance If one goes to a covariant gauge as the Lorentz one thetheory is covariant but one has to work with a big Hilbert space with non denitepositive metric and where the physical states are extracted through a supplementarycondition The way in which the Goldstone theorem is evaded is that the Goldstonebosons disappear and at the same time the gauge bosons corresponding to thebroken symmetries acquire mass This is the famous Higgs mechanism

Let us start with a scalar theory invariant under O

L

and let us analyze the spontaneous symmetry breaking mechanism If thesymmetry is broken and we can choose the vacuum as the state

v v

r

After the translation v with hjji we get the potential m

V m

m

rm

In this case the group O is completely broken except for the discrete symmetry The Goldstone eld is This has a peculiar way of transformingunder O In fact the original elds transform as

from which v

We see that the Goldstone eld undergoes a rotation plus a translation v Thisis the main reason for the Goldstone particle to be massless In fact one can haveinvariance under translations of the eld only if the potential is &at in the corre sponding direction This is what happens when one moves in a way which is tangentto the surface of the degenerate vacuums in this case a circle How do things changeif our theory is gauge invariant, In that case we should have invariance under atransformation of the Goldstone eld given by

x xx xv

Since x is an arbitrary function of the space time point it follows that we canchoose it in such a way to make x vanish In other words it must be possibleto eliminate the Goldstone eld from the theory This is better seen by using polarcoordinates for the elds that is

q sin

p

Under a nite rotation the new elds transform as

It should be also noticed that the two coordinate systems coincide when we are closeto the vacuum as when we are doing perturbation theory In fact in that case wecan perform the following expansion

q v v v

v

v

Again if we make the theory invariant under a local transformation we will haveinvariance under

x x x

By choosing x x we can eliminate this last eld from the theory The onlyremaining degree of freedom in the scalar sector is x

Let us study the gauging of this model It is convenient to introduce complexvariables

p i y

p i

The O transformations become phase transformations on

ei

and the lagrangian can be written as

L y y y

We know that it is possible to promote a global symmetry to a local one by intro ducing the covariant derivative

igA

from which

L igAy igA y y

FF

In terms of the polar coordinates we have

pei y

pei

By performing the following gauge transformation on the scalars

ei

and the corresponding transformation on the gauge elds

A A A

g

the lagrangian will depend only on the elds and A we will put again A

A

L

igA

igA

FF

In this way the Goldstone boson disappears We have now to translate the eld

v hjji

and we see that this generates a bilinear term in A coming from the covariantderivative given by

gvAA

Therefore the gauge eld acquires a mass

mA gv

It is instructive to count the degrees of freedom before and after the gauge trans formation Before we had degrees of freedom two from the scalar elds and twofrom the gauge eld After the gauge transformation we have only one degree offreedom from the scalar sector but three degrees of freedom from the gauge vectorbecause now it is a massive vector eld The result looks a little bit strange butthe reason why we may read clearly the number of degrees of freedom only after thegauge transformation is that before the lagrangian contains a mixing term

A

between the Goldstone eld and the gauge vector which makes complicate to readthe mass of the states The previous gauge transformation realizes the purpose ofmaking that term vanish The gauge in which such a thing happens is called theunitary gauge

We will consider now the further example of a symmetry ON The lagrangianinvariant under local transformations is

L

DijjD

ikk

ii

ii

whereDij ij i

g

TABijW

AB

where TABlm iAl Bm Am

Bl In the case of broken symmetry we

choose again the vacuum along the direction N with v dened as in

i viN

Recalling that

T abij j

min

T aNij j

min

ivai a b N

the mass term for the gauge eld is given by

gTAB

ij j

min

TCDikk

min

WAB W CD

gT aN

ij j

min

T bNikk

min

W aN W bN

gvai

biW

aN W bN

gvW aN

W bN

Therefore the elds W aN associated to the broken directions T aN acquire a mass

gv whereas W ab associated to the unbroken symmetry ON remain mass

lessIn general if G is the global symmetry group of the lagrangian H the subgroup

of G leaving invariant the vacuum and GW the group of local gauge symmetriesGW G one can divide up the broken generators in two categories In the rstcategory fall the broken generators lying in GW they have associated massive vectorbosons In the second category fall the other broken generators they have associatedmassless Goldstone bosons Finally the gauge elds associated to generators of GW

lying in H remain massless From the previous derivation this follows noticing thatthe generators of H annihilate the minimum of the elds leaving the correspondinggauge bosons massless whereas the non zero action of the broken generators generatea mass term for the other gauge elds

The situation is represented in Fig

G

H G W

Fig This gure shows the various groups G the global symmetry of thelagrangian H G the symmetry of the vacuum and GW the group of local symmetries The broken generators in GW correspond to massive vector bosons Thebroken generators do not belonging to GW correspond to massless Goldstone bosonsTh unbroken generators in GW correspond to massless vector bosons

We can now show how to eliminate the Goldstone bosons In fact we can denenew elds a and as

i

eiT aNa

iN

v

where a N that is the sum is restricted to the broken directions Theother degree of freedom is in the other factor The correspondence among the elds

and a can be seen easily by expanding around the vacuumeiT aN a

iN

iN iT aN iNa iN ai a

from whichi a v

showing that the as are really the Goldstone elds The unitary gauge is denedthrough the transformation

i eiT

aNaij

j iN v

W eiTaNaWe

iT aN a i

g

e

iT aNaeiT aN a

This transformation eliminates the Goldstone degrees of freedom and the resultinglagrangian depends on the eld on the massive vector elds W aN

and on themassless eldW ab

Notice again the counting of the degrees of freedom NNN N in a generic gauge and N N N N in theunitary gauge

Quantization of a spontaneously broken gauge

theory in the Rgauge

In the previous Section we have seen that in the unitary gauge some of the gaugeelds acquire a mass The corresponding quadratic part of the lagrangian will bewe consider here the abelian case corresponding to a symmetry U or O

L

FF

m

W

where W is the massive gauge eld called also a Yang Mills massive eld Thefree equations of motion are

mW W

Let us study the solutions of this equation in momentum space We dene

Wx

ZdWke

ikx

It is convenient to introduce a set of four independent four vectors By puttingk Ek we introduce

i ni i

m

jkj E

k

jkj

withk ni jnij

Then we can decompose Wk as follows

Wk ak kbk

Since k from the equation of motion we get

k mak kbk kkbk

implyingk mak mbk

Therefore for m the eld has three degrees of freedom with k m corre sponding to transverse and longitudinal polarization When the momentum is onshell we can dene a fourth unit vector

k

m

In this way we have four independent unit vectors which satisfy the completenessrelation

X

g g

Therefore the sum over the physical degrees of freedom gives

X

g kk

m

The propagator is dened by the equation m

Gx ig

x

Solving in momentum space we nd

Gk

Zdk

eikx i

k m i

g kk

m

Therefore the corresponding Feynmans rule for the massive gauge eld propagatoris

i

k m i

g kk

m

Notice that the expression in parenthesis is nothing but the projector over the phys ical states given in eq This propagator has a very bad high energy be haviour In fact it goes to constant for k As a consequence the argument

for the renormalizability of the theory based on power counting does not hold anymore Although in the unitary gauge a spontaneously broken gauge theory does notseem to be renormalizable we have to take into account that the gauge invarianceallows us to make dierent choices of the gauge In fact we will show that thereexist an entire class of gauges where the theory is renormalizable by power countingWe will discuss this point only for the abelian case with gauge group O but theargument cab be easily extended to the non abelian case Let us consider the Otheory as presented in Section Let us recall that under a transformation of Othe two real scalar eld of the theory transform as

Therefore in order to make it a gauge theory we introduce covariant derivatives

D gW

D gW

with W transforming under a local transformation as

Wx

gx

Assuming that the spontaneous breaking occurs along the direction we introducethe elds

v

with

v

Remember that is the Higgs eld whereas is the Goldstone eld In terms ofthe new variables the lagrangian is

L

gW

gW gvW

V phi

FF

where the potential up to quadratic terms is given by see eq

V m

m

with m We see that the eect of the shift on is to produce the terms

gvW gvW

gvW

The last term is an interaction term and it does not play any role in the followingconsiderations The rst term is the mass term for the W

mW gv

whereas the second term is the mixing between the W and the Goldstone eld thatwe got rid of by choosing the unitary gauge However this choice is not unique Infact let us consider a gauge xing of the form

f B f W gv

where is an arbitrary parameter and Bx is an arbitrary function By proceedingas in Section we know that the eect of such a gauge xing is to add to thelagrangian a term

f

W gv

W

gvW

gv

The rst term is the usual covariant gauge xing The second term after integrationby parts reads

gvW

and we see that with the choice

it cancels the unwanted mixing term Finally the third contribution

gv

is a mass term for the Goldstone eld

m gv m

W

However this mass is gauge dependent it depends on the arbitrary parameter and this is a signal of the fact that in this gauge the Goldstone eld is ctitiousIn fact remember that it disappears in the unitary gauge We will see later in anexample that evaluating a physical amplitude the contribution of the Goldstonedisappears as it should be Finally we have to consider the ghost contribution fromthe Faddeev Popov determinant For this purpose we need to know the variation ofthe gauge xing f under an innitesimal gauge transformation

fx

gx gvxx v

givingfx

y

gx y gvx vx y

Therefore the ghost lagrangian can be written as

cc gvcc gvcc

Notice that in contrast with QED the ghosts cannot be ignored since they are cou pled to the Higgs eld We are now in the position of reading the various propagatorsfrom the quadratic part of the lagrangian

Higgs propagator i

k m

Goldstone propagator i

k mW

Ghost propagator c i

k mW

The propagator for the vector eld is more involved let us collect the variousquadratic terms in W

FF

m

WW

W

In momentum space wave operator is

gk kkmWg

kk

which can be rewritten asg kk

k

k m

W kk

k

k m

W

The propagator in momentum space is the inverse of the wave operator except fora factor i To invert the operator we notice that it is written as a combination oftwo orthogonal projection operators P and P

P P

The inverse of such an operator is simply

P

P

Therefore the W propagator is given by

i

k mW

g kk

k

i

k mW

kk

k

i

k mW

g kk

k mW

We see that for this class of gauges the asymptotic behaviour of the propagator isk and we can apply the usual power counting Notice that the unitary gaugeis recovered in the limit In this limit the W propagator becomes the onediscussed at the beginning of the Section Of course the singularity of the limitdestroys the asymptotic behaviour that one has for nite values of Also for the ghost and the Goldstone elds decouple since their mass goes to innity

To end this Section we observe that with this approach one is able to prove at thesame time that the theory is renormalizable and that it does not contain spuriousstates In fact the theory is renormalizable for nite values of and it does notcontain spurious states for But since the gauge invariant quantities do notdepend on the gauge xing they do not depend in particular on and thereforeone gets the wanted result

cancellation in perturbation theory

In this Section we will discuss the cancellation of the dependent terms in thecalculation of a simple amplitude at tree level To this end we consider again theO model of the previous Section and we add to the theory a fermion eld Wewill also mimic the standard model by making transform under the gauge grouponly the left handed part of the fermion eld That is we decompose as

L R

where

L

R

and we assume the following transformation under the gauge group

L eiL R R

A fermion mass term would destroy the gauge invariance since it couples left handedand right handed elds In fact

!L

y

!

and therefore

! !

!LR !RL

Therefore we will introduce a Yukawa coupling between the scalar eld and thefermions in such a way to be gauge invariant and to gives rise to a mass term forthe fermion when the symmetry is spontaneously broken We will write

Lf !Li "DL !Ri"R f !LR# !RL#

withD igW

Here we have used again the complex notation for the scalar eld as in eq and we remind that it transforms as

# ei#

Therefore Lf is gauge invariant The lagrangian can be rewritten in terms of theeld by writing explicitly the chiral projectors We get

Lf !i"

!i"

g !

W

fp

!

i !

i

from which

L !i" g !

W fp

!v i !

where v and This expression shows that the fermion eld acquiresa mass through the Higgs mechanism given by

mf fvp

The previous expression shows also that the Higgs eld is a scalar parity whereas the Goldstone eld is a pseudoscalar parity Consider now the fermion fermion scattering in this model at tree level The amplitude gets contribution fromthe diagrams of Fig

p

p' k

k'

W

q

φ χ+ +

Fig The fermionfermion scattering in the U Higgs model

We want to show that the contribution cancels out The dependence arises fromthe gauge eld and the Goldstone eld exchange Therefore we will consider onlythese two contributions Let us start with the Goldstone exchange The Feynmansrule for the vertex is given in Fig

φ

Fig Goldstonefermion vertexfp

The corresponding amplitude is

Mf

fp

!up upi

q mW

!uk uk

where q p p k k The Feynman rule for the W fermion vertex is given inFig

Fig Wfermion vertex ig

The corresponding amplitude is

MW ig!up

up

iq m

W

g qq

q mW

!uk

uk

It is convenient to isolate the dependence by rewriting the gauge eld propagatoras

iq m

W

g qq

mW

qq

mW

q mW

i

q mW

g qq

mW

iq m

W

qq

mW

The rst term is independent and it is precisely the W propagator in the unitarygauge The second term can be simplied by using the identity

q!up

up p p!up

up

!up

"p "p

up mf !up

up

By using this relation we have

qq!up

up!uk

uk

mf!up upmf!uk

uk

fp

v!up up!uk uk

where we have made use of eq Substituting this result in MW we get

MW ig!up

up

iq m

W

g qq

mW

!uk

uk

i

q mW

gv

mW

fp

!up up!uk uk

Finally using mW gv we see that the last term cancels exactly the M ampli

tude Therefore the result is the same that we would have obtained in the unitarygauge where there is no Goldstone contribution

Now let us make some comment about the result In QED we have used thefact that the qq terms in the propagator do not play a physical rolesince theyare contracted with the fermionic current which is conserved In the present casethis does not happens but we get a term proportional to the fermion mass Infact the fermionic current ! it is not conserved because the fermionacquires a mass through the spontaneous breaking of the symmetry On the otherhand the term that one gets in this way it is just the one necessary to give rise tothe cancellation with the Goldstone boson contribution It is also worth to comparewhat happens for dierent choices of For the gauge propagator is transverse

ik m

W

g kk

k

This corresponds to the Lorentz gauge The propagator has a pole term at k which however it is cancelled by the Goldstone pole which also is at k For thechoice the gauge propagator has the very simple form

igk m

W

This is the Feynman t Hooft gauge Here the role of the Goldstone boson whichhas a propagator

i

k mW

it is to cancel the contribution of the time like component of the gauge eld whichis described by the polarization vector

In fact the gauge propagator can be

rewritten as

igk m

W

i

k mW

X

g i

k mW

X

i

k mW

kk

mW

The rst piece is the propagator in the unitary gauge whereas the second piece iscancelled by the Goldstone boson contribution

Chapter

The Standard model of the

electroweak interactions

The Standard Model of the electroweak in

teractions

We will dscribe now the Weinberg Salam model for the electroweak interactionsLet us recall that the Fermi theory of weak interactions is based on a lagrangianinvolving four Fermi elds This lagrangian can be written as the product of twocurrents which are the sum of various pieces

LF GFpJ J

Limiting for the moment our considerations to the charged current for the electronand its neutrino here e the current is given by

J !e J

J

y ! e

To understand the symmetry hidden in this couplings it is convenient to introducethe following spinors with components

L

LeL

e

where we have introduced left handed elds The right handed spinors projectedout with dont feel the weak interaction This is in fact the physicalmeaning of the V A interaction By using

!L !L !eL

! !e

we can write the weak currents in the following way

J !e

!L !eL

L

!L !eL

LeL

!LL

where we have dened

i

the is being the Pauli matrices Analogously

J !LL

In the fties the hypothesis of the intermediate vector bosons was advanced Ac cording to this idea the Fermi theory was to be regarded as the low energy limitof a theory where the currents were coupled to vector bosons very much as QEDHowever due to the short range of the weak interactions these vector bosons had tohave mass The interaction among the vector bosons and the fermions is given by

Lint g

p

J W J

W

gp!LW

WL

The coupling g can be related to the Fermi constant through the relation

GFp

g

mW

Introducing real elds in place of W

W W iWp

we get

Lint g

!LW

W

L

Let us now try to write the electromagnetic interaction within the same formalism

Lem e !eQeA e !eQ

eA

e !eLQeL !eRQeRA

where

eR

e !eR !e

where Q is the electric charge of the electron in unit of the electric charge ofthe proton e It is also convenient to denote the right handed components of theelectron as

R eR

We can write

!eLeL !L

L !L

L

and !eReR !RR

from which using Q

Lem e

!LL !RR !L

L

A ejem A

From this equation we can write

jem j

jY

withj !L

L

andjY !LL !RR

These equations show that the electromagnetic current is a mixture of j which isa partner of the charged weak currents and of jY which is another neutral currentThe following notations unify the charged currents and j

ji !LiL

Using the canonical anticommutators for the Fermi elds it is not dicukt to provethat the charges associated to the currents

Qi

Zdx jix

span the Lie algebra of SU

Qi Qj iijkQk

The charge QY associated to the current jY commutes with Qi as it can be easilyveried noticing that the right handed and left handed elds commute with eachother Therefore the algebra of the charges is

Qi Qj iijkQk Qi QY

This is the Lie algebra of SU U since the Qis generate a group SUwhereas QY generates a group U with the two groups commuting among them selves To build up a gauge theory from these elements we have to start with aninitial lagrangian possessing an SU U global invariance This will produce

gauge vector bosons Two of these vectors are the charged one already introducewhereas we would like to identify one of the the other two with the photon Byevaluating the commutators of the charges with the elds we can read immediatelythe quantum numbers of the various particles We have

Lc Qi

Zdx Lc L

ya

i

abLb

i

cbLb

andRQi

Lc QY Lc RQY R

These relations show that L transforms as an SU spinor or as the representation where we have identied the representation with its dimensionality R belongs tothe trivial representation of dimension Putting everything together we have thefollowing behaviour under the representations of SU U

L R

The relation gives the following relation among the dierent charges

Qem Q

QY

As we have argued if SU U has to be the fundamental symmetry ofelectroweak interactions we have to start with a lagrangian exhibiting this globalsymmetry This is the free Dirac lagrangian for a massless electron and a left handedneutrino

L !Li"L !Ri"R

Mass terms mixing left and right handed elds would destroy the global symmetryFor instance a typical mass term gives

m ! my

m !RL !LR

We shall see later that the same mechanism giving mass to the vector bosons can beused to generate fermion masses A lagrangian invariant under the local symmetrySU U is obtained through the use of covariant derivatives

L !Li ig

iW i

ig

Y

L !Ri igY R

The interaction term is

Lint g !L

W

L g

!LLY

g !RRY

In this expression we recognize the charged interaction with W We have also asexpected two neutral vector bosons W and Y The photon must couple to theelectromagnetic current which is a linear combination of j and jY The neutralvector bosons are coupled to these two currents so we expect the photon eldA to be a linear combination of W

and Y It is convenient to introduce twoorthogonal combination of these two elds Z and A The mixing angle can beidentied through the requirement that the current coupled to A is exactly theelectromagnetic current with coupling constant e Let us consider the part of Lint

involving the neutral couplings

LNC gjW

g

jY Y

By putting is called the Weinberg angle

W Z cos A sin

Y Z sin A cos

we get

LNC

g sin j g cos

jY

A

g cos j g sin

jY

Z

The electromagnetic coupling is reproduced by requiring the two conditions

g sin g cos e

These two relations allow us to express both the Weinberg angle and the electriccharge e in terms of g and g

In the low energy experiments performed before the LEP era the fun damental parameters used in the theory were e or rather the ne structure constant and sin We shall see in the following how the Weinberg angle can be elim inated in favor of the mass of the Z which is now very well known By using eq and jem j

jY we can write

LNC ejem A g cos j g sin jem jZ

Expressing g through g g tan the coecient of Z can be put in the form

g cos g sin j g sin jem g

cos j

g

cos sin jem

from which

LNC ejem A g

cos j sin jem Z ejem A

g

cos jZZ

where we have introduced the neutral current coupled to the Z

jZ j sin jem

The value of sin was evaluated initially from processes induced by neutral currentsas the scattering e at low energy The approximate value is

sin

This shows that both g and g are of the same order of the electric charge Noticethat the eq gives the following relation among the electric charge and thecouplings g and g

e

g

g

The Higgs sector in the Standard Model

According to the discussion made in the previous Section we need three brokensymmetries in order to give mass to W and Z Since SU U has four gen erators we will be left with one unbroken symmetry that we should identify withthe group U of the electromagnetism Uem in such a way to have the corre sponding gauge particle the photon massless Therefore the group Uem mustbe a symmetry of the vacuum that is the vacuum should be electrically neutralTo realize this aim we have to introduce a set of scalar elds transforming in a con venient way under SU U The simplest choice turns out to be a complexrepresentation of SU of dimension Higgs doublet As we already noticed thevacuum should be electrically neutral so one of the components of the doublet mustalso be neutral Assume that this component is the lower one than we will write

#

To determine the weak hypercharge QY of # we use the relation

Qem Q

QY

for the lower component of # We get

QY

Since QY and Qi commute both the components of the doublet must have the samevalue of QY and using again eq we get

Qem

which justies the notation for the upper component of the doublet

The most general lagrangian for # with the global symmetry SU U andcontaining terms of dimension lower or equal to four in order to have a renormal izable theory is

LHiggs #y# #y# #y#

where and The potential has innitely many minima on the surface

j#jminimum

v

with

v

Let us choose the vacuum as the state

hj#ji

vp

By putting

#

vp

h ip

# #

withhj#ji

we get

j#j

v vh jj

h

from which

VHiggs

vh vh

jj

h

jj

h

From this expression we read immediately the particle masses y

m m

m

andm

h v

It is also convenient to express the parameters appearing in the original form of thepotential in terms of m

h e v mh m

hv In this way we get

VHiggs mh

mh

h

mh

vh

jj

h

m

h

v

jj

h

Summarizing we have three massless Goldstone bosons and and a massivescalar h This is called the Higgs eld

As usual by now we promote the global symmetry to a local one by introducingthe covariant derivative Recalling that # of SU U we have

DH i

g

W i

g

Y

andLHiggs DH

#yDH# #y# #y#

To study the mass generation it is convenient to write # in a way analogous to theone used in eq

# ei v

v hp

According to our rules the exponential should contain the broken generators In factthere are and which are broken Furthermore it should contain the combinationof and which is broken remember that is the electric charge of thedoublet # and it is conserved But at any rate is a broken generator so theprevious expression gives a good representation of # around the vacuum In factexpanding around this state

ei v

v hp

iv i iv

i iv iv

v hp

p

i i

v h i

and introducing real components for # ip

# p

iv h i

we get the relation among the two sets of coordinates

h h

By performing the gauge transformation

# ei v#

W ei v W

ei v

i

g

ei v

ei v

Y Y

the Higgs lagrangian remains invariant in form except for the substitution of # with vh

p We will also denote old and new gauge elds with the same symbol

Dening

#d

#u

the mass terms for the scalar elds can be read by substituting in the kinetic term# with its expectation value

# vp#d

We get

ig

W

g

Y

vp#d i

gpW W

g

W

g

Y

vp#d

i vp

gpW#u

gW gY #d

Since #d and #u are orthogonals the mass term is

v

g

jWj

gW gY

From eq we have tan gg and therefore

sin gp

g g cos

gpg g

allowing us to write the mass term in the form

v

g

jWj g g

W cos Y sin

From eq we see that the neutral elds combination is just the Z eldorthogonal to the photon In this way we get

gv

jWj

g g

vZ

Finally the mass of the vector bosons is given by

MW

gv M

Z

g gv

Notice that the masses of the W and of the Z are not independent since theirratio is determined by the Weinberg angle or from the gauge coupling constants

MW

MZ

g

g g cos

Let us now discuss the parameters that we have so far in the theory The gaugeinteraction introduces the two gauge couplings g and g which can also be expressedin terms of the electric charge or the ne structure constant and of the Weinbergangle The Higgs sector brings in two additional parameters the mass of the Higgs

mh and the expectation value of the eld # v The Higgs particle has not been yetdiscovered and at the moment we have only an experimental lower bound on mhfrom LEP which is given by mh GeV The parameter v can be expressed interms of the Fermi coupling constant GF In fact from eq

GFp

g

MW

using the expression for MW we get

v pGF

GeV

Therefore the three parameters g g and v can be traded for e sin and GF Another possibility is to use the mass of the Z

MZ

pGF

e

sin cos

pGF sin cos

to eliminate sin

sin

s p

GFMZ

The rst alternative has been the one used before LEP However after LEP themass of the Z is very well known

MZ GeV

Therefore the parameters that are now used as input in the SM are GF and MZ

The last problem we have to solve is how to give mass to the electron since it isimpossible to construct bilinear terms in the electron eld which are invariant underthe gauge group The solution is that we can build up trilinear invariant terms inthe electron eld and in # Once the eld # acquires a non vanishing expectationvalue due to the breaking of the symmetry the trilinear term generates the electronmass Recalling the behaviour of the elds under SU U

L R #

we see that the following coupling Yukawian coupling is invariant

LY ge !L#R hc ge !L !eL

eR hc

In the unitary gauge which is obtained by using the transformation bothfor # and for the lepton eld L we get

LY ge !L !eL

v hp

eR hc

from whichLY

gevp !eLeR

gep !eLeRh hc

Since

!eLeRy !e

ey ye

e !e

e !eReL

we get

LY gevp!ee

gep!eeh

Therefore the symmetry breaking generates an electron mass given by

me gevp

In summary we have been able to reproduce all the phenomenological featuresof the V A theory and its extension to neutral currents This has been done with atheory that before spontaneous symmetry breaking is renormalizable In fact alsothe Yukawian coupling has only dimension The proof that the renormalizabilityof the theory holds also in the case of spontaneous symmetry breaking is absolutely non trivial In fact the proof was given only at the beginning of theseventies by t Hooft

The electroweak interactions of quarks and

the KobayashiMaskawaCabibbo matrix

So far we have formulated the SM for an electron neutrino pair but it can beextended in a trivial way to other lepton pairs The extension to the quarks is alittle bit less obvious and we will follow the necessary steps in this Section Firstof all we recall that in the quark model the nucleon is made up of three quarkswith composition p uud n udd Then the weak transition n p e !e isobtained through

d u e !e

and the assumption is made that quarks are point like particles as leptons There fore it is natural to write quark current in a form analogous to the leptonic currentsee eqs and

Jd !+L

+L

with

+L

uLdL

ud

Phenomenologically from the hyperon decays it was known that it was necessary afurther current involving the quark s given by

Js !+L

+L

with

+L

uLsL

us

Furthermore the total current contributing to the Fermi interaction had to be of theform

Jh Jd cos C Js sin C

where C is the Cabibbo angle and

sin C

The presence of the angle in the current was necessary in order to explain why theweak decays strangeness violating that is the ones corresponding to the transitionu s as K where suppressed with respect to the decays strangenessconserving Collecting together the two currents we get

Jh !QL

QL

with

QL

uL

dL cos C sL sin C

uLdCL

anddCL dL cos C sL sin C

This allows us to assign QL to the representation of SU U In factfrom

Qem Q

QY

we get

QY uL

and Qems Qemd

QY dCL

As far as the right handed components are concerned we assign them to SUsinglets obtaining

uR

dCR

Therefore the hadronic neutral currents contribution from quarks u d s are givenby

j !QLQL

jY

!QLQL

!uRuR

!dCRd

CR

Since the Z is coupled to the combination j sin jY it is easily seen that thecoupling contains a bilinear term in the eld dC given by

cos

Z !dCLd

CL

sin Z !dCRd

CR

This gives rise to terms of the type !ds and !sd which produce strangeness changingneutral current transitions However the experimental result is that these transitionsare strongly suppressed In a more general way we can get this result from the simpleobservation that W is coupled to the current j which has an associated charge Q

which can be obtained by commuting the charges Q and Q

Q Q

Zdxdy Qy

LQL QyLQL

ZdxQy

L QL

i

ZdxQy

LQL i

ZdxuyLuL dCyL dCL

The last term is just the charge associated to a Flavour Changing Neutral CurrentFCNC We can compare the strength of a FCNC transition which from

dCydC dyd cos C sys sin C dys syd sin C cos C

is proportional to sin C cos C with the strength of a &avour changing chargedcurrent transition as u s which is proportional to sin C From this comparisonwe expect the two transitions to be of the same order of magnitude But if wecompare K induced by the neutral current with K induced bythe charged current see Fig using BRK we nd

)K )K

This problem was solved in by the suggestion Glashow Illiopoulos andMaiani GIM that another quark with charge % should exist the charm Thenone can form two left handed doublets

QL

uLdCL

Q

L

cLsCL

wheresCL dL sin C sL cos C

K0d

s–

Z µ+

µ–

K+u

s–

W+ µ+

νµ

Fig The neutral current avour changing process K and hecharged current avour changing process K

is the combination orthogonal to dCL As a consequence the expression of Q getsmodied

Q

Zdx uyLuL cyLcL dCyL dCL sCyL sCL

ButdCyL dCL sCyL sCL dyLdL syLsL

since the transformation matrixdCLsCL

cos C sin C sin C cos C

dLsL

is orthogonal The same considerations apply toQY This mechanism of cancellationturns out to be eective also at the second order in the interaction as it shouldbe since we need a cancellation at least at the order G

F in order to cope withthe experimental bounds In fact one can see that the two second order graphscontributing to K cancel out except for terms coming from the massdierence between the quarks u and c One gets

AK GF m

c m

u

allowing to get a rough estimate of the mass of the charm quark mc GeV The charm was discovered in when it was observed the bound state !cc withJPC known as J The mass of charm was evaluated to be around GeV In there was the observation of a new vector resonance the - interpreted asa bound state !bb where b is a new quark the bottom or beauty with mb GeV Finally in the partner of the bottom the top t was discovered at Fermi LabThe mass of the top is around GeV We see that in association with three

leptonic doublets ee

L

L

L

there are three quark doubletsud

L

cs

L

tb

L

We will show later that experimental evidences for the quark b to belong to a doubletcame out from PEP and PETRA much before the discovery of top Summarizingwe have three generations of quarks and leptons Each generation includes twoleft handed doublets

AeA

L

uAdA

L

A

and the corresponding right handed singlets Inside each generation the total chargeis equal to zero In fact

Qtoti

Xfgen

Qf

where we have also taken into account that each quark comes in three colors Thisis very important because it guarantees the renormalizability of the SM In fact ingeneral there are quantum corrections to the divergences of the currents destroyingtheir conservation However the conservation of the currents coupled to the Yang Mills elds is crucial for the renormalization properties In the case of the SM forthe electroweak interactions one can show that the condition for the absence of thesecorrections Adler Bell Jackiw anomalies is that the total electric charge of theelds is zero

We will complete now the formulation of the SM for the quark sector showingthat the mixing of dierent quarks arises naturally from the fact that the quarkmass eigenstates are not necessarily the same elds which couple to the gauge eldsWe will denote the last ones by

AL

AeA

L

qAL

uAdA

L

A

and the right handed singlets by

eAR uAR dAR

We assume that the neutrinos are massless and that they do not have any right handed partner The gauge interaction is then

L XfL

!fAL

i g

W g

QY fL

Y

fAL

XfR

!fAR

i g

QY fR

fAR

where fAL AL qAL and fAR eAR u

AR d

AR We recall also the weak hyper

charge assignments

QY AL QY qAL

QY eAR QY uAR

QY dAR

Let us now see how to give mass to quarks We start with up and down quarks anda Yukawian coupling given by

gd!qL#dR hc

where # is the Higgs doublet When this takes its expectation value h#i v

p a mass term for the down quark is generated

gdvp

!dLdR !dRdL

gdvp!dd

corresponding to a mass md gdvp In order to give mass to the up quark we

need to introduce the conjugated Higgs doublet dened as

# i#

Observing that QY # another invariant Yukawian coupling is given by

gu!qL #uR hc

giving mass to the up quark mu guvp The most general Yukawian coupling

among quarks and Higgs eld is then given by

LY XAB

geAB

! AL#eBR guAB!q

AL

#uBR gdAB!qAL#d

BR

where giAB with i e u d are arbitrary matrices When we shift # by itsvacuum expectation value we get a mass term for the fermions given by

Lfermion mass vp

XAB

geAB

! ALeBR guAB!u

ALu

BR gdAB

!dALdBR

Therefore we obtain three mass matrices

M iAB vp

giAB i e u d

These matrices give mass respectively to the charged leptons and to the up anddown of quarks They can be diagonalized by a biunitary transformation

M id SyM iT

where S and T are unitary matrices depending on i and M id are three diagonal

matrices Furthermore one can take the eigenvalues to be positive This followsfrom the polar decomposition of an arbitrary matrix

M HU

where Hy H pMM y is a hermitian denite positive matrix and U y U is a

unitary matrix Therefore if the unitary matrix S diagonalizes H we have

H id SyHS SyMUS SyMT

with T US is a unitary matrix Each of the mass terms has the structure!LMR Then we get

!LMR !LSSyMT T yR !LMdR

where we have dened the mass eigenstates

L SL R TR

We can now express the charged current in terms of the mass eigenstates We have

Jh !qALq

AL !uALd

AL !uALS

uSdABdBL

DeningV SuSd V y V

we getJh !uALd

V AL

withdV AL VABdBL

The matrix V is called the Cabibbo Kobayashi Maskawa CKM matrix and itdescribes the mixing among the down type of quarks this is conventional we couldhave chosen to mix the up quarks as well We see that its physical origin is thatin general there are no relations between the mass matrix of the up quarks Muand the mass matrix of the down quarks Md The neutral currents are expressionswhich are diagonal in the (primed( states therefore their expression is the same alsoin the basis of the mass eigenstates due to the unitarity of the various S matricesIn fact

jh !uLuL !dLd

L !uLS

uSuuL !dLSdSddL !uLuL !dLdL

For the charged leptonic current since we have assumed massless neutrinos we get

J !ALSeABeBL ! ALeAL

where L SeyL is again a massless eigenstate Therefore there is no mixingin the leptonic sector among dierent generations However this is tied to ourassumption of massless neutrinos By relaxing this assumption we may generate amixing in the leptonic sector producing a violation of the dierent leptonic numbers

It is interesting to discuss in a more detailed way the structure of the CKMmatrix In the case we have n generations of quarks and leptons the matrix V being unitary depends on n parameters However one is free to choose in anarbitrary way the phase for the n up and down quark elds But an overall phasedoes not change V Therefore the CKM matrix depends only on

n n n

parameters We can see that in general it is impossible to choose the phases in sucha way to make V real In fact a real unitary matrix is nothing but an orthogonalmatrix which depends on

nn

real parameters This means that V will depend on a number of phases given by

number of phases n nn

n n

Then in the case of two generations V depends only on one real parameter theCabibbo angle For three generations V depends on three real parameters andone phase Since the invariance under the discrete symmetry CP implies that allthe couplings in the lagrangian must be real it follows that the SM in the caseof three generations implies a CP violation A violation of CP has been observedexperimentally by Christensen et al in These authors discovered that theeigenstate of CP with CP

jKi

p

jKi j !Ki

decays into a two pion state with CP with a BR

BRK

It is not clear yet if the SM is able to explain the observed CP violation in quanti tative terms

To complete this Section we give the experimental values for some of the matrixelements of V The elements Vud and Vus are determined through nuclear Kand hyperon decays using the muon decay as normalization One gets

jVudj

andjVusj

The elements Vcd and Vcs are determined from charm production in deep inelasticscattering N cX The result is

jVcdj

andjVcsj

The elements Vub and Vcb are determined from the b u and b c semi leptonicdecays as evaluated in the spectator model and from the B semi leptonic exclusivedecay B !D One gets

jVubjjVcbj

andjVcbj

More recently from the branching ratio t Wb CDF has measured jVtbjjVtbj

A more recent comprehensive analysis gives the following . CL range for thevarious elements of the CKM matrix

V

A

The parameters of the SM

It is now the moment to count the parameters of the SM We start with the gaugesector where we have the two gauge coupling constants g and g Then the Higgssector is specied by the parameters and the self coupling We will rather usev

p and mh Then we have the Yukawian sector that is that

part of the interaction between fermions and Higgs eld giving rise to the quarkand lepton masses If we assume three generations and that the neutrinos are allmassless we have three mass parameters for the charged leptons six mass parametersfor the quarks assuming the color symmetry and four mixing angles In order totest the structure of the SM the important parameters are those relative to thegauge and to the Higgs sectors As a consequence one assumes that the mass matrixfor the fermions is known This was not the case till two years ago before thetop discovery and its mass was unknown The masses of the vector bosons can beexpressed in terms of the previous parameters However the question arises about

the better choice of the input parameters Before LEP the better choice was touse quantities known with great precision related to the parameters g g v Thechoice was to use the ne structure constant

and the Fermi constant

GF GeV

as measured from the decay of the muon Then most of the experimental researchof the seventies and eighties was centered about the determination of sin Theother parameters as mt mh aect only radiative corrections which in this type ofexperiments can be safely neglected due to the relatively large experimental errorsTherefore in order to relate the set of parameters GF sin

to g g v we canuse the tree level relations The situation has changed a lot after the beginning ofrunning of LEP In fact the mass of the Z has been measured with great precisionMZMZ In this case the most convenient set is GF MZ with

MZ GeV

Also the great accuracy of the experimental measurements requires to take intoaccount the radiative corrections We will discuss this point in more detail at the endof these lectures However we notice that one gets informations about parametersas mt mh just because they aect the radiative corrections In particular theradiative corrections are particularly sensitive to the top mass In fact LEP wasable to determine the top mass in this indirect way before the top discovery

Appendix A

A Properties of the real antisymmetric matri

ces

Given a real and antisymmetric matrix n n A we want to show that it is possibleto put it in the form by means of an orthogonal transformation To thisend let us notice that iA is a hermitian matrix and therefore it can be diagonalizedthrough a unitary transformation U

UiAU y Ad A

where Ad is diagonal and real The eigenvalues of iA satisfy the equation

detjiA j A

Since AT A it follows

detjiA j detjiA T j detj iA j A

Therefore if is an eigenvalue the same is for lo e It follows that Ad can bewritten in the form

Ad

A

Notice that if n is odd Ad has necessarily a zero eigenvalue that we will write asAdnn We will assume also to order the matrix in such a way that all the i arepositiveEach submatrix of the type

i i

A

can be put in the form ii

ii

A

through the unitary matrix

V p

i i

A

In fact

V

i i

V y

ii

ii

A

Dening

V

V V

A

with Vnn for n odd we get

V AdVy

i i i i

iAs A

with As dened in Therefore

V UiAV Uy iAs A

orV UAV Uy As A

But A and As are real matrices and therefore also V U must be real However beingV U real and unitary it must be orthogonal