Advanced Quantum Field Theory Chapter 6 Non-Abelian ......Advanced Quantum Field Theory Chapter 6...

Advanced Quantum Field Theory

Chapter 6

Non-Abelian Gauge Theories

Jorge C. RomaoInstituto Superior Tecnico, Departamento de Fısica & CFTP

A. Rovisco Pais 1, 1049-001 Lisboa, Portugal

Fall 2020

Lecture 11

Lecture 11

Vacuum Polarization

S-Matrix

WT Identities in QED

Unitarity and WI

Jorge C. Romao TCA – 2

Example: Transversality of vacuum polarization

Lecture 11

Vacuum Polarization

•Formal Method

•Practical Method

S-Matrix


Unitarity and WI


❐ We are going to give an example of the application of the Ward identities. Forthis we will show that the vacuum polarization is transversal. As the puregauge theory is already non-trivial, we only consider this case, thegeneralizations being straightforward.

❐ To show the details of the calculations that will shed some light on the moreformal expression we just proved, we are going to do this example using twomethods.

❐ The first one, that we will call formal method, will use the general expressionfor the Ward identities satisfied by the generating functional of the irreducibleGreen functions, Γ.

❐ The second method, which we call practical method, will use the results of oneof the theorems on the BRS transformations that we proved before.

❐ The comparison between the two methods will be important to clarify themeaning of the expressions.

Transversality of vacuum polarization: formal method

Lecture 11

Vacuum Polarization

•Formal Method

•Practical Method

S-Matrix


Unitarity and WI


❐ For the pure gauge theory case, the expression for the Ward identities for thegenerating functional Γ is,

∫

d4x

[

δΓ

δKaµ(x)

δΓ

δAµa(x)− δΓ

δLa(x)

δΓ

δωa(x)− 1

ξF a(x)

δΓ

δωa(x)

]

= 0 [WI]

where we will choose a covariant linear gauge,

F a(x) = ∂µAaµ(x)

❐ To proceed it is necessary to know what is the meaning of the functionalderivatives, δΓ

δKaµand δΓ

δLa . From their definition we have

δΓ

δKaµ(x)

=δW

δKaµ

=δ

iδKaµ

lnZ =1

Z

δZ

iδKaµ(x)

=1

Z

∫

D(· · · )sAaµ(x)ei(Σ+sources)

Transversality of vacuum polarization: formal method

Lecture 11

Vacuum Polarization

•Formal Method

•Practical Method

S-Matrix


Unitarity and WI


❐ As sAaµ(x) = Dabµ ω

b = ∂µωa(x) + gfabcωb(x)Acµ(x), we then get

δΓ

δKaµ(x)

= ∂xµ1

Z

δZ

iδηa(x)+ gfabc

1

Z

δ2Z

iδJcµ(x)iδηb(x)

❐ Introducing now Z ≡ exp(iW ), the previous expression becomes,

δΓ

δKaµ(x)

= ∂µxδ(iW )

iδηa(x)+ gfabc

[

δ2iW

iδJcµ(x)iδηb(x)

+δiW

iδJcµ(x)

δiW

iδηb(x)

]

❐ This has the following diagrammatic representation

a µµ

bb

cc

iW

iWiW

iW

δΓ

δKaµ(x)

= ∂µx + gfabc + gfabc

where W is the generating functional for the connected Green functions.

Transversality of vacuum polarization: formal method · · ·

Lecture 11

Vacuum Polarization

•Formal Method

•Practical Method

S-Matrix


Unitarity and WI


❐ In a similar way we can show that (sωa = − 12gf

abcωbωc)

δΓ

δLa(x)=− 1

2g fabc

1

Z

δ2Z

iδηc(x)iδηb(x)

=− 1

2g fabc

[

δ2(iW )

iδηc(x)iδηb(x)+

δ(iW )

iδηc(x)

δ(iW )

iδηb(x)

]

❐ In diagrammatic form this gives

bb

cc

iW

iW

iW

δΓ

δLa(x)=−1

2g fabc − 1

2g fabc


Lecture 11

Vacuum Polarization

•Formal Method

•Practical Method

S-Matrix


Unitarity and WI


❐ We want to apply δ2

δωb(y)δAcν(z)to the original equation. We get [WI]

δ2

δωb(y)δAcν(z)

(

δΓ

δKaµ(x)

δΓ

δAµa(x)

)∣

∣

∣

∣

=0

=δ2Γ

δωb(y)δKaµ(x)

∣

∣

∣

∣

=0

δ2Γ

δAcν(z)δAµa(x)

∣

∣

∣

∣

=0

❐ But we have

δ2Γ

δωb(y)δKaµ(x)

∣

∣

∣

∣

=0

=

∫

d4w

(

−i δ2Γ

δωb(y)δωf (w)

) (

δ2Γ

iδηf (w)δKaµ(x)

)∣

∣

∣

∣

=0

= ∂µx

∫

d4w

(

−i δ2Γ

δωb(y)δωf (w)

) (

δ2(iW )

iδηf (w)iδηa(x)

)∣

∣

∣

∣

=0

+g fab′c

∫

d4w

(

−i δ2Γ

δωb(y)δωf (w)

)

(

δ3iW

iδηf (w)iδηb′

(x)iδJcµ(x)

)∣

∣

∣

∣

∣

=0

= ∂µx δ4(x− y)δab + gfab

′c

∫

d4w

(

−i δ2Γ

δωb(y)δωf (w)

)

(

δ3iW

iδηf (w)iδηb′

(x)iδJcµ(x)

)∣

∣

∣

∣

∣

=0


Lecture 11

Vacuum Polarization

•Formal Method

•Practical Method

S-Matrix


Unitarity and WI


❐ In a similar way we have for the second term,

δ2

δωb(y)δAcν(z)

(

δΓ

δLaδΓ

δωa

)∣

∣

∣

∣

=0

= 0

and for the third

δ2

δωb(y)δAcν(z)

(

1

ξ∂ρA

ρa(x)δΓ

δωa(x)

)∣

∣

∣

∣

=0

=1

ξ∂νxδ

4(x− z)δ2Γ

δωb(y)δωa(x)

∣

∣

∣

∣

=0

❐ Using these results we get

−∂yµδ2Γ

δAbµ(y)δAcν(z)

+ gfade∫

d4xd4w

(

−i δ2Γ

δωb(y)δωf (w)

)

(

δ3iW

iδηf (w)iδηd(x)iδJeµ(x)

)

(

δ2Γ

δAaµ(x)δAcν(z)

)

+1

ξ∂νz

δ2Γ

δωb(y)δωc(z)= 0


Lecture 11

Vacuum Polarization

•Formal Method

•Practical Method

S-Matrix


Unitarity and WI


❐ We now apply the Fourier transform, with the conventions shown in the Figure

p

z y

We get

−ipµ(i)G−1cbνµ(p) + gfadeiG−1ca

νµ(p)∆−1fbXµdef + (−ipν) i

ξ∆−1cb(p) = 0

❐ This can be written as

pµG−1cbνµ = −1

ξ∆−1cbpν − ig fadeG−1ca

νµ(p) ∆−1fbXµdef [1][3]

where

Xµdef = FT[

< 0|Tωd(x)ωf (w)Aµe(x)|0 >c]

≡ µd

e

fiW


Lecture 11

Vacuum Polarization

•Formal Method

•Practical Method

S-Matrix


Unitarity and WI


❐ To prove the Transversality we also need the equation of motion for theghosts. For our case this is

δΓ

δωa(z)= −∂µz

δΓ

δKµa(z)

❐ Applying the operator δδωb(y)

, we get

δ2Γ

δωb(y)δωa(z)=− ⊔⊓ δabδ4(y − z)

− gfadc∫

d4w

(

−i δ2Γ

δωb(y)δωf (w)

)

∂µz

(

δ3iW

iδJcµ(z)iδηf (w)iδηd(z)

)

❐ Applying now the Fourier transform, we get

i∆−1ab = p2δab − gfadc(−ipµ)Xdcfµ ∆−1fb[2][4]


Lecture 11

Vacuum Polarization

•Formal Method

•Practical Method

S-Matrix


Unitarity and WI


❐ The previous equations allow now to complete the proof of the transversality ofthe vacuum polarization. For this we write,

G−1abµν = G−1

Tabµν + i

a

ξδabpµpν

where pµG−1T

abµν = 0.

❐ For the free propagator we have a = 1. To show the transversality we just haveto show that the longitudinal part is not renormalized and that therefore thevalue of a remains always a = 1.

❐ Using

pµG−1abµν = i

a

ξδabp2pν

and multiplying equation[3] by pν we obtain

ia

ξp4δcb = −1

ξp2∆−1cb +

a

ξp2g f cdepµX

µdef∆−1fb


Lecture 11

Vacuum Polarization

•Formal Method

•Practical Method

S-Matrix


Unitarity and WI


❐ Using now equation [4] we get after some trivial algebra

0 = −1

ξp2∆−1cb +

a

ξp2∆−1cb

❐ This implies

a = 1

as we wanted to shown.

Transversality of vacuum polarization: practical method

Lecture 11

Vacuum Polarization

•Formal Method

•Practical Method

S-Matrix


Unitarity and WI


❐ Now we are going to use the so-called practical method based in Theorem 5

❐ Using

sωb(x) =1

ξ∂µA

µb(x)

and

sAaν = ∂νωa + gfadcωdAcν

it is easy to see that the starting Green function should be⟨

0|TAaν(x)ωb(y)|0⟩

.

❐ Then Theorem 5 tells us that

s⟨

0|TAaµ(x)ωb(y)|0⟩

= 0

that is

1

ξ

⟨

0|TAaν(x)∂µAµb(y)|0⟩

=⟨

0|T∂νωa(x)ωb(y)|0⟩

+gfadc⟨

0|Tωd(x)Acν(x)ωb(y)|0⟩

Transversality of vacuum polarization: practical method · · ·

Lecture 11

Vacuum Polarization

•Formal Method

•Practical Method

S-Matrix


Unitarity and WI


❐ We now take the Fourier transform obtaining

i

ξpρGabνρ(p) = −ipν∆ab(p) + gfadcXdcb

ν

where Xdcbν has been defined before. Multiplying by G−1νµ∆−1 we then get

pµG−1acνµ = −1

ξpν∆

−1ac − igffdeχdebν ∆−1bcG−iνµaf

which is precisely equation [1]. The equation [2] can be easily obtainedknowing that the only vertex of the ghosts is

−gfabcpµ

µ, c

a b

p

Transversality of vacuum polarization: practical method · · ·

Lecture 11

Vacuum Polarization

•Formal Method

•Practical Method

S-Matrix


Unitarity and WI


❐ Then

µ

daa abb b

cp

iW iW= +

which means

∆ab(p) =i

p2δab − i

p2gfadcpµXdcb

µ

or in another form

i∆−1ab = p2δab + igfadcpµXdcb′

µ ∆−1b′b

which is precisely equation[2]

❐ The proof of transversality follows now the same steps as in the formal case.

Gauge invariance of the S matrix

Lecture 11

Vacuum Polarization

S-Matrix


Unitarity and WI


❐ We have shown before the gauge invariance of the S matrix using theequivalence theorem and the fact that the generating functionals correspondingto different gauge conditions only differ in the source terms

❐ The proof used some properties of the Coulomb gauge and this can raise somedoubts about the general validity of the argument

❐ We are going to show here, using the Ward identities, that the functionals ZFand ZF+∆F corresponding to the gauge conditions F and F +∆F ,respectively, only differ in the source terms. As F and ∆F are arbitrary theproof is general.

❐ We have

ZF [Jaµ , Ji] =

∫

D(· · · )ei[Seff+∫d4x(Ja

µAµa+Jiφi)]

❐ Then

ZF+∆F − ZF =

∫

D(· · · )∫

d4x i

[

−1

ξF a∆F a − ωa

∫

d4yδ∆F a(x)δAbµ(y)

sAbµ(y)

−ωa∫

d4yδ∆F a(x)

δφi(y)sφi(y)

]

ei(Seff+sources)

Gauge invariance of the S matrix · · ·

Lecture 11

Vacuum Polarization

S-Matrix


Unitarity and WI


❐ We use now the Ward identities in the form that corresponds to the generatingfunctional Z, that is

0 =

∫

D(· · · )∫

d4x[JµasAaµ + J isφi + ηsω − sωη] e{i(Seff+JaµA

µa+Jiφi+ωη+ηω)}

❐ Taking the derivative in order to ηa(x) and after setting the ghost sources tozero, we get

0 =

∫

D(· · · )[

1

ξF a(x) + iωa(x)

∫

d4y[JµbsAbµ + J isφi]

]

ei[Seff+sources

or

−1ξF a[

δiδJ

]

∫

D(· · · )ei(Seff+sources) =

=

∫

D(· · · )iωa(x)∫

d4y[JµbsAbµ + Jisφi]ei(Seff+sources)


Lecture 11

Vacuum Polarization

S-Matrix


Unitarity and WI


❐ Then

∫

D(· · · )(

−1ξF a∆F a

)

ei(Seff+sources) =

= ∆F a[

δiδJ

] (

−1ξF a[

δiδJ

])

∫

D(· · · )ei(Seff+sources)

= ∆F a[

δiδJ

]

∫

D(· · · )iωa(x)∫

d4y[JµbsAbµ + J isφi]ei(Seff+sources)

=

∫

D(· · · ){

ωa(x)

∫

d4y

[

δ∆F a(x)

δAbµ(y)sAbµ(y) +

δ∆F a(x)

δφi(y)sφi(y)

]

+iωa(x)∆F a(x)

∫

d4y[JµbsAbµ + J isφi]

}

ei(Seff+sources)


Lecture 11

Vacuum Polarization

S-Matrix


Unitarity and WI


❐ We get therefore∫

D(· · · )(

−1

ξF a∆F a−ωa(x)

∫

d4y

[

δ∆F a(x)

δAbµ(y)sAbµ(η)+

δ∆F a

δφi(y)sφi(y)

])

ei(Seff+sources)

=

∫

D(· · · )iωa(x)∆F a(x)∫

d4y[

JµbsAbµ + J isφi]

ei(Seff+sources)

❐ We can then write

ZF+∆F − ZF =

=

∫

D(· · · )i∫

d4x[

iωa(x)∆F a(x)

∫

d4y(JµbsAbµ + Jisφi)]

ei(Seff+sources)

=

∫

D(· · · )ei{

Seff+

∫

d4y[Jaµ(y)Aµa(y) + JiΦi(y)]}


Lecture 11

Vacuum Polarization

S-Matrix


Unitarity and WI


❐ where

Φi(y) ≡ φi(y) + i

∫

d4x[ωa(x)∆F a(x)sφi(y)]

and

Aaµ(y) ≡ Aaµ(y) + i

∫

d4x[ωb(x)∆F b(x)sAaµ(y)]

❐ The difference between the generating functionals ZF+∆F and ZF is only inthe functional form of the source terms. We can then use the equivalencetheorem to show that the renormalized S matrix are equal in both cases.

SRF+∆F = SRF

Ward-Takahashi identities for the functional Z[J ]

Lecture 11

Vacuum Polarization

S-Matrix


•WI for Z

•WI for W & Γ

•WI for the Vertex

•Ghosts in QED

•Photon Propagator

•WI for Vertex

Unitarity and WI


❐ We will now derive again the Ward identities for QED, that we found in ourstudy of renormalization, using now the functional methods.

❐ The generating functional for the Green functions for QED is given by, in alinear gauge,

Z(Jµ, η, η) =

∫

D(Aµ, ψ, ψ)ei(Seff+

∫d4x(JµA

µ+ηψ+ψη)

where Jµ, η e η are the sources for Aµ, ψ and ψ respectively. The effectiveaction is given by,

Seff =

∫

d4x

[

LQED − 1

2ξ(∂ ·A)2

]

= SQED + SGF

where

LQED = −1

4FµνF

µν + ψ(iγµDµ −m)ψ .

Ward-Takahashi identities for the functional Z[J ] · · ·

Lecture 11

Vacuum Polarization

S-Matrix


•WI for Z

•WI for W & Γ


•Ghosts in QED


•WI for Vertex

Unitarity and WI


❐ SQED is invariant under local gauge transformation of the group U(1) that wewrite as,

δAµ = ∂µΛ

δψ = ieΛψ

δψ = −ieΛψFA = ∂µΛ, Fψ = −ieΛψ, Fψ = ieΛψ

❐ The Seff contains the part of the gauge fixing that it is not invariant underthese transformations. Therefore the Ward identities take the form,

(

δSGFδφi

[

δ

iδJ

]

+ Ji

)

Fi[

δiδJ

]

Z(J) = 0

❐ This can be written in our case as, putting back the explicit integrations,

0 =

∫

d4x

[

1

ξ∂µ∂ν

(

δ

iδJν

)

∂µΛ + Jµ∂µΛ + ieΛηδ

iδη− ieΛη

δ

iδη

]

Z(Jµ, η, η)

Ward-Takahashi identities for the functional Z[J ] · · ·

Lecture 11

Vacuum Polarization

S-Matrix


•WI for Z

•WI for W & Γ


•Ghosts in QED


•WI for Vertex

Unitarity and WI


❐ After an integration by parts we get,

∫

d4x Λ

[

−1

ξ⊔⊓∂ν

(

δ

iδJν

)

− ∂µJµ + ieη

δ

iδη− ieη

δ

iδη

]

Z(Jµ, η, η) = 0

❐ This can be written as

[

1

ξ⊔⊓∂µ

(

δ

iδJµ

)

+ ∂µJµ − ieη

(

δ

iδη

)

+ ieη

(

δ

iδη

)]

Z(J, η, η) = 0

Ward-Takahashi identities for the functionals W and Γ

Lecture 11

Vacuum Polarization

S-Matrix


•WI for Z

•WI for W & Γ


•Ghosts in QED


•WI for Vertex

Unitarity and WI


❐ From the point of view of the applications it is more useful the Ward identityfor the generating functional of the irreducible Green functions. This problemis simpler than in the case of non-abelian gauge theories, that we just discuss,as the the previous equation is linear in the functional derivatives with respectto the different sources (we notice that if we had chosen a non-linear gaugefixing this would not be true, even in QED).

❐ The linearity allow us to write immediately

∂µJµ +

[

1

ξ⊔⊓∂µ

(

δ

iδJµ

)

− ieηδ

iδη+ ieη

δ

iδη

]

W (J, η, η) = 0

where W is the generating functional for the connected Green functions,

Z(Jµ, η, η) ≡ eiW (Jµ,η,η)

❐ As we saw the generating functional for the irreducible Green functions is givenby,

Γ(Aµ, ψ, ψ) =W (Jµ, η, η)−∫

d4x[JµAµ + ηψ + ψη]

Ward-Takahashi identities for the functionals W and Γ

Lecture 11

Vacuum Polarization

S-Matrix


•WI for Z

•WI for W & Γ


•Ghosts in QED


•WI for Vertex

Unitarity and WI


❐ We also have the relations

Aµ =δW

iδJµ; ψ =

δW

iδη; ψ = −δW

iδη

and

Jµ = − δΓ

δAµ; η = − δΓ

δψ; η =

δΓ

δψ

where, as usual, the fermionic derivatives are left derivatives.

❐ We can them write

1

ξ⊔⊓∂µAµ − ∂µ

δΓ

δAµ− ie

δΓ

δψψ − ieψ

δΓ

δψ= 0

❐ This equation is the starting point to generate all the Ward identities in QED.Its application it is much easier than the equivalent expression that was provedusing the canonical formalism. The functional methods make this expressionsparticularly simple.

Example: Ward identity for the QED vertex

Lecture 11

Vacuum Polarization

S-Matrix


•WI for Z

•WI for W & Γ


•Ghosts in QED


•WI for Vertex

Unitarity and WI


❐ To convince ourselves that this equation reproduces the Ward identities thatwe already know, let us derive the Ward identity for the vertex in QED.

❐ We applyδ2

δψα(y)δψβ(z)to the master equation. We get then

∂µxδ3Γ

δψα(y)δψβ(z)δAµ(x)

=− ie

[

δ2Γ

δψα(y)δψβ(x)δ4(z − x)− δ2Γ

δψα(x)δψβ(z)δ4(y − x)

]

❐ This equation means

∂µxΓµβα(x, z, y) = −ie[

Γβα(x, y)δ4(z − x)− Γβα(z, x)δ

4(y − x)]

Example: Ward identity for the QED vertex · · ·

Lecture 11

Vacuum Polarization

S-Matrix


•WI for Z

•WI for W & Γ


•Ghosts in QED


•WI for Vertex

Unitarity and WI


❐ Taking now the Fourier transform to both sides of the equation, with themomenta defined as the Figure below,

µ

β α

pp′

q = p′ − p

we get,

qµΓµ(p′, p) = −ie[S−1(p)− S−1(p′)]

❐ This is precisely the well known Ward identity.

Ghosts in QED

Lecture 11

Vacuum Polarization

S-Matrix


•WI for Z

•WI for W & Γ


•Ghosts in QED


•WI for Vertex

Unitarity and WI


❐ We said before that the generating functional for QED was given by,

Z(Jµ, η, η) =

∫

D(Aµ, ψ, ψ)ei∫d4x[LQED+LGF+JµA

µ+ηψ+ψη]

where LQED is the usual Lagrangian for QED and the gauge fixing term was,

LGF = − 1

2ξ(∂ ·A)2 .

❐ In fact this is not strictly true. If we use the prescription for the gauge theories,we would get instead,

Z(Jµ, η, η, ζ, ζ) =

∫

D(Aµ, ψ, ψ, ω, ω)ei∫d4x[Leff+J

µAµ+ηψ+ψη+ωζ+ζω]

❐ In this expression ω e ω are anti-commuting scalar fields known as theFaddeev-Popov ghosts as we saw before. Although in physical process theynever appear as external states, it is useful to introduce also sources for themto discuss the Ward identities.

Ghosts in QED · · ·

Lecture 11

Vacuum Polarization

S-Matrix


•WI for Z

•WI for W & Γ


•Ghosts in QED


•WI for Vertex

Unitarity and WI


❐ In the previous action, the Lagrangian Leff is

Leff = LQED + LGF + LG

where

LG = −ω ⊔⊓ ω

❐ The reason why in QED we can work with the functional Z instead of Z isbecause the ghosts do not have interactions with the gauge fields and can beintegrated out (Gaussian integration) and absorbed in the normalization.

❐ Nevertheless, for the Ward identities it is useful to keep them. The effectiveLagrangian, Leff , is invariant under the BRS transformations given by,

δψ = ieωθψ

δψ = −ieψωθδAµ = ∂µωθ

δω = 1ξ(∂ ·A)θ

δω = 0

Ghosts in QED · · ·

Lecture 11

Vacuum Polarization

S-Matrix


•WI for Z

•WI for W & Γ


•Ghosts in QED


•WI for Vertex

Unitarity and WI


❐ The parameter θ is an anti-commuting (Grassmann variable). The BRStransformations on the physical fields are gauge transformations withparameter Λ = ωθ and therefore LQED is left invariant. The transformationsin the ghosts ω and ω are such that the variation of LGF cancels that of LG,just like in the non-abelian case.

❐ The invariance of the integration measure and of Seff allows us to writeimmediately the Ward identities for the generating functionals.

❐ The BRS transformations allow us to obtain the Ward identities in a quick waywithout having to resort to the functional Γ. This method is based on the fact,as we saw in Theorem 5, that the application of the operator δBRS to anyGreen function gives zero, that is

δBRS 〈0|TAµ1· · ·ω · · ·ω · · ·ψ · · ·ψ · · · |0〉 = 0

❐ Let us show two simple applications of the method in QED.

The non-renormalization of the longitudinal photon propagator

Lecture 11

Vacuum Polarization

S-Matrix


•WI for Z

•WI for W & Γ


•Ghosts in QED


•WI for Vertex

Unitarity and WI


❐ This result is equivalent, as we have seen, to the statement that the vacuumpolarization is transversal. It is proved easily starting with the Green function,〈0|TAµω |0〉, and using

δBRS 〈0|TAµω |0〉 = 0

❐ This gives

1

ξ〈0| |TAµ∂νAν |0〉 θ − 〈0|T∂µωω |0〉 θ = 0

❐ After taking the Fourier transform we get

1

ξkµGµν(k) = −kν∆(k)

where the ghost propagator is the free propagator

∆(k) =i

k2

because the ghosts have no interactions.

The non-renormalization of the longitudinal photon propagator

Lecture 11

Vacuum Polarization

S-Matrix


•WI for Z

•WI for W & Γ


•Ghosts in QED


•WI for Vertex

Unitarity and WI


❐ Multiplying by the inverse propagator of the photon we get

1

ξkµ = −ikν

k2G−1νµ(k)

❐ Therefore

kνG−1νµ(k) =

i

ξkµk2 = kνG

−1νµ(0) (k)

❐ This shows that the longitudinal part of the photon propagator is equal to thefree longitudinal part and therefore does not get any renormalization.

Ward Identity for the Vertex

Lecture 11

Vacuum Polarization

S-Matrix


•WI for Z

•WI for W & Γ


•Ghosts in QED


•WI for Vertex

Unitarity and WI


❐ For the vertex we start from

δBRS 〈0|Tωψψ |0〉 = 0

❐ This means

1

ξ〈0|T∂µAµψψ |0〉 = −ie 〈0|Tωωψψ |0〉+ ie 〈0|Tωψψω |0〉

❐ After taking the Fourier transform we get

i

ξqµTµ = T

Ward Identity for the Vertex · · ·

Lecture 11

Vacuum Polarization

S-Matrix


•WI for Z

•WI for W & Γ


•Ghosts in QED


•WI for Vertex

Unitarity and WI


❐ Where we have defined

iTµ = = Gµν(q)S(p′)iΓνS(p)µ

p

qp′

iT = ie − ie

= ie∆(q)S(p)− ie∆(q)S(p′)

p

q

p′

p

q

p′

Ward Identity for the Vertex · · ·

Lecture 11

Vacuum Polarization

S-Matrix


•WI for Z

•WI for W & Γ


•Ghosts in QED


•WI for Vertex

Unitarity and WI


❐ The last equality results from the fact that ghosts have no interactions in QEDin a linear gauge. Putting everything together we get

i

ξqµGµν(q)S(p

′)iΓνS(p) = ie∆(q)S(p)− ie∆(q)S(p′)

❐ Using

1

ξkµGµν(k) = −kν∆(k)

and multiplying by the inverse of the fermion propagators we get again the wellknown the Ward identity for the vertex,

qµΓµ(p′, p) = −ie

[

S−1(p)− S−1(p′)]

Optical Theorem

Lecture 11

Vacuum Polarization

S-Matrix


Unitarity and WI

•Optical Theorem

•Cutkosky rules

• Scalars & Fermions

•Gauge Fields


❐ The S matrix, (Heisenberg 1942), can be written in the form

S = 1 + iT

❐ Then its unitarity SS† = 1 implies,

2 ImT = TT †

❐ If we insert this relation between the same initial and final state (elasticscattering) we get

2 Im 〈i|T |i〉 =⟨

i|TT †|i⟩

=∑

f

| 〈f |T |i〉 |2

where we have introduced a complete set of states.

Optical Theorem · · ·

Lecture 11

Vacuum Polarization

S-Matrix


Unitarity and WI

•Optical Theorem

•Cutkosky rules


•Gauge Fields


❐ This relation can still be written in the form,

σtotal = 2 Im T elasticforward

known as the optical theorem

❐ What we call here σtotal it is not exactly the cross section, because the fluxfactors are missing. It is for our purpose the quantity defined by

σtotal ≡∑

f

| 〈f |T |i〉 |2

❐ Unitarity establishes therefore a relation between the total cross section andthe imaginary part of the elastic amplitude in the forward direction (the initialand final state have to be the same).

Cutkosky rules

Lecture 11

Vacuum Polarization

S-Matrix


Unitarity and WI

•Optical Theorem

•Cutkosky rules


•Gauge Fields


❐ To show that unitarity is obeyed in a given process we have to know how tocalculate the imaginary part of Feynman diagrams.

❐ Of course there is always the possibility of doing explicitly the calculations andretrieve the imaginary part, but this only possible for simple diagrams (seebelow).

❐ Therefore it is useful to have rules, known as Cutkosky rules, that give us theimaginary part of any diagram. We will state them now.

❐ Rule 1

The imaginary part of an amplitude is obtained using the expression

2 Im T = −∑

cuts

T

❐ Rule 2

The cut is obtained by writing the amplitude iT = · · · and substituting in thisexpression the propagators of the lines we cut by the following expression,

Cutkosky rules · · ·

Lecture 11

Vacuum Polarization

S-Matrix


Unitarity and WI

•Optical Theorem

•Cutkosky rules


•Gauge Fields


❐ Scalar fields

∆(p) ⇒ 2πθ(po)δ(p2 −m2)

❐ Fermion fields

S(p) ⇒ (p/+m)2πθ(po)δ(p2 −m2)

❐ Vector gauge fields (in the Feynman gauge)

Gµν(p) ⇒ −gµν2πθ(p0)δ(p2 −m2)

❐ In these expression the θ functions ensure the energy flux. The Cutkosky rulesare complicated to prove in general (see G. ’t Hooft, ”Diagrammar”, CERNReport 1972) but we are going to show in the two explicit examples how theywork.

Cutkosky rules: Free propagator

Lecture 11

Vacuum Polarization

S-Matrix


Unitarity and WI

•Optical Theorem

•Cutkosky rules


•Gauge Fields


❐ For the free propagator of a scalar field the amplitude is

iT =i

p2 −m2 + iε

❐ The imaginary part is obtained using

1

x+ iε= P

(

1

x

)

− iπδ(x)

❐ Therefore

T = P

(

1

p2 −m2

)

− iπδ(p2 −m2)

Cutkosky rules: Free propagator

Lecture 11

Vacuum Polarization

S-Matrix


Unitarity and WI

•Optical Theorem

•Cutkosky rules


•Gauge Fields


❐ The imaginary part is then

2 ImT = −2πδ(p2 −m2)

❐ Using the Cutkosky rule we get

2 ImT = −2πδ(p2 −m2)θ(p0)

which is precisely the same result.

❐ The function θ(p0) tell us that the flux of energy is from left to right.

Cutkosky rules: Self-energy in φ3

Lecture 11

Vacuum Polarization

S-Matrix


Unitarity and WI

•Optical Theorem

•Cutkosky rules


•Gauge Fields


❐ Let us consider the self-energy in the theory given by the Lagrangian,

L =1

2∂µφ∂

µφ− 1

2m2φ2 − λ

3!φ3

❐ The self-energy is given by the diagram in the Figure below,

p

kk

p− k

❐ The corresponding amplitude is

iT = (−iλ)2∫

d4p

(2π)4i

p2 −m2 + iε

i

(p− k)2 −m2 + iε

Cutkosky rules

Lecture 11

Vacuum Polarization

S-Matrix


Unitarity and WI

•Optical Theorem

•Cutkosky rules


•Gauge Fields


❐ Let us calculate the imaginary part of T by two methods, first doing theexplicit calculation and second using the Cutkosky rule.

❐ i) Explicit Calculation

iT =λ2∫

d4p

(2π)41

(p2 −m2 + iε)[(p− k)2 −m2 + iε]

=λ2∫

d4p

(2π)4

∫ 1

0

dx1

(p2 + 2p · P −M2 + iε)2

=λ2∫

d4p

(2π)4

∫ 1

0

dx1

[(p+ P )2 −∆]2

where

P = −x k∆ = P 2 +M2 = m2 − k2x(1− x)− iε

Cutkosky rules

Lecture 11

Vacuum Polarization

S-Matrix


Unitarity and WI

•Optical Theorem

•Cutkosky rules


•Gauge Fields


❐ The amplitude is then

iT = λ2∫

d4p

(2π)4

∫ 1

0

dx1

(p2 −∆)2

❐ The integral is divergent. Using dimensional regularization we get

T =λ

16π2µεΓ

(

2− d

2

)∫ 1

0

dx ∆−(2− d2 )

❐ Choosing on-shell renormalization, TR(k2 = m2) = 0, we get

TR =T − T (k2 = m2)

=λ2

16π2Γ(ε

2

)

∫ 1

0

dx

[

(

∆(k2)

µ2

)− ε2

−(

∆(k2 = m2)

µ2

)− ε2

]

=λ2

16π2

(

2

ε− C +O(ε)

)∫ 1

0

dx

[

1− 1− ε

2lnm2 − k2x(1− x)− iε

m2 −m2x(1− x)− iε

]

=− λ2

16π2

∫ 1

0

dx ln

[

1− βx(1− x)− iε

1− x(1− x)− iε

]

= − λ2

16π2[L(β)− L(1)]

Cutkosky rules

Lecture 11

Vacuum Polarization

S-Matrix


Unitarity and WI

•Optical Theorem

•Cutkosky rules


•Gauge Fields


❐ In this expression β = k2

m2 and the function L(β) is given by

L(β) ≡∫ 1

0

dx ln[

1− β(1− x)x− iε]

❐ It satisfies

ImL(β) = −π√

1− 4

βθ(β − 4)

❐ Therefore

ImT = − λ2

16π2

[

ImL(β)− ImL(1)]

and we get finally,

ImT =λ2

16π

√

1− 4m2

k2θ

(

1− 4m2

k2

)

❐ The θ functions ensures that there is only imaginary part when the intermediatestate could also be a final state (production of two particles of mass m).

Cutkosky rules

Lecture 11

Vacuum Polarization

S-Matrix


Unitarity and WI

•Optical Theorem

•Cutkosky rules


•Gauge Fields


❐ ii) Using the Cutkosky rules

Using the rules we get

2ImT =− (iλ)2∫

d4p

(2π)4(2π)2θ(p0)θ(k0 − p0)δ(p2 −m2)δ((p2 − k2)−m2)

=λ2∫

d4p

(2π)4d4p′(2π)2θ(p0)θ(k0 − p0)δ(p2 −m2)δ(p′2 −m2)δ4(p′ − k + p)

❐ Using now the result

∫

d4p θ(p0)δ(p2 −m2) =

∫

d3p1

2p0

Cutkosky rules

Lecture 11

Vacuum Polarization

S-Matrix


Unitarity and WI

•Optical Theorem

•Cutkosky rules


•Gauge Fields


❐ We get

2ImT = λ2∫

d3p

(2π)3d3p′

1

2p01

2p′02πδ4(p′ − k + p)

or

2ImT = λ2∫

d3p

(2π)31

2p01

2p′02πδ(k0 − p0 − p′0)

❐ In the CM frame

k = (√s,~0) ; p = (

√

|~p|2 +m2, ~p) ; p′ = (√

|~p′|2 +m2,−~p)

Cutkosky rules

Lecture 11

Vacuum Polarization

S-Matrix


Unitarity and WI

•Optical Theorem

•Cutkosky rules


•Gauge Fields


❐ Therefore we get

2ImT =λ2∫

d3p

(2π)31

4(|~p|2 +m2)2πδ(

√s− 2

√

|~p|2 +m2)

=λ2

4π

∫

d|~p| |~p|2|~p|2 +m2

δ(|~p| −√

s4 −m2)

2|~p|√|~p|2+m2

θ

(

1− 4m2

s

)

=λ2

8π

√

1− 4m2

sθ

(

1− 4m2

s

)

❐ Using s = k2 we get

ImT =λ2

16π

√

1− 4m2

k2θ

(

1− 4m2

k2

)

which is the same result as we got in the explicit calculation.

Example of Unitarity: scalars and fermions

Lecture 11

Vacuum Polarization

S-Matrix


Unitarity and WI

•Optical Theorem

•Cutkosky rules


•Gauge Fields


❐ As an example of checking the unitarity let us consider a theory described bythe Lagrangian,

L = iψ∂/ψ −mψψ +1

2∂µφ∂

µφ− 1

2M2φ2 + gψψφ

❐ We will show unitarity in two cases (cutting fermions lines):

i) Scalar Self-energy

The self-energy of the scalars is given by the diagram in the Figure, to whichcorresponds the amplitude,

p

kk

p− k

iT = g2∫

d4p

(2π)4Tr

[

i

p/−m+ iε

i

p/− k/−m+ iε

]

Example of Unitarity: scalars and fermions · · ·

Lecture 11

Vacuum Polarization

S-Matrix


Unitarity and WI

•Optical Theorem

•Cutkosky rules


•Gauge Fields


❐ Applying Cutkosky rules we get,

2 ImT =−∑

cuts

T

=− g2∫

d4p

(2π)4Tr[(p/+m)(p/− k/+m)](2π)θ(p0)δ(p2 −m2)

(2π)θ(k0 − p0)δ((p− k)2 −m2)

❐ To show the unitarity we calculate the cross section,

σ =∑

f

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

2

k

p

p′


Lecture 11

Vacuum Polarization

S-Matrix


Unitarity and WI

•Optical Theorem

•Cutkosky rules


•Gauge Fields


❐ We get

σ =∑

f

|igu(p)v(p′)|2 = −g2∑

f

Tr[(p/+m)(−p/′ +m)]

where we have used∑

spins v(p′)v(p) = −(−p/′ +m) and

∑

spins u(p)u(p′) = p/+m.

❐ Therefore

σ = −g2∫

dρ2Tr[(p/+m)(−p/′ +m)]

where dρ2 is the phase space of two particles, that is,

∫

dρ2 ≡∫

d3p

(2π)3d3p′

(2π)31

2p01

2p′0 (2π)

4δ4(k − p− p′)

=

∫

d4p

(2π)4d4p′

(2π)4(2π)θ(p0)δ(p2 −m2)(2π)θ(p′

0)δ(p′2 −m2)(2π)4δ4(k−p−p′)


Lecture 11

Vacuum Polarization

S-Matrix


Unitarity and WI

•Optical Theorem

•Cutkosky rules


•Gauge Fields


❐ We conclude then that

σ =− g2∫

d4p

(2π)4(2π)θ(p0)δ(p2 −m2)(2π)θ(k0 − p0)δ((p− k)2 −m2)

Tr[(p/+m)(p/− k/+m)]

❐ Comparing we obtain

2ImT = σ

as we wanted to show.


Lecture 11

Vacuum Polarization

S-Matrix


Unitarity and WI

•Optical Theorem

•Cutkosky rules


•Gauge Fields


❐ ii) General case

Let us consider the general case of two internal fermion lines. The amplitudeiT is represented by the diagram

k1k1

k2k2

knkn

p

−p′≡ iT

p′ =n∑

i=1

ki − p

❐ The amplitude iT is given by

iT = −∫

d4p

(2π)4Tr[

T ′S(p)T ′S(−p′)]


Lecture 11

Vacuum Polarization

S-Matrix


Unitarity and WI

•Optical Theorem

•Cutkosky rules


•Gauge Fields


❐ Where we have defined the amplitude iT ′ by

≡ u(p)iT ′v(p′)

k1

k2

kn

p

−p′

❐ Therefore

2 ImT =−∫

d4p

(2π)4(2π)2δ(p2 −m2)θ(p0)δ(p′2 −m2)θ(p′0)

Tr[

T ′(p/+m)T ′(−p/′ +m)]

=−∫

dρ2Tr[

T ′(p/+m)T ′(−p/′ +m)]


Lecture 11

Vacuum Polarization

S-Matrix


Unitarity and WI

•Optical Theorem

•Cutkosky rules


•Gauge Fields


❐ On the other hand

σ =∑

f

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

2

σ =∑

f

|u(p)T ′v(p′)|2

= −∫

dρ2Tr[

(p/+m)T ′(−p/′ +m)T ′]

k1

k2

kn

p

−p′


Lecture 11

Vacuum Polarization

S-Matrix


Unitarity and WI

•Optical Theorem

•Cutkosky rules


•Gauge Fields


❐ Therefore

σ = 2 ImT

❐ If the lines to be cut were scalars the result would be the same. In this casethere will be no minus sign from the loop but there will be no minus sign fromthe spins sum. The proof is left as an exercise.

k1k1 k1

k2k2 k2

knkn kn

p p

−p′ −p′2 Im =∑

f

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

2

Unitarity and gauge fields

Lecture 11

Vacuum Polarization

S-Matrix


Unitarity and WI

•Optical Theorem

•Cutkosky rules


•Gauge Fields


❐ In the previous slides we have shown that unitarity holds for theories withscalar and fermion fields.

❐ We are now going to show that the proof of unitarity for gauge theories ismore complicated and requires the use of the Ward identities

❐ The problem resides in the fact that the gauge fields in internal lines haveunphysical polarizations while the final states should have only physical degreesof freedom

❐ This difference would lead to a violation of unitarity in gauge theories.However we will show that the ghosts in the internal lines will compensate forthe this and will make the theory unitary as it should

Unitarity and gauge fields · · ·

Lecture 11

Vacuum Polarization

S-Matrix


Unitarity and WI

•Optical Theorem

•Cutkosky rules


•Gauge Fields


❐ Let us define the following amplitudes

iT =

iT abµν =

iT ab =

+

k1k1

k2k2

p1p1p1p1

p2p2p2p2

k1

k2

p1

p2

µ, a

ν, b

k1

k2

p1

p2

a

b

k2 = p1 + p2 − k1


Lecture 11

Vacuum Polarization

S-Matrix


Unitarity and WI

•Optical Theorem

•Cutkosky rules


•Gauge Fields


❐ Using these definitions we can write the amplitude in the form (the factor 1/2is a symmetry factor for the gauge fields and the minus sign is for the loop ofghosts)

iT =

∫

d4k1(2π)4

{

1

2T abµνG

aa′

µµ′(k1)Gbb′

νν′(k2)T∗a′b′µ′ν′ − T ab∆aa′(k1)∆

bb′(k2)T∗a′b′

}

❐ Applying the Cutkosky rules we find for the imaginary part

2 ImT =

∫

d4k1(2π)4

(2π)2θ(k01)θ(k02)δ(k

21)δ(k

22)

{

1

2T abµνT

∗abµν − T abT ∗ab

}

≡∫

dρ2

[

1

2T abµνT


]


Lecture 11

Vacuum Polarization

S-Matrix


Unitarity and WI

•Optical Theorem

•Cutkosky rules


•Gauge Fields


❐ Now we have to evaluate σtotal. As the ghosts are not physical we have

σ =∑

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

2

σ =1

2

∫

dρ2∑

Pol

∣

∣εµ(k1)εν(k2)T

abµν

∣

∣

2

k1

k2

p1

p2

µ, a

ν, b

where the factor 1/2 comes now from identical particles in the final state.

❐ Writing

∑

Pol

εµ(k1)εµ′∗(k1) = Pµµ

′

(k1)

we get

σ =

∫

dρ21

2T abµνT

∗abµ′ν′Pµµ

′

(k1)Pνν′

(k2)


Lecture 11

Vacuum Polarization

S-Matrix


Unitarity and WI

•Optical Theorem

•Cutkosky rules


•Gauge Fields


❐ We now use the result (see problems)

Pµν(k) = −gµν + kµην + kνηµ

k · η

where ηµ is a four-vector that satisfies η · ε and η2 = 0. We get

1

2T abµνT


′

(k1)Pνν′

(k2) =

=1

2T abµνT

∗abµν − 1

2(T ab · k2) · (T ∗ab · η) 1

k2 · η

− 1

2(T ab · η) · (T ∗ab · k2)

1

k2 · η− 1

2(k1 · T ab) · (η · T ∗ab)

1

k1 · η

− 1

2(η · T ab) · (k1 · T ∗ab)

1

k1 · η+

[

1

2(k1 · T ab · η)(η · T ∗ab · k2)+

+1

2(k1 · T ab · k2)(η · T ∗ab · η) + 1

2(η · T ab · η)(k1 · T ∗ab · k2)

+1

2(η · T ab · k2)(k1 · T ∗ab · η)

]

1

(k1 · η)(k2 · η)


Lecture 11

Vacuum Polarization

S-Matrix


Unitarity and WI

•Optical Theorem

•Cutkosky rules


•Gauge Fields


❐ Using the following Ward identities (see problems),

kµ1Tabµν = k2νT

ab

kµ2Tabµν = k1νT

ab=⇒ k1 · T ab · k2 = 0

we get

1

2T abµνT


′

(k1)Pνν′

(k2) =

=1

2T abµνT

∗abµν − 1

2T ab(k1 · T ∗ab · η) 1

k2 · η

− 1

2T ∗ab(k1 · T ab · η)

1

k2 · η− 1

2T ab(η · T ∗ab · k2)

1

k1 · η

− 1

2(η · T ab · k2)T ∗ab 1

k1 · η+

1

2T abT ∗ab +

1

2T abT ∗ab

=1

2T abµνT



Lecture 11

Vacuum Polarization

S-Matrix


Unitarity and WI

•Optical Theorem

•Cutkosky rules


•Gauge Fields


❐ Therefore after the sum over polarizations is correctly taken in account weobtain,

σ =

∫

dρ2

[

1

2T abµνT


]

❐ Comparing with the expression for 2 ImT we get

σ = 2 ImT

as we wanted to show.

❐ It should be clear that the ghosts with their minus sign in the loop played acrucial role in subtracting the extra degrees of freedom.

❐ Also the Ward identities were necessary to relate the gauge field amplitudeswith the ghost amplitude.

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Advanced Quantum Field Theory Chapter 6 Non-Abelian ......Advanced Quantum Field Theory Chapter 6...

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