Review of Generating functional and Green's...

Review of Generating functional and Green’sfunctions

Zhiguang Xiao

March 26, 2017

Contents

1 Full Green’s Function

2 Connected Green’s function & Generating Functional

3 One particle irreducible Green’s function

4 Amputated Green’s function: G(n)Amp(x1, . . . , xn)

5 Renormalized Green’s function & Bare Green’s function

6 Equations For Green’s Functions— Schwinger-Dyson Eq.

7 Global symmetry and Ward Id.

8 Appendix: Legendre transformation and IPI(Zinn Justin)

Full Green’s function

G(n)(x1, x2, . . . , xn) = ⟨Ω|T Φ(x1) . . .Φ(xn)|Ω⟩

=

∫[DΦ]Φ(x1) . . .Φ(xn) exp iℏS[Φ]∫

[DΦ] exp iℏS[Φ]

Remarks: We have divided out the bubble diagrams.

Full Green’s function

Example. 4-point Full Green’s Function: ⟨Ω|T Φ(x1) . . .Φ(xn)|Ω⟩

Generating functional for full Green’s functions

Z[J ] =

∫[DΦ] exp iℏ (S[Φ] + ℏ

∫d4xJ(x)Φ(x))∫

[DΦ] exp iℏS[Φ]

=

∞∑n=0

in

n!

∫d4x . . . d4xnG

(n)(x1, . . . , xn)J(x1) . . . J(xn)

=⟨Ω|Ω⟩J⟨Ω|Ω⟩ (1)

G(n)(x1, . . . , xn) =1

inδnZ[J ]

δJ(x1) . . . J(xn)

∣∣∣∣J=0

• Expansion of Z[J ] around J = 0, generating the full Green’s functions.

• Z[0] = 1, G(0) = 1.

Connected Green’s functions

G(n)conn(x1, . . . , xn) only includes connected diagrams

Example: Four-point connected Green’s function in ϕ4:

Generating functional for Connected Green’s functions

Generating functional for Connected Green’s functions: using the fullgenerating functional Z[J ]

W [J ] =ℏilnZ[J ]

=ℏi

∞∑n=1

in

n!

∫d4x1 . . . d

4xnG(n)conn(x1, . . . , xn)J(x1) . . . J(xn)

G(n)conn(x1, . . . , xn) =

1

ℏ1

in−1

δnW [J ]

δJ(x1) . . . J(xn)

∣∣∣∣J=0

We now take ℏ = 1, G(n)conn(x1, . . . , xn) is symmetric with respect to idential

J(xi).

Symmetric factor for a connected diagram

L =1

2∂µϕ∂

µϕ− 1

2m2ϕ2 +

1

3!λϕ3 = L0 + L1

Z[J ] ∝ exp

i

∫d4xL1

(1

i

δ

δJ

)∫DϕeiS0+Jϕ

∝∞∑V=0

1

V !

[i

3!λ

∫d4x

(δ

iδJ(x)

)3]V

×∞∑P=0

1

P !

(1

2

∫d4x(iJ(x))(−i∆(x, y))(iJ(y))

)P• Combine equal terms: Permutations of vertices, cancel V !; permutationsof propagators, cancel P !; permutations of δ/δJ , cancel 3!; permutationsof J, cancel 1/2.

• Overcounting must be divided — symmetric factor

• Feynman rules — already including all the contractions (permutations) forthe same field, eg. ϕ3 vertex −iλ, 1/3! cancelled.

Symmetric factor for a connected diagram

• For the coefficients with J in the generating functional, the external legswith J ’s — permutations of external J , overcounted, should be divided;for amplitude, or greens function, extra external different δ/δJ — countedas different, not overcounted.

iW generates all connected GF (Srednicki chapt 9)

CI connected components, nI times in one disconnected diagrams: the term inZ[J ] for this diagram

DnI =1

SD

∏I

(CI)nI

SD =∏I

nI !

Exchange all vertices, propagators . . . for nI CI components. Overcount nI !times.The full Z[J ]

Z[J ] ∝∑nI

DnI ∝∑nI

∏I

1

nI !(CI)

nI

∝∏I

∞∑nI=0

1

nI !(CI)

nI ∝∏I

exp(CI)

∝ exp∑I

CI

∝ expiW [J ]

Since we normalize Z[0] = 1, all the ∝ should be =. So iW [J ] =∑I CI ,

generating all the connected diagrams.

Example: 2-point conn. GF in ϕ3 theory

⟨Ω|TΦ(x1)Φ(x2)|Ω⟩conn =δ

iδJ(x1)

δ

iδJ(x2)lnZ[J ]

∣∣∣∣J=0

=

(δ

iδJ(x1)

⟨Ω|Φ(x2)|Ω⟩J⟨Ω|Ω⟩J

) ∣∣∣∣J=0

=

(⟨Ω|Φ(x1)Φ(x2)|Ω⟩J

⟨Ω|Ω⟩J− ⟨Ω|Φ(x1)|Ω⟩J

⟨Ω|Ω⟩J⟨Ω|Φ(x2)|Ω⟩J

⟨Ω|Ω⟩J

) ∣∣∣∣J=0

One particle irreducible Green’s function

1PI: amputated diagrams remains connected after cutting an arbitrary internalpropagator.

in ϕ4 theory, 1PI 4-point function:

Γ(x1, . . . , x4) =

Generating functional for 1PI

We expect the Generating functional to be:

Γ[Φ] =

∞∑n=1

1

n!

∫d4x1 . . . d

4xnΓ(n)(x1 . . . xn)Φ(x1) . . .Φ(xn)

Γ(n)(x1 . . . xn) =δΓ[Φ]

δΦ(x1) . . . δΦ(xn)

∣∣∣∣Φ=0

In fact, there are some subtleties for the generating functional we will construct:

• The generating funtional generates 1PI for n ≥ 3 at Φ = 0 for vev⟨Φ(x)⟩ = 0.

• for n = 2, Γ(2)(x, y) = (G(2)conn)

−1, inverse propagator (as operatorinverse).

• for n = 1, ???.

Construct Γ from W

Legendre transformation:

1 Recall the classcial mechanics:From Lagrangian to Hamiltonian L(q, q) → H(q, p), we define p ≡ ∂L[q,q]

∂q,

solve q = q(p, q) and insert into

H(q, p) ≡ pq − L[q, q(p, q)].

We change from q, q to p, q formalism.

2 For J [x] = 0, (L[q, q] →W [J ], q → J ,) we define classical field:

φcl(x) =⟨Ω|Φ(x)|Ω⟩J

⟨Ω|Ω⟩J=δW [J ]

δJ(x).

φcl(x) is a functional of J(x). Solve J(x) = Jx[φcl].

3 define

Γ[φcl] ≡W [J [φcl]]−∫d4xJ [φcl]φcl(x),

Γ is the generating functional we need.

Generating functional Γ

Γ[φcl] ≡W [J [φcl]]−∫d4xJ [φcl]φcl(x), φcl(x) =

⟨Ω|Φ(x)|Ω⟩J⟨Ω|Ω⟩J

=δW [J ]

δJ(x).

1

δΓ[φcl]

δφcl=

∫d4y

δJ(y)

δφcl(x)

δW [J ]

δJ(y)− J(x)−

∫d4y

δJ(y)

δφcl(x)φcl(y)

= −J(x)

2 if J = 0, φcl =⟨Ω|Φ(x)|Ω⟩

⟨Ω|Ω⟩ = δW [J]δJ[x]

∣∣∣∣J=0

≡ ϕc, the v.e.v. of Φ.

δΓ[φcl]

δφcl(x)

∣∣∣∣ϕc

= 0

gives the equation of motion for ϕc, the vev for Φ.

3 Γ[φcl] is also called Quantum Effective Action.

4 Vacuum preserves Poincare symmetry, and no SSB, ϕc = 0.

5 Perturbation theory: perturbation around φ = ϕc = 0.

Γ(1) =δΓ[φcl]

δφcl

∣∣∣ϕc

= 0.


1 n = 2, from −J = δΓ[φcl]δφcl

−δ(4)(y − x) = −δJ(x)δJ(y)

=δ2Γ[φcl]

δJ(y)δφcl(x)

=

∫d4z

δφcl(z)

δJ(y)

δ2Γ[φcl]

δφcl(z)δφcl(x)

=

∫d4z

δ2W [J ]

δJ(y)δJ(z)

δ2Γ[φcl]

δφcl(z)δφcl(x)

Γ(2)(x, y) =δ2Γ[φcl]

δφcl(x)δφcl(y)

∣∣∣∣φcl=ϕc

=

[− δ2W [J ]

δJ(x)δJ(y)

]−1 ∣∣∣∣J=0

= [−iG(2)conn]

(−1)

(x,y)

2 in momentum space:

G(2)(k2)Γ(2)(k2) = i

G(2)(k2) =i

k2 −m2 −M(k2)

Γ(2)(k2) = k2 −m2 −M(k2)

−iΓ(2) inverse propagator.


1 Expand around ϕc

Γ[Φ] =

∞∑n=2

1

n!

∫d4x1 . . . d

4xnΓ(n)(x1 . . . xn)(Φ(x1)− ϕc) . . . (Φ(xn)− ϕc)

2 n = 3,

G(3)(x, y, z) =iδW [J ]

(iδJ(x))(iδJ(y))(iδJ(z))

∣∣∣∣J=0

= −∫x′,y′,z′

iG(2)(x− x′)iG(2)(y − y′)iG(2)(z − z′)

× δ3Γ[φcl]

δφcl(x′)δφcl(y′)δφcl(z′)

∣∣∣∣ϕc

iΓ(3)(x, y, z) is 1PI.

3 for general n, iΓ(n)(x1, . . . , xn) = i δnΓ[φcl]δφcl(x1)...δφcl(xn)

∣∣∣ϕc

is 1PI, for n ≥ 3.

Amputated Green’s function: G(n)Amp(x1, . . . , xn)

G(n)conn(x1, . . . , xn) =

∫d4y1 . . . d

4ynG(2)(x1, y1) . . . G

(2)(xn, yn)G(n)Amp(y1, . . . , yn)

• In general, amputated Green’s function is not 1PI.

• In momentum space:

G(n)conn(p1, . . . , pn) = G(2)(p1) . . . G

(2)(pn)G(n)Amp(p1, . . . , pn)

Renormalized Green’s function & Bare Green’s function

• Bare field ΦB(x) and renormalized field ΦR(x): ΦB(x) = Z1/2ΦR(x)Bare Green’s function:

G(n)B (x1, . . . , xn) ≡ ⟨Ω|ΦB(x1) . . .ΦB(xn)|Ω⟩

= (Z1/2)n⟨Ω|ΦR(x1) . . .ΦR(xn)|Ω⟩= Zn/2G

(n)R (x1, . . . , xn)

If Φ’s are different, differnt Z’s are used.

• Generating functional for Bare .

SB =

∫d4xL[ΦB ] =

∫d4xLR[ΦR] + Lct = SR + Sct (2)

ZB [JB ] =

∫[DΦB ] expi(S[ΦB ] +

∫d4xJB(x)ΦB(x))∫

[DΦB ] expiS[ΦB ](3)

G(n)B (x1, . . . , xn) =

1

inδnZB [JB ]

δJB(x1) . . . JB(xn)

∣∣∣∣JB=0

Renormalized Green’s function & Bare Green’s function

• Renormalized Green’s funct: define JR = Z1/2JB ,∫JBΦB =

∫JRΦR

ZR[JR] = ZB [JB ] =

∫[DΦR] expi(SR[ΦR] + Sct +

∫d4xJR(x)ΦR(x))∫

[DΦR] expiS[ΦR] + Sct

G(n)R (x1, . . . , xn) =

1

inδnZR[JR]

δJR(x1) . . . JR(xn)

∣∣∣∣JR=0

=1

inδnZB [JB ]

Zn/2δJB(x1) . . . JB(xn)

∣∣∣∣JB=0

= Z−n/2G(n)B (x1, . . . , xn)

Similar for Connected Green’s function.For on-shell renormalization scheme: in momentum space, near physicalpole

G(2)R (p2) → i

p2 −m2ph + iϵ

G(2)B (p2) → iZ

p2 −m2ph + iϵ

Renormalized & Bare Amputated Green’s function

• Amputated Green’s funct:

G(n)B,conn

(x1, . . . , xn) =

∫d4y1 . . . d

4ynG

(2)B

(x1, y1) . . . G(2)B

(xn, yn)G(n)B,Amp

(y1, . . . , yn)

G(n)R,conn

(x1, . . . , xn) =

∫d4y1 . . . d

4ynG

(2)R

(x1, y1) . . . G(2)R

(xn, yn)G(n)R,Amp

(y1, . . . , yn)

using G(n)B (x1, . . . , xn) = Zn/2G

(n)R (x1, . . . , xn), G

(2)B = ZG

(2)R :

G(n)B,Amp(y1, . . . , yn) = Z−n/2G

(n)R,Amp(y1, . . . , yn)

• Bare & Renormalized 1PI: similar as before JR = Z1/2JB

ΓB [φB,cl] ≡ WB [JB [φB,cl]]−∫d4xJB [φB,cl]φB,cl(x)

= WR[JR[φR,cl]]−∫d4xJR[φR,cl]φR,cl(x) ≡ ΓR[φR,cl]

φB,cl =δWB [JB ]

δJB= Z1/2 δWR[JR]

δJR= Z1/2φR,cl(x)

Γ(n)R (x1, . . . , xn) =

δnΓR[φR,cl]

δφR,cl(x1) . . . δφR,cl(xn)= Zn/2Γ

(n)B (x1, . . . , xn)

Equations For Green’s Functions— Schwinger-Dyson Eq.

Consider path integral:(Bare fields) Z[J ] =∫[Dϕ]ei(S[ϕ]+

∫d4yJa(y)ϕa(y)).

Change the integration variable: ϕa → ϕ′a(x) = ϕa(x) + δϕa(x), the integral isinvariant. Assume [Dϕ] = [Dϕ′]. Just like d(x+ a) = dx.

0 = δZ[ϕ]

=

∫[Dϕ′]ei(S[ϕ

′]+∫d4yJa(y)ϕ′a(y)) −

∫[Dϕ]ei(S[ϕ]+

∫d4yJa(y)ϕa(y))

=

∫d4x

∫[Dϕ]ei(S[ϕ]+

∫d4yJa(y)ϕa(y))

(i

δS

δϕa(x)+ iJ(x)

)δϕa(x)

Take n functional derivative w.r.t iJaj (xj), set J = 0

0 =

∫[Dϕ]eiS

∫d4x

[i

δS

δϕa(x)

n∏i=1

ϕai (xi)

+n∑j=1

ϕai (x1) . . . δa,aj δ4(x− xj) . . . ϕan (xn)

δϕa(x) .Since δϕ(x) is arbitrary,

0 = ⟨Ω|iTδS

δϕa(x)

n∏i=1

ϕai (xi)|Ω⟩

+n∑j=1

⟨Ω|Tϕai (x1) . . . δa,aj δ4(x− xj) . . . ϕan (xn)|Ω⟩

Equations For Green’s Functions— Schwinger-Dyson Eq.

Schwinger-Dyson Eq.

0 = ⟨Ω|iTδS

δϕa(x)

n∏i=1

ϕai (xi)|Ω⟩

+n∑j=1

⟨Ω|Tϕai (x1) . . . δa,aj δ4(x− xj) . . . ϕan (xn)|Ω⟩

• δSδϕ(x)

= 0, the classical EOM. Free scalar: (∂2x +m2)ϕ = 0

• Two point function, scalar: i(∂2x +m2)⟨Ω|Tϕ(x)ϕ(y)|ω⟩ = −δ4(x− y)

• For x = xi,

⟨Ω|iT δS

δϕa(x)

n∏i=1

ϕai(xi)|Ω⟩ = 0.

For more detailed discussion, see Itzykson and Zuber,Quantum field theory, orArxiv:1008.4337, Swanson, A primer on functional methods and theSchwinger-Dyson Equation.

Global symmetry and Ward Id.

Noether theorem: Continuous symmetry ⇒ conserved current. Classical globalcontinuous symmetry: ϕa → ϕ′a(x) = ϕa(x) + iα∆ϕa(x), α : an infinitesimalconstant. L[ϕ′(x)] = L[ϕ(x)] + iα∂µK

µ

If α→ α(x), depends on x, only nonzero in a finite region

δL = L[ϕ′(x)]− L[ϕ(x)]

=∂L

∂ϕa(x)iα(x)∆ϕa(x) +

∂L∂(∂µϕa(x))

i(α(x)∂µ∆ϕa(x) + (∂µα(x))∆ϕ

a(x)))

= iα(x)∂µKµ + i(∂µα(x))

∂L∂(∂µϕa(x))

∆ϕa(x)

= i∂µα(x) jµ + iα(x)∂µK

µ

For ϕ satisfying EOM

0 = δS[ϕ] = i

∫d4x ∂µα(x) j

µ + α(x)∂µKµ

(integrate by part) = i

∫d4xα(x) ∂µ(−jµ +Kµ) = −

∫d4xα(x)∂µJ

µ

Since α(x) is arbitrary, define Jµ = i(jµ −Kµ), we have conserved current:

∂µJµ = 0.

Current conservation is a local property: α(x) can be non-zero in any arbitrarysmall region. Charge conservation is a global property.


Consider correlation function:⟨∏ni=1 ϕ(xi)⟩ =

1Z

∫[Dϕ]

∏ni=1 ϕ(xi)e

iS[ϕ].A global symmetric trans: ϕa → ϕ′a(x) = ϕa(x) + iα∆ϕa(x), S[ϕ] = S[ϕ′].α→ α(x), S[ϕ′] = S[ϕ]−

∫d4xα(x)∂µJ

µ.

0 =

∫[Dϕ′]

n∏i=1

ϕ′(xi)eiS[ϕ′] −

∫[Dϕ]

n∏i=1

ϕ(xi)eiS[ϕ]

=

∫[Dϕ]

(n∏i=1

ϕ′(xi)eiS[ϕ′] −

n∏i=1

ϕ(xi)eiS[ϕ]

)

=

∫[Dϕ]

(n∑i=1

ϕ(x1) . . . iα(xi)∆ϕ(xi) . . . ϕ(xn)− i

∫d4xα(x)∂µJ

µn∏i=1

ϕ(xi)

)eiS[ϕ]

i∂µ⟨TJµ(x)n∏i=1

ϕ(xi)⟩ =n∑i=1

⟨Tϕ(x1) . . . δ4(x− xi)(i∆ϕ(xi)) . . . ϕ(xn)⟩

In momentum space: ϕ(x) =∫

d4k(2π)4

eik·xφ(k). (outgoing)

−kµ⟨TJµ(k)n∏i=1

ϕ(ki)⟩ =n∑i=1

⟨Tϕ(k1) . . . i∆ϕ(ki + k) . . . ϕ(kn)⟩

For S-Matrix: from LSZ, for each external leg, limp2→m2p2−m2+iϵ√

Zi, there is no

on-shell pole for leg ∆ϕi(k + ki), the right hand side → 0, kµMµ = 0


QED: consider U(1) symmetry ψ → eieαψ, Jµ = −eψγµψ,correlation funtion ⟨Jµ(k)ψ(−(p+ k))ψ(p)⟩,Ward Id:

−kµ⟨Jµ(k)ψ(−(p+ k))ψ(p)⟩ =ie(⟨ψ(−p)ψ(p)⟩ − ⟨ψ(−(p+ k))ψ(p+ k)⟩

)S(p+ k)[−ikµΓµ(p+ k, p)]S(p) = S(p)− S(p+ k),

where

S(p) = ⟨ψ(−p)ψ(p)⟩ = i

p/−m− Σ(p).

−ikµΓµ(p+ k, p) = S−1(p+ k)− S−1(p) .

Γµ(p+ k, p) → Z−11 γµ as kµ → 0, S(p) ∼ i Z2

p/−m , at around p2 = m2, k → 0

−iZ−11 k/ = −iZ−1

2 k/

Z1 = Z2

Note: ⟨(ψγµψ)ψ(−(p+ k))ψ(p)⟩ → γµαβ⟨ψ(−(p+ k))ψαψβψ(p)⟩, no minussign

Local gauge symmetry and Ward Id. in generatingfunctional formulation

L =−1

4(F i0,µν)

2 + i(ψ0∂/ψ0 + ie0q1ψ0A/0,µψ0)−mψ,0ψ0ψ0 −1

2ξ0(∂µA

µ0 )

2

=− Z31

4(F iµν)

2 + iZψ2 (ψ∂/ψ + ieq1Zψ1

Zψ2ψA/ψ)−mZψ0 ψψ −

1

2ξ(∂µA

µ)2

• e0 = e Z1

Z2Z1/23

= e

Z1/23

. Only depends on Z3.

• Bare gauge trans: ψ0 → eiαq1ψ0, A0,µ → A0,µ − 1e0∂µα.

• Renormalized gauge trans: ψ → eiαq1ψ, Aµ → Aµ − 1e∂µα. e =

Zψ1

Zψ2

e = e

Consistent with A0,µ = Z1/23 Aµ.

• the gauge fixing term is not invariant.

• We will see that there is no counterterm for gauge fixing ξ term.


Adding source term for bare fields: (omit the subscript 0.) (α, α : grassmanvariables.) L[A, j, α, α] = L+ αψ + ψα+ jµA

µ

Under gauge transformations A→ A′ = Aµ + 1e∂µϵ, ψ → ψ′ = e−iϵq1ψ:

source terms and gauge fixing terms not invariant, all others including the pathintegral measure are inv.

Z[A,ψ; j, α, α] =N

∫[dAdψdψ] expiS[A,ψ, ψ] + i

∫(αψ + ψα+ jµA

µ)

(rename ) =N

∫[dA′dψ′dψ′] exp

iS[A′, ψ′, ψ′] + i

∫(αψ′ + ψ′α+ jµA

′µ)

(Gauge inv) =N

∫[dAdψdψ] exp

iS[A,ψ, ψ] + i

∫(αψ′ + ψ′α+ jµA

′µ)

−1

2ξ0∂µA

′µ∂νA′ν

0 = δZ =N

∫[dAdψdψ]

∫i(αδψ + δψα+ jµδA

µ −1

ξ0∂µA

µ∂νδAν)expiS + . . .

=N

∫[dAdψdψ]

∫dx ϵ(x)

(− iqαψ + iqψα−

1

e∂µjµ −

1

eξ0∂2∂µA

µ)exp. . .

⇒⟨−iqeαψ + iqeψα− ∂µjµ −1

ξ0∂2∂µA

µ⟩j,α,α = 0

⇒− iqeαδZ

iδα− iqe

δZ

iδαα− ∂µjµZ −

1

ξ0∂2∂µ

δZ

iδjµ= 0


−iqeα δZiδα

− iqeδZ

iδαα− ∂µjµZ − 1

ξ0∂2∂µ

δZ

iδjµ= 0

In connected Green Functional

iqeαδW

δα+ iqe

δW

δαα+ ∂µjµ +

1

ξ0∂2∂µ

δW

δjµ= 0

In 1PI generating functional:

iqeδΓ

δψcψc + iqeψc

δΓ

δψc+ ∂µ

δΓ

δAµc+

1

ξ0∂2∂µAcµ = 0

where Γ =W −∫jµA

µ −∫αψ −

∫ψα,

Aµc =δW

δjµ, ψc =

δW

δα, ψc = −δW

δα;

jµ = − δΓ

δAcµ, α = − δΓ

δψc, α =

δΓ

δψc.


iqeδΓ

δψc(z)ψc(z) + iqeψc(z)

δΓ

δψc(z)+ ∂µ

δΓ

δAµc (z)+

1

ξ0∂2∂µAcµ(z) = 0

δψc(x)ψc(y)

, Aµc , ψc, ψc → 0,:

− iqeδΓ

δψc(x)δψc(y)δ(x− z) + iqe

δΓ

δψc(x)δψc(y)δ(y − z) + ∂µ

δΓ

δAµc (z)δψc(x)δψc(y)= 0

⇒ iqe(−iS−1(x, y)δ(x− z) + iS−1(x, y)δ(y − z)) + iqe∂µz Γµ(z, x, y) = 0

⇒ −ikµΓµ(p+ k, p) = S−1(p+ k)− S−1(p)


eg:

iqeαδW

δα+ iqe

δW

δαα+ ∂µjµ +

1

ξ0∂2∂µ

δW

δjµ= 0

δ

δjν(z)

∣∣∣α,α,jµ→0

: ∂µx (gµνδ(x− y)) +1

ξ0∂2∂µx

δW

δjµ(x)δjν(y)

∣∣∣jµ=0

= 0

δW

δjµ(x)δjν(y)=i⟨TAµ(x)Aν(y)⟩ = iG(2)

µν (x− y) = i

∫d4k

(2π)4e−ik·(x−y)G(2)

µν (k)

=i

∫d4k

(2π)4e−ik·(x−y)

[(gµν −

kµkνk2

)A(k2) +B(k2)

kµkνk2

]We then have:

−ikν + iB(k2)ikνk

2

ξ0= 0 ⇒ B(k2) = − iξ0

k2

⇒ G(2)µν (k) =

(gµν −

kµkνk2

)A(k2) +

−iξ0k2

kµkνk2

Notice: Free photon propagator: G(2)µν = − i

k2

(gµν − kµkν

k2

)− iξ0kµkν

k4.

The ξ0 term is not renormalized. Renormalized propagator GR(2)µν = Z−1

3 G(2)µν ,

ξ = ξ0Z−13 .

the Legendre transformation and IPI

Consider the action:

Sϵ(ϕ) =1

2

∫dx dyϕ(x)

[K(x, y) + ϵ

]ϕ(y) + V (ϕ)

ϵ: a small parameter, expanded to the first order

Propagator ∆ϵ(x, y) :

∫∆ϵ(x, z)

[K(z, y) + ϵ

]dz = δ(x− y)

∆ϵ(x, y) =∆(x, y)− ϵη(x)η(y) +O(ϵ2) , η(x) =

∫∆(x, z)dz

• Expand the Feynman diagrams w.r.t ϵ, (using ∆ϵ(x, y)): the ϵ1 order

terms, replace ∆(x, y) → −η(x)η(y), cut one internal propagator.

• Higher order term: irreducible when cut 2,3,. . . ,lines.

1PI ⇔ ϵ1 term is connected


The generating functional

Zϵ[J ] =

∫[dϕ] expi(Sϵ + Jϕ)

=(1 + i

1

2ϵ

∫dx dy

δ

iδJ(x)

δ

iδJ(y)

)∫[dϕ] expi(S + Jϕ)+O(ϵ2)

=(1 + i

1

2ϵ

∫dx dy

δ

iδJ(x)

δ

iδJ(y)

)expiW [J ]+O(ϵ2)

=

(1 + i

1

2ϵ(∫

dxδ

δJ(x)W [J ]

)2+i

2ϵ

∫dx dy

δ

iδJ(x)

δ

iδJ(y)iW [J ]

)expiW [J ]

=Z[J ]

(1 + i

1

2ϵ(∫

dxδ

δJ(x)W [J ]

)2+i

2ϵ

∫dx dy

δ

iδJ(x)

δ

iδJ(y)iW [J ]

)+O(ϵ2)

⇒Wϵ[J ] =W [J ] + i1

2ϵ(∫

dxδ

δJ(x)W [J ]

)2

+i

2ϵ

∫dx dy

δ

iδJ(x)

δ

iδJ(y)iW [J ]

The second term just means the disconnectness of the W [J ] after cutting apropagator.


Legendre transformation:

Γϵ[φcl] ≡Wϵ[J [φcl]]−∫d4xJ [φcl]φcl(x), φcl(x) =

⟨Ω|Φ(x)|Ω⟩J⟨Ω|Ω⟩J

=δWϵ[J ]

δJ(x).

∂Γϵ∂ϵ

∣∣∣∣φcl

=−∫dx∂J(x)

∂ϵ

∣∣∣∣φcl

φcl(x) +∂Wϵ

∂ϵ

∣∣∣∣J

+

∫dx

δWϵ

δJ(x)

∂J(x)

∂ϵ

∣∣∣∣φcl

=∂Wϵ

∂ϵ

∣∣∣∣J

Thus,

Γϵ[φcl] =Γ[φcl] +iϵ

2

[(∫dxφcl(x)

)2

+

∫dx dy

( δ2iΓ[J ]

δφcl(x)δφcl(y)

)−1

+O(ϵ2)

]The first term in the ϵ term is just from the term ϵϕ2 we added in the actionand the second term contains just connected diagrams. Thus, Γ[φcl] is 1PI.

Review of Generating functional and Green's...

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