Review of Generating functional and Green s functions

Transcript

1 Review of Generating functional and Green s functions Zhiguang Xiao March 26, 2017

2 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 (x 1,..., x n ) 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)

3 Full Green s function G (n) (x 1, x 2,..., x n) = Ω T Φ(x 1)... Φ(x n) Ω [DΦ]Φ(x1)... Φ(x n) exp{ i ħ = S[Φ]} [DΦ] exp{ i S[Φ]} ħ Remarks: We have divided out the bubble diagrams.

4 Full Green s function Example. 4-point Full Green s Function: Ω T Φ(x 1)... Φ(x n) Ω

5 Generating functional for full Green s functions Z[J] = = [DΦ] exp{ i (S[Φ] + ħ d 4 xj(x)φ(x))} ħ [DΦ] exp{ i S[Φ]} ħ i n d 4 x... d 4 x n G (n) (x 1,..., x n)j(x 1)... J(x n) n! n=0 = Ω Ω J Ω Ω (1) G (n) (x 1,..., x n) = 1 i n δ n Z[J] δj(x 1)... J(x n) J=0 Expansion of Z[J] around J = 0, generating the full Green s functions. Z[0] = 1, G (0) = 1.

6 Connected Green s functions G (n) conn(x 1,..., x n) only includes connected diagrams Example: Four-point connected Green s function in ϕ 4 :

7 Generating functional for Connected Green s functions Generating functional for Connected Green s functions: using the full generating functional Z[J] W [J] = ħ ln Z[J] i = ħ i n i n! G (n) n=1 conn(x 1,..., x n) = 1 ħ d 4 x 1... d 4 x n G (n) conn(x 1,..., x n)j(x 1)... J(x n) 1 i n 1 δ n W [J] δj(x 1)... J(x n) J=0 We now take ħ = 1, G (n) conn(x 1,..., x n) is symmetric with respect to idential J(x i).

8 Symmetric factor for a connected diagram L = 1 2 µϕ µ ϕ 1 2 m2 ϕ ! λϕ3 = L 0 + L 1 { Z[J] exp i V =0 [ 1 i V! P =0 1 P! ( )} d 4 1 δ xl 1 Dϕe is 0+Jϕ i δj ( ) ] 3 V 3! λ d 4 δ x iδj(x) ( 1 d 4 x(ij(x))( i (x, y))(ij(y)) 2 Combine equal terms: Permutations of vertices, cancel V!; permutations of propagators, cancel P!; permutations of δ/δj, cancel 3!; permutations of J, cancel 1/2. Overcounting must be divided symmetric factor Feynman rules already including all the contractions (permutations) for the same field, eg. ϕ 3 vertex iλ, 1/3! cancelled. ) P

9 Symmetric factor for a connected diagram For the coefficients with J in the generating functional, the external legs with J s permutations of external J, overcounted, should be divided; for amplitude, or greens function, extra external different δ/δj counted as different, not overcounted.

10 iw generates all connected GF (Srednicki chapt 9) C I connected components, n I times in one disconnected diagrams: the term in Z[J] for this diagram 1 D ni = (C I) n I S D S D = I Exchange all vertices, propagators... for n I C I components. Overcount n I! times. The full Z[J] I n I! Z[J] D ni {ni } {ni } I n I =0 exp{ C I} I exp{iw [J]} I 1 n I! (CI)n I I 1 n I! (CI)n I exp(c I) Since we normalize Z[0] = 1, all the should be =. So iw [J] = I CI, generating all the connected diagrams.

11 Example: 2-point conn. GF in ϕ 3 theory δ δ Ω T Φ(x 1)Φ(x 2) Ω conn = ln Z[J] iδj(x 1) iδj(x 2) ( ) δ Ω Φ(x 2) Ω J = iδj(x 1) Ω Ω J J=0 ( Ω Φ(x1)Φ(x 2) Ω J = Ω Φ(x1) Ω J Ω Ω J Ω Ω J J=0 Ω Φ(x 2) Ω J Ω Ω J ) J=0

12 One particle irreducible Green s function 1PI: amputated diagrams remains connected after cutting an arbitrary internal propagator. in ϕ 4 theory, 1PI 4-point function: Γ(x 1,..., x 4) =

13 Generating functional for 1PI We expect the Generating functional to be: 1 Γ[Φ] = d 4 x 1... d 4 x nγ (n) (x 1... x n)φ(x 1)... Φ(x n) n! n=1 Γ (n) δγ[φ] (x 1... x n) = δφ(x 1)... δφ(x n) 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 operator inverse). for n = 1,???.

14 Construct Γ from W Legendre transformation: 1 Recall the classcial mechanics: From Lagrangian to Hamiltonian L(q, q) H(q, p), we define p solve q = q(p, q) and insert into H(q, p) p q 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: L[q, q] q, φ cl (x) = Ω Φ(x) Ω J Ω Ω J = δw [J] δj(x). φ cl (x) is a functional of J(x). Solve J(x) = J x[φ cl ]. 3 define Γ[φ cl ] W [J[φ cl ]] Γ is the generating functional we need. d 4 xj[φ cl ]φ cl (x),

15 Generating functional Γ Γ[φ cl ] W [J[φ cl ]] d 4 xj[φ cl ]φ cl (x), φ cl (x) = Ω Φ(x) Ω J Ω Ω J = δw [J] δj(x). 1 δγ[φ cl ] δφ cl = = J(x) 2 if J = 0, φ cl = Ω Φ(x) Ω Ω Ω d 4 y δj(y) δw [J] δφ cl (x) δj(y) J(x) δw [J] = δj[x] ϕ c, the v.e.v. of Φ. J=0 δγ[φ cl ] δφ cl (x) = 0 ϕc gives the equation of motion for ϕ c, the vev for Φ. 3 Γ[φ cl ] is also called Quantum Effective Action. 4 Vacuum preserves Poincaré symmetry, and no SSB, ϕ c = 0. 5 Perturbation theory: perturbation around φ = ϕ c = 0. Γ (1) = δγ[φ cl] δφ cl ϕc = 0. d 4 y δj(y) δφ cl (x) φ cl(y)

16 Generating functional Γ 1 n = 2, from J = δγ[φ cl] δφ cl δ (4) (y x) = δj(x) δj(y) = δ2 Γ[φ cl ] δj(y)δφ cl (x) = d 4 z δφ cl(z) δ 2 Γ[φ cl ] δj(y) δφ cl (z)δφ cl (x) = d 4 δ 2 W [J] δ 2 Γ[φ cl ] z δj(y)δj(z) δφ cl (z)δφ cl (x) [ ] Γ (2) δ 2 Γ[φ cl ] 1 (x, y) = δφ cl (x)δφ cl (y) = δ2 W [J] φcl =ϕ c δj(x)δj(y) J=0 2 in momentum space: iγ (2) inverse propagator. G (2) (k 2 )Γ (2) (k 2 ) = i G (2) (k 2 i ) = k 2 m 2 M(k 2 ) Γ (2) (k 2 ) = k 2 m 2 M(k 2 ) = [ ig (2) conn] ( 1) (x,y)

17 Generating functional Γ 1 Expand around ϕ c 1 Γ[Φ] = d 4 x 1... d 4 x nγ (n) (x 1... x n)(φ(x 1) ϕ c)... (Φ(x n) ϕ c) n! n=2 2 n = 3, G (3) iδw [J] (x, y, z) = (iδj(x))(iδj(y))(iδj(z)) J=0 = ig (2) (x x )ig (2) (y y )ig (2) (z z ) x,y,z δ 3 Γ[φ cl ] δφ cl (x )δφ cl (y )δφ cl (z ) iγ (3) (x, y, z) is 1PI. 3 for general n, iγ (n) (x 1,..., x n) = i δ n Γ[φ cl ] δφ cl (x 1 )...δφ cl (x n) ϕc is 1PI, for n 3. ϕc

18 Amputated Green s function: G (n) Amp (x 1,..., x n ) G (n) conn(x 1,..., x n) = d 4 y 1... d 4 y ng (2) (x 1, y 1)... G (2) (x n, y n)g (n) Amp (y1,..., yn) In general, amputated Green s function is not 1PI. In momentum space: G (n) conn(p 1,..., p n) = G (2) (p 1)... G (2) (n) (p n) G Amp (p1,..., pn)

19 Renormalized Green s function & Bare Green s function Bare field Φ B(x) and renormalized field Φ R(x): Φ B(x) = Z 1/2 Φ R(x) Bare Green s function: G (n) B (x1,..., xn) Ω ΦB(x1)... ΦB(xn) Ω If Φ s are different, differnt Z s are used. = (Z 1/2 ) n Ω Φ R(x 1)... Φ R(x n) Ω = Z n/2 G (n) (x1,..., xn) R Generating functional for Bare. S B = d 4 xl[φ B] = d 4 xl R[Φ R] + L ct = S R + S ct (2) Z B[J B] = [DΦB] exp{i(s[φ B] + d 4 xj B(x)Φ B(x))} [DΦB] exp{is[φ B]} (3) G (n) 1 δ n Z B[J B] B (x1,..., xn) = i n δj B(x 1)... J B(x n) JB =0

20 Renormalized Green s function & Bare Green s function Renormalized Green s funct: define J R = Z 1/2 J B, J BΦ B = J RΦ R [DΦR] exp{i(s R[Φ R] + S ct + d 4 xj R(x)Φ R(x))} Z R[J R] = Z B[J B] = [DΦR] exp{is[φ R] + S ct} G (n) 1 δ n Z R[J R] R (x1,..., xn) = i n δj R(x 1)... J R(x n) JR =0 = 1 δ n Z B[J B] i n Z n/2 δj B(x 1)... J B(x n) = Z n/2 G (n) (x1,..., xn) B JB =0 Similar for Connected Green s function. For on-shell renormalization scheme: in momentum space, near physical pole G (2) i R (p2 ) p 2 m 2 ph + iϵ G (2) iz B (p2 ) p 2 m 2 ph + iϵ

21 Renormalized & Bare Amputated Green s function Amputated Green s funct: G (n) B,conn (x 1,..., x n) = G (n) R,conn (x 1,..., x n) = using G (n) B d 4 y 1... d 4 y ng (2) B (x 1, y 1 )... G (2) (xn, yn)g(n) B B,Amp (y 1,..., y n) d 4 y 1... d 4 y ng (2) R (x 1, y 1 )... G (2) (xn, yn)g(n) R R,Amp (y 1,..., y n) (x1,..., xn) = Zn/2 G (n) (x1,..., xn), G(2) R G (n) B,Amp (y1,..., yn) = Z n/2 G (n) R,Amp (y1,..., yn) B = ZG(2) R : Bare & Renormalized 1PI: similar as before J R = Z 1/2 J B Γ B[φ B,cl ] W B[J B[φ B,cl ]] d 4 xj B[φ B,cl ]φ B,cl (x) = W R[J R[φ R,cl ]] d 4 xj R[φ R,cl ]φ R,cl (x) Γ R[φ R,cl ] φ B,cl = δwb[jb] = Z 1/2 δw R[J R] = Z 1/2 φ R,cl (x) δj B δj R Γ (n) R (x1,..., xn) = δ n Γ R[φ R,cl ] δφ R,cl (x 1)... δφ R,cl (x = n) Zn/2 Γ (n) B (x1,..., xn)

22 Equations For Green s Functions Schwinger-Dyson Eq. Consider path integral:(bare fields) Z[J] = [Dϕ]e i(s[ϕ]+ d 4 yj a (y)ϕ a (y)). Change the integration variable: ϕ a ϕ a (x) = ϕ a (x) + δϕ a (x), the integral is invariant. Assume [Dϕ] = [Dϕ ]. Just like d(x + a) = dx. 0 = δz[ϕ] = [Dϕ ]e i(s[ϕ ]+ d 4 yj a (y)ϕ a (y)) ( ) = d 4 x [Dϕ]e i(s[ϕ]+ d 4 yj a (y)ϕ a (y)) δs i δϕ a (x) + ij(x) δϕ a (x) [Dϕ]e i(s[ϕ]+ d 4 yj a (y)ϕ a (y)) Take n functional derivative w.r.t ij aj (x j), set J = 0 [ 0 = [Dϕ]e is d 4 δs n x i δϕ a ϕ ai (x i ) (x) i=1 n + ϕ ai (x 1 )... δ a,aj δ 4 (x x j )... ϕ an (x n) δϕ a(x). j=1 Since δϕ(x) is arbitrary, 0 = Ω it δs δϕ a (x) + n ϕ ai (x i ) Ω i=1 n Ω Tϕ ai (x 1 )... δ a,aj δ 4 (x x j )... ϕ an (x n) Ω j=1

23 Equations For Green s Functions Schwinger-Dyson Eq. Schwinger-Dyson Eq. 0 = Ω it δs δϕ a (x) + n ϕ ai (x i ) Ω i=1 n Ω Tϕ ai (x 1 )... δ a,aj δ 4 (x x j )... ϕ an (x n) Ω j=1 δs δϕ(x) = 0, the classical EOM. Free scalar: ( 2 x + m 2 )ϕ = 0 Two point function, scalar: i( 2 x + m 2 ) Ω T ϕ(x)ϕ(y) ω = δ 4 (x y) For x x i, Ω it δs n ϕ δϕ a ai (x i) Ω = 0. (x) For more detailed discussion, see Itzykson and Zuber,Quantum field theory, or Arxiv: , Swanson, A primer on functional methods and the Schwinger-Dyson Equation. i=1

24 Global symmetry and Ward Id. Noether theorem: Continuous symmetry conserved current. Classical global continuous symmetry: ϕ a ϕ a (x) = ϕ a (x) + iα ϕ a (x), α : an infinitesimal constant. 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 L ϕ a (x) iα(x) ϕa (x) + ( µϕ a (x)) i(α(x) µ ϕa (x) + ( µα(x)) ϕ a (x))) = iα(x) µk µ L + i( µα(x)) ( µϕ a (x)) ϕa (x) = i µα(x) j µ + iα(x) µk µ For ϕ satisfying EOM 0 = δs[ϕ] = i d 4 x µα(x) j µ + α(x) µk µ (integrate by part) = i d 4 x α(x) µ( j µ + K µ ) = d 4 xα(x) µj µ Since α(x) is arbitrary, define J µ = i(j µ K µ ), we have conserved current: µj µ = 0. Current conservation is a local property: α(x) can be non-zero in any arbitrary small region. Charge conservation is a global property.

25 Global symmetry and Ward Id. Consider correlation function: n i=1 ϕ(xi) = 1 Z [Dϕ] n i=1 ϕ(xi)eis[ϕ]. A global symmetric trans: ϕ a ϕ a (x) = ϕ a (x) + iα ϕ a (x), S[ϕ] = S[ϕ ]. α α(x), S[ϕ ] = S[ϕ] d 4 xα(x) µj µ. n n 0 = [Dϕ ] ϕ (x i )e is[ϕ ] [Dϕ] ϕ(x i )e is[ϕ] i=1 ( n = [Dϕ] ϕ (x i )e is[ϕ ] i=1 i=1 ) n ϕ(x i )e is[ϕ] i=1 i=1 ( n n ) = [Dϕ] ϕ(x 1 )... iα(x i ) ϕ(x i )... ϕ(x n) i d 4 xα(x) µj µ ϕ(x i ) e is[ϕ] i µ TJ µ (x) n ϕ(x i) = i=1 In momentum space: ϕ(x) = k µ TJ µ (k) i=1 n Tϕ(x 1)... δ 4 (x x i)(i ϕ(x i))... ϕ(x n) i=1 n ϕ(k i) = i=1 d 4 k (2π) 4 e ik x φ(k). (outgoing) n Tϕ(k 1)... i ϕ(k i + k)... ϕ(k n) i=1 For S-Matrix: from LSZ, for each external leg, lim p 2 m 2 p2 m 2 +iϵ Zi, there is no on-shell pole for leg ϕ i(k + k i), the right hand side 0, k µm µ = 0

26 Global symmetry and Ward Id. QED: consider U(1) symmetry ψ e ieα ψ, J µ = e ψγ µ ψ, correlation funtion J µ (k)ψ( (p + k)) ψ(p), Ward Id: k µ J µ (k)ψ( (p + k)) ψ(p) ( =ie ψ( p) ψ(p) ψ( (p + k)) ψ(p ) + k) where S(p + k)[ ik µγ µ (p + k, p)]s(p) = S(p) S(p + k), S(p) = ψ( p) ψ(p) = i p/ m Σ(p). ik µγ µ (p + k, p) = S 1 (p + k) S 1 (p). Γ µ (p + k, p) Z 1 1 γµ as k µ 0, S(p) i Z 2 p/ m, at around p2 = m 2, k 0 iz 1 1 k/ = iz 1 2 k/ Z 1 = Z 2 Note: ( ψγ µ ψ)ψ( (p + k)) ψ(p) γ µ αβ ψ( (p + k)) ψ αψ β ψ(p), no minus sign

27 Local gauge symmetry and Ward Id. in generating functional formulation L = 1 4 (F i 0,µν) 2 + i( ψ 0 /ψ 0 + ie 0 q 1 ψ0 A/ 0,µ ψ 0 ) m ψ,0 ψ0 ψ 0 1 2ξ 0 ( µa µ 0 )2 = Z (F i µν )2 + iz ψ 2 ( ψ /ψ + ieq 1 Z ψ 1 Z ψ 2 e 0 = e Z 1 = e Z 2 Z 1/2 3 Z 1/2 3. Only depends on Z 3. ψa/ψ) mz ψ 0 ψψ 1 2ξ ( µaµ ) 2 Bare gauge trans: ψ 0 e iαq 1 ψ 0, A 0,µ A 0,µ 1 e 0 µα. Renormalized gauge trans: ψ e iαq 1 ψ, A µ A µ 1 µα. ē = Zψ 1 e = e ē Z ψ 2 Consistent with A 0,µ = Z 1/2 3 A µ. the gauge fixing term is not invariant. We will see that there is no counterterm for gauge fixing ξ term.

28 iqeᾱ δz δz iqe iδᾱ iδα α µ j µz 1 2 µ δz ξ 0 iδj µ = 0 Local gauge symmetry and Ward Id. in generating functional formulation Adding source term for bare fields: (omit the subscript 0.) (α, ᾱ : grassman variables.) L[A, j, α, ᾱ] = L + ᾱψ + ψα + j µa µ Under gauge transformations A A = A µ + 1 µϵ, ψ e ψ = e iϵq 1 ψ: source terms and gauge fixing terms not invariant, all others including the path integral measure are inv. Z[A, ψ; j, α, ᾱ] =N [dad ψdψ] exp{is[a, ψ, ψ] + i (ᾱψ + ψα + j µa µ )} (rename ) =N (Gauge inv) =N 0 = δz =N [dad ψdψ] =N [dad ψdψ] [da d ψ { dψ ] exp [dad ψdψ] { exp i S[A, ψ, ψ] + i 1 µa µ νa ν} 2ξ 0 is[a, ψ, ψ ] + i (ᾱψ + ψ α + j µa µ ) (ᾱψ + ψ } α + j µa µ ) ( i ᾱδψ + δ ψα + j µδa µ 1 µa µ νδa ν) exp{is +... } ξ 0 ( dx ϵ(x) iqᾱψ + iq ψα 1 e µ j µ 1 2 µa µ) exp{... } eξ 0 iqeᾱψ + iqe ψα µ j µ 1 ξ 0 2 µa µ j,α,ᾱ = 0

29 Local gauge symmetry and Ward Id. in generating functional formulation iqeᾱ δz δz iqe iδᾱ iδα α µ j µz 1 2 µ δz ξ 0 iδj = 0 µ In connected Green Functional iqeᾱ δw δᾱ In 1PI generating functional: + iqe δw δα α + µ j µ + 1 ξ 0 2 µ δw δj µ = 0 iqe δγ ψ c + iqe δψ ψ δγ c c δ ψ + µ c where Γ = W j µa µ ᾱψ ψα, A µ c = δw δj µ, j µ = δγ δa cµ, δγ δa µ c δw ψc = δᾱ, α = δγ δ ψ c, + 1 ξ 0 2 µ A cµ = 0 ψc = δw δα ; ᾱ = δγ δψ c.

30 Local gauge symmetry and Ward Id. in generating functional formulation δγ iqe δψ ψc(z) + iqe ψ δγ c(z) c(z) δ ψ + δγ c(z) µ δa µ c (z) µ A cµ(z) = 0 ξ 0 δ ψ c(x) ψ c(y), Aµ c, ψ c, ψ c 0,: δγ δγ δγ iqe δψ c(x)δ ψ δ(x z) + iqe c(y) δψ c(x)δ ψ δ(y z) + µ c(y) δa µ c (z)δψ c(x)δ ψ = 0 c(y) iqe( is 1 (x, y)δ(x z) + is 1 (x, y)δ(y z)) + iqe µ z Γ µ(z, x, y) = 0 ik µγ µ (p + k, p) = S 1 (p + k) S 1 (p)

31 Local gauge symmetry and Ward Id. in generating functional formulation eg: δ δj ν (z) iqeᾱ δw δᾱ α,ᾱ,j µ 0 + iqe δw δα α + µ j µ + 1 ξ 0 2 µ δw δj µ = 0 : x µ (g µνδ(x y)) µ δw x = 0 ξ 0 δj µ (x)δj ν (y) j µ =0 δw =i TAµ(x)Aν(y) = δj µ (x)δj ν ig(2) (y) We then have: =i d 4 k µν (x y) = i e ik (x y) G (2) (2π) 4 [ d 4 k ( ) e ik (x y) g (2π) 4 µν kµkν A(k 2 ) + B(k 2 ) kµkν k 2 k 2 ik ν + ib(k 2 ) ikνk2 G (2) µν (k) = ξ 0 = 0 B(k 2 ) = iξ0 k 2 ( ) g µν kµkν A(k 2 ) + iξ0 k 2 k 2 k µk ν k 2 ( ) Notice: Free photon propagator: G (2) µν = i g k 2 µν kµkν iξ 0k µk ν. k 2 k 4 µν (k) ] The ξ 0 term is not renormalized. Renormalized propagator G R(2) µν = Z 1 3 G(2) µν, ξ = ξ 0Z 1 3.

32 the Legendre transformation and IPI Consider the action: S ϵ(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(ϵ 2 ), η(x) = (x, z)dz Expand the Feynman diagrams w.r.t ϵ, (using ϵ(x, y)): the ϵ 1 order terms, replace (x, y) η(x)η(y), cut one internal propagator. Higher order term: irreducible when cut 2,3,...,lines. 1PI ϵ 1 term is connected

33 the Legendre transformation and IPI The generating functional Z ϵ[j] = [dϕ] exp{i(s ϵ + Jϕ)} = (1 + i 1 2 ϵ δ δ ) dx dy [dϕ] exp{i(s + Jϕ)} + O(ϵ 2 ) iδj(x) iδj(y) = (1 + i 1 2 ϵ δ δ ) dx dy exp{iw [J]} + O(ϵ 2 ) iδj(x) iδj(y) ( = 1 + i 1 ( 2 ϵ δ δ δ dx dx dy ( =Z[J] 1 + i 1 ( 2 ϵ dx δj(x) W [J] ) 2 + i 2 ϵ δ ) 2 δj(x) W [J] i + 2 ϵ iδj(x) iδj(y) δ δ dx dy iδj(x) iδj(y) ) iw [J] exp{iw [J]} ) iw [J] + O(ϵ 2 ) W ϵ[j] =W [J] + i 1 ( 2 ϵ dx δ ) 2 δj(x) W [J] i + 2 ϵ δ δ dx dy iw [J] iδj(x) iδj(y) The second term just means the disconnectness of the W [J] after cutting a propagator.

34 the Legendre transformation and IPI Legendre transformation: Γ ϵ[φ cl ] W ϵ[j[φ cl ]] d 4 xj[φ cl ]φ cl (x), φ cl (x) = Ω Φ(x) Ω J Ω Ω J = δwϵ[j] δj(x). Γ ϵ ϵ = dx J(x) φcl ϵ φ cl (x) + Wϵ φcl ϵ + J = Wϵ ϵ J dx δwϵ δj(x) J(x) ϵ φcl Thus, Γ ϵ[φ cl ] =Γ[φ cl ] + iϵ 2 [ ( ) 2 dxφ cl (x) + ( δ 2 iγ[j] ) 1 dx dy + O(ϵ )] 2 δφ cl (x)δφ cl (y) The first term in the ϵ term is just from the term ϵϕ 2 we added in the action and the second term contains just connected diagrams. Thus, Γ[φ cl ] is 1PI.