1 Effective action

Quantum Field Theory-II Solutions Set n. 3

UZH and ETH, FS-2020 Prof. G. Isidori

Assistants: C. Cornella, D. Faroughy, F. Kirk, J. Pag`es, A. Rolandi www.physik.uzh.ch/en/teaching/PHY552

1 Effective action

Starting from the Generating Functional of a scalar field theory W[J] =e^iZ[J]=N

!

Dφ e^{i!d4x [L(φ)+Jφ]} , (1)

and defining

φc(x) = δZ[J]

δJ = 〈0|φ(x)|0〉^J

〈0|0〉J

, (2)

we can use Eq. (2) to replace J(x) in terms of φ_c(x). This allow us to perform a functional Legendre transformation and define the functional

Γ[φc] =Z[J]−

!

d⁴xJ(x)φc(x) , (3)

(acting on φc) called the Effective Action.

I. Show that for a free theory, L0 = ¹₂(∂µφ)² −¹₂m²φ², Γ0[φc] =

! d⁴x

"

1

2∂µφc(x)∂^µφc(x)−1

2m²φc(x)²

#

. (4)

II. Assuming

V(φ) = λ

4!φ⁴ , (5)

and inserting the appropriate ! factors, show that J(x) = (∂²+m²)φc(x) + λ

3!φ³_c(x) +O(!) , (6)

and

Γ[φc] =

! d⁴x

"

1

2∂µφc(x)∂^µφc(x)− 1

2m²φc(x)²− λ 4!φ⁴_c(x)

#

+O(!) . (7)

III. Assuming

V(φ) = λ

4!φ⁴ , (8)

solve Eq. (2) iteratively in λ up to the first order, without expanding in !, and show that Γ[φc] =

! d⁴x

$1

2∂µφc(x)∂^µφc(x)−

"

Vλ+ 1

2(m²+∆m²_λ)φc(x)²+ λ 4!φc(x)⁴

#%

+O(λ²), (9) whereVλ and ∆m²_λ are constants of O(λ). Determine the explicit expression of ∆m²_λ.

1

Solution In the following we adopt the notation:

!

d⁴xif(xi, xj)≡ 〈fij〉i . (10) I. Generating Functional of a free scalar field theory as shown inLecture notes 4 is¹

W₀[J] =e^iZ0[J]=W₀[0]e^{−12〈J1D12J2〉12} , (11) where the Feynman propagator is

D(x1, x2)≡D(x2, x1) =

! d⁴p (2π)⁴

i

p²−m²+i&e^{−ip(x1−x2)} . (12) Using the above relations, we find

φ⁽⁰⁾_c (x) = δZ0[J]

δJ(x) =−iδlnW0[J]

δJ(x) =i〈J1Dx1〉₁ . (13) Since the Feynman propagator satisfies

(∂_x²+ ˜m²)D_x1 =−iδ⁴(x−x₁) , (14) where for simplicity we defined ˜m² ≡m² −i&, then

(∂²+ ˜m²)φ⁽⁰⁾_c (x) = J(x) . (15) Performing Legendre transformation and replacing J with φ⁽⁰⁾c

Γ0[φ⁽⁰⁾_c ] = Z0[J]−&

J1

'φ⁽⁰⁾_c (

1

)

1 ,

= i

2〈J1D12J2〉₁₂−&

J1

'φ⁽⁰⁾_c (

1

)

1 , (16)

= i 2

&'

(∂²+ ˜m²)φ⁽⁰⁾_c (

1D12

'(∂²+ ˜m²)φ⁽⁰⁾_c (

2

)

12−&'

(∂²+ ˜m²)φ⁽⁰⁾_c (

1

'φ⁽⁰⁾_c (

1

)

1 . Performing integration by parts and using Eq. (14), we find

Γ0[φ⁽⁰⁾_c ] =−1 2

&' φ⁽⁰⁾_c (

1

'(∂²+ ˜m²)φ⁽⁰⁾_c (

1

)

1 . (17)

Again integrating by parts and sending&→0 we find Eq. (4).

II. In the first part of Problem Set n. 2 we have shown that the generating functional for connected Green functions up to order O(!) is simply given by

Z[J] =S[φ0, J], (18)

1If we properly normalize our generating functionalW0[J] then the factorW0[0] disappears in equation (11). This is why we do not consider it in (16). This term−iln (W0[0]) is anyway irrelevant, independently of the normalisation, since when taking the functional derivative in the correlation functions, constant terms (independent of the source) will always vanish.

2

where the action is

S[φ, J] =

*1 2

'(∂µφ)²(

1− 1 2m²'

φ²(

1−[V(φ)]₁+J1[φ]₁ +

1

, (19)

and φ0 is defined through the classical equations of motion δS[φ, J]

δφ(x) ,, ,,

φ=φ0

= (∂²+m²)φ0(x) +V^′(φ0(x))−J(x) = 0 . (20) On the other hand

φ_c(x) = δZ[J]

δJ = δS[φ₀, J] δφ0

δφ₀

δJ +δS[φ₀, J]

δJ =φ₀ . (21)

With the last two equations we prove Eq. (6) after replacing V^′(φc(x)) = _3!^λφ³_c(x). Furthermore, up to orderO(!),

Γ[φc] =S[φc, J]− 〈J1[φc]₁〉₁ , (22) which proves Eq. (7).

III. As shown in the Lecture notes 5,

W[J] =e^iZ[J]=W0[0]e⁻ⁱ

"

V(−i_δJ^δ

1)#

1 e^{−12〈J1D12J2〉12} , (23)

and

iZ[J] =−1

2〈J1D12J2〉12+ ln

"

1 +e^{12〈J1D12J2〉12} -

e⁻ⁱ

"

V(−i_δJ^δ

1)#

1 −1 .

e^{12〈J1D12J2〉12}

#

. (24) Assuming V(φ) = _4!^λφ⁴ and expanding up to order O(λ²), we get (see Lecture notes 5 for derivation)

Z[J] = i

2〈J1D12J2〉₁₂−λ 4!

/3&

D₃₃² )

3−6〈D33D13D23J1J2〉₁₂₃+〈D13D23D43D53J1J2J4J5〉₁₂₃₄₅0

+O(λ²). (25) Now, let us calculate

φc(x) = δZ[J]

δJ(x) =i〈J1Dx1〉1+λ

2 〈D33Dx3D23J2〉23− λ

3!〈Dx3D23D43D53J2J4J5〉2345+O(λ²). (26) Assuming

J(x) = J⁽⁰⁾(x) +λJ⁽¹⁾(x) +O(λ²) , (27) and solving iteratively we find

J⁽⁰⁾(x) = (∂²+ ˜m²)φc(x) , J⁽¹⁾(x) = −i

3!

12D2xD4xD5xJ₂⁽⁰⁾J₄⁽⁰⁾J₅⁽⁰⁾3

245−32

DxxD2xJ₂⁽⁰⁾3

2

4 . (28)

Using integration by parts and Eq. (14), we find J⁽¹⁾(x) = 1

2φ_c(x)D_xx+ 1

3!φ³_c(x). (29)

3

Performing Legendre transformation and replacing J with φc

Γ0[φc] = Z[J]− 〈J1[φc]₁〉₁ ,

= i 2

2J₁⁽⁰⁾D₁₂J₂⁽⁰⁾3

12−2

J₁⁽⁰⁾[φ_c]₁3

1

+ iλ2

J₁⁽¹⁾D₁₂J₂⁽⁰⁾3

12−λ2

J₁⁽¹⁾[φ_c]₁3

1 (30)

− λ 4!

13&

D²₃₃)

3−62

D₃₃D₁₃D₂₃J₁⁽⁰⁾J₂⁽⁰⁾3

123+2

D₁₃D₂₃D₄₃D₅₃J₁⁽⁰⁾J₂⁽⁰⁾J₄⁽⁰⁾J₅⁽⁰⁾3

12345

4

+ O(λ²) .

Using Eq. (14), Eq. (28), Eq. (29) and integration by parts we finally get² Γ[φc] =

! d⁴x

$1

2∂µφc(x)∂^µφc(x)−

"

Vλ+ 1

2(m²+∆m²_λ)φc(x)²+ λ

4!φc(x)⁴

#%

+O(λ²) , (31) where

Vλ = λ

8D²₃₃ , (32)

and

∆m²_λ = λ

2D33 . (33)

A small remark on the normalisation N in (1) and the vacuum bubbles found in (32). The generating functional W[J] is by definition the vacuum-to-vacuum transition amplitude in the presence of the source J so it is sensible to use the normalisation condition

W[0] = 1 (34)

In this case one finds by definition

N⁻¹ =

!

Dφei(S[φ] +〈J1φ1〉1),,,,

J=0

= e i Sint

"

δ iδJ

# e

i

2〈J1D12J2〉12!

Dφei S⁰[φ]

,, ,, ,, ,J=0

(35)

whereS[φ] is the action without the sources and S₀[φ] the action of the free field. One can check explicitly that this term in the denominator will cancel exactly the vacuum bubble terms found in the numerator ofW[J] and thus they will not appear inZ[J] andΓ[φc]. The conclusion is that by fixing the normalisation factor as in (34), the vacuum shift is by definition cancelled and the vacuum bubble terms like (32) drop from expression (31).

2Useful relation: !

Dx2J₂⁽⁰⁾"

2=−iφc(x).

4