e¬ective action has associated a factor a’1 as has the source, which in this context we

also refer to as a vertex (on a par with “1 ). A diagram with V vertices (of either kind)

and P propagator lines is thus proportional to aP ’V . Since the diagrams generated

by the path integral in Eq. (10.80) are vacuum diagrams they are loop diagrams, the

tree diagrams being those with zero number of loops. Since it takes two times two

protruding lines from vertices (or one vertex) to form one loop, the number of loops

L is speci¬ed by L = P ’ V + 1, and an L-loop diagram carries an overall factor

proportional to aL’1 .15 The theory de¬ned by the functional integral in Eq. (10.80)

can thus be described at the diagrammatic level in terms of the diagrams for the

theory where a is unity, a = 1, according to

˜ ˜

aL’1 W (L) [J] ,

Wa [J] = (10.81)

L=0

˜

where W (L) [J] comprises the sum of connected vacuum diagrams with L loops for

the theory de¬ned by the action “[•], i.e. the theory speci¬ed by Eq. (10.80) for the

case a = 1. We note that the tree diagrams singled out in Eq. (10.79) correspond to

˜

the zero loop term W (0) [J].

In the limit of vanishing a, the value of the functional integral, Eq. (10.80), is

determined by the ¬eld at which the exponent is stationary, denote it •, according

to ’1

˜

eiWa [J] ∝ eia (“[•] +• J) , (10.82)

where the prefactor (a horrendous determinant term) involves the square root of

a(G’1 ’ Σ) and therefore its lowest power is a0 and will therefore turn out to be

0

15 This

observation gives, for the e¬ective action, a characterization of its diagrammatic structure,

and a controlled approximation scheme, the loop expansion. Say for a quantum ¬eld theory, the

diagrammatic representation of the e¬ective action corresponds to an expansion in L , where L is

the number of loops in a diagram.

332 10. E¬ective action

harmless when a eventually is set to zero. The stationary ¬eld, •, is determined as

the solution of the equation

δ“[•]

+ J1 = 0 (10.83)

δ•1

thus making contact with the original theory, since this is the equation satis¬ed by

the e¬ective action, Eq. (9.84). According to Eq. (10.81), in the limit of vanishing

a we have Wa [J] a’1 W (0) [J], and by taking the logarithm of Eq. (10.82) we get

˜ ˜

(noting that in this limit, the constant prefactor in Eq. (10.82) gives no contribution)

˜

W (0) [J] = “[•] + • J . (10.84)

But according to the Legendre transformation, Eq. (9.85), this implies

˜

W (0) [J] = W [J] (10.85)

and we have shown the validity of Eq. (10.79). That is, we have shown that the gener-

ator of connected Green™s functions can be expressed as the sum of all connected tree

diagrams where the vertices are one-particle irreducible.16 In diagrammatic terms,

the generator of connected Green™s functions, Eq. (10.79), can thus be displayed as

depicted in Figure 10.1.

=

+ +

1 1 + ···

2! 3!

Figure 10.1 The tree diagram expansion of the generator of connected amplitudes

in terms of the one-particle irreducible vertices.

The sum of all connected vacuum diagrams in the presence of the source is thus

captured by keeping only the tree diagrams if at the same time the bare vertices are

exchanged by the one-particle irreducible vertices.

The e¬ective action “[φa ], Eq. (9.80), taken for an arbitrary ¬eld value φa can

also be expressed in terms of a restricted functional integral, viz.

Dφ eiS[φ+φa ] ,

“[φa ] = (10.86)

1PICVD

16 This provides a proof in terms of the functional integral method, that i“ consists of the one-

particle irreducible vertices. We already knew this because of its diagrammatic construction accord-

ing to Section 9.6.

10.4. 1PI loop expansion of the e¬ective action 333

where 1PICVD indicates that in the perturbation expansion, only the connected

one-particle irreducible vacuum diagrams should be kept of the connected diagrams

generated by the perturbative expansion of the path integral, since upon expanding

in φa the prescription on the restricted functional integral generates “[φa ] according

to Eq. (9.80). In particular we have shown that “[0] is the quantity represented by

the sum of all connected one-particle irreducible vacuum diagrams for the theory (in

the absence of the source).

Since W [J] is related to “[•] by a Legendre transformation, the above observation

for “[0] corresponds to the statement that “[0] equals W [J] for the value of the

source for which the ¬eld δW [J]/δJ1 vanishes. Since the vanishing of δW [J]/δJ1 is

equivalently to δZ[J]/δJ1 vanishing, we can state the observation as

W [J] = sum of one-particle irreducible connected (10.87)

vacuum diagrams (1PICVD).

δ Z [J ]

δ J 1 =0

We shall make use of this observation when we consider the loop expansion of the

e¬ective action.

10.4 1PI loop expansion of the e¬ective action

In this section we shall use the path integral representation of the generators to

get a useful path integral expression for the e¬ective action which has an explicit

diagrammatic expansion. We follow Jackiw, and show how to express the e¬ective

action in terms of the one-particle irreducible connected vacuum diagrams for a

theory with a shifted action [52].

Consider a ¬eld theory speci¬ed by the action S[φ] and the corresponding path

integral expression for the generating functional

Dφ eiS[φ]+if φ ,

Z[f ] = (10.88)

where we have used the notation f for the one-particle source. In fact, in the next

chapter, when we consider non-equilibrium phenomena in classical statistical dynam-

ics the source will not be set equal to zero by the end of the day as it will contain

the classical force coupled to the classical degree of freedom of interest.

The path integral is invariant with respect to an arbitrary shift of the ¬eld, recall

Eq. (10.3),

φ ’ φ + φ0 (10.89)

giving for the generating functional

Dφ eiS[φ+φ0 ]+if (φ+φ0 ) = eiS[φ0 ]+if φ0 Z1 [f ] ,

Z[f ] = (10.90)

334 10. E¬ective action

where

Z1 [f ] = Dφ ei(S[φ+φ0 ]’S[φ0 ])+if φ . (10.91)

The subscript on Z1 [f ] is not a state label but just discriminates the generator from

the original generating functional Z[f ]. State labels in the functional di¬erentiations

are in the following suppressed throughout, and matrix multiplication is implied.

The generator of connected Green™s functions then becomes

’i ln Z = S[φ0 ] + f φ0 ’ i ln Z1 [f ]

W [f ] =

= S[φ0 ] + f φ0 + W1 [f ] , (10.92)

where

iW1 [f ] = ln Dφ ei(S[φ+φ0 ]’S[φ0 ])+if φ . (10.93)

To make the so-far arbitrary function φ0 a functional of f , we choose φ0 to be the

average ¬eld which e¬ects the Legendre transformation to the e¬ective action, “[φ],

i.e.

δW [f ]

φ0 ≡ φ = , (10.94)

δf

where a bar now speci¬es the average ¬eld, φ = •, for visual clarity in the following

equations. Recalling that this vice versa gives f implicitly as a functional of φ,

f = f [φ], we have according to Eq. (10.94) and Eq. (10.92)

δS[φ] δW1 δφ

+f + =0 (10.95)

δf

δφ δφ

and thereby, since the second factor on the left is the full Green™s function,

δS[φ] δW1

+f + = 0. (10.96)

δφ δφ

The e¬ective action can, according to Eq. (10.92) and Eq. (10.94), be expressed

as

“[φ] = W [f ] ’ φf = S[φ] + W1 [f ] = S[φ] + W1 [φ] , (10.97)

where in the last equality we have been sloppy, using the same notation for W1 as

a functional of the implicit function of φ as that of f , W1 [f ]. But by employing

Eq. (10.96) in Eq. (10.93) we can in fact eliminate the explicit dependence on f , and

get the expression for W1 as a functional of the average ¬eld, φ, as speci¬ed by the

functional integral

δS[φ] δW1

iW1 [φ] = ln Dφ exp i(S[φ + φ] ’ S[φ]) ’ iφ + . (10.98)

δφ δφ

10.4. 1PI loop expansion of the e¬ective action 335

The aim is now to evaluate W1 [φ], or rather to show that it can be expressed in

terms of one-particle irreducible connected vacuum graphs. We therefore introduce

the generating functional

δS[φ]

˜

Z[φ; J] = Dφ exp i(S[φ + φ] ’ S[φ]) ’ iφ + iJφ (10.99)

δφ

for the theory governed by the action

δS[φ]

˜

S[φ, φ] = S[φ + φ] ’ S[φ] ’ φ , (10.100)

δφ

i.e. the action for the original theory expanded around the average ¬eld but keeping

only second- and higher-order terms. Correspondingly for the generator of connected

Green™s functions in this theory we have

˜ ˜

iW [φ; J] = ln Z[φ; J] (10.101)

and evidently by comparing Eq. (10.99) and Eq. (10.98)

˜

W1 [φ] = W [φ; J] . (10.102)

J=’δW1 /δφ

We shortly turn to show that for this particular choice of the source as speci¬ed

˜

in Eq. (10.102), J = ’δW1 /δφ, the generator Z vanishes

˜

δ Z[φ; J]

=0 (10.103)

δJ

J=’δW1 /δφ

or equivalently for the generator of connected Green™s functions

˜

δ W [φ; J]

=0, (10.104)

δJ

J=’δW1 /δφ

i.e. the average ¬eld

˜

δ W [φ; J]

•= (10.105)

δJ

˜

vanishes for the theory governed by the action S for the value of the source J =

’δW1 /δφ. The statement in Eq. (10.102) thus becomes equivalent to the statement

˜

that W1 [φ] is identical to the e¬ective action for the theory governed by S[φ, φ]

˜

for vanishing average ¬eld, “[φ; • = 0]. We then use the result of Section 10.3,

that in general “[• = 0] is given by the one-particle irreducible connected vacuum

diagrams, or equivalently for the generator of connected Green™s functions the ex-

pression Eq. (10.87), viz. that W [f ]δW/δf =0 consists of the sum of all the one-particle

irreducible vacuum diagrams. The functional W1 thus in diagrammatic terms only

consists of the sum of all the one-particle irreducible vacuum diagrams for the theory

˜

governed by S[φ, φ]. These diagrammatic identi¬cations will be exploited shortly.

336 10. E¬ective action

To establish the validity of Eq. (10.102) we di¬erentiate Eq. (10.98) with respect

to the average ¬eld φ

δ2

δW1 1 δS[φ + φ] δS[φ]

Dφ ’ ’φ

= (S[φ] + W1 [φ])

Z1

δφ δφ δφ δφ δφ

δS[φ] δW1

— exp i S[φ + φ] ’ S[φ] ’ φ ’φ . (10.106)

δφ δφ

The term originating from the ¬rst term in the parenthesis on the right-hand side

can be rewritten as

δS[φ + φ] δS[φ] δW1

Dφ exp i S[φ + φ] ’ S[φ] ’ φ ’φ

δφ δφ δφ

δ δS[φ] δW1

’i Dφ exp i S[φ + φ] ’ S[φ] ’ φ ’φ

=

δφ δφ δφ

δS[φ] δW1 δS[φ] δW1

Dφ exp i S[φ + φ] ’ S[φ] ’ φ ’φ

+ +

δφ δφ δφ δφ

(10.107)

and since the ¬rst term on the right is an integral of a total derivative it vanishes,

giving

δ2

δW1 1 δW1

Dφ ’φ

= (S[φ] + W1 [φ])

Z1

δφ δφ δφδφ

δS[φ] δW1

— exp i S[φ + φ] ’ S[φ] ’ φ ’φ . (10.108)

δφ δφ

The ¬rst term on the right in Eq. (10.108) is equal to the term on the left, giving the

equation

δ 2 (S[φ] + W1 [φ]) δS[φ] δW1

Dφ φ exp i S[φ + φ] ’ S[φ] ’ φ ’φ = 0.

δφ δφ δφ δφ

(10.109)

The ¬rst factor

δ2 δ 2 “[φ]

S[φ] + W1 [φ] = (10.110)

δφ δφ δφ δφ

is according to Eq. (9.95) the inverse Green™s function and therefore nonzero, and we

have the sought after statement of Eq. (10.103)

δS[φ] δW1

Dφ φ exp i S[φ + φ] ’ S[φ] ’ φ ’φ =0. (10.111)

δφ δφ

10.4. 1PI loop expansion of the e¬ective action 337

We have thus according to Eq. (10.102) shown that

W1 [φ] = sum of all one-particle irreducible connected vacuum (10.112)

diagrams (1PICVD) for the theory de¬ned by the

˜

action S[φ, φ].

˜

Dividing the action S[φ, φ] into its quadratic part and the interaction part

˜ ˜ ˜

S[φ; φ] = S0 [φ; φ] + Sint [φ; φ] (10.113)

we have

1 δ 2 S[φ] 1

φ ≡ φD’1 [φ] φ

˜

S0 [φ; φ] = φ (10.114)

2 δφδφ 2

and

∞

1 δ N S[φ]

˜ φ1 · · · φN .

Sint [φ; φ] = (10.115)

N ! δφ1 · · · δφN

N =3

˜

In the path integral expression for the generator Z, Eq. (10.99), the normalization

factor

√

’1

i

Dφ e 2 φD φ = i det D (10.116)

was kept implicit, but by exposing it the expression for the e¬ective action, Eq. (10.98),

can ¬nally be written as

i

“[φ] = S[φ] ’ Tr ln iD’1 [φ] ’ i ln eiSint [φ;φ] 1PICVD ,

˜

(10.117)

2

where the last term should be interpreted as

˜ ˜ ˜

Dφ eiS0 [φ;φ] eiSint [φ;φ]

eiSint [φ;φ] = (10.118)

1PICVD

and the subscript “1PICVD” indicates the restriction to the one-particle irreducible

connected vacuum diagrams resulting from the functional integral. We have explicitly

displayed the one-loop contribution, the second term on the right in Eq. (10.117),

and consequently we have the normalization

1 1PICVD = 1. (10.119)

The ¬rst term on the right in Eq. (10.117), the zero loop or tree approximation,

speci¬es the classical limit, determined by the stationarity of the action, and the

second term gives the contribution from the Gaussian ¬‚uctuations. The last term,

the higher loop contributions, gives the quantum corrections due to interactions,

radiative corrections. Reinstating gives the result that the contribution for a given

loop order is proportional to raised to that power.

For a 3- plus 4-vertex theory, the e¬ective action has the series expansion in

terms of one-particle irreducible vacuum diagrams as depicted (explicitly to three

loop order) in Figure 10.2, where we have reinstated the notation • = φ.

338 10. E¬ective action

“[•] = + +

+

+

+ +

+ +

+ + ···

Figure 10.2 The 1PI vacuum diagram expansion of the e¬ective action.