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We note that the one-particle reducible diagram depicted in Figure 10.3, which
contributes in the sum of vacuum diagrams to W [J = 0], is absent in the series
expansion of the e¬ective action.




Figure 10.3 One-particle reducible diagram contributing to W [J = 0].




The above functional evaluation of the e¬ective action generates the one-particle
irreducible loop expansion in terms of skeleton diagrams, and in¬nite partial sum-
mation of naive perturbation theory diagrams is thus already done. A virtue of the
above expansion is that at each loop level for the e¬ective action it contains far fewer
diagrams than the naive perturbation expansion.
In Section 10.6, where the e¬ective action approach is applied to a Bose gas,
and in Chapter 12, where the theory of classical statistical dynamics is applied to
vortex dynamics in a superconductor, we shall need to take the loop expansion to
the next level where only two-particle irreducible vacuum diagrams will appear. We
therefore ¬rst go back to the generating functional technique, but now we will include
a two-particle source.
10.5. Two-particle irreducible e¬ective action 339


10.5 Two-particle irreducible e¬ective action
The e¬ective action can be taken to the next level in irreducibility in which only two-
particle irreducible vertices appear. To construct such a description, we introduce
a two-particle source K12 in addition to the one-particle source J1 , and a generator
of Green™s functions where the connected Green™s functions of the theory now are
contracted on both types of sources, i.e. de¬ned according to the diagrammatic
expansion in terms of the two sources as depicted in Figure 10.4.




+ 1
, + ···
=
2




+ +
1 1
2 2




+ 1
+ 1
+ ···
2 2




+ 1
+ ···
2




Figure 10.4 Diagrammatic expansion of the generator, W [J, K], in the presence of
one- and two-particle sources.




The diagrammatic notation for the two-particle source is thus as displayed in
Figure 10.5.




K12 = 2
1




Figure 10.5 Diagrammatic notation for the two-particle source.
340 10. E¬ective action


The diagrammatic notation for the generator makes explicit the feature that it
depends on both a one- and a two-particle source as stipulated in Figure 10.6.




,
W [J, K] =


Figure 10.6 Diagrammatic notation for the generator in the presence of one- and
two-particle sources.




The generator consists of the same one-particle source terms as the previous
generator of Section 9.5, and therefore generates the connected amplitudes or Green™s
functions of the theory according to

δ N W [J, K]
G12...N = . (10.120)
δJ1 δJ2 · · · δJN
J=0,K=0

In addition the generator contains two-particle source terms and mixed terms.
We notice the new feature of the presence of the two-particle source, that dif-
ferentiating with respect to the two-particle source can lead to the appearance of
disconnected diagrams; for example, see Figure 10.7.



1


1


+
δW [J,K]
2 =
δK12 J=K=0

2
2




Figure 10.7 Removing a two-particle source can create disconnected diagrams.




Taking the derivative with respect to the one-particle source

δW [J, K]
•1 = (10.121)
δJ1
10.5. Two-particle irreducible e¬ective action 341


we can, analogous to the procedure of Section 9.6, exploit the topological features
of diagrams to construct the diagrammatic expansion in terms of two-particle irre-
ducible vertices, as depicted in Figure 10.8. In the following we leave out in the
diagrammatic notation the implicit source dependences of quantities.




= + +




+ +




+ +
1 1
2 3!




+ ···
+ +




Figure 10.8 Two-particle irreducible expansion of the 1-state Green™s function.




The topological arguments for the diagrammatic equation displayed in Figure 10.8
is: the particle state exposed can propagate directly to either a one-particle source
or a two-particle source. In the latter case its other state can end up in anything
342 10. E¬ective action


connected, and these two classes of diagrams are depicted as the two ¬rst diagrams
on the right in Figure 10.8. Or the exposed state can enter into a two-particle
irreducible diagram, giving the class of diagrams represented by the third diagram
on the right, or into a two-particle reducible diagram. A two-particle irreducible
vertex is by de¬nition a vertex diagram which can not be cut in two by cutting only
two lines, otherwise it is two-particle reducible. In the case of entering into a two-
particle reducible diagram, the exposed state can enter into a two-particle irreducible
vertex which emerges by one line into anything connected, accounting for the fourth
diagram on the right containing the self-energy in the skeleton representation where
it is two-particle irreducible (recall the topological discussion of diagrams in Section
4.5.2). Or it can enter into a two-particle irreducible vertex which emerges by two
lines into anything connected, which can be done in the two ways as depicted in the
¬fth and sixth displayed diagrams on the right, or three or four, etc., lines as depicted
in Figure 10.8. We note that, from a two-particle irreducible vertex, three lines can
not emerge into a connected 3-state diagram since such a part is already included in
the vertex owing to its two-particle irreducibility.
Analytically we have, according to the diagrammatic equation depicted in Figure
10.8, the equation


1
(0)
•1 = G12 J2 + K23 •3 + “2 + Σ23 •3 + “2(34) G34 + “234 •3 •4
2


1 (0)
+ “2345 •3 •4 •5 + “23(45) •3 G45 + “2(34)(56) G34 G56 + ... .
3!

(10.122)

Operating on both sides of Eq. (9.79) with the inverse free propagator gives


1
J1 + K12 •2 + “1 + (’G’1 + Σ)12 •2 + “1(23) G23 +
0= “123 •2 •3
0
2


1
“1234 •2 •3 •4 + “12(34) •2 G34 + “1(23)(45) G23 G45 + · · ·
+ ,
3!

(10.123)

which corresponds to the diagrammatic equation depicted in Figure 10.9.
10.5. Two-particle irreducible e¬ective action 343




0 = + +




+ +




+1 +
2




+ ···
+




1 1 δ“[φ,G]
≡ +
+ δφ1



Figure 10.9 Two-particle irreducible vertices and source relation.


The last equality de¬nes the ¬eld-derivative of the two-particle irreducible e¬ective
action, i.e, just as the diagrams in Figure 9.48 lead to the introduction of the one-
particle irreducible e¬ective action, we collect the two-particle irreducible vertex
344 10. E¬ective action


functions into the two-particle irreducible e¬ective action

1
“[•, G] ≡ “12...N •1 •2 · · · •N
N!
N


“1(23) •1 G23 + “1(23)(45) •1 G23 G45 + · · · ,
+ (10.124)

which in addition to the ¬eld is a functional of the full propagators.
In the two-particle irreducible action, we encounter two di¬erent types of vertices,
viz. only ¬eld attachment vertices

δ N “[•, G]
“12...N = (10.125)
δ•1 · · · δ•N
φ=0,G=0

which are two-particle irreducible and for which we introduce the diagrammatic no-
tation depicted in Figure 10.10.

2
1
“12...N =
N


Figure 10.10 The 2PI vertex with only ¬eld attachments.


In addition we encounter vertices with also propagator attachments, for example
δ δ δ δ
“1(23)4(56) = “[•, G] (10.126)
δ•1 δG23 δ•4 δG56
φ=0,G=0

for which we introduce the diagrammatic notation depicted in Figure 10.11.

4

5
3
“1(23)4(56) =
2 6
1


Figure 10.11 Vertex with both ¬eld and propagator attachments.


In terms of the two-particle irreducible e¬ective action, we can diagrammatically
represent the equation depicted in Figure 10.9 as depicted in Figure 10.12 (rede¬ning
“12 ≡ (’G’1 + Σ)12 ).
0
10.5. Two-particle irreducible e¬ective action 345




0 = + +


Figure 10.12 Sources and 2PI e¬ective action relation.


Analytically, the diagrammatic relationship depicted in Figure 10.12 is

δ“[•, G]
= ’J1 ’ K12 •2 . (10.127)
δ•1
By diagrammatic construction we have analogously to the one-particle irreducible
case, G1 ≡ •1 ,
δW [J, K]
G1 = (10.128)
δJ1
but now in addition
δW [J, K] 1
= (G12 + •1 •2 ) (10.129)
K12 2
and these two relationships give implicitly the sources as functions of the ¬eld and
the full Green™s function
J = J[•, G] (10.130)
and
K = K[•, G] . (10.131)
Since the sources are independent, so are • and G. We then have the two generators
being related by the double Legendre transformation, i.e. with respect to two sources,

1 1
W [J, K] ’ • J ’ •K • ’ GK
“[•, G] = (10.132)
2 2
J=J[•,G],K=K[•,G]

and we obtain the second relation for the two-particle irreducible e¬ective action and
the sources
δ“[•, G] 1
= ’ K12 . (10.133)
δG12 2
For K = 0 we encounter the usual Legendre transformation and e¬ective action,
i.e. “[•] = “[•, G(0) ] for the value of the Green™s function for which

δ“[•, G(0) ]
= 0. (10.134)
(0)
δG12

By construction “[•, G] is the generator, in the ¬eld variable •, of the two-
particle irreducible vertices with lines representing the full Green™s function, G, and
346 10. E¬ective action


“[• = 0, G] is thus the sum of all two-particle irreducible connected vacuum diagrams.
Using Eq. (10.132) and Eq. (10.133) we have
δ“[0, G] δ“[0, G]
’ i ln Dφ φ exp i S[φ] + φJ (0) ’ φ
“[0, G] = Tr G φ
δG δG


i ln Dφ φ exp {iS0 [φ]} ,
+ (10.135)

where J (0) is the value of the source for which δW [J, K]/δJ vanishes, i.e. tadpoles

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