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The perturbative expansion of truncated functions is expressed in terms of truncated diagrams, i.e., that have no self-energy part on their external lines. Moreover, in the Feynman rules, no factor or propagator is attributed to the external lines (see Fig. 6-23 for illustration). Finally, if we restore >the factors /1, as indicated in the last subsection, L-loop truncated diagrams get a factor /1L-1. We finally define proper or one-particle irreducible diagrams. Those are truncated connected diagrams which remain connected when an arbitrary internal line is cut (see Fig. 6-23). The proper functions, defined by their perturbative expansion in terms of proper diagrams, are the building blocks of perturbation theory, since the integrations over internal momenta may be carried out independently in each proper subdiagram of a given diagram. For the same reason, they playa central role
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Figure 6-23 Examples of a nontruncated diagram (a), of a truncated but not proper diagram (b), and of a proper diagram (c). In cases (b) and (c), no factor is ascribed to the external lines>
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in the renormalization program (see Chap. 8), since it is necessary and sufficient to make them finite to get rid of all ultraviolet divergences. Beside this topological definition, the proper functions may be defined algebraically. As shown by Jona-Lasinio, the generating functional of proper vertices is the Legendre transform of the generating functional of connected diagrams. The latter, denoted Gc(j), has been defined in Chap. 5 as the logarithm of the generating functional of all Green's functions
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Z(j) = <01 Texp
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d4 x <P(X)j(X)J!0> = eGAj)
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(6-71a)
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We proved in Sec. 6-1 that it corresponds indeed to connected Feynman diagrams with
(6-71b)
We now construct the Legendre transform of Gc as follows. Let <Pc(x,j) be the functional of j defined through
J <Pc(x,j) = iJj(x) Gc(j)
(6-72)
and let us assume that the relation <Pc(x) = <pix,j) may be inverted to yieldj(x) = jc(x, <Pc). This is possible at least as a formal series provided (JGc/Jj)lj=o = 0, which means that the one-point function vanishes and [J 2 Gc/Jj(x)Jj(y)Jlj=o:;6 0. This is what we assume in the following. The subscript c in <Pc aims at reminding us that <Pc is an ordinary c-number function, not to be confused with the quantum field <po The functional r( <Pc) is by definition
W(<Pc) = [Gc(j) - i fd4Xj(X)<Pc(X)JI. _.
J(X) - J,(X.'P,)
(6-73)
The factor i has been mtroduced for later convenience. From (6-73), it follows by differentiation with respect to <Pc(x) that . J I ~() T'( <Pc) = .o<Pc X
d4Y {[JGc(j) - l<Pc(Y) . . . JMy,() - IJc (X, <Pc) <Pc)} .. y-() ~ 'J Y J= J,('P,) u<pc X
The first term of the right-hand side vanishes, owing to (6-72), and
. Jix, <Pc)
Jr( <Pc) J<pix)
(6-74)
As is well known in the analogous instances of classical mechanics or thermodynamics, the Legendre transformation is involutive. We also notice that iT'(<pc) may be regarded as the value of Gc(j) - i Jd 4xj(x)<pc(x) (j and <Pc independent) at its stationary point inj.
PERTURBATION THEORY
We want to show that r( ((>c) is the generating functional of proper functions r(xr, ... ,xn):
r(({>c)
L~ n.
d4X l ... d4xn r(n)(Xl,".' Xn)({>iXl) ((>c(xn)
(6-75)
where in the last equation use has been made of Eq. (6-74). Hence, the kernel [J2r/J({>c(y)J({>c(z)] is the inverse of i [J 2 GjJj(z)Jj(x)] !i=i,' We may now set ({>c = 0, according to the assumption that (JGc/Jj)!i=O = 0. The previous identity tells us that the connected two-point function 2 )(z - x) = - [J 2 Gc/Jj(z)Jj(x)] !i= 0 is the inverse (for convolution) of the function - ir(2)(y - z) = - i[J 2r/J({>c(y)J({>c(z)]cp,=o
From translational invariance r(2) depends only on the difference y - z and
d4z r(2l(y -
Z)G~2)(Z -
x) = iJ 4(x - y)
(6-77)
In momentum space, this reads
G(2)(p, - p)r(2)(p, - p)
where the Fourier transforms of the r are defined as in Eqs. (6-20) and (6-21). Henceforth, we shall use shorter notations for the two-point function G(2)(p) == G(2)(P, -p), r(2)(p) == r(2)(p, -p). If we write the former as
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