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cpP V(P)(CPo)}) p! (9-109)
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d x icp[O
+ m 2 + VI/(cpo)]~
(9-110)
QUANTUM FIELD THEORY
and leads to a nontrivial propagator through its dependence on <po. To obtain the h expansion let us rescale the field as <p -+ h 1/ 2 <p so that
Z(j) =
eO,Ul/h
= e(i/h)I('Po,j) f0(<p) exp (i
d4 x {-!-(O<P -
~2 [m
+ VI/(<po)]
(9-111)
p ::3
hP2 /
<pP V(P)( <po)}) p!
Wick's theorem applied to (9-111) yields non vanishing contributions only for even polynomials in <po Only integer powers of h will therefore occur in the loop expansion. From Eq. (9-111) we read that the leading term (order hO) to Gc(j) is I( <Po, j). Let us compute the next term. The integral over the quadratic part yields
f0(<p) exp {- i
d4 x <p[D
+ m2 + VI/(<po)]<P} =
(Det Ko 1 K V )-1/2
(9-112)
{Dx+m2+ VI/[<Po(x)]}t5 4 (x-y)
K o = (Ox
+ m2)t5 4 (x -
As in Chap. 4, we use capital letters for determinants and traces of operators of infinite dimension. The inverse of Ko, introduced by the normalization, has to be chosen as GF(x - y). Hence
Ko1Kv = t5 4 (x - y)
+ GF(x = eTr(InA)
y)VI/[<po(Y)]
Since a determinant may be written Det A we find
- iGc(j)
(9-113)
= I(<Po,j) +:2 h Tr In [1 + GFVI/(<po)] + O(h2)
(9-114)
To obtain 1(<p) we have to invert the relation
<p(x, J) = it5j(x)
t5Gc(j)
According to (9-114) and (9-108) <p is given to leading order by <Po up to corrections of order h. Moreover, since I is stationary at <Po we have I(<p,j) - I(<Po,j) = O(h2). Finally, we have to subtract Jd4 x <p(x)j(x) from -iGc(j) to obtain r, given therefore by
(9-115)
FUNCTIONAL METHODS
Figure 9-4 The effective potential to the one-loop order.
The perturbative interpretation of the second term is clear if we expand it as
~ Trln [1 + GFV"(cp)]
00 ( 1)"-1 ih L ----Tr {[GFV"(cp)]n} n~ 1 2n
This is the sum of the contributions of one-loop diagrams made of n propagators - iGF(x - Y) and n vertices -iV"(cp). It is depicted on Fig. 9-4 in the case of V(cp) = Acp4j4!. Notice that the factor 1j2n in front of each term of the sum is the symmetry factor of the corresponding diagram (n stands for the rotations, 2 for the reflection); similarly in the case of cp4 theory, the factor i in V"(cp) = Acp2j2 takes into account the symmetry between the two external legs attached to each vertex. This expansion may be carried out to all orders. The successive terms of Gc are represented by connected Feynman diagrams generated by the interaction term -Lp;;'3hP/2-I(cpPjp!)V(p)(cpo) with propagators obtained by the inversion of the kernel 0 + m 2 + V"(CPo). As far as qcp) is concerned, the result of the Legendre transformation is to select among the previous diagrams only the one-particle irreducible ones, and to replace CPo everywhere by the arbitrary argument cp. Of course, any actual calculation has to face ultraviolet subtractions.
In short, the steepest-descent or stationary phase method leads elegantly to the semiclassical expansion according to the number of loops. To go beyond perturbation theory requires either to expand around nontrivial extrema or to approximate the path integral in some utterly different way. We return to 1(<p). We know that its expansion in <p generates the oneparticle irreducible Green functions. As far as particle physics is concerned it is generally this aspect which is relevant. We may also insist on the role of 1(<p) as an effective action. Taking into account translation invariance we can find an expansion involving higher and higher derivatives in the field <p in the form
(9-116)
QUANTUM FIELD THEORY
In (9-116) the first term involves the sum of all proper functions at zero external momentum, the second sums all second derivatives at the same point, and so on. In principle the function q>(x) remains arbitrary. However, if we wish to compute Yeff only, we may satisfy ourselves with a calculation for a constant q> provided we can unambiguously factorize the four-volume divergent integration over x. As an example let us extract Yeff up to order h from Eqs. (9-114) and (9-115). In general,
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