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Figure 13-13 The connected Green functions G(4) and G~l.
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QUANTUM FIELD THEORY
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TI (- P)JI
Y E :F2
(13-137)
We can organize the summation as follows. We sum over the renormalization part r including v, the forests ff'2 of r excluding r itself, and over the forests ,~\ of the reduced diagram G/r. The latter is a possible diagram for a function r~1. Let (J be the part of the diagram pertaining to G,(n+2) corresponding to r. It is an irreducible four-point function except for the propagators and possible self-energy insertions on the two lines which join at v in r. We call it 1"(4). For each r the integrand I is factorized in I = la/tIt. In an intermediate renormalization, the operators P represent the Taylor series at zero momentum. The forests ff'2 are those of (J, ff'1 those of G/r, and T amounts to restricting 9 ((J) to its constant term 9 o((J). This operation is illustrated on Fig. 13-14 where one shows a contribution to [,(4) with the two-lines candidate to be joined at v. The operation T sets the external momenta equal to zero except for the wouldbe circulating momentum k in r. Translated into symbols this means
L 9 q>2(G/r)9 o((J)
(13-138)
so that, from Eq. (13-135),
9 = 9 q>2
+ L 9 q>2(G/r)9 o((J)
(13-139)
The renormalized integrand for G'(n+2) is expressed in a way showing the subtractions implying v which are, of course, compensated by the second term on the right-hand side. The notation 9 q>2 is a little ambiguous since it still involves n + 2 arguments and is relative to a Green function which can be denoted <0\ T[ <p(x)<p(y)] 2 <p(zr) <p(Zn) \0). After integration over loop momenta and summation over diagrams we may write the Green functions as
G(n+ 2)(X, 'y, Zl, ... , Zn) =
<0 \T<p(X)<p(Y)<P(Zl)'" <p(Zn)\O) d4q1 d4q2 TI d4Pa
(2n)4(n:2)
(2n)4(j4(q1
+ q2 + ~Pa)
(13-140a)
ei(q, x+q,' y+L p,' z,)
rTI (2n)4 : It(q'" k' G) d4kb p
We define the quantities
<0\ T[<p(X)<P(Y)]2<P(Zl)'" <p(Zn)\O) = idem with: lt -4 9 q>2 <0\T[<p 2(x)]2<p(zl)"'<p(zn)\0)=idemwith: lt-49 q>2 and X=Y (13-140b) (13-140c)
/-ql-q2
ASYMPTOTIC BEHAVIOR
-k+ql +q2
[,(4)
Figure 13-14 Zero-momentum subtractions on
The notation [J2 is to remind us that dimension two is attributed to the operator between brackets. For notational simplicity we have used the minkowskian timeordering symbol. Setting
(13-141)
for the zero-momentum Fourier transform, we conclude from the identity (13-139) that
<0 [T<p(x)<p(y)<p(zl) <p(zn) [0) = <0 [T[ <p(x)<P(y)J2 <p(z I) <p(zn) [0)
+ ~ <Or T<P(
y) <p(y;
<p(0) <p(0) [oY
x <0[T[<p2(x;Y)1 <P(ZI) <P(Zn)[O)
(13-142)
The index P recalls that the quantity in question is obtained from the irreducible r(4) by adding complete propagators on two external lines. The reader might find it useful to tryout the above operations on an example in order to be convinced of the presence of the symmetry factor i in (13-142). This algebraic construction is well suited to study the limit x --+ y. Indeed, renormalization theory implies the convergence of the subtracted integral. In particular, Green functions involving [<p(X)<P(Y)]2 tend order by order to those of [<p 2(X)J2 up to corrections to the type (x - y)2(ln [x - y [)a, in the euclidean region at least. Consequently, Eq. (13-142) enables us to isolate the most singular part of G(n+ 2) in the form
<0 [T<p(x)<p(y)<P(ZI) <p(zn) [0)
C(x - y) <0 [T [<p2 (x
+ y)]2 <P(ZI) <p(zn) [0) 2
(13-143)
C(x - y) =
1 + ~ <Or T<p (x; y) <p(y ~ x ) <p(0) <p(0) [O)P
To each order C(x) behaves as a polynomial in In [x [. For instance, to lowest
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order C(x) is proportional to
rdp C(x) ~ J(2n)4 e
+ m2 = (2n
Ko(m x
I I)
(13-144)
1 ~ (2n)3
[mlxl + O(m -In -2-
One can refine the above derivation in order to exhibit the successive terms in the Wilson expansion. We shall skip this tedious task and, assuming the result, will study the consequences of the renormalization group on the coefficient C. According to Eq. (13-143) this implies the analysis of a Green function at exceptional momenta. From the above example it is clear that C(x) might contain subdominant terms in its perturbative behavior. We have therefore the choice to follow either the original Callan-Symanzik method with zero-momentum mass insertion or the Weinberg approach of mass-independent normalization conditions. We take for definiteness the first option and call Co(x) the asymptotic part of C(x) obtained by dropping perturbative subdominant terms. We want to show that Co(x) satisfies an equation of the form
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