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(13-145)
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a a a a } { x ox + y. oy + Za OZa + f3(g) og + (n + 2) [1 + y",(g)] G(n+ 2)(X, y, Za)
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(13-146) We have absorbed the factor 2[1 + <5(g)] in the definition of L\. We substitute Wilson's expansion when x --+ y and keep only the dominant term. In this limit the left-hand side reads
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A Callan-Symanzik equation also holds for G~n! of the form
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:x +
Za O!a
+ f3(g) :g + n + 2 + ny",(g) + y,.,(g) ] G~n!(x, za)
Gi~~2(X, za)
(13-147)
ASYMPTOTIC BEHAVIOR
The desired result will follow, provided that in the limit x side of Eq. (13-146) can be identified with Co Gr;<p' :
(n+2)( x,y, GLl.
Za x-:y
y the right-hand
x - Y)G(n) Ll.,<p'
(x +
~2~' Za
(13-148)
That this is true is a consequence of the same analysis as the one sketched above, generalized to Green functions involving a mass insertion ~. The latter is outside the renormalization part T which contributes to C(x - y) since the only divergent function with two <p2 insertions is a vacuum-to-vacuum amplitude. We conclude that (13-148) holds and we have therefore proved that Co(x) obeys the renormalization group equation (13-145). All this may be extended to the successive terms of the operator expansion. Returning to a general product A(x)B(y) we see that Eq. (13-132) must be corrected to read (13-149) with dA, dB, dON the canonical dimensions and YA, YB, YON the anomalous dimensions of the operators assumed to be multiplicatively renormalizable. Strictly speaking, Eqs. (13-145) and (13-149) are only valid in the euclidean domain. They hold in the Minkowski case provided Feynman's it; is kept finite. The genuine minkowskian limit as implied in e+ e- annihilation requires a careful discussion of possible oscillations at large momenta.
Wilson's expansion is only established in the weak sense. For instance, it holds for each function
G;;'h with n elementary fields cp and the operators A and B. It is not guaranteed that (13-131) can
be naively applied to other Green functions containing extra composite operators. For instance, it is not true that lim <O[ Tcp(X)cp(y)cp2(Z) [0>
is equal to
In this case, since the global function is primitively divergent there will exist new contributions to the short-distance behavior apart from those involving two fields cp generating the coefficient C(x - y). We must also account for the subtraction of the complete diagram, which will add a new function C(x - y) independent of z: lim <O[ Tcp(X)cp(y)cp2(Z) [0>
C(x - y)<O[ Tcp2 (x
+ Y)cp2(Z)[0> + C(x
- y)
(13-150)
The conclusions to be drawn from the expansion depend on the nonperturbative existence or nonexistence of ultraviolet fixed points. If such a point exists we recover the results of a modified dimensional analysis with (13-151)
In an asymptotically free theory where the functions y(g) are of order g2 we
678 QUANTUM FIELD THEORY
write, as in Sec. 13-3,
P(g)
_bg 3
y(g) = cg 2
+ ... + '"
(13-152)
(13-153) and therefore predicts logarithmic deviations from canonical scaling. To apply the above techniques to the concrete examples we have to specify the relevant operators, study their conservation laws, and extend the analysis to light-like separations.
13-5-2 Dominant and Subdominant Operators, Operator Mixing, and Conservation Laws
In the simple example of the product <p(x)<p(y) the operators ON of the expansion are local monomials of the fields and their derivatives compatible with the symmetry properties. For instance, the next subdominant terms in a theory invariant under the change of <p into - <p are of canonical dimension four and read
<p(~)<p( -~) =
C 2(X)<p2(O)
+ C~)(x)(o<p (O) + C~)(x)<pD<p(O)
(13-154)
+ C~3)(X)<p4(O) + .. ,
The notation C 2 (x) stands here for what was previously called C(x). The number of operators grows with the canonical dimension. Even when dealing with the first subdominant terms new difficulties arise. We recall that renormalization mixes operators of the same canonical dimension and the same quantum numbers. Moreover, renormalization is not exactly multiplicative since the insertion of an operator of dimension d necessitates counterterms of dimension smaller or equal to d. Special conventions are required to disentangle m2x 2 corrections to C 2(x) from contributions arising in C~)(x). In a massless theory, where such problems do not arise, we have only to deal with a multiplicative matrix renormalization of these operators. The dominant behavior of the CM) is therefore governed by the equation
{[x :x + P(g) :g + dA + YA(g) + dB + YB(g) 0
j dN JOi -
(Ykyj(g)}c~\X) = 0
(13-155)
where (y~)ij(g) = y{.j(g) is the transposed anomalous dimension matrix of the operators oM) mixed under renormalization [see Eq. (13-75)]. In the case of an ultraviolet fixed point goo, a diagonalization of y{.j(goo) will give the observed anomalous dimensions. A similar diagonalization is necessary for the matrix dj generalizing the constant c in Eq. (13-152) for an asymptotically free theory.
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