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We have poorly mistreated the beautiful and numerous experimental results on deep inelastic phenomena. Our aim was simply to illustrate an area of particle physics where modern ideas allow quantitative predictions which may be severely tested. For nonasymptotically free theories, such as quantum electrodynamics, little is known of the short-distance behavior.
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13-5 OPERATOR PRODUCT EXPANSIONS
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The preceding discussion illustrates the interest of studying the behavior of matrix elements of products of composite operators in several well-defined limits:
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1. The space-like separation tends to zero (euclidean case).
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QUANTUM FIELD THEORY
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2. The time-like separation tends to zero (minkowskian case). 3. The square separation tends to zero (light-like limit). Problems 2 and 3 are specific of particle physics. In general, the operators involved are conserved or partially conserved currents. Problem 1 is the one that can be analyzed most thoroughly. The results can be directly applied in statistical mechanics. Crucial contributions were made by Wilson and Zimmermann.
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13-5-1 Short-Distance Expansion
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Consider the product of two local operators. For simplicity, we indicate only their dependence upon the configuration variable. Wilson has suggested a shortdistance expansion of the form
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A(x)B(y) ~
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X-'y
CN(x - y)ON ( - 2
(13-131)
The ON are a sequence of local regular operators, while the c-number coefficients CN(x - y) are singular in the limit x --+ y. Perturbatively their behavior is dictated by the canonical dimension of the corresponding operators up to logarithms
X-'O
lim CN(x) ~ xYN(mod In Ix j) (13-132)
dON - d A
The higher the dimension of ON the faster the C N go to zero. Of course, these notions will be somehow modified by renormalization group effects. When dealing with the euclidean theory the name operator is somehow abusive. We have in mind the possibility of constructing generalized Green functions G~k(x, y; Z1, ... , zn) such that the n arguments Zl, ... , Zn refer to fundamental fields and the remaining two to A and B. The meaning of Eq. (13-131) is that when ordering the ON according to their dimension
{[G(n) ( . ) AB x,y,Z1,"',Zn -
,\,N L..
m ..
C N (x- Y)G(n) ON
(x +
-2-,Zl"",Zn
)] (13-133)
CNm~,(X - y)} = 0
The Wilson expansion is characterized by the fact that the singularities generated in the limit x --+ yare given by the c-number coefficients CN(x - y) independently of the arguments and types of elementary field appearing in Green functions. In Minkowski space Eq. (13-133) is understood as an asymptotic series in the weak sense of matrix elements between physical states. Clearly this is a generalization from the case considered previously, when all separations were tending simultaneously to zero. We shall see that it is possible to write for the coefficients C N renormalization group equations. Instead of giving a cumbersome general proof, we shall satisfy ourselves with
ASYMPTOTIC BEHAVIOR
a simple example from a scalar field theory. The operators A and B are taken as elementary fields and we look for the dominant coefficient
(13-134)
We expect from (13-132) that C(X - y) behaves at a given order as a polynomial in In [x - y[, up to terms of order (x - y)2. To prove this, we consider the two sets of connected Green's functions G(n+ 2)(ql, q2, Pb .. . , Pn) and G~l(ql + q2 ; Pb . .. , Pn) and study the additional subtractions required in the construction of G~l as compared to those already necessary to define G(n+2). An example involving G(4) and G~l is shown in Fig. 13-13. A Feynman integrand relative to G~l may be obtained by identifying two external vertices pertaining to G(n+2) as a unique one and adjusting the symmetry coefficient. The subtractions to be performed are, however, different. For the sake of renormalization, it suffices to consider one-particle irreducible functions n:i. However, the two external lines to be contracted in G(n+2) need not be truncated. Let G'(n+ 2)(ql, q2 ; Pl, ... , Pn) denote the one-particle irreducible (n + 2)-point function with complete propagators on the external lines of momentum ql and q2. Let fl1<p2 and fl1 be the renormalized integrands of r~~) and G,(n+2). They are related through (13-135) where Y' comes from the subtractions on the renormalization parts containing the vertex v of the <p2 operator. The latter are irreducible Green functions with two <p external lines; they are of superficial degree zero if <pl. is considered of dimension two. Following Chap. 8, denote by P the forests of renormalization parts, one of them containing v. If T is the smallest renormalization part of P containing the vertex v, we write (13-136) with P2 the set of renormalization parts included in T and Pl those containing r or disjoint with it. Let I be the un subtracted integrand common to r<;~ and G,(n+ 2),
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