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1 [(P _ k)2 _ m2](k2 _ m2)
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where some regularization is understood and the superscript 1 refers to the one-loop correction. A straightforward computation yields
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and The renormalized function r) )[l J satisfying (8-58) reads
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rlinI1J(p2) = - - 3
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fl dec {[m 2 0
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ce(1 - ce)P2] In
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m - ce
2 2 m - ce(1 - ce)P } 2 (1 ) 2 + ce(1 - ce)(P2 - /1 2 ) - ce /1
(8-60)
where we do not bother to perform the ce integration explicitly. If we now change /1 into Ii, it is easy to see that r) ) = p 2 - m 2 + r) )[l J satisfies Eq. (8-59) with
RENORMALIZATION
(8-61)
The function X would be determined by the calculation of the one-loop three-point function. This will be left as an exercise to the reader.
This may be generalized to arbitrary conditions, in any renormalizable theory, with a conclusion similar to Eq. (8-59). The information conveyed in Eq. (8-59), namely, the equivalence of changing renormalization points or renormalized parameters, is referred to as renormalization group invariance. The implications of this equation or of its infinitesimal form - the so-called renormalization group equation-will be studied in Chap. 13. We shall see that the innocent-looking freedom in the choice of normalization conditions has nontrivial and important consequences.
8-2-6 Composite Operators
The Green functions considered so far involved only elementary fields, i.e., dynamical variables entering the lagrangian. The renormalization procedure performed either through Bogoliubov's subtraction R operation or through the introduction of counterterms extends to a wider class of functions involving composite operators. By this we mean local monomials offield operators and their derivatives. A prototype of such an operator is the electromagnetic current ljIYp.ljJ. Composite operators play an important role in many developments. For the sake of simplicity, we shall again discuss only the scalar <p4 theory. Composite operators are of the form <p2, (o<p , <p 0 <p, <p4, <p6, etc., but also <pop.<P, op.<p ov<p, ... , if we construct vector- or tensor-like operators. For power counting, these operators have clearly the dimensions Wi = 2, 4, 4, 4, 6, ... , 3, 4, ... respectively. To deal with Green functions containing insertions of these operators Oi(X), it is convenient to add sources Xi coupled to them in the action. Consequently,
Z(j, X) =
<01 Texp {i
d4 x [j(x)<p(x)
+ Xi(X)Oi(X)]}
(8-62)
will generate these new functions. Connected Green functions will be obtained from the logarithm of Z, as in Eq. (6-71), while the Legendre transformation of Chap. 6, performed on the source j only, will generate the proper functions, i.e., one-particle irreducible but with an arbitrary number of Oi insertions. When restricting our attention to a finite number of those, we shall perform a finite number of derivatives with respect to the Xi and then set Xi = O. If we consider a diagram with N insertions of operators of dimensions Wi, a mere application of Eq. (8-13c) reveals that the new superficial degree of
QUANTUM FIELD THEORY
divergence Wi differs from the one in the absence of insertions by an amount
Wi - W
= L (Wi - 4)
(8-63)
Insertions of operators of degree less or equal to four in a superficially convergent diagram preserve the convergence, whereas insertions of degree larger than four deteriorate the power counting. However, whatever the (finite) number of insertions of composite operators, there exists a subtraction prescription or, equivalently, counterterms that make proper Green's functions finite. For instance, assume that the two-point (two external <p) function (8-64) is superficially divergent with degree Wi. There exists a local counterterm, quadratic in <p and proportional to Xl (x) XN(X), with a polynomial of derivatives of degree less or equal to Wi. Clearly this counterterm will contribute only to the function (8-64). This will now be exemplified in simple instances.
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