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Insertion of the <p2 operator. This operator has a dimension W(<p2) = 2 and in VS .NET
1. Insertion of the <p2 operator. This operator has a dimension W(<p2) = 2 and Scan PDF417 2d Barcode In VS .NET Using Barcode Control SDK for VS .NET Control to generate, create, read, scan barcode image in .NET applications. Painting PDF 417 In .NET Using Barcode drawer for VS .NET Control to generate, create PDF417 2d barcode image in .NET applications. therefore its insertion improves power counting. There are two superficially divergent proper functions with <p2 insertions. They are depicted in Fig. 89. Both have w = O. To the first, we associated a counterterm quadratic in X but <p independent, while the second requires a counterterm of the form h<p2. The original term h<p2 in the lagrangian is thus changed into Decoding PDF417 2d Barcode In VS .NET Using Barcode decoder for .NET Control to read, scan read, scan image in VS .NET applications. Paint Bar Code In .NET Using Barcode creator for VS .NET Control to generate, create barcode image in VS .NET applications. !(l + (j1)x<p2 + (j2X 2
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Scanning Bar Code In Java Using Barcode Control SDK for BIRT Control to generate, create, read, scan barcode image in BIRT applications. Bar Code Generation In None Using Barcode creator for Software Control to generate, create barcode image in Software applications. (865) Here q stands for the momentum entering the diagram at the <p2 vertex. We see that <p2 is multiplicatively renormalized, as <p, with a wavefunction renormalization constant Z'I'2. The insertion of several operators is a straightforward generalization r~]'PZ,R(qb q2; Pb"" Pn, A, m) (866) +(a) Figure 89 Divergent diagrams with cp2 insertions.
RENORMALIZATION
where the last term takes account of the vacuum diagrams (n = 0) of Fig. 89a. Of course, these new subtractions require new normalization conditions to be fully specified. For instance, we may impose conditions at zero momentum r(~) (0' 0) 'P,R ' r(~) 2 R(O, 0) 'P 'P , (867) in agreement with lowest order.
The renormalization of this cp2 operator is not independent of mass renormalization. Since the addition of a term h(X)cp2(X) in the lagrangian may be regarded as an xdependent variation of the mass m2 > m2  x(x), it follows that r~~)R(O;p;,{,m)= Z""~2 nl)(p;'{,m)IAo , omo
where the derivative is taken at fixed ,{O. Therefore, if subtractions are performed at zero momentum as in Eqs. (836) and (867) we conclude that om~ I am
(868) 2. Insertion of the <p4 operator. The insertion of operators of dimension four does not affect power counting. The counterterms linear in the source X",4, that is, counterterms relevant for a single insertion, will be combinations of {Oi' i = 1, ... , 4} == {<p2(X), <p4(X), (o<p (x), <p D <p(x)} (the last two are equivalent when integrated over x, but differ in general). These operators are in fact mixed by renormalization. Consequently, <p4 may not be considered independently from those. To be more specific, the generalization ofEq. (865) reads r&~R(q; PI, , Pn; A, rn) 1:: zn/2 Zijr~:reg(q; PI, , Pn; ..1,0, rno) (869) This amounts to saying that the renormalized insertion of <p4 requires the following source term in the lagrangian: X",4 [ Z21 Z
2 + Z22 Z2 4! + Z23 Z
(O<p)2 + Z24 Z
<p D <pJ 2 where the Zij are determined after introduction of suitable normalization conditions. Finally, from the previous study, we know that <p2 is multiplicatively renormalizable, which means that Zlj = 0 for j # 1 and Zl1 = Z",2. More generally, in a renormalizable theory, a complete set of composite operators Oi of dimension less or equal to a given number D, and with the same quantum numbers, is multiplicatively renormalizable in the previous matrix form, at least as long as we deal with a single insertion. Moreover, Zij = 0 if dim Oi < dim OJ (the matrix Zij is not symmetric). Both results are obvious consequences of the power counting of Eq. (863). As an exercise, the reader may investigate the relation between the Zij for dimension four operators and the bare constants ,{O, mo, Z. He or she may also carry out the analysis of the renormalization of dimension six operators, or the one of the tensor operators, such as the energy momentum tensor. 402 QUANTUM FIELD THEORY
In some instances, it may be interesting to assign to an operator a dimension greater than the one coming from dimensional counting, and to renormalize it accordingly. For instance, if instead of considering cp2 as an operator of dimension two we assign to it the dimension four, more subtractions (and normalization conditions) will be necessary. The new (or "hard") operator, denoted N 4(cp2) to distinguish it from the old (or "soft") one N2(cp2), will be on the same footing as cp4, (Ocp)2, and cp 0 cp. In particular, Z I j will no longer vanish. This generalization, useful in some applications, has been introduced by Zimmermann. In the case of several insertions, with the rule (863) as Ariadne's clue, we find that counterterms (multilinear in the sources) keep dimension four as long as the initial operators Oi have dimension four, but that their dimension increases as dim (Oi) > 4. For instance, a double insertion of <p4 and <p D <p requires again <p2, <p4, <p D <p, and (O<p)2 counterterms, while the insertion of <p4 and <p6 also involves all operators of dimension six and a double insertion of <p6 leads to counterterms of dimension eight!

