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(gool/2
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nontrivial fixed point asymptotic freedom (13-177)
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The last case seems to be favored by the experimental findings.
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The reader may wonder whether the observed scaling is not consistent with the existence of a nontrivial fixed point goo, provided that all YON(goo) vanish. However, this would imply that the anomalous dimension of the fundamental field itself vanishes:
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Together with positivity this entails that the theory is free. Therefore we are left with asymptotic freedom as the only possibility. We return briefly to gauge theories required for asymptotic freedom in order to survey some specific intricacies arising as usual in this case. Products of physically observable operators, and hence gauge invariant, can however involve unphysical ones, in particular ghost fields, in their shortdistance expansion. Such operators occur in the counterterms to Green functions and are therefore indispensible to compute physical anomalous dimensions. These additional operators are charac-. terized using the method of Ward identities developed in Chap. 12 for Green functions without insertions. The result of Eq. (12-163) is generalized as follows. Using the same notations as in this chapter, a gauge-invariant operator 0 of dimension d generates counterterms of dimension smaller or equal to d with the same quantum numbers as 0, which are either gauge invariant or of the form (JO'. The second class of operators is stable under renormalization since (J2 = O. Consequently, we can organize a computation of anomalous dimensions in a basis of the form {Oinv' (JO'j}. The renormalization matrix is then upper triangular by blocks. Only the submatrix in the subspace O;nv enters the calculation of gauge-invariant physical anomalous dimensions. In favorable cases some arguments allow us to simplify the analysis by computing directly this submatrix. Finally, physical matrix elements of gauge-invariant operators, and their anomalous dimensions, do not depend on the gauge parameter. The term in ((ojf))') introduced in Eq. (13-74) may therefore be dropped.
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We close this discussion with a summary of results obtained in applying gauge theories of strong interactions to leptoproduction. They were derived by Georgi and Politzer, on the one hand, and Gross and Wilczek, on the other. For electro(or muon)-production, the previous analysis applies to the structure functions mW1 and vW2/mx, denoted collectively !a(q2, x), and to their moments
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M~Nl(q2)=
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dXX N- 1fa(q2,X)
(13-178)
In that case, the last two series of operators in (13-170) contribute. We have to face the problem of mixing of the two types at fixed N. Since we deal with an asymptotically free theory, naive scaling is violated by computable powers of logarithms
M(Nl(q2)
aN,a(1n _q2)-AN .
(13-179)
where the sum runs over the eigenvector operators of the renormalization matrix. In general, the coefficients aN in (13-179) remain unknown. The lowest moment N = 2 is the most easily handled. It implies the conserved energy momentum tensor Tllv of zero anomalous dimension, together with another subleading operator. Moreover, the diagonal matrix element of Tllv between proton states is known:
<p[ Tllv[p)
This results in the following sum rules: lim
-~~oo
2PIlPv
l dx x [2m Wi (q2, x)] =
lip1
-r~oo
fl dx [VW (q2 X)] =
(13-180)
where a is given in terms of the average square charge <Q2) of fermion constituents through
(13-181)
The last expression applies to f flavor multiplets of fermions belonging to the same representation of dimension n of the gauge group SU(N). For instance, for the three triplets of SU(3) with charges j, - j, - j, the value of a is -fo" with four triplets of charges j, - j, - j, j, a = ~ ~ 0.12. In the free parton model, the corresponding value was a = <Q2) from (13-110). In the asymptotically free model, the reduction of a is due to the fact that part of the energy momentum is carried by neutral gluons. For N larger or equal to four, it is a reasonable approximation to retain only the eigenoperator of lowest anomalous dimension. For large N, AN behaves as In N. Although it is hopeless to reconstruct the structure functions from their moments without further information, some results may be derived from positivity
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