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QUANTUM FIELD THEORY
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(13-182) Since AN ~ In N as N ~ 00, we expect f to decrease for fixed x faster than any power of In _q2. This property, together with the sum rule (13-180), implies that as - q2 grows the structure function has to increase in the vicinity of x = O. The asymptotic regime for the moments is slowly reached since subdominant terms are only suppressed by factors of order In [In ( -q2j,u2)Jjln( _q2j,u2). This results from the two-loop contribution to the effective charge [Eq. (13-90)]. The rate of approach depends on the unknown scale for which the effective charge becomes small. Moreover, for finite q2, mass effects may be sizable. On the other hand, the moments are not easy to obtain from the experimental data, since they involve measurements at very large energy (small x), and for any finite q2 the change of the x variable into x' = x + O(1jq2) may somehow modify the results. In spite of these difficulties, the comparison between theory and experiment seems to encounter a reasonable success. It is convenient to rewrite the effective coupling of Eq. (13-129) in terms of a single parameter A which sets the scale
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A = 0.2 A=0.5
J.-' "'"
o ~----_~I----_~I~--_~I----~I------~I~-----------.
1 2 4 8 15 30
Iq21(GeV 2 )
Figure 13-16 The theoretical prediction for the second moment of the muon-production structure function compared with the experimental data ofR. L. Anderson et aI., Phys. Rev. Lett., ser. B, vol. 38, p. 1450, 1977. This drawing and the one of Fig. 13-17 were communicated by G. AltareIIi.
ASYMPTOTIC BEHAVIOR
in the case of the color group SU(3) (C = 3) and for ftriplets of quarks (ffiavors). For f = 4, the best fits are obtained for A '" 400 200 MeV, as of 1977. In Fig. 13-16, the experimental data for the muon-production moment M(2) are compared with the theoretical prediction. The arrow shows the limit (13-180) (a = 0.12 for f = 4). For higher moments (Fig. 13-17) only the ratio M(N)(q2)/M(N)(q'6) may be compared with experiment. For inelastic neutrino scattering a similar analysis applies to the light-cone
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"'" J.,' 2 :;
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"'" J.,'
"'" J.,' 2
'" ~
2-- ___ _
Iq2 i(Gey2)
~'1~5
______
~2~O
Figure 13-17 Fourth and sixth moments for muon production, normalized at _q2 = 9 Gey2. The experimental data are from Anderson et al. (Communication at the Hambourg Conference, 1977.)
QUANTUM FIELD THEORY
expansion of the product of weak currents (VI' - AI')l+iZ(X)(lI" - Av)l-iZ(O) in terms of the twist two operators:
(a) (b) (c)
iN - 1 !/JYI',DI'2 ... DI')l
Ys)!/J
(13-184)
A third structure function f3 = vW3/m [see Eq. (13-106)] is also involved. By combining amplitudes for neutrinos and antineutrinos on proton and neutron targets, it is possible to isolate a particular channel with given quantum numbers. For instance, only the octet operator (13-184b) enters the difference W(v) - W(V). The sum rules arising from hadronic symmetries commuting with the gauge group continue to be satisfied. Thus Adler's sum rule (11-105) is verified for all qZ:
dw (vwz(vn) - vWz(V P )
(13-185)
Others are only asymptotic and approached logarithmically. An example is the Callan-Gross sum rule which reads
fro (dw/w N+ 1) WL ( w, qZ)
(dW/WN+1)VWz(w,qZ)
-,'Z..
0(In _qz)
(13-186)
where the quantity in the numerator WL = (2/W)Wl - vWz/m z vanishes in the parton model [Eq. (13-109)].
NOTES
Early work on the renormalization group was done by E. C. G. Stueckelberg and A. Peterman, Helv. Phys. Acta, vol. 26, p. 499, 1953, and M. Gell-Mann and F. E. Low, Phys. Rev., vol. 95, p. 1300, 1954. Applications to quantum electrodynamics were discussed by M. Baker and K. Johnson, Phys. Rev., vol. 183, p. 1292, 1969, who computed the third-order contribution to the !/J function, and by S. L. Adler, Phys. Rev., ser. D, vol. 5, p. 3021, 1972. L. D. Landau, A. A. Abrikosov, and 1. M. Khalatnikov studied consistency questions in Doklady Akad. Nauk SSSR, vol. 95, p. 1177, 1954. The f3 function for electrodynamics to third order is from E. de Rafael and J. L. Rosner, Ann. of Phys. (New York), vol. 82, p. 369, 1974. Ward identities for broken-scale in variance appear in the work ofK. Symanzik, Comm. Math. Phys., vol. 18, p. 227, 1970, and C. G. Callan, Phys. Rev., ser. D, vol. 2, p. 1541, 1970. An early discussion of the dilatation current is given in