The Perturbative QCD Series. The Parameters of QCD
A large number of the more reliable results in QCD come from perturbative expansions at large momenta, and are due to the asymptotic freedom property. This justifies devoting a section to presenting a summary of our knowledge of the basic functions β and γm. In supersymmetric extensions of QCD, both functions are related and, for some specific supersymmetric theories, they can be calculated exactly. Actually, and as proved by Mandelstam (1983), there are renormalization schemes in which both ß, γm vanish identically, and in others they can be found to all orders, as remarked first by Shifman, Vainshtein and Zakharov (1983). We will not discuss these theories here. The interested reader may find information, and trace the relevant literature, from the monumental papers of Seiberg and Witten (1994); we turn now to ordinary QCD.
KeywordsPeris Mandel Stam
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- 1.A discussion in the case of deep inelastic scattering may be found in Bardeen, Buras, Duke and Muta (1978).Google Scholar
- 2.Another example of this has already been encountered when we discussed lattice QCD.Google Scholar
- 3.This is similar to the replacement of D by the expression (3.3.22b), with 11 to second order.Google Scholar
- 4.Moreover, part of the renormalon singularity is spurious. If we consider an observable quantity, such as the mass of a quarkonium state, we have to add to V the rest energy, 2m with m the pole mass, which also has a renormalon ambiguity that partially cancels that in V; see Beneke (1998) and Hoang, Smith, Stelzer and Willenbrock (1998) for details.Google Scholar
- 5.It is not easy to connect directly some of our results as given here with those presented in the last edition of the Particle Data Group (1996) tables. The reason is that these authors have chosen to employ definitions both for ß n and of ∧(3 loop) which are at variance with the ones used here. Of course, we have verified that the figures given there for αs agree with ours.Google Scholar
- 6.González-Arroyo, López and Ynduráin (1979).Google Scholar
- 7.Cf. the reviews of Aoki et al. (1998) and especially Bhattacharya and Gupta (1998).Google Scholar
- 8.See, for example, Di Giacomo and Rossi (1981).Google Scholar
- 9.Of course, we shall not be able to evaluate all orders in perturbation theory; actually, we we will sum the one-gluon exchange to all orders (which can be done explicitly in the nonrelativistic regime) and add one loop corrections to this.Google Scholar