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The coupling in QCD depends on the renormalization point used to define it. Although the complete rate for any process must be independent of , it may be that a suitable choice of will allow many of the higher order terms in the perturbation series for the rate to be absorbed into the coupling . In this paper I shall consider primarily processes which depend on only one kinematic invariant. For a particular value of , all terms in these processes up to may be absorbed into an effective coupling , but in higher orders, there exist terms which cannot be accounted for in this way. The choice of which causes the most terms in the perturbation series to be absorbed into the effective coupling is proportional to the value of the kinematic invariant. However, the constant of proportionality, and hence the magnitude of the effective coupling for a particular value of the invariant, depends sensitively on the details of the process considered. I shall discuss at length processes of the form ( is a gluon)

and then give a briefer discussion of several other processes. The non-Abelian nature of QCD introduces inessential complications into the discussion, and so I shall ignore it throughout. With this simplification (which is equivalent to considering QED rather than QCD), the square of the amplitude for (1) to order is given by the classes of diagrams in Fig. 1. I assume that there can be no gluon corrections to the or lines (the gluon may be considered to carry conserved quantum numbers possessed by but not by ). I take the effective ABG vertex to be of the form (1)

where is the invariant mass of the gluon, and .

[ Figure 1 ] Typical diagrams contributing to to order .

If the quark mass () were taken to be non-zero, than the gluon propagator could be renormalized on shell, so that the diagrams giving corrections to single real gluon emission would not contribute. In that case the total rate for (1) is given by (removing a trivial phase space factor for the diagram )

where is the reduced gluon vacuum polarization operator, including improper vacuum polarization diagrams such as . To lowest order, as usual

To order , therefore, taking , the diagrams and give

for . This rate clearly diverges as . For , however, one must perform renormalization at . In that case, the diagram gives a contribution (a pair with zero invariant mass cannot be distinguished from a real gluon)

if . Adding (5) and (6) the complete rate for the process (1) to in the limit becomes

where if .

In eq. (7) I have displayed the fact that the couplings depend on the renormalization point ; they obey the renormalization group equation (RGE) . Of course, the renormalizability of QCD guarantees that at each order in the rate cannot depend on , so that . Nevertheless, by a suitable choice of the expression in braces in (7) may be absorbed into the effective coupling . Keeping only the term in one has (e.g., [1])

where is a renormalization group invariant mass (2) used to set the scale of the effective coupling. If the constant in eq. (7) were zero, then by choosing the rate could be rewritten (at least to ) as . This expression for the rate could be written in terms of again by using the first form in eq. (8), giving

which agrees with (7) (with ) to , but contains further higher-order terms. The form (9) is a conventional `renormalization group improved' estimate for .

For any physical value of , however, is non-zero. Hence a re-expansion of in terms of will not give the correct form (7); the `constant' term proportional to will be missing. A simple trick may, however, be used to remedy this deficiency. Instead of choosing , take and arrange so that the term in brackets in eq. (7) vanishes (i.e., ). In that case, the `renormalization group improved' estimate will agree with the full perturbation theory result, at least to order . Instead of taking , one could take , but replace the in the eq. (8) for by an effective

At (in the `unimproved' perturbation series) the coupling is fixed. Higher-order terms may be interpreted as inducing a dependence of the coupling on . Since this dependence is governed by the RGE, some of the higher-order terms in the perturbation series for are determined. In particular, the coefficient of the in eq. (7) is determined. However, the constant is not, and depends on the value of which characterizes the ABG vertex. Hence the choice of in which gives the correct non-logarithmic term at depends on . Alternatively, the effective value of if is sensitive to constant terms at and hence to . (3) Table 1 gives the values of and for various . (4) Note the large difference of from one in all cases, and the strong dependence of on . Let us temporarily ignore and higher terms. Then eq. (7) gives the exact rate for the process (1) which could, in principle, be measured experimentally. The experimental measurement could be used to determine the effective QCD coupling. Of course, the measured size of the coupling could be fit to the formula of eq. (8) and an effective value of deduced. However, because of the presence of constant terms, this effective will differ from the true . It is clear from Table 1 that the difference is rather large, and depends considerably on . To the rates for all processes which depend on only one invariant will have the form (7), so that the same phenomena will occur. For example, if the measured cross-section for is fit to the naive QCD prediction then the effective deduced will not necessarily be even close to the true value of . To obtain a better estimate of the true from the experimental , one must compute the constant in ( is the effective number of quark flavors)

Agreement between values of deduced from experimental measurements of various processes without making at least this correction is clearly not a direct prediction of QCD. Rather it will occur only if (unlike the case of changing ) the constant does not differ significantly between the processes considered.

[ Table 1 ] The corrections to the effective value of resulting from the inclusion of constant terms at for various forms of transition.

I return now to process (1), and discuss the contributions to its rate. Examples of the relevant classes of diagrams are given in Fig. 1. The iterated (`improper') vacuum polarization diagrams (, ) contribute

to , while the diagrams involving true vacuum polarization contribute

The coefficient of the in (11) corresponds to the term in the expansion of the effective coupling (8) deduced from the term in , while the coefficient of the log in (12) corresponds to the term [2] in . (5) The coefficient of the log in (11) depends on the constant at . If the effective coupling is computed from keeping up to in and then this coupling, evaluated at , is used to estimate the rate for (1) using , then not only will the constant term at be missed, but also a part of the at . Of course, the same choice of which accounts for the constant at will also account for the at . However, the constant terms at remain unaccounted for even when . If , then and the full form of to is [3]

I showed above that to , a suitable choice of allows the rate for (1) to be written simply as . The value of is chosen so that the expansion of the `renormalization group improved' effective coupling (obtained from ) in terms of agrees with the explicit perturbation theory result (7) for to . However, if is expanded to there is no longer a choice of such that to . This is because the equation to determine both depends on and in general has no real roots (if only the terms are considered, the quadratic equation for has no real roots). Hence the trick of changing the renormalization point or of modifying the effective to account for the constant terms at cannot be used at , and the naive prescription for including higher-order effects by replacing the encountered in a lowest-order calculation by has failed. Nevertheless, whereas the effective value of is very sensitive to the terms in , it is not particularly affected by the terms, at least for sufficiently small that perturbation theory should be reliable. The change in due to terms is given very approximately by , where is the constant term in . For reasonable values of , the change is less than about 25%.

I shall now discuss briefly the effective coupling in quark-quark scattering through one-gluon exchange, again ignoring non-Abelian effects. The amplitude for this process is proportional to the real part of the improper gluon vacuum polarization, which is related to by a once-subtracted dispersion relation (assuming , and taking ):

This is very closely related to the formula (3) for when . In the limit , to , and in fact, for all terms in of the form the principal part of the integral in (14) vanishes as , so that result holds. At , contains a term . In that case the results for and differ by . The result for the scattering amplitude (which may be deduced directly from the calculation of Källén and Sabry [4] (6) ) is

where is the invariant mass transferred along the gluon. It is clear that the conclusions reached above for the process (1) also apply here.

I now give several further examples of the phenomena discussed above. The first is the anomalous magnetic moment of a charged lepton in QED. Using the lowest-order result, , the RGE gives [6]

Explicit perturbation theory calculations show that (e.g., [7])

If one makes the conventional choice then the logarithms in (16) become . If , they therefore do not contribute. However, if , they will contribute and generate a large part of the difference between and . In practice there are also terms arising from the presence of new diagrams in the case, containing both and loops, and in addition terms of order , which are not amenable to a RGE analysis. Nevertheless the coefficients of the logarithmic terms at are determined by the RGE [6]. From eq. (16) one may deduce the value of which would allow the complete term to be absorbed into the effective coupling. The result is .

The second example is the decay of positronium. The rate for the decay of the state is given by [8]

yielding . For the state, [9]

so that . It is clear from this example that the effective value of deduced from decays may differ considerably from the true . The complete cross-section (averaged over the initial spins) for annihilation in the nonrelativistic limit is of the form (17), but with [10] , corresponding to .

Finally it should be pointed out that in QED there exist low-energy theorems which show that the cross-sections for certain processes (for example, [11]) in the nonrelativistic limit are proportional simply to , and contain no constant terms. In QCD, however, analogous low-energy theorems are rendered useless by the strong coupling of the theory in the low-energy domain.

In this paper I have considered only processes which depend on one kinematic invariant. For processes which involve two invariants (e.g., the total cross-section for deep inelastic lepton-hadron scattering) similar results should hold, but the value of which causes all terms to be absorbed into the effective coupling will then consist of a dimensionless combination of the invariants.

I am grateful to several people for discussions, especially G. C. Fox, H. D. Politzer and A. E. Terrano.

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