Perturbative Corrections to the Leading Pole Approximation

To , the decay is described by the two-body decay probability . At , terms describing emission of one gluon must be added. In the leading pole approximation, the relevant quark decay is independent of the decay, and the differential cross-section is given by the product . However, when the invariant mass of the virtual quark approaches its kinematic maximum , corrections may become important, and invalidate the independent emission approximation. Such corrections may be included by introducing a three-body decay probability defined simply as the piece of the complete differential cross-section not accounted for by the successive independent two-body decays . The contribution of to integrated cross-sections is typically relative to the dominant independent decay term . It turns out that with the identification for made here (see above), is identically zero away from . However, if one considers the decay of a scalar photon , then the independent two-body decay term remains as for a vector photon, but the three-body decay term becomes (where , refer to , , respectively, , and is the , invariant mass). Even at the kinematic boundary , so that the three-body decay term provides a rather small correction away from the end point, and the sharp cutoff assumed for the two-body decay probabilities is adequate.

Just as for , the approximation of successive independent two-body parton decays may receive significant corrections when a parton produced in the first decay does not have an invariant mass much smaller than its parent. To account for these corrections, one may again introduce a three-body decay probability (6). The exact differential cross-section for production of partons by decay is then given by a sum of terms: the first results from independent two-body decays; the second from one three-body decay and two-body decays; the final term represents a single -body decay. In most cases, the successive terms in this series should be progressively much smaller; the first term, corresponding to the simple leading pole approximation should then provide an adequate estimate for the complete differential cross-section. Observable properties of the parton final state are given as integrals of this differential cross-section. It appears that the results obtained by using the leading pole approximation for the differential cross-section but keeping the exact kinematic boundaries agree (apart from overall normalization discussed below) well with relevant regions of available complete explicit calculations (7) , suggesting that may indeed be neglected. Note that in many discussions (e.g., [12]) of the ``leading log approximation'', further kinematic approximations are made on integrating the leading pole approximation differential cross-section (typically, all interdependence in the limits of and integrations is ignored, and the cutoff is implemented only in , but not , integrals). The results of this procedure are often inaccurate: a good estimate of the differential cross-section is wasted by maltreatment of the kinematic limits of integration. Inclusive observables in the parton final state (e.g., single parton energy or distributions, or distributions) often receive divergent contributions close to the kinematic boundaries at each order in (e.g., ) terms arise in the relative transverse momentum distribution of the leading which dominates the distribution). When summed to all orders in using the leading pole approximation, the contributions exponentiate to provide ``radiation damping'' at the kinematic boundary (e.g., the relative transverse momentum distribution between the leading becomes , as discussed in [1]). All contributions are formally relative to the leading pole terms (in practice they appear to have small coefficients): since perturbative methods apply only when , this factor probably ensures small contributions in the relevant region. In addition, multiple decay probabilities corresponding to ``Bose-Einstein correlations'' between gluons should be suppressed by powers of because a larger number of colors gives a smaller probability for a set of gluons to be indistinguishable. Note that for small , the two body decay probabilities receive corrections from finite light quark current masses, and presumably suffer ``higher twist'' corrections from the onset of hadron formation.

I now discuss the origin and form of the effective coupling appearing in the decay probabilities of eqs. (1, 2). The give leading pole approximations to the probabilities for two-body decays of off-shell partons, summed over all subsequent interactions of the decay products. All parton interactions receive virtual corrections which exhibit ultraviolet divergences; such divergences may be renormalized by a subtraction at an invariant mass . Renormalized quantities, such as the physical coupling constant , depend on the value of this renormalization mass: different choices for leave measurable cross-sections unchanged by altering contributions from explicit virtual corrections so as to cancel the changes in . At , the probability for the decay . At , the decay probability receives its leading corrections from the diagrams (8). In (3a), the outgoing gluon is not on its mass-shell, but may have an invariant mass up to the kinematic limit imposed by the mass of the decaying quark. The total correction due to diagrams (3a) is roughly

In diagram (3b), the final gluon must be on-shell, so that the result toes not depend on : it serves simply to cancel the divergences from (3a), yielding total correction to the decay probability where is the renormalization mass introduced in subtracting the divergences of (3b). In higher orders, the dominant diagrams involve several virtual corrections followed by real pair production: the diagrams form a geometric series whose sum is . Hence the probability for the decay , summing over all subsequent fates for the produced . If is chosen to be , then explicit higher order diagrams will provide no (leading pole) corrections to : all such corrections will have been included implicitly in the effective coupling (9).

The argument of appearing in the decay probabilities (1, 2) is roughly the relative transverse momentum (squared) between the products of the parton decay. If rather than were used in then the three-body decay probabilities would contain terms which become large near the kinematic boundary: such terms are summed and accounted for by use of in the two-body decay probabilities. In obtaining the usual form from the diagrams (3), one assumes that intermediate gluons may have arbitrarily small invariant masses. As discussed above, the used in the calculation become inaccurate for : to be consistent with the treatment of real emissions, one should assume that no parton may have an invariant mass . In this case, the higher order corrections imply a form (10) : this freezes at a large fixed value for small , and toes not diverge at the Lantau point . For some purposes, this behavior is analogous to providing an effective mass for gluons. (Note that the exact form of the corrected decay probabilities could in principle be obtained by Monte Carlo integration over all possible final states accessible from each decay considered: the procedure is, however, quite unwieldy, and probably unnecessary in view of the small changes expected (see below).)

In addition to the logarithmic terms from (3) accounted for by use of , there are also constant, nonlogarithmic terms which provide contributions to both two- and three-body decay probabilities. The values of these terms depend on the renormalization prescription used: arbitrary constant terms may be removed with the divergences in (3b): whenever divergences, such as those in (3b), are subtracted (renormalized) away, a nondivergent remainder is in general left. This remainder may be removed by redefinitions of ``bare'' parameters, such as the coupling constant . The value of to be inserted into the Lagrangian is, of course, not known a priori (and, in fact, must be divergent to cancel the divergences in the perturbation series), but may be determined by fitting theoretical calculations to experimental data. The fitted numerical value for as will depend on the remainder removed (in such a way that the output experimental prediction used for the fit remains unchanged). The value of the remainder is thus determined by the ``renormalization prescription'' used to remove the divergences, and to define physical parameters such as . Theories such as QCD possess the properties of being renormalizable and infrared factorizable, whose meaning is that the number of distinct divergences (due to large and to small momentum configurations) at a given order in perturbation theory is limited: once prescriptions have been devised and applied to subtract these divergences in a limited number of processes, definite parameter-free physical predictions may be obtained for any other processes to the same order in perturbation theory (11). Hence the ``value of '' to be used in (or ) in annihilation at a given order could be deduced (in principle), for example, from the experimental total cross-section. If, say, both processes are calculated only to , then higher order terms in each perturbation series will lead to errors in the predictions. Such errors can be corrected for by modifying the to be used by a calculated correction, thereby absorbing the higher order terms into the numerical value of . Of course, the correction will, in general, be different for different processes. Note that all renormalization prescriptions introduce some renormalization mass : the logarithmic dependence of on was discussed in the preceding paragraph. To , all nonlogarithmic terms can be accounted for by using effective differing by numerical factors determined from explicit calculations in different processes: this procedure fails, however, beyond .

As a simple, but revealing, example of higher-order nonlogarithmic corrections to parton production, I consider the QED-like processes which modify . Diagrams with only this structure may be selected by taking the formal limit that the number of fermion flavors goes to infinity. Then a simple calculation [5] reveals multiplicative corrections to the lowest-order of the form, even), where is the number of loops. The basic origin of the embarrassing terms is with the Landau singularity. The fermion vacuum polarization corrections to a gluon propagator with invariant mass may be summarized (in the limit ) by the leading log effective coupling . Then the finite part of the diagrams (4) involves roughly , which introduces . The alternating sign in these corrections makes them formally amenable to Borel summation (with result . However, if the integral over had run over the Landau singularity (at , where the denominator in the leading vanishes), then all terms would have had the same sign, and no reordering would allow the divergence as to be removed. This behavior occurs in, for example, corrections to due to multiple electron loops in QED [6]. In QCD (forsaking the limit ), the Landau singularity is at small, rather than large, invariant masses: in a purely perturbative calculation (with no imposed), the integration in evaluating, for example, corrections to parton decay probabilities runs across the singularity, and terms are expected. As discussed in the preceding paragraph, any corrections would be irrelevant if they could be absorbed universally by a change of renormalization prescription. Unfortunately, diagrams requiring the same renormalization may or may not involve integration over the Landau singularity, so that the corrections cannot be absorbed universally. (Perhaps by defining separate ``Landau divergent'' and ``Landau convergent'' , this particular class of corrections could be avoided.)

At low orders, corrections, e.g., or may be important [7]. Such terms arise in comparing processes with incoming and outgoing partons or spacelike and timelike (some are visible in (4) if , and result from unitarization corrections: for some external kinematic configurations, intermediate lines may reach their mass shells, thereby sampling the second term in the propagator . Clearly such terms are usually associated with logarithms, and may then mathematically be obtained by changing signs in arguments of logarithms, (unitarity specifies the relevant Riemann sheet) according to the signs of external kinematic invariants. Hence they may often be summed in parallel with the logarithms, usually forming exponential or geometric series (12).

Despite these indications, one might hope that on summing all diagrams to a given order in , tolerable corrections would result. Even if each diagram gave (13) (as would be obtained if its internal loop integration was uniform up to kinematic and renormalization cutoffs) with random sign, the total would be since the number of diagrams is . (Indications from in QED that diagrams tend to cancel [8] are probably accidental, since individual gauge invariant diagram sets may grow (as in the example discussed above), and available QCD calculations (e.g., [9]) reveal mainly constructive, rather than destructive arrangements of signs, and large numerical coefficients.) The large observed value of means that higher order terms in the perturbation series will not become small for many orders before eventual divergence (as they presumably do in QED) and reliable truncation of the (perhaps asymptotic) perturbation series becomes impossible. In the face of these apparently insuperable difficulties, I shall use only the lowest order , with the hope that, as appears phenomenologically for the case of very high orders in QED perturbation theory, the correction terms will eventually conspire to be small. To make some allowance for higher order corrections, I allow an arbitrary normalization correction to , but assume the lowest-order kinematic structure (14). (For , this corresponds simply to treating as a free parameter, unconstrained by other determinations of .) Note that existing higher order calculations (e.g., for ) have found large corrections only in the lowest-order kinematic configuration: the next order calculations will, however, undoubtedly exhibit large corrections to any kinematic configuration accessible at the lower order.

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