Light-cone gauge and the calculation of the two-loop splitting functions

Abstract

We present calculations of next-to-leading order QCD splitting functions, employing the light-cone gauge method of Curci, Furmanski, and Petronzio ͑CFP͒. In contrast with the ''principal-value'' prescription used in the original CFP paper for dealing with the poles of the light-cone gauge gluon propagator, we adopt the Mandelstam-Leibbrandt prescription which is known to have a solid field-theoretical foundation. We find that indeed the calculation using this prescription is conceptually clear and avoids the somewhat dubious manipulations of the spurious poles required when the principal-value method is applied. We reproduce the wellknown results for the flavor nonsinglet splitting function and the N C 2 part of the gluon-to-gluon singlet splitting function, which are the most complicated ones, and which provide an exhaustive test of the ML prescription. We also discuss in some detail the xϭ1 end point contributions to the splitting functions.

On regularizing the infrared singularities in QCD NLO splitting functions with the new Principal Value prescription

“…Namely, the 1/ 3 and 1/ 2 (d V ) and (d R ) terms are given in Table 3.10 of [9] and the 1/ terms are given in Table 1 of [5] as a sum of real and virtual graphs. We observe, that: • The 1/ virtual plus real terms given in NPV Eq.…”

“…More detailed results in the standard PV prescription can be found in [9,6]. One can see there that in the standard PV prescription the triple poles, 1/ 3 , appear only in the four real and virtual interference graphs of the type "(d)", shown in Fig.…”

“…Instead, one has to isolate the integral over the plus component of m-momentum and leave it for the very end. The appropriate formulae are available in the literature [6] (see [9] for details of derivation). Let us quote here Eq.…”

“…The simplest way is to use the Principal Value (PV) prescription [5,6,9,10]. The other option is the Mandelstam-Leibbrandt (ML) prescription [11,12], which is better justified from the field-theoretical point of view, but leads to more complicated calculations, especially for the real-emission-graphs [9]. Other methods of avoiding the problem of spurious singularities can be found in [13,14].…”

“…These unphysical singularities cancel at the end, but in the intermediate stages of the calculations one has to somehow regularize them. The simplest way is to use the Principal Value (PV) prescription [5,6,9,10]. The other option is the Mandelstam-Leibbrandt (ML) prescription [11,12], which is better justified from the field-theoretical point of view, but leads to more complicated calculations, especially for the real-emission-graphs [9].…”