Analytic continuation of the Sudakov form factor in QCD

Abstract

We exhibit a solution to the evolution equation for the Sudakov form factor in QCD with massless quarks, which exponentiates infrared poles in dimensional continuation as well as logarithms of momentum transfer. We use this solution to construct an expression for the absolute value of the ratio of timelike to spacelike form factors, in which the infrared finiteness of the ratio is manifest. Finally, we compare this result to explicit calculations of the form factor available in the literature. Most of the large two-loop corrections to the absolute value of the ratio come from the exponentiation of one-loop corrections, including the effect of the running coupling.

Exponentiation of the Drell-Yan cross section near partonic threshold in the DIS and MS-bar schemes

“…At this point one can already observe that ω DY (N, ǫ) exponentiates up to corrections suppressed by powers of N : the exponentiation of the form factor in dimensional regularization was proven in Ref. [7], while the exponentiation of ψ R and U R to this accuracy was proven in Ref. [3].…”

“…[3,7] this form of the refactorization is sufficient to prove the exponentiation of the full cross section up to corrections suppressed by powers of N . In fact F 2 now involves, to this accuracy, only the form factor, and a product of real functions which have been shown to exponentiate by using their respective evolution equations.…”

“…[18,19,20], it was shown in Ref. [7] that the dimensionally regularized Sudakov form factor may be written as an exponential of integrals over functions only of α s , Γ(Q 2 , ǫ) = exp 1 2…”

“…[7] showing that the absolute value of the ratio is finite to all orders and exponentiates. To illustrate these results, note that the (timelike) Sudakov form factor Γ(Q 2 , ǫ) for the electromagnetic coupling of a massless quark of charge e q is defined via…”

“…The resulting resummation of constant terms established to all orders the earlier observation [5] that, in the DIS-scheme Drell-Yan cross section, the largest constants are related to the ratio of the timelike Sudakov form factor to its spacelike continuation. By solving an appropriate evolution equation [6], an exponential representation for this form factor, and thus also for the ratio, was derived [7], generalizing the observation of [5] to a full nonabelian exponentiation. Comparison with exact two-loop results [8] showed in that case that N -independent contributions at two loops are indeed dominated by the exponentiation of the one-loop result, combined with running coupling effects [9,10].…”