Resummed small-x and first-moment evolution of fragmentation functions in perturbative QCD

Abstract

We study the splitting functions for the evolution of fragmentation distributions and the coefficient functions for single-hadron production in semi-inclusive e + e − annihilation in massless perturbative QCD for small values of the momentum fraction and scaling variable x, where their fixed-order approximations are completely destabilized by huge double logarithms of the form α n s x −1 ln 2n−a x. Complete analytic all-order expressions in Mellin-N space are presented for the resummation of these terms at the next-to-next-to-leading logarithmic accuracy. The poles for the first moments, related to the evolution of hadron multiplicities, and the small-x instabilities of the next-to-leading order splitting and coefficient functions are removed by this resummation, which leads to an oscillatory small-x behaviour and functions that can be used at N = 1 and down to extremely small values of x. First steps are presented towards extending these results to the higher accuracy required for an all-x combination with the state-of-the-art next-to-next-to-leading order large-x results.If the constants up to F (m) n,ℓ are known for all n and ℓ, then the splitting functions and coefficient functions can be determined at N m LL accuracy at all orders of the strong coupling. As observed in Ref. [19], the n th order small-x contributions to F T, φ are built up from n terms of the form(2.4)Since the terms with ε −2n+1 , . . . , ε −n−1 have to cancel in sum (2.1), there are n−1 relations between the LL coefficients A n,k which lead to the constants F (0) n,ℓ in Eq. (2.3), n−2 relations between the NLL coefficients B n,k etc. As discussed above, a N m LO calculation fixes the (nonvanishing) coefficients of ε −n , . . . , ε −n+m at all orders n, adding m + 1 more relations between the coefficients in Eq. (2.4). Consequently the highest m+1 double logarithms, i.e., the N m LL approximation, can be determined order by order from the N m LO results. Finally the resulting series, here calculated to order α 18 s using FORM and TFORM [24], can be employed to find their generating functions via over-constrained systems of linear equations. The whole procedure is analogous to, if computationally more involved than, the large-x resummation in Ref. [25].

Timelike structure functions and hadron multiplicities

“…In the following after reviewing the history which lead to the latest important improvements in "timelike" QCD of Refs. [12,9] and sketching our formalism, we present, as anticipated, the result of a slightly different global fit than in [8] where also the strong coupling contstant has been added among the free parameters confirming the good quality of the fit.…”

“…[38,39] and finally the NNLL accuracy has been reached in Ref. [9]. Here thanks to the approach used by A. Vogt in Ref.…”

“…[8], at the time of the contribution to this conference we have done a study where we have extended our analysis adding as a fit parameter also the strong coupling constant. Thanks to the new outstanding results in small-x timelike resummation obtained in [9] in the MS scheme, we were able to compute exact next-to-next-to-leading-logarithm (NNLL) contributions to the evolution of the multiplicity with with approximated NNNLO normalization factors in the √ α s expansion. Previously published results where available up to the NLL accuracy but in a massive gluon scheme which in general is unfortunately not suitable to combine resummation with fixed higher order corrections which are naturally given in the MS scheme.…”

“…[12,9] represent the beginning of the happy end of a quite long story: In the 1972 it was realized in Ref. [13] that at the lowest order the "timelike" splitting functions (occuring in Semi-Inclusive electron positron Annihilation (SIA)) are the same as their "spacelike" counterparts (occuring in Deep Inelastic Scattering (DIS)).…”

“…Then the highly non trivial part of this approach is the solution of these sequences that are obtained for the large logarithms and this has been successfully obtained in Refs. [38,9].…”