Resummation effects in the bottom-quark fragmentation function

Abstract

We compute the perturbative component of the fragmentation function of the b quark to the best of the present theoretical knowledge. The fixed-order calculation to order $$ {\alpha}_s^2 $$ α s 2 of the fragmentation function at the initial scale is matched with soft-emission logarithm resummation to next-to-next-to-leading logarithmic accuracy, so that order-$$ {\alpha}_s^2 $$ α s 2 corrections are accounted for exactly, and logarithmically enhanced contributions from loops of b quarks are included. This requires the calculation of the Mellin transform of the order-$$ {\alpha}_s^2 $$ α s 2 result in the whole complex plane for the Mellin variable, which we provide for the first time for all the fragmenting partons. Evolution is performed to next-to-next-to-leading log accuracy, and mixing with the gluon fragmentation function is taken into account. The perturbative fragmentation functions are made available via LHAPDF grids.

Heavy quark fragmentation in e+e− collisions to NNLO+NNLL accuracy in perturbative QCD

“…Both the coefficient functions and the initial condition D Q→Q exhibit large-logarithmic behaviour of soft origin in the x → 1 region (or, equivalently, large-N in moment space, α n s log k N with k ≤ 2n). Such logarithms can be resummed to all orders, and results are known up to NNLL accuracy for the initial conditions [9,19,29,30] and up to N3LL for the coefficient function [9,39,40]. Explicit expressions are given in appendix A.3 and A.…”

“…We start by considering the bottom quark case, for easier comparison with the existing recent literature [29,30]. In particular, we study in some detail the D b→b (N, µ 0F , m) component.…”

“…Despite fixed order calculations to next-to-next-to-leading-order accuracy having been available since quite some time [22][23][24][25][26][27], it is only quite recently that they were implemented into codes and tools useful for studying them numerically and for doing heavy quark fragmentation phenomenology. Notably, bottom quark fragmentation at next-to-next-to-leading order (NNLO) plus next-to-next-to-leading logarithmic (NNLL) collinear and soft accuracy has been considered in [21,[28][29][30], but the implications of the higher order perturbative accuracy have probably not been fully explored, and charm quark fragmentation has not been studied.…”

A consistent resummation of mass and soft logarithms in processes with heavy flavours

“…Nevertheless, it would be interesting to combine our formalism with recent high-precision studies of heavy-quark fragmentation, see e.g. [29,30,72], and investigate our approach beyond NLL, where because of soft-gluon interference effects, it is no longer possible to factorise the resummation formula into the product of two independent jet functions.…”

“…The initial condition of the evolution is, by definition, free of mass logarithms, but it is affected by soft logarithms, that can be resummed to all orders too. [27][28][29][30] Virtues of the massive and massless schemes can be combined together by matching the fixed-order calculation performed in the massive scheme, with the all-order resummation of mass logarithms achieved by the fragmentation function approach [31][32][33][34][35][36][37]. Since soft-gluon resummation is available for both the massive and the massless schemes, it is natural to JHEP11(2023)120 explore the possibility of matching the two calculations at the resummed level, obtaining a theoretical prediction that fully accounts for all mass and soft logarithms.…”