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

TL;DR

This paper tackles the instability of fixed-order fragmentation-function evolution at small x caused by double logarithms, by deriving analytic NNLL resummations of timelike splitting functions and fragmentation-coefficient functions in Mellin-N space. The resummation yields finite first moments and an oscillatory small-x behavior describable by Bessel functions, enabling all-x evolution when combined with NNLO large-x results. It provides NNLL expressions for P_ji^T and for F_T, F_L, F_φ coefficient functions, and extends to F_L via analytic-continuation relations, with explicit N^3LL inputs in certain sectors. The work also connects timelike and spacelike evolution through the DMS relation, offers x-space representations, and outlines next steps toward N^4LL accuracy and full all-x matching.

Abstract

We study the splitting functions for the evolution of fragmentation distributions and the coefficient functions for single-hadron production in semi-inclusive electron-positron 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 alpha_s^n 1/x 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.

Figures (4)

• Figure 1: The timelike gluon-quark and gluon-gluon splitting functions (multiplied by $x\,$), shown for a very wide range of the momentum fraction $x$ at a typical value of the strong coupling constant $\alpha_{\rm s}$. The all-$x$ (minimal $N\!=\!1$ finite) 'LO$\,+\,$resummed' and 'NLO$\,+\,$resummed' approximations are compared to the corresponding LO and NLO results valid only at large $x$, e.g., $x \raisebox{-0.07cm}{$\:\:\stackrel{>}{{ \sim}}\:\:$} 10^{\,-2}$ for NLO.
• Figure 2: As Fig. \ref{['fig:Pgitres']}, but for the timelike quark-quark and quark-gluon splitting functions, for which the LO contributions does not include $1/x$ terms and the resummation starts at NLL level.
• Figure 3: The all-$x$ gluon coefficient functions for the fragmentation functions $F_T$ and $F_L$ (multiplied by $x\,$), down to extremely small values of the scaling variable $x$. As discussed above Eqs. (\ref{['cTg-cl']}) and (\ref{['AC-KTL']}), the respective '(N)LO$\,+\,$resummed' approximations are defined by resumming as many logarithms as required to remove all $1/x$ terms due to the (N)LO contributions.
• Figure 4: As Fig. \ref{['fig:cagtres']}, but for the quark coefficient functions which are suppressed by one power of $\ln x\,$ w.r.t. the gluon quantities. As the diagonal splitting functions, $C_{T,\rm q}$ includes a $\delta(1-x)$ term.