Soft-Gluon Resummation for the Fragmentation of Light and Heavy Quarks at Large x

TL;DR

The paper develops next-to-leading logarithmic (NLL) soft-gluon resummation for fragmentation processes of both light and heavy quarks in the large-x region, employing a factorization framework that isolates universal perturbative fragmentation functions. For light quarks, it derives all-order LL and NLL resummed expressions for one- and two-particle inclusive distributions in $e^+e^-$ and DIS, with explicit matching to fixed-order results. For heavy quarks, it extends the perturbative fragmentation-function formalism to include soft-gluon effects in the initial condition and demonstrates process-independence in the $m/Q\to 0$ limit, yielding softened high-x distributions and reduced scale dependence. Numerically, the resummation improves theoretical stability and predictive power, providing a clearer separation between perturbative and non-perturbative fragmentation effects and guiding phenomenological fits to experimental data.

Abstract

We present a QCD study of fragmentation processes for light and heavy quarks in the semi-inclusive region of large x. Large logarithmic terms, due to soft-gluon radiation, are evaluated and resummed to all perturbative orders in the QCD coupling alpha_s. Complete analytical results to next-to-leading logarithmic accuracy are given for one-particle and two-particle inclusive distributions in e+e- annihilation and DIS. Factorization of parton radiation at low transverse momenta is exploited to identify the universal (process-independent) perturbative fragmentation function that controls heavy-quark processes, and to perform next-to-leading logarithmic resummation of its soft-gluon contributions. To gauge the quantitative impact of resummation, we perform numerical studies of light- and heavy-quark fragmentation in e+e- collisions.

Figures (11)

• Figure 1: Inclusive production of a hadron $h$ with momentum $p$ in $e^+e^-$ annihilation.
• Figure 2: Effect of Sudakov resummation on a typical light-hadron fragmentation spectrum in $e^+e^-$ collisions. The single-particle distribution is computed at NLO (dashed lines) and at NLO+NLL accuracy (solid lines). The factorization and renormalization scales are varied in the range $Q/4 \leq \mu_F=\mu \leq 4Q$.
• Figure 3: Dependence of the light-hadron fragmentation spectrum shown in Fig. \ref{['fig:light']} on the factorization/renormalization scales $\mu=\mu_F$, at three different valules of $x$.
• Figure 4: Scaling violations at $x=0.5$ and $x=0.9$, and uncertainty bands due to variations of the renormalization/factorization scales ($\mu = \mu_F$). The fragmentation function at the input scale $\mu_0$ is the same as in Fig. \ref{['fig:light']}.
• Figure 5: Schematic picture of matrix element factorization in the collinear and quasi-collinear limits. The thicker line denotes the particle (massless parton or heavy quark) that undergoes the collinear or quasi-collinear splitting process.