Soft Gluon Resummation for Heavy Quark Electroproduction

TL;DR

This paper develops threshold resummation for heavy-quark electroproduction in deep-inelastic scattering at next-to-leading logarithmic accuracy within single-particle inclusive kinematics, focusing on the gluon-initiated channel. It provides the resummed differential cross section, its finite-order expansions to NLO and NNLO, and analyzes the resulting partonic coefficient functions and hadronic observables, notably the charm structure function F2^charm and its pT distribution. The study demonstrates that NLL resummation significantly improves perturbative predictions and reduces scale dependence, with NNLO approximations showing sizable but stabilizing corrections, especially at small x. Appendices extend the framework to pair-invariant-mass kinematics and summarize essential Laplace transforms, enabling broader application to heavy-quark production across kinematics.

Abstract

We present the threshold resummation for the cross section for electroproduction of heavy quarks. We work to next-to-leading logarithmic accuracy, and in single-particle inclusive kinematics. We provide next-to-leading and next-to-next-to-leading order expansions of our resummed formula, and examine numerically the quality of these finite order approximations. For the case of charm we study their impact on the structure function $F_2$ and its differential distribution with respect to the charm transverse momentum.

Figures (17)

• Figure 1: Factorization of heavy quark electroproduction near threshold. The double lines denote eikonal propagators.
• Figure 2: One-loop corrections to the soft function $S$ for heavy-quark production in photon-gluon fusion. The double lines denote eikonal propagators. The lines labelled $k$ corresponds to a gluonic eikonal line, and those labelled $p_1$ and $p_2$ denote quark and antiquark eikonal lines, respectively.
• Figure 3: (a): The $\eta$-dependence of the coefficient functions $c^{(k,0)}_{2,g}(\eta,\xi),\;k=0,1$ for $Q^2=0.1\,{\rm GeV}^2$ with $m=1.5\,{\rm GeV}$. Plotted are the exact results for $c^{(k,0)}_{2,g},\;k=0,1$ (solid lines), the LL approximation to $c^{(1,0)}_{2,g}$ (dotted line) and the NLL approximation to $c^{(1,0)}_{2,g}$ (dashed line). (b): The $\eta$-dependence of the coefficient function $c^{(2,0)}_{2,g}(\eta,\xi)$ for $Q^2=0.1\,{\rm GeV}^2$ with $m=1.5\,{\rm GeV}$. Plotted are the LL approximation (dotted line) and the NLL approximation (dashed line).
• Figure 4: (a): The $\eta$-dependence of the coefficient functions $c^{(k,0)}_{2,g}(\eta,\xi),\;k=0,1$ for $Q^2=10\,{\rm GeV}^2$ with $m=1.5\,{\rm GeV}$. The notation is the same as in Fig. \ref{['plot-onem1']}a. (b): The $\eta$-dependence of the coefficient function $c^{(2,0)}_{2,g}(\eta,\xi)$ for $Q^2=10\,{\rm GeV}^2$ with $m=1.5\,{\rm GeV}$. The notation is the same as in Fig. \ref{['plot-onem1']}b.
• Figure 5: (a): The NLL approximation to the coefficient function $c^{(1,0)}_{2,g}(\eta,\xi)$ for $Q^2=10\,{\rm GeV}^2$ and $m=1.5\,{\rm GeV}$ with restrictions to the small $\eta$-region. Plotted are the unmodified NLL result (solid line), the NLL result with a factor $\theta(m^2-s_4)$ included (dotted line) and the NLL result multiplied with a damping factor $1/\sqrt{1+\eta}$ (dashed line). (b): The $\eta$-dependence of the NLL approximation to the coefficient function $c^{(2,0)}_{2,g}(\eta,\xi)$ for $Q^2=10\,{\rm GeV}^2$ and $m=1.5\,{\rm GeV}$. The notation is the same as in Fig. \ref{['plot-onep1veto']}a.