Resummation of transverse momentum and mass logarithms in DIS heavy-quark production

TL;DR

The paper develops a unified framework to resum both heavy-quark mass logarithms and transverse-momentum logarithms in deep inelastic scattering with heavy-quark production. Building on the ACOT (and S-ACOT) factorization schemes and the Collins-Soper-Sterman (CSS) formalism, it extends the $b$-space resummation to include mass effects, introducing mass-dependent scaling variables and kinematics to ensure correct threshold behavior and smooth interpolation between regimes. The authors apply the method to bottom-quark production at HERA, demonstrating consistent predictions and highlighting the reduced need for nonperturbative large-$b$ inputs due to the heavy mass, while outlining future work for charm and higher-order improvements. This approach provides a more accurate, differential description of heavy-flavor DIS observables across a wide range of $Q^2$, with implications for precise structure-function determinations and global PDF analyses.

Abstract

Differential distributions for heavy quark production depend on the heavy quark mass and other momentum scales, which can yield additional large logarithms and inhibit accurate predictions. Logarithms involving the heavy quark mass can be summed in heavy quark parton distribution functions in the ACOT factorization scheme. A second class of logarithms involving the heavy-quark transverse momentum can be summed using an extension of Collins-Soper-Sterman (CSS) formalism. We perform a systematic summation of logarithms of both types, thereby obtaining an accurate description of heavy-quark differential distributions at all energies. Our method essentially combines the ACOT and CSS approaches. As an example, we present angular distributions for bottom quarks produced in parity-conserving events at large momentum transfers Q at the ep collider HERA.

Figures (7)

• Figure 1: The parity-conserving semi-inclusive production $e+p\rightarrow e+H+X$ of heavy hadrons in the $\gamma ^{*}p$ c.m. reference frame. The resummation effects considered here are important in the current fragmentation region $\theta _{H}\rightarrow 0$, i.e., when the final-state heavy quark $h$ closely follows the direction of its escape in the ${\cal O}(\alpha _{S}^{0})$ flavor-excitation process $\gamma ^{*}+h\rightarrow h$.
• Figure 2: Basic subprocesses in the ACOT scheme: (a) flavor excitation $\gamma ^{*}+h\rightarrow h$ at ${\mathcal{O}}(\alpha _{S}^{0})$; (b) gluon flavor creation (photon-gluon fusion) $\gamma ^{*}+G\rightarrow h+\bar{h}$ at ${\mathcal{O}}(\alpha _{S}^{1})$; (c) light-quark flavor creation $\gamma ^{*}+q\rightarrow (\gamma ^{*}+G)+q\rightarrow (h+\bar{h})+q$ at ${\cal O}(\alpha _{S}^{2})$. The thick and thin solid lines correspond to the heavy quark $h$ and light quarks $q=u,d,s$, respectively.
• Figure 3: Balance of various terms in the total resummed cross section $d\sigma _{TOT}/dq_{T}$: (a) away from the threshold ($Q\gg M$); (b) near the threshold ($Q\approx M$). In each plot the thick curves correspond to the 'active' cross section (TOT, FO, W or ASY), and the thin curves correspond to the other three cross sections.
• Figure 4: Plots of $q_{T}/W$ vs. $\theta _{H}$ at various values of $\lambda \equiv M_{H}/E_{H}=0.999$ (lower curve), $\lambda =0.5$ (middle curve), and $\lambda =0.001$ (upper curve).
• Figure 5: The angular distributions of the bottom hadrons in the $\gamma ^{*}p$ c.m. frame at (a) $Q=5$ GeV, (b) $Q=15$ GeV, (c) $Q=50$ GeV without the Sudakov factor, and (d) $Q=50$ GeV with the Sudakov factor. At $Q=50$ GeV, an additional cut $E_{H}>0.1(W/2)$ is made to suppress contributions at $z<0.1$, i.e., from the region where the conventional factorization may be inapplicable. The plots show the finite-order cross section $\sigma _{FO}$ (long-dashed line), the $b$-space integral $\sigma _{\widetilde{W }}$ (dot-dashed line), the asymptotic piece $\sigma _{ASY}$ (dotted line), and the full resummed cross section $\sigma _{TOT}$ (solid line).