Differential heavy quark pair production at small $x$

TL;DR

This work extends the High-Energy Large Logarithms (HELL) framework to differential observables at LL accuracy in small-$x$ for proton-proton collisions, using $k_t$-factorization with unintegrated PDFs and running-coupling resummation. It derives differential off-shell coefficient functions for gluon-gluon initiated processes and implements them in a momentum-space formulation, including all partonic channels and a fixed-order matching scheme. The heavy-quark sector is used as a practical application, showing that differential small-$x$ resummation yields sizable effects, especially in forward kinematics, and improves perturbative stability, with results sensitive to the small-$x$ gluon PDF. The new HELL release provides a tool for phenomenology and PDF studies at small $x$, enabling better predictions for heavy-flavour production at the LHC and related forward experiments.

Abstract

We consider the production of a heavy quark pair in proton-proton collisions. For bottom and charm quarks, the final state invariant mass is typically much smaller than the collider energy (e.g. at the LHC), so that high-energy logarithms may spoil the perturbativity of the theoretical prediction at fixed order. The resummation of these logarithms to all orders is thus needed to obtain reliable predictions. In this work, we extend previous results on high-energy (or small-$x$) resummation to differential distributions in rapidity, transverse momentum and invariant mass, and implement them in the public code HELL.

Figures (8)

• Figure 1: The auxiliary Eq. \ref{['eq:resCaux']} and regular Eq. \ref{['eq:resCreg']} functions as a function of partonic rapidity $y$ for single quark production of mass $m=4.6$ GeV at $p_t=2$ GeV and $x=10^{-5}$ (left plot). The resummed coefficient functions at parton level for each partonic channel constructed according to Eq. \ref{['eq:resC_reg_aux']} for the same kinematics (right plot).
• Figure 2: The double differential distribution in rapidity and transverse momentum of the bottom quark, plotted as a function of the rapidity for $p_t=2$ GeV, for bottom pair production at LHC $13$ TeV. The left plots are obtained using NNPDF31sx at fixed order, while in the right plot the resummed result is computed with the resummed PDFs from the same family. The uncertainty band represents an estimate of NLL corrections.
• Figure 3: Breakdown of the individual contributions to the resummed result from the $gg$, $gq+qg$ and $qq$ channels separating the regular and auxiliary parts. The left plot focuses on the resummed contribution to be matched to the LO, while the right plot focuses on the resummed contribution to be matched to NLO. The results in these plots are obtained using NNPDF31sx with resummation.
• Figure 4: Scale uncertainty for the double differential distribution in rapidity and transverse momentum of the bottom quark, plotted as a function of the rapidity for $p_{\rm t}=2$ GeV, for bottom pair production at LHC $13$ TeV. The left plot shows factorization scale uncertainty only, while the right plot shows the standard 7-point uncertainty envelope.
• Figure 5: The double differential distribution in rapidity and transverse momentum of the bottom quark, plotted as a function of $p_t$ for central rapidity $Y=0$, for bottom pair production at LHC $13$ TeV.