Resummation of threshold double logarithms in inclusive production of heavy quarkonium

TL;DR

The paper tackles the problem of unphysical negative cross sections in fixed-order NRQCD predictions for heavy quarkonium production at large $p_T$ caused by threshold logarithms near the phase-space boundary. It develops a soft-factorization framework that resums threshold double logarithms to all orders at leading double logarithmic level, yielding a resummed fragmentation function that exponentiates and remains positive. Numerical results for $J/\psi$ and $\chi_{cJ}$ show that the resummed predictions agree with ATLAS data at $\sqrt{s}=13$ TeV, in contrast to fixed-order NLO predictions which can be negative; this demonstrates that threshold resummation is essential for reliable NRQCD predictions. The work sets the stage for higher-order soft-function improvements and broader applications to differential observables and NLP contributions, strengthening the use of heavy quarkonia as QCD probes.

Abstract

We resum threshold double logarithms in inclusive production of heavy quarkonium that arise from singularities near the boundary of phase space. This resolves the catastrophic failure in the conventional approach based on fixed-order perturbation theory calculations in nonrelativistic QCD, where quarkonium cross sections at large transverse momentum can turn negative. We identify the root cause of this negative cross section problem as the appearance of threshold logarithms in radiative corrections, and resum them to all orders in perturbation theory at the leading double logarithmic level. We find that resummation of threshold logarithms is imperative for describing measured $J/ψ$ production rates at large transverse momentum.

Figures (3)

• Figure 1: Comparison of $\sigma_{Q \bar{Q} ({\cal N})}^{\rm LP}$ (lines) and $\sigma_{Q \bar{Q} ({\cal N})}$ (points) computed at $\alpha_s^4$ accuracy for ${\cal N} = {}^3S_1^{[8]}$, $^3P_J^{[8]}$, $^3P_1^{[1]}$, and $^3P_2^{[1]}$ in $pp$ collisions at $\sqrt{s}=7$ TeV, $|y|<1.2$. Error bars correspond to numerical uncertainties in $\sigma_{Q \bar{Q} ({\cal N})}$. For $P$-wave channels, negative values are shown because the cross sections turn negative at large $p_T$.
• Figure 2: Gluon FFs with resummed threshold double logarithms times $z^3$ for production of $J/\psi$ (top) and $\chi_{cJ}$ (bottom) for $J=1$ and 2. Central values of FO results are also shown for comparison. ${\rm Br}_J \equiv {\rm Br}_{\chi_{cJ} \to J/\psi + \gamma}$ is the branching fraction for decays of $\chi_{cJ}$ into $J/\psi+\gamma$. Taken from ref. Chung:2024jfk.
• Figure 3: Prompt $J/\psi$ production rates from $pp$ collisions at $\sqrt{s}=13$ TeV computed from resummed SDCs compared to ATLAS data. Central values of FO NLO results are shown for comparison. ${\rm B} \equiv {\rm Br}_{J/\psi \to \mu^+ \mu^-}$ is the $J/\psi$ dimuon branching fraction. Taken from ref. Chung:2024jfk.