Resummation of threshold double logarithms in quarkonium fragmentation functions

TL;DR

This work develops an all-orders resummation framework for threshold double logarithms in quarkonium fragmentation functions within NRQCD. By applying soft-factorization via the Grammer-Yennie approach, it isolates endpoint singularities into universal soft functions defined with adjoint Wilson lines and field-strength insertions, enabling exponentiation in Mellin space to sum leading double logarithms. Leading and next-to-leading order soft functions for key channels ($^3S_1^{[8]}$, $^3P^{[8]}$, and $^3P_J^{[1]}$) are computed, demonstrating IR cancellation and identifying the cusp anomalous-dimension origin of the threshold logarithms; this yields a resummed, well-behaved, positive-definite fragmentation function. The resummation improves theoretical control over large-$p_T$ quarkonium production and informs NRQCD LDME phenomenology, with potential extensions to single-pole accuracy and next-to-leading power effects.

Abstract

We develop a formalism for resumming threshold double logarithms that appear in fragmentation functions for production of heavy quarkonia. Threshold singularities appear in fixed-order calculations of quarkonium fragmentation functions in the nonrelativistic QCD factorization formalism due to radiation of soft gluons. Because of this, at fixed order quarkonium fragmentation functions are not positive definite, and can lead to unphysically negative cross sections. This problem can be resolved by resumming threshold logarithms to all orders in perturbation theory, which renders the fragmentation functions finite and ensures the positivity of cross sections. We present a detailed derivation of the resummation formalism and derive the formula for resummed quarkonium fragmentation functions, which can be computed entirely within perturbation theory without the need for nonperturbative model functions.

Figures (4)

• Figure 1: Feynman diagrams for soft functions $S_{^3S_1^{[8]}}(z)$ (left), $S_{^3P^{[8]}}(z)$ (middle), and $S_{^3P^{[1]}}(z)$ (right) at leading nonvanishing order in the strong coupling. Black solid lines are timelike Wilson lines in the $p$ direction, dotted lines are lightlike Wilson lines in the $n$ direction, vertical dashed lines are final-state cuts, filled circles are the spacetime origins, curly lines are gluons, and the symbol $\otimes$ stands for insertion of the field-strength tensor.
• Figure 2: Feynman diagrams that appear in NLO calculation of the soft function $S_{^3S_1^{[8]}}(z)$. There are additional diagrams that can be obtained by complex conjugation.
• Figure 3: Planar Feynman diagrams that appear in NLO calculation of the soft function $S_{^3P^{[8]}}(z)$. There are additional diagrams that can be obtained by complex conjugation.
• Figure 4: Feynman diagrams for amplitudes that give rise to the doubly real contributions in the soft function $S_{^3P^{[8]}}(z)$ at NLO. See Eq. (\ref{['eq:Amps']}) and text for definitions of ${\cal A}_1$--${\cal A}_7$.