Order-$v^4$ corrections to heavy quark fragmentation to S-wave heavy quarkonium

TL;DR

This work extends NRQCD factorization-based fragmentation theory by computing the complete $\ ext{O}(v^{4})$ relativistic corrections to heavy-quark fragmentation into equal-mass $S$-wave quarkonia ($^{1}S_{0}^{[1]}$ and $^{3}S_{1}^{[1]}$). Using the gauge-invariant Collins–Soper fragmentation function and axial-gauge calculations, the authors derive explicit short-distance coefficients for the $\,v^{4}$ terms and perform a perturbative NRQCD matching, also providing unequal-mass results in Appendix B. Numerically, the $\, ext{O}(v^{2})$ corrections are large and negative, while the $\, ext{O}(v^{4})$ corrections are positive but significantly smaller, demonstrating good convergence of the NRQCD relativistic expansion and enabling more precise predictions at high transverse momentum. The results fill a gap in the literature by delivering the full $ ext{O}(v^{4})$ SDCs for heavy-quark fragmentation to $S$-wave quarkonia and offer analytical expressions for the unequal-mass case, with direct implications for $$-quarkonium production phenomenology.

Abstract

Within the framework of nonrelativistic quantum chromodynamics (NRQCD) factorization, we compute the $\mathcal{O}(v^{4})$ relativistic corrections to the fragmentation of a heavy quark into the color-singlet $^{1}S_{0}^{[1]}$ and $^{3}S_{1}^{[1]}$ quarkonium states. Using the Collins--Soper definition of the fragmentation function, we reproduce the known $\mathcal{O}(v^{2})$ results. We find that the $\mathcal{O}(v^{4})$ correction gives a positive contribution relative to the leading order result over a wide range of the light-cone momentum fraction $z$, while its magnitude remains much smaller than that of the $\mathcal{O}(v^{2})$ correction. This behavior indicates a good convergence of the NRQCD relativistic expansion in this process. We further extend the calculation to the fragmentation functions in the unequal-mass case at $\mathcal{O}(v^{4})$ and obtain the corresponding analytical expressions.

Figures (1)

• Figure 1: (Color online) The heavy-quark fragmentation functions $D(c \to \eta_c)$, $D(b \to \eta_b)$, $D(c \to J/\psi)$, and $D(b \to \Upsilon(1S))$ as functions of the momentum fraction $z$ in different $v^2$ ranges. The black, blue, and red curves correspond to the theoretical predictions at leading order (LO), next-to-leading order (NLO) $\mathcal{O}(v^2)$, and next-to-next-to-leading order (NNLO) $\mathcal{O}(v^4)$, respectively. The shaded bands accompanying each curve represent the theoretical uncertainties due to variations of $v^2$ within its adopted ranges. The calculations employ $v_{\eta_c}^2$, $v_{\eta_b}^2$, $v_{J/\psi}^2$, and $v_{\Upsilon(1S)}^2$ with heavy-quark masses fixed to $m_c = 1.4 \pm 0.05\,\text{GeV}$ and $m_b = 4.6 \pm 0.05\,\text{GeV}$, respectively. The normalization factor is taken as $C = 10^{-2}\, \alpha_s^2 \langle \mathcal{O} \rangle$.