Higher-order relativistic corrections to gluon fragmentation into spin-triplet S-wave quarkonium

We compute the relative-order-ν₄ contribution to gluon fragmentation into quarkonium in the 3S1 color-singlet channel, using the nonrelativistic QCD (NRQCD) factorization approach. The QCD fragmentation process contains infrared divergences that produce single and double poles in ε in 4-2ε dimensions. We devise subtractions that isolate the pole contributions, which ultimately are absorbed into long-distance NRQCD matrix elements in the NRQCD matching procedure. The matching procedure involves two-loop renormalizations of the NRQCD operators. The subtractions are integrated over the phase space analytically in 4-2ε dimensions, and the remainder is integrated over the phase-space numerically. We find that the order-v₄ contribution is enhanced relative to the order-ν₀ contribution. However, the order-ν₄ contribution is not important numerically at the current level of precision of quarkonium-hadroproduction phenomenology. We also estimate the contribution to hadroproduction from gluon fragmentation into quarkonium in the 3PJ color-octet channel and find that it is significant in comparison to the complete next-to-leading-order-in-αs contribution in that channel.