Order-ν⁴ relativistic corrections to gluon fragmentation into ³S₁ quarkonium

We compute the relativistic corrections to the color-singlet contribution to gluon fragmentation into a J/ψ at relative order ν⁴, making use of the nonrelativistic QCD (NRQCD) factorization approach. The corresponding full-QCD process exhibits infrared divergences that manifest themselves as single and double poles in ε in 4-2ε dimensions. We isolate the infrared-divergent contributions and treat them analytically. In the matching of full QCD to NRQCD, the pole contributions are absorbed into long-distance NRQCD matrix elements. The renormalizations of the ultraviolet divergences of the long-distance NRQCD matrix elements involve Born and one-loop single-pole counterterm contributions and Born double-pole counterterm contributions. While the order-ν⁴ contribution enhances the J/ψ hadroproduction rate for the color-singlet channel substantially, this contribution is not important numerically in comparison with the color-octet contributions. We also find that the ³P_J color-octet channel in the gluon fragmentation function contributes to J/ψ hadroproduction significantly in comparison with the complete contribution of next-to-leading order in α_s in that channel.