Next-to-next-to-leading-order QCD corrections to ${}^3S_1^{(8)}$ gluon fragmentation function for quarkonium

Abstract

We present the first computation of the next-to-next-to-leading-order (NNLO) QCD corrections to the ${}^3S_1^{(8)}$ gluon fragmentation function for quarkonium within the nonrelativistic QCD (NRQCD) factorization framework, accurate to the lowest order in the velocity expansion. The calculation is performed with high numerical precision and encompasses both polarized and unpolarized cases. We find that the NNLO corrections are positive and substantial across most of the $z$ region. Furthermore, the logarithmic singularities near the endpoint $z\to 1$ are fully reconstructed, providing essential inputs for future threshold resummation beyond leading-logarithmic accuracy. Combined with threshold-resummed formulas in the large-$z$ region, our results yield phenomenologically viable inputs for the $^3S_1^{(8)}$ gluon fragmentation function. This enables a more reliable description of large-$p_T$ $J/ψ$ ($ψ'$) and $χ_{cJ}$ production and polarization at hadron colliders, representing a crucial step toward a definitive test of the color-octet mechanism.