Order-v4 corrections to heavy quark fragmentation to S-wave heavy quarkonium

A bstract Within the framework of nonrelativistic quantum chromodynamics (NRQCD) factorization, we compute the O(v⁴) O v 4 relativistic corrections to the fragmentation of a heavy quark into the color-singlet ¹S₀[1] S 0 1 1 and ³S₁[1] S 1 1 3 quarkonium states. Using the Collins-Soper definition of the fragmentation function, we reproduce the known O(v²) O v 2 results. We find that the O(v⁴) 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 O(v²) 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 O(v⁴) O v 4 and obtain the corresponding analytical expressions.