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

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.