Order-$v^2$ relativistic corrections to heavy-quark fragmentation into $P$-wave quarkonium states

Within the framework of nonrelativistic QCD (NRQCD) factorization,and based on the Collins--Soper operator definition of fragmentation functions, we present a systematic calculation of the fragmentation functions for a heavy quark fragmenting into color-singlet $P$-wave quarkonium states. After reproducing and confirming the known leading-order results, we further compute the relativistic corrections up to order $\mathcal{O}(v^{2})$. Our analysis applies both to quarkonium systems composed of heavy quarks with the same flavor and to $B_c$-type mesons formed by heavy quarks of different flavors. Numerical results show that, for all color-singlet $P$-wave channels, the $\mathcal{O}(v^{2})$ relativistic corrections give sizable negative contributions over most of the momentum-fraction $z$ region. We further compute inclusive cross sections for $P$-wave quarkonium plus charmed hadrons in $e^+e^-$ annihilation via the single photon process up to $\mathcal{O}(v^{2})$ by applying our obtained fragmentation functions, and the resulting predictions are consistent with the full fixed-order results in the high-energy region.