Order-v2 relativistic corrections to heavy-quark fragmentation into P-wave quarkonium states

A bstract 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 $$ \mathcal{O}\left({v}^2\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>O</mml:mi> <mml:mfenced> <mml:msup> <mml:mi>v</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mfenced> </mml:math> . 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 $$ \mathcal{O}\left({v}^2\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>O</mml:mi> <mml:mfenced> <mml:msup> <mml:mi>v</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mfenced> </mml:math> 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 $$ \mathcal{O}\left({v}^2\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>O</mml:mi> <mml:mfenced> <mml:msup> <mml:mi>v</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mfenced> </mml:math> by applying our obtained fragmentation functions, and the resulting predictions are consistent with the full fixed-order results in the high-energy region.