Fragmentation functions for gluon into $P$-wave $B_c$ mesons

TL;DR

This paper computes the fragmentation functions of a gluon into $P$-wave $B_c$ mesons within NRQCD factorization, including both color-singlet and color-octet channels and addressing ultraviolet divergences via dimensional regularization and $\overline{\rm MS}$ renormalization. Short-distance coefficients are extracted at LO in $\alpha_s$ by matching perturbative $g\to (c\bar{b})$ states using projection operators, with HQSS linking the nonperturbative LDMEs to two independent quantities and one LDME related to the derivative of the $P$-wave radial wavefunction. The final fragmentation functions are presented both graphically and as fitted analytic expressions, and exhibit strong sensitivity to the factorization scale while the mean momentum fraction $\langle z \rangle$ remains comparatively scale-stable; color-singlet contributions dominate across most $z$ except at small $z$ for certain $2P$ states. The results provide essential inputs for precision predictions of $P$-wave $B_c$ production and its feed-down at high-energy colliders, enabling improved modeling of $B_c$ spectra at large $p_T$.

Abstract

We calculate the fragmentation functions for a gluon into $P$-wave $B_c$ mesons within the nonrelativistic QCD factorization framework, incorporating color-singlet and color-octet contributions. Ultraviolet divergences arising from phase-space integrals are removed via operator renormalization in the modified minimal subtraction scheme. The resulting fragmentation functions are presented in both graphical form and as fitted analytic expressions.

Figures (3)

• Figure 1: Four sample cut diagrams for the fragmentation functions $D_{g \to (c\bar{b})[n]}(z,\mu_F)$.
• Figure 2: Fragmentation functions for a gluon into the $2P$$B_c$ states as a function of $z$, with $\mu_F=2M$ and $\mu_R=2m_c$.
• Figure 3: Fragmentation functions for $\mu_F=M$, $\mu_F=2M$ and $\mu_F=4M$, where the strong coupling constant is fixed as $\alpha_s(2m_c)=0.259$.