Reciprocity in Heavy Quark Fragmentation Function

TL;DR

The paper investigates whether a reciprocity relation between the heavy-quark distribution inside a heavy hadron and the soft fragmentation function of the heavy quark into that hadron holds in the fast-velocity frame, defining $f_Q(x)$ and $f_H(x)$ with $f_H(x)=f_Q(x)$ in the soft regime. It presents a two-stage fragmentation framework: a soft stage yielding $f_H(x)\approx f_Q(x)$ via the light-lump reciprocity and a hard, perturbative evolution of the fragmentation function $D_Q^H(x,\mu)$ governed by a DGLAP-like equation with splitting kernels $\mathscr P_{p\to Q}$. The work discusses heavy-quarkonia as a testing ground, invoking NRQCD/pNRQCD to describe vector vs pseudoscalar fragmentation and highlighting that vector states are harder to fragment into, consistent with the reciprocity idea, while real data require disentangling competing mechanisms. Overall, the reciprocity offers a qualitative bridge between intrinsic heavy-quark distributions and observable fragmentation patterns, but quantitative verification demands careful separation of hard production, soft fragmentation, and competing production channels.

Abstract

At high energies a form of fragmentation function for a heavy quark into a hadron is substantiated to agree with a reciprocity relation to the distribution of heavy quark as the virtual parton in the hadron. The relevance of the relation is analysed in its application to empirical data.

Figures (4)

• Figure 1: Specular on mass-shell distributions of light and heavy partons $f_l(x)$ and $f_Q(x)$, respectively, in the naive model of heavy meson mark by superscript $M$ and the heavy baryon marked by superscript $B$.
• Figure 2: Composing the heavy hadron by virtual light lump and virtual heavy quark.
• Figure 3: Fragmenting the virtual heavy quark into the on mass-shell heavy hadron and virtual light lump.
• Figure 4: The reciprocity as the specular relation for the light and heavy virualiries and momentum fractions.