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

TL;DR

Problem: improve the precision of heavy-quark fragmentation into color-singlet P-wave quarkonia by including relativistic corrections up to $O(v^{2})$. Approach: employ the Collins-Soper gauge-invariant fragmentation function definition within NRQCD factorization, reproduce LO results and derive complete $O(v^{2})$ corrections for both equal-mass quarkonia and unequal-mass mesons via matching to perturbative QQbar states. Contributions: analytic $O(v^{2})$ short-distance coefficients for all $^{1}P_{1}$ and $^{3}P_{J}$ channels in equal- and unequal-mass cases, with numerical results showing negative and sizable corrections, especially for charmonium, and consistent high-energy behavior with full fixed-order calculations. Significance: enhances predictive power for high-$p_T$ production of P-wave quarkonia and $B_c$-type mesons, and lays groundwork for future higher-order relativistic and QCD corrections within a unified fragmentation framework.

Abstract

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.

Figures (5)

• Figure 1: Feynman diagrams for heavy-quark fragmentation into heavy quarkonium at leading-order in $\alpha_s$, where the shaded blob denotes the heavy quarkonium state and the double line represents the Wilson line.
• Figure 2: (Color online) The heavy-quark fragmentation functions $D(c \to h_c)$, $D(c \to \chi_{c0})$, $D(c \to \chi_{c1})$, and $D(c \to \chi_{c2})$ as functions of the momentum fraction $z$. The black solid curves denote the leading-order results, while the red dashed curves include the $\mathcal{O}(v^{2})$ relativistic corrections. The shaded bands correspond to the variation of $\langle v^{2} \rangle_{c\bar{c}} = 0.23 \pm 0.05$ in the charmonium system. The charm-quark mass is taken as $m_c = 1.5~\mathrm{GeV}$. The normalization factor is chosen as $C = 10^{-2}\,\alpha_s^2 \langle \mathcal{O} \rangle$.
• Figure 3: (Color online) The heavy-quark fragmentation functions $D(b \to h_b)$, $D(b \to \chi_{b0})$, $D(b \to \chi_{b1})$, and $D(b \to \chi_{b2})$ as functions of the momentum fraction $z$. The notation and color coding follow those in Figure \ref{['fig:frag_charmonium']}, but with the bottom-quark mass fixed to $m_b = 4.7~\mathrm{GeV}$. The shaded bands correspond to the variation of $\langle v^{2}\rangle_{b\bar{b}} = 0.10 \pm 0.05$ in the bottomonium system. The normalization factor is taken as $C = 10^{-4}\,\alpha_s^2 \langle \mathcal{O} \rangle$.
• Figure 4: Feynman diagrams for $e^+ + e^- \to \gamma^* \to H + X_{c\bar{c}}$ at quark level.
• Figure 5: (Color online) Ratios of cross sections $\sigma(e^+e^- \to H + X_{c\bar{c}})$ and $\sigma(e^+e^- \to c\bar{c})$ as a function of the center-of-mass energy $E_{\mathrm{cm}}$. Here, "Frag LO" and "Frag NLO" denote the leading-order and next-to-leading-order results in the fragmentation approximation, while "Full LO" and "Full NLO" denote the leading-order and next-to-leading-order results from the full fixed-order calculation, respectively.