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Color-singlet relativistic correction to inclusive $J/ψ$ production associated with light hadrons at $B$ factories

Yu Jia

TL;DR

The paper develops a novel NRQCD matching framework that uses physical hadronic kinematics and the Gremm-Kapustin relation to compute the first-order relativistic ($v^2$) correction to inclusive $J/\u001bpsi$ production in $e^+e^-$ annihilation at $B$ factories, focusing on $e^+e^-\to J/\u001bpsi+gg$ in the color-singlet channel. It provides analytic and numerical results for the integrated cross section and differential distributions (energy, polarization, and angular) at $\sqrt{s}=10.58$ GeV, finding a modest ~30% enhancement from the $O(v^2)$ term and only mild changes to distributions, with endpoint logarithms signaling the need for resummation or higher-order effects. The work also clarifies the relative merits of the proposed matching strategy, demonstrates consistency with orthodox approaches at the integrated level, and discusses implications for color-octet contributions and future refinements. Overall, it advances a practical method to incorporate relativistic corrections in complex quarkonium production kinematics and highlights theoretical uncertainties and endpoint issues that guide future improvements.

Abstract

We study the first-order relativistic correction to the associated production of $J/ψ$ with light hadrons at $B$ factory experiments at $\sqrt{s}=10.58$ GeV, in the context of NRQCD factorization. We employ a strategy for NRQCD expansion that slightly deviates from the orthodox doctrine, in that the matching coefficients are not truly of ``short-distance" nature, but explicitly depend upon physical kinematic variables rather than partonic ones. Our matching method, with validity guaranteed by the Gremm-Kapustin relation, is particularly suited for the inclusive quarkonium production and decay processes with involved kinematics, exemplified by the process $e^+e^-\to J/ψ+gg$ considered in this work. Despite some intrinsic ambiguity affiliated with the order-$v^2$ NRQCD matrix element, if we choose its value as what has been extracted from a recent Cornell-potential-model-based analysis, including the relative order-$v^2$ effect is found to increase the lowest-order prediction for the integrated $J/ψ$ cross section by about 30\%, and exert a modest impact on $J/ψ$ energy, angular and polarization distributions except near the very upper end of the $J/ψ$ energy. The order-$v^2$ contribution to the energy spectrum becomes logarithmically divergent at the maximum of $J/ψ$ energy. A consistent analysis may require that these large end-point logarithms be resummed to all orders in $α_s$.

Color-singlet relativistic correction to inclusive $J/ψ$ production associated with light hadrons at $B$ factories

TL;DR

The paper develops a novel NRQCD matching framework that uses physical hadronic kinematics and the Gremm-Kapustin relation to compute the first-order relativistic () correction to inclusive production in annihilation at factories, focusing on in the color-singlet channel. It provides analytic and numerical results for the integrated cross section and differential distributions (energy, polarization, and angular) at GeV, finding a modest ~30% enhancement from the term and only mild changes to distributions, with endpoint logarithms signaling the need for resummation or higher-order effects. The work also clarifies the relative merits of the proposed matching strategy, demonstrates consistency with orthodox approaches at the integrated level, and discusses implications for color-octet contributions and future refinements. Overall, it advances a practical method to incorporate relativistic corrections in complex quarkonium production kinematics and highlights theoretical uncertainties and endpoint issues that guide future improvements.

Abstract

We study the first-order relativistic correction to the associated production of with light hadrons at factory experiments at GeV, in the context of NRQCD factorization. We employ a strategy for NRQCD expansion that slightly deviates from the orthodox doctrine, in that the matching coefficients are not truly of ``short-distance" nature, but explicitly depend upon physical kinematic variables rather than partonic ones. Our matching method, with validity guaranteed by the Gremm-Kapustin relation, is particularly suited for the inclusive quarkonium production and decay processes with involved kinematics, exemplified by the process considered in this work. Despite some intrinsic ambiguity affiliated with the order- NRQCD matrix element, if we choose its value as what has been extracted from a recent Cornell-potential-model-based analysis, including the relative order- effect is found to increase the lowest-order prediction for the integrated cross section by about 30\%, and exert a modest impact on energy, angular and polarization distributions except near the very upper end of the energy. The order- contribution to the energy spectrum becomes logarithmically divergent at the maximum of energy. A consistent analysis may require that these large end-point logarithms be resummed to all orders in .

Paper Structure

This paper contains 22 sections, 84 equations, 4 figures.

Table of Contents

  1. Introduction
  2. NRQCD factorization and long-distance matrix elements
  3. Perturbative matching strategy
  4. Orthodox matching strategy motivating the shape-function method
  5. Matching strategy adopted in this work
  6. Three-body phase space for $\bm{e^+}\bm{e^-} \bm{\to} \bm{c}\bar{\bm{c}}\bm{({}^3S_1)}\bm{+}\bm{gg}$
  7. Outline of matching calculations for $\bm{e^+ e^- \to} \bm{c}\bar{\bm{c}} \bm{({}^3S_1)} \bm{+} \bm{gg}$
  8. Matching calculation for $\bm{\gamma^\ast \to} \bm{c}\bar{\bm{c}} \bm{({}^3S_1)} \bm{+} \bm{gg}$
  9. Matching calculation for $\bm{\gamma^\ast \to} \bm{c}\bar{\bm{c}} \bm{({}^3S_1, \lambda=0)} \bm{+} \bm{gg}$
  10. Matching calculation for $\bm{e^+ e^- \to} \bm{c}\bar{\bm{c}} \bm{({}^3S_1)} \bm{+} \bm{gg}$
  11. Inclusive $\bm{J/\psi}$ distributions at $\bm{B}$ factory
  12. Choice of input parameters
  13. The integrated production rate for $\bm{J/\psi}$ associated with light hadrons
  14. Energy spectrum of unpolarized $\bm{J/\psi}$
  15. Polarization distribution of $\bm{J/\psi}$
  16. ...and 7 more sections

Figures (4)

  • Figure 1: Lowest-order Feynman diagrams for $e^+ e^-\to J/\psi+gg$.
  • Figure 2: The energy spectra of the unpolarized $J/\psi$ (left panel) and longitudinally polarized $J/\psi$ (right panel) associated with light hadrons at the energy of $B$ factory. The dot-dashed curve represents $d\sigma^{(0)}/dz$, the dashed curve represents $d\sigma^{(2)}/dz$, and the solid curve represents their sum. For simplicity, in all the figures in this work, we have taken only the central values of the input parameters and not drawn the error band.
  • Figure 3: Profile of the $J/\psi$ polarization parameter $\alpha(z)$ in associated production with light hadrons at $\sqrt{s}=10.58$ GeV. The dot-dashed curve represents the leading order prediction $\alpha^{(0)}$, whereas the solid curve represents the corresponding one including the $O(v^2)$ effect.
  • Figure 4: Profile of the $J/\psi$ angular distribution parameter $A(z)$ in associated production with light hadrons at $\sqrt{s}=10.58$ GeV. The dot-dashed curve represents the leading-order prediction $A^{(0)}$, whereas the solid curve represents $A^{v^2}(z)$ defined in (\ref{['A:corr:NLO:v2']}), which has included the order-$v^2$ effect.