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Quarkonium momentum distributions in photoproduction and B decay

M. Beneke, G. A. Schuler, S. Wolf

TL;DR

This work addresses the challenge of predicting quarkonium energy distributions near the kinematic endpoint, where the NRQCD velocity expansion fails due to soft-gluon fragmentation. The authors introduce a shape-function framework that encodes soft emission via a nonperturbative radiation function, enabling a unified convolution with short-distance production amplitudes. They apply the model to $B\to J/\u001bpsi X$ and inelastic $J/\u001bpsi$ photoproduction, finding that a simple ansatz with a scale $\Lambda \sim 300$ MeV describes the $J/\u001bpsi$ momentum spectrum in $B$ decay and yields a turnover in photoproduction spectra consistent with endpoint physics, though strong experimental cuts are required for reliable tests of color-octet contributions. The results support a quasi-universal description of soft fragmentation effects within NRQCD and highlight the importance of endpoint modeling and kinematic cuts for interpreting quarkonium production data.

Abstract

According to our present understanding many $J/ψ$ production processes proceed through a coloured $c\bar{c}$ state followed by the emission of soft particles in the quarkonium rest frame. The kinematic effect of soft particle emission is usually a higher-order effect in the non-relativistic expansion, but becomes important near the kinematic endpoint of quarkonium energy (momentum) distributions. In an intermediate region a systematic resummation of the non-relativistic expansion leads to the introduction of so-called `shape functions'. In this paper we provide an implementation of the kinematic effect of soft gluon emission which is consistent with the non-relativistic shape function formalism in the region where it is applicable and which models the extreme endpoint region. We then apply the model to photoproduction of $J/ψ$ and $J/ψ$ production in $B$ meson decay. A satisfactory description of $B$ decay data is obtained. For inelastic charmonium photoproduction we conclude that a sensible comparison of theory with data requires a transverse momentum cut larger than the currently used 1 GeV.

Quarkonium momentum distributions in photoproduction and B decay

TL;DR

This work addresses the challenge of predicting quarkonium energy distributions near the kinematic endpoint, where the NRQCD velocity expansion fails due to soft-gluon fragmentation. The authors introduce a shape-function framework that encodes soft emission via a nonperturbative radiation function, enabling a unified convolution with short-distance production amplitudes. They apply the model to and inelastic photoproduction, finding that a simple ansatz with a scale MeV describes the momentum spectrum in decay and yields a turnover in photoproduction spectra consistent with endpoint physics, though strong experimental cuts are required for reliable tests of color-octet contributions. The results support a quasi-universal description of soft fragmentation effects within NRQCD and highlight the importance of endpoint modeling and kinematic cuts for interpreting quarkonium production data.

Abstract

According to our present understanding many production processes proceed through a coloured state followed by the emission of soft particles in the quarkonium rest frame. The kinematic effect of soft particle emission is usually a higher-order effect in the non-relativistic expansion, but becomes important near the kinematic endpoint of quarkonium energy (momentum) distributions. In an intermediate region a systematic resummation of the non-relativistic expansion leads to the introduction of so-called `shape functions'. In this paper we provide an implementation of the kinematic effect of soft gluon emission which is consistent with the non-relativistic shape function formalism in the region where it is applicable and which models the extreme endpoint region. We then apply the model to photoproduction of and production in meson decay. A satisfactory description of decay data is obtained. For inelastic charmonium photoproduction we conclude that a sensible comparison of theory with data requires a transverse momentum cut larger than the currently used 1 GeV.

Paper Structure

This paper contains 17 sections, 95 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Shape function model
  3. Factorization and the shape function in the Coulomb limit
  4. The general case
  5. Derivation of the smeared spectrum
  6. The shape function limit
  7. Form of $\Phi_n(k;p_R,P)$
  8. Computation of the shape function in the Coulomb limit
  9. Momentum spectrum in $B\to J/\psi X$
  10. Energy distribution in $b$ quark decay
  11. Normalization difficulty
  12. Fermi motion effects
  13. Final result and comparison with CLEO data
  14. Inelastic $J/\psi$ photoproduction
  15. Kinematics of photoproduction
  16. ...and 2 more sections

Figures (9)

  • Figure 1: Diagrammatic representation of the amplitude that leads to Eq. (\ref{['start']}).
  • Figure 2: A contribution to the $\gamma g\to J/\psi g g$ amplitude discussed in the text.
  • Figure 3: Left: one of four NRQCD diagrams which contribute to the M1 (E1) transition from an $^1\!S_0^{(8)}$ ($^3\!P_0^{(8)}$) state, specified by $\Gamma_n^{(')}$, to $J/\psi$. Right: an example for a double E1 transition from a $^3\!S_1^{(1,\,8)}$ state.
  • Figure 4: Dependence of $k_0/(4\pi^2)\,S_n(k;p_R,P)$ for $n={}^1\!S_0^{(8)}$ and $n={}^3\!P_0^{(8)}$ on the emitted gluon energy $k_0$. The parameters are chosen as $m_c=1.5\,$GeV, $\alpha_s=0.4$, $\lambda=-1/8$.
  • Figure 5: Sum of colour octet modes $d{\rm BR}_{[8]}/dp_R[n]$ with $n = \{^1\!S_0^{(8)}, \, ^3\!P_0^{(8)}, \, ^3\!S_1^{(8)} \}$ to the differential branching ratio $d{\rm BR}/dp_R$ of the decay $B \to J/\psi X$ for various values of the shape function parameter $\Lambda$. The ACCMM model parameters are fixed at $p_F = 300$ GeV and $m_{sp} = 150$ GeV.
  • ...and 4 more figures