Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Spin correlations and velocity-scaling in color-octet NRQCD matrix elements

Geoffrey T. Bodwin, Jungil Lee, D. K. Sinclair

TL;DR

This study probes NRQCD factorization and the $v$-scaling rules for heavy-quarkonium by computing color-octet, spin-dependent decay matrix elements on the lattice. Using lattice NRQCD on quenched configurations with multiple heavy-quark Green's-function schemes, the authors extract ratios of color-octet to color-singlet matrix elements for $S$-wave bottomonium and charmonium. They find that, for bottomonium, the leading transitions (singlet$ ightarrow$triplet and triplet$ ightarrow$singlet) dominate as predicted by $v$-scaling, while diagonal and spin-flip octet contributions are suppressed; for charmonium, large UV-cutoff effects due to $oldsymbol{ abla} ightarrow m_c$ produce deviations, though the qualitative hierarchy remains and improves when the effective cutoff is lowered. The results support the usefulness of the $v$-expansion in organizing NRQCD matrix elements and highlight the need for perturbative matching between lattice and continuum NRQCD to connect to phenomenology of high-$p_T$ quarkonium production. Overall, the work informs interpretations of Tevatron data and motivates further perturbative studies of lattice-continuum matching. Key quantities involve the heavy-quark mass $m$, the UV cutoff $oldsymbol{ abla}$, the velocity parameter $v$, and large transverse momentum $p_T$, with color-octet matrix elements linked to production via crossing relations.

Abstract

We compute spin-dependent decay matrix elements for S-wave charmonium and bottomonium in lattice nonrelativistic quantum chromodynamics (NRQCD). Particular emphasis is placed upon the color-octet matrix elements, since the corresponding production matrix elements are expected to appear in the dominant contributions to the production cross sections at large transverse momenta. We use three slightly different versions of the heavy-quark lattice Green's functions in order to minimize the contributions that scale as powers of the ultraviolet cutoff. The lattice matrix elements that we calculate obey the hierarchy that is suggested by the velocity-scaling rules of NRQCD.

Spin correlations and velocity-scaling in color-octet NRQCD matrix elements

TL;DR

This study probes NRQCD factorization and the -scaling rules for heavy-quarkonium by computing color-octet, spin-dependent decay matrix elements on the lattice. Using lattice NRQCD on quenched configurations with multiple heavy-quark Green's-function schemes, the authors extract ratios of color-octet to color-singlet matrix elements for -wave bottomonium and charmonium. They find that, for bottomonium, the leading transitions (singlettriplet and tripletsinglet) dominate as predicted by -scaling, while diagonal and spin-flip octet contributions are suppressed; for charmonium, large UV-cutoff effects due to produce deviations, though the qualitative hierarchy remains and improves when the effective cutoff is lowered. The results support the usefulness of the -expansion in organizing NRQCD matrix elements and highlight the need for perturbative matching between lattice and continuum NRQCD to connect to phenomenology of high- quarkonium production. Overall, the work informs interpretations of Tevatron data and motivates further perturbative studies of lattice-continuum matching. Key quantities involve the heavy-quark mass , the UV cutoff , the velocity parameter , and large transverse momentum , with color-octet matrix elements linked to production via crossing relations.

Abstract

We compute spin-dependent decay matrix elements for S-wave charmonium and bottomonium in lattice nonrelativistic quantum chromodynamics (NRQCD). Particular emphasis is placed upon the color-octet matrix elements, since the corresponding production matrix elements are expected to appear in the dominant contributions to the production cross sections at large transverse momenta. We use three slightly different versions of the heavy-quark lattice Green's functions in order to minimize the contributions that scale as powers of the ultraviolet cutoff. The lattice matrix elements that we calculate obey the hierarchy that is suggested by the velocity-scaling rules of NRQCD.

Paper Structure

This paper contains 6 sections, 18 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. NRQCD
  3. Continuum NRQCD
  4. Lattice NRQCD
  5. Calculations and results
  6. Summary and conclusions

Figures (2)

  • Figure 1: The effective quarkonium "mass" (energy) as measured with point and Gaussian-smeared (extended) sources for the $J/\psi$ and the $\Upsilon$. E is the effective energy, and T is the time separation between the source and the sink. The effective energies shown are measured in the nrqcd updating scheme that is described in Sec. \ref{['sec:NRQCD']}. The plots in the hybrid scheme and for the $\eta_c$ and the $\eta_b$ are similar.
  • Figure 2: Values of ratios $R$ for charmonium as a function of $T=T'$. The upper plot shows $R$ for the singlet $\rightarrow$ triplet spin transition in the hybrid scheme. The lower plot shows $R$ for the longitudinal $\rightarrow$ transverse spin transition in the coulomb scheme.