Potential-model calculation of an order-v^2 NRQCD matrix element

TL;DR

This work tackles the nonperturbative NRQCD matrix elements tied to the second derivative of the heavy-quarkonium wave function at the origin, ψ^{(2)}(0), which govern leading relativistic corrections. It develops two calculation paths: a general hard-cutoff approach that subtracts the regulator difference to dimensional regularization and a direct potential-model method that reproduces the Gremm–Kapustin relation and permits resummation of higher-v contributions. Applying these to the J/ψ in the Cornell model, the authors obtain a robust determination of ψ^{(2)}(0) and the related ⟨v^2⟩, with quantified uncertainties dominated by neglected v^2 terms and potential-model choices. The results provide improved NRQCD inputs for quarkonium production and decay and highlight caveats for lattice extractions due to power corrections and two-loop short-distance effects.

Abstract

We present two methods for computing dimensionally-regulated NRQCD heavy-quarkonium matrix elements that are related to the second derivative of the heavy-quarkonium wave function at the origin. The first method makes use of a hard-cutoff regulator as an intermediate step and requires knowledge only of the heavy-quarkonium wave function. It involves a significant cancellation that is an obstacle to achieving high numerical accuracy. The second method is more direct and yields a result that is identical to the Gremm-Kapustin relation, but it is limited to use in potential models. It can be generalized to the computation of matrix elements of higher order in the heavy-quark velocity and can be used to resum the contributions to decay and production rates that are associated with those matrix elements. We apply these methods to the Cornell potential model and compute a matrix element for the J/psi state that appears in the leading relativistic correction to the production and decay of that state through the color-singlet quark-antiquark channel.

Figures (2)

• Figure 1: Feynman diagram corresponding to $\tilde{I}^{(2)}(\bm{p})$, which is the one-loop renormalization of the operators in Eq. (\ref{['psi20']}). The solid lines represent the heavy quark and antiquark, and the dotted line represents the potential between them.
• Figure 2: $\psi_\Lambda^{(2)}(0)$ and $\psi_{\textrm{DR,}\Lambda}^{(2)}(0)\equiv\psi^{(2)}_{\Lambda}(0)-\Delta\psi^{(2)}_{\textrm{NB}}(0)$ as a function of $\Lambda/m$. The left figure corresponds to the LO potential-model parameters of Table \ref{['table:ma-lo']} for $\lambda=0.6$, $0.7$, and $0.8$. The right figure corresponds to the NLO potential-model parameters of Table \ref{['table:ma-nlo']} for $\lambda=0.9$, $1.0$, and $1.1$. In each figure, the three curves that are nearly linear are $0.1\times \psi_\Lambda^{(2)}(0)$, and the three curves that reach a plateau at large $\Lambda/m$ are $\psi_{\textrm{DR,}\Lambda}^{(2)}(0)$.