[Paper Review] Charmonium-nucleon interaction from lattice QCD with a relativistic heavy quark action

Taichi Kawanai, Shoichi Sasaki|arXiv (Cornell University)|Nov 5, 2010

Quantum Chromodynamics and Particle Interactions6 references3 citations

TL;DR

This study computes the low-energy charmonium-nucleon interaction in lattice QCD using two novel methods: a relativistic heavy quark action and effective Schrödinger equation from equal-time Bethe-Salpeter amplitudes, and extended Lüscher's formula with twisted boundary conditions. It finds a weakly attractive, exponentially screened potential and determines scattering parameters, revealing a slightly larger scattering length for $J/\psi$-N ($\sim 0.35$ fm) than $\eta_c$-N ($\sim 0.25$ fm), indicating marginally stronger attraction in the $J/\psi$-N channel.

ABSTRACT

Detailed information of the low-energy interaction between the charmonia (ηc and J/ψ) and the nucleon is indispensable for exploring the formation of charmonium bound to nuclei. In order to investigate the charmonium-nucleon interactions at low energies, we adopt two essentially different approaches in lattice QCD simulations. The charmonium-nucleon potential can be calculated from the equal-time Bethe-Salpeter amplitude through the effective Schrödinger equation. This novel method is based on the same idea originally applied for the nucleon force by Aoki- Hatsuda-Ishii. Another approach is to utilize extended Lüscher's formula with partially twisted boundary conditions, which allows us to calculate the s-wave phase shift at any small value of the relative momentum even in a finite box. We then extract model independent information of the scattering length and the effective range from the phase shift through the effective-range expansion. Our simulations are carried out at a lattice cutoff of $1/a \approx$ 2 GeV in a spatial volume of (3 fm)^3 with the non-perturbatively O(a)-improved Wilson fermions for the light quarks and a relativistic heavy quark action for the charm quark. Although our main results are calculated in quenched lattice calculations, we also present a preliminary full QCD result by using the 2+1 flavor gauge configurations generated by PACS-CS Collaboration. We have found that the charmonium-nucleon potential is weakly attractive at short distances and exponentially screened at large distances. We have also successfully evaluated both the scattering length and effective range from the charmonium-nucleon scattering phase shift.

Motivation & Objective

• To compute the low-energy interaction between charmonia ($\eta_c$, $J/\psi$) and nucleons from first principles in QCD.
• To investigate whether charmonium can form bound states with nuclei, particularly in few-body systems like $^3$He.
• To extract model-independent scattering parameters—scattering length and effective range—from lattice QCD simulations.
• To compare results between quenched and full QCD frameworks to assess dynamical quark effects on the interaction.

Proposed method

• Employ a relativistic heavy quark action for charm quarks and $O(a)$-improved Wilson fermions for light quarks on a $(3\,\text{fm})^3$ lattice with $1/a \approx 2$ GeV cutoff.
• Calculate the equal-time Bethe-Salpeter amplitude from four-point correlation functions to extract the hadron-hadron potential via an effective Schrödinger equation.
• Use partially twisted boundary conditions to access low-momentum scattering states and compute $s$-wave phase shifts in finite volume.
• Apply the extended Lüscher formula to relate finite-volume energy levels to scattering phase shifts.
• Perform effective-range expansion on $p\cot\delta_0(p)$ using polynomial fits to extract scattering length $a_0 = 1/d_0$ and effective range $r_0 = 2d_1$.
• Conduct both quenched and preliminary full QCD simulations using PACS-CS 2+1 flavor gauge configurations to assess quark mass and dynamical quark dependence.

Experimental results

Research questions

• RQ1 What is the nature of the charmonium-nucleon potential at low energies, and how does it depend on the charmonium state ($\eta_c$ vs $J/\psi$)?
• RQ2 Can the $s$-wave scattering length and effective range for $c\bar{c}$-nucleon scattering be extracted from lattice QCD with high precision using twisted boundary conditions?
• RQ3 How do dynamical quarks affect the charmonium-nucleon interaction compared to the quenched approximation?
• RQ4 Is the $J/\psi$-nucleon system more attractive than the $\eta_c$-nucleon system at low energies, as suggested by potential and scattering parameter differences?

Key findings

• The charmonium-nucleon potential is weakly attractive at short distances and exponentially screened at large distances, with no significant quark mass dependence in the $640$–$870$ MeV pion mass range.
• The $J/\psi$-nucleon scattering length ($a_0^{J/\psi\text{-}N}$) is found to be approximately $0.35$ fm, while the $\eta_c$-nucleon scattering length ($a_0^{\eta_c\text{-}N}$) is approximately $0.25$ fm, indicating slightly stronger attraction in the $J/\psi$-N channel.
• Both $\eta_c$-N and $J/\psi$-N channels yield a similar effective range of $r_0 \sim 1.0$ fm, albeit with large uncertainties.
• The scattering parameters extracted via the effective-range expansion from phase shifts show consistency between linear and quadratic fits, validating the analysis method.
• Preliminary full QCD results at $M_\pi \approx 410$ MeV show no appreciable dynamical quark effects on the potential or scattering parameters, suggesting stability of results under inclusion of sea quarks.
• The channel dependence in scattering length is consistent with earlier potential-based observations, reinforcing the idea of spin-dependent attraction in the $c\bar{c}$-N system.