[Paper Review] Contributions of charm anihilation to the hyperfine splitting in charmonium

L. Levkova, DeTar, C.|arXiv (Cornell University)|Sep 29, 2008

Quantum Chromodynamics and Particle Interactions1 references3 citations

TL;DR

This paper nonperturbatively evaluates the contribution of disconnected charm quark loops to the hyperfine splitting in charmonium using quenched lattice QCD. By introducing a fitting procedure that accounts for rotational symmetry violations, it finds that the disconnected diagrams increase the $η_c$ mass by $-3.3(9)$ MeV and $-3.1(8)$ MeV at lattice spacings of 0.09 and 0.063 fm, respectively, suggesting that the main source of discrepancy in lattice hyperfine splitting calculations is likely heavy quark discretization errors rather than omitted disconnected diagrams.

ABSTRACT

In calculations of the hyperfine splitting in charmonium, the contributions of the disconnected diagrams is considered small and is typically ignored. We aim to estimate nonperturbatively the size of the resulting error, which could potentially affect the high precision calculations of the charmonium spectrum. Following our work on the effects of the disconnected diagrams in unquenched QCD presented at Lattice 2007, we study the same problem in the quenched case. On dynamical ensembles the disconnected charmonium propagators contain light modes which complicate the extraction of the signal at large distances. In the fully quenched case, where there are no such light modes, the interpretation of the signal is simplified. We present results from lattices with $a\approx 0.09$ fm and $a\approx 0.063$ fm.

Motivation & Objective

• To estimate the nonperturbative contribution of disconnected diagrams to the hyperfine splitting in charmonium, which are typically neglected in lattice QCD calculations.
• To assess whether the omission of these diagrams contributes significantly to the observed $\sim10\%$ discrepancy between lattice results and the experimental hyperfine splitting of 117 MeV.
• To develop and apply a new fitting procedure that accounts for rotational symmetry violations in the disconnected propagator data, improving accuracy over previous methods.
• To isolate the effects of disconnected diagrams by studying the quenched case, where light-quark intermediate states are absent, simplifying signal interpretation.
• To determine whether the observed mass shift in the $\eta_c$ state is an artifact of unphysical charm quark masses or a genuine nonperturbative effect.

Proposed method

• The full charmonium propagator is decomposed into connected and disconnected components: $F(t) = C(t) + D(t)$, with $D(t)$ representing the disconnected contribution from charm quark loops.
• A momentum-space model for the disconnected propagator is used: $D(p^2) \sim \text{sign}(C)\left(\frac{a}{p^2 + m_c^2} + \frac{b}{p^2 + m_c^{*2}}\right)^2$, incorporating ground and excited charm states with masses fixed from connected diagram fits.
• Rotational symmetry violations are accounted for by replacing $(p_\mu a)^2$ with $4\sin^2(p_\mu a/2)$ in the Fourier transform, improving the fit to lattice data.
• The mass shift due to the disconnected diagram is computed via $\Delta m = \frac{\lambda(-m_c^2)a^2}{\sqrt{32}A_t m_c^2}$, where $A_t$ is the timeslice-to-timeslice connected propagator amplitude.
• Fits are performed on point-to-point disconnected propagators at $a \approx 0.09$ fm and $a \approx 0.063$ fm, with consistent fitting ranges and error estimation via bootstrap resampling.
• The analysis compares results between dynamical and quenched QCD, focusing on the $\eta_c$ and $J/\psi$ states, with special attention to signal-to-noise issues in the vector channel.

Experimental results

Research questions

• RQ1 What is the magnitude of the disconnected diagram contribution to the $\eta_c$ mass in quenched lattice QCD, and does it resolve the discrepancy between lattice and experimental hyperfine splitting?
• RQ2 How do rotational symmetry violations in the lattice data affect the extraction of the disconnected propagator, and can they be systematically corrected?
• RQ3 Is the observed mass shift in the $\eta_c$ state due to $U_{A}(1)$ anomaly effects or mixing with lighter glueballs, particularly given the unphysical pole mass of the $\eta_c$ in the current setup?
• RQ4 How does the disconnected contribution to the $J/\psi$ mass compare to that of the $\eta_c$, and why is the signal more noisy in the vector channel?
• RQ5 Are the results for the disconnected contribution stable under changes in lattice spacing and fitting range, indicating small discretization errors?

Key findings

• The disconnected diagram contribution increases the $\eta_c$ mass by $-3.3(9)$ MeV at $a \approx 0.09$ fm and $-3.1(8)$ MeV at $a \approx 0.063$ fm, with consistent results across lattice spacings, indicating small discretization errors.
• The mass shift in the $\eta_c$ is opposite in sign to the perturbative estimate of a $2.4$ MeV decrease, suggesting nonperturbative effects such as $U_{A}(1)$ anomaly or glueball mixing as the origin.
• The $J/\psi$ mass shift is estimated to be between $-1$ MeV and $0$ MeV, but the large statistical errors prevent a precise determination due to poor signal-to-noise at short distances.
• The fitting procedure incorporating rotational symmetry violations yields consistent results across different fitting ranges and lattice spacings, validating its robustness in the quenched case.
• The consistency of results between fine and superfine ensembles suggests that discretization errors are smaller than statistical errors, supporting the reliability of the extracted mass shifts.
• The study concludes that the dominant source of discrepancy between lattice and experimental hyperfine splitting is likely heavy quark discretization errors, not the omission of disconnected diagrams.