[Paper Review] Effects of the disconnected flavor singlet corrections on the hyperfine splitting in charmonium

DeTar, C., L. Levkova|arXiv (Cornell University)|Oct 5, 2007

Muon and positron interactions and applications4 citations

TL;DR

This study computes the disconnected flavor singlet contributions to the charmonium hyperfine splitting using high-precision, fine-lattice QCD simulations with 2+1 dynamical quarks. By employing point-to-point correlators and unbiased stochastic subtraction, the authors reduce noise by three orders of magnitude and find a 1–4 MeV mass shift for the ηc due to disconnected diagrams, suggesting a small but non-negligible correction to the hyperfine splitting.

ABSTRACT

Experimentally the charmonium hyperfine splitting is $M_{J/ψ}-M_{η_c}=117$ MeV and current lattice results are generally below this value. The difference could be due to the effects of the disconnected flavor singlet diagrams which have not been included in these calculations. Previous attempts to determine the disconnected flavor singlet corrections have led just to rough estimates in the range of $\pm 20$ MeV. We present preliminary results for these corrections calculated on fine ($a\approx 0.09$ fm) Asqtad 2+1 flavor lattices provided by the MILC Collaboration.

Motivation & Objective

• To investigate whether disconnected flavor singlet diagrams—previously neglected in lattice calculations—contribute significantly to the discrepancy between lattice and experimental hyperfine splitting in charmonium.
• To address the long-standing issue that current lattice results for the J/ψ–ηc splitting are ~10% below the experimental value of 117 MeV.
• To improve the signal-to-noise ratio in disconnected diagram calculations using advanced stochastic techniques and fine lattices.
• To extract the ground state mass shift of the ηc from the disconnected pseudoscalar correlator using a model-dependent fit to the spatial correlation function.

Proposed method

• The study uses 2+1 flavor Asqtad fermions on fine lattices (a ≈ 0.09 fm) from the MILC collaboration, with physical charm quark mass tuned via the Fermilab clover action.
• Disconnected correlators are computed using point-to-point (PP) propagators instead of time-slice-to-time-slice (TS) methods, reducing statistical noise by three orders of magnitude.
• An unbiased subtraction technique based on a truncated hopping expansion (up to third order in κc) is applied to suppress noise in stochastic estimators.
• The spatial behavior of the disconnected correlator D(r) is modeled as D(r) ∼ B r^{-1/2} e^{-m_c r}, with a fit to r ∈ [5,11] to extract the amplitude ratio B/A.
• A simplified ansatz for the disconnected correlator includes contributions from the ηc ground state, an excited ηc state, and a light state (e.g., η, η′, glueballs), with amplitudes fitted to data.
• The mass shift is extracted via the relation B/A = m_c - m_f, where m_f is the full mass including disconnected contributions.

Experimental results

Research questions

• RQ1 What is the magnitude and sign of the disconnected flavor singlet contribution to the ηc mass in charmonium?
• RQ2 Can the signal-to-noise problem in disconnected diagram calculations be sufficiently mitigated to allow reliable extraction of the ηc mass shift?
• RQ3 Does the disconnected contribution to the hyperfine splitting significantly reduce the discrepancy between lattice QCD and experiment?
• RQ4 How sensitive are the results to the choice of fitting model and the inclusion of excited and light hadronic states in the correlator decomposition?

Key findings

• The disconnected diagram contribution increases the ηc mass by 1–4 MeV, as inferred from the amplitude ratio B/A = m_c - m_f ∈ [−4, −1] MeV.
• The use of point-to-point correlators reduces statistical noise by approximately three orders of magnitude compared to traditional time-slice methods.
• The unbiased subtraction technique with a third-order κc expansion reduces the standard deviation of the disconnected correlator by a factor of four.
• The fitted light state mass is m_l = 0.43(1) GeV, consistent with expectations for light hadronic modes such as η and η′.
• The results are preliminary and sensitive to model assumptions; further work is needed to confirm robustness against variations in fitting form and lattice spacing.
• If the J/ψ mass shift is negligible—as expected—the hyperfine splitting would be slightly reduced, partially resolving the lattice-experiment discrepancy.