[Paper Review] $\eta'$-$\eta_c$-mixing with improved stochastic estimators

Christian Ehmann, Gunnar Bali|arXiv (Cornell University)|Mar 17, 2009

Quantum Chromodynamics and Particle Interactions9 references3 citations

TL;DR

This study investigates $η'$--$η_c$ mixing in lattice QCD using improved stochastic all-to-all propagator techniques to compute disconnected quark-line diagrams. The authors employ staggered spin dilution (SSD) and hopping parameter acceleration (HPA) to reduce statistical noise, finding no significant mixing between the $η_c$ and $η'$ states within statistical uncertainties, despite the inclusion of quark-antiquark annihilation effects.

ABSTRACT

Charmonia are flavour singlet mesons and thus in principle contributions from disconnected quark line diagrams might affect their masses, either directly or via mixing with other flavour singlet channels. We present a first study that takes both effects into account. We employ improved stochastic all-to-all propagator techniques (including new methods) to calculate the diagrams that appear within the mixing matrix between the $\\eta'$ and the $\\eta_c$. The runs are initially performed on $N_f=2$ $16^3\ imes 32$ configurations with the non-perturbatively improved Sheikholeslami-Wilson action, both for valence and sea quarks.

Motivation & Objective

• To assess the impact of disconnected quark-line diagrams on the $η_c$ mass, particularly via mixing with the $η'$ meson.
• To implement and test advanced stochastic all-to-all propagator techniques to reduce noise in disconnected diagram calculations.
• To determine whether quark-antiquark annihilation or $η'$--$η_c$ mixing contributes to the underestimation of the charmonium hyperfine splitting in lattice QCD.
• To evaluate the numerical stability and efficiency of new noise reduction schemes like SSD and HPA in heavy quark systems.

Proposed method

• Uses $N_f=2$ $16^3\times32$ lattices with non-perturbatively improved Sheikholeslami-Wilson fermions and tuned charm quark mass to match experimental spin-averaged charmonium mass.
• Applies improved stochastic estimators with $N=200$ noise sources per configuration to compute all-to-all propagators, reducing statistical noise via unbiased sampling.
• Introduces Staggered Spin Dilution (SSD), a novel partitioning scheme that alternates spinor components spatially to minimize noise from nearest-neighbor couplings.
• Employs Hopping Parameter Acceleration (HPA) to enhance convergence of stochastic estimates, achieving a net gain of up to 11.8× in computational efficiency.
• Constructs a 6×6 correlation matrix including both light and charm quark interpolators, using variational methods to extract eigenvalues and eigenvectors.
• Performs generalized eigenvalue analysis to extract effective masses and mixing components, focusing on the $(c\bar{c})_{10}, (c\bar{c})_{80}, (u\bar{u})_5, (u\bar{u})_{40}$ basis for numerical stability.

Experimental results

Research questions

• RQ1 Does quark-antiquark annihilation via disconnected diagrams significantly affect the $η_c$ mass?
• RQ2 Is there measurable mixing between the $η_c$ and the $η'$ meson in the charmonium sector?
• RQ3 How effective are the new stochastic estimators—SSD and HPA—in reducing noise for heavy quark systems?
• RQ4 To what extent do disconnected contributions and mixing with flavor-singlet states contribute to the underestimation of the charmonium hyperfine splitting?

Key findings

• No significant mixing is observed between the $η_c$ and $η'$ states, as the eigenvector components of the $η_c$ ground state show no statistically significant admixture from the charm sector.
• The eigenvector components of the $η$ state also show no significant contribution from the $η_c$ sector, indicating negligible mixing.
• Effective masses from the full 4-state basis are consistent with those from isolated submatrices within statistical errors, supporting the stability of the results.
• The SSD and HPA techniques yield a net computational gain of up to 11.8 times, significantly improving the feasibility of disconnected diagram calculations.
• The study concludes that neither $η'$--$η_c$ mixing nor disconnected quark-line contributions are responsible for the hyperfine splitting underestimation in this setup.
• The conclusion may be affected by the large pion mass ($m_\pi \approx 1007$ MeV), suggesting that future simulations with lighter pions are needed to confirm the absence of mixing.