[Paper Review] Finite-volume effects and the electromagnetic contributions to kaon and pion masses

Subhasish Basak, Alexei Bazavov+15 more|arXiv (Cornell University)|Sep 25, 2014

Quantum Chromodynamics and Particle Interactions10 references4 citations

TL;DR

This paper presents a lattice QCD calculation of electromagnetic contributions to kaon and pion masses using the MILC Collaboration's asqtad staggered fermions with quenched photons, correcting for finite-volume effects via one-loop staggered chiral perturbation theory. The key result is a refined determination of the parameter $\epsilon = 0.84(5)_{\rm stat}(18)_{a^{2}}(6)_{\rm FV}$, reducing the electromagnetic uncertainty in $m_u/m_d$ by over a factor of two compared to prior work.

ABSTRACT

We report on the MILC Collaboration calculation of electromagnetic effects on light pseudoscalar mesons. The simulations employ asqtad staggered dynamical quarks in QCD plus quenched photons, with lattice spacings varying from 0.12 to 0.06 fm. Finite volume corrections for the MILC realization of lattice electrodynamics have been calculated in chiral perturbation theory and applied to the lattice data. These corrections differ from those calculated by Hayakawa and Uno because our treatment of zero modes differs from theirs. Updated results for the corrections to "Dashen's theorem" are presented.

Motivation & Objective

• To reduce the dominant uncertainty in the lattice determination of the $m_u/m_d$ ratio by accurately calculating electromagnetic contributions to kaon and pion masses.
• To address the significant finite-volume (FV) effects that previously hindered precise extraction of electromagnetic splittings in lattice simulations.
• To apply one-loop staggered chiral perturbation theory to model and correct FV effects in the MILC framework with quenched photons.
• To improve the precision of Dashen's theorem tests by determining the correction parameter $\epsilon$ with controlled systematic errors.
• To provide a more accurate estimate of the electromagnetic uncertainty in $m_u/m_d$ by correcting for FV effects and lattice artifacts.

Proposed method

• Simulations use (2+1)-flavor asqtad staggered fermions with dynamical quarks and quenched photons on lattices with spacings from 0.06 to 0.12 fm and volumes from $12^3 \times 64$ to $48^3 \times 144$.
• Finite-volume corrections are derived using one-loop staggered chiral perturbation theory, differing from prior treatments due to a distinct zero-mode handling in the MILC formulation.
• Meson masses are extracted from correlation functions using correlated fits, with a central uncorrelated fit used due to poor p-values in correlated fits.
• The electromagnetic splitting $\epsilon$ is computed as $\epsilon = (M^2_{K^\pm} - M^2_{K^0})^\gamma / (M^2_{\pi^\pm} - M^2_{\pi^0})^{\rm expt}$, with $\pi^0$ defined as the RMS average of $u\bar{u}$ and $d\bar{d}$ states.
• Systematic errors are estimated by varying fit procedures, including data thinning, cutoffs, inclusion of higher-order chiral terms, and correlation handling, with a final error estimate based on the difference between experimental and lattice pion splittings.
• Continuum extrapolation is performed using NLO SU(3) chiral perturbation theory, with adjustments for sea quark charges and physical values.

Experimental results

Research questions

• RQ1 What is the size of finite-volume effects on electromagnetic splittings of kaons and pions in lattice QCD with quenched photons?
• RQ2 How do the finite-volume corrections in the MILC staggered formulation differ from those in previous treatments, such as Hayakawa and Uno’s?
• RQ3 What is the corrected value of the parameter $\epsilon$ in Dashen’s theorem, accounting for finite-volume and discretization effects?
• RQ4 To what extent do finite-volume effects limit the precision of $m_u/m_d$ determinations from kaon mass splittings?
• RQ5 How do the lattice results for $\epsilon$ compare to the experimental pion splitting when systematic errors are properly accounted for?

Key findings

• The finite-volume corrections in the MILC framework are successfully modeled and applied using one-loop staggered chiral perturbation theory, reducing the residual systematic error to $\sim 0.06$.
• The corrected value of $\epsilon$ is determined to be $0.84(5)_{\rm stat}(18)_{a^2}(6)_{\rm FV}$, a significant improvement over the uncorrected result of $0.65(7)$.
• The electromagnetic uncertainty in the $m_u/m_d$ ratio is reduced by more than a factor of two compared to previous estimates, due to the improved control of $\epsilon$.
• The dominant source of systematic error in $\epsilon$ arises from the difference between the experimental pion splitting and the lattice-calculated electromagnetic splitting, estimated at $\pm 0.18$.
• Preliminary results from additional ensembles at $a \approx 0.06$ fm and $a \approx 0.045$ fm suggest that lattice artifacts in $\epsilon$ will be further reduced upon full analysis.
• The final estimate for $m_u/m_d$ is $0.4482(48)_{\rm stat}({}^{+21}_{-115})_{a^2}(1)_{\rm FV_{QCD}}(165)_{\rm EM}$, with the $\rm EM$ error now the largest component, reflecting improved control of other uncertainties.