[Paper Review] SU(2) chiral fits to light pseudoscalar masses and decay constants

Alexei Bazavov, C. Bérnard+13 more|arXiv (Cornell University)|Nov 3, 2009

Quantum Chromodynamics and Particle Interactions7 references3 citations

TL;DR

This paper presents a next-to-next-to-leading order (NNLO) SU(2) chiral perturbation theory fit to MILC asqtad lattice QCD data for light pseudoscalar mesons, incorporating continuum chiral logarithms and analytic terms. The analysis yields a pion decay constant of $ f_\pi = 128.3(9)^{+20}_{-8} \, \text{MeV} $, consistent with experiment, and determines SU(2) low-energy constants and the two-flavor chiral condensate in the chiral limit with controlled systematic errors.

ABSTRACT

We present the results of fits to recent asqtad data in the light pseudoscalar sector using SU(2) partially-quenched staggered chiral perturbation theory. Superfine (a~0.06 fm) and ultrafine (a~0.045 fm) ensembles are used, where light sea quark masses and taste splittings are small compared to the strange quark mass. Our fits include continuum NNLO chiral logarithms and analytic terms. We give preliminary results for the pion decay constant, SU(2) low-energy constants and the chiral condensate in the two-flavor chiral limit.

Motivation & Objective

• To perform a controlled extrapolation of lattice QCD results for light pseudoscalars to the physical point using SU(2) chiral perturbation theory.
• To extract SU(2) low-energy constants (LECs) such as $ l_3 $ and $ l_4 $, which are key parameters in the two-flavor chiral effective theory.
• To compare the convergence and systematic errors of SU(2) vs. SU(3) chiral fits by analyzing data with strange quark masses near physical values.
• To determine the pion decay constant and the chiral condensate in the two-flavor chiral limit using fine and ultrafine lattice ensembles.

Proposed method

• The study uses SU(2) partially-quenched staggered chiral perturbation theory (SχPT) to model the masses and decay constants of light pseudoscalars from MILC asqtad ensembles.
• The fit includes continuum NNLO chiral logarithms and analytic terms, with lattice spacing effects parameterized via $ a^2 $-dependent terms and taste-splittings.
• The analysis incorporates data from fine (a ≈ 0.06 fm) and ultrafine (a ≈ 0.045 fm) ensembles with small sea quark masses and taste splittings.
• The scale is set using $ r_1 = 0.318(7) \, \text{fm} $ from $ \Upsilon $-splittings, and finite-volume corrections are applied with a small (≤0.3%) correction.
• The fit function includes NLO and NNLO terms in the quark mass expansion, with explicit dependence on valence and sea quark masses, and includes the fourth-root procedure for staggered fermions.
• Systematic errors are estimated by varying fit ranges, including higher-order analytic terms, and incorporating residual finite-volume effects.

Experimental results

Research questions

• RQ1 How well does SU(2) chiral perturbation theory converge compared to SU(3) when applied to lattice QCD data with strange quark masses near physical values?
• RQ2 What are the values of the SU(2) low-energy constants $ l_3 $ and $ l_4 $, and how do they compare to SU(3) fits?
• RQ3 What is the pion decay constant $ f_\pi $ in the two-flavor chiral limit, and how does it compare to the PDG value?
• RQ4 How do the chiral condensate and light quark mass $ \hat{m} $ in the $ \overline{\text{MS}} $ scheme at 2 GeV compare to SU(3) results?
• RQ5 What is the impact of including NNLO chiral logarithms and $ a^2 $-dependent terms on the fit quality and systematic error estimates?

Key findings

• The pion decay constant is determined as $ f_\pi = 128.3(9)^{+20}_{-8} \, \text{MeV} $, in good agreement with the PDG 2008 value of $ 130.4 \pm 0.2 \, \text{MeV} $.
• The SU(2) low-energy constant $ \bar{l}_3 = 3.0(6)^{+9}_{-6} $ and $ \bar{l}_4 = 3.9(2)(3) $ are extracted with small uncertainties.
• The average up and down quark mass in the $ \overline{\text{MS}} $ scheme at 2 GeV is $ \hat{m} = 3.21(3)(5)(16) \, \text{MeV} $, with the third error from perturbative matching.
• The chiral condensate in the two-flavor limit is $ \langle \bar{u}u \rangle_2 = -[280(2)^{+4}_{-7}(4) \, \text{MeV}]^3 $, consistent with SU(3) results.
• The NNLO SU(2) fit shows significantly better convergence than SU(3) fits: NNLO corrections are 0.3% for $ f_\pi $ and 2.6% for $ m_\pi^2/(m_x + m_y) $, compared to 2.9% and 15.6% in SU(3) fits.
• The results are consistent with those from SU(3) SχPT fits when including higher-order analytic terms, validating the SU(2) approach for near-physical strange quark masses.