Enhanced chiral logarithms in partially quenched QCD

TL;DR

This paper analyzes partially quenched QCD with valence masses $m_V$ and sea masses $m_S$ in the presence of lattice fermions with residual chiral symmetry. Using partially quenched chiral perturbation theory, it computes the full one-loop corrections to the masses and decay constants of pions made from two non-degenerate quarks, showing that leading-order results predicting independence from $m_S$ are spoiled by enhanced non-analytic chiral logarithms of relative size $m_S\ln m_V$ as $m_V\to 0$. The work provides explicit expressions for $[M_{12}^2]_{\rm 1-loop}$, $[M_{VV}^2]_{\rm 1-loop}$, $[M_{SS}^2]_{\rm 1-loop}$, and the corresponding decay constants, highlighting which combinations cancel analytic low-energy constants and how observables like $R_{BG}$ can isolate non-analytic effects. It also discusses extensions to baryons and practical testing strategies with staggered, domain-wall, or Wilson fermions, including non-perturbative quark-mass definitions to control discretization errors. Overall, the results reveal where partially quenched extrapolations can mislead about physical pions but provide sensitive tests for chiral-logarithm dynamics in QCD-like theories.

Abstract

I discuss the properties of pions in ``partially quenched'' theories, i.e. those in which the valence and sea quark masses, $m_V$ and $m_S$, are different. I point out that for lattice fermions which retain some chiral symmetry on the lattice, e.g. staggered fermions, the leading order prediction of the chiral expansion is that the mass of the pion depends only on $m_V$, and is independent of $m_S$. This surprising result is shown to receive corrections from loop effects which are of relative size $m_S \ln m_V$, and which thus diverge when the valence quark mass vanishes. Using partially quenched chiral perturbation theory, I calculate the full one-loop correction to the mass and decay constant of pions composed of two non-degenerate quarks, and suggest various combinations for which the prediction is independent of the unknown coefficients of the analytic terms in the chiral Lagrangian. These results can also be tested with Wilson fermions if one uses a non-perturbative definition of the quark mass.

Figures (3)

• Figure 1: Predictions for pion masses using values for the parameters discussed in the text. The solid and short-dashed curves show $M_{VV}^2$ and $M_{VS}^2$, including one-loop contributions, plotted against $m_V$. The three sets of curves correspond to $m_S=m_{\rm st}$, $m_{\rm st}/2$, and $m_{\rm st}/4$ as one moves from top to bottom. The long-dashed curve is the result for $M_{SS}^2$ at one-loop plotted against $m_S$.
• Figure 2: Predictions for $\ln M_{VV}^2/2 \mu m_V$, $\ln M_{VS}^2/\mu(m_V+m_S)$ and $\ln M_{SS}^2/2\mu m_S$, including one-loop contributions, plotted against the average quark mass. Notation as in Fig. \ref{['fig:mpi']}. The three sets of curves for ${VV}$ and ${VS}$ mesons correspond to $m_S=m_{\rm st}/4$, $m_{\rm st}/2$ and $m_{\rm st}$ as one moves from left to right.
• Figure 3: Results for decay constants plotted against average quark mass. Notation as in Fig. \ref{['fig:mpi']}. As one moves from top to bottom, the three sets of curves correspond to $m_S=m_{\rm st}$, $m_{\rm st}/2$, and $m_{\rm st}/4$.