Isospin breaking and chiral symmetry restoration

Abstract

We analyze quark condensates and chiral (scalar) susceptibilities including isospin-breaking effects at finite temperature $T$. These include ${m}_{u}\ensuremath{\ne}{m}_{d}$ contributions as well as electromagnetic ($e\ensuremath{\ne}0$) corrections, both treated in a consistent chiral Lagrangian framework to leading order in $SU(2)$ and $SU(3)$ chiral perturbation theory, so that our predictions are model-independent. The chiral restoration temperature extracted from $⟨\overline{q}q⟩=⟨\overline{u}u+\overline{d}d⟩$ is almost unaffected, while the isospin-breaking order parameter $⟨\overline{u}u\ensuremath{-}\overline{d}d⟩$ grows with $T$ for the three-flavor case $SU(3)$. We derive a sum rule relating the condensate ratio $⟨\overline{q}q⟩(e\ensuremath{\ne}0)/⟨\overline{q}q⟩(e=0)$ with the scalar susceptibility difference $\ensuremath{\chi}(T)\ensuremath{-}\ensuremath{\chi}(0)$, directly measurable on the lattice. This sum rule is useful also for estimating condensate errors in staggered lattice analysis. Keeping ${m}_{u}\ensuremath{\ne}{m}_{d}$ allows one to obtain the connected and disconnected contributions to the susceptibility, even in the isospin limit, whose temperature, mass, and isospin-breaking dependence we analyze in detail. The disconnected part grows linearly, diverging in the chiral (infrared) limit as $T/{M}_{\ensuremath{\pi}}$, while the connected part shows a quadratic behavior, infrared regular as ${T}^{2}/{M}_{\ensuremath{\eta}}^{2}$, and coming from ${\ensuremath{\pi}}^{0}\ensuremath{\eta}$ mixing terms. This smooth connected behavior suggests that isospin-breaking correlations are weaker than critical chiral ones near the transition temperature. We explore some consequences in connection with lattice data and their scaling properties, for which our present analysis for physical masses, i.e. beyond the chiral limit, provides a useful model-independent description for low and moderate temperatures.