Isospin breaking and chiral symmetry restoration

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We analyze quark condensates and chiral (scalar) susceptibilities including isospin-breaking effects at finite temperature T. These include m(u) not equal m(d) contributions as well as electromagnetic (e not equal 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 <(q) over barq > = <(u) over baru + (d) over bard > is almost unaffected, while the isospin-breaking order parameter <(u) over baru - (d) over bard > grows with T for the three-flavor case SU(3). We derive a sum rule relating the condensate ratio <(q) over barq >(e not equal 0)/<(q) over barq >(e = 0) with the scalar susceptibility difference (x)(T) - (x)(0), directly measurable on the lattice. This sum rule is useful also for estimating condensate errors in staggered lattice analysis. Keeping m(u) not equal 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(pi), while the connected part shows a quadratic behavior, infrared regular as T(2)/M(eta)(2), and coming from pi(0)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.
© 2011 American Physical Society. We are grateful to W. Unger for useful comments. Work partially supported by Spanish research Contracts No. FPA2008-00592, FIS2008-01323, UCM-BSCH GR58/08 910309, and the FPI program (BES-2009-013672).
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