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 Þ m d contributions as well as electromagnetic (e Þ 0) corrections, both treated in a consistent chiral Lagrangian framework to leading order in SUð2Þ and SUð3Þ chiral perturbation theory, so that our predictions are model-independent. The chiral restoration temperature extracted from h " qqi ¼ h " uu þ " ddi is almost unaffected, while the isospin-breaking order parameter h " uu À " ddi grows with T for the three-flavor case SUð3Þ. We derive a sum rule relating the condensate ratio h " qqiðe Þ 0Þ=h " qqiðe ¼ 0Þ with the scalar susceptibility difference ðTÞ À ð0Þ, directly measurable on the lattice. This sum rule is useful also for estimating condensate errors in staggered lattice analysis. Keeping m u Þ 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 , while the connected part shows a quadratic behavior, infrared regular as T 2 =M 2 , and coming from 0 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.

Patterns and partners for chiral symmetry restoration

“…Consequently, its connected part χ con S would also peak at Oð4Þ × Uð1Þ A restoration. Actually, in the physical massive case, χ con S ¼ χ δ S =2 grows with T below the transition [5,37,40] and, because χ π P decreases like hqqi l [31], their degeneracy would give rise to a maximum for χ δ S near U A ð1Þ restoration. A hint of that behavior for χ con S is seen at around T ∼ 190 MeV in the lattice [4].…”

Pion masses and chiral scalar susceptibilities at finite temperature in SU(2) ChPT

“…being p the external pion three-momentum, Z = C/F 4 and Σ γEx ( p) the self energy contribution from the one-photon exchange on the mass shell [18,16]. The differences between considering the static limit ( p = 0) and values of the three-momentum compatible with the temperature range considered are tiny [16], so that we will restrict ourselves to this limit to define the pion masses.…”

“…The differences between considering the static limit ( p = 0) and values of the three-momentum compatible with the temperature range considered are tiny [16], so that we will restrict ourselves to this limit to define the pion masses.…”

“…The modifications to the expression of Das et al for non-zero quark masses and temperatures are discussed in [16].…”

“…Since non-zero quark masses invalidate by definition the use of the soft pion limit leading to such connection, the analysis for non-zero masses and finite temperature is pertinent and the role of resonances can be established by the use of effective models. The detailed analysis, results and discussion will be the subject of a forthcoming work [16].…”