Weinberg sum rules at finite temperature

TL;DR

This paper investigates the validity of Weinberg sum rules at finite temperature using QCD sum rule results, finding they hold up to near the critical temperature where deconfinement and chiral symmetry restoration occur.

Contribution

It provides a detailed analysis of the thermal behavior of hadronic parameters and the validity of Weinberg sum rules near the QCD phase transition.

Findings

Sum rules are well satisfied up to 0.7-0.8 T_c.

Near T_c, hadronic widths diverge indicating deconfinement.

Sum rules are trivially satisfied at T_c due to chiral symmetry restoration.

Abstract

The saturation of the two Weinberg sum rules is studied at finite temperature, using recent independent QCD sum rule results for the thermal behaviour of hadronic parameters in the vector and axial-vector channels. Both sum rules are very well satisfied from $T=0$ up to $T/T_c \simeq 0.7-0.8$. At higher temperatures close to $T_c$ a hadronic, pion-loop contribution in the space-like region proportional to $T^2$, present at leading order in the vector but not in the axial-vector channel, induces an asymmetry leading to a small deviation. In this region, though, QCD sum rules for the hadronic parameters begin to have no solutions, as the hadronic widths of the $\rho$ and the $a_1$ mesons diverge signalling deconfinement. Close to, and at $T=T_c$ there are no pions left in the medium and chiral symmetry is restored, so that the sum rules are trivially satisfied.