Abstract
We derive the formal Ward identities relating pseudoscalar susceptibilities and quark condensates in three-flavor QCD, including consistently the $\eta$-$\eta'$ sector and the $U_A(1)$ anomaly. These identities are verified in the low-energy realization provided by ChPT, both in the standard $SU(3)$ framework for the octet case and combining the use of the $U(3)$ framework and the large-$N_c$ expansion of QCD to account properly for the nonet sector and anomalous contributions. The analysis is performed including finite temperature corrections as well as the calculation of $U(3)$ quark condensates and all pseudoscalar susceptibilities, which together with the full set of Ward identities, are new results of this work. Finally, the Ward identities are used to derive scaling relations for pseudoscalar masses, which explain the behavior with temperature of lattice screening masses near chiral symmetry restoration.
Highlights
Properties have been studied in [14] and unitarized interactions have allowed to describe properly thermal resonances and transport coefficients [15]
In a recent analysis [16], we have shown that an operator Ward identity between the pseudoscalar susceptibility and the quark condensate allows to understand the scaling of lattice screening masses near Tc
We will see that the Ward identities between pseudoscalar susceptibilities and quark condensates allow to understand the behavior with temperature of lattice screening masses in the pion, kaon and ss channels, connecting it with chiral symmetry restoration
Summary
We will first proceed to the formal derivation of the relevant Ward identities from QCD. We start by writing the expected value of a local operator O(x1, · · · , xn) from the QCD generating functional as: O(x1, · · · , xn) = Z−1 [dG][dψ][dψ]O(x1, · · · , xn)eSQCD,. Where Gaμ, ψ are gluon and quark fields respectively, Z = [dG][dψ][dψ]eSQCD is the partition function and SQCD = i d4xLQCD in Minkowski space-time, where the fermion QCD. To the order we are considering here, all temperature corrections will show up through the T -correction to the tadpole function coming from the finite part of the meson propagators at equal space-time points (we follow the same dimensional regularization scheme as in [11]): μi(T ).
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