Pseudo-Scalar $$\mathbf{q}\bar{\mathbf{q}}$$ q q ¯ Bound States at Finite Temperatures Within a Dyson-Schwinger–Bethe-Salpeter Approach

Abstract

The combined Dyson-Schwinger--Bethe-Salpeter equations are employed at non-zero temperature. The truncations refer to a rainbow-ladder approximation augmented with an interaction kernel which facilitates a special temperature dependence. At low temperatures, $T \to 0$, we recover a quark propagator from the Dyson-Schwinger (gap) equation which delivers, e.g. mass functions $B$, quark renormalization wave function $A$, and two-quark condensate $\la q \bar q \ra$ smoothly interpolating to the $T = 0$ results, despite the broken O(4) symmetry in the heat bath and discrete Matsubara frequencies. Besides the Matsubara frequency difference entering the interaction kernel, often a Debye screening mass term is introduced when extending the $T = 0$ kernel to non-zero temperatures. At larger temperatures, however, we are forced to drop this Debye mass in the infra-red part of the longitudinal interaction kernel to keep the melting of the two-quark condensate in a range consistent with lattice QCD results. Utilizing that quark propagator for the first few hundred fermion Matsubara frequencies we evaluate the Bethe-Salpeter vertex function in the pseudo-scalar $ q \bar q$ channel for the lowest boson Matsubara frequencies and find a competition of $ q \bar q$ bound states and quasi-free two-quark states at $T = {\cal O}$ (100 MeV). This indication of pseudo-scalar meson dissociation below the anticipated QCD deconfinement temperature calls for an improvement of the approach, which is based on an interaction adjusted to the meson spectrum at $T = 0$.