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

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, $$T \rightarrow 0$$ , 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 $$\langle q \bar{q} \rangle $$ 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 $$T = \mathcal{O}$$ (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 kernel adjusted to the meson spectrum at zero temperature.