  Using general properties of the $Q=0$ topological sector we previously argued
that a vector-like theory, with chiral $U(1)_A$ anomaly, and exact non-Abelian
chiral symmetry, should exhibit divergent susceptibilities in the chiral limit,
the two-flavor Schwinger model being a paradigmatic example of the realization
of this scenario. Two flavor $QCD$ at $T>T_c$ satisfies all the above
conditions, and it is also expected that the $U(1)_A$ axial symmetry remains
effectively broken in its high temperature phase. Therefore we would expect a
non-analyticity in the quark mass dependence of the free energy density, in
contrast with the Dilute Instanton Gas Approximation (DIGA) prediction. We
investigate in this work whether the aforementioned results can also be
reproduced making only use of standard properties of the spectral density of
the Dirac operator, without having to resort to general properties of the $Q=0$
topological sector. We show that the only way to derive a non-trivial
$\theta$-dependence, and an analytical free energy density in $QCD$ with two
degenerate flavors is that the spectral density, $\rho\left(\lambda,m\right)$,
of the absolute value of the non-zero modes of the Dirac-Ginsparg-Wilson
operator develops a $m^2\delta(\lambda)$ function in the thermodynamic limit.
This is the expected result in the DIGA, where interactions between instantons
in the dilute gas are fully neglected. However, at temperatures close to $T_c$
the interaction between instantons should become non-negligible, and the
splitting from zero of the near-zero modes, which has been neglected in the
DIGA, should be taken into account. Therefore we expect that the
$m^2\delta(\lambda)$ contribution to the spectral density is no longer correct
at these temperatures, and that the free energy density becomes a non-analytic
function of the quark mass.some clarifications added
