I'm trying to understand the $\theta$-dependence of the following expression for the quark condensate in QCD,
$$ \langle \bar{\psi}\psi\rangle = - \Sigma \cos(\theta)$$
taken from Eqs. (5) and (7) of the paper "Dirac spectrum and chiral condensate for QCD at fixed $\theta$-angle" by M. Kieburg et al. Here, $\theta$ is the QCD vacuum angle and $\Sigma$ is the absolute value of the chiral condensate in the limit $m= 0$ and $\theta = 0$.
The expression above implies that the quark condensate is independent of the bare quark mass $m$ but dependent on the vacuum angle $\theta$. However, for $m=0$, the $\theta$-angle can be rotated away by a chiral quark rotation. Thus, how is it possible that the expression above stays the same, as claimed in the said reference and also in Eq. (69) of the paper "Massive Schwinger model within mass perturbation theory" by C. Adam on the analogous Schwinger model? In other words, how is it possible that the condensate depends on the angle $\theta$, even though $\theta$ should become unphysical when $m=0$?
Edit, in response to the comments below: Physical observables in QCD should become independent of $\theta$ in the chiral limit. Thus, is $\langle \bar{\psi}\psi\rangle$ is not a physical quantity, but only $|\langle \bar{\psi}\psi\rangle|$ which enters the pion masses and is independent of $\theta$?