Chiral condensate and Dirac spectrum of one- and two-flavor QCD at nonzero $\theta$-angle

TL;DR

This paper derives exact formulas for the Dirac eigenvalue density in one- and two-flavor QCD at nonzero theta, revealing how spectral contributions ensure a continuous chiral condensate despite sign changes in fixed topological charge scenarios.

Contribution

It provides the first exact analytical expressions for the Dirac spectrum at nonzero theta in the epsilon domain of QCD, clarifying the role of zero modes in chiral condensate continuity.

Findings

Exact eigenvalue density expressions at nonzero theta for one and two flavors.

Zero modes are crucial for the continuity of the chiral condensate.

Spectral density contributions cancel divergences, ensuring physical consistency.

Abstract

In the $\epsilon$-domain of QCD we have obtained exact analytical expressions for the eigenvalue density of the Dirac operator at fixed $\theta \ne 0$ for both one and two flavors. These results made it possible to explain how the different contributions to the spectral density conspire to give a chiral condensate at fixed $\theta$ that does not change sign when the quark mass (or one of the quark masses for two flavors) crosses the imaginary axis, while the chiral condensate at fixed topological charge does change sign. From QCD at nonzero density we have learnt that the discontinuity of the chiral condensate may move to a different location when the spectral density increases exponentially with the volume with oscillations on the order of the inverse volume. This is indeed what happens when the product of the quark masses becomes negative, but the situation is more subtle in this case: the contribution of the "quenched" part of the spectral density diverges in the thermodynamic limit at nonzero $\theta$, but this divergence is canceled exactly by the contribution from the zero modes. We conclude that the zero modes are essential for the continuity of the chiral condensate and that their contribution has to be perfectly balanced against the contribution from the nonzero modes. Lattice simulations at nonzero $\theta$-angle can only be trusted if this is indeed the case.