Exploring the Aoki regime

TLDR: In this paper, next-to-leading order (NLO) corrections to the phase boundary between the Aoki phase and the Sharpe-Singleton scenario in the thermodynamical limit were derived in the ϵ-regime of Wilson (WChPT).

Abstract: We compute next-to-leading order (NLO) corrections in the ϵ-regime of Wilson (WChPT) and Staggered Chiral Perturbation Theory (SChPT). A difference between the two is that in WChPT already at NLO, that is at $ \mathcal{O}\left( {{\epsilon^2}} \right) $ , new low energy constants (LECs) contribute, whereas in SChPT they only enter at $ \mathcal{O}\left( {{\epsilon^4}} \right) $ . We first determine the NLO corrections in WChPT for SU(2), and for U(N f ) at fixed index. This implies corrections to the phase boundary between the Aoki phase and the Sharpe-Singleton scenario in the thermodynamical limit via corrections to the mean field potential. We also compute NLO corrections to the two-point function in the scalar and pseudo-scalar sector in WChPT. Turning to SChPT we determine the NLO corrections to the LECs and their effect on the taste splitting. Here the NLO partition function can be written as the leading order one with renormalized couplings, thus preserving the equivalence to staggered chiral random matrix theory at NLO for any number of flavors N f . In WChPT this relation only appears to hold for SU(2).