Overlap-Dirac fermions with a small hopping parameter

We consider overlap-Dirac fermions at non-zero bare coupling and for a smallhopping parameter, or, equivalently, large $|M|$ with $M$ the domain-wallheight. We prove the existence of a phase at large positive $M$ where theabelian axial group $U_A(1)$ is a symmetry, and the corresponding pseudo-scalaris an exact Goldstone boson. We also provide a conjecture for the phase diagramof asymptotically free gauge theories with overlap-Dirac fermions. Inparticular, we suggest that, for large gauge coupling, the massive-fermionphase at negative $M$ possibly extends to all $M<4$.Comment: 14 pages, 1 figure, minor change