New term in effective field theory at fixed topology

A random matrix model for lattice QCD which takes into account the positive definite nature of the Wilson term is introduced. The corresponding effective theory is derived to next to leading order. It reveals a new term proportional to the topological index of the Wilson Dirac operator and the lattice spacing. We conjecture that this term is the first in a new family of such terms. Within the effective field theory we derive the universal parametric correlation relevant for the axial anomaly and show that, the contribution from the measure to the axial anomaly, is completely dominated by the would be zero modes of the Wilson Dirac operator. This effective theory also naturally explains the effect of order $a$-improvement observed in the spectrum of the Wilson Dirac operator of lattice QCD simulations.