Understanding CP violation in lattice QCD

It is pointed out that any CP violation which may be found in lattice QCD with a chiral phase in the fermion mass term cannot be relevant for the continuum theory. CP is classically conserved in the corresponding continuum theory and is non-anomalous. A symmetry in a classical field theory is said to develop an anomaly in the quantized theory if it is not possible to find any regularization to preserve it. It is, of course, not difficult to find regularizations which violate a given symmetry. To prove that a classical symmetry is preserved in the quantum theory, it is sufficient to find one regularization which preserves the symmetry, whereas to prove the existence of an anomaly, it would be necessary to show that all regularizations break it. This principle will be applied here to the question of parity or CP violation in QCD with reference to lattice regularization. The strong interactions per se do not violate parity, but the Lagrangian of QCD may contain a θtr FF̃ term which violates parity and CP. Further, the quark mass term ψ̄me5 ′ ψ has an unknown chiral phase θ which also appears to violate these. There is no experimental evidence of such violations, and it is difficult to estimate directly the effect of the gluonic θ term, but calculations and proposals are often made with θ. In what follows, we shall consider only this term and not the gluonic θ-term. While it used to be thought that the θ-term violates CP classically [1, 2], it was later realized [3] that a classically and perturbatively conserved parity can [email protected]