Time-reversal and parity conservation for gravitating quarks

The complex mass term of a quark does not violate time-reversal or parity in gravitational interactions, in spite of an axial anomaly. Time-reversal is known to be violated in the weak interactions. Mass terms q̄LMqR + ̄̃qLM̃q̃R+hc are generated for quarks q having charge + 2 3 and quarks q̃ having charge − 1 3 in the shape of complex matrices M, M̃ from electroweak symmetry breaking. On diagonalization of M, M̃ through flavour matrices, qL → A L qL, qR → A −1 R qR (1) q̃L → Ã L q̃L, q̃R → Ã −1 R q̃R, (2) charged current weak interactions pick up a matrix ALÃ −1 L ≡ C, the CabibboKobayashi-Maskawa matrix, which is complex and violates time-reversal. The diagonalized mass terms too may be complex. One can write ψ̄Lme ψR + hc = ψ̄me 5ψ = cos θψ̄mψ + i sin θψ̄mγ5ψ. (3) This looks like a combination of a scalar and a pseudoscalar, suggesting parity violation. More important, the phase factor e → e under antilinear operations, suggesting additional time-reversal violation in the standard model beyond the weak interaction violation mentioned above. A chiral transformation may be used to remove θ: ψ → e5ψ, ψ̄ → ψ̄e5. (4) θ does get removed from the mass term, but may reappear as a topological term in the gluon sector [1] through the Jacobian of the fermion measure which is not invariant under the chiral transformation. This is because of the axial