Track theta phases through topology and mass redefinitions

Use a Hermitian SU(N) connection with $D=d-iA$, $F=dA-iA\wedge A$, and fundamental trace $\operatorname{tr}(T^aT^b)=\delta^{ab}/2$. The coupling is absorbed into A; it would reappear explicitly after a canonical-field rescaling. For smooth ordinary SU(N) bundles on closed oriented four-manifolds, or finite-action fields on Euclidean space with suitable compactification, the normalized instanton number k is integer. Quotient gauge groups and additional background twists can change the allowed fractional sectors, so that qualification matters.

The Euclidean action includes $-i\theta k$, giving a weight $e^{i\theta k}$ in the path integral. Integer k makes theta periodic modulo $2\pi$ in this specified setting. The topological density is locally a total derivative, but the potential used to write its primitive need not be globally defined across sectors. Dropping the term by a local integration-by-parts argument therefore loses physical information. A semiclassical instanton expansion is one controlled regime for studying sector weights, not a universally reliable computation at strong coupling.

Write the quark mass term as $-\bar q_LMq_R+\mathrm{h.c.}$ with nonsingular complex mass matrix M. Use the change-of-variables convention in which old fields equal $e^{i\alpha\gamma_5}$ times new fields. The transformed mass matrix acquires phase $e^{2i\alpha}$, while the axial measure shifts theta by minus $2N_f\alpha$ for fundamental Dirac quarks. These shifts cancel in $\bar\theta=\theta+\arg\det M$. The sign declaration is necessary because reversing the direction of the field redefinition reverses both parameter shifts.

Thus setting theta to zero by a convenient field basis does not generally remove strong CP violation: a mass-determinant phase appears instead. CP reverses topological charge and sends the invariant angle to its negative. Angles zero and pi are special modulo periodicity, but CP can be spontaneously broken at pi in some theories, so periodicity alone is not a vacuum-uniqueness theorem. The strong-CP problem asks why the physical angle is so small near the observed CP-preserving regime despite no generic requirement that two independent bare phases cancel.

An exactly massless quark would make the associated axial redefinition able to remove theta without leaving a phase in that zero mass, but this is a conditional theoretical mechanism, not an assertion about the measured quark spectrum. The Peccei–Quinn mechanism instead introduces a dynamical axion whose anomalous coupling makes the effective angle a field. When the relevant QCD potential dominates and additional explicit breaking is sufficiently controlled, its minimum relaxes the effective angle. Arbitrary axion-like fields do not automatically solve the same problem.

Topological susceptibility measures the curvature of vacuum free energy with respect to theta near zero. It depends on the sector weights and hence on dynamics, masses, temperature, and volume. A small finite sum can illustrate interference and derivatives exactly, but cannot claim a universal QCD cosine potential or a strong-coupling instanton rate. Keeping that boundary between algebra and dynamics makes the theta-sector formulation useful without overstating what topology alone predicts.