Preserve the vector current and expose the axial anomaly

Work in four-dimensional Minkowski spacetime with $\hbar=c=1$ and $D_\mu=\partial_\mu-iqA_\mu$. The dimensionless coupling q is the fermion's electromagnetic charge in this normalization. The vector number current is $j_V^\mu=\bar\psi\gamma^\mu\psi$, and the electromagnetic current is $qj_V^\mu$. Define axial charge as right-handed number minus left-handed number. Choose the chirality and field orientation convention in which parallel positive electric and magnetic fields produce positive axial charge. This fixes the sign used below.

For zero fermion mass, both currents are conserved by the classical equations. A nonzero mass already breaks axial symmetry explicitly, adding a pseudoscalar mass term to its divergence. That explicit breaking must be separated from the quantum anomaly. If electromagnetism is dynamical, its gauge Ward identity must remain exact so that redundant photon polarizations decouple consistently. An anomalous axial global current is compatible with that requirement; an uncanceled anomaly in the gauged vector current would not be.

The relevant three-point function contains one axial-current vertex and two vector-current vertices. Its superficially linearly divergent loop integral cannot be shifted as though it were absolutely convergent. Different routings, or equivalent local subtraction choices, redistribute local terms among the three Ward identities. Enforcing the two vector identities fixes this freedom and leaves the axial divergence. The anomaly is therefore not removed by a clever integration variable change; the proposed change must respect the regulator and the required gauge identities.

With the stated vector-preserving prescription, a single Dirac fermion contributes $q^2\mathbf E\cdot\mathbf B/(2\pi^2)$ to axial-number production. The square arises from the two vector couplings, while the axial current has no extra electromagnetic charge factor. A complementary derivation counts the lowest Landau-level degeneracy per transverse area and one-dimensional spectral flow along the field. Higher levels pair opposite contributions. The anomaly coefficient has a protected meaning in this relativistic theory, but does not by itself determine a material's steady transport current or relaxation rate.

Integrating a local divergence includes flux through the spatial boundary. A measured change in axial population can therefore differ from the spacetime integral of the anomaly source when axial charge escapes, when a mass term matters, or when interactions relax the population. Parallel fields give a nonzero source, perpendicular fields give zero dot product, and reversing one field reverses the source. Reversing the sign of q alone does not reverse axial-number production because its coefficient contains the square.

One can formally subtract a gauge-potential-dependent Chern–Simons current to write a conserved-looking axial expression. The price is loss of gauge invariance, so this does not restore a gauge-invariant conserved axial charge. Similarly, assigning all triangle vertices an identical anomalous Ward identity changes the current prescription rather than contradicting the vector-preserving result. Every comparison should identify whether it concerns vector, axial, consistent chiral, or covariant chiral current before comparing numerical coefficients.