Audit a chiral spectrum for gauge consistency

Use the left-handed anomaly-polynomial sign and normalized Abelian charge convention of the preceding lesson. Replace a right-handed Weyl fermion in representation R and charge q by its left-handed charge conjugate in the conjugate representation and charge minus q. Then sum all left-handed contributions. Keeping a right-handed field with its original charge and adding rather than subtracting its cubic contribution would falsely make a vectorlike Dirac pair anomalous. Charge conjugation changes the representation as well as the handedness label.

Multiplicity matters. An Abelian anomaly sum counts every independent component of a multiplet under the other gauge factors, including color and weak-isospin dimension. A mixed non-Abelian-squared–Abelian sum instead contains the quadratic index of the non-Abelian representation and dimensions of any remaining factors. Distinguish this index from representation dimension. For the fundamental of SU(N), the convention $\operatorname{tr}(T^aT^b)=\delta^{ab}/2$ gives index one half, not N.

For a simple non-Abelian gauge factor, the local four-dimensional anomaly is proportional to the symmetrized cubic trace. The sum over all left-handed representations must vanish. Real or pseudoreal representations have no such cubic local anomaly, but that statement does not eliminate possible global anomalies. For product groups, mixed terms impose further conditions. A single non-Abelian generator has zero trace in ordinary irreducible representations of a simple group, explaining why some apparent mixed triangles vanish automatically while those with two non-Abelian vertices need explicit checking.

The Standard Model provides a useful multiplicity diagnostic. In one generation, normalized hypercharge $y=6Y$ gives left-handed multiplets with multiplicities and charges $(6,1),(3,-4),(3,2),(2,-3),(1,6)$. The negative charge of the conjugated right-handed up quark is particularly easy to miss. Both their linear and cubic charge sums vanish, as do the independent color-squared–hypercharge and weak-squared–hypercharge coefficients. This cancellation links quark and lepton contributions rather than occurring separately for every individual species.

A vectorlike pair cancels its own local anomaly because the conjugate representation has the opposite cubic coefficient. A genuinely chiral anomaly-free spectrum can also exist without such pairs, so anomaly cancellation does not require every fermion to admit an invariant mass. Conversely, adding arbitrary scalar fields does not cancel a Weyl triangle contribution. Additional transforming fields and Wess–Zumino or Green–Schwarz terms can participate in cancellation in a larger effective theory, but those structures must be specified rather than silently assumed.

Passing the local tests is necessary for an ordinary stand-alone gauge theory, but not a proof of every global or dynamical property. Large gauge transformations can reveal a further sign obstruction, and the global form of the gauge group and allowed spacetime structures affect the complete audit. Nor does cancellation establish confinement, a mass gap, or phenomenological viability. It establishes the absence of the tested quantum gauge obstruction, leaving those other questions to additional analysis.