A mixed-background counterterm shifts the local representative

Let $A$ be a smooth compactly supported anti-Hermitian $\mathfrak{s}\mathfrak{u}(3)$ connection one-form on trivial ${\mathbb{R}}^{1,3}$, with curvature $F_A=dA+A\wedge A$, and let $B$ be a smooth real $U(1{)}_V$ background one-form with compactly supported gauge parameter $\alpha$. In natural units, write $A=A_{\mu }d{x}^{\mu }$, $B=B_{\mu }d{x}^{\mu }$, and $F_A=\frac{1}{2}(F_A{)}_{\mu \nu }d{x}^{\mu }\wedge d{x}^{\nu }$, summing repeated indices from $0$ to $3$. The components satisfy $[A_{\mu }]=[B_{\mu }]=\text{mass}$ and $[(F_A{)}_{\mu \nu }]={\text{mass}}^2$; since $[d{x}^{\mu }]={\text{mass}}^{-1}$, the forms $A,B,F_A$ are dimensionless, as are $\alpha$, $K_3(A)$, and $C_4$ below. Define the local three-form

$K_3(A):=\mathrm{tr}(A\wedge dA+\frac{2}{3}A\wedge A\wedge A)$

and add the four-dimensional local counterterm $C_4=\lambda B\wedge K_3(A)$ to the effective action, with dimensionless coefficient $\lambda$. Under the $U(1{)}_V$ background transformation $B\mapsto B+d\alpha$ with $A$ fixed, its local variation is

${\delta }_{\alpha }{C}_4=\lambda d(\alpha K_3(A))-\lambda \alpha \mathrm{tr}(F_A\wedge F_A)\mathrm{.}$

The first term integrates to a boundary term and vanishes for compactly supported $\alpha$. Thus this counterterm shifts the $U(1{)}_V$ component of the local anomaly representative by $-\lambda \alpha \mathrm{tr}(F_A\wedge F_A)$ while the full relative class is unchanged modulo the admitted counterterm space. This calculation does not assert that the full $SU(3)\times U(1{)}_V$ anomaly vanishes or that the $SU(3)$ component of the representative is unchanged.

Bilal, §§8.3 and 9.2–9.4

Alvarez, §§2.1–2.4