Heavy-chiral threshold Wess–Zumino anomaly match

On natural-unit Minkowski space $M={\mathbb{R}}^{1,3}$ with $\eta =\mathrm{diag}(-1,1,1,1)$ and orientation $d{x}^0\wedge d{x}^1\wedge d{x}^2\wedge d{x}^3$, take the trivial $H=SU(3)\times U(1{)}_V$ bundle, zero $SU(3)$ background, and a smooth compactly supported real nondynamical $U(1{)}_V$ connection $B$ with $G=dB$. Let $\alpha ,\theta \in C_c^{\mathrm{\infty }}(M,\mathbb{R})$. Use the model's source-signed chirality convention ${\gamma }_5=-i{\gamma }^0{\gamma }^1{\gamma }^2{\gamma }^3=i{\gamma }_0{\gamma }_1{\gamma }_2{\gamma }_3$, with positive chirality meaning eigenvalue $+1$. Use the model's Grassmann-odd coefficient algebra and conjugation for both Weyl fields: one smooth compactly supported positive-chirality Weyl field in $(\mathbf{1},+1)$, one smooth compactly supported neutral negative-chirality mass partner, and the charge-one scalar $\mathrm{\Phi }=v{e}^{i\theta }$ in the Yukawa term $-y\mathrm{\Phi }{\bar{\psi }}_{+}{\psi }_{-}+\mathrm{h}\mathrm{.}\mathrm{c}\mathrm{.}$, where $v>0$, $y\mathrm{\ne }0$, and $M_H=\mathrm{\mid }y\mathrm{\mid }v$. Fix the transformations $B\mapsto B+d\alpha$, ${\psi }_{+}\mapsto e^{-i\alpha }{\psi }_{+}$, ${\psi }_{-}\mapsto {\psi }_{-}$, and $\theta \mapsto \theta -\alpha$, and use the consistent-anomaly representative with no non-invariant local counterterm. For any fixed integer $n\ge 1$, consider a specified $n$-point background-field vertex of ${\mathrm{\Gamma }}_H$ with insertions from components of $B$ and/or $\theta$, and incoming real coordinate covectors $p_j=(p_{j0},p_{j1},p_{j2},p_{j3})\in {\mathbb{R}}^{1,3}$ satisfying ${\sum }_{j=1}^n{p}_j=0$; set $E={\max }_{1\le j\le n,0\le \mu \le 3}\mathrm{\mid }{p}_{j\mu }\mathrm{\mid }$. No on-shell condition is imposed. Require $E\mathrm{/}{M}_H\ll 1$ only for the low-energy effective-field-theory interpretation of that vertex. This condition applies to its finite set of external legs, not to the Fourier support of the compactly supported backgrounds, and gives no expansion-remainder bound; the exact local anomaly identity below is independent of it.

Adopt the source-derived consistent-anomaly formula for a positive-chirality Weyl field in representation $R$, written in Bilal's Hermitian convention,

${\delta }_{ϵ}{\mathrm{\Gamma }}_R=-\frac{1}{24{\pi }^2}{\int }_M{\mathrm{tr}}_R\text{\hspace{0.17em}⁣}\left(ϵd(\mathcal{A}\wedge d\mathcal{A}-\frac{i}{2}\mathcal{A}\wedge \mathcal{A}\wedge \mathcal{A})\right)\mathrm{.}$

For the selected $U(1{)}_V$ representation, set $\mathcal{A}=-B$ and $ϵ=-\alpha$, so $d-i\mathcal{A}=d+iB$ and the positive-chirality field transforms as specified above. Since $R=(\mathbf{1},+1)$ is one-dimensional, ${\mathrm{tr}}_R(1)=1$; since $\mathcal{A}$ is Abelian, its cubic wedge term vanishes. The massless chiral-triangle calculation gives ${\mathcal{A}}_{\mathrm{t}\mathrm{r}\mathrm{i},0}(\alpha ,B)=+\frac{1}{24{\pi }^2}{\int }_M\alpha G\wedge G$. Use $Z_H=\exp (i{\mathrm{\Gamma }}_H)$ and the model data's local field-redefinition relation ${\mathrm{\Gamma }}_H={\mathcal{F}}_M[B+d\theta ,M_H]-{\mathrm{\Delta }}_M[B,\theta ]$, where ${\mathcal{F}}_M$ is invariant and ${\mathrm{\Delta }}_M={\mathrm{\Gamma }}_R^M[B+d\theta ]-{\mathrm{\Gamma }}_R^M[B]$. Thus ${\delta }_{\alpha }{\mathrm{\Delta }}_M=-{\mathcal{A}}_{\mathrm{t}\mathrm{r}\mathrm{i},0}$ and the effective-action contribution $-{\mathrm{\Delta }}_M$ transfers the positive local variation to the Yukawa-massive sector independently of $M_H$, so its local variation is ${\mathcal{A}}_H(\alpha ,B)=+\frac{1}{24{\pi }^2}{\int }_M\alpha G\wedge G$. If ${\mathrm{\Gamma }}_L$ is any specified remaining-light-sector functional with infinitesimal variation ${\mathcal{A}}_L$, and the low-energy theory retains $\theta$ and is required to match this perturbative background anomaly, then

${\mathrm{\Gamma }}_{\mathrm{W}\mathrm{Z}}[B,\theta ]=-\frac{1}{24{\pi }^2}{\int }_M\theta G\wedge G$

satisfies ${\delta }_{\alpha }{\mathrm{\Gamma }}_{\mathrm{W}\mathrm{Z}}={\mathcal{A}}_H$ and hence ${\delta }_{\alpha }({\mathrm{\Gamma }}_L+{\mathrm{\Gamma }}_{\mathrm{W}\mathrm{Z}})={\mathcal{A}}_L+{\mathcal{A}}_H={\delta }_{\alpha }({\mathrm{\Gamma }}_L+{\mathrm{\Gamma }}_H)$. Its coefficient has no $M_H^{-1}$ factor. This is an exact local perturbative anomaly identity in the adopted formal model and fixed representative; the low-energy interpretation is restricted to the specified finite-leg vertex above. It asserts no global determinant phase, dynamical-gauge consistency, gravitational contribution, non-Abelian generalization, quantitative EFT remainder, or continuum interacting QFT.

Selected SU(3)-singlet model, U(1)_V charge, mass, regulator, and threshold conventions

Local perturbative background-anomaly class modulo counterterms

Selected H=SU(3)×U(1)_V representation, regulator, and descent data

Bilal, full Lectures on Anomalies PDF

Preskill, full Gauge Anomalies in an Effective Field Theory lecture notes

The retained $\theta$ is the smooth transforming order-parameter phase specified in the model. The functional depends on $\theta$ as well as $B$, so ${\mathrm{\Gamma }}_{\mathrm{W}\mathrm{Z}}[B,\theta ]$ is not a background-only counterterm in the quotient of [F2]. Its variation reproduces the heavy-sector anomaly; this does not show that the background anomaly class vanishes.