Heavy chiral U(1)_V threshold model data

Use natural units $c=\mathrm{\hslash }=1$ on the four-dimensional affine Minkowski space $M={\mathbb{R}}^{1,3}$ with metric $\eta =\mathrm{diag}(-1,1,1,1)$ and orientation $d{x}^0\wedge d{x}^1\wedge d{x}^2\wedge d{x}^3$, as in the selected HEFT spacetime convention. Use the trivial spin structure, spinor bundle $S=M\times {\mathbb{C}}^4$, gamma matrices satisfying $\{{\gamma }^{\mu },{\gamma }^{\nu }\}=2{\eta }^{\mu \nu }I$, chirality operator ${\gamma }_5=-i{\gamma }^0{\gamma }^1{\gamma }^2{\gamma }^3=i{\gamma }_0{\gamma }_1{\gamma }_2{\gamma }_3$, and eigensubbundles $S_{\pm }=\ker ({\gamma }_5\mp I)$. Here ${\gamma }_0=-{\gamma }^0$, ${\gamma }_j={\gamma }^j$ for spatial $j$, and $\bar{\psi }={\psi }^{†}i{\gamma }^0$; choose $(i{\gamma }^0{)}^{†}=i{\gamma }^0$ and $({\gamma }^j{)}^{†}={\gamma }^j$. This matches the shared anomaly-matching data and Bilal's mostly-plus positive-chirality convention in Eq. (2.20), which fixes the sign of his anomaly formula. Take the trivial bundle for the fixed global symmetry group $H=SU(3)\times U(1{)}_V$, set the $SU(3)$ background connection to zero, and keep only the nondynamical $U(1{)}_V$ connection $B\in {\mathrm{\Omega }}_c^1(M,\mathbb{R})$ with curvature $G=dB$. Here $U(1{)}_V=\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }=1\}$ and its charge-$q$ representation is ${\rho }_q(z)=z^q$ on $\mathbb{C}$. The background $B$ is not integrated over as a quantum gauge redundancy.

Fix the complex exterior algebra $\mathrm{\Lambda }={\bigwedge }_{\mathbb{C}}{\mathrm{span}}_{\mathbb{C}}\{{\xi }_a,{\bar{\xi }}_a:a\in \mathbb{N}\}$, with algebraic finite sums, and let ${\mathrm{\Lambda }}_{\mathrm{o}\mathrm{d}\mathrm{d}}$ be its odd-degree subspace. Conjugation is the antilinear involution exchanging ${\xi }_a$ and ${\bar{\xi }}_a$ and reversing product order. Let ${\psi }_{+}\in C_c^{\mathrm{\infty }}(M,S_{+}\otimes {\mathbb{C}}_{(\mathbf{1},+1)}){\otimes }_{\mathbb{C}}{\mathrm{\Lambda }}_{\mathrm{o}\mathrm{d}\mathrm{d}}$ and ${\psi }_{-}\in C_c^{\mathrm{\infty }}(M,S_{-}\otimes {\mathbb{C}}_{(\mathbf{1},0)}){\otimes }_{\mathbb{C}}{\mathrm{\Lambda }}_{\mathrm{o}\mathrm{d}\mathrm{d}}$ be Weyl fields: finite sums of smooth compactly supported sections with Grassmann-odd coefficients in the indicated representations. Odd coefficients anticommute; the Dirac adjoint uses this conjugation and the spinor adjoint. This implements the shared anomaly-matching data's fermionic statistics for both the charged field and its neutral mass partner; the determinant and measure below remain formal perturbative inputs, not integrals over a fixed finite set of generators. Use $D_q=d+iqB$ on charge-$q$ fields and the fixed Lorentz-scalar chiral Dirac pairing ${\bar{\psi }}_{+}{\psi }_{-}$. Let $\mathrm{\Phi }=v{e}^{i\theta }$ be a nonvanishing order-parameter field in $(\mathbf{1},+1)$ with fixed modulus $v>0$ and smooth real phase $\theta \in C_c^{\mathrm{\infty }}(M,\mathbb{R})$. A dimensionless nonzero Yukawa parameter $y$ gives the mass term $-y\mathrm{\Phi }{\bar{\psi }}_{+}{\psi }_{-}+\mathrm{h}\mathrm{.}\mathrm{c}\mathrm{.}$, where h.c. denotes Hermitian conjugation, and heavy mass $M_H=\mathrm{\mid }y\mathrm{\mid }v>0$. The fermion fields have mass dimension $3\mathrm{/}2$, $v$ and $M_H$ have mass dimension one, $B_{\mu }$ has mass dimension one, $G_{\mu \nu }$ has mass dimension two, and $\theta$ is dimensionless.

For $\alpha \in C_c^{\mathrm{\infty }}(M,\mathbb{R})$, use $B\mapsto B+d\alpha$, ${\psi }_{+}\mapsto e^{-i\alpha }{\psi }_{+}$, ${\psi }_{-}\mapsto {\psi }_{-}$, and $\mathrm{\Phi }\mapsto e^{-i\alpha }\mathrm{\Phi }$, hence $\theta \mapsto \theta -\alpha$. These transformations leave the Yukawa term invariant. In the anti-Hermitian product-group convention, the selected singlet has connection $iB$ and covariant derivative $d+iB$; in Bilal's Hermitian convention write $\mathcal{A}=-B$ and $ϵ=-\alpha$, so $d-i\mathcal{A}=d+iB$ and the transformations agree. Let ${\mathrm{\Gamma }}_H[B,\theta ]$ denote the formal one-loop effective-action contribution of this Yukawa-massive pair, whose fermion mass is $M_H=\mathrm{\mid }y\mathrm{\mid }v>0$. For determining its local consistent-anomaly coefficient, separately use the Pauli–Villars-regulated one-loop triangle with one gauge-current insertion and two $U(1{)}_V$ background-connection insertions for a massless positive-chirality Weyl field in $(\mathbf{1},+1)$. In this triangle calculation only, the physical loop has $M_0=0$ and weight ${\eta }_0=1$, while the three regulator loops have weights ${\eta }_1={\eta }_3=-1$ and ${\eta }_2=1$, a positive scale $M_{\mathrm{P}\mathrm{V}}$, and masses $M_1=M_3=M_{\mathrm{P}\mathrm{V}}$, $M_2=\sqrt{2}{M}_{\mathrm{P}\mathrm{V}}$, with $M_{\mathrm{P}\mathrm{V}}\to \mathrm{\infty }$ after subtraction. The $M_0=0$ physical loop belongs only to this massless triangle calculation; it is not the physical loop of ${\mathrm{\Gamma }}_H$, whose mass remains $M_H>0$. Let ${\mathcal{A}}_{\mathrm{t}\mathrm{r}\mathrm{i},0}(\alpha ,B)$ denote the local consistent-anomaly variation computed from this massless triangle. It fixes the massless chiral representation's coefficient and is not the variation of the massive pair or a definition of its full determinant. Define the consistent anomaly as an infinitesimal effective-action variation, rather than as a covariant-current divergence. Add no non-invariant local counterterm to ${\mathrm{\Gamma }}_H$; gauge-invariant local counterterms may be added. For an undetermined real coefficient $c$, the candidate Wess–Zumino functional is ${\mathrm{\Gamma }}_c[B,\theta ]=c{\int }_M\theta G\wedge G$; the theorem on this page fixes $c$.

After a constant rephasing of the neutral field absorbs the phase of $y$, the phase-dependent mass term is $-M_H{e}^{i\theta }{\bar{\psi }}_{+}{\psi }_{-}+\mathrm{h}\mathrm{.}\mathrm{c}\mathrm{.}$. The pointwise invertible change of variables ${\psi }_{+}^{\mathrm{\prime }}=e^{-i\theta }{\psi }_{+}$, ${\bar{\psi }}_{+}^{\mathrm{\prime }}=e^{i\theta }{\bar{\psi }}_{+}$, and ${\psi }_{-}^{\mathrm{\prime }}={\psi }_{-}$ removes $\theta$ from this mass term and changes the charged kinetic connection to $B_{\theta }:=B+d\theta$. Both ${\psi }_{+}^{\mathrm{\prime }}$ and $B_{\theta }$ are invariant under the simultaneous background transformation.

Use the Minkowski effective-action convention $Z_H[B,\theta ]=\exp (i{\mathrm{\Gamma }}_H[B,\theta ])$, as a formal perturbative generating-functional notation. Let ${\mathrm{\Gamma }}_R^M[B]$ denote the massless charged species' effective action in the selected consistent representative, with ${\delta }_{\alpha }{\mathrm{\Gamma }}_R^M={\mathcal{A}}_{\mathrm{t}\mathrm{r}\mathrm{i},0}(\alpha ,B)$. For compactly supported real $\theta$, define the local finite difference along the connected Abelian orbit by ${\mathrm{\Delta }}_M[B,\theta ]:={\mathrm{\Gamma }}_R^M[B+d\theta ]-{\mathrm{\Gamma }}_R^M[B]={\int }_0^1{\mathcal{A}}_{\mathrm{t}\mathrm{r}\mathrm{i},0}(\theta ,B+td\theta )dt\mathrm{.}$ This is a local perturbative difference, not a chosen global determinant phase. Adopt the scope-limited source-backed change-of-variables premise that the regulated old measure equals $J_M[B,\theta ]$ times the new measure, with $J_M=\exp (-i{\mathrm{\Delta }}_M)$ in this Minkowski convention. Its contribution to the effective action is therefore ${\mathrm{\Gamma }}_J=(1\mathrm{/}i)\log J_M=-{\mathrm{\Delta }}_M$, on the perturbative logarithm branch vanishing at $\theta =0$. The transformed massive determinant is a functional ${\mathcal{F}}_M[B_{\theta },M_H]$ of the invariant connection and constant mass, so the adopted local threshold relation is ${\mathrm{\Gamma }}_H[B,\theta ]={\mathcal{F}}_M[B_{\theta },M_H]-{\mathrm{\Delta }}_M[B,\theta ],{\delta }_{\alpha }{\mathcal{F}}_M[B_{\theta },M_H]=0.$ This source input concerns the anomalous local variation and its independence of $M_H$, not a construction or local expansion of the entire determinant. To compare signs with Preskill's Euclidean convention, continue as ${\mathrm{\Gamma }}_E=-i{\mathrm{\Gamma }}_M$ with oriented differential forms continued as in Bilal §10.1, and put ${\mathrm{\Delta }}_E=-i{\mathrm{\Delta }}_M$. Preskill's Eqs. (6.34), (6.43)–(6.45) then read $J_E=\exp ({\mathrm{\Delta }}_E)$ and ${\mathrm{\Gamma }}_{J,E}=-\log J_E=-{\mathrm{\Delta }}_E$. Neither $J_E$ nor $\log J_E$ is the Minkowski effective-action term. Since $B_{\theta }$ is invariant, ${\delta }_{\alpha }{\mathrm{\Delta }}_M=-{\mathcal{A}}_{\mathrm{t}\mathrm{r}\mathrm{i},0}(\alpha ,B)$, whereas ${\delta }_{\alpha }{\mathrm{\Gamma }}_J=+{\mathcal{A}}_{\mathrm{t}\mathrm{r}\mathrm{i},0}(\alpha ,B)$. These opposite signs distinguish the finite difference from its effective-action contribution. No global or large-transformation claim about the Jacobian is made.

For low-energy use, a specified $n$-point background-field vertex of ${\mathrm{\Gamma }}_H$, with $n\ge 1$, is the coefficient multilinear in $n$ components chosen from $B_{\mu }$ and $\theta$ in the formal expansion about $B=0$, $\theta =0$, after Fourier-transforming each argument and removing the overall momentum-conservation delta. Its incoming real covectors $p_j=(p_{j0},p_{j1},p_{j2},p_{j3})\in {\mathbb{R}}^{1,3}$ satisfy ${\sum }_{j=1}^n{p}_j=0$; no on-shell condition is imposed. For that specified vertex set $E={\max }_{1\le j\le n,0\le \mu \le 3}\mathrm{\mid }{p}_{j\mu }\mathrm{\mid }$. The threshold condition $E\mathrm{/}{M}_H\ll 1$ concerns the finite set of external legs of that vertex, not the Fourier support of $B,\theta \in C_c^{\mathrm{\infty }}(M)$, and supplies no Lorentz-invariant norm or quantitative expansion-remainder bound. The exact local anomaly variation below is independent of $E$. All determinant and anomaly data here are formal perturbative inputs; no exact functional integral or continuum limit is claimed.

Adopted Minkowski spacetime and mostly-plus metric convention

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

Bilal, full Lectures on Anomalies PDF

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