Consistent and Covariant Anomalies

The consistent and covariant currents are two local representatives of the same perturbative anomaly data, designed to answer different questions. The consistent current is obtained by differentiating one effective action. It therefore obeys functional integrability and its anomalous variation realizes the gauge algebra. The covariant current is obtained by a local Bardeen–Zumino improvement. It transforms homogeneously under gauge transformations, but in an anomalous theory it is generally not the functional derivative of any four-dimensional effective action.

Neither current is universally preferable. The consistent representative is the one relevant to gauging, source differentiation, and Wess–Zumino consistency. The covariant representative is useful when a locally covariant operator or response equation is required. Their anomaly coefficients can differ—even by a factor of three in the pure four-dimensional Abelian example—although their vanishing conditions agree. This page treats that local perturbative distinction on a closed spacetime; boundary and global data require additional input.

Required background. Perturbative Chiral and Gauge Anomalies supplies the all-left-handed convention, representation coefficients, local cancellation tests, and the warning that a covariantly regulated current is not yet an effective-action Ward identity.

One effective action defines the consistent current

Work first on a closed oriented Euclidean spin four-manifold $X$. Continue the site’s physical Lorentzian left-handed field to negative Euclidean chirality, as on the prerequisite page. Let $\mathcal A$ be a Hermitian matrix-valued source connection with all couplings and charges absorbed, and write

$$\mathcal F=\mathrm d\mathcal A-i\mathcal A^2, \qquad \delta_\Lambda\mathcal A=D\Lambda =\mathrm d\Lambda-i[\mathcal A,\Lambda].$$

All products of differential forms below are wedge products. Define the Euclidean effective action $W_E=-\log Z$ and its consistent current three-form $\mathcal J_{\mathrm{cons}}$ operationally by

$$\delta W_E =\int_X\operatorname{tr}_R \bigl(\delta\mathcal A\,\mathcal J_{\mathrm{cons}}\bigr).$$

In components, this is the familiar functional derivative

$$J_{\mathrm{cons}}^{\mu a}(x) =\frac{1}{\sqrt g}\frac{\delta W_E} {\delta\mathcal A_\mu^a(x)},$$

with the trace pairing understood. Because both derivatives act on the same functional, commuting variations give the integrability, or functional-curl, condition

$$\begin{aligned} \mathfrak C_{ab}^{\mu\nu}(x,y) &:=\frac{\delta\!\left[\sqrt{g(x)} J_{\mathrm{cons}}^{\mu a}(x)\right]} {\delta\mathcal A_\nu^b(y)} -\frac{\delta\!\left[\sqrt{g(y)} J_{\mathrm{cons}}^{\nu b}(y)\right]} {\delta\mathcal A_\mu^a(x)} \\ &=0. \end{aligned}$$

This is the functional form of Bose symmetry among current insertions. It is not an optional property of a current claimed to come from $W_E$. The equality is distributional: contact terms must be retained in the same renormalization scheme on both sides.

Fix the anomaly-form convention by defining the integrated anomaly functional

$$\mathfrak A_{\mathrm{cons}}(\Lambda) :=\delta_\Lambda W_E =-i\int_X\mathscr I_{4,\mathrm{cons}}(\Lambda).$$

On a closed $X$, covariant integration by parts then gives

$$\frac1i\int_X\operatorname{tr}_R \bigl(\Lambda D\mathcal J_{\mathrm{cons}}\bigr) =\int_X\mathscr I_{4,\mathrm{cons}}(\Lambda).$$

The commutator of two transformations closes with $\Lambda_{12}=-i[\Lambda_1,\Lambda_2]$. Applying this closure to the same functional $W_E$ yields

$$\begin{aligned} &\delta_{\Lambda_1}\mathfrak A_{\mathrm{cons}}(\Lambda_2) -\delta_{\Lambda_2}\mathfrak A_{\mathrm{cons}}(\Lambda_1) \\ &\hspace{7em} =\mathfrak A_{\mathrm{cons}} \bigl(-i[\Lambda_1,\Lambda_2]\bigr). \end{aligned}$$

This is the Wess–Zumino consistency condition in the present convention. It follows from effective-action integrability; it does not say that $J_{\mathrm{cons}}$ itself transforms covariantly. In an anomalous theory it generally does not. The current, functional-curl, and consistency chain is developed explicitly in Takeuchi and Endo 2017, § 2, article pp. 3–6, eqs. (2.8)–(2.27), especially (2.11)–(2.14), (2.23)–(2.25), Open PDF, while the gauge-algebra derivation of the consistency condition appears in Bilal 2008, § 9.1, arXiv v1, pp. 69–71, eqs. (9.2)–(9.9), Open PDF. The original condition is due to Wess and Zumino 1971, pp. 95–97. The full descent construction is deferred to the next page.

A local Bardeen–Zumino shift defines the covariant current

A different regularization can make the current transform homogeneously in the adjoint representation. Equivalently, start from the consistent current and add a local connection polynomial:

$$\mathcal J_{\mathrm{cov}} =\mathcal J_{\mathrm{cons}}+\mathcal J_{\mathrm{BZ}}, \qquad \delta_\Lambda\mathcal J_{\mathrm{cov}} =i[\Lambda,\mathcal J_{\mathrm{cov}}].$$

For the physical left-handed convention above, one convention-matched four-dimensional Bardeen–Zumino three-form is

$$\mathcal J_{\mathrm{BZ}} =-\frac{i}{24\pi^2} \left( \mathcal F\mathcal A +\mathcal A\mathcal F +\frac{i}{2}\mathcal A^3 \right).$$

Its covariant divergence changes the anomaly representative:

$$\frac1i\operatorname{tr}_R \bigl(\Lambda D\mathcal J_{\mathrm{BZ}}\bigr) =\mathscr I_{4,\mathrm{cov}}(\Lambda) -\mathscr I_{4,\mathrm{cons}}(\Lambda).$$

The consistent and covariant four-forms in this representative are

$$\begin{aligned} \mathscr I_{4,\mathrm{cons}}^L(\Lambda) &=-\frac{1}{24\pi^2} \operatorname{tr}_R\!\left[ \Lambda\,\mathrm d\!\left( \mathcal A\,\mathrm d\mathcal A -\frac{i}{2}\mathcal A^3 \right) \right] +\frac{p_1(TX)}{24}\operatorname{tr}_R\Lambda, \\[3pt] \mathscr I_{4,\mathrm{cov}}^L(\Lambda) &=-\frac{1}{8\pi^2} \operatorname{tr}_R(\Lambda\mathcal F^2) +\frac{p_1(TX)}{24}\operatorname{tr}_R\Lambda. \end{aligned}$$

These expressions use the standard representative that preserves diffeomorphism and local-Lorentz covariance and places the mixed violation in the gauge-current Ward identity. A local counterterm can redistribute that mixed violation, so the displayed placement is part of the convention.

The covariant anomaly transforms in the adjoint. If $\mathfrak A_{\mathrm{cov}}(\Lambda):=-i\int_X \mathscr I_{4,\mathrm{cov}}(\Lambda)$ and the test parameters are held fixed, then

$$\begin{aligned} &\delta_{\Lambda_1}\mathfrak A_{\mathrm{cov}}(\Lambda_2) -\delta_{\Lambda_2}\mathfrak A_{\mathrm{cov}}(\Lambda_1) \\ &\hspace{7em}=2\mathfrak A_{\mathrm{cov}}(\Lambda_{12}). \end{aligned}$$

The factor of two differs from the consistent Wess–Zumino equation. It disappears only when the relevant commutator contribution vanishes, including for an Abelian group Cohen, Lu, and Zhang 2023, § 3.4, arXiv v1, pp. 18–21, eqs. (3.40)–(3.42), Open PDF.

The price of covariance is that the functional curl of $J_{\mathrm{cov}}$ is generally nonzero. Thus $J_{\mathrm{cov}}$ normally cannot equal $\delta\widetilde W/\delta\mathcal A$ for any single four-dimensional functional $\widetilde W$. This remains a real distinction even for an Abelian group: the elementary Abelian Wess–Zumino commutator is trivial, but the Hessian symmetry required of a functional derivative can still fail. The original local construction is given in Bardeen and Zumino 1984, §§ 1–3, pp. 423–436; a heat-kernel calculation of the current difference and functional curl appears in Takeuchi and Endo 2017, §§ 2 and 4, article pp. 3–6 and 8–11, eqs. (2.17)–(2.27) and (4.1)–(4.25), Open PDF.

A four-dimensional Weyl fermion fixes the relative coefficients

Specialize to one physical Lorentzian left-handed fermion of compact $U(1)$ charge $q$. Write $\mathcal A=q a$, $\Lambda=q\lambda$, $f=\mathrm da$, and

$$c_1=\frac{f}{2\pi}, \qquad p_1(TX)=-\frac{1}{8\pi^2} \operatorname{tr}_{\mathrm{vec}}(\mathcal R^2).$$

After factoring out the gauge parameter, the two local Ward representatives are

$$\begin{aligned} \mathscr I_{4,\mathrm{cons}}^L(q) &=-\frac{q^3}{6}c_1^2 +\frac{q}{24}p_1(TX), \\ \mathscr I_{4,\mathrm{cov}}^L(q) &=-\frac{q^3}{2}c_1^2 +\frac{q}{24}p_1(TX). \end{aligned}$$

The pure $U(1)^3$ coefficient in the covariant divergence is therefore three times the consistent coefficient. The mixed gauge–gravity terms agree in this particular diffeomorphism- and local-Lorentz-preserving representative; that agreement should not be detached from the representative choice. Opposite chirality reverses both lines.

The corresponding Abelian current improvement gives a direct sign and factor check:

$$\star J_{\mathrm{BZ},E} =-\frac{i q^3}{12\pi^2}\,a f,$$

and hence

$$\begin{aligned} \frac1i\,\mathrm d\star J_{\mathrm{BZ},E} &=-\frac{q^3}{12\pi^2}f^2 \\ &=-\frac{q^3}{3}c_1^2 =\mathscr I_{4,\mathrm{cov}}^L(q) -\mathscr I_{4,\mathrm{cons}}^L(q). \end{aligned}$$

The Abelian functional curl exposes the distinction that the commuting Wess– Zumino equation cannot. Define the field-space one-form

$$\Theta_{\mathrm{BZ}}[\delta a] :=\int_X\delta a\,\star J_{\mathrm{BZ},E}.$$

For two independent variations on a closed $X$,

$$\begin{aligned} &\delta_1\Theta_{\mathrm{BZ}}[\delta_2a] -\delta_2\Theta_{\mathrm{BZ}}[\delta_1a] \\ &\hspace{3em} =\frac{i q^3}{4\pi^2} \int_X\delta_1a\,\delta_2a\,f, \end{aligned}$$

which is generally nonzero. Thus the Abelian covariant current is not the gradient of one effective action even though the Abelian Wess–Zumino commutator vanishes identically. The general relation between this functional curl and the current/anomaly difference is Takeuchi and Endo 2017, § 2.2, article pp. 4–6, eqs. (2.17)–(2.27), Open PDF.

The factor of three is special to the pure Abelian cubic term. The non-Abelian consistent expression contains $\mathrm d(\mathcal A\,\mathrm d\mathcal A-i\mathcal A^3/2)$, whereas the covariant expression contains $\mathcal F^2$; they are not related by multiplying the entire functional by a universal number. What is universal is the pure-gauge zero test: both use the same symmetrized cubic tensor. With $U(1)$ factors, cancellation of the mixed gauge–gravity term additionally requires the linear charge trace to vanish. A modern regulator comparison that exhibits the covariant and Wess–Zumino-consistent prescriptions, their relative coefficients, and their common cubic-trace cancellation condition is Cohen, Lu, and Zhang 2023, §§ 3.4 and 4.2, arXiv v1, pp. 18–21 and 24–27, eqs. (3.36)–(3.51) and (4.15)–(4.20), Open PDF. The characteristic-form normalization and chirality reversal are fixed in Álvarez-Gaumé and Vázquez-Mozo 2024, §§ 2–3, arXiv v2, pp. 4–8, eqs. (6)–(11), (17)–(19), and (22), Open PDF.

The two representatives answer different questions

The comparison below is local and perturbative, at fixed chirality, regulator, and counterterm scheme. Read across each row: construction from a single effective action forces integrability and Wess–Zumino consistency, whereas the Bardeen–Zumino improvement generally loses integrability and, in the non-Abelian case, Wess–Zumino consistency in exchange for a homogeneously transforming local current. For commuting Abelian transformations the Wess–Zumino condition is vacuous, but the integrability distinction remains.

Consistent and covariant anomaly representatives in a fixed local perturbative scheme

Question	Consistent representative	Covariant representative
Construction	Functional derivative of one regulated effective action	Consistent current plus a local Bardeen–Zumino current
Gauge transformation	Generally not homogeneous when the theory is anomalous	Transforms homogeneously in the adjoint representation
Functional integrability	Yes: its functional curl vanishes	Generally no: it need not be a derivative of an effective action
Ward condition	Obeys Wess–Zumino consistency	Has a covariant divergence; non-Abelian Wess–Zumino consistency generally fails
Local freedom	An allowed local action counterterm gives another integrable representative	A local current improvement gives covariance but is not generally an action counterterm
Appropriate use	Gauging, generating-functional Ward identities, descent, and source differentiation	Covariant local response equations and covariantly transforming operator insertions
Shared limit	Does not decide large-gauge phases, torsion, or boundary charge data	Does not decide large-gauge phases, torsion, or boundary charge data

The table does not identify two independent anomaly classes. It compares two representatives connected by local data. In particular, choosing the covariant current cannot turn a nonzero dynamical gauge anomaly into a consistent gauge theory.

Bardeen counterterms redistribute mixed Ward identities

Three local operations are easy to conflate. First, an admissible local action counterterm changes the effective action,

$$W_E[\mathcal A]\longmapsto W_E[\mathcal A]+C[\mathcal A].$$

Its current shift is integrable by construction:

$$J_{\mathrm{cons}}\longmapsto J_{\mathrm{cons}}+\frac{\delta C}{\delta\mathcal A}, \qquad \delta_\Lambda W_E\longmapsto \delta_\Lambda W_E+\delta_\Lambda C.$$

This changes the consistent representative but not a nontrivial anomaly class. Second, adding $J_{\mathrm{BZ}}$ converts the current to a covariant one; because that current is generally nonintegrable, this operation is not usually generated by a four-dimensional action counterterm. Third, a Bardeen counterterm involving several backgrounds can redistribute a mixed consistent anomaly among their Ward identities.

A simple closed-manifold calculation makes the third operation concrete. Take globally defined potentials $a,b$ on trivial $U(1)$ bundles, with $f_a=\mathrm da$ and $f_b=\mathrm db$. For real $\eta$, choose the imaginary Euclidean counterterm

$$C_\eta =2\pi i\eta\int_X \frac{a}{2\pi}\frac{b}{2\pi}\frac{f_a}{2\pi}.$$

Under $a\mapsto a+\mathrm d\lambda$ and $b\mapsto b+\mathrm d\chi$, integration by parts gives

$$\begin{aligned} \delta_\lambda C_\eta &=-2\pi i\eta\int_X \frac{\lambda}{2\pi} \frac{f_b}{2\pi} \frac{f_a}{2\pi}, \\ \delta_\chi C_\eta &=+2\pi i\eta\int_X \frac{\chi}{2\pi} \left(\frac{f_a}{2\pi}\right)^2. \end{aligned}$$

Changing $\eta$ therefore moves a fixed mixed variation between the two consistent Ward identities; it does not remove the total obstruction. In a mixed dynamical/background problem, the allowed choice is constrained by the dynamical gauge Ward identity. After that identity is preserved, a remaining $G_{\mathrm{dyn}}^2U(1)_{\mathrm{source}}$ divergence means that the putative continuous $U(1)$ current is ABJ-broken, perhaps to a subgroup. It should not automatically be called an ‘t Hooft anomaly of an exact global $U(1)$ before the surviving symmetry has been identified. If all connections instead probe genuine exact global symmetries and remain nondynamical, the remaining class is background ‘t Hooft-anomaly data.

The distinction between a current improvement and an effective-action counterterm is emphasized in Bilal 2008, § 4.2, arXiv v1, pp. 20–22, eqs. (4.17)–(4.19), Open PDF. Redistribution in the standard vector–axial example, including the counterterm relating two assignments, is shown in Cohen, Lu, and Zhang 2023, § 5.3, arXiv v1, pp. 32–33, eqs. (5.17)–(5.24), Open PDF. Redistribution of the mixed gauge–gravity representative is discussed in Bilal 2008, § 11.3.2, arXiv v1, pp. 92–93, eqs. (11.29)–(11.38), Open PDF. The physical distinction between a dynamical gauge obstruction and background anomaly data is reviewed in Bhardwaj et al. 2024, §§ 4.1–4.2.1, arXiv v2, pp. 60–69, Open PDF.

Local current shifts do not change global data

The derivations above used a closed manifold. On a region with boundary, the integration-by-parts step instead reads

$$\begin{aligned} \delta_\Lambda W_E &=\int_X\operatorname{tr}_R (D\Lambda\,\mathcal J_{\mathrm{cons}}) \\ &=\int_{\partial X}\operatorname{tr}_R (\Lambda\mathcal J_{\mathrm{cons}}) -\int_X\operatorname{tr}_R (\Lambda D\mathcal J_{\mathrm{cons}}). \end{aligned}$$

The surface term is not part of the bulk divergence. Boundary conditions determine which transformations preserve the field domain; transformations with nonzero boundary values may carry charges rather than represent gauge redundancies. At fixed Lorentzian time on a bounded spatial region $\Omega$, the improved charge differs by the bulk Bardeen–Zumino charge,

$$Q_{\mathrm{cov}}^\Omega-Q_{\mathrm{cons}}^\Omega =\int_\Omega\mathrm d^3x\,J_{\mathrm{BZ}}^0.$$

Its balance law includes both boundary flux and the difference of the two bulk divergences:

$$\begin{aligned} \frac{\mathrm d}{\mathrm dt} (Q_{\mathrm{cov}}^\Omega-Q_{\mathrm{cons}}^\Omega) &=-\int_{\partial\Omega}\mathrm d\Sigma_i\,J_{\mathrm{BZ}}^i \\ &\quad+\int_\Omega\mathrm d^3x \left(\partial_\mu J_{\mathrm{cov}}^\mu -\partial_\mu J_{\mathrm{cons}}^\mu\right). \end{aligned}$$

Equality of bulk local classes therefore does not imply equality of bounded-region charges without specified boundary conditions, boundary counterterms, edge degrees of freedom, or inflow. The support assumption behind the usual integration by parts is explicit in Bilal 2008, § 9, arXiv v1, p. 69, eq. (9.1), Open PDF; a Maxwell example separating boundary-vanishing gauge transformations from charged boundary symmetries is given in Harlow and Wu 2020, Introduction and § 3.3, arXiv v4, pp. 1–3 and 26–27, eqs. (3.18)–(3.25), Open PDF.

A local perturbative Bardeen–Zumino polynomial cannot by itself determine or erase a nontrivial determinant-line holonomy, a torsion phase under a large transformation, the global form of the gauge group, or the charge lattice. Globally admissible counterterms can change representatives, but triviality of the global anomaly remains a separate question. These are not refinements of the factor of three. Similarly, constitutive relations and anomaly-induced transport require a hydrodynamic effective theory, not just the local Ward representative derived here.

Common pitfalls

Calling the covariant current more physical in every setting. Covariance is useful, but gauging and source differentiation require the integrable consistent representative. The appropriate current is selected by the question being asked.

Treating the Bardeen–Zumino current as an action counterterm. It is a local current improvement and is generally nonintegrable. Only shifts of the form $\delta C/\delta\mathcal A$ arise from a local four-dimensional counterterm.

Multiplying every consistent anomaly by three. The factor of three holds for the pure four-dimensional Abelian cubic term. The non-Abelian expressions have different polynomial structure, and the mixed gauge–gravity coefficient agrees in the representative used here.

Mistaking a representative shift for cancellation. A counterterm may move a mixed variation between Ward identities, and a Bardeen–Zumino polynomial may make a current covariant. Neither operation removes a nontrivial class while preserving all required dynamical gauge symmetries.

Ignoring boundaries or large transformations. The local divergence identity assumes the surface term vanishes. It neither fixes boundary charges nor detects global and torsion anomalies.

Check your understanding

1. Why must the functional curl of $J_{\mathrm{cons}}$ vanish?

   Solution

   The consistent current is one functional derivative of $W_E$. Its functional curl is the antisymmetric part of the second derivative of the same bosonic functional. Commuting those derivatives makes that antisymmetric part zero. A nonzero curl therefore rules out interpreting the current as $\delta W_E/\delta\mathcal A$.

2. Starting from the two $U(1)$ anomaly forms, compute their difference and reproduce it from $\star J_{\mathrm{BZ},E}$.

   Solution

   Subtraction gives $(-q^3/2+q^3/6)c_1^2=-q^3c_1^2/3$; the mixed terms cancel in the chosen representative. Since $\mathrm d(a f)=f^2$,

   $$\frac1i\mathrm d\!\left( -\frac{i q^3}{12\pi^2}a f \right) =-\frac{q^3}{12\pi^2}f^2 =-\frac{q^3}{3}c_1^2.$$

3. A left-handed spectrum has charges $\{1,5,-7,-8,9\}$. What happens to the consistent and covariant Abelian gauge anomalies?

   Solution

   The cubic sum is $1+125-343-512+729=0$. The pure Abelian consistent coefficient therefore vanishes, and the covariant coefficient—three times the same cubic sum—vanishes as well. The linear sum also vanishes, so the displayed mixed gauge–gravity coefficient is zero. This checks only the declared local Abelian anomalies, not global obstructions.

4. Why can a Bardeen–Zumino improvement change an integrated charge on a bounded region even though it is local?

   Solution

   The charge difference is the bulk integral $\int_\Omega\mathrm d^3x\,J_{\mathrm{BZ}}^0$; locality does not force that integral to vanish. Its time derivative is controlled jointly by the Bardeen–Zumino flux through $\partial\Omega$ and by the integrated difference of the two bulk divergences. Boundary conditions, boundary currents, or inflow are therefore needed before the two bounded-region charges can be compared.

5. In the two-background counterterm example, what does changing $\eta$ do?

   Solution

   It changes the $a$ Ward identity by a term proportional to $-f_bf_a$ and the $b$ Ward identity by a term proportional to $+f_a^2$. Thus it redistributes one mixed consistent variation between the two currents. The total nontrivial anomaly class is unchanged, and any dynamical gauge Ward identity still has to be exact.

What to carry forward

• Wess–Zumino Consistency and Descent derives the consistency equation and representative shifts from the descent sequence.
• Anomaly Polynomials and Inflow explains how a complete bulk–boundary system cancels a boundary variation without declaring the isolated boundary theory anomaly-free.
• Global and Torsion Anomalies treats large transformations and phases invisible to every local current polynomial.
• Local Anomaly Descent and Wess–Zumino Consistency gives the theorem-first relative local-cohomology classification.
• Charged and Anomalous Hydrodynamics uses current representatives inside constitutive relations and transport.

References

• Álvarez-Gaumé, Luis, and Miguel Á. Vázquez-Mozo. “Anomalies and the Green–Schwarz Mechanism.” In Handbook of Quantum Gravity, edited by Cosimo Bambi, Leonardo Modesto, and Ilya L. Shapiro, 2241–2284. Singapore: Springer, 2024. DOI. Open PDF, arXiv v2.
• Bardeen, William A., and Bruno Zumino. “Consistent and Covariant Anomalies in Gauge and Gravitational Theories.” Nuclear Physics B 244, no. 2 (1984): 421–453. DOI.
• Bhardwaj, Lakshya, Lea E. Bottini, Ludovic Fraser-Taliente, Liam Gladden, Dewi S. W. Gould, Arthur Platschorre, and Hannah Tillim. “Lectures on Generalized Symmetries.” Physics Reports 1051 (2024): 1–87. DOI. Open PDF, arXiv v2.
• Bilal, Adel. “Lectures on Anomalies.” LPTENS-08/05, arXiv:0802.0634v1, 2008. Stable record. Open PDF.
• Cohen, Timothy, Xiaochuan Lu, and Zhengkang Zhang. “Anomalies from the Covariant Derivative Expansion.” Physical Review D 107, no. 11 (2023): 116015. DOI. Open PDF, arXiv v1.
• Harlow, Daniel, and Jie-qiang Wu. “Covariant Phase Space with Boundaries.” Journal of High Energy Physics 2020, no. 10 (2020): 146. DOI. Open PDF, arXiv v4.
• Takeuchi, Masaharu, and Ryusuke Endo. “Heat Kernel Approach to the Relations between Covariant and Consistent Currents in Chiral Gauge Theories.” Progress of Theoretical and Experimental Physics 2017, no. 3 (2017): 033B03. DOI. Open PDF.
• Wess, Julius, and Bruno Zumino. “Consequences of Anomalous Ward Identities.” Physics Letters B 37, no. 1 (1971): 95–97. DOI.