Spurions, Local Counterterms, and Symmetry Response

A spurion is a coupling promoted to a transforming external field. It makes the covariance of a family of theories explicit, but it does not restore a conserved charge in any member whose fixed coupling breaks the symmetry. Once all sources are present, functional derivatives encode current response; finite local functionals of those sources can shift contact terms and momentum-space polynomials. Universal response is therefore what survives the allowed local shifts after the current normalization and improvement convention have been fixed.

The object is the renormalized, time-ordered functional

$$W[a,\lambda]=-i\log\mathcal Z[a,\lambda],$$

where $a$ is a nondynamical connection and $\lambda^I(x)$ denotes other operator sources. The page treats algebraic Ward constraints and local ambiguities only; retarded correlators, hydrodynamic limits, material response, and transport coefficients require additional state and causal data.

Required background. Current Sources and Generating Functionals supplies the definition and Lorentzian differentiation rules for $W$. Localized Transformations and Ward–Takahashi Identities supplies the insertion Ward identity and its distributional contact terms.

Helpful background. Coupling to Background Gauge Fields and Bundles supplies the global bundle-and-connection meaning of the background source.

Spurion covariance is a statement about a family

Suppose a fixed coupling $\lambda^I$ explicitly breaks $G$. Promote it temporarily to a spacetime-dependent source and prescribe a transformation $\delta_\alpha\lambda^I$. If the regulated source-dependent theory is covariant and nonanomalous, then

$$\overline J_A^\mu(x;a) \equiv\frac{\delta W}{\delta a_\mu^A(x)} =\left\langle [\mathscr J_A^\mu(x;a)]_R \right\rangle_a$$

is the current one-point response, and

$$\begin{aligned} 0=\delta_\alpha W =\int\mathrm d^d x\Bigg[ &\left\langle \overline J^\mu, D_\mu^{\mathrm{ad}}\alpha \right\rangle_{\mathfrak g} \\ &+\frac{\delta W}{\delta\lambda^I} \delta_\alpha\lambda^I \Bigg]. \end{aligned}$$

This equation compares different source values related by the transformation. Holding a noninvariant $\lambda^I$ fixed removes the compensating variation and leaves an explicit breaking insertion. For transformations linear or affine in the fields, inheritance by the ordinary effective action is established at Weinberg 1996, Vol. II, § 16.4, pp. 75–78; nonlinear transformations generally require additional composite sources or an induced quantum transformation. Schwartz develops connected and 1PI effective actions in the background-field framework and exhibits their dependence on counterterm choices at Schwartz 2014, §§ 34.1–34.2, pp. 735–752. The spacetime-dependent spurion equation and the general source-local argument below are derived directly here; neither citation is being used as a substitute for those derivations.

For the threaded scalar,

$$\Delta\mathcal L =h\phi^N+h^*(\phi^\dagger)^N, \qquad N\geq2,$$

spurion covariance assigns

$$\delta_\alpha h=-iN\alpha h, \qquad \delta_\alpha h^*=+iN\alpha h^*.$$

The local functional identity is

$$\partial_\mu\frac{\delta W}{\delta a_\mu} +iNh\frac{\delta W}{\delta h} -iNh^*\frac{\delta W}{\delta h^*} =0.$$

At a fixed nonzero constant $h$, this equation records the breaking insertion rather than a conserved continuous charge. Assuming no other breaking interaction, the faithful exact symmetry is the finite subgroup $\mathbb Z_N$.

The complete current-response kernel

Let the source-dependent current insertion be

$$\mathscr J^\mu(x;a) \equiv\frac{\delta S}{\delta a_\mu(x)}.$$

Define the connected response kernel by differentiating $W$ itself:

$$\Pi^{\mu\nu}(x,y) \equiv \frac{\delta^2W} {\delta a_\mu(x)\,\delta a_\nu(y)}.$$

The Lorentzian path integral gives

$$\begin{aligned} \Pi^{\mu\nu}(x,y) ={}&i\Bigl\langle \mathrm T\{ [\mathscr J^\mu(x;a)]_R \\ &\qquad\quad\times [\mathscr J^\nu(y;a)]_R \} \Bigr\rangle_{c,a} \\ &+\left\langle \frac{\delta[\mathscr J^\mu(x;a)]_R} {\delta a_\nu(y)} \right\rangle_a. \end{aligned}$$

The second line is the seagull or contact contribution. Further local terms arise from the renormalization prescription. A Ward identity constrains the complete $\Pi^{\mu\nu}$, not the separated current-current term with its contacts discarded.

Relative to a fixed baseline source action and composite-product prescription, it is useful to keep three sources of locality distinct:

• nonlinear source coupling forces seagulls such as the scalar $a^2\phi^\dagger\phi$ term;
• time ordering and coincident operator products generate distributional contacts required by the Ward identity;
• finite symmetry-compatible background counterterms shift the chosen contact-term scheme.

The first two are not freely adjustable merely because they are local. Only the third is the finite counterterm ambiguity discussed below. A finite $S_{\mathrm{loc}}$ can be folded into a new baseline action and then appear algebraically inside $\delta\mathscr J/\delta a$; the invariant distinction is between contacts forced by the chosen covariant completion and Ward identity, and the remaining allowed local shift—not the line of a formula on which a term is recorded.

For the scalar at $a=0$, the classical source completion contains

$$\left. \frac{\delta\mathscr J^\mu(x;a)} {\delta a_\nu(y)} \right|_{a=0} =2\eta^{\mu\nu}\phi^\dagger\phi(x) \delta^{(d)}(x-y).$$

This local term is required to combine with the regulated current-current distribution. Testing transversality after omitting it is not a valid symmetry check.

Transverse and longitudinal constraints

Differentiate the spurion identity with respect to $a_\nu(y)$. Introduce the mixed kernels

$$\begin{aligned} \Xi_h^\nu(x,y) &\equiv \frac{\delta^2W} {\delta h(x)\,\delta a_\nu(y)}, \\ \Xi_{h^*}^\nu(x,y) &\equiv \frac{\delta^2W} {\delta h^*(x)\,\delta a_\nu(y)}. \end{aligned}$$

At vanishing $a$, the exact source identity gives

$$\begin{aligned} \partial_\mu\Pi^{\mu\nu}(x,y) &+iNh(x)\Xi_h^\nu(x,y) \\ &-iNh^*(x)\Xi_{h^*}^\nu(x,y) =0. \end{aligned}$$

Thus an exact $U(1)$ point, $h=0$, has a transverse complete kernel,

$$\partial_\mu\Pi^{\mu\nu}(x,y)=0.$$

At nonzero $h$, the longitudinal part is fixed by mixed responses to the breaking operators. Spurion covariance does not set that longitudinal part to zero.

For a translation-invariant vacuum, define

$$\widetilde\Pi^{\mu\nu}(p) =\int\mathrm d^d z\, e^{+ip\cdot z}\Pi^{\mu\nu}(z).$$

With the transform defined above, $\partial_\mu\mapsto-ip_\mu$, so exact symmetry implies

$$p_\mu\widetilde\Pi^{\mu\nu}(p)=0.$$

In a Lorentz-invariant, parity-even sector and away from singular zero-momentum limits, write

$$\begin{aligned} \widetilde\Pi^{\mu\nu}(p) ={}&\left( p^2\eta^{\mu\nu}-p^\mu p^\nu \right)\Pi_T(p^2) \\ &+\widetilde\Pi_{\mathrm{loc}}^{\mu\nu}(p). \end{aligned}$$

This split requires a convention: polynomial multiples of $p^2\eta^{\mu\nu}-p^\mu p^\nu$ can be moved between the two terms. Here all momentum polynomials are assigned to $\widetilde\Pi_{\mathrm{loc}}^{\mu\nu}$, so $\Pi_T$ is understood modulo polynomials and carries the nonpolynomial poles, logarithms, and branch cuts. The local term includes forced seagulls and coincident operator contacts as well as adjustable finite-counterterm shifts; only the last category is freely scheme dependent. Parity-odd tensor structures, boundaries, and singular infrared phases require their own decompositions.

Finite local counterterms

Two renormalized source functionals can describe the same separated-point local physics while differing by a finite local functional:

$$W'[a,\lambda] =W[a,\lambda]+S_{\mathrm{loc}}[a,\lambda].$$

For a nonanomalous symmetry, $S_{\mathrm{loc}}$ may be chosen symmetry compatible. Functional derivatives of it are supported when insertion points coincide; after Fourier transformation they are polynomials in momenta.

In four dimensions, a simple Abelian example is

$$S_{\mathrm{loc}}[a] =\frac{c}{4} \int\mathrm d^4x\, f_{\mu\nu}f^{\mu\nu}.$$

It shifts the two-point kernel by

$$\Delta\widetilde\Pi^{\mu\nu}(p) =c\left( p^2\eta^{\mu\nu}-p^\mu p^\nu \right),$$

which is polynomial and exactly transverse. The coefficient $c$ cannot be recovered from the separated-point Ward identity alone.

Spurion backgrounds allow further invariant local terms. For the charge-$-N$ scalar spurion,

$$D_\mu h=(\partial_\mu+iNa_\mu)h,$$

so, when dimensions and power counting permit,

$$S_{\mathrm{loc}}[a,h] =c_h\int\mathrm d^d x\, (D_\mu h)^*D^\mu h$$

is allowed. At constant nonzero $h$ it contains a local $N^2|h|^2a_\mu a^\mu$ term and shifts both current and mixed spurion response. The full differentiated spurion identity remains valid because the counterterm is covariant.

On flat finite-group backgrounds, local topological phases need not be visible as ordinary curvature polynomials. They can alter global partition-function phases. Whether two such choices are called scheme-equivalent depends on the declared equivalence relation: this requires allowing stacking with an invertible $G$-protected phase. If that stacking is not quotiented out, the phase is physical global-response data. After gauging, distinct Dijkgraaf–Witten or discrete-torsion weights generally define inequivalent gauged theories rather than two schemes for one gauged theory. This bounded finite-group distinction is described at Gaiotto et al. 2015, § 2, pp. 7–10, arXiv PDF. Which terms exist depends on the spacetime dimension, the global symmetry group, and the permitted background structures.

What is invariant under local shifts

After fixing the current normalization and representative, background-only local counterterms cannot change:

• correlators with all insertion points separated in a contractible patch;
• nonanalytic momentum dependence such as isolated poles, branch cuts, and logarithms;
• a Ward identity written for the complete response functional;
• an anomaly class, although a noninvariant local term can change its representative.

They can change coincident distributions, polynomial momentum terms, and global phases on nontrivial backgrounds. Quantized topological counterterms can leave a discrete ambiguity rather than a continuously adjustable one. Any claim of “universal response” must state which local functionals were allowed and which quotient by them was taken.

Current improvements are a separate choice

For an Abelian current, an improvement

$$j'^\mu=j^\mu+\partial_\nu B^{[\nu\mu]}$$

changes the linear source coupling by

$$\begin{aligned} \int\mathrm d^d x\,a_\mu j'^\mu ={}&\int\mathrm d^d x\,a_\mu j^\mu \\ &-\frac12\int\mathrm d^d x\, f_{\nu\mu}B^{[\nu\mu]}, \end{aligned}$$

up to a boundary term. This is not merely a background-only counterterm because $B^{[\nu\mu]}$ is a dynamical operator. An improvement can change separated current correlators even when the integrated charge is unchanged under the required falloff conditions. Quantum Currents, Improvements, and Conservation fixes the operator-side distinction; detailed renormalized mixing belongs to Symmetry-Protected Operators, Currents, and Improvement. For a non-Abelian current, adjoint indices and a covariant improvement are required; replacing $\partial$ by a symbol after the fact does not reproduce the Abelian integration-by-parts identity.

The four representative types have the following local and integrated consequences. The comparison uses current forms $\mathcal J_\epsilon$ on a hypersurface $\Sigma$ with $S=\partial\Sigma$; it assumes that $Y_\epsilon$ is regular and globally defined. A background-only functional of $a_\mu$ is not a row in the table: it changes contact terms in the source response rather than adding the dynamical operator $\mathrm dY_\epsilon$ to the current.

Canonical, improved, identically conserved, and boundary-shifted current representatives

Representative	Local statement	Integrated charge	Boundary or corner consequence
Canonical current 𝓙ε	d𝓙ε ≈ 0 in the declared theory.	𝓠ε[Σ] = ∫Σ𝓙ε.	Its value still depends on the declared boundary problem and flux conditions.
Improved representative 𝓙ε + dYε	It has the same local divergence when Yε is regular and globally defined.	It agrees with 𝓠ε only when ∫SYε vanishes under the stated conditions.	Local matrix elements can change even when the integrated charge does not.
Identically conserved contribution dYε	d(dYε) = 0 off shell.	Its entire hypersurface integral is the surface term ∫SYε.	It is physically trivial only after that surface term and any singular or global obstruction are checked.
Boundary-shifted representative	The bulk local equation is unchanged.	𝓠′ε − 𝓠ε = ∫SYε survives.	The surface charge and possibly its algebra must be recomputed; facewise choices can leave a corner mismatch.

The boundary-charge realization of the same comparison derives the Stokes shift and explains when a facewise choice leaves corner data. Here the table’s role is narrower: it prevents an operator representative change from being absorbed into the finite background-local counterterm ambiguity of the response functional.

Structural response is not transport

The source functional determines time-ordered vacuum response subject to its declared prescription. Before comparing two response calculations, check:

• the sign of the source coupling and the definition of $W$;
• whether the kernel includes seagulls and other contact terms;
• the current normalization and improvement representative;
• the allowed finite local counterterms;
• the Fourier and analytic-continuation conventions;
• the state and the order of low-frequency and low-momentum limits.

Only after choosing a real-time contour and a state does one obtain retarded response. Sources, Linear Response, and Kubo Formulae develops that step. Vertices, Response, and Ward Identities treats conserving approximations and material response.

Common pitfalls

Calling a spurion a restored symmetry. The source family is covariant; a member with a fixed noninvariant coupling still lacks the continuous conserved charge.

Testing only the separated current product. The Ward identity applies to the complete functional derivative, including seagulls and renormalized contacts.

Calling every transverse polynomial universal. Gauge-invariant local counterterms can shift transverse polynomial terms without changing separated-point physics.

Treating improvement and scheme changes as identical. A background-only local counterterm changes contacts. A current improvement changes the operator coupled to the source and may change separated correlators.

Reading time-ordered response as a Kubo coefficient. Causal response requires a retarded prescription, a state, and controlled limits.

Forgetting global counterterms. Flat finite backgrounds can support topological phases even when local curvature vanishes.

Check your understanding

For the four-dimensional counterterm

$$S_{\mathrm{loc}}[a] =\frac{c}{4}\int\mathrm d^4x\, f_{\mu\nu}f^{\mu\nu},$$

derive its contribution to the current two-point response and verify transversality using the transform defined above.

Check

Varying once and integrating by parts gives

$$\frac{\delta S_{\mathrm{loc}}}{\delta a_\nu} =-c\,\partial_\mu f^{\mu\nu}.$$

With $\partial_\mu\mapsto-ip_\mu$, a second variation gives

$$\Delta\widetilde\Pi^{\mu\nu}(p) =c\left( p^2\eta^{\mu\nu}-p^\mu p^\nu \right).$$

Therefore

$$p_\mu\Delta\widetilde\Pi^{\mu\nu} =c\left(p^2p^\nu-p^2p^\nu\right)=0.$$

The shift is polynomial in $p$, so its inverse Fourier transform is a differential operator acting on a delta function. It is a contact contribution, not new separated-point propagation.

What to carry forward

Spurions expose explicit breaking without disguising it as an exact physical symmetry. Complete source derivatives obey transverse or mixed longitudinal Ward constraints, while local background counterterms define an equivalence relation on contact and polynomial response. Background Fields versus Dynamical Gauging next separates these fixed-background statements from changing the theory by summing over backgrounds. Renormalization-scheme dependence beyond source-local terms belongs to Renormalization Conditions, Schemes, and Finite Parts.

References

• Gaiotto, Davide, Anton Kapustin, Nathan Seiberg, and Brian Willett. “Generalized Global Symmetries.” Journal of High Energy Physics 02 (2015): 172. DOI. Open PDF
• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge: Cambridge University Press, 2014. DOI
• Weinberg, Steven. The Quantum Theory of Fields. Volume II: Modern Applications. Cambridge: Cambridge University Press, 1996. DOI