Conformal Ward Identities

• Conformal Ward identities are differential equations that enforce scale and special conformal invariance, fixing tensor structures in QFT correlators.
• They integrate anomaly-induced modifications in parity-odd sectors, introducing unique features like anomaly poles in chiral and trace anomaly correlators.
• Momentum-space solutions uniquely determine allowed form factors in 4D theories, impacting diverse applications from condensed matter to cosmology.

Conformal Ward identities are a fundamental set of constraints on correlation functions in quantum field theories exhibiting conformal symmetry. These identities express how correlators must transform under the full conformal group, including scale and special conformal transformations, and serve as a powerful tool for determining the tensor structures and form factors permitted in both even and odd (parity-violating) sectors. In the presence of anomalies—especially chiral and parity anomalies—the structure of conformal Ward identities is modified, typically introducing nontrivial longitudinal components, “anomaly poles,” and unique analytic features in correlators. The conformal bootstrap, trace and divergence anomalies, and various parity-odd phenomena in condensed matter and high-energy physics are all governed or tightly constrained by these identities.

1. Mathematical Structure of Conformal Ward Identities

Conformal Ward identities (CWIs) are differential equations in coordinate or momentum space that encode the symmetry properties of correlation functions under the conformal group. For $n$-point functions in momentum space, the CWIs comprise:

• Dilatation Identity:

$\left[\sum_{i=1}^n (\Delta_i - d - p_i^\mu \partial_{p_i^\mu})\right] \langle \mathcal{O}_1 \hdots \mathcal{O}_n \rangle = 0$

expressing scale invariance, where %%%%2%%%% are scaling dimensions.

• Special Conformal Identities:

$\sum_{i=1}^n K_i^\kappa \langle \mathcal{O}_1 \hdots \mathcal{O}_n \rangle = 0$

with

$$K_i^\kappa = 2(\Delta_i - d)\frac{\partial}{\partial p_{i,\kappa}} - 2 p_i^\alpha \frac{\partial}{\partial p_i^\alpha} \frac{\partial}{\partial p_{i,\kappa}} + p_i^\kappa \frac{\partial^2}{\partial p_{i}^\lambda \partial p_{i\lambda}} + \text{spin part}$$

encoding invariance under special conformal transformations. The solution space of these equations fixes allowed tensor structures and constrains the possible form factors to a finite-dimensional space.

In the parity-odd (CP-violating) sector, the CWIs are generally supplemented by anomaly-induced inhomogeneous terms or boundary conditions, reflecting the non-invariance of certain classical symmetries at the quantum level (Corianò et al., 2023, Corianó et al., 5 Aug 2024, Corianó et al., 16 Sep 2024, Lionetti, 18 Dec 2025).

2. Implementation in Parity-Odd Correlators: Chiral and Conformal Anomalies

Parity-odd or CP-violating correlators are distinguished by the presence of the antisymmetric Levi-Civita tensor $\varepsilon^{\mu\nu\rho\sigma}$ and are closely tied to quantum anomalies:

• Chiral (Axial) Anomaly: The divergence of the axial current in four dimensions acquires an anomalous contribution:

$$\partial_\mu J_5^\mu = a_1\, \varepsilon^{\mu\nu\rho\sigma} F_{\mu\nu} F_{\rho\sigma} + a_2\, \varepsilon^{\mu\nu\rho\sigma} R^\alpha_{\ \beta\mu\nu} R^\beta_{\ \alpha\rho\sigma}$$

where $a_1$, $a_2$ are anomaly coefficients. This anomalous divergence modifies the corresponding CWI, leading to non-zero three-point functions such as $\langle JJ J_5 \rangle$ with unique longitudinal “anomaly pole” terms (i.e., a structure proportional to $p^{\mu_3}/p^2$) (Corianò et al., 2023, Corianó et al., 16 Sep 2024, Lionetti, 18 Dec 2025).

• Conformal (Trace) Anomaly: The trace of the stress tensor in a CFT is anomalously non-vanishing in curved backgrounds, introducing terms like

$$T^\mu_\mu = c_1 \varepsilon^{\mu\nu\rho\sigma} F_{\mu\nu} F_{\rho\sigma} + c_2 \varepsilon^{\mu\nu\rho\sigma} R_{\mu\nu\alpha\beta} R_{\rho\sigma}^{\ \ \, \alpha\beta}$$

These anomalies are encoded in the inhomogeneous part of the special conformal Ward identity for correlators involving $T^{\mu\nu}$.

• In both cases, the conformal Ward identities determine that only specific three-point functions with particular operator dimensions and tensor structures are nonzero, and their form factors are essentially fully fixed by the anomaly coefficients (Corianó et al., 5 Aug 2024, Corianò et al., 2023, Corianó et al., 16 Sep 2024).

3. Solutions in Momentum Space and the Uniqueness of Parity-Odd Form Factors

For three-point functions $\langle JJO \rangle_{\text{odd}}$ and $\langle TTO \rangle_{\text{odd}}$ in $d=4$, the conformal Ward identities and current/stress-tensor conservation imply a decomposition into projectors (enforcing transversality and tracelessness) and a single allowed parity-odd tensor structure built with $\varepsilon^{\mu\nu\rho\sigma}$ (Corianó et al., 5 Aug 2024, Corianò et al., 2023). These correlators are only nonvanishing if the scalar/pseudoscalar operator $O$ has conformal dimension $\Delta_O=4$ (or equivalently, if it coincides with the divergence of the axial current or the trace of the stress tensor).

Explicitly, for $$\langle J^{\mu_1}(p_1) J^{\mu_2}(p_2) O(p_3) \rangle_{\text{odd}}$$,

$$\langle J^{\mu_1} J^{\mu_2} O \rangle_{\text{odd}} = C_{JJO}\, \pi^{\mu_1}_{\alpha_1}(p_1)\, \pi^{\mu_2}_{\alpha_2}(p_2)\, \varepsilon^{\alpha_1 \alpha_2 p_1 p_2}$$

with $$\pi^{\mu}_\alpha(p) = \delta^\mu_\alpha - p^\mu p_\alpha/p^2$$, and $C_{JJO}$ determined by the anomaly.

For the three-point function $\langle T T O \rangle_{\text{odd}}$, an analogous structure holds (Corianó et al., 5 Aug 2024, Corianò et al., 2023).

In the case of the VVA correlator (two vector currents and one axial current), the conformal Ward identities not only fix the anomaly pole in the longitudinal channel but also fully determine the transverse part up to a normalization given by the anomaly coefficient, matching the result of the one-loop triangle diagram (Corianò et al., 2023, Corianó et al., 16 Sep 2024, Lionetti, 18 Dec 2025).

4. Analytic Features and Physical Content

The solution of the CWIs in the parity-odd sector exhibits several salient analytic features:

• Anomaly Pole: The nonlocal term $p^{\mu}/p^2$ in momentum space represents exchange of a massless pseudoscalar (an “axion pole”), and all CFT and perturbative solutions reproduce the same anomaly pole structure in $\langle JJJ_5 \rangle$ (Corianó et al., 16 Sep 2024, Corianò et al., 2023).
• Non-renormalization: The anomaly part of these correlators is one-loop exact and is protected from renormalization by virtue of the Wess–Zumino consistency conditions; the residue of the pole is exact to all orders (Corianó et al., 16 Sep 2024, Corianò et al., 2023, Lionetti, 18 Dec 2025).
• Dimension Selection Rule: The vanishing of the correlators for operator dimensions $\Delta \neq 4$ is a consequence of the singularities of the triple-K integrals that arise in the momentum-space CWI solution; only for the “anomaly” dimension is there a surviving, finite, nonzero contribution (Corianó et al., 5 Aug 2024, Corianò et al., 2023).
• Absence of Additional Form Factors: Unlike parity-even sectors where several independent tensor structures appear, in the parity-odd channel, conformal symmetry and anomalies uniquely specify the result once the symmetry-breaking term is given.

5. Surface and Boundary Parity-Odd Anomalies

Parity-odd contributions in conformal field theories with boundaries or defects are determined by the anomalous response to background fields, encoded via the conformal Ward identities and anomaly inflow:

• d=4 Pontryagin Anomaly: For a four-dimensional bulk with a conical defect, the surface (boundary) term is given by the outer curvature scalar, originating from the “pushed-down” Pontryagin bulk anomaly (Cvitan et al., 2015). The resulting surface anomaly is of the form

$$A^{(2)}_{odd}[\omega] = \int_E \omega \, \left(k_1 \Omega + k_2 \varepsilon^{\mu\nu\alpha\beta} K_{\mu\rho\,\alpha} K_{\nu}^{\ \rho}{}_\beta\right)$$

where $k_1,k_2$ are determined by the form of the singularity/defect.

• Contact Terms: The resulting contact terms in flat-space correlators have lower tensorial rank than the bulk anomaly, reflecting the interplay of anomaly descent and the “rank reduction” mechanism (Cvitan et al., 2015).

6. Physical and Phenomenological Implications

The implications of conformal Ward identities and their parity-violating, anomaly-induced modifications are far-reaching:

• Quantum Hall effect and Surfaces: In 4D systems with boundaries (e.g., topological insulators), the parity anomaly fixes the quantized Chern–Simons response on the boundary, determining a ¼-quantized Hall conductivity per surface and ensuring gauge invariance under large gauge transformations (Kurkov et al., 2017).
• Cosmological Signatures: The nonlocal parity-odd couplings $\int \phi F\tilde{F}$ (axion-photon) and $\int \tau R\tilde{R}$ (dilaton-gravity) implied by the anomaly structures of $\langle JJO \rangle_{\text{odd}}$ and $\langle TTO \rangle_{\text{odd}}$ have pivotal roles in generating cosmological helicity asymmetries, stochastic gravitational wave backgrounds, and scenarios of baryogenesis (Corianó et al., 5 Aug 2024, Corianò et al., 2023).
• Bootstrap Bounds and Positivity: In d=3 CFTs, parity-odd exchanges controlled by the CWI generate new operator towers and collider bounds—elliptical constraints $a_2^2 + \alpha_j^2 \leq 4$—with analytic features (e.g., square-root branch cuts in blocks) reflecting the structure of the anomalous conformal blocks (Chowdhury et al., 2018).

7. Summary Table: Parity-Odd Conformal Ward Identity Implications

Context / Observable	CWI-Imposed Parity-Odd Structure	Anomaly Input
4D $\langle JJO \rangle_{\text{odd}}$	Unique $\varepsilon^{\alpha_1\alpha_2p_1p_2}$ structure (only if $\Delta=4$)	Chiral anomaly
4D $\langle TTO \rangle_{\text{odd}}$	Unique pair of tensor structures (only if $\Delta=4$)	Conformal anomaly
$\langle JJJ_5\rangle$ in $d=4$	Nonlocal “anomaly pole” in longitudinal channel	Chiral anomaly
CFT with boundary / conical defect	Parity-odd surface contact term from outer curvature / extrinsic curvature	Pontryagin anomaly
3D CFT (bootstrap)	New parity-odd operator towers, collider bounds, square-root branch cuts	Parity-odd stress

Conformal Ward identities thus provide a rigid, anomaly-driven structure for parity-odd sectors in quantum field theories, ensuring that all such contributions to correlators, effective actions, and physical observables are uniquely fixed by the underlying anomaly coefficients and the demands of conformal invariance (Corianò et al., 2023, Corianó et al., 5 Aug 2024, Corianó et al., 16 Sep 2024, Lionetti, 18 Dec 2025, Chowdhury et al., 2018, Cvitan et al., 2015, Kurkov et al., 2017).