Current and Stress-Tensor CFT Data

Conserved currents and the stress tensor are distinguished CFT operators: their dimensions are fixed, their normalizations measure intrinsic degrees of freedom, and their three-point functions control Ward identities and energy-flow observables. This page fixes a concrete normalization, separates separated-point data from contact terms, and explains which statements require unitarity or a choice of tensor-structure basis.

Required background. Currents and the stress tensor supplies the shortening conditions and Ward identities. Embedding-space formalism supplies the transverse tensor structures used below. Helpful background. Conformal-collider bounds develops the Lorentzian positivity argument to which the last section points.

Two-point normalizations

Work at separated Euclidean points in $d>2$. Define

$$I_{\mu\nu}(x)=\delta_{\mu\nu}-2\frac{x_\mu x_\nu}{x^2},$$

and its symmetric-traceless lift

$$\mathcal I_{\mu\nu,\rho\sigma}(x) =\frac12\left(I_{\mu\rho}I_{\nu\sigma}+I_{\mu\sigma}I_{\nu\rho}\right) -\frac1d\delta_{\mu\nu}\delta_{\rho\sigma}.$$

For Hermitian currents $J_\mu^a$ whose Lie-algebra generators have already been fixed, and for the conformal stress tensor $T_{\mu\nu}$, use

$$\begin{aligned} \langle J_\mu^a(x)J_\nu^b(0)\rangle &=\frac{C_J\,\delta^{ab}I_{\mu\nu}(x)}{(x^2)^{d-1}},\\ \langle T_{\mu\nu}(x)T_{\rho\sigma}(0)\rangle &=\frac{C_T\,\mathcal I_{\mu\nu,\rho\sigma}(x)}{(x^2)^d}. \end{aligned}$$

Reflection positivity gives $C_J>0$ and $C_T>0$ in a unitary theory. Their numerical values are not meaningful until the generator normalization and the overall definition of $T_{\mu\nu}$ are specified. In particular, rescaling $J^a$ while inversely rescaling the generators changes $C_J$ without changing the symmetry action.

The integrated charges remove that ambiguity. With outward-oriented surface element $dS^\mu$ around the insertion,

$$Q^a=\int_{S^{d-1}}dS^\mu J_\mu^a, \qquad P_\nu=\int_{S^{d-1}}dS^\mu T_{\mu\nu},$$

and the conventions are completed by requiring $Q^a$ and $P_\nu$ to generate the chosen internal transformation and translation. Shrinking the sphere through an operator insertion then fixes one linear combination of the relevant three-point coefficients in terms of the operator’s charge or scaling dimension. The signs depend on whether the Ward identity is written with an active or passive transformation, so a calculation should test the integrated identity rather than compare an isolated sign.

Three-point data and conservation

Conformal covariance first produces a finite-dimensional space of tensor structures. Conservation is a linear map from that space to descendant structures. The physical separated-point coefficients lie in its kernel, while Ward identities impose inhomogeneous conditions associated with contact terms. In general $d$, conservation leaves three parity-even structures for $\langle TTT\rangle$; an identity special to $d=3$ reduces this to two. The complete parity-even construction and Ward identities are derived in Osborn and Petkou 1994, §§2–9. Three-dimensional theories can also admit a parity-odd structure, as exhibited for conserved-current correlators in Maldacena and Zhiboedov 2012, §§3–5. These dimension-dependent sectors and degeneracies must be imposed before fitting data, not inferred from an accidentally singular generic-$d$ basis.

The same procedure applies to $\langle JJJ\rangle$, $\langle JJT\rangle$, and correlators containing charged primaries:

1. enumerate conformal structures for the specified representations and permutation symmetries;
2. impose Bose or Fermi exchange signs and group-theory tensors;
3. impose current or stress-tensor conservation at separated points;
4. add the integrated Ward constraint in the fixed charge normalization;
5. quotient dimension-specific identities and transform the surviving coefficients to the desired basis.

A coefficient vector by itself is therefore incomplete. It must be accompanied by the ordered tensor basis, the two-point normalizations, the parity convention, and the map used to impose conservation.

Improvement and contact-term choices

If the theory contains a scalar $L$ of dimension $d-2$, the conserved tensor can sometimes be shifted by

$$T_{\mu\nu}\longmapsto T_{\mu\nu} +\xi\left(\partial_\mu\partial_\nu-\delta_{\mu\nu}\partial^2\right)L.$$

The conformal stress tensor is the symmetric, conserved choice whose trace vanishes at separated points, subject to anomalies and equations of motion. An improvement can change local contact terms and some apparent three-point coefficients, but it cannot be treated as an arbitrary redefinition once the Ward normalization and conformal-primary condition have been fixed.

Distributional terms deserve separate notation. Equations such as $\partial^\mu J_\mu=0$ and $\partial^\mu T_{\mu\nu}=0$ hold away from insertions; derivatives of time-ordered or Euclidean correlators also produce delta functions that implement the symmetry action. Dropping them makes the separated-point conservation test pass while the integrated Ward identity fails. Regularization can move local terms between correlators and counterterms, but it cannot alter the nonlocal separated-point CFT data.

In four dimensions, central-charge language is tied to a particular trace-anomaly convention. If

$$\langle T^\mu{}_\mu\rangle =\frac{c}{16\pi^2}W_{\mu\nu\rho\sigma}^2 -\frac{a}{16\pi^2}E_4+\text{local scheme-dependent terms},$$

then the two-point convention above gives $C_T=40c/\pi^4$, while $a$ is an independent combination of stress-tensor three-point data Osborn and Petkou 1994, §8. In odd dimensions there is no analogous local Weyl anomaly on a smooth closed manifold, so $C_T$ should not be renamed as an anomaly coefficient.

From Euclidean coefficients to collider constraints

After analytic continuation with a stated $i\epsilon$ prescription, a normalized state created by $J$ or $T$ has an energy one-point function on the celestial sphere. Its angular dependence is a linear combination of the same three-point coefficients. In a unitary Lorentzian CFT satisfying the assumptions of the averaged null-energy argument, nonnegative detected energy restricts those combinations. This is the origin of conformal-collider bounds; it is not a consequence of Euclidean conservation alone. The construction, including the relation between stress-tensor structures and energy distributions, is given in Hofman and Maldacena 2008, §§2–3.

To use a quoted collider region, first transform its basis and $C_T$ convention to the conventions above. Then check whether parity-odd structures, supersymmetry, additional conserved currents, or dimension-specific identities alter the allowed parameter space. The conformal-collider chapter carries out the positivity analysis; this page supplies its local CFT input.

Failure tests

Ward test. Integrate the divergence over a small sphere. The result must reproduce the chosen generator on every charged insertion and the translation generator for $T_{\mu\nu}$.

Separated-point test. Contract the proposed three-point function with a derivative at a point distinct from all other insertions. Any remainder signals a missing conservation constraint or a basis identity.

Basis test. Evaluate the structures at generic configurations and verify their rank in the physical dimension. A rank drop identifies an overcomplete generic-$d$ basis.

Positivity test. Do not infer $C_J>0$, $C_T>0$, or collider inequalities in a nonunitary continuation. Conservation fixes dimensions and relations, not Hilbert-space positivity.

Exercises

Show directly that $\partial^\mu I_{\mu\nu}(x)/(x^2)^{d-1}=0$ for $x\ne0$.

Solution

Differentiate the two terms. The derivative of $\delta_{\mu\nu}(x^2)^{1-d}$ is $2(1-d)x_\nu(x^2)^{-d}$. Using $\partial^\mu(x_\mu x_\nu)= (d+1)x_\nu$ and differentiating $(x^2)^{-d}$, the second term contributes $-2(1-d)x_\nu(x^2)^{-d}$, so the sum vanishes away from the origin.

Explain why an improvement cannot be added when the spectrum contains no scalar primary of dimension $d-2$.

Solution

The differential operator contributes two units of dimension. To preserve $\Delta_T=d$, its scalar input must have dimension $d-2$. Without such an operator, no local shift with the required spin, dimension, and conservation law exists.

References

• Hofman, D. M., and Maldacena, J. (2008), “Conformal collider physics: Energy and charge correlations,” Journal of High Energy Physics 2008(05), 012. doi:10.1088/1126-6708/2008/05/012. Open PDF
• Maldacena, J., and Zhiboedov, A. (2012), “Constraining conformal field theories with a slightly broken higher spin symmetry,” Classical and Quantum Gravity 30, 104003. doi:10.1088/0264-9381/30/10/104003. Open PDF
• Osborn, H., and Petkou, A. C. (1994), “Implications of conformal invariance in field theories for general dimensions,” Annals of Physics 231, 311–362. doi:10.1006/aphy.1994.1045. Open PDF