Conserved Currents and the Stress Tensor

Conserved currents and the stress tensor are short conformal multiplets with additional normalization data fixed by Ward identities. Representation theory fixes their dimensions, $\Delta_J=d-1$ and $\Delta_T=d$, but it does not fix the choice of internal generators, the normalization of the conserved charges, improvement terms, contact terms, or the two-point coefficients $C_J$ and $C_T$. Those choices must be aligned before an OPE coefficient or central-charge ratio can be compared between sources.

Required background. Primaries, Descendants, and Conformal Multiplets supplies shortening and descendants. Localized Transformations and Ward–Takahashi Identities supplies distributional Ward identities. Helpful background. Unitarity Bounds and Null States derives the positive-energy shortening bounds. Contact Terms, Equal-Time Commutators, and Schwinger Terms supplies the contact-term qualifications.

Conserved conformal multiplets

Let $J_\mu^a$ be a spin-one primary in a reflection-positive $d\geq3$ CFT. The spin-one bound and its saturation condition are

$$\Delta_J\geq d-1, \qquad \Delta_J=d-1 \quad\Longleftrightarrow\quad \partial^\mu J_\mu^a=0,$$

where the reverse implication assumes an ordinary irreducible positive-energy conformal module. The null state is the level-one scalar $P^\mu\lvert J_\mu^a\rangle$. The index $a$ labels a basis of the internal Lie algebra and is independent of spacetime spin.

Similarly, a conformal stress tensor is a symmetric traceless spin-two primary satisfying

$$T_{\mu\nu}=T_{\nu\mu}, \qquad \delta^{\mu\nu}T_{\mu\nu}=0, \qquad \partial^\mu T_{\mu\nu}=0, \qquad \Delta_T=d.$$

Conservation removes the level-one vector divergence. Symmetry and tracelessness select the spin-two irrep at level zero. In a theory initially presented only with a conserved symmetric tensor whose trace is a double derivative, an improvement may be needed before this conformal primary is obtained.

The equivalence between saturation and conservation is representation-theoretic Poland, Rychkov, and Vichi 2019, § III.E. The existence of a current with a specified charge, or of a stress tensor generating translations, is dynamical input encoded by Ward identities. A conserved spin-one operator that integrates to zero on every physical state is not automatically a nontrivial global-symmetry current.

Distributional Ward identities

Away from other insertions, conservation is an operator equation. At coincident points it becomes a distributional statement. Choose internal generators $t^a$ by $\delta^a\mathcal O_i=(t^a)_i{}^j\mathcal O_j$. With the Euclidean sign convention used here,

$$\partial^\mu \left\langle J_\mu^a(x)\prod_{k=1}^n\mathcal O_k(x_k)\right\rangle =-\sum_{k=1}^n\delta^{(d)}(x-x_k) \left\langle \mathcal O_1\cdots(t^a\mathcal O_k)(x_k)\cdots\mathcal O_n \right\rangle.$$

Integrating over a small sphere gives the charge action. If $S_d=2\pi^{d/2}/\Gamma(d/2)$ is the area of the unit $(d-1)$-sphere, the leading current–operator OPE is

$$J_\mu^a(x)\mathcal O_i(0) \sim -\frac{x_\mu}{S_d\,\lvert x\rvert^d} (t^a)_i{}^j\mathcal O_j(0)+\cdots.$$

Changing the sign used to define $\delta^a$ changes both displayed signs and no observable conclusion. Rescaling $t^a$ rescales $J^a$ and therefore $C_J$; a quoted $C_J$ is meaningless without the generator normalization.

For translations, the Ward identity is

$$\partial^\mu \left\langle T_{\mu\nu}(x)\prod_k\mathcal O_k(x_k)\right\rangle =-\sum_k\delta^{(d)}(x-x_k)\partial_{x_k^\nu} \left\langle\prod_k\mathcal O_k(x_k)\right\rangle$$

for scalar insertions; spinning insertions add the local rotation response under a general metric variation. Dilatation and special-conformal identities follow by contracting $T_{\mu\nu}$ with the corresponding conformal Killing vector when the improved trace vanishes. Osborn and Petkou give the general separated-point forms together with the contact analysis in Osborn and Petkou 1994, §§ 2–4.

Two-point normalizations

Define the inversion tensor

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

and the symmetric-traceless projector transported between two points,

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

At separated Euclidean points, conformal symmetry fixes

$$\langle J_\mu^a(x)J_\nu^b(0)\rangle =C_J\,\kappa^{ab}\frac{I_{\mu\nu}(x)}{(x^2)^{d-1}},$$

where $\kappa^{ab}$ is the declared invariant metric on the internal algebra, and

$$\langle T_{\mu\nu}(x)T_{\rho\sigma}(0)\rangle =C_T\frac{\mathcal I_{\mu\nu,\rho\sigma}(x)}{(x^2)^d}.$$

Reflection positivity gives $C_J>0$ for nonzero Hermitian currents in a positive internal channel and $C_T>0$ for a nonzero stress tensor. Some literature extracts factors of $S_d^{-2}$ from these definitions; numerical values must be converted before comparison. Unlike a freely rescalable current basis, the stress tensor normalization is fixed by the translation Ward identity, so $C_T$ is physical once coordinates and the metric-response convention

$$\delta S=-\frac12\int d^dx\,\sqrt g\, T_{\mu\nu}\,\delta g^{\mu\nu}$$

are fixed. The standard two-point conventions are reviewed in Simmons-Duffin 2017, § 5.

The scalar–scalar–stress-tensor coefficient

Let a scalar primary have

$$\langle\phi(x)\phi(0)\rangle =\frac{C_\phi}{(x^2)^\Delta}.$$

For $r_{ij}=\lvert x_i-x_j\rvert$, define

$$Z_\mu=\frac{(x_{13})_\mu}{x_{13}^2} -\frac{(x_{23})_\mu}{x_{23}^2}.$$

Conformal symmetry permits one parity-even $\phi\phi T$ structure,

$$\left\langle\phi(x_1)\phi(x_2)T_{\mu\nu}(x_3)\right\rangle =f_{\phi\phi T} \frac{Z_\mu Z_\nu-\delta_{\mu\nu}Z^2/d} {r_{12}^{2\Delta-d+2}r_{23}^{d-2}r_{31}^{d-2}}.$$

Integrate $x^\nu T_{\mu\nu}$ over a small sphere around one scalar. With the outward-normal convention for the punctured correlator, the dilatation Ward identity fixes

$$\boxed{ f_{\phi\phi T} =-\frac{d\Delta}{(d-1)S_d}C_\phi. }$$

Equivalently, the leading stress-tensor–scalar OPE is

$$T_{\mu\nu}(x)\phi(0) \sim -\frac{d\Delta}{(d-1)S_d} \frac{x_\mu x_\nu-\delta_{\mu\nu}x^2/d}{(x^2)^{(d+2)/2}} \phi(0)+\cdots.$$

The sphere integral of $-x^\nu T_{\mu\nu}$ is then $\Delta$ on $\phi$. This directly checks the sign and factor $d/(d-1)$. For $C_\phi=1$ and the unit-normalized operator $\widehat T=T/\sqrt{C_T}$ in the two-point convention above, the corresponding OPE coefficient is

$$\lambda_{\phi\phi\widehat T} =-\frac{d\Delta}{(d-1)S_d\sqrt{C_T}}.$$

Thus the magnitude of the stress-tensor contribution is not independent CFT data: it is fixed by $\Delta$ and $C_T$. The derivation and formula appear in Simmons-Duffin 2017, Eqs. (74)–(75).

Improvements and what they change

A conserved symmetric stress tensor can be shifted by the scalar improvement

$$T'_{\mu\nu} =T_{\mu\nu} +\left(\partial_\mu\partial_\nu -\delta_{\mu\nu}\partial^2\right)L.$$

The added term is identically conserved, while its trace is $-(d-1)\partial^2L$. If

$$T^\mu{}_{\mu}=(d-1)\partial^2L,$$

then $T'$ is traceless. Such an improvement exists only if the theory contains an appropriate scalar operator $L$ of dimension $d-2$ with the required locality and symmetry properties. It is not licensed merely because the integrated trace vanishes. The improved stress tensor and its renormalization role originate in Callan, Coleman, and Jackiw 1970.

Likewise,

$$J'^a_\mu=J^a_\mu+\partial^\nu B^a_{\nu\mu}, \qquad B^a_{\nu\mu}=-B^a_{\mu\nu},$$

is identically conserved. Under boundary conditions for which the surface term vanishes, it generates the same integrated charge, but local correlators and contact terms can change. Stress-tensor improvements can also change separated-point $C_T$ if they mix $T$ with another dimension-$d$ primary descendant; one must first choose the conformal primary that obeys the Ward normalization.

Separated points, contact terms, and anomalies

The formulas for $C_J$, $C_T$, and $f_{\phi\phi T}$ are separated-point statements. Coincident correlators may contain derivatives of delta functions. Their coefficients can depend on local counterterms, while anomaly coefficients can be scheme independent under the allowed counterterms. In even dimensions, Weyl anomalies modify trace Ward identities on curved backgrounds even when $T^\mu{}_{\mu}=0$ at separated points in flat space. In parity-violating odd-dimensional theories, parity-odd contact terms may be quantized only modulo counterterm shifts.

Consequently, a complete comparison records:

Quantity	Required convention	What can alter it
$C_J$	Internal generator metric $\kappa^{ab}$ and current two-point prefactor	Rescaling generators or mixing currents
$C_T$	Stress response to the metric and whether $S_d^{-2}$ is extracted	Stress-tensor improvement and convention conversion
$f_{\phi\phi T}$	Scalar two-point coefficient, $T$ Ward normalization, tensor-structure sign	Only linked convention changes; it is Ward fixed
Contact terms	Distributional extension and local counterterm scheme	Finite local counterterms and improvements
Anomaly coefficient	Background fields, allowed counterterms, parity and dimension	Scheme changes only within the appropriate cohomology class

Scale versus Conformal Invariance tests whether a virial current permits the required stress-tensor improvement. Spinning Correlators and Tensor Structures carries $C_J$, $C_T$, and Ward-fixed OPE coefficients into correlators, while the general derivation of Ward identities and anomaly contact terms remains in Symmetry and Gauge Structure.

Common pitfalls

Conservation means zero divergence inside every correlator. It holds at separated points. Ward identities require delta-function terms when the current crosses charged insertions.

A dimension-$d$ spin-two primary is automatically the stress tensor. It is conserved under the unitary shortening hypotheses, but it generates translations only if its Ward identity has the required normalization. Product CFTs can contain several conserved spin-two operators while the physical stress tensor is the sum that translates all sectors.

$C_J$ values can be compared without generator conventions. Rescaling the internal generators and currents changes $C_J$. Specify $\kappa^{ab}$ or a charge normalization first.

Improvements are harmless at every level. They preserve integrated charges under suitable boundary conditions, but can change local correlators, contact terms, and which operator is a conformal primary.

Exercises

Recover the coefficient in the leading $T_{\mu\nu}(x)\phi(0)$ OPE by integrating the dilatation charge.

Solution

Assume the coefficient is $A$ multiplying $(x_\mu x_\nu-\delta_{\mu\nu}x^2/d)/(x^2)^{(d+2)/2}$. On a sphere of radius $r$,

$$n^\mu x^\nu \left(x_\mu x_\nu-\frac1d\delta_{\mu\nu}x^2\right) =r^3\frac{d-1}{d}.$$

The denominator and area element cancel the remaining powers of $r$, leaving $A S_d(d-1)/d$. The charge is $-\int dS^\mu x^\nu T_{\mu\nu}$ in the chosen convention, so setting its action equal to $\Delta$ gives $A=-d\Delta/[(d-1)S_d]$.

Show that the scalar improvement is identically conserved and compute its trace.

Solution

Commuting derivatives gives

$$\partial^\mu(\partial_\mu\partial_\nu-\delta_{\mu\nu}\partial^2)L=0.$$

Contracting indices gives $(1-d)\partial^2L$. Therefore a trace $(d-1)\partial^2L$ is removed by adding the displayed improvement.

References

• Callan, Curtis G., Sidney Coleman, and Roman Jackiw. “A New Improved Energy–Momentum Tensor.” Annals of Physics 59 (1970): 42–73. DOI
• Osborn, Hugh, and Andreas Petkou. “Implications of Conformal Invariance in Field Theories for General Dimensions.” Annals of Physics 231 (1994): 311–362. DOI; Open PDF
• Poland, David, Slava Rychkov, and Alessandro Vichi. “The Conformal Bootstrap: Theory, Numerical Techniques, and Applications.” Reviews of Modern Physics 91 (2019): 015002. DOI; Open PDF
• Simmons-Duffin, David. “TASI Lectures on the Conformal Bootstrap.” In New Frontiers in Fields and Strings, 1–74. Singapore: World Scientific, 2017. DOI; Open PDF