Spinning Correlators and Tensor Structures

A spinning correlator is a vector in a finite-dimensional space of conformally covariant tensor structures. Its dimension depends on spacetime dimension, Spin cover, representations, parity, chirality, statistics, conservation, and permutation symmetry. The safe procedure is to enumerate a redundant invariant set, impose representation and dimension identities, test linear independence, and only then apply conservation and exchange constraints.

Required background. Spin and Tensor Representations supplies null-vector and auxiliary-spinor encodings. Scalar Two- and Three-Point Functions fixes the scalar normalization convention. Helpful background. Multiplets, Invariants, and Selection Rules supplies internal invariant tensors and exchange sectors.

Embedding-space building blocks

For a symmetric traceless primary of spin $\ell_i$, represent point $x_i$ by a projective null vector $P_i$ and its indices by $Z_i$, with

$$P_i^2=Z_i^2=P_i\mathbin{\cdot}Z_i=0, \qquad Z_i\sim Z_i+\alpha_iP_i.$$

Gauge-invariant polynomial building blocks for three points are

$$\begin{aligned} H_{ij}&=(Z_i\mathbin{\cdot}Z_j)(P_i\mathbin{\cdot}P_j) -(Z_i\mathbin{\cdot}P_j)(Z_j\mathbin{\cdot}P_i),\\ V_{i,jk}&= \frac{(Z_i\mathbin{\cdot}P_j)(P_i\mathbin{\cdot}P_k) -(Z_i\mathbin{\cdot}P_k)(P_i\mathbin{\cdot}P_j)} {P_j\mathbin{\cdot}P_k}. \end{aligned}$$

Overall signs depend on the embedding metric and the convention $P_{ij}=-2P_i\mathbin{\cdot}P_j$; they must be changed coherently. A parity-even three-point structure is a monomial

$$\prod_{i<j}H_{ij}^{n_{ij}} \prod_{i=1}^3V_{i,jk}^{m_i}, \qquad m_i+\sum_{j\neq i}n_{ij}=\ell_i.$$

The nonnegative integer solutions give a candidate basis. Schouten identities, Gram-determinant relations, Hodge dualities, and null structures can reduce it in fixed dimension. Direct evaluation at generic numerical configurations is a useful independence test, but it does not replace an analytic account of exceptional dimensions. The construction is developed in Costa et al. 2011, §§ 3–4.

The same conventions fix normalization already at two points. For symmetric-traceless primaries with the same $(\Delta,\ell)$ and conjugate internal quantum numbers,

$$\langle\mathcal O_a(P_1,Z_1)\mathcal O_b(P_2,Z_2)\rangle =G_{ab}\frac{H_{12}^{\ell}}{P_{12}^{\Delta+\ell}}.$$

The power $\Delta+\ell$ cancels the $P_i^{\ell}$ homogeneity of $H_{12}^{\ell}$, leaving projective weight $-\Delta$ at each point. In a reflection-positive non-null Hermitian sector, $G$ can be made the identity; otherwise it must remain in all contractions. Rescaling $H_{12}$ or changing its sign rescales the spin-$\ell$ metric by the corresponding power, so a “unit-normalized” spinning operator is meaningful only together with the declared $H$ convention.

Stress tensor with two scalars

Let $T(P_1,Z_1)$ have $(\Delta_1,\ell_1)=(d,2)$ and let $\phi_2,\phi_3$ be scalars. The degree constraints allow only

$$\langle T(P_1,Z_1)\phi_2(P_2)\phi_3(P_3)\rangle =\lambda_{T23} \frac{V_{1,23}^{\,2}} {P_{12}^{(d+2+\Delta_2-\Delta_3)/2} P_{13}^{(d+2+\Delta_3-\Delta_2)/2} P_{23}^{(\Delta_2+\Delta_3-d-2)/2}}.$$

The powers use the embedding homogeneity parameter $\tau_i=\Delta_i+\ell_i$ (not the physical twist $\Delta_i-\ell_i$): the denominator at points 1 and 2 carries $(\tau_1+\tau_2-\tau_3)/2$, and cyclically. This check is essential because using $\Delta_i$ in place of $\tau_i$ would give the wrong projective homogeneity at the spinning point.

Conservation at $P_1$ makes the separated-point expression vanish unless $\Delta_2=\Delta_3$ and their two-point pairing is allowed. If $\phi_2=\phi_3=\phi$ and $T$ is the Ward-normalized stress tensor, the remaining coefficient is not independent: it is fixed by $\Delta_\phi$ and the scalar two-point normalization Osborn and Petkou 1994, §§ 6–7. The explicit coefficient in the site’s physical-space conventions is derived on Conserved Currents and the Stress Tensor. A different normalization of $V$ changes $\lambda_{T\phi\phi}$, so the basis definition must accompany the number.

This example illustrates the correct order:

1. homogeneity and transversality give the candidate structure;
2. trace removal is built into $Z_1^2=0$;
3. conservation imposes a differential constraint;
4. the translation and dilatation Ward identities fix the coefficient; and
5. contact terms are added only in the distributional Ward identity.

Fermion–fermion–scalar structures

For spinors the orthogonal representation is insufficient: one must choose $\operatorname{Spin}(d)$ in Euclidean signature or $\operatorname{Spin}(d-1,1)$ in Lorentzian signature, together with gamma matrices, chirality, and conjugation. In a conformal frame with points at $0$, a unit vector $e$, and infinity, complex Dirac spinors admit candidate singlets

$$\bar s_1s_2, \qquad \bar s_1\gamma(e)s_2.$$

Here $\gamma(e)=e_\mu\gamma^\mu$. Their covariant lifts are independent only if the actual restricted representations contain both singlets. Weyl chirality can remove one; Majorana or symplectic reality can relate coefficients; gamma duality and Fierz identities can collapse structures in special dimensions. If the two fermions are identical, exchanging points gives a minus sign and also acts on the tensor basis. The allowed coefficient vector is therefore an eigenspace of the full exchange matrix, not a sign attached after the calculation. A general-dimensional polarization-spinor construction is given in Isono 2017, §§ 2–3.

Conservation as a linear system

Write an $n$-point correlator in a declared basis,

$$\langle\mathcal O_1\cdots\mathcal O_n\rangle =\sum_{A=1}^{N}\lambda_A\,\mathcal T_A(P_i,Z_i).$$

At a unitarity-shortening value, conservation acts through a differential operator $\mathscr D_i$ appropriate to the representation:

$$\mathscr D_i\mathcal T_A =\sum_B C_{i,BA}\,\widehat{\mathcal T}_B, \qquad C_i\boldsymbol\lambda=0.$$

The independent conserved structures span $\ker C_i$ after all dimension identities have been imposed. Computing the null space before removing linearly dependent $\mathcal T_A$ creates spurious solutions. At coincident points, $\mathscr D_i$ also produces contact terms fixed by Ward identities; the homogeneous equation applies only at separated points.

Permutations, parity, and basis covariance

A permutation $\pi$ acts by

$$\mathcal T_A(\pi\cdot P,\pi\cdot Z) =\mathsf P(\pi)_A{}^B\mathcal T_B(P,Z).$$

For identical bosons, the coefficient vector is invariant; for identical fermions, it acquires the parity of the permutation after internal-index exchange. The matrices must satisfy the permutation-group relations, such as $\mathsf P((12))^2=\mathbf1$. This is a strong implementation check.

Parity-odd structures use an epsilon tensor in physical or embedding space. They exist only when enough independent vectors are available and can become dependent in low dimension. Color alone must never distinguish parity sectors in a later plot; the basis labels must state “even” or “odd.”

Obligation	Analytic check	Common false inference
Spin cover	Every bilinear is an intertwiner of the chosen Spin representations	An $SO(d)$ tensor formula is applied to a spinor
Trace and gauge quotient	Replace $Z_i\to Z_i+\alpha_iP_i$ and verify invariance	A longitudinal structure is counted
Independence	Reduce Schouten/Fierz identities at the actual $d$	A generic-$d$ count is used in $d=3$
Conservation	Solve the separated-point differential system	Conservation is imposed by deleting a coefficient by inspection
Statistics	Diagonalize the full exchange matrix including internal tensors	A fermionic sign is attached to each scalar coefficient
Normalization	Record basis rescalings and two-point metrics	Coefficients from different bases are compared numerically

Mixed Correlators and Global-Symmetry Sectors combines these permutation matrices with internal group projectors. Spinning conformal blocks require the same three-point bases at the left and right OPE vertices; changing either basis conjugates the block matrix and the OPE vectors together.

Common pitfalls

A candidate monomial is automatically independent. Low-dimensional Gram and Fierz identities can make it zero or dependent. Test the actual dimension and representation.

Conservation fixes every coefficient. It gives homogeneous separated-point relations. Ward identities and contact terms are required to determine charge-normalized coefficients such as $T\phi\phi$.

A parity label is dimension independent. The existence and number of epsilon-tensor structures depend on dimension and on whether parity belongs to the chosen global group.

Exercises

Count the parity-even three-point structures for two vectors and one scalar before conservation.

Solution

The degree constraints allow either $H_{12}$ or $V_{1,23}V_{2,13}$. They are independent for generic $d\geq3$, giving two structures. Identical exchange, current conservation, parity, or low-dimensional identities may reduce this number afterward.

References

• Costa, Miguel S., João Penedones, David Poland, and Slava Rychkov. “Spinning Conformal Correlators.” Journal of High Energy Physics 2011, no. 11 (2011): 071. DOI; Open PDF
• Isono, Hiroshi. “On Conformal Correlators and Blocks with Spinors in General Dimensions.” Physical Review D 96 (2017): 065011. DOI; Open PDF
• 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