Spinning and Tensor Witten Diagrams

Spinning Witten diagrams must satisfy both bulk gauge constraints and the finite set of boundary conformal tensor structures. Polarization variables make these conditions algebraic, but pure-gauge terms, derivative vertices, and boundary contributions must still cancel. The physical result is the coefficient of a conserved boundary structure, not an unprojected index expression.

Required background. Spinning bulk fields and boundary operators supplies the representation map. AdS propagators supplies normalized internal and external legs.

Helpful background. Spinning tensor structures supplies the boundary basis.

Polarizations and gauge constraints

In embedding space a spin-$J$ primary is encoded as $O(P,Z)$ with $P^2=Z^2=P\cdot Z=0$ and homogeneity $O(\lambda P,\alpha Z)=\lambda^{-\Delta}\alpha^JO(P,Z)$. The redundancy $Z\sim Z+\beta P$ imposes transversality. Bulk symmetric tensors admit an analogous auxiliary polarization $W$, and their propagators are defined only up to pure-gauge terms when coupled to conserved sources.

For a bulk gauge field interacting through $A_MJ^M$, the replacement $A_M\to A_M+\nabla_M\Lambda$ changes the diagram by

$$\delta\mathcal W=\int dX\sqrt g\,\nabla_M\Lambda\,J^M =-\int dX\sqrt g\,\Lambda\nabla_MJ^M+\int_{\partial}\!d\Sigma_M\Lambda J^M.$$

Both terms must vanish or be matched to the boundary Ward identity. Dropping the surface term before specifying sources is not a proof of gauge invariance. Embedding-space propagators and differential operators organize this calculation efficiently Costa et al. 2014.

First application: a current-current-scalar structure

For two conserved currents $J(P_1,Z_1)$, $J(P_2,Z_2)$ and a scalar $O(P_3)$, conformal invariance permits a small tensor basis built from

$$H_{12}=(Z_1\!\cdot Z_2)(P_1\!\cdot P_2) -(Z_1\!\cdot P_2)(Z_2\!\cdot P_1)$$

and the corresponding $V_i$ structures. A Witten diagram with two gauge bulk-to-boundary propagators and one scalar kernel is reduced onto this basis. Current conservation fixes a linear relation among coefficients; in a parity-even minimal coupling the surviving coefficient is proportional to the bulk cubic coupling after two-point normalizations are divided out.

Replacing $Z_1$ by $P_1$ makes $H_{12}$ and the physical combination vanish. The same result must follow from the integrated bulk Ward identity. Agreement of these algebraic and integral tests detects gauge-dependent propagator pieces and missed boundary terms.

Adversarial control: a pure-gauge external state

Insert a longitudinal polarization and retain only the bulk integration-by-parts term. A nonzero answer signals either a nonconserved source, an anomalous boundary Ward identity, or a missing boundary contact contribution. Projecting the raw expression numerically onto one tensor component can hide this failure; all independent structures and conservation equations must be checked.

The evidence ceiling is a gauge-invariant AdS correlator contribution in a declared tensor basis and quantization. It does not determine a unique higher-spin theory or guarantee causality at energies near a higher-spin gap. Lorentzian diagrams next add ordering and state dependence.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Costa, M. S., Gonçalves, V., and Penedones, J. (2014), “Spinning AdS Propagators,” Journal of High Energy Physics 2014(09), 064. arXiv:1404.5625.
• Costa, M. S., Penedones, J., Poland, D., and Rychkov, S. (2011), “Spinning Conformal Correlators,” Journal of High Energy Physics 2011(11), 071. arXiv:1109.6321.