Mellin Contact Polynomials and Exchange Poles

Local AdS contact interactions give polynomial Mellin amplitudes, while exchange of a bulk particle gives an infinite family of poles whose residues factorize into lower-point data. The polynomial degree measures derivative order only after equations of motion and crossing are imposed. Exchange amplitudes remain ambiguous up to contact polynomials.

Required background. Contact Witten diagrams and exchange Witten diagrams supply the position-space objects. Mellin conventions separates measure poles.

Helpful background. Large-N crossing and contact ambiguities supplies the CFT freedom.

Polynomial and meromorphic structures

A local vertex with $2k$ derivatives yields a crossing polynomial of degree at most $k$ in $s,t,u_M$, modulo $s+t+u_M=\sum_i\Delta_i$. For exchange of dimension $\Delta_\chi$ and spin $J$ in the $s$ channel,

$$M_{\mathrm{exch}}(s,t)= \sum_{m=0}^{\infty} \frac{Q_{J,m}(t)}{s-(\Delta_\chi-J+2m)}+P(s,t),$$

where the Mack-polynomial residues $Q_{J,m}$ encode spin and descendant level. Their leading normalization factorizes into the two cubic OPE coefficients. $P$ is a contact polynomial fixed only by additional conventions or crossing data Fitzpatrick et al. 2011.

First application: derivative contact versus scalar exchange

Compare a four-derivative scalar contact vertex with $J=0$ exchange. The contact amplitude is quadratic, for example a crossing combination proportional to $s^2+t^2+u_M^2$. It has no dynamical poles. Scalar exchange instead has poles at $s=\Delta_\chi+2m$ with residues of bounded polynomial degree in $t$, plus an allowed contact polynomial.

The first pole identifies the primary $\chi$ and the later poles its AdS descendants, not new particles. In the flat limit the poles condense into the ordinary propagator singularity. Residue factorization checks the normalization against $C_{12\chi}C_{34\chi}$ Penedones 2011.

Adversarial control: promote Gamma poles to particles

Read the universal Gamma poles at $s=\Delta_1+\Delta_2+2n$ as poles of $M$. A pure contact diagram would then appear to contain infinitely many exchanged particles, contradicting its constant or polynomial dynamical amplitude. Removing the Gamma measure restores the correct distinction between double-trace kinematics and single-trace dynamics.

The evidence ceiling is the spectrum and derivative information visible at the measured correlator order, with contact freedom stated. Pole data do not determine a unique off-shell Lagrangian, and polynomial boundedness at finite order does not prove an infinite higher-spin gap. Double-trace data extracts the binding information these structures induce.

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

• Fitzpatrick, A. L., Kaplan, J., Penedones, J., Raju, S., and van Rees, B. C. (2011), “A Natural Language for AdS/CFT Correlators,” Journal of High Energy Physics 2011(11), 095. arXiv:1107.1499.
• Penedones, J. (2011), “Writing CFT Correlation Functions as AdS Scattering Amplitudes,” Journal of High Energy Physics 2011(03), 025. arXiv:1011.1485.