Imported Mellin Representations and Bulk-Normalization Conventions

A Mellin representation splits a correlator into a universal Gamma-function measure and a dynamical Mellin amplitude. The measure already contains double-trace pole families. Moving those Gamma factors into the object called “the amplitude” changes the apparent spectrum and polynomial degree, so a bulk interpretation begins with an explicit convention map.

Required background. The GKPW generating functional supplies the bulk normalization. Mellin-space CFT correlators supplies the imported representation.

Helpful background. Diagram normalization benchmarks supplies coefficient fixtures.

Identical-scalar Mellin convention

For four identical scalars of dimension $\Delta$, choose dimensionless variables $s,t,u_M$ with $s+t+u_M=4\Delta$ and write

$$\mathcal G_{\mathrm{conn}}(u,v)= \int\frac{ds\,dt}{(4\pi i)^2} u^{s/2}v^{t/2-\Delta} \prod_{x=s,t,u_M}\Gamma^2\!\left(\frac{2\Delta-x}{2}\right) M(s,t).$$

The contours separate left and right pole families in a domain where the integral converges, followed by analytic continuation. Crossing permutes $s,t,u_M$. Alternative conventions shift variables or absorb Gamma functions and normalization factors; they are valid only with an explicit translation Mack 2009.

The Gamma poles at $s=2\Delta+2n$ encode the mean-field double-trace sequence and are present even when $M$ is constant. Poles of $M$ instead signal dynamical exchanged single-trace families, subject to possible cancellations with zeros or contact additions.

First application: a scalar contact correlator

A zero-derivative $\phi^4$ contact diagram has

$$M_{\phi^4}(s,t)=\lambda_{\mathrm{Mellin}}$$

in this reduced convention. Substitution still yields all double-trace powers and logarithms because the Gamma measure is nontrivial. The constant is related to the canonically normalized bulk coupling only after the four external two-point coefficients and powers of $L$ are included Penedones 2011.

This example fixes three checks: $M$ is crossing symmetric, has polynomial degree zero, and has no dynamical exchange poles. The position-space result must agree with the normalized $D$-function.

Adversarial control: absorb a Gamma factor

Define $\widetilde M=\Gamma^2((2\Delta-s)/2)M$ but continue to interpret $\widetilde M$ with the original measure. The poles at $s=2\Delta+2n$ now appear to be a new exchanged tower and the constant contact amplitude appears meromorphic. Comparing with the original $D$-function and the known absence of a single-trace internal line rejects the false spectrum.

The evidence ceiling is a convention-correct representation of a CFT correlator. Mellin meromorphy alone does not prove a local bulk, and large Mellin variables are not yet flat-space momenta. The Penedones limit supplies the additional scaling.

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

• Mack, G. (2009), “D-Independent Representation of Conformal Field Theories in D Dimensions via Transformation to Auxiliary Dual Resonance Models,” arXiv:0907.2407.
• Penedones, J. (2011), “Writing CFT Correlation Functions as AdS Scattering Amplitudes,” Journal of High Energy Physics 2011(03), 025. arXiv:1011.1485.