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 strongest conclusion 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 compares these assumptions and limitations.

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.