Mellin Amplitudes and Penedones-Type Flat-Space Limits

The Penedones flat-space limit extracts a scattering amplitude by sending the AdS radius $L$ to infinity while Mellin variables scale as $L^2$ times fixed flat Mandelstam invariants. An integral transform removes external wavefunction normalization and the AdS energy spread. Taking Mellin variables large at fixed $L$ probes a high-energy AdS correlator, not automatically an S-matrix.

Required background. Genus counting and the string regime supplies the perturbative order. Mellin conventions supplies the normalized input.

Helpful background. S- and T-matrix normalization supplies the target. Bulk-point and flat-space limits supplies the boundary result.

The large-radius scaling transform

For fixed flat invariants $S_{ij}=-(p_i+p_j)^2$, the Mellin variables scale schematically as

$$\delta_{ij}=\frac{L^2S_{ij}}{4\alpha}, \qquad L\to\infty,$$

inside an inverse-Laplace integral over $\alpha$. The exact transform contains a known factor $\mathcal N(L,\Delta_i,d)$ from external wavefunctions and Newton/bulk-field normalization:

$$\mathcal T(S_{ij})= \lim_{L\to\infty}\mathcal N^{-1} \int_{c-i\infty}^{c+i\infty}\frac{d\alpha}{2\pi i} e^\alpha\alpha^\beta M_L\!\left(\delta_{ij}=\frac{L^2S_{ij}}{4\alpha}\right).$$

The exponent $\beta$ and $\mathcal N$ depend on $d$ and the external dimensions and must be copied with the Mellin convention, not guessed Penedones 2011.

First application: a polynomial contact interaction

Define a canonically normalized flat bulk vertex $g_{2k}(\partial^{2k}\phi^4)$ and its AdS Mellin polynomial by

$$M_L(s,t)=\mathcal N_L\,g_{2k}L^{-2k}P_k(s,t,u_M),$$

where $P_k$ is homogeneous of degree $k$. Substituting $s=L^2S/(2\alpha)$ gives

$$L^{-2k}P_k(s,t,u_M) \longrightarrow (2\alpha)^{-k}P_k(S,T,U).$$

The $\alpha$ integral combines this factor with the external normalization and returns the flat polynomial contact amplitude $g_{2k}P_k(S,T,U)$. For $k=0$, a constant Mellin amplitude becomes a constant flat $\phi^4$ amplitude. This makes every derivative-related $L$ power explicit; the remaining $\mathcal N_L$ is fixed by the two-point benchmark.

Adversarial control: reverse or omit the limits

Send $s,t\to\infty$ at fixed $L$. AdS remains a finite box, energy levels do not coalesce, and repeated boundary reflections are not removed. Alternatively take $L\to\infty$ before dividing by $\mathcal N_L$; the answer vanishes or diverges according to external normalization rather than yielding an LSZ amplitude. Large $N$ must also be ordered consistently with the desired loop order.

The evidence ceiling is a flat-space amplitude for the sector, external states, perturbative order, and scaling covered by the transform. It does not prove that the full CFT has a local bulk or fix nonperturbative $e^{-N}$ effects. Contact polynomials and exchange poles interpret the finite-$L$ analytic structures first.

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.