$p$-adic Mellin Amplitudes

TL;DR

The paper develops a p-adic analogue of Mellin amplitudes for scalar operators in p-adic AdS/CFT, leveraging local zeta functions ζ_p and the Bruhat–Tits tree. It establishes a p-adic Mellin framework, derives a split representation for bulk-to-bulk propagators, and computes explicit closed-form Mellin amplitudes for N-point contact diagrams and tree-level diagrams with up to three internal lines, revealing a remarkable simplification relative to the real case due to the absence of descendants. The results reproduce the expected position-space structures upon inverse Mellin transform and mirror real-Mellin constructions with ζ_p in place of Euler gamma factors, while offering tractable, finite expressions. The work opens avenues for recursion, extensions to derivatives and spins, and potential loop-level analyses, suggesting a robust, computationally efficient bridge between p-adic and real holography. Overall, it solidifies the utility of Mellin space in the p-adic AdS/CFT context and provides a concrete toolkit for analyzing arbitrary-tree-level bulk diagrams.

Abstract

In this paper, we propose a $p$-adic analog of Mellin amplitudes for scalar operators, and present the computation of the general contact amplitude as well as arbitrary-point tree-level amplitudes for bulk diagrams involving up to three internal lines, and along the way obtain the $p$-adic version of the split representation formula. These amplitudes share noteworthy similarities with the usual (real) Mellin amplitudes for scalars, but are also significantly simpler, admitting closed-form expressions where none are available over the reals. The dramatic simplicity can be attributed to the absence of descendant fields in the $p$-adic formulation.

Figures (5)

• Figure 1: Left: $\mathcal{N}$-point bulk contact diagram. Right: Arbitrary-point bulk exchange diagram.
• Figure 2: The split representation (\ref{['HarmExp']}).
• Figure 3: (a) The bulk 4-point contact Feynman diagram for scalar fields with scaling dimensions $\Delta_i$. (b) The coordinate configuration on the Bruhat--Tits tree. Solid lines are geodesics on the Bruhat–Tits tree, tracing the path joining together the four points on the boundary of the tree, which is the projective line over the degree $n$ unramified extension of $\mathbb{Q}_p$. The figure is drawn for $u < 1$ where $u, v$ are defined in (\ref{['uvDef']}). For the $u=v=1$ configuration, the vertices on the Bruhat--Tits tree, labeled $v_l$ and $v_r$, become coincident.
• Figure 4: Integration contour for $\gamma_{12}$ for computing the position space 4-point contact amplitude starting from its Mellin representation (\ref{['pMel4pt']}). The circumference of the cylinder is $\frac{\pi}{\log p}$.
• Figure 5: The characteristic function $\gamma_p(x)$ can be expressed in terms of a closed contour integral running around a cylinder with a circumference of $\frac{2\pi}{k\log p}$.