$p$-adic Mellin Amplitudes

TL;DR

This paper introduces a $p$-adic version of Mellin amplitudes for scalar operators, providing simpler closed-form expressions and new formulas for contact and tree-level amplitudes, highlighting the unique features of $p$-adic holography.

Contribution

It develops the first $p$-adic Mellin amplitude framework, including contact and tree-level amplitudes, and derives the $p$-adic split representation formula.

Findings

$p$-adic Mellin amplitudes are simpler than their real counterparts.

Closed-form expressions for amplitudes are obtained where real versions lack them.

The absence of descendant fields explains the simplicity of $p$-adic amplitudes.

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.