# Recursion Relations in $p$-adic Mellin Space

## TL;DR

This paper introduces recursive rules for calculating $p$-adic Mellin amplitudes at tree-level, providing closed-form expressions and connecting them with Feynman rules and flat-space amplitude recursion methods.

## Contribution

It develops a recursive framework for $p$-adic Mellin amplitudes, linking them to Feynman rules and on-shell recursion techniques, and extends the approach to real Mellin amplitudes and pre-amplitudes.

## Key findings

- Derived recursive rules for $p$-adic Mellin amplitudes.
- Established connection with Feynman rules for real Mellin amplitudes.
- Showed pre-amplitudes can be expressed as products of Mellin amplitudes with complexified dimensions.

## Abstract

In this work, we formulate a set of rules for writing down $p$-adic Mellin amplitudes at tree-level. The rules lead to closed-form expressions for Mellin amplitudes for arbitrary scalar bulk diagrams. The prescription is recursive in nature, with two different physical interpretations: one as a recursion on the number of internal lines in the diagram, and the other as reminiscent of on-shell BCFW recursion for flat-space amplitudes, especially when viewed in auxiliary momentum space. The prescriptions are proven in full generality, and their close connection with Feynman rules for real Mellin amplitudes is explained. We also show that the integrands in the Mellin-Barnes representation of both real and $p$-adic Mellin amplitudes, the so-called pre-amplitudes, can be constructed according to virtually identical rules, and that these pre-amplitudes themselves may be re-expressed as products of particular Mellin amplitudes with complexified conformal dimensions.