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Recursion Relations in $p$-adic Mellin Space

Christian Baadsgaard Jepsen, Sarthak Parikh

TL;DR

The work develops a unified, recursive framework to construct tree-level $p$-adic Mellin amplitudes for arbitrary scalar diagrams and shows that these rules mirror, in key structural ways, the real Mellin amplitude Feynman rules. It introduces a twofold recursive approach: (i) a graph-level recursion on the number of internal lines and (ii) an on-shell, BCFW-like recursion in auxiliary momentum space, both proven to yield closed-form expressions and to reproduce known results. A universal pre-amplitude construction is established, valid for both $p$-adic and real theories, with pre-amplitudes expressible as products of Mellin amplitudes with complexified dimensions, and contour integrals that reconstruct full amplitudes by gluing undressed diagrams. The findings suggest deep structural parallels between real and $p$-adic AdS/CFT Mellin space, offer practical computational tools, and point toward broader adelic or loop-level generalizations that could impact holographic CFT analyses.

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.

Recursion Relations in $p$-adic Mellin Space

TL;DR

The work develops a unified, recursive framework to construct tree-level -adic Mellin amplitudes for arbitrary scalar diagrams and shows that these rules mirror, in key structural ways, the real Mellin amplitude Feynman rules. It introduces a twofold recursive approach: (i) a graph-level recursion on the number of internal lines and (ii) an on-shell, BCFW-like recursion in auxiliary momentum space, both proven to yield closed-form expressions and to reproduce known results. A universal pre-amplitude construction is established, valid for both -adic and real theories, with pre-amplitudes expressible as products of Mellin amplitudes with complexified dimensions, and contour integrals that reconstruct full amplitudes by gluing undressed diagrams. The findings suggest deep structural parallels between real and -adic AdS/CFT Mellin space, offer practical computational tools, and point toward broader adelic or loop-level generalizations that could impact holographic CFT analyses.

Abstract

In this work, we formulate a set of rules for writing down -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 -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.

Paper Structure

This paper contains 28 sections, 213 equations, 1 figure.

Table of Contents

  1. Introduction and Summary
  2. Mellin Amplitudes and Pre-Amplitudes
  3. Construction of Mellin Amplitudes
  4. Recursive prescription for $p$-adic Mellin amplitudes
  5. Feynman rules for real Mellin amplitudes
  6. Construction of Pre-Amplitudes
  7. Obtaining pre-amplitudes from Mellin amplitudes
  8. Pre-amplitude examples
  9. Integrating the pre-amplitude to obtain the amplitude
  10. On-Shell Recursion for Mellin Amplitudes
  11. Factorization property of undressed diagrams
  12. $p$-adic on-shell recursion relation and proof
  13. Outlook
  14. Useful Formulae and Conventions
  15. Inductive Proof of Prescription \ref{['pres:Preamp']} for Pre-Amplitudes
  16. ...and 13 more sections

Figures (1)

  • Figure 1: (a) A five-point diagram built out of cubic interaction vertices. The Mandelstam variables $s_A$ and $s_B$ are defined in (\ref{['sAsBDef']}). (b) A five-point diagram built out of cubic and quartic interaction vertices.

Theorems & Definitions (13)

  • Example 1
  • Example 2
  • Example 3
  • Example 4
  • Example 5
  • Example 6
  • Example 7
  • Example 8
  • Example 9
  • Example 10
  • ...and 3 more