Recursion Relations in $p$-adic Mellin Space

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.

45 pages + appendices, several figures

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• Holographic dual of the five-point conformal block
• On spinning loop amplitudes in Anti-de Sitter space
• Opacity from Loops in AdS
• The boundary theory of a spinor field theory on the Bruhat-Tits tree
• Effective field theory on a finite boundary of the Bruhat-Tits tree
• Finite Temperature at Finite Places
• Toward Holography on Biregular Trees