p -adic Mellin amplitudes
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Springer
Received: January 12, Revised: March 24, Accepted: April 8, Published: April 15,
2019 2019 2019 2019
Christian Baadsgaard Jepsena and Sarthak Parikhb a
Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544, U.S.A. b Division of Physics, Mathematics and Astronomy, California Institute of Technology, Pasadena, CA 91125, U.S.A.
E-mail: [email protected], [email protected] 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 arbitrarypoint 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. Keywords: AdS-CFT Correspondence, Scattering Amplitudes, Classical Theories of Gravity, Lattice Models of Gravity ArXiv ePrint: 1808.08333
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP04(2019)101
JHEP04(2019)101
p-adic Mellin amplitudes
Contents 1 Introduction 1.1 Mellin space and local zeta functions
1 3 7 7 11 13
3 From Mellin space to position space: an example
15
4 p-adic Mellin amplitudes 4.1 N -point contact diagram 4.2 The split representation of the bulk-to-bulk propagator 4.3 Exchange diagrams 4.4 Diagrams with two internal lines 4.5 Diagrams with three internal lines
18 18 22 24 28 31
5 Discussion 5.1 Comparison between p-adic and real Mellin amplitudes 5.2 Outlook
37 37 41
A Barnes lemmas: real and p-adic
43
1
Introduction
Anti-de Sitter/conformal field theory (AdS/CFT) duality [1–4] provides a powerful framework for investigating the properties of correlators, the basic observables, in strongly coupled CFTs. Early work in the subject [5–15] laid the foundation for computational techniques, especially in the context of the holographic evaluation of correlators via bulk Feynman diagram methods. Traditionally, CFT correlators are obtained in position space, which though physically intuitive, often falls short of utilizing the full power of conformal symmetry. Consequently, despite major advances in evaluating holographic correlators in position space, the study and computation of arbitrarily complicated bulk diagrams remained a challenging task. But beginning with the work of Mack [16], developed further in refs. [17–26] in the holographic context, Mellin amplitudes emerged as an effective tool in this regard. Analogous to momentum space for flat space scattering amplitudes, Mellin space can be regarded as the natural space for studying scattering amplitudes in AdS, one reason being that it manifestly takes into account the conformal symmetry of the underlying theory. While position space correlators are wri
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