Weights, recursion relations and projective triangulations for positive geometry of scalar theories
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Springer
Received: August 8, 2020 Accepted: September 6, 2020 Published: October 7, 2020
Renjan Rajan John,a,b Ryota Kojimac and Sujoy Mahatod a
Universit` a del Piemonte Orientale, Dipartimento di Scienze e Innovazione Tecnologica, Viale T. Michel 11, I-15121 Alessandria, Italy b INFN — Sezione di Torino, Via P. Giuria 1, I-10125 Torino, Italy c KEK Theory Center, Tsukuba, Ibaraki, 305-0801, Japan d Institute of Mathematical Sciences, Homi Bhabha National Institute (HBNI), IV Cross Road, C.I.T. Campus, Taramani, Chennai, 600113 Tamil Nadu, India
E-mail: [email protected], [email protected], [email protected] Abstract: The story of positive geometry of massless scalar theories was pioneered in [1] in the context of bi-adjoint φ3 theories. Further study proposed that the positive geometry for a generic massless scalar theory with polynomial interaction is a class of polytopes called accordiohedra [2]. Tree-level planar scattering amplitudes of the theory can be obtained from a weighted sum of the canonical forms of the accordiohedra. In this paper, using results of the recent work [3], we show that in theories with polynomial interactions all the weights can be determined from the factorization property of the accordiohedron. We also extend the projective recursion relations introduced in [4, 5] to these theories. We then give a detailed analysis of how the recursion relations in φp theories and theories with polynomial interaction correspond to projective triangulations of accordiohedra. Following the very recent development [6] we also extend our analysis to one-loop integrands in the quartic theory. Keywords: Differential and Algebraic Geometry, Scattering Amplitudes ArXiv ePrint: 2007.10974
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP10(2020)037
JHEP10(2020)037
Weights, recursion relations and projective triangulations for positive geometry of scalar theories
Contents 1
2 Review of the accordiohedron 2.1 D-accordiohedron and the canonical form 2.2 Planar scattering form and amplitudes
3 3 4
3 Weights for the mixed vertices 3.1 Weights and the factorization 3.2 Explicit calculations
6 6 7
4 Recursion relation for φp amplitudes 4.1 Proof for no pole at z = 0 4.2 Proof for no pole at infinity
10 12 13
5 Explicit computations in φp theories 5.1 n = 10 in φ4 5.1.1 Cube 5.1.2 Snake 5.1.3 Lucas type 5.1.4 Mixed type 5.2 Mixed vertices
13 13 14 15 16 17 18
6 Projective triangulations 6.1 Quartic interaction 6.2 Polynomial interactions
19 21 26
7 1-loop in φ4
27
8 Conclusion
29
A n = 8 in φ3 + φ4 + φ5
30
1
Introduction
In recent years, tremendous progress has been made in relating scattering amplitudes to interesting mathematical and geometric structures [7–11]. One of the issues with this progress was that the relevant geometry lived in an auxiliary space as opposed to the kinematic space where the S-matrix lives. This was overcome in [1] where the tree-level amplitudes of the bi-adjoint φ3 theory [12] were related to canonical forms of a po
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