Algebraic branch points at all loop orders from positive kinematics and wall crossing

Abstract

There is a remarkable connection between the boundary structure of the positive kinematic region and branch points of integrated amplitudes in planar N = 4 SYM. A long standing question has been precisely how algebraic branch points emerge from this picture. We use wall crossing and scattering diagrams to systematically study the boundary structure of the positive kinematic regions associated with MHV amplitudes. The notion of asymptotic chambers in the scattering diagram naturally explains the appearance of algebraic branch points. Furthermore, the scattering diagram construction also motivates a new coordinate system for kinematic space that rationalizes the relations between algebraic letters in the symbol alphabet. As a direct application, we conjecture a complete list of all algebraic letters that could appear in the symbol alphabet of the 8-point MHV amplitude.

Symbol alphabets from plabic graphs

“…The fact that cluster algebras [1][2][3][4] govern the symbol alphabets [5] of multiloop nparticle amplitudes in planar maximally supersymmetric Yang-Mills (SYM) theory is by now well-established for n = 6, 7 [6] (see [7] for a review of recent progress on the computation of these amplitudes via bootstrap). Starting at n = 8 qualitatively new features arise, which have been studied via several different approaches (see for example [8][9][10][11][12][13][14][15][16]).…”

Notes on cluster algebras and some all-loop Feynman integrals

“…Starting n = 8, the cluster algebras become infinite and the (finite) symbol alphabet involves algebraic letters which go beyond usual cluster coordinates. Recently, the two-loop NMHV amplitudes have been computed for n = 8 [17] and higher [18] using the method of Q equations [19], and the alphabet has been explained using tropical positive Grassmannian [20][21][22] (see also [23]), as well as Yangian invariants/plabic graphs [24][25][26].…”

The elliptic double box and symbology beyond polylogarithms

“…In the case of MPLs with rational arguments, the symbol alphabets occurring for amplitudes as well as their adjacency conditions can be understood in terms of cluster algebras [12,13,16], and a similar understanding is currently being developed in the case of the Feynman integrals [80][81][82] and amplitudes including algebraic letters [83][84][85][86][87][88]. It would be very interesting to use the data we provide in this work to extend the cluster program to the elliptic case.…”