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 $$ \mathcal{N} $$ 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.

Singularities of eight- and nine-particle amplitudes from cluster algebras and tropical geometry

“…As a check of our formalism, in subsection 3.2 we reapply it to the pTr + (4, 8) case, and confirm that it provides two rays associated to 18 square-root letters, as was previously found in [47]. Then, in subsection 3.3 we compare these results with a more recent refinement of the works [47][48][49], based on the framework of wall-crossing and scattering diagrams [69]. While this approach naively predicts a large number of additional non-rational letters, very interestingly we find that these are in fact only two: the inequivalent realisations of the four-mass box Gram determinant by eight cyclically ordered massless legs, ∆ 1,3,5,7 and ∆ 2,4,6,8 .…”

“…1 In this article, building on our previous work, we make an important step towards generalising these exciting developments to arbitrary multiplicity n. First, we analyse infinite mutation sequences of rank-two affine cluster algebras with general coefficients, which allow us to trivially obtain predictions for square-root letters for any such subalgebra of Gr(4, n). As a cross-check, we then apply our procedure to the known eight-particle case, not only finding agreement with the earlier analysis, but also comparing it to more recent predictions based on the closely related scattering diagram approach [69]. As essentially all proposals for n-particle alphabets to date correspond to different compactifications of the region of positive kinematics of Gr(4, n), concretely this approach amounts to a refinement of the tropical compactification, which at first sight seems to predict another 34 squareroot letters on top of the two-loop NMHV ones.…”

“…As a first cross check however, in the remainder of this subsection we will use our method to confirm the results reported in [47] for the eightparticle alphabet. Let us also comment that while the prescription (3.19) may currently seem ad-hoc and only justified by the agreement of its symbol alphabet predictions with explicit computations, in the next subsection we will provide further evidence about its correctness by comparing it with a more recent approach based on scattering diagrams and wall-crossing [69].…”

Symbol alphabets from plabic graphs III: n = 9

“…The fact that cluster algebras [1][2][3][4][5][6] govern the symbol alphabets [7] of multiloop n-particle amplitudes in planar maximally supersymmetric Yang-Mills (SYM) theory is by now wellestablished for n = 6, 7 [8] (see [9] 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 [10][11][12][13][14][15][16][17][18]).…”

Bootstrapping octagons in reduced kinematics from A2 cluster algebras

“…Starting at eight points, G(4, n)/T cluster algebras are infinite and the (finite) symbol alphabets involve algebraic letters that go beyond the usual cluster coordinates. Recently, the two-loop NMHV amplitudes 1 have been computed for n = 8 [24] and higher [25] using the method of Q equations [26], and the alphabet has been explained using tropical positive Grassmannians [27][28][29] (see also [30]) as well as Yangian invariants and plabic graphs [31][32][33]. There has also been new progress on the cluster algebra structures for individual Feynman integrals in N = 4 SYM [34,35] and in a broader context [36].…”