All-helicity symbol alphabets from unwound amplituhedra

Abstract

We review an algorithm for determining the branch points of general amplitudes in planar N = 4 super-Yang-Mills theory from amplituhedra. We demonstrate how to use the recent reformulation of amplituhedra in terms of 'sign flips' in order to streamline the application of this algorithm to amplitudes of any helicity. In this way we recover the known branch points of all one-loop amplitudes, and we find an 'emergent positivity' on boundaries of amplituhedra. arXiv:1711.11507v2 [hep-th] 15 May 2018 7 Singularities and Symbology 32 8 Conclusion 34 -1 -

Boundaries of amplituhedra and NMHV symbol alphabets at two loops

“…The possibility to do so exists because of the simple fact pointed out in [5] that the locus in the space of external data Conf n (P 3 ) where the symbol letters of a given amplitude vanish should be the same as the locus where the corresponding Landau equations [6,7] admit solutions. A slight refinement of this statement, to account for the fact that amplitudes in general have algebraic branch cuts in addition to logarithmic cuts, was discussed in section 7 of [3].…”

“…In section 2 we develop a procedure for constructing certain boundaries of two-loop amplituhedra by "merging" one-loop configurations of the type classified in the prequel [3]. In section 3 we organize the results according to helicity and codimensionality (the number of on-shell conditions satisfied by each configuration) and discuss some subtleties about overconstrained configurations that require resolution.…”

“…This analysis builds heavily on sections 3-5 of [3], and in particular we show how to recycle the one-loop boundaries classified there by "merging" pairs of one-loop boundaries into two-loop boundaries. We find that two different formulations of the amplituhedron -the original formulation in terms of C and D matrices [18], and the reformulation in terms of sign flips [19] -play two complementary roles, exactly as in [3]. Specifically, the former is useful for establishing the existence of boundaries by constructing explicit C and D matrix representatives, while the latter is useful for establishing the non-existence of any other boundaries.…”

“…In planar N = 4 super-Yang-Mills theory (which we refer to as SYM theory), the current state of the art for carrying out explicit computations of multi-loop amplitudes is a bootstrap program that relies fundamentally on assumptions about the location of branch points of certain amplitudes. The aim of the research program initiated in [1,2] for MHV amplitudes and generalized to non-MHV amplitudes in [3] (to which this paper should be considered a sequel) is to…”

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Abstract:In this sequel to [3] we classify the boundaries of amplituhedra relevant for determining the branch points of general two-loop amplitudes in planar N = 4 superYang-Mills theory. We explain the connection to on-shell diagrams, which serves as a useful cross-check.

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