Bootstrapping solutions of scattering equations

Liu, Zhengwen; Zhao, Xiaoran

doi:10.1007/jhep02(2019)071

Cited by 8 publications

(5 citation statements)

References 65 publications

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“…The approach rests on a theorem by Huh [74] which identifies this dimension as the number of solutions to a system of rational critical point equations. A similar technique has been recently applied to tree-level scattering amplitudes in [75,76] (see also [77] for previous work).…”

Section: Counting the Number Of Master Integralsmentioning

confidence: 99%

Landau Discriminants

Mizera¹,

Telen²

2021

Preprint

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Scattering amplitudes in quantum field theories have intricate analytic properties as functions of the energies and momenta of the scattered particles. In perturbation theory, their singularities are governed by a set of nonlinear polynomial equations, known as Landau equations, for each individual Feynman diagram. The singularity locus of the associated Feynman integral is made precise with the notion of the Landau discriminant, which characterizes when the Landau equations admit a solution. In order to compute this discriminant, we present approaches from classical elimination theory, as well as a numerical algorithm based on homotopy continuation. These methods allow us to compute Landau discriminants of various Feynman diagrams up to 3 loops, which were previously out of reach. For instance, the Landau discriminant of the envelope diagram is a reducible surface of degree 45 in the three-dimensional space of kinematic invariants. We investigate geometric properties of the Landau discriminant, such as irreducibility, dimension and degree. In particular, we find simple examples in which the Landau discriminant has codimension greater than one. Furthermore, we describe a numerical procedure for determining which parts of the Landau discriminant lie in the physical regions. In order to study degenerate limits of Landau equations and bounds on the degree of the Landau discriminant, we introduce Landau polytopes and study their facet structure. Finally, we provide an efficient numerical algorithm for the computation of the number of master integrals based on the connection to algebraic statistics. The algorithms used in this work are implemented in the open-source Julia package Landau.jl available at https://mathrepo.mis.mpg.de/Landau/.

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Section: Counting the Number Of Master Integralsmentioning

confidence: 99%

Landau Discriminants

Mizera¹,

Telen²

2021

Preprint

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“…In fact, already the localization formula (2.80) is not practical for direct computations beyond n ≤ 5, since by the Abel-Ruffini theorem positions of the saddle points cannot be determined analytically. There has been considerable progress in solving the scattering equations numerically [29,51,[95][96][97][98][99][100][101][102][103][104], as well as evaluating the CHY formula without solving them explicitly [85,96,. In Section 3 we will present recursion relations for intersection numbers which-in the massless limit-allow to evaluate CHY formulae analytically.…”

Section: Scattering Equationsmentioning

confidence: 99%

Aspects of Scattering Amplitudes and Moduli Space Localization

Mizera

2019

Preprint

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We elaborate on the recent proposal that intersection numbers of certain cohomology classes on the moduli space of genus-zero Riemann surfaces with n punctures, M 0,n , compute tree-level scattering amplitudes in quantum field theories with a finite spectrum of particles. The relevant cohomology groups are twisted by representations of the fundamental group π 1 (M 0,n ) that describes how punctures braid around each other on the Riemann surface. Such a structure can be used to link the space of Riemann surfaces with the space of kinematic invariants. Intersection numbers of said cohomology classes-whose representatives we call twisted forms-can be shown to fully localize on the boundaries of M 0,n , which are in one-to-one map with Feynman diagrams. In this work we develop systematic approaches towards accessing such boundary information. We prove that when twisted forms are logarithmic, their intersection numbers have a simple expansion in terms of trivalent Feynman diagrams weighted by residues, allowing only for massless propagators on the internal and external lines. It is also known that in the massless limit intersection numbers have a different localization formula on the support of so-called scattering equations. Nevertheless, for physical applications one also needs to study non-logarithmic forms as they are responsible for propagation of massive states. We utilize the natural fibre bundle structure of M 0,n -which allows for a direct access to the boundaries-to introduce recursion relations for intersection numbers that "integrate out" puncture-by-puncture. The resulting recursion involves only linear algebra of certain matrices describing braiding properties of M 0,n and evaluating one-dimensional residues, thus paving a way for explicit analytic computations of scattering amplitudes. Together with the previous reformulation of the tree-level S-matrix of string theory in terms of twisted forms, the results of this work complete a unified geometric framework for studying scattering amplitudes from genus-zero Riemann surfaces. We show that a web of dualities between different homology and cohomology groups allows for deriving a host of identities among various types of amplitudes computed from the moduli space, which in this setup become a consequence of linear algebra. Throughout this work we emphasize that algebraic computations can be supplemented-or indeed replaced-by combinatorial, geometric, and topological ones.

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“…On the other hand, several methods have been developed to compute the CHY contour integral given in (1.1), most of them are applied to φ 3 or focused on solving the scattering equations [7,[10][11][12][13][14][15][16][17][18][19][20][21][22][23][24]. In this work, from the double-cover representation, we have been able to achieve a graphic off-shell algorithm to carry out any color-ordered scattering of n-gluons and interactions with scalar fields, resulting in an expansion in terms of three-point amplitudes 4 .…”

Section: Introductionmentioning

confidence: 99%

Scattering equations and a new factorization for amplitudes. Part I. Gauge theories

Gomez

2019

J. High Energ. Phys.

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In this work we show how a double-cover (DC) extension of the Cachazo, He and Yuan formalism (CHY) can be used to provide a new realization for the factorization of the amplitudes involving gluons and scalar fields. First, we propose a graphic representation for a color-ordered Yang-Mills (YM) and special Yang-Mills-Scalar (YMS) amplitudes within the scattering equation formalism. Using the DC prescription, we are able to obtain an algorithm (integration-rules) which decomposes amplitudes in terms of three-point building-blocks. It is important to remark that the pole structure of this method is totally different to ordinary factorization (which is a consequence of the scattering equations). Finally, as a byproduct, we show that the soft limit in the CHY approach, at leading order, becomes trivial by using the technology described in this paper. arXiv:1810.05407v2 [hep-th]

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Bootstrapping solutions of scattering equations

Cited by 8 publications

References 65 publications

Landau Discriminants

Landau Discriminants

Aspects of Scattering Amplitudes and Moduli Space Localization

Scattering equations and a new factorization for amplitudes. Part I. Gauge theories

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