Two-loop Feynman integrals involve Riemann spheres, elliptic curves, hyperelliptic curves of genus 2 and 3, K3 surfaces, and a rationalizable Del Pezzo surface of degree 2.
Arkani-Hamed, J.L
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abstract
Recently, an explicit, recursive formula for the all-loop integrand of planar scattering amplitudes in N=4 SYM has been described, generalizing the BCFW formula for tree amplitudes, and making manifest the Yangian symmetry of the theory. This has made it possible to easily study the structure of multi-loop amplitudes in the theory. In this paper we describe a remarkable fact revealed by these investigations: the integrand can be expressed in an amazingly simple and manifestly local form when represented in momentum-twistor space using a set of chiral integrals with unit leading singularities. As examples, we present very-concise expressions for all 2- and 3-loop MHV integrands, as well as all 2-loop NMHV integrands. We also describe a natural set of manifestly IR-finite integrals that can be used to express IR-safe objects such as the ratio function. Along the way we give a pedagogical introduction to the foundations of the subject. The new local forms of the integrand are closely connected to leading singularities---matching only a small subset of all leading singularities remarkably suffices to determine the full integrand. These results strongly suggest the existence of a theory for the integrand directly yielding these local expressions, allowing for a more direct understanding of the emergence of local spacetime physics.
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Cluster algebras for planar conformal kinematics are identified as G(4,n) subalgebras and used to bootstrap the symbol of an 8-point three-loop wheel integral via D3 and new algebraic letters.
First numerical evaluation of planar two-loop helicity amplitudes for W-boson plus four partons using finite-field reduction and sector decomposition on a subset of master integrals.
Feynman integrals selected for unit leading singularities in complex geometries satisfy epsilon-factorized differential equations with new transcendental functions corresponding to periods and differential forms in the Gauss-Manin connection.
One-cycle negative geometries in N=4 SYM have singularities only at z=-1, 0, and infinity to all loop orders.
A basis of six uniform-transcendental five-point off-shell conformal integrals is constructed and mapped to known families, yielding symbol-level two-loop results for half-BPS correlators.
citing papers explorer
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The spectrum of Feynman-integral geometries at two loops
Two-loop Feynman integrals involve Riemann spheres, elliptic curves, hyperelliptic curves of genus 2 and 3, K3 surfaces, and a rationalizable Del Pezzo surface of degree 2.
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Kinematics, cluster algebras and Feynman integrals
Cluster algebras for planar conformal kinematics are identified as G(4,n) subalgebras and used to bootstrap the symbol of an 8-point three-loop wheel integral via D3 and new algebraic letters.
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A numerical evaluation of planar two-loop helicity amplitudes for a W-boson plus four partons
First numerical evaluation of planar two-loop helicity amplitudes for W-boson plus four partons using finite-field reduction and sector decomposition on a subset of master integrals.
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Integrand Analysis, Leading Singularities and Canonical Bases beyond Polylogarithms
Feynman integrals selected for unit leading singularities in complex geometries satisfy epsilon-factorized differential equations with new transcendental functions corresponding to periods and differential forms in the Gauss-Manin connection.
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Landau Analysis of One-Cycle Negative Geometries
One-cycle negative geometries in N=4 SYM have singularities only at z=-1, 0, and infinity to all loop orders.
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Notes on off-shell conformal integrals and correlation functions at five points
A basis of six uniform-transcendental five-point off-shell conformal integrals is constructed and mapped to known families, yielding symbol-level two-loop results for half-BPS correlators.