Reducing the computation of Feynman integrals to integrals in the Wiener measure applying analyticcontinuation

Abstract We extend the recently developed Yangian bootstrap for Feynman integrals to Minkowski space, focusing on the case of the one-loop box integral. The space of Yangian invariants is spanned by the Bloch-Wigner function and its discontinuities. Using only input from symmetries, we constrain the functional form of the box integral in all 64 kinematic regions up to twelve (out of a priori 256) undetermined constants. These need to be fixed by other means. We do this explicitly, employing two alternative methods. This results in a novel compact formula for the box integral valid in all kinematic regions of Minkowski space.

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Essential aspects of Wiener-measure regularization for quantum-mechanical path integrals

Nonlinear Analysis ◽

10.1016/j.na.2005.03.012 ◽

2005 ◽

Vol 63 (5-7) ◽

pp. e1253-e1261

Author(s):

John R. Klauder

Keyword(s):

Path Integrals ◽

Wiener Measure ◽

Quantum Mechanical

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Pentagon functions for scattering of five massless particles

Journal of High Energy Physics ◽

10.1007/jhep12(2020)167 ◽

2020 ◽

Vol 2020 (12) ◽

Author(s):

D. Chicherin ◽

V. Sotnikov

Keyword(s):

Phase Space ◽

Numerical Evaluation ◽

Public Library ◽

Basis Set ◽

Particle Scattering ◽

Minimal Basis ◽

Feynman Integrals ◽

Analytic Expressions ◽

Transcendental Functions ◽

Branch Cuts

Abstract We complete the analytic calculation of the full set of two-loop Feynman integrals required for computation of massless five-particle scattering amplitudes. We employ the method of canonical differential equations to construct a minimal basis set of transcendental functions, pentagon functions, which is sufficient to express all planar and nonplanar massless five-point two-loop Feynman integrals in the whole physical phase space. We find analytic expressions for pentagon functions which are manifestly free of unphysical branch cuts. We present a public library for numerical evaluation of pentagon functions suitable for immediate phenomenological applications.

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Notes on cluster algebras and some all-loop Feynman integrals

Journal of High Energy Physics ◽

10.1007/jhep06(2021)119 ◽

2021 ◽

Vol 2021 (6) ◽

Author(s):

Song He ◽

Zhenjie Li ◽

Qinglin Yang

Keyword(s):

Configuration Space ◽

Strong Evidence ◽

Wilson Loop ◽

Cluster Algebra ◽

Cluster Algebras ◽

Gauge Invariant ◽

Feynman Integrals ◽

Cluster Function ◽

Invariant Description ◽

Cluster Configuration

Abstract We study cluster algebras for some all-loop Feynman integrals, including box-ladder, penta-box-ladder, and double-penta-ladder integrals. In addition to the well-known box ladder whose symbol alphabet is $$ {D}_2\simeq {A}_1^2 $$ D 2 ≃ A 1 2 , we show that penta-box ladder has an alphabet of D3 ≃ A3 and provide strong evidence that the alphabet of seven-point double-penta ladders can be identified with a D4 cluster algebra. We relate the symbol letters to the u variables of cluster configuration space, which provide a gauge-invariant description of the cluster algebra, and we find various sub-algebras associated with limits of the integrals. We comment on constraints similar to extended-Steinmann relations or cluster adjacency conditions on cluster function spaces. Our study of the symbol and alphabet is based on the recently proposed Wilson-loop d log representation, which allows us to predict higher-loop alphabet recursively; by applying it to certain eight-point and nine-point double-penta ladders, we also find D5 and D6 cluster functions respectively.

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