Numerical Evaluation of Multi-loop Feynman Integrals

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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Numerical evaluation of tensor Feynman integrals in Euclidean kinematics

The European Physical Journal C ◽

10.1140/epjc/s10052-010-1516-y ◽

2011 ◽

Vol 71 (1) ◽

Cited By ~ 38

Author(s):

J. Gluza ◽

K. Kajda ◽

T. Riemann ◽

V. Yundin

Keyword(s):

Numerical Evaluation ◽

Feynman Integrals

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A semi-analytic method to compute Feynman integrals applied to four-loop corrections to the $$ \overline{\mathrm{MS}} $$-pole quark mass relation

Journal of High Energy Physics ◽

10.1007/jhep09(2021)152 ◽

2021 ◽

Vol 2021 (9) ◽

Author(s):

Matteo Fael ◽

Fabian Lange ◽

Kay Schönwald ◽

Matthias Steinhauser

Keyword(s):

Quark Mass ◽

Numerical Evaluation ◽

Dimensionless Parameter ◽

Series Expansions ◽

Analytic Method ◽

Coefficient Function ◽

Mass Relation ◽

Kinematic Range ◽

Numerical Accuracy ◽

Feynman Integrals

Abstract We describe a method to numerically compute multi-loop integrals, depending on one dimensionless parameter x and the dimension d, in the whole kinematic range of x. The method is based on differential equations, which, however, do not require any special form, and series expansions around singular and regular points. This method provides results well suited for fast numerical evaluation and sufficiently precise for phenomenological applications. We apply the approach to four-loop on-shell integrals and compute the coefficient function of eight colour structures in the relation between the mass of a heavy quark defined in the $$ \overline{\mathrm{MS}} $$ MS ¯ and the on-shell scheme allowing for a second non-zero quark mass. We also obtain analytic results for these eight coefficient functions in terms of harmonic polylogarithms and iterated integrals. This allows for a validation of the numerical accuracy.

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Numerical Evaluation of Feynman Integrals by a Direct Computation Method

10.22323/1.070.0122 ◽

2009 ◽

Author(s):

Fukuko Yuasa ◽

Tadashi Ishikawa ◽

Junpei Fujimoto ◽

Nobuyuki HAMAGUCHI ◽

Elise de Doncker ◽

...

Keyword(s):

Numerical Evaluation ◽

Computation Method ◽

Direct Computation ◽

Feynman Integrals

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Causal representation of multi-loop Feynman integrands within the loop-tree duality

Journal of High Energy Physics ◽

10.1007/jhep01(2021)069 ◽

2021 ◽

Vol 2021 (1) ◽

Cited By ~ 1

Author(s):

J. Jesús Aguilera-Verdugo ◽

Roger J. Hernández-Pinto ◽

Germán Rodrigo ◽

German F. R. Sborlini ◽

William J. Torres Bobadilla

Keyword(s):

Scattering Amplitudes ◽

Finite Fields ◽

Numerical Evaluation ◽

Simple Structure ◽

Space Time ◽

Dual Representation ◽

Feynman Integrals ◽

Causal Representation ◽

Analytic Expressions ◽

Do So

Abstract The numerical evaluation of multi-loop scattering amplitudes in the Feynman representation usually requires to deal with both physical (causal) and unphysical (non-causal) singularities. The loop-tree duality (LTD) offers a powerful framework to easily characterise and distinguish these two types of singularities, and then simplify analytically the underling expressions. In this paper, we work explicitly on the dual representation of multi-loop Feynman integrals generated from three parent topologies, which we refer to as Maximal, Next-to-Maximal and Next-to-Next-to-Maximal loop topologies. In particular, we aim at expressing these dual contributions, independently of the number of loops and internal configurations, in terms of causal propagators only. Thus, providing very compact and causal integrand representations to all orders. In order to do so, we reconstruct their analytic expressions from numerical evaluation over finite fields. This procedure implicitly cancels out all unphysical singularities. We also interpret the result in terms of entangled causal thresholds. In view of the simple structure of the dual expressions, we integrate them numerically up to four loops in integer space-time dimensions, taking advantage of their smooth behaviour at integrand level.

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