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We derive an analytic representation of the ten-particle, two-loop double-box integral as an elliptic integral over weight-three polylogarithms. To obtain this form, we first derive a four-fold, rational (Feynman-)parametric representation…

High Energy Physics - Theory · Physics 2018-03-28 Jacob L. Bourjaily , Andrew J. McLeod , Marcus Spradlin , Matt von Hippel , Matthias Wilhelm

We present an algorithm for the analytical evaluation of dimensionally regularized massless on-shell double box Feynman diagrams with arbitrary polynomials in numerators and general integer powers of propagators. Recurrence relations…

High Energy Physics - Phenomenology · Physics 2015-06-25 V. A. Smirnov , O. L. Veretin

We introduce a circular restricted charged three-body problem on the plane. In this model, the gravitational and Coulomb forces, due to the primary bodies, act on a test particle; the net force exerted by some primary body on the test…

Dynamical Systems · Mathematics 2015-03-31 Abimael Bengochea , Claudio Vidal

In this paper we continue and improve the analysis of the effective actions obtained by integrating out a scalar and a fermion field coupled to external symmetric sources, started in the previous paper. The first subject we study is the…

High Energy Physics - Theory · Physics 2018-02-14 L. Bonora , M. Cvitan , P. Dominis Prester , S. Giaccari , T. Štemberga

We present the calculation of massless two-loop Master Integrals relevant to five-point amplitudes with one off-shell external leg and derive the complete set of planar Master Integrals with five on-mass-shell legs, that contribute to many…

High Energy Physics - Phenomenology · Physics 2016-05-04 Costas G. Papadopoulos , Damiano Tommasini , Christopher Wever

We study the problem of calculating two-loop three-point diagrams with irreducible numerators (i.e. numerators which cannot be expressed in terms of the denominators). For the case of massless internal particles and arbitrary (off-shell)…

High Energy Physics - Phenomenology · Physics 2014-11-17 Natalia I. Ussyukina , Andrei I. Davydychev

Starting at 3-loop order, the massive Wilson coefficients for deep-inelastic scattering and the massive operator matrix elements describing the variable flavor number scheme receive contributions of Feynman diagrams carrying quark lines…

High Energy Physics - Phenomenology · Physics 2017-08-23 J. Ablinger , J. Blümlein , A. De Freitas , A. Hasselhuhn , C. Schneider , F. Wißbrock

Different mathematical methods have been applied to obtain the analytic result for the massless triangle Feynman diagram yielding a sum of four linearly independent hypergeometric functions $F_4$. In this paper I work out the diagram and…

High Energy Physics - Theory · Physics 2008-08-12 Alfredo Takashi Suzuki

The approach of direct integration of the three-dimensional Faddeev equations with respect to the breakup T-matrix in momentum space for three bodies of different masses is presented. The Faddeev equations are written out explicitly without…

Quantum Physics · Physics 2025-02-03 Mikhail Egorov

Presented in this paper is the kinematic analysis of a symmetrical three-degree-of-freedom planar parallel manipulator. In opposite to serial manipulators, parallel manipulators can admit not only multiple inverse kinematic solutions, but…

Robotics · Computer Science 2007-05-23 Damien Chablat , Philippe Wenger

The main goal of the present research is to make a summary of analytical equations that can be found to calculate a swing-by maneuver in the three-dimensional space. Analytical equations based in the patched conics approximation are showed…

Earth and Planetary Astrophysics · Physics 2024-12-24 Antonio Fernando Bertachini de Almeida Prado

In this paper, we propose a new method for evaluating scalar one-loop Feynman integrals in generalized D-dimension. The calculations play an important building block for two-loop and higher-loop corrections to the processes at future…

High Energy Physics - Phenomenology · Physics 2017-07-10 Khiem Hong Phan

We calculate the three-loop master integrals contributing to the three-loop five-point amplitude on the special Coulomb branch of $\mathcal{N}=4$ SYM theory. For the genuine pentagon integrals, we follow the approach of Ref. [JHEP 12 (2025)…

High Energy Physics - Theory · Physics 2026-05-19 Andrei V. Belitsky , Leonid V. Bork , Roman N. Lee , Andrei I. Onishchenko , Vladimir A. Smirnov

We study a 3-loop 5-propagator Feynman Integral, which we call the vacuum seagull, with arbitrary masses and spacetime dimension using the Symmetries of Feynman Integrals method. It is our first example with potential numerators. We…

High Energy Physics - Theory · Physics 2018-01-03 Philipp Burda , Barak Kol , Ruth Shir

We compute the master integrals containing 2 and 3 massive propagators entering the planar amplitudes of the 2-loop electroweak form factor. The masses of the $W$, $Z$ and Higgs bosons are assumed to be degenerate. This work is a…

High Energy Physics - Phenomenology · Physics 2009-11-10 U. Aglietti , R. Bonciani

It is well known that forward limits of tree-level amplitudes (and those trivalent diagrams they consist of) produce one-loop amplitudes and trivalent diagrams with propagators linear in the loop momentum. They naturally arise from one-loop…

High Energy Physics - Theory · Physics 2022-09-07 Bo Feng , Song He , Yong Zhang , Yao-Qi Zhang

We summarize two geometrical approaches to analytically evaluate higher-fold Mellin-Barnes (MB) integrals in terms of hypergeometric functions. The first method is based on intersections of conic hulls, while the second one, which is more…

High Energy Physics - Theory · Physics 2024-07-30 Sumit Banik , Samuel Friot

We propose that the concept of multidimensional residues can be used to directly extracting the coefficients of scalar master integrals (with single propagators only) from one-loop Feynman integrals with generic power of propagators. Unlike…

High Energy Physics - Theory · Physics 2011-12-20 Jian-Hui Zhang

The problem of evaluating potential integrals on planar triangular elements has been addressed using a polar coordinate decomposition. The resulting formulae are general, exact, easily implemented, and have only one special case, that of a…

Numerical Analysis · Mathematics 2013-03-01 Michael Carley

A short review of expansion by regions is presented. It is a well-known strategy to obtain an expansion of a given multiloop Feynman integral in a given limit where some kinematic invariants and/or masses have certain scaling measured in…

High Energy Physics - Theory · Physics 2024-06-18 Vladimir A. Smirnov