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We review the structure of gauge theory scattering amplitudes at tree level and describe how a compact expression can be found which encodes all the tree-level amplitudes in the maximally supersymmetric N=4 theory. The expressions for the…

High Energy Physics - Theory · Physics 2015-05-28 J. M. Drummond

In this paper we study the calculation of multiloop Feynman integrals that cannot be expressed in terms of multiple polylogarithms. We show in detail how certain types of two- and three-point functions at two loops, which appear in the…

High Energy Physics - Phenomenology · Physics 2019-06-26 Johannes Broedel , Claude Duhr , Falko Dulat , Brenda Penante , Lorenzo Tancredi

The integrand of any multi-loop integral is characterised after Feynman parametrisation by two polynomials. In this review we summarise the properties of these polynomials. Topics covered in this article include among others: Spanning trees…

High Energy Physics - Phenomenology · Physics 2015-05-18 Christian Bogner , Stefan Weinzierl

The two-loop QCD corrections to vector boson pair production at hadron colliders involve a new class of Feynman integrals: two-loop four-point functions with two off-shell external legs. We describe their reduction to a small set of master…

High Energy Physics - Phenomenology · Physics 2013-08-26 Thomas Gehrmann , Lorenzo Tancredi , Erich Weihs

We present the mixed QCD-EW two-loop virtual amplitudes for the neutral current Drell-Yan production, one of the bottlenecks for the complete calculation of the NNLO mixed QCD-EW corrections. We present the computational details and the…

High Energy Physics - Phenomenology · Physics 2022-08-17 Simone Devoto

We present the analytic expressions of the three-loop virtual corrections to the helicity amplitudes of 2 -> 2 four-fermion scattering processes in massless QED. The contributing Feynman diagrams are grouped into integrand families…

High Energy Physics - Phenomenology · Physics 2026-02-12 Giulio Crisanti , Thomas Dave , Pierpaolo Mastrolia , Jonathan Ronca , Sid Smith , William J. Torres Bobadilla

We propose an alternative approach based on series representation to directly reduce multi-loop multi-scale scattering amplitude into set of freely chosen master integrals. And this approach avoid complicated calculations of inverse matrix…

High Energy Physics - Phenomenology · Physics 2020-05-06 Yefan Wang , Zhao Li , Najam ul Basat

Problems occurring in physically important non-trivial examples of loop calculations are discussed. A procedure of deriving expansions of two-loop self-energy diagrams with different masses is constructed. The cases of small and large…

High Energy Physics - Phenomenology · Physics 2007-05-23 A. I. Davydychev

We present a new method for numerically computing generic multi-loop Feynman integrals. The method relies on an iterative application of Feynman's trick for combining two propagators. Each application of Feynman's trick introduces a…

High Energy Physics - Phenomenology · Physics 2022-06-30 Martijn Hidding , Johann Usovitsch

We identify two families of ten-point Feynman diagrams that generalize the elliptic double box, and show that they can be expressed in terms of the same class of elliptic multiple polylogarithms to all loop orders. Interestingly, one of…

High Energy Physics - Theory · Physics 2023-06-05 Andrew McLeod , Roger Morales , Matt von Hippel , Matthias Wilhelm , Chi Zhang

Multiloop gauge-theory amplitudes written in the Feynman-parameter representation are poised to take advantage of two important developments of the last decade: the spinor-helicity technique and the superstring reorganization. The former…

High Energy Physics - Phenomenology · Physics 2009-10-22 C. S. Lam

We show how studying leading singularities of Feynman diagrams, when all momenta are complex, gives a simple way of writing multi-loop and multi-particle scattering amplitudes in N=4 super Yang-Mills. The simplicity of the method is…

High Energy Physics - Theory · Physics 2008-03-14 Freddy Cachazo

We present recent developments on the topic of the integrand reduction of scattering amplitudes. Integrand-level methods allow to express an amplitude as a linear combination of Master Integrals, by performing operations on the…

High Energy Physics - Phenomenology · Physics 2013-12-06 Hans van Deurzen , Gionata Luisoni , Pierpaolo Mastrolia , Edoardo Mirabella , Giovanni Ossola , Tiziano Peraro , Ulrich Schubert

The presence of strong electromagnetic fields adds huge complexity to QED Feynman diagrams, such that new methods are required to calculate higher-loop and higher-multiplicity scattering amplitudes. Here we use the worldline formalism to…

High Energy Physics - Theory · Physics 2023-11-27 Patrick Copinger , James P. Edwards , Anton Ilderton , Karthik Rajeev

We present a method for the integrand-level reduction of two-loop helicity amplitudes in both $d=4-2\epsilon$ and $d=4$ dimensions. The amplitude is expressed in terms of a set of Feynman integrals and their coefficients that depend on the…

High Energy Physics - Phenomenology · Physics 2025-11-14 Giuseppe Bevilacqua , Dhimiter Canko , Costas Papadopoulos , Aris Spourdalakis

Starting from the parametric representation of a Feynman diagram, we obtain it's well defined value in dimensional regularisation by changing the integrals over parameters into contour integrals. That way we eventually arrive at a…

High Energy Physics - Phenomenology · Physics 2007-05-23 K. Knecht , H. Verschelde

We compute the complete set of two-loop master integrals for the scattering of four massless particles and a massive one. Our results are ready for phenomenological applications, removing a major obstacle to the computation of complete…

High Energy Physics - Phenomenology · Physics 2024-05-03 Samuel Abreu , Dmitry Chicherin , Harald Ita , Ben Page , Vasily Sotnikov , Wladimir Tschernow , Simone Zoia

The two point integrals contributing to the self energy of a particle in a three dimensional quantum field theory are calculated to two loop order in perturbation theory as well as the vacuum ones contributing to the effective potential to…

High Energy Physics - Phenomenology · Physics 2009-10-28 Arttu K. Rajantie

We calculate 3-loop master integrals for heavy quark correlators and the 3-loop QCD corrections to the $\rho$-parameter. They obey non-factorizing differential equations of second order with more than three singularities, which cannot be…

High Energy Physics - Theory · Physics 2018-08-01 J. Ablinger , J. Blümlein , A. De Freitas , M. van Hoeij , E. Imamoglu , C. G. Raab , C. -S. Radu , C. Schneider

Understanding the cancellation of ultraviolet and infrared singularities in perturbative quantum field theory is of central importance for the development and automation of various theoretical tools that make accurate predictions for…

High Energy Physics - Theory · Physics 2024-11-11 David F. Rentería-Estrada