Two-Loop Form Factors in QED

We evaluate the on shell form factors of the electron for arbitrary momentum transfer and finite electron mass, at two loops in QED, by integrating the corresponding dispersion relations, which involve the imaginary parts known since a long time. The infrared divergences are parameterized in terms of a fictitious small photon mass. The result is expressed in terms of Harmonic Polylogarithms of maximum weight 4. The expansions for small and large momentum transfer are also givenComment: 13 pages, 1 figur

The analytic value of a 3-loop sunrise graph in a particular kinematical configuration

We consider the scalar integral associated to the 3-loop sunrise graph with a massless line, two massive lines of equal mass $M$, a fourth line of mass equal to $Mx$, and the external invariant timelike and equal to the square of the fourth mass. We write the differential equation in $x$ satisfied by the integral, expand it in the continuous dimension $d$ around $d=4$ and solve the system of the resulting chained differential equations in closed analytic form, expressing the solutions in terms of Harmonic Polylogarithms. As a byproduct, we give the limiting values of the coefficients of the $(d-4)$ expansion at $x=1$ and $x=0$.Comment: 9 pages, 3 figure

Multiloop Integrand Reduction for Dimensionally Regulated Amplitudes

We present the integrand reduction via multivariate polynomial division as a natural technique to encode the unitarity conditions of Feynman amplitudes. We derive a recursive formula for the integrand reduction, valid for arbitrary dimensionally regulated loop integrals with any number of loops and external legs, which can be used to obtain the decomposition of any integrand analytically with a finite number of algebraic operations. The general results are illustrated by applications to two-loop Feynman diagrams in QED and QCD, showing that the proposed reduction algorithm can also be seamlessly applied to integrands with denominators appearing with arbitrary powers.Comment: Published version. 5 pages, 2 figure

Six-Photon Amplitudes

We present analytical results for all six-photon helicity amplitudes. For the computation of this loop induced process two recently developed methods, based on form factor decomposition and on multiple cuts, have been used. We obtain compact results, demonstrating the applicability of both methods to one-loop amplitudes relevant to precision collider phenomenology.Comment: replaced by published versio

Analytic evaluation of Feynman graph integrals

We review the main steps of the differential equation approach to the analytic evaluation of Feynman graphs, showing at the same time its application to the 3-loop sunrise graph in a particular kinematical configuration.Comment: 5 pages, 1 figure, uses npb.sty. Presented at RADCOR 2002 and Loops and Legs in Quantum Field Theory, 8-13 September 2002, Kloster Banz, Germany. Revised version: minor typos corrected, one reference adde

Cartan, Schouten and the search for connection

In this paper we provide an analysis, both historical and mathematical, of two joint papers on the theory of connections by \uc3\u89lie Cartan and Jan Arnoldus Schouten that were published in 1926. These papers were the result of a fertile collaboration between the two eminent geometers that flourished in the two-year period 1925-1926. We describe the birth and the development of their scientific relationship especially in the light of unpublished sources that, on the one hand, offer valuable insight into their common research interests and, on the other hand, provide a vivid picture of Cartan's and Schouten's different technical choices. While the first part of this work is preeminently of a historical character, the second part offers a modern mathematical treatment of some contents of the two contributions

Generalised Unitarity for Dimensionally Regulated Amplitudes

We present a novel set of Feynman rules and generalised unitarity cut-conditions for computing one-loop amplitudes via d-dimensional integrand reduction algorithm. Our algorithm is suited for analytic as well as numerical result, because all ingredients turn out to have a four-dimensional representation. We will apply this formalism to NLO QCD corrections.Comment: Presented at SILAFAE 2014, 24-28 Nov, Ruta N, Medellin, Colombi

The Integrand Reduction of One- and Two-Loop Scattering Amplitudes

The integrand-level methods for the reduction of scattering amplitudes are well-established techniques, which have already proven their effectiveness in several applications at one-loop. In addition to the automation and refinement of tools for one-loop calculations, during the past year we observed very interesting progress in developing new techniques for amplitudes at two- and higher-loops, based on similar principles. In this presentation, we review the main features of integrand-level approaches with a particular focus on algebraic techniques, such as Laurent series expansion which we used to improve the one-loop reduction, and multivariate polynomial division which unveils the structure of multi-loop amplitudes.Comment: 7 pages, v2: fixed typos, added references. Presented at "Loops and Legs in Quantum Field Theory", Wernigerode, Germany, 15-20 April 201

The analytic value of the sunrise self-mass with two equal masses and the external invariant equal to the third squared mass

We consider the two-loop self-mass sunrise amplitude with two equal masses $M$ and the external invariant equal to the square of the third mass $m$ in the usual $d$-continuous dimensional regularization. We write a second order differential equation for the amplitude in $x=m/M$ and show as solve it in close analytic form. As a result, all the coefficients of the Laurent expansion in $(d-4)$ of the amplitude are expressed in terms of harmonic polylogarithms of argument $x$ and increasing weight. As a by product, we give the explicit analytic expressions of the value of the amplitude at $x=1$, corresponding to the on-mass-shell sunrise amplitude in the equal mass case, up to the $(d-4)^5$ term included.Comment: 11 pages, 2 figures. Added Eq. (5.20) and reference [4