QCD Amplitudes: new perspectives on Feynman integral calculus

I analyze the algebraic patterns underlying the structure of scattering amplitudes in quantum field theory. I focus on the decomposition of amplitudes in terms of independent functions and the systems of differential equations the latter obey. In particular, I discuss the key role played by unitarity for the decomposition in terms of master integrals, by means of generalized cuts and integrand reduction, as well as for solving the corresponding differential equations, by means of Magnus exponential series.Comment: Presented at Rencontres de Moriond 201

CSW Diagrams and Electroweak Vector Bosons

Based on the joined work performed together with Z. Bern, D. Forde, and D. Kosower [1], in this talk it is recalled the (twistor-motivated) diagrammatic formalism describing tree-level scattering amplitudes presented by Cachazo, Svr\v{c}ek and Witten, and it is discussed an extension of the vertices and accompaining rules to the construction of vector-boson currents coupling to an arbitrary source.Comment: 8 pages, 2 figures, Talk given at the workshop QCD at Work 2005, Conversano (BA), Italy, June 16-20, 200

Unitarity-Cuts, Stokes' Theorem and Berry's Phase

Two-particle unitarity-cuts of scattering amplitudes can be efficiently computed by applying Stokes' Theorem, in the fashion of the Generalised Cauchy Theorem. Consequently, the Optical Theorem can be related to the Berry Phase, showing how the imaginary part of arbitrary one-loop Feynman amplitudes can be interpreted as the flux of a complex 2-form.Comment: presented at RADCOR 2009 - 9th International Symposium on Radiative Corrections, October 25 - 30 2009, Ascona, Switzerlan

Feynman Integrals and Intersection Theory

We introduce the tools of intersection theory to the study of Feynman integrals, which allows for a new way of projecting integrals onto a basis. In order to illustrate this technique, we consider the Baikov representation of maximal cuts in arbitrary space-time dimension. We introduce a minimal basis of differential forms with logarithmic singularities on the boundaries of the corresponding integration cycles. We give an algorithm for computing a basis decomposition of an arbitrary maximal cut using so-called intersection numbers and describe two alternative ways of computing them. Furthermore, we show how to obtain Pfaffian systems of differential equations for the basis integrals using the same technique. All the steps are illustrated on the example of a two-loop non-planar triangle diagram with a massive loop.Comment: 13 pages, published versio

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