Numerical implementation of the loop–tree duality method

Abstract

We present a first numerical implementation of the loop-tree duality (LTD) method for the direct numerical computation of multi-leg one-loop Feynman integrals. We discuss in detail the singular structure of the dual integrands and define a suitable contour deformation in the loop threemomentum space to carry out the numerical integration. Then we apply the LTD method to the computation of ultraviolet and infrared finite integrals, and we present explicit results for scalar and tensor integrals with up to eight external legs (octagons). The LTD method features an excellent performance independently of the number of external legs.

To $${d}$$ d , or not to $${d}$$ d : recent developments and comparisons of regularization schemes

“…As explained in Refs. [114][115][116], the intersection of forward and backward hyperboloids defined by the on-shell conditions allows one to identify the IR (and threshold) singularities. Moreover, this study is crucial to prove the compactness of the region developing IR divergences [103][104][105], which constitutes a very important result by itself.…”

From Jacobi off-shell currents to integral relations

“…In order to extract these identities, we use the Loop-Tree duality (LTD) formalism [35][36][37][38][39]. To…”

The Loop-Tree Duality: Progress Report

“…To integrate the dual contributions over requires most of the times a contour deformation due to the presence of the so-called ellipsoid and hyperboloid singularities [9] that in general are present at the integrand level.…”

“…The Loop-Tree Duality (LTD) method [1][2][3][4][5][6][7][8][9][10][11][12][13][14] turns N-leg loop quantities (integrals and amplitudes) into a sum of connected tree-level-like diagrams with a remaining integration measure that is similar to the (N + 1)-body phase-space [1]. Therefore, loop and tree-level corrections of the same order, may in principle be treated under a common integral sign with the use of a proper numerical integrator (usually a Monte Carlo routine) [11,12].…”