Loop-Tree Duality for Multiloop Numerical Integration

Loop-tree duality (LTD) offers a promising avenue to numerically integrate multiloop integrals directly in momentum space. It is well established at one loop, but there have been only sparse numerical results at two loops. We provide a formal derivation for a novel multiloop LTD expression and study its threshold singularity structure. We apply our findings numerically to a diverse set of up to four-loop finite topologies with kinematics for which no contour deformation is needed. We also lay down the ground work for constructing such a deformation. Our results serve as an important stepping stone towards a generalized and efficient numerical implementation of LTD, which is applicable to the computation of virtual corrections.

Figure 1

Numerical LTD results obtained for the scalar massless one-, two-, and three-loop ladder box diagram with external kinematics satisfying (in ${\mathrm{GeV}}^2$) $p_1^2=-5$, $p_{i=2,3,4}^2=s=-1$ and values of the Mandelstam invariant $t$ ranging from $t=-7$ (loop threshold) to $t=100$. The analytic results are taken from Ref. [34].