An Approach Toward the Numerical Evaluation of Multi-Loop Feynman Diagrams
G. Passarino
TL;DR
The paper introduces a numerically robust scheme to evaluate multi-loop Feynman diagrams by leveraging the Bernstein-Tkachov (BT) theorem to smooth Feynman-parameter integrands, and then extends this approach to two-loop topologies via a minimal BT strategy that applies the BT operation to the largest one-loop subdiagram. By combining contour deformation in the complex parameter space with careful handling of infrared and Landau singularities, the method yields stable results for the sunset ($S_3$) topology and related tensor integrals, with extensive comparisons to existing analytic results. A key contribution is showing how to navigate normal thresholds with an alternative power-raising procedure and how to compute both scalar and tensor two-loop integrals without relying on Gram-determinant reductions. The work aims to enable automatic, high-precision two-loop Standard Model predictions (e.g., for observables like $\sin^2\theta^{l}_{\rm eff}$) by providing practical, adaptable formulas and numerical strategies that can be extended to all two-loop topologies.
Abstract
A scheme for systematically achieving accurate numerical evaluation of multi-loop Feynman diagrams is developed. This shows the feasibility of a project aimed to produce a complete calculation for two-loop predictions in the Standard Model. As a first step an algorithm, proposed by F. V. Tkachov and based on the so-called generalized Bernstein functional relation, is applied to one-loop multi-leg diagrams with particular emphasis to the presence of infrared singularities, to the problem of tensorial reduction and to the classification of all singularities of a given diagram. Successively, the extension of the algorithm to two-loop diagrams is examined. The proposed solution consists in applying the functional relation to the one-loop sub-diagram which has the largest number of internal lines. In this way the integrand can be made smooth, a part from a factor which is a polynomial in the vector of Feynman parameters needed for the complementary sub-diagram with the smallest number of internal lines. Since the procedure does not introduce new singularities one can distort the integration hyper-contour into the complex hyper-plane, thus achieving numerical stability. The algorithm is then modified to deal with numerical evaluation around normal thresholds. Concise and practical formulas are assembled and presented, numerical results and comparisons with the available literature are shown and discussed for the so-called sunset topology.