Four Loop Massless Propagators: a Numerical Evaluation of All Master Integrals

TL;DR

The paper tackles the verification of all non-trivial master integrals for four-loop massless propagators by applying numerical sector decomposition via the FIESTA2 framework. It leverages the α-representation with $U$ and $F$ polynomials to systematically resolve singularities and perform ε-expansions, using CubaVegas for high-precision integration. The authors compute 13 complicated master integrals (in addition to related relations) and provide an extra ε-term, obtaining results that agree with the analytic BaCh results and thereby offering an independent cross-check of the previous reductions. They also highlight the computational challenges, notably the particularly difficult planar integral $M_{61}$, and demonstrate FIESTA2's capabilities and scalability for validating high-loop calculations. The work underlines the value of numerical cross-checks in complex multiloop computations and positions FIESTA2 as a powerful tool for future advances, including five-loop evaluations.

Abstract

We present numerical results which are needed to evaluate all non-trivial master integrals for four-loop massless propagators, confirming the recent analytic results of[1]and evaluating an extra order in $\ep$ expansion for each master integral.

Figures (1)

• Figure 1: $M_{61}$ --- $M_{43}$: the thirteen complicated four-loop master integrals according to BaCh. The integrals are ordered (if read from left to right and then from top to bottom) according to their complexity. The two MI's $M_{52}$ and $M_{43}$ can be identically expressed through the three-loop nonplanar MI $N_0$.