Analytic result of a three-loop integral family in the Higgs decay to four massive bottom quarks

Abstract

When calculating the decay width of $H\to b\bar{b}b\bar{b}$ induced by the effective Higgs-gluon-gluon interaction using the optical theorem, the three-loop Feynman diagrams are encountered. Among them, there is an interesting integral family, which involves not only the sectors evaluated to multiple polylogarithms but also those related to an elliptic curve and a K3 surface. We demonstrate how to construct and solve the canonical differential equation for the entire family, which includes mixing between different sectors, by using the recently proposed algorithm in~\cite{e-collaboration:2025frv, Bree:2025tug}. Analytical results for all master integrals in this family up to the first two orders of $\epsilon$ are given.