Phase-space integrals through Mellin-Barnes representation

This letter introduces a novel analytical approach to calculating phase-space integrals, crucial for precision in particle physics. We develop a method to compute angular components using multifold Mellin-Barnes integrals, yielding results in terms of Goncharov polylogarithms for integrals involving three denominators. Our results include expressions for massless momenta up to $\cal{O}(\epsilon^2)$ and for one massive momentum up to $\cal{O}(\epsilon)$. Additionally, we derive recursion relations that reduce integrals with higher powers of denominators to simpler ones. We detail how to combine the angular part with the radial one which requires a careful handling of singularities.