On the finite basis of two-loop `t Hooft-Veltman Feynman integrals

In this work, we investigate the finite basis topologies of two-loop dimensionally regularized Feynman integrals in the `t Hooft-Veltman scheme in the Standard Model. We present a functionally distinct finite basis of Master Integrals which spans the whole transcendental space of all two-loop Feynman integrals with external momenta in four dimensions. We also indicate that all the two-loop Master Integrals, in an appropriate basis, with more than 8 denominators do not contribute to the finite part of any two-loop scattering amplitude. In addition, we elaborate on the application of the `t Hooft-Veltman decomposition to improve the performance of numerical evaluation of Feynman integrals using AMFlow and DCT packages. Moreover, we analyze the spectrum of special functions and the corresponding geometries appearing in any two-loop scattering amplitude. Our work will allow for a reduction in the computational complexity required for providing high-precision predictions for future high-multiplicity collider observables, both analytically and numerically.