Scalar 1-loop Feynman integrals in arbitrary space-time dimension $d$ - an update

The study and use of analyticity of scattering amplitudes was founded by R. Eden, P. Landshoff, D. Olive and J. Polkinghorn in their famous book “The Analytic S-Matrix” in 1966 [1]. Indeed, already in 1969 J. Schwinger quotes: “One of the most remarkable discoveries in elementary particle physics has been that of the complex plane,” “... the theory of functions of complex variables plays the role not of a mathematical tool, but of a fundamental description of nature inseperable from physics. ...” [2]. It took many years to make the use of analyticity and unitarity, together with renormalizability and gauge invariance of quantum field theory a practical tool for the calculation of cross sections at real colliders. When the analysis of LEP 1 data, around 1989, was prepared it became evident that the S-matrix language helps to efficiently sort the various perturbative contributions of the Standard Model.