Numerical calculation of one-loop integration with hypergeometric functions

Understand

One-loop two-, three- and four-point scalar functions are analytically integrated directly such that they are expressed in terms of Lauricella's hypergeometric function $F_D$.

• For two- and three-point functions, exact expressions are obtained with arbitrary combination of kinematic and mass parameters in arbitrary space-time dimension.
• Four-point function is expressed in terms of $F_D$ up to the finite part in the expansion around 4-dimensional space-time with arbitrary combination of kinematic and mass parameters.
• Since the location of the possible singularities of $F_D$ is known, information about the stabilities in the numerical calculation is obtained.