A Subtraction Scheme for Feynman Integrals

We present a subtraction scheme for ultraviolet (UV) divergent, infrared (IR) safe scalar Feynman integrals in dimensional regularization with any number of scales. This is done by the introduction of $u$-variables, which are a suitable generalization of dihedral coordinates on the open string moduli space to Feynman integrals. The subtraction scheme furnishes subtraction terms which are products of lower loop Feynman integrals deformed by order $\epsilon$ powers of $u$-variables and deformations of the degree of divergence. The result is a canonical and algorithmic prescription to express the Feynman integral as a sum of convergent integrals dressed with inverse powers of $\epsilon$.