Fast evaluation of Feynman diagrams - art. no. 125001

We develop a new representation for the integrals associated with Feynman d iagrams. This leads directly to a novel method for the numerical evaluation of these integrals, which avoids the use of Monte Carlo techniques. Our ap proach is based on the theory of generalized sine [sin(x)/x] functions, fro m which we derive an approximation to the propagator that is expressed as a n infinite sum. When the propagators in the Feynman integrals are replaced with the approximate form all integrals over internal momenta and vertices are converted into Gaussians, which can be evaluated analytically. Performi ng the Gaussians yields a multi-dimensional infinite sum which approximates the corresponding Feynman integral. The difference between the exact resul t and this approximation is set by an adjustable parameter, and can be made arbitrarily small. We discuss the extraction of regularization independent quantities and demonstrate, both in theory and practice, that these sums c an be evaluated quickly, even for third or fourth order diagrams. Last, we survey strategies for numerically evaluating the multi-dimensional sums. We illustrate the method with specific examples, including the second order s unset diagram from quartic scalar field theory, and several higher-order di agrams. In this initial paper we focus upon scalar field theories in Euclid ean spacetime, but expect that this approach call be generalized to fields with spin.