Fermions, Gauge Theories, and the Sinc Function Representation for Feynman Diagrams

We extend our new approach for numeric evaluation of Feynman diagrams to integrals that include fermionic and vector propagators. In this initial discussion we begin by deriving the Sinc function representation for the propagators of spin-1/2 and spin-1 fields and exploring their properties. We show that the attributes of the spin-0 propagator which allowed us to derive the Sinc function representation for scalar field Feynman integrals are shared by fields with non-zero spin. We then investigate the application of the Sinc function representation to simple QED diagrams, including first order corrections to the propagators and the vertex.Comment: 10 pages, Latex, 9 figure

Fast Evaluation of Feynman Diagrams

We develop a new representation for the integrals associated with Feynman diagrams. This leads directly to a novel method for the numerical evaluation of these integrals, which avoids the use of Monte Carlo techniques. Our approach is based on based on the theory of generalized sinc ($\sin(x)/x$) functions, from which we derive an approximation to the propagator that is expressed as an 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. Performing the Gaussians yields a multi-dimensional infinite sum which approximates the corresponding Feynman integral. The difference between the exact result 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 can be evaluated quickly, even for third or fourth order diagrams. Lastly, we survey strategies for numerically evaluating the multi-dimensional sums. We illustrate the method with specific examples, including the the second order sunset diagram from quartic scalar field theory, and several higher-order diagrams. In this initial paper we focus upon scalar field theories in Euclidean spacetime, but expect that this approach can be generalized to fields with spin.Comment: uses feynmp macros; v2 contains improved description of renormalization, plus other minor change