Examining the analytic structure of Green’s functions: Massive parallel complex integration using GPUs

Graphics Processing Units (GPUs) are employed for a numerical determination of the analytic structure of two-point correlation functions of Quantum Field Theories. These functions are represented through integrals in d-dimensional Euclidean momentum space. Such integrals can in general not be solved analytically, and therefore one has to rely on numerical procedures to extract their analytic structures if needed. After describing the general outline of the corresponding algorithm we demonstrate the procedure by providing a completely worked-out example in four dimensions for which an exact solution exists. We resolve the analytic structure by highly parallel evaluation of the correlation functions momentum space integral in the complex plane. The (logarithmically) divergent integral is regularized by applying a BPHZ-like Taylor subtraction to the integrand. We find perfect agreement with the exact solution. The fact that each point in the complex plane does not need any information from other points makes this a perfect candidate for GPU treatment. A significant gain in speed as compared to sequential execution is obtained. We also provide typical running times on several GPUs.