Abstract:
Feynman diagrams are used as a graphical way to represent and calculate various contributions to particle interactions (probabilities of certain processes, decay rates, etc.). If the number of interacting particles is N, this is described by an N-point diagram (having N external legs). Whenever a diagram contains a closed cycle (a "loop"), a four-dimensional integration with respect to the corresponding particle momentum (or coordinate) is assumed. Straightforward integration of separate contributions may lead to singularities, which are commonly regulated by adding a small epsilon-parameter to the space-time dimension, such a procedure being known as "dimensional regularization". In physically-relevant quantities, the singularities (represented by the poles in epsilon) should cancel, but one still needs to evaluate the finite contributions. In this talk, a geometrical way to calculate dimensionally-regulated Feynman diagrams is reviewed. In the one-loop N-point case, the results can be related to certain volume integrals in non-Euclidean geometry. For example, the result for the four-point function can be associated with the content of a spherical or hyperbolic tetrahedron in three-dimensional spherical or hyperbolic space. Analytical continuation of the results to other regions of kinematical variables (momenta and masses of the particles) is discussed. In a number of cases, analytic results can be presented in terms of the (generalized) polylogarithms and associated functions.
This seminar is easily accessible to persons with disabilities. For more information or for assistance, please contact the Mathematics Department at 743-3500.