Summing Feynman diagrams in the worldline formalism

Abstract

The worldline formalism shares with string theory the property that it allows one to write down master integrals that e(cid:27)ectively combine the contributions of many Feynman diagrams. While at the one-loop level these diagrams di(cid:27)er only by the position of the external legs along a (cid:28)xed line or loop, at multiloop they generally involve di(cid:27)erent topolo-gies. Here we summarize various e(cid:27)orts that have been made over the years to exploit this property in a computationally meaningful way. As a (cid:28)rst example, we show how to generalize the Landau-Khalatnikov-Fradkin formula for the non-perturbative gauge transformation of the fermion propagator in QED to the general 2 n - point case by pure manipulations at the path-integral level. At the parameter-integral level, we show how to integrate out individual photons in the low-energy expansion, and then sketch a recently introduced general framework for the analytical evaluation of such worldline integrals involving a reduction to quantum mechanics on the circle and the relation between inverse derivatives and Bernoulli polynomials.