One-loop amplitudes in the worldline formalism

We summarize recent progress in applying the worldline formalism to the analytic calculation of one-loop N-point amplitudes. This string-inspired approach is well-adapted to avoiding some of the calculational inefficiencies of the standard Feynman diagram approach, most notably by providing master formulas that sum over diagrams differing only by the position of external legs and/or internal propagators. We illustrate the mathematical challenge involved with the low-energy limit of the N-photon amplitudes in scalar and spinor QED, and then present an algorithm that, in principle, solves this problem for the much more difficult case of the N-point amplitudes at full momentum in ϕ3 theory. The method is based on the algebra of inverse derivatives in the Hilbert space of periodic functions orthogonal to the constant ones, in which the Bernoulli numbers and polynomials play a central role.