Discontinuity Method for Evaluating Scattering Amplitudes within the Worldline Formalism

Abstract

The worldline formalism offers an alternative framework to the standard diagrammatic approach in quantum field theory, grounded in first-quantized relativistic path integrals. Over recent decades, this formalism has attracted growing interest due to its potential applications and computational advantages. As a result, a collection of worldline master integrals has been derived within this approach. However, efficient mathematical tools for evaluating these integrals remain limited. Motivated by unitarity methods used in the conventional formalism of quantum field theory, this work proposes a novel framework for evaluating one-loop worldline master integrals, up to a rational function in the kinematical external invariants, through the computation of their discontinuities. Similarly to standard unitarity techniques, the method involves decomposing an $n$-point one loop amplitude in $D=4$ dimensions into a linear combination of tadpole, bubble, triangle, and box master integrals. The coefficients of this decomposition are rational functions of the external kinematic invariants. Each master integral exhibits a characteristic discontinuity with respect to a specific kinematic invariant. Consequently, the rational coefficients can be determined by directly computing the discontinuities of the corresponding worldline master integrals. In the three-point and four-point on-shell cases, this computation is particularly tractable, as the discontinuities involve Dirac delta functions arising from the Sokhotski-Plemelj formula, which simplify the resulting integrals. For the four-point on-shell case, a set of master formulas for the discontinuities is derived. The practical implementation of the proposed discontinuity method is demonstrated through several illustrative examples.