FUCHSIAN STRUCTURE AND LOCAL EXPONENTS OF DISPERSION AMPLITUDES OF TWO LOOPS WITH PHYSICAL MASSES

Abstract

This article analyzes the analytical structure of the Feynman integrals that contribute to the two-loop scattering amplitudes in quantum field theory, focusing specifically on the presence of multiple non-degenerate internal physical masses. By using and exploiting integration-by-parts identities, the infinite family of multi-loop integrals is projected and reduced to a minimal, finite set of independent master integrals. From this basis vector, the coupled system of linear ordinary differential equations with respect to the external Mandelstam kinematic invariants is derived. It is rigorously shown that, after applying a rational norm transformation based on a modified variant of the Moser reduction algorithm, the differential system can be uniquely mapped to a strictly Fuchsian structure characterized exclusively by first-order logarithmic poles. The regular singular loci of the system are analytically associated with the geometric configurations of physical particle production thresholds and pseudo-thresholds. Furthermore, the local analytic exponents in the vicinity of these singular points are calculated through the exact solution of the algebraic-matrix indices derived from the residue matrices. This demonstrates how the spectrum of these exponents uniquely determines the analytic branching of the amplitudes, their physical discontinuities, and their monodromy properties in the complex kinematic plane. Finally, the robustness of the algebraic framework is validated by implementing it on the nontrivial sector of the three-mass asymmetric Sunrise diagram, contrasting the efficiency of the resulting Taylor-Frobenius series local expansions against traditional numerical sector decomposition techniques. The results obtained offer an exact, divergence-free, and algorithmically efficient analytical alternative for evaluating high-precision observations in contemporary particle collider phenomenology.