Abstract: The numerical evaluation of multi-loop scattering amplitudes in the Feynman representation usually requires to deal with both physical (causal) and unphysical (non-causal) singularities. The loop-tree duality (LTD) offers a powerful framework to easily characterise and distinguish these two types of singularities, and then simplify analytically the underling expressions. In this paper, we work explicitly on the dual representation of multi-loop Feynman integrals generated from three parent topologies, which we refer to as Maximal, Next-to-Maximal and Next-to-Next-to-Maximal loop topologies. In particular, we aim at expressing these dual contributions, independently of the number of loops and internal configurations, in terms of causal propagators only. Thus, providing very compact and causal integrand representations to all orders. In order to do so, we reconstruct their analytic expressions from numerical evaluation over finite fields. This procedure implicitly cancels out all unphysical singularities. We also interpret the result in terms of entangled causal thresholds. In view of the simple structure of the dual expressions, we integrate them numerically up to four loops in integer space-time dimensions, taking advantage of their smooth behaviour at integrand level.

Abstract: We investigate the analytic structure of the quark propagator in the Landau gauge by dynamically coupling the standard gap equation to the non-perturbative quark-gluon vertex. Employing the full vertex basis, we demonstrate that for sub-GeV time-like momenta, the proper inclusion of the underlying dynamics leads to a pair of real poles with opposite-sign residues. In particular, in stark contradistinction to the results obtained in widely used approximations, we see no sign of complex conjugate poles. This distinctive analytic structure evades conceptual shortcomings frequently associated with complex conjugate poles while remaining fully compatible with the aspects of color confinement related to positivity violation. Crucially, this novel behavior is governed by a dominant triplet of vertex form factors: the tree-level component, the anomalous chromomagnetic moment, and a component we label as “spin-momentum curvature”. By gradually tuning the individual strengths of these components, we demonstrate that while they contribute in distinct ways to the quark propagator, their joint action is vital for stabilizing the system. Together, they place the low-lying poles onto the real axis while producing a robust constituent quark mass of 350 MeV.