Reconstructing propagators of confined particles in the presence of complex singularities

Propagators of confined particles, especially the Landau-gauge gluon propagator, may have complex singularities as suggested by recent numerical works as well as several theoretical models, e.g., motivated by the Gribov problem. In this paper, we study formal aspects of propagators with complex singularities in reconstructing Minkowski propagators starting from Euclidean propagators by the analytic continuation. We derive the following properties rigorously for propagators with arbitrary complex singularities satisfying some boundedness condition. The two-point Schwinger function with complex singularities violates the reflection positivity. In the presence of complex singularities, while the holomorphy in the usual tube is maintained, the reconstructed Wightman function on the Minkowski spacetime becomes a non-tempered distribution and violates the positivity condition. On the other hand, the Lorentz symmetry and locality are kept intact under this reconstruction. Finally, we argue that complex singularities can be realized in a state space with an indefinite metric and correspond to confined states. We also discuss consequences of complex singularities in the BRST formalism. Our results could open up a new way of understanding a confinement mechanism, mainly in the Landau-gauge Yang-Mills theory.