Physical spectrum from confined excitations in a Yang-Mills-inspired toy model

We study a toy model for an interacting scalar field theory in which the fundamental excitations are confined in the sense of having unphysical, positivity-violating propagators, a fact tracing back to a decomposition of these in propagators with complex conjugate mass poles (the so-called $i$-particles). Similar two-point functions show up in certain approaches to gluon or quark propagators in Yang-Mills gauge theories. We investigate the spectrum of our model and show that suitable composite operators may be constructed having a well-defined K\"all\'{e}n-Lehmann spectral representation, thus allowing for a particle interpretation. These physical excitations would correspond to the "mesons" of the model, the latter being bound states of two unphysical $i$-particles. The meson mass is explicitly estimated from the pole emerging in a resummed class of diagrams. The main purpose of this paper is thus to explicitly verify how a real mass pole can and does emerge out of constituent $i$-particles that have complex masses.