Källén-Lehmann spectroscopy for (un)physical degrees of freedom

We consider the problem of ``measuring'' the K\"all\'en-Lehmann spectral density of a particle (be it elementary or bound state) propagator by means of 4D lattice data. As the latter are obtained from operations at (Euclidean momentum squared) p²≥0, we are facing the generically ill-posed problem of converting a limited data set over the positive real axis to an integral representation, extending over the whole complex p² plane. We employ a linear regularization strategy, commonly known as the Tikhonov method with the Morozov discrepancy principle, with suitable adaptations to realistic data, e.g. with an unknown threshold. An important virtue over the (standard) maximum entropy method is the possibility to also probe unphysical spectral densities, for example, of a confined gluon. We apply our proposal here to ``physical'' mock spectral data as a litmus test and then to the lattice $SU(3)$ Landau gauge gluon at zero temperature.