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

Abstract

We consider the problem of "measuring" the Källén-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) p 2 ≥ 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 p 2 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 SUð3Þ Landau gauge gluon at zero temperature.

Spectral densities from the lattice

“…This study leaded us to try to find out a clever way to compute the spectral density, and hopefully an updated method arose -see [12] deep infrared region due to the different choices of the IR cut-off µ 0 , the results are much more stable if compared with previous results -see Figure 5. Moreover, the IR cut-off is now interrelated with the Tikhonov parameter λ -see [12] for details.…”

“…Here we use an alternative approach, based on Tikhonov regularization with Morozov discrepancy principle. For details see our previous papers [10,11,12].…”

Analytic structure of QCD propagators in Minkowski space

“…Nevertheless, by a linear regularization strategy, a Källen-Lehmann spectral function was reconstructed in Ref. [34] from the lattice data of the gluon propagator, giving some direct evidence for positivity violation and the absence of any discrete mass pole on the physical real axis.…”

Untitled

“…Here we also report on our investigation of the gluon and ghost spectral density computed as described in [14].…”