Probing the singularities of the Landau-gauge gluon and ghost propagators with rational approximants

TL;DR

This study uses model-independent Padé approximants to analyze lattice data of Landau-gauge gluon and ghost propagators, confirming complex poles for the gluon and a pole at zero for the ghost, with evidence of branch cuts.

Contribution

The paper introduces the application of Padé and D-Log Padé approximants to lattice propagator data, providing a more rigorous and error-aware analysis of their analytic structures.

Findings

Confirmation of a pair of complex poles in the gluon propagator.

Evidence of a pole at zero in the ghost propagator.

Indications of a branch cut along the negative real axis for the ghost.

Abstract

We employ Pad\'e approximants in the study of the analytic structure of the four-dimensional $SU(2)$ Landau-gauge gluon and ghost propagators in the infrared regime. The approximants, which are model independent, serve as fitting functions for the lattice data. We carefully propagate the uncertainties due to the fitting procedure, taking into account all possible correlations. For the gluon-propagator data, we confirm the presence of a pair of complex poles at $p_{\rm pole}^2 = \left[(-0.37 \,\pm\, 0.05_{\rm stat}\,\pm\, 0.08_{\rm sys}) \pm i\,(0.66\, \pm\, 0.03_{\rm stat}\, \pm\, 0.02_{\rm sys})\right]\, \mathrm{GeV}^2$, where the first error is statistical and the second systematic. The existence of this pair of complex poles, already hinted upon in previous works, is thus put onto a firmer basis, thanks to the model independence and to the careful error propagation of our analysis. For the ghost propagator, the Pad\'es indicate the existence of a single pole at $p^2 = 0$, as expected. In this case, our results also show evidence of a branch cut along the negative real axis of $p^2$. This is corroborated with another type of approximant, the D-Log Pad\'es, which are better suited to studying functions with a branch cut and are applied here for the first time in this context. Due to particular features and limited statistics of the gluon-propagator data, our analysis is inconclusive regarding the presence of a branch cut in the gluon case.