# Reflection positivity and complex analysis of the Yang-Mills theory from a viewpoint of gluon confinement

## TL;DR

This paper investigates the confining properties of the Yang-Mills theory by analyzing the massive Yang-Mills model, demonstrating reflection positivity violation, and exploring the complex structure of the gluon propagator through both numerical and analytical methods.

## Contribution

It establishes the equivalence of the massive Yang-Mills model with a gauge-invariant extension and confirms its ability to reproduce lattice results and explain gluon confinement features.

## Key findings

- Gluon and ghost propagators match lattice data with one-loop corrections.
- Reflection positivity is violated in the model due to complex conjugate poles.
- The gluon propagator's complex structure explains the Gribov-Stingl form fit.

## Abstract

In order to understand the confining decoupling solution of the Yang-Mills theory in the Landau gauge, we consider the massive Yang-Mills model which is defined by just adding a gluon mass term to the Yang-Mills theory with the Lorentz-covariant gauge fixing term and the associated Faddeev-Popov ghost term. First of all, we show that massive Yang-Mills model is obtained as a gauge-fixed version of the gauge-invariantly extended theory which is identified with the gauge-scalar model with a single fixed-modulus scalar field in the fundamental representation of the gauge group. This equivalence is obtained through the gauge-independent description of the Brout-Englert-Higgs mechanism proposed recently by one of the authors. Then, we reconfirm that the Euclidean gluon and ghost propagators in the Landau gauge obtained by numerical simulations on the lattice are reproduced with good accuracy from the massive Yang-Mills model by taking into account one-loop quantum corrections. Moreover, we demonstrate in a numerical way that the Schwinger function calculated from the gluon propagator in the Euclidean region exhibits violation of the reflection positivity at the physical point of the parameters. In addition, we perform the analytic continuation of the gluon propagator from the Euclidean region to the complex momentum plane towards the Minkowski region. We give an analytical proof that the reflection positivity is violated for any choice of the parameters in the massive Yang-Mills model, due to the existence of a pair of complex conjugate poles and the negativity of the spectral function for the gluon propagator to one-loop order. The complex structure of the propagator enables us to explain why the gluon propagator in the Euclidean region is well described by the Gribov-Stingl form.