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

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. We try to understand these results in light of the Fradkin-Shenker continuity between confinement-like and Higgs-like regions in a single confinement phase in the complementary gauge-scalar model.

Some remarks on the spectral functions of the Abelian Higgs model

“…As such, it is laying the groundwork for future work on these properties in the non-Abelian SU (2) Higgs case as well as in the massive model of [37], eq. (1), investigating the origin of the complex poles structure reported in [54,58,59].…”

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

“…The connection with the present work is obvious from the Källén-Lehmann representation of propagators, in terms of positive-definite spectral functions. 1 As we will see, the Padés to the gluon propagator D(p 2 ) have poles with an imaginary part incompatible with zero, which is in contradiction with the usual Källén-Lehmann representation for D(p 2 ) [68,69]. Following ref.…”

The analytic structure of the lattice Landau gauge gluon and ghost propagators

“…We would like to call the readers attention that if the studies performed herein and in [10,11,23,36,37,[42][43][44][45][46] suggest or assume that the gluon propagator has pairs of complex poles singularities, this is not always the case. For example, in [30] the authors solved the coupled set of Dyson-Schwinger equations for the gluon and ghost propagators, using a particular truncation, and found no evidence of complex conjugate poles.…”