Complex poles and spectral function of Yang-Mills theory

Abstract

We derive general relationships between the number of complex poles of a propagator and the sign of the spectral function originating from the branch cut in the Minkowski region under some assumptions on the asymptotic behaviors of the propagator. We apply this relation to the massdeformed Yang-Mills model with one-loop quantum corrections, which is identified with a low-energy effective theory of the Yang-Mills theory, to show that the gluon propagator in this model has a pair of complex conjugate poles or "tachyonic" poles of multiplicity two, in accordance with the fact that the gluon field has a negative spectral function, while the ghost propagator has at most one "unphysical" pole. Finally, we discuss implications of these results for gluon confinement and other nonperturbative aspects of the Yang-Mills theory.

Some remarks on the spectral functions of the Abelian Higgs model

“…Most articles on massive Yang-Mills models employ the renormalizable Landau gauge, although it was noticed that this gauge might not be the preferred gauge in non-perturbative calculations [53]. In fact, as was recently established in [54], the use of the Landau gauge in the massive Yang-Mills model (1) leads to complex pole masses, which will obstruct a calculation of the Källén-Lehmann spectral function. Indeed, if at some order in perturbation theory (one-loop as in [54] for example) a pair of Euclidean complex pole masses appear, at higher order these poles will generate branch points in the complex p 2 -plane at unwanted locations, i.e.…”

“…In fact, as was recently established in [54], the use of the Landau gauge in the massive Yang-Mills model (1) leads to complex pole masses, which will obstruct a calculation of the Källén-Lehmann spectral function. Indeed, if at some order in perturbation theory (one-loop as in [54] for example) a pair of Euclidean complex pole masses appear, at higher order these poles will generate branch points in the complex p 2 -plane at unwanted locations, i.e. away from the negative real axis, deep into the complex plane, thereby invalidating a Källén-Lehmann spectral representation.…”

Extracting topological information from momentum space propagators

“…Here, again, we point out that our topological invariant assumes the same absolute values as the one of [37], despite that they do not necessarily represent the same topological quantity. Therefore, following a completely different approach from the one in [37] (although both are aimed to the momentum space topological analysis of the gauge field two-point Green's function), we could find a phase transition between different regimes of the gauge propagator, in agreement with [37].…”

“…2), while avoiding every possible real pole, branch point and branch cut. This new topological object is a relative of the topological object N W defined in [37], with the difference that their object is defined in terms of the propagator itself, while our N Γ depends on the mass function M(z), which in general can be defined by writing in full generality a fermion propagator as in (3), or for a bosonic propagator (stripping off all possible color/Lorentz tensorial structures)…”

“…Interestingly, the authors of [37] found N W = 0 for gauge propagators displaying only real poles; and N W = −2 for gauge propagators with a set of complex conjugate poles.…”