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 mass-deformed 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 non-perturbative aspects of the Yang-Mills theory.
Highlights
Color confinement is one of the central features of the strong interactions, which implies that colored particles are absent in the observed spectrum
We apply this relation to the mass-deformed Yang-Mills model with one-loop quantum corrections, which is identified with a low-energy effective theory of the YangMills 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
III, we show by using the argument principle how the number of complex poles of a propagator is determined under assumptions on the sign of the spectral function and the asymptotic behaviors of the propagator at the small and large momenta in the complex momentum plane
Summary
Color confinement is one of the central features of the strong interactions, which implies that colored particles are absent in the observed spectrum. We pay special attention to complex poles and the branch cut of the gluon and ghost propagators in the Landau gauge Yang-Mills theory. We apply one of these relations to the massive Yang-Mills model with one-loop quantum corrections to show that the gluon propagator has a pair of complex conjugate poles or “tachyonic” poles (in the Euclidean region) of multiplicity two, as a consequence of the fact that the gluon field has a negative spectral function in the one-loop level. We investigate the pole structure of the gluon propagator in the whole parameter space of the massive Yang-Mills model to examine the possible crossover from confining region to Higgs-like region in the single confined phase, which has been recently discussed in [6,8,17] from the different viewpoints. In Appendix B, we analyze the other models available in the literature to show the consistency with the general relation
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