Abstract
We consider the unitary Abelian Higgs model and investigate its spectral functions at one-loop order. This analysis allows to disentangle what is physical and what is not at the level of the elementary particle propagators, in conjunction with the Nielsen identities. We highlight the role of the tadpole graphs and the gauge choices to get sensible results. We also introduce an Abelian Curci-Ferrari action coupled to a scalar field to model a massive photon which, like the non-Abelian Curci-Ferarri model, is left invariant by a modified non-nilpotent BRST symmetry. We clearly illustrate its non-unitary nature directly from the spectral function viewpoint. This provides a functional analogue of the Ojima observation in the canonical formalism: there are ghost states with nonzero norm in the BRST-invariant states of the Curci-Ferrari model.
Highlights
In recent years there has been an increasing interest in the properties of the spectral function (Källen-Lehmann density) of two-point correlation functions, especially in nonAbelian gauge theories such as quantum chromodynamics (QCD)
We introduce an Abelian Curci-Ferrari action coupled to a scalar field to model a massive photon which, like the non-Abelian Curci-Ferarri model, is left invariant by a modified non-nilpotent BRST symmetry
It was found [1,2,3,4,5], in lattice simulations for the minimal Landau gauge, that the spectral function of the gluon propagator is not non-negative everywhere, which means that there is no physical interpretation for this propagator like there is for the photon propagator in quantum electrodynamics (QED)
Summary
In recent years there has been an increasing interest in the properties of the spectral function (Källen-Lehmann density) of two-point correlation functions, especially in nonAbelian gauge theories such as quantum chromodynamics (QCD). Unitarity of the gauge bosons sector is not so much an issue, as one expects them to be undetectable anyhow, due to confinement Within this perspective the existence of an exact nilpotent BRST symmetry becomes a quite relevant property when trying to generalize the action (1) to other gauges than Landau gauge, as next to unitary one should expect that the correlation functions of gauge-invariant observables are gauge-parameter independent. 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 p2plane at unwanted locations, i.e., away from the negative real axis, deep into the complex plane, thereby invalidating a Källen-Lehmann spectral representation This can be appreciated by rewriting the Feynman integrals in terms of Schwinger or Feynman parameters, whose analytic properties can be studied through the Landau equations [55].
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