Working towards a gauge-invariant description of the Higgs model: from local composite operators to spectral density functions - Egmond, Duive Maria Van (2020)
We analyze diﬀerent BRST invariant solutions for the introduction of a mass termin Yang-Mills (YM) theories. First, we analyze the non-local composite gauge-invariantﬁeld Ahµ(x), which can be localized by the Stueckelberg-like ﬁeld ξa(x). This enables usto introduce a mass term in the SU(N) YM model, a feature that has been indicated ata non-perturbative level by both analytical and numerical studies. The conﬁguration ofAh(x) , obtained through the minimization ofRd4xA2along the gauge orbit, gives riseto an all orders renormalizable action, a feature which will be illustrated by means of aclass of covariant gauge ﬁxings which, as much as ’t Hooft’s Rξ-gauge of spontaneouslybroken gauge theories, provide a mass for the Stueckelberg-like ﬁeld. Then, we considerthe 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 theelementary particle propagators, in conjunction with the Nielsen identities. We highlightthe role of the tadpole graphs and the gauge choices to get sensible results. We alsointroduce an Abelian Curci-Ferrari action coupled to a scalar ﬁeld to model a massivephoton which, like the non-Abelian Curci-Ferarri model, is left invariant by a modiﬁednon-nilpotent BRST symmetry. We clearly illustrate its non-unitary nature directly fromthe spectral function viewpoint. This provides a functional analogue of the Ojima ob-servation in the canonical formalism: there are ghost states with nonzero norm in theBRST-invariant states of the Curci-Ferrari model. Finally, the spectral properties of aset of local gauge-invariant composite operators are investigated in the U(1) and SU(2)Higgs model quantized in the ’t Hooft Rξgauge. These operators enable us to give agauge-invariant description of the spectrum of the theory, thereby surpassing certain in-commodities when using the standard elementary ﬁelds. The corresponding two-pointcorrelation functions are evaluated at one-loop order and their spectral functions are ob-tained explicitly. It is shown that the spectral functions of the elementary ﬁelds suﬀerfrom a strong unphysical dependence from the gauge parameter ξ, and can even exhibitpositivity violating behaviour. In contrast, the BRST invariant local operators exhibit awell deﬁned positive spectral density.