Interacting quasiparticle positivity and propagator structure

Determine whether loop corrections preserve positivity of the spectral functions and the Osterwalder–Schrader axioms for higher-point correlation functions of the Refined Gribov–Zwanziger quasiparticle fields, and whether interactions generate a nonzero mixed propagator between the quasiparticles \(\lambda_\mu^a\) and \(\eta_\mu^a\) or induce longitudinal components in their propagators.

Background

At quadratic order, the transverse Refined Gribov–Zwanziger formulation yields quasiparticle fields λμa\lambda_\mu^a and ημa\eta_\mu^a with vanishing mixed propagator and transverse two-point functions. Their spectral functions are nonnegative when the mass poles are real and positive. Once interactions are included, however, the action contains numerous vertices involving both quasiparticles and the remaining Zwanziger auxiliary-field components.

The paper identifies several unresolved consequences of radiative corrections: preservation or violation of spectral positivity, validity of the Osterwalder–Schrader axioms beyond two-point functions, reappearance of a mixed ⟨λη⟩\langle\lambda\eta\rangle propagator, and generation of longitudinal propagator components. The authors state that a systematic analysis of these questions is deferred to future work. A subsequent one-loop calculation in the appendix does show a divergent mixed contribution, but the broader questions concerning its renormalization and the full interacting quasiparticle interpretation remain unresolved.

References

An important motivation for such an analysis would be to test whether loop corrections spoil or not the positivity of the spectral function for the quasiparticle propagators. Furthermore, we wish to know if higher correlation functions of the quasiparticle fields of the theory also respect the Osterwalder-Schrader axioms, at least to some extent, and not just the two-point functions. In this context, other valid questions are whether or not a mixed propagator \langle\lambda_\mua\eta_\nub\rangle\not=0 could appear, blurring the quasiparticle picture in the interacting theory, or if the quasiparticle propagators would acquire a longitudinal part through radiative corrections. As one may anticipate, a proper analysis of these questions would be quite demanding and we defer it to future work.

— On Quasiparticles within the Refined Gribov-Zwanziger Model  (2609.04423 - Garcia et al., 3 Sep 2026) in Appendix A, Section A.1, “Some difficulties with quasiparticles in the interacting theory”