Positivity issues for the pinch-technique gluon propagator and their resolution

TL;DR

This paper investigates the positivity issues of the gluon propagator in the pinch technique, attributing the sign change to asymptotic freedom, and proposes a factorization approach that aligns with the K"allen-Lehmann representation.

Contribution

It introduces a factorization of the PT gluon propagator into components satisfying the KL representation, resolving positivity issues linked to asymptotic freedom.

Findings

The product of the coupling and scalar part of the PT propagator can be factored into two KL-satisfying parts.

Analytic models with a large gauge boson mass support the proposed factorization.

PT Schwinger-Dyson equations depend on the factorized component, not directly on the propagator.

Abstract

Although gauge-boson propagators in asymptotically-free gauge theories satisfy a dispersion relation, they do not satisfy the K\"allen-Lehmann (KL) representation because the spectral function changes sign. We argue that this is a simple consequence of asymptotic freedom. On the basis of the QED-like Ward identities of the pinch technique (PT) we claim that the product of the coupling $g^2$ and the scalar part $\hat{d}(q^2)$ of the PT propagator, which is both gauge invariant and renormalization-group invariant, can be factored into the product of the running charge $\bar{g}^2(q^2)$ and a term $\hat{H}(q^2)$ both of which satisfy the KL representation although their product does not. We show that this behavior is consistent with some simple analytic models that mimic the gauge-invariant PT Schwinger-Dyson equations (SDE) provided that the dynamic gauge boson mass is sufficiently large. The PT SDEs do not depend directly on the PT propagator through $\hat{D}$ but only through $\hat{H}$.