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

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 $\stackrel{^}{d}({q}^{2})$ of the PT propagator, which is both gauge invariant and renormalization-group invariant, can be factored into the product of the running charge ${\overline{g}}^{2}({q}^{2})$ and a term $\stackrel{^}{H}({q}^{2})$ 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 $\stackrel{^}{d}$ but only through $\stackrel{^}{H}$.