  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 $\hat{H}$.
