Vector-scalar mixing to all orders, for an arbitrary gauge model in the generic linear gauge

I give explicit fromulae for full propagators of vector and scalar fields in a generic spin-1 gauge model quantized in an arbitrary linear covariant gauge. The propagators, expressed in terms of all-order one-particle-irreducible correlation functions, have a remarkably simple form because of constraints originating from Slavnov-Taylor identities of Becchi-Rouet-Stora symmetry. I also determine the behavior of the propagators in the neighborhood of the poles, and give a simple prescription for the coefficients that generalize (to the case with an arbitrary vector-scalar mixing) the standard $\sqrt{\mathcal{Z}}$ factors of Lehmann, Symanzik and Zimmermann. So obtained generalized $\sqrt{\mathcal{Z}}$ factors, are indispensable to the correct extraction of physical amplitudes from the amputated correlation functions in the presence of mixing. The standard $R_\xi$ guauges form a particularly important subclass of gauges considered in this paper. While the tree-level vector-scalar mixing is, by construction, absent in $R_\xi$ gauges, it unavoidably reappears at higher orders. Therefore the prescription for the generalized $\sqrt{\mathcal{Z}}$ factors given in this paper is directly relevant for the extraction of amplitudes in $R_\xi$ gauges.