Abstract
I give explicit formulae 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 Z factors of Lehmann, Symanzik and Zimmermann. So obtained generalized Z factors, are indispensable to the correct extraction of physical amplitudes from the amputated correlation functions in the presence of mixing.The standard Rξ gauges form a particularly important subclass of gauges considered in this paper. While the tree-level vector-scalar mixing is, by construction, absent in Rξ gauges, it unavoidably reappears at higher orders. Therefore the prescription for the generalized Z factors given in this paper is directly relevant for the extraction of amplitudes in Rξ gauges.
Highlights
Bosonic fields can mix with each other, unless the symmetries tell us otherwise
In the Standard Model (SM) of particle physics, there are four neutral elementary bosonic fields: scalar h0, vector γμ, vector Zμ and the scalar would-be Goldstone field GZ associated with Zμ
While the mixing between h0 and the other fields requires a transfer of CP-violation from the quark sector, there are dozens of SM extension in which Zμ is mixed with physical scalars already at one-loop; the singlet Majoron model [2] is perhaps the simplest extension of this sort
Summary
Bosonic fields can mix with each other, unless the symmetries tell us otherwise. In the Standard Model (SM) of particle physics (see e.g. [1]), there are four neutral elementary bosonic fields: scalar h0, vector γμ (photon), vector Zμ and (in renormalizable gauges) the scalar would-be Goldstone field GZ associated with Zμ. 2 it would be welcomed to have a simple prescription that gives, in the presence of a generic mixing, the physical amplitudes directly it terms of amputated correlation functions. In order to make the present paper as self-contained as possible, I will recapitulate here the generalized LSZ algorithm in purely scalar theories. From the representation (3) of the propagator it is clear how to generalize the LSZ algorithm to the case of (purely scalar) mixing. (Strictly speaking, asymptotic fields Φlr exist only for real pole masses mS(l), and I put the prime on the first sum in (7).) In other words, to obtain the correctly normalized (i.e. consistent with unitarity) amplitude of the process involving a particle corresponding to Φlr , one has to contract the eigenvector ζSj [lr] with the amputated correlation functions Aj...(p, . I will explain how to generalize this algorithm to the mixing in gauge theories, beyond the Landau gauge, what is the main subject of this paper
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