Abstract
We analyze in the Landau gauge mixing of bosonic fields in gauge theories with exact and spontaneously broken symmetries, extending to this case the Lehmann–Symanzik–Zimmermann (LSZ) formalism of the asymptotic fields. Factorization of residues of poles (at real and complex values of the variable p2) is demonstrated and a simple practical prescription for finding the “square-rooted” residues, necessary for calculating S-matrix elements, is given. The pseudo-Fock space of asymptotic (in the LSZ sense) states is explicitly constructed and its BRST-cohomological structure is elucidated. Usefulness of these general results, obtained by investigating the relevant set of Slavnov–Taylor identities, is illustrated on the one-loop examples of the Z0-photon mixing in the Standard Model and the GZ-Majoron mixing in the singlet Majoron model.
Highlights
We analyze in the Landau gauge mixing of bosonic fields in gauge theories with exact and spontaneously broken symmetries, extending to this case the Lehmann-Symanzik-Zimmermann (LSZ) formalism of the asymptotic fields
S-operator (see e.g. [6] and the formula (13) below) are normalized so as to reproduce the behavior of the corresponding two-point functions near the poles associated with stable particles, and ii) poles at complex values of p2 are associated with no asymptotic states, i.e. unstable particles contribute to the S-matrix only through the internal lines
In case of generic mixing, the unphysical components of these asymptotic fields create states which combine into the Kugo-Ojima quartet representations of the BRST algebra [20] what is essential for unitarity of the S-matrix [6, 20]
Summary
We analyze in the Landau gauge mixing of bosonic fields in gauge theories with exact and spontaneously broken symmetries, extending to this case the Lehmann-Symanzik-Zimmermann (LSZ) formalism of the asymptotic fields. The pseudo-Fock space of asymptotic (in the LSZ sense) states is explicitly constructed and its BRST-cohomological structure is elucidated. Usefulness of these general results, obtained by investigating the relevant set of Slavnov-Taylor identities, is illustrated on the one-loop examples of the Z0-photon mixing in the Standard Model and the GZ-Majoron mixing in the singlet Majoron model. The proper way of extracting S-matrix elements is provided by the Lehmann-Symanzik-Zimmermann (LSZ) asymptotic approach which basically consist of analyzing the pole structure of the relevant two-point functions of the fields which mix, and reconstructing on this basis the Fock space of the true asymptotic states.
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