Summary
We have shown that expansions for the fourth-rank vacuum polarization tensor for a virtual four-photon system can be built up in general, in terms of seventeen linearly independent explicitly gauge-invariant polynomial tensors. A special choice of the tensor basis corresponds to the one obtained by Karplus and Neuman. We have also dealt with the real four-photon system. An expression for the tensor amplitude which describes this process has been found in general, in terms of five linearly independent manifestly gauge-invariant polynomial tensors. Both in the case of massive and real photons, we have investigated the kinematical singularities of the scalar amplitudes relative to the chosen tensor bases. For the real case we have also obtained the low-energy limits of the helicity amplitudes in the centre-of-mass system. Finally, we have looked for the effective Lagrangian densities related to the helicity amplitudes in the low-energy region.
Riassunto
Si costruiscono, in generale, decomposizioni del tensore di polarizzazione del vuoto di rango quattro, relative a un sistema di quattro fotoni virtuali, in diciassette tensori polinomiali invarianti di gauge linearmente indipendenti. Una scelta particolare della base tensoriale riproduce la base di Karplus e Neuman. Si considera anche il sistema di quattro fotoni reali. Si determina, in generale, un’espressione dell’ampiezza tensoriale corrispondente a questo processo, in funzione di cinque tensori polinomiali invarianti di gauge linearmente indipendenti. Sia nel caso di fotoni virtuali che in quello di fotoni reali, si studiano le singolarità cinematiche delle ampiezze scalari relative alle basi tensoriali scelte. Nel caso di fotoni reali, si ottengono i limiti di bassa energia delle ampiezze di elicità nel centro di massa. Si scrivono, infine, le densità di lagrangiana effettiva relative alle ampiezze di elicità nella regione di bassa energia.
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We use here the notation of ref. (4).
We note that, att =0, the relations (12) and (13) become trivial identities (0=0). This is the reason whyBrown andMuzinich (ref. (2)) have no trouble of redundancy.
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ℵ is the electron mass.
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Leo, R.A., Soliani, G. & Minguzzi, A. Tensor amplitudes for elastic photon-photon scattering. Nuov Cim A 30, 270–286 (1975). https://doi.org/10.1007/BF02730173
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DOI: https://doi.org/10.1007/BF02730173