Tensor amplitudes for elastic photon-photon scattering

It is shown that the 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 explicity gauge-invariant polynomial tensors. A special choice of the tensor basis corresponds to the one obtained by Karplus and Neuman. The papers deal also with the real four-photon system. An expression for the tensor amplitude which describes this process have been found in general, in terms of five linearly independent manifestly gauge-invariant polynomial tensors. Both in the case of massive and real photons, the kinematical singularities of the scalar amplitudes relative to the chosen tensor bases have been investigated. For the real case the low energy limits of the helicity amplitudes in the centr-of-mass system have been obtained. Finally the effective Lagrangian densities related to the helicity amplitudes in the low energy region has been looked for