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Relativistic Corrections to the Sunyaev-Zel'dovich Effect for Clusters of Galaxies. III. Polarization Effect

We extend the formalism of the relativistic thermal and kinematic Sunyaev-Zel'dovich effects and include the polarization of the cosmic microwave background photons. We consider the situation that a cluster of galaxies is moving with a velocity \vec{v} with respect to the cosmic microwave background radiation. In the present formalism, polarization of the scattered cosmic microwave background radiation caused by the proper motion of a cluster of galaxies is naturally derived as a special case of the kinematic Sunyaev-Zel'dovich effect. The relativistic corrections are included also in a natural way. Our results are in complete agreement with the recent results of the relativistic corrections obtained by Challinor, Ford and Lasenby with an entirely different method as well as the nonrelativistic limit obtained by Sunyaev and Zel'dovich. The relativistic correction becomes to be significant in the Wien region.