International Journal of Modern Physics A c © World Scientific Publishing Company A NUMERICAL EVALUATION OF VACUUM POLARIZATION

Hattori-Itakura have recently derived the full Landau-level summation form for the pho-ton vacuum polarization tensor in constant external magnetic fields at the one-loop level. The Landau-level summation form is essential when the photon momentum exceeds the threshold of the pair creation of charged particles in a magnetic field stronger than the squared mass of the charged particle. The tensor has three different form factors depend-ing on the tensor direction with respect to the external magnetic field. The renormaliza-tion is nontrivial because these form factors are expressed in terms of double or triple summation forms. We give a numerical UV subtraction method which can be applied to numerically evaluate the form factors in constant external magnetic fields. We numeri-cally investigate the photon vacuum polarization tensor in the form of the Landau-level summation and estimate the systematic errors coming from truncation of the Landau-level summation in a parameter region realized in heavy ion collision experiments. We find that the error is practically controllable at an O(10−2) level for electrons and muons in strong magnetic fields expected in heavy ion collisions in the experimentally feasible kinematic parameter regions