Electrostatic self-energy and Bekenstein entropy bound in the massive   Schwinger model
  We obtain the electrostatic energy of two opposite charges near the horizon of stationary black-holes in the massive Schwinger model. Besides the confining aspects of the model, we discuss the Bekenstein entropy upper bound of a charged object using the generalized second law. We show that despite the massless case, in the massive Schwinger model the entropy of the black hole and consequently the Bekenstein bound are altered by the vacuum polarization. 
H. Mohseni Sadjadi and M. Alimohammadi