Quantum entropy of a Schwarzschild black hole due to the Dirac field

The quantum entropy of the massive Dirac field outside a Schwarzschild black hole is investigated. The cutoff in the brick-wall model is removed by introducing some fictitious regulator fields, and the entropy is regulated by the large mass of these fictitious regulators. The one-loop effective action for the Dirac field is also obtained by using these regulators. The divergent entropy can be renormalized to the standard Bekenstein-Hawking formula, in terms of Newton's constant appearing in the renormalization of the action for the gravitational-quantum matter fields. The renormalized Newton constant is different from the case of the scalar field, at least in form.