Quantum corrections to the Schwarzschild metric and reparametrization transformations

Quantum corrections to the Schwarzschild metric generated by loop diagrams have been considered by Bjerrum-Bohr, Donoghue, and Holstein (BHD) [ Phys. Rev. D 68 084005 (2003)], and Khriplovich and Kirilin (KK) [ J. Exp. Theor. Phys. 98 1063 (2004)]. Though the same field variables in a covariant gauge are used, the results obtained differ from one another. The reason is that the different sets of diagrams have been used. Here we will argue that the quantum corrections to metric must be independent of the choice of field variables, i.e. must be reparametrization invariant. Using simple reparametrization transformation, we will show that the contribution considered by BDH, is not invariant under it. Meanwhile the corrections to the Schwarzschild metric considered by KK satisfies the requirement of the invariance.