Background-field method versus normal field theory in explicit examples: One-loop divergences in theSmatrix and Green's functions for Yang-Mills and gravitational fields

The gauge dependence in the background-field method of counterterms on and off the mass shell is investigated in several gauges for Yang-Mills and gravitational fields, including the axial gauge. The results agree with theorems of DeWitt and Kallosh. The relation between the quantum and classical gauge-fixing terms is discussed. It is shown how to determine specific S-matrix elements in the background-field method. A complete calculation using normal field theory of the one-loop divergences of scalar-scalar scattering through graviton exchange is presented. The results agree with those obtained in the background-field method by 't Hooft and Veltman.