Renormalization group and nonlocal terms in the curved-spacetime effective action: Weak-field results

Renormalization-group methods are used to study interacting quantum field theory in curved spacetime. Details are presented explicitly showing how the short-distance behavior of the theory is linked to the existence of nonlocal terms in the effective action. We show how the usual value for the running electric charge in QED in flat spacetime may be obtained in this way by using the background-field method. A similar technique is then used to find nonlocal terms in the effective action arising from a real scalar field on a weakly curved gravitational background. It is shown how the results for the effective gravitational coupling constants obtained from analysis of these nonlocal terms agree with those found in our previous work.