Effective Lagrangian forλφ4theory in curved spacetime with varying background fields: Quasilocal approximation

Abstract

By means of a Riemann normal coordinate expansion for the metric and a momentum-space representation for the Green's function, we derive in analytic form the one-loop effective Lagrangian for a $\ensuremath{\lambda}{\ensuremath{\varphi}}^{4}$ theory in curved spacetime which is exact to all orders in $\ensuremath{\lambda}$ and includes variation of the background field up to the second order. Ultraviolet divergences are removed by small-proper-time expansion and dimensional regularization. We obtain a generalized expression for the ${a}_{2}$ Minakshisundaram-DeWitt coefficient of the scalar wave operator with spacetime-dependent background field. A set of renormalization-group equations for the coupling constants of the theory is obtained, which can be used for analyzing their curvature and energy dependence. An alternative derivation of the effective Lagrangian is presented via the heat-kernel technique for anisotropic harmonic oscillators. Our result is useful for the study of quantum processes in the early universe or black holes under conditions where spacetime curvature and dynamical field effects are important. When suitably generalized, the effective Lagrangian obtained here in the form of a quasipotential should provide an improvement over the flat-space Coleman-Weinberg potential assumed in most discussions of the new inflationary universe. Possible directions for developing our method for more general problems are also discussed.