Quantum Effective Action in Odd Dimensions and Zeta-Function Regularization

In this work, we mainly study the one-loop effective action for real scalar theories in non-homogeneous backgrounds in odd dimensions. It is shown that through the method studied in Ref. [1], it is possible to obtain a unified result for the one-loop correction to the Euclidean effective action in 2+1 dimensions. Our result simplifies into the ones given in the literature calculated via different techniques based on some assumptions of field and its derivative. The one-loop correction is obtained in (2n+1)-dimensional Euclidean space-times, as well. Moreover, its generic behavior is discussed for non-homogeneous backgrounds. Afterwards, the structure of the divergence for scalar field theories at one-loop is discussed by studying the one-loop correction in 2n-dimensional Euclidean space-times. The beta function of the O(N)-invariant nonlinear sigma-model in d=2+1 is calculated at leading order in the 1/N expansion without using the epsilon-expansion method.