One-loop effective action: nonlocal form factors and renormalization group

We review and present full detail of the Feynman diagram - based and heat-kernel method - based calculations of the simplest nonlocal form factors in the one-loop contributions of a massive scalar field. The paper has a pedagogical and introductory purposes and is intended to help the reader in better understanding the existing literature on the subject. The functional calculations are based on the solution by Avramidi and Barvinsky & Vilkovisky for the heat kernel and are performed in curved spacetime. One of the important points is that the main structure of non-localities is the same as in the flat background.