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Non-local form factors in flat and curved spacetime

This chapter is devoted to the direct explicit calculations of non-local form factors in two-point functions in real scalar field theory. Two simple examples in flat spacetime demonstrate the relationship between logarithmic ultraviolet (UV) divergences in the cut-off and dimensional regularizations, which is used for deriving the form factors. The chapter then shows how one can establish the direct relation between logarithmic UV divergences and the logarithmic behavior of the momentum-dependent non-local form factors in the UV. In the low-energy (infrared) limit, it is possible to observe quadratic decoupling with respect to the mass of the quantum field. In curved space, analogous results are reproduced using the generally covariant heat-kernel solution. Calculations are given in full details.