Abstract
We calculate and analyse non-local gravitational form factors induced by quantum matter fields in curved two-dimensional space. The calculations are performed for scalars, spinors and massive vectors by means of the covariant heat kernel method up to the second order in the curvature and confirmed using Feynman diagrams. The analysis of the ultraviolet (UV) limit reveals a generalized “running” form of the Polyakov action for a nonminimal scalar field and the usual Polyakov action in the conformally invariant cases. In the infrared (IR) we establish the gravitational decoupling theorem, which can be seen directly from the form factors or from the physical beta function for fields of any spin.
Highlights
The decoupling of quantum massive field at the energies which are much smaller than the mass of the field is the cornerstone of the effective approach to quantum field theory
The nonlocal form factors of the vacuum effective action have been derived for several types of 2D massive fields in an approximation which includes all the terms up to the second order in curvature
An equivalent derivation based on Feynman diagrams for the scalar field case is deferred to Appendix B
Summary
The decoupling of quantum massive field at the energies which are much smaller than the mass of the field is the cornerstone of the effective approach to quantum field theory. A practical realization of this program meets serious obstacles, especially because of the technical difficulties of making calculations in higher derivative gravity within a mass-dependent renormalization scheme In this situation it is tempting to start from the simplest model of quantum gravity, and in this respect the two dimensional (2D) quantum gravity is a very useful model to test general concepts. Regardless of its relative simplicity compared to the 4D vacuum calculations, we shall observe that the 2D results may be interesting, since they enable one to explore general issues as the relation between massless limit and the anomaly-induced effective action in the UV from one side and the decoupling of massive degrees of freedom at the IR end of the energy scale.
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