Effective Action from the Functional Renormalization Group

Abstract

We study the quantum gravitational system coupled to a charged scalar, Dirac fermions, and electromagnetic fields. We use the "exact" or "functional" renormalization group equation to derive the effective action Γ 0 by integrating the flow equation from the ultraviolet scale down to k = 0. The resulting effective action consists of local terms and nonlocal terms with unique coefficients, which could be tested by comparing it with observation.

Nonlocal Spacetime-Matter

“…Therefore, we remind the reader to [22], but a short proof is given in the appendix D. In particular, it deserves to be notice that in the perturbative expansion in of equation ( 8) the exponential form factor always contributes at the zero order, namely 0 . Moreover, for the standard model of particle physics the Hessian resulting from the local action is diagonal (or constant) at the order 0 (see [23]). Hence, the inversion ( 9) is actually trivial at any perturbative order.…”

“…Last comment is about the non-diagonal elements of the operator ∆. Indeed, one can easily see that such components are at least linear in the fields [23] and, therefore, can not affect the propagators around the Minkowski spacetime.…”

“…For the reader who needs to see the explicit form of the Hessian, we refer to the paper[23], where the components of ∆ are provided for gravity coupled to a complex scalar, a dirac fermion, and an abelian gauge vector.…”

Functional Renormalization and $\overline{\text{MS}}$

“…By integrating these formulae from some ultraviolet scale Λ down to k = 0, one obtains the effective action for the metric. See [24,25] for some calculations of this type.…”