A proper fixed functional for four-dimensional Quantum Einstein Gravity

Abstract

Realizing a quantum theory for gravity based on Asymptotic Safety hinges on the existence of a non-Gaussian fixed point of the theory's renormalization group flow. In this work, we use the functional renormalization group equation for the effective average action to study the fixed point underlying Quantum Einstein Gravity at the functional level including an infinite number of scale-dependent coupling constants. We formulate a list of guiding principles underlying the construction of a partial differential equation encoding the scale-dependence of f (R)-gravity. We show that this equation admits a unique, globally well-defined fixed functional describing the non-Gaussian fixed point at the level of functions of the scalar curvature. This solution is constructed explicitly via a numerical double-shooting method. In the UV, this solution is in good agreement with results from polynomial expansions including a finite number of coupling constants, while it scales proportional to R 2 , dressed up with non-analytic terms, in the IR. We demonstrate that its structure is mainly governed by the conformal sector of the flow equation. The relation of our work to previous, partial constructions of similar scaling solutions is discussed.

Renormalization group equation and scaling solutions for f(R) gravity in exponential parametrization

“…On the other hand, no ghosts are present in [22]. In spite of these differences, the equation of [22] was shown in [23] to have a global scaling solution whose general shape is quite similar to ours. The equation for unimodular gravity has only been analyzed at polynomial level so far.…”

“…In the Hessian on the four-sphere (2.11), the operator = −∇ 2 appears everywhere and is a natural choice. However, in order to gain some additional freedom, we follow [22,23] and add to terms proportional to the scalar curvature, with coefficients −α, −γ , and −β for spin two, one, and zero, respectively. These parameters should not be confused with the gauge fixing parameters which do not appear in the following.…”

“…The heat kernel coefficients b 2n for acting on spin two, one, and zero are given in [12,23] for type I cutoff. We extend the calculation to our case and give the results in Appendix C. Substituting these heat kernel coefficients and Eqs.…”

“…This is a general issue that goes beyond the f (R) truncations and progress in this direction has been made in [19,20]. In the meantime solutions were found in simplified (lower-dimensional and/or conformally reduced) settings [21,22] and recently also in the full four-dimensional case [23].…”

Fate of nonpolynomial interactions in scalar field theory

“…However this is also an area where there is little guidance from current experimental observation or other techniques, and therefore one must place particular reliance on a rigorous understanding of the mathematical structure that the exact RG exposes, in so far as this is possible. This is especially so with recent work on "functional truncations" [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36].…”