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Register a new user$d=4$ as the critical dimensionality of asymptotically safe interactions
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We explore the question why our universe is four dimensional from an asymptotically safe vantage point. We find hints that asymptotically safe quantum fluctuations of gravity can only solve the $U(1)$ Landau-pole problem in the Standard Model in four dimensions. This could single out the observed dimensionality of the universe as the critical dimensionality of asymptotically safe interactions.
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The effect of gravitational fluctuations on the quantum effective potential for scalar fields is a key ingredient for predictions of the mass of the Higgs boson, understanding the gauge hierarchy problem and a possible explanation of an---asymptotically---vanishing cosmological constant. We find that the quartic self-interaction of the Higgs scalar field is an irrelevant coupling at the asymptotically safe ultraviolet fixed point of quantum gravity. This renders the ratio between the masses of the Higgs boson and top quark predictable. If the flow of couplings below the Planck scale is approximated by the Standard Model, this prediction is consistent with the observed value. The quadratic term in the Higgs potential is irrelevant if the strength of gravity at short distances exceeds a bound that is determined here as a function of the particle content. In this event, a tiny value of the ratio between the Fermi scale and the Planck scale is predicted.
We comment on Weinbergs interesting analysis of asymptotically safe inflation (arXiv:0911.3165). We find that even if the gravity theory exhibits an ultraviolet fixed point, the energy scale during inflation is way too low to drive the theory close to the fixed point value. We choose the specific renormalization groupflow away from the fixed point towards the infrared region that reproduces the Newtons constant and todays cosmological constant. We follow this RG flow path to scales below the Planck scale to study the stability of the inflationary scenario. Again, we find that some fine tuning is necessary to get enough efolds of infflation in the asymptotically safe inflationary scenario.
We study the consistency of dynamical fermionic matter with the asymptotic safety scenario of quantum gravity using the functional renormalisation group. Since this scenario suggests strongly coupled quantum gravity in the UV, one expects gravity-induced fermion self-interactions at energies of the Planck-scale. These could lead to chiral symmetry breaking at very high energies and thus to large fermion masses in the IR. The present analysis which is based on the previous works cite{Christiansen:2015rva, Meibohm:2015twa}, concludes that gravity-induced chiral symmetry breaking at the Planck scale is avoided for a general class of NJL-type models, regardless of the number of fermion flavours. This suggests that the phase diagram for these models is topologically stable under the influence of gravitational interactions.
The renormalization group flow of unimodular quantum gravity is investigated within two different classes of truncations of the flowing effective action. In particular, we search for non-trivial fixed-point solutions for polynomial expansions of the $f(R)$-type as well as of the $F(R_{mu u}R^{mu u})+R,Z(R_{mu u}R^{mu u})$ family on a maximally symmetric background. We close the system of beta functions of the gravitational couplings with anomalous dimensions of the graviton and Faddeev-Popov ghosts treated according to two independent prescriptions: one based on the so-called background approximation and the other based on a hybrid approach which combines the background approximation with simultaneous vertex and derivative expansions. For consistency, in the background approximation, we employ a background-dependent correction to the flow equation which arises from the proper treatment of the functional measure of the unimodular path integral. We also investigate how different canonical choices of the endomorphism parameter in the regulator function affect the fixed-point structure. Although we have found evidence for the existence of a non-trivial fixed point for the two classes of polynomial projections, the $f(R)$ truncation exhibited better (apparent) convergence properties. Furthermore, we consider the inclusion of matter fields without self-interactions minimally coupled to the unimodular gravitational action and we find evidence for compatibility of asymptotically safe unimodular quantum gravity with the field content of the Standard Model and some of its common extensions.
The asymptotic safety scenario in gravity is accessed within the systematic vertex expansion scheme for functional renormalisation group flows put forward in cite{Christiansen:2012rx,Christiansen:2014raa}, and implemented in cite{Christiansen:2015rva} for propagators and three-point functions. In the present work this expansion scheme is extended to the dynamical graviton four-point function. For the first time, this provides us with a closed flow equation for the graviton propagator: all vertices and propagators involved are computed from their own flows. In terms of a covariant operator expansion the current approximation gives access to $Lambda$, $R$, $R^2$ as well as $R_{mu u}^2$ and higher derivative operators. We find a UV fixed point with three attractive and two repulsive directions, thus confirming previous studies on the relevance of the first three operators. In the infrared we find trajectories that correspond to classical general relativity and further show non-classical behaviour in some fluctuation couplings. We also find signatures for the apparent convergence of the systematic vertex expansion. This opens a promising path towards establishing asymptotically safe gravity in terms of apparent convergence.
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