Fixed points of quantum gravity in extra dimensions

We study quantum gravity in more than four dimensions with renormalisation group methods. We find a non-trivial ultraviolet fixed point in the Einstein-Hilbert action. The fixed point connects with the perturbative infrared domain through finite renormalisation group trajectories. We show that our results for fixed points and related scaling exponents are stable. If this picture persists at higher order, quantum gravity in the metric field is asymptotically safe. We discuss signatures of the gravitational fixed point in models with low scale quantum gravity and compact extra dimensions. PACS numbers: 04.60.-m, 04.50.+h, 11.15.Tk, 11.25.Mj The physics of gravitational interactions in more than four space-time dimensions has received considerable interest in recent years. The possibility that the fundamental Planck mass – within a higher dimensional setting – may be as low as the electroweak scale [1, 2, 3] has stimulated extensive model building and numerous investigations aiming at signatures of extra spatial dimensions ranging from particle collider experiments to cosmological and astrophysical settings. Central to these scenarios is that gravity lives in higher dimensions, while standard model particles are often confined to the four dimensional brane (although the latter is not crucial in what follows). In part, these models are motivated by string theory, where additional spatial dimensions arise naturally [4]. Then string theory would, at least in principle, provide for a short distance definition of these theories which presently have to be considered as effective rather than fundamental ones. In the absence of an explicit ultraviolet completion, gravitational interactions at high energies including low scale gravity can be studied with effective field theory or semi-classical methods, as long as quantum gravitational effects are absent, or suppressed by some ultraviolet cutoff of the order of the fundamental Planck mass, e.g. [5]. One may then wonder whether a quantum theory of gravity in the metric degrees of freedom can exist in four and more dimensions as a cutoff-independent, well-defined and non-trivial local theory down to arbitrarily small distances. It is generally believed that the above requirements imply the existence of a non-trivial ultraviolet fixed point under the renormalisation group, governing the short-distance physics. The corresponding fixed point action then provides a microscopic starting point to access low energy phenomena of quantum gravity. This ultraviolet completion does apply for quantum gravity in the vicinity of two dimensions, where an ultraviolet fixed point has been identified with ǫ-expansion techniques [6, 7, 8]. In the last couple of years, a lot of efforts have been put forward to access the four-dimensional case, and a number of independent studies have detected an ultraviolet fixed point using functional and renormalisation group methods in the continuum [9, 10, 11, 12, 13, 14, 15, 16, 17, 18] and Monte Carlo simulations on the lattice [19, 20]. Continuity in the dimension suggests that a non-trivial fixed point – if it exists in four dimensions and below – should persist at least in the vicinity and above four dimensions. Furthermore, the critical dimension of quantum gravity – the dimension where the gravitational coupling has vanishing canonical mass dimension – is two. For any dimension above the critical one, the mass dimension of the gravitational coupling is negative. Hence, from a renormalisation group point of view, four dimensions are not special. More generally, one expects that the local structure of quantum fluctuations, and hence local renormalisation group properties of quantum theories of gravity, are qualitatively similar for all dimensions above the critical one, modulo topological effects in specific dimensions. In this Letter, we perform a fixed point search for quantum gravity in more than four dimensions [15] (see also [13]). An ultraviolet fixed point, if it exists, should already be visible in the purely gravitational sector, to which we confine ourselves. Matter degrees of freedom and gauge interactions can equally be taken into account. We employ a functional renormalisation group based on a cutoff effective action Γk for the metric field [9, 10, 11, 12, 13, 14, 15, 16, 17, 21], see [22] and [23] for reviews in scalar and gauge theories. In Wilson’s approach, the functional Γk comprises momentum fluctuations down to the momentum scale k, interpolating between ΓΛ at some reference scale k = Λ and the full quantum effective action at k → 0. The variation of the effective action with the cutoff scale (t = ln k) is given by an exact functional flow ∂tΓk = 1 2Tr 1 Γ (2) k + Rk ∂tRk . (1) The trace is a sum over fields and a momentum integration, and Rk is a momentum cutoff for the propagating fields. The flow relates the change in Γk with a loop integral over the full cutoff propagator. By construction, the flow (1) is finite and, together with the boundary condition ΓΛ, defines the theory. In renormalisable theories, the cutoff Λ can be removed, Λ → ∞, and ΓΛ → Γ∗ remains well-defined for arbitrarily short distances. In