Abstract
We study an f(R) approximation to asymptotic safety, using a family of non-adaptive cutoffs, kept general to test for universality. Matching solutions on the four-dimensional sphere and hyperboloid, we prove properties of any such global fixed point solution and its eigenoperators. For this family of cutoffs, the scaling dimension at large n of the nth eigenoperator, is λn ∝ b n ln n. The coefficient b is non-universal, a consequence of the single-metric approximation. The large R limit is universal on the hyperboloid, but not on the sphere where cutoff dependence results from certain zero modes. For right-sign conformal mode cutoff, the fixed points form at most a discrete set. The eigenoperator spectrum is quantised. They are square integrable under the Sturm-Liouville weight. For wrong sign cutoff, the fixed points form a continuum, and so do the eigenoperators unless we impose square-integrability. If we do this, we get a discrete tower of operators, infinitely many of which are relevant. These are f(R) analogues of novel operators in the conformal sector which were used recently to furnish an alternative quantisation of gravity.
Highlights
We study an f (R) approximation to asymptotic safety, using a family of non-adaptive cutoffs, kept general to test for universality
Solutions of this equation determine the flow of the infinite number of effective couplings that parametrise the most general effective action. (The space spanned by all these couplings is known as theory space.) Exact solutions would require solving an infinite number of coupled differential equations and seem out of reach in any realistic setting, so for quantum gravity
One analytical approach is powerful, namely to solve the RG equations asymptotically at large curvature R [34, 35]. This technique is sufficient on its own to allow one to draw definitive conclusions about both the nature of the fixed points and their eigenoperator spectra in a given f (R) approximation [34, 35]. It was adapted from studies of scalar field theories in derivative expansion approximation, where it had already proved to be powerful [36,37,38,39,40,41], and in ref. [42] it was applied to the so-called conformal sector of quantum gravity
Summary
We start on the four-sphere, which was the space-time explicitly treated in [15]. It has space-time volume V = 384π2/R2, and there the space-time traces are sums over the discrete set of eigenvalues of the corresponding Laplacian:. As is clear from the cutoff profile formula, (2.8), the αs parameters allow us to shift the action of the cutoff up or down relative to the tower of eigenvalues, so as to ensure all the modes are passed as k is lowered to k → 0+ This requires that the lowest mode λnφ,s + αsR is positive. [15], it is safe to choose α2 = 0 and α1 = 0, but to implement this condition in the physical scalar (a.k.a. conformal factor) sector we need to choose α0 > 1/3 At this point we recognise the need to specialise to smooth (infinitely differentiable) cutoff functions r(z). Given that the eigenvalues are discrete set, proportional to R, cutoff functions that are not smooth, for example the optimised one r(z) = (1 − z) θ(1 − z) [51], will lead to points of limited differentiability which accumulate as R → 0. There, the sums go over to an integral and the equations go over to ones in flat space
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