The gradient-flow formulation of the energy-momentum tensor of QCD is extended to NNLO perturbation theory. This means that the Wilson coefficients which multiply the flowed operators in the corresponding expression for the regular energy-momentum tensor are calculated to this order. The result has been obtained by applying modern tools of regular perturbation theory, reducing the occurring two-loop integrals, which also include flow-time integrations, to a small set of master integrals which can be calculated analytically.

We study quantum effects in higher curvature extensions of general relativity using the functional renormalisation group. New flow equations are derived for general classes of models involving Ricci scalar, Ricci tensor, and Riemann tensor interactions. Our method is applied to test the asymptotic safety conjecture for quantum gravity with polynomial Riemann tensor interactions of the form ∼, and functions thereof. Interacting fixed points, universal scaling dimensions, gaps in eigenvalue spectra, quantum equations of motion, and de Sitter solutions are identified by combining high order polynomial approximations, Padé resummations, and full numerical integration. Most notably, we discover that quantum-induced shifts of scaling dimensions can lead to a four-dimensional ultraviolet critical surface. Increasingly higher-dimensional interactions remain irrelevant and show near-Gaussian scaling and signatures of weak coupling. Moreover, a new equal weight condition is put forward to identify stable eigenvectors to all orders in the expansion. Similarities and differences with results from the Einstein-Hilbert approximation, f (R) approximations, and f (R, Ric 2 ) models are highlighted and the relevance of findings for quantum gravity and the asymptotic safety conjecture is discussed.

We derive all heat kernel coefficients for Laplacians acting on scalars, vectors, and tensors on fully symmetric spaces, in any dimension. Final expressions are easy to evaluate and implement, and confirmed independently using spectral sums. We also obtain the Green's function for Laplacians acting on transverse traceless tensors in any dimension. Applications to quantum gravity and the functional renormalisation group are indicated. Contents

Two typos in Ref. 1 are corrected.

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