Quantum Distortion Gravity: Hidden Matter Sectors and the Fixed Point of the Einstein–Hilbert Flow

A first version of this work argued that asymptotic safety of Quantum Distortion Gravity (QDG) requires a sufficiently large hidden sector. That conclusion was drawn from a phenomenological one-loop flow whose gravitational functions are not derived from the Einstein–Hilbert action, and it does not survive a derivation from the Wetterich equation. We repeat the analysis with the flow derived in the Einstein–Hilbert truncation (optimised cutoff, harmonic gauge $\alpha = 1$, linear split, background $S^4$), with minimally coupled matter characterised by two integers $(X,Y)$ that count scalars, Dirac fermions and vectors. Three results are obtained. (i) No hidden sector is needed for a non-Gaussian fixed point to exist: it exists for QDG-like content ($n_f$ Dirac fermions and two Proca fields) for all $n_f < 10$, and for the full Standard-Model content, but for $n_f = 3$ and for the Standard Model it lies outside the range $g_* < 2$, $\lambda_* > -1$ in which the truncation is plausible. (ii) A hidden sector acts as a displacement in the $(X,Y)$ plane: massless vectors, and for QDG-like content also scalars, move the fixed point back towards the pure-gravity one (one hidden vector suffices for $n_f = 3$; seven are needed for the Standard Model with the optimised cutoff), whereas hidden Dirac fermions move it away and, in the fermion-rich direction, destroy it by run-away. (iii) These statements depend on the scheme chosen for the matter cutoff; with a cutoff that is a function of $-\bar{D}^2$ only, the fermion and vector coefficients change and the fermion run-away is absent in the cases tested. The Weyl-gauged operators of QDG are not part of the flow, so none of this is a statement about the asymptotic safety of QDG itself.