Quantum Distortion Gravity: A Complete One-Loop Analysis of the Minimal Truncation

We present a complete, self-contained analysis of the one-loop functional renormalization group (FRG) flow of Quantum Distortion Gravity (QDG), a metric-affine framework with gauged Weyl symmetry. We derive the classical foundations from first principles: the irreducible decomposition of the distortion tensor with an explicit inversion of the trace matrix, the Weyl-covariant distortion $\tilde D$, the no-go theorem for bare distortion invariants, and the Weyl-invariant ultraviolet action with distortion entering through $\phi^2 I_i[\tilde D]$. We then establish the constraint structure of the truncated theory (8 physical degrees of freedom, no spin-2 ghost) via a complete Dirac--Bergmann analysis, and analyze the BRST--BV quantization of the gauge algebra $Diff(\mathcal M) \ltimes Weyl$, including the antifield spectrum, the nilpotency proof, and the classical master equation. We compute the Weyl anomaly coefficients from heat-kernel data and prove the anomaly is cancellable by a Wess--Zumino term. In the main part of the paper we derive all one-loop beta functions from first principles, keeping every intermediate step. We show that in unitary gauge, the dilaton is a gauge mode and does not contribute; Proca fields contribute with three transverse polarizations and no Faddeev--Popov ghosts; Dirac fermions contribute via the Litim fermionic regulator. We extract the matter coefficients $a_S = +1/2$, $a_D = -2$, and $a_V = +3/2$ (the last for a Proca vector in unitary gauge), and we verify each by two independent routes. We derive the gravitational anomalous dimension from the explicit trace over TT graviton, trace mode, and Faddeev--Popov ghosts, reproducing the Litim result $A(\lambda)$ and $B(\lambda)$, and we display the ADM decomposition that yields the mode multiplicities. We show that the fixed-point equation $\beta_{\tilde G} = 0$ is genuinely quadratic in $\tilde G$ at fixed $\lambda$, we solve it explicitly, and we give the resulting discriminant. For the Standard Model matter content ($n_f = 3$, $n_V = 2$), we find a non-trivial UV fixed point at $\tilde G^* \approx 0.43$, $\lambda^* \approx -0.02$, with two relevant directions in the $(\tilde G, \lambda)$ sector, and we compute the stability matrix, critical exponents, and their numerical uncertainties. We verify robustness with a sensitivity scan, showing the fixed point exists for all $C_{\rm bl} > C_{\rm bl}^{\rm crit} \approx -1.02$, well above the QDG value $C_{\rm bl} = -3/(2\pi) \approx -0.477$; we give the analytic form of the critical line. We also analyze the heat kernel in a background field strength $F_{\mu\nu}$, showing that the mixing between graviton and Proca in the $(h^{01}, V^1)$ sector gives a negative contribution to the FRG coefficient $\hat c_V$ (distinct from the Weyl-anomaly coefficient $c^{\rm W}$), and we exhibit the second-order heat-kernel trace explicitly. Finally, we discuss scale generation via Coleman--Weinberg in a two-scalar extension, derive the flat-direction condition, and compare our results with the recent Proca--gravity fixed point of Pastor-Marcos et al. All symbolic and numerical calculations are performed in Python (SymPy, NumPy, SciPy), and the complete code is provided in the appendices, together with additional verification scripts.