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

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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 D, the no-go theorem for bare distortion invariants, and the Weyl-invariant ultraviolet action with distortion entering through φ² Iᵢ[ 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( M) 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 aS = +1/2, aD = -2, and aV = +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(λ) and B(λ), and we display the ADM decomposition that yields the mode multiplicities. We show that the fixed-point equation βG = 0 is genuinely quadratic in G at fixed λ, we solve it explicitly, and we give the resulting discriminant. For the Standard Model matter content (nf = 3, nV = 2), we find a non-trivial UV fixed point at G^* ≈ 0.43, λ^* ≈ -0.02, with two relevant directions in the ( G, λ) 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 Cbl > Cblᶜʳⁱᵗ ≈ -1.02, well above the QDG value Cbl = -3/(2π) ≈ -0.477; we give the analytic form of the critical line. We also analyze the heat kernel in a background field strength Fμν, showing that the mixing between graviton and Proca in the (h⁰¹, V¹) sector gives a negative contribution to the FRG coefficient cV (distinct from the Weyl-anomaly coefficient cW), 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.