From the Bach Tensor to Traversable Wormholes in Conformal Gravity: Exact Computation for the Zero-Tidal-Force Ansatz and the Hamada Running Coupling Resolution

We present a two-part investigation of wormhole solutions in four-dimensional conformal (Weyl) gravity. Part I: We compute the Bach tensor Bμν for the Morris–Thorne wormhole metric with zero tidal force (Φ=0) and constant shape function b(r)=r₀, using a two-step covariant divergence method implemented in Python/SymPy. The result is verified to satisfy the tracelessness identity gμνBμν=0. All four diagonal components are proportional to r₀² and vanish only when r₀=0 (flat Minkowski space). Hence the simplest Morris–Thorne wormhole is not a vacuum solution of pure conformal gravity. Part II: We show that Hamada's radial running coupling, which replaces the constant coupling t² by the running quantity t̄²(r)=1/[β₀ ln(r²/r₀²)], modifies the Bach equation to α(r)Bμν+Dμν=0 with α(r)=β₀ ln(r²/r₀²). Near the throat (r→r₀⁺) the effective coefficient α→0⁺ and the induced quantum stress-energy diverges, automatically satisfying the flare-out condition b'(r₀)<1. A linearised boundary value problem confirms the existence of a non-trivial shape function perturbation. Thus conformal gravity à la Hamada admits traversable wormholes without exotic matter, with the throat radius r₀∼ξ_Λ=1/Λ_QG∼10⁻³⁵ m.