Large Spatial Variation of the Fine-Structure Constant Driven by Scalar Potential Relaxation: A Modified BSBM Framework

We propose a theoretical modification to the Bekenstein-Sandvik-Barrow-Magueijo (BSBM) theory by introducing an effective stabilization potential $V(\psi)$ for the scalar field governing the fine-structure constant $\alpha$. While the position of the potential minimum may be motivated by anthropic considerations (e.g., nuclear stability), the dynamics are treated as a classical relaxation process. In this framework, the spatial dipole in $\alpha$ reported by Webb et al. is interpreted not as a static anisotropy, but as a snapshot of an ultra-light scalar field undergoing slow-roll relaxation across the Hubble volume. We discuss the necessary mass scale of the field and the implications for local equivalence principle tests.