Mathematical Anatomy of Neutrino Decoherence in Red Turbulence: Fractional Calculus and Convergence of the Perturbative Series

Abstract: We present a comprehensive study of perturbation theory for neutrino decoherence in power-law correlated turbulent matter, establishing rigorous convergence criteria and providing detailed numerical validation. Starting from the exact Laplace-space solution derived from the generalized master equation with memory kernel $K(t) \propto t^{-ν}e^{-t/τ_c}$, we develop a systematic perturbation expansion in the turbulence strength parameter $ξ$. The expansion is performed around the high-density matter basis where $\cos 2θ_m \approx 1$, rather than the vacuum basis, reflecting the physical conditions in supernova cores. We obtain explicit expressions for the first few perturbation terms and derive the general structure of higher-order terms, which involve Mittag-Leffler functions that emerge from the fractional dynamics. Using high-precision numerical methods (multiple-precision arithmetic and asymptotic expansions), we analyze the convergence radius and rate of the perturbation series. We find that for $ξ< ξ_c \approx 0.8$, the series converges with error decreasing geometrically. For $ξ= 0.16$ (typical supernova conditions), third-order perturbation theory achieves $\sim 1\%$ accuracy. We provide practical guidelines for applying perturbation theory to realistic astrophysical scenarios and validate all analytical results with extensive numerical computations. Our work clarifies several subtle aspects of the perturbative approach that were not addressed in the original literature, offering a complete toolkit for researchers studying neutrino-turbulence interactions. In particular, we systematically address the ultraviolet divergence of the correlation function at short time scales for spectral indices $ν\geq 1$, introducing a physical small-scale cutoff regularization that ensures mathematical rigor for the fractional master equation and perturbative expansion.