Mathematical Anatomy of Neutrino Decoherence in Red Turbulence: A Fractional Calculus Approach

We develop an exact framework for neutrino decoherence in power-law correlated turbulent matter, as encountered in core-collapse supernovae. Employing the Nakajima–Zwanzig projection technique, we derive an exact non-Markovian master equation for the neutrino density matrix. For kernels K ( t ) ∝ t − ν , the red-noise sector in our convention corresponds to ν < 0, while ν = 1 is the white-noise boundary. To treat ultraviolet singularities for ν ≥ 1 without spoiling the fractional structure, we use a renormalization prescription based on Hadamard finite parts and analytic continuation. The exact Laplace-space solution for the survival probability is obtained. In the high-density matter basis relevant to supernovae, the solution is expressed through Mittag–Leffler functions, establishing a direct link to anomalous diffusion phenomena. For red spectra ( ν < 0), the memory integral corresponds to a higher-order fractional operator. Our work clarifies how spectral index, renormalization scale, and decoherence efficiency interrelate, providing a complete analytical description and practical tools for supernova neutrino simulations. The fractional calculus formulation reveals fundamental mathematical connections between neutrino flavor evolution and other systems governed by long-range temporal correlations.