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

Abstract

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 \( K(t) \propto t^{-\nu} \), the spectral index \(\nu\) characterizes the correlation structure: smaller (including negative) values of \(\nu\) correspond to stronger long-range correlations. To treat ultraviolet singularities for \( \nu \geq 1 \) 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 negative spectral indices (\( \nu < 0 \)), 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.