Fractional-Order Regularization of Quantum Vacuum Energy Via The Mittag-Leffler Function

Abstract

The challenge of the vacuum catastrophe arises from the discrepancy between quantum field theoretical predictions and the observed vacuum energy density. Using analytic continuation techniques and asymptotic analysis to ensure physical consistency, we explored the Mittag-Leffler function (MLF), a generalized exponential function applied in fractional calculus and anomalous diffusion, as a potential framework to address the vacuum catastrophe. We computed the MLFregularized vacuum energy integral, evaluated renormalization group equations, derived modified field equations for different parameter choices and provided numerical solutions of the modified Friedmann equations to track the evolution of the scale factor. Unlike conventional approaches relying on arbitrary cutoffs and standard QFT predictions, which exhibit uncontrolled growth at high energies, MLF regulates coupling divergences by attenuating high-energy contributions while preserving Lorentz invariance and renormalization group consistency. The field propagation profile exhibited suppression of high-energy components, consistent with modified dispersion relations predicted by fractional-order formulations. The scale factor evolution indicated a reduced contribution from vacuum energy, aligning with the expectation that MLF diminishes the effective cosmological constant over cosmic timescales. We found minimal deviations in the cosmic microwave background power spectrum relative to the standard cosmological model, consistent with current observational constraints.