Fractional-Order Regularization of Quantum Vacuum Energy via the Mittag-Leffler Function

The challenge of the vacuum catastrophe arises from the vast discrepancy between quantum field theoretical predictions and 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 widely applied in fractional calculus and anomalous diffusion, as a potential framework to address vacuum catastrophe. We computed the MLF-regularized vacuum energy integral, evaluated renormalization group equations, derived modified field equations for different parameter choices, and provided numerical solutions to 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 smoothly regulates coupling divergences by naturally attenuating high-energy contributions while preserving Lorentz invariance and renormalization group consistency. The field propagation profile exhibited suppression of high-energy components, consistent with the modified dispersion relation predicted by fractional-order differential formulations. The scale factor evolution indicated a reduced contribution from vacuum energy, aligning with the expectation that MLF diminishes the effective cosmological constant. We found minimal deviations in the cosmic microwave background power spectrum, suggesting that MLF regularization induces subtle modifications to the ΛCDM model. Our findings suggest that the vacuum may follow fractional-order statistical behaviour rather than purely Gaussian statistics, accounting for long-range correlations and memory effects while naturally suppressing high-energy divergences. While not a definitive resolution, MLF regularization represents a significant advancement in addressing the vacuum catastrophe, offering a physically motivated approach to vacuum energy regularization.