Multifractal Analysis & the Dynamics of Effective Field Theories

Abstract

We show that the flow from the ultraviolet to the infrared sector of any multidimensional nonlinear field theory approaches chaotic dynamics in a universal way. This result stems from several independent routes to aperiodic behavior and implies that the infrared attractor of effective field theories is likely to replicate the geometry of multifractal sets. In particular, we find that the Einstein-Hilbert Lagrangian is characterized by a single generalized dimension (D_GR = 4), while the Standard Model (SM) Lagrangian is defined by a triplet of generalized dimensions (D_SM = 2, 3 & 4). On the one hand, this finding disfavors any naïve field-theoretic unification of SM and General Relativity (GR). On the other, it hints that the continuous spectrum of generalized dimensions lying between D = 2 and D = 4 may naturally account for the existence of non-baryonic Dark Matter.