  Phase transitions in active fluids attracted significant attention within the
last decades. Recent results show [L. Chen et al., New J. Phys. 17, 042002
(2015)] that an order-disorder phase transition in incompressible active fluids
belongs to a new universality class. In this work, we further investigate this
type of phase transition and focus on the effect of long-range interactions.
This is achieved by introducing a non-local shear stress into the hydrodynamic
description, which leads to superdiffusion of the velocity field, and can be
viewed as a result of the active particles performing Levy walks. The universal
properties in the critical region are derived by performing a perturbative
renormalization group analysis of the corresponding response functional within
the one-loop approximation. We show that the effect of non-local shear stress
decreases the upper critical dimension of the model, and can lead to the
irrelevance of the active fluid self-advection with the resulting model
belonging to an unusual 'long-range Model A' universality class not reported
before. Moreover, when the degree of non-locality is sufficiently high all
non-linearities become irrelevant and the mean-field description is valid in
any spatial dimension.
