  We consider stochastic dynamics of self-propelled particles with nonlocal
normalized alignment interactions subject to phase lag. The role of the lag is
to indirectly generate chirality into particle motion. To understand large
scale behavior, we derive a continuum description of an active Brownian
particle (ABP) flow with macroscopic scaling in the form of a partial
differential equation (PDE) for a one-particle probability density function
(DF). Due to indirect chirality, we find a new spatially homogeneous
nonstationary analytic solution for this class of equations. Our development of
kinetic and hydrodynamic theories towards such a solution reveals the existence
of a wide variety of spatially nonhomogeneous patterns reminiscent of the
traveling bands, clouds, and vortical structures of linear active matter. Our
model may thereby serve as the basis for understanding the nature of chiral
active media and designing multiagent swarms with designated behavior.
