  The recent inflow of empirical data about the collective behaviour of
strongly correlated biological systems has brought field theory and the
renormalization group into the biophysical arena. Experiments on bird flocks
and insect swarms show that social forces act on the particles' velocity
through the generator of its rotations, namely the spin, indicating that
mode-coupling field theories are necessary to reproduce the correct dynamical
behaviour. Unfortunately, a theory for three coupled fields - density, velocity
and spin - has a prohibitive degree of intricacy. A simplifying path consists
in getting rid of density fluctuations by studying incompressible systems. This
requires imposing a solenoidal constraint on the primary field, an unsolved
problem even for equilibrium mode-coupling theories. Here, we perform an
equilibrium dynamic renormalization group analysis of a mode-coupling field
theory subject to a solenoidal constraint; using the classification of Halperin
and Hohenberg, we can dub this case as a solenoidal Model G. We demonstrate
that the constraint produces a new vertex that mixes static and dynamical
coupling constants, and that this vertex is essential to grant the closure of
the renormalization group structure and the consistency of dynamics with
statics. Interestingly, although the solenoidal constraint leads to a
modification of the static universality class, we find that it does not change
the dynamical universality class, a result that seems to represent an exception
to the general rule that dynamical universality classes are narrower than
static ones. Our results constitute a solid stepping stone in the admittedly
large chasm towards developing an off-equilibrium mode-coupling theory of
biological groups.
