  Motivated by the collective behaviour of biological swarms, we study the
critical dynamics of field theories with coupling between order parameter and
conjugate momentum in the presence of dissipation. By performing a dynamical
renormalization group calculation at one loop, we show that the violation of
momentum conservation generates a crossover between a conservative yet
IR-unstable fixed point, characterized by a dynamic critical exponent $z=d/2$,
and a dissipative IR-stable fixed point with $z=2$. Interestingly, the two
fixed points have different upper critical dimensions. The interplay between
these two fixed points gives rise to a crossover in the critical dynamics of
the system, characterized by a crossover exponent $\kappa=4/d$. Such crossover
is regulated by a conservation length scale, $\mathcal R_0$, which is larger
the smaller the dissipation: beyond $\mathcal R_0$ the dissipative fixed point
dominates, while at shorter distances dynamics is ruled by the conservative
fixed point and critical exponent, a behaviour which is all the more relevant
in finite-size systems with weak dissipation. We run numerical simulations in
three dimensions and find a crossover between the exponents $z=3/2$ and $z=2$
in the critical slowing down of the system, confirming the renormalization
group results. From the biophysical point of view, our calculation indicates
that in finite-size biological groups mode-coupling terms in the equation of
motion can significantly change the dynamical critical exponents even in the
presence of dissipation, a step towards reconciling theory with experiments in
natural swarms. Moreover, our result provides the scale within which fully
conservative Bose-Einstein condensation is a good approximation in systems with
weak symmetry-breaking terms violating number conservation, as quantum magnets
or photon gases.
