  Active matter is not only relevant to living matter and diverse
nonequilibrium systems, but also constitutes a fertile ground for novel
physics. Indeed, dynamic renormalization group (DRG) analyses have uncovered
many new universality classes (UCs) in polar active fluids (PAFs) - an
archetype of active matter systems. However, due to the inherent technical
difficulties in the DRG methodology, almost all previous studies have been
restricted to polar active fluids in the incompressible or infinitely
compressible (i.e., Malthusian) limits, and, when the $\epsilon$-expansion was
used in conjunction, to the one-loop level. Here, we use functional
renormalization group (FRG) methods to bypass some of these difficulties and
unveil for the first time novel critical behavior in compressible polar active
fluids, and calculate the corresponding critical exponents beyond the one-loop
level. Specifically, we investigate the multicritical point of compressible
PAFs, where the critical order-disorder transition coincides with critical
phase separation. We first study the critical phenomenon using a DRG analysis
and find that it is insufficient since two-loop effects are important to obtain
a nontrivial correction to the scaling exponents. We then remedy this defect by
using a FRG analysis. We find three novel universality classes and obtain their
critical exponents, which we then use to show that at least two of these
universality classes are out of equilibrium because they violate the
fluctuation-dissipation relation.
