  We study the analytical structure of the effective action for spin- and
mass-imbalanced Fermi mixtures at the onset of the superfluid state. Of our
particular focus is the possibility of suppressing the tricritical temperature
to zero, so that the transition remains continuous down to $T=0$ and the phase
diagram hosts a quantum critical point. At mean-field level we analytically
identify such a possibility in a regime of parameters in dimensionality $d=3$.
In contrast, in $d=2$ we demonstrate that the occurrence of a quantum critical
point is (at the mean-field level) excluded. We show that the Landau expansion
of the effective potential remains well-defined in the limit $T\to 0^+$ except
for a subset of model parameters which includes the standard BCS limit. We
calculate the mean-field asymptotic shape of the transition line. Employing the
functional renormalization group framework we go beyond the mean field theory
and demonstrate the stability of the quantum critical point in $d=3$ with
respect to fluctuations.
