  When studying the collective motion of biological groups a useful theoretical
framework is that of ferromagnetic systems, in which the alignment interactions
are a surrogate of the effective imitation among the individuals. In this
context, the experimental discovery of scale-free correlations of speed
fluctuations in starling flocks poses a challenge to the common statistical
physics wisdom, as in the ordered phase of standard ferromagnetic models with
$\mathrm{O}(n)$ symmetry, the modulus of the order parameter has finite
correlation length. To make sense of this anomaly a novel ferromagnetic theory
has been proposed, where the bare confining potential has zero second
derivative (i.e.\ it is marginal) along the modulus of the order parameter. The
marginal model exhibits a zero-temperature critical point, where the modulus
correlation length diverges, hence allowing to boost both correlation and
collective order by simply reducing the temperature. Here, we derive an
effective field theory describing the marginal model close to the $T=0$
critical point and calculate the renormalization group equations at one loop
within a momentum shell approach. We discover a non-trivial scenario, as the
cubic and quartic vertices do not vanish in the infrared limit, while the
coupling constants effectively regulating the exponents $\nu$ and $\eta$ have
upper critical dimension $d_c=2$, so that in three dimensions the critical
exponents acquire their free values, $\nu=1/2$ and $\eta=0$. This theoretical
scenario is verified by a Monte Carlo study of the modulus susceptibility in
three dimensions, where the standard finite-size scaling relations have to be
adapted to the case of $d>d_c$. The numerical data fully confirm our
theoretical results.
