  Velocity field provides a complementary avenue to constrain cosmological
information, either through the peculiar velocity surveys or the kinetic
Sunyaev Zel'dovich effect. One of the commonly used statistics is the mean
radial pairwise velocity. Here, we consider the three-point mean relative
velocity, i.e. the mean relative velocities between pairs in a triplet. Using
halo catalogs from the Quijote suite of N-body simulations, we first showcase
how the analytical prediction for the mean relative velocities between pairs in
a triplet achieve better than 4-5% accuracy using standard perturbation theory
at leading order for triangular configurations with a minimum separation of $r
\geq 50\ h^{-1}$Mpc. Furthermore, we present the three-point relative velocity
as a novel probe of neutrino mass estimation. We explore the full cosmological
information content of the halo mean pairwise velocities, and the mean relative
velocities between halo pairs in a triplet. We undertake this through the
Fisher-matrix formalism using 22,000 simulations from the Quijote suite, and
considering all triangular configurations with a minimum and a maximum
separation of $20\ h^{-1}$Mpc and $120\ h^{-1}$Mpc, respectively. We find that
the mean relative velocities in a triplet allows a 1$\sigma$ neutrino mass
($M_\nu$) constraint of 0.065 eV, that is roughly 13 times better than the mean
pairwise velocity constraint (0.877 eV). This information gain is not limited
only to neutrino mass, but extends to other cosmological parameters:
$\Omega_{\mathrm{m}}$, $\Omega_{\mathrm{b}}$, $h$, $n_{\mathrm{s}}$ and
$\sigma_{8}$ achieving a gain of 8.9, 11.8, 15.5, 20.9 and 10.9 times
respectively. These results illustrate the possibility of exploiting the mean
three-point relative velocities for constraining the cosmological parameters
accurately from future cosmic microwave background experiments and peculiar
velocity surveys.
