  New particles with masses in the sub-eV range have been predicted by various
theories beyond the Standard Model. Some can induce new spin-spin interactions
between fermions. Existing constraints on such interactions between nucleons
with mesoscopic ranges (millimeters to nanometers) are quite poor. Polarized
$^{3}$He gas is an especially clean system to use to constrain these possible
new spin-spin interactions because the spin-independent atomic potential
between helium atoms is well-characterized experimentally. The small effects
from binary atomic collisions in a polarized gas from magnetic dipole-dipole
and other possible weak spin-spin interactions which lead to spin relaxation
can be calculated perturbatively. We compare existing measurements of the
longitudinal spin relaxation rate $\Gamma_{1}$ of polarized $^3$He gas with
theoretical calculations and set a $1\sigma$ upper bound on the pseudoscalar
coupling strength $g_p$ for possible new neutron-neutron dipole-dipole
interactions of $g_p^{(n)}g_p^{(n)}/4\pi \le 1.7\times 10^{-3}$ for distances
larger than 100 nm. We also set new direct limits on possible gravitational
torsion interactions between neutrons.
