Limits on possible new spin-spin interactions between neutrons from measurements of the Longitudinal Spin Relaxation Rate of Polarized $^{3}$He Gas

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.