Large-momentum distribution of a polarized Fermi gas and $p$-wave contacts

We present a derivation of the adiabatic energy relations as well as the large momentum distribution of a polarized Fermi gas near $p$-wave Feshbach resonances. The leading asymptotic behavior $k^{-2}$ and subleading behavior $k^{-4}$ of the large momentum distribution have recently been predicted by Z. Yu et al. [Phys. Rev. Lett. 115, 135304 (2015)] and by M. Y. He et al. [Phys. Rev. Lett. 116, 045301 (2016)] using two different approaches. Here, we show that the subleading asymptotic behavior ($\sim k^{-4}$) cannot fully be captured by the contact defined from the adiabatic energy relation related to the $p$-wave effective range, and there should be an extra term resulting from the center-of-mass motion of the pairs. The omission of this extra term is perhaps a reasonable approximation at zero temperature. However, it should be taken into account at finite temperature and should have significant importance to understand the recently measured momentum distribution in a resonant $p$-wave Fermi gas of ultracold ${}^{40}\mathrm{K}$ atoms [C. Luciuk et al., Nat. Phys. 12, 599 (2016)].