Universal relations for hybridized $s$- and $p$-wave interactions from spin-orbital coupling

In this work, we study the universal relations for one-dimensional spin-orbital-coupled fermions near both $s$- and $p$-wave resonances using effective field theory. Since the spin-orbital coupling mixes different partial waves, a contact matrix is introduced to capture the non-trivial correlation between dimers. We find the signature of the spin-orbital coupling appears at the leading order for the off-diagonal components of the momentum distribution matrix, which is proportional to $1/q^{3}$ ($q$ is the relative momentum). We further derive the large frequency behavior of the Raman spectroscopy, which serves as an independent measurable quantity for contacts. Finally, we give an explicit example of contacts by considering a two-body problem.