Covariant nucleon-nucleon contact Lagrangian up to order $\mathcal{O}\left(q^4\right)$

We adopt a covariant version of the naive dimensional analysis and construct the two-nucleon contact Lagrangian constrained by Lorentz, parity, charge conjugation, and Hermitian conjugation symmetries. We show that at $\mathcal{O}\left(q^0\right)$, $\mathcal{O}\left(q^2\right)$, and $\mathcal{O}\left(q^4\right)$, where $q$ denotes a generic small momentum, there are 4, 13, and 23 terms, respectively. We find that by performing $1/m_N$ expansions, the covariant Lagrangian reduces to the conventional nonrelativistic one, which includes 2, 7, and 15 terms at each corresponding order.