Momentum dependent nucleon-nucleon contact interaction from a relativistic Lagrangian

A complete set of parity- and time-reversal conserving relativistic nucleon-nucleon contact operators is identified up to the order $O(p^4)$ of the expansion in soft momenta $p$. A basis is also provided for the corresponding non-relativistic operators contributing in the general reference frame. We show that the non-relativistic expansions of the relativistic operators involve twenty-six independent combinations, two starting at $O(p^0)$, seven at order $O(p^2)$ and seventeen at order $O(p^4)$. This gives supporting evidence to the existence of two free low-energy constants which parametrize an interaction depending on the total nucleon pair momentum ${\bf P}$, and were recently found to be instrumental for the resolution of the long standing $A_y$ problem in low-energy $p-d$ scattering. Furthermore, all remaining ${\bf P}$-dependent interactions at the same order are uniquely determined as relativistic corrections.