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

## TL;DR

This paper constructs a covariant chiral two-nucleon contact Lagrangian up to order b5(q^4), detailing the number of terms at each order and showing its reduction to the non-relativistic form via expansion.

## Contribution

It provides a systematic covariant formulation of the chiral nucleon-nucleon contact Lagrangian up to b5(q^4), including the enumeration of terms and connection to non-relativistic models.

## Key findings

- Number of terms at each order: 4, 13, 23.
- Reduction to non-relativistic Lagrangian with fewer terms.
- Framework constrained by multiple symmetries.

## Abstract

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