# Non-relativistic expansion of single-nucleon Dirac equation: Comparison between Foldy-Wouthuysen transformation and similarity renormalization group

## TL;DR

This paper compares two non-relativistic expansion methods, Foldy-Wouthuysen transformation and similarity renormalization group, for the single-nucleon Dirac equation, analyzing their differences up to the fourth order in 1/M.

## Contribution

It provides exact analytic expressions for the Dirac equation expansion up to 1/M^4 order and compares two different non-relativistic methods within covariant density functional theory.

## Key findings

- Derived explicit 1/M^4 order expressions for the Dirac equation.
- Identified the origin of differences between FW and SRG results.
- Enhanced understanding of non-relativistic expansions in nuclear physics.

## Abstract

By following the Foldy-Wouthuysen (FW) transformation of the Dirac equation, we work out the exact analytic expressions up to the $1/M^4$ order for the general cases in the covariant density functional theory. These results are further compared with the corresponding ones derived from another novel non-relativistic expansion method, the similarity renormalization group (SRG). Based on that, the origin of the difference between the results obtained by the FW transformation and SRG method is explored.