Three Dimensional SRG Evolution of the $NN$ Interaction Using Picard Iteration

We solve the similarity renormalization group (SRG) flow equations in a Three Dimensional (3D) helicity representation (without partial wave decomposition) for realistic nucleon-nucleon ($NN$) interactions. During the 3D SRG evolution, the flow equations become extremely stiff for far off diagonal matrix elements (e.g. $|{\bf k}| \gg |{\bf k}^\prime|$). We alleviate this by transforming the differential form of the SRG flow equation into an integral equation that is solved using Picard iteration. The evolved $NN$ interactions are obtained from realistic potentials by solving a single integral equation for total spin $0$ and four coupled integral equations for total spin $1$. We demonstrate the efficiency and accuracy of the Picard integral approach for the Bonn-B and Chiral-N2LO $NN$ potentials. The successful 3D implementation paves the path to consider a 3D evolution of three-nucleon forces.