Fixed points of the SRG evolution and the on-shell limit of the nuclear force

Abstract

We study the infrared limit of the similarity renormalization group (SRG) using a simple toy model for the nuclear force aiming to investigate the fixed points of the SRG evolution with both the Wilson and the Wegner generators. We show how a fully diagonal interaction at the similarity cutoff $\lambda \rightarrow 0$ may be obtained from the eigenvalues of the hamiltonian and quantify the diagonalness by means of operator norms. While the fixed points for both generators are equivalent when no bound-states are allowed by the interaction, the differences arising from the presence of the Deuteron bound-state can be disentangled very clearly by analyzing the evolved interactions in the infrared limit $\lambda \to 0$ on a finite momentum grid. Another issue we investigate is the location on the diagonal of the hamiltonian in momentum-space where the SRG evolution places the Deuteron bound-state eigenvalue once it reaches the fixed point. This finite momentum grid setup provides an alternative derivation of the celebrated trace identities, as a by product. The different effects due to either the Wilson or the Wegner generators on the binding energies of $A=2,3,4$ systems are investigated and related to the ocurrence of a Tjon-line which emerges as the minimum of an avoided crossing between $E_\alpha= 4 E_t - 3 E_d$ and $E_\alpha= 2 E_t $. All infrared features of the flow equations are illustrated using the toy model for the two-nucleon $S$-waves.