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.