Decoupling in the similarity renormalization group for nucleon-nucleon forces

Decoupling via the similarity renormalization group (SRG) of low-energy nuclear physics from high-energy details of the nucleon-nucleon interaction is examined for two-body observables and few-body binding energies. The universal nature of this decoupling is illustrated and errors from suppressing high-momentum modes above the decoupling scale are shown to be perturbatively small.

Figure 1

Absolute value of the matrix element $\langle k\left|V_s\right|k^{\text{'}}\rangle$ for a representative sampling of off-diagonal $(k,k^{\text{'}})$ pairs as a function of $s$, compared with the simple solutions from Eq. (11), which are straight lines (and agree at $s=0$). Two partial waves are shown.

Figure 2

Phase shifts and relative errors in the ${}^1\mathrm{S}$${}_0$ channel for SRG potentials evolved from the N${}^3\mathrm{LO}$ (500-MeV) potential of Ref. 7. The upper left graph shows the phase shifts vs energy for the uncut $\lambda =2\hspace{0.5em}{\mathrm{fm}}^{-1}$ potential and several cut versions with $n=8$. The other panels show the relative error as a function of the momentum cut parameter Λ at various values of energy $E,\lambda$, and $n$, respectively.

Figure 3

The phase shift errors computed in select partial waves, by using a regulator with $n=8$. Other channels exhibit the same power-law dependence of the error for $\Lambda >\lambda$.

Figure 4

The relative error vs cut parameter Λ of the deuteron energy (left) and rms radius (right) with several values of the regulator parameter $n$ indicated in the legends. In each case, the nearby solid line is the estimate of the error calculated by using first-order perturbation theory with Eq. (13).

Figure 5

Calculations of the ${}^4\mathrm{He}$ ground-state energy by using the NCSM. On the left is the energy obtained from the NCSM for potentials evolved to several different λ values as a function of the cut (regulator) momentum Λ with $n=8$. On the right is the relative error of the energy for the $\lambda =2\hspace{0.5em}{\mathrm{fm}}^{-1}$ case as a function of the cut momentum (with $n=8$) for several different harmonic oscillator basis sizes. Also shown is the slope of the error in the decoupling region predicted from perturbation theory (dotted line).

Figure 6

Calculations of the ${}^6\mathrm{Li}$ ground-state energy by using the NCSM. On the left is the energy obtained from the NCSM for potentials evolved to several different λ values as a function of the cut (regulator) momentum Λ with $n=8$. On the right is the relative error of the energy for the same λ values as a function of the cut momentum for the same λ values but with two values of $n$.