Infrared length scale and extrapolations for the no-core shell model

We precisely determine the infrared (IR) length scale of the no-core shell model (NCSM). In the NCSM, the $A$-body Hilbert space is truncated by the total energy, and the IR length can be determined by equating the intrinsic kinetic energy of $A$ nucleons in the NCSM space to that of $A$ nucleons in a $3(A-1)$-dimensional hyper-radial well with a Dirichlet boundary condition for the hyper radius. We demonstrate that this procedure indeed yields a very precise IR length by performing large-scale NCSM calculations for ${}^6\mathrm{Li}$. We apply our result and perform accurate IR extrapolations for bound states of ${}^4\mathrm{He},{}^6\mathrm{He},{}^6\mathrm{Li}$, and ${}^7\mathrm{Li}$. We also attempt to extrapolate NCSM results for ${}^{10}\mathrm{B}$ and ${}^{16}\mathrm{O}$ with bare interactions from chiral effective field theory over tens of MeV.

Figure 1

Comparison of kinetic energy spectra for three fermions in a hyper-radial well (left), three fermions in a one-dimensional NCSM basis (middle), and three fermions in a one-dimensional infinite square well (right).

Figure 2

Discrete intrinsic kinetic energy spectra for $A=4,6$ particles in the NCSM (right pane) and for the corresponding $D=3(A-1)$ dimensional hyper-radial infinite well. In each case, we plot in units of the smallest eigenvalue.

Figure 3

Ground-state energy of ${}^6\mathrm{Li}$ plotted as a function of the IR scale determined by either $L_2$ [8] (left panel) or $L_{\mathrm{eff}}$ from this work (right panel). The color of each circular marker indicates the UV cutoff of that calculation, with darker colors corresponding to larger cutoffs. Insets show $\mathrm{\Delta }E=E(L,\mathrm{\Lambda })-E_{\infty }$ on a semilogarithmic scale.

Figure 4

Extrapolations of the binding energy per particle for several $p$-shell nuclei computed with the NCSM. The color of each circular marker indicates the UV cutoff of that calculation with darker colors corresponding to larger cutoffs. Markers with a black border are included in the extrapolation. The solid red (gray) curve shows the exponential fit (16), and the horizontal red (gray) line marks the value of $E_{\infty }$ with uncertainty estimates indicated as blue (gray) bands. The dashed black line marks the variational minimum $E_{\mathrm{varmin}}$ for the largest model space included in the fit.