Effective field theory in the harmonic oscillator basis

We develop interactions from chiral effective field theory (EFT) that are tailored to the harmonic oscillator basis. As a consequence, ultraviolet convergence with respect to the model space is implemented by construction and infrared convergence can be achieved by enlarging the model space for the kinetic energy. In oscillator EFT, matrix elements of EFTs formulated for continuous momenta are evaluated at the discrete momenta that stem from the diagonalization of the kinetic energy in the finite oscillator space. By fitting to realistic phase shifts and deuteron data we construct an effective interaction from chiral EFT at next-to-leading order. Many-body coupled-cluster calculations of nuclei up to ${}^{132}\mathrm{Sn}$ converge fast for the ground-state energies and radii in feasible model spaces.

Figure 1

Computed $np$ phase shifts of the $NN$ potential ${\mathrm{N}}^3{\mathrm{LO}}_{\mathrm{EM}}$ [60] projected onto finite oscillator spaces with $N_{\max }=10$ and $N_{\max }=20$ compared to the phase shifts from the full (i.e., not projected) interaction in selected partial waves as a function of the relative momentum. In the top three panels the oscillator frequencies were chosen as $\hslash \omega =40$ and 23 MeV, respectively, to obtain an oscillator cutoff ${\mathrm{\Lambda }}_{\mathrm{UV}}\approx 700$ MeV that well exceeds the chiral cutoff ${\mathrm{\Lambda }}_{\chi }=500$ MeV. The solid circles mark the phase shifts at the momenta corresponding to the eigenenergies of the scattering channel in the truncated oscillator space. In the bottom panel $\hslash \omega$ is increased to 80 MeV (for $N_{\max }=10$) and 46 MeV (for $N_{\max }=20$), respectively. This yields a UV cutoff ${\mathrm{\Lambda }}_{\mathrm{UV}}\approx 1000$ MeV in the oscillator basis and significantly reduces the phase-shift oscillations. Units of the LECs as in Ref. [2].

Figure 2

Phase shifts of ${\mathrm{N}}^3{\mathrm{LO}}_{\mathrm{EM}}$ obtained from a projection onto a $N_{\max }=10, \hslash \omega =40$ MeV oscillator space using (i) exact integration in Eq. (19) (blue dashed line) and (ii) Gauss-Laguerre quadrature Eq. (29) (green crosses), compared to the unprojected ${\mathrm{N}}^3{\mathrm{LO}}_{\mathrm{EM}}$ phase shifts (black solid line). The solid circles mark the phase shifts at the momenta corresponding to the eigenenergies of the scattering channel in the truncated oscillator space.

Figure 3

Phase shifts of NLO interactions obtained from oscillator EFT through fits to ${\text{NLO}}_{\text{sim}}$ phase shifts compared to the ${\text{NLO}}_{\text{sim}}$ data [27]. The momentum-space interaction matrix elements used in the construction of the oscillator EFT interactions employ a fixed chiral cutoff ${\mathrm{\Lambda }}_{\chi }=500$ MeV. We fix $N_{\max }=10$ and consider the two oscillator frequencies $\hslash \omega =20$ and 34 MeV, yielding UV cutoffs ${\mathrm{\Lambda }}_{\mathrm{UV}}=500$ and 650 MeV, respectively. For ${\mathrm{\Lambda }}_{\mathrm{UV}}=650$ MeV, the oscillations in the phase shifts are notably reduced.

Figure 4

Phase shifts from oscillator EFT with $N_{\max }=10$ (solid blue circles and blue lines) compared to ${\mathrm{NLO}}_{\mathrm{sim}}$ phase shifts (black lines) for partial-wave channels, as indicated. Also shown (green crosses) are results from oscillator EFT in a large model space with $N_{\max }=80$.

Figure 5

Phase shift from oscillator EFT at NLO with $N_{\max }=10$ (solid blue circles and blue lines) compared to those of the CD Bonn interaction (solid black lines) for partial-wave channels, as indicated. The green crosses show results from a large model space with $N_{\max }=80$.

Figure 6

IR convergence of the ground-state energy (top) and squared radius (bottom) of ${}^{132}\mathrm{Sn}$ computed from CCSD calculations using the oscillator EFT interaction. Data are shown as blue points, the extrapolations as red dashed lines, and the asymptotic energy and radius as solid black lines. The inset on the top shows that the ground-state energy is approached exponentially in the IR length $L$.