Effective operators in two-nucleon systems

Effective Hamiltonians and effective electroweak operators are calculated with the Okubo-Lee-Suzuki formalism for two-nucleon systems. Working within a harmonic oscillator basis, first without and then with a confining harmonic oscillator trap, we demonstrate the effects of renormalization on observables calculated for truncated basis spaces. We illustrate the renormalization effects for the root-mean-square point-proton radius, electric quadrupole moment, magnetic dipole moment, Gamow-Teller transition, and neutrinoless double-$\beta$ decay operator using nucleon-nucleon interactions from chiral effective field theory. Renormalization effects tend to be larger in the weaker traps and smaller basis spaces, suggesting applications to heavier nuclei with transitions dominated by weakly bound nucleons would be subject to more significant renormalization effects within achievable basis spaces.

Figure 1

The fractional differences, where Fract. Diff. of an observable $=(\mathrm{model}-\mathrm{exact})/\left|\mathrm{exact}\right|$, for the deuteron ground-state energy at three values of the HO energy $\hslash \mathrm{\Omega }$ as a function of the $P$-space limit $N_{\max }$. The model results from diagonalizing the $P$-space truncated Hamiltonian matrix produce Fract. Diff. curves that decrease towards zero with increasing $N_{\max }$ in accordance with the variational principle. The model results from diagonalizing the OLS–renormalized Hamiltonian matrix reproduce the exact results at each $N_{\max }$ to high precision, yielding flat and overlapping green lines for their Fract. Diff. plots in all cases. Panels (a), (b) and (c) correspond to the Hamiltonians constructed with the LENPIC chiral EFT interactions [10, 11, 12, 13, 14] at NLO, $\mathrm{N}{}^2\mathrm{LO}$, and $\mathrm{N}{}^3\mathrm{LO}$, respectively. We employ the LENPIC interactions with coordinate-space regulator $R=1.0$ fm. Panel (d) corresponds to the Hamiltonian constructed with the Idaho–$\mathrm{N}{}^3\mathrm{LO}$ potential [19] with momentum-space regulator $500\mathrm{MeV}$. The exact ground-state energies are given in Table 1 of the Appendix.

Figure 2

The fractional differences for a selection of deuteron properties at three values of the HO energy basis parameter $\hslash \mathrm{\Omega }$ as a function of the $P$-space limit $N_{\max }$. Following the scheme of Fig. 1, we present the Fract. Diff. for the truncated basis calculations (three colored curves approaching zero at high $N_{\max }$) and for the OLS renormalized calculations (green curves all coincident with zero). All results are obtained with the LENPIC–$\mathrm{N}{}^2\mathrm{LO}$ interaction with regulator $R=1.0$ fm. The ground-state energy in panel (a) is an expanded version of the ground-state energy in panel (b) of Fig. 1. The rms point-proton radius $r_{\mathrm{rms}}$ appears in panel (b), the electric quadrupole moment $Q$ in panel (c) and the magnetic moment $\mu$ in panel (d). The model results using the OLS transformation method reproduce the exact results at each $N_{\max }$ to high precision, yielding flat and overlapping green lines for their Fract. Diff. plots in all cases. The short dashed lines provide envelopes for results with sawtooth patterns. The exact values used for the observables here are given in Table 2 of the Appendix.

Figure 3

Fractional differences between model and exact results as a function of the $P$ space for selected ground-state observables of the two-nucleon system in the ${}^{3}S_1-{}^{3}D_1$ channel for three different HO traps. The $NN$ interaction is the LENPIC–$\mathrm{N}{}^2\mathrm{LO}NN$ interaction with regulator $R=1.0$ fm. HO energies $\hslash \mathrm{\Omega }$ for the bases correspond to the HO energies of the traps. The observables correspond to the eigenenergy (a), the rms point-proton radius (b), the electric quadrupole moment (c), and the magnetic dipole moment (d). The exact values used for the observables here are given in Table 3 of the Appendix.

Figure 4

Fractional differences between model and exact results as a function of the $P$ space for selected ground-state transitions from the lowest state of the ${}^{1}S_{0}nn$ system in three different HO traps. Panels (a) and (b) are for the allowed GT transition to the ground state of the ${}^{3}S_1-{}^{3}D_1$ channel. Panels (c) and (d) are for the $0\nu 2\beta$ decay to the ground state of the ${}^{1}S_{0}pp$ system. The $NN$ interaction for cases (a) and (c) are taken to be the LENPIC–NLO potential, while we adopt the LENPIC–$\mathrm{N}{}^2\mathrm{LO}$ potential for cases (b) and (d). All results shown employ LENPIC $NN$ interactions with coordinate space regulator $R=1.0$ fm. The exact values used for the observables here are given in Table 4 of the Appendix.

Figure 5

Quenching factor defined as exact/model for GT-decay and $0\nu 2\beta$-decay matrix elements as a function of the $P$ space for ground-state transitions from the lowest state of the ${}^{1}S_{0}nn$ system in three different HO traps. Panels (a) and (b) are for the allowed GT-transition to the ground state of the ${}^{3}S_1-{}^{3}D_1$ channel. Panels (c) and (d) are for the $0\nu 2\beta$ decay to the ground state of the ${}^{1}S_0 pp$ system. The $NN$ interaction for cases (a) and (c) are taken to be the LENPIC–NLO potential, while we adopt the LENPIC–$\mathrm{N}{}^2\mathrm{LO}$ potential for cases (b) and (d). All results shown employ LENPIC $NN$ interactions with coordinate space regulator $R=1.0$ fm. A quenching factor greater than unity signals an enhancement of the model results is required to arrive at the exact results. The exact values used for the observables here are given in Table 4 of the Appendix.