Magnetic moments of $A=3$ nuclei obtained from chiral effective field theory operators

Chiral effective field theory ($\chi \mathrm{EFT}$) provides a framework for obtaining internucleon interactions in a systematically improvable fashion from first principles, while also providing for the derivation of consistent electroweak current operators. In this work, we apply consistently derived interactions and currents towards calculating the magnetic dipole moments of the $A=3$ systems ${}^3\mathrm{H}$ and ${}^3\mathrm{He}$. We focus here on LENPIC interactions obtained using semilocal coordinate-space (SCS) regularization. Starting from the momentum-space representation of the LENPIC $\chi \mathrm{EFT}$ vector current, we derive the SCS-regularized magnetic dipole operator up through next-to-next-to-leading order (${\mathrm{N}}^2\mathrm{LO}$). We then carry out no-core shell-model calculations for ${}^3\mathrm{H}$ and ${}^3\mathrm{He}$ systems using the SCS LENPIC interaction at ${\mathrm{N}}^2\mathrm{LO}$ in $\chi \mathrm{EFT}$ and evaluate the magnetic dipole moments obtained using the consistently derived one-nucleon and two-nucleon electromagnetic currents. As anticipated by prior results with $\chi \mathrm{EFT}$ currents, the current corrections through ${\mathrm{N}}^2\mathrm{LO}$ provide improved, but not yet complete, agreement with experiment for the ${}^3\mathrm{H}$ and ${}^3\mathrm{He}$ magnetic dipole moments.

Figure 1

Calculated ground-state energies (left), magnetic dipole moments (center), and $2N$ MEC corrections (right), for ${}^3\mathrm{H}$ (top) and ${}^3\mathrm{He}$ (bottom), illustrating their convergence with respect to basis parameters $N_{\text{max}}$ (successive curves) and $\hslash \omega$. For the magnetic dipole moment, ${\mu }^{\text{IA}}$ represents the contribution from the $1N$ impulse approximation (IA) operator ${\mathit{\mu }}^{1N}$ in Eq. (16), ${\mu }^{\text{MEC}}$ represents the contribution from the $2N$ MEC operator ${\mathit{\mu }}^{2N}$ in Eq. (21), and ${\mu }^{\text{IA}+\text{MEC}}={\mu }^{\text{IA}}+{\mu }^{\text{MEC}}$. Wave functions are obtained using the $2N$ LENPIC SCS potential with $R=1.0\mathrm{fm}$ and no SRG transformation of the potential.

Figure 2

Magnetic moments of the $A=3$ nuclides (a) ${}^3\mathrm{H}$ and (b) ${}^3\mathrm{He}$, calculated with the LENPIC potentials and currents, along with results of prior calculations [23, 24, 25] (see Table 1 caption for details of these calculations). Both IA and IA $+$ MEC results are shown (connected by arrow), where the $\chi \mathrm{EFT}$ current contains contributions from terms appearing through NLO. For the prior calculations, results including contributions through ${\mathrm{N}}^3\mathrm{LO}$ are also shown (connected by dotted line); however, these results involve new LECs which are chosen to replicate (at least approximately) the experimental $A=3$ moments, and are thus not predictions per se (see text). The SRG-unevolved LENPIC $2N$ results are shown for $N_{\text{max}}=14$, 16, and 18 (increasing symbol size), as an indicator of convergence. The SRG-unevolved LENPIC $2N$ results are shown for both $R=0.9\mathrm{fm}$ and $R=1.0\mathrm{fm}$, while all other LENPIC results are shown for $R=1.0\mathrm{fm}$. Experimental values [53] are provided for reference (horizontal bars). Note that the magnetic dipole moment axis for ${}^3\mathrm{He}$ (right) is inverted, to facilitate comparison of the pattern of MEC contributions (of approximately equal magnitude but opposite sign, as noted in the text) across mirror nuclides.