Three-Nucleon Low-Energy Constants from the Consistency of Interactions and Currents in Chiral Effective Field Theory

The chiral low-energy constants $c_D$ and $c_E$ are constrained by means of accurate ab initio calculations of the $A=3$ binding energies and, for the first time, of the triton $\beta$ decay. We demonstrate that these low-energy observables allow a robust determination of the two undetermined constants, a result of the surprising fact that the determination of $c_D$ depends weakly on the short-range correlations in the wave functions. These two- plus three-nucleon interactions, originating in chiral effective field theory and constrained by properties of the $A=2$ system and the present determination of $c_D$ and $c_E$, are successful in predicting properties of the $A=3$ and 4 systems.

Figure 1

Contact and one-pion exchange plus contact interaction (a), and contact MEC (b) terms of $\chi \mathrm{PT}$.

Figure 2

$c_D\mathrm{\text{−}}{c}_E$ trajectories fitted to reproduce ${}^3\mathrm{H}$ and ${}^3\mathrm{He}$ experimental BE. The dotted lines show the region for which $|1-⟨E_1^A{⟩}_{\mathrm{theor}}/⟨E_1^A{⟩}_{\mathrm{emp}}|$ is within the $\pm 0.54\%$ error bars.

Figure 3

The ratio $⟨E_1^A{⟩}_{\mathrm{theor}}/⟨E_1^A{⟩}_{\mathrm{emp}}$ using the ${\mathrm{N}}^3\mathrm{LO}$ $NN$ potential [22] with and without the local ${\mathrm{N}}^3\mathrm{LO}$ $NNN$ interaction [23], and the axial-vector current with and without MEC ($c_D$, $c_E$ are varied along the averaged trajectory of Fig. 2). The shaded area is twice the experimental uncertainty. Also shown: results (without $NNN$ force) for the phenomenological AV18 potential (with $\Lambda =500\mathrm{MeV}$ imposed in the current), and for the ${\mathrm{N}}^3\mathrm{LO}$ $NN$ potential of Epelbaum et al. [30] (with $\Lambda =450,600\mathrm{MeV}$, and a 700 MeV spectral-function cutoff in the two-pion exchange).