Local chiral interactions, the tritium Gamow-Teller matrix element, and the three-nucleon contact term

The Gamow-Teller (GT) matrix element contributing to tritium $\beta$ decay is calculated with trinucleon wave functions obtained from hyperspherical-harmonics solutions of the Schrödinger equation with the chiral two- and three-nucleon interactions including $\mathrm{\Delta }$ intermediate states that have recently been constructed in configuration space. Predictions up to next-to-next-to-next-to-leading order (N3LO) in the chiral expansion of the axial current (with $\mathrm{\Delta }$'s) overestimate the empirical value by 1–4%. By exploiting the relation between the low-energy constant (LEC) in the contact three-nucleon interaction and two-body axial current, we provide new determinations of the LECs $c_D$ and $c_E$ that characterize this interaction by fitting the trinucleon binding energy and tritium GT matrix element. Some of the implications that the resulting models of three-nucleon interactions have on the spectra of light nuclei and the equation of state of neutron matter are briefly discussed. We also provide a partial analysis, which ignores $\mathrm{\Delta }$'s, of the contributions due to loop corrections in the axial current at next-to-next-to-next-to-next-to-leading order (N4LO). Finally, explicit expressions for the axial current up to N4LO have been derived in configuration space, which other researchers in the field may find useful.

Figure 1

OPE and TPE contributions at LO (a), NLO [(b)–(g)], and N2LO [(h)–(p)]. Nucleons, $\mathrm{\Delta }$ isobars, and pions are denoted, respectively, by the solid, thick solid, and dashed lines; note that both direct and crossed box contributions are retained in diagrams (d), (f)–(g), (k), and (n)–(p). The open circles denote $\pi N$ and $\pi N\mathrm{\Delta }$ couplings from the subleading chiral Lagrangians ${\mathcal{L}}_{\pi N}^{\left(2\right)}$ [36] and ${\mathcal{L}}_{\pi N\mathrm{\Delta }}^{\left(2\right)}$ [37].

Figure 2

Diagrams illustrating the contributions to the axial current up to N3LO ($Q^0$). Nucleons, $\mathrm{\Delta }$ isobars, pions, and external fields are denoted by solid, thick solid, dashed, and wavy lines, respectively. The squares in panels (c) and (d) represent relativistic corrections, while the dots in panels (i) and (j) denote vertices implied by the ${\mathcal{L}}_{\pi N}^{\left(2\right)}$ chiral Lagrangian. Only a single time ordering is shown. Note that the contact contributions in panels (g) and (h) vanish.

Figure 3

Upper panel: The calculated ratio ${\mathrm{GT}}_{\mathrm{th}}/{\mathrm{GT}}_{\exp }$ as function of $c_D$ (solid line; each point on this line reproduces the trinucleon binding energies). Lower panel: The $c_D\text{−}{c}_E$ trajectories obtained by fitting the experimental trinucleon binding energies (solid line) and $nd$ doublet scattering length (dashed line) (the intercept of these two lines gives the $c_D$ and $c_E$ values that reproduce these two observables simultaneously). The NV2+3-Ia chiral interactions are used here for illustration. The values of 8.475 and 7.725 MeV and $0.645\pm 0.010$ fm [44] are used for the ${}^3\mathrm{H}$ and ${}^3\mathrm{He}$ binding energies and $nd$ scattering length, respectively. Note that these energies have been corrected for the small contributions ($+7$ keV in ${}^3\mathrm{H}$ and $-7$ keV in ${}^3\mathrm{He}$) due to the $n\text{−}p$ mass difference [45]. The band (left panel) results from experimental uncertainty ${\mathrm{GT}}_{\mathrm{EXP}}$, which has conservatively been doubled.

Figure 4

Integration contour.