Implications of Efimov physics for the description of three and four nucleons in chiral effective field theory

In chiral effective field theory the leading order (LO) nucleon-nucleon potential includes two contact terms, in the two spin channels $S=0,1$, and the one-pion-exchange potential. When the pion degrees of freedom are integrated out, as in the pionless effective field theory, the LO potential includes two contact terms only. In the three-nucleon system, the pionless theory includes a three-nucleon contact term interaction at LO whereas the chiral effective theory does not. Accordingly arbitrary differences could be observed in the LO description of three- and four-nucleon binding energies. We analyze the two theories at LO and conclude that a three-nucleon contact term is necessary at this order in both theories. In turn this implies that subleading three-nucleon contact terms should be promoted to lower orders. Furthermore, this analysis shows that one single low-energy constant might be sufficient to explain the large values of the singlet and triplet scattering lengths.

Figure 1

RG trajectories of the coupling constant ${\tilde{C}}_{\lambda }$ as a function of the inverse cutoff parameter $r_0$ in units of the singlet ($\lambda =0$, negative values) and triplet ($\lambda =1$, positive values) scattering length $a_{\lambda }$ for different values of the regulator $\beta$.

Figure 2

The triton energy $B\left({}^3\mathrm{H}\right)$ as a function of the regulator $\beta$ for different values of the potential range $r_0$. The (red) squares indicate the results obtained with $C_1=C_0$ corresponding to the $\text{SU}{\left(4\right)}_W$ symmetry.

Figure 3

The $B\left({}^4\mathrm{He}\right)$ binding energy as a function of the regulator $\beta$ for different values of the potential range $r_0$ with and without the three-body interaction. In the first case, the $r_0$ values of the different curves are given in the box. The curves are very close to each other and are almost indistinguishable. In the second case, the curves are well separated and the $r_0$ values are explicitly shown on the curves. The (red) squares indicate the results obtained with $C_1=C_0$ corresponding to the $\text{SU}{\left(4\right)}_W$ symmetry.