Large-cutoff behavior of local chiral effective field theory interactions

Interactions from chiral effective field theory have been successfully employed in a broad range of ab initio calculations of nuclei and nuclear matter, but it has been observed that most results of few- and many-body calculations experience a substantial residual regulator and cutoff dependence. In this work, we investigate the behavior of local chiral potentials at leading order under variation of the cutoff scale for different local regulators. When varying the cutoff, we require that the resulting interaction produces no spurious bound states in the deuteron channel. We find that, for a particular choice of leading-order operators, nucleon-nucleon phase shifts and the deuteron ground-state energy converge to cutoff-independent plateaus for all regulator functions we investigate. This observation may enable improved calculations with chiral Hamiltonians that also include three-nucleon interactions.

Figure 1

Spin-isospin LECs $C_{10}$ (left panel, logarithmic scale) and $C_{01}$ (middle panel) as functions of the inverse cutoff $R_0^{-1}$ for local chiral interactions at LO with different local regulators characterized by $n_1$, $n_2,n$. Right panel: Spin-isospin LEC $C_{10}$ as function of the inverse cutoff $R_0^{-1}$ for local chiral interactions at LO allowing one, two, and three bound states in the coupled ${}^{3}S_1\text{−}{}^{3}D_1$ channel.

Figure 2

Chiral LO potentials in the ${}^{3}S_1$ partial wave as well as the corresponding wave functions for one or two bound states for three cutoff values and $n_1=n_2=n=2$. In addition, we compare the chiral LO potentials with a potential where the short-range interaction is replaced by an $\omega$-meson-exchange potential.

Figure 3

Phase shifts in the ${}^{3}P_0$ partial wave at laboratory energies $E_{\text{lab}}=$ 10, 50, 100, and 150 MeV vs the inverse cutoff $R_0^{-1}$ for different LO operator structures and $n_1=n_2=n=2$. The first panel shows the phase shifts for the operator pair $\mathbb{1},{\sigma }_{12}$, where the spin-isospin LEC $C_{11}$ is equal to $C_{10}$. The second and third panels show the operator combinations ${\sigma }_{12},{\tau }_{12}$ and ${\tau }_{12}$, ${\sigma }_{12}{\tau }_{12}$, where $C_{11}\sim -C_{10}$ (see Table 1). In the lower panels, we show the interaction ${\mathrm{LO}}_{\mathrm{n}P}$, where $C_{11}=0$ and which is closest to an interaction that respects Fierz rearrangement freedom, as well as the operator combinations $\mathbb{1},{\tau }_{12}$ and ${\sigma }_{12}$, ${\sigma }_{12}{\tau }_{12}$, where $C_{11}\sim C_{01}$ with $C_{01}\to 0$ (see Fig. 1). In the lower panels, we find a behavior similar to that in Ref. [8].

Figure 4

Phase shifts in the ${}^{1}S_0,{}^{3}S_1,{\epsilon }_1,{}^{3}D_2,{}^{1}P_1$, ${}^{3}P_1$, and ${}^{3}P_2$ partial waves for laboratory energies $E_{\text{lab}}=$ 10, 50, 100, and 150 MeV as well as the deuteron ground-state energy $E_d$ (upper right panel) as functions of the inverse cutoff $R_0^{-1}$ for the operators ($\mathbb{1},{\sigma }_{12}$) and $n_1=n_2=n=2$.

Figure 5

Phase shifts in the ${}^{3}S_1,{}^{3}P_0$, and ${}^{3}D_1$ partial waves for different laboratory energies and for different regulator functions, as functions of the inverse cutoff $R_0^{-1}$ for the operators $(\mathbb{1},{\sigma }_{12})$.

Figure 6

Phase shifts in the ${}^{1}S_0,{}^{3}S_1,{\epsilon }_1,{}^{3}D_1$, ${}^{1}P_1,{}^{3}P_{0,1,2},{}^{1}F_3,{}^{3}F_2,{\epsilon }_2$, and ${}^{3}D_2$ channels as function of the laboratory energy for the inverse cutoffs $R_0^{-1}=1{\mathrm{fm}}^{-1}$ and $R_0^{-1}=10{\mathrm{fm}}^{-1}$ for $n_1=n_2=n=2$ in comparison to the PWA phase shifts.