Maximally local two-nucleon interactions at ${\mathrm{N}}^3\mathrm{LO}$ in $\mathrm{\Delta }$-less chiral effective field theory

We present new maximally local two-nucleon interactions derived in $\mathrm{\Delta }$-less chiral effective field theory up to next-to-next-to-next-to-leading order (${\mathrm{N}}^3\mathrm{LO}$), which include all contact and pion-exchange contributions to the nuclear Hamiltonian up to this order. Our interactions are fit to nucleon-nucleon phase shifts using a Bayesian statistical approach, and explore a wide cutoff range from $0.6–0.9$ fm ($\approx 660–440\mathrm{MeV}$). These interactions can be straightforwardly employed in quantum Monte Carlo methods, such as the auxiliary field diffusion Monte Carlo method. Together with local three-nucleon forces, calculations with these new interactions will provide improved benchmarks for the structure of atomic nuclei and serve as crucial input to analyses of astrophysical phenomena of neutron stars, such as binary neutron-star mergers.

Figure 1

(a) The local part of our ${\mathrm{N}}^3{\mathrm{LO}}_{\mathrm{LA}}$ interactions in the ${}^{1}S_0$ channel (i.e., 17 local contacts and all the pion-exchange pieces) for the least-squares fit as a function of particle separation $r$. The LECs are obtained via least-squares fits as described in Sec. 3. We show the interactions for different cutoff values, i.e., ${\mathrm{N}}^3{\mathrm{LO}}_{\mathrm{LA}}$-09 to ${\mathrm{N}}^3{\mathrm{LO}}_{\mathrm{LA}}$-06. (b) Cutoff dependence of the LO spectral LEC in the ${}^{1}S_0$ partial wave at $\mathrm{N}{}^3\mathrm{LO}$. The solid blue line shows the least-squares results whereas the band represents the 95% confidence level (CL) calculated using a Bayesian analysis; see Sec. 3.

Figure 2

Phase shifts in the ${}^{1}S_0$ and ${}^{3}S_1\text{−}{}^{3}D_1$ partial waves. In panels (a)–(d), the chiral EFT order is fixed to be $\mathrm{N}{}^3\mathrm{LO}$ and we show results for different cutoffs as indicated in the legend. In panels (e)–(h) [panels (i)–(l)], we show results for different chiral EFT orders at $R_0=0.6$ fm [$R_0=0.9$ fm]. The bands correspond to the 95% CL. The solid lines represent the solutions that maximize the Bayesian posterior distributions while the dashed lines represent the least-squares fit results. For comparison, we show the Nijmegen partial-wave analysis (NPWA) [44] (black squares) and the phase shifts of the AV18 potential [45] (blue stars), which has been used in past QMC calculations.

Figure 3

Same as Fig. 2, but for the ${}^{1}P_1, {}^{3}P_0, {}^{3}P_1$, and ${}^{3}P_2\text{−}{}^{3}F_2$ partial waves.

Figure 4

Same as Fig. 2, but for the ${}^{1}D_2, {}^{3}D_2$, and ${}^{3}D_3$ partial waves.

Figure 5

Objective function defined in Eq. (41) divided by the number of fit parameters $N$ (21 at $\mathrm{N}{}^3\mathrm{LO}$) as a function of cutoff for our ${\mathrm{N}}^3{\mathrm{LO}}_{\mathrm{LA}}$ interactions. The calculated values are written explicitly in the figure.

Figure 6

Deuteron binding energy as a function of cutoff for different orders, as indicated in the legend. The bands correspond to the 95% CL obtained from the Bayesian analysis whereas the dashed lines represent the least-squares results. The dashed black line shows the experimental binding energy.

Figure 7

Deuteron wave function in coordinate space [panels (a) and (b)] and momentum space [panels (c) and (d)]. The results for different $R_0$ correspond to the least-squares phase shifts analyses. The AV18 [45] analysis is shown as a reference.

Figure 8

Differences between the phase shifts obtained using the full interaction and the predictions with all nonlocal parts set to zero. We show results for the ${\mathrm{N}}^3{\mathrm{LO}}_{\mathrm{LA}}$-06 interaction as well as the interactions defined in Eqs. (C1) to (C4), i.e., Set 1 to Set 4.

Figure 9

Impact of changing the SFR cutoff $\tilde{\mathrm{\Lambda }}$. The solid blue line corresponds to our ${\mathrm{N}}^3{\mathrm{LO}}_{\mathrm{LA}}$-06 interaction with $\tilde{\mathrm{\Lambda }}=1$ GeV and the dashed orange curve is obtained with the same interaction but with $\tilde{\mathrm{\Lambda }}=2$ GeV.

Figure 10

Comparison of the BUQEYE estimate of the EFT truncation uncertainties with the EKM estimate in different $S$-wave channels. This serves as input to our Bayesian analysis—see Eq. (40)–and similar error estimates will result in similar Bayesian fit posteriors. The blue distribution is obtained using Eq. (E1) and the solid black lines are the corresponding 68% CL. The EKM uncertainty, i.e., Eq. (36) with $k=4$, is shown as dashed red lines. Different panels correspond to different partial waves and laboratory energies as written in the upper left of each panel.

Figure 11

Same as Fig. 10 but for $P$ waves.

Figure 12

Same as Fig. 10 but for $D$ waves.