Description of light nuclei in pionless effective field theory using the stochastic variational method

We construct a coordinate-space potential based on pionless effective field theory (EFT) with a Gaussian regulator. Charge-symmetry breaking is included through the Coulomb potential and through two- and three-body contact interactions. Starting with the effective field theory potential, we apply the stochastic variational method to determine the ground states of nuclei with mass number $A\le 4$. At next-to-next-to-leading order, two out of three independent three-body parameters can be fitted to the three-body binding energies. To fix the remaining one, we look for a simultaneous description of the binding energy of ${}^4\mathrm{He}$ and the charge radii of ${}^3\mathrm{He}$ and ${}^4\mathrm{He}$. We show that at the order considered we can find an acceptable solution, within the uncertainty of the expansion. We find that the EFT expansion shows good agreement with empirical data within the estimated uncertainty, even for a system as dense as ${}^4\mathrm{He}$.

Figure 1

Running of the parameters of the $\mathit{NN}$ potential as a function of the range $\sigma$. (a) $A_1$ (black solid curve), $A_3{\sigma }^2$ (red dashed curve), $A_5/{\sigma }^2$ (blue dotted curve). (b) $A_2$ (green solid curve), $A_4{\sigma }^2$ (orange dashed curve), $A_6/{\sigma }^2$ (cyan dotted curve). (c) $A_1^{\mathrm{CSB}}$ (black solid curve), $A_3^{\mathrm{CSB}}{\sigma }^2$ (red dashed curve), $A_5^{\mathrm{CSB}}/{\sigma }^2$ (blue dotted curve). (d) $A_7$ (solid magenta curve). All parameters have been multiplied with appropriate powers of $\sigma$ so that they have units of ${\mathrm{fm}}^{-1}$.

Figure 2

Low-energy $pn$ phase shifts as described by our $\mathit{NN}$ potentials. The panels show the phase shifts as follows: (a) ${}^1{S}_0$, (b) ${}^3{S}_1$, (c) ${\epsilon }_1$, (d) ${}^3{D}_1$, (e) ${}^1{P}_1$, (f) ${}^3{P}_{0,1,2}$. Curves corresponding to the different values of $\sigma$ are red solid: $\sigma =1.2\mathrm{fm}$, blue dashed: $\sigma =1.0\mathrm{fm}$, green dash-dotted: $\sigma =0.8\mathrm{fm}$, orange double-dot-dashed: $\sigma =0.6\mathrm{fm}$. The black dotted line shows the results of the Nijmegen PWA93 analysis [43, 44]. Note that our calculation gives identical results for the ${}^3{P}_{0,1,2}$ phase shifts, see panel (f); the PWA93 curves for these phase shifts are labeled in the figure.

Figure 3

Low-energy $pp$ phase shifts as described by our $\mathit{NN}$ potentials. The panels show the phase shifts as follows: (a) ${}^1{S}_0$, (b) ${}^3{P}_{0,1,2}$. The curves are denoted as in Fig. 2.

Figure 4

Graphical representation of the relationship between the parameters $B_1$ and $B_2$ that reproduces the triton binding energy $E\left({}^{3}H\right)=-8.48$ MeV. Panels (a)–(d) correspond to $\sigma =0.6,0.8,1.0$, and 1.2 fm, in order.

Figure 5

(a) Correlation between the contributions of the two three-nucleon potentials, $B_1$ and $B_2$, to the triton ground-state energy for the values shown in Fig. 4. The behavior for different values of $\sigma$ is represented by color and line type as in Fig. 2. (b) Correlation between the contributions of $B_1$ to the triton and ${}^4\mathrm{He}$ ground-state energies. As explained in the main text, no calculation for $\sigma =0.6$ fm has been performed.

Figure 6

Correlation between the results for the ground-state energies of ${}^3\mathrm{He}$ and ${}^4\mathrm{He}$, calculated without the CSB three-nucleon force. The lines for different values of $\sigma$ are denoted as in Fig. 2. The cross marks the experimental values.

Figure 7

Correlation of the strength $B_{\mathrm{CSB}}$ of the CSB three-nucleon potential with the value of $B_1$, for potentials fitted to the binding energies of ${}^3\mathrm{H}$ and ${}^3\mathrm{He}$. Panels (a)–(c) correspond to $\sigma =0.8$, 1.0, and 1.2 fm, in order.

Figure 8

(a) Correlation between ${}^4\mathrm{He}$ and ${}^3\mathrm{H}$ charge radii. (b) Correlation between ${}^4\mathrm{He}$ and ${}^3\mathrm{He}$ charge radii. The lines for different values of $\sigma$ are denoted as in Fig. 2. The experimental values are taken from Ref. [45].

Figure 9

Correlation between ${}^4\mathrm{He}$ charge radius and ground state energy. The notation is as in Fig. 4.