Accurate determination of the interaction between $\Lambda$ hyperons and nucleons from auxiliary field diffusion Monte Carlo calculations

Background: An accurate assessment of the hyperon-nucleon interaction is of great interest in view of recent observations of very massive neutron stars. The challenge is to build a realistic interaction that can be used over a wide range of masses and in infinite matter starting from the available experimental data on the binding energy of light hypernuclei. To this end, accurate calculations of the hyperon binding energy in a hypernucleus are necessary.

Purpose: We present a quantum Monte Carlo study of $\Lambda$ and $\Lambda \Lambda$ hypernuclei up to $A=91$. We investigate the contribution of two- and three-body $\Lambda$-nucleon forces to the $\Lambda$ binding energy.

Method: Ground state energies are computed solving the Schrödinger equation for nonrelativistic baryons by means of the auxiliary field diffusion Monte Carlo algorithm extended to the hypernuclear sector.

Results: We show that a simple adjustment of the parameters of the $\Lambda NN$ three-body force yields a very good agreement with available experimental data over a wide range of hypernuclear masses. In some cases no experiments have been performed yet, and we give new predictions.

Conclusions: The newly fitted $\Lambda NN$ force properly describes the physics of medium-heavy $\Lambda$ hypernuclei, correctly reproducing the saturation property of the hyperon separation energy.

Figure 1

Meson exchange processes between nucleons and hyperons. (a) and (b) represent the $\Lambda N$ channels. (c)–(e) are the three-body $\Lambda NN$ channels included in the potential by Usmani et al. See [41] and references therein.

Figure 2

$\Lambda$ separation energy for ${}_{\Lambda }^5$He as a function of strengths $W_D$ and $C_P$ of the three-body $\Lambda NN$ interaction. The red grid represents the experimental $B_{\Lambda }=3.12\left(2\right)$ MeV [12]. The dashed yellow curve is the interception between the expected result and the $B_{\Lambda }$ surface in the $W_D\text{−}{C}_P$ parameter space. Statistical error bars on AFDMC results (solid black dots) are of the order of $0.10–0.15$ MeV.

Figure 3

Projection of Fig. 2 on the $W_D\text{−}{C}_P$ plane. Error bars come from a realistic conservative estimate of the uncertainty in the determination of the parameters due to the statistical errors of the Monte Carlo calculations. Blue and green dashed, long-dashed, and dot-dashed lines (lower curves) are the variational results of Ref. [41] for different $\varepsilon$ and $\overline{v}$ (two-body $\Lambda N$ potential). The dashed box corresponds to the parameter domain of Fig. 2. Black dots and the red band (upper curve) are the projected interception describing the possible set of parameters reproducing the experimental $B_{\Lambda }$.

Figure 4

$\Lambda$ separation energy as a function of $A^{-2/3}$. Solid green dots (dashed curve) are the available $B_{\Lambda }$ experimental or semiempirical values. Empty blue dots (upper banded curve) refer to the AFDMC results for the two-body $\Lambda N$ interaction alone. Empty red diamonds (middle banded curve) and empty black triangles (lower banded curve) are the results with the inclusion also of the three-body hyperon-nucleon force, respectively for the set of parameters (I) and (II). In the inset, the same data plotted as a function of $A$.