Single-particle potential of the $\mathbf{\Lambda }$ hyperon in nuclear matter with chiral effective field theory NLO interactions including effects of $YNN$ three-baryon interactions

Adopting hyperon-nucleon and hyperon-nucleon-nucleon interactions parametrized in chiral effective field theory, single-particle potentials of the $\mathrm{\Lambda }$ and $\mathrm{\Sigma }$ hyperons are evaluated in symmetric nuclear matter and in pure neutron matter within the framework of lowest-order Bruckner theory. The chiral NLO interaction bears strong $\mathrm{\Lambda }N-\mathrm{\Sigma }N$ coupling. Although the $\mathrm{\Lambda }$ potential is repulsive if the coupling is switched off, the $\mathrm{\Lambda }N-\mathrm{\Sigma }N$ correlation brings about the attraction consistent with empirical data. The $\mathrm{\Sigma }$ potential is repulsive, which is also consistent with empirical information. The interesting result is that the $\mathrm{\Lambda }$ potential becomes shallower beyond normal density. This provides the possibility of solving the hyperon puzzle without introducing ad hoc assumptions. The effects of the $\mathrm{\Lambda }NN-\mathrm{\Lambda }NN$ and $\mathrm{\Lambda }NN-\mathrm{\Sigma }NN$ three-baryon forces are considered. These three-baryon forces are first reduced to normal-ordered effective two-baryon interactions in nuclear matter and then incorporated in the $G$-matrix equation. The repulsion from the $\mathrm{\Lambda }NN-\mathrm{\Lambda }NN$ interaction is of the order of 5 MeV at normal density and becomes larger with increasing density. The effects of the $\mathrm{\Lambda }NN-\mathrm{\Sigma }NN$ coupling compensate the repulsion at normal density. The net effect of the three-baryon interactions on the $\mathrm{\Lambda }$ single-particle potential is repulsive at higher densities.

Figure 1

Nucleon s.p. potentials as a function of the nucleon momentum $k$ obtained from the lowest-order Brueckner theory calculations in SNM with the ChEFT NN potential [25] of the cutoff $\mathrm{\Lambda }=550$ MeV, including the effects from the 3BF. These potentials are used for the propagator in the YN $G$-matrix equation.

Figure 2

Same as Fig. 1, but for neutron and proton s.p. potentials in PNM.

Figure 3

Real part of the $\mathrm{\Lambda }$ and $\mathrm{\Sigma }$ s.p. potentials in SNM with chiral NLO YN interactions [19]. Solid and dot-dashed curves represent the results of the calculation with the $\mathrm{\Lambda }N-\mathrm{\Sigma }N$ coupling switched on and off, respectively. The potentials multiplied by the regularization factor $e^{-{(k^2/\left(2{\mathrm{\Lambda }}^2\right))}^3}$ with $\mathrm{\Lambda }=550$ MeV are shown by dotted curves. The results of the calculation in which the prescription of the regularization is not applied are shown by the dashed curves for $k_F=1.35$ and $1.60{\mathrm{fm}}^{-1}$.

Figure 4

Imaginary part of the $\mathrm{\Lambda }$ and $\mathrm{\Sigma }$ s.p. potentials in SNM with chiral NLO YN interactions [19]. Short-dashed and dot-dashed curves represent the results including, in addition, the effects from $\mathrm{\Lambda }NN-\mathrm{\Lambda }\mathrm{NN}3\text{BFs}$ and $\mathrm{\Lambda }NN-\mathrm{\Sigma }NN3\text{BFs}$, respectively, which are discussed in Sec. 3.

Figure 5

Density dependence of the partial-wave contributions to the $\mathrm{\Lambda }$ s.p. potential in SNM with NLO YN interactions [19] only.

Figure 6

$\mathrm{\Lambda }$ and $\mathrm{\Sigma }$ s.p. potentials in PNM with chiral NLO YN interactions [19]. Solid curves represent the results including the cutoff form factor for the intermediate potentials in the denominator of the $G$-matrix equation. The latter potentials are shown by dotted curves. Calculated results with the $\mathrm{\Lambda }N-\mathrm{\Sigma }N$ coupling switched off are shown by the dashed curves.

Figure 7

$\mathrm{\Lambda }NN-\mathrm{\Lambda }NN$ diagrams. The dotted line represents pion exchange. (a) $\mathrm{\Lambda }NN-\mathrm{\Lambda }NN$ three-body interaction $V_{\mathrm{TPE}}^{\mathrm{\Lambda }NN}$, (b) direct term in one-nucleon folding of $V_{\mathrm{TPE}}^{\mathrm{\Lambda }NN}$, and (c) exchange term in one-nucleon folding of $V_{\mathrm{TPE}}^{\mathrm{\Lambda }NN}$. ${\mathbf{k}}_h$ indicates an occupied state.

Figure 8

$\mathrm{\Lambda }NN-\mathrm{\Sigma }NN$ diagrams. The dotted line represents pion exchange. (a) 3BF $V_{\mathrm{TPE},a}^{\mathrm{\Lambda }-\mathrm{\Sigma }}$ with $\pi \pi \mathrm{\Lambda }\mathrm{\Sigma }$ vertex, (b) 3BF $V_{\mathrm{TPE},b}^{\mathrm{\Lambda }-\mathrm{\Sigma }}$ with $\pi \mathrm{\Lambda }\mathrm{\Sigma }$ vertex, (c) direct term in one-nucleon folding of $V_{\mathrm{TPE},a}^{\mathrm{\Lambda }-\mathrm{\Sigma }}$, (d) exchange term in one-nucleon folding of $V_{\mathrm{TPE},a}^{\mathrm{\Lambda }-\mathrm{\Sigma }}$, (e) direct term in one-nucleon folding of $V_{\mathrm{TPE},b}^{\mathrm{\Lambda }-\mathrm{\Sigma }}$, (f) exchange term in one-nucleon folding of $V_{\mathrm{TPE},b}^{\mathrm{\Lambda }-\mathrm{\Sigma }}$, and (g) exchange term in one-nucleon folding of $V_{\mathrm{TPE},a}^{\mathrm{\Lambda }-\mathrm{\Sigma }}$. ${\mathbf{k}}_h$ indicates an occupied state.

Figure 9

$k_F$ dependence of $\mathrm{\Delta }{U}_{\mathrm{\Lambda }}$, Eq. (12), in SNM and in PNM. Coupling constants estimated by Petschauer et al. [31] are employed.

Figure 10

Real part of the $\mathrm{\Lambda }$ s.p. potentials in SNM with chiral NLO YN interactions [19]. Solid curves are the same as in Fig. 3, that is, without the effects of 3BFs. Dotted curves represent the results including the effects of $\mathrm{\Lambda }NN-\mathrm{\Lambda }NN3\text{BFs}$. Dashed curves represent the results, including the effects of both $\mathrm{\Lambda }NN-\mathrm{\Lambda }NN$ and $\mathrm{\Lambda }NN-\mathrm{\Sigma }NN3\text{BFs}$.

Figure 11

Same as Fig. 10, but for PNM.

Figure 12

$k_F$ dependence of $U_{\mathrm{\Lambda }}\left(0\right)$ and ${\mu }_n-\mathrm{\Delta }m$ in PNM. Solid and dashed curves for the $U_{\mathrm{\Lambda }}\left(0\right)$ represent the results with and without the 3BF contribution, respectively. The dashed curve is based on the energy density parameterized by Akmal, Pandharipande, and Ravenhall [34], and the short-dashed curve is evaluated by the parametrization of Schulze and Rijken [32]. The dot-dashed curve and the dot-dot-dashed curve for $U_{\mathrm{\Lambda }}\left(0\right)$ are calculated using the energy densities of Schulze and Rijken [32].