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The hyperon puzzle is one of the biggest problems in neutron star physics. Using two-body interactions based on hypernuclear data, $\Lambda$ appears in neutron star matter at $(2-4)\rho_0$. This would soften the equation of state and make it difficult to explain the existence of $2 M_\odot$ neutron stars. Therefore, $\Lambda$ should feel a strong repulsive potential in dense nuclear matter. Phenomenologically, it is possible to introduce repulsion at high density by introducing $\Lambda NN$ 3-body repulsion. It is also known that repulsion appears at high density in model calculations based on chiral EFT [1], but there is no experimental data to support this repulsion.

Now, let us focus on the directed flow of $\Lambda$, which is considered to be sensitive to the potential at high densities. It was shown that the proton directed flow in the beam energy of $2\, \mathrm{GeV} < \sqrt{s_{NN}} < 20\, \mathrm{GeV}$ can be explained by the hadron transport model [2]. The repulsive EOS contributes positively to the slope in the early stage (compression stage), while the tilted ellipsoid contributes negatively in the late stage (expansion stage). Sum of these contributions causes the $dv_1/dy$ sign change at $\sqrt{s_{NN}}\simeq 10\, \mathrm{GeV}$ [3]. This mechanism predicts that the $\Lambda$ directed flow slope becomes negative at lower beam energies since $\Lambda$ does not exist before the collision, but the data [4] show the balance energy of $\Lambda$ is similar to that of proton. Then the positive contribution needs to be larger and the repulsion should be stronger for $\Lambda$.

In this work, we study the directed flow of $\Lambda$ by using the transport model (JAM2-RQMDv) [2] and estimate the potential of $\Lambda$ at high densities. We use the Fermi momentum expansion of the potential given in [5], and constrain the slope ($L_\Lambda$) and the curvature ($K_\Lambda$) parameters of the $\Lambda$ potential.

[1] D. Gerstung, N. Kaiser, W. Weise, Eur. Phys. J. A 56 (2020), 175.

[2] Y. Nara, A. Ohnishi, arXiv:2109.07594 [nucl-th].

[3] L. Adamczyk et al. [STAR], Phys. Rev. Lett. 112 (2014), 162301.

[4] P. Shanmuganathan et al. [STAR], Nucl. Phys. A 956 (2016), 260.

[5] I. Tews, J. M. Lattimer, A. Ohnishi, E. E. Kolomeitsev, Astrophys. J. 848 (2017), 105.