Quasiparticle properties of a single $\Lambda$ impurity in symmetric nuclear matter with a regulated $N\Lambda$ interaction

We explore the quasiparticle properties of a single $\Lambda$ hyperon propagating through symmetric nuclear matter using the Green's function formalism. The $N\Lambda$ interaction is described by a non-local regulated low-momentum contact potential with a leading-order constant term and a next-to-leading-order derivative correction. The two coupling constants in the ${}^1S_0$ and ${}^3S_1$ channels are fixed by matching the vacuum on-shell $T$ matrix to the scattering length and effective range obtained from modern next-to-next-to-leading-order chiral effective field theory. Using this effective interaction, we calculate the retarded $\Lambda$ self-energy from the in-medium $N\Lambda$ ladder $T$ matrix, which sums repeated $N\Lambda$ scattering in the nucleonic medium. At saturation density, the zero-momentum quasiparticle pole is found at $E_{\rm qp}(0,\rho_{\rm sat})=-29.55~{\rm MeV}$, in good agreement with the empirical depth of the single $\Lambda$ potential in nuclear matter. The self-energy decomposition gives a static Born contribution $\Sigma_\Lambda^{\rm Born}(0)=-26.36~{\rm MeV}$ and a dynamical correlation contribution ${\rm Re}\,\Sigma_\Lambda^{\rm corr,R}(0,E_{\rm qp})=-3.19~{\rm MeV}$, showing that repeated in-medium $N\Lambda$ scattering is needed to reproduce the empirical binding scale. The quasiparticle remains narrow and well defined, with a large residue $Z(0)=0.98$, a small damping width $\Gamma(0)=0.023~{\rm MeV}$, and a sharp spectral peak near the quasiparticle energy. At finite momentum, the $\Lambda$ quasiparticle becomes less bound, with $E_{\rm qp}(k,\rho_{\rm sat})$ increasing from $-29.55~{\rm MeV}$ at $k=0$ to $-6.49~{\rm MeV}$ at $k=1~{\rm fm}^{-1}$, while the residue and width change only weakly. A low-momentum fit gives $m_\Lambda^*/m_\Lambda=0.747$, consistent with the range obtained in Brueckner calculations with Nijmegen hyperon--nucleon potentials.