Beyond-mean-field study of the hyperon impurity effect in hypernuclei with shape coexistence

[Background] The hyperon impurity effect in nuclei has been extensively studied in different mean-field models. Recently, there is a controversy about whether the $\Lambda$ hyperon is more tightly bound in the normal deformed (ND) states than that in the superdeformed (SD) states.[Purpose] This article is aimed to provide a beyond-mean-field study of the low-lying states of hypernuclei with shape coexistence and to shed some light on the controversy.[Method] The models of relativistic mean-field and beyond based on a relativistic point-coupling energy functional are adopted to study the low-lying states of both $^{37}_\Lambda$Ar and $^{36}$Ar. The wavefunctions of low-lying states are constructed as a superposition of a set of relativistic mean-field states with different values of quadrupole deformation parameter. The projections onto both particle number and angular momentum are considered.[Results] The $\Lambda$ binding energies in both ND and SD states of $^{37}_{\Lambda}$Ar are studied in the case of the $\Lambda$ hyperon occupying $s, p$, or $d$ state in the spherical limit, respectively. For comparison, four sets of nucleon-hyperon point-coupling interactions are used respectively. Moreover, the spectra of low-lying states in $^{36}$Ar and $^{37}_{\Lambda_s}$Ar are calculated based on the same nuclear energy density functional. The results indicate that the SD states exist in $^{37}_{\Lambda}$Ar for all the four effective interactions. Furthermore, the $\Lambda_s$ reduces the quadrupole collectivity of ND states to a greater extent than that of SD states. For $^{37}_{\Lambda}$Ar, the beyond-mean-field decreases the $\Lambda_s$ binding energy of the SD state by 0.17 MeV, but it almost has no effect on that of the ND state. [Conclusions] In $^{37}_{\Lambda_s}$Ar, the $\Lambda_p$ and $\Lambda_d$ binding energies of the SD states ...