Strength of pairing interaction for hyperons in multistrangeness hypernuclei

Pairing correlations play a very important role in atomic nuclei. Although several effective pairing interactions have been used in mean field calculations for nucleons, little is known about effective pairing interactions for hyperons. Based on the quark model, we propose a relationship between effective pairing interactions for hyperons and for nucleons; e.g., for $\Lambda$s, the strength of the pairing interaction is 4/9 of that for nucleons. A separable pairing force of finite range which has been widely applied to describing pairing correlations in normal nuclei is used to investigate pairing effects in multi-$\Lambda$ Ca, Sn and Pb hypernuclei.

Hypernuclei are unique quantum many-body systems for the investigation of hyperon-nucleon (YN) and hyperon-hyperon (YY) interactions which are, in turn, crucial for understanding the hypernuclear structure as well as the hypernuclear matter and properties of neutron stars. With the strangeness degree of freedom, a hyperon can move deep inside the nucleus and serve as an impurity for probing nuclear properties that are not accessible by conventional methods developed for normal nuclei. Lots of many-body techniques for normal nuclei, including various mean-field models , have been extended to hypenuclei. In particular, the relativistic mean-field (RMF) models [40][41][42][43][44][45][46][47][48][49] which have been very successful in describing normal nuclei in the whole nuclear chart are also used extensively to study hypernuclei.
In the RMF models, one needs effective interactions both for the particle-hole (ph) and particle-particle (pp) channels. For normal nuclei, a large amount of effective interactions for the ph channel have been proposed, see, e.g., Refs. [50][51][52][53][54][55][56][57][58]. Meanwhile, both zero-range and finite-range pairing forces have been used in the pp channel [59][60][61]. For hypernuclei, YN and YY interactions in the ph channel can be either obtained by fitting experimental data or estimated with the naive quark model [29,[62][63][64][65][66][67]. However, effective interactions for hyperons in the pp channel are much less known. In this Letter, we propose a way to estimate the strength of pairing interactions for hyperons based on the meson exchange picture and the naive quark model. From the quark model we know that nucleons consist of three u/d quarks and hyperons consist of, besides u/d quarks, one or more s quarks. Next we use n u/d to label the number of u/d quarks in a baryon (a nucleon or a hyperon) and define g BM to be the coupling constant of a non-strangeness meson M (σ, ω, ρ, · · · ) to a baryon B (N, Λ, · · · ). According to the OZI rule only u/d quarks are involved in the coupling of a non-strangeness meson to a baryon at the tree level. Therefore the following relation holds: g Y M = n u/d /3 · g N M . Similar discussions have been made with the quark-meson coupling model [68]. If g N M is known, one can readily get g Y M . For example, since n u/d = 2 in Λs, g ΛM = 2/3 · g N M ; this has been proposed and used in the study of Λ-hypernuclei [62,63,[69][70][71].
The exchange of the meson M between two baryons B 1 and B 2 results in an interaction with the strength proportional to g B 1 M g B 2 M [72]. For single-Λ hypernuclei, the central potential for the Λ in the mean field generated by nucleons is proportional to g N M g ΛM while that for nucleons is proportional to g 2 N M , leading to the well known observation that the depth of the potential for the Λ is roughly 2/3 of that for nucleons [73][74][75][76].
The strength for the YY interaction is proportional to g 2 Y M , i.e., n 2 u/d /9 = g 2 Y M /g 2 N M of that for the NN interaction. Since the pairing force is the residual of the two-body BB interaction, the ratio n 2 u/d /9 holds also between the strength of the pairing interaction for hyperons and that for nucleons. Note that mesons consisting of strange quarks may be exchanged between hyperons and result in possible deviations of this ratio from n 2 u/d /9. In this work we restrict our discussions in the framework of conventional RMF models with non-strangeness mesons.
As an illustration, we use the relativistic Hartree-Bogoliubov (RHB) theory to study the effects of the ΛΛ pairing in multistrangeness hypernuclei. The RHB theory provides a unified description of the relativistic mean field and the pairing correlations via the Bogoliubov transformation. The RHB equation consists of the single particle Hamiltonian h B and the pairing field ∆ [42,77] where λ is the Fermi energy and E k and (U k , V k ) T are the quasi-particle energy and wave function, respectively. For the ph channel, the Dirac Hamiltonian for nucleons has been established [40][41][42][43][44][45][46][47][48][49] and that for Λs reads [71] where the scalar, vector and tensor potentials are The tensor potential T Λ (r) is included to achieve the small spin-orbit splitting for the Λ [78][79][80]. We adopt two effective interactions NLSH-A [51,63] and PK1-Y1 [54,65] which have been extensively used in the study of Λ hypernuclei.
In the pp channel, the pairing potential reads where V is the effective pairing interaction and κ is the pairing tensor We use the separable pairing force of finite range proposed by Tian et al. [60,[81][82][83]] where G is the pairing strength and R = (r 1 + r 2 )/2 and r = r 1 − r 2 are the center of mass and relative coordinates, respectively. P(r) denotes the Gaussian function, where a is the effective range of the pairing force. For nucleons, the pairing strength G N = 728 MeV·fm 3 and the effective range a = 0.644 fm have been obtained by fitting the momentum dependence of the pairing gap in the nuclear matter calculated from the Gogny force. According to our proposal discussed before, the pairing strength for Λs is taken to be G Λ = 4/9 · G N . We have carried out calculations with the multidimensionally-constrained (MDC) RHB theory [84], one of the recently developed MDC covariant density functional theories (MDC-CDFTs) [49,[84][85][86][87]. For simplicity, we choose doubly-magic 40 Ca, 132 Sn and 208 Pb as the core nuclei and study even-even-even hypernuclei 40 Fig. 1, the two-Lambda separation energies S 2Λ are shown for them. One can find that S 2Λ decreases monotonically with the number of Λs increasing. As far as the two-Lambda separation energy is concerned, at least 20, 40 and 70 Λs can be bound to the core nuclei 40 Ca, 132 Sn and 208 Pb, respectively. There are sudden drops in S 2Λ when −S = 2, 8, 20, 34, 40 and 58. These numbers are magic numbers for Λs. Since the spin-orbit splitting is very small in the single Λ spectrum, these numbers actually correspond to shell closures or sub-closures in the single particle level scheme of a harmonic oscillator potential. The pairing gap is one of the typical quantities to characterize pairing effects. We have calculated the average pairing gap as [88] where ∆ k is the pairing gap corresponding to a single Λ state k in the canonical basis and u 2 k and v 2 k give the empty and occupation probabilities, respectively.   There have not been much work on the pairing effects of hyperons in finite nuclei, though some efforts were made to the study of double-Λ and multistrangeness (−S ≥ 3) hypernuclei [13][14][15][16][17][18][19][20]. In Ref. [19], Güven et al. have investigated multistrangeness hypernuclei with the Hartree-Fock-Bogoliubov (HFB) theory and obtained interesting results concerning pairing effects of Λs. Next we make a brief comparison of our results with Ref. [19]. In Fig. 3(a), the pairing gaps for Λs in three typical multistrangeness hypernuclei 46 6Λ Ca, 160 28Λ Sn and 272 64Λ Pb are compared with the HFB results [19]. It can be seen that ΛΛ pairing gaps of these three nuclei in the present work are smaller than those given in Ref. [19]. Furthermore, the ∆ Λ from the MDC-RHB theory decreases faster with A than the HFB predictions. This conclusion holds also for the average pairing gap , as seen in Fig. 3(b). In Ref. [19], a zero-range δ force is adopted for the ΛΛ pairing and its strength for 40 [19]. In the present work, however, a global finite-range pairing force [Eq. (6)] is adopted and, thus, there is no hard cut-off for the pairing window.
In normal nuclei, it has been well known that the pairing gap for nucleons ∆ N declines more or less with √ A [89]. This decreasing tendency is roughly consistent with the dependence of ΛΛ pairing gaps with respect to the number of Λs obtained in the present work. Nevertheless, in Ref. [19], the decrease of the ΛΛ pairing gaps is much gentler with the number of Λs. This is quite interesting and should be investigated further.
To summarize, we have proposed a relationship between effective pairing interactions for hyperons and for nucleons based on the quark model. Namely, the ratio between the strength of the effective pairing interaction for the hyperon Y consisting of n u/d u/d quarks and that for the nucleon is n 2 u/d /9. For Λs, this ratio is simply 4/9. A separable pairing force of finite range has been implemented in the MDC-RHB theory to investigate ΛΛ pairing effects in multi-Λ Ca, Sn and Pb hypernuclei. By examining the two-Λ separation energy S 2Λ and the pairing gap ∆ Λ , it is revealed that −S = 2, 8, 20, 34, 40 and 58 are magic or semi-magic numbers for Λs. The ∆ Λ decreases when the mass number of the core nucleus increasing. It is also found that the pairing effects of Λs are weaker than nucleons due to the supression of the pairing strength by a factor of 4/9.
Finally, let us make two further remarks. First, one may notice that the ratio n 2 u/d /9 is probably very rough. Other factors such as the violation of the OZI rule [91,92], medium effects [68], possible different couplings of ρ to baryons [93] and mass splittings for baryons [79] may alternate this ratio or make the relation more complex between pairing interactions of Λs and nucleons. Second, although in the present work the ratio n 2 u/d /9 between the strength of the pairing interaction for hyperons and that for nucleons has been used in the framework of the RMF models, we expect it is also applicable to non-relativistic mean field models.