Hypernuclei and nuclear matter in a chiral SU(3) RMF model

EPJ manuscript No. (will be inserted by the editor) Hypernuclei and nuclear matter in a chiral SU(3) RMF model Kohsuke Tsubakihara, Hideki Maekawa and Akira Ohnishi Department of Physics, Faculty of Science, Hokkaido University, Sapporo 060-0810, Japan. e-mail: [email protected] Received: date / Revised version: date Abstract. We develop a chiral SU(3) RMF model for octet baryons, as an extension of the recently developed chiral SU(2) RMF model with logarithmic sigma potential. For Σ-meson coupling, strong repulsion(SR) and weak repulsion(WR) cases are examined in existing atomic shift data of Σ − . In both of these cases, we need an attractive pocket of a few MeV depth around nuclear surface. PACS. 21.65.+f Nuclear matter – 21.80.+a Hypernuclei 1 Introduction In this paper, we determine the hyperon-meson coupling constants in this chiral SUf (3) RMF model by fitIn constructing the dense matter equation of state (EOS), ting existing data. We show that we can reproduce the (SΛ ) [9] and it is strongly desired to respect both of hypernuclear physics separation energies of single Λ hypernuclei 6 the ΛΛ bond energy (∆B ) in He [10] by choosing ΛΛ information and chiral symmetry. Strangeness is expected ΛΛ the coupling constants appropriately in a reasonable pato play a decisive role and the partial restoration of chiral symmetry would modify the hadron properties in dense rameter range. The EOS of symmetric matter is found to matter. One of the promising approaches is to apply chi- be softened by the scalar meson with hidden strangeness, ral symmetric relativistic mean field (RMF) models [1–7]. ζ = s̄s, which couples with σ through the determinant We have recently developed a chiral SU(2) symmetric interaction. We also discuss the strength of repulsion in medium and attraction around nuclear surface in RMF model [6] with logarithmic sigma potential in the nuclear − potential by comparing the calculated results form of − log σ, which is derived in the strong coupling Σ -nucleus − limit (SCL) of the lattice QCD [8]. In this model, the with Σ atomic shift data [11]. energy density in vacuum at zero temperature is evaluated in the mean field approximation as, Uσ = −a log(det M M † ) + b tr(M M † ) + cσ σ σ 1 ∼ −2a fSCL ( ) + m2σ σ 2 , (1) fπ 2 f2 x2 , a = π (m2σ − m2π ) , fSCL (x) = log(1 − x) + x + 2 4 where √ M denotes the SU(2) meson matrix, M = (σ + iπ · τ )/ 2. In the second line of Eq. (1), σ field is replaced with its fluctuation around the vacuum expectation value, σ → fπ − σ. In this SCL model [6], we can describe bulk properties of finite nuclei, we have neither the chiral collapse at low densities [1] nor instability at large σ [2], and the nuclear matter EOS is not very stiff [3]. Compared to previously proposed chiral RMF models [4,5] and a more recently proposed one [7], this model has an advantage that the vacuum energy density is derived based on QCD. It is straightforward to extend this chiral SU(2) RMF model to an SU(3) version which contains strangeness degrees of freedom. We expect that this extension enables us to get detailed information on Λ, Σ and Ξ hypernuclei. 2 Chiral SU(3) RMF model In extending the chiral SU(2) RMF model to SU(3), it is necessary to include mesons with hidden strangeness (s̄s) such as ζ and φ in addition to σ, ω and ρ. The chiral SU(2) RMF model [6] tells us the form of chiral potential of σ and ζ by a simple extension written as, Uσζ = − a log(det M M † ) + b tr(M M † ) + cσ σ + cζ ζ + d (det M + det M † ), (2) where the last term in rhs is introduced to take care of the UA (1) anomaly. When the chiral symmetry is spontaneously broken and meson mass terms are generated, this effective interaction is written as, # " ζ σ Uσζ = − a 2fSCL ( ) + fSCL ( 0 ) fπ fζ 1 1 + m2σ σ 2 + m2ζ σ 2 + ξσζ σζ , 2 2 (3) Tsubakihara et al.,: Hypernuclei and nuclear matter in a chiral SU(3) RMF model where fζ0 = fζ +ms and ms is related to the strange quark mass. We have six parameters in this interaction (a, b, cσ , cζ , d and ms ), and five out of six are fixed by fitting experimental masses of π, K and ζ, and vacuum expectation values of σ and ζ. There remains only one parameter, mσ . With this scalar meson effective interaction, the RMF Lagrangian is given as, 0.6 E/A(MeV/fm-3) 0.4 0.2 ζ/fζ 2 cω L =LFree (ψi , ψ̄i , σ, ζ, ω, ρ, φ) + LEM − Uσζ + ω 4 4 X + ψ̄i [gσi σ + gζi ζ − γµ (gωi ω µ + gρi ρµ + gφi φµ )] ψi , 0 -0.2 -0.4 -0.6 i (4) -0.4 -0.3 -0.2 -0.1 4 Following the Okubo-Zweig-Iizuka (OZI) rule [13], we assume that nucleons do not couple with s̄s mesons (ζ and φ). Then there are two independent parameters, gωN and gρN , and hyperon-vector meson coupling constants are found to be represented by gωN and gρN as follows, √ 1 2 5 (gωN + 3gρN ) , (6) gωΛ = gωN − gρN , gφΛ = 6 2 3 1 gφΞ gωΣ = gρΣ = √ = (gωN + gρN ) , (7) 2 2 1 gφΣ (8) gωΞ = gρΞ = √ = (gωN − gρN ) . 2 2 In the later discussion, we try to keep the above relations as far as possible. In the scalar and pseudo scalar sector, it is necessary to include negative parity baryons or we only have Dtype when the chiral SU(3) symmetry is required [5,18]. This problem is out of the scope of this proceedings, and hyperon-scalar meson coupling constants are regarded as parameters. When the Λ-scalar meson couplings are obtained and SUf (3) symmetry works also for scalar couplings, we can evaluate the Ξ-scalar couplings as, √ √ 2 1 2 2 gζΛ , gζΞ = gσN + gζΛ . (9) gσΞ = gσN − 3 2 3 2 3 Nuclear matter and hypernuclei 3.1 Normal nuclei and nuclear matter In the present chiral RMF model, bulk properties of normal nuclei are well described, and these results are reported elsewhere. The strangeness degrees of freedom are found to soften the nuclear matter EOS, and thus have 20 0.1 0.2 0.3 0.4 Equation of state 15 Chiral SU(3)(mσ = 690) Chiral SU(3)(mσ = 710) TM1 Chiral SU(2) 10 E/A (MeV) where the ω term is phenomenologically introduced to simulate the high density behavior of the vector self-energy in the RBHF theory as in Ref. [12]. In determining hyperon-vector meson couplings, we start from the SUf (3) symmetric interaction, √

LBM = 2{gs tr (M ) tr B̄B + g1 tr B̄M B
+ g2 tr B̄BM } . (5) 0 σ/fπ 5 0 -5 -10 -15 -20 0 0.1 0.2 0.3 -3 ρB (fm ) 0.4 0.5 Fig. 1. Energy surface and EOS in chiral SU(3) model. effects also on normal nuclei. The interaction in Eq. (3) contains the σζ mixing term, which gives rise to a correlation in σ and ζ along the softest valley in the vacuum energy surface as shown in the upper panel of Fig. 1. Since the matter can evolve along this valley as the density increases, EOS is softened than in the chiral SU(2) RMF model [6], in which there is no ζ degree of freedom. The incompressibility is found to be K ∼ 220 MeV when we fit the bulk properties of normal nuclei and nuclear matter saturation point, as shown in the lower panel of Fig. 1. 3.2 Λ hypernuclei Next we study Λ hypernuclei with this chiral SU(3) RMF Lagrangian. There appear four additional parameters, gσΛ , gζΛ , gωΛ and gφΛ . We fix the vector coupling constants, gωΛ and gφΛ by using the SU(3) symmetry relation in Eq. (6). Two remaining parameters are determined by fitting SΛ and ∆BΛΛ data. As shown in the upper panel of Fig. 2, we can explain SΛ nicely in a wide mass region by giving the Λ potential depth around 30 MeV, which is represented by a linear combination of gσΛ and gζΛ . By fitting ∆BΛΛ in 6ΛΛ He simultaneously with SΛ , both of gσΛ and gζΛ are determined as shown in Fig. 2. Tsubakihara et al.,: Hypernuclei and nuclear matter in a chiral SU(3) RMF model 30 SΛ from A+1ΛZ SΛ(MeV) 20 15 10 5 0 0 0.05 2.5 0.15 Acore-2/3 0.2 0.25 6 1.5 1 0.5 Chiral SU(3) (SΛ fit) 0 SR WR Exp. 50 6 8 10 1000 NAGARA 12 14 0 0.5 1 1.5 gζΛ/gσΛ 2 2.5 Fig. 2. Λ separation energy and ∆BΛΛ of 3 6 ΛΛ He. 3.3 Σ hyper atom Recent analyses of quasi-free Σ − production spectra [14, 15] suggest that Σ − -nucleus potential should be repulsive in nuclear medium. On the other hand, Σ − -nucleus potential needs to possess a few MeV attractive pocket around nuclear surface to explain Σ − atomic shift data [16, 17]. Here we would like to extract Σ-meson coupling constants which explain Σ − atomic shifts. In the present RMF model, we have four additional parameters for Σ, gσΣ , gζΣ , gωΣ and gρΣ . First we set gωΣ , which determines the strength of repulsion in nuclear medium. We have examined two cases. (i) Strong Repulsion (SR) case: From the flavor SU(3) symmetry and OZI rule, gωΣ is given as gωΣ = (gωN +gρN )/2 ∼ 2gωN /3. (ii) Weak Repulsion (WR) case: gωΣ ∼ gωN /3 which is also adopted in Ref. [17]. Secondly, scalar meson couplings (gσΣ and gζΣ ), which determine the attractive pocket depth around nuclear surface, are chosen so as to reproduce atomic shifts of symmetric N = Z core nuclei (O, Si, S). Finally, gρΣ is adjusted to get a correct atomic shift in Pb. In Fig. 3, we show calculated atomic shifts and conversion widths of O, Mg, Al, Si, S, W and Pb for n = 4 → 3(O), n = 5 → 4(Mg, Al, Si and S) and n = 10 → 9(W and Pb) transitions. Atomic shift results are in good agreement except for W and the total χ2 / dof is around 16 Z 18 70 5->4 75 80 85 10->9 100 SR WR Exp. 10 -0.5 -1 10->9 5->4 100 10 0.3 ∆BΛΛ of ΛΛHe(MeV) 2 ∆BΛΛ 0.1 Width (eV) -5 4->3 500 Atomic Shift(eV) 25 1000 Chiral SU(3) exp. 3 1 10 12 14 Z 16 70 75 80 85 Fig. 3. Atomic shift and conversion width of Σ − . 1.3. The conversion width is calculated as the expectation value of ImVopt = tρp . Imaginary parts are found to be −15 ∼ −20 MeV. 4 Summary and conclusion We have developed a chiral SU(3) relativistic mean field (RMF) model with a logarithmic chiral potential for σ and ζ(= s̄s) mesons derived in the strong coupling limit of lattice QCD [8], as an extension of the chiral SU(2) RMF model [6]. The chiral symmetry and the mass generation by the spontaneous chiral symmetry breaking give severe constraints on parameters. After fitting several meson masses and vacuum expectation values, mσ is left unfixed in this chiral potential. Nucleon parameters (N meson coupling constants, mσ and the coefficient of ω 4 term) are determined to reproduce the vacuum nucleon mass, the nuclear matter saturation point, and bulk properties (binding energies and charge rms radii) of normal nuclei from C to Pb isotopes. Λ-meson coupling constants are determined by fitting hypernuclear data (Λ separation energies SΛ and ΛΛ bond energy BΛΛ ) under the constraints of SUf (3) symmetry for vector couplings. By fitting the Σ − atomic shifts, we find that the attractive pocket in the Σ-nucleus potential around the nu- Tsubakihara et al.,: Hypernuclei and nuclear matter in a chiral SU(3) RMF model 4 50 Vopt of Σ in Si 40 30 Vopt(MeV) be strong or relatively weak. It is also interesting to investigate Ξ hypernuclei and hyperatoms. If the SUf (3) relations in Eqs. (8) and (9) approximately hold in Ξ-meson couplings, we have smaller ambiguities in the Ξ-nucleus potential. Predictions along this line are in progress. 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Conf. on Hypernuclear & tigated Σ − quasi-free spectrum with DWIA+Local OptiStrange Particle Physics, Jefferson Lab., Newport News, Virginia, USA, Oct. 14-18, 2003 (unpublished). mized Fermi Average t-matrix [15]. With this method, it Vopt(MeV) 30 would be possible to judge whether Σ − repulsion should View publication stats