Few-Body Systems Suppl. 99, 1{?? (1995)
FewBody Systems
c by Springer-Verlag 1995 Printed in Austria
Strange Baryonic Systems J. Maresy , B. K. Jennings TRIUMF, 4004 Wesbrook Mall, Vancouver, B.C., Canada V6T 2A3
Abstract. Strange baryonic systems are investigated by using the relativistic
mean (RMF) approach. Present discussion ranges from ordinary hypernuclei to multiply strange objects. Application to the shell model as well as optical model calculations allows to study various aspects of hyperon-nucleus interactions.
1 Introduction This contribution is concerned with recent eorts to describe hyperon-nucleus interactions within the relativistic mean eld theory [1, 2, 3]. This approach has been extremely successful in nuclear structure as well as nuclear dynamics calculations [4, 5]. Attempts for its extrapolation to more general baryonic systems are thus justi ed. Hypernuclei, as bound objects of dierent types of baryons, represent a sound generalization of traditional nuclear matter to a many-body baryonic system. Their study provides a direct test of various models for the baryon-baryon and baryon-nucleus interactions. Recent investigations taking into account the Lorentz-tensor coupling of the ! meson to the hyperon [6, 7, 1] have proved that a consistent description of both nuclear and hypernuclear systems can be achieved within the RMF model. This is regarded a great success of the Dirac approach. Once the RMF model accounts for the hypernuclear data it is rather straightforward to extend considerations to more \strange" objects { multiply strange baryonic systems [8, 9, 10, 11]. Strange matter, either quark or baryonic, is of much interest in astrophysics [12] and in the physics of relativistic heavy ion collisions [13]. Present calculations of atoms not only revealed that the RMF model is capable of high quality ts to the data but also demonstrated that the data are sucient to constrain the couplings of mesons to a hyperon [3]. This has important consequences for the spectroscopy of hypernuclei.
Permanent address: Nuclear Physics Institute, 250 68 R ez, Czech Republic y E-mail addresses:
[email protected],
[email protected],
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Hyperon-nucleus scattering [2] is still a rather unexplored eld. Currently there is no experimental data available, however, this may change soon [14]. Recent calculations [2] thus should be considered just a rst qualitative estimate of the hyperon-nucleus scattering observables. The RMF model predicts a large eect of the Lorentz-tensor coupling on the predictions for the analyzing power. There are other applications of the RMF approach to the physics of baryonic systems with strange particles which could not be discussed here due to limited space. Hypernuclear magnetic moments and currents [15], and decays of hypernuclei [16] should be mentioned among others. Interested reader will nd relevant information and references in the contributions of Bennhold [17] and Ramos [18] to this conference and in the recent review [19]. In the next section, we introduce the underlying RMF model (subsect.2.1) and discuss the choice of parameters (subsect.2.2). Application to the spectroscopy of hypernuclei is presented in Sect.3. A brief summary of the RMF predictions for multi-strange baryonic systems follows in Sect.4. In section 5, we apply Dirac phenomenology to costructing the optical potential for hyperons. First, we make use of the Sigma atom data to extract some information about the -nucleus interaction and discuss implications for hypernuclei. Then, we apply the RMF model to description of a hyperon-nucleus scattering. Conclusions are drawn in Sect.6.
2 Relativistic Mean Field Model 2.1 Lagrangian The RMF formalism describes baryons as Dirac spinors interacting via meson elds in the mean eld approximation [4]. The underlying Lagrangian density appropriate for strange baryonic systems can be expressed as:
L = LN + LY ; Y = ; ; ; (1) LY = Y [ i @ g!Y V (MY + gY )] Y + LY + LAY + LT : (2) The detailed form of the standard nuclear part LN can be found elsewhere [4]. In general, the Lagrangian density LY includes interactions of a hyperon with the isoscalar (, !) meson elds, as well as contributions from the meson (LY ) and Coulomb (LAY ) elds. For a particular hyperon LY and LAY acquire the following form: L = LA = 0 ; (3) L + LA = 21 g b + 2e (3; 1) A ; (4)
L + LA = where
+ e A ( ) ; ij g
ki 3 jk jk 2 2
= p 2 0
p
2 + 0
;
(5)
(6)
3
and
p
p0 2+ : (7) 0 2 Finally, the Lagrangian density LT describes the !-Y anomalous coupling, f!Y @ V : LT = 2M (8) Y Y Y It is to be noted that similar tensor coupling term for nucleons is omitted since the coupling constant f!N is small. We neglect the Y Y and NN coupling terms, as well. There eect was found to be negligible [1]. The Euler-Lagrange equations lead to a system of equations of motion for both baryon and meson elds which was solved fully self-consistently (for more details see [1]). This appeared important particularly for calculating the hypernuclear magnetic moments [20] and for evaluation of the isovector contribution to the hyperon binding energies [1].
=
2.2 Parametrization The parameters of the nucleonic part LN of the Lagrangian from Eq.(1) are usually tted to the properties of spherical nuclei. Here, we adopted parameter sets of Sharma et al [21], Horowitz and Serot [22] and Reinhard [23]. The couplings (characterized via the ratios i = gi =giN , i = ; !) that describe the experimental single particle spectra reasonably well are located along a line in the ( , ! ) plane [9]. This linear dependence results from the constraint to obtain the correct well depth U 28 MeV in nuclear matter [11]: 2 MN ) g!N U = ( SN + ! VN ) = MN (1 M m2! 0 ! ; N where SN and VN are the scalar and vector potentials for nucleon, respectively. If the tensor coupling contribution LT (Eq.(8)) is not considered, the empirically known small spin orbit splitting is achieved at a price of an extremely weak coupling (i 1=3) [8]. This value, however, is in contradiction with the constituent quark model predictions, as well as, with the bounds obtained from neutron star mass calculations [12], namely i 0:5. In this contribution we demonstrate that a consistent description of hypernuclei can be achieved even with the quark model values of the hyperon-meson couplings provided the tensor coupling term LT is introduced in the Lagrangian LY (Eq.(2)) [6, 7, 1]. We used the constituent quark model to determine the values of the coupling constants g!Y , f!Y and gY . The g couplings were tted to the experimental hypernuclear data. For hypernuclei, we chose g to roughly obtain the potential well depth U 20 25 MeV which is compatible with a scarce data from hypernuclei [24]. For hyperon, we used quark model and/or QCD sum rules predictions [25] for ! . Corresponding as well as were then tted to atomic level shifts and widths [3]. In view of new results
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from atoms (presented in Sect.5.1), the case of hypernuclei deserves more attention and will be discussed in separate subsection 5.1.1. The values of Y = gY =gN , !Y = g!Y =g!N , Y = gY =gN and TY = f!Y =g!Y used in the following sections are presented in Table 1.
Table 1. The couplings used for hyperonic sector. The ratios of hyperon to nucleon
couplings for Y = , and are presented here. For TY the values of Nijmegen models F [26], D [27], and the values from Dover and Gal [28] are used for comparison.
Y
!Y Y
TY
0.621 2/3
0.0
0:0
-0:541[26] -1:0
0.544 2/3 0.77 1.0
2/3 2/3
0:0
0:76 [28]
0.375 1/3
1.0
0:0
-0:4 [27] -2:0
1:0
3 Hypernuclear Shell Model The RMF model introduced in Sect.2 yields reasonable description of known hypernuclear characteristics { hyperon binding in nuclear matter, spin orbit interaction, single particle spectra [1]. In this section, we select only a few examples illustrating the role of dierent terms in the Lagrangian LY from Eq.(2). Figure 1 demonstrates the competing eects of the Coulomb and isovector ( meson) interactions in hypernuclei. The attractive Coulomb potential leads to a considerably stronger binding of in the nuclear medium when compared with neutral hyperons (, 0 ). The meson eld signi cantly reduces the eect of the Coulomb interaction in nuclei with a neutron excess. In Zr and Pb the meson contribution is repulsive (attractive) for ( 0 ) and thus tends to compensate the contribution from the Coulomb potential. The same role is played by the above elds in the -nucleus interaction. The tensor coupling (Eq.(9)) is crucial for evaluating the hypernuclear spin orbit interaction. In Fig.2, 17 Y O serves as an example of a quite dierent contribution of the tensor coupling term LT to the spin orbit splitting in a case of the three kinds of hyperons (Y = ; 0 ; 0). It is to be stressed that for
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Figure 1. Comparison of the and 0 single-particle levels in hypernuclei for
= 0 and = 1.
sake of comparison we adopted here the well depth of -nucleus potential of the same size as that for , U U . The results of Fig. 2 become apparent from Schrodinger equivalent spin orbit potential: VlsY l s = 2M1 2 r1 g!Y V00 gY 0 + 2f!Y MMeff V00 l s ; Y e 1 Me = MY 2 (g!Y V0 gY ) :
For f!Y = 0 the spin-orbit splitting for hyperons is reduced when compared to the nuclear case due to a larger mass Me in the denominator and due to smaller couplings to and ! mesons. The quark model values of f!Y for , and hyperons dier in their strengths and signs. Consequently, the tensor
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Figure 2. The hyperon single-particle levels in 17Y O as a function of TY = f!Y =g!Y . coupling contribution is comparable in magnitude with the original ! part and is negative (positive) for (). It is negative and even larger than ! term for . The resulting spin orbit interaction for nearly vanishes, it is almost doubled for , and changes sign for . It should be noted, however, that although the eects of tensor coupling are relatively large, the absolute shifts of energy levels are probably still beyond the reach of contemporary experimental resolution.
4 Multiply Strange Baryonic Systems Baryonic objects with larger contents of strangeness are expected to be produced in the relativistic heavy ion collisions [13]. In addition, sertain form of strange (baryonic) matter is predicted to be present in the cores of neutron stars [29, 12]. Investigations of these multi-strange systems are expected to provide unique information about the baryon-baryon interaction. This could help to distinguish between various models more or less equivalent in describing the Y N scattering or single hypernuclei. Unfortunately, our empirical knowledge 6 He, 10 Be, 13 B. The is limited to a few events of double- hypernuclei: calculations revealed that the RMF model, as formulated in section 2, has to be improved by additional meson exchanges (scalar f0 and vector ) acting exclusively between hyperons in order to get stronger Y Y interaction in agreement with the hypernuclear data and predictions of a one-boson exchange potential [10, 11].
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The RMF models predict a possibility of forming bound systems with an appreciable number of hyperons. The dependence of the binding energies per particle, density distributions and rms radii of such systems on a number of hyperons nY , results from a delicate interplay between the eect of Pauli blocking (hyperon is distinguishable from N) and the weaker Y N interaction compared to the NN one. First studies of multi-strange objects have limited themselves to calculations of N systems[8, 9]. However, the considerations can not be restricted to nucleons and particles only as far as 0 and are necessary constituents of the strange baryonic matter [10]. This is because the process ! N becomes energetically favourable due to the Pauli blocking of 's in some multi- hypernuclei. This greatly increases the possible amount of strangeness in the bound systems (stable against strong decay). Figure 3 adopted from ref.[11] serves as an example. Here, the binding energies per particle (EB =A) for 56Ni + Y (Y = , ) are calculated.
Figure 3. The binding energy per particle (EB =A) in 56 Ni + Y as a function of A. The squares correspond to Y = only, the circles are obtained by adding 's to a system. (The gure is adopted from ref.[11].)
The RMF calculations yield baryonic (N, , ) systems with densities (2 3)0 , jS j=A 1 and jZ j=A 1. These values resemble the speculations about droplets of strange quark matter \strangelets"[29]. However, the baryonic objects are more loosely bound, jEB=Aj 10 20 MeV. Since it is much less than the N mass dierence ( 177 MeV), these objects will decay by weak interaction with lifetimes 10 10 s. An excellent review on the (N, , ) systems can be found in ref.[11].
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5 Optical Potential for Hyperons So far we have discussed the RMF calculations of bound nuclear systems, either hypernuclei or strange hadronic matter. However, the Dirac approach is a valuable tool to also describe the optical model as we illustrate here on two examples. First, we apply the RMF approach to constructing the -nucleus optical potential to be used in calculations of -atoms. Then, we extend further our considerations to higher energies and apply Dirac phenomenology to and scattering o nuclei.
5.1 What We Have Learnt from Atoms Recent phenomenological analyses of level shifts and widths in atoms by Batty et al. [30] suggest that the real part of the -nucleus potential ReVopt is attractive only at the nuclear surface, changing into a repulsive potential as density increases in the interior (for more details see contribution of E. Friedman). The shallow attractive pocket of such potential does not provide sucient binding to form hypernuclei. These conclusions are seemingly in 25 30 MeV. contradiction with previous analyses [31, 32], yielding ReVopt Since calculations of hypernuclei have often been based on the above analyses it is desirable to apply the RMF approach directly to determining the -nucleus optical potential by tting to atom data. In our work [3] we used as a real part of the -nucleus potential the Schrodinger equivalent potential constructed out of the scalar () and vector (!, ) meson mean elds. A purely phenomenological imaginary part of = tp (r) was chosen in order to account for the conversion the form ImVopt p ! n. While the proton density p (r) was calculated within the RMF model, the parameter t was tted to the atomic data. The other free parameters of the model were the scalar meson coupling ratio and isovector meson coupling ratio . The values of the coupling ratio ! were adopted from constituent quark model [6] (! = 2=3) and QCD sum rules evaluations [25] ((! = 1). Since a rather detailed description of our analysis can be found in the contribution of E. Friedman to this conference and in ref.[3] we limit ourselves here to mere summary of the results: 1) The RMF approach is capable of very good quality ts to the atomic data. 2) The best ts were obtained for larger values of ! (2=3 ! 1). 3) The - coupling ratio 2=3 holds unambiguously for all the parametrizations used. with a volume repulsion in the nuclear inte4) The RMF model yields ReVopt rior and a shallow attractive pocket at the surface in agreement with the latest density dependent phenomenological analyses [30]. For illustration, we show in Fig.4 one of the nucleus potentials compatible with the data.
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(solid line) and ImV (dashed line) as a function of r for the Figure 4. ReVopt opt
optical potential in Pb. Linear RMF model with ! = 2=3 was used.
5.1.1 Do Hypernuclei exist? Up to now, no hypernuclear bound state has been clearly established [33, 34], except perhaps a J = 0+ , I = 1=2 4 He bound state [35]. The volume repulsion obtained in the previous section for the isoscalar in fact precludes binding for 0 hypernuclei. component of Vopt generally does not overcome For + , the attractive isovector part of Vopt the repulsion from the isoscalar and Coulomb interactions. It is thus unlikely to bind a + in nuclei. It is to be noted, that above considerations could be modi ed in very light hypernuclei (4 He) where isovector potential dominates over the Coulomb interaction [36, 37]. For and high Z nuclear cores, the attractive Coulomb potential gives rise to bound states that might be called nuclear as the corresponding wave is included functions are located within the nucleus or at its surface. When Vopt these Coulomb levels are shifted to lower binding energies and acquire strong interaction widths. These states may be considered an example of \Coulomb assisted states" [37]. Their location could provide a valuable information about . However, since these states are \pushed" outside the nucleus, the cross Vopt sections to excite them by a nuclear reaction are not expected to be sizeable. The chances of establishing a meaningful hypernuclear spectroscopy are thus vanishingly small at present.
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5.2 Hyperon-Nucleus Scattering The Dirac approach yields remarkably accurate results for proton-nucleus scattering observables at medium energies [5]. Similar investigations of a hyperon optical potential have been missing till recently as there are no data on hyperon scattering o nuclei. The situation could change, however, as an experiment at KEK [14] may soon provide data on + -12 C. In addition, knowledge of the hyperon-nucleus optical potential will become necessary for description of quasi-free hyperon production ( K + ) proposed at CEBAF [38]. With this motivation we have developed an optical potential describing hyperon-nucleus interaction at medium energies. We used the global optical model for nucleon-nucleus scattering [39] to get the shape and energy dependence of the potential. The strength of the potential was determined according the information coming from quark model, hypernuclei and atom data [3]. We made several assumptions which remain to be veri ed: 1) the energy dependence of the Y -nucleus interaction is that of nucleon- nucleus potential; 2) the imaginary strengths of the potentials for hyperon are obtained from the imaginary parts for protons by multiplying the latter by the squares of the ratios for the real potentials; 3) the absorption due to the N ! N conversion is energy independent. The above optical potential was used in calculations of the dierential cross sections and analyzing powers for and scattering o nuclei.
Figure 5. The eect of 40the tensor coupling on the observables for 300 MeV 's elastically scattered from Ca. The predictions for T = 1 (dashed line), T = 0 (dotted line) and T = 1(solid line) are compared.
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Figure 5 illustrates the dominant role of the tensor coupling in predictions of the analyzing power Ay . Scattering of from 40Ca at 300 MeV serves here as an example. The cross sections are qualitatively similar for dierent values of the tensor coupling ratio T (T = 1; 0; +1) with maxima and minima at roughly same angles. On the contrary, the predictions for Ay strongly depend on the value of the tensor coupling. Particularly T = f=g = 1 (quark model value for ) yields considerably smaller values of analyzing power; Ay is close to zero for forward angles. This result suggests that measurements of Ay would give information about the tensor coupling of the vector meson to a hyperon. In order to get closer to experiment [14] we have calculated the scattering of + from 12C at 200 MeV using coupling ratios determined by tting the atom data [3]. The results presented in Fig.6 demonstrate the sensitivity of Ay to the strengths of -meson couplings. For comparison we have included also results for the case of U = U , which was assumed in our previous calculations [2].
Figure 6. The observables for 200 MeV + elastically scattered from 12 C calculated
for potentials compatible with the atom data. The predictions for ! = 1 (dashed line) and ! = 2=3 (solid line) are compared. The prediction for i = i (dotted line) is included for comparison.
In view of the assumptions made the present calculations should be considered just a rst qualitative estimate of the hyperon-scattering observables. There is still a lot of theoretical as well as experimental work to be done.
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6 Concluding Remarks This contribution aimed to demonstrate that the relativistic mean eld approach provides a natural description of hyperon-nucleus interactions: With reasonable (quark model inspired) values of the meson-hyperon couplings it is possible to reproduce the hypernuclear data. Extrapolation of the RMF theory from ordinary hypernuclei to multistrange systems predicts rather weakly bound stable objects composed of N, , 0 and baryons, of arbitrarily large A, high strangeness content and small charge. The atom data are sucient to signi cantly constrain the possible hyperon-nucleus couplings. The analysis yields potentials with a repulsive real part in the nuclear interior. Consequently, the chances of forming bound nuclear states of (except perhaps the lightest systems) are very limited. Hyperon-nucleus scattering experiments could become another important source of information about hyperon-nucleon interaction. In particular, we predict a strong sensitivity of the analyzing power Ay to the tensor coupling. Experimental programs in BNL, KEK, CEBAF, DANE promise new perspectives in a study of hyperon-nucleus interactions. Acknowledgement. Calculations and investigations which formed basis for
this contribution were done under fruitful and stimulating collaboration with E.D. Cooper, A. Gal and E. Friedman. To them belongs our gratitude. B.K.J. would like to thank the Natural Sciences and Engineering Research Council of Canada for nancial support.
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