Faddeev calculations for light Ξ-hypernuclei
hhttp://mmg.tversu.ru ISSN 2311-1275 MM G Mathematical Model lingand Geometry
Volume 5, No 2, p. 1 – 11 (2017) Special issue
Faddeev calculations for light Ξ -hypernuclei Igor Filikhin a , Vladimir M. Suslov and Branislav Vlahovic Department of Mathematics and Physics, North Carolina Central University, 1801 FayettevilleStreet, Durham, NC 27707, USA e-mail: a ifi[email protected] Received 19 April 2017, Published 3 May 2017.
Abstract.
The hypernuclear systems
N N
Ξ and ΞΞ N are considered as an ana-logue of nnp ( H) nuclear system (with the notation as
AAB system). We use therecently proposed modification for the s -wave Malfliet-Tjon potential. The modi-fication simulates the Extended-Soft-Core model (ESC08c) for baryon-baryon in-teractions. The Ξ N spin/isospin triplet ( S, I ) = (1 ,
1) potential generates a boundstate with the energy B ( AB )=1.56 MeV. Three-body binding energy B for thestates with maximal total isospin is calculated employing the configuration-spaceFaddeev equations. Comparison with the results obtained within the integralrepresentation for the equations is presented. The different types of the relationbetween B and B ( V AA = 0) are discussed. Keywords:
Few-body systems, Hypernuclei, Nuclear forces, Faddeev equations
PACS numbers: (cid:13)
The author(s) 2017. Published by Tver State University, Tver, Russia a r X i v : . [ nu c l - t h ] M a y I. Filikhin et al . The first Ξ-hypernuclear bound state has been reported in Ref. [1]. The lifetime ofa Ξ-hypernucleus is long enough to enable the hypernuclear state to be well defined.According the current experimental data, the Ξ-nucleus interactions are attractive[2]. In particular, the hyprnucleus
Be can be interpreted by assuming a nucleusWood-Saxon potential with a strength parameter of about -14 MeV [3]. Anotherhypernucleus
C is considered to be the cluster system N(ground state) + Ξ,where Ξ can be in s or p -wave state [4].The stable states in the systems Ξ N N and ΞΞ N were recently predicted inRefs. [5, 6, 7] based on the recent update of the Extended-Soft-Core (ESC08c)model [8, 9, 10] for baryon-baryon interactions. This model has predicted the Ξ N bound spin/isospin triplet ( S, I ) = (1 ,
1) state with three-body energy B to beequal to 1.56 MeV. This bound state of proton and Ξ or neutron and Ξ − hasmaximal isospin of the Ξ N pair. For the three-body systems when all pairs N N ,Ξ N , and ΞΞ are in triplet isospin states, the strong decay N Ξ → ΛΛ is forbidden.Such three-body systems can be stable under the strong interaction. The firstcalculations [6, 7] based on the assumption yield the existence of bound states forthe Ξ
N N and ΞΞ N systems.In the presented work, we use the differential Faddeev equations to mathemat-ically formulate the bound state problems for the Ξ N N and ΞΞ N systems. Thealternative treatment is presented in Refs. [6, 7] where the integral Faddeev equa-tions were applied. Our calculations for the systems are generally in the agreementwith the results [6, 7]. However, we found that a small correction for the results isneeded. We present our results along with the correction [11] of the results pub-lished in Refs. [6, 7]. Additionally, the binding energy for the spin, isospin (0, 1)bound state for the ΞΞ α system is calculated. This state was not considered inRefs. [6, 7].The models for Ξ N N and ΞΞ N (ΞΞ α ) are restricted by the s -wave approach.The coupling to higher-mass channels, ΣΛ and ΣΣ, does not taken into accountassuming that their contributions have the second order of smallness to the bindingenergy of three-body system. The calculations do not also take into account theCoulomb force. . The differential Faddeev equations [12] can be reduced to a simpler form for thecase of two identical particles (like an
AAB system). In this case the total wavefunction of the system is decomposed into the sum of the Faddeev components U and W corresponding to the ( AA ) B and ( AB ) B types of rearrangements: Ψ = U + W ± P W , where P is the permutation operator for two identical particles.In the latter expression the sign ”+” corresponds to two identical bosons, while ight Ξ hypernuclei − ” corresponds to two identical fermions, respectively. The set of theFaddeev equations is written as following:( H + V AA − E ) U = − V AA ( W ± P W ) , ( H + V AB − E ) W = − V AB ( U ± P W ) . (1)Here, H is the operator of kinetic energy of the Hamiltonian taken for correspond-ing Jacobi coordinates. The functions V AA and V AB describe the pair interactionsbetween the particles. The model space is restricted to the states with the totalangular momentum L = 0, the momentum of pair l = 0, and momentum λ = 0 ofthe third particle respectively to the center of mass of the pair. s -wave approach The description of the above mentioned
AAB systems is distinguished by the massesof particles and the type of AA and AB interactions. We use s -wave V AA and V AB potentials, which are spin-isospin dependent. This requires to write Eq. (1) withthe corresponding spin-isospin configurations.The separation of spin-isospin variables leads to the Faddeev equations for theconsidering systems in the following form:( H + V AA − E ) U = − V AA D (1 + p ) W , ( H + V AB − E ) W = − V AB ( D T U + Gp W ) , (2)where matrices D and G are defined by the nuclear system under consideration, W is a column matrix with the singlet and triplet parts of the W component ofthe wave function of a nuclear system, and the exchange operator p acts on thecoordinates of identical particles.For the H nucleus, considered as pnn system in the state (
S, I )=(1 / , − ), weapplied the isospin-less approach proposed in Ref. [13]. The inputs into Eq. (2) arethe following: the spin singlet nn potential V AA = v snn and V AB = diag { v snp , v tnp } that is a diagonal 2 × v snp and spin triplet v tnp np potentials, respectively, and D = ( − , √
32 ) , G = (cid:32) − − √ − √
32 12 (cid:33) , W = (cid:18) W s W t (cid:19) , U = U s , (3)where W s and W t are the spin singlet and spin triplet parts of the W component.Within the isospin formalism when the protons and neutron are identical particles,instead of Eq. (2), which is a set of three equations, one has the set of two equationsfor the state ( S, I )=(1 / , /
2) of the three nucleon system
N N N :( H + V NN − E )Φ = − V NN B ( p + + p − )Φ , (4)where V NN = diag { v sNN , v tNN } , B = (cid:18) − −
34 14 (cid:19) , Φ = (cid:18) Φ s Φ t (cid:19) and p ± are the operators of cyclical permutations for coordinates of the particles.6 I. Filikhin et al
In Eq. (1), the Faddeev component U (and W ) of the total wave-function is ex-pressed in terms of spin and isospin: U = U χ spin η isospin . The graphical representation of the spin-isospin configurations in the ΞΞ N and N N
Ξ systems is given in Fig. 1. Here, we have taken into account that the spinFigure 1:
The spin-isospin configurations in
AAB systems: a) nnp , (
S, I )= (1 / , − ), b) N N
Ξ, (
S, I )=(3 / , / N N
Ξ, (
S, I )=(1 / , / N , ( S, I )=(1 / , / α ,( S, I )=(0 , s and t ). The two-body bound states are noted by ovals. (isospin) basis of the spin (isospin) 3 / AAB system is formedby a single basis element. Thus, the Faddeev equations for each system consideredhave the form (2)-(3). The equation for the state ΞΞ α ( S, I )=(0 ,
1) has a ”scalarform” instead the form (2)-(3):( H + V AA − E ) U = − V AA (1 + p ) W , ( H + V AB − E ) W = − V AB ( U + p W ) , (5)where the V AB and W are scalars: V AB = v AB . Here, we used what the spin-isospinpart of the wave function of the fermion pair ΞΞ is antisymmetric relatively to thepermutation P in Eq. (1).Let us assume that V AA = 0, then Eq. (5) is reduced to a single equation:( H + V AB − E ) W = − V AB p W . (6) ight Ξ hypernuclei N N
Ξ, (
S, I )=(3 / , / V AA = 0 corresponds to the situation when N N potential can be neglected for the spin/isospin triplet (
S, I ) = (1 ,
1) state.The differential Eq. (6) shows that the right hand side term is attractive (includingattractive Ξ α potential) and can give additional contribution into the binding en-ergy coming from the left hand side term. The corresponding term for the N N
Ξ,(
S, I )=(3 / , /
2) state is repulsive due to symmetry of 3 / P in Eq. (1). The state N N
Ξ, (
S, I )=(3 / , /
2) is unbound[6].
In this section we consider the two-body interactions, which are the inputs to ourpresent study. To describe a nucleon-nucleon interaction, we use the semi-realisticMalfliet and Tjon MT I-III [14] potential with the modification from Ref. [15]. TheMT I-III model has the Yukawa-type form: S = 0, I = 1: V NN ( r ) = ( − . exp ( − . r ) + 1438 . exp ( − . r )) /r,S = 1, I = 0: V NN ( r ) = ( − . exp ( − . r ) + 1438 . exp ( − . r )) /r, where the strength parameters are given in MeV and range parameters are given infm − . The parameters were chosen in Ref. [14] to reproduce the experimental datafor np -scattering. It has to be noted that we do not use isospin formalism for the nnp system. Thus, the protons and neutrons are not identical. The details of suchtreatment are presented in Ref. [13]. To take into account that the nn interactionis not equivalent to np interaction (that is known as the charge dependence of N N interaction), we have made modification of the spin singlet (
S, I ) = (0 ,
1) componentof the MT I-III potential according Ref. [13] and have defined spin singlet nn potential. The modification was performed by scaling strength parameter. Thescaling parameter γ is fixed as γ =0.975 to reproduce experimental nn scatteringlength for which we used the value of -18.8 fm [16, 17]. By this way, we haveobtained three potentials v snn , v snp and v tnp needed for Eq. (2). Note that the MTI-III potential is not defined for the spin/isospin triplet ( S, I ) = (1 ,
1) and singlet(
S, I ) = (0 ,
0) states. The corresponding potentials are taken to be equal zero.The Ξ N and ΞΞ potentials simulating the ESC08c Nijmegen model are writtenin the form [7]: S = 0, I = 1: V Ξ N ( r ) = ( − . exp ( − . r ) + 155 . exp ( − . r )) /r, I. Filikhin et al S = 1, I = 0: V Ξ N ( r ) = ( − . exp ( − . r ) + 425 . exp ( − . r )) /r,S = 0, I = 1: V ΞΞ ( r ) = ( − . exp ( − . r ) + 490 . exp ( − . r )) /r. The parameters of the potentials were fixed to reproduce the scattering lengthsand effective radii given by the ESC08c Nijmegen model for the baryon-baryoninteraction [8, 9, 10].For the Ξ α interaction we use the Isle-type potential [18] which has the Gaussianform: V Ξ α ( r ) = 450 . exp ( − ( r/ . ) − . exp ( − ( r/ . ) . with parameters from Ref. [19]. . The ground state binding energies B of the N N N , nnp , N N
Ξ, ΞΞ N , ΞΞ α sys-tems were calculated using the models suggested above. The numerical results arepresented in Table 1. For each system, we show the spin-isospin state ( S, I ) andtwo-body energies B ( AA ) and B ( AB ) for AA and AB pairs. Additionally, wepresent the three-body binding energy calculated under the condition V AA = 0.Our results are compared with ones obtained within the integral representation ofRefs. [7, 11]. One can see that results of both approaches are in the agreementwith high accuracy.Table 1: Binding energy B (in MeV) calculated for various systems AAB withinthe differential (DFE) and integral Faddeev (IFE) equations. The B ( V AA = 0)is shown in brackets. Binding energy B (in MeV) for AA and AB pairs are alsopresented. Here m N =938.91 MeV, m Ξ =1318.07 MeV, m α =3727.38 MeV.System (Spin, Isospin) B ( AA ) B ( AB ) B , DFE B , IFE[7, 11] N N N (1 /
2, 1 /
2) 2.23 – 8.58 [13] – nnp (1 /
2, –) – 2.23 8.38[13] (3.40) –
N N
Ξ (3 /
2, 1 /
2) 2.23 1.67 17.205 (2.213) 17.203
N N
Ξ (1 /
2, 3 /
2) – 1.67 2.886 (1.785) 2.8855ΞΞ N (1 /
2, 3 /
2) – 1.67 4.512 (3.408) 4.5119ΞΞ α (0, 1) – 2.09 7.635 (4.335) –The ”spin/isospin complication” [20] of the Faddeev equations for the consideredsystems is appeared by the matrix form of Eq. (3) and leads to the following ight Ξ hypernuclei N N
Ξ system in the spin-isospinstates (
S, I )=(3 / , / / , / B ( AB ) < B ( V AA = 0) < B ( AB ) . (7)The value of B ( V AA = 0) is restricted by 3.34 MeV. The similar result we have forthe nnp system. In this case, B ( V AA = 0) is restricted by 4.46 MeV. In contrast,the scalar form (5) of Eq. (2) for the case ΞΞ α ( S, I )=(0 ,
1) leads to the relation: B ( V AA = 0) > B ( AB ) . (8)This relation is known as the mass polarization effect which takes place when m B /m A > S, I )=(0 ,
1) of ΞΞ α system,the mass polarization energy can be evaluated [20]. The contribution of this en-ergy ( B ( V AA = 0) − B ( AB )) /B ( V AA = 0) in the three-body bound energy isequal 3.6% that is compatible with the values of 2%-4% [22, 20] for the similarnuclear system ΛΛ α . The similarity takes place due to approximate equality ofthe masses of non-identical particles: m B /m A ∼ α and m B /m A ∼ α . In the limit m B /m A >> B ( V AA = 0) = 2 B ( AB ). The case m B /m A < N (1 / , / B ( V AA = 0) and2 B ( AB ) for the case. One can define the incremental binding energy ∆ B ΞΞ forthe system ΞΞ N (1 / , /
2) as ∆ B ΞΞ = B − B (Ξ N ) according the analoguewith the He hypernucleus. For
He, the incremental binding energy is definedas ∆ B ΛΛ = B ΛΛ ( A ΛΛ Z ) − B Λ ( ( A − Z ) [21, 22]. Calculating the incremental energy,one can evaluate the strength of the ΛΛ interaction. For the system ΞΞ N (1 / , / m N /m Ξ ∼
1. Regardless that the relation (7) is not satisfied, the more appropriatevalue for an evaluation of the strength of the ΞΞ interaction in ΞΞ N (1 / , /
2) isthe value of B − B ( V ΞΞ = 0). The corresponding evaluation can be obtained fromTable 1. The ΞΞ interaction is attractive in ΞΞ N (1 / , / N N interaction is also attractive in the
N N
Ξ (1 / , /
2) system. Theseattractive forces add about 1 Mev to the binding energies of the mirror systems.Thus, the matrix elements < Ψ | V AA | Ψ > have the close values for the systems. Itis possible, because the ΞΞ N system is more compact (larger B value) and the ΞΞpotential has a minimum closer to the origin as is shown in Fig. 2. The Faddeevcomponents U , W for the N N
Ξ (1 / , /
2) and ΞΞ N (1 / , /
2) systems are pre-sented in Fig. 3. From the figure, one can see that the system ΞΞ N (1 / , /
2) ismore compact than the
N N
Ξ (1 / , /
2) system. For both systems, the rearrange-ment channel A + ( AB ) dominates due to existence of the isospin singlet Ξ N boundstate.The mirror N N
Ξ (1 / , /
2) and ΞΞ N (1 / , /
2) systems under the condition V AA = 0 can be transformed ”one into another” by changing the particle masses.The parameter ξ ≥ m ξA = (1 + ξ ) m A , m ξB = (1 − ξm A /m B ) m B . The results of calculations for 2 E and E ( V AA = 0)0 I. Filikhin et al
Figure 2:
The
N N ( S, I )=(0 ,
1) and ΞΞ (
S, I )=(0 ,
1) potentials. as a function of ξ are shown in Fig. 4a). The transformation N N
Ξ (1 / , /
2) toΞΞ N (1 / , /
2) replaces the ratio m B /m A > m B /m A <
1. Onecan see that the relation (7) is well satisfied up to ξ =0.2 when m ξB /m ξA ≥
1. Weconclude that the relation (7) is not guaranteed when m ξB /m ξA (cid:28)
1. The affect ofthe AB potential on the relation (7) is obvious. To show this we have repeated thecalculations for more deep spin triplet N Ξ potential. The potential has been scaledby the factor of 1.05. The result is shown in Fig. 4b). The relation (7) is satisfiedfor all possible values ξ for this case.It has to be noted that, as follows from Table 1 for the N N
Ξ (3 / , /
2) state,the three-body system having two bound subsystems has a deep bound state. Thevalue of this
N N
Ξ (3 / , /
2) binding energy is related with two-body energies as B >> B ( AB ) + B ( AA ). Obviously, the V AA potential plays a key role forformation of the bound state. We assume that it is a general property of suchthree-body systems. . We studied the hypernuclear system
N N
Ξ (and ΞΞ N ) based on the configuration-space Faddeev equations. The baryon-baryon potential of ESC08c model, whichgenerates the Ξ N ( S, I ) = (1 , s -wave bound state, results in the stable states forthese three-body systems. The stability relatively N Ξ → ΛΛ conversion is providedby fixing the states with maximal isospin. Our results and ones obtained within theintegral Faddeev equation formalism [7, 11] are in agreement with high accuracy.Additionally, we have calculated the binding energy of the ΞΞ α ( S, I ) = (0 ,
1) state.The relations between B and B ( V AA = 0) were proposed for the ”spin/isospincomplicated” and ”scalar” states. The corresponding relations are significantlydifferent. ight Ξ hypernuclei The contour plots of the Faddeev components U a) and W b),c) for the N N
Ξ (1 / , /
2) (Left) and ΞΞ N (1 / , /
2) (Right) bound states. The Jacobi coordinatescorresponding to the components U and W are presented as x , y x , y I. Filikhin et al
Figure 4:
The transformation
N N
Ξ (1 / , /
2) to ΞΞ N (1 / , /
2) when V AA = 0.The 2 E (solid line) and E ( V AA = 0) (dashed line) as a function of ξ are shown. Theparameter ξ is related to the N N
Ξ (1 / , /
2) system, when ξ =0, and - to the ΞΞ N (1 / , /
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