BIHEP-TH-97-50 H-DIHYPERON IN QUARK CLUSTER MODEL 1 P.N.Shena,b,c, Z.Y.Zhanga, Y.W.Yua, X.Q.Yuana, S.Yanga 9 9 9 1 a. Institute of High Energy Physics, Chinese Academy of Sciences, P.O.Box 918(4), n a Beijing 100039, China J 4 2 b. China Center of Advanced Science and Technology (World Laboratory), 1 P.O.Box 8730, Beijing 100080, China v 8 6 c. Institute of Theoretical Physics, Chinese Academy of Sciences, P.O.Box 2735, 0 1 Beijing 100080, China 0 9 9 / h t - l c u Abstract n The H dihyperon (DH) is studied in the framework of the SU(3) chiral quark model. It : v is shown that except the σ chiral field, the overall effect of the other SU(3) chiral fields is i destructive in forming a stable DH. The resultant mass of DH in a three coupled channel X calculation is ranged from 2225 MeV to 2234 MeV. r a 1This work was partly supported bythe National Natural Science Foundation of China In 1977, Jaffe predicted DH [1], a six-quark state with strangeness (s) being -2 and JP = 0+(S = 0,T = 0), by using a simple color magnetic interaction in the MIT bag model. Since then, many theoretical [2, 4, 3, 5, 6, 7] and experimental [8] (and references therein) efforts have been devoted to the DH study. There were so many theoretical predictions of DH which are quite different in different models. For instance, by using the MIT bag model, Jaffe gave a bindingenergy of about80MeV below thetwo Λthreshold (E ) [1]; in terms of ΛΛ the Skyrme model, Balachandran et al. even showed a larger binding energy of about several hundred MeV [2]; by employing the cluster model, Yazaki et al. predicted the energy of DH from about 10MeV above E to about 10MeV below E [4, 14]; also, by using the cluster ΛΛ ΛΛ model but different interaction with Yazaki’s, Straub et al. announced a binding energy around 20MeV [3]; in terms of Quantum Chromodynamics(QCD) sum rules, Kodama et al. gave a binding energy around 40MeV although the error bar was quite large; considering mutually the two-cluster and six-quark cluster configurations, Wolfe et al. obtained a deeply bound DH with a binding energy about several hundred MeV [6]; and by employing a quark model without the one-gluon-exchange interaction (OGE), Glozman et al. announced the non-existence of a bound DH [7]. On the other hand, there is no experimental evidence showing the existence of DH up to now. The provided lower limit of the DH mass is about 2200MeV [8, 15]. Thereasonforcarryingoutsuchresearchesisstraightforward. Accordingtothefeatureof the color magnetic interaction (CMI) in the one-gluon-exchange potential (OGE), when the strangeness of the system concerned is equal to -2, S=0 and (λµ) = (0 0), the expectation f value of CMI presents more attractive feature than those contributed by two Λ baryons. As a consequence, six quarks could be squeezed in a small region, the typical short-range QCD behavior would be demonstrated and some new physics might be revealed. Therefore, studying DH will be rather significant in understanding the quark characteristics of the wave function of the multiquark system and the short-range behavior of the QCD theory. On the other hand, till now, there still exist some uncertainties in the nucleon-hyperon (NY) interaction on the baryon-meson degrees of freedom, especially in the short-range part, so as the hyperon-hyperon (YY) interaction and the prediction of DH. Therefore, the next generation, the quark-gluon degrees of freedom, may bea more effective base to establish the relations among the nucleon-nucleon (NN), NY and YY interactions, and consequently to give a morereliable prediction of DH. As well known, mostnuclear phenomenaare justof the low energy approximation of QCD. There exist lots of nonperturbative QCD (NPQCD) effects. Unfortunately, nowadays one still cannot solve NPQCD directly, and has to employs certainQCDinspiredmodels. TheSU(3)chiralquarkmodelisjustoneofthemostsuccessful 1 ones. In that model, the couplings between chiral fields and quark fields were introduced to describe the short- and medium-range NPQCD effects, and a more reasonable quark-quark interaction V (q = u,d, and s)[11] was obtained. With that V , one could mutually q−q q−q describe the experimental NN scattering phase shifts, available YN scattering cross sections [11] and some properties of single baryons including the empirical masses of single baryon ground states [12, 13]. Extrapolating that model to the s = 2 system, one can study the − YY interaction, double strangeness hypernuclei, DH and etc.. In this letter, we would choose DH as a target, because it is a simplest case with two strangeness and its structure is relative simple so that the interaction can be preliminarily examined by the present experimental finding although it is only a lower limit of the DH mass, and the short-range behavior can be revealed. It is clear that both the DH structure and V would affect the theoretical prediction of q−q the DH mass. Here, we first briefly introduce the employed interaction. The Hamiltonian of a six-quark system in the SU(3) chiral quark model reads H = T + (VCONF + VOGE + VPS + VS), (1) ij ij ij ij i<j X where T denotes the kinetic energy operator of the system and VCONF, VOGE, VPS and VS ij ij ij ij represent the confinement, one-gluon exchange, pseudo-scalar chiral field induced and scalar chiralfieldinducedpotentialsbetweenthei-thandj-thquarks,respectively. Theconfinement potential is phenomenologically taken as VCONF = (λaλa) (a +a r2), (2) ij i j c 0 ij ij ij − whichdescribesthelongrangenonperturbativeQCDeffect. Theintroduceda termswhich 0 ij takedifferentvaluesfordifferentq qpairsarecalledzero-pointenergyterms. Theyguarantee − that theempirical thresholdsof considered channels can more accurately bereproduced. The shortrangeperturbativeOGEpotentialischosentobethecommonlyusedform[9,10,11,12]. In order to restore the important symmetry of strong interaction, the chiral symmetry, we introduce SU(3) chiral fields coupling to quark fields so that the medium-range NPQCD effects can be described [9, 10, 11]. The pseudoscalar-field-induced potentials are: m2 VPS = C(g , m , Λ) πa f (m , Λ, r ) (σ~ σ~ ) (λaλa) , (3) ij ch πa 12m m · 1 πa ij i· j · i j f i j 2 and the scalar-field-induced potentials are: VS = C(g , m , Λ) f (m , Λ, r ) (λaλa) , (4) ij − ch σa · 2 σa ij · i j f where the subscript f denotes that the operators in parentheses are in flavor space. The expressions of f , Y, and C are shown in Ref.[11]. To retain the important chiral symmetry i as much as possible, we take all chiral-quark coupling constants to be the same value gc2h = 9 gN2Nπ mq 2. (5) 4π 25 4π M N (cid:16) (cid:17) In Eq.(3), π with (a= 1,2,...,8, and 0) correspond to the pseudoscalar fields π, K, η and a 8 η , respectively, and η and η′ are the linear combinations of η and η with the mixing angle 0 0 8 θ. In Eq.(4), σ with (a = 1,2,...,8, and 0) correspond to the scalar fields σ′, κ, ǫ and σ, a respectively. Insolvingthissix-quark systemproblem,thefirstselected setofmodelparametersis that used in Ref.[11]. This is because that with this set of parameters, almost all empirical partial wave phase shifts of the N N scattering can be well re-produced, meanwhile the available − cross sections of N Y processes can reasonably beexplained, some masses of baryon ground − states can accurately be obtained, and some properties of baryons such as EM transition rates and etc. can better be understood [13, 12]. Thus, the predicted mass of DH is based on a more solid ground, and the reliability of the prediction is increased. Furthermore, the possiblerangeoftheDHmasscanbetestedbyshiftingthevaluesofmodelparameterswithin reasonable regions. Then, we show how to choose the model space in solving the bound state problem of a six-quark system. There are two types of possible configurations in studying the structure of DH. (1) Six-quark cluster configuration. In this configuration, the trial wave function can be expressed as the linear combination of differently sized basis functions: Ψ = C Φ (ω ), (6) (λµ)f T S i (λµ)f T S i i X with the basis function Φ (ω ) = φ [ (0s)6, ω ] χfσ χc , (7) (λµ)f T S i i (λµ)f (00) 3 where φ [ (0s)6, ω ] is the orbital wave function (ω = 1 ), and χfσ and χc denote the i mb2 (λµ)f (00) wave functions in the flavor-spin and color spaces, respectively. This trial wave function is in the pure symmetry-basis-function space. In this configuration space, (0s)6 configurations which describe the breath mode are considered only. (2) Two-cluster configuration. In this configuration, there exist three possible channels: ΛΛ, NΞ and ΣΣ. In the frame- work of Resonating Group Method (RGM), the trial wave function of DH can be written as Ψ = α Λ Λ + β N Ξ + γ Σ Σ , (8) | i | i | i with the two-cluster wave function B B = [ φ φ χ ] , (9) | 1 2 i A B1 B2 rel Rcm ST=00 where stands for the antisymmetrizer, φ denotes the wave function of the cluster B A 1 (2) B , χ represents the trial wave function of the relative motion between clusters B and 1 (2) rel 1 B and is the wave function of the total center of mass motion. This trial wave function 2 cm R is in the physics-basis-function space. The physical picture of this configuration is that in the compound region of two-interacting clusters (or composite particles), there might exist a bound state or a resonance. Let us define a quantity E = M M , (10) H H ΛΛ − where M and M denote the mass of DH and two Λ’s, respectively. Apparently, E < 0 H ΛΛ H stands for a stable DH against weak decay. E (or M ) can be obtained by solving the H H Schro¨dinger equation in which the above mentioned potentials are employed. The results with different configurations are discussed in the following. (1) Six-quark cluster case. In this case, ω are taken as variational parameters. By using the variational method, one i can minimize the Hamiltonian matrix element with respect to ω . The resultant masses of i DH are tabulated in Table I. 4 Table I. E (MeV)† in the six-quark cluster case H VOGE + VCONF 311 VOGE + VCONF 127 +Vπ + Vσ VOGE + VCONF 276 +VPS + VS Parameters used are those in ref.[11]. † It is shown that if one only employ OGE and confinement potentials, the mass of six-quark cluster is 311MeV heavier than those of two Λ’s, M . When one additionally employs Vπ ΛΛ and Vσ, the mass of DH would decrease, but it still 127MeV heavier than M . However, ΛΛ when one includes all SU(3) chiral pseudoscalar and scalar fields, namely employs the other chiral fields in additional to the π and σ fields, the corresponding mass becomes larger again, which is 276MeV heavier than M . This means that the couplings between σ chiral field ΛΛ and quark fields cause additional attraction, which is helpful to reduce the mass of the six- quark system. On the contrary, the overall effect of the contributions from other SU(3) chiral fields provides a repulsive feature so that DH is quite hard to form. Moreover, no matter in which case, the mass of DH is heavier than M , namely DH is not a stable particle against ΛΛ to strong decay to ΛΛ and/or NΞ. Therefore, a model space with (0s)6 six-quark cluster structure only may not be a favored model space in studying the DH structure. (2) Two-cluster case. In RGM, to solve the bound state problem, one usually expands the unknown relative wave function χ by using locally peaked Gaussian basis functions rel χ = c χ , (11) rel i i i X where c ’s are variational parameters [17]. i Due to the Pauli principle, there exists a forbidden degree in the six-quark trial function [16] 1 1 Ψforbidden = ΛΛ + NΞ + ΣΣ . A √3 | i √3 | i | i (cid:16) (cid:17) This forbidden degree and almost forbidden degrees can be detected by examining the zero andalmostzeroeigenvaluesofthenormalizationkernel,respectively[18]. Inparticular,inthe 5 bound state RGM calculation, a component with the inter-cluster distance to be zero in the trial wave function, which is just a six-quark cluster configuration with the [6] symmetry, has to be included so that the behaviors of two clusters at the shorter inter-cluster distance can be well described and the stable and reliable solutions can be obtained. As a side-effect, the disturbances from the forbidden and almost forbidden degrees become serious. Sometimes, these disturbances would spoil the numerical calculation and produce non-physical results. Therefore, only after all the non-physical degrees are completely eliminated, the resultant energy of the bound state can be trusted. Due to the aforesaid reasons, at this moment, it may not be necessary to further carry out the mixing of the configurations (1) and (2). Moreover, in practice, eliminating the non-physical degree can be realized by performing the off-shell transformation. Then, carrying out the variational procedure, one can obtain the mass of DH or E . The resultant E ’s are tabulated in Table II. H H Table II. E (MeV) and (fm) in the two-cluster case†. H R |ΛΛi |ΛΛi ΛΛ + |NΞi | i + |NΞi (cid:18) (cid:19) + |ΣΣi ! (cid:0) (cid:1) E 9.41 8.95 6.77 H VOGE + VCONF 1.89 1.85 1.68 R E 5.47 -38.71 -65.80 H VOGE + VCONF +Vπ + Vσ 1.66 0.75 0.72 R E 4.39 4.02 1.98 H VOGE + VCONF +VPS + VS 1.59 1.57 1.41 R Parameters used are those in ref.[11] and R denotes theroot-mean-squared radius of DH. † It is shown that in the first(VOGE+VCONF) and third (VOGE+VCONF+VPS+VS) cases, the results do not supporta bound DH. In the second case (VOGE+VCONF +Vπ+Vσ), the 6 one-channel calculation ( ΛΛ ) result(E = 5.47MeV) also does not supporta boundstate, H | i but, the two- channel ( ΛΛ + NΞ ) and three-channel ( ΛΛ + NΞ + ΣΣ ) calculations | i | i | i | i | i showaboundstatewiththebindingenergiesof38.71MeV and65.80MeV, respectively. These results can roughly be explained by the interaction matrix elements. The contribution from OGE shows repulsive feature in the ΛΛ channel and attractive features in both NΞ and ΣΣ channels, and the net contribution from π and σ fields at the short distance demonstrates the more attractive character in the NΞ and ΣΣ channels than that in the ΛΛ channel. However, due to the existences of two strange quarks in the DH system, the chiral clouds with strangeness surrounding interacting baryons become important. Thus, in our opinion, all the SU(3) chiral fields should be considered. In fact, after including these fields, the aforesaid over-strong attraction disappears, and the results in all cases become smooth. This phenomenon can be understood by the matrix elements of the spin-flavor-color operators of chiral fields, namely sfc coefficients, which show that the contributions from σ and η 0 present the attractive character and those from the other mesons, i.e., π, K, η , σ′, κ and ε, 8 are repulsive. Moreover, in the third case, although all coupled channel calculations do not show a bound DH, in comparison with the result in the six-quark cluster case, the inclusion of additional channel would reduce the mass of DH. Eventually, the resultant mass of DH is around the ΛΛ threshold. To further understand the calculated E , we also list the corresponding root-mean- H squared radii ( ) of DH in different cases in Table II. These numbers indicate that in R the two- and three-channel calculations, if one considers OGE, π and σ only, the root-mean- squared radii of DH are 0.75fm and 0.72fm for the two and three coupled channel cases, respectively. Therefore, DH is a bound state. In all the other cases, the resultant values R are greater than 1.4fm, thus DH is no longer bound. How thevalues ofmajor modelparameters affect theresultantmassof DHisalso studied. Three coupled channel results are used as samples to demonstrate these effects. When the width parameter b increases, the corresponding E value changes to a smaller value. De- H creasing the mass of σ, would make the DH mass lighter. Moreover, a smaller s quark mass m corresponds to a lower DH mass, and decreasing the mixing angle θ would just lower the s DH mass in a very small amount. Among these parameters, the mass of σ would give the biggest effect on E . H Finally, we give the possible mass range of DH. As mentioned above, the NN scattering phase shifts and the NY scattering cross sections as well as the mass of DH depend on the 7 values of model parameters. Our calculation showed that except the set of model parameters used in Ref.[11], another set of model parameters, say b = 0.53fm, m = 600MeV, m = σ s 430MeV, θ = 23◦, ΛPS, σ′ = 987MeV, Λκ, ǫ, σ′ = 1381MeV, which are almost the − values in limits, can also fit the experimental NN and NY data [19]. With this set of model parameters,theresultantE is 6.9MeV. Furtherextendingtheseparametervaluestotheir H − physicallimits withwhichtheempirical NN andNY datacannoteven beenreproduced,one can obtain the upper and lower bounds of the DH mass. When b = 0.6fm, m = 550MeV, σ the lower bound of E is 9.1MeV. On the other direction, if one takes b = 0.48fm, m = H σ − 675MeV, the upper bound of DH is around 4.9MeV. Fromabovecalculations, onefindsthatintheframeworkofourSU(3)chiral-quarkmodel, as long as one picks up a set of model parameters which satisfy the stability conditions, the masses of the ground states of baryons and meanwhile can be used to fit the experimental NN and NY scattering data, the resultant mass of DH would be rather stable and would be ranged in a very small region. This mass is consistent with the present experimental finding and reflects that the SU(3) chiral-quark model is reasonable. Asasummery,onemayhavefollowingconclusions. Thesix-quarksystemwithstrangeness being -2, JP = 0+ (S=0, T=0) is studied in two possible model spaces. One is in a six-quark cluster configuration space with breath mode, and the other is in a two-cluster configuration space with three possible channels. It is shown that the (0s)6 model space is not larger enough even the breath mode is considered. Therefore, the mass of DH in this model space is generally 100 300 MeV heavier than that in the two-cluster model space. The similar ∼ result also appears in the other six-quark system calculation, say Deltaron (d∗) [10]. In the two-cluster configurationcase, theresultshowsthatthemassofDHisrangedfrom2225MeV to 2234MeV if the experimental NN and NY data should simultaneously be reproduced. It seems that all the SU(3) chiral fields must be considered in studying DH. In this case, the SU(3) chiral fields surroundthe baryons and make the baryons more stable and independent. 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