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From "Super-Hypernuclei" to Strange Stars

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From EG0600173 21"1 Conference on Nuclear and Particle Physics, 13 -17 Nov. 1999, Cairo, Egypt From "Super-Hypernuclei" to Strange Stars Hidezumi Terazawa Institute of Particle and Nuclear Studies, HEARO (Tanashi) Midori-cho, Tanashi, Tokyo 188-8501, Japan Abstract Both theoretical investigations of and experimental searches for not only super-hypernuclei (or "long lived hyperstrange multiquark droplets", "strangelets ", or "strange quark matter") consisting of roughly equal numbers of up, down, and strange quarks, but also super-hypernuclear matter in bulk (or "quark nuggets" or "strange matter") (in the early Universe or inside neutron stars) and strange stars made of super-hypernuclear matter are reviewed and discussed in some detail. The contents include: I. Introduction II. Quark-Shell Model of Nuclei III. Nuclear Mass Formula in the Quark-Shell Model IV. Mass Spectrum of Super-Hypernuclei V. Charge-to-Mass-Number Ratio of Super-Hypernuclei VI. Super-Hypernuclei or Other Exotic Nuclei in Cosmic Rays VII. Strange Stars in the Universe VIII. Conclusion, Further Discussions, and Future Prospects. 28 I. Introduction A "super-hypernucleus" is a nucleus which consists of many strange quarks as well as up and down quarks. In 1979 [1], I proposed the quark-shell model of nuclei in quantum chromodynamics (QCD) [2], presented the effective two-body potential be- tween quarks in a nucleus, pointed out violent breakdown of isospin-invariance and importance of U-spin invariance in superheavy nuclei, and predicted possible creation of super-hypernuclei in heavy-ion collisions at high energies, based on the natural expectation that not only the Fermi energy but also the Coulomb repulsive energy is reduced in such nuclei. A similar idea was presented independently and almost simultaneously by Chin and Kerman [3], who called super-hypernuclei "long lived hy- perstrange multiquark droplets". Five years later in 1984, the possible creation of such super-hypernuclear matter in bulk (in a much larger scale of both mass number and space size) .in the early Universe or inside neutron stars was discussed in detail in QCD by Witten [4], who called super-hypernuclear matter "quark nuggets" while the properties of super-hypernuclei were investigated in detail in the Fermi-gas model by Farhi and Jaffe [5], who called super-hypernuclei "strange matter". In 1988 [6], the CERN SPS opened up a new energy range of order hundred GeV/nucleon for ion collisions, which might be high enough to produce super-hypernuclei. It seemed, therefore, worthwhile to reinvestigate theoretically in more details various properties of super-hypernuclei. In a series of papers published in 1989 and 1990 [7-9], I reported an important part of the results of my investigation on the mass spectrum and other prop- erties of super-hypernuclei in the quark-shell model. Now in 1999, since the BNL RHIC is about to open up an even much higher energy range of order hundred GeV/nucleon for heavy-ion-heavy-ion colliding beams, it seems more relevant to review both theo- retical investigations of and experimental searches for not only super-hypernuclei (or "long lived hyperstrange multiquark droplets", "strangelets", or "strange quark mat- ter") consisting of roughly equal numbers of up, down, and strange quarks, but also super-hypernuclear matter in bulk (or "quark nuggets" or "strange matter") (in the early Universe or inside neutron stars) and strange stars made of super-hypernuclear matter, which I am going to do in this talk. Before doing that, let me make the following short but key note: Suppose that a super-hypernucleus consists of Nu up quarks (u's) Nj down quarks (GTS), and Ns strange quarks (s's). Then, a nucleus of (Nu, Nd, Ns) has the atomic number (or electric charge in the unit of e), the mass number (or baryon number), and the strangeness given by Z = (1/3)(2NU -Nd- N,),A = (1/3)(JVU + Nd + Ns), and S = -N3. By noting the existing similarity between the effective potential for the nuclear force and that for the quark force and the additional three color-degrees of freedom [10], I have predicted that the magic numbers in the quark-shell model are three times the famous magic numbers in the nucleon-shell model (Z, A — Z = 2,8,20,28,50,82,126, • • •), which are Nu, Nd, Ns = 6,24,60,84,150,246,378, • • •. Therefore, the magic nuclei such as ^He and J f 0 are doubly magic and superstable also in the quark-shell model since Nu — Nd = 6 1 for jHe and Nv = N4 = 24 for f0. What is new in the quark-shell model is the expectation that not only certain exotic nuclei with a single magic number such as 29 ++++ the "dideltas", D8 with Nu = 6 and D8~~ with Nd = 6, but also certain super- hypernuclei with a triple magic number such as the "hexalambda", H\ with Nu = Nd = Ns = 6 [11], and the "vigintiquattuoralambda", VqX with Nu = Nd = Na = 24, may appear as quasi-stable nuclei. In fact, in the MIT bag model [12], one can easily estimate the mass of HX to be as small as 6.3 GeV [8], which is smaller than 6mA (= 6.7 GeV). However, in the quark-shell model, there is no qualitative reason why the "dihyperon" or "H dibaryon", H with Nu = Nd = Ns = 2 [13], should be quasi-stable or even stable. I would like to organize this talk in the following way: II. Quark-Shell Model of Nuclei, III. Nuclear Mass Formula in the Quark-Shell Model, IV. Mass Spectrum of Super-Hypernuclei, V. Charge-to-Mass-Number Ratio of Super-Hypernuclei, VI. Super-Hypernuclei or Other Exotic Nuclei in Cosmic Rays, VII. Strange Stars in the Universe, VIII. Conclusion, Further Discussions, and Future Prospects. II. Quark-Shell Model of Nuclei [7] Let us now discuss the quark-shell model in QCD. In QCD [2], the strong force between quarks is originated from the Yang-Mills interactions [14] of the color-triplet quarks q with the color-octet vector gluons Ga(a = 1—8) and of gluons with themselves, LY-M = gqY^qGl - \id.Gl - dvGl + gr^Glf + \id.Gl - dvGfi\ (1) where g is the gluon coupling constant, A°'s are the Gell-Mann's color SU(3) matrices, and /a6c's are the SU(3) structure constants. This interaction Lagrangian approxi- mately and effectively gives in non-relativistic limit the two-body Coulomb potential for the strong interaction between quarks due to the one-gluon exchange, with where A,a/2(z = 1,2) are the quark color-spins and g(fi) is the effective or running gluon coupling constant at the renormalization energy point fi defined in the familiar renormalization group approach [15]. For the three flavors of quarks (u, d, and s), the running gluon coupling constant can be phenomenologically parameterized by [16] with 30 A £ 200 -300 MeV [17]. (5) Furthermore, the renormalization point fj. may be related, though not rigorously, with the distance between the two quarks r by 1 (6) This Coulomb potential is, however, a good approximation only for the strong inter- action between quarks at a short distance (r <$; 1 fermi) where the effective coupling constant is small (as(v) ^ !)• At a long distance (r,>l fermi), the strong interac- tion can be caused by the Nambu-Susskind string [18] spanned between two quarks, probably due to exchange of an infinite number of gluons. It may be, then, simply approximated by the Nambu-Susskind linear potential, for example, for a color singlet state of a quark and an antiquark, VNS(T) = Ar + B. (7) By analyzing especially the data on the Regge recurrence of hadrons, the parameters A and B have been determined to be [19] 2 2 A 2 0.2 GeV and B £ 0.0 GeV . (8) The effective two-body potential between quarks in a nucleus, however, can not be approximated simply by the sum of these two potentials (2) and (7). It is not only because at a long distance the attractive Coulomb force becomes dominant between a color-triplet quark and a color-antitriplet diquark which is made of two nearby quarks in the nucleus but also because at a longer distance the Nambu-Susskind string can be formed between these quark and diquark. For an even longer distance the string formation would be disturbed and eventually destroyed by many other quarks located close to the diquark. In order to obtain a better approximation, one should also include spin-dependent and velocity-dependent forces in the Coulomb potential which have been given by De Riijula, Georgi, and Glashow [20] in QCD. Taking care of these things together, I therefore have proposed to use the following potential for the effective two-body force between quarks in a nucleus: TT " "T") SD-G-G[T) for r<0.\ fermi, 1 y • y ) SD-G.a{r) for 0.1 fermi <r<0.3 fermi, (9) 1 y • y J (AT + B) oxp (-K,T) for r>Q.S fermi, with 1 1 [Pi • P2 (r-Pi)(r-pa)| r J --8\v) \— + — + 16(S1'S2) 2 \m\ m\ 31 H 2(r x pi) • s2 - 2(r x p2) • Si - 2si • s2 r2 where m's, p's, and s's are the quark masses, momenta, and spins, respectively, \^z/2 is the color-spin of the diquark state, and K is the damping parameter. The magnitude of K"1 should be of order of the average distance between quarks inside a nucleus so that K-1 £ 0.3-0.4 fermi . (11) Note that there exists a close similarity between this effective potential for the quark force in a nucleus and the Hamada-Johnston potential for the nuclear force except for their difference lying in the scales of distance and energy.
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