BLED WORKSHOPS IN PHYSICS VOL. 1, NO. 1 Proceedins of the Mini-Workshop Few-Quark Problems (p. 56) Bled, Slovenia, July 8-15, 2000 NJL Model and the Nuclear Tightrope Steve A. Moszkowski? UCLA, Los Angeles, USA 1 Introduction Tightrope is balancing act. There are actually two aspects to this: I. Large two nucleon scattering lengths and II. Small Nuclear Binding Energies relative to Rest Energy. Both of these were known since the 1930’s. However, the NJL Model can help to get more basic understanding. 2 Large Two Nucleon Scattering Lengths Large scattering length = small binding (or antibinding). For T=0, S=1 (d), we get binding = 2.22 MeV, a = 5.4 fm , while for T=1, S=0 (pp), we get antibinding = 0.1 MeV, a = -23 fm. Clearly, it requires only a slight change in the potential to get zero binding. Splitting (to both sides of tightrope!) is due to spin-dependence. Without it we would not be here! But its role in quark-nuclear physics is unclear. Neglect spin- dependence for now. 3 Scalar Meson Exchange with NJL Model For a review of the NJL model, see Klevansky [1] and Vogl andWeise [2]. We will not discuss the model here, but only mention two important consequences for the Sigma (Scalar Meson) Exchange Interaction: 1. The mass of the sigma is: m = 2mq = 2M=N (1) so that the q- q̄ forms a state with zero binding relative to the constituent quarks. (This is if we neglect any explicit chiral symmetry breaking, which means that the current quark mass, and thus also the pion mass, is neglected.) 2. The strength of the equivalent Yukawa interaction is:g24  N ; (2)? E-mail: stevemos@ucla.edu NJL Model and the Nuclear Tightrope 57 (provided the NJL Cutoff is atm) This is not far from the strength required to get a = 1 and Nb = N -12 deeply bound states. Some other points: OPEP (with empirical pion mass) gives only 30 percent of binding. We need a repulsion to get rid of deeply bound states. Goldstone-Boson exchange can lead to such a repulsion, see Bartz and Stancu [3], though it is not the only possible explanation. 4 Small Nuclear Binding Energies Relative to Rest Energy 4.1 Known Results BE/A of nuclei ranges up to 8.5 MeV. BE/A of nuclear matter  16 MeV. Rest Energy/A = 938 MeV. Binding energies are only about 1 percent of rest energies! 4.2 NJL Model For Nuclear Matter We are actually describing quark matter. There is no confinement or quark clus- tering in the NJL model. Consider first a toy model in two dimensions.T = g 2 for small  (3)W = -(g- g )2 (4) This expression forW applies for  up to the valuewheremq = 0. g is the critical value of g necessary to just give two body binding in 2D.W = 11=2 - 2 + 3-1 for larger (5) We get saturation, but with zero quark mass! For a more realistic model in three dimensions, the calculations are more compli- cated, but one still gets saturation with zero quark mass, similar to 2D. 4.3 Generalized NJL Model (With J. da Providencia) Assume q - q̄ coupling gets stronger with density:gs = g[1-b(g-g )22℄ This still preserves chiral symmetry (with dependence on2). Effective scalar coupling gs = (b+1)(g-g ) but we need vector meson with coupl. gv = b(g-g ) to get same low  result. We (somewhat arbitrarily) identify b with b = Nb = N -12 We can solve the Generalized NJL model numerically. Note that the correction opposes chiral symmetry restoration. 58 S. A. Moszkowski We can make a low density expansion. For energy per particle and neglect- ing all kinetic energies: Wm  -g2 + bg22 + ::: (6)WM  - g2N + (N - 1)g222N + ::: (7) Here m denotes the constituent quark mass andM = N m the nucleon mass. For the effective mass, which is the ratio of either mass in the medium to that in free space, we have: m = 1 - g + ::: (8)g0 = 12(N - 1) = (1 -m) (9) Apart from kinetic energies, the saturation energy per nucleon is:W0M = - 18N (N - 1) (10) For N = 3;W0 = -20MeV (CLOSE to empirical value!) 4.4 Connection with RelativisticMean Field Theory at Low Density In the relativistic mean field approach, the nuclear matter energy per particle, (neglecting kinetic energy) is given by:W(m; ̂)M = m - 1 + Bv ̂2 + (1 -m)22Bŝm s (11) Herem denotes the effectivemass in units of the free nucleonmass. TheWalecka and Zimanyi-Moszkowski derivative coupling models [4] correspond to s = 0; 2 respectively. If B 1, then Bv  Bs  B. We then obtain, for small densities:m = 1 - B̂ + ::: (12)WM = sB22 (-2̂ + ̂2) + ::: (13) Comparing the effective mass, with that from the generalized NJL model, we see that: B = 12(N - 1) (14)W0 = -M sB22 = -M s8(N - 1)2 (15) For s = 1, we reproduce the results of the generalized NJL model, at least for large N . This is intermediate between the original Walecka model and the derivative coupling model and is close to the hybrid model used by Glenden- ning, Weber and S.M. [5]. Of course, the mean field models, unlike the general- ized NJL model, lead to finite energies at all densities, but the GNJL is slightly less phenomenological. NJL Model and the Nuclear Tightrope 59 5 Open Problems NJL is like a quark shell model, see Petry et. al. [6] and Talmi [7]. How to include effect of quark clustering, without losing NJL simplifications? Relation of Effective Vector repulsion to short range correlations? Can Goldstone Boson Exchange do the job, or do we need non-localities, as in Moscow potential? Where does the density dependence of gs come from? References 1. S. P. Klevansky, Rev. Mod. Phys. 64 (1992) 649. 2. U. Vogl, W. Weise, Prog. Part. Nucl. Phys. 27 (1991) 195. 3. D.Bartz, Fl. Stancu, Phys. Rev. C 59 (1999) 1756. 4. J. Zimanyi, S. A. Moszkowski, Phys. Rev. C 42 (1990) 1416. 5. N. K. Glendenning, F. Weber, S. A. Moszkowski, Phys. Rev. C 45 (1992) 844. 6. H. R. Petry et al., Phys. Lett. B 159 (1985) 363. 7. I. Talmi, Phys. Lett. B 205 (1988) 140.