BLED WORKSHOPS IN PHYSICS VOL. 1, NO. 1 Proceedins of the Mini-Workshop Few-Quark Problems (p. 82) Bled, Slovenia, July 8-15, 2000 Nucleon-Nucleon Scattering in a Chiral Constituent Quark Model Floarea Stancu? Institute of Physics, B.5, University of Liege, Sart Tilman, B-4000 Liege 1, Belgium Abstract. We study the nucleon-nucleon interaction in the chiral constituent quark model of Refs. [1,2] by using the resonating group method, convenient for treating the interac- tion between composite particles. The calculated phase shifts for the 3S1 and 1S0 channels show the presence of a strong repulsive core due to the combined effect of the quark in- terchange and the spin-flavour structure of the effective quark-quark interaction. Such a structure stems from the pseudoscalar meson exchange between quarks and is a conse- quence of the spontaneous breaking of the chiral symmetry. We perform single and cou- pled channel calculations and show the role of coupling of the  and hidden colour CC channels on the behaviour of the phase shifts. The addition of a -meson exchange quark- quark interaction brings the 1S0 phase shift closer to the experimental data. We intend to include a tensor quark-quark interaction to improve the description of the 3S1 phase shift. In this talk I shall mainly present results obtained in collaboration with Daniel Bartz [3,4] for the nucleon-nucleon (NN) scattering phase shifts calculated in the resonating group method. The study of the NN interaction in the framework of quark models has al- ready some history. Twenty years ago Oka and Yazaki [5] published the first L = 0 phase shifts with the resonating group method. Those results were obtained from models based on one-gluon exchange (OGE) interaction between quarks. Based on such models one could explain the short-range repulsion of the NN interaction potential as due to the chromomagnetic spin-spin interaction, com- bined with quark interchanges between 3q clusters. In order to describe the data, long- and medium-range interactions were added at the nucleon level. During the same period, using a cluster model basis as well, Harvey [6] gave a classi- fication of the six-quark states including the orbital symmetries [6℄O and [42℄O. Mitja Rosina, Bojan Golli and collaborators [7] discussed the relation between the resonating group method and the generator coordinate method and introduced effective local NN potentials. Here we employ a constituent quark model where the short-range quark- quark interaction is entirely due to pseudoscalar meson exchange, instead of one-gluon exchange. This is the chiral constituent quark model of Ref. [1], para- metrized in a nonrelativistic version in Ref. [2]. The origin of thismodel is thought to lie in the spontaneous breaking of chiral symmetry in QCD which implies the existence of Goldstone bosons (pseudoscalar mesons) and constituent quarks? E-mail: fstancu@ulg.ac.be Nucleon-Nucleon Scattering in a Chiral Constituent Quark Model 83 with dynamical mass. If a quark-pseudoscalar meson coupling is assumed this generates a pseudoscalar meson exchange between quarks which is spin and flavour dependent. The spin-flavour structure is crucial in reproducing the cor- rect order of the baryon spectra [1,2]. The present status of this model is presented by L. Glozman and W. Plessas at this workshop. Hereafter this model will be called the Goldstone boson exchange (GBE) model. It is important to correctly describe both the baryon spectra and the baryon- baryon interactionwith the samemodel. Themodel [1,2] gives a good description of the baryon spectra and in particular the correct order of positive and negative parity states, both in nonstrange and strange baryons, in contrast to the OGE model. In fact the pseudoscalar exchange interaction has two parts : a repulsive Yukawa potential tail and an attractive contact Æ-interaction. When regularized, the latter generates the short-range part of the quark-quark interaction. This dom- inates over the Yukawa part in the description of baryon spectra. The whole in- teraction contains the main ingredients required in the calculation of the NN po- tential, and it is thus natural to study the NN problem within the GBE model. In addition, the two-meson exchange interaction between constituent quarks re- inforces the effect of the flavour-spin part of the one-meson exchange and also provides a contribution of a -meson exchange type [8] required to describe the middle-range attraction. Preliminary studies of the NN interaction with the GBE model have been made in Refs. [9–11]. They showed that the GBE interaction induces a short-range repulsion in theNN potential. In Refs. [9,10] this is concluded from studies at zero separation between clusters and in [11] an adiabatic potential is calculated explic- itly. Here we report on dynamical calculations of the NN interaction obtained in the framework of the GBE model and based on the resonation group method [3,4]. In Ref. [3] the 3S1 and 1S0 phase shifts have been derived in single and three coupled channels calculations. It was found that the coupling to the  and CC (hidden colour) channels contribute very little to the NN phase shift. These studies show that the GBEmodel can explain the short-range repulsion, as due to the flavour-spin quark-quark interaction and to the quark interchange between clusters. However, to describe the scattering data and the deuteron properties, inter- mediate- and long-range attraction potentials are necessary. In Ref. [4] a -meson exchange interaction has been added at the quark level to the six-quark Hamilto- nian. This interaction has the formV = -g2q4 (e-rr - e-rr ) ; (1) An optimal set of values of the parametres entering this potential has been found to be g2q4 = g2q4 = 1:24;  = 0:60 GeV ;  = 0:83 GeV : (2) As one can see from Fig. 1, with these values the theoretical phase shift for 1S0 gets quite close to the experimental points without altering the good short-range behaviour, and in particular the change of sign of the phase shift at Elab  260 84 Fl. Stancu -60 -40 -20 0 20 40 60 80 0 50 100 150 200 250 300 350 de lta ( de g) Elab (MeV) 1S0 exp no scalar with scalar Fig. 1. The 1S0 NN scattering phase shift obtained in the GBE model as a function of Elab. The solid line is without and the dashed linewith the -meson exchange potential between quarks with  = 0:60 GeV and  = 0:83 GeV. Experimental data are from Ref. [12]. MeV. Thus the addition of a -meson exchange interaction alone leads to a good description of the phase shift in a large energy interval. One can argue that the still existing discrepancy at low energies could possibly be removed by the coupling of the 5D0 N- channel. To achieve this coupling, as well as to describe the 3S1 phase shift, the introduction of a tensor interaction is necessary. References 1. L.Ya. Glozman and D.O. Riska, Phys. Rep. 268, 263 (1996) 2. L.Ya. Glozman, Z. Papp, W. Plessas, K. Varga and R. Wagenbrunn, Nucl.Phys. A623 (1997) 90c 3. D. Bartz and Fl. Stancu, e-print nucl-th/0009010 4. D. Bartz and Fl. Stancu, e-print hep-ph/0006012 5. M. Oka and K. Yazaki, Phys. Lett. 90B 41 (1980); Progr. Theor. Phys 66 556 (1981); ibid 66 572 (1981). 6. M. Harvey, Nucl. Phys. A352 (1981) 301; A481 (1988) 834. 7. M. Cvetic, B. Golli, N. Mankoc-Borstnik and M. Rosina, Nucl. Phys. A395 (1983) 349 8. D. O. Riska and G. E. Brown, Nucl. Phys. A653 (1999) 251 9. Fl. Stancu, S. Pepin and L. Ya. Glozman, Phys. Rev. C56 (1997) 2779; C59 (1999) 1219 (erratum). 10. D. Bartz, Fl. Stancu, Phys. Rev. C59 (1999) 1756. 11. D. Bartz and Fl. Stancu, Phys. Rev. C60 (1999) 055207 12. V. G. J. Stoks, R. A. M. Klomp, M. C. M. Rentmeester and J. J. de Swart, Phys. Rev. C48 (1993) 792; V. G. J. Stoks, R. A. M. Klomp, C. P. F. Terheggen and J. J. de Swart, Phys. Rev. C49 (1994) 2950. BLED WORKSHOPS IN PHYSICS VOL. 1, NO. 1 Proceedins of the Mini-Workshop Few-Quark Problems (p. 85) Bled, Slovenia, July 8-15, 2000 Description of nucleon excitations as decaying states Bojan Golli? Faculty of Education, University of Ljubljana, and J. Stefan Institute, Ljubljana, Slovenia Abstract. Two methods to describe excited states of baryons as decaying states are pre- sented: the Analytic Continuation in Coupling Constant and the Kohn variational prin- ciple for the K-matrix. The methods are applied to a simple model of the  resonance consisting of the pion coupled to three valence quarks. The work has been done in collaboration with Vladimir Kukulin and Simon Širca. 1 Motivation Baryons are usually computed as bound states neglecting possible decay chan- nels. The inclusion of strongly decaying channels may considerably influence the position of the state as well as some other properties. The aim of the present work is to estimate this effect in a simplified model and to discuss two possible approaches to describe decaying states. The methods determine the position and the width of the resonance, and furthermore, provide a suitable tool to calculate new observables, which cannot be obtained in a bound state calculation, such as non-resonant contributions to production amplitudes. In this work we shall focus on the decay of the  resonance. 2 The model The decay of the  resonance into the nucleon and the pion is most naturally de- scribed in models with chiral symmetry, such as the linear  model (LSM), the chromodielectric model (CDM), the cloudy bag model CBM, etc. Here we use a simplified model which contains the main features of these models. It assumes frozen quark profiles and neglects meson-self interaction. Furthermore, it does not take into account additional scalar fields (sigma mesons in the LSM, chro- modielectric field and sigmamesons in the CDM, or the bag potential in the CBM) since their main role is to fix the quark profiles and generate a constant energy shift for all baryons. In the present calculation, the quarks profiles are taken over from the ground state calculation in the LSM[1].We know that the profiles do not change considerably from one model to the other, so this is not a very severe re- striction. The inclusion of meson self-interactionmay, however, more importantly alter the results.? E-mail: Bojan.Golli@ijs.si 86 B. Golli For the quark-pion interaction we assume the usual pseudoscalar form:Hquark-meson = ig Z dr3q̄  ̂ 5q : (1) In models with spontaneous symmetry breaking, such as the LSM, the parame- ter g is related to the ‘constituent’ quark mass by Mq = gf. From 350 MeV 1 the state becomes bound. Let’s denote the threshold value as th. The method [3] is based on the fact that it much easier to solve the bound state problem than the continuum case. It consists of the following steps: Determine th and calculate E as a function of  for  > th. Introduce a variable x = p- th; calculate k(x) = ip-2mE in the bound state region. Fit k(x) by a polynomial:k(x) = i( 0 + 1x+ 2x2 + : : : + 2Mx2M) : Construct a Padé approximant:k(x) = i a0 + a1x+ : : : + aMxM1 + b1x+ : : :+ bMxM : (5) 88 B. Golli 1 2 3 40.0 0.51.0 1.52.0 2.53.0 Æ ÆÆ Æ ÆÆ Æ Æ ÆÆÆÆÆ ÆÆÆ Æ Æ Æ ÆÆ Æ                 !0=m Æ33 Fig. 1. The phase shift in the P33 channel: Æ are the experimental values,  values from the variational calculation using g = 4:3 and " = 273 MeV, and  those for g = 5 and" = 253MeV. Analytically continue k(x) to the region  < th (i.e. to imaginary x) wherek(x) becomes complex. Determine the position and the width of the resonance as analytic continua- tion in : Er = 12m Re cont!1k2 ; = -2 12m Im cont!1k2 : (6) This method does not provide only the position and the width of the resonance; the matrix element of an operator O between the resonant state j ri and a bound state ji can be calculated ash rjOji = cont!1h b()jOji : In our implementation of the method, we relate the coupling constant  to the parameter of the phenomenological hyperfine interaction:V(r)! "P(3q) ; x = p"th - " (7) where "th is the value of " at the threshold: E("th) - EN = m. For sufficiently high ", the real part of the energy eventually reaches the experimental position of the resonance; this value of " then corresponds to  = 1 of the original formula- tion of the method. In our very preliminary calculation we treat the pion non-relativistically. For" < "th we calculatek(x) = ip2m(Eth - E); E = E(x) - EN ; Description of nucleon excitations as decaying states 89 fit k(x) using a Padé approximant (5) and continue k(x) to the resonance region. The energy difference, E-EN, and the width of the resonance are then obtained by (6). The ‘physical value’ of x (and " from (7)) is determined as ReE(= E-EN) reaches the experimental value 293 MeV. The corresponding value of ImE(= ) then predicts the width of  and is to be compared with the experimental value 120MeV. Fig. 2 shows the behaviour of E - EN and as functions of x for two vales of g. For higher order of the Padé approximant, M  3, the method becomes numerically instable and the determination of E and is no more reliable. Forg = 4:3 andM = 1 and 2, the experimental splitting is reached for x2  230MeV (and corresponding " = 300 MeV). This yields  60MeV which is only half of the experimental value, most probably due to the non-relativistic treatment. Forg = 5 the value of is larger (in accordance with Fig. 1) but its determination is less reliable. In order to be able predict reliable results it is necessary formulate the ap- proach relativistically and to understand the origin of numerical instabilities for higherM. 20i 15i 10i 5i 0 5 10050 100150 200250 300350 ................................................ ................................................................................... ......................................... ............................................ ................................................................................................... ... ...................................... M = 1 .... ..................... .............. ................................... ..... .... . .... . .... . ... M = 2 ......... ..................... ..................... ........................... ........................... .......... ...................... M = 3 x E EN (exp)E EN  (a) 20i 15i 10i 5i 0 5 100 50100 150200 250300 350 ....................................................................................... ................................................... ............................................ ...... .................................................................................................. ........................ ...................................... M = 1....................... .................. .................................... ..................... .... . .... . .... . ... M = 2............................ ...................... ............................ ......................... .......................... ............ ...................... M = 3 x E EN (exp)E EN  (b) Fig. 2. N splitting and  width (in MeV) as functions of x (in units pMeV) for g = 4:3 (a), and g = 5 (b). References 1. B. Golli and M. Rosina, Phys. Lett. B 165 (1985) 347; M. C. Birse, Phys. Rev. D 33 (1986) 1934. 2. B. Golli, M. Rosina, J. da Providência, Nucl. Phys. A436 (1985) 733 3. V.M. Krasnopolsky and V.I.Kukulin, Phys. Lett, 69A (1978) 251, V.M. Krasnopolsky and V.I.Kukulin, Phys. Lett, 96B (1980) 4, N. Tanaka et al. Phys. Rev. C59 (1999) 1391