Does the quark cluster model predict any isospin two dibaryon. resonance? (1) Grupo defsica Nuclear
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1 FUSAL - 4/95 Does the quark cluster odel predict any isospin two dibaryon resonance? A. Valcarce (1), H. Garcilazo (),F.Fernandez (1) and E. Moro (1) (1) Grupo defsica uclear Universidad de Salaanca, E Salaanca, Spain () Escuela Superior de Fsica y Mateaticas Instituto Politecnico acional, Edicio 9, Mexico D.F., Mexico Abstract We analyze the possible existence of a resonance in the J P =0 channel with isospin two by eans of nucleon- interactions based on the constituent quark odel. We solve the bound state and the scattering proble using two dierent potentials, a local and a non-local one. The non-local potential results to be the ore attractive, although not enough to generate the experientally predicted resonance. 1
2 The existence of bound states of negative pions and neutrons (pineuts) was predicted theoretically years ago [1]. A syste of neutrons and negative pions gives rise to a structure siilar to an ordinary nucleus, where the protons have been replaced by negative pions. Since these systes can only decay through weak interactions, they should be stable and have lifeties coparable to that of the pion. Then, a question arises iediately as to if one could observe even the siple of these possible pineuts, a bound state of a pion and two neutrons, just like anucleon- bound state. This proble has been studied fro the theoretical point of view by eans of dierent ethods [,3] and with dierent conclusions, but never denitively excluding the possibility of a resonance. Mainly due to this controversial situation, a lot of experients were done to nd evidences of such a resonance. After soe signatures in experients with poor resolutions [4], an experient with high intensity proton beas alost excluded the existence of these structures [5]. The situation has been recently renewed fro the experiental point of view. In Ref. [6], a J P =0 resonance has been proposed to explain a sharp peak seen on the pionic double charge exchange cross section in several nuclei fro 14 C to 48 Ca. The narrow width of these peaks suggested that the resonance ust have isospin even, otherwise decay into nucleon-nucleon () would be allowed, causing a uch large width. Besides, based on QCD string odels, they assued that the resonance has isospin zero. However, in Refs. [3,7] was pointed out that the narrow width of this structure could be related with the vicinity of the nucleon- ( ) threshold and therefore the resonance ost likely ust have isospin two (the syste cannot coupled to isospin zero). Our ai in this paper is to present the predictions of a quark odel based potential about the possibility ofa resonance with isospin two. Previous calculations based on the three-body foralis of the syste predict a large attraction in the 0 channel [], the sae proposed in Ref. [6]. Therefore, the 0 channel is the ideal candidate to posses a resonance and we will concentrate on it. Moreover, we will see that in case the resonance exists there is a strong correlation between its ass and its width due to the proxiity of
3 the threshold. We have derived a interaction using the sae two-center quark-cluster odel of Ref. [9]. Quarks acquire a dynaical ass as a consequence of the chiral syetry breaking. To restore this syetry one has at least to introduce the exchange of a pseudoscalar (pion) and a scalar (siga) boson between quarks. Besides, a perturbative contribution is obtained fro the non-relativistic reduction of the one-gluon exchange diagra in QCD. Therefore, the ingredients of the quark-quark interaction are the conning potential (CO), the one-gluon exchange (OGE), the one-pion exchange (OPE) and the one-siga exchange (OSE). The explicit for of these interactions is given by (see Ref. [9] for details), V CO (~r ij )= a c ~ i ~ j r ij ; (1) V OGE (~r ij )= 1 ( 4 ~ s i ~ 1 j 1+ ) r ij 3 ~ 3 i~ j (~r ij ) S ij ; () V OP E (~r ij )= 1 3 ch V OSE (~r ij )= " H( r ij ) ch 4 q 3 q ("Y( r ij ) # ) S ij Y ( r ij ) 4 q r3 ij # ~ i ~ j + H( r ij ) ~ i ~ j ; (3) Y ( r ij ) Y ( r ij ) : (4) The ain advantage of this odel coes fro the fact that it works with a single qq-eson vertex. Therefore, its paraeters (coupling constants, cut-o asses,...) are independent of the baryon to which the quarks are coupled, the dierence aong the being generated by SU() scaling. This akes the generalization of the interaction to any other non-strange baryonic syste straightforward, and in particular to the syste. Once the quark-quark interaction is chosen, an eective nucleon- potential can be obtained as the expectation value of the energy of the six-quark syste inus the selfenergies of the two clusters, which can be coputed as the energy of the six-quark syste when the two quark clusters do not interact: where, V (LST)!(L0 S 0 T )(R; R 0 S )= L0 0 T LST (R; R0 S ) L0 0 T LST (1;1); (5) 3
4 L0 S 0 T LST (R; R 0 )= rd L0 D L0 S 0 T S 0 T ( R ~ 0 )j P 6 i<j=1 V LST qq(~r ij ) j ( ~ R 0 ) j L 0 S 0 T ( ~ R) E E rd E : (6) ( R ~ 0 ) LST ( R) ~ LST j ( R) ~ The paraeters of the odel are those of Ref. [9]. As local potential we will assue R = R 0. In order to deterine the nature (attractive or repulsive) of the 0 channel, we will rst calculate the Fredhol deterinant of that channel as a function of energy assuing a stable delta and nonrelativistic kineatics. That eans, we will use the Lippann- Schwinger equation T ij (q; q 0 )=V ij (q; q 0 )+ X k Z 1 0 q 0 dq 0 V ik (q; q 0 )G 0 (E;q 0 )T kj (q 0 ;q 0 ); (7) where the two-body propagator is with reduced ass G 0 (E;q)= 1 E q =+i ; (8) = The energy and on-shell oentu are related as + : (9) E = q 0 =; (10) and we will restrict ourselves to the region E 0. If we replace the integration in Eq. (7) by a nuerical quadrature, the integral equations take the for T ij (q n ;q 0 )=V ij (q n ;q 0 )+ X k X w q V ik(q n ;q )G 0 (E;q )T kj (q ;q 0 ); (11) where q and w are the abscissas and weights of the quadrature (we use a 40-point Gauss quadrature). Eq. (11) gives rise to the set of inhoogeneous linear equations with X X k M ik n(e)t kj (q ;q 0 )=V ij (q n ;q 0 ); (1) 4
5 M ik n(e) = ik n w q V ik(q n ;q )G 0 (E;q ): (13) If a bound state exists at an energy E B, the deterinant of the atrix M ik n(e B ) (the Fredhol deterinant) ust vanish, i.e., M ik n(e B ) =0: (14) Even if there is no bound state, the Fredhol deterinant is a very useful tool to deterine the nature of a given channel. If the Fredhol deterinant is larger than one that eans that channel is repulsive. If the Fredhol deterinant is less than one that eans the channel is attractive. Finally, if the Fredhol deterinant passes through zero that eans there is a bound state at that energy. In Figure 1 we copare the Fredhol deterinant generated by the local and nonlocal quark odel based potentials. The non-locality of the interaction generates additional attraction, enough to produce a resonance (it goes through zero). To deterine the exact location of the resonance, we calculate Argand diagras between a stable and an unstable particle using the foralis of Ref. [8]. In this case, however, we will use relativistic kineatics and will include the width of the delta. That eans, instead of the propagator (8) we will use [8] G 0 (S; q) = s +i (s; q) ; (15) where S is the invariant ass squared of the syste, while s is the invariant ass squared of the subsyste (those are the decay products of the ) and is given by q s = S + S( + q ) : (16) The width of the is taken to be [8] (s; q) = 3 0:35 p3 0 q + q p ; (17) s where p 0 is the pion-nucleon relative oentu given by 5
6 ! p 0 = [s ( + ) ][s ( ) 1= ] : (18) 4s We show in Figure the phase shifts for the 0 channel. As it can be seen fro this gure, the attraction is only strong enough to produce a resonance with the non-local potential (it reaches 90 degrees). This resonance lies at MeV and has a width of MeV for the ass of the siga predicted by chiral syetry requireents 675 MeV. The proposed resonance has a ass of 065 MeV and a very sall width of 0.51 MeV [6]. It is therefore very interesting to investigate whether the nucleon- syste exhibit the features of this resonance, and particularly such a tiny width. In order to do this, wehave articially varied the ass of the eson with both potential odels, such as to increase the aount of attraction. We show intable I the ass and width of the resonance and the corresponding ass of the siga eson necessary to generate it. The width of the resonance drops draatically when its ass approaches the threshold (017 MeV). This result can be understood fro siple angular oentu barrier considerations. If we call q and L to the relative oentu and relative orbital angular oentu between a nucleon and the -nucleon pair, respectively, then since L = 1 the width of the resonance will be proportional to q L+1 = q 3, so that it will drop very fast as one approaches the threshold since there q! 0. In both local and non local potential odels, when the ass of the siga is taken to reproduce the predicted ass of the resonance (065 MeV) the width is very narrow, whichis in very good agreeent with the predictions extracted by Bilger and Cleent [6]. Therefore, the sharp peak seen in the double charge exchange reactions could be justied as a nucleon- resonance in the isospin channel, without resorting to other ore exotic processes. As a suary, we have studied the nucleon- syste in the 0 channel with isospin two, within the quark cluster odel of the baryon-baryon interaction. We have used a local and a non-local potential. We found that the non-local eects generate additional attraction, although not enough to reproduce the resonance predicted in Ref. [6]. However, due to the proxiity of the nucleon- threshold, if we force the resonance ass to reach 6
7 the experiental predicted value of 065 MeV, then its width is very narrow, in very good agreeent with the width extracted in Ref. [6]. ACKOWLEDGMETS This work has been partially funded by EU project ERBCHBICT941800, DGICYT Contract o. PB and by COFAA-IP (Mexico). 7
8 REFERECES [1] H. Garcilazo, Phys. Rev. Lett. 50, 1567 (1983). [] W. A. Gale and I. M. Duck, ucl. Phys. B8, 109 (1968). G. Kalberann and J.M. Eisenberg, J. Phys. G 5, 35 (1977). H. Garcilazo and L. Mathelitsch, Phys. Rev. C 34, 145 (1986). [3] A. Valcarce, H. Garcilazo and F. Fernandez, Phys. Rev. C (1995), in press. [4] D. Ashery et al, Phys Lett 15B, 41 (1988). [5] F.W.. de Boer et al, Phys. Rev. Lett. 53, 43 (1984). [6] R. Bilger, H. A. Cleent, and M. G. Schepkin, Phys. Rev. Lett. 71, 4 (1993); 7 97 (1994). [7] H. Garcilazo and L. Mathelitsch, Phys. Rev. Lett. 7, 971 (1994). [8] H. Garcilazo and M. T. Pe~na, Phys. Rev. C 44, 311 (1991). [9] F. Fernandez, A. Valcarce, U. Straub, and A. Faessler, Jour. Phys. G 50, 46 (1993). 8
9 TABLES TABLE I. Mass and width of the 0 resonance with the corresponding ass of the siga eson using the local and non-local nucleon- potentials based on the constituent quark odel. Potential odel (MeV) M Res (MeV) Res(MeV) Local on local
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