23 February
1995
PHYSICS Physics
Letters
LETTERS
B
6 345 ( 19015) 367-371
Influence of the ‘Do nucleon-A channel on the ‘Sonucleon-nucleon phase shift in a quark model A. Valcarcc a.b;Amand Faessler‘, F. Fernindez b a Ins&z/ h Grupo
fir T/zeorcfische Plzyik. Universi~iit Tiibizzgen. de Fhca Nucleus Uzziversidad de Sulamanca.
Received
3 Novemtxr
1994: revised Editor:
manuscript C. Mahaux
D-72076 E-37008
received
Tiibinpzt. German) Salamcznczz. Spnitz
14 December
1994
Abstract
i’he problem Additional
:nfluence of the “Do Nucleon-A channel on the ‘.!$I nucleon-nucleon phase shift is studied as ir. the framework of the Resonating Group Method Group using the quark cluster model with attraction is generated in the ‘.%I nucleon-nucleon interaction by the coupling to the ‘01 (NA)
Up to now, all quark model calculations (potential, bag and soliton models) of the nuclear force have difficulties to obtain the intermediate range attraction. In meson exchange models (Bonn and Nijmegen potentials) this intermediate range attraction is fitted by sigma( cr)-meson exchange, Hybrid quark models get also a main part of this attraction from the g meson. But using chiral symmetry, which fixes the m-quark c’bupling constant by the r-quark and finally by the Ir-r&con interaction, one finds not enough attraction in the ‘& channel, although the ‘SJ -’ D1 and higher partial wave ph&e shifts are reproduced correctly. Wc show here that this ciiCza!!y can be resolved if the ‘WNN) --+ ‘Do(NA) co u P Iin’t is included, which never has been consldered on the quark level. The admixture to the AA and hidden color sta!cs has been studied in the past, but it was found that its effect in ’ LJ=L and “LJ=L,J+~ NN channels is negligible [ I ,2] and this is why we will not consider them in tile present calculation. These admixtures have been considered through the one-gluon exchange between quarks and v exchange between nucleons. [ ! ,2]. Recently it has 0370-2693/95/$09.50 SSD/O370-2693(94)01663-i
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a coupled
channel chir:B symmetry. chanlOfl
been consldered through the OC~~-:~UOI, :In(i one-pion exch*nge between quarks with the same conclusio,l [ 31. The purpose of the present lette* is to study the influence of the 5D~ (NA) coupling on the ‘SO (NN) phase shift as a coupled channel problem (NN, A’A) in the framework of the Resonating Group Method (RGM). The constituent quark model for the NN interaction developed by the Tiibingen group [ I j during thL last deczdc has been rcccntly extended in a series ot’papers [3-S] by introducing a scalar meson (the a-meson) exchange between quarks and not between the nuclcons as done previously. In this WC.Y, the u-meson parameters are related to those of the pion exchange and to the constituent quark mass through chiral symmetry requirements. Within this framework, the freedom associated to ths scalar particle (I%,, g&.,,, and A,,) is lost and therefore the number of fret parameters is reduced. This model has been applied to the description of the nucleon-nucleon (NN) scattering phase shifts [ 41,. the statrc and electromagnetic properties of the deutcron
368
A, Vdcarce
et al. /Physics
[ 51 and the hyperon-nucleon cross sections [ 31 in the framework of the Resonating Group Method (RGM). However, as has been pointed out in Ref. [4]. the main drawback of the model is the lack of attraction in the isotriplet (T = 1) sector, specially in the ‘So partial wave. The good predictions of the model for all the properties related with isosinglet (T = 0) partial waves, suggests the possibility that some isospindependent mechanism could have been missed. In particular, it has been argued [ 41 that the coupling to the NA system, an isospin-dependent mechanism, could provide the necessary additional attraction [ 6) in this sector. Therefore, our aim in this letter is to study the influence of the sDc (AN) channel on the ‘50 (NN) phase shift within the constituent quark model. This singlet (NN)-quintet (NA) coupling has never been studied bcforc on the quark level. In Ref. [6] it is shown th.tt this coupling is important on the nucleon level. The model has been introduced in Refs. [4,5]. The ingredients of the quark-quark interaction are the confining potential, the one-gluon exchange (OGE), the one-pion exchange (OPE) and the one-sigma exchange (OSE). The explicit form of these interactions is given by (See Refs. [4,5] for details.), k’3N
(‘ii)
=
-0,
Ai
. Aj
ffi
,
(1)
1 Ui
* Uj
(3) 4m2 hss(rljj
x
d-m,
= -Qch
Y(m,,r;j)
A2
mf A2-mi
-
(4)
The main advantage of this model comes from the fact that it works with a single qq-meson vertex. Therefore, its parameters (coupling constants, cut-off
Letters
B 345 (199.7) 367-37F
Table I Quark modelparameters m, CMeV) h (fm) WV a,( MeV ‘%h
mdfm-‘) m,(fm-‘) Acsdfm-‘)
313 0.518 fm-‘)
0.485 46.938 0.027 3.421 0.70 4.2
masses, ... ) are independent of the baryon to which the quarks are coupled, the difference among them being generated by SU(2) scaling. This makes the generalization of the NN interaction to any other nonstrange baryonic system straightforward. In particular, the transition potential ‘&( NN) --) ‘Da(NA), and the elastic interaction ‘Da( NA) ---) 5Do( NA) can he immediately calculated within the same framework [ 7.81. Moreover, as explained in Refs. [ 4.51 the parameters associated with the scalar field ‘ae related to those of the pion exchange and the constituent quark mass through: g&,,/47r = 36cqhmi/m$ A, = 11, and rni = (2m,) 2 + m?;. We will use the parameters of Ref. [ 51, which reproduce correet:y the isosinglet NN scattering phase shifts and the static and electromagnetic properties of the deuteron. However, as pointed out in they yield to little attraction for the ‘So phase shift. These parameters’are shown in Table 1. In the energy regimen 0 < Et& i 300 MeV which we consider, all the inelastic channels are closed, and the amplitudes for finding any state other than a free NN state goes to zero exponentially for large separations. Within the region of interaction, however, there will be a nonzero r~nptitude for finding virtual excited states of the NJ svs!zm. The purpose of this work is to investigate the ei’fcct on the elastic t&( NN) partial wave of one of the most important virtual excited states present, the 5D~ (NA) [ 61. We are only interested in the state ‘SO. because the inelastic process NN -+ NA is restricted to T = 1. The most important effects are expected in the lower partial waves, where the overlap of the two baryons is not supressed by the centrifugal barrier. We consider the full coupling of the two nucleon$ in the ’ SOchannel includingtheelastic ‘&(NN) + !$(,VN). ‘Da(NA) + sDa(NA) and the inelastic ‘&(NN) - SDo(NA)
A. Valcarce
0.0
1.0
2.0 r
Fig.
3.0
el al. /Physics
4.0
(fm)
I.
Strength of the adiabatic tensor coupling --+ 5 DO ( NA ) (solid line) in comparison to that of the ‘So (NW purely nucleonic sector fSt (NN) -+ >DI ( NN) (dashed line).
terms. In this configuration the two nucleons are originally in the state (L = 0, S = 0). Therefore, the NA system is formed in the state (L = 2, S = 2). This transition only takes place through the tensor interaction and therefore only the OGE and the OPE potentials will contribute. The second being dominant. In meson-exchange models the theory of isobars in nuclei became much more reliable when the rho-meson exchange was included [6]. It regularizes the si:lgular OPE tensor force both in the purely nuclccnic sector and’ in the NN --) NA transiticn potential, and it reduces the strong dependence on the value of the cut-off mass. In the quark model, it has been shown [ 71 that the quark antisymmetrization (two quark exchange) plays a similar role. The transition potential ‘WNW + ‘Da( NA) and the spin-iscspin-color coefficients needed are discussed in detail in Ref. [ 71. As we are using fzN,,/4r - 0.0749, we would obtain &J47r = $jf&/4n = 0.22. This value is usually considered to be an underestimation, because the one obtained from the decay of the A resonance is fzNd/4v = 0.35. However, fcr the sake of consistency we will use all the parameters as predicted by the quark model. For the calculation we need also the elastic potential sDa( NA) -+ sDa( NA). This potential has been tiiscussed in detail within the framework of our model in Ref. [ 81. Although being important to satisfy unitarity, its quantitative effect is small. In order to gain some qualitative insight of the impoxtance of the ‘So ( NN) -+ “Da ( NA ) tensor coupling, we compare in Fig. 1 the strength of the adia-
larrers
B 345 (1995)
-60
d----r-0
367-371
369
/ too
----
-I
200 &.a
300
WV)
Fig. 2. ‘.Q NN scattering phase shifts as a function of the lab. cnergy. ‘De dotted line shows the result without the sigma-meson exchange between quarks and without the couplkg to the “D,, (NA) channel; the dashed line inckles the effect of lhc sigma-meson exchvlge between quarks (the results corresponding to Ref. 15 I ). The solid line shows the result of the full coupled-channel calculation. Experimental data are irom Ref. 1 I I I.
batic potential for this transition to that of the tensor interaction JSt (NW) -+ ‘Dt (NN) in the purely nucleonic sector, that is known IO bc very important to reproduce the “St phase shifts and the binding energy of the deuteron. We have neglected the width o!’ the A ( 1232) in this calculation and we treat it on the same footing as the nucleon. We observe that both couplings are of the same order of magnitude, although being ar >ciated to a different process. To evaluate the effect of the coupling to the sD~(NA) channel on the ‘SO (NN) phase shift, we have performed a two-chaanel RGM calculatio~r. As it is well known, the coupling of a channel lower in energy (NN) to a channel of higher energy (NJ) produces additional attraction in the lower energy channel [ 91. The results are shown in Fig. 2. In order to clarify th,: role played by the different terms contributing to the intermediate range attraction, we have plotted by the dotted line the results obtained using only the NN char;:d (without coupling to the NA system), and without the effect of the sigma-meson exchange between quarks. The dashed line incorpolates the effect of the sigma-meson exchange hut still in a single-channel calculation (those results corresponding to Ref. [ 51). Finally, the solid line shows th- results of the full two-channel calculation. As can be seen, the agreement with the expcrimcntal data is
I
90
‘-r-‘----.-----.~I---‘
i
-, -1..
i
-30
..____
j
-6O+----
-~---~...-..._._____~~~~~~ /
0
r-4 100
200 Elab
300
(MeV)
Fig. 3. ‘St XIV scakring phnsr $hifts tis ;I i’unclion of rhc lab. cncvy. The dotted line shows the resuh without the Ggma-meson exchange IWwccn quarks and without the tensor coupling to the 7D~ CN,V) partial WIVC: the daphcd line includrs the cffecr of the sigma-meson cxchrngc bctwecn quarks. The solid line rhnws the rc~lt ol’ ‘hr full cal~ulnlion (the results corrcrpnndmg to Ref. ) 5 I ). NW lhnr the h’A sysrent does not couple to the isosingler &‘N panial waves. Experimental data arc from Ret: 1 I I J.
rcasonahly. In Ref. [ IO] it has been shown that the coupling to the NA channel and the sigma-meson exchange being completely different processes, they have a similar cffeet on the NN scattering phase shifts: both produce intcrmcdiate range attraction. Obviously, the coupling to the NA system. which has only T = I or T = 2, cannot :affcct the results in isosinglet partial waves (‘S, and ‘DI). AA admixture is possible for the ‘5, and ‘D1 nuc!eon-nucleon channels, but turns out to ‘*e not important. The same is true for the hidden color states [ I ,2]. For comparison, we show in Fig. 3 the results for the ‘SI partial wave. Note that due IO isospin conservation in strong interactions, the isosinglet (T=O) partial waves are not affected by the coupling to the NA system. WC have done a similar analysis for this partial wave. We show in Fig. 3 by the dotted line the result without the sigma-meson exchange hctween quarks. Simultaneously we have switched-off the tensor coupling to die 3 DI NN partial wave. In the daqhcd line WC: incorporate the effect of the sigma-meson exchange and the solid line shows the full calculation (those results corresponding to Ref. [ 5) ). In summary, we would like to emphasize that the model of Refs. [ 3-51 supplemented with the effect of the sDo (NA) intermediate state, is able with chiral symmetry to reproduce correctly the ‘SO NN partial
wave phase shifts without modifying the quality of the results in the isosingiet partial waves. Our model includes simultaneously the cxchangc of the sigmameson and the coupling to the “DO (NA) channel. Both mechanisms, the fl exchange and the s DO ( NA ) admixture in the ‘& (NN) channel yield attraction. In Ref. [ 6] it was shown 3n the baryon level. that the ‘Do (NA) admixture in the ’ & (NN) channel can be simulated by rr exchange. But different “DO (NA) and g-meson exchange contributions give very different results for the potential energy of neutron matter. However, in a quark mode! with chiral syrnnretry, the a-q lark parameters are fixed by the r-quark intcraction and the constituent quark mass, and therefore WC do not have a freedom to fit the data. Agreement with experiment for the ‘& (NN) partial wave can only be reached by incltiding both, the (r exchange dctcrmined hy chiral symmetry and the “Do (NA) admixture. The influcncc of the AA and hidden colorhidden color admixture has been studied in the past and has been found to he negligible. The effect of the coupling to the sD~ (NA) channel in the ‘So (NN) partial wave suggests further studies on higher NN partial waves. This is now under investigation. In partirular, due to the different coupling of the “PJ (NN) partial waves to the NA system, it could heip to solve some of the remaining problems of the model related to the spin-orbit interaction. Howcvcr, due to the fact that the virtual N.A states can only exist in the overlap region of the IWO haryons, the A admixture below the pion threshold has a smaller effect in higher partial waves. where the centrifugal barrier reduces strongly the overlap between the Iwo baryons. One of the authors (A. V.) thanks the Theoretical Physics Group at the University of Tiibingen for the kind hospitality extended to him. He alsc thanks European Community for support in the program Hunrorr Cupiful and Mobility CHRX-CT93-0323. This work has been partially l’undcd by Dircccidn General dc Invcstigaci6n Cientilica y TCcnica (DGICYT) under the Contract No. F691-01 I!%CO2-02 and by the Gcrman Federal Ministcry for Research and Technology (BMFT! under Contract No. 06Tii736.
A. hicarce
et al. /Physics
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