Effects of baryon resonances on nucleon–nucleon interactions in a quark model

17 February 2000

Physics Letters B 474 Ž2000. 255–261

Effects of baryon resonances on nucleon–nucleon interactions in a quark model Ichiro Maeda a b

a,1

, Masaki Arima

a,2

, Keiichi Masutani

b,3

Department of Physics, Osaka City UniÕersity, Osaka 558-8585, Japan Faculty of Engineering, Yamanashi UniÕersity, Kofu 400-8511, Japan

Received 15 September 1999; received in revised form 14 December 1999; accepted 4 January 2000 Editor: W. Haxton

Abstract Nucleon–nucleon S-wave scattering is studied by using the resonating group method, which includes not only the nucleon and D but also their excited states. As quark–quark interactions, the one-gluon-exchange and one-pion-exchange potentials are employed with phenomenological confinement potential. Compared with the N and D, coupling effects induced by excited-state baryons are small. q 2000 Published by Elsevier Science B.V. All rights reserved. PACS: 12.39.jh; 12.39.pn; 13.75.Cs; 14.20.G Keywords: Resonating group method

There have been many attempts to explain the nucleon–nucleon Ž NN . interaction by using the resonating group method ŽRGM. in the nonrelativistic constituent quark models w1–11x. In these works, the one-gluon-exchange potential ŽOGEP. andror the one-meson-exchange potential ŽOMEP. have been mainly used as quark–quark Ž qq . interactions, supplemented with phenomenological confinement potential. The repulsive core of the NN potential has been successfully explained by the Pauli principle for the constituent quarks and the color magnetic interaction of the OGEP w1,2x. The medium- and long-range

1

[email protected] [email protected] 3 [email protected] 2

attractive parts of the NN interaction, however, cannot be reproduced by the OGEP. In order to obtain this attraction, additional one-meson-exchange potentials ŽOMEP. have been introduced w1–4,7,10,11x. It recently has been shown that the OMEP also can induce the short-range NN repulsion w8,9x. The standard RGM calculations, which usually contain only the ‘‘ground-state’’ baryons, i.e., the N and D, have been in a reasonably good agreement with scattering data w1–4,7x. Particular channel-couplings, such as the 3 S1 NN– 3 D 1 NN and 1S0 NN– 5 D 0 ND, have played a crucial role in reproducing the experimental NN phase shifts. These couplings are mainly induced by the tensor interaction of the onepion-exchange potential ŽOPEP.. There still remain some issues to be clarified. One of these issues is the role of excited states of the nucleon and D, i.e., N ) and D ) , in the NN interac-

0370-2693r00r$ - see front matter q 2000 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 0 0 . 0 0 0 3 1 - 9

I. Maeda et al.r Physics Letters B 474 (2000) 255–261

256

tion. Several authors w10,11x have partially taken account of these states and shown their influence on the NN scattering. However, detailed analyses of their role in the NN interaction have not been performed yet. Moreover, using the Born-Oppenheimer approximation, Stancu et al. w12x recently have claimed that the commonly used RGM wave functions are not appropriate in a study of the short-range NN interaction if the OPEP is employed. It has been well known, on the other hand, that in the case of the OGEP excited-state baryons do not cause any significant changes in the NN interaction w1–4x. The purpose of this paper is therefore to clarify the effect of baryon resonances on the NN interaction. To this end, we systematically study the NN S-wave scattering by using the coupled-channel RGM, which includes not only the N and D but also their excited states. Besides the confinement potential, both OGEP and OPEP are employed as qq interactions. Other mesons, such as the s meson, are not included, in order to make the model simple. Although the nature of the s meson is still an open problem, many authors have claimed that this meson is required to reproduce the intermediate attraction of the NN interaction w3,4,7,10,13x. Because baryon resonances are explicitly taken into account by the coupled-channel procedure in this work, some scattering processes, such as the 2p-exchange process with intermediate NN ) states, can mimic a part of the one-sigma-exchange NN interaction. If the contribution of the s meson were included in the present model, the coupling strength should be different from that used in previous works that have not included baryon resonances. The model hamiltonian for a single-baryon is 3

H3 s

Ý K i y tG y Ý a Ž l i P l j . ri j q Ý Vi j q 3m, is1

i-j

i-j

Ž 1. where m is the constituent u- and d-quark mass, and K i and tG are the nonrelativistic kinetic energies of the i-th quark and the center of mass of the baryon, respectively. The strength of the linear confinement potential is denoted by a, the SUŽ3. color generator for the i-th quark by l i , and the distance between the i-th and j-th quarks by ri j .

The OGEP is necessary to reproduce the N– D mass splitting if the OPEP has the quark–pion coupling constant gp q q derived from the experimental value of the pion–nucleon coupling constant Gp N N ; the observed N– D mass splitting is about 290 MeV, while the OPEP provides 190 MeV in the present model. The qq interaction Vi j is assumed as EP Vi j s Vi OG q Vi OPEP , j j

Vi OGEP s j

as 4

gp2 q q

y ri j 1

2

2p

1

Ž liPlj.

=eyk r i j y

Vi OPEP s j

Ž 2. 3m 1

4 m ri3j

1

4p 12 m 2

2

k

2

Ž siPsj.

Ž 3.

Ž t i P t j . Sp Ž ri j . Ž s i P s j . Ž 4.

ym p r i j

Tp Ž ri j . s

ž

e

ri j

3 ri3j

q

žp/

Si j ,

qTp Ž ri j . Si j , Sp Ž ri j . s mp2

3r2

y 4p

3mp ri2j

q

kp

3r2

žp/

mp2 ri j

/

2

ey kp r i j ,

eymp r i j ,

Ž 5. Ž 6.

where mp is the pion mass and a s is the quark–gluon coupling constant. The spin and isospin generators for the i-th quark are denoted by s i and t i , respectively. Si j stands for the standard tensor operator. The d-function terms stemming from the contact interactions in the OGEP and OPEP are smeared out by introducing the range parameters k and kp w14x. It should be noted that there is a controversy over the use of the OPEP as the short-range part of qq interactions w15–19x; this problem has not been solved yet. Since there are many papers explaining how to construct single-baryon wave functions w20x, only a brief description is given here. For a baryon with total spin j and total isospin T, the wave function F is schematically written as

F Ž j . s A3

½

fnl m x s

j

x Tx C ,

5

Ž 7.

where j stands for the spatial and internal coordinates, and A3 is the three-quark antisymmetrizer. The isospin part is denoted by x T, and the color part by x C , which is always color-singlet. The spatial-spin

I. Maeda et al.r Physics Letters B 474 (2000) 255–261

part is represented by w fnl m x s x j; fnl is the spatial wave function with radial node n and orbital angular momentum l, and it is combined properly with the spin wave function x s, which has the total intrinsicspin s of quarks. In order to make the RGM calculation tractable, we assume the single-baryon wave function is an eigenfunction of a simple hamiltonian that contains only the harmonic-oscillator potential 2 Ži.e., m6v Ýri2j . instead of the linear confinement and the residual qq interaction Vi j in Eq. Ž1.. As in the previous RGM calculations w1x, the baryon mass is defined by m B s d jF ) Ž j . H3F Ž j . .

Ž 8.

H

The harmonic-oscillator parameter b Žs m v . is determined by the variational condition for the nucleon mass m N ;

E mN Eb

s 0.

Ž 9.

In addition to the N and D, which have been usually taken into account, ground-state baryons’ excited states up to 2 " v excitations are also considered in this work. These states are summarized in Table 1, together with their symmetric structure, expressed by a shorthand notation of the Young diagram. The symbols O˜ and R˜ stand for the orbital and radial excitations, respectively. Excited states that belong to the same energy-shell are distinguished by their

257

calculated masses and labeled by subscripts Ž1, 2, PPP ., from the lightest one. Since these excited states have various combinations of the orbital, spin and isospin symmetries, the present model has much richer symmetry patterns than the conventional RGM calculations. This is an advantage of our study. We now fix the model parameters. The value of gp q q is derived from the experimental value of Gp2 N N r4p s 14.3 w21x by using the spin–isospin symmetry relation of the constituent quark model. ŽThe results change little if a newly reported value of Gp2 N N r4p s 13.7 w22x is used.. The pion mass mp s 140 MeV is also taken from the observed value. The oscillator parameter b is chosen by the variational condition Ž9.. The other parameters are determined by fitting the N– D mass splitting; other mass splittings, such as N–N1) , D – D ) and so on, are also used as subsidiary conditions. All the parameters obtained are summarized in Table 2. Note that there are no adjustable parameters in the following calculations of the NN scattering. Before discussing the NN interaction, we briefly comment on baryon mass spectra. Our calculations do not reproduce the observed spectra satisfactorily, as seen in Fig. 1. In order to improve the result, we must diagonalize the hamiltonian Ž1. in terms of the single-baryon wave functions and then readjust the parameters. Although this procedure is in principle desirable, the RGM calculation with these diagonalized single-

Table 1 Properties of baryons Baryon

N N1) N2) N1) ) N2) ) N3) ) N4) )

D D) D1) ) D2) ) D3) )

Shell

Symmetry

Excitation

orbit

spin

isospin

0

w3x

1

w21x w21x

w21x w21x

1

w21x w21x

2

w3x

2 2

w21x w111x

w3x w21x w21x

2 0

w21x w3x

w21x w3x w3x

1 2

w21x w3x

w21x w3x

2

w3x

2

w21x

Parity

Total spin

–

q

1r2

O˜

y

1r2

w21x w21x w21x

O˜

y

1r2

R˜

q

1r2

O˜ q R˜

q

1r2

w21x

O˜

q

1r2

w21x w3x w3x

O˜ –

q q

1r2 3r2

O˜

y

3r2

w3x

R˜

q

3r2

w3x

w3x

O˜

q

3r2

w21x

w3x

O˜

q

3r2

I. Maeda et al.r Physics Letters B 474 (2000) 255–261

258 Table 2 Model parameters m ŽMeV. b Ž1rfm2 . a ŽMeVrfm. as k Ž1rfm2 . mp ŽMeV. kp Ž1rfm2 . gp q q

energy of the center of mass of the two-baryon system. The RGM wave function is constructed in terms of the two color-singlet baryon wave functions, Fa and F b , as follows;

340 4.34 87.0 0.420 10.0 140 7.00 2.91

Ca Ž j a , j b , R . sA

baryon wave functions becomes quite complicated and time-consuming. Fortunately, it has been found numerically that our simple wave functions are the dominant components of the diagonalized singlebaryon wave functions, and that similar results are finally obtained for the coupling effects of the N ) and D ) even if the model parameters are largely varied. It is therefore expected that this approximation for the single-baryon wave functions works at least for qualitative arguments. We proceed to consider a two-baryon system, i.e., a six-quark system. The model hamiltonian is given as 6

Hs

Ý K i y TG y Ý a Ž l i P l j . ri j q Ý Vi j q 6 m, is1

i-j

i-j

Ž 10 . where the two-body qq interactions are the same as those for a single-baryon, and TG is the kinetic

½

Fa Ž j a . m F b Ž j b . m xa Ž R .

a

5,

where xa is the relative-motion wave function and a stands for a set of quantum numbers that completely specifies the two-baryon scattering channel. The relative coordinate between the two baryons is denoted by R. The six-quark antisymmetrizer is expressed by A. The RGM equation of motion w1x can be reduced to the Lippmann-Schwinger equation for the T-matrix. In the present model, the calculation of the transition potential Va Xa is rather involved because various two-baryon channels are dealt with. In order to calculate Va Xa systematically by using the TalmiMoshinsky technique, we expand the relative-motion wave function xa in terms of the harmonic-oscillator eigenfunction with the same oscillator parameter b used in the single-baryon wave functions F w24,25x. In Table 3 are listed the two-baryon channels examined in this work; they are classified according to the sum of excitation energies of the two baryons. When the total isospin, total spin and relative angular momentum of the two-baryon system are specified, there are 42 channels coupling to the 3 S1 NN channel and 34 channels coupling to the 1 S0 NN channel. Among these channels, we first search for several channels that may affect the NN S-wave scattering; in the case of the 3 S1 NN scattering, for example, we solve the two-channel Lippmann-Schwinger equation that contains one of the 42 channels and the 3S1 NN channel, and compare the result with the single-channel 3S1 NN calculation.

Table 3 Two-baryon channels up to 2 " v excitations )

)

Fig. 1. Mass spectra of N and D . The experimental values are taken from Ref. w23x. Note that the theoretical value of the nucleon mass is 1735 MeV; in the figure, the theoretical values are shifted by 796 MeV. N4) ) and D3) ) , which are not shown here, correspond to the so-called missing resonances.

Ž 11 .

Excitation energy

Two-baryon channel

0 "v 1 "v 2 "v

NN, ND, DD NN ) , N )D, ND ) , DD ) NN ) ) , N ) N ) , N ) )D, N )D ) ND ) ) , DD ) ) , D )D )

I. Maeda et al.r Physics Letters B 474 (2000) 255–261

After choosing such effective channels, we carry out multi-channel calculations and examine channel-coupling effects on the NN interaction. Here we make a brief comment on the hidden color channels, which we do not consider explicitly in this work. The explicit inclusion of the hidden color channels generally causes the problem of a long-range color van der Waals force due to the two-body confinement interaction Ž1.. It is also known that the RGM wave function Ž11. with all possible combinations of color-singlet baryons provides a complete set of channels for a totally colorsinglet six-quark system, and that a lot of highly excited states are required to express a configuration of two well-separated colored-baryons w1x. Since only low-lying excited states are taken into account, the present RGM wave function is affected little by the hidden color channels. Our model is therefore expected to be almost free from this undesirable force. We first discuss the two-channel calculations, which are shown in Fig. 2. Quite large effects are observed in the cases of the 3S1 NN– 3 D 1 NN and 1 S0 NN– 5 D 0 ND calculations; however, these results are not presented in Fig. 2, and their effects can be recognized by comparing Fig. 2 with Fig. 3. This fact was already known on the baryon level and has also been reproduced on the quark level w3,4,7x. Besides these two channels, we have selected several channels that are expected to have some effects on the NN interaction. They are the 7D 1 DD, 1 P1 NN2) , 3 D 1 NN1) ) and 3S1 DD2) ) channels for the 3S1 NN scattering, and the 5 D 0 DD and 1 S0 DD2) ) channels for the 1 S0 NN scattering. The DD channels, which have been intensively studied in Refs. w3,4x, have larger effects than the other channels containing excited baryons. The selected channels Ži.e., 3 D 1 NN1) ) , 3S1 DD2) ) and 1S0 DD2) ) . that are newly considered in this work contain the ‘‘ground-state’’ baryons Ži.e., the N and D . and their positive-parity excited states Ži.e., the N1) ) and D2) ) .. These excited states have the same combination of the orbital, intrinsic-spin and isospin symmetries of the corresponding ‘‘ground states’’; the N1) ) is a radial excited state and the D2) ) is an orbital excited state Žsee Table 1.. Note that this is not the case for the 1 P1 NN2) channel. The effects seen in Fig. 2 are mainly induced by the tensor interaction of the OPEP. On the other

259

Fig. 2. Ža.: 3 S1 NN phase shifts as a function of Erel , the energy of the relative motion in the center of mass system. The solid line corresponds to the one-channel Ž 3 S1 NN . calculation. The other lines are the results of the two-channel calculations: the dashed line for 3 S1 NN – 7D1 DD, the dot-dashed line for 3 S1 NN – 3 D1 NN1) ) , and the dotted line for 3 S1 NN – 3 S1 DD2) ) . The result for 3 S1 NN – 1 P1 NN2) is omitted since it almost coincides with that for 3 S1 NN – 3 D1 NN1) ) . Žb.: Same as Ža. except for 1 S0 NN. The solid line corresponds to the one-channel Ž1 S0 NN . calculation. The other lines are the results of the two-channel calculations: the dotted line for 1 S0 NN – 5 D 0 DD and the dot-dashed line for 1 S0 NN – 1 S0 DD2) ) .

hand, the OGEP does not play an essential role; for example, the OGEP affects the channels containing D2) ) , but these effects are much smaller than those of the OPEP. It also has been verified that channel couplings are weak in the pure OGEP model w1–4x, in which the parameters are readjusted to fit baryon spectra, as was stated before. Concerning channelcoupling effects, we therefore conclude that the OPEP is more important than the OGEP in the present model. Due to the operator l i P l j , the effects of the OGEP are limited within the short distance between baryons. Contrastingly, the OPEP plays a crucial role not only in the medium- and long-range region but also in the short-range region. It should be empha-

260

I. Maeda et al.r Physics Letters B 474 (2000) 255–261

Fig. 3. Ža.: 3 S1 NN phase shifts as a function of Erel . The dot-dashed line shows the result of the two-channel Ž 3 S1 NN – 3 D1 NN . calculation. The dotted line includes the additional contribution of 7D1 DD. The solid line represents the full calculation. The crosses show the results of the partial wave analysis w26x. Žb.: Same as Ža. except for 1 S0 NN phase shifts. The dot-dashed line shows the result of the two-channel Ž1 S0 NN – 5 D 0 ND . calculation. The calculation of the dotted line includes 5 D 0 DD. The solid line is the full calculation. See text for details.

sized that the short-range part of the OPEP is known to be quite important in the analysis of single-baryons w15,16x. Before closing the discussion on two-channel calculations, we will comment on channels other than those mentioned above. They show quite small effects, that is, they vary the single-channel result by only a few percent; especially, the channels containing the N3) ) or N4) ) do not provide any visible effects. In Fig. 3 are presented the full calculations, which include all the channels selected above, i.e., the 3 D 1 NN, 7D 1 DD, 1 P1 NN2) , 3 D 1 NN1) ) and 3S1 DD2) ) channels for the 3 S1 NN scattering, and the 5 D 0 ND, 5 D 0 DD and 1S0 DD2) ) channels for the 1S0 NN scattering.

Channel couplings induce an attractive interaction on the NN scattering all over the energy range. Apart from the huge effects of the 3 D 1 NN and 5 D 0 ND channels, the 7D 1 DD and 5 D 0 DD channels have relatively large contributions. In contrast, the other channels containing excited baryons provide small effects. It is remarkable, however, that the channel couplings with these excited states have some influence even at low energies. To show the coupling effects in the low energy region, the deuteron binding energy and the scattering length are calculated and tabulated in Table 4. Although our results roughly reproduce the energy dependence of the experimental NN phase shifts, there still remain some discrepancies. In particular, the calculation provides less attractive interaction in the 1 S0 NN scattering. At this stage, however, we do not expect a perfect fit to the experimental data since we have adopted various approximations in the present calculations. We are now refining the model in several aspects: using more appropriate wave functions for single-baryons, enlarging the number of two-baryon channels, including other OMEP and so on. In this work we have analyzed the NN S-wave scattering by solving the coupled-channel RGM equation with the OGEP, OPEP and confinement potential. The potential parameters have been completely fixed by the single-baryon mass spectra. We have confirmed the large coupling effects due to the channels containing the ‘‘ground-state’’ baryons, i.e., the 3 D 1 NN and 7D 1 DD channels for the 3S1 NN scattering, and the 5 D 0 ND and 5 D 0 DD channels for the 1 S0 NN scattering. On the other hand, only small effects have been found for the channels containing excited-state baryons, such as the 1 P1 NN2) ,

Table 4 Deuteron binding energy ŽB.E.. and scattering length Ž a0 . State

Coupled channel

B.E. ŽMeV.

a 0 Žfm.

3

S1

3

D1 NN 3 D1 NN, 7D1 DD full

– 0.58 1.1

y53 9.1 7.0

1

S0

5

D 0 ND D 0 ND,5 D 0 DD full

– – –

y1.1 y1.5 y1.5

5

I. Maeda et al.r Physics Letters B 474 (2000) 255–261 3

D 1 NN1) ) and 3S1 DD2) ) for the 3S1 NN scattering, and the 1S0 DD2) ) for the 1 S0 NN scattering. These channel-coupling effects are mainly induced by the tensor interaction of the OPEP.

Acknowledgements The authors thank Dr. M. Oka and Dr. T. Sato for useful discussions and suggestions. One of them ŽK.M.. acknowledges the warm hospitality of the Institute for Theoretical Physics, University of Erlangen-Nurnberg, where parts of this work were done. ¨ References w1x M. Oka, K. Yazaki, in: W. Weise ŽEd.., Quarks and nuclei, World Scientific, Singapore, 1984, p. 489, and references therein. w2x K. Shimizu, Rep. Prog. Phys. 52 Ž1989. 1, and references therein. w3x J. Burger, R. Muller, K. Tragl, H.M. Hofmann, Nucl. Phys. ¨ A 493 Ž1989. 427. w4x J. Burger, Thesis, 1985, Erlangen-Nurnberg Universitiy. ¨ w5x M. Oka, S. Takeuchi, Phys. Rev. Lett. 63 Ž1989. 1780. w6x F. Wang, G. Wu, L. Teng, T. Goldman, Phys. Rev. Lett. 69 Ž1992. 2901.

261

w7x A. Valcarce, A. Faessler, F. Fernandez, Phys. Lett. B 345 ´ Ž1995. 367. w8x C. Nakamoto, H. Toki, Prog. Theor. Phys. 99 Ž1998. 1001. w9x K. Shimizu, L.Ya. Glozman, nucl-thr9906008. w10x Z. Zhang, K. Brauer, A. Faessler, K. Shimizu, Nucl. Phys. A ¨ 443 Ž1985. 557. w11x A.M. Kusainov, V.G. Neudatchin, I.T. Obukhovsky, Phys. Rev. C 44 Ž1991. 2343. w12x Fl. Stancu, S. Pepin, L.Ya. Glozman, Phys. Rev. C 56 Ž1997. 2779; C 59 Ž1999. 1219 ŽE.. w13x D. Bartz, Fl. Stancu, Phys. Rev. C 60 Ž1999. 055207. w14x H. Kowata, M. Arima, K. Masutani, Prog. Theor. Phys. 100 Ž1998. 339. w15x L.Ya. Glozman, D.O. Riska, Phys. Rep. 268 Ž1996. 263. w16x L.Ya. Glozman, W. Plessas, K. Varga, R.F. Wagenbrunn, Phys. Rev. D 58 Ž1998. 094030. w17x A.W. Thomas, G. Krein, Phys. Lett. B 456 Ž1999. 5. w18x L.Ya. Glozman, hep-phr9902278. w19x C.D. Carone, hep-phr9907412. w20x S. Capstick, in: D.V. Bugg ŽEd.., Hadron spectroscopy and the confinement problem, Plenum Press, New York, 1996, p. 311. w21x O. Dumbrajs, R. Koch, H. Pilkuhn, G.C. Oades, H. Behrens, J.J. de Swart, P. Kroll, Nucl. Phys. B 216 Ž1983. 277. w22x M.M. Pavan, R.A. Arndt, I.I. Strakovsky, R.L. Workman, nucl-thr9910040. w23x Particle Data Group, Phys. Rev. D 54 Ž1996. 47. w24x L. Trlifaj, Phys. Rev. C 5 Ž1972. 1534. w25x I. Maeda, M. Arima, K. Masutani, in preparation. w26x V.G.J. Stoks, R.A.M. Klomp, M.C.M. Rentmeester, J.J. de Swart, Phys. Rev. C 48 Ž1993. 792.