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NUCLEON-NUCLEON AND HYPERON-NUCLEON INTERACTIONS IN THE QUARK MODEL*,**)
Amand Faessler lnstitute of Theoretical Physics, University of Tiibingen, Auf der Morgenstelle 14,
D-7400 T~bingen, West Germany
The resonating group method is used to calculate the 3S, IS, and P wave phase shifts of the nucleon-nucleon interaet ion in the six-quark model. For large distances, the model is supplement- ed by re, 0, Q, and t0-meson exchange. We show that for meson exchanges which mediate the long-range behaviour we t a n now use the SU 3 flavour rat ios of the meson-nucleon coupling constants even for the r coupling. For the co-meson one had to use in the OBEP's an c0-N coupling twice to three times as large as predicted by the SU 3 flavour to describe the short-range repulsion. We also calculated the nucleon-hyperon interaction and described the NA-scattering in agreement with the data. The model is also applied to the study of the EMC effect.
1. I N T R O D U C T I O N
The phase shifts [1] of the nucleon-nucleon (N-N) scattering are characteristic for a strong short-range repulsion. This short-range repulsion has been described in the past by the exchange of the o-meson between the nucleons [2]. But to obtain the observed repulsion one needed to increase the c0 meson-nucleon coupling con-
2 stant 0o, Nrq/4rr to twice or even three rimes the value predicted by SU s flavour in relation to the Q meson-nucleon coupling n (g,0NN/4rr = 900ZNN/4rr = 4"5). Ail other meson-nucleon coupling constants follow roughly the SUa-flavour relations. The discrepancy for the oe-meson-nucleon coupling reflects the fact that the co exchange is not the mechanism primarily responsible for the short-range repulsion.
With the advent of the quark model and QCD it was suggested that the intrinsic quark structure of the two interacting nucleons could explain this short-range repulsion [3, 4]. These early trials have serious shortcomings: (i) They used [ 3 - 5 ] the Born-Oppenheimer approximation, which would be justified only if the effective mass of the quarks were small compared to that of the nucleon. This is hOt the case in the constituent quark model. Even in the current quark model the energy eigen- value of the quark is about one third of the nucleon mass. (il) Another serious short- coming of these calculations is the neglect of the orbital [42]r symmetry for the six quarks at distance zero between the two nucleons. The importance of the [42]r symmetry has first been pointed out by Neudatchin and coworkers [6, 7].
Oka and Yazaki [8], and Faessler and coworkers [9, 10] showed that the [42]~ symmetry yields hard core phase shifts for the ~S and 3S interaction between two nucleons. This short-range repulsion is strongly influenced by the colour magnetic interaction.
*) Supported by the Deutsche Forschungsgemeinschaft . **) Invited talk to the symposium "Mesons and Light Nuclei IV", Bechynœ Czechoslovakia,
September 5-- 10, 1988.
Czech. J, ehys. B 39 (1989) 9 3 3
A. Faessler: Baryonobaryon interactions in the quark model . . .
In chapter two we summarize shortly the model in detail and discuss a possible mechanism for the short-range repulsion. In chapter three we give the results for the nucleon-nucleon interaction. In chapter four we treat the nucleon-lambda inter- action and compare it with the available data. The nucleon-sigma interaction, which is then determined parameter - free, is treated in chapter rive. In chapter six we then apply the same model to the investigation of the quark momentum distribution in nucleons which are embedded in nuclei (the EMC effect). Finally, in chapter seven we summarize the main conclusions.
2. I N T E R A C T I O N OF TWO N U C L E O N S IN THE Q U A R K M O D E L
At short distances between two nucleons, we describe the nucleon-nucleon (NN) interaction by the exchange of gluons and quarks. The gluon exchange is determined by the quark glu0n vertex.
The one-gluon exchange between two quarks in the non-relativistic reduction is given by [11]:
~s).~a~ ,~,a~[~.l i ~ 6(r,j)(1 + ~y171 + (i) v~176 = ~ : "y �9 2, ~176
mq is the constituent quark mass and OEs = g 2 / 4 n is the strong fine structure constant. The dots indicate terms of tensor and two-body spin-orbit nature of the quark- quark interaction. They do not play a role, if we restrict ourselves to the 1S and 3S interaction between nucleons, but they have to be included for the higher partial waves. The quark-quark interaction (1) is the leading term only for large momentum transfer and, therefore, short distance between nucleons. But even there one should not take (1) literally. Probably it gives only a rough dependence. Its quality is impro- ved by fitting the parameters m u = md, ms, and OEs to the nucleon and the A mass.
The total six-quark Hamiltonian (for different quark masses see eq. (6))
(2) n~=:~ m,+:~]+ 2[voo~'(~,:)-(~,.~~)~~,~] ~= 1 2 m q . I i < j
must include also a colour confinement terre. The parameter a [MeV fm -1] is adjusted to the charge root mean square radius of the proton including the pion cloud [213.
For large distances one cannot exchange colour objects like quarks and gluons. Thus, we go back there to meson exchange. But we have to guarantee asymptotic freedom. This is done by allowing the coupling of the mesons with the quarks only near the surface of the nucleon.
(3) gqq,(r) = cltr 2 .
The free parameters c, will be adjusted for each meson Ix to the meson-nucleon
934 Czech. J. Phys, B 39 (t989)
A. Faessler: Baryon-barvon interactions in the quark model.. .
™ constants [13] at zero momentum transfer g~NN(q = 0)/4r~ = 14"1, #ZNN(q = 0)/4~ = 5"65, and g~ZNN(q = 0)/arc = 0"5. For the co-meson coupling we shall sec that the SU a flavour value 2 2 9,~NN = 99~NN yields a good agreement for the NN phase shifts. This is opposed to the OBEP's [2, 13] where the c0-nucleon r constant has to be blown up by a factor of two to three to describe the short- range repulsion.
In a first step we include only the exchange of the two lightest mesons (m, = = 140/MeV; my = 520 MeV). The ansatz for the resonating group wave function is
(4) ~/6q = A{]NN) ZN(r) + ISA> za(r) + ]oE) zc(r)} �9
The Kohn-Hulthen variational principle
(5) ™ - El~6q) = 0
yields for IS and 3S channels three coupled integral equations for the relative wave functions gN(r), zA(r), and �9
The above basis states [NN), [AA), and ]CC) contain no dependence on r. ST and colour C are coupled only with the symmetry conjugate to the orbital one. The states [NN), [~�9 and [CC) contain also the internal spatial variables for each nucleon. A solution of the resonating group problem (5) yields for each symmetry its own radial dependence.
One can show that the orbital symmetry [42]r is forbidden for the Os oscillator part of the orbital function iN(r). This requires anode in the orbital [42] part of the relative wave functions in the NN channel. If we had only [42]r, the node would guarantee a hard core phase shift. (One should stress that the relative wave function is not unique, For the [42]r symmetry, one can add any admixture of Os oscillator functions without changing any observable since the antisymmetrization removes any Os contribution. This can remove the n6de in the relative wave function [14].)
3. RESULTS
The solution of the coupled system of resonating group equations yields the relative "wave functions zN(r), zA(r), and Zc(r) for the channels with two asymptotic nucleons, Xwo asymptotic A's and the six-quark hidden colou r state, respectively. The asymptotic form of the relative wave function in the NN-channel gives the phase shift for the nucleon-nucleon scattering. This calculation is performed including quark and gluon ~exchange H 6 and the exchange of ~- and er-mesons. The parameters are adjusted as described in chapter 2. The results are given for 1S and •S in figures 1 and 2, Tespectively.
The results presented in figures 1 and 2 include only the exchange of two lightest mesons, the n- and y I t i s interesting to sec what happens if in addition, we include also, the ~- and the c0-mesons. This question is especially exciting, since
(~ztch, J, Phys. B 39 (1989) 9 3 5
1oo
50
¤
J~
o .
o 0 0 9
A. Faessler: Baryon-baryon interactions in the quark model. , .
.,'\
'\\ ",
\ ', \ ', \ ', \ ', \ ",, \,,
0 I ( )0
i I .... r i
15 0
--- Arrœ et et . . . . . Ar'ndf
. . . . . . . . . . T h e o r y
- ZL ' : .
i
200 3| & ~o
EIy MeVI
100
eu
o OE
i
0
i
r r : =
3S I
- - - A r n d t er et
- . . . . . . . T h e o r y
k',
\ - . \ - .
\ , .
\ . . .
"-2L..... r i ~ "i ~'--
100 200 300 ~00 5(X~
E 1 0 b (MeV }
Fig. I. Fig. 2.
Fig. 1. IS nucleon-nucleon phase shifts as a function of the laboratory energy. The dotted curve gives the results of the theory presented in [12]. The dashed and the dashed-dotted lines are two different sets of experimental phase shifts [15]. (mq = 336 MeV, OEs = 1"3, and ar = 41 MeV _
�9 fm- z).
Fig. 2. 3S nucleon-nucleon phase shifts as a function of the laboratory energy of the nucleor~ projectile. The dotted curve is the present calculation [12] and the dashed curve represents the
experimental phase shifts [15]. (For parameters see caption of fig. 1).
in the one-meson exchange potentials the co-meson is solely responsible for the short-range repulsion. For this reason, one adjusts in the OBEP's the co-nucleon coupling constant to a value which is by a factor o f two to three rimes larger than the flavour S U 3 value derived from the Q-nucleon coupling. If we now had the correct nature of the short-range repulsion, we should get a satisfactory fit to the phase shifts, if we used the flavour S U 3 value g~NN/4rC = 4"5. Indeed, this is the case. These results support strongly our conviction that we found really the correct nature o f the short-range nucleon-nueleon repulsion.
Recently, we have extended this model to include also the finite size o f the pions . This has the advantage of taking into account the pion cloud in determining th› energy o f the proton, and still keeping the nucleon stable at a finite radius [ 1 5 - 1 7 ] . F o r a point pion the pion self-energy exerts such a pressure on the nucleon that it is~ compressed to zero radius. In addition, we calculated also the 1P l and averaged 3p: partial waves in good agreement with the data.
9 3 6 Czer J. P h i . B 39 (19'~9)~
A. Faessler." Baryon-baryon interactions in the quark model...
4. THE LAMBDA-NUCLEON INTERACTION
The lambda-nucleon (A-N) can be calculated in the same way as the nucleon- nucleon interaction. The one-gluon exchange (1) must now include the different masses for the up and down quarks and for the strange quark. We choose this mass by 150 MeV larger than the up and down quark masses [18].
[ ~ ~ - - ( y ' -~ y (.) 1 rc~5(ri1 ) 1 + ~ + 4 . ( 6 ) vOGEP(l'j) = --4 " "~i 2 mj 3 mdny /A
The quark and gluon exchange between the hyperon and the nucleon yield the short-range part of the interaction. One has to add the confinement potential (2) and,
�9 for large distances, the meson exchange. The meson exchange potential is a generaliza- tion of the pion potential of ref. [19] that include all pseudoscalar mesons and also mixing of the octet qs with the singlet 1 h . A mixing angle of 0 = - 2 3 ~ is assumed.
(7) vPSM(i,j ) 1 qqqM #2 A 2 -- e-.U2b2/3 3 4r~ 4 M A M B A 2 _ /./2
with
• ~ [e-"'__�9 L #r~j
A 3 e -Ar i j 1 O F
#3 Ar~j A y " y " ~J
(8) 3 = _3 14"2, qqql -- gNN~t ---- - ~ gNNn �9 gqq8 = ? qNN~ 5
Here p i s the mass of the pseudoscalar meson in question, A is a size parameter describing the finite size of the meson, M A and M B are the masses of the baryons A, B, respectively, between which the mesons are exchanged, and O~~ is an operator in flavour space according to table 1.
There are two independent coupling constants 9qqS and 9qq 1 describing the coupling ,of the ocrer or singlet mesons with the quarks. Because the coupling constants are assumed to be independent of flavour, they can be determined by looking at the
Table 1. Meson parameters.
/t A OF. (MeV) (MeV) ~~
K
n q,
3
138 832 E z�9 ~,~(j) K = I
7
495 832 E Z�9162 2�9
549 4000 eos 0 ~F(i) 2F(]) - - sin 0
958 4000 sin 0 ).8F(i) �9 F(./) + COS 0
~Czech. J. Phys. B 39 (1989) 937
A . Faessler: B a r y o n - b a r y o n i n t e r a c t i o n s in the q u a r k m o d e l . . .
long-range part of (7) and comparing it with the one-meson exchange potential fo r the NN interaction. In this way, one finds the relations given in eq. (8 ) .
In addition to these potentials acting on the quark level, we add a phenomenological o-meson potential on the baryon level with a microscopically calculated form fac tor [20] representing the contribution of two-pion exchange.
1 -- e -mr -- e - 2 m R sinh (mr)
(9) V~(r) = y 1 for r < 2R, an 2m2R2r rcosh (2mR) - 1] e -mr
for r < 2R.
Here m = 520 MeV is the mass of the y and R = 0.72 fm is a size parametev fitted to the form factor.
Table 2.
Parameter sets for OGE.
b (fm) 0"50 0"55 0"50 0"55
m u = m a (MeV) 313 313 313 313
m s (MeV) 535"2 535"2 425.4 443-2
~s 0"82 1"09 0"26 0"48
ar (MeV/fm 2) 85"2 47" 0 58" 3 33-2
( M A - - MN)meso n (MeV) 0 0 74"7 61-9
(M z -- MN)meso n (MeV) 0 0 135"3 114"2
g ~ / 4 ~ 0 0 2"3 2"6
A p e last ic c ross sec t ion
\ \b=o.~f~ ~oo \ ~ o~~ . . . . ~ o , .
\ \ 400 ~ ~ ~Sech,-Zorn68 i
~ ' \ \ : ,~,:od~k I ]
loo
. . . . . . . . . . . . . . . . . . , t , , , , . . . . . . . i - , , , ,
o so ~oo lso 200 2so 300 350 400 450 soo sso
PTQb rMeV//cI
Fig. 3. Total elastic cross-section for the 5p-scattering as a function ofthe incident A-momentun~
in the laboratory frame. Two theorctical curves are compared with the experimental data frorrL
refs. [21--23]. The theoretical curves are distinguished by the sets of parameters adjusted fo r two different oscillator lengths b = 0"5 fro and b = 0"55 fro.
938 Czey J. Phys. B 39 (1989)~
A. Faessler: Baryon-baryon interactions in the quark model. . .
In the kinetic energy, we use an average quark mass. These approximations that use a mean mass in the kinetic energy are unavoidable, if one uses the saine symmetry space as for the nucleon-nucleon interaction. The parameters of t h e short-range part o f the interaction are fixed by requiring that the mass differences MA -- MN = = 177 MeV, ME -- Mr~ = 253 MeV be reproduced, and that the stability condit ion d M N [ d b = 0 be satisfied. The parameters obtained by this procedure are shown in table 2 for two values o f the oscillator length b without and with the meson cloud included. One sees that the meson cloud contributes to about one half o f the mass splittings.
Figure 3 shows the total elastic cross-section for the Ap-scattering as a function of the incident A-momentum in the laboratory frame. The theoretical results are compared with the experimental data given in the references [ 2 1 - 23].
The comparison between theory and experiment for the A-proton scattering shows a surprisingly good agreement.
5. THE SIGMA-NLICLEON INTERACTION
For the Y,-nucleon interaction we use the same parameters as for the A-nucleon interaction. Thus, this part o f the calculation is totally parameter-free. Figures 4 and 5 show the results for the T = 1/2 3S 1 and 1S o and T = 3/2 3S~ and 1S o phase
' I i I i I i I
60 ~N T--~
20 ~ b--o~o~ .... ~ -b=O_~Sfm
3 $1
- I00 f f ~ ~ - 1 0 0
0 50 1 0 0 1 5 0 200
z.~ [Mv]
-20
- 6 0
l
ci ,r- 0-
C~ ,..o,
I
c7 r-
60
20
-20
-60
I I I I
~N T=3/2
............ ~ ........................................... - x .
S 0
~~ ~~t 50 100 150 200
E=. [~,v]
Fig. 4. Fig. 5.
Fig. 4. Phase shifts of the Y.-nucleon scattering for the T = 1/2 3S 1 and 1S o partial waves. The results are given for two different oscillator lengths b = 0"50 fro and b = 0"55 fm, corre-
sponding to two different root rnean square radii of the quark content of the baryons. Fig. 5. Z-nueleon T = 3/2 3S l and 1S 1 phase shifts for b = 0"50 fm and b = 0"55 fm.
C z e c h . J . P h y s . B 3 9 ( 1 9 8 9 ) 939
A. Faessler: Baryon-baryon interactions in the quark model.. .
shifts, respectively. The results are given for the parameters in the last two columns of table 2. For the A-N and the Y~-N splittings they include also the contributions of the meson clouds. The results are given in table 2 and in figures 4 and 5 for two different oscillator lengths b - - -0 .50 fm and b = 0.55 fm, corresponding to two different root mean square radii of the quark content of the baryon b = (re) ~/z.
6. THE EMC EFFECT
We have also extended [24] the method described above to the investigation of the EMC effect [25, 26]. The quark momentum distribution in nucleons inside is calculated for the nucleon-nucleon scattering in the nucleus by solving a generalized Bethe-Goldstone equation with quark degrees of freedom in a nonrelativistic quark model, including Pauli blocking due to other nucleons and six-quark bags with different radii [24]. The structure functions [26] of a nucleon in 56Fe and ~~ are calculated including also the binding effects [27] due to the average binding energy [e] of a nucleon, which leads to rescaling of x.
(10) x A = xfree MN = Q2
M~-I~1 2Pq"
0
0
' ' 1 ' ' ' I ' ' ' I ' ' '
with correl. without eorrel.
tf ',§
ff
, . . I , , . I , , , I , , , I , . .
0.0 0.2 0.4 0.6 0.8 1.0
X Fig. 6. The ratio of the structure function of a nucleon in iron to that of a nucleon in deuteron. The solid line and dashed line denote results with and without NN correlations, respectively.
9 4 0 Czech. J. Phys. B 39 (1989)
A. Faessler: Bao, on-baryon interactions in the quark model. . .
Figure 6 gives the ratio of the structure functions F 2 of Fe and Ag to that of the deuteron, as a function of the scaling variable x.
The comparison of the results with the data in figure 6 shows that the binding energy of the nucleons H is important. But it also shows that the Pauli effect and the short-range correlations are decisive for the good agreement. They are described by a Quark-Bethe-Goldstone equation on the quark level.
7. CONCLUSIONS
The NN phase shifts are calculated using the quark model with a QCD-inspired quark-quark force. The short-range part of the NN force is given by quark and gluon exchanges. The long-range part is described by zr and o-meson exchange. The data fitted to the model are rive values connected with three quarks only: The nucleon mass, the A mass, the root mean square radius of the charge distribution of the proton including the pion cloud, the zc-N and ~-N coupling constants at zero momentum transfer. The 1S and 3S phase shifts are nicœ reproduced. The short-range repulsion is decisively influenced by the node in the [42]r relative wave �9 The colour magnetic quark-quark force which enlarges the [42]r admixture is very important. We are also able to describe the higher phase shifts.
In the OBEP's the short-range repulsion is connected with the exchange of the oe-meson. But to reproduce the short-range repulsion one had to blow up the c0-N coupling constant by a factor of 2 to 3 as compared to the SU3 flavour. With quark and gluon exchanges, the best fit to the co-N coupling constant lies close to the SU3 flavour value. This fact strongly supports the notion that we have round the real nature of the short-range repulsion of the NN interaction.
In chapter 4 we have applied the same theoretical description to the A-nucleon scattering including quark-gluon exchange for the short-range interaction and the exchange of the whole pseudo-scalar octet and singlet mesons (g, K, 11, 11') for the long range. For the elastic A-proton cross-section we obtain a good agreement with the data.
In chapter rive, the same ideas are applied to the nucleon-sigma interaction. The nucleon-sigma cross-sections can be reproduced without additional parameters. In chapter six, we investigated the momentum distribution of the quarks in nuclei (the EMC effect). These calculations contain the effect of the nuclear binding and the modifications of the quark wave functions by the interaction between the nucleons during the collisions. The results show that a large part of the modification of the quark momentum distribution can be explained by these effects.
I would like to thank Dr. Brfiuer, Dr. Fernandez, Dr. Shimizu, and Dipl.-Phys. Straub, with whom I was working on the results reported above.
Received 1 November 1988
Czech. J. Phu B 39 (1989) 94] .
A. Faessler: Baryon-bao'on interactions" in the quark model. . .
ReJ•
[1] Arndt R. et al.: Phys. Rev. C 15 (1977) 1002. Arndt R.: in Nucleon-Nucleon Interaction. 1977 Vancouver Conference, p. 117,
[2] Holinde K.: Phys. Rep. 68 (1981) 191. [3] Liberman D. A.: Phys. Rev. D 16 (1977) 1542. [4] De Tar C. E.: Phys. Rev. D 17 (1978) 323. [5] Harvey M.: Nucl. Phys. A 352 (1980) 301 and 326. [6] Neudatchin V. G., Obukhovsky I. T., Kukulin V. 1., Golanova N. F.: Phys. Rev. C 11
(1975) 128.
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[15] Strobel G., Brfiuer K., Faessler A.: Nucl. Phys. A 347 (1985) 605. [16] Furui S., Khadkikar S. B., Faessler A.: Nuel. Phys. A 347 (1985) 619. [17] Br~iuer K., Faessler A., Fernandez F., Shimizu K.: Z. Phys. A 320 (1985) 609, [18] Faessler A., Straub U.: Phys. Lett. B 183 (1987) 10. [19] Fernandez F., Oset E.: Nucl. Phys. A 455 (1%6) 720. [20] Faessler A., Fernandez F., BrRuer K., Kuyucak S., Shimizu K.: T/ibingen preprint, I985~
unpublished. [21] Ale• G, et al.: Phys. Rev. 173 (I96b) 1452. [22] Sechi-Zorn B. et al.: Phys. Rev. 175 (1968) 1735. [23] Kadyk J. A. et al.: Nucl. Phys. B 27 (1971) 13. [24] Kurihara Y., Faessler A.: Nucl. Phys. A 467 i19~7) 621. [25] Aubert J. J. et al.: Phys. Lett. B 123 (1983) 275. [26] Arnold R. et al.: Phys. Rev. Lett. 52 (1984) 727. [27] Birbrair B. L. et al.: Phys. Lett. B 166 (1986) 119.
942 Czech™ L Phys, B 39 (1989)
