Chiral constituent quark model predictions of electroproduction of the Δ on the nucleon and 3He

N ELSEVIER

NUCLEAR PHYSICS A Nuclear Physics A629 (1998) 180c-189c

Chiral Constituent Quark Model Predictions of Electroproduction of the A on the Nucleon and SHe T.-S. H. Lee a b

~Argonne National Laboratory, Argonne IL 60439-4843 USA bResearch Center for Nuclear Physics, Osaka University, Osaka, Japan We show t h a t the phenomenological A isobar models can be related to the chiral constituent quark model. The parameters needed for predicting A dynamics in nuclei can therefore be determined, as has been done in a recent investigation of 7rN and 7 N reactions. A preliminary prediction for SHe(e, e ' A ++) reaction is presented for testing the A components in SHe predicted by an 87-channel Faddeev calculation for three-nucleon systems. 1. I N T R O D U C T I O N The phenomenological models of A excitations in nuclei have been fairly successful in understanding very extensive d a t a of intermediate energy nuclear reactions and nuclear properties at high densities. In the first part of this talk, I will briefly review the main features of these phenomenological models. I then will discuss how these models can be related to a chiral constituent quark model we have developed [1,2] recently in a s t u d y of 7rN and 7 N reactions. In the last p a r t of the talk, I will discuss how this finding can be used to quantify the predictions of A components in nuclei and how these predictions can be tested by an on-going experiment [3] on SHe(e, e'Jr+p) reaction, as suggested by Lipkin and me in Ref. [4]. The results for SHe are from a calculation involving a collaboration with Chmielewski and Sauer of the University of Hannover. 2. P H E N O M E N O L O G I C A L

ISOBAR MODELS

The need of the A degree of freedom to describe nuclear properties was already suggested in the 1960's. However, most of our understanding of A dynamics were obtained in the s t u d y of pion-nucleus interactions. At intermediate energies, the dominant pionnucleus interaction is the excitation of a nucleon to a A resonance. The simplest approach for describing this resonant nuclear dynamics is the isobar model in which a fictituous bare A particle with spin and isospin equM to 3/2 is introduced to couple with 7rN or 7 N channels. W i t h appropriate nonresonant mechanisms, these vertex interactions can be a d j u s t e d to describe the d a t a of 7rN and 7 N reactions. The Hamiltonian for such a phenomenological isobar model takes the following form = Ho + H~, 0375-9474/98/$19 1998Elsevier Science B.V. PII S0375-9474(97)00684-2

(i)

T-S.H. Lee~Nuclear Physics A629 (1998) 180c-189c

181c

where H0 is the sum of the free energy operators for ~, N, and A degrees of freedom, and

H, ----E h A ~ + E v~,z,

(2)

with c~,/3 -- ~N, 7 N channels. Assuming that the nonresonant mechanism v~,z can be described by meson-exchange mechanisms, such a model has been constructed recently by Sato and me [1]. As illustrated in Fig. 1, the model can describe very well the data of ~/N --~ ~rN reactions.

40

3oi

' ~o

I

I

I

(a) '60 o'

i

ANL-P-22,155 I

I

I

9~,~o_

~" 20 1o I

I

120°

0

I

I

~o

I

150°

!

1.0 (b) '60 ° 0.80.6

,

,

,

|

I

I

0.8 ~ 0.6 0.4 0.20

I

90°

0.4 0.2 o

I

~o

~:o

-~

I

,

,

,

I

I

I

I

200 250 300 350 200 250 300 350 400 E7 (MeV) Figure 1. The differential cross sections ( d a / d ~ ) and photon-asymmetry ratio (/~) of the 3'N --* 7rN reaction predicted in Ref. [1] are compared with the data.

The details of our model can be found in Ref. [I]. Here I only want to point out that an essential feature of the model is that the resonant structure of the predicted cross section is due to a A propagator with an energy-dependent mass shift E(E) GA(E) = E -

1 MR-- ~(E) "

(3)

T-S.H. Lee~Nuclear Physics A629 (1998) 180c-189c

182c

The self-energy E ( E ) is generated dynamically by the interaction H± of Eq. (2). It is real at energies below the ~rN threshold, and becomes complex above threshold. The mass of the bare A particle, MR = 1300 MeV, is considerably larger than the value 1236 MeV of the observed A resonance energy. Of course the mass shift E(E) will make up the difference and give the correct resonance position. At E = ER = 1236 MeV, a correct model of //i of Eq. (2) must yield MR + ReaI(E(ER)) = ER. With the nonresonant mechanisms constrained by the well-studied meson-exchange models, the main outcome of the model is the determination of the bare A mass MR and the vertex interactions hA~N and hA*--*•N. To describe A excitations in nuclei, we need to extend the interaction term Eq. (2) to account for two-baryon interactions. Such a model, called ~ N N model, has been developed by several groups. In our approach [5], the hadronic part of Eq. (2) is simply extended to

HI = •

[hA~N(i) + V~N,~N(i)] +

E

VB,B2,B'~B~+ H~NN~NN,

(4)

B1B2,B~B~

i=1,2

where B = N or A. The transition operator HIrNN~NN is introduced to account for the pion production through non-A mechanisms, such as that due to pion rescattering in non-P33 channels.

I

'

I

'

20 - On =14.5

I

ANL-P-22,152 I

'

PP "'~ n P g +

9~=42

~ t

., ...C}

i

!p ..... ~

0

I

Op

i =14.5

I

'

I

_

,

I

?",,, , y . \

400

600

Figure 2. The differential cross sections of compared with the data.

800 P ( MeV/c ) pp ~

npTr +

1000 reaction predicted in Ref. [5] are

Assuming that the two-baryon interactions can be described by the meson-exchange mechanisms, a 7rNN Hamiltonian of the form of Eq. (4) was constructed [5] to give a fairly

T-S.H. Lee~Nuclear Physics A629 (1998) 180c-189c

183c

good description of very extensive data of NN and ~rd reactions up to the A excitation energy region. As an example, we show in Fig. 2 that the model can describe very well the pp --+ nplr + reaction in both the regions dominanted by the A excitation(solid curve) and np final state interaction (dashed curve). Here I should mention that the success of the model depends crucially on the dynamics contained in the A propagator which has a form similar to Eq. (3) but with an additional dependence on the momentum of a spectator nucleon. Again the resonant position is determined by the interplay between the mass MR of the bare A and the self-energy that is generated dynamically from the Hamiltonian Eq. (4). The Hamiltonian Eq. (4) implies that a nuclear system can have A components. By using a similar model with pion degree of freedom suppressed, the Hannover Group [6] had carried out an 87-channel Faddeev calculation for A = 3 bound states. They found that the A component in 3He is about 2.4~0. By using the model Hamiltonian Eq. (4) to generate the Bruckner G-matrix, the A components in heavy nuclei have also been calculated [7] recently within the contex of the nuclear shell-model. We found that the A component in a heavy nucleus, like 9°Zr, is only about 170, but can be increased to about 10% when the system is compressed to high densities accessible to relativistic heavy-ion collisions. To facilitate the study of such a A-rich system, the model has also been used [8] to predict the binary collison cross sections involving a A in either the entrance or the exit channels. These cross sections can not be measured experimentally but are crucial input to tranport equation calculations of relativistic heavy-ion collisions. In Fig. 3, we show some of our predictions of total cross sections.

ANL-P-21,874

0

40 ~'30

~10

'

I

. . . .

!

--

"

I

. . . .

I

'

I

I

I

"-...Nh*--+ NN Nz~--> N,~ ""-

20 10

0

5OO

1000 E L (MeV)

1500

2000

Figure 3. The t o t a l cross sections for N N and N A collisions predicted in Ref. [8].

T-S.H. Lee~Nuclear Physics A629 (1998) 180c-189c

184c 3. C H I R A L

CONSTITUENT

QUARK

MODEL

The question I would like to address now is whether the phenomenolocal isobar models described above can be understood in terms of quark substructure of N and A. The first step to answer this question was taken in Ref. [1] by comparing the phenomenologically determined A +-+ ~/N transition amplitudes with the constituent quark model predictions. W i t h i n the model defined by Eqs. (1)-(2), the A excitation has two components. The first one is the direct excitation due to the bare vertex interaction h~,yN, The second one is due to the nonresonant mechanism V,./N,~N followed by the 7rN -~ A transition which is also dressed by the ~ N scattering states and the nonresonant interaction V~N,7~N. This is illustrated in Fig, 4.

ANL-P-22,154

N

= dressed

+

x

bare

Figure 4. The A excitation in ~/N reaction. See text for explanation.

The calculated helicity amplitudes for the A --+ 7N transition are listed in Table 1 and compared with the predictions from two constituent quark model calculations [9,10]. We observe that the bare values are in good agrement with the constituent quark model predictions, while the dressesd values are close to the values listed by Particle Data Group (PDG). The differences can be understood [1] by noting that the contributions due to the dressing by 7rN scattering states (second term in the right-hand side of Fig. 4) are included in the empirical PDG values from amplitude analyses, but are not included in the existing constituent quark model calculations. Our results have provided an explanation of a longstanding puzzle and had indicated that the bare A in the phenomenological isobar model can be identified with the A state of the constituent quark model. Table 1 The predicted A --. 7N helicity amplitudes are compared model predictions [9,10]. A

A3/2 A1/2

PDG -257 ~ 8 -141 i 5

Dressed -228 -118

Bare -153 -84

with two constituent quark

Ref. [9] -157 -91

Ref. [10] -186 -108

The next step is to see whether the determined bare mass and the vertex h A ~ N can also be identified with a constituent quark model. This was explored in Ref. [2] within the

T-S.H. Lee~Nuclear Physics A629 (1998) 180c-189c

185c

chiral constituent quark model in which pions are treated as Goldstone bosons resulting from QCD dynamics and are coupled directly with the constituent quarks inside hadrons. This model was discussed by Dr. Glozman in his talk at this conference. Here I just mention the essential steps in relating this model to QCD. Because the masses of up, down, and strange quarks are small compared with the typical hadron scale about 1 GeV, QCD is approxiinatelly invariant under [SU(3)]L × [SU(3)]R chiral symmetry. Because parity-doubling spectra are not observed experimentally, this symmetry must be broken spontaneously. The hadron structure can then be described equivalently by an effective Lagrangian for constituent quarks interacting with octect mesons. It also contains a part including all complications involving gluons. In the low energy region, one can integrate out the short-range or high-momentum dynamics and obtain a Hamiltonian with glnonexchange and pion-exchange interactions between constituent quarks. The above rationale led us [2] to assume that a Hamiltonian for describing the ~N and ")IN reactions in terms of the quark substructure of N and A is of the following form

B,B'

where B,/3' =- N, A are eigenstates of a one-baryon Hamiltonian HB which is defined by p? HB

"7 \

2raq]

i>j

where V °aE, V c°ny, and V ~ are respectively the gluon-exchange, confinement, and pionexchange potentials. The vertex interactions in Eq. (5) are calculated from the baryon wave functions and quark operators

fB,~B' = ( B I Z f~q,q(i)lB' ) ,

(7)

i

fB,.~B' = ( B IY~ f~q,q(i)lB').

(8)

i

Following the previous constituent quark models, the quark operators are defined by the following matrix elements +,

1

ei (/~ ÷ / 7 ~ - i g i x / 7 c ) . e~*~,(k" )a(Pi ~' - ~ - k ),

1

(Pi I f~q,q I / ~ ) = x / ~ V/(27r)3 2mq

(9)

where ea(fc) is the photon polarization vector, and

1

1

f~qq (icri •/~)e(y', -- ~ -- k ) .

(10)

m~r

We assume that the vertices defined by Eqs. (7)-(10) for B = A and B' = N are the vertices hA,.~N and hA,~N of the phenomenological isobar model defined by Eqs. (1)(2). Accordingly the A mass generated from the one-baryon Hamiltonian Eq. (6) must be identified with the mass 1300 MeV of the bare A. If this conjecture is correct, the

T.-S.H. Lee~Nuclear Physics A629 (1998) 180c-189c

186c

predicted A --+ 7 N transition amplitudes must be close to the bare values listed in Table 1. To establish this, we need to define the p a r a m e t e r s of the Hamiltonian Eq. (6) for solving the three-quark b o u n d state problem. The model defined by Eqs. (6)-(10) has four parameters: quark masss mq, a~ of the confinement potential, as of the one-gluon-exchange potential, and the coupling constant f~qq which also defines the strength of the one-pion-exchange potential Vi~. We choose the nucleon mass as the zero-point of the Hamiltonian. Furthermore we assume t h a t the nucleon internal wavefucntion only has s-wave components. Then the quark mass has to be about 350 MeV to reproduce the proton magnetic moment. The coupling constant f~qq is fixed by using Eq. (7) to reproduce the known value of 7rNN coupling constant. The remaining two parameters ac and as are t h e n adjusted to obtain a mass 1300 MeV for the lowest J = T = 3/2 A state. This bound state calculation yields the baryon wavefuctions which are needed to evaluate the m a t r i x elements Eqs. (7)-(8). We employed a variational m e t h o d to solve this bound state problem within a model space spanned by harmonic oscillator basis wavefunctions up to N = 4 excitation. Our results for the magnetic MI+ and electric El+ transitions are listed in Table 2. Note t h a t the magnitude of EI+ measures the L = 2 components in the N and A states. W i t h i n the constituent quark model, it was known t h a t very little L = 2 component can be induced by the one-gluon-exchange potential. This turns out to be also the case here and the nonvanishing value of El+ reflects mainly the tensor coupling of pions with constituent quarks. We see that the calculated 11/i+ and El+ amplitudes are in good agreement with the bare values of the A isobar model defined by Eqs. (i)-(2). Furthermore, the phenomenologically determined vertex function hA~N is found to be of the same range of the A --+ ~rN form factor calculated by using Eq. (7). These results are important since the predictions, such as the A component in 3He [6] and the AN cross sections shown in Fig. 3, made previously based on the Hamiltonian Eq. (4) can now be related to the quark substructure of N and ZX. This is a step, perhaps a very small step, toward understanding nuclear dynamics within QCD.

Table 2 The magnetic MI+ and electric El+ predicted by the chiral constituent quark model axe compared with the bare vaues determined in Ref. [I]. I~A--+TN Bare values of Ref. [1] Chiral Constituent Quark Model

4. P R E D I C T I O N S

MI+ 175

176

El+ -2.28 -1.92

REM -1.3% -I.09%

FOR 3He(e, e'A++) REACTION

To make further progress, we need to look for some decisive experimental tests. There are two possibilities. The first one is to extend the baryon structure calculation based on Eqs. (6)-(10) to investigate higher mass N* nucleon resonances and examine whether the predicted N* --~ 7 N and N* --+ ~ N are in agreement with the d a t a of 7rN and

T.-S.H. Lee~Nuclear Physics A629 (1998) 180c-189c

187c

~N reactions. This is being pursued in a collaboration with Arima, Sato, Ogizai and Yushimoto. The second one is to follow Ref. [4] to use the 3He(e, e'zc+p) reaction to test the A component in 3He that had been predicted by the Hannover 87-channel Faddeev calculation [6]. It is important to note that the the calculation of the A components in nuclei is sensitive to the 7r - NN and 7~ - NA form factors in the NN --+ NA transition potential of the Hamiltonian Eq. (4). These form factors used in the Hannover calculation are very close to what we can now calculate by using Eqs. (7) and (I0). Thus the 2.4 percent A component in 3He predicted by Hannover group can be considered as the consequence of the chiral constituent quark model. The experimental verification of this prediction is therefore more interesting than what was originally suggested in Ref. [4]. ANL-P-22,075

3He()

A*÷

j.

/

(a)

f

A+

"I" / ~'

(

p

n

(b) Figure 5. Basic mechanisms

for 3He(e, e'Tr+p) reaction. See text for explanation.

Let us now follow Ref. [4] and consider electron scattering from 3He with the production of a A++ which then decays into a 9r+p state. There are two basic mechanisms for this reaction, as illustrated in Fig. 5. The first one is the direct knockout of a A ++ which then decays into a ~r+p state. The second one is the excitation of a proton in 3He to a A + (or a proton) which is then converted into a A ++ by charge-exchange with the second proton. The second process can occcur in the absence of A components in 3He. Note

T.-S.H. Lee~Nuclear Physics A629 (1998) 180c-189c

188c

that the relative importance of these two competing processes is completely fixed since all vertex interactions in Fig. 5 can be calculated from the same chiral constituent quark model. Of course we need to use the full Hamiltonian defined by Eq. (4) to account for the NN and NA multiple scattering effects. This can be done by solving coupled NN ® NA 0 7~NN scattering equations, as developed in Ref. [?]. The calculation of 3He(e, e~A ++) cross sections thus does not involve any adjustable parameters. Clearly the prediction of the A component in 3He can be tested only in the kinematic region where the A ++ knockout process dominants. Our preliminary results are shown in Fig. 6. We see that the 3He(e, e~A ++) corss section is the largest at forward angles with respect to the photon direction and is about I0 #b/sr. The charge-exchange process is much weaker although its effect must be included in a realistic calculation. At large momentum-transfer, the A ++ knockout process completely dominants. However the cross sections may be too small for any experiment to measure in the near future.

102

i

I

I

ANL-P-22,151 I OA++ -

=

w

,~

5

o

:

.eo

"C" 10 "~

--~

10 -1 10"2

~

\

lO-3 10"4~10"51 0

\

" i

\ \

\i

2 3 O2((GeV/c)2)

i 4

1 5

Figure 6. The predicted differential cross section of 3He(e, e ' A ++) reeaction at 0A++ = 5 ° with respect to the direction of virtual photon. See text for explanation.

In conclusion, I have shown that the phenomenological isobar models for the A excitations in 7rN, ~N, and NN interactions can be identified with the chiral constituent quark model. The A-nucleus dynamics predicted by using the model Hamiltonians Eq. (4) can now be related to the quark substructure of N and A. A crucial test of this conclusion is the experimental verification of our prediction of 3He(e, e~A ++) reaction. The extension of this work to investigate higher mass N* nucleon resonances is being pursued. This work is supported by the U.S. Department of Energy, Nuclear Physics Division, under contract W-31-109-ENG-38.

T-S.H. Lee~Nuclear Physics A629 (1998) 180c-189e

189c

REFERENCES

1. T. Sato and T.-S. H. Lee, Phys. Rev. C54 (1996) 2660. 2. P.N. Shen, Y.-B. Dong, Z.-Y. Zhang, Y.-W. Wu, and T.-S. H. Lee, Phys. Rev. C55 (1997) 2024. 3. M. Cuss, contributed paper to QUEN97, these proceedings. 4. H . J . Lipkin and T.-S. H. Lee, Phys. Lett. B183 (1987) 22. 5. T.-S. H. Lee and A. Matsuyama, Phys. Rev. C32 (1985) 516; C34 (1986) 1900; C36 (1987) 1459. 6. C. Hajduk, P. U. Saner, and W. Strueve, Nucl. Phys. A405 (1983) 581. 7. M.A. Hansan, T.-S. H. Lee and J. P. Vary, in preparation. 8. T.-S. H. Lee, Phys. Rev. C54 (1996) 350. 9. R. Bijker, F. Iachello, and A. Leviatan, Ann. Phys.(N.Y.) 236 (1994) 69. 10. S. Capstick, Phys. Rev. D46 (1992) 2864.