the nuclear shell model - of nucleons or quarks?

Volume 205, number 2,3 PHYSICS LETTERS B 28 April 1988

T H E N U C L E A R S H E L L M O D E L - OF N U C L E O N S OR QUARKS?

Igal TALMI The Weizmann Institute of Science, 76100 Rehovot, Israel

Received 15 June 1987

A nuclear shell model of independent quarks is critically examined. Differences between the quark shell model and the conventional shell model are pointed out. The uncorrelated quark wave functions extending over the entire nucleus differ markedly from those in physical )-nucleons. Results of some reactions seem to contradict the description of nuclei in terms of the quark shell model.

The internal structure o f nucleons in nuclei is of great interest and has been the subject o f many ex- perimental and theoretical investigations. A few years ago a rather extreme model was suggested in which the 3A quarks move independently in the whole volume of the nucleus [ 1-4 ]. The individual quarks are assumed to occupy the various j-orbits in a central potential. The order of the j-orbits is that o f the Mayer-Jensen shell model which is due to a strong spin-orbit interaction. The u and d quarks occupy the various orbits under the restriction that every nucleus is in a color singlet state. There are still many states of the quarks in unfilled shells. Among those there are some which have the same quantum numbers as states of nucleons in the j-orbits. These states are singled out by a special quark-quark interaction which makes them the lowest states. Thus, the nuclear shell model seems to arise f rom a shell model based on quarks. Recently, magnetic moments of nuclei have been calculated in the framework of this model [5 ]. There are some disagreements with ex- perimental data and possible improvements of the model were suggested.

The aim of this note is to examine the actual composition of states in the quark shell model. The quark-quark interaction introduced in refs. [ 1-4 ] is a generalization of the usual pairing interaction. Hence, results obtained in the seniority scheme for the conventional shell model may help in clarifying the situation in the quark shell model. It is perhaps worthwhile to point out that the model of refs. [ 1-4]

has been applied to spherical nuclei. In trying to ap- ply it to strongly detormed nuclei it will have to be modified in a non-trivial way.

The quark-quark interaction acts only in states with T = 0 , J = 0

~ j 1/2 A,*I0) = Y~ X ( - - 1 ) j-m+l/2-rm

k , l = 1 m = - - j rn~ = - - l / 2

X ~ik;aLnmTaL . . . . )10 ) ( 1 )

In ( 1 ), i, k and l are color indices and a~m,+_ 1/2 cre- ates a quark state in the j-orbit with color k, m j = m and rn~= + 1/2. The state ( 1 ) is fully symmetric in the spin and isospin variables and antisymmetric in the color degree o f freedom. The quark-quark interaction is "color blind" and is defined by

3 P J = - g J E At, A, • (2)

i=l

It has in the state ( 1 ) the eigenvalue - g j ( 4 j + 2). States of three quarks with vanishing color can be

obtained by using the operator in ( 1 ) and coupling it to another quark creation operator

3 B t - ~ a~mm,A~ (3) j m , m r - -

k=l

Among all three quark states which are color singlets the state B~mm, I 0 ) has a non-vanishing eigenvalue o f (2) equal to - g j ( 4 j + 4 ) . Other color singlet states

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have vanishing eigenvalues of the pairing interaction (2). They correspond to excited states of the quark triplet with various spins and isospins T = 1/2 and T= 3/2. Among those, there is one state with T= 3/2, J = j which corresponds to a state of A baryon in the j-orbit.

The following states can now be constructed:

B~jmm,B?jm,m,BJm,,m,,... 10 ) . (4)

The operators in (4) anticommute and hence not more than 2 (2j + 1 ) of them can be put in the j-orbit. The allowed states of any number of them correspond exactly to states in which there are colorless j- fermions in states with m m , m' m ' , m" mT , .... They all have vanishing color and have the same quantum numbers as states of nucleons in the j-orbit. All states of 3N quarks in the )'-orbit belong to the fully antisymmetric representation of the group S U ( 3 ) c ® U ( 2 ( 2 j + I ) ) . The allowed irreducible representations of U (2 ( 2j + 1 ) ) are characterized by Young ta- bleaux with three columns whose lengths are n~, n2, n 3 with n~ + n2 + n3 = n = 3N. The irreducible representations corresponding to color singlets are characterized by

nl =//2 =n3 = N . (5)

The interaction (2) is invariant under transfor- mations which leave the state Ij2T= 0 J = 0) invariant. Hence, its eigenstates can be characterized also by the irreducible representations of the orthogonal group in 2 ( 2 j + 1 ) dimensions, O ( 2 ( 2 j + 1 ) ) , to which they belong. The latter are characterized by the lengths v~, v2, v3 of the columns of their Young ta- bleaux. The states (4) belong to the irreducible representations given by Vl =N, v2 = v3 = 0. Other color singlet states of 3N quarks can be obtained by start- ing with a state with 3L quarks no two of which are in state ( 1 ). Operating on such a state with N - L different operators B~,,m¢ the state obtained belongs to the irreducible representation of O (2 (2j + 1 ) ) with Vl = N, v2 = v3 = L. The eigenvatues of (2) are given in terms of N and L by the expression [ 1-4 ]

- g j [N(4 j+ 5 - N ) - L ( a j + 5 - L ) ] . (6)

Looking at (6) we see that the states of 3N quarks with L = 0 given by (4) are indeed the lowest ones. For N = 1, L = 0 the eigenvalue (6) becomes equal to - -g j (4 j+4) .

The lowest states in the model of refs. [ 1-4 ] can thus be mapped, in a unique way, onto states of nucleons in j-orbits. It is therefore important to empha- size the big difference between the quark shell model and the shell model based on nucleons. In the conventional picture of nuclear structure quark triplets in color singlet states with isospin T= 1/2 and spin S = 1/2 have very definite and strong spatial correla- tions. The individual quarks in such a triplet move together within the nuclear volume. On the other hand, in the quark shell model states of quark triplets are only correlated by the coupling of their angular momenta in the j-orbit. The radial part of the quark triplet state is a product of three single quark radial functions.

In ref. [ 4 ] it is mentioned that the pairing nature of the quark-quark attraction is due to a short range ~-potential. The spectrum due to a short range potential is indeed similar to that due to the pairing interaction. This is true, however, only within a j-orbit or a group of j-orbits in a major shell. A sufficiently strong short range interaction would mix states in many orbits yielding genuine short range quark cor- relations. It is difficult to imagine that the strong interaction yielding the large binding energy difference between states of a nucleon and a A baryon would have negligible matrix elements between states in major shells of the Mayer-Jensen model.

We can still examine whether the quark shell model as presented in refs. [ 1-4 ] leads to an acceptable description of nuclei. In that model the uncorrelated wave functions of all quarks extend over the entire volume of the nucleus. Hence, its predictions for any reaction that probes the structure of the nucleus will be different from those in the conventional picture. An interesting example is offered by high energy muon and electron scattering from nuclei.

Here I would like to discuss other reactions that probe the nucleon contents of nuclei. For that we should first calculate the actual composition of states in the quark shell model. This may be conveniently done by considering the eigenvalues (6) in various nuclear states.

The eigenvalue (6) in the state corresponding to two nucleons, with N=2 , L = 0 is equal to - g j 2 ( 4 j + 3). Thus, the eigenvalue of (2) in a state of two nucleons in the j-orbit is less than twice the eigenvalue for one nucleon. The eigenvalues (6) are

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not a linear function of N, their maximum absolute value is in the middle of the j-orbit, N = 2j+ 1. For the completely filled j-orbit, N = 4 j + 2 , the eigenvalue is only -g j3 (4j+ 2). It is less than three times the binding energy of one nucleon. It is precisely three times the binding energy of the pair ( 1 ). The reason for this behavior is simple. In the closed shell, all states with m, mr and color are occupied. To obtain a pair state with given T and J, we must choose a linear combination with two different colors of states given by

Imm~; m'm'~> = ~ (jmjm' IjjJM)

t!~., , ' ½½TM~) × ~ 2 .... ~m~ I Ij2TMT JM> (7)

The weight of the state ( 1 ) with T= 0, J = 0 in any state (7) is given by

(jmjm' IjjOO )2( ½m,½m'1½ ½00 ) 2

1 -~- 2 (2j-l- 1 ) t~m.--rn'£~mr,--m~r • (8)

Hence, the total weight of the pair state with T = 0, J = 0 , and given two colors is just 1. Since there are three choices of two different colors the total weight of a pair state like ( 1 ) is 3. This leads to the eigenvalue -gj3 (4j+ 2) for the closed j-orbit.

Each pair state ( 1 ) in the closed shell can be com- bined with the state I m, mT> of the third color to form a color singlet state with L = 0 , J=j, M = m and M,=m~. These 2 (2 j+ 1) states correspond to the possible states of a single nucleon in the j-orbit. There is, however, some overlap of states if this is repeated again and again. Hence, the total weight of states corresponding to nucleon states in the closed j-orbit is slightly less than 3. All other states correspond to excited states of quark triplets, either with T= 1/2 or with T = 3 / 2 . Some of these are color singlets but many more states of quark triplets have hidden color.

Thus, a major difficulty of the model is the very poor contents of quark triplets which correspond to j-nucleon states. This correspondence, as explained above, is only in the quantum numbers. As will be discussed below in detail the color singlet states of three quarks with quantum numbers of j-nucleons are very different from those in the conventional shell model. Let me point out that the eigenvalues of (2) as given by (6) were used only to learn about the composition of states. The criticism is not based on

the actual values of (6), since the binding energy itself could be easily modified to yield a linear behavior in N. Adding to the pairing interaction (2) a constant quark-quark attraction equal to -9g j /2 cancels the repulsive quadratic term in (6). Still, the states of the system remain unchanged.

It may seem surprising that having tried to put in the j-orbit only quark triplets which correspond to nucleon states, we realize that only a few remain in nucleon-like states. This is due to the overlap of single (independent) quark wave functions. As a result, the antisymmetrization strongly affects all quark triplets. They are no longer confined to small vol- umes or shielded by individual bags.

In various nuclear reactions nucleons are ejected from nuclei. That in itself does not constitute proof that nucleons are indeed the building blocks of nuclei. The amplitude of the state of a real nucleon cou- pled to the nucleus with A - 1 nucleons could still be small as in the case of the quark shell model. This would be analogous to ejection of a-particles and heavier fragments in conventional nuclear structure theory. The rate of nucleon pick-up reactions, however, should be proportional to the number of "pre- formed" nucleons actually occupying the j-orbit. I f there is one nucleon occupying the j-orbit the spectroscopic factor for removing it is 1 in the conventional shell model. In the quark shell model it depends on the amplitude of a physical j-nucleon state in the given state of three j-quarks.

States of a physical j-nucleon can be fairly well ap- proximated as follows. The wave function of the center of mass of the three quarks has the orbital angular momentum l of the j-orbit. Their internal motion is in its lowest possible state and their S = 1/2, T= 1/2 spin isospin state is fully symmetric. Hence, the amplitude to be evaluated is of such a state in the given state of three quarks in the j-orbit. It can be evaluated if harmonic oscillator wave functions are adopted. In that case, the spatial part of the wave function of three j-quarks can be expanded in terms of products of wave functions of the center-of-mass coordinate R = (r~ + r2 + r3) / 3 and functions of two independent relative coordinates. The lowest state of the relative motion is that of three quarks in the lowest l= 0 oscillator orbit. The wave function of the center of mass should have the/-value of the j-orbit. The number of nodes of that wave function can be determined as in

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the case of two nucleon wave functions [ 6 ]. The lowest state of internal motion of quark trip-

lets considered above, is obtained from the lowest l= 0 states in the oscillator potential well of the nucleus. The oscillator constant of the nuclear potential is different from the one which determines the charge distribution of the proton. Hence, the internal state of the physical nucleon, being the lowest state in the nucleon potential, has non-vanishing overlap with sev- eral internal states of the quark triplet in the nuclear potential. These are l= 0 states of each of the two relative coordinates. Let n~, n2 and Nbe the number of nodes (apart from the zeroes at r= 0 and r= oo of the radial functions of the relative coordinates and the center-of-mass coordinate. If the radial functions of the j-quarks have no nodes, the following relation holds:

hog(31+9)=hog(2N+l+3+2n~ + 2 n 2 + 2 × ~ ) . (9)

The functions which may contribute to a nucleon pick-up reaction are thus given by n~= n 2 = 0, N----l; n, = 1, n 2 = 0 , N = l - 1; or n~ =0, n2= 1, N = l - 1, etc!

The result of a pick-up reaction strongly depends on the value of the radial function of the nucleon at the nuclear surface. Among all states which satisfy the relation (9), the radial function with N = l nodes is the largest at the nuclear surface. The spectroscopic factor extracted from experiments in which a j-nucleon (without radial nodes) is removed is com- puted for a nodeless radial function with given l. The value of the center-of-mass radial function with N = l at the nuclear surface R = Ro should be compared with the one without nodes. In the conventional model, the radial function of the nucleon is determined by the oscillator constant of the nuclear potential well. The oscillator constant of the center-of-mass radial function is, however, three times bigger.

Let us consider, for instance, the 1 d3/2 orbit in 4oCa. With the accepted RMS radius of the 4°Ca charge distribution (3.5 fm), the center-of-mass radial function with N = l= 2 nodes at the nuclear radius is larger by a factor of 1.85 from the radial function of ld nucleon. Hence, in order to compare with experiment, the amplitude of the function with N = 2 should be multiplied by this factor 1.85. The amplitude of the wave function with n~ =n2=0 , N = 2 in the state of three ld3/2 quarks is 0.06. The amplitude of the lowest state of internal motion whose oscillator constant

yields the proton charge distribution (RMS radius 0.8 fm) in that n~=nz=O, N = 2 wave function is 0.57. Hence, the amplitude which contributes to the reaction is given by 0.06 × 0.57 X 1.85 = 0.063.

To that amplitude the contribution of other states should be added. The next largest value at the nuclear surface is that of the radial function with nt = 1, n2 = 0, N = 1 or nl =0, n2-- 1, N = 1. It is smaller by a factor 5 than the one with n~ = n 2 = 0, N2 = 2. The amplitude due to these should be multiplied by 1.85/5 =0.37. The amplitude of the internal motion in a physical nucleon in the state with n~ = 1, n2 = 0 or n~ = 0, n2 = 1, is 0.39. We see that there is no need to calculate the amplitudes of the n~=l , n2=0, N = I and n~=0, n2 = 1, N = 1 in the state of three 1 d3/2 quarks. Even if each of them had the maximum possible amplitude 0.7, their contribution would amount to 2 X 0.7 × 0.39 X 0.37 = 0.202. The spectroscopic factor thus calculated in the quark shell model cannot exceed the value of (0.202 + 0.063) 2 = 0.07.

The comparison with experiment becomes even more significant for nucleon removal from closed shells. The spectroscopic factor for removing a proton or a neutron from the closed j-orbit is 2j+ 1 in the conventional shell model. In the quark shell model it would be about 3 / 2 = 1.5 multiplied by the factor to the poor overlap. The measured spectroscopic fac- tors in closed shell nuclei are much larger than those of a single nucleon. The spectroscopic factor for removing a ld3/z proton or neutron from 4°Ca, calculated to be 4 in the conventional model, is experi- mentally even larger than 4. The quark shell model prediction for j = 3/2 as obtained above is definitely less than (2.4/2) ×0.07=0.084. This fact, as well as the results of other similar reactions, indicate that the basic constituents of the nuclear shell model are indeed protons and neutrons.

The author would like to thank M.W. Kirson for helpful discussions. This work was supported in part by the Minerva Foundation, Munich, Fed. Rep. Germany.

References

[ 1 ] K. Bleuler, H. Hofest~idt, S. Merk and H.R. Petty, Z. Natur- forsch. 38a (1983) 705.

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[2] H.R. Petry, Lecture Notes in Physics, Vol. 197 (Springer, Berlin, 1983) p. 236.

[3] K. Bleuler, in: Perspectives in nuclear physics (World Sci- entific, Singapore, 1984) p. 455.

[4] H.R. Perry, H. Hofestiidt, S. Merk, K. Bleuler, H. Bohr and K.S. Narain, Phys. Lett. B 159 (1985) 363.

[5] A. Arima, K. Yazaki and H. Bohr, Phys. Lett. B 183 (1987) 131.

[6] I. Talmi, Helv. Phys. Acta 25 (1952) 185.

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the nuclear shell model - of nucleons or quarks?

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