Progress of Theoretical Physics, Vol. 61, No. 5, May 1979

Di-Nucleon Exotics in a Joined Spring Model and Statistics of Quarks

Shin ISHIDA and Masuho ODA *

Lltomic Energy Research Institute

College of Science and Technology, Nihon University, Tol::yo 101

*Faculty of Engineering, Kolmshikan University, Tol.zyo 154

(Received November 11, 1978)

1401

To take account of the effects due to triality phenomenologically, we set up, following a similar line to in the string-junction model, a joined-spring mechanism into the quark model, where a junction plays a role of neutralizer of forces acting on it. Then we reinvestigate the level structures of eli-nucleon exotics in the two promising unified model, the "standard" color quark model and the Bose quark model. The results, which are drastically changed from the simple case of shell-type harmonic oscillator in a previous work, seem to show that the Bose Q. M. is much favored over the color Q. M. by present experiments. Feasible experiments to further clarify this are also proposed.

§ I. Introduction

Recently the confined color quark model seems to be widely accepted as a

"standard" model describing the world of hadrons. As a matter of fact, it has

many attractive features. However, we have quite a perplexing feeling on the

notion of confined color, remembering an important role of "observable" played

1n a stage of discovery of quantum mechanics. It, moreover, makes us lose one

of the most important reasons for introduction of the color freedom, since the spin­

statistics connection is needed only for particles which can be free.

Actually in another line of unified models, the Bose quark model,n~31 \Vhich

seems to us more promising, the Fermi statistics for a baryon is consistently sup­

posed with the symmetric Bose-like behavior of constituent quarks without the

color freedom. The essential point is that there hadrons are described2l,<J,sl by

multi-local fields, which play a role of wave functions in the usual approach and

are, at the same time, second-quantized as Q-number fields.

For non-exotic hadrons these two models give essentially the same results, but

they give different level structures') for exotic hadrons, especially different for

ground six-quark exotics. This may be comprehensible from the fact that in the

color Q. M. the number 3 has a special meaning as the color symmetry of SU (3) ',

while in the Bose Q. M. there is no such special number.

In previous works'J,sl we examined the level structures of exotic hadrons,

especially those of eli-nucleon exotics (that is a system of six nuclear quarks), and

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1402 S. Ishida and M. Oda

pointed out both facts which seem to favor the Bose or the color Q. M. respec­tively, by comparing them with the present experimental candidates from eli-nucleon resonances. However, there we adopted the simple shell-type9J, 2J harmonic oscilla­tor model as the level scheme, and no consideration was given upon the triality problem.

In this connection an approach of the string-junction model 10l~Jzl is interesting, where possible effects due to triality constraint and duality are tried to take pheno­menologically into account. In this work we shall introduce a joined spring mecha­nism into both the Bose Q. M. and the color Q. M., where a string in the string­junction model is replaced by a massless spring as a simple idealization, and re­examme the level structures for eli-nucleon exotics with the aim of discriminating between the two models, the Bose Q. M. and the color Q. M.

§ 2. Joined spring model and six-quark system

The basic setup for our joined spring quark model is as follows: Pl. A hadron is a system of universal oriented springs which are joined, similarly

0

to in the string-junction

6

x, s,

J,

J,

s 4

s,

model/ol~121 so as to produce only triality-zero haclrons, and carry quarks at their free ends.

P2. Quarks obey their own statistics, Bose (Fermi) statistics in the Bose (color) Q. M.

P3. In the color Q. M. a junction has a trans­formation property of E:ijk in the color SU (3)­space so as to give only color-singlet haclrons, while it is ignorant of the other quantum number of quarks such as flavor and spin in both the color and Bose quark models. Thus our relevant system of six quarks is

Fig. 1. Six-quark system in a joined spring model. Open circles (bold lines) denote quarks (springs) and

described schematically as in Fig. 1. To get a physical insight we first treat the sys­tem classically, of which Lagrangian Is given

J's junctions. as

6 3 G

L= L;tm.:t/-(2.:; !kr/+ L:iks/), (1) i=l j=l L-=1

where m is quark mass, k an elastic constant of the sprmg, and the .T/s (i = 1, 2, ... , 6) are the coordinates of the i-th quark which are represented as

(2)

m terms of the center-of-mass (relative) coordinate*) x 0 (the ri's and s;'s) (see

*l At the beginning x 0 is simply the coordinate of junction Jo, which is proved to be the C. M. of system afterwards.

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Di-Nucleon Exotics in a Joined Spring ]Model 1403

Fig. 1 as for their definition), and A means the time derivative dA/ dt. From

(1) we derive a set of Lagrange's equations of motion;

3 6

m (6.ia + 2 :z.= r rl- :z.= si) = 0, j~l 1~1

m(2x0 +2r1 +sl+sz) = -krr etc.,

m(.io+j\+:51.2) = -ks1. 2 etc.

(3)

From these equations we find the restrictions at the junctions J0 , and the J/s;

Sum of forces at J 0 =k(r1 +r2+rs) =0,

Sum of forces at J1 = k (s1 + s2 - r 1) = 0 etc. (4)

These equations are also derivable directly as Newton's equations of motion for

the junctions ·with zero-mass. Thus in our scheme junctions play a role of neutrali­

zer of forces worked on them. From Eqs. (3) we can easily obtain the solutions

as

X 0 =Vt+C;

(0) 3 (0)

rj=rjsin(uJ,t+iJ1) with l::ri=O; j=l

(0)

5 1-52 = S12 sin (o)zt + 01) etc.,

(0) (0)

(5)

where ·v, C, rh 5 12 , r), and 01 are constant parameters, and we see that the internal

motion consists of tvvo types of harmonic oscillations, "tnmk"-oscillation and "limb"­

one, vvith respective angular frequencies

(6)

The Hamiltonian for our system is given as

111 terms of the independent normal coordinates; x 0 is center-of-mass one and R, S

and ui (i=l, 2, 3) are relative ones, defined by

:r0 =1/G· (.r1 +x,+···+x,),

R=r1 +r2 = 1/9· (x1 +.r,+.r3 +.r.,) -2/9· (x5 +x6), (8)

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1404 S. Ishida and M. Oda

where P and the Px's are their conjugate momenta.

§ 3. Relativistic quantized six-quark system

Following a similar procedure to in previous works"' we can easily extend our scheme to the relativistic quantized one. We set up a Klein-Gordon-type equation of motion for the relevant multi-local field (dependent on six space-time points) describing the general six-quark system as

(9)

where X denotes R, S, Ui in (8), and mass squared term is identified, aside from a dimensional factor d, with the Hint in (7) for the internal motion with the replace­ment*' of Px~-iha/aX" and of X (three-vector)~X" (Lorentz four-vector).

Masses for the Fierz-componene3' hadrons are determined as an eigenvalue problem;

(10)

The mass operator (10) consists of, corresponding to the two types of classical oscillations (6), two types of harmonic oscillator with respective oscillator quanta

(11)

which lead to two linear orbital Regge trajectories with slope of v3 S20 -I and Q0 -~, respectively. It is interesting that the trunk oscillation has a steeper slope ( =v3 S20 - 1) than the usual one ( = S20 -~, see the following), which reflects an effect of the joined spring mechanism, even though we have started with all univer­sal springs.

We can easily apply our procedure to non-exotic mesons (baryons), which are systems of two (three) springs being joined together with one ends and carrying quarks at free ends. This physical setup is almost the same as in our previous unified model,"' and it is easily shown that the mesons and baryons have the universal slope ( = Q0 - 1) of orbital Regge trajectory. Although we have regarded the quan­tity (/) in (9) and (10) as a multi-local field in the above, it can also be regarded as a B-S amplitude from the usual bound state picture. In the following we call it merely a wave function.

§ 4. Symmetry of wave function and quark statistics

Now we give attention to the ground state wave function of (10). Since the

*' As for prescription for the problem concerning a relative time freedom in a relativistic system, see previous works.''

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Di-Nucleon Exotics in a Joined Spring 1'\fodel 1405

ground state is non-degenerate (and the kinetic energy term of c'1YL 2 111 (10) is

completely symmetric), the wave function has the same symmetry, with regards to

exchange of coordinates xi> as the potential energy term of joined spring c5!1 2 in (10). Thus it is symmetric, as easily seen from Fig. 1, for exchange of

(12a)

This is confirmed directly by the explicit form *J of the ground state solution;

(/)G~-cexp jl_ Ko_ [-L3 ( [ (x1 + :c2) - (x3 + x4) J 2 + [ (xa-1- x4) - (xd- Xa) r 2 9

+ [(x5+xo)- (x1+x2)r) + (x1-x2) 2+ (x3-x,) 2-l- (xs-xa) 2]}. (13)

Then the supposition of quark statistics (P2) and of transformation property of junction (P3) requires that our ground ·wave function

(14)

where ~l; denotes thus far omitted spin and :flavor of the i-th quark, has the symmetry vvith regards to spin- and :flavor-suffices as follows:

(L) It is symmetric for exchange of

(T) It 1s symmetric (anti-symmetric) for exchange of

C-lb A2) k-> (A3, A4), (As, A 4) H C-15 , A 6), (L15, A. 6) H (.:1!> il2)

in the Bose (color) Q. M.

(15a)

(15b)

These may be understood by considering the corresponding exchange of constit­uents, ·where quark statistics makes a (no) difference in (L) (in (T)) because of one (two) -particle exchange and there a junction gives an extra minus sign in the

color Q. M. To get a clear physical insight into the above situations it is useful to consider expressions**) for the wave function like, 111 Bose Q. M.

(16a)

m color Q. M.

*1 The expression A 2 for a 4·vector A 1, in (13) should be taken as A'- Ao2 and A'+ 110 2 I or the indefinite-metric type and the definite-metric type solution, respectively.

**J As for a rigorous definition of (16a) in the Bose Q. M., see Ref. 2), while (16b) in the color Q. M. can be regarded as a B. S. amplitude.

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1406 S. Ishida and M. Oda

(16b)

where i, j, k etc. denote the color indices. The difference (T) of symmetry property between the Bose and the color Q.

M. will lead to a decisive difference in their level structures.

§ 5. Level structures of di-nucleon exotics

Now, following a similar (and even easier) procedure to in the previous work,8l

we can determine the level structures. The results are given in Table I for ground, 1st- and 2nd-excited*) di-nucleon exotics, where the number means the degeneracy of corresponding state (in the symmetric limit) and the limb- and the trunk-excitation correspond to the two types of harmonic oscillation with the quan­tum Qt and .fJ1 in (11), respectively.

From this table we see the general features: i) The level structures in the ground states are decisively different between the

Bose and color Q. M.

Table I. Level structures of the di-nucleon exotics in the joined-spring Bose and Color quark models. The numbers denote degeneracy of the corresponding levels (the ones in parentheses are in the case of shell-type harmonic oscillator).

I

J

ground levels

Bose Q. M.

(shell type)

Color Q. M.

(shell type)

1st excited levels trunk excitation

Bose Q. M.

Color Q. M.

limb excitation

Bose Q. M.

Color Q. M.

2nd excited levels trunk excitation

Bose Q. M.

Color Q. M.

0

0 1 2 3 4 5

I

il3 (1)

I 2 I Cl)

2 5

2 5

5 6

2 6

1

4

4

7

4

1

(1)

2

2

2

2

14 20 27 13

8 26 22 15

---

1

6

5 1

1

0 1 2 3 4 5

4 1 1 I (1)

1 2 1 2 ! (1) (1)

5 10 9 4 1 I

5 10 9 4 1

4 13 8 4

6 11 10 4 1

I

16 53 47 33 11 3 22 48 50 29 12 2

2 3

0123451012345

1 1 2 (1)

2 1 (1)

3 7 6

3 7 6

4 7 7

3 7 5

3

3

3

2

16 32 35 21

13 34 33 22

1 1

1

1

1

I 9 2 i 3 8 2 1 5

1

2 2

2 1

2 1

1 1

12 10

9 11

1

(1)

1

1

1

8

6

---

---

3 1

3

*l Out of three possible 2nd-excited states the one with only trunk-oscillator quanta, which has the lowest mass, is considered.

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Di-Nucleon Exotics in a Joined Spring Model 1407

ii) The more complicated the level structures become rapidly, the more excited

in both the models, as naturally expected for multi-body systems. In comparison with our previous results8) in the case of shell-type harmonic

oscillator, the present level structures become more complicated. Actually all

levels in the previous case are also contained in the present level scheme, as is

inferred from the situation that the full space-symmetry among quark coordinates

in the previous case is now limited within that of joined spring mechanism. In Table I also given is the previous result for the ground levels to show an effect

of the mechanism.

§ 6. Comparison with experiments

At present there are four candidates for the eli-nucleon exotics from possible eli-nucleon resonances, DI (mass in Ge V) = D 1 (2 ·17), D 1 (2 · 22), and Do (2 · 38) and D 1 (2 · 43), which may, due to their mass and/ or parity, be classified into the ground, 1st-excited, and 2nd-excited exotics, respectively. As for the I= 1 eli-nu­cleon states we have rather rich experimental knowledge (referred to as HP here­after) due to the extensive phase shift analysis by Hoshizaki. ll)~ 161

6.1. Ground levels

In Table II we have recapitulated our theoretical level structures for the ground states, and have given the corresponding quantum numbers 2s+JLJ in eli­

nucleon systems for the states with I= 0 and 1. The resonance D 1 (2 ·17) was rather a controversial*) one, but now it seems to be established16l.m as a eli-nucleon resonance in the 1 D 2 state (so (I, JP) = (1, 2+)) by HP. In both the models it has its right seat. However, in the color Q. M. we expect fine structures in this state (since the degeneracy= 2), while not in the Bose Q. M. In HP there seems to be no evidence for it. In the color Q. M. we expect an existence of another eli-nucleon resonance with (I,JP) = (1,0 1 ) of a similar mass ("-'2.17GeV), while not in the Bose Q. M. In the corresponding 1S0 state of HP there seems to be no evidence for it. The above two facts seem to favor much the Bose Q. M.

Table II. Level structures of the ground di-nucleon exotics and its coupling with di-nucleon systems.

I 0 1 2 3 ----------- ------~----

J ! 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 0 1 2 3 ----------

Bose Q. M. 3 X 1 X X X >< 4 1 1 X X I 1 1 2 X X 1 X 1

Color Q. M. X 2 X 1 X X I 2 1 2 X X X I X 2 1 X I 1 X X X

Nlf'S+1LJ X 'S, 'D, 'D, 'G, 'G, 1 'So X 'D, X 'G, X

HP I No? Yes!

*l See Ref. 16) on this point and on the list of related papers.

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1408 S. Ishida and M. Oda

over the color Q. M. It is very interesting to check experimentally the existence of the other expected ground particles.

6.2. 1st excited levels

The candidate D1 (2 · 22) 18) is now established so as to be a di-nucleon resonance m the 3F 3 state15l, 19) with (I, JP) = (1, 3-). In both the models this can be assigned to a corresponding trunk-excited state. However, we expect a fine struc­ture due to 4-fold degeneracy in both the models, of which no evidence is seen in HP. Many other trunk-excited states of a similar mass ( '"'--'2.3 Ge V) and limb­excited states of slightly higher mass ( '"'--'2.4 Ge V) with negative parity are also expected in both the models. Some structures15) in all 3P 0, 1, 2 states seen in HP may correspond to them. It is interesting that both the models predict the exis­tence of a simple resonance with a similar mass to D 1 (2 · 22) in the 3F 4 di-proton state.

6.3. 2nd excited levels

The candidate Do (2 · 38) 20) 1s inferred to have (I, JP) = (0, 3+). As for the D 1 (2 · 43) some signals for it are seen15) in the 18 0 and 1G4 state of HP so (I, JP) = (1, o+) and (1, 4 +). All of them have their right seats in the trunk-excited states in both the models. However, the level structures are too complicated to be compared in detail.

6.4. Level spacing

mnss(Gevl

2.8

2.6

2.4 2.41 _l_

t' 2·63 --} D,( 2.43) 2.54. ~ 15 0

1G4 (NN)

2_45 _t_ 00 ( 2.38)

2.31 _t- 0,(2.22)

3 G,(NN)

2. 2

2.0

ll?-- 01 ( 2.17) 1D1(NN)

3

ground 1st exc .

F3(NN)

2nd exc.

Fig. 2. A schematic illustration of our mass spectra in comparison with experiments. t(l) denotes a trunk (limb) harmonic oscillation with the quanta !J,= (1/v'Z)!Jo (!J,=!Jo). Q,-'= (1.1 GeV')- 1 is the slope of Regge trajectories for non-exotic hadrons.

Our level scheme is determined by the two types of harmonic oscillator, the trunk- and limb­oscillator with an oscillator quan­tum .Q, = 1/v'3 · .!20 and .!21 = t20 , res­pectively. In Fig. 2 schematically shown is the mass spectrum, which is determined by taking the mass of D1 (2 ·17) as an input and by using the usual value .!20 = 1.1 (Ge V) 2, in comparison with present experimental candidates assigned above.

From Fig. 2 our level scheme seems to be consistent with present experiments. It is interesting that the trunk-excited states with a

steeper Regge slope than the usual one seems to be realized actually.

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Di-Nucleon Exotics in a Joined Spring i\1odel 1409

§ 7. Proposal of crucial experiments

As has been shown in the previous section, general features of our scheme are

supported experimentally and, as far as the present experimental candidates are

concerned, the Bose Q. M. seems to be much favored over the color Q. M. Then

it is desirable to test further experimentally the predictions of our scheme to discriminate between the two models. For this it is most effective to check the

level structures of ground particles, \vhich are rather simple and show a clear

difference between the t\vo. Here \Ye propose two experimentally feasible pro­

cesses, *l which investigate the existence of the ground particles with I= 1 and 2,

and they are described in Figs. 3 and 4, respectively.

The first process is the one which was proposed in the previous work,8) to

which detailed discussion should be referred. Now our theoretical expectation is

the existence of a series of particles with the properties given in the table of Fig. 3. There we have also given the expected (detailed) mass values which are obtained by using a regularity') on the spin-dependence noticed previously, and by

using the mass value 2.17 Ge V of the 1D 2-di-nucleon resonance as an input. In investigating the existence of the particle with J = 3 (expected in the Bose Q. M., while not in the color Q. M.) from the mass distribution of final three particles P, P and 17-, it is better to first select the events corresponding to the p.J0-decay

TT+ _______ -,-_________ IT+ TT+ _______ ~---------TT

______,. time

I 1 P + = =

J

Bose 0-.

Color 0. . mass (GeV)

0 X

2

2.03

p

~-} !::.0

·v 1 2 3

4 1 1

1 2 X

2.08 2.17 2.30 --

Fig. 3. A feasible process for investigating the expected ground particles with I= 1 and with respective properties given in the table in both the models. For a particle with J=3 a selec· tion of decay process D+ --7pL1' may be useful to get riel of Pomeron exchange process.

x* _______,.. time

p d+ p

D+t+ TT+ 1=2 P=+ t

T 0 2

Bose 0.. 1 2

Color 0--. X 2 mass

2.2 (GeV)

Fig. 4. A feasible process for investigating the expected ground particles with 1=2 and with respective properties given in the table in both the models. The existence of particle ex++) with 1=2 is presupposed here.

*l The authors deeply thank Professor M. Fukawa, Professor A Masaike and Professor T. Hirose for their useful discussions and interest in these processes.

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1410 S. Ishida and M. Oda

of the particle (note the expected mass value 2.30 Ge V and the N-J threshold value 2.17 Ge V). This assures to get rid of possible background effects due to Pomeron exchange. The cross section for this process is expected to be of the same order of the magnitude ()rv25,ub as for a similar process.2!l

In the second process for investigation of particles with I= 2, we presuppose the existence of the so-called baryonium pa:~;ticle with double charge. The momen­tum of incident rr+ may be, corresponding to the expected mass of the ground baryonium m'"'-' 1.3 Ge V, appropriate to being taken around Pin= 1.5rv3.0 Ge V /c. The cross section for this process may have the same order of ()rv 10,ub as the previous one, or may22l be much suppressed as () = 1rv0.1,ub, as the line diagram is not planar but of the X-type. The properties of expected ground particles in the two models are recapitulated in the table of Fig. 4.

§ 8. Concluding remarks

In this work we have explicitly supposed a joined spring mechanism for con­structing a composite hadron. However, our result on the level structure is rather of general character, only depending upon the space-symmetry of wave functions. Thus, for example, the same level structure as for the trunk-oscillator is, aside from the relation Q, = (1/y':.f) !20, also derivable in a di-quark modeP3) where three di-quarks are bound together with a harmonic potential.

In order to introduce further a possible effect due to duality a spring in the present scheme should be replaced by a string. There as far as the lower mass region is concerned, the same level structure as the present one is expected in a joined string model, since a string freedom is equivalent to that of an infinite num­ber of springs with the normal frequencies.

In this work we have simply assumed the recent possible di-nucleon resonances with mass around 2 Ge V to be candidates of di-nucleon exotics, although there seems to be some general belief against it. In the previous work7l some evidences for this were given by a consideration on mass spectrum of general hadrons. In this connection also notable is a recent work24l which expects di-baryon Regge trajectories, of which pole position has a similarity to that of recent di-baryon resonances.

The authors acknowledge encouragement given by Professor 0. Hara. They also appreciate useful discussions with the members of their laboratories. One of the authors (S. I.) is indebted to Professor M. Uehara and Professor S. Y. Tsai for useful discussions, through which a motivation for this work was raised.

References

1) 0. Hara and T. Goto, Prog. Theor. Phys. 44 (1970), 1383. 0. Hara, T. Goto, S. Y. Tsai and H. Yabuki, Prog. Theor. Phys. 39 (1968), 203.

2) S. Ishida, Prog. Theor. Phys. 46 (1971), 1570, 1905.

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Di-Nucleon Exotics in a Joined Spring Model 1411

S. Ishida, M. Oda and Y. Yamazaki, Prog. Theor. Phys. 50 (1973), 2000.

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As for the list of related works see, I. Sogami, Soryusiron Kenkyu (Kyoto) 57 (1978), B63.

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7) S. Ishida and M. Oda, Prog. Theor. Phys. 59 (1978), 959. 8) S. Ishida and M. Oda, Prog. Theor. Phys. 60 (1978), 828. 9) 0. W. Greenberg, Phys. Rev. Letters 13 (1964), 598.

10) X. Artru, Nucl. Phys. B85 (1975), 442. 11) M. Imachi, S. Otsuki and F. Toyoda, Prog. Theor. Phys. 55 (1976), 551; 57 (1977), 517.

12) F. Toyoda and M. Uehara, Prog. Theor. Phys. 58 (1977), 1456.

M. Uehara, Prog. Theor. Phys. 59 (1978), 1587. 13) M. Fierz, Helv. Phys. Acta 23 (1950), 412. 14) N. Hoshizaki, Prog. Theor. Phys. 57 (1977), 335, 1099; 58 (1977), 716. 15) N. Hoshizaki, Kyoto Univ. Preprint (NEAP-18), July 1978. 16) N. Hoshizaki, Kyoto Univ. Preprint (NEAP-19), August 1978. 17) I. P. Auer et al., ANL-HEP-PR-78-13 (1978).

A. Yokosawa, ANL-HEP-CP-78-11. 18) L P. Auer et al., Phys. Letters 67B (1977), 113. 19) Hidaka et al., Phys. Letters 70B (1977), 479. 20) T. Kamae et al., Phys. Rev. Letters 38 (1977), 468.

T. Kamae and T. Fujita, Phys. Rev. Letters 38 (1977), 47L 21) 0. Braun et al., Nucl. Phys. B124 (1977), 45. 22) M. Uehara, Saga Univ. Preprint (SAGA-78-2), June 1978. 23) D. B. Lichtenberg, Phys. Rev. 178 (1969), 2197. 24) I. Ohba, Waseda Univ. Preprint (WU-HEP-78-5), October 1978.

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