Dibaryons in a quantum-chromodynamics-based consistent quark model

IL NUOVO CIMENTO VOL. 101 A, N. 2 Febbraio 1989

Dibaryons in a Quantum-Chromodynamics-Based Consistent Quark Model (*).

C. S. KALMAN and S. BARBARI

Concordia University, Elementary Particle Physics Group 1455 de Maisonneuve Blvd. West, Montreal, P.Q. Ca~w~da H3G IM8

(ricevuto il 25 Marzo 1988)

Summary. - - A QCD-based quark potential model is applied to predict masses of S-wave dibaryons in (q3) (q3) clustering, using parameters derived from a baryon fit.

PACS 21.10.Dr - Bindings energy and masses. PACS 21.30 - Nuclear forces.

1. - I n t r o d u c t i o n .

If quantum chromodynamics (QCD) is the definitive theory of the strong interactions, it follows that nuclear physics can be understood in terms of quarks and gluons.

Six-quark configurations have aroused considerable experimental interest('~). Theoretically, they have been examined both in the context of the ~J~ interaction and as resonant q6 states or dibaryons. The ~,V interaction has been investigated by DeTar (5), Libermann (6), Ribiero (7), Maltman and Isgur (8), and others (9). Bag

(*) Supported in part by the Natural Sciences and Engineering Research Council of Canada. (1) T. KAMhE: Nucl. Phys. A, 374, 25 (1982). (2) D. BUGO: Nucl. Phys. A, 374, 95 (1982). (s) M.M. MAKAROV, G. Z. OBRANT and V. V. SARANTSEV: Phys. Lett. B, 122,343 (1983). (4) I. AUER, E. COLTON, H. HALPERN, D. HILL, n. SPINKA, G. THEODOSIU, D. UNDERWOON, Y. WATANABE and A. YOKOSAWA: Phys. Rev. Lett., 51, 1411 (1983). (5) C. DETAR: Phys. Rev. D, 17, 323 (1978); 19, 1451 (1979). (6) D. A. LIBERMAN: Phys. Rev. B, 16, 1542 (1977). (7) j . E. F. T. RIBIERO: Z. Phys. C, 5, 27 (1980). (s) K. MALTMAN and N. ISGUR: Phys. Rev. D, 29, 952 (1984). (9) G . W . BARRY: Phys. Rev. B, 16, 2886 (1977); C. S. WARKE and R. SHANKER: Phys. Rev. C, 21, 2643 (1980); D. ROBSON: Nucl. Phys. A, 308, 381 (1978).

193

194 c . s . KALMAN and s. BARBARI

model calculations for dibaryon resonances in (q2)(q4) and (q~)(qS) clustering have been performed by Jaffe (lo), Aerts and Dover (11), H0gaassen and Sorba (12), and others (13), QCD-based potential models have been applied by Maltman C4), 0ka and Yazaki (15) and Cvetic (16).

In this paper we have applied a consistent quark model to dibaryons in (q3)(q3) clustering. This being motivated by the success of the model in describing baryon structures and by its success in describing other quark systems in terms of baryon parameters.

2. - The dibaryon Hamiltonian.

In the ground state there are no tensor nor spin-orbit interactions and the Isgur-Karl Hamiltonian is written as

6

(2.1) H = • mi + H0 + Hcon~et, i~ l

where m~ = m is the common constituent quark mass. The Ho term includes the harmonic-oscillator confining potential together

with the anharmonic U potential. This is written as

(2.2)

Pi

Ho = ~'~--~- + ~ V~o~ ~<j 12m

where

(2.3) Vr = Ai " Aj Viy

and

1 2 (2.4) gij=[-~Krij+ U(rij)l.

(,o) R. L. JAFFE: Phys. Rev. Lett., 38, 195 (1977); Phys. Rev. Lett., 38, 617 (1977). (n) A. T. M. AERTS and C. B. DOVER: Phys. Rev. D, 29, 433 (1984); 28, 450 (1983); Phys. Rev., 49, 1752 (1980). (~) H. HOGAASEN and P. SORBA: Nucl. Phys. B, 150, 427 (1979). (t3) A. W. M. AERTS, P. J. G. MULDERS and J. J. DE SWART: Phys. Rev. D, 17,260 (1978). (14) K. MALTMAN: Nucl. Phys. A, 438, 669 (1985). (15) M. OKh and K. YAZAKI: Phys. Lett. B, 93, 489 (1980). (16) M. CVETIC, B. GOLLI, N. MANKOC, and M. ROSINA: Phys. Lett. B, 93, 489 (1980).

DIBARYONS IN A QUANTUM CHROMODYNAMICS ETC.

The Fermi contact term is written as

8 ~ 0 ~ s (2.5) Hr162 = ~ Z Z (S, . S;) A? A; ~a)(ru)"

= z< 3

195

3. - The c o l o u r fac tors for three -quark c lus ter ing .

If we consider a dibaryon to be made up of two clusters of three quarks each, then for each cluster in colour SU(3)

(3.1) 3 | 1 7 4 1 "~-8MA'4-SMs-]- 10.

We find that colour singlets in a dibaryon arise only from the 1 @ 1 and 8 | 8 terms. The singlet arising from the 1 | i leads to in-cluster colour factors similar to those of baryons and no inter-cluster colour coupling. The singlet in the

(3.2) 8 | 8 = 1 + 8Ms + 8Ms + 10 + 10 + 27

can arise from one of three possible cases

8M S | 8Ms, 8M A | 8MA, 8M A | 8M A

and each case corresponds to a different set of colour factors. The composite state must be a colour singlet.

6

(3.3) ... ~A~Z=0. i=1

We label the clusters as A and B, where dus ter A is made up of quarks 1, 2 and 3, and dus ter B is made up of quarks 4, 5 and 6. Each cluster is a eolour octet.

(3.4) (A1 + A2 + A3) z = (A4 + As + A6) z = 3

and each quark is a eolour triplet

4 (3.5) ... A~2 = ~ , i = 1 ,2 ,3 ,4 ,5 ,6 .

We decompose (3.1) to show the origins of the 8MA and 8Ms

(3.6) 3 @ 3 = 6 ( ~ 3

196 C. S. KAIA~AN and S. BARBARI

and

(3.7) A

3 | (6 | 3) = (8M~ ~ 10) @ (1 @ 8MA).

We see that the 8Ms arises from the 3 | 6 and the 8Ma form the 3 | 3. Therefore in the 8Ms a quark ,,sees,, the other two quarks in a 6 of colour, while in the 8MA they appear as a 3. Next we calculate the colour factor for the three types of octet combinations.

1) 8Ms | 8Ms. A 3 in one cluster sees the other two quarks in that cluster as a 6. Therefore within each cluster

(3.8) (A~ + Aj)~j = 1 0 3 '

Hence

1 (3.9) Ai. Aj,,j = --~.

From the condition that the composite state be a colour singlet

(3.10) (A1 at- A2 + Aa + A4 "+- A5 + A6) 2 = 0 .

Due to the symmetry of the dibaryon we can write

6 (3.11) ~ ~ a i . Aj = 9 a a " AB,

i=i j=4

where A = 1, 2,3 and B = 4,5, 6. From (3.10) and (3.11) we have

2 (3.12) Aa" A8 = - ~ ,

which is the inter-cluster colour coupling factor.

i , j = 1, 2, 3 or 4, 5, 6.

4 (3.13) (A~ + Aj)~,, s = i , j = 1 , 2 , 3 or 4 ,5 ,6 ,

and

2 (3.14) A~. A~,,, = - -~.

2) 8Ma ~) 8MA. In this case a 3 in one cluster sees the other two quarks in that cluster as a 3. Therefore, within each cluster

DIBARYONS IN A QUANTUM CHROMODYNAMICS ETC. 197

And as in the previous case we evaluate the inter-cluster coupling. We have

(3.15) AA" AB = 0.

3) 8M~ | 8~. The colour factors within each cluster are different and have previously been evaluated. The inter-cluster colour factors are

(3.16) AA" AB = -- 1 3"

Next we observe that the symmetry of the dibaryon system requires that the spatial expectation values of the potential be such that

VI----- <V12> = <V13> -~- <V23> = <V45> = <V56> --- <V46> (3.17)

and

(3.18) V I I = (V14) --- (V15) = (V16) = (V24) = (V25) =

Therefore for the four colour compositions of dibaryons (E Vg~) is given by

(3.19)

a) 1 | -~ 4V~,

b) 8M,, | 8Ms ---) 2VI - 6VH,

C) 8MAQ8MA -~-4V~,

d) 8M~ | 8Ms -+ - VI - 3VII.

4. - The harmonic-oscillator term.

We write the harmonic-oscillator Hamiltonian

1 ) (4.1) Y H o = ~ P ~ - P~ + ill

1 2 1 + -~fAg(r12 + r~8 + r~3) + -~fsg(r45 + r~6 + r~6) +

1 2 + ~ f c g ( r 1 4 + r~5 + r~6 + r~4 + r~5 + r~6 + r'~4 + r]5 + r]6),

where fA, f8 and fc are colour factors.

198 c . s . KALMAN and s. BARBARI

We incorporate a - 2/3 factor in the spring constant K for ease of comparison with the Isgur-Karl baryon model and we write the colour factors (fA, f s , fc) as follows:

(4.2)

a) 1@1 -~ (1, 1, 0),

b) 8M~| ---)( 1 1 1) 2 ' 2 ' '

C) 8M n | 8MA ---) (1, 1, 0),

d) 8 ~ A N a ~ ~ l , - - g , .

The Hamiltonian HHo separates in terms of a set of orthogonal relative co- ordinates (,7,1s).

(4.3)

03

1

v~ 1

v~ 0

0

1

v~

- 1 0 0 0 0

v~ 1 - 2 0 0 0

v~ v~ 1 - 1 0 0 0

v~ ~ 1 1 - 2

0 0 v~ v~ v~

1 1 - 1 - 1 - 1

v~ v~ v~ v~ v~

/'1

r2

r3

r4

r5

r6

with the corresponding relative momentum =i--ihVpi and

(4.4) Hno = ~ m (z2 + z~ + z~ + z2 + z~) +

3 o + + 3 K ( f A +fc)(~:~ + ~) + ~K( fA +fc)(P~ + P~) 3fcKz2s.

(17) R. L. HALL and B. SCHWESINGER: J. Math. Phys. (N.Y), 20, 2481 (1979). ('~) C. S. KALMAN: Phys. Rev. D, 26, 2326 (1982); Nuovo Cimento A, 94, 219 (1986).


HHo is now diagonal and we may write down the energies and corresponding eigenstates. First, however, we relate the dibaryon parameters to the Isgur- Karl baryon parameters. We write

(4.5)

where

f 9~2 = 5f2S1 , 0)1 = 0)S1,

~ ~2S2, 0)2=0)$2,

[~ 2 ~2S3, 0) 3 -- 0)S3 ,

and

(4.6)

2 = (3mK)1/2,

Sl = ( fA "~- f c ) 1/2 ,

0) = (3K/m) 1/2

$2 = (f~ +fc) v2 , S~ = (2fc) 1/2 .

The ground-state eigenfunction can now be written as

(4.7) r p~, p4, ps, p6) =

_ 2 3 exp - ~ t a ~ p 2 + ~p3

and the ground-state energy is

(4.8) ( EHo=3 0)1+0)2+~0)3 =30) S I + S 2 + ~ - �9

(4.9)

The factors ($1, $2, $8) are given by

a) 1 | (1, 1, 0),

c) 8 ~ | ~ ( 1 , 1 , 0 ) ,

200 c .S . KALMAN and S. BARBARI

Using the baryon parameters of Kalman('8) we obtain for co = 274.02MeV

EHo (1 | 1)

(4.10) E~o (8MA | 8M~)

EHO(8MA | 8Ms)

= 6o, = 1644 MeV,

= 9o,/V2 = 1744 MeV,

= 6o, = 1644 MeV,

= 3co + = 1418 MeV.

5. - The a n h a r m o n i c potent ia l .

F rom (3.1)-(3.3) the anharmonic U potential is given by

(5.1) U =fa [V(r12) + U(r13) + U(r23)] + f 8 [U(r45) -t- U(r46) + U(rsG)] +

+ f c [ U (r14) + U (r15) + U (r,6) + U (r16) +

+ U(r24) + U(r26) + U(ra4) + U(r35) + U(r36)].

The first-order contribution of U to the energy is (~booo] U kbooo>. We evaluate the in-cluster contributions <~boool U (r~2) kbooo> and < ~oool U (r45)1r

(5.2) (~oool U(r12)l~ooo> - - - ~16 ~266~33 fd p2d a ~ d 3 ~4d 3,~5d 3 p6U (V2,~) "

"exp [ - (a2t :2+a21P32 "{- ~22~42 "{- ~2D22, 5 "~ ~3~6)]22 ---

a(S,) = aasl~fdap2 U(V2p2) exp [ - S, a2:~] = 3

The calculation of this integral has been performed by Kalman (,8):

a(S2) (5.3) <~ooolV(r45)l~ooo> - 3

In order to evaluate the inter-cluster contributions we transform to a new set


of relative co-ordinates (a2, as, ~4, as, a6), where

(5.4)

P3

P4

P5

P6

1 1 1 2 o o

1 1 1 0 . / 2 ~ 3

1 1 1 2 2 0 0

1 1 1 0 . ] 2 ~ 3

1 1 1 0 0

0"3

0" 4

where

and

r l , = V 2 a 2 , r 2 5 = V ~ a 3 , r.~e = V~a4, 1 ~(r12 + r45) ---- a5

(r13 -}-/'23 -}-/'46 -}- r56) = ~6.

We substi tute the new set of relative co-ordinates in the exponent of the ground- state wave function and then using the Lagrange method remove the cross- product terms by suitable transformations and we express the quadratic form as a sum of squares.

The exponent -(1/2)(a~p~+ ~,~32+a 2 ~ + ~ 2+a3%~) transforms to 2,~4 2t~5 , _(1 /2 ) ( f l~a~+f l~D~+ 2 2 2 2 2 2 f13~r~4 "~/~4~5"~ fl5~6) where ~3 , ~'~4, ~r~5, and t9~ are a new set of relative co-ordinates and fl~ ,f12 ,fl],fl42 ,fl~ are coefficients given by

(5.5) fl~ _ 3a z sl s2 s8 8 ls3-}-8283-~-8182 '

4a281 82(s183...~-82 83+8182) (5.6) ~ -

(81+82)(8183-}-8283-}-48182)'

(5.7)

(5.8)

(5.9)

~2 _ ~ 83 -~- 8283 "~ 4sl s2)

3 ( 8 1 + S 2 )

/~4~ = ~2(sl + s2) 2 '

fl~ _ 2a2(d1 + d2)

3

202 c.S. KALMAN and s. BARBARI

We then wri te the ground-state wave function in t e rms of the new coordinates

(5.10) r O.3, Y24, ~5, t9~) =

15/283/21 S 23/2 S 33/4

ff15/2 1 " 22.~_f~2~2_~_~2~2_{_o2~r~2.~_ 2 2 ] e x p - ~ ( f l 1 ~ 2 ~2 3 ~4 5 ~5~6) �9

Therefore ,

(5.11) <~ooolU(r14)l~ooo> = - ~ - a - . 81 82 --{- 82 83 --{- 81 83

F r o m the permutat ion symmet ry of the ground state we have from (5.1)

(5.12) <~ooolUkbooo> =3fA(~ooolU(r12)l~booo> +

+ 3fB<~0001U(r45)l~ooo> + 9fc< Jzoool U (r14)l~ooo > .

We evaluate the matr ix elements of the anharmonic potential. We have in MeV

<fA U (r12)> <fB U(r45)> (fc U(r14)>

1 | 1 - 282 - 282 0 8Ms (~ 8Ms 112 112 - 164 8M~ | 8M, -- 282 -- 282 0 8MA | 8M$ -- 323 33 -- 21

The total energy in S-wave excluding hyperfine interaction is given by

(5.13) Eto~ = 6M + Euo + ('~o0olU I~0o0>

and we have for the ground-state energy for m = 385.69MeV (,8)

a) 1 | = 2 2 6 4 M e V ,

b) 8M s | 8Ms = 3252 MeV,

c) 8MA | 8MA = 2264 MeV,

d) 8MA | 8Ms = 2668 MeV.

6. - T h e h y p e r f i n e i n t e r a c t i o n .

The one-gluon exchange interaction between quarks in a dibaryon gives rise to a Fermi contact interaction in the ground state. The Hamiltonian has


the form

Heontac t - 87~as 3 m 2 ~ ~,.(si .sj)A~A~8(a)(rij) . l < j

The spatial in tegrals requi red for the contact t e r m are reduced to th ree by the spatial s y m m e t r y of the d ibaryon sys tem. We have

(6. i) ( 8 (3) (rl 2) > = (~(3) (r13) > = (~(3) (r23) >,

(6.2) <~(r45)> = <~(r46)> = (~(r56)>,

(6.3) (~(r,4)> - - (8(r~5)> -- <~(r,6)> =

= <8(r24)> = <8(r2~)> <8(rz6)> = <~(r34)> = <8(r33)> = <8(r36)>.

We can then wri te the Hamil tonian as

(6.4) Heont = _ 8=38 <~(8)(r14) ) E 2m 2

. ~ ce si sjAi "Aj - i<j

i = 1 , 2 , 3

j=4,5,6

- 3 8rra~ A ( ," A2) [ < ~(3) (rl e) > - (~a (rl 4) > ]"

" [S l" 82A1" A2 + 81" 83A1" A3 + 82" 83A1" A 3 ] ,

since r,2 = V 2 p and r14 - - - -V~O" we can wri te

(6.5)

and

(6.6)

F r o m (6.5) and (4.7) we have

(6.7)

8 (3) (r12) = 2 - 3 ~ c?(3)(p)

c~ (3) (r14) = 2 -ar~ 8 (3) ( a ) .

< 8 ( 3 ) ( r 1 2 ) > - 813/253

(2=) az

and f rom the s y m m e t r y of the clusters we can wri te

_ 8~12~ 3 (6.8) (8 (3)(r45)> 2~3~ .

14 - ll Nuovo Cimento A.

204 c . s . KALMAN and S. BARBARI

F r o m (6.7) and (5.10) we have

(6.9) <am) (r14)> = [ 9 ] 3~ a~ (2~-) 3/2 [8182 818283 13/2 s,s j �9

In gene ra l (8 3 (ri/)> can be w r i t t e n as

(6 .10) < 83 (rij) > ----f(81,82,83) (2=)3~.

Also [8~a3/3m 2] can be w r i t t e n as

(6.11) 8na.~_,, _ 8 ~ ' z ( ) . 2 = . 3m 2 ~3 ,

w h e r e 8 is the 5-• mass difference. T h e r e f o r e

(6.12) 8na, <8(m (r~j)) = f ( s l s2 s3) 8. 3m 2

Us ing 8 = 365 (18) we t abu la te below the p roduc t s f (s l s2 s3) 8 for the d i f fe ren t colour s y m m e t r y conf igura t ions in MeV.

8haS < 8(3) (rl 3) > 3m 2

8 = ~ S ( a ( 3 ) ( r 4 5 ) ) 3m 2

8 ~ S (8(a) (r, 4) > 3m 2

1 | 265 265 0 8Ms| 158 158 135 8MA| 265 265 0 8MA| 359 0 0

N e x t we t abu la te the colour fac tor products .

A 1 �9 A 2 A 4 �9 A 5

2 2 1 | . . . . 3 3

1 1 8~ s | 8Ms w w

2 2 8MA | 8MA -- ~ --

2 1 8MA | 8Ms -- --

3


The spin states for each cluster are given in SU(2)s by

2 | 2 N 2 = 4s + 2Ms + 2Ms,

where the 4s is a symmetric spin-3/2 multiplet and the 2Ms and 2Ms are mixed- symmetry spin-l/2 doublets.

The SU(2) Casimir operator S 2 for three quarks (a, b, c) in a cluster may be written as

therefore

S 2 = (So + Sb + S~) 2 ,

(6.13) Sa'Sb+ Sa.S~+ Sb.S~=I(s(s+ 1) - 9 )

and we have the following table:

S(S + 1)

4s 15 3_ 4 4

2Ms 3-- _ --3 �9 4 4

2,,~ 3- - 3 - 4 4

Following Jaffe (10) and H0gaasen and Sorba (12) we note that the products A'S k are among the generators of colour-spin SU(6)~. The generators of SU(6)~ are defined as follows:

(6.14) 1 2(2/3)1a S ~ , k = 1 ,2 ,3 ,

a = 1 , 2 , . . . , 8 ,

The 35 generators are normalized to Tr~ 2= 4. We then express the colour-spin products of the Fermi contact interaction in

terms of the quadratic Casimir operators of SU(3)c, SU(2)s and SU(6)cs.

1 1 (6.15) - 4 ~ ~,(S~.Sj)A~A~=48+ S ( S + I)+-~C3c--~C6cs. i<j


The Casimir operators are defined as follows:

(~1 )2 / 6 y / 6 k\2 C6o~ = ,=,~ ~ , Cao = ~,., (~lAi) , S(S + 1)= ~k=l li~-lSi) "

For colour singlets C3c = 0 and (6.15) becomes

1 1 (6.16/ - 2, ~j(Si. Sj)ATA~ = 12 + S (S + 11 - gC6cs-

In S-wave the product of the colour-spin and flavour representations is antisymmetric as a result of the generalized Pauli principle. A rotation by 90 ~ of the Young tableaux of the colour-spin representation gives us the associated (conjugate) Young tableaux for the flavour representation. A symmetry (antisymmetry) in the colour-spin indices then induces the corresponding ant isymmetry (symmetry) in flavour.

A quark is a [6] in SU(6)cs and thus a three-quark cluster is given by

6 | 6 | 6 = 56s + 70Ms + 70MA + 20A.

The SU (3)c--~ S U (2)s decompositions of the three-quark S U (6)cs multiplets are as follows:

56 = (10, 4) + (8, 2), 70 = (10, 2) + (8, 4) + (8, 2) + (1, 2), 20 = (1, 4) + (8, 2).

Here we use the notation (de, d,), where dc is the dimension of the SU(3)c representation and d~ is the dimension of the SU(2)s representation.

The isospin group SU(2)F allows the following representations: for three quarks

2|174174174

The antisymmetric colour-spin-flavour combinations of the two three-quark clusters are then limited to

([20]cs [4]F)([20]cs [4]~),

([20]cs [4]F)([70]CS [2]~),

([70]CS [2]F)([70]CS [2]~).

The SU(6)cs colour-spin representations are shown below:

20 | 20 = 1 + 35 + 175 + 189,

20 | 70 = 35 + 189 + 280 + 896,

70 | 70 = 175 + 189 + 280 + 490 + 840 + 896 + 896 + 1134.


The S U (3)c | S U (2)s decompositions of the six quark S U (6)cs multiplets are as follows:

1 = (1, 1),

35 = (8, 1) + (8, 3) + (1, 3),

175 = (27, 3) + (8, 5) + (1, 7) + (10, 1) + (10, 1) + (8, 3) + (1, 3),

189 = (1, 1) + (1, 5) + (8, 1) + (8, 5) + (8, 3) + (8, 3) + (10, 3) + (10, 3) + (27, 1),

280 = (27, 3) + (10, 5) + (10, 3) + (8, 5) + (10, 1) + (10, 1) +

+ (8, 3) + (8, 3) + (8, 1) + (1, 3),

490 = (10, 7) + (27, 5) + (8, 5) + (35, 3) + (10, 3) + (10, 3) +

+ (8, 3) + (28, 1) + (27, 1) + (1, 1),

840 = (35, 5) + (35, 3) + (10, 7) + (27, 5) + (27, 3) + (10, 5) +

+ (27, 1) + (10, 3) + (10, 3) + (10,3) + (8, 5) + (10, 1) + (8, 3) +

+ (8, 3) + (8, 1) + (1, 1),

896 = (35, 3) + (35, 1) + (27, 5) + (27, 3) + (27, 3) + (10, 5) +

+ (10, 5) + (8, 7) + (27, 1) + (10, 3) + (10, 3) + (10, 3) + (8, 5) +

+ (8, 5) + (10, 1) + (8, 3) + (8, 3) + (8, 3) + (8, 1) + (8, 1) + (1, 5) + (1, 3),

1134 -- (28, 3) + (27, 7) + (35, 5) + (35, 3) + (27, 5) + (10, 5) +

+ (35, 1) + (10, 5) + (8, 5) + (27, 3) + (27, 3) + (10, 3) + (8, 3) +

+ (8, 3) + (10, 1) + (10, 1) + (8, 1) + (1, 3).

We now construct the eigenstates of Hcont for each of the allowed cluster combinations.

A) ([20]cs[4]~)([20]cs[4]F). The SU(6)cs representat ions for these 16- dimensional flavour multiplets contain the following colour singlets:

(1, 1) c 1, (1, 3) r 35, (1, 3), (1, 7) r 175, (1, 1), (1, 5) r 189.

Only the (1,4)| (1,4) colour-spin decompositions of the three-quark clusters can contribute to the J = 3 and J = 2 dibaryons. The wave functions are then given by

13 +, 16[175]> = I(1, 4)4; (1, 4)4; (1, 7)>

208

and


12 +, 161189]) = I(1, 4)4; (1, 4)4; (1, 5)>.

The J = 1 wave functions are orthogonal linear combinations of (1,4) | (1,4) and (8,2) | (8,2).

and

(1)1o I1 +, 16135]} = I(1, 4)4; (1, 4)4; (1, 3)} + 1(8, 2)4; (8, 2)4;(1, 3)}

(1)1 I1+, 16[175]> = / ~ ) I(1, 4)4; (1, 4)4; (1, 3) > - ](8,2)4;(8,2)4;(1,3)).

Since both these states arise from the same combinations of colour-spin representations and since they both have the same total spin, Heont mixes these two states and the eigenstates are given by

and

I1 +, 16) = 0.60111 +, 161351) - 0.79811 +, 161175])

I1 § 16"> = 0.79811 § 16135]) + 0.60111 § 161175]).

Similarly the J = 0 wave functions are orthogonal linear combinations of (1,4) | (1,4) and (8,2) | (8,2)

]0*, 1611]) = I(1, 4)4; (1, 4)4; (1, 1)> + 1(8, 2)4; (8, 2)4; (1, 1)>

and

I1+, 16[189])--/-g ) 1(1,4)4;(1,4)4;(1,1))- 1(8,2)4;(8,2)4;(1,1)>.

Again Hcont mixes these two states and the eigenstates are

l0 § 16> = 0.92810 § 1611]) + 0.37310 § 1611891)

and

D + , 1 6 " ) = O.aTarO +, 16111) - 0 . 9 2 8 1 0 § , 1611891).

The eigenvalues are shown in table I.

DIBARYONS IN A QUANTUM CHROMODYNAMICS ETC.

TABLE I. - Masses and magnetic interaction energies of S-wave dibaryons.

209

State Magnetic Mass Baryonic interaction content

13 § 161175]) 265 2529 hA ]2 § 161189]) 265 2529 AA I1 +, 16) 134 2580 AA ]1 +, 16") 322 2842 AA [0 +, 16) 438 3024 AA ]0 +, 16") 150 2542 AA 2 +, 81189] S) - 160 2722 AN 2 § 81189] A ) 80 2523 AN 1 +, 8[35] S ) - 127 2692 AN 1 +, 8135] A ) - 8 2536 AN 0 +, 81189] S) 41 2754 AN 0 +, 81189] S) 2 2715 /%V 3 +, 41175] SS) - 551 2700 NN 3 § 41175] SA ) 179 2847 NN 3 § 41175] AS) 179 2847 NN 3 ~, 4[ 175] AA) 265 2529 XN 2 +, 4[ 189] SS) - 235 2892 NN 2 +, 41189] SA ) - 51 2742 A~N 2 +, 4[189]AS) 64 2645 N~" 2 § 4[ 189] AA ) 190 2540 A~N 1 +, 4[175] SS) - 63 2951 NN 1 +, 41175] SA) 17 2778 NN 1 +, 41175] AS) 31 2632 NN 1 § 4[175] AA) 99 2525 NN 0 +, 4[ 189] SS ) - 204 2836 NN 0 +, 41189] SA) 104 2770 NN 10 +, 4{1891 AS ) 104 2770 NN 10 § 41189] AA ) 88 2495 NN

The notation we use to describe the states is [JP, Dr ICS]MIM2), where J is the angular momentum, p is the parity, Dr is the dimension of the flavour multiplet, [CS] is the colour-spin representation, MtM2 are the type of mixed-symmetry eolour-spin representations of the clusters.

W e no te h e r e t h a t in e v a l u a t i n g the m a t r i x e l e m e n t s of Hcont we r equ i r e an e x p r e s s i o n for the con tac t i n t e rac t ion which is i n d e p e n d e n t of the to ta l colour- spin Cas imi r ope ra to r . This is because the off-diagonal t e r m s of the m a t r i x mix s t a t e s of d i f fe ren t to ta l colour-spin r e p r e s e n t a t i o n s . I t is e a s y to def ine an

e x p r e s s i o n equ iva l en t to (6.16) which is a funct ion only of the in -c lus te r and

i n t e r c l u s t e r colour fac to r s and the c lus t e r and to ta l spin Cas imi r ope ra to r s . This

210

is given by

(6.17)


-- E E (S~ .Sj)A~A~ = i<j

=-lfA~'A2(S~2a(S~2a+ 1) - 9 ) +A4"As(S4s6(S4ss+ 1 ) - 9 1 +

+ AI" A4((Stot(Stot + 1) - S123(S123 -~- 1) - $456(S~56 + 1))].

B) ([20]cs [4]r)([70]cs [2]~.). The SU(6)cs representations for these flavour octets contain the following colour singlets:

(1, 3) c 35, (1, 1), (1, 5) c 189.

The 280- and 896-dimensional SU(6)cs representations are ignored because for two flavours it is impossible to make a sufficiently antisymmetric flavour combination to correspond to the conjugate of the SU(6)cs Young tableaux.

The wave function of the J = 2 dibaryons is constructed from linear combinations of (1, 4) | (1, 2) and (8, 2) @ (8, 4) we write

12 +, 81189]) = I(1, 4)4; (1, 2)2; (1, 5)) + 1(8, 2)4; (8, 4)2; (1, 5)).

For the J = 1 dibaryons we have

11 +, 8135]) /43] 1(1,4)4;(1,2/2;(1, 3)) +

f16 1 + ~-~] 1(8, 2)4; (8, 4)2; (1, 3)) + ~--~] 1(8, 2)4; (8, 2)2; (1, 3)).

And for the J = 0 dibaryons

[0 § 81189]) = 1(8, 2)4; (8, 2)2; (1, 1)).

The eigenvalues are shown in table I.

C) ([70]cs[2]F)([70]cs[2]r). The SU(6)cs representation for these four- dimensional flavour multiplets contain the following colour singlets:

(1, 3), (1, 7) c 175, (1, 1), (1, 5) c 189.


As before the higher-dimensional SU(6)cs representations are ignored. The wave functions are

13 +, 41175]> = 1(8, 4)2; (8, 4)2; (1, 7)>,

12+, 4[189]> = l(8, 4)2;(8, 4)2;(1, 5)> + [(8, 4)2;(8, 2)2;(1, 5)>,

/ 12', 1/2 /16\'~ / ~ ) 1(8,4)2;(8,2)2;(1,3)> + ]1+,4[175]> =/~-~) 1(8,4)2;(8,4)2;(1,3))+

~-~-] !(8,2)2;(8,2)2;(1,3)) + ~-~ !(1,2)2;(1,2)2;(1,3)>,

~16\,~ 10+,41189]> =(~-~) 1(8,4)2;(8,4)2;(1,1))+

+ gg I(a, 2)2; (8, 2)2; (1,1)> + \ ~ ]

Again the eigenvaiues are shown in table I.

l(1,2)2;(1,2)2;(1, 1)>.

7. - R e s u l t s a n d c o n c l u s i o n s .

The magnetic interaction energies, the masses and the baryonic content of the dibaryon states are listed in table I below. Our results suggest the following:

a) No stable dibaryons exist in the u, d sector. The predicted masses are large enough to permit strong decay.

b) Aa dibaryons experience a short-range repulsion due to the magnetic interaction.

c) 2/A c and 5A c dibaryons in mixed symmetry states experience a weak short-range attractive magnetic interaction.

The future 4 GeV Continuous Electron Beam Accelerator Facility (CEBAF) will be largely devoted to an understanding of nuclear physics in terms of QCD. It is expected that CEBAF will be used to search for and characterize dibaryon states, the predictions of this work can then be tested.


�9 R I A S S U N T O (*)

Si applica un modello di potenziale di quark basato sulle QCD per prevedere masse dei dibarioni con onda S nel clustering (q3)(q3) usando i parametri derivati da un'ap- prossimazione barionica.

(*) Traduzione a cura della Redazione.

~H6apHOHbI B KaalITOBOK XpOMO~MHaMHKe, OCHOBaHIIO~ aa KBapKoBOi MO~e~M.

Pe3RoMe(*): - - KBaHTOBag XpOMO/IItHaMHKa, OCHOBaH'Itafl Ha MO,IIeJIH ~ff.rlg IIOTelllIHaJia KBapKoa, npn.~lemseTcs ~3Lq npeacra3amm Macc S-ao.arloBbtx ~H6aprtoHoB B (q3)(q3) r .nacTepnaat~, Hcnoab3y~ IlapaMeTpbt, nonyqeHnbm npn noJlronKVt a cayqae 6apnonoa.

(*) Hepe6ec)euo pei:)aKt4ue5.

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Dibaryons in a quantum-chromodynamics-based consistent quark model