Pseudobaryons

Volume 74B, number 3 PttYSICS LETTERS 10 April 1978

PSEUDOBARYONS

M. FUKUGITA 1 and K. KONISH1 Rutherfi)rd Laboratory, Chilton, DMcot, Oxon OXl l OQX, England

and

T.H. HANSSON 2 Institute of Theoretical Physics, CTH, 20 Gdteborg 5, Sweden

Received 3 February 1978

We point out the possibility of observing narrow peaks of bound states of qq with qqC t. They are expected to be produced diffractively in pp ~ (PplS)forward + anything, with (p~) forming baryonium (qq-qCt) states.

Several narrow high-naass hadrons recently observed [1,2], notably at 2.95 and 2.2 GeV I l l , h a v e been in- terpreted as candidates for the so-called baryonium [3], a bound state o f a diquark (qq) and an antidiquark (qCt) [4]. If this is really the case, it is the begin- ning of a rich multiquark spectroscopy and many narrow states belonging to other quark configurations are expected to exist. An interesting possibility is the bound state of (qq) with (qqT:t), which is an analogue of baryonium but with baryon number one.

In this note we investigate the bound state of (qq) and (qq~t) which we shall call "pseudobaryon". The pseudobaryon is analogous to baryonium in the sense that the former is obtained by replacing an antiquark in the latter with a diquark. The model we consider follows closely that of Chan and H6gaasen [3], as it gives an attractive explanation of the narrow width and decay patterns of the observed states including the one at 2.95 GeV.

We point out the possibility of observing certain pseudobaryons with narrow widths (P ~< 20 MeV). These states are expected to decay into a baryon and a baryonium as well as into a pion and another pseudo-

1 On leave from National Laboratory for High Energy Physics, Tsukuba, Ibaraki 300-32, Japan.

2 Partially supported by the Swedish Natural Science Research Council under Contract R8326-002.

baryon. They may be seen in tile pp~ channel, diffractively produced in processes such as pp -+ (PPP)forward + anything, with a cross section times branching ratio of order ~ (0 .1-0.5) / lb . We estimate the lowest of these to be in the mass range 3.3 3.5 GeV.

There is another class of multiquark bound states with baryon number one, namely, (qq)eolourS- (qqq)colourS" These states, if they exist at all,would mainly decay into a baryon plus nmltimesons, but not into BBB. Experimental observation of these states would be more difficult, and they will not be dis- cussed here.

Let us first summarize briefly the model of baryonium [3]. The approach is based on the MIT bag model [5,6] and on the dual unitarization scheme [7]. The baryonium is a string-like object with a diquark (qq) and an antidiquark (q~) on the two ends. There are two kinds of baryonium states, depending on whether the diquark (antidiquark) forms 3* (3) or 6 (_6_*) representation of colour SU(3) (called T- and M-baryonia, respectively). The T-baryonium has a normal decay width into a baryon-ant ibaryon pair, although decays into mesons are suppressed. Candidates for this kind of baryonium are S(1936), T(2.19) and U(2.31) [21. On the other hand, the narrow states at 2.9 and 2.2 GeV (and possibly at 2.6 and 2.2 GeV) [ 1 ] can be as- signed as M-baryonia. The M-baryonia, which we hereafter denote by c/?f, cannot simply split into a BB pair

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because of colour confinement. The decay into mesons is also shown to be suppressed from the viewpoint of dual unitarization.

In the bag model Regge slopes are proportional to the inverse of the flux in the tube [5],

c~'~x 1/x/C, (1)

where C is the eigenvalue of the Casimir operator of colour SU(3) for the cluster of quarks (and/or antiquarks) at one end of the "string". It then follows that, in the case of M-baryonium,

! t ~ = X / 2 ~ e , (2)

where a ' ~ 0.9 (GeV/c) -2 is the Regge slope of ordinary meson trajectories. The relation a~t/a' > ½, makes the M-baryonium stable against the superallowed decay * 1 into two excited mesons.

The long-range confining force is assumed to be spin independent [5]. The spin-dependent force be- comes effective only among the quarks in the same cluster at an end, as the orbital angular momentum L increases. It was found in ref. [3], that this holds true for L ~> 2 (it fairly does even for L = 1). The two quarks in the diquark are assumed to stay in the S- wave. Then the mass splitting among the diquarks in different colour and spin representations, is given by the colour-magnetic hyperfine interaction,

= %x;) %x) ), (3) o~ '~>] ' I

with/3 ~ 20 MeV from the A - N mass splitting. Combining the results for the diquark state with

those for the antidiquark state, one finds the spectrum of baryonium up to the absolute value of the Regge intercepts * 2

With these preparations we now discuss the pseudobaryon, the bound state of (qq) and (qqq). The all- quark is in either a colour 3-* or 6- representation; the ( q q ~ system should then be in a 3- and a 6* representation, respectively, in order to make a colour singlet

,1 Superallowed decays are those which involve no creation of any new quark pairs, nor any annihilation of existing quark pairs.

,2 For the states with small L (L m 0) the hyperfine interaction gives rise to mixing between T- and M-baryonium states (thus allowing ~ to decay into BB). The diagonal- ization would depend on the details of the dynamics, and will not be treated here.

as a whole. Some of the pseudobaryons with clusters of quarks in a colour 6 representation will be narrow, while those with clusters in a 3 representation will have large decay widths into a baryon plus mesons in analogy with T-baryonia [3]. In this note we shall con- centrate on the colour 6- states, which are more interesting from the experimental point of view.

Let us again assume that the (qq) and the (qq?t) sys- tems are both in S-wave states. Then the (qq?t) states with respect to [(colour SU(3),

spin SU(2))colour_spi n SU(6), flavour SU(3)] are:

[(6", 2 )84 ,3 • l S ] , [(6", 4)84, _3 • 1-5_1, - - - - ( 4 )

[(6", 2)120,3- ~9 6-*l-

These are to be combined with the two diquark states [3,6]:

[(6-,/--)15,6-], [(6-, 3 ) 2 1 , 3 * ] . (5)

Thus we obtain six families of pseudobaryons shown in table 1. Also shown in table 1 is the hyperfine mass splitting due to the colour-magnetic force, which is evaluated using eq. (3). The coefficient of eq. (3) for the (qq~l) system may be different from that for (qq) (/3' and 13 in table 1). We shall, however, assume/3' ~ 13 ~-- 20 MeV, for simplicity.

The spectrum of the pseudobaryon can be obtained in a manner similar to the case of baryonium. The slope of the Regge trajectory is given by eq. (2). In order to estimate the intercept, we take the following simple model [8]. The mass of the cluster at an end, before taking account of the hyperfine mass splitting, is given by,

m = N/~ 0 + g2 ~ ~ X~.X~ = Nl~ + 2g2C, (6) i>] t l

where N is the total number of quarks and antiquarks in the cluster, g the effective gluon coupling to quarks, /a the effective quark mass. In eq. (6) C is the same Casimir operator as in eq. (I) . Furthermore, we assume that the mass of a pseudobaryon (without hyperfine mass splitting) at L = 0 is given by a sum of two contri- butions of the form, eq.(6), one for each end ,a . The parameters in eq. (6) are estimated from the masses of p, n, A, N and those of c'~ (L = 0) obtained in ref. [3] :

#3 We interpret the above mass at L = 0 as the extrapolation to L = 0 of the asymptotic straight Regge trajectory.

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Volume 74B, number 3 PHYSICS LETTERS

Table I Six families of pseudobaryon ( I -VI) and four families of baryonium (i-iv).

10 April 1978

(SU(3) C, SU(2)spin)SU(6 ) Spin Elavour SU(3) Isospin (non-strange members)

(qq) X (qq~)

I (6, 1)is x (6", 2)84 1/2 2 × (8 8 1_0) 8 1_00" 8 27 8 35 5/2, 3/2 x 2, 1/2 × 2

11 (6, 1)is X (6", 4)84 3/2 2 x (8 8 10) 8 10" 8 27 8 35 5/2, 3/2 × 2, l /2 × 2

I I I (6, 1) is x (6", ])12__o 1/2 l 8 8 8 _8 8 i_0 8 2-7 1/2, 3/2 IV (6,3)21 X (6" ,2)s4 1/2,3/2 1 8 8 8 8 8 10 82_7 1/2,3/2

V (6 ,3)21× ( 6 " , 4 ) 8 j 1/2,3/2,5/2 1 8 8_ 8 8 8 1 0 82-7 1/2,3/2

VI (6, 3)21 x (6*, 2)x2o 1/2, 3/2 1 8 8 • 8 8 10" 1/2

A rtl

16 , 4/3 + -T~

4 , 4[3 + 7/3 4/3 - 8/3'

4 / 3 + 3"/316,

4 # + 3/34,

_ 4 # _ 8#'

(qq) × (EIE[)

i (6, 1)t s × ( 6 " , 1 ) 1 5 0 1 8 8 8 2 7 0 , 1 , 2 8/3 8

ii (6, 1)15 X (6_*, 3)21 1 8 8 10 1 -~/3 8

iii (6, 3)21 X (6", 1)15 1 8 8 10" 1 fir 8

iv (6, 3)2 ] X (6_*, 3)21 0, ], 2 1 8 8 0 --5/3

J

13/2

11/2

9 / 2

7/2

5/2

3/2

1/2

Ol

All

DIZI

o r V

A~

a n ] / 2

I= 3/2.1/2

: 312,112

/,/ i I I I i J f I i 6 7 8 9 10 11 12 13 14

M 2 ( G e V 2)

Fig. 1. Spectrum of pseudobaryons with L ~ 1. The orbital angular momentum and parity of each state are shown as L P. Some low lying trajectories (lower than the parent trajectory of I) are omitted. For the notation I -VI , see table 1.

263


~k

=:~> ~ " - " ~ b]

'=I:::> ~E::::~ (E::::~ c)

d)

ring with a q~ pair creation into a baryon B(L1) and a baryonium(L2) (L 1 +L 2 ~ L ) is forbidden since the flux has colour 6 quantum number (fig. 2c); (iv) the decay into a meson plus a baryon by rearrangement of quarks (fig. 2d) or by annihilation of qq pair should be of the order of exchange-degeneracy breaking or OZ| rule violation; therefore it is suppressed for L ~> 1 [3] ; (v) cascade: emission of a meson or a baryon from one of the clusters (fig. 2e),

q3-+ q3 + 7r,

L ,.,

e)

Fig. 2. Possible decay patterns of pseudobaryons. Solid and open circles denote quarks and antiquarks, respectively.

= 0.30-0.40 (GeV) g2 = 0.03-0.06 (GeV). (7)

Using the hyperfine splitting in table 1, we obtain the spectrum of non-strange pseudobaryons shown in fig. 1 (with ta = 0.36 GeV and g2 = 0.051 GeV). In ob- taining the spectrum we assumed for the (qqTt) system "ideal mixing" of 3 • 15 and of 3 e 6, e.g., that u(u~ + dd), are eigenstates of the mass hamiltonian.

Note that the states with small masses tend to have non-exotic flavour quantum numbers (see table 1). This is a consequence of the fact that the colour-magnetic hyperfine interaction is most attractive for states in the largest representation of colour-spin SU(6), as found in ref. [6].

We next discuss the decay of the pseudobaryon (we hereafter denote it by ~ ). We classify the possible decay patterns (see fig. 2) and discuss them in order: (i) dissociation (superallowed decay) ofC~ (L)into meson M(~ L) and baryon B(~ L) is forbidden just as in the case of M-baryonium [3] because of ct'q0/c( >-~ (fig. 2a); (ii) another dissociation, Oral (L) -~ B(L) + M(L = 0) is forbidden by the colour quantum number 6_ of the (qq~) cluster ,4 (fig. 2b); (iii) the split-

q6 ~ O'/~ +N. (9) L 1~(+ ~'s)

If the final pseudobaryon (or baryonium) has the same L as the initial pseudobaryon (AL = 0), no suppression mechanism works and the decay has a normal width provided the phase space is sufficient. Cascade decays changing L (AL v~ 0) are caused by quantum fluctuations followed by the emission of 7r (or N) and they should be relatively suppressed.

The arguments in (i)-(v) above lead to the conclu- sion that the dominant decay of pseudobaryons is the cascade mode * s (v), which we discuss below in more detail. We assume that one of the two clusters emits 7r or N, while the other remains in the initial state,

e.g., l a l , a 2 ) ~ ]o~] + rr, a2)or Lal, ~2)-+ l a l , a ~ +rr) with a i the quantum number of each cluster.

We shall estimate the cascade decay width in a simple potential model. A partial width of 7r/N emission is given by,

F ~ T A L ( q / M ) B I ( q R ) , (10)

where q is the decay momentum and M the mass of the decaying particle. We take the Blatt-Weisskopf barrier factor [9] with interaction radius R ~ 1 fro. In eq. (10) 7AL denotes the reduced decay width. The reduced width for ~L = 0 is estimated from A -+ N~ to give T~L=0 ~ 1340 MeV. The reduced width for AL 4= 0 should be estimated from the cascade decay between M-baryonia, e.g., [2.95] -+ ~ + [2.20] [1]. Adopting the assignment that [2.95] belongs to L

4:4 The classes (ii) and (iii) are allowed for the pseudobaryon with clusters in colour 3.

, s For the discussions on this point , we are indebted to Chan Hong-Mo.

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Table 2 Summary of the total width and the branching ratio into N + ~rr( of pseudobaryon for 3 < L < 5.

States l II III IV V VI

150-300 30-80 20-40 200-400 20-60 5 -15 Total widths (MeV)

BR (%) into 5-20 <20 <10 5-30 5 4 0 10-30 for L = 4, 5 N+~" (L > 1) ( < 2 0 f o r L = 3 )

= 3(J PC = 4 - - ) and [2.20] t o L = l(J Pc = 3 - - ) [31,

we obtain 7AL = 2 ~- 40 MeV, much smaller than 7aL =0' This suppression is in accordance with the discussion in (v) above. For other AL we simply assume

a geometric ratio TaL+I/'YAL ~ const ~ 1/6. The results of the calculations for parent trajecto-

ries for the pseudobaryons with L ~> 3, are summarized in table 2. Among the six states, I and IV are broad as they have a phase volume sufficient for pion emission without changing L. For the other states, pion emission with AL = 0 is either forbidden (1II, VI) or suppressed by phase volume (II, V). In consequence the states II, Il l , V and VI should be seen as narrow peaks. The states VI are narrow (Fro t <~ 20 MeV), and for L ~> 4 the mode

cB ~ N + c'~ (L ~ 1)

LN~ has a substantial branching ratio (10-32%) . The states V have similar properties (I ' to t ~ 50 MeV, BR = 25-40%). The states II and I l l are also narrow, but have a smaller branching ratio to N +qY( (L ~> 1)(BR ~< 10%). These features are insensitive to changes of the parameters.

The above discussion of the decays of pseudobaryons suggests that the most interesting objects to look

for are those states labelled V1 in table 1 (and proba- bly the states V). The non-strange members of the

1 I 3 states VI (V) have isospin T (7 and 7 ) and may appear as an ordinary baryon. A clean signature of the pseudobaryon, however, will be the decay into pp~ with the pp pair forming one of the narrow baryonium states X(2.2), X(2.0), etc. The estimated masses of these states are shown in fig. 1. In particular the lowest state decaying into pp~ is expected to be in the range, m = 3 .3 -3 .5 GeV.

Another interesting feature of our pseudobaryons is that the states with exotic quantum numbers ( " Z ~ " [ I = 0, S = 1, Q = 0] , "~+"[3 /2 , - 2 , +1] and " Z . . . . [3/2, - 2 , - 2 ] ) appear in the class VI, and therefore they should be narrow. In particular, the narrow peak of the S -- +1 state is expected to be seen in the (NKmr ...)+ or (ANN) + channel but not in the (NKnTr ...)0 or (S, NN) 0.

The product ion of pseudobaryons is quite analogous to that of baryonia, which are produced by dif- fraction [3] in the process

¢r-p ~ ~ forward + anything

Lpo (fig. 3a). We expect that the pseudobaryon is also produced diffractively in the proton fragmentation region, e.g., in

PP +c13 forward + anything

L PpI5

FIG. 3a • FIG.3b :

Fig. 3. Quark diagram of the Pomeron couplings to (a) M~ and to (b) Bq0.

(~g. 36). We can estimate the production of pseudobaryons

by using an empirical cross section of the M-baryonium [2.95] [1]. The lat ter was found to be o .BR ~ 1 ~tb with BR the branching ratio of [2.95] -~ lrp~. Such a large cross section is not surprising, since the mixing between ordinary meson q~l and four-quark states qq~l?:t can be large for space-like momentum (t ~< 0) [10]. This proper ty of quark mixing is an outstanding

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Volume 74B, number 3 PttYS1CS LETTERS 10 April 1978

result of the dun unitarization scheme [7] ~6. Since productions of c/g and q3 are both mediated by the

same mixing of qq and qqqr:t (fig. 3), one gets o(pp -+ cl~ + anything)/o(rrp -+ ~ + anything) ~ oT(pp)/

OT(np). This gives o(q3 ) ' B R ( ~ -+ ppp) = 0 .1-0 .5/Jb . We believe that the experimental observation of q8 should not be too difficult, as a fairly small back- ground is expected in the pp~ channel. Experimental confirmation of these states is not only interesting by itself, but it would also verify the present picture of multiquark hadron states.

We thank Chan Hong-Mo for illuminating discussions. We thank him and Dick Roberts for reading the manuscript. One of us (T.H.H.) thanks R.J.N. Phillips for his kind hospitality at the Theory Division of the Rutherford Laboratory.

~6 This explains, for instance, the smallness of the OZI rule violation and at the same time the large contribution of the Pomeron [7].

References

[ 1 ] C. Evangelista et al., Phys. Lett. 72B (1977) 139. [ 2] For a review, see: L. Montanet, CERN preprint CERN/

EP/Phys. 77-22 (1977).

I3] Chan H.-M. and H. H~bgaasen, Phys. Lett. 72B (1977) 121,400; Rutherford Lab. preprint RL-77-144/A T 209 (1977).

[4} G.I.'. Chew, LBL preprint LBL-5391 (1976); G.C. Rossi and G. Veneziano, Nucl. Phys. B123 (1977) 507; M. hnachi, S. Otsuki and F. Toyoda, Prog. Theor. Phys. 57 (1977) 517.

[5] T. de Grand, R.L. Jaffe, K. Johnson and J. Kiskis, Phys. Rev. D12 (1975) 2060; K. Johnson and C.B. Thorn, Phys. Rev. D13 (1976) 1934.

[6] R.L. daffe, Phys. Rev. D15 (1977) 267; 281; M1T preprint CTP # 657 (1977).

[7] Chan H.-M., J.E. Paton, Tsou S.T. and Ng S.W., Nucl. Phys. B92 (1975) 13; Chan H.-M. and Tsou S.T., Rutherford Lab. preprint RL-76-080 (1976); G. Veneziano, Nucl. Phys. B74 (1974) 365; Phys. Lett. 52B (1974) 220; C. Rosenzweig and G.F. ('hew, Phys. Lett. 58B (1975) 93; C. Schmid and C. Sorensen, Nucl. Phys. B96 (1975) 209; M. Fukugita, T. lnami, N. Sakai and S. Yazaki, Nucl. Phys. B121 (1977) 93.

[8] Y. Nambu and M.-Y. Han, Phys. Rev. 10 (1974) 674. [9] J.M. Blatt and V.F. Weisskopf, Theoretical nuclear

physics (Wiley, New York, 1952). [10] Chan H.-M. and Tsou S.T., Nucl. Phys. Bl18 (1977)413;

T.H. Hansson, in preparation.

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Pseudobaryons