Multiquark statesKerkyra September 11thFranco Buccella, Napoli

1) Historical introduction

2) Spectrum given by the chromo-magnetic interaction

3) Selection rules

4) Conclusions

D. Falcone, H. Hogaasen, P. Sorba e F. Tramontano

The search for SU(3) flavour exotic hadrons (27, 10 + mesons and , 27, 35 barions) is very old.

R. Jaffe, Chang-Hong-Mo, Hogaasen and Sorba.

Encouraged by the success of deriving mass splittings within SU(6) flavour-spin multiplets (De Rujula, Georgi and Glashow), Jaffe considered mesons.

The chromo-magnetic interaction

Gives contribution proportional to

r

aa 21

43

1

2

1236 dCdCdC

and therefore the transformation properties with respect to SU(6) colour-spin and its subalgebras dictate the spectrum

1010

qqqq

For qq interactions the lighter states correspond to the higher SU(6) colour-spin representations (with larger Casimir), while the opposite happens for .

In conclusion the lighter states are expected to be the ones, with 2q (2q) transforming as a 21 (21) SU(6) colour-spin representation and (2q 2q) as a singlet of SU(6) colour-spin.

So we expect a nonet of SU(3) flavour

with and to be the lightest states.

It is attractive to identify the (u or d s) states with the

resonances.

As long as for barions, Y=2 states have been found in bubble chambers: or

4321 qqqq

4321 qqqq

)980(0 00 Af

21502074186517801720 05031311 DDPP

qqqq

) or ( sdu

qq

More recently, the discovery of a Θ+ at 1540 MeV brought new interst in the subject, since that value of mass, lower than generally expected, was predicted in the Skyrme model where a is expected to be in the same supermultiplet with the more firmly established states of the 56 flavour-spin L=0 ( and ).

To get positive parity baryons with low mass, Jaffe and Wilczeck assumed the state to be built with two pairs of light qq S=0 3 of colour and flavour in P-wave, which combine with the s to give the proper quantum numbers of the Θ+. We have constructed 4q L=1 states with similar properties and the baryons obtained by combining them with a

s in S-wave with respect to them have been found.

The colour triplets, one can build with 4q L=1 states have the SU(6) colour-spin SU(3) flavour SU(2) spin behaviour

2

110

2

18

2

310

FCSFCS

qqLqq 3,213,21 1

70

113470566210

)810,2,1(

)3()2()3(

3,2,310533,3

3,2,321063,3

3,2,321061,3

)6()3()2()3(

1,33,611,33,6

FSC

FSCSC

FSCSC

FSCSC

CSFSC

SCSCSCSC

SUSUSU

qSUSUSUSU

L

lighter,1134 heavier since the chromo-magnetic interaction gives a contribution proportional to

So we expect rather light S=1/2, L=1, J=1/2+3/2 multiplets to be splitted by the L·S term

3

4

3

1

3

12266 tCpCtCpC

The complete mass formula is

1

2266,4

4

1

21

3

4

3

1

3

1

KSLa

tCpCtCpCC

mmmmpm

q

qq

iqqqqqqi

One can write a similar formula for the positive parity pentaquarks with 4q L=0 and the in P-wave with respect to them.

Finally for the negative parity states with all the constituent in S-wave one gets the formula

s

3

4

3

1

3

1

3

26

3

1

2266,4

26

4

1

tCpCtCpCC

tCtCCmmsm

qq

iqqqqi

To get low masses one should have large and small

For 4q in S-wave one has the following correlation

tC6 pC6

FCS SUSU )3()6(

2'15,3,15

115,531,'105

06,3,105

3,31,210

2)3()2()6(

I

I

I

stateno

YSUSUSU FSCS

By composing with the antiquark one gets the following states

)5,2,70()5,4,20(

)3,531,540()3,2,70()3,4,20(

)1,4,560()1,2,70(

)2()2(6

ISCS SUSUSU

In his study of mesons Jaffe remarked that some states are "open door", since they can go into two mesons just by separation dueto an overlap of the ( ) and the ( )( ) wave functions.This property can be easily understood in terms of the transformationproperties of the octet of pseudoscalar mesons, a 35 of SU(6) flavour-spin ( SU(6)FS )and a singlet of SU(6) colour-spin ( SU(6)CS ).In the hypothesis that the decay occurs as the separation of the ( ) into two pseudoscalars, only the SU(6)CS singlets have non-vanishing amplitudes.

For a similar reason only the ( ) states, which transform as a 35 of SU(6)CS may decay by separation into a final state consisting of a pseudoscalar and a vector meson.The chromomagnetic interaction gives lower mass to the ( )states, which transform as SU(6)CS singlets; these particles have larger amplitudes and broader widths.

qqqq

qqqq qq qq

qqqq

qqqq

qqqq

This favours the interpretation of the very light and broad f0 as a( ) state. For a similar reason the negative parity pentaquarks easier to be

identified are the 1/2-, which do not transform as a 70 of SU(6)CS, as

the 1/2+octet, and the 3/2-, which do not transform as a 20 of SU(6)CS, as the 3/2+ decuplet, as it is the case of the D03 and D15, which transform as a 560 and 540 of SU(6)CS, respectively.For the positive parity pentaquarks, which decay into a pseudoscalarmeson and a baryon of the 35 of SU(6)FS in P-wave, one has to look for the states, which transform as a 70 of SU(6)CS, which may decay by separation into KN or as a 20 to decay into K Δ .Beyond the state, we have identified with the Θ+ at 1540, we predict the existence of two I=1 states with J = 1/2 and 3/2 at 1707 and 1767, which compare well with the P11(1720) and P13(1780) found experimentally.The Ξ 1862 I=3/2 Y=1 state found by NA49 may be identified with the partner of the P11(1720) in a 27F 1/2+.

qqqq

uddss

Conclusion

SU(6)CS plays a central role in the study of multiquark states, by characterizing the spectrum dictated by the chromomagnetic interaction and supplying selection rule for the amplitudes in the approximation that the decay happens by separation.

For S-wave resonance the decays allowed may be toobroad to be identified, while for the P-wave decays the allowed ones are expected to be enough broad to be detected.

The comparison with the present experimental information ispromising.