A study of pentaquark $Theta$ state in the chiral SU(3) quark model
a r X i v :h e p -p h /0310040v 1 3 O c t 2003
A study of pentaquark Θstate in the chiral SU(3)quark model
1
F.Huang,Z.Y.Zhang,Y.W.Yu
Institute of High Energy Physics,Beijing 100039,P.R.China
B.S.Zou
CCAST (World Laboratory),P.O.Box 8730,Beijing 100080;Institute of High Energy Physics,Beijing 100039,P.R.China
Abstract
The structure of the pentaquark state uudd -ˉs is studied in the chiral SU (3)quark model as well as in the extended chiral SU (3)quark model,in which the vector meson exchanges are included.Four con?gurations of J π=
1
2
+are considered.The results show that the isospin T =0state is
always the lowest one for both J π=
1
2+
cases in various models.
But the theoretical value of the lowest one is still about 200?300MeV higher than the experimental mass of Θ.It seems that a dynamical calculation should be done for the further study.
Key words:Pentaquark state,Quark Model,Chiral Symmetry.
1Introduction
Recently,LEPS Collaboration at SPring 8[1],DIANA Collaboration at ITEP [2],CLAS Collaboration at Je?erson Lab [3]and SAPHIR Collaboration at ELSA [4]report that they observed a new resonance Θ,with strangeness quantum number S =+1.The mass of this Θparticle is around M Θ=1540MeV and the upper limit of the width is about ΓΘ<25MeV .Since it has strangeness quantum number S =+1,it must
be a5-quark system.The interesting problem is whether it is a strange meson-baryon
molecule like state or a pentaquark state.If it is really a pentaquark state,it will be the?rst multi-quark state people found.There are already many theoretical works to try to explain its properties with various quark models[5,6,7]or other approaches [8].A re-analysis[9]of older experimental data on the K+-nucleon elastic scattering process put a more stringent constraint on the width to beΓΘ<1MeV.Since the mass ofΘ,MΘ,is larger than the sum of nucleon mass and kaon mass,M N+M K,it is not easy to understand why its width is so narrow,unless it has very special quantum numbers.As to the mass ofΘ,although it is predicted by the original chiral soliton model[10]quite well,there is no concrete calculation from quark model available yet.
In this work,we calculate the energies of the pentaquark states in chiral quark model.Four con?gurations of Jπ=1
2
+are considered.Some qualitative information is obtained:(1)The isoscalar state,T=0,is always the lowest
one for both cases of Jπ=1
2
+.(2)The calculated results of the extended chiral SU(3)quark model are quite similar to those of the chiral SU(3)quark model, when the parameters are taken as what we used in the N?N scattering calculation [11,12].(3)When the size parameter is adjusted to be0.6fm,the energy of the lowest
state([4]orb[31]σf ts=01ˉs,LST=01
2
?),is1670MeV,still about130MeV higher than the experimental value of theΘmass.A dynamical calculation will be done for getting quantitative information of theΘparticle’s structure.
2Theoretical framework
For a4q?ˉq color singlet system,the4q wave function includes three parts:orbital,?avor-spin SU(3)×SU(2)and color SU(3)part.InΘparticle case,its strangeness is+1,4q part only includes u and d quarks,and the anti-quark isˉs.Four con?gu-
rations for Jπ=1
2
0,Jπ=
1
2
1,Jπ=1
21,Jπ=1
2
2,Jπ=1
2
+:([31]
orb
[4]σf ts=00ˉs,LST=11
2
+),
([31]orb[4]σf ts=11ˉs,LST=11
2+),([31]
orb
[4]σf ts=11ˉs,LST=13
2
+)
and([31]orb[4]σf ts=22ˉs,LST=13
2
+).The color part of them is[211]c,
i.e.(λμ)c=(10),combining(01)ofˉs,the total quantum number in color space is singlet.For Jπ=1
2+states,[31]
orb
replaces[31]σf
to make the anti-symmetrization.
In the chiral SU(3)quark model the Hamiltonian of the system can be written as
H=
i T i?T G+
i V ij+ i=1?4 V i5,(1) where i T i?T G is the kinetic energy of the system,V ij,i,j=1?4and V i5,i=1?4 represent the interactions between quark-quark(q?q)and quark-anti-quark(q?ˉq) respectively. V ij=V conf ij +V OGE ij +V ch ij,(2) V conf ij is the con?nement potential taken as the quadratic form, V conf ij =?a c ij(λc i·λc j)r2ij?a c0ij(λc i·λc j),(3) and V OGE ij is the one gluon exchange(OGE)interaction, V OGE ij = 1 r ij ? π m2qi + 1 3 1 with V ch qˉq ( r)= i (?1)G i V ch,i qq ( r).(8) Here(?1)G i describes the G parity of the ith meson.For theΘparticle case,qˉq can only annihilate into K and K?mesons,thus V ann i5 can be expressed as: V ann i5=V K ann+V K? ann ,(9) with V K ann=?g2ch 1 2 )spin( 2+3λq·λ?ˉq 9+ 1 (?m+?m s)2?m2K? ( 3+ σq· σˉq 6 )color (19 6 λq·λ?ˉq)flavorδ( r q? rˉq).(11) Where?g ch and?g chv are the coupling constants of pseudo-scalar-scalar chiral?eld and vector chiral?eld in the annihilation case respectively.?m represents the e?ective quark mass.Actually,?m is quark momentum dependent,here we treat it as an e?ective mass. Using these two models,we did an adiabatic approximation calculation to study the energies of the(uudd-ˉs)system. 3Results and discussions First,we carry on the calculation by taking the parameters which can reasonably reproduce the experimental data of N?N and Y?N scattering[11,12].In the chiral SU(3)quark model,besides pseudo-scalar and scalar?elds coupling,OGE interaction is still there to o?er part of the short range repulsion,as well as in the extended chiral SU(3)quark model,the OGE interaction is almost replaced by the vector meson exchanges.About the annihilation interaction between u(d)?ˉs,it is a complicated problem,in Eqs.(10)and(11),the quark e?ective masses?m and?m s,as well as the annihilation coupling constants?g ch and?g chv are subject to signi?cant uncertainties.In our calculation,we treat(?m+?m s),?g ch and?g chv as parameters,and adjust them to?t the masses of K and K?mesons,named case I.In case II,we omitted the annihilation interaction in the calculation to see its e?ects.All results of4con?gurations of Jπ=1 2 +are listed in Table1. ?From Table1,one can see that:(1)The isoscalar state(T=0)is always the lowest state both in Jπ=1 2+cases,and([4] orb [31]σf ts=01ˉs,LST= 01 2 ?)is always the lowest one in di?erent models.(2)The results of the chiral SU(3)quark model and the extended chiral SU(3)quark model are quite similar,although the short range interactions of these two models are di?erent,one is from OGE and the other is from vector meson exchanges.(3)The annihilation interactions o?er attraction to the states of Jπ=1 2 +states. (4)When the annihilation interaction is considered,the energy of the lowest state, ([4]orb[31]σf ts=01ˉs,LST=01 2 ?),is about250?300MeV higher than the experimental value of theΘmass. We tried to adjust the size parameter b u to be larger to see the in?uence.As an example,the results of b u=0.6fm in the chiral SU(3)quark model are given in Table 2.In this case,the energies of all states become smaller,caused by the kinetic energy of the system is reduced for larger b u.When the annihilation interaction is included (case I),the energy of the lowest state,([4]orb[31]σf ts=01ˉs,LST=01 2 ?),is 1670MeV,about130MeV higher than theΘ’mass. In our results,the states of Jπ=1 2 +,even in the extended chiral SU(3)quark model,in which the OGE interaction is almost totally replaced by vector meson exchanges.According to Stancu and Riska’s argument[6], the state of T=0,Jπ=1 2 ?,because the spin-?avor dependent interactions from Goldstone-Boson exchange potential o?er more attractions to the state of T=0,Jπ=1 2 +, but when the interactions between u(d)andˉs are included,especially the annihilation terms are considered,the state of T=0,Jπ=1 2 ?has1 Table1:Energies of pentaquark states in di?erent chiral quark model Ex.Chiral SU(3)Quark Model con?guration b u=0.45fm ?I II 2 (MeV) 18011957 20492128 21172190 23232369 Jπ=1I II (MeV) [31]orb[4]σf ts=00ˉs2******* [31]orb[4]σf ts=11ˉs2******* (S=1 23622282 ) 2 [31]orb[4]σf ts=22ˉs2******* Table2:Energies of pentaquark states in chiral SU(3)quark model with b u=0.6fm (b u=0.60fm) 2 ? (MeV) 167218672027 [4]orb[31]σf ts=10ˉs 198320262051 [4]orb[31]σf ts=21ˉs Jπ=1I II III [31]orb[4]σf ts=00ˉs 212420512018 (S=1 [31]orb[4]σf ts=11ˉs 2 ) 217721222051 pair u?ˉs of(0s)2with spin s=0and color singlet(00)c(i.e.K meson’s quantum numbers)and1 2+only has1 4 pair of(0s0p)s=0(00)c, 1 4 pair of(0s0p)s=1(00)c and the other part is color octet.If we take the annihilation interaction to?t the masses of K and K?,the state of T=0,Jπ=1 2 +can be the lowest one,but its energy (1997MeV)is much higher than theΘ’s mass. 4Conclusions. The structures of pentaquark states are studied by an adiabatic approximation calcula-tion in the chiral quark model.When the interactions between4q andˉs are considered, especially the parameters in the annihilation interactions are?xed by?tting the masses of K and K?,our results show that state T=0,Jπ=1 +can be lower then state T=0,Jπ=1 2 ?con?guration has a potential s-wave KN 2 fall-apart mode[7]and hence is very di?cult to explain the very narrow width of the observed width ofΘ.It seems that it is impossible to reproduce the observed low mass and narrow width ofΘby quark models with reasonable model parameters in the adiabatic approximation and a dynamical calculation may be necessary for the further study. References [1]T.Nakano,D.S.Ahn,et.al.,Phys.Rev.Lett.,91,2003,012002. [2]V.V.Barmin et.al.,hep-ex/0304040. [3]S.Stepanyan,K.Hicks,et.al.,hep-ex/0307018v3,16Jul2003. [4]J.Barth,W.Braun,et.al.,hep-ex/0307083v3,6Aug2003. 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