Photoproduction of pentaquark cascades from nucleons

a r X i v :n u c l -t h /0312119v 1 29 D e c 2003

W.Liu 1and C.M.Ko 1

Cyclotron Institute and Physics Department,

Texas A &M University,College Station,Texas 77843-3366,USA

(Dated:February 8,2008)

Abstract

The cross sections for production of pentaquark Ξ+5from the reaction γp →K 0K 0Ξ+5and

Ξ??5from the reaction γn →K +K +Ξ??5are evaluated in a hadronic model that includes their couplings to both ΣˉK

and ΣˉK ?.With these coupling constants determined from the empirical πNN (1710)and ρNN (1710)coupling constants by assuming that Ξ+5,Ξ??

5,and N (1710)belong to

the same antidecuplet of spin 1/2and positive parity,and using form factors at strong interaction vertices similar to those for pentaquark Θ+production in photonucleon reactions,we obtain a cross section of about 0.03-0.6nb for the reaction γp →K 0K 0Ξ+5and about 0.1-0.6nb for the reaction γn →K +K +Ξ??5at photon energy E γ=4.5GeV,depending on the value of the coupling constant g K ?ΣΞ5.

PACS numbers:13.60.-r,13.60.Rj

I.INTRODUCTION

The observation of the pentaquarkΘ+(uuddˉs)baryon[1]in nuclear reactions induced by photons[2,3,4,5]and kaons[6]has prompted extensive theoretical studies on both its properties[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]as well as production [23,24,25,26,27,28,29,30]and decay mechanisms[31,32].Although most models predict thatΘ+has spin1/2and isospin0,their predictions on its parity vary widely.While the soliton model gives a positive parity and the lattice QCD study favors a negative parity, the quark model can give either positive or negative parities,depending on whether quarks are correlated or not.The parity ofΘ+a?ects both the magnitude of its production cross section in these reactions and the photon asymmetry in photonucleon reactions.With a positiveΘ+parity,the production cross section is almost an order-of-magnitude larger than that for a negative parityΘ+as a result of a smaller KNΘcoupling constant in the latter case[25,26,27].The positive parity also leads to a large positive photon asymmetry,which becomes negative if theΘ+parity is negative[28,30].

There are other pentaquark baryonsΞ+5(uussˉd)andΞ??5(ddssˉu)in the same antide-cuplet asΘ+.Although theΞ??5has already been observed recently in p+p collisions at √

center-of-mass energy

γp

K K K K K K K K K K ΞΞΞΞ0

0ΣΣ+

++++ΣΣΣ+

(1a)

(1b)

(1d)

(1e)

(1c)

K Σ?0

K ?0

+Ξ+5

Ξ5

FIG.1:Diagrams for pentaquark Ξ+5production from the reaction γp →K 0K 0Ξ+5.

K K ΞΣγ

K K ΞΣK γ

K K ΞΣΣγK K ΞΣγ

K K ΞΣγ

K K ΞΣK n

+++??

K +

_ _

(2a)

(2b)

(2c)

(2d)

(2e)

(2f)

(2g)

?++

+ 5

5 5

Ξ_ _

FIG.2:Diagrams for pentaquark Ξ??5production from the reaction γn →K +K +Ξ??5.II.

PENTAQUARK CASCADE PRODUCTION IN PHOTON-NUCLEON REAC-

TIONS

Possible reactions for pentaquark Ξ5production in photon-nulceon reactions near thresh-old are γp →K 0K 0Ξ+5and γn →K +K +Ξ??

5.To evaluate their cross sections,we use the

hadronic model introduced in Refs.[23,24,25]for studying Θ+production in photon-nucleon

reactions.This model is a generalization of the SU(3)?avor-invariant Lagrangian for the interactions between octet pseudoscalar mesons and baryons[34]to include the interactions ofΘ+with nucleons and K as well as K?.Photon in this model is introduced as U em(1) gauge particle,and the symmetry breaking e?ects are taken into account phenomenologically by using empirical hadron masses and coupling constants.In the present study,we extend the model to include interactions among the pentaquarkΞ5,Σ,and K or K?.Since only three-point interactions appear in the interaction Lagrangians,relevant Feynman diagrams that contribute to these reactions are those shown in Figs.1and2.

A.Interaction Lagrangians

The interaction Lagrangians relevant for evaluating the amplitudes shown in Figs.1and 2are:

L KNΣ=ig KNΣˉNγ5 Σ· τK+H.c.,

L KΣΞ5=ig KΣΞˉΞ5γ5 Σ· T K+H.c.,

L K?NΣ=g K?NΣˉNγμ Σ· τK?μ+H.c.,

L K?ΣΞ5=g K?ΣΞˉΞ5γμ Σ· T K?μ+H.c.,

LγNN=?eˉN γμ1+τ34m N[κp+κn+τ3(κp?κn)]σμν?νAμ N,

LγΣΣ=?eˉΣγμQ1ΣAμ,

LγKK=ie[ˉKQ2?K?(?μˉK)Q2K]Aμ,

LγKK?=gγKK??αβμν?αAβ[(?μˉK?ν)K+ˉK?μK?ν].(1) In the above,N,Σ,andΞ5denote,respectively,the nucleon isospin doublet,the sigma isospin triplet,and the pentaquarkΞ5isospin quartet;K and K?are the pseudoscalar and vector kaon isospin doublets,respectively;and Aμdenotes the photon.The Pauli matrices are given by τ,while T is an isopsin transition operator represented by a4×2matrix

with matrix elements given by 3

2n =(3

n),whereλ=0,±1.The operators

Q1=diag(1,0,?1)and Q2=diag(1,0)are diagonal charge operators,and?αβμνdenotes the antisymmetric tensor with the usual convention?0123=1.

For the coupling constants g KNΣand g K?NΣ,their values can be related to gπNN and gρNN, which has empirical values of gπNN=13.5[35]and gρNN=3.25[36],by SU(3)symmetry,

i.e.,g KNΣ=(1?2α)gπNN=?3.78and g K?NΣ=gρNN=3.25if we use the ratioα=D/(D+ F)=0.64[37]for the D?and F?type interaction Lagrangians between pseudoscalar mesons

and baryons.Similarly,we have g KΣΞ

5=g KNΘ/

√2=1.27

[38],where the values for g KNΘ=3.06and g K?NΘ=1.8have been determined in[25]from

the empiricalπNN(1710)andρNN(1710)coupling constants using the SU(3)symmetry. Since the sign of g K?NΘrelative to g KNΘand g KNΣis not?x by the SU(3)symmetry,we shall consider both signs for the coupling constant g K?ΣΞ

=±1.27as well as the case of

g K?ΣΞ

=0.

For photon coupling to nucleon,we include also its interaction with the anomalous mag-netic moment of nucleons with empirical values ofκp=1.79andκn=?1.91.Since the anomalous magnetic moment ofΘ+is not known,we neglect its coupling to photon.

The coupling constant gγKK?denotes the photon anomalous parity interaction with kaons and has the dimension of inverse of energy.Its value is gγK0K?0=0.388GeV?1and gγK±K?±=0.254GeV?1[24]using the decay widthΓK?0→K0γ=0.117MeV of K?0and ΓK?±→K±γ=0.05MeV of K?±to kaon and photon[39].Although the sign of gγKK?relative to other coupling constants in the interaction Lagrangians is not known either,it is not rele-vant for our study as both constructive and destructive interferences among the diagrams in Fig.1or Fig.2are automatically taken into account by using di?erent signs for the coupling

constant g K?ΣΞ

B.Amplitudes for the reactionsγp→K0K0Ξ+5andγn→K+K+Ξ??5

With the above interaction Lagrangians,we can write the amplitudes for the?ve diagrams in Fig.1as follows:

M1a=i√((p

2?p5)2?m2K?)(t?m2Σ)

×?αβμνpα2pβ5ˉΞ5(p4)γν(p/1?p/3+mΣ)γ5n(p1)?μ,

M1b=?√(s

1?m2Σ)(t?m2Σ)

×ˉΞ5(p4)(p/4+p/5?mΣ)γμ(p/1?p/3?mΣ)n(p1)?μ,

M1c=?√(t?m2

)((p2?p4)2?m2Ξ5)

×ˉΞ5(p4)γμ(p/4?p/2+mΞ5)(p/1?p/3?mΣ)n(p1)?μ,

M1d=√(s

1?m2Σ)(s?m2N)

×ˉΞ(p4)(p/4+p/5?mΣ)(p/1+p/2+m N) γμ+iκp

2gγK0K?0g K?NΣg KΣΞ

2gγK+K?+g K?ΣΞ

5g KNΣ

2eg KΣΞ

5g KNΣ

2eg KNΣg KΣΞ

2eg KΣΞ

5g KNΣ

2eκn

(s1?m2Σ)(s?m2N)

×ˉΞ5(p4)(p/4+p/5?mΣ)(p/1+p/2+m N)σμνp2νn(p1)?μ,

M2f=?√(s

1?m2Σ)((p2?p3)2?m2K)

×ˉΞ5(p4)(p/4+p/5?mΣ)n(p1)(2p3?p2)μ?μ,

M2g=i√(s

1?m2Σ)((p2?p3)2?m2K?)

×ˉΞ5(p4)γ5(p/4+p/5+mΣ)γνn(p1)?αβμνpα2pβ3?μ.(3) In the above,p1and p2denote,respectively,the momenta of nucleon and photon in the initial states;p3,p4,and p5are,respectively,those of the kaon on the left,the cascade,and the kaon on the right in the?nal states of the Feynman diagrams in Figs.1and2.We have also introduced the following de?nitions:s=(p1+p2)2,t=(p1?p3)2,and s1=(p4+p5)2.

C.Form factors

To take into account the e?ects due to hadron internal structure,a form factor is intro-duced to each amplitude,and it is taken to have the following form[40]:

F(m x,m y)= Λ4Λ4+(q2y?m2y)2 ,(4) where q x and q y are four momenta of the intermediate o?-shell particles with masses m x and m y in each diagram,andΛis the cuto?parameter that characterizes the o?-shell momentum above which hadron internal structure becomes important.Since the amplitudes for diagram(a)and(e)in Fig.1for the reactionγp→K0K0Ξ+5are individually gauge invariant,including form factors does not a?ect their gauge invariance.This is di?erent for the amplitudes for diagrams(b),(c),and(d)as only their sum is gauge invariant. Using di?erent form factors for these amplitudes thus leads to a violation of the gauge invariance.To recover the gauge invariance for the total amplitude,one can either drop the gauge violating terms in the amplitude[41]or use an averaged form factor[42]or other combinations of form factors[43]for these terms.For simplicity,we use here a common form factor,which is given by the average of the original form factors for diagrams(b),(c), and(d),for all these three diagrams.A similar consideration is applied to the diagrams in Fig.2for the reactionγn→K+K+Ξ??5.In the following,we summarize the form factors used in the present study:

F(1a)=F(2a)=F(mΣ,m K?)

F(1b)=F(1c)=F(1d)=

4[F(mΣ,m K)+F(mΣ,mΣ)+F(mΣ,mΞ

)+F(mΣ,m N)]

F(2e)=F(mΣ,m N),(5)

where the subscripts refer to the diagrams in Figs.1and2.

The value of the cuto?parameter is taken to beΛ=1.7GeV,which is determined from ?tting the measured cross section for charmed hadron production with three-body?nal states from photon-proton reactions at center-of-mass energy of6GeV[44]using a similar hadronic model based on SU(4)?avor-invariant Lagrangians with empirical hadron masses

and coupling constants.This cuto?parameter is smaller than the value of1.2GeV used in our previous study of the reactionγp→K?0Θ+[25],where the form factor is taken to be F=Λ4/(Λ4+m4x)and the cuto?parameter was determined from?tting the cross section for charmed hadron production with two-body?nal states in photon-proton reactions.

D.Cross sections forγp→K0K0Ξ+5andγn→K+K+Ξ??5

As shown in the Appendix,the total cross section for these reactions can be expressed as

σγN→KKΞ

32sp2i

dtds1

×k s

sinθ4dθ4 π0dφ4| i M i(γN→KKΞ5)|2,(6)

where s,t,and s1were de?ned after Eq.(3).We have also introduced by p i the momenta of initial-state particles in their center-of-mass system,and k the momenta of?nal-state particles4and5in their center-of-mass system.The anglesθ4andφ4are polar angles of the three-momentum of particle4in the center-of-mass system of particles4and5as shown in Fig.5of the Appendix.

III.RESULTS

We?rst show in the upper panel of Figs.3and4the cross sections for the reactions γp→K0K0Ξ+5andγn→K+K+Ξ??5as functions of photon energy in the laboratory frame without including form factors at the strong interaction vertices.Both cross sections are seen to increase rapidly with increasing photon energy.Since the cross sections for the case

of g K?ΣΞ

5=0is between those with g K?ΣΞ

=1.27and g K?ΣΞ

=-1.27,the contribution from

diagrams involving the coupling g K?ΣΞ

is less important than those involving the coupling

g KΣΞ

.In the lower panels,the cross sections with form factors are shown,and they are much smaller than those without form factors.Depending on the value of the coupling

constant g K?ΣΞ

,the cross section for the reactionγp→K0K0Ξ+5has values of0.03-0.6nb at Eγ=4.5GeV,while that for the reactionγn→K+K+Ξ??5has values of0.1-0.6nb.We note that these values are signi?cantly smaller that those predicted forΘ+production in photonucleon reactions using the same hadronic model[23,24,25].

-1

101

102

without form factor

γp K 0K 0Ξ5

σ (n b )

3.5

4.0 4.5

5.0 5.5

6.0

-4

-3

10-2

10-1

100

with form factor

g K *

N Ξ5

g K *

N Ξ5

=1.27

g K *

N Ξ5

=-1.27

E γ (GeV)

FIG.3:(Color online)Total cross section for Ξ+5production from the reaction γp →K 0K 0Ξ+5as

a function of photon energy and for the coupling constant g K ?ΣΞ5=1.27(dotted curve),0(solid curve),and -1.27(dashed curve).Upper and lower panels are for the cases without and with form factors,respectively.IV.

SUMMARY

Using a hadronic model that includes the coupling of pentaquark Ξ5to usual Σand K or K ?,we have evaluated the cross sections for their production in the reactions γp →K 0K 0Ξ+5and γn →K +K +Ξ??5by assuming that Ξ5has spin 1/2and positive https://www.360docs.net/doc/6315712755.html,ing coupling constants related to those for pentaquark Θ+couplings to N and K or K ?,and also including form factors at the strong interaction vertices with empirical cuto?parameters,these cross sections are found in the range of 0.03-0.6nb for γp →K 0K 0Ξ+5and 0.1-0.6nb for γn →K +K +Ξ??5at photon energy E γ=4.5GeV.Since the coupling constant g KN Θis much smaller for a negative parity Θ+than for a positive parity one,the coupling constant g K ΣΞ5also becomes smaller if Ξ5has a negative parity.As a result,the cross sections for producing

negative pentaquark Ξ+5and Ξ??

are much smaller than for positive ones as in the case of Θ+production.Although the cross sections for photoproduction of pentaquark Ξ5is an order of magnitude smaller than those for producing the Θ+in these reactions,measurements of Ξ5production will be useful for understanding the properties of pentaquark baryons.

-1

101

102

without form factor

γn K +K +Ξ--

σ (n b )

3.5

4.0 4.5

5.0 5.5

6.0

-4

-3

10-2

-1

100

with form factor

g K *

N Ξ5

g K *

N Ξ5

=1.27

g K *

N Ξ5

=-1.27

E γ (GeV)

FIG.4:(Color online)Total cross section for Ξ??5production from the reaction γn →K +K +Ξ??5as a function of photon energy and for the coupling constant g K ?ΣΞ5=1.27(dotted curve),0(solid curve),and -1.27(dashed curve).Upper and lower panels are for the case without and with form factors,respectively.Acknowledgments

This paper was based on work supported in part by the US National Science Foundation under Grant No.PHY-0098805and the Welch Foundation under Grant No.A-1358.

Appendix

In this appendix,we describe the derivation of the cross section formula shown in Eq.(6)

for the reactions γp →K 0K 0Ξ+5and γn →K +K +Ξ??5

involving two particles in the initial state and three particles in the ?nal state,i.e.,1+2→3+4+5.It essentially follows the method given in Ref.[45].

In the center-of-mass frame of particles 1and 2,the cross section for such a reaction generally reads as

σ1+2→3+4+5=

4s 1/2p i

d 4p 3d 4p ?δ(p 23?m 23)δ(4)

(p ??p 1+p 3)

× d4p4d4p5δ(p24?m24)δ(p25?m25)δ(4)(p?p4?p5)

×|M(p1+p2→p4+p4+p5)|2,(7) with s=(p1+p2)2and p i denoting the momentum of initial particles in their center-of-mass

system.

FIG.5:(Color online)Coordinate system used for the three momenta of the particles in the reaction1+2→3+4+5.

To evaluate the phase space integral in Eq.(7),we choose the coordinate system in the

center-of-mass frame of particles4and5as shown in Fig.5,in which the z-axis is along the

three momentum p2of particle2,p1and p3are in the x?z plane.In terms of the angles

shown in the?gure,the four momenta of these particle can be expressed as

p1=(E1,|p3|sinθ3,0,|p3|cosθ3?|p2|)

p2=(E2,0,0,|p2|)

p3=(E3,|p3|sinθ3,0,|p3|cosθ3)

p4=(E4,|p4|sinθ4sinφ4,|p4|sinθ4cosφ4,|p4|cosθ4)

p5=(E5,?|p4|sinθ4sinφ4,?|p4|sinθ4cosφ4,?|p4|cosφ4).(8) From four-momentum conservation and on-shell constraints,the following identities can

be derived:

E1=t+s?m22?m23

,E2=

s1?t+m22

,E3=

s?s1?m23

E4=s1+m24?m25

,E5=

s1+m25?m24

,cosθ3=

m21?E21+|p3|2+|p2|2

E22?m22,|p3|= E24?m24,(9)

where t=(p1?p3)2and s1=(p4+p5)2.The two-body phase-space integral can be rewritten as

d4p4d4p5δ(p24?m24)δ(p25?m25)δ(4)(p?p4?p5)=k s1 π0sinθ4dθ4 π0dφ4,(10)

where k is the momentum of particle4or5in their center-of-mass frame.

After integrating over the four-momentum p3,we have

σ1+2→3+4+5=

4s1/2p i d4p?δ((p1?p?)2?m23)

×k s

sinθ4dθ4 π0dφ4|M(p1+p2→p3+p4+p5)|2.(11)

In the center-of-mass system of particles1and2,we can rewrite d4p?in terms of ds1 dp2?= ds1 dt by using the identities

s1=m23?m21+m22+2√

(2π)4

2√

[5]J.Barth et al.,SAPHIR Collaboration,Phys.Lett.B572,127(2003).

[6]V.V.Barmin et al.,Phys.At.Nucl.66,1715(2003).

[7]M.Praszalowicz,Phys.Lett.B575,234(2003).

[8]M.V.Polyakov and A.Rathke,Eur.Phys.J.A18,691(2003).

[9]H.Walliser and V.B.Kopeliovich,J.Exp.Theor.Phys.97,433(2003).

[10] B.K.Jennings and K.Maltman,hep-ph/0308286.

[11] D.Borisyuk,M.Faber,and A.Kobushkin,hep-ph/0307370.

[12]N.Itzhaki,I.R.Klebanov,P.Ouyang,and L.Rastelli,hep-ph/0309305.

[13]Fl.Stancu and D.O.Riska,Phys.Lett.B575,242(2003).

[14]M.Karliner and H.J.Lipkin,hep-ph/0307243.

[15]R.L.Ja?e and F.Wilczek,Phys.Rev.Lett.91,232003(2003).

[16] A.Hosaka,Phys.Lett.B55,571(2003).

[17]L.Y.Glozman,Phys.Lett.B575,18(2003).

[18]S.L.Zhu,Phys.Rev.Lett.91,232002(2003).

[19]R.D.Matheus,F.S.Navarra,M.Nielsen,R.Rodrigues da Silva,and S.H.Lee,Phys.Lett.

B578,323(2004).

[20]J.Sugiyama,T.Doi,and M.Oka,hep-ph/0309271.

[21]S.Sasaki,hep-lat/0310014.

[22] F.Ciskor,Z.Fodor,S.D.Katz,and T.G.Kov′a cs,JHEP0311,070(2003).

[23]W.Liu and C.M.Ko,Phys.Rev.C68,045203(2003);

[24]W.Liu and C.M.Ko,nucl-th/0309023.

[25]W.Liu,C.M.Ko,and V.Kubarovsky,nucl-th/0310087,Phys.Rev.C(to be published).

[26]Y.Oh,H.Kim,and S.H.Lee,hep-ph/0310019,Phys.Rev.D(to be published);

hep-ph/0311054.

[27]S.I.Nam,A.Hosaka,and H.C.Kim,Phys.Lett.B579,43(2004).

[28]Q.Zhao,hep-hp/0310350.

[29]T.Hyodo,A.Hosaka,and E.Oset,nucl-th/0307105,Phys.Lett.B(to be published).

[30]K.Nakayama and K.Tsushima,hep-ph/0311112.

[31] C.E.Garlson,C.D.Carone,H.J.Kwee,and V.Nazaaryan,Phys.Lett.B573,101(2003).

[32]X.Chen,Y.Mao,and B.Q.Ma,hep-ph/0307381.

[33] C.Alt et al.,NA49Collaboration,hep-ex/0310014.

[34] C.H.Li and C.M.Ko,Nucl.Phys.A712,110(2002).

[35] B.Holzenkamp,K.Holinde,and J.Speth,Nucl.Phys.A500,485(1989).

[36]G.Janssen,K.Holinde,and J.speth,Phys.Rev.C54,2218(1996).

[37]R.A.Adelseck and B.Saghai,Phys.Rev.C42,108(1990).

[38]Y.Oh,H.Kim,and S.H.Lee,hep-ph/0310117.

[39]Particle Data Group,K.Hagiwara et a.,Phys.Rev.D66,010001(2002).

[40]W.S.Chung,G.Q.Li,and C.M.Ko,Phys.Lett.B401,1(1997).

[41]K.Ohta,Phys.Rev.C40,1335(1989).

[42]H.Haberzettl,C.Bennhold,T.Mart,and T.Feuster,Phys.Rev.C58,R40(1998).

[43]R.M.Davidson and R.Workman,Phys.Rev.C63,058201(2001).

[44]W.Liu,S.H.Lee,and C.M.Ko,Nucl.Phys.A724,375(2003).

[45]W.Beenakker,H.Kuijf,and W.L.van Neerven,and J.Smith,Phys.Rev.D40,54(1989).