Production of the pentaquark exotic baryon Xi_5 in bar{K} N scattering : bar{K} N -> K Xi_5 and bar{K} N -> K* Xi_5

0.22 MB·English

by  Seung-Il Nam

Production of the Pentaquark Exotic Baryon Ξ in K¯N Scattering:
5
K¯N KΞ and K¯N K∗Ξ
5 5
→ →
Seung-il Nam,1,∗ Atsushi Hosaka,2,† and Hyun-Chul Kim3,4,‡
1Yukawa Institute for Theoretical Physics (YITP), Kyoto University, Kyoto 606-8502, Japan
2Research Center for Nuclear Physics (RCNP), Ibaraki, Osaka 567-0047, Japan
3Department of Physics, Inha University, Incheon 402-751, Republic of Korea.
4Department of Physics and Nuclear Physics & Radiation Technology Institute (NuRI),
Pusan National University, Busan 609-735, Korea
(Dated: February 1, 2008)
WeinvestigatetheproductionofthenewlyfoundpentaquarkexoticbaryonΞ5intheK¯N →KΞ5
and the K¯N →K∗Ξ5 reactions at the tree level. We consider both positive- and negative-parities
of the Ξ5. The reactions are dominated by the s- and the u-channel processes, and the resulting
8 crosssectionsareobservedtodependverymuchontheparityofΞ5 andonthetypeofformfactor.
0 Wehaveseen thatthecross sectionsforthepositive-parity Ξ5 aregenerally aboutahundredtimes
0 largerthanthoseofthenegative-parityone. Thislargedifferenceinthecrosssectionswillbeuseful
2 for further studyof thepentaquarkbaryons.

n
PACSnumbers: 13.60.Rj,13.75.Jz,13.85.Fb
a
J Keywords: PentaquarkexoticbaryonΞ5,Parityofthepentaquarkstates

2
2

2
I. INTRODUCTION
v
7
2 The experimental observation of the Θ+ performed by the LEPS collaborationat SPring-8 [1], which is motivated
2 by Diakonov et al. [2], has paved the way for intensive studies on the exotic five-quark baryon states, also known as
5 pentaquarks, experimentally[3]aswellastheoretically[4, 5, 6, 7, 8, 9,10, 11,12, 13,14, 15, 16,17, 18,19, 20,21, 22,
0 23, 24, 25, 26, 27, 28, 29, 30]. As a consequence of the finding of the Θ+, the existence of other pentaquark baryons,
4
such as the N , Σ , and Ξ , which have also been predicted theoretically, is anticipated.
0 5 5 5
TheNA49[31]collaborationreportedasignalforthepentaquarkbaryonΞ ,whichwasalsopredictedtheoretically.
/ 5
h The Ξ was found to have a mass of 1862MeV, a strangeness S = 2, and an isospin I = 3/2. It is characterized
5
p by its narrow decay width of 18MeV, like that of the Θ+. Howe−ver, we have thus far no concrete experimental
- ∼
p evidence for its quantum numbers such as spin and parity.
e As for the parity of the Θ+, a consensus has not been reached. For example, the chiral soliton model [2, 4], the
h diquarkmodel[5],thechiralpotentialmodel[6],andconstituentquarkmodelswithspin-flavorinteractions[10,11,12]
v: preferapositive-parityfortheΘ+ whereastheQCDsumruleapproach[7,8]1 andthequenchedlatticeQCD[13,14]
i havesupportedanegative-parity. Inthemeanwhile,variousreactionsforΘ+ production[15,16,17,18,19,20,21,22,
X
23,24,25,26,27,28,29,30]havebeeninvestigated,wherethedeterminationoftheparityofΘ+ hasbeenemphasized.
r In many cases, the total cross-sections of the positive-parity Θ+ production is typically about ten times larger than
a
those of the negative-parity one. Liu et al. [32] evaluated the γN KΞ reactions, assuming the positive-parity Ξ
5 5
→
and its spin J = 1/2. However, since the parity of the Ξ is not known yet, it is worthwhile studying the dynamics
5
of Ξ production with two different parities taken into account.
5
However, we note that negative results for the pentaquark baryons have emerged recently. Especially, the CLAS
collaborationat Jefferson laboratory could not see any obvious evidence for the Θ+ pentaquark, which was expected
to have a peak at about 1530 1540 MeV in the reaction γp K¯0Θ+ [33]. Moreover, the existence of an S = 2
penataquark, such as the Ξ or∼the charmed pentaquark (Θ+),→has not been completely confirmed yet. −
5 c
Inthepresentwork,nonetheless,fortheunclearstatusofthepentaquark,wewanttoinvestigatetheΞ production
5
from the K¯N KΞ and the K¯N K∗Ξ reactions. Due to the exotic strangeness quantum number the Ξ has,
5 5 5
→ →
the reaction process at the tree level becomes considerably simplified. This is a very specific feature of the process
∗E-mail: [email protected]
†E-mail: [email protected]
‡E-mail: [email protected]
1 Recently, Ref.[9]pointed outthatthe exclusionofthe non-interactingKN statefromthe two-pointcorrelationfunctionmayreverse
theparityofΘ+.
Typeset by REVTEX
2
K(p1) KK(cid:3)(p3) K(p1) KK(cid:3)(p3)
N(p2) (cid:6)(q) (cid:4)3=2(p4) N(p2) (cid:6)(q) (cid:4)3=2(p4)
FIG. 1: Born diagrams, s–(left) and u–channels (right) for Ξ5 productions
containingthe exoticstrangenessquantumnumber. We willfollowthe same frameworkas inRefs.[17,18, 19, 20,21,
25, 26]. We assume that the spin of the Ξ is 1/2 [32]. Then, we estimate the total and the differential cross-sections
5
for the production of Ξ with positive and negative parities.
5
This paper is organized as follows: In Section II, we define the effective Lagrangians and construct the invariant
amplitudes. In Section III, we present the numerical results for the total and the differential cross-sections for both
positive- and negative-parity Ξ . Finally, in Section IV, we briefly summarize our discussions.
5
II. EFFECTIVE LAGRANGIANS AND AMPLITUDES
We study the reactions K¯N KΞ and K¯N K∗Ξ by using an effective Lagrangian at the tree level of Born
5 5
→ →
diagrams. The reactions are schematically presented in Fig. 1, where we define the four momenta of each particle
for the reactions by p . There is no t-channel contribution because strangeness-two (S =2) mesons do not exist.
1,···4
As discussed in Ref. [34], we do not include the B1¯0M8B10 coupling because it is forbidden in exact SU(3) flavor
symmetry. Hence, the interaction Lagrangians can be written as
= ig Σ¯γ KN + (h.c.),
KNΣ KNΣ 5
L
= ig Ξ¯ Γ KΣ + (h.c.),
LKΣΞ5 KΣΞ5 5 5
K∗NΣ = gK∗NΣΣ¯γµK∗µN + (h.c.),
L
LK∗ΣΞ5 = gK∗ΣΞ5Ξ¯5γµΓˆ5K∗µΣ + (h.c.), (1)
where Σ, Ξ , N, K, and K∗ denote the corresponding fields for the octet Σ, the antidecuplet Ξ , the nucleon, the
5 5
pseudo-scalar K, and the vector K∗, respectively. The isospin operators are dropped because we treat the isospin
states of the fields explicitly. We define Γ =γ for the positive-parity Ξ whereas Γ =1 for the negative-parity
5 5 5 5 4×4
one. Γˆ5 is also defined by Γ5γ5 for the vector meson K∗. The values of the coupling constants gKNΣ and gK∗NΣ are
taken from the new Nijmegen potential [35] as gKNΣ = 3.54 and gK∗NΣ = 2.99 whereas we assume SU(3) flavor
−
symmetry for g so that we obtain the relation g = g [34]. Employing the decay width Γ = 15
KΣΞ5 KΣΞ5 KNΘ Θ→KN
MeV and M =1540 MeV, we obtain g =g =3.77(0.53)for the positive (negative) parity. The remaining
Θ KNΘ KΣΞ5
one, gK∗ΣΞ5, is not known, which we will discuss in the next section.
The invariant scattering amplitude for K¯N KΞ can be written as
5
→
i =ig g F2(1q2)u¯(p )Γ qx+MΣγ u(p ), (2)
Mx,K KΣΞ5 KNΣ x 4 5q2 M2 5 2
x− Σ
wherexlabelseitherthes-channelortheu-channel,andthecorrespondingmomentaareq =p +p andq =p p .
s 1 2 u 2 3
For K¯N K∗Ξ , we have −
5
→
/q +M
iMs,K∗ = gK∗ΣΞ5gKNΣFs2(q2)u¯(p4)ǫ/Γˆ5qs2 MΣ2γ5u(p2),
s − Σ
/q +M
iMu,K∗ = gK∗NΣgKΣΞ5Fu2(q2)u¯(p4)Γ5qu2 MΣ2/ǫu(p2). (3)
u− Σ
As indicated in Eq. (2), the coupling constants are commonly factored out for the s- and the u-channels in the K
production. Therefore,thereis no ambiguitydue tothe signofthe couplingconstants. Onthe contrary,there is such
an ambiguity due to the unknown sign of gK∗ΣΞ5 in the case of K∗ production,
Since the baryon has an extended structure, we need to introduce a form factor. We employ the form factor [20]
Λ2
F (x)= 1 (4)
1
pΛ4+(x M2)2
1 − Σ
3
(a) (b)
0.4 0.4
Threshold F2(s) F2(s) : 2.4GeV
0.3 2.36 GeV FF11222((uq2)) s 0.3 FFF111222(((ssu))) ::: 222...684GGGeeeVVV
ss es F12(u) : 2.6GeV
mensionle 0.2 ensionl 0.2 F112(u) : 2.8GeV
Di Dim
0.1 0.1
0.0 0.0
2.0 2.5 3.0 3.5 4.0 0 30 60 90 120 150 180
E [GeV] θ [deg]
cm
FIG. 2: Energy dependence of the squared form factors F2(s), F2(u) and the F2(~q2) (a), and angular dependence of the
1 1 2
squared form factor F2 for thes-and the u-channelsat three CM energies. The typesof curvesare explained bythelabels in
1
thefigure.
in such a way that the singularities appearing in the pole diagrams can be avoided. Here, Λ and M stand for the
1 Σ
cutoff parameterandthe Σ mass,respectively. We setthe cutoff parameterΛ =0.85GeV asin Ref. [20]. This value
1
was used to reproduce the cross sections of γp K+Λ. In order to verify the dependence of the form factor, we
→
consider also the three-dimensional form factor
Λ2
F (~q2)= 2 , (5)
2 Λ2+ ~q2
2 | |
where~q denotesthethreemomentumofthe externalmeson. As forthe cutoffparameter,wesetΛ =0.5GeV,which
2
was deduced from the πN KΛ reaction [15].
→
In the left panel (a) of Fig. 2, we show the dependence of the two form factors F and F on the CM energy
1 2
while in the left panel (b) the angular dependence of the F form factor is drawn. The F form factor does not have
1 2
any angular dependence. Obviously, they show very different behaviors. For instance, the F decreases much faster
2
than F as the center-of-mass (CM) energy grows. The form factor F in the u-channel shows a strong enhancement
1 1
in a backward direction as the CM energy increases. As we will see, this feature has a great effect on the angular
dependence of the differential cross-sections.
III. NUMERICAL RESULTS
A. K¯N →KΞ5
Inthis subsection,wediscussthe resultsforthe reactionK¯N KΞ . Due toisospinsymmetry,wecanverifythat
5
the twopossible reactionsK¯0p K0Ξ+ andK−n K+Ξ−− a→reexactly the same inthe isospinlimit. InFig.3, we
→ 5 → 5
present the total and the differential cross-sections in the left and the right panels, respectively. The average values
of the total cross-sections are σ 2.6µb with the F form factor and σ 1.5µb with the F in the energy range
1 2
Eth = 2.35GeV E 3.35∼GeV (from the threshold to the point of∼1 GeV larger). Though the average total
CM ≤ CM ≤
cross-sectionsforthedifferentformfactorsaresimilarinorderofmagnitude,the energyandthe angulardependences
are very different from each other. They are largely dictated by the form factor, as shown in Fig. 2. The angular
distributions are drawn in the right panel (b) of Fig. 3, where θ represents the scattering angle between the incident
and the final kaons in the CM system. We show the results at E =2.4, 2.6, and 2.8 GeV. As shown there, when
CM
F is used, the backward production is strongly enhanced while the cross sections are almost flat apart from a tiny
1
increase in the backwardregion,when F is employed. Note that the angular dependence of the latter is the same as
2
that of the bare cross section without the form factor.
In Fig. 4, we plot the total and differential cross-sections for the negative-parity Ξ . The energy dependence of
5
the total cross section looks similar to that for the positive-parity one. We find that σ 26nb for F and 12nb
1
for F in average for the CM energy region Eth E 3.35GeV. We see that∼the total cross-secti∼ons are
2 CM ≤ CM ≤
4
(a) (b)
10 15
2.4 GeV : F
1
F 2.6 GeV : F
8 1 12 1
F 2.8 GeV : F
2 1
b] 2.4 GeV : F2
µb] 6 θµ [ 9 22..68 GGeeVV :: FF22
σ [T 4 dcos 6
σ/
d
2 3
0 0
2.0 2.5 3.0 3.5 4.0 0 30 60 90 120 150 180
E [GeV] θ [deg]
cm
FIG. 3: Cross sections for production of the positive parity Ξ5 in the reaction K¯0p → K0Ξ+5. (a) The left panel shows the
total cross-sections as functions of the center-of-mass energy ECM. (b) The right panel shows the differential cross-sections as
functions of the scattering angle θ for incident energies ECM = 2.4, 2.6, and 2.8 GeV. In both cases, results using the form
factors F1 and F2 are shown as indicated by thelabels in thefigures.
(a) (b)
60 60
F 2.4 GeV : F
1 1
50 F 50 2.6 GeV : F
2 1
2.8 GeV : F
1
40 b]40 2.4 GeV : F2
n 2.6 GeV : F
] [ 2
σ [nbT30 θdcos 30 2.8 GeV : F2
20 σ/ 20
d
10 10
0 0
2.0 2.5 3.0 3.5 4.0 0 30 60 90 120 150 180
E [GeV] θ [deg]
cm
FIG. 4: Cross sections for production of the negative parity Ξ5 in the reaction K¯0p→ K0Ξ+5. Notations are the same as in
Fig. 3.
almost a hundred times smaller than that for the positive-parity Ξ . The difference between the results of the
5
two parities is even more pronounced than in the previously investigated reactions, such as γN, KN, and NN
scattering [15, 16, 17, 18, 19, 20, 21, 24, 25, 26, 27, 28, 29, 30], where typically the difference was about an order of
ten. Inthepresentreaction,theinterferencebetweenthes-andtheu-channelsbecomesimportant,inadditiontothe
kinematicaleffectinthe p-wavecouplingfor the positive-parity(butnotin the s-waveforthe negative-parity),which
is proportional to ~σ ~q and enhance the amplitude at high momentum transfers. In the case of the positive parity,
·
the two terms which are kinematically enhanced are interfered constructively, while for the negative-parity Ξ , the
5
relativelysmallamplitudeswithouttheenhancementfactorisdonedestructively. Thesetwoeffectsaresimultaneously
responsible for the large difference in the cross sections.
In the right panel (b) of Fig. 4, the angular distributions for the production of the negative-parity Ξ are plotted.
5
Here, the angular dependence changes significantly as compared with the positive-parity case. When the form factor
F isused,forwardscatteringsignificantlyincreasesbecausethebareamplitudeshowsanenhancementintheforward
2
direction. When using F , however, due to its strong enhancement in the backward direction, the cross sections get
1
quite larger in the backward direction, except for those in the vicinity of the threshold, i.e., E 2.45 GeV.
CM
≤
5
(a) (b)
3 12
+,1.89
1.89,2.8GeV
−,1.89
1.89,3.0GeV
+,6.53
1.89,3.2GeV
−,6.53
2 b] 8 6.53,2.8GeV
µb] θ [µ 66..5533,,33..02GGeeVV
σ [T dcos
1 σ/ 4
d
0 0
2.5 3.0 3.5 4.0 4.5 0 30 60 90 120 150 180
E [GeV] θ [deg]
cm
FIG.5: CrosssectionsforproductionofthepositiveparityΞ5 inthereactionK¯0p→K∗0Ξ+5 withtheF1 formfactoremployed.
Thetotalcross-sectionsin(a)arecalculatedforfourdifferentgK∗ΣΞ5 couplingconstantsasindicatedbythelabels. Theangular
distributions in (b)are calculated for threedifferentCM energies and twodifferent gK∗ΣΞ5 coupling constants as indicated by
thelabels.
B. K¯N →K∗Ξ5
In this subsection, we discuss the K∗ production. As explained in the previous section, the appearance of the
coupling constant gK∗ΣΞ5 raises the problem of the relative sign in the amplitude. First, we briefly discuss possible
relations to determine the magnitude of the gK∗ΣΞ5 coupling. If we use the SU(3) relation this coupling may be set
equal to gK∗NΘ. There are several discussions on the gK∗NΘ coupling. For example, a small value of the gK∗NΘ
was chosen according to the relation gK∗NΘ/gKNΘ = 1/2 as inferred from a phenomenological study of the hyperon
coupling constants [36], while in the quark model, the decay of the pentaquark states predicts a positive parity Θ+
by using the relation gK∗NΘ/gKNΘ = √3 [37]. In the meanwhile, we find gK∗NΘ/gKNΘ = 1/√3 for the negative
parity. Since we are not able to determine the sign of the coupling constant in this study, we will present the results
for four different cases: gK∗ΣΞ5 = ±√3gKNΘ = ±6.53 and gK∗ΣΞ5 = ±1/2gKNΘ = ±1.89 for positive parity, and
gK∗ΣΞ5 =±√3gKNΘ =±0.91 and gK∗ΣΞ5 =±1/2gKNΘ=±0.27 for negative parity.
Figures 5 and 6 show the total and the differential cross-sectionsfor the positive parity Ξ with the F and the F
5 1 2
formfactors,respectively. Wepresenttheresultswiththefourdifferentcouplingconstantsforthetotalcross-sections
while for the differential cross-sections, we present those with the two positive coupling constants at three different
energies, E = 2.8, 3.0, and 3.2 GeV. The results for the negative coupling constants are qualitatively similar to
CM
eachother. WhentheF1 formfactorisused,theresultsdonotdependonthechoiceofgK∗ΣΞ5 becausetheu-channel
is the dominantcomponent. Inthis case,similardiscussionscanbe made asinthe previouscaseofthe K production
reaction. However, when F2 is used, the results are very sensitive to the sign of gK∗ΣΞ5, which determines whether
the s- and the u-channels interfere constructively or not.
Fig.7andFig.8showthe results forthe negative-parityΞ with the F andthe F formfactorsused,respectively.
5 1 2
Similar discussions apply for this case as for the positive parity case, but the values of cross sections are reduced by
about a factor of a hundred.
IV. SUMMARY AND DISCUSSION
We have studied the production of the pentaquark exotic baryon Ξ (mass = 1862MeV, I = 3/2S = 2, spin =
5
1/2(assumed)) in the reactions K¯N KΞ and K¯N K∗Ξ . We have employed two different phenom−enological
5 5
→ →
form factor Eqs. (4) and (5), with appropriate parameters for the coupling strengths and the cutoff parameters. In
the present reactions, since two units of strangeness are transferred, only s- and u- channel diagrams are allowed at
the tree level. On one hand, this fact simplifies the reaction mechanism and, hence, the computation. Furthermore,
there is no ambiguity in the relative signs of coupling constants for the case of K production. On the other hand,
the cross sections strongly depend on the choice of form factors. In fact, Fig. 3 8 show that we have found a rather
∼
differentenergyandangulardependencewhenusingdifferentformfactors. Atthismoment,itisdifficulttheoretically
to say which is better. Nevertheless, it would be useful to summarize the present result for the total cross-sectionsin
6
(a) (b)
8 4
+,1.89
−,1.89 1.89,2.8GeV
6 +,6.53 3 1.89,3.0GeV
−,6.53 b] 1.89,3.2GeV
µ
σµ [b]T4 θ [dcos2 666...555333,,,233...802GGGeeeVVV
σ/
2 d1
0 0
2.5 3.0 3.5 4.0 4.5 0 30 60 90 120 150 180
E [GeV] θ [deg]
cm
FIG.6: CrosssectionsforproductionofthepositiveparityΞ5 inthereactionK¯0p→K∗0Ξ+5 withtheF2 formfactoremployed.
For notations, see thecaption of Fig. 5.
(a) (b)
30 50
+,0.27 0.27,2.8GeV
−,0.27 40 0.27,3.0GeV
+,0.91 0.27,3.2GeV
20 −,0.91 b] 0.91,2.8GeV
] [n30 0.91,3.0GeV
σ [nbT θ dcos20 0.91,3.2GeV
10 σ/
d
10
0 0
2.5 3.0 3.5 4.0 4.5 0 30 60 90 120 150 180
E [GeV] θ [deg]
cm
FIG. 7: Cross sections for production of the negative parity Ξ5 in the reaction K¯0p → K∗0Ξ+5 with the F1 form factor
employed. Thetotalcross-sections in(a)arecalculated forfourdifferentgK∗ΣΞ5 couplingconstantsasindicatedbythelabels.
The angular distributions in (b) are calculated for threedifferent CM energies and two different gK∗ΣΞ5 coupling constantsas
indicated bythe labels.
Table I. There, we see once again that the total cross-sections are, generally, much larger for positive-parity Ξ than
5
for positive-parity one by about factor of a hundred because there is a cancellation due to destructive interference.
Reaction F1 F2 Reaction F1 F2
σK¯N→KΞ5(P =+1) 2.6 µb 1.5 µb σK¯N→K∗Ξ5(P =+1) 1.6 µb <∼ 2 µb
σK¯N→KΞ5(P =−1) 26 nb 12 nb σK¯N→K∗Ξ5(P =−1) 14 nb <∼ 20 nb
TABLEI:Summaryfortheaveragetotalcross-sectionsintheCMenergyregion2.35GeV ≤ ECM ≤ 3.35GeVforK¯N →KΞ5
and 2.75GeV ≤ ECM ≤ 3.75GeV for K¯N →K∗Ξ5. For K∗ production with the F2 form factor used, only theupper values
are quoted because the interference suppresses them.
7
(a) (b)
80 40
+,0.27 0.27,2.8GeV
−,0.27 0.27,3.0GeV
60 +,0.91 30 0.27,3.2GeV
−0.91 b] 0.91,2.8GeV
n
] [ 0.91,3.0GeV
σ [nbT40 θ dcos20 0.91,3.2GeV
σ/
d
20 10
0 0
2.5 3.0 3.5 4.0 4.5 0 30 60 90 120 150 180
E [GeV] θ [deg]
cm
FIG. 8: Cross sections for production of the negative parity Ξ5 in the reaction K¯0p → K∗0Ξ+5 with the F2 form factor
employed. For notations, see thecaption of Fig. 7.
Acknowledgements
TheauthorsaresupportedbyaKoreaResearchFoundationgrant(KRF-2003-041-C20067). TheworkofS.i.N.was
supported by a scholarship endowed by the Ministry of Education, Science, Sports and Culture of Japan. He is also
supported by a grant for Scientific Research (Priority Area No. 17070002)from the Ministry of Education, Culture,
Science and Technology, Japan.
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