A narrow "peanut" pentaquark

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by  Dmitri Melikhov

A narrow "peanut" pentaquark
Dmitri Melikhov
NuclearPhysicsInstitute,MoscowStateUniversity,119992,Moscow,Russia
5
Abstract. WeanalysethedecayQ (1/2+) NKinanon-relativisticFockspacedescriptionusing
0 s
→
threeandfiveconstituentquarksforthenucleonandthepentaquark,respectively.FollowingJaffe
0
2 andWilczek[1],weassumethatquark-quarkcorrelationsinspin-zerostateplayanimportantrole
for the pentaquarkinternal structure. Within this scenario, a strong dynamicalsuppression of the
n
decay width is shown to be possible only if the pentaquarkhas an asymmetric "peanut"structure
a
withthestrangeantiquarkinthecenterandthetwoextendedcompositediquarksrotatingaround.
J
Inthiscaseadecaywidthof 1MeVmaybeanaturalpossibility.
9 ≃
1

1 Theexistenceofpentaquarksisnotyetundoubtedlyestablished.Butiftheseparticles
v
exist, the exotic members of the pentaquark multiplet must have a very small decay
6
7 width of order 1 MeV or even lower. For the possible origin of the small pentaquark
1 width many qualitative suggestions have been put forward. In a scenarios proposed by
1
Jaffe and Wilczek [1] the positive-parity spin-1/2 pentaquark consists of an antiquark
0
5 and two scalar diquarks in a relative P-wave state. In this talk I present the results
0 of a fully dynamical quark-model calculation of the pentaquark width done together
/
h withB.Stech andS.Simula[2]usinganon-relativisticFockspacerepresentationforthe
p JP = 1+ pentaquark intheJaffe-Wilczek scenario.
- 2
p Thedecay amplitudeT(Q KN) isrelated to thematrixelement
e →
h
: N(p′) s¯g m g5d Q (p) = gA(q2)u¯N(p′)g m g5uQ (p)+gP(q2)qm u¯N(p′)g5uQ (p)
v h | | i
Xi +gT(q2)u¯N(p′)s mn qn g5uQ (p), q= p−p′.
r Here the form factors g contain poles at q2 >0 due to strange meson resonances with
a i
theappropriatequantumnumbers.Theresidueofthepoleing atq2 =M2 isrelatedto
P K
theamplitudeofinterest T(Q NK): forq2 M2
K
→ →
(MK2 q2)gP(q2)u¯N(p′)g5uQ (p) fKT(Q NK),
− → →
where f =160 MeV is the kaon decay constant. The form factor g contains the pole
K A
at q2 = (K )2, but at q2 = M2 it is a regular function. Making use of the relationship
A∗ K
betweentheformfactorsg andg emerginginthelimitofspontaneouslybrokenchiral
A P
symmetry[2]gives
T(Q →NK)= MQ +f MNgA(MK2)·u¯N(p′)ig5uQ (p)
K
and
1 ~q 3
G (Q )=G (Q K+n)+G (Q K0p) | | g2(M2).
→ → ≃ p f2 A K
K
ForMQ =1540MeVonefinds ~q =270MeVandG (Q )=240g2 MeV.Fortransitions
| | A
betweenhadronsofthesamequarkstructureg 1(e.g.forthenucleong 1.23).So
A A
for a normal resonance one would expect G (Q )≃ 200 MeV. To obtain a wid≃th of 10
≃ ≤
MeVoneneeds astronglysuppressedvalueg 0.2.
A
In [2] we calculated the amplitude N s¯g m g5d≤Q and the form factor gA(q2) using a
h | | i
non-relativisticequal-timeFock spacerepresentation.
The nucleon in this framework is described by its coordinate wave function depending
ontherelativecoordinates~r =~r ~r and~l = 1(~r +~r ) ~r ,forwhichwetakethe
N 2− 3 N 2 2 3 − 1
Gaussianfunction
1 2
Y (r r ,r ) exp ~r 2 ~l 2 .
N 1| 2 3 ∼ −2a 2 N−3a 2 N
r N l N !
The pentaquark coordinate wave function depends on the relative coordinates~r =
23
~r2 −~r3, ~R23 = 21(~r2 +~r3), ~r45 =~r4 −~r5, ~R45 = 21(~r4 +~r5), ~r Q = ~R23 −~R45, ~l Q =
1(~R +~R ) ~r , where~r is the position of the strange particle, ~R and ~R are the
2 23 45 − 1 1 23 45
positionsofthetwodiquarks.Asrequiredbythequark-diquarkscenario,thepentaquark
coordinatewavefunction factorizes intothediquark wavefunctionsand thewavefunc-
tionofthethree-particlequark-diquar-diquarksystem,forwhichwetakeagainGaussian
parameterizations
1 2 ~r2 ~r2
Y Q (r1|r2,r3|r4,r5)∼exp −2a 2 ~r Q2 −3a 2 ~l Q2 exp −2a232 exp −2a452 .
r Q l Q ! (cid:18) D(cid:19) (cid:18) D(cid:19)
The formfactorg can beexpressedthroughthefollowingvectoroverlapamplitude
A
24 ~r +~r +~r
d~r2d~r4d~r5exp i~q 2 4 5 ~r Q Y Q (rs r2,rd r4,r5)
√3 3 | |
Z (cid:18) (cid:19)
2Y (r r ,r )+Y (r r ,r ) ,
N 2 4 5 N 4 2 5
×{ | | }
1
~r Q = (~r2+~rd ~r4 ~r5).
2 − −
Detailsofthiscalculationcan befoundinourpaper[2].
Numericalestimates.Wepresentnownumericalresultsforthepentaquarkwidth.Two
assumptionsreducethenumberofparameters:
1. The structure of the diquark in the nucleon and in the pentaquark coincide, i.e. the
size-parametera D ofthediquarkwavefunctionF D isequaltotheparametera r N ofthe
nucleonwavefunction,a D =a r N.
2. The parameters of the nucleon wave function are chosen such that the experimen-
tal nucleon electromagnetic form factor is reproduced for small momentum transfers,
a l2N/16+a r2N/48=1/Mr2. Wefirst take asymmetricwavefunctiona l N =a r N =0.9
fm. The diquark size parameter is then a D =a r N =0.9 fm. Now only the two free pa-
rameters of the pentaquark wave function a r Q and a l Q remain to be fixed. Recall that
a r Q determines the average distance between the two extended diquarks, and a l Q de-
terminestheaveragedistancebetweenthes-antiquarkandthecenter-of-massofthetwo
diquarks.Littleis knownabout thedetailsof thepentaquark structure. Therefore weal-
lowtheparametersa l Q anda r Q tovaryinabroadrange0.6 fm<a l Q , a r Q <1.6 fm
andstudythedependenceofg and thewidthontheseparameters.
A
Fig. 1(a) shows G (Q ) vs the pentaquark size parameters a l Q and a r Q . If both pa-
rameters are 1 fm, then g 0.8 and the width is 150 MeV. No suppression due to a
A
≃ ≃
possible mismatch of color and flavour quantum numbers in the initial and final states
takesplace.However,astrongdynamicalsuppressionoccurs ifthestructureofthepen-
taquarkisasymmetric:Forinstance,fora l Q =0.6 fm,a r Q =1.4 fm,wegetgA=0.05
andG (Q )=1 MeV.
Fig.1(b)presentsG (Q )vsthediquarksizea forfixedvaluesofthepentaquarksize-
D
parameters a r Q = a l Q = 1 fm. A sizeable reduction of the pentaquark width occurs
only for a very small diquark size which corresponds to implausibly large deviations
from a symmetric nucleon wave function. Such compact diquarks are not supported by
asuccessfuldescriptionofthenucleon propertieswithasymmetricwavefunction.
V30 V
Me a Ql =0.8 fm Me200
), 25 ),
Q( Q(175
G G
20 a Ql =0.7 fm 150
125
15
100
a Ql =0.6 fm
10 75
50
5
25
0 0
0.75 1 1.25 1.5 1.75 0.2 0.4 0.6 0.8 1
a , fm a , fm
Qr D
FIGURE 1. Left (a): G (Q ) vs the pentaquark size parameters a r Q and a l Q . Right (b): G (Q ) vs the
diquarksizeparametera D forasymmetricpentaquarka Q r =a l Q =1fm.
Summing up, the pentaquark decay width G (Q ) is found to depend strongly on the
pentaquarkconfiguration:whenallsize-parametersofthepentaquarkwavefunctionare
close to 1 fm, one obtains a width of about 150 MeV, i.e. a typical hadronic value. The
color-flavourstructureof thepentaquarkcausesno suppressionof thewidth.1
A strong dynamical suppression of the amplitude occurs for a "peanut"-shaped pen-
taquark, i.e. when it has an asymmetric structure with a l Q a r Q . For instance,
a l Q =0.6fm anda r Q =1.4fmbringsthewidthdownto1 MeV≪.
Wethereforeconcludethatifthepentaquarkcanbedescribedasafive-quarksystem,
in which two composite spin-zero diquarks are in the relative P-wave state, the small
width requires a rather asymmetric "peanut" structure with two extended diquarks
rotatingaboutthestrangeantiquarklocalizednearthecenter.
1 Foradiscussionofthepentaquarkwidthinthechirallimitwereferto[3].
ACKNOWLEDGMENTS
I would like to thank my friends Berthold Stech and Silvano Simula for the most
enjoyable collaboration, and Nora Brambilla for the invitation to participate in this
excitingConference.
REFERENCES
1. R.Jaffe,F.Wilczek,Phys.Rev.Lett.91,232003(2003).
2. D.Melikhov,S.Simula,B.Stech,Phys.Lett.B594,265(2004).
3. D.Melikhov,B.Stech,hep-ph/0409015,inprintinPhys.Lett.B;hep-ph/0501108.