Spin and parity of a possible baryon antidecuplet

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by  C. Dullemond

Spin and parity of a possible baryon antidecuplet
C. Dullemond
Institute for Theoretical Physics
University of Nijmegen
Nijmegen, The Netherlands
Abstract
Therecentlypostulatedexistenceofabaryonantidecuplet[1]canbereproduced
instrongcouplingtheoryinwhichabarebaryonspin1/2octetinteractswithan
4
0 octetofpseudoscalarmesons. WhenasuitablemixtureofF-andD-typeYukawa
0 couplingsischosenthedressedbaryonsgroupthemselvesintoaninfinitenumber
2 ofSU(3)multiplets ofwhichthe groundstateturnsouttobe aspin1/2baryon

n octet and the first excited state is a spin 3/2 baryon decuplet. Then follows a
a spin1/2baryonantidecuplet. Allstatesofthespectrumhavepositiveparity. If
J
the hypothetical baryonantidecuplet can be identified with the antidecuplet in

6 the strong coupling spectrum then a positive parity and a spin 1/2 is predicted

for this multiplet.
1
v
3 1 Introduction
2
0
1 Recently new baryon resonances have been found which are “exotic” in the
0 sense that they cannot be consideredas three-quarkstates [1]. As “pentaquark
4 baryons” they may fit into a baryon antidecuplet. It is the purpose of this
0
Lettertoremindofanearlierattempttogeneratebaryonspectra,namelystrong
/
h couplingtheory. Nonrelativisticstrongcouplingtheoryhasplayedanimportant
p role in the early days of field theory and nuclear physics [2]. With the advent
-
p of flavor SU(3) particle multiplets, it has been tried to explain the existence of
e the spin 1/2 baryon octet and the spin 3/2 baryon decuplet from field theory
h models with strong coupling betweena “bare”baryonoctet anda mesonoctet.
:
v Although the physical ideas underlying the model are presently unacceptable,
i the obtained baryon spectrum was in remarkable agreement with experimental
X
data at that time and may still be of some relevance. Results were first given
r
a for a spinless baryon octet in interaction with a scalar meson octet [3], later
followed the results for a spin 1/2 baryonoctet interacting with a pseudoscalar
meson octet [4,5]. The interactions in the model are of Yukawa type and are a
mixtureofF-andD-typecouplingsuchthatbreakingofmultiplet-antimultiplet
symmetry in the final spectrum is guaranteed. While in the spinless case the
spectrumis acontinuousfunctionofthe F/Dratio,inthe morerealisticcaseof
spin 1/2 baryons in interaction with pseudoscalar mesons the spectrum turns
out to be much more rigid.
2 The method
Starting point is the following hamiltonian:
1
H = 2 pαβjipβαji +µ2qβαjiqαβij +g1B¯δγiqεδjiBγεj +g2B¯εδiqδγjiBγεj (1)
(cid:16) (cid:17)
1
Relative level height Multiplet structure (SU(3), spin)
72 {35*} 1/2
57 {35} 3/2 {35*} 3/2
40 {27} 1/2
32 {35} 5/2
25 {27} 3/2
24 {10*} 1/2
9 {10} 3/2
0 {8} 1/2
Figure 1: Mass spectrum of baryon SU(3) multiplets (arbitrary units)
Heregreekindices runfrom1to 3andlatinindices from1 to2. Forbothtypes
of indices the Einstein summation convention is adopted. The 24 q-variables
represent the pseudoscalar field in a P-state with respect to the bare baryons.
The 24 p-variables are the associated momenta. The B¯- and B-variables (16
of both) are the baryon creation and annihilation operators. The variables are
traceless:
qαi =pαi =qαi =pαi =Bαi =B¯α =0 (2)
βi βi αj αj α αi
If the index K (= 1,...,24) distinguishes between the different field and mo-
mentum variables the following relations are valid:
[pK,qK′]=−iδKK′ (3)
Finally, if |i denotes the baryon vacuum we have:
1
BγjB¯µ |i= δγδµ− δγδµ δj |i (4)
δ νk (cid:18) ν δ 3 δ ν(cid:19) k
the δi and the δα being the Kronecker delta symbols in 2 and 3 dimensions
j β
respectively.
Eigenstates are sought of the form
ξ(qαi) vµkB¯ν |i (5)
βj ν µk
(cid:0) (cid:1)
2
where ξ has the index structure characteristic of the multiplet under consider-
ation.
In first instance, when g1 and g2 are large, the kinetic energy term can be
neglected. The first problem is then to find the value of qβαji, with q2 and g1/g2
constant, for which the operator
1
H′ = 2µ2qβαjiqαβij +g1B¯δγiqεδjiBγεj +g2B¯εδiqδγjiBγεj (6)
hasthelowestpossibleeigenvalue. Ifonewritesqαi intheformofa3×8matrix,
βj
writtenasthecombinationofa3×3anda3×5matrix,thentheminimumoccurs
when the first matrix is proportional to the identity and the second matrix is
zero. Thisisthestandardform. Anyother3×8matrixforwhichthisminimum
occurs can be obtained from it by applying symmetry transformations of both
kinds. The simplicity of the standard form and the existence of symmetry
transformations leaving this form invariant guarantee that the final spectrum
of energy eigenstates is simple. To find this spectrum the kinetic term in the
hamiltonian, previously omitted, must be taken into account. Like in ordinary
quantummechanics,itmustbesplitintoa“radial”partandan“angular”part.
However,there are now13 “radius-like”variablesand11 “angle-like”variables.
Still, the splitting can be carried out. In the strong coupling limit the radial
waveequationsbecomeirrelevant. Thedifferentialequationsassociatedwiththe
angularpartofthe hamiltonianleadto the desiredspectrum. This is presented
in the figure.
The abovemethod has been describedin detail in reference [4]. An alterna-
tive method leading to the same results has been presented by Goebel [5].
References
[1]S.Stepanyanetal.,CLAScollaboration,2003hep-ex/0307018;C.Altetal.,
2003 hep-ex/0310014
[2] W. Pauli and S.M. Dancoff, Phys. Rev. 62, 85 (1942); G. Wentzel, Rev.
Mod. Phys. 19, 1 (1947), Phys. Rev. 125, 771 (1962) and Phys. Rev. 129,
1367 (1963)
[3] C. Dullemond, Ann. Phys. (N.Y.) 33, 214 (1965);G. Wentzel, Suppl. Prog.
Theoret. Phys. Commemoration Issue, 108 (1965)
[4]F.J.M.vonderLindenandC.Dullemond,Ann. Phys. (N.Y.)41,372(1967)
[5] C. Goebel, Phys. Rev. Lett. 16, 1130 (1966)
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