Pentaquark04 Spring8, Aioi, 2004

1
Exotic pentaquarks, crypto-heptaquarks and linear
three-hadronic molecules.
Pedro Bicudo
Dep Física IST CFIF , Lisboa

• Pentaquark04 _at_ Spring-8, Aioi, 2004

2
Exotic pentaquarks, crypto-heptaquarks and linear
three-hadronic molecules.
Pedro Bicudo
Dep Física IST CFIF , Lisboa
1. The excitation of pentaquarks Q , X-- and
D- . 2. A Quark Model criterion for short range
repulsion/attraction 3. The Q , X-- and D- as
heptaquarks and linear molecules. 4. Prediction
of strange, charmed and bottomed exotic
pentaquarks.

• Pentaquark04 _at_ Spring-8, Aioi, 2004

3
1. The excitation of pentaquarks Q , X-- and
D- p.
4
1. The excitation of pentaquarks Q , X-- and
D- p.
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1. The excitation of pentaquarks Q , X-- and
D- p.
T. Nakano al, hep-ex/0301020,
Phys.Rev.Lett.91012002 (2003)
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Recent breakthrough in the program (P.B, Ribeiro
) to include c Symmetry Breaking in the Quark
Model The Axial Ward Identity shows that the
quark-antiquark annihilation interaction is
identical to the V- of p Salpeter equation
lt f A f gt mp - (2/3)(2mN-mD)
Technical
K
K
Relevant for crypto-multiquarks, Possible ex
the Ds(2317)
A
D
D
P. B , S. Cotanch, F. Llanes-Estrada , P. Maris,
G. Marques, E. Ribeiro, A. Szczepaniak
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In the Quark Model it is well known that
pentaquarks cannot be simple s-wave groundstates.
An s-wave uudds would have a mass of 1535 MeV,
seen in the Quark Model(see Lipkin, Hosaka,
Enyo, Hiyama, Carlson) and in the Lattice and sum
rules(see Chiu, Ishii, Takahashi, Okiharu,
Sasaki, Lee, Oka)
It would have parity - however it would be very
unstable!
! The pentaquark sould be an excited state !
K-N exotic s-wave phase shifts are repulsive!
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Ex Chiral soliton
The exotic anti-decuplet, including the Q (with
the correct mass and with), and the X-- , was
predicted in 1997 by Diakonov, Petrov and
Polyakov in the Chiral Soliton Model.
This is based on the skyrmion (1962), where the
pion is an effective field and the baryons are
topological solitons. Skyrme showed that a
divergent pion field is equivalent to a spin 1/2
nucleon. The chiral soliton was upgraded to
flavour SU(3) by Guadagnini by Mazur, Nowak and
Praszalowicz and by Diakonov and Petrov
(1984).
p
p
N
p
p
p
p
p
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The first rotational excitation of the SU(3)
topological soliton, composed of the p, K and h
fields produces flavour multiplets. The
vertices of the anti- decuplet are exotic
pentaquarks!
Diakonov, Petrov, Polyakov, Z. Phys. A 359, 305
(1997) arXivhep-ph/9703373
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Ex Diquark string
Jaffe and Wilczeck Phys. Rev. Lett. 91, 232003
(2003) arXiv hep-ph/0307341 suggest an
anti-baryon structure for the Q where two (ud)
diquarks play the role of anti-quarks, in the
sense that the diquarks are tightly bound and
have color 3.
Indeed the quark model supports an attractive
interaction for a diquark with color 3. The
antisymmetry of the wave-function implies that
one of the space coordinates is excited to a
p-wave, resulting in the same parity of the
chiral soliton model. The masses of the other
pentaquarks are also predicted, Q M 1540 X--
M 1750 , G X-- is 50 wider than G Q D-
p M2710
ud
s
ud
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The model of Karliner and Lipkin, Phys. Lett. B
575, 249 (2003) arXivhep-ph/0307343 0402260
ressembles a double-bag structure for the Q . In
this case a color triplet (ud) plays the role of
the anti-quark, and a color anti-triplet (uds)
plays the role of the quark.
The p-wave prevents the overlap of the two
coloured sub-bags. This again produces a total
parity .
p-wave
ud s
ud
string
L(l1)/2 mr2
The confining linear potential, together with a
repulsive centrifugal barrier indeed suggests
that a resonance above threshold may exist.
V(r)
r
12
However,
a criticism may still be applied to the p-wave
advocated by the topological Chiral soliton (see
Weigel, Harada, Diakonov, Praszalowicz ) diquark
and string, (see Takeuchi, Lipkin, Suganuma,
Sugamoto ) flavour-spin potential, (see Stancu
) The p-wave excitation is usually large in
hadrons, M f1 - M r 500 MeV M N 1/2- - M
N 1/2 500 MeV M J/y 1 - M J/y 1-- 400
MeV so it is not straightforward to get a light
mass for the Q , M Q - (M K M N ) 100
MeV .
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The 5th Quark Model excitation
1st. excitation, radial M r 1-- - M r 1--
700 MeV 2nd excitation, angular M f1 1 -
M r 1-- 500 MeV 3rd excitation, spin M
K 1-- - M K 0- 400 MeV 4th excitation,
flavour M K 1-- - M r 1-- 150 MeV 5th
excitation, quark- antiquark creation M p 140
MeV The quark-antiquark pair creation is
effectively used in chiral lagrangians and chiral
solitons, and it is equivalent to the chiral
doubling of Nowak, Rho and Zahed, Phys. Rev. D
48, 4370 (1993) arXivhep-ph/9209272. Possibly
rotational excitations of the chiral solitons
partly correspond, in the Quark Model, to pion
creation.
Close to the Q -K N mass?
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2. A Quark Model criterion for short range
repulsion/attraction
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2. A Quark Model criterion for short range
repulsion/attraction
We use a standard Quark Model Hamiltonian. The
Resonating Group Method is a convenient method to
compute the energy of multiquarks and to study
hadronic coupled channels. The RGM was first
used by Ribeiro (1978), Oka, Toki to explain the
N_N hard-core repulsion. Deus and Ribeiro (1980)
also found that the RGM may lead to hard-core
attraction .
Technical
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2. A Quark Model criterion for short range
repulsion/attraction
We use a standard Quark Model Hamiltonian. The
Resonating Group Method is a convenient method to
compute the energy of multiquarks and to study
hadronic coupled channels. The RGM was first
used by Ribeiro (1978), Oka, Toki to explain the
N_N hard-core repulsion. Deus and Ribeiro (1980)
also found that the RGM may lead to hard-core
attraction .
meson a
q1
r12
q2
rab
q4
q3
r34
Technical
meson b
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We compute the matrix element of the
Hamiltonian... in an antizymmetrized. basis
of hadrons....
lt fa fb cab ( E - Si Ti - Siltj Vij - Si j Ai j
) (1- P13 )(1 Pab)
fa fb cab gt
Technical
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We compute the matrix element of the
Hamiltonian... in an antizymmetrized. basis
of hadrons....
Annihilation interaction
lt fa fb cab ( E - Si Ti - Siltj Vij - Si j Ai j
) (1- P13 )(1 Pab)
fa fb cab gt
Technical
This is the standard quark model potential Vij
li.lj V0 li.lj Si.Sj Vss ...
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We compute the matrix element of the
Hamiltonian... in an antizymmetrized. basis
of hadrons....
Annihilation interaction
lt fa fb cab ( E - Si Ti - Siltj Vij - Si j Ai j
) (1- P13 )(1 PAB)
fa fb cab gt
The antisymmetrizer produces the states
color-octet x color-octet, expected in
multiquarks
Technical
Relative coordinate
This is the standard quark model potential Vij
li.lj V0 li.lj Si.Sj Vss ...
color singlet meson
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Implementing chiral symmmetry in the quark Model
Bicudo, Ribeiro 1989 --gt2003
quark
fp
hadron
Technical
f-p
fa
fa
fa
fa
Hadronic interactions
fb
fb
fb
fb
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We arrive at the criterion for the short-range
interaction of ground-state hadrons - whenever
the two interacting ground-sate hadrons have a
common flavour, the repulsion is increased, -
when the two interacting hadrons have a matching
quark and antiquark the attraction is
enhanced. Exs
u d s u
u d s u
u d u s
Annihilation attraction Veff. a -(2/3)(2mN-mD)
u d u s
Exchange repulsion Veff. a (4/3)(mD-mN)
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For instance, let us apply the criterion to the
S1 I0 pentaquark, uud ds or ddu us
we find repulsion! All other systems are even
more repulsive or unstable. The Q is not a
uudds pentaquark! In other words, the s1
s-wave K-N are repelled! Indeed we arrive at the
separable K-N potential VK-N 2
-(4/3)tK.tN (mD-mN) Nb2 fbgtlt fb
(5/4) (1/3) tK.tN 3 Na2
Technical
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We get a 1.535 GeV Mass for an extremely wide P-
state.
And we get the repulsive K-N exotic s-wave phase
shifts, which have been understood long ago, by
Bender al, B. Ribeiro and Barnes Swanson.
And we get the repulsive K-N exotic s-wave phase
shifts, which have been understood long ago, by
Bender al, B. Ribeiro and Barnes Swanson.
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3. The Q , X-- and D- as heptaquarks and
linear molecules.
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Q
Suppose that a q-q pair is added to the uudds .
udd ud ds or uud du us Then the new system
may bind. Moreover the heptaquark has a different
parity and therefore it is an independent system
(a chiral partner). The only possible decay
channel is to a p-wave KN, with a low energy p
annihilation which is naturally suppressed. We
proposed hep-ph/0308073 that the Q is in fact
a linear crypto-heptaquark with the strong
overlap of a KpN, where the p is bound by the
I1/2 pK and pN attractive interactions(see
Oset, Vacas, Nagahiro).
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Q
The crypto-heptaquark or linear tri-hadronic
molecule model
p du
K us
uud N
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Q
The crypto-heptaquark or linear tri-hadronic
molecule model
p du
K us
uud N
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Q
The crypto-heptaquark or linear tri-hadronic
molecule model
p du
K us
uud N
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Q
The crypto-heptaquark or linear tri-hadronic
molecule model
p du
K us
uud N
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Q
Proceeding with a quantitative study we arrive at
the separable potentials for the different
2-body systems, VK-N 2 -(4/3)tK.tN
(mD-mN) Nb2 fbgtlt fb (5/4)
(1/3) tK.tN 3 Na2 Vp-N
2 (2mN-mD) Nb2 tp.tN fbgtlt fb
9 Na2 Vp-K
8 (2mN-mD) Nb2 tp.tK fbgtlt fb
27 Na2 (where the a and
b parameters may differ from exchange to
annihilation channels)
Repulsive attractive when I1/2 attractive
when I1/2
Technical
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Q
We move on. Because the pion is quite light we
start by computing the pion energy in an
adiabatic K-N system. This is our parameter
set, tested in 2-body channels,
Technical
Where all numbers are in units of Fm-1
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Q
The only favourable flavour combination
is, Total I1 I0
p I1/2 I1/2
K I1 N
I1/2 I1/2

Technical
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Q
Again we use the T matrix, in this case with a
relativistic pion under the action of the K and
N potentials centered in two different points.
x
a N
-a K
z
Technical
0
rNp
rKp
rKp
p
y
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Q
We get for the pion energy as a function of the
K-N distance,
Technical
Indeed we get quite a bound pion, but it only
binds at very short K-N distances. I also
overcomes the K-N repulsion.
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Q
However when we remove the adiabaticity, by
allowing the K and N to move in the pion field,
we are not yet able to overcome the the K-N
kinetic energy. This suggests that other
relevant effects should be included in the pKN
study, say the 3-body pKN interaction, p
p K K N N the attractive
medium range two pion exchange interaction and
the coupling to the KN p-wave decay channel.
Technical
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X--
Extending the pentaquark and the molecular
heptaquark picture to the full SU(3)
anti-decuplet we arrive at the following picture,
-The X--(1860) discovered at NA49 cannot be a
ddssu pentaquark, because this suffers from
repulsion. - Adding a q-q pair we arrive at a
I1/2 K-N-K where the the K-N system has isospin
I1, an attractive system. We find that the K-N-K
molecule is bound, although we are not yet able
to arrive at a binding energy of -60 MeV. -
Then the non-exotic I1/2 elements of the
anti-decuplet are K-K-N molecules. - Only the
I1 elements are pentaquarks, or equivalently
overlapping K-N systems
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X--
Q !

K-p-N
N ?

K-K-N
parity
S ...
-
K-N
X ...

K-N-K
This figure summarises the anti-decuplet spectrum
of the pentaquarks/heptaquark
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X--
In the quantitative computation of the binding
energy for the K-N-K molecule, we indeed find
binding. Actually this system is easier to bind
than the Q, because the pion is very
light. However it is difficult to produce a
binding energy of 60 MeV, unless one assumes
an unnatural longer range for the short range
interaction. This suggests that the attractive
medium range interaction (two pion exchange)
should be included. We then expect that a
separation of the interaction in a medium range
attraction and a hard core, with similar
distances to the N-N interaction, may correct the
binding energies.
V N-N
Hard core repulsion (Ribeiro)
R Fm
1
0
Long Range One Pion Exchange Potential (Yukawa)
Sigma or pp exchange potential
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D- p
In what concerns anti-charmed pentaquarks like
the very recently observed Dp, or anti-bottomed
ones, this extends the anti-decuplet to flavour
SU(4) or SU(5). Anti-charmed pentaquarks were
predicted by many authors, replacing the s by a
c. Again the pentaquark uuddc is unbound. A
linear-crypto-heptaquark may exist with a mass
close to the observed resonance at H1. We
consider the molecule, D - p -N M3.10
GeV with a positive energy of 15 MeV above
threshold. We understand qualitatively that this
system is less bound than the Q .
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4. Prediction of strange, charmed and bottomed
exotic pentaquarks.
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4. Prediction of strange, charmed and bottomed
exotic pentaquarks.
It has been advocated long ago by Lipkin, J.-M.
Richard, that exotic multiquarks are favoured by
the presence of several different flavours. (see
Stancu, Lipkin, Chiu,) Indeed we find a large
number of flavoured molecules where a central
hadron attracts the two external ones (the two
external hadrons repel each other). We are not
yet able to compute exactly the binding energies
of the tri-hadron linear molecules. Nevertheless
the binding energy is much smaller than the total
sum of masses, and indeed our estimates are close
to the experimental pentaquark candidates. This
motivates us to produce all plausible exotic
flavour pentaquarks in our crypto-heptaquark
model.
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5. Conclusion
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5. Conclusion
- We conclude that the Q(1540), X(1860) and
Dp(3100) hadrons very recently discovered cannot
really be s-wave pentaquarks. - We also find
that they may be heptaquark states, say with two
repelled clusters K and N clusters bound by a
third p cluster. - We fail by 50 MEV to
reproduce the experimental binding energies. More
effects need to be included, say medium range
interactions, K-N p-wave coupled channel, and
exact Fadeev eqs. - This is a difficult subject
with the interplay of many effects. The
theoretical models should not just explain the
pentaquarks, they should be more comprehensive.
They should try to reproduce all the known
hadrons p ,K, N, D and their interactions pp,
pN, KN, NN, LL...
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Some more references This work The Theta
(1540) as a heptaquark with the overlap of a
pion, a kaon and a nucleon, P. B. , G. M. Marques
Phys. Rev. D69 rap. Com. (2004) 011503 ,
hep-ph/0308073 The anti-decuplet candidate
Xi--(1862) as a heptaquark with the overlap of
two anti-kaons and a nucleon , P.B.,
hep-ph/0403146 Are the anti-charmed and
bottomed pentaquarks molecular heptaquarks? P.
B., hep-ph/0403295 Prediction of the masses and
decay processes of strange, charmed and bottomed
pentaquarks from the linear molecular
crypto-heptaquark model, P. B.,
hep-ph/0405086 Other tests of this ideaOn the
possible nature of the Theta as a K pi N bound
state, F. Llanes-Estrada, E. Oset and V. Mateu,
nucl-th/0311020. Chiral doubling Chiral
effective action with heavy-quark symmetry, M.A.
Nowak, M. Rho and I. Zahed, Phys. Rev.D 48, 4370
(1993) hep-ph/9209272.