Are the Q(1540), X(1860) and D*p(3100) Pentaquarks or Heptaquarks?

1
Are the Q(1540), X(1860) and Dp(3100)
Pentaquarks or Heptaquarks?
Pedro Bicudo
Dep Física IST CFIF , Lisboa

• Rencontres de Moriond 2004

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Are the Q(1540), X(1860) and Dp(3100)
Pentaquarks or Heptaquarks?
Pedro Bicudo
Dep Física IST CFIF , Lisboa
1. A QM criterion for hard core attraction and
repulsion 2. Why the Q cannot be a simple uudds
or K-N state 3. The p-K, p -N and K- p -N
systems 4. SU(4) flavour the K-N-K and
anti-charmed systems 5. Conclusion

• Hadron 2003 Aschaffengurg

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We study the Q(1540) discovered at SPring-8.
We apply Quark Model techniques, that explain
with success the repulsive hard core of
nucleon-nucleon, kaon-nucleon exotic scattering,
and the short range attraction present in
pion-nucleon and pion-pion non-exotic scattering.
We find that a K-N repulsion which excludes the
Q as a K-N s-wave pentaquark. We explore the
Q as a heptaquark, equivalent to a K-p-N
borromean boundstate, with positive parity and
total isospin I0. The attraction is provided by
the pion-nucleon and pion-kaon interaction. The
other candidates to pentaquarks X- - , observed
at NA49, and Dp, observed at H1, are also
studied as linear heptaquarks.
4
The Pentaquark uudds was discovered at LEPS T.
Nakano al, hep-ex/0301020, Phys.Rev.Lett.91
012002 (2003) DIANA V.V. Barmin al,
hep-ex/0304040, Phys.Atom.Nucl.661715-1718,(
2003) Yad.Fiz.661763-1766, (2003) After the
Jefferson Lab confirmation, it was observed in
several different experiences, with a mass of
1540 -10 MeV and a decay width of 15-15 MeV.
Recently the ddssu pentaquark X--(1860) was
observed at, NA49 C. Alt al.,
hep-ex/0310014, Phys.Rev.Lett.92042003,2004 and
the uuddc pentaquark was Dp(3100) observed
at, H1 hep-ex/0403017
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1. A Quark Model criterion for repulsion/attractio
n
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1. A Quark Model criterion for repulsion/attractio
n
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) to explain the N_N
hard-core repulsion. Deus and Ribeiro (1980) also
found that the RGM may lead to hard-core
attraction .
7
1. A Quark Model criterion for repulsion/attractio
n
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) 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
meson b
8
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
9
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
This is the standard quark model potential Vij
li.lj V0 li.lj Si.Sj Vss ...
10
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
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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T1
V12
T2
x
T3
E -
-
T4
V34
1

1-
Relative energy overlap (E-Ta-Tb) (1-n faabgtlt
faab )
12
The exchange overlap results in a separable
potential
p
-Kp/2
faa
faa
q
Kp/2
-p
-Kp/2
-q
fab
fab
K-p/2
faba (q) faab (p)
13



x
V23
V14
V24
V13

1

1-
Repulsive, a qq hyperfine potential cst.
(2/3)(mD-mN) faabgtlt faab
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With no exchange the li.lj potential
cancels With exchange only the hyperfine part
of the potential contributes
l1
l3
V13
0
l1
l1
l3

l4
0
15

x
1
Attractive a qq spin independent potential -cst.
(2/3)(2mN-mD) fbabgtlt fbab
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fa
fa
Recent breakthrough in Quark Model c Symmetry
Breaking RGM Using the Axial Ward Identity
we can show that the annihilation interaction is
identical to the V- of p Salpeter equation lt f
A f gt mp - (2/3)(2mN-mD)
fb
fb
fp
f-p
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We arrive at the criterion for the interaction of
ground-state hadrons - whenever the two
interacting 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
Annihilation attraction Veff. a -(2/3)(2mN-mD)
u d u s
Exchange repulsion Veff. a (4/3)(mD-mN)
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2. Why the Q cannot be a simple uudds or K-N
state
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Applying 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
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And we get the repulsive K-N exotic s-wave phase
shifts, which have been undestood long ago, by
Bender al, Bicudo Ribeiro and Barnes
Swanson.
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Because we checked all our only approximations,
say the use a variational method, and neglecting
the meson exchange interactions, we estimate that
something even more exotic is probably
occuring! Suppose that a q-q pair is added to
the system. Then the new system may bind.
Moreover the hepatquark had a different parity
and therefore it is an independent system (a
chiral partner). Here we propose that the Q is
in fact a heptaquark with the strong overlap of a
KpN, where the p is bound by the I1/2 pK and
pN attractive interactions.
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2. The K-N, p-K, p -N and p -K-N systems
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2. The K-N, p-K, p -N and p -K-N systems
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 differ from
exchange to annihilation channels
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Because the potential is separable, it is simple
to compute the scattering T matrix. Here we show
the 2-body non-relativistic case T fbgt (1-v
g0 )-1 lt fb , g0(E,m,b) lt fb E-p 2 /(2m )
ie -1 fbgt
-0.5
0
E
E
The binding energy is determined from the pole
position of the T matrix We have binding
if -4 m v gt b2
-1
0
1/v
-2
g0(E,1,1)
-4
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We move on. Because the pion is quite light we
start by computing the pion energy in an
adiabatic K-N system. Again we use the T matrix,
in this case with a relativistic pion. This is
our parameter set, tested in 2-body channels,
Where all numbers are in units of Fm-1
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The only favorable flavor combination is, Total
I1 I0 p
I1/2 I1/2
K I1 N I1/2
I1/2
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Again we use the T matrix, in this case with a
relativistic pion under the action of two
separable potentials centered in two different
points.
x
a N
-a K
z
0
rNp
rKp
rKp
p
y
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We get for the pion energy as a function of the
K-N distance,
Indeed we get quite a bound pion, but it only
binds at very short K-N distances.
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However when we remove the adiabaticity, by
allowing the K and N to move in the pion field,
we find that the pion attraction overcomes the
K-N repulsion but not yet the the K-N kinetic
energy. We are planning to include other
relevant effects to the pKN system, starting by
the 3-body pKN interaction p p
K K N N and the coupling to the
KN p-wave channel.
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4. SU(4) flavour the K-N-K and anti-charmed
systems
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Extending the pentaquark and the molecular
heptaquark picture to the full SU(3)
anti-decuplet we arrive at the following
picture, -The X--(1860) 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 I1/2 elements of the exotic
anti-decuplet are K-K-N molecules. - Only the
I1 elements are pentaquarks, or equivalently
overlapping K-N systems
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This figure summarizes the anti-decuplet spectrum
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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, and we
are researching the possible molecular
heptaquarks that may exist in these systems.
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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 a heptaquark states, with two
repelled clusters K and N clusters bound third p
cluster. - More effects need to be included,
say exact Fadeev eq., the K-N p-wave coupled
channel, and medium range interactions. - 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 at least explain all
the ground-state hadrons and their interactions.
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Some references
This work The Theta (1540) as a heptaquark with
the overlap of a pion, a kaon and a
nucleon P.Bicudo, G. M. Marques Phys. Rev. D69
rapid communication (2004) 011503 ,
hep-ph/0308073 The family of strange
multiquarks related to the Ds(2317) and
Ds(2457) P. Bicudo, hep-ph/0401106 The
anti-decuplet candidate Xi--(1862) as a
heptaquark with the overlap of two anti-kaons and
a nucleon P. Bicudo, hep-ph/0403146 Other
tests of this ideaOn the possible nature of the
Theta as a K pi N bound state F. J.
Llanes-Estrada, E. Oset and V. Mateu,
nucl-th/0311020. Chiral doubling M.A. Nowak,
M. Rho and I. Zahed, Phys. Rev.D 48, 4370
(1993) hep-ph/9209272.