SEARCH OF THR RADIAL EXCITATED STATES OF CHARMONIUM IN EXPERIMENTS USING ANTIPROTON BEAMS WITH MOMENTUM RANGING FROM 1 GeV/C TO 15 GeV/c

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SEARCH OF THR RADIAL EXCITATED STATES OF
CHARMONIUM IN EXPERIMENTS USING ANTIPROTON BEAMS
WITH MOMENTUM RANGING FROM 1 GeV/C TO 15 GeV/c
Barabanov M.Yu. 1, Vodopianov A.S.1, Dodokhov
V.Kh.1, Babkin V.A.1, Chukanov S.N. 2 , Nartov
B.K. 2
1) Veksler-Baldin Laboratory of High Energy
Physics, JINR, Dubna 2) Sobolev
Institute for Mathematics, Siberian Department of
Russian Academy of Sciences
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PREAMBLE

1. STUDY OF THE MAIN CHARACTERISTIC OF CHARMONIUM
   SPECTRUM (MASS, WIDTH BRANCH RATIOS) BASED ON
   THE QUARKONIUM POTENTIAL MODEL, HADRON RESONANCE
   CONCEPTION AND RELATIVISTIC SPHERICAL SYMMETRIC
   TOP MODEL FOR CHARMONIUM DECAY PRODUCTS.
2. ANALYSIS OF SPECTRUM OF SCALAR AND VECTOR
   CHARMONIUM STATES IN MASS REGION MAINLY ABOVE
   DD-THRESHOLD. ESPECIALLY PAY ATTENTION AT THE
   NEW STATES WITH HIDDEN CHARM DISCOVERED RECENTLY
   DURING THE LAST SEVERAL YEARS (THE EXPERIMENTAL
   DATA FROM CLEO, BELLE BaBar COLLABORATIONS WERE
   USED).
3. DISCUSSION OF THE RESULTS OF CULCULATION FOR THE
   RADIAL EXCITED STATES OF CHARMONIUM (SCALAR AND
   VECTOR STATES) COMPARISON WITH RECENTLY
   REVEALED EXPERIMENTAL DATA.
4. APPLICATION OF THE INTEGRAL FORMALISM FOR DECAY
   OF HADRON RESONANCES FOR CALCULATION THE WIDTHS
   OF RADIAL EXCITED STATES OF CHARMONIUM.

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Coupling strength between two quarks as a
function of their distance. For small distances
( 10-16 m) the strengths as is 0.1, allowing a
theoretical description by perturbative QCD. For
distances comparable to the size of the nucleon,
the strength becomes so large (strong QCD) that
quarks can not be further separated they remain
confined within the nucleon. For charmonium
states as 0.3 and ltv2/c2gt 0.2. The size of
charmonium is of an order of 0.2 Fm (rQas mq),
mq mass of charmed quark.
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Why namely charmonium!? Charmonium is an
excellent testbench for QCD

• Charmonium is the simplest two-particle
  system consists of quark antiquark
• Charmonium(a) are compact bound systems
  with small widths varying from several MeV to
  several tens of MeV compared to the light
  unflavored mesons
• Charm quark c has large mass (1.25 0.09 GeV),
  compared to the masses of u, d s ( 0.1 GeV)
  quarks
• Quark motion velocities in charmonium are
  non-relativistic (the coupling constant, as 0.3
  is not too large, and relativistic effects are
  manageable ( v2/c2 0.2))
• The size of charmonium is of an order of 0.2
  Fm (rQ as mq ) so that one of the main
  doctrines of QCD asymptotic freedom is
  emerging
• Therefore
• charmonium studies are promising for
  understanding the dynamics of quark interaction
  at small distances, and charmonium spectroscopy
  is a good testing ground for the theories of
  strong interactions
• QCD in both perturbative and nonperturbative
  regimes
• QCD inspired purely phenomenological potential
  models
• NRQCD and Lattice QCD

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According to the non-relativistic potential model
of quarkonium the spectrum and wave functions
defines from the Schrodinger-type equation
where - is reduced mass of
-system. In central symmetric potential
field V(r)
where U(0) 0 and U(0) R(0) and U(r) rR(r),
GeV, R(r) radial
wave function, r distance between quark and
antiquark in charmonium (quarkonium).
Potential deals with one-gluon exchange
or dominates in
potential where
g constant of colour interaction -
three-dimensional momentum
transferred between quark and antiquark. At
small distances interaction reduces and manifests
via the dependence (as via q2 or r)
(when ) or
(when )
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where ? is QCD parameter. This dependence
defines the phenomenon of asymptotic freedom and
emerges from renormgroup approach (A.A. Bykov
et al. Physics Uspekhi, V.143, N1, 1 (1984))
QCD doesnt applicable at large distances.
From LQCD we have
or corresponding interaction between
quarks with the strength
or
Cornell Potential
Izmestev A.A. shown /Nucl. Phys., V.52, N.6
(1990) Nucl. Phys., V.53, N.5 (1991)/ that in
the case of curved coordinate space with radius a
(confinement radius) and dimension N the
quark-antiquark potential defines via Gauss
equations (considering compact space sphere S3)
where R(r), D(r) and GN(r) are scaling factor,
gauging and determinant of metric tensor Gµ?(r).
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The relativistic corrections can be
considered via relativistic Bethe-Solpeter
equations
where a and b quark and antiquark indexes, Jab
formation amplitude of bound state by quark a
and antiquark b ( and their
propogators), Gab - two particle irreducible
nucleus of Bethe-Solpeter equation describing
interaction between quark and antiquark that
reduces to their bound state. In momentum
representation (in the center mass system of
quark and antiquark) and assuming instantaneous
interaction Bether-Solpeter equation can be
written
where means transition to the instantaneous
interaction, projective operators onto
the states with positive and negative energy
accordingly, ?0 Dirac matrix, E energy of
bound state, Ea and Eb quark antiquark
energies. To study quarkonium spectroscopy one
must define the quark - antiquark interaction,
i.e. define G or equivalent V(r).
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The graphic representation of Bethe Solpeter
equation.
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Let us define the set of generators of SO(4) group
Translation operator on the sphere S3 has
the form
The linear combinations of these orthonormal
operators
contribute two set of generators of the SU(2)
group. Thus the SU(2) group generates the action
on a three-dimensional sphere S3. This action
consists of the translation with whirling around
the direction of translation. We get
The spectrum is
The wave function
was taken as eigenfunction of whole momentum
of the top.
Advances in Applied Clifford Algebras, V.8, N.2,
p.235-254 (1998) V.8, N.2, p.255-270 (1998) .
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Let us generalize this concept to the
relativistic case
were ma and mb are the masses of resonance decay
products. The spectrum is
The formula for resonance mass spectrum has the
form
where - a binary decay
channel (we used the system where
), ma and mb the masses of resonance binary
decay products, P0 basic momentum, Pn
asymptotic momentum of their relative motion.
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• The feature of all charmonium states is their
  narrowness in compared for example, with light
  unflavored mesons. The spectrum has low density,
  and thus arises the possibility for experimental
  reveal of charmonium states.
• The earliest studies of the charmonium system
  were performed at ee- colliders, where the
  charmonium system was created through the
  intermediate virtual photon. The quantum numbers
  of the final state were thus limited to those of
  the photon, JPC 1--. The higher laying vector
  states (?, ?? and ?(3770)) are easily produced as
  narrow resonances in such experiments. The
  advantage of ee- experiments is the high peak
  yield of the resonance compared to the underlying
  hadronic background. Other charmonium states like
  scalar and P-wave states ?c, ?0,1,2 can only be
  found in the cascade decays of vector states ?s.
  While the uncertainty in the beam energy
  determines the precision of the mass measurement
  of the ?s, the mass resolution for all other
  states is limited by the detector resolution for
  the low-energy photons. This fact leads to an
  unsatisfactory precision for the measurements.
• In contrast to the limitations of ee- colliders,
  the pp annihilation process allows the direct
  formation of all charmonium states. Their
  spectrum is sensitive against the shape of the
  confining potential. So, the study of charmonium
  in pp annihilations allows much more precise
  measurements than can be achieved in ee-
  experiments.

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Charmonium states and their decay modes.
Undiscovered and poorly known states are marked
by dashes.
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The charmonium spectrum. Black boxes indicate
established states, hatched boxes unknown or
badly known states.
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The charmonium system has been investigated in
great detail first in electron-positron
reactions, and afterwards on a restricted scale,
but with high precision, in antiproton-proton
annihilation. The number of unsolved questions
dealing with charmonium still remains
- the radial excited scalar states of
charmonium (except ??c) are not found yet,
hc-state is poorly studied - properties of the
higher radial excited vector states of charmonium
? are poorly known - only few partial widths of
3PJ-states are known some of the measured decay
widths dont fit into theoretical schemes and
additional experimental checks need to be made
and more data on different decay modes are
desirable to clarify the situation - radial
excitations of 3PJ-states are not established -
little is known on charmonium states above the
the DD threshold - many recently discovered
states above DD - threshold (NEW STATES) wait for
their verification explanation
OF PARTICULAR INTEREST ARE THE FOLLOWING
CHARMONIUM DECAYS

• ? ? ?p, ?c ??p, ? ? barion-antibarion, ?c ?
  barion-antibarion
• ?c0 ? baryon-antibaryon (hadron helicity
  non conserving process)
• - ?? pp-, ?p, ?p (G-parity violating decays)
• - ? ? ? ?p, ??, ... (radiative decays)
• - ?cJ ? ??, ff, ...

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ee?J/? X(3940)
X(3872)?J/???
Y(3940)?J/??
Many new charmonium states 8 above DD
threshold 2 below (?c(2S) and hc) for the
recent years were revealed in experiment. Most
of heavy charmonium states (above DD threshold
) are not explained by theory.
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• The XYZ particles
• X(3872) B ? Kpp-J/?
• Z(3930) ?? ? DD
• Y(3940) B ? K?J/?
• X(3940) ee- ? J/?X ee- ? J/? DD
• X(4160) ee- ? J/?X ee- ? J/ ? DD
• Y(4260) ee- ? ? pp-J/?
• Y(4350) ee- ? ? pp-?(2S)
• Y(4660) ee- ? ? p p- ?(2S)

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• Theory complains for many years for lack of new
  data in spectroscopy especially over DD -
  threshold.
• Now theory does not know where to put the
  discovered new states
• presented by Prof. Luciano Maiani, INFN, XII
  Conference on Hadron Spectroscopy, Frascati,
  Italy, 2007
• X(3872) JPC 1 possibly D0D0 molecule or
• diquark-antidiquark bound state (cq)
  (cq)S-wave (q u, d)
• Y(3940) possibly hybrid
• Z(3930) possibly ??c2
• X(3940) probably ?c(3S)
• X(4160) possibly ?c(4S) or what ? (very NEW
  STATE)
• Y(4260) probably ??? or hybrid (ccg) ( JPC
  1- - not 0 - ) possibly diquark-antidiquark
  bound state (cs) (cs)P-wave or baryonium ?c
  ?-c
• Y(4350) probably ????
• Y(4660) probably ????? (very NEW STATE)
• Most of these assignments are still not
  confident!!!

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SUMMARY OF THE CHARMONIUM SPECTRUMpresented by
Prof. Antimo Palano, INFN, XII Conference on
Hadron Spectroscopy, Frascati, Italy, 2007
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POSSIBLE SPECTRUM OF SCALAR AND VECTOR STATES OF
CHARMONIUM
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The integral formalism (or in other words
integral approach) is based on the possibility of
appearance of the discrete quasi stationary
states with finite width and positive values of
energy in the barrier-type potential. This
barrier is formed by the superposition of two
type of potentials short-range attractive
potential V1(r) and long-distance repulsive
potential V2(r). Thus, the width of a quasi
stationary state in the integral approach is
defined by the following expression (integral
formula)
where where FL(r) is the regular decision
in the V2(r) potential, normalized on the energy
delta-function ?L(r) normalized wave function
of the resonance state. This wave function
transforms into irregular decision in the V2(r)
potential far away from the internal turning
point.


The integral can be estimated with the well known
approximately methods for example, the
saddle-point technique or the other numerical
method.
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SUMMARY 1. AUTHORS HAVE PROPOSED THE APPROACH
FOR CALCULATION OF THE MAIN CHARACTERISTICS OF
CHARMONUM SPECTRUM. IT WAS DEMONSTRATED THAT THIS
APPROACH DESCRIBES THE EXISITING EXPERIMENTAL
DATA WITH HIGH ACCURACY. 2. IT WAS
DEMONSTRATED THAT THE POTENTIAL MODELS WITH
CORNEL-LIKE QUARK-ANTIQUARK POTENTIAL ARE
SUITABLE TO PROVIDE THE CONNECTION BETWEEN QCD
AND THE MORE PHENOMENOLOGICAL TREATMENTS AT
DISTANCE SCALES COMPARABLE TO THE NUCLEON
RADIUS. 3. THE SCALAR AND VECTOR STATES OF
CHARMONIUM HAVE BEEN ANALYZED. THE POSSIBILITY OF
EXISTENCE OF THEIR RADIAL EXCITATIONS WAS
DEMONSTRATED. SO, IT BECOMES POSSIBLE TO PREDICT
NEW RADIAL EXCITED STATES (SCALAR AND VECTOR) OF
CHARMONIUM WITH QUANTUM NUMBERS DETERMINED
BEFORAHAND. 4. NEW RECENTLY DISCOVERED STATES
ABOVE DD - THRESHOLD HAVE BEEN ANALYZED. SOME OF
THESE STATES CAN BE INTERPRETED AS HIGHER LAYING
RADIAL EXCITED SCALAR VECTOR STATES OF
CHARMONIUM. THIS TREATMENT NEEDS TO BE CAREFULLY
VERIFED IN THE ONCOMING PANDA EXPERIMENT. 5.
THE STUDY OF CHARMONIUM SPECTROSCOPY SEEMS
PERSPECTIVE IN THE EXPERIMENTS USING LOW ENERGY
ANTIPROTON BEAMS WITH THE MOMENTUM RANGING FROM 1
GeV/c TO 15 GeV/C. THEREFORE THE FUTURE PANDA
EXPERIMENT WITH ITS HIGH QUALITY ANTIPROTON BEAM
SEEMS TO BE THE EXCELLENT TOOL FOR CHARMONIUM
SPECTROSCOPY STUDIES.