Transcript Slides - Agenda INFN

THE RADIAL EXCITATIONS OF
CHARMONIUM IN THE RELATIVISTIC
SPHERICAL SYMMETRIC TOP MOIDEL
Barabanov M.Yu. 1, Chukanov S.N. 2 , Nartov B.K. 2 ,
Vodopianov A.S. 1, Yamaleev R.M. 3 , Babkin V.A.1
1)
Veksler-Baldin Laboratory of High Energies, JINR, Dubna,
2) Sobolev Institute for Mathematics, Siberian Department of
Russian Academy of Sciences
3) Laboratory of Information Technologies, JINR, Dubna
THE MAIN ASPECTS OF THE TOPIC PRESENTED ON THE XII
INTERNATIONAL CONFERENCE ON HADRON SPECTROSCOPY
1. STUDY OF
THE MAIN CHARACTERISTIC OF CHARMONIUM
SPECTRUM (MASS & WIDTH) BASED ON THE QUARKONIUM
POTENTIAL MODEL, HADRON RESONANCE CONCEPTION AND
RELATIVISTIC SPHERICAL SYMMETRIC TOP MODEL FOR
CHARMONIUM DECAY PRODUCTS.
2. ANALYSIS OF SCALAR AND VECTOR CHARMONIUM STATES IN
MASS RANGE FROM 2980 MeV UP TO 5200 MeV (UP TO BOOTOM
MESONS DOMAIN).
3. DISCUSSION OF THE RESULTS OF CULCULATION FOR THE RADIAL
EXCITED STATES OF CHARMONIUM (SCALAR AND VECTOR
STATES).
4. APPLICATION OF THE INTEGRAL FORMALISM FOR DECAY OF
HADRON RESONANCES FOR CALCULATION THE WIDTHS OF
RADIAL EXCITED STATES OF CHARMONIUM.
Coupling strength between two quarks as a function of their distance. For small
distances (≤ 10-16 m) the strengths αs 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 αs ≈ 0.3 and < v2/c2> ≈ 0.2. The
“size” of charmonium is of an order of 0.1 Fm (rQ=αs · mq), mq– mass of charmed
quark.
According to the non-relativistic potential model of quarkonium the spectrum and wave
functions defines from the Schrodinger-type equation:
where m 


2 2 
  ψ r   V r   Eψ r   0
2m
m1m2
- is reduced mass of
m1  m2
cc -system. In central symmetric potential
field V(r):
2m
l (l  1) 2
U ' ' r   2 {E  V r  
}U (r )  0

2mr 2
where U(0) = 0 and U’(0) = R(0) and U(r) = rR(r), mc  mc  1.5 GeV, R(r) – radial
wave function, r – distance between quark and antiquark in charmonium (quarkonium).
Potential of cc deals with one-gluon exchange:
 
 
 2 α s q 2 or
V q ~
q2
V r ~
α s r 
r

where αs  g / 4π; g – constant of colour interaction q - three-dimensional momentum
2
transferred between quark and antiquark.
At small distances interaction reduces and manifests via the dependence (αs via q2 or r):
αs ~
1
1
(when q 2   ) or αs ~
(when r  0 )
2
2
2 2
ln q /Λ
ln 1 / r Λ




where Λ – is QCD parameter. This dependence defines the phenomenon of asymptotic
free and emerges from renormgroup approach.
QCD doesn’t applicable at big distances.
 


1 2
q  0 or V r  ~ r r  
4
q
r
corresponding interaction between quarks with the strength F  dV
 const
dr γ
When R→∞ quantum fluctuations of the string present: V r |r   kr  ;
r
These properties underlies for choice most of potentials:
From LQCD we have: V q 2 ~
1
V r  |r 0  1 / r or
;
r ln 1 / r 

V r  |r   kr;
V r   
a
 k r; a = 0.52 GeV; k = 0.18 GeV
r
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):
Cornell Potential:


VN r   const GN1 2 r δ r 
VN r   V0  Dr R1 N r dr / r , V0  const  0
V3 (r )  V0 tg (r / a)  B; V0 , B  0
V 3 r   V0 ctg r / a   B; V0 , B  0
ctg (r / a)  a / r  r / 3a
Rr   sin r / a Dr   r / a
r 2 / a 2   2
where R(r), D(r) and GN(r) are scaling factor, gauging and determinant of metric tensor Gμν(r).
The relativistic corrections can be considered via relativistic Bethe-Solpeter equations:
J ab x1 , x2    d 4 x3d 4 x4 d 4 x5 d 4 x6 S 'aF x1  x3 S 'bF x2  x4 
 Gab x3 , x4 , x5 , x6   J ab x5 , x6 
where a and b – quark and antiquark indexes, Jab – formation amplitude of bound state
by quark a and antiquark b ( S'aF and S'bF – 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:

 
 

 a b

 
Λ
Λ
Λ
3 ~
2
~
~
a b
a Λb
ψab q   

γ0 γ0  2πi   d kGab  k  ψab q  k
 E  Ea  Eb E  Ea  Eb 



where “~” means transition to the instantaneous interaction, Λa,b – projective operators
onto the states with positive and negative energy accordingly, γ0 – 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).

a
S' F
Gab
J ab
b
S' F
The graphic representation of Bethe –Solpeter equation.
The visual interpretation of motion trajectory of the relativistic top over the
spherical surface embedded in four-dimensional Euclidean space
Let us define the set of generators of SO(4) group

Translation operator N on the sphere S3 has the form
The linear combinations of these orthonormal operators

  
 
M  r  p , N  r4 p  r p4

  
N  Rp  r r p  / R
 

  M  N


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:
 
 
1
2    , 2    , 
H
2mR2
 


R    / R  M  N / R   p


 
 
1
1   2








 p,  
2



,

2



,




2
2m
2mR
The spectrum is:
The wave function
2

2


Hn 
n

1
n , n  0, 1, 2...
n  LSJM J
2
2mR
H
was taken as eigenfunction of whole momentum
2


J   μ  s 2 of the top.
Advances in Applied Clifford Algebras, V.8, N.2, p.235-270 (1998).
Let us generalize this concept to the relativistic case:
H  ma2 
1
R2
μ σ   2 2
 mb2 
1
R2
μ σ   2 2
were ma and mb are the masses of resonance decay products. The spectrum is:
2
2
2
2





n

1

n

1
2
E  m a2 

m
, n  0, 1, 2...
b
2
2
R
R
The formula for resonance mass spectrum has the form

E  M th  m  P  m  P  m  n  P0
2
a
2
n
2
b
2
n
2
a

2
 mb2  n  P0 
2
where res  a  b - a binary decay channel (we used the system where   c  1 ), ma
and mb – the masses of resonance binary decay products, P0 – basic momentum, Pn –
asymptotic momentum of their relative motion, conjugated to the parameter R.
Instead the parameter with dimension of length R equivalent to the range potential we
will consider experimentally observed parameter Pn = n ·P0
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 spectrum
remains:
- the radial excited scalar states of charmonium (except η‫׳‬c) not found yet,
hc-state is poorly studied;
- properties of the radial excited vector states of charmonium Ψ are poorly
known;
- only few partial widths of 3PJ-states known; some of the measured decay
widths don’t fit into any theoretical scheme and additional experimental
checks need to be made and more data on different decay modes are
desirable to clarify the situation;
- little is known on charmonium states above the the DD -threshold
OF PARTICULAR INTEREST ARE THE FOLLOWING DECAYS OF
CHARMONIUM STATES:
- Ψ → ρπ, ηc →ρπ, Ψ → barion-antibarion, ηc → barion-antibarion, χc0 → baryonantibaryon (hadron helicity non conserving process);
- Ψ→ ππ, ωπ, ρπ (G-parity violating decays);
- Ψ‫ →׳‬γ+π, η, ... (radiative decays);
- χcJ → ρρ, φφ, ...
Charmonium states and their decay modes. Undiscovered and poorly known states are
marked by dashes.
The charmonium spectrum. Black boxes indicate established states, hatched boxes
unknown or badly known states.
eeJ/ X(3940)
X(3872)J/
Y(3940)J/
eeY(4260)
c2’
(2S)
eeY(4350)

Many new charmonium states: 6 above DD – threshold +
+ 2 below (c(2S) and hc) for last 4 years were revealed in experiment.
Most of heavy charmonium states are not explained by theory.
• Theory complains for many years for lack of new data
in spectroscopy
• Now theory does not know where to put the plenty of
new states:
– X(3872) – JPC=1++ --- most probable D0D*0 molecule
– Y(3940) – hybrid?
– Z(3940) – c2
– X(3940) – c(3S)
– Y(4260)
– Y(4350)
• Most of these assignments are still not confident.
The XYZ particles
• X(3872) – B→ Kπ+π-J/ψ
• Z(3930) – γγ → DD
• Y(3940) – B → KωJ/ψ
• X(3940) – e+e- → J/ψX & e+e- → J/ψ DD*
• Y(4260) – e+e- → γ π+π-J/ψ
• Y(4350) – e+e- → γ π+π-ψ*
Unusual strong decay into hidden charm
The possible spectrum of scalar and vector states of charmonium.
Dependence of 2-distribution on
momentum of relative motion of
resonance decay products
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):
2
  2  L ( R)V1 ( R) FL ( R) R dR ,
2
where FL – is the regular decision in the V2(R) potential, normalized on the
energy delta-function; φL(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.
CONCLUSIONS
1. THE PROPOSED APPROACH FOR CALCULATION OF THE MAIN
CHARACTERISTIC OF CHARMONUM SPECTRUM (MASS AND WIDTH)
DESCRIBES THE EXISITING EXPERIMENTAL DATA WITH HIGH ACCURACY.
THE POSSIBILITY OF PREDICTION FOR THE NEW RADIAL EXCITED STATES
OF CHARMONIUM ARISES.
2. THE CHARMONIUM DECAY CHANNELS CAN BE JOINT WITH THE GROUPS
HAVING A CHARACTERISTIC LENTH (BASIC MOMENTUM). INSIDE EACH
GROUP THESE CHANNELS ARE CLASSIFIED BY THE QUANTUM NUMBERS
R=/p0 AND n=pn/p0. THE CHARACTERISTIC LENGTH IN THE RELATIVISTIC
TOP MODEL CORRESPONS TO A PREDISSOCIATION RADIUS EQUIVALENT
TO THE RANGE OF THE POTENTIAL.
3. THE SCALAR AND VECTOR STATES OF CHARMONIUM HAVE BEEN
ANALYZED. THE POSSIBILITY OF THE 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. THE STUDY OF CHARMONIUM SPECTROSCOPY SEEMS PERSPECTIVE IN
THE EXPERIMENTS USING LOW ENERGY ANTIPROTON BEAMS WITH THE
MOMENTUM VARING FROM 1 GeV/c TO 15 GeV/C.