Excited States of Charmonium in Antiproton Beam Experiments

1. 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

2. PREAMBLE • 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. • 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). • DISCUSSION OF THE RESULTS OF CULCULATION FOR THE RADIAL EXCITED STATES OF CHARMONIUM (SCALAR AND VECTOR STATES) & & COMPARISON WITH RECENTLY REVEALED EXPERIMENTAL DATA. • APPLICATION OF THE INTEGRAL FORMALISM FOR DECAY OF HADRON RESONANCES FOR CALCULATION THE WIDTHS OF RADIAL EXCITED STATES OF CHARMONIUM.

3. 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 ≈ 0.3and <v2/c2> ≈ 0.2. The “size” of charmonium is of an order of 0.2 Fm (rQ=αs · mq), mq– mass of charmed quark.

4. 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, αs ≈ 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 =αs · 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

5. 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 (αs via q2 or r): (when ) or (when )

6. a = 0.52 GeV; k = 0.18 GeV When R→∞ quantum fluctuations of the string present: ; These properties underlies for choice most of potentials: whereΛ– 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 doesn’t 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 Nthe 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 Gμν(r).

7. 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).

8. 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 andmb – the masses of resonance binary decay products, P0 – basic momentum, Pn – asymptotic momentum of their relative motion.

9. 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 e+e- 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 (ψ, ψ׳ and ψ(3770)) are easily produced as narrow resonances in such experiments. The advantage of e+e- experiments is the high peak yield of the resonance compared to the underlying hadronic background. Other charmonium states like scalar and P-wave states: ηc, χ0,1,2 can only be found in the cascade decays of vector states ψ’s. While the uncertainty in the beam energy determines the precision of the mass measurement of the ψ’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 e+e- 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 e+e- experiments.

10. Charmonium states and their decay modes. Undiscovered and poorly known states are marked by dashes.

11. The charmonium spectrum. Black boxes indicate established states, hatched boxes unknown or badly known states.

12. The charmonium system has been investigated in great detail first in electron-positron reactions, and afterwards ona 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 η׳c) are not found yet, hc-state is poorly studied; - properties of the higher radial excited vector states of charmonium Ψ are poorly known; -only few partial widths of 3PJ-states are known; some of the measured decay widths don’t 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: • Ψ → ρπ, ηc →ρπ, Ψ → barion-antibarion, ηc → barion-antibarion & • & χc0 → baryon-antibaryon (hadron helicity non conserving process); • - Ψ→ π+π-, ωπ, ρπ (G-parity violating decays); • - Ψ ׳ → γπ, γη, ... (radiative decays); • - χcJ → ρρ, φφ, ...

13. eeY(4260) (2S) eeY(4350) Z(3930) 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.

14. 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 D*D • • X(4160) – e+e- → J/ψX &e+e- →J/ ψ D*D* • • Y(4260) – e+e- → γ π+π-J/ψ • • Y(4350) – e+e- → γ π+π-ψ(2S) • • Y(4660) –e+e-→ γπ+ π- ψ(2S) • Unusual strong decay into hidden charm

15. 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: presentedby Prof. Luciano Maiani, INFN, XII Conference on Hadron Spectroscopy, Frascati, Italy, 2007 • X(3872) – JPC = 1++ – possibly D0D*0 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 ψ׳׳ ·· or hybrid (ccg) ( JPC =1-- not 0 - +); possiblydiquark-antidiquark bound state [(cs) (cs)]P-wave or baryonium Λ+c Λ-c –Y(4350) – probably ψ׳׳׳ ··· • Y(4660) – probably ψ׳׳׳׳ ··· (very NEW STATE) Most of these assignments are still not confident!!!

16. SUMMARY OF THE CHARMONIUM SPECTRUMpresentedby Prof. Antimo Palano, INFN, XII Conference on Hadron Spectroscopy, Frascati, Italy, 2007

20. POSSIBLE SPECTRUM OF SCALAR AND VECTOR STATES OF CHARMONIUM

21. 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.

22. 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 EXPERIMENTWITH ITS HIGH QUALITY ANTIPROTON BEAM SEEMS TO BE THE EXCELLENT TOOL FOR CHARMONIUM SPECTROSCOPY STUDIES.