1. Theoretical description of the charmonium structure in QCD
Gabi Hoffmeister

2. Summary
1. Introduction
2. Charmonium spectroscopy and theoretical potential models
3. Transitions and decays of cc
4. New states above the DD-threshold
5. Conclusion

3. Charmonium production
1. Introduction
Charmonium production
J/Y, Y(2S) hc, cc,...
Color-suppressed b  c decay
Predominantly from B-meson decays
e+e- annihilation/Initial State Radiation (ISR)
e+e- collision below nominal cm energy
JPC = 1--
Double charmonium production
Typically one J/Y or Y, plus second cc state
Two-photon production
Access to C = +1 states
pp annihilation
All quantum numbers available
K-, K0,K*, K- resonances…
Untagged gg: Charmonium states with JPC = 0+, 2+
J = 0,2
J = 1

4. 1. Introduction History of discovered charmonium states
1974: first charmonium state J/Y with mJ/Y = 3096 MeV discovered (SLAC: e+e- → Y → e+e-, m+m-, hadrons and BNL: p + Be → J → e+e- + X)
1974: discovery of Ψ´ (excited 3S1 state) with mΨ´ = GeV and Γ ≤ 2.7 MeV at SLAC
Studying of radiative decays of Ψ´: BR (Ψ´ → J/Ψ + p- + p+) = 0.32
BR (Ψ´ → J/Ψ → neutrals) = 0.25
No other narrow resonances found from reactions e+e- → hadrons
1976: cc,1,2,3 (triplet states 3P0,1,2) discovered from radiative decays of Y´→ g cc,J
1980: discovery of 1S0 singlet hc with mass m = 2.98 GeV in decay Y´→ g hc
1982: hc´ (excited state of hc) seen at Crystall Ball (SPEAR) with mhc = 3594 MeV
1977: Discovery of upsilon meson U (bottonium bb with JPC = 1--) at Fermi Lab with mU ≈ 9.46 GeV via p-Cu interaction again with very narrow width ~52 keV
Many excited states of the U like in case of J/Ψ (similar energy levels)
Bound state tt non observed: top-quark decays before building a bound state
(t → W+ + b)

5. 2. Charmonium spectroscopy and theoretical potential modelshc cc
J/Ψ ηc
Singlet S-states (spin 0): hc, hc´ Singlet P-states (spin 0): hc
Triplet S-states (spin 1): J/Y, Y´,Y´´,… Triplet p-states (spin 1): c1,2,3
Charmonia:

6. 2. Charmonium spectroscopy and theoretical potential models
Charmonium states
Experimental data can be used to compare results to the expected values of different theoretical potential models
charmonium_1.pdf –

7. 2. Charmonium spectroscopy and theoretical potential models
c,b are heavy quarks can be treated in nonrelativistic approximations (Schrödinger equation + static potential) because relativistic corrections are small
At small distances: one-gluon exchange dominates (asymptotic freedom): V ~ 1/r
At large distances confining potential:
Coulomb + linear potential:
„Cornell-Potential“
Vector part Vv
Scalar part Vs
=> Fits to the data show that Vv is small
Contributions to the cc-potential:
k = (2pa´) -1 ≈ 0.18 GeV² is the string tension (energy density of qq pair in string model of hadrons) with typical slope a´= 1 GeV-² of a hadronic Regge trajectory

8. 2. Charmonium spectroscopy and theoretical potential models
Fine structur splitting (spin orbit interaction):
Vs: scalar part from confining term
Vv: vector part from one-gluon (vector boson) exchange
Spin spin (splitting of singlet and triplet states):
→ no contribution from Vs
Tensor term:
By computating the various expectation values one obtains mass splitting relations:
1 (3P2)
-1 (3P1)
-2 (3P0)
¼ (3S1)
-¾ (1S0)
-1/5 (3P2)
(3P1)
(1P0)

9. 2. Charmonium spectroscopy and theoretical potential models
The resulting mass relations for the triplet are:
→ testing if long-range potential transforms as a 4-scalar Vs or a 4-vector Vv considering a modification of the Cornell model (V = br - a/r):
with
Inserting this potential model and setting
(hb(1P))
(hb(2P))
experimental data on U-P-wave
for vector confinement (x ≈ 1) formula in accord with experimental data only for l ≈ 0, whereas scalar confinement (x ≈ 0) larger range 0.4 ≤ l ≤ 1.0 in accord with exp. values
Conclusion: confinement produced by a long-range 4-scalar interaction

10. 3. Transitions and decays of cc
Annihilation:
Generally suppressed for bound state
Decay to leptons is a clean experimental signal
Strong interaction:
Dominant above ~3.72 GeV (D mesons)
Suppressed below this mass threshold
Radiative transition:
EM radiative transition emitting photon
Emission of gluons producing light quarks
Features:
Suppression of strong decays leads to (relatively) long lifetimes, narrow widths
Radiative decays are competitive; often most accessible transitions
Selection rules:
Conservation of J
Conservation of P,C in strong and electromagnetic decays

11. 3. Transitions and decays of cc
All quarkonia are unstable and decay through: 1) annihilation processes and
2) radiative transitions
1) Annihilation processes (electromagnetic and hadronic decays):
for a bound state with wavefunction Yn(x) in electromagnetic decay
Including QCD radiative corrections and substituting the electric charge by ec = (2/3)e for the charm-quark charge and a color factor of 3: c
el.mag. decay
gg or gg c
hadronic decay c f c
g *
ggg or ggg
for 3S1 state c c f

12. 3. Transitions and decays of cc
Decays from the 3S1-system with 3 final particles or a lepton pair including QCD radiative corrections :
electromagnetic decay
hadronic decay
el.mag. decay
Problems:
- factor lΨn(0)l² comes from non relativistic approximation, can be modified by relativistic corrections
- second order terms O(as²) could play an important role

13. 3. Transitions and decays of cc
hadronic transitions: J/Ψ, Ψ´→ PV, PP, VV
(P: pseudoscalar and V: vector mesons)
with quark flavor basis:
el.magn. J/Y and Y´ decays into meson pairs
mixing mechanism for charmonium decays into meson pairs
G-parity and isospin violating transitions with BR ~ , supressed by factor ~ compared to G-parity and isospin allowed J/Ψ decays
Charmonium state possesses Fock components of light quarks, can therefore decay through these by a soft mechanism; node in 2S radial function leads to suppression of mechanism in Ψ´decays
mixings:

14. 3. Transitions and decays of cc
branching ratios of decays of J/Ψ and Ψ´ into meson pairs from experimental data (Beijing Electron Spectrometer Collaboration)
„12%-rule“
G parity violating transition
flavor symmetry breaking mixing
Isospin violating transition

15. 3. Transitions and decays of cc
2) radiative transitions (M1 and E1 dipole transitions):
M1 transitions (no parity change, spin flip: DL = 0, DS = 1):
J/Ψ → hc + g, Ψ´ → hc + g, Ψ´ → hc´ + g → J/Ψ + g
with w = Ei - Ef
Dipole approximation:
Schrödinger wave function for charmonium: Y(r) = Yspin·Ylm (J, f) Rnl(r)
for E1, M1
where
Relativistic corrections and anomalous magnetic moment for quarks neglected!
2p·(phase space)
where j0 is the spheric Bessel function (jo(x) = sin(x)/x )
Relativistic corrections and anomalous magnetic moment for quarks are neglected!

16. 3. Transitions and decays of cc
E1 transitions (parity changes, no spin flip: DL = 1, DS = 0):
Ψ´ → g + cc,J → J/Ψ + g
1 for spin singlet transition
3 for spin triplet transitions
Jf : spin of the final state and Sfi =
where
estimation of decay width by building ratios:
Determination of as(mc):
from experimental decay width one gets: as(mF) ≈ as(mJ/Ψ) ≈ 0.21, as(mU) ≈ 0.18

17. 3. Transitions and decays of cc
Higher multipole contributions in charmonium
Magnetic quadrupole (M2) amplitudes provide indirect measure of charmed quark´s anomalous magnetic moment and are sensitive to D-wave mixtures in S-wave states (Ψ´´ – Ψ´)
Affect angular distributions in decays Ψ´→ cc,J g and cc,J → J/Ψ g (experimentally accessible through interference with dominant E1 amplitudes)
Radiative widths given by helicity amplitudes Al, Al´ with l labelling the projection of the spin of cc,J parallel (Al) or antiparallel (Al´) to the photon
setting e ≡ x·Eg/(4mc) where x = -1 for Ψ´→ g cc,J and x = +1 for cc,J → J/Ψ
kc: quark anomalous magnetic moment
(deviation from Dirac magnetic moment mc = ⅔ ec/(2mc))
Searching for interferences with dominant E1 amplitudes (cc,J → g J/Y): expected normalized M2/E1 ratios a2:

18. 3. Transitions and decays of cc
Hadronic transitions [QQ → (QQ)´+ light hadrons]
examples:
theoretical description uses multipole expansion for gluon emission, very similar to usual multipole expansion for photonic transitions:
(color electric and color magnetic emission from a heavy quark)
Single interaction of HI changes color singlet QQ initial state i into some color octett QQ state, second interaction HI is required to return to a color singlet QQ final state (f) -> at least two gluons have to be emitted
Ordering of amplitudes in powers of velocity with leading contribution from color electric gluon emissions:
above DD-threshold: C(3872) → J/Ψ p+p- and Y(3940) → J/Ψ w
ta: generator of color SU(3),(a = 1,…,8)
sum over all allowed QQ octett intermediate states nO
S-wave 2p-system
D-wave 2p-system
lowest mass light hadron state: p

19. 3. Transitions and decays of cc
Properties of Ψ(2S) → g cc,J E1 radiative transition with Gtot [Ψ(2S)] = 33713 keV
Properties of transitions cc,J → g J/Ψ E1 radiative transition
Partial widths and BR for spin-singlet states,
O = r (GeV-1) for E1 and O = j0(kr/2) for M1 transitions
Ψ´→ g hc´/ hc
hc´ → g hc
Ψ´→ g J/Ψ
hc → g hc
Phys. Rev. D 66, (2002)

20. 3. Transitions and decays of cc
Decay to g hc(1S):
forbidden M1 transition (would vanish in the limit of Eg = 0 because of orthogonality of 1S and 2S wave functions) at photon energy of 638 MeV → ≠ averaged BR = (3.00.5)·10-3 => G[Ψ(2S) → g hc(1S)] = (1.00 0.16) keV
Decay to g hc(2S):
allowed M1 transition characterized by ≈ 1 for small photon energies
Assumption: matrix elements for Y(2S) → ghc´(2S) and J/Y(1S) → ghc(1S) are equal
=> (2S-2S)-rate = times (1S-1S)-rate leading to a BR = (2.6 0.7)·10-4 and therefore G[Ψ(2S) → g hc´(2S)] = (87 25) eV
Hadronic transitions to J/Ψ:
Via electric dipole emission of gluon pair followed by its hadronization into pp
dominating decay mode in pions

21. 3. Transitions and decays of cc
Ψ(2S) → p0 hc → p0 g hc hc
CLEO data with
MeV
background function plus signal

22. 4. New states above the DD-threshold
Discovery of a new signal X(3872) in B+X K+, XJ/Ψ p+p- at Belle in 2003 with narrow width G < 2.3 MeV and mass mX = 0.6 MeV
Confirmed by CDF, D0 and BaBar
X  J/Ψ g radiative decay confirmed by BaBar determines C = +1
Belle/CDF dipion angular analysis in XJ/Ψp+p- favours JPC = 1++
not seen in X  J/Ψ p0p => neutral state

23. 4. New states above the DD-threshold
Interpretation of X(3872)
similar to charmonium: narrow state decaying to J/Ψp+p-
above DD threshold should be wide and XDD dominant
Quantum numbers established: 1++
It does not fit into the charmonium model!
m(X) ≈ m(D) + m(D*0) => X could be a bound state of 2 D mesons, a D0D*0 molecule assumption supported by predictions of mass, decay modes, JPC, branching fractions and small binding energy (deuteron like)
Other exotic predictions: - “tetraquark” 4 quark bound state
- “glueball” gluon bound state, charmonium-gluon hybrid ccg
Further new states discovered:
X(3940):
- discovered by Belle in double charmonium production e+e-J/Ψ X(3940)
- Decays to DD* but not DD and J/Ψ w
- Likely excited charmonium state (hc’’’ or cc1’)
- JPC = 0-+,1++ ?
PRL 98, (2007)
XDD*

24. 4. New states above the DD-threshold
Z(3930)
- observed in the two-photon decays gg  Z(3930)  DD
- Predicted mass and width match charmonium assignment of cc2’
- JPC = 2++
Y(3940)
- discovered by Belle the decay BKY, Y (w J/Ψ)
- Possible cc1’ charmonium state but
requires further investigation
- not found in DD or DD* final states
- JPC = 1++, …
ZDD
YJ/y w
If X=Y, difficult to explain absence of Y  open charm => Hybrid?

25. 4. New states above the DD-threshold
Y(4260)
new peak in ISR events discovered at Babar, found in decay Y(4260)J/Ψp+p-
e+e- requires quantum numbers JPC = 1--
However, all of the 1-- charmonium states have already been discovered!
Very difficult to accommodate as cc, unless previous assignments are wrong
for Y(4260)J/Ψ p+p-, Belle reproduces BaBar’s signal:
Broad second peak at slightly lower mass:

26. 4. New states above the DD-threshold
Candidates for hybrids
Ψ(4415): IG(JPC) = 0-(1--)
Ψ(4160): IG(JPC) = 0-(1--)
Ψ(4040): IG(JPC) = 0-(1-- )

27. 5. Conclusion
Charmonium states and decay widths can be calculated quite well in NRQCD but in order to obtain a higher precision relativistic corrections have to be included
Determination of as(mc) from various rations of decay widths
Charmonium model with has great success below the DD-threshold
Above DD threshold, several states remain undiscovered or need further study
A recent flood of experimental results from the B-factories is challenging our understanding of the strong force:
- What is the nature of the new “Y” states?
Meson molecules? Tetraquarks? Hybrids? Glueballs? Something else?
Rich new spectroscopy?
What excited unknown states do exist? => waiting for data of (upgraded) B-factories like Babar, Belle, CLEO, BES
searching for resonances with non-quarkonium JPC (1-+, …)

28. Thanks for your attention!

29. References
„A modern introduction to Particle Physics“, chapter 8, Fayyazuddin Riazuddin, World Scientific
„Dynamics of the Standard Model“, chapter 13, J.F. Donoghue E. Golowich B.R. Holstein, Cambridge Monographs on particle physics, Nuclear physics and cosmology
Lecture notes Università di Pisa, Prof. V. Cavasinni, Particelle Elementari I, „modello a quark“, 2006/07
charmonium_1.pdf –
charmonium_2.pdf –
theory.gsi.de/~leupold/lecture-1-13_7_07.pdf
„Implications of light-quark admixtures on charmonium decays into meson pairs“, Phys. Rev. D, Vol. 62, , T. Feldmann, P. Kroll
www-rnc.lbl.gov/ISMD/talks/Aug9/1130_Fulsom.ppt
„Production of singlet P-wave cc and bb states“, Phys. Rev. D 66, (2002), S. Godfrey, J.L. Rosner
„Two-pion transitions in quarkonium revisited“. Phys. Rew. D 74, (2006), M.B. Voloshin

30. Determinating as(mc)
Extraction from partial decay widths ratios from J/Ψ:
Extraction from hc:
Extraction from cc,J:
Extraction from J/Ψ:
≈ 1.8 => large correction => caution with the value
for cc,2 and
for cc,0

31. New charmonium states
Further resonances observed in e+e-  YgISR (certainly JPC=1--)
Most of these 1-- states should preferentially decay into D(*)D(*) states.
Ψ(3770), Ψ(4040), Ψ(4415) [regular charmonia]
clearly visible, nothing else
Y(4350)y(2S)pp
y(4160)
y(4040) y’
4660
4350
4260
4008
Only seen in Ψ(2S)pp
y(3770)
4 Ψs to place!
J/y
Y(4260) can be fit by a tetraquark model (decaying into J/yf0 …) or a hybrid (with gpp)

32. Multipole expansion in QCD
Chromo-electric dipole transition:
Chromo-magnetic dipole transition:
For Y2 →Y1p+p-
where
so with effective Hamiltonian
S-wave
D-wave
mixing of 3D1 – 3S1 in 3S1 states
for 3S1-states
for 1S0-states
where
supressed by (v/c)²
F1, F2 : coordinate parts of S-wave functions