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Theoretical description of the charmonium structure in QCD

This article provides a theoretical description of the structure of charmonium in Quantum Chromodynamics (QCD). It covers topics such as charmonium spectroscopy, transitions and decays of cc, and the discovery of new states above the DD-threshold.

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Theoretical description of the charmonium structure in QCD

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  1. Theoretical description of the charmonium structure in QCD Gabi Hoffmeister 06.12.2007

  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. 1. Introduction Charmonium production • 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 J/Y, Y(2S) hc, cc,... 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/Ywith 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´ (excited3S1 state) with mΨ´ = 3.686 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 models Y´ ηc‘ hc 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 www.e18.physik.tu-muenchen.de/teaching/struktur-dynamik-hadronen/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) 0 (3P1) -2 (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 0.66 (hb(1P)) 0.70 (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 ~ 10-4 - 10-3, supressed by factor ~10-2 - 10-1 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): M1transitions (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 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) ≈ 0.44as(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,Jg and cc,J → J/Ψg (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 = -1forΨ´→ g cc,J and x = +1 for cc,J → J/Ψ kc:quark anomalous magnetic moment (deviation from Dirac magnetic momentmc = 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/Ψw 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) → gcc,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´→ ghc´/ hc hc´ → g hc Ψ´→ g J/Ψ hc→ ghc Phys. Rev. D 66, 014012 (2002)

  20. 3. Transitions and decays of cc Ψ(2S) Decay to ghc(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 → ≠ 0 averaged BR = (3.00.5)·10-3 => G[Ψ(2S) → ghc(1S)] = (1.00 0.16) keV Decay to ghc(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 thereforeG[Ψ(2S) → ghc´(2S)] = (87 25) eV Hadronic transitions to J/Ψ: • Via electric dipole emission of gluon pair followed by its hadronization intopp dominating decay mode in pions

  21. 3. Transitions and decays of cc Ψ(2S) → p0hc→ p0ghc 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/Ψp+p- at Belle in 2003 with narrow widthG < 2.3MeV and mass mX = 3871.2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*0molecule 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/ΨX(3940) - Decays to DD* but not DD and J/Ψw - Likely excited charmonium state (hc’’’ or cc1’) - JPC = 0-+,1++ ? PRL 98, 082001 (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 • www.e18.physik.tu-muenchen.de/teaching/struktur-dynamik-hadronen/charmonium_1.pdf – • www.e18.physik.tu-muenchen.de/teaching/struktur-dynamik-hadronen/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, 074006, T. Feldmann, P. Kroll • http://uk.arxiv.org/PS_cache/arxiv/pdf/0711/0711.1927v2.pdf • http://arxiv.org/abs/hep-ph/0701208 • www-rnc.lbl.gov/ISMD/talks/Aug9/1130_Fulsom.ppt • „Production of singlet P-wave cc and bb states“, Phys. Rev. D 66, 014012 (2002), S. Godfrey, J.L. Rosner • „Two-pion transitions in quarkonium revisited“. Phys. Rew. D 74, 05022 (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 partsof S-wave functions

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