1. Some Issues in Charmonium Physics Some Issues in Charmonium Physics K-T Chao Peking University

2. 1. Puzzles in Double Charm Production in e + e  Annihilation – Inclusive J/  cc{\bar} production – Exclusive J/   C (  C0,  C (2S),…) production 2. D-wave Charmonium production – in e + e  Annihilation – in B meson decay – S-D Mixing 3. Search for h C in B exclusive decays and infrared divergences

3. Puzzles in Double Charm Production in e + e  Annihilation INCLUSIVE PRODUCTION : e + e   J/  cc Theory: via ONE virtual photon  Cho-Leibovich (1996) Yuan-Qiao-Chao (1997) Baek-Ko-Lee-Song (1998)  pQCD predicts: cross section at  s  10.6 GeV  0.10-0.15pb  Belle data  0.9pb, PRL89(2002)142001 larger than theory by almost one order of magnitude. Higher order corrections expected not large enough.

4. Puzzles in Double Charm Production in e + e  Annihilation EXCLUSIVE PROCESS e + e   J/   C (  C0,  C (2S),…) Theory: via ONE virtual photon (Braaten-Lee (2003) PRD67, 054007) (Liu-He-Chao (2003) PLB557, 45) (Hagiwara-Kou-Qiao (2003) PLB570, 39) pQCD prediction smaller again by an order of magnitude than Belle cross section  0.033 pb for e + e   J/  C (decaying to  4 charged) (PRL89, 142001)

5. Puzzles in Double Charm Production in e + e  Annihilation  Theory: via TWO photons  Enhanced by photon fragmentation (small photon virtuality 4m c 2  s )  Suppressed by QED over QCD couplings  Exclusive J/  +J/  enhanced (Bodwin-Braaten-Lee, PRL90, 162001), the same order as for J/  +  C (but ruled out later by data)  Inclusive J/  cc\bar via two photons prevail over via one photon when  s  20GeV ( Liu-He-Chao, PRD68, R031501 )

6. Puzzles in Double Charm Production in e + e  Annihilation  Annihilation into TWO photons can NOT solve problems for both inclusive and exclusive double charm production  Both data larger than pQCD predictions by about an order of magnitude  Color octet contributions are negligible  pQCD factorization fails(?)  C=+ glueballs misidentified as  C (?) (Brodsky et al.) Search for C=+ glueballs near 3 GeV from  (2S) decay @ CLEOc & BESIII

7. D-wave Charmonium production in e + e  Annihilation and B decay  New finding by Belle: D-wave charmonium is observed in B decay for the first time (hep-ex/0307061)  B +   (3770)K +, BR = (0.48 ±0.11± 0.12) x 10 -3, very large, even comparable to B +   (2S)K +, BR=(0.66±0.06) x 10 -3  If this implies large 2S-1D mixing?  S-D mixing vs. Color-Octet mechanism in D-wave charmonium production in B meson decay and in e + e  Annihilation

8. S-D mixing between  ’ =  (2S) &  ’’ =  (3770) If ignoring D-wave contribution to leptonic widths  mixing angle   ± 19º

9. Detailed calculations (including tensor force and coupled channel effects) indicating  absolutely value smaller than 10 º (Eichten et al, Kuang-Yan, Moxhay-Rosner, … ) Including D-wave contribution to leptonic widths    – 10 º or   +30 º

10.   +30 º disfavored because it would give E1 transition width 5 times larger than the observed value of  (2S)   co  (Ding-Qin-Chao, PRD44(1991) Measurement of  (3770)   cJ  at CLEOc & BESIII will be another helpful check for the S-D mixing   +30 º disfavored because it would give E1 transition width 5 times larger than the observed value of  (2S)   co  (Ding-Qin-Chao, PRD44(1991) Measurement of  (3770)   cJ  at CLEOc & BESIII will be another helpful check for the S-D mixing

11. tan 2  =0.11, if   ± 19º tan 2  =0.03, if   -- 10º Small S-D mixing can hardly explain the Belle data if only the Color-Singlet (CS) S-wave component contributes (via CS V- A currents) B +   (3770)K +, BR = (0.48 ±0.11± 0.12) x 10 -3, B +   (2S)K +, BR=(0.66±0.06) x 10 -3

12. D-wave heavy quarkonium production may be a crucial test of NRQCD color- octet mechanism. In certain processes (e.g. gluon fragmentation, B meson decay, … ) the D-wave charmonium signal could be as strong as  (2S) (Qiao-Yuan-Chao, PRD55(1997)4001) (Yuan-Qiao-Chao, PRD56(1997)329)

13. Color-Octet (CO) mechanism may play important role for D-wave charmonium production in B decay due to large Wilson coefficient for CO effective V-A Hamiltonian and the NRQCD Fock state Expansion. CO coefficient >> CS coefficient

14. The inclusive decay branching ratio was predicted BR(B   (3770)X)=0.28% (Yuan-Qiao-Chao,1997), [c.f. BR(B +   (2S)X)=(0.35±0.05)%] [see also Ko-Lee-Song(1997)]

15. NRQCD velocity scaling rules (with some uncertainties)

16. D-wave charmonium production in e + e  Annihilation Color-Octet insignificant for double charm Color-singlet contributes 2-4 fb to e + e    (3770)cc\bar (Hao-Liu-Chao, PLB546(2002)216)  Color-octet suppressed by color factor of 3/32, no significant contribution  S-D mixing will much help, since the observed rate of e + e    (1S)cc\bar  0.9pb,  (2S)cc\bar expected to be about a half of  (1S)cc\bar.

17. D-wave Charmonium production in e + e  Annihilation and B meson decay  Observed large rate of B +   (3770)K + could be a strong support to either the Color-Octet mechanism or the large S-D mixing.  e + e    (3770)cc\bar could be another test of S-D mixing (no Color-Octet contamination).   (3770)   cJ  at CLEOc & BESIII will be another helpful check for the S-D mixing.

18. Infrared Divergences in B   CJ K and B  h C K Decays in QCD Factorization (Song, Chao, Phys.Lett. B568 (2003)127) (Song, Meng, Gao, Chao, hep-ph/0309105) BBNS (Beneke et al.) QCD factorization: Good for B  pi pi, B  D pi. OK (infrared safe) for B  J/  K (Chay-Kim, Cheng-Yang) B   C K (Song-Meng-Chao) ; Color transparency, small cc-bar size, viewed as color dipole. Infrared divergences for B   CJ K and B  h C K Decays in QCD factorization and NRQCD

36. Z= M 2 /M B 2  4m c 2 /m b 2 is the gluon mass for infrared regularization

43. If using the infrared divergence term to estimate the decay widths as in the case of hadronic widths:  (h c  ggg)=5/6  (  c1  qqg)

45. Very large branching ratio obtained for B  h c K obtained! But new method based on NRQCD still expected to remove infrared divergences!