1. Exotic States in QCD Sum Rules
乔从丰
中国科学院大学
@2015 BESIII Charm Physics Workshop

2. Outline Ds Molecular States Background & Motivation
Ds Molecular States via QCD Sum Rules
Summary of Ds Molecular States
Oddballs
Background & Motivation
Oddballs in QCD Sum Rules
Summary of Oddballs

3. Part I: Ds Molecular States
C.-F. Qiao and L. Tang,   EPL107, (2014).

4. Background & Motivation
Zc(3900) M(+J/)
Zc(4020) M(+hc)
PRL,111, (2013).
PRL, 110, (2013).
Zc(3885) M(DD*)
Zc(4025) M(D*D*)
PRL,112,022001(2014).
PRL,112,132001(2014).

5. Background & Motivation
Zc0(4020) M(0hc)
Zc0(3900) M(0J/)
PRL,115,112003(2015).
PRL,113,212002(2014).
Zc0(3885) M(DD*)
Zc0(4025) M(D*D*)
PRL,115,222002(2015).
PRL,115,182002(2015).

6. Background & Motivation
Summary of the observed Zc states at BESIII
P. Liu [BESIII Collaboration], arXiv:
Zc+
Zc0
Zc0

7. Background & Motivation
Possible quark configurations
Molecualr states
Tetraquark states
Q.Wang, C.Hanhart, Q.Zhao, PRL 111, (2013);
H.W.Ke, Z.T.Wei and X.Q.Li, EPJC 73,2561 (2013);
J.He, X.Liu, Z.F.Sun and S.L.Zhu,  EPJC 73,2635 (2013);
And so on…
L.Maiani, et al.,PRD87, (2013);
C.F.Qiao and L.Tang,  EPJC74, 3122(2014), EPJC74, 2810(2014);
L.Zhao, W.Z.Deng and S.L.Zhu,   PRD 90, (2014);
And so on…

8. Background & Motivation
Interpretation of molecular states via QCDSR
Zc(3900) was successfully interpreted as a 𝐷 𝐷 ∗ molecular state.
J.R.Zhang, PRD 87, (2013); C.Y.Cui, Y.L.Liu,
W.B.Chen and M.Q.Huang,  JPG41, (2014).
Zc(4025) was successfully interpreted as a 𝐷 ∗ 𝐷 ∗ molecular state.
C.Y.Cui, Y.L.Liu and M.Q.Huang,  EPJC 73, 2661 (2013);
W.Chen, T.G.Steele, M.L.Du and S.L.Zhu,  EPJC 74, 2773 (2014).
It is reasonable that there exist Ds molecular states.
Our motivation is to explore these states via QCDSR.

9. Ds Molecular States via QCDSR
The calculations of the QCD Sum Rules are based on the correlator constructed by two hadronic currents:
As 𝐽 𝜇 (𝑥) is not a conserved current, the two-point correlation function has two independent Lorentz structures:
Corresponding to the axial-vector state

10. Ds Molecular States via QCDSR
Phenomenological side
𝜆 𝐻 is defined by 0 𝑗 𝜇 𝑍 𝑐𝑠 = 𝜆 𝐻 𝜖 𝜇 , where 𝑍 𝑐𝑠 is the lowset lying 1 +− 𝐷 𝑠 𝐷 𝑠 ∗ or 𝐷 𝑠 ∗ 𝐷 𝑠 ∗ molecular state.
Quark-gluon side
where

11. Ds Molecular States via QCDSR
The mass of the Ds molecular state is expressed as
where,

12. Ds Molecular States via QCDSR
Input parameters

13. Ds Molecular States via QCDSR
The interpolating currents of 𝐷 𝑠 𝐷 𝑠 ∗ & 𝐷 𝑠 ∗ 𝐷 𝑠 ∗ are constructed as
The interpolating currents of 𝐷 𝑠 𝐷 ∗ & 𝐷 𝑠 ∗ 𝐷 ∗ are constructed as

14. Ds Molecular States via QCDSR
OPE convergence in the region 0.30 ≤𝜏≤ 0.65 𝐺𝑒 𝑉 −2 for the 𝐷 𝑠 𝐷 𝑠 ∗ molecular state with √ 𝑠 0 = 4.6 GeV, where 𝜏= 𝑀 𝐵 −2 .
The relative pole contribution of 𝐷 𝑠 𝐷 𝑠 ∗ state with √ 𝑠 0 = 4.6 GeV.

15. Ds Molecular States via QCDSR
The mass of the 𝐷 𝑠 𝐷 𝑠 ∗ state as a function of the sum rule parameter 𝜏, for different values of √ 𝑠 0 .

16. Ds Molecular States via QCDSR
OPE convergence in the region 0.25 ≤𝜏≤ 0.50 𝐺𝑒 𝑉 −2 for the 𝐷 𝑠 ∗ 𝐷 𝑠 ∗ molecular state with √ 𝑠 0 = 4.9 GeV, where 𝜏= 𝑀 𝐵 −2 .
The relative pole contribution of 𝐷 𝑠 ∗ 𝐷 𝑠 ∗ state with √ 𝑠 0 = 4.9 GeV.

17. Ds Molecular States via QCDSR
The mass of the 𝐷 𝑠 ∗ 𝐷 𝑠 ∗ state as a function of the sum rule paramter 𝜏, for different values of √ 𝑠 0 .

18. Ds Molecular States via QCDSR
The mass of the 𝐷 𝑠 ∗ 𝐷 ∗ state as a function of the sum rule paramter 𝜏, for different values of √ 𝑠 0 .

19. Ds Molecular States via QCDSR
The mass of the 𝐷 𝑠 ∗ 𝐷 ∗ state as a function of the sum rule paramter 𝜏, for different values of √ 𝑠 0 .

20. Ds Molecular States via QCDSR
Numerical Results of Ds Molecular States
State
Lower MB2
(GeV2)
Upper MB2
𝑠 0
(GeV)
Mass
DsD*s
1.80
3.10
4.60±0.10
3.98±0.15
D*sD*s
2.50
3.60
4.90±0.10
4.38±0.16
DsD*
1.70
2.90
4.50±0.10
3.93±0.15
D*sD*
2.40
3.30
4.80±0.10
4.33±0.15

21. Ds Molecular States via QCDSR
Comparison with their Molecular Thresholds
The central value of the 𝐷 𝑠 𝐷 𝑠 ∗ < 𝐸 𝑡ℎ𝑟𝑒 [ 𝐷 𝑠 𝐷 𝑠 ∗ ] (4.08 GeV).
Ds and Ds* form a bound state.
The center value of the 𝐷 𝑠 ∗ 𝐷 𝑠 ∗ > 𝐸 𝑡ℎ𝑟𝑒 [ 𝐷 𝑠 ∗ 𝐷 𝑠 ∗ ] (4.22 GeV).
𝐷 𝑠 ∗ and 𝐷 𝑠 ∗ form a resonance.
The central value of the 𝐷 𝑠 𝐷 ∗ < 𝐸 𝑡ℎ𝑟𝑒 [ 𝐷 𝑠 𝐷 ∗ ] (3.98 GeV).
𝐷 𝑠 and 𝐷 ∗ form a bound state.
The center value of the 𝐷 𝑠 ∗ 𝐷 ∗ > 𝐸 𝑡ℎ𝑟𝑒 [ 𝐷 𝑠 ∗ 𝐷 ∗ ] (4.12 GeV).
𝐷 𝑠 ∗ and 𝐷 ∗ form a resonance.

22. Ds Molecular States via QCDSR
Decay Modes of Hidden Charm and Hidden Strange Molecular States
Possible decay modes for 𝐷 𝑠 𝐷 𝑠 ∗ bound state:
Possible decay modes for 𝐷 𝑠 ∗ 𝐷 𝑠 ∗ resonance:

23. Ds Molecular States via QCDSR
Decay Modes of Hidden Charm and Open Strange Molecular States
Possible decay modes for 𝐷 𝑠 𝐷 ∗ bound state:
Possible decay modes for 𝐷 𝑠 ∗ 𝐷 ∗ resonance:

24. Summary of Ds Molecular States
Four Ds molecular states, 𝐷 𝑠 𝐷 𝑠 ∗ , 𝐷 𝑠 ∗ 𝐷 𝑠 ∗ , 𝐷 𝑠 𝐷 ∗ , 𝐷 𝑠 𝐷 ∗ , are predicted via QCDSR.
The possible decay modes to ascertain these Ds molecular states are suggested.
The BESIII and forthcoming BelleII will facilitate searching for such Ds molecular states.

25. Part II: Oddballs C.-F. Qiao and L. Tang, PRL 113, 221601 (2014) .
L. Tang and C.-F. Qiao, arXiv: , to appear in NPB.

26. Background & Motivation
Color structure
Quark= fundamental representation 3
Gluon= Adjoint representation 8
Observable particles=color singlet 1
Mesons
Baryons
Glueballs
Glueballs are allowed by QCD.
No definite observations in the experiment until now.
lack knowledge of their production & decay properties
mixing with quark states adds difficulty to isolate them.

27. Background & Motivation
Good evidence exists for a scalar glueball ( 0 ++ ) which is mixed with nearby mesons, but a full understanding is still missing.
Evidence for tensor (2 ++ ) and pseudoscalar ( 0 − + ) glueballs are weak.
The study of the oddballs in experiments is still lacking.
Therefore, the glueball has not yet been definitely observed by experiments.
V. Crede and C. A. Meyer, Prog.Part.Nucl.Phys. 63 (2009) , and ref. therein.

28. Background & Motivation
Theoretical Approaches
Lattice QCD
Flux tube model
Constituent Models
MIT bag model
Coulomb gauge model
QCD Sum Rules (QCDSR)
V.Mathieu, N.Kochelev&V.Vento, Int.J.Mod.Phys. E18,1(2009)

29. Background & Motivation
Oddballs
Oddballs: glueballs with exotic quantum numbers
Physics at BESIII, Editors Kuang-Ta Chao & Yifang Wang,
Int. JMPA24,1,(2009).
Trigluon glueballs
V.Mathieu, N.Kochelev&V.Vento, Int.J.Mod.Phys. E18,1(2009).
Trigluon glueballs: 𝐶=+ 𝑜𝑟 𝐶=−;
Two-gluon glueballs: 𝐶=+.
Proofs can be found in 《粒子物理导论》written by 杜东生 杨茂志
Exotic quantum numbers Do not mix with

30. Oddballs in QCDSR Interpolating currents of 0 −− oddballs
Constraints: quantum number, gauge invariance, Lorentz invariance and SUc(3) symmetry
where 𝑔 𝛼𝛽 𝑡 = 𝑔 𝛼𝛽 − 𝜕 𝛼 𝜕 𝛽 𝜕 2 , 𝐺 𝜇𝜈 𝑎 = 1 2 𝜖 𝜇𝜈𝜅𝜏 𝐺 𝜅𝜏 𝑎

31. Oddballs in QCDSR Interpolating currents of 0 +− oddballs

32. Oddballs in QCDSR Interpolating currents of 2 +− oddballs
The two-point correlation function

33. Oddballs in QCDSR Typical Feynman diagrams of trigluon glueballs Fig-1

34. Oddballs in QCDSR
The QCD side of the correlation function
The phenomenological side of the correlation function
The dispersion relation

35. Oddballs in QCDSR The main function The Borel transformation
The quark-hadron duality approximation

36. Oddballs in QCDSR The moments The mass function
Ratio to constrain 𝜏 & 𝑠 0 by the pole contribution (PC)

37. Oddballs in QCDSR Ratio to constrain 𝜏 & 𝑠 0 by convergence of the OPE
Input parameters

38. Oddballs in QCDSR 0 −− -Current A 0 −− -Current B 0 −− -Current A

39. Oddballs in QCDSR 0 −− -Current D 0 −− -Current C 0 −− -Current C

40. Oddballs in QCDSR 0 +− -Current A 2 +− -Current B 2 +− -Current B

41. Oddballs in QCDSR 2 +− -Current C 2 +− -Current D 2 +− -Current C

42. Oddballs in QCDSR Comparison with other methods, in unit of GeV.
[1] N.Isgur and J.E.Paton, PRD 31, 2910 (1985).
[2] E.Gregory, et al., JHEP1210, 170 (2012).
[3] C.J.Morningstar and M.J.Peardon,PRD60, (1999).
[4] Y.Chen et al.,PRD73, (2006).
[5] K.Ishikawa, et al., PLB120, 387 (1983).
[6] L. Bellantuono, et al., JHEP 1510 (2015) 137.
[7] Y.-D Chen and Mei Huang, arXiv:

43. Oddballs in QCDSR
Proposed production channels of 0 −− oddball (Taking the lighter 0 −− as an example)
Proposed decay channels of 0 −− oddball

44. Oddballs in QCDSR
Proposed production channels of 0 +− and 2 +− oddballs
Proposed decay channels of 0 +− and 2 +− oddballs

45. Summary of Oddballs Four oddballs are predicted via QCDSR.
The possible production & decay modes to ascertain these oddballs are suggested.
These decay modes are expected to be measured in BESIII, BELLEII, Super-B, PANDA, and LHCb experiments.

46. 谢谢大家！