1. Scalar Meson σ(600) in the QCD Sum Rule
轻标量介子的求和规则研究
Hua-Xing Chen, Atsushi Hosaka, Hiroshi Toki, Shi-Lin Zhu
Department of Physics and State Key Laboratory of
Nuclear Physics and Technology, Peking University
RCNP, Osaka University
April 18, 2010 南昌

2. Contents Motivations QCD sum rule Tetraquark currents
Light scalar mesons
Two-pion continuum
Summary

3. 1. Motivation Conventional mesons and baryons
QCD allows much richer hadron spectrum
Exotic hadrons:
glueballs ,
multiquark states ,
hybrids
molecular states

4. Quark Model

5. 1. Motivation Exotic in quantum numbers:
mesons : JPC=0--,0+-,1-+,2+-, etc.
baryons : S=+1 & B=1, I=5/2, etc.
Candidates: π1(1400), π1(1600) and π1(2000) with IGJPC =1-1-+
pentaquark (NAKANO 03)
Exotic in structure:
hadron molecule
tetraquark, pentaquark
Λ(1405), X(3872)
(JAFFE 77)

6. 1. Motivation Tetraquark is complicated:
For each state, there may exist more than one currents. Previous QCD sum rule calculations usually used just one or two currents, which are not complete.
We try to do a systematical study on tetraquark currents.
SU(3)F
Flavor structure

7. 2. QCD Sum Rule QCD non-perturbative dynamics (Low energy side)
Standard perturbation theory does not simply apply.
Therefore, we need some special methods.
Phenomenological models
Lattice QCD
QCD sum rule

8. 2. QCD Sum Rule
In sum rule analyses, we consider two-point correlation functions:
where η is the current which can couple to hadronic states.
By using the dispersion relation, we can obtain the spectral density
In QCD sum rule, we can calculate these matrix elements from QCD (OPE) and relate them to observables by using dispersion relation.

9. 2. QCD Sum Rule current with JPC=0++ (it couples to κ)
where u, d and s are up, down and strange quarks, and contain color indices a, b.
Spectral density:
ρ(s) is in powers of s, and its convergence is important.

10. Quark and Gluon Level Hadron Level SVZ sum rule (Shifman 1979)
(Convergence of OPE)
dispersion relation
s = -q2
Quark-Hadron Duality
Hadron Level
(Positivity) ρ
(for meson case)
(Sufficient amount of Pole contribution) M s0 s

11. 2. QCD Sum Rule
Borel transformation to suppress the higher order terms:
Two Parameters
MB , s0
We need to choose certain region of (MB, s0).
Criteria
1. Stability
2. Convergence of OPE
3. Positivity of spectral density
4. Sufficient amount of pole contribution

12. 2. QCD Sum Rule
Ideally, if we could calculate the OPE exactly, we would be able to extract all information of hadron n.
In practice, calculation is approximate. We find that we need to choose a good current.

13. 3. Tetraquark Currents Meson Currents Scalar JP=0+ Vector JP=1-
Tensor JP=1+-
Axial-vector JP=1+
Pseudo-scalar JP=0-
A, B are the flavor indices; a is the color index. By adding δAB and λAB, we can obtain singlet and octet, respectively.

14. Diquark Currents

15. Tetraquark Currents
SU(3)F
Flavor structure 1 2 3
Ideal mixing 1
Back

16. 3. Tetraquark Currents
We find that there are five independent currents for σ(600) JPC=0++ states:

17. Fierz Transformations

18. 4. Light scalar mesons (JPC=0++)
Weight Diagrams
Mass
Tetraquark Scheme

19. 4. Scalar mesons σ(600) (JPC=0++)
The search for good Borel window (MB,s0) is done by using all five independent diquark-antidiquark currents and their linear combinations out of two
QCD sum rule works for some currents, while it does not work for some other currents. We find the following mixed currents leads to a reasonable result

20. Single current V3κ Mixed current ηκ Spectral densities OPE convergence
About 40%
No definition (sometimes negative)
Pole Contribution

21. 5. Two-Pion Continuum
In this work, we study the linear combination of five independent currents
where ti and θi are ten mixed parameters.
We can not determine them in advance. Therefore, we will choose them randomly, and the results of QCD sum rule would tell which values are better.

23. 5. Two-Pion Continuum
We have tested more than fifty sets. Some of them lead to negative spectral densities. These results are non-physical.
There are fifteen sets which lead to positive spectral densities. We use them to perform the QCD sum rule analysis.

24. 5. Two-Pion Continuum
Some of them lead to mass curves where the Borel mass and s0 dependence are both very weak.

25. 5. Two-Pion Continuum
While some of them lead to mass curves where the s0 dependence are not very weak.

26. 5. Two-Pion Continuum The spectral density of the two-π continuum is
The two-π continuum is probably the largest contribute to this threshold value dependence.
The spectral density of the two-π continuum is
After doing some try and error tests, we find that the following function leads to a reasonable QCD sum rule result

27. 5. Two-Pion Continuum
After adding the contribution of two-π continuum, the mass curves again have weak Borel mass and s0 dependence.

28. 5. Two-Pion Continuum
By fixing and , we find that the good cases lead to a pole contribution around 40% - 50%.
This is relatively larger than the bad cases.

29. 6. Summary
We studied light scalar mesons with JPC =0++. Their masses are in the region of MeV, with the ordering
while the conventional currents lead to a mass around 1.2 GeV, which is considerably heavier. So the light scalar mesons are interpreted as tetraquark states in our QCD sum rule analysis.
We use a general tetraquark currents which are linear combinations of five independent currents. We find some cases where the Borel mass and threshold value dependence are both weak. The mass is around
The continuum contribution exists in the back ground of the σ(600) meson. We introduce a function of the two-π continuum to describe it, and again obtain a weak threshold value dependence.

30. 谢谢!

31. Light scalar mesons (JPC=0++)M s0 s M s0 s
S. H. Lee, Phys.Rev.D69:074016,2004
S. L. Zhu, hep-ph/

32. Light scalar mesons (JPC=0++)