Twist-3 distribution amplitudes of scalar mesons from QCD sum rules

1. Twist-3 distribution amplitudes of scalar mesons from QCD sum rules Y.M Wang In collaboration with C.D Lu and H. Zou Institute of High Energy Physics, CAS

2. Outline • Introduction • Sum rules for the moments of twist-3 DAs of scalar mesons • Numerical calculations • Summary and outlook

3. Introduction • Structures of scalar mesons are still not clear. two-quark state, multi-quark state, meson-meson states, glueball… Two nonets above or below/near 1 GeV : 1. f_0(600), f_0(980), K_0^{*} (800), a_0(980) 2. f_0(1370), f_0(1500)/ f_0(1710), K_0^{*} (1430), a_0(1450) It has been suggested that the scalar mesons above 1 GeV can be identified as conventional two-quark states, even with some possible gluon component. The light scalar mesons below or near 1GeV are dominated by multi-quark states. F. Close and N.A. Tornqvist, J. Phys. G 28, 249 (2002)

4. Introduction • Study on production of scalar mesons in B meson decay can provide much useful information about their inner structure of scalar mesons. Wei Wang, Yue-Long Shen, Ying Li and Cai-Dian Lu, hep-ph/0609082 However, only twist-2 distribution amplitudes (DAs) of scalar mesons are available now. So it is necessary to calculate the moments of twist-3 DAs in order to give more accuracy predictions. • Twist-3 DAs of pion are very important for B decays to pion pion, so it is necessary to investigate the effects of twist-3 DAs of scalar mesons in B decays to scalar mesons.

5. Introduction The idea of the QCDSR formulism is to approach the bound state problem in QCD from the asymptotic freedom side, i.e., to start at short distances and move to larger distances where confinement effects become important, asymptotic freedom starts to break down and resonances emerge as a reflection of the fact that quarks and gluons are permanently confined within hadrons. The breakdown of asymptotic freedom is signalled by emergence of power corrections due to non-perturbative effects in the QCD vacuum. These are introduced via nonvanishing vacuum expectation values of quark and gluon condensates operators. L.J. Reinders, H.R. Rubinstein, S. Yazaki, Phys. Rep. 127, 1 (1985).

6. Sum rules for the moments of scalar mesons • Definition of twist-3 DAs for scalar mesons: The decay constants \bar{f}_s is defined as: • Here .

7. Sum rules for the moments of scalar mesons In general, the above two twist-3 DAs have the following form: • Setting y=-x=z and expanding the above definitions around z^2=0, we have

8. Sum rules for the moments of scalar mesons • In order to calculate the moments of distribution amplitudes, we consider the following two correlation functions • The correlation functions can be calculated by virtue of OPE in deep Euclidean region (-q^2>>0). The results can be written as

9. Sum rules for the moments of scalar mesons

10. Sum rules for the moments of scalar mesons

11. Sum rules for the moments of scalar mesons • The correlation functions can also be calculated in hadron level.

12. Sum rules for the moments of scalar mesons • We can match two different forms of correlation functions by dispersion relation, • In order to suppress the contribution from excited states and continuum states, we apply Borel transformations to both sides of above equations

13. Sum rules for the moments of scalar mesons • Then, we can find the sum rules of moments for twist-3 DAs of scalar mesons below.

14. Sum rules for the moments of scalar mesons

15. Sum rules for the moments of scalar mesons • RG evolution of decay constant, quark mass and condensate

16. Numerical calculations • Moments for twist-3 DAs of a_0 • Moments for twist-3 DAs of K^{*}_0 • Moments for twist-3 DAs of f_0

17. Moments for twist-3 DAs of a_0 • Moments for twist-3 DA \phi_{a_0}^{s} of a_0 • Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0

18. Moments for twist-3 DA \phi_{a_0}^{s} of a_0 • The numerical parameters at 1 GeV scale used in this paper are taken as

19. Moments for twist-3 DA \phi_{a_0}^{s} of a_0 • Here a_0 indicate that the scalar meson is composed of u \bar{d}. In order to obtain the value of its moments from Eq. (16), we should calculate the mass and decay constant for a_0 before. • The mass can be obtained by taking logarithm of both sides of Eq. (16), and then applying differential operator to both sides of Eq. (16), while the decay constant can be immediately calculated from Eq. (16) once the mass is known. • For the sum rules of mass, the threshold value and Borel parameter are taken as .

20. Moments for twist-3 DA \phi_{a_0}^{s} of a_0 The mass within Borel window can be plotted as below. the mass of a_0 within Borel window at s_0 = 4.5 GeV^2

21. Moments for twist-3 DA \phi_{a_0}^{s} of a_0 • Similarly, we can display the decay constant within Borel window [1.3,1.6] GeV^2 below. • From the above figure, we can find that the mass of a_0 is in the range of

22. Moments for twist-3 DA \phi_{a_0}^{s} of a_0 • The value of decay constant is within Borel window. • Making use of the above mass and decay constant, we can plot the first two moments of a_0 below. the second moment of a_0 (\xi^{s}_{2,a_0}) within Borel window

23. Moments for twist-3 DA \phi_{a_0}^{s} of a_0 the forth moment of a_0 (\xi^{s}_{2,a_0}) within Borel window The values of the first two moments for a_0 from sum rules (16) are [0.30,0.35] and [0.18,0.22] respectively.

24. Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0 • The mass and decay constant can also be calculated from (18). The results are • The first two moments for \phi_{a_0}^{\sigma} of a_0 within Borel window [1.2,1.5]GeV^2 and [1.1,1.4]GeV^2 are showed below. From the figures, we can obtain the number of these two moments: \xi_{\sigma}^{2,a_0} =[0.21,0.23], \xi_{\sigma}^{4,a_0} =[0.099,0.107] .

25. Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0 the second moment of a_0 (\xi^{\sigma}_{2,a_0}) within Borel window

26. Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0 the forth moment of a_0 (\xi^{\sigma}_{4,a_0}) within Borel window

27. Moments for twist-3 DAs of k^{*}_0 • Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_0 • Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0

28. Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_0 • Here k^{*}_0 indicates that the flavor content of scalar meson is s \bar{u}. • Following the same procedure as a_0, we can derive the mass and decay constant within Borel window [1.9,2.1]GeV^2 and [1.3, 1.7] GeV^2 from (16) as Here the threshold value is chosen as (5.4 \pm 0.3) GeV^2. • The first moment of \phi_{k^{*}_0}^{s} is not zero due to SU(3) symmetry breaking effect.

29. Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_0 • The first two moments for \phi_{k^{*}_0}^{s} within Borel window are showed below. the first moment of k^{*}_0 ( \xi^{s}_{1,k_0}) within Borel window

30. Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_0 second moment of k^{*}_0 ( \xi^{s}_{2, k_0}) within Borel window

31. Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_0 • From the above figures, we can find that the value of these two moments are in the range of [0.0017,0.0023] and [0.21,0.29] respectively. It is obvious that the first moment corresponding to SU(3) symmetry breaking effect is tiny.

32. Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 • The mass and decay constant of k^{*}_0 from (18) are in the range of [1449,1543] MeV and [350,376] MeV corresponding to Borel window [2.1,2.3] GeV^2 and [1.3,1.6]] GeV^2 respectively. • The first two moments for \phi_{k^{*}_0}^{\sigma} within Borel window [1.8,2.6] GeV^2 and [1.0,1.2] GeV^2 are showed below. the first moment of k^{*}_0 ( \xi^{\sigma}_{1,k_0}) within Borel window

33. Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 second moment of k^{*}_0 ( \xi^{\sigma}_{2,k_0}) within Borel window • The number of these two moments are in the range of [0.02,0.036] and [0.13,0.17] .

34. Moments for twist-3 DAs of f_0 • Moments for twist-3 DA \phi_{f_0}^{s} of f_0 • Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_0

35. Moments for twist-3 DA \phi_{f_0}^{s} of f_0 • Here f_0 refers to the scalar meson which is constitute of s \bar{s} quarks. • The mass and decay constant of f_0 from sum rules (16) are [1629,1713] MeV and [371,393] MeV corresponding to threshold value s_0=(6.5 \pm 0.3) GeV^2 with Borel window [2.5, 2.7]GeV^2 and [1.7, 2.0]GeV^2. • The second moment of f_0 (\xi^{s}_{2,f_0}) within Borel window [1.3, 1.6]GeV^2 is showed below. The value of this moment is located at the range of [0.18,0.26]. • The forth moment of f_0 (\xi^{s}_{4,f_0}) could not be obtained due to unstable platform.

36. Moments for twist-3 DA \phi_{f_0}^{s} of f_0 the second moment of f_0 ( \xi^{s}_{2,f_0}) within Borel window

37. Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_0 • The mass and decay constant of f_0 from sum rules (18) are [1616,1703] MeV and [380,427] MeV within Borel window [2.5, 2.7]GeV^2 and [1.2, 1.6]GeV^2. • The second moment of f_0 (\xi^{\sigma}_{2,f_0}) within Borel window [1.5, 1.8]GeV^2 is showed below. the second moment of f_0 ( \xi^{\sigma}_{2,f_0}) within Borel window

38. Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_0 • The value of this moment is located at the range of [0.13,0.18]. • The forth moment of f_0 (\xi^{\sigma}_{4,f_0}) could not be obtained due to unstable platform.

39. Summary and Outlook • In this work, we have calculated the moments of twist-3 DAs for scalar mesons. For convenience, we collect the results of mass, decay constant and Gegenbauer moments at 1 GeV scale as follows.

40. Summary and Outlook • Our results indicates that the Gegenbauer moments of twist-3 DAs for scalar mesons are small. • The twist-3 DAs of scalar mesons can be applied to various approaches, such as PQCD, QCDF, LCSR, to explore the inner structure of scalar mesons in the exclusive process of scalar meson production in heavy flavor hadron decay. • We can also investigate the DAs of glueball component for scalar mesons to help us discover the mystery of scalar meson.

41. Thanks!