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 \barf_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-xz and expanding the above
  definitions around z20, 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
  (-q2gtgt0). 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_0s of a_0
• Moments for twist-3 DA \phi_a_0\sigma of
  a_0

18
Moments for twist-3 DA \phi_a_0s 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_0s of a_0

• Here a_0 indicate that the scalar meson is
  composed of u \bard. 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_0s 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 GeV2
21
Moments for twist-3 DA \phi_a_0s of a_0

• From the above figure, we can find that the mass
  of a_0 is in the range of

• Similarly, we can display the decay constant
  within Borel window 1.3,1.6 GeV2 below.

22
Moments for twist-3 DA \phi_a_0s 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 (\xis_2,a_0) within
Borel window
23
Moments for twist-3 DA \phi_a_0s of a_0

the forth moment of a_0 (\xis_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 1.2,1.5GeV2 and
  1.1,1.4GeV2 are showed below. From the
  figures, we can obtain the number of these two
  moments
• \xi_\sigma2,a_0
  0.21,0.23,
• \xi_\sigma4,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_0s of
  k_0
• Moments for twist-3 DA \phi_k_0\sigma of
  k_0

28
Moments for twist-3 DA \phi_k_0s of
k_0

• Here k_0 indicates that the flavor content of
  scalar meson is s \baru.
• Following the same procedure as a_0, we can
  derive the mass and decay constant within Borel
  window 1.9,2.1GeV2 and 1.3, 1.7 GeV2 from
  (16) as

• Here the threshold value is chosen as (5.4
  \pm 0.3) GeV2.
• The first moment of \phi_k_0s is not
  zero due to SU(3) symmetry breaking effect.

29
Moments for twist-3 DA \phi_k_0s of
k_0

• The first two moments for \phi_k_0s
  within Borel window are showed below.

the first moment of k_0 (
\xis_1,k_0) within Borel window
30
Moments for twist-3 DA \phi_k_0s of
k_0
second moment of k_0 ( \xis_2, k_0)
within Borel window
31
Moments for twist-3 DA \phi_k_0s 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 GeV2
  and 1.3,1.6 GeV2 respectively.
• The first two moments for \phi_k_0\sigma
  within Borel window 1.8,2.6 GeV2 and
  1.0,1.2 GeV2 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_0s of f_0
• Moments for twist-3 DA \phi_f_0\sigma of
  f_0

35
Moments for twist-3 DA \phi_f_0s of f_0

• Here f_0 refers to the scalar meson which is
  constitute of s \bars 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) GeV2 with Borel window 2.5, 2.7GeV2 and
  1.7, 2.0GeV2.
• The second moment of f_0 (\xis_2,f_0) within
  Borel window 1.3, 1.6GeV2 is showed below. The
  value of this moment is located at the range of
  0.18,0.26.
• The forth moment of f_0 (\xis_4,f_0) could
  not be obtained due to unstable platform.

36
Moments for twist-3 DA \phi_f_0s of f_0
the second moment of f_0 ( \xis_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, 2.7GeV2 and 1.2,
  1.6GeV2.
• The second moment of f_0 (\xi\sigma_2,f_0)
  within Borel window 1.5, 1.8GeV2 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!