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,

Twist-3 distribution Twist-3 distribution amplitudes of scalar mesons amplitudes of scalar mesons

from QCD sum rulesfrom QCD sum rules

Y.M Wang In collaboration with C.D Lu and H. Zou

Institute of High Energy Physics, CAS

OutlineOutline

Introduction Sum rules for the moments of twist-3

DAs of scalar mesons Numerical calculations Summary and outlook

IntroductionIntroduction

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)

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.

IntroductionIntroduction

IntroductionIntroduction The idea of the QCDSR formulism is to approach the bound sta

te 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).

Sum rules for the moments of scalar Sum rules for the moments of scalar mesonsmesons

Definition of twist-3 DAs for scalar mesons:

The decay constants \bar{f}_s is defined as:

Here .


Setting y=-x=z and expanding the above definitions around z^2=0, we have

In general, the above two twist-3 DAs have the following form:

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



The correlation functions can also be calculated in hadron level.


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

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



RG evolution of decay constant, quark mass and condensate


Numerical calculationsNumerical 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

Moments for twist-3 DAs of a_0Moments 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

The numerical parameters at 1 GeV scale used in this paper are taken as

Moments for twist-3 DA \phi_{a_0}^{s} of Moments for twist-3 DA \phi_{a_0}^{s} of a_0 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 .

the mass of a_0 within Borel window at s_0 = 4.5 GeV^2the mass of a_0 within Borel window at s_0 = 4.5 GeV^2


The mass within Borel window can be plotted as below.

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] GeV^2 below.


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


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.


Moments for twist-3 DA \phi_{a_0}^{\Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0sigma} 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] .


the second moment of a_0 (\xi^{\sigma}_{2,a_0}) within Borel window


the forth moment of a_0 (\xi^{\sigma}_{4,a_0}) within Borel window

Moments for twist-3 DAs of k^{*}_0Moments 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

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

Moments for twist-3 DA \phi_{k^{*}_0}^{s} Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_0of k^{*}_0

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.

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


second moment of k^{*}_0 ( \xi^{s}_{2, k_0}) within Borel window

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.


Moments for twist-3 DA \phi_{k^{*}_0}^{\Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of 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

Moments for twist-3 DA \phi_{k^{*}_0}^{\Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 sigma} of k^{*}_0

The number of these two moments are in the range of

[0.02,0.036] and [0.13,0.17] .

second moment of k^{*}_0 ( \xi^{\sigma}_{2,k_0}) within Borel window

Moments for twist-3 DAs of f_0Moments 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

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.

Moments for twist-3 DA \phi_{f_0}^{s} of Moments for twist-3 DA \phi_{f_0}^{s} of f_0f_0

Moments for twist-3 DA \phi_{f_0}^{s} of Moments for twist-3 DA \phi_{f_0}^{s} of f_0f_0

the second moment of f_0 ( \xi^{s}_{2,f_0}) within Borel window

Moments for twist-3 DA \phi_{f_0}^{\Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_0sigma} 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

Moments for twist-3 DA \phi_{f_0}^{\Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_0sigma} 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.

Summary and OutlookSummary 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.

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.

Summary and OutlookSummary and Outlook

Thanks!Thanks!

Documents

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,