^{*}

^{3}.

Within the framework of the perturbative QCD approach based on

The study of weak decays of the

In the quark model, the

In particular, in Refs.

The light scalar mesons considered in this paper include the isosinglet

In the factorization hypothesis, for these considered

The remainder of the paper is organized as follows. The framework of PQCD, as well as the distribution amplitudes and decay constants of the mesons, are given in Sec.

In this work, we shall describe the meson’s momenta by using the light-cone coordinate. In the rest frame of the

It is known that in studying exclusive hadron decays the main theoretical uncertainties are from the calculations of matrix elements. The key point of the PQCD approach is to keep the intrinsic transverse momenta of the inner quarks, which is the so-called

There are several typical scales in the

In the PQCD approach, the universal nonperturbative wave functions are the most important inputs. Unlike

Given

For the charmed

Due to experimental developments, many scalar states have been discovered. Theoretically, as mentioned above, there are two different scenarios for describing the scalar mesons in the quark model. S1 is the typical two-quark model: the nonet mesons below 1 GeV [including

In S2, the nonet mesons near 1.5 GeV are viewed as the lowest-lying states, while the mesons below 1 GeV may be viewed as four-quark bound states. Because of the difficulty in dealing with four-quark states, we only do the calculation about the heavier nonet in S2.

Now, we shall discuss the decay constants and the distribution amplitudes of the scalar mesons. The two decay constants of scalar mesons are defined as

In the two-quark model, the wave function of the scalar meson is given by

For the considered decays, the weak effective Hamiltonian

Current-current (tree) operators,

QCD penguin operators,

Electroweak penguin operators

According to the effective Hamiltonian, we can draw the possible lowest-order diagrams, as shown in Fig.

Leading-order Feynman diagrams contributing to the

As mentioned above, the annihilation-type diagrams can be calculated reliably in the PQCD approach. For the factorizable annihilation diagrams in Figs.

For nonfactorizable annihilation diagrams in Figs.

Similarly, the formulas for the

For the total decay amplitudes, the Wilson coefficients and the CKM elements are the same as for the corresponding

In this section, we first list the other input parameters we used in the numerical calculations, such as the masses and lifetimes of mesons and the CKM matrix elements

Within the above parameters, we calculated the

The

The

The

The

As we know, the quark components and physical properties of

The calculated branching fractions of

As stated in Ref.

On the basis of the numerical results we obtained, we provide the following discussion.

For the decays with an emitted scalar, the contribution

Inevitably, there are large theoretical uncertainties in the numerical calculations; in particular, the properties of the scalar meson are not well understood, and the wave function of the

The

As expected, the branching fractions of the four “C”-type decays are much smaller than the “A”-type decays, and the contributions of the penguin operators are negligible. Although the four decays also have a large Wilson coefficient (

From Tables

We now discuss the decay modes involving

From the tables, it is apparent that the direct

In this work, within the PQCD approach, we studied the branching fractions and

This work was supported in part by the National Science Foundation of China under the Grant Nos. 11575151, 11705159, 11235005, 11765012, 11447032, by the Natural Science Foundation of Shandong province (ZR2014AQ013 and ZR2016JL001), and by the Research Fund of Jiangsu Normal University under Grant No. HB2016004. Y. L. is grateful to the Institute of High Energy Physics (IHEP) for hospitality where this work was initiated.

In this appendix, we summarize the functions that appear in the analytic formulas in Sec.