arXiv subjects

Wen-Fei Wang

Publications and source records attributed to Wen-Fei Wang.

At least 19 recordsLinked to original sources

arXiv2026

Contributions of the subprocesses $ρ(770,1450,1700)\to K \bar{K}$ and $ω(782,1420,1650)\to K \bar{K}$ for the three-body decays $B\to η^{(\prime)} K\bar{K}$

As an extension of our prior work, we analyze the resonance contributions for the kaon pair originating from the intermediate $ρ(770)$, $ω(782)$ and their excited states in the three-body decays $B\to η^{(\prime)} K\bar{K}$ within the perturbative QCD approach. The information of subprocesses $ρ(770,1450,1700)\to K\bar K$ and $ω(782,1420,1650)\to K\bar K$ are included in the distribution amplitudes for $K\bar K$ system by using the kaon vector time-like form factors. We calculate the $CP$ averaged branching fractions and the direct $CP$ asymmetries for the relevant quasi-two-body $B$ meson decays. The branching fractions of the virtual contributions for $K\bar K$ from the Breit-Wigner formula tails of $ρ(770)$ and $ω(782)$ for these decays are found comparable to the corresponding contributions from the resonances $ρ(1450,1700)$ and $ω(1420,1650)$. Consequently, they constitute a significant component that should be accounted for in the considered three-body decays. All the predictions in this work are expected to be tested by the LHCb and Belle-II experiments in the future.

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arXiv2025

The virtual contributions from $K^$ in the $B \to K^πh$ and $B \to Kρh$ decays

Inspired by the significant virtual contributions of the subprocesses $D^{*}/B^{*}\to Dπ$ and $ ρ\toωπ/K\bar{K}$ in the three-body hadronic $B$ meson decays, we study the off-shell effects of the resonance $K^*$ decaying into the $K^*π$ or $Kρ$ system in the decays $B \to K^*πh$ and $B \to Kρh$ (with $h=π, K$) within the perturbative QCD approach. The strong coupling constants $g_{K^*K^*π}$ and $g_{K^*Kρ}$ involved in this work are derived from the coupling constant $g_{ρωπ}$ under the flavor SU$(3)$ symmetry. The $CP$ averaged branching fractions for the quasi-two-body decays $B\to K^*[\to K^*π] h$ and $B\to K^*[\to Kρ] h$ are predicted to be on the order of $10^{-9}$ to $10^{-7}$ in this study. And the branching fractions for the decays $B\to K^*[\to Kρ] h$ are found to be around half of the corresponding results for the $B\to K^*[\to K^*π] h$ channels, mainly due to the phase space difference between the $Kρ$ and $K^*π$ pairs originating from the intermediate state $K^*$. Experimental data for the $K^*$ decaying into the final states $K^*π$ and $Kρ$ in the $B$ meson decays is expected to be obtained by the LHCb and Belle-II experiments in the near future.

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arXiv2025

Contributions of $ρ(770,1450)\to ωπ$ for the Cabibbo-favored $D \to hωπ$ decays

Recently, the BESIII Collaboration has observed the three-body decays $D_s^+\to ηωπ^+$, $D^+\to K^0_Sπ^+ω$ and $D^0\to K^-π^+ω$. In this work, we investigate the contributions of the subprocesses $ρ^+\to ωπ^+$ in these Cabibbo-favored decays $D \to hωπ$, with $ρ^+= \{ρ(770)^+, ρ(1450)^+, ρ(770)^+\&ρ(1450)^+\}$ and $h=\{ η, K^0_S, K^-\}$, by introducing these subprocesses into the decay amplitudes of relevant decay processes via the vector form factor $F_{ωπ}$ which has measured in the related $τ$ and $e^+e^-$ processes; we provide the first theoretical predictions for the branching fractions of the quasi-two-body decays $D_s^+\toη[ρ^+\to]ωπ^+$, $D^+\to K^0_S[ρ^+\to]ωπ^+$ and $D^0\to K^-[ρ^+\to]ωπ^+$. Our findings reveal that the contributions from the subprocess $ρ(770)^+\toωπ^+$ are significant in these observed three-body decays $D_s^+\toηωπ^+$, $D^+\to K^0_S ωπ^+$ and $D^0\to K^- ωπ^+$, notwithstanding the contributions originating from the Breit-Wigner tail effect of $ρ(770)^+$. The numerical results of this study suggest that the dominant resonance contributions for the three-body decays $D_s^+\toηωπ^+$ and $D^+\to K^0_S ωπ^+$ are originated from the $P$-wave intermediate states $ρ(770)^+$, $ρ(1450)^+$ and their interference effects.

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arXiv2024

Low-mass enhancement of kaon pairs in $B^+\to\bar{D}^{()0}K^+\bar{K}^0$ and $B^0\to D^{()-}K^+\bar{K}^0$ decays

Very recently, the Belle~II Collaboration presented a measurement for the decays $B^+\to\bar{D}^{(*)0} K^+\bar{K}^0$ and $B^0\to D^{(*)-}K^+\bar{K}^0$, the bulk of observed $m(K^+ K_S^0)$ distributions showing low-mass structures in all four channels. In this work, we study the contributions of $ρ(770,1450)^+$, $a_2(1320)^+$ and $a_0(980,1450)^+$ resonances to these decay processes. The intermediate states $ρ(770,1450)^+$ are found to dominate the low-mass distribution of kaon pairs roughly contributing to half of the total branching fraction in each of the four decay channels. The contribution of the tensor $a_2(1320)^+$ meson is found to be negligible. Near the threshold of the kaon pair, the state $a_0(980)^+$ turns out to be much less important than expected, not being able to account for the enhancement of events in that energy region observed in the $B^+\to\bar{D}^{(*)0} K^+\bar{K}^0$ decays. Further studies both from the theoretical and experimental sides are needed to elucidate the role of the non-resonant contributions governing the formation of $K^+\bar{K}^0$ pairs near their threshold in these decay processes.

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arXiv2024

Contributions of the subprocess $K^*_0(1430) \to Kη^{\prime}$ in the charmless three-body $B$ meson decays

We study the contributions for $Kη^{\prime}$ pair originating from the scalar intermediate state $K_0^{*}(1430)$ in the three-body decays $B\to Kη^{\prime} h$ ($h=π, K$) within the perturbative QCD approach. The contribution of $K^*_0(1430)\to Kη^{\prime}$ is described by the Flatt${\rm \acute{e}}$ formula with coupled channels $Kπ$, $Kη$ and $Kη^{\prime}$. The strong coupling constants $g_{K^*_0Kη^{(\prime)}}$ are extracted from $g_{K^*_0 Kπ}$ within flavor SU$(3)$ symmetry. In spite of the strong depression by phase space near the threshold of $Kη^\prime$, the $CP$ averaged branching fractions for the $B\to K^*_0(1430) h \to Kη^\prime h$ decays are predicted to be on the order of $10^{-8}$ to $10^{-5}$, which are non-negligible for the corresponding three-body $B$ decays. Since the K$η$ system is almost decoupled from the even-spin strange mesons under flavor SU$(3)$ symmetry, those quasi-two-body $B$ decays with subprocess $K^*_0(1430) \to K η$ shall have quite small branching ratios and are not taken into account in this work. We also estimate that the branching fraction for $K_0^{*}(1430)\to Kη^{\prime}$ is about one fifth of that for $K_0^{*}(1430)\to Kπ$. The predictions for the relevant decays are expected to be tested by the LHCb and Belle-II experiments in the future.

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arXiv2024

The $ρ(770,1450)\to ωπ$ contributions for three-body decays $B\to\bar{D}^{(*)} ωπ$

The decays $B\to\bar{D}^{(*)} ωπ$ are very important for the investigation of $ρ$ excitations and the test of factorization hypothesis for $B$ meson decays. The $B^{+}\to \bar{D}^{(*)0}ωπ^+$ and $B^{0}\to D^{(*)-}ωπ^+$ have been measured by different collaborations but without any predictions for their observables on theoretical side. In this work, we study the contributions of $ρ(770,1450)\to ωπ$ for the cascade decays $B^{+}\to \bar{D}^{(*)0} ρ^+ \to \bar{D}^{(*)0}ωπ^+$, $B^{0}\to D^{(*)-} ρ^+ \to D^{(*)-}ωπ^+$ and $B_s^{0}\to D_s^{(*)-} ρ^+ \to D^{(*)-}ωπ^+$. We introduce $ρ(770,1450)\to ωπ$ subprocesses into the distribution amplitudes for $ωπ$ system via the vector form factor $F_{ωπ}(s)$ and then predict the branching fractions for the first time for concerned quasi-two-body decays with $ρ(770,1450)\to ωπ$, as well as the corresponding longitudinal polarization fractions $Γ_L/Γ$ for the cases with the vector $\bar{D}^{*0}$ or $D_{(s)}^{*-}$ in their final states. The branching fractions of these quasi-two-body decays are predicted at the order of $10^{-3}$, which can be detected at the LHCb and Belle-II experiments. The predictions for the decays ${B}^0 \to{D}^{*-} ρ(770)^+\to {D}^{*-} ωπ^+$ and ${B}^0 \to {D}^{*-} ρ(1450)^+\to {D}^{*-} ωπ^+$ agree well with the measurements from Belle Collaboration. In order to avoid the pollution from annihilation Feynman diagrams, we recommend to take the $B_s^0 \to D_s^{*-}ρ(770,1450)^+$ decays, which have only emission diagrams at quark level, to test the factorization hypothesis for $B$ decays.

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arXiv2023

Traces of the new $a_0(1780)$ resonance in the $J/ψ\to ϕK^+K^-(K^0\bar{K}^0)$ reaction

We study the $J/ψ\to ϕK \bar K$ decay, looking for differences in the production rates of $K^+ K^-$ or $K^0 \bar K^0$ in the region of 1700-1800 MeV, where two resonances appear dynamically generated from the vector-vector interaction. Two resonances are known experimentally in that region, the $f_0(1710)$ and a new resonance reported by the BABAR and BESIII collaborations. The $K \bar K$ should be produced with $I=0$ in that reaction, but due to the different $K^{*0}$ and $K^{*+}$ masses some isospin violation appears. Yet, due to the large width of the $K^*$, the violation obtained is very small and the rates of $K^+ K^-$ or $K^0 \bar K^0$ production are equal within $5\%$. However, we also find that due to the step needed to convert two vectors into $K \bar K$, a shape can appear in the $K \bar K$ mass distribution that can mimic the $a_0$ production around the $K^* \bar K^*$ threshold, and is simply a threshold effect.

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arXiv2023

A new look at the $P_{cs}$ states from a molecular perspective

We have a look at the $P_{cs}$ states generated from the interaction of $\bar D^{(*)} Ξ^{(\prime*)}_c$ coupled channels. We consider the blocks of pseudoscalar-baryon $({\frac12}^+, {\frac32}^+)$ and vector-baryon $({\frac12}^+, {\frac32}^+)$, and find $10$ resonant states coupling mostly to $\bar D Ξ_c, \bar D^* Ξ_c,\bar D Ξ'_c, \bar D^* Ξ'_c,\bar D Ξ^*_c$ and $\bar D^* Ξ^*_c$. A novel aspect of the work is the realization that the $\bar D Ξ_c,\bar D_sΛ_c$ or $\bar D^* Ξ_c,\bar D^*_sΛ_c$ channels, with a strong transition potential, collaborate to produce a larger attraction than the corresponding states $\bar D Σ_c,\bar DΛ_c$ or $\bar D^* Σ_c,\bar D^*Λ_c$ appearing in the generation of the strangenessless $P_{c}$ states, since in the latter case the transition potential between those channels is zero. The extra attraction obtained in the $\bar D Ξ_c,\bar D^* Ξ_c$ pairs preclude the association of these channels to the $P_{cs}(4338)$ and $P_{cs}(4459)$ states respectively. Then we find a natural association of the $P_{cs}(4338)$ state coupling mostly to $\bar D^* Ξ_c$ while the $P_{cs}(4459)$ is associated to the state found that couples mostly to $\bar D Ξ'_c$. Four more states appear, like in other molecular pictures, and some of the states are degenerate in spin. Counting different spin states we find $10$ states, which we hope can be observed in the near future.

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arXiv2022

Molecular $Ω_{cc}$ , $Ω_{bb}$ and $Ω_{bc}$ states

We study the interaction of meson-baryon coupled channels carrying quantum numbers of $Ω_{cc}$ , $Ω_{bb}$ and $Ω_{bc}$ presently under investigation by the LHCb collaboration. The interaction is obtained from an extension of the local hidden gauge approach to the heavy quark sector that has proved to provide accurate results compared to experiment in the case of $Ω_{c}$ , $Ξ_{c}$ states and pentaquarks, $P_c$ and $P_{cs}$. We obtain many bound states, with small decay widths within the space of the chosen coupled channels. The spin-parity of the states are $J^P={\frac{1}{2}}^-$ for coupled channels of pseudoscalar-baryon (${\frac{1}{2}}^+$), $J^P={\frac{3}{2}}^-$ for the case of pseudoscalar-baryon (${\frac{3}{2}}^+$), $J^P={\frac{1}{2}}^-,{\frac{3}{2}}^-$ for the case of vector-baryon (${\frac{1}{2}}^+$) and $J^P={\frac{1}{2}}^-,{\frac{3}{2}}^-,{\frac{5}{2}}^-$ for the vector-baryon (${\frac{3}{2}}^+$) channels. We look for poles of the states and evaluate the couplings to the different channels. The couplings obtained for the open channels can serve as a guide to see in which reaction the obtained states are more likely to be observed.

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arXiv2022

Resonances $ρ(1450)^+$ and $ρ(1700)^+$ in $B \to D KK$ decays

The contributions for the kaon pair from the intermediate states $ρ(1450)^+$ and $ρ(1700)^+$ in the decays $B^+ \to \bar{D}^0 K^+ \bar{K}^0$, $B^0 \to D^- K^+ \bar{K}^0$, and $B_s^0 \to D_s^-K^+ \bar{K}^0$ are analyzed within the perturbative QCD factorization approach. The decay amplitudes for all concerned decays in this work are dominated by the factorizable Feynman diagrams with the emission of the kaon pair, and the charged $ρ$ mesons should be of great importance in the $KK$ channel of the related three-body $B$ decays. Moreover, these quasi-two-body decays are CKM-favored, and the relevant branching ratios are predicted to be in the order of $10^{-5}$, which have the potential to be measured by experiments. It is also shown that the contributions of the subprocesses $ρ(1450, 1700)^+ \to KK$ for the three-body $B$ meson decays are considerable according to the total three-body branching fractions presented by Belle. Therefore, the decays $B^+ \to \bar{D}^0 K^+ \bar{K}^0$, $B^0 \to D^- K^+ \bar{K}^0$, and $B_s^0 \to D_s^-K^+ \bar{K}^0$ can be employed to study the properties of $ρ(1450)$ and $ρ(1700)$ in the LHCb and Belle-II experiments.

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arXiv2021

Subprocesses $ρ(770,1450)\to K\bar{K}$ for the three-body hadronic $D$ meson decays

We construct the theoretical framework for quasi-two-body $D$ meson decays with the help of pion and kaon electromagnetic form factors, and with which we study the contributions of the subprocesses $ρ(770,1450)\to K\bar{K}$ for the three-body $D$ decays within the flavour $SU(3)$ symmetry. Because of the limitations imposed by phase space and strong coupling, the contributions for kaon pair from the virtual bound state $ρ(770)$ are channel-dependent and generally small for the concerned three-body $D$ decays. But some quasi-two-body processes could still be observed in the Dalitz plot analyses for related decays, such as $D^0 \to K^-ρ(770)^+ \to K^- K^+K_S^0$ and $D^+ \to K_S^0ρ(770)^+ \to K_S^0 K^+K_S^0$, they are predicted to have the branching fractions $\mathcal{B}=(0.82\pm0.04)\times 10^{-4}$ and $\mathcal{B}=0.47^{+0.05}_{-0.03}\times 10^{-4}$, which are $(1.86\pm0.16)\%$ and $(1.84^{+0.21}_{-0.16})\%$, respectively, of the total branching fractions for the corresponding three-body $D$ decays. We find in this work that the normal subprocesses like $ρ(1450)^+\to π^+π^0$ or $ρ(1450)^+\to K^+\bar{K}^0$, which are bound by the masses of decaying initial states, will provide virtual contributions in some special decays.

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arXiv2021

Contributions for the kaon pair from $ρ(770)$, $ω(782)$ and their excited states in the $B\to K\bar K h$ decays

We study the resonance contributions for the kaon pair originating from the intermediate states $ρ(770,1450,1700)$ and $ω(782,1420,1650)$ for the three-body hadronic decays $B\to K\bar K h$ in the perturbative QCD approach, where $h=(π, K)$. The branching fractions of the virtual contributions for $K\bar K$ from the Breit-Wigner formula tails of $ρ(770)$ and $ω(782)$ which have been ignored in experimental and theoretical studies for these decays are found larger than the corresponding contributions from the resonances $ρ(1450,1700)$ and $ω(1420,1650)$. The differential branching fractions for $B\to ρ(770) h\to K\bar K h$ and $B\toω(782) h \to K\bar K h$ are found nearly unaffected by the quite different values of the full widths for $ρ(770)$ and $ω(782)$ in this paper. The predictions in this work for the branching fractions of the quasi-two-body decays $B^+\to π^+ ρ(1450)^0\to π^+K^+K^-$ and $B^+\to π^+ ρ(1450)^0\to π^+π^+π^-$ meet the requirement of $SU(3)$ symmetry relation.

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arXiv2021

The role of $D_{(s)}^\ast$ and their contributions in $B_{(s)}\to D_{(s)} h h^\prime$ decays

We demonstrate the roles of $D_{(s)}^\ast$ and their contributions in the quasi-two-body decays $B_{(s)} \to D_{(s)} h h^\prime$ ($h, h^\prime = \{π, K\}$) in the perturbative QCD approach, stemming from the quark flavour changing $\bar{b} \to \bar{c} \, q_2 \, \bar{q}_1$ and $\bar{b} \to c \, \bar{q_1} \, \bar{q_2}$ with $q_1, q_2 = \{s/d, u\}$. The main motivation of this study is the measurements of significant derivations from the simple phase-space model in the channels $B_{(s)} \to D_{(s)} h h^\prime$ at $B$ factories and LHC, which is now clarified as the Breit-Wigner-tail effects from the corresponding intermediate resonant states $D_{(s)}^\ast$. We confirm that these effect from $D^\ast$ is small ($\sim 5\%$) in the quasi-two-body $B_{(s)} \to D ππ(K)$ decaying channels, and predict the tiny ($< 1 \%$) contributions from $D^\ast$ in the $B_{(s)} \to D_s K π(K)$ decaying channels, our result for the $B_s \to D K π(K)$ decaying channels contributed only from the Breit-Wigner-tail effect of $D_s^\ast$ is in agreement with the current LHCb measurement. We recommend the Belle-II and the LHCb collaborations to restudy the processes $B^+\to \bar D^{\ast 0}π^+(K^+) \to D^-π^+π^+(K^+)$ to reveal the structure of $D^{\ast 0}$ and the strong decay $D^{\ast 0} \to D^+ π^-$.

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arXiv2021

Contributions of the kaon pair from $ρ(770)$ for the three-body decays $B \to D K\bar{K}$

We study the contributions of the kaon pair originating from the resonance $ρ(770)$ for the three-body decays $B \to D K\bar{K}$ by employing the perturbative QCD approach. According to the predictions in this work, the contributions from the intermediate state $ρ(770)^0 $ are relatively small for the three-body decays such as $B^0 \to \bar{D}^0 K^+ K^-$, $B_s^0 \to \bar{D}^0 K^+ K^-$, and $B^+ \to D_s^+ K^+K^-$, while about $20\%$ of the total three-body branching fraction for $B^+ \to \bar{D}^0 K^+ \bar{K}^0$ could possibly come from the subprocess $ρ(770)^+\to K^+ \bar{K}^0$. We also estimate the branching fractions for $ρ(770)^\pm$ decay into the kaon pair to be about $1\%$, and that for the neutral $ρ(770)$ decay into $K^+K^-$ or $K^0\bar{K}^0$ to be about $0.5\%$, which will be tested by future experiments.

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arXiv2020

Resonance contributions $ϕ(1020,1680)\to K\bar K$ for the three-body decays $B\to K\bar K h$

We study the contributions for the $K^+K^-$ and $K^0\bar K^0$ originated from the intermediate states $ϕ(1020)$ and $ϕ(1680)$ in the charmless three-body decays $B\to K\bar K h$, with $h=(π, K)$, in the perturbative QCD approach. The subprocesses $ϕ(1020,1680)\to K\bar K$ are introduced into the distribution amplitudes of $K\bar K$ system via the kaon electromagnetic form factors with the coefficients taken from the fitted results. The predictions of the branching fractions for the decays $B\toϕ(1680)h$ with the intermediate state $ϕ(1680)$ decays into $K^+K^-$ or $K^0\bar K^0$ are about $6\%$-$8\%$ of the corresponding results for the quasi-two-body decays $B\toϕ(1020)h \to K^+ K^- h$ in this work.

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arXiv2020

Will the subprocesses $ρ(770,1450)^0\to K^+K^-$ contribute large branching fractions for $B^\pm \to π^\pm K^+K^-$ decays?

We analyze the quasi-two-body decays $B^+\toπ^+ρ(770,1450)^0 \toπ^+ K^+K^-$ in the perturbative QCD approach. The results in this work do not support that large branching fractions contributed by the resonances $ρ(770,1450)^0$ in the $B^\pm \to π^\pm K^+K^-$ decays. The virtual contribution for $K^+K^-$ from the tail of the resonance $ρ(770)^0$ which has been ignored in the experimental studies is about $1.5$ times of the $ρ(1450)^0 \to K^+K^-$ contribution, with the predicted branching fractions ${\mathcal B}_v=(1.31\pm0.27)\times10^{-7}$ and ${\mathcal B}=(8.96\pm2.61)\times10^{-8}$, respectively, for these two subprocesses in the $B^\pm \to π^\pm K^+K^-$ decays. The absence of $ρ(770)^0\to K^+K^-$ for the decay amplitude of a three-body hadronic $B$ decay involving charged kaon pair could probably result in a larger proportion for the contribution from the resonance $ρ(1450)^0$ in the experimental analysis.

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arXiv2020

Contributions of $K^_0(1430)$ and $K^_0(1950)$ in the three-body decays $B\to Kπh$

We study the contributions of the resonant states $K_0^{*}(1430)$ and $K_0^{*}(1950)$ in the three-body decays $B\to Kπh$ (with $h=π, K$) in the perturbative QCD approach. The crucial nonperturbative input $F_{Kπ}(s)$ in the distribution amplitudes of the $S$-wave $Kπ$ system is derived from the matrix element of vacuum to $Kπ$ pair. The $CP$ averaged branching fraction of the quasi-two-body decay process $B\to K^*_0(1950)h\to Kπh$ is about one order smaller than that of the corresponding decay $B\to K^*_0(1430)h\to Kπh$. In view of the important contribution from the $S$-wave $Kπ$ system for the $B\to Kπh$ decays, it is not appropriate to neglect the $K_0^{*}(1950)$ in the theoretical or experimental studies for the relevant three-body $B$ meson decays. The predictions in this work for the relevant decays are consistent with the existing experimental data.

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arXiv2019

Quasi-two-body decays $B_{(s)} \to P D_0^*(2400) \to P D π$ in the perturbative QCD approach

We study the quasi-two-body decays $B\to P D^{\ast}_0(2400) \to P Dπ$ with $P=(π, K, η, η^{\prime})$ in the perturbative QCD factorization approach. The predicted branching fractions for the considered decays are in the range of $10^{-9}$-$10^{-4}$. The strong Cabibbo-Kobayashi-Maskawa (CKM) suppression factor $R_{CKM}\approx λ^4 (\barρ^2 + \barη^2) \approx 3\times 10^{-4}$ results in the great difference of the branching ratios for the decays with $D_0^*$ and $\bar{D}_0^*$ as the intermediate states. The ratio $R_{\bar{D}_0^{*0}}$ between the decays $B^0 \to \bar{D}_0^{*0} K^0\to D^-π^+K^0$ and $B^0 \to \bar{D}_0^{*0}π^0 \to D^-π^+π^0$ is about $0.091^{+0.003}_{-0.005}$, consistent with the flavour-$SU$(3) symmetry result. The ratio for the branching fractions is found to be $1.10^{+0.05}_{-0.02}$ between $\mathcal{B}(B_s^0\to D_0^{*+}K^-\to D^0π^+K^-)$ and $\mathcal{B}(B^0\to D_0^{*+} π^-\to D^0π^+ π^-)$ and to be $1.03^{+0.06}_{-0.07}$ between $\mathcal{B}(B_s^0\to\bar{D}_0^{*0} \bar{K}^0\to D^-π^+ \bar{K}^0)$ and $2\mathcal{B}(B^0\to \bar{D}_0^{*0}π^0\to D^-π^+π^0)$. The predictions in this work can be tested by the future experiments.

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