arXiv subjects

Da-Cheng Yan

Publications and source records attributed to Da-Cheng Yan.

15 recordsLinked to original sources

arXiv2026

Polarization fractions and helicity-dependent CP asymmetries in $B_{(s)} \to \rho\rho, \rho K^\ast$ and $K^\ast K^\ast$ decays

In this paper, we present a phenomenological analysis of $B_{(s)} \to \rho\rho, \rho K^\ast$ and $K^\ast K^\ast$ decays using state-of-the-art perturbative QCD (pQCD) calculations. Our study is primarily motivated by recent polarization measurements from the LHCb and Belle II collaborations, which have significantly improved the precision of polarization fractions and enabled the first full determination of polarization-dependent CP asymmetries. This work extends the comprehensive pQCD study of charmless two-body $B$ decays reported in our previous paper [Chin. Phys. C 46 (2022) 123103], with a particular focus on polarization observables, especially the CP asymmetries in each helicity state, which reflect distinct orbital angular momentum configurations between the two vector mesons. Our predictions for the branching ratios and longitudinal polarization fractions in the $B^0 \to K^{\ast 0} {\bar K}^{\ast 0}$ and $B^+ \to \rho^0 K^{\ast +}$ modes are in good agreement with the new experimental data. However, the calculated longitudinal polarization fraction for $B_s \to K^{\ast 0} {\bar K}^{\ast 0}$ is significantly larger than the LHCb measurement. Moreover, the predicted (helicity-dependent) CP asymmetries in $B^+ \to \rho^0 K^{\ast +}$ are about $30 \%$ smaller than the observed values. These discrepancies point to a rich interplay between different topological amplitudes, highlighting the need for further theoretical investigation to resolve the long-standing polarization puzzle in two-body $B$ decays into vector mesons.

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arXiv2026

$B_c$ meson decays into $S$-wave charmonium plus light meson pairs in the perturbative QCD approach

In this work, we explore the $P$-wave resonance contributions to the three-body charmonium decays of $B_c\to \Psi (V\to) P_1P_2$ using the perturbative QCD formalism at leading order, where $\Psi$ denotes a $S$-wave charmonium state, such as $\eta_c(1S,2S),J/\psi$, and $\psi(2S)$. Here, $P_1P_2$ represents a collinear $\pi\pi$ ($K\pi$) pair in the final state, which was primarily produced through the vector resonance $\rho(770)$ ($K^*(892)$ ). With the improved two-meson distribution amplitudes determined from our previous works, we examined the $CP$-averaged branching ratios and polarization fractions of the considered three-body decays. The longitudinal polarization fractions of the $B_c\to [J/\psi, \psi(2S)] (V\to) P_1P_2$ decays are found to be as large as $\sim 90\%$, since the transverse amplitudes from the dominant factorizable emission diagrams are always power suppressed with respect to the longitudinal ones. The direct $CP$ violations in $B_c\to \Psi (V\to) P_1P_2$ decays are predicted naturally to be zero as they solely receive contributions from tree diagrams. Several interesting relative ratios among the branching fractions of the concerned processes are investigated. In particular, the obtained ratio $R^{\rm PQCD}_{2\pi/\pi}\equiv \mathcal{B}(B^+_c \to J/\psi(\rho\to)\pi^+\pi^0)/{\mathcal{B}(B^+_c \to J/\psi\pi^+)}=2.67^{+0.21}_{-0.14}$ is consistent well with the LHCb measurement $R^{\rm exp}_{2\pi/\pi}=2.80\pm0.25$. Other similar ratios proposed in this work can be tested by LHCb experiments in the near future.

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arXiv2025

Improved global determination of two-meson distribution amplitudes from multi-body $B$ decays

We improve the perturbative QCD (PQCD) formalism for multi-body charmless hadronic $B$ meson decays, such as $B\to VP_3\to P_1P_2P_3$, by resolving the possible discrepancy in parametrizing the contribution of the $P$-wave resonance $V$ to the two-meson distribution amplitudes (DAs) associated with the pairs $P_1P_2=\pi\pi, K\pi, KK$. The determination of the Gegenbauer moments in the two-meson DAs is then updated in the global fit of the improved PQCD factorization formulas at leading order in the strong coupling $\alpha_s$ to available data for branching ratios and polarization fractions of three- and four-body $B$ decays. The convergence of the Gegenbauer expansion of the resultant two-meson DAs is manifest. The satisfactory quality of the fit implies the consistency of the PQCD framework for multi-body $B$ decays and the universality of the nonperturbative two-meson DAs. In particular, the predicted longitudinal polarization fraction $f_0(B_s^0\to K^{*0} {\bar K}^{*0})=28.2^{+8.8}_{-9.5} \%$ with the updated Gegenbauer moments matches well the measurement. The observable $L_{K^{*0}{\bar K}^{*0}}=7.7^{+4.9}_{-3.8}$, defined as the ratio of the longitudinal amplitudes of the two $U$-spin related channels $B_s^0\to K^{*0} {\bar K}^{*0}$ and $B^0\to K^{*0} {\bar K}^{*0}$, accommodates the current data within errors. It is found that the direct $CP$ asymmetries ${\cal A}^{0,||,\bot}_{\rm CP}$ in the polarization states of some four-body decays $B\to V_1V_2\to (P_1P_2)(P_3P_4)$ might be large, but the destruction among them result in small net $CP$ violation. Our predictions can be confronted with LHCb and Belle-II data in the future.

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arXiv2024

$CP$-violating observables of four-body $B_{(s)} \to (\pi\pi)(K\bar{K})$ decays in perturbative QCD

In this work, we investigate six helicity amplitudes of the four-body $B_{(s)} \to (\pi\pi)(K\bar{K})$ decays in the perturbative QCD (PQCD) approach. The $\pi\pi$ invariant mass spectrum is dominated by the vector resonance $\rho(770)$ together with scalar resonance $f_0(980)$, while the vector resonance $\phi(1020)$ and scalar resonance $f_0(980)$ are expected to contribute in the $K\bar{K}$ invariant mass range. We extract the two-body branching ratios ${\cal B}(B_{(s)}\to \rho\phi)$ from the corresponding four-body decays $B_{(s)}\to \rho\phi\to (\pi\pi)(K \bar K)$. The predicted ${\cal B}(B^0_{s}\to \rho\phi)$ agrees well with the current experimental data within errors. The longitudinal polarization fractions of the $B_{(s)}\to \rho\phi$ decays are found to be as large as $90\%$, basically consistent with the previous two-body predictions within uncertainties. In addition, the triple-product asymmetries (TPAs) of the considered decays are also presented for the first time. Since the $B_s^0\to \rho^0\phi\to(\pi^+\pi^-)(K^+K^-)$ decay is induced by both tree and penguin operators, the values of the ${\cal A}^{\rm CP}_{\rm dir}$ and ${\cal A}^{1}_{\text{T-true}}$ are calculated to be $(21.8^{+2.7}_{-3.3})\%$ and $(-10.23^{+1.73}_{-1.56})\%$ respectively. While for pure penguin decays $B^0\to \rho^0\phi\to(\pi^+\pi^-)(K^+K^-)$ and $B^+\to \rho^+\phi\to(\pi^+\pi^0)(K^+K^-)$, both the direct $CP$ asymmetries and ``true" TPAs are naturally expected to be zero in the standard model (SM). The ``fake" TPAs requiring no weak phase difference are usually none zero for all considered decay channels. The sizable ``fake" ${\cal A}^{1}_{\text{T-fake}}=(-20.92^{+6.26}_{-2.80})\%$ of the $B^0\to \rho^0\phi\to(\pi^+\pi^-)(K^+K^-)$ decay is predicted in the PQCD approach, which provides valuable information on the final-state interactions.Our predictions can be tested by the future experiments.

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arXiv2024

Two-body hadronic weak decays of bottomed hadrons

The structure of light diquarks plays a crucial role in the formation of exotic hadrons beyond the conventional quark model, especially in their line shapes of bottomed hadron decays. We study the two-body hadronic weak decays of bottomed baryons and bottomed mesons to probe the light diquark structure and pin down the quark-quark correlations in the diquark picture. We find that the light diquark does not favor a compact structure. For instance, the isoscalar diquark $[ud]$ in $\Lambda_{b}^{0}$ can be easily split and rearranged to form $\Sigma_{c}^{(*)}\bar{D}^{(*)}$ via the color-suppressed transition. This provides a hint that the hidden charm pentaquark states produced in $\Lambda^0_b$ decays could be the $\Sigma_{c}^{(*)}\bar{D}^{(*)}$ hadronic molecular candidates. This quantitative study resolves the apparent conflicts between the production mechanism and molecular nature of these $P_c$ states observed in experiment.

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arXiv2023

Study of four-body decays $B_{(s)} \to (\pi\pi)(\pi\pi)$ in the perturbative QCD approach

In this work, we make a systematical study on the four-body $B_{(s)} \to (\pi\pi)(\pi\pi)$ decays in the perturbative QCD approach, where the $\pi\pi$ invariant mass spectra are dominated by the vector resonance $\rho(770)$ and the scalar resonance $f_0(980)$. We improve the Gengenbauer moments for the longitudinal $P$-wave two-pion distribution amplitudes (DAs) by fitting the PQCD factorization formulas to measured branching ratios of three-body and four-body $B$ decays. With the fitted Gegenbauer moments, we make predictions for the branching ratios and direct $CP$ asymmetries of four-body $B_{(s)} \to (\pi\pi)(\pi\pi)$ decays. We extract the branching ratios of two-body $B_{(s)} \to \rho\rho$ from the corresponding four-body decay modes and calculate the relevant polarization fractions. We find that the ${\cal B}(B^0 \to \rho^+\rho^-)$ is consistent with the previous theoretical predictions and data. The leading-order PQCD calculations of the ${\cal B}(B^+\to \rho^+\rho^0)$, ${\cal B}(B^0\to \rho^0\rho^0)$ and the $f_0(B^0\to \rho^0\rho^0)$ are a bit lower than the experimental measurements, which should be further examined. In addition, the "true" and "fake" triple-product asymmetries (TPAs) in the $B_{(s)}\to (\pi\pi)(\pi\pi)$ decays are also analyzed. The sizable averaged TPA ${\cal A}_{\text{T-true}}^{1, \text{ave}}=25.26\%$ of the color-suppressed decay $B^0\to \rho^0\rho^0 \to (\pi^+\pi^-)(\pi^+\pi^-)$ is predicted for the first time, which deviates a lot from the so-called "true" TPA $\mathcal{A}_\text{T-true}^1=7.92\%$ due to the large direct $CP$ violation. A large "fake" TPA $\mathcal{A}_\text{T-fake}^1=24.96\%$ of the decay $B^0\to \rho^0\rho^0 \to (\pi^+\pi^-)(\pi^+\pi^-)$ is also found, which indicates the significance of the final-state interactions. The predictions in this work can be tested by LHCb and Belle-II experiments in the near future.

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arXiv2022

Study of $B_{(s)}^0 \to \phi\phi \to (K^+K^-)(K^+K^-)$ decays in the perturbative QCD approach

In this work, we make a detailed analysis on the penguin-dominant processes $B_{(s)}^0 \to \phi\phi \to (K^+K^-)(K^+K^-)$ in the perturbative QCD (PQCD) approach. In addition to the dominant $P$-wave resonance, the scalar background $f_0(980) \to K^+K^-$ is also accounted for. We improve the Gegenbauer moments in $KK$ two-meson distribution amplitudes by fitting the PQCD factorization formulas to measured branching ratios of three-body and four-body $B$ decays. We extract the branching ratios of two-body $B_{(s)}^0 \to \phi\phi$ decays from the corresponding four-body decay modes and calculate the relevant polarization fractions together with two relative phases $\phi_{\parallel,\perp}$, which are consistent with the previous theoretical predictions. The PQCD predictions for the "true" triple product asymmetries (TPAs) are zero which are expected in the standard model due to the vanishing weak phase difference, and support the current data reported by the CDF and LHCb Collaborations. A large "fake" TPA $\mathcal{A}_\text{T-fake}^1=30.4\%$ of the decay $B^0_s \to \phi\phi \to (K^+K^-)(K^+K^-)$ is predicted for the first time, which indicates the presence of the significant final-state interactions. The TPAs of the rare decay channel $B^0 \to \phi\phi\to (K^+K^-)(K^+K^-)$ are also predicted and can be tested in the near future.

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arXiv2021

Study of $B_{(s)} \to (\pi\pi)(K\pi)$ decays in the perturbative QCD approach

In this work, we provide estimates of the branching ratios, direct $CP$ asymmetries and triple product asymmetries in $B_{(s)} \to (\pi\pi)(K\pi)$ decays in the perturbative QCD approach, where the $\pi\pi$ and $K\pi$ invariant mass spectra are dominated by the vector resonances $\rho(770)$ and $K^*(892)$, respectively. Some scalar backgrounds, such as $f_0(500,980) \to \pi\pi$ and $K^*_0(1430) \to K\pi$ are also accounted for. The $\rho(700)$ is parametrized by the Gounaris-Sakurai function. The relativistic Breit-Wigner formula for the $f_0(500)$ and Flatt\'e model for the $f_0(980)$ are adopted to parameterize the time-like scalar form factors $F_S(\omega^2)$. We also use the D.V. Bugg model to parameterize the $f_0(500)$ and compare the relevant theoretical predictions from different models. While in the region of $K\pi$ invariant mass, the $K^*_0(1430)$ is described with the LASS lineshape and the $K^*(892)$ is modeled by the Breit-Wigner function. We find that the decay rates for the considered decay modes agree with currently available data within errors. As a by-product, we extract the branching ratios of two-body decays $B_{(s)} \to \rho(770)K^*(892)$ from the corresponding four-body decay modes and calculate the relevant polarization fractions. Our prediction of longitudinal polarization fraction for $B^0\to \rho(770)^0 K^*(892)^0$ decay deviates a lot from the recent LHCb measurement, which should be resolved. It is shown that the direct $CP$ asymmetries are large due to the sizable interference between the tree and penguin contributions, but they are small for the tree-dominant or penguin-dominant processes. The PQCD predictions for the triple product asymmetries are small which are expected in the standard model, and consistent with the current data reported by the LHCb Collaboration.Our results can be tested by the future precise data from the LHCb and Belle II experiments.

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arXiv2021

Global determination of two-meson distribution amplitudes from three-body $B$ decays in the perturbative QCD approach

We perform a global analysis of three-body charmless hadronic decays $B\to VP_3\to P_1P_2P_3$ in the perturbative QCD (PQCD) approach, where $V$ denotes an intermediate vector resonance, and $P_i$, $i=1,2,3$, denote final-state pseudoscalar mesons. Fitting the PQCD factorization formulas at leading order in the strong coupling $\alpha_s$ to measured branching ratios and direct $CP$ asymmetries, we determine the Gegenbauer moments in the two-meson distribution amplitudes (DAs) for the meson pairs $P_1P_2=\pi\pi, K\pi, KK$. The fitted Gegenbauer moments are then employed to make predictions for those observables, whose data are excluded in the fit due to larger experimental uncertainties. A general consistency between our predictions and data is achieved, which hints the validity of the PQCD formalism for the above three-body $B$ meson decays and the universality of the nonperturbative two-meson DAs. The obtained two-meson DAs can be applied to PQCD studies of other multi-body $B$ meson decays involving the same meson pairs. We also attempt to determine the dependence of the Gegenbauer moments on the meson-pair invariant mass, and demonstrate that this determination is promising, when data become more precise.

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arXiv2020

Resonant contributions to three-body $B_{(s)} \to [ D^{()}, \bar{D}^{()} ] K^+K^-$ decays in the perturbative QCD approach

In this work, we study the $S$, $P$ and $D$ wave resonance contributions to three-body decays $B_{(s)} \to [ D^{(*)}, \bar{D}^{(*)} ] K^+K^-$ by employing the perturbative QCD (PQCD) approach, where the kaon-kaon invariant mass spectra are dominated by the $f_0(980),f_0(1370),$ $\phi(1020),\phi(1680),f_2(1270),f^{\prime}_2(1525),f_2(1750)$ and $f_2(1950)$ resonances. The $KK$ $S$-wave component $f_0(980)$ is modeled with the Flatt\'e formalism, while other resonances are described by the relativistic Breit-Wigner (BW) line shape. The corresponding decay channels are studied by constructing the kaon-kaon distribution amplitude $\Phi_{KK}$, which captures important final state interactions between the kaon pair in the resonant region. We found that the PQCD predictions for the branching ratios for most considered decays agree with currently available data within errors. The associated polarization fractions of those vector-vector and vector-tensor decay modes are also predicted, which are expected to be tested in the near future experiments. The invariant mass spectra for the corresponding resonances in the $B_{(s)} \to [D^{(*)}, \bar{D}^{(*)} ] K^+K^-$ decays are well established, which can be confronted with the precise data from the LHCb and Belle II experiments.

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arXiv2019

$S$, $P$ and $D$-wave resonance contributions to $B_{(s)} \to \eta_c(1S,2S) K\pi$ decays in the perturbative QCD approach

In this work, we analyze the three-body $B_{(s)} \to \eta_c(1S,2S) K \pi$ decays within the framework of the perturbative QCD approach (PQCD) under the quasi-two-body approximation, where the kaon-pion invariant mass spectra are dominated by the $K_0^*(1430)^0,K_0^*(1950)^0,K^*(892)^0,K^*(1410)^0,K^*(1680)^0$ and $K_2^*(1430)^0$ resonances. The time-like form factors are adopted to parametrize the corresponding $S$, $P$, $D$-wave kaon-pion distribution amplitudes for the concerned decay modes, which describe the final-state interactions between the kaon and pion in the resonant region. The $K\pi$ $S$-wave component at low $K\pi$ mass is described by the LASS line shape, while the time-like form factors of other resonances are modeled by the relativistic Breit-Wigner function. We find the following main points: (a) the PQCD predictions of the branching ratios for most considered $B \to \eta_c(1S)(K^{*0}\to )K^+\pi^-$ decays agree well with the currently available data within errors; (b) for ${\cal B}(B^0 \to \eta_c (K_0^*(1430)\to )K^+\pi^-)$ and ${\cal B}(B^0 \to \eta_c K^+\pi^-({\rm NR}))$ (here NR means nonresonant), our predictions of the branching ratios are a bit smaller than the measured ones; and (c) the PQCD results for the $D$-wave contributions considered in this work can be tested once the precise data from the future LHCb and Belle-II experiments are available.

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arXiv2019

$B_{(s)} \to \eta_c(P,V)$ decays and effects of the next-to-leading order contributions in the perturbative QCD approach

By employing the perturbative QCD (PQCD) factorization approach, we studied the sixteen $B/B_s \to \eta_c (\pi, K, \eta^{(\prime)},\rho,K^*,\omega,\phi)$ decays with the inclusion of the currently known next-to-leading order (NLO) contributions. We found the following main points: (a) for the five measured $B \to \eta_c (K,K^*)$ and $B_s\to \eta_c\phi$ decays, the NLO contributions can provide $ (80-180)\%$ enhancements to the leading order (LO) PQCD predictions of their branching ratios, which play an important role to help us to interpret the data; (b) for the seven ratios $R_{1,\cdots,7}$ of the branching ratios defined among the properly selected pair of the considered decay modes, the PQCD predictions for the values of $R_{3,4,5}$ agree well with those currently available measurements from BaBar and Belle Collaboration; (c) for $B^0 \to \eta_c K_S^0$ decay, the PQCD predictions for both the direct and mixing induced CP asymmetries do agree very well with the measured values within errors; and (d) the PQCD predictions for ratios $R_{1,2}$ and $R_{6,7}$ also agree with the general expectations and will be tested by the future experiments.

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arXiv2019

Anatomy of $B_s \to PP $ decays and effects of the next-to-leading order contributions in the perturbative QCD approach

By employing the perturbative QCD (PQCD) factorization approach, we made a systematic investigation for the CP-averaged branching ratios and the CP-violating asymmetries of the thirteen $\bar{B}^0_s \to PP $ decays ( here $P=(\pi,K, \eta, \eta^{\prime})$) with the inclusion of all currently known next-to-leading order (NLO) contributions, and compared our results with the measured values or the theoretical predictions from other different approaches. We focused on the examination of the effects of the NLO contributions and found the following points: (a) for $\bar{B}_s^0 \to (K^0 \bar{K}^0, K^+ K^-, \eta^\prime \eta^\prime,\pi^- K^+)$ decays, the NLO contributions can provide a $(40-150) \%$ enhancement or a $20\%$ reduction to the corresponding leading order (LO) PQCD predictions for their decay rates and result in a much better agreement between the PQCD predictions and the experimental measurements; (b) for the pure annihilation decay $\bar{B}_s^0 \to \pi^+\pi^-$, the PQCD prediction still remain consistent with the data after the inclusion of the small NLO reduction; (c) the PQCD predictions for the ratio $(f_s/f_d)\cdot {\cal B}(B_s^0\to \pi^+\pi^-)/{\cal B}(B^0\to K^+\pi^-)$ and $(f_s/f_d)\cdot {\cal B}(B_s^0\to \pi^+ K^-)/{\cal B}(B^0\to K^+\pi^-)$ become agree very well with the measured ones after the inclusion of a $40\%$ NLO reduction; (d) for $\bar{B}_s^0 \to K^+ K^-$ and $\bar{B}_s^0 \to \pi^- K^+$ decays, the NLO PQCD predictions for their CP-violating asymmetries do agree very well with the measured values in both the sign and the magnitude; and (e) for all $\bar{B}_s^0 \to P P$ decays, we also compared our results with those obtained in the QCD factorization approach and soft-collinear effective theory and discussed their similarities and differences.

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arXiv2018

Anatomy of $B_s \to VV $ decays and effects of next-to-leading order contributions in the perturbative QCD factorization approach

By employing the perturbative QCD (PQCD) factorization approach, we calculated the branching ratios, CP-violating asymmetries, the longitudinal and transverse polarization fractions and other physical observables of the thirteen charmless hadronic $\bar{B}^0_s \to V V $ decays with the inclusion of all currently known next-to-leading order (NLO) contributions. We focused on the examination of the effects of all those currently known NLO contributions and found that: (a) for the measured decays $\bar{ B}_s^0 \to \phi \phi, K^{*0} \phi,\bar K^{*0} K^{*0}$ and $\rho^0 \phi$, the NLO contributions can provide $\sim 20\%$ to $\sim 40\%$ enhancements to the leading order (LO) PQCD predictions of their CP-averaged branching ratios, and consequently the agreement between the PQCD predictions and the measured values are improved effectively after the inclusion of the NLO contributions; (b) for the measured decays, the NLO corrections to the LO PQCD predictions for $(f_L,f_\perp)$ and $(\phi_{\|},\phi_{\bot})$ are generally small in size, but the weak penguin annihilation contributions play an important role in understanding the data about their decay rates, $f_L$ and $f_\perp$; (c) the NLO PQCD predictions for above mentioned physical observables do agree with the measured ones and the theoretical predictions from the QCDF, SCET and FAT approaches; (d) for other considered $B_s^0 \to VV$ decays, the NLO PQCD predictions for their decay rates and other physical observables are also basically consistent with the theoretical predictions from other popular approaches, future precision measurements could help us to test or examine these predictions.

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arXiv2017

Anatomy of $B_s \to PV $ decays and effects of next-to-leading order contributions in the perturbative QCD factorization approach

In this paper, we will make systematic calculations for the branching ratios and the CP-violating asymmetries of the twenty one $\bar{B}^0_s \to PV $ decays by employing the perturbative QCD (PQCD) factorization approach. Besides the full leading-order (LO) contributions, all currently known next-to-leading order (NLO) contributions are taken into account. We found numerically that: (a) the NLO contributions can provide $\sim 40\%$ enhancement to the LO PQCD predictions for ${\cal B}(\bar{B}_s^0 \to K^0 \bar{K}^{*0})$ and $ {\cal B}(\bar{B}_s^0 \to K^{\pm}K^{*\mp})$, or a $\sim 37\% $ reduction to $ \calb(\bar{B}_s^0 \to \pi^{-} K^{*+})$, and we confirmed that the inclusion of the known NLO contributions can improve significantly the agreement between the theory and those currently available experimental measurements, (b) the total effects on the PQCD predictions for the relevant $B\to P$ transition form factors after the inclusion of the NLO twist-2 and twist-3 contributions is generally small in magnitude: less than $ 10\%$ enhancement respect to the leading order result, (c) for the "tree" dominated decay $\bar B_s^0\to K^+ \rho^- $ and the "color-suppressed-tree" decay $\bar B_s^0\to \pi^0 K^{*0}$, the big difference between the PQCD predictions for their branching ratios are induced by different topological structure and by interference effects among the decay amplitude ${\cal A}_{T,C}$ and ${\cal A}_P$: constructive for the first decay but destructive for the second one, and (d) for $\bar{B}_s^0 \to V(\eta, \etar)$ decays, the complex pattern of the PQCD predictions for their branching ratios can be understood by rather different topological structures and the interference effects between the decay amplitude $\cala(V\eta_q)$ and $\cala(V\eta_s)$ due to the $\eta-\etar$ mixing.

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