arXiv subjects

Ai-Jun Ma

Publications and source records attributed to Ai-Jun Ma.

At least 19 recordsLinked to original sources

arXiv2026

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

As an extension of our prior work, we analyze the resonance contributions for the kaon pair originating from the intermediate $\rho(770)$, $\omega(782)$ and their excited states in the three-body decays $B\to \eta^{(\prime)} K\bar{K}$ within the perturbative QCD approach. The information of subprocesses $\rho(770,1450,1700)\to K\bar K$ and $\omega(782,1420,1650)\to K\bar K$ 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 $\rho(770)$ and $\omega(782)$ for these decays are found comparable to the corresponding contributions from the resonances $\rho(1450,1700)$ and $\omega(1420,1650)$. 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 $B \to K^\pi h$ and $B \to K\rho h$ decays

Inspired by the significant virtual contributions of the subprocesses $D^{*}/B^{*}\to D\pi$ and $ \rho\to\omega \pi/K\bar{K}$ in the three-body hadronic $B$ meson decays, we study the off-shell effects of the resonance $K^*$ decaying into the $K^*\pi$ or $K\rho$ system in the decays $B \to K^*\pi h$ and $B \to K\rho h$ (with $h=\pi, K$) within the perturbative QCD approach. The strong coupling constants $g_{K^*K^*\pi}$ and $g_{K^*K\rho}$ involved in this work are derived from the coupling constant $g_{\rho\omega\pi}$ under the flavor SU$(3)$ symmetry. The $CP$ averaged branching fractions for the quasi-two-body decays $B\to K^*[\to K^*\pi] h$ and $B\to K^*[\to K\rho] h$ are predicted to be on the order of $10^{-9}$ to $10^{-7}$ in this study. And the branching fractions for the decays $B\to K^*[\to K\rho] h$ are found to be around half of the corresponding results for the $B\to K^*[\to K^*\pi] h$ channels, mainly due to the phase space difference between the $K\rho$ and $K^*\pi$ pairs originating from the intermediate state $K^*$. Experimental data for the $K^*$ decaying into the final states $K^*\pi$ and $K\rho$ 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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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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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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arXiv2022

Probing isovector scalar mesons in the charmless three-body $B$ decays

We propose to study the multiparticle configurations of isovector scalar mesons, saying $a_0(980)$ and $a_0(1450)$, in the charmless three-body $B$ decays by considering the width effects. Two scenarios of $a_0$ configurations are assumed, in which the first one take $a_0(980)$ as the lowest-lying $q{\bar q}$ state and $a_0(1450)$ as the first radial excited state, the second one take $a_0(1450)$ as the lowest-lying $q{\bar q}$ state and $a_0(1950)$ as the first radial excited state while $a_0(980)$ is not a $q{\bar q}$ state. Within these two scenarios, we do the PQCD calculation for the quasi-two-body $B \to a_0 \left[\to K{\bar K}/πη\right] h$ decays and extract the corresponding branching fractions of two-body $B \to a_0 h$ decays under the narrow width approximation. Our predictions show that the first scenario of $a_0(980)$ configuration can not be excluded by the available measurements in $B$ decays, the contributions from $a_0(1450)$ to the branching fractions in most channels are comparable in the first and second scenarios. Several channels are suggested for the forthcoming experimental measurements to reveal the multiparticle configurations of $a_0$, such as the channel $B^0 \to a_0^-(980) \left[\to π^-η\right] π^+$ with the largest predicted branching fraction, the channels $B^0 \to a_0^{\pm}(1450) \left[\to K^\pm{\bar K}^0, π^\pmη\right] π^\mp$ whose branching fractions obtained in the second scenario is about three times larger in magnitude than that obtained in the first scenario, and also the channels $B^+ \to a_0^+(1950) \left[ K^+{\bar K}^0/π^+η\right] K^0$ whose branching fractions are linear dependent on the partial width $Γ_{a_0(1950) \to KK/πη}$.

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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

Resonances $ϕ(1020)$ and $ϕ(1680)$ contributions for the three-body decays $B^+ \to D_s^+K\bar K$

We study the resonances $ϕ(1020)$ and $ϕ(1680)$ contributions for the three-body decays $B^+ \to D_s^+K\bar K$ in the perturbative QCD approach. The branching ratios for $B^+ \to D_s^+ϕ(1020) \to D_s^+ K^+K^-$ and $B^+ \to D_s^+ϕ(1020) \to D_s^+ K^0\bar{K}^0$ are predicted to be $(1.53\pm0.23) \times 10^{-7}$ and $(1.02^{+0.19}_{-0.13} )\times 10^{-7}$, respectively. The decay $B^+ \to D_s^+ϕ(1680)$ with $ϕ(1680)$ decays into $K^+K^-$ or $K^0\bar{K}^0$, has the branching fraction $(6.94^{+1.83}_{-2.02})\times 10^{-9}$, which is about $5\%$ of the result for $B^+ \to D_s^+ϕ(1020) \to D_s^+ K^+K^-$.

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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 $B \to D K^*(892) \to D K π$ in the perturbative QCD approach

We study the quasi-two-body decays $B\to D K^*(892) \to D Kπ$ by employing the perturbative QCD approach. The two-meson distribution amplitudes $Φ_{Kπ}^{\text{$P$-wave}}$ are adopted to describe the final state interactions of the kaon-pion pair in the resonance region. The resonance line shape for the $P$-wave $Kπ$ component $K^*(892)$ in the time-like form factor $F_{Kπ}(s)$ is parameterized by the relativistic Breit-Wigner function. For most considered decay modes, the theoretical predictions for their branching ratios are consistent with currently available experimental measurements within errors. We also disscuss some ratios of the branching fractions of the concerned decay processes. More precise data from LHCb and Belle-II are expected to test our predictions.

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arXiv2019

Quasi-two-body decays $B_{(s)}\to K^*(892)h\to Kπh$ in perturbative QCD approach

We extend our recent works on the $P$-wave two-pion resonant contributions to the kaon-pion cases in the hadronic charmless $B$ meson decays by employing the perturbative QCD approach. The concerned decay modes are analysed in the quasi-two-body framework by parameterizing the kaon-pion distribution amplitude $Φ_{K π}^{\rm P}$, which contains the final state interactions between the kaon and pion in the resonant region. The relativistic Breit-Wigner formula for the $P$-wave resonant state $K^*(892)$ is adopted to parameterize the time-like form factor $F_{Kπ}$. We calculate the $CP$-averaged branching ratios and direct $CP$-violating asymmetries of the quasi-two-body decays $B_{(s)}\to K^*(892) h\to Kπh$, with $h=(π, K)$, in this work. It is shown that the agreement of theoretical results with the experimental data can be achieved, through which Gegenbauer moments of the $P$-wave kaon-pion distribution amplitudes are determined. The predictions in this work will be tested by the precise data from the LHCb and the future Belle II experiments.

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arXiv2018

Quasi-two-body decays $B_{(s)}\to P f_2(1270)\to Pππ$ in the perturbative QCD approach

In this work, we calculate the $CP$-averaged branching ratios and direct $CP$-violating asymmetries of the quasi-two-body decays $B_{(s)}\to P f_2(1270)\to Pππ$ with the two-pion distribution amplitude $Φ_{ππ}^{\rm D}$ by using the perturbative QCD factorization approach, where $P$ represents a light pseudoscalar meson $K, π, η$ and $η^{\prime}$. The relativistic Breit-Wigner formula for the $D$-wave resonance $f_2(1270)$ is adopted to parameterize the timelike form factor $F_π$, which contains the final state interactions between the pions in the resonant regions. The consistency of theoretical results with data can be achieved by determining the Gegenbauer moments of the $D$-wave two-pion distribution amplitudes. The decay rates for the considered decay modes are generally in the order of $10^{-9}$ to $ 10^{-6}$. The integrated direct $CP$ asymmetries for the charged modes agree with the {\it BABAR} and Belle measurements. As a by-product, we extract the branching ratios of $B_{(s)}\to Pf_2(1270)$ from the corresponding quasi-two-body decay modes, which still need experimental tests at the ongoing and forthcoming experiments.

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arXiv2017

Quasi-two-body decays $B_c \to D_{(s)} [ρ(770),ρ(1450),ρ(1700) \to ] ππ$ in the perturbative QCD factorization approach

In this paper, we studied the quasi-two-body $B_c \to D_{(s)} [ ρ(770),ρ(1450),ρ(1700) \to ] ππ$ decays by employing the perturbative QCD (PQCD) factorization approach. The two-pion distribution amplitudes $Φ_{ππ}$ are applied to include the final-state interactions between the pion pair, while the time-like form factors $F_π(w^2)$ associated with the $P$-wave resonant states $ρ(770)$, $ρ(1450)$ and $ρ(1700)$ are extracted from the experimental data of the $e^+e^-$ annihilation. We found that: (a) the PQCD predictions for the branching ratios of the quasi-two-body $B_c \to D_{(s)} [ρ(770), ρ(1450), ρ(1700) \to ] ππ$ decays are in the order of $10^{-9}$ to $ 10^{-5}$ and the direct $CP$ violations around $(10-40)\%$ in magnitude; (b) the two sets of the large hierarchy $R_{1a,1b,1c}$ and $R_{2a,2b,2c}$ for the ratios of the branching ratios of the considered decays are defined and can be understood in the PQCD factorization approach, while the self-consistency between the quasi-two-body and two-body framework for $B_{c} \to D_{(s)} [ρ(770) \to ] ππ$ and $B_{c} \to D_{(s)} ρ(770)$ decays are confirmed by our numerical results; (c) taking currently known ${\cal B}(ρ(1450)\toππ)$ and ${\cal B}(ρ(1700) \to ππ)$ as input, we extracted the theoretical predictions for ${\cal B}(B_{c} \to D ρ(1450)) $ and ${\cal B}(B_{c} \to D ρ(1700))$ from the PQCD predictions for the decay rates of the quasi-two-body decays $B_{c} \to D [ρ(1450), ρ(1700) \to ] ππ$. All the PQCD predictions will be tested in the future experiments.

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arXiv2017

Quasi-two-body decays $B_{(s)} \to D (ρ(1450),ρ(1700)) \to D ππ$ in the perturbative QCD factorization approach

By employing a framework for the quasi-two-body decays in the perturbative QCD (PQCD) factorization approach, we calculate the branching ratios of the decays $B_{(s)} \to D (ρ(1450),ρ(1700))\to D ππ$ with $D=(D_{(s)}, \bar{D}_{(s)})$. The pion vector form factor $F_π$, acquired from a {\it BABAR} Collaboration analysis of $e^+ e^- \to π^+ π^-(γ)$ data, is involved in the two-pion distribution amplitudes $Φ^{I=1}_{ππ}$. The PQCD predictions for the branching ratios of the considered quasi-two-body decays are in the range of $10^{-10} \sim 10^{-4}$. The PQCD predictions for ${\cal B}(B^0\to \bar{D}^0 (ρ^0(1450),ρ^0(1700)) \to \bar{D}^0 π^+π^-)$ agree well with the measured values as reported by LHCb if one takes still large theoretical errors into account. Unlike the traditional way of the PQCD approach, one can extract the decay rates for the two-body decays $B_{(s)} \to D (ρ(1450),ρ(1700))$ from the results of the corresponding quasi-two-body decays. The PQCD predictions for ${\cal B}(B_{(s)} \to D ρ(1450))$ and ${\cal B}(B_{(s)} \to D ρ(770))$ are similar in magnitude: an interesting relation to be tested by future experimental measurements.

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arXiv2017

Quasi-two-body decays $B \to η_c {(1S ,2S)}\;[ρ(770),ρ(1450),ρ(1700) \to ]\; ππ$ in the perturbative QCD approach

In this paper, we calculated the branching ratios of the quasi-two-body decays $B \to η_c (1S ,2S)$ $[ρ(770), ρ(1450),ρ(1700)\to ] ππ$ by employing the perturbative QCD (PQCD) approach. The contributions from the $P$-wave resonances $ρ(770)$, $ρ(1450)$ and $ρ(1700)$ were taken into account. The two-pion distribution amplitude $Φ_{ππ}^{\rm P}$ is parameterized by the vector current time-like form factor $F_π$ to study the considered decay modes. We found that (a) the PQCD predictions for the branching ratios of the considered quasi-two-body decays are in the order of $10^{-7} \sim 10^{-6}$, while the two-body decay rates ${\cal B}(B \to η_c{(1S,2S)} (ρ(1450),ρ(1700)))$ are extracted from those for the corresponding quasi-two-body decays; (b) the whole pattern of the pion form factor-squared $|F_π|^2$ measured by the BABAR Collaboration could be understood based on our theoretical results; (c) the general expectation based on the similarity between $B \to η_c ππ$ and $B \to J/ψππ$ decays are confirmed: $R_2(η_c)\approx 0.45$ is consistent with the measured $R_2(J/ψ)\approx 0.56\pm 0.09$ within errors; and (d) new ratios $R_3(η_c(1S))$ and $R_4(η_c(2S))$ among the branching ratios of the considered decay modes are defined and could be tested by future experiments.

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arXiv2017

The quasi-two-body decays $B_{(s)} \to (D_{(s)},\bar{D}_{(s)}) ρ\to (D_{(s)}, \bar{D}_{(s)})ππ$ in the perturbative QCD factorization approach

In this paper, we studied the $B_{(s)} \to (D_{(s)},\bar{D}_{(s)}) ρ\to (D_{(s)}, \bar{D}_{(s)})ππ$ decays by employing a framework for the quasi-two-body decays in the perturbative QCD (PQCD) factorization approach. We use the two-pion distribution amplitudes $Φ_{ππ}$, which contains both resonant and nonresonant contributions from the pion pair, to describe the final state interactions (FSIs) between the pions in the resonant region. We found that (a) for all considered decays, the PQCD predictions for their branching ratios based on the quasi-two-body and the two-body framework agree well with each other due to ${\cal B}(ρ\to ππ) \approx 100\%$; For $B^+\to \bar{D^0}ρ^+\to \bar{D^0}π^+ π^0$ and other four considered decay modes, the PQCD predictions do agree well with the measured values within errors; (b) the great difference between the PQCD predictions for ${\cal B}(B\to \bar{D} ρ\to \bar{D} ππ)$ and ${\cal B}(B\to D ρ\to D ππ)$ can be understood by the strong CKM suppression factor $ R_{\rm CKM} \approx 3 \times 10^{-4}$; (c) for the $B_s\to D ρ\to D ππ$ and $B_s\to \bar{D} ρ\to \bar{D} ππ$ decays, however, the PQCD predictions of $R_{\rm s1}\approx 0.13$ and $R_{\rm s2}\approx 0.14$ do agree very well with the moderate CKM suppression factor $R^s_{\rm CKM}\approx 0.14$; and (d) the PQCD predictions for the ratios $R_{Dρ}$ and the strong phase difference $\cosδ_{Dρ}$ of the three $B \to \bar{D} ρ$ decay modes agree well with the LHCb measurements within one standard deviation.

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arXiv2017

Quasi-two-body decays $B_{(s)}\to Pρ^\prime(1450), Pρ^{\prime\prime}(1700)\to Pππ$ in the perturbative QCD approach

In this work, we calculate the $CP$-averaged branching ratios and direct $CP$-violating asymmetries of the quasi-two-body decays $B_{(s)}\to P ρ^\prime(1450), Pρ^{\prime\prime}(1700)\to P ππ$ by employing the perturbative QCD (PQCD) factorization approach, where $P$ is a light pseudoscalar meson $K, π, η$ and $η^{\prime}$. The considered decay modes are studied in the quasi-two-body framework by parameterizing the two-pion distribution amplitude $Φ_{ππ}^{\rm P}$. The $P$-wave time-like form factor $F_π$ in the resonant regions associated with the $ρ^\prime(1450)$ and $ρ^{\prime\prime}(1700)$ is estimated based on available experimental data. The PQCD predictions for the $CP$-averaged branching ratios of the decays $B_{(s)}\to Pρ^\prime(1450), Pρ^{\prime\prime}(1700)\to Pππ$ are in the order of $10^{-7} \sim 10^{-9}$. The branching ratios of the two-body decays $B_{(s)}\to Pρ^\prime(1450), Pρ^{\prime\prime}(1700)$ are extracted from the corresponding quasi-two-body decay modes. The whole pattern of the pion form factor-squared $|F_π|^2$ measured by {\it BABAR} Collaboration could also be understood based on our studies. Our PQCD predictions will be tested by the precise data from the LHCb and the future Belle II experiments.

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