Perturbative QCD Factorization Approach Research Articles

Quasi-two-body decays B→Pf0(500)→Pπ+π− in the perturbative QCD approach

In this paper, we study the quasi-two-body decays B→Pf0(500)→Pπ+π− [with P=(π,K,η,η′)] within framework of perturbative QCD (PQCD) factorization approach. With the help of π−π distribution amplitude and scalar form factor Fππ(ω2), we calculate the CP averaged branching fraction and the CP asymmetry for the quasi-two-body decays B→Pf0(500)→Pπ+π−. Taking the quasi-two-body decay B+→π+f0(500)→π+π+π− as an explicit example, we present the behavior of differential branching fraction and direct CP violation versus the π−π invariant mass. The total branching fraction and direct CP violation are B(B+→π+[σ→]π+π−)=(1.78±0.41±0.51)×10−6 and ACP(B+→π+[σ→]π+π−)=(29.8±11.1±13.0)% respectively. Our results could be tested by further experiments. Published by the American Physical Society 2024

Mixing effects of η−η′ in Λb→Λη(′) decays

We perform a thorough analysis of the $\eta-\eta'$ mixing effects on the $\Lambda_b\rightarrow \Lambda \eta^{(')}$ decays based on the perturbative QCD (PQCD) factorization approach. Branching ratios, up-down and direct $CP$ asymmetries are computed by considering four popular mixing schemes, such as $\eta-\eta'$, $\eta-\eta'-\eta_c$, $\eta-\eta'-G$, and $\eta-\eta'-G-\eta_c$ mixing formalism, where $G$ represents the physical pseudoscalar gluball. The PQCD predictions with the four mixing schemes does not change much for the $\eta$ channel but changes significantly for the $\eta'$ one. In particular, the value of $\mathcal{B}(\Lambda_b\rightarrow \Lambda \eta^{'})$ in the $\eta-\eta'-G-\eta_c$ mixing scheme exceeds the present experimental bound by a factor of 2, indicates the related mixing angles may be overestimated. Because of the distinctive patterns of interference between $S$-wave and $P$-wave amplitudes, the predicted up-down asymmetries for the two modes differ significantly. The obvious discrepancies among different theoretical analyses should be clarified in the future. The direct $CP$ violations are predicted to be at the level of a few percent mainly due to the tree contributions of the strange and nonstrange amplitudes suffer from the color suppression and CKM suppression. Finally, as a byproduct, we investigate the $\Lambda_b\rightarrow \Lambda \eta_c$ process, which has a large branching ratio of order $10^{-4}$, promising to be measured by the LHCb experiment. Our findings are useful for constraining the mixing parameters, comprehending the $\eta^{(')}$ configurations, and instructing experimental measurements.

Charmless two-body B meson decays in the perturbative QCD factorization approach* *Supported by the National Science Foundation of China (11805060, 11975112, 12105028, 12070131001, 11775117), the Joint Large-Scale Scientific Facility Funds of the NSFC and CAS (U1932110), the Natural Science Foundation of Jiangsu Province (BK20200980) and the National Key Research and Development Program of China (2020YFA0406400)

The perturbative quantum chromodynamics (PQCD) approach based on factorization has resulted in great achievements in the QCD calculation of hadronic B decays. By regulating the endpoint divergence by the transverse momentum of quarks in the propagators, one can perform the perturbation calculation for various diagrams, including annihilation type diagrams. In this paper, we review the current status of the PQCD factorization calculation of two-body charmless U decays up to next-to-leading order (NLO) QCD corrections. Two new power suppressed terms in the decaying amplitudes are also considered. Using universal input (non-perturbative) parameters, we collect the branching ratios and asymmetry parameters of all charmless two body B decays, which are calculated in the PQCD approach up to NLO. The results are compared with those of the QCD factorization approach, soft-collinear effective theory approach, and current experimental measurements. For most of the considered B meson decays, the PQCD results for branching ratios agree well with those of other approaches and experimental data. The PQCD predictions for the asymmetry parameters of many of the decay channels do not agree with those of other approaches but have better agreement with experimental data. The longstanding puzzle regarding the pattern of the direct CP asymmetries of penguin-dominated decays can be understood after the inclusion of NLO contributions in PQCD. The NLO corrections and power suppressed terms play an important role in color suppressed and pure annihilation type B decay modes. These rare decays are more sensitive to different types of corrections, providing an opportunity to examine the factorization approach with more precise experimental measurements.

Resonances ρ(1450)+ and ρ(1700)+ in B → DKKdecays * *Supported by the National Natural Science Foundation of China (11947011), the Natural Science Foundation of Jiangsu Province (BK20191010) and the Scientific Research Foundation of Nanjing Institute of Technology (YKJ201854)

The contributions for the kaon pair from the intermediate states and in the decays , , and 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 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 , which have the potential to be measured by experiments. It is also shown that the contributions of the subprocesses for the three-body B meson decays are considerable according to the total three-body branching fractions presented by Belle. Therefore, the decays , , and can be employed to study the properties of and in the LHCb and Belle-II experiments.

Pure annihilation decays of B s 0 → a 0 + a 0 − and B d 0 → in the PQCD approach

We study the CP-averaged branching fractions and the CP-violating asymmetries in the pure annihilation decays of and , where denotes the scalar a 0(980) and a 0(1450) [ and ], with the perturbative QCD factorization approach under the assumption of two-quark structure for the a 0 and states. The numerical results show that the branching ratios of the decays are in the order of 10−6, while the decay rates of the modes are in the order of 10−5. In light of the measured modes with the same quark components in the pseudoscalar sector, namely, and , the predictions for the considered decay modes in this work are expected to be measured at the Large Hadron Collider beauty and/or Belle-II experiments in the (near) future. Meanwhile, it is of great interest to find that the twist-3 distribution amplitudes ϕ S and ϕ T with inclusion of the Gegenbauer polynomials for the scalar a 0(1450) and states in scenario 2 contribute slightly to the branching ratios while significantly to the CP violations in the and decays, which indicates that, compared to the asymptotic ϕ S and ϕ T , these Gegenbauer polynomials could change the strong phases evidently in these pure annihilation decay channels. These predictions await for the future confirmation experimentally, which could further provide useful information to help explore the inner structure of the scalars and shed light on the annihilation decay mechanism.

Power expansion for heavy quarkonium production at next-to-leading order in e+e\u2212 annihilation

We study heavy quarkonium production associated with gluons in e+e− annihilation as an illustration of the perturbative QCD (pQCD) factorization approach, which incorporates the first nonleading power in the energy of the produced heavy quark pair. We show how the renormalization of the four-quark operators that define the heavy quark pair fragmentation functions using dimensional regularization induces “evanescent” operators that are absent in four dimensions. We derive closed forms for short-distance coefficients for quark pair production to next-to-leading order ( {alpha}_s^2 ) in the relevant color singlet and octet channels. Using non-relativistic QCD (NRQCD) to calculate the heavy quark pair fragmentation functions up to v4 in the velocity expansion, we derive analytical results for the differential energy fraction distribution of the heavy quarkonium. Calculations for {}^3{S}_1^{left[1right]} and {}^1{S}_0^{left[8right]} channels agree with analogous NRQCD analytical results available in the literature, while several color-octet calculations of energy fraction distributions are new. We show that the remaining corrections due to the heavy quark mass fall off rapidly in the energy of the produced state. To explore the importance of evolution at energies much larger than the mass of the heavy quark, we solve the renormalization group equation perturbatively to two-loop order for the {}^1{S}_0^{left[8right]} case.

P-wave contributions to B(s)→ψKπ decays in perturbative QCD approach * * Supported by the National Natural Science Foundation of China (11947013, 11605060, 11775117, 11547020); Ya Li is also Supported by the Natural Science Foundation of Jiangsu Province (BK20190508) and the Research Start-up Funding of Nanjing Agricultural University; Zhou Rui is Supported in part by the Natural Science Foundation of Hebei

We study the quasi-two-body decays by employing the perturbative QCD (PQCD) factorization approach, where the charmonia represents and . The corresponding decay channels are studied by constructing the kaon-pion distribution amplitude (DA) , which comprises important final state interactions between the kaon and pion in the resonant region. Relativistic Breit-Wigner formulas are adopted to parameterize the time-like form factor appearing in the kaon-pion DAs. The SU(3) flavor symmetry breaking effect resulting from the mass difference between the kaon and pion is taken into account, which makes significant contributions to the longitudinal polarizations. The observed branching ratios and the polarization fractions of are accommodated by tuning hadronic parameters for the kaon-pion DAs. The PQCD predictions for modes from the same set of parameters can be tested by precise data obtained in the future from LHCb and Belle II experiments.

B(s) → ηc(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/Bs→ηc(π,K,η(′),ρ,K⁎,ω,ϕ) 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→ηc(K,K⁎) and Bs→ηcϕ 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 R1,⋯,7 of the branching ratios defined among the properly selected pair of the considered decay modes, the PQCD predictions for the values of R3,4,5 agree well with those currently available measurements from BaBar and Belle Collaboration; (c) for B0→ηcKS0 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 R1,2 and R6,7 also agree with the general expectations and will be tested by the future experiments.

Anatomy of Bs → 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 B¯s0→PP decays (here P=(π,K,η,η′)) 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 B¯s0→(K0K¯0,K+K−,η′η′,π−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 B¯s0→π+π−, the PQCD prediction still remain consistent with the data after the inclusion of the small NLO reduction; (c) the PQCD predictions for the ratio (fs/fd)⋅B(Bs0→π+π−)/B(B0→K+π−) and (fs/fd)⋅B(Bs0→π+K−)/B(B0→K+π−) become agree very well with the measured ones after the inclusion of a 40% NLO reduction; (d) for B¯s0→K+K− and B¯s0→π−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 B¯s0→PP 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.

Quasi-two-body decays B(s)→PD0*(2400)→PDπ in the perturbative QCD approach

We study the quasi-two-body decays $B\to P D^{\ast}_0(2400) \to P D\pi $ with $P=(\pi, K, \eta, \eta^{\prime})$ 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 $R_{CKM}\approx \lambda^4 (\bar{\rho}^2 + \bar{\eta}^2) \approx 3\times 10^{-4}$ 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 $B^0 \to \bar{D}_0^{*0} K^0\to D^-\pi^+K^0$ and $B^0 \to \bar{D}_0^{*0}\pi^0 \to D^-\pi^+\pi^0$ 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 $\mathcal{B}(B_s^0\to D_0^{*+}K^-\to D^0\pi^+K^-)$ and $\mathcal{B}(B^0\to D_0^{*+} \pi^-\to D^0\pi^+ \pi^-)$ and to be $1.03^{+0.06}_{-0.07}$ between $\mathcal{B}(B_s^0\to\bar{D}_0^{*0} \bar{K}^0\to D^-\pi^+ \bar{K}^0)$ and $2\mathcal{B}(B^0\to \bar{D}_0^{*0}\pi^0\to D^-\pi^+\pi^0)$. The predictions in this work can be tested by the future experiments.

Anatomy of Bs → 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 B¯s0→VV 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 B¯s0→ϕϕ,K⁎0ϕ,K¯⁎0K⁎0 and ρ0ϕ, the NLO contributions can provide ∼20% to ∼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 (fL,f⊥) and (ϕ‖,ϕ⊥) are generally small in size, but the weak penguin annihilation contributions play an important role in understanding the data about their decay rates, fL and f⊥; (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 Bs0→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.

Subleading-power corrections to the radiative leptonic B \u2192 \u03b3\u2113\u03bd decay in QCD

Applying the method of light-cone sum rules with photon distribution amplitudes, we compute the subleading-power correction to the radiative leptonic B → γℓν decay from the twist-two hadronic photon contribution at next-to-leading order in QCD; and further evaluate the higher-twist “resolved photon” corrections at leading order in αs, up to twist-four accuracy. QCD factorization for the vacuum-to-photon correlation function with an interpolating current for the B-meson is established explicitly at leading power in Λ/mb employing the evanescent operator approach. Resummation of the parametrically large logarithms of mb2/Λ2 entering the hard function of the leading-twist factorization formula is achieved by solving the QCD evolution equation for the light-ray tensor operator at two loops. The leading-twist hadronic photon effect turns out to preserve the symmetry relation between the two B → γ form factors due to the helicity conservation, however, the higher-twist hadronic photon corrections can yield symmetry-breaking effect already at tree level in QCD. Using the conformal expansion of photon distribution amplitudes with the non-perturbative parameters estimated from QCD sum rules, the twist-two hadronic photon contribution can give rise to approximately 30% correction to the leading-power “direct photon” effect computed from the perturbative QCD factorization approach. In contrast, the subleading-power corrections from the higher-twist two-particle and three-particle photon distribution amplitudes are estimated to be of mathcal{O}left(3sim 5%right) with the light-cone sum rule approach. We further predict the partial branching fractions of B → γℓν with a photon-energy cut Eγ ≥ Ecut, which are of interest for determining the inverse moment of the leading-twist B-meson distribution amplitude thanks to the forthcoming high-luminosity Belle II experiment at KEK.

Anatomy of Bs → 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 B¯s0→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 ∼40% enhancement to the LO PQCD predictions for B(B¯s0→K0K¯⁎0) and B(B¯s0→K±K⁎∓), or a ∼37% reduction to B(B¯s0→π−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 Bs0→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 B¯s0→K+ρ− and the “color-suppressed-tree” decay B¯s0→π0K⁎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 AT,C and AP: constructive for the first decay but destructive for the second one; and (d) for B¯s0→V(η,η′) 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 A(Vηq) and A(Vηs) due to the η–η′ mixing.

Quasi-two-body decays Bc → D(s)[ρ(770),ρ(1450),ρ(1700)→]ππ in the perturbative QCD factorization approach

In this paper, we studied the quasi-two-body Bc→D(s)[ρ(770),ρ(1450),ρ(1700)→]ππ decays by employing the perturbative QCD (PQCD) factorization approach. The two-pion distribution amplitudes Φππ are applied to include the final-state interactions between the pion pair, while the time-like form factors Fπ(w2) 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 Bc→D(s)[ρ(770),ρ(1450),ρ(1700)→]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 R1a,1b,1c and R2a,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 Bc→D(s)[ρ(770)→]ππ and Bc→D(s)ρ(770) decays are confirmed by our numerical results; (c) taking currently known B(ρ(1450)→ππ) and B(ρ(1700)→ππ) as input, we extracted the theoretical predictions for B(Bc→Dρ(1450)) and B(Bc→Dρ(1700)) from the PQCD predictions for the decay rates of the quasi-two-body decays Bc→D[ρ(1450),ρ(1700)→]ππ. All the PQCD predictions will be tested in the future experiments.

Quasi-two-body decays B(s)→D(ρ(1450), ρ(1700))→Dππ 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 $B_{(s)} \to D (\rho(1450),\rho(1700))\to D \pi \pi$ with $D=(D_{(s)}, \bar{D}_{(s)})$. The pion vector form factor $F_\pi$, acquired from a {\it BABAR} Collaboration analysis of $e^+ e^- \to \pi^+ \pi^-(\gamma)$ data, is involved in the two-pion distribution amplitudes $\Phi^{I=1}_{\pi\pi}$. 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 ${\cal B}(B^0\to \bar{D}^0 (\rho^0(1450),\rho^0(1700)) \to \bar{D}^0 \pi^+\pi^-)$ 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 $B_{(s)} \to D (\rho(1450),\rho(1700))$ from the results of the corresponding quasi-two-body decays. The PQCD predictions for ${\cal B}(B_{(s)} \to D \rho(1450))$ and ${\cal B}(B_{(s)} \to D \rho(770))$ are similar in magnitude: an interesting relation to be tested by future experimental measurements.

Branching ratios, CP asymmetries and polarizations of Brightarrow psi (2S) V decays

We analyze the non-leptonic decays B/B_srightarrow psi (2S) V with V=(rho , omega , K^{*}, phi ) by employing the perturbative QCD (pQCD) factorization approach. Here the branching ratios, the CP asymmetries and the complete set of polarization observables are investigated systematically. Besides the traditional contributions from the factorizable and non-factorizable diagrams at the leading order, the next-to-leading order (NLO) vertex corrections could also provide considerable contributions. The pQCD predictions for the branching ratios of the B_{(s)}rightarrow psi (2S)K^{*}, psi (2S)phi decays are consistent with the measured values within errors. As for Brightarrow psi (2S) rho , psi (2S) omega decays, the branching ratios can reach the order of 10^{-5} and could be measured in the LHCb and Belle-II experiments. The numerical results show that the direct CP asymmetries of the considered decays are very small. Thus the observation of any large direct CP asymmetry for these decays will be a signal for new physics. The mixing-induced CP asymmetries in the neutral modes are very close to sin 2beta _{(s)}, which suggests that these channels can give a cross-check on the measurement of the Cabbibo–Kobayashi–Maskawa (CKM) angle beta and beta _s. We find that the longitudinal polarization fractions f_0 are suppressed to sim 50% due to the large non-factorizable contributions. The magnitudes and phases of the two transverse amplitudes mathcal {A}_{parallel } and mathcal {A}_{perp } are roughly equal, which is an indication for the approximate light-quark helicity conservation in these decays. The overall polarization observables of Brightarrow psi (2S) K^{*0} and B_srightarrow psi (2S) phi channels are also in good agreement with the experimental measurements as reported by LHCb and BaBar. Other results can also be tested by the LHCb and Belle-II experiments.

Quasi-two-body decays B(s)→Pρ′(1450),Pρ′′(1700)→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 $B_{(s)}\to P \rho^\prime(1450), P\rho^{\prime\prime}(1700)\to P \pi\pi$ by employing the perturbative QCD (PQCD) factorization approach, where $P$ is a light pseudoscalar meson $K, \pi, \eta$ and $\eta^{\prime}$. The considered decay modes are studied in the quasi-two-body framework by parameterizing the two-pion distribution amplitude $\Phi_{\pi\pi}^{\rm P}$. The $P$-wave time-like form factor $F_{\pi}$ in the resonant regions associated with the $\rho^\prime(1450)$ and $\rho^{\prime\prime}(1700)$ is estimated based on available experimental data. The PQCD predictions for the $CP$-averaged branching ratios of the decays $B_{(s)}\to P\rho^\prime(1450), P\rho^{\prime\prime}(1700)\to P\pi\pi$ are in the order of $10^{-7} \sim 10^{-9}$. The branching ratios of the two-body decays $B_{(s)}\to P\rho^\prime(1450), P\rho^{\prime\prime}(1700)$ are extracted from the corresponding quasi-two-body decay modes. The whole pattern of the pion form factor-squared $|F_\pi|^2$ 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.

S-wave resonance contributions to in the perturbative QCD factorization approach**Supported by National Natural Science Foundation of China (11235005, 11547038)

By employing the perturbative QCD (PQCD) factorization approach, we study the quasi-two-body decays, where the pion pair comes from the S-wave resonance f0(X). The Breit–Wigner formula for the f0(500) and f0(1500) resonances and the Flatté model for the f0(980) resonance are adopted to parameterize the time-like scalar form factors in the two-pion distribution amplitudes. As a comparison, Bugg's model is also used for the wide f0(500) in this work. For decay rates, we found the following PQCD predictions: (a) when the contributions from f0(980) and f0(1500) are all taken into account; (b) in the Breit-Wigner model and in Bugg's model.

The quasi-two-body decays [formula omitted] in the perturbative QCD factorization approach

In this paper, we studied the B(s)→(D(s),D¯(s))ρ→(D(s),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 Φππ, 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 B(ρ→ππ)≈100%; For B+→D0¯ρ+→D0¯π+π0 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 B(B→D¯ρ→D¯ππ) and B(B→Dρ→Dππ) can be understood by the strong CKM suppression factor RCKM≈3×10−4; (c) for the Bs→Dρ→Dππ and Bs→D¯ρ→D¯ππ decays, however, the PQCD predictions of Rs1≈0.13 and Rs2≈0.14 do agree very well with the moderate CKM suppression factor RCKMs≈0.14; and (d) the PQCD predictions for the ratios RDρ and the strong phase difference cos⁡δDρ of the three B→D¯ρ decay modes agree well with the LHCb measurements within one standard deviation.

Decays with perturbative QCD approach

Nonleptonic two-body weak decays are studied phenomenologically with a perturbative QCD factorization approach. It is found that the , and decays have branching ratios 10−9, and might be promisingly measurable in the running of LHC and forthcoming SuperKEKB experiments in the future.