$B_{d,s}^0 \to f_1 f_1$ decays with $f_1(1285)-f_1(1420)$ mixing in the perturbative QCD approach

We investigate the $B_{d,s}^0 \to f_1 f_1$ decays in the framework of perturbative QCD(PQCD) approach with a referenced value $\phi_{f_1} \sim 24^\circ$. Here, $f_1$ denotes the axial-vector mesons $f_1(1285)$ and $f_1(1420)$ with mixing angle $\phi_{f_1}$ in the quark-flavor basis. The observables such as branching ratios, direct CP violations, and polarization fractions of the $B_s^0 \to f_1 f_1$ decays are predicted for the first time. We find that: (a) the almost pure penguin modes $B_s^0 \to f_1 f_1$ have large branching ratios in the order of $10^{-6} \sim 10^{-5}$ due to the Cabibbo-Kobayashi-Maskawa enhancement and generally constructive interferences between the amplitudes of $B_s^0 \to f_n f_s$ and $B_s^0 \to f_s f_s$ with $f_n$ and $f_s$ being the quark-flavor states of $f_1$ mesons; (b) the observables receive important contributions from the weak annihilation diagrams in the PQCD approach. In particular, without the annihilation contributions, the $B_s^0 \to f_1(1420) f_1(1420)$ branching ratio will decrease about 81% and its longitudinal polarization fraction will reduce around 43%; (c) the dependence of the $B_{d,s}^0 \to f_1 f_1$ decay rates on $\phi_{f_1}$ exhibits some interesting line-shapes, whose confirmations would be helpful to constrain the determination of $\phi_{f_1}$ inversely. All the PQCD predictions await for the (near) future examinations at Large Hadron Collider beauty and/or Belle-II experiments to further understand the properties of the axial-vector mesons and the perturbative dynamics released from the considered decay modes.

with a mixing angle φ f1 , which is correlated with the angle θ f1 in the singlet-octet basis via the following relation, Here, θ i is the "ideal" mixing angle with the value θ i = 35.3 • . It is therefore clear to see that φ f1 could be as a probe to examine the deviation from ideal mixing. On one hand, the definite understanding of this φ f1 (or θ f1 ) could shed light on the structure of these two f 1 mesons; On the other hand, it is of great interest to note that, as one of the three important mixing angles in the sector of axial-vector mesons, φ f1 (or θ f1 ) has the potential to help constrain the distinct mixing between K 1A and K 1B states with angle θ K1 [1,5], where the former is a 3 P 1 state while the latter is a 1 P 1 one. It means that the good constraints on φ f1 (or θ f1 ) could indirectly pin down the θ K1 to better investigate the structure of K 1 (1270) and K 1 (1400) mesons [6][7][8][9][10].
Up to now, there are several explorations on the φ f1 (or θ f1 ) at both theoretical and experimental aspects [4-6, 9, 11-24]. One cannot yet determine definitely its value due to limited understanding on the nature of these two f 1 states, although, about seven years ago, the Large Hadron Collider beauty(LHCb) collaboration extracted experimentally φ f1 = (24.0 +3.1+0. 6 −2.6−0.8 ) • with a two-fold ambiguity from the B 0 d,s → J/ψf 1 (1285) decays for the first time [4]. Because there are no interferences between the flavor f n and f s states in this type of decay modes, this ambiguity is expected to be settled in the decay modes with significantly constructive or destructive interferences between those two flavor states, for example, in the B (s) → f 1 P decays [10], the B (s) → f 1 V [25] modes, and other B (s) /D (s) → f 1 M (M stands for the possible mesons) channels. However, it is worth pointing out that the D (s) → f 1 M decays cannot yet be perturbatively calculated based on the QCD theory. Hence, those relevant D (s) meson decays have to be left for future studies elsewhere. In this work, we will study the B 0 d,s → f 1 f 1 decays in the perturbative QCD(PQCD) approach [26] based on the k T factorization theorem at leading order 2 . The significant interferences among the B 0 d,s → f n f n , f n f s , and f s f s decay amplitudes could be observed in the considered modes, just like those in the pseudoscalar B 0 d,s → η (′) η (′) cases [32,33]. As discussed in Ref. [10], due to the consistency between the latest calculations from Lattice QCD [24] and the current measurement from LHCb [4], we will adopt φ f1 = 24 • as a referenced value to make quantitative evaluations and phenomenological discussions.
In the literature, the B 0 d → f 1 f 1 decays have been investigated in the QCDF approach, and the decay rates and the longitudinal polarization fractions have been collected in the Table X of Ref. [22]. However, the predicted branching ratios are too small to be measured in the near future at LHCb and/or Belle-II experiments. Compared to these Cabibbo-Kobayashi-Maskawa(CKM) suppressed B 0 d → f 1 f 1 modes, the CKM favored B 0 s → f 1 f 1 ones are expected to be measurable with possibly large decay rates due to the naive enhancement of | Vts V td | 2 ∼ 20 for both penguin-dominated channels or of | VtsV tb V ub V ud | 2 ∼ 100 for the penguindominated B 0 s while the tree-dominated B 0 d decays, apart from the possibly constructive interferences in the B 0 s → f 1 f 1 decays. To our best knowledge, the B 0 s → f 1 f 1 decays presented in this work are studied theoretically for the first time in the literature. Moreover, as discussed in Ref. [22], power corrections in QCDF always involve troublesome end-point divergences. Therefore, more parameters are introduced to parametrize the contributions arising from the non-factorizable emission and the annihilation diagrams [34], which results in large theoretical uncertainties. Objectively speaking, the QCDF approach is a powerful tool for analyzing the B meson decays by global fitting to the data. But, the data-fitting and/or model-dependent parametrization always make it lose the predictive power more or less.
The PQCD approach we adopted in this work is one of the important and popular factorization methods based on QCD dynamics. It is known that the PQCD approach, based on the k T factorization theorem, is free of end-point divergences by keeping quarks' transverse momentum and the Sudakov formalism makes it more self-consistent. Thus, the PQCD approach doesn't need to introduce any other parameters, except for the essential non-perturbative inputs, namely, wave functions or distribution amplitudes for the initial and final mesons. Note that, these inputs are universal and are usually computed in the nonperturbative techniques such as QCD sum rules and Lattice QCD, or extracted from the available experimental data. A distinct advantage of the PQCD approach is that one can really do the quantitative calculations of form factor, non-factorizable emission and annihilation type diagrams, apart from the factorizable emission ones. It is worth addressing that one have realized the importance of annihilation contributions in the heavy flavor B and D meson decays, for example, the predictions of CP-violating asymmetries of B 0 d → π ± π ∓ , K ± π ∓ decays [26,35], the explanations to polarization problem of B → φK * modes [36][37][38], and the explorations of phenomenologies of D 0 → π ± π ∓ , K ± K ∓ channels [39], and so forth. And what's more, the confirmation from LHCb experiment on the pure annihilation B 0 d → K + K − and B 0 s → π + π − decay rates predicted in the PQCD approach are very exciting [40,41]. Actually, the PQCD predictions for the B → P P , P V , and V V decays have shown good consistency globally with the existing data within errors. It means that the PQCD approach has the unique advantage and general reliability at the aspects of calculating the hadronic matrix elements in the heavy B meson decays. The interested readers could refer to the review article [26] for more details about this PQCD approach.
The decay amplitude for B 0 d,s → f 1 f 1 decays in the PQCD approach can be conceptually written as follows: in which, x i (i = 1, 2, 3) is the momentum fraction of the valence quark in the initial and final state mesons; b i is the conjugate space coordinate of the transverse momentum k iT ; Tr denotes the trace over Dirac and SU(3) color indices; C(t) stands for the Wilson coefficients including the large logarithms ln(m W /t) [26]; t is the largest running energy scale in hard kernel H(x i , b i , t); and Φ is the wave function describing the hadronization of quark and anti-quark to a meson (The explicit form of the involved wave functions associated with the distribution amplitudes can be found later in the Appendix A ). The jet function S t (x i ) comes from threshold resummation, which exhibits a strong suppression effect in the small x region [42,43], while the Sudakov factor e −S(t) arises from k T resummation, which provides a strong suppression in the small k T (or large b) region [44,45]. These resummation effects therefore guarantee the removal of the end-point singularities. The detailed expressions for S t (x i ) and e −S(t) can be easily found in Refs. [42][43][44][45]. Note that, to keep the consistency, we will use the leading order Wilson coefficients in the following calculations. For the renormalization group evolution of the Wilson coefficients from higher scale to lower scale, we will adopt the formulas in Ref. [26] directly. For the B 0 d,s → f 1 f 1 decays, the related weak effective Hamiltonian H eff can be read as [46] in which, q = d or s, G F = 1.16639 × 10 −5 GeV −2 is the Fermi constant, V denotes the CKM matrix elements, and C i (µ) stands for Wilson coefficients at the renormalization scale µ. The local four-quark operators O i (i = 1, · · · , 10) are written as (2) QCD penguin operators (3) Electroweak penguin operators with the color indices α, β and the notations ( The index q ′ in the summation of the above operators runs through u, d, s, c, and b. As illustrated in Fig. 1, it is easy to find that the considered B 0 d,s → f 1 f 1 decays contain two kinds of topologies of the diagrams, namely, the emission one and the annihilation one, which include eight types of diagrams in the PQCD approach at leading order: (i) factorizable [non-factorizable] emission diagrams Fig. 1 7), we can straightforwardly calculate the contributions in the PQCD approach. Hereafter, for the sake of simplicity, we will adopt F and F P1 (M and M P1 ) to denote the factorizable (non-factorizable) Feynman amplitudes induced by the (V − A)(V − A) and (V − A)(V + A) operators, and F P2 (M P2 ) to denote the factorizable (non-factorizable) Feynman amplitudes from the (S − P )(S + P ) operators, which are resulted from a Fierz transformation of the (V − A)(V + A) ones.
A remark is in order for the Feynman amplitudes: The Feynman amplitudes for the B meson decaying into two axial-vector mesons have been collected in [47]. In this work, it is not necessary for us to list the same calculations existed in the literature. The interested readers could refer to Eqs. (25)-(60) [47] for detail. In Ref. [48], the authors studied the B (s) → V V decays by keeping the higher power terms proportional to r 2 V = m 2 V /m 2 B in the denominator of propagators for virtual quarks and gluons, which resulted in the predictions for most branching ratios and polarization fractions in the PQCD approach being in good agreement with the existing measurements. In light of this success, we would like to retain the terms proportional to in the B 0 d,s → f 1 f 1 decays too. In fact, we have also taken this strategy into account in the studies of B → f 1 V decays [25].
Together with various contributions from different diagrams as presented in Eqs. (25)-(60) [47] and the quark-flavor mixing scheme as shown in Eq. (1), the decay amplitudes of six B 0 d,s → f 1 f 1 channels can thus be written in terms of the combinations of B 0 d,s → f n f n , f n f s , and f s f s with different coefficients as follows (the superscript h in the following formulas stands for the helicity amplitudes with longitudinal(L), normal(N ), and transverse(T ) polarizations, respectively): The decay amplitudes for the B 0 d meson decaying into the flavor states f n f n , f n f s , and f s f s can be easily written as follows, In the above formulas, i.e., Eqs. (8)-(10), the subscripts "(n)f e" and "(n)f a" are the abbreviations of (non-)factorizable emission and (non-)factorizable annihilation, and a i is the standard combination of the Wilson coefficients C i defined as follows: 5,7,9), where C 2 ∼ 1 is the largest one among all the Wilson coefficients.
The decay amplitudes for the physical states are then 2. For B 0 s → f 1 f 1 decays Analogously, the decay amplitudes of B 0 s → f n f n , f n f s , and f s f s can be written as, Then, we could give the decay amplitudes for the physical states similarly, Now, we will perform the numerical calculations in the PQCD approach on the experimental observables such as the CPaveraged branching ratios(B), the direct CP-violating asymmetries(A dir CP ), and the CP-averaged polarization fractions, etc. for the considered B 0 d,s → f 1 f 1 decays. Some essential comments on the input parameters are in order: (a) Distribution amplitudes for the flavor states f n and f s As discussed in [23], the 3 P 1 -axial-vector meson has the similar behavior to the vector one. Meanwhile, it is noted that, for the distribution amplitudes, the flavor η n and η s states of η (′) usually took the same form as pion but with different decay constants f ηn and f ηs in the pseudoscalar sector. Therefore, for the flavor states f n and f s in this work, we shall adopt the the same distribution amplitudes as those of the a 1 (1260) meson. The decay constants f fn and f fs , and the relevant Gegenbauer moments can be easily found in Refs. [9,23,25,49,50].
A. CP-averaged branching ratios Similar to the B → f 1 V decays [25], the B 0 d,s → f 1 f 1 decay rate can also be written as where |P c | ≡ |P 2z | = |P 3z | is the momentum of either the outgoing axial-vector meson and A h can be found, for example, in Eqs. (19)- (21). The corresponding branching ratios B can thus be easily obtained through the relation B = τ B 0 d,s Γ.  (1420) decays obtained in the PQCD approach, where the errors are sequentially from the shape parameter ωB, the decay constants fM , the Gegenbauer moments a f 1 , the mixing angle φ f 1 , the higher order corrections factor at, and the CKM parameters V .

Decay Modes
The numerical results predicted in the PQCD approach for the observables, specifically, branching ratios, direct CP violations, and polarization fractions associated with the theoretical errors are collected in Tables I-III. As for the errors, they are mainly induced by the uncertainties of the shape parameter ω B = 0.40 ± 0.04 (ω B = 0.50 ± 0.05) GeV in the B 0 d (B 0 s ) meson distribution amplitude, of the combined decay constants f M from the 3 P 1 -axial-vector state as f fn = 0.193 +0.043 −0.038 GeV and f fs = 0.230 ± 0.009 GeV, of the combined Gegenbauer moments a f1 from a 2 and a ⊥ 1 in the f n and f s state distribution amplitudes, of the mixing angle φ f1 = (24.0 +3.2 −2.7 ) • for the f 1 (1285) − f 1 (1420) mixing system in the quark-flavor basis, of the maximal running hard scale t max = (1.0 ± 0.2) t 3 , and of the combined CKM matrix elements V from the parametersρ andη, respectively.
Based on the effective Hamiltonian as shown in Eq. (4), it is clear to see that, at the quark level, the B 0 d → f 1 f 1 decays are the ∆S = 0 (here, the capital S describes the strange flavor number) type modes with theb →d transition, while the B 0 s → f 1 f 1 ones are the ∆S = 1 type channels with theb →s transition, where the former is CKM suppressed, and the latter is, however, CKM favored. Then, as generally expected, the B 0 s → f 1 f 1 decay rates are much larger than the B 0 d → f 1 f 1 ones with different extents due to the CKM enhancement and the constructive/destructive interferences among the flavor f n f n , f n f s , and f s f s final states. The B(B 0 d,s → f 1 f 1 ) predicted in the PQCD approach can confirm this expectation numerically. One can see the predictions as presented explicitly in the Tables I-III. Within a bit large theoretical uncertainties, the branching ratios of B 0 d,s → f 1 f 1 decays in the PQCD approach can be read as follows, where all the errors arising from the input parameters have been added in quadrature.  As aforementioned, the B 0 d → f 1 f 1 decays have been investigated in the QCDF approach [22]. The numerical results with large errors presented in [22] can be read as follows 4 : The largest errors are from the parametrized hard spectator scattering and annihilation diagrams, as mentioned in [22]. Note that the parametrization of these contributions are inferred from those in the B → V V decays in the QCDF approach due to the similar behavior between the vector meson and the 3 P 1 axial-vector one. One can see that the B 0 d → f 1 f 1 decay rates predicted in the PQCD and QCDF approaches are roughly consistent with each other within large uncertainties, although, in terms of the central values, the branching ratios of the latter two modes in the QCDF approach are smaller than those in the PQCD approach with one order.
According to the mixing scheme in Eq. (1) with referenced value φ f1 = 24 • , one can easily find that the mode is tree-(penguin-)diagramdominant, but with only a few percent of penguin (tree) contaminations. For the B 0 d → f 1 (1285)f 1 (1420) channel, both of the tree diagrams and the penguin ones contribute evidently to the decay rate simultaneously. The decay amplitudes induced by the tree diagrams and the penguin diagrams for the B 0 d → f 1 f 1 decays have been collected and can be seen clearly in the Table IV. To clarify the above expectations, we present the CP-averaged branching ratios in the PQCD approach without considering the tree contributions for the considered B 0

Decay Modes
Tree diagrams Penguin diagrams Tree diagrams Penguin diagrams Tree diagrams Penguin diagrams In principle, the B 0 d → f 1 f 1 decay rates could be accessible at the LHCb and/or Belle-II experiments with accumulating a large number of B 0 dB 0 d events in the near future, after all, the B 0 d → K + K − with decay rate 1.3 ± 0.5 × 10 −7 and B 0 s → π + π − with branching ratio 7.6 ± 1.9 × 10 −7 [1,40,53] have been measured at LHCb. But, by considering the secondary decay process of f 1 (1285), namely, B(f 1 (1285) → ηπ + π − ) ∼ 35% [1] or B(f 1 (1285) → 2π 0 π + π − ) ∼ 22.3% [1], then the with the branching ratios under the narrow width approximation, It seems that the above two results are too small to be measured experimentally in the near future. However, the B 0 s → f 1 f 1 decays have large branching ratios in the order of 10 −6 ∼ 10 −5 , which are expected to be measured in the near future at LHCb and Belle-II experiments. Unlike the B 0 d → f 1 f 1 decays, the B 0 s → f 1 f 1 ones are almost dominated by the pure penguin contributions just with the tiny while negligible tree pollution, which can be clearly seen from the decay amplitudes presented in the Table V. Furthermore, when the contributions from tree diagrams are turned off for the B 0 s → f 1 f 1 decays, the CP-averaged branching ratios, in terms of the central values, will change slightly as follows, Relative to the dominant f 1 (1285) → ηππ decay, the decay rate for the dominant f 1 (1420) → KKπ process is not yet available now 5 . Therefore, it is not easy to exactly estimate the branching ratios of B 0 s → f 1 (1285)f 1 (1420) and f 1 (1420)f 1 (1420) decays via the resonant channels B 0 . Fortunately, as discussed in Ref. [54], the only decay modes of the f 1 (1420) were assumed asKK * , a 0 (980)π, and φγ, and the decay rate of f 1 (1420) → K * K was given as about 96%, then the branching ratio could be naively assumed as B(f 1 (1420) → K 0 S K ± π ∓ ) ≈ 64%. Then, similarly, under the narrow width approximation, the B 0 processes have the branching ratios as follows, Certainly, these two large values as given in Eqs. (35) and (36) are believed to be detectable at LHCb experiments, as well as at Belle-II ones in the near future.
Tree diagrams Penguin diagrams Tree diagrams Penguin diagrams Tree diagrams Penguin diagrams Different from the ideal mixing between ω and φ in the vector sector, both of f 1 (1285) and f 1 (1420) mesons have some admixtures of f s and f n correspondingly. Therefore, though the similarity of the distribution amplitudes between the f 1 states and the ω(φ) mesons has been observed [22], relative to the B 0 d,s → ω(φ)ω(φ) and B 0 d,s → f 1 ω(φ) decays, the more complicated interferences among the B 0 d,s → f n f n , f n f s , and f s f s are involved in the B 0 d,s → f 1 f 1 decays, as presented explicitly in the Eqs. (13)- (15) and Eqs. (19)- (21). In other words, it is not easy to naively anticipate the constructive or destructive interferences in the B 0 d,s → f 1 f 1 decays just like those in the B 0 d,s → ω(φ)ω(φ), and B 0 d,s → f 1 ω(φ) ones. But, as observed in the Table VI, the B 0 s → f n f n channel has a small longitudinal while two tiny and negligible transverse amplitudes, which would make the interferences in the B 0 s → f 1 f 1 decays more easy to be explored. Therefore, it can still be expected that the nearly pure penguin B 0 s → f 1 (1420)f 1 (1420) decay with a large branching ratio could provide useful information to constrain the B 0 s −B 0 s mixing phase, even to find new physics signal beyond the standard model complementarily.
As clearly seen in the Tables I-III, the numerical results calculated in the PQCD approach suffer from large uncertainties induced by the less constrained distribution amplitudes of the involved hadrons. At the same time, frankly speaking, it is worthy of stressing that the large errors presented in the Tables I-III for the considered B 0 d,s → f 1 f 1 decays are mainly induced by the decay constants, Gegenbauer moments, even the mixing angle of the f 1 mesons. In light of these large uncertainties, we then define some interesting ratios of the branching ratios for the decay modes. As generally expected, if the modes in a ratio have 5 Due to the currently unknown nature [1] and the different understanding [55] of the f 1 (1420), we just take the absolutely dominant mode, i.e., f 1 (1420) → KK * into account. Then, by including the secondary decay chain K * → Kπ under the narrow width approximation, the branching ratio of the strong decay f 1 (1420) →KK * → K 0 S K ± π ∓ could be naively estimated as B(f 1 (1420) →KK * ) · B(K * → K ± π ∓ ).  similar dependence on a specific input parameter, the error induced by the uncertainty of this input parameter will be largely canceled in the ratio, even if one cannot make an explicit factorization for this parameter. Furthermore, from the experimental side, we know that the ratios of the branching ratios generally could be measured with a better accuracy than that for the individual branching ratios. The relevant ratios about the decay rates of the considered B 0 d,s → f 1 f 1 decays can be read as follows: Generally speaking, it should be noted that the errors arising from the parameters, in particular, f fn and f fs , and a || 2 and a ⊥ 1 , cannot be reduced effectively in the ratios because of the significant interferences among the decay amplitudes of B 0 d,s → f n f n , f n f s , and f s f s . Nevertheless, we still expect that the LHCb and/or Belle-II experiments could perform the measurements with enough precision on these ratios in the future, in order to give some essential constraints on the input parameters or the mixing angle φ f1 .  Table VI, the decay amplitudes from B 0 d → f s f s and B 0 s → f n f n in the longitudinal polarization also contribute to the B 0 d,s → f 1 f 1 decays. Therefore, it is expected that, if the precise f n and f s distribution amplitudes are available, then the mixing angle φ f1 could be constrained by the near future measurements on the large B 0 s → f 1 f 1 decay rates associated with the interferences as exhibited in Fig. 2, and vice versa.

B. CP-averaged polarization fractions
Now, we turn to the calculations for the polarization fractions of the B 0 d,s → f 1 f 1 decays. Based on the helicity amplitudes A i (i = L, N, T ), we can equivalently define the amplitudes in the transversity basis as follows: for the longitudinal, parallel, and perpendicular polarizations, respectively, with the normalization factor ξ = G 2 F P c /(16πm 2 B Γ) and the ratio r = P 2 · P 3 /(m f1 · m f1 ). These amplitudes satisfy the relation, following the summation in Eq. (23). Since the transverse-helicity contributions can manifest themselves through polarization observables, we therefore define CP-averaged fractions in three polarizations f L , f , and f ⊥ as the following, With the above transversity amplitudes shown in Eq. (46), the relative phases φ and φ ⊥ can be defined as From the Tables I-III, one can clearly find that four of the considered B 0 d,s → f 1 f 1 decays are dominated by the transverse contributions, while the other two are governed by the longitudinal ones, whose values for the polarization fractions f L and f T (= 1 − f L ) can be explicitly read as follows, and in which, all the errors from various parameters have been added in quadrature. These predicted CP-averaged polarization fractions will be tested at LHCb and/or Belle-II to further explore the decay mechanism with helicities associated with experimental confirmations on the decay rates. In Ref. [22], the longitudinal polarization fractions of the B 0 d → f 1 f 1 decays have been calculated in the QCDF approach, As far as the central values are considered, the results in the QCDF approach exhibit the dominance of the longitudinal decay amplitudes, which are very contrary to those in the PQCD approach at leading order. However, when the large uncertainties are taken into account, then one can find that the transverse contributions can also govern these B 0 d → f 1 f 1 decays possibly. In order to show explicitly the interferences among different flavor states contributing to the B 0 d,s → f 1 f 1 decays in three polarizations, we collect their corresponding decay amplitudes, namely, Table VI,  Unfortunately, no data or theoretical predictions for the considered B 0 s → f 1 f 1 decays are available nowadays. It is therefore expected that our predictions in the PQCD approach would be confronted with future LHCb and/or Belle-II experiments, as well as the theoretical comparisons within the framework of QCDF, soft-collinear effective theory [56], and so forth.
Although, as aforementioned, the global agreement with data for B → V V decays has been greatly improved in the PQCD approach theoretically [48] by picking up higher power r 2 i terms that were previously neglected, it seems that the predictions about the polarization fractions for the B 0 d,s → f 1 f 1 decays cannot be understood similarly as the B 0 d,s → ωω, φφ decays [48] due to the constructive and/or destructive interferences with different extents.
Let us take the B 0 d,s → f 1 (1285)f 1 (1285) and B 0 d,s → f 1 (1420)f 1 (1420) decays as examples to clarify the differences from the vector decays of B 0 d,s → ωω and B 0 d,s → φφ with ideal mixing in the PQCD approach at leading order [48]. As we know, with the referenced value φ f1 = 24 • , the physical states f 1 (1285) and (b) Without the interferences from the f n component, The ideal mixing, i.e., φ f1 = 0  (1285) and B 0 d → f 1 (1285)f 1 (1420) decays received the contributions arising from the colorsuppressed tree amplitudes with different extents. To our knowledge, the B 0 d → ρ 0 ρ 0 decay dominated by the color-suppressed tree amplitude has a small decay rate and a similarly small longitudinal polarization fraction in the PQCD approach at leading order that cannot be comparable to the measurements. Nevertheless, the partial next-to-leading order contributions from vertex corrections, quark loop, and chromomagnetic penguin diagrams [58], and the evolution from the Glauber-gluon associated with the transverse-momentum-dependent wave functions [28] could remarkably enhance the branching ratio and the longitudinal polarization fractions simultaneously. Then the theoretical predictions could be consistent well with the measurements given by BABAR [59] and LHCb [60] experiments within errors 6 . Therefore, the future examinations at LHCb and/or Belle-II experiments on these two mentioned channels sensitive to the above mentioned color-suppressed tree amplitudes could help to identify the needs of the possible next-to-leading order corrections. Of course, this issue has to be left for future study elsewhere.
In addition, we present the relative phases(in units of rad) φ and φ ⊥ of the B 0 d,s → f 1 f 1 decays for the first time. The numerical results can be seen explicitly in the Tables I-III. Of course, these predictions have to await for the future tests because no any measurements are available till now.

C. Direct CP-violating asymmetries
Now we come to the evaluations of direct CP-violating asymmetries of the B 0 d,s → f 1 f 1 decays in the PQCD approach. As for the direct CP violation A dir CP , it is defined as where Γ and A final stand for the decay rate and the decay amplitude of B 0 d,s → f 1 f 1 , while Γ and A final denote the charge conjugation ones, correspondingly. Meanwhile, according to Ref. [62], the direct-induced CP asymmetries can also be studied with the help of helicity amplitudes. Usually, we need to combine three polarization fractions, as shown in Eq. (48), with those corresponding conjugation ones of B decays and then to quote the resultant six observables to define direct CP violations of B 0 d,s → f 1 f 1 decays in the transversity basis as follows: where ℓ = L, , ⊥ and the definition off is the same as that in Eq.(48) but for the correspondingB 0 d,s decays. Using Eq. (63), we calculate the direct CP-violating asymmetries in the B 0 d,s → f 1 f 1 decays and present the results as shown in Tables I-III. Based on these numerical values, some comments are in order: (1) The direct CP-violating asymmetries within still large theoretical errors for the B 0 d,s → f 1 f 1 decays could be read straightforwardly from the Tables I-III as follows, where all the errors from various parameters as specified previously have been added in quadrature. For the former decays with ∆S = 0, as exhibited in the Table IV, the considerable tree or penguin contaminations lead to the large direct CP asymmetries. While for the latter decays with ∆S = 1, as displayed in the Table V, the negligible tree pollution result in the very small direct CP violations. Currently, all these direct CP violations seem hard to be detected in the near future experimentally due to the small decay rates for the former decays while the tiny CP asymmetries for the latter ones.
(2) For the B 0 s → f 1 (1285)f 1 (1285) and B 0 s → f 1 (1285)f 1 (1420) decays, there exist the large direct CP-violating asymmetries in both transverse polarizations, namely, parallel and perpendicular, with still large theoretical errors as follows, and A dir,|| which may be easily accessible associated with the large decay rates in the order of 10 −6 ∼ 10 −5 within theoretical errors. Other predictions about the direct CP violations in every polarization for the considered B 0 d,s → f 1 f 1 decays could be found out in the Tables I-III explicitly, we here will not list them individually. These results could be tested in the (near) future at LHCb, Belle-II, and other facilities such as Circular Electron-Positron Collider.
At last, we shall give some remarks on the important annihilation contributions. In particular, the penguin annihilation contributions was proposed to explain the polarization anomaly in the B → φK * decays in standard model [65]. Subsequently, more studies about the B → V V decays based on the rich data were made in a systematic manner, and the penguin-dominated channels were further found to need the annihilation contributions to a great extent [22,48,[62][63][64]66]. It is worth pointing out that, because of similar behavior between the vector and the 3 P 1 -axial-vector mesons, the authors proposed the similar annihilation contributions, as in the B → V V decays, to estimate the B → A( 3 P 1 )V and A( 3 P 1 )A( 3 P 1 ) decay rates and polarization fractions [22]. Therefore, we shall explore the important contributions from weak annihilation diagrams to the B 0 d,s → f 1 f 1 decays considered in this work. In order to clearly examine the important annihilation contributions, we present the explicit decay amplitudes in the Tables IV and V decomposed as tree diagrams and penguin diagrams with and without annihilation contributions in three polarizations. To show the variations of the considered B 0 d,s → f 1 f 1 decays with no inclusion of the contributions from annihilation diagrams, we shall list the observables numerically such as the CP-averaged branching ratios, the polarization fractions, and the direct CP-violating asymmetries by taking only the factorizable emission plus the non-factorizable emission decay amplitudes into account in the PQCD approach. For the sake of simplicity, we will present the central values of the above mentioned PQCD predictions with φ f1 = 24 • for clarifications.
• Branching ratios Without the contributions from annihilation diagrams, then the branching ratios will turn out to be, (b) Indeed, the annihilation diagrams can modify the polarization fractions of the B 0 d,s → f 1 f 1 decays with different extents. Without the contributions from the annihilation diagrams, it is found that the B 0 s → f 1 (1285)f 1 (1420) channel remains longitudinal-polarization-dominated but with a 26% enhancement, the B 0 d → f 1 (1285)f 1 (1285), B 0 d → f 1 (1285)f 1 (1420), B 0 s → f 1 (1420)f 1 (1420) and B 0 d → f 1 (1420)f 1 (1420) decays remains transverse-polarizationdominated but with a 40%, 26%, 42% reduction of f L , and a near 70% enhancement of f L , respectively, and the B 0 s → f 1 (1285)f 1 (1285) mode goes from a large longitudinal polarization fraction to a slightly larger transverse one than one half.
(c) As claimed in [67], the annihilation diagrams in the heavy B meson decays could contribute a large imaginary part, as shown in the Tables IV and V, and act as the main source of large strong phase in the PQCD approach. Therefore, the absence of the contributions from annihilation diagrams change the interferences highly between the weak and strong phases in the B 0 d,s → f 1 f 1 decays and finally results in the significant variations of the direct CP-violating asymmetries, even the positive or negative signs.
In short, we have analyzed the B 0 s → f 1 f 1 decays for the first time in the quark-flavor basis with the PQCD approach. We obtained the small decay rates that are hard to be measured in the CKM suppressed B 0 d → f 1 f 1 decays while the large branching ratios that are easy to be accessible in the CKM favored B 0 s → f 1 f 1 ones due to the interferences with different extents among the flavor decay amplitudes B 0 d,s → f n f n , f n f s , and f s f s . Particularly, the B 0 s → f 1 (1420)f 1 (1420) decay with a large branching ratio in the order of 10 −5 is expected to be measured through the B 0 s → (K 0 S K ± π ∓ ) f1(1420) (K 0 S K ± π ∓ ) f1(1420) channel. Our numerical results of the observables such as the CP-averaged branching ratios, the polarization fractions, and the direct CPviolating asymmetries indicate that the weak annihilation diagrams play important roles to understand the dynamics in these B 0 d,s → f 1 f 1 decays in the PQCD approach. Of course, these predictions in the PQCD approach await for the confirmations from the future examinations, which could help us to understand the annihilation decay mechanism in vector-vector and vectoraxial-vector B decays in depth. We explored the dependence of B(B 0 d,s → f 1 f 1 ) on the mixing angle φ f1 in the quark-flavor basis and found the interesting line-shapes to hint useful information. In light of the large theoretical errors induced by the unconstrained inputs, we also defined nine ratios of the B 0 d,s → f 1 f 1 decay rates to await for the (near) future measurements at LHCb and/or Belle-II, even other facilities, e.g., Circular Electron-Positron Collider. Note that the large uncertainties of the predicted branching ratios are canceled to a large extent in several ratios. Then the mixing angle φ f1 between the flavor states f n and f s could be further constrained, which would finally help pin down the θ K1 angle to understand the properties of the light axial-vector mesons more precisely.
For the twist-3 ones, we use the following form as in Ref. [49]: