Phenomenological studies on neutral B -meson decays into J/ψf 1 and η c f 1

We study the branching ratios ( B ) and the direct CP asymmetries of decays B 0 → J/ψf 1 and B 0 → η c f 1 in the perturbative QCD approach at the known next-to-leading order accuracy. Here, f 1 denotes the axial-vector mesons f 1 (1285) and f 1 (1420), the mixtures of ﬂavor states f n and f s with mixing angle ϕ in the quark-ﬂavor basis. It is found that the B ( B 0 d,s → J/ψf 1 (1420)) are very sensitive to the mixing angle ϕ , and future measurements with high precision can be used to constrain this angle. In comparison with the observed B ( B 0 d,s → J/ψf 1 (1285)), the decays B 0 s → ( J/ψ, η c ) f 1 (1420)) have large branching ratios, which could be measured more promisingly through f 1 (1420) → K 0 S K ± π ∓ at LHCb and Belle-II experiments. We also deﬁne several ratios, such as R J/ψd,s and R η c d,s , and they are useful to constrain the mixing angle ϕ and to explore the eﬀects of SU(3) ﬂavor symmetry breaking. Moreover, for the decays B 0 d → J/ψf n and η c f n , due to the eﬀects from penguin operators, the direct CP asymmetries of these two decays arrive at (a) few percent within errors. on the interesting polarization fractions in the decays


I. INTRODUCTION
In the quark model, two nonets of J P = 1 + axial-vector mesons are expected as the orbital excitation of the qq system.In terms of the spectroscopic notation 2S+1 L J , there are two types of P -wave axial-vector mesons, namely, 3 P 1 and 1 P 1 .These two nonets have distinctive C quantum numbers for the corresponding neutral mesons, C = + and C = −, respectively [1][2][3].The light axial-vector f 1 states, namely, f 1 (1285) and f 1 (1420), accompanied with a 1 (1260) and K 1A , are categorized as the 1 ++ multiplets, while the 1 +− multiplets incorporate b 1 (1235), h 1 (1170), h 1 (1380) and K 1B [4,5].Although lots of efforts have been made to investigate these light axial-vectors , our understanding on their natures is still far from satisfactory [6].Similar to the η − η mixing in the pseudoscalar sector, two physical f 1 mesons (For convenience, we will adopt f 1 to denote the f 1 (1285) and f 1 (1420) mesons in the following context, unless otherwise stated.)are generally viewed as the mixtures of two flavor states f n (≡ (ūu + dd)/ √ 2) and f s (≡ ss) with a single mixing angle ϕ, which can be written as However, both the magnitude and the sign of the mixing angle ϕ have not been determined yet.In addition, in light of Gell-mann−Okubo mass relation, this mixing angle ϕ could provide constraints to the unique mixing between the K 1A (1 3 P 1 ) and K 1B (1 1 P 1 ) states for the axial-vector strange K 1 mesons [1].In order to study the mixing angle ϕ, besides the f 1 mesons decays, the productions of f 1 mesons in the nonleptonic decays of heavy mesons could also be used.With this strategy, many B decays to f 1 have been explored in the literature [13,14,[20][21][22][23][24][25][26].
In 2013, LHCb collaboration reported their first measurements of the branching fractions of B 0 d,s → J/ψf 1 (1285) decays as follows [27], B(B 0 s → J/ψf 1 (1285)) Exp = (7.14+1.36 −1.41 ) × 10 −5 , where the uncertainties from different sources have been added in quadrature.It is found that the uncertainties are still large, and are expected to be reduced in the on-going LHCb and Belle-II experiments.Using the SU(3) symmetry and neglecting the contributions from penguin operators, the mixing angle ϕ was extracted to be ϕ Exp = ±(24.0+3.1+0.6 −2.6−0.8 ) • [27].In Ref. [20], two of us (Liu and Xiao) had studied B 0 s → J/ψf 1 (1285) decay and obtained |ϕ Theo | ∼ 15 • , where the large errors arose from large theoretical uncertainties of the branching fraction.Although both |ϕ Theo | and |ϕ Exp | locate in the range proposed by Stone and Zhang [18], the evident discrepancy still demands further explorations.
It is the purpose of this article to analyze the B 0 d,s → J/ψf 1 decays and to search for new observables for determination of the mixing angle, and the quark level Feynman diagrams for these decays are illustrated in Figure .1.The B-meson decays with a charmonium state have been studied extensively in the QCD-inspired approaches [28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43].In this article we focus on the perturbative QCD (PQCD) approach to non-leptonic B-decays [44][45][46].It is a model-independent framework that systematically disentangles short-distance (perturbative) from long-distance (non-perturbative) effects based on the k T factorization, and the basic concepts will be given in next section.It is found that for the color-suppressed modes, the contributions beyond leading order (LO) play important roles in explaining the experimental data.For instance, B → J/ψV and B → η c V decays have been studied in PQCD approach in Refs.[42,43] associated with the next-to-leading order (NLO) contributions, namely, the vertex corrections and the NLO Wilson coefficients, and the theoretical predictions are improved to basically agree with current data [4,5].Recently, the Sudakov factor for charmonium that plays critical roles in suppressing the long-distance contributions was derived, which affects the observables of the decays with charmonium remarkably [47,48].With above new ingredients, we will reexamine the decays B 0 s → J/ψf 1 [20] and evaluate the modes B 0 d → J/ψf 1 and B 0 → η c f 1 for the first time, and will provide predictions of the branching fractions, polarization fractions, relative phases and CP asymmetries.Within the experimental data, the branching fractions could help us constrain |ϕ| effectively, while other observables are helpful to further understand the QCD of f 1 .Moreover, as pointed out in Refs.[12][13][14][15], the QCD behavior of 1 ++ axial-vector meson is very similar to that of 1 −− vector, then it is natural to expect some important information provided by the considered B 0 s decay modes, relative to the golden channel B 0 s → J/ψφ.For example, B 0 s → J/ψf 1 (1420) and B 0 s → η c f 1 (1420) decays might serve as the alternative channels to explore the B 0 s − B0 s mixing phase φ s in a supplementary manner, provided that the mixing angle has been well determined by other ways.After all, it is an unarguable fact that the exclusive decays of neutral B-meson into charmonium have attracted great attention in the past decades at both theoretical and experimental aspects, as they can play special roles in studies of CP asymmetries [49] and B 0 − B0 mixing phases.
The paper is organized as follows.In Sect.II, we shortly review the formalism of PQCD approach in association with the meson wave functions, and then present the perturbative calculations of considered decays.The analytic expressions of decay amplitudes are also collected in this section.In Sect.III, we perform the numerical evaluations and discuss the theoretical results.Finally, a brief summary of this work is given in Sect.IV.

II. FORMALISM AND PERTURBATIVE QCD CALCULATIONS
We consider the B meson at rest for simplicity.For the decays B 0 → M cc f 1 with M cc denoting the J/ψ and η c , M cc and f 1 are assumed to move in the plus and minus z-directions, respectively.In the light-cone coordinate, the B meson momentum P 1 , the charmonium momentum P 2 and f 1 meson momentum P 3 are taken to be: and the polarization vectors 2 of J/ψ and 3 of f 1 as, and where the ratios r 2 = m Mcc /m B and r 3 = m f1 /m B .Due to the conservation of the angular momentum, only the longitudinal polarization vector 3L of f 1 contributes to the B 0 → η c f 1 decays.Denoting the (light-)quark momenta in the B 0 , M cc and f 1 mesons as k 1 , k 2 and k 3 correspondingly, we then have For the considered decays B 0 → M cc f 1 , the effective Hamiltonian H eff could be read as [50] H where the light quark q = d or s, the Fermi constant G F = 1.16639 × 10 −5 GeV −2 , V ij represents the Cabibbo-Kobayashi-Maskawa (CKM) matrix element, and C i (µ) is Wilson coefficients corresponding to the effective operator O i at the renormalization scale µ.The local four-quark operators O i (i = 1, • • • , 10) are given as • QCD penguin operators • Electroweak penguin operators where α and β are the color indices and q are the active quarks at the scale m b , i.e. q = (u, d, s, c, b).The left handed current is defined as (q α q β ) V −A = q α γ ν (1−γ 5 )q β and the right handed current (q α q β ) V +A = q α γ ν (1+γ 5 )q β .For convenience, the combination a i of the Wilson coefficients is defined as [46,51] Under the factorization hypothesis, the amplitude of B 0 → M cc f 1 decay in PQCD can be written conceptually as follows [44][45][46], where C(t) stands for the corresponding Wilson coefficient.x i (i = 1, 2, 3) denotes the fraction of momentum carried by (light-)quark inside the meson, and b i is the conjugate space coordinate of transverse momentum k iT .The wave function Φ describes the hadronization of quark and antiquark into a meson, and it is nonperturbative in nature but universal.The hard kernel H(x i , b i , t) involves the dynamics associated with the effective "six-fermion interaction" by exchanging a hard gluon [44,45,52,53], where t is the largest energy scale involved in the hard part.The rest two factors, namely, the Sudakov factor e −S(t) and the jet function S t (x i ) as shown in the above Eq.( 13), play important roles on the effective evaluations of B-meson decay amplitude in PQCD, which is based on the k T factorization theorem.The Sudakov factor e −S(t) suppresses the soft dynamics effectively, which makes perturbative calculation of the hard part H applicable at intermediate scale [54,55].The jet function S t (x i ) smears the endpoint singularities with threshold resummation technique [56,57].Recently, the Sudakov factor for charmonium up to the next-to-leading-logarithm accuracy has been derived in Refs.[47,48], where the effects of the charm quark mass are also included.In this work, we will adopt the new factor.Besides, more concepts of PQCD can be found in Ref. [58].In recent years, several developments on this approach have been obtained, for a review, see, e.g.[59,60].

A. Meson Wave Functions
As aforementioned, the nonperturbative meson wave functions and the related distribution amplitudes are the most important inputs in PQCD, which are universal and usually determined within the experimental data or the nonperturbative techniques, such as QCD sum rule or Lattice QCD.
For the B meson wave function, we adopt the form used widely in the literature [44,45,52,53,58], which is expressed as N c = 3 being the color factor.φ B is the leading-twist B-meson distribution amplitude, and x and k T are the momentum fraction and the intrinsic transverse momentum of light quark in B meson, respectively.The subscripts α and β are the color indices.
For charmonium states J/ψ and η c , their wave functions have been studied within the non-relativistic QCD approach [61].For the vector J/ψ meson, the longitudinal and transverse wave functions are given as, Here, L and T are the two polarization vectors of J/ψ, φ L J/ψ (x) and φ T J/ψ (x) are the twist-2 distribution amplitudes, while φ t J/ψ (x) and φ v J/ψ (x) are the twist-3 ones.For the pseudoscalar η c meson, its wave function could be read as, where φ v ηc (x) and φ s ηc (x) are the twist-2 and twist-3 distribution amplitudes, respectively.For the light axial-vector f q with q = n or s, its wave function could be written as follows [12,62], where x denotes the momentum fraction carried by quark in f q , n = (1, 0, 0 T ) and v = (0, 1, 0 T ) are the dimensionless lightlike vectors, the Levi-Cività tensor µναβ is conventionally taken as 0123 = 1.It should be stressed that m fq stands for the mass of f q obtained through the following mass relations [63], with m f1(1285) and m f1(1420) being the masses of physical states f 1 (1285) and f 1 (1420), respectively.For the sake of simplicity, the distribution amplitudes in the above wave functions have been collected in Appendix.A. With the above essential hadronic inputs, we can then proceed with perturbative calculations of the B 0 → M cc f 1 decay amplitudes within the framework of PQCD approach.

B. Perturbative calculations
It can be seen from Figure . 1 that for the B 0 d → M cc f 1 decays, the spectator d quark enters the final axial vector meson f 1 , therefore only f n component contributes to the decays with mixing angles.Similarly, only f s component contributes to the B 0 s → M cc f 1 decays.According to the effective Hamiltonian eq. ( 8), the lowest-order (LO) Feynman diagrams are summarized in Figure .2 for B → M cc f 1 decays, where the first two diagrams are called factorizable and the last two diagrams are the non-factorizable diagrams.Due to the fact that the behavior of axial vector in B → M cc f 1 decays is very similar to that of vectors, the amplitudes F for the factorizable diagrams and M for the nonfactorizable ones are almost same as those of decays B → (J/ψ, η c )V decays.In this case, we will not list them in current work, and the readers are referred to Refs.[42,43] for detail.It should be stressed that the term 1 − r 2 2 in the denominator of the longitudinal polarization vector 3L is kept in Eq. ( 6), while it has been neglected in Ref. [42].In Refs.[28][29][30][31][32][33][34][35]37], it was found that for the color-suppressed processes, such as B-meson decays into charmonium states, the vertex corrections play so important roles in explaining the experimental data that cannot be neglected.For this reason, the vertex corrections in B 0 → M cc f 1 decays as illustrated in Figure .3 should be included, and their effects are embodied by modifying the Wilson coefficients in the factorizable emission diagrams, leading to the effective Wilson coefficients ãh i (i = 2, 3, 5, 7, 9) as follows, In above formulae, the function f h I with helicities h = 0, ± are defined as: where the explicit expressions for the functions f I and g I can be found in Ref. [29].
We also note that in the LO calculations we used the LO Wilson coefficients C i (m W ) and the LO renormalization group evolution matrix U (t, m) (0) for the Wilson coefficient associating with the LO running coupling α s , where β 0 = (33 − 2N f )/3.In the NLO contributions, it is natural for us to adopt the NLO Wilson coefficients C i (m W ) and the NLO renormalization group evolution matrix U (t, m, α) with the running coupling α s (t) at twoloop [50], where QCD = 0.287 GeV (0.326 GeV) could be arrived within Λ (5) QCD = 0.225 GeV for the LO (NLO) case.In addition, we set µ 0 = 1.0 GeV [64] as the lower cut-off for the hard scale t.
With the amplitudes of each diagrams in Figure .2, the amplitude of B 0 → J/ψf q decay could be written as with ξ = √ 2 and ξ = 1 for f n and f s , respectively.The superscript σ(= L, N, T ) denotes the helicity state of the final states.Combining above amplitude and the quark-flavor mixing scheme as shown in Eq. ( 1), we finally obtain the amplitudes of the B 0 → J/ψf 1 decays, (1) For B 0 d → J/ψf 1 decays, (2) For B 0 s → J/ψf 1 decays, Now, we turn to the amplitudes of B 0 → η c f 1 decays.Similarly, the amplitudes of B 0 → η c f 1 decays at NLO could be obtained straightforwardly by replacing the information of vector J/ψ with that of pseudoscalar η c in Eqs. ( 30)- (34).Of course, only the longitudinal contributions of f 1 mesons contribute to the B 0 → η c f 1 decays, because of the conservation of the angular momentum.Therefore, we have Here, the effective Wilson coefficients ãi have included the related vertex corrections with new functions f I and g I that arise from η c emission [30].The explicit expressions of f ( ) I and g I can be found in Refs.[29,30,42,43].Also, the amplitudes of B 0 → η c f 1 decays could be read as follows, (1) (2) For B 0 s → η c f 1 decays, As a matter of fact, using these decay channels, we can only extract the absolute value |ϕ| of the mixing angle.In order to determine its sign, we have to resort to other decays that the interference between f n and f s is involved, such as B → M f 1 with M being the open-charmed and charmless mesons [41].

III. NUMERICAL RESULTS AND DISCUSSIONS
In this section, we will perform numerical calculations based on the given analytic expressions to estimate the experimental observables in the B 0 → M cc f 1 decays, such as the branching fractions and the direct CP asymmetries.Furthermore, the results of the polarization fractions in the decays B 0 → J/ψf 1 are also discussed.

B. B → J/ψf1
We firstly focus on the B 0 → J/ψf 1 decays.The branching fraction of B 0 → J/ψf 1 decay can be written as where τ B is the lifetime of neutral B-meson, |p c | ≡ |p 2z | = |p 3z | is the three-momentum of the two outgoing final states in the center-of-mass frame of B meson and A σ denotes the helicity amplitudes of B 0 → J/ψf 1 modes as given in Eqs. ( 31)- (34).
For the mixing angle ϕ, we take both |ϕ Theo | ∼ 15 • and |ϕ Exp | ∼ 24 • as typical values to predict the B(B 0 → J/ψf 1 ) and further discuss them phenomenologically.By employing the decay amplitudes and various inputs, the CP -averaged branching fractions in the PQCD approach at NLO with uncertainties are presented in Table.I. We acknowledge that there are many uncertainties in our calculations, and four kinds of uncertainties are included here.
The first uncertainties are from the wave function of B meson, and we adopt the variations of shape parameters ω B 0 d = 0.40±0.04GeV and ω B 0 s = 0.50±0.05GeV for B 0 d and B 0 s , respectively.The second uncertainties come from the nonperturbative parameters in the wave functions of the final states, such as decay constants and the Gegenbauer moments, which are given in Appendix.A. The third uncertainties arise from the higher-order and higher-power corrections, which are characterized by varying the running hard scale t max with 20%, namely, from 0.8t to 1.2t in the hard kernel.The last errors are induced by the variations of |ϕ Theo | = (15.0± 1.5) • (|ϕ Exp | = (24.0+3.2 −2.7 ) • ), respectively.From this table, one finds that all theoretical results agree roughly with the available measurements in 2σ standard deviations.Additionally, the changes induced by the mixing angle |ϕ| are plagued by uncertainties arising from other parameters.In this regard, in order to determine the mixing angle |ϕ|, more stringent constraints from other observables are required.We also note that the large branching fractions of the B 0 → J/ψf 1 decays are expected to be tested at LHCb and Belle-II experiments [66] in future.The decays B 0 s → J/ψf 1 (1285, 1420) have been investigated previously by two of us (Liu and Xiao) in Ref. [20] by including the vertex corrections.Comparing the new results with previous ones, we find all results agree with each other with uncertainties, and the acceptable differences are from the effects from the new ingredients in Sudakov factor.Furthermore, for comparison, we also take the decay B 0 s → J/ψf 1 (1285) as an example and calculate its branching fractions for different values of |ϕ| without vertex corrections and new ingredients in Sudakov factor, and the results are given as It is obvious that the vertex corrections can enhance the branching fractions remarkably, because the Wilson coefficient ã2 is much larger than a 2 , which has been also shown in Refs.[28,32].We also note that the uncertainties from the scale t are declined from 30% to 5%, as we expected.
We also acknowledge that our NLO calculation are incomplete.In principle, the NLO contributions should contain both the vertex corrections and hard spectator scattering (HS) amplitudes.For the charmless B meson decays, the NLO effects of HS have been explored partly [59,67,68].However, for the decays with charm quark, the calculations of NLO corrections of HS are very complicated because the new scale m c is involved, and we shall left it as our future work.
In PDG [4], the branching fractions of B 0 d → J/ψρ 0 and B 0 s → J/ψφ have been well measured as These branching fractions with high precision can be used for normalizations in studying B → J/ψf 1 decays.Theoretically, both branching fractions have been calculated in PQCD at NLO [42] are updated, and the results are given as where the uncertainties arising from different sources have been added in quadrature.Considering B(φ → K + K − ) = 0.492 ± 0.005 and B(ρ 0 → π + π − ) ∼ 100% [4], we then obtain the branching fractions of B 0 s → J/ψφ(→ K + K − ) and B 0 d → J/ψρ 0 (→ π + π − ) decays as follows, Then, four ratios could be defined as follows, and All the above results could be tested in LHCb and Belle-II experiments.As far as the central values are concerned, because the behaviors of f 1 meson are very similar to those of ρ/φ [12][13][14][15], the results indicate that f n (f s ) state should predominately govern the f 1 (1285)(f 1 (1420)) meson, that is, the small |ϕ| is preferred.In Ref. [27], after neglecting the contributions from penguin operators and assuming the SU(3) flavor symmetry, LHCb estimated the angle ϕ with the relation where the phase space factor is given as In fact, the corrections from penguin operators are about 15% [33], and SU(3) asymmetry is particularly estimated to be about 30%.Without above uncertainties, we define two ratios and evaluate them as • cot 2 ϕ ≈ 12.81 +0.17 Such two ratios with small uncertainties could also be used to determine |ϕ| in future, if f 1 (1285) and f 1 (1420) are believed to be two-quark states.Now we turn to other observables such as the polarization fractions and the direct CP asymmetries in B 0 → J/ψf 1 decays.With the helicity amplitudes shown in Eqs. ( 31)-( 34), a set of transversity amplitudes, namely, the longitudinal one A L , the parallel one A , and the perpendicular one A ⊥ , can be defined respectively as follows, with the ratio κ = P 2 • P 3 /(m J/ψ m f1 ).The polarization fractions f L and f T in B 0 → J/ψf 1 decays can be defined as follows [69], which satisfy the relation of f L + f T = 1.The relative phases φ and φ ⊥ (in units of rad) are thus obtained as follows, In PQCD, these observables of B 0 → J/ψf 1 decays are calculated, and the results are given as and in which various errors have been added in quadrature.Though these quantities are the ratios of (squared) decay amplitudes, the nonperturbative parameters, especially the Gegenbauer moment a ⊥ 1 in the distribution amplitudes of f n and f s and the charm quark mass m c , take large theoretical uncertainties.We also note that the longitudinal polarization fractions and transverse ones are almost equal, due to the helicity flip.Such phenomenon has been confirmed in B s → J/ψφ and B d → J/ψK * decays [42,70,71].All these observables will also be tested in experiments.
The direct CP asymmetry A dir CP of B 0 → J/ψf 1 decays is defined as where A stand for the decay amplitudes of B 0 → J/ψf 1 , while A describe the corresponding charge conjugation ones.With the obtained amplitudes, we calculate the direct CP asymmetries and present the results as Meanwhile, the direct CP asymmetries in each polarization can also be studied as [72] A where fα is the polarization fraction for the corresponding B decays in Eq. (60).In this work, the direct CP asymmetries for B 0 → J/ψf 1 decays at each polarization are collected as follows, • for B 0 d → J/ψf 1 modes (in units of 10 • for B 0 s → J/ψf 1 modes (in units of 10 −4 ), A dir,L CP = 1.69 All uncertainties from various parameters in above results have been added in quadrature.It is obvious that the direct CP violations for B 0 d → J/ψf 1 decays are a few percent within errors, however, which are still too small to be detected in the running LHCb and Belle-II experiments.

C. B → ηcf1
Different from the decays of B 0 → J/ψf 1 , B 0 → η c f 1 decays have only the longitudinal contributions because of the conservation of the angular momentum.By employing the decay amplitudes as presented in Eqs. ( 36)- (39), the corresponding branching ratio can be expressed as follows, It is noticeable that these ratios are all about 4, which attributes to the large contributions from the transverse polarization.The measurements of such kinds of ratios in future could help us to test the polarization mechanism.Analogous to the decays B 0 → J/ψf 1 , we also define four ratios in B 0 → η c f 1 modes as follows, which can be used to constrain the absolute value of the mixing angle ϕ.In PQCD, these above ratios could be calculated from where all uncertainties are added in quadrature.
Since the branching fractions of B 0 d,s → J/ψf 1 (1285) decays were measured with B(f 1 (1285) → 2π + 2π − ) = 0.109 +0.006 −0.006 , we also propose the analogous measurements on B 0 d,s → η c f 1 (1285)(→ 2π + 2π − ), whose branching fractions are estimated to be Meanwhile, with the large widths of f 1 (1285) → ηπ + π − and f 1 (1420) → K 0 S K + π − , the branching fractions of It is obvious that the decays ) with large decay rates could be observed in the running LHCb and Belle-II experiments.Once these predictions would be confirmed experimentally, then the mixing angle ϕ between the f 1 (1285) and f 1 (1420) mixing could receive more constraints, though they still suffer from large theoretical uncertainties induced by the hadronic parameters in the f n and f s light-cone distribution amplitudes.
In measuring the branching fractions of B 0 d,s → η c f 1 decays, we can also select the modes B 0 d → η c ρ 0 and B 0 s → η c φ for normalization.Theoretically, the B(B 0 d → η c ρ 0 ) and B(B 0 s → η c φ) in Ref. [43] are updated, and the results are presented as follows, These ratios are also expected to be examined in future.Lastly, we shall study the direct CP asymmetries of B 0 → J/ψf 1 decays.Based on the definition in Eq. ( 64), the numerical results can be given as It is shown that the direct CP asymmetries of B → η c f 1 are as large as a few percent, and are an order of magnitude larger than those of B 0 → J/ψf 1 decays.The measurements of such asymmetries in the running LHCb and Belle-II experiments are helpful for understanding the nature of f 1 states and testing the PQCD approach.
IV. SUMMARY We have made reexaminations on B 0 → J/ψf 1 and the first time studies on B 0 → η c f 1 decays in the framework of PQCD, where f 1 = f 1 (1280) and f 1 (1420) are viewed as the mixtures of f n and f s with mixing angle ϕ.In this work, the contributions of vertex corrections, nonfactorizable diagrams and penguin operators are all included.It is found that the branching fractions of these decays are large enough to be measured in the running LHCb and Belle-II experiments.We also note that the B 0 → η c f 1 (1285) and B 0 s → (J/ψ, η c )f 1 (1420) decays could be analyzed within the secondary decay chains f 1 (1285) → 2π + 2π − , f 1 (1285) → ηπ + π − and f 1 (1420) → K 0 S K + π − .In addition, we proposed several ratios that could be used to constrain the mixing angle ϕ, but its sign cannot be determined in these decays.We also studied the direct CP asymmetries in these decays, and results indicate the large penguin pollution in the B 0 d → (J/ψ, η c )f 1 decays.It should be emphasized that there are large theoretical uncertainties arising from the nonperturbative parameters, especially from the distribution amplitudes of axial-vector mesons and charmonium states, and more precise parameters from nonperturbative QCD approaches are needed.The comparisons between our results and future experimental data would help us to understand the nature of f 1 states and to test the PQCD approach.

FIG. 2 .
FIG. 2. (Color online) TypicalFeynman diagrams for neutral B-meson decays into J/ψf1 and ηcf1 at LO in the PQCD approach.

TABLE I .
The CP-averaged branching ratios for neutral B-meson decays into J/ψf1 in the PQCD approach −3 ),