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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 BaBar and Belle measurements; (c) for B0 → ηcK S 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.

On the theory side, such kinds of B/B s meson decays have been studied intensively by employing rather different theocratical methods, such as the naive factorization approach(NFA) [10], the QCD-improved factorization(QCDF) approach [11][12][13][14][15], the final-state interactions (FSI) [16][17][18], and the light-cone sum rules(LCSR) [19][20][21]. At the quark level, all considered decay modes are induced by the b → ccq(q = d, s) transitions in the framework of the standard model(SM) and belong to the color-suppressed category, as illustrated by the leading Feynman diagram in Fig. 1. In Ref. [13], for example, the authors studied B → η c K 0 decay and found a small decay rate: which is only abour 20% of the world average (8.0 ± 1.2) × 10 −4 as given in PDG 2018 [9]. In Ref. [16], on the other hand, the author studied B 0 → η c K * decay and found that the FSI correction could be comparable with the contribution from the naive factorizable amplitude, and the prediction with the inclusion of the FSI part was increased significantly to the value B(B → η c K * ) = (4.83 − 6.94) × 10 −4 , which is well consistent with the experimental data.
In this paper, we will make a comprehensive study for the sixteen B (s) → η c (P, V ) ( where P = (π, K, η (′) ) and V = (ρ, K * , ω, φ) are the light charmless mesons ) decays by employing the PQCD approach. Apart from the full leading-order (LO) contributions, the next-to-leading order (NLO) vertex corrections are also taken into account. Besides, the NLO twist-2 and twist-3 contributions to the form factors of B (s) → P transitions are also included in B (s) → η c P decays. This paper is organized as follows. In Sec. II, we give a brief review about the PQCD factorization approach and then calculate analytically the relevant Feynman diagrams and present the various decay amplitudes for the considered decay modes at the LO and NLO level. In Sec. III, we will show the PQCD predictions for the branching ratios and CP violating asymmetries of all sixteen B (s) → η c (P, V ) decays and make some phenomenological discussions about these results. A short summary is given in the last section.

II. DECAY AMPLITUDES AT LO AND NLO LEVEL
In the PQCD approach, we treat the B meson 1 as a heavy-light system and consider it at rest for simplicity. By employing the light-cone coordinates, we define the B meson with momentum P 1 , the emitted meson M 2 = η c with the momentum P 2 along the direction of n = (1, 0, 0 T ), and the recoiled meson M 3 = (P, V ) with the momentum P 3 in the direction of v = (0, 1, 0 T ) ( here n and v are the light-like dimensionless unit vectors) , in the following form: The longitudinal polarization vector of the final state vector meson can then be parameterized as: where r 2 = m ηc /M B and r 3 = m 3 /M B are the ratios of the meson masses, M B is the initial B/B s meson mass, m ηc and m 3 are the masses of the final state mesons. The momenta k i (i = 1, 2, 3) carried by the light anti-quark in the initial B/B s and the final M 2,3 mesons are chosen as follows: For the considered B → η c M 3 decays, the integration over k − 1,2 and k + 3 will lead conceptually to the decay amplitudes in the PQCD approach, (8) in which, b is the conjugate space coordinate of transverse momentum k T , C(t) stands for the Wilson coefficients evaluated at the scale t, and Φ i denotes the wave functions of the initial and final state mesons. The kernel H(x i , b i , t) describes the hard dynamics associated with the effective "six-quark interaction" with a hard gluon. The Sudakov factors e −S(t) and S t (x i ) together can suppress the soft dynamics in the endpoint region effectively [25].

A. Wave functions and decay amplitudes
For the wave function of the B meson, we adopt its wave function as being widely used, for example, in Refs. [25,32,33] where the distribution amplitude (DA) φ B can be parameterized in the following form 2 [25]: with ω B being the shape parameter. According to the discussions in Ref. [25,32,34], we here take ω B = 0.40 ± 0.04 GeV for the B u,d mesons [25], and ω B = 0.50 ± 0.05 GeV for the B s meson [32,34]. The normalization factor N B will be determined through the normalization condition: . For the pseudoscalar charmonium state η c , its wave function can be written in the form of where the twist-2 and twist-3 asymptotic distribution amplitudes (DAs), φ v and φ s , can be read as [35], For the light pseudo-scalar mesons M = (π, K, η q , η s ), their wave functions are the same ones as those in Refs. [36][37][38][39][40][41][42][43][44]: where m p 0 is the chiral mass of the relevant meson M, P and x are the momentum and the fraction of the momentum of M. The parameter ζ = 1 or −1 when the momentum fraction of the quark (anti-quark) of the meson is set to be x. The DAs of the meson M can be found easily, for example, in Refs. [37,38,[42][43][44]: For η − η ′ mixing, we adopt the quark-flavor basis: η q = (uū + dd)/ √ 2 and η s = ss as being used for example in Refs. [26,44,45]. The physical η and η ′ can then be written in the form of where the angle φ is the mixing angle between η q and η s . The relation between the decay constants (f q η , f s η , f q η ′ , f s η ′ ) and (f q , f s ) can be found for example in Ref. [44]. The chiral masses m ηq 0 and m ηs 0 have been defined in Ref. [45] by assuming the exact isospin symmetry m q = m u = m d . The parameters (f q , f s ) and mixing angle φ in Eq. (18) have been extracted from the data [47,48] With f π = 0.13 GeV, the chiral masses m ηq 0 and m ηs 0 will take the values of m ηq 0 = 1.07 GeV and m ηs 0 = 1.92 GeV [45]. Analogously, we adopt the ideal form for the ω − φ mixing as ω = (uū + dd)/ √ 2 and φ = ss. For the considered B (s) → η c V decays, only the longitudinal polarization component of the involved vector mesons contributes to the decay amplitudes. Therefore we choose the wave functions of the vector mesons as in Refs. [32,[49][50][51]: where P and m V are the momentum and the mass of the light vector mesons (ρ, K * , φ, ω), and ǫ L is the longitudinal polarization vector of these vector mesons. The twist-2 DA φ V (x) and the twist-3 DAs φ t V (x) and φ s V (x) in Eq. (20) can be written in the following form [49][50][51] is the decay constant of the vector meson with longitudinal (transverse) polarization. The Gegenbauer moments in Eq. (21) are the same as those in Refs. [49,50]:

B. Example of the LO decay amplitudes
In the SM, for the considered B (s) → η c (P, V ) decays induced by the b → q transition with q = (d, s), the weak effective Hamiltonian H ef f can be written as [52], where the Fermi constant G F = 1.16639 × 10 −5 GeV −2 , and V ij is the CKM matrix element, C i (µ) are the Wilson coefficients and O i (µ) are the local four-quark operators. For convenience, the combinations a i of the Wilson coefficients are defined as usual [32,33]: where the upper(lower) sign applies, when i is odd(even).
In the leading order PQCD approach, as illustrated in Fig. 2, there are only four types of the Feynman diagrams contributing to the B (s) → η c (P, V ) decays, which can be classified into two types: (a) the factorizable emission diagrams ( Fig. 2(a) and 2(b) ), and (b) the nonfactorizable emission (hard-spectator) diagrams ( Fig. 2(c) and 2(d) ). By evaluating and combining the contributions from the different Feynman diagrams as illustrated in Fig. 2, one can get the total decay amplitudes for the B (s) → η c (P, V ) decays: (26) where the terms F and M describes the contributions from the factorizable and nonfactorizable diagrams respectively. The superscript LL, LR and SP refers to the contributions from the where C F = 4/3 and α s (t i ) is the strong coupling constant. In the above functions, r V = m V /m B and r 0p = m p 0 /m B with m p 0 the chiral mass of the pseudoscalar meson. The explicit expression of the Sudakov factors (S ab (t a ), S ab (t b ), S cd (t f )) and S t (x i ), the hard scales t i , the hard functions h i (x i , b i ) can be found in Refs. [27,28,30,31,53].
In this work, beyond the full LO contributions, the following two currently known NLO corrections to the considered B (s) → η c (P, V ) decays are also taken into account: (1) The NLO vertex corrections to the factorizable amplitudes F eP and F eV , as shown in Fig. 3.
(2) The NLO twist-2 and twist-3 contributions to the form factors of B (s) → P transitions, as shown in Fig. 4.
According to Refs. [14,27], the vertex corrections can be absorbed into the redefinition of the Wilson coefficients a i (µ) by adding an additional term to them: where the function f I describes the vertex corrections [11,12]: with r 2 = m ηc /m B . For more discussions about the properties of function f I , one can see Refs. [11][12][13][14][15]28]. The NLO twist-2 and twist-3 contributions to the form factors of B → π transition have been calculated very recently in Refs. [54,55]. Based on the approximation of the SU(3) flavor symmetry, we can extend the formulas for B → π transitions as given in Refs. [54,55] to the cases for B (s) → (K, η q , η s ) transition form factors directly, after making appropriate replacements for some relevant parameters. The NLO form factor f + (q 2 ) for B s → K transition, for example, can be written in the following form: where η = 1 − q 2 /m 2 Bs with q 2 = (P Bs − P 3 ) 2 and P 3 is the momentum of the meson M 3 which absorbed the spectators quark of theB 0 s meson, µ (µ f ) is the renormalization (factorization ) scale, the hard scale t 1,2 are chosen as the largest scale of the propagators in the hard b-quark decay diagrams [54,55]. The explicit expressions of the threshold Sudakov function S t (x) and the hard function h(x i , b j ) can be found in Refs. [54,55]. The NLO correction factor F (1) T2 and F (1) T3 appeared in Eq. (33) describe the NLO twist-2 and twist-3 contributions to the form factor f +,0 (q 2 ) of the B s → K transition respectively, and can be written in the following form [54,55]: where r i = m 2 Bs /ξ 2 i with the choice of ξ 1 = 25m Bs and ξ 2 = m Bs . For the B (s) → η c (P, V ) decays, the large recoil region corresponds to the energy fraction η ∼ O(1−r 2 ηc ). The factorization scale µ f is set to be the hard scales corresponding to the largest energy scales in Fig. 2(a) and 2(b), respectively. The renormalization scale µ is defined as [44,54,55] The explicit expressions of the coefficients C 1,2,3 in above equation can be found in Refs. [54,55].

III. NUMERICAL RESULTS
In the numerical calculations, the following input parameters will be used implicitly. The masses, decay constants and QCD scales are in units of GeV For the CKM matrix elements, we adopt the Wolfenstein parametrization up to O(λ 5 ) with the updated parameters as presented in Ref. [9] λ = 0.22453, A = 0.836 ± 0.015,ρ = 0.122 +0.018 −0.017 ,η = 0.355 +0.012 −0.011 .
For the considered two-body B (s) → η c (P, V ) decays, the branching ratios can be expressed as: where τ B is the lifetime of the B meson. For the two B → η c π decay modes, for example, the PQCD predictions for the CP-averaged branching ratios (in units of 10 −5 ) with the inclusion of the known NLO contributions are the following: The first error is from the two-kinds of input hadronic parameters: (a) the shape parameter ω B = 0.40 ± 0.04 GeV or ω Bs = 0.50 ± 0.05 GeV ; and (b) the Gegenbauer moments such as a π 2 = 0.25 ± 0.15 as given in Eq. (17) . The second error arises from the variation of the had scale t from 0.8t to 1.2t, which characterizes the effects of the NLO QCD contributions. The last error is the combined uncertainty from the errors of the CKM matrix elements, as given in Eq. (38). It is easy to see that the first uncertainty of the theoretical predictions in Eq. (40) is the dominant one. For other considered decay modes, we also found the similar relations among the uncertainties from different sources. TABLE I. The LO and NLO PQCD predictions for the CP-averaged branching ratios of the five measured B → η c K ( * )− andB 0 s → η c φ decay modes. As a comparison, we also list the theoretical predictions as given in Refs. [13,16,20,27,28,56] and the measured values as given in PDG 2018 [9].

Modes
In Table I, we list the PQCD predictions for the CP-averaged branching ratios of the considered sixteen B/B s → η c (P, V ) decays together with currently available experimental measurements for five decay modes [1- 7,9]. The label "LO" denote the PQCD predictions at the full leading order, while the label "+VC" means that the additional NLO vertex corrections are included. The label "NLO" means that the contributions from the NLO twist-2 and twist-3 corrections to the form factors of B (s) → P transitions are also taken into account. For B (s) → η c V decays, unfortunately, such NLO twist-2 and twist-3 corrections to B (s) → V transition form factors are still not known. In Table I, we show the total theoretical uncertainties for the NLO PQCD predictions, obtained by adding the individual errors in quadrature.
As comparison, we also listed the previous LO PQCD predictions for the four B → η c (K, K * ) decays as given in Ref. [27] and the PQCD predictions for the two B → η c K decays with the inclusion of the NLO vertex corrections as given in Ref. [28]. The central values of the theoretical predictions obtained from the QCDF approach [13], the Light-cone sum rule (LCSR) [20] and the final state interaction (FSI) [16] are also listed. Those currently available experimental measurements as given in PDG 2018 [9] are also presented in last column of Table I.
From the numerical results and the experimental data as listed in Table I, we find the following points: (1) For all considered decays, the NLO vertex corrections can provide large enhancements to the LO PQCD predictions of their branching ratios, about 80% − 180% in magnitude. For the NLO Twist-2 and Twist-3 contributions, however, play a minor role only: resulting an enhancement or a decrease less than10% to B/B s → η c P decay modes. Among the five measured decays, the central values of the LO PQCD predictions for their decay rates are clearly much smaller than the measured ones. The large NLO contributions can provide a great help for us to interpret the data as listed in last column of Table I. It is easy to see that the NLO PQCD predictions for B(B → η c (K, K * , φ) agree well with the measured values [9] within two standard deviations. For three B → η c (K, K * − ) decays, specifically, the central values of the decay rates are smaller than the measured ones, and there seems some space left for still unknown higher order corrections or the non-perturbative contributions to these decays, which would be further studied and tested in the future.
(2) At the quark level, all considered decays can be classified into two types. The type-1 decays include the CKM-favored B → η c (K, K * ) and B s → η c (η s , φ) decays, corresponding to the b → (cc)s transition at the quark level, and have the decay rates proportional to |V * cb V cs | 2 ∼ λ 4 . The type-2 ones are the CKM-suppressed B → η c (π, η q , ρ, ω) decays, corresponding to the b → (cc)d transitions, and have the decay rates proportional to |V * cb V cd | 2 ∼ λ 6 . The PQCD predictions for the branching ratios of the type-1 decays are about 20 − 30 times larger than the ones for type-2 decays mainly due to the CKM enhancement |V cs /V cd | 2 ∼ λ −2 ≈ 21.
Considering the η − η ′ mixing as defined by Eq. (18) , we have the expression for η and η ′ : where φ = 39.3 0 is the mixing angle of η − η ′ system [47,48]. For the CKM-suppressed B 0 → η c (η, η ′ ) decays, only the dd component of η q contributes, and we can define and evaluate the ratio R 1 : For the CKM-faveredB 0 s → η c (η, η ′ ) decays, only the η s contributes, and we can define and evaluate the ratio R 2 : These two ratios could be measured and tested in the LHCb and Belle-II experiments. As a primary estimation for the ratios R 1 and R 2 , the possible effects of the different phase space factors for η and η ′ meson are not large in magnitude and have been neglected in this paper.
Besides the decay rates, some ratios of the branching fractions for the decay modes involving K and K * mesons have also been defined and measured by the BaBar and Belle Collaborations [3,4,7]. As is well-known, one major advantage of studying the ratios of the branching ratios for the properly selected pair of the decay modes is the large cancelation of the theoretical uncertainties.
The three relative ratios measured by Babar and Belle [3,4,7] and the corresponding PQCD predictions are the following: It is easy to see that the PQCD predictions for both R 3 and R 4 agree very well with the measured values within one standard deviation. The theoretical errors of the PQCD predictions for the ratios R 3,4,5 are around ten percent, which have been smaller than the uncertainties of currently available experimental measurements ( from 13% to 37%) [3,4,7]. The ratio R 3 is mainly governed by the difference between the lifetime of theB 0 and B − mesons: The ratio R 4 has a dependence on the distribution amplitudes of the K and K * mesons. For the ratio R 5 , the central value of our theoretical prediction is slightly larger than the measured one. In fact, this ratio satisfy the relation of R 5 = R 3 · R 4 by definition. These ratios will be tested by experiments when more precise data from Belle-II and LHCb become available in the near future. Analogous to the ratio R 1 , we can also define the ratio R 6 for the decays involving (π, ρ) mesons: Based on the similarity betweenB 0 andB 0 s meson decays and the small SU(3) breaking effect, it is reasonable for us to define the ratio R 7 between B(B 0 s → η cK * 0 ) and B(B 0 s → η cK 0 ) and expect a similar PQCD prediction with R 4 . Direct numerical calculation tell us that: which is actually close to R 2 = 1.04 +0.08 −0.06 . Now we turn to the evaluations of the CP-violating asymmetries for the considered decay modes. For the charged B ± meson decays, there exists the direct CP violation asymmetry A dir CP only, which can be defined as usual: For the neutral B 0 decays, the mixing effects should be taken into account. For B 0 decays, the very small ratio ∆Γ d /Γ d = −0.002 ± 0.010 [57] can be neglected safely. The direct and mixing-induced CP violation A dir CP and A mix CP can then be defined in the following form: with the CP violating parameter λ f : where η f = ±1 for a CP-even or CP-odd final state f , and β = arg For the neutral B 0 s decays, the ratio ∆Γ s /Γ s ≈ 0.13 [57] is large and should be taken into account in our calculations for the CP violating asymmetries. For B 0 s decays, the CP asymmetries A dir CP , S f and H f are constrained physically by the relation |A dir CP | 2 + |S f | 2 + |H f | 2 = 1, and can be defined in the usual way: with the CP violating parameter λ f : is the phase angle for B 0 s system.
Among the sixteen B (s) → η c (P, V ) decays considered in this work, only the CP asymmetries of the decay B 0 → η c K 0 S have been measured now [9], as listed in Eq. (2). In Table II and III, we list the PQCD predictions for the CP violating asymmetries of the considered B u,d and B 0 s decay modes respectively. The errors here are defined in the same way as those for the branching ratios. For the direct CP asymmetries, the error from the wave function parameters is largely cancelled between the numerator and denominator, and the dominant uncertainty is from the variation of the hard scale t. For the mixing induced CP asymmetries, the errors from the input hadronic quantities and CKM matrix elements are actually very small, and we only list the total errors by adding the individual errors in quadrature. From the numerical results as listed in Table II and III, one can see the following points: (1) For the seven CKM-favored b → ccs transition decays of B → η c (K, K * ) and B s → η c (η (′) , φ), their decay amplitudes are all proportional to the CKM factor V cb V * cs and there exist no weak phase in it at the NLO level 3 , which lead to the zero direct CP asymmetries for these decay modes. For the remaining nine b → ccd transition decays, since the corresponding weak phase is very small in size due to a strong suppression of λ 7 , their direct CP violating parameters are therefore very small: less than 2% in size as listed in Table II and  III. (2) For the neutral B 0 /B 0 s meson decays, because of the zero or very small A dir CP , the mixing induced CP asymmetries A mix CP are approximately proportional to the sin 2β or sin 2β s , specially for the decays ofB 0 → η cK 0 andB 0 s → η c (η (′) , φ). The PQCD predictions agree very well numerically with the current world average values sin 2β and −2β s [9].
(3) For B 0 → η c K 0 S decay, the PQCD predictions for both direct and mixing induced CP asymmetries as listed in Table II do agree well with the measured values [9] within errors. It is easy to see that the direct CP violation has not been seen by experiment up to now. In other words, any observation of large direct CP asymmetries for these considered decays will be a signal for new physics beyond the SM. Besides the A mix CP (B 0 → η c K 0 S ) exp , the large mixing induced CP asymmetry (∼ 70%) for other decays with similar b → ccd transition are also measurable in the near future LHCb and Belle-II experiment.

IV. SUMMARY
In summary, we studied the sixteen B (s) → η c (P, V ) decays by employing the PQCD factorization approach with the inclusion of the all currently known NLO contributions. We calculated the branching ratios and CP-violating asymmetries of the considered decay modes, defined several ratios of the decay rates, and compared our PQCD predictions with the measured values or the previous theoretical predictions based on the PQCD approach or other methods.
From our numerical calculations and phenomenological analysis, we found the following points: (1) The NLO vertex corrections can provide about 80%−180% enhancements to the LO PQCD predictions of the branching ratios of the considered decay modes. The NLO Twist-2 and Twist-3 contributions to the form factors of B/B s → P transitions, however, can only 3 In the Wofenstein parametrization up to O(λ 5 ), we have V cb = Aλ 2 , V cs = 1 − λ 2 2 − λ 4 1 8 + A 2 2 and V cd = λ + A 2 λ 5 1 2 − ρ − iη [9].
(2) We defined seven ratios of the branching ratios for properly selected pairs of considered decay modes. For the three measured ratios R 3,4,5 , the PQCD predictions agree well with currently available BaBar and Belle measurements. For other four ratios R 1,2 and R 6,7 , the PQCD predictions also agree with the general expectations and will be tested by the future experiments..
(3) For all considered decays, the PQCD predictions for the CP-violating asymmetries agree with the general expectations. For the only measured B 0 → η c K 0 S decay, the PQCD predictions for both the direct and mixing induced CP asymmetries do agree very well with the measured values within errors: A dir CP (B 0 → η c K 0 S ) = 0, PQCD, 0.08 ± 0.13, PDG2018, A mix CP (B 0 → η c K 0 S ) = 0.71 ± 0.01, PQCD, 0.93 ± 0.17, PDG2018.
The large mixing induced CP asymmetries (∼ 70%) for other similar CKM-suppressed b → ccd transition decays could be measured in the future LHCb and Belle-II experiments.