The decays $B\to \Psi(2S)\pi(K),\eta_c(2S)\pi(K)$ in the pQCD approach beyond the leading-order

Two body $B$ meson decays involving the radially excited meson $\psi(2S)/\eta_c(2S)$ in the final states are studied by using the perturbative QCD (pQCD) approach. We find that: (a) The branching ratios for the decays involving $K$ meson are predicted as $Br(B^+\to\psi(2S)K^+)=(5.37^{+1.85}_{-2.22})\times10^{-4}, Br(B^0\to\psi(2S)K^0)=(4.98^{+1.71}_{-2.06})\times10^{-4}, Br(B^+\to\eta_c(2S)K^+)=(3.54^{+3.18}_{-3.09})\times10^{-4}$, which are consistent well with the present data when including the next-to-leading-order (NLO) effects. Here the NLO effects are from the vertex corrections and the NLO Wilson coefficients. The large errors in the decay $B^+\to\eta_c(2S)K^+$ are mainly induced by using the decay constant $f_{\eta_c(2S)}=0.243^{+0.079}_{-0.111}$ GeV with large uncertainties. (b) While there seems to be some room left for other higher order corrections or the non-perturbative long distance contributions in the decays involving $\pi$ meson, $Br(B^+\to\psi(2S)\pi^+)=(1.17^{+0.42}_{-0.50})\times10^{-5}, Br(B^0\to\psi(2S)\pi^0)=0.54^{+0.20}_{-0.23}\times10^{-5}$, which are smaller than the present data. The results for other decays can be tested at the running LHCb and forthcoming Super-B experiments. (c) There is no obvious evidence of the direct CP violation being seen in the decays $B\to \psi(2S)\pi(K), \eta_c(2S)\pi(K)$ in the present experiments, which is supported by our calculations. If a few percent value is confirmed in the future , it would indicate new physics definitely.

Furthermore, the direct CP-violating asymmetries of the two charged decay channels are given by PDG [1], though with large uncertainties: A CP (B + → ψ(2S)π + ) = (3.0 ± 6.0)%. (6) In fact, B meson exclusive decays into charmonia have been received a lot of attentions for many years. They are regarded as the golden channels in researching CP violation and exploring new physics. At the same time, they play the important roles in testing the unitarity of the Cabibbo-Kobayashi-Maskawa (CKM) triangle. Moreover, these decays are ideal modes to check the different factorization approaches, such as the naive factorization assumption (FA) [2], QCD-improved factorization (QCDF) [3,4], light-cone sum rules (LCSR) [5], and the perturbtive QCD (pQCD) approach [6]. Most of these approaches can not work well when describing these decays, such as B → (J/Ψ, η c , χ c0 , χ c1 )K ( * ) . Because their predictions about the branching ratios are too small to explain the data. For example, the branching ratios of the decays B + → J/ΨK + and B 0 → J/ΨK 0 measured by Babar were about (10.61 ± 0.50) × 10 −4 and (8.69 ± 0.37) × 10 −4 [7], respectively, which imply a larger Wilson coefficient a 2 ≈ 0.20 ∼ 0.30 [9]. So the naive factorization assumption does not work with the Wilson coefficient a 2 ∼ 0. As for the QCDF approach, it has been found that the end point singularities exist in the amplitudes from the hard spectator scattering and annihilation diagrams, where the contributions are parametrized into respectively. These non-universal and uncontrollable parameters ρ H,A and δ H,A will induce large theoretical uncertainties. What's worse, when the emitted meson is heavy, such as D and charmonia states J/Ψ, Ψ(2S), η c (2S), the QCDF will break down. For example, in theB 0 → D 0 π 0 decay, since the D 0 meson is not a compact object with small transverse extension, it will strongly interact with the Bπ system, which makes the factorization fail. Fortunately, the transverse size of the charmonium is small in the heavy quark limit, so the authors of Ref. [8] considered that the QCDF method could be used to the decay B → J/ΨK, while they found that the leading-twist (twist-2) contributions were too small to explain the data. Then the authors of Ref. [9] calculated the twist-3 contributions, where the divergent integral was parametrized as Eq. (7). But they still could not explain the experimental data. The end-point singularities were also found in the twist-3 amplitudes for the decays B → η c (1S)K, η c (2S)K [10], where the QCDF approach was used. The LCSR approach was also insufficient to account for the data of these B meson decays into charmonia [5]. While under the pQCD approach, the spectator quark and other quarks are connected by one hard gluon (shown in Fig.1). Unlike the QCDF approach, the hard part of the pQCD approach consists of six quarks rather than four. Certainly, there also exist the soft and collinear gluon exchanges between quarks. So the double logarithms ln 2 P b will arise from the overlap of the soft and collinear divergences in radiative corrections to the meson wave functions, P being the dominant light-cone component of a meson momentum, b being the conjugate variable of parton transverse momentum k T . One can use the k T resummation [11] to organize these leading double logarithms for all loop diagrams into a Sudakov factor, which suppresses the long distance contributions in the large b region. When the end-point region with a momentum fraction x → 0, 1 is important for the hard amplitude, the corresponding large double logarithms α s ln 2 x shall appear in the hard amplitude. One can use the threshold resummation [12] to organize this type of double logarithms for all loop diagrams into a jet function, which suppresses the endpoint behavior of the hard amplitude. With the Sudakov factor and the jet function, one can evaluate all possible Feynman diagrams for the six-quark amplitude directly, including the nonfactorizable emission diagrams and annihilation type diagrams. But it is difficult to calculate these two kinds of contributions in QCDF approach. The pQCD approach has been used to B meson decays into charmonia such as B → J/ΨK ( * ) in Refs. [6,13], where the consistent results with the experimental data were obtained.
So we would like to use the pQCD approach to study B → ψ(2S)P, η c (2S)P (here P refers to the pseudo-scalar meson π or K) decays. Except the full leading-order (LO) contributions, the next-to-leading-order (NLO) contributions, which are mainly from the NLO Wilson coefficients and the vertex corrections to the hard kernel, are also included. Certainly, other NLO contributions, such as the quark loops and the magnetic penguin corrections, are available in the literatures [14,15], but they will not contribute to these considered decays.
We review the LO predictions for the B → ψ(2S)P, η c (2S)P decays including those for the main NLO contributions, in Sec. II. We perform the numerical study in Section IV, where the theoretical uncertainties are also considered. Section V is the conclusion. It is noticed that we will use the abbreviation ψ and η c to denote the mesons ψ(2S) and η c (2S) unless specified in the following.

II. THE LEADING-ORDER PREDICTIONS AND THE MAIN NEXT-TO-LEADING ORDER CORRECTIONS
As previously stated, the pQCD factorization approach has been used to calculate many two-body charmed B meson decays, and has obtained consistent results with experiments. So we use this approach to consider the decays B → ψ(2S)π(K), η c (2S)π(K), and the corresponding effective Hamiltonian can be written as: FIG. 1: Feynman Diagrams contributing to the decay B 0 → ψ(η c )π 0 at leading order. By replacing the spectator quark d with s quark, one can obtain the Feynman Diagrams for the where C i (µ) are Wilson coefficients at the renormalization scale µ, V represents for the Cabibbo-Kobayashi-Maskawa (CKM) matrix element, and the four fermion operators O i are given as: with i, j being the color indices. At the leading order, the relevant Feynman diagrams only include the factorizable and non-factorizable emission diagrams, as shown in Fig.1, where we take the decays B 0 → ψ(η c )π 0 as examples. If the emission particle is the vector meson ψ(2S), then the amplitude for the factorizable emission diagrams Fig.1(a) and Fig.1(b) can be written as: where r ψ = m ψ /m B , r P = m P 0 /m B (P = π, K), the color factor C F = 4/3, and φ A(P,T ) P (P =π,K) are the twist-2 (twist-3) distribution amplitudes for the meson π or K. x 1 and x 3 are the light quark momentum fractions in the B and P mesons, respectively. The evolution factors evolving the Sudakov factors and the jet function S t (x) are listed as: with c = 0.3, and S B (t), S π (t) being the Sudakov factors, which can be found in Ref. [16]. The hard functions h e is given as: The amplitude for the non-factorizable spectator diagrams Fig.1(c) and Fig.1(d) is given as: where the twist-2(twist-3) distribution amplitudes ψ L(t) (x 2 , b 2 ) can be found in the next section. x 2 is the c quark momentum fraction in ψ(2S) meson. The evolution factor E d (t) and the hard function h d are listed as: with the variable A 2 d = r 2 c + (x 1 − x 2 )(x 2 r 2 ψ + x 3 (1 − r 2 ψ )), and K 0 , I 0 and H 0 being the modified Bessel functions.
If the emission particle is the pseudo-scalar meson η c (2S), then the corresponding amplitude where r ηc = m ηc /m B , and the twist-3 distribution amplitudes of η c meson do not contribute to the amplitude. It is different from the case of M V −A ψP . While the evolution factor E d (t) and the hard function h d are similar with those given in Eq. (18) and Eq. (19).
By combining the amplitudes from the different Feynman diagrams, one can get the total decay amplitude for the decays B → ψ(η c )π: where V ij is the CKM matrix element and the combinations of Wilson coefficients a 2 = C 1 +C 2 /3, a i = C i +C i+1 /3 with i = 3, 5, 7, 9. The amplitudes F V −A ψ(ηc)π and M V −A ψ(ηc)π are given in Eqs. (14) and (17), respectively. As for the decays B → ψ(η c )K, the total amplitude can be obtained by replacing , V cs and V ts , respectively in Eq. (21).
As stated before, the NLO corrections to the hard kernel for the decays B → ψ(2S)P and B → η c (2S)P are simpler compared with other B meson decays such as B → πK, ρK.
Here only the vertex corrections are need to be considered. Since the vertex corrections can reduce the dependence of the Wilson coefficients on the renormalization scale µ, they play the important roles in the NLO analysis. It is well known that the nonfactorizable contributions are small [17], we concentrate only on the vertex corrections to the factorizable amplitudes, as shown in Fig.2. Furthermore, the infrared divergences from the soft and collinear gluons in these Feynman diagrams can be canceled each other. That is to say, these corrections are free from the end-point singularity in collinear factorization theorem, so we can simply quote the QCDF expressions for the vertex corrections: their effects can be combined into the Wilson coefficients, with the function f I defined as [18]: where we have neglected the terms proportional to r 4 ψ(ηc) . Certainly, in the following numerical analysis, the NLO Wilson coefficients will be used in the NLO calculations.

III. NUMERICAL RESULTS AND DISCUSSIONS
We use the following input parameters for the numerical calculations [1,19]: For the CKM matrix elements, we adopt the Wolfenstein parametrization and the updated values A = 0.814, λ = 0.22537,ρ = 0.117 ± 0.021 andη = 0.353 ± 0.013 [1]. With the total amplitudes, one can write the decay width as: The wave functions of B, π and K have been well defined in many works, while those of the two excited charmonia states exist many uncertainties. Here we take the harmonicoscillator model [19]: with where f 2S refers to the decay constant f ψ or f ηc , the free parameter ω = 0.2±0.1 GeV and the c quark mass m c = 1.275 ± 0.025 GeV. The main errors come from these parameters and ω b = 0.4 ± 0.1 for B meson wave funcion in our calculations. Using the input parameters and the wave functions as specified in this section, we give the LO and NLO predictions for the considered decays in Table I. For these colorsuppressed decays, the amplitudes associated with the Wilson coefficient C 1 + C 2 /3 usually play the dominant roles. It is instructive to check the contributions from these amplitudes at the leading-order and the next-to-leading-order. Here we take the decay B 0 → ψ(2S)K 0 as an example, the value of the factorizable color-suppressed amplitude F V −A ψK (a 2 ) is about −3.55 × 10 −2 , which will be partly canceled by the real part of nonfactorizable one Re(M V −A ψK (C 2 )) = 1.16 × 10 −2 . And the imagine part Im(M V −A ψK ) is small and only about −5.59 × 10 −3 . When the vertex corrections are included, the factorizable color-suppressed amplitude F V −A ψK (a 2 ) becomes a complex number. It's real part reduces to −1.49 × 10 −2 , which will be largely canceled by Re(M V −A ψK (C 2 )) = 1.09 × 10 −2 (the difference from the leading order value is because of using the NLO Wilson coefficient). While the imagine part induced by the vertex corrections is large and about −3.42 ×10 −2 . So the total amplitude from the tree operators increases after including the NLO contributions. From the numerical results, we find that the contribution (the square of the  amplitudes) from the penguin operators is very small compared with that from the tree operators, about 1.2% at the leading order and 0.05% at the next-to-leading order. There is the similar situation in the decay B 0 → J/ΨK 0 [13,20]. That is to say the penguin pollution is very small in these decays. From our results, we can find that the branching ratios of the channels B → ψ(2S)K, η c (2S)K are larger than those of the decays B → ψ(2S)π, η c (2S)π by one even two orders. This is mainly because of the CKM suppression factor λ = 0.22 within the latter. Just like the argument given in Ref. [4] that the soft gluon contribution is suppressed by a factor Λ QCD /(m b α s ) rather than Λ QCD /m b in this type of decay, so the perturbative and power corrections can be sizeable. Our predictions shown in Table I support this argument: the NLO contributions can provide a (52 ∼ 55)% enhancement for the decays involving η c (2S) meson. And the enhancement for the decays involving ψ(2S) meson is much more large. Furthermore, the charmed meson rescattering effects [21] in the decays B → ψ(2S)K, η c (2S)K might not be very important. While the branching ratios for the decays B + → ψ(2S)π + , B 0 → ψ(2S)π 0 are still smaller than the data even with the vertex corrections and the NLO Wilson coefficients included. Maybe other possible higher order contributions or the contributions from the Glauber gluons [22] in the spectator diagrams play the important roles. In the following we will discuss the CP-violating asymmetries of B → ψ(η c )π, ψ(η c )K decays. For the charged decays B + → ψ(η c )K + , there is no weak phase in their decay amplitudes, so the direct CP asymmetries of these decays are zero, and it is in agreement with the experimental value A CP (B + → ψ(2S)K + ) = (−2.4 ± 2.3)% within 1σ error. The direct CP asymmetries of the charged decay B + → ψ(2S)π + predicted by pQCD approach are listed as: which are consistent with the experimental data (2.2±8.5±1.6)% determined by Belle [23] and (4.8 ± 9.0 ± 1.1)% by LHCb [24] with no evidence of direct CP violation being seen. As for the neutral decay channels, the effects of B 0 −B 0 mixing should be considered. The direct and mixing induced CP violating asymmetries are defined as: where the CP-violating parameter λ CP is for b → d(b → s) transition and η f is the CP-eigenvalue of the final states. From the left panel in Fig.3, one can find that the dependence of the mixing CP-asymmetry A mix CP (B 0 → ψ(2s)π 0 ) on the CKM weak phase α. If taking the CKM weak phase α = (85.4 +3.9 −3.8 ) • [1], we find that the value of A mix CP (B 0 → ψ(2s)π 0 ) is (−31.3 +26.4 −22.3 )%. The mixing induced CP violating asymmetry for the channel B 0 → ψ(2s)π 0 is very sensitive to the angle α. As to another CKM weak phase β, there are much more uncertainties: by including earlier sin(2β) measurements [25] and recent results from LHCb [26] and Belle [27], the Heavy Flavor Averaging Group (HFAVG) gives two possible solutions 2β = (43.8±1.4) • and 2β = (136.2 ± 1.4) • [28], which is very consistent with our predictions shown in the right panel of Fig.3. If the assumption that A dir CP = 0 is relaxed, then A mix CP = −η f 1 − A dir CP sin 2β. There are similar results between the decays B 0 → ψ(2S)K 0 S (π 0 ) and B 0 → η c (2S)K 0 S (π 0 ) about the mixing induced CP violating asymmetries. As for the direct CP asymmetries of the decays B 0 → η c (2S)π 0 and B + → η c (2S)π + , they are very small and at 10 −5 order.

IV. SUMMARY
We study the B meson decays B → ψ(2S)K(π), η c (2S)K(π) within the pQCD approach, where the radially excited charmonia states are involved. With the wave functions of these two mesons ψ(2S) and η c (2S) derived from the harmonic-oscillator-model, we find that the branching ratios for the decays B + → ψ(2S)K + , η c (2S)K + and B 0 → ψ(2S)K 0 can agree well with the data within errors after including the NLO corrections. While there is still some room left for other high order corrections or the non-perturbative long distance contributions for the decays B + → ψ(2S)π + and B 0 → ψ(2S)π 0 . The pQCD predictions for the direct CP-violating asymmetries support the present experimental opinion, that is no evidence of direct CP violation being observed in these decays. If a few percent value is confirmed in the future, it would indicate new physics definitely.