𝐵→𝐽/𝜓𝐾(1270) and 𝐵→𝐽/𝜓𝐾(1400) Decays in QCD Factorization Approach

Abstract

The hadronic 𝐵→𝐽/𝜓𝐾(1270) and 𝐵→𝐽/𝜓𝐾(1400) has been analyzed in “QCD factorization” approach and generalized factorization. The effective coefficients 𝑎^ℎ_𝑖 have been calculated for three helicity states ℎ=0,+,− which give three different contribution of amplitudes. We consider that J/ψ behaves as a light meson in compared to 𝐵 meson. For 𝐵⁺→𝐽/𝜓𝐾⁺(1270) and 𝐵⁺→𝐽/𝜓𝐾⁺(1400), experimental data of branching ratios are (1.8 ± 0.5) × 10⁻³ and <5 × 10⁻⁴, respectively. Our best obtained results are (1.79 ± 0.01) × 10⁻³ at θ = 58° and (1.42 ± 0.01) × 10⁻³ at θ = 32° for 𝐵⁺→𝐽/𝜓𝐾⁺(1270) decay. And we have 4.27 × 10⁻⁴ at θ = 58° and 3.45 × 10⁻⁴ at θ = 32° for 𝐵⁺→𝐽/𝜓𝐾⁺(1400), which are in agreement with experiment.

1. Introduction

Recent experimental results sobtained by BABAR, Belle, and CLEO have opened an interesting area of research about production of axial-vector mesons in 𝐵 decays. Two body 𝐵 decays look for CP violation and overconstrain the CKM parameters in the Standard Model. Exclusive modes containing 𝐵→𝑃𝑃, 𝑃𝑉, and 𝑉𝑉, which have been extensively discussed in the literature have confirmed such expectation [1]. In this research, the two-body hadronic decays of 𝐵 meson into 𝑉𝐴 (𝑉: vector, 𝐴: axial-vector meson) are studied in QCD factorization method. First we introduce the structure of axial-vector meson. Then we study the hadronic decays 𝐵→𝐽/𝜓𝐾(1270) and 𝐵→𝐽/𝜓𝐾(1400), particularly. There are two distinct types of axial-vector meson, namely, ³𝑃₁ and ¹𝑃₁. In the quark model, two nonets of 𝐽^𝑃=1⁺ axial-vector meson are expected as the orbital excitation of the 𝑞𝑞 system. In terms of the spectroscopic notation ^2𝑆+1𝐿_𝐽, there are two types of 𝑝-wave mesons (³𝑃₁ and ¹𝑃₁). These two nonets have distinctive 𝐶 quantum numbers, 𝐶=+ and 𝐶=−, respectively. Experimentally, the 𝐽^𝑃𝐶=1⁺⁺ nonet consists of 𝑎₁(1260), 𝑓₁(1285), 𝑓₁(1420), and 𝐾_1𝐴, while the 1⁺⁻ nonet has 𝑏₁(1235), ℎ₁(1170), ℎ₁(1380), and 𝐾_1𝐵. There are two mixing effects for axial-vector meson: one is the mixing between ³𝑃₁ and ¹𝑃₁ states, for example, 𝐾_1𝐴 and 𝐾_1𝐵, and the other is mixing among ³𝑃₁ or ¹𝑃₁ states themselves [2]. The ³𝑃₁ meson behaves similarly to the vector meson this is not the case for the ¹𝑃₁ meson. For the latter, its decay constant vanishes in 𝑆𝑈(3) limit. Their light-cone distribution amplitudes are given by using the QCD sum rule method, and the chiral-even two-parton light-cone distribution amplitudes of the ³𝑃₁ (¹𝑃₁) meson are symmetric (antisymmetric) under the exchange of quark and antiquark momentum fractions in the 𝑆𝑈(3) limit due to 𝐺-parity. The 𝐵 decays involving an axial-vector meson and vector meson in final state have three polarization states. We have studied the two body 𝐵 decays involving axial-vector meson 𝐾(1270) or 𝐾(1400), and a vector meson 𝐽/𝜓 in final state. The simplest approach to obtain the hadronic matrix elements in decay amplitudes is naive factorization, where the matrix elements have been parameterized into the product of decay constant and form factors. In generalized factorization, it is well established that nonfactorizable contributions must be present in the matrix elements in order to cancel the energy scale 𝜇 and renormalization scheme dependence of Wilson coefficients. In QCD factorization approach, the nonfactorizable effects such as vertex corrections, hard spectator interactions, and annihilation contributions are calculable in hard scattering approach, and they have been studied in [3, 4] for the decays 𝐵→𝑎₁(1260)𝜋, 𝑎₁(1260)𝐾, 𝐵→ℎ₁(1235)𝐾^∗, 𝑏₁(1235)𝐾^∗ and 𝑏₁(1235)𝜌 [5].

Using the product expansion, the low-energy effective Hamiltonian for 𝐵→𝑀₁𝑀₂ decays can be written generally as [6, 7]𝐻_eff=𝐺_𝐹√2_𝑝=𝑢,c𝜆^𝐷_𝑝𝐶₁𝑂^𝑝₁+𝐶₂𝑂^𝑝₂+¹⁰_𝑖=3𝐶_𝑖𝑂_𝑖+h.c., (1.1) where 𝜆^𝐷_𝑝=𝑉_𝑝𝑏𝑉^∗_𝑝𝐷(𝐷=𝑑,𝑠) are the product of CKM elements and 𝐶_𝑖's are Wilson coefficients which have been evaluated at next leading order (NLO). The expressions of local operators are [8].

Current-current operators:
𝑂^𝑝₁=𝑝_𝛼𝑏_{𝛽𝑉−𝐴}𝑞_𝛽𝑝_{𝛼𝑉−𝐴},𝑂^𝑝₂=𝑝_𝛼𝑏_{𝛼𝑉−𝐴}𝑞_𝛽𝑝_{𝛽𝑉−𝐴}. (1.2)

QCD penguin operators:
𝑂₃=𝑞_𝛼𝑏_{𝛼𝑉−𝐴́𝑞}𝑞_𝛽́𝑞_{𝛽𝑉−𝐴},𝑂₄=𝑞_𝛼𝑏_{𝛽𝑉−𝐴́𝑞}𝑞_𝛽́𝑞_{𝛼𝑉−𝐴},𝑂₅=𝑞_𝛼𝑏_{𝛼𝑉−𝐴́𝑞}𝑞_𝛽́𝑞_{𝛽𝑉+𝐴},𝑂₆=𝑞_𝛼𝑏_{𝛽𝑉−𝐴́𝑞}𝑞_𝛽́𝑞_{𝛼𝑉+𝐴}. (1.3)

Electroweak penguin operators:
𝑂₇=𝑞_𝛼𝑏_{𝛼𝑉−𝐴́𝑞}32𝑒_́𝑞𝑞_𝛽́𝑞_{𝛽𝑉−𝐴},𝑂₈=𝑞_𝛼𝑏_{𝛽𝑉−𝐴́𝑞}32𝑒_́𝑞𝑞_𝛽́𝑞_{𝛼𝑉+𝐴},𝑂₉=𝑞_𝛼𝑏_{𝛼𝑉−𝐴́𝑞}32𝑒_́𝑞𝑞_𝛽́𝑞_{𝛽𝑉−𝐴},𝑂₁₀=𝑞_𝛼𝑏_{𝛽𝑉−𝐴́𝑞}32𝑒_́𝑞𝑞_𝛽́𝑞_{𝛼𝑉+𝐴}. (1.4)

Here (𝑞₁𝑞₂)_𝑉±𝐴=𝑞₁(1±𝛾₅)𝑞₂. For hadronic weak decays, the short distance effects can be calculated by perturbative theory but the computation of nonperturbative long distance effects is difficult. The fundamental problem in computation of hadronic matrix elements is due to nonperturbative effects arising from the strong interactions. The simplest approach to hadronic matrix elements is naive factorization hypothesis, where the hadronic matrix elements have been parameterized into the product of the decay constants and the form factors. Beneke et al. suggested QCD formula to compute the hadronic matrix elements in the heavy quark limit, combining the hard scattering approach with power counting in 1/𝑚_𝑏 [9, 10]. In heavy quark limit 𝑚_𝑏≫Λ_QCD, up to power corrections of order of Λ_QCD/𝑚_𝑏, the QCD formula for 𝐵→𝑀₁𝑀₂ can be written as𝑀₁𝑀₂𝑂_𝑖𝐵=_𝑗𝐹^𝐵→𝑀1_𝑗𝑚²₂¹₀𝑑𝑢𝑇^𝐼_𝑖𝑗(𝑢)𝜙_𝑀2+𝑀₁⟷𝑀₂+¹₀𝑑𝜉𝑑𝑢𝑑𝑣𝑇_𝑖^𝐼𝐼(𝜉,𝑢,𝑣)𝜙_𝐵(𝜉)𝜙_𝑀1(𝑣)𝜙_𝑀2(𝑢). (1.5)𝑀₁ and 𝑀₂ are both light. Which 𝐹^𝐵→𝑀 is form factor for 𝐵 to 𝑀 transition and 𝜙_𝑀 is the light cone distribution amplitude for meson. 𝜉,𝑢,𝑣 are momentum fractions of constituent quarks in 𝐵, 𝑀₁ and 𝑀₂ mesons, respectively. 𝑇_𝑖 and 𝑇_𝑖𝑗 are hard-scattering kernels arising from hard gluon exchanges. Hence, the hadronic matrix elements can be separated into short distance part (hard-scattering kernels) and long distance part. In QCD factorization, nonfactorizable loop effects and spectator scattering contribution have been considered. At leading order, there is a single diagram with no hard gluon interaction, and the contribution to 𝑇_𝑖𝑗(𝑢) in (1.5) is independent of 𝑢, and 𝑢-integral reduces to the normalization condition for the 𝑀₂ wave function. consequently, the factorization formula (1.5) reproduces naive factorization, if we neglect gluon exchange. In naive factorization, one neglect all corrections of order Λ_QCD/𝑚_𝑏 and of order 𝛼_𝑠. In QCD factorization, one computes systematically corrections to higher order in 𝛼_𝑠, but still neglects power correction of order Λ_QCD/𝑚_𝑏. In factorization, we have factorizable and non factorizable contributions. There are soft gluons in the factorizable contributions which are absorbed into the physical form factor. The diagrams also have hard contributions, which go into the short distance coefficient. The diagrams containing gluon exchanges which do not belong to the form factor are called nonfactorizable contributions. At order 𝛼_𝑠, these contributions can be divided into four groups: vertex corrections, penguin diagrams, hard spectator interactions, and annihilation diagrams. A detailed discussion of QCD factorization approach can be found in [8–10].

This method works well for the case with two light mesons in which the final-state mesons carry large momenta. When there is a heavy quark in the final state such as 𝐵→𝐷⁺𝜋⁻, this method still works when a spectator quark of 𝐵 meson is absorbed by 𝐷 meson. However, when the spectator quark is absorbed by a light quark, for example, in 𝐵→𝐷𝜋⁰, nonfactorizable contributions are infrared divergent, and the factorization breaks down. When we consider 𝐵→𝐽/𝜓𝐾(1270) and 𝐵→𝐽/𝜓𝐾(1400), it looks ambiguous at first sight whether we can apply the same method used in 𝐵→𝜋𝜋, or 𝐵→𝐷⁺𝜋⁻, since the spectator quark in the 𝐵 meson goes into light 𝐾(1270) or 𝐾(1400) meson. However, what is special about 𝐽/𝜓 is that the size of the charmonium is so small (~1/𝛼_𝑠𝑚_𝐽/𝜓) that the charmonium has a negligible overlap with the (𝐵,𝐾) system [11].

2. Input Parameters

2.1. Light-Cone Distribution Amplitude (LCDA)

The QCD corrections can change the local quark-anti quark operators into a series of nonlocal operators. If we assume that the 𝐽/𝜓 behave as a light meson, we can describe the light cone distribution. Note that in QCDF, the LCDAs of the light vector meson are written as [12] 𝐽/𝜓𝑝,𝜀𝑐_1𝛼𝑦𝑐_2𝛽(𝑥0𝑖)=−4¹₀𝑑𝑢exp𝑖𝑢𝑝⋅𝑦+×𝑓𝑢𝑝⋅𝑥_𝐽/𝜓𝑚_𝐽/𝜓╱𝑝𝜀^∗𝑧𝜙𝑝𝑧^‖𝐽/𝜓(𝑢)+𝜀_¡^∗_⟂𝑔_⟂^{𝐽/𝜓(𝑣)}(𝑢)+14𝜀_{𝜇𝜈𝜌𝜎}𝜀^∗𝜇𝑝^𝜌𝑧^𝜎𝛾^𝜈𝛾₅𝑔_⟂^{𝐽/𝜓(𝑎)}(𝑢)+𝑓𝜀^⟂_{𝐽/𝜓¡⟂╱}𝑝𝜙_⟂^𝐽/𝜓𝑚(𝑢)−𝑖^2𝐽/𝜓(𝜀^∗𝑧)𝑝⋅𝑧²𝜎_𝜇𝜈𝑝^𝜇𝑧^𝜈ℎ_‖^{𝐽/𝜓(𝑡)}(𝑢)−𝑖𝑚²_𝐽/𝜓𝜀^∗𝑧ℎ_‖^{𝐽/𝜓(𝑠)}(𝑢)2_𝛼𝛽, (2.1) where 𝑧=𝑦−𝑥 with 𝑧²=0. 𝜙_‖(𝑢), 𝜙_⟂(𝑢) are twist-2 DAs; 𝑔_⟂^(𝑣)(𝑢), 𝑔_⟂^(𝑎)(𝑢), ℎ_‖^(𝑠)(𝑢) and ℎ_‖^(𝑡)(𝑢) are twist-3 ones. 𝑢 is momentum fraction of 𝑐 quark in 𝐽/𝜓, 𝑢=1−𝑢. 𝑓_𝐽/𝜓, 𝑓^⟂_𝐽/𝜓 are vector and tensor decay constants, respectively. Similarly, for axial-vector meson ³⁽¹⁾𝑃₁, we have³⁽¹⁾𝑃₁𝑝,𝜀𝑑_1𝛼𝑦𝑢_2𝛽(𝑥0𝑖)=−4¹₀𝑑𝑢exp𝑖𝑣𝑝⋅𝑦+×𝑓𝑣𝑝⋅𝑥_3(1)𝑃1𝑚_3(1)𝑃1╱𝑝𝛾₅𝜀^∗𝑧𝜙𝑝𝑧^3(1)𝑃1‖(𝑣)+𝜀_¡^∗_⟂𝛾₅𝑔^3(1)𝑃1_⟂^(𝑣)(𝑣)+14𝜀_{𝜇𝜈𝜌𝜎}𝜀^∗𝜇𝑝^𝜌𝑧^𝜎𝛾^𝜈𝑔^3(1)𝑃1_⟂^(𝑎)(𝑣)+𝑓^⟂₃₍₁₎𝑃₁×𝜀_¡⟂╱𝑝𝛾₅𝜙^3(1)𝑃1_⟂𝑚(𝑣)−𝑖^23(1)𝑃1(𝜀^∗𝑧)𝑝⋅𝑧²𝜎_𝜇𝜈𝑝^𝜇𝑧^𝜈𝛾₅ℎ^3(1)𝑃1_‖^(𝑡)(𝑣)−𝑖𝑚²_3(1)𝑃1(𝜀^∗𝑧)𝛾₅ℎ^3(1)𝑃1_‖^(𝑝)(𝑣)2_𝛼𝛽. (2.2) Applying equation of motion to LCDAs, one can obtain the Wandzura-Wilczek relations in which twist-3 LCDAs related to the twist-2 ones [2]. Also 𝜙_±(𝑥) are defined as𝜙₊(𝑥)=𝑔_⟂^′(𝑎)(𝑥)4+𝑔_⟂^(𝑎)(𝑥)=¹_𝑥𝜙_‖(𝑢)𝑢𝜙𝑑𝑢,₋(𝑥)𝑔=−^⟂′(𝑎)(𝑥)4−𝑔_⟂^(𝑎)(𝑥)=^𝑥₀𝜙_‖(𝑢)1−𝑢𝑑𝑢, (2.3) which are transverse components of twist-3 LCDAs. While 𝜙_𝑣(𝑎) are longitudinal components of twist-3 ones for vector (axial-vector) meson. We specify the light-cone distribution amplitudes (LCDAs) as𝜙_𝑉𝑥,𝜇=6𝑥(1−𝑥)1+^∞_𝑛=1𝑎^𝑉_𝑛𝜇𝐶_𝑛^3/2(,𝜙2𝑥−1)_𝑣𝑥,𝜇=32𝑥−1+^∞_𝑛=1𝑎_𝑛^⟂,𝑉𝜇𝑃_𝑛+1(2𝑥−1), (2.4) for the vector meson, where 𝐶_𝑛(𝑥) are Gegenbauer polynomials and 𝑃_𝑛(𝑥) are Legendre polynomials. The normalization of LCDAs for vector meson is¹₀𝑑𝑥𝜙_𝑉(𝑥)=1,₁₀𝑑𝑥𝜙_𝑣(𝑥)=0. (2.5) Also,𝜙^𝐴_‖(𝑥)=6𝑥(1−𝑥)1+3𝑎^‖,3𝑃1₁(,𝜙2𝑥−1)_𝐴⟂(𝑥)=6𝑥(1−𝑥)𝑎^⟂,3𝑃1₀+3𝑎^⟂,3𝑃1₁(,𝜙2𝑥−1)_𝐴𝑎𝑥,𝜇=3𝑎^⟂,3𝑃1₀(2𝑥−1)+^∞_𝑛=1𝑎^⟂,3𝑃1_𝑛𝜇𝑃_𝑛+1(2𝑥−1), (2.6) for ³𝑃₁ axial-vector meson and𝜙^𝐴_‖(𝑥)=6𝑥(1−𝑥)𝑎^‖,1𝑃1₀+3𝑎^‖,1𝑃1₁(,𝜙2𝑥−1)_𝐴⟂(𝑥)=6𝑥(1−𝑥)1+3𝑎^⟂,1𝑃1₁(,𝜙2𝑥−1)_𝐴𝑎𝑥,𝜇=3(2𝑥−1)+^∞_𝑛=1𝑎^⟂,1𝑃1_𝑛𝜇𝑃_𝑛+1(2𝑥−1), (2.7) for ¹𝑃₁ axial-vector meson. The normalization conditions are¹₀𝑑𝑥𝜙_‖(𝑥)=1,₁₀𝑑𝑥𝜙_⟂(𝑥)=𝑎^⟂,3𝑃1₀,₁₀𝑑𝑥𝜙_𝑎(𝑥)=0, (2.8) for ³𝑃₁ axial-vector meson and¹₀𝑑𝑥𝜙_‖(𝑥)=𝑎^‖,1𝑃1₀,₁₀𝑑𝑥𝜙_⟂(𝑥)=1,₁₀𝑑𝑥𝜙_𝑎(𝑥)=0, (2.9) for ¹𝑃₁ axial-vector meson.

Here 𝜙_𝑉, 𝜙^𝐴_‖, and 𝜙^𝐴_⟂ are twist-2 LCDAs; likewise 𝜙_𝑣 and 𝜙^𝐴_𝑎 are twist-3 ones for vector and axial-vector mesons, respectively. The parameters 𝑎^‖₂ and 𝑎^⟂₂ are defined by the matrix element of a twist-2 conformal operator with conformal spin  3 in [13]. These parameters are unknown for 𝐽/𝜓; then we consider asymptotic forms of amplitudes for 𝐽/𝜓. Also, the Gegenbauer moments for 𝐾_1𝐴 and 𝐾_1𝐵 are given in [14, 15].

For the 𝐵 meson, we can write the projection as0𝑏_𝛼(𝑥)𝑞_𝛽(0)𝐵𝑝=−𝑖𝑓_𝐵4𝜙_𝐵𝜌_╱𝑝+𝑚_𝑏𝛾_5𝛽𝛾. (2.10) A detailed dissection of wave function of 𝐵 meson can be found [9, 10, 16–18]0𝑏_𝛼(𝑥)𝑞_𝛽(0)𝐵𝑝_{(𝑥+=𝑥⟂=0)}𝑓=−𝑖^𝐵4^╱𝑝+𝑚_𝑏𝛾_5𝛽𝛾×¹₀𝑑𝜌𝑒^{−𝜌𝑝+𝑥−}𝜙^𝐵₁𝜌+_&𝑛₋𝜙^𝐵₂𝜌_𝛾𝛼, (2.11) with 𝑛₋=(1,0,0,−1), and the normalization conditions are¹₀𝑑𝜌𝜙^𝐵₁𝜌=1,₁₀𝑑𝜌𝜙^𝐵₂𝜌=0. (2.12) The 𝐵 wave function corresponds to 𝜆_𝐵 defined by ∫¹₀𝑑𝜌𝜙^𝐵₁(𝜌)/𝜌≡𝑚_𝐵/𝜆_𝐵. 𝜆_𝐵 is the first inverse moment of the 𝐵 meson's distribution which is of order Λ_QCD.

2.2. Decay Constant

Decay constants of vector and axial-vector mesons are defined as𝑉𝑝,𝜀𝑞₂𝛾_𝜇𝑞₁0=−𝑖𝑓_𝑉𝑚_𝑉𝜀^∗_𝜇,𝐴𝑝,𝜀𝑞₂𝛾_𝜇𝛾₅𝑞₁0=𝑖𝑓_𝐴𝑚_𝐴𝜀^∗_𝜇. (2.13) Also, the transverse decay constants are defined via the tensor current as𝑉𝑝,𝜀𝑞₂𝜎_𝜇𝜈𝑞₁0=−𝑓^⟂_𝑉𝜀^∗_𝜇𝑝^𝜈−𝜀^∗_𝜈𝑝^𝜇,𝐴𝑝,𝜀𝑞₂𝛾_𝜇𝛾₅𝑞₁0=𝑖𝑓_𝐴𝑚_𝐴𝜀^∗_𝜇. (2.14) In general, the decay constants 𝑓_1𝑝1 and 𝑓^⟂_3𝑝1 are zero in the 𝑆𝑈(3) limit.

2.3. Form Factor

The form factors of 𝐵→𝐴 transitions are defined as [2]𝐴𝑝,𝜀𝐴_𝜇𝐵𝑝_𝐵2=𝑖𝑚_𝐵−𝑚_𝐴𝜀_{𝜇𝜈𝛼𝛽}𝜀^∗𝜈𝑝^𝛼_𝐵𝑝^𝛽𝐴^𝐵𝐴𝑞²,𝐴𝑝,𝜀𝑉_𝜇𝐵𝑝_𝐵=−𝑚_𝐵−𝑚_𝐴𝜀^∗_𝜇𝑉₁^𝐵𝐴𝑞²−𝜀^∗⋅𝑝_𝐵𝑝_𝐵+𝑝_𝜇𝑉₂^𝐵𝐴𝑞²𝑚_𝐵−𝑚_𝐴−2𝑚_𝐴𝜀^∗⋅𝑝_𝐵𝑞²𝑞^𝜇𝑉₃^𝐵𝐴𝑞²−𝑉₀^𝐵𝐴𝑞², (2.15) and for 𝐵→𝑉 transitions, we have⟨𝑉𝑝,𝜀𝑉_𝜇𝐵𝑝_𝐵2⟩=𝑚_𝐵+𝑚_𝑉𝜀_{𝜇𝜈𝛼𝛽}𝜀^∗𝜈𝑝^𝛼𝑝^𝛽_𝐵𝑉^𝐵𝑉𝑞²,⟨𝑉𝑝,𝜀𝐴_𝜇𝐵𝑝_𝐵⟩=𝑖𝑚_𝐵+𝑚_𝑉𝜀^∗_𝜇𝐴₁^𝐵𝑉𝑞²−𝜀^∗⋅𝑝_𝐵𝑚_𝐵+𝑚_𝑉𝑝_𝐵+𝑝_𝜇𝐴₂^𝐵𝑉𝑞²−2𝑚_𝑉𝜀^∗⋅𝑝_𝐵𝑞²𝑞_𝜇𝐴₃^𝐵𝑉𝑞²−𝐴₀^𝐵𝑉𝑞², (2.16) where 𝑞=𝑝_𝐵−𝑝, and for 𝑞²=0 we have 𝐴₃^𝐵𝑉=𝐴₀^𝐵𝑉 and 𝑉₃^𝐵𝐴=𝑉₀^𝐵𝐴. 𝜀_𝜇 is the polarization four-vector, and 𝑉, 𝐴₁, 𝐴₂, 𝐴₃, and 𝐴₀ are form factors and𝑉₃^𝐵𝐴𝑞²=𝑚_𝐵−𝑚_𝐴2𝑚_𝐴𝑉₁^𝐵𝐴𝑞²−𝑚_𝐵+𝑚_𝐴2𝑚_𝐴𝑉₂^𝐵𝐴𝑞²,𝐴₃^𝐵𝑉𝑞²=𝑚_𝐵+𝑚_𝑉2𝑚_𝑉𝐴₁^𝐵𝑉𝑞²−𝑚_𝐵−𝑚_𝑉2𝑚_𝑉𝐴₂^𝐵𝑉𝑞². (2.17) The form factors of 𝐵→𝐾_1𝐴 and 𝐵→𝐾_1𝐵 transitions have been calculated in the light-cone sum rule (LCSR) [19]. The momentum dependence of form factors is calculated in the LCSR approach which is parameterized in three-parameter form:𝐹𝑞²=𝐹(0)1−𝑎𝑞²/𝑚²_𝐵+𝑏𝑞⁴/𝑚⁴_𝐵, (2.18) which 𝑞²=𝑚²_𝐽/𝜓, and two parameters 𝑎 and 𝑏 are given in Table 1.

Table 1 

Form factors for 𝐵→𝐾_1𝐴 and 𝐵→𝐾_1𝐵 transitions which are obtained in light-cone sum rule models.

2.4. 𝐵→𝐽/𝜓𝐾(1270) and 𝐵→𝐽/𝜓𝐾(1400) Decays in QCD Factorization

Now, we want to calculate the branching ratios of 𝐵→𝐽/𝜓𝐾(1270) and 𝐵→𝐽/𝜓𝐾(1400) decays in QCD factorization method. First, we have calculated the branching ratios in generalized factorization which the effective Wilson coefficients are not dependent on renormalization scale 𝜇. To solve the issue of scale 𝜇 dependence, but not the renormalization scheme dependence [20], it is proposed in [21, 22], to isolate from the matrix element of four operators the 𝜇 dependence and link with the 𝜇 dependence in the Wilson coefficients 𝐶_𝑖(𝜇) to form 𝑐^eff_𝑖, effective Wilson coefficients independence of 𝜇. The formula for effective Wilson coefficients and their numerical values has been given explicitly in [22]. For 𝑏→𝑠 transition (in naive dimensional regularization): 𝑐^eff₁=1.1680, 𝑐^eff₂=−0.3652, 𝑐^eff₃=0.0233+0.0043𝑖, 𝑐^eff₄=−0.0482−0.0129𝑖, 𝑐^eff₅=0.0140+0.0043𝑖, 𝑐^eff₆=−0.0504−0.0129𝑖, 𝑐^eff₇/𝛼=−0.0312−0.0357𝑖, 𝑐^eff₈/𝛼=0.0551, 𝑐^eff₉/𝛼=−1.4277−0.0357𝑖, 𝑐^eff₁₀/𝛼=0.4804 [1]. 𝛼 is the fine structure constant. Wilson coefficients appear in decay amplitudes as linear combinations. So, the 𝑎^eff_𝑖 coefficients are𝑎^eff_𝑖≡𝑐^eff_𝑖+𝑐^eff_𝑖+1𝑁_𝑐(𝑖=odd),𝑎^eff_𝑖≡𝑐^eff_𝑖+𝑐^eff_𝑖−1𝑁_𝑐(𝑖=even). (2.19) As it known the physical states 𝐾₁(1270) and 𝐾₁(1400) are mixture of states ¹𝑃₁ and ³𝑃₁, whose relations could be parameterized by𝐾₁(1270)=𝐾_1𝐴sin𝜃+𝐾_1𝐵𝐾cos𝜃,₁(1400)=𝐾_1𝐴cos𝜃−𝐾_1𝐵sin𝜃, (2.20) where 𝐾_1𝐴 and 𝐾_1𝐵 are ³𝑃₁ and ¹𝑃₁ axial-vector mesons, respectively, and 𝜃 is the mixing angle. From experimental date on masses and partial ratios of 𝐾₁(1270) and 𝐾₁(1400), it is found two solutions for the mixing angle with a two-fold ambiguity, 𝜃=±32^∘ and 𝜃=±58^∘ [1]. Hence, the physical decaying amplitudes are given by𝐴𝐵⟶𝐽/𝜓𝐾₁(1270)_𝑝=𝐴𝐵⟶𝐽/𝜓𝐾_1𝐴sin𝜃+𝐴𝐵⟶𝐽/𝜓𝐾_1𝐵𝐴cos𝜃,𝐵⟶𝐽/𝜓𝐾₁(1400)_𝑝=𝐴𝐵⟶𝐽/𝜓𝐾_1𝐴cos𝜃−𝐴𝐵⟶𝐽/𝜓𝐾_1𝐵sin𝜃. (2.21) Since the final states of 𝐵→𝑉𝐴 carry spin degrees of freedom, we have calculated the helicity amplitudes, separately. In our work, the longitudinal amplitude ℎ=0 and the transverse amplitudes ℎ=± are calculated for 𝐵→𝐽/𝜓𝐾_1𝐴 and 𝐵→𝐽/𝜓𝐾_1𝐵 decays [22]. If we assume that 𝐽/𝜓 behaves as a light meson, due to its size, we can describe the light-cone distribution. For 𝐵→𝐽/𝜓𝐾(1270) and 𝐵→𝐽/𝜓𝐾(1400) decays, we have colour suppressed contribution 𝑎₂ and penguin contributions 𝑎₃, 𝑎₅, 𝑎₇, and 𝑎₉. Then, the contributions of vertex corrections for these decays are only for 𝑖=2,3,5,7,9. Hence, three different helicity amplitudes have been written as𝐴^ℎ𝐵⟶𝐽/𝜓𝐾_1𝐴=𝑉_𝑐𝑏𝑉^∗_𝑐𝑠𝑎^ℎ₂−𝑉_𝑡𝑏𝑉^∗_𝑡𝑠𝑎^ℎ₃+𝑎^ℎ₅+𝑎^ℎ₇+𝑎^ℎ₉𝑋^{(𝐵𝐾1𝐴}_ℎ^,𝐽/𝜓),𝐴^ℎ𝐵⟶𝐽/𝜓𝐾_1𝐵=𝑉_𝑐𝑏𝑉^∗_𝑐𝑠𝑎^ℎ₂−𝑉_𝑡𝑏𝑉^∗_𝑡𝑠𝑎^ℎ₃+𝑎^ℎ₅+𝑎^ℎ₇+𝑎^ℎ₉𝑋^{(𝐵𝐾1𝐵}_ℎ^,𝐽/𝜓), (2.22) where the factorizable amplitude is written as𝑋^{(𝐵𝐾1𝐴(1𝐵),𝐽/𝜓)}≡𝐽/𝜓(𝑐𝑐)_𝑉−𝐴0𝐾_1𝐴(1𝐵)𝑏𝑠_𝑉−𝐴𝐵. (2.23) The helicity dependencies read as𝑋₀^{(𝐵𝐾,𝐽/𝜓)}=𝑖𝑓_𝐽/𝜓2𝑚_𝐾𝑚²_𝐵−𝑚²_𝐾−𝑚²_𝐽/𝜓𝑚_𝐵−𝑚_𝐾𝑉₁^𝐵𝐾𝑞²−4𝑚²_𝐵𝑝²_𝑐𝑚_𝐵−𝑚_𝐾𝑉₂^𝐵𝐾𝑞²,𝑋_±^{(𝐵𝐾,𝐽/𝜓)}=−𝑖𝑓_𝐽/𝜓𝑚_𝐵𝑚_𝐽/𝜓𝑚1−^𝐾𝑚_𝐵𝑉₁^𝐵𝐾𝑞²∓2𝑝_𝑐𝑚_𝐵−𝑚_𝐾𝐴^𝐵𝐾𝑞², (2.24) where 𝑝_𝑐 is 𝑐.𝑚. momentum of decays in the 𝐵 rest frame𝑝_𝑐=𝑚²_𝐵−𝑚₁+𝑚₂²𝑚²_𝐵−𝑚₁−𝑚₂²2𝑚_𝐵, (2.25)𝑚₁ and 𝑚₂ are the mass of final state mesons. The branching ratio is given by [18]()=𝑝𝐵𝑅𝐵⟶𝑉𝐴^𝑐8𝜋𝑚²_𝐵𝜏_𝐵𝐺_𝐹√2²|𝐴(𝐵→𝑉𝐴)|², (2.26) where 𝐺_𝐹 is Fermi constant and 𝜏_𝐵 is life time of 𝐵 meson. If the final states are two light vector mesons, it is expected that |𝐴₀|>|𝐴₊|>|𝐴₋| for 𝐵 meson decays. (For 𝐵 decays, exchange −↔+.) It is because of that the amplitude |𝐴₊| is suppressed by a factor of √2𝑚₂/𝑚_𝐵, while the |𝐴₋| amplitude is further suppression in 𝑚₁/𝑚_𝐵. But for 𝐵 to heavy-light final states, √2𝑚_heavy/𝑚_𝐵 is of order unity [23]. In QCD factorization, the effective coefficients 𝑎^ℎ_𝑖 basically are Wilson coefficients in conjunction with short distance nonfactorizable corrections such as vertex corrections and hard spectator interactions. Then, we have [2]𝑎^ℎ_𝑖𝐽/𝜓𝐾_1𝐴𝐾_1𝐵=𝐶_𝑖+𝐶_𝑖±1𝑁_𝑐𝑁_𝑖𝐽/𝜓¹₀𝜙^{𝐽/𝜓,ℎ}(𝑥)+𝐶𝑑𝑥_𝑖±1𝑁_𝑐𝐶_𝐹𝛼_𝑠𝑉4𝜋^ℎ𝑖𝐽/𝜓+4𝜋²𝑁_𝑐𝐻^ℎ_𝑖𝐽/𝜓𝐾_1𝐴𝐾_1𝐵, (2.27) where 𝑖=1,…,10, the lower (upper) signs apply when 𝑖 is even (odd), 𝐶_𝑖 are Wilson coefficients (we use the next to leading order coefficients, NLO, obtained in the naive dimensional regularization scheme, NDR, at the energy scale 𝜇=𝑚_𝑏(𝑚_𝑏).), 𝐶_𝐹=(𝑁²_𝑐−1)/(2𝑁_𝑐) with 𝑁_𝑐=3, 𝐽/𝜓 is emitted meson and 𝐾_1𝐴(𝐾_1𝐵) have the spectator quark of the meson. 𝑉^ℎ_𝑖(𝐽/𝜓) is the vertex corrections and 𝐻^ℎ_𝑖(𝐽/𝜓𝐾_1𝐴(𝐾_1𝐵)) is the hard-scattering interactions, obtained due to exchange soft and hard gluons. The superscript ℎ denotes the helicity of the final-state meson, that is, ℎ=0,±. The LCDA 𝜙^{𝐽/𝜓,ℎ} is 𝜙_‖^𝐽/𝜓 for ℎ=0 and 𝜙_⟂^𝐽/𝜓 for ℎ=±. And 𝑁^ℎ_𝑖(𝐽/𝜓)=1. We can take the infinite mass limit of the 𝑏 quark in which 𝑚_𝑏 goes to infinity while 𝑚_𝐽/𝜓 is fixed (𝑚_𝐽/𝜓/𝑚_𝑏→0). The vertex corrections are given by𝑉⁰_𝑖𝐽/𝜓=¹⁰𝑑𝑥𝜙_‖^𝐽/𝜓(𝑥𝑚)12𝐿𝑛^𝑏𝜇−18+𝑔(𝑥),𝑖=2,3,9,₁₀𝑑𝑥𝜙_‖^𝐽/𝜓(𝑥𝑚)−12𝐿𝑛^𝑏𝜇+6−𝑔(1−𝑥),𝑖=5,7,𝑉^±_𝑖𝐽/𝜓=¹⁰𝑑𝑥𝜙_±^𝐽/𝜓(𝑥𝑚)12𝐿𝑛^𝑏𝜇−18+𝑔_𝑇(𝑥),𝑖=2,3,9,₁₀𝑑𝑥𝜙_∓^𝐽/𝜓(𝑥𝑚)−12𝐿𝑛^𝑏𝜇+6−𝑔_𝑇(1−𝑥),𝑖=5,7. (2.28)(2.29) With𝑔(𝑥)=31−2𝑥+1−𝑥ln𝑥−𝑖𝜋2𝐿𝑖₂(𝑥)−𝐿𝑛²𝑥+2𝐿𝑛𝑥−(,𝑔1−𝑥𝑥⟷1−𝑥)_𝑇(𝑥)=𝑔(𝑥)+𝐿𝑛𝑥,1−𝑥 (2.30) we consider asymptotic function for 𝜙_‖^𝐽/𝜓(𝑥)=6𝑥(1−𝑥), and also we have 𝜙₊(𝑥)=3(1−𝑥)² and 𝜙₋(𝑥)=3𝑥². Hard spectator corrections are given by𝐻⁰_𝑖𝐽/𝜓𝐾_1𝐴(1𝐵)=𝑖𝑓_𝐵𝑓_𝐽/𝜓𝑓_{𝐾1𝐴(1𝐵)}𝑋^{(𝐵𝐾1𝐴(1𝐵)}₀^,𝐽/𝜓)𝑚_𝐵𝜆_𝐵×¹₀𝜙𝑑𝑦𝑑𝑥^{𝐾1𝐴(1𝐵)‖}𝑦𝜙_‖^𝐽/𝜓(𝑥)1−𝑦(1−𝑥)−𝑟^{𝐾1𝐴(1𝐵)}_𝜒𝜙^{𝐾1𝐴(1𝐵)}𝑦𝜙_‖^𝐽/𝜓(𝑥)𝑥,1−𝑦 (2.31) for 𝑖=2,3,9, and𝐻⁰_𝑖𝐽/𝜓𝐾_1𝐴(1𝐵)=−𝑖𝑓_𝐵𝑓_𝐽/𝜓𝑓_{𝐾1𝐴(1𝐵)}𝑋^{(𝐵𝐾1𝐴(1𝐵)}₀^,𝐽/𝜓)𝑚_𝐵𝜆_𝐵×¹₀𝜙𝑑𝑦𝑑𝑥^{𝐾1𝐴(1𝐵)‖}𝑦𝜙_‖^𝐽/𝜓(𝑥)1−𝑦𝑥−𝑟^{𝐾1𝐴(1𝐵)}_𝜒𝜙^{𝐾1𝐴(1𝐵)}𝑦𝜙_‖^𝐽/𝜓(𝑥)(),1−𝑦1−𝑥 (2.32) for 𝑖=5,7. 𝑟^{𝐾1𝐴(1𝐵)}_𝜒=2(𝑚_{𝐾1𝐴(1𝐵)}𝑓^⟂_{𝐾1𝐴(1𝐵)})/(𝑚_𝑏(𝜇)𝑓_{𝐾1𝐴(1𝐵)}) are chirally enhanced parameters. The transverse hard spectator terms are𝐻⁻_𝑖𝐽/𝜓𝐾_1𝐴(1𝐵)=−2𝑖𝑓_𝐵𝑓^⟂_{𝐾1𝐴(1𝐵)}𝑓_𝐽/𝜓𝑚_𝐽/𝜓𝑚_𝐵𝑋^{(𝐵𝐾1𝐴(1𝐵)}₋^,𝐽/𝜓)𝑚_𝐵𝜆_𝐵×¹₀𝜙𝑑𝑦𝑑𝑥^{𝐾1𝐴(1𝐵)⟂}𝑦𝜙₋^𝐽/𝜓(𝑥)1−𝑦²𝑥,𝐻⁺_𝑖𝐽/𝜓𝐾_1𝐴(1𝐵)=−2𝑖𝑓_𝐵𝑓_{𝐾1𝐴(1𝐵)}𝑓_𝐽/𝜓𝑚_𝐽/𝜓𝑚_{𝐾1𝐴(1𝐵)}𝑚²_𝐵𝑋^{(𝐵𝐾1𝐴(1𝐵)}₊^,𝐽/𝜓)𝑚_𝐵𝜆_𝐵×¹₀𝜙𝑑𝑦𝑑𝑥1−𝑦−𝑥^{𝐾1𝐴(1𝐵)}₊𝑦𝜙₊^𝐽/𝜓(𝑥)1−𝑦²(1−𝑥)², (2.33) for 𝑖=2,3,9, and𝐻⁻_𝑖𝐽/𝜓𝐾_1𝐴(1𝐵)=2𝑖𝑓_𝐵𝑓^⟂_{𝐾1𝐴(1𝐵)}𝑓_𝐽/𝜓𝑚_𝐽/𝜓𝑚_𝐵𝑋^{(𝐵𝐾1𝐴(1𝐵)}₋^,𝐽/𝜓)𝑚_𝐵𝜆_𝐵×¹₀𝜙𝑑𝑦𝑑𝑥^{𝐾1𝐴(1𝐵)⟂}𝑦𝜙₊^𝐽/𝜓(𝑥)1−𝑦²(),1−𝑥𝐻⁺_𝑖𝐽/𝜓𝐾_1𝐴(1𝐵)=−2𝑖𝑓_𝐵𝑓_{𝐾1𝐴(1𝐵)}𝑓_𝐽/𝜓𝑚_𝐽/𝜓𝑚_{𝐾1𝐴(1𝐵)}𝑚²_𝐵𝑋^{(𝐵𝐾1𝐴(1𝐵)}₊^,𝐽/𝜓)𝑚_𝐵𝜆_𝐵×¹₀𝜙𝑑𝑦𝑑𝑥𝑦−𝑥^{𝐾1𝐴(1𝐵)}₊𝑦𝜙₋^𝐽/𝜓(𝑥)1−𝑦²𝑥², (2.34)(2.35) for 𝑖=5,7. In our calculations, the transverse hard spectator term 𝐻⁺_𝑖(𝐽/𝜓𝐾_1𝐴) is zero. There are some divergences in other hard spectator terms. We have used the unknown parameter 𝑋_𝐻 to eliminate these divergences, defined as𝑋_𝐻=¹₀𝑑𝑥𝑚1−𝑥=𝐿𝑛^𝐵Λ_QCD(1+𝑟), (2.36) where Λ_QCD is hadronic parameter and 𝑟=𝜌_𝐻𝑒^𝑖𝜑𝐻 which 𝜌_𝐻 and 𝜑_𝐻 are phases parameters (0≤𝜌_𝐻≤1, −180≤𝜑_𝐻≤180). We consider 𝑟=0 in these decays and 𝐵→𝐽/𝜓𝐾 [24]. This parameter phenomenologically is significant. Hence, the non factorizable contributions (vertex corrections and hard spectator interactions) have been calculated by (2.28)–(2.35).

In the other hand, we know that 𝐽/𝜓 is a heavy vector meson, although in [10] has been supposed that quarkonium is light meson relative to the 𝐵 meson. If we consider other limit in which 𝑚_𝑏 goes to infinity, 𝑚_𝐽/𝜓/𝑚_𝑏 is held fixed. In this scenario, the amplitude wave function of 𝐽/𝜓 has represented like 𝐵 meson. Also, to the leading order in 1/𝑚_𝑐, the 𝐽/𝜓 wave function is [18]𝐽/𝜓𝑝,𝜀𝑐_𝛼(𝑥)𝑐_𝛽(0)0_{(𝑥+=𝑥⟂=0)}=𝑓_𝐽/𝜓4𝜀^∗𝑝+𝑚_𝑏𝛾_5𝛽𝛾¹₀𝑑𝜉𝑒^{−𝑖𝜉𝑝+𝑥−}𝜙₁^𝐽/𝜓(𝜉)+𝑛₋𝜙₂^𝐽/𝜓(𝜉)_𝛾𝛼. (2.37) Using [18], the effect of higher twist for 𝐽/𝜓 may not converge enough. Because the charmed quark carries a momentum fraction of order ~𝑚_𝑐/𝑚_𝐽/𝜓, the distribution amplitudes of 𝐽/𝜓 vanish in the end point region. 𝜙_‖ is adapted as the DA of nonlocal vector current of 𝐽/𝜓 rather 𝑔^𝑣_⟂ as the DA of 𝜀_⟂ component since the latter does not vanish at the end point. Comparing (2.1) with (2.37) the leading twist DA of 𝐽/𝜓 is𝜙₁^𝐽/𝜓(𝑢)=𝜙_⟂^𝐽/𝜓(𝑢)∼6𝑢𝑓𝑢,_𝐽/𝜓=𝑓^⟂_𝐽/𝜓. (2.38) Then, for vertex corrections [18], we have𝑉⁰_𝑖𝐽/𝜓=¹⁰𝑑𝑥𝜙_‖^𝐽/𝜓(𝑥𝑚)12𝐿𝑛^𝑏𝜇−18+𝑓(𝑥,𝑧)+¹₀𝑑𝑥𝜙_⟂^𝐽/𝜓(𝑥)𝑔(𝑥,𝑧),𝑖=2,3,9,₁₀𝑑𝑥𝜙_‖^𝐽/𝜓(𝑥𝑚)−12𝐿𝑛^𝑏𝜇+6−𝑓(𝑥,𝑧)+¹₀𝑑𝑥𝜙_⟂^𝐽/𝜓(𝑥)𝑔()𝑉𝑥,𝑧,𝑖=5,7,_±𝑖𝐽/𝜓=¹⁰𝑑𝑥𝜙_‖^𝐽/𝜓(𝑥𝑚)12𝐿𝑛^𝑏𝜇−18+ℎ(𝑥,𝑧)+¹₀𝑑𝑥𝜙_⟂^𝐽/𝜓(𝑥)𝐼(𝑥,𝑧),𝑖=2,3,9,₁₀𝑑𝑥𝜙_‖^𝐽/𝜓(𝑥𝑚)−12𝐿𝑛^𝑏𝜇+6−ℎ(𝑥,𝑧)+¹₀𝑑𝑥𝜙_⟂^𝐽/𝜓(𝑥)𝐼(𝑥,𝑧),𝑖=5,7, (2.39) with𝑓(𝑥,𝑧)=2𝑧𝑥()+()1−𝑧1−𝑥3−2𝑥ln𝑥+−31−𝑥+11−𝑧𝑥()−1−𝑧1−𝑥2𝑧𝑥(1−𝑧(1−𝑥))²+3()𝑧𝑥ln𝑧𝑥1−𝑧+2𝑧𝑥+2𝑧²𝑥²()()1−𝑧1−𝑥ln1−𝑧−𝑖𝜋()+()×1−𝑧1−𝑥1−𝑧−4𝑥(1−𝑧)(1−𝑥)ln𝑥+𝑧𝑥(1−𝑧(1−𝑥))²()+1ln1−𝑧(1−𝑧𝑥)²−1(1−𝑧(1−𝑥))²+2(1+𝑧−2𝑧𝑥)(1−𝑧)(1−𝑧𝑥)²𝑧𝑥ln𝑧𝑥−𝑖𝜋𝑧𝑥(1−𝑧(1−𝑥))²,𝑔(𝑥,𝑧)=−4𝑟ln𝑥+1−𝑥4𝑧𝑟ln𝑧𝑥()()1−𝑧1−𝑥−4𝑧𝑟ln1−𝑧−𝑖𝜋()+(1−𝑧1−𝑥1−𝑧)4𝑟(1−𝑧)(1−𝑥)ln𝑥−4𝑟𝑧(),ℎ()=1−𝑧)(1−𝑧𝑥ln𝑧𝑥𝑥,𝑧2𝑧𝑥()+()1−𝑧1−𝑥3−2𝑥ln𝑥+−31−𝑥+11−𝑧𝑥()−1−𝑧1−𝑥2𝑧𝑥(1−𝑧(1−𝑥))²+3()𝑧𝑥ln𝑧𝑥1−𝑧+2𝑧𝑥+2𝑧²𝑥²()()1−𝑧1−𝑥ln1−𝑧−𝑖𝜋(),𝐼()1−𝑧1−𝑥𝑥,𝑧=−4𝑟ln𝑥+1−𝑥4𝑧𝑟ln𝑧𝑥()()1−𝑧1−𝑥−4𝑧𝑟ln1−𝑧−𝑖𝜋(),1−𝑧1−𝑥 (2.40) where 𝑧=𝑚²_𝐽/𝜓/𝑚²_𝐵 and 𝑟=𝑓^⟂_𝐽/𝜓𝑚_𝑐/𝑓_𝐽/𝜓𝑚_𝐽/𝜓. In [11] by contracting (2.14) for 𝐽/𝜓 with 𝑝_𝜈 and applying the equation of motion and (2.13) for 𝐽/𝜓, it finds that 𝑓^⟂_𝐽/𝜓𝑚_𝐽/𝜓=2𝑓_𝐽/𝜓𝑚_𝑐. Therefore𝑓^⟂_𝐽/𝜓𝑚_𝑐𝑓_𝐽/𝜓𝑚_𝐽/𝜓𝑚=2^𝑐𝑚_𝐽/𝜓². (2.41) In [25], the charmed quark carries a momentum fraction of order ~𝑚_𝑐/𝑚_𝐽/𝜓, which is the same fraction momentum 𝑥, then 𝑟=2𝑥². The contributions of hard spectator interaction have been written as𝐻⁰_𝑖𝐽/𝜓𝐾_1𝐴(1𝐵)=𝑖𝑓_𝐵𝑓_𝐽/𝜓𝑓_{𝐾1𝐴(1𝐵)}𝑋^{(𝐵𝐾1𝐴(1𝐵)}₀^,𝐽/𝜓)𝑚_𝐵𝜆_𝐵×¹₀𝜙𝑑𝑦𝑑𝑥^{𝐾1𝐴(1𝐵)‖}𝑦𝜙_‖^𝐽/𝜓(𝑥)1−𝑦(1−𝑥)−𝑟^{𝐾1𝐴(1𝐵)}_𝜒𝜙_{𝐾1𝐴(1𝐵)}𝑦𝜙_‖^𝐽/𝜓(𝑥)𝑥,1−𝑦 (2.42) for 𝑖=2,3,9,𝐻⁰_𝑖𝐽/𝜓𝐾_1𝐴(1𝐵)=−𝑖𝑓_𝐵𝑓_𝐽/𝜓𝑓_{𝐾1𝐴(1𝐵)}𝑋^{(𝐵𝐾1𝐴(1𝐵)}₀^,𝐽/𝜓)𝑚_𝐵𝜆_𝐵×¹₀𝜙𝑑𝑦𝑑𝑥^{𝐾1𝐴(1𝐵)‖}𝑦𝜙_‖^𝐽/𝜓(𝑥)1−𝑦𝑥−𝑟^{𝐾1𝐴(1𝐵)}_𝜒𝜙_{𝐾1𝐴(1𝐵)}𝑦𝜙_‖^𝐽/𝜓(𝑥)(),1−𝑦1−𝑥 (2.43) for 𝑖=5,7. The transverse hard spectator terms are𝐻⁻_𝑖𝐽/𝜓𝐾_1𝐴(1𝐵)=−2𝑖𝑓_𝐵𝑓^⟂_{𝐾1𝐴(1𝐵)}𝑓_𝐽/𝜓𝑚_𝐽/𝜓𝑚_𝐵𝑋^{(𝐵𝐾1𝐴(1𝐵)}₋^,𝐽/𝜓)𝑚_𝐵𝜆_𝐵1×1−𝑧₁₀𝜙𝑑𝑦𝑑𝑥^{𝐾1𝐴(1𝐵)⟂}𝑦𝜙₋^𝐽/𝜓(𝑥)1−𝑦²𝑥,𝐻⁺_𝑖𝐽/𝜓𝐾_1𝐴(1𝐵)=−2𝑖𝑓_𝐵𝑓_{𝐾1𝐴(1𝐵)}𝑓_𝐽/𝜓𝑚_𝐽/𝜓𝑚_{𝐾1𝐴(1𝐵)}𝑚_𝐵𝑋^{(𝐵𝐾1𝐴(1𝐵)}₊^,𝐽/𝜓)𝑚_𝐵𝜆_𝐵1×1−𝑧₁₀𝜙𝑑𝑦𝑑𝑥1−𝑦−𝑥^{𝐾1𝐴(1𝐵)}₊𝑦𝜙₊^𝐽/𝜓(𝑥)1−𝑦²(1−𝑥)², (2.44) for 𝑖=2,3,9, and𝐻⁻_𝑖𝐽/𝜓𝐾_1𝐴(1𝐵)=2𝑖𝑓_𝐵𝑓^⟂_{𝐾1𝐴(1𝐵)}𝑓_𝐽/𝜓𝑚_𝐽/𝜓𝑚_𝐵𝑋^{(𝐵𝐾1𝐴(1𝐵)}₋^,𝐽/𝜓)𝑚_𝐵𝜆_𝐵1×1−𝑧₁₀𝜙𝑑𝑦𝑑𝑥^{𝐾1𝐴(1𝐵)⟂}𝑦𝜙₊^𝐽/𝜓(𝑥)1−𝑦²(),1−𝑥𝐻⁺_𝑖𝐽/𝜓𝐾_1𝐴(1𝐵)=−2𝑖𝑓_𝐵𝑓_{𝐾1𝐴(1𝐵)}𝑓_𝐽/𝜓𝑚_𝐽/𝜓𝑚_{𝐾1𝐴(1𝐵)}𝑚_𝐵𝑋^{(𝐵𝐾1𝐴(1𝐵)}₊^,𝐽/𝜓)𝑚_𝐵𝜆_𝐵1×1−𝑧₁₀𝜙𝑑𝑦𝑑𝑥𝑦−𝑥^{𝐾1𝐴(1𝐵)}₊𝑦𝜙₋^𝐽/𝜓(𝑥)1−𝑦²𝑥², (2.45)(2.46) for 𝑖=5,7. Hence, the nonfactorizable contributions which are the same vertex corrections and hard spectator interactions have calculated by (2.28)–(2.35) for first scenario and by (2.39)–(2.46) for second one. Finally, the branching ratios of 𝐵→𝐽/𝜓𝐾(1270) and 𝐵→𝐽/𝜓𝐾(1400) have calculated at 𝜇=𝑚_𝑏 and at mixing angles 𝜃=32^∘,58^∘ in Tables 2 and 3.

Table 2 

Branching ratios of 𝐵→𝐽/𝜓𝐾(1270) in generalized factorization and QCD Factorization at 𝜇=𝑚_𝑏.

Table 3 

Branching ratios of 𝐵→𝐽/𝜓𝐾(1400) in generalized factorization and QCD Factorization at 𝜇=𝑚_𝑏.

3. Input Quantities

In this section, we have been introduced the essential input quantities [26]:𝑚_𝐵=5.28GeV,𝑚_𝐽/𝜓=3.1GeV,𝑚_𝐾1𝐴=1.367GeV,𝑚_𝐾1𝐵=1.310GeV,𝑓_𝐵=210±20MeV,𝑓_𝐽/𝜓=405MeV,𝑓_𝐾1𝐴=0.250GeV,𝑓^⟂_𝐾1𝐴𝜇=1GeV=0.065GeV,𝑎^⟂_0𝐾1𝐴=0.26,𝑎^⟂_1𝐾1𝐴=−1.08,𝜆_𝐵(1GeV)=225MeV,𝑓_𝐾1𝐵=−0.028GeV,𝑓^⟂_𝐾1𝐵𝜇=1GeV=0.190GeV,𝑎^‖_1𝐾1𝐵𝐶=−1.95,_𝐹=𝑁²−12𝑁,𝐺_𝐹=1.166×10⁻⁵𝑉,𝑁=3,_𝑐𝑏𝑉^∗_𝑐𝑠=0.039,𝑉_𝑡𝑏𝑉^∗_𝑡𝑠=0.041,𝜏_𝐵±=(1.638±0.011)×10⁻¹²S. (3.1) From (2.19), the magnitudes of factorizable amplitudes are obtained as𝑋^{(𝐵𝐾1𝐴}₀^,𝐽/𝜓)=3.54𝑖,𝑋^{(𝐵𝐾1𝐴}₊^,𝐽/𝜓)=−1.25𝑖,𝑋^{(𝐵𝐾1𝐴}₋^,𝐽/𝜓)𝑋=−5.56𝑖,_{(𝐵𝐾1𝐵0,𝐽/𝜓)}=−5.36𝑖,𝑋^{(𝐵𝐾1𝐵}₊^,𝐽/𝜓)=1.21𝑖,𝑋^{(𝐵𝐾1𝐵}₋^,𝐽/𝜓)=5.09𝑖. (3.2)

4. Discussion

In generalized factorization, the branching ratios of 𝐵→𝐽/𝜓𝐾(1270) and 𝐵→𝐽/𝜓𝐾(1400) decays are small in order to experiment, although they are in range of experimental data for these decays in mixing angles as shown in Tables 2 and 3. In this work, we have been tried to analyze these decays within “QCD factorization” method. Hence, the factorizable amplitudes in (2.24) have been calculated by using the input data of form factors, decay constants, and the non-factorizable contributions corresponding to the vertex corrections, and hard spectator interactions in 𝑎^ℎ_𝑖 have been calculated at two scenarios. First, 𝐽/𝜓 behaves as a light meson in compared to 𝐵 meson (𝑧→0). And in second, we have considered mass of 𝐽/𝜓 in which 𝑚_𝐽/𝜓/𝑚_𝑏 is held fixed. The effective coefficients 𝑎^ℎ_𝑖 have been calculated at three helicity states ℎ=0,+,−, which give three different contributions of amplitude. The branching ratios are calculated at these conditions and for the mixing angles 𝜃=32,58, where given in Tables 2 and 3. As it is shown, the obtained results in QCD factorization for these decays are not suitable assumption for all of the mixing angles. For 𝐵→𝐽/𝜓𝐾(1270) and 𝐵→𝐽/𝜓𝐾(1400), the experimental branching ratios are (1.8±0.5)×10⁻³ and <5×10⁻⁴, respectively [26]. In first scenario, the best obtained results for 𝐵→𝐽/𝜓𝐾(1270) decay are 1.79×10⁻³ at 𝜃=58^∘ and 1.42×10⁻³ at 𝜃=32^∘; for 𝐵→𝐽/𝜓𝐾(1400), these are 1.79×10⁻³ at 𝜃=58^∘ and 1.42×10⁻³ at 𝜃=32^∘, while in second one, as shown in Tables 2 and 3.