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

Authors: Mahboobeh Sayahi; Hossein Mehraban
Journal: Advances in High Energy Physics
 (2012)

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## Abstract

The hadronicB→J/ψK(1270) and B→J/ψK(1400) has been analyzed in “QCD factorization” approach and generalized factorization. The effective coefficients aih 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.


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## Body

## 1. Introduction

Recent experimental results sobtained by BABAR, Belle, and CLEO have opened an interesting area of research about production of axial-vector mesons inB 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, P13 and P11. In the quark model, two nonets of JP=1+ axial-vector meson are expected as the orbital excitation of the qq¯ system. In terms of the spectroscopic notation 2S+1LJ, there are two types of p-wave mesons (P13 and P11). These two nonets have distinctive C quantum numbers, C=+ and C=-, respectively. Experimentally, the JPC=1++ nonet consists of a1(1260), f1(1285), f1(1420), and K1A, while the 1+- nonet has b1(1235), h1(1170), h1(1380), and K1B. There are two mixing effects for axial-vector meson: one is the mixing between P13 and P11 states, for example, K1A and K1B, and the other is mixing among P13 or P11 states themselves [2]. The P13 meson behaves similarly to the vector meson this is not the case for the P11 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 P13 (P11) 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 forB→M1M2 decays can be written generally as [6, 7](1.1)Heff=GF2∑p=u,cλpD(C1O1p+C2O2p+∑i=310CiOi)+h.c.,
where λpD=VpbVpD*(D=d,s) are the product of CKM elements and Ci's are Wilson coefficients which have been evaluated at next leading order (NLO). The expressions of local operators are [8].Current-current operators:
(1.2)O1p=(p¯αbβ)V-A(q¯βpα)V-A,O2p=(p¯αbα)V-A(q¯βpβ)V-A.QCD penguin operators:
(1.3)O3=(q¯αbα)V-A∑q´(q¯β´qβ)V-A,O4=(q¯αbβ)V-A∑q´(q¯β´qα)V-A,O5=(q¯αbα)V-A∑q´(q¯β´qβ)V+A,O6=(q¯αbβ)V-A∑q´(q¯β´qα)V+A.Electroweak penguin operators:
(1.4)O7=(q¯αbα)V-A∑q´32eq´(q¯β´qβ)V-A,O8=(q¯αbβ)V-A∑q´32eq´(q¯β´qα)V+A,O9=(q¯αbα)V-A∑q´32eq´(q¯β´qβ)V-A,O10=(q¯αbβ)V-A∑q´32eq´(q¯β´qα)V+A.Here(q1¯q2)V±A=q¯1(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/mb [9, 10]. In heavy quark limit mb≫ΛQCD, up to power corrections of order of ΛQCD/mb, the QCD formula for B→M1M2 can be written as(1.5)〈M1M2|Oi|B〉=∑jFjB→M1(m22)∫01duTijI(u)ϕM2+(M1⟷M2)+∫01dξdudvTiII(ξ,u,v)ϕB(ξ)ϕM1(v)ϕM2(u).M1 and M2 are both light. Which FB→M is form factor for B to M transition and ϕM is the light cone distribution amplitude for meson. ξ,u,v are momentum fractions of constituent quarks in B, M1 and M2 mesons, respectively. Ti and Tij 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 Tij(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 ΛQCD/mb and of order αs. In QCD factorization, one computes systematically corrections to higher order in αs, but still neglects power correction of order ΛQCD/mb. 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 αs, 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 asB¯→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/αsmJ/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 theJ/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]
(2.1)〈J/ψ(p,ɛ)|c¯1α(y)c2β(x)|0〉=-i4∫01duexp⁡[i(up⋅y+u¯p⋅x)]×{∫fJ/ψmJ/ψ(pɛ*zpzϕ∥J/ψ(u)+ɛ⊥*g⊥J/ψ(v)(u)+14ɛμνρσɛ*μpρzσγνγ5g⊥J/ψ(a)(u))+fJ/ψ⊥(ɛ⊥pϕ⊥J/ψ(u)-imJ/ψ2(ɛ*z)(p⋅z)2σμνpμzνh∥J/ψ(t)(u)-imJ/ψ2(ɛ*z)h∥J/ψ(s)(u)2)}αβ,
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. fJ/ψ, fJ/ψ⊥ are vector and tensor decay constants, respectively. Similarly, for axial-vector meson P13(1), we have(2.2)〈P13(1)(p,ɛ)|d¯1α(y)u2β(x)|0〉=-i4∫01duexp⁡[i(vp⋅y+v¯p⋅x)]×{(ɛ⊥pγ5ϕ⊥3(1)P1(v)-im3(1)P12(ɛ*z)(p⋅z)2σμνpμzνγ5h∥3(1)P1(t)(v)-im3(1)P12(ɛ*z)γ5h∥3(1)P1(p)(v)2)f3(1)P1m3(1)P1(pγ5ɛ*zpzϕ∥3(1)P1(v)+ɛ⊥*γ5g⊥3(1)P1(v)(v)+14ɛμνρσɛ*μpρzσγνg⊥3(1)P1(a)(v))+fP13(1)⊥×(ɛ⊥pγ5ϕ⊥3(1)P1(v)-im3(1)P12(ɛ*z)(p⋅z)2σμνpμzνγ5h∥3(1)P1(t)(v)-im3(1)P12(ɛ*z)γ5h∥3(1)P1(p)(v)2)}αβ.
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 as(2.3)ϕ+(x)=g⊥′(a)(x)4+g⊥(a)(x)=∫x1ϕ∥(u)udu,ϕ-(x)=-(g⊥′(a)(x)4-g⊥(a)(x))=∫0xϕ∥(u)1-udu,
which are transverse components of twist-3 LCDAs. While ϕv(a) are longitudinal components of twist-3 ones for vector (axial-vector) meson. We specify the light-cone distribution amplitudes (LCDAs) as(2.4)ϕV(x,μ)=6x(1-x)[1+∑n=1∞anV(μ)Cn3/2(2x-1)],ϕv(x,μ)=3[2x-1+∑n=1∞an⊥,V(μ)Pn+1(2x-1)],
for the vector meson, where Cn(x) are Gegenbauer polynomials and Pn(x) are Legendre polynomials. The normalization of LCDAs for vector meson is(2.5)∫01dxϕV(x)=1,∫01dxϕv(x)=0.
Also,(2.6)ϕ∥A(x)=6x(1-x)[1+3a1∥,3P1(2x-1)],ϕ⊥A(x)=6x(1-x)[a0⊥,3P1+3a1⊥,3P1(2x-1)],ϕaA(x,μ)=3[a0⊥,3P1(2x-1)+∑n=1∞an⊥,3P1(μ)Pn+1(2x-1)],
for P13 axial-vector meson and(2.7)ϕ∥A(x)=6x(1-x)[a0∥,1P1+3a1∥,1P1(2x-1)],ϕ⊥A(x)=6x(1-x)[1+3a1⊥,1P1(2x-1)],ϕaA(x,μ)=3[(2x-1)+∑n=1∞an⊥,1P1(μ)Pn+1(2x-1)],
for P11 axial-vector meson. The normalization conditions are(2.8)∫01dxϕ∥(x)=1,∫01dxϕ⊥(x)=a0⊥,3P1,∫01dxϕa(x)=0,
for P13 axial-vector meson and(2.9)∫01dxϕ∥(x)=a0∥,1P1,∫01dxϕ⊥(x)=1,∫01dxϕa(x)=0,
for P11 axial-vector meson.HereϕV, ϕ∥A, and ϕ⊥A are twist-2 LCDAs; likewise ϕv and ϕaA are twist-3 ones for vector and axial-vector mesons, respectively. The parameters a2∥ and a2⊥ 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 K1A and K1B are given in [14, 15].For theB meson, we can write the projection as(2.10)〈0|b¯α(x)qβ(0)|B(p)〉=-ifB4ϕB(ρ)[(p+mb)γ5]βγ.
A detailed dissection of wave function of B meson can be found [9, 10, 16–18](2.11)〈0|b¯α(x)qβ(0)|B(p)〉|(x+=x⊥=0)=-ifB4[(p+mb)γ5]βγ×∫01dρ¯e-ρ¯p+x-[ϕ1B(ρ¯)+n-ϕ2B(ρ¯)]γα,
with n-=(1,0,0,-1), and the normalization conditions are(2.12)∫01dρ¯ϕ1B(ρ¯)=1,∫01dρ¯ϕ2B(ρ¯)=0.
The B wave function corresponds to λB defined by ∫01dρ¯ϕ1B(ρ¯)/ρ¯≡mB/λB. λB is the first inverse moment of the B meson's distribution which is of order ΛQCD.


### 2.2. Decay Constant

Decay constants of vector and axial-vector mesons are defined as(2.13)〈V(p,ɛ)|q¯2γμq1|0〉=-ifVmVɛμ*,〈A(p,ɛ)|q¯2γμγ5q1|0〉=ifAmAɛμ*.
Also, the transverse decay constants are defined via the tensor current as(2.14)〈V(p,ɛ)|q¯2σμνq1|0〉=-fV⊥(ɛμ*pν-ɛν*pμ),〈A(p,ɛ)|q¯2γμγ5q1|0〉=ifAmAɛμ*.
In general, the decay constants f1p1 and f3p1⊥ are zero in the SU(3) limit.


### 2.3. Form Factor

The form factors ofB¯→A transitions are defined as [2](2.15)〈A(p,ɛ)|Aμ|B¯(pB)〉=i2mB-mAɛμναβɛ*νpBαpβABA(q2),〈A(p,ɛ)|Vμ|B¯(pB)〉=-{(mB-mA)ɛμ*V1BA(q2)-(ɛ*⋅pB)(pB+p)μV2BA(q2)mB-mA-2mAɛ*⋅pBq2qμ[V3BA(q2)-V0BA(q2)]V2BA(q2)mB-mA},
and for B¯→V transitions, we have(2.16)〈V(p,ɛ)|Vμ|B¯(pB)〉=2mB+mVɛμναβɛ*νpαpBβVBV(q2),〈V(p,ɛ)|Aμ|B¯(pB)〉=i{(mB+mV)ɛμ*A1BV(q2)-ɛ*⋅pBmB+mV(pB+p)μA2BV(q2)-2mVɛ*⋅pBq2qμ[A3BV(q2)-A0BV(q2)]},
where q=pB-p, and for q2=0 we have A3BV=A0BV and V3BA=V0BA. ɛμ is the polarization four-vector, and V, A1, A2, A3, and A0 are form factors and(2.17)V3BA(q2)=mB-mA2mAV1BA(q2)-mB+mA2mAV2BA(q2),A3BV(q2)=mB+mV2mVA1BV(q2)-mB-mV2mVA2BV(q2).
The form factors of B→K1A and B→K1B 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:(2.18)F(q2)=F(0)1-a(q2/m2)B+b(q4/mB4),
which q2=mJ/ψ2, and two parameters a and b are given in Table 1.Table 1
Form factors forB→K1A and B→K1B transitions which are obtained in light-cone sum rule models.
F(0)abABK1A0.27±0.051.600.974V0BK1A0.22±0.042.401.78V1BK1A0.56±0.110.6350.211V2BK1A0.25±0.051.511.18ABK1B-0.22-0.04+0.061.720.912V0BK1B-0.45-0.08+0.121.340.690V1BK1B-0.48-0.08+0.130.7290.074V2BK1B-0.01-0.02+0.030.9190.855


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

Now, we want to calculate the branching ratios ofB→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 Ci(μ) to form cieff, 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): c1eff=1.1680, c2eff=-0.3652, c3eff=0.0233+0.0043i, c4eff=-0.0482-0.0129i, c5eff=0.0140+0.0043i, c6eff=-0.0504-0.0129i, c7eff/α=-0.0312-0.0357i, c8eff/α=0.0551, c9eff/α=-1.4277-0.0357i, c10eff/α=0.4804 [1]. α is the fine structure constant. Wilson coefficients appear in decay amplitudes as linear combinations. So, the aieff coefficients are(2.19)aieff≡cieff+ci+1effNc(i=odd),aieff≡cieff+ci-1effNc(i=even).
As it known the physical states K1(1270) and K1(1400) are mixture of states P11 and P13, whose relations could be parameterized by(2.20)K1(1270)=K1Asin⁡θ+K1Bcos⁡θ,K1(1400)=K1Acos⁡θ-K1Bsin⁡θ,
where K1A and K1B are P13 and P11 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 by(2.21)A(B⟶J/ψK1(1270))p=A(B⟶J/ψK1A)sin⁡θ+A(B⟶J/ψK1B)cos⁡θ,A(B⟶J/ψK1(1400))p=A(B⟶J/ψK1A)cos⁡θ-A(B⟶J/ψK1B)sin⁡θ.
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/ψK1A and B→J/ψK1B 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 as(2.22)Ah(B⟶J/ψK1A)=[VcbVcs*a2h-VtbVts*(a3h+a5h+a7h+a9h)]Xh(BK1A,J/ψ),Ah(B⟶J/ψK1B)=[VcbVcs*a2h-VtbVts*(a3h+a5h+a7h+a9h)]Xh(BK1B,J/ψ),
where the factorizable amplitude is written as(2.23)X(BK1A(1B),J/ψ)≡〈J/ψ|(c¯c)V-A|0〉〈K1A(1B)|(b¯s)V-A|B〉.
The helicity dependencies read as(2.24)X0(BK,J/ψ)=ifJ/ψ2mK[(mB2-mK2)-mJ/ψ2(mB-mK)V1BK(q2)-4mB2pc2mB-mKV2BK(q2)],X±(BK,J/ψ)=-ifJ/ψmBmJ/ψ[(1-mKmB)V1BK(q2)∓2pcmB-mKABK(q2)],
where pc is c.m. momentum of decays in the B rest frame(2.25)pc=(mB2-(m1+m2)2)(mB2-(m1-m2)2)2mB,m1 and m2 are the mass of final state mesons. The branching ratio is given by [18](2.26)BR(B⟶VA)=pc8πmB2τB(GF2)2|A(B→VA)|2,
where GF is Fermi constant and τB 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/mB, while the |A-| amplitude is further suppression in m1/mB. But for B to heavy-light final states, 2mheavy/mB is of order unity [23]. In QCD factorization, the effective coefficients aih basically are Wilson coefficients in conjunction with short distance nonfactorizable corrections such as vertex corrections and hard spectator interactions. Then, we have [2](2.27)aih(J/ψK1A(K1B))=(Ci+Ci±1Nc)Ni(J/ψ)∫01ϕJ/ψ,h(x)dx+Ci±1NcCFαs4π[Vih(J/ψ)+4π2NcHih(J/ψK1A(K1B))],
where i=1,…,10, the lower (upper) signs apply when i is even (odd), Ci 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=mb(mb).), CF=(Nc2-1)/(2Nc) with Nc=3, J/ψ is emitted meson and K1A(K1B) have the spectator quark of the meson. Vih(J/ψ) is the vertex corrections and Hih(J/ψK1A(K1B)) 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 ϕJ/ψ,h is ϕ∥J/ψ for h=0 and ϕ⊥J/ψ for h=±. And Nih(J/ψ)=1. We can take the infinite mass limit of the b quark in which mb goes to infinity while mJ/ψ is fixed (mJ/ψ/mb→0). The vertex corrections are given by(2.28)Vi0(J/ψ)={∫01dxϕ∥J/ψ(x)[12Lnmbμ-18+g(x)],i=2,3,9,∫01dxϕ∥J/ψ(x)[-12Lnmbμ+6-g(1-x)],i=5,7,(2.29)Vi±(J/ψ)={∫01dxϕ±J/ψ(x)[12Lnmbμ-18+gT(x)],i=2,3,9,∫01dxϕ∓J/ψ(x)[-12Lnmbμ+6-gT(1-x)],i=5,7.
With(2.30)g(x)=3(1-2x1-xln⁡x-iπ)+[2Li2(x)-Ln2x+2Lnx1-x-(x⟷1-x)],gT(x)=g(x)+Lnx1-x,
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 by(2.31)Hi0(J/ψK1A(1B))=ifBfJ/ψfK1A(1B)X0(BK1A(1B),J/ψ)mBλB×∫01dydx(ϕ∥K1A(1B)(y)ϕ∥J/ψ(x)(1-y)(1-x)-rχK1A(1B)ϕK1A(1B)(y)ϕ∥J/ψ(x)(1-y)x),
for i=2,3,9, and(2.32)Hi0(J/ψK1A(1B))=-ifBfJ/ψfK1A(1B)X0(BK1A(1B),J/ψ)mBλB×∫01dydx(ϕ∥K1A(1B)(y)ϕ∥J/ψ(x)(1-y)x-rχK1A(1B)ϕK1A(1B)(y)ϕ∥J/ψ(x)(1-y)(1-x)),
for i=5,7. rχK1A(1B)=2(mK1A(1B)fK1A(1B)⊥)/(mb(μ)fK1A(1B)) are chirally enhanced parameters. The transverse hard spectator terms are(2.33)Hi-(J/ψK1A(1B))=-2ifBfK1A(1B)⊥fJ/ψmJ/ψmBX-(BK1A(1B),J/ψ)mBλB×∫01dydxϕ⊥K1A(1B)(y)ϕ-J/ψ(x)(1-y)2x,Hi+(J/ψK1A(1B))=-2ifBfK1A(1B)fJ/ψmJ/ψmK1A(1B)mB2X+(BK1A(1B),J/ψ)mBλB×∫01dydx(1-y-x)ϕ+K1A(1B)(y)ϕ+J/ψ(x)(1-y)2(1-x)2,
for i=2,3,9, and(2.34)Hi-(J/ψK1A(1B))=2ifBfK1A(1B)⊥fJ/ψmJ/ψmBX-(BK1A(1B),J/ψ)mBλB×∫01dydxϕ⊥K1A(1B)(y)ϕ+J/ψ(x)(1-y)2(1-x),(2.35)Hi+(J/ψK1A(1B))=-2ifBfK1A(1B)fJ/ψmJ/ψmK1A(1B)mB2X+(BK1A(1B),J/ψ)mBλB×∫01dydx(y-x)ϕ+K1A(1B)(y)ϕ-J/ψ(x)(1-y)2x2,
for i=5,7. In our calculations, the transverse hard spectator term Hi+(J/ψK1A) is zero. There are some divergences in other hard spectator terms. We have used the unknown parameter XH to eliminate these divergences, defined as(2.36)XH=∫01dx1-x=Ln(mBΛQCD)(1+r),
where ΛQCD is hadronic parameter and r=ρHeiφH which ρH and φH are phases parameters (0≤ρH≤1, -180≤φH≤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 thatJ/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 mb goes to infinity, mJ/ψ/mb 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/mc, the J/ψ wave function is [18](2.37)〈J/ψ(p,ɛ)|c¯α(x)cβ(0)|0〉|(x+=x⊥=0)=fJ/ψ4[ɛ*(p+mb)γ5]βγ∫01dξ¯e-iξp+x-[ϕ1J/ψ(ξ)+n-ϕ2J/ψ(ξ)]γα.
Using [18], the effect of higher twist for J/ψ may not converge enough. Because the charmed quark carries a momentum fraction of order ~mc/mJ/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 ɛ⊥ component since the latter does not vanish at the end point. Comparing (2.1) with (2.37) the leading twist DA of J/ψ is(2.38)ϕ1J/ψ(u)=ϕ⊥J/ψ(u)~6uu¯,fJ/ψ=fJ/ψ⊥.
Then, for vertex corrections [18], we have(2.39)Vi0(J/ψ)={∫01dxϕ∥J/ψ(x)[12Lnmbμ-18+f(x,z)]+∫01dxϕ⊥J/ψ(x)g(x,z),i=2,3,9,∫01dxϕ∥J/ψ(x)[-12Lnmbμ+6-f(x,z)]+∫01dxϕ⊥J/ψ(x)g(x,z),i=5,7,Vi±(J/ψ)={∫01dxϕ∥J/ψ(x)[12Lnmbμ-18+h(x,z)]+∫01dxϕ⊥J/ψ(x)I(x,z),i=2,3,9,∫01dxϕ∥J/ψ(x)[-12Lnmbμ+6-h(x,z)]+∫01dxϕ⊥J/ψ(x)I(x,z),i=5,7,
with(2.40)f(x,z)=2zx1-z(1-x)+(3-2x)ln⁡x1-x+(-31-zx+11-z(1-x)-2zx(1-z(1-x))2)zxln⁡zx+(3(1-z)+2zx+2z2x21-z(1-x))ln⁡(1-z)-iπ1-z(1-x)+(1-z)×{-4x(1-z)(1-x)ln⁡x+zx(1-z(1-x))2ln⁡(1-z)+(1(1-zx)2-1(1-z(1-x))2+2(1+z-2zx)(1-z)(1-zx)2)zxln⁡zx-iπzx(1-z(1-x))2},g(x,z)=-4rln⁡x1-x+4zrln⁡zx1-z(1-x)-4zrln⁡(1-z)-iπ1-z(1-x)+(1-z){4r(1-z)(1-x)ln⁡x-4rz(1-z)(1-zx)ln⁡zx},h(x,z)=2zx1-z(1-x)+(3-2x)ln⁡x1-x+(-31-zx+11-z(1-x)-2zx(1-z(1-x))2)zxln⁡zx+(3(1-z)+2zx+2z2x21-z(1-x))ln⁡(1-z)-iπ1-z(1-x),I(x,z)=-4rln⁡x1-x+4zrln⁡zx1-z(1-x)-4zrln⁡(1-z)-iπ1-z(1-x),
where z=mJ/ψ2/mB2 and r=fJ/ψ⊥mc/fJ/ψmJ/ψ. In [11] by contracting (2.14) for J/ψ with pν and applying the equation of motion and (2.13) for J/ψ, it finds that fJ/ψ⊥mJ/ψ=2fJ/ψmc. Therefore(2.41)fJ/ψ⊥mcfJ/ψmJ/ψ=2(mcmJ/ψ)2.
In [25], the charmed quark carries a momentum fraction of order ~mc/mJ/ψ, which is the same fraction momentum x, then r=2x2. The contributions of hard spectator interaction have been written as(2.42)Hi0(J/ψK1A(1B))=ifBfJ/ψfK1A(1B)X0(BK1A(1B),J/ψ)mBλB×∫01dydx(ϕ∥K1A(1B)(y)ϕ∥J/ψ(x)(1-y)(1-x)-rχK1A(1B)ϕK1A(1B)(y)ϕ∥J/ψ(x)(1-y)x),
for i=2,3,9,(2.43)Hi0(J/ψK1A(1B))=-ifBfJ/ψfK1A(1B)X0(BK1A(1B),J/ψ)mBλB×∫01dydx(ϕ∥K1A(1B)(y)ϕ∥J/ψ(x)(1-y)x-rχK1A(1B)ϕK1A(1B)(y)ϕ∥J/ψ(x)(1-y)(1-x)),
for i=5,7. The transverse hard spectator terms are(2.44)Hi-(J/ψK1A(1B))=-2ifBfK1A(1B)⊥fJ/ψmJ/ψmBX-(BK1A(1B),J/ψ)mBλB11-z×∫01dydxϕ⊥K1A(1B)(y)ϕ-J/ψ(x)(1-y)2x,Hi+(J/ψK1A(1B))=-2ifBfK1A(1B)fJ/ψmJ/ψmK1A(1B)mBX+(BK1A(1B),J/ψ)mBλB11-z×∫01dydx(1-y-x)ϕ+K1A(1B)(y)ϕ+J/ψ(x)(1-y)2(1-x)2,
for i=2,3,9, and(2.45)Hi-(J/ψK1A(1B))=2ifBfK1A(1B)⊥fJ/ψmJ/ψmBX-(BK1A(1B),J/ψ)mBλB11-z×∫01dydxϕ⊥K1A(1B)(y)ϕ+J/ψ(x)(1-y)2(1-x),(2.46)Hi+(J/ψK1A(1B))=-2ifBfK1A(1B)fJ/ψmJ/ψmK1A(1B)mBX+(BK1A(1B),J/ψ)mBλB11-z×∫01dydx(y-x)ϕ+K1A(1B)(y)ϕ-J/ψ(x)(1-y)2x2,
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=mb and at mixing angles θ=32∘,58∘ in Tables 2 and 3.Table 2
Branching ratios ofB→J/ψK(1270) in generalized factorization and QCD Factorization at μ=mb.
FactorizationMixing angleθ∘BR(B→J/ψK(1270))General32(0.44±0.02)×10-458(0.42±0.01)×10-4QCDF (z→0)32(1.42±0.01)×10-358(1.79±0.01)×10-3QCDF (z≠0)32(3.18±0.001)×10-358(4.08±0.01)×10-3Exp.(1.8±0.5)×10-3Table 3
Branching ratios ofB→J/ψK(1400) in generalized factorization and QCD Factorization at μ=mb.
FactorizationMixing angleθ∘BR(B→J/ψK(1400))General321.27×10-6583.35×10-6QCDF (z→0)323.45×10-4584.27×10-4QCDF (z≠0)325.06×10-4588.51×10-4Exp.<5×10-4


## 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 theJ/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]
(2.1)〈J/ψ(p,ɛ)|c¯1α(y)c2β(x)|0〉=-i4∫01duexp⁡[i(up⋅y+u¯p⋅x)]×{∫fJ/ψmJ/ψ(pɛ*zpzϕ∥J/ψ(u)+ɛ⊥*g⊥J/ψ(v)(u)+14ɛμνρσɛ*μpρzσγνγ5g⊥J/ψ(a)(u))+fJ/ψ⊥(ɛ⊥pϕ⊥J/ψ(u)-imJ/ψ2(ɛ*z)(p⋅z)2σμνpμzνh∥J/ψ(t)(u)-imJ/ψ2(ɛ*z)h∥J/ψ(s)(u)2)}αβ,
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. fJ/ψ, fJ/ψ⊥ are vector and tensor decay constants, respectively. Similarly, for axial-vector meson P13(1), we have(2.2)〈P13(1)(p,ɛ)|d¯1α(y)u2β(x)|0〉=-i4∫01duexp⁡[i(vp⋅y+v¯p⋅x)]×{(ɛ⊥pγ5ϕ⊥3(1)P1(v)-im3(1)P12(ɛ*z)(p⋅z)2σμνpμzνγ5h∥3(1)P1(t)(v)-im3(1)P12(ɛ*z)γ5h∥3(1)P1(p)(v)2)f3(1)P1m3(1)P1(pγ5ɛ*zpzϕ∥3(1)P1(v)+ɛ⊥*γ5g⊥3(1)P1(v)(v)+14ɛμνρσɛ*μpρzσγνg⊥3(1)P1(a)(v))+fP13(1)⊥×(ɛ⊥pγ5ϕ⊥3(1)P1(v)-im3(1)P12(ɛ*z)(p⋅z)2σμνpμzνγ5h∥3(1)P1(t)(v)-im3(1)P12(ɛ*z)γ5h∥3(1)P1(p)(v)2)}αβ.
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 as(2.3)ϕ+(x)=g⊥′(a)(x)4+g⊥(a)(x)=∫x1ϕ∥(u)udu,ϕ-(x)=-(g⊥′(a)(x)4-g⊥(a)(x))=∫0xϕ∥(u)1-udu,
which are transverse components of twist-3 LCDAs. While ϕv(a) are longitudinal components of twist-3 ones for vector (axial-vector) meson. We specify the light-cone distribution amplitudes (LCDAs) as(2.4)ϕV(x,μ)=6x(1-x)[1+∑n=1∞anV(μ)Cn3/2(2x-1)],ϕv(x,μ)=3[2x-1+∑n=1∞an⊥,V(μ)Pn+1(2x-1)],
for the vector meson, where Cn(x) are Gegenbauer polynomials and Pn(x) are Legendre polynomials. The normalization of LCDAs for vector meson is(2.5)∫01dxϕV(x)=1,∫01dxϕv(x)=0.
Also,(2.6)ϕ∥A(x)=6x(1-x)[1+3a1∥,3P1(2x-1)],ϕ⊥A(x)=6x(1-x)[a0⊥,3P1+3a1⊥,3P1(2x-1)],ϕaA(x,μ)=3[a0⊥,3P1(2x-1)+∑n=1∞an⊥,3P1(μ)Pn+1(2x-1)],
for P13 axial-vector meson and(2.7)ϕ∥A(x)=6x(1-x)[a0∥,1P1+3a1∥,1P1(2x-1)],ϕ⊥A(x)=6x(1-x)[1+3a1⊥,1P1(2x-1)],ϕaA(x,μ)=3[(2x-1)+∑n=1∞an⊥,1P1(μ)Pn+1(2x-1)],
for P11 axial-vector meson. The normalization conditions are(2.8)∫01dxϕ∥(x)=1,∫01dxϕ⊥(x)=a0⊥,3P1,∫01dxϕa(x)=0,
for P13 axial-vector meson and(2.9)∫01dxϕ∥(x)=a0∥,1P1,∫01dxϕ⊥(x)=1,∫01dxϕa(x)=0,
for P11 axial-vector meson.HereϕV, ϕ∥A, and ϕ⊥A are twist-2 LCDAs; likewise ϕv and ϕaA are twist-3 ones for vector and axial-vector mesons, respectively. The parameters a2∥ and a2⊥ 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 K1A and K1B are given in [14, 15].For theB meson, we can write the projection as(2.10)〈0|b¯α(x)qβ(0)|B(p)〉=-ifB4ϕB(ρ)[(p+mb)γ5]βγ.
A detailed dissection of wave function of B meson can be found [9, 10, 16–18](2.11)〈0|b¯α(x)qβ(0)|B(p)〉|(x+=x⊥=0)=-ifB4[(p+mb)γ5]βγ×∫01dρ¯e-ρ¯p+x-[ϕ1B(ρ¯)+n-ϕ2B(ρ¯)]γα,
with n-=(1,0,0,-1), and the normalization conditions are(2.12)∫01dρ¯ϕ1B(ρ¯)=1,∫01dρ¯ϕ2B(ρ¯)=0.
The B wave function corresponds to λB defined by ∫01dρ¯ϕ1B(ρ¯)/ρ¯≡mB/λB. λB is the first inverse moment of the B meson's distribution which is of order ΛQCD.


## 2.2. Decay Constant

Decay constants of vector and axial-vector mesons are defined as(2.13)〈V(p,ɛ)|q¯2γμq1|0〉=-ifVmVɛμ*,〈A(p,ɛ)|q¯2γμγ5q1|0〉=ifAmAɛμ*.
Also, the transverse decay constants are defined via the tensor current as(2.14)〈V(p,ɛ)|q¯2σμνq1|0〉=-fV⊥(ɛμ*pν-ɛν*pμ),〈A(p,ɛ)|q¯2γμγ5q1|0〉=ifAmAɛμ*.
In general, the decay constants f1p1 and f3p1⊥ are zero in the SU(3) limit.


## 2.3. Form Factor

The form factors ofB¯→A transitions are defined as [2](2.15)〈A(p,ɛ)|Aμ|B¯(pB)〉=i2mB-mAɛμναβɛ*νpBαpβABA(q2),〈A(p,ɛ)|Vμ|B¯(pB)〉=-{(mB-mA)ɛμ*V1BA(q2)-(ɛ*⋅pB)(pB+p)μV2BA(q2)mB-mA-2mAɛ*⋅pBq2qμ[V3BA(q2)-V0BA(q2)]V2BA(q2)mB-mA},
and for B¯→V transitions, we have(2.16)〈V(p,ɛ)|Vμ|B¯(pB)〉=2mB+mVɛμναβɛ*νpαpBβVBV(q2),〈V(p,ɛ)|Aμ|B¯(pB)〉=i{(mB+mV)ɛμ*A1BV(q2)-ɛ*⋅pBmB+mV(pB+p)μA2BV(q2)-2mVɛ*⋅pBq2qμ[A3BV(q2)-A0BV(q2)]},
where q=pB-p, and for q2=0 we have A3BV=A0BV and V3BA=V0BA. ɛμ is the polarization four-vector, and V, A1, A2, A3, and A0 are form factors and(2.17)V3BA(q2)=mB-mA2mAV1BA(q2)-mB+mA2mAV2BA(q2),A3BV(q2)=mB+mV2mVA1BV(q2)-mB-mV2mVA2BV(q2).
The form factors of B→K1A and B→K1B 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:(2.18)F(q2)=F(0)1-a(q2/m2)B+b(q4/mB4),
which q2=mJ/ψ2, and two parameters a and b are given in Table 1.Table 1
Form factors forB→K1A and B→K1B transitions which are obtained in light-cone sum rule models.
F(0)abABK1A0.27±0.051.600.974V0BK1A0.22±0.042.401.78V1BK1A0.56±0.110.6350.211V2BK1A0.25±0.051.511.18ABK1B-0.22-0.04+0.061.720.912V0BK1B-0.45-0.08+0.121.340.690V1BK1B-0.48-0.08+0.130.7290.074V2BK1B-0.01-0.02+0.030.9190.855


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

Now, we want to calculate the branching ratios ofB→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 Ci(μ) to form cieff, 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): c1eff=1.1680, c2eff=-0.3652, c3eff=0.0233+0.0043i, c4eff=-0.0482-0.0129i, c5eff=0.0140+0.0043i, c6eff=-0.0504-0.0129i, c7eff/α=-0.0312-0.0357i, c8eff/α=0.0551, c9eff/α=-1.4277-0.0357i, c10eff/α=0.4804 [1]. α is the fine structure constant. Wilson coefficients appear in decay amplitudes as linear combinations. So, the aieff coefficients are(2.19)aieff≡cieff+ci+1effNc(i=odd),aieff≡cieff+ci-1effNc(i=even).
As it known the physical states K1(1270) and K1(1400) are mixture of states P11 and P13, whose relations could be parameterized by(2.20)K1(1270)=K1Asin⁡θ+K1Bcos⁡θ,K1(1400)=K1Acos⁡θ-K1Bsin⁡θ,
where K1A and K1B are P13 and P11 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 by(2.21)A(B⟶J/ψK1(1270))p=A(B⟶J/ψK1A)sin⁡θ+A(B⟶J/ψK1B)cos⁡θ,A(B⟶J/ψK1(1400))p=A(B⟶J/ψK1A)cos⁡θ-A(B⟶J/ψK1B)sin⁡θ.
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/ψK1A and B→J/ψK1B 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 as(2.22)Ah(B⟶J/ψK1A)=[VcbVcs*a2h-VtbVts*(a3h+a5h+a7h+a9h)]Xh(BK1A,J/ψ),Ah(B⟶J/ψK1B)=[VcbVcs*a2h-VtbVts*(a3h+a5h+a7h+a9h)]Xh(BK1B,J/ψ),
where the factorizable amplitude is written as(2.23)X(BK1A(1B),J/ψ)≡〈J/ψ|(c¯c)V-A|0〉〈K1A(1B)|(b¯s)V-A|B〉.
The helicity dependencies read as(2.24)X0(BK,J/ψ)=ifJ/ψ2mK[(mB2-mK2)-mJ/ψ2(mB-mK)V1BK(q2)-4mB2pc2mB-mKV2BK(q2)],X±(BK,J/ψ)=-ifJ/ψmBmJ/ψ[(1-mKmB)V1BK(q2)∓2pcmB-mKABK(q2)],
where pc is c.m. momentum of decays in the B rest frame(2.25)pc=(mB2-(m1+m2)2)(mB2-(m1-m2)2)2mB,m1 and m2 are the mass of final state mesons. The branching ratio is given by [18](2.26)BR(B⟶VA)=pc8πmB2τB(GF2)2|A(B→VA)|2,
where GF is Fermi constant and τB 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/mB, while the |A-| amplitude is further suppression in m1/mB. But for B to heavy-light final states, 2mheavy/mB is of order unity [23]. In QCD factorization, the effective coefficients aih basically are Wilson coefficients in conjunction with short distance nonfactorizable corrections such as vertex corrections and hard spectator interactions. Then, we have [2](2.27)aih(J/ψK1A(K1B))=(Ci+Ci±1Nc)Ni(J/ψ)∫01ϕJ/ψ,h(x)dx+Ci±1NcCFαs4π[Vih(J/ψ)+4π2NcHih(J/ψK1A(K1B))],
where i=1,…,10, the lower (upper) signs apply when i is even (odd), Ci 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=mb(mb).), CF=(Nc2-1)/(2Nc) with Nc=3, J/ψ is emitted meson and K1A(K1B) have the spectator quark of the meson. Vih(J/ψ) is the vertex corrections and Hih(J/ψK1A(K1B)) 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 ϕJ/ψ,h is ϕ∥J/ψ for h=0 and ϕ⊥J/ψ for h=±. And Nih(J/ψ)=1. We can take the infinite mass limit of the b quark in which mb goes to infinity while mJ/ψ is fixed (mJ/ψ/mb→0). The vertex corrections are given by(2.28)Vi0(J/ψ)={∫01dxϕ∥J/ψ(x)[12Lnmbμ-18+g(x)],i=2,3,9,∫01dxϕ∥J/ψ(x)[-12Lnmbμ+6-g(1-x)],i=5,7,(2.29)Vi±(J/ψ)={∫01dxϕ±J/ψ(x)[12Lnmbμ-18+gT(x)],i=2,3,9,∫01dxϕ∓J/ψ(x)[-12Lnmbμ+6-gT(1-x)],i=5,7.
With(2.30)g(x)=3(1-2x1-xln⁡x-iπ)+[2Li2(x)-Ln2x+2Lnx1-x-(x⟷1-x)],gT(x)=g(x)+Lnx1-x,
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 by(2.31)Hi0(J/ψK1A(1B))=ifBfJ/ψfK1A(1B)X0(BK1A(1B),J/ψ)mBλB×∫01dydx(ϕ∥K1A(1B)(y)ϕ∥J/ψ(x)(1-y)(1-x)-rχK1A(1B)ϕK1A(1B)(y)ϕ∥J/ψ(x)(1-y)x),
for i=2,3,9, and(2.32)Hi0(J/ψK1A(1B))=-ifBfJ/ψfK1A(1B)X0(BK1A(1B),J/ψ)mBλB×∫01dydx(ϕ∥K1A(1B)(y)ϕ∥J/ψ(x)(1-y)x-rχK1A(1B)ϕK1A(1B)(y)ϕ∥J/ψ(x)(1-y)(1-x)),
for i=5,7. rχK1A(1B)=2(mK1A(1B)fK1A(1B)⊥)/(mb(μ)fK1A(1B)) are chirally enhanced parameters. The transverse hard spectator terms are(2.33)Hi-(J/ψK1A(1B))=-2ifBfK1A(1B)⊥fJ/ψmJ/ψmBX-(BK1A(1B),J/ψ)mBλB×∫01dydxϕ⊥K1A(1B)(y)ϕ-J/ψ(x)(1-y)2x,Hi+(J/ψK1A(1B))=-2ifBfK1A(1B)fJ/ψmJ/ψmK1A(1B)mB2X+(BK1A(1B),J/ψ)mBλB×∫01dydx(1-y-x)ϕ+K1A(1B)(y)ϕ+J/ψ(x)(1-y)2(1-x)2,
for i=2,3,9, and(2.34)Hi-(J/ψK1A(1B))=2ifBfK1A(1B)⊥fJ/ψmJ/ψmBX-(BK1A(1B),J/ψ)mBλB×∫01dydxϕ⊥K1A(1B)(y)ϕ+J/ψ(x)(1-y)2(1-x),(2.35)Hi+(J/ψK1A(1B))=-2ifBfK1A(1B)fJ/ψmJ/ψmK1A(1B)mB2X+(BK1A(1B),J/ψ)mBλB×∫01dydx(y-x)ϕ+K1A(1B)(y)ϕ-J/ψ(x)(1-y)2x2,
for i=5,7. In our calculations, the transverse hard spectator term Hi+(J/ψK1A) is zero. There are some divergences in other hard spectator terms. We have used the unknown parameter XH to eliminate these divergences, defined as(2.36)XH=∫01dx1-x=Ln(mBΛQCD)(1+r),
where ΛQCD is hadronic parameter and r=ρHeiφH which ρH and φH are phases parameters (0≤ρH≤1, -180≤φH≤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 thatJ/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 mb goes to infinity, mJ/ψ/mb 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/mc, the J/ψ wave function is [18](2.37)〈J/ψ(p,ɛ)|c¯α(x)cβ(0)|0〉|(x+=x⊥=0)=fJ/ψ4[ɛ*(p+mb)γ5]βγ∫01dξ¯e-iξp+x-[ϕ1J/ψ(ξ)+n-ϕ2J/ψ(ξ)]γα.
Using [18], the effect of higher twist for J/ψ may not converge enough. Because the charmed quark carries a momentum fraction of order ~mc/mJ/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 ɛ⊥ component since the latter does not vanish at the end point. Comparing (2.1) with (2.37) the leading twist DA of J/ψ is(2.38)ϕ1J/ψ(u)=ϕ⊥J/ψ(u)~6uu¯,fJ/ψ=fJ/ψ⊥.
Then, for vertex corrections [18], we have(2.39)Vi0(J/ψ)={∫01dxϕ∥J/ψ(x)[12Lnmbμ-18+f(x,z)]+∫01dxϕ⊥J/ψ(x)g(x,z),i=2,3,9,∫01dxϕ∥J/ψ(x)[-12Lnmbμ+6-f(x,z)]+∫01dxϕ⊥J/ψ(x)g(x,z),i=5,7,Vi±(J/ψ)={∫01dxϕ∥J/ψ(x)[12Lnmbμ-18+h(x,z)]+∫01dxϕ⊥J/ψ(x)I(x,z),i=2,3,9,∫01dxϕ∥J/ψ(x)[-12Lnmbμ+6-h(x,z)]+∫01dxϕ⊥J/ψ(x)I(x,z),i=5,7,
with(2.40)f(x,z)=2zx1-z(1-x)+(3-2x)ln⁡x1-x+(-31-zx+11-z(1-x)-2zx(1-z(1-x))2)zxln⁡zx+(3(1-z)+2zx+2z2x21-z(1-x))ln⁡(1-z)-iπ1-z(1-x)+(1-z)×{-4x(1-z)(1-x)ln⁡x+zx(1-z(1-x))2ln⁡(1-z)+(1(1-zx)2-1(1-z(1-x))2+2(1+z-2zx)(1-z)(1-zx)2)zxln⁡zx-iπzx(1-z(1-x))2},g(x,z)=-4rln⁡x1-x+4zrln⁡zx1-z(1-x)-4zrln⁡(1-z)-iπ1-z(1-x)+(1-z){4r(1-z)(1-x)ln⁡x-4rz(1-z)(1-zx)ln⁡zx},h(x,z)=2zx1-z(1-x)+(3-2x)ln⁡x1-x+(-31-zx+11-z(1-x)-2zx(1-z(1-x))2)zxln⁡zx+(3(1-z)+2zx+2z2x21-z(1-x))ln⁡(1-z)-iπ1-z(1-x),I(x,z)=-4rln⁡x1-x+4zrln⁡zx1-z(1-x)-4zrln⁡(1-z)-iπ1-z(1-x),
where z=mJ/ψ2/mB2 and r=fJ/ψ⊥mc/fJ/ψmJ/ψ. In [11] by contracting (2.14) for J/ψ with pν and applying the equation of motion and (2.13) for J/ψ, it finds that fJ/ψ⊥mJ/ψ=2fJ/ψmc. Therefore(2.41)fJ/ψ⊥mcfJ/ψmJ/ψ=2(mcmJ/ψ)2.
In [25], the charmed quark carries a momentum fraction of order ~mc/mJ/ψ, which is the same fraction momentum x, then r=2x2. The contributions of hard spectator interaction have been written as(2.42)Hi0(J/ψK1A(1B))=ifBfJ/ψfK1A(1B)X0(BK1A(1B),J/ψ)mBλB×∫01dydx(ϕ∥K1A(1B)(y)ϕ∥J/ψ(x)(1-y)(1-x)-rχK1A(1B)ϕK1A(1B)(y)ϕ∥J/ψ(x)(1-y)x),
for i=2,3,9,(2.43)Hi0(J/ψK1A(1B))=-ifBfJ/ψfK1A(1B)X0(BK1A(1B),J/ψ)mBλB×∫01dydx(ϕ∥K1A(1B)(y)ϕ∥J/ψ(x)(1-y)x-rχK1A(1B)ϕK1A(1B)(y)ϕ∥J/ψ(x)(1-y)(1-x)),
for i=5,7. The transverse hard spectator terms are(2.44)Hi-(J/ψK1A(1B))=-2ifBfK1A(1B)⊥fJ/ψmJ/ψmBX-(BK1A(1B),J/ψ)mBλB11-z×∫01dydxϕ⊥K1A(1B)(y)ϕ-J/ψ(x)(1-y)2x,Hi+(J/ψK1A(1B))=-2ifBfK1A(1B)fJ/ψmJ/ψmK1A(1B)mBX+(BK1A(1B),J/ψ)mBλB11-z×∫01dydx(1-y-x)ϕ+K1A(1B)(y)ϕ+J/ψ(x)(1-y)2(1-x)2,
for i=2,3,9, and(2.45)Hi-(J/ψK1A(1B))=2ifBfK1A(1B)⊥fJ/ψmJ/ψmBX-(BK1A(1B),J/ψ)mBλB11-z×∫01dydxϕ⊥K1A(1B)(y)ϕ+J/ψ(x)(1-y)2(1-x),(2.46)Hi+(J/ψK1A(1B))=-2ifBfK1A(1B)fJ/ψmJ/ψmK1A(1B)mBX+(BK1A(1B),J/ψ)mBλB11-z×∫01dydx(y-x)ϕ+K1A(1B)(y)ϕ-J/ψ(x)(1-y)2x2,
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=mb and at mixing angles θ=32∘,58∘ in Tables 2 and 3.Table 2
Branching ratios ofB→J/ψK(1270) in generalized factorization and QCD Factorization at μ=mb.
FactorizationMixing angleθ∘BR(B→J/ψK(1270))General32(0.44±0.02)×10-458(0.42±0.01)×10-4QCDF (z→0)32(1.42±0.01)×10-358(1.79±0.01)×10-3QCDF (z≠0)32(3.18±0.001)×10-358(4.08±0.01)×10-3Exp.(1.8±0.5)×10-3Table 3
Branching ratios ofB→J/ψK(1400) in generalized factorization and QCD Factorization at μ=mb.
FactorizationMixing angleθ∘BR(B→J/ψK(1400))General321.27×10-6583.35×10-6QCDF (z→0)323.45×10-4584.27×10-4QCDF (z≠0)325.06×10-4588.51×10-4Exp.<5×10-4


## 3. Input Quantities

In this section, we have been introduced the essential input quantities [26]:(3.1)mB=5.28GeV,mJ/ψ=3.1GeV,mK1A=1.367GeV,mK1B=1.310GeV,fB=210±20MeV,fJ/ψ=405MeV,fK1A=0.250GeV,fK1A⊥(μ=1GeV)=0.065GeV,a0K1A⊥=0.26,a1K1A⊥=-1.08,λB(1GeV)=225MeV,fK1B=-0.028GeV,fK1B⊥(μ=1GeV)=0.190GeV,a1K1B∥=-1.95,CF=N2-12N,GF=1.166×10-5,N=3,|VcbVcs*|=0.039,|VtbVts*|=0.041,τB±=(1.638±0.011)×10-12S.
From (2.19), the magnitudes of factorizable amplitudes are obtained as(3.2)X0(BK1A,J/ψ)=3.54i,X+(BK1A,J/ψ)=-1.25i,X-(BK1A,J/ψ)=-5.56i,X0(BK1B,J/ψ)=-5.36i,X+(BK1B,J/ψ)=1.21i,X-(BK1B,J/ψ)=5.09i.


## 4. Discussion

In generalized factorization, the branching ratios ofB→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 aih 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 mJ/ψ/mb is held fixed. The effective coefficients aih 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.



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