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Insights into the nature of X(3872) through B meson decays

  • We study the Bc,u,dX(3872)P decays in the perturbative QCD (PQCD) approach, involving the puzzling resonance X(3872), where P represents a light pseudoscalar meson (K or π). Assuming X(3872) to be a 1++ charmonium state, we obtain the following results. (a) The branching ratios of the B+cX(3872)π+ and B+cX(3872)K+ decays are consistent with the results predicted by the covariant light-front approach within errors; however, they are larger than those given by the generalized factorization approach. (b) The branching ratio of the B+X(3872)K+ decay is predicted as (3.8+1.11.0)×104, which is smaller than the previous PQCD calculation result but still slightly larger than the upper limits set by Belle and BaBar. Hence, we suggest that theB0,+X(3872)K0,+ decays should be precisely measured by the LHCb and Belle II experiments to help probe the inner structure of X(3872). (c) Compared with the Bu,dX(3872)Kdecays, the Bu,dX(3872)π decays have significantly smaller branching ratios, which drop to values as low as 106. (d) The direct CP violations of these considered decays are small (103102) because the penguin contributions are loop suppressed compared to the tree contributions. The mixing-induced CP violation of the BX(3872)K0S decay is highly consistent with the current world average value sin2β=(69.9±1.7)%. Experimentally testing the results for the branching ratios and CP violations, including the implicit SU(3) and isospin symmetries of these decays, helps probe the nature of X(3872).
  • The hadron X(3872) has attracted considerable attention since it was first observed by Belle in the exclusive decay B±K±π+πJ/Ψ [1]. Though X(3872) has been confirmed by many experimental collaborations, such as the CDF [2], D0 [3], Babar [4], and LHCb [5], with quantum numbers JPC=1++ and isospin I=0, there are still many uncertainties. Because the mass of X(3872) is close to the D0ˉD0 threshold, several authors interpret it as a loosely bound molecular state [610], in which the building blocks are hadrons [11]. Others regard X(3872) as a compact tetraquark state [1215], in which the building blocks are quarks and anti-quarks. There are other explanations, such as cˉcg hybrid meson [16, 17] and glueballs [18]. Though there are many different exotic hadron state interpretations of X(3872), it has not been ruled out that the first radial excitation of the 1P charmonium state χc1(1P) is the most natural assignment [1921]. Note that X(3872) was renamed χc1(3872) by the Particle Data Group (PDG) [22].

    Many studies on the production and decays of X(3872)have been performed to investigate the inner structure ofX(3872) [2326]. In Ref. [26], the authors calculated Γ(X(3872)J/ψπ+π) using QCD sum rules and concluded that X(3872) is approximately 97% a charmonium state with a small molecular component. Many B meson decays with X(3872) in the final states have been studied using different approaches [2733]. In Ref. [28], the authors studied the Bχc1(1P,2P)K decays using the QCD factorization (QCDF) approach and argued that X(3872) has a dominant cˉc component but mixes with the D0ˉD0+D0ˉD0 continuum component. The BcX(3872)π(K) decays were studied both using the covariant light-front (CLF) approach [29] and generalized factorization (GF) approach [30], respectively. In the former, X(3872) was identified as a 1++ charmonium state, whereas a tetraquark state was assumed in the latter. One may expect different results for the same decays under the different structure hypotheses ofX(3872). The BX(3872)K decay has also received significant attention from many researchers. In Refs. [31, 32], the authors assumed X(3872) to be a loosely bound S-wave molecular state of D0ˉD0(D0ˉD0) and estimated the branching ratio of the B+X(3872)K+ decay to be (0.071)×104. Furthermore, they considered the branching ratio of the B0X(3872)K0 decay to be suppressed by more than one order of magnitude compared with that of the B+X(3872)K+decay, which indicates that there is large isospin symmetry between the B+X(3872)K+ and B0X(3872)K0decays. If this type of large isospin symmetry is observed in experiments, any charmonium interpretation of X(3872) will be disfavored. Two years later, the branching ratio of the B+X(3872)K+ decay was calculated using the perturbative QCD (PQCD) approach, assuming X(3872) as a regular cˉc charmonium state, in Ref. [33]; a large value of Br(B+X(3872)K+)=(7.88+4.873.76)×104was obtained. Clearly, this result is significantly larger than the present experimental upper limits given by Belle [34] and BaBar [35] at the 90 % C.L.,

    Br(B+X(3872)K+)<2.6×104(Belle),

    (1)

    Br(B+X(3872)K+)<3.2×104(BaBar).

    (2)

    Here, we conduct a systematic study of the Bc,u,dX(3872)P decays using the PQCD approach, where P represents a light pseudoscalar meson (K or π). The layout of this paper is as follows. We present the analytic calculations of the amplitudes of the Bc,u,dX(3872)P decays in Section II. The numerical results and discussions are given in Section III, where we compare our results with other theoretical predictions and experimental data. The conclusions are presented in Section IV.

    Because the PQCD approach based on kT factorization has been successfully applied to many two-body charmed B meson decays [3639], we use this approach to investigate the Bc,u,dX(3872)P decays in this study. First, the effective Hamiltonian for the B+cX(3872)π+(K+) decays can be written as [40]

    Heff=GF2VcbVuq[C1(μ)O1(μ)+C2(μ)O2(μ)]+H.c.,

    (3)

    where the Fermi coupling constant GF1.166×105 GeV2 [22], VcbVuq is the product of the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements with q=d (q=s) for B+cX(3872)π+ (B+cX(3872)K+) decay, Ci(μ)(i=1,2) are the Wilson coefficients at the renormalization scale μ, and Oi(i=1,2) are the local four-quark operators,

    O1=ˉdαγμ(1γ5)uβˉcβγμ(1γ5)bα,O2=ˉdαγμ(1γ5)uαˉcβγμ(1γ5)bβ,

    (4)

    where α and β are the SU(3) color indices, and the summation convention over repeated indices is understood. Because the four quarks in the operators are different, there is no penguin contribution and thus no CP violation. Here, we analyze the B+cX(3872)π+ decay as an example, and its Feynman diagrams are given in Fig. 1, where only the factorizable and non-factorizable emission diagrams need to be considered at the leading order. The amplitude for the factorizable emission diagrams in Fig. 1(a) and Fig. 1(b) can be written as

    Figure 1

    Figure 1.  Feynman diagrams contributing to the B+cX(3872)π+ decay at the leading order.

    FLLBcX=223πCFm4BcfπfBc1r2X10dx20b1db1b2db2exp(ω2Bcb212){[ΨL(x2)(x22rb)+Ψt(x2)(rb2x2)]Ee(ta)h(α,βa,b1,b2)St(x2)ΨL(x2)(rc+r2X(x11))Ee(tb)h(α,βb,b2,b1)St(x1)},

    (5)

    where the superscript LL denotes the contribution from the (VA)(VA) operators, the color factor CF=4/3,fπ(Bc) is the decay constant for the meson π(Bc), the mass ratio rX(b,c)=mX(mb,mc)/mBc, the exponent exp(ω2Bcb21/2) originates from the Bc meson wave function, and ΨL,t(x2) are the distribution amplitudes for X(3872) (given in Sec. III).

    The amplitude for the non-factorizable spectator diagrams in Fig. 1(c) and Fig. 1(d) is given as

    MLLBcX=83πCFm4BcfBc1r2X10dx20b1db1b3db3exp(ω2Bcb212)ϕAπ(x3){[ΨL(x2)(x3x1)(1r2X)+rXΨt(x2)(1x1x2)]Ecd(tc)h(α,βc,b1,b3)+[ΨL(x2)(r2X(x2x3)+2x1+x2+x32)+rXΨt(x2)(1x1x2)]Ecd(td)h(α,βd,b1,b3)}.

    (6)

    Note that the hard function h originates from the Fourier transform of the virtual quark and gluon propagators, which is defined as

    h(α,β,b1,b2)=h1(α,b1)×h2(β,b1,b2),

    (7)

    h1(α,b1)={K0(αb1),α>0,K0(iαb1),α<0,

    (8)

    h1(β,b1,b2)={θ(b1b2)I0(βb2)K0(βb1)+(b1b2),β>0,θ(b1b2)J0(βb2)K0(iβb1)+(b1b2),β<0,

    (9)

    where J0 is the Bessel function, and K0,I0 are the modified Bessel functions with K0(ix)=π(N0(x)+iJ0(x))/2. In Eqs. (5) and (6), α and βa,b,c,d in the hard function h are the invariant masses of the internal quarks and gluons, respectively. The hard scales ta,b,c,d are given as the maximum energy scale appearing in each Feynman diagram to remove the large logarithmic radiative corrections. Their expressions are listed in the appendix. The evolution factors Ee(t),Ecd(t) evolving the Sudakov exponent and jet function St(x) can be found in Refs. [38, 41]. For the reader's convenience, their explicit forms are also summarized in the appendix.

    Second, the effective Hamiltonian for the Bu,dX(3872)π(K) decays is written as

    Heff=GF2[VcbVcq(C1(μ)Oc1(μ)+C2(μ)Oc2(μ))VtbVtq10i=3Ci(μ)Oi(μ)],

    (10)

    where Vc(t)bVc(t)q is the product of the CKM matrix elements, q=d or s. The local four-quark operators Oi(μ) and corresponding QCD-corrected Wilson coefficients Ci(μ) can be found in Ref. [40]. Here, we analyze B+X(3872)π+ as an example, and its Feynman diagrams are given in Fig. 2. The amplitudes for the factorizable and nonfactorizable emission diagrams from the (VA)(VA) operators are denoted as FLLBπ and MLLBπ, respectively. Their analytical expressions are given as

    Figure 2

    Figure 2.  Feynman diagrams contributing to the B+X(3872)π+ decay at the leading order.

    FLLBπ=8πCFm4BfX1r2X10dx1dx30b1db1b3db3ϕB(x1,b1){[(r2X1)ϕAπ(x3)((r2X1)x31)+(r2X1)ϕPπ(x3)rπ(2x31)+ϕTπ(x3)rπ(2x3(r2X1)+1+r2X)]×Ee(ta)h(α,βa,b1,b3)St(x3)2rπ(1r2X)ϕPπ(x3)Ee(tb)h(α,βb,b3,b1)St(x1)},

    (11)

    MLLBπ=3261(1r2X)πCFm4B10dx1dx20b1db1b2db2ϕB(x1,b1){[ΨL(x2)(ϕAπ(x3)(r2X1)+2ϕTπ(x3)rπ)(r2X(x1+x32x2)+x1x3)+4rπrXrcϕTπ(x3)Ψt(x2)]}Ec(tc)h(α,βc,b1,b2),

    (12)

    where the evolution factors Ee(t),Ec(t) evolving the Sudakov exponent are given in the appendix. Besides the upper two (VA)(VA) type amplitudes, there are factorizable and nonfactorizable emission diagram contributions from the (VA)(V+A) and (SP)(S+P) operators, which are expressed as FLRBπ and MSPBπ, respectively.

    FLRBπ=FLLBπ,

    (13)

    MSPBπ=3261(1r2X)πCFm4B10dx1dx20b1db1b2db2ϕB(x1,b1)×{[ΨL(x2)(ϕAπ(x3)(r2X1)+2ϕTπ(x3)rπ)(r2X(x1+x32x2)+x1x3)4rπrXrcϕTπ(x3)Ψt(x2)]}Ec(tc)h(βc,α,b2,b1),

    (14)

    where α,βa,b,c in the upper hard function and the hard scales ta,b,c are defined in the appendix.

    By combining the amplitudes from the different Feynman diagrams, the total decay amplitudes for the considered decays are given as

    A(BcX(3872)P)=VcbVuq[a1FLLBcX+C1MLLBcX],

    (15)

    A(Bu,dX(3872)P)=VcbVcq[a2FLLBP+C2MLLBP]VtbVtq[(a3+a9a5a7)FLLBP+(C4+C10)MLLBP+(C6+C8)MSPBP],

    (16)

    where the combinations of the Wilson coefficients a1=C1/3+C2,a2=C1+C2/3,ai=Ci+Ci+1/3 with i= 3, 5, 7, and 9, and q=d (q=s) corresponds to the decays induced by the bd (bs) transition.

    We use the following input parameters for the numerical calculations [22, 29, 30]:

    fBc=0.398+0.0540.055 GeV,  fB=0.19 GeV,fX=0.234±0.052 GeV,

    (17)

    MBc=6.275 GeV,  MB=5.279 GeV,MX=3.87169 GeV,

    (18)

    τBc=0.510×1012s,τ±B=1.638×1012s,τB0=1.519×1012s.

    (19)

    For the CKM matrix elements, we adopt the Wolfenstein parameterization and the updated values A=0.814, λ=0.22537, ˉρ=0.117±0.021, and ˉη=0.353±0.013 [22]. With the total amplitudes, the decay width can be expressed as

    Γ(BX(3872)P)=G2F32πmB(1r2X)|A(BX(3872)P)|2.

    (20)

    The wave functions of B,π, and K have been well defined in many studies, whereas those of Bc and X(3872) still have many uncertainties. For the Bc meson, we use its wave function in the nonrelativistic limit [42],

    ΦBc(x)=ifBc4NC[(Bc+MBc)γ5δ(xrc)]exp(b2ω2Bc2),

    (21)

    where b is the conjugate space coordinate of the parton transverse momentum kT, and the shape parameter ωBc=0.6 GeV. The last exponent term reveals the kT dependence.

    For the light cone distribution amplitude of X(3872), we adopt a similar formula to that of the χc1 meson [33, 43],

    X(p,ϵL)|ˉcα(z)cβ(0)|0=12Ncdxeixpz{mX[γ5ϵ̸L]βαϕLX(x)+[γ5ϵ̸L]βαϕtX(x)},

    (22)

    where ϵL is the longitudinal polarization vector, and mX is the X(3872) mass. Here, only the longitudinal polarization contributes to the considered decays, and the asymptotic models of the twist-2 distribution amplitude ϕLX(x) and twist-3 distribution amplitude ϕtX(x) are given as

    ϕLX(x)=24.68fX22Ncx(1x)×{x(1x)(12x)2[14x(1x)][13.47x(1x)]3}0.7,

    (23)

    ϕtX(x)=13.53fX22Nc(12x)2×{x(1x)(12x)2[14x(1x)][13.47x(1x)]3}0.7,

    (24)

    where fX is the X(3872) decay constant.

    Using the input parameters and wave functions specified in this section, we present the branching ratios of the B+cX(3872)π+(K+) decays as follows:

    Br(B+cX(3872)π+)=(2.7+1.4+0.9+0.7+0.21.00.60.50.1)×104,

    (25)

    Br(B+cX(3872)K+)=(2.5+1.3+0.8+0.6+0.21.00.60.40.1)×105,

    (26)

    where the first error arises from the X(3872) decay constant, fX=0.234±0.052 GeV, the second and third uncertainties are caused by the shape parameter ωBc=0.6±0.1 GeV and decay constant fBc=0.398+0.0540.055 GeV, respectively, and the final error is from the variation in the hard scale from 0.8t to 1.2t, which characterizes the size of the next-to-leading-order QCD contributions. The branching ratios are sensitive to the decay constant fX because the dominant contributions for these two channels are from the factorization emission amplitudes, which are proportional to fX. The branching ratio of B+cX(3872)π+ is approximately one order of magnitude larger than that of BcX(3872)K, which is mainly induced by the difference between the CKM elements Vud=1λ2/2 and Vus=λ. From Table 1, it is shown that our predictions are consistent with the results given in the covariant light-front quark model within errors [29]; however, they are significantly larger than those calculated using the generalized factorization approach [30].

    Table 1

    Table 1.  Our predictions for the branching ratios of the B+cX(3872)π+(K+)decays, along with the results from the covariant light-front (CLF) approach [29] and generalized factorization (GF) approach [30].
    ModeThis studyCLF [29]GF [30]
    B+cX(3872)π+(×104)2.7+1.4+0.9+0.7+0.21.00.60.50.11.7+0.7+0.1+0.40.60.20.40.60+0.22+0.140.180.07
    B+cX(3872)K+(×105)2.5+1.3+0.8+0.6+0.21.00.60.40.11.3+0.5+0.1+0.30.50.20.30.47+0.17+0.110.140.05
    DownLoad: CSV
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    Similarly, the branching ratios of the BX(3872)P decays are calculated as follows:

    Br(B+X(3872)K+)=(3.8+0.9+0.6+0.30.80.50.2)×104,

    (27)

    Br(B0X(3872)K0)=(3.5+0.7+0.5+0.30.60.40.2)×104,

    (28)

    Br(B+X(3872)π+)=(9.3+1.5+0.9+0.51.30.80.4)×106,

    (29)

    Br(B0X(3872)π0)=(4.3+0.7+0.5+0.30.60.40.3)×106,

    (30)

    where the first uncertainty arises from the shape parameter ωB=0.4±0.04 GeV in the B meson wave function, the second error is from the decay constant fX=0.234±0.052 GeV of X(3872), and the third error arises from the choice of hard scales, which vary from 0.8t to 1.2t. From the results, we find that the branching ratios of the B+X(3872)K+ and B0X(3872)K0 decays are similar because they differ only in the lifetimes between B+ and B0 in our formalism. Our prediction for the branching ratio of the B+X(3872)K+decay is less than the previous PQCD calculation result (7.88+4.873.76)×104 [33]. However, it is still slightly larger than the upper limits 2.6×104 given by Belle [34] and 3.2×104 given by BaBar [35]. If the present experimental upper limits are confident, a pure charmonium assignment for X(3872) may not be suitable under the PQCD approach. We expect that the branching ratios of the B0,+X(3872)K0,+ decays can be precisely measured at the current LHCb and SuperKEKB experiments, which will help probe the inner structure of X(3872).

    However, note that X(3872) was renamed χc1(3872) by the current PDG [22], which seems to assume it is a radial excited state of χc1(1P). As we know, the χc1(1P) meson is another P-wave charmonium state with the same quantum numbers J(PC)=1++ and a slightly lighter mass of 3.511 GeV. In this case, they should have similar properties in B meson decays. For example, the branching ratio of the B+χc1(1P)K+ decay is measured as (4.85±0.33)×104 [22], which is consistent with the result predicted using the PQCD approach (4.4+1.91.6)×104 [43]. The corresponding decay B+X(3872)K+ should have a similar but slightly smaller branching ratio. Comparisons of the branching ratios of the BX(3872)π(K) and Bχc1(1P)π(K) decays can be found in Table 2, where the theoretical predictions for the branching ratios of the Bχc1(1P)π(K)) decays are taken from other PQCD calculations [43]. From Table 2, we know that calculations for the BX(3872)P decays using the PQCD approach are under control and credible. Therefore, we suggest that experimental researchers measure these decays at LHCb and Belle II to help discriminate the inner structure of X(3872) from different assumptions.

    Table 2

    Table 2.  Comparison of Br(BX(3872)π(K)) (this study) and Br(Bχc1(1P)π(K)) [43] calculated using the PQCD approach. The data are taken from the Particle Data Group 2020 [22].
    Mode(×104)B+X(3872)K+B+χc1(1P)K+B0X(3872)K0B0χc1(1P)K0
    PQCD3.8+0.9+0.6+0.30.80.50.24.4+1.91.63.5+0.7+0.5+0.30.60.40.24.1+1.81.6
    Exp.4.85±0.333.95±0.27
    Mode (×105)B+X(3872)π+B+χc1(1P)π+B0X(3872)π0B0χc1(1P)π0
    PQCD0.93+0.15+0.09+0.050.130.080.041.7±0.60.43+0.07+0.05+0.030.060.040.030.8±0.3
    Exp.2.2±0.51.12±0.28
    DownLoad: CSV
    Show Table

    In Table 3, we compare our predictions with the results calculated using the generalized factorization approach [30]. It is interesting that the branching ratios of the BX(3872)π(K) decays calculated with these two different approaches are consistent with each other within errors. We find that Br(B+X(3872)π+)2Br(B0X(3872)π0), which is supported by the isospin symmetry.

    Table 3

    Table 3.  Our predictions for the branching ratios of the BX(3872)π(K)decays, along with the results from the generalized factorization (GF) approach [30].
    ModeThis studyGF [30]
    B+X(3872)K+(×104)3.8+0.9+0.6+0.30.80.50.22.3+1.10.9±0.1
    B0X(3872)K0(×104)3.5+0.7+0.5+0.30.60.40.22.1+1.00.8±0.1
    B+X(3872)π+(×106)9.3+1.5+0.9+0.51.30.80.411.5+5.74.5±0.3
    B0X(3872)π0(×106)4.3+0.7+0.5+0.30.60.40.35.3+2.62.1±0.2
    DownLoad: CSV
    Show Table

    In the following we discuss the CP asymmetries in the BX(3872)Pdecays. As we know, CP asymmetry arises from the interference between the tree and penguin amplitudes; however, there are no contributions from the penguin amplitudes for the B+cX(3872)π+(K+)decays. Therefore, the corresponding direct CP violation is zero. For the charged decays B+ X(3872)π+(K+), we only need to consider the direct CP violation AdirCP, which is defined as

    AdirCP=|ˉA|2|A|2|ˉA|2+|A|2,

    (31)

    where ˉA is the CP-conjugate amplitude of A. For neutral B meson decays, there is another type of CP violation that must be considered, known as as time-dependent CP asymmetry, which is induced by interference between the direct decay and the decay via oscillation. Time-dependent CP violation can be defined as

    A(t)CP=Afcos(Δmt)+Sfsin(Δmt),

    (32)

    where the subscript f represents a CP eigenstate, Δm is the mass difference of the two neutral B meson mass eigenstates, and the direct CP asymmetry Af and mixing-induced CP asymmetry Sf are expressed as

    Af=|λf|21|λf|2+1,Sf=2Im(λf)|λf|2+1,

    (33)

    with

    λf=ηfe2iβˉAA,

    (34)

    where ηf is 1(1) for a CP-even (CP-odd) final state f, and β is the CKM angle [22]. Because the charged decay channel and corresponding neutral mode are the same, except for the lifetime and isospin factor in the amplitudes, they have the same direct CP asymmetries. Therefore, we only need to consider the neutral decays, whose direct CP asymmetries are calculated as

    AX(3872)K0=(1.2+0.0+0.0+0.20.00.00.3)×103,

    (35)

    AX(3872)π0=(2.7+0.1+0.0+0.40.20.00.4)×102,

    (36)

    where the errors are induced by the same sources as those for the branching ratios; however, the direct CP violations are less sensitive to the nonperturbative parameters within their uncertainties, except for the hard scale t. Compared to the tree contributions, the penguin amplitudes are loop suppressed by one to two orders of magnitude. At the same time, the product of the CKM matrix elements associated with the tree amplitudes is approximately four times larger than that of penguin amplitudes. Hence, direct CP violations, which arise from interference between the tree and penguin contributions, are very small. Because the final state X(3872)K0 and its CP conjugate state are flavor-specific, we should use the CP-odd eigenstate X(3872)K0S to analyze the mixing-induced CP violations. The results for the mixing-induced CP violations are calculated as

    SX(3872)K0S=(70.3+0.0+0.0+0.90.00.01.2)%,

    (37)

    SX(3872)π=(60.8+0.0+0.0+1.50.00.01.4)%,

    (38)

    where the errors are similar to those listed in the direct CP violations and are not sensitive to the nonperturbative parameters given in the wave functions. We find that SX(3872)K0S is highly consistent with the current world average value sin2β=0.699±0.017 [44], which is obtained from B0 decays to charmonium and K0S. Therefore, we can check the nature of X(3872) by extracting the CKM phase β from future experimental data on the B0X(3872)K0Sdecay. Conversely, the mixing-induced CP asymmetry of the B0X(3872)π0 decay exhibits a significant deviation from the world average value of sin2β because the imaginary parts of the total amplitudes for this channel and its CP-conjugate process exhibit a large difference. Our results can be tested in future experiments.

    In this study, we analyze the Bc,u,dX(3872)π(K) decays using the PQCD approach by assuming X(3872) to be a 1++ charmonium state. Comparing our predictions for the branching ratios and CP asymmetries of the considered decays with other theoretical results and available experimental data, we find the following results:

    (1) The branching ratios of the BcX(3872)π and BcX(3872)K decays can reach orders of 104 and 105, respectively, which are consistent with the results obtained via the covariant light-front approach within errors but larger than those given by the generalized factorization approach. These results can be discriminated at the current LHCb and Belle II experiments.

    (2) Our predictions for the branching ratio of the BX(3872)K and BX(3872)π decays are consistent with the results given by the generalized factorization approach. The branching ratio of the BX(3872)K) decay can reach the order of 104, which is significantly larger than that of the BX(3872)π decay induced by the bd transition. On the experimental side, it is helpful to probe the inner structure of X(3872) by measuring the branching ratios and testing the SU(3) and isospin symmetries of these considered decays.

    (3) The direct CP violations of the BX(3872)π(K) decays are small (only 103102). The mixing-induced CP violation of the BX(3872)K0S decay agrees with the current world average value sin2β=(69.9±1.7) %. However, it is different for the value of SX(3872)π0 because the imaginary parts of the total amplitudes of the BX(3872)π0 decay and its CP-conjugate process exhibit a large difference.

    The invariant masses of virtual quarks and gluons are given as follows:

    α=(x2+x11)(r2X(1x2)x1)m2Bc,

    βa=(r2bx2(1r2X(1x2))m2Bc,

    βb=(r2c(1x1)(r2Xx1))m2Bc,

    βc=(1x1x2)(r2X(1x2x3)+x3x1)m2Bc,

    βd=(1x1x2)(r2X(x3x2)+1x3x1)m2Bc,

    α=x1x3(1r2X)m2B,

    βa=x3(1r2X)m2B,

    βb=x1(1r2X)m2B,

    βc=(r2c+(x1x2)(x3+r2X(x2x3)))m2B.

    The hard scale t is chosen as the maximum of the virtuality of the internal momentum transition in each amplitude, including 1/bi(i=1,2,3)

    ta(b)=max

    \begin{eqnarray} t_{c(d)}=\max(\sqrt{|\alpha|},\sqrt{|\beta_{c(d)}|},1/b_1,1/b_3), \end{eqnarray}\tag{A11}

    \begin{eqnarray} t^\prime_{a(b)}=\max(\sqrt{|\alpha'|},\sqrt{|\beta'_{a(b)}|},1/b_1,1/b_3), \end{eqnarray}\tag{A12}

    \begin{eqnarray} t'_c=\max(\sqrt{|\alpha'|},\sqrt{|\beta'_c|},1/b_1,1/b_2). \end{eqnarray}\tag{A13}

    The functions E_{e(c,d)}(t) are defined by

    \begin{eqnarray} E_e(t)=\alpha_s(t)\exp[-S_{B}(t)-S_X(t)], \end{eqnarray}\tag{A14}

    \begin{eqnarray} E_{cd}=\alpha_s(t)\exp[-S_B(t)-S_X(t)-S_\pi(t)]|_{b_1=b_2}, \end{eqnarray} \tag{A15}

    \begin{eqnarray} E_{e'}(t)=\alpha_s(t)\exp[-S_{B}(t)-S_\pi(t)], \end{eqnarray} \tag{A16}

    \begin{eqnarray} E_c=\alpha_s(t)\exp[-S_B(t)-S_X(t)-S_\pi(t)]|_{b_1=b_3}, \end{eqnarray}\tag{A17}

    where the Sudakov factors can be written as

    \begin{eqnarray} S_B(t)=s \left(x_1\frac{m_B}{\sqrt2},b_1\right)+2\int^t_{1/b_1}\frac{{\rm d}\bar\mu}{\bar\mu}\gamma_q(\alpha_s(\bar\mu)), \end{eqnarray} \tag{A18}

    \begin{aligned}[b] S_{X}(t)=s\left(x_2\frac{m_B}{\sqrt2},b_2\right)+s\left((1-x_2)\frac{m_B}{\sqrt2},b_2\right)\end{aligned}

    \begin{aligned}[b] \;\;\;\;\;\quad\quad +2\int^t_{1/b_2}\frac{{\rm d}\bar\mu}{\bar\mu}\gamma_q(\alpha_s(\bar\mu)), \end{aligned} \tag{A19}

    \begin{aligned}[b] S_{\pi}(t)=&s\left(x_3\frac{m_B}{\sqrt2},b_3\right)+s\left((1-x_3)\frac{m_B}{\sqrt2},b_3\right)\\&+2\int^t_{1/b_3}\frac{{\rm d}\bar\mu}{\bar\mu}\gamma_q(\alpha_s(\bar\mu)), \end{aligned}\tag{A20}

    where the quark anomalous dimension \gamma_q=-\alpha_s/\pi , and the expression for s(Q,b) in the one-loop running coupling constant is used,

    \begin{aligned}[b] s(Q,b)=&\frac{A^{(1)}}{2\beta_1}\hat{q}\ln\left(\frac{\hat{q}}{\hat{b}}\right)-\frac{A^{(1)}}{2\beta_1}(\hat{q}-\hat{b}) +\frac{A^{(2)}}{4\beta^2_1}\left(\frac{\hat{q}}{\hat{b}}-1\right)\\ &-\left[\frac{A^{(2)}}{4\beta^2_1}-\frac{A^{(1)}}{4\beta_1}\ln\left(\frac{{\rm e}^{2\gamma_E-1}}{2}\right)\right] \ln\left(\frac{\hat{q}}{\hat{b}}\right), \end{aligned}\tag{A21}

    with the variables defined by \hat{q}= \ln[Q/(\sqrt2\Lambda)], \; \hat{q}= \ln[1/(b\Lambda)] and the coefficients A^{(1,2)} and \beta_{1} expressed as

    \begin{eqnarray} \beta_1=\frac{33-2n_f}{12},\quad A^{(1)}=\frac{4}{3}, \end{eqnarray} \tag{A22}

    \begin{eqnarray} A^{(2)}=\frac{67}{9}-\frac{\pi^2}{3} -\frac{10}{27}n_f+\frac{8}{3}\beta_1\ln\left(\frac{1}{2}{\rm e}^{\gamma_{\rm E}}\right), \end{eqnarray}\tag{A23}

    n_f is the number of quark flavors, and \gamma_{\rm E} is Euler's constant.

    As we know, the double logarithms \alpha_s\ln^2x produced by the radiative corrections are not small expansion parameters when the end point region is important. To improve the perturbative expansion, the threshold resummation of these logarithms to all orders is required, which leads to a quark jet function

    \begin{eqnarray} S_t(x)=\frac{2^{1+2c}\Gamma(3/2+c)}{\sqrt{\pi}\Gamma(1+c)}[x(1-x)]^c, \end{eqnarray} \tag{A24}

    with c=0.3 . It is effective to smear the end point singularity with a momentum fraction x\to0 .

    From now on, we will use \begin{document}$ X $\end{document} to denote \begin{document}$ X(3872) $\end{document} for simply in some places.

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    [42] X. Liu, Z. J. Xiao, and C. D. Lu, Phys. Rev. D 81, 014022 (2010) doi: 10.1103/PhysRevD.81.014022
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  • [1] S. K. Choi et al. (Belle Collaboration), Phys. Rev. Lett. 91, 262001 (2003) doi: 10.1103/PhysRevLett.91.262001
    [2] D. Acosta et al. (CDF Ⅱ Collaboration), Phys. Rev. Lett. 93, 072001 (2004) doi: 10.1103/PhysRevLett.93.072001
    [3] V. M. Abazov et al. (D0 Collaboration), Phys. Rev. Lett. 93, 162002 (2004) doi: 10.1103/PhysRevLett.93.162002
    [4] B. Aubert et al. (BaBar Collaboration), Phys. Rev. D 71, 071103 (2005) doi: 10.1103/PhysRevD.71.071103
    [5] R. Aaij et al. (LHCb Collaboration), Phys. Rev. D 102, 092005 (2020) doi: 10.1103/PhysRevD.102.092005
    [6] F. E. Close and P. R. Rage, Phys. Lett. B 578, 119 (2004) doi: 10.1016/j.physletb.2003.10.032
    [7] C. Y. Wong, Phys. Rev. C 69, 055202 (2004) doi: 10.1103/PhysRevC.69.055202
    [8] E. Braaten and M. Kusunoki, Phys. Rev. D 69, 074005 (2004) doi: 10.1103/PhysRevD.69.074005
    [9] E. S. Swanson, Phys. Lett. B 588, 189 (2004) doi: 10.1016/j.physletb.2004.03.033
    [10] Z. Y. Lin, J. B. Cheng, and S. L. Zhu, arXiv: 2205.03572
    [11] H. Chen, W. Q. Niu, and H. Q. Zheng, arXiv: 2205.14628
    [12] T. W. Chiu et al. (TWQCD Collaboration), Phys. Lett. B 646, 95 (2007) doi: 10.1016/j.physletb.2007.01.019
    [13] N. Barnea, J. Vijande, and A. Valcarce, Phys. Rev. D 73, 054004 (2006) doi: 10.1103/PhysRevD.73.054004
    [14] L. Maiani, F. Piccinini, A. D. Polosa et al., Phys. Rev. D 71, 014028 (2005) doi: 10.1103/PhysRevD.71.014028
    [15] H. Hogaasen, J. M. Richard, and P. Sorba, Phys. Rev. D 73, 054013 (2006) doi: 10.1103/PhysRevD.73.054013
    [16] F. E. Close and S. Godfrey, Phys. Lett. B 574, 210 (2003) doi: 10.1016/j.physletb.2003.09.011
    [17] B. A. Li, Phys. Lett. B 605, 306 (2005) doi: 10.1016/j.physletb.2004.11.062
    [18] K. K. Seth, Phys. Lett. B 612, 1 (2005) doi: 10.1016/j.physletb.2005.02.057
    [19] T. Barnes and S. Godfrey, Phys. Rev. D 69, 054008 (2004) doi: 10.1103/PhysRevD.69.054008
    [20] E. J. Eichten, K. Lane, and C. Quigg, Phys. Rev. D 69, 094019 (2004) doi: 10.1103/PhysRevD.69.094019
    [21] C. Quigg, Nucl. Phys. Proc. Suppl. 142, 87 (2005) doi: 10.1016/j.nuclphysbps.2005.01.016
    [22] P. A. Zyla et al. (Particle Data Group Collaboration), Prog. Theor. Exp. Phys. 2020, 083C01 (2020) doi: 10.1093/ptep/ptaa104
    [23] H. X. Chen, Commun. Theor. Phys. 74, 025201 (2022) doi: 10.1088/1572-9494/ac449e
    [24] L. Meng, G. J. Wang, B. Wang et al., Phys. Rev. D 104, 094003 (2021) doi: 10.1103/PhysRevD.104.094003
    [25] Q. Wu, D. Y. Chen, and T. Matsuki, Eur. Phys. J. C 81, 193 (2021) doi: 10.1140/epjc/s10052-021-08984-2
    [26] R. D. Matheus, F. S. Navarra, M. Nielsen et al., Phys. Rev. D 80, 056002 (2009) doi: 10.1103/PhysRevD.80.056002
    [27] C. Meng, J. J. Sanz-Cillero, M. Shi et al., Phys. Rev. D 92, 034020 (2015) doi: 10.1103/PhysRevD.92.034020
    [28] C. Meng, Y. J. Gao, and K. T. Chao, Phys. Rev. D 87, 074035 (2013) doi: 10.1103/PhysRevD.87.074035
    [29] W. Wang, Y. L. Shen, and C. D. Lu, Eur. Phys. J. C 51, 841 (2007) doi: 10.1140/epjc/s10052-007-0334-3
    [30] Y. K. Hsiao and C. Q. Geng, Chin. Phys. C 41, 013101 (2017) doi: 10.1088/1674-1137/41/1/013101
    [31] E. Braaten, M. Kusunoki, and S. Nussinov, Phys. Rev. Lett. 93, 162001 (2004) doi: 10.1103/PhysRevLett.93.162001
    [32] E. Braaten and M. Kusunoki, Phys. Rev. D 71, 074005 (2005) doi: 10.1103/PhysRevD.71.074005
    [33] X. Liu and Y. M. Wang, Eur. Phys. J. C 49, 643 (2007) doi: 10.1140/epjc/s10052-006-0135-0
    [34] Y. Kato et al. (Belle Collaboration), Phys. Rev. D 97, 012005 (2018) doi: 10.1103/PhysRevD.97.012005
    [35] B. Aubert et al. (BaBar Collaboration), Phys. Rev. Lett 96, 052002 (2006) doi: 10.1103/PhysRevLett.96.052002
    [36] C. H. Chen and H. n. Li, Phys. Rev. D 71, 114008 (2005) doi: 10.1103/PhysRevD.71.114008
    [37] Y. Li, C. D. Lu, and C. F. Qiao, Phys. Rev. D 73, 094006 (2006) doi: 10.1103/PhysRevD.73.094006
    [38] Z. Rui and Z. T. Zou, Phys. Rev. D 90, 114030 (2014) doi: 10.1103/PhysRevD.90.114030
    [39] Z. Q. Zhang, Phys. Lett. B 772, 719 (2017) doi: 10.1016/j.physletb.2017.07.049
    [40] G. Buchalla, A. J. Buras, and M. E. Lautenbacher, Rev. Mod. Phys. 68, 1125 (1996) doi: 10.1103/RevModPhys.68.1125
    [41] Z. Q. Zhang, S. Y. Wang, and X. K. Ma, Phys. Rev. D 93, 054034 (2016) doi: 10.1103/PhysRevD.93.054034
    [42] X. Liu, Z. J. Xiao, and C. D. Lu, Phys. Rev. D 81, 014022 (2010) doi: 10.1103/PhysRevD.81.014022
    [43] R. Zhou, Q. Zhao, and L. L. Zhang, Eur. Phys. J. C 78, 463 (2018) doi: 10.1140/epjc/s10052-018-5958-y
    [44] Y. S. Amhis et al., Eur. Phys. J. C 81, 226 (2021) and the online update at https://hflav.web.cern.ch/
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Zhi-Qing Zhang, Zhi-Lin Guan, Yan-Chao Zhao, Zi-Yu Zhang, Zhi-Jie Sun, Na Wang and Xiao-Dong Ren. Insights into the nature of X(3872) through B meson decays[J]. Chinese Physics C. doi: 10.1088/1674-1137/ac945a
Zhi-Qing Zhang, Zhi-Lin Guan, Yan-Chao Zhao, Zi-Yu Zhang, Zhi-Jie Sun, Na Wang and Xiao-Dong Ren. Insights into the nature of X(3872) through B meson decays[J]. Chinese Physics C.  doi: 10.1088/1674-1137/ac945a shu
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Insights into the nature of X(3872) through B meson decays

  • Department of Physics, Henan University of Technology, Zhengzhou, Henan 450052, China

Abstract: We study the B_{c,u,d}\to X(3872)P decays in the perturbative QCD (PQCD) approach, involving the puzzling resonance X(3872) , where P represents a light pseudoscalar meson (K or π). Assuming X(3872) to be a 1^{++} charmonium state, we obtain the following results. (a) The branching ratios of the B^+_c\to X(3872)\pi^+ and B^+_c\to X(3872) K^+ decays are consistent with the results predicted by the covariant light-front approach within errors; however, they are larger than those given by the generalized factorization approach. (b) The branching ratio of the B^+\to X(3872)K^+ decay is predicted as (3.8^{+1.1}_{-1.0})\times10^{-4} , which is smaller than the previous PQCD calculation result but still slightly larger than the upper limits set by Belle and BaBar. Hence, we suggest that the B^{0,+}\to X(3872)K^{0,+} decays should be precisely measured by the LHCb and Belle II experiments to help probe the inner structure of X(3872) . (c) Compared with the B_{u,d}\to X(3872)K decays, the B_{u,d}\to X(3872)\pi decays have significantly smaller branching ratios, which drop to values as low as 10^{-6} . (d) The direct CP violations of these considered decays are small ( 10^{-3}\sim 10^{-2} ) because the penguin contributions are loop suppressed compared to the tree contributions. The mixing-induced CP violation of the B\to X(3872)K^0_S decay is highly consistent with the current world average value \sin2\beta=(69.9\pm1.7)%. Experimentally testing the results for the branching ratios and CP violations, including the implicit S U(3) and isospin symmetries of these decays, helps probe the nature of X(3872) .

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    I.   INTRODUCTION
    • The hadron X(3872) has attracted considerable attention since it was first observed by Belle in the exclusive decay B^{\pm}\to K^{\pm}\pi^+\pi^-J/\Psi [1]. Though X(3872) has been confirmed by many experimental collaborations, such as the CDF [2], D0 [3], Babar [4], and LHCb [5], with quantum numbers J^{PC}=1^{++} and isospin I=0 , there are still many uncertainties. Because the mass of X(3872) is close to the D^0\bar D^{*0} threshold, several authors interpret it as a loosely bound molecular state [610], in which the building blocks are hadrons [11]. Others regard X(3872) as a compact tetraquark state [1215], in which the building blocks are quarks and anti-quarks. There are other explanations, such as c\bar c g hybrid meson [16, 17] and glueballs [18]. Though there are many different exotic hadron state interpretations of X(3872) , it has not been ruled out that the first radial excitation of the 1P charmonium state \chi_{c1}(1P) is the most natural assignment [1921]. Note that X(3872) was renamed \chi_{c1}(3872) by the Particle Data Group (PDG) [22].

      Many studies on the production and decays of X(3872) have been performed to investigate the inner structure of X(3872) [2326]. In Ref. [26], the authors calculated \Gamma(X(3872)\to J/\psi \pi^+\pi^-) using QCD sum rules and concluded that X(3872) is approximately 97% a charmonium state with a small molecular component. Many B meson decays with X(3872) in the final states have been studied using different approaches [2733]. In Ref. [28], the authors studied the B\to \chi_{c1}(1P,2P)K decays using the QCD factorization (QCDF) approach and argued that X(3872) has a dominant c\bar c component but mixes with the D^0\bar D^{*0}+D^{*0}\bar D^0 continuum component. The B_c\to X(3872)\pi(K) decays were studied both using the covariant light-front (CLF) approach [29] and generalized factorization (GF) approach [30], respectively. In the former, X(3872) was identified as a 1^{++} charmonium state, whereas a tetraquark state was assumed in the latter. One may expect different results for the same decays under the different structure hypotheses of X(3872) . The B\to X(3872)K decay has also received significant attention from many researchers. In Refs. [31, 32], the authors assumed X(3872) to be a loosely bound S-wave molecular state of D^0\bar D^{*0}(D^{*0}\bar D^0) and estimated the branching ratio of the B^+\to X(3872)K^+ decay to be (0.07\sim1)\times10^{-4} . Furthermore, they considered the branching ratio of the B^0\to X(3872)K^0 decay to be suppressed by more than one order of magnitude compared with that of the B^+\to X(3872)K^+ decay, which indicates that there is large isospin symmetry between the B^+\to X(3872)K^+ and B^0\to X(3872)K^0 decays. If this type of large isospin symmetry is observed in experiments, any charmonium interpretation of X(3872) will be disfavored. Two years later, the branching ratio of the B^+\to X(3872)K^+ decay was calculated using the perturbative QCD (PQCD) approach, assuming X(3872) as a regular c\bar c charmonium state, in Ref. [33]; a large value of {\rm Br}(B^+\to X(3872)K^+)= (7.88^{+4.87}_{-3.76})\times10^{-4} was obtained. Clearly, this result is significantly larger than the present experimental upper limits given by Belle [34] and BaBar [35] at the 90\ % C.L.,

      {\rm Br}(B^+\to X(3872)K^+)<2.6\times10^{-4} \;\;\;(\rm{Belle}) ,

      (1)

      {\rm Br}(B^+\to X(3872)K^+)<3.2\times10^{-4} \;\;\;(\rm{BaBar}).

      (2)

      Here, we conduct a systematic study of the B_{c,u,d}\to X(3872)P decays using the PQCD approach, where P represents a light pseudoscalar meson (K or π). The layout of this paper is as follows. We present the analytic calculations of the amplitudes of the B_{c,u,d}\to X(3872)P decays in Section II. The numerical results and discussions are given in Section III, where we compare our results with other theoretical predictions and experimental data. The conclusions are presented in Section IV.

    II.   AMPLITUDES OF THE \boldsymbol{B_{c,u,d}} \boldsymbol{\to X(3872)P} DECAYS
    • Because the PQCD approach based on k_T factorization has been successfully applied to many two-body charmed B meson decays [3639], we use this approach to investigate the B_{c,u,d}\to X(3872)P decays in this study. First, the effective Hamiltonian for the B^+_c\to X(3872)\pi^+ (K^+) decays can be written as [40]

      \begin{eqnarray} {H}_{\rm eff}=\frac{G_{\rm F}}{\sqrt2}V^*_{cb}V_{uq}\left[C_1(\mu)O_1(\mu)+C_2(\mu)O_2(\mu)\right]+ {\rm H.c.}, \end{eqnarray}

      (3)

      where the Fermi coupling constant G_{\rm F}\simeq1.166\times 10^{-5} GeV ^{-2} [22], V^*_{cb}V_{uq} is the product of the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements with q=d ( q=s ) for B^+_c\to X(3872)\pi^+ ( B^+_c\to X(3872)K^+ ) decay, C_{i}(\mu)(i=1,2) are the Wilson coefficients at the renormalization scale μ, and O_{i}(i=1,2) are the local four-quark operators,

      \begin{aligned}[b] O_1=&\bar d_{\alpha}\gamma_{\mu}(1-\gamma_5)u_{\beta}\otimes\bar c_{\beta}\gamma_{\mu}(1-\gamma_5)b_{\alpha},\\ O_2=&\bar d_{\alpha}\gamma_{\mu}(1-\gamma_5)u_{\alpha}\otimes\bar c_{\beta}\gamma_{\mu}(1-\gamma_5)b_{\beta},\; \end{aligned}

      (4)

      where α and β are the S U(3) color indices, and the summation convention over repeated indices is understood. Because the four quarks in the operators are different, there is no penguin contribution and thus no CP violation. Here, we analyze the B^+_c\to X(3872)\pi^+ decay as an example, and its Feynman diagrams are given in Fig. 1, where only the factorizable and non-factorizable emission diagrams need to be considered at the leading order. The amplitude for the factorizable emission diagrams in Fig. 1(a) and Fig. 1(b) can be written as

      Figure 1.  Feynman diagrams contributing to the B^+_c\to X(3872)\pi^+ decay at the leading order.

      \begin{aligned}[b] F^{LL}_{B_c\to X}=&2\sqrt{\frac{2}{3}}\pi C_{\rm F} m^4_{B_c}f_{\pi}f_{B_c}\sqrt{1-r_X^2}\int_0^1 {\rm d} x_2 \int_0^\infty b_1 {\rm d} b_1 b_2 {\rm d} b_2 \exp\left(-\frac{\omega^2_{B_c}b^2_1}{2}\right)\\ & \left\{\left[ \Psi^L(x_2)(x_2-2r_b)+\Psi^t(x_2)(r_b-2x_2)\right]E_e(t_a)h(\alpha,\beta_a,b_1,b_2)S_t(x_2)\right.\\ &\left. -\Psi^L(x_2)(r_c+r_X^2(x_1-1))E_e(t_b)h(\alpha,\beta_b,b_2,b_1)S_t(x_1) \right\}, \end{aligned}

      (5)

      where the superscript LL denotes the contribution from the (V-A)(V-A) operators, the color factor C_F=4/3, f_{\pi(B_c)} is the decay constant for the meson \pi(B_c) , the mass ratio r_{X(b,c)}=m_X(m_b,m_c)/m_{B_c} , the exponent \exp(-\omega^2_{B_c}b^2_1/2) originates from the B_c meson wave function, and \Psi^{L,t}(x_2) are the distribution amplitudes for X(3872) (given in Sec. III).

      The amplitude for the non-factorizable spectator diagrams in Fig. 1(c) and Fig. 1(d) is given as

      \begin{aligned}[b]\\ M^{LL}_{B_c\to X}=&\frac{8}{3}\pi C_{\rm F} m^4_{B_c}f_{B_c}\sqrt{1-r_X^2} \int_0^1 {\rm d}x_2 \int_0^\infty b_1 {\rm d} b_1 b_3 {\rm d} b_3 \exp\left(-\frac{\omega^2_{B_c}b^2_1}{2}\right) \phi^A_\pi(x_3)\\ &\left\{[\Psi^L(x_2)(x_3-x_1)(1-r_X^2) +r_X\Psi^t(x_2)(1-x_1-x_2)]E_{cd}(t_c)h(\alpha,\beta_c,b_1,b_3) \right.\\ &\left. +[\Psi^L(x_2)(r_X^2(x_2-x_3)+2x_1+x_2+x_3-2)+r_X\Psi^t(x_2)(1-x_1-x_2)]\right. E_{cd}(t_d)h(\alpha,\beta_d,b_1,b_3)\Big\}. \end{aligned}

      (6)

      Note that the hard function h originates from the Fourier transform of the virtual quark and gluon propagators, which is defined as

      h(\alpha,\beta,b_1,b_2)=h_1(\alpha,b_1)\times h_2(\beta,b_1,b_2),

      (7)

      \begin{eqnarray} h_1(\alpha,b_1)=\left\{\begin{matrix}K_0(\sqrt{\alpha}b_1), & \alpha > 0,\\ K_0({\rm i}\sqrt{-\alpha}b_1), & \alpha < 0, \\\end{matrix}\right. \end{eqnarray}

      (8)

      \begin{aligned}[b]& h_1(\beta,b_1,b_2)\\=&\left\{\begin{matrix}\theta(b_1-b_2) I_0(\sqrt{\beta}b_2) K_0(\sqrt{\beta}b_1)+(b_1\leftrightarrow b_2), & \beta > 0,\\ \theta(b_1-b_2) J_0(\sqrt{-\beta}b_2) K_0({\rm i}\sqrt{-\beta}b_1)+(b_1\leftrightarrow b_2), & \beta < 0, \\\end{matrix}\right. \end{aligned}

      (9)

      where J_0 is the Bessel function, and K_0,I_0 are the modified Bessel functions with K_0({\rm i}x)=\pi(-N_0(x)+{\rm i}J_0(x))/2. In Eqs. (5) and (6), α and \beta_{a,b,c,d} in the hard function h are the invariant masses of the internal quarks and gluons, respectively. The hard scales t_{a,b,c,d} are given as the maximum energy scale appearing in each Feynman diagram to remove the large logarithmic radiative corrections. Their expressions are listed in the appendix. The evolution factors E_e(t),E_{cd}(t) evolving the Sudakov exponent and jet function S_t(x) can be found in Refs. [38, 41]. For the reader's convenience, their explicit forms are also summarized in the appendix.

      Second, the effective Hamiltonian for the B_{u,d}\to X(3872)\pi(K) decays is written as

      \begin{aligned}[b] {H}_{\rm eff}=&\frac{G_{\rm F}}{\sqrt2}\Bigg[V^*_{cb}V_{cq}(C_1(\mu)O^c_1(\mu)+C_2(\mu)O^c_2(\mu))\\&- V^{*}_{tb}V_{tq}\sum^{10}_{i=3}C_i(\mu)O_i(\mu)\Bigg], \end{aligned}

      (10)

      where V^*_{c(t)b}V_{c(t)q} is the product of the CKM matrix elements, q=d or s. The local four-quark operators O_i(\mu) and corresponding QCD-corrected Wilson coefficients C_i(\mu) can be found in Ref. [40]. Here, we analyze B^+\to X(3872)\pi^+ as an example, and its Feynman diagrams are given in Fig. 2. The amplitudes for the factorizable and nonfactorizable emission diagrams from the (V-A)(V-A) operators are denoted as F^{LL}_{B\to \pi} and M^{LL}_{B\to\pi} , respectively. Their analytical expressions are given as

      Figure 2.  Feynman diagrams contributing to the B^+\to X(3872)\pi^+ decay at the leading order.

      \begin{aligned}[b] F^{LL}_{B\to \pi}=&\frac{8\pi C_{\rm F} m^4_Bf_{X}}{\sqrt{1-r^2_X}}\int_0^1 {\rm d} x_1 {\rm d} x_3 \int_0^\infty b_1 {\rm d} b_1 b_3 {\rm d} b_3\phi_B(x_1,b_1) \left\{\left[ \left(r^2_X-1)\phi^A_\pi(x_3)((r^2_X-1) x_3-1\right)\right.\right. \\ &\left.\left. +(r^2_X-1)\phi^P_\pi(x_3)r_\pi(2x_3-1)+\phi^T_\pi(x_3)r_\pi(2x_3(r_X^2-1)+1+r^2_X)\right] \right.\\ &\left.\times E_{e'}(t'_a)h(\alpha',\beta'_a,b_1,b_3)S_t(x_3) -2r_\pi(1-r_X^2)\phi^P_\pi(x_3)\right.\left. E_{e'}(t'_b)h(\alpha',\beta'_b,b_3,b_1)S_t(x_1) \right\}, \end{aligned}

      (11)

      \begin{aligned}[b] M^{LL}_{B\to\pi}=&\frac{32}{\sqrt6}\frac{1}{\sqrt{(1-r_X^2)}}\pi C_{\rm F} m^4_B \int_0^1 {\rm d} x_1 {\rm d} x_2 \int_0^\infty b_1 {\rm d} b_1 b_2 {\rm d} b_2\phi_B(x_1,b_1)\Big\{\Big[\Psi^L(x_2)(\phi^A_\pi(x_3)(r_X^2-1)\\ &+2\phi^T_\pi(x_3)r_\pi)(r_X^2(x_1+x_3-2x_2)+x_1-x_3) +4r_\pi r_Xr_c\phi^T_\pi(x_3)\Psi^t(x_2)\Big]\Big\}E_c(t'_c)h(\alpha',\beta'_c,b_1,b_2), \end{aligned}

      (12)

      where the evolution factors E_{e'}(t),E_{c}(t) evolving the Sudakov exponent are given in the appendix. Besides the upper two (V-A)(V-A) type amplitudes, there are factorizable and nonfactorizable emission diagram contributions from the (V-A)(V+A) and (S-P)(S+P) operators, which are expressed as F^{LR}_{B\to\pi} and M^{SP}_{B\to\pi} , respectively.

      F^{LR}_{B\to\pi}=-F^{LL}_{B\to \pi},

      (13)

      \begin{aligned}[b] M^{SP}_{B\to\pi}=&-\frac{32}{\sqrt6}\frac{1}{\sqrt{(1-r_X^2)}}\pi C_{\rm F} m^4_B \int_0^1 {\rm d} x_1 {\rm d} x_2 \int_0^\infty b_1 {\rm d} b_1 b_2 {\rm d} b_2\phi_B(x_1,b_1)\\ & \times \left\{\left[\Psi^L(x_2)(\phi^A_\pi(x_3)(r_X^2-1)+2\phi^T_\pi(x_3)r_\pi)(r_X^2(x_1+x_3-2x_2)+x_1-x_3) -4r_\pi r_Xr_c\phi^T_\pi(x_3)\Psi^t(x_2)\right]\right\}E_c(t'_c)h(\beta'_c,\alpha',b_2,b_1), \end{aligned}

      (14)

      where \alpha',\beta'_{a,b,c} in the upper hard function and the hard scales t'_{a,b,c} are defined in the appendix.

      By combining the amplitudes from the different Feynman diagrams, the total decay amplitudes for the considered decays are given as

      \begin{eqnarray} \mathcal{A}(B_c\to X(3872)P)=V^*_{cb}V_{uq}\left[a_1F^{LL}_{B_c\to X}+C_1M^{LL}_{B_c\to X}\right], \end{eqnarray}

      (15)

      \begin{aligned}[b] \mathcal{A}(B_{u,d}\to X(3872)P)=&V^*_{cb}V_{cq}\left[a_2F^{LL}_{B\to P}+C_2M^{LL}_{B\to P}\right]\\&-V^*_{tb}V_{tq}\left[(a_3+a_9-a_5-a_7)F^{LL}_{B\to P}\right.\\ &\left.+(C_4 + C_{10})M^{LL}_{B\to P} + (C_6 + C_8)M^{SP}_{B\to P}\right], \end{aligned}

      (16)

      where the combinations of the Wilson coefficients a_1=C_1/3+C_2, a_2=C_1+C_2/3, a_i=C_i+C_{i+1}/3 with i= 3, 5, 7, and 9, and q=d ( q=s ) corresponds to the decays induced by the b\to d ( b\to s ) transition.

    III.   NUMERICAL RESULTS AND DISCUSSIONS
    • We use the following input parameters for the numerical calculations [22, 29, 30]:

      \begin{aligned}[b]f_{B_c}= & 0.398^{+0.054}_{-0.055}~ {\rm GeV},~~ f_{B}=0.19~{\rm GeV},\\ f_{X}= & 0.234\pm0.052~{\rm GeV}, \end{aligned}

      (17)

      \begin{aligned}[b] M_{B_{c}}=& 6.275~{\rm GeV}, ~~M_B=5.279~{\rm GeV},\\ M_{X}= & 3.87169~{\rm GeV}, \end{aligned}

      (18)

      \begin{aligned}[b] \tau_{B_c}=&0.510\times 10^{-12} {\rm s},\quad \tau_B^\pm=1.638\times 10^{-12} {\rm s},\\\tau_{B^0}=&1.519\times 10^{-12} {\rm s}. \end{aligned}

      (19)

      For the CKM matrix elements, we adopt the Wolfenstein parameterization and the updated values A=0.814 , \lambda=0.22537,~ \bar\rho=0.117\pm0.021 , and \bar\eta=0.353\pm0.013 [22]. With the total amplitudes, the decay width can be expressed as

      \begin{eqnarray} \Gamma(B\to X(3872)P)=\frac{G^2_{\rm F}}{32\pi m_B}(1-r^2_{X})|\mathcal{A}(B\to X(3872)P)|^2. \end{eqnarray}

      (20)

      The wave functions of B, \pi , and K have been well defined in many studies, whereas those of B_c and X(3872) still have many uncertainties. For the B_c meson, we use its wave function in the nonrelativistic limit [42],

      \begin{eqnarray} \Phi_{B_c}(x)=\frac{{\rm i}f_{B_c}}{4N_C}\left[( \not p _{B_{c}} +M_{B_{c}})\gamma_5\delta(x-r_c)\right] \exp\left(-\frac{b^2\omega^2_{B_c}}{2}\right), \end{eqnarray}

      (21)

      where b is the conjugate space coordinate of the parton transverse momentum k_T , and the shape parameter \omega_{B_c}=0.6 GeV. The last exponent term reveals the k_T dependence.

      For the light cone distribution amplitude of X(3872) , we adopt a similar formula to that of the \chi_{c1} meson [33, 43],

      \begin{aligned}[b]& \langle X(p,\epsilon_L)|\bar c_\alpha(z)c_{\beta}(0)|0\rangle\\=&\frac{1}{\sqrt{2N_c}}\int {\rm d} x {\rm e}^{{\rm i} x p\cdot z} \left\{m_X[\gamma_5 \not \epsilon _L]_{\beta\alpha}\phi^L_X(x)+[\gamma_5 \not \epsilon _L {\not p} ]_{\beta\alpha}\phi^t_X(x)\right\}, \end{aligned}

      (22)

      where \epsilon_L is the longitudinal polarization vector, and m_X is the X(3872) mass. Here, only the longitudinal polarization contributes to the considered decays, and the asymptotic models of the twist-2 distribution amplitude \phi^{L}_X(x) and twist-3 distribution amplitude \phi^{t}_X(x) are given as

      \begin{aligned}[b] \phi^{L}_X(x)=&24.68\frac{f_X}{2\sqrt{2N_c}}x(1-x)\\&\times\left\{\frac{x(1-x)(1-2x)^2\left[1-4x(1-x)\right]}{[1-3.47x(1-x)]^3} \right\}^{0.7}, \end{aligned}

      (23)

      \begin{aligned}[b] \phi^{t}_X(x)=&13.53\frac{f_X}{2\sqrt{2N_c}}(1-2x)^2\\&\times\left\{\frac{x(1-x)(1-2x)^2\left[1-4x(1-x)\right]}{[1-3.47x(1-x)]^3} \right\}^{0.7}, \end{aligned}

      (24)

      where f_X is the X(3872) decay constant.

      Using the input parameters and wave functions specified in this section, we present the branching ratios of the B^+_c\to X(3872)\pi^{+}(K^+) decays as follows:

      {\rm Br}(B^+_c\to X(3872)\pi^+)=(2.7^{+1.4+0.9+0.7+0.2}_{-1.0-0.6-0.5-0.1})\times10^{-4},

      (25)

      {\rm Br}(B^+_c\to X(3872)K^+)=(2.5^{+1.3+0.8+0.6+0.2}_{-1.0-0.6-0.4-0.1})\times10^{-5},

      (26)

      where the first error arises from the X(3872) decay constant, f_{X}=0.234\pm0.052 GeV, the second and third uncertainties are caused by the shape parameter \omega_{B_c}=0.6\pm0.1 GeV and decay constant f_{B_c}=0.398^{+0.054}_{-0.055} GeV, respectively, and the final error is from the variation in the hard scale from 0.8t to 1.2t , which characterizes the size of the next-to-leading-order QCD contributions. The branching ratios are sensitive to the decay constant f_{X} because the dominant contributions for these two channels are from the factorization emission amplitudes, which are proportional to f_{X} . The branching ratio of B^+_c\to X(3872)\pi^{+} is approximately one order of magnitude larger than that of B^-_c\to X(3872)K^{-} , which is mainly induced by the difference between the CKM elements V_{ud}=1-\lambda^2/2 and V_{us}=\lambda . From Table 1, it is shown that our predictions are consistent with the results given in the covariant light-front quark model within errors [29]; however, they are significantly larger than those calculated using the generalized factorization approach [30].

      ModeThis studyCLF [29]GF [30]
      B^+_{c}\to X(3872)\pi^{+}(\times10^{-4}) 2.7^{+1.4+0.9+0.7+0.2}_{-1.0-0.6-0.5-0.1} 1.7^{+0.7+0.1+0.4}_{-0.6-0.2-0.4} 0.60^{+0.22+0.14}_{-0.18-0.07}
      B^+_{c}\to X(3872)K^{+}(\times10^{-5}) 2.5^{+1.3+0.8+0.6+0.2}_{-1.0-0.6-0.4-0.1} 1.3^{+0.5+0.1+0.3}_{-0.5-0.2-0.3} 0.47^{+0.17+0.11}_{-0.14-0.05}

      Table 1.  Our predictions for the branching ratios of the B^+_{c}\to X(3872)\pi^{+}(K^{+}) decays, along with the results from the covariant light-front (CLF) approach [29] and generalized factorization (GF) approach [30].

      Similarly, the branching ratios of the B\to X(3872)P decays are calculated as follows:

      {\rm Br}(B^+\to X(3872)K^+)=(3.8^{+0.9+0.6+0.3}_{-0.8-0.5-0.2})\times10^{-4},

      (27)

      {\rm Br}(B^0 \to X(3872)K^{0})=(3.5^{+0.7+0.5+0.3}_{-0.6-0.4-0.2})\times10^{-4},

      (28)

      {\rm Br}(B^+\to X(3872)\pi^+)=(9.3^{+1.5+0.9+0.5}_{-1.3-0.8-0.4})\times10^{-6},

      (29)

      {\rm Br}( B^0\to X(3872)\pi^{0})=(4.3^{+0.7+0.5+0.3}_{-0.6-0.4-0.3})\times10^{-6},

      (30)

      where the first uncertainty arises from the shape parameter \omega_{B}=0.4\pm0.04 GeV in the B meson wave function, the second error is from the decay constant f_{X}=0.234\pm0.052 GeV of X(3872) , and the third error arises from the choice of hard scales, which vary from 0.8t to 1.2t . From the results, we find that the branching ratios of the B^+\to X(3872)K^{+} and B^0 \to X(3872)K^{0} decays are similar because they differ only in the lifetimes between B^+ and B^0 in our formalism. Our prediction for the branching ratio of the B^+\to X(3872)K^{+} decay is less than the previous PQCD calculation result (7.88^{+4.87}_{-3.76})\times10^{-4} [33]. However, it is still slightly larger than the upper limits 2.6\times10^{-4} given by Belle [34] and 3.2\times10^{-4} given by BaBar [35]. If the present experimental upper limits are confident, a pure charmonium assignment for X(3872) may not be suitable under the PQCD approach. We expect that the branching ratios of the B^{0,+}\to X(3872)K^{0,+} decays can be precisely measured at the current LHCb and SuperKEKB experiments, which will help probe the inner structure of X(3872) .

      However, note that X(3872) was renamed \chi_{c1}(3872) by the current PDG [22], which seems to assume it is a radial excited state of \chi_{c1}(1P) . As we know, the \chi_{c1}(1P) meson is another P-wave charmonium state with the same quantum numbers J^{(PC)}=1^{++} and a slightly lighter mass of 3.511 GeV. In this case, they should have similar properties in B meson decays. For example, the branching ratio of the B^{+}\to \chi_{c1}(1P) K^+ decay is measured as (4.85\pm 0.33)\times 10^{-4} [22], which is consistent with the result predicted using the PQCD approach (4.4^{+1.9}_{-1.6})\times10^{-4} [43]. The corresponding decay B^{+}\to X(3872) K^+ should have a similar but slightly smaller branching ratio. Comparisons of the branching ratios of the B\to X(3872)\pi(K) and B\to \chi_{c1}(1P)\pi(K) decays can be found in Table 2, where the theoretical predictions for the branching ratios of the B\to \chi_{c1}(1P)\pi(K)) decays are taken from other PQCD calculations [43]. From Table 2, we know that calculations for the B\to X(3872)P decays using the PQCD approach are under control and credible. Therefore, we suggest that experimental researchers measure these decays at LHCb and Belle II to help discriminate the inner structure of X(3872) from different assumptions.

      Mode (\times10^{-4}) B^+\to X(3872)K^{+} B^+\to \chi_{c1}(1P)K^{+} B^0\to X(3872)K^{0} B^0\to \chi_{c1}(1P)K^{0}
      PQCD 3.8^{+0.9+0.6+0.3}_{-0.8-0.5-0.2} 4.4^{+1.9}_{-1.6} 3.5^{+0.7+0.5+0.3}_{-0.6-0.4-0.2} 4.1^{+1.8}_{-1.6}
      Exp. 4.85\pm0.33 3.95\pm0.27
      Mode ( \times10^{-5}) B^+\to X(3872)\pi^{+} B^+\to \chi_{c1}(1P)\pi^{+} B^0\to X(3872)\pi^{0} B^0\to \chi_{c1}(1P)\pi^{0}
      PQCD 0.93^{+0.15+0.09+0.05}_{-0.13-0.08-0.04} 1.7\pm0.6 0.43^{+0.07+0.05+0.03}_{-0.06-0.04-0.03} 0.8\pm0.3
      Exp. 2.2\pm0.5 1.12\pm0.28

      Table 2.  Comparison of {\rm Br}(B\to X(3872)\pi(K)) (this study) and {\rm Br}(B\to \chi_{c1}(1P)\pi(K)) [43] calculated using the PQCD approach. The data are taken from the Particle Data Group 2020 [22].

      In Table 3, we compare our predictions with the results calculated using the generalized factorization approach [30]. It is interesting that the branching ratios of the B\to X(3872)\pi(K) decays calculated with these two different approaches are consistent with each other within errors. We find that {\rm Br}(B^+\to X(3872)\pi^{+})\simeq 2 {\rm Br}(B^0\to X(3872)\pi^{0}), which is supported by the isospin symmetry.

      ModeThis studyGF [30]
      B^+\to X(3872)K^{+}(\times10^{-4}) 3.8^{+0.9+0.6+0.3}_{-0.8-0.5-0.2} 2.3^{+1.1}_{-0.9}\pm0.1
      B^0\to X(3872)K^{0}(\times10^{-4}) 3.5^{+0.7+0.5+0.3}_{-0.6-0.4-0.2} 2.1^{+1.0}_{-0.8}\pm0.1
      B^+\to X(3872)\pi^{+}(\times10^{-6}) 9.3^{+1.5+0.9+0.5}_{-1.3-0.8-0.4} 11.5^{+5.7}_{-4.5}\pm0.3
      B^0\to X(3872)\pi^{0}(\times10^{-6}) 4.3^{+0.7+0.5+0.3}_{-0.6-0.4-0.3} 5.3^{+2.6}_{-2.1}\pm0.2

      Table 3.  Our predictions for the branching ratios of the B\to X(3872)\pi(K) decays, along with the results from the generalized factorization (GF) approach [30].

      In the following we discuss the CP asymmetries in the B\to X(3872)P decays. As we know, CP asymmetry arises from the interference between the tree and penguin amplitudes; however, there are no contributions from the penguin amplitudes for the B^+_c\to X(3872)\pi^+(K^+) decays. Therefore, the corresponding direct CP violation is zero. For the charged decays B^+\to X(3872)\pi^{+}(K^+), we only need to consider the direct CP violation A^{\rm dir}_{CP}, which is defined as

      \begin{eqnarray} A^{\rm dir}_{CP}=\frac{|\mathcal{\bar A}|^2-|\mathcal{A}|^2}{|\mathcal{ \bar A}|^2+|\mathcal{A}|^2}, \end{eqnarray}

      (31)

      where \mathcal{ \bar A} is the CP-conjugate amplitude of \mathcal{A} . For neutral B meson decays, there is another type of CP violation that must be considered, known as as time-dependent CP asymmetry, which is induced by interference between the direct decay and the decay via oscillation. Time-dependent CP violation can be defined as

      \begin{eqnarray} A(t)_{CP}=A_f\cos(\Delta mt)+S_f\sin(\Delta mt), \end{eqnarray}

      (32)

      where the subscript f represents a CP eigenstate, \Delta m is the mass difference of the two neutral B meson mass eigenstates, and the direct CP asymmetry A_f and mixing-induced CP asymmetry S_f are expressed as

      \begin{eqnarray} A_f=\frac{|\lambda_f|^2-1}{|\lambda_f|^2+1}, \;\;S_f=\frac{2{\rm Im}(\lambda_f)}{|\lambda_f|^2+1}, \end{eqnarray}

      (33)

      with

      \begin{eqnarray} \lambda_f=\eta_f {\rm e}^{-2{\rm i}\beta}\frac{\mathcal{\bar A}}{\mathcal{A}}, \end{eqnarray}

      (34)

      where \eta_f is 1(-1) for a CP-even (CP-odd) final state f, and β is the CKM angle [22]. Because the charged decay channel and corresponding neutral mode are the same, except for the lifetime and isospin factor in the amplitudes, they have the same direct CP asymmetries. Therefore, we only need to consider the neutral decays, whose direct CP asymmetries are calculated as

      A_{X(3872)K^0}=(1.2^{+0.0+0.0+0.2}_{-0.0-0.0-0.3})\times10^{-3},

      (35)

      A_{X(3872)\pi^0}=(2.7^{+0.1+0.0+0.4}_{-0.2-0.0-0.4})\times10^{-2},

      (36)

      where the errors are induced by the same sources as those for the branching ratios; however, the direct CP violations are less sensitive to the nonperturbative parameters within their uncertainties, except for the hard scale t. Compared to the tree contributions, the penguin amplitudes are loop suppressed by one to two orders of magnitude. At the same time, the product of the CKM matrix elements associated with the tree amplitudes is approximately four times larger than that of penguin amplitudes. Hence, direct CP violations, which arise from interference between the tree and penguin contributions, are very small. Because the final state X(3872)K^0 and its CP conjugate state are flavor-specific, we should use the CP-odd eigenstate X(3872)K^0_S to analyze the mixing-induced CP violations. The results for the mixing-induced CP violations are calculated as

      S_{X(3872)K^0_S}=(70.3^{+0.0+0.0+0.9}_{-0.0-0.0-1.2}){\text{%}},

      (37)

      S_{X(3872)\pi}=(-60.8^{+0.0+0.0+1.5}_{-0.0-0.0-1.4}){\text{%}},

      (38)

      where the errors are similar to those listed in the direct CP violations and are not sensitive to the nonperturbative parameters given in the wave functions. We find that S_{X(3872)K^0_S} is highly consistent with the current world average value \sin2\beta=0.699\pm0.017 [44], which is obtained from B^0 decays to charmonium and K^0_S . Therefore, we can check the nature of X(3872) by extracting the CKM phase β from future experimental data on the B^0\to X(3872)K^0_S decay. Conversely, the mixing-induced CP asymmetry of the B^0\to X(3872)\pi^0 decay exhibits a significant deviation from the world average value of \sin2\beta because the imaginary parts of the total amplitudes for this channel and its CP-conjugate process exhibit a large difference. Our results can be tested in future experiments.

    IV.   SUMMARY
    • In this study, we analyze the B_{c,u,d}\to X(3872)\pi(K) decays using the PQCD approach by assuming X(3872) to be a 1^{++} charmonium state. Comparing our predictions for the branching ratios and CP asymmetries of the considered decays with other theoretical results and available experimental data, we find the following results:

      (1) The branching ratios of the B^-_c\to X(3872)\pi^- and B^-_c\to X(3872) K^- decays can reach orders of 10^{-4} and 10^{-5} , respectively, which are consistent with the results obtained via the covariant light-front approach within errors but larger than those given by the generalized factorization approach. These results can be discriminated at the current LHCb and Belle II experiments.

      (2) Our predictions for the branching ratio of the B\to X(3872) K and B\to X(3872)\pi decays are consistent with the results given by the generalized factorization approach. The branching ratio of the B\to X(3872) K) decay can reach the order of 10^{-4} , which is significantly larger than that of the B\to X(3872) \pi decay induced by the b\to d transition. On the experimental side, it is helpful to probe the inner structure of X(3872) by measuring the branching ratios and testing the S U(3) and isospin symmetries of these considered decays.

      (3) The direct CP violations of the B\to X(3872)\pi(K) decays are small (only 10^{-3}\sim 10^{-2} ). The mixing-induced CP violation of the B\to X(3872)K^0_S decay agrees with the current world average value \sin2\beta=(69.9\pm1.7)\ %. However, it is different for the value of S_{X(3872)\pi^0} because the imaginary parts of the total amplitudes of the B\to X(3872)\pi^0 decay and its CP-conjugate process exhibit a large difference.

    APPENDIX A
    • The invariant masses of virtual quarks and gluons are given as follows:

      \begin{eqnarray} \alpha=(x_2+x_1-1)(r_X^2(1-x_2)-x_1)m^2_{B_c}, \end{eqnarray}\tag{A1}

      \begin{eqnarray} \beta_a=(r_b^2-x_2(1-r^2_X(1-x_2))m^2_{B_c}, \end{eqnarray}\tag{A2}

      \begin{eqnarray} \beta_b=(r_c^2-(1-x_1)(r^2_X-x_1))m^2_{B_c}, \end{eqnarray} \tag{A3}

      \begin{eqnarray} \beta_c=-(1-x_1-x_2)(r_X^2(1-x_2-x_3)+x_3-x_1)m^2_{B_c}, \end{eqnarray}\tag{A4}

      \begin{eqnarray} \beta_d&=&-(1-x_1-x_2)(r_X^2(x_3-x_2)+1-x_3-x_1)m^2_{B_c}, \end{eqnarray}\tag{A5}

      \begin{eqnarray} \alpha'=x_1x_3(1-r_X^2)m^2_{B}, \end{eqnarray} \tag{A6}

      \begin{eqnarray} \beta'_a=x_3(1-r_X^2)m^2_{B}, \end{eqnarray} \tag{A7}

      \begin{eqnarray} \beta'_b=x_1(1-r_X^2)m^2_{B}, \end{eqnarray} \tag{A8}

      \begin{eqnarray} \beta'_c=(r_c^2+(x_1-x_2)(x_3+r_X^2(x_2-x_3)))m^2_B. \end{eqnarray}\tag{A9}

      The hard scale t is chosen as the maximum of the virtuality of the internal momentum transition in each amplitude, including 1/b_i(i=1,2,3)

      \begin{eqnarray} t_{a(b)}=\max(\sqrt{|\alpha|},\sqrt{|\beta_{a(b)}|},1/b_1,1/b_2), \end{eqnarray}\tag{A10}

      \begin{eqnarray} t_{c(d)}=\max(\sqrt{|\alpha|},\sqrt{|\beta_{c(d)}|},1/b_1,1/b_3), \end{eqnarray}\tag{A11}

      \begin{eqnarray} t^\prime_{a(b)}=\max(\sqrt{|\alpha'|},\sqrt{|\beta'_{a(b)}|},1/b_1,1/b_3), \end{eqnarray}\tag{A12}

      \begin{eqnarray} t'_c=\max(\sqrt{|\alpha'|},\sqrt{|\beta'_c|},1/b_1,1/b_2). \end{eqnarray}\tag{A13}

      The functions E_{e(c,d)}(t) are defined by

      \begin{eqnarray} E_e(t)=\alpha_s(t)\exp[-S_{B}(t)-S_X(t)], \end{eqnarray}\tag{A14}

      \begin{eqnarray} E_{cd}=\alpha_s(t)\exp[-S_B(t)-S_X(t)-S_\pi(t)]|_{b_1=b_2}, \end{eqnarray} \tag{A15}

      \begin{eqnarray} E_{e'}(t)=\alpha_s(t)\exp[-S_{B}(t)-S_\pi(t)], \end{eqnarray} \tag{A16}

      \begin{eqnarray} E_c=\alpha_s(t)\exp[-S_B(t)-S_X(t)-S_\pi(t)]|_{b_1=b_3}, \end{eqnarray}\tag{A17}

      where the Sudakov factors can be written as

      \begin{eqnarray} S_B(t)=s \left(x_1\frac{m_B}{\sqrt2},b_1\right)+2\int^t_{1/b_1}\frac{{\rm d}\bar\mu}{\bar\mu}\gamma_q(\alpha_s(\bar\mu)), \end{eqnarray} \tag{A18}

      \begin{aligned}[b] S_{X}(t)=s\left(x_2\frac{m_B}{\sqrt2},b_2\right)+s\left((1-x_2)\frac{m_B}{\sqrt2},b_2\right)\end{aligned}

      \begin{aligned}[b] \;\;\;\;\;\quad\quad +2\int^t_{1/b_2}\frac{{\rm d}\bar\mu}{\bar\mu}\gamma_q(\alpha_s(\bar\mu)), \end{aligned} \tag{A19}

      \begin{aligned}[b] S_{\pi}(t)=&s\left(x_3\frac{m_B}{\sqrt2},b_3\right)+s\left((1-x_3)\frac{m_B}{\sqrt2},b_3\right)\\&+2\int^t_{1/b_3}\frac{{\rm d}\bar\mu}{\bar\mu}\gamma_q(\alpha_s(\bar\mu)), \end{aligned}\tag{A20}

      where the quark anomalous dimension \gamma_q=-\alpha_s/\pi , and the expression for s(Q,b) in the one-loop running coupling constant is used,

      \begin{aligned}[b] s(Q,b)=&\frac{A^{(1)}}{2\beta_1}\hat{q}\ln\left(\frac{\hat{q}}{\hat{b}}\right)-\frac{A^{(1)}}{2\beta_1}(\hat{q}-\hat{b}) +\frac{A^{(2)}}{4\beta^2_1}\left(\frac{\hat{q}}{\hat{b}}-1\right)\\ &-\left[\frac{A^{(2)}}{4\beta^2_1}-\frac{A^{(1)}}{4\beta_1}\ln\left(\frac{{\rm e}^{2\gamma_E-1}}{2}\right)\right] \ln\left(\frac{\hat{q}}{\hat{b}}\right), \end{aligned}\tag{A21}

      with the variables defined by \hat{q}= \ln[Q/(\sqrt2\Lambda)], \; \hat{q}= \ln[1/(b\Lambda)] and the coefficients A^{(1,2)} and \beta_{1} expressed as

      \begin{eqnarray} \beta_1=\frac{33-2n_f}{12},\quad A^{(1)}=\frac{4}{3}, \end{eqnarray} \tag{A22}

      \begin{eqnarray} A^{(2)}=\frac{67}{9}-\frac{\pi^2}{3} -\frac{10}{27}n_f+\frac{8}{3}\beta_1\ln\left(\frac{1}{2}{\rm e}^{\gamma_{\rm E}}\right), \end{eqnarray}\tag{A23}

      n_f is the number of quark flavors, and \gamma_{\rm E} is Euler's constant.

      As we know, the double logarithms \alpha_s\ln^2x produced by the radiative corrections are not small expansion parameters when the end point region is important. To improve the perturbative expansion, the threshold resummation of these logarithms to all orders is required, which leads to a quark jet function

      \begin{eqnarray} S_t(x)=\frac{2^{1+2c}\Gamma(3/2+c)}{\sqrt{\pi}\Gamma(1+c)}[x(1-x)]^c, \end{eqnarray} \tag{A24}

      with c=0.3 . It is effective to smear the end point singularity with a momentum fraction x\to0 .

Reference (44)

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