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Effects from hadronic structure of photon on Bϕγ and Bs(ρ0,ω)γ decays

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Yun Li, Zhi-Tian Zou, Yue-Long Shen and Ying Li. Effects from Hadronic Structure of Photon on Bϕγ and Bs(ρ0,ω)γ Decays[J]. Chinese Physics C. doi: 10.1088/1674-1137/aca38f
Yun Li, Zhi-Tian Zou, Yue-Long Shen and Ying Li. Effects from Hadronic Structure of Photon on Bϕγ and Bs(ρ0,ω)γ Decays[J]. Chinese Physics C.  doi: 10.1088/1674-1137/aca38f shu
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Received: 2022-09-30
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Effects from hadronic structure of photon on Bϕγ and Bs(ρ0,ω)γ decays

  • 1. Shaanxi Collaborative Innovation Center of Industrial Auxiliary Chemistry & Technology, Shaanxi University of Science & Technology, Xi'an 710021, China
  • 2. Department of Physics, Yantai University, Yantai 264005,China
  • 3. College of Physics and Photoelectric Engineering, Ocean University of China, Qingdao 266100, China

Abstract: Using the perturbative QCD approach, we studied the effects of the hadronic structure of photons on the pure annihilation rediative decays Bϕγ and Bs(ρ0,ω)γ. These decays have small branching fractions due to the power suppression by Λ/mB, which makes them very sensitive to next-leading power corrections. The quark components and the related two-particle distribution amplitudes of a final state photon are introduced. The branching fractions can be enhanced remarkably by factorizable and nonfactorizable emission diagrams. The branching fraction of Bϕγ increases by approximately 40 times, and those of Bsρ0γ and Bsωγ are on the order of O(1010). We also note that the ratio of branching fractions of Bsρ0γ and Bsωγ is very sensitive to the effects of the hadronic structure of photons. All these results can be tested in future.

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    I.   INTRODUCTION
    • The B meson system, being a bound state that consists of a b quark and a light antiquark, provides an ideal laboratory for precise study of the Standard Model (SM) of particle physics and thus facilitates the search for new physics (NP). Because the b quark mass is much larger than the typical scale of the strong interaction, the otherwise troublesome long-distance strong interactions are generally less important and are under better control than in other lighter meson systems. Radiative decays BVγ are of particular interest in this respect. For example, the isospin-asymmetry parameter Δ±0=Γ(B±ρ±γ)/[2Γ(B0(ˉB0)ρ0γ)]1 and the direct CP asymmetry parameter ACP=[B(Bργ)B(B+ρ+γ)]/[B(Bργ)+B(B+ρ+γ)] allow us to extract the angle α of the Cobibbo-Kobayashi-Maskawa (CKM) unitary triangle [1]. The radiative BKγ decays are usually viewed as probes of the NP [2, 3] because they are induced by the flavor-changing-neutral-current bsγthat only occurs by loops. Therefore, it is meaningful to improve the theoretical predictions of these radiative decays, so as to match the accurate measurements to be obtained in the ongoing LHCb and Belle II experiments.

      In the past few years, many efforts [48] have been devoted to improving the theoretical predictions of these exclusive radiative decays by including various corrections, such as next-to-leading order in the strong coupling αs, non-factorizable corrections, and charm loop contributions. Each correction may affect the experimental observables, such as the branching fractions, CPasymmetries, and forward-backward asymmetries. Some special radiative decays such as Bϕγ and Bs(ρ0,ω)γ, where the quarks in the final state are different from those in the initial B meson, are called pure annihilation radiative decays. In particular, the annihilation diagrams are viewed as power suppressed by Λ/mB with a typical hadronic scale Λ; therefore, the branching fractions are estimated to be small. For the decay B0ϕγ, which is induced by penguin operators, the decay amplitude can be factorized into the simple matrix ϕ|ˉsγμs|0 and the transition form factor γ|ˉdγμ(1γ5)b|B0 in the naive factorization. Its branching fraction is estimated to be on the order of O(1013), and it can be O(1012) [9] by adding the QCD corrections. Within the perturbative QCD (PQCD) approach, its branching fraction has been predicted to be 1.6×1012 [10] by one of us (Li). These predictions are still much smaller than the current available upper limit 1.0×107 [11]. As a decay induced only by the penguin operators, the branching fraction might be enhanced remarkably by the effects of NP, such as in the R-parity violation supersymmetry model [9] and the non-universal Z model [12]. In this respect, before judging the effects of NP, we should calculate the branching fractions in the SM as precisely as possible.

      As we know, in addition to the high order corrections of αs, power corrections are important for the finite b quark mass. In the B meson decays, there are many types of power corrections, such as that from the high-twist distribution amplitudes of the B mesons [1315], the high-twist light-cone distribution amplitudes (LCDAs) of the final state mesons [16, 17], and the hadronic structure of the photon (HSP) [18, 19] in radiative decays. In the framework of PQCD, the power corrections for Bγlν decay have also been investigated [19], wherein it was indicated that both the contribution from the high-twist B meson wave functions and the HSP can change the leading power result by approximately 20%. In view of the power counting rules, the corrections from high-twist LCDAs of light mesons are no more than 10% for nonleptonic B decays [20]. For the radiative nonleptonic B decays that only occur through annihilation diagrams at leading power, when the HSP is considered, the contributions from emission diagrams are involved, and the branching ratio might be enhanced. Hence, the pure annihilation decays are perhaps sensitive to the power corrections. Motivated by this, we employ the PQCD approach to investigate the corrections from the HSP to the Bϕγ and Bs(ρ,ω)γ amplitudes, so as to improve the precision of the theoretical prediction.

      The outline of this paper is as follows. In Sec. II, we briefly review the theoretical background and summarize the expressions for the Bϕγ and Bs(ρ,ω)γ amplitudes. In Sec. III, the numerical results and discussion are presented. We conclude this study in Sec. IV.

    II.   FRAMEWORK AND INPUTS
    • It is well known that, when facing annihilation type contributions, in collinear factorization such as QCD factorization, the singularities destroy the perturbative calculations, and then, parametrization is adopted to evaluate this type of contribution, which leads to a decrease in predictive ability. Using the PQCD approach [21, 22] based on kT factorization, one can calculate the annihilation diagrams perturbatively, because the end-point singularities are smeared by keeping the intrinsic transverse momenta of the inner quarks. The pure annihilation decay Bsππ was first calculated in 2004 [23] and the predicted branching fraction was confirmed by LHCb in 2006 [24], which implies that the results of pure annihilation decays based on the PQCD approach are reliable.

      In this study, we work in light-cone coordinates. In the B meson rest frame, the momenta of the initial B meson, the final vector meson (V), and the photon are expressed as

      PB=mB2(1,1,0),PV=mB2(r2,1,0),Pγ=mB2(1r2,0,0),

      (1)

      with r=mV/mB, mV and mB being the masses of the vector meson and the B meson, respectively. For the light quark in the B meson, we denote its momentum as

      k1=(x1mB2,0,k1T),

      (2)

      where x1 is the momentum fraction of the light quark, and k1T is the transverse momentum of the light quark. Similarly, the momentum of the quark of the final vector can be written as

      k2=(0,x2mB2,k2T).

      (3)

      In this study, we do not regard the photon as a point-like particle any more and consider its hadronic structure; then, the momentum of the light quark in the photon can be written as

      k3=(mB2(1r2)x3,0,k3T).

      (4)

      In PQCD, the decay amplitude of the process BM1M2 is factorized into the convolution of the Wilson coefficients (WCs) C(t), the hard kernel H(xi,bi,t), and the wave functions of the initial and final states, which can be expressed as

      A=10dx1dx2dx30b1db1b2db3b3db3C(t)ΦB(x1,b1)×Φ2(x2,b2)Φ3(x3,b3)H(xi,bi,t)exp[S(xi,bi,t)],

      (5)

      with bi being the conjugate variable of the transverse momentum kiT. The physics above the scale (mb) have already been absorbed into the WCs C(t). The hard kernel H(xi,bi,t) is governed by exchanging one hard gluon and can be calculated perturbatively. The parameter t is the largest scale appearing in the hard kernel H(xi,bi,t). The wave functions describing the soft physics below the factorization scale are not perturbatively calculable but universal, which can be studied in some nonperturbative approaches. The last exponential term is the so-called Sudakov form factor, which arises from the resummation on the double logarithmic terms containing the additional energy scale introduced by the transverse momentum.

      In PQCD, the most important inputs are the wave functions, which describe the inner dynamics of the initial and final particles. In the past decades, the wave functions of the B meson and vector mesons have been well studied in two-body nonleptonic B decays, such as in Refs. [2123, 2528]. In general, the wave function ΦM,αβ with Dirac indices α,β are decomposed into 16 independent components, 1αβ, γμαβ, σμναβ, (γμγ5)αβ, and γ5αβ. For the heavy pseudo-scalar B meson, the structure (γμγ5)αβ and γ5αβ components remain as leading contributions. Therefore, ΦB,αβ is written as

      ΦB,αβ=i2Nc{(Bγ5)αβϕAB+γ5αβϕPB},

      (6)

      where Nc=3 is the color's degree of freedom, and ϕA,PB are Lorentz scalar wave functions. According to the heavy quark effective theory, we obtain ϕPBmBϕAB. Hence, the B meson's wave function can be simplified as

      ΦB,αβ(x,b)=i2Nc[(1γ5)αβ+mBγ5αβ]ϕB(x,b).

      (7)

      For the Lorentz scalar wave function ϕB(x,b), there is a sharp peak at the small x region; we use

      ϕB(x,b)=NBx2(1x)2exp[m2B x22ω2b12(ωbb)2],

      (8)

      which is adopted in Refs. [21, 22]. It is noted that ϕB is normalized by the decay constant fB,

      10dx ϕB(x,b=0)=fB22Nc.

      (9)

      As mentioned above, the parameters ωb=0.40±0.08 for theB0 meson and ωb=0.50±0.10 for the B0s meson are almost best fits from the well measured results of the Bd,sKπ, ππ decays [21, 22, 28, 29], including the branching fractions and the CP asymmetries. For the vector mesons ϕ, ρ0, and ω, we also pursue the same strategy and adopt the same wave functions obtained in QCD sum rules [30]. Very recently, the Gegenbauer moments in the wave functions of the light mesons have been fitted to available data of branching fractions and direct CP asymmetries globally [31], and the fitted results are in good agreement with those in [30].

      When studying the hard exclusive processes involving photon emission in QCD, a specific feature is that a real photon contains both a pointlike, electromagnetic (EM) component and a soft, hadronic component. Similar to a transversely polarized vector meson, the two-particle distribution amplitudes of a final state photon can be defined as [32]

      γ(p,λ)|ˉq(z)σαβq(0)|0=iQqχ(μ)ˉqq(pβεγαpαεγβ)10dueiupzϕγ(u,μ),

      (10)

      γ(p,λ)|ˉq(z)γαq(0)|0=Qqf3γεγα10dueiupzψ(v)γ(u,μ),

      (11)

      γ(p,λ)|ˉq(z)γαγ5q(0)|0=14Qqf3γϵαβρσpρzσεβγ10dueiupzψ(a)γ(u,μ),

      (12)

      where λ and εγ denote the polarization and the related polarization vector. The Lorentz scalar wave functions ϕγ(u,μ) and ψ(a,v)γ(u,μ) are twist-2 and twist-3 distribution amplitudes (DAs), respectively. ˉqq and χ(μ) are the quark condensate and the corresponding magnetic susceptibility. f3γ is the decay constant of the photon, which appears in the twist-3 DAs. It should be emphasized that ˉqq, χ(μ) and f3γ are all scale dependent, and their evolution equations can be found in Refs. [18, 19]. Finally, we write the momentum space projector for the two-particle LCDAs (up to twist-3) as

      Mγαβ=14Qq{ˉqq(ε̸γ)χ(μ)ϕγ(u,μ)f3γ(ε̸)ψ(v)γ(u,μ)18f3γϵμνρσ(γμγ5)ˉnρεν[nσdduψ(a)γ(u,μ)2Eγψ(a)γ(u,μ)kσ]}αβ,

      (13)

      and n=(1,0,0T) and ˉn=(0,1,0T) are two unit vectors with opposite directions. In our numerical calculations, the contributions from transverse momentum dependence of the photon wave functions are very small, which indicates that the last part in the third term can be neglected safely.

      The distribution amplitudes ϕγ(u,μ), ψv(u,μ) and ψa(u,μ) have been systematically studied in Ref. [32], and their expressions are written as

      ϕγ(u,μ)=6uˉu[1+n=2bn(μ0)C3/2n(ξ)],

      (14)

      ψ(v)(u,μ)=10P2(ξ)+158[3ωVγ(μ)ωAγ(μ)]P4(ξ),

      (15)

      ψ(a)(u,μ)=53C3/22(t)[1+916ωVγ(μ)316ωAγ(μ)],

      (16)

      with ξ=2u1. C3/2n(ξ) and Pn(ξ) are Gegenbauer polynomials and Legendre polynomials, respectively. Since the scale dependence of the parameters bn(μ), ωVγ(μ) and ωAγ(μ) have been well studied in Refs. [19, 32], we do not show them here. Typically, when μ=1 GeV, we have

      b2=0.07±0.07,ωVγ=3.8±1.8,ωAγ=2.1±1.0.

      (17)

      Because the massless photon is transverse polarized only, the decay amplitude of the radiative decay BVγ can only be decomposed into two parts as

      A=(εVεγ)AS+1PVPγϵμνρσεμγενVPργPσVAP,

      (18)

      with the polarization vectors of the vector meson and the photon εV and εγ, respectively. With the calculated AS and AP in the PQCD approach, the branching fraction of the BVγ decay is

      B=τBAS2+AP28πmB(1r2),

      (19)

      where τB is the lifetime of the B meson. At the leading order and the leading power, the decay amplitudes AS and AP have been calculated in detail in Ref. [10], and we do not present them further.

      Now, we study the contributions from the HSP to decays Bϕγ and Bs(ρ0,ω)γ. In the following, we take B0ϕγ as an example for illustration, which is governed by the bdˉss transition. According to the effective Hamiltonian [33], the corresponding diagrams are plotted in Fig. 1. The decay amplitude with the contribution of the HSP can be expressed as

      Figure 1.  (color online) Leading order Feynman diagrams of B0ϕγ decay with the hadronic structure of the photon in PQCD.

      AiH(Bϕγ)=VtdVtb[(a3+a5a72a92)Ai(LL)H,ef+(C4C102)Ai(LL)H,enf+(C6C82)Ai(SP)H,enf+(a3a92)(Ai(LL)H,af1+Ai(LL)H,af2)+(a5a72)(Ai(LR)H,af1+Ai(LR)H,af2)+(C4C102)(Ai(LL)H,anf1+Ai(LL)H,anf2)+(C6C82)(Ai(SP)H,anf1+Ai(SP)H,anf2)],

      (20)

      with i=S,P. The combined coefficients ai are defined as

      a3=C3+C4/3,a5=C5+C6/3,a7=C7+C8/3,a9=C9+C10/3.

      (21)

      The subscripts "ef" and "enf" stand for the factorizable and nonfactorizable diagrams with the emission of a ϕ meson, respectively. Similarly, "af1(2)" and "anf1(2)" mean the factorizable and nonfactorizable annihilation diagrams, where the number "1 (2)" means the case in which the produced strange quark enters the ϕ meson (γ). The superscripts "LL," "LR," and "SP" represent the inserted (VA)(VA), (VA)(V+A), and (SP)(S+P) currents. The expressions of the diagrams with different operators are given in the Appendix. Combining Eqs. (19) and (20), we calculate the branching fraction of this decay.

    III.   RESULTS AND DISCUSSION
    • In this section, we first list some parameters in our numerical calculations as [11, 30]

      fB=210±20MeV,  fBs=230±20MeV,fTϕ=186±9MeV,  fTρ=165±9MeV,fTω=151±9MeV,Vtb=0.999172+0.0000240.000035,  Vts=0.03978+0.000820.00060,Vtd=0.00854+0.000230.00016,τB=1.519×1012s,  τBs=1.527×1012s.

      (22)

      Using the above input parameters, we calculate the branching fractions of B0ϕγ,Bsρ0γ, and Bsωγ including corrections from the HSP at leading order and list the results in Table 1. In addition, the results of the leading power are presented for comparison. In our calculations, three types of errors have been studied, which are from uncertainties of the nonperturbative physics, unknown QCD corrections, and CKM matrix parameters. The first errors are from the uncertainties of the parameters in the wave functions, such as the decay constants and the inner parameters in the distribution amplitudes of the initial and final mesons. It should be emphasized that these types of errors are dominant, and more precise results from nonperturbative approaches are needed in future. The second uncertainty comes from the QCD scale ΛQCD and the hard kernel scale t, whose variants reflect the effects of the higher order QCD corrections. In this study, we set ΛQCD=(0.25±0.05) GeV and vary t from 0.8t to 1.2t. The final errors are caused by the CKM matrix elements.

      Decay modesLO(1012)IBR(1011)
      B0ϕγ0.9+0.2+0.3+0.00.20.40.03.57+2.12+0.25+0.191.580.310.14
      Bsρ0γ72.8+26.5+4.6+2.421.04.43.312.4+4.98+1.04+0.703.870.550.60
      Bsωγ8.6+5.3+6.9+0.23.44.20.235.1+31.8+2.13+1.7021.24.141.90

      Table 1.  Leading order (LO) branching ratios and improved branching ratios (IBR) of the pure annihilation type BVγ decays in the PQCD approach.

      From Table 1, one finds that, at the leading order, the branching fractions of the Bs(ρ0,ω)γ decays are much larger than that of B0ϕγ; this is because the former decays have the ˉuucomponent and thus gain contributions from the tree operators with larger WCs. It is important to note that, when the contributions from HSP are included, the branching fractions of all concerned decays are enhanced remarkably, and the branching fraction of Bϕγ increases by approximately 40 times compared with that of leading power. At leading power,Bϕγ is governed by the transition bdˉss, and the produced ˉs and s quarks form the final ϕ meson. The photon can be radiated from any quark participating in the weak interaction. Due to the symmetry, the contributions of the photon radiated from the strange and anti-strange quarks are canceled by each other; therefore, only the diagrams with photons emitted from the beauty and down quarks contribute to the amplitudes. The explicit amplitudes are given in Ref. [10]. When the contributions of the HSP are included, more diagrams contribute to the amplitudes AS and AP, as shown in Fig. 1. Compared with diagrams of leading power, both factorizable and nonfactorizable emission diagrams contribute to the amplitudes without cancellations. Furthermore, for the nanfactorizable diagrams, besides the (VA)(VA) operators, the (SP)(S+P) operator that results from the Fierz transformation of the (VA)(V+A) takes large contributions. This picture is very similar to that of the Bρ0ϕ decay. After summing all contributions, we find that the amplitude from the HSP is much larger than that of the leading power, leading to a larger branching fraction.

      Generally, the power expansion (1/mB) in B meson decays is an effective expansion in most decay modes, such as two-body non-leptonic decays. As stated in Sec. I, there are many sources of power corrections, such as the high-twist DAs of B and light mesons, the soft corrections to the leading twist contributions, and the subleading power “hadronic” photon correction. For the power corrections from B and light mesons, their contributions are less than 20%. Such corrections could be plagued by large theoretical uncertainties, and we have not discussed these effects yet in the current work. However, for the pure annihilation radiative BVγ decays, the leading power contributions (annihilation diagrams) are already power suppressed compared with decays dominated by the emission type diagrams. When the HSP is taken into account, the emission diagrams are involved, which enhances the next-to-leading power contributions and makes the higher power corrections significantly larger than those of the leading power. All these can be summarized as follows: for some special decay modes, if the leading power contributions are suppressed by some mechanism, the higher power contributions could be larger than those of the leading power, which may lead to the branching fraction being enhanced significantly.

      We also note that the branching fractions of Bsρ0γ and Bsωγ are on the order of O(1010), and that of Bϕγ is on the order of O(1011). Unfortunately, such small branching fractions cannot be measured in current experiments. To highlight the effects of the corrections from HSP, we define a ratio as follows:

      Rρω=B(Bsρ0γ)B(Bsωγ),

      (23)

      which can reduce of the dependence on the nonperturbative parameters effectively. At leading power, Rρω8 is much larger than 1, whereas this ratio becomes 0.35 when the HSP is involved. It is apparent that this ratio decreases by approximately 40 times; thus, it may be a good probe for testing the effects of HSP when the data are available. At leading order, because of the interference between theuˉu and dˉd components, Bsρ0γ has larger penguin contributions than Bsωγ, leading to the branching fractions of Bsρ0γ being much larger than that of Bsωγ. When including the HSP corrections, although they are suppressed by the CKM matrix elements and the power, the newly introduced tree contributions by the ˉuu component in the photon are comparable with those of penguin contributions, due to the large WCs, especially in the nonfactorizable diagrams. Because the signs of ˉdd in ρ0 and ω are different, the interferences between contributions of tree operators and penguin ones are different for Bsρ0γ and Bsωγ decays. Again, due to the interference between the tree and penguin contributions, the HSP corrections to Bsρ0γ are much smaller than those to Bsωγ, leading to a small Rρω.

      As mentioned above, the HSP corrections could increase the branching fractions of these pure annihilation decays remarkably. In Ref. [19], it is found that the contribution of the HSP plays an important role in the Bγlν decay and the branching faction can be decreased by 20%. In Ref. [34], the contributions of electromagnetic penguin operators to Bs(ρ0,ω)γ decays have also been examined in QCDF, and the results indicate that they are suppressed by the electromagnetic coupling constant αe. Very recently, these three decays have been comprehensively analyzed at leading power in the framework of soft-collinear effective theory in [35], where the authors found that the ϕωρ0 mixing effect could enhance the branching fractions of B0ϕγ and Bsωγ by approximately three orders of magnitude. If their conclusion holds in PQCD, the branching fractions of B0ϕγ and Bsωγ are roughly estimated to be on the order of 108 and 107, respectively, which might be measurable in Belle-II and LHCb experiments. Future measurements on these decays will be helpful in discerning all the above physics mechanisms.

    IV.   SUMMARY
    • As pure annihilation type radiative B decays suppressed by Λ/mB, the decays Bϕγ and Bs(ρ0,ω)γ governed by flavor changing neutral currents have small branching fractions, which makes them very sensitive to the effects of the new physics beyond the SM. Before studying the effects of new physics, it is necessary for us to calculate the observables in the SM with high precision. In this study, we mainly investigate the power corrections from the HSP in these pure annihilation decays. Because the quark components in photons are taken in account, emission diagrams are involved, which are not suppressed in comparison to the annihilation diagrams. Thus, the branching fraction of Bϕγ is enhanced by approximately 40 times, and the branching fractions of Bsρ0γ and Bsωγ are on the order of O(1010). Moreover, to shed light on these types of corrections and reduce theoretical uncertainties, we define a ratio of branching fractions of Bsρ0γ and Bsωγ, namely Rρω, and find that this ratio is changed significantly by the HSP. All the above results can be tested in high luminosity accelerators in the future.

    APPENDIX A: AMPLITUDES
    • The amplitudes of factorizable emission diagrams (a and b) with the (VA)(VA) current are given as

      AS(LL)H,ef=32πCFfTϕrVQdm3BVtbVtd10dx1dx30b1db1b3db3ϕB(x1,b1)×{[f3γ(x3ϕaγ(x3)+4(x3+2)ϕvγ(x3))4mBχ(μ)ˉddϕγ(x3)]C(ta)Eef(ta)hef[x1,x3(1r2V),b1,b3]f3γ[ϕaγ(x3)4ϕvγ(x3)]C(tb)Eef(tb)hef[x3,x1(1r2V),b3,b1]},

      AP(LL)H,ef=32πCFfTϕrVQdm3BVtbVtd10dx1dx30b1db1b3db3ϕB(x1,b1)×{[f3γ((2+x3)ϕaγ(x3)+4x3ϕvγ(x3))+4mBχ(μ)ˉddϕγ(x3)]C(ta)Eef(ta)hef(x1,x3(1r2V),b1,b3)+f3γ[ϕaγ(x3)4ϕvγ(x3)]C(tb)Eef(tb)hef[x3,x1(1r2V),b3,b1]},

      with the color factor CF=4/3 and the ϕ(1020) meson decay constant fTϕ. Qd is the charge of the down quark in the hadronic structure of the photon. The Sudakov form factors Eef, hard functions hef, and scales ta,b can be found in Ref. [27] because the behaviors of the photon are very similar to a vector.

      For the nonfactorizable emission diagrams (c and d), not only the (VA)(VA) current but also the (SP)(S+P) current can be inserted, and the corresponding amplitudes are written as

      AS(LL)H,enf=πCFrVm3BQd10dx1dx2dx30b1db1b2db2ϕB(x1,b1)×{[8(x21)ϕγ(x3)χ(μ)mBˉdd(ϕaϕ(x2)+ϕvϕ(x2))]Eenf(ta)henfa[x1,x2,x3,b1,b2]4[2x2ϕγ(x3)χ(μ)mBˉdd(ϕaϕ(x2)+ϕvϕ(x2))f3γ(x2+x3)(ϕaγ(x3)ϕaϕ(x2)4ϕvγ(x3)ϕvϕ(x2))]×Eenf(tb)henfb[x1,x2,x3,b1,b2]},

      AP(LL)H,enf=πCFrVm3BQd10dx1dx2dx30b1db1b2db2ϕB(x1,b1)×{[8(1x2)ϕγ(x3)χ(μ)mBˉdd(ϕaϕ(x2)+ϕvϕ(x2))]Eenf(ta)henfa[x1,x2,x3,b1,b2]+4[2x2ϕγ(x3)χ(μ)mBˉdd(ϕaϕ(x2)+ϕvϕ(x2))+f3γ(x2+x3)(4ϕvγ(x3)ϕvϕ(x2)ϕaγ(x3)ϕaϕ(x2))]×Eenf(tb)henfb[x1,x2,x3,b1,b2]},

      AS(SP)H,enf=πCFrVm3BQd10dx1dx2dx30b1db1b2db2ϕB(x1,b1)×{4[2(x21)ϕγ(x3)χ(μ)mBˉdd(ϕaϕ(x2)ϕvϕ(x2))f3γ(x2x31)(4ϕvγ(x3)ϕvϕ(x2)+ϕaγ(x3)ϕaϕ(x2))]×Eenf(ta)henfa[x1,x2,x3,b1,b2]+[8x2ϕγ(x3)χ(μ)mBˉdd(ϕvϕ(x2)ϕaϕ(x2))]Eenf(tb)henfb[x1,x2,x3,b1,b2]},

      AP(SP)H,enf=πCFrVm3BQd10dx1dx2dx30b1db1b2db2ϕB(x1,b1)×{4[2(1x2)ϕγ(x3)χ(μ)mBˉdd(ϕaϕ(x2)ϕvϕ(x2))+f3γ(1x2+x3)(4ϕvγ(x3)ϕvϕ(x2)+ϕaγ(x3)ϕaϕ(x2))]×Eenf(ta)henfa[x1,x2,x3,b1,b2]+[8x2ϕγ(x3)χ(μ)mBˉdd(ϕaϕ(x2)ϕvϕ(x2))]Eenf(tb)henfb[x1,x2,x3,b1,b2]}.

      For the annihilation diagrams (e-h), because the produced strange quark can enter not only the ϕ meson but also the photon, we have to discuss two cases. If the strange quark enters the photon, the corresponding factorizable and nonfactorizable annihilation amplitudes with possible currents are given as

      AS(LL)H,af1=32πCFfbQsrVf3γm3B10dx2dx30b2db2b3db3×{[x3(ϕaγ(x3)+4ϕvγ(x3))(ϕaϕ(x2)ϕvϕ(x2))2ϕaγ(x3)ϕaϕ(x2)+8ϕvγ(x3)ϕvϕ(x2)]Eaf(te)hafe[x2,x3,b2,b3]+[x2(ϕaγ(x3)4ϕvγ(x3))(ϕaϕ(x2)+ϕvϕ(x2))+(ϕaγ(x3)+4ϕvγ(x3))(ϕaϕ(x2)ϕvϕ(x2))]Eaf(tf)haff[x2,x3,b2,b3]},

      AP(LL)H,af1=32πCFfbQsrVf3γm3B10dx2dx30b2db2b3db3×{[ϕaγ(x3)((x32)ϕvϕ(x2)x3ϕaϕ(x2))4ϕvγ(x3)((x32)ϕaϕ(x2)x3ϕvϕ(x2))]Eaf(te)hafe[x2,x3,b2,b3]+[x2(ϕaγ(x3)4ϕvγ(x3))(ϕaϕ(x2)+ϕvϕ(x2))(ϕaγ(x3)+4ϕvγ(x3))(ϕaϕ(x2)ϕvϕ(x2))]Eaf(tf)haff[x2,x3,b2,b3]},

      AS(LR)H,af1=AS(LL)H,af1,

      AP(LR)H,af1=AP(LL)H,af1;

      AS(LL)H,anf1=4πCFQsrVm3B10dx1dx2dx30b1db1b2db2ϕB(x1,b1)×{[f3γ(ϕaγ(x3)ϕaϕ(x2)4ϕvγ(x3)ϕvϕ(x2))+2(x21)rVϕγ(x3)ϕtϕ(x2)χ(μ)mBˉss]×Eanf(tg)hanfg[x1,x2,x3,b1,b2]2[x2rVϕγ(x3)ϕtϕ(x2)χ(μ)mB]Eanf(th)hanfh[x1,x2,x3,b1,b2]},

      AP(LL)H,anf1=4πCFQsrVm3B10dx1dx2dx30b1db1b2db2ϕB(x1,b1)×{[f3γ(4ϕvγ(x3)ϕaϕ(x2)ϕaγ(x3)ϕvϕ(x2))+2(x21)rVϕγ(x3)ϕtϕ(x2)χ(μ)mBˉss]×Eanf(tg)hanfg[x1,x2,x3,b1,b2]2[x2rVϕγ(x3)ϕtϕ(x2)χ(μ)mB]Eanf(th)hanfh[x1,x2,x3,b1,b2]},

      AS(SP)H,anf1=AS(LL)H,anf1,

      AP(SP)H,anf1=AP(LL)H,anf1.

      When the strange quark enters the ϕ meson, the amplitudes are expressed as

      AS(LL)H,af2=AS(LR)H,af2=AS(LL)H,af1,

      AP(LL)H,af2=AP(LR)H,af2=AP(LL)H,af2,

      AS(LL)H,anf2=4πCFQsrVm3B10dx1dx2dx30b1db1b2db2ϕB(x1,b1)×{[f3γ(ϕaγ(x2)ϕaϕ(x3)4ϕvγ(x2)ϕvϕ(x3))2x3rVϕγ(x2)ϕtϕ(x3)χ(μ)mBˉss]Eanf(tg)hanfg[x1,x2,x3,b1,b2]+2[(x31)rVϕγ(x2)ϕtϕ(x3)χ(μ)mB]Eanf(th)hanfh[x1,x2,x3,b1,b2]},

      \begin{aligned}[b] \mathcal{A}^{P(LL)}_{H,anf2}=&4\pi C_F Q_s r_V m_B^3\int_0^1{\rm d}x_1{\rm d}x_2{\rm d}x_3\int_0^{\infty}b_1{\rm d}b_1b_2{\rm d}b_2 \phi_B(x_1,b_1) \\&\times \Bigg\{\left[f_{3\gamma}\Big(4\phi_{\gamma}^v(x_2)\phi_{\phi}^a(x_3)-4\phi_{\gamma}^a(x_2)\phi_{\phi}^v(x_3)\Big) +2x_3 r_V\phi_{\gamma}(x_2)\phi_{\phi}^t(x_3)\chi(\mu)m_B\langle\bar{s}s\rangle\right]E_{anf}(t_g)h_{anfg}[x_1,x_2,x_3,b_1,b_2] \\& -2\left[(x_3-1)r_V\phi_{\gamma}(x_2)\phi_{\phi}^t(x_3)\chi(\mu)m_B\right] E_{anf}(t_h)h_{anfh}[x_1,x_2,x_3,b_1,b_2]\Bigg\}, \end{aligned}\tag{A18}

      \mathcal{A}^{S(SP)}_{H,anf2} =\mathcal{A}^{S(LL)}_{H,anf2}, \tag{A19}

      \mathcal{A}^{P(SP)}_{H,anf2} =-\mathcal{A}^{P(LL)}_{H,anf2}. \tag{A20}

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