# Study ofBs⟶ϕℓ+ℓ− Decays in the PQCD Factorization Approach with Lattice QCD Input

Authors: Su-Ping Jin; Zhen-Jun Xiao
Journal: Advances in High Energy Physics
 (2021)

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## Abstract

In this paper, we studied systematically the semileptonic decaysBs⟶ϕl+l− with l−=e−,μ−,τ− by using the perturbative QCD (PQCD) and the “PQCD+Lattice” factorization approach, respectively. We first evaluated all relevant form factors Fiq2 in the low-q2 region using the PQCD approach, and we also took the available lattice QCD results at the high-q2 region as additional input to improve the extrapolation of Fiq2 from the low-q2 region to the endpoint qmax2. We then calculated the branching ratios and many other physical observables AFBl, FLϕ, S3,4,7, and A5,6,8,9 and the clean angular observables P1,2,3 and P4,5,6,8′. From our studies, we find the following points: (a) the PQCD and “PQCD+Lattice” predictions of BBs⟶ϕμ+μ− are about 7×10−7, which agree well with the LHCb measured values and the QCD sum rule prediction within still large errors; (b) we defined and calculated the ratios of the branching ratios Rϕeμ and Rϕμτ; (c) the PQCD and “PQCD+Lattice” predictions of the longitudinal polarization FL, the CP-averaged angular coefficients S3,4,7, and the CP asymmetry angular coefficients A5,6,8,9 agree with the LHCb measurements in all considered bins within the still large experimental errors; and (d) for those currently still unknown observables Rϕeμ,Rϕμτ,AFBl,P1,2,3, and P4,5,6,8′, we suggest LHCb and Belle-II Collaboration to measure them in their experiments.


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## Body

## 1. Introduction

In the standard model (SM) of particle physics, one treats these three generations of the charged leptonsℓ−=e−,μ−,τ− as exact copies of each other. These charged leptons behave in the same way but differ only in the masses determined by their Yukawa coupling to the Higgs boson. The lepton flavor universality (LFU), i.e., the equality of the coupling to the all electroweak gauge bosons among three families of leptons, has been regarded as an exact symmetry for quite a long time [1]. In recent years, however, some physics observables associated with the flavor-changing neutral current (FCNC) transitions b⟶sℓℓ have exhibited deviations from the SM expectations. These include the LFU-violating(LFUV) ratios RK and RK∗ [2, 3], whose measurements deviate from μ‐e universality [4–6] by around 2.5σ. More notably, the measurements of the angular observable P5′ of B⟶K∗μ+μ− decay in the large recoil region [7–11] as reported by the LHCb [12, 13] and Belle Collaboration [14] point to a deviation of about 3σ with respect to the SM prediction [15].As is well known, the FCNCb⟶s transition is forbidden at tree-level, but proceeds by way of loop diagrams with a very low rate. Due to the strong suppression within SM, such kinds of FCNC decays may be sensitive to the possible new physics (NP) effects. Therefore, the semileptonic b⟶sℓℓ decay has received striking attentions by means of measurements of the inclusive B⟶Xsℓ+ℓ− and/or the exclusive B⟶K∗ℓ+ℓ− decays and their comparison with the SM predictions. Besides the decay rates, many angular observables of the semileptonic B⟶K∗μ+μ− decays have also been measured previously [12–14]. The precision of the experimental measurements will also be expected to upgrade remarkably in the forthcoming year.The semileptonic decayBs⟶ϕμ+μ−, which is closely relevant to the decay B⟶K∗μ+μ−, offers an alternative scene to check out the same fundamental quark process, in a different hadronic background. On the theoretical side, various studies on the quark level b⟶s transition and the exclusive Bs⟶Vℓ+ℓ− decays by using rather different theories or models have been performed within the SM, such as the constituent quark model or covariant quark model [16, 17], the light front quark model [18], the QCD factorization (QCDF) [19], and the light-cone sum rule (LCSR) [20–25], and beyond the SM, such as the universal extra dimension [26, 27] and the supersymmetric theory [28]. On the experimental side, the Bs⟶ϕμ+μ− decay mode was first observed and studied by the CDF Collaboration [29] and subsequently by the LHCb Collaboration [30–34]. Beyond the measurement of the branching ratio, a rich phenomenology of various kinematical distributions can be presented. While the angular distributions were found to be consistent with the SM expectations obtained in References [22, 23], however, LHCb also observed a deficit with respect to the SM prediction for the branching ratio Bs0⟶ϕμ+μ− in the low-q2 region: the tension between the theory and experiment is about 3σ in the region 1.0≤q2≤6.0GeV2, where the form factors are evaluated by using the combined fit of lattice and the LCSR results [22, 23].In a previous paper [35], the semileptonic Bs⟶K∗ℓ+ℓ+ decays have been studied by us using the perturbative QCD (PQCD) factorization approach [36–47]. In this paper, we will make systematic studies for the semileptonic Bs⟶ϕℓ+ℓ− and present the theoretical predictions for many physical observables:
(1)
ForBs⟶ϕℓ+ℓ− decays, we treat them as a four-body decay Bs⟶ϕ⟶K−K+ℓ+ℓ− described by four kinematic variables: the lepton invariant mass squared q2 and three angles θK,θℓ,Φ. We defined and calculated the full angular decay distribution, the transverse amplitudes, the partially integrated decay amplitudes over the angles θK,θℓ,Φ, the CP-averaged differential branching, the ratios Reμϕ and Rμτϕ of the branching ratios, the forward-backward asymmetry AFBq2, the ϕ polarization fraction FLq2, the CP-averaged (asymmetry) angular coefficients Si (Ai), and the optimized observables Pi and Pi′. Following Reference [48], where the authors approved that the possible S-wave correction to the branching fractions of Bs⟶ϕ⟶K−K+ℓ+ℓ− decays is small and may modify the differential decay widths by about 5% only, we therefore will take the S-wave correction to the branching fractions as an additional uncertainty of 5% in magnitude(2)
We used both the PQCD factorization approach and the “PQCD+Lattice” approach to determine the values and theirq2-dependence of the Bs⟶ϕ transition form factors. We used the z-series parametrization to make the extrapolation for all form factors from the low-q2 region to the endpoint qmax2. We will calculate the branching ratios and all other physical observables by using the PQCD approach itself and the “PQCD+Lattice” approach, respectively, and compare their predictions with those currently available experimental measurementsThe paper is organized as follows: In Section2, we give a short review for the kinematics of the Bs⟶ϕℓ+ℓ− decays including distribution amplitudes of Bs and ϕ mesons and the effective Hamiltonian for the quark level b⟶sℓ+ℓ−. In Section 3, we define explicitly all physical observables for Bs⟶ϕℓ+ℓ− decays. In Section 4, we present our theoretical predictions of all relevant physical observables of the considered decay modes, compare these predictions with those currently available experimental measurements, and make some phenomenological analysis. A short summary is given in the last section.


## 2. Kinematics and Theoretical Framework

### 2.1. Kinematics and Wave Functions

We treat theBs meson at rest as a heavy-light system. The kinematics of the semileptonic Bs⟶ϕℓ+ℓ− decays in the large-recoil (low-q2) region will be discussed below, where the PQCD factorization approach is applicable to the considered decays. In the rest frame of B¯s0 meson, we define the B¯s0 meson momentum p1 and the ϕ momentum p2 in the light-cone coordinates as Reference [41]. We also use xi to denote the momentum fraction of light antiquark in each meson and set the momentum pi and ki (the momenta carried by the spectator quark in Bs and ϕ meson) in the following forms:
(1)p1=mBs21,1,0⊥,p2=rmBs2η+,η−,0⊥,k1=0,x1mBs2,k1⊥,k2=mBs2x2rη+,x2rη−,k2⊥,where the mass ratio r=mϕ/mBs and the factor η± is defined in the following form:
(2)η±=η±η2−1,withη=12r1+r2−q2mBs2,where q=p1−p2 is the lepton-pair four-momentum. For the final state ϕ meson, its longitudinal and transverse polarization vector εL,T can be written in the form of εL=η+,−η−,0⊥/2 and εT=0,0,1.For theBs meson wave function, we use the same kind of parameterizations as in References [42–44].
(3)ΦBs=i2NcpBs+mBsγ5ϕBsk1.Here, only the contribution of the Lorentz structureϕBsk1 is taken into account, since the contribution of the second Lorentz structure ϕ¯Bs is numerically small [49, 50] and has been neglected. We adopted the distribution amplitude of the Bs meson in the similar form as that of B-meson in the SU3f limit being widely used in the PQCD approach [42–44]
(4)ϕBsx,b=NBsx21−x2exp−mBs2x22ωBs2−12ωBsb2.In order to estimate the theoretical uncertainties induced by the variations ofϕBsx,b, one usually take ωBs=0.50±0.05GeV for Bs0 meson [38, 42]. The normalization factor NBs depends on the values of the shape parameter ωBs and the decay constant fBs and defined through the normalization relation: ∫01dxϕBsx,0=fBs/26 [42, 44].For the vector mesonϕ, the longitudinal and transverse polarization components can both provide the contribution. Here, we adopt the wave functions of the vector ϕ as in Reference [42]:
(5)Φϕ∣p,εL=i6εLmϕϕϕx+εLpϕϕtx+mϕϕϕsx,(6)Φϕ⊥p,εT=i6εTmϕϕϕvx+εTpϕϕTx+mϕiεωυρσγ5γωεTvnρvσϕϕax,where p and mϕ are the momentum and the mass of the ϕ meson, εL and εT correspond to the longitudinal and transverse polarization vectors of the vector meson ϕ, respectively. The twist-2 DAs ϕϕ and ϕϕT in Equations (5) and (6) can be reconstructed as a Gegenbauer expansion [42]:
(7)ϕϕx=3fϕ6x1−x1+∑n=12anϕ∣Cn3/2t,(8)ϕϕTx=3fϕT6x1−x1+∑n=12anϕ⊥Cn3/2t,where t=2x−1 and a1,2∣,⊥ are the Gegenbauer moments, while C1,23/2 are the Gegenbauer polynomials as given in Reference [42]. fϕ and fϕT are the longitudinal and transverse components of the decay constants of the vector meson ϕ with fϕ=0.231±0.004GeV and fϕT=0.20±0.01GeV as given in Reference [42]. For the relevant Gegenbauer moments, we use the same ones as those in References [42, 46, 47, 51].
(9)a1∣,⊥=0,a2ϕ∣=0.18±0.08,a2ϕ⊥=0.14±0.07.The twist-3 DAsϕϕs,t and ϕϕv,a in Equations (5) and (6) are the same ones as those defined in Reference [42]:
(10)ϕϕt=3fϕT26t2,ϕϕs=3fϕT26−t,ϕϕv=3fϕ861+t2,ϕϕa=3fϕ46−t,where t=2x−1.


### 2.2. Effective Hamiltonian forb⟶sℓ+ℓ− Decays

The effective Hamiltonian for the considered semileptonic decayBs⟶ϕℓ+ℓ− is defined by the same one as in References [52–56]:
(11)Heff=−4GF2VtbVts∗C1μO1cμ+C2μO2cμ+∑i=310CiμOiμ+VubVus∗C1μO1cμ−O1uμ+C2μO2cμ−O2uμ+h.c.,where GF=1.16638×10−5GeV−2 is the Fermi constant and Vij are the CKM matrix elements. For the operators Oi, we adopt those as defined in the so-called γ5-free basis [57, 58]. Following Reference [59], the operators Oi can be written in the following form:
(12)O1c=s¯γμTaPLcc¯γμTaPLb,O2c=s¯γμPLcc¯γμPLb,O1u=s¯γμTaPLuu¯γμTaPLb,O2u=s¯γμPLuu¯γμPLb,O3=s¯γμPLb∑qq¯γμq,O4=s¯γμTaPLb∑qq¯γμTaq,O5=s¯γμγνγρPLb∑qq¯γμγνγρq,O6=s¯γμγνγρTaPLb∑qq¯γμγνγρTaq,O7=eg2mbs¯σμνPRbFμν,O8=1gmbs¯σμνTaPRbGμνa,O9=e2g2s¯γμPLb∑ℓℓ¯γμℓ,O10=e2g2s¯γμPLb∑ℓℓ¯γμγ5ℓ,where O1,2c,u are the current-current operators, O3‐6 are the QCD penguin operators, O7,8 are the electromagnetic and chromomagnetic penguin operators, respectively, and finally, O9,10 are the semileptonic operators. The inclusion of the factors 4π/g2=1/αs in the definition of the operators O7,8,9,10 serves to allow a more transparent organisation of the expansion of the relevant Wilson coefficients as defined in References [59, 60] up to next-to-next-to-leading order (NNLO). They are then evolved from the scale μ=mW down to the scale μ=mb using the renormalization group equations.Since the contributions from the subleading chromomagnetic penguin, quark loop, and annihilation diagrams are highly suppressed for the consideredb⟶sℓ+ℓ− decays [55], we will neglect them in our calculations. Using the effective Hamiltonian in Equation (11), the decay amplitude for b⟶sℓ+ℓ− loop transition can be decomposed as a product of a short-distance contributions through Wilson coefficients and long-distance contribution which is further expressed in terms of form factors:
(13)Ab⟶sℓ+ℓ−=GF2αemπVtbVts∗C9effq2s¯γμPLbℓ¯γμℓ+C10s¯γμPLbℓ¯γμγ5ℓ−2mbC7effs¯iσμνqνq2PRbℓ¯γμℓ,where C7effμ and C9effμ are the effective Wilson coefficients, defined as in References [44, 61].
(14)C7effμ=C7μ+Cb⟶sγ′μ,(15)C9effμ,q2=C9μ+Ypertq2+Yresq2.The termCb⟶sγ′ in Equation (14) is the absorptive part of b⟶sγ transition and was given in Reference [61].
(16)Cb⟶sγ′μ=iαs29η14/23xtxt2−5xt−28xt−13+3xt2lnxt4xt−14−0.1687−0.03C2μ,where xt=mt2/mW2 and η=αsmW/αsμ. The explicit expressions of the term Ypertq2 and Yresq2 in Equation (15) are of the following form [56, 62–65]:
(17)Ypertq2=0.124ωs^+gm^c,s^C0+λugm^c,s^−gm^u,s^3C1+C2−12gm^b,s^C3+3C4−12gm^b,s^4C3+4C4+3C5+C6+293C3+C4+3C5+C6,(18)Yresq2=−3παem2C0·∑V=J/Ψ,Ψ′⋯mVBV⟶l+l−ΓtotVq2−mV2+imVΓtotV−λugm^u,s^3C1+C2·∑V=ρ,ω,ϕmVBV⟶l+l−ΓtotVq2−mV2+imVΓtotV,where C0=3C1+C2+3C3+C4+3C5+C6, s^=q2/mb2, m^q=mq/mb, and the CKM ratio λu=VubVus∗/VtbVts∗. In Equation (17), the function ωs^ is the soft-gluon correction to the matrix element of operator O9. The function gm^q,s^ in Equations (17) and (18) is related to the basic fermion loop. The contributions from four quark operators O1‐O6 are usually combined with coefficient C9 into an “effective” one. One can find the explicit expressions of the function ωs^ and gm^q,s^ easily, for example, in Reference [35] and references therein.The termYpertq2 in Equations (15) and (17) defines the short-distance perturbative part that involves the indirect contributions from the matrix element of the four quark operators ∑i=110ℓ+ℓ−sOib [62–65] and lies at the place far away from cc¯ resonance regions.The termYresq2 in Equations (15) and (18) describes the long-distance resonant contributions related with the Bs⟶ϕV⟶ϕV⟶l+l− transitions in the resonance regions, where V=ρ,ω,ϕ,J/Ψ,ψ′,⋯ are the light vector mesons and cc¯ charmonium states. Up to now, the term Yresq2 cannot be calculated from the first principle of QCD and may also introduce the double-counting problem with the term Ypertq2. For more details about such kinds of double-counting problem, one can see the discussions as given in References [66, 67]. In this paper, we checked the possible effects on the theoretical predictions for the branching ratios and other considered physical observables by including the term Yresq2 or not in our numerical calculations, and we found that the resulted variations of the theoretical predictions are less than 5%. It is much smaller than the total theoretical errors, say around 30‐40%. According to the argument in Reference [68], the term Yresq2 is also generally small. Because of its smallness and the possible double-counting problem, we here simply drop the term Yresq2 out in our numerical evaluations for all physical observables considered in this paper.


### 2.3.Bs⟶ϕ Transition Form Factors

For the vector mesonϕ with polarization vector ε∗, as usual, the relevant form factors for Bs⟶ϕ transitions are Vq2 and A0,1,2q2 of the vector and axial vector currents and T1,2,3 of the tensor currents. Between the form factors A0,1,2q2 at the point q2=0, there is an exact relation 2mϕA00=mBs+mϕA10−mBs−mϕA20 in order to avoid the kinematical singularity. Between the form factor T1,2, there also exists a relation T10=T20 in an algebraic manner which is implied by the identity σμνγ5=−i/2εμναβσαβ with the ε0123=+1 convention for the Levi-Civita tensor.Using the well-studied wave functions as given in previous subsection, the PQCD factorization formulas for the relevant form factors ofBs⟶ϕℓ+ℓ− decays can be calculated and written in the following form:
(19)Vq2=8πmBs2CF1+r∫dx1dx2∫b1db1b2db2ϕBsx1×−x2rϕϕvx2+ϕϕTx2+1+x2rηη2−1ϕϕax2·H1t1+r+x12η2−1ϕϕvx2−x1−2rη2η2−1ϕϕax2·H2t2,A0q2=8πmBs2CF∫dx1dx2∫b1db1b2db2ϕBsx1×1+x2r2η−rϕϕx2+1−2x2rϕϕtx2+1−rη−2x2rη−rη2−1ϕϕsx2·H1t1+x1η2−1η+r2−rη2+x12−x1rη+r2ϕϕx2−x11−rη+2rr−ηη2−1−x1rϕϕsx2·H2t2,A1q2=16πmBs2CFr1+r∫dx1dx2∫b1db1b2db2ϕBsx1×1+x2rηϕϕvx2+η−2x2rϕϕTx2+x2rη2−1ϕϕax2·H1t1+rη−x12ϕϕvx2+rη2−1+x12ϕϕax2·H2t2,A2q2=1+r2η−r2rη2−1A1q2−8πmBs2CF1+rη2−r∫dx1dx2∫b1db1b2db2ϕBsx1×η1−x2r2+rx22η2−1−1ϕϕx2+1+2x2r2−1+2x2rηϕϕtx2+r1−2x2η2−1ϕϕsx2·H1t1+rη−12x1η2−1−rrη−1−x1η2+x1r+η2ϕϕx2+x1rη−1+x2−2rη2−1ϕϕsx2·H2t2,T1q2=8πmBs2CF∫dx1dx2∫b1db1b2db2ϕBsx1×1−2x2rϕϕvx2+1+2x2rη−x2r2ϕϕTx2+1+2x2r2−1+2x2rηη2−1ϕϕax2·H1t1+1−x12r−x1rη−12η2−1ϕϕvx2+rη−rη2−1+x12r+rη−1η2−1ϕϕax2·H2t2,T2q2=16πmBs2CFr1−r2∫dx1dx2∫b1db1b2db2ϕBsx1×1−1+2x2rη+2x2r2ϕϕvx2+x2rη2η−r−x2r+η−rϕϕTx2+1−2x2rη2−1ϕϕax2·H1t1+x221+ηη2−1rη−1+r+x12η2−1η−rϕϕvx2+1−x12rη2−1+x121−rηϕϕax2·H2t2,T3q2=1−r2η+r2rη2−1T2q2−8πmBs2CF1−r2η2−1∫dx1dx2∫b1db1b2db2ϕBsx1×η2−1+2x2rη+2x2r2η−rϕϕx2+1+x2rηϕϕtx2+x2rη2−1ϕϕsx2·H1t1+r−x12η+η2−1ϕϕx2+x1+2rη2−1ϕϕsx2·H2t2,where r=mϕ/mBs, the twist-2 DAs ϕϕ,ϕϕT, and the twist-3 DAs ϕϕs,t,ϕϕv,a have been defined in Equations (7) and (10). The function Hiti in above equations is of the following form:
(20)Hiti=hix1,x2,b1,b2·αsti·Stx2exp−Sabti,fori=1,2.The hard functionsh1,2x1,x2,b1,b2 come form the Fourier transform of virtual quark and gluon propagators and they can be defined by the following:
(21)h1=K0β1b1θb1−b2I0α1b2K0α1b1+θb2−b1I0α1b1K0α1b2,h2=K0β2b1θb1−b2I0α2b2K0α2b1+θb2−b1I0α2b1K0α2b2,where K0 and I0 are the modified Bessel functions and
(22)α1=mBsx2rη+,α2=mBsx1rη+−r2+rs2,β1=β2=mBsx1x2rη+,where r=mϕ/mBs and rs=ms/mBs. The hard scales ti in Equation (20) are chosen as the largest scale of the virtuality of the internal particles in the hard b-quark decay diagram, including 1/bii=1,2:
(23)t1=maxα1,1b1,1b2,t2=maxα2,1b1,1b2.The threshold resummation factorStx in Equation (20) is adopted from [49]:
(24)St=21+2cΓ3/2+cπΓ1+cx1−xc,with a fitted parameter cQ2=0.04Q2−0.51Q+1.87 [39] and Q2=mBs21−r2 [69]. The function Stx is normalized to unity. The function exp−Sabt in Equation (20) contains the Sudakov logarithmic corrections and the renormalization group evolution effects of both the wave functions and the hard scattering amplitude; for more details of function exp−Sabt, one can see References [37, 49].


## 2.1. Kinematics and Wave Functions

We treat theBs meson at rest as a heavy-light system. The kinematics of the semileptonic Bs⟶ϕℓ+ℓ− decays in the large-recoil (low-q2) region will be discussed below, where the PQCD factorization approach is applicable to the considered decays. In the rest frame of B¯s0 meson, we define the B¯s0 meson momentum p1 and the ϕ momentum p2 in the light-cone coordinates as Reference [41]. We also use xi to denote the momentum fraction of light antiquark in each meson and set the momentum pi and ki (the momenta carried by the spectator quark in Bs and ϕ meson) in the following forms:
(1)p1=mBs21,1,0⊥,p2=rmBs2η+,η−,0⊥,k1=0,x1mBs2,k1⊥,k2=mBs2x2rη+,x2rη−,k2⊥,where the mass ratio r=mϕ/mBs and the factor η± is defined in the following form:
(2)η±=η±η2−1,withη=12r1+r2−q2mBs2,where q=p1−p2 is the lepton-pair four-momentum. For the final state ϕ meson, its longitudinal and transverse polarization vector εL,T can be written in the form of εL=η+,−η−,0⊥/2 and εT=0,0,1.For theBs meson wave function, we use the same kind of parameterizations as in References [42–44].
(3)ΦBs=i2NcpBs+mBsγ5ϕBsk1.Here, only the contribution of the Lorentz structureϕBsk1 is taken into account, since the contribution of the second Lorentz structure ϕ¯Bs is numerically small [49, 50] and has been neglected. We adopted the distribution amplitude of the Bs meson in the similar form as that of B-meson in the SU3f limit being widely used in the PQCD approach [42–44]
(4)ϕBsx,b=NBsx21−x2exp−mBs2x22ωBs2−12ωBsb2.In order to estimate the theoretical uncertainties induced by the variations ofϕBsx,b, one usually take ωBs=0.50±0.05GeV for Bs0 meson [38, 42]. The normalization factor NBs depends on the values of the shape parameter ωBs and the decay constant fBs and defined through the normalization relation: ∫01dxϕBsx,0=fBs/26 [42, 44].For the vector mesonϕ, the longitudinal and transverse polarization components can both provide the contribution. Here, we adopt the wave functions of the vector ϕ as in Reference [42]:
(5)Φϕ∣p,εL=i6εLmϕϕϕx+εLpϕϕtx+mϕϕϕsx,(6)Φϕ⊥p,εT=i6εTmϕϕϕvx+εTpϕϕTx+mϕiεωυρσγ5γωεTvnρvσϕϕax,where p and mϕ are the momentum and the mass of the ϕ meson, εL and εT correspond to the longitudinal and transverse polarization vectors of the vector meson ϕ, respectively. The twist-2 DAs ϕϕ and ϕϕT in Equations (5) and (6) can be reconstructed as a Gegenbauer expansion [42]:
(7)ϕϕx=3fϕ6x1−x1+∑n=12anϕ∣Cn3/2t,(8)ϕϕTx=3fϕT6x1−x1+∑n=12anϕ⊥Cn3/2t,where t=2x−1 and a1,2∣,⊥ are the Gegenbauer moments, while C1,23/2 are the Gegenbauer polynomials as given in Reference [42]. fϕ and fϕT are the longitudinal and transverse components of the decay constants of the vector meson ϕ with fϕ=0.231±0.004GeV and fϕT=0.20±0.01GeV as given in Reference [42]. For the relevant Gegenbauer moments, we use the same ones as those in References [42, 46, 47, 51].
(9)a1∣,⊥=0,a2ϕ∣=0.18±0.08,a2ϕ⊥=0.14±0.07.The twist-3 DAsϕϕs,t and ϕϕv,a in Equations (5) and (6) are the same ones as those defined in Reference [42]:
(10)ϕϕt=3fϕT26t2,ϕϕs=3fϕT26−t,ϕϕv=3fϕ861+t2,ϕϕa=3fϕ46−t,where t=2x−1.


## 2.2. Effective Hamiltonian forb⟶sℓ+ℓ− Decays

The effective Hamiltonian for the considered semileptonic decayBs⟶ϕℓ+ℓ− is defined by the same one as in References [52–56]:
(11)Heff=−4GF2VtbVts∗C1μO1cμ+C2μO2cμ+∑i=310CiμOiμ+VubVus∗C1μO1cμ−O1uμ+C2μO2cμ−O2uμ+h.c.,where GF=1.16638×10−5GeV−2 is the Fermi constant and Vij are the CKM matrix elements. For the operators Oi, we adopt those as defined in the so-called γ5-free basis [57, 58]. Following Reference [59], the operators Oi can be written in the following form:
(12)O1c=s¯γμTaPLcc¯γμTaPLb,O2c=s¯γμPLcc¯γμPLb,O1u=s¯γμTaPLuu¯γμTaPLb,O2u=s¯γμPLuu¯γμPLb,O3=s¯γμPLb∑qq¯γμq,O4=s¯γμTaPLb∑qq¯γμTaq,O5=s¯γμγνγρPLb∑qq¯γμγνγρq,O6=s¯γμγνγρTaPLb∑qq¯γμγνγρTaq,O7=eg2mbs¯σμνPRbFμν,O8=1gmbs¯σμνTaPRbGμνa,O9=e2g2s¯γμPLb∑ℓℓ¯γμℓ,O10=e2g2s¯γμPLb∑ℓℓ¯γμγ5ℓ,where O1,2c,u are the current-current operators, O3‐6 are the QCD penguin operators, O7,8 are the electromagnetic and chromomagnetic penguin operators, respectively, and finally, O9,10 are the semileptonic operators. The inclusion of the factors 4π/g2=1/αs in the definition of the operators O7,8,9,10 serves to allow a more transparent organisation of the expansion of the relevant Wilson coefficients as defined in References [59, 60] up to next-to-next-to-leading order (NNLO). They are then evolved from the scale μ=mW down to the scale μ=mb using the renormalization group equations.Since the contributions from the subleading chromomagnetic penguin, quark loop, and annihilation diagrams are highly suppressed for the consideredb⟶sℓ+ℓ− decays [55], we will neglect them in our calculations. Using the effective Hamiltonian in Equation (11), the decay amplitude for b⟶sℓ+ℓ− loop transition can be decomposed as a product of a short-distance contributions through Wilson coefficients and long-distance contribution which is further expressed in terms of form factors:
(13)Ab⟶sℓ+ℓ−=GF2αemπVtbVts∗C9effq2s¯γμPLbℓ¯γμℓ+C10s¯γμPLbℓ¯γμγ5ℓ−2mbC7effs¯iσμνqνq2PRbℓ¯γμℓ,where C7effμ and C9effμ are the effective Wilson coefficients, defined as in References [44, 61].
(14)C7effμ=C7μ+Cb⟶sγ′μ,(15)C9effμ,q2=C9μ+Ypertq2+Yresq2.The termCb⟶sγ′ in Equation (14) is the absorptive part of b⟶sγ transition and was given in Reference [61].
(16)Cb⟶sγ′μ=iαs29η14/23xtxt2−5xt−28xt−13+3xt2lnxt4xt−14−0.1687−0.03C2μ,where xt=mt2/mW2 and η=αsmW/αsμ. The explicit expressions of the term Ypertq2 and Yresq2 in Equation (15) are of the following form [56, 62–65]:
(17)Ypertq2=0.124ωs^+gm^c,s^C0+λugm^c,s^−gm^u,s^3C1+C2−12gm^b,s^C3+3C4−12gm^b,s^4C3+4C4+3C5+C6+293C3+C4+3C5+C6,(18)Yresq2=−3παem2C0·∑V=J/Ψ,Ψ′⋯mVBV⟶l+l−ΓtotVq2−mV2+imVΓtotV−λugm^u,s^3C1+C2·∑V=ρ,ω,ϕmVBV⟶l+l−ΓtotVq2−mV2+imVΓtotV,where C0=3C1+C2+3C3+C4+3C5+C6, s^=q2/mb2, m^q=mq/mb, and the CKM ratio λu=VubVus∗/VtbVts∗. In Equation (17), the function ωs^ is the soft-gluon correction to the matrix element of operator O9. The function gm^q,s^ in Equations (17) and (18) is related to the basic fermion loop. The contributions from four quark operators O1‐O6 are usually combined with coefficient C9 into an “effective” one. One can find the explicit expressions of the function ωs^ and gm^q,s^ easily, for example, in Reference [35] and references therein.The termYpertq2 in Equations (15) and (17) defines the short-distance perturbative part that involves the indirect contributions from the matrix element of the four quark operators ∑i=110ℓ+ℓ−sOib [62–65] and lies at the place far away from cc¯ resonance regions.The termYresq2 in Equations (15) and (18) describes the long-distance resonant contributions related with the Bs⟶ϕV⟶ϕV⟶l+l− transitions in the resonance regions, where V=ρ,ω,ϕ,J/Ψ,ψ′,⋯ are the light vector mesons and cc¯ charmonium states. Up to now, the term Yresq2 cannot be calculated from the first principle of QCD and may also introduce the double-counting problem with the term Ypertq2. For more details about such kinds of double-counting problem, one can see the discussions as given in References [66, 67]. In this paper, we checked the possible effects on the theoretical predictions for the branching ratios and other considered physical observables by including the term Yresq2 or not in our numerical calculations, and we found that the resulted variations of the theoretical predictions are less than 5%. It is much smaller than the total theoretical errors, say around 30‐40%. According to the argument in Reference [68], the term Yresq2 is also generally small. Because of its smallness and the possible double-counting problem, we here simply drop the term Yresq2 out in our numerical evaluations for all physical observables considered in this paper.


## 2.3.Bs⟶ϕ Transition Form Factors

For the vector mesonϕ with polarization vector ε∗, as usual, the relevant form factors for Bs⟶ϕ transitions are Vq2 and A0,1,2q2 of the vector and axial vector currents and T1,2,3 of the tensor currents. Between the form factors A0,1,2q2 at the point q2=0, there is an exact relation 2mϕA00=mBs+mϕA10−mBs−mϕA20 in order to avoid the kinematical singularity. Between the form factor T1,2, there also exists a relation T10=T20 in an algebraic manner which is implied by the identity σμνγ5=−i/2εμναβσαβ with the ε0123=+1 convention for the Levi-Civita tensor.Using the well-studied wave functions as given in previous subsection, the PQCD factorization formulas for the relevant form factors ofBs⟶ϕℓ+ℓ− decays can be calculated and written in the following form:
(19)Vq2=8πmBs2CF1+r∫dx1dx2∫b1db1b2db2ϕBsx1×−x2rϕϕvx2+ϕϕTx2+1+x2rηη2−1ϕϕax2·H1t1+r+x12η2−1ϕϕvx2−x1−2rη2η2−1ϕϕax2·H2t2,A0q2=8πmBs2CF∫dx1dx2∫b1db1b2db2ϕBsx1×1+x2r2η−rϕϕx2+1−2x2rϕϕtx2+1−rη−2x2rη−rη2−1ϕϕsx2·H1t1+x1η2−1η+r2−rη2+x12−x1rη+r2ϕϕx2−x11−rη+2rr−ηη2−1−x1rϕϕsx2·H2t2,A1q2=16πmBs2CFr1+r∫dx1dx2∫b1db1b2db2ϕBsx1×1+x2rηϕϕvx2+η−2x2rϕϕTx2+x2rη2−1ϕϕax2·H1t1+rη−x12ϕϕvx2+rη2−1+x12ϕϕax2·H2t2,A2q2=1+r2η−r2rη2−1A1q2−8πmBs2CF1+rη2−r∫dx1dx2∫b1db1b2db2ϕBsx1×η1−x2r2+rx22η2−1−1ϕϕx2+1+2x2r2−1+2x2rηϕϕtx2+r1−2x2η2−1ϕϕsx2·H1t1+rη−12x1η2−1−rrη−1−x1η2+x1r+η2ϕϕx2+x1rη−1+x2−2rη2−1ϕϕsx2·H2t2,T1q2=8πmBs2CF∫dx1dx2∫b1db1b2db2ϕBsx1×1−2x2rϕϕvx2+1+2x2rη−x2r2ϕϕTx2+1+2x2r2−1+2x2rηη2−1ϕϕax2·H1t1+1−x12r−x1rη−12η2−1ϕϕvx2+rη−rη2−1+x12r+rη−1η2−1ϕϕax2·H2t2,T2q2=16πmBs2CFr1−r2∫dx1dx2∫b1db1b2db2ϕBsx1×1−1+2x2rη+2x2r2ϕϕvx2+x2rη2η−r−x2r+η−rϕϕTx2+1−2x2rη2−1ϕϕax2·H1t1+x221+ηη2−1rη−1+r+x12η2−1η−rϕϕvx2+1−x12rη2−1+x121−rηϕϕax2·H2t2,T3q2=1−r2η+r2rη2−1T2q2−8πmBs2CF1−r2η2−1∫dx1dx2∫b1db1b2db2ϕBsx1×η2−1+2x2rη+2x2r2η−rϕϕx2+1+x2rηϕϕtx2+x2rη2−1ϕϕsx2·H1t1+r−x12η+η2−1ϕϕx2+x1+2rη2−1ϕϕsx2·H2t2,where r=mϕ/mBs, the twist-2 DAs ϕϕ,ϕϕT, and the twist-3 DAs ϕϕs,t,ϕϕv,a have been defined in Equations (7) and (10). The function Hiti in above equations is of the following form:
(20)Hiti=hix1,x2,b1,b2·αsti·Stx2exp−Sabti,fori=1,2.The hard functionsh1,2x1,x2,b1,b2 come form the Fourier transform of virtual quark and gluon propagators and they can be defined by the following:
(21)h1=K0β1b1θb1−b2I0α1b2K0α1b1+θb2−b1I0α1b1K0α1b2,h2=K0β2b1θb1−b2I0α2b2K0α2b1+θb2−b1I0α2b1K0α2b2,where K0 and I0 are the modified Bessel functions and
(22)α1=mBsx2rη+,α2=mBsx1rη+−r2+rs2,β1=β2=mBsx1x2rη+,where r=mϕ/mBs and rs=ms/mBs. The hard scales ti in Equation (20) are chosen as the largest scale of the virtuality of the internal particles in the hard b-quark decay diagram, including 1/bii=1,2:
(23)t1=maxα1,1b1,1b2,t2=maxα2,1b1,1b2.The threshold resummation factorStx in Equation (20) is adopted from [49]:
(24)St=21+2cΓ3/2+cπΓ1+cx1−xc,with a fitted parameter cQ2=0.04Q2−0.51Q+1.87 [39] and Q2=mBs21−r2 [69]. The function Stx is normalized to unity. The function exp−Sabt in Equation (20) contains the Sudakov logarithmic corrections and the renormalization group evolution effects of both the wave functions and the hard scattering amplitude; for more details of function exp−Sabt, one can see References [37, 49].


## 3. Observables forBs⟶ϕℓ+ℓ− Mode Decays

In experimental analysis, theB¯s⟶ϕℓ+ℓ− decay is treated as the four body differential decay distribution B¯s⟶ϕ⟶K+K−ℓ+ℓ− and has been described in terms of the four kinematic variables [10, 11, 13, 19]: the lepton invariant mass squared q2 and the three decay angles Ω⟶=cosθK,cosθℓ,Φ. The angle θK is the angle between the direction of flight of K+ and Bs meson in the rest frame of ϕ, θℓ is the angle made by ℓ− with respect to the Bs meson in the dilepton rest frame, and Φ is the azimuthal angle between the two planes formed by dilepton and K+K−.With the hadronic and leptonic amplitudes defined in Equation (13), we write down the fourfold differential distribution of four-body B¯s⟶ϕ⟶K+K−ℓ+ℓ− decay [13, 55, 70, 71]:
(25)d4Γdq2dΩ⟶=932πIq2,Ω⟶,dΩ⟶=dcosθKdcosθℓdΦ,where the functions Iq2,Ω⟶ can be written in terms of a set of angular coefficients and trigonometric functions [70]:
(26)Iq2,Ω⟶=∑iIiq2fiΩ⟶=I1ssin2θK+I1ccos2θK+I2ssin2θK+I2ccos2θKcos2θℓ+I3sin2θKsin2θℓcos2Φ+I4sin2θKsin2θℓcosΦ+I5sin2θKsinθℓcosΦ+I6ssin2θKcosθℓ+I7sin2θKsinθℓsinΦ+I8sin2θKsin2θℓsinΦ+I9sin2θKsin2θℓsin2Φ.For the CP-conjugated modeBs⟶ϕ⟶K−K+ℓ+ℓ−, the corresponding expression of the angular decay distribution is as follows:
(27)d4Γ¯dq2dΩ⟶=932πI¯q2,Ω⟶,where the function I¯q2,Ω⟶ is obtained from Iq2,Ω⟶ in Equation (26) by making the complex conjugation for all weak phases in Ii [70] and numerically by the following substitution:
(28)I1c,s,2c,s,3,4,7⟶I¯1c,s,2c,s,3,4,7,I5,6s,8,9⟶−I¯5,6s,8,9.The minus sign in Equation (28) is a result of the convention that, under the previous definitions of three angles θK,θl,Φ, a CP transformation interchanges the lepton and antilepton, i.e., leading to the transformation θℓ⟶θℓ−π and Φ⟶−Φ.The angular coefficientsIi, which are functions of q2 only, are usually expressed in terms of the transverse amplitudes [8, 13]. In the limit of massless leptons, there are six such complex amplitudes: A0L,R, A∥L,R, and A⊥L,R, where L and R refer to the chirality of the leptonic current. For the massive case, an additional complex amplitude At is required, where the timelike component of the virtual gauge boson (which can later decay into dilepton) couples to an axial vector current.In Table1, we summarize the treatment of the angular distribution by decomposition of the angular coefficients Iiq2 into seven transverse amplitude A⊥,∥,0L,R and At as well as the corresponding trigonometric factor fiΩ⟶. Here, we will not consider scalar contribution to facilitate the comparison with Reference [35]. Notice that the distribution including lepton masses (but neglecting scalar I6c=0) contains eleven Ii where only 10 of them are independent [8, 72]. In the limit of massless leptons, it is easy to obtain the relations I1s=3I2s and I1c=−I2c [70].Table 1
The explicit expressions of the angular coefficientsIiq2 and fiΩ⟶ appeared in Equation (26).
iIiq2fiΩ⟶1s34−m^ℓ2A∥L2+A⊥L2+A∥R2+A⊥R2+4m^ℓ2ReA⊥LA⊥R∗+A∥LA∥R∗sin2θK1cA0L2+A0R2+4m^ℓ2At2+2ReA0LA0R∗cos2θK2s14βℓ2A∥L2+A⊥L2+A∥R2+A⊥R2sin2θKcos2θℓ2c−βℓ2A0L2+A0R2cos2θKcos2θℓ312βℓ2A⊥L2−A∥L2+A⊥R2−A∥R2sin2θKsin2θℓcos2Φ412βℓ2ReA0LA∥L∗+A0RA∥R∗sin2θKsin2θℓcosΦ52βℓReA0LA⊥L∗−A0RA⊥R∗sin2θKsinθℓcosΦ6s2βℓReA∥LA⊥L∗−A∥RA⊥R∗sin2θKcosθℓ72βℓImA0LA∥L∗−A0RA∥R∗sin2θKsinθℓsinΦ812βℓ2ImA0LA⊥L∗+A0RA⊥R∗sin2θKsin2θℓsinΦ9βℓ2ImA∥L∗A⊥L+A∥R∗A⊥Rsin2θKsin2θℓsin2ΦThe seven transverse amplitudesA0L,R, A∥L,R, A⊥L,R, and At of Bs⟶ϕℓ+ℓ− decay, in turn, can be parameterized by means of the relevant form factors [70, 73]:
(29)A⊥L,R=−Nℓ2NϕλC9eff∓C10Vq2mBs+mϕ+2m^bC7effT1q2,A∥L,R=Nℓ2NϕC9eff∓C10mBs+mϕA1q2+2m^bC7effmBs2−mϕ2T2q2,A0L,R=NℓNϕ2mϕq2C9eff∓C10mBs2−mϕ2−q2mBs+mϕA1q2−λmBs+mϕA2q2+2mbC7effmBs2+3mϕ2−q2T2q2−λmBs2−mϕ2T3q2,At=2NℓNϕλq2C10A0q2,where λ=mBs2−mϕ2−q22−4mϕ2q2, m^b=mb/q2, and the normalization constants are given as follows:
(30)Nℓ=iαemGF42πVtbVts∗,Nϕ=8λq23×256π3mBs31−4mℓ2q2Bϕ⟶K+K−.In numerical calculations, we takeBϕ⟶K+K−=0.492 from PDG 2018 [74]. It is easy to see that the narrow width approximation works well in the case of ϕ meson since Γϕ/mϕ=4.17×10−3~0.Analogous to Reference [70], to separate CP-conserving and CP-violating effects, one can define the CP-averaged angular coefficients Si and CP asymmetry angular coefficients Ai normalized by the differential (CP-averaged) decay rate to reduce the theoretical uncertainties:
(31)Si=Ii+I¯idΓ+Γ¯/dq2,Ai=Ii−I¯idΓ+Γ¯/dq2,where Ii and I¯i have been defined in Equations (26), (27), and (28) and Table 1, and the differential decay rate reads (analogously for Γ¯)
(32)dΓdq2=143I1c+6I1s−I2c−2I2s.Based on the definition ofSi, one can find the relation 3S1c+6S1s−S2c−2S2s=4. Consequently, all established observables can be expressed in terms of Si and Ai:
(1)
The CP asymmetry:(33)ACPq2=dΓ/dq2−dΓ¯/dq2dΓ/dq2+dΓ¯/dq2=143A1c+6A1s−A2c−2A2s.(2)
The lepton forward-backward (CP) asymmetry:(34)AFBq2=∫01−∫−10dcosθℓd2Γ−Γ¯/dq2dcosθℓdΓ+Γ¯/dq2=34S6s,AFBCPq2=∫01−∫−10dcosθℓd2Γ+Γ¯/dq2dcosθℓdΓ+Γ¯/dq2=34A6s.(3)
Theϕ polarization fractions:(35)FLq2=143S1c−S2c,FTq2=123S1s−S2s.In the massless limit, since the CP-averaged observableS1c,s,2c,s obey the relations S1s=3S2s and S1c=−S2c, the definitions of the polarization fractions can be simplified directly as follows:
(36)FLq2=S1c=−S2c,FTq2=43S1s=4S2s.(4)
The clean (no S-wave pollution) observablesP1,2,3 and P4,5,6′ in the natural basis can be defined in terms of the coefficients Si through the following relations [71, 75, 76]:(37)P1=S32S2s,P2=βℓS6s8S2s,P3=−S94S2s,(38)P′4=S4S1cS2s,P′5=βℓS52S1cS2s,P′6=−βℓS72S1cS2s,P′8=−S8S1cS2s,where βℓ=1−4mℓ2/q2.
(5)
In the massless limit of leptons, the optimized observablesPi′ [8] can be transformed as the following form:(39)P1=2S3FT,P2=S6s2FT,P3=−S9FT,P′4=2S4FL1−FL,P′5=S5FL1−FL,P′6=−S7FL1−FL,P′8=−2S8FL1−FL.One should know that our definitions of the CP-averaged angular coefficientsSi, the CP asymmetry angular coefficients Ai, and the clean observable P1,2,3 and P4,5,6′ differ from those adopted by the LHCb Collaboration. To be specific, the reasons are the following:
(1)
Our conventions for the angles to define theBs⟶ϕℓ+ℓ− kinematics are identical to the Reference [70] but different from the LHCb choices [13, 31]. The corresponding relations are the following:(40)θKLHCb=θK,θℓLHCb=π−θℓ,ΦLHCb=−Φ.Some angular coefficientsIi, Si, and Ai, consequently, will have different signs:
(41)I4,6,7,9LHCb=−I4,6s,7,9,S4,6,7,9LHCb=−S4,6s,7,9,A4,6,7,9LHCb=−A4,6s,7,9.Other remaining coefficientsIi (Si and Ai), however, have the same sign in both conventions.
(2)
Our definitions of the clean observablesP1,2,3 and P4,5,6,8′ in Equation (38) in terms of Si may be different from those defined and used by the LHCb Collaboration, for example, in Reference [13]. The resultant differences of the sign and normalization are of the following:(42)P1LHCb=P1,P2,3LHCb=−P2,3,P4,8′LHCb=−12P4,8′,P5,6′LHCb=P5,6′.For more details about the angular conventions of the angular observables of the semileptonic decaysBs⟶Vl+l−, one can see Reference [77].


## 4. Numerical Results and Discussions

In the numerical calculations, we use the following input parameters (here, masses and decay constants are in units of GeV) [42, 74]:
(43)ΛMS¯f=4=0.250,τBs0=1.509ps,mb=4.8,mW=80.38,mϕ=1.019,mBs=5.367,me=0.000511,mμ=0.105,mτ=1.777,fBs=0.23,fϕ=0.2314,fϕ⊥=0.201,a2ϕ∣=0.188,a2ϕ⊥=0.147.For the CKM matrix elements and angles, we adopt the following values as given in Reference [74]:
(44)Vtb=1.01925,Vus=0.22435,∣Vts∣=39.4±2.3×10−3,∣Vub∣=3.94±0.36×10−3,2βs=0.02131,γ=73.5−5.1+4.2°.


### 4.1. The Form Factors

For the considered semileptonic decays, the differential decay rates and other physical observables strongly rely on the value and the shape of the relevant form factorsVq2,A0,1,2q2 and T1,2,3q2 for Bs⟶ϕℓ+ℓ− decays. These form factors have been calculated in rather different theories or models [23, 36–38, 42]. Since the PQCD predictions for the considered form factors are valid only at the large hadronic recoil (low-q2) region, we usually calculate explicitly the values of the relevant form factors at the low-q2 region, say 0≤q2≤mτ2, and then make an extrapolation for all relevant form factors from the low-q2 towards the high-q2 region by using the pole model parametrization [78, 79] or other different methods.In References [80–82], we developed a new method: the so-called “PQCD+Lattice” approach. Here, we still use the PQCD approach to evaluate the form factors at the low-q2 region, but take those currently available lattice QCD results for the relevant form factors at the high-q2 region as the lattice QCD input to improve the extrapolation of the form factors up to qmax2. In References [81, 82], we used the Bourrely-Caprini-Lellouch (BCL) parametrization method [83, 84] instead of the traditional pole model parametrization.In Table2, we list the PQCD predictions for all seven relevant form factors V0, A0,1,20, and T1,2,30 for Bs⟶ϕ transition at the point q2=0. The dominant theoretical errors come from the uncertainties of the parameter ωBs=0.50±0.05 [42], the Gegenbauer moments a2ϕ∣=0.18±0.08 and a2ϕ⊥=0.14±0.07 [42, 46, 47, 51], and the decay constants fϕ=0.231±0.004GeV and fϕT=0.20±0.01GeV [42].Table 2
The PQCD predictions of the seven form factorsFBs⟶ϕi0 with the theoretical uncertainties from the variations of the parameters ωBs, a2ϕ∣,⊥, fϕ, andfϕT.
FBs⟶ϕi0PQCD predictionsV00.311ωBs−0.051+0.063a2ϕ⊥−0.007+0.008±0.004fϕ±0.004fϕTA000.262ωBs−0.038+0.046a2ϕ∣−0.008+0.009±0.002fϕ±0.008fϕTA100.247−0.041+0.051ωBsa2ϕ⊥−0.006+0.005±0.003fϕ±0.003fϕTA200.239−0.042+0.054ωBs±0.009a2ϕ∣,⊥±0.004fϕ±0.001fϕTT1,200.264−0.042+0.052ωBs±0.006a2ϕ⊥±0.003fϕ±0.005fϕTT300.196ωBs−0.033+0.041±0.008a2ϕ∣,⊥±0.003fϕ±0.001fϕTIn Table3, as a comparison, we also list the central values of the theoretical predictions for the form factors FBs⟶ϕi0 at q2=0 evaluated in the PQCD approach [42, 85] and in other different theories or models [17, 20, 21, 23, 86–91]. One can see that there exist always some differences between different authors, even among the authors using the same approach. Taking the calculations based on the LCSR method as an example, the authors of Reference [23] introduced the hadronic input parameters, Ball and Zwicky considered the one-loop radiative corrections [20], and Yilmaz included the radiative and higher twist corrections and SU(3) breaking effects [86].Table 3
The central values of the theoretical predictions for the form factors ofBs⟶ϕ transitions at q2=0 from different theories or models [17, 20, 21, 23, 42, 85–91].
This workV0A00A10A20T1,20T300.3110.2620.2470.2390.2640.196PQCD [42]0.250.300.19——PQCD [85]0.260.310.180.120.230.19LCSR [23]0.3870.3890.296—0.309—LCSR [20]0.4340.4740.3110.2340.3490.175LCSR [86]0.4330.3820.2960.2550.3480.254QCDSR [87]0.450.300.320.300.330.22RDA [88]0.440.420.340.310.380.26RQM [89]0.4060.3220.3200.3180.2750.133SCET [90]0.3290.2790.2320.2100.2760.170HQEFT [21]0.3390.2690.2710.2120.2990.191SQEH [91]0.2590.3110.194———CQM [17]0.310.280.270.270.270.18In Table4, for the Bs⟶ϕ transition form factors V,A0,1,2,T1,2,3, we quote directly the values of the lattice QCD results at two or three reference points of the high-q2 region, say q2=12,16GeV2 and qmax2=mBs−mϕ2≈18.9GeV2, as listed in Table XXXI of Reference [92]. In Reference [92], the authors defined the helicity form factors A12q2 and T23q2 from the ordinary form factors A1,2q2 and T2,3q2:
(45)A12q2=mBs+mϕ2mBs2−mϕ2−q2A1q2−λq2A2q216mBsmϕ2mBs+mϕ,(46)T23q2=mBs+mϕ8mBsmϕ2mBs2+3mϕ2−q2T2q2−λq2T3q2mBs2−mϕ2,where the kinematic variable λq2=t+−q2t−−q2 with t±=mBs±mϕ2. From above two equations and the numerical values of A1q2,T2q2,A12q2,T23q2 as given in Reference [92], we can find the corresponding lattice QCD results of A2q2 and T3q2 at the two points q2=12,16GeV2 by direct numerical calculations. When q2⟶qmax2=mBs−mϕ2, however, the parameter λqmax2 in Equation (45) is also approaching zero simultaneously; one therefore cannot determine A2qmax2 and T3qmax2 reliably from the values of A12qmax2 and T23qmax2 as given in Reference [92]. Consequently, A2qmax2 and T3qmax2 are absent in Table 4.Table 4
The values for the lattice QCD results of the relevantBs⟶ϕ transition form factors at two or three reference points [92].
q2Vq2A0q2A1q2A2q2T1q2T2q2T3q2120.77 (6)0.90 (6)0.44 (3)0.48 (4)0.69 (4)0.45 (3)0.46 (4)161.19 (7)1.32 (7)0.52 (3)0.54 (4)0.99 (5)0.53 (3)0.70 (5)18.91.74 (10)1.85 (10)0.62 (3)—1.36 (8)0.62 (3)—In this work, we will use both the PQCD factorization approach and the “PQCD+Lattice” approach to evaluate all relevant form factors over the whole range ofq2.
(1)
In the PQCD approach, we use the definitions and formulae to calculate the values of all relevant form factorsVq2, A0,1,2q2, and T1,2,3q2 at some points in the region of 0≤q2≤mτ2. We then make the extrapolation for these form factors to the large q2 region up to qmax2 by using the selected parametrization method directly(2)
In the “PQCD+Lattice” approach, we take the lattice QCD results of the form factors at the points ofq2=12,16,18.9GeV2 as the new input in the high-q2 region and then make a combined fit of the PQCD results in low-q2 region and the lattice QCD results in high-q2 region to determine the relevant parameters bki in the z-series expansion and then complete the extrapolation(3)
For both approaches, we always use the model-independentz-series parametrization, which is based on a rapidly converging series in the parameter z, as in References [23, 55] to make the extrapolation. The entire cut q2-plane will be mapped onto the unit disc zq2≤1 under the conformal transformation as [93](47)zq2=t+−q2−t+−t0t+−q2+t+−t0,where t±=mBs±mϕ2 and 0≤t0<t− is an auxiliary parameter which can be optimized to reduce the maximum value of zq2 in the physical range of the form factors and will be taken in the same way as in Reference [94]: t0=t+1−1−t−/t+. The form factors are finally parameterized in the BCL version of the z-series expansion [83].
(48)FBs⟶ϕiq2=FBs⟶ϕi01−q2/mi,pole21+∑k=1Nbkizq2,t0k−z0,t0k=FBs⟶ϕi01−q2/mi,pole21+b1izq2,t0−z0,t0+⋯.Since the termzq2,t02≤0.04 in the whole considered q2 region, the high-order N≥2 terms in Equation (48) should be very small in magnitude and therefore can be neglected. After the truncation at N=1, the coefficient b1i for the corresponding form factor FBs⟶ϕiq2 can be determined by fitting to the PQCD predictions at low-q2 region and the lattice QCD results in the high-q2 region. Taking the form factor A2q2 as an example, we calculate A2q2 first by employing the PQCD approach in the sixteen points in the low 0≤q2≤mτ2 region, take the lattice QCD results A212=0.48±0.04 and A216=0.54±0.04 as additional input, and finally make the fitting for the parameter b1A2 and find that
(49)b1A2=−1.820±0.148standarderror,with the goodness-of-fit R2=0.9978. For other form factors Fiq2, we find the results by following the same kind of procedure. The input values of the various s¯b-resonance mass mi,pole in Equation (48) can be found from Reference [74] and are collected in Table 5. For further discussions on the systematic uncertainties due to the dependence of truncation schemes and on the implementation of the strong unitary constraints, one can see References [94, 95].
(4)
In Figure1, we show the theoretical predictions of the form factors Vq2, A0,1,2q2, and T1,2,3q2 for Bs⟶ϕ transition based on the PQCD approach (red curves) and the “PQCD+Lattice” approach (blue curves) with the extrapolation from q2=0 to qmax2=mBs−mϕ2 by applying the z-series parameterizations. The shaded bands represent the total theoretical error obtained by adding in quadrature of the separate errors from the uncertainty of the parameter ωBs, a2ϕ∣,⊥, fϕ, and fϕ⊥. The black error bars in the low-q2 region correspond to the PQCD predictions of the corresponding form factors, while the error bars in the high-q2 region are the currently known lattice QCD results as collected in Table 3Table 5
The massesmi,pole in Equation (48) for the form factors FBs⟶ϕiq2 [23].
FBs⟶ϕiq2s¯bJPmi,poleGeVA0q2Bs0−5.366Vq2, T1q2Bs∗1−5.415A1,2q2, T2,3q2Bs11+5.829Figure 1
Theoretical predictions of the relevant form factors forBs⟶ϕ transition in the PQCD approach (red curves) and the “PQCD+Lattice” approach (blue curves). The red (blue) shaded band represents the theory uncertainties. The black error bars in the low-q2 region correspond to the PQCD predictions. The error bars in high-q2 region are the lattice QCD results.


### 4.2. Numerical Results

We now proceed to explore the phenomenological aspects of the cascade decaysBs⟶ϕ⟶K−K+ℓ+ℓ−, which allow us to define and compute a number of physical observables and compare them with those measured by experiments. We first compare our results for the branching ratios and angular observables with the experimental data reported by the LHCb Collaboration [31, 33]. As studied systematically in the last section, the physical observables accessible in the semileptonic decays Bs⟶ϕℓ+ℓ− are the CP-averaged differential branching fraction dB/dq2 [31, 33], the CP-averaged ϕ meson longitudinal polarization fraction FL, the forward-backward asymmetry AFB, the angular coefficients Si and Ai, and the optimized observables Pi and Pj′ [25]. The CP asymmetry angular coefficients A5,6,8,9 in the SM are induced by the weak phase from the CKM matrix. For the b⟶s transition, the CP asymmetries proportional to ImVubVus∗/VtbVts∗, which is of order 10−2 [19] as measured by the LHCb Collaboration (see Table 3 in Reference [31]), but the statistical uncertainties are still large. For these reasons, we will focus on the CP-averaged quantities when taking the binned observables into consideration.We begin with the branching ratios of the decaysBs⟶ϕℓ+ℓ−. From the differential decay rates as defined in Equation (32), it is straightforward to make the integration over the range of 4mℓ2≤q2≤mBs−mϕ2. In order to be consistent with the choices made by LHCb Collaboration in their data analysis, we here also cut off the regions of dilepton-mass squared around the charmonium resonances J/ψ1S and ψ2S; i.e., 8.0<q2<11.0GeV2 and 12.5<q2<15.0GeV2 for ℓ=e,μ,τ cases. We display the PQCD and “PQCD+Lattice” predictions for the differential branching ratios dB/dq2 in Figure 2 for the cases of l=μ,τ, including currently available LHCb results in six or eight bins of q2 [31, 33] indicated by the crosses for Bs⟶ϕμ+μ− decay. From Figure 2, one can see that both PQCD and “PQCD+Lattice” predictions for the differential branching ratios do agree well with the LHCb results within the still large errors. Since the theoretical prediction for the differential branching ratio of the electron mode is almost identical with the one of the muon, we do not draw the figure of dBBs⟶ϕe+e−/dq2 in Figure 2.Figure 2
Theoretical predictions for theq2-dependence of differential branching fraction dB/dq2 for the semileptonic decays Bs⟶ϕμ+μ− in the PQCD (red band) and “PQCD+Lattice” (blue band) approach, respectively. The crosses show the LHCb measurements in different bins as given in Reference [31] (orange ones) and in Reference [33] (black ones). The vertical grey blocks are the experimental veto regions.In Table6, we present the theoretical predictions of the total branching fractions for Bs⟶ϕℓ+ℓ− with ℓ=e,μ,τ obtained by the integration over the six q2 bins using the PQCD (the first row) and “PQCD+Lattice” approach (the second row), respectively. The major theoretical errors from different sources, such as the form factors (FFs) as listed in Table 2, the scale μ, and the CKM matrix element Vtb and Vts, are also listed. As in Reference [31], a correction factor fveto=1.52 is applied to account for the contribution in the veto q2 bins for ℓ=e,μ cases. As a comparison, we also show the LHCb measured value BBs⟶ϕμ+μ−=7.97−0.80+0.81×10−7 [31] and 8.14−0.47+0.47×10−7 [33] and the QCDSR predictions BBs⟶ϕℓ+ℓ− for all three decay modes [87]. For Bs⟶ϕμ+μ− decay, for instance, the theoretical predictions and the LHCb measurement [31, 33] (in unit of 10−7) are the following:
(50)BBs⟶ϕμ+μ−=7.07−2.34+3.43,inPQCD,6.76−1.25+1.52,inPQCD+Lattice,7.06−1.59+1.59,inQCDSR87,7.97−0.80+0.81,LHCb31,8.14−0.47+0.47,LHCb33.Table 6
Theoretical predictions for the total branching fractionsBBs⟶ϕℓ+ℓ− (in units of 10−7) in the PQCD (the first row) and “PQCD+Lattice” (the second row) approaches. As a comparison, we also list the LHCb measured value for muon channel corresponding to an integrated luminosity of 3fb−1 [31] and 9fb−1 [33] and the QCDSR predictions for all three channels [87].
BFsPQCD/“PQCD+Lattice”QCDSR [87]LHCbBBs⟶ϕe+e−8.55−2.69+4.02FFs±0.15μ±0.42Vtb±0.65Vts7.12±1.408.24−1.53+2.03FFs±0.14μ±0.41Vtb±0.63VtsBBs⟶ϕμ+μ−7.07−2.25+3.37FFs±0.12μ±0.38Vtb±0.53Vts7.06±1.597.97−0.80+0.81 [31]6.76−1.09+1.39FFs±0.11μ±0.33Vtb±0.52Vts8.14±0.47 [33]BBs⟶ϕτ+τ−0.81−0.27+0.42FFs±0.02μ±0.04Vtb±0.06Vts0.35±0.170.68−0.06+0.06FFs±0.02μ±0.03Vtb±0.05VtsFrom the numerical results in above equation and Table6, one can see that
(1)
The PQCD and “PQCD+Lattice” predictions for the branching ratioBBs⟶ϕℓ+ℓ− with ℓ=e,μ,τ do agree well with each other within the errors, while the “PQCD+ Lattice” predictions of BBs⟶ϕℓ+ℓ− have smaller errors than those of the PQCD predictions(2)
Both PQCD and “PQCD+Lattice” predictions ofBBs⟶ϕμ+μ− do agree well with currently available LHCb measured values [31, 33] within errors. For the electron and tau mode, however, we have to wait for the future experimental measurements(3)
For all three decay modes, our theoretical predictions of the branching ratios do agree well with the theoretical predictions obtained from the QCD sum rule [87]Since the large theoretical uncertainties of the branching ratios could be largely canceled in the ratio of the branching ratios ofBs⟶ϕℓ+ℓ− decays, one can define and check the physical observables Rϕeμ and Rϕμτ [5]. In the region q2<4mμ2, where only the e+e− modes are allowed, there is a large enhancement due to the 1/q2 scaling of the photon penguin contribution [96]. In order to remove the phase space effects in the ratio Rϕeμ and keep consistent with other analysis [5], we here also use the lower cut of 4mμ2 for both the electron and muon modes in the definition of the ratio Rϕeμ as in Reference [5]:
(51)Rϕeμ=∫4mμ2qmax2dq2dBBs⟶ϕμ+μ−/dq2∫4mμ2qmax2dq2dBBs⟶ϕe+e−/dq2=0.992±0.002,inPQCD,0.991±0.002,inPQCD+Lattice.For the case of the ratioRϕμτ, we have
(52)Rϕμτ=∫4mτ2qmax2dq2dBBs⟶ϕτ+τ−/dq2∫4mμ2qmax2dq2dBBs⟶ϕμ+μ−/dq2=0.115±0.004,inPQCD,0.100±0.009,inPQCD+Lattice,where the total error is the combination of the individual errors in quadrature. We suggest the LHCb and Belle-II to measure these two ratios.ForBs⟶ϕμ+μ− decay, we show our theoretical predictions for the q2-dependence of the longitudinal polarization FLq2, the CP-averaged angular coefficients S3,4,7q2, and the CP asymmetry angular coefficients A5,6,8,9q2 in Figure 3. As a comparison, the currently available LHCb measurements for these observables of Bs⟶ϕμ+μ− decay in the six q2 bins [31] are also shown by those crosses explicitly. One can see from Figure 3 that
(1)
For the longitudinal polarizationFLq2, although both PQCD and “PQCD+Lattice” predictions all agree well with the LHCb measurements in the six bins, our theoretical predictions in the region of the fourth and fifth bin are little larger than the measured ones(2)
For the CP-averaged angular coefficientsS3,4,7q2, the PQCD and “PQCD+Lattice” predictions agree very well with each other and are consistent with the LHCb results within the still large experimental errors. For the last two high-q2 bins, the LHCb results of S3 (S7) are a little larger (smaller) than our theoretical predictions(3)
For the CP asymmetry angular coefficientsA5,6,8,9q2, the PQCD and “PQCD+Lattice” predictions are very small: in the range of 10−4 to 10−2. For the LHCb measurements in the six bins, they are clearly consistent with our theoretical predictions due to still large experimental errorsFigure 3
Theoretical predictions for theq2-dependence of the observables FLq2, S3,4,7q2, and A5,6,8,9q2 for the decay Bs⟶ϕμ+μ− in the PQCD (red lines) and “PQCD+Lattice” (blue lines) approach. The crosses in each figure represent the LHCb measurements in six q2 bins [31]. The vertical grey blocks are the two experimental veto regions.In Figure4, we show our theoretical predictions for the q2-dependence of the forward-backward asymmetry AFBq2, the optimized observables P1,2,3q2, and P4,5,6,8′q2 for Bs⟶ϕμ+μ− decay. Unfortunately, there exist no any experimental measurements for these observables. We have to wait for future LHCb and Belle-II measurements. Analogous to Figure 3, the vertical grey blocks in Figure 4 also denote the two experimental veto regions of q2: 8.0<q2<11.0GeV2 and 12.5<q2<15.0GeV2.Figure 4
Theoretical predictions for theq2-dependence of the observables AFBq2, P1,2,3q2, and P4,5,6,8′q2 for Bs⟶ϕμ+μ− decay in the PQCD (red lines) and “PQCD+Lattice” (blue lines) approach. The vertical grey blocks are the two experimental veto regions.In Table7, we list the theoretical predictions for the values of the observables FLϕ, AFB, S3,4,7, A5,6,8,9, P1,2,3, and P4,5,6,8′, obtained after the integrations over the whole kinematic region of q2 for the semileptonic decays Bs⟶ϕℓ+ℓ− with ℓ=e,μ,τ in the PQCD (the first row) and “PQCD+Lattice” (the second row) approaches, respectively. Of course, the regions corresponding to resonance J/ψ1S and ψ2S, say 8.0<q2<11.0GeV2 and 12.5<q2<15.0GeV2 numerically, are also cut off here. The total errors are the combinations of the individual errors from the form factors, the renormalization scales, and the relevant CKM matrix elements. The above theoretical predictions should be tested in the near future LHCb and Belle-II experiments. For the considered Bs meson decays, one should consider the effects from the Bs-B¯s mixing [6]. The theoretical framework for examining the time-dependent decays with the inclusion of such mixing effects can be found in Reference [97]. The authors of Reference [97] proved that the mixing effects on the values of decay rates and CP-averaged observables are generally within a few percent and could be neglected.Table 7
Theoretical predictions for the observablesFLϕ, AFB, S3,4,7, A5,6,8,9, P1,2,3, and P4,5,6,8′ integrated over the whole kinematic region for Bs⟶ϕℓ+ℓ− decays in the PQCD (the first row) and “PQCD+Lattice” (the second row) approaches, respectively.
Obs.ℓ=eℓ=μℓ=τFLϕ0.383−0.004+0.0030.454−0.007+0.0060.396−0.003+0.0020.446−0.013+0.0120.533−0.001+0.0010.442−0.002+0.002−S30.120−0.004+0.0050.144−0.005+0.0050.080−0.001+0.0010.102−0.008+0.0090.124−0.006+0.0070.075−0.001+0.001S40.210−0.004+0.0040.258−0.004+0.0030.100−0.001+0.0010.201−0.013+0.0120.249−0.006+0.0070.098−0.001+0.001103S70.350−0.011+0.0100.371−0.013+0.0120.022−0.000+0.0010.392−0.024+0.0260.423−0.035+0.0400.030−0.000+0.001−103A50.489−0.009+0.0100.585−0.013+0.0120.040−0.000+0.0000.518−0.020+0.0200.623−0.046+0.0410.038−0.001+0.001−103A60.482−0.001+0.0010.577−0.004+0.0020.068−0.001+0.0000.429−0.021+0.0200.518−0.043+0.0380.058−0.001+0.001104A80.541−0.027+0.0300.241−0.020+0.0170.011−0.000+0.0000.550−0.095+0.0880.237−0.070+0.0600.014−0.000+0.000104A90.040−0.004+0.0020.054−0.004+0.0020.013−0.001+0.0010.041−0.008+0.0060.056−0.007+0.0040.015−0.001+0.001−AFB0.192−0.004+0.0030.233−0.004+0.0040.173−0.002+0.0020.152−0.008+0.0100.186−0.004+0.0050.151−0.003+0.002−P10.399−0.011+0.0140.555−0.012+0.0100.795−0.004+0.0070.381−0.039+0.0410.564−0.024+0.0260.817−0.005+0.007−P20.213−0.003+0.0020.299−0.001+0.0010.281−0.004+0.0030.189−0.013+0.0150.282−0.004+0.0050.268−0.004+0.004102P30.103−0.006+0.0030.142−0.007+0.0020.122−0.006+0.0030.118−0.012+0.0120.174−0.008+0.0070.158−0.003+0.002P′41.049−0.015+0.0131.111−0.012+0.0111.338−0.002+0.0021.033−0.037+0.0331.098−0.027+0.0241.345−0.003+0.002−P′50.482−0.004+0.0020.524−0.005+0.0030.427−0.006+0.0050.470−0.010+0.0080.515−0.003+0.0020.408−0.006+0.006−103P′60.875−0.030+0.0310.780−0.029+0.0280.069−0.001+0.0011.007−0.101+0.0890.910−0.089+0.0780.097−0.001+0.001−102P′80.643−0.003+0.0090.640−0.008+0.0040.287−0.002+0.0020.749−0.004+0.0040.748−0.006+0.0060.374−0.001+0.001


### 4.3. Theq2-Binned Predictions

ForBs⟶ϕμ+μ− decay mode, the LHCb Collaboration has reported their experimental measurements for many physical observables in several q2 bins [31, 33]. In order to compare our theoretical predictions with the LHCb results bin by bin, we make the same choices of the q2 bins as LHCb did, calculate, and show our theoretical predictions for the branching ratio BBs⟶ϕℓ+ℓ− and the asymmetry FLϕ with ℓ=μ,τ in Tables 8 and 9 and the observables S3,4,7 with ℓ=μ in Table 10. For observables S7 and A5,6,8,9, in fact, our theoretical predictions for their values are very small, say in the range of 10−3‐10−4 in magnitude, but still agree with the LHCb measurements in different bins [31] due to still large experimental errors. For the observables P3 and P6,8′, they are also very small in size: in the range of 10−3‐10−4 and there exist no corresponding data at present. For observables P1,2 and P4,5′, on the other hand, although there exist no experimental measurements for them at present, they are relatively large in size and may be measured in the near future LHCb and Belle-II experiments, so we calculate and list the theoretical predictions of these observables bin by bin for the cases of ℓ=μ,τin Tables 11 and 12. Very recently, LHCb reported some new measurements for the angular observables of Bs⟶ϕμ+μ− decay [34] in the q2 bins different from those in their previous work [31], which will be studied in our next work.Table 8
Theoretical predictions for theq2-binned observables BBs⟶ϕμ+μ− (in unit of 10−7) in the PQCD (the first low) and “PQCD+Lattice” (the second row) approach, respectively. For a comparison, we also list the new LHCb measurements [33].
q2 bin GeV2Bℓ=μLHCb [33]q2 bin GeV2Bℓ=μLHCb [33]0.10,0.980.25−0.07+0.100.68±0.061.10,2.500.19−0.05+0.080.44±0.050.25−0.07+0.100.21−0.05+0.072.50,4.000.23−0.07+0.090.35±0.044.00,6.000.41−0.12+0.180.62±0.060.27−0.06+0.090.50−0.11+0.146.00,8.000.61−0.19+0.280.63±0.0611.0,12.50.78−0.25+0.380.72±0.060.69−0.13+0.180.76−0.11+0.1415.00,17.001.26−0.42+0.641.05±0.0817.0,19.00.86−0.30+0.450.84±0.071.04−0.10+0.110.67−0.06+0.051.10,6.000.83−0.24+0.351.41±0.1015.0,19.02.12−0.71+1.091.85±0.130.98−0.22+0.301.70−0.16+0.17Table 9
Theoretical predictions for theq2-binned observables BBs⟶ϕℓ+ℓ− (in unit of 10−7) in the PQCD (the first low) and “PQCD+Lattice” (the second row) approach, respectively. For a comparison, we also list the new LHCb measurements [33].
q2binGeV2Bl=μLHCbBl=τFLϕl=μLHCbFLϕl=τ0.1,2.00.39−0.11+0.161.11±0.16 [31]—0.441−0.008+0.0070.20±0.09—0.40−0.11+0.16—0.472−0.012+0.011—2.0,5.00.48−0.14+0.200.77±0.14 [31]—0.738−0.009+0.0080.68±0.15—0.58−0.13+0.18—0.796−0.007+0.007—5.0,8.00.84−0.25+0.370.96±0.15 [31]—0.584−0.009+0.0080.54±0.10—0.96−0.19+0.25—0.682−0.008+0.010—11.0,12.50.78−0.25+0.380.71±0.12 [31]—0.433−0.005+0.0030.29±0.11—0.76−0.11+0.140.72±0.06 [33]—0.524−0.008+0.008—15.0,17.01.12−0.40+0.610.90±0.13 [31]0.45−0.14+0.220.368−0.001+0.0010.23±0.090.421−0.004+0.0030.99−0.10+0.111.05±0.08 [33]0.39−0.04+0.040.412−0.004+0.0040.479−0.003+0.00317.0,19.00.80−0.27+0.410.75±0.13 [31]0.36−0.12+0.180.346−0.001+0.0010.40±0.140.365−0.002+0.0010.67−0.05+0.050.82±0.07 [33]0.29−0.02+0.020.363−0.001+0.0010.391−0.001+0.0011.0,6.00.93−0.27+0.291.29±0.19 [31]—0.708−0.009+0.0070.63±0.09—1.10−0.25+0.34—0.777−0.006+0.008—15.0,19.01.99−0.71+1.021.62±0.20 [31]0.82−0.26+0.400.359−0.001+0.0020.29±0.070.396−0.003+0.0031.60−0.16+0.161.85±0.13 [33]0.68−0.06+0.060.394−0.003+0.0030.442−0.001+0.002Table 10
Theoretical predictions for theq2-binned observables S3,4,7 of the decays Bs⟶ϕμ+μ− in the PQCD (the first low) and “PQCD+Lattice” (the second row) approaches. For a comparison, we also list the LHCb measured values [31].
S3S4S7q2 bin GeV2Theor.LHCbTheor.LHCbTheor. (10−3)LHCb0.1,2.00.003−0.000+0.000−0.05±0.13−0.054−0.001+0.001−0.27±0.231.571−0.002+0.001−0.04±0.120.002−0.001+0.000−0.053−0.000+0.0001.551−0.006+0.0032.0,5.0−0.024−0.002+0.002−0.06±0.210.191−0.004+0.0040.47±0.371.065−0.003+0.0030.03±0.21−0.021−0.001+0.0010.177−0.003+0.0030.979−0.018+0.0175.0,8.0−0.057−0.003+0.003−0.10±0.250.270−0.002+0.0020.10±0.170.453−0.003+0.003−0.04±0.18−0.050−0.003+0.0040.259−0.003+0.0030.453−0.005+0.00411.0,12.5−0.124−0.003+0.004−0.19±0.210.296−0.001+0.0010.47±0.250.153−0.002+0.0020.00±0.16−0.115−0.001+0.0020.303−0.001+0.0010.185−0.001+0.00115.0,17.0−0.219−0.003+0.003−0.06±0.180.314−0.001+0.0010.03±0.150.052−0.001+0.001−0.12±0.15−0.213−0.001+0.0010.323−0.001+0.0010.071−0.001+0.00117.0,19.0−0.283−0.001+0.001−0.07±0.250.325−0.001+0.0010.39±0.300.019−0.001+0.001−0.20±0.26−0.281−0.001+0.0010.329−0.001+0.0010.027−0.001+0.0011.0,6.0−0.026−0.002+0.002−0.02±0.130.180−0.004+0.0040.19±0.141.063−0.001+0.0010.03±0.14−0.023−0.001+0.0010.169−0.001+0.0010.985−0.022+0.02015.0,19.0−0.245−0.002+0.002−0.09±0.120.318−0.001+0.0010.14±0.110.038−0.002+0.002−0.13±0.11−0.239−0.001+0.0010.325−0.001+0.0010.054−0.001+0.001Table 11
Theoretical predictions for theq2-binned observables AFBμ and P1,2 of the decays Bs⟶ϕℓ+ℓ− with ℓ=μ,τ in the PQCD (the first low) and “PQCD+Lattice” (the second row) approaches.
q2 bin GeV2AFBμAFBτP1ℓ=μP1ℓ=τP2ℓ=μP2ℓ=τ0.1,2.00.131−0.003+0.003—0.015−0.001+0.001—0.206−0.001+0.001—0.122−0.002+0.003—0.013−0.001+0.001—0.204−0.001+0.001—2.0,5.0−0.047−0.003+0.004—−0.197−0.007+0.007—−0.128−0.003+0.003—−0.038−0.002+0.001—−0.225−0.006+0.005—−0.132−0.002+0.002—5.0,8.0−0.265−0.005+0.006—−0.280−0.008+0.009—−0.431−0.001+0.001—−0.200−0.006+0.006—−0.323−0.004+0.003—−0.427−0.001+0.001—11.0,12.5−0.365−0.001+0.001—−0.440−0.008+0.010—−0.431−0.003+0.002—−0.298−0.005+0.005—−0.489−0.003+0.002—−0.420−0.001+0.001—15.0,17.0−0.326−0.001+0.001−0.188−0.001+0.001−0.697−0.006+0.009−0.707−0.005+0.009−0.345−0.004+0.003−0.341−0.004+0.003−0.290−0.005+0.005−0.160−0.003+0.003−0.730−0.005+0.007−0.738−0.006+0.006−0.330−0.003+0.003−0.326−0.004+0.00317.0,19.0−0.226−0.003+0.002−0.153−0.002+0.001−0.869−0.003+0.005−0.875−0.003+0.005−0.231−0.003+0.002−0.225−0.004+0.003−0.208−0.003+0.004−0.137−0.002+0.003−0.884−0.004+0.004−0.890−0.003+0.005−0.219−0.004+0.003−0.213−0.004+0.004Table 12
Theoretical predictions for theq2-binned optimized observables P′4 and P′5 of the decays Bs⟶ϕℓ+ℓ− with ℓ=e,μ,τ in the PQCD (the first low) and “PQCD+Lattice” (the second row) approach.
q2 bin GeV2P′4ℓ=eP′4ℓ=μP′4ℓ=τP′5ℓ=eP′5ℓ=μP′5ℓ=τ0.1,2.0−0.322−0.005+0.005−0.289−0.005+0.005—0.484−0.002+0.0010.459−0.002+0.001—−0.318−0.001+0.001−0.285−0.001+0.001—0.475−0.001+0.0020.449−0.001+0.001—2.0,5.00.903−0.007+0.0070.903−0.007+0.007—−0.606−0.002+0.001−0.607−0.002+0.001—0.916−0.001+0.0010.916−0.001+0.001—−0.608−0.003+0.004−0.608−0.004+0.004—5.0,8.01.110−0.005+0.0031.110−0.005+0.004—−0.816−0.006+0.005−0.816−0.006+0.005—1.127−0.001+0.0021.127−0.001+0.002—−0.796−0.001+0.003−0.796−0.002+0.002—11.0,12.51.198−0.005+0.0021.198−0.005+0.002—−0.727−0.007+0.005−0.727−0.007+0.005—1.217−0.001+0.0011.217−0.001+0.001—−0.698−0.001+0.001−0.698−0.001+0.002—15.0,17.01.302−0.004+0.0021.302−0.004+0.0021.306−0.004+0.002−0.533−0.007+0.005−0.533−0.007+0.005−0.525−0.008+0.0061.314−0.003+0.0021.314−0.003+0.0021.317−0.002+0.003−0.506−0.006+0.005−0.506−0.006+0.005−0.498−0.006+0.00617.0,19.01.366−0.002+0.0011.366−0.002+0.0011.369−0.002+0.001−0.341−0.006+0.004−0.341−0.006+0.004−0.331−0.006+0.0041.371−0.001+0.0011.371−0.001+0.0011.374−0.002+0.001−0.323−0.006+0.005−0.323−0.006+0.005−0.314−0.006+0.006The definitions of theq2-binned observables are the following:
(53)Bq12,q22=∫q12q22dq2dBBs⟶ϕℓ+ℓ−dq2,FLϕq12,q22=∫q12q22dq23I1c+I¯1c−I2c+I¯2c4∫q12q22dq2dΓ+Γ¯/dq2,AFBq12,q22=3∫q12q22dq2I6s+I¯6s4∫q12q22dq2dΓ+Γ¯/dq2,S3,4,7q12,q22=∫q12q22dq2I3,4,7+I¯3,4,7∫q12q22dq2dΓ+Γ¯/dq2,P1q12,q22=∫q12q22dq2S32∫q12q22dq2S2s,P2q12,q22=∫q12q22dq2βℓS6s8∫q12q22dq2S2s,P4′q12,q22=∫q12q22dq2S4−∫q12q22dq2S2cS2s,P5′q12,q22=∫q12q22dq2βℓS52−∫q12q22dq2S2cS2s.From the numerical values as shown in Figure3 and in Tables 8–10, we find the following points about the relevant physical observables of the considered Bs⟶ϕμ+μ− decays in bins:
(1)
ForBs⟶ϕμ+μ− decay, besides the good consistency between the theory and the LHCb data for the integrated total branching ratio BBs⟶ϕμ+μ− as listed in Equation (50), the PQCD and “PQCD+Lattice” predictions for BBs⟶ϕμ+μ− in most bins do agree well with the measured ones within 2σ errors. For the first low-q2 bin 0.1<q2<2GeV2, however, the central value of the LHCb result 1.11±0.16 is larger than the theoretical ones by roughly a factor of three. The LHCb results of BBs⟶ϕμ+μ− in different bins of q2 as listed in the third column of Table 9 are obtained from the results as given in References [31, 33] by multiplying the LHCb measured values of differential decay rate dBBs⟶ϕμ+μ−/dq2 with the width of the corresponding bin q22−q12. The theoretical errors of our theoretical predictions of the branching ratios in bins are still relatively large, while the differences between the PQCD and “PQCD+Lattice” predictions for BBs⟶ϕℓ+ℓ− with ℓ=μ,τ are small(2)
In the first low-q2 bin 0.1<q2<2GeV2, both the PQCD and “PQCD+Lattice” predictions for FLϕℓ=μ are larger than the LHCb measured results FLϕℓ=μLHCb=0.20−0.09+0.08±0.02 [31]. For other bins, both PQCD and “PQCD+Lattice” predictions of FLϕ for muon mode do agree very well with currently available LHCb measured values [31] within 2σ errors. It is worth of remaining that our theoretical predictions of FLϕ have a little error of ~2% due to the strong cancellation of the theoretical errors in the ratios. The theoretical predictions for BBs⟶ϕτ+τ− and FLϕℓ=τ in different bins of q2 as listed in Table 9 will be tested by future experimental measurements(3)
For the observablesS3,4,7, as listed in Table 10, the PQCD and “PQCD+Lattice” predictions for their values in all bins are in the range of 10−3‐10−1 and show a good agreement with the LHCb measured values [31]. The errors of the theoretical predictions are also very small, ~2% in magnitude, because of their nature of the ratios. In all bins, the LHCb measured values of S3,4,7 are still consistent with zero due to their still large errors, which is a clear feature as can be seen easily from the numerical values in Table 10 and the crosses in Figure 3In Tables11 and 12, we show the PQCD and “PQCD+Lattice” predictions for the physical observables AFBμ,τ,P1,2μ,τ and P4,5′ℓ=e,μ,τ in six bins. These physical observables could be tested in the near future LHCb and Belle-II experiments.


## 4.1. The Form Factors

For the considered semileptonic decays, the differential decay rates and other physical observables strongly rely on the value and the shape of the relevant form factorsVq2,A0,1,2q2 and T1,2,3q2 for Bs⟶ϕℓ+ℓ− decays. These form factors have been calculated in rather different theories or models [23, 36–38, 42]. Since the PQCD predictions for the considered form factors are valid only at the large hadronic recoil (low-q2) region, we usually calculate explicitly the values of the relevant form factors at the low-q2 region, say 0≤q2≤mτ2, and then make an extrapolation for all relevant form factors from the low-q2 towards the high-q2 region by using the pole model parametrization [78, 79] or other different methods.In References [80–82], we developed a new method: the so-called “PQCD+Lattice” approach. Here, we still use the PQCD approach to evaluate the form factors at the low-q2 region, but take those currently available lattice QCD results for the relevant form factors at the high-q2 region as the lattice QCD input to improve the extrapolation of the form factors up to qmax2. In References [81, 82], we used the Bourrely-Caprini-Lellouch (BCL) parametrization method [83, 84] instead of the traditional pole model parametrization.In Table2, we list the PQCD predictions for all seven relevant form factors V0, A0,1,20, and T1,2,30 for Bs⟶ϕ transition at the point q2=0. The dominant theoretical errors come from the uncertainties of the parameter ωBs=0.50±0.05 [42], the Gegenbauer moments a2ϕ∣=0.18±0.08 and a2ϕ⊥=0.14±0.07 [42, 46, 47, 51], and the decay constants fϕ=0.231±0.004GeV and fϕT=0.20±0.01GeV [42].Table 2
The PQCD predictions of the seven form factorsFBs⟶ϕi0 with the theoretical uncertainties from the variations of the parameters ωBs, a2ϕ∣,⊥, fϕ, andfϕT.
FBs⟶ϕi0PQCD predictionsV00.311ωBs−0.051+0.063a2ϕ⊥−0.007+0.008±0.004fϕ±0.004fϕTA000.262ωBs−0.038+0.046a2ϕ∣−0.008+0.009±0.002fϕ±0.008fϕTA100.247−0.041+0.051ωBsa2ϕ⊥−0.006+0.005±0.003fϕ±0.003fϕTA200.239−0.042+0.054ωBs±0.009a2ϕ∣,⊥±0.004fϕ±0.001fϕTT1,200.264−0.042+0.052ωBs±0.006a2ϕ⊥±0.003fϕ±0.005fϕTT300.196ωBs−0.033+0.041±0.008a2ϕ∣,⊥±0.003fϕ±0.001fϕTIn Table3, as a comparison, we also list the central values of the theoretical predictions for the form factors FBs⟶ϕi0 at q2=0 evaluated in the PQCD approach [42, 85] and in other different theories or models [17, 20, 21, 23, 86–91]. One can see that there exist always some differences between different authors, even among the authors using the same approach. Taking the calculations based on the LCSR method as an example, the authors of Reference [23] introduced the hadronic input parameters, Ball and Zwicky considered the one-loop radiative corrections [20], and Yilmaz included the radiative and higher twist corrections and SU(3) breaking effects [86].Table 3
The central values of the theoretical predictions for the form factors ofBs⟶ϕ transitions at q2=0 from different theories or models [17, 20, 21, 23, 42, 85–91].
This workV0A00A10A20T1,20T300.3110.2620.2470.2390.2640.196PQCD [42]0.250.300.19——PQCD [85]0.260.310.180.120.230.19LCSR [23]0.3870.3890.296—0.309—LCSR [20]0.4340.4740.3110.2340.3490.175LCSR [86]0.4330.3820.2960.2550.3480.254QCDSR [87]0.450.300.320.300.330.22RDA [88]0.440.420.340.310.380.26RQM [89]0.4060.3220.3200.3180.2750.133SCET [90]0.3290.2790.2320.2100.2760.170HQEFT [21]0.3390.2690.2710.2120.2990.191SQEH [91]0.2590.3110.194———CQM [17]0.310.280.270.270.270.18In Table4, for the Bs⟶ϕ transition form factors V,A0,1,2,T1,2,3, we quote directly the values of the lattice QCD results at two or three reference points of the high-q2 region, say q2=12,16GeV2 and qmax2=mBs−mϕ2≈18.9GeV2, as listed in Table XXXI of Reference [92]. In Reference [92], the authors defined the helicity form factors A12q2 and T23q2 from the ordinary form factors A1,2q2 and T2,3q2:
(45)A12q2=mBs+mϕ2mBs2−mϕ2−q2A1q2−λq2A2q216mBsmϕ2mBs+mϕ,(46)T23q2=mBs+mϕ8mBsmϕ2mBs2+3mϕ2−q2T2q2−λq2T3q2mBs2−mϕ2,where the kinematic variable λq2=t+−q2t−−q2 with t±=mBs±mϕ2. From above two equations and the numerical values of A1q2,T2q2,A12q2,T23q2 as given in Reference [92], we can find the corresponding lattice QCD results of A2q2 and T3q2 at the two points q2=12,16GeV2 by direct numerical calculations. When q2⟶qmax2=mBs−mϕ2, however, the parameter λqmax2 in Equation (45) is also approaching zero simultaneously; one therefore cannot determine A2qmax2 and T3qmax2 reliably from the values of A12qmax2 and T23qmax2 as given in Reference [92]. Consequently, A2qmax2 and T3qmax2 are absent in Table 4.Table 4
The values for the lattice QCD results of the relevantBs⟶ϕ transition form factors at two or three reference points [92].
q2Vq2A0q2A1q2A2q2T1q2T2q2T3q2120.77 (6)0.90 (6)0.44 (3)0.48 (4)0.69 (4)0.45 (3)0.46 (4)161.19 (7)1.32 (7)0.52 (3)0.54 (4)0.99 (5)0.53 (3)0.70 (5)18.91.74 (10)1.85 (10)0.62 (3)—1.36 (8)0.62 (3)—In this work, we will use both the PQCD factorization approach and the “PQCD+Lattice” approach to evaluate all relevant form factors over the whole range ofq2.
(1)
In the PQCD approach, we use the definitions and formulae to calculate the values of all relevant form factorsVq2, A0,1,2q2, and T1,2,3q2 at some points in the region of 0≤q2≤mτ2. We then make the extrapolation for these form factors to the large q2 region up to qmax2 by using the selected parametrization method directly(2)
In the “PQCD+Lattice” approach, we take the lattice QCD results of the form factors at the points ofq2=12,16,18.9GeV2 as the new input in the high-q2 region and then make a combined fit of the PQCD results in low-q2 region and the lattice QCD results in high-q2 region to determine the relevant parameters bki in the z-series expansion and then complete the extrapolation(3)
For both approaches, we always use the model-independentz-series parametrization, which is based on a rapidly converging series in the parameter z, as in References [23, 55] to make the extrapolation. The entire cut q2-plane will be mapped onto the unit disc zq2≤1 under the conformal transformation as [93](47)zq2=t+−q2−t+−t0t+−q2+t+−t0,where t±=mBs±mϕ2 and 0≤t0<t− is an auxiliary parameter which can be optimized to reduce the maximum value of zq2 in the physical range of the form factors and will be taken in the same way as in Reference [94]: t0=t+1−1−t−/t+. The form factors are finally parameterized in the BCL version of the z-series expansion [83].
(48)FBs⟶ϕiq2=FBs⟶ϕi01−q2/mi,pole21+∑k=1Nbkizq2,t0k−z0,t0k=FBs⟶ϕi01−q2/mi,pole21+b1izq2,t0−z0,t0+⋯.Since the termzq2,t02≤0.04 in the whole considered q2 region, the high-order N≥2 terms in Equation (48) should be very small in magnitude and therefore can be neglected. After the truncation at N=1, the coefficient b1i for the corresponding form factor FBs⟶ϕiq2 can be determined by fitting to the PQCD predictions at low-q2 region and the lattice QCD results in the high-q2 region. Taking the form factor A2q2 as an example, we calculate A2q2 first by employing the PQCD approach in the sixteen points in the low 0≤q2≤mτ2 region, take the lattice QCD results A212=0.48±0.04 and A216=0.54±0.04 as additional input, and finally make the fitting for the parameter b1A2 and find that
(49)b1A2=−1.820±0.148standarderror,with the goodness-of-fit R2=0.9978. For other form factors Fiq2, we find the results by following the same kind of procedure. The input values of the various s¯b-resonance mass mi,pole in Equation (48) can be found from Reference [74] and are collected in Table 5. For further discussions on the systematic uncertainties due to the dependence of truncation schemes and on the implementation of the strong unitary constraints, one can see References [94, 95].
(4)
In Figure1, we show the theoretical predictions of the form factors Vq2, A0,1,2q2, and T1,2,3q2 for Bs⟶ϕ transition based on the PQCD approach (red curves) and the “PQCD+Lattice” approach (blue curves) with the extrapolation from q2=0 to qmax2=mBs−mϕ2 by applying the z-series parameterizations. The shaded bands represent the total theoretical error obtained by adding in quadrature of the separate errors from the uncertainty of the parameter ωBs, a2ϕ∣,⊥, fϕ, and fϕ⊥. The black error bars in the low-q2 region correspond to the PQCD predictions of the corresponding form factors, while the error bars in the high-q2 region are the currently known lattice QCD results as collected in Table 3Table 5
The massesmi,pole in Equation (48) for the form factors FBs⟶ϕiq2 [23].
FBs⟶ϕiq2s¯bJPmi,poleGeVA0q2Bs0−5.366Vq2, T1q2Bs∗1−5.415A1,2q2, T2,3q2Bs11+5.829Figure 1
Theoretical predictions of the relevant form factors forBs⟶ϕ transition in the PQCD approach (red curves) and the “PQCD+Lattice” approach (blue curves). The red (blue) shaded band represents the theory uncertainties. The black error bars in the low-q2 region correspond to the PQCD predictions. The error bars in high-q2 region are the lattice QCD results.


## 4.2. Numerical Results

We now proceed to explore the phenomenological aspects of the cascade decaysBs⟶ϕ⟶K−K+ℓ+ℓ−, which allow us to define and compute a number of physical observables and compare them with those measured by experiments. We first compare our results for the branching ratios and angular observables with the experimental data reported by the LHCb Collaboration [31, 33]. As studied systematically in the last section, the physical observables accessible in the semileptonic decays Bs⟶ϕℓ+ℓ− are the CP-averaged differential branching fraction dB/dq2 [31, 33], the CP-averaged ϕ meson longitudinal polarization fraction FL, the forward-backward asymmetry AFB, the angular coefficients Si and Ai, and the optimized observables Pi and Pj′ [25]. The CP asymmetry angular coefficients A5,6,8,9 in the SM are induced by the weak phase from the CKM matrix. For the b⟶s transition, the CP asymmetries proportional to ImVubVus∗/VtbVts∗, which is of order 10−2 [19] as measured by the LHCb Collaboration (see Table 3 in Reference [31]), but the statistical uncertainties are still large. For these reasons, we will focus on the CP-averaged quantities when taking the binned observables into consideration.We begin with the branching ratios of the decaysBs⟶ϕℓ+ℓ−. From the differential decay rates as defined in Equation (32), it is straightforward to make the integration over the range of 4mℓ2≤q2≤mBs−mϕ2. In order to be consistent with the choices made by LHCb Collaboration in their data analysis, we here also cut off the regions of dilepton-mass squared around the charmonium resonances J/ψ1S and ψ2S; i.e., 8.0<q2<11.0GeV2 and 12.5<q2<15.0GeV2 for ℓ=e,μ,τ cases. We display the PQCD and “PQCD+Lattice” predictions for the differential branching ratios dB/dq2 in Figure 2 for the cases of l=μ,τ, including currently available LHCb results in six or eight bins of q2 [31, 33] indicated by the crosses for Bs⟶ϕμ+μ− decay. From Figure 2, one can see that both PQCD and “PQCD+Lattice” predictions for the differential branching ratios do agree well with the LHCb results within the still large errors. Since the theoretical prediction for the differential branching ratio of the electron mode is almost identical with the one of the muon, we do not draw the figure of dBBs⟶ϕe+e−/dq2 in Figure 2.Figure 2
Theoretical predictions for theq2-dependence of differential branching fraction dB/dq2 for the semileptonic decays Bs⟶ϕμ+μ− in the PQCD (red band) and “PQCD+Lattice” (blue band) approach, respectively. The crosses show the LHCb measurements in different bins as given in Reference [31] (orange ones) and in Reference [33] (black ones). The vertical grey blocks are the experimental veto regions.In Table6, we present the theoretical predictions of the total branching fractions for Bs⟶ϕℓ+ℓ− with ℓ=e,μ,τ obtained by the integration over the six q2 bins using the PQCD (the first row) and “PQCD+Lattice” approach (the second row), respectively. The major theoretical errors from different sources, such as the form factors (FFs) as listed in Table 2, the scale μ, and the CKM matrix element Vtb and Vts, are also listed. As in Reference [31], a correction factor fveto=1.52 is applied to account for the contribution in the veto q2 bins for ℓ=e,μ cases. As a comparison, we also show the LHCb measured value BBs⟶ϕμ+μ−=7.97−0.80+0.81×10−7 [31] and 8.14−0.47+0.47×10−7 [33] and the QCDSR predictions BBs⟶ϕℓ+ℓ− for all three decay modes [87]. For Bs⟶ϕμ+μ− decay, for instance, the theoretical predictions and the LHCb measurement [31, 33] (in unit of 10−7) are the following:
(50)BBs⟶ϕμ+μ−=7.07−2.34+3.43,inPQCD,6.76−1.25+1.52,inPQCD+Lattice,7.06−1.59+1.59,inQCDSR87,7.97−0.80+0.81,LHCb31,8.14−0.47+0.47,LHCb33.Table 6
Theoretical predictions for the total branching fractionsBBs⟶ϕℓ+ℓ− (in units of 10−7) in the PQCD (the first row) and “PQCD+Lattice” (the second row) approaches. As a comparison, we also list the LHCb measured value for muon channel corresponding to an integrated luminosity of 3fb−1 [31] and 9fb−1 [33] and the QCDSR predictions for all three channels [87].
BFsPQCD/“PQCD+Lattice”QCDSR [87]LHCbBBs⟶ϕe+e−8.55−2.69+4.02FFs±0.15μ±0.42Vtb±0.65Vts7.12±1.408.24−1.53+2.03FFs±0.14μ±0.41Vtb±0.63VtsBBs⟶ϕμ+μ−7.07−2.25+3.37FFs±0.12μ±0.38Vtb±0.53Vts7.06±1.597.97−0.80+0.81 [31]6.76−1.09+1.39FFs±0.11μ±0.33Vtb±0.52Vts8.14±0.47 [33]BBs⟶ϕτ+τ−0.81−0.27+0.42FFs±0.02μ±0.04Vtb±0.06Vts0.35±0.170.68−0.06+0.06FFs±0.02μ±0.03Vtb±0.05VtsFrom the numerical results in above equation and Table6, one can see that
(1)
The PQCD and “PQCD+Lattice” predictions for the branching ratioBBs⟶ϕℓ+ℓ− with ℓ=e,μ,τ do agree well with each other within the errors, while the “PQCD+ Lattice” predictions of BBs⟶ϕℓ+ℓ− have smaller errors than those of the PQCD predictions(2)
Both PQCD and “PQCD+Lattice” predictions ofBBs⟶ϕμ+μ− do agree well with currently available LHCb measured values [31, 33] within errors. For the electron and tau mode, however, we have to wait for the future experimental measurements(3)
For all three decay modes, our theoretical predictions of the branching ratios do agree well with the theoretical predictions obtained from the QCD sum rule [87]Since the large theoretical uncertainties of the branching ratios could be largely canceled in the ratio of the branching ratios ofBs⟶ϕℓ+ℓ− decays, one can define and check the physical observables Rϕeμ and Rϕμτ [5]. In the region q2<4mμ2, where only the e+e− modes are allowed, there is a large enhancement due to the 1/q2 scaling of the photon penguin contribution [96]. In order to remove the phase space effects in the ratio Rϕeμ and keep consistent with other analysis [5], we here also use the lower cut of 4mμ2 for both the electron and muon modes in the definition of the ratio Rϕeμ as in Reference [5]:
(51)Rϕeμ=∫4mμ2qmax2dq2dBBs⟶ϕμ+μ−/dq2∫4mμ2qmax2dq2dBBs⟶ϕe+e−/dq2=0.992±0.002,inPQCD,0.991±0.002,inPQCD+Lattice.For the case of the ratioRϕμτ, we have
(52)Rϕμτ=∫4mτ2qmax2dq2dBBs⟶ϕτ+τ−/dq2∫4mμ2qmax2dq2dBBs⟶ϕμ+μ−/dq2=0.115±0.004,inPQCD,0.100±0.009,inPQCD+Lattice,where the total error is the combination of the individual errors in quadrature. We suggest the LHCb and Belle-II to measure these two ratios.ForBs⟶ϕμ+μ− decay, we show our theoretical predictions for the q2-dependence of the longitudinal polarization FLq2, the CP-averaged angular coefficients S3,4,7q2, and the CP asymmetry angular coefficients A5,6,8,9q2 in Figure 3. As a comparison, the currently available LHCb measurements for these observables of Bs⟶ϕμ+μ− decay in the six q2 bins [31] are also shown by those crosses explicitly. One can see from Figure 3 that
(1)
For the longitudinal polarizationFLq2, although both PQCD and “PQCD+Lattice” predictions all agree well with the LHCb measurements in the six bins, our theoretical predictions in the region of the fourth and fifth bin are little larger than the measured ones(2)
For the CP-averaged angular coefficientsS3,4,7q2, the PQCD and “PQCD+Lattice” predictions agree very well with each other and are consistent with the LHCb results within the still large experimental errors. For the last two high-q2 bins, the LHCb results of S3 (S7) are a little larger (smaller) than our theoretical predictions(3)
For the CP asymmetry angular coefficientsA5,6,8,9q2, the PQCD and “PQCD+Lattice” predictions are very small: in the range of 10−4 to 10−2. For the LHCb measurements in the six bins, they are clearly consistent with our theoretical predictions due to still large experimental errorsFigure 3
Theoretical predictions for theq2-dependence of the observables FLq2, S3,4,7q2, and A5,6,8,9q2 for the decay Bs⟶ϕμ+μ− in the PQCD (red lines) and “PQCD+Lattice” (blue lines) approach. The crosses in each figure represent the LHCb measurements in six q2 bins [31]. The vertical grey blocks are the two experimental veto regions.In Figure4, we show our theoretical predictions for the q2-dependence of the forward-backward asymmetry AFBq2, the optimized observables P1,2,3q2, and P4,5,6,8′q2 for Bs⟶ϕμ+μ− decay. Unfortunately, there exist no any experimental measurements for these observables. We have to wait for future LHCb and Belle-II measurements. Analogous to Figure 3, the vertical grey blocks in Figure 4 also denote the two experimental veto regions of q2: 8.0<q2<11.0GeV2 and 12.5<q2<15.0GeV2.Figure 4
Theoretical predictions for theq2-dependence of the observables AFBq2, P1,2,3q2, and P4,5,6,8′q2 for Bs⟶ϕμ+μ− decay in the PQCD (red lines) and “PQCD+Lattice” (blue lines) approach. The vertical grey blocks are the two experimental veto regions.In Table7, we list the theoretical predictions for the values of the observables FLϕ, AFB, S3,4,7, A5,6,8,9, P1,2,3, and P4,5,6,8′, obtained after the integrations over the whole kinematic region of q2 for the semileptonic decays Bs⟶ϕℓ+ℓ− with ℓ=e,μ,τ in the PQCD (the first row) and “PQCD+Lattice” (the second row) approaches, respectively. Of course, the regions corresponding to resonance J/ψ1S and ψ2S, say 8.0<q2<11.0GeV2 and 12.5<q2<15.0GeV2 numerically, are also cut off here. The total errors are the combinations of the individual errors from the form factors, the renormalization scales, and the relevant CKM matrix elements. The above theoretical predictions should be tested in the near future LHCb and Belle-II experiments. For the considered Bs meson decays, one should consider the effects from the Bs-B¯s mixing [6]. The theoretical framework for examining the time-dependent decays with the inclusion of such mixing effects can be found in Reference [97]. The authors of Reference [97] proved that the mixing effects on the values of decay rates and CP-averaged observables are generally within a few percent and could be neglected.Table 7
Theoretical predictions for the observablesFLϕ, AFB, S3,4,7, A5,6,8,9, P1,2,3, and P4,5,6,8′ integrated over the whole kinematic region for Bs⟶ϕℓ+ℓ− decays in the PQCD (the first row) and “PQCD+Lattice” (the second row) approaches, respectively.
Obs.ℓ=eℓ=μℓ=τFLϕ0.383−0.004+0.0030.454−0.007+0.0060.396−0.003+0.0020.446−0.013+0.0120.533−0.001+0.0010.442−0.002+0.002−S30.120−0.004+0.0050.144−0.005+0.0050.080−0.001+0.0010.102−0.008+0.0090.124−0.006+0.0070.075−0.001+0.001S40.210−0.004+0.0040.258−0.004+0.0030.100−0.001+0.0010.201−0.013+0.0120.249−0.006+0.0070.098−0.001+0.001103S70.350−0.011+0.0100.371−0.013+0.0120.022−0.000+0.0010.392−0.024+0.0260.423−0.035+0.0400.030−0.000+0.001−103A50.489−0.009+0.0100.585−0.013+0.0120.040−0.000+0.0000.518−0.020+0.0200.623−0.046+0.0410.038−0.001+0.001−103A60.482−0.001+0.0010.577−0.004+0.0020.068−0.001+0.0000.429−0.021+0.0200.518−0.043+0.0380.058−0.001+0.001104A80.541−0.027+0.0300.241−0.020+0.0170.011−0.000+0.0000.550−0.095+0.0880.237−0.070+0.0600.014−0.000+0.000104A90.040−0.004+0.0020.054−0.004+0.0020.013−0.001+0.0010.041−0.008+0.0060.056−0.007+0.0040.015−0.001+0.001−AFB0.192−0.004+0.0030.233−0.004+0.0040.173−0.002+0.0020.152−0.008+0.0100.186−0.004+0.0050.151−0.003+0.002−P10.399−0.011+0.0140.555−0.012+0.0100.795−0.004+0.0070.381−0.039+0.0410.564−0.024+0.0260.817−0.005+0.007−P20.213−0.003+0.0020.299−0.001+0.0010.281−0.004+0.0030.189−0.013+0.0150.282−0.004+0.0050.268−0.004+0.004102P30.103−0.006+0.0030.142−0.007+0.0020.122−0.006+0.0030.118−0.012+0.0120.174−0.008+0.0070.158−0.003+0.002P′41.049−0.015+0.0131.111−0.012+0.0111.338−0.002+0.0021.033−0.037+0.0331.098−0.027+0.0241.345−0.003+0.002−P′50.482−0.004+0.0020.524−0.005+0.0030.427−0.006+0.0050.470−0.010+0.0080.515−0.003+0.0020.408−0.006+0.006−103P′60.875−0.030+0.0310.780−0.029+0.0280.069−0.001+0.0011.007−0.101+0.0890.910−0.089+0.0780.097−0.001+0.001−102P′80.643−0.003+0.0090.640−0.008+0.0040.287−0.002+0.0020.749−0.004+0.0040.748−0.006+0.0060.374−0.001+0.001


## 4.3. Theq2-Binned Predictions

ForBs⟶ϕμ+μ− decay mode, the LHCb Collaboration has reported their experimental measurements for many physical observables in several q2 bins [31, 33]. In order to compare our theoretical predictions with the LHCb results bin by bin, we make the same choices of the q2 bins as LHCb did, calculate, and show our theoretical predictions for the branching ratio BBs⟶ϕℓ+ℓ− and the asymmetry FLϕ with ℓ=μ,τ in Tables 8 and 9 and the observables S3,4,7 with ℓ=μ in Table 10. For observables S7 and A5,6,8,9, in fact, our theoretical predictions for their values are very small, say in the range of 10−3‐10−4 in magnitude, but still agree with the LHCb measurements in different bins [31] due to still large experimental errors. For the observables P3 and P6,8′, they are also very small in size: in the range of 10−3‐10−4 and there exist no corresponding data at present. For observables P1,2 and P4,5′, on the other hand, although there exist no experimental measurements for them at present, they are relatively large in size and may be measured in the near future LHCb and Belle-II experiments, so we calculate and list the theoretical predictions of these observables bin by bin for the cases of ℓ=μ,τin Tables 11 and 12. Very recently, LHCb reported some new measurements for the angular observables of Bs⟶ϕμ+μ− decay [34] in the q2 bins different from those in their previous work [31], which will be studied in our next work.Table 8
Theoretical predictions for theq2-binned observables BBs⟶ϕμ+μ− (in unit of 10−7) in the PQCD (the first low) and “PQCD+Lattice” (the second row) approach, respectively. For a comparison, we also list the new LHCb measurements [33].
q2 bin GeV2Bℓ=μLHCb [33]q2 bin GeV2Bℓ=μLHCb [33]0.10,0.980.25−0.07+0.100.68±0.061.10,2.500.19−0.05+0.080.44±0.050.25−0.07+0.100.21−0.05+0.072.50,4.000.23−0.07+0.090.35±0.044.00,6.000.41−0.12+0.180.62±0.060.27−0.06+0.090.50−0.11+0.146.00,8.000.61−0.19+0.280.63±0.0611.0,12.50.78−0.25+0.380.72±0.060.69−0.13+0.180.76−0.11+0.1415.00,17.001.26−0.42+0.641.05±0.0817.0,19.00.86−0.30+0.450.84±0.071.04−0.10+0.110.67−0.06+0.051.10,6.000.83−0.24+0.351.41±0.1015.0,19.02.12−0.71+1.091.85±0.130.98−0.22+0.301.70−0.16+0.17Table 9
Theoretical predictions for theq2-binned observables BBs⟶ϕℓ+ℓ− (in unit of 10−7) in the PQCD (the first low) and “PQCD+Lattice” (the second row) approach, respectively. For a comparison, we also list the new LHCb measurements [33].
q2binGeV2Bl=μLHCbBl=τFLϕl=μLHCbFLϕl=τ0.1,2.00.39−0.11+0.161.11±0.16 [31]—0.441−0.008+0.0070.20±0.09—0.40−0.11+0.16—0.472−0.012+0.011—2.0,5.00.48−0.14+0.200.77±0.14 [31]—0.738−0.009+0.0080.68±0.15—0.58−0.13+0.18—0.796−0.007+0.007—5.0,8.00.84−0.25+0.370.96±0.15 [31]—0.584−0.009+0.0080.54±0.10—0.96−0.19+0.25—0.682−0.008+0.010—11.0,12.50.78−0.25+0.380.71±0.12 [31]—0.433−0.005+0.0030.29±0.11—0.76−0.11+0.140.72±0.06 [33]—0.524−0.008+0.008—15.0,17.01.12−0.40+0.610.90±0.13 [31]0.45−0.14+0.220.368−0.001+0.0010.23±0.090.421−0.004+0.0030.99−0.10+0.111.05±0.08 [33]0.39−0.04+0.040.412−0.004+0.0040.479−0.003+0.00317.0,19.00.80−0.27+0.410.75±0.13 [31]0.36−0.12+0.180.346−0.001+0.0010.40±0.140.365−0.002+0.0010.67−0.05+0.050.82±0.07 [33]0.29−0.02+0.020.363−0.001+0.0010.391−0.001+0.0011.0,6.00.93−0.27+0.291.29±0.19 [31]—0.708−0.009+0.0070.63±0.09—1.10−0.25+0.34—0.777−0.006+0.008—15.0,19.01.99−0.71+1.021.62±0.20 [31]0.82−0.26+0.400.359−0.001+0.0020.29±0.070.396−0.003+0.0031.60−0.16+0.161.85±0.13 [33]0.68−0.06+0.060.394−0.003+0.0030.442−0.001+0.002Table 10
Theoretical predictions for theq2-binned observables S3,4,7 of the decays Bs⟶ϕμ+μ− in the PQCD (the first low) and “PQCD+Lattice” (the second row) approaches. For a comparison, we also list the LHCb measured values [31].
S3S4S7q2 bin GeV2Theor.LHCbTheor.LHCbTheor. (10−3)LHCb0.1,2.00.003−0.000+0.000−0.05±0.13−0.054−0.001+0.001−0.27±0.231.571−0.002+0.001−0.04±0.120.002−0.001+0.000−0.053−0.000+0.0001.551−0.006+0.0032.0,5.0−0.024−0.002+0.002−0.06±0.210.191−0.004+0.0040.47±0.371.065−0.003+0.0030.03±0.21−0.021−0.001+0.0010.177−0.003+0.0030.979−0.018+0.0175.0,8.0−0.057−0.003+0.003−0.10±0.250.270−0.002+0.0020.10±0.170.453−0.003+0.003−0.04±0.18−0.050−0.003+0.0040.259−0.003+0.0030.453−0.005+0.00411.0,12.5−0.124−0.003+0.004−0.19±0.210.296−0.001+0.0010.47±0.250.153−0.002+0.0020.00±0.16−0.115−0.001+0.0020.303−0.001+0.0010.185−0.001+0.00115.0,17.0−0.219−0.003+0.003−0.06±0.180.314−0.001+0.0010.03±0.150.052−0.001+0.001−0.12±0.15−0.213−0.001+0.0010.323−0.001+0.0010.071−0.001+0.00117.0,19.0−0.283−0.001+0.001−0.07±0.250.325−0.001+0.0010.39±0.300.019−0.001+0.001−0.20±0.26−0.281−0.001+0.0010.329−0.001+0.0010.027−0.001+0.0011.0,6.0−0.026−0.002+0.002−0.02±0.130.180−0.004+0.0040.19±0.141.063−0.001+0.0010.03±0.14−0.023−0.001+0.0010.169−0.001+0.0010.985−0.022+0.02015.0,19.0−0.245−0.002+0.002−0.09±0.120.318−0.001+0.0010.14±0.110.038−0.002+0.002−0.13±0.11−0.239−0.001+0.0010.325−0.001+0.0010.054−0.001+0.001Table 11
Theoretical predictions for theq2-binned observables AFBμ and P1,2 of the decays Bs⟶ϕℓ+ℓ− with ℓ=μ,τ in the PQCD (the first low) and “PQCD+Lattice” (the second row) approaches.
q2 bin GeV2AFBμAFBτP1ℓ=μP1ℓ=τP2ℓ=μP2ℓ=τ0.1,2.00.131−0.003+0.003—0.015−0.001+0.001—0.206−0.001+0.001—0.122−0.002+0.003—0.013−0.001+0.001—0.204−0.001+0.001—2.0,5.0−0.047−0.003+0.004—−0.197−0.007+0.007—−0.128−0.003+0.003—−0.038−0.002+0.001—−0.225−0.006+0.005—−0.132−0.002+0.002—5.0,8.0−0.265−0.005+0.006—−0.280−0.008+0.009—−0.431−0.001+0.001—−0.200−0.006+0.006—−0.323−0.004+0.003—−0.427−0.001+0.001—11.0,12.5−0.365−0.001+0.001—−0.440−0.008+0.010—−0.431−0.003+0.002—−0.298−0.005+0.005—−0.489−0.003+0.002—−0.420−0.001+0.001—15.0,17.0−0.326−0.001+0.001−0.188−0.001+0.001−0.697−0.006+0.009−0.707−0.005+0.009−0.345−0.004+0.003−0.341−0.004+0.003−0.290−0.005+0.005−0.160−0.003+0.003−0.730−0.005+0.007−0.738−0.006+0.006−0.330−0.003+0.003−0.326−0.004+0.00317.0,19.0−0.226−0.003+0.002−0.153−0.002+0.001−0.869−0.003+0.005−0.875−0.003+0.005−0.231−0.003+0.002−0.225−0.004+0.003−0.208−0.003+0.004−0.137−0.002+0.003−0.884−0.004+0.004−0.890−0.003+0.005−0.219−0.004+0.003−0.213−0.004+0.004Table 12
Theoretical predictions for theq2-binned optimized observables P′4 and P′5 of the decays Bs⟶ϕℓ+ℓ− with ℓ=e,μ,τ in the PQCD (the first low) and “PQCD+Lattice” (the second row) approach.
q2 bin GeV2P′4ℓ=eP′4ℓ=μP′4ℓ=τP′5ℓ=eP′5ℓ=μP′5ℓ=τ0.1,2.0−0.322−0.005+0.005−0.289−0.005+0.005—0.484−0.002+0.0010.459−0.002+0.001—−0.318−0.001+0.001−0.285−0.001+0.001—0.475−0.001+0.0020.449−0.001+0.001—2.0,5.00.903−0.007+0.0070.903−0.007+0.007—−0.606−0.002+0.001−0.607−0.002+0.001—0.916−0.001+0.0010.916−0.001+0.001—−0.608−0.003+0.004−0.608−0.004+0.004—5.0,8.01.110−0.005+0.0031.110−0.005+0.004—−0.816−0.006+0.005−0.816−0.006+0.005—1.127−0.001+0.0021.127−0.001+0.002—−0.796−0.001+0.003−0.796−0.002+0.002—11.0,12.51.198−0.005+0.0021.198−0.005+0.002—−0.727−0.007+0.005−0.727−0.007+0.005—1.217−0.001+0.0011.217−0.001+0.001—−0.698−0.001+0.001−0.698−0.001+0.002—15.0,17.01.302−0.004+0.0021.302−0.004+0.0021.306−0.004+0.002−0.533−0.007+0.005−0.533−0.007+0.005−0.525−0.008+0.0061.314−0.003+0.0021.314−0.003+0.0021.317−0.002+0.003−0.506−0.006+0.005−0.506−0.006+0.005−0.498−0.006+0.00617.0,19.01.366−0.002+0.0011.366−0.002+0.0011.369−0.002+0.001−0.341−0.006+0.004−0.341−0.006+0.004−0.331−0.006+0.0041.371−0.001+0.0011.371−0.001+0.0011.374−0.002+0.001−0.323−0.006+0.005−0.323−0.006+0.005−0.314−0.006+0.006The definitions of theq2-binned observables are the following:
(53)Bq12,q22=∫q12q22dq2dBBs⟶ϕℓ+ℓ−dq2,FLϕq12,q22=∫q12q22dq23I1c+I¯1c−I2c+I¯2c4∫q12q22dq2dΓ+Γ¯/dq2,AFBq12,q22=3∫q12q22dq2I6s+I¯6s4∫q12q22dq2dΓ+Γ¯/dq2,S3,4,7q12,q22=∫q12q22dq2I3,4,7+I¯3,4,7∫q12q22dq2dΓ+Γ¯/dq2,P1q12,q22=∫q12q22dq2S32∫q12q22dq2S2s,P2q12,q22=∫q12q22dq2βℓS6s8∫q12q22dq2S2s,P4′q12,q22=∫q12q22dq2S4−∫q12q22dq2S2cS2s,P5′q12,q22=∫q12q22dq2βℓS52−∫q12q22dq2S2cS2s.From the numerical values as shown in Figure3 and in Tables 8–10, we find the following points about the relevant physical observables of the considered Bs⟶ϕμ+μ− decays in bins:
(1)
ForBs⟶ϕμ+μ− decay, besides the good consistency between the theory and the LHCb data for the integrated total branching ratio BBs⟶ϕμ+μ− as listed in Equation (50), the PQCD and “PQCD+Lattice” predictions for BBs⟶ϕμ+μ− in most bins do agree well with the measured ones within 2σ errors. For the first low-q2 bin 0.1<q2<2GeV2, however, the central value of the LHCb result 1.11±0.16 is larger than the theoretical ones by roughly a factor of three. The LHCb results of BBs⟶ϕμ+μ− in different bins of q2 as listed in the third column of Table 9 are obtained from the results as given in References [31, 33] by multiplying the LHCb measured values of differential decay rate dBBs⟶ϕμ+μ−/dq2 with the width of the corresponding bin q22−q12. The theoretical errors of our theoretical predictions of the branching ratios in bins are still relatively large, while the differences between the PQCD and “PQCD+Lattice” predictions for BBs⟶ϕℓ+ℓ− with ℓ=μ,τ are small(2)
In the first low-q2 bin 0.1<q2<2GeV2, both the PQCD and “PQCD+Lattice” predictions for FLϕℓ=μ are larger than the LHCb measured results FLϕℓ=μLHCb=0.20−0.09+0.08±0.02 [31]. For other bins, both PQCD and “PQCD+Lattice” predictions of FLϕ for muon mode do agree very well with currently available LHCb measured values [31] within 2σ errors. It is worth of remaining that our theoretical predictions of FLϕ have a little error of ~2% due to the strong cancellation of the theoretical errors in the ratios. The theoretical predictions for BBs⟶ϕτ+τ− and FLϕℓ=τ in different bins of q2 as listed in Table 9 will be tested by future experimental measurements(3)
For the observablesS3,4,7, as listed in Table 10, the PQCD and “PQCD+Lattice” predictions for their values in all bins are in the range of 10−3‐10−1 and show a good agreement with the LHCb measured values [31]. The errors of the theoretical predictions are also very small, ~2% in magnitude, because of their nature of the ratios. In all bins, the LHCb measured values of S3,4,7 are still consistent with zero due to their still large errors, which is a clear feature as can be seen easily from the numerical values in Table 10 and the crosses in Figure 3In Tables11 and 12, we show the PQCD and “PQCD+Lattice” predictions for the physical observables AFBμ,τ,P1,2μ,τ and P4,5′ℓ=e,μ,τ in six bins. These physical observables could be tested in the near future LHCb and Belle-II experiments.


## 5. Summary

In this paper, we made a systematic study of the semileptonic decaysBs⟶ϕℓ+ℓ− with ℓ−=e−,μ−,τ− using the PQCD and the “PQCD+Lattice” factorization approach, respectively. We first evaluated all relevant form factors in the low-q2 region using the PQCD approach, and we also took currently available lattice QCD results at the high-q2 points q2=12,16,18.9GeV2 as additional input to improve the extrapolation of the form factors from the low to the high-q2 region. We calculated the branching ratios BBs⟶ϕℓ+ℓ−, the CP-averaged ϕ longitudinal polarization fraction FLq2, the forward-backward asymmetry AFBq2, the CP-averaged angular coefficients S3,4,7q2, the CP asymmetry angular coefficients A5,6,8,9q2, the optimized observables P1,2,3q2, and P4,5,6,8′q2. For Bs⟶ϕμ+μ− decay mode, we calculated the binned values of the branching ratio BBs⟶ϕμ+μ− and the observables FLϕ and S3,4,7 in the same bins as defined by LHCb Collaboration [31] in order to compare our theoretical predictions with those currently available LHCb measurements bin by bin directly.Based on the analytical evaluations, the numerical results, and the phenomenological analysis, we found the following main points:(1)
For the branching ratioBBs⟶ϕμ+μ−, the PQCD and “PQCD+Lattice” predictions are 7.07−2.34+3.43×10−7 and 6.76−1.25+1.52×10−7, respectively, which agree well with the LHCb measured values [31, 33] and the QCDSR prediction within still large errors. For the electron and tau mode, our theoretical predictions for their decay rates are also well consistent with the corresponding QCDSR predictions and to be tested by future experimental measurements(2)
For the ratios of the branching ratiosRϕeμ and Rϕμτ, the PQCD and “PQCD+Lattice” predictions agree with each other and with small theoretical errors because of the strong cancellation of the theoretical errors in such ratios. We suggest the LHCb and Belle-II Collaboration to measure these ratios(3)
For the longitudinal polarizationFL, both PQCD and “PQCD+Lattice” predictions agree with the LHCb measurements in the considered bins within the errors. For the CP-averaged angular coefficients S3,4,7, the PQCD and “PQCD+Lattice” predictions in all bins are small in magnitude, in the range of 10−3‐10−1, and agree well with the LHCb results within the still large experimental errors. For the CP asymmetry angular coefficients A5,6,8,9, the PQCD and “PQCD+Lattice” predictions are very small, in the range of 10−4‐10−2, and clearly consistent with the LHCb measurements in the six bins(4)
For the physical observablesAFBl, P1,2,3, and P4,5,6,8′, the experimental measurements are still absent now; we think that the PQCD and “PQCD+Lattice” predictions for these physical observables will be tested in the near future LHCb and Belle-II experiments



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