# Hunting for Direct CP Violation inB¯s0⟶π+π−K∗0

Authors: Sheng-Tao Li; Gang Lü
Journal: Advances in High Energy Physics
 (2022)

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## Abstract

In perturbative QCD approach, based on the first order of isospin symmetry breaking, we study the directCP violation in the decay of B¯s0⟶ρωK∗0⟶π+π−K∗0. An interesting mechanism is applied to enlarge the CP violating asymmetry involving the charge symmetry breaking between ρ and ω. We find that the CP violation is large by the ρ−ω mixing mechanism when the invariant masses of the π+π− pairs are in the vicinity of the ω resonance. For the decay process of B¯s0⟶ρωK∗0⟶π+π−K∗0, the maximum CP violation can reach −59.12%. Furthermore, taking ρ−ω mixing into account, we calculate the branching ratio for B¯s0⟶ρωK∗0. We also discuss the possibility of observing the predicted CP violation asymmetry at the LHC.


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## Body

## 1. Introduction

Charge-parity (CP) violation is an open problem, even though it has been known in the neutral kaon systems for more than five decades [1]. The study of CP violation in the heavy quark systems is important to our understanding of both particle physics and the evolution of the early universe. Within the standard model (SM), CP violation is related to the nonzero weak complex phase angle from the Cabibbo-Kobayashi-Maskawa (CKM) matrix, which describes the mixing of the three generations of quarks [2, 3]. Theoretical studies predicted large CP violation in the B meson system [4–6]. In recent years, the LHCb collaboration has measured sizable direct CP asymmetries in the phase space of the three-body decay channels of B±⟶π±π+π− and B±⟶K±π+π− [7–9]. These processes are also valuable for studying the mechanism of multibody heavy meson decays. Hence, more attention has been focused on the nonleptonic B meson three-body decay channels in searching for CP violation, both theoretically and experimentally.The directCP violation in the b hadron decays occurs through the interference of at least two amplitudes with different weak phase ϕ and strong phase δ. The weak phase difference ϕ is determined by the CKM matrix elements, while the strong phase can be produced by the hadronic matrix elements and interference between the intermediate states. The hadronic matrix elements are not still well determined by the theoretical approach. The mechanism of two-body B decays is still not quite clear, although many physicists are devoted to this field. Many factorization approaches have been developed to calculate the two-body hadronic decays, such as the naive factorization approach [10–13], the QCD factorization (QCDF) [14–18], perturbative QCD (pQCD) [19–21], and soft-collinear effective theory (SCET) [22–24]. Most factorization approaches are based on heavy quark expansion and light-cone expansion in which only the leading power or part of the next to leading power contributions are calculated to compare with the experiments. However, the different methods may present different strong phases so as to affect the value of the CP violation. Meanwhile, in order to have a large signal of CP violation, we need an appeal to some phenomenological mechanism to obtain a large strong phase δ. In Refs. [25–30], the authors studied the direct CP violation in hadronic B (including Bs and Λb) decays through the interference of tree and penguin diagrams, where ρ-ω mixing was used for this purpose in the past few years and focused on the naive factorization and QCD factorization approaches. This mechanism was also applied to generalize the pQCD approach to the three-body nonleptonic decays in B0,±⟶π0,±π+π− and Bc⟶Ds+π+π−, where even larger CP violation may be possible [31, 32]. In this paper, we will investigate direct CP violation of the decay process B¯s0⟶ρωK∗0⟶π+π−K∗0 involving the same mechanism in the pQCD approach.The three-body decays of heavyB mesons are more complicated than the two-body decays as they receive both resonant and nonresonant contributions. Unlike the two-body case, to date, we still do not have effective theories for hadronic three-body decays, though attempts along the framework of pQCD and QCDF have been used in the past [33–36]. As a working starting point, we intend to study ρ-ω mixing effect in three-body decays of the B meson. The ρ−ω mixing mechanism is caused by the isospin symmetry breaking from the mixing between the u and d flavors [37, 38]. In Ref. [39], the authors studied the ρ−ω mixing and the pion form factor in the time-like region, where ρ−ω mixing comes from three part contributions: two from the direct coupling of the quasi-two-body decay of B¯s0⟶ρK∗0⟶π+π−K∗0 and B¯s0⟶ωK∗0⟶π+π−K∗0 and the other from the interference of B¯s0⟶ωK∗0⟶ρK∗0⟶π+π−K∗0 mixing. Generally speaking, the amplitudes of their contributions are as follows: B¯s0⟶ρK∗0⟶π+π−K∗0>B¯s0⟶ωK∗0⟶ρK∗0⟶π+π−K∗0>B¯s0⟶ωK∗0⟶π+π−K∗0. ω⟶π+π− and ω⟶ρ⟶π+π− were used to obtain the (effective) mixing matrix element Π~ρωs [40–42]. The magnitude has been determined by the pion form factor through the data from the cross section of e+e−⟶π+π− in the ρ and ω resonance region [39, 42–45]. Recently, isospin symmetry breaking was discussed by incorporating the vector meson dominance (VMD) model in the weak decay process of the meson [27, 32, 46–49]. However, one can find that ρ−ω mixing produces the large CP violation from the effect of isospin symmetry breaking in the three and four bodies decay process. Hence, in this paper, we shall follow the method of Refs. [27, 32, 46–49] to investigate the decay process of B¯s0⟶ρωK∗0⟶π+π−K∗0 by the isospin symmetry breaking.The remainder of this paper is organized as follows. In Sec.2, we will present the form of the effective Hamiltonian and briefly introduce the pQCD framework and wave functions. In Sec. 3, we give the calculating formalism and details of the CP violation from ρ−ω mixing in the decay process B¯s0⟶ρωK∗0⟶π+π−K∗0. In Sec. 4, we calculate the branching ratio for the decay process of B¯s0⟶ρ0ωK∗0. In Sec. 5, we show the input parameters. We present the numerical results in Sec. 6. Summary and discussion are included in Sec. 7. The related functions defined in the text are given in the Appendix.


## 2. The Framework

Based on the operator product expansion, the effective weak Hamiltonian for the decay processesB¯s0⟶ρ0ωK∗0 can be expressed as [50]
(1)Heff=GF2VubVud∗C1μQ1uμ+C2μQ2uμ−VtbVtd∗∑i=310CiμQiμ+H.c.,where GF represents the Fermi constant, Ciμ (i=1,⋯,10) are the Wilson coefficients, and Vub, Vud, Vtb, and Vtd are the CKM matrix elements. The operators Oi have the following forms:
(2)O1u=d¯αγμ1−γ5uβu¯βγμ1−γ5bα,(3)O2u=d¯αγμ1−γ5uαu¯βγμ1−γ5bβ,(4)O3=d¯αγμ1−γ5bα∑q′q′¯βγμ1−γ5q′β,(5)O4=d¯αγμ1−γ5bβ∑q′q′¯βγμ1−γ5q′α,(6)O5=d¯αγμ1−γ5bα∑q′q′¯βγμ1+γ5q′β,(7)O6=d¯αγμ1−γ5bβ∑q′q′¯βγμ1+γ5q′α,(8)O7=32d¯αγμ1−γ5bα∑q′eq′q′¯βγμ1+γ5q′β,(9)O8=32d¯αγμ1−γ5bβ∑q′eq′q′¯βγμ1+γ5q′α,(10)O9=32d¯αγμ1−γ5bα∑q′eq′q′¯βγμ1−γ5q′β,(11)O10=32d¯αγμ1−γ5bβ∑q′eq′q′¯βγμ1−γ5q′α,where α and β are the SU(3) color indices, eq′ is the electric charge of quark q′ in the unit of ∣e∣, and the sum extend over q′=u,d,s,c or b quarks. In Eq. (11) O1u and O2u are the tree operators, O3–O6 are the QCD penguin operators, and O7–O10 are the operators associated with electroweak penguin diagrams.The Wilson coefficientCiμ in Eq. (1) describes the coupling strength for a given operator and summarizes the physical contributions from scales higher than μ [51]. They are calculable perturbatively with the renormalization group improved perturbation theory. Usually, the scale μ is chosen to be of order Omb for B meson decays. Since we work in the leading order of perturbative QCD (Oαs), it is consistent to use the leading order Wilson coefficients. So, we use the numerical values of Cimb as follow [19, 21]:
(12)C1=−0.2703,C2=1.1188,C3=0.0126,C4=−0.0270,C5=0.0085,C6=−0.0326,C7=0.0011,C8=0.0004,C9=−0.0090,C10=0.0022,The combinationsa1–a10 of the Wilson coefficients are defined as usual [52–55]:
(13)a1=C2+C13,a2=C1+C23,ai=Ci+Ci±13i=3−10,where the upper (lower) sign applies, when i is odd (even).For the two-body decay processes ofB¯s0⟶M2M3, we denote the emitted or annihilated meson as M2, while the recoiling meson is M3. The meson M2 (ρ or ω) and the final state meson M3 (K∗0) move along the direction of n=1,0,0T and v=0,1,0T in the light-cone coordinates, respectively. The decay amplitude can be expressed as the convolution of the wave functions ϕBs, ϕM2, and ϕM3 and the hard scattering kernel TH in the pQCD. The pQCD factorization theorem has been developed for the two-body nonleptonic heavy meson decays, based on the formalism of Botts, Lepage, Brodsky, and Sterman [56–59]. The basic idea of the pQCD approach is that it takes into account the transverse momentum of the valence quarks in the hadrons which results in the Sudakov factor in the decay amplitude. Then, the decay channels of B¯s0⟶ρ0ωK∗0 are conceptually written as the following:
(14)AB¯s0⟶ρ0ωK∗0=∫d4k1d4k2d4k3TrCtϕBsk1ϕM2k2ϕM3k3THk1,k2,k3,t,where kii=1,2,3 is the momentum of light quark in each meson. Tr denotes the trace over Dirac structure and color indices. Ct is the short distance Wilson coefficients at the hard scale t. The meson wave functions ϕBs and ϕMm=2,3, including all nonperturbative contribution during the hadronization of mesons, can be extracted from the experimental data or other nonperturbative methods. The hard kernel THk1,k2,k3,andt describes the four quark operator and the spectator quark connected by a hard gluon, which can be perturbatively calculated including all possible Feynman diagrams of the factorizable and nonfactorizable contributions without endpoint singularity.Theρ0ω and K∗0 mesons are treated as a light-light system. At the Bs meson rest frame, they are moving very fast. We define the ratios rK∗0=MK∗0/MBs, rρ=Mρ/MBs, and rω=Mω/MBs. In the limits MK∗0, Mρ, and Mω⟶0, one can drop the terms of proportional to rK∗02, rρ2, and rω2 safely. The symbols PB, P2, and P3 refer to the B¯s0 meson momentum, the ρ0ω meson momentum, and the final state K∗0 meson momentum, respectively. The momenta of the participating mesons in the rest frame of the Bs meson can be written as:
(15)PB=MBs21,1,0T,P2=MBs21,0,0T,P3=MBs20,1,0T.One can denote the light (anti-)quark momentak1, k2, and k3 for the initial meson B¯s0 and the final mesons ρ0ω and K∗0, respectively. We can choose
(16)k1=x1MBs2,0,k1⊥,k2=x2MBs2,0,k2⊥,k3=0,x3MBs2,k3⊥,where x1, x2, and x3 are the momentum fractions. k1⊥, k2⊥, and k3⊥ refer to the transverse momentum of the quark, respectively. To extract the helicity amplitudes, we parameterize the following longitudinal polarization vectors of the ρ0ω and K∗0 as the following:
(17)ϵ2L=P2Mρω−MρωP2·vv,ϵ3L=P3MK∗0−MK∗0P3·nn,which satisfy the orthogonality relationship of ϵ2L·P2=ϵ3L·P3=0, and the normalization of ϵ22L=ϵ32L=−1. The transverse polarization vectors can be adopted directly as
(18)ϵ2T=0,0,1T,ϵ3T=0,0,1T.Within the pQCD framework, both the initial and the final state meson wave functions and distribution amplitudes are important as nonperturbative input parameters. For theBs meson, the wave function of the meson can be expressed as
(19)ΦBs=i6PBs+MBsγ5ϕBsk,where the distribution amplitude ϕBs is shown in Refs. [60–62]:
(20)ϕBsx,b=NBsx21−x2exp−MBs222ωb2−12ωbb2.The shape parameterωb is a free parameter and NBs is a normalization factor. Based on the studies of the light-cone sum rule, lattice QCD or be fitted to the measurements with good precision [63], we take ωb=0.50GeV for the Bs meson. The normalization factor NBs depends on the values of the shape parameter ωb and decay constant fBs, which is defined through the normalization relation ∫01dxϕBsx,0=fBs/26.The distribution amplitudes of vector meson (V=ρ, ω, or K∗), ϕV, ϕVT, ϕVt, ϕVs, ϕVv, and ϕVa, can be written in the following forms [64, 65]:
(21)ϕρx=3fρ6x1−x1+0.15C23/2t,ϕωx=3fω6x1−x1+0.15C23/2t,ϕK∗x=3fK∗6x1−x1+0.03C13/2t+0.11C23/2t,ϕρTx=3fρT6x1−x1+0.14C13/2t,ϕωTx=3fωT6x1−x1+0.14C13/2t,ϕK∗Tx=3fK∗T6x1−x1+0.04C13/2t+0.10C23/2t,ϕVtx=3fVT26t2,ϕVsx=3fVT26−t,ϕVvx=3fV861+t2,ϕVax=3fV46−t,where t=2x−1. Here, fVT is the decay constant of the vector meson with longitudinal (transverse) polarization. The Gegenbauer polynomials Cnνt can be defined as [66, 67]
(22)C13/2t=3t,C23/2t=325t2−1.


## 3.CP Violation in B¯s0⟶ρ0ωK∗0⟶π+π−K∗0 Decay Process

### 3.1. Formalism

The decay widthΓ for the processes of B¯s0⟶ρ0ωK∗0 is given by
(23)Γ=Pc8πMBs2∑σ=L,TAσ†Aσ,where Pc is the absolute value of the three-momentum of the final state mesons. The decay amplitude Aσ which is decided by QCD dynamics will be calculated later in the pQCD factorization approach. The superscript σ denotes the helicity states of the two vector mesons with the longitudinal (transverse) components L(T). The amplitude Aσ for the decays BsPBs⟶VρωP2,ε2μ∗+VK∗0P3,ε3μ∗ can be decomposed as follows [67–70]:
(24)Aσ=MBs2AL+MBs2ANϵ2∗σ=T·ϵ3∗σ=T+iATεαβγρε2α∗σε3β∗σP2γP3ρ,where ε∗ is the polarization vector of the vector meson. The amplitude Ai (i refers to the three kinds of polarizations, longitudinal (L), normal (N), and transverse (T)) can be written as
(25)MBs2AL=aε2∗L·ε3∗L+bM2M3ε2∗L·P3ε3∗L·P2,MBs2AN=a,AT=cM2M3,where a, b, and c are the Lorentz-invariant amplitudes. M2 and M3 are the masses of the vector mesons ρ0ω and K∗0, respectively.The longitudinalH0 and transverse H± of helicity amplitudes can be expressed
(26)H0=MBs2AL,(27)H±=MBs2AN∓M2M3κ2−1AT,where H0 and H± are the penguin level and tree level helicity amplitudes of the decay process B¯s0⟶ρωK∗0⟶π+π−K∗0 from the three kinds of polarizations, respectively. The helicity summation satisfies the relation
(28)∑σ=L,RAσ†Aσ=H02+H+2+H−2.In the vector meson dominance model [71, 72], the photon propagator is dressed by coupling to vector mesons. Based on the same mechanism, ρ−ω mixing was proposed and later gradually applied to B meson physics [29, 39, 47, 73]. According to the effective Hamiltonian, the amplitude A (A¯) for the three-body decay process B¯s0⟶π+π−K∗0 (Bs0⟶π+π−K¯∗0) can be written as follows [47]:
(29)A=<π+π−K∗0HTB¯s0>+<π+π−K∗0HPB¯s0>,(30)A¯=<π+π−K¯∗0HTBs0>+<π+π−K¯∗0HPBs0>,where HT and HP are the Hamiltonian for the tree and penguin operators, respectively.The relative magnitude and phases between the tree and penguin operator contribution are defined as follows:(31)A=<π+π−K∗0HTB¯s0>1+reiδ+ϕ,(32)A¯=<π+π−K¯∗0HTBs0>1+reiδ−ϕ,where δ and ϕ are the strong and weak phase differences, respectively. The weak phase difference ϕ can be expressed as a combination of the CKM matrix elements, and it is ϕ=argVtbVtd∗/VubVud∗ for the b⟶d transition. The parameter r is the absolute value of the ratio of the tree and penguin amplitudes:
(33)r≡<π+π−K∗0HPB¯s0><π+π−K∗0HTB¯s0>.The parameter ofCP violating asymmetry, ACP, can be written as
(34)ACP=A2−A¯2A2+A¯2=−2T02r0sinδ0+T+2r+sinδ++T−2r−sinδ−sinϕ∑i=0+−Ti21+ri2+2ricosδicosϕ,where Tii=0,+,− represent the tree level helicity amplitudes of the decay process B¯s0⟶π+π−K∗0 from H0, H+, and H− of Eq. (27), respectively. rjj=0,+,− refer to the absolute value of the ratio of the tree and penguin amplitude for the three kinds of polarizations, respectively. δkk=0,+,− are the relative strong phases between the tree and penguin operator contributions from the three kinds of the helicity amplitudes, respectively. We can see explicitly from Eq. (34) that both weak and strong phase differences are needed to produce CP violation. In order to obtain a large signal for the direct CP violation, we intend to apply the ρ−ω mixing mechanism, which leads to large strong phase differences in hadron decays.With theρ−ω mixing mechanism, the process of the B¯s0⟶ρωK∗0⟶π+π−K∗0 decay is shown in Figure 1. In the isospin representation, the decay amplitude MB¯s0⟶ρω⟶π+π− in Figure 1 can be written as [31, 39, 48, 74]
(35)MB¯s0⟶π+π−=MρI⟶ππ1sρMB¯s0⟶ρI+MρI⟶ππ1sρΠρω1sωMB¯s0⟶ωI.Figure 1
The diagram for theB¯s0⟶π+π−K∗0 decay with the ρ−ω mixing mechanism for the first order of the isospin violation in the isospin representation.Introducing theε=Πρω/sρ−sω [31, 39, 48, 74], we have identified the physical amplitudes as
(36)Mρ⟶ππ=MρI⟶ππ,Mω⟶ππ=εMρI⟶ππ,MB¯s0⟶ρ=MB¯s0⟶ρI−εMB¯s0⟶ωI,MB¯s0⟶ω=MB¯s0⟶ωI+εMB¯s0⟶ρI.From the physical representation, we can obtain the decay amplitude:(37)MB¯s0⟶π+π−=Mρ⟶ππ1sρMB¯s0⟶ρ+Mρ⟶ππΠρωsρsρ−sω1sρsωMB¯s0⟶ω,where Oε2 corrections are neglected and Mρ⟶ππ=gρ, MB¯s0⟶ρ=tρi or pρi, and MB¯s0⟶ω=tωi or pωi are used. So, we can get Π~ρω=Πρωsρ/sρ−sω. Π~ρω is the effective ρ−ω mixing amplitude which also effectively includes the direct coupling ω⟶π+π−. At the first order of the isospin violation, we have the following tree and penguin amplitudes when the invariant mass of π+π− pair is near the ω resonance mass [26, 47]:
(38)<π+π−K∗0HTB¯s0>=gρsρsωΠ~ρωtωi+gρsρtρi,(39)<π+π−K∗0HPB¯s0>=gρsρsωΠ~ρωpωi+gρsρpρi,where tρipρi and tωipωi are the tree (penguin) level helicity amplitudes for B¯s0⟶ρ0K∗0 and B¯s0⟶ωK∗0, respectively. The amplitudes tρi, tωi, pρi, and pωi can be found in Sec. 3.2. gρ is the coupling constant for the decay process ρ0⟶π+π−. sV, mV, and ΓV(V=ρ or ω) are the inverse propagator, mass, and decay width of the vector meson V, respectively. sV can be expressed as
(40)sV=s−mV2+imVΓV,with s being the invariant masses of the π+π− pairs. The ρ−ω mixing parameters Π~ρωs=ReΠ~ρωmω2+ImΠ~ρωmω2 are [75]
(41)ReΠ~ρωmω2=−4760±440MeV2,ImΠ~ρωmω2=−6180±3300MeV2.From Eqs. (29), (31), (38), and (39), one has
(42)reiδieiϕ=Π~ρωpωi+sωpρiΠ~ρωtωi+sωtρi,defining [25, 76]
(43)pωitρi≡r′eiδqi+ϕ,tωitρi≡αeiδαi,pρipωi≡βeiδβi,where δαi, δβi, and δqi are the strong phases of the decay process B¯s0⟶ρ0ωK∗0⟶π+π−K∗0 from the three kinds of polarizations, respectively. One finds the following expression from Eqs. (42) and (43):
(44)reiδi=r′eiδqiΠ~ρω+βeiδβisωΠ~ρωαeiδαi+sω.αeiδαi, βeiδβi, and r′eiδqi will be calculated in the perturbative QCD approach. In order to obtain the CP violating asymmetry in Eq. (34), ACP, sinϕ, and cosϕ are needed. ϕ is determined by the CKM matrix elements. In the Wolfenstein parametrization [77], the weak phase ϕ comes from VtbVtd∗/VubVud∗. One has
(45)sinϕ=ηρ1−ρ−η22+η2,(46)cosϕ=ρ1−ρ−η2ρ1−ρ−η22+η2,where the same result has been used for b⟶d transition from Ref. [28, 78].


### 3.2. Calculation Details

We can decompose the decay amplitudes for the decay processesB¯s0⟶ρ0ωK∗0 in terms of the tree and penguin contributions depending on the CKM matrix elements of VubVud∗ and VtbVtd∗. From Eqs. (34), (42), and (43), in the leading order to obtain the formulas of the CP violation, we need to calculate the amplitudes tρ, pρ, tω, and pω in the perturbative QCD approach. The relevant function can be found in the Appendix from the perturbative QCD approach.In the pQCD, there are eight types of the leading order Feynman diagrams contributing toB¯s0⟶ρ0ωK∗0 decays, which are shown in Figure 2. The first row is for the emission-type diagrams, where the first two diagrams in Figures 2(a) and 2(b) are called factorizable emission diagrams and the last two diagrams in Figures 2(c) and 2(d) are called nonfactorizable emission diagrams [68, 79]. The second row is for the annihilation-type diagrams, where the first two diagrams in Figures 2(e) and 2(f) are called factorizable annihilation diagrams and the last two diagrams in Figures 2(g) and 2(h) are called nonfactorizable annihilation diagrams [62, 80]. The relevant decay amplitudes can be easily obtained by these hard gluon exchange diagrams, and the Lorenz structures of the mesons wave functions. Through calculating these diagrams, the formulas of B¯s0⟶ρK∗0 or B¯s0⟶ωK∗0 are similar to those of B⟶ϕK∗ and Bs⟶K∗−K∗+ [79, 81]. We just need to replace some corresponding Wilson coefficients, wave functions, and corresponding parameters.Figure 2
Leading order Feynman diagrams forB¯s0⟶ρ0ωK∗0.
(a)(b)(c)(d)(e)(f)(g)(h)With the Hamiltonian Equation (1), depending on CKM matrix elements of VubVud∗ and VtbVtd∗, the tree dominant decay amplitudes Ai for B¯s0⟶ρK∗0 in pQCD can be written as
(47)2AiB¯s0⟶ρ0K∗0=VubVud∗Tρi−VtbVtd∗Pρi,where the superscript i denote the different helicity amplitudes L, N, and T. The longitudinal tρω0 and transverse tρω± of helicity amplitudes satisfy relationship from Eq. (27). The amplitudes of the tree and penguin diagrams can be written as Tρi=tρi/VubVud∗ and Pρi=pρi/VtbVtd∗, respectively. The formula for the tree level amplitude is
(48)Tρi=GF2fρFBs⟶K∗LL,ia2+MBs⟶K∗LL,iC2,where fρ refers to the decay constant of ρ meson. The penguin level amplitudes are expressed in the following:
(49)Pρi=−GF2fρFBs⟶K∗LL,i−a4+32a7+32a9+12a10−MBs⟶K∗LR,i−C5+12C7+MBs⟶K∗LL,i−C3+12C9+32C10−MBs⟶K∗SP,i32C8+fBsFannLL,i−a4+12a10−fBsFannSP,i−a6+12a8+MannLL,i−C3+12C9−MannLR,i−C5+12C7.The tree dominant decay amplitude forB¯s0⟶ωK∗0 can be written as
(50)2AiB¯s0⟶ωK∗0=VubVud∗Tωi−VtbVtd∗Pωi,where Tωi=tωi/VubVus∗ and Pωi=pωi/VtbVts∗ refer to the tree and penguin amplitude, respectively. We can give the tree level contribution in the following:
(51)Tωi=GF2fωFBs⟶K∗LL,ia2+MBs⟶K∗LL,iC2,where fω refers to the decay constant of ω meson. The penguin level contribution is given as follows:
(52)Pωi=GF2fωFBs⟶K∗LL,i2a3+a4+2a5+12a7+12a9−12a10−MBs⟶K∗LR,iC5−12C7+MBs⟶K∗LL,iC3+2C4−12C9+12C10−MBs⟶K∗SP,i2C6+12C8+fBsFannLL,ia4−12a10−fBsFannSP,ia6−12a8+MannLL,iC3−12C9−MannLR,iC5−12C7Based on the definition of Eq. (43), we can get
(53)αeiδαi=tωitρi,(54)βeiδβi=pρipωi,(55)r′eiδqi=PωiTρi×VtbVtd∗VubVud∗,where
(56)VtbVtd∗VubVud∗=ρ1−ρ−η22+η21−λ2/2ρ2+η2.From above equations, the new strong phasesδαi, δβi, and δqi are obtained from the tree and penguin diagram contributions by the ρ−ω interference. Substituting Eqs. (53), (54), and (55) into (44), we can obtain total strong phase δi in the framework of the pQCD. Then in combination with Eqs. (45) and (46), the CP violating asymmetry can be obtained.


## 3.1. Formalism

The decay widthΓ for the processes of B¯s0⟶ρ0ωK∗0 is given by
(23)Γ=Pc8πMBs2∑σ=L,TAσ†Aσ,where Pc is the absolute value of the three-momentum of the final state mesons. The decay amplitude Aσ which is decided by QCD dynamics will be calculated later in the pQCD factorization approach. The superscript σ denotes the helicity states of the two vector mesons with the longitudinal (transverse) components L(T). The amplitude Aσ for the decays BsPBs⟶VρωP2,ε2μ∗+VK∗0P3,ε3μ∗ can be decomposed as follows [67–70]:
(24)Aσ=MBs2AL+MBs2ANϵ2∗σ=T·ϵ3∗σ=T+iATεαβγρε2α∗σε3β∗σP2γP3ρ,where ε∗ is the polarization vector of the vector meson. The amplitude Ai (i refers to the three kinds of polarizations, longitudinal (L), normal (N), and transverse (T)) can be written as
(25)MBs2AL=aε2∗L·ε3∗L+bM2M3ε2∗L·P3ε3∗L·P2,MBs2AN=a,AT=cM2M3,where a, b, and c are the Lorentz-invariant amplitudes. M2 and M3 are the masses of the vector mesons ρ0ω and K∗0, respectively.The longitudinalH0 and transverse H± of helicity amplitudes can be expressed
(26)H0=MBs2AL,(27)H±=MBs2AN∓M2M3κ2−1AT,where H0 and H± are the penguin level and tree level helicity amplitudes of the decay process B¯s0⟶ρωK∗0⟶π+π−K∗0 from the three kinds of polarizations, respectively. The helicity summation satisfies the relation
(28)∑σ=L,RAσ†Aσ=H02+H+2+H−2.In the vector meson dominance model [71, 72], the photon propagator is dressed by coupling to vector mesons. Based on the same mechanism, ρ−ω mixing was proposed and later gradually applied to B meson physics [29, 39, 47, 73]. According to the effective Hamiltonian, the amplitude A (A¯) for the three-body decay process B¯s0⟶π+π−K∗0 (Bs0⟶π+π−K¯∗0) can be written as follows [47]:
(29)A=<π+π−K∗0HTB¯s0>+<π+π−K∗0HPB¯s0>,(30)A¯=<π+π−K¯∗0HTBs0>+<π+π−K¯∗0HPBs0>,where HT and HP are the Hamiltonian for the tree and penguin operators, respectively.The relative magnitude and phases between the tree and penguin operator contribution are defined as follows:(31)A=<π+π−K∗0HTB¯s0>1+reiδ+ϕ,(32)A¯=<π+π−K¯∗0HTBs0>1+reiδ−ϕ,where δ and ϕ are the strong and weak phase differences, respectively. The weak phase difference ϕ can be expressed as a combination of the CKM matrix elements, and it is ϕ=argVtbVtd∗/VubVud∗ for the b⟶d transition. The parameter r is the absolute value of the ratio of the tree and penguin amplitudes:
(33)r≡<π+π−K∗0HPB¯s0><π+π−K∗0HTB¯s0>.The parameter ofCP violating asymmetry, ACP, can be written as
(34)ACP=A2−A¯2A2+A¯2=−2T02r0sinδ0+T+2r+sinδ++T−2r−sinδ−sinϕ∑i=0+−Ti21+ri2+2ricosδicosϕ,where Tii=0,+,− represent the tree level helicity amplitudes of the decay process B¯s0⟶π+π−K∗0 from H0, H+, and H− of Eq. (27), respectively. rjj=0,+,− refer to the absolute value of the ratio of the tree and penguin amplitude for the three kinds of polarizations, respectively. δkk=0,+,− are the relative strong phases between the tree and penguin operator contributions from the three kinds of the helicity amplitudes, respectively. We can see explicitly from Eq. (34) that both weak and strong phase differences are needed to produce CP violation. In order to obtain a large signal for the direct CP violation, we intend to apply the ρ−ω mixing mechanism, which leads to large strong phase differences in hadron decays.With theρ−ω mixing mechanism, the process of the B¯s0⟶ρωK∗0⟶π+π−K∗0 decay is shown in Figure 1. In the isospin representation, the decay amplitude MB¯s0⟶ρω⟶π+π− in Figure 1 can be written as [31, 39, 48, 74]
(35)MB¯s0⟶π+π−=MρI⟶ππ1sρMB¯s0⟶ρI+MρI⟶ππ1sρΠρω1sωMB¯s0⟶ωI.Figure 1
The diagram for theB¯s0⟶π+π−K∗0 decay with the ρ−ω mixing mechanism for the first order of the isospin violation in the isospin representation.Introducing theε=Πρω/sρ−sω [31, 39, 48, 74], we have identified the physical amplitudes as
(36)Mρ⟶ππ=MρI⟶ππ,Mω⟶ππ=εMρI⟶ππ,MB¯s0⟶ρ=MB¯s0⟶ρI−εMB¯s0⟶ωI,MB¯s0⟶ω=MB¯s0⟶ωI+εMB¯s0⟶ρI.From the physical representation, we can obtain the decay amplitude:(37)MB¯s0⟶π+π−=Mρ⟶ππ1sρMB¯s0⟶ρ+Mρ⟶ππΠρωsρsρ−sω1sρsωMB¯s0⟶ω,where Oε2 corrections are neglected and Mρ⟶ππ=gρ, MB¯s0⟶ρ=tρi or pρi, and MB¯s0⟶ω=tωi or pωi are used. So, we can get Π~ρω=Πρωsρ/sρ−sω. Π~ρω is the effective ρ−ω mixing amplitude which also effectively includes the direct coupling ω⟶π+π−. At the first order of the isospin violation, we have the following tree and penguin amplitudes when the invariant mass of π+π− pair is near the ω resonance mass [26, 47]:
(38)<π+π−K∗0HTB¯s0>=gρsρsωΠ~ρωtωi+gρsρtρi,(39)<π+π−K∗0HPB¯s0>=gρsρsωΠ~ρωpωi+gρsρpρi,where tρipρi and tωipωi are the tree (penguin) level helicity amplitudes for B¯s0⟶ρ0K∗0 and B¯s0⟶ωK∗0, respectively. The amplitudes tρi, tωi, pρi, and pωi can be found in Sec. 3.2. gρ is the coupling constant for the decay process ρ0⟶π+π−. sV, mV, and ΓV(V=ρ or ω) are the inverse propagator, mass, and decay width of the vector meson V, respectively. sV can be expressed as
(40)sV=s−mV2+imVΓV,with s being the invariant masses of the π+π− pairs. The ρ−ω mixing parameters Π~ρωs=ReΠ~ρωmω2+ImΠ~ρωmω2 are [75]
(41)ReΠ~ρωmω2=−4760±440MeV2,ImΠ~ρωmω2=−6180±3300MeV2.From Eqs. (29), (31), (38), and (39), one has
(42)reiδieiϕ=Π~ρωpωi+sωpρiΠ~ρωtωi+sωtρi,defining [25, 76]
(43)pωitρi≡r′eiδqi+ϕ,tωitρi≡αeiδαi,pρipωi≡βeiδβi,where δαi, δβi, and δqi are the strong phases of the decay process B¯s0⟶ρ0ωK∗0⟶π+π−K∗0 from the three kinds of polarizations, respectively. One finds the following expression from Eqs. (42) and (43):
(44)reiδi=r′eiδqiΠ~ρω+βeiδβisωΠ~ρωαeiδαi+sω.αeiδαi, βeiδβi, and r′eiδqi will be calculated in the perturbative QCD approach. In order to obtain the CP violating asymmetry in Eq. (34), ACP, sinϕ, and cosϕ are needed. ϕ is determined by the CKM matrix elements. In the Wolfenstein parametrization [77], the weak phase ϕ comes from VtbVtd∗/VubVud∗. One has
(45)sinϕ=ηρ1−ρ−η22+η2,(46)cosϕ=ρ1−ρ−η2ρ1−ρ−η22+η2,where the same result has been used for b⟶d transition from Ref. [28, 78].


## 3.2. Calculation Details

We can decompose the decay amplitudes for the decay processesB¯s0⟶ρ0ωK∗0 in terms of the tree and penguin contributions depending on the CKM matrix elements of VubVud∗ and VtbVtd∗. From Eqs. (34), (42), and (43), in the leading order to obtain the formulas of the CP violation, we need to calculate the amplitudes tρ, pρ, tω, and pω in the perturbative QCD approach. The relevant function can be found in the Appendix from the perturbative QCD approach.In the pQCD, there are eight types of the leading order Feynman diagrams contributing toB¯s0⟶ρ0ωK∗0 decays, which are shown in Figure 2. The first row is for the emission-type diagrams, where the first two diagrams in Figures 2(a) and 2(b) are called factorizable emission diagrams and the last two diagrams in Figures 2(c) and 2(d) are called nonfactorizable emission diagrams [68, 79]. The second row is for the annihilation-type diagrams, where the first two diagrams in Figures 2(e) and 2(f) are called factorizable annihilation diagrams and the last two diagrams in Figures 2(g) and 2(h) are called nonfactorizable annihilation diagrams [62, 80]. The relevant decay amplitudes can be easily obtained by these hard gluon exchange diagrams, and the Lorenz structures of the mesons wave functions. Through calculating these diagrams, the formulas of B¯s0⟶ρK∗0 or B¯s0⟶ωK∗0 are similar to those of B⟶ϕK∗ and Bs⟶K∗−K∗+ [79, 81]. We just need to replace some corresponding Wilson coefficients, wave functions, and corresponding parameters.Figure 2
Leading order Feynman diagrams forB¯s0⟶ρ0ωK∗0.
(a)(b)(c)(d)(e)(f)(g)(h)With the Hamiltonian Equation (1), depending on CKM matrix elements of VubVud∗ and VtbVtd∗, the tree dominant decay amplitudes Ai for B¯s0⟶ρK∗0 in pQCD can be written as
(47)2AiB¯s0⟶ρ0K∗0=VubVud∗Tρi−VtbVtd∗Pρi,where the superscript i denote the different helicity amplitudes L, N, and T. The longitudinal tρω0 and transverse tρω± of helicity amplitudes satisfy relationship from Eq. (27). The amplitudes of the tree and penguin diagrams can be written as Tρi=tρi/VubVud∗ and Pρi=pρi/VtbVtd∗, respectively. The formula for the tree level amplitude is
(48)Tρi=GF2fρFBs⟶K∗LL,ia2+MBs⟶K∗LL,iC2,where fρ refers to the decay constant of ρ meson. The penguin level amplitudes are expressed in the following:
(49)Pρi=−GF2fρFBs⟶K∗LL,i−a4+32a7+32a9+12a10−MBs⟶K∗LR,i−C5+12C7+MBs⟶K∗LL,i−C3+12C9+32C10−MBs⟶K∗SP,i32C8+fBsFannLL,i−a4+12a10−fBsFannSP,i−a6+12a8+MannLL,i−C3+12C9−MannLR,i−C5+12C7.The tree dominant decay amplitude forB¯s0⟶ωK∗0 can be written as
(50)2AiB¯s0⟶ωK∗0=VubVud∗Tωi−VtbVtd∗Pωi,where Tωi=tωi/VubVus∗ and Pωi=pωi/VtbVts∗ refer to the tree and penguin amplitude, respectively. We can give the tree level contribution in the following:
(51)Tωi=GF2fωFBs⟶K∗LL,ia2+MBs⟶K∗LL,iC2,where fω refers to the decay constant of ω meson. The penguin level contribution is given as follows:
(52)Pωi=GF2fωFBs⟶K∗LL,i2a3+a4+2a5+12a7+12a9−12a10−MBs⟶K∗LR,iC5−12C7+MBs⟶K∗LL,iC3+2C4−12C9+12C10−MBs⟶K∗SP,i2C6+12C8+fBsFannLL,ia4−12a10−fBsFannSP,ia6−12a8+MannLL,iC3−12C9−MannLR,iC5−12C7Based on the definition of Eq. (43), we can get
(53)αeiδαi=tωitρi,(54)βeiδβi=pρipωi,(55)r′eiδqi=PωiTρi×VtbVtd∗VubVud∗,where
(56)VtbVtd∗VubVud∗=ρ1−ρ−η22+η21−λ2/2ρ2+η2.From above equations, the new strong phasesδαi, δβi, and δqi are obtained from the tree and penguin diagram contributions by the ρ−ω interference. Substituting Eqs. (53), (54), and (55) into (44), we can obtain total strong phase δi in the framework of the pQCD. Then in combination with Eqs. (45) and (46), the CP violating asymmetry can be obtained.


## 4. Branching Ratio ofB¯s0⟶ρ0ωK∗0

Based on the relationship of Eqs. (23) and (28), we can calculate the decay rates for the processes of B¯s0⟶ρ0ωK∗0 by using the following expression:
(57)Γ=Pc8πMBs2H02+H+2+H−2,where
(58)Pc=MBs−Mρ/ω+MK∗02MBs−Mρ/ω−MK∗022MBsis the c.m. momentum of the product particle and Hii=0,+,− are the helicity amplitudes.In this case, we take into account theρ−ω mixing contribution to the branching ratio, since we are working on the first order of isospin violation. The derivation is straightforward, and we can explicitly express the branching ratio for the processes B¯s0⟶ρ0ωK∗0 [28, 82]:
(59)BRB¯s0⟶ρ0ωK∗0=τBs0Pc8πMBs2Hρω02+Hρω+2+Hρω−2,where τBs0 is the lifetime of the Bs meson and
(60)Hρωii=0,+,−=VubVud∗Tρi−VtbVtd∗Pρi+VubVud∗Tωi−VtbVtd∗PωiΠ~ρωsρ−Mω2+iMωΓωtake into account the helicity amplitudes of the ρ meson and ω meson contribution involved in the tree and penguin diagrams.


## 5. Input Parameters

The CKM matrix, which elements are determined from experiments, can be expressed in terms of the Wolfenstein parametersA, ρ, λ, and η [77, 83]:
(61)1−12λ2λAλ3ρ−iη−λ1−12λ2Aλ2Aλ31−ρ−iη−Aλ21,where Oλ4 corrections are neglected. The latest values for the parameters in the CKM matrix are [84]
(62)λ=0.22650±0.00048,A=0.790−0.012+0.017,(63)ρ¯=0.141−0.017+0.016,η¯=0.357±0.011,where
(64)ρ¯=ρ1−λ22,η¯=η1−λ22.From Eqs. (63) and (64), we have
(65)0.127<ρ<0.161,0.355<η<0.378.The other parameters and the corresponding references are listed in Table1.Table 1
Input parameters.
ParametersInput dataReferencesFermi constant (inGeV−2)GF=1.16638×10−5.[84]MBs0=5366.88,τBs0=1.515×10−12s.Mρ0770=775.26,Γρ0770=149.1.Masses and decay widths (in MeV)Mω782=782.65,Γω782=8.49.[84]Mπ=139.57,MK∗=895.55.fρ=215.6±5.9,fρT=165±9.Decay constants (in MeV)fω=196.5±4.8,fωT=145±10.[65, 85, 86]fK∗=217±5,fK∗T=185±10.


## 6. The Numerical Results ofCP Violation and Branching Ratio

### 6.1.CP Violation via ρ−ω Mixing in B¯s0⟶ρ0ωK∗0⟶π+π−K∗0

We have investigated theCP violating asymmetry, ACP, for the B¯s0⟶ρ0ωK∗0⟶π+π−K∗0 of the three-body decay process in the perturbative QCD. The numerical results of the CP violating asymmetry are shown for the B¯s0⟶ρ0ωK∗0⟶π+π−K∗0 decay process in Figure 3. It is found that the CP violation can be enhanced via ρ−ω mixing for the decay channel B¯s0⟶ρ0ωK∗0⟶π+π−K∗0 when the invariant mass of π+π− pair is in the vicinity of the mω resonance within perturbative QCD scheme.Figure 3
TheCP violation, Acp, as a function of s for different CKM matrix elements. The solid line, dotted line, and dashed line correspond to the maximum, middle, and minimum CKM matrix elements for the decay channel of B¯s0⟶ρ0ωK∗0⟶π+π−K∗0, respectively.TheCP violating asymmetry depends on the weak phase difference ϕ from CKM matrix elements and the strong phase difference δ in the Eq. (34). The CKM matrix elements, which relate to ρ¯, A, η¯, and λ, are given in Eq. (63). The uncertainties due to the CKM matrix elements are mostly from ρ and η since λ is well determined. Hence, we take the central value of λ=0.226 in Eq. (65). In the numerical calculations for the B¯s0⟶ρ0ωK∗0⟶π+π−K∗0 decay process, we use ρ, η, and λ=0.226 which vary among the limiting values. The numerical results are shown from Figure 3 with the different parameter values of CKM matrix elements. The solid line, dotted line, and dashed line correspond to the maximum, middle, and minimum CKM matrix element for the decay channel of B¯s0⟶ρ0ωK∗0⟶π+π−K∗0, respectively. We find that the numerical results of the CP violation are not sensitive to the CKM matrix elements for the different values of ρ and η. In Figure 3, we show the plot of CP violation as a function of s in the perturbative QCD. From the figure, one can see the CP violation parameter is dependent on s and changes rapidly by the ρ−ω mixing mechanism when the invariant mass of π+π− pair is in the vicinity of the mω resonance. From the numerical results, it is found that the CP violating asymmetry is large and ranges from -50.19% to 43.02% via the ρ−ω mixing mechanism for the process. The maximum CP violating parameter can reach -48.22−2.04+1.97% for the decay channel of B¯s0⟶π+π−K∗0 in the case of (ρ, η). This error corresponds to the CKM parameters.From Eq. (34), one can find that the CP violating parameter is related to r and sinδ. In Figures 4 and 5, we show the plots of sinδ0 (sinδ+ and sinδ−) and r0 (r+ and r−) as a function of s, respectively. We can see that the ρ−ω mixing mechanism produces a large sinδ0 (sinδ+ and sinδ−) in the vicinity of the ω resonance. As can be seen from Figure 4, the plots vary sharply in the cases of sinδ0, sinδ+ and sinδ− in the range of the resonance. Meanwhile, sinδ+ and sinδ− change weakly compared with the sinδ0. It can be seen from Figure 5 that r0, r+, and r− change more rapidly when the π+π− pairs in the vicinity of the ω resonance. Since amplitude AT is a small quantity, it contributes so little to the transverse H+ and H− of the helicity amplitudes in which they are almost equal in Eq. (27). So, we can see that the two curves sin (δ+) and (δ−) of Figure 4 are the different in the region of resonance but almost the same in other regions.Figure 4
Plot ofsinδ as a function of s corresponding to central parameter values of CKM matrix elements for B¯s0⟶ρ0ωK∗0⟶π+π−K∗0. The solid line, dotted line, and dashed line correspond to sinδ0, sinδ+, and sinδ−, respectively.Figure 5
Plot ofr as a function of s corresponding to central parameter values of CKM matrix elements for B¯s0⟶ρ0ωK∗0⟶π+π−K∗0. The solid line, dotted line, and dashed line correspond to r0, r+, and r−, respectively.We have shown that theρ−ω mixing does enhance the direct CP violating asymmetry and provide a mechanism for large CP violation in the perturbative QCD factorization scheme. In other words, it is important to see whether it is possible to observe this large CP violating asymmetry in the experiments. This depends on the branching ratio for the decay channel of B¯s0⟶ρ0K∗0. We will study this problem in the next section.


### 6.2. Branching Ratio viaρ−ω Mixing in B¯s0⟶ρ0ωK∗0

In the pQCD, we calculate the value of the branching ratio viaρ−ω mixing mechanism for the decay channel B¯s0⟶ρ0ωK∗0. The numerical result is shown for the decay process in Figure 6. Based on a reasonable parameter range, we obtain the maximum branching ratio of B¯s0⟶ρ0ωK∗0 as (3.05−0.20+0.25)×10−7, which is consistent with the result in [81, 87]. The error comes from CKM parameters. On the one hand, although we calculate the branching ratio due to ρ−ω mixing in the pQCD factorization scheme, we find that the contribution of ρ−ω mixing to the branching ratio of B¯s0⟶ρ0ωK∗0 is small and can be neglected. This is because branching ratio is proportional to the contribution of Hρωii=0,+,−2 in Eq. (59). The square of the ρ−ω mixing parameter Π~ρω2 is about 10−6 GeV2, which leads to large suppression for the contribution of the branching ratio of B¯s0⟶ωK∗0 and the effect of the width Γρ of the resonances ρ from sρ in Eq. (60). On the other hand, the ρ−ω mixing mechanism produces new strong phase differences. In B¯s0⟶ρ0ωK∗0 decay, we take into account the ρ−ω mixing contribution to the branching ratio, and we find that the two decays interfere with each other to generate a new resonance region around 1.1 GeV. By expanding the factor Π~ρω/sρ−Mω2+iMωΓω from Eq. (60), the denominator sρ−Mω2+iMωΓω produces a new propagator s−Mρ2+Mω2+iMρΓρ+MωΓω. This is the reason why the dips of the curves in Figure 6 does not appear in the region of resonance but around 1.1 GeV Mρ2+Mω2≈1.1GeV.Figure 6
The branching ratio,BRB¯s0⟶ρωK∗0, as a function of s for the different CKM matrix elements. The solid line, dotted line, and dashed line correspond to the maximum, middle, and minimum CKM matrix elements for the decay channel of B¯s0⟶ρ0ωK∗0, respectively.The Large Hadron Collider (LHC) is a proton-proton collider that has started at the European Organization for Nuclear Research (CERN). With the designed center-of-mass energy 14 TeV and luminosityL=1034cm−2s−1, the LHC provides a high energy frontier at TeV-level scale and an opportunity to further improve the consistency test for the CKM matrix. LHCb is a dedicated heavy flavor physics experiments and one of the main projects of LHC. Its main goal is to search for indirect evidence of new physics in CP violation and rare decays in the interactions of beauty and charm hadrons systems, by looking for the effects of new particles in decay processes that are precisely predicted in the SM. Such studies can help us to comprehend the matter-antimatter asymmetry of the universe. Recently, the LHCb collaboration found clear evidence for direct CP violation in some three-body decay channels of B meson. Large CP violation is obtained for the decay channels of B±⟶π±π+π− in the localized phase spaces region mπ+π−low2<0.4 GeV2 and mπ+π−high2>15 GeV2 [7, 88]. A zoom of the π+π− invariant mass from the B+⟶π+π+π− decay process is shown the region mπ+π−low2<1 GeV2 zone in Ref. [88]. In addition, the branching ratio of B¯s0⟶π+π−ϕ is probed in the π+π− invariant mass range 400<mπ+π−<1600 MeV/c2 [89]. In the next years, we expect the LHCb collaboration to collect date for detecting our prediction of CP violation from the B¯s0⟶ρ0ωK∗0⟶π+π−K∗0 decay process when the invariant mass of π+π− is in the vicinity of the ω resonance.At the LHC, theb hadrons come from pp collisions. The possible asymmetry between the numbers of the b hadrons Hb and those of their antiparticles H¯b has been studied by using the intrinsic heavy quark model and the Lund string fragmentation model [90, 91]. It has been shown that this asymmetry can only reach values of a few percent. In the following discussion, we will ignore this small asymmetry and give the numbers of HbH¯b pairs needed for observing our prediction of the CP violating asymmetries. These numbers depend on both the magnitudes of the CP violating asymmetries and the branching ratios of heavy hadron decays which are model dependent. For one-standard deviation (1σ) signature and three-standard deviation (3σ) signature and the numbers of HbH¯b pairs, we need [92–94]
(66)NHbH¯b1σ~1BRB¯s0⟶ρ0K∗0ACP21−ACP2,NHbH¯b3σ~9BRB¯s0⟶ρ0K∗0ACP21−ACP2,where ACP is the CP violation in the process of B¯s0⟶ρ0ωK∗0⟶π+π−K∗0. Now, we can estimate the possibility to observe CP violation. The branching ratio for B¯s0⟶ρ0ωK∗0 is of the order 10−7, and then, the number NHbH¯b1σ is 106 for 1σ signature and 107 for 3σ signature, theoretically, in order to achieve the current experiments on b hadrons, which can only provide about 107BsB¯s pairs. Therefore, it is very possible to observe the large CP violation for B¯s0⟶ρ0ωK∗0⟶π+π−K∗0 when the invariant masses of π+π− pairs are in the vicinity of the ω resonance in experiments at the LHC.


## 6.1.CP Violation via ρ−ω Mixing in B¯s0⟶ρ0ωK∗0⟶π+π−K∗0

We have investigated theCP violating asymmetry, ACP, for the B¯s0⟶ρ0ωK∗0⟶π+π−K∗0 of the three-body decay process in the perturbative QCD. The numerical results of the CP violating asymmetry are shown for the B¯s0⟶ρ0ωK∗0⟶π+π−K∗0 decay process in Figure 3. It is found that the CP violation can be enhanced via ρ−ω mixing for the decay channel B¯s0⟶ρ0ωK∗0⟶π+π−K∗0 when the invariant mass of π+π− pair is in the vicinity of the mω resonance within perturbative QCD scheme.Figure 3
TheCP violation, Acp, as a function of s for different CKM matrix elements. The solid line, dotted line, and dashed line correspond to the maximum, middle, and minimum CKM matrix elements for the decay channel of B¯s0⟶ρ0ωK∗0⟶π+π−K∗0, respectively.TheCP violating asymmetry depends on the weak phase difference ϕ from CKM matrix elements and the strong phase difference δ in the Eq. (34). The CKM matrix elements, which relate to ρ¯, A, η¯, and λ, are given in Eq. (63). The uncertainties due to the CKM matrix elements are mostly from ρ and η since λ is well determined. Hence, we take the central value of λ=0.226 in Eq. (65). In the numerical calculations for the B¯s0⟶ρ0ωK∗0⟶π+π−K∗0 decay process, we use ρ, η, and λ=0.226 which vary among the limiting values. The numerical results are shown from Figure 3 with the different parameter values of CKM matrix elements. The solid line, dotted line, and dashed line correspond to the maximum, middle, and minimum CKM matrix element for the decay channel of B¯s0⟶ρ0ωK∗0⟶π+π−K∗0, respectively. We find that the numerical results of the CP violation are not sensitive to the CKM matrix elements for the different values of ρ and η. In Figure 3, we show the plot of CP violation as a function of s in the perturbative QCD. From the figure, one can see the CP violation parameter is dependent on s and changes rapidly by the ρ−ω mixing mechanism when the invariant mass of π+π− pair is in the vicinity of the mω resonance. From the numerical results, it is found that the CP violating asymmetry is large and ranges from -50.19% to 43.02% via the ρ−ω mixing mechanism for the process. The maximum CP violating parameter can reach -48.22−2.04+1.97% for the decay channel of B¯s0⟶π+π−K∗0 in the case of (ρ, η). This error corresponds to the CKM parameters.From Eq. (34), one can find that the CP violating parameter is related to r and sinδ. In Figures 4 and 5, we show the plots of sinδ0 (sinδ+ and sinδ−) and r0 (r+ and r−) as a function of s, respectively. We can see that the ρ−ω mixing mechanism produces a large sinδ0 (sinδ+ and sinδ−) in the vicinity of the ω resonance. As can be seen from Figure 4, the plots vary sharply in the cases of sinδ0, sinδ+ and sinδ− in the range of the resonance. Meanwhile, sinδ+ and sinδ− change weakly compared with the sinδ0. It can be seen from Figure 5 that r0, r+, and r− change more rapidly when the π+π− pairs in the vicinity of the ω resonance. Since amplitude AT is a small quantity, it contributes so little to the transverse H+ and H− of the helicity amplitudes in which they are almost equal in Eq. (27). So, we can see that the two curves sin (δ+) and (δ−) of Figure 4 are the different in the region of resonance but almost the same in other regions.Figure 4
Plot ofsinδ as a function of s corresponding to central parameter values of CKM matrix elements for B¯s0⟶ρ0ωK∗0⟶π+π−K∗0. The solid line, dotted line, and dashed line correspond to sinδ0, sinδ+, and sinδ−, respectively.Figure 5
Plot ofr as a function of s corresponding to central parameter values of CKM matrix elements for B¯s0⟶ρ0ωK∗0⟶π+π−K∗0. The solid line, dotted line, and dashed line correspond to r0, r+, and r−, respectively.We have shown that theρ−ω mixing does enhance the direct CP violating asymmetry and provide a mechanism for large CP violation in the perturbative QCD factorization scheme. In other words, it is important to see whether it is possible to observe this large CP violating asymmetry in the experiments. This depends on the branching ratio for the decay channel of B¯s0⟶ρ0K∗0. We will study this problem in the next section.


## 6.2. Branching Ratio viaρ−ω Mixing in B¯s0⟶ρ0ωK∗0

In the pQCD, we calculate the value of the branching ratio viaρ−ω mixing mechanism for the decay channel B¯s0⟶ρ0ωK∗0. The numerical result is shown for the decay process in Figure 6. Based on a reasonable parameter range, we obtain the maximum branching ratio of B¯s0⟶ρ0ωK∗0 as (3.05−0.20+0.25)×10−7, which is consistent with the result in [81, 87]. The error comes from CKM parameters. On the one hand, although we calculate the branching ratio due to ρ−ω mixing in the pQCD factorization scheme, we find that the contribution of ρ−ω mixing to the branching ratio of B¯s0⟶ρ0ωK∗0 is small and can be neglected. This is because branching ratio is proportional to the contribution of Hρωii=0,+,−2 in Eq. (59). The square of the ρ−ω mixing parameter Π~ρω2 is about 10−6 GeV2, which leads to large suppression for the contribution of the branching ratio of B¯s0⟶ωK∗0 and the effect of the width Γρ of the resonances ρ from sρ in Eq. (60). On the other hand, the ρ−ω mixing mechanism produces new strong phase differences. In B¯s0⟶ρ0ωK∗0 decay, we take into account the ρ−ω mixing contribution to the branching ratio, and we find that the two decays interfere with each other to generate a new resonance region around 1.1 GeV. By expanding the factor Π~ρω/sρ−Mω2+iMωΓω from Eq. (60), the denominator sρ−Mω2+iMωΓω produces a new propagator s−Mρ2+Mω2+iMρΓρ+MωΓω. This is the reason why the dips of the curves in Figure 6 does not appear in the region of resonance but around 1.1 GeV Mρ2+Mω2≈1.1GeV.Figure 6
The branching ratio,BRB¯s0⟶ρωK∗0, as a function of s for the different CKM matrix elements. The solid line, dotted line, and dashed line correspond to the maximum, middle, and minimum CKM matrix elements for the decay channel of B¯s0⟶ρ0ωK∗0, respectively.The Large Hadron Collider (LHC) is a proton-proton collider that has started at the European Organization for Nuclear Research (CERN). With the designed center-of-mass energy 14 TeV and luminosityL=1034cm−2s−1, the LHC provides a high energy frontier at TeV-level scale and an opportunity to further improve the consistency test for the CKM matrix. LHCb is a dedicated heavy flavor physics experiments and one of the main projects of LHC. Its main goal is to search for indirect evidence of new physics in CP violation and rare decays in the interactions of beauty and charm hadrons systems, by looking for the effects of new particles in decay processes that are precisely predicted in the SM. Such studies can help us to comprehend the matter-antimatter asymmetry of the universe. Recently, the LHCb collaboration found clear evidence for direct CP violation in some three-body decay channels of B meson. Large CP violation is obtained for the decay channels of B±⟶π±π+π− in the localized phase spaces region mπ+π−low2<0.4 GeV2 and mπ+π−high2>15 GeV2 [7, 88]. A zoom of the π+π− invariant mass from the B+⟶π+π+π− decay process is shown the region mπ+π−low2<1 GeV2 zone in Ref. [88]. In addition, the branching ratio of B¯s0⟶π+π−ϕ is probed in the π+π− invariant mass range 400<mπ+π−<1600 MeV/c2 [89]. In the next years, we expect the LHCb collaboration to collect date for detecting our prediction of CP violation from the B¯s0⟶ρ0ωK∗0⟶π+π−K∗0 decay process when the invariant mass of π+π− is in the vicinity of the ω resonance.At the LHC, theb hadrons come from pp collisions. The possible asymmetry between the numbers of the b hadrons Hb and those of their antiparticles H¯b has been studied by using the intrinsic heavy quark model and the Lund string fragmentation model [90, 91]. It has been shown that this asymmetry can only reach values of a few percent. In the following discussion, we will ignore this small asymmetry and give the numbers of HbH¯b pairs needed for observing our prediction of the CP violating asymmetries. These numbers depend on both the magnitudes of the CP violating asymmetries and the branching ratios of heavy hadron decays which are model dependent. For one-standard deviation (1σ) signature and three-standard deviation (3σ) signature and the numbers of HbH¯b pairs, we need [92–94]
(66)NHbH¯b1σ~1BRB¯s0⟶ρ0K∗0ACP21−ACP2,NHbH¯b3σ~9BRB¯s0⟶ρ0K∗0ACP21−ACP2,where ACP is the CP violation in the process of B¯s0⟶ρ0ωK∗0⟶π+π−K∗0. Now, we can estimate the possibility to observe CP violation. The branching ratio for B¯s0⟶ρ0ωK∗0 is of the order 10−7, and then, the number NHbH¯b1σ is 106 for 1σ signature and 107 for 3σ signature, theoretically, in order to achieve the current experiments on b hadrons, which can only provide about 107BsB¯s pairs. Therefore, it is very possible to observe the large CP violation for B¯s0⟶ρ0ωK∗0⟶π+π−K∗0 when the invariant masses of π+π− pairs are in the vicinity of the ω resonance in experiments at the LHC.


## 7. Summary and Conclusion

In this paper, we have studied the directCP violation for the decay process of B¯s0⟶ρ0ωK∗0⟶π+π−K∗0 in perturbative QCD. It has been found that, by using ρ−ω mixing, the CP violation can be enhanced at the area of ω resonance. There is the resonance effect via ρ−ω mixing which can produce large strong phase in this decay process. As a result, one can find that the maximum CP violation can reach -50.19% when the invariant mass of the π+π− pair is in the vicinity of the ω resonance. Furthermore, taking ρ−ω mixing into account, we have calculated the branching ratio of the decays of B¯s0⟶ρ0ωK∗0. We have also given the numbers of BsB¯s pairs required for observing our prediction of the CP violating asymmetries at the LHC experiments.Theρ−ω mixing is a small effect duo to the isospin violation. One can estimate the contributions by comparing the two terms on the right-hand side of Eqs. (38) and (39). However, when the invariant mass squared of the π+π− pair is in the vicinity of the omega, sω~imωΓω, it becomes comparable with Π~ρω. In other words, the ρ−ω mixing becomes important in the vicinity of the omega. This is also the reason why we only see large CP violation in the vicinity of ω. At the same time, the mixing parameter has determined the magnitude, and s dependence of the effective ρ−ω mixing matrix element Π~ρωs fits to e+e−⟶π+π− data in the vicinity of the ω resonance. We can make the Taylor expansion for Π~ρωs at s=mw2 and ignore the s dependence of Π~ρωs in practice. It will cause a large resonance near invariant masses of the π+π− pairs, and the effect will be negligible in distant regions. In the ρ−ω mechanism, we will ignore the contribution of the final state interaction between the pions in the resonant regions associated with ρ and ω in the decay process of B¯s0⟶ρ0ωK∗0⟶π+π−K∗0.In our calculation, there are some uncertainties. The major uncertainties come from the input parameters. In particular, these include the CKM matrix element, the particle mass, the perturbative QCD approach, and the hadronic parameters (decay constants, the wave functions, the shape parameters, etc.). We expect that our predictions will provide useful guidance for future experiments.



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