# DirectCP Violation from Isospin Symmetry Breaking Effects for the Decay Process B-s→P(V)π0 in PQCD

Authors: Gang Lü; Qin-Qin Zhi
Journal: Advances in High Energy Physics
 (2019)

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## Abstract

We investigate the directCP violation for the decay process of B-s→P(V)π0 (P,V refer to the pseudoscalar meson and vector meson, resp.) via isospin symmetry breaking effects from the π0-η-η′ mixing mechanism in PQCD factorization approach. Isospin symmetry breaking arises from the electroweak interaction and the u-d quark mass difference by the strong interaction, which are known to be tiny. However, we find that isospin symmetry breaking at the leading order shifts the CP violation due to the new strong phases.


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## Body

## 1. Introduction

The measurement ofCP violation is an important area in understanding Standard Model (SM) and exploring new physics signals. Cabibbo-Kobayashi-Maskawa (CKM) matrix [1, 2] due to the quark flavour mixing provides us with the weak phases. The weak phase associated with the strong phase is responsible for the source of the CP violation. The strong phase comes from the dynamics of QCD and the other mechanism.The hadronic matrix elements of the nonleptonic weak decay are known to be associated with the strong phase. We can estimate the power contribution by the factorization method in the limit of1/mb (mb refers to b quark mass) in B meson decay process. Based on the QCD correction and taking into account transverse momenta, PQCD factorization method safely avoids the infrared divergence by introducing the Sudakov factor which is applied to deal with the decay amplitude related with the hadronic matrix elements. The decay amplitude can be written as the convolution of the meson wave functions and the hard kernel, which show the contributions of the non-perturbative and the perturbative parts, respectively [3–11].Isospin symmetry plays an important part in the weak decay process of B meson. We can infer sum rule associated with the isospin symmetry to form a triangular shape on a complex plane for the decay amplitude. One can eliminate uncertainty from the penguin diagram by the isospin analysis in B decays [12]. Isospin symmetry breaking via ρ-ω mixing produces the strong phase to lead to the large CP violation in the three bodies decay process [13, 14]. Isospin symmetry is approximate symmetry due to identical u and d quark masses in Standard Model (SM). The mixing of pseudoscalar mesons π0-η-η′ is from the isospin symmetry breaking within QCD. Isospin symmetry breaking plays a significant role for the decays of B→ππ, which breaks the triangle relationship in the framework of generalized factorization [15]. π0-η-η′ mixing is discussed by the model-independent way in B→ππ decay process using flavor SU(3) symmetry [16]. The quark-flavor mixing produces the π0-η-η′ mixing due to the isospin symmetry breaking [17]. Recently, isospin symmetry breaking is discussed by incorporating the Nambu-Jona-Lasinio model in a generalized multiquark interaction scheme [18]. However, one can find that the research rarely pays attention to the CP violation from the effect of isospin symmetry breaking via π0-η-η′ mixing. The strong phase may be introduced to affect the value of CP violation accordingly which is similar to the contribution from the isospin symmetry breaking by the ρ-ω mixing [13, 14].The remainder of this paper is organized as follows. In Section2 we present the form of the effective Hamiltonian. In Section 3 we give the calculating formalism of CP violation from isospin symmetry breaking in B-s→P(v)π0. Input parameters are presented in Section 4. We present the numerical results in Section 5. Summary and discussion are included in Section 6. The related functions defined in the text are given in the Appendix.


## 2. The Effective Hamiltonian

With the operator product expansion, the effective weak Hamiltonian can be written as [19](1)HΔB=1=GF2VubVud∗C1O1u+C2O2u-VtbVtd∗∑i=310CiOi+H.C.,where GF represents Fermi constant, Ci (i=1,…,10) are the Wilson coefficients, Vub, Vud, Vtb and Vtd are the CKM matrix elements. The operators Oi have the following forms: (2)O1u=d-αγμ1-γ5uβu-βγμ1-γ5bα,O2u=d-γμ1-γ5uu-γμ1-γ5b,O3=d-γμ1-γ5b∑q′q-′γμ1-γ5q′,O4=d-αγμ1-γ5bβ∑q′q-β′γμ1-γ5qα′,O5=d-γμ1-γ5b∑q′q-′γμ1+γ5q′,O6=d-αγμ1-γ5bβ∑q′q-β′γμ1+γ5qα′,O7=32d-γμ1-γ5b∑q′eq′q-′γμ1+γ5q′,O8=32d-αγμ1-γ5bβ∑q′eq′q-β′γμ1+γ5qα′,O9=32d-γμ1-γ5b∑q′eq′q-′γμ1-γ5q′,O10=32d-αγμ1-γ5bβ∑q′eq′q-β′γμ1-γ5qα′.where α and β are color indices, and q′=u,d or s quarks. In Eq.(2) O1u and O2u are tree operators, O3–O6 are QCD penguin operators and O7–O10 are the operators associated with electroweak penguin diagrams.We can obtain numerical values ofCi. When Ci(mb) [11], (3)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.One can obtain numerical values ofai. The combinations ai of Wilson coefficients are defined as [6–8](4)a1=C2+C13,a2=C1+C23,a3=C3+C43,a4=C4+C33,a5=C5+C63,a6=C6+C53,a7=C7+C83,a8=C8+C73,a9=C9+C103,a10=C10+C93.


## 3.CP Violation from Isospin Symmetry Breaking Effects

### 3.1. Formalism

It is convenient to introduce isospin vector tripletπ3, isospin scalar ηn and isospin scalar ηs which can be distinguished by including strange quark or not. The SU(3) singlet η0 and octet η8 can be well described by the translation ηn=2η0+η8/3 and ηs=1/3η0-2/3η8. The states of π3, ηn and ηs are identified by π3=1/2uu--dd-, ηn=1/2uu-+dd- and ηs=ss- which are obtained from the quark model, respectively. The physical meson states can be transformed from the π3, ηn and ηs by unitary matrix U [17]: (5)π0ηη′=Uε1,ε2,ϕπ3ηnηs,where (6)Uε1,ε2,ϕ=1ε1+ε2cos⁡ϕ-ε2sin⁡ϕ-ε2-ε1cos⁡ϕcos⁡ϕ-sin⁡ϕ-ε1sin⁡ϕsin⁡ϕcos⁡ϕ,ε1, ε2∝O(λ),λ≪1 and the higher order terms are neglected. In the isospin limit of ε1→0,ε2→0, we can find that the formula is expressed as the η-η′ mixing in Eq.(7): (7)ηη′=U′ϕηnηs=cos⁡ϕ-sin⁡ϕsin⁡ϕcos⁡ϕηnηs,where ϕ is the mixing angle [20]. The η and η′ mixing depends on the quark flavor basises ηn and ηs.The relevant decay constants can be written as [21, 22] (8)0n-γμγ5nηnP=i2fnPμ,0s-γμγ5sηsP=ifsPμ,where P refers to the momenta of ηn or ηs.One can understand that isospin symmetry breaking comes from the electroweak interaction andu-d quark mass difference in Standard Model. We can calculate the isospin symmetry breaking correction by chiral perturbative theory which induces the π0-η-η′ mixing. To the leading order of isospin symmetry breaking, the physical eigenstate π0, η and η′ from Eq.(5)(6) can be written as (9)π0=π3+ε1+ε2cos⁡ϕηn-ε2sin⁡ϕηs,η=-ε2-ε1cos⁡ϕπ3+cos⁡ϕηn-sin⁡ϕηs,η′=-ε1sin⁡ϕπ3+sin⁡ϕηn+cos⁡ϕηs,One can define ε=ε2+ε1cos⁡ϕ, ε′=ε1sin⁡ϕ. π3 refer to the isospin I=1 component in the triplet. We use the values of ε=0.017±0.002, ε′=0.004±0.001, ϕ=39.0∘ [17].The mixing is generated by the strong interaction fromO(md-mu), where we assert ε, ε′~O((md-mu)/Λhad) or O(α) neglecting higher-order terms for the isospin breaking. The pseudoscalar meson octet in current algebra[23] and the lowest order chiral perturbation theory[24] discuss the π0-η8 mixing angle which can be written as 3/4(md-mu/ms-m^) with m^=(mu+md)/2, so that the scale is expeted as Λhad~O(ms).Generally, the mesonsη and η′ are composed of u-u, d-d, s-s linear combination. One believes that there is gluon in the η′ meson for explaining the large branching ratios for the decay processes B→Kη′ and J/ψ→η′γ [25, 26]. H.N. Li et al. find the tiny contribution from the gluonic admixture [27, 28]. Z.J. Xiao et al. calculate the branching ratios and CP violations in PQCD for the decay processes B→Mη′ and show that the theoretical result is in agreement with the experimental one neglecting the contribution of gluon [29–31]. Thomas shows the contribution from composition of gluon does not make obvious impact on fitted values on the basis of existing experiment data [32].For theBs meson function, we use the model [20, 33] (10)ϕBsx,b=NBsx21-x2exp⁡-MBs2x22ωb2-12ωbb2,where the normalization factor NBs is dependent on the free parameter ωb. b is the conjugate variable of the parton transverse momenta kT. MBs refers to the mass of the Bs meson. For the Bs meson, one can obtain the value of ωb=0.50±0.05 from the light cone sum rule [34].The wave functions ofηn and ηs are the same in form and can be defined as [26] (11)ΦP,x,ζ≡12NCγ5PϕAx+m0ϕPx+ζnυ-υ·nm0ϕTxwhere P and x are the momentum and the momentum fraction of ηn(ηs), respectively. Following Ref.[26], we assumed here that the wave function of ηn(ηs) is the same as the π wave function based on SU(3) flavor symmetry. Depending on the assignment of the momentum fraction x, the parameter ζ should be chosen as +1 or -1. In this paper, we will use those distribution amplitudes [20]: (12)ϕπAx=3fπ6x1-x1+0.44C23/2t,ϕπPx=fπ261+0.43C21/2t,ϕπTx=-fπ26C11/2t+0.55C31/2t,ϕKAx=3fK6x1-x1+0.17C13/2t+0.2C23/2t,ϕKPx=fK261+0.24C21/2t,ϕKTx=-fK26C11/2t+0.35C31/2t,ϕϕ=3fϕ6x1-x1+0.18C23/2t,where t=2x-1. fP(v) are the decay constants of scalar (vector) mesons, respectively. The pseudoscalar mesons π,η and η′ have the similar wave functions. The expressions of amplitudes can be obtained by the replacements ϕπ→ϕη,ϕπp→ϕηp,ϕπt→ϕηt. Gegenbauer polynomials are defined as (13)C11/2t=t,C13/2t=3tC21/2t=123t2-1,C23/2t=325t2-1,C31/2t=12t5t2-3.The wave function of the meson is applied to describe the formation of hadron from the positive and negative quarks, which provides distribution of the momentum carried by the parton. It is non-perturbative and process independent from partons to hadrons. The wave function of the meson is transverse-momentum-dependent wave fucnction in PQCD. The results show that there is large contribution from transverse-momentum for the heavy meson function. However, the wave function of light meson is less reliant on transverse-momentum. Currently, it is reasonable that the results form Light-cone QCD sum rule and Lattice QCD. The wave function of light pseudoscalar meson is usually obtained by Light-cone QCD sum rule. The branching ratios and CP violations are calculated in the framework of PQCD. The theoretical prediction is in good agreement with the experimental results for the most decay processes ofB/Bs→M2M3. Hence, the wave function is credible. The transverse-momentum-dependent (TMD) wave fucnction is given with simpler soft subtraction in [35] which presents the more accurate results.The transverse momentum cannot be neglected for eliminating the divergence in the endpoints. The double logarithms will be obtained when collinear divergent is overlapping with soft divergen considering radiative corrections for the form factor. For the effective perturbative expansion, we need to sum the double logarithmic terms by the resummation technique [36]. The wave function and hard kernel can absorb collinear divergent and infrared divergence, respectively. The other infrared divergences are cancelled out. At the range of large b, we need to sum the double logarithmic terms by renormalization group. Hence, the sudakov factor is introduced. Due to the suppression of sudakov factor, the non-perturbative contribution becomes very small. The amplitude of hard kernal produces the double logarithmic terms from radiative corrections. The threshold resummation introduces the sudakov factor which drops rapidly at the endpoints. Hence, perturbative calculation plays the dominant role when the endpoint singularity is suppressed.Recently, an evolution for the B meson wave fucntions are constructed in thekt factorization theorem, whose solutions sum the double logarithms with the light-cone singularities, namely, the rapidity logarithms. The effective heavy-quark field is involved for an energetic light hadron comparing with the Sudakov resummation. The resummation effect maintians the normalization of the B meson wave functions and strengths their convergent behavior at small momentum, which combines the threshold and Kt resummations [37]. Due to the resummation technique with off-light-cone Wilson lines, an evolution for the pion meson wave fucntions is introduced in the Kt factorization theorem, whose solutions sum the mixed logarithm ln⁡xln⁡kt to all orders, with x(kt) being a parton momentum fraction (transverse momentum). This joint resummation shows strong suppression of the pion wave fucnction in the small x and large b regions, b being the impact parameter conjugate to kt. The next-to-leading-order contributions are different from those under the conventional resummations at high energy. The scheme-independent formalism can be extended to the kt factorization of more complicated exclusive processes. [38]. The improvement of the wave function can be applied to a more accurate discussion for the next step.


### 3.2. Calculation Details

In the framework of PQCD, we can calculate theCP violation for the decay process B-s→P(V)π0 via π0-η-η′ mixing. Firstly, we calculate the amplitudes T and P, which can be decomposed in terms of tree and penguin contributions depending on the CKM matrix elements VubVud∗ and VtbVtd∗. Next, we take the decay process B-s→K0π0 and B-s→π0η(′) as examples for the study of the π0-η-η′ mixing mechanism.


#### 3.2.1. TheCP Violation for the Decay Modes of B-s→P(V)π0 Except B-s→π0η′

We take the decay process ofB-s→K0π0 as example to introduce the CP violation via π0-η-η′ mixing. The decay amplitude A of B-s→K0π0 in PQCD can be written as (14)2AB-s→K0π0=VubVud∗T1-VtbVtd∗P1,where T1 and P1 are the amplitudes form tree and penguin contributions, respectively. The tree level amplitude T1 can be given as (15)T1=fπFBs→KLLa2+MBs→KLLC2,and the penguin level amplitude P1 can be written as (16)P1=fπFBs→KLL-a4-32a7+32a9+12a10+fπFBs→KSP-a6+12a8+MBs→KLL-C3+32c8+12C9+32c10+fBsFannSP-a6+12a8+fBsFannLL-a4+12a10+MannLL-C3+12C9+MannLR-C5+12C7,where the fi refers to the decay constant. The individual decay amplitudes in the above equations, such as FBs→KLL, FBs→KSP, MBs→KLL, FannSP, Fann, MannLL and MannLR arise from the (V-A)(V-A), (V-A)(V+A) and (S-P)(S+P) operators, respectively, and will be given in Appendix.Based on the CKM matrix elements ofVubVud∗ and VtbVtd∗, we can express the decay amplitudes as follows: (17)2AB-s→ηK0=VubVud∗Tn-VtbVtd∗Pn,(18)2AB-s→η′K0=VubVud∗Ts-VtbVtd∗Ps.The contributions of Tn and Pn for the decay amplitudes B-s→ηK0 can be written as (19)Tn=fnFBs→KLLa2+MBs→KLLC2,(20)Pn=fnFBs→KSPa6-12a8+fnFBs→KLL2a3+a4-2a5-12a7+12a9-12a10+fBsFannLLa4-12a10+MBs→KLLC3+2C4+12C8-12C9+12C10+fBsFannSPa6-12a8+MannLLC3-12C9+MannLRC5-12C7,(21)Ts=0,(22)Ps=fsFBs→KLLa3-a5+12a7-12a9+fKFBs→ηsLLa4-12a10+fKFBs→ηsSPa6-12a8+MBs→KLLC4+C4-12C8-12C10+MBs→ηsLLC3-12C9+MBs→ηsLRC5-12C7+fBsFannLLa4-12a10+fBsFannSPa6-12a8+MannLLC3-12C9+MannLRC5-12C7,for the formula of 2A(B-s→η′K0)=VubVud∗Ts-VtbVtd∗Ps.The amplitudes T and P from the decay process ofB-s→K0π0 with π0-η-η′ mixing can be written as (23)T=T1+ε1+ε2cos⁡ϕTn-ε2sin⁡ϕTs,P=P1+ε1+ε2cos⁡ϕPn-ε2sin⁡ϕPs.One can see that the Eq.(23) without π0-η-η′ mixing is reduced to (24)T=T1,P=P1,which are expressed in Eq.(15) and Eq.(16).The relevant weak phaseθ and strong phase δ are obtained as follows: (25)reiδeiθ=PT×VtbVtd∗VubVud∗,where the parameter r represents the absolute value of the ratio of penguin and tree amplitudes: (26)r≡K0π0B¯s0HPB¯s0B¯s0K0π0B¯s0HTB¯s0B¯s0.The strong phase associated with r can be given as (27)reiδ=PT×VtbVtd∗VubVud∗=rcos⁡δ+irsin⁡δ,where (28)VtbVtd∗VubVud∗=ρ1-ρ-η22+η21-λ2/2ρ2+η2.where ρ, η, λ are the Wolfenstein parameters.TheCP violation, ACP, can be written as (29)ACP≡A2-A-2A2+A-2=-2rsin⁡δsin⁡θ1+2rcos⁡δcos⁡θ+r2.


#### 3.2.2. TheCP Violation of B-s→π0η(′) via π0-η-η′ Mixing

Due to the interference betweenπ0 and η(′), the effect of the isospin symmetry breaking is more significant for the decay process of B-s→π0η(′). Hence, the π0-η-η′ mixing, which including η or η′ meson, may shift the phase larger so as to have a bigger impact on CP violation. The decay amplitudes of B-s→π0η(′) with isospin symmetry are defined as (30)AB-s→π0η=AB-s→π0ηncos⁡ϕ-AB-s→π0ηssin⁡ϕ,(31)AB-s→π0η′=AB-s→π0ηnsin⁡ϕ+AB-s→π0ηscos⁡ϕ.Taking into account ofπ0-η-η′ mixing, the decay amplitudes A for B-s→π0η in Eq.(30) can be written as (32)A=π0ηHeffB-s=π3ηHeffB-s+ε1+ε2cos⁡ϕηnηB¯sHB¯sB¯s-ε2sin⁡ϕηsηB¯sHeffB¯sB¯s.We can define (33)A1=π3ηHeffB-s=-ε2-ε1cos⁡ϕπ3π3HeffB-s+cos⁡ϕπ3ηnHeffB-s-sin⁡ϕπ3ηsHeffB-s,(34)A2=ε1+ε2cos⁡ϕηnηB¯sHeffB¯sB¯s=ε1+ε2cos⁡ϕcos⁡ϕηnηnB¯sHeffB¯sB¯s-ε1+ε2cos⁡ϕsin⁡ϕηnηsB¯sHeffB¯sB¯s+Oε,(35)A3=-ε2sin⁡ϕηsηHeffB¯s=-ε2sin⁡ϕcos⁡ϕηnηsHeffB¯s+ε2sin⁡ϕcos⁡ϕηsηsHeffB¯s+Oε,where (36)Oε=Oε1+Oε2,and we have ignored the higher order term of ε. One can express A=A1+A2+A3.In the same way, we can present the decay amplitudesA′ for B-s→π0η′ with π0-η-η′ mixing in Eq.(31) (37)A′=π0η′HeffB-s=π3η′HeffB-s+ε1+ε2cos⁡ϕηnη′HeffB-s-ε2sin⁡ϕηsη′HeffB-s=-ε1sin⁡ϕπ3π3HeffB-s+sin⁡ϕπ3ηnHeffB-s+cos⁡ϕπ3ηsHeffB-s+ε1+ε2cosϕsin⁡ϕηnηnB¯sHeffB¯sB¯s+ε1+ε2cosϕcos⁡ϕηnηsB¯sHeffB¯sB¯s-ε2sin⁡ϕsin⁡ϕηsηnB¯sHeffB¯sB¯s-ε2sin⁡ϕcos⁡ϕηsηsB¯sHeffB¯sB¯s.Hence, depending on the CKM matrix elementsVubVud∗ and VtbVtd∗, we can express the decay amplitudes A(B-s→π0η(′) as follows: (38)AB-s→π0η′=VubVud∗T-VtbVtd∗P,where T and P refer to the tree and penguin contributions from A and A′ in Eq.(32),(37), respectively. The relevant amplitudes can be obtained from the decay processes of B-s→π0π0, B-s→π0ηn, B-s→π0ηs, B-s→ηnηn and B-s→ηsηs. Combined with Eq.(32), (33), (34), (35), (37), (38) we can also obtain CP violation from the Eqs.(26), (27), (28) and (29).


## 3.1. Formalism

It is convenient to introduce isospin vector tripletπ3, isospin scalar ηn and isospin scalar ηs which can be distinguished by including strange quark or not. The SU(3) singlet η0 and octet η8 can be well described by the translation ηn=2η0+η8/3 and ηs=1/3η0-2/3η8. The states of π3, ηn and ηs are identified by π3=1/2uu--dd-, ηn=1/2uu-+dd- and ηs=ss- which are obtained from the quark model, respectively. The physical meson states can be transformed from the π3, ηn and ηs by unitary matrix U [17]: (5)π0ηη′=Uε1,ε2,ϕπ3ηnηs,where (6)Uε1,ε2,ϕ=1ε1+ε2cos⁡ϕ-ε2sin⁡ϕ-ε2-ε1cos⁡ϕcos⁡ϕ-sin⁡ϕ-ε1sin⁡ϕsin⁡ϕcos⁡ϕ,ε1, ε2∝O(λ),λ≪1 and the higher order terms are neglected. In the isospin limit of ε1→0,ε2→0, we can find that the formula is expressed as the η-η′ mixing in Eq.(7): (7)ηη′=U′ϕηnηs=cos⁡ϕ-sin⁡ϕsin⁡ϕcos⁡ϕηnηs,where ϕ is the mixing angle [20]. The η and η′ mixing depends on the quark flavor basises ηn and ηs.The relevant decay constants can be written as [21, 22] (8)0n-γμγ5nηnP=i2fnPμ,0s-γμγ5sηsP=ifsPμ,where P refers to the momenta of ηn or ηs.One can understand that isospin symmetry breaking comes from the electroweak interaction andu-d quark mass difference in Standard Model. We can calculate the isospin symmetry breaking correction by chiral perturbative theory which induces the π0-η-η′ mixing. To the leading order of isospin symmetry breaking, the physical eigenstate π0, η and η′ from Eq.(5)(6) can be written as (9)π0=π3+ε1+ε2cos⁡ϕηn-ε2sin⁡ϕηs,η=-ε2-ε1cos⁡ϕπ3+cos⁡ϕηn-sin⁡ϕηs,η′=-ε1sin⁡ϕπ3+sin⁡ϕηn+cos⁡ϕηs,One can define ε=ε2+ε1cos⁡ϕ, ε′=ε1sin⁡ϕ. π3 refer to the isospin I=1 component in the triplet. We use the values of ε=0.017±0.002, ε′=0.004±0.001, ϕ=39.0∘ [17].The mixing is generated by the strong interaction fromO(md-mu), where we assert ε, ε′~O((md-mu)/Λhad) or O(α) neglecting higher-order terms for the isospin breaking. The pseudoscalar meson octet in current algebra[23] and the lowest order chiral perturbation theory[24] discuss the π0-η8 mixing angle which can be written as 3/4(md-mu/ms-m^) with m^=(mu+md)/2, so that the scale is expeted as Λhad~O(ms).Generally, the mesonsη and η′ are composed of u-u, d-d, s-s linear combination. One believes that there is gluon in the η′ meson for explaining the large branching ratios for the decay processes B→Kη′ and J/ψ→η′γ [25, 26]. H.N. Li et al. find the tiny contribution from the gluonic admixture [27, 28]. Z.J. Xiao et al. calculate the branching ratios and CP violations in PQCD for the decay processes B→Mη′ and show that the theoretical result is in agreement with the experimental one neglecting the contribution of gluon [29–31]. Thomas shows the contribution from composition of gluon does not make obvious impact on fitted values on the basis of existing experiment data [32].For theBs meson function, we use the model [20, 33] (10)ϕBsx,b=NBsx21-x2exp⁡-MBs2x22ωb2-12ωbb2,where the normalization factor NBs is dependent on the free parameter ωb. b is the conjugate variable of the parton transverse momenta kT. MBs refers to the mass of the Bs meson. For the Bs meson, one can obtain the value of ωb=0.50±0.05 from the light cone sum rule [34].The wave functions ofηn and ηs are the same in form and can be defined as [26] (11)ΦP,x,ζ≡12NCγ5PϕAx+m0ϕPx+ζnυ-υ·nm0ϕTxwhere P and x are the momentum and the momentum fraction of ηn(ηs), respectively. Following Ref.[26], we assumed here that the wave function of ηn(ηs) is the same as the π wave function based on SU(3) flavor symmetry. Depending on the assignment of the momentum fraction x, the parameter ζ should be chosen as +1 or -1. In this paper, we will use those distribution amplitudes [20]: (12)ϕπAx=3fπ6x1-x1+0.44C23/2t,ϕπPx=fπ261+0.43C21/2t,ϕπTx=-fπ26C11/2t+0.55C31/2t,ϕKAx=3fK6x1-x1+0.17C13/2t+0.2C23/2t,ϕKPx=fK261+0.24C21/2t,ϕKTx=-fK26C11/2t+0.35C31/2t,ϕϕ=3fϕ6x1-x1+0.18C23/2t,where t=2x-1. fP(v) are the decay constants of scalar (vector) mesons, respectively. The pseudoscalar mesons π,η and η′ have the similar wave functions. The expressions of amplitudes can be obtained by the replacements ϕπ→ϕη,ϕπp→ϕηp,ϕπt→ϕηt. Gegenbauer polynomials are defined as (13)C11/2t=t,C13/2t=3tC21/2t=123t2-1,C23/2t=325t2-1,C31/2t=12t5t2-3.The wave function of the meson is applied to describe the formation of hadron from the positive and negative quarks, which provides distribution of the momentum carried by the parton. It is non-perturbative and process independent from partons to hadrons. The wave function of the meson is transverse-momentum-dependent wave fucnction in PQCD. The results show that there is large contribution from transverse-momentum for the heavy meson function. However, the wave function of light meson is less reliant on transverse-momentum. Currently, it is reasonable that the results form Light-cone QCD sum rule and Lattice QCD. The wave function of light pseudoscalar meson is usually obtained by Light-cone QCD sum rule. The branching ratios and CP violations are calculated in the framework of PQCD. The theoretical prediction is in good agreement with the experimental results for the most decay processes ofB/Bs→M2M3. Hence, the wave function is credible. The transverse-momentum-dependent (TMD) wave fucnction is given with simpler soft subtraction in [35] which presents the more accurate results.The transverse momentum cannot be neglected for eliminating the divergence in the endpoints. The double logarithms will be obtained when collinear divergent is overlapping with soft divergen considering radiative corrections for the form factor. For the effective perturbative expansion, we need to sum the double logarithmic terms by the resummation technique [36]. The wave function and hard kernel can absorb collinear divergent and infrared divergence, respectively. The other infrared divergences are cancelled out. At the range of large b, we need to sum the double logarithmic terms by renormalization group. Hence, the sudakov factor is introduced. Due to the suppression of sudakov factor, the non-perturbative contribution becomes very small. The amplitude of hard kernal produces the double logarithmic terms from radiative corrections. The threshold resummation introduces the sudakov factor which drops rapidly at the endpoints. Hence, perturbative calculation plays the dominant role when the endpoint singularity is suppressed.Recently, an evolution for the B meson wave fucntions are constructed in thekt factorization theorem, whose solutions sum the double logarithms with the light-cone singularities, namely, the rapidity logarithms. The effective heavy-quark field is involved for an energetic light hadron comparing with the Sudakov resummation. The resummation effect maintians the normalization of the B meson wave functions and strengths their convergent behavior at small momentum, which combines the threshold and Kt resummations [37]. Due to the resummation technique with off-light-cone Wilson lines, an evolution for the pion meson wave fucntions is introduced in the Kt factorization theorem, whose solutions sum the mixed logarithm ln⁡xln⁡kt to all orders, with x(kt) being a parton momentum fraction (transverse momentum). This joint resummation shows strong suppression of the pion wave fucnction in the small x and large b regions, b being the impact parameter conjugate to kt. The next-to-leading-order contributions are different from those under the conventional resummations at high energy. The scheme-independent formalism can be extended to the kt factorization of more complicated exclusive processes. [38]. The improvement of the wave function can be applied to a more accurate discussion for the next step.


## 3.2. Calculation Details

In the framework of PQCD, we can calculate theCP violation for the decay process B-s→P(V)π0 via π0-η-η′ mixing. Firstly, we calculate the amplitudes T and P, which can be decomposed in terms of tree and penguin contributions depending on the CKM matrix elements VubVud∗ and VtbVtd∗. Next, we take the decay process B-s→K0π0 and B-s→π0η(′) as examples for the study of the π0-η-η′ mixing mechanism.


### 3.2.1. TheCP Violation for the Decay Modes of B-s→P(V)π0 Except B-s→π0η′

We take the decay process ofB-s→K0π0 as example to introduce the CP violation via π0-η-η′ mixing. The decay amplitude A of B-s→K0π0 in PQCD can be written as (14)2AB-s→K0π0=VubVud∗T1-VtbVtd∗P1,where T1 and P1 are the amplitudes form tree and penguin contributions, respectively. The tree level amplitude T1 can be given as (15)T1=fπFBs→KLLa2+MBs→KLLC2,and the penguin level amplitude P1 can be written as (16)P1=fπFBs→KLL-a4-32a7+32a9+12a10+fπFBs→KSP-a6+12a8+MBs→KLL-C3+32c8+12C9+32c10+fBsFannSP-a6+12a8+fBsFannLL-a4+12a10+MannLL-C3+12C9+MannLR-C5+12C7,where the fi refers to the decay constant. The individual decay amplitudes in the above equations, such as FBs→KLL, FBs→KSP, MBs→KLL, FannSP, Fann, MannLL and MannLR arise from the (V-A)(V-A), (V-A)(V+A) and (S-P)(S+P) operators, respectively, and will be given in Appendix.Based on the CKM matrix elements ofVubVud∗ and VtbVtd∗, we can express the decay amplitudes as follows: (17)2AB-s→ηK0=VubVud∗Tn-VtbVtd∗Pn,(18)2AB-s→η′K0=VubVud∗Ts-VtbVtd∗Ps.The contributions of Tn and Pn for the decay amplitudes B-s→ηK0 can be written as (19)Tn=fnFBs→KLLa2+MBs→KLLC2,(20)Pn=fnFBs→KSPa6-12a8+fnFBs→KLL2a3+a4-2a5-12a7+12a9-12a10+fBsFannLLa4-12a10+MBs→KLLC3+2C4+12C8-12C9+12C10+fBsFannSPa6-12a8+MannLLC3-12C9+MannLRC5-12C7,(21)Ts=0,(22)Ps=fsFBs→KLLa3-a5+12a7-12a9+fKFBs→ηsLLa4-12a10+fKFBs→ηsSPa6-12a8+MBs→KLLC4+C4-12C8-12C10+MBs→ηsLLC3-12C9+MBs→ηsLRC5-12C7+fBsFannLLa4-12a10+fBsFannSPa6-12a8+MannLLC3-12C9+MannLRC5-12C7,for the formula of 2A(B-s→η′K0)=VubVud∗Ts-VtbVtd∗Ps.The amplitudes T and P from the decay process ofB-s→K0π0 with π0-η-η′ mixing can be written as (23)T=T1+ε1+ε2cos⁡ϕTn-ε2sin⁡ϕTs,P=P1+ε1+ε2cos⁡ϕPn-ε2sin⁡ϕPs.One can see that the Eq.(23) without π0-η-η′ mixing is reduced to (24)T=T1,P=P1,which are expressed in Eq.(15) and Eq.(16).The relevant weak phaseθ and strong phase δ are obtained as follows: (25)reiδeiθ=PT×VtbVtd∗VubVud∗,where the parameter r represents the absolute value of the ratio of penguin and tree amplitudes: (26)r≡K0π0B¯s0HPB¯s0B¯s0K0π0B¯s0HTB¯s0B¯s0.The strong phase associated with r can be given as (27)reiδ=PT×VtbVtd∗VubVud∗=rcos⁡δ+irsin⁡δ,where (28)VtbVtd∗VubVud∗=ρ1-ρ-η22+η21-λ2/2ρ2+η2.where ρ, η, λ are the Wolfenstein parameters.TheCP violation, ACP, can be written as (29)ACP≡A2-A-2A2+A-2=-2rsin⁡δsin⁡θ1+2rcos⁡δcos⁡θ+r2.


### 3.2.2. TheCP Violation of B-s→π0η(′) via π0-η-η′ Mixing

Due to the interference betweenπ0 and η(′), the effect of the isospin symmetry breaking is more significant for the decay process of B-s→π0η(′). Hence, the π0-η-η′ mixing, which including η or η′ meson, may shift the phase larger so as to have a bigger impact on CP violation. The decay amplitudes of B-s→π0η(′) with isospin symmetry are defined as (30)AB-s→π0η=AB-s→π0ηncos⁡ϕ-AB-s→π0ηssin⁡ϕ,(31)AB-s→π0η′=AB-s→π0ηnsin⁡ϕ+AB-s→π0ηscos⁡ϕ.Taking into account ofπ0-η-η′ mixing, the decay amplitudes A for B-s→π0η in Eq.(30) can be written as (32)A=π0ηHeffB-s=π3ηHeffB-s+ε1+ε2cos⁡ϕηnηB¯sHB¯sB¯s-ε2sin⁡ϕηsηB¯sHeffB¯sB¯s.We can define (33)A1=π3ηHeffB-s=-ε2-ε1cos⁡ϕπ3π3HeffB-s+cos⁡ϕπ3ηnHeffB-s-sin⁡ϕπ3ηsHeffB-s,(34)A2=ε1+ε2cos⁡ϕηnηB¯sHeffB¯sB¯s=ε1+ε2cos⁡ϕcos⁡ϕηnηnB¯sHeffB¯sB¯s-ε1+ε2cos⁡ϕsin⁡ϕηnηsB¯sHeffB¯sB¯s+Oε,(35)A3=-ε2sin⁡ϕηsηHeffB¯s=-ε2sin⁡ϕcos⁡ϕηnηsHeffB¯s+ε2sin⁡ϕcos⁡ϕηsηsHeffB¯s+Oε,where (36)Oε=Oε1+Oε2,and we have ignored the higher order term of ε. One can express A=A1+A2+A3.In the same way, we can present the decay amplitudesA′ for B-s→π0η′ with π0-η-η′ mixing in Eq.(31) (37)A′=π0η′HeffB-s=π3η′HeffB-s+ε1+ε2cos⁡ϕηnη′HeffB-s-ε2sin⁡ϕηsη′HeffB-s=-ε1sin⁡ϕπ3π3HeffB-s+sin⁡ϕπ3ηnHeffB-s+cos⁡ϕπ3ηsHeffB-s+ε1+ε2cosϕsin⁡ϕηnηnB¯sHeffB¯sB¯s+ε1+ε2cosϕcos⁡ϕηnηsB¯sHeffB¯sB¯s-ε2sin⁡ϕsin⁡ϕηsηnB¯sHeffB¯sB¯s-ε2sin⁡ϕcos⁡ϕηsηsB¯sHeffB¯sB¯s.Hence, depending on the CKM matrix elementsVubVud∗ and VtbVtd∗, we can express the decay amplitudes A(B-s→π0η(′) as follows: (38)AB-s→π0η′=VubVud∗T-VtbVtd∗P,where T and P refer to the tree and penguin contributions from A and A′ in Eq.(32),(37), respectively. The relevant amplitudes can be obtained from the decay processes of B-s→π0π0, B-s→π0ηn, B-s→π0ηs, B-s→ηnηn and B-s→ηsηs. Combined with Eq.(32), (33), (34), (35), (37), (38) we can also obtain CP violation from the Eqs.(26), (27), (28) and (29).


## 3.2.1. TheCP Violation for the Decay Modes of B-s→P(V)π0 Except B-s→π0η′

We take the decay process ofB-s→K0π0 as example to introduce the CP violation via π0-η-η′ mixing. The decay amplitude A of B-s→K0π0 in PQCD can be written as (14)2AB-s→K0π0=VubVud∗T1-VtbVtd∗P1,where T1 and P1 are the amplitudes form tree and penguin contributions, respectively. The tree level amplitude T1 can be given as (15)T1=fπFBs→KLLa2+MBs→KLLC2,and the penguin level amplitude P1 can be written as (16)P1=fπFBs→KLL-a4-32a7+32a9+12a10+fπFBs→KSP-a6+12a8+MBs→KLL-C3+32c8+12C9+32c10+fBsFannSP-a6+12a8+fBsFannLL-a4+12a10+MannLL-C3+12C9+MannLR-C5+12C7,where the fi refers to the decay constant. The individual decay amplitudes in the above equations, such as FBs→KLL, FBs→KSP, MBs→KLL, FannSP, Fann, MannLL and MannLR arise from the (V-A)(V-A), (V-A)(V+A) and (S-P)(S+P) operators, respectively, and will be given in Appendix.Based on the CKM matrix elements ofVubVud∗ and VtbVtd∗, we can express the decay amplitudes as follows: (17)2AB-s→ηK0=VubVud∗Tn-VtbVtd∗Pn,(18)2AB-s→η′K0=VubVud∗Ts-VtbVtd∗Ps.The contributions of Tn and Pn for the decay amplitudes B-s→ηK0 can be written as (19)Tn=fnFBs→KLLa2+MBs→KLLC2,(20)Pn=fnFBs→KSPa6-12a8+fnFBs→KLL2a3+a4-2a5-12a7+12a9-12a10+fBsFannLLa4-12a10+MBs→KLLC3+2C4+12C8-12C9+12C10+fBsFannSPa6-12a8+MannLLC3-12C9+MannLRC5-12C7,(21)Ts=0,(22)Ps=fsFBs→KLLa3-a5+12a7-12a9+fKFBs→ηsLLa4-12a10+fKFBs→ηsSPa6-12a8+MBs→KLLC4+C4-12C8-12C10+MBs→ηsLLC3-12C9+MBs→ηsLRC5-12C7+fBsFannLLa4-12a10+fBsFannSPa6-12a8+MannLLC3-12C9+MannLRC5-12C7,for the formula of 2A(B-s→η′K0)=VubVud∗Ts-VtbVtd∗Ps.The amplitudes T and P from the decay process ofB-s→K0π0 with π0-η-η′ mixing can be written as (23)T=T1+ε1+ε2cos⁡ϕTn-ε2sin⁡ϕTs,P=P1+ε1+ε2cos⁡ϕPn-ε2sin⁡ϕPs.One can see that the Eq.(23) without π0-η-η′ mixing is reduced to (24)T=T1,P=P1,which are expressed in Eq.(15) and Eq.(16).The relevant weak phaseθ and strong phase δ are obtained as follows: (25)reiδeiθ=PT×VtbVtd∗VubVud∗,where the parameter r represents the absolute value of the ratio of penguin and tree amplitudes: (26)r≡K0π0B¯s0HPB¯s0B¯s0K0π0B¯s0HTB¯s0B¯s0.The strong phase associated with r can be given as (27)reiδ=PT×VtbVtd∗VubVud∗=rcos⁡δ+irsin⁡δ,where (28)VtbVtd∗VubVud∗=ρ1-ρ-η22+η21-λ2/2ρ2+η2.where ρ, η, λ are the Wolfenstein parameters.TheCP violation, ACP, can be written as (29)ACP≡A2-A-2A2+A-2=-2rsin⁡δsin⁡θ1+2rcos⁡δcos⁡θ+r2.


## 3.2.2. TheCP Violation of B-s→π0η(′) via π0-η-η′ Mixing

Due to the interference betweenπ0 and η(′), the effect of the isospin symmetry breaking is more significant for the decay process of B-s→π0η(′). Hence, the π0-η-η′ mixing, which including η or η′ meson, may shift the phase larger so as to have a bigger impact on CP violation. The decay amplitudes of B-s→π0η(′) with isospin symmetry are defined as (30)AB-s→π0η=AB-s→π0ηncos⁡ϕ-AB-s→π0ηssin⁡ϕ,(31)AB-s→π0η′=AB-s→π0ηnsin⁡ϕ+AB-s→π0ηscos⁡ϕ.Taking into account ofπ0-η-η′ mixing, the decay amplitudes A for B-s→π0η in Eq.(30) can be written as (32)A=π0ηHeffB-s=π3ηHeffB-s+ε1+ε2cos⁡ϕηnηB¯sHB¯sB¯s-ε2sin⁡ϕηsηB¯sHeffB¯sB¯s.We can define (33)A1=π3ηHeffB-s=-ε2-ε1cos⁡ϕπ3π3HeffB-s+cos⁡ϕπ3ηnHeffB-s-sin⁡ϕπ3ηsHeffB-s,(34)A2=ε1+ε2cos⁡ϕηnηB¯sHeffB¯sB¯s=ε1+ε2cos⁡ϕcos⁡ϕηnηnB¯sHeffB¯sB¯s-ε1+ε2cos⁡ϕsin⁡ϕηnηsB¯sHeffB¯sB¯s+Oε,(35)A3=-ε2sin⁡ϕηsηHeffB¯s=-ε2sin⁡ϕcos⁡ϕηnηsHeffB¯s+ε2sin⁡ϕcos⁡ϕηsηsHeffB¯s+Oε,where (36)Oε=Oε1+Oε2,and we have ignored the higher order term of ε. One can express A=A1+A2+A3.In the same way, we can present the decay amplitudesA′ for B-s→π0η′ with π0-η-η′ mixing in Eq.(31) (37)A′=π0η′HeffB-s=π3η′HeffB-s+ε1+ε2cos⁡ϕηnη′HeffB-s-ε2sin⁡ϕηsη′HeffB-s=-ε1sin⁡ϕπ3π3HeffB-s+sin⁡ϕπ3ηnHeffB-s+cos⁡ϕπ3ηsHeffB-s+ε1+ε2cosϕsin⁡ϕηnηnB¯sHeffB¯sB¯s+ε1+ε2cosϕcos⁡ϕηnηsB¯sHeffB¯sB¯s-ε2sin⁡ϕsin⁡ϕηsηnB¯sHeffB¯sB¯s-ε2sin⁡ϕcos⁡ϕηsηsB¯sHeffB¯sB¯s.Hence, depending on the CKM matrix elementsVubVud∗ and VtbVtd∗, we can express the decay amplitudes A(B-s→π0η(′) as follows: (38)AB-s→π0η′=VubVud∗T-VtbVtd∗P,where T and P refer to the tree and penguin contributions from A and A′ in Eq.(32),(37), respectively. The relevant amplitudes can be obtained from the decay processes of B-s→π0π0, B-s→π0ηn, B-s→π0ηs, B-s→ηnηn and B-s→ηsηs. Combined with Eq.(32), (33), (34), (35), (37), (38) we can also obtain CP violation from the Eqs.(26), (27), (28) and (29).


## 4. Input Parameters

The CKM matrix, the elements of which are determined from experiments, can be expressed in terms of the Wolfenstein parametersA, ρ, λ and η [39, 40]:(39)1-12λ2λAλ3ρ-iη-λ1-12λ2Aλ2Aλ31-ρ-iη-Aλ21,where O(λ4) corrections are neglected. The latest values for the parameters in the CKM matrix are [41] (40)λ=0.22506±0.00050,A=0.811±0.026,ρ-=0.124-0.018+0.019,η-=0.356±0.011.where (41)ρ-=ρ1-λ22,η-=η1-λ22.From Eqs. (40) (41) we have (42)0.109<ρ<0.147,0.354<η<0.377.The other parameters are given as follows [39–41]: (43)fπ=0.13GeV,fK=0.16GeV,mBs0=5.37GeV,fBs=0.23GeV,fs=0.17GeV,fn=0.14GeV,mπ=0.14GeV,mW=80.39GeV,mt=173.21GeV,mb=4.8MeV.


## 5. Numerical Results

The CP violation depends on the weak phase differences from the CKM matrix elements and the strong phase differences associated with QCD. The CKM matrix elements are determined by the parameters ofA, ρ, λ and η. We find that the results for the CP violation are less reliant on A and λ in the course of calculations. Hence, we present the CP violation from the weak phases associated with the ρ and η in the CKM matrix elements while the A and λ are assigned for the central values. In Table 1, we show the values of CP violation of Bs decay modes from isospin symmetry and isospin symmetry breaking via π0-η-η′ mixing. From Table 1, it can be seen that the increasing rate of the CP violation, which is defined |x2|-|x1|/|x1|×100% (where x1, x2 represent the CP violation values from isospin symmetry and isospin symmetry breaking, respectively.), is larger in B-s→π0η(′) decay process compared with the other decay channels we are considering. It is intelligible that the final states for the decay process include the η or η′ meson. Due to the isospin symmetry breaking, the interference between the π0 and η(′) mesons is stronger than other decay channels whose final states do not contain η or η′ meson. Hence, these decay channels including η or η′ meson make the strong phase larger resulting in a great impact on CP violation. We can find that the CP violation of the decay mode B-s→K0π0 has not changed much in Table 1. The CP violation of the decay mode B-s→K0∗π0 is changed from -23.58% to -36.88%. From Table 1, one can also see that the isospin symmetry breaking changes the sign of the CP violation, for example, from 8.32% to -6.33% for the decay channel of B-s→ϕπ0, from -8.76% to 15.48% for the decay channel of B-s→π0η. The increasing rate of CP violation for the decay mode B-s→π0η′ is -65.99% for the central value.Table 1
The CP violation ofBs decay mode via isospin symmetry and isospin symmetry breaking via π0-η-η′ mixing. The increasing rate is defined |x2|-|x1|/|x1|×100%, where x1, x2 represent the values of CP violation(%) from isospin symmetry and isospin symmetry breaking, respectively. The fluctuation numerical values refer to the contribution of the limiting parameters from the CKM matrix elements.
decay mode isospin symmetry π 0 - η - η ′ mixing the increasing rate B-s→K0π0 53.4 3 - 2.14 + 2.26 % 51.4 4 - 3.00 + 3.30 % - 3.7 2 - 1.87 + 2.01 % B-s→K0∗π0 - 23.5 8 - 1.22 + 1.12 % - 36.8 8 - 0.70 + 0.73 % 56 . 4 - 4.87 + 4.55 % B-s→ϕπ0 8.3 2 - 0.47 + 0.48 % - 6.3 3 - 0.20 + 0.18 % - 23.9 2 - 6.42 + 6.85 % B-s→π0η - 8.7 6 - 0.25 + 0.28 % 15.4 8 - 0.73 + 0.81 % 76.7 1 - 13.00 + 15.39 % B-s→π0η′ 27.4 3 - 1.05 + 1.09 % 9.3 3 - 0.60 + 0.67 % - 65.9 9 - 0.92 + 1.05 %From Table1, we can find great changes between the values of CP violation from isospin symmetry and isospin symmetry breaking via π0-η-η′ mixing. In order to study the influence of the weak phase on CP violation and understand the π0-η-η′ mixing mechanism, we present the CP violation as a function of ρ and η in Figure 1 while taking the mixing parameters ε1 and ε2 as central values. We vary (ρ,η) from the limiting values (ρmin,ηmin) to (ρmax,ηmax), respectively, in Figure 1. Due to the effect of weak phases from CKM matrix elements, the value of CP violation for the decay process of B-s→π0η changes from 14.75% to 16.29% taking into account of isospin symmetry breaking.Figure 1
The directCP violation as a function of ρ and η from the CKM matrix element with isospin symmetry breaking for the decay process of B-s→π0η. The horizontal axis and vertical axis refer to the values of ρ and η, respectively.It can be seen from the Eq.(29) that the value of direct CP violation is also dependent on sin⁡δ and r. We take the decay channel of B-s→π0η as an example. When ρ and η are taken as the central value for the CKM matrix elements, we present the direct CP violation as a function of ε1, ε2 in Figure 2(a) from the isospin symmetry and in Figure 2(b) from isospin symmetry breaking via π0-η-η′ mixing. Comparing the Figure 2(a) with the Figure 2(b), the CP violation value has a great change. Only considering the central value, the value of CP violation changes from -8.76% in Figure 2(a) to 15.48% in Figure 2(b) and shifts the sign. In Figures 3 and 4, we give the numerical result of sin⁡δ and r for the decay process of B-s→π0η. Comparing Figure 3(a) with Figure 3(b), we can find that the value of sin⁡δ changes the sign from 0.276 in Figure 3(a) to the central value -2.219 in Figure 3(b). In Figure 4, the central value of r changes large comparing the result of isospin symmetry breaking in Figure 4(b) to the value from isospin symmetry in Figure 4(a). Based on the changes of sin⁡δ and r, large CP violation is obtained from isospin symmetry breaking via π0-η-η′ mixing.Figure 2
(a) The directCP violation as a function of ε1 and ε2 from the effects of isospin symmetry for the decay process of B-s→π0η. (b) The same as (a) from the effects of isospin symmetry breaking. The horizontal axis and vertical axis refer to the values of ε1 and ε2, respectively.
(a) (b)Figure 3
(a) The value ofsin⁡δ as a function of ε1 and ε2 from the effects of the isospin symmetry for the decay process of B-s→π0η. (b) The same as (a) from the effects of isospin symmetry breaking. The horizontal axis and vertical axis refer to the values of ε1 and ε2, respectively.
(a) (b)Figure 4
(a) The value ofr as a function of ε1 and ε2 from the effects of isospin symmetry for the decay process of B-s→π0η. (b) The same as (a) from the effects of isospin symmetry breaking. The horizontal axis and vertical axis refer to the values of ε1 and ε2, respectively.
(a) (b)


## 6. Summary and Conclusion

In this paper, we study theCP violation for the decay process of B-s→P(V)π0 in Perturbative QCD. It is found that the CP violation can be shifted via π0-η-η′ mixing from the isospin symmetry breaking. The CP violation arises from the weak phase difference in CKM matrix and the strong phase difference. The CP violation changes small for the decay mode B-s→K0π0 via π0-η-η′ mixing and the central value of the increasing rate is -3.72%. The rate of increase of the CP violation is larger for the decay process of B-s→π0η(′) than other decay channels. Due to the breaking of isospin symmetry, the interference between the π0 and η(′) mesons is stronger than other decay channels which in the final state does not contain η or η′ meson. For the decay process B-s→ϕπ0 and B-s→π0η, the isospin symmetry breaking changes the sign of the CP violation.In order to achieve the required energy and luminosity requirements, the Large Hadron Collider (LHC), which has currently started at CERN, has been upgraded many times. The LHC Run I data started in 2010. The peak instantaneous luminosity documentary during Run I was8.0×1032cm-2s-1. The center-of mass energy was primarily s = 7 TeV and was raised to 8 TeV in 2012 [42]. This was followed by the first long shutdown period (LS1), which was devoted to upgrades essential for increasing beam energy to s = 13 TeV centre of mass energy and peak instantaneous luminosity 1.7×1034cm-2s-1 [43, 44]. In the following years, there are two primary detector (CMS and ATLAS) upgrades happening after Run II and Run III. Phase-I and II upgrade prepares for an instantaneous luminosity of 2-3×1034cm-2s-1 and 5-7×1034cm-2s-1 [45], respectively. With a series of modifications and upgrades, the LHC gives access to high energy frontier at TeV scale and an occasion to further improve the consistency test for the CKM matrix. The production rates for heavy quark flavors will be great at the LHC, and the bb- production cross section will be of the order 0.5 mb, providing as many as 1012 bottom events per year [42, 46]. The heavy quark physics is one of the major topics of LHC experiments. Especially, the LHCb experiment exploits amounts of b mesons, produced in proton-proton collisions at the LHC to search for CP violation. Recently, LHCb Collaboration presents observation of the decay Bs0→ϕπ+π- meson. Obtaining more data from LHC, it is possible to make further analysis for CP violation of Bs0 decays [47]. We expect that our results is valuable for measurement of CP violation of Bs0 decays in the following LHCb experiments.



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