$CP$ violation induced by the double resonance for pure annihilation decay process in Perturbative QCD

In Perturbative QCD (PQCD) approach we study the direct $CP$ violation in the pure annihilation decay process of $\bar{B}^0_{s}\rightarrow\pi^+\pi^-\pi^+\pi^-$ induced by the $\rho$ and $\omega$ double resonance effect. Generally, the $CP$ violation is small in the pure annihilation type decay process. However, we find that the $CP$ violation can be enhanced by double $\rho-\omega$ interference when the invariant masses of the $\pi^+\pi^-$ pairs are in the vicinity of the $\omega$ resonance. For the decay process of $\bar{B}^0_{s}\rightarrow\pi^+\pi^-\pi^+\pi^-$, the maximum $CP$ violation can reach 28.64{\%}.


I. INTRODUCTION
CP violation is an important area in searching new physics signals beyond the standard model(SM). It is generally believed that the B meson system provides rich information about CP violation. The theoretical work has been done in this direction in the past few years. CP violation arises from the weak phase in the Cabibbo-Kobayasgi-Maskawa (CKM) matrix [1,2] in SM. Meanwhile, it is remarkable that CP violation can still be produced by the interference effects between the tree and penguin amplitudes. Since the kinematic suppression, the strong phase associated with long distance rescattering is generally neglected during the past decades. Recently, the LHCb Collaboration found the large CP violation in the three-body decay channels of B ± → π ± π + π − and B ± → K ± π + π − [3][4][5]. Hence, the nonleptonic B meson decay from the three-body and four-body decay channels has been become an important area in searching for CP violation.
A mixing between the u and d flavor leads to the breaking of isospin symmetry for the ρ − ω system. The chiral dynamics has been shown restore the isospin symmetry [6]. The ρ − ω mixing matrix elementΠ ρω (s) gives rise to isospin violation, where s is the Mandelstam variable. The magnitude has been extracted by the pion form factor through the cross section of e + e − → π + π − . We can separate theΠ ρω (s) into two contribution of the direct coupling of ω → 2π and the mixing of ω → ρ → 2π. The emergence ofΠ ρω (s) arises from the inclusion of a nonresonant contribution to ω → 2π. The appearance of the ρ and ω resonance is associated with complex strong phase from relatively broad ρ resonance region. Especially, there is perhaps larger strong phase from double ρ and ω interference.
The CP violation origins from the weak phase difference and the strong phase difference. Hence, the decay process ofB 0 s → π + π − π + π − is a great candidate for studying the origin of the CP violation.
Meanwhile, it is known that the CP violation is extremely tiny from the pure annihilation decay process in experiment. There is relatively large error in dealing with the decay amplitudes from the QCD factorization approach [7]. The perturbative QCD (PQCD) factorization approach [8][9][10][11] is based on k T factorization. The amplitude can be divided into the convolution of the Wilson coefficients, the light cone wave function, and hard kernels by the low energy effective Hamiltonian. The endpoint singularity can be eliminated by introducing the transverse momentum.
However, The transverse momentum integration leads to the double logarithm term which is resummed into the Sudakov form factor. The nonperturbative dynamics are included in the meson wave function which can be extracted from experiment. The hard one can be calculated by perturbation theory.
The remainder of this paper is organized as follows. In Sec. II we present the form of the effective Hamiltonian.
In Sec. III we give the calculating formalism and calculation details of CP violation from ρ − ω mixing in thē In Sec. IV we show input parameters. We present the numerical results in Sec. V. Summary and discussion are included in Sec. VI. The related function defined in the text are given in the Appendix.

II. THE EFFECTIVE HAMILTONIAN
With the operator product expansion, the effective weak Hamiltonian can be written as [12] where q = (d, s), G F represents Fermi constant, C i (i=1,...,10) are the Wilson coefficients, V q1q2 (q 1 and q 2 represent quarks) is the CKM matrix element, and O i is the four quark operator. The operators O i have the following forms: where α and β are color indices, and q ′ = u, d, s, c or b quarks. In Eq.
So, we can obtain numerical values of a i . The combinations a i of Wilson coefficients are defined as usual [9]: a 9 = C 9 + C 10 /3, a 10 = C 10 + C 9 /3.
The amplitudes A σ of the processB s (p) → V 1 (p 1 , ǫ 1 ) + V 2 (p 2 , ǫ 2 ) can be written [13] where σ is the helicity of the vector meson. ǫ 1 (p 1 ) and ǫ 2 (p 2 ) are the polarization vectors (momenta) of V 1 and V 2 , respectively. m 1 and m 2 refer to the masses of the vector mesons V 1 and V 2 . The invariant amplitudes a, b, c are associated with the amplitude A i ( i refer to the three kind of polarizations, longitudinal (L), normal (N) and transverse (T)). Then we have The longitudinal H 0 , transverse H ± of helicity amplitudes can be expressed The decay width is written The interaction of the photon and the hadronic matter can be described by the vector meson dominance model (VMD) [14]. The photon can couple to the hadronic field through a ρ meson. The mixing matrix element Π ρω (s) is extracted from the data of the cross section for e + e − → π + π − [15,16]. The nonresonant contribution of ω → π + π − has been effectively absorbed into Π ρω which leads to the explicit s dependence of Π ρω [17]. We can make the . However, one can neglect the s dependence of Π ρω in practice. The ρ − ω mixing parameters were determined in the fit of Gardner and O'Connell [18]: The formalism of the CP violation is presented for theB 0 s meson decay process in the following. The amplitude A (Ā) for the decay processB 0 s → π + π − π + π − (B 0 s → π + π − π + π − ) can be written as: where H T and H P refer to the tree and penguin operators in the Hamiltonian, respectively. We define the relative magnitudes and phases between the tree and penguin operator contributions as follows: where δ and φ are strong and weak phases, respectively. The weak phase difference φ can be expressed as a combination of the CKM matrix elements: The parameter r is the absolute value of the ratio of tree and penguin amplitudes: The parameter of CP violating asymmetry, A cp , can be written as where and T i (i = 0, +, −) represent the tree-level helicity amplitudes. We can see explicitly from Eq. (14) that both weak and strong phase differences are responsible for CP violation. ρ − ω mixing introduces the strong phase difference and well known in the three body decay processes of the bottom hadron [19][20][21][22][23][24][25]. Due to ρ − ω interference from the u and d quark mixing, we can write the following formalism in an approximate from the first order of isospin violation: where t ρρ (p ρρ ) and t ρω (p ρω ) are the tree (penguin) amplitudes forB s → ρ 0 ρ 0 andB s → ρ 0 ω, respectively, g ρ is the coupling for ρ 0 → π + π − , Π ρω is the effective ρ − ω mixing amplitude which also effectively includes the direct is the inverse propagator, mass and decay rate of the vector meson V , respectively.

B. Calculation details
We can decompose the decay amplitude for the decay processB 0 s → ρ 0 (ω)ρ 0 (ω) in terms of tree-level and penguinlevel contributions depending on the CKM matrix elements of V ub V * us and V tb V * ts . Due to the equations (14)(19)(20), we calculate the amplitudes t ρρ , t ρω , p ρρ and p ρω in perturbative QCD approach. The F and M function associated with the decay amplitudes can be found in the appendix from the perturbative QCD approach.
There are four types of Feynman diagrams contributing toB s → M 2 M 3 (M 2 ,M 3 =ρ or ω) annihilation decay mode at leading order. The pure annihilation type process can be classified into factorizable diagrams and non-factorizable diagrams [29,30]. Through calculating these diagrams, we can get the amplitudes A (i) , where i = L, N, T standing for the longitudinal and two transverse polarizations. Because these diagrams are the same as those of B → K * φ and B → K * ρ decays [29,30], the formulas ofB s → ρρ orB s → ρω are similar to those of B → K * φ and B → K * ρ. We just need to replace some corresponding wave functions, Wilson coefficients and corresponding parameters.
With the Hamiltonian (1), depending on CKM matrix elements of V ub V * us and V tb V * ts , the decay amplitudes A (i) (i = L, N, T ) forB 0 s → ρ 0 ρ 0 in PQCD can be written as The tree level amplitude t ρρ can written as where f Bs refers to the decay constant ofB s meson.
The penguin level amplitude are expressed in the following The decay amplitude forB 0 s → ρ 0 ω can be written as We can give the tree level the contribution in the following and the penguin level contribution are given as following Based on the definition of (20), we can get where .

IV. INPUT PARAMETERS
The CKM matrix, which elements are determined from experiments, can be expressed in terms of the Wolfenstein parameters A, ρ, λ and η [28]: The other parameters and the corresponding references are listed in Table.1.

V. THE NUMERICAL RESULTS OF CP VIOLATION INB
In the numerical results, we find that the CP violation can be enhanced via double ρ − ω mixing for the pure annihilation type decay channelB 0 s → ρ 0 (ω)ρ 0 (ω) → π + π − π + π − when the invariant mass of π + π − is in the vicinity of the ω resonance within perturbative QCD scheme. The CP violation depends on the weak phase difference from  ofB 0 s → ρ 0 (ω)ρ 0 (ω) → π + π − π + π − , respectively. We find the results are not sensitive to the values of ρ, η, λ and A. In Fig. 1, we give the plot of CP violating asymmetry as a function of √ s. From the Fig. 1, one can see the CP violation parameter is dependent on √ s and changes rapidly due to ρ − ω mixing when the invariant mass of π + π − is in the vicinity of the ω resonance. From the numerical results, it is found that the maximum CP violating parameter reaches 28.64% in the case of (ρ mini , η mini ).
From Eq. (14), one can see that the CP violating parameter depend on both sinδ and r. The plots of sin δ and r as a function of √ s are shown in Fig. 2, and Fig. 3, respectively. It can be seen that sin δ 0 (sinδ − and sinδ + ) vary sharply at the range of the resonance in Fig. 2. One can see that r change largely in the vicinity of the ω resonance. s → ρ 0 (ω)ρ 0 (ω) → π + π − π + π − . The dash line, dot line and solid line corresponds to r 0 , r + and r − , respectively.

VI. SUMMARY AND CONCLUSION
In this paper, we study the CP violation for the pure annihilation type decay process ofB 0 s → π + π − π + π − in perturbative QCD. It has been found that the CP violation can be enhanced greatly at the area of ρ − ω resonance.
The maximum CP violation value can reach 28.64% due to double ρ and ω resonance.
The theoretical errors are large which follows to the uncertainties of results. Generally, power corrections beyond the heavy quark limit give the major theoretical uncertainties. This implies the necessity of introducing 1/m b power corrections. Unfortunately, there are many possible 1/m b power suppressed effects and they are generally nonperturbative in nature and hence not calculable by the perturbative method. There are more uncertainties in this scheme.
The first error refers to the variation of the CKM parameters, which are given in Eq. (34). The second error comes from the hadronic parameters: the shape parameters, form factors, decay constants, and the wave function of the B s meson. The third error corresponds to the choice of the hard scales, which vary from 0.75t to 1.25t, which character-izing the size of next-to-leading order QCD contributions. Therefore, the results for CP violating asymmetrie of the decay processB 0 s → π + π − π + π − is given as following: A CP (B 0 s → π + π − π + π − ) = 28.43 +0.21+0.25+5.62 −0.25−0.16−3.98 %, (37) where the first uncertainty is corresponding to the CKM parameters, the second comes from the hadronic parameters, and the third is associated with the hard scales. The LHC experiment may detect the large CP violation for the decay processB 0 s → π + π − π + π − in the region of the ω resonance.

VII. APPENDIX: RELATED FUNCTIONS DEFINED IN THE TEXT
In this appendix we present explicit expressions of the factorizable and non-factorizable amplitudes with Perturbative QCD in Eq.(23) and Eq.(26) [10,11,34,35]. The factorizable amplitudes F LL,i ann (a i ), and F SP,i ann (a i ) (i=L,N,T) are written as f Bs F LL,T ann (a i ) = −f Bs F LR,T ann (a i ) (40) The hard functions h are written as [36] h e (x 1 , where J 0 and Y 0 are the Bessel function with H (1) 0 (z) = J 0 (z) + i Y 0 (z). The threshold re-sums factor S t follows the parameterized [37] S t (x) = 2 1+2c Γ(3/2 + c) where the parameter c = 0.4. In the nonfactorizable contributions, S t (x) gives a very small numerical effect to the amplitude [38]. Therefore, we drop S t (x) in h n and h na .
in which the Sudakov exponents are defined as where γ q = −α s /π is the anomalous dimension of the quark. The explicit form for the function s(Q, b) is: where the variables are defined byq and the coefficients A (i) and β i are with n f is the number of the quark flavors and γ E is the Euler constant. We will use the one-loop expression of the running coupling constant.
In this study, we use the model function where the share parameter ω b = 0.5 ± 0.05 GeV, and the normalization constant N Bs = 63.5688 GeV is related to the B s decay constant f Bs = 0.23 ± 0.03 GeV.