Effect of resonance for $CP$ asymmetry of the decay process $\bar{B}_{s}\rightarrow P\pi^+\pi^-$ in perturbative QCD

In the framework of Perturbative QCD (PQCD) approach we study the direct $CP$ asymmetry for the decay channel $\bar{B}_{s}\rightarrow P\pi^+\pi^-$ around the resonance range via the $\rho-\omega$ mixing mechanism (where P refer to pseudoscalar meson). We find that the $CP$ asymmetry can be enhanced by $\rho-\omega$ mixing when the masses of the $\pi^+\pi^-$ pairs are at the area of $\rho-\omega$ resonance, and the maximum $CP$ asymmetry can reach 59{\%} for the relevant decay channels.


I. INTRODUCTION
The rich data from B meson factories make the study of B physics a very hot topic. A lot of research has been made, especially for CP asymmetry. CP asymmetry is an important area in test of the Standard Model (SM) and searching new physics signals. The detection of Cabibbo-Kobayashi-Maskawa (CKM) matrix elements play an important role in understanding of CP asymmetry. The nonleptonic decay of B meson is expected to be ideal decay process in searching CP asymmetry. Direct CP asymmetry in B meson decay channel arises from weak phase and strong phase differences. In SM, the weak phase is responsible for the CP asymmetry by CKM matrix [1,2]. Meanwhile, the large strong phase is needed for producing CP asymmetry which comes from QCD correction. Recently, the large CP asymmetry was found by the LHCb Collaboration in the three-body decay channels of B ± → π ± π + π − and B ± → K ± π + π − [3]. Hence, more attention about CP asymmetry has been focused on the three body decay channels of B meson.
Direct CP asymmetry arises from the weak phase difference and the strong phase difference. The weak phase difference is determined by the CKM matrix elements, while the strong phase can be produced by the hadronic matrix and interference between intermediate states. The vacuum polarisation of photon are described by coupling the vector meson in the vector meson dominance (VMD) model. The strength of coupling of the ω meson to the photon is weak comparing with the ρ meson [4]. However, the strong interaction enhances the π + π − pair production amplitudes in the ρ and ω resonance region. ρ − ω interference presents the large contribution for the process of e + e − → π + π − due to the isospin-breaking effects. Since the strong phase exist, the ρ and ω interference can affect the direct CP asymmetry and present the sizeable contribution.
The direct CP asymmetry is discussed via ρ − ω interference in B decays by the the naive factorization approach [5]. But the method bases on the assumption of no strong rescattering, and can not predict direct CP asymmetry effectively. Recently, the CP asymmetry of charmless three-body B-decay is presented in the leading term of QCD factorization by model dependent approach, where focus on the local CP asymmetry [6]. The direct CP asymmetry of the quasi-two-body decay of B → P ρ → P ππ is calculated in perturbative QCD approach, where does not taking into account the resonance effects [7]. In our opinion, B → P ππ have effectively three contributions around the ρ resonance: (a) B → P ρ → P ππ, (b)B → P ω → P ρ → P ππ, and (c)B → P ω → P ππ. Roughly speaking, the amplitudes of their contributions: a > b > c. We have absorbed (c) into (b) effectively, which is just the (effective) ρ − ω mixing parameter:Π ρω .
The hadronic matrix elements can be calculated by the factorization approach introducing the strong phase. Adding the QCD corrections, the different dynamic methods are given based on the leading power of 1/m b (m b is b quark mass). The non-leptonic weak decay amplitudes of B mesons can be calculated by the perturbative QCD (PQCD) approach taking into account transverse momenta [8][9][10][11]. In the PQCD approach, the hard interaction consisting of six quark operator dominants the decay amplitude from short distance. The nonperturbative dynamics are included in the meson wave function which can be extracted from experiment. Finally, we obtain new large strong phases by the phenomenological mechanism of ρ− ω mixing and the dynamics of the PQCD approach. The large CP asymmetry may be obtained by the resonant region due to the strong phase.
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 of CP asymmetry from ρ − ω mixing inB s → P π + π − . Input parameters are presented in Sec.V. We present the numerical results in Sec.VI. Summary and discussion are included in Sec. VII.
The related function defined in the text are given in the Appendix.

II. THE EFFECTIVE HAMILTONIAN
Based on the expansion of the operator product, the effective weak Hamiltonian can be written as [12] where G F represents Fermi constant, c i (i=1,...,10) are the Wilson coefficients, V ub , V ud , V tb and V td are the CKM matrix elements. The operators O i have the following forms: where α and β are color indices, and q ′ = u, d or s quarks. In Eq.
One can obtain numerical values of a i including Wilson coefficients and the color index N c [9]: a 9 = C 9 + C 10 /N c , a 10 = C 10 + C 9 /N c .
The relative amplitudes and phases of H T and H P can be expressed as follows [13]: with δ and φ are strong and weak phases, respectively. φ is the weak phase in the CKM matrix that causes the CP asymmetry, which is arg[V tb V * tq /(V ub V * uq )](q = d, s). The parameter r represents the absolute value of the ratio of penguin and tree amplitudes: The CP violating asymmetry, A CP , can be written as From Equation (10), it can be seen that the CP asymmetry depends on the weak phase difference and the strong phase difference. The weak phase is determined for a particular decay process. Hence, in order to obtain a large CP asymmetry, we need some mechanism to increase δ. It has been found that ρ − ω mixing can lead to a large strong phase difference [4,[14][15][16][17][18][19][20]. Based on ρ − ω mixing and working to the first order of isospin violation, we have the following results [13]: where t ρ (p ρ ) and t ω (p ω ) are the tree (penguin) amplitudes forB 0 s → ρ 0 P andB 0 s → ωP , respectively; g ρ is the coupling constant of ρ 0 → π + π − decay process; Π ρω is the effective ρ − ω mixing amplitude which also effectively absorbed into the direct coupling ω → π + π − . s V , m V and Γ V (V =ρ or ω) represent the inverse propagator, mass and decay rate of the vector meson V , respectively.
The ρ − ω mixing paraments were recently determined precisely by Wolfe and Maltnan [21,22] ReΠ ρω (m 2 ρ ) = −4470 ± 250 model ± 160 data MeV 2 , One can find that the mixing parameter is the momentum dependence including the non-resonant contribution that absorbs the direct decay ω → π + π − . We introduce the momentum dependence of the mixing parameter Π ρω (s) for ρ − ω mixing, which leads to the explicit s dependence. It is reasonable to devote one's energies to search the mixing contribution at the region of ω mass where the two pions can be produced. We write Π ρω (s) = Re Π ρω (m 2 ω ) + Im Π ρω (m 2 ω ), and update the values as follows [23]: In fact, the contribution of the s dependence of Π ρω is negligible. We can make the expansion Π ρω (s) = Π ρω (m 2 ω ) + (s − m ω ) Π ′ ρω (m 2 ω ). From Eqs. (5)(7)(11)(12) one has Defining with δ α , δ β and δ q are strong phases. It is available from Eqs. (16)(17): In order to obtain the CP violating asymmetry in Eq. (10), sinφ and cosφ are necessary. The weak phase φ is fixed by the CKM matrix elements. In the Wolfenstein parametrization [24], one has where the same result has been found for b → d transition from Λ b decay process [14].

IV. CALCULATION
For the simplification, we take the decay process ofB 0 s → ρ 0 (ω)K 0 → π + π − K 0 as example for the study of the ρ − ω interference. The other decay channels can be obtained similarly. According to the Hamiltonian(1), based on where t ρ and p ρ refer to the tree and penguin contributions respectively. We write: and The decay amplitude forB 0 s → ωK 0 can be written as One can also present the contributions of t ω and p ω as well. t p The function F and M are given in Sec.IX. The index LL, LR and SP arise from the ( and (S − P )(S + P ) operators, respectively. where From above equations, the new strong phases δ α , δ β and δ q are introduced by the interference of ρ − ω mesons. The strong phase δ are obtained by the equations (17) and (18) in the framework of PQCD.
For the pure annihilation type decay process, one can also divides the amplitudes into t ρ , t ω , p ρ , and p ω depending on V ub V * us and V tb V * ts . The amplitudes can be given as following for the channelB 0 s → π 0 ρ 0 (ω):

V. 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 [24]: The other parameters are given as following [24,28,29]:

VI. NUMERICAL RESULTS
In the numerical results, we find the CP asymmetry can be enhanced when the masses of the π + π − pairs are in the area around the ρ − ω resonance, and the maximum CP asymmetry for our considering the decay channels can reach 59%. We also discuss the numerical results from the case of tree and penguin dominated type decay and the case of pure annihilation type decay in the framework of Perturbative QCD. The CP violation is associated with the CKM matrix elements and √ s. In our numerical calculations, we find that the CP asymmetry depend weakly on the variation of the CKM matrix elements. Hence, we let (ρ, η) vary between the central values (ρ central , η central ).

A. The case of tree and penguin dominated type decay
We refer to the decay processes ofB 0 s → ρ 0 (ω)K 0 → π + π − K 0 ,B 0 s → ρ 0 (ω)η → π + π − η andB 0 s → ρ 0 (ω)η ′ → π + π − η ′ as the case of tree and penguin dominated type decay. In Fig.1, we show the plot of CP asymmetry as a function of √ s. One can find the CP asymmetry varies sharply when the masses of the π + π − pairs are in the area around the ρ−ω resonance range. For the decay process ofB 0 s → ρ 0 (ω)K 0 → π + π − K 0 , the maximum CP asymmetry can reach 40%. For the decay channels ofB 0 s → ρ 0 (ω)η ′ → π + π − η ′ andB 0 s → ρ 0 (ω)η → π + π − η, we obtain the maximum CP asymmetry is 59% and 21%, respectively. From Equation (10), one can find the CP asymmetry is affected by the weak phase difference, the strong phase difference and r. The weak phase depends on the CKM matrix elements. Hence, the change of CP asymmetry is derived from the variation of strong phase δ and r except the CKM matrix. We take the central values from the parameters of (ρ central , η central ). Taking into account of ρ − ω mixing, we can see that sin δ oscillate considerably at the area of ρ − ω resonance from Fig.2 for the considering decay processes.
The plot of r as a function of √ s is presented in Fig.3. One can see that the r change sharply for the process of B. The case of pure annihilation decay type In Fig.4, we present the plot of CP asymmetry parameter as a function √ s corresponding to central parameter values of CKM matrix elements for the pure annihilation decay type ofB 0 s → ρ 0 (ω)π 0 → π + π − π 0 . One can find the maximum CP asymmetry reach 28% when the masses of the π + π − pairs are in the area around the ρ − ω resonance range. The plots of sin δ and r as a function of √ s are given in Fig.5 and Fig.6, respectively. We can see that sin δ and r oscillate sharply taking into account ρ − ω resonance. Generally, the CP asymmetry is tiny in the case of pure annihilation decay. However, the maximum CP asymmetry can reach 28% at the area of ρ − ω resonance, which give us a chance to search CP asymmetry from the pure annihilation decay type.

VII. SUMMARY AND CONCLUSION
In this paper, we study the CP asymmetry for the decay process ofB s → P π + π − in Perturbative QCD. It has been found the CP asymmetry can be enhanced greatly at the area of ρ − ω resonance. The maximum CP asymmetry can reach 40% for the process ofB 0 s → ρ 0 (ω)K 0 → π + π − K 0 . However, the paper has also discussed the CP asymmetry of the decay process ofB s → ρ 0 (ω)K 0 → π + π − K 0 from b → d transition in QCD factorization. The maximum CP asymmetry reach 46% when the invariant mass of the π + π − pair is in the vicinity of the ω resonance from QCD factorization [19]. The difference of CP asymmetry mainly comes from the strong phase difference between QCD factoriztion and Perturbative QCD. The hadronic matrix elements can be calculated from first principles in the decays of B-meson. Due to the power expansion of 1/m b (m b is b quark mass), all of the theories of factorization are shown to deal with the hadronic matrix elements in the leading power of 1/m b . But these methods are different significantly due to the collinear degree or transverse momenta. The power counting is different from the hard kernels between QCDF and PQCD. It is important to extract the strong phase difference for CP violation. The more different feature of QCDF and PQCD is the strong interaction scale at which of PQCD is low, typically of order 1 ∼ 2 GeV, the case of QCDF is order O(m b ) for the Wilson coefficients.
Meanwhile, we find that the CP asymmetry associated with the case of pure annihilation type decay process of B 0 s → ρ 0 (ω)π 0 → π + π − π 0 can be enhanced and the maximum value reach 28%. Hence, one can search for the large CP asymmetry at the area of ρ− ω resonance from pure annihilation type decay process ofB 0 s → ρ 0 (ω)π 0 → π + π − π 0 . In this work, we have take the Perturbative QCD approximation which add the QCD correction to the naive factorization which is based on the power expansion of 1/m b . The final state interaction is also neglected in this approximation which may give some uncertainties. There are some uncertainties from the input parameters, the hard scattering scale and CKM matrix elements. The theoretical results can be improved by high order correction from α s and 1/m b . The functions associated with the tree and penguin contributions are presented for the factorization and nonfactorization amplitudes in PQCD approach [10,11,30]. The functions of the case of tree and penguin dominated type decay are written as

VIII. ACKNOWLEDGMENTS
• (V − A)(V + A) operators: • (S − P )(S + P ) operators: • (S − P )(S + P ) operators: re-summation [31], the other is the propagator of virtual quark and gluon. They are defined by where H (1) 0 (z) = J 0 (z) + i Y 0 (z). The S t re-sums the threshold logarithms ln 2 x appearing in the hard kernels to all orders and it has been parameterized as with c = 0.4. In the nonfactorizable contributions, S t (x) gives a very small numerical effect to the amplitude [32].
Therefore, we drop S t (x) in h n and h na .