Direct $CP$ violation for $\bar{B}_s^0 \to \phi {\pi^+}{\pi^-}$ in Perturbative QCD

In perturbative QCD approach, we study the direct $CP$ violation in the decay of $\bar{B}_s^0\to\rho(\omega )\phi\to {\pi^+}{\pi^-}\phi$ via isospin symmetry breaking. An interesting mechanism involving the charge symmetry violating between $\rho$ and $\omega$ is applied to enlarge the $CP$ violating asymmetry. We find that the $CP$ violation can be enhanced by the $\rho-\omega$ mixing mechanism when the invariant masses of the $\pi^+\pi^-$ pairs are in the vicinity of the $\omega$ resonance. For the decay process of $\bar{B}^0_{s}\to\rho(\omega )\phi\to{\pi^+}{\pi^-}\phi$, the maximum $CP$ violation can reach $5.98\%$. The possibility of detecting the $CP$ violation is also presented.

Based on the power expansion in 1/m b (m b is the 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 . However, these methods pertain to whether one takes into account the collinear degrees of freedom or the transverse momenta. Meanwhile, in order to have a large signal of CP violation, we need appeal to some phenomenological mechanism to obtain a large strong phase δ. ρ − ω mixing has been used for this purpose in the past few years and focuses on the naive factorization and QCD factorization approaches [25][26][27][28][29][30]. Recently, Lü et al. attempted to generalize the pQCD approach to the three-body non-leptonic decay via ρ − ω mixing in B 0,± → π 0,± π + π − and B c → D + (s) π + π − decays [31,32]. In this paper, we will focus on the CP violation of the decay processB 0 s → ρ(ω)φ → π + π − φ via ρ − ω mixing in the pQCD approach. Isospin symmetry breaking plays a significant role in the ρ − ω mixing mechanism. The mixing between the u and d flavors leads to the breaking of isospin symmetry for the ρ − ω system [33,34]. In Refs. [35,36], the authors studied the ρ − ω mixing and the pion form factor in the time-like region, where ρ − ω mixing was used to obtain the (effective) mixing matrix element Π ρω (s), which consists of two part contributions: one from the direct coupling of ω → 2π and the other from ω → ρ → 2π mixing [37][38][39]. The magnitude has been determined by the pion form factor through the data for the cross section of e + e − → π + π − in the ρ and ω resonance region [36,[39][40][41][42]. Recently, isospin symmetry breaking was discussed by incorporating the vector meson dominance (VMD) model in the weak decay process of the meson [27,32,[43][44][45]. 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,[43][44][45] to investigate the decay process ofB 0 s → ρ(ω)φ → π + π − φ via isospin symmetry breaking.
The remainder of this paper is organized as follows. In Sec. II we will briefly introduce the pQCD framework and present the form of the effective Hamiltonian and wave functions. In Sec. III we give the calculating formalism and details of the CP violation from ρ − ω mixing in the decay processB 0 s → ρ(ω)φ → π + π − φ. In Sec. IV we show the 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 Appendix.

II. THE FRAMEWORK
For the decay process ofB s → M 2 M 3 , integrated over the longitudinal and the transverse momenta, the emitted or annihilated particle M 2 can be factored out. The rest of the amplitude can be expressed as the convolution of the wave functions φ Bs , φ M3 and the hard scattering kernel T H . The pQCD factorization theorem has been developed for non-leptonic heavy meson decays, based on the formalism of Lepage, Brodsky, Botts and Sterman [46][47][48][49]. 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, it is conceptually written as the following: where k i (i = 1, 2, 3) are momenta of light quarks in the mesons. Tr denotes the trace over Dirac and color indices.
is Wilson coefficient which comes from the radiative corrections at short distance. φ M (m = 2, 3) is wave function which describes non-perturbative contribution during the hadronization of mesons, which should be universal and channel independent. The hard part T H is rather process-dependent.
With the operator product expansion, the effective weak Hamiltonian in bottom hadron decays is [50] where G F is the Fermi constant, C i (µ) (i=1,...,10) are the Wilson coefficients, V q1q2 (q 1 and q 2 represent quarks) is the CKM matrix element. The operators O i have the following forms: where α and β are color indices, the sum index q runs over the "active" flavors quarks at the scale m b , and e q is the electric charge of the quark q (q = u, d, s, c or b quarks). In Eq. The Wilson coefficients, C i (µ), represent the power contributions from scales higher than µ (which refer to the long-distance contributions) [51]. Since the QCD has the property of asymptotic freedom, they can be calculated in perturbation theory. The Wilson coefficients include the contributions of all heavy particles, such as the top quark, the W ± bosons, and so on. Usually, the scale µ is chosen to be of order O(m b ) for B meson decays. Since we work in the leading order of perturbative QCD (O(α s )), it is consistent to use the leading order Wilson coefficients. So, we use numerical values of C i (m b ) as follow [19,21]: The Wilson coefficients a 1 -a 10 are defined as usual [52][53][54][55]: a 9 = C 9 + C 10 /3, a 10 = C 10 + C 9 /3.
For the decay channel ofB 0 s → M 2 M 3 , we denote the emitted meson as M 2 while the recoiling meson is M 3 . The M 2 (ρ or ω) and the final-state M 3 (φ) move along the direction of n + = (1, 0, 0 T ) and n − = (0, 1, 0 T ) in the light-cone coordinates, respectively. We denote the ratios r φ = M φ M Bs , r ρ = Mρ M Bs and r ω = Mω M Bs . In the limit M φ , M ρ , M ω → 0, one can drop the terms of proportional to r 2 φ , r 2 ρ , r 2 ω safely. The symbols P B , P 2 and P 3 refer to thē B s meson momentum, the ρ(ω) meson momentum, and the final-state φ momentum, respectively. Under the above approximation, the momenta can be written as: One can denote the light (anti-)quark momenta k 1 , k 2 and k 3 for the mesons B s , ρ(ω) and φ, respectively. We can write: where x 1 , x 2 and x 3 are the momentum fraction. k 1⊥ , k 2⊥ and k 3⊥ refer to the transverse momentum of the quark, respectively. The longitudinal polarization vectors of the ρ(ω) and φ are given as which satisfy the orthogonality relationship of 2 (L)·P 2 = 3 (L)·P 3 = 0, and the normalization of 2 2 (L) = 2 3 (L) = −1. The transverse polarization vectors can be adopted diredtly as The wave function of B s meson can be expressed as where the distribution amplitude φ Bs is shown in Refs. [56][57][58]: The shape parameter ω b is a free parameter. Based on lattice QCD and the light-cone sum rule [59], we take ω b = 0.50 GeV for the B s meson. The normalization factor N Bs depends on the values of ω b and the decay constant f Bs , which is defined through the normalization relation The distribution amplitudes of vector meson(V=ρ, and φ a V , are calculated using light-cone QCD sum rule [60,61]: where t = 2x − 1. Here f V is the decay constant of the vector meson with longitudinal polarization. The Gegenbauer polynomials C ν n (t) can be found easily in Refs. [62,63].
The hadronic decay rare for the process ofB 0 s → ρ(ω)φ is written as where P c = |P 2z | = |P 3z | is the momentum of the vector meson. The superscript σ denotes the helicity states of the two vector mesons with the longitudinal (transverse) components L(T). The amplitude A (σ) is decomposed into with the convention 0123 = 1. The amplitude A i (i refer to the three kind of polarizations, longitudinal (L), normal (N) and transverse (T)) can be written as where a, b and c are the Lorentz-invariant amplitudes. M 2 , M 3 refer to the masses of the vector mesons ρ(ω) and φ, respectively.
The longitudinal H 0 , transverse H ± of helicity amplitudes can be expressed where H 0 , H + and H − are the tree-level and penguin-level helicity amplitudes of the decay processB 0 s → ρ(ω)φ → π + π − φ from the three kind of polarizations, respectively. The helicity summation satisfy the relation, In the VMD model [66,67], the vacuum polarisation of the photons are assumed to be coupled through the vector meson (ρ meson). Based on the same mechanism, ρ − ω mixing was proposed and later gradually applied to B meson physics. The formalism for the CP violation in B hadronic decays can be generalized to B s in a straightforward manner [25,27,43]. According to the effective Hamiltonian, the amplitude A (Ā) for the decay process ofB 0 can be written as [43]: where H T and H P 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: 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, and it is φ = arg[(V tb V * ts )/(V ub V * us )] for the b → s transition. 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 T i (i = 0, +, −) are the tree-level helicity amplitudes of the decay processB 0 s → π + π − φ from H 0 , H + and H − of the Eq. (23), respectively. r j (j = 0, +, −) refer to the absolute value of the ratio of tree and penguin amplitude for the three kind of polarizations, respectively. δ k (k = 0, +, −) represent the relative strong phases between the tree and penguin operator contributions from three kind of helicity amplitudes. We can see explicitly from Eq. (30) that both weak and strong phase differences are responsible for CP violation. In order to obtain a large signal for direct CP violation, we need some mechanism to change either sin δ or r. With this mechanism, working at the first order of isospin violation, we have the following results when the invariant mass of π + π − is near the ω resonance mass [26,43]: where t i ρ (p i ρ ) and t i ω (p i ω ) are the tree (penguin)-level helicity amplitudes forB s → ρ 0 φ andB s → ωφ, respectively.The amplitudes t i ρ , p i ρ , t i ω and p i ω can be found in Sec. III B. 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 coupling ω → π + π − [39]. s V , m V and Γ V (V =ρ or ω) is the inverse propagator, mass and decay rate of the vector meson V , respectively. s V can be expressed as with √ s being the invariant masses of the π + π − pairs. The numerical values for the ρ − ω mixing parameter From Eqs. (25), (27), (31) and (32) one has Defining [25,69] where δ i α , δ i β and δ i q are strong phases form the three kind of polarizations, respectively. One finds the following expression from Eqs. (35) and (36): αe iδ i α , βe iδ i β , and r e iδ i q will be calculated later. In order to obtain the CP violating asymmetry in Eq. (30), A cp , sinφ and cosφ are needed, where φ is determined by the CKM matrix elements. In the Wolfenstein parametrization [70], where the same result has been used for b → s transition from Ref. [44].

B. Calculation details
We can decompose the decay amplitude for the decay processB 0 s → ρ 0 (ω)φ in terms of tree-level and penguin-level contributions depending on the CKM matrix elements of V ub V * us and V tb V * ts . From Eqs. (30), (35) and (36), in order to obtain the formulas of the CP violation, we need calculate the amplitudes t ρ , t ω , p ρ and p ω in perturbative QCD approach. The F and M function can be found in the Appendix from the perturbative QCD approach. There are four types of Feynman diagrams contributing toB 0 s → ρ 0 (ω)φ emission decay mode. The leading order diagrams in pQCD approach are shown in Fig.1. The first two diagrams in Fig.1 (a)(b) are called factorizable diagrams and the last two diagrams in Fig.1 (c)(d) are called non-factorizable diagrams [71,72]. 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 ofB 0 s → ρφ orB 0 s → ωφ are similar to those of B → φK * and B s → K * − K * + [72,73]. We just need to replace some corresponding wave functions, Wilson coefficients and corresponding parameters.
With the Hamiltonian Eq. (2), depending on CKM matrix elements of V ub V * us and V tb V * ts , the electroweak penguin dominant decay amplitudes A (i) forB 0 s → ρ 0 φ in pQCD can be written as where the superscript i denote different helicity amplitudes L, N and T . The longitudinal t 0 ρ(ω) , transverse t ± ρ(ω) of helicity amplitudes satisfy relationship of t 0 ρ(ω) = t L ρ(ω) and t ± ρ(ω) = (t N ρ(ω) ∓ t T ρ(ω) )/( √ 2). The amplitudes from tree and penguin diagrams can be written as T i ρ = t i ρ /V ub V * us and P i ρ = p i ρ /V tb V * ts , respectively. The formula for the tree level amplitude is where f ρ refers to the decay constant of ρ meson. The penguin level amplitude are expressed in the following The QCD penguin dominant decay amplitude forB 0 s → ωφ can be written as where T i ω = t i ω /V ub V * us and P i ω = p i ω /V tb V * ts which refer to the tree and penguin amplitude, respectively. We can give the tree level contribution in the following where f ω refers to the decay constant of ω meson. The penguin level contribution are given as following Based on the definition of Eq. (36), we can get where .
From above equations, the new strong phases δ i α , δ i β and δ i q are obtained from tree and penguin diagram contributions by the ρ − ω interference. The total strong phase δ i are obtained by the Eqs. (36) and (37) in the framework of pQCD.
The other parameters and the corresponding references are listed in Table I.
We have investigated the CP violating asymmetry, A CP , for theB 0 s → ρ 0 (ω)φ → π + π − φ decay process. The numerical results of the CP violating asymmetry are shown for the decay process in Fig. 2. It is found that the CP violation can be enhanced via ρ − ω mixing for the decay channelB 0 s → ρ 0 (ω)φ → π + π − φ 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 CKM matrix elements and the strong phase difference.
The CKM matrix elements, which relate to ρ, η and λ, are given in Eq. (50). 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 = 0.224 in Eq. (52). In our numerical calculations, we let ρ, η and λ = 0.224 vary among the limiting values. The numerical results are shown from Fig. 2 to Fig. 4 with the different parameter values of CKM matrix elements. The dash line, dot line and solid line corresponds to the maximum, middle, and minimum CKM matrix element for the decay channel ofB 0 s → ρ 0 (ω)φ → π + π − φ, respectively. We find the CP violation is not sensitive to the CKM matrix elements for the different values of ρ and η. In Fig. 2, we give the plot of CP violating asymmetry as a function of √ s. From the Fig. 2, 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 5.98% for the decay channel ofB 0 s → π + π − φ in the case of (ρ max , η max ). From Eq. (30), one can see that the CP violating parameter is related to sinδ and r. The plots of sin δ 0 (sin δ + and sin δ − ) and r 0 (r + and r − ) as a function of √ s are shown in Fig. 3 and Fig. 4. We can see that the ρ − ω mixing mechanism produces a large sin δ 0 (sin δ + and sin δ − ) at the ω resonance. As can be seen from Fig. 3, the plots vary sharply in the cases of sin δ 0 , sin δ + and sin δ − . Meanwhile, sin δ 0 and sin δ − change weakly compared with the sin δ + .
It can be seen from Fig. 4 that r + change more rapidly than r 0 and r − when the π + π − pairs in the vicinity of the ω resonance.
The Large Hadron Collider (LHC) is a proton-proton collider which has currently started at the European Organization for Nuclear Research (CERN). In order to achieve the required energy and luminosity, the technology and equipment has been upgraded many times. The LHC Run I first data-taking period lasted from 2010 to 2013 [77]. In the next few years, there are two major detector (ATLAS and CMS) upgrades happening after Run II and Run III. With a series of upgrades and modifications, the LHC provides a TeV-level high energy frontier and an opportunity to further improve conformance testing of the CKM matrix. The production rates for heavy quark flavors will be much at the LHC, and the bb production cross section will be of the order of 0.5 mb, providing about 10 12 bottom quark events per year [77,78]. The heavy flavour physics experiment is one of the main projects of LHC experiments.
Especially, LHCb is a specialized B-physics experiment, designed primarily to precisly measure the parameters of new physics in CP violation and rare decays in the interactions of beauty and charm hadrons systems. Such studies can help to explain the Matter-Antimatter asymmetry of the Universe. Recently, the LHCb collaboration found clear evidence for direct CP violation in some three-body decay channels in charmless decays of B meson. Meanwhile, large CP violation is obtained in B ± → π ± π + π − in the region 0.6 GeV 2 < m 2 π + π − low < 0.8 GeV 2 and m 2 π + π − high > 14 GeV 2 [8,79]. A zoom of the π + π − invariant mass from the B ± → π ± π + π − decay process is shown the region 0.6 GeV 2 < m 2 π + π − low < 0.8 GeV 2 zone in the Figure 8 of the Ref. [79]. In addition, the branching fractions is probed in the π + π − invariant mass range 400 < m(π + π − ) < 1600 MeV/c 2 forB 0 s → π + π − φ [80]. In the next years, we expect the LHCb Collaboration to focus our prediction of CP violation from theB 0 s → ρ 0 (ω)φ → π + π − φ decay process when the invariant mass of π + π − is in the vicinity of the ρ resonance [80]. Theoretically, in order to achieve the current experiments on b-hadrons, which can only provide about 10 7 BB pairs [81]. Therefore, it is very convenient to observe the CP violation forB 0 s → ρ 0 (ω)φ → π + π − φ when the invariant masses of π + π − pairs are in the vicinity of the ω resonance at the LHC experiments.

VI. SUMMARY AND CONCLUSION
In this paper, we study the CP violation for the decay process ofB 0 s → ρ 0 (ω)φ → π + π − φ due to the interference of ρ-ω mixing in perturbative QCD. It has been found that the CP violation can be enhanced at the area of ρ − 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 5.98% when the invariant mass of the π + π − pair is in the vicinity of the ω resonance.
In the calculation, we need the renormalization scheme independent Wilson coefficients for the tree and penguin operators at the scale m b . The major uncertainties is from the input parameters. In particular, these include the CKM matrix element parameters, the perturbative QCD approach and the hadronic parameters (the shape parameters, decay constants, the wave function and etc). We expect that our predictions will provide useful guidance for future investigations in B s decays.
The hard scale t are chosen as the maximum of the virtuality of the internal momentum transition in the hard amplitudes, including 1/b i : The function h, coming from the Fourier transform of hard part H, are written as [82], 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 [83] S t (x) = 2 1+2c Γ(3/2 + c) where the parameter c = 0.4. In the non-factorizable contributions, S t (x) gives a very small numerical effect to the amplitude [84]. 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.