Direct $CP$ violation of three bodies decay process from the resonance effect

The physical state of $\rho$-$\omega$-$\phi$ mesons can be mixed by the unitary matrix. The decay processes of $\omega \rightarrow \pi^{+}\pi^{-}$ and $\phi \rightarrow \pi^{+}\pi^{-}$ are from the isospin symmetry breaking. The $\rho-\omega$, $\rho-\phi$ and $\omega-\phi$ interferences lead to resonance contribution to produce the strong phases. The $CP$ asymmetry is considered from above isospin symmetry breaking due to the new strong phase for the first order. It has been found the $CP$ asymmetry can be enhanced greatly for the decay process of $B^{0}\rightarrow\pi^+\pi^{-}\eta^{(')}$ when the invariant masses of the $\pi^+\pi^{-}$ pairs are in the area around the $\omega$ resonance range and the $\phi$ resonance range in perturbative QCD. We also discuss the possibility to search the predicted $CP$ violation at the LHC.

is suggested that the re-scattering of final-state particles should play an equally important role in the other three-body non-charm-decay processes of B-meson. Since the isospin is conserved in the decay process of ρ 0 → π + π − and the decay rate is 100%, the large contribution of S-wave amplitude has been observed [15,16]. At the same time, CP violation was found in the low invariant mass region of S wave, which established that CP violation was related to the interference of S wave and P wave amplitudes [13]. Some progress has also been made in the measurement of CP violating phase angle and for the analysis of amplitudes in the B meson four-body decay process. In the invariant mass regions of K ± π ∓ , the LHCb Collaboration analyzed the time-dependent amplitudes and phase angles of CP violation by resonance of K * 0 (800) 0 and K * 0 (1430) 0 , K * 0 (892) 0 and K * 2 (1430) 0 [17]. Considering the resonance of ρ, ω, f 0 (500), f 0 (980) in the invariant mass regions of π + π − , and the main contribution of K * (892) 0 decay in the invariant mass regions of K + π − , one analyses the decay amplitude of B 0 → (π + π − )(K + π − ) [18].
Theoretically, three-body or multi-body decay is a relatively complex calculation process, and more studies have been done recently [19][20][21][22][23][24][25][26][27][28][29]. Under the perturbation QCD framework, the final state interaction is described by the two-particle distribution amplitude in the resonance region. The three-body decay process is treated as a quasi-twobody decay process in the form of intermediate resonant states [25,26]. Recently, the narrow width approximation is applied to extract the branching fraction of the quasi-two-body decay processes by an intermediate resonant state.
The correction will be considered when the resonance has a sufficiently large width. Since the widths ω(782) and φ(1020) are relatively small, one can neglect the effects safely as quasi-two-body decay processes. It has been show that the correction is generally less than 10% for vector resonances. It is known that the decay width of ρ(770) is large. The correction factor is at 7% level for the decay process of B − → ρ(770)π − → π + π − π − in the frame of QCD factorization Under different factorization frameworks, the numerical results may vary greatly within the error range. Particularly, we notice that the parameter η R is introduced to identify the degree of approximation for Γ(B → RP 3 )B(R → P 1 P 2 ) = η R Γ(B → RP 3 → P 1 P 2 P 3 ) in narrow width approximation [30,31]. One can find that the η R can be divided out for the calculation of CP violation. Based on the above considerations, we focus on vector meson resonance and ignore the effect of this correction in this work.
Considering the influence of isospin symmetry breaking, CP violations for the decay processes of the three-body or four-body decay from B meson have been studied via ρ − ω mixing [27][28][29]. The mechnism of ρ − ω mixing produces strong phase to change the CP violation. Hence, the ρ − ω, ρ − φ and ω − φ interferences may lead to resonance contribution to produce the new strong phases. The CP violation is considered from above isospin symmetry breaking due to the new strong phase for the first order. We will focus on the CP violations from the ρ − ω − φ interferences.
The paper is organized as follows. In the second part, we give our theoretical derivation process for the resonance effects in detail. In the third part, we give the values of CP violation in the decay process of individual decay process under our theoretical framework. Summary and discussion are presented in forth part.

A. Formalism
According to the vector mesons dominance (VMD) [32], e + e − annihilate into photons, which are polarized in vacuum to form vector particles ρ 0 (770), ω(782), φ(1020) and then decay into π + π − pairs. The mixed amplitude parameters of the corresponding two or three particles can be obtained by the electromagnetic form factor of π meson, and the values are given by combining with the experimental results [33]. The intermediate state particle is a non-physical state, which is transformed into a physical field through an isospin field and connected by the unitary matrix R. The mixed amplitude parameters can be expressed as Π ρω , Π ρφ and Π ωφ , and the contribution of higher order terms are ignored.
The momentum is transmitted by the vector meson by VMD model. These amplitudes should be related to the square of momentum. The transformation amplitudes are dependent on s associated with the square of momentum. The unitary matrix R(s) relates the isopin field ρ 0 I , ω I , φ I to the physical field ρ 0 , ω, φ by the relation: where which F ρω (s), F ρφ (s), F ωφ (s) is order O(λ), (λ 1). The transformation of two representations is related to each other through unitary matrices R. Based on isospin ρ 0 I , ω I , φ I field, we can construct the isospin basis vector |I, I 3 >. Thus, the physical particle state can be represented as a linear combination of the above basis vectors. We use M and N to represent the physical state and the isospin basis vector of the particle, respectively. According to the orthogonal normalization relation, we can get: and One can write |M >=  [33]. From the translation of the two representations, the physical states can be written as We define Ignoring the contribution of higher order terms, we can diagonalize the equation W I by the matrix R in the physical representation.
From the Eqs. (9) and (10), we can neglect the high order terms of F 2 , F < ρ I |W |ω I > and F < ρ I |W |φ I > for simplification. We can obtain the symmetry relationship of F < ρ I |W |ω I >= F < ω I |W |ρ I > and F < ρ I |W |φ I >= F < φ I |W |ρ I > : The square of the complex mass can be written as [33,34] where Γ ρ , Γ ω and Γ φ are the decay width of the mesons ρ 0 , ω, φ, respectively. Hence, In the physical representation, the propagator of intermediate state particle from vector meson can be expressed as D V1V2 and D I V1V2 refer to the propagator D V1V2 =< 0|T V 1 V 2 |0 > and D I V1V2 =< 0|T V I 1 V I 2 |0 > in the representations of physics and isospin, respectively. We can obtain In the same way, D ωρ = D ρω , D ρφ = D φρ and D ωφ = D φω .
In the state of physics, there are not the ρ − ω − φ mixing so that D ρω , D ρφ and D ωφ are equal to zero. One can The parameters of Π ρω , Π ωφ , Π ρφ , F ρω , F ρφ and F ωφ are order of O(λ) (λ 1). Any two or three terms multiplied together are of higher order and can be ignored. Hence, we can get and where we can define s V , m V , and Γ V (V = ρ, ω or φ) refer to the inverse propagator, mass and decay rate of the vector meson V , respectively. We can write where the √ s denotes the invariant mass of the π + π − pairs [35].
The ρ − ω mixing paraments were recently determined precisely by Wolfe and Maltnan [36,37] ReΠ ρω (m 2 ρ ) = − 4470 ± 250 model ± 160 data MeV 2 , The ρ − φ mixing paraments have been given near the φ meson [38] The mixing parameter depends on the momentum including both the resonant and non-resonant contribution which absorbs the direct decay processes ω → π + π − and φ → π + π − from the isospin symmetry breaking effects. The mixing parameters Π ρω (s) and Π ρφ (s) are the momentum dependence for ρ − ω mixing and ρ − φ mixing, respectively. We expect to search for the contribution of this mixing mechanism in the resonance region of ω and φ mass where two pions are also produced by isospin symmetry breaking. One can express Π ρω (s) = Re Π ρω (m 2 ω ) + Im Π ρω (m 2 ω ) and Π ρφ (s) = Re Π ρφ (m 2 φ ) + Im Π ρφ (m 2 φ ) and update the values: and The experiments of e + e − → hadrons are measured for the cross section to determine the parameters of vector mesons in the energy range of ρ 0 , ω and φ from the reactions. The processes of ω → π + π − and φ → π + π − from the isospin symmetry breaking can provide dynamics information about the interference of ρ 0 , ω and φ mesons. CP violation depends on the CKM matrix elements associated with weak phase and strong phase. The effect of the isospin symmetry breaking can provide the strong phase to change the CP violation from intermediate vector meson mixing.
The decay amplitude A(Ā) for the process of B 0 → π + π − η ( ) can be expressed as: where π + π − η ( ) |H T |B 0 and π + π − η ( ) |H P |B 0 refer to the contribution from the tree level and penguin level due to the operators of Hamiltonian, respectively. The ratio of the penguin diagram contribute to the tree diagram contribution produces the phase angle, which affects the CP violation in the decay process. The formalism of amplitude can be expressed as follows: The weak phase φ is from the CKM matrix. The strong phase δ and parameter r are dependent on the interference of the two level contribution and other mechanism. We can define We can provide the decay amplitude from the isospin field. Then, the physical decay amplitude is obtained by the translation of the two representation from the B → V I and V I → π + π − by the unitary matrix R. One can find the propagators of intermediate vector mesons become physical states from the diagonal matrix. To the leading order approximation of isospin violation, one can provide the following results: respectively. The coupling constant g ρ is from decay process of ρ 0 → π + π − . Then, we can obtain Defining with δ α , δ β and δ q are strong phases. It is available from Eqs. (40) (41): We need give sinφ and cosφ to obtain the CP violation The weak phase φ comes from the CKM matrix elements.
In the Wolfenstein parametrization [39], one has One can use perturbation theory to calculate the decay amplitudes by introducing the Sudakov factor to eliminate endpoint divergence.
For simplification, we take the decay process B 0 → ρ 0 (ω, φ)η → π + π − η as an example to illustrate the mechanism in detail. One need obtain the formalisms of t ρ , t ω , t φ and p ρ , p ω , p φ to calculate the CP violations which are from the tree level and penguin level contributions, respectively. C i are the Wilson coefficients. One can find the formalisms of the functions F and M by Appendix.
Based on CKM matrix elements of V ub V * ud and V tb V * td , the decay amplitude of B 0 → ρ 0 η in perturbation QCD approach can be written as where t ρ and p ρ refer to the tree and penguin contributions respectively. The formalisms can be obtained by the perturbative QCD method.

D. INPUT PARAMETERS AND WAVE FUNCTIONS
The CKM matrix, which elements are determined from experiments, can be expressed in terms of the Wolfenstein parameters A, ρ, λ and η [39]: The other parameters are given as following [39,43]: For the B meson wave function, we adopt the model where ω b is a free parameter and we take ω b = 0.4 ± 0.04GeV and N B = 91.7456 is the normalization factor for [44,45]. This is the best fit for most of the measured hadronic B decays. For the light meson wave function, we neglect the b dependent part, which is not important in numerical analysis. We choose the wave function of ρ(ω, φ) meson similar to the pion case for φ ρ(ω,φ) , φ t ρ(ω,φ) , and φ s ρ(ω,φ) [46][47][48]. The relevant Gegenbauer polynomials are defined by C  [46]. The two input parameters f q and f s , in the quark-flavor basis have been extracted from various related experiments [41,42]. The other parameters can be found in [49][50][51][52][53].

E. Numerical results
In the framework of perturbative QCD, we find that the CP violation is changed sharply for the decay processes of B 0 → π + π − η and B 0 → π + π − η from the ρ − ω − φ resonance in the vicinity of ω and φ mass. The results are shown in Fig.1, Fig.2 and Fig.3, respectively. The plot of the CP violation as a function of √ s is presented in Fig.1.
One can find the CP violation varies sharply when the invariant masses of the π + π − pairs are in the area around the ω resonance range and changes slightly around the φ resonance range. For the decay channel of B 0 → π + π − η, we obtain the CP violation varies from 99.6% to −14.2% and from −18.8% to −6.3% at the ρ − ω resonance range and the ρ − φ resonance range, respectively. For the decay channel of B 0 → π + π − η , the CP violation varies from 95.9% to −18.8% and from 47.1% to 59.5% at the ρ − ω resonance range and the ρ − φ resonance range, respectively.
One can find the CP violation is affected by the weak phase difference, the strong phase difference and r. The considerably at the area of ω resonance, and changes slightly at the area of φ resonance. The plot of r as a function of √ s is presented in Fig.3. The r changes sharply for the ω resonance range and slightly for the φ resonance range.

III. SUMMARY AND DISCUSSION
In this paper, we introduce the formalism for the ρ−ω −φ mesons interferences from the isospin symmetry breaking.
The new strong phase can be produced by the resonance contributions of ρ − ω, ρ − φ and ω − φ. The mechanism is applied to the decay process of B 0 → π + π − η ( ) . It has been found the CP asymmetry oscillates greatly for the resonance range. The maximum CP asymmetry can reach 99.6% and −18.2% in the vicinity of the ω resonance range and the φ resonance range for the decay process of B 0 → π + π − η, respectively. For the decay process of B 0 → π + π − η , the maximum CP asymmetry is 95.9% and 59.5% at the area of ω resonance and φ resonance. Our formalism can be used to calculate the other decay process.
Detection of CP violation signal is an important field in the B meson decay process. For the three bodies final states, the CP violation is often dominated by quasi-two-body decay channels and depends on the relative phase between the two quasi-two-body amplitudes. The numbers needed for observing the large CP violation depend on both the magnitudes of the CP violation and the branching ratios of heavy B meson decays. We find that the contribution of three meson mixing has little effect on the branching ratio and can be ignored safely because the mechanism can only provide the strong phase. For one (three) standard deviation signature, the number of BB pairs we need is [54][55][56] where BR is the branching ratio for B → ρ 0 η ( ) . We present the numbers of BB pairs for observing the large CP violation at LHC. For the channel B 0 → ρ 0 (ω, φ)η → π + π − η, the numbers of BB pairs are 10 4 (10 5 ) and 10 8 (10 9 ) in the resonance ranges of ω and φ for 1σ (3σ) signature. We need 10 5 (10 6 ) and 10 7 (10 8 ) BB pairs to observe the CP violation from the two resonance ranges in the decay process of B 0 → ρ 0 (ω, φ)η → π + π − η for 1σ (3σ) signature, respectively.
The  [57,58]. To extend its discovery potential, the LHC made a major upgrade and increased its luminosity by a factor of five beyond its design value recently. Hence, it is very possible to observe the large CP violation in small energy range of ρ 0 ∼ ω and ρ 0 ∼ φ resonances at the peak values of CP violation from the LHC experiment due to the high luminosity large hadron collider (HL-LHC) even though the branching fractions in these regions may be tiny. For the experiments, it is possible to reconstruction π + , π − and η ( ) mesons when the invariant masses of π + π − pairs are in the vicinity of the ω or φ resonances. Therefore, it is very possible to observe the large CP violation in B 0 → ρ 0 (ω, φ)η ( ) → π + π − η ( ) at the LHC.

IV. ACKNOWLEDGMENTS
This work was supported by National Natural Science Foundation of China (Project Numbers 11605041).

V. APPENDIX: RELATED FUNCTIONS DEFINED IN THE TEXT
The functions related with the tree and penguin contributions are presented with PQCD approach [44,45,59].
The hard scales t are chosen as The function h coming from the Fourier transformations of the function H (0) [60]. They are defined by where J 0 is the Bessel function and K 0 , I 0 are the modified Bessel functions K 0 (−ix) = − π 2 y 0 (x) + i π 2 J 0 (x), and F (j) 's are defined by 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 = 0.3. In the nonfactorizable contributions, S t (x) gives a very small numerical effect on the amplitude [61].
The Sudakov exponents are defined as The explicit form for the function s(k, b) is [45]: where the variables are defined byq and the coefficients A (i) and β i are where n f is the number of the quark flavors and γ E is the Euler constant.
The decay amplitude F e , F eρ and F eω induced by inserting the (V − A)(V − A) operators are [62] F From the (S + P )(S − P ) operators, we can get The decay amplitude for (V − A)(V − A) and (V − A)(+A) operators can be written as follows