$CP$ violation for $B^{0}\rightarrow \rho^{0}(\omega)\rho^{0}(\omega) \rightarrow \pi^+\pi^-\pi^+\pi^-$ in QCD factorization

In the QCD factorization (QCDF) approach we study the direct $CP$ violation in $\bar{B}^{0}\rightarrow\rho^0(\omega)\rho^0(\omega)\rightarrow\pi^+\pi^-\pi^+\pi^-$ via the $\rho-\omega$ mixing mechanism. We find that the $CP$ violation can be enhanced by double $\rho-\omega$ mixing when the masses of the $\pi^+\pi^-$ pairs are in the vicinity of the $\omega$ resonance, and the maximum $CP$ violation can reach 28{\%}. We also compare the results from the naive factorization and the QCD factorization.


Introduction
CP violation is an extensive research topics in recent years. In Standard Model (SM), CP violation is related to the weak complex phase in the Cabibbo-Kobayashi-Maskawa (CKM) matrix [1,2]. In the past few years more attention has been focused on the decays of B meson system both theoretically and experimentally. Recently, the large CP violation was found by the LHCb Collaboration in the three-body decay channels of B ± → π ± π + π − and B ± → K ± π + π − [3,4]. Hence, the theoretical mechanism for the three or four-body decays become more and more interesting. In this paper, we focus on the interference from intermediate ρ and ω mesons in the four-body decay.
It is known that the naive factorization [5,6], the QCD factorization (QCDF) [7,8,9], the perturbative QCD (PQCD) [10,11,12], and the soft-collinear effective theory (SCET) [13,14] are the most extensive approaches for calculating the hadronic matrix elements. These factorization approaches present different methods for dealing with the hadronic matrix elements in the leading power of 1/m b (m b is the b-quark mass). Direct CP violation occurs through the interference of two amplitudes with different weak phases and strong phases. The weak phase difference is directly determined by the CKM matrix elements, while the strong phase is usually difficult to control. However, and not well a e-mail:ganglv66@sina.com b e-mail:18236915407@139.com c e-mail: xhguo@bnu.edu.cn d e-mail:zhangzh@iopp.ccnu.edu.cn e e-mail:weikw@hotmail.com determined from a theoretical approach. The B meson decay amplitude involves the hadronic matrix elements which computation is not trivial. Different methods may present different strong phases. Meanwhile, we can also obtain a large strong phase difference by some phenomenological mechanism. ρ − ω mixing has been used for this purpose in the past few years [15,16,17,18,19,20,21,22,23,24,25]. In this paper, we will investigate the CP violation via double ρ − ω mixing in the QCDF approach.
In the QCDF approach, at the rest frame of the heavy B meson, B meson can decay into two light mesons with large momenta. In the heavy-quark limit, QCD corrections can be calculated for the non-leptonic two-body Bmeson decays. The decay amplitude can be obtained at the next-to-leading power in α s and the leading power in Λ QCD /m b . In the QCD factorization, there is cancellation of the scale and renormalizaion scheme dependence between the Wilson coefficients and the hadronic matrix elements. However, this does not happen in the naive factorization. The hadronic matrix elements can be expressed in terms of form factors and meson light-cone distribution amplitudes including strong interaction corrections.
The remainder of this paper is organized as follows. In Sec. 2 we present the form of the effective Hamiltonian. In Sec. 3 we give the calculating formalism of CP violation from ρ − ω mixing in B 0 → ρ 0 (ω)ρ 0 (ω) → π + π − π + π − . Input parameters are presented in Sec.4. We present the numerical rusults in Sec.5. Summary and discussion are included in Sec. 6.

The effective hamiltonian
With the operator product expansion, the effective weak Hamiltonian can be written as [26] where G F represents the Fermi constant, c i (i = 1, ...., 10, 7γ, 8g) are the Wilson coefficients, V pb , V pq are the CKM matrix elements. The operators O i have the following forms: where α and β are color indices, O p 1 and O p 2 are the tree operators, O 3 − O 6 are QCD penguin operators which are isosinglets, O 7 −O 10 arise from electroweak penguin operators which have both isospin 0 and 1 components. O 7γ and O 8g are the electromagnetic and chromomagnetic dipole operators, e q ′ are the electric charges of the quarks and q ′ = u, d, s, c, b is implied.
For the decay channel B 0 → ρ 0 (ω)ρ 0 (ω), neglecting power corrections of order Λ QCD /m b , the transition matrix element of an operator O i in the weak effective Hamiltonian is given by [8,9] Here F B→V1,2 j (m 2 V2,1 ) denotes B → V 1,2 (V 1,2 represent ρ 0 and ω mesons) form factor, and Φ V (u) is the lightcone distribution amplitude for the quark-antiquark Fock state of mesons ρ 0 and ω. T I ij (u) and T II i (ξ, u, v) are hardscattering functions, which are perturbatively calculable.
The hard-scattering kernels and light-cone distribution amplitudes (LCDA) depend on the factorization scale and the renormalization scheme. m V1,2 denote the ρ 0 and ω masses, respectively.
We match the effective weak Hamiltonian onto a transition operator, the matrix element is given by (λ where T p,h A denotes the contribution from vertex correction, penguin amplitude and spectator scattering in terms of the operators a p,h i , T p,h B refers to annihilation terms contribution by operators b p,h i . h is the helicity of the final state.
The flavor operators a p i are defined in [8,9] as follows: where N c is the number of colors, the upper (lower) signs apply when i is odd (even), and C F = 2Nc . It is understood that the superscript 'p' is to be omitted for i = 1, 2. The quantities V h i (V 2 ) account for one-loop vertex corrections, H h i (V 1 V 2 ) for hard spectator interactions, and P p,h i (V 2 ) for penguin contractions. N h i (V 2 ) is given by The coefficients of the flavor operators α p,h i can be expressed in terms of the coefficients a p,h i . We will present the form in the following section. Using the unitarity relation we can get where the sums extend over q = u, d, s, andq s denotes the spectator antiquark.
Next we need to change the annihilation part into the following form [8,9]: where b p,h i , b p,h i,EW and B will be given in the following section.

Formalism
The B → V 1 (ǫ 1 , P 1 )V 2 (ǫ 2 , P 2 ) (ǫ 1 (P 1 ) and ǫ 2 (P 2 ) are the polarization vectors (momenta) of V 1 and V 2 , respectively) decay rate is written as where P c refers to the c.m. momentum. A (σ) is the helicity amplitude for each helicity of the final state. The decay amplitude, A, can be decomposed into three components H 0 , H + , H − according to the helicity of the final state.
With the helicity summation, we can get In the vector meson dominance model [27], the photon propagator is dressed by coupling to vector mesons. Based on the same mechanism, ρ − ω mixing was proposed [28,29]. The formalism for CP violation in the decay of a bottom hadron,B, will be reviewed in the following. The amplitude forB → V π + π − , A, can be written as where H T and H P are the Hamiltonians for the tree and penguin operators, respectively. We define the relative magnitude and phases between these two contributions as follows: where δ and φ are strong and weak phase differences, respectively. The weak phase difference φ arises from the appropriate combination of the CKM matrix elements: The parameter r is the absolute value of the ratio of tree and penguin amplitudes, The amplitude for B →V π + π − is Then, the 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. (16) that both weak and strong phase differences are needed to produce CP violation. ρ − ω mixing has the dual advantages that the strong phase difference is large and well known [15,16]. In this scenario one has where t V (V = ρ or ω) is the tree amplitude and p V is the penguin amplitude for producing a vector meson, V . t a V (V = ρ or ω) is the tree annihilation amplitude and p a V is the penguin annihilation amplitude. g ρ is the coupling for ρ 0 → π + π − ,Π ρω is the effective ρ − ω mixing amplitude, and s V is from the inverse propagator of the vector meson V, with √ s being the invariant mass of the π + π − pair. The direct ω → π + π − is effectively absorbed intoΠ ρω , leading to the explicit s dependence ofΠ ρω [30,31]. Making the , the ρ − ω mixing parameters were determined in the fit of Gardner and O'Connell [32]: ReΠ ρω (m 2 ω ) = −3500 ± 300 MeV 2 , ImΠ ρω (m 2 ω ) = −300 ± 300 MeV 2 , andΠ ′ ρω (m 2 ω ) = 0.03 ± 0.04. In practice, the effect of the derivative term is negligible. From Eqs. (16)(18), one has Defining where δ α , δ β , and δ q are strong phases, one finds the following expression from Eq. (21): αe iδα , βe iδ β , and re iδ will be calculated in the QCD factorization approach in the next section. With Eq. (25), we can obtain r sin δ and r cos δ. In order to get the CP violating asymmetry, A CP , in Eq. (16), sin φ and cos φ are needed. φ is determined by the CKM matrix elements. In the Wolfenstein parametrization [33,34], one has .

The calculation details
In the QCD factorization approach, α i associated with the coefficients a i can be written as follows (helicity indices are neglected) [8,9]: where we have used the notation with f ⊥ V , f V referring to the transverse decay constant and decay constant of the vector meson, respectively.
The flavor operators a p,h i include short-distance nonfactorizable corrections such as vertex corrections and hard spectator interactions. V 2 is the emitted meson and V 1 shares the same spectator quark with the B meson.

Hard spectator terms
arise from hard spectator interactions with a hard gluon exchange between the emitted meson and the spectator quark of the B meson. H 0 i (V 1 V 2 ) have the expressions [8,9]: for i = 1 − 4, 9, 10, for i = 5, 7, and H 0 i (V 1 V 2 ) = 0 for i = 6, 8. The transverse hard spectator terms for i = 1 − 4, 9, 10, for i = 5, 7, and for i = 6, 8. One can find that the expressions for H ± i (V 1 V 2 ) are independent of the choice for transverse polarization vectors.
The helicity dependent factorizable amplitudes defined by have the expressions where A BV1 i (i = 1, 2) and V BV1 are weak form factors.

Penguin terms
At order α s , corrections from penguin contractions are present only for i = 4, 6. For i = 4 we have [8,9] where s i = m 2 i /m 2 b and the function G h V2 (s) is given by with Φ V2,0 = Φ V2 , Φ V2,± = Φ V2 ± . For i = 6, the result for the penguin contribution is where the functionĜ V2 (s) is defined aŝ contributes to the decay amplitude in the product r V2 χ a 0 6 ≈ r V2 χ P 0 6 [8,9]. For i = 8, 10, The relevant integrals for the dipole operators O g,γ are [8,9] The dipole operators Q 8g and Q 7γ do not contribute to the transverse penguin amplitudes at O(α s ) due to angular momentum conservation [35].

Annihilation contributions
The annihilation contributions to the decay B → V 1 V 2 can be described in terms of b p,h i and b p,h i,EW The building blocks have the expressions where we have omitted the superscripts p and h in above expressions for simplicity. The subscripts 1,2,3 of A i,f n denote the annihilation amplitudes induced from and (S − P )(S + P ) operators, respectively, and the superscripts i and f refer to gluon emission from the initial and final-state quarks, respectively. V 1 contains an antiquark from the weak vertex and V 2 contains a quark from the weak vertex [8,9]. The explicit expressions of weak annihilation amplitudes are: and = 0. V 1 contains an antiquark from the weak vertex with longitudinal fraction y, while V 2 contains a quark from the weak vertex with momentum fraction x [8,9].
A i,± 1,2 are suppressed by a factor of m 1 m 2 /m 2 B relative to other terms, so only the annihilation contributions due to A f,0 3 , A f,− 3 , A i,0 1,2,3 and A i,− 3 are considered.

The calculation of CP violation
In order to obtain the CP violation ofB → ρ 0 (ω)ρ 0 (ω) → π + π − π + π − in Eq.(16), we calculate the amplitudes t ρ , t a ρ , t ω , t a ω , p ρ , p a ρ , p ω and p a ω in Eqs. (18) (19) in the QCDF approach, which are tree-level and penguin-level amplitudes. The decay amplitudes for the processB → ρ 0 ρ 0 (ω) are in the QCD factorization as follows: where From Eq. (22), one can get where In a similar way, with the aid of the Fierz identities, we can evaluate the penguin operator contributions p ρ and p ω . From Eq. (23) we have where Form Eq. (24) we have where . (80)

Input parameters
In the numerical calculations, we should input distribution amplitudes and the CKM matrix elements in the Wolfenstein parametrization. For the CKM matrix elements, which are determined from experiments, we use the results in Ref. [36]: The general expressions of the helicity-dependent amplitudes can be simplified by considering the asymptotic distribution amplitudes for Φ V , Φ v :   Fig. 2. Plot of sin δ as a function of √ s corresponding to central parameter values of CKM matrix elements forB 0 → ρ 0 (ω)ρ 0 (ω) → π + π − π + π − . The solid (dashed and dotted) line corresponds to sin δ0 (sin δ− and sin δ+) respectively.
Power corrections in QCDF always involve endpoint divergences which produce some uncertainties. The endpoint divergence X ≡ 1 0 dx/x in the annihilation and hard spectator scattering diagrams is parameterized as with the unknown real parameters ρ A,H and φ A,H [8,9]. For simplicity, we shall assume that X h A and X h H are helicity independent:

Numerical results
In the numerical results, we find that for the decay channel we are considering the CP violation can be enhanced  Fig. 3. Plot of r as a function of √ s corresponding to central parameter values of CKM matrix elements forB 0 → ρ 0 (ω)ρ 0 (ω) → π + π − π + π − . The solid (dashed and dotted) line corresponds to r0 (r− and r+) respectively. via ρ − ω mixing when the invariant mass of π + π − is in the vicinity of the ω resonance. The uncertainties of the CKM matrix elements mainly come from ρ and η. In our numerical results, we let ρ and η vary between the limiting values. We find the results are not sensitive to the values of ρ and η. Hence, the numerical results are shown in Fig.1, Fig.2 and Fig.3 with the central parameter values of CKM matrix elements. From the numerical results, it is found that there is a maximum CP violating parameter value, A max CP , when the masses of the π + π − pairs are in the vicinity of the ω resonance. In Fig.1, one can find that the maximum CP violating parameter reaches 28% in the case of (ρ central , η central ).
From the Eq.(16) one can find that the CP violating parameter is related to sin δ and r. In Fig.2, we show the plot of sin δ 0 (sin δ − and sin δ + ) as a function of √ s. 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.2, the plots vary sharply in the cases of sin δ 0 and sin δ − . Meanwhile, sin δ + changes weakly comparing with the sin δ 0 and sin δ − . It can be seen from the Fig.3 that r 0 and r − change more rapidly than r + when the masses of the π + π − pairs are in the vicinity of the ω resonance.
In the paper [23], we studied the enhanced CP violation for the decay channelB 0 → π + π − π + π − in the naive factorization. Since non-factorizable contribution can not be calculated in the naive factorization, N c was treated as an effective parameter. We found that the CP violating asymmetry was large and ranges from −82% to −98% via the ρ − ω mixing mechanism strongly depending on the value N c when the invariant mass of the π + π − pair is in the vicinity of the ω resonance. However, the maximum CP violation only can reach 28% via double ρ − ω mixing in the QCD factorization. The naive factorization scheme has been shown to be the leading order result in the framework of QCD factorization when the radiative QCD corrections O(α s (m b )) and the order O(1/m b ) ef-fects are neglected. The QCD factorization can evaluate systematically corrections to the results from the naive factorization. The distinction between the naive factorization and the QCDF mainly come from the strong phases of the QCD corrections. In the calculating process, we find that the annihilation contributions in QCDF which introduce the unknown parameters are small. Hence, the uncertainties of the results from the QCDF become small.

Summary and conclusions
In this paper, we studied the CP violation for the decay processB 0 → π + π − π + π − due to the interference of ρ − ω mixing in the QCDF approach. This process induces two ρ − ω interference. It was found the CP violation can be enhanced at the region of ρ − ω resonance. As a result, the maximum CP violation could reach 28%. ρ − ω mixing is small due to the isospin violation. However, the mixing can produce a large strong phase, δ, in Eq. (21). This is because when the invariant masses of the π + π − pairs are in the vicinity of ω, s ω ∼ im ω Γ ω , and it becomes comparable withΠ ρω in Eq. (21). In other words, ρ − ω mixing becomes important in the vicinity of ω. This is the reason why we can see large CP violation in the vicinity of ω. Beyond the ρ − ω interference region, the noticeable values of CP violation are caused by the strong phases provided by the Wilson coefficients.
The LHC experiments are designed with the center-ofmass energy 14 TeV and the luminosity L = 10 34 cm −2 s −1 . The heavy quark physics is one of the main topics of LHC experiments. Especially, LHCb detector is designed to make precise studies on CP asymmetries and rare decays of b-hadron systems. Recently, the LHCb Collaboration found clear evidence for direct CP violation in some three-body decay channels in charmless decays of B meson. Large CP violation is observed in B + → K + K − π + , B ± → π ± π + π − in the region m 2 π + π − low < 0.4 GeV 2 and m 2 π + π − low > 15 GeV 2 [3]. LHCb experiment may collect data in the region of the invariant masses of π + π − associated the ω resonance for detecting our prediction of CP violation.
In our calculations there are some uncertainties. The QCD factorization scheme provides a framework in which we can evaluate systematically corrections to the results obtained in the naive factorization scheme. However, when we take into account the nonfactorizable and chirally enhanced hard-scattering spectator and annihilation contributions which appear at order O(α s (m b )) and O(1/m b ), respectively, the involvement of the twist-3 hadronic distribution amplitudes leads to logarithmical divergence coming from the endpoint integrals. This brings large uncertainties in the predictions of the CP violating asymmetries in the QCD factorization scheme. Furthermore, in addition to the model dependence appearing in the factorized hadronic matrix elements just as in the naive factorization scheme, we cannot avoid the model dependence and process dependence of the hard-scattering spectator and annihilation contributions due to their dependence on the hadronic distribution amplitudes and dependence on different processes. Such dependence will also appear if one tries to include other 1/m b corrections and even higher order corrections. This leads to uncertain of our results.