CP violation in non-leptonic $B_c$ decays to excited final states

We study the CP violation in two-body nonleptonic decays of $B_c$ meson. We concentrate on the decay channels which contain at least one excited heavy meson in the final states. Specifically, the following channels are considered: $B_c\to c\bar c(2S, 2P)+\bar cq(1S, 1P)$, $B_c\to c\bar c(1S)+\bar cq(2S, 2P)$, $B_c\to c\bar c(1P)+\bar cq(2S)$, $B_c\to c\bar c(1D)+\bar cq(1S, 1P)$, and $B_c\to c\bar c(3S)+\bar cq(1S)$. The improved Bethe-Salpeter method is applied to calculate the hadronic transition matrix element. Our results show that some decay modes have large branching ratios, which is of the order of $10^{-3}$. The CP violation effect in $B_c \rightarrow \eta_c(1S)+D(2S)$, $B_c \rightarrow \eta_c(1S)+D_0^{*}(2P)$, and $B_c \rightarrow J/\psi+D^{*}(2S)$ are most likely to be found. If the detection precision of the CP asymmetry in such channels can reach the $3\sigma$ level, at least $10^7$ $B_c$ events are needed.


I. INTRODUCTION
The B c meson was discovered by CDF more than two decades ago [1]. Since then, there have been a lot of studies both theoretical and experimental on this particle. The reason why it's so interesting is that the B c meson is the lowest bound state which consists of two heavy quarks with different flavors. It cannot decay through strong interaction or electromagnetic interaction, and only the weak decay channels are allowed. This makes the B c meson an ideal platform to study the properties of heavy quarks and test accurately the Standard Model (SM) predictions. Experimentally, about 5 × 10 10 B c mesons per year could be produced at the LHC [2], which provides us the opportunity to get more information of this particle, such as the CP violation effect in two-body nonleptonic decay modes.
The CP asymmetry in the non-leptonic B decays has been extensively studied by using different methods, such as the QCD factorization approach [3,4], the soft-collinear effective theory (SCET) [5], and the perturbative QCD (pQCD) method [6][7][8]. For B c meson, the CP violation effects have also attracted some attentions. Based on the pQCD method, Ref. [9][10][11][12] studied the direct CP asymmetry parameter of two-body decays of B c meson with one light final state. In Ref. [13], a relativistic quark model was applied to explore the decay channels of B + c → D + sD 0 and B + c → D + s D 0 . In Ref. [14], with a relativistic independent quark model the authors predicted that there were significant CP violation in the B c → D * D * , D * D * s channels. In Refs. [15][16][17][18][19], the model-independent method was used to estimate the Cabibbo-Kobayashi-Maskawa (CKM) angle γ. In Ref. [15], the authors pointed out that to observe the CP violation, about 10 8 B c events were needed.
In Refs. [20,21], the Bethe-Salpeter (BS) method was used to study the CP violation of B c and B s mesons. In this formalism, by solving the instantaneous BS equations, we get the wave functions of heavy mesons, which are used to calculate the hadronic transition matrix elements (Ref. [22] applied a similar formalism to study the CP violation of B c with S-wave final states). In this paper, we consider the direct CP violation in two-body nonleptonic decays of B c meson, where a radial excited state is included in the final states.
According to Ref. [23], the relativistic effects are important in such cases. As we did in Ref. [24], an improved BS method which writes the amplitude in a more covariant form will be applied. To our knowledge, the CP violation effects in such decay channels have not been studied yet. However, as more and more excited D and D s states are discovered, such as D * s1 (2710) ± , D(2550) 0 , D * (2640) ± , and D * 1 (2680) 0 [25], which can be identified as D * s (2S), D(2S) 0 , D * (2S) ± , and D * (2S) 0 , respectively, those calculations will become necessary.
As the direct CP violation comes from the interference of different diagrams, we will consider the color-favored tree diagram, color-suppressed tree diagram, and the time-like penguin diagram. The annihilation diagram and the space-like penguin diagram are helicity suppressed and will not be considered. The soft strong phase arising from the rescattering effects of final states may also bring considerable contribution (see Ref. [26]), but it will not be discussed in this work.
The paper is organized as follows. In Section II, we present the theoretical formalism of the CP asymmetry in nonleptonic decays of B c meson. In Section III, we use the improved BS method to calculate the hadronic transition matrix elements which are expressed as the overlap integral of the wave functions of heavy mesons. In Section IV, we give the numerical values of the CP asymmetry and make discussions of the results.
Q 1 and Q 2 are the tree diagram operators. Q 3 , Q 4 , Q 5 , and Q 6 are the QCD penguin diagram operators. Q 7 , Q 8 , Q 9 , and Q 10 are the electroweak penguin diagram operators.
Specifically, these operators have the following forms [15] where e q is the electric charge of the quark q which can be u, d, s, c, or b; the subscripts α and β are color indices; (q 1α q 2β ) V ±A ≡q 1α γ µ (1 ± γ 5 )q 2β . As Ref. [28] did, we will use the factorization approximation, under which the amplitude can be factorized into the product of hadronic transition matrix element and decay constants. The Feynman diagrams we will consider are shown in Fig. 1.
FIG. 1. The Feynman diagrams for B − c decaying into two mesons X 0 and X − in the spectator approximation, where q, q v = u, c and q , q v = d, s. The subscript v denotes 'vacuum'.
The amplitudes corresponding to the three Feynman diagrams are [15] A 1 = a 1 A, where A and B are the factorized hadronic matrix elements for B − c → X − X 0 , which have the forms The a k in Eq. (3) are defined by Wilson coefficients at the renormalization scale µ ∼ m b where N c is the number of colors. In the actual calculation, one often uses the effective number of colors [29] 1 where δ i stands for the nonfactorizable part. In this paper we will choose N c = 3 for all operators to get the results, and we will also discuss the effect of different N ef f c . The parameter ξ f in the Eq. (3) comes from the Fiertz rearrangement, which transforms the (V ∓ A)(V ± A) currents into the (S ± P )(S ∓ P ) currents. For decay channels with different final states, this parameter takes different values [21] where X 1 is the final meson which contains the spectatorc and X 2 is the other one. Their J P numbers are presented in the parentheses.
where we take α s (m Z ) = 0.1176, α e (m Z ) = 1/128, m W = 91.1876 GeV, and Λ (f =5) QCD = 220.9 MeV. The function G(m q , k 2 ) is defined as [15] G m q , k 2 = 3 2 where F q (k 2 ) is the penguin loop-integral function of the squared momentum k 2 carried by the virtual gluon at the renormalization scale µ ∼ m b , According to Ref. [30], we can rewrite the function G(m q , k 2 ) as where r q = 4m 2 q /k 2 . As Ref. [20] did, we will take k 2 to be its average valuek 2 , which is defined as In the calculation of the QCD penguin diagram, we will take the current quark masses [25] m u = 0.0022 GeV, m d = 0.0047 GeV, m s = 0.095 GeV, Finally the amplitude of two-body nonleptonic decays of B c meson can be written as where λ q = V qb V * qd . By combining terms with the same λ q , we can write M as (14) The CP asymmetry of two-body decays of B c meson is defined as By calculating |M(B + c →f )| 2 and |M(B + c →f )| 2 and inserting them into Eq. (16), we get According to Ref. [31], the CKM matrix can be parameterized as Inserting Eq. (18) into Eq. (17), we get where we have defined γ ≡ arg −

III. THE IMPROVED BS METHOD
In this section, we give a brief introduction to the improved BS method, which is used to calculate the hadronic transition matrix element More details about this method can be found in our previous work [24].
According to the Mandelstam formalism, the hadronic matrix can be written as where Γ µ = γ µ (1 − γ 5 ); χ is the BS wave function; p i (q) and p f i (q f ) are the (relative) momenta of quark and antiquark within initial and final mesons, respectively; S i is the quark propagator. By applying the instantaneous approximation, Eq. (20) can be reduced into the three-dimensional form, where are the positive energy projection of wave functions; the symbol L r is expressed as In the above equations, we have definited the projection operator as follows where J = 1 and −1 for quark and antiquark, respectively; m i and m if are quark masses.
The quark energies ω i , ω if , and ω if have the forms where with s r = m 2 The symbols ⊥ and ⊥ ⊥ mean projecting onto the momenta of the initial and final mesons, respectively.
The instantaneous BS wave functions of heavy mesons with different quantum numbers (0 − , 0 + , 1 − , 1 +− , and 1 ++ ) have the following forms where f i , g i , h i , r i , and s i are the radial wave functions; µ is the polarization vector of the vector or axial vector mesons. The 1 + (P 1/2 1 ) and 1 + (P 3/2 1 ) states are the mixing of 1 P 1 and 3 P 1 states, which can be written as [32] P 1/2 1 = 1 P 1 cos θ nP + 3 P 1 sin θ nP , According to Ref. [32], the mixing angle for different states takes the values:

IV. NUMERICAL RESULTS AND DISCUSSIONS
The numerical results of the wave functions are achieved by solving the corresponding instantaneous BS equations. The Cornell-like potential is adopted, whose detailed forms can be found in Ref. [24]. The values of the related parameters are fixed by fitting the masses of the ground states. The constituent quark masses used in this work, which are different from those of the current quarks when calculating the QCD penguin diagrams, have the values: The PDG values of the CKM matrix elements are [25]: For the Wilson coefficients, we use the results in Ref. [33]: The masses and decay constants of mesons in the ground and excited states are presented in Table I. Here we have used the method in Refs. [34][35][36] to calculate the decay constants except those of D ± (1S) and D ± s (1S) for which the experimental data are adopted. By using the numerical results of wave functions and the parameter values mentioned above, we get the branching ratios (Br) and CP asymmetry (A cp ) of different decay channels, which are presented in Table II, Table III, and Table IV. The values of D 1,2 defined in Eq. (19) are also given. Following Refs. [15,20], we also give an estimation of how many B c events are needed to observe the CP violation effect. If it is observed at three standard deviation (3σ) level, the B ± c events needed are f N ∼ 9 BrA 2 cp , where f is the detecting efficiency of the final state [20]. For the cases when the radial excited state is the charmonium (Table II and IV), one can see that the ψ(2S)D * s channel has the largest branching ratio but a small A cp ; the ψ(1D)D * 0 channel has the largest A cp but a small branching ratio; the f N for η c (2S)D(1S) and ψ(2S)D * (1S) channels are of the order of 10 8 , which is possible to be observed by the current experiments. For the cases when the radial excited state is the heavy-light meson (Table III), there are five channels whose f N is of the order of 10 7 ∼ . One also notices that the decay channels J/ψD * s (2S) and η c D(2S) have the largest branching ratio and A cp , respectively.
The CP asymmetry is related to the weak CP phase γ by Eq. (19). Experimentally, γ is constrained by the nonleptonic decays of B meson, the latest results of which in PDG2020 is (72.1 +4.1 −4.5 ) • [25]. In Fig. 2 we draw |A cp | as a function of γ. The eight decay channels, whose |A cp | are of the order of 10 7 ∼ Here we only consider the decay channels with f N ∼ 10 7 or 10 8 . One can see that as N ef f c changes, A cp and Br can change at most several times. The changes of k 2 have even little effect on these quantities. For example, the branching ratio of η c (2S)D * 0 (1S) channel (see Fig. 7) changes less than 50% when k 2 changes from 0.35m 2 b to 0.80m 2 b . At last, we study how the P -wave mixing angle affects the CP asymmetry. The results are shown in Fig. 10 and Fig. 11. One can see that except B c → J/ψD ( ) s1 (2P ), all the other channels with a 1 +( ) final state have a critical angle around which A cp changes severely.
In conclusion, we have calculated the CP violation in two-body nonleptonic decays of B c , where excited states are included in the final states. Some decay modes have large branching ratios, which is of the order of 10 −3 . We studied in detail seven decay channels whose CP asymmetry could be detected by the current experiments. Among these channels, , and B c → J/ψD * (2S) are the most promising ones, for which, about 10 7 B c events are needed, if the CP violation is observed at 3σ level.     .