Estimated of C P violation in B 0 meson decays into D ∗+ and D − mesons

The decay B 0 → D ∗+ D − is favorable mode for studying C P violation in the interference between mixing and decay for B 0 and ¯ B 0 mesons. The latest analysis of the C P parameters has been performed by the LHCb Collaboration values of S D ∗ D = − 0 . 861 ± 0 . 077 ± 0 . 019, C D ∗ D = − 0 . 059 ± 0 . 092 ± 0 . 020, (cid:4) S D ∗ D = 0 . 019 ± 0 . 075 ± 0 . 012, (cid:4) C D ∗ D = − 0 . 031 ± 0 . 092 ± 0 . 016, and A CPD ∗ D = 0 . 008 ± 0 . 014 ± 0 . 006 ± 0 . 003. We have been estimated the parameters S D ∗ D and C D ∗ D of the B 0 → D ∗+ D − decay as − 0 . 709 ± 0 . 024 and − 0 . 051 ± 0 . 004. In the following, we have obtained the values of (cid:4) S D ∗ D = 0 . 054 ± 0 . 003 and (cid:4) C D ∗ D = 0 . 020 ± 0 . 001 and direct C P violation of 0 . 008 ± 0 . 001. Also, we have calculated the branching ratio of B 0 → D ∗+ D − decay. The values obtained in this work are comparable with the corresponding experimental values.


Introduction
The standard model (SM) is a relativistic quantum field theory that involves the search for fundamental particles and the fundamental interactions that occurring among them.To perform such searches through high-precision measurements of the parameters of the quark-flavour of the SM sector with b-and c-hadron decays is developed.In this way, possible inconsistencies with the SM predictions are revealed.The increasing amount of data makes it necessary to consider higher-order the SM corrections [1].One way to do this is to examine decays that involve b → c cd transitions, such as B 0 → D * + D − .Neutral meson mixing is one important effect that allows access to parameters in the flavour sector [2].The mesons composed of a different quarks and antiquarks type decay weakly, allowing C P violation and mixing.Mixing describes the transformation of a neutral meson a e-mail: eliamirkhanlou@yahoo.com b e-mail: be.mohammadi@urmia.ac.ir (corresponding author) into an antiparticle state and vice versa, and is also called meson oscillation.The time-dependent oscillation between the particle and antiparticle states appears [3].C P violation in general could lead to the excess of a matter-antimatter in our universe, but the smallness of the observed C P violation is not sufficient to explain the observations [4].Nevertheless, the fact that the C P violation is a relatively small non-zero value is interesting and allows for further studies on its properties.Also, new sources of C P violation beyond the SM that account for the difference between measured values and SM predictions can be considered as a research idea for the yet-undiscovered physics [5].In the case of C P symmetry in the B meson system, we can study the processes in which the B mesons decay into a C P-eigenstate state.In a general way, we can compare the rate at which a B meson decays into a C P-eigenstate with the rate at which a B meson decays into a C P-conjugate final state ( f ), to the rate at which a B meson decays into the C P final state ( f ) and to the rate at which a B meson decays into the f .These different final states provide additional information about the system, and only by combining such information from different measurements can we get a complete picture of the subject as well as accurate results.The difference between the B 0 and B0 meson decays appears only in the time-dependent decay rate, and this time corresponds to the time when the B meson freely propagates before it decays to the C P-eigenstate [6].In Tables 1 and 2 an overview of existing measurements and the world average  branching ratio for the B 0 → D * ± D ∓ decay.Under the factorization approach, the amplitudes of B 0 → D * ± D ∓ decay can be obtained as separate factorizable contributions that include the current-current and penguin contributions.In the case of We explained the theoretical background of C P violation in the B 0 meson system using the SM of particle physics and its constructed theoretical framework.We then presented an overview of the field of flavour physics, including the basic ideas of quark mixing and C P violation in the B meson.Now we want to calculate the direct C P violation.The direct C P violation arises in the ratio of the amplitude In this case, two types of phases occur in these amplitudes [13].The first type of phase is created in complex parameters in the Lagrangian.In the SM, these phases occur only in the CKM matrix and are called weak phases (φ i ) [14].
The CKM matrix elements are in the unitarity triangle rela- ) and weak phases are introduced as φ 1 = arg(V cq ) and φ 2 = −arg(V * tb ).Another type of phase can appear in the scattering or decay amplitudes that are called the strong phases (δ i ).these phases occur even when the Lagrangian is real.Such phases do not violate C P because they appear in amplitudes (A f and Ā f ) with the same sign.Their origin is the possible contribution of the mode the intermediates on-shell states in the process of decay.In fact, it is an absorptive part of an amplitude that has contributions from coupled channels.The dominant rescattering is due to strong interactions and this is the reason for naming these phases.The C P violation will not occur unless we have different strong phases in addition to different weak phases [14].The strong phase δ 1 is obtained from |A 1 |e iδ 1 and δ 2 from |A 2 |e iδ 2 .The B 0 → D * + D − decay (and B0 → D * − D + decay), with two contributing amplitudes A 1 and A 2 .This means that the decay can be done by two different paths and those are tree (A 1 ) and penguin (A 2 ) diagrams.For the total decay amplitude we have [15]: where Feynman tree diagrams have the largest amplitude contribution compared to penguin diagrams.The Feynman diagrams of B 0 → D * + D − decay are shown in Fig. 1 and the decay amplitude can be expressed as where the tree and penguin level amplitudes are as follows, respectively and [16].The form factor F 1 is obtained form [17] here the ξ + (ω) and ξ − (ω) are under the heavy quark symmetry to be equal ξ(ω) and zero respectively.We use the Isgur-Wise function ξ(ω) = 1 − ρ 2 D (ω − 1) for the transition B → D, where ω = (m 2 B + m 2 D − q 2 )/(2m B m D ) and ρ 2 D = 0.90 ± 0.06.The ρ 2 D is called the slope parameter.The basis of QCD lagrangian is quark mass, although it cannot be directly related to measurable physical quantities.The masses depend on the renormalization scheme and, in a given scheme, on the renormalization scale μ.The most important issue in obtaining the amplitude is the calculation of the Wilson coefficients in the NLO or LO approximation.Therefore, we must know the appropriate value of Wilson's coefficients (C j ) in μ = O(M W ). Also, the C j 's are quantities dependent on the renormalization scheme.The dependence of the renormalization scheme is felt in next-to-leading order (NLO) but not significant in the leading order (LO).In this calculation, the evolution of the renormalization group to the low energy scales μ M W related to the decays is considered.In fact, the Wilson coefficients are the coupling constants for the interaction terms of the effective Hamiltonian operators, transformed into non-computable functions α s , M W and the renormalization scale μ [18].
For the Wilson parameter a j (j = 1, …, 10), we have We used the next-to-leading logarithm in the naive dimensional regularization (NDR) scheme for the Wilson coefficients C j (μ) at the scale μ that are shown in Table 3.
In this paper, we take the decay constants, quark, and meson masses (in units of MeV) [12] We calculated the branching fractions for the B 0 → D * + D − decay is written as here the tot B 0 is (4.33 ± × 10 −13 GeV.The direct C P violation can be expressed as [21] A We obtained the strong phases with values δ 1 = − 90.01 • and δ 2 = 67.47• .Also, we calculated for weak phases φ 1 = 0.03 • and φ 2 = 22.58 • .In the SM, C P violation occurs when more than one of the Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix elements is complex.Here, we use the CKM matrix elements at order λ 5 that is [10] We adopt the Wolfenstein parameterization and choose the parameters A, ρ, η and λ as [12] λ = 0.22650 ± 0.00048, A = 0.790 +0.017 −0.012 , ρ = 0.141 +0.016 −0.017 , η = 0.357 ± 0.011, (12) with ρ = ρ(1 − 1/2λ 2 ) and η = η(1 − 1/2λ 2 ).Therefore, the CKM matrix elements are obtained as follows (in units of 10 −3 ) V cb = 40.529,V cd = −226.368− 0.136i, Another type of C P violation that occurs in the B 0 meson decay, is the violation from interference between decay with and without mixing (without any of the other types of C P violation).We have [6] λ where A( Ā) is the decay amplitude for B 0 ( B0 ) and q/ p is the ratio of the flavor contributions to the mass eigenstates.Since the t quark has more mass, only hadrons with c or u quarks are allowed to transition to physical states.In this case, we have two probability restrictions for these transitions: first, the decay of both B 0 and B0 are Cabibbo-suppressed, second, the decay for B 0 is Cabibbo-allowed, and for B0 mesons doubly Cabibbo-suppressed, or vice versa.Therefore, the decay width difference is small compared to the mass difference, which allows us to express q/ p in terms of CKM matrix elements as The M 12 and M * 12 are denote mass matrices.If |λ| = 1, C P violation is manifest through either decay or mixing, but if I mλ = 0, C P violation is manifest through the interference between decays with and without mixing.The decay timedependent C P asymmetry, A C P D * D (t), can be defined [22] where m d = 0.510h ps −1 and with [23].
For them also applies [24] (S and this constraint may or may not imposed to fits.To calculate the mixing-induced and direct C P violation, we use S D * + D − and C D * + D − parameters, respectively.parameter |A D * + D − | introduces another observable for neutral meson systems.In the B 0 decay, the expression for the timedependent amplitude A C P D * D (t) is simplified because of the low oscillation frequency.Therefore, the Eq. ( 16) becomes [25]   The

Numerical results and conclusion
The C P parameters resulting from the fit to the decay time, direct C P violation and branching ratio for the B 0 → D * + D − decay are shown in Table 4. The is provided for the B 0 → D * ± D ∓ decays by the different collaborations.Recently, the first measurement of C P violation in the B 0 → D * ± D ∓ decay has been reported in the LHCb experiment.They have measured the C P parameters as S D * D = − 0.861 ± 0.077 ± 0.019, S D * D = 0.019 ± 0.075 ± 0.012, C D * D = − 0.059 ± 0.092 ± 0.020, C D * D = − 0.031 ± 0.092 ± 0.016 and A C P D * D = 0.008 ± 0.014 ± 0.006 ± 0.003 [9].In this work, we have estimated the C P parameters and where the matrix elements B 0 to D − ( B0 to D + ) transition multiplying D * + (D * − ) arising from the vacuum.We have obtained the branching fraction using the decay amplitude that is to be B(B 0 → D * + D − ) = (5.20 ± 1.25) × 10 −4 at μ = 2m b scale.This value is well compatible with the value of B(B 0 → D * + D − ) = (6.03±0.50)×10−4 reported by HFLAV [10].We have estimated the C P violation as A C P D * D = 0.008 ± 0.001 and we have obtained other parameters of C P violation, such as S D * D = − 0.709 ± 0.024, S D * D = 0.054 ± 0.003, C D * D = − 0.051 ± 0.004 and C D * D = 0.020 ± 0.001. 2 Branching fraction and C P violation in B 0 → D * + D − decay

4
The C P violation parameters and branching ratio for B → D * + D − decay at three different choices of μ scale Parameters μ = m b /2 μ = m b μ = 2m b Exp.
S D * D is mixing induced C P violation However S D * D is insensitive to C P violation because is related to the strong phase.In the case of C P invariance, S D * + D − = −S D * − D + is fulfilled.The C D * D is direct C P violation and C D * D define the asymmetry between the rates (B 0 → D * + D − )+ ( B0 → D * − D + ) and (B 0 → D * − D + ) + ( B0 → D * + D − ) [27].The C D * D = ±1 denotes a flavour-specific decay, where no C P violation in the interference between decay and decay after mixing is feasible, while decays with C D * D = 0 have the highest sensitivity to mixing induced C P violation.
main our goal of the analysis of B 0 → D * + D − decay was to calculate the C P parameters (S D * D , C D * D , S D * D , C D * D , and A C P D * D ).Studying decays that involve C P violation is a good way to verify the theoretical principles in the quark-flavour of the SM.The B 0 → D * + D − decay, involves b → c cd transitions, which are CKM suppressed.The contributions of higher-order are not Cabibbo-suppressed so the analysis of the B 0 → D * + D − decay helps to constrain these contributions in order to distinguish them from the effects of new physics.Here we have obtained the direct C P violation and parameters C P violation from interference between decay with and without mixing.The uncertainty of the calculated parameters is due to the mass of quarks and mesons, the decay constant and CKM matrix elements.The most important value in the theoretical uncertainty is related to the decay constant.We have calculated the C P parameters as A C P D * D = 0.008±0.001.Also, we have found S D * D = −0.709±0.024,S D * D = 0.054 ± 0.003, C D * D = −0.051± 0.004 and C D * D = 0.020 ± 0.001.From the sum of the amplitudes, we have calculated the total amplitude and obtained comparable result with experimental value for the branching ratio as B(B 0 → D * + D − ) = (5.20 ± 1.25) × 10 −4 at μ = 2m b scale.

Table 1
Experimentally values C P violation parameters for B 0 → D * ± D ∓ decay

Table 2
Measured results of the time-dependent C P violatin parameters for B 0 → D * ± D ∓ decay

Table 3
[19]on coefficients C j in the NDR scheme (α = 1/129)[19] Data Availability StatementThis manuscript has no associated data or the data will not be deposited.[Authors'comment:Thiswork is purely theoretical.Therefore, no data has been used except experimental results.The results obtained in this article are theoretical values and can be used by the public.]OpenAccessThisarticle is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material.If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.To view a copy of this licence, visit http://creativecomm ons.org/licenses/by/4.0/.Funded by SCOAP3.SCOAP 3 supports the goals of the International Year of Basic Sciences for Sustainable Development.