Imprints of CP violation asymmetries in rare $\Lambda_{b}\to \Lambda\ell^{+}\ell^{-}$ decay in family non-universal $Z^{\prime}$ model

We investigate the exclusive rare baryonic $\Lambda_{b}\to\Lambda\ell^{+}\ell^{-}$ in a family non-universal $Z^{\prime}$ model, which is one of the natural extension of standard model. Using transition form factors, calculated in the framework of light cone QCD sum rules, we analyze the effects of polarized and unpolarized CP violation asymmetries for the said decay. Our results indicate that the value of unpolarized and polarized CP-violation asymmetries are considerable in both $\Lambda_{b}\to\Lambda\mu^{+}\mu^{-}$ and $\Lambda_{b}\to\Lambda\tau^{+}\tau^{-}$ channels and hence they give clear indication of new physics arising from the neutral $Z^{\prime}$ gauge boson. It is hoped that the measurements of these CP-violating asymmetries will not only help us to relate new physics, but also help us to determine the precise values of the parameters of new gauge boson $Z^{\prime}$.


I. INTRODUCTION
Despite the discovery of the last missing chunk of the Standard Model (SM), the Higgs boson, and its phenomenological success, there are hints that leads to new physics (NP). The flavour sector is one of the key area which comes in these paths. Because of the joint efforts at the hadron colliders and at the B factories which provided us with data of unprecedented precision in this sector, which are not sensitive to small effects in theoretical calculations that are essential in comparison with the experimental measurements and to see if there are any hints of the NP.
There are two approaches which are mainly considered to investigate physics beyond the Standard model. The first approach is the direct search of new particles, where to produce the particles corresponding to different NP models, the most alluring in this class are different supersymmetric models, the energy of the colliders is raised. The second approach is the indirect search, i.e. to increase the experimental precision on the data of different SM processes where NP effects can manifest themselves. The focus of the two major detectors ATLAS and CMS experiments at the Large Hadron Collider (LHC) at CERN is to detect the possible new particles produced at at sufficiently large energy. However, in the indirect searches, flavour physics plays an important role to investigate physics within and beyond the SM, and the experiment which represents the precision frontier are the LHCb at LHC, the Belle II at the super KEKB and different planned super-B factories will join this arena in future.
In the precision approach, the processes that are suitable to investigate physics within and beyond the SM are the rare decays, particularly the decays which are described by the b → s(d) transitions. The attractive feature of such kind of decays is that they are not allowed at tree level in the SM and possible only at loop level [1]. Therefore, these decays serve as an excellent candidates to chalk out the status of new physics beyond the SM. In mesonic sector, rare decays of B mesons has been widely studied both theoretically and experimentally in detail [2,3].
It is well known that the predications of the SM results are in good agreement with the current experimental data, however there are still unanswered questions in this elegant model, e.g. CP violation, hierarchy puzzle, neutrino oscillations, the few to name. To answer these questions a large number of NP models such as extra dimension models, different versions of supersymmetric models, etc exist in literature and the extensive studies on the exclusive semileptonic decays of B mesons and Λ b baryonic decays both has also been made [4][5][6][7][8][9].
In grand unified theories such as SU (5) or string inspired E 6 models [10][11][12][13][14], one of the most relevant is the Z scenarios that include the family non-universal Z [15,16] and leptophobic Z models [17,18]. Experimental searches of an extra Z boson is an important task of the Tevatron [19] and LHC [20] experiments. On the other hand to get the constraints on the Z gauge boson couplings through low energy processes are crucial and complementary for direct searches Z → e + e − at Tevatron [21]. The most interesting thing about the family non universal Z model is the new CP-violating phase which have large effects on many FCNC processes [16,22], such as B s −B s mixing [23][24][25][26][27], and rare hadronic B-meson decays [28][29][30].
In baryonic sector exclusive Λ b → Λ + − decays, at quark level are described by b → s + − transition. The main difference between these and other mesonic decays are that they can give information about the helicity structure of the effective Hamiltonian for the FCNC b → s process in the SM and beyond [31]. On the experimental side, first observation on rare baryonic Λ b → Λµ + µ − decay has been observed by CDF Collaboration [32], and recently this decay was also studied by LHCb collaboration [33]. The experimental investigation motivates theoreticians to do more deep analysis of the different physical observables such as branching ratio, forward backward asymmetry, single and double lepton polarization asymmetries and CP violation asymmetry in the these decay modes. It will be hoped that such studiers are useful to distingish various extensions of the SM.
In the present work, we analyze the effects of polarized and unpolarized CP violation asymmetries for Λ b → Λ + − decay in the family non universal Z model developed in [16]. From the CP violation asymmetry view point it is important to emphasize here that b → s transition matrix elements are proportional to three quark coupling matrix elements usually called CKM matrix elements, V tb V * ts ,V cb V * cs and V ub V * us ; however due to the unitarity condition, and neglecting the matrix elements V ub V * us in comparision with V tb V * ts and V cb V * cs the CP asymmetry is highly suppressed in the SM. Therefore, the measurements of CP violating asymmetries in b → s decays play an important role to find the imprints of the Z model.
The structure of the paper is as follows. In section II we develop a theoretical tool box in which we present the effective Hamiltonian for the decay b → s + − . In the same section we present the transition matrix element for the decay Λ b → Λ + − decay, and the expressions for unpolarized and polarized CP violation for the said decay in family non-universal Z model. In Section III we discuss the numerical results of the said physical observables. The concluding remarks are also presented in the same section.

II. THEORETICAL TOOL BOX
At quark level the decay Λ b → Λ + − ( = µ, τ ) is governed by the transition b → s + − , the effective Hamiltonian for such kind of decays at O(m b ) scale can be written as where G F is Fermi coupling constant and V ij are the matrix elements of the CKM matrix. In Eq. (1), O i (µ) are the local quark operators and C i (µ) are the corresponding Wilson coefficients at energy scale µ. The explicit expressions for of the Wilson coefficients at next to leading logrithim order and next to next leading logrithim are given in ref [34][35][36][37][38][39][40][41][42][43]. The operators responsible for such kind of decays are O 7 , O 9 and O 10 which are summarized in [7]. In terms of the effective Hamiltonian given in Eq. (1), the quark level amplitude for the said decay in the SM can be written as where q 2 is the square of the momentum transfer and α is the fine structure constant. A family non-universal Z boson could be derived naturally in many extensions of the SM, the most economical way to get it is to include an additional U (1) gauge symmetry. This model has been formulated in detail by Langacker and Plümacher [16]. In a family non-universal Z model, FCNC transitions b → s + − could be induced at tree level because of the non-diagonal chiral coupling matrix. Assuming that the couplings of right handed quark flavors with Z boson are diagonal and ignoring Z − Z mixing, the effective Hamiltonian of Z part for the decay b → s + − can be written as [44][45][46] One can also write Eq. (3) in the following way where and The most economical feature of the family non-universal Z -model is that operator basis remains the same as in the SM and the only modifications come are in the Wilson coefficients, where C 9 and C 10 get modification while the Wilson coefficient C ef f 7 remains unchanged. The total amplitude for the decay Λ b → Λ + − is the sum of SM and Z contribution, and can be written as follows where C tot 9 = C ef f 9 + Λ sb C Z 9 and C tot 10 = C SM 10 + Λ sb C Z

10
The matrix elements for the decay Λ b → Λ + − can be straightforwardly parameterized in terms of the form factors as follows [47] where f i , g i and f T i , g T i are the transition form factors for the decay Λ b → Λ. By using the matrix elements which are parameterized in terms of transition form factors [Eqs. (9) and (10)] with expression (8), the decay amplitude for the decay Λ b → Λ + − can be written as The hadronic functions τ 1 µ and τ 2 µ are given by The auxiliary functions from A(q 2 ) to L(q 2 ) given in Eqs. (12) and (13) contains both short and long distance effects which are encapsulated in terms of Wilson coefficients and form factors. The explicit form of these functions can be The matrix element for the decay Λ b → Λ + − given in Eq. (11) is useful to calculate the physical observables. The formula for double differential decay rate can be written as Also s is the square of the momentum transfer q and θ is the angle between lepton and final state baryon in the rest frame of Λ b . By using the expression of amplitude given in Eq. (11) and integration over cos θ, one can get the expression of the dilepton invariant mass specturm as with Following the recipe given in ref. [7], one can define the CP -violation asymmetry for the decay Λ b → Λ + − for both the cases, i.e. with polarized and unpolarized leptons as The analogous expression for CP conjugated differential decay width is given in ref. [7]. The expression for CP violation asymmetry can be obtained by using Eq. (16) and Eq. (17), so one gets where i represents the longitudinal (L), normal (N ) and transverse (T ) polarization of the final state leptons. Also one can write the polarized and unpolarized CP asymmetry as The normalized CP violation asymmetry can be defined by using the above definition as follows In Eq.(21) the positive sign in the second term represents to L and N polarizations, and the negative sign is for T polarization.
The following are the results of unpolarized The explicit form of Q(s) and the Q i (s) are given below.
The explicit form of the functions H 1 ,H 2 and H 3 can be written as The expressions for polarized CP asymmetry are given below.

A. Longitudinal CP violation
The longitudinal lepton polarization can be written as with where (17) and the terms H L 4 and H L 5 are given as follows

III. NUMERICAL ANALYSIS
In this section we will discuss the numerical analysis of the unpolarized and polarized CP violation asymmetries for Λ b → Λ + − with = µ, τ decays. In order to see the imprints of the family non-universal Z gauge boson on the said physical observables, first we have to summarize the numerical values of various input parameters used in calculations such as masses of particles, life time, quark coupling CKM matrix etc., in Table I, while the values of Wilson coefficents are presented in Table-II. The most important input parameters which are important in any hadronic decays are the non perturbative quantities, called form factors, and for the said decay we rely on light cone QCD sum rules approach [47]. The parametrization of the form factors f 1,2,3 , g 1,2,3 , f T 2,3 and g T 2,3 are given by while the form factors f T 1 and g T 1 are of the form The numerical values of the light cone QCD sum rules form factors along with the different fitting parameters [47] are summarized in Table III and IV. Regarding to the couplings of family non universal Z -model there are some strong constraint from both inclusive and exclusive B meson decays [23]. The numerical values of quarks and leptons coupligns parameters of Z model are given in Table V, where S1 and S2 represents the two different fitting values for B s −B s mixing data by the UTfit collaboration [25] and the numerical values of S3 are chosen from [48,49] and are also summarized in Table V. It has been already mentioned that B sb = |B sb |e −iφ sb is the off diagonal left handed coupling of Z boson with quarks and φ sb corresponds to a new weak phase, whereas S LL and D LL represent the combination of left and right handed couplings of Z with the leptons [c.f. Eq. (6)]. In order to fully scan the three scenarios, let us make a Unpolarized CP violation asymmetry: • Figs.1 and 2 represents the unpolarized CP violation asymmetries for the decay Λ b → Λµ + µ − (τ + τ − ) as a function of D LL and S LL respectively. In standard model CP violation asymmetry is zero hence the non-zero value will give us a clue of physics beyond the standard model which is commonly known as New Physics (NP). It is evident from Eq.(22) that the A CP is proportional to Z parameters which comes through the imaginary part of the Wilson coefficents as well as of weak phase φ sb which conceals in Λ sb (c.f.Eq. (5)). Therefore the dependence on new weak phase φ sb is expected and is evident from Figs.1 and 2 where band in each case depicits the variation of new phase φ sb in respective scenarios. In Fig.(1) A CP is plotted vs D LL by changing the values of S LL , φ sb and B sb . In case of µ's as a final state leptons, the value of A CP is positive in both scenarios S1 and S2 but for positive values of S LL . However, the value of A CP reaches to −0.05 when D LL = −1.6 × 10 −2 and corresponding S LL = −6.7 × 10 −2 depicited by red band. Similarly for the case of τ 's as final state leptons, the value of A CP is positive in both scenarios for positive values of S LL . However, the value of A CP is around −0.08 for D LL = −1.6 × 10 −2 and S LL = −6.7 × 10 −2 shown by red bands.
• Fig. 2 presents the behavior of A CP with S LL by varying the values of D LL , φ sb and B sb in the range given in Table V. It can be immediately noticed that in case of µ's the value is small compared to the case when τ 's    [23,25,48,49].
1.09 ± 0.22 −72 ± 7 −2.8 ± 3.9 −6.7 ± 2.6 S2 2.20 ± 0.15 −82 ± 4 −1.2 ± 1.4 −2.5 ± 0.9 Longitudinal polarized CP violation asymmetry: • The longitudinal polarized CP violation asymmetry A L CP is plotted in Figs. 3 and 4. From Eq. (29) it can be noticed that Q L is proportional to the imaginary part of the combination of Wilson coefficients which involve C 7 , C 9 and C 10 both in the SM as well as in the Z model. Even though, the Wilson coefficient C 7 does not get contribution from the Z , but the change in the Wilson coefficients C 9 and C 10 due to the parameters of Z model will make the A L CP sensitive to the change arising due to extra neutral boson Z . In Fig. 3(a) and 3(b), we have plotted the A L CP vs D LL by fixing the values of S LL and other Z parameters in the range given in Table V. We can see that the value of A L CP increases from 0.008 to 0.053 when µ's are the final state leptons and from 0.004 to 0.030 in case of τ 's as final state leptons which can be visualized from the colour bands that corresponds to scenario S1(S2). The situation when the longitudinal polarized CP violation asymmetry is plotted with S LL by taking other parameters in the range given in Table V and it is displayed in Fig.4. Here we can see that it is an increasing function of S LL where in S1 and S2 the value increase from 0.01(0.005) to Unpolarized CP violation asymmetry ACP as function of DLL for Λ b → Λµ + µ − (τ + τ − ) for scenarios S1, S2 and S3 . The red, blue, grey and yellow bands correspnds to S1. Green, orange, pink and purple band correspnds to S2. The dots of different colors corresponds to S3. The band in each case depicts the variations of φ sb in respective scenario.  0.05(0.025) when we have µ s(τ s) as final state leptons and it is clearly visible from the red(pink) band. It can also be seen in Fig. 4a, that value of the longitudinal polarized CP violation asymmetry in scenario S3 is much suppressed when we have µ's as final state leptons. However, in case of the τ 's the value of the longitudinal CP violation asymmetry is around 0.030 when φ sb = −140 • and |B sb | = 5 × 10 −3 . It is shown with the green dot in Fig. 4b. It can be noticed that the value of longitudinal polarized CP violation asymmetry in Λ b → Λτ + τ − is significantly different from its value in the S1 and S2. Hence, by measuring A L CP one can not only segregate the NP coming through the Z boson but can also distinguish the three scenarios.
Normal polarized CP violation asymmetry: • Contrary to the A CP and A L CP , the normal polarized CP violation asymmetry (A N CP ) is an order of magnitude smaller in case of µ's compared to the τ 's as final state leptons. By looking at the Eq. (33). The A N CP comes from the function Q N which contains H 1 ..., H 6 . In Eqs. (34 -38) it is clear that these asymmetries are proportional to the lepton mass and their suppression in case of muon is obvious and Figs. 5(a) and 6(a) depict this fact. Coming to the Figs. 5(b) and 6(b) we can see that the A N CP is very sensitive to the parameters of Z both in the S1 and S2. In Fig. 5(b), the value of A N CP decreases from 0.040 to 0.018 in the parameter range of Z in S1 and from 0.048 to 0.028 in S2. The situation remains the same as in Fig. 5 when A N CP is plotted with S LL  in Figs. 6(a) and 6b. It can also be noted that average value of the A N CP increases from 0.020 to 0.045 in S1 and 0.035 to 0.05 in S2.
What comes out to be more interesting is the impact of parametric space of scenario S3 in case of µ's and final state leptons. In this scenario, the value of the normal CP violation asymmetry in Λ b → Λµ + µ − is an order of magnitude larger than the corresponding values in S1 and S2. Here, the maximum value is 0.018 (the green dot) when φ sb = −140 • and |B sb | = 5 × 10 −3 . While in case of τ 's as final state leptons the order of asymmetries remains the same as in S1 and S2.
Transverse polarized CP violation asymmetry: • Just like the normal polarized CP violation asymmetry, the different terms in transverse polarized CP violation asymmetry A T CP are also m l suppressed which is visible from H's appearing in the function Q T in Eq. (40). The graphs given in Figs. 7(a) and 8(a) depict the fact that in the presence of NP, the maximum value of A T CP is around 0.016 (shown by the green band) in Λ b → Λµ + µ − , while Figs. 7(b) and 8(b) are shown that in case of the τ 's as final state leptons the value of the A T CP reaches upto 0.08 in certain parametric space of the Z scenario S1.
By varying the Z parameters in the range given in Eq. (51) the trend of transverse CP violation asymmetry is shown by different colors of dots in Fig.8. In case of µ's as final state leptons, we can see that for φ sb = −140 • , |B sb | = 5 × 10 −3 in scenario S3 the value of transverse polarized CP violation asymmetry is slightly higher than the first two scenarios (shown by the green dot). However, in Λ b → Λτ + τ − decay the effects coming through the parametric space of S3 are smaller than that of the first two scenarios.  In short, we have analyzed the imprints of NP coming through the neutral Z boson on the unpolarized and polarized CP violation asymmetries in Λ b → Λ + − decays. In addition motivated by the fact that the CP violation asymmetry is negligible in the SM, we have chosen this observable to explore the effects of Z in Λ b → Λ + − decays. It has been noticed that the value of unpolarized and polarized CP violation asymmetry is considerable in both Λ b → Λµ + µ − and Λ b → Λτ + τ − channels and hence it gives a clear message of NP arising from the neutral Z boson. Though the detection of leptons' polarization effects in semileptonic decays is really a daunting task at the experiments such as the ATLAS, CMS and at LHCb, but the fact that these CP violation asymmetries which suffer less from hadronic uncertainties provide a useful probe to establish the NP coming through the Z model.  FIG. 8: Transversally polarized CP violation asymmetry A T CP as function of SLL for Λ b → Λµ + µ − (τ + τ − ) for scenarios S1, S2 and S3. The color and band description is same as in Fig. 1.