Analysis of angular observables of (cid:2) b → (cid:2)( → p π )μ + μ − decay in the standard and Z (cid:3) models

In 2015, the LHCb collaboration measured the differential branching ratio d B dq 2 , the lepton- and hadron-side forward–backward asymmetries, denoted by A (cid:2) FB and A (cid:3) FB , respectively, in the range 15 < q 2 ( = s ) < 20 GeV 2 with 3 fb − 1 of data. Motivated by these measurements, we perform an analysis of q 2 -dependent (cid:3) b → (cid:3)( → p π)μ + μ − angular observables at large- and low-recoil in the standard model (SM) and in a family non-universal Z (cid:4) model. The exclusive (cid:3) b → (cid:3) transition is governed by the form factors, and in the present study we use the recently performed high-precision lattice QCD calculations that have well-controlled uncertainties, especially in the 15 < s < 20 GeV 2 bin. Using the full four-folded angular distribution of (cid:3) b → (cid:3)( → p π)μ + μ − decay, ﬁrst of all we focus on calculations of the experimentally measured d B ds , A (cid:2) FB , and A (cid:3) FB in the SM and compare their numerical values with the measurements in appropriate bins of s . In case of a possible discrepancy between the SM prediction and the measurements, we try to see if these can be accommodated though the extra neutral Z (cid:4) boson. We ﬁnd that in the dimuon momentum range 15 < s < 20 GeV 2 the value of d B ds and central value of A (cid:2) FB in the Z (cid:4) model is compatible with the measured values. In addition, the fraction of longitudinal polarization of the dimuon F L was measured to be 0.61 + 0.11 − 0.14 ± 0.03 in 15 < s < 20 GeV 2 at the LHCb. We ﬁnd that in this bin the value found in the Z (cid:4) model is close to the observed values. After comparing the results of these observables, we have proposed other observables such as α i and α ( (cid:4) ) i with i = θ (cid:2) , θ (cid:3) , φ , L , U and coefﬁcients of different foldings P 1, ... ,9 in different bins of s in the SM and Z (cid:4) model. We illustrate that the experimental observations of the s -dependent angular observables calculated here in several bins of s can help to test the predictions of the SM and unravel new physics contributions arising due to the Z (cid:4) model in (cid:3) b → (cid:3)( → p π)μ + μ − decays. decays, we calculated combinations of different angular observables in (cid:3) b


Introduction
Rare decays involving b-quarks, such as b → (s , d)γ , b → (s , d) + − , have been of immense interest in recent decades. This is because these decays are induced by flavor-changing neutral current transitions (FCNC) involving the quantum number transitions | Q| = 0 and | B| = 1. In the standard model (SM), FCNC transitions are not allowed at the tree level but occur at loop level because of the Glashow-Iliopoulos-Maiani (GIM) mechanism [1]. This makes them sensitive to the masses of particles that run in the loop, e.g. m t and m W in the SM. As a consequence, these decays play a pivotal role in the determination of Cabibbo-Kobayashi-Masakawa (CKM) [2,3] matrix elements in an indirect way. In different extensions of the SM, there is a possibility that the new particles can also run in the SM loop diagrams, making these rare decays sensitive to the masses and couplings PTEP 2018, 043B08 A. Nasrullah et al. of the new particles. Hence, rare decays provide a rich laboratory to test the predictions of the SM and help us to establish possible new physics (NP) indirectly [4,5].
As long as the inclusive radiative and semi-leptonic decays are concerned, there are hardly any open issues that could lead us towards evidence of NP. However, experimental precision is limited at present and it is expected that these bounds will be improved significantly at Belle II [6]. The situation for exclusive semi-leptonic B-meson decays is different, with a lot of open issues. Among them, the most pertinent is lepton flavor universality (LFU), i.e., the couplings of gauge bosons in the SM are the same for different families of leptons. This important prediction of the SM can be tested by measuring the ratio of the decay widths of B → K ( * ) μ + μ − and B → K ( * ) e + e − , defined as in specific bins of the dilepton invariant mass squared, written as s ∈ [s min , s max ] from here onwards.
As this ratio involves the same B → K ( * ) transition, the hadronic uncertainties arising from the form factors cancel out to a good approximation. Therefore, any possible deviations from the SM predictions, i.e., a value of the ratio different from one, will hint towards the NP. In 2014, the LHCb collaboration observed more than a 2 σ mismatch between experimental observations and SM predictions in different bins of the square of momentum transfer s = q 2 [7]. This hints at the breakdown of the SM LFU, i.e., the couplings of gauge bosons with μ and e are not the same [8,9]. There are also some other areas where tensions between SM predictions and experimental observations are found, such as the P 5 anomaly (3.5 σ in one bin s ∈ [4.30, 8.68] GeV 2 [10], which corresponds to a certain coefficient in the angular distribution of the B → K * (→ Kπ)μ + μ − decay [10][11][12]. This anomaly was again observed at 3 σ in the data with 3 fb −1 luminosity in the two bins s ∈ [4, 6] GeV 2 and s ∈ [6,8] GeV 2 [13], and this was later confirmed by Belle in the bin s ∈ [4,8] GeV 2 [14]. This anomaly was accompanied by a 2.9 σ tension in the second bin of another angular observable called P 2 [15]. In addition, a small but noticeable difference was found in the branching ratio of B → K * μ + μ − [16][17][18] and B s → φμ + μ − (2.0 σ larger than the SM prediction both in low-and large-φ recoil) [19][20][21]. Making use of the available data and motivated by these tantalizing anomalies observed in these B decays, in addition to explaining them in different beyondthe-SM scenarios [22][23][24], global analyses have also been carried out [15,[25][26][27][28][29][30][31][32][33]. Incorporating the factorizable (absorbed in the form factors) and non-factorizable contributions, these global analyses favor a negative shift in the Wilson coefficient C 9 to explain most of the data. However, before we could claim that these are indications of NP, we have to get full control of the possible hadronic uncertainties arising due to form factors in the exclusive decays [34][35][36][37][38][39][40]. In order to establish the hints of NP, on the experimental side we need to have the improved statistical data that is expected at Belle II and the LHCb, whereas on the theoretical side we can study some other decays that are governed at quark level by b → s + − ( = μ, τ ) transitions.
In the present study, we have considered the b → (→ pπ) + − decay that is interesting to its own regard. On the experimental side, this decay was first studied by the CDF collaboration [41], and the LHCb later published the first measurement of the differential branching ratio as well as the forward-backward asymmetry of the final state muon, i.e., A FB [42,43]. Recently, the LHCb collaboration has made an observation of CP violation and the asymmetries arising due to the angle between the μ + μ − and pK − planes (aT odd CP ) in b → pK − μ + μ − by analyzing the data available at an integrated luminosity of 3 fb −1 [44]. On the theoretical front, at first in the decay b → + − , the hadrons involved in the initial and final states are the baryons, therefore the study of such decays 2/30 PTEP 2018, 043B08 A. Nasrullah et al. will help us to understand the helicity structure of the underlying effective Hamiltonian [45][46][47].
Another added benefit is that analysis of the angular asymmetries in the sequential decay b → (→ pπ)μ + μ − is expected to complement the different angular asymmetries in the corresponding B → K * (→ Kπ) + − decays [48][49][50]. One important aspect is the stability of under strong interactions, and the decay b → (→ pπ) + − is theoretically cleaner than the decay B → K * (→ Kπ) + − . Hence, the decay b → + − has been theoretically studied in a number of papers . Just like the exclusive decays of B-mesons, the decay b → + − is prone to the uncertainties arising due to form factors. However, at present the b → transition form factors are calculated using lattice QCD calculations with high precision [89]; to have their profile in the full q 2 range, these form factors are extrapolated using the Bourrely-Caprini-Lellouch parametrization [90]. The lattice results are quite consistent with the recent QCD light-cone sum rule calculation [53], with the added benefit of a much smaller uncertainty in most of the kinematic range. However, in contrast to the B decays, the QCD factorization is not fully developed for the b-baryon decays; therefore, we will not include these non-factorizable contributions in the present study. After having control of the hadronic uncertainties in the form factors, the next choice is to find observables that are relatively clean. In line with the B → K * (→ Kπ)μ + μ − decays, we have calculated combinations of different angular observables in b → (→ pπ)μ + μ − decays, namely, forward-backward asymmetries (A FB , A FB , A FB ), the longitudinal (F L ) and transverse (F T ) fractions of the dimuon, the longitudinal asymmetry α L , the transverse asymmetry α U , and the observables named as P i that are derived from different foldings, in the SM at its first right. It has been observed that in order to explain the R K anomaly in B → K + − decays, a possible candidate is the Z model [91][92][93]. The economy of these Z models is that they can be accommodated to the SM just by extending the electroweak SM group by an additional U (1) gauge group with which the extra gauge boson Z is associated. Also, in the grand unification theories (GUTs) such as SU (5) or string-inspired E 6 models [94][95][96][97][98], relevant scenarios are the family non-universal Z model [99,100] and the leptophobic Z models [101,102]. The direct signature of an extra Z boson is still missing in the analysis of data taken so far at the LHC [103] experiment, but we already have some indirect constraints on the couplings of the Z gauge boson through low-energy processes that are crucial and complementary for direct searches Z → e + e − at Tevatron [104]. The additional interesting thing that the family non-universal Z models have in their favor is the new CP-violating phase, which has large effects on various FCNC processes [100,105,106], such as B s -B s mixing [107][108][109][110][111][112][113][114][115][116][117][118][119][120][121] and rare hadronic and B-meson decays [122][123][124][125][126][127][128][129][130][131][132][133][134][135][136][137][138][139][140]. As extending the SM group by an extra U (1) gauge group does not change the operator basis of the SM, the Z model therefore belongs to a class of minimal flavor-violating models having its imprints in the Wilson coefficients that correspond to the SM operators. Keeping in view that among the different hadrons produced at the LHCb, almost 20% will be b baryons, it is expected that in future the results of decay distributions and different angular asymmetries will be available with much better statistics. Therefore, in addition to the SM calculation of the different observables mentioned above, we have studied the impact of different Z parameters on these observables in different bins of s.
The paper is organized as follows: In Sect. 2, the theoretical framework for the decay b → (→ pπ) + − is discussed. Helicity amplitudes for the decay are written in terms of transition form factors and four-fold differential decay distributions. After summarizing the Wilson coefficients and other parameters of the Z model in Sect. 3, we present the calculation of several observables that have been obtained using four-folded angular distributions in Sect. 4. Section 5 presents numerical 3/30 PTEP 2018, 043B08 A. Nasrullah et al. analysis of the observables done in the SM and in the Z model, and here we compare the results of certain asymmetries with the measurements available from the LHCb experiment. In addition to the tabular form of the results, these are also plotted graphically here. Finally, the main findings are summarized in the last section.

Effective Hamiltonian formalism for the SM and Z model
The quark-level decay governing b → (→ pπ)μ + μ − is b → sμ + μ − . In this decay of the bbaryon, the short-distance effects are encoded in the Wilson coefficients, whereas the long-distance contributions are incorporated through the four-quark operators. After integrating out the heavy degrees of freedom, W ± , Z bosons, and top quark, the SM effective Hamiltonian for these decays is 9,10 where G F is the Fermi coupling constant, V tb V * ts are the CKM matrix elements, α e is the fine structure constant, and C i (μ) with i = 7, 9, 10 are the Wilson coefficients corresponding to the electromagnetic operator O 7 and semi-leptonic operators O 9,10 that are defined as It has already been mentioned that QCD factorization at low-q 2 is not fully developed for the hadronic b-baryon decay; therefore, we have ignored the non-factorizable contributions here. 1 The factorizable non-local matrix elements of the four-quark operators O 1-6 and O g 8 are encoded into effective Wilson coefficients C eff 7 (s) and C eff 9 (s), where s is the dilepton squared mass q 2 (q μ = p μ 1 − p μ 2 ). In the high-q 2 region, the Wilson coefficients C eff 7 (s) and C eff 9 (s) can be written as [31] C eff 7 (s) = C 7 − 1,c (s) + C 8 F 8 (s) , where h(m q , s) with q = b, c corresponds to the fermionic loop functions. These h(m q , s) along with the functions F (7,9) 8 and F (7,9) (1,2,c) are calculated in Refs. [54,141]. Long ago, Langacker and Plümacher included a family non-universal Z boson through additional U (1) gauge symmetry [100]. In contrast to the SM, having a non-diagonal chiral coupling matrix, 1 In the case of B → K * μ + μ − decay, it is evident that the non-factorizable charm-loop effects (i.e., corrections that are not described using hadronic form factors) play a sizeable role in the low-q 2 region [40] and the same is expected in case of the decay under consideration. However, in the present study we shall neglect their contributions because there is no systematic framework available in which these non-factorizable charm-loop effects can be calculated in baryonic decays [53]. Therefore, our results at low-q 2 are affected by the uncertainities due to these contributions. In the whole q 2 range, the effective Wilson coefficients are given in Eq. (3). According to Ref. [89], we use Eq. (3) in the low-and high-q 2 regions by increasing the 5% uncertainty. Thus, having a control on the non-factorizable contributions in baryonic decays will help us to hunt for deviations from the SM predictions. in a family non-universal Z model, the FCNC transitions b → s + − could be induced at tree level. Ignoring Z-Z mixing, along with the assumption that the couplings of right-handed quark flavors with Z boson are diagonal, the effective Hamiltonian for the b → s + − transition corresponding to the Z boson becomes [142][143][144][145][146] In Eq. (4), S L and S R represent the couplings of the Z boson with the left-and right-handed leptons, respectively. The corresponding off-diagonal left-handed coupling of quarks with the new Z boson is taken care of by B sb = |B sb |e −iφ sb , with φ sb a new weak phase. In a more sophisticated form, Eq. (4) can be written as and By comparing Eqs. (2) and (5) it can be noticed that except for C eff 7 , which is absent in the Z model, the operator basis of the family non-universal Z model is the same as that of the SM for O 9,10 . Hence, the contribution arising due to the extra Z boson is absorbed in the Wilson coefficients C eff 9 and C 10 .
The total amplitude for the decay b → + − is the sum of the SM and Z contributions, and it can be formulated in terms of b → matrix elements as where C tot 9 = C eff 9 + sb C Z 9 and C tot 10 = C SM 10 + sb C Z 10 , with C eff 9 defined in Eq. (3).

Helicity amplitudes and form factors for b → transitions
The matrix elements for the b → transition for different possible currents can be straightforwardly parameterized in terms of the form factors. The helicity formalism provides a convenient way to describe these transformations.
where p 1 (s 1 ) and p 2 (s 2 ) are the momentum (spin) of b and , respectively. The dilepton polarization vector is written as * μ (λ) with λ = t, 0, ±; their explicit definitions are given in Ref. [48] and summarized in the appendix.
In Eqs. (9)- (12), the functions f i (s), g i (s), h i (s), and h i (s) with i = 0, +, ⊥ are the transition form factors. In the heavy quark spin symmetry, the symmetry where the spin of a spectator diquark remains the same in the initial and final states, the number of form factors is reduced. The tensor form factors can be written in terms of vector and axial-vector form factors, and with this symmetry we can also equate the longitudinal and transverse form factors. Thus it reduces the number of independent form factors to two, i.e., the Isuger-Wise relations ξ 1 and ξ 2 ; the form factors being the non-perturbative quantities needed to be calculated in some model. In the decay under consideration here, we will use the form factors that are calculated in lattice QCD with much better control on the various uncertainties. In the full dilepton mass squared range these can be expressed as [89] where the inputs a A. Nasrullah et al. 2.4782 ± 0.9549 Table 2. Pole masses for different form factors [89].

Angular distribution and physical observables
The four-folded angular distribution of the four-body b → (→ pπ)μ + μ − decay, with an unpolarized b , can be written in terms of K l,m , where l and m denote the relative angular momentum and its third component for pπ and μ + μ − systems, respectively, as [48] In Eq. (15), θ and θ are the helicity angles, φ is the azimuthal angle, and s is the dilepton mass squared (see also Fig. 1). The different kinematic relations are defined in Ref. [48]. The different angular coefficients correspond to the particular values of (l, m): e.g., the coefficients of cos 2 θ l , sin 2 θ l , and cos θ correspond to K 0,0 , whereas the coefficients of cos 2 θ cos θ , sin 2 θ cos θ , and cos θ cos θ correspond to K 1,0 and the last four terms correspond to K 1,1 . These angular parameters

Differential decay rate and different asymmetry parameters
One of the most important observables, from both the theoretical and experimental points of view, is the differential decay distribution. By integrating over θ ∈ [0, π ], θ ∈ [0, π ], and φ ∈ [0, 2π], the expression for the differential decay rate becomes In addition to the decay rate, we can extract a number of asymmetry parameters that correspond to different angles and they can be separated out by doing different integrations one by one. For example, by integrating on θ ∈ [0, π] and φ ∈ [0, 2π], the expression for the differential decay rate takes the form where α θ is the asymmetry parameter for the longitudinal polarization of the baryon. It can be noticed that if we integrate Eq. (17) on θ ∈ [0, π], we get back Eq. (16). In terms of the helicity parameters K ij , the asymmetry parameter α θ can be expressed as follows: Here, α is the asymmetry parameter corresponding to the parity-violating → pπ − decay, and its experimental value is α = 0.642 ± 0.013 [151].
Similarly, by performing an integration on θ ∈ [0, π] and φ ∈ [0, 2π] and leaving the angle θ , we will have asymmetries corresponding to the angle θ . In terms of α θ and α θ , the differential decay rate can be formulated as with On the same lines, if we perform integration on the helicity angles θ ∈ [0, π ] and θ ∈ [0, π], Eq. (15) can be written in terms of asymmetries corresponding to the angle φ as where From Eq. (15), the s dependence of the transverse (α U ) and longitudinal (α L ) asymmetry parameters is written in the following form [147]: Even though one of the important observables is the decay rate, it is affected by the uncertainties arising from different input parameters, where the major contributors are the form factors. It is a well-established fact that the zero position of the forward-backward asymmetry in the different semi-leptonic decays of the B-meson have a minimal dependence on the form factors [148][149][150]. Based on these observations the different forward-backward asymmetries are exploited in the b decays [48][49][50]75,76]. The forward-backward asymmetries corresponding to the lepton angle θ is defined as A FB = (F −B)/(F +B). Similarly, the hadron-side forward-backward asymmetry, i.e., the asymmetry corresponding to the hadronic angle θ , is In both cases, F and B are the forward and backward hemispheres, respectively. From Eq. (15), these forward-backward asymmetries become We take this opportunity to mention that in the case of the b → (→ pπ)μ + μ − decay, the sequential decay → pπ is parity violating. Therefore, the helicity components with the polarizations of the proton being ± 1 2 are not the same, and hence the hadron-side forward-backward asymmetry is non-zero in these b-baryon decays. This is contrary to what we have seen in the B → K * (→ Kπ)μ + μ − decay. In addition to this, the combined lepton-hadron forward-backward asymmetry can be expressed as According to the experimental point of view, the other interesting observables are the fractions of longitudinal (F L ) and transverse (F T ) polarized dimuons in b → μ + μ − decay, and these have already been measured in different bins by the LHCb Collaboration [153]. In order to achieve the mathematical formula for these helicity fractions we have to integrate the four-folded differential decay rate given in Eq. (15) on θ ∈ [0, π ] and φ ∈ [0, 2π ]. Their explicit expressions in terms of K ij are 9/30 PTEP 2018, 043B08 A. Nasrullah et al. Table 3. Foldings required for P i for which θ ∈ [0, π 2 ], θ l ∈ [0, π 2 ], and φ vary in different ranges corresponding to different observables [15].
The following things can be noticed from Eq. (27): • The coefficients of sin 2 θ and sin 2 θ cos θ correspond to the angular coefficients named as P 1 and P 2 , respectively. • The coefficient of cos θ cos θ corresponds to the angular coefficient P 3 , and that of sin θ cos θ sin θ sin φ is P 4 . • P 5 is the coefficient of sin θ cos θ sin θ cos φ, whereas P 6 is the coefficient of sin θ sin θ cos φ. • P 7 , P 8 , and P 9 are the coefficients of sin θ sin θ sin φ, cos θ , and cos θ , respectively. In terms of the different helicity components, the angular coefficients P i , i = 1, . . . , 9 are where = d ds . It is worth mentioning that while obtaining the different P i we have used the first six foldings defined in Table 3, because the last two foldings do not add any new observable.
In order to compare the results with some of the experimentally measured observables and to propose possible candidates that might be useful to establish new physics, the interesting quantities are the normalized fractions calculated in different bins of the square of the dimuon momentum, i.e., s = q 2 . The normalized branching ratio, various asymmetry observables, and different angular coefficients can be calculated as

Numerical analysis
In this section we discuss the numerical results obtained for the different observables defined in Sect. 4 in both the standard and Z models for the b → (→ pπ)μ + μ − decay. In b → decays, the final state → pπ − is a parity-violating decay and the corresponding asymmetry parameter (α ) has been measured experimentally [151]. This is really helpful in disentangling the direct b → pπ − μ + μ − from the one that occurs through the intermediate decay that subsequently decays to pπ − . This is contrary to B → K * (→ Kπ)μ + μ − decay, where the final-state K * meson decays to Kπ via the strong interaction. Therefore, the angular analysis of b → (→ pπ − )μ + μ − decay is quite interesting from both theoretical and experimental points of view [48,49]. In addition to the input parameters given above, the other important ingredient in the numerical calculations in b decays is the form factors. In the numerical calculation, we will use one of the most accurately calculated form factors at the QCD lattice [89] with 2+1-flavor dynamics (cf. Table 1) along with the next-to-next-to-leading (NNLL) corrections to the form factors for the SM that are given in [141,152].
In addition to the form factors, the numerical values of the other input parameters that correspond to the standard and Z models are given in Tables 4 and 5, respectively. Using these values a quantitative analysis of the above-calculated observables in various bins of s is presented in Tables 6, 7, and 8. In the whole analysis, we have observed that the results are not sensitive to the different scenarios of the Z model; therefore, we have used only the scenario S 1 to generate the results in various bins of s.
The first observable that is of prime interest from both theoretical and experimental points of view is the branching ratio in different bins of s that can be set up by the experimentalists. From Eq.   Table 5. Numerical values of the parameters corresponding to the different scenarios of the Z model [139,140].
12/30 PTEP 2018, 043B08 A. Nasrullah et al. By looking at Eqs. (30) and (31), we can say that the deviations from the measured value in this bin are quite large in the SM and even larger in the Z model. One possible reason for such a large deviation is that the form factors are not very precisely calculated in this region. Contrary to this, the calculation of form factors is more precise in s ∈ [15,20] GeV 2 (low-recoil). In this bin the average 13/30 PTEP 2018, 043B08 A. Nasrullah et al.
This can be reconciled because in this region, the deviations from the measured values are small compared to that of the large-recoil bin; in this case, the deviations are 3.2 σ and 0.1 σ in the SM and Z model, respectively. Hence, the results of the branching ratio in the Z model for the low-recoil bin look more promising when compared with the corresponding experimental value. In future, when we have more data from the LHCb experiment and Belle II, on one hand it will give us a chance to see possible hints of the extra neutral Z boson and on the other hand it will help us to test the SM predictions with better accuracy. It is a well-known fact that the branching ratio is prone to uncertainties arising due to the form factors. In order to cope with some of the uncertainties, there are observables such as the baryon forward-backward asymmetry (A FB ) and lepton forward-backward asymmetry (A FB ) that are measured with respect to the baryon angle θ and lepton angle θ l , respectively. The asymmetry A FB can be expressed in terms of the ratio of a linear combination of the angular coefficients K 2ss and K 2cc to a linear combination of the angular coefficients K 1ss and K 1cc , as given in Eq. (24). Due to the change in the value of Wilson coefficient C 9 in the Z model, K 2ss and K 2cc get more contribution It can easily be seen that at low-recoil, the SM prediction is close to the experimentally measured value and the deviation is 0.2 σ . The Z value of (A FB ) exceeds the experimental result by 2.2 σ . From the above discussion, it is clear that in the first large-recoil bin both the SM and Z model values deviate significantly from the experimental result for this bin, whereas at low-recoil the SM prediction is much closer to the experimental result compared with the Z model. We hope that in the future, when more data comes from the LHCb, the results of measurements will become more concrete to compare with the SM and to see if the deviations can be accommodated with the Z model. Another observable which is clean from the QCD uncertainties and that has been experimentally measured is the lepton forward-backward asymmetry (A FB ), which is an asymmetry with respect to the lepton scattering angle (θ l ); its mathematical expression is given in Eq. (24). Here, it can be noticed that (A FB ) depends on the angular coefficient K 1c and its denominator is same as that of A FB . The angular coefficient K 1c is higher for the SM than the Z model for s < 4 GeV 2 , whereas its behavior reverses when s > 4 GeV 2 . For s < 4 GeV 2 K 1c is dominated by C Z 9 , whereas for s > 4 GeV 2 the terms containing C Z 7 dominate over the one that contains C Z 9 . Therefore, A FB increases with s at the start of the large-recoil and then it starts decreasing and crosses the zero point at around 4 GeV 2 . Our results for A FB in the SM and Z model calculated in the experimentally set-up bin [0.1, 2] GeV 2 are The experimental value of (A FB ) in the corresponding bin is [153] A FB exp = 0.37 +0.37 −0.48 .
In Eq. (34), one can see that the errors are significant, and this is likely to improve with future data from LHCb. However, the current central values are significantly away from the SM and Z values, respectively. In the low-recoil region (s ∈ [15,20] GeV 2 ) the results for this asymmetry are It can be deduced that in this particular bin the average value of A FB in Z is comparable to the lower limit of the experimentally measured value, i.e., −0.14.
The last in the category of the forward-backward asymmetry is the combined forward-backward asymmetry A FB , which mainly depends on the angular coefficient K 2c [cf. Eq. (25)]. Compared to the SM, the value of K 2c is higher in the Z model. At large-recoil our results in the SM and Z model are whereas at low-recoil, the combined hadron-lepton forward-backward asymmetry is It can be seen that at large-recoil the deviations between the SM and Z model are small, and it grows significantly in the low-recoil region. The next observable to be discussed here is the fraction of longitudinal polarization (F L ) of the dilepton system. Due to linear combinations of the same angular coefficients (K 1ss and K 1cc ) in both numerator and denominator of F L , the Z model is not much different from the SM. The values in one of the large-recoil bins, [0.1, 2], for the SM and Z model are In contrast to the large-recoil, at low-recoil the results of F L in the Z model are closer to the experimentally measured results. Therefore, to uncover the imprints of the neutral boson in the longitudinal helicity fraction of the dimuon system in b → (→ pπ)μ + μ − decays, the low-recoil bin might provide fertile ground.
Having compared the SM and Z model with the experimentally measured values of the different observables as discussed above, we will now exploit some other observables that may be of interest in future at the LHCb and different B-factories. In connection with F L , the fraction of transverse polarization F T depends on K 1cc and K 1ss and its value at the large-recoil is where it can be seen that the value in the Z model is very close to the SM result. However, at low-recoil, the results for the Z model significantly differ from those of the SM. Hence, measurement of the fraction of transverse polarization in the low-recoil region will help us to see the possible contribution of the neutral Z boson in these b-baryon decays.
It is well known that in the case of b → J /ψ the different asymmetries have been experimentally measured. Motivated by this fact, let us explore the asymmetries corresponding to the hadronic angle θ and θ l one by one. The asymmetry arising due to the angle θ is defined as α θ and its explicit expression is given in Eq. (18); the corresponding numerical values in low-and large-recoil bins are tabulated in Table 6. In the large-recoil bin s ∈ [1, 6] GeV 2 the value reads Here we can see that α θ differs in the Z model from the SM results significantly in both low-and large-recoil bins. Likewise, the asymmetry α θ that corresponds to angle θ given in Eq. (20) depends on the angular coefficient K 1c , and therefore its behavior is similar to A FB . The results in the large-recoil bin s ∈ [1, 6] GeV 2 for the SM and Z model are It can be noticed that the results in the low-recoil bin are an order of magnitude large than the corresponding values in the large-recoil bin both in the SM and in the Z model. These values are quite large to be measured at the LCHb experiment to test the predictions of the SM. We now discuss α θ , which depends upon the angular coefficients K 1ss and K 1cc . This is not significantly affected by the couplings of the Z model and hence show little deviation from the SM, especially in the large-recoil region. In this region the numerical values are In comparison with the low-s region, here the values of α θ in the SM and Z model differ significantly. Therefore, to establish the possible new physics arising in the Z model, analysis of α θ in the high-s region will serve as a useful probe.
Looking at α φ discloses that it depends upon K 4s . At very low-s, the C 7 term dominates in the SM which results in negative K 4s , but for s > 2 GeV 2 the Wilson coefficient C 9 term dominates, giving positive results. For the Z model C Z 9 gets affected much more than C Z 7 for the entire range of s and hence α φ is expected to change significantly with s in the Z model from the corresponding SM result. The values of α φ in the bin s ∈ [1, 6] GeV 2 for the SM and Z model are Hence, it can be revealed that in the SM the value of α φ is almost the same in the low-and largerecoil bins, which is not the case for the Z model where a large deviation is observed in both bins. Also, in both these bins the results of the Z model are quite large compared to the SM results and experimental observation of α φ will act as a useful observable. The longitudinal (transverse) asymmetry parameter α L (α U ) is the ratio of the helicity combinations K 2ss (K 2cc ) to K 1ss as depicted in Eq. (23). Their values in the large-recoil region are It can be deduced that the value of α L (α U ) in the Z model is half that of the SM model in this bin. With the current luminosity of the LHCb experiment, the values of these observables are in the measurable range. Hence, experimental observation of these observables will give us a chance to test the predictions of the SM and the possibility of exploring the imprints of the Z boson in It is a well-established fact that certain asymmetries, such as P ( ) 5 , that correspond to different foldings in B → K * μ + μ − have shown significant deviations from the SM predictions. This make them a fertile hunting ground to dig for the various beyond-SM scenarios that give possible explanations, and Z is one of them [154]. Motivated by this fact, we have calculated such foldings in the decay under consideration; their expressions in terms of the helicity combinations are given in Eq. (28). Among them the first one is P 1 , which behaves very similarly to F T . The average values of P 1 at large-recoil in the SM and Z model are It can be observed that just like P 1 , for the asymmetry defined by P 3 the average values in the SM and Z model are comparable at large-recoil but differ significantly at low-recoil. We have observed that with 3 fb −1 of data, the LHCb Collaboration has measured A h FB , which is of the same order as P 3 . Therefore, it is expected that in future P 3 will be measured.
Average values of P 5 at large-recoil are This case is similar to P 1 and P 3 as the values in both models are very close at large-recoil and deviations started to appear in the low-recoil region of s. Now we come to P 6 , which depends on the angular coefficient K 4s and hence behaves as α φ . The values of the observable in the SM and Z at large-recoil become From the above results, it can be easily deduced that the value of P 6 in the Z model differs significantly from the SM results both at large-and low-recoil, which is also the case for α φ . In particular, in the low-recoil region, the value of an asymmetry is an order of magnitude larger from that in the large-recoil bin and it is in the experimentally measurable range with the current luminosity of the LHCb experiment. The next observable to be discussed is P 8 , which mainly depends on the angular coefficient K 1c and therefore its behavior is exactly the same as A l FB . Its results in the large-recoil bin are  We can see that there is an order of magnitude difference between the results in the large-and lowrecoil regions. Therefore, the number of events required to see the deviations in the low-recoil region is much smaller compared to the large-recoil region. The last observable in this list is P 9 , which depends on the angular coefficient K 2cc . Its values at large-recoil are P 9 SM = −0.160 +0.001 −0.023 , P 9 Z = −0.076 +0.024 −0.007 , and at low-recoil the results become P 9 SM = −0.308 +0.013 −0.008 , P 9 Z = −0.161 +0.003 −0.004 .
We can see that the value of the SM is almost twice that of the Z model in both regions.
In the case of b → (→ pπ)μ + μ − decay, the LHCb experiment has measured the values of the branching ratio, forward-backward asymmetries, and longitudinal dimuon helicity fraction in small bins of s. Therefore, we have tabulated the values of the abovementioned observables in the large-and low-recoil regions in Table 6, and various small bins in Tables 7 and 8. In addition, to see the profile of these asymmetries we have plotted them graphically in Figs  momentum s. We hope that in future, when more precise results for various asymmetries come from the LHCb, it will give us a chance to compare the profile of various asymmetries calculated here with the experiments for both the SM and the Z model.

Conclusion
In this study we have investigated the full four-folded angular distributions for the semi-leptonic b-baryon decay b → (→ pπ)μ + μ − in the SM and Z model. At the quark level, this decay is mediated by the quark-level transition b → sμ + μ − , which is same for the well-studied meson decay B → K * μ + μ − . For b → transitions, we have used the high-precision form factors calculated in the lattice QCD using 2 + 1 dynamical flavors along with the factorizable non-local matrix elements of the four-quark operators O 1-6 and O g 8 encoded into effective Wilson coefficients C eff 7 (s) and C eff 9 (s). By using them we have numerically calculated the differential branching ratio dB ds , the lepton, hadron, and combined hadron-lepton forward-backward asymmetries (A FB , A FB , A FB ), the various asymmetry parameters (α), the fractions of longitudinal (F L ) and transverse (F T ) polarized dimuons, and different angular asymmetry observables denoted by P in different bins of s.  • In the large-recoil region the results of hadron-side forward-backward asymmetry (A FB ) are significantly away from the experimental observations for both the SM and Z model. However, in the low-recoil region the results of the SM lie close to the experimental observations. • The experimental measurements of lepton-side forward-backward asymmetry (A FB ) in both low-and large-recoil regions have significant errors. However, in the bin s ∈ [15, 20] GeV 2 the lower limit is comparable to the Z model. We hope that in the future, when the statistics of the data are improved, it will help us to find the signatures of the extra neutral Z boson.
• We have also predicted the values of the lepton-hadron combined forward-backward asymmetry (A FB ) both in the SM and the Z model. It has been found that in the low-recoil bin the value of the Z model deviates significantly from the SM result.
• The longitudinal polarization fraction F L of the dimuon system is measured experimentally where the statistics is not good enough in the large-recoil bin as compared to the low-recoil region. In the region s ∈ [1, 6] GeV 2 the central value of the SM is compatible with the central value of the experimental measurements. However, in the bin s ∈ [15,20] GeV 2 , where uncertainties in the form factors are better controlled, the experimental observations favor the results of the Z model. • In line with these asymmetries, we have also calculated the transverse polarization fraction of the dimuon system F T , the asymmetry parameters α and different angular asymmetry observables 22 P i for i = 1, . . . , 9 in the SM and the Z model. We have found significantly large values of some of these observables that can be measured in the future at LHCb and Belle II.
In the end we would like to emphasize that some of the asymmetries calculated here were also reported in the SM and aligned 2HDM in Ref. [155], and our SM results match these results. We hope that in future, the precise measurement of some of the asymmetries reported here in the fourfolded distribution of b → (→ pπ)μ + μ − decay, in fine bins of s, at the LHCb and Belle II will help us to test the SM predictions in b decays with significantly improved statistics. It will also give us a chance to hunt for the indirect signals of NP arising due to the neutral Z boson, especially where the SM is mismatched with the experimental predictions.

Appendix. Definitions
In the rest frame of the decaying b baryon, the momentum of the daughter baryon is defined as where m b is the mass of the b baryon. The lepton polarization vectors in the dilepton rest frame are given as