Probing effects of new physics in $\Lambda^0_{b}\to\Lambda^+_{c}\mu^{-}\bar{\nu}_{\mu}$ decays

We present, for the first time, the six-fold differential decay density expression for $\Lambda^0_b\to\Lambda^+_{c} l^- \bar{\nu}_{l}$, taking into account the polarisation of the $\Lambda^0_b$ baryon and a complete basis of new physics operators. Using the expected yield in the current dataset collected at the LHCb experiment, we present sensitivity studies to determine the experimental precision on the Wilson coefficients of the new physics operators with $\Lambda^0_{b}\to\Lambda^+_{c}\mu^{-}\bar{\nu}_{\mu}$ decays in two scenarios. In the first case, unpolarised $\Lambda^0_{b}\to\Lambda^+_{c}\mu^{-}\bar{\nu}_{\mu}$ decays with $\Lambda^+_c\to p K^+ \pi^-$ are considered, whereas polarised $\Lambda^0_{b}\to\Lambda^+_{c}\mu^{-}\bar{\nu}_{\mu}$ decays with $\Lambda^+_c \to p K^0_S$ are studied in the second. For the latter scenario, the experimental precision that can be achieved on the determination of $\Lambda^0_b$ polarisation and $\Lambda^+_c$ weak decay asymmetry parameter is also presented.


Introduction
Semileptonic b-hadron decays are highly promising avenues to search for New Physics (NP) due to their large signal yields and controllable theoretical uncertainties. The hint of lepton flavour universality violation in B → D ( * ) ν decays [1][2][3][4][5][6][7][8] 1 has led to the proposal of various NP scenarios that could affect decays involving b → c ν transitions [9][10][11]. In addition, numerous studies involving B → D ( * ) τ ν decays have shown the effects of these NP contributions on the corresponding angular distributions [12][13][14][15][16][17][18]. Global fits to b → cτ ν transitions have also been conducted to determine the Wilson coefficients of the NP operators [19][20][21][22]. A recent global fit to b → cµν and b → ceν transitions [22] has proven that a good sensitivity to various different NP operators can be achieved through studies of b-meson decays involving lighter leptons in the final state.
In this study, we present for the first time an expression for the six-fold differential decay density of Λ 0 b → Λ + c − ν transitions, including the effects of Λ 0 b polarisation and all the relevant NP contributions which are encapsulated by Wilson coefficients. These decays can currently be studied only at the LHCb experiment and present several experimental challenges. On one side, in the Λ 0 b → Λ + c τ − ν τ case multiple missing neutrinos in the final state drastically dilute the resolution on the kinematic variables in addition to contributions from irreducible backgrounds (such as feed-down Λ 0 b → Λ + c * − ν and Λ 0 b → Λ + c X c , where X c is a charmed meson). Furthermore, Λ 0 b → Λ + c e − ν e decays are challenging to reconstruct at LHCb due to the poor electron reconstruction efficiency [35]. Therefore, we conduct sensitivity studies to determine the experimental precision on the Wilson coefficients with Λ 0 b → Λ + c µ − ν µ channel in two different scenarios, using the expected yield from Run I and II data collected at the LHCb experiment.
In the first scenario, the Λ 0 b is unpolarised and the Λ + c decay kinematics are integrated over and is assumed to be reconstructed using the Λ + c →pK − π + channel. This is an experimentally favourable signature due to the presence of three charged particles in the final state and the large branching fraction, which ensure a cleaner reconstruction with small background contributions at LHCb.
In the second scenario we allow for a non-zero Λ 0 b polarisation, with a subsequent Λ + c → pK 0 S decay accounting for the involved kinematics of the process. The interest in this case lies in the achievable sensitivity not only to the polarisation of Λ 0 b (P Λ 0 b ), but also to the Λ + c decay asymmetry parameter (α Λ + c ). So far, at the LHC no hint for a non-zero value of P Λ 0 b has been observed [36][37][38], whereas the Λ + c decay asymmetry has been very recently measured at the BESIII experiment, but with a large uncertainty [39]. Therefore, we present an estimate on the experimental precision which could be achieved on P Λ 0 b and α Λ + c at the LHCb, relying on the large signal yields of semileptonic decays. The paper is organised in the following way. In section 2, the effective Lagrangian expression for b → clν transitions is reported, including all the relevant NP operators. The decay amplitude of Λ 0 b →Λ + c (→p K 0 S )( − ν ) is presented in section 3. Section 4 contains the expression for the six-fold differential decay density for the polarised Λ 0 b → (pK 0 S )( − ν ) decays in the context of NP. In section 5, the results of the sensitivity studies undertaken on the Wilson coefficients in the two working assumptions are reported. The conclusions of this work are given in section 6.

Effective Lagrangian for b → clν
The most generic effective Lagrangian of the four-fermion interaction, extending the Standard Model (SM) within the NP scenario and governing semileptonic b → clν transitions, is given by: where the four-fermion operators O C i are defined as: Here the factors C V L,R , C S L,R , C T L denote the Wilson coefficients of their respective operators, that take a value of zero within the SM. The symbol represents the lepton flavour involved in the interaction. It is noted here that the right-handed tensor operator O T R =q L σ µν b RτR σ µν ν L vanishes [13]. As in the case of SM, we assume the absence of right-handed ν and left-handed ν 2 . Since the flavour of the neutrino is not observed, neutrino mixing effects are also not considered.

Decay amplitude
The transition matrix elements for Λ 0 b → (pK 0 S )( − ν ) can be expressed as the product of amplitudes of unstable particles involved in the decay, i.e. Λ 0 b , Λ + c , W * − : In Eq.(2), the term q 2 denotes the squared transferred four momentum, defined as q 2 = (P Λ b − P Λc ) 2 = (P − + P ν ) 2 ; m pK 0 S is the combined mass of the system pK 0 S ; G F represents the Fermi constant; the index λ refers to the helicity of the particle involved in the transition. The propagator term for intermediate Λ + c particle is parametrised as the relativistic Breit-Wigner and is denoted by BW . Using the narrow-width approximation for BW Λc , the m 2 pK 0 S dependence is integrated out in the expression of the differential decay density. In Eq.(2), we have also summed over i, denoting the operator O i (Eq.(1)) involved in the transition, and the helicities of the intermediate unstable particles. In the following the helicity index λ K 0 S is dropped as it is null and λ ν is fixed to 1/2 for Λ 0 b decays (or λ ν = −1/2 for Λ 0 b decays). Since the weak decay of Λ + c → pK 0 S involves the charged current transition of c →sud, we express the total decay density in terms of the weak decay asymmetry parameter, α Λ + c [39]. The transition amplitude shown in Eq.(2) can be expanded in terms of the helicity amplitudes of the involved decay processes as follows: Here H, L and G represent the helicity amplitudes of Λ 0 b , W * − and Λ + c decays, respectively, retaining dependence on all the angular degrees of freedom. In this study the two lowest allowed spins for W * − , i.e. (J W = 0, λ W = 0) and (J W = 1, λ W = −1, 0, 1), are being considered. To distinguish the former helicity configuration from the latter, we denote λ W = t when J W = 0. The term η λ denotes a metric factor that originates when we replace the metric tensor with the polarisation vectors of the virtual W * − i.e. g µν = λ η λ †µ (λ) ν (λ) where η 0,±1 = −η t = −1.
The hadronic amplitudes expressed above in Eq.(3) are related to those involving vector (V), axial-vector (A), scalar (S), pseudo-scalar (PS), tensor (T) and pseudo-tensor (PT) currents through the following relations: In Appendix A the expressions for these amplitudes are provided in the rest frame of Λ b , where Λ c momentum has spherical coordinates (p = 0). In further discussions, we drop superscript '[Λ b ]' for brevity, specifying that the quantity has been defined in the Λ 0 b rest frame. The polar angle and momentum of Λ + c in this frame are depicted in Figure 1. It is worth noting that these hadronic helicity amplitudes are functions of q 2 and θ Λc . The leptonic amplitudes, shown in Eq.(3), are defined as: Hereū l , νν l and µ represents the particle helicity spinor of the lepton, the anti-particle helicity spinor for neutrino and the polarisation vector of W , respectively. In Appendix B, we present the expressions for the leptonic amplitudes defined in the helicity frame of W * − 3 , (x ,ŷ ,ẑ ), where the − momentum has spherical coordinates (p [W ] , θ [W ] , φ [W ] ). As before, the superscript '[W ]' is dropped in further discussions for brevity. The angles and momentum of the − , defined in the W * − helicity frame, are shown in Figure 2. The leptonic helicity amplitudes expressed above are functions of q 2 , θ l and φ l . Figure 2: Pictorial representation of W * − helicity frame (x l ,ŷ l ,ẑ l ) and Λ + c helicity frame (x p ,ŷ p ,ẑ p ). (Top) The unit vectorsp Λc denote the direction of propagation of W * − and Λ + c in the Λ 0 b rest frame, respectively. (Bottom) The unit vectorp The amplitudes corresponding to the weak decay Λ + c → pK 0 S are given as: where g λp denotes the rotationally invariant amplitude of the Λ c decay, defined in the rest frame of Λ c with the proton moving in positive direction of the z-axis. The Wigner-D elements, D 1/2 * λ Λc ,λp , specify the rotation of the helicity states into the helicity frame of Λ c . In this frame, the proton momentum has spherical coordinates (p in further discussions is omitted for brevity. As noted above, after incoherent sum over λ p , we can express the decay density in terms of the weak decay asymmetry parameter α Λ + c , through the substitution: The expressions for the amplitude shown in Eq. (13), when expanding out the Wigner-D elements, are given in Appendix C.
The transition amplitudes shown in Eq.(2) apply to the Λ 0 b → (pK 0 S )( − ν ) decay channel. To obtain the amplitude of the CP conjugate process, we complex conjugate all the Wilson co-coefficients ({C}) that carry the weak phase, change the sign of all the azimuthal angles involved ({φ}) and change the set of final state particles helicities to those of their CP conjugate partner {λ} [12], i.e.:

Decay density
The full six-fold normalised angular differential decay density considering the Λ 0 b polarisation effects is given by: and More details are provided in Appendix D. In Eq. 17, B Λc denotes the branching fraction of Λ + c → pK 0 S decay; m Λ b and m Λc are the masses of Λ 0 b and Λ + c ; p Λc , p p and p l denote the three-momentum magnitudes of Λ + c , proton and lepton respectively, all defined in the rest frame of their parent particle and expressed in terms of Lorentz invariant quantities in Eq.(31). In Eq. 18, P Λ b refers to the polarisation of Λ 0 b ; the terms K i depend on all the phase space observables with the exception of θ Λc , which are given in Appendix E.
The expression of the decay density intrinsically depends on the assumptions made on Λ 0 b polarisation and Λ + c decay kinematics. When Λ 0 b is produced unpolarised and the two-body decay Λ + c → pK 0 S is considered, the decay density is independent of cos θ Λc and φ p (Eq.(36)), and the variable φ l can be expressed in terms of χ, defined as the angle between pK 0 S and − ν decay planes. Conversely, if Λ 0 b is produced polarised and the degrees of freedom related to Λ + c decay are integrated out, the decay density exhibits dependence on q 2 , cos θ Λc , cos θ l and φ l (Eq.(37)), where φ l can be expressed in terms of the angle between the Λ 0 b polarisation plane (i.e. the one containingn andp Λc ) and − ν decay plane. In the case that the Λ 0 b baryon is unpolarised and degrees of freedom related to the Λ + c decay are integrated out, the decay density depends only on q 2 and cos θ l .

Experimental sensitivity
In this section the sensitivity that can be achieved on the Wilson coefficients, by studying the differential decay density in different scenarios, is presented.
In the first case, Λ 0 b baryons are considered to be produced unpolarised. The angular distribution of the Λ + c decay is integrated over and the q 2 and cos θ l distributions are measured. The expression for the employed fit model is shown in Eq.(39) of Appendix F. The expected signal yield is determined from Ref. [26], where the Λ + c → pK + π − decay mode is adopted. When extrapolated to the current LHCb dataset of 9 fb −1 , this gives 7.5M expected signal candidates The abundant signal yield suggests that the Λ + c → pK + π − decay mode is the most sensitive to NP operators, as favoured by the experimental signature.
For the second scenario the Λ + c → pK 0 S decay is reconstructed and a non-zero polarisation of Λ 0 b is foreseen. In this case, the angles cos θ p and φ p are additionally included in the differential decay density and thus the four-dimensional distribution is fitted. The expression for the complete fit model is shown in Eq.(40) of Appendix F. It is worth noting that the low reconstruction efficiency of long-lived K 0 S mesons and the smaller Λ + c branching fraction translates into a signal yield which is reduced by approximately a factor 20 with respect to the Λ + c → pK + π − three-body decay case. The background level for Λ 0 b → Λ + c µ − ν µ decays is small, as reported in Ref. [26]. Furthermore, the acceptance is not expected to be a strong function of the decay variables as the muon trigger selection at LHCb is efficient [26]. As a result, these effects are neglected in the following studies. One aspect that cannot be ignored, however, is the dilution of resolution on q 2 and cos θ l variables due the unreconstructed neutrino. To take this into account, we generate the Λ 0 b → Λ + c µ − ν µ signal sample using Pythia [42,43], requiring the signal events to be approximately within the LHCb acceptance (i.e. 2 < η < 5). The Λ 0 b vertex position is then smeared according a spatial resolution inspired in Ref. [44]. Data migration between kinematic bins in q 2 and cos θ l is included in the fit by convoluting the decay density with a migration matrix, which is described in more detail in Appendix. G.
In order to assess the sensitivity to the Wilson coefficients, pseudo-experiments have been generated with the expected signal yield and experimental resolution. A binned maximum likelihood fit, with 20 bins employed in each dimension, is then performed. A Gaussian constraint is applied to the Λ 0 b →Λ + c hadronic form factors, using lattice QCD results from [24,30]. The 95% CL intervals obtained from this study are compared with those inferred from B → D ( * ) ν decays [22]. No effects of CP violation have been considered, therefore the Wilson coefficients in this study are assumed to be real.
At first, only one Wilson coefficient at a time is varied; the results are shown in Fig. 3. As the production fraction of Λ 0 b decays has a relatively large uncertainty [45], the normalised differential distribution is fitted which has no sensitivity to the Wilson coefficient C V L and the CKM matrix element V cb . The sensitivity to other NP operators is expected to be significantly better than that of the current constraints, mainly due to the huge signal yields expected at LHCb.
The interplay between different Wilson coefficients is explored in Fig. 4 and compared to Ref. [22]. Large non-Gaussian correlations are observed, affecting both the 2D and 4D differential widths. When all the Wilson coefficients are fitted at once, then the observed correlations between the tensor and scalar currents become even larger, implying that contributions from these operators are difficult to disentangle from the explored distributions alone.
The four-dimensional decay density distribution involving Λ + c → pK 0 S is sensitive to both the P Λ 0 b and α Λ + c . A comparison of the results of this study with existing measurements from BES III [39] and LHCb [36] is illustrated in Fig. 5. The expected sensitivity to α Λ + c is currently a world-leading value, whereas the sensitivity to P Λ 0 b is slightly less precise than previous measurements [36][37][38], but could be improved in the future with a full angular analysis of Λ 0 b →Λ + c (→pK 0 S )µ − ν µ decays. A summary of the sensitivity for the various cases can be found in Table 1. and α Λ + c as obtained from a four-dimensional fit to the Λ 0 b → Λ + c (→ pK 0 S )µν differential distribution. Free parameters pK 0 S case pK − π + case

Conclusions
In this study a general expression for the effective Lagrangian governing b → clν l transitions has been considered, including NP contributions through a complete basis of dimension-six operators and assuming only left-handed neutrinos. Using this formalism, we presented for the first time an expression for the six-fold differential decay density for polarised Λ 0 b → Λ + c − ν decays, with subsequent Λ + c →pK 0 S decay. In addition, we carried out sensitivity studies to determine the experimental precision on the Wilson coefficients which can be achieved through the analysis of Λ 0 b → Λ + c µ − ν µ decays at the LHCb experiment. When considering the integrated Run I and Run II data samples collected at LHCb. The first case considered was the decay channel Λ 0 b →Λ + c (→pK − π + )µ − ν µ , where the 2D distribution in q 2 and cos θ l was studied. The second explored scenario focused on Λ 0 b →Λ + c (→pK 0 S )µ − ν µ decays, including polarisation effects on the production of Λ 0 b . At this purpose, the 4D distributions in q 2 , cos θ l , cos θ p and φ p variables were inspected. Since a missing unreconstructed neutrino in the final state spoils the experimental resolution on q 2 and cos θ l , in both of the mentioned cases the resolution effects were folded into the fit model through a migration matrix.
The results of the sensitivity studies show that the best precision on the Wilson coefficients can be achieved by probing the q 2 and cos θ l distributions of Λ 0 b →Λ + c (→pK − π + )µ − ν µ decays collected at LHCb, leading to a good sensitivity to C V R , C T L , C S R and C S L . No sensitivity is expected to overall global factors, such as V cb and C V L , as the present study is performed on the normalised differential decay distributions.
Although no enhanced sensitivity to the Wilson coefficients could be achieved through the investigation of 4D kinematic distributions of Λ 0 b →Λ + c (→pK 0 S )µ − ν µ decay channel, they do however provide a prospect of measuring α Λ + c and P Λ 0 b . The parameter α Λ + c is particularly promising, with a precision which is two orders of magnitude smaller than that measured by the BESIII experiment. The precision on P Λ 0 b could also be improved by performing a full six-dimensional angular analysis. That would require large signal yields expected at the LHCb upgrades and a different treatment of the resolution.
In this paper we have assumed CP conservation and considered the Wilson coefficients to be real variables. To further distinguish NP models, one could easily extend the present study allowing Wilson coefficients to be complex to probe CP violation in The results of the sensitivity studies have been compared to the model-independent constraints obtained in global fits to B → D ( * ) − ν semileptonic decays, presented in Ref. [22], and have shown a significant improvement in the achievable experimental precision. As a consequence, it can be concluded that studying Λ 0 b → Λ + c µ − ν µ decays at the LHCb experiment will not only lead to a more precise measurement on Λ + c decay asymmetry parameter, but also allows to place stringent world leading constraints on the Wilson coefficients of the corresponding NP operators.
In this section, we give expressions of the hadronic helicity amplitudes for Λ 0 b →Λ + c W * − decay. The hadronic matrix elements shown in Eqs.(4-9) are expressed as: Here ' †µ ' denotes the polarisation vector of W * − . The definitions of the matrix elements in terms of the form factors are given in Ref. [30].
When considering the transverse polarisation of Λ b , the common choice of a Λ b rest frame is the one where the polarisation axis (n) is perpendicular to the production plane, i.e.:ẑ whereˆrefers to the unit vector, the superscript 'lab' indicates the lab frame andp Λc is the unit vector of Λ c momentum in Λ b rest frame (Figure 1). In such a frame, the choice of azimuthal angle forp Λc is arbitrary and is set to zero. Therefore, Λ c momentum is oriented in the direction (θ Λc , φ Λc = 0) with magnitude p Λc and the virtual W * − moves in the opposite direction, which is (π − θ Λc , π) with p W * − = p Λc .
The generic expression for the polarisation vector, helicity spinors and the representation of gamma matrices used in this work follow Ref. [46,47]. We show below the expression for helicity spinors of

The vector (H
λ Λc ,λ W ) and axial vector (H λ Λc ,λ W ) amplitudes can then be expressed as: where the q 2 dependent quantities are expressed in terms of the form factors F 0,+,⊥ and G 0,+,⊥ as: Similarly, the scalar (H The tensor (H ) and pseudo-tensor (H ) amplitudes can be expressed as follows: In the above expressions, the q 2 dependent quantities are given in terms of the tensor form factors h + , h ⊥ ,h + andh ⊥ as: It is worth noting that when θ Λc = 0 we recover the expressions as quoted in Ref. [27,28,30].
The helicity amplitudes presented in this section are expressed in terms of the form factors F 0,+,⊥ , G 0,+,⊥ , h +,⊥ andh +,⊥ , which are defined in such a way that they correspond to time-like (scalar), longitudinal and transverse polarisation with respect to the momentum-transfer q µ . An alternate parameterisation of form factors (denoted by f 1,2,3 , g 1,2,3 , f T 1,2 and g T 1,2 ) that are based on the large and small projections of massive fermion spinors, can often be found in the literature [27,28]. The relation between these two form factor parameterisation is given in Appendix B of Ref. [48].

B Leptonic amplitudes of W * − → − ν decay
In this section, we give the expressions of the leptonic helicity amplitudes for W * − → − ν decay, shown in Eqs. (10)(11)(12). The choice of the lepton azimuthal angle in the W * rest frame is now fixed byn, i.e. the Λ b polarisation axis (Eq. 19): Therefore, the lepton − momentum is oriented in the direction (θ l , φ l ) with magnitude p l , whereas the neutrino ν moves in the opposite direction, which is (π − θ l , π + φ l ) with p ν = p l . This frame of reference is depicted in Figure 2.
We show below the expressions for helicity spinors of − (u l ) with mass m l , ν (uν l ) and polarisation vector of W * − ( µ ) in the above defined frame: The vector and axial-vector amplitudes (L SM ;λ W λ l ,λ ν = 1 2 ) are then given by: = −e −iφ l cos θ l a l , The scalar and pseudo-scalar leptonic helicity amplitudes (L S L λ l ) become: The tensor amplitudes (L T L ;λ W ,λ W λ l ) are given by: The rest of the tensor amplitudes can be obtained using the relations: The q 2 dependent terms that appear in the above equations are given by: We note that the relations obtained here match that of Ref. [30] when φ l = 0. As done in Ref. [30], in the definition of the polarisation vector the Euler angle has been set to γ = −φ l , contrary to γ = 0 as done in Ref. [14]. As a result, the expressions presented in Ref. [14], differ from ours and those presented in Ref. [30] by an unimportant overall phase factor, e −π .
C Hadronic amplitudes of Λ + c → pK 0 S decay In this section, we expand out the Wigner-D elements and provide expressions of the hadronic amplitudes for Λ + c → pK 0 S decay, which is shown in Eqs. (13). These amplitudes are defined in the Λ + c rest frame where the choice of azimuthal angle for the proton is fixed byn defined in Eq. (19): Therefore, in this frame, p momentum is oriented in the direction (θ p , φ p ) with magnitude p p and K 0 S moves in the opposite direction i.e. (π − θ p , π + φ p ) with p K 0 S = p p . A transformation from this frame to the frame defined in Eq.(23) can be achieved through rotation by angle π about y p -axis. This frame of reference is depicted in Figure 2.
The hadronic amplitudes (G

D Phase space
The differential decay rate can be written as: where T denotes the complex transition amplitude, 'P A ' is the four-momentum of the particle 'A' in Λ b rest frame, m Λ b is mass of Λ 0 b and dΦ 4 is the four-body phase space element that can be written as the product of two-body phase space elements as follows: dΦ 2 (P Λ b ; P Λc , q) dΦ 2 (P Λc ;P p ,P K 0 S ) dΦ 2 (q;P l ,P ν ) .
Here m 2 pK 0 S = (P p + P K 0 S ) 2 , q 2 = (P l + P ν ) 2 andˆdenotes that the four-momenta are now defined in the rest frame of the parent particle. The two-body phase space is given by: where p i denotes the magnitude of three-momentum of particle i in the ij rest frame. The full four-body phase space element then becomes: where p Λc , p p , p l with their corresponding angles are defined in the Λ 0 b , Λ + c and W * − rest frames, respectively. These momenta can be expressed as: where λ(a, b, c) = a 2 + b 2 + c 2 − 2(ab + ac + bc).
In the main text, it is highlighted that the choice of φ Λc is arbitrary and has been set to zero, removing its dependence from |T | 2 . It can also be seen in Eq.(3) that, for a given helicity of initial and final state, |T | 2 would have dependence on m 2 pK 0 S through the propagation terms BW Λc BW † Λc . Since the total width of Λ + c is far below its mass (Γ Λc << m Λc ), we can use here the narrow width approximation to give: We can also factor out the term (|g + 1 2 | 2 +|g − 1 2 | 2 ) from |T | 2 and normalise Λ + c decay density using the relation: Substituting all the above three relations in Eq.(32) and integrating over m 2 pK 0 S and φ Λc , the differential decay density becomes: where dΩ = dq 2 d cos θ Λc d cos θ p dφ p d cos θ l dφ l and B Λc denotes the branching fraction of Λ + c → pK 0 S decay. When the Λ 0 b is unpolarised (P Λ b = 0), it is clear from Eq. (18), that the dependence of |T | 2 on θ Λc is inherently removed. Also the choice of the azimuthal angle φ p that was previously fixed by the definition of the Λ 0 b polarisation axis (n), now becomes arbitrary. We therefore set the Λ + c helicity frame (x p , y p , z p ) in such a way that angle φ p = 0, i.e. the proton momentum always lies in the x p -z p plane and, as before,x l = −x p . In this case, the differential decay density becomes: where dΩ = dq 2 d cos θ p d cos θ l dφ l . Note that with the choice of φ p = 0, one can also express angle φ l ( Figure 2) in terms of the relative angle between the (pK 0 S ) and ( − ν ) decay planes, χ. Either of the two relations that are often employed in the literature can be used, i.e. either φ l = π + mod(2π) − χ [49] or φ l = χ [14] (However, when chosen the definition should be adopted consistently throughout the analysis).
When integrating out the Λ + c dynamics, the three-body phase space element is considered. This case is very similar to setting Λ + c decay asymmetry to zero (α Λc = 0), where the dependency of |T | 2 on cos θ p and φ p is inherently removed. The differential decay density then takes the form: where dΩ = dq 2 d cos θ Λc d cos θ l dφ l . The choice of angle φ l in the above case is now fixed through the choice of W * − helicity frame, defined with respect to the Λ 0 b polarisation plane (i.e.ŷ l =n ×p Λc ). Additionally, if Λ 0 b is unpolarised, the dependence of |T | 2 α Λc =0 on cos θ Λc and φ l is inherently removed, leaving the dependence of decay density on only q 2 and cos θ l (with an additional factor 4π from integration over the element d cos θ Λc dφ l ).

8
, where the q 2 dependent terms I i are defined Eqs. (38). It can be seen from the above equation that the decay density is independent of P Λ b and α Λc . The shape of this decay density is similar to the case when Λ + c is considered stable and Λ 0 b is unpolarised. For the second case studied in this paper, we present below the decay density as a function of q 2 , cos θ l , cos θ p and φ p , after integrating Eq.(15) over all other phase space variables d 4 Γ dω = N Γ 1 8 π − α Λc 1 − cos θ 2 p e −iφp P Λ b − 2πe 2iφp (cos θ 2 l − 1)I 7 I * 8 + (cos θ l I 6 + I 10 )(cos θ l I * 5 + I * for cos(θ l ) the migration effects are prominent at lower cos(θ l ) reco and cos(θ l ) true values. In Figure 7, the effects of migration on SM-like Monte Carlo sample are illustrated.