Angular distribution of polarised $\Lambda_b$ baryons decaying to $\Lambda \ell^+\ell^-$

Rare $b \to s\ell^+\ell^-$ flavour-changing-neutral-current processes provide important tests of the Standard Model of particle physics. Angular observables in exclusive $b \to s\ell^+\ell^-$ processes can be particularly powerful as they allow hadronic uncertainties to be controlled. Amongst the exclusive processes that have been studied by experiments, the decay $\Lambda_b\to \Lambda\ell^+\ell^-$ is unique in that the $\Lambda_b$ baryon can be produced polarised. In this paper, we derive an expression for the angular distribution of the $\Lambda_b\to \Lambda\ell^+\ell^-$ decay for the case where the $\Lambda_b$ baryon is produced polarised. This extends the number of angular observables in this decay from 10 to 34. Standard Model expectations for the new observables are provided and the sensitivity of the observables is explored under a variety of new physics models. At low-hadronic recoil, four of the new observables have a new short distance dependence that is absent in the unpolarised case. The remaining observables depend on the same short distance contributions as the unpolarised observables, but with different dependence on hadronic form-factors. These relations provide possibilities for novel tests of the SM that could be carried out with the data that will become available at the LHC or a future $e^+e^-$ collider.


Introduction
Rare b → s + − have been studied extensively by experiments at the B-factories as well as experiments at the Tevatron and Large Hadron Collider (LHC). Amongst the b → s + − processes that have been studied, the decay Λ b → Λµ + µ − is unique for two reasons: it is the only baryonic decay that has been studied; and the Λ baryon decays weakly leading to new hadron-side observables. The angular distribution of Λ b → Λµ + µ − decays has been studied in Refs. [1,2] for the case of unpolarised Λ b baryons. The resulting angular distribution is described by 10 angular observables. The decay rate and lepton side angular distribution has also been studied in the SM and in several extensions of the SM (NP models) in Refs. [3][4][5][6][7][8][9][10][11][12]. If the Λ b is produced polarised, a much larger number of observables are measurable. These observables are explored in this paper. The exploitation of production polarisation in radiative Λ b → Λ ( * ) γ decays has previously been studied in Refs. [13][14][15][16].
In e + e − collisions, Λ b baryons can be produced with large longitudinal polarisations. The longitudinal polarisation of Λ b baryons and b-quarks produced via e + e − → Z 0 (→ bb) decays has been studied by the LEP experiments in Refs. [17][18][19]. The production of Λ b baryons with longitudinal polarisation is forbidden in strong interactions, due to parity conservation. The Λ b can, however, be produced with transverse polarisation in pp collisions. In this paper, we focus on the transverse polarisation of the Λ b baryon. The transverse polarisation of Λ b baryons produced in pp collisions at √ s = 7 and 8 TeV has been studied by the LHCb and CMS experiments in Refs. [20] and [21], respectively. The LHCb experiment measures P Λ b = 0.06 ± 0.07 ± 0.02 at √ s = 7 TeV.
The CMS experiment measures P Λ b = 0.00 ± 0.06 ± 0.02 combining data from √ s = 7 and 8 TeV. In both cases, the production polarisation is determined from the observed angular distribution of Λ b → J/ψ Λ decays. Whilst the measured transverse production polarisation is small, polarisations of O(10%) cannot be excluded. Polarised Λ b baryons can also be obtained from decays of heavier b-baryons, for example in decays of the Σ ( * ) b [22]. The only existing measurements of the angular distribution of the Λ b → Λ + − decay come from the LHCb experiment [23]. Due to the limited size of their dataset, LHCb only studied a subset of the angular distribution that could be accessed from single angle projections on the lepton-and hadron-side. With the much larger data sets that will be available at the LHC experiments after run 2 of the LHC, the experiments will be able to probe the full angular distribution. However, the sheer number of observables involved will most likely require an analysis of the moments of the angular distribution (see for example Ref. [24]) rather than the conventional approach of fitting for the angular observables. This approach is discussed in Sec. 6, where we provide the weighting functions needed to extract the observables.

Angular distribution
The angular distribution of the Λ b → Λ + − decay has been previously studied in Refs. [1,2]. In this paper we extend those studies to include the case where the Λ b baryon is produced with a transverse polarisation. We start by expanding the differential decay rate for the Λ b → Λ + − decay in terms of generalised helicity amplitudes d 6 Γ dq 2 d Ω ∝ λ 1 ,λ 2 ,λp,λ ,λ , J,J ,m,m ,λ Λ ,λ Λ , which depends on five angles, Ω = (θ l , φ l , θ b , φ b , θ), and the dilepton invariant mass squared, q 2 . The angular basis is illustrated in Fig. 1. The helicity basis is defined starting from the normal vector between the direction of the Λ b baryon in the lab-frame and the beam-axis of the experiment (n =p Λ b ×p beam ). This is an appropriate choice when considering transverse production polarisation of the Λ b baryon. Equation 1 involves three sets of helicity amplitudes: H m,J λ Λ ,λ (q 2 ) describing the decay of the Λ b baryon into a Λ baryon with helicity λ Λ and a dilepton pair with helicity λ ; h m,J λ 1 ,λ 2 describing the decay of the dilepton system to leptons with helicities λ 1 and λ 2 ; and h Λ λp,0 describing the decay Λ → pπ to a proton with helicity λ p . The index J refers to the spin of the dilepton system, which can either be zero or one. When J = 0, λ = 0, and when J = 1, λ can take the values −1, 0, +1. The helicity labels λ p , λ Λ , λ 1 and λ 2 can take the values ±1/2. Angular momentum conservation in the Λ b decay requires |λ Λ − λ | = 1/2. The factor (−1) J+J originates from the structure of the Minkowski metric tensor, see Ref. [25] for details. The remaining index, m = V, A, denotes the decay of the dilepton system by either a vector or an axial-vector current. The term ρ λ Λ −λ ,λ Λ −λ is the polarisation density matrix for the transverse polarisation of the Λ b . The matrix is a two-by-two matrix (with Tr(ρ) = 1) given by Finally, the D j m,m (φ, θ, −φ) are Wigner-D functions. An explicit form of the Wigner-D functions is given in Appendix A. Figure 1: The Λ b → Λ + − decay is described by five angles: the angle, θ, between the direction of the Λ baryon and the normal vectorn in the Λ b rest-frame; and two sets of helicity angles, describing the decays of the Λ baryon (θ b , φ b ) and the dilepton system (θ l , φ l ). For transverse production polarisationn is chosen to bep Λ b ×p beam . The helicity angles are then defined with respect to this normal vector through the coordinate systems (x Λ ,ŷ Λ ,ẑ Λ ) and (x ¯ ,ŷ ¯ ,ẑ ¯ ). Theẑ axis points in the direction of the Λ/dilepton system in the Λ b rest-frame. The angle between the two decay planes in the Λ b rest frame is χ = φ l + φ b . The angles θ l , θ b and χ are sufficient to parameterise the angular distribution of the decay in the case of zero production polarisation

Lepton system amplitudes
There are two sets of amplitudes for the dilepon system, with either a vector or an axial-vector current, where γ µ is a Dirac γ-matrix and ε µ is a polarisation vector. These amplitudes evaluate to [1] where m l is the lepton mass and β l is the lepton velocity in the dilepton rest frame The amplitudes with J = 0 vanish in the case that the lepton mass is zero (when β l = 1). Under the Parity transformation

Hadron system amplitudes
On the hadron side, the Λ decay amplitudes can be expressed in terms of the well known Λ asymmetry parameter [26] The hadron side amplitudes are normalised such that

Helicity and transversity amplitudes
After replacing the lepton and hadron-side amplitudes with the expressions given in Secs. 2.2 and 2.1, the angular distribution can be expanded in terms of 10 helicity amplitudes, , α Λ and a set of kinematic factors that come from the lepton-side amplitudes. For the remainder of this paper it is convenient to absorb a common factor of q 2 from the lepton-side amplitudes into these helicity amplitudes, i.e.
By absorbing this factor, the only kinematic dependence outside of H m,J λ Λ ,λ (q 2 ) comes from factors of β l .
The helicity amplitudes can be replaced by a corresponding set of transversity amplitudes for the decay that separate the vector and axial-vector contributions on the hadron-side: the amplitudes A R, L 1 and A R, L 0 depend only on the vector contribution to H λ Λ ,λ (i.e. on Λ|sγ µ b|Λ b ); and the amplitudes A R, L ⊥1 and A R, L ⊥0 depend only on the axial-vector contribution to H λ Λ ,λ (i.e. on Λ|sγ µ γ 5 b|Λ b ). To do this, we start by re-writing the original helicity amplitudes as where the indices L and R refer to left-and right-handed chiralities of the dilepton system, respectively. This is followed by the replacements Here, the subscript t refers to the time-like polarisation vector of the dilepton system.

Observables
Expanding out the sum in Eq. 1, gives 34 different angular terms (K 9 sin θ l cos θ l + K 10 sin θ l ) sin θ b sin (φ b + φ l ) + K 11 sin 2 θ l + K 12 cos 2 θ l + K 13 cos θ l cos θ+ K 14 sin 2 θ l + K 15 cos 2 θ l + K 16 cos θ l cos θ b cos θ+ (K 21 cos θ l sin θ l + K 22 sin θ l ) sin φ l sin θ+ (K 23 cos θ l sin θ l + K 24 sin θ l ) cos φ l sin θ+ (K 25 cos θ l sin θ l + K 26 sin θ l ) sin φ l cos θ b sin θ+ (K 27 cos θ l sin θ l + K 28 sin θ l ) cos φ l cos θ b sin θ+ Integrating this expression over Ω yields the differential decay rate as a function of q 2 , dΓ This can be used to define a set of normalised angular observables 4 Angular terms The first ten angular terms are These terms are accessible even if the Λ b baryon is unpolarised and have been previously studied in Refs. [2,27]. There is a straightforward relationship between our observables and those of Ref. [2], with K 1ss = K 1 , K 1cc = K 2 , K 1c = K 3 , K 2ss = K 4 , K 2cc = K 5 , K 2c = K 6 , K 4sc = K 7 , K 4s = K 8 , K 3sc = K 9 and K 3s = K 10 . The remaining 24 terms are only non-vanishing if P Λ b is non-zero. Terms K 11 through K 16 have a similar dependence to K 1 through K 6 . These are The observables K 13 and K 16 are trivially related to K 6 and K 3 through K 13 = −P Λ b K 6 and K 16 = −P Λ b K 3 and can therefore be used as an experimental consistency check or to determine P Λ b . The observables K 11 , K 12 , K 14 and K 15 have a similar structure to K 1 , K 2 , K 4 and K 5 but, unlike in those observables, the amplitudes with λ = 0 enter with a different relative sign to those with λ = ±1. The observables K 17 through K 34 also involve new combinations of amplitudes that are not accessible if the Λ b baryon is unpolarised. They are The angular terms K 29 and K 31 are zero in the massless lepton limit.

Angular distribution of Λ b → J/ψ Λ
The angular distribution of the Λ b → J/ψ Λ decay is a limiting case of Eq. 1, with a pure vector current in the dilepton system. In this limit, the expression collapses to the one given in Refs. [28,29] with β l ∼ 1. The amplitudes a ± and b ± in Ref. [28,29] are related to the ones in this paper by

Weighting functions
The values of the normalised angular observables can be determined experimentally from an analysis of the moments of the angular distribution, if the weighting functions g i ( Ω) are chosen such that they satisfy In this case, the moments can be extracted from data using Monte Carlo integration. The statistical uncertainty and correlation between the moments can be determined from the single sample covariance or by bootstrapping the measurement (see for example Ref. [30]). The weighting functions for M 1 -M 10 are These weighting functions have been previously derived in Ref. [24]. The weighting functions for the polarisation-dependent terms can be derived in a similar manner, they are g 11 ( Ω) = 3 4 (3 − 5 cos 2 θ l ) cos θ , g 23 ( Ω) = 15 2 cos θ l sin θ l sin θ cos φ l , g 12 ( Ω) = 3 2 (5 cos 2 θ l − 1) cos θ , g 24 ( Ω) = 3 2 sin θ sin θ l cos φ l , g 13 ( Ω) =3 cos θ l cos θ , g 25 ( Ω) = 45 2 cos θ l sin θ l cos θ b sin θ sin φ l , g 14 ( Ω) = 9 4 (3 − 5 cos 2 θ l ) cos θ b cos θ , g 26 ( Ω) = 9 2 sin θ sin θ l cos θ b sin φ l , g 15 ( Ω) = 9 2 (5 cos 2 θ l − 1) cos θ b cos θ , g 27 ( Ω) = 45 2 cos θ l sin θ l cos θ b sin θ cos φ l , g 16 ( Ω) =9 cos θ cos θ l cos θ b , g 28 ( Ω) = 9 2 sin θ sin θ l cos θ b cos φ l , The weighting functions are not unique and a more compact set can be formed by exploiting the fact that the integral of sin θ b over dcos θ b is π/2 e.g. the weighting functions for M 33 and M 34 can be written in a shorter form as More compact expressions can also be found for many of the other observables. Note, the different sets of weighting functions can lead to different experimental precision on the normalised moments. In general, the longer form of the weighting functions provides the best precision.

Standard Model predictions
In order to describe the SM contribution to the decay amplitudes, an effective field theory approach is used. The Hamiltonian for the decay is factorised into local four-fermion operators and Wilson coefficients (see for example Ref. [31]). The Wilson coefficients describe the short-distance contributions from the heavy SM particles. Numerical values for the SM predictions, in the case that P Λ b = 1, are provided in Appendix B in two q 2 ranges: at large hadronic recoil, in the range 1 < q 2 < 6 GeV 2 /c 4 , and at low hadronic recoil, in the range 15 < q 2 < 20 GeV 2 /c 4 . To evaluate SM predictions for the different angular observables we use the EOS flavour tool [32]. At low hadronic recoil, the SM calculations employ an operator product expansion of the fourquark contributions to the matrix element in powers of Λ QCD / q 2 [33]. At large recoil, EOS uses some of the known α s corrections to charm loop processes. However, potentially large contributions from hard spectator scattering [34] and soft gluon emission [35] are neglected. The form-factors for the Λ b → Λ transition are taken from a recent Lattice QCD calculation in Ref. [27]. These form-factors enable the observables to be computed with high-precision. The form-factors at large hadronic recoil have also been calculated in the framework of light-cone-sum-rules, see for example Refs. [36] and [37]. The SM Wilson coefficients are computed in EOS to NNLO in QCD. The Λ b lifetime and CKM matrix elements are taken from the latest experimental values [26]. The quark masses are taken in the MS scheme.
Tables 2 and 3 in Appendix B also provide 68% confidence level intervals for the SM predictions. To evaluate these intervals: the form-factors from Ref. [27] have been varied within their full covariance matrix; the Λ b lifetime, the Λ asymmetry parameter and CKM matrix elements are varied within their experimental precision [26,38]; the scale dependence of Wilson coefficients C i (µ) is explored by varying the scale, µ, in the range m b /2 < µ < 2m b ; and in keeping with Ref. [39] a 3% correction to the amplitudes from hadronic matrix elements is considered (see also Ref. [40]).

Low-hadronic recoil
At low hadronic recoil the observables are precisely predicted in the SM. The uncertainties on the predictions are worse at large recoil, where a large extrapolation in q 2 of the form-factors is needed. Figures 2-9 in Appendix C demonstrate how the observables depend on NP contributions to the Wilson coefficients. In the large-recoil region there is sensitivity to C NP 9 from both the polarised and unpolarised observables. Interestingly, the observables M 23 and M 27 can also distinguish between two of the possibilities that are favoured by global fits to b → s + − processes: where C NP 9 −1 with C NP 10 = 0 and where C NP 9 = −C NP 10 −1 [41][42][43]. In the low-recoil range the sensitivity to C NP 9 is reduced.
In Ref. [2], the authors point out that the observables at low hadronic recoil place constraints on six combinations of Wilson coefficients where C V contains contributions from C 7 and C 9 . The primed coefficients correspond to right-handed currents whose contribution is vanishingly small in the SM. The shortdistance dependence of K 1 -K 34 on ρ ± 1 , ρ ± 3 , ρ 2 and ρ 4 is provided for completeness in Appendix D.
If the Λ b is unpolarised, the decay rate is insensitive to the short-distance contribution Im(ρ 2 ) but provides sensitivity to ρ ± 1 , Re(ρ 2 ), ρ ± 3 , Re(ρ 4 ) and Im(ρ 4 ). The polarised observables also depend on these short-distance contributions but have different formfactor dependencies. This permits a new set of checks of the OPE and the form-factors. The short-distance combination Im(ρ 2 ) can also be determined from M 19 , M 25 , M 30 and M 34 . Furthermore, in K 1 -K 10 the short-distance contributions ρ + 1 and ρ − 1 always appear together as a sum. Using the polarised observables , ρ + 1 and ρ − 1 can be separated, e.g. by using where f V ⊥ and f A ⊥ are helicity form-factors (see for example Ref. [44]). A similar trick can be used to separate ρ + 3 and ρ − 3 using K 24 and K 8 . It is also possible to form new short-distance relationships, in which the form-factors cancel by taking ratios of the K i , The short-distance combinations ρ 2 and ρ 4 can then be determined up-to their overall normalisation, independent of the hadronic form-factors, using Eq. 26 and the relationship from Ref. [2]. Similarly, one can form short-distance relationships that depend only on ρ ± 1 and ρ ± 3 Alternatively, it is possible to form ratios that depend only on the form-factors and not on the short-distance physics. For example, allow the ratios

Photon-polarisation at large hadronic-recoil
At very large hadronic recoil (q 2 1 GeV 2 /c 4 ), the angular distribution of the Λ b → Λµ + µ − decay is sensitive primarily to the Wilson coefficients C 7 and C 7 due to a pole-like enhancement of the amplitudes. The observable K 33 is proportional to Re(C 7 C 7 ) and can therefore provide a null test of the size of C 7 (in the same way as the S 3 observable in the B 0 → K * 0 µ + µ − decay). In this case, however, the observable is suppressed by the size of P Λ b . Table 1 indicates the typical precision on the angular moments that could be achieved at the LHCb experiment. The experimental precision has been estimated using pseudoexperiments corresponding approximately to the expected signal yield in the current and in a future LHCb dataset. Experimental backgrounds and non-uniform angular acceptance have been neglected in this estimate. However, these are expected to have only a small impact on the experiments sensitivity. The sensitivity that can be achieved with the large datasets that will be available at an upgraded LHCb experiment is interesting event for modest values of P Λ b . Table 1: Expected experimental precision on the angular moments of the Λ b → Λµ + µ − decay at the LHCb experiment. The four columns correspond to: the observed yield of 300 Λ b → Λµ + µ − candidates with 15 < q 2 < 20 GeV 2 /c 4 in the LHC run 1 dataset [23]; an expected yield of ∼1000 candidates at the end of run 2 of the LHC; an expected yield of ∼8 000 candidates in 50 fb −1 of integrated luminosity with an upgraded LHCb experiment; and an expected yield of ∼50 000 candidates in 300 fb −1 with the proposed LHCb phase II upgrade.

Expected experimental precision
Obs

Conclusion
In this paper we have derived an expression for the angular distribution of the Λ b → Λµ + µ − in the case of non-zero production polarisation. This extends the number of observables in the decay from 10 to 34. These observables can be determined from moments of the Λ b → Λµ + µ − angular distribution. Explicit expressions have been provided for the observables in terms of the angular moments to enable an experiment to determine the new observables from their dataset. A phenomenological analysis has also been performed to illustrate how these observables might vary in extensions of the Standard Model. The analysis shows that there is interesting new sensitivity that can be gained if the Λ b baryon is produced polarised.

Acknowledgements
We

Appendices A Wigner D-functions
The Wigner D-functions are where the α, β and γ correspond to the Euler rotation angles needed to rotate between the reference frame of the mother particle and the helicity frame of its daughters. The relevant small d-functions are

B Numerical results
Standard Model predictions for the angular observables with P Λ b = 1 are provided in Tables 2 and 3. Predictions are provided in two q 2 ranges: at large hadronic recoil, in the range 1 < q 2 < 6 GeV 2 /c 4 , and at low hadronic recoil, in the range 15 < q 2 < 20 GeV 2 /c 4 . The SM predictions are evaluated using the EOS flavour-tool. For any other choice of P Λ b , predictions for M 11 -M 34 can be achieved by multiplying the values in Tables 2 and  3 by the new value of P Λ b .    Figure 3: Variation of the polarisation dependent angular observables of the Λ b → Λµ + µ − decay from their SM central values in the large-recoil region (1 < q 2 < 6 GeV 2 /c 4 ) with a NP contribution to Re(C 9 ) or Re(C 10 ). The SM point is at (0, 0). To illustrate the size of the effects, P Λ b = 1 is used.  Figure 7: Variation of the polarisation dependent angular observables of the Λ b → Λµ + µ − decay from their SM central values in the large-recoil region (1 < q 2 < 6 GeV 2 /c 4 ) with a NP contribution to Re(C 9 ) or Re(C 9 ). The SM point is at (0, 0). To illustrate the size of the effects, P Λ b = 1 is used.  Figure 9: Variation of the polarisation dependent angular observables of the Λ b → Λµ + µ − decay from their SM central values in the low-recoil region (15 < q 2 < 20 GeV 2 /c 4 ) with a NP contribution to Re(C 9 ) or Re(C 9 ). The SM point is at (0, 0). To illustrate the size of the effects, P Λ b = 1 is used.

C Variation of observables with NP contributions
D Short-distance dependence at low hadronic recoil and The remaining K i 's vanish in the low-recoil and zero lepton mass limits. In Eqs. 33 and 34: f V 0 , f A 0 , f V ⊥ and f A ⊥ are the vector and axial-vector helicity form-factors for the Λ b → Λ transition; m Λ b and m Λ are the masses of the Λ b and Λ baryon, respectively; and s ± = (m Λ b ± m Λ ) 2 − q 2 . The four contributing tensor form-factors have been removed by exploiting Isgur-Wise relationships [44] to relate the tensor form-factors to the vector and axial-vector form-factors.