Angular Analysis of \boldmath $\Lambda_b\to \Lambda_c (\to \Lambda \pi)\ell\bar\nu$

We revisit the decay $\Lambda_b^0\to \Lambda_c^+ \ell^-\bar\nu$ ($\ell = e,\mu,\tau$) with a subsequent two-body decay $\Lambda_c^+ \to \Lambda^0 \pi^+$ in the Standard Model and in generic New Physics models. The decay's joint four-differential angular distribution can be expressed in terms of ten angular observables, assuming negligible polarization of the initial $\Lambda_b$ state. We present compact analytical results for all angular observables, which enables us to discuss their possible New Physics reach. We find that the decay at hand probes more and complementary independent combinations of Wilson coefficients compared to its mesonic counter parts $\bar{B}\to D^{(*)}\ell^-\bar\nu$. Our result for the angular distribution is at variance with some of the results on scalar-vector interference terms in the literature. We provide numerical estimates for all angular observables based on lattice-QCD results for the $\Lambda_b \to \Lambda_c$ form factors and account for a recent measurement of the parity-violating parameter in $\Lambda_c^+\to \Lambda^0\pi^+$ decays by BESIII. A numerical implementation of our results is made publicly available as part of the EOS software.

The remainder of this document is structured as follows. We present and discuss our analytical results for the four-body differential decay rate and the ten angular observables arising from the latter in section 2. In section 3 we provide numerical results for the angular observables based on lattice QCD results for the full set of Λ b → Λ c form factors [19,25] at mass dimension six, and based on our own average of the parity violating parameter α in the secondary decay Λ + c → Λ 0 π + [26]. We conclude in section 4.

Analytical Results
We work within an effective field theory for semileptonic flavour-changing |∆B| = |∆C| = 1 transitions. Its effective Hamiltonian can be expressed as where a basis of operators up to mass dimension six and with only left-handed neutrinos can be chosen as This approach is the standard way to account model-independently for the effects of New Physics (NP). The SM corresponds to the parameter point C V,L = 1 and C j = 0 for all remaining operators. In the SM the parameterṼ cb coincides with the CKM matrix element V cb . Beyond the SM, it merely provides a common normalization for the effective operators. Throughout this work the value ofṼ cb is not relevant, since it cancels in the angular distribution. We note in passing that any interpretation of NP constraints in terms of the Wilson coefficients C i within the SM Effective Field Theory requires a careful interpretation ofṼ cb [27].

Angular Distribution
Following ref. [22], we express the kinematics of the four-body decay distribution in terms of q 2 , the square of the dilepton mass; cos θ , the helicity angle of the charged lepton in the dilepton center-of-mass frame; cos θ Λc , the helicity angle of the Λ 0 baryon in the Λ 0 π + center-of-mass frame; and φ, the azimuthal angles between the to decay planes. For more details, we refer to appendix A. The fourfold differential distribution takes the form The physical ranges of the kinematical variables are The distribution can be decomposed in terms of a set of trigonometric functions K(q 2 , cos θ , cos θ Λc , φ) = K 1ss sin 2 θ + K 1cc cos 2 θ + K 1c cos θ + K 2ss sin 2 θ + K 2cc cos 2 θ + K 2c cos θ cos θ Λc + K 3sc sin θ cos θ + K 3s sin θ sin θ Λc sin φ + K 4sc sin θ cos θ + K 4s sin θ sin θ Λc cos φ , giving rise to ten angular observables K i ≡ K i (q 2 ). Our choice of the normalization in eq. (2.3) imposes an exact relation between two of the angular observables of the first row, 2K 1ss +K 1cc = 1. The structure of eq. (2.4) is imposed by angular momentum conservation [22]. It is generally compatible with the findings of refs. [20,21], that is, we find the same number of independent angular terms based on the results for the angular distribution of refs. [20,21] as in eq. (2.4).
In the presence of all five effective operators of eq. (2.2), we can express each angular observable as a sesquilinear form of ten amplitudes Here λ =⊥, denotes the transversity state; m = t denotes time-like dilepton state; m = 0, 1 denotes the magnitude of the z-component of the dilepton angular momentum in a vector dilepton state; and the T superscript indicates that an amplitude arises only in the presence of tensor operators. For the first row of eq. (2.4) we find: For the second row of eq. (2.4) we find: For the third row of eq. (2.4) we find: For the fourth row of eq. (2.4) we find: Comparing our results to the literature, our findings are summarized as follows: • We find complete agreement with the results of ref. [19], which discusses the noncascade decay in the presence of all dimension-six operators.
• Our findings are at variance with the results of ref. [20], which discusses the cascasde decay for all dimension-six operators except for the tensor operator. We find interference terms of the type V × S and A × P in our angular observables K 3s and K 4s that are absent from the corresponding term C int 4 of ref. [20,arXiv v4]. We believe our results to be correct, since they pass a crucial cross check: the interference between vector and scalar operators can be completely described by a shift of the time-like transversity amplitudes A ⊥t and A t . This shift is consistent with a Ward-like identity for the Λ b → Λ c matrix elements: The term C int 4 in eq. (19) of ref. [20] cannot be expressed through this shift. Additionally, we find a lack of an overall multiplicative factor B(Λ + c → Λ 0 π + ) from the secondary decay. Furthermore, we find agreement only by redefining the angles θ → π − θ , θ Λc → θ s and φ → −χ.
• Comparing our results to the SM results for the fourfold distribution in ref. [21], we find complete agreement when redefining the angles as θ → π − θ , θ Λc → θ s and φ → −χ. We note the absence of any terms corresponding to the angular observables K 3sc and K 3s , which vanish in the SM.
In the definition of seven out of the ten observables, we explicitly factor out the quantity , which is an angular asymmetry parameter arising in Λ + c → Λ 0 π + decays. It emerges as the hadronic matrix element of the parity-violating weak decay of the Λ + c . The present world average of this parameter by the Particle Data Group (PDG) [28] reads: It includes measurements by the CLEO [29] and ARGUS [30] experiments, in addition to the statistically dominating measurements by the CLEO-2 [31] e + e − collider experiment, and by the FOCUS (Fermilab E-831) [32] fixed target experiment, With the recent measurement of α by the BESIII collaboration in e + e − collisions, we compute our own average including also the CLEO-2 and FOCUS results. We obtain: In the following we will use this average in lieu of an updated world average by the PDG.

Transversity Amplitudes
The transversity amplitudes arising in the angular observables K i are further decomposed into Λ b → Λ c helicity form factors, Wilson coefficients, and kinematical factors. Equations of motion for the b and c-quark fields relate the hadronic matrix elements of scalar (pseudoscalar) currents to timelike vector (axial-vector) form factors. In the absence of the tensor operator we find six independent transversity amplitudes:

24)
We note that the timelike leptonic polarization state requires a non-vanishing lepton mass. Hence, the transversity amplitudes A ⊥t and A t always enter observabes with a factor m such that all physical observables are well-defined in the limit m → 0. In the above we introduce a phenomenologically useful basis of effective form factors F λm and F S λ , and abbreviate common linear combinations of the Wilson coefficients: We also introduce an overall normalization factor with the Källén function λ(a, b, c) = a 2 In the presence of the tensor operator we find four additional transversity amplitudes and hence four additional effective form factors F T λm :

Phenomenology
To understand the NP reach of any of the angular observables or their combinations, it is instrumental to understand the sesquilinear combinations of Wilson coefficients entering the observables. To facilitate this understanding we define five real-valued and ten complexvalued quantities σ: (2.34) This constitutes the complete set of sesquilinear combinations of the Wilson coefficients, and corresponds to 25 real-valued degrees of freedom. In terms of these combinations and effective form factors, we find in the limit m → 0 The result above illustrates that the cascade decay for = e, µ is sensitive to 12 out of the 25 real-valued combinations of Wilson coefficients: This sensitivity should be compared to the sensitivity exhibited byB → D ( * ) −ν decays.
Presently, the only prospect for measurement of the Λ 0 b → Λ + c µ −ν decays is the LHCb experiment. Within LHCb, reconstruction of theν momentum from the remaining initial state and decay kinematics is difficult, but possible [33]. We do not expect a measurement of all ten angular observables before a sufficiently large data set is available. In the mean time, we focus on identifying observables that can be more readily measured than the full angular distribution. Analysis of eq. (2.4) suggests only one angular observable whose determination can be achieved without reconstruction of either the dilepton helicity angle or the azimuthal angle. This observable is the hadronic forward-backward asymmetry, Measurement of this observable can be achieved through a counting experiment with respect to the sign of cos θ Λc , normalized to the sum of all events. The normalization through the decay rate is helpful in two ways. First, it reduces the inherent theoretical uncertainties by partial cancellation of the (correlated) form factor uncertainties. Second, it reduces systematic experimental uncertainties in the measurement, including the poorly-known Λ b fragmentation fraction at LHCb. The ability to determine A Λc FB regardless of the reconstruction of the lepton helicity angle will also make possible to probe LFU through the ratio: (2.46) In the above, the = τ and = µ subscripts denote that the forward-backward asymmetry is to be extracted from decays into the specific lepton species . By virtue of taking the LFU ratio as defined in eq. (2.46) the sensitivity to the parity violating parameter α is completely removed.
For further phenomenological applications we provide the full set of angular observables for the full basis of dimension-six operators as part of the EOS software [34] as of version 0.2.5.

SM
NP obs.

Numerical Results
For the numerical illustration we define a NP benchmark point:  This point is inspired by and compatible with the best-fit point labelled "minimum 1" in a recent global analysis of the available data on b → cτν processes [35].
We provide numerical results for the entire set of angular observables integrated over their full q 2 phase space in table 1. Our results include SM predictions for both = µ and = τ , as well as predictions for the NP benchmark point ( = τ only). We illustrate the q 2 dependence of the hadronic forward-backward asymmetry and the NP reach of its LFU ratio in figure 1. For all predictions we estimate the theoretical uncertainties based on two sets of inputs. First, for the parity violating parameter α-following the discussion in section 2.1-we use an average that includes the new BESIII measurement in lieu of the world average. Second, for the hadronic form factors we use the lattice QCD results of the full set of form factors based on the tables provided in refs. [19,36]. We use the EOS software [34] for the computation of all numerical values and plots shown in this work.
Benefiting from the correlated results for the Λ b → Λ c form factors and from cancellations due to the normalization to the decay rate, we find small uncertainties of the order of 11% for A Λc FB that are dominated by the uncertainty in the parameter α. This becomes much more visible in the LFU ratio where α cancels completely. For that observable the relative uncertainty is reduced to ∼ 1%, providing an excellent opportunity for a highprecision probe of New Physics in semileptonic b → c transitions. We therefore encourage a sensitivity study to determine the experimental precision that the LHCb experiment can achieve for the projected size of the upcoming data sets.    [36] and the parity violating parameter α (see text). Due to the absence of any threshold effects with respect to the dilepton mass, the two bands for the hadronic forward-backward asymmetry are virtually indistinguishable. The LFU ratio hence exhibits this large sensitivity to the NP benchmark point.

Summary
We present the first study of the full angular distribution of the cascade decay Λ 0 b → Λ + c (→ Λ 0 π + ) −ν using the complete basis of dimension-six operators, assuming only left-handed neutrinos, in the weak effective field theory for b → c ν transitions. As a cross check, we reproduce the rate of the non-cascade decay. Our findings are at variance with some scalar/vector and pseudoscalar/axialvector interference terms of the four-differential decay rate in the literature. However, our results pass a crucial cross check for amplitudes to time-like polarized dilepton states, which is not fulfilled by the results in the literature.
We express the four-differential rate through a set of ten angular observables, nine of which are independent. The full set of angular observables is shown to be sensitive to more combinations of NP couplings than the decaysB → D ν andB → D * ν taken together. This highlights the usefulness of the cascade decay in constraining potential NP effects in b → c ν transitions for all lepton species = e, µ, τ . For convenience, we provide computer code for the numerical evaluation of all angular observables as part of the EOS software.
We suggest to measure the hadronic forward-backward asymmetry, which is the only angular observable that can be extracted from the angular distribution without either knowledge of the lepton helicity angle or the azimuthal angle between the decay planes. We expect good prospects for its measurement at the LHCb experiment for all three lepton species. We find that the LFU ratio of the hadronic forward-backward asymmetry features very small hadronic uncertainties in the Standard Model and beyond. It is therefore a prime candidate to cross check the present anomalies in the LFU ratios R(D) and R(D * ).