Angular analysis of Λb→ Λc(→ Λπ) ℓν¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \overline{\nu} $$\end{document}

We revisit the decay Λb0→Λc+ℓ−v¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\varLambda}_b^0\to {\varLambda}_c^{+}{\ell}^{-}\overline{v} $$\end{document} (ℓ = e, μ, τ) with a subsequent two-body decay Λc+→Λ0π+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\varLambda}_c^{+}\to {\varLambda}^0{\pi}^{+} $$\end{document} 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 Λb 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 B¯→D∗ℓ−v¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \overline{B}\to {D}^{\left(\ast \right)}{\ell}^{-}\overline{v} $$\end{document}. Our result for the angular distribution disagrees 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 Λb→ Λc form factors and account for a recent measurement of the parity-violating parameter in Λc+→Λ0π+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\varLambda}_c^{+}\to {\varLambda}^0{\pi}^{+} $$\end{document} decays by BESIII. A numerical implementation of our results is made publicly available as part of the EOS software.


Introduction
Anomalous measurements [1][2][3][4][5][6][7][8][9] in observables that probe Lepton Flavour Universality (LFU) inB → D ( * ) τ −ν decays motivate careful reappraisal of the theoretical inputs [10][11][12][13][14][15][16][17][18][19][20][21][22] entering the present Standard Model (SM) predictions of these decays. At the same time, they also motivate further phenomenological analyses to uncover new observables that complement the LFU probes [23][24][25][26][27], either in terms of independent systematic uncertainties, or in terms of independent constraints on the origin of LFU effects. This work aims to achieve the latter objective by studying the four-differential decay rate of the cascade process 1 Λ 0 b → Λ + c (→ Λ 0 π + ) −ν , with any charged lepton species = e, µ, τ . To achieve this goal, we work within the most general effective theory of local b → c ν operators with left-handed neutrinos and up to mass dimension six, which includes scalar and tensor interactions besides the SM contributions. Our work extends earlier studies of the threebody decay Λ 0 b → Λ + c −ν [31], and of the four-body decay Λ 0 b → Λ + c (→ Λ 0 π + ) −ν [32,33]. Our analysis follows closely previous works on the flavour-changing neutral-current de- [34][35][36], which has a qualitatively similar angular distribution. The LHCb experiment can only reconstruct the decay mode at hand by reconstructing Λ 0 → pπ − decays, which correspond to 64% of the Λ 0 branching fraction [37]. To assess LHCb's ability to measure the four-differential rate, we estimate the expected signal yield of cascade decays by relating it to the signal yield of Λ 0 b → Λ + c (→ pK − π + )µ −ν [29]. Generally, reconstruction of a three-particle final state is less efficient than of a two-particle final JHEP12(2019)082 state. For the purpose of our estimate, we will treat the reconstruction efficiencies for both Λ + c decay modes as equal. (Any more sophisticated estimate would require LHCb-internal information that is unavailable to us.) Using the world average for the Λ + c branching fractions, we obtain a rough estimate of the signal yield as: c → pK − π + ) 0.5 · 10 6 .
(1.1) This corresponds to ten times the signal yield of Λ b → Λ * c (2625)µ −ν decays, for which a sensitivity study established that a full angular analysis is possible [26]. Hence, we expect excellent prospects for a precision angular analysis of the cascade decay at LHCb in case of the muon mode.
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 [31,38] 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 + [39]. 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 product of parametersG FṼcb coincides with the product of the Fermi constant as extracted from leptonic µ decays and the CKM matrix element V cb . Beyond the SM, they merely provide a common normalization for the effective operators. Throughout this work the value ofG F ×Ṽ cb is not relevant for numerical estimates, since it cancels in the angular distribution. However, we emphasize that value becomes relevant when interpreting external inputs for the Wilson coefficients, such as benchmark NP points taken from other works. 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 [40].

Angular distribution
Following ref. [34], 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 angle between the two 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 [34]. It is generally compatible with the findings of refs. [32,33], that is, we find the same number of independent angular terms based on the results for the angular distribution of refs. [32,33] 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 timelike 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: Note that the two observables K 3sc and K 3s vanish when analyzing the combined Λ 0 b and Λ 0 b angular distribution; their extraction therefore requires separation of the two modes. Comparing our results to the literature, our findings are summarized as follows: • We find complete agreement with the results of ref. [31], which discusses the noncascade decay in the presence of all dimension-six operators.
• Our findings disagree with the results of ref. [32], 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. [32,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 timelike 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. [32] 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. [33], 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.
Our results therefore provide for the first time the full angular distribution of the cascade decay in the presence of all dimension-six operators with left-handed neutrinos.
In the definition of seven out of the ten angular observables in eq. (2.4), we explicitly factor out the quantity α ≡ α(Λ + c → Λ 0 π + ), which is an angular asymmetry parameter arising in Λ + c → Λ 0 π + decays. It emerges as the hadronic matrix element of the parityviolating weak decay of the Λ + c . The present world average of this parameter by the Particle Data Group (PDG) [37] reads: (2.16)

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It includes measurements by the CLEO [41] and ARGUS [42] experiments, in addition to the statistically dominating measurements by the CLEO-2 [43] e + e − collider experiment, and by the FOCUS (Fermilab E-831) [44] 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 as defined in ref. [45], Wilson coefficients, and kinematical factors. Equations of motion for the b and c-quark fields relate the hadronic matrix elements of scalar (pseudo-scalar) currents to timelike vector (axial-vector) form factors. In the absence of the tensor operator we find six independent transversity amplitudes:

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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 observables 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 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)

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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. Σ ≡ Σ(m = 0) ∪ {Re σ ± V S,1 , Im σ ± V S,1 , Re σ V S,2 , Re σ SV,2 , Re σ ± V T,1 , Im σ ± V T,1 } . 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 [46]. 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. The LHCb experiment provides projections of the angular resolution inB → D * (→ Dπ) −ν decays [47, section 5.3.1 and figure 5.4], which clearly show that the Dπ helicity angle is the best candidate for reconstruction. Under the assumption that LHCb's performance in the cascade decay reflects the projected performanceB → D * (→ Dπ) −ν , we search for observables that can be determined without reconstruction of either the dilepton helicity angle or the azimuthal angle. Analysis of eq. (2.4) suggests only one angular observable that meets this criterium: 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 [48] as of version 0.2.5.

Numerical results
For the numerical illustration we define three NP benchmark points (BMPs). BMP #1 reads: C   BMP #2 reads: All BMPs are inspired by and compatible with the best-fit points labelled "minimum 1" through "minimum 3" in a recent global analysis of the available data on b → cτν processes [49]. 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 2 of the full set of form factors based on the tables provided in refs. [31,50]. We use the EOS software [48] 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 2 Note that the numerical results for the tensor form factors obtained from lattice QCD [50] seem to be inconsistent with the numerical results for the remaining form factors obtained using the same method in ref. [31] according to a recent analysis of the heavy-quark expansion to order 1/m 2 c [51]. We emphasize that our numerical implementation is agnostic of the parametrization and numerical inputs for the full set of form factors. Our predictions for the benchmark points involving a non-zero C    [50] 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. 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.

JHEP12(2019)082 4 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 lefthanded 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 disagree 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 timelike 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 * ).

B Correlation matrices
We collect here the correlation matrices between the integrated angular observables K i in the SM and in the selected benchmark points. The individual correlation coefficients have

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been obtained from the sample covariance using 2.5 · 10 5 samples. The sequence in which the observables appear is the same as in Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.