Testing lepton flavour universality in semileptonic Λb → Λc* decays

Lepton Flavour Universality tests with semileptonic Λb → Λc* decays are important to corroborate the present anomalies in the similar ratios RD∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {R_D}_{{}^{\left(\ast \right)}} $$\end{document}, and can provide complementary constraints on possible origins of these anomalies beyond the Standard Model. In this paper we provide — for the first time — all the necessary theoretical ingredients to perform and interpret measurements of RΛc*\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {R}_{\Lambda_c^{*}} $$\end{document} at the LHCb experiment. For this, we revisit the heavy-quark expansion of the relevant hadronic matrix elements, and provide their expressions to order αs and 1/m accuracy. Moreover, we study the sensitivity to the form factor parameters given the projected size and purity of upcoming and future LHCb datasets of Λb→Λc*μν¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\Lambda}_b\to {\Lambda}_c^{*}\mu \overline{\nu} $$\end{document} decays. We demonstrate explicitly the need to perform a simultaneous fit to both Λc* final states. Finally, we provide projections for the uncertainty of RΛc*\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {R}_{\Lambda_c^{*}} $$\end{document} based on the form factors analysis from semimuonic decays and theoretical relations based on the heavy-quark expansion.


Introduction
Tests of lepton flavour universality in semileptonic decays of b quarks are presently in focus of both experimental as well as theoretical particle physics. This interest has been sparked by deviations between Standard Model (SM) estimates and measurements in both -1 -

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charged-current [1][2][3][4][5][6][7][8][9] and neutral-current [10,11] semileptonic b quark decays. Deviations in both sectors are at the level of three to four standard deviations, which is at present intriguing but does not yet provide conclusive evidence for particles beyond the SM. It is therefore important to extend the current tests to new decay modes to provide measurements with orthogonal experimental and theoretical systematic uncertainties as well as a complementary sensitivity to new physics.
In this paper we will concentrate on Lepton Flavour Universality (LFU) in b → cτν versus b → cµν decays, in particular for Λ 0 b decays. At the LHC, Λ 0 b baryons are copiously produced, at approximately half the rate of B 0 mesons [12,13]. The decay involving the ground state charmed baryon, Λ 0 b → Λ + c −ν has been studied in lattice QCD in ref. [14] and precise predictions for the LFU ratio R Λc are provided in the SM and beyond [14,15]. In addition, the LHCb collaboration has recently measured the slope of the leading order Isgur-Wise (IW) function of the decay Λ 0 b → Λ + c µ −ν [16]. While studying backgrounds to this decay, large samples of Λ c (2595) + and Λ c (2625) + candidates were reconstructed as background, which demonstrates the potential of precise LFU tests in these decays. Therefore, we propose to investigate the LFU ratios where Λ * + c denotes either the Λ c (2595) + (with J P = 1/2 − ) or the Λ c (2625) + (with J P = 3/2 − ) charmed baryon.
The challenge in exploiting these modes for LFU tests is controlling uncertainties related to the hadronic matrix elements, which are genuinely non-perturbative objects. As a consequence of both baryons forming a doublet under Heavy Quark Spin Symmetry (HQSS), the hadronic matrix elements for the Λ b → Λ * c transitions can be expressedin the infinite mass limit -through a single IW function ζ [17] at leading power in 1/m. The power suppressed contributions at the 1/m level -where m = m b , m c -have been previously calculated in [18].
The purpose of this paper is to provide for the first time all the necessary ingredients to carry out a LFU study of these decays. In section 2, we first revisit the definition of the hadronic form factors, and provide a helicity decomposition that is convenient for the description of the decay observables. Subsequently, we provide formulae for these hadronic form factors in the Heavy Quark Expansion (HQE) up to order α s and 1/m, beyond what has been done in the literature so far. Continuing in section 3, we model the kinematic dependence of the leading and subleading IW functions, and then provide a set of benchmark points based on inputs from non-perturbative approaches. Afterwards, we calculate the differential decay width, including the finite lepton-mass contributions that are necessary for testing LFU. The following section 4 shows the impact of using LHCb data for constraining the relevant form factor parameters, and control the theory uncertainties for the prediction of the LFU ratios. We conclude in section 5. In the following we investigate form factors for the transitions where p and k denote the four momenta of the initial and final state respectively, and J P indicates both angular momentum and parity eigenvalues of the Λ * + c states. The states' rest-frame helicities are denoted as s b and J z . Note that, for the J P = 3/2 − state, J z can be decomposed into the rest-frame helicity of a 1/2 + spinor (s c ), and the polarisation of a polarisation vector η ≡ η(λ c ). For later use we also define the momentum transfer to the leptons q µ ≡ p µ − k µ .

Helicity form factors
We define the hadronic matrix elements for vector and axialvector transitions to the Λ c (2595) + state as:

3) whereū
(3/2) α is the spin 3/2 projection of a Rarita-Schwinger object; see also appendix A. A possible basis of Dirac structures for the vector current is given in [19]. We choose a different basis for both vector and axialvector currents. We compile the list of all Dirac structures Γ αµ V (A),i in appendix B. We define the helicity amplitudes for the two currents Γ µ = γ µ , γ µ γ 5 as JHEP06(2018)155 more possible combinations of λ c and s c than physically permitted. We identify the helicity amplitudes with total angular moment J = 1/2 as (2.5) The complementary set of J = 3/2 amplitudes reads (2.6) For transitions to J = 1/2 the set of amplitudes in eq. (2.6) is required to vanish identically, and similarly for transitions to J = 3/2 the set in eq. (2.5) needs to be zero. We explicitly verify this to be the case for the structures listed in appendix B.
Our Dirac structures Γ αµ V (A),i have been chosen such that the form factors F 1/2,λq and G 1/2,λq , λ q ∈ {t, 0, ⊥}, correspond to transitions into Λ c (2595) + states with |J z | = 1/2, while the Λ c (2625) + states with |J z | = 3/2 are only produced via the form factors F 3/2,⊥ and G 3/2,⊥ . Note that all helicity amplitudes depend only on one single form factor; see eqs. (C.31)-(C.33), eqs. (C.34)-(C.36), eqs. (C.73)-(C.76), and eqs. (C.77)-(C.80). We have therefore achieved a decomposition of the (axial)vector hadronic matrix elements in terms of helicity form factors as inspired by [20]. We note that our definitions of the form factors differ from the one adopted in [18], where the decomposition of the vector and axial vector hadronic matrix elements do not yield form factors for transitions with well-defined angular momentum of the final states. In particular in the conventions of [18] the time-like polarisation, which is relevant for the LFU ratio R Λ * c , depends on linear combinations of multiple form factors instead of one form factor per current.

Heavy-quark expansion
In ref. [18], the usual basis of form factors has been studied in the HQE up to 1/m contributions. We cross-check their results, and adapt them to our choice of a helicity basis for the form factors. In particular, we study the hadronic matrix elements in and beyond the heavy quark limit m b → ∞, m c → ∞ with m c /m b = const. Following [17], we use that the transition matrix elements can be written at leading power in the expansion as , v and v are the four-velocities of the initial and final states, respectively, and Γ denotes a Dirac structure. Here the most general decomposition of the light-state transition amplitude ζ reads As a consequence, at leading power all form factors can be expressed in terms of the single amplitude ζ(w), which must vanish at the zero hadronic recoil w = 1, which corresponds to In order to include also 1/m and α s corrections, we use for the vector current (and similarly for the axialvector current) with perturbative coefficients C i and power corrections ∆J µ V . The perturbative functions C i are the Wilson coefficients arising in the matching of HQET onto QCD. Their argumentw is the recoil parameter as experienced by the heavy quarks within the hadrons. Note that for a decay to orbitally excited hadronsw is not the same as defined for transitions among ground-state baryons. Instead, we usē whereΛ andΛ are the usual HQET parameters in the infinite mass limit. In the following we choose to use the pole mass scheme to determine the Λ ( ) parameters from the respective baryon masses. Our eq. (2.10) yields the product of heavy-quark velocities as defined in [21] in the limitΛ →Λ. We use the matching coefficients C i to order α s , which are given in eq. (3.111) of [21]. At the precision that we aim for, we do not require the renormalizationgroup improved matching coefficients, which can be extracted from [21], eq. (3.121).
In eq. (2.9) we use only power corrections ∆J µ V O 1 and ∆J µ V O 8 , arising from the local operators O 1 and O 8 as defined in [21], respectively. The remaining local operators only contribute at the order α s /m and are therefore beyond the precision we aim for. The hadronic matrix elements of O 1 and O 8 can be parametrised as: and -5 -

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After some algebra, we obtain the following for the contributions from ∆J V µ O 1 and ∆J V µ O 8 : (2.13) We can follow the very same steps also with the axial vector current. In this case we have: (2.14) where the subleading contributions ∆J µ (2.16) The subleading IW functions are related by the equations of motion. In particular we have that v β ζ αβ (b) = 0, and v β ζ αβ (c) = 0. This leads to the following relations: , where we denote h b(c) as the usual HQET fields. This identity allows us to write the following relations: With these 5 relations we can reduce the initial 6 subleading IW functions to one independent subleading IW function. We find it convenient to use ζ (b) 3 : Beside the effects on local operators, we also need to consider effects from non-local insertions of the HQET Lagrangian at power 1/m. Following the discussion in [18,21], non-local insertions of the kinetic operator give rise to an w-dependent shift η kin (w) to the leading-power IW function ζ(w). We can absorb this shift into the definition of ζ: The w-dependent shift due to the chromomagnetic operator is more delicate. The two contributions are: In [18], it is argued that the two functions η (q) mag (w) must vanish at zero recoil, and are expected to be small compared to the size of Λ QCD . We follow this argument, and therefore choose to not consider contributions from either η Thus, at leading power in 1/m only the (J, J z ) = (3/2, ±1/2) form factors receive contributions from the leading-power IW function. As a consequence, the sum rule at zero recoil (w = 1 or s − = 0) as discussed later will be less sensitive to the contributions from the J = 3/2 amplitudes.

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We note in passing that our results for the HQE of the form factors fulfil the relations (2.40) as required by analyticity; i.e., any spurious poles of the hadronic matrix elements in the limit q 2 → 0 do not correspond to any physical states with quantum numbers B = −C = 1, and therefore must be cancelled due to the above relations.

Parametrisation of the Isgur-Wise functions
Determining the parameters of the leading and subleading IW functions is a crucial point to evaluate the form factors. Unfortunately, there are no first principles in HQET which allow us to estimate the q 2 dependence of the IW functions. In light of this, we need to infer a functional form for ζ(q 2 ) and ζ SL (q 2 ) through some other means. For the ground-state transition Λ b → Λ c and in the large N c limit, it has been motivated in [22] to express the IW functions as exponential functions. Inspired by this, one of the models we consider here for the parametrisation of the leading and subleading IW function ζ(q 2 ) and ζ SL (q 2 ) is where the normalisation ζ(q 2 max ), the relative normalisation δ SL and the two shape parameters ρ and ρ SL are to be determined.
We can also use a Taylor expansion of ζ(q 2 ) and ζ SL (q 2 ) around q 2 q 2 max . For our purposes we use an expansion up to the first order in q 2 : In the following we will refer to eq. (3.2) as the nominal parametrisation.
Both parametrisations have been chosen such that they share their complete parameter set, and such that both the leading and the subleading IW functions have a common normalisation ζ(q 2 max ).

Benchmarking the form factors' parameters from Zero Recoil Sum Rules
The kinematic point of zero hadronic recoil is a special one for bottom-to-charm transitions.
In this point the hadronic form factors for Λ b → X c transitions, where X c denotes a -10 -

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singly-charmed baryonic state, are minimally sensitive to the dynamics of the light degrees of freedom within the respective hadrons; see e.g. [23]. As a consequence, the inclusive spectral density for the forward matrix elements of two bi-local insertions of the weak current can be expressed in terms of Λ b → X c form factors. Inference of weighted sum of squares for the form factor normalisations follows in what is known as a Zero Recoil Sum Rule (ZRSR) [24,25]. This is only possible since the spectral density consists of a sum of positive-definite exclusive terms. The ZRSR is well established for B → D and B → D * transitions, with OPE contributions known up to order α 2 s [26]. After the first lattice QCD results for the Λ b → Λ c form factors appeared [14], they were scrutinised in the ZRSR framework [27]. The conclusion of the latter analysis is as follows. Given our present knowledge of the Λ b forward matrix elements, and given the lack of mixed α s /m results for the ZRSR, the lattice results for Λ b → Λ c transition lead to a negative contribution from non-ground state transitions. As mentioned above, negative contributions to the spectral density are not possible by construction. Hence, either the inclusive calculation of the spectral density yields too small a value, or the lattice results are too large.
For the discussion at hand, we will assume that the inclusive calculation underestimates the magnitude of the spectral density. Specifically, we assume that 1/m 4 and 1/m 5 terms in the Heavy-Quark-Expansion, which have not been taken into account due to lack of information on the relevant hadronic matrix elements, will increase the magnitude. A priori it is not intuitive that terms at order 1/m 4 or beyond can make a qualitative difference to the ZRSR. However, there is precedent for numerically relevant shifts in the case of B → D * [28]. In the latter study, it was observed that -based on rather precise knowledge of the HQE parameters for B mesons -the sum of 1/m 4 and 1/m 5 terms yields roughly a third of the 1/m 2 and 1/m 3 terms.
In the absence of further information on the Λ b forward matrix elements, we will therefore proceed as follows. We will rescale the estimate of the 1/m 2 and 1/m 3 terms by a factor of 1.33, thereby copying the situation in B → D * decays. 1 The corresponding shift can now accommodate fully the lattice results for the Λ b → Λ c form factors, as well as form factors for Λ b decays to excited charm baryons. The setup of the ZRSR involves an upper bound on the excitation energies ε ≡ M Xc − M Λc of the contributing charm baryons. For the analysis at hand, ε ≤ 0.7 GeV. Based on the known spectrum of charmed baryons [29,Ch. 109 Charmed Baryons], the ZRSR covers -beside the ground stateform factors for Λ b decays into Σ c (2455), Σ c (2520), Λ c (2595), Λ c (2625), and Σ c (2800). 2 The Σ c states form an isospin triplet and therefore carry isospin I = 1. Consequently, the 1 We stress that this rescaling, and the corresponding shift to the inclusive upper bound on the form factor normalisations, is based on a supposition rather than data, and will only be used for the purpose of benchmarking the experimental sensitivity. Ultimately, only improved knowledge of the hadronic matrix elements will settle the discrepancy between the ZRSR and lattice results. 2 We do not consider here the states of roughly 2.8 GeV to 2.9 GeV for which there exists no definite assignment as either a Λc, or a Σc state, or as a kinematical artifact in the Λcππ spectrum. A recent LHCb analysis of Λ b → Λc ν [16] suggests that the yield of Λcππ background stemming from this kinematic region corresponds to roughly 10% of the first orbitally excited Λ * c states. Given the overall accuracy of our analysis, this further supports our decision not to consider these states. transitions Λ b → Σ c violate isospin conservation, and we will assume them to be further suppressed with respect to the Λ b → Λ * c transitions. This supposition is corroborated by the non-observation of Λ b → Σ c ν decays in the recent LHCb study [16]. Under the above assumptions, the inelastic parts of the ZRSR can be recast as matrix elements involving only Λ b → Λ * c transitions. Following the definitions and analysis of ref. [27], applying the assumptions above we arrive at the following constraints at zero recoil: The individual contributions from the orbitally-excited Λ * c states for the vector current read: and where N V = 1. For the axialvector current, including the normalisation factor N A = 3, the individual contributions read: and In the zero-recoil point, both parametrisation eq. (3.1) and eq. (3.2) yield the same expressions, involving only the parameters ζ(q 2 max ) and δ SL .

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Using two uncorrelated gaussian distributions for F inel and G inel and using symmetrised 68% intervals based on eq. (3.3) we obtain correlated distributions for ζ(q 2 max ) and δ SL . The ζ(q 2 max ) distribution is highly non-gaussian, and due to the large set of assumptions on which our results are founded, both distributions are not instructive for physics analyses. However, they can be used to define a benchmark point for further phenomenological analyses, in particular for the sensitivity study later on in this article. For later applications, we define the normalisation parameters of our benchmark point to be compatible with these distributions: corresponding to a subleading contribution of 14% of the leading-power IW function. This is fully in line with naive power-counting expectations for the subleading-power IW function.
Since the ZRSR cannot provide us with any information on the slopes of either IW function, we have to draw inspiration from elsewhere. Given the lower bound on the slope of the leading-power IW function for B → D ( * ) transitions, we assume ρ, ρ SL 0.25. On the other hand, in order to avoid unphysical zero crossings of the IW functions in the semileptonic region in the nominal parametrisation, we need to impose ρ, ρ SL 0.75. We choose to use the boundaries to define the slope parameters of our benchmark points as: We emphasise again that these values are not viable for any physics analysis, and are merely used when studying the sensitivity to the IW function parameters for upcoming LHCb analyses.

Observables
The fully differential decay rate of an unpolarised Λ b to a Λ * c with total angular momentum J can be written as for the specific final-state lepton flavour ∈ {e, µ, τ }. The momentum transfer q 2 is defined as the invariant mass of the leptons in the final state, and θ is the helicity angle of the charged lepton with the -ν momentum in the Λ b rest frame. Our choice of normalisation reads which should not be confused with the total decay width Γ ( )

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From the double-differential rate, we can construct two angular observables in addition to the q 2 -differential decay rate: first, the forward-backward asymmetry which arises from the term linear in cos θ . And secondly, the flat term which arises from a linear combination of the coefficients a (J) and c (J) that differs from the one comprising the decay rate eq. (3.19). The weight functions for both observables read: In the above, P n denotes the nth Legendre polynomial.
Note that the definition of the flat term F H in eq. (3.21) is similar to the one proposed for e.g. the decay B → K + − ; see ref. [30]. However, contrary to what happens in the mesonic decays in the limit m → 0, the baryonic F H does not vanish in the SM. This is due to the fact that the Λ b → Λ * c transitions are also mediated by perpendicular polarisation states of the virtual W , which is impossible in the mesonic transitions.
For the decay to the J = 1/2 final state the coefficients are

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For the J = 3/2 we have We can compare them to the results for the fully differential decay rate in the limit m → 0 as presented in [18]. We find complete agreement between our limit and the results of [18] when converting to the different basis of form factors as shown in eq. (B.6).
4 Prospects for the determination of the Λ 0 b → Λ * + c form factors using LHCb data Similarly to the mesonic B → D ( * ) transitions, the most precise SM prediction for R Λ * c will arise from a combination of theoretical and experimental input. In this section, we investigate the sensitivity to the IW parameters from the decay Λ 0 b → Λ * + c µ −ν in the present and future LHCb datasets when assuming a SM-like distribution. 3 To achieve this, we first produce a series of toy ensembles and subsequently fit the decay distribution to the simulated pseudo events. Estimates for the theoretical uncertainty on R Λ * c within the SM are then produced based on our fits.

Experimental situation
Two aspects of the experimental situation are needed to assess the experimental sensitivity. The reconstructed and selected signal yields of the decays Λ 0 b → Λ c (2625) + µ −ν and Λ 0 b → Λ c (2595) + µ −ν and the resolution in q 2 and cos θ l . We estimate the expected signal yields for a given luminosity by extrapolating from the numerical values quoted in ref. [16], taking into the account the increased bb cross-section at 13 TeV [31]. We explore the sensitivity to parameters of interest as a function of the luminosity, starting from the current LHCb dataset, up to the luminosity expected at the end of the first LHCb upgrade [32].
A key factor which limits the precision of the experimental measurements is the resolution in q 2 and cos θ l , induced by the unreconstructed neutrino. The resolution determines how finely the data is binned and introduces a statistical correlation between adjacent bins. Purity as a function of q 2 and cos θ l , defined as the fraction of candidates which belong in a particular kinematic bin. The purity for cos θ l is better than for q 2 due to the better resolution.
At a hadron collider, the momentum of the neutrino can be deduced using the information of the Λ 0 b flight direction and its mass, up to a two-fold ambiguity. The dominant effects on the resulting resolution originate from the measurement of the primary pp collision and Λ 0 b vertices, as well the effect of choosing the wrong kinematic solution from the two available. In order to approximate the resolution of the LHCb detector, a sample of Λ 0 b → Λ * + c µ −ν candidates are simulated using Pythia at 13 TeV [33,34], with a required pseudo-rapidity of 2 < η < 5, approximately corresponding to the LHCb acceptance. The vertices of the pp collision and Λ 0 b decay are varied according to a resolution inspired from ref. [35] and used in ref. [36]. The resolutions of ±20 µm in the x and y directions and ±200 µm in the z direction (defined as the direction aligned with the LHC beam line) is used for the Λ 0 b vertex. For the pp collision vertex, a resolution of ±13 µm in x and y and ±70 µm in z is assumed. With these new vertex positions the two kinematic solutions for the neutrino are then calculated, and one is chosen randomly.
The resulting purities with 4 q 2 bins and 4 cos θ l bins are shown in figure 1, where the purity is defined as the fraction of the number of candidates reconstructed correctly for a given q 2 bin. There is a better purity at negative cos θ l , which is due to the interplay between q 2 and cos θ l : at high q 2 the cos θ l resolution is poor, and in this region there is a positive cos θ l distribution. The resolution limits the number of bins and induces a statistical correlation between neighbouring bins, which is calculated based on the number of candidates which migrate between those two bins. In the 4 × 4 bins configuration, this correlation is around 10-30% in both q 2 and cos θ l .
In addition to the above, precision measurements of b-hadrons branching fractions at the LHC require a well-measured normalisation channel to cancel the uncertainties related to the production. In principle one could normalise to a well measured B meson decay and take the ratio of production fractions. However, this method would inherit substantial systematic uncertainties, and therefore for this study the decay rate is normalised and only the shape information is used to determine the parameters of interest. This means that the absolute normalisation of the form factors cannot be constrained experimentally. As a consequence we do not report any sensitivity for the form factor parameter ζ(q 2 max ), which corresponds to this absolute normalisation.

Fits to the differential decay rate
For the purpose of this analysis we fix the two HQE parametersΛ = m Λ b − m b and Λ = m Λ * c − m c in the fits. 4 We start by fitting the one-dimensional q 2 distribution of the Λ 0 b → Λ c (2625) + µ −ν decay, Λ 0 b → Λ c (2595) + µ −ν decay or a combination thereof. We generate about 300 pseudoexperiments for each parametrisation and benchmark points, and for each pseudoexperiment we generate 50000 Λ 0 b → Λ c (2625) + µ −ν and 20000 Λ 0 b → Λ c (2595) + µ −ν events, corresponding to the expected size of the LHCb dataset at the end of the LHC Run II. The resulting one-dimensional distributions of the form factor parameters are shown in figure 2 for the benchmark point described in eq. (3.13). All benchmark points yield similar results. When fitting a single decay mode, we find that there is a degeneracy between the two slope parameters ρ and ρ SL due to a strong correlation that is positive for the Λ 0 b → Λ c (2625) + µ −ν decay and negative for the Λ 0 b → Λ c (2595) + µ −ν decay. Only by combining both states in a single fit can the interference between the positive and negative correlation break this degeneracy.
In order to maximise the sensitivity to all three form factor parameters and make full use of the LHCb dataset, we investigate fits to the two-dimensional q 2 and cos θ l . The resulting one-dimensional and two-dimensional distributions of the parameters are shown in appendix G. A comparison between the distributions of the IW parameters for the oneand two-dimensional fits is shown in figure 3. The results show that a two-dimensional fit improves the precision on all three parameters with reduced correlations between them, as shown in figure 6. This strongly motivates a full two-dimensional fit to both Λ * + c states simultaneously for any future LHCb analysis to give the best possible precision on the form factor parameters.  derived from the same parameters and therefore have similar uncertainties. We carry out our study for each of the two paramatrisations of the IW functions given in section 3.1, and each of the common benchmark points defined in section 3.2. The precision as a function of the luminosity collected by the LHCb experiment is shown in figure 4, where in order to be conservative and ensure the legibility of our results we only show the worst case of our studies. Assuming the exponential model 5 describes the data well, a statistical precision of ∼ 7% can be expected from run I+II data. A reduction to ∼ 2% can be expected after upgrade 1 of the LHCb detector. For the linear model, we find in general smaller uncertainties than for the exponential model. Our estimates for the uncertainties ignore power suppressed terms in the HQET expansion and experimental systematic uncertainties, which could become relevant at that level of precision.

Projected precision on the R Λ *
Similar to what has been done in the literature for R D * , we can estimate the impact of the dominant unknown 1/m 2 c corrections to the HQET relations on the theory predictions 5 With exponential model we indicate the exponential parametrisation described in section 3.1 together with the benchmark points obtained in section 3.2.

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for the R Λ * c . Following the discussion [37], we wish to separate the term involving the timelike form factors from the term that can be taken directly from data on the semimuonic decay mode. We therefore decompose Note here that the (τ, 1) terms are taken directly from data, while the (τ, 2) terms rely on the HQET relations between the form factors for theoretical predictions. Correspondingly, we then decompose We find that the relative contribution by the (τ, 1) term is both dominant and stable under variation of the slope parameters across our four benchmark points in the exponential model. We find that R Λc(2595),1 0.76 · R Λc(2595) + , and R Λc(2625),1 0.77 · R Λc(2625) + . For a conservative estimate, we can assume that the 1/m 2 c contributions yield 30% corrections to the HQET relations as estimated in [37]. Consequently, we would face an inherent theory uncertainty of ∼ 8% for R Λc(2595) and up to ∼ 7% for R Λc(2625) . 6 Given that projected statistical uncertainty in figure 4 are of similar size already with the full run II dataset, we come to the conclusion that our theoretical uncertainty estimates strongly motivate dedicated lattice QCD studies of the Λ b → Λ * c form factors.

Conclusion
Motivated by the recent deviations in LFU in semileptonic b → s and b → c decays, we have provided the theoretical ingredients needed to constrain the theoretical uncertainty of the lepton universality ratios R Λc(2595) + and R Λc(2625) + , collectively denoted as R Λ * c . To this end, we have improved and extended upon the work in [18]. We provide a new definition of the hadronic form factors, convenient for the decay observables, and work out 6 Switching the b and c quark mass schemes from the pole to the kinetic scheme yields a shift in RΛ * c by less then 4%. The scheme dependence, and therefore the values of the heavy-quark expansion parameters Λ andΛ are presently inconsequential compared to the inherent 1/m 2 c uncertainty.

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formulae for O(α s ) corrections to HQE. We then propose a parameterisation of the Isgur-Wise function informed from previous studies on the ground state Λ 0 b → Λ + c transition [22] and perform a zero recoil sum rule to provide a benchmark point for these parameters to be used in a study of the sensitivity to these parameters for a future analysis of LHCb data. Last but not least, we provide the finite lepton mass terms for the two double differential decay distributions.
We investigated the benefits of fitting the two-dimensional q 2 − cos θ l distribution over fitting only the q 2 distribution, for either of the Λ * + c hadronic states and their combination. We find that fitting the angular information in addition to the q 2 spectrum is crucial to obtain sensitivity to the sub-leading Igsur-Wise function. In addition, we stress that a combined analysis of both Λ * + c states is necessary to break the degeneracy between the slopes of the leading and sub-leading Igsur-Wise functions. Finally, we show that by measuring the differential decay rate of Λ 0 b → Λ * + c µ −ν , small statistical uncertainty for a data driven determination of the R Λ * c ratios can be achieved. Our results therefore motivate an LHCb analysis of the Λ 0 b → Λ * + c µ −ν double-differential decay rate and the subsequent experimental measurement of the R Λ * c ratios. On the other hand, we also demonstrate that the unknown 1/m 2 terms in the form factors' expansion produce at present an irreducible uncertainty that is of the same order as the statistical uncertainty. This motivates further theoretical studies of the form factors, e.g. from lattice QCD.

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In the above, u(k, s c ) denotes a spin-1/2 + spinor of four momentum k and rest-frame helicity s c = ±1/2, and η denotes a polarisation vector with J P = 1 − . Likewise, we can also characterise the J P = 1/2 − state in term of the projection onto the spin-1/2 component as: The Rarita-Schwinger object fulfills the equation of motion By virtue of the equations of motions, the following identities hold while for the spin 3/2 projection u α of a Rarita-Schwinger object, the following relations are also true: The completeness relation for the 3/2 spinor read λ( ),sc( ) while for the 1/2 spinor we have:

B Details on the form factor definitions
The spin structures Γ αµ J,i that contribute to the transition Λ b → Λ * c are listed in the following. For the final state Λ c (2595) + and for the vector current (J = V ) we find:

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while for the axialvector current (J = A) we obtain: In the case of the final state Λ c (2625) + , for the vector current (J = V ) we obtain: while for the axialvector current (J = A) we use

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C Helicity amplitudes we find the following non vanishing terms: For the pseudoscalar current, defined as one finds: For the vector current For the transverse polarisation we find:

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For the longitudinal polarisation we find: Similarly for the axialvector current For the transverse polarisation we find For the longitudinal polarisation we find

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Using the above expressions, we can now list the helicity amplitudes for the transition Λ b → Λ c (2595) + . For the vector current we find the following non-zero helicity amplitudes: For the axialvector current we find similarly (C. 36) In the heavy quark expansion, if we use eq. (2.9) for the vector current, we calculated the following helicitity amplitudes:

G Additional material on the sensitivity study
We show in figure 5 the distributions of the Isgur-Wise parameters resulting from a twodimensional fit to both q 2 and cos θ l , comparing ensembles of pseudo-experiments using only the Λ c (2595) + , only the Λ c (2625) + , or both. In figure 6 we investigate the correlations between the Isgur-Wise parameters resulting from a two-dimensional fit to q 2 and cos θ l of the three sets of pseudo-experiments. In particular, the leftmost plots demonstrate how only a simultaneous fit to both Λ * + c states can solve the degeneracy between the two slope parameters. Moreover, both Λ c (2595) + and Λ c (2625) + data sets are individually sensitive to the δ SL parameter, but a simultaneous fit provides much better precision. 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.