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

TL;DR

This paper provides a detailed angular analysis of the decay mbda_b, deriving analytical expressions for angular observables to explore potential New Physics effects and comparing with existing literature.

Contribution

It offers the first comprehensive analytical framework for angular observables in mbda_b decay, enabling enhanced sensitivity to New Physics beyond mesonic decay analyses.

Findings

All angular observables derived analytically with compact formulas.

Decay probes more independent Wilson coefficient combinations than mesonic decays.

Numerical estimates based on lattice QCD and recent experimental data.

Abstract

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.

Figures

Tables

Table 1. Table 1: Predictions for the angular observables in the SM and in the NP benchmark point. The observable K 1 ​ s ​ s subscript 𝐾 1 𝑠 𝑠 K_{1ss} can be obtained as ( 1 − K 1 ​ c ​ c ) / 2 1 subscript 𝐾 1 𝑐 𝑐 2 (1-K_{1cc})/2 and is hence not listed. The observables K 3 ​ s ​ c subscript 𝐾 3 𝑠 𝑐 K_{3sc} and K 3 ​ s subscript 𝐾 3 𝑠 K_{3s} are zero in the SM and in all NP models without new CP-violating phases in the b → c ​ ℓ ​ ν ¯ → 𝑏 𝑐 ℓ ¯ 𝜈 b\to c\ell\bar{\nu} Wilson coefficients.

	SM		NP
obs.	$\ell =\mu$	$\ell =\tau$	$\ell =\tau$
$K_{1cc}$	$+0.206\pm 0.004$	$+0.310\pm 0.001$	$+0.311\pm 0.000$
$K_{1c}$	$-0.134\pm 0.004$	$+0.016\pm 0.003$	$+0.037\pm 0.002$
$K_{2ss}$	$+0.288\pm 0.032$	$+0.221\pm 0.024$	$+0.228\pm 0.025$
$K_{2cc}$	$+0.115\pm 0.013$	$+0.183\pm 0.020$	$+0.193\pm 0.021$
$K_{2c}$	$-0.164\pm 0.018$	$-0.031\pm 0.004$	$-0.017\pm 0.003$
$K_{4sc}$	$+0.063\pm 0.008$	$+0.023\pm 0.003$	$+0.022\pm 0.002$
$K_{4s}$	$+0.125\pm 0.015$	$+0.063\pm 0.007$	$+0.065\pm 0.007$

Equations

$${\mathcal{H}}^{\text{eff}}=\frac{4G_F}{\sqrt{2}}{\tilde{V}}_{cb}\left[\sum _i{\mathcal{C}}_i{\mathcal{O}}_i\right],$$

$${\mathcal{O}}_{V,L(R)}$$

$${\mathcal{O}}_T$$

$$K(q^2,\cos {\theta }_{\ell },\cos {\theta }_{{\Lambda }_c},\varphi )$$

$$\displaystyle+\big{(}K_{2ss}\sin^{2}\theta_{\ell}+\,K_{2cc}\cos^{2}\theta_{\ell}+K_{2c}\cos\theta_{\ell}\big{)}\cos\theta_{\Lambda_{c}}$$

$$\displaystyle+\big{(}K_{3sc}\sin\theta_{\ell}\cos\theta_{\ell}+K_{3s}\sin\theta_{\ell}\big{)}\sin\theta_{\Lambda_{c}}\sin\phi$$

$$\displaystyle+\big{(}K_{4sc}\sin\theta_{\ell}\cos\theta_{\ell}+K_{4s}\sin\theta_{\ell}\big{)}\sin\theta_{\Lambda_{c}}\cos\phi\,,$$

$$\{A_{{\lambda }_m},A_{{\lambda }_m}^T\}=\{A_{{\perp }_t},A_{{\perp }_0},A_{{\perp }_1},A_{{\parallel }_t},A_{{\parallel }_0},A_{{\parallel }_1},A_{{\perp }_0}^T,A_{{\perp }_1}^T,A_{{\parallel }_0}^T,A_{{\parallel }_1}^T\}.$$

$$\frac{\text{d}\Gamma }{\text{d}{q}^2}{K}_{1ss}$$

$$\displaystyle\quad-2\frac{m_{\ell}}{\sqrt{q^{2}}}\operatorname{Re}\bigg{\{}A^{T}_{\perp_{0}}A^{*}_{\perp_{0}}+A^{T}_{\perp_{1}}A^{*}_{\perp_{1}}+(\perp\,\leftrightarrow\,\parallel)\bigg{\}}$$

$$+\left[\left(1+\frac{m_{\ell }^2}{q^2}\right)|A_{{\perp }_1}^T{|}^2+2\frac{m_{\ell }^2}{q^2}|A_{{\perp }_0}^T{|}^2+(\perp \leftrightarrow \parallel )\right],$$

$$\frac{\text{d}\Gamma }{\text{d}{q}^2}{K}_{1cc}$$

$$\displaystyle\quad-2\frac{m_{\ell}}{\sqrt{q^{2}}}\operatorname{Re}\bigg{\{}A^{T}_{\perp_{0}}A^{*}_{\perp_{0}}+A^{T}_{\perp_{1}}A^{*}_{\perp_{1}}+(\perp\,\leftrightarrow\,\parallel)\bigg{\}}$$

$$+2\left[|A_{{\perp }_0}^T{|}^2+\frac{m_{\ell }^2}{q^2}|A_{{\perp }_1}^T{|}^2+(\perp \leftrightarrow \parallel )\right],$$

$$\frac{\text{d}\Gamma }{\text{d}{q}^2}{K}_{1c}$$

$$-2\frac{m_{\ell }}{\sqrt{q^2}}\mathrm{Re}\left\{A_{{\perp }_0}^T{A}_{{\perp }_t}^{\ast }+A_{{\perp }_1}^T{A}_{{\parallel }_1}^{\ast }+(\perp \leftrightarrow \parallel )\right\}$$

$$+4\frac{m_{\ell }^2}{q^2}\mathrm{Re}\left\{A_{{\perp }_1}^T{A}_{{\parallel }_1}^{T\ast }\right\}.$$

$$\frac{\text{d}\Gamma }{\text{d}{q}^2}{K}_{2ss}$$

$$-2\alpha \frac{m_{\ell }}{\sqrt{q^2}}\mathrm{Re}\left\{A_{{\perp }_0}^T{A}_{{\parallel }_0}^{\ast }+A_{{\perp }_1}^T{A}_{{\parallel }_1}^{\ast }+(\perp \leftrightarrow \parallel )\right\}$$

$$\displaystyle\quad+2\,\alpha\operatorname{Re}\left\{\Big{(}1+\frac{m_{\ell}^{2}}{q^{2}}\Big{)}A^{T}_{\perp_{1}}A^{T*}_{\parallel_{1}}+2\frac{m_{\ell}^{2}}{q^{2}}A^{T}_{\perp_{0}}A^{T*}_{\parallel_{0}}\right\}\,,$$

$$\frac{\text{d}\Gamma }{\text{d}{q}^2}{K}_{2cc}$$

$$-2\alpha \frac{m_{\ell }}{\sqrt{q^2}}\mathrm{Re}\left\{A_{{\perp }_0}^T{A}_{{\parallel }_0}^{\ast }+A_{{\perp }_1}^T{A}_{{\parallel }_1}^{\ast }+(\perp \leftrightarrow \parallel )\right\}$$

$$+4\alpha \mathrm{Re}\left\{A_{{\perp }_0}^T{A}_{{\parallel }_0}^{T\ast }+\frac{m_{\ell }^2}{q^2}{A}_{{\perp }_1}^T{A}_{{\parallel }_1}^{T\ast }\right\},$$

$$\frac{\text{d}\Gamma }{\text{d}{q}^2}{K}_{2c}$$

$$-2\alpha \frac{m_{\ell }}{\sqrt{q^2}}\mathrm{Re}\left\{A_{{\perp }_0}^T{A}_{{\parallel }_t}^{\ast }+A_{{\perp }_1}^T{A}_{{\perp }_1}^{\ast }+(\perp \leftrightarrow \parallel )\right\}$$

$$\displaystyle\quad+2\,\alpha\frac{m_{\ell}^{2}}{q^{2}}\bigg{[}|A^{T}_{\perp_{1}}|^{2}+(\perp\,\leftrightarrow\,\parallel)\bigg{]}\,.$$

$$\frac{\text{d}\Gamma }{\text{d}{q}^2}{K}_{3sc}=$$

$$\displaystyle+2\,\sqrt{2}\,\alpha\bigg{(}1-\frac{m_{\ell}^{2}}{q^{2}}\bigg{)}\operatorname{Im}\left\{A^{T}_{\perp_{0}}A^{T*}_{\perp_{1}}-(\perp\,\leftrightarrow\,\parallel)\right\}\,,$$

$$\frac{\text{d}\Gamma }{\text{d}{q}^2}{K}_{3s}=$$

$$+\sqrt{2}\alpha \frac{m_{\ell }}{\sqrt{q^2}}\mathrm{Im}\left\{A_{{\perp }_0}^T{A}_{{\parallel }_1}^{\ast }+A_{{\perp }_1}^T{A}_{{\parallel }_0}^{\ast }+A_{{\perp }_1}^T{A}_{{\perp }_t}^{\ast }-(\perp \leftrightarrow \parallel )\right\}$$

$$-2\sqrt{2}\alpha \frac{m_{\ell }^2}{q^2}\mathrm{Im}\left\{A_{{\perp }_0}^T{A}_{{\parallel }_1}^{T\ast }-(\perp \leftrightarrow \parallel )\right\}.$$

$$\frac{\text{d}\Gamma }{\text{d}{q}^2}{K}_{4sc}=$$

$$\displaystyle+2\,\sqrt{2}\,\alpha\bigg{(}1-\frac{m_{\ell}^{2}}{q^{2}}\bigg{)}\operatorname{Re}\left\{A^{T}_{\perp_{0}}A^{T*}_{\parallel_{1}}-(\perp\,\leftrightarrow\,\parallel)\right\}\,,$$

$$\frac{\text{d}\Gamma }{\text{d}{q}^2}{K}_{4s}=$$

$$+\sqrt{2}\alpha \frac{m_{\ell }}{\sqrt{q^2}}\mathrm{Re}\left\{A_{{\perp }_0}^T{A}_{{\perp }_1}^{\ast }+A_{{\perp }_1}^T{A}_{{\perp }_0}^{\ast }+A_{{\perp }_1}^T{A}_{{\parallel }_t}^{\ast }-(\perp \leftrightarrow \parallel )\right\}$$

$$-2\sqrt{2}\alpha \frac{m_{\ell }^2}{q^2}\mathrm{Re}\left\{A_{{\perp }_0}^T{A}_{{\perp }_1}^{T\ast }-(\perp \leftrightarrow \parallel )\right\}.$$

$$⟨{\Lambda }_c|\bar{c}({\gamma }_5)b|{\Lambda }_b⟩=\frac{q^{\mu }}{m_b\mp m_c}⟨{\Lambda }_c|\bar{c}({\gamma }_5){\gamma }_{\mu }b|{\Lambda }_b⟩.$$

$${\alpha }^{\text{PDG}}=-0.91\pm 0.15.$$

$${\alpha }^{\text{CLEO-2}}=-0.94\pm 0.21(\text{stat.})\pm 0.12(\text{syst.}),$$

$${\alpha }^{\text{FOCUS}}=-0.78\pm 0.16(\text{stat.})\pm 0.13(\text{syst.}).$$

$${\alpha }^{\text{BESIII}}=-0.80\pm 0.11(\text{stat.})\pm 0.02(\text{syst.}),$$

$$\alpha =-0.82\pm 0.09$$

$$A_{{\perp }_1}$$

Full text

††preprint: EOS-2019-01, TUM-HEP-1213/19, MPP-2019-152

1 Introduction

Anomalous measurements Lees:2012xj ; Lees:2013uzd ; Huschle:2015rga ; Aaij:2015yra ; Hirose:2016wfn ; Hirose:2017dxl ; Aaij:2017uff ; Aaij:2017deq ; Abdesselam:2019dgh in observables that probe Lepton Flavour Universality (LFU) in $\bar{B}\to D^{(*)}\tau^{-}\bar{\nu}$ decays motivate careful reappraisal of the theoretical inputs Neubert:1992pn ; Ligeti:1993hw ; Faller:2008tr ; Fajfer:2012vx ; Na:2015kha ; Lattice:2015rga ; Neubert:1992wq ; Wang:2017jow ; Gubernari:2018wyi 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, 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 by studying the four-differential decay rate of the cascade process $\Lambda_{b}^{0}\to\Lambda_{c}^{+}(\to\Lambda^{0}\pi^{+})\ell^{-}\bar{\nu}$, with any charged lepton species $\ell=e,\mu,\tau$. To achieve this goal, we work within the most general effective theory of local $b\to c\ell\bar{\nu}$ 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 three-body decay $\Lambda_{b}^{0}\to\Lambda_{c}^{+}\ell^{-}\bar{\nu}$ Datta:2017aue , and of the four-body decay $\Lambda_{b}^{0}\to\Lambda_{c}^{+}(\to\Lambda^{0}\pi^{+})\ell^{-}\bar{\nu}$ Shivashankara:2015cta ; Gutsche:2015mxa . Our analysis follows closely previous works on the flavour-changing neutral-current decay $\Lambda_{b}^{0}\to\Lambda^{0}(\to p\pi^{-})\ell^{+}\ell^{-}$ Boer:2014kda ; Das:2018sms ; Das:2018iap , which has a qualitatively similar angular distribution.

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 $\Lambda_{b}\to\Lambda_{c}$ form factors Meinel:2016dqj ; Datta:2017aue at mass dimension six, and based on our own average of the parity violating parameter $\alpha$ in the secondary decay $\Lambda_{c}^{+}\to\Lambda^{0}\pi^{+}$ Ablikim:2019zwe . We conclude in section 4.

2 Analytical Results

We work within an effective field theory for semileptonic flavour-changing $|\Delta B|=|\Delta C|=1$ transitions. Its effective Hamiltonian can be expressed as

[TABLE]

where a basis of operators up to mass dimension six and with only left-handed neutrinos can be chosen as

[TABLE]

This approach is the standard way to account model-independently for the effects of New Physics (NP). The SM corresponds to the parameter point $\mathcal{C}_{V,L}=1$ and $\mathcal{C}_{j}=0$ for all remaining operators. In the SM the parameter $\tilde{V}_{cb}$ 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 of $\tilde{V}_{cb}$ 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 $\mathcal{C}_{i}$ within the SM Effective Field Theory requires a careful interpretation of $\tilde{V}_{cb}$ Descotes-Genon:2018foz .

2.1 Angular Distribution

Following ref. Boer:2014kda , we express the kinematics of the four-body decay distribution in terms of $q^{2}$, the square of the dilepton mass; $\cos\theta_{\ell}$, the helicity angle of the charged lepton in the dilepton center-of-mass frame; $\cos\theta_{\Lambda_{c}}$, the helicity angle of the $\Lambda^{0}$ baryon in the $\Lambda^{0}\pi^{+}$ center-of-mass frame; and $\phi$, the azimuthal angles between the to decay planes. For more details, we refer to appendix A. The fourfold differential distribution takes the form

[TABLE]

The physical ranges of the kinematical variables are $m_{\ell}^{2}\leq q^{2}\leq(m_{\Lambda_{b}}-m_{\Lambda_{c}})^{2}$, $\cos\theta_{\ell,\Lambda_{c}}\in\left[-1,1\right]$ and $\phi\in\left[0,2\pi\right]$. The distribution can be decomposed in terms of a set of trigonometric functions

[TABLE]

giving rise to ten angular observables $K_{i}\equiv K_{i}(q^{2})$. Our choice of the normalization in eq. (3) imposes an exact relation between two of the angular observables of the first row, $2K_{1ss}+K_{1cc}=1$. The structure of eq. (4) is imposed by angular momentum conservation Boer:2014kda . It is generally compatible with the findings of refs. Shivashankara:2015cta ; Gutsche:2015mxa , that is, we find the same number of independent angular terms based on the results for the angular distribution of refs. Shivashankara:2015cta ; Gutsche:2015mxa as in eq. (4).

In the presence of all five effective operators of eq. (2), we can express each angular observable as a sesquilinear form of ten amplitudes

[TABLE]

Here $\lambda=\perp,\parallel$ 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. (4) we find:

[TABLE]

For the second row of eq. (4) we find:

[TABLE]

For the third row of eq. (4) we find:

[TABLE]

For the fourth row of eq. (4) we find:

[TABLE]

Comparing our results to the literature, our findings are summarized as follows:

• •

We find complete agreement with the results of ref. Datta:2017aue , which discusses the non-cascade decay in the presence of all dimension-six operators.

• •

Our findings are at variance with the results of ref. Shivashankara:2015cta , which discusses the cascasde decay for all dimension-six operators except for the tensor operator. We find interference terms of the type $V\times S$ and $A\times P$ in our angular observables $K_{3s}$ and $K_{4s}$ that are absent from the corresponding term $C_{4}^{int}$ of ref. (Shivashankara:2015cta, , 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 ${A_{\perp_{t}}}$ and ${A_{\|_{t}}}$. This shift is consistent with a Ward-like identity for the $\Lambda_{b}\to\Lambda_{c}$ matrix elements:

[TABLE]

The term $C_{4}^{int}$ in eq. (19) of ref. Shivashankara:2015cta cannot be expressed through this shift. Additionally, we find a lack of an overall multiplicative factor $\mathcal{B}(\Lambda_{c}^{+}\to\Lambda^{0}\pi^{+})$ from the secondary decay. Furthermore, we find agreement only by redefining the angles $\theta_{\ell}\to\pi-\theta_{\ell}$, $\theta_{\Lambda_{c}}\to\theta_{s}$ and $\phi\to-\chi$.

• •

Comparing our results to the SM results for the fourfold distribution in ref. Gutsche:2015mxa , we find complete agreement when redefining the angles as $\theta_{\ell}\to\pi-\theta_{\ell}$, $\theta_{\Lambda_{c}}\to\theta_{s}$ and $\phi\to-\chi$. 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 $\alpha\equiv\alpha(\Lambda_{c}^{+}\to\Lambda^{0}\pi^{+})$, which is an angular asymmetry parameter arising in $\Lambda_{c}^{+}\to\Lambda^{0}\pi^{+}$ decays. It emerges as the hadronic matrix element of the parity-violating weak decay of the $\Lambda_{c}^{+}$. The present world average of this parameter by the Particle Data Group (PDG) Tanabashi:2018oca reads:

[TABLE]

It includes measurements by the CLEO Avery:1990ya and ARGUS Albrecht:1991vs experiments, in addition to the statistically dominating measurements by the CLEO-2 Bishai:1995gp $e^{+}e^{-}$ collider experiment,

[TABLE]

and by the FOCUS (Fermilab E-831) Link:2005ft fixed target experiment,

[TABLE]

With the recent measurement of $\alpha$ by the BESIII collaboration in $e^{+}e^{-}$ collisions,

[TABLE]

we compute our own average including also the CLEO-2 and FOCUS results. We obtain:

[TABLE]

In the following we will use this average in lieu of an updated world average by the PDG.

2.2 Transversity Amplitudes

The transversity amplitudes arising in the angular observables $K_{i}$ are further decomposed into $\Lambda_{b}\to\Lambda_{c}$ 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 (pseudo-scalar) currents to timelike vector (axial-vector) form factors. In the absence of the tensor operator we find six independent transversity amplitudes:

[TABLE]

with

[TABLE]

We note that the timelike leptonic polarization state requires a non-vanishing lepton mass. Hence, the transversity amplitudes $A_{\perp_{t}}$ and $A_{\parallel_{t}}$ always enter observabes with a factor $m_{\ell}$ such that all physical observables are well-defined in the limit $m_{\ell}\to 0$. In the above we introduce a phenomenologically useful basis of effective form factors $F_{\lambda_{m}}$ and $F^{S}_{\lambda}$, and abbreviate common linear combinations of the Wilson coefficients:

[TABLE]

We also introduce an overall normalization factor

[TABLE]

with the Källén function $\lambda(a,b,c)=a^{2}+b^{2}+c^{2}-2ab-2ac-2bc$.

In the presence of the tensor operator we find four additional transversity amplitudes and hence four additional effective form factors $F^{T}_{\lambda_{m}}$:

[TABLE]

2.3 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 complex-valued quantities $\sigma$:

[TABLE]

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 $m_{\ell}\to 0$

[TABLE]

The result above illustrates that the cascade decay for $\ell=e,\mu$ is sensitive to 12 out of the 25 real-valued combinations of Wilson coefficients:

[TABLE]

This sensitivity should be compared to the sensitivity exhibited by $\bar{B}\to D^{(*)}\ell^{-}\bar{\nu}$ decays. For the pseudoscalar final state meson, the coefficients $\mathcal{C}_{V,-}$ and $\mathcal{C}_{S,-}$ do not enter at all by virtue of vanishing matrix elements. For the vector final state meson, the coefficient $\mathcal{C}_{S,+}$ does not enter for the same reason. Hence, not all elements of $\Sigma(m_{\ell}=0)$ are accessible in the $B$-meson decays, and therefore the $\Lambda_{b}$ cascade decay provides new and complementary constraints on the Wilson coefficients. We refrain from providing lengthy expressions for the massive case $\ell=\tau$, which can be readily obtained from our results in section 2.1. We find that the cascade decay with a massive lepton is sensitive to a larger set of 22 (out of 25) real-valued combinations of Wilson coefficients. The full set of combinations reads:

[TABLE]

Presently, the only prospect for measurement of the $\Lambda_{b}^{0}\to\Lambda_{c}^{+}\mu^{-}\bar{\nu}$ decays is the LHCb experiment. Within LHCb, reconstruction of the $\bar{\nu}$ momentum from the remaining initial state and decay kinematics is difficult, but possible Ciezarek:2016lqu . 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. (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,

[TABLE]

Measurement of this observable can be achieved through a counting experiment with respect to the sign of $\cos\theta_{\Lambda_{c}}$, 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 $\Lambda_{b}$ fragmentation fraction at LHCb. The ability to determine $A_{\text{FB}}^{\Lambda_{c}}$ regardless of the reconstruction of the lepton helicity angle will also make possible to probe LFU through the ratio:

[TABLE]

In the above, the $\ell=\tau$ and $\ell=\mu$ subscripts denote that the forward-backward asymmetry is to be extracted from decays into the specific lepton species $\ell$. By virtue of taking the LFU ratio as defined in eq. (46) the sensitivity to the parity violating parameter $\alpha$ 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 EOS as of version 0.2.5.

3 Numerical Results

For the numerical illustration we define a NP benchmark point:

[TABLE]

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 $b\to c\tau\bar{\nu}$ processes Murgui:2019czp .

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 $\ell=\mu$ and $\ell=\tau$, as well as predictions for the NP benchmark point ($\ell=\tau$ 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 $\alpha$—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. Detmold:2015aaa ; Datta:2017aue . We use the EOS software EOS for the computation of all numerical values and plots shown in this work.

Benefiting from the correlated results for the $\Lambda_{b}\to\Lambda_{c}$ form factors and from cancellations due to the normalization to the decay rate, we find small uncertainties of the order of $11\%$ for $A_{\text{FB}}^{\Lambda_{c}}$ that are dominated by the uncertainty in the parameter $\alpha$. This becomes much more visible in the LFU ratio where $\alpha$ cancels completely. For that observable the relative uncertainty is reduced to $\sim 1\%$, providing an excellent opportunity for a high-precision probe of New Physics in semileptonic $b\to c$ 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.

4 Summary

We present the first study of the full angular distribution of the cascade decay $\Lambda_{b}^{0}\to\Lambda_{c}^{+}(\to\Lambda^{0}\pi^{+})\ell^{-}\bar{\nu}$ using the complete basis of dimension-six operators, assuming only left-handed neutrinos, in the weak effective field theory for $b\to c\ell\bar{\nu}$ 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 decays $\bar{B}\to D\ell\bar{\nu}$ and $\bar{B}\to D^{*}\ell\bar{\nu}$ taken together. This highlights the usefulness of the cascade decay in constraining potential NP effects in $b\to c\ell\bar{\nu}$ transitions for all lepton species $\ell=e,\mu,\tau$. 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 $R(D^{*})$.

Acknowledgements.

This work is supported by the DFG within the Emmy Noether Programme under grant DY-130/1-1 and the DFG Collaborative Research Center 110 “Symmetries and the Emergence of Structure in QCD”.

Appendix A Kinematics

We label the momenta in the cascade decay as follows:

[TABLE]

The $z$-axis is chosen in the $\Lambda_{b}$ rest frame such that the $\Lambda_{c}$ travels in positive and the $W$ boson in negative $z$-direction. Within the $\Lambda_{c}$ rest frame, we define the angle $\theta_{\Lambda_{c}}$ as the angle enclosed by the flight direction of the $\Lambda$ baryon and the $z$-axis. Analogously, within the dilepton rest frame, $\theta_{\ell}$ is defined as the angle between the charged lepton momentum and the $z$-axis. Finally, we define $\phi$ as the azimuthal angle of the $\pi^{+}$, i.e., if $\phi=0$, the coordinate system is chosen such that the charged lepton and the $\pi^{+}$ both travel in positive $x$-direction.

The polarization vectors in the dilepton rest frame we choose to be

[TABLE]

Furthermore, we define

[TABLE]

With these definitions, some of the kinematical Lorentz invariants are

[TABLE]

with $\epsilon^{0123}=-\epsilon_{0123}=+1$. We defined

[TABLE]