Precise predictions for $\Lambda_b \to \Lambda_c$ semileptonic decays

TL;DR

This paper provides precise, model-independent predictions for $ ext{Lambda}_b o ext{Lambda}_c$ semileptonic decays, including potential new physics effects, by calculating form factors and decay rates with higher-order corrections and minimal unknown parameters.

Contribution

It introduces a systematic approach to predict $ ext{Lambda}_b o ext{Lambda}_c$ decay observables with reduced theoretical uncertainties, incorporating new physics and higher-order effects in heavy quark effective theory.

Findings

Predictions for decay rates and form factors with higher-order corrections.

Model-independent estimates for $R( ext{Lambda}_c)$ including new physics.

Emphasis on measuring differential decay rates to test factorization.

Abstract

We calculate the $\Lambda_b \to \Lambda_c \ell \nu$ form factors and decay rates for all possible $b\to c \ell\bar\nu$ four-Fermi interactions beyond the Standard Model, including nonzero charged lepton masses and terms up to order $\alpha_s\, \Lambda_\text{QCD}/m_{c,b}$ and $\Lambda_\text{QCD}^2/m_c^2$ in the heavy quark effective theory. At this order, we obtain model independent predictions for semileptonic $\Lambda_b \to \Lambda_c$ decays in terms of only two unknown sub-subleading Isgur-Wise functions, which can be determined from fitting LHCb and lattice QCD data. We thus obtain model independent results for $\Lambda_b\to \Lambda_c\ell\bar\nu$ decays, including predictions for the ratio $R(\Lambda_c) = {\cal B}(\Lambda_b\to \Lambda_c \tau\bar\nu) / {\cal B}(\Lambda_b\to \Lambda_c \mu\bar\nu)$ in the presence of new physics, that are more precise than prior results in the literature, and systematically improvable with better data on the decays with $\mu$ (or $e$) in the final state. We also explore tests of factorization in $\Lambda_b \to \Lambda_c\pi$ decays, and emphasize the importance of measuring at LHCb the double differential rate $d^2\Gamma(\Lambda_b\to\Lambda_c\ell\bar\nu) / (d q^2\, d\cos\theta)$, in addition to the $q^2$ spectrum.

Tables

Table 1. Table 1: HQET parameters extracted from the two fits discussed in the text. Predictions for R ​ ( Λ c ) 𝑅 subscript Λ 𝑐 R(\Lambda_{c}) for each fit are shown in the last row. The b ^ 1 , 2 subscript ^ 𝑏 1 2 \hat{b}_{1,2} values marked with an asterisk were fixed to zero in the fit; see text for details.

	LHCb + LQCD	LHCb + LQCD
${\zeta }^{\prime }$	$-2.04\pm 0.08$	$-2.06\pm 0.08$
${\zeta }^{\prime \prime }$	$3.16\pm 0.38$	$3.28\pm 0.36$
${\hat{b}}_1$/GeV2	$-0.46\pm 0.15$	$0^{\ast }$
${\hat{b}}_2$/GeV2	$-0.39\pm 0.39$	$0^{\ast }$
$m_b^{1S}$/GeV	$4.72\pm 0.05$	$4.69\pm 0.04$
$\delta m_{bc}$/GeV	$3.40\pm 0.02$	$3.40\pm 0.02$
${\chi }^2/\text{ndf}$	$7.20/20$	$18.8/22$
$R\left({\Lambda }_c\right)$	$0.3237\pm 0.0036$	$0.3252\pm 0.0035$

Equations

$$R({\Lambda }_c)=\frac{\Gamma ({\Lambda }_b\to {\Lambda }_c\tau \bar{\nu })}{\Gamma ({\Lambda }_b\to {\Lambda }_c\mu \bar{\nu })}=0.324\pm 0.004,$$

$$R(D^{(\ast )})=\frac{\Gamma (B\to D^{(\ast )}\tau \bar{\nu })}{\Gamma (B\to D^{(\ast )}l\bar{\nu })},l=\mu ,e,$$

$$O_V$$

$$O_S$$

$$⟨{\Lambda }_c(p^{\prime },s^{\prime })|\bar{c}{\gamma }_{\nu }b|{\Lambda }_b(p,s)⟩$$

$$⟨{\Lambda }_c(p^{\prime },s^{\prime })|\bar{c}{\gamma }_{\nu }{\gamma }_{5}b|{\Lambda }_b(p,s)⟩$$

$$⟨{\Lambda }_c(p^{\prime },s^{\prime })|\bar{c}b|{\Lambda }_b(p,s)⟩$$

$$⟨{\Lambda }_c(p^{\prime },s^{\prime })|\bar{c}{\gamma }_{5}b|{\Lambda }_b(p,s)⟩$$

$$⟨{\Lambda }_c(p^{\prime },s^{\prime })|\bar{c}{\sigma }_{\mu \nu }b|{\Lambda }_b(p,s)⟩$$

$$\displaystyle\qquad\quad+i\,h_{4}(v_{\mu}v^{\prime}_{\nu}-v_{\nu}v^{\prime}_{\mu})\big{]}u(p,s)\,.$$

$$⟨{\Lambda }_c(p^{\prime },s^{\prime })|\bar{c}{\gamma }_{\mu }b|{\Lambda }_b(p,s)⟩$$

$$⟨{\Lambda }_c(p^{\prime },s^{\prime })|\bar{c}{\gamma }_{\mu }{\gamma }_{5}b|{\Lambda }_b(p,s)⟩$$

$$m_{{\Lambda }_Q}=m_Q+{\bar{\Lambda }}_{\Lambda }-\frac{{\lambda }_1^{\Lambda }}{2m_Q}+\dots ,Q=b,c,$$

$$⟨{\Lambda }_c(v^{\prime },s^{\prime })|\bar{c}\Gamma b|{\Lambda }_b(v,s)⟩=\zeta (w)\bar{u}(v^{\prime },s^{\prime })\Gamma u(v,s),$$

$$f_1(w)=g_1(w)=h_S(w)=h_P(w)=h_1(w)=\zeta (w),$$

$$f_2(w)=f_3(w)=g_2(w)=g_3(w)=h_2(w)=h_3(w)=h_4(w)=0.$$

$$\zeta (w)+({\epsilon }_c+{\epsilon }_b){\zeta }_{\mathrm{ke}}(w)\to \zeta (w),$$

$$\big{\{}\hat{f}_{i}(w)\,,\ \hat{g}_{i}(w)\,,\ \hat{h}_{i}(w),\ \hat{b}_{i}(w)\big{\}}=\big{\{}f_{i}(w)\,,\ g_{i}(w)\,,\ h_{i}(w)\,,\ b_{i}(w)\big{\}}\big{/}\zeta(w)\,.$$

$${\hat{f}}_1$$

$${\hat{f}}_2$$

$$\displaystyle\qquad+2(w-1)C^{\prime}_{V_{2}}(\varepsilon_{c}+\varepsilon_{b})\bigg{]}+\frac{\hat{b}_{2}}{4m_{c}^{2}}+\ldots,$$

$${\hat{f}}_3$$

$$\displaystyle\qquad+2(w-1)C^{\prime}_{V_{3}}(\varepsilon_{c}+\varepsilon_{b})\bigg{]}+\ldots\,,$$

$${\hat{g}}_1$$

$${\hat{g}}_2$$

$$\displaystyle\qquad+2(w-1)C^{\prime}_{A_{2}}(\varepsilon_{c}+\varepsilon_{b})\bigg{]}+\frac{\hat{b}_{2}}{4m_{c}^{2}}+\ldots,$$

$${\hat{g}}_3$$

$$\displaystyle\qquad+2(w-1)C^{\prime}_{A_{3}}(\varepsilon_{c}+\varepsilon_{b})\bigg{]}+\ldots\,,$$

$${\hat{h}}_S$$

$${\hat{h}}_P$$

$${\hat{h}}_1$$

$${\hat{h}}_2$$

$$\displaystyle\qquad+2(w-1)C^{\prime}_{T_{2}}(\varepsilon_{c}+\varepsilon_{b})\bigg{]}+\frac{\hat{b}_{2}}{4m_{c}^{2}}+\ldots\,,$$

$${\hat{h}}_3$$

$$\displaystyle\qquad+2(w-1)C^{\prime}_{T_{3}}(\varepsilon_{c}+\varepsilon_{b})\bigg{]}+\ldots\,,$$

$${\hat{h}}_4$$

$$\frac{d^2\Gamma }{dwd\cos \theta }$$

$$\displaystyle-3\sqrt{w^{2}-1}\,\bigg{(}\!2f_{1}\,g_{1}\,\hat{q}^{2}-\frac{\rho_{\ell}}{\hat{q}^{2}}\,{\cal H}_{+0}\bigg{)}\cos\theta+\bigg{(}1-\frac{\rho_{\ell}}{\hat{q}^{2}}\bigg{)}\Big{(}\hat{q}^{2}\,{\cal H}_{1}-{\cal H}_{+}\Big{)}\frac{3\cos^{2}\theta-1}{2}\,\bigg{]},$$

$${\mathcal{H}}_1$$

$${\mathcal{H}}_0$$

$${\mathcal{F}}_{+}$$

$${\mathcal{G}}_{+}$$

$${\mathcal{F}}_0$$

$${\mathcal{G}}_0$$

Full text

1. I Introduction

2. II HQET expansion of the form factors

3. II.1 Form factor definitions

4. II.2 Form factors in HQET

5. II.3 Differential decay rates and forward-backward asymmetry

6. III Fits to LHCb and lattice QCD data

7. III.1 SM form factor fits

8. III.2 Tensor form factors

9. III.3 $R(\Lambda_{c})$ predictions with new physics

10. IV Factorization and $\Lambda_{b}\to\Lambda_{c}\pi$

11. V Conclusions

12. A Form factor definitions, conversions, relations

13. B Amplitudes

I Introduction

In a recent paper Bernlochner et al. (2018a), it was shown that LHCb data for the semileptonic $\Lambda_{b}\to\Lambda_{c}\mu\nu$ decays Aaij et al. (2017) combined with lattice QCD calculations Detmold et al. (2015), provide sensitivity for the first time to sub-subleading $\mathcal{O}(\Lambda_{\text{QCD}}^{2}/m_{c}^{2})$ terms in the heavy quark effective theory (HQET) expansion Georgi (1990); Eichten and Hill (1990) of the $\Lambda_{b}\to\Lambda_{c}$ semileptonic decay form factors, independent of $|V_{cb}|$. The $\mathcal{O}(\Lambda_{\text{QCD}}^{2}/m_{c}^{2})$ corrections were found to have their expected characteristic size, suggesting that the expansion in $\Lambda_{\text{QCD}}/m_{c}$ for baryon form factors is well-behaved up to $\Lambda_{\text{QCD}}^{2}/m_{c}^{2}$ terms. The same framework also resulted in a new standard model (SM) prediction for the ratio

[TABLE]

which is significantly more precise than prior results Detmold et al. (2015); Woloshyn (2013); Shivashankara et al. (2015); Gutsche et al. (2015); Dutta (2016); Azi (2018); Di Salvo et al. (2018).

The ratio in Eq. (1) is of particular interest in light of the persistent hints of deviations from the SM, in the ratios

[TABLE]

at approximately the $4\sigma$ level, once the measurements for the $D$ and $D^{*}$ final states are combined Amhis et al. (2017). The $\Lambda_{b}\to\Lambda_{c}\mu\bar{\nu}$ decays involve the same underlying $b\to c\tau\nu$ new physics (NP) operators as $B\to D^{(*)}\tau\bar{\nu}$, but the HQET expansion for the ground-state baryon form factors is simpler than for mesons. The “brown muck” surrounding the heavy quark is in a spin and isospin zero ground state. A consequence of this is a simpler expansion of the form factors, in which the $\mathcal{O}(\Lambda_{\text{QCD}}/m_{c,b}\,,\,\alpha_{s}\,\Lambda_{\text{QCD}}/m_{c,b})$ subleading contributions are determined by the leading order Isgur-Wise function, reducing the number of free parameters in the form factor fits, and thereby providing sensitivity to $\mathcal{O}(\Lambda_{\text{QCD}}^{2}/m_{c}^{2})$ terms.

The spread in the uncertainties quoted for theoretical predictions for $R(D^{*})$ in the SM are largely due to different estimates of $\mathcal{O}(\Lambda_{\text{QCD}}^{2}/m_{c}^{2})$ effects Bernlochner et al. (2017a); Bigi et al. (2017a); Jaiswal et al. (2017). The very same hadronic matrix elements are also crucial to resolve tensions between inclusive and exclusive determinations of $|V_{cb}|$ Bernlochner et al. (2017a); Bigi et al. (2017b); Grinstein and Kobach (2017); Bernlochner et al. (2017b); Bailey et al. (2015); BDs (2018); Kaneko et al. (2018); Bigi et al. (2017a); Jaiswal et al. (2017). The abundant sample of $\Lambda_{b}$ baryons produced at the LHC may therefore provide a complementary and theoretically cleaner laboratory to study the behavior of the heavy quark expansion, identify possible NP effects, and extract $|V_{cb}|$.

In this paper, we expand and generalize the study of Ref. Bernlochner et al. (2018a) beyond the SM, to include all $b\to c\tau\bar{\nu}$ four-Fermi operators, including those containing right-handed (sterile) neutrinos. We compute the relevant form factors including $\mathcal{O}(\Lambda_{\text{QCD}}^{2}/m_{c}^{2})$ terms, and compare the fit results of Ref. Bernlochner et al. (2018a) to the lattice QCD determinations of not only the three vector and three axial vector SM form factors, but also the four NP tensor current form factors. We further emphasize the importance of measuring at LHCb the double differential rate $d^{2}\Gamma(\Lambda_{b}\to\Lambda_{c}\ell\bar{\nu})/(dq^{2}\,d\cos\theta)$ in addition to the $q^{2}$ spectrum, and also explore tests of factorization in $\Lambda_{b}\to\Lambda_{c}\pi$ decay.

II HQET expansion of the form factors

II.1 Form factor definitions

We are interested in the $\Lambda_{b}\to\Lambda_{c}$ matrix elements of operators with all possible Dirac structures, for which we choose the basis

[TABLE]

with $\sigma_{\mu\nu}=(i/2)\,[\gamma_{\mu},\gamma_{\nu}]$. As done in Refs. Leibovich et al. (1997, 1998); Bernlochner and Ligeti (2017); Bernlochner et al. (2018b) for excited charm mesons, we use the conventions $\text{Tr}[\gamma^{\mu}\gamma^{\nu}\gamma^{\sigma}\gamma^{\rho}\gamma^{5}]=-4i\epsilon^{\mu\nu\rho\sigma}$, so that $\sigma^{\mu\nu}\gamma^{5}\equiv+(i/2)\epsilon^{\mu\nu\rho\sigma}\sigma_{\rho\sigma}$. (This is the opposite of the common convention in the $\,\overline{\!B}\to D^{(*)}\ell\bar{\nu}$ literature, which typically chooses $\text{Tr}[\gamma^{\mu}\gamma^{\nu}\gamma^{\sigma}\gamma^{\rho}\gamma^{5}]=+4i\epsilon^{\mu\nu\rho\sigma}$, so that $\sigma^{\mu\nu}\gamma^{5}\equiv-(i/2)\epsilon^{\mu\nu\rho\sigma}\sigma_{\rho\sigma}$.)

The semileptonic $\Lambda_{b}\to\Lambda_{c}\ell\bar{\nu}$ form factors in HQET are conventionally defined for the SM currents as Isgur and Wise (1991); Falk and Neubert (1993); Manohar and Wise (2000)

[TABLE]

where $p=m_{\Lambda_{b}}v$, $p^{\prime}=m_{\Lambda_{c}}v^{\prime}$, and the $f_{i}$ and $g_{i}$ are functions of $w=v\cdot v^{\prime}=(m_{\Lambda_{b}}^{2}+m_{\Lambda_{c}}^{2}-q^{2})/(2m_{\Lambda_{b}}m_{\Lambda_{c}})$. The spinors are normalized to $\bar{u}(p,s)u(p,s)=2m$. We further define the NP form factors,

[TABLE]

In the definition of the NP tensor current, the conventions are chosen to simplify the $\alpha_{s}$ corrections when expressed in terms of the standard coefficient functions.

In full QCD, the form factors of the SM currents were instead traditionally defined as Falk and Neubert (1993),

[TABLE]

Our notation for the form factors follows Ref. Manohar and Wise (2000); the notation of Ref. Falk and Neubert (1993) corresponds to an exchange of upper and lowercase symbols, $F_{i}\leftrightarrow f_{i}$ and $G_{i}\leftrightarrow g_{i}$, in Eqs. (II.1) and (II.1). The relations between the form factors in Eqs. (II.1) and (II.1) are given in the Appendix A.

II.2 Form factors in HQET

The ground state baryons are singlets of heavy quark spin symmetry, because the light degrees of freedom, the “brown muck”, are in the spin-0 state. Hence, the baryon masses can be written as

[TABLE]

where the ellipsis denote terms suppressed by more powers of $\Lambda_{\rm QCD}/m_{Q}$. The parameter $\bar{\Lambda}_{\Lambda}$ is the energy of the light degrees of freedom in the $m_{Q}\to\infty$ limit. The $\lambda_{1}^{\Lambda}$ parameter is related to the heavy quark kinetic energy in the $\Lambda$ baryon. We use $m_{\Lambda_{b}}=5.620\,$GeV, $m_{\Lambda_{c}}=2.286$ GeV Tanabashi et al. (2018), and employ the $1S$ short distance mass scheme Hoang et al. (1999a, b); Hoang (2000) to eliminate the leading renormalon ambiguities in the definition of the quark masses and $\bar{\Lambda}_{\Lambda}$. Details of the $1S$ scheme treatment can be found in Ref. Bernlochner et al. (2017a). In particular, we treat $m_{b}^{1S}=(4.71\pm 0.05)\,\text{GeV}$ and $\delta m_{bc}=m_{b}-m_{c}=(3.40\pm 0.02)\,\text{GeV}$ as independent parameters Ligeti and Tackmann (2014). (The latter is well constrained by $B\to X_{c}\ell\bar{\nu}$ spectra Bauer et al. (2004, 2003).) We match HQET onto QCD at scale $\mu=\sqrt{m_{b}m_{c}}$, so that $\alpha_{s}\simeq 0.26$. For example, using Eq. (7) for both $\Lambda_{b}$ and $\Lambda_{c}$ to eliminate $\lambda_{1}^{\Lambda}$, at $\mathcal{O}(\alpha_{s})$ we obtain $\bar{\Lambda}_{\Lambda}=(0.81\pm 0.05)\,\text{GeV}$ and $\lambda_{1}^{\Lambda}=-(0.24\pm 0.08)\,\text{GeV}^{2}$. (Similar HQET-based discussions can be found for other decay modes, $B\to D^{(*)}\ell\bar{\nu}$ Bernlochner et al. (2017a), $B\to D^{**}\ell\bar{\nu}$ Leibovich et al. (1997, 1998); Bernlochner and Ligeti (2017); Bernlochner et al. (2018b), and $\Lambda_{b}\to\Lambda_{c}^{*}\ell\bar{\nu}$ Leibovich and Stewart (1998); Boer et al. (2018).)

Making the transition to HQET Georgi (1990); Eichten and Hill (1990), at leading order in $\Lambda_{\text{QCD}}/m_{c,b}$,

[TABLE]

where $u(v,s)$ satisfies $\not{v}\,u(v,s)=u(v,s)$ and $\zeta(w)$ is the Isgur-Wise function for ground state baryons Isgur and Wise (1991), satisfying $\zeta(1)=1$. At leading order, one finds

[TABLE]

At order $\Lambda_{\text{QCD}}/m_{c,b}$ a remarkable simplification occurs compared to meson decays. The $\mathcal{O}(\Lambda_{\text{QCD}}/m_{c,b})$ corrections from the matching of the $\bar{c}\,\Gamma b$ heavy quark current onto HQET Falk et al. (1990); Falk and Grinstein (1990); Neubert (1992) can be expressed in terms of $\bar{\Lambda}_{\Lambda}$ and the leading order Isgur-Wise function $\zeta(w)$ Georgi et al. (1990). In addition, for $\Lambda_{b}\to\Lambda_{c}$ transitions, i.e., between the ground state baryons, there are no ${\cal O}(\Lambda_{\text{QCD}}/m_{c,b})$ contributions from the chromomagnetic operator. The kinetic energy operator in the ${\cal O}(\Lambda_{\text{QCD}}/m_{c,b})$ HQET Lagrangian gives rise to a heavy quark spin symmetry conserving subleading term, parametrized by $\zeta_{\rm ke}(w)$, which can be absorbed into the leading order Isgur-Wise function by redefining $\zeta$ via

[TABLE]

where $\varepsilon_{c,b}=\bar{\Lambda}_{\Lambda}/(2\,m_{c,b})$. Luke’s theorem Luke (1990) implies $\zeta_{\rm ke}(1)=0$, so the normalization $\zeta(1)=1$ is preserved. Thus, no additional unknown functions beyond $\zeta(w)$ are needed to parametrize the ${\cal O}(\Lambda_{\text{QCD}}/m_{c,b})$ corrections. Perturbative corrections to the heavy quark currents can be computed by matching QCD onto HQET Falk et al. (1990); Falk and Grinstein (1990); Neubert (1992), and introduce no new hadronic parameters. The same also holds for the order $\alpha_{s}\,\Lambda_{\text{QCD}}/m_{c,b}$ corrections Neubert (1993, 1994).

The $\mathcal{O}(\Lambda_{\text{QCD}}^{2}/m_{c,b}^{2})$ corrections are parametrized by six linear combinations of sub-subleading Isgur-Wise functions, $b_{1,\,\ldots\,,\,6}$ Falk and Neubert (1993), which are functions of $w$. Only two of these, $b_{1,2}(w)$, occur at $\mathcal{O}(\Lambda_{\text{QCD}}^{2}/m_{c}^{2})$. The redefinition in Eq. (10) introduces additional $\epsilon_{c}^{2}\,\zeta_{\text{ke}}(w)$ terms, which can be reabsorbed into $b_{1,2}(w)$. We may then define

[TABLE]

Thus, including $\alpha_{s}$, $\Lambda_{\text{QCD}}/m_{c,b}$, $\alpha_{s}\,\Lambda_{\text{QCD}}/m_{c,b}$, and $\Lambda_{\text{QCD}}^{2}/m_{c}^{2}$ corrections, the SM form factors are Bernlochner et al. (2018a)

[TABLE]

where the $C_{\Gamma_{i}}$ are functions of $w$, and ${\hat{\alpha}_{s}}=\alpha_{s}/\pi$. (We use the notation of Ref. Manohar and Wise (2000); explicit expressions for $C_{\Gamma_{i}}$ are in Ref. Bernlochner et al. (2017a).) In Eq. (II.2), primes denote $\partial/\partial w$ and the ellipses denote ${\cal O}(\varepsilon_{c}\varepsilon_{b},\,\varepsilon_{b}^{2},\,\varepsilon_{c}^{3})$ and higher order terms in $\Lambda_{\text{QCD}}/m_{Q}$ and/or $\alpha_{s}$. Equation (II.2) agrees with Eq. (4.75) in Ref. Neubert (1994) (where a redefinition different from Eq. (10) was used).

For the expansions of the form factors parametrizing the BSM currents, we obtain,

[TABLE]

Similar to $f_{3}$ and $g_{3}$, neither of the $h_{3}$ and $h_{4}$ form factors receive $\Lambda_{\text{QCD}}^{2}/m_{c}^{2}$ corrections. The structure of $h_{1,2,3}$ is similar to $g_{1,2,3}$, while $h_{4}$ is non-zero only at $\mathcal{O}(\alpha_{s}\,\Lambda_{\text{QCD}}/m_{c,b})$.

II.3 Differential decay rates and forward-backward asymmetry

In Appendix B, we collect explicit expressions for the $\Lambda_{b}\to\Lambda_{c}\ell\nu$ amplitudes for all NP operators, including contributions from massless right-handed sterile neutrinos Asadi et al. (2018); Greljo et al. (2018). Including the charged lepton mass dependence, and defining $\theta$ as the angle between the lepton and the $\Lambda_{c}$ momentum in the dilepton rest frame,111This angle is not measurable in the $\tau$ channel by present experiments, because neither the $\Lambda_{b}$ nor $\tau$ momentum can be precisely reconstructed. In principle, if the $\Lambda_{b}$ momentum was known and the $\tau\to 3\pi\nu$ decay mode was used to reconstruct the $\tau$ vertex, then $\theta$ could be reconstructed. the SM double differential decay rate is

[TABLE]

where $\rho_{\ell}=m_{\ell}^{2}/m_{\Lambda_{b}}^{2}$, $r_{\Lambda}=m_{\Lambda_{c}}/m_{\Lambda_{b}}$, $\hat{q}^{2}\equiv q^{2}/m_{\Lambda_{b}}^{2}=1-2r_{\Lambda}w+r_{\Lambda}^{2}$,

[TABLE]

and

[TABLE]

The double differential rate in Eq. (II.3) can be at most a degree-two polynomial in $\cos\theta$, and it was written in Eq. (II.3) in the Legendre polynomial basis, so that only the zeroth order term in the first line contributes to the $d\Gamma/dq^{2}$, after integration over $d\cos\theta$.

The single differential rate in the SM is correspondingly

[TABLE]

and the forward-backward asymmetry is given by

[TABLE]

Our result in Eq. (17) agrees with those in Refs. Boyd et al. (1997); Detmold et al. (2015). Including all possible NP current operators and a nonzero charged lepton mass, our result for $d\Gamma/dw$ as derived from Appendix B agrees with the result for SM neutrinos in Eq. (2.51) of Ref. Datta et al. (2017). We see from Eqs. (II.3) or (II.3) that the $\theta$ distribution in the light lepton modes gives sensitivity to the product $f_{1}\,g_{1}$, which is not present in $d\Gamma/dw$. The quadratic term in $\cos\theta$ in the angular distribution provides sensitivity to the combination $\hat{q}^{2}{\cal H}_{1}-{\cal H}_{+}$. Thus, just like in the case of $b\to s\ell^{+}\ell^{-}$ Lee et al. (2007), measuring the dependencies on all three polynomials of $\cos\theta$, gives information on the form factors beyond measuring only $d\Gamma/dq^{2}$ and $dA_{\rm FB}/dq^{2}$.

To gain more information than obtainable from Eq. (II.3), the distribution of the $\Lambda_{c}$ decay products would have to be studied. Such an analysis would be simplest for two-body decays, such as $\Lambda_{c}\to\Lambda(p\pi^{-})\pi^{+}$ Shivashankara et al. (2015). This channel loses an order of magnitude in statistics compared to the commonly used $\Lambda_{c}\to pK\pi$ reconstruction, however, a model independent description of this three-body decay amplitude is not currently available. With much higher statistics and using $\Lambda_{c}\to\Lambda\pi^{+}$, the measurement of all $\Lambda_{b}\to\Lambda_{c}$ form factors would be similar to that for $\Lambda_{c}\to\Lambda e\nu$ Korner and Kramer (1992); Crawford et al. (1995); Hinson et al. (2005), requiring measuring distributions in three angles (as for $B\to(D^{*}\to D\pi)l\bar{\nu}$).

If NP only modifies the (axial)vector interactions (see e.g. Refs. Shivashankara et al. (2015); Dutta (2016); Li et al. (2017) for other cases), which may be the most plausible scenario, then Eqs. (II.3) – (II.3) are simply modified via the replacements

[TABLE]

and, in particular,

[TABLE]

In the $m_{l}=0$ limit, i.e., in the $\Lambda_{c}\mu\nu$ and $\Lambda_{c}e\nu$ modes, the forward-backward asymmetry only receives further contributions from tensor–(pseudo)scalar interference, even in the presence of arbitrary NP. The relation in Eq. (20) is then valid in the light lepton modes, as long as NP does not simultaneously generate (pseudo)scalar and tensor operators.

III Fits to LHCb and lattice QCD data

III.1 SM form factor fits

The methods used to fit $d\Gamma(\Lambda_{b}\to\Lambda_{c}\mu\bar{\nu})/dq^{2}$ measured by LHCb Aaij et al. (2017) and lattice QCD (LQCD) calculation of the (axial)vector form factors Detmold et al. (2015) were described in Ref. Bernlochner et al. (2018a), and are only briefly recapitulated here. LHCb measured the $q^{2}$ spectrum in 7 bins, normalized to unity Aaij et al. (2017), reducing the effective degrees of freedom in the spectrum from 7 to 6. This measurement is shown as the data points in the left plot in Fig. 1. Our fits to the LHCb data use the measured and predicted partial rates in each bin. This procedure differs slightly from the fits performed by LHCb Aaij et al. (2017), which used the square root of $dN_{\rm corr}/dw$ evaluated at the midpoint in the seven unfolded $w$ bins. The right plot in Fig. 1 shows our prediction for $1/\Gamma\times d\Gamma(\Lambda_{b}\to\Lambda_{c}\tau\bar{\nu})/dq^{2}$, normalized to $R(\Lambda_{c})$.

The lattice QCD results Detmold et al. (2015) for the six (axial)vector form factors are published as fits to the BCL parametrization Bourrely et al. (2009), using either 11 or 17 parameters. We derive predictions for $f_{1,2,3}$ and $g_{1,2,3}$ using the 17 parameter result at three $q^{2}$ values, q^{2}=\big{\{}1\,\text{GeV}^{2},\ q^{2}_{\rm max}/2,\ q^{2}_{\rm max}-1\,\text{GeV}^{2}\big{\}} for a total of eighteen form factor values, constructing a covariance matrix from their correlation structure. The values of $q^{2}$ are chosen to sample both ends and the middle of the $q^{2}$ spectrum. Adding more $q^{2}$ values from the BCL fit of the LQCD result to our sampling does not noticeably affect the fit results. The difference in the form factor values obtained using the 17 or the 11 BCL parameter results is added as an uncorrelated uncertainty. This slightly differs from the prescription in Ref. Detmold et al. (2015), which used the maximal differences of the form factor values between the two parametrizations, and cannot preserve the correlation structure between the form factor values. The 18 form factor values used in our fits are shown as data points in Fig. 2. The LQCD predictions, following the prescription of Ref. Detmold et al. (2015), are shown as gray bands. The uncertainties are in good agreement. Similarly, the gray band in Fig. 1 (left plot) shows the LQCD prediction for the normalized spectrum, using the BCL parametrization.

In our fits, $m_{b}^{1S}$ and $\delta m_{bc}$ are constrained using Gaussian uncertainties. The leading order Isgur-Wise function is fitted to quadratic order in $w-1$

[TABLE]

Alternative expansions using the conformal parameters $z$ or $z^{*}$ Boyd et al. (1996, 1997); Boyd and Savage (1997); Bourrely et al. (2009) instead of $w$ yield nearly identical fits. Therefore, we do not explore the differences in the unitarity bounds between meson and baryon form factors Boyd and Lebed (1997). Fits with $\zeta$ linear in either $w$, $z$, or $z^{*}$ are poor, while adding more $q^{2}$ values to our sampling indicates no preference for the inclusion of higher order terms in $w-1$. In the fits $\hat{b}_{1,2}$ are assumed to be constants, which is appropriate at the current level of sensitivity. With better experimental and lattice constraints in the future, the sensitivity to lifting these assumptions should be tested.

Fit results combining the LHCb and LQCD results are shown in Table 1, and in Fig. 2 by red bands. To test the importance of the $\Lambda_{\text{QCD}}^{2}/m_{c}^{2}$ terms, we also perform a fit with the order $\Lambda_{\text{QCD}}^{2}/m_{c}^{2}$ terms, parametrized by $\hat{b}_{1,2}$, set to zero. These fits are shown in Fig. 2 as blue bands, and the corresponding fit values are provided in Table 1. This is a much poorer fit, changing $\chi^{2}/{\rm ndf}$ from $7.2/20$ to $18.8/22$.

We do not include explicitly an uncertainty for neglected higher order terms in Eqs. (II.2) and (II.2). Four form factors, $f_{3}$, $g_{3}$, $h_{3}$, and $h_{4}$ receive no $\Lambda_{\text{QCD}}^{2}/m_{c}^{2}$ corrections, so the agreement of $f_{3}$ and $g_{3}$ with the LQCD results in the plots in the bottom row in Fig. 2 indicates that these higher order corrections are probably small. The order $\varepsilon_{c}\,\varepsilon_{b}$ corrections to $f_{3}$ and $g_{3}$ are given by two new functions of $w$, $b_{5}$ and $b_{6}$ Falk and Neubert (1993), while the $\varepsilon_{c}^{3}$ corrections to $f_{3}$ and $g_{3}$ also vanish. Thus, including such corrections, $b_{5}$ and $b_{6}$ would simply accommodate the $0.5\,\sigma-1\,\sigma$ differences between the LQCD results and our fit for $f_{3}$ and $g_{3}$. The impact of this is small, for example, setting $f_{3}=0$ does not perceptibly change the SM prediction for $R(\Lambda_{c})$ compared to Eq. (1), while setting $g_{3}=0$ changes the SM prediction from $R(\Lambda_{c})=0.324\pm 0.004$ in Eq. 1 by about $1\sigma$, to $0.320\pm 0.003$.

In Fig. 3 show our fit results for ratios of form factors (red bands) and the LQCD predictions (gray bands). The top plot shows $f_{1}/g_{1}$, which HQET predicts to be ${\cal O}(1)$, whereas the four ratios $f_{2}/f_{1}$ and $g_{2}/g_{1}$ (second row) and $f_{3}/f_{1}$ and $g_{3}/g_{1}$ (third row) are predicted to be ${\cal O}(\varepsilon_{c,b},\,\alpha_{s})$. The ratio, $f_{1}/g_{1}\,(=f_{\perp}/g_{\perp})$, is determined by Eq. (II.2) as

[TABLE]

so the enhancement of $f_{1}$ relative to $g_{1}$ is a model independent prediction of HQET, as seen in the top plot in Fig. 3.

III.2 Tensor form factors

LQCD results Datta et al. (2017) for the tensor form factors are available, and may be compared to HQET predictions from our fits to the (axial)vector form factors, via Eqs. (II.2).222 In Ref. Datta et al. (2017) the equations of motions were used to express the scalar and pseudoscalar current matrix elements in terms of the axial and vector currents. The resulting expressions depend on the quark masses, $m_{b,c}$. It is inconsistent beyond leading order in $\alpha_{s}$ to use in such expressions the $\overline{\text{MS}}$ masses $\overline{m}_{b}(\overline{m}_{b})$ and $\overline{m}_{c}(\overline{m}_{c})$ Datta et al. (2017) to evaluate the decay rates. Instead, one must use $\overline{m}_{c}(\mu)$ and $\overline{m}_{b}(\mu)$ at the same $\mu$. The correspondence between the four form factors used in this paper for the tensor current, \big{\{}h_{1},\,h_{2},\,h_{3},\,h_{4}\big{\}}, defined in Eq. (II.1), and those used in the LQCD calculation Datta et al. (2017), \big{\{}h_{+},\,h_{\perp},\,\widetilde{h}_{+},\,\widetilde{h}_{\perp}\big{\}}, are given in Appendix A. In the former basis, only one form factor, $h_{1}$, is nonzero in the heavy quark limit, while the four form factors of the LQCD basis are equal to one another in this limit. Note in particular that $h_{1}=\widetilde{h}_{+}$.

The LQCD results Datta et al. (2017) are presented using the BCL parametrization, including the correlations of the parameters. These results are computed at the scale $\mu=m_{b}$, while in this paper we match HQET onto QCD at $\mu=\sqrt{m_{c}m_{b}}$. Since the tensor current has a nonzero anomalous dimension, we use the multiplicative renormalization factor \big{[}\alpha_{s}(m_{b})/\alpha_{s}(\sqrt{m_{b}m_{c}})\big{]}^{4/25}\simeq 0.97 Dorsner et al. (2013); Freytsis et al. (2015), in order to scale the form factors to $\mu=\sqrt{m_{b}\,m_{c}}$.

In Fig. 4 the gray bands show the LQCD results for the tensor form factors converted to the $h_{1,2,3,4}$ basis. Our prediction from the fit to the (axial)vector SM form factors and the LHCb data are overlaid as red bands. The LQCD uncertainties are large for $h_{2,3,4}$ at both ends of the spectrum. This is an artifact of the $1/(w-1)$ and $1/q^{2}$ factors in the transformation from the LQCD basis in Eq. (A). (The same information in the \big{\{}h_{+},\,h_{\perp},\,\widetilde{h}_{+},\,\widetilde{h}_{\perp}\big{\}} basis is shown in Fig. 7 in Appendix A. In this basis the uncertainties are not strongly $q^{2}$ dependent.) Unlike the fits in Sec. III.1, the LQCD results for the tensor form factors are not an input to our fits, so there is no free parameter in these comparisons. Figure 7 shows that the order $\varepsilon_{c}$ terms, which are fully determined by HQET in Eq. (II.2), combined with the definitions in Eq. (A), account for the near equality of $\widetilde{h}_{\perp}$ and $\widetilde{h}_{+}$, the slight enhancement of $h_{\perp}$, and the substantial enhancement of $h_{+}$. The top left plot in Fig. 4 shows a tension between our fit and the LQCD determination of $h_{1}=\widetilde{h}_{+}$, visible in all plots in Fig. 7. In addition, the LQCD result for $h_{1}$ prefers a slightly smaller curvature than our prediction. This is similar to what is seen for $f_{1}$ and $g_{1}$ in the top row of Fig. 2: The LQCD results prefer a smaller curvature at small $q^{2}$. This is related to the observation that LQCD rate in Fig. 1 falls more quickly at small $q^{2}$ than the LHCb measurement.

III.3 $R(\Lambda_{c})$ predictions with new physics

LHCb expects that the precision of the measurement of $R(\Lambda_{c})$ can compete with that of $R(D^{(*)})$ in the future Alvarez Cartelle (2017). For the SM prediction we obtained Bernlochner et al. (2018a)

[TABLE]

Our form factor fit, combined with the expressions for the NP rates in Appendix B and the HQET predictions in Eqs. (II.2), allows for precision computation of $R(\Lambda_{c})$ for arbitrary NP contributions (see e.g. Refs Shivashankara et al. (2015); Dutta (2016); Li et al. (2017) for prior analyses). To gain a sense of the sensitivity of $R(\Lambda_{c})$, in Fig. 5, we show the allowed regions in the $R(\Lambda_{c})-R(D)$ and $R(\Lambda_{c})-R(D^{*})$ planes, as any one of the five NP couplings in Eq. (II.1) are turned on. The boundary of each region corresponds to real NP Wilson coefficients, while the interior requires a relative phase between the SM and NP. The $V-A$ NP interaction cannot have a physical phase relative to the SM, and therefore spans a line in the $R(\Lambda_{c})-R(D^{(*)})$ planes. Possibly by numerical coincidence, the scalar operator exhibits a very large correlation between $R(\Lambda_{c})$ and $R(D)$, resulting in a very narrow $R(\Lambda_{c})-R(D)$ region for this operator. Note that the (pseudo)scalar contributions vanish for the $D\,(D^{*})$ modes, respectively, and are not shown.

In Fig. 6 we compare the variation in $R(\Lambda_{c})/R(\Lambda_{c})_{\text{SM}}$ with the corresponding ratios for $D^{(*)}$, as a function of each NP coupling, assuming they are real. An error band, corresponding to the uncertainties in the fit of Ref. Bernlochner et al. (2018a), is also shown. In some cases the errors are imperceptible. We see that the NP sensitivity of $R(\Lambda_{c})$ is typically between the $R(D^{*})$ and $R(D)$ variations.

IV Factorization and $\Lambda_{b}\to\Lambda_{c}\pi$

The LHCb measurement of the $d\Gamma(\Lambda_{b}\to\Lambda_{c}^{+}\mu^{-}\bar{\nu})/dq^{2}$ spectrum Aaij et al. (2017) is normalized to unity, and the LCQD results for the $\Lambda_{b}\to\Lambda_{c}$ form factors are also independent of $|V_{cb}|$. Thus, our fit is sensitive to hadronic parameters, but it cannot be combined with the present LHCb data to extract $|V_{cb}|$. One may, however, use the LHCb measurement of $d\Gamma(\Lambda_{b}\to\Lambda_{c}^{+}\mu^{-}\bar{\nu})/dq^{2}$ to test factorization in $\Lambda_{b}\to\Lambda_{c}\pi$, or to extract $|V_{cb}|$ assuming factorization (see also Ref. Leibovich et al. (2004)). For $B\to D^{(*)}\pi$ decays, it has long been known that the ratios ${\cal B}(B^{-}\to D^{0}\pi^{-})/{\cal B}(\bar{B}^{0}\to D^{+}\pi^{-})\simeq 1.9$ and ${\cal B}(B^{-}\to D^{*0}\pi^{-})/{\cal B}(\bar{B}^{0}\to D^{*+}\pi^{-})\simeq 1.8$ Tanabashi et al. (2018) deviate substantially from unity, the prediction in the heavy quark limit. This implies that $\mathcal{O}(\Lambda_{\text{QCD}}/m_{c})$ contributions to the amplitudes enter at the $30\%$ level, and deviations from factorization in the heavy quark limit are substantial.

At leading order in the heavy quark expansion, the $\Lambda_{b}\to\Lambda_{c}\pi$ matrix element factorizes such that the nonleptonic rate is related to the semileptonic rate at $q^{2}=m_{\pi}^{2}$ via

[TABLE]

where $f_{\pi}=131\,$MeV is the pion decay constant, and $C_{1,2}$ are the usual Wilson coefficients in the effective Hamiltonian, satisfying $(C_{1}+C_{2}/3)\,|V_{ud}|\simeq 1$. (Uncertainties in this linear combination, $f_{\pi}$, and $\tau_{\Lambda_{b}}$ are neglected.) In Eq. (24), we write the $\Lambda_{c}e\bar{\nu}$ final state to emphasize that the semileptonic rate has to be evaluated neglecting lepton masses. In $\Lambda_{b}\to\Lambda_{c}\mu\bar{\nu}$ decay, measured by LHCb, the impact of $m_{\mu}\neq 0$ is substantial at $q^{2}=m_{\pi}^{2}$.

Combining the factorization relation in Eq. (24), our fit for the form factors, and |V_{cb}|=\big{(}4.22\pm 0.08\big{)}\times 10^{-2} Tanabashi et al. (2018) predicts ${\cal B}(\Lambda_{b}\to\Lambda_{c}\pi)=(3.6\pm 0.3)\times 10^{-3}$, where this uncertainty is from the fit and $|V_{cb}|$. By comparison, the measured nonleptonic branching ratio Tanabashi et al. (2018) is333This PDG average for ${\cal B}(\Lambda_{b}\to\Lambda_{c}\pi)$ includes an uncertainty scale factor of 1.5 Tanabashi et al. (2018), and is based on two LHCb Aaij et al. (2014a, b) and one CDF Abulencia et al. (2007) measurements. Reproducing this is not easy, as it involves rescaling the CDF result from ${\cal B}(\Lambda_{c}\to pK^{-}\pi^{+})=(5.0\pm 1.3)\%$ to the latest values: ${\cal B}(\Lambda_{c}\to pK^{-}\pi^{+})=(6.84\pm 0.24^{+0.21}_{-0.27})\%$ Zupanc et al. (2014) and ${\cal B}(\Lambda_{c}\to pK^{-}\pi^{+})=(5.87\pm 0.27\pm 0.23)\%$ Ablikim et al. (2016). The LHCb measurements also preceded Ref. Ablikim et al. (2016), and lifetime and other data also changed.

[TABLE]

Conversely, assuming factorization, one could use Eqs. (25) in Eq. (24) to extract $|V_{cb}|=(4.9\pm 0.3)\times 10^{-2}$, where this uncertainty is only from our form factor fit and the measured branching fraction, without an uncertainty assigned to the factorization relation itself. Thus we observe an $\mathcal{O}(15\text{--}20\%)$ deviation from the factorization relation in Eq. (24), consistent with this deviation arising from a $\Lambda_{\text{QCD}}/m_{c}$ suppressed correction Bauer et al. (2001). 444Regarding the behavior of the heavy quark expansion, the decay constants also satisfy the HQET scaling better than was thought in the 1990s. The $N_{f}=2+1+1$ FLAG [][; andupdatesat\url{http://flag.unibe.ch/}.]Aoki:2016frl averages, $f_{B}=(186\pm 4)\,\text{MeV}$ and $f_{D}=(212\pm 1.5)\,\text{MeV}$, yield $f_{B}/f_{D}\simeq 0.88$, which is not inconsistent with the leading order HQET relation Politzer and Wise (1988); Shifman and Voloshin (1987) $\sqrt{m_{D}/m_{B}}\,[\alpha_{s}(m_{b})/\alpha_{s}(m_{c})]^{-6/25}\simeq 0.68$, plus $\Lambda_{\text{QCD}}/m_{c,b}$ corrections.

V Conclusions

Fitting the LHCb measurement of the normalized $q^{2}$ spectrum for $\Lambda_{b}\to\Lambda_{c}\mu\nu$ decay Aaij et al. (2017), and the six (axial)vector form factors calculated in lattice QCD Detmold et al. (2015), one can test HQET relations and the applicability of power counting. In Ref. Bernlochner et al. (2018a) we found that the $\Lambda_{\text{QCD}}^{2}/m_{c}^{2}$ corrections were constrained by the fit to be of the expected magnitude, without any signs of enhancements or breakdown of the power counting at the $m_{c}$ scale, as is sometimes claimed in the literature. Compared to the lattice QCD only determination of the SM prediction of $R(\Lambda_{c})$, by fitting the LHCb measurement as well, we further found that the uncertainty of the SM prediction may be substantially reduced, generating the most precise SM prediction for $R(\Lambda_{c})$ to date, $R(\Lambda_{c})=0.324\pm 0.004$.

We expanded and generalized the results of Ref. Bernlochner et al. (2018a) in several ways. First, we calculated $\Lambda_{b}\to\Lambda_{c}$ semileptonic form factors for all four-Fermi NP operators, including the ${\cal O}(\Lambda_{\text{QCD}}^{2}/m_{c}^{2})$ corrections (as well as the corresponding helicity amplitudes for use in the Hammer library Duell et al. (2017)). Using our fit of the LHCb measurement and the LQCD prediction for the six (axial)vector SM form factors, we obtained parameter-free predictions for the four tensor form factors at $\mathcal{O}(\Lambda_{\text{QCD}}^{2}/m_{c}^{2})$. We observed some tension between our results based on HQET and those in Ref. Datta et al. (2017), at a magnitude greater than the $\Lambda_{\text{QCD}}^{2}/m_{c}^{2}$ corrections (see the top left figure for the $h_{1}=\widetilde{h}_{+}$ form factor in Fig. 4).

The small uncertainties in our fit to the (axial)vector form factors, combined with HQET predictions for the form factors at $\mathcal{O}(\Lambda_{\text{QCD}}^{2}/m_{c}^{2})$ allowed us to derive precise predictions for $R(\Lambda_{c})$ for arbitrary NP. We studied the NP impacts on $R(\Lambda_{c})$, including their correlations with $R(D^{(*)})$. The NP sensitivity of $R(\Lambda_{c})$ typically falls between those of $R(D^{*})$ and $R(D)$. We also explored tests of factorization in $\Lambda_{b}\to\Lambda_{c}\pi$ decay. Factorization in the heavy quark limit, combined with $|V_{cb}|$ measurements and our fit to the semileptonic form factors, implies a mildly lower nonleptonic rate than is measured, consistent with corrections to the factorization relations arising at $\mathcal{O}(\Lambda_{\text{QCD}}/m_{c})$.

LHCb measurements of the double differential rate $d^{2}\Gamma(\Lambda_{b}\to\Lambda_{c}\ell\bar{\nu})/(dq^{2}\,d\cos\theta)$, in addition to the $q^{2}$ spectrum, will provide the most differential information measurable in the massless lepton channels ($\mu$ and $e$), if the details of the $\Lambda_{c}$ decay are ignored. Besides the $q^{2}$ spectrum and the ($q^{2}$ dependent) forward-backward asymmetry, this double differential distribution involves a third function of $q^{2}$, which can help constrain form factors and test heavy quark symmetry. If the absolute normalization and the double differential rate of semileptonic $\Lambda_{b}\to\Lambda_{c}$ decays can be measured, it will provide a fully complementary path to extract $|V_{cb}|$, explore the $b\to c\tau\nu$ anomalies, and test HQET. We look forward to these developments.

Acknowledgements.

We thank Marina Artuso, Sheldon Stone, and Mark Wise for helpful conversations. We thank the Aspen Center of Physics, supported by the NSF grant PHY-1607611, where parts of this work were completed. FB and WS were supported by the DFG Emmy-Noether Grant No. BE 6075/1-1. ZL was supported in part by the U.S. Department of Energy under contract DE-AC02-05CH11231. The work of DR was supported in part by NSF grant PHY-1720252.

Appendix A Form factor definitions, conversions, relations

The form factors in Eqs. (II.1) and (II.1) are related via Falk and Neubert (1993)

[TABLE]

or in the opposite direction,

[TABLE]

The form factors used in the lattice QCD calculation Detmold et al. (2015) and in the LHCb analysis Aaij et al. (2017) follow the definitions in Ref. Feldmann and Yip (2012),

[TABLE]

where $q=p-p^{\prime}$, and $s_{\pm}=(m_{\Lambda_{b}}\pm m_{\Lambda_{c}})^{2}-q^{2}=2\,m_{\Lambda_{b}}m_{\Lambda_{c}}(w\pm 1)$. These form factors are related to the HQET form factors defined in Eq. (II.1) via

[TABLE]

At $w=1$, corresponding to $q^{2}_{\rm max}$, the form factors satisfy $g_{+}(q^{2}_{\rm max})=g_{\perp}(q^{2}_{\rm max})$.

In the heavy quark limit, $f_{0}=f_{+}=f_{\perp}=g_{0}=g_{+}=g_{\perp}=\zeta+{\cal O}(\alpha_{s},\,\Lambda_{\text{QCD}}/m_{c,b})$. The lattice QCD results in Fig. 12 in Ref. Detmold et al. (2015) show that $f_{0}\,,\ f_{+}\,,\ g_{0}\,,\ g_{+}\,,\ g_{\perp}$ differ from one another by less than ${\cal O}(10\%)$, however, $f_{\perp}$ is substantially enhanced, consistent with the HQET prediction in Eq. (22).

The form factors in Eq. (A), expressed in terms of the HQET definitions in Eq. (II.1), are

[TABLE]

Finally, the translation between the $h_{1,2,3,4}$ tensor form factors used in this paper, defined in Eq. (II.1), and those defined in Eq. (2.14) in Ref. Datta et al. (2017) are

[TABLE]

and in the opposite direction,

[TABLE]

In the heavy quark limit, the tensor form factors calculated in LQCD and shown in Fig. 2 of Ref. Datta et al. (2017) satisfy $h_{+}=h_{\perp}=\widetilde{h}_{+}=\widetilde{h}_{\perp}=\zeta+{\cal O}(\alpha_{s},\,\Lambda_{\text{QCD}}/m_{c,b})$.

Appendix B Amplitudes

In this appendix we collect explicit expressions for the $\Lambda_{b}\to\Lambda_{c}\ell\nu$ amplitudes, including mass terms and right-handed sterile neutrino contributions. These amplitudes correspond to those used in the Hammer code Duell et al. (2017).

As in Ref. Ligeti et al. (2017), we write explicit expressions for the $\bar{b}\to\bar{c}$ amplitudes rather than $b\to c$, defining the basis of NP operators to be

[TABLE]

The lower index of $\beta$ denotes the $\nu$ chirality and the lower index if $\alpha$ is that of the $c$ quark. Operators for the CP conjugate $b\to c$ processes follow by Hermitian conjugation. (The correspondence between the $\alpha$, $\beta$ coefficients and the basis typically chosen for $b\to c$ operators can be found in Ref. Bernlochner et al. (2018b).) The $\Lambda_{b}\to\Lambda_{c}\ell\nu$ process has four external spins: $s_{b}=\pm$, $s_{c}=1,2$, $s_{\ell}=1,2$ and $s_{\nu}=\pm$. (We label the $\Lambda_{c}$ and $\ell$ spin by $1$ and $2$, to match the conventions of Ref. Ligeti et al. (2017) for massive spinors on internal lines.)

Helicity angles and momenta are similarly defined with respect to the $\bar{b}\to\bar{c}$ process. Definitions for the conjugate process follow by replacing all particles with their antiparticles. The single physical polar helicity angle, $\theta_{\ell}$, defines the orientation of the lepton momenta in their center of mass reference frame, with respect to $-\bm{p}_{\Lambda_{b}}$, as shown in Fig. 4 of Ref. Bernlochner et al. (2018b). Note that $\theta_{\ell}=\pi-\theta$, for $\theta$ defined in Eq. (II.3).

If subsequent $\Lambda_{c}\to\Lambda Y$ decays are included coherently, one further defines $\phi_{\ell}$ and $\phi_{\Lambda}$ as twist angles of the $\ell$–$\nu$ and $\Lambda$–$Y$ decay planes, with the combination $\phi_{\ell}-\phi_{\Lambda}$ becoming a physical phase. Our phase conventions match the spinor conventions of Ref. Ligeti et al. (2017) for not only $\tau$ but also $\Lambda_{c}$ decay amplitudes. This amounts to requiring the inclusion in the $\tau$ and/or $\Lambda_{c}$ decay amplitudes of an additional spinor phase function, $h_{s_{\ell}}(s_{\nu})$ and $h_{s_{c}}(s_{b})$, defined with respect to $s_{\nu}$ and $s_{b}$, such that $h_{1}(-)=1=h_{2}(+)$, $h_{1}(+)=e^{i\phi_{\ell}}$ and $h_{2}(-)=e^{-i\phi_{\ell}}$. Under these conventions, the $\Lambda_{b}\to\Lambda_{c}\ell\nu$ amplitudes themselves are independent of $\phi_{\ell}-\phi_{\Lambda}$.

For compact expression of the amplitudes, it is convenient to define

[TABLE]

along with

[TABLE]

The $\Lambda_{b}\to\Lambda_{c}\ell\nu$ amplitudes obey the conjugation relation

[TABLE]

in which the exchange $\sqrt{w^{2}-1}\to-\sqrt{w^{2}-1}$ implies also $w_{-}\leftrightarrow w_{+}$. One then need only write the $s_{b}=-$ amplitudes, with the $s_{b}=+$ amplitudes following via Eq. (36). Further writing $\mathcal{A}=2\sqrt{2}G_{F}m_{\Lambda_{b}}^{2}\sqrt{r_{\Lambda}(\hat{q}^{2}-\rho_{\ell})}\times A$, the explicit amplitudes are

[TABLE]

The total differential rate for $\Lambda_{b}\to\Lambda_{c}\ell\nu$ is obtained from these expressions via

[TABLE]