Lepton flavor violating $\Lambda _b\rightarrow \Lambda \ell _1\ell _2$ decay

Abstract

Inspired by the recent hints of lepton flavor universality violation in $b\rightarrow s\ell \ell$ and $b\rightarrow c\ell \nu$ transitions, we study lepton flavor violating exclusive $\Lambda _b\rightarrow \Lambda \ell _1^+\ell _2^-$ ($\ell _1\ne \ell _2$) decay, which is forbidden in the Standard Model. Starting from a general effective Hamiltonian for a $b\rightarrow s\ell _1^+\ell _2^-$ transition that includes vector and axial-vector operators, and scalar and pseudo-scalar operators, we derive a two-fold decay distribution of $\Lambda _b\rightarrow \Lambda \ell _1^+\ell _2^-$. The distribution helps us to construct the differential branching ratio and the lepton-side forward–backward asymmetry, which are studied in a vector leptoquark model. The parameter space of the vector leptoquark model is constrained by low energy observables.

1 Introduction

Though an ${{{\mathcal {O}}}}(1)$ signal of new physics (NP) is still at large, the recent results by the Belle and LHCb Collaborations in the neutral and charged current transitions of b-flavored mesons are intriguing hints of lepton flavor universality (LFU) violation, which is absent in the Standard Model (SM). In the flavor changing neutral current transition $b\rightarrow s\ell \ell$ the observables that probe LFU are

$$\begin{aligned} R_{K^{(*)}} = \frac{\mathcal {B}(B\rightarrow K^{(*)}\mu ^+\mu ^-)}{\mathcal {B}(B\rightarrow K^{(*)} e^+ e^-)}. \end{aligned}$$

(1)

The LHCb Collaboration has measured $R_K$ and the most recent result is [1]

$$\begin{aligned} R_K = 0.846^{+0.060+0.016}_{-0.054-0.014},\quad 1 \le q^2 \le 6.0 \mathrm{GeV}^2, \end{aligned}$$

(2)

where $q^2$ is the invariant mass squared of the final state dilepton pair. This result is lower than the SM prediction $R_K^\mathrm{SM}=1.00\pm 0.01$ [2] by about $2.5\sigma$. On the other hand, the most recent measurements of $R_{K^{(*)}}$ by the LHCb [3] in the two dilepton invariant mass squared bins

$$\begin{aligned} R_{K^*} = {\left\{ \begin{array}{ll} 0.66^{0.11}_{-0.17} \pm 0.03, &{}\quad 0.045 \le q^2 \le 1.1 \mathrm{GeV}^2, \\ 0.69^{0.11}_{-0.07} \pm 0.05, &{}\quad 1.1 \le q^2 \le 6.0 \mathrm{GeV}^2, \end{array}\right. } \end{aligned}$$

(3)

deviate from the SM predictions $R_{K^*}^{\mathrm{SM}} =0.906 \pm 0.028$ and $R_{K^*}^{\mathrm{SM}} = 1.00 \pm 0.01$ by 2.3$\sigma$ and $2.5\sigma$, respectively. Belle has also presented [4] their results of $R_K$ and $R_{K^*}$ which are closer to the SM predictions but has large uncertainties.

Independently of the results in the $b\rightarrow s\ell \ell$ transitions, hints of LFU violation have also been found in the charged current transition $b\rightarrow c\ell \nu$. The observables in which deviations from the SM predictions have been observed are $R_D$ and $R_{D^*}$,

$$\begin{aligned} R_{D^{(*)}} = \frac{\mathcal {B}(B\rightarrow D^{(*)}\tau \nu )}{\mathcal {B}(B\rightarrow D^{(*)}\ell \nu )},\quad \ell = e, \mu . \end{aligned}$$

(4)

$R_{D^{*}}$ has been measured by Belle [5,6,7] LHCb [8] and BaBar [9]. The new measurement by Belle [10] using semi-leptonic tagging gives

$$\begin{aligned}&R_D = 0.307 \pm 0.37 \pm 0.016, \end{aligned}$$

(5)

$$\begin{aligned}&R_{D^*} = 0.283 \pm 0.018 \pm 0.14. \end{aligned}$$

(6)

HFLAV has combined the most recent results and their averages [11] exceed the SM predictions $R_D^{\mathrm{SM}} = 0.299 \pm 0.003$ [12] and $R_{D^*}^{\mathrm{SM}} = 0.258 \pm 0.005$ [13] by $2.3\sigma$ and $3.4\sigma$, respectively.

A number of NP models with new particle content has been constructed that can explain these deviations. Shortly after the first hints of LFU violation were announced [14] it was shown in Ref. [15] that LFU violation implies lepton flavor violating (LFV) interactions. Despite several counter examples to this observation [16, 17], most models that generate LFU violation also generate LFV processes which are strictly forbidden in the SM. Therefore, the observation of LFV decay will be a smoking gun signal of NP. Some of the LFV processes that have been extensively looked for are leptonic decays $\tau \rightarrow 3\mu$, $\mu \rightarrow 3e$ and $\ell \rightarrow \ell ^\prime M$, where M is a meson, radiative decays $\mu \rightarrow e\gamma$, and $\mu \rightarrow e$ conversion. Interestingly, in the Higgs sector $h\rightarrow \mu \tau$ was studied and an apparent excess was also reported by CMS [18], which disappeared in subsequent measurements.

In this paper we discuss LFV baryonic decay $\Lambda _b\rightarrow \Lambda \ell _1^+\ell _2^-$, which proceeds through a $b\rightarrow s\ell _1^+\ell _2^-$ transition where $\ell _1^+$ and $\ell _2^-$ are charged leptons of different flavors. Though its SM counterpart $\Lambda _b\rightarrow \Lambda \ell \ell$ has been measured by the LHCb [19, 20], to the best of our knowledge currently there are no experimental data on $\Lambda _b\rightarrow \Lambda \ell _1^+\ell _2^-$. Unlike $\Lambda _b\rightarrow \Lambda \ell \ell$, the advantage with $\Lambda _b\rightarrow \Lambda \ell _1^+\ell _2^-$ decay is that it does not suffer from long-distance QCD and charmonium resonance backgrounds. The $\Lambda _b\rightarrow \Lambda \ell _1^+\ell _2^-$ decay was earlier discussed in [21] in the context of scalar leptoquark model where only vector and axial-vector type effective operators were considered. In this paper we include in addition scalar and pseudo-scalar operators and present a double differential distribution. From this distribution we study the differential branching ratio and the forward–backward asymmetry. These observables are studied in a vector leptoquark model $U_1\equiv (\mathbf 3,1 )_{2/3}$. We use several low energy observables to constrain the model parameters.

The paper is organized as follows. We begin by describing in Sect. 2 the effective Hamiltonian for a $b\rightarrow s\ell _1^+\ell _2^-$ transition. The differential decay distribution of the exclusive $\Lambda _b\rightarrow \Lambda \ell _1^+\ell _2^-$ is calculated in Sect. 3 followed by a numerical analysis in Sect. 4. We summarize our discussions in Sect. 5.

2 Effective Hamiltonian

We start with the following effective Hamiltonian for the lepton flavor violating $b\rightarrow s\ell _1^+\ell _2^-$ transition:

$$\begin{aligned} {\mathcal {H}}^{\mathrm{eff}} = - \frac{1}{2v^2}V_{tb}V_{ts}^*\frac{\alpha _e}{4\pi } \sum _i\bigg ( {\mathcal {C}}_i {{\mathcal {O}}}_i + {\mathcal {C}}^\prime _i {\mathcal {O}}^\prime _i \bigg ), \end{aligned}$$

(7)

where $v^2=1/(\sqrt{2}G_F)\approx 246$ GeV is the SM vacuum expectation value, and $i = V, A, S, P$ correspond to vector, axial-vector, scalar, and pseudo-scalar operators, which read

$$\begin{aligned} \begin{aligned}&{\mathcal {O}}^{(\prime )}_V = \big [\bar{s}\gamma ^\mu P_{L(R)}b \big ]\big [\ell _2\gamma _\mu \ell _1 \big ],\\&{\mathcal {O}}^{(\prime )}_{A} = \big [\bar{s}\gamma ^\mu P_{L(R)}b \big ]\big [\ell _2\gamma _\mu \gamma _5\ell _1 \big ],\\&{\mathcal {O}}_S^{(\prime )} = \big [\bar{s}P_{R(L)}b \big ]\big [\ell _2\ell _1 \big ],\quad {\mathcal {O}}_P^{(\prime )} = \big [\bar{s}P_{R(L)}b \big ]\big [\ell _2\gamma _5\ell _1 \big ]. \end{aligned} \end{aligned}$$

(8)

Here $\alpha _e$ is the fine structure constant, $V_{tb}V_{ts}^*$ are the Cabibbo–Kobayashi–Maskawa matrix elements, $P_{L,R}=(1{\mp }\gamma _5)/2$ are the chiral projectors. The ${\mathcal {C}}_{V,A,S,P}^{(\prime )}$ are the short-distance Wilson coefficients that vanish in the SM but can be non-zero in many scenarios beyond the SM. In the SM $\ell _1,\ell _2$ are leptons of the same flavor, say $\ell$, and it is customary to denote the operators ${\mathcal {O}}_{V,A}$ as ${\mathcal {O}}_{9,10}$ with the corresponding Wilson coefficients ${\mathcal {C}}_{9,10}$. Additionally, in the SM there is also a dipole operator ${\mathcal {O}}_7$ that contributes to the $b\rightarrow s\ell \ell$ transition. The long-distance part of the decay is encoded in the $\Lambda _b\rightarrow \Lambda$ transition matrix elements (see [22] for definitions) which are parametrized in terms of six $q^2$ dependent form factors $f^V_{t,0,\perp }$, $f^A_{t,0,\perp }$ [23]. For our numerical analysis the form factors are taken from calculations in lattice QCD [24].

3 Exclusive $\Lambda _b \rightarrow \Lambda \ell _1^+\ell _2^-$ decay

To set up the kinematics of the decay we assume that the $\Lambda _b$ is at rest while the $\Lambda$ and the dilepton pair travel along the $+z$- and $-z$-axis, respectively. We assign $p,k,q_1$ and $q_2$ as the momenta of the $\Lambda _b, \Lambda , \ell _1$, and $\ell _2$, and $s_p, s_k$ are the spins of $\Lambda _b, \Lambda$ on the z-axis in their respective rest frames. We also introduce two kinematic variables; $q^\mu = q_1^\mu + q_2^\mu$ is the four-momentum of the dielpton pair, and $\theta _\ell$ is the angle that the lepton $\ell _1$ makes with respect to the z-axis in the dilepton rest frame. The decay amplitudes can be written as

$$\begin{aligned} {\mathcal {M}}^{\lambda _2,\lambda _1}(s_p,s_k)&= - \frac{V_{tb}V_{ts}^*}{2v^2}\frac{\alpha _e}{4\pi } \sum _{i=L,R}\left[ \sum _{\lambda } \eta _\lambda H^{i,s_p,s_k}_\mathrm{VA, \lambda } L^{\lambda _2,\lambda _1}_{i,\lambda }\, \right. \nonumber \\&\quad \left. +H^{i,s_p,s_k}_{\mathrm{SP}} L^{\lambda _2,\lambda _1}_i \right] . \end{aligned}$$

(9)

Here $H^{i,s_p,s_k}_{\mathrm{VA, \lambda }}$ and $H^{i,s_p,s_k}_{\mathrm{SP}}$ are the hadronic helicity amplitudes corresponding to vector and axial-vector (VA), and scalar and pseudo-scalar (SP) operators, and the $L^{\lambda _2,\lambda _1}_{i,\lambda }, L^{\lambda _2,\lambda _1}_i$ are the leptonic helicity amplitudes. Here $i={ L,R}$ corresponds to the chiralities of the lepton current and the $\lambda =t,\pm 1,0$ are the helicity states of the virtual gauge boson that decay into the dilepton pair. The $\lambda _{1,2}$ are the helicities of the leptons and $\eta _t=1,\eta _{\pm 1,0}=-1$. The definitions and the expressions of $H^{i,s_p,s_k}_{\mathrm{VA, \lambda }}$ and $H^{i,s_p,s_k}_{\mathrm{SP}}$ in terms of Wilson coefficients and form factors can be found in [25]. In the literature, instead of the hadronic helicity amplitudes, transversity amplitudes $A^i_{\perp (\Vert )_{\mathrm{1}}}, A^i_{\perp (\Vert )_{\mathrm{0}}}$ and $A_\mathrm{S\perp (\Vert )}, A_{\mathrm{P\perp (\Vert )}}$ are often used. Following [22] the expressions of the transversity amplitudes are collected in Appendix A.

The $L^{\lambda _2,\lambda _1}_{i,\lambda }$ and $L^{\lambda _2,\lambda _1}_{i}$ amplitudes are defined as

$$\begin{aligned}&L^{\lambda _2,\lambda _1}_{L(R)} = \langle {\bar{\ell }}_2(\lambda _2)\ell _1(\lambda _1) | {\bar{\ell }}_2 (1{\mp }\gamma _5) \ell _1 | 0\rangle , \nonumber \\&L^{\lambda _2,\lambda _1}_{L(R),\lambda } = {\bar{\epsilon }}^\mu (\lambda ) \langle {\bar{\ell }}_2(\lambda _2) \ell _1(\lambda _1) | {\bar{\ell }}_2 \gamma _\mu (1{\mp }\gamma _5) \ell _1 | 0\rangle , \end{aligned}$$

(10)

where $\epsilon ^{\mu }$ is the polarization vector of the virtual gauge boson that decays in to the dilepton pair. The details of the calculations of $L^{\lambda _2,\lambda _1}_{i,\lambda }$ and $L^{\lambda _2,\lambda _1}_{i}$ are given in Appendix B. Based on these calculations we obtain the differential branching ratio of $\Lambda _b\rightarrow \Lambda \ell _1\ell _2$ as

$$\begin{aligned} \frac{d\mathcal {B}}{dq^2 d\cos \theta _\ell } = \frac{3}{2} \bigg ( K_{1ss} \sin ^2\theta _\ell + K_{1cc} \cos ^2\theta _\ell + K_{1c} \cos \theta _\ell \bigg ). \end{aligned}$$

(11)

Each of the angular coefficients $K_{1ss,1cc,1c}$ can be written in the following way:

$$\begin{aligned} K_{1ss, 1cc} = K_{1ss, 1cc}^{\mathrm{VA}} + K_{1ss, 1cc}^{\mathrm{SP}} +K_{1ss, 1cc}^{\mathrm{int}}, \end{aligned}$$

(12)

where $K_{1ss, 1cc,1c}^{\mathrm{VA}}, K_{1ss, 1cc,1c}^{\mathrm{SP}}$ are contributions from VA and SP operators, and $K_{1ss, 1cc,1c}^\mathrm{int}$ includes their interference terms. In terms of the transversity amplitudes the expressions of $K_{1ss, 1cc,1c}^\mathrm{VA}, K_{1ss, 1cc,1c}^{\mathrm{SP}}$ read

$$\begin{aligned} K_{1ss}^{\mathrm{VA}}&= \frac{1}{4} \bigg ( 2|{A^R_{\Vert _{0}}}|^2 + |{A^R_{\Vert _{1}}}|^2 + 2|{A^R_{\perp _{0}}}|^2 + |{A^R_{\perp _{1}}}|^2 + \{ R \leftrightarrow L \} \bigg )\nonumber \\&\quad - \frac{m_+^2+m_-^2}{4q^2} \bigg [ \bigg ( \left| A^R_{\Vert _0}\right| ^2 + \left| A^R_{\perp _0}\right| ^2 + \{ R \leftrightarrow L \} \bigg ) \, \nonumber \\&\quad - \bigg ( |A_{\perp t}|^2 + \{ \perp \leftrightarrow \Vert \}\bigg ) \bigg ] \nonumber \\&\quad + \frac{m_+^2-m_-^2}{4q^2} \bigg [ 2\mathrm{Re}\bigg ( A^R_{\perp _0}A^{*L}_{\perp _0}\, + A^R_{\perp _1}A^{*L}_{\perp _1} + \{ \perp \leftrightarrow \Vert \} \bigg ) \bigg ] \nonumber \\&\quad -\frac{m_+^2m_-^2}{4q^4} \bigg [ \bigg (\left| {A^R_{\Vert _{1}}}\right| ^2 + \left| {A^R_{\perp _{1}}}\right| ^2 \nonumber \\&\quad + \{ R \leftrightarrow L \} \bigg ) + 2|A_{\Vert t}|^2 + 2|A_{\perp t}|^2\bigg ] \, , \end{aligned}$$

(13)

$$\begin{aligned} K_{1cc}^{\mathrm{VA}}&= \frac{1}{2}\bigg ( \left| {A^R_{\Vert _{1}}}\right| ^2 + \left| {A^R_{\perp _{1}}}\right| ^2 + \{R \leftrightarrow L \} \bigg ) +\frac{m_+^2+m_-^2}{4q^2}\,\nonumber \\&\quad \times \bigg [ \bigg ( \left| A^R_{\Vert _0}\right| ^2 - \left| A^R_{\Vert _1}\right| ^2 + \left| A^R_{\perp _0}\right| ^2 - \left| A^R_{\perp _1}\right| ^2 + \{ R \leftrightarrow L \} \bigg )\,\nonumber \\&\quad + \bigg ( |A_{\perp t}|^2 + |A_{\Vert t}|^2 \bigg ) \bigg ] + \frac{m_+^2-m_-^2}{4q^2} \bigg [ 2\mathrm{Re}\bigg ( A^R_{\perp _0}A^{*L}_{\perp _0}\, \nonumber \\&\quad + A^R_{\perp _1}A^{*L}_{\perp _1} + \{\perp \leftrightarrow \Vert \} \bigg ) \bigg ]\nonumber \\ {}&\quad - \frac{m_+^2 m_-^2}{2q^4} \bigg [ \bigg (\left| {A^R_{\Vert _{0}}}\right| ^2 + \left| {A^R_{\perp _{0}}}\right| ^2 + \{ R \leftrightarrow L \} \bigg )\nonumber \\&\quad + |A_{\Vert t}|^2 + |A_{\perp t}|^2\bigg ] \, , \end{aligned}$$

(14)

$$\begin{aligned} K_{1c}^{\mathrm{VA}}&= -\beta _\ell \beta _\ell ^\prime \left( A^R_{\perp _1}A^{*R}_{\Vert _1} - \{ R \leftrightarrow L \} \right) \,\nonumber \\&\quad + \beta _\ell \beta _\ell ^\prime \frac{m_+ m_-}{q^2} \mathrm{Re}\left( {A^L_{\Vert _{0}}} A_{\Vert t}^*+ {A^L_{\perp _{0}}} A_{\perp t}^*\right) \, ,\end{aligned}$$

(15)

$$\begin{aligned} K_{1ss}^{\mathrm{SP}}&= \frac{1}{4}\left( |A_{\mathrm{S\perp }}|^2 + |A_{\mathrm{P\perp }}|^2 + \{ \perp \leftrightarrow \Vert \} \right) \,\nonumber \\&- \frac{m_+^2}{4q^2}\big (|A_{S \Vert }|^2 + |A_{S \perp }|^2\big ) \nonumber \\&\quad - \frac{m_-^2}{4q^2}\big (|A_{P \Vert }|^2 + |A_{P \perp }|^2\big ) \, ,~~~~\end{aligned}$$

(16)

$$\begin{aligned} K_{1cc}^{\mathrm{SP}}&= \frac{1}{4}\left( |A_{\mathrm{P\perp }}|^2 + |A_{\mathrm{S\perp }}|^2 + \{\perp \leftrightarrow \Vert \} \right) \, \nonumber \\&\quad - \frac{m_+^2}{4q^2}\big (|A_{S \Vert }|^2 + |A_{S \perp }|^2\big ) \nonumber \\&\quad - \frac{m_-^2}{4q^2}\big (|A_{P \Vert }|^2 + |A_{P \perp }|^2\big )\, ,\end{aligned}$$

(17)

$$\begin{aligned} K_{1c}^{\mathrm{SP}}&= 0\, . \end{aligned}$$

(18)

The interference terms read

$$\begin{aligned} K_{1ss}^{\mathrm{int}}&= \frac{m_+}{2\sqrt{q^2}} \mathrm{Re}\left( {A_{\Vert t}}{A^*_{\mathrm{P \Vert }}}+ {A_{\perp t}}{A^*_{\mathrm{P \perp }}}\right) \nonumber \\&\quad + \frac{m_-}{2\sqrt{q^2}} \mathrm{Re}\left( {A_{\Vert t}}{A^*_{\mathrm{S \Vert }}}+ {A_{\perp t}}{A^*_{\mathrm{S \perp }}}\right) \nonumber \\&\quad - \frac{m_+^2m_-}{2q^2\sqrt{q^2}} \mathrm{Re}\left( {A_{\Vert t}}{A^*_{\mathrm{S \Vert }}}+ {A_{\perp t}}{A^*_{\mathrm{S \perp }}}\right) \nonumber \\ {}&\quad - \frac{m_+m_-^2}{2q^2\sqrt{q^2}}\mathrm{Re}\left( {A_{\Vert t}}{A^*_{\mathrm{P \Vert }}}+ {A_{\perp t}}{A^*_{\mathrm{P \perp }}}\right) ,\end{aligned}$$

(19)

$$\begin{aligned} K_{1cc}^{\mathrm{int}}&= \frac{m_+}{2\sqrt{q^2}}\mathrm{Re}\left( {A_{\Vert t}}{A^*_{\mathrm{P \Vert }}}+ {A_{\perp t}}{A^*_{\mathrm{P \perp }}}\right) \nonumber \\&\quad + \frac{m_-}{2\sqrt{q^2}}\mathrm{Re}\left( {A_{\Vert t}}{A^*_{\mathrm{S \Vert }}}+ {A_{\perp t}}{A^*_{\mathrm{S \perp }}}\right) \nonumber \\&\quad - \frac{m_+^2m_-}{2q^2\sqrt{q^2}}\mathrm{Re}\left( {A_{\Vert t}}{A^*_{\mathrm{S \Vert }}}+ {A_{\perp t}}{A^*_{\mathrm{S \perp }}}\right) \,\nonumber \\&\quad - \frac{m_+ m_-^2}{2q^2\sqrt{q^2}}\mathrm{Re}\bigg ( {A_{\Vert t}}{A^*_{\mathrm{P \Vert }}}+ {A_{\perp t}}{A^*_{\mathrm{P \perp }}}\bigg ),\end{aligned}$$

(20)

$$\begin{aligned} K_{1c}^{\mathrm{int}}&= \frac{\beta _\ell \beta _\ell ^\prime }{2\sqrt{q^2}} \mathrm{Re}\left( {A_{\mathrm{S \Vert }}}{A^{L *}_{\Vert _{0}}} + {A_{\mathrm{S \perp }}}{A^{L *}_{\perp _{0}}} + {A_{\mathrm{S \Vert }}}{A^{R *}_{\Vert _{0}}} + {A_{\mathrm{S \perp }}}{A^{R *}_{\perp _{0}}} \right) \nonumber \\&\quad + \frac{\beta _\ell \beta _\ell ^\prime }{2\sqrt{q^2}} \mathrm{Re}\left( {A_{\mathrm{P \Vert }}}{A^{L *}_{\Vert _{0}}} + {A_{\mathrm{P \perp }}}{A^{L *}_{\perp _{0}}} - {A_{\mathrm{P \Vert }}}{A^{R *}_{\Vert _{0}}} - {A_{\mathrm{P \perp }}}{A^{R *}_{\perp _{0}}} \right) . \end{aligned}$$

(21)

We have defined $m_\pm = m_1 \pm m_2$ where $m_1, m_2$ are the masses of $\ell _1, \ell _2$, respectively, and the factors $\beta _{\ell }^{(\prime )}$ are defined in Appendix A. From the differential decay distribution we define two observables [22]; the differential branching ratio

$$\begin{aligned} \frac{d\mathcal {B}}{dq^2} = 2 K_{1ss} + K_{1cc}, \end{aligned}$$

(22)

and the forward–backward asymmetry

$$\begin{aligned} A^{\ell }_{\mathrm{FB}} = \frac{3}{2} \frac{K_{1c}}{K_{1ss} + K_{1cc}}. \end{aligned}$$

(23)

The available phase space in the dilepton invariant mass squared $q^2$ is

$$\begin{aligned} (m_1+m_2)^2 \le q^2 \le ({m_{\Lambda _b}}- {m_{\Lambda }})^2. \end{aligned}$$

(24)

4 Numerical analysis

Among many leptoquark models proposed to explain flavor anomalies, the vector leptoquark $U_1\equiv (\mathbf{3},\mathbf{1})_{2/3}$ has emerged as an excellent candidate that can simultaneously alleviate the tensions between theory and experiments in both the charged and the neutral current sectors. In fact, $U_1$ can accommodate a large number of low energy data and high-$p_T$ searches without too much fine-tuning of the model parameters [26]. Early work reconciling these anomalies by coupling the $U_1$ with the third generation quarks and leptons can be found in Refs. [17, 27]. The UV completion of this model has also recently been discussed in Ref. [28]. The SM gauge symmetry allows couplings of the $U_1$ leptoquark to both left- and right-handed fermions and the Lagrangian reads

$$\begin{aligned} \mathcal {L} \supset \frac{U^\mu _1}{\sqrt{2}} \bigg [ \beta ^{ij}_L \left( \bar{Q}^i_L \gamma _\mu L^j_L \right) + \beta ^{ij}_R \left( \bar{d}^i_R \gamma _\mu \ell ^j_R \right) \bigg ]. \end{aligned}$$

(25)

Here the $Q^i_L = (V^*_{ji}u^j_L~d^i_L)^T$ and $L^i_L = (\nu ^i_L~\ell ^i_L)^T$ are $SU(2)_L$ doublets, and the $\beta _{L,R}$ are $3\times 3$ Yukawa matrices. To address the flavor anomalies we assume the following flavor ansatz:

$$\begin{aligned} \beta _L = \begin{pmatrix} 0 &{}&{} 0 &{}&{} 0 \\ 0 &{}&{} \beta _L^{s\mu } &{}&{} \beta _L^{s\tau } \\ 0 &{}&{} \beta _L^{b\mu } &{}&{} \beta _L^{b\tau } \end{pmatrix},\quad \beta _R = \begin{pmatrix} 0 &{}&{} 0 &{}&{} 0 \\ 0 &{}&{} 0 &{}&{} 0 \\ 0 &{}&{} 0 &{}&{} \beta _R^{b\tau } \end{pmatrix}. \end{aligned}$$

(26)

With the couplings to the first generation set to zero the experimental limits on atomic parity violation, $\mu$–e conversion on nuclei, and ${\mathcal {B}}(K\rightarrow \pi {\bar{\nu }}\nu )$ are evaded. An important feature of the vector leptoquark model is the absence of the tree level $b\rightarrow s\nu {\bar{\nu }}$ transition evading the current experimental constraints coming from $B\rightarrow K^*\nu {\bar{\nu }}$ [29]. There is also a “flavor protection” mechanism in the $U_1$ loops due to which the purely leptonic processes $\tau \rightarrow 3\mu$, $\tau \rightarrow \mu \nu {\bar{\nu }}$ and $b\rightarrow s\nu {\bar{\nu }}$ have little phenomenological significance [26, 30, 31]. These processes aside, we consider a number of low energy flavor observables to constrain the flavor structure (26).

The Lagrangian (25) generates the following VA and SP operators for $b\rightarrow s\ell _1^+\ell _2^-$:

$$\begin{aligned}&{\mathcal {C}}^{\ell _1\ell _2}_V = - {\mathcal {C}}_A^{\ell _1\ell _2} = - \frac{\pi v^2}{2V_{tb}V_{ts}^*\alpha _e m_{\mathrm{LQ}}^2} \beta ^{s\ell _2}_L \left( \beta ^{b\ell _1}_L\right) ^*, \end{aligned}$$

(27)

$$\begin{aligned}&{\mathcal {C}}^{\prime \ell _1\ell _2}_V = {\mathcal {C}}_A^{\prime \ell _1\ell _2} = -\frac{\pi v^2}{2V_{tb}V_{ts}^*\alpha _e m_{\mathrm{LQ}}^2} \beta ^{s\ell _2}_R \left( \beta ^{b\ell _1}_R\right) ^*,\end{aligned}$$

(28)

$$\begin{aligned}&{\mathcal {C}}^{\ell _1\ell _2}_S = - {\mathcal {C}}_P^{\ell _1\ell _2} = \frac{\pi v^2}{V_{tb}V_{ts}^*\alpha _e m_{\mathrm{LQ}}^2} \beta ^{s\ell _2}_L \left( \beta ^{b\ell _1}_R\right) ^*,\end{aligned}$$

(29)

$$\begin{aligned}&{\mathcal {C}}^{\prime \ell _1\ell _2}_S = {\mathcal {C}}_P^{\prime \ell _1\ell _2} = \frac{\pi v^2}{V_{tb}V_{ts}^*\alpha _e m_{\mathrm{LQ}}^2} \beta ^{s\ell _2}_R \left( \beta ^{b\ell _1}_L\right) ^*. \end{aligned}$$

(30)

For the given flavor ansatz (26) $R_{K^{(*)}}$ receives the following modifications [32] through the NP Wilson coefficients ${\mathcal {C}}_{V,A}^{\mu \mu }$:

$$\begin{aligned}&R_K^{[1,6]\mathrm{GeV}^2} \approx 1 + 0.46 C_V^{\mu \mu },\end{aligned}$$

(31)

$$\begin{aligned}&R_{K^*}^{[1.1,6]\mathrm{GeV}^2} \approx 1 + 0.47 C_V^{\mu \mu }. \end{aligned}$$

(32)

Global fits to the most recent $b\rightarrow s\mu \mu$ data have been performed by several groups and we take the range $-0.59\le {\mathcal {C}}_V^{\mu \mu }=-{\mathcal {C}}_A^{\mu \mu }\le -0.40$ [33, 34] in our analysis. For a large $\beta _L^{s\tau }$ there are additional flavor-universal contributions to the $\rightarrow s\ell \ell$ in the direction of ${\mathcal {C}}_V^{\mu \mu }$ due to the off-shell photon penguins [31]

$$\begin{aligned} \Delta {\mathcal {C}}_V \approx -\frac{v^2}{6m_{\mathrm{LQ}}^2V_{tb}V_{ts}^*} \beta _L^{s\tau } (\beta _L^{s\tau })^*\log \bigg (\frac{m_b^2}{m_{\mathrm{LQ}}^2}\bigg ). \end{aligned}$$

(33)

Experiments yield $\Delta {\mathcal {C}}_V =-0.73 \pm 0.23$ [33, 34].

While the contributions of $U_1$ leptoquark to $b\rightarrow s\mu \mu$ processes are through vector and axial-vector operators only, in the presence of a right-handed coupling $\beta _R^{b\tau }$ scalar and pseudo-scalar currents can contribute to $b\rightarrow s\tau \tau$ processes $B_s\rightarrow \tau ^+\tau ^-$ and $B\rightarrow K\tau ^+\tau ^-$. The $B_s\rightarrow \tau \tau$ branching ratio reads

$$\begin{aligned} {\mathcal {B}}(B_s\rightarrow \tau ^+\tau ^-)&= {\mathcal {B}}(B_s\rightarrow \tau ^+\tau ^-)_{\mathrm{SM}}\, \nonumber \\&\quad \times \bigg |1 + \frac{\pi v^2}{2 V_{tb}V_{ts}^*\alpha m_{\mathrm{LQ}}^2} \frac{\beta _L^{s\tau } }{C_{10}^{\mathrm{SM}}}\nonumber \\&\quad \times \bigg (\beta _L^{b\tau } - \frac{ m_{B_s}^2}{m_\tau (m_s+m_b)} (\beta _R^{b\tau })^*\bigg ) \bigg |^2\, \nonumber \\&\quad + \left( 1-\frac{4m_\tau ^2}{m^2_{B_s}} \right) \left| \frac{\pi v^2}{2 V_{tb}V_{ts}^*\alpha m_{\mathrm{LQ}}^2C_{10}^\mathrm{SM}} \frac{ m_{B_s}^2 \beta _L^{s\tau }\left( \beta _R^{b\tau }\right) ^*}{m_\tau (m_s+m_b)} \right| ^2. \end{aligned}$$

(34)

The present experimental upper limit is ${\mathcal {B}}(B_s\rightarrow \tau ^+\tau ^-)<0.0(3.4)\times 10^{-3}$ [35] and the SM prediction read ${\mathcal {B}}(B_s\rightarrow \tau ^+\tau ^-)<(7.73\pm 0.49)\times 10^{-7}$ [36]. The SM branching ratio of $B\rightarrow K\tau ^+\tau ^-$ is ${\mathcal {B}}(B\rightarrow K\tau ^+\tau ^-)=1.44(0.28)\times 10^{-7}$ where we use hadronic inputs from [37], and the experimental upper bound is ${\mathcal {B}}(B\rightarrow K\tau ^+\tau ^-)=(1.36\pm 0.71)\times 10^{-3}$ [38].

The leptoquark also contributes to the LFV observables ${\mathcal {B}}(B^+\rightarrow K^+\tau ^{\pm }\mu ^{\mp })$ and ${\mathcal {B}}(\tau \rightarrow \mu \phi )$. Following the simplified expressions given in [39] we get

$$\begin{aligned}&{\mathcal {B}}(B^+\rightarrow K^+\tau ^+\mu ^-) \nonumber \\&\quad \approx \frac{v^4}{m_{\mathrm{LQ}}^4}\bigg ( 0.50\big |\beta _L^{s\mu }\left( \beta _L^{b\tau }\right) ^*\big |^2 + 2.83 \big |\beta _L^{s\mu }\left( \beta _R^{b\tau }\right) ^*\big |^2,\nonumber \\&\qquad - 1.39 \mathrm{Re}\left[ \beta _L^{b\tau } \left( \beta _R^{b\tau }\right) ^*\right] \left| \beta _L^{s\mu }\right| ^2 \bigg ),\end{aligned}$$

(35)

$$\begin{aligned}&{\mathcal {B}}(B^+\rightarrow K^+\tau ^-\mu ^+) \approx \frac{v^4}{m_{\mathrm{LQ}}^4}0.50 \left| \beta _L^{b\mu }\left( \beta _L^{s\tau }\right) ^*\right| ^2 . \end{aligned}$$

(36)

The experimental upper limit is ${\mathcal {B}}(B^+\rightarrow K^+\tau ^+\mu ^-) \le 2.8\times 10^{-5}$ and ${\mathcal {B}}(B^+\rightarrow K^+\mu ^+\tau ^-) \le 4.5\times 10^{-5}$ [40]. For the $\tau \rightarrow \mu \phi$ decay, following [41] we get after neglecting the mass of the muon

$$\begin{aligned} {\mathcal {B}}(\tau \rightarrow \mu \phi )&= \frac{f_\phi ^2 m_\tau ^3}{32\pi \Gamma _\tau }\cdot \frac{1}{16m_{\mathrm{LQ}}^4} \left( 1-\frac{m^2_\phi }{m^2_\tau } \right) ^2 \left( 1+2\frac{m^2_\phi }{m^2_\tau } \right) \, \nonumber \\&\quad \times \left| \beta _L^{s\tau } \left( \beta _L^{s\mu }\right) ^*\right| ^2. \end{aligned}$$

(37)

The experimental upper limit from Belle [42] is ${\mathcal {B}}(\tau \rightarrow \mu \phi )\le (0.0\pm 5.1)\times 10^{-8}$. In the presence of right-handed coupling, $\tau \rightarrow \mu \gamma$ is also induced:

$$\begin{aligned} {\mathcal {B}}(\tau \rightarrow \mu \gamma ) = \frac{1}{\Gamma _\tau }\frac{\alpha _e}{64\pi ^4} \frac{m_\tau ^3m_b^2}{16m_{\mathrm{LQ}}^4} \left| \beta _R^{b\tau }\left( \beta _L^{b\mu }\right) \right| ^2\, . \end{aligned}$$

(38)

The experimental upper bound is ${\mathcal {B}}(\tau \rightarrow \mu \gamma )=0.0(3.0)\times 10^{-8}$ [11].

The charged current transition $b\rightarrow c\ell \nu$ also receives contributions from the vector leptoquark. Here the flavor of the final state neutrino in general may have flavors different from the accompanying lepton. The most general effective Hamiltonian for this transition is

$$\begin{aligned} \mathcal {H}^{b\rightarrow c\ell {\bar{\nu }}}_{\mathrm{eff}}&= \frac{2 V_{cb}}{v^2} \bigg ( left(1+{\mathcal {C}}_{V_1}^{\ell }right) {\mathcal {O}}_{V_1} + {\mathcal {C}}_{V_2}^{\ell }{\mathcal {O}}_{V_2}\,\nonumber \\&\quad + {\mathcal {C}}_{S_1}^{\ell }{\mathcal {O}}_{S_1} + {\mathcal {C}}_{S_2}^{\ell }{\mathcal {O}}_{S_2} + {\mathcal {C}}_T^{\ell }{\mathcal {O}}_T \bigg ), \end{aligned}$$

(39)

where the operators are given by

$$\begin{aligned} {\mathcal {O}}_{V_1}&= (\bar{c}_L\gamma ^\mu b_L)({\bar{\ell }}_L\gamma _\mu \nu _{L}),\quad {\mathcal {O}}_{V_2} = (\bar{c}_R\gamma ^\mu b_R)({\bar{\ell }}_L\gamma _\mu \nu _{L}),\nonumber \\ {\mathcal {O}}_{S_1}&= (\bar{c}_L b_R)({\bar{\ell }}_R \nu _{L})\, ,\quad {\mathcal {O}}_{S_2} = (\bar{c}_R b_L)({\bar{\ell }}_R \nu _{L}) ,\nonumber \\ {\mathcal {O}}_T&= (\bar{c}_R\sigma ^{\mu \nu }b_L)({\bar{\ell }}_R\sigma _{\mu \nu }\nu _{L}). \end{aligned}$$

(40)

In the SM all the Wilson coefficients ${\mathcal {C}}^{\ell }_{i} =0$. In the $U_1$ leptoquark model the only non-vanishing Wilson coefficients are

$$\begin{aligned}&{\mathcal {C}}^{\ell }_{V_1} = \frac{v^2}{4m_{\mathrm{LQ}}^2} \left( \beta ^{b\ell }_L\right) ^*\bigg (\beta _L^{b\ell } + \frac{V_{cs}}{V_{cb}} \beta _L^{s\ell } \bigg ),\end{aligned}$$

(41)

$$\begin{aligned}&{\mathcal {C}}^{\ell }_{S_1} = -\frac{v^2}{2m_{\mathrm{LQ}}^2} \left( \beta ^{b\ell }_L\right) ^*\bigg (\beta _L^{b\ell } + \frac{V_{cs}}{V_{cb}} \beta _L^{s\ell } \bigg ). \end{aligned}$$

(42)

The set of observables that we consider in this category are $R_D, R_{D^*}$ and the branching ratio ${\mathcal {B}}(B_c\rightarrow \tau \nu )$. The expressions for $R_D$ and $R_{D^*}$ are [43]

$$\begin{aligned} R_D&\approx R_D^{\mathrm{SM}}\bigg \{\left| 1+{\mathcal {C}}^{\tau }_{V_1}\right| ^2 + 1.54 \mathrm{Re}\big [\left( 1+{\mathcal {C}}^{\tau }_{V_1}\right) \left( {\mathcal {C}}^{\tau }_{S_1}\right) ^*\big ]\,\nonumber \\&\quad + 1.09\left| {\mathcal {C}}^{\tau }_{S_1}\right| ^2 \bigg \}, \end{aligned}$$

(43)

$$\begin{aligned} R_{D^*}&\approx R_{D^*}^{\mathrm{SM}}\bigg \{\left| 1+{\mathcal {C}}^{\tau }_{V_1}\right| ^2 + 0.13 \mathrm{Re}\big [\left( 1+{\mathcal {C}}^{\tau }_{V_1}\right) \left( {\mathcal {C}}^{\tau }_{S_1}\right) ^*\big ]\,\nonumber \\&\quad + 0.05\left| {\mathcal {C}}^{\tau }_{S_1}\right| ^2 \bigg \}. \end{aligned}$$

(44)

The HFLAV averages that use the most recent measurements of these two observables are $R_D=0.340\pm 0.030$ and $R_{D^*} = 0.295\pm 0.013$ [11]. The SM prediction of $R_{D}^{\mathrm{SM}}$ and $R_{D^*}^{\mathrm{SM}}$ are given in the Introduction section.

The $B_c\rightarrow \tau \nu$ branching ratio reads

$$\begin{aligned}&{\mathcal {B}}(B_c\rightarrow \tau \nu ) = \frac{\tau _{B_c} m_{B_c} f^2_{B_c} |V_{cb}|^2 }{16\pi v^4} m_\tau ^2 \left( 1 - \frac{m_\tau ^2}{m^2_{B_c}} \right) ^2 \Bigg | 1\nonumber \\&\quad + \frac{v^2}{4m_{\mathrm{LQ}}^2} \left( \beta _L^{b\tau }-\frac{2 m^2_{B_s} \beta _R^{b\tau }}{m_\tau (m_b+m_c)} \right) ^*\left( \beta _L^{b\tau } + \frac{V_{cs}}{V_{cb}} \beta _L^{s\tau } \right) \Bigg |^2. \end{aligned}$$

(45)

The most stringent constraint on $B_c\rightarrow \tau \nu$ come from LEP data from which Ref. [44] put the limit ${\mathcal {B}}(B_c\rightarrow \tau \nu )\le 10\%$. Another charged current observable in the $b\rightarrow u$ transition that we consider is

$$\begin{aligned} {\mathcal {B}}(B\rightarrow \tau \nu )&= {\mathcal {B}}(B\rightarrow \tau \nu )_{\mathrm{SM}} \bigg | 1+ \frac{v^2}{4m_{\mathrm{LQ}}^2} \nonumber \\&\quad \bigg ( \beta _L^{b\tau }\, -\frac{2 m^2_{B_s}}{m_\tau (m_b+m_c)} \beta _R^{b\tau } \bigg )^*\bigg ( \beta _L^{b\tau } + \frac{V_{cs}}{V_{ub}} \beta _L^{s\tau } \bigg ) \bigg |^2. \end{aligned}$$

(46)

Table 1 List of observables included in the fit

According to [45] ${\mathcal {B}}(B\rightarrow \tau \nu )\le (1.09 \pm 0.24)\times 10^{-4}$ and ${\mathcal {B}}(B\rightarrow \tau \nu )_\mathrm{SM} = (0.812 \pm 0.054)\times 10^{4}$ [46].

We now perform a $\chi ^2$ analysis to find the parameter space allowed by the above low energy observables listed in Table 1. The $\chi ^2$ is defined as

$$\begin{aligned} \chi ^2 = \sum _i \bigg ( \frac{({\mathcal {O}}_i^{\mathrm{expt}} - {\mathcal {O}}_i^{\mathrm{th}})}{\Delta {\mathcal {O}}_i^{\mathrm{expt}}} \bigg )^2\, , \end{aligned}$$

(47)

where ${\mathcal {O}}_i^{\mathrm{expt,(th)}}$ are the experimental (theoretical) values of the observables and $\Delta {\mathcal {O}}_i^{\mathrm{expt}}$ are the experimental errors. We minimize the $\chi ^2$ and choose a $2\sigma$ region about $\chi ^2_{\mathrm{min}}$. In this analysis we set mass of the leptoquark $m_{\mathrm{LQ}}=1.5$ TeV. In Fig. 1 the obtained parameter space is shown. For this parameter space, the $q^2$ distribution of the differential branching ratio and the lepton-side forward–backward asymmetry is shown in Fig. 2 for a set of benchmark values of the couplings. The plots are obtained for the central values of the form factors and other inputs. Due to our choice of the flavor structure (26) the $\Lambda _b\rightarrow \mu ^+\tau ^-$ receives contributions from VA type operators only while the $\Lambda _b\rightarrow \tau ^+\mu ^-$ mode receives contributions from both VA and SP operators. Since in our model ${\mathcal {C}}_V=-{\mathcal {C}}_A$, in the $\Lambda _b\rightarrow \Lambda \mu ^+\tau ^-$ mode the $A^\ell _{\mathrm{FB}}$ is independent of the couplings $\beta _L$ and the forward–backward asymmetry is entirely determined by the form factors and kinematic variables. Interestingly, the $\Lambda _b\rightarrow \Lambda \tau ^+\mu ^-$ mode has a $A^\ell _{\mathrm{FB}}$ zero-crossing which is absent in the $\Lambda _b\rightarrow \Lambda \mu ^+\tau ^-$ mode. For the obtained parameter space we also calculate the maximum and the minimum values of the branching ratio and $A^\ell _{\mathrm{FB}}$ integrated over the entire $q^2$ phase space,

$$\begin{aligned} \langle {\mathcal {B}}(\Lambda _b\rightarrow \Lambda \tau ^+\mu ^-)\rangle&= [1.55\times 10^{-9} ,7.83\times 10^{-6}],\end{aligned}$$

(48)

$$\begin{aligned} \langle {\mathcal {B}}(\Lambda _b\rightarrow \Lambda \mu ^+\tau ^-)\rangle&= [5.01\times 10^{-9}, 1.78\times 10^{-5}],\end{aligned}$$

(49)

$$\begin{aligned} \langle A^\ell _{\mathrm{FB}}(\Lambda _b\rightarrow \Lambda \tau ^+\mu ^-)\rangle&= [-0.2504,-0.003],\end{aligned}$$

(50)

$$\begin{aligned} \langle A^\ell _{\mathrm{FB}}(\Lambda _b\rightarrow \Lambda \mu ^+\tau ^-)\rangle&= -0.4040. \end{aligned}$$

(51)

Fig. 1

The parameter space (in blue) allowed by low energy observables for the vector leptoquark mass $m_{\mathrm{LQ}}=1.5$ TeV

Fig. 2

The $q^2$ distribution of the differential branching ratio and the lepton-side forward–backward asymmetry is shown for a set ($\beta ^{s\mu }_L=-0.031$, $\beta ^{s\tau }_L=0.433$, $\beta ^{b\mu }_L=-0.112$, $\beta ^{b\tau }_L=-0.957$, $\beta ^{b\tau }_R=-0.128$) of benchmark values of the $U_1$ leptoquark model parameters that are allowed by the low energy observables. The blue and the orange lines correspond to $\Lambda _b\rightarrow \Lambda \tau ^+\mu ^-$ and $\Lambda _b\rightarrow \Lambda \mu ^+\tau ^-$ modes, respectively

The large branching ratios of the order ${\mathcal {O}}(10^{-5},10^{-6})$ are induced by the large ranges of $\beta ^{q\tau }$ allowed by the current data. Such large ranges arise due to poor experimental bounds on modes such as $B_s\rightarrow \tau ^+\tau ^-, B^+\rightarrow K\tau ^+\tau ^-$. These branching ratios are accessible in the LHCb.

5 Summary

Lepton flavor violating decays are strictly forbidden in the Standard Model and therefore any observation is a smoking gun signal of physics beyond the Standard Model. In recent years a number of lepton flavor universality violating decays has been observed albeit of low statistical significance. Many physics beyond the Standard Models that has been constructed to explain the origin of flavor universality violating couplings can also give rise to flavor violating decays. Motivated by these results, in this paper we have explored lepton flavor violating $b\rightarrow s\ell _1^+\ell _2^-$ transition in $\Lambda _b\rightarrow \Lambda \ell _1^+\ell _2^-$ decay. In this paper we have presented a double differential distribution of the decay in terms of dilepton invariant mass squared $q^2$ and lepton angle $\theta _\ell$. From this distribution we have obtained the differential branching ratio and the lepton-side forward–backward asymmetry. We have studied these two observables in the vector leptoquark model $U_1\equiv (\mathbf 3,1 )_{2/3}$. The parameter space of the model has been constrained by low energy observables. Our predicted range of the branching ratio in the vector leptoquark model may be accessible by the LHCb.

Appendices

Appendix A: Transversity amplitudes

Corresponding to the effective Hamiltonian (7) the expressions of the transversity amplitudes read [22]

$$\begin{aligned} A^{L,(R)}_{\perp _1}= & {} -\sqrt{2}N \bigg ( f^V_\perp \sqrt{2s_-} {\mathcal {C}}^{L,(R)}_{\mathrm{VA+}} \bigg ), \end{aligned}$$

(A1)

$$\begin{aligned} A^{L,(R)}_{\Vert _1}= & {} \sqrt{2}N \bigg ( f^A_\perp \sqrt{2s_+} {\mathcal {C}}^{L,(R)}_{\mathrm{VA-}} \bigg ),\end{aligned}$$

(A2)

$$\begin{aligned} A^{L,(R)}_{\perp _0}= & {} \sqrt{2}N \bigg ( f^V_0 ({m_{\Lambda _b}}+ {m_{\Lambda }}) \sqrt{\frac{s_-}{q^2}} {\mathcal {C}}^{L,(R)}_{\mathrm{VA+}} \bigg ),\end{aligned}$$

(A3)

$$\begin{aligned} A^{L,(R)}_{\Vert _0}= & {} -\sqrt{2}N \bigg ( f^A_0 ({m_{\Lambda _b}}- {m_{\Lambda }}) \sqrt{\frac{s_+}{q^2}} {\mathcal {C}}^{L,(R)}_{\mathrm{VA-}} \bigg ),\end{aligned}$$

(A4)

$$\begin{aligned} A_{\perp t}= & {} -2\sqrt{2}N f^V_t ({m_{\Lambda _b}}- {m_{\Lambda }}) \sqrt{\frac{s_+}{q^2}} \left( {\mathcal {C}}_{A} + {\mathcal {C}}_A^\prime \right) ,\end{aligned}$$

(A5)

$$\begin{aligned} A_{\Vert t}= & {} 2\sqrt{2}N f^A_t ({m_{\Lambda _b}}+ {m_{\Lambda }}) \sqrt{\frac{s_-}{q^2}} \left( {\mathcal {C}}_{A} - {\mathcal {C}}_A^\prime \right) . \end{aligned}$$

(A6)

Here the normalization constant $N(q^2)$ is given by

$$\begin{aligned} N(q^2)&= \frac{V_{tb}V_{ts}^*\alpha _e}{\sqrt{2}v^2} \sqrt{ \tau _{\Lambda _b} \frac{q^2 \sqrt{\lambda \left( {m^2_{\Lambda _b}},{m^2_{\Lambda }},q^2\right) } }{2^{15} m^3_{\Lambda _b} \pi ^5} \beta _\ell \beta _\ell ^\prime },\nonumber \\ \beta _\ell&= \sqrt{1 - \frac{(m_1+m_2)^2}{q^2}}\,,\quad \beta _\ell ^\prime = \sqrt{1 - \frac{(m_1-m_2)^2}{q^2}}, \end{aligned}$$

(A7)

where $\lambda (a,b,c)=a^2 + b^2 + c^2 - 2(ab + bc + ca)$ and the Wilson coefficients are

$$\begin{aligned}&{\mathcal {C}}_{\mathrm{VA,+}}^{L(R)} = ({\mathcal {C}}_V{\mp } {\mathcal {C}}_A)+\left( {\mathcal {C}}_V^\prime {\mp } {\mathcal {C}}_A^\prime \right) , \end{aligned}$$

(A8)

$$\begin{aligned}&{\mathcal {C}}_{\mathrm{VA,-}}^{L(R)} = ({\mathcal {C}}_V{\mp } {\mathcal {C}}_A)-\left( {\mathcal {C}}_V^\prime {\mp } {\mathcal {C}}_A^\prime \right) . \end{aligned}$$

(A9)

The transversity amplitudes corresponding to the SP operators are [22]

$$\begin{aligned} A_{\mathrm{S\perp }}= & {} 2\sqrt{2}N f^V_t \frac{{m_{\Lambda _b}}- {m_{\Lambda }}}{m_b} \sqrt{s_+} \left( {\mathcal {C}}_S + {\mathcal {C}}_S^\prime \right) , \end{aligned}$$

(A10)

$$\begin{aligned} A_{\mathrm{S\Vert }}= & {} -2\sqrt{2}N f^A_t \frac{{m_{\Lambda _b}}+ {m_{\Lambda }}}{m_b} \sqrt{s_-}\left( {\mathcal {C}}_S - {\mathcal {C}}_S^\prime \right) ,\end{aligned}$$

(A11)

$$\begin{aligned} A_{\mathrm{P\perp }}= & {} -2\sqrt{2}N f^V_t \frac{{m_{\Lambda _b}}- {m_{\Lambda }}}{m_b} \sqrt{s_+} \left( {\mathcal {C}}_P + {\mathcal {C}}_P^\prime \right) ,\end{aligned}$$

(A12)

$$\begin{aligned} A_{\mathrm{P\Vert }}= & {} 2\sqrt{2}N f^A_t \frac{{m_{\Lambda _b}}+ {m_{\Lambda }}}{m_b} \sqrt{s_-} \left( {\mathcal {C}}_P - {\mathcal {C}}_P^\prime \right) . \end{aligned}$$

(A13)

Appendix B: Spinors in dilepton rest frame

We assume that the lepton $\ell _2^-$ is negatively charged and has four-momentum is $q_2^\mu =(E_1,\mathbf {q})$, while $\ell _1^+$ is positively charged and has four-momentum $q_1^\mu =(E_1,-\mathbf {q})$

$$\begin{aligned}&q_1^\mu \Big |_{2\ell } = (E_2, -|q_{2\ell }|\sin \theta _\ell , 0, -|q_{2\ell }|\cos \theta _\ell ), \end{aligned}$$

(B1)

$$\begin{aligned}&q_2^\mu \Big |_{2\ell } = (E_1, |q_{2\ell }|\sin \theta _\ell , 0, |q_{2\ell }|\cos \theta _\ell ), \end{aligned}$$

(B2)

with

$$\begin{aligned} |q_{2\ell }|= & {} \frac{\lambda ^{1/2}\left( q^2,m_1^2,m_2^2\right) }{2\sqrt{q^2}}\, ,\quad \quad E_1 = \frac{q^2+m_1^2-m_2^2}{2\sqrt{q^2}}\,,\nonumber \\ E_2= & {} \frac{q^2+m_2^2-m_1^2}{2\sqrt{q^2}}. \end{aligned}$$

(B3)

The explicit expressions of the lepton helicity amplitudes require us to calculate

$$\begin{aligned} \bar{u}_{\ell _2} (1{\mp }\gamma _5) v_{\ell _1},\quad {\bar{\epsilon }}^\mu (\lambda ) \bar{u}_{\ell _2} \gamma _\mu (1{\mp }\gamma _5) v_{\ell _1}. \end{aligned}$$

(B4)

Following [47] the explicit expressions of the spinor for the lepton $\ell _2^-$ is

$$\begin{aligned} u_{\ell _2}(\lambda )&= \begin{pmatrix} \sqrt{E_\ell +m_\ell } \chi ^u_\lambda \\ 2 \lambda \sqrt{E_\ell -m_\ell } \chi ^u_\lambda \end{pmatrix}, \quad \chi ^u_{+\frac{1}{2}} = \begin{pmatrix} \cos \frac{\theta _\ell }{2} \\ \sin \frac{\theta _\ell }{2} \end{pmatrix}\, , \nonumber \\ \chi ^u_{-\frac{1}{2}}&= \begin{pmatrix} -\sin \frac{\theta _\ell }{2} \\ \cos \frac{\theta _\ell }{2} \end{pmatrix}. \end{aligned}$$

(B5)

For the lepton $\ell _1^+$ which is moving in the opposite direction to $\ell _2$, the two component spinor $\chi ^v$ looks like

$$\begin{aligned} \chi ^v_{-\lambda } = \xi _\lambda \chi ^u_{\lambda },\quad \xi _\lambda = 2\lambda e^{-2i\lambda \phi }. \end{aligned}$$

(B6)

Hence we have

$$\begin{aligned} v_{\ell _1}(\lambda )&= \begin{pmatrix} \sqrt{E_\ell -m_\ell } \chi ^v_{-\lambda } \\ -2 \lambda \sqrt{E_\ell +m_\ell } \chi ^v_{-\lambda } \end{pmatrix}, \quad \chi ^v_{+\frac{1}{2}} = \begin{pmatrix} \sin \frac{\theta _\ell }{2} \\ -\cos \frac{\theta _\ell }{2} \end{pmatrix}\, \nonumber \\ \chi ^v_{-\frac{1}{2}}&= \begin{pmatrix} \cos \frac{\theta _\ell }{2} \\ \sin \frac{\theta _\ell }{2} \end{pmatrix}. \end{aligned}$$

(B7)

With these choices of lepton spinors we get the following expressions of the lepton helicity amplitudes $L^{\lambda _2,\lambda _1}_{L(R)}$ and $L^{\lambda _2,\lambda _1}_{L(R),\lambda }$:

$$\begin{aligned}&L_L^{{+\frac{1}{2}+\frac{1}{2}}} = \sqrt{q^2}\left( \beta _\ell ^\prime +\beta _\ell \right) ,\quad L_L^{{+\frac{1}{2}-\frac{1}{2}}} = 0,\quad L_L^{{-\frac{1}{2}+\frac{1}{2}}} = 0\, \nonumber \\&L_L^{{-\frac{1}{2}-\frac{1}{2}}} = \sqrt{q^2}\left( \beta _\ell ^\prime -\beta _\ell \right) ,\end{aligned}$$

(B8)

$$\begin{aligned}&L_R^{{+\frac{1}{2}+\frac{1}{2}}} = -\sqrt{q^2}\left( \beta _\ell ^\prime - \beta _\ell \right) ,\quad L_R^{{+\frac{1}{2}-\frac{1}{2}}} = 0,\quad L_R^{{-\frac{1}{2}+\frac{1}{2}}} = 0,\nonumber \\&L_R^{{-\frac{1}{2}-\frac{1}{2}}} = -\sqrt{q^2}\left( \beta _\ell ^\prime + \beta _\ell \right) ,\end{aligned}$$

(B9)

$$\begin{aligned}&L_{L,+1}^{{+\frac{1}{2}+\frac{1}{2}}} = \frac{1}{\sqrt{2}}\big [m_1\left( \beta _\ell ^\prime +\beta _\ell \right) + m_2\left( \beta _\ell ^\prime - \beta _\ell \right) \big ]{\sin \theta _\ell },\end{aligned}$$

(B10)

$$\begin{aligned}&L_{L,+1}^{{+\frac{1}{2}-\frac{1}{2}}} = -\sqrt{\frac{q^2}{2}} \left( {\beta ^\prime _\ell }-{\beta _\ell }\right) (1-{\cos \theta _\ell })\, ,\nonumber \\&L_{L,+1}^{{-\frac{1}{2}+\frac{1}{2}}} = \sqrt{\frac{q^2}{2}}\left( {\beta ^\prime _\ell }+{\beta _\ell }\right) (1+{\cos \theta _\ell })\, ,\end{aligned}$$

(B11)

$$\begin{aligned}&L_{L,+1}^{{-\frac{1}{2}-\frac{1}{2}}} = -\frac{1}{\sqrt{2}}\big [m_1({\beta ^\prime _\ell }- {\beta _\ell }) + m_2({\beta ^\prime _\ell }+ {\beta _\ell }) \big ],\end{aligned}$$

(B12)

$$\begin{aligned}&L_{R,+1}^{{+\frac{1}{2}+\frac{1}{2}}} = \frac{1}{\sqrt{2}}\bigg [ m_1\left( {\beta ^\prime _\ell }-{\beta _\ell }\right) + m_2\left( {\beta ^\prime _\ell }+{\beta _\ell }\right) \bigg ]{\sin \theta _\ell },\end{aligned}$$

(B13)

$$\begin{aligned}&L_{R,+1}^{{+\frac{1}{2}-\frac{1}{2}}} = -\sqrt{\frac{q^2}{2}} \left( {\beta ^\prime _\ell }+{\beta _\ell }\right) (1-{\cos \theta _\ell })\, ,\nonumber \\&L_{R,+1}^{{-\frac{1}{2}+\frac{1}{2}}} = \sqrt{\frac{q^2}{2}} \left( {\beta ^\prime _\ell }-{\beta _\ell }\right) (1+{\cos \theta _\ell })\, , \end{aligned}$$

(B14)

$$\begin{aligned}&L_{R,+1}^{{-\frac{1}{2}-\frac{1}{2}}} = -\frac{1}{\sqrt{2}} \big [ m_1\left( {\beta ^\prime _\ell }+{\beta _\ell }\right) + m_2\left( {\beta ^\prime _\ell }-{\beta _\ell }\right) \bigg ]{\sin \theta _\ell },\end{aligned}$$

(B15)

$$\begin{aligned}&L_{L,-1}^{{+\frac{1}{2}+\frac{1}{2}}} = -\frac{1}{\sqrt{2}} \big [m_1\left( {\beta ^\prime _\ell }+{\beta _\ell }\right) +m_2\left( {\beta ^\prime _\ell }-{\beta _\ell }\right) \big ],\end{aligned}$$

(B16)

$$\begin{aligned}&L_{L,-1}^{{+\frac{1}{2}-\frac{1}{2}}} = -\sqrt{\frac{q^2}{2}} \left( {\beta ^\prime _\ell }-{\beta _\ell }\right) (1+{\cos \theta _\ell })\, ,\nonumber \\&L_{L,-1}^{{-\frac{1}{2}+\frac{1}{2}}} = \sqrt{\frac{q^2}{2}} \left( {\beta ^\prime _\ell }+{\beta _\ell }\right) (1-{\cos \theta _\ell })\, ,\end{aligned}$$

(B17)

$$\begin{aligned}&L_{L,-1}^{{-\frac{1}{2}-\frac{1}{2}}} = \frac{1}{\sqrt{2}} \big [m_1\left( {\beta ^\prime _\ell }-{\beta _\ell }\right) + m_2\left( {\beta ^\prime _\ell }+{\beta _\ell }\right) \big ]{\sin \theta _\ell },\end{aligned}$$

(B18)

$$\begin{aligned}&L_{R,-1}^{{+\frac{1}{2}+\frac{1}{2}}} = -\frac{1}{\sqrt{2}} \big [m_1\left( {\beta ^\prime _\ell }-{\beta _\ell }\right) + m_2\left( {\beta ^\prime _\ell }+{\beta _\ell }\right) \big ]{\sin \theta _\ell },\end{aligned}$$

(B19)

$$\begin{aligned}&L_{R,-1}^{{+\frac{1}{2}-\frac{1}{2}}} = \sqrt{\frac{q^2}{2}}\left( {\beta ^\prime _\ell }+{\beta _\ell }\right) (1+{\cos \theta _\ell }),\nonumber \\&L_{R,-1}^{{-\frac{1}{2}+\frac{1}{2}}} = \sqrt{\frac{q^2}{2}}\left( {\beta ^\prime _\ell }-{\beta _\ell }\right) (1-{\cos \theta _\ell }),\end{aligned}$$

(B20)

$$\begin{aligned}&L_{R,-1}^{{-\frac{1}{2}-\frac{1}{2}}} = \frac{1}{\sqrt{2}}\big [m_1\left( {\beta ^\prime _\ell }+{\beta _\ell }\right) + m_2\left( {\beta ^\prime _\ell }-{\beta _\ell }\right) \big ]{\sin \theta _\ell },\end{aligned}$$

(B21)

$$\begin{aligned}&L_{L,0}^{{+\frac{1}{2}+\frac{1}{2}}} = -\big [m_1\left( {\beta ^\prime _\ell }+{\beta _\ell }\right) -m_2\left( {\beta ^\prime _\ell }-{\beta _\ell }\right) \big ]{\cos \theta _\ell },\end{aligned}$$

(B22)

$$\begin{aligned}&L_{L,0}^{{+\frac{1}{2}-\frac{1}{2}}} = \sqrt{q^2}\left( {\beta ^\prime _\ell }-{\beta _\ell }\right) {\cos \theta _\ell },\nonumber \\&L_{L,0}^{{-\frac{1}{2}+\frac{1}{2}}}=\sqrt{q^2}\left( {\beta ^\prime _\ell }+{\beta _\ell }\right) {\cos \theta _\ell },\end{aligned}$$

(B23)

$$\begin{aligned}&L_{L,0}^{{-\frac{1}{2}-\frac{1}{2}}} = \big [m_1\left( {\beta ^\prime _\ell }-{\beta _\ell }\right) + m_2\left( {\beta ^\prime _\ell }+{\beta _\ell }\right) \big ]{\cos \theta _\ell }\, ,\end{aligned}$$

(B24)

$$\begin{aligned}&L_{R,0}^{{+\frac{1}{2}+\frac{1}{2}}} = -\big [m_1\left( {\beta ^\prime _\ell }-{\beta _\ell }\right) + m_2\left( {\beta ^\prime _\ell }+{\beta _\ell }\right) \big ]{\cos \theta _\ell },\end{aligned}$$

(B25)

$$\begin{aligned}&L_{R,0}^{{+\frac{1}{2}-\frac{1}{2}}} = \sqrt{q^2}\left( {\beta ^\prime _\ell }+{\beta _\ell }\right) {\cos \theta _\ell }\,,\nonumber \\&L_{R,0}^{{-\frac{1}{2}+\frac{1}{2}}} =\sqrt{q^2}\left( {\beta ^\prime _\ell }-{\beta _\ell }\right) {\sin \theta _\ell }\,,\end{aligned}$$

(B26)

$$\begin{aligned}&L_{R,0}^{{-\frac{1}{2}-\frac{1}{2}}} = \big [m_1\left( {\beta ^\prime _\ell }+{\beta _\ell }\right) + m_2\left( {\beta ^\prime _\ell }-{\beta _\ell }\right) \big ]{\cos \theta _\ell },\end{aligned}$$

(B27)

$$\begin{aligned}&L_{L,0}^{{+\frac{1}{2}+\frac{1}{2}}} = \big [m_1\left( {\beta ^\prime _\ell }+{\beta _\ell }\right) + m_2\left( {\beta ^\prime _\ell }-{\beta _\ell }\right) \big ]\, ,\nonumber \\&L_{L,0}^{{+\frac{1}{2}-\frac{1}{2}}} = L_{L,0}^{{-\frac{1}{2}+\frac{1}{2}}} = 0,\end{aligned}$$

(B28)

$$\begin{aligned}&L_{L,0}^{{-\frac{1}{2}-\frac{1}{2}}} = \big [m_1\left( {\beta ^\prime _\ell }-{\beta _\ell }\right) + m_2\left( {\beta ^\prime _\ell }+{\beta _\ell }\right) \big ],\end{aligned}$$

(B29)

$$\begin{aligned}&L_{R,0}^{{+\frac{1}{2}+\frac{1}{2}}} = -\big [m_1\left( {\beta ^\prime _\ell }-{\beta _\ell }\right) + m_2\left( {\beta ^\prime _\ell }+{\beta _\ell }\right) \big ]\, ,\nonumber \\&L_{R,0}^{{+\frac{1}{2}-\frac{1}{2}}} = L_{R,0}^{{-\frac{1}{2}+\frac{1}{2}}} = 0,\end{aligned}$$

(B30)

$$\begin{aligned}&L_{R,0}^{{-\frac{1}{2}-\frac{1}{2}}} = -\left[ m_1\left( {\beta ^\prime _\ell }+{\beta _\ell }\right) + m_2\left( {\beta ^\prime _\ell }-{\beta _\ell }\right) \right] . \end{aligned}$$

(B31)