# Time-reversal asymmetries in $\Lambda_{b}\to\Lambda(\to
p\pi^{-})\ell^{+}\ell^{-}$

Chao-Qiang Geng, Chia-Wei Liu, Zheng-Yi<EMAIL_ADDRESS>School of Fundamental Physics and Mathematical Sciences, Hangzhou Institute
for Advanced Study, UCAS, Hangzhou 310024, China
University of Chinese Academy of Sciences, 100190 Beijing, China

###### Abstract

We study the decays of $\Lambda_{b}\to\Lambda(\to p\pi^{-})\ell^{+}\ell^{-}$
with $\ell=(e,\mu,\tau)$. In particular, we examine the full angular
distributions with polarized $\Lambda_{b}$ and identify the time-reversal
asymmetries or T-odd observables. By using the homogeneous bag model, we find
that the decay branching fractions of $\Lambda_{b}\to\Lambda\ell^{+}\ell^{-}$
are $(9.1\pm 2.5,7.9\pm 1.8,2.1\pm 0.2)\times 10^{-7}$ for
$\ell=(e,\mu,\tau)$, respectively. In addition, we obtain that
$A_{FB}^{\ell}=-0.369\pm 0.007$ and $A_{FB}^{h}=-0.333\pm 0.004$, averaged in
the range of $15\leq q^{2}\leq 20~{}\text{GeV}^{2}$. These results are well
consistent with the current experimental data. We also explore the T-odd
observables in $\Lambda_{b}\to\Lambda(\to p\pi^{-})\mu^{+}\mu^{-}$, which are
sensitive to new physics (NP). Explicitly, we illustrate that the current
experimental measurement from one of the T-odd observables favors the
existence of NP, such as the extra $Z$-boson model.

##  I Introductions

The CP violating observables in $b\to s\ell^{+}\ell^{-}$ with
$\ell=(e,\mu,\tau)$ play important roles to search for new physics (NP) as
they are highly suppressed in the standard model (SM) Bobeth:2011gi ;
Kruger:1999xa ; Altmannshofer:2008dz ; Bobeth:2008ij ; LHCb:2017slr ;
Fleischer:2017ltw ; Kindra:2018ayz . In recent years, special attentions have
been given to the decays of $B\to K^{(*)}\mu^{+}\mu^{-}$ and
$B_{s}\to\phi\mu^{+}\mu^{-}$ LHCb:2013tgx ; LHCb:2014cxe ; LHCb:2013ghj .
Benefited by the experimental developments, precise measurements of the
angular observables are now accessible CMS:2015bcy ; ATLAS:2018gqc ;
LHCb:2020lmf ; LHCb:2021xxq ; LHCb:2020gog ; CMS:2017rzx ; LHCb:exp ;
LHCb:2015svh ; LHCb:2018angular . These observables are useful in
disentangling the helicities, providing reliable methods to probe the Lorentz
structure of NP Buchalla:1995vs ; Mott:2011cx ; Roy:2017dum ; Das:2018iap ;
Aliev:2002nv ; Huang:1998ek ; gutsche ; Boer:2014kda . Besides, the ratios of
$R_{K^{(*)}}\equiv\Gamma(B\to K^{(*)}\mu^{+}\mu^{-})/\Gamma(B\to
K^{(*)}e^{+}e^{-})$ were measured, where discrepancies against the SM were
given. In particular, 3.1$\sigma$ and 2.5$\sigma$ deviations have been found
in $R_{K}(1.1\text{GeV}^{2}\leq q^{2}\leq 6.0\text{GeV}^{2})$ and
$R_{K^{*}}(0.045\text{GeV}^{2}\leq q^{2}\leq 6.0\text{GeV}^{2})$ LHCb:2021trn
; LHCb:2017avl , showing that the lepton universality may be violated by NP.
Very recently, a global fit of $b\to s\ell^{+}\ell^{-}$ with the $B$ meson
experiments has been performed SinghChundawat:2022zdf , and the large complex
Wilson coefficients have been demonstrated to be permitted by the current
experimental data.

The baryonic decays of $\Lambda_{b}\to\Lambda(\to p\pi^{-})\ell^{+}\ell^{-}$
are interesting for several reasons. For polarized $\Lambda_{b}$, the decays
of $\Lambda_{b}\to\Lambda(\to p\pi^{-})\ell^{+}\ell^{-}$ provide dozens of
angular observables, which are three times more than those in $B\to
K\mu^{+}\mu^{-}$. The polarization fraction $(P_{b})$ of $\Lambda_{b}$ is
reported as $(6\pm 7)\%$ at the center of mass energy 7 TeV of $pp$ collisions
LHCb:2013hzx . The full angular distribution of $\Lambda_{b}\to\Lambda(\to
p\pi^{-})\mu^{+}\mu^{-}$ has been measured at LHCb LHCb:2018angular . Notably,
the experiment obtains that one of the physical observables is given by

$K_{10}=-0.045\pm 0.037\pm 0.006\,,$ (1)

which deviates to the SM prediction of $K_{10}\approx 0$ by $1.2\sigma$. It is
reasonable to expect that the precision will be improved in the forthcoming
update. In this work, we will show explicitly that $K_{10}$ is an T-odd
quantity, which can be sizable in the presence of NP.

On the theoretical aspect, the angular distributions of
$\Lambda_{b}\to\Lambda\mu^{+}\mu^{-}$ have been studied intensively gutsche ;
Boer:2014kda ; Blake:2017angular . In particular, an analysis of NP with real
Wilson coefficients has been performed in Ref. Blake:2019guk , in which
$P_{b}=(0\pm 5)\%$ is found at $1\sigma$ confidence level. In this work, we
would like to focus on the time-reversal (T) violating observables induced by
the complex NP Wilson coefficients. In comparison to the CP violating
quantities, the T violating ones do not require strong phases. In the leptonic
decays, this feature is very useful as strong phases are often negligible.

This paper is organized as follows. In Sec. II , we decompose
$\Lambda_{b}\to\Lambda\ell^{+}\ell^{-}$ into products of two-body decays. In
Sec. III , we construct T-odd observables. In Sec. VI, we briefly review the
angular distributions of $\Lambda_{b}\to\Lambda(\to
p\pi^{-})\ell^{+}\ell^{-}$, and identify the T-odd observables. In Sec. V, we
give the numerical results from the homogeneous bag model (HBM). We conclude
the study in Sec. VI.

##  II Helicity amplitudes

The amplitudes of $\Lambda_{b}\to\Lambda\ell^{+}\ell^{-}$, induced by the
transitions of $b\to s\ell^{+}\ell^{-}$ at the quark level, are given as
Buchalla:2000sk

$\displaystyle\frac{G_{F}}{\sqrt{2}}\frac{\alpha
V^{*}_{ts}V_{tb}}{2\pi}\left[\langle\Lambda|\bar{s}j_{1}^{\mu}b|\Lambda_{b}\rangle\bar{\ell}\gamma_{\mu}\ell+\langle\Lambda|\bar{s}j_{2}^{\mu}b|\Lambda_{b}\rangle\bar{\ell}\gamma_{\mu}\gamma_{5}\ell\right],$
(2)

where $G_{F}$ is the Fermi constant, $V_{ts,tb}$ are the Cabibbo-Kobayashi-
Maskawa (CKM) matrix elements,

$\displaystyle
j_{1}^{\mu}=(C_{9}^{eff}+C^{\text{NP}}_{9})L^{\mu}-\frac{2m_{b}}{q^{2}}C_{7\gamma}^{eff}i\sigma^{\mu
q}(1+\gamma_{5})+(C_{L}+C_{R})R^{\mu}\,,$ (3) $\displaystyle
j_{2}^{\mu}=(C_{10}+C^{\text{NP}}_{10})L^{\mu}+(C_{R}-C_{L})R^{\mu}\,,$

$C^{(eff)}$ are the (effective) Wilson coefficients, $\sigma^{\mu
q}=i(\gamma^{\mu}\gamma^{\nu}-\gamma^{\nu}\gamma^{\mu})q_{\nu}/2$ with
$q=(q^{0},\vec{q}~{})$ the four-momentum of $\ell^{+}\ell^{-}$,
$L^{\mu}=\gamma^{\mu}(1-\gamma_{5})$, $R^{\mu}=\gamma^{\mu}(1+\gamma_{5})$,
and $m_{q}$ stands for the quark mass. The first (second) term in Eq. (2) can
be interpreted as $\Lambda_{b}\to\Lambda j_{eff}^{1(2)}$ followed by
$j_{eff}^{1(2)}\to\ell^{+}\ell^{-}$, where $j_{eff}^{1(2)}$ is an effective
off-shell (axial) vector boson, conserving the parity in its cascade decays,
and $j^{\mu}_{1,2}$ are the couplings of $b-s-j_{eff}^{1,2}$. Alternatively,
the interpretation can also be rephrased as $\Lambda_{b}\to\Lambda
j_{eff}^{R,L}$, where $j_{eff}^{R(L)}$ couples only to the right-handed (left-
handed) leptons, given as

$\displaystyle\frac{G_{F}}{\sqrt{2}}\frac{\alpha
V^{*}_{ts}V_{tb}}{2\pi}\left[\langle\Lambda|\bar{s}j_{+}^{\mu}b|\Lambda_{b}\rangle\bar{\ell}R_{\mu}\ell+\langle\Lambda|\bar{s}j_{-}^{\mu}b|\Lambda_{b}\rangle\bar{\ell}L_{\mu}\ell\right],$
(4)

where $j_{\pm}^{\mu}=(j_{1}^{\mu}\pm j^{\mu}_{2})/2$. In the SM,
$C^{\text{NP}}_{9,10}=C_{L,R}=0$ and the others are gutsche ; faustov

$\displaystyle C_{7\gamma}^{eff}$ $\displaystyle=-0.313,$ (5) $\displaystyle
C_{9}^{eff}$
$\displaystyle=C_{9}+h(\frac{m_{c}}{m_{b}},\frac{q^{2}}{m_{b}^{2}})-\frac{1}{2}h(1,\frac{q^{2}}{m_{b}^{2}})(4C_{3}+4C_{4}+3C_{5}+C_{6})$
$\displaystyle-\frac{1}{2}h(0,\frac{q^{2}}{m_{b}^{2}})(C_{3}+3C_{4})+\frac{2}{9}(3C_{3}+C_{4}+3C_{5}+C_{6}),$

where

$\displaystyle\begin{split}h(\frac{m_{c}}{m_{b}},\frac{q^{2}}{m_{b}})&=-\frac{8}{9}\ln\frac{m_{c}}{m_{b}}+\frac{8}{27}+\frac{4}{9}x-\frac{2}{9}(2+x)\\\
&\times|1-x|^{1/2}\left\\{\begin{aligned}
\left(\ln\big{|}\frac{\sqrt{1-x}+1}{\sqrt{1-x}-1}\big{|}-i\pi\right),&~{}~{}\text{for}~{}~{}x\
<1,\\\
2\text{arctan}\frac{1}{\sqrt{x-1}},&~{}~{}\text{for}~{}~{}x>1,\end{aligned}\right.\\\
h(0,\frac{q^{2}}{m_{b}})&=\frac{8}{27}-\frac{4}{9}\ln\frac{q^{2}}{m_{b}}+\frac{4}{9}i\pi,\end{split}$
(6)

and $x=4m_{c}^{2}/q^{2}$. Their explicit values can be found in Ref. faustov .

As the parity is conserved in $j_{eff}^{1,2}\to\ell^{+}\ell^{-}$, it is easier
to obtain the angular distributions with the $j_{eff}^{1,2}$ interpretations.
However, to examine NP, the second interpretation with $j_{eff}^{R,L}$ is more
preferable as NP is likely to couple with the leptons with the same
handedness. We note that physical quantities are of course independent of the
interpretations. For our purpose, the angular distributions are studied with
$j_{eff}^{1,2}$, whereas NP with $j_{eff}^{R,L}$.

By decomposing the Minkowski metric as

$g^{\mu\nu}=\epsilon_{t}^{\mu}\epsilon_{t}^{*\nu}-\sum_{\lambda=0,\pm}\epsilon_{\lambda}^{\mu}\epsilon_{\lambda}^{*\nu}\,,$
(7)

we arrive at

$\frac{G_{F}}{\sqrt{2}}\frac{\alpha
V^{\dagger}_{ts}V_{tb}}{2\pi}\sum_{m=1,2}\left(L_{t}^{m}B_{t}^{m}-\sum_{\lambda=0,\pm}L_{\lambda}^{m}B_{\lambda}^{m}\right)\,,$
(8)

where

$\displaystyle
B_{\lambda_{m}}^{m}=\epsilon_{\lambda_{m}}^{\ast\mu}\langle\Lambda|\bar{s}j_{m}^{\mu}b|\Lambda_{b}\rangle\,,$
$\displaystyle~{}~{}~{}L_{\lambda_{m}}^{1}=\epsilon_{\lambda_{m}}^{\mu}\bar{u}_{\ell}\gamma_{\mu}v\,,$
$\displaystyle~{}~{}~{}L_{\lambda_{m}}^{2}=\epsilon_{\lambda_{m}}^{\mu}\bar{u}_{\ell}{\gamma_{\mu}\gamma_{5}}v\,,$
(9)

${\lambda_{m}}=(t,0,\pm)$ is the helicity of $j_{eff}^{m}$ with $t$ indicating
spin-0 off-shell contributions, and $\epsilon$ are the polarization vectors of
$j_{eff}^{m}$, given as TRSEMI

${\epsilon}^{\mu}_{\pm}=\frac{1}{\sqrt{2}}(0,\pm
1,i,0)^{T}\,,\quad{\epsilon}_{0}^{\mu}=(0,0,0,-1)^{T}\,,\quad{\epsilon}^{\mu}_{t}=(-1,0,0,0)^{T}\,,$
(10)

and

${\epsilon}^{\mu}_{\pm}=\frac{1}{\sqrt{2}}(0,\mp
1,i,0)^{T}\,,\quad{\epsilon}_{0}^{\mu}=\frac{1}{\sqrt{q^{2}}}(|\vec{q}\,|,0,0,-q^{0})^{T}\,,\quad{\epsilon}^{\mu}_{t}=-\frac{1}{\sqrt{q^{2}}}q^{\mu},$
(11)

in the center of mass (CM) frames of $j_{eff}^{m}$ and $\Lambda_{b}$,
respectively. In Eq. (8), the amplitudes are decomposed as the products of
Lorentz scalars, where $B_{\lambda_{m}}$ and $L_{\lambda_{m}}$ describe
$\Lambda_{b}\to\Lambda j_{eff}^{m}$ and $j_{eff}^{m}\to\ell^{+}\ell^{-}$,
respectively, reducing the three-body problems to two-body ones.

To deal with the spins, we adopt the helicity approach. The projection
operators in the $SO(3)$ rotational $(SO(3)_{R})$ group are given by

$|J,M\rangle\langle J,N|=\frac{2J+1}{8\pi^{2}}\int d\phi d\theta d\psi
R_{z}(\phi)R_{y}(\theta)R_{z}(\psi)D^{J\dagger}(\phi,\theta,\psi)^{N}\,_{M}\,,$
(12)

where $N$ and $M$ are the angular momenta toward the $\hat{z}$ direction, the
Wigner-$D$ matrices are defined by

$D^{J}(\phi,\theta,\psi)^{M}\,_{N}\left\langle
J,N|J,N\right\rangle=\left\langle
J,M\left|R_{z}(\phi)R_{y}(\theta)R_{z}(\psi)\right|J,N\right\rangle\,,$ (13)

and $R_{y(z)}$ are the rotation operator pointing toward $\hat{y}(\hat{z})$.
We note that it is important for Eq. (12) to be a linear superposition of
$R_{y,z}$, which commutes with scalar operators. In the following, we take the
shorthand notation of $D^{J}(\phi,\theta)\equiv D^{J}(\phi,\theta,0)$.

The simplest two-particle state with a nonzero momentum is defined by

$|p\hat{z},\lambda_{1},\lambda_{2}\rangle\equiv
L_{z}|\vec{p}=0,J_{z}=\lambda_{1}\rangle_{1}\otimes
L_{z}^{\prime}|\vec{p}=0,J_{z}=-\lambda_{2}\rangle_{2}\,,$ (14)

where $\lambda_{1,2}$ are the helicities, the subscripts denote the particles,
and $L^{(\prime)}_{z}$ is the Lorentz boost, which brings the first (second)
particle to $(-)p\hat{z}$. As $L_{z}^{(\prime)}$ commutes with $R_{z}$, the
state defined by Eq. (14) is an eigenstate of $J_{z}=\lambda_{1}-\lambda_{2}$.
Plugging Eq. (12) into Eq. (14) with $N=\lambda_{1}-\lambda_{2}$, we arrive at

$\displaystyle|\vec{p}\,^{2},\lambda_{1},\lambda_{2};J,J_{z}\rangle=\frac{2J+1}{4\pi}\int
d\phi d\cos\theta
R_{z}(\phi)R_{y}(\theta)|p\hat{z},\lambda_{1},\lambda_{2}\rangle_{1,2}D^{J*}(\phi,\theta)^{J_{z}}\,_{N}\,,$
(15)

which expresses the angular momentum eigenstate as the linear superposition of
the three-momentum ones. Conversely, we have

$|p\hat{z},\lambda_{1},\lambda_{2}\rangle=\sum_{J}|\vec{p}\,^{2},\lambda_{1},\lambda_{2};J,N\rangle\,.$
(16)

Note that the identities of Eqs. (15) and (16) purely come from the
mathematical consideration. The simplification happens when the angular
momentum conservation is considered. At the CM frames of $\Lambda_{b}$ and
$j_{eff}^{m}$, it is clear that only $J=1/2$ and $J=(0,1)$ need to be
considered for the $\Lambda j_{eff}^{m}$ and $\ell^{+}\ell^{-}$ systems,
respectively.

Utilizing Eq. (16), we have that

$\langle\vec{p}\,^{2},\lambda_{1},\lambda_{2};J,N|{\cal
S}|J,J_{z};i\rangle=\langle p\hat{z},\lambda_{1},\lambda_{2}|{\cal
S}|J,J_{z};i\rangle\,,$ (17)

where ${\cal S}$ is an arbitrary scalar operator, and $|J,J_{z};i\rangle$
stands for an arbitrary initial state. In Eq. (17), the final state in the
left side possesses a definite angular momentum, which is irreducible under
$SO(3)_{R}$, i.e. it contains only the dynamical details. On the contrary, the
one in the right side is three-momentum eigenstate, containing less physical
insights but providing a way to compute the helicity amplitude.

Let us return to $\Lambda_{b}\to\Lambda j_{eff}^{m}$ and
$j_{eff}^{m}\to\ell^{+}\ell^{-}$. We take the uppercase and lowercase of $H$
and $h$ for the helicity amplitudes of $\Lambda_{b}\to\Lambda j_{eff}^{m}$ and
$j_{eff}^{m}\to\ell^{+}\ell^{-}$, respectively. To be explicit, we have

$\displaystyle H_{\lambda_{\Lambda}\lambda_{m}}^{m}$
$\displaystyle=B_{\lambda_{m}}\left(\lambda_{\Lambda_{b}}=\lambda_{\Lambda}-\lambda_{m},\lambda_{\Lambda},\vec{p}_{\Lambda}=-\vec{q}=|\vec{p}_{\Lambda}|\hat{z}\right)\,,$
(18) $\displaystyle h_{0,\lambda_{+}\lambda_{-}}^{m}$
$\displaystyle=L_{t}^{m}(\lambda_{+},\lambda_{-},\vec{q}=0,\vec{p}_{+}=-\vec{p}_{-}=|\vec{p}_{+}|\hat{z})\,,$
$\displaystyle h_{1,\lambda_{+}\lambda_{-}}^{m}$
$\displaystyle=L_{\lambda_{+}-\lambda_{-}}^{m}(\lambda_{+},\lambda_{-},\vec{q}=0,\vec{p}_{+}=-\vec{p}_{-}=|\vec{p}_{+}|\hat{z})\,,$

where $\lambda_{\Lambda_{b}}$ corresponds to the angular momentum of
$\Lambda_{b}$, $(\lambda_{\Lambda},\lambda_{\pm})$ are the helicities of
$(\Lambda,\ell^{\pm})$, and $\vec{p}_{\Lambda}$ and $\vec{p}_{\pm}$ are the
3-momentua of $\Lambda$ and $\ell^{\pm}$ in the CM frame of $\Lambda_{b}$ and
$j_{eff}^{m}$, respectively. Theoretically speaking, the dynamical parts of
the amplitudes are extracted by Eq. (17), whereas the kinematic dependencies
are governed by $D^{J}$.

For compactness, we take the abbreviations

$\displaystyle|a^{m}_{\pm}\rangle=|\vec{p}\,^{2},\pm 1/2,0;J,J_{z}\rangle,$
$\displaystyle|b^{m}_{\pm}\rangle=|\vec{p}\,^{2},\mp 1/2,\mp
1;J,J_{z}\rangle,{}{}{}$ $\displaystyle|c^{m}_{\pm}\rangle=|\vec{p}\,^{2},\pm
1/2,t;J,J_{z}\rangle~{}~{}~{}$ (19) $\displaystyle
a^{m}_{\pm}=H^{m}_{\pm\frac{1}{2}0}=\langle a_{\pm}^{m}|{\cal
S}_{eff}|\Lambda_{b}\rangle,$ $\displaystyle
b^{m}_{\pm}=H^{m}_{\mp\frac{1}{2}\mp 1}=\langle a_{\pm}|{\cal
S}_{eff}|\Lambda_{b}\rangle,$
$\displaystyle~{}~{}c^{m}_{\pm}=H^{m}_{\pm\frac{1}{2}t}=\langle
c_{\pm}^{m}|{\cal S}_{eff}|\Lambda_{b}\rangle,$

where ${\cal S}_{eff}$ is the transition operator responsible for
$\Lambda_{b}\to\Lambda j_{eff}^{m}$, and $J_{z}$ is not written down
explicitly. The artificial ${\cal S}_{eff}$ is needed to interpret
$\Lambda_{b}\to\Lambda\ell^{+}\ell^{-}$ as products of two-body ones. For the
$\Lambda_{b}\to\Lambda j_{eff}^{R,L}$ interpretation, the helicity amplitudes
are

$\displaystyle
a_{\pm}^{R}=\frac{1}{\sqrt{2}}(a_{\pm}^{1}+a_{\pm}^{2})\,,~{}~{}~{}a_{\pm}^{L}=\frac{1}{\sqrt{2}}(a_{\pm}^{1}-a_{\pm}^{2})\,,$
$\displaystyle
b_{\pm}^{R}=\frac{1}{\sqrt{2}}(b_{\pm}^{1}+b_{\pm}^{2})\,,~{}~{}~{}~{}b_{\pm}^{L}=\frac{1}{\sqrt{2}}(b_{\pm}^{1}-b_{\pm}^{2})\,,$
$\displaystyle
c_{\pm}^{R}=\frac{1}{\sqrt{2}}(c_{\pm}^{1}+c_{\pm}^{2})\,,~{}~{}~{}~{}c_{\pm}^{L}=\frac{1}{\sqrt{2}}(c_{\pm}^{1}-c_{\pm}^{2})\,.$
(20)

##  III T-odd observables

From Eq. (LABEL:eq3), we see that the NP contributions are absorbed into the
couplings of $b-s-j_{eff}^{r}$, while the Lorentz structures of
$j_{eff}^{r}\to\ell^{+}\ell^{-}$ are simple with $r=(1,2,R,L)$. Thus, to
discuss the NP effects, it is sufficient to study $\Lambda_{b}\to\Lambda
j_{eff}^{r}$.

The most simple T-odd operator in $\Lambda_{b}\to\Lambda j_{eff}^{m}$ is
defined as TRlamV

$\hat{T}=(\vec{s}_{\Lambda}\times\vec{s}_{m})\cdot\hat{p}_{\Lambda},$ (21)

$\vec{s}_{\Lambda}$ and $\vec{s}_{m}$ are the spin operators of $\Lambda$ and
$j^{m}_{eff}$, respectively, and $\hat{p}_{\Lambda}$ is the unit vector of
$\vec{p}_{\Lambda}$. The spin operators can only be defined for the massive
objects, given as

$M\vec{s}=P^{0}\vec{J}-\vec{p}\times\vec{K}-\frac{1}{P^{0}+M}\vec{p}(\vec{p}\cdot\vec{J})\,,$
(22)

where $M$ is the particle mass, and $P^{0}$, $\vec{p}$, $\vec{J}$ and
$\vec{K}$ are the time translation, space translation, rotation and Lorentz
boost generators, respectively. As $(\vec{p},\vec{J})$ and $\vec{K}$ are T-odd
and T-even, respectively, $\vec{s}$ is T-odd. In addition, $\vec{s}$ satisfies
the relations

$\displaystyle\vec{s}\cdot\vec{p}=\vec{J}\cdot\vec{p}\,,~{}~{}~{}[s_{i},s_{j}]=i\epsilon^{ijk}\epsilon_{k}\,,~{}~{}~{}[s_{i},p_{j}]=0\,,$
(23)
$\displaystyle\vec{s}\exp(i\vec{K}\cdot\vec{\omega})|\vec{p}=0,J_{z}=M\rangle=\exp(i\vec{K}\cdot\vec{\omega})\vec{J}|\vec{p}=0,J_{z}=M\rangle\,,$

with arbitrary $\vec{\omega}$. The key of solving the eigenstates of $\hat{T}$
relies on that $\hat{T}$ is a scalar operator. We have

$\displaystyle\hat{T}|\vec{p}\,^{2},\lambda_{1},\lambda_{2};J,J_{z}\rangle$
(24) $\displaystyle~{}~{}~{}=\frac{2J+1}{4\pi}\int d\phi d\cos\theta
R_{z}(\phi)R_{y}(\theta)\hat{T}|p\hat{z},\lambda_{1},\lambda_{2}\rangle_{1,2}D^{J*}(\phi,\theta)^{J_{z}}\,_{\lambda_{1}-\lambda_{2}}\,,$

and

$\hat{T}|p\hat{z},\lambda_{1},\lambda_{2}\rangle=\frac{i}{2}(s_{\Lambda}^{+}s_{m}^{-}-s_{\Lambda}^{-}s_{m}^{+})|p\hat{z},\lambda_{1},\lambda_{2}\rangle\,,$
(25)

with $s^{\pm}=s_{x}\pm is_{y}$. It is then straightforward to show that

$\hat{T}|a^{m}_{\pm}\rangle=\pm\frac{i}{\sqrt{2}}|b^{m}_{\pm}\rangle,~{}~{}~{}\hat{T}|b^{m}_{\pm}\rangle=\mp\frac{i}{\sqrt{2}}|a^{m}_{\pm}\rangle,$
(26)

resulting in the eigenstates

$\displaystyle|\lambda_{T}^{m}=\pm\frac{1}{\sqrt{2}},\lambda_{\text{tot}}=\frac{1}{2}\rangle=\frac{1}{\sqrt{2}}(|a^{m}_{+}\rangle\mp
i|b^{m}_{+}\rangle),$ (27)
$\displaystyle|\lambda_{T}^{m}=\pm\frac{1}{\sqrt{2}},\lambda_{\text{tot}}=-\frac{1}{2}\rangle=\frac{1}{\sqrt{2}}(|a^{m}_{-}\rangle\pm
i|b^{m}_{-}\rangle)\,,$

where $\lambda_{T}^{m}$ and $\lambda_{\text{tot}}$ are the eigenvalues of
$\hat{T}$ and $\vec{J}\cdot\vec{p}$, respectively. They are also the
eigenstates of $\vec{J}\cdot\vec{p}$, as $\hat{T}$ commutes with both
$\vec{J}$ and $\vec{p}$. Note that $c_{\pm}^{m}$ are not involved since they
are contributed by spinless $j_{eff}^{m}$.

Because $\hat{T}$ and $\vec{J}\cdot\vec{p}$ are T-odd and T-even,
respectively, we have

${\cal
I}_{t}|\lambda_{T}^{m},\lambda_{\text{tot}}\rangle=e^{i\theta_{T}}|-\lambda_{T}^{m},\lambda_{\text{tot}}\rangle\,,~{}~{}~{}{\cal
I}_{s}|\lambda_{T}^{m},\lambda_{\text{tot}}\rangle=e^{i\theta_{m}}|-\lambda_{T}^{m},-\lambda_{\text{tot}}\rangle\,,$
(28)

where ${\cal I}_{t(s)}$ is the time-reversal (space-inversion) operator, and
$\theta_{T,m}$ depend on the conventions. On the other hand, ${\cal I}_{s}$
would interchange $j_{eff}^{R}$ and $j_{eff}^{L}$, given as

${\cal
I}_{s}|\lambda_{T}^{R},\lambda_{\text{tot}}\rangle=e^{i\theta_{R}}|-\lambda_{T}^{L},-\lambda_{\text{tot}}\rangle\,,~{}~{}~{}{\cal
I}_{s}|\lambda_{T}^{L},\lambda_{\text{tot}}\rangle=e^{-i\theta_{R}}|-\lambda_{T}^{R},-\lambda_{\text{tot}}\rangle\,,$
(29)

with

$\small|\lambda_{T}^{R},\lambda_{\text{tot}}\rangle=\frac{1}{\sqrt{2}}\left(|\lambda_{T}^{1},\lambda_{\text{tot}}\rangle+|\lambda_{T}^{2},\lambda_{\text{tot}}\rangle\right)\,,~{}~{}~{}|\lambda_{T}^{L},\lambda_{\text{tot}}\rangle=\frac{1}{\sqrt{2}}\left(|\lambda_{T}^{1},\lambda_{\text{tot}}\rangle-|\lambda_{T}^{2},\lambda_{\text{tot}}\rangle\right)\,,$
(30)

since $j_{eff}^{1}$ and $j_{eff}^{2}$ have opposite parity.

For each combinations of $\lambda_{\text{tot}}$ and $j_{eff}^{r}$, we define
an T-odd quantity

${\cal
T}_{\lambda_{\text{tot}}}^{\,r}\equiv|\langle\lambda_{T}^{r}=1/\sqrt{2},\lambda_{\text{tot}}|{\cal
S}_{eff}|\lambda_{b}\rangle|^{2}-|\langle\lambda_{T}^{r}=-1/\sqrt{2},\lambda_{\text{tot}}|{\cal
S}_{eff}|\lambda_{b}\rangle|^{2}\,,$ (31)

which vanishes if ${\cal S}_{eff}$ is invariant under ${\cal I}_{t}$.
Explicitly, we find

$\displaystyle{\cal
T}_{+}^{\,r}=-2\text{Im}\left(a_{+}^{r}\overline{b_{+}^{r}}\right)\,,~{}~{}$
$\displaystyle{\cal
T}_{-}^{\,r}=2\text{Im}\left(a_{-}^{r}\overline{b_{-}^{r}}\right)\,,$ (32)

which are proportional to the relative complex phase. They are called as T-odd
quantities as ${\cal I}_{t}$ interchanges the final states of the two terms in
Eq. (31).

The operator of $\hat{T}$ contains $\vec{s}_{\Lambda}$, which is difficult to
be measured directly. To probe the spin of $\Lambda$, it is plausible to study
the cascade decays of $\Lambda\to p\pi^{-}$. Subsequently, the final states
involve four particles $p\pi^{-}\ell^{+}\ell^{-}$, containing three
independent three-momenta. It is then possible to observe the triple product

$\alpha(\vec{p}_{+}\times\vec{p}_{p})\cdot\vec{p}_{\Lambda},$ (33)

where $\alpha$ is the polarization asymmetry in $\Lambda\to p\pi^{-}$, and
$\vec{p}_{p}$ is the three-momentum of the proton. Notice that $\alpha$ is a
necessary component in Eq. (33) as $\vec{s}_{\Lambda}$ does not affect
$\vec{p}_{p}$ if $\alpha=0$. Observe that Eq. (33) is P-even. Therefore, we
have to construct P-even observables out of Eq. (32). From the transformation
rules, it is easy to see that

${\cal T}^{R}\equiv{\cal T}_{-}^{R}-{\cal T}_{+}^{L}\,,~{}~{}~{}{\cal
T}^{L}\equiv{\cal T}_{-}^{L}-{\cal T}_{+}^{R}\,,$ (34)

which are both T-odd and P-even.

##  IV Angular distributions

The lepton helicity amplitudes are calculated as

$\displaystyle h_{0,++}^{1}$ $\displaystyle=0\,,~{}~{}~{}$ $\displaystyle
h_{1,++}^{1}=2M_{\ell}\,,$ (35) $\displaystyle h_{0,++}^{2}$
$\displaystyle=2M_{\ell}\,,~{}~{}~{}$ $\displaystyle h_{1,++}^{2}=0\,,$
$\displaystyle h_{1,+-}^{1}$ $\displaystyle=-\sqrt{2q^{2}}\,,~{}~{}~{}$
$\displaystyle h_{1,+-}^{2}=\sqrt{2q^{2}(1-\delta_{\ell})}\,,$

where $\delta_{\ell}=4M_{\ell}^{2}/q^{2}$ and $M_{\ell}$ is the lepton mass.
On the other hand, the baryonic matrix elements are conventionally
parameterized by the form factors, given by

$\displaystyle\langle\Lambda|\bar{s}\gamma^{\mu}b|\Lambda_{b}\rangle$
$\displaystyle=\bar{u}_{\Lambda}\big{[}f_{1}^{V}(q^{2})\gamma^{\mu}-f_{2}^{V}(q^{2})i\sigma^{\mu\nu}\frac{q_{\nu}}{M_{\Lambda_{b}}}+f_{3}^{V}(q^{2})\frac{q^{\mu}}{M_{\Lambda_{b}}}\big{]}u_{\Lambda_{b}},$
(36)
$\displaystyle\langle\Lambda|\bar{s}\gamma^{\mu}\gamma_{5}b|\Lambda_{b}\rangle$
$\displaystyle=\overline{u}_{\Lambda}\big{[}f_{1}^{A}(q^{2})\gamma^{\mu}-f_{2}^{A}(q^{2})i\sigma^{\mu\nu}\frac{q_{\nu}}{M_{\Lambda_{b}}}+f_{3}^{A}(q^{2})\frac{q^{\mu}}{M_{\Lambda_{b}}}\big{]}\gamma_{5}u_{\Lambda_{b}},$
$\displaystyle\langle\Lambda|\bar{s}i\sigma^{\mu q}b|\Lambda_{b}\rangle$
$\displaystyle=\bar{u}_{\Lambda}\left[\frac{f_{1}^{TV}(q^{2})}{M_{\Lambda_{b}}}\left(\gamma^{\mu}q^{2}-q^{\mu}\not{q}\right)-f_{2}^{TV}(q^{2})i\sigma^{\mu
q}\right]u_{\Lambda_{b}},$ $\displaystyle\langle\Lambda|\bar{s}i\sigma^{\mu
q}\gamma_{5}b|\Lambda_{b}\rangle$
$\displaystyle=\bar{u}_{\Lambda}\left[\frac{f_{1}^{TA}(q^{2})}{M_{\Lambda_{b}}}\left(\gamma^{\mu}q^{2}-q^{\mu}\not{q}\right)-f_{2}^{TA}(q^{2})i\sigma^{\mu
q}\right]\gamma_{5}u_{\Lambda_{b}},$

where $u_{\Lambda_{(b)}}$ and $M_{\Lambda_{(b)}}$ are the Dirac spinor and
mass of $\Lambda_{(b)}$. In turn, we find that

$\displaystyle H^{Vm}_{\frac{1}{2},\,0}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{Q_{-}}{q^{2}}}\left[M_{+}F^{Vm}_{1}(q^{2})+\frac{q^{2}}{M_{\Lambda_{b}}}F^{Vm}_{2}(q^{2})\right]\,,$
(37) $\displaystyle H^{Vm}_{\frac{1}{2},\,1}$ $\displaystyle=$
$\displaystyle\sqrt{2Q_{-}}\left[F^{Vm}_{1}(q^{2})+\frac{M_{+}}{M_{\Lambda_{b}}}F^{Vm}_{2}(q^{2})\right]\,,$
(38) $\displaystyle H^{Vm}_{\frac{1}{2},\,t}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{Q_{+}}{q^{2}}}\left[M_{-}F^{Vm}_{1}(q^{2})+\frac{q^{2}}{M_{\Lambda_{b}}}F^{Vm}_{3}(q^{2})\right]\,,$
(39) $\displaystyle H^{Am}_{\frac{1}{2},\,0}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{Q_{+}}{q^{2}}}\left[M_{-}F^{Am}_{1}(q^{2})-\frac{q^{2}}{M_{\Lambda_{b}}}F^{Am}_{2}(q^{2})\right]\,,$
(40) $\displaystyle H^{Am}_{\frac{1}{2},\,1}$ $\displaystyle=$
$\displaystyle\sqrt{2Q_{+}}\left[F^{Am}_{1}(q^{2})+\frac{M_{-}}{M_{\Lambda_{b}}}F^{Am}_{2}(q^{2})\right]\,,$
(41) $\displaystyle H^{Am}_{\frac{1}{2},\,t}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{Q_{-}}{q^{2}}}\left[M_{+}F^{Am}_{1}(q^{2})-\frac{q^{2}}{M_{\Lambda_{b}}}F^{Am}_{3}(q^{2})\right]\,,$
(42)

where $M_{\pm}=M_{\Lambda_{b}}\pm M_{\Lambda}$, $Q_{\pm}=(M_{\pm})^{2}-q^{2}$,
and

$\displaystyle F^{V1}_{1}(q^{2})$ $\displaystyle=$
$\displaystyle[C_{9}^{eff}+C^{\text{NP}}_{9}+(C_{L}+C_{R})]f^{V}_{1}(q^{2})-\frac{2m_{b}}{M_{\Lambda_{b}}}C_{7\gamma}^{eff}f^{TV}_{1}(q^{2})\,,$
(43) $\displaystyle F^{V1}_{2}(q^{2})$ $\displaystyle=$
$\displaystyle[C_{9}^{eff}+C^{\text{NP}}_{9}+(C_{L}+C_{R}))]f^{V}_{2}(q^{2})-\frac{2m_{b}M_{\Lambda_{b}}}{q^{2}}C_{7\gamma}^{eff}f^{TV}_{2}(q^{2})\,,$
(44) $\displaystyle F^{V1}_{3}(q^{2})$ $\displaystyle=$
$\displaystyle[C_{9}^{eff}+C^{\text{NP}}_{9}+(C_{L}+C_{R})]f^{V}_{3}(q^{2})+\frac{2m_{b}M_{-}}{q^{2}}C_{7\gamma}^{eff}f^{TV}_{1}(q^{2})\,,$
(45) $\displaystyle F^{A1}_{1}(q^{2})$ $\displaystyle=$
$\displaystyle[C_{9}^{eff}+C^{\text{NP}}_{9}-(C_{L}+C_{R})]f^{A}_{1}(q^{2})+\frac{2m_{b}}{M_{\Lambda_{b}}}C_{7\gamma}^{eff}f^{TA}_{1}(q^{2})\,,$
(46) $\displaystyle F^{A1}_{2}(q^{2})$ $\displaystyle=$
$\displaystyle[C_{9}^{eff}+C^{\text{NP}}_{9}-(C_{L}+C_{R}))]f^{A}_{2}(q^{2})+\frac{2m_{b}M_{\Lambda_{b}}}{q^{2}}C_{7\gamma}^{eff}f^{TA}_{2}(q^{2})\,,$
(47) $\displaystyle F^{A1}_{3}(q^{2})$ $\displaystyle=$
$\displaystyle[C_{9}^{eff}+C^{\text{NP}}_{9}-(C_{L}+C_{R})]f^{V,A}_{3}(q^{2})+\frac{2m_{b}M_{+}}{q^{2}}C_{7\gamma}^{eff}f^{TA}_{1}(q^{2})\,,$
(48) $\displaystyle F^{V2}_{i}(q^{2})$ $\displaystyle=$
$\displaystyle[C_{10}+C^{\text{NP}}_{10}+(C_{R}-C_{L})]f^{V}_{i}(q^{2})\,,$
(49) $\displaystyle F^{A2}_{i}(q^{2})$ $\displaystyle=$
$\displaystyle[C_{10}+C^{\text{NP}}_{10}-(C_{R}-C_{L})]f^{A}_{i}(q^{2})\,,$
(50)

with $i=(1,2,3).$ Combining the relations

$H^{m}_{\lambda_{\Lambda}\lambda_{m}}=H^{Vm}_{\lambda_{\Lambda}\lambda_{m}}-H^{Am}_{\lambda_{\Lambda}\lambda_{m}}\,,~{}~{}~{}H^{Vm}_{-\lambda_{\Lambda},\,-\lambda_{m}}=H^{Vm}_{\lambda_{\Lambda},\,\lambda_{m}}\,,~{}~{}~{}H^{Am}_{-\lambda_{\Lambda},\,-\lambda_{m}}=-H^{Am}_{\lambda_{\Lambda},\,\lambda_{m}},$

the evaluations of $H$ are completed once the form factors are given.

Figure 1:  Definitions of the angles

The angular distributions of $\Lambda_{b}\to\Lambda(\to
p\pi^{-})\ell^{+}\ell^{-}$, related to the kinematic parts, are given by
piling $D^{J}$ to be

$\displaystyle{\cal
D}(q^{2},\vec{\Omega})\equiv\frac{\partial^{6}\Gamma(\Lambda_{b}\to\Lambda(\to
p\pi^{-})\ell^{+}\ell^{-})}{\partial
q^{2}\partial\cos\theta\partial\cos\theta_{b}\partial\cos\theta_{\ell}\partial\phi_{b}\partial\phi_{\ell}}=\mathcal{B}(\Lambda\to
p\pi^{-})\frac{\zeta(q^{2})}{32\pi^{2}}\sum_{\lambda_{p}\,,\lambda_{\pm}\,,\lambda_{b}}\rho_{\lambda_{\Lambda_{b}}\lambda_{\Lambda_{b}}}\left|A_{\lambda_{p}}\right|^{2}$
$\displaystyle\left|\sum_{m}\sum_{\lambda_{m},\lambda_{\Lambda}}(-1)^{J_{m}}H_{\lambda_{\Lambda}\lambda_{m}}^{m}D^{\frac{1}{2}*}(0,\theta)^{\lambda_{b}}\,_{\lambda_{\Lambda}-\lambda_{m}}D^{\frac{1}{2}*}(\phi_{b},\theta_{b})^{\lambda_{\Lambda}}\,_{\lambda_{p}}h^{m}_{J_{m},\lambda_{+}\lambda_{-}}D^{J_{m}*}(\phi_{\ell},\theta_{\ell})^{\lambda_{m}}\,_{\lambda_{+}-\lambda_{-}}\right|^{2},$
$\displaystyle\zeta(q^{2})=\frac{\alpha^{2}G_{F}^{2}|V_{ts}^{\dagger}V_{tb}|^{2}}{32\pi^{5}}\frac{q^{2}|\vec{p}_{\Lambda}|}{24M_{\Lambda_{b}}^{2}}\sqrt{1-\delta_{\ell}},$
(51)

where $\rho_{\pm,\pm}=(1\pm P_{b})/2$, $|A_{\pm}|^{2}=(1\pm\alpha)/2$,
$\lambda_{p}=\pm 1/2$,
$|\vec{p}_{\Lambda}|=\sqrt{Q_{+}Q_{-}}/2M_{\Lambda_{b}}$, and $J_{m}=0~{}(1)$
for $\lambda_{m}=t~{}(\pm,0)$. The angles are defined in FIG. 1, where
$\theta,\theta_{b}$ and $\theta_{\ell}$ are defined in the CM frames of
$\Lambda_{b},\Lambda$ and $\ell^{+}\ell^{-}$, respectively, and
$\phi_{b,\ell}$ are the azimuthal angles between the decay planes.

The physical meaning of Eq. (IV) is decomposed as follows:

  * •

The
$H^{m}_{\lambda_{\Lambda}\lambda_{m}}D^{\frac{1}{2}*}(0,\theta)^{\lambda_{b}}\,_{\lambda_{\Lambda}-\lambda_{m}}$
is responsible for $\Lambda_{b}\to\Lambda j_{eff}^{m}$, where $H$ and $D$
describe the dynamical and kinematic parts of the amplitudes, respectively.

  * •

The kinematic part of the $\Lambda\to p\pi^{-}$ is described by
$D^{\frac{1}{2}*}(\phi_{b},\theta_{b})^{\lambda_{\Lambda}}\,_{\lambda_{p}}$,
while the dynamical part by $|A_{\lambda_{p}}|$.

  * •

The terms of $h^{m}_{J_{m},\lambda_{+}\lambda_{-}}$ and
$D^{J_{m}*}(\phi_{\ell},\theta_{\ell})^{\lambda_{m}}\,_{\lambda_{+}-\lambda_{-}}$
describe the dynamical and kinematic parts of
$j^{m}_{eff}\to\ell^{+}\ell^{-}$, respectively.

The derivation is similar to those of the appendices in Ref. TRSEMI . We
cross-check our results of $\mathcal{D}(\vec{\Omega})$ with Ref.
Blake:2017angular  and find that they are matched. For practical purposes,
$\mathcal{D}(\vec{\Omega})$ is expanded as LHCb:2018angular

$\displaystyle{\cal
D}(q^{2},\vec{\Omega})=\frac{3}{32\pi^{2}}\Big{(}\left(K_{1}\sin^{2}\theta_{l}+K_{2}\cos^{2}\theta_{l}+K_{3}\cos\theta_{l}\right)+\left(K_{4}\sin^{2}\theta_{l}+K_{5}\cos^{2}\theta_{l}+K_{6}\cos\theta_{l}\right)\cos\theta_{b}+$
(52)
$\displaystyle\left(K_{7}\sin\theta_{l}\cos\theta_{l}+K_{8}\sin\theta_{l}\right)\sin\theta_{b}\cos\left(\phi_{b}+\phi_{l}\right)+\left(K_{9}\sin\theta_{l}\cos\theta_{l}+K_{10}\sin\theta_{l}\right)\sin\theta_{b}\sin\left(\phi_{b}+\phi_{l}\right)+$
$\displaystyle\left(K_{11}\sin^{2}\theta_{l}+K_{12}\cos^{2}\theta_{l}+K_{13}\cos\theta_{l}\right)\cos\theta+\left(K_{14}\sin^{2}\theta_{l}+K_{15}\cos^{2}\theta_{l}+K_{16}\cos\theta_{l}\right)\cos\theta_{b}\cos\theta+$
$\displaystyle\left(K_{17}\sin\theta_{l}\cos\theta_{l}+K_{18}\sin\theta_{l}\right)\sin\theta_{b}\cos\left(\phi_{b}+\phi_{l}\right)\cos\theta+\left(K_{19}\sin\theta_{l}\cos\theta_{l}+K_{20}\sin\theta_{l}\right)\sin\theta_{b}\sin\left(\phi_{b}+\phi_{l}\right)\cos\theta$
$\displaystyle+\left(K_{21}\cos\theta_{l}\sin\theta_{l}+K_{22}\sin\theta_{l}\right)\sin\phi_{l}\sin\theta+\left(K_{23}\cos\theta_{l}\sin\theta_{l}+K_{24}\sin\theta_{l}\right)\cos\phi_{l}\sin\theta+$
$\displaystyle\left(K_{25}\cos\theta_{l}\sin\theta_{l}+K_{26}\sin\theta_{l}\right)\sin\phi_{l}\cos\theta_{b}\sin\theta+\left(K_{27}\cos\theta_{l}\sin\theta_{l}+K_{28}\sin\theta_{l}\right)\cos\phi_{l}\cos\theta_{b}\sin\theta+$
$\displaystyle\left(K_{29}\cos^{2}\theta_{l}+K_{30}\sin^{2}\theta_{l}\right)\sin\theta_{b}\sin\phi_{b}\sin\theta+\left(K_{31}\cos^{2}\theta_{l}+K_{32}\sin^{2}\theta_{l}\right)\sin\theta_{b}\cos\phi_{b}\sin\theta+$
$\displaystyle\left(K_{33}\sin^{2}\theta_{l}\right)\sin\theta_{b}\cos\left(2\phi_{l}+\phi_{b}\right)\sin\theta+\left(K_{34}\sin^{2}\theta_{l}\right)\sin\theta_{b}\sin\left(2\phi_{l}+\phi_{b}\right)\sin\theta\Big{)}~{}\,,$

where the definitions of $K_{i}(i=1\sim 34)$ can be found in Appendix A. We
note that $K_{11\sim 34}$ are proportional to $P_{b}$, imposing difficulties
to extract physical meanings since $P_{b}$ depends on the productions.
Interestingly, $K_{9}$ and $K_{10}$ are found to be

$\displaystyle K_{9}$
$\displaystyle=\frac{\sqrt{2}\alpha\left(1-\delta_{\ell}\right)}{4}\left({\cal
T}^{R}+{\cal T}^{L}\right)\,,$ (53) $\displaystyle K_{10}$
$\displaystyle=-\frac{\sqrt{2}\alpha\sqrt{1-\delta_{\ell}}}{4}\left({\cal
T}^{R}-{\cal T}^{L}\right)\,,$

which are T-odd according to Eq. (34). We note that $K_{19,20}$, $K_{21,22}$,
$K_{25,26}$, $K_{29,30}$ and $K_{34}$ are also sensitive to the complex phases
of NP as they are proportional to the imaginary parts of the helicity
amplitudes.

##  V Numerical results

In this work, we estimate the form factors by the HBM, where the calculation
details are given in Ref. centerofmass . The bag parameters adopted in this
work are given as

$(m_{s}\,,m_{b})=(0.28,4.8)~{}\text{GeV}\,,~{}~{}0.313~{}\text{GeV}<E_{u,d}<0.368~{}\text{GeV}\,,$
(54)

where $R=4.8~{}\text{GeV}^{-1}$ and $E_{q}$ are the bag radius and quark
energy, respectively. Recently, $\alpha$ has been updated by BESIII bes32018 ;
bes3  with remarkable precision. We take $\alpha=0.732\pm 0.014$,
$M_{\Lambda_{b}}=5.6196$ GeV and the $\Lambda_{b}$ lifetime of
$\tau_{b}=1.471\times 10^{-12}$s from the particle data group pdg2022 . The
main uncertainties of the HBM come from $E_{q}$, affected the form factors
largely at the low $q^{2}$ region.

Table 1: ${\cal B}_{\ell}$ in units of $10^{-6}$

| HBM | CQM | LCSR | LCSR | BSE | CQM | LCSR | RQM | Data
---|---|---|---|---|---|---|---|---|---
|  | gutsche  | aliev  | ymwang  | liu2019  | mott2015  | gan2012  | faustov  | pdg2022
${\cal B}_{e}$ | 0.91(25) | 1.0 | 4.6(1.6) |  | $0.660\sim 1.208$ |  | $2.03(^{26}_{9})$ | 1.07 | 1.08(28)
${\cal B}_{\mu}$ | 0.79(18) | 1.0 | 4.0(1.2) | $6.1(^{5.8}_{1.7})$ | $0.812\sim 1.445$ | 0.70 |  | 1.05
${\cal B}_{\tau}$ | 0.21(2) | 0.2 | 0.8(3) | $2.1(^{2.3}_{0.6})$ | $0.252\sim 0.392$ | 0.22 |  | 0.26

The total branching fractions are obtained by integrating $\vec{\Omega}$ and
$q^{2}$ in Eq. (IV), given as

${\cal B}_{\ell}={\cal
B}(\Lambda_{b}\to\Lambda\ell^{+}\ell^{-})=\tau_{b}\int^{M_{-}^{2}}_{4m_{\ell}^{2}}\zeta(K_{1}+2K_{2})dq^{2}\,.$
(55)

The computed values and the ones in the literature within the SM are listed in
Table 1. In the literature, Refs. gutsche ; mott2015  consider the covariant
quark model (CQM), Refs. aliev ; ymwang ; gan2012  light-cone QCD sum rules
(LCSR), Ref. faustov  relativistic quark model (RQM), and Ref. liu2019  Bethe-
Salpeter equation (BSE). We see that our results of ${\cal B}_{\ell}$ are
consistent with those of the CQM, RQM and current experimental data but
systematically smaller than LCSR. Notably, we find that ${\cal B}_{e}>{\cal
B}_{\mu}$, which are consistent with Refs. faustov  and aliev . Explicitly, we
obtain ${\cal B}_{e}/{\cal B}_{\mu}=1.15$ with little uncertainties due to the
correlations. The future experiments on ${\cal B}_{e}/{\cal B}_{\mu}$ may
discriminate the approaches.

Some of the angular observables ($K_{i}$) bear special names. In the
following, we concentrate on $\ell=\mu$. The integrated $K_{i}$ are defined as

$\langle
K_{i}\rangle=\frac{1}{\Gamma_{\kappa}}\int^{\kappa^{\prime}}_{\kappa}\zeta
K_{i}dq^{2}\,,~{}~{}~{}\Gamma_{\kappa}=\int^{\kappa^{\prime}}_{\kappa}\zeta(K_{1}+2K_{2})dq^{2}\,.$
(56)

The integrated hadron (lepton) forward-backward asymmetry of
$A_{FB}^{h}~{}(A_{FB}^{\ell})$ is related to $\langle K_{i}\rangle$ through

$\displaystyle A_{FB}^{h}=\langle K_{4}\rangle+\frac{1}{2}\langle
K_{5}\rangle\,,~{}~{}~{}A_{FB}^{\ell}=\frac{3}{2}\langle K_{3}\rangle\,,$ (57)

while

$A_{FB}^{\ell h}=\frac{3}{4}\langle K_{6}\rangle\,,~{}~{}~{}F_{L}=2\langle
K_{1}\rangle-\langle K_{2}\rangle\,,$ (58)

are the combined forward-backward asymmetry and longitudinal polarized
fraction, respectively. The average decay branching fraction is defined as

$\left\langle\frac{\partial{\cal B}}{\partial
q^{2}}\right\rangle\equiv\frac{\tau_{b}}{\kappa^{\prime}-\kappa}\Gamma_{\kappa}\,.$
(59)

Note that the $q^{2}$ regions of $[\kappa,\kappa^{\prime}]=[8,11]$ and
$[12.5,15]$ in units of GeV2 are contaminated largely by the charmonium
resonances, so are not considered.

Our results within the HBM are given in Table 2, along with the ones from the
literature and experimental data LHCb:exp ; LHCb:2018angular . Our values of
$A_{FB}^{h,\ell,h\ell}$ and $F_{L}$ have little uncertainties as $K_{i}$ are
correlated in the model calculations. In the literature, Ref. detmold  employs
the lattice QCD, and Ref. faustov  includes the contributions from the
charmonium resonances. We see that the angular observables in the literature
and this work are basically consistent. Our results of $\langle
A^{h}_{FB}\rangle$ and $\langle A^{\ell h}_{FB}\rangle$ are slightly larger
than the others due to the updated $\alpha$222They used $\alpha=0.642\pm
0.013$ pdg2016 , in sharp contrast to $\alpha=0.732\pm 0.014$ adopted in this
work.. Notably, the experimental values of $A_{FB}^{\ell h}$ are nearly twice
larger than the theoretical predictions.

Table 2:  Decay observables, where $\langle\partial{\cal B}/\partial
q^{2}\rangle$ and $\kappa^{(\prime)}$ are in units of $10^{-7}$ GeV-2 and
GeV2, respectively.

| $[\kappa,\kappa^{\prime}]$ | HBM |  RQM faustov  |  lattice detmold  |  LHCb LHCb:exp ; LHCb:2018angular
---|---|---|---|---|---
$\left\langle\frac{\partial{\cal B}}{\partial q^{2}}\right\rangle$ | $[0.1,2]$ | 0.25(11) | 0.34 | 0.25(23) | $0.36(^{14}_{13})$
$[2,4]$ | 0.16(7) | 0.31 | 0.18(12) | $0.11(^{12}_{9})$
$[4,6]$ | 0.20(8) | 0.40 | 0.23(11) | $0.02(^{9}_{1})$
$[6,8]$ | 0.26(9) | 0.57 | 0.307(94) | $0.25(^{13}_{12})$
$[11,12.5]$ | 0.44(11) | 0.65 |  | 0.75(21)
$[15,16]$ | 0.61(10) | 0.72 | 0.796(75) | 1.12(30)
$[16,18]$ | 0.65(8) | 0.68 | 0.827(76) | 1.22(29)
$[1.1,6]$ | 0.18(7) | 0.34 | 0.20(12) | $0.09(^{6}_{5})$
$[15,20]$ | 0.60(6) | 0.61 | 0.756(70) | $1.20(^{26}_{27})$
$A_{FB}^{\ell}$ | $[0.1,2]$ | 0.076(0) | 0.067 | 0.095(15) | $0.37(^{37}_{48})$
$[11,12.5]$ | $-0.357(6)$ | $-0.35$ |  | $0.01(^{20}_{19})$
$[15,16]$ | $-0.403(8)$ | $-0.41$ | $-0.374(14)$ | $-0.10(^{18}_{16})$
$[16,18]$ | $-0.396(9)$ | $-0.36$ | $-0.372(13)$ | $-0.07(^{14}_{13})$
$[18,20]$ | $-0.320(9)$ | $-0.32$ | $-0.309(15)$ | $0.01(^{16}_{15})$
| $[15,20]$ | $-0.369(7)$ | $-0.33$ | $-0.350(13)$ | $-0.39(4)$
$A_{FB}^{h}$ | $[0.1,2]$ | $-0.294(2)$ | $-0.26$ | $-0.310(18)$ | $-0.12(^{34}_{32})$
$[11,12.5]$ | $-0.408(2)$ | $-0.30$ |  | $-0.50(^{11}_{4})$
$[15,16]$ | $-0.384(4)$ | $-0.32$ | $-0.3069(83)$ | $-0.19(^{14}_{16})$
$[16,18]$ | $-0.358(6)$ | $-0.31$ | $-0.2891(90)$ | $-0.44(^{10}_{6})$
$[18,20]$ | $-0.275(6)$ | $-0.25$ | $-0.227(10)$ | $-0.13(^{10}_{12})$
| $[15,20]$ | $-0.333(4)$ | $-0.29$ | $-0.2710(92)$ | $-0.30(5)$
$A_{FB}^{h\ell}$ | $[0.1,2]$ | $-0.028(0)$ | $-0.021$ | $-0.0302(51)$ |
$[2,4]$ | $-0.001(1)$ | 0.010 | $-0.0169(99)$ |
$[4,6]$ | 0.047(2) | 0.045 | 0.021(13) |
$[6,8]$ | 0.084(1) | 0.072 | 0.053(13) |
$[15,20]$ | 0.179(1) | 0.129 | 0.1398(43) | 0.25(4)
$F_{L}$ | $[0.1,2]$ | 0.541(4) | 0.66 | 0.465(84) | $0.56(^{24}_{56})$
$[11,12.5]$ | 0.615(0) | 0.51 |  | $0.40(^{37}_{36})$
$[15,16]$ | 0.507(1) | 0.41 | 0.454(20) | $0.49(30)$
$[16,18]$ | 0.469(0) | 0.38 | 0.417(15) | $0.68(^{15}_{21})$
$[18,20]$ | 0.416(1) | 0.35 | 0.3706(79) | $0.62(^{24}_{27})$

After showing that our results in the HBM are compatible with those in the
literature, we are ready to estimate the NP contributions to the T-odd
observables. From the global fit in the $B$ meson decays
SinghChundawat:2022zdf , the permitted imaginary parts of the NP Wilson
coefficients are found in TABLE 3 with four different scenarios333See FIG. 1
of Ref. SinghChundawat:2022zdf . It is clear that the signs of NP Wilson
coefficients are barely determined. . As an illustration, we calculate
$\langle K_{j}\rangle$ with $K_{j}\in\\{K_{9},K_{10},K_{19},K_{30}\\}$ and
$(\kappa,\kappa^{\prime})=(15~{}\text{GeV}^{2},20~{}\text{GeV}^{2})$ in
different scenarios given in TABLE 3. We fit $P_{b}$ from the data of
$K_{1-34}$ and find that $P_{b}$ is consistent with zero regardless to the
presence of NP.

Table 3:  The Wilson coefficients and $\langle K_{j}\rangle$ in units of $10^{-3}$, with four NP scenarios.  Scenarios | $\text{Im}(C_{9}^{NP})$ | $\text{Im}(C_{10}^{NP})$ | $\text{Im}(C_{L})$ | $\text{Im}(C_{R})$ | $K_{9}$ | $K_{10}$ | $K_{19}$ | $K_{30}$ | $P_{b}$
---|---|---|---|---|---|---|---|---|---
Scenario #1 | $\pm 0.73$ | 0 | 0 | 0 | $0$ | $\mp 4$ | $0$ | $0$ | $-0.022(72)$
Scenario #2 | 0 | $\pm 1.86$ | 0 | 0 |  |  |  |  |
Scenario #3 | $\pm 1.66$ | $\mp 1.66$ | 0 | 0 | $0$ | $\pm 3$ | $0$ | $0$ | $-0.021(65)$
Scenario #4 | $\pm 0.77$ | 0 | $\mp 0.77$ | $\mp 0.77$ | $\mp 1$ | $\mp 42$ | $\mp 1$ | $0$ | $-0.019(64)$

In the SM, due to lacking of relative complex phases, $\langle K_{j}\rangle$
are found to be less than $10^{-4}$. Therefore, they provide excellent
opportunities to test the SM. Since $K_{j}$ are proportional to the imaginary
parts of the NP Wilson coefficients, which have not been determined yet, their
signs remain unknown. However, nonzero values in the experiments would be a
smoking gun of NP. Scenario #1 affects little to $\langle K_{j}\rangle$, and
the results are not listed. In addition, $\langle K_{9}\rangle$ is found to be
very small in all the scenarios, which is consistent with the experimental
searches. Remarkably, the experimental result of $\langle K_{10}\rangle$ can
be explained by Scenario #4, which can be provided by the $Z^{\prime}$ model
Chao:2021qxq ; Li:2021cty ; Alok:2022pjb . The reason can be traced back to
$C_{L}$ as it interferes largely with the left-handed particles produced by
the SM. On the other hand, $K_{19}$ and $K_{30}$ are highly suppressed by
$P_{b}$.

##  VI conclusions

We have derived the angular distributions of $\Lambda_{b}\to\Lambda(\to
p\pi^{-})\ell^{+}\ell^{-}$ based on the effective schemes of
$\Lambda_{b}\to\Lambda(\to p\pi^{-})j_{eff}^{m}(\to\ell^{+}\ell^{-})$. We have
shown that our results are consistent with those in the literature. By
studying the effective two-body decays of $\Lambda_{b}\to\Lambda j_{eff}^{m}$,
we have explored the time-reversal asymmetries by identifying the T-odd
correlations in the form of
$(\vec{s}_{\Lambda}\times\vec{s}_{m})\cdot\hat{p}$. For the numerical
estimations, we have adopted the HBM and found that
$\mathcal{B}_{e}=0.91(25)\times 10^{-6}$, $\mathcal{B}_{\mu}=0.79(18)\times
10^{-6}$, and $\mathcal{B}_{\tau}=0.21(2)\times 10^{-6}$. For
$\Lambda_{b}\to\Lambda(\to p\pi^{-})\mu^{+}\mu^{-}$, $A_{FB}^{\ell}$ and
$A_{FB}^{h}$, averaged in $15\leq q^{2}\leq 20\text{GeV}^{2}$, have been
evaluated as $-0.369(7)$ and $-0.333(4)$, respectively. These results are
consistent with those in the literature and experiments, showing that the HBM
is suitable for estimating $\Lambda_{b}\to\Lambda\ell^{+}\ell^{-}$. We have
demonstrated that $K_{9}$ and $K_{10}$ are related to
$(\vec{s}_{\Lambda}\times\vec{s}_{m})\cdot\hat{p}_{\Lambda}$, in which
$K_{10}$ is sensitive to the complex phases generated by NP. We have found
that $C_{L}=-0.77i$ can explain the $K_{10}$ puzzle. We recommend the
experiment to revisit $K_{10}$ for a stringent constraint.

###### Acknowledgements.

This work is supported in part by the National Key Research and Development
Program of China under Grant No. 2020YFC2201501 and the National Natural
Science Foundation of China (NSFC) under Grant No. 12147103.

##  Appendix A Angular observables

All $K_{i}$ are real, which are given as

$\displaystyle
K_{1}=\frac{1}{4}\Big{(}-\delta_{\ell}a^{2}_{+}\overline{a^{2}_{+}}-\delta_{\ell}a^{2}_{-}\overline{a^{2}_{-}}+\frac{\delta_{\ell}b^{1}_{+}\overline{b^{1}_{+}}}{2}-\frac{\delta_{\ell}b^{2}_{+}\overline{b^{2}_{+}}}{2}+\frac{\delta_{\ell}b^{1}_{-}\overline{b^{1}_{-}}}{2}-\frac{\delta_{\ell}b^{2}_{-}\overline{b^{2}_{-}}}{2}+\delta_{\ell}c^{2}_{+}\overline{c^{2}_{+}}$
(60)
$\displaystyle+\delta_{\ell}c^{2}_{-}\overline{c^{2}_{-}}+a^{1}_{+}\overline{a^{1}_{+}}+a^{2}_{+}\overline{a^{2}_{+}}+a^{1}_{-}\overline{a^{1}_{-}}+a^{2}_{-}\overline{a^{2}_{-}}+\frac{b^{1}_{+}\overline{b^{1}_{+}}}{2}+\frac{b^{2}_{+}\overline{b^{2}_{+}}}{2}+\frac{b^{1}_{-}\overline{b^{1}_{-}}}{2}+\frac{b^{2}_{-}\overline{b^{2}_{-}}}{2}\Big{)},$
$\displaystyle
K_{2}=\frac{1}{4}\Big{(}\delta_{\ell}a^{1}_{+}\overline{a^{1}_{+}}+\delta_{\ell}a^{1}_{-}\overline{a^{1}_{-}}-\delta_{\ell}b^{2}_{+}\overline{b^{2}_{+}}-\delta_{\ell}b^{2}_{-}\overline{b^{2}_{-}}+\delta_{\ell}c^{2}_{+}\overline{c^{2}_{+}}+\delta_{\ell}c^{2}_{-}\overline{c^{2}_{-}}+b^{1}_{+}\overline{b^{1}_{+}}$
$\displaystyle+b^{2}_{+}\overline{b^{2}_{+}}+b^{1}_{-}\overline{b^{1}_{-}}+b^{2}_{-}\overline{b^{2}_{-}}\Big{)},$
$\displaystyle
K_{3}=-\frac{K_{16}}{P_{b}}=\frac{\sqrt{1-\delta_{\ell}}}{4}\Big{(}b^{1}_{+}\overline{b^{2}_{+}}+b^{2}_{+}\overline{b^{1}_{+}}-b^{1}_{-}\overline{b^{2}_{-}}-b^{2}_{-}\overline{b^{1}_{-}}\Big{)}$
$\displaystyle
K_{4}=\frac{1}{4}\alpha\Big{(}-\delta_{\ell}a^{2}_{+}\overline{a^{2}_{+}}+\delta_{\ell}a^{2}_{-}\overline{a^{2}_{-}}-\frac{\delta_{\ell}b^{1}_{+}\overline{b^{1}_{+}}}{2}+\frac{\delta_{\ell}b^{2}_{+}\overline{b^{2}_{+}}}{2}+\frac{\delta_{\ell}b^{1}_{-}\overline{b^{1}_{-}}}{2}-\frac{\delta_{\ell}b^{2}_{-}\overline{b^{2}_{-}}}{2}+\delta_{\ell}c^{2}_{+}\overline{c^{2}_{+}}$
$\displaystyle-\delta_{\ell}c^{2}_{-}\overline{c^{2}_{-}}+a^{1}_{+}\overline{a^{1}_{+}}+a^{2}_{+}\overline{a^{2}_{+}}-a^{1}_{-}\overline{a^{1}_{-}}-a^{2}_{-}\overline{a^{2}_{-}}-\frac{b^{1}_{+}\overline{b^{1}_{+}}}{2}-\frac{b^{2}_{+}\overline{b^{2}_{+}}}{2}+\frac{b^{1}_{-}\overline{b^{1}_{-}}}{2}+\frac{b^{2}_{-}\overline{b^{2}_{-}}}{2}\Big{)},$
$\displaystyle
K_{5}=\frac{1}{4}\alpha\Big{(}\delta_{\ell}a^{1}_{+}\overline{a^{1}_{+}}-\delta_{\ell}a^{1}_{-}\overline{a^{1}_{-}}+\delta_{\ell}b^{2}_{+}\overline{b^{2}_{+}}-\delta_{\ell}b^{2}_{-}\overline{b^{2}_{-}}+\delta_{\ell}c^{2}_{+}\overline{c^{2}_{+}}-\delta_{\ell}c^{2}_{-}\overline{c^{2}_{-}}-b^{1}_{+}\overline{b^{1}_{+}}$
$\displaystyle-b^{2}_{+}\overline{b^{2}_{+}}+b^{1}_{-}\overline{b^{1}_{-}}+b^{2}_{-}\overline{b^{2}_{-}}\Big{)},$
$\displaystyle
K_{6}=-\frac{K_{13}}{P_{b}}=\frac{\alpha\sqrt{1-\delta_{\ell}}}{4}\Big{(}-b^{1}_{+}\overline{b^{2}_{+}}-b^{2}_{+}\overline{b^{1}_{+}}-b^{1}_{-}\overline{b^{2}_{-}}-b^{2}_{-}\overline{b^{1}_{-}}\Big{)},$
$\displaystyle
K_{7}-iK_{9}=\frac{\sqrt{2}\alpha\left(1-\delta_{\ell}\right)}{4}\Big{(}a^{1}_{-}\overline{b^{1}_{-}}+a^{2}_{-}\overline{b^{2}_{-}}-b^{1}_{+}\overline{a^{1}_{+}}-b^{2}_{+}\overline{a^{2}_{+}}\Big{)},$
$\displaystyle
K_{8}-iK_{10}=-\frac{\sqrt{2}\alpha\sqrt{1-\delta_{\ell}}}{4}\Big{(}a^{1}_{-}\overline{b^{2}_{-}}+a^{2}_{-}\overline{b^{1}_{-}}+b^{1}_{+}\overline{a^{2}_{+}}+b^{2}_{+}\overline{a^{1}_{+}}\Big{)},$
$\displaystyle
K_{11}=\frac{P_{b}}{4}\Big{(}-\delta_{\ell}a^{2}_{+}\overline{a^{2}_{+}}+\delta_{\ell}a^{2}_{-}\overline{a^{2}_{-}}+\frac{\delta_{\ell}b^{1}_{+}\overline{b^{1}_{+}}}{2}-\frac{\delta_{\ell}b^{2}_{+}\overline{b^{2}_{+}}}{2}-\frac{\delta_{\ell}b^{1}_{-}\overline{b^{1}_{-}}}{2}+\frac{\delta_{\ell}b^{2}_{-}\overline{b^{2}_{-}}}{2}+\delta_{\ell}c^{2}_{+}\overline{c^{2}_{+}}$
$\displaystyle-\delta_{\ell}c^{2}_{-}\overline{c^{2}_{-}}+a^{1}_{+}\overline{a^{1}_{+}}+a^{2}_{+}\overline{a^{2}_{+}}-a^{1}_{-}\overline{a^{1}_{-}}-a^{2}_{-}\overline{a^{2}_{-}}+\frac{b^{1}_{+}\overline{b^{1}_{+}}}{2}+\frac{b^{2}_{+}\overline{b^{2}_{+}}}{2}-\frac{b^{1}_{-}\overline{b^{1}_{-}}}{2}-\frac{b^{2}_{-}\overline{b^{2}_{-}}}{2}\Big{)},$
$\displaystyle
K_{12}=\frac{P_{b}}{4}\Big{(}\delta_{\ell}a^{1}_{+}\overline{a^{1}_{+}}-\delta_{\ell}a^{1}_{-}\overline{a^{1}_{-}}-\delta_{\ell}b^{2}_{+}\overline{b^{2}_{+}}+\delta_{\ell}b^{2}_{-}\overline{b^{2}_{-}}$
$\displaystyle+\delta_{\ell}c^{2}_{+}\overline{c^{2}_{+}}-\delta_{\ell}c^{2}_{-}\overline{c^{2}_{-}}+b^{1}_{+}\overline{b^{1}_{+}}+b^{2}_{+}\overline{b^{2}_{+}}-b^{1}_{-}\overline{b^{1}_{-}}-b^{2}_{-}\overline{b^{2}_{-}}\Big{)},$
$\displaystyle
K_{14}=\frac{P_{b}}{4}\alpha\Big{(}-\delta_{\ell}a^{2}_{+}\overline{a^{2}_{+}}-\delta_{\ell}a^{2}_{-}\overline{a^{2}_{-}}-\frac{\delta_{\ell}b^{1}_{+}\overline{b^{1}_{+}}}{2}+\frac{\delta_{\ell}b^{2}_{+}\overline{b^{2}_{+}}}{2}-\frac{\delta_{\ell}b^{1}_{-}\overline{b^{1}_{-}}}{2}+\frac{\delta_{\ell}b^{2}_{-}\overline{b^{2}_{-}}}{2}+\delta_{\ell}c^{2}_{+}\overline{c^{2}_{+}}$
(61)
$\displaystyle+\delta_{\ell}c^{2}_{-}\overline{c^{2}_{-}}+a^{1}_{+}\overline{a^{1}_{+}}+a^{2}_{+}\overline{a^{2}_{+}}+a^{1}_{-}\overline{a^{1}_{-}}+a^{2}_{-}\overline{a^{2}_{-}}-\frac{b^{1}_{+}\overline{b^{1}_{+}}}{2}-\frac{b^{2}_{+}\overline{b^{2}_{+}}}{2}-\frac{b^{1}_{-}\overline{b^{1}_{-}}}{2}-\frac{b^{2}_{-}\overline{b^{2}_{-}}}{2}\Big{)},$
$\displaystyle K_{15}=\frac{P_{b}}{4}\alpha\Big{(}\delta_{\ell}a^{1}_{+}\
\overline{a^{1}_{+}}+\delta_{\ell}a^{1}_{-}\overline{a^{1}_{-}}+\delta_{\ell}b^{2}_{+}\overline{b^{2}_{+}}+\delta_{\ell}b^{2}_{-}\overline{b^{2}_{-}}+\delta_{\ell}c^{2}_{+}\overline{c^{2}_{+}}$
$\displaystyle+\delta_{\ell}c^{2}_{-}\overline{c^{2}_{-}}-b^{1}_{+}\overline{b^{1}_{+}}-b^{2}_{+}\overline{b^{2}_{+}}-b^{1}_{-}\overline{b^{1}_{-}}-b^{2}_{-}\overline{b^{2}_{-}}\Big{)},$
$\displaystyle
K_{17}-iK_{19}=-\frac{\sqrt{2}P_{b}\alpha\left(1-\delta_{\ell}\right)}{4}\Big{(}a^{1}_{-}\overline{b^{1}_{-}}+a^{2}_{-}\overline{b^{2}_{-}}+b^{1}_{+}\overline{a^{1}_{+}}+b^{2}_{+}\overline{a^{2}_{+}}\Big{)},$
$\displaystyle
K_{18}-iK_{20}=-\frac{\sqrt{2}P_{b}\alpha\sqrt{1-\delta_{\ell}}}{4}\Big{(}-a^{1}_{-}\overline{b^{2}_{-}}-a^{2}_{-}\overline{b^{1}_{-}}+b^{1}_{+}\overline{a^{2}_{+}}+b^{2}_{+}\overline{a^{1}_{+}}\Big{)},$
$\displaystyle
K_{23}-iK_{21}=\frac{P_{b}\sqrt{2}(1-\delta_{\ell})}{4}\Big{(}b^{1}_{+}\overline{a^{1}_{-}}-a^{1}_{+}\overline{b^{1}_{-}}-a^{2}_{+}\overline{b^{2}_{-}}+b^{2}_{+}\overline{a^{2}_{-}}\Big{)},$
$\displaystyle
K_{24}-iK_{22}=-\frac{P_{b}\sqrt{2}\sqrt{(1-\delta_{\ell})}}{4}\Big{(}a^{1}_{+}\overline{b^{2}_{-}}+a^{2}_{+}\overline{b^{1}_{-}}+b^{1}_{+}\overline{a^{2}_{-}}+b^{2}_{+}\overline{a^{1}_{-}}\Big{)},$
$\displaystyle
K_{27}-iK_{25}=-\frac{P_{b}\alpha\sqrt{2}(1-\delta_{\ell})}{4}\Big{(}-a^{1}_{+}\overline{b^{1}_{-}}-a^{2}_{+}\overline{b^{2}_{-}}-b^{1}_{+}\overline{a^{1}_{-}}-b^{2}_{+}\overline{a^{2}_{-}}\Big{)},$
$\displaystyle
K_{28}-iK_{26}=-\frac{P_{b}\alpha\sqrt{2}\sqrt{(1-\delta_{\ell})}}{4}\Big{(}a^{1}_{+}\overline{b^{2}_{-}}+a^{2}_{+}\overline{b^{1}_{-}}-b^{1}_{+}\overline{a^{2}_{-}}-b^{2}_{+}\overline{a^{1}_{-}}\Big{)},$
$\displaystyle
K_{31}-iK_{29}=-\frac{P_{b}\alpha\delta_{\ell}}{2}\Big{(}a^{1}_{-}\overline{a^{1}_{+}}+c^{2}_{-}\overline{c^{2}_{+}}\Big{)},$
$\displaystyle
K_{32}-iK_{30}=-\frac{P_{b}\alpha}{2}\Big{(}-a^{1}_{-}\overline{a^{1}_{+}}-a^{2}_{-}\overline{a^{2}_{+}}\Big{)}+\delta_{\ell}\Big{(}a^{2}_{-}\overline{a^{2}_{+}}-c^{2}_{-}\overline{c^{2}_{+}}\Big{)},$
$\displaystyle
K_{33}-iK_{34}=\frac{P_{b}\alpha}{4}b^{1}_{+}\overline{b^{1}_{-}}.$

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