# Studying
$\mathcal{B}_{1}(\frac{1}{2}^{+})\to\mathcal{B}_{2}(\frac{1}{2}^{+})\ell^{+}\ell^{-}$
Semi-leptonic Weak Baryon Decays
with the SU(3) Flavor Symmetry

Ru-Min Wang1,†, Yuan-Guo Xu1,‡, Chong Hua1,♯, Xiao-Dong Cheng2,§
1College of Physics and Communication Electronics, JiangXi Normal University,
NanChang, JiangXi 330022, China
2College of Physics and Electronic Engineering, XinYang Normal University,
XinYang, Henan 464000, China
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Motivated by recent anomalies in flavor changing neutral current $b\to
s\ell^{+}\ell^{-}$ transitions, we study
$\mathcal{B}_{1}\to\mathcal{B}_{2}\ell^{+}\ell^{-}\leavevmode\nobreak\
(\ell=e,\mu,\tau)$ semileptonic weak decays with the SU(3) flavor symmetry,
where $\mathcal{B}_{1,2}$ are the spin-$\frac{1}{2}$ baryons of single
bottomed antitriplet $T_{b3}$, single charmed antitriplet $T_{c3}$, or light
baryons octet $T_{8}$. Using the SU(3) irreducible representation approach, we
first obtain the amplitude relations among different decay modes and then
predict the relevant not-yet measured observables of $T_{b3}\to
T_{8}\ell^{+}\ell^{-}$, $T_{c3}\to T_{8}\ell^{+}\ell^{-}$, and $T_{8}\to
T^{\prime}_{8}\ell^{+}\ell^{-}$ decays. (a) We calculate the branching ratios
of the $T_{b3}\to T_{8}\mu^{+}\mu^{-}$ and $T_{b3}\to T_{8}\tau^{+}\tau^{-}$
decay modes in the whole $q^{2}$ region and in the different $q^{2}$ bins by
the measurement of $\Lambda^{0}_{b}\to\Lambda^{0}\mu^{+}\mu^{-}$. Many of them
are obtained for the first time. In addition, the longitudinal polarization
fractions and the leptonic forward-backward asymmetries of all $T_{b3}\to
T_{8}\ell^{+}\ell^{-}$ decays are very similar to each other in certain
$q^{2}$ bins due to the SU(3) flavor symmetry. (b) We analyze the upper limits
of $\mathcal{B}(T_{c3}\to T_{8}\ell^{+}\ell^{-})$ by using the experimental
upper limits of $\mathcal{B}(\Lambda^{+}_{c}\to p\mu^{+}\mu^{-})$ and
$\mathcal{B}(\Lambda^{+}_{c}\to pe^{+}e^{-})$, and find the experimental upper
limit of $\mathcal{B}(\Lambda^{+}_{c}\to p\mu^{+}\mu^{-})$ giving the
effective bounds on the relevant SU(3) flavor symmetry parameters. The
predictions of $\mathcal{B}(\Xi^{0}_{c}\to\Xi^{0}e^{+}e^{-})$ and
$\mathcal{B}(\Xi^{0}_{c}\to\Xi^{0}\mu^{+}\mu^{-})$ will be different between
the single-quark transition dominant contributions and the $W$-exchange
dominant ones. (c) As for $T_{8}\to T^{\prime}_{8}\ell^{+}\ell^{-}$ decays, we
analyze the single-quark transition contributions and the $W$-exchange
contributions by using the two experimental measurements of
$\mathcal{B}(\Xi^{0}\to\Lambda^{0}e^{+}e^{-})$ and $\mathcal{B}(\Sigma^{+}\to
p\mu^{+}\mu^{-})$, and give the branching ratio predictions by assuming either
single-quark transition dominant contributions or the $W$-exchange dominant
contributions. According to our predictions, some observables are accessible
to the experiments at BESIII, LHCb and Belle-II.

##  I INTRODUCTION

Flavor changing neutral current (FCNC) processes, such as $b\to
s\ell^{+}\ell^{-}$, give access to important tests of the standard model (SM)
and searches for new physics beyond the SM. Recently, some discrepancies with
the SM are reported in several observables in $B$ meson decays, for example,
the angular-distribution observable $P^{\prime}_{5}$ of $B^{0}\to
K^{*0}\mu^{+}\mu^{-}$ DescotesGenon:2012zf ; Aaij:2015oid ; ATLAS:2017dlm ;
Sirunyan:2017dhj  and the lepton flavor universality observables $R_{K^{+}}$
and $R_{K^{*0}}$ with $R_{M}=\frac{\mathcal{B}(B\to
M\mu^{+}\mu^{-})}{\mathcal{B}(B\to Me^{+}e^{-})}$ Aaij:2014ora ; Aaij:2017vbb
. Semileptonic baryon decays are quite different to $B,D,K$ meson ones, for
instance, the initial baryons may be polarized, the transitions involve a
diquark system as a spectator rather than a single-quark, and the $W$-exchange
contributions of two-quark and three-quark transitions might appear in baryon
decays. Therefore, the baryon decays provide the important additional tests of
the SM predictions, which can be used to improve the understanding of recent
anomalies in $B$ meson decays. Recently significant experimental progress has
been achieved in studying rare $\Lambda_{b}$ decays. The
$\Lambda_{b}\to\Lambda\mu^{+}\mu^{-}$ baryon decay is the only one measured
among the $T_{b3}\to T_{8}\ell^{+}\ell^{-}$ decays at present.
$\mathcal{B}(\Lambda_{b}\to\Lambda\mu^{+}\mu^{-})$ was first measured by the
CDF Collaboration Aaltonen:2011qs  and then greatly improved by LHCb
Aaij:2015xza ; Aaij:2013hna . For $T_{c3}\to T_{8}\ell^{+}\ell^{-}$ decays,
only $\mathcal{B}(\Lambda_{c}\to pe^{+}e^{-})$ and $\mathcal{B}(\Lambda_{c}\to
p\mu^{+}\mu^{-})$ have been upper limited by $BABAR$ and LHCb Lees:2011hb ;
Aaij:2017nsd . As for $T_{8}\to T^{\prime}_{8}\ell^{+}\ell^{-}$ decays,
$\Xi^{0}\to\Lambda^{0}e^{+}e^{-}$ and $\Sigma^{+}\to p\mu^{+}\mu^{-}$ have
been measured by NA48 Batley:2007hp  and HyperCP Park:2005eka , respectively.
With the experiment development, some
$\mathcal{B}_{1}\to\mathcal{B}_{2}\ell^{+}\ell^{-}$ decays will be improved or
detected by the BESIII, LHCb, and Belle-II Collaborations in the near future,
so it is necessary to study
$\mathcal{B}_{1}\to\mathcal{B}_{2}\ell^{+}\ell^{-}$ decays theoretically.

The theoretical challenge in the study of
$\mathcal{B}_{1}\to\mathcal{B}_{2}\ell^{+}\ell^{-}$ decays is calculating the
hadronic $\mathcal{B}_{1}\to\mathcal{B}_{2}$ form factors in the hadronic
matrix elements. Form factors for $\Lambda_{b}\to\Lambda$ have been estimated
in lattice QCD Detmold:2016pkz ; Boer:2014kda ; Gutsche:2013pp , QCD light
cone sum rules Wang:2015ndk , the soft-collinear effective theory
Feldmann:2011xf , and perturbative QCD He:2006ud . Form factors for
$\Lambda_{b}\to n$ have been estimated in the relativistic quark diquark
picture Faustov:2017ous  and the context of light cone QCD sum rules
Azizi:2010zzb ; Aliev:2010uy . Nevertheless, other form factors of $T_{b3}\to
T_{8}\ell^{+}\ell^{-}$, such as the ones for $\Xi^{0}_{b}\to\Xi^{0}$,
$\Xi^{-}_{b}\to\Xi^{-}$, $\Xi^{0}_{b}\to\Lambda^{0}$,
$\Xi^{0}_{b}\to\Sigma^{0}$ and $\Xi^{-}_{b}\to\Sigma^{-}$, have not been
calculated yet. Similarly in $T_{c3}\to T_{8}\ell^{+}\ell^{-}$ and $T_{8}\to
T^{\prime}_{8}\ell^{+}\ell^{-}$ decays, only some form factors are calculated,
for example, ones for $\Lambda^{+}_{c}\to p$ transition Azizi:2010zzb ;
Sirvanli:2016wnr ; Meinel:2017ggx ; Faustov:2018dkn .

Theoretical calculations of the hadronic matrix elements are not well
understood due to our poor understanding of QCD at low energy regions. The
SU(3) flavor symmetry approach is independent of the detailed dynamics
offering us an opportunity to relate different decay modes. Nevertheless, it
cannot determine the sizes of the amplitudes by itself. However, if
experimental data are enough, one may use the data to extract the amplitudes,
which can be viewed as predictions based on symmetry. Although SU(3) flavor
symmetry is only an approximate symmetry because $u$, $d$ and $s$ quarks have
different masses, it still provides some useful information about the decays.
One popular way of predicting the SU(3) flavor symmetry is to construct the
SU(3) irreducible representation amplitude by decomposing the effective
Hamiltonian, in which one only focuses on the SU(3) flavor structure of the
initial states and finial states, but does not involve the details about the
interaction dynamics.

Some
$\mathcal{B}_{1}(\frac{1}{2}^{+})\to\mathcal{B}_{2}(\frac{1}{2}^{+})\ell^{+}\ell^{-}$
semileptonic baryon decays have been well studied, for instance, semileptonic
$\Lambda^{0}_{b}$ decays in Refs. Detmold:2016pkz ; Boer:2014kda ;
Gutsche:2013pp ; Wang:2008ni ; Hu:2017qxj ; Roy:2017dum ; Das:2018iap ;
Aslam:2008hp ; Faustov:2017wbh , semileptonic $\Lambda^{+}_{c}$ decays in
Refs. deBoer:2017way ; Faustov:2018dkn ; Sirvanli:2016wnr , and semileptonic
$\Sigma^{+}$ decays in Refs. He:2006fr ; He:2005yn ; He:2018yzu ;
Xiangdong:2007vv ; Bergstrom:1987wr . In this work, we will study all weak
$\mathcal{B}_{1}(\frac{1}{2}^{+})\to\mathcal{B}_{2}(\frac{1}{2}^{+})\ell^{+}\ell^{-}$
decays by using the SU(3) irreducible representation approach. We first obtain
the amplitude relations among different decay modes, then use the available
data to extract the SU(3) irreducible amplitudes and finally predict the not-
yet-measured modes for further tests in experiments.

This paper is organized as follows. In Sec. II, we will collect the
representations for the baryon multiplets of $\frac{1}{2}$-spin and the
observable expressions of relevant baryon decays. In Sec. III, we will analyze
the semileptonic weak decays of $T_{b3}\to T_{8}\ell^{+}\ell^{-}$, $T_{c3}\to
T_{8}\ell^{+}\ell^{-}$, and $T_{8}\to T^{\prime}_{8}\ell^{+}\ell^{-}$. Our
conclusions are given in Sec. IV.

##  II Theoretical Frame

###  II.1 Baryon multiplets with $\frac{1}{2}$ spin

The light baryons octet $T_{8}$ under the SU(3) flavor symmetry of $u,d,s$
quarks can be written as

$\displaystyle T_{8}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}\frac{\Lambda^{0}}{\sqrt{6}}+\frac{\Sigma^{0}}{\sqrt{2}}&\Sigma^{+}&p\\\
\Sigma^{-}&\frac{\Lambda^{0}}{\sqrt{6}}-\frac{\Sigma^{0}}{\sqrt{2}}&n\\\
\Xi^{-}&\Xi^{0}&-\frac{2\Lambda^{0}}{\sqrt{6}}\end{array}\right)\,.$ (4)

The single charmed antitriplet $T_{c3}$ is given as

$\displaystyle T_{c3}$ $\displaystyle=$
$\displaystyle(\Xi^{0}_{c},\leavevmode\nobreak\
-\Xi^{+}_{c},\leavevmode\nobreak\ \Lambda^{+}_{c}).$ (5)

The antitriplet $T_{b3}$ with a heavy b quark is

$\displaystyle T_{b3}$ $\displaystyle=$
$\displaystyle(\Xi^{-}_{b},\leavevmode\nobreak\
-\Xi^{0}_{b},\leavevmode\nobreak\ \Lambda^{0}_{b}).$ (6)

###  II.2 Helicity amplitudes for semileptonic decays

In the SM, the low energy effective Hamiltonians for $b\to
s/d\ell^{+}\ell^{-}$, $c\to u\ell^{+}\ell^{-}$ and $s\to d\ell^{+}\ell^{-}$
FCNC transitions have similar forms and can be written as Buchalla:1995vs ;
Meinel:2017ggx ; deBoer:2015boa ; Shifman:1976de ; He:2005yn

$\displaystyle\mathcal{H}(q_{1}\to
q_{2}\ell^{+}\ell^{-})=-\frac{\alpha_{e}G_{F}}{\sqrt{2}\pi}\lambda_{q_{1}q_{2}}\left(C^{eff}_{9}\bar{q}_{2}\gamma^{\mu}P_{L}q_{1}\bar{\ell}\gamma_{\mu}\ell+C_{10}\bar{q}_{2}\gamma^{\mu}P_{L}q_{1}\bar{\ell}\gamma_{\mu}\gamma_{5}\ell-\frac{2m_{q_{1}}}{q^{2}}C^{eff}_{7}\bar{q}_{2}iq_{\nu}\sigma^{\mu\nu}P_{R}q_{1}\bar{\ell}\gamma_{\mu}\ell\right),$
(7)

where $G_{F}$ denotes the Fermi constant, the fine structure constant
$\alpha_{e}=\frac{e^{2}}{4\pi}$, the chiral projection operators
$P_{L,R}=(1\mp\gamma_{5})/2$,
$\sigma_{\mu\nu}=\frac{i[\gamma_{\mu},\gamma_{\nu}]}{2}$,
$\lambda_{q_{1}q_{2}}$ denotes the Cabibbo-Kobayashi-Maskawa (CKM) elements,
and $C_{i}$ denote Wilson coefficients. For the $b\to s/d\ell^{+}\ell^{-}$
transitions via the $u\bar{u},c\bar{c}$ loops,
$\lambda_{bs(bd)}=V_{tb}V^{*}_{ts}$($V_{tb}V^{*}_{td}$), and the expressions
of Wilson coefficients $C^{eff}_{7,9}$ and $C_{10}$ are given in Ref.
Buchalla:1995vs . For the $c\to u\ell^{+}\ell^{-}$ transition via the
$d\bar{d},s\bar{s}$ loops,
$\lambda_{cu}C^{eff}_{7,9}=\frac{4\pi}{\alpha_{s}}\Big{[}V^{*}_{cd}V_{ud}C^{eff(d)}_{7,9}(q^{2})+V^{*}_{cs}V_{us}C^{eff(s)}_{7,9}(q^{2})\Big{]}$
as well as $C_{10}=0$ and the expressions of $C^{eff(s,d)}_{7,9}(q^{2})$ can
be found in Refs. Meinel:2017ggx ; deBoer:2015boa . For the $s\to
d\ell^{+}\ell^{-}$ transition via the $u\bar{u}$ loop,
$\lambda_{sd}=V_{us}V^{*}_{ud}$,
$C^{eff}_{7}\approx\frac{V_{cs}V^{*}_{cd}}{2V_{us}V^{*}_{ud}}c^{c}_{7\gamma}$,
$C^{eff}_{9}=(z_{7V}-\frac{V_{ts}V^{*}_{td}}{V_{us}V^{*}_{ud}}y_{7V})\frac{2\pi}{\alpha_{e}}$,
and
$C_{10}=-\frac{V_{ts}V^{*}_{td}}{V_{us}V^{*}_{ud}}\frac{2\pi}{\alpha_{e}}y_{7A}$
with $z_{7V}=-0.046\alpha_{e}$, $y_{7V}=0.735\alpha_{e}$,
$y_{7A}=-0.700\alpha_{e}$ as well as $c^{c}_{7\gamma}=0.13\alpha_{e}$ from
Refs. Buchalla:1995vs ; Shifman:1976de ; He:2005yn . For $T_{b3}\to
T_{8}\ell^{+}\ell^{-}$ and $T_{c3}\to T_{8}\ell^{+}\ell^{-}$ decays, the
Wilson coefficient $C^{eff}_{9}$ receives not only from the four quark
operators but also from the long distance (LD) contributions coming from
$c\bar{c}$ for $T_{b3}\to T_{8}\ell^{+}\ell^{-}$ and $d\bar{d},s\bar{s}$ for
$T_{c3}\to T_{8}\ell^{+}\ell^{-}$. Note that the $T_{c3}\to
T_{8}\ell^{+}\ell^{-}$ decays are dominated by LD contributions. For $T_{8}\to
T^{\prime}_{8}\ell^{+}\ell^{-}$ decays, the LD contribution arises mainly from
the photon-mediated process $T_{8}\to T^{\prime}_{8}\gamma^{*}\to
T^{\prime}_{8}\ell^{+}\ell^{-}$ He:2018yzu ; Bergstrom:1987wr .

The helicity amplitudes for
$\mathcal{B}_{1}\to\mathcal{B}_{2}\ell^{+}\ell^{-}$ can be obtained from Eq.
(7),

$\mathcal{M}^{\lambda_{1},\lambda_{2}}_{VA}(s_{p},s_{k})=-\frac{G_{F}}{\sqrt{2}}\frac{\alpha_{e}}{4\pi}\lambda_{q_{1}q_{2}}\sum_{\lambda}\eta_{\lambda}\left[H^{L,s_{p},s_{k}}_{VA,\lambda}L^{\lambda_{1},\lambda_{2}}_{L,\lambda}+H^{R,s_{p},s_{k}}_{VA,\lambda}L^{\lambda_{1},\lambda_{2}}_{R,\lambda}\right],$
(8)

with

$\displaystyle L^{\lambda_{1}\lambda_{2}}_{L(R),\lambda}$ $\displaystyle=$
$\displaystyle\bar{\epsilon}^{\mu}(\lambda)\left\langle\bar{\ell}(\lambda_{1})\ell(\lambda_{2})\left|\bar{\ell}\gamma_{\mu}(1\mp\gamma_{5})\ell\right|0\right\rangle,$
$\displaystyle H^{L(R),s_{p},s_{k}}_{VA,\lambda}$ $\displaystyle=$
$\displaystyle\bar{\epsilon}^{*}_{\mu}(\lambda)\left\langle\mathcal{B}_{2}(k,s_{k})\left|\left[\Big{(}C^{eff}_{9}\mp
C_{10}\Big{)}\bar{q}_{2}\gamma^{\mu}(1-\gamma_{5})q_{1}-\frac{2m_{q_{1}}}{q^{2}}C^{eff}_{7}\bar{q}_{2}iq_{\nu}\sigma^{\mu\nu}(1+\gamma_{5})q_{1}\right]\right|\mathcal{B}_{1}(p,s_{p})\right\rangle,$
(9)

where $q^{2}\equiv(p-k)^{2}$ bounded in physical region as
$(2m_{\ell})^{2}\leq q^{2}\leq(m_{\mathcal{B}_{1}}-m_{\mathcal{B}_{2}})^{2}$,
the polarization of the gauge boson $\lambda=t,\pm 1,0$, the helicities of the
final state leptons are $\lambda_{1,2}$, and $\eta_{t}=1,\eta_{\pm 1,0}=-1$.

The nonvanishing leptonic helicity amplitudes
$L^{\lambda_{1}\lambda_{2}}_{L(R),\lambda}$ are

$\displaystyle L^{-\frac{1}{2}+\frac{1}{2}}_{L,+1}$ $\displaystyle=$
$\displaystyle-L^{+\frac{1}{2}-\frac{1}{2}}_{R,-1}=\sqrt{\frac{q^{2}}{2}}(1+\beta_{\ell})(1+cos\theta_{\ell}),$
$\displaystyle L^{+\frac{1}{2}-\frac{1}{2}}_{R,+1}$ $\displaystyle=$
$\displaystyle-L^{-\frac{1}{2}+\frac{1}{2}}_{L,-1}=-\sqrt{\frac{q^{2}}{2}}(1+\beta_{\ell})(1-cos\theta_{\ell}),$
$\displaystyle L^{-\frac{1}{2}+\frac{1}{2}}_{L,0}$ $\displaystyle=$
$\displaystyle
L^{+\frac{1}{2}-\frac{1}{2}}_{R,0}=\sqrt{q^{2}}(1+\beta_{\ell})sin\theta_{\ell},$
(10)

with $\beta_{\ell}=\sqrt{1-\frac{4m^{2}_{\ell}}{q^{2}}}$.

The $\mathcal{B}_{1}\to\mathcal{B}_{2}$ hadronic matrix elements are
calculated in the frameworks of soft-collinear effective theory
Feldmann:2011xf  and lattice QCD Detmold:2016pkz ; Boer:2014kda ;
Gutsche:2013pp . The helicity-based definition of the form factors are
presented as Detmold:2016pkz ; Boer:2014kda ; Gutsche:2013pp

$\displaystyle\langle\mathcal{B}_{2}(k,s_{k})|\bar{q}_{2}\gamma^{\mu}q_{1}|\mathcal{B}_{1}\rangle$
$\displaystyle=$
$\displaystyle\bar{u}(k,s_{k})\Bigg{[}f_{0}(q^{2})(m_{B_{1}}-m_{B_{2}})\frac{q^{\mu}}{q^{2}}+f_{+}(q^{2})\frac{m_{B_{1}}+m_{B_{2}}}{s_{+}}\left\\{p^{\mu}+k^{\mu}-\frac{q^{\mu}}{q^{2}}(m_{B_{1}}-m_{B_{2}})\right\\}$
(11) $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+f_{\perp}(q^{2})\left\\{\gamma^{\mu}-\frac{2m_{B_{2}}}{s_{+}}p^{\mu}-\frac{2m_{B_{1}}}{s_{+}}k^{\mu}\right\\}\Bigg{]}u(p,s_{p})\,,$
$\displaystyle\langle\mathcal{B}_{2}(k,s_{k})|\bar{q}_{2}\gamma^{\mu}\gamma_{5}q_{1}|\mathcal{B}_{1}(p,s_{p})\rangle$
$\displaystyle=$
$\displaystyle-\bar{u}(k,s_{k})\gamma_{5}\Bigg{[}g_{0}(q^{2})(m_{B_{1}}+m_{B_{2}})\frac{q^{\mu}}{q^{2}}+g_{+}(q^{2})\frac{m_{B_{1}}-m_{B_{2}}}{s_{-}}\left\\{p^{\mu}+k^{\mu}-\frac{q^{\mu}}{q^{2}}(m_{B_{1}}-m_{B_{2}})\right\\}$
(12) $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
+g_{\perp}(q^{2})\left\\{\gamma^{\mu}+\frac{2m_{B_{2}}}{s_{-}}p^{\mu}-\frac{2m_{B_{1}}}{s_{-}}k^{\mu}\right\\}\Bigg{]}u(p,s_{p})\,,$
$\displaystyle\langle\mathcal{B}_{2}(p^{\prime},s^{\prime})|\bar{q}_{2}\,i\sigma_{\mu\nu}q^{\nu}\,q_{1}|\mathcal{B}_{1}(p,s)\rangle$
$\displaystyle=$
$\displaystyle-\bar{u}_{\mathcal{B}_{2}}(p^{\prime},s^{\prime})\left\\{h_{+}(q^{2})\frac{q^{2}}{s_{+}}\left(p_{\mu}+p_{\mu}^{\prime}-\frac{q_{\mu}}{q^{2}}(m_{\mathcal{B}_{1}}^{2}-m_{\mathcal{B}_{2}}^{2})\right)\right.$
(13)
$\displaystyle\left.+(m_{\mathcal{B}_{1}}+m_{\mathcal{B}_{2}})h_{\perp}(q^{2})\left(\gamma_{\mu}-\frac{2m_{\mathcal{B}_{2}}}{s_{+}}p_{\mu}-\frac{2m_{\mathcal{B}_{1}}}{s_{+}}p^{\prime}_{\mu}\right)\right\\}u_{\mathcal{B}_{1}}(p,s),$
$\displaystyle\langle\mathcal{B}_{2}(p^{\prime},s^{\prime})|\bar{q}_{2}\,i\sigma_{\mu\nu}\gamma_{5}q^{\nu}q_{1}|\mathcal{B}_{1}\rangle$
$\displaystyle=$
$\displaystyle-\bar{u}_{\mathcal{B}_{2}}(p^{\prime},s^{\prime})\gamma_{5}\left\\{\tilde{h}_{+}(q^{2})\frac{q^{2}}{s_{-}}\left(p_{\mu}+p_{\mu}^{\prime}-\frac{q_{\mu}}{q^{2}}(m_{\mathcal{B}_{1}}^{2}-m_{\mathcal{B}_{2}}^{2})\right)\right.$
(14)
$\displaystyle\left.+(m_{\mathcal{B}_{1}}-m_{\mathcal{B}_{2}})\tilde{h}_{\perp}(q^{2})\left(\gamma_{\mu}+\frac{2m_{\mathcal{B}_{2}}}{s_{-}}p_{\mu}-\frac{2m_{\mathcal{B}_{1}}}{s_{-}}p^{\prime}_{\mu}\right)\right\\}u_{\mathcal{B}_{1}}(p,s),$

where $s_{\pm}=(m_{\mathcal{B}_{1}}\pm m_{\mathcal{B}_{2}})^{2}-q^{2}$ and
$f^{V,A,T,T_{5}}_{0,\perp}$ are the form factors. And then we obtain the
nonvanishing hadronic helicity amplitudes $H^{L(R),s_{p},s_{k}}_{VA,\lambda}$

$\displaystyle H^{L(R),+\frac{1}{2}+\frac{1}{2}}_{VA,0}$ $\displaystyle=$
$\displaystyle
f_{+}(q^{2})(m_{\mathcal{B}_{1}}+m_{\mathcal{B}_{2}})\sqrt{\frac{s_{-}}{q^{2}}}\mathcal{C}^{L(R)}_{VA}-g_{+}(q^{2})(m_{\mathcal{B}_{1}}-m_{\mathcal{B}_{2}})\sqrt{\frac{s_{+}}{q^{2}}}\mathcal{C}^{L(R)}_{VA}+\frac{2m_{b}}{q^{2}}\bigg{(}h_{+}(q^{2})\sqrt{q^{2}s_{-}}-\tilde{h}_{+}(q^{2})\sqrt{q^{2}s_{+}}\bigg{)}\mathcal{C}_{7}^{eff}\,,$
$\displaystyle H^{L(R),-\frac{1}{2}-\frac{1}{2}}_{VA,0}$ $\displaystyle=$
$\displaystyle
f_{+}(q^{2})(m_{\mathcal{B}_{1}}+m_{\mathcal{B}_{2}})\sqrt{\frac{s_{-}}{q^{2}}}\mathcal{C}^{L(R)}_{VA}+g_{+}(q^{2})(m_{\mathcal{B}_{1}}-m_{\mathcal{B}_{2}})\sqrt{\frac{s_{+}}{q^{2}}}\mathcal{C}^{L(R)}_{VA}+\frac{2m_{b}}{q^{2}}\bigg{(}h_{+}(q^{2})\sqrt{q^{2}s_{-}}+\tilde{h}_{+}(q^{2})\sqrt{q^{2}s_{+}}\bigg{)}\mathcal{C}_{7}^{eff}\,,$
$\displaystyle H^{L(R),-\frac{1}{2}+\frac{1}{2}}_{VA,+}$ $\displaystyle=$
$\displaystyle-
f_{\perp}(q^{2})\sqrt{2s_{-}}\mathcal{C}^{L(R)}_{VA}+g_{\perp}(q^{2})\sqrt{2s_{+}}\mathcal{C}^{L(R)}_{VA}-\frac{2m_{b}}{q^{2}}\bigg{(}h_{\perp}(q^{2})(m_{\mathcal{B}_{1}}+m_{\mathcal{B}_{2}})\sqrt{2s_{-}}-\tilde{h}_{\perp}(q^{2})(m_{\mathcal{B}_{1}}-m_{\mathcal{B}_{2}})\sqrt{2s_{+}}\bigg{)}\mathcal{C}_{7}^{eff}\,,$
$\displaystyle H^{L(R),+\frac{1}{2}-\frac{1}{2}}_{VA,-}$ $\displaystyle=$
$\displaystyle-
f_{\perp}(q^{2})\sqrt{2s_{-}}\mathcal{C}^{L(R)}_{VA}-g_{\perp}(q^{2})\sqrt{2s_{+}}\mathcal{C}^{L(R)}_{VA}-\frac{2m_{b}}{q^{2}}\bigg{(}h_{\perp}(q^{2})(m_{\mathcal{B}_{1}}+m_{\mathcal{B}_{2}})\sqrt{2s_{-}}+\tilde{h}_{\perp}(q^{2})(m_{\mathcal{B}_{1}}-m_{\mathcal{B}_{2}})\sqrt{2s_{+}}\bigg{)}\mathcal{C}_{7}^{eff}\,,$

with
$\mathcal{C}^{L(R)}_{VA}\equiv(\mathcal{C}_{9}^{eff}\mp\mathcal{C}_{10})$.

In addition, in terms of the SU(3) flavor symmetry, baryon states and quark
operators can be parametrized into SU(3) tensor forms, while the polarization
vectors $\bar{\epsilon}^{*}(\lambda)$ and leptonic helicity amplitudes
$L^{\lambda_{1},\lambda_{2}}_{L(R),\lambda}$ are invariant under SU(3) flavor
symmetry. The hadronic helicity amplitude relations of
$\mathcal{B}_{1}\to\mathcal{B}_{2}\ell^{+}\ell^{-}$ are similar to ones of
$\mathcal{B}_{1}\to\mathcal{B}_{2}\gamma$ as given in Ref. Wang:2020wxn , and
will be given in next section for convenience.

###  II.3 Observables for $\mathcal{B}_{1}\to\mathcal{B}_{2}\ell^{+}\ell^{-}$

In the rest frame of the baryon $\mathcal{B}_{1}$, the double differential
decay branching ratio is Das:2018sms

$\displaystyle\frac{d\mathcal{B}(\mathcal{B}_{1}\to\mathcal{B}_{2}\ell^{+}\ell^{-})}{dq^{2}d\cos\theta_{\ell}}$
$\displaystyle=$
$\displaystyle\frac{\tau_{\mathcal{B}_{1}}}{2m^{3}_{\Lambda_{b}}}\frac{2\sqrt{\lambda(m^{2}_{\mathcal{B}_{1}},m^{2}_{\mathcal{B}_{2}},q^{2})}}{(8\pi)^{3}}\frac{1}{2s_{p}+1}\sum_{\lambda_{1},\lambda_{2}}\sum_{s_{p},s_{k}}\big{|}\mathcal{M}^{\lambda_{1},\lambda_{2}}_{VA}(s_{p},s_{k})\big{|}^{2}\,$
(16) $\displaystyle=$ $\displaystyle
N^{2}(q^{2})\bigg{[}(1-cos^{2}\theta_{\ell})\bigg{(}\big{|}H_{VA,0}^{L,+\frac{1}{2}+\frac{1}{2}}\big{|}^{2}+\big{|}H_{VA,0}^{L,-\frac{1}{2}-\frac{1}{2}}\big{|}^{2}+\big{|}H_{VA,0}^{R,+\frac{1}{2}+\frac{1}{2}}\big{|}^{2}+\big{|}H_{VA,0}^{R,-\frac{1}{2}-\frac{1}{2}}\big{|}^{2}\bigg{)}$
$\displaystyle+\frac{1}{2}(1-cos\theta_{\ell})^{2}\bigg{(}\big{|}H_{VA,-}^{L,+\frac{1}{2}-\frac{1}{2}}\big{|}^{2}+\big{|}H_{VA,+}^{R,-\frac{1}{2}+\frac{1}{2}}\big{|}^{2}\bigg{)}$
$\displaystyle+\frac{1}{2}(1+cos\theta_{\ell})^{2}\bigg{(}\big{|}H_{VA,-}^{R,+\frac{1}{2}-\frac{1}{2}}\big{|}^{2}+\big{|}H_{VA,+}^{L,-\frac{1}{2}+\frac{1}{2}}\big{|}^{2}\bigg{)}\bigg{]},$

with

$N(q^{2})=G_{F}\lambda_{q_{1}q_{2}}\alpha_{e}\left(1+\sqrt{1-\frac{4m^{2}_{\ell}}{q^{2}}}\right)\sqrt{\tau_{\mathcal{B}_{1}}\frac{q^{2}\sqrt{\lambda(m_{\mathcal{B}_{1}},m_{\mathcal{B}_{2}},q^{2})}}{2^{15}m^{3}_{\mathcal{B}_{1}}\pi^{5}}},$
(17)

where $\lambda(a,b,c)=a^{2}+b^{2}+c^{2}-2(ab+bc+ca)$. And the differential
decay branching ratio is

$\displaystyle\frac{d\mathcal{B}(\mathcal{B}_{1}\to\mathcal{B}_{2}\ell^{+}\ell^{-})}{dq^{2}}=\frac{4}{3}N^{2}(q^{2})H_{M}(q^{2}),$
(18)

with

$\displaystyle H_{M}(q^{2})$ $\displaystyle=$
$\displaystyle\bigg{(}\big{|}H_{VA,-}^{L,+\frac{1}{2}-\frac{1}{2}}\big{|}^{2}+\big{|}H_{VA,+}^{R,-\frac{1}{2}+\frac{1}{2}}\big{|}^{2}\bigg{)}+\bigg{(}\big{|}H_{VA,-}^{R,+\frac{1}{2}-\frac{1}{2}}\big{|}^{2}+\big{|}H_{VA,+}^{L,-\frac{1}{2}+\frac{1}{2}}\big{|}^{2}\bigg{)}$
(19)
$\displaystyle+\bigg{(}\big{|}H_{VA,0}^{L,+\frac{1}{2}+\frac{1}{2}}\big{|}^{2}+\big{|}H_{VA,0}^{L,-\frac{1}{2}-\frac{1}{2}}\big{|}^{2}+\big{|}H_{VA,0}^{R,+\frac{1}{2}+\frac{1}{2}}\big{|}^{2}+\big{|}H_{VA,0}^{R,-\frac{1}{2}-\frac{1}{2}}\big{|}^{2}\bigg{)}.$

The longitudinal polarization fraction can be obtained by Eq. (16)

$F_{L}(q^{2})=\frac{\displaystyle\int_{-1}^{+1}d\cos\theta_{\ell}(2-5\cos^{2}\theta_{\ell})\frac{d^{2}\mathcal{B}}{dq^{2}d\cos\theta_{\ell}}}{\displaystyle\int_{-1}^{+1}d\cos\theta_{\ell}\frac{d^{2}\mathcal{B}}{dq^{2}d\cos\theta_{\ell}}}\,.$
(20)

and the concrete expression is

$\displaystyle F_{L}(q^{2})$ $\displaystyle=$
$\displaystyle\bigg{(}\big{|}H_{VA,0}^{L,+\frac{1}{2}+\frac{1}{2}}\big{|}^{2}+\big{|}H_{VA,0}^{L,-\frac{1}{2}-\frac{1}{2}}\big{|}^{2}+\big{|}H_{VA,0}^{R,+\frac{1}{2}+\frac{1}{2}}\big{|}^{2}+\big{|}H_{VA,0}^{R,-\frac{1}{2}-\frac{1}{2}}\big{|}^{2}\bigg{)}\left[H_{M}(q^{2})\right]^{-1}.$
(21)

The leptonic forward-backward asymmetry

$A^{\ell}_{FB}(q^{2})=\frac{\displaystyle\int_{0}^{+1}d\cos\theta_{\ell}\frac{d^{2}\mathcal{B}}{dq^{2}d\cos\theta_{\ell}}-\displaystyle\int_{-1}^{0}d\cos\theta_{\ell}\frac{d^{2}\mathcal{B}}{dq^{2}d\cos\theta_{\ell}}}{\displaystyle\int_{-1}^{0}d\cos\theta_{\ell}\frac{d^{2}\mathcal{B}}{dq^{2}d\cos\theta_{\ell}}+\displaystyle\int_{0}^{+1}d\cos\theta_{\ell}\frac{d^{2}\mathcal{B}}{dq^{2}d\cos\theta_{\ell}}}\,,$
(22)

and the concrete expression is

$\displaystyle A^{\ell}_{FB}(q^{2})=$
$\displaystyle\frac{3}{4}\left[\bigg{(}\big{|}H_{VA,-}^{R,+\frac{1}{2}-\frac{1}{2}}\big{|}^{2}+\big{|}H_{VA,+}^{L,-\frac{1}{2}+\frac{1}{2}}\big{|}^{2}\bigg{)}-\bigg{(}\big{|}H_{VA,-}^{L,+\frac{1}{2}-\frac{1}{2}}\big{|}^{2}+\big{|}H_{VA,+}^{R,-\frac{1}{2}+\frac{1}{2}}\big{|}^{2}\bigg{)}\right]\left[H_{M}(q^{2})\right]^{-1}.$
(23)

The lepton flavor universality in baryon weak decays $T_{b3}\to
T_{8}\ell^{+}\ell^{-}$ is defined in a manner identical $R_{K^{(*)}}$ as

$\displaystyle R_{T_{b3}\to
T_{8}}\equiv\frac{\int^{q_{max}}_{q_{min}}d\mathcal{B}(T_{b3}\to
T_{8}\mu^{+}\mu^{-})/ds}{\int^{q_{max}}_{q_{min}}d\mathcal{B}(T_{b3}\to
T_{8}e^{+}e^{-})/ds}.$ (24)

For $q^{2}$ integration of $X(q^{2})=F_{L}(q^{2})$ and $A^{\ell}_{FB}(q^{2})$,
following Ref. Bobeth:2010wg , two ways of integration are considered. The
normalized $q^{2}$-integrated observables $\langle X\rangle$ are calculated by
separately integrating the numerators and denominators with the same $q^{2}$
bins. The “naively integrated” observables are obtained by

$\displaystyle\overline{X}=\frac{1}{q^{2}_{max}-q^{2}_{min}}\int^{q^{2}_{max}}_{q^{2}_{min}}dq^{2}X(q^{2}).$
(25)

Note that, besides the single-quark transition contributions, the $W$-exchange
contributions via the two-quark and three-quark transitions as well as the
internal radiation transition, which contribute to the radiative baryon decays
$\mathcal{B}_{1}\to\mathcal{B}_{2}\gamma$ Verma:1988gf ; Azimov:1996uf ;
Lach:1995we , may also contribute to the semileptonic baryon decays
$\mathcal{B}_{1}\to\mathcal{B}_{2}\ell^{+}\ell^{-}$. In some decays, for
example, $\Sigma^{+}\to p\ell^{+}\ell^{-}$ decays, the $W$-exchange
contributions with the two-quark transition will play a major role
Bergstrom:1987wr . So we will consider these W-exchange contributions in the
later analysis of SU(3) flavor symmetry.

##  III Results and Analysis

The theoretical input parameters and the experimental data within the
$1\sigma$ error from the Particle Data Group PDG2020  will be used in our
numerical results. To obtain SU(3) IRA amplitudes, one just needs to contract
all upper and lower indices of the hadrons and the Hamiltonian to form all
possible SU(3) singlets and associate each with a parameter which lumps up the
Wilson coefficients and unknown hadronization effects Pan:2020qqo . These
parameters can be determined theoretically and experimentally. In this work,
we will determine these parameters by relevant experimental data, and then
give the predictions for other not-yet-measured decay modes. For $T_{b3}$
semileptonic decays, there are enough phase spaces to allow for
$e^{+}e^{-},\mu^{+}\mu^{-}$, and $\tau^{+}\tau^{-}$ decays. $T_{c3}$ and
$T_{8}$ semileptonic decays only have enough phase spaces to allow for both
$e^{+}e^{-}$ and $\mu^{+}\mu^{-}$ decays. The results of $T_{b3}\to
T_{8}e^{+}e^{-},T_{8}\mu^{+}\mu^{-},T_{8}\tau^{+}\tau^{-}$ decays, $T_{c3}\to
T_{8}e^{+}e^{-},T_{8}\mu^{+}\mu^{-}$ decays, and $T_{8}\to
T^{\prime}_{8}e^{+}e^{-},T^{\prime}_{8}\mu^{+}\mu^{-}$ decays are given in the
following subsections A, B, and C, respectively.

###  III.1 $T_{b3}\to T_{8}\ell^{+}\ell^{-}$ weak decays

For $T_{b3}\to T_{8}\ell^{+}\ell^{-}$ decays, the single-quark transition
contributions are strongly dominant so that other contributions like the
$W$-exchange contributions are usually omitted. The SU(3) flavor structure
Hamiltonian with $b\to s,d$ transitions can been found, for instance, in Refs.
Zeppenfeld:1980ex ; Savage:1989ub ; Deshpande44 , and the SU(3) IRA hadronic
helicity amplitudes for $T_{b3}\to T_{8}\ell^{+}\ell^{-}$ via $b\to
s/d\ell^{+}\ell^{-}$ can be parametrized as

$\displaystyle H(T_{b3}\to
T_{8}\ell^{+}\ell^{-})^{L(R),s_{p},s_{k}}_{VA,\lambda}$ $\displaystyle=$
$\displaystyle
e_{1}(T_{b3})^{[ij]}T(\bar{3})^{k}(T_{8})_{[ij]k}+e_{2}(T_{b3})^{[ij]}T(3)^{k}(T_{8})_{[ik]j},$
(26)

which are similar to the decay amplitudes of corresponding $T_{b3}\to
T_{8}\gamma$ modes in Ref. Wang:2019alu . In Eq. (26), $T(\bar{3})=(0,1,1)$
denoted the transition operators $(\bar{q}_{2}b)$ with $q_{2}=s,d$, and the
model as well as scale independent parameters
$e_{i}\equiv(e_{i})^{L(R),s_{p},s_{k}}_{VA,\lambda}(q^{2})$. The parameters
$e_{i}$ contain information about QCD dynamics, and could include the long
distance (LD) contributions from hadron resonances. The SU(3) IRA amplitudes
of the $T_{b3}\to T_{8}\ell^{+}\ell^{-}$ weak decays are given in Tab. 2, and
for a better understanding, the information of relevant CKM matrix elements
are also listed in Tab. 2. From Tab. 2, one can see that
$\Lambda^{0}_{b}\to\Sigma^{0}\ell^{+}\ell^{-}$ decays are not allowed by the
SU(3) flavor symmetry, and other decay modes via $b\to s/d\ell^{+}\ell^{-}$
can be related by only one parameter $E\equiv e_{1}+e_{2}$.

Table 1: The SU(3) IRA amplitudes of the $T_{b3}\to T_{8}\ell^{+}\ell^{-}$
weak decays by the $b\to s/d\ell^{+}\ell^{-}$ transitions, and $E\equiv
e_{1}+e_{2}$.

Decay modes |  $A(T_{b3}\to T_{8}\ell^{+}\ell^{-})$
---|---
$T_{b3}\to T_{8}\ell^{+}\ell^{-}$ via the $b\to s\ell^{+}\ell^{-}$ transition: |
$\Lambda^{0}_{b}\to\Lambda^{0}\ell^{+}\ell^{-}$ |  $-2\lambda_{bs}\leavevmode\nobreak\ E/\sqrt{6}$
$\Lambda^{0}_{b}\to\Sigma^{0}\ell^{+}\ell^{-}$ |  $0$
$\Xi^{0}_{b}\to\Xi^{0}\ell^{+}\ell^{-}$ |  $-\lambda_{bs}\leavevmode\nobreak\ E$
$\Xi^{-}_{b}\to\Xi^{-}\ell^{+}\ell^{-}$ |  $\lambda_{bs}\leavevmode\nobreak\ E$
$T_{b3}\to T_{8}\ell^{+}\ell^{-}$ via the $b\to d\ell^{+}\ell^{-}$ transition: |
$\Lambda^{0}_{b}\to n\ell^{+}\ell^{-}$ |  $\lambda_{bd}\leavevmode\nobreak\ E$
$\Xi^{0}_{b}\to\Lambda^{0}\ell^{+}\ell^{-}$ |  $-\lambda_{bd}\leavevmode\nobreak\ E/\sqrt{6}$
$\Xi^{0}_{b}\to\Sigma^{0}\ell^{+}\ell^{-}$ |  $-\lambda_{bd}\leavevmode\nobreak\ E/\sqrt{2}$
$\Xi^{-}_{b}\to\Sigma^{-}\ell^{+}\ell^{-}$ |  $\lambda_{bd}\leavevmode\nobreak\ E$

Table 2: Branching ratios for $T_{b3}\to T_{8}\ell^{+}\ell^{-}$ decays with
$1\sigma$ error in the whole $q^{2}$ region within $S_{1}$ and $S_{2}$ cases.

Decay modes |  Experimental data PDG2020  |  Our results in $S_{1}$ |  Our results in $S_{2}$ | Other predictions
---|---|---|---|---
$\mathcal{B}(\Lambda^{0}_{b}\to\Lambda^{0}\mu^{+}\mu^{-})(\times 10^{-6})$ | $1.08\pm 0.28$ | $1.08\pm 0.28$ | $1.08\pm 0.28$ |  $1.05$ Faustov:2020thr
$\mathcal{B}(\Xi^{0}_{b}\to\Xi^{0}\mu^{+}\mu^{-})(\times 10^{-6})$ | $\cdots$ | $1.55^{+0.45}_{-0.43}$ | $1.77^{+0.49}_{-0.53}$ |
$\mathcal{B}(\Xi^{-}_{b}\to\Xi^{-}\mu^{+}\mu^{-})(\times 10^{-6})$ | $\cdots$ | $1.65^{+0.49}_{-0.46}$ | $1.87^{+0.56}_{-0.54}$ |
$\mathcal{B}(\Lambda^{0}_{b}\to\Lambda^{0}\tau^{+}\tau^{-})(\times 10^{-7})$ | $\cdots$ | $2.30\pm 0.60$ | $2.74^{+0.85}_{-0.71}$ |  $2.60$ Faustov:2020thr
$\mathcal{B}(\Xi^{0}_{b}\to\Xi^{0}\tau^{+}\tau^{-})(\times 10^{-7})$ | $\cdots$ | $3.23^{+0.94}_{-0.89}$ | $4.42^{+1.36}_{-1.21}$ |
$\mathcal{B}(\Xi^{-}_{b}\to\Xi^{-}\tau^{+}\tau^{-})(\times 10^{-7})$ | $\cdots$ | $3.42^{+1.01}_{-0.95}$ | $4.76^{+1.44}_{-1.36}$ |
$b\to d\ell^{+}\ell^{-}:$ |  |  |  |
$\mathcal{B}(\Lambda^{0}_{b}\to n\mu^{+}\mu^{-})(\times 10^{-8})$ | $\cdots$ | $8.15^{+2.44}_{-2.30}$ | $7.77^{+2.42}_{-2.28}$ |  ${}^{\big{(}4.1^{+5.4}_{-1.2}\big{)}}_{\leavevmode\nobreak\ \leavevmode\nobreak\ 3.75}$ Liu:2019rpm ; Faustov:2020thr
$\mathcal{B}(\Xi^{0}_{b}\to\Lambda^{0}\mu^{+}\mu^{-})(\times 10^{-8})$ | $\cdots$ | $1.34^{+0.43}_{-0.39}$ | $1.45^{+0.44}_{-0.45}$ |
$\mathcal{B}(\Xi^{0}_{b}\to\Sigma^{0}\mu^{+}\mu^{-})(\times 10^{-8})$ | $\cdots$ | $3.77^{+1.22}_{-1.10}$ | $4.13^{+1.36}_{-1.24}$ |
$\mathcal{B}(\Xi^{-}_{b}\to\Sigma^{-}\mu^{+}\mu^{-})(\times 10^{-8})$ | $\cdots$ | $8.00^{+2.56}_{-2.40}$ | $8.61^{+3.06}_{-2.52}$ |
$\mathcal{B}(\Lambda^{0}_{b}\to n\tau^{+}\tau^{-})(\times 10^{-8})$ | $\cdots$ | $2.07^{+0.62}_{-0.58}$ | $2.46^{+0.78}_{-0.70}$ |  ${}^{\big{(}2.9^{+3.7}_{-0.8}\big{)}}_{\leavevmode\nobreak\ \leavevmode\nobreak\ 1.21}$ Liu:2019rpm ; Faustov:2020thr
$\mathcal{B}(\Xi^{0}_{b}\to\Lambda^{0}\tau^{+}\tau^{-})(\times 10^{-9})$ | $\cdots$ | $3.42^{+1.11}_{-1.00}$ | $4.63^{+1.71}_{-1.35}$ |
$\mathcal{B}(\Xi^{0}_{b}\to\Sigma^{0}\tau^{+}\tau^{-})(\times 10^{-9})$ | $\cdots$ | $8.97^{+2.91}_{-2.62}$ | $12.23^{+4.12}_{-3.62}$ |
$\mathcal{B}(\Xi^{-}_{b}\to\Sigma^{-}\tau^{+}\tau^{-})(\times 10^{-8})$ | $\cdots$ | $1.91^{+0.61}_{-0.57}$ | $2.60^{+0.87}_{-0.77}$ |

Table 3: Branching ratios for $T_{b3}\to T_{8}\mu^{+}\mu^{-}$ weak decays in
different $q^{2}$ bins with $1\sigma$ error in $S_{1}$ and $S_{2}$ cases ( in
unit of $10^{-7}$).

$[q^{2}_{min},q^{2}_{max}](\mbox{GeV}^{2})$ | $[0.1,2.0]$ | $[2.0,4.3]$ | $[0.1,4.3]$ | $[4.0,6.0]$ | $[1.0,6.0]$ | $[6.0,8.0]$ | $[4.3,8.68]$ | $[10.09,12.86]$ | $[14.18,16.0]$ | $[0.1,16.0]$ | $[18.0,20.0]$ | $[15.0,20.0]$
---|---|---|---|---|---|---|---|---|---|---|---|---
$\mathcal{B}(\Lambda^{0}_{b}\to\Lambda^{0}\mu^{+}\mu^{-})^{Exp.}$ | $0.71\pm 0.27$ | $0.28^{+0.28}_{-0.21}$ | $2.7\pm 2.7$ | $0.04^{+0.18}_{-0.02}$ | $0.47^{+0.31}_{-0.27}$ | $0.50^{+0.26}_{-0.24}$ | $0.5\pm 0.7$ | $2.2\pm 0.6$ | $1.7\pm 0.5$ | $7.0\pm 2.9$ | $2.44\pm 0.57$ | $6.0\pm 1.3$
$\mathcal{B}(\Xi^{0}_{b}\to\Xi^{0}\mu^{+}\mu^{-})_{S_{1}}$ | $1.03^{+0.42}_{-0.41}$ | $0.41^{+0.43}_{-0.31}$ | $3.91^{+4.00}_{-3.85}$ | $0.058^{+0.270}_{-0.030}$ | $0.68^{+0.47}_{-0.40}$ | $0.73^{+0.40}_{-0.36}$ | $0.73^{+1.06}_{-1.02}$ | $3.21^{+0.97}_{-0.92}$ | $2.47^{+0.80}_{-0.77}$ | $10.18^{+4.53}_{-4.35}$ | $3.23^{+0.85}_{-0.81}$ | $8.46^{+2.12}_{-2.01}$
$\mathcal{B}(\Xi^{0}_{b}\to\Xi^{0}\mu^{+}\mu^{-})_{S_{2}}$ | $1.14^{+0.49}_{-0.46}$ | $0.45^{+0.48}_{-0.34}$ | $4.39^{+4.58}_{-4.32}$ | $0.065^{+0.302}_{-0.033}$ | $0.76^{+0.54}_{-0.45}$ | $0.81^{+0.45}_{-0.39}$ | $0.81^{+1.20}_{-0.81}$ | $3.60^{+1.14}_{-1.06}$ | $2.83^{+0.91}_{-0.89}$ | $11.46^{+5.29}_{-4.96}$ | $3.82^{+1.01}_{-1.03}$ | $9.81^{+2.45}_{-2.55}$
$\mathcal{B}(\Xi^{-}_{b}\to\Xi^{-}\mu^{+}\mu^{-})_{S_{1}}$ | $1.09^{+0.46}_{-0.43}$ | $0.43^{+0.45}_{-0.32}$ | $4.15^{+4.28}_{-4.08}$ | $0.062^{+0.289}_{-0.032}$ | $0.72^{+0.51}_{-0.42}$ | $0.77^{+0.43}_{-0.38}$ | $0.77^{+1.13}_{-1.09}$ | $3.40^{+1.04}_{-0.99}$ | $2.62^{+0.86}_{-0.82}$ | $10.80^{+4.89}_{-4.65}$ | $3.40^{+0.91}_{-0.87}$ | $8.94^{+2.30}_{-2.15}$
$\mathcal{B}(\Xi^{-}_{b}\to\Xi^{-}\mu^{+}\mu^{-})_{S_{2}}$ | $1.23^{+0.52}_{-0.49}$ | $0.48^{+0.51}_{-0.37}$ | $4.57^{+4.92}_{-4.50}$ | $0.069^{+0.324}_{-0.035}$ | $0.81^{+0.58}_{-0.47}$ | $0.86^{+0.50}_{-0.42}$ | $0.86^{+1.28}_{-0.86}$ | $3.84^{+1.17}_{-1.13}$ | $3.00^{+0.99}_{-0.94}$ | $12.22^{+5.56}_{-5.38}$ | $3.98^{+1.14}_{-1.03}$ | $10.30^{+2.77}_{-2.56}$
$\mathcal{B}(\Lambda^{0}_{b}\to n\mu^{+}\mu^{-})_{S_{1}}$ | $0.047^{+0.020}_{-0.019}$ | $0.019^{+0.020}_{-0.014}$ | $0.180^{+0.189}_{-0.177}$ | $0.0027^{+0.0127}_{-0.0014}$ | $0.031^{+0.022}_{-0.019}$ | $0.034^{+0.019}_{-0.017}$ | $0.034^{+0.050}_{-0.047}$ | $0.152^{+0.048}_{-0.045}$ | $0.123^{+0.041}_{-0.039}$ | $0.482^{+0.225}_{-0.208}$ | $0.236^{+0.065}_{-0.062}$ | $0.491^{+0.128}_{-0.121}$
$\mathcal{B}(\Lambda^{0}_{b}\to n\mu^{+}\mu^{-})_{S_{2}}$ | $0.044^{+0.020}_{-0.018}$ | $0.017^{+0.019}_{-0.013}$ | $0.169^{+0.176}_{-0.166}$ | $0.0025^{+0.0116}_{-0.0012}$ | $0.029^{+0.021}_{-0.017}$ | $0.030^{+0.017}_{-0.015}$ | $0.031^{+0.045}_{-0.031}$ | $0.132^{+0.042}_{-0.040}$ | $0.107^{+0.036}_{-0.034}$ | $0.423^{+0.204}_{-0.181}$ | $0.215^{+0.059}_{-0.056}$ | $0.445^{+0.117}_{-0.114}$
$\mathcal{B}(\Xi^{0}_{b}\to\Lambda^{0}\mu^{+}\mu^{-})_{S_{1}}$ | $0.008\pm 0.003$ | $0.003\pm 0.003$ | $0.029^{+0.031}_{-0.029}$ | $0.0000^{+0.0025}_{-0.0000}$ | $0.005^{+0.004}_{-0.003}$ | $0.005\pm 0.003$ | $0.005\pm 0.008$ | $0.025\pm 0.008$ | $0.020\pm 0.007$ | $0.079^{+0.038}_{-0.035}$ | $0.039^{+0.012}_{-0.011}$ | $0.081^{+0.023}_{-0.021}$
$\mathcal{B}(\Xi^{0}_{b}\to\Lambda^{0}\mu^{+}\mu^{-})_{S_{2}}$ | $0.008^{+0.004}_{-0.003}$ | $0.003^{+0.003}_{-0.002}$ | $0.030^{+0.034}_{-0.029}$ | $0.0004^{+0.0021}_{-0.0002}$ | $0.005^{+0.004}_{-0.003}$ | $0.005\pm 0.003$ | $0.005^{+0.008}_{-0.005}$ | $0.024\pm 0.008$ | $0.020^{+0.007}_{-0.006}$ | $0.077^{+0.039}_{-0.034}$ | $0.041^{+0.012}_{-0.011}$ | $0.083^{+0.024}_{-0.022}$
$\mathcal{B}(\Xi^{0}_{b}\to\Sigma^{0}\mu^{+}\mu^{-})_{S_{1}}$ | $0.023^{+0.010}_{-0.009}$ | $0.009^{+0.010}_{-0.007}$ | $0.087^{+0.093}_{-0.085}$ | $0.0013^{+0.0062}_{-0.0013}$ | $0.015^{+0.011}_{-0.009}$ | $0.016^{+0.010}_{-0.008}$ | $0.016^{+0.025}_{-0.023}$ | $0.073^{+0.024}_{-0.023}$ | $0.058^{+0.022}_{-0.019}$ | $0.231^{+0.113}_{-0.103}$ | $0.103^{+0.031}_{-0.028}$ | $0.225^{+0.063}_{-0.057}$
$\mathcal{B}(\Xi^{0}_{b}\to\Sigma^{0}\mu^{+}\mu^{-})_{S_{2}}$ | $0.024^{+0.011}_{-0.010}$ | $0.010^{+0.011}_{-0.007}$ | $0.093^{+0.101}_{-0.092}$ | $0.0014^{+0.0065}_{-0.0007}$ | $0.016^{+0.012}_{-0.009}$ | $0.017^{+0.010}_{-0.009}$ | $0.017^{+0.025}_{-0.017}$ | $0.075^{+0.025}_{-0.023}$ | $0.060^{+0.023}_{-0.020}$ | $0.237^{+0.120}_{-0.103}$ | $0.114^{+0.034}_{-0.032}$ | $0.245^{+0.069}_{-0.064}$
$\mathcal{B}(\Xi^{-}_{b}\to\Sigma^{-}\mu^{+}\mu^{-})_{S_{1}}$ | $0.048^{+0.022}_{-0.020}$ | $0.019^{+0.021}_{-0.014}$ | $0.184^{+0.200}_{-0.181}$ | $0.0027^{+0.0133}_{-0.0014}$ | $0.032^{+0.024}_{-0.019}$ | $0.034^{+0.021}_{-0.017}$ | $0.034^{+0.053}_{-0.049}$ | $0.155^{+0.053}_{-0.047}$ | $0.124^{+0.045}_{-0.041}$ | $0.491^{+0.240}_{-0.218}$ | $0.220^{+0.066}_{-0.061}$ | $0.478^{+0.140}_{-0.122}$
$\mathcal{B}(\Xi^{-}_{b}\to\Sigma^{-}\mu^{+}\mu^{-})_{S_{2}}$ | $0.051^{+0.024}_{-0.020}$ | $0.020^{+0.023}_{-0.015}$ | $0.196^{+0.213}_{-0.193}$ | $0.0029^{+0.0140}_{-0.0015}$ | $0.034^{+0.026}_{-0.020}$ | $0.036^{+0.021}_{-0.018}$ | $0.036^{+0.054}_{-0.036}$ | $0.159^{+0.056}_{-0.049}$ | $0.129^{+0.047}_{-0.043}$ | $0.508^{+0.260}_{-0.225}$ | $0.243^{+0.077}_{-0.066}$ | $0.515^{+0.151}_{-0.130}$

Table 4: Branching ratios for $T_{b3}\to T_{8}\tau^{+}\tau^{-}$ weak decays in
different $q^{2}$ bins with $1\sigma$ error in $S_{1}$ and $S_{2}$ cases ( in
unit of $10^{-7}$).

$[q^{2}_{min},q^{2}_{max}](\mbox{GeV}^{2})$ |  $[14.18,16.0]$ in $S_{1}$ |  $[14.18,16.0]$ in $S_{2}$ |  $[18.0,20.0]$ in $S_{1}$ |  $[18.0,20.0]$ in $S_{2}$ |  $[15.0,20.0]$ in $S_{1}$ |  $[15.0,20.0]$ in $S_{2}$
---|---|---|---|---|---|---
$\mathcal{B}(\Lambda^{0}_{b}\to\Lambda^{0}\tau^{+}\tau^{-})$ | $0.83\pm 0.25$ | $0.84^{+0.27}_{-0.25}$ | $1.51\pm 0.36$ | $1.52\pm 0.37$ | $3.41\pm 0.76$ | $3.44^{+0.86}_{-0.80}$
$\mathcal{B}(\Xi^{0}_{b}\to\Xi^{0}\tau^{+}\tau^{-})$ | $1.21^{+0.39}_{-0.38}$ | $1.40^{+0.46}_{-0.45}$ | $2.00^{+0.53}_{-0.50}$ | $2.32^{+0.66}_{-0.59}$ | $4.79^{+1.20}_{-1.14}$ | $5.65^{+1.43}_{-1.44}$
$\mathcal{B}(\Xi^{-}_{b}\to\Xi^{-}\tau^{+}\tau^{-})$ | $1.29^{+0.42}_{-0.40}$ | $1.49^{+0.50}_{-0.48}$ | $2.11^{+0.57}_{-0.54}$ | $2.50\pm 0.67$ | $5.07^{+1.30}_{-1.22}$ | $5.92^{+1.58}_{-1.48}$
$\mathcal{B}(\Lambda^{0}_{b}\to n\tau^{+}\tau^{-})$ | $0.060^{+0.020}_{-0.019}$ | $0.053\pm 0.018$ | $0.147^{+0.041}_{-0.039}$ | $0.133^{+0.038}_{-0.034}$ | $0.282^{+0.074}_{-0.070}$ | $0.257^{+0.068}_{-0.064}$
$\mathcal{B}(\Xi^{0}_{b}\to\Lambda^{0}\tau^{+}\tau^{-})$ | $0.010^{+0.004}_{-0.003}$ | $0.0095^{+0.0039}_{-0.0030}$ | $0.024\pm 0.007$ | $0.026\pm 0.007$ | $0.047^{+0.013}_{-0.012}$ | $0.048^{+0.014}_{-0.013}$
$\mathcal{B}(\Xi^{0}_{b}\to\Sigma^{0}\tau^{+}\tau^{-})$ | $0.029^{+0.011}_{-0.009}$ | $0.030^{+0.011}_{-0.010}$ | $0.064^{+0.019}_{-0.017}$ | $0.071^{+0.021}_{-0.019}$ | $0.129^{+0.036}_{-0.033}$ | $0.140^{+0.041}_{-0.037}$
$\mathcal{B}(\Xi^{-}_{b}\to\Sigma^{-}\tau^{+}\tau^{-})$ | $0.061^{+0.022}_{-0.020}$ | $0.063^{+0.023}_{-0.021}$ | $0.137^{+0.041}_{-0.038}$ | $0.152^{+0.047}_{-0.041}$ | $0.273^{+0.080}_{-0.070}$ | $0.301^{+0.086}_{-0.078}$

Table 5: Longitudinal polarization fractions and forward-backward asymmetries
for $\Lambda^{0}_{b}\to\Lambda^{0}\mu^{+}\mu^{-},\Lambda^{0}\tau^{+}\tau^{-}$
decays in different $q^{2}$ bins with $1\sigma$ error in $S_{2}$ case.

$[q^{2}_{min},q^{2}_{max}](\mbox{GeV}^{2})$ | $[0.1,2.0]$ | $[2.0,4.3]$ | $[0.1,4.3]$ | $[4.0,6.0]$ | $[1.0,6.0]$ | $[6.0,8.0]$ | $[4.3,8.68]$ | $[10.09,12.86]$ | $[14.18,16.0]$ | $[0.1,16.0]$ | $[18.0,20.0]$ | $[15.0,20.0]$ |  whole $q^{2}$ regions
---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\overline{f_{L}}(\Lambda^{0}_{b}\to\Lambda^{0}\mu^{+}\mu^{-})$ | $0.64^{+0.04}_{-0.01}$ | $0.86\pm 0.01$ | $0.77^{+0.06}_{-0.03}$ | $0.81\pm 0.01$ | $0.83^{+0.02}_{-0.01}$ | $0.73\pm 0.01$ | $0.77\pm 0.02$ | $0.57^{+0.00}_{-0.02}$ | $0.46\pm 0.01$ | $0.66^{+0.02}_{-0.03}$ | $0.36\pm 0.01$ | $0.39^{+0.02}_{-0.01}$ | $0.60\pm 0.02$
$\langle f_{L}\rangle(\Lambda^{0}_{b}\to\Lambda^{0}\mu^{+}\mu^{-})$ | $0.36^{+0.04}_{-0.02}$ | $0.86^{+0.06}_{-0.04}$ | $0.42^{+0.04}_{-0.02}$ | $0.81^{+0.03}_{-0.02}$ | $0.82^{+0.05}_{-0.02}$ | $0.73\pm 0.01$ | $0.75\pm 0.03$ | $0.56\pm 0.01$ | $0.45\pm 0.01$ | $0.47^{+0.04}_{-0.02}$ | $0.36\pm 0.01$ | $0.40^{+0.01}_{-0.02}$ | $0.34^{+0.03}_{-0.02}$
$\overline{A^{\ell}_{FB}}(\Lambda^{0}_{b}\to\Lambda^{0}\mu^{+}\mu^{-})$ | $0.12\pm 0.01$ | $0.05^{+0.00}_{-0.01}$ | $0.08^{+0.00}_{-0.01}$ | $-0.05^{+0.00}_{-0.01}$ | $0.03^{+0.00}_{-0.01}$ | $-0.15^{+0.00}_{-0.01}$ | $-0.12^{+0.00}_{-0.01}$ | $-0.29\pm 0.01$ | $-0.36\pm 0.00$ | $-0.15\pm 0.01$ | $-0.31^{+0.01}_{-0.00}$ | $-0.35^{+0.01}_{-0.02}$ | $-0.19^{+0.00}_{-0.01}$
$\langle A^{\ell}_{FB}\rangle(\Lambda^{0}_{b}\to\Lambda^{0}\mu^{+}\mu^{-})$ | $0.08^{+0.00}_{-0.01}$ | $0.06^{+0.00}_{-0.01}$ | $0.07\pm 0.00$ | $-0.05^{+0.00}_{-0.01}$ | $0.06^{+0.00}_{-0.01}$ | $-0.15^{+0.00}_{-0.01}$ | $-0.12^{+0.00}_{-0.01}$ | $-0.29^{+0.01}_{-0.02}$ | $-0.37^{+0.02}_{-0.01}$ | $-0.03^{+0.00}_{-0.01}$ | $-0.30\pm 0.01$ | $-0.34^{+0.01}_{-0.02}$ | $-0.04^{+0.00}_{-0.01}$
$\overline{f_{L}}(\Lambda^{0}_{b}\to\Lambda^{0}\tau^{+}\tau^{-})$ |  |  |  |  |  |  |  |  | $0.46^{+0.00}_{-0.01}$ |  | $0.36^{+0.01}_{-0.00}$ | $0.39^{+0.02}_{-0.01}$ | $0.42\pm 0.01$
$\langle f_{L}\rangle(\Lambda^{0}_{b}\to\Lambda^{0}\tau^{+}\tau^{-})$ |  |  |  |  |  |  |  |  | $0.46\pm 0.01$ |  | $0.36\pm 0.01$ | $0.39^{+0.02}_{-0.01}$ | $0.40^{+0.03}_{-0.01}$
$\overline{A^{\ell}_{FB}}(\Lambda^{0}_{b}\to\Lambda^{0}\tau^{+}\tau^{-})$ |  |  |  |  |  |  |  |  | $-0.36\pm 0.00$ |  | $-0.31\pm 0.01$ | $-0.34^{+0.01}_{-0.02}$ | $-0.34\pm 0.01$
$\langle A^{\ell}_{FB}\rangle(\Lambda^{0}_{b}\to\Lambda^{0}\tau^{+}\tau^{-})$ |  |  |  |  |  |  |  |  | $-0.36^{+0.01}_{-0.02}$ |  | $-0.30\pm 0.01$ | $-0.34^{+0.01}_{-0.02}$ | $-0.33^{+0.02}_{-0.01}$

Among $\Lambda^{0}_{b}\to\Lambda^{0}\ell^{+}\ell^{-}$,
$\Xi^{0}_{b}\to\Xi^{0}\ell^{+}\ell^{-}$
$\Xi^{-}_{b}\to\Xi^{-}\ell^{+}\ell^{-}$, $\Lambda^{0}_{b}\to
n\ell^{+}\ell^{-}$, $\Xi^{0}_{b}\to\Lambda^{0}\ell^{+}\ell^{-}$,
$\Xi^{0}_{b}\to\Sigma^{0}\ell^{+}\ell^{-}$, and
$\Xi^{-}_{b}\to\Sigma^{-}\ell^{+}\ell^{-}$ decays, only
$\Lambda^{0}_{b}\to\Lambda^{0}\mu^{+}\mu^{-}$ decay has been measured, and its
branching ratios in the whole $q^{2}$ region and in different $q^{2}$ bins are
listed in Tab. 2 and Tab. 5, respectively. One can constrain the relevant
SU(3) flavor parameters by the experimental data within $1\sigma$ error bar
and then predict other not-yet-measured branching ratios. Two cases will be
considered in our analysis of $T_{b3}\to T_{8}\ell^{+}\ell^{-}$ decays.

  * $S_{1}$ :

The SU(3) flavor symmetry parameters without the baryonic momentum-transfer
$q^{2}$ dependence. We treat the SU(3) flavor parameters
$(E)^{L(R),s_{p},s_{k}}_{VA,\lambda}(q^{2})$ as constants without $q^{2}$
dependence, which will lead $H_{M}(q^{2})$ in Eq. (18) to a constant, too. We
use the $1\sigma$ error experimental data of
$\mathcal{B}(\Lambda^{0}_{b}\to\Lambda^{0}\mu^{+}\mu^{-})$ to constrain
$H_{M}(q^{2})$ ($i.e.$, $|E|^{2}$), and then predict other
$\mathcal{B}(T_{b3}\to T_{8}\ell^{+}\ell^{-})$ by the amplitude relations in
Tab. 2.

  * $S_{2}$:

The SU(3) flavor symmetry parameters with the baryonic momentum-transfer
$q^{2}$ dependence. In order to obtain more precise predictions, we use the
hadronic helicity amplitude expressions in Eq. (LABEL:EQ:H), which are $q^{2}$
dependent and can be expressed by the Wilson coefficients and the form
factors. The expressions of the Wilson coefficients without the LD
contributions are taken from Ref. Buras:1994dj . As for the $q^{2}$ dependent
form factors involving the $T_{b3}\to T_{8}$ transitions, we use the recent
lattice QCD results of $\Lambda^{0}_{b}\to\Lambda^{0}$ Detmold:2016pkz , in
which the form factors are parametrized by

$\displaystyle
f(q^{2})=\frac{f(0)}{1-q^{2}/(m^{f}_{pole})^{2}}\left[1+\frac{a^{f}_{1}}{a^{f}_{0}}z(q^{2})+\frac{a^{f}_{2}}{a^{f}_{0}}[z(q^{2})]^{2}\right],$
(27)

where
$f=f_{+},f_{\perp},f_{0},g_{+},g_{\perp},g_{0},h_{+},h_{\perp},\widetilde{h}_{+},\widetilde{h}_{\perp}$,
and the details of $z(q^{2})$ and $m^{f}_{pole}$ can be found in Ref.
Detmold:2016pkz . We keep $f_{+}(0)$ as an undetermined constant without
$q^{2}$ dependence, and other $f(0)$ can be expressed as
$\frac{a^{f}_{0}}{a^{f_{+}}_{0}}f_{+}(0)$. The central values of $a^{f}_{i}$
in Tab. V of Ref. Detmold:2016pkz  will be used in our analysis. Since these
form factors also preserve the SU(3) flavor symmetry, the same relations in
Tab. 2 will be used for $f_{+}(0)$. We use the $1\sigma$ error experimental
data of $\mathcal{B}(\Lambda^{0}_{b}\to\Lambda^{0}\mu^{+}\mu^{-})$ to
constrain $f_{+}(0)$ and then predict other $\mathcal{B}(T_{b3}\to
T_{8}\ell^{+}\ell^{-})$.

Using the experimental data of
$\mathcal{B}(\Lambda^{0}_{b}\to\Lambda^{0}\mu^{+}\mu^{-})$ in the whole
$q^{2}$ region, one can obtain the branching ratios for $T_{b3}\to
T_{8}\mu^{+}\mu^{-}$ and $T_{b3}\to T_{8}\tau^{+}\tau^{-}$ weak decays in the
whole $q^{2}$ region, which are listed in the third and forth columns of Tab.
2 for $S_{1}$ case and $S_{2}$ case, respectively. Noted that, the amplitude
relations listed in Tab. 2 are obtained from the SU(3) flavor symmetry;
nevertheless, the different baryon masses in the same baryon multiplets are
considered in the branching ratio predictions, and the below is same.

Previous predictions are also listed in the last column of Tab. 2 for
comparing. Since the results of $T_{b3}\to T_{8}e^{+}e^{-}$ decays are quite
similar to ones of $T_{b3}\to T_{8}\mu^{+}\mu^{-}$ decays, we only show
$T_{b3}\to T_{8}\mu^{+}\mu^{-}$ in this work. We have the following remarks
for the results in Tab. 2.

  * •

Comparing the branching ratios in $S_{1}$ and $S_{2}$ cases, one can see that
the predictions are slightly different between $S_{1}$ and $S_{2}$ cases,
which are mainly due to the $q^{2}$ dependence of the hadronic helicity
amplitudes.

  * •

Comparing our predictions for
$\mathcal{B}(\Lambda^{0}_{b}\to\Lambda^{0}\tau^{+}\tau^{-}),\leavevmode\nobreak\
\mathcal{B}(\Lambda^{0}_{b}\to n\tau^{+}\tau^{-})$ and
$\mathcal{B}(\Lambda^{0}_{b}\to n\mu^{+}\mu^{-})$ with previous ones in the
relativistic quark model Faustov:2020thr  and the Bethe-Salpeter equation
approach Liu:2019rpm , one can see that our predicted
$\mathcal{B}(\Lambda^{0}_{b}\to\Lambda^{0}\tau^{+}\tau^{-})$ and
$\mathcal{B}(\Lambda^{0}_{b}\to n\mu^{+}\mu^{-})$ are quite consistent with
previous ones, nevertheless, our central value of
$\mathcal{B}(\Lambda^{0}_{b}\to n\tau^{+}\tau^{-})$ is about 2 times larger
than theirs.

  * •

Many branching ratio predictions for $T_{b3}\to T_{8}\ell^{+}\ell^{-}$ are
obtained for the first time. The not-yet-measured $\mathcal{B}(T_{b3}\to
T_{8}\ell^{+}\ell^{-})$ are on the order of $\mathcal{O}(10^{-8}-10^{-6})$,
and some of them could be reached by the LHCb or Belle-II experiments.

Using the experimental data of
$\mathcal{B}(\Lambda^{0}_{b}\to\Lambda^{0}\mu^{+}\mu^{-})$ in different
$q^{2}$ bins, one can get the branching ratios of $T_{b3}\to
T_{8}\mu^{+}\mu^{-}$ and $T_{b3}\to T_{8}\tau^{+}\tau^{-}$ weak decays in
different $q^{2}$ bins within $S_{1}$ and $S_{2}$ cases, which are collected
for reference in Tab. 5 and Tab. 5, respectively.

The longitudinal polarization fractions and the leptonic forward-backward
asymmetries with two ways of integration for $T_{b3}\to T_{8}\ell^{+}\ell^{-}$
decays could also be obtained in the $S_{2}$ case. As shown in Eq. (21) and
Eq. (23), the $N(q^{2})$ terms are canceled in the ratios; therefore, the
longitudinal polarization fractions and the leptonic forward-backward
asymmetries only depend on the hadronic helicity amplitudes, which preserve
the SU(3) flavor symmetry in $T_{b3}\to T_{8}\ell^{+}\ell^{-}$ decays. So the
longitudinal polarization fractions and the leptonic forward-backward
asymmetries of all $T_{b3}\to T_{8}\mu^{+}\mu^{-}$ ($T_{b3}\to
T_{8}\tau^{+}\tau^{-}$) decays are very similar to each other in certain
$q^{2}$ bins. We take ones of
$\Lambda^{0}_{b}\to\Lambda^{0}\mu^{+}\mu^{-},\Lambda^{0}\tau^{+}\tau^{-}$ as
examples, which are given in Tab. 5. Excepting in $[0.1,2.0]$, $[0.1,4.3]$,
$[0.1,16.0]$ and the whole $q^{2}$ regions, $\overline{f_{L}}$ and $\langle
f_{L}\rangle$ ($\overline{A^{\ell}_{FB}}$ and $\langle A^{\ell}_{FB}\rangle$)
with different $q^{2}$ integration ways are quite similar in other certain
$q^{2}$ bins. So the obvious differentiation between $\overline{f_{L}}$ and
$\langle f_{L}\rangle$ ($\overline{A^{\ell}_{FB}}$ and $\langle
A^{\ell}_{FB}\rangle$) mainly appears in the quite low $q^{2}$ regions. Note
that the normalized longitudinal polarization fraction and normalized leptonic
forward-backward asymmetry of $\Lambda^{0}_{b}\to\Lambda^{0}\mu^{+}\mu^{-}$ in
$q^{2}\in[15,20]$ GeV2 have been measured by the LHCb experiment PDG2020 ,

$\displaystyle\langle
f_{L}\rangle(\Lambda^{0}_{b}\to\Lambda^{0}\mu^{+}\mu^{-})_{[15,20]}$
$\displaystyle=$ $\displaystyle 0.61^{+0.11}_{-0.14},$ $\displaystyle\langle
A^{\ell}_{FB}\rangle(\Lambda^{0}_{b}\to\Lambda^{0}\mu^{+}\mu^{-})_{[15,20]}$
$\displaystyle=$ $\displaystyle-0.39\pm 0.04\pm 0.01.$ (28)

We do not impose the above experimental bounds but leave them as predictions.
Comparing with the experimental results for $\langle
f_{L}\rangle(\Lambda^{0}_{b}\to\Lambda^{0}\mu^{+}\mu^{-})_{[15,20]}$ and
$\langle
A^{\ell}_{FB}\rangle(\Lambda^{0}_{b}\to\Lambda^{0}\mu^{+}\mu^{-})_{[15,20]}$,
our prediction of $\langle
f_{L}\rangle(\Lambda^{0}_{b}\to\Lambda^{0}\mu^{+}\mu^{-})_{[15,20]}$ and
$\langle
A^{\ell}_{FB}\rangle(\Lambda^{0}_{b}\to\Lambda^{0}\mu^{+}\mu^{-})_{[15,20]}$
are agreeable with their experimental data within $1.5\sigma$ and $1\sigma$
error ranges, respectively.

In addition, the lepton flavor universality $R_{T_{b3}\to T_{8}}$ in three
$q^{2}$ bins within the $S_{2}$ case are given in Tab. 6. One can see that all
predictions in three $q^{2}$ bins are virtually indistinguishable from unity;
$i.e.$, the lepton mass effects on all $R_{\mathcal{B}_{1}\to\mathcal{B}_{2}}$
are small in both the low-$q^{2}$ region and high-$q^{2}$ region in the SM.

Table 6: Lepton flavor universality of $T_{b3}\to T_{8}\ell^{+}\ell^{-}$ baryon weak decays in different $q^{2}$ bins with $1\sigma$ error in the $S_{2}$ case.  $[q^{2}_{min},q^{2}_{max}](\mbox{GeV}^{2})$ |  $[1,6]$ |  $[0.1,16.0]$ |  $[15.0,20.0]$
---|---|---|---
$R_{\Lambda^{0}_{b}\to\Lambda^{0}}$ |  $0.99^{+0.05}_{-0.03}$ |  $0.98^{+0.07}_{-0.04}$ |  $1.02^{+0.04}_{-0.07}$
$R_{\Xi^{0}_{b}\to\Xi^{0}}$ |  $0.99\pm 0.04$ |  $0.99^{+0.06}_{-0.05}$ |  $0.99^{+0.08}_{-0.06}$
$R_{\Xi^{-}_{b}\to\Xi^{-}}$ |  $0.99\pm 0.04$ |  $0.99^{+0.06}_{-0.05}$ |  $1.03^{+0.03}_{-0.09}$
$R_{\Lambda^{0}_{b}\to n}$ |  $1.00\pm 0.03$ |  $0.98^{+0.07}_{-0.04}$ |  $1.01^{+0.03}_{-0.05}$
$R_{\Xi^{0}_{b}\to\Lambda^{0}}$ |  $1.01^{+0.02}_{-0.05}$ |  $1.01^{+0.04}_{-0.08}$ |  $1.00\pm 0.04$
$R_{\Xi^{0}_{b}\to\Sigma^{0}}$ |  $0.98^{+0.04}_{-0.03}$ |  $0.98^{+0.07}_{-0.04}$ |  $1.00\pm 0.03$
$R_{\Xi^{-}_{b}\to\Sigma^{-}}$ |  $1.00^{+0.03}_{-0.04}$ |  $1.03^{+0.02}_{-0.09}$ |  $1.00^{+0.03}_{-0.02}$

###  III.2 $T_{c3}$ semileptonic weak decays

Similar to $T_{c3,8}\to T^{\prime}_{8}\gamma$ radiative decays Singer:1995is ;
Verma:1988gf ; Lach:1995we ; Azimov:1996uf , $T_{c3}\to T_{8}\ell^{+}\ell^{-}$
decays receive single-quark, two-quark, and three-quark transition
contributions with the $W$-exchange and internal radiation contributions. The
internal radiation contributions are suppressed by the two $W$ propagators and
can be safely neglected. The SU(3) IRA hadronic helicity amplitudes for
$T_{c3}\to T_{8}\ell^{+}\ell^{-}$ may be parametrized as

$\displaystyle H(T_{c3}\to
T_{8}\ell^{+}\ell^{-})^{L(R),s_{p},s_{k}}_{VA,\lambda}$ $\displaystyle=$
$\displaystyle
f_{1}(T_{c3})^{[ij]}T^{\prime}(\bar{3})^{k}(T_{8})_{[ij]k}+f_{2}(T_{c3})^{[ij]}T^{\prime}(3)^{k}(T_{8})_{[ik]j}$
(29)
$\displaystyle+\Big{(}\widetilde{f}_{1}H(\bar{6})^{lk}_{j}+\widetilde{f}_{4}H(15)^{lk}_{j}\Big{)}(T_{c3})^{[ij]}(T_{8})_{[il]k}$
$\displaystyle+\Big{(}\widetilde{f}_{2}H(\bar{6})^{lk}_{j}+\widetilde{f}_{5}H(15)^{lk}_{j}\Big{)}(T_{c3})^{[ij]}(T_{8})_{[ik]l}$
$\displaystyle+\Big{(}\widetilde{f}_{3}H(\bar{6})^{lk}_{j}+\widetilde{f}_{6}H(15)^{lk}_{j}\Big{)}(T_{c3})^{[ij]}(T_{8})_{[lk]i},$

where the SU(3) flavor symmetry parameters
$f_{i}\equiv(f_{i})^{L(R),s_{p},s_{k}}_{VA,\lambda}$ and
$\widetilde{f}_{i}\equiv(\widetilde{f}_{i})^{L(R),s_{p},s_{k}}_{VA,\lambda}$.
The $f_{i}$ terms in Eq. (29) and the later $g_{i}$ terms in Eq. (32) denote
the short distance (SD) and the LD contributions via the single-quark
transitions. The $\widetilde{f}_{i}$ terms in Eq. (29) and the later
$\widetilde{g}_{i}$ terms in Eq. (32) denote the $W$-exchange contributions of
the two-quark and three-quark transitions. $T^{\prime}(\bar{3})=(1,0,0)$
denotes the transition operators $(\bar{q}_{2}c)$ with $q_{2}=u$, and
$H(\bar{6})^{lk}_{j}$ ($H(15)^{lk}_{j}$) related to the
$(\bar{q}_{l}q^{j})(\bar{q}_{k}c)$ operator is antisymmetric (symmetric) in
upper indices. The nonvanishing $H(\bar{6})^{lk}_{j}$ and $H(15)^{lk}_{j}$ for
$c\to su\bar{d},du\bar{s},u\bar{d}d,u\bar{s}s$ transitions can be found in
Ref. Wang:2017azm . Using $l,k$ antisymmetric in $H(\bar{6})^{lk}_{j}$ and
$l,k$ symmetric in $H(15)^{lk}_{j}$, we have

$\displaystyle\widetilde{f}_{2}=-\widetilde{f}_{1},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\widetilde{f}_{5}=\widetilde{f}_{4},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \widetilde{f}_{6}=0,$ (30)

which will be used in the following discussion.

For the $W$-exchange transitions, there are three kinds of charm quark
decaying into light quarks

$\displaystyle d+c\to u+s+\ell^{+}\ell^{-},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ d+c\to
u+d+\ell^{+}\ell^{-}\leavevmode\nobreak\ \leavevmode\nobreak\ (s+c\to
u+s+\ell^{+}\ell^{-}),\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ s+c\to u+d+\ell^{+}\ell^{-},$ (31)

which are related to $H(\bar{6},15)^{13}_{2}$,
$H(\bar{6},15)^{12}_{2}\leavevmode\nobreak\
\big{(}H(\bar{6},15)^{13}_{3}\big{)}$ and $H(\bar{6},15)^{12}_{3}$, and they
are proportional to $V^{*}_{cs}V_{ud}\approx 1$,
$V^{*}_{cd}V_{ud}\leavevmode\nobreak\ (V^{*}_{cs}V_{us})\approx-
s_{c}\leavevmode\nobreak\ (s_{c})$ and $V^{*}_{cd}V_{us}\approx-s_{c}^{2}$
with $s_{c}\equiv sin\theta_{C}\approx 0.22453$, respectively. So three kinds
decays given in Eq. (31) are called Cabibbo allowed, singly Cabibbo
suppressed, and doubly Cabibbo suppressed decays, respectively.

The SU(3) IRA amplitudes of the $T_{c3}\to T_{8}\ell^{+}\ell^{-}$ weak decays
are given in the third column of Tab. 8, and for a better understanding, the
information of relevant CKM matrix elements is also listed in this Table.

Table 7: The SU(3) IRA amplitudes of the $T_{c3}\to T_{8}\ell^{+}\ell^{-}$
weak decays in the $S_{1}$ case, $F_{1}\equiv f_{1}+f_{2}$,
$\widetilde{F}_{1}\equiv\widetilde{f}_{1}-\widetilde{f}_{3}+\widetilde{f}_{4}$,
$\widetilde{F}_{2}\equiv\widetilde{f}_{1}-\widetilde{f}_{3}-\widetilde{f}_{4}$,
and $\widetilde{F}\equiv\widetilde{f}_{1}-\widetilde{f}_{3}$.

Decay modes | $A(T_{c3}\to T_{8}\ell^{+}\ell^{-})$ |  Approximative $A(T_{c3}\to T_{8}\ell^{+}\ell^{-})$
---|---|---
Cabibbo allowed $T_{c3}\to T_{8}\ell^{+}\ell^{-}$: |  |
$\Lambda^{+}_{c}\to\Sigma^{+}\ell^{+}\ell^{-}$ | $-\widetilde{F}_{1}$ | $-\widetilde{F}$
$\Xi^{0}_{c}\to\Xi^{0}\ell^{+}\ell^{-}$ | $-\widetilde{F}_{2}$ | $-\widetilde{F}$
singly Cabibbo suppressed $T_{c3}\to T_{8}\ell^{+}\ell^{-}$: |  |
$\Lambda^{+}_{c}\to p\ell^{+}\ell^{-}$ | $\big{[}F_{1}-\big{(}\frac{5}{8}\widetilde{F}_{1}-\frac{1}{8}\widetilde{F}_{2}\big{)}\big{]}s_{c}$ | $\big{[}F_{1}-\frac{1}{2}\widetilde{F}\leavevmode\nobreak\ \big{]}s_{c}$
$\Xi^{+}_{c}\to\Sigma^{+}\ell^{+}\ell^{-}$ | $\big{[}-F_{1}-\big{(}\frac{5}{8}\widetilde{F}_{1}-\frac{1}{8}\widetilde{F}_{2}\big{)}\big{]}s_{c}$ | $-\big{[}F_{1}+\frac{1}{2}\widetilde{F}\leavevmode\nobreak\ \big{]}s_{c}$
$\Xi^{0}_{c}\to\Lambda^{0}\ell^{+}\ell^{-}$ | $\big{[}F_{1}+3\big{(}\frac{1}{8}\widetilde{F}_{1}-\frac{5}{8}\widetilde{F}_{2}\big{)}\big{]}s_{c}/\sqrt{6}$ | $\big{[}F_{1}-\frac{3}{2}\widetilde{F}\leavevmode\nobreak\ \big{]}s_{c}/\sqrt{6}$
$\Xi^{0}_{c}\to\Sigma^{0}\ell^{+}\ell^{-}$ | $\big{[}-F_{1}+\big{(}\frac{1}{8}\widetilde{F}_{1}-\frac{5}{8}\widetilde{F}_{2}\big{)}\big{]}s_{c}/\sqrt{2}$ | $-\big{[}F_{1}+\frac{1}{2}\widetilde{F}\leavevmode\nobreak\ \big{]}s_{c}/\sqrt{2}$
doubly Cabibbo suppressed $T_{c3}\to T_{8}\ell^{+}\ell^{-}$: |  |
$\Xi^{+}_{c}\to p\ell^{+}\ell^{-}$ | $\widetilde{F}_{1}s^{2}_{c}$ | $\widetilde{F}s^{2}_{c}$
$\Xi^{0}_{c}\to n\ell^{+}\ell^{-}$ | $\widetilde{F}_{2}s^{2}_{c}$ | $\widetilde{F}s^{2}_{c}$

Table 8: Branching ratios of $T_{c3}\to T_{8}\ell^{+}\ell^{-}$ decays within
$1\sigma$ theoretical error in the $S_{1}$ case.

Decay modes |  Exp. UL PDG2020  |  Our predictions without $F_{1}$ | Others without LD | Others with LD
---|---|---|---|---
$\mathcal{B}(\Lambda^{+}_{c}\to\Sigma^{+}e^{+}e^{-})(\times 10^{-6})$ | $\cdots$ | $\leq 2.63$ |  |
$\mathcal{B}(\Xi^{0}_{c}\to\Xi^{0}e^{+}e^{-})(\times 10^{-6})$ | $\cdots$ | $\leq 2.35$ |  |
$\mathcal{B}(\Lambda^{+}_{c}\to pe^{+}e^{-})(\times 10^{-8})$ | $\leq 550$ | $\leq 7.95$ |  ${}^{(3.8\pm 0.5)\times 10^{-4}}_{(4.05\pm 2.37)\times 10^{-6}}$ Faustov:2018dkn ; Sirvanli:2016wnr  |  ${}^{37\pm 8}_{420\pm 73}$ Faustov:2018dkn ; Sirvanli:2016wnr
$\mathcal{B}(\Xi^{+}_{c}\to\Sigma^{+}e^{+}e^{-})(\times 10^{-7})$ | $\cdots$ | $\leq 1.29$ |  |
$\mathcal{B}(\Xi^{0}_{c}\to\Lambda^{0}e^{+}e^{-})(\times 10^{-8})$ | $\cdots$ | $\leq 8.69$ |  |
$\mathcal{B}(\Xi^{0}_{c}\to\Sigma^{0}e^{+}e^{-})(\times 10^{-8})$ | $\cdots$ | $\leq 2.22$ |  |
$\mathcal{B}(\Xi^{+}_{c}\to pe^{+}e^{-})(\times 10^{-8})$ | $\cdots$ | $\leq 5.55$ |  |
$\mathcal{B}(\Xi^{0}_{c}\to ne^{+}e^{-})(\times 10^{-8})$ | $\cdots$ | $\leq 1.92$ |  |
$\mathcal{B}(\Lambda^{+}_{c}\to\Sigma^{+}\mu^{+}\mu^{-})(\times 10^{-6})$ | $\cdots$ | $\leq 2.50$ |  |
$\mathcal{B}(\Xi^{0}_{c}\to\Xi^{0}\mu^{+}\mu^{-})(\times 10^{-6})$ | $\cdots$ | $\leq 2.25$ |  |
$\mathcal{B}(\Lambda^{+}_{c}\to p\mu^{+}\mu^{-})(\times 10^{-8})$ | $\leq 7.7$ | $\leq 7.7$ |  ${}^{(2.8\pm 0.4)\times 10^{-4}}_{(3.77\pm 2.28)\times 10^{-6}}$ Faustov:2018dkn ; Sirvanli:2016wnr  |  ${}^{37\pm 8}_{230\pm 66}$ Faustov:2018dkn ; Sirvanli:2016wnr
$\mathcal{B}(\Xi^{+}_{c}\to\Sigma^{+}\mu^{+}\mu^{-})(\times 10^{-7})$ | $\cdots$ | $\leq 1.25$ |  |
$\mathcal{B}(\Xi^{0}_{c}\to\Lambda^{0}\mu^{+}\mu^{-})(\times 10^{-8})$ | $\cdots$ | $\leq 8.42$ |  |
$\mathcal{B}(\Xi^{0}_{c}\to\Sigma^{0}\mu^{+}\mu^{-})(\times 10^{-8})$ | $\cdots$ | $\leq 2.15$ |  |
$\mathcal{B}(\Xi^{+}_{c}\to p\mu^{+}\mu^{-})(\times 10^{-8})$ | $\cdots$ | $\leq 5.41$ |  |
$\mathcal{B}(\Xi^{0}_{c}\to n\mu^{+}\mu^{-})(\times 10^{-8})$ | $\cdots$ | $\leq 1.87$ |  |

From Tab. 8, one can see that singly Cabibbo suppressed $\Lambda^{+}_{c}\to
p\ell^{+}\ell^{-},\Xi^{+}_{c}\to\Sigma^{+}\ell^{+}\ell^{-},\Xi^{0}_{c}\to\Lambda^{0}\ell^{+}\ell^{-},\Xi^{0}_{c}\to\Sigma^{0}\ell^{+}\ell^{-}$
decays receive both the single-quark transition and the $W$-exchange
contributions, nevertheless, Cabibbo allowed
$\Lambda^{+}_{c}\to\Sigma^{+}\ell^{+}\ell^{-},\Xi^{0}_{c}\to\Xi^{0}\ell^{+}\ell^{-}$
decays and doubly Cabibbo suppressed $\Xi^{+}_{c}\to
p\ell^{+}\ell^{-},\Xi^{0}_{c}\to n\ell^{+}\ell^{-}$ decays only receive the
$W$-exchange contributions.

In addition, the contribution of $H(\bar{6})$ to the decay branching ratio is
about 5.5 times larger than one of $H(15)$ due to Wilson coefficient
suppressed; for example, see Refs. Geng:2017esc ; Geng:2018plk . If ignoring
the Wilson coefficient suppressed $H(15)$ term contributions, there are only
two parameters, $F_{1}$ and
$\widetilde{F}\equiv\widetilde{f}_{1}-\widetilde{f}_{3}$, in the decay
amplitudes of $T_{c3}\to T_{8}\ell^{+}\ell^{-}$. The simplified results are
listed in the last column of Tab. 8. One can see that the Cabibbo allowed and
doubly Cabibbo suppressed eight decay modes of $T_{c3}\to
T_{8}\ell^{+}\ell^{-}$ are related by only one parameter $\widetilde{F}$;
nevertheless, the singly Cabibbo suppressed eight decays are related by two
parameters $\widetilde{F}$ and $F_{1}$. Moreover, there are amplitude
relations
$A(\Xi^{+}_{c}\to\Sigma^{+}\ell^{+}\ell^{-})=\sqrt{2}A(\Xi^{0}_{c}\to\Sigma^{0}\ell^{+}\ell^{-})$.

In these $T_{c3}\to T_{8}\ell^{+}\ell^{-}$ decays, only
$\mathcal{B}(\Lambda^{+}_{c}\to pe^{+}e^{-})$ and
$\mathcal{B}(\Lambda^{+}_{c}\to p\mu^{+}\mu^{-})$ are upper limited by
experiment. We list their experimental upper limits (exp. UL) in the second
column of Tab. 8. Due to the lack of the experiment in $T_{c3}\to
T_{8}\ell^{+}\ell^{-}$ decays and the complex amplitude expressions included
the $W$-exchange contributions, we will only analyze the $T_{c3}\to
T_{8}\ell^{+}\ell^{-}$ decays in the $S_{1}$ case. We assume that $W$-exchange
contributions noted by $\widetilde{F}$ or the single-quark transition
contributions noted by $F_{1}$ play a dominant role in these decays. They are
separately discussed as follows.

  * •

Only considering the $W$-exchange contributions by setting $F_{1}=0$, all the
$T_{c3}\to T_{8}\ell^{+}\ell^{-}$ decays are related by one parameter
$\widetilde{F}$ as shown in Tab. 8. The upper limit predictions of $T_{c3}\to
T_{8}e^{+}e^{-}$ and $T_{c3}\to T_{8}\mu^{+}\mu^{-}$ decays in the $S_{1}$
case are listed in the third column of Tab. 8. One can see that
$\mathcal{B}(\Lambda^{+}_{c}\to p\mu^{+}\mu^{-})$ gives an effective
constraint on these upper limit predictions of the branching ratios.

  * •

Only considering the single-quark transition contributions by setting
$\widetilde{F}=0$, as shown in Tab. 8, all eight singly Cabibbo suppressed
$T_{c3}\to T_{8}e^{+}e^{-}$ and $T_{c3}\to T_{8}\mu^{+}\mu^{-}$ decays are
related by the parameter $F_{1}$. Then the six upper limit predictions of
$\mathcal{B}(\Lambda^{+}_{c}\to pe^{+}e^{-})$,
$\mathcal{B}(\Xi^{+}_{c}\to\Sigma^{+}e^{+}e^{-})$,
$\mathcal{B}(\Xi^{0}_{c}\to\Sigma^{0}e^{+}e^{-})$,
$\mathcal{B}(\Lambda^{+}_{c}\to p\mu^{+}\mu^{-})$,
$\mathcal{B}(\Xi^{+}_{c}\to\Sigma^{+}\mu^{+}\mu^{-})$, and
$\mathcal{B}(\Xi^{0}_{c}\to\Sigma^{0}\mu^{+}\mu^{-})$ in the $S_{1}$ case have
the same predictions as the ones listed in the third column of Tab. 8.
Nevertheless, the predictions of other two singly Cabibbo suppressed
$\mathcal{B}(\Xi^{0}_{c}\to\Lambda^{0}e^{+}e^{-})$ and
$\mathcal{B}(\Xi^{0}_{c}\to\Lambda^{0}\mu^{+}\mu^{-})$ are different from the
ones listed in Tab. 8. We obtain that
$\mathcal{B}(\Xi^{0}_{c}\to\Lambda^{0}e^{+}e^{-})\leq 9.66\times 10^{-9}$ and
$\mathcal{B}(\Xi^{0}_{c}\to\Lambda^{0}\mu^{+}\mu^{-})\leq 9.36\times 10^{-9}$.
The previous other predictions for $\mathcal{B}(\Lambda^{+}_{c}\to
pe^{+}e^{-})$ and $\mathcal{B}(\Lambda^{+}_{c}\to p\mu^{+}\mu^{-})$, which
considered the single-quark contributions with/without LD contributions, are
also listed in the last two columns of Tab. 8 for comparing. The predicted
upper limits of $\mathcal{B}(\Lambda^{+}_{c}\to p\ell^{+}\ell^{-})$ are about
5 orders larger than the predictions with the SD contributions, smaller than
the LD contributing ones, which might mean that the $W$-exchange transitions
cancel out with the LD contributions.

###  III.3 $T_{8}$ semileptonic weak decays

Similar to $T_{8}\to T^{\prime}_{8}\gamma$ radiative decays, $T_{8}\to
T^{\prime}_{8}\ell^{+}\ell^{-}$ semileptonic decays receive the single-quark
transition contributions and the $W$-exchange contributions Bergstrom:1987wr .
The SD contributions come from the $Z^{0}$ and electromagnetic penguin
diagrams as well as the $Z^{0}$ box diagrams, and the LD contributions arise
from an intervirtual photon in the $T_{8}\to T^{\prime}_{8}\gamma^{*}$
processes. The LD contributions are much larger than the SD ones in the
single-quark transition contributions in the $T_{8}\to
T^{\prime}_{8}\ell^{+}\ell^{-}$ decays. So the LD contributions and the
W-exchange contributions might play the major roles in $T_{8}\to
T^{\prime}_{8}\ell^{+}\ell^{-}$ decays He:2005yn ; Bergstrom:1987wr . The
SU(3) flavor structure of the $s\to d$ Hamiltonian can be found in Ref.
Wang:2019alu . The SU(3) IRA hadronic helicity amplitudes for $T_{8}\to
T^{\prime}_{8}\ell^{+}\ell^{-}$ decays via $s\to d\ell^{+}\ell^{-}$ are

$\displaystyle H(T_{8}\to
T^{\prime}_{8}\ell^{+}\ell^{-})^{L(R),s_{p},s_{k}}_{VA,\lambda}$
$\displaystyle=$ $\displaystyle
g_{1}(T_{8})^{[ij]n}T^{\prime\prime}(\bar{3})^{k}(T^{\prime}_{8})_{[ij]k}+g_{2}(T_{8})^{[ij]n}T^{\prime\prime}(3)^{k}(T_{8})_{[ik]j}$
(32) $\displaystyle+$ $\displaystyle
g_{3}(T_{8})^{[in]j}T^{\prime\prime}(\bar{3})^{k}(T^{\prime}_{8})_{[ij]k}+g_{4}(T_{8})^{[in]j}T^{\prime\prime}(\bar{3})^{k}(T^{\prime}_{8})_{[ik]j}$
$\displaystyle+$ $\displaystyle
g_{5}(T_{8})^{[in]j}T^{\prime\prime}(\bar{3})^{k}(T^{\prime}_{8})_{[jk]i}+\widetilde{g}_{1}(T_{8})^{[ij]n}(T^{\prime}_{8})_{[il]k}H(4)^{lk}_{j}$
$\displaystyle+$
$\displaystyle\widetilde{g}_{2}(T_{8})^{[in]j}(T^{\prime}_{8})_{[il]k}H(4)^{lk}_{j}+\widetilde{g}_{3}(T_{8})^{[jn]i}(T^{\prime}_{8})_{[il]k}H(4)^{lk}_{j},$

where the model and scale independent parameters
$g_{i}\equiv(g_{i})^{L(R),s_{p},s_{k}}_{VA,\lambda}$ and
$\widetilde{g}_{i}\equiv(\widetilde{g}_{i})^{L(R),s_{p},s_{k}}_{VA,\lambda}$,
$T^{\prime\prime}(\bar{3})=(0,1,0)$ related to the transition operator
$(\bar{d}s)$, and $H(4)^{lk}_{j}$ related to
$(\bar{q}_{l}q^{j})(\bar{q}_{k}q^{n})$ operator with $n\equiv 3$ for $s$ quark
is symmetric in upper indices Wang:2019alu . The SU(3) IRA hadronic helicity
amplitudes of $T_{8}\to T^{\prime}_{8}\ell^{+}\ell^{-}$ weak decays are given
in Tab. 9, in which the information of the same CKM matrix elements
$V_{us}V^{*}_{ud}$ is not shown.

Table 9: The SU(3) IRA amplitudes of the $T_{8}\to T^{\prime}_{8}\ell^{+}\ell^{-}$ weak decays, $G_{1}\equiv g_{1}+g_{2}+g_{3}-g_{5}$, $G_{2}\equiv g_{4}+g_{5}$, $\widetilde{G}_{A}\equiv\widetilde{g}_{1}-\widetilde{g}_{3}$, and $\widetilde{G}_{B}\equiv\widetilde{g}_{2}+\widetilde{g}_{3}$. Decay modes | $A(T_{8}\to T^{\prime}_{8}\ell^{+}\ell^{-})$
---|---
$\Xi^{-}\to\Sigma^{-}\ell^{+}\ell^{-}$ | $G_{1}$
$\Xi^{0}\to\Lambda^{0}\ell^{+}\ell^{-}$ | $(G_{1}+2G_{2})/\sqrt{6}$
$\Xi^{0}\to\Sigma^{0}\ell^{+}\ell^{-}$ | $(G_{1}+2\widetilde{G}_{A})/\sqrt{2}$
$\Lambda^{0}\to n\ell^{+}\ell^{-}$ | $-\big{[}(G_{1}+2G_{2})+(C_{1}+2\widetilde{G}_{A})-(G_{2}-\widetilde{G}_{B})\big{]}/\sqrt{6}$
$\Sigma^{0}\to n\ell^{+}\ell^{-}$ | $-(G_{2}-\widetilde{G}_{B})/\sqrt{2}$
$\Sigma^{+}\to p\ell^{+}\ell^{-}$ | $-(G_{2}+\widetilde{G}_{B})$
Table 10: Branching ratios of $T_{8}\to T^{\prime}_{8}\ell^{+}\ell^{-}$ decays within $1\sigma$ theoretical error in the $S_{1}$ case.  Decay modes |  Experimental data PDG2020  |  Our predictions without $\widetilde{G}_{A,B}$ in $S_{1}$
---|---|---
$\mathcal{B}(\Xi^{-}\to\Sigma^{-}e^{+}e^{-})(\times 10^{-6})$ | $\cdots$ | $2.49^{+0.31}_{-0.29}$
$\mathcal{B}(\Xi^{0}\to\Lambda^{0}e^{+}e^{-})(\times 10^{-6})$ | $7.6\pm 0.6$ | $7.6\pm 0.6$
$\mathcal{B}(\Xi^{0}\to\Sigma^{0}e^{+}e^{-})(\times 10^{-6})$ | $\cdots$ | $2.05\pm 0.17$
$\mathcal{B}(\Lambda^{0}\to ne^{+}e^{-})(\times 10^{-5})$ | $\cdots$ | $2.06^{+0.25}_{-0.23}$
$\mathcal{B}(\Sigma^{0}\to ne^{+}e^{-})(\times 10^{-17})$ | $\cdots$ | $7.61^{+6.59}_{-4.44}$
$\mathcal{B}(\Sigma^{+}\to pe^{+}e^{-})(\times 10^{-7})$ | $<70$ | $1.60^{+1.14}_{-0.87}$
$\mathcal{B}(\Sigma^{0}\to n\mu^{+}\mu^{-})(\times 10^{-17})$ | $\cdots$ | $1.22^{+1.05}_{-0.71}$
$\mathcal{B}(\Sigma^{+}\to p\mu^{+}\mu^{-})(\times 10^{-8})$ | $2.4^{+1.7}_{-1.3}$ | $2.40^{+1.70}_{-1.30}$

There are four complex parameters $G_{1},G_{2},\widetilde{G}_{A}$ and
$\widetilde{G}_{B}$ in $T_{8}\to T^{\prime}_{8}\ell^{+}\ell^{-}$ weak decays.
Since the initial baryon $\Xi^{-}$ does not contain $u$ quark and the
$W$-exchange contributions are canceled in
$\Xi^{0}\to\Lambda^{0}\ell^{+}\ell^{-}$ decays, the amplitudes of
$\Xi^{-}\to\Sigma^{-}\ell^{+}\ell^{-}$ and
$\Xi^{0}\to\Lambda^{0}\ell^{+}\ell^{-}$ listed in Tab. 9 only contain
coefficients $G_{1,2}$, which means that the $W$-exchange transitions do not
contribute to the $\Xi^{-}\to\Sigma^{-}\ell^{+}\ell^{-}$ and
$\Xi^{0}\to\Lambda^{0}\ell^{+}\ell^{-}$ decays. Therefore,
$\Xi^{-}\to\Sigma^{-}\ell^{+}\ell^{-}$ and
$\Xi^{0}\to\Lambda^{0}\ell^{+}\ell^{-}$ decays could be used to explore the LD
contributions. Other decay amplitudes contained both $G_{1,2}$ and
$\widetilde{G}_{A,B}$ could proceed from the LD contributions and the
W-exchange contributions.

Only two branching ratios of $\Xi^{0}\to\Lambda^{0}e^{+}e^{-}$ and
$\Sigma^{+}\to p\mu^{+}\mu^{-}$ decays have been measured at present, which
are listed in the second column of Tab. 10. We may constrain $|G_{1}+2G_{2}|$
and $|G_{2}+\widetilde{G}_{B}|$ from the experimental data of
$\mathcal{B}(\Xi^{0}\to\Lambda^{0}e^{+}e^{-})$ and $\mathcal{B}(\Sigma^{+}\to
p\mu^{+}\mu^{-})$, respectively, and we obtain that
$\frac{|G_{1}+2G_{2}|}{|G_{2}+\widetilde{G}_{B}|}\approx 12$.
$\mathcal{B}(\Sigma^{+}\to pe^{+}e^{-})$ is obtained by using the constrained
$|G_{2}+\widetilde{G}_{B}|$,

$\displaystyle\mathcal{B}(\Sigma^{+}\to
pe^{+}e^{-})=(1.60^{+1.14}_{-0.87})\times 10^{-7},$ (33)

which is 1 order smaller than its experimental upper limits
$\mathcal{B}(\Sigma^{+}\to pe^{+}e^{-})\leq 7\times 10^{-6}$ at the 90%
confidence level PDG2020  and is also smaller than its SM predictions with the
single-quark transition LD contributions, $9.1\times
10^{-6}\leq\mathcal{B}(\Sigma^{+}\to pe^{+}e^{-})\leq 10.1\times 10^{-6}$ in
Ref. He:2005yn .

It is difficult to estimate which term gives the main contribution among
$G_{1},G_{2},\widetilde{G}_{A}$, and $\widetilde{G}_{B}$ now. Nevertheless,
the LD contributions noted by $G_{1,2}$ can not been entirely ignored via the
experimental measurement of $\Xi^{0}\to\Lambda^{0}\ell^{+}\ell^{-}$. So we
will give the following discussions.

  * •

If only considering the single-quark transition contributions, $i.e.$,
$\widetilde{G}_{A}=\widetilde{G}_{B}=0$, one obtains
$\left|\frac{G_{1}}{G2}+2\right|\approx 12$; $i.e.$, $G_{1}\approx 10G_{2}$ or
$-14G_{2}$. After ignoring the small $G_{2}$ terms in $G_{1}+2G_{2}$ and
$2G_{1}+G_{2}$, one gets all branching ratios of relevant $T_{8}\to
T^{\prime}_{8}\ell^{+}\ell^{-}$ weak decays in $S_{1}$ case, which are given
in the last column of Tab. 10.

  * •

If $|\widetilde{G}_{B}|\gg|G_{2}|$,
$\frac{|G_{1}+2G_{2}|}{|G_{2}+\widetilde{G}_{B}|}\approx\frac{|G_{1}|}{|\widetilde{G}_{B}|}\approx
12$, the predictions of
$\mathcal{B}(\Xi^{-}\to\Sigma^{-}e^{+}e^{-}),\leavevmode\nobreak\
\mathcal{B}(\Xi^{0}\to\Lambda^{0}e^{+}e^{-}),\leavevmode\nobreak\
\mathcal{B}(\Sigma^{0}\to ne^{+}e^{-}),\leavevmode\nobreak\
\mathcal{B}(\Sigma^{+}\to pe^{+}e^{-}),\leavevmode\nobreak\
\mathcal{B}(\Sigma^{0}\to n\mu^{+}\mu^{-})$, and $\mathcal{B}(\Sigma^{+}\to
p\mu^{+}\mu^{-})$ are the same as given in Tab. 10. Nevertheless,
$B\mathcal{}(\Xi^{0}\to\Sigma^{0}\ell^{+}\ell^{-})$ and
$\mathcal{B}(\Lambda^{0}\to n\ell^{+}\ell^{-})$ can not be predicted in this
case.

  * •

In the case of $\tilde{G}_{B}\approx G_{2}$, the predicted results are similar
to above ones with $|\widetilde{G}_{B}|\gg|G_{2}|$ except
$\mathcal{B}(\Sigma^{0}\to n\ell^{+}\ell^{-})\approx 0$.

  * •

If $\widetilde{G}_{B}\approx-G_{2}$, $i.e.$, the contributions between $G_{2}$
and $\widetilde{G}_{B}$ are largely canceled in $G_{2}+\widetilde{G}_{B}$
term, the situations are complex, which is beyond the scope of this paper.

All predicted branching ratios except $\mathcal{B}(\Sigma^{0}\to ne^{+}e^{-})$
and $\mathcal{B}(\Sigma^{0}\to n\mu^{+}\mu^{-})$ in above first three cases
are on the order of $\mathcal{O}(10^{-8}-10^{-5})$, some of them might be
observed by BESIII and Belle-II experiments in the near future. The
measurement of $\mathcal{B}(\Xi^{-}\to\Sigma^{-}e^{+}e^{-})$ and
$\mathcal{B}(\Xi^{0}\to\Sigma^{0}e^{+}e^{-})$ in the future could further help
us to understand the LD contributions and the $W$-exchange contributions,
respectively.

##  IV SUMMARY

Semileptonic baryon decays induced by flavor changing neutral current
transitions play very important roles in testing the SM and probing new
physics. We have studied the semileptonic decays of baryons with $\frac{1}{2}$
spin via the single-quark transitions $b\to s/d\ell^{+}\ell^{-}$, $c\to
u\ell^{+}\ell^{-}$ and $s\to d\ell^{+}\ell^{-}$ as well as relevant
$W$-exchange transitions by using the SU(3) flavor symmetry, which is a
powerful tool to test the physics and to connect the physical quantities
without knowing the underlying dynamics. Our main results can be summarized as
follows.

  * •

$T_{b3}\to T_{8}\ell^{+}\ell^{-}$ decays: Decay
$\Lambda^{0}_{b}\to\Sigma^{0}\ell^{+}\ell^{-}$ is not allowed by the SU(3)
irreducible representation approach, and all other 21 decay amplitudes of the
$T_{b3}\to T_{8}\ell^{+}\ell^{-}$ decay modes via the $b\to
s/d\ell^{+}\ell^{-}$ transitions could be related by only one SU(3) flavor
symmetry parameter, which could be constrained by the present experimental
data of $\mathcal{B}(\Lambda^{0}_{b}\to\Lambda^{0}\mu^{+}\mu^{-})$. Using the
constrained parameter, we have predicted the not-yet-measured observables in
the whole $q^{2}$ region and in different $q^{2}$ bins within the $S_{1}$ and
$S_{2}$ cases. The predicted branching ratios are on the order of
$\mathcal{O}(10^{-8}-10^{-6})$, many of them are obtained for the first time,
and some of them could be reached by the LHCb or Belle-II experiments. The
longitudinal polarization fractions and the leptonic forward-backward
asymmetries of all $T_{b3}\to T_{8}\ell^{+}\ell^{-}$ decays are very similar
to each other in certain $q^{2}$ bins by the SU(3) flavor symmetry. The
predictions of $\langle
f_{L}\rangle(\Lambda^{0}_{b}\to\Lambda^{0}\mu^{+}\mu^{-})_{[15,20]}$ and
$\langle
A^{\ell}_{FB}\rangle(\Lambda^{0}_{b}\to\Lambda^{0}\mu^{+}\mu^{-})_{[15,20]}$
are agreeable with their experimental data within $1.5\sigma$ and $1\sigma$
error ranges, respectively.

  * •

$T_{c3}\to T_{8}\ell^{+}\ell^{-}$ decays: $T_{c3}\to T_{8}\ell^{+}\ell^{-}$
decays are quite different from $T_{b3}\to T_{8}\ell^{+}\ell^{-}$ decays,
since the former may receive both the single-quark $c\to u\ell^{+}\ell^{-}$
transition contributions and the $W$-exchange contributions. After ignoring
the Wilson coefficient suppressed $H(15)$ terms, all decay amplitudes of
$T_{c3}\to T_{8}\ell^{+}\ell^{-}$ have been related by two SU(3) flavor
symmetry parameters. Using the 90% experimental upper limit of
$\mathcal{B}(\Lambda^{+}_{c}\to p\mu^{+}\mu^{-})$, we have obtained the upper
limit predictions of the not-yet-measured $\mathcal{B}(T_{c3}\to
T_{8}\ell^{+}\ell^{-})$ by considering only one kind of dominant contributions
from either single-quark transition LD contributions or the $W$-exchange
contributions.

  * •

$T_{8}\to T^{\prime}_{8}\ell^{+}\ell^{-}$ decays: Decays $T_{8}\to
T^{\prime}_{8}\ell^{+}\ell^{-}$ are more complicated than both $T_{b3}\to
T_{8}\ell^{+}\ell^{-}$ and $T_{c3}\to T_{8}\ell^{+}\ell^{-}$ ones, since the
quarks are antisymmetric in both the initial states $T_{8}$ and the final
states $T^{\prime}_{8}$. We have predicted $\mathcal{B}(\Sigma^{+}\to
pe^{+}e^{-})$ in the $S_{1}$ case. Moreover, we have analyzed the single-quark
transition LD contributions and the $W$-exchange contributions, and found that
$\mathcal{B}(\Xi^{-}\to\Sigma^{-}e^{+}e^{-})$,
$\mathcal{B}(\Xi^{0}\to\Sigma^{0}e^{+}e^{-})$, and $\mathcal{B}(\Lambda^{0}\to
ne^{+}e^{-})$ are on the order of $\mathcal{O}(10^{-6}-10^{-5})$ in most cases
except $\widetilde{G}_{B}\approx-G_{2}$.

According to our predictions, many results in this work can be tested by the
experiments at BESIII, LHCb, and Belle-II. And these results can be used to
test SU(3) flavor symmetry approach in $T_{b3,c3,8}\to
T^{\prime}_{8}\ell^{+}\ell^{-}$ by the future experiments.

## ACKNOWLEDGEMENTS

The work was supported by the National Natural Science Foundation of China
(Contract No. 11675137).

## References

## References

  * (1) S. D. Genon, J. Matias, M. Ramon and J. Virto, JHEP 01, 048 (2013) [arXiv:1207.2753 [hep-ph]].
  * (2) R. Aaij et al. [LHCb Collaboration], JHEP 02, 104 (2016) [arXiv:1512.04442 [hep-ex]].
  * (3) [ATLAS Collaboration], ATLAS-CONF-2017-023.
  * (4) A. M. Sirunyan et al. [CMS Collaboration], Phys. Lett. B 781, 517-541 (2018) [arXiv:1710.02846 [hep-ex]].
  * (5) R. Aaij et al. [LHCb Collaboration], Phys. Rev. Lett. 113, 151601 (2014) [arXiv:1406.6482 [hep-ex]].
  * (6) R. Aaij et al. [LHCb Collaboration], JHEP 08, 055 (2017) [arXiv:1705.05802 [hep-ex]].
  * (7) T. Aaltonen et al. [CDF Collaboration], Phys. Rev. Lett. 107, 201802 (2011) [arXiv:1107.3753 [hep-ex]].
  * (8) R. Aaij et al. [LHCb Collaboration], JHEP 06, 115 (2015) [erratum: JHEP 09, 145 (2018)] [arXiv:1503.07138 [hep-ex]].
  * (9) R. Aaij et al. [LHCb Collaboration], Phys. Lett. B 725, 25-35 (2013) [arXiv:1306.2577 [hep-ex]].
  * (10) R. Aaij et al. [LHCb Collaboration], Phys. Rev. D 97, 091101 (2018) [arXiv:1712.07938 [hep-ex]].
  * (11) J. P. Lees et al. [BaBar Collaboration], Phys. Rev. D 84, 072006 (2011) [arXiv:1107.4465 [hep-ex]].
  * (12) J. R. Batley et al. [NA48 Collaboration], Phys. Lett. B 650, 1-8 (2007) [arXiv:hep-ex/0703023 [hep-ex]].
  * (13) H. Park et al. [HyperCP Collaboration], Phys. Rev. Lett. 94, 021801 (2005) [arXiv:hep-ex/0501014 [hep-ex]].
  * (14) W. Detmold and S. Meinel, Phys. Rev. D 93, 074501 (2016) [arXiv:1602.01399 [hep-lat]].
  * (15) P. Böer, T. Feldmann and D. van Dyk, JHEP 01 (2015), 155, [arXiv:1410.2115 [hep-ph]].
  * (16) T. Gutsche et al.,V. E. Lyubovitskij and P. Santorelli, Phys. Rev. D 87 (2013), 074031, [arXiv:1301.3737 [hep-ph]].
  * (17) Y. M. Wang and Y. L. Shen, JHEP 02, 179 (2016) [arXiv:1511.09036 [hep-ph]].
  * (18) T. Feldmann and M. W. Y. Yip, Phys. Rev. D 85, 014035 (2012) [erratum: Phys. Rev. D 86, 079901 (2012)] [arXiv:1111.1844 [hep-ph]].
  * (19) X. G. He, T. Li, X. Q. Li and Y. M. Wang, Phys. Rev. D 74, 034026 (2006) [arXiv:hep-ph/0606025 [hep-ph]].
  * (20) R. N. Faustov and V. O. Galkin, Mod. Phys. Lett. A 32, 1750125 (2017) [arXiv:1706.01379 [hep-ph]].
  * (21) T. M. Aliev, K. Azizi and M. Savci, Phys. Rev. D 81 (2010), 056006 [arXiv:1001.0227 [hep-ph]].
  * (22) K. Azizi, M. Bayar, Y. Sarac and H. Sundu, J. Phys. G 37, 115007 (2010)
  * (23) R. N. Faustov and V. O. Galkin, Eur. Phys. J. C 78, 527 (2018) [arXiv:1805.02516 [hep-ph]].
  * (24) B. B. Sirvanli, Phys. Rev. D 93, 034027 (2016).
  * (25) S. Meinel, Phys. Rev. D 97, 034511 (2018) [arXiv:1712.05783 [hep-lat]].
  * (26) R. N. Faustov and V. O. Galkin, Phys. Rev. D 96, 053006 (2017) [arXiv:1705.07741 [hep-ph]].
  * (27) Y. M. Wang, M. J. Aslam and C. D. Lu, Eur. Phys. J. C 59, 847-860 (2009) [arXiv:0810.0609 [hep-ph]].
  * (28) Q. Y. Hu, X. Q. Li and Y. D. Yang, Eur. Phys. J. C 77, no.4, 228 (2017) [arXiv:1701.04029 [hep-ph]].
  * (29) S. Roy, R. Sain and R. Sinha, Phys. Rev. D 96, 116005 (2017) [arXiv:1710.01335 [hep-ph]].
  * (30) D. Das, JHEP 07, 063 (2018) [arXiv:1804.08527 [hep-ph]].
  * (31) M. J. Aslam, Y. M. Wang and C. D. Lu, Phys. Rev. D 78, 114032 (2008) [arXiv:0808.2113 [hep-ph]].
  * (32) S. de Boer, Eur. Phys. J. C 77, 801 (2017) [arXiv:1707.00988 [hep-ph]].
  * (33) X. G. He, J. Tandean and G. Valencia, Phys. Rev. D 72, 074003 (2005) [arXiv:hep-ph/0506067 [hep-ph]].
  * (34) X. G. He, J. Tandean and G. Valencia, JHEP 1810, 040 (2018) [arXiv:1806.08350 [hep-ph]].
  * (35) X. Gao, C. S. Li, Z. Li and H. Zhang, Eur. Phys. J. C 55, 317 (2008) [arXiv:0712.0257 [hep-ph]].
  * (36) X. G. He, J. Tandean and G. Valencia, Phys. Rev. Lett. 98, 081802 (2007) [hep-ph/0610362].
  * (37) L. Bergstrom, R. Safadi and P. Singer, Z. Phys. C 37, 281 (1988)
  * (38) S. de Boer and G. Hiller, Phys. Rev. D 93, 074001 (2016) [arXiv:1510.00311 [hep-ph]].
  * (39) M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, Phys. Rev. D 18, 2583-2599 (1978)
  * (40) G. Buchalla, A. J. Buras and M. E. Lautenbacher, Rev. Mod. Phys. 68, 1125-1144 (1996) [arXiv:hep-ph/9512380 [hep-ph]].
  * (41) R. M. Wang, X. D. Cheng, Y. Y. Fan, J. L. Zhang and Y. G. Xu, arXiv:2008.06624 [hep-ph].
  * (42) D. Das, Eur. Phys. J. C 78, 230 (2018) [arXiv:1802.09404 [hep-ph]].
  * (43) C. Bobeth, G. Hiller and D. van Dyk, JHEP 1007, 098 (2010) [arXiv:1006.5013 [hep-ph]].
  * (44) J. Lach and P. Zenczykowski, Int. J. Mod. Phys. A 10, 3817 (1995).
  * (45) Y. I. Azimov, Z. Phys. A 359, 75 (1997) [hep-ph/9611315].
  * (46) R. C. Verma and A. Sharma, Phys. Rev. D 38, 1443 (1988).
  * (47) P. A. Zyla et al. (Particle Data Group), Prog. Theor. Exp. Phys. 2020, 083C01 (2020).
  * (48) J. Pan, Y. K. Hsiao, J. Sun and X. G. He, Phys. Rev. D 102, 056005 (2020) [arXiv:2007.02504 [hep-ph]].
  * (49) D. Zeppenfeld, Z. Phys. C 8, 77 (1981).
  * (50) M. J. Savage and M. B. Wise, Phys. Rev. D 39, 3346 (1989) Erratum: [Phys. Rev. D 40, 3127 (1989)].
  * (51) N. G. Deshpande and X. G. He, Phys. Rev. Lett. 75, 1703-1706 (1995) [arXiv:hep-ph/9412393 [hep-ph]].
  * (52) R. M. Wang, M. Z. Yang, H. B. Li and X. D. Cheng, Phys. Rev. D 100, 076008 (2019) [arXiv:1906.08413 [hep-ph]].
  * (53) R. N. Faustov and V. O. Galkin, Particles 3, 208 (2020).
  * (54) L. L. Liu, C. Wang, X. W. Kang and X. H. Guo, Eur. Phys. J. C 80, 193 (2020) [arXiv:1912.12622 [hep-ph]].
  * (55) A. J. Buras and M. Munz, Phys. Rev. D 52, 186 (1995) [hep-ph/9501281].
  * (56) P. Singer, Nucl. Phys. B Proc. Suppl. 50, 202-210 (1996) [arXiv:hep-ph/9512308 [hep-ph]].
  * (57) W. Wang, Z. P. Xing and J. Xu, Eur. Phys. J. C 77, 800 (2017) [arXiv:1707.06570 [hep-ph]].
  * (58) C. Q. Geng, Y. K. Hsiao, Y. H. Lin and L. L. Liu, Phys. Lett. B 776, 265 (2018) [arXiv:1708.02460 [hep-ph]].
  * (59) C. Q. Geng, Y. K. Hsiao, C. W. Liu and T. H. Tsai, Phys. Rev. D 97, 073006 (2018) [arXiv:1801.03276 [hep-ph]].