# Sizable Time-reversal Violating Effects in Bottom Baryon Decays

Chia-Wei Liu and Chao-Qiang Geng  School of Fundamental Physics and
Mathematical Sciences, Hangzhou Institute for Advanced Study, UCAS, Hangzhou
310024, China

###### Abstract

We study time-reversal (T) asymmetries in the charmless two-body decays of
antitriplet bottom baryons. We find that in ${\bf B}_{b}\to{\bf B}_{n}P$ with
${\bf B}_{b}=(\Xi_{b}^{-},\,\Xi_{b}^{0},\,\Lambda_{b})$ and ${\bf B}_{n}(P)$ a
octet baryon (pseudoscalar meson), the positive and negative helicity
amplitudes are both sizable, resulting in the large T-violating effects.
Particularly, the T-odd parameters, $\beta_{w}$, for the color-enhanced
channels are expected to be around $-40\%$ and $25\%$ in $b\to s$ and $b\to d$
transitions for the standard model, which can be measured by the experiments
at LHCb, respectively. On the other hand, in ${\bf B}_{b}\to{\bf
B}_{n}V(\gamma)$ with $V(\gamma)$ a vector meson (photon), the decays are
predominated by a single helicity amplitude, and the T violating effects by
the correlations among spins are suppressed. We also explore the T violating
observables for the $\Xi_{b}$ decays based on the angular distributions.

##  I Introduction

It is well known that time-reversal (T) and CP violations are equivalent due
to the CPT theorem CPT . The indirect and direct evidences of CP violation
(CPV) were found in 1964 Christenson:1964fg  and 2001 BaBar:2001ags ;
Belle:2001zzw , respectively. On the other hand, the first direct evidence on
T violation (TV) was long-awaited until 2012 via the $B^{0}-\overline{B}^{0}$
oscillation BaBar:2012bwc . As baryon-antibaryon oscillations are forbidden, a
direct measurement on TV in baryon decays is still lacking. Nevertheless, we
can study TV with the triple-vector product asymmetries (TPAs), which can be
viewed as indirect evidences on TV. The TPAs have been intensively studied in
the mesonic weak decays ref1 ; ref4 ; ref7 ; ref5 ; ref6 ; ref8 ; Li:2010ra ;
Geng:BtoVV ; Datta:2003mj ; Kramer:1993yu ; Valencia:1988it ; Chiang:1999qn ;
London:2004ws ; LondonBtoVT ; Gronau:2011cf ; Ali:1998gb ; Kagan:2004uw . In
terms of angular analyses, there are over 40 experimental values related to
TPAs in $B\to VV^{\prime}$ with $V^{(\prime)}=(\rho,\omega,K^{*},\phi)$
LHCb:2012mid ; pdg ; LHCb:2014xzf ; LHCb:2018hsm ; LHCb:2019jgw . However,
there is still no experimental evidence on a TPA in the bottom baryon decays.
Our goal in this work is to seek some measurable large T violating effects
(TVEs) in the bottom baryon decays.

Compared to the spinless mesons, the antitriplet bottom baryons (${\bf
B}_{b}$) are polarized, providing a possibility for observing TPAs even in the
two body decays of ${\bf B}_{b}\to{\bf B}_{n}(P,V,\gamma)$ with ${\bf B}_{n}$,
$P(V)$ and $\gamma$ being the octet baryon, pseudoscalar (vector) meson and
photon, respectively. Recently, the lifetimes of ${\bf B}_{b}$ have been
updated with high precision at LHCb XibTau ; XibmTau ; CMS:2017ygm ; pdg , and
CP violating effects have been studied XibBr ; XibCP ; LbPK . In particular, a
full angular analysis has been carried out in $\Lambda_{b}\to J/\psi\Lambda$
ExpPolarized1 ; ExpPolarized2 ; ExpPolarized3 . Clearly, the experimental
progresses on ${\bf B}_{b}$ result in a great opportunity for us to explore TV
in the two-body decays of ${\bf B}_{b}$. On the theoretical side for ${\bf
B}_{b}$ decays, many efforts have been made Dery:2020lbc ; Franklin:2020bvb ;
Han:2021gkl ; Roy:2019cky ; Roy:2020nyx ; GeneralGeng:2016kjv ;
GeneralGeng:2021nkl ; GeneralHsiao:2017tif ; GeneralModifiedBagModel ;
NaiveFacZhao:2018zcb ; NaiveKhodjamirian:2011jp ; KornerSM ; KornerSM2 ;
Boer:2019zmp ; Zhu:2018jet ; Lu:2009cm , but only few of them have discussed
TVEs ref2 ; ref3 ; Lambdab to LambdaV ; French0 ; French1 ; French2 ;
Gronau:2015gha . For the weak transitions described by the effective
Hamiltonian (${\cal H}_{eff}$) Buras:1991jm , based on the Hermitian nature of
${\cal H}_{eff}$ and T-symmetry, we have the complex phase approach with

$H_{\lambda}=\langle\lambda;\text{``out''}|{\cal H}_{eff}|{\bf
B}_{b}\rangle=\langle{\bf B}_{b}|{\cal
H}_{eff}|\lambda;\text{``in''}\rangle\,,$ (1)

where $\lambda$ are the helicities of the final states. By ignoring the final
state interactions (FSIs), “in” and “out” are interchangable, such that Eq.
(1) demands $H_{\lambda}$ to be real due to the anti-liner property of the
T-transformation. This approach is a common method in exploring TV, which has
been shown closely related to the TPAs involving momentum and spin ref2 ; ref3
; French0 ; Lambdab to LambdaV ; Gronau:2015gha ; French1 ; French2 . However,
there are some defects in the complex phase approach. The relation between
phases and TV is not intuitively understandable. In addition, the analyses of
the TPAs carried out in the literature suffer a theoretical inconsistency
Lambdab to LambdaV .

In this study, we use a different approach to define an asymmetry parameter
with a T-odd scalar operator $\hat{T}$ in the ${\bf B}_{b}$ decays, given by

$\Delta_{\hat{T}}\equiv\frac{\Gamma(\lambda_{t}>0)-\Gamma(\lambda_{t}<0)}{\Gamma(\lambda_{t}>0)+\Gamma(\lambda_{t}<0)}\,,$
(2)

where $\lambda_{t}$ are the eigenvalues of $\hat{T}$. If ${\cal H}_{eff}$
respects T-symmetry, $\pm\lambda_{t}$ are related by

$|\langle\lambda_{t};\text{``out''}|{\cal H}_{eff}|{\bf
B}_{b}\rangle|^{2}-|\langle{\bf B}_{b}|{\cal
H}_{eff}|-\lambda_{t};\text{``in''}\rangle|^{2}=0\,.$ (3)

Note that if “in” and “out” are interchangable, the left-handed side of Eq.
(3) is proportional to the numerator in $\Delta_{\hat{T}}$, leading to that a
nonzero value of $\Delta_{\hat{T}}$ would be a signal of TV. Nonetheless,
under FSIs, we have

$\langle\mp\lambda_{t};\text{``out''}|\pm\lambda_{t};\text{``in''}\rangle\neq
0\,,$ (4)

describing the rescattering effects between $\pm\lambda_{t}$. In this case,
$\Delta_{\hat{T}}$ is referred to as a naive T-odd observable. To subtract the
effects of FSIs, we utilize the CPT symmetry and thereby define a true T
violating observable, which is also a CP violating one, as Gronau:2011cf ;
Gronau:2015gha

${\cal T}\equiv(\Delta_{\hat{T}}-(\pm)\overline{\Delta}_{\hat{T}})/2\,,$ (5)

where the overline denotes the charge conjugate and the sign corresponds to
the parity of ${\hat{T}}$. The most simple cases for $\hat{T}$ are the triple
products of the spins ($\vec{s}$) and 3-momenta ($\vec{p}$) of the particles
in the decays. Since there is only one independent 3-momentum in the two-body
decays of ${\bf B}_{b}\to{\bf B}_{n}(P,V,\gamma)$, $\hat{T}$ contains at least
two spins, called as the spin correlations. Consequently, to observe a TVE, a
decay must involve at least two different helicities. In this work, we explore
TV by concentrating on the T violating observables in ${\bf B}_{b}\to{\bf
B}_{n}M$.

This paper is organized as follows. In Sec. II, we discuss TV in the two-body
decays of ${\bf B}_{b}\to{\bf B}_{n}(P,V,\gamma)$. We also provide the
numerical results in the standard model for ${\bf B}_{b}\to{\bf B}_{n}P$. In
Sec. III, we study the angular distributions of the ${\bf B}_{b}$ decays. We
give our conclusions in Sec. IV.

##  II T-odd observables

For the ${\bf B}_{b}$ decays, the effective Hamiltonian is given by
Buras:1991jm

$\displaystyle{\cal
H}_{eff}=\frac{G_{F}}{\sqrt{2}}\sum_{f=d,s}\left[V_{ub}V_{uf}^{*}\left(C_{1}O^{f}_{1}+C_{2}O^{f}_{2}\right)-V_{tb}V_{tq}^{*}\sum_{i=3}^{10}C_{i}O^{f}_{i}\right]\,,$
(6)

with

$\displaystyle
O_{1}^{f}=(\overline{u}_{\alpha}\gamma^{\mu}Lb_{\alpha})(\overline{f}_{\beta}\gamma_{\mu}Lu_{\beta})\,,\,\,\,\,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}O_{2}^{f}=(\overline{u}_{\alpha}\gamma^{\mu}Lb_{\beta})(\overline{f}_{\beta}\gamma_{\mu}Lu_{\alpha})\,,$
$\displaystyle
O_{3}^{f}=(\overline{f}_{\alpha}\gamma^{\mu}Lb_{\alpha})\sum_{q}(\overline{q}_{\beta}\gamma_{\mu}Lq_{\beta})\,,\,\,\,\,~{}~{}~{}~{}~{}O_{4}^{f}=(\overline{f}_{\alpha}\gamma^{\mu}Lb_{\beta})\sum_{q}(\overline{q}_{\beta}\gamma_{\mu}Lq_{\alpha})\,,$
$\displaystyle
O_{5}^{f}=(\overline{f}_{\alpha}\gamma^{\mu}Lb_{\alpha})\sum_{q}(\overline{q}_{\beta}\gamma_{\mu}Rq_{\beta})\,,\,\,\,\,~{}~{}~{}~{}~{}O_{6}^{f}=(\overline{f}_{\alpha}\gamma^{\mu}Lb_{\beta})\sum_{q}(\overline{q}_{\beta}\gamma_{\mu}Rq_{\alpha})\,,$
$\displaystyle
O_{7}^{f}=\frac{3}{2}(\overline{f}_{\alpha}\gamma^{\mu}Lb_{\alpha})\sum_{q}e_{q}(\overline{q}_{\beta}\gamma_{\mu}Rq_{\beta})\,,\,\,\,\,O_{8}^{f}=\frac{3}{2}(\overline{f}_{\alpha}\gamma^{\mu}Lb_{\beta})\sum_{q}e_{q}(\overline{q}_{\beta}\gamma_{\mu}Rq_{\alpha})\,,$
$\displaystyle
O_{9}^{f}=\frac{3}{2}(\overline{f}_{\alpha}\gamma^{\mu}Lb_{\alpha})\sum_{q}e_{q}(\overline{q}_{\beta}\gamma_{\mu}Lq_{\beta})\,,\,\,\,\,O_{10}^{f}=\frac{3}{2}(\overline{f}_{\alpha}\gamma^{\mu}Lb_{\beta})\sum_{q}e_{q}(\overline{q}_{\beta}\gamma_{\mu}Lq_{\alpha})\,,$
$\displaystyle
O^{f}_{7\gamma}=\frac{e}{8\pi^{2}}m_{\mathrm{b}}\bar{f}_{\alpha}\sigma^{\mu\nu}Rb_{\alpha}F_{\mu\nu}\,,\,\,\,\,O_{8G}=\frac{g}{8\pi^{2}}m_{\mathrm{b}}\bar{f}_{\alpha}\sigma^{\mu\nu}T_{\alpha\beta}^{a}Rb_{\beta}G_{\mu\nu}^{a}\,,$
(7)

where $O_{1,2}$, $O_{3,4,5,6}$, $O_{7,8,9,10}$ and $O_{7\gamma,8G}$ correspond
to tree, penguin, electroweak-penguin and magnetic-penguin operators,
respectively, $G_{F}$ is the Fermi constant, $V$ represents the CKM quark
mixing matrix, $\alpha$ and $\beta$ denote the color, $q=(u,d,s)$, and
$L~{}(R)$ stands for the chiral structure of $1\mp\gamma_{5}$.

In the radiative decays of ${\bf B}_{b}\to{\bf B}_{n}\gamma$, the relevant
operator is $O^{f}_{7\gamma}$, in which $f$ is essentially left handed. As the
decays are predominated by a single helicity state of $\lambda_{n}=-1/2$, the
T-violating effects are expected to be suppressed. In the factorization
approach for ${\bf B}_{b}\to{\bf B}_{n}(P,V)$, $O_{1,2,3,4,9,10}$ can only
produce left-handed ${\bf B}_{n}$ since they involve only left-handed
currents. On the other hand, for $O_{5,6,7,8}$, based on the Fierz
transformation, one has that

$(\overline{f}\gamma^{\mu}Lb)\sum_{q}(\overline{q}\gamma_{\mu}Rq)=-2(\overline{q}Lb)(\overline{f}Rq)\,,$
(8)

in which the chiral structure of $(\overline{q}Lb)$ leads to a right-handed
${\bf B}_{n}$, given by

$\langle{\bf B}_{n}|\overline{q}Lb|{\bf
B}_{b}\rangle=\overline{u}_{n}(f_{s}-g_{a}\gamma_{5})u_{b}\,,$ (9)

where $u_{b(n)}$ represents the Dirac spinor for ${\bf B}_{b(n)}$, and
$f_{s}(g_{a})$ corresponds to the (pseudo)scalar form factor with
$f_{s}=g_{a}$ in the heavy quark limit. Here, the color index has not been
explicitly shown. Clearly, the amplitudes between $\lambda_{n}=-1/2$ and
$\lambda_{n}=1/2$ from $O_{1,2,3,4,9,10}$ and $O_{5,6,7,8}$, respectively, can
potentially have interferences. Furthermore, in ${\bf B}_{b}\to{\bf B}_{n}V$,
the vector meson matrix elements of (pseudo)scalar operators vanish, read as

$\langle V|\overline{f}Rq|0\rangle=0\,,$ (10)

resulting in that the TVEs are suppressed because the decays are dominated by
a single helicity state Lambdab to LambdaV . In contrast, in ${\bf
B}_{b}\to{\bf B}_{n}P$, the pseudoscalar meson matrix elements are enhanced by
the smallness of the light quark masses, given by

$i\langle P|\overline{f}Rq|0\rangle=\frac{m_{P}^{2}}{m_{f}+m_{q}}f_{P}\,,$
(11)

with $m$ standing for the mass and $f_{P}$ the meson decay constants. Since
the denominators of the right side of Eq. (11) are made of the light quark
masses, $O_{5,6}$ cause a significant contribution to $\lambda_{n}=1/2$,
providing a interference with $\lambda_{n}=-1/2$ from $O_{1,2}$. We conclude
that the decays of ${\bf B}_{b}\to{\bf B}_{n}P$ are the most promising modes
to observe TVEs.

Figure 1:  Topological diagrams for the color enhanced processes in ${\bf
B}_{b}\to{\bf B}_{n}P$, where the left and right figures are associated with
the CKM coefficients of $V_{ub}V_{us}^{*}$ and $V_{tb}V_{ts}^{*}$,
respectively.

In ${\bf B}_{b}\to{\bf B}_{n}P$, as $P$ are spinless, the spin correlations
only happen between ${\bf B}_{b}$ and ${\bf B}_{n}$. The most simple T-odd
observables are then described by

$\hat{T}\equiv(\hat{p}\times\vec{s})\cdot\vec{J}\,,$ (12)

where $\hat{p}~{}(\vec{s})$ are the 3-momenta (spins) of ${\bf B}_{n}$, and
$\vec{J}$ are the angular momenta in the final states. Since $\vec{J}$ are
always conserved, one can identify them as the spins of ${\bf B}_{b}$. The
eigenstates of $\hat{T}$ are related to the helicity ones as Lambdab to
LambdaV

$|\lambda_{t}=\pm\frac{1}{2}\rangle=\frac{1}{\sqrt{2}}\left(|H_{+}\rangle\pm
i|H_{-}\rangle\right)\,,$ (13)

with $J=1/2$ in both sides. The naive T-odd observable is then given by the
asymmetry between $\lambda_{t}=\pm 1/2$, read as HXG

$\beta\equiv\frac{\Gamma(\lambda_{t}=\frac{1}{2})-\Gamma(\lambda_{t}=-\frac{1}{2})}{\Gamma(\lambda_{t}=\frac{1}{2})+\Gamma(\lambda_{t}=-\frac{1}{2})}=\frac{2\Im(H_{+}H_{-}^{*})}{|H_{+}|^{2}+|H_{-}|^{2}}=\frac{2\Im({\cal
P}{\cal S}^{*})}{|{\cal S}|^{2}+|{\cal P}|^{2}}\,,$ (14)

where ${\cal S}$ and ${\cal P}$ are the amplitudes with negative and positive
parities, respectively, and $H_{\pm}$ stand for the amplitudes with
$\lambda_{n}=\pm 1/2$. Thereby, from Eq. (5), we can define the true and fake
T-odd observables as Gronau:2015gha

$\beta_{w}=(\beta+\overline{\beta})/2\,,\,\,\,\,\,\,\,\beta_{s}=(\beta-\overline{\beta})/2\,,$
(15)

related by the CP-odd and even phases, respectively. We emphasize that a
nonzero value of $\beta_{w}$ implies not only TV but also CPV. To have a
complete study in ${\bf B}_{b}\to{\bf B}_{n}P$, we also define the direct CP
and up-down asymmetries by HXG

$\displaystyle A_{\text{CP}}$ $\displaystyle\equiv$
$\displaystyle\frac{\Gamma-\overline{\Gamma}}{\Gamma+\overline{\Gamma}}\,,$
(16) $\displaystyle\alpha_{b}$ $\displaystyle\equiv$
$\displaystyle\frac{\Gamma(\lambda_{n}=\frac{1}{2})-\Gamma(\lambda_{n}=-\frac{1}{2})}{\Gamma(\lambda_{n}=\frac{1}{2})+\Gamma(\lambda_{n}=-\frac{1}{2})}=\frac{2\Re({\cal
P}{\cal S}^{*})}{|{\cal S}|^{2}+|{\cal P}|^{2}}\,,$ (17)

respectively. The linear combinations of $\alpha_{b}$ and
$\overline{\alpha}_{b}$ with the overline denoting the charge conjugate are
given by

$\displaystyle\alpha_{\text{av}}\equiv(\alpha_{b}-\overline{\alpha}_{b})/2\,,\,\,\,\,\,\,\,\,\,\,\,\Delta\alpha=\frac{\alpha_{b}+\overline{\alpha}_{b}}{\alpha_{b}-\overline{\alpha}_{b}}\,,$
(18)

in which $\Delta\alpha$ is a CP-odd quantity.

It is well known that to have nonzero values of $A_{\text{CP}}$ and
$\Delta\alpha$, nonvanishing strong phases are both required. Explicitly, one
has that

$A_{\text{CP}},\Delta\alpha\sim\sin\phi_{s}\sin\phi_{w}\,,$ (19)

where $\phi_{s}$ and $\phi_{w}$ correspond to the CP-even and CP-odd phases
between ${\cal S}$ and ${\cal P}$, respectively. On the other hand, one can
show that

$\beta_{w}\propto\cos\phi_{s}\sin\phi_{w}\,,\,\,\,\,\,\,\beta_{s}\propto\cos\phi_{w}\sin\phi_{s}\,.$
(20)

Clearly, a nonzero strong phase is not necessary in $\beta_{w}$. In fact, when
$\phi_{s}\to 0$, $\beta_{w}$ becomes maximal, whereas
$(A_{\text{CP}},\Delta\alpha)\to 0$. Since the strong phases are much less
understood compared to the weak ones, $\beta_{w}$ could be a more promising T
and CP violating observable compared to the others.

In ${\bf B}_{b}\to{\bf B}_{n}P$, TV depends on the relative CP-odd phase
between $V_{ub}V_{uf}^{*}$ and $V_{tb}V_{tf}^{*}$ in the different helicity
amplitudes (see Eq. (14) and FIG. 1), related to the interferences between
$O_{1,2}^{f}$ and $O_{5,6}^{f}$. Since $C_{1,6}$ are much larger compared to
$C_{2,5}$, respectively, one expects that large T-asymmetries can be induced
in the color-enhanced decays.

In practice, the decays are often dominated by one of the helicities. For
$|H_{\pm}|^{2}/|H_{\mp}|^{2}\gg 0$, we have the approximations, given by

$\beta\approx 2\Im\left(\frac{H_{+}H_{-}^{*}}{H_{\pm}H_{\pm}^{*}}\right)=\mp
2\Im\left(\frac{H_{\mp}}{H_{\pm}}\right)\,.$ (21)

As we will see subsequently, in ${\bf B}_{b}\to{\bf B}_{n}P$, the transitions
associated with $b\to s$ and $b\to d$ are predominated by $H_{+}$ and $H_{-}$,
respectively. Using the estimations of $H_{-}\propto c_{1}V_{ub}V_{uf}^{*}$
and $H_{+}\propto-c_{6}V_{tb}V_{tf}^{*}$, we have

$\beta_{w}\approx
2\Im\left(\frac{V_{ub}V_{us}^{*}c_{1}}{V_{tb}V_{ts}^{*}c_{6}}\right)=\frac{2c_{1}}{c_{6}}\lambda^{2}\eta\approx-0.7\,,$
(22)

and

$\beta_{w}\approx-2\Im\left(\frac{V_{tb}V_{td}^{*}c_{6}}{V_{ub}V_{ud}^{*}c_{1}}\right)\approx-\frac{2c_{6}}{c_{1}}\frac{\eta}{\rho^{2}+\eta^{2}}\approx
0.2\,,$ (23)

for $b\to s$ and $b\to d$ transitions, respectively. Here, we have used
$c_{6}/c_{1}=0.04$ Buras:1991jm  and the Wolfenstein parametrization for the
CKM matrix, given by Wolfenstein:1983yz

$\left[\begin{array}[]{lll}V_{ud}&V_{us}&V_{ub}\\\ V_{cd}&V_{cs}&V_{cb}\\\
V_{td}&V_{ts}&V_{tb}\end{array}\right]=\left[\begin{array}[]{ccc}1-\lambda^{2}/2&\lambda&A\lambda^{3}(\rho-i\eta)\\\
-\lambda&1-\lambda^{2}/2&A\lambda^{2}\\\
A\lambda^{3}(1-\rho-i\eta)&-A\lambda^{2}&1\end{array}\right]+O(\lambda^{4})\,.$
(24)

It is interesting to point out that although
$|V_{ub}V_{us}^{*}/V_{tb}V^{*}_{ts}|$ and
$|V_{ub}V_{ud}^{*}/V_{tb}V^{*}_{td}|$ are very different, $|\beta_{w}|$ are in
the same size for $b\to s$ and $b\to d$ transitions. It is due to that they
are applied at the different limits of Eq. (21), allowing them to be both
sizable.

It is known that the factorization approach works well in the color-enhanced
channels EffectiveWilson . To get the more precise results, we adopt the
generalized factorization approach, in which the renormalization dependency in
the naive factorization has been absorbed into the effective Wilson
coefficients to the next to leading log precision EffectiveWilson .
Accordingly, the color-enhanced amplitudes are given as
GeneralModifiedBagModel ; GeneralGeng:2021nkl ; GeneralGeng:2016kjv ;
GeneralHsiao:2017tif

$\langle{\bf B}_{n}P|{\cal H}_{eff}|{\bf
B}_{b}\rangle=i\overline{u}_{n}(A+B\gamma_{5})u_{b}\,,$ (25)

with

$\displaystyle
A=\frac{G_{F}}{\sqrt{2}}f_{P}\left\\{m_{b}V_{ub}V_{uf}^{*}a_{1}-V_{tb}V_{tf}^{*}\left[m_{b}(a_{4}+a_{10})+\frac{2m_{P}^{2}}{m_{u}+m_{f}}(a_{6}+a_{8})\right]\right\\}f_{s}\,,$
$\displaystyle
B=\frac{G_{F}}{\sqrt{2}}f_{P}\left\\{m_{b}V_{ub}V_{uf}^{*}a_{1}-V_{tb}V_{tf}^{*}\left[m_{b}(a_{4}+a_{10})-\frac{2m_{P}^{2}}{m_{u}+m_{f}}(a_{6}+a_{8})\right]\right\\}g_{a}\,,$
(26)

where $a_{i}$ are the effective Wilson coefficients GeneralGeng:2021nkl ;
GeneralGeng:2016kjv ; GeneralHsiao:2017tif . The helicity amplitudes with
definite angular momenta are then given as Korner:1992wi

$H_{\pm}=\sqrt{Q_{+}}A\mp\sqrt{Q_{-}}B\,,$ (27)

with $Q_{\pm}=\sqrt{(m_{{\bf B}_{b}}\pm m_{{\bf B}_{n}})^{2}-m_{P}^{2}}$, and
the decay widths are given by

$\Gamma=\frac{|\vec{p}_{c}|}{16\pi m_{{\bf
B}_{b}^{2}}}\left(|H_{+}|^{2}+|H_{-}|^{2}\right)\,,$ (28)

where $\vec{p}_{c}$ are the 3-momenta of ${\bf B}_{n}$ in the central mass
frames. At the heavy quark limit, we have that $Q_{+}=Q_{-}$. It is straight
forward to see that the terms associated with $(a_{6}+a_{8})$ contribute to
$H_{+}$, while the others $H_{-}$ only.

We evaluate the form factors of $f_{s}$ and $g_{a}$ in Eq. (II) from the
modified bag model GeneralModifiedBagModel  with the numerical results given
in Table I footnote1 , where the first and second uncertainties in the
branching ratios come from the bag radius of $R=(4.8\pm 0.2$ GeV)-1
GeneralModifiedBagModel  and CKM elements pdg , respectively.

Table 1: Numerical results in ${\bf B}_{b}\to{\bf B}_{n}P$, where the first and second uncertainties in the branching ratios come from the bag radius GeneralModifiedBagModel  and CKM elements pdg , respectively.  channels | $10^{6}{\cal B}$ | $\alpha_{\text{av}}$ | $\beta_{s}$ | $\phi_{s}$
---|---|---|---|---
$A_{\text{CP}}$ | $\Delta\alpha$ | $\beta_{w}$ | $\phi_{w}$
$\Xi_{b}^{0}\to$ | $\Sigma^{+}K^{-}$ | $7.52\pm 1.29\pm 0.27$ | $0.40\pm 0.01$ | $-0.04\pm 0.02$ | $0.15\pm 0.00$
$0.06\pm 0.00$ | $0.18\pm 0.00$ | $-0.38\pm 0.01$ | $0.76\pm 0.01$
$\Sigma^{+}\pi^{-}$ | $6.62\pm 1.09\pm 0.86$ | $-0.93\pm 0.01$ | $0.08\pm 0.04$ | $0.08\pm 0.00$
$-0.05\pm 0.00$ | $-0.01\pm 0.00$ | $0.25\pm 0.02$ | $-2.88\pm 0.02$
$\Xi_{b}^{-}\to$ | $\Lambda^{0}K^{-}$ | $1.10\pm 0.27\pm 0.04$ | $0.41\pm 0.01$ | $-0.04\pm 0.02$ | $0.15\pm 0.00$
$0.06\pm 0.00$ | $0.18\pm 0.00$ | $-0.39\pm 0.01$ | $0.76\pm 0.01$
$\Sigma^{0}K^{-}$ | $4.00\pm 0.68\pm 0.14$ | $0.40\pm 0.01$ | $-0.04\pm 0.02$ | $0.15\pm 0.00$
$0.06\pm 0.00$ | $0.18\pm 0.00$ | $-0.38\pm 0.01$ | $0.76\pm 0.01$
$\Lambda^{0}\pi^{-}$ | $0.98\pm 0.24\pm 0.13$ | $-0.93\pm 0.01$ | $0.08\pm 0.04$ | $0.08\pm 0.00$
$-0.05\pm 0.00$ | $-0.01\pm 0.00$ | $0.25\pm 0.02$ | $-2.88\pm 0.02$
$\Sigma^{0}\pi^{-}$ | $3.52\pm 0.58\pm 0.46$ | $-0.93\pm 0.01$ | $0.08\pm 0.00$ | $0.08\pm 0.00$
$-0.05\pm 0.00$ | $-0.01\pm 0.00$ | $0.25\pm 0.02$ | $-2.88\pm 0.02$
$\Lambda_{b}^{0}\to$ | $pK^{-}$ | $5.52\pm 0.65\pm 0.19$ | $0.37\pm 0.01$ | $-0.04\pm 0.02$ | $0.15\pm 0.00$
$0.06\pm 0.00$ | $0.18\pm 0.00$ | $-0.35\pm 0.01$ | $0.76\pm 0.01$
$p\pi^{-}$ | $4.62\pm 0.52\pm 0.60$ | $-0.91\pm 0.006$ | $0.09\pm 0.04$ | $0.08\pm 0.00$
$-0.05\pm 0.00$ | $-0.03\pm 0.00$ | $0.25\pm 0.02$ | $-2.88\pm 0.02$

In particular, the T violating asymmetries are found to be

$\displaystyle\beta_{w}(\Xi_{b}^{0}\to\Sigma^{+}K^{-}\,,\Xi_{b}^{-}\to\Lambda^{0}K^{-}\,,\Xi_{b}^{-}\to\Sigma^{0}K^{-}\,,\Lambda_{b}^{0}\to
pK^{-})$ $\displaystyle=$ $\displaystyle-0.38\pm 0.01\,,-0.39\pm
0.01\,,-0.38\pm 0.01\,,-0.35\pm 0.01\,,$
$\displaystyle\beta_{w}(\Xi_{b}^{0}\to\Sigma^{+}\pi^{-}\,,\Xi_{b}^{-}\to\Lambda^{0}\pi^{-}\,,\Xi_{b}^{-}\to\Sigma^{0}\pi^{-}\,,\Lambda_{b}^{0}\to
p\pi^{-})$ $\displaystyle=$ $\displaystyle 0.25\pm 0.02\,,0.25\pm
0.02\,,0.25\pm 0.01\,,0.25\pm 0.02\,,$ (30)

for $b\to d$ and $b\to s$ transitions in the standard model, respectively. As
the absolute values of the asymmetries can be as large as $40\%$ and $25\%$,
they can be measured in the experiments at LHCb, which provide a great
opportunity to directly observe TV. The main differences compared to the naive
estimations in Eqs. (22) and (23) are caused by the corrections from the
effective Wilson coefficients, given by $|a_{6}/c_{6}|\approx 1.5$
GeneralModifiedBagModel ; GeneralGeng:2021nkl ; GeneralGeng:2016kjv ;
GeneralHsiao:2017tif ; EffectiveWilson . Note that the results of $\alpha$ and
$\beta$ have small uncertainties due to the correlations between $f_{s}$ and
$g_{a}$, i.e. their ratios depend little on the model parameters. Nonetheless,
it must be emphasized that the uncertainties from the nonfactorizable effects
are not considered as they can not be reliably calculated. However, the
$\Xi_{b}^{-}$ decays are free from the W-exchange diagrams, making the results
reliable.

##  III Cascade decay angular distributions

We now concentrate on the effects of the T-odd spin correlations. Although the
cascade decays of ${\bf B}_{n}$ conserve T-symmetry, the T-odd spin
correlations in ${\bf B}_{b}\to{\bf B}_{n}P$ are handed down to the angular
distributions of the sequential particles. For the level-2 cascade decays of
$\Xi_{b}^{-}\to\Lambda(\to p\pi^{-})K^{-}/\pi^{-}$ and
$\Xi_{b}^{0}\to\Sigma^{+}(\to p\pi^{0})K^{-}/\pi^{-}$, the angular
distributions are given by

$\displaystyle{\cal
D}_{2}(\vec{\Omega})=\frac{1}{\Gamma}\frac{\partial^{3}\Gamma}{\partial\cos\theta\partial\cos\theta_{1}\partial\phi}=$
$\displaystyle\frac{1}{8\pi}\left[1+P_{b}\alpha\cos\theta\cos\theta_{1}+\alpha_{b}(\alpha\cos\theta_{1}+P_{b}\cos\theta)+P_{b}\alpha(\beta\sin\phi-\gamma^{\prime}\cos\phi)\sin\theta\sin\theta_{1}\right],\,$
(31)

where $P_{b}$ describe the $\Xi_{b}$ polarization fractions,
$\alpha(=\alpha_{\Lambda,\Sigma^{+}})$ represent the up-down asymmetry
parameters in $\Lambda\to p\pi^{-}$ and $\Sigma^{+}\to p\pi^{0}$, and

$\gamma^{\prime}\equiv\frac{|{\cal S}|^{2}-|{\cal P}|^{2}}{|{\cal
S}|^{2}+|{\cal P}|^{2}}\,.$ (32)

Here, $\theta$ and $\phi$ stand for the polar and azimuthal angles in the
level-1 decays, while $\theta_{1}$ is the polar angle in the level-2 decays
(see FIGs. 2a and 2b, in which we have taken $P=K^{-}$ as an example).

(a) Cascade decays with $\Xi_{b}^{-}\to\Lambda(\to p\pi^{-})K^{-}$.

(b) Cascade decays with $\Xi_{b}^{0}\to\Sigma^{+}(\to p\pi^{0})K^{-}$.

(c) Cascade decays with $\Xi_{b}^{-}\to\Sigma^{0}(\to\Lambda\gamma)K^{-}$.

(d) Cascade decays with $\Sigma^{0}\to\Lambda(\to p\pi^{-})\gamma$.

Figure 2:  Angles in ${\cal D}(\vec{\Omega})$ for cascade decays.

We see that the T-odd spin correlations indeed manifest themselves in ${\cal
D}_{2}(\vec{\Omega})$ and therefore can be observed in the experiments at
LHCb. They are described by the asymmetries of the dihedral angles between the
initial and cascade decay planes. To probe them, it is sufficient to consider
the $\phi$-dependence only. Based on the numbers of events, we define the
naive T-odd observable as

${\cal T}(N_{+},N_{-})=\frac{N_{+}-N_{-}}{N_{+}+N_{-}}\,,$ (33)

where $N_{\pm}$ are the numbers of events with $\pi>\phi\geq 0$ and
$2\pi>\phi\geq\pi$, respectively. For sufficient data points, we have

$\lim\limits_{N_{\pm}\to\infty}{\cal
T}(N+,N_{-})=\frac{1}{\Gamma}\left(\int^{\pi}_{0}\frac{\partial\Gamma}{\partial\phi}-\int^{2\pi}_{\pi}\frac{\partial\Gamma}{\partial\phi}\right)=\frac{\pi}{8}P_{b}\alpha\beta\,.$
(34)

Nonetheless, $\beta$ are always followed by $P_{b}$, requiring that $\Xi_{b}$
should be polarized. This is a reasonable result since $\beta$ shall be able
to trace back to the spin correlations, and naturally demands that $P_{b}\neq
0$.

On the other hand, $\Xi_{b}^{-}\to\Sigma^{0}(\to\Lambda(\to
p\pi^{-})\gamma)K^{-}/\pi^{-}$ is a level-3 cascade decay with the angular
distribution, given by

$\displaystyle{\cal D}_{3}(\vec{\Omega})$ $\displaystyle=$
$\displaystyle\frac{1}{\Gamma}\frac{\partial^{5}\Gamma}{\partial\cos\theta\partial\cos\theta_{1}\partial\cos\theta_{2}\partial\phi\partial\phi_{1}}$
(35) $\displaystyle=$
$\displaystyle\frac{1}{32\pi^{2}}\left[1-P_{b}\alpha_{\Lambda}\cos\theta\cos\theta_{1}\cos\theta_{2}+\alpha_{b}(P_{b}\cos\theta-\alpha_{\Lambda}\cos\theta_{1}\cos\theta_{2})\right.$
$\displaystyle\left.+P_{b}\alpha_{\Lambda}(\gamma^{\prime}\cos\phi-\beta\sin\phi)\sin\theta\sin\theta_{1}\cos\theta_{2}\right]\,,$

where the definitions of $\theta,\,\theta_{1}$ and $\phi$ are same as the
previous ones, while $\phi_{1}~{}(\theta_{2})$ stands for the polar
(azimuthal) angle in the level-2 (3) decays, shown in FIGs. 2c and 2d. Due to
the angular momentum conservation, ${\cal D}_{3}(\vec{\Omega})$ is independent
of $\phi_{1}$.

Similar to ${\cal D}_{2}(\vec{\Omega}),$ $\beta$ are also followed by $P_{b}$
in ${\cal D}_{3}(\vec{\Omega})$, which is a sensible result. Notably, to
observe $\beta$, it is necessary to include the cascade decays of $\Lambda$.
To see this, by integrating ${\cal D}_{3}(\vec{\Omega})$ over $\phi_{1}$ and
$\cos\theta_{2}$, we find that

$\int^{2\pi}_{0}\int^{1}_{-1}{\cal
D}_{3}(\vec{\Omega})d\cos\theta_{2}d\phi_{1}=\frac{1}{8\pi}\left(1+\alpha_{b}P_{b}\cos\theta\right)\,,$
(36)

which is identical to ${\cal D}_{2}(\Omega)$ by setting $\alpha=0$. Due to
this reason, as opposed to ${\cal D}_{2}(\vec{\Omega})$, it is not sufficient
to merely consider the $\phi$-dependence for observing TVEs, and there is no
simple T violating observable as the one defined in Eq. (33).

Note that $\alpha_{\Lambda,\Sigma^{+}}$ can be determined by the experiments
in the hyperon decays, given by pdg ; BESIII:2018cnd

$\alpha_{\Lambda}=0.731\pm 0.014\,,\,\,\,\,\,\,\,\alpha_{\Sigma^{+}}=-0.982\pm
0.014\,.$ (37)

However, the values of $P_{b}$ depend on the production method of the bottom
baryons, which are ${\cal O}(10\%)$ at LHCb ExpPolarized3 ; Hiller:2007ur .
Theoretically, it has been suggested that $P_{b}$ can be as large as 20%
French0 . Their actual values could be measured from the decays with enough
data points BESIII:2017kqg ; BESIII:2019odb . For instances, we can obtain
$P_{b}$ from the angular distributions of $\Xi_{b}^{0,+}\to\Xi^{0,+}J/\psi(\to
l^{+}l^{-})$, similar to $\Lambda_{b}\to\Lambda J/\psi(\to l^{+}l^{-})$
ExpPolarized1 ; ExpPolarized2 ; ExpPolarized3 , in which $P_{b}$ is found to
be $[-0.06,0.05]$, $[-0.04,0.05]$ and $[-0.01,0.07]$ at the central energies
of 7, 8 and 13 TeV of proton-proton collisions, respectively, with 68%
credibility level intervals ExpPolarized3 . Finally, we note that the
branching ratios of $\Xi_{b}^{0,+}\to\Xi^{0,+}J/\psi$ are found to be around
$10^{-4}$ in terms of the $SU(3)$ flavor symmetry with the experimental input
of ${\cal B}(\Lambda_{b}\to\Lambda J/\psi)=(5.8\pm 0.8)\times 10^{-5}$ pdg . A
large decay branching ratio can enhance the experiments to extract $P_{b}$
with high precision.

##  IV Conclusions

Aiming on probing TV, we have analyzed the most simple cases in the charmless
two-body decays of ${\bf B}_{b}\to{\bf B}_{n}M$ with $M=(P,V,\gamma)$, in
which the TVEs are described by the T-odd spin correlations. We have found
that the T violating parameters of
$\beta_{w}(\Xi_{b}^{0,-}\to\Sigma^{+,0}K^{-}\,,\Xi_{b}^{-,0}\to(\Lambda^{0},p)K^{-})$
and
$\beta_{w}(\Xi_{b}^{0,-}\to\Sigma^{+,0}\pi^{-}\,,\Xi_{b}^{-,0}\to(\Lambda^{0},p)\pi^{-})$
are around $-40\%$ and $25\%$ with small uncertainties in the standard model,
which can be measured by the experiments at LHCb, respectively. We emphasize
that in contrast to other CP asymmetries, the signs of $\beta_{w}$ are
insensitive to the strong phases which can not be reliably calculated. Hence,
if the experiments obtain opposite signs of $\beta_{w}$, it would be a smoking
gun for new physics, such as the right-handed current beyond the standard
model.

We have studied the angular distributions of the $\Xi_{b}$ cascade decays and
identified the TVEs, in which polarized $\Xi_{b}$ are needed in order to
observe TV. We have also shown that $P_{b}$ can be obtained from the angular
distributions of $\Xi_{b}\to\Xi J/\psi(\to l^{+}l^{-})$ in the experiments at
LHCb since their decay branching ratios are predicted to be around $10^{-4}$
by virtual of the $SU(3)$ flavor symmetry.

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