††institutetext: School of Physics, Huazhong University of Science and
Technology, Wuhan 430074, China

# Double-mixing CP violation in $B$ decays

Wen-Jie Song  Yin-Fa Shen  and Qin Qin<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study a long-overlooked CP violation observable, termed double-mixing CP
violation, which arises from the interference between two neutral meson
oscillating paths involved in a decay chain. The double-mixing CP violation is
beneficial for the precise test of the Standard Model CKM mechanism, as it
offers the potential to extract weak phases without pollution from strong
dynamics. To provide a comprehensive understanding of the double-mixing CP
violation, we perform phenomenological analyses on the cascade decays of the
$B^{0}_{d}$ and $B^{0}_{s}$ mesons. Our results show that the double-mixing CP
violation can be very significant in certain decay channels, such as
$B^{0}_{s}\to\rho^{0}K\to\rho^{0}(\pi^{-}\ell^{+}\nu_{\ell})$. In addition, we
employ the decay process $B^{0}_{d}\to D^{0}K\to(K^{-}\pi^{+})(\pi\ell\nu)$ to
demonstrate that the involved strong and weak phases can be directly
determined from the experimental data without any theoretical inputs.

##  1 Introduction

CP violation plays a crucial role in flavor physics. It is one of the
fundamental criteria for explaining the matter-antimatter asymmetry of the
universe Sakharov:1967dj . However, the standard model (SM) does not account
for a sufficiently large CP violation to explain the observed asymmetry
Bernreuther:2002uj ; Canetti:2012zc . This suggests the presence of additional
sources of CP violation beyond the standard Cabibbo-Kobayashi-Maskawa (CKM)
mechanism Cabibbo:1963yz ; Kobayashi:1973fv . Consequently, the quest for non-
standard CP violation is a vital undertaking in the quark flavor sector. One
systematic approach involves testing the unitarity of the CKM matrix, which
relies on measurements of CP violation in quark flavor processes to determine
the values of the CKM phase angles. Therefore, it is imperative to conduct
comprehensive investigations into CP violation in flavor physics, to precisely
test and explore potential sources of CP violation beyond the framework of the
CKM mechanism.

Since the unexpected discovery of indirect CP violation in the neutral $K$
meson system in 1964 Christenson:1964fg , significant theoretical and
experimental advancements have been achieved NA31:1988eyf ; NA48:1999szy ;
KTeV:1999kad ; BaBar:2001pki ; Belle:2001zzw ; Belle:2004nch ; BaBar:2004gyj ;
BaBar:2010hvw ; Belle:2010xyn ; LHCb:2011aa ; LHCb:2012xkq ; LHCb:2013syl ;
LHCb:2014kcb ; LHCb:2019hro , including the detection of CP violation in the
$B$ and $D$ meson decays BaBar:2001pki ; Belle:2001zzw ; LHCb:2019hro . These
studies have successfully observed three types of CP violation under the CKM
mechanism in meson systems, along with measurements of CP-violating phases and
testing of the unitarity of the CKM matrix. As no evidence of non-standard CP
violation has been observed yet, collaborative efforts in both theoretical and
experimental domains are essential to conduct more precise tests. In general,
experimental precision of channels takes advantage of powerful flavor
experiments such as BESIII BESIII:2020nme , Belle II Belle-II:2018jsg , and
LHCb LHCb:2012myk ; LHCb:2018roe ; LHCb:2023hlw . The primary theoretical
challenge in improving precision lies in disentangling the effects of strong
interactions within the SM and non-standard contributions. Despite numerous
studies dedicated to strong interactions (see e.g. Beneke:1999br ;
Beneke:2000ry ; Keum:2000wi ; Lu:2000em ; Bauer:2000yr ; Bauer:2001yt ;
Li:2012cfa ; Qin:2013tje ; Qin:2014xta ; Qin:2021tve ), the theoretical
predictions arising from strong interactions still struggle to align well with
the current experimental precision. Therefore, there is an anticipation to
provide CP violation observables that are less affected or unaffected by
strong interactions.

Of the three types of observed CP violation—direct CP violation, CP violation
in mixing, and CP violation in interference between a decay without mixing
$M^{0}\to f$ and a decay with mixing $M^{0}\to\bar{M}^{0}\to f$—the last one
has the potential to avoid contamination from strong dynamics. The
visualization of CP violation induced by the CKM mechanism typically involves
interference between amplitudes with distinct CKM phases and CP-conserving
phases, often including strong phases. However, the $M^{0}\to f$ and
$M^{0}\to\bar{M}^{0}\to f$ amplitudes have a relative CP-conserving phase
induced by different time evolution, making strong phases not a necessity. In
fact, the first significant CP violation effect observed by $B$ factories in
$B^{0}_{d}\to J/\psi K^{0}_{S}$ BaBar:2001pki ; Belle:2001zzw  serves as a
representative example of this type. The CP violation in the interference
between $i\to M^{0}$ and $i\to\bar{M}^{0}\to M^{0}$ Yu:2017oky ; Meca:1998ee ;
Amorim:1998pi  also capitalizes on the same advantage provided by neutral
meson mixing. It motivates us to develop more CP violation observables that
involve mixing.

In our recent study Shen:2023nuw , we introduced a long-overlooked form of CP
violation arising from the interference between two neutral meson oscillating
paths involved in a decay chain. Such a decay, involving two neutral mesons
$M^{0}_{1,2}$, typically occurs via two amplitudes: $M^{0}_{1}\to
M^{0}_{2}\to\bar{M}^{0}_{2}\to f$ and
$M^{0}_{1}\to\bar{M}^{0}_{1}\to\bar{M}^{0}_{2}\to f$, as depicted in Figure 1,
where the particles associated with $M_{2}$ are hidden for convenience. We
refer to this type of CP violation as the double-mixing CP violation. This
introduces a new experimental approach, allowing for measuring the dependence
of CP violation on two time variables, specifically the evolution periods of
$M_{1,2}$. Our previous research Shen:2023nuw  revealed that the existence of
double-mixing CP violation does not necessarily require a nonzero strong
phase, offering the potential to extract weak phases without the influence of
strong dynamics. In some cases where a strong phase is involved, it is also
feasible to directly extract the strong phase from experimental data, thereby
avoiding significant theoretical uncertainties. To provide a more
comprehensive understanding, we will explore the double-mixing CP violation in
typical $B^{0}_{d}$ and $B_{s}^{0}$ meson cascade decays in this study.

The rest of the paper is organized as follows. In section II, we provide the
general formulas for the double-mixing CP violation in the process
$M^{0}_{1}(t_{1})\to M_{2}(t_{2})\to f$, where $M_{2}$ is either $M^{0}_{2}$
or $\bar{M}^{0}_{2}$. In section III, we introduce a numerical analysis
strategy to calculate the double-mixing CP violation in cascade decays of
$M^{0}_{1}$, where $M^{0}_{1}$ can be either $B^{0}_{d}$ or $B^{0}_{s}$.
Different initial and final states exhibit markedly distinct patterns. With
frequently oscillating $B^{0}_{(s)}$ and $K^{0}$ involved, the double-mixing
CP violation effects are generally more promising. In the case of semi-
leptonic final states, such as $B^{0}_{d}\to J/\psi K\to J/\psi(\pi\ell\nu)$,
the total CP violation predominantly stems from contributions of the double-
mixing CP violation. Conversely, the contribution of the double-mixing CP
violation is exceedingly small when $D^{0}$ is involved, primarily due to the
negligible mixing of $D^{0}$. We employ certain decay processes, such as
$B^{0}_{s}\to\rho^{0}K\to\rho^{0}(\pi^{\pm}\ell^{\mp}\nu_{\ell})$, to reveal
that the double-mixing CP violation is possibly significantly large, exceeding
50%. These decay modes could serve as the initial experimental attempts to
search for the double-mixing CP violation. Of greater significance, the
double-mixing CP violation in certain decay channels under investigation can
be utilized for CKM phase extraction, such as in the case of $B^{0}_{d}\to
D^{0}K\to(K^{-}\pi^{+})(\pi\ell\nu)$. Despite involving both strong and weak
phases simultaneously, both phases can be extracted through experimental
measurements of the double-mixing CP violation without relying on any
theoretical inputs. In section IV, we summarize our results and present a
conclusion.

##  2 Formulae

We adopt the conventional formulae that the mass eigenstates $M_{H,L}$ of the
flavored neutral mesons are superpositions of their flavor eigenstates $M^{0}$
and $\bar{M}^{0}$ Workman:2022ynf ,

$\displaystyle\ket{M_{\mathrm{H,L}}}=p\ket{M^{0}}\mp q\ket{\bar{M}^{0}}\;,$
(1)

where $q,p$ are complex coefficients. Then, the time evolution starting from a
flavor eigenstate in the case of CPT conservation is given by,

$\displaystyle\ket{M^{0}(t)}$
$\displaystyle=g_{+}(t)\ket{M^{0}}-\frac{q}{p}g_{-}(t)\ket{\bar{M}^{0}},$
$\displaystyle\ket{\bar{M}^{0}(t)}$
$\displaystyle=g_{+}(t)\ket{\bar{M}^{0}}-\frac{p}{q}g_{-}(t)\ket{M^{0}},$ (2)

where the functions $g_{\pm}(t)$ of time are given by

$\displaystyle
g_{\pm}(t)=\frac{1}{2}\left[\mathrm{exp}\left(-im_{\mathrm{H}}t-\frac{1}{2}\Gamma_{\mathrm{H}}t\right)\pm\mathrm{exp}\left(-im_{\mathrm{L}}t-\frac{1}{2}\Gamma_{\mathrm{L}}t\right)\right],$
(3)

with $m_{\mathrm{H(L)}}$ and $\Gamma_{\mathrm{H(L)}}$ being the mass and decay
width of the ‘Heavy(Light)’ mass eigenstate $M_{\mathrm{H(L)}}$, respectively.

Figure 1: Two oscillation paths in the cascade decay $M_{1}^{0}\to M_{2}\to f$
with only $M_{1}^{0}\to M_{2}^{0}$, $\bar{M}_{2}^{0}\to f$ and their CP
conjugate processes being allowed. The decay products associated with $M_{2}$
are not displayed.

The double-mixing CP violation will occur in the cascade decay containing at
least two different neutral mesons. The process illustrated in Figure 1 serves
as a clear example. During the time interval $t_{1}$, a primary neutral meson
$M^{0}_{1}$ either remains in its original state or oscillates into its
antiparticle $\bar{M}^{0}_{1}$, which subsequently decays into a secondary
neutral meson $\bar{M}^{0}_{2}(M^{0}_{2})$. Other accompanying particles are
not shown in the figure for simplicity. Similarly, the intermediate state
$M_{2}(t_{2})$, representing a time-evolved neutral meson, $M^{0}_{2}(t_{2})$
or $\bar{M}^{0}_{2}(t_{2})$, decays into a final state $f$. Here, $t_{2}$
denotes the time for the oscillation of $M^{0}_{2}(\bar{M}^{0}_{2})$ in the
rest frame of $M^{0}_{2}(\bar{M}^{0}_{2})$. The time-dependent CP asymmetry in
the chain $M^{0}_{1}(t_{1})\to M_{2}(t_{2})\to f$ can be defined as

$\displaystyle A_{\rm CP}(t_{1},t_{2})\equiv$ $\displaystyle\
\frac{\absolutevalue{\mathcal{M}}^{2}(t_{1},t_{2})-\absolutevalue{\bar{\mathcal{M}}}^{2}(t_{1},t_{2})}{\absolutevalue{\mathcal{M}}^{2}(t_{1},t_{2})+\absolutevalue{\bar{\mathcal{M}}}^{2}(t_{1},t_{2})}=\frac{N(t_{1},t_{2})}{D(t_{1},t_{2})},$
(4) $\displaystyle N(t_{1},t_{2})=$ $\displaystyle\
e^{-\Gamma_{1}t_{1}}\Big{[}C_{h}(t_{2})\cosh{\frac{\Delta\Gamma_{1}t_{1}}{2}}+C_{n}(t_{2})\cos{\Delta
m_{1}t_{1}}$ $\displaystyle+S_{n}(t_{2})\sin{\Delta
m_{1}t_{1}}+S_{h}(t_{2})\sinh{\frac{\Delta\Gamma_{1}t_{1}}{2}}\Big{]},$
$\displaystyle D(t_{1},t_{2})=$ $\displaystyle\
e^{-\Gamma_{1}t_{1}}\Big{[}C_{h}^{\prime}(t_{2})\cosh{\frac{\Delta\Gamma_{1}t_{1}}{2}}+C_{n}^{\prime}(t_{2})\cos{\Delta
m_{1}t_{1}}$ $\displaystyle+S_{n}^{\prime}(t_{2})\sin{\Delta
m_{1}t_{1}}+S_{h}^{\prime}(t_{2})\sinh{\frac{\Delta\Gamma_{1}t_{1}}{2}}\Big{]},$

where the amplitude $\mathcal{M}(t_{1},t_{2})$ is the sum of amplitudes of all
possible paths, while $\bar{\mathcal{M}}(t_{1},t_{2})$ represents the CP
conjugate of $\mathcal{M}(t_{1},t_{2})$. The differences in widths and masses
are defined as $\Delta\Gamma\equiv\Gamma_{\mathrm{H}}-\Gamma_{\mathrm{L}}$ and
$\Delta m\equiv m_{\mathrm{H}}-m_{\mathrm{L}}$, respectively, and the average
of widths is $\Gamma\equiv(\Gamma_{\mathrm{H}}+\Gamma_{\mathrm{L}})/2$. Here,
the subscripts 1 and 2 denote the primary meson $M_{1}$ and the secondary
meson $M_{2}$, respectively. Evidently, according to the $t_{1}$ dependence,
$N(t_{1},t_{2})$ and $D(t_{1},t_{2})$ can be written in four terms.
Alternatively, a different $t_{1}$ function basis can be chosen, for example,
rewriting the two terms in $N(t_{1},t_{2})$ as

$\displaystyle
e^{-\Gamma_{1}t_{1}}\Big{[}C_{h}(t_{2})\cosh{\frac{\Delta\Gamma_{1}t_{1}}{2}}+C_{n}(t_{2})\cos{\Delta
m_{1}t_{1}}\Big{]}\to
C_{+}(t_{2})\absolutevalue{g_{+,1}(t_{1})}^{2}+C_{-}(t_{2})\absolutevalue{g_{-,1}(t_{1})}^{2},$
(5)

and the same substitution also applies to $D(t_{1},t_{2})$.

We will provide the explicit expressions of the terms in (4) for different
cases. We categorize all cases into three groups based on the number of
possible oscillation and decay paths. Besides the two-path scenario shown in
Figure 1, there are also four- and eight-path ones, which will be discussed in
more detail below.

Category 1

In the first category, we consider decay processes that occur via two possible
paths, as exemplified in Figure 1. In this study, we are only interested in
cases where direct CP violation is negligible, so we have
$\innerproduct{\bar{M}^{0}_{2}}{\bar{M}^{0}_{1}}=\innerproduct{M^{0}_{2}}{M^{0}_{1}}e^{i\omega}$
with where $\omega$ being a pure weak phase, and
$\innerproduct{f}{\bar{M}^{0}_{2}}=\innerproduct{\bar{f}}{M^{0}_{2}}$. The
mixing parameters of $M_{1,2}$ are written as
$(q/p)_{1,2}=\absolutevalue{(q/p)}_{1,2}e^{-i\phi_{1,2}}$, with $\phi_{1,2}$
complex phases.

The amplitude $\mathcal{M}(t_{1},t_{2})$ of the process $M_{1}^{0}(t_{1})\to
M_{2}(t_{2})\to f$ and its CP conjugate $\bar{\mathcal{M}}(t_{1},t_{2})$ are
given by

$\displaystyle\mathcal{M}(t_{1},t_{2})$
$\displaystyle=\innerproduct{f}{M^{0}_{2}(t_{2})}\innerproduct{M^{0}_{2}}{M^{0}_{1}(t_{1})}+\innerproduct{f}{\bar{M}^{0}_{2}(t_{2})}\innerproduct{\bar{M}^{0}_{2}}{M^{0}_{1}(t_{1})},$
(6) $\displaystyle\bar{\mathcal{M}}(t_{1},t_{2})$
$\displaystyle=\innerproduct{\bar{f}}{M^{0}_{2}(t_{2})}\innerproduct{M^{0}_{2}}{\bar{M}^{0}_{1}(t_{1})}+\innerproduct{\bar{f}}{\bar{M}^{0}_{2}(t_{2})}\innerproduct{\bar{M}^{0}_{2}}{\bar{M}^{0}_{1}(t_{1})}.$

Considering the time evolution of the neutral mesons (2) - (3), we can make
the following substitutions

$\displaystyle\innerproduct{M^{0}_{2}}{M^{0}_{1}(t_{1})}$
$\displaystyle=g_{+,1}(t_{1})\innerproduct{M^{0}_{2}}{M^{0}_{1}},\qquad\innerproduct{\bar{M}^{0}_{2}}{M^{0}_{1}(t_{1})}=-\frac{q_{1}}{p_{1}}g_{-,1}(t_{1})\innerproduct{\bar{M}^{0}_{2}}{\bar{M}^{0}_{1}},$
$\displaystyle\innerproduct{\bar{M}^{0}_{2}}{\bar{M}^{0}_{1}(t_{1})}$
$\displaystyle=g_{+,1}(t_{1})\innerproduct{\bar{M}^{0}_{2}}{\bar{M}^{0}_{1}},\qquad\innerproduct{M^{0}_{2}}{\bar{M}^{0}_{1}(t_{1})}=-\frac{p_{1}}{q_{1}}g_{-,1}(t_{1})\innerproduct{M^{0}_{2}}{M^{0}_{1}},$
$\displaystyle\innerproduct{f}{M^{0}_{2}(t_{2})}$
$\displaystyle=-\frac{q_{2}}{p_{2}}g_{-,2}(t_{2})\innerproduct{f}{\bar{M}^{0}_{2}},\qquad\innerproduct{\bar{f}}{M^{0}_{2}(t_{2})}=g_{+,2}(t_{2})\innerproduct{\bar{f}}{M^{0}_{2}},$
$\displaystyle\innerproduct{\bar{f}}{\bar{M}^{0}_{2}(t_{2})}$
$\displaystyle=-\frac{p_{2}}{q_{2}}g_{-,2}(t_{2})\innerproduct{\bar{f}}{M^{0}_{2}},\qquad\innerproduct{f}{\bar{M}^{0}_{2}(t_{2})}=g_{+,2}(t_{2})\innerproduct{f}{\bar{M}^{0}_{2}}.$
(7)

Then, it arrives that the total time-dependent CP asymmetry (4).

The corresponding double-mixing CP asymmetry is reflected by the $S_{h}$ and
$S_{n}$ terms, which are induced by the interference between the upper and
lower paths in Figure 1. The results are given by

$\displaystyle S_{h}(t_{2})=$ $\displaystyle\
\frac{e^{-\Gamma_{2}t_{2}}}{2}\Big{[}\sinh\frac{\Delta\Gamma_{2}t_{2}}{2}\cos{(\omega-\phi_{1}+\phi_{2})}\pqty{\absolutevalue{\frac{q_{1}}{p_{1}}}\absolutevalue{\frac{q_{2}}{p_{2}}}-\absolutevalue{\frac{p_{1}}{q_{1}}}\absolutevalue{\frac{p_{2}}{q_{2}}}}$
$\displaystyle+\sin{\Delta
m_{2}t_{2}}\sin{(\omega-\phi_{1}+\phi_{2})}\pqty{\absolutevalue{\frac{q_{1}}{p_{1}}}\absolutevalue{\frac{q_{2}}{p_{2}}}+\absolutevalue{\frac{p_{1}}{q_{1}}}\absolutevalue{\frac{p_{2}}{q_{2}}}}\Big{]},$
(8) $\displaystyle S_{n}(t_{2})=$ $\displaystyle\
\frac{e^{-\Gamma_{2}t_{2}}}{2}\Big{[}-\sinh{\frac{\Delta\Gamma_{2}t_{2}}{2}}\sin{(\omega-\phi_{1}+\phi_{2})}\pqty{\absolutevalue{\frac{p_{1}}{q_{1}}}\absolutevalue{\frac{p_{2}}{q_{2}}}+\absolutevalue{\frac{q_{1}}{p_{1}}}\absolutevalue{\frac{q_{2}}{p_{2}}}}$
$\displaystyle+\sin{\Delta
m_{2}t_{2}}\cos{(\omega-\phi_{1}+\phi_{2})}\pqty{\absolutevalue{\frac{q_{1}}{p_{1}}}\absolutevalue{\frac{q_{2}}{p_{2}}}-\absolutevalue{\frac{p_{1}}{q_{1}}}\absolutevalue{\frac{p_{2}}{q_{2}}}}\Big{]}.$
(9)

The common factors $\absolutevalue{\innerproduct{M^{0}_{2}}{M^{0}_{1}}}^{2}$
and $\absolutevalue{\innerproduct{f}{\bar{M}^{0}_{2}}}^{2}$ are omitted in the
above expression. The remaining terms in the numerator, which are not the
focus of our current discussion, are given by

$\displaystyle
C_{+}(t_{2})=\pqty{\absolutevalue{\frac{q_{2}}{p_{2}}}^{2}-\absolutevalue{\frac{p_{2}}{q_{2}}}^{2}}\absolutevalue{g_{-,2}(t_{2})}^{2},\qquad
C_{-}(t_{2})=\pqty{\absolutevalue{\frac{q_{1}}{p_{1}}}^{2}-\absolutevalue{\frac{p_{1}}{q_{1}}}^{2}}\absolutevalue{g_{+,2}(t_{2})}^{2}\;.$
(10)

The $C_{+}(t_{2})$ term is contributed by purely the upper path in Figure 1,
where $M^{0}_{2}\to\bar{M}^{0}_{2}$ oscillation is the only CP violation
source, while the $C_{-}(t_{2})$ term is contributed purely by the lower path,
where CP violation is induced by $M^{0}_{1}\to\bar{M}^{0}_{1}$ oscillation.
The terms contributing to $D(t_{1},t_{2})$ are given by

$\displaystyle C_{+}^{\prime}(t_{2})=$
$\displaystyle\pqty{\absolutevalue{\frac{q_{2}}{p_{2}}}^{2}+\absolutevalue{\frac{p_{2}}{q_{2}}}^{2}}\absolutevalue{g_{-,2}(t_{2})}^{2},\qquad
C_{-}^{\prime}(t_{2})=\pqty{\absolutevalue{\frac{q_{1}}{p_{1}}}^{2}+\absolutevalue{\frac{p_{1}}{q_{1}}}^{2}}\absolutevalue{g_{+,2}(t_{2})}^{2},$
(11) $\displaystyle S_{h}^{\prime}(t_{2})=$
$\displaystyle\frac{e^{-\Gamma_{2}t_{2}}}{2}\Big{[}\sinh\frac{\Delta\Gamma_{2}t_{2}}{2}\cos{(\omega-\phi_{1}+\phi_{2})}\pqty{\absolutevalue{\frac{q_{1}}{p_{1}}}\absolutevalue{\frac{q_{2}}{p_{2}}}+\absolutevalue{\frac{p_{1}}{q_{1}}}\absolutevalue{\frac{p_{2}}{q_{2}}}}$
$\displaystyle+\sin{\Delta
m_{2}t_{2}}\sin{(\omega-\phi_{1}+\phi_{2})}\pqty{\absolutevalue{\frac{q_{1}}{p_{1}}}\absolutevalue{\frac{q_{2}}{p_{2}}}-\absolutevalue{\frac{p_{1}}{q_{1}}}\absolutevalue{\frac{p_{2}}{q_{2}}}}\Big{]},$
(12) $\displaystyle S_{n}^{\prime}(t_{2})=$
$\displaystyle\frac{e^{-\Gamma_{2}t_{2}}}{2}\Big{[}-\sinh{\frac{\Delta\Gamma_{2}t_{2}}{2}}\sin{(\omega-\phi_{1}+\phi_{2})}\pqty{\absolutevalue{\frac{q_{1}}{p_{1}}}\absolutevalue{\frac{q_{2}}{p_{2}}}-\absolutevalue{\frac{p_{1}}{q_{1}}}\absolutevalue{\frac{p_{2}}{q_{2}}}}$
$\displaystyle+\sin{\Delta
m_{2}t_{2}}\cos{(\omega-\phi_{1}+\phi_{2})}\pqty{\absolutevalue{\frac{q_{1}}{p_{1}}}\absolutevalue{\frac{q_{2}}{p_{2}}}+\absolutevalue{\frac{p_{1}}{q_{1}}}\absolutevalue{\frac{p_{2}}{q_{2}}}}\Big{]}.$
(13)

For the convenience of further analysis, we define the two-dimensional time-
dependent double-mixing CP asymmetry $A_{dm}(t_{1},t_{2})$, which is
represented as

$\displaystyle A_{dm}(t_{1},t_{2})$
$\displaystyle=\frac{e^{-\Gamma_{1}t_{1}}}{D(t_{1},t_{2})}\pqty{S_{h}(t_{2})\sinh{\frac{\Delta\Gamma_{1}t_{1}}{2}}+S_{n}(t_{2})\sin{\Delta
m_{1}t_{1}}}$ (14)
$\displaystyle=A_{h,dm}(t_{1},t_{2})+A_{n,dm}(t_{1},t_{2}).$ (15)

It represents the double-mixing component within the $A_{\rm CP}(t_{1},t_{2})$
in (4). In fact, the remaining part is negligible.

For decay processes occurring via $M^{0}_{1}\to\bar{M}^{0}_{2}\to M_{2}^{0}\to
f$ and $M^{0}_{1}\to\bar{M}^{0}_{1}\to M_{2}^{0}\to f$, the results can be
directly obtained by applying the replacement $p_{2}/q_{2}\to q_{2}/p_{2}$
from (11) - (13), i.e., $|p_{2}/q_{2}|\to|q_{2}/p_{2}|$ and
$\phi_{2}\to-\phi_{2}$.

Category 2

Figure 2: Four oscillation paths in the cascade decay $M_{1}^{0}\to M_{2}\to
f$ where $f$ is a CP eigenstate, with $M_{1}^{0}\to M_{2}^{0}$,
$\bar{M}_{2}^{0}\to f$ and their CP conjugate processes being allowed.

In the second category, we consider that $f$ is a CP eigenstate that can decay
from both $M_{2}^{0}$ and $\bar{M}_{2}^{0}$. Unlike Category 1, there are four
possible paths, as shown in Figure 2. Assuming negligible direct CP
asymmetries, we can express
$\innerproduct{\bar{M}^{0}_{2}}{\bar{M}^{0}_{1}}=\innerproduct{M^{0}_{2}}{M^{0}_{1}}e^{i\omega}$,
and $\innerproduct{f}{M^{0}_{2}}=\innerproduct{f}{\bar{M}^{0}_{2}}$ under a
suitable convention. It is important to note that in practical terms,
$M_{2}^{0}$ here must be $K^{0}$. If it were $D^{0}$ decaying from $B$ mesons,
its antiparticle $\bar{D}^{0}$ could also decay from the same $B$ meson,
rendering this category inapplicable. This justifies the assumption
$\innerproduct{f}{M^{0}_{2}}=\innerproduct{f}{\bar{M}^{0}_{2}}$ as CP
violation is tiny in strange decays. The result for the CP asymmetry (4) is
listed below.

To clarify the physical meaning, we separate the $S_{h}$ and $S_{n}$ terms in
(4) into two parts, which are given by

$\displaystyle S_{n,1}(t_{2})=$ $\displaystyle\
-\frac{e^{-\Gamma_{2}t_{2}}}{2}\left[\pqty{\absolutevalue{\frac{q_{2}}{p_{2}}}+\absolutevalue{\frac{p_{2}}{q_{2}}}}\sin{\Phi}\sinh{\frac{\Delta\Gamma_{2}t_{2}}{2}}-\pqty{\absolutevalue{\frac{q_{2}}{p_{2}}}-\absolutevalue{\frac{p_{2}}{q_{2}}}}\cos{\Phi}\sin{\Delta
m_{2}t_{2}}\right]\times$
$\displaystyle\pqty{\absolutevalue{\frac{q_{1}}{p_{1}}}+\absolutevalue{\frac{p_{1}}{q_{1}}}},$
(16) $\displaystyle S_{h,1}(t_{2})=$ $\displaystyle\
\frac{e^{-\Gamma_{2}t_{2}}}{2}\left[\pqty{\absolutevalue{\frac{q_{2}}{p_{2}}}+\absolutevalue{\frac{p_{2}}{q_{2}}}}\cos{\Phi}\sinh{\frac{\Delta\Gamma_{2}t_{2}}{2}}+\pqty{\absolutevalue{\frac{q_{2}}{p_{2}}}-\absolutevalue{\frac{p_{2}}{q_{2}}}}\sin{\Phi}\sin{\Delta
m_{2}t_{2}}\right]\times$
$\displaystyle\pqty{\absolutevalue{\frac{q_{1}}{p_{1}}}-\absolutevalue{\frac{p_{1}}{q_{1}}}},$
(17) $\displaystyle S_{n,2}(t_{2})=$ $\displaystyle\
-\pqty{\absolutevalue{\frac{p_{1}}{q_{1}}}+\absolutevalue{\frac{q_{1}}{p_{1}}}}\Big{[}\sin{\pqty{\Phi-\phi_{2}}}\absolutevalue{g_{+,2}(t_{2})}^{2}+\sin{\pqty{\Phi+\phi_{2}}}\absolutevalue{g_{-,2}(t_{2})}^{2}\Big{]},$
(18) $\displaystyle S_{h,2}(t_{2})=$ $\displaystyle\
\pqty{\absolutevalue{\frac{q_{1}}{p_{1}}}-\absolutevalue{\frac{p_{1}}{q_{1}}}}\Big{[}\cos{\pqty{\Phi-\phi_{2}}}\absolutevalue{g_{+,2}(t_{2})}^{2}+\cos{\pqty{\Phi+\phi_{2}}}\absolutevalue{g_{-,2}(t_{2})}^{2}\Big{]},$
(19)

where $\Phi$ is a shorthand for $\omega-\phi_{1}+\phi_{2}$. The
$S_{n,1}(t_{2})$ and $S_{h,1}(t_{2})$ terms are induced by the interference
between two oscillation paths $M^{0}_{1}\to M^{0}_{2}\to\bar{M}^{0}_{2}$ and
$M^{0}_{1}\to\bar{M}^{0}_{1}\to\bar{M}^{0}_{2}$, as well as the interference
between $M^{0}_{1}\to M^{0}_{2}$ and
$M^{0}_{1}\to\bar{M}^{0}_{1}\to\bar{M}^{0}_{2}\to M^{0}_{2}$. The
$S_{n,2}(t_{2})$ and $S_{h,2}(t_{2})$ terms are induced by the interference
between $M^{0}_{1}\to M^{0}_{2}$ and
$M^{0}_{1}\to\bar{M}^{0}_{1}\to\bar{M}^{0}_{2}$ and also the interference
between $M^{0}_{1}\to M^{0}_{2}\to\bar{M}^{0}_{2}$ and
$M^{0}_{1}\to\bar{M}^{0}_{1}\to\bar{M}^{0}_{2}\to M_{2}^{0}$. The other terms
involved in (4) are

$\displaystyle C_{n}(t_{2})=$ $\displaystyle\
\frac{e^{-\Gamma_{2}t_{2}}}{2}\left\\{\sinh{\frac{\Delta\Gamma_{2}t_{2}}{2}}\cos{\phi_{2}}\left[\absolutevalue{\frac{q_{2}}{p_{2}}}\pqty{1+\absolutevalue{\frac{p_{1}}{q_{1}}}^{2}}-\absolutevalue{\frac{p_{2}}{q_{2}}}\pqty{1+\absolutevalue{\frac{q_{1}}{p_{1}}}^{2}}\right]\right.$
$\displaystyle\left.+\sin{\Delta
m_{2}t_{2}}\sin{\phi_{2}}\left[\absolutevalue{\frac{q_{2}}{p_{2}}}\pqty{1+\absolutevalue{\frac{p_{1}}{q_{1}}}^{2}}+\absolutevalue{\frac{p_{2}}{q_{2}}}\pqty{1+\absolutevalue{\frac{q_{1}}{p_{1}}}^{2}}\right]\right\\}$
$\displaystyle+\frac{1}{2}\left[\absolutevalue{\frac{q_{2}}{p_{2}}}^{2}\pqty{1+\absolutevalue{\frac{p_{1}}{q_{1}}}^{2}}-\absolutevalue{\frac{p_{2}}{q_{2}}}^{2}\pqty{1+\absolutevalue{\frac{q_{1}}{p_{1}}}^{2}}\right]\absolutevalue{g_{-,2}(t_{2})}^{2}$
$\displaystyle-\frac{1}{2}\pqty{\absolutevalue{\frac{q_{1}}{p_{1}}}^{2}-\absolutevalue{\frac{p_{1}}{q_{1}}}^{2}}\absolutevalue{g_{+,2}(t_{2})}^{2},$
(20) $\displaystyle C_{h}(t_{2})=$ $\displaystyle\
\frac{e^{-\Gamma_{2}t_{2}}}{2}\left\\{\sinh{\frac{\Delta\Gamma_{2}t_{2}}{2}}\cos{\phi_{2}}\left[\absolutevalue{\frac{q_{2}}{p_{2}}}\pqty{1-\absolutevalue{\frac{p_{1}}{q_{1}}}^{2}}-\absolutevalue{\frac{p_{2}}{q_{2}}}\pqty{1-\absolutevalue{\frac{q_{1}}{p_{1}}}^{2}}\right]\right.$
$\displaystyle\left.+\sin{\Delta
m_{2}t_{2}}\sin{\phi_{2}}\left[\absolutevalue{\frac{q_{2}}{p_{2}}}\pqty{1-\absolutevalue{\frac{p_{1}}{q_{1}}}^{2}}+\absolutevalue{\frac{p_{2}}{q_{2}}}\pqty{1-\absolutevalue{\frac{q_{1}}{p_{1}}}^{2}}\right]\right\\}$
$\displaystyle+\frac{1}{2}\left[\absolutevalue{\frac{q_{2}}{p_{2}}}^{2}\pqty{1-\absolutevalue{\frac{p_{1}}{q_{1}}}^{2}}-\absolutevalue{\frac{p_{2}}{q_{2}}}^{2}\pqty{1-\absolutevalue{\frac{q_{1}}{p_{1}}}^{2}}\right]\absolutevalue{g_{-,2}(t_{2})}^{2}$
$\displaystyle+\frac{1}{2}\pqty{\absolutevalue{\frac{q_{1}}{p_{1}}}^{2}-\absolutevalue{\frac{p_{1}}{q_{1}}}^{2}}\absolutevalue{g_{+,2}(t_{2})}^{2},$
(21) $\displaystyle S_{n}^{\prime}(t_{2})=$
$\displaystyle-\frac{1}{2}\pqty{\absolutevalue{\frac{q_{1}}{p_{1}}}-\absolutevalue{\frac{p_{1}}{q_{1}}}}\left\\{2\sin{\pqty{\Phi-\phi_{2}}}\absolutevalue{g_{+,2}(t_{2})}^{2}+2\sin{\pqty{\Phi+\phi_{2}}}\absolutevalue{g_{-,2}(t_{2})}^{2}\right.$
$\displaystyle\left.+e^{-\Gamma_{2}t_{2}}\left[\pqty{\absolutevalue{\frac{q_{2}}{p_{2}}}+\absolutevalue{\frac{p_{2}}{q_{2}}}}\sin{\Phi}\sinh{\frac{\Delta\Gamma_{2}t_{2}}{2}}-\pqty{\absolutevalue{\frac{q_{2}}{p_{2}}}-\absolutevalue{\frac{p_{2}}{q_{2}}}}\cos{\Phi}\sin{\Delta
m_{2}t_{2}}\right]\right\\},$ (22) $\displaystyle S_{h}^{\prime}(t_{2})=$
$\displaystyle\
\frac{1}{2}\pqty{\absolutevalue{\frac{q_{1}}{p_{1}}}+\absolutevalue{\frac{p_{1}}{q_{1}}}}\left\\{2\cos{\pqty{\Phi-\phi_{2}}}\absolutevalue{g_{+,2}(t_{2})}^{2}+2\cos{\pqty{\Phi+\phi_{2}}}\absolutevalue{g_{-,2}(t_{2})}^{2}\right.$
$\displaystyle\left.+e^{-\Gamma_{2}t_{2}}\left[\pqty{\absolutevalue{\frac{q_{2}}{p_{2}}}+\absolutevalue{\frac{p_{2}}{q_{2}}}}\cos{\Phi}\sinh{\frac{\Delta\Gamma_{2}t_{2}}{2}}+\pqty{\absolutevalue{\frac{q_{2}}{p_{2}}}-\absolutevalue{\frac{p_{2}}{q_{2}}}}\sin{\Phi}\sin{\Delta
m_{2}t_{2}}\right]\right\\},$ (23) $\displaystyle C_{n}^{\prime}(t_{2})=$
$\displaystyle\
\frac{e^{-\Gamma_{2}t_{2}}}{2}\left\\{\sinh{\frac{\Delta\Gamma_{2}t_{2}}{2}}\cos{\phi_{2}}\left[\absolutevalue{\frac{q_{2}}{p_{2}}}\pqty{1-\absolutevalue{\frac{p_{1}}{q_{1}}}^{2}}+\absolutevalue{\frac{p_{2}}{q_{2}}}\pqty{1-\absolutevalue{\frac{q_{1}}{p_{1}}}^{2}}\right]\right.$
$\displaystyle\left.+\sin{\Delta
m_{2}t_{2}}\sin{\phi_{2}}\left[\absolutevalue{\frac{q_{2}}{p_{2}}}\pqty{1-\absolutevalue{\frac{p_{1}}{q_{1}}}^{2}}-\absolutevalue{\frac{p_{2}}{q_{2}}}\pqty{1-\absolutevalue{\frac{q_{1}}{p_{1}}}^{2}}\right]\right\\}$
$\displaystyle+\frac{1}{2}\left[\absolutevalue{\frac{q_{2}}{p_{2}}}^{2}\pqty{1-\absolutevalue{\frac{p_{1}}{q_{1}}}^{2}}+\absolutevalue{\frac{p_{2}}{q_{2}}}^{2}\pqty{1-\absolutevalue{\frac{q_{1}}{p_{1}}}^{2}}\right]\absolutevalue{g_{-,2}(t_{2})}^{2}$
$\displaystyle-\frac{1}{2}\pqty{\absolutevalue{\frac{q_{1}}{p_{1}}}^{2}+\absolutevalue{\frac{p_{1}}{q_{1}}}^{2}-2}\absolutevalue{g_{+,2}(t_{2})}^{2},$
(24) $\displaystyle C_{h}^{\prime}(t_{2})=$ $\displaystyle\
\frac{e^{-\Gamma_{2}t_{2}}}{2}\left\\{\sinh{\frac{\Delta\Gamma_{2}t_{2}}{2}}\cos{\phi_{2}}\left[\absolutevalue{\frac{q_{2}}{p_{2}}}\pqty{1+\absolutevalue{\frac{p_{1}}{q_{1}}}^{2}}+\absolutevalue{\frac{p_{2}}{q_{2}}}\pqty{1+\absolutevalue{\frac{q_{1}}{p_{1}}}^{2}}\right]\right.$
$\displaystyle\left.+\sin{\Delta
m_{2}t_{2}}\sin{\phi_{2}}\left[\absolutevalue{\frac{q_{2}}{p_{2}}}\pqty{1+\absolutevalue{\frac{p_{1}}{q_{1}}}^{2}}-\absolutevalue{\frac{p_{2}}{q_{2}}}\pqty{1+\absolutevalue{\frac{q_{1}}{p_{1}}}^{2}}\right]\right\\}$
$\displaystyle+\frac{1}{2}\left[\absolutevalue{\frac{q_{2}}{p_{2}}}^{2}\pqty{1+\absolutevalue{\frac{p_{1}}{q_{1}}}^{2}}+\absolutevalue{\frac{p_{2}}{q_{2}}}^{2}\pqty{1+\absolutevalue{\frac{q_{1}}{p_{1}}}^{2}}\right]\absolutevalue{g_{-,2}(t_{2})}^{2}$
$\displaystyle+\frac{1}{2}\pqty{\absolutevalue{\frac{q_{1}}{p_{1}}}^{2}+\absolutevalue{\frac{p_{1}}{q_{1}}}^{2}+2}\absolutevalue{g_{+,2}(t_{2})}^{2}.$
(25)

Again, if the decay occurs via $M^{0}_{1}\to\bar{M}^{0}_{2}$ instead of
$M^{0}_{1}\to M^{0}_{2}$, the results can be directly obtained from (16) -
(25) by replacing $p_{2}/q_{2}$ with $q_{2}/p_{2}$.

Category 3

Figure 3: Eight oscillation paths in the cascade decay $M_{1}^{0}\to M_{2}\to
f$, with $M_{1}^{0}\to M_{2}^{0},\bar{M}_{2}^{0}$, $\bar{M}_{2}^{0}\to
f,\bar{f}$ and their CP conjugate processes being allowed.

In the third category, $M^{0}_{1}$ can decay into both $M^{0}_{2}$ and
$\bar{M}^{0}_{2}$, so the number of possible paths for $M_{1}^{0}\to M_{2}\to
f$ is doubled compared with the second category, as shown in Figure 3. To
evaluate the CP asymmetry of such a process, the following parameters need to
be introduced,

$\displaystyle\frac{\innerproduct{M^{0}_{2}}{M^{0}_{1}}}{\innerproduct{\bar{M}^{0}_{2}}{M^{0}_{1}}}=r_{1}e^{i(\delta_{1}+\theta_{1})},\qquad\frac{\innerproduct{f}{\bar{M}^{0}_{2}}}{\innerproduct{f}{M^{0}_{2}}}=r_{2}e^{i\pqty{\delta_{2}+\theta_{2}}},\qquad\frac{\innerproduct{M^{0}_{2}}{\bar{M}^{0}_{1}}}{\innerproduct{\bar{M}^{0}_{2}}{M^{0}_{1}}}=e^{i\theta_{3}},$
(26)

where $\delta_{1,2}$ are the relative strong phases, $\theta_{1,2,3}$ are the
relative weak phases, and $r_{1,2}$ are the magnitude ratios. All complete
results can be found in Appendix 1, where (65) and (67) correspond to the
double-mixing CP violation terms. This is the most complex and comprehensive
scenario, and by making appropriate approximations for this category, we can
derive the complete results of the preceding two categories.

##  3 Numerical analysis

In this section, we will conduct numerical analyses of the double-mixing CP
violation for various channels. As $B_{d}^{0}$ and $B_{s}^{0}$ have very
different oscillation rates, we will discuss their decays separately. The
input parameters involved are detailed in Table 1.

Parameter | Value | Parameter | Value
---|---|---|---
$|q_{B_{d}}/p_{B_{d}}|$ |  $1.0010\pm 0.0008$ HFLAV:2022pwe  | $x_{B_{d}}$ |  $0.769\pm 0.004$ Workman:2022ynf
$|q_{B_{s}}/p_{B_{s}}|$ |  $1.0003\pm 0.0014$ HFLAV:2022pwe  | $x_{B_{s}}$ |  $27.03\pm 0.09$ Workman:2022ynf
$|q_{K}/p_{K}|$ |  $0.996774\pm 0.000019$ Workman:2022ynf  | $x_{K}$ |  $0.946\pm 0.002$ Workman:2022ynf
$|q_{D}/p_{D}|$ |  $0.995\pm 0.016$ HFLAV:2022pwe  | $x_{D}$ |  $\pqty{4.1\pm 0.5}\times 10^{-3}$ HFLAV:2022pwe
$\phi_{B_{d}}$ |  $\pqty{44.4\pm 1.4}^{\circ}$ Workman:2022ynf  | $y_{B_{d}}$ |  $-0.0005\pm 0.0050$ Workman:2022ynf
$\phi_{B_{s}}$ |  $-\pqty{2.106\pm 0.135}^{\circ}$ Workman:2022ynf  | $y_{B_{s}}$ |  $-0.064\pm 0.003$ Workman:2022ynf
$\phi_{K}$ |  $\pqty{0.176\pm 0.001}^{\circ}$ Workman:2022ynf  | $y_{K}$ |  $-0.996506\pm 0.000016$ Workman:2022ynf
$\phi_{D}$ |  $-\pqty{177.5\pm 1.2}^{\circ}$ Workman:2022ynf  | $y_{D}$ |  $\pqty{6.2\pm 0.6}\times 10^{-3}$ HFLAV:2022pwe
$\omega_{B_{d}}$ |  $\pqty{0.00372^{+0.00025}_{-0.00023}}^{\circ}$ Workman:2022ynf  | $\omega_{B_{s}}$ |  $\pqty{-0.070^{+0.004}_{-0.005}}^{\circ}$ Workman:2022ynf
$\omega_{K}$ |  $\pqty{43.52\pm 0.05}^{\circ}$ Workman:2022ynf  | $\omega_{D}$ |  $\pqty{83.4446^{+32.2995}_{-36.5375}}^{\circ}$ Workman:2022ynf
$\omega_{1}$ |  $0^{\circ}$ Workman:2022ynf  | $\omega_{2}$ |  $\pqty{65.54\pm 1.55}^{\circ}$ Workman:2022ynf
$\omega_{3}$ |  $0^{\circ}$ Workman:2022ynf  | $\omega_{4}$ |  $\pqty{-114.419\pm 1.549}^{\circ}$ Workman:2022ynf
$r_{D}$ |  $-\pqty{5.857\pm 0.017}\times 10^{-2}$ HFLAV:2022pwe  | $\delta_{D}$ |  $\pqty{7.2^{+7.9}_{-9.2}}^{\circ}$ HFLAV:2022pwe
Table 1: The input parameters and their values, with $x_{M}\equiv\Delta
m_{M}/\Gamma_{M}$ and $y_{M}\equiv\Delta\Gamma_{M}/(2\Gamma_{M})$,
respectively.

###  3.1 $B^{0}_{d}$ decays

In this subsection, we discuss the double-mixing CP violation in cascade
$B_{d}^{0}$ decays. For the $B^{0}_{d}$ meson, it is a reasonable
approximation to assume that the decay width difference, denoted as
$\Delta\Gamma_{B_{d}}$, is negligible HFLAV:2022pwe ; Workman:2022ynf .
Consequently, all terms involving $\sinh{\Delta\Gamma_{1}t_{1}/2}$ in (4) can
be disregarded while $\cosh{\Delta\Gamma_{1}t_{1}/2}$ effectively equals 1.
Furthermore, the magnitude of ratio $q_{B_{d}}/p_{B_{d}}$ is approximated to
be 1 HFLAV:2022pwe ; Workman:2022ynf . Under these approximations, the double-
mixing CP violation $A_{dm}$ is simplified to $A_{n,dm}$. Here, $A_{n,dm}$
represents the component of CP violation induced by double-mixing interference
and it is proportional to $\sin{\Delta m_{1}t_{1}}$. This simplification aids
in a more straightforward analysis of the double-mixing CP violation effect in
$B^{0}_{d}$ decays.

####  3.1.1 $B^{0}_{d}\to J/\psi K\to J/\psi(\pi^{\pm}\ell^{\mp}\nu)$

We initially explore the decay channel $B^{0}_{d}\to J/\psi K\to
J/\psi(\pi^{+}\ell^{-}\bar{\nu}_{\ell})$. The decay mode $B^{0}_{d}\to J/\psi
K^{0}$, corresponding to the $b\to c\bar{c}s$ transition at the quark level,
is frequently employed for the extraction of the CKM angle $\beta$ because of
the tiny pollution of strong interaction and the relatively substantial CP
violation effect Workman:2022ynf . The cascading mixing effect in this decay
channel where the secondary decay is semileptonic has been previously
investigated in paper Azimov:1997ix .

In this decay channel, only the paths $B_{d}^{0}\to J/\psi K^{0}\to
J/\psi\bar{K}^{0}\to J/\psi(\pi^{+}\ell^{-}\bar{\nu}_{\ell})$ and
$B_{d}^{0}\to\bar{B}_{d}^{0}\to J/\psi\bar{K}^{0}\to
J/\psi(\pi^{+}\ell^{-}\bar{\nu}_{\ell})$ can contribute. This case aligns with
the first category defined previously. As a reasonable approximation, we have
neglected the direct CP violation in neutral $K$ meson decay, assuming
$\innerproduct{\pi^{+}\ell^{-}\bar{\nu}_{\ell}}{\bar{K}^{0}}=\innerproduct{\pi^{-}\ell^{+}\nu_{\ell}}{K^{0}}$.
The primary decay amplitudes are related as
$\innerproduct{J/\psi\bar{K}^{0}}{\bar{B}^{0}_{d}}=-\innerproduct{J/\psi
K^{0}}{B^{0}_{d}}e^{i\omega_{B_{d}}}$, where $\omega_{B_{d}}$ is a pure weak
phase, with the latest result given by
$\arg(V_{cb}V^{\star}_{cs}/V^{\star}_{cb}V_{cs})=\pqty{0.00372^{+0.00025}_{-0.00023}}^{\circ}$
Workman:2022ynf . It is important to note that, the minus sign originates from
the convention $CP\ket{J/\psi K^{0}}=-\ket{J/\psi\bar{K}^{0}}$. Then, all the
components in (4) are calculated as:

$\displaystyle A_{n,dm}(t_{1},t_{2})=$ $\displaystyle\
\frac{e^{-\pqty{\Gamma_{B_{d}}t_{1}+\Gamma_{K}t_{2}}}}{2D(t_{1},t_{2})}\Big{\\{}\sinh{\frac{\Delta\Gamma_{K}t_{2}}{2}}\sin{\Phi_{1}}\pqty{\absolutevalue{\frac{p_{K}}{q_{K}}}+\absolutevalue{\frac{q_{K}}{p_{K}}}}$
$\displaystyle-\sin{\Delta
m_{K}t_{2}}\cos{\Phi_{1}}\pqty{\absolutevalue{\frac{q_{K}}{p_{K}}}-\absolutevalue{\frac{p_{K}}{q_{K}}}}\Big{\\}}\sin{\Delta
m_{B_{d}}t_{1}},$ (27) $\displaystyle A_{non-dm}(t_{1},t_{2})=$
$\displaystyle\
\frac{1}{D(t_{1},t_{2})}\pqty{\absolutevalue{\frac{q_{K}}{p_{K}}}^{2}-\absolutevalue{\frac{p_{K}}{q_{K}}}^{2}}\absolutevalue{g_{-,K}(t_{2})}^{2}\absolutevalue{g_{+,B_{d}}(t_{1})}^{2},$
(28)

where $\Phi_{1}$ is defined as $\omega_{B_{d}}-\phi_{B_{d}}+\phi_{K}$,
approximated as -2$\beta$. The denominator $D(t_{1},t_{2})$ is given by (4)
with the $t_{2}$ functions substituted with (11) - (13). Hence, by measuring
the double-mixing CP violation of this channel, the $\beta$ angle can be
extracted. The double-mixing CP violation $A_{n,dm}(t_{1},t_{2})$ is induced
by the interference between the decay paths $B_{d}^{0}\to J/\psi K^{0}\to
J/\psi\bar{K}^{0}\to J/\psi(\pi^{+}\ell^{-}\bar{\nu}_{\ell})$ and
$B_{d}^{0}\to\bar{B}_{d}^{0}\to J/\psi\bar{K}^{0}\to
J/\psi(\pi^{+}\ell^{-}\bar{\nu}_{\ell})$, corresponding to the $S_{n}(t_{2})$
component in (4). Meanwhile, $A_{non-dm}(t_{1},t_{2})$ corresponds to the
$C_{+}(t_{2})$ term within $N(t_{1},t_{2})$ as defined in (4).

$t_{1}/\tau_{B_{d}}$ | $t_{2}/\tau_{K}$ | $A_{\rm CP}$
---|---|---
0$\sim$3 | 0$\sim$ 2 | 40.52%
0$\sim$2 | 0$\sim$ 2 | 39.76%
0$\sim$3 | 0$\sim$ 1 | 42.09%
0$\sim$2 | 0$\sim$ 1 | 48.04%
0$\sim$3 | 0$\sim$ 0.5 | 29.28%
0$\sim$2 | 0$\sim$ 0.5 | 38.76%
Table 2: The time integrated $A_{\rm CP}$ in $B^{0}_{d}\to J/\psi K\to
J/\psi(\pi^{+}\ell^{-}\bar{\nu}_{\ell})$.

To investigate the double-mixing CP violation effect, we conducted a numerical
analysis of both the total CP asymmetry $A_{\rm CP}$ and the double-mixing CP
asymmetry $A_{dm}$. Notably, $A_{dm}$ plays a crucial role in $A_{\rm CP}$, as
evidenced by the mathematical formula in (27) and the corresponding numerical
results in Figure 4. The left panel clearly shows that the maximum value of
the two-dimensional time-dependent double-mixing CP asymmetry can even exceed
50%. When integrating $t_{2}$ from 0 to $2\tau_{K}$, as depicted in the middle
panel, the double-mixing effect almost overlaps with the total CP asymmetry
$A_{\rm CP}$, indicating its dominance. Its magnitude initially increases with
$t_{1}$ and then decreases, a pattern attributed to its sine dependence on
$t_{1}$. Within an interval of 4$\tau_{B_{d}}$, it completes only half an
oscillation period, consistent with the small value of $x_{B_{d}}$ listed in
Table 1. Similar trends are observed when considering the dependence on
$t_{2}$, as shown in the right panel, where $t_{1}$ is integrated from 0 to
$3\tau_{B_{d}}$. It rapidly increases within a period shorter than $\tau_{K}$,
and then decreases before stabilizing. This phenomenon can be explained by the
fact that the numerator is mainly dominated by the first term in (27), which
increases from 0 to $\tau_{K}$ and then stabilizes. and that the denominator
is dominated by $C^{\prime}_{+}(t_{2})$ and $C^{\prime}_{-}(t_{2})$, which
decreases from 0 to 0.5$\tau_{K}$ and then steadily increases until around
$t_{2}\approx 2.5\tau_{K}$. A comprehensive two-dimensional time integration
of $A_{\rm CP}(t_{1},t_{2})$ yields the results presented in Table 2. For
experimental convenience, we perform the integration over different $t_{1}$
and $t_{2}$ intervals. The integrated $A_{\rm CP}$ fluctuates around 30-40%
with different configurations.

The yield of $B^{0}_{d}\to J/\psi K^{0}_{S}$ has been previously observed at
LHCb LHCb:2015brj  using data from Run 1 of the LHC at a center-of-mass energy
$\sqrt{s}=7\,\rm{TeV}$ with an integrated luminosity 3 $\rm{fb}^{-1}$. This
observation involved the final states of $J/\psi\to\mu^{+}\mu^{-}$ and
$K^{0}_{S}\to\pi^{+}\pi^{-}$. Based on this study, we estimate the yield of
$B^{0}_{d}\to J/\psi K\to J/\psi(\pi^{+}\ell^{-}\bar{\nu}_{\ell})$ with 300
$\rm{fb}^{-1}$ to be approximately $3.6\times 10^{4}$.

Figure 4: Time dependence of the double-mixing CP asymmetry $A_{\rm CP}$ in
$B^{0}_{d}(t_{1})\to J/\psi K(t_{2})\to
J/\psi(\pi^{+}\ell^{-}\bar{\nu}_{\ell})$. The left panel displays the two-
dimensional time dependence. The middle panel and the right panel display the
dependence on $t_{1}$ (with $t_{2}$ integrated from 0 to $2\tau_{K}$) and
$t_{2}$ (with $t_{1}$ integrated from $0$ to $3\tau_{B_{d}}$), respectively.

Substituting the semileptonic final state with $\pi^{-}\ell^{+}\nu_{\ell}$
instead of $\pi^{+}\ell^{-}\bar{\nu}_{\ell}$ resulting in two viable decay
paths, denoted as $B^{0}_{d}\to J/\psi K^{0}\to
J/\psi(\pi^{-}\ell^{+}\nu_{\ell})$ and $B^{0}_{d}\to\bar{B}^{0}_{d}\to
J/\psi\bar{K}^{0}\to J/\psi K^{0}\to J/\psi(\pi^{-}\ell^{+}\nu_{\ell})$. The
corresponding CP asymmetries are then calculated to be

$\displaystyle A_{n,dm}(t_{1},t_{2})=$ $\displaystyle\
\frac{e^{-\pqty{\Gamma_{B_{d}}t_{1}+\Gamma_{K}t_{2}}}}{2D(t_{1},t_{2})}\Big{\\{}\sinh{\frac{\Delta\Gamma_{K}t_{2}}{2}}\sin{\Phi_{1}}\pqty{\absolutevalue{\frac{p_{K}}{q_{K}}}+\absolutevalue{\frac{q_{K}}{p_{K}}}}-\sin{\Delta
m_{K}t_{2}}\cos{\Phi_{1}}$
$\displaystyle\times\pqty{\absolutevalue{\frac{q_{K}}{p_{K}}}-\absolutevalue{\frac{p_{K}}{q_{K}}}}\Big{\\}}\sin{\Delta
m_{B_{d}}t_{1}},$ (29) $\displaystyle A_{non-dm}(t_{1},t_{2})=$
$\displaystyle\
\frac{1}{D(t_{1},t_{2})}\pqty{-\absolutevalue{\frac{q_{K}}{p_{K}}}^{2}+\absolutevalue{\frac{p_{K}}{q_{K}}}^{2}}\absolutevalue{g_{-,K}(t_{2})}^{2}\absolutevalue{g_{-,B_{d}}(t_{1})}^{2}.$
(30)

The denominator $D(t_{1},t_{2})$ here can be obtained by combining (4) with
(11) - (13), while also performing the substitution
$C^{\prime}_{+}(t_{2})\leftrightarrow C^{\prime}_{-}(t_{2})$, and adding an
extra negative sign in front of $S^{\prime}_{n}(t_{2})$.

Compared to the final state $\pi^{+}\ell^{-}\bar{\nu}_{\ell}$, a key
difference for the final state $\pi^{-}\ell^{+}\nu_{\ell}$ lies in an
additional negative sign in the numerator of the non-double-mixing CP
violation term $A_{non-dm}(t_{1},t_{2})$, as illustrated in Figure 5. The
double-mixing CP violation remains significant, with the peak occurring later
compared to the decay process with the final state
$\pi^{+}\ell^{-}\bar{\nu}_{\ell}$. For instance, the peak in the middle panel
of Figure 5 occurs at around $t_{1}=3\tau_{B_{d}}$, in contrast to about
$1.5\tau_{B_{d}}$ for the other final state. This difference is due to an
additional negative sign in the term $S^{\prime}_{n}(t_{2})$ in (4). Moreover,
the magnitude of CP violation increases significantly over time, reflecting
the later occurrence of its peak in both $t_{1}$ and $t_{2}$.

Figure 5: Time dependence of the double-mixing CP asymmetry $A_{\rm CP}$ in
$B^{0}_{d}(t_{1})\to J/\psi K(t_{2})\to J/\psi(\pi^{-}\ell^{+}\nu_{\ell})$.
The left panel displays the two-dimensional time dependence. The middle panel
and the right panel display the dependence on $t_{1}$ (with $t_{2}$ integrated
from 0 to $2\tau_{K}$) and $t_{2}$ (with $t_{1}$ integrated from $0$ to
$3\tau_{B_{d}}$), respectively.

####  3.1.2 $B^{0}_{d}\to J/\psi K\to J/\psi f_{+}$

We now analyze the double-mixing CP violation in the decay chain $B^{0}_{d}\to
J/\psi K\to J/\psi f_{+}$. In this context, $f_{+}$ refers to CP-even final
states, such as $\pi^{+}\pi^{-}$. Notably, the CP-even state under
consideration is common to $K^{0}$ and $\bar{K^{0}}$. This will lead to four
conceivable decay paths: $B^{0}_{d}\to K^{0}\to f_{+}$, $B^{0}_{d}\to
K^{0}\to\bar{K}^{0}\to f_{+}$, $B^{0}_{d}\to\bar{B}^{0}_{d}\to\bar{K}^{0}\to
f_{+}$ and $B^{0}_{d}\to\bar{B}^{0}_{d}\to\bar{K}^{0}\to K^{0}\to f_{+}$.
Hence, it belongs to the second category in our classification. Considering
the negligible direct CP violation in the neutral $K$
meson($\innerproduct{f_{+}}{K^{0}}=\innerproduct{f_{+}}{\bar{K}^{0}}$) and the
suppressed penguin amplitudes in $B^{0}_{d}\to J/\psi K^{0}$ (both Cabbibo-
suppressed and loop-suppressed as indicated in Workman:2022ynf ), we obtain
all components of the two-dimensional time-dependent CP asymmetry $A_{\rm
CP}(t_{1},t_{2})$ in (4) as follows:

$\displaystyle A_{n,dm}(t_{1},t_{2})=$ $\displaystyle\
\frac{e^{-\pqty{\Gamma_{B_{d}}t_{1}+\Gamma_{K}t_{2}}}}{D(t_{1},t_{2})}\left\\{\sin{\Delta
m_{K}t_{2}}\pqty{\absolutevalue{\frac{q_{K}}{p_{K}}}-\absolutevalue{\frac{p_{K}}{q_{K}}}}\cos{\Phi_{1}}\right.$
$\displaystyle\left.-\sinh{\frac{\Delta\Gamma_{K}t_{2}}{2}}\pqty{\absolutevalue{\frac{q_{K}}{p_{K}}}+\absolutevalue{\frac{p_{K}}{q_{K}}}}\sin{\Phi_{1}}\right\\}\sin{\Delta
m_{B_{d}}t_{1}},$ (31) $\displaystyle A_{non-dm}(t_{1},t_{2})=$
$\displaystyle\
\frac{2e^{-\Gamma_{B_{d}}t_{1}}}{D(t_{1},t_{2})}\left\\{\sin{(\phi_{B_{d}}-\omega_{B_{d}})}\absolutevalue{g_{+,K}(t_{2})}^{2}\right.$
$\displaystyle\left.+\sin{(\phi_{B_{d}}-\omega_{B_{d}}+2\phi_{K})}\absolutevalue{g_{-,K}(t_{2})}^{2}\right\\}\sin{\Delta
m_{B_{d}}t_{1}}.$ (32)

Figure 6: Time dependence of the CP asymmetry $A_{\rm CP}$ in
$B^{0}_{d}(t_{1})\to J/\psi K(t_{2})\to J/\psi(\pi^{+}\pi^{-})$. The left
panel displays the two-dimensional time dependence. The middle panel and the
right panel display the dependence on $t_{1}$ (with $t_{2}$ integrated from 0
to $2\tau_{K}$) and $t_{2}$ (with $t_{1}$ integrated from $0$ to
$3\tau_{B_{d}}$), respectively.

In this situation, $A_{n,dm}$ and $A_{non-dm}$ constitute $S_{n}$ in (4).
Notably, $\Phi_{1}\approx-2\beta,\,\phi_{B_{d}}-\omega_{B_{d}}\approx 2\beta$
and $\phi_{K}\approx 0$. Under the approximation that
$\absolutevalue{q_{K}/p_{K}}\approx 1$, $S_{n}(t_{2})$ simplifies to

$\displaystyle S_{n}(t_{2})=2\sin{2\beta}e^{-\Gamma_{S}t_{2}}.$ (33)

By taking $q_{K}/p_{K}\to 1$, we neglect the indirect CP violation of neutral
kaon, so only $K_{S}^{0}\to f_{+}$ contributes to the decay and it explains
why only the $e^{-\Gamma_{S}t_{2}}$ term survives. As a result, the total CP
asymmetry $A_{\rm CP}$ is expressed as

$\displaystyle A_{\rm CP}(t_{1},t_{2})=\sin{2\beta}\cdot\sin{\Delta
m_{B_{d}}t_{1}},$ (34)

which is consistent with the formulas (13.74) and (13.82) in the “ CP
Violation in the Quark Sector ” review of the PDG Workman:2022ynf .

$t_{1}/\tau_{B_{d}}$ | $t_{2}/\tau_{K}$ | $A_{\rm CP}$
---|---|---
0$\sim$3 | 0$\sim$ 2 | -35.36%
0$\sim$2 | 0$\sim$ 2 | -32.35%
0$\sim$3 | 0$\sim$ 1 | -35.30%
0$\sim$2 | 0$\sim$ 1 | -32.28%
0$\sim$3 | 0$\sim$ 0.5 | -35.31%
0$\sim$2 | 0$\sim$ 0.5 | -32.27%
Table 3: Time integrated $A_{\rm CP}$ in $B^{0}_{d}\to J/\psi K\to
J/\psi(\pi^{+}\pi^{-})$.

We conducted a numerical analysis of $A_{\rm CP}$ and $A_{dm}$ for this
process. The two-dimensional time-dependent CP violation result is shown in
the left panel of Figure 6, with its peak appearing around
$t_{1}=2\tau_{B_{d}}$ and $t_{2}>0.5\tau_{K}$. By integrating $t_{2}$ from 0
to $2\tau_{K}$, the $t_{1}$-dependent result, displayed in the middle panel,
reaches its peak at $t_{1}=2\tau_{B_{d}}$. Since both $A_{\rm CP}$ and
$A_{n,dm}$ exhibit sine dependence on $t_{1}$, their oscillations are similar,
but with different magnitudes and signs. It can be concluded that the double-
mixing contribution and the non-double-mixing contribution have opposite
signs, with the former having a smaller magnitude. The right panel of Figure 6
shows that $A_{\rm CP}$ with $t_{1}$ integrated out is almost independent of
$t_{2}$. On the other hand, the $A_{n,dm}$ part, with its hyperbolic sine
dependence on $t_{2}$, increases with $t_{2}$ ranging from 0 to $\tau_{K}$.
The two-dimensional time integration of $A_{\rm CP}(t_{1},t_{2})$ for this
channel is given in Table 3, demonstrating its insensitivity to changes in
both $t_{1}$ and $t_{2}$, owing to the fact that most of the decays occur
earlier.

####  3.1.3 $B^{0}_{d}\to
DK\to(K^{-}\pi^{+})(\pi^{+}\ell^{-}\bar{\nu}_{\ell})$

The decay $B^{0}_{d}\to DK^{\star}$ is well-known for its precision in
measuring the angle $\gamma$. First attempts to constraint $\gamma$ using
$B^{0}_{d}\to DK^{\star}$ decays have adopted neutral $D$ meson decays into
$K^{0}_{S}\pi^{+}\pi^{-}$ BaBar:2008mds  and suppressed final states such as
$K^{-}\pi^{+}$, with relevant branching ratio and parameters measured in
several studies Belle:2002jgk ; BaBar:2005osj ; BaBar:2006ftr . Here, we
consider a similar process $B^{0}_{d}\to DK$, which was also briefly discussed
in our previous work Shen:2023nuw . This channel, comprising two oscillating
paths, $B^{0}_{d}\to D^{0}K^{0}\to D^{0}\bar{K}^{0}$ and
$B^{0}_{d}\to\bar{B}^{0}_{d}\to D^{0}\bar{K}^{0}$, falls within the first
category. The following relations are employed to evaluate the CP asymmetry,

$\displaystyle\frac{A\pqty{\bar{B}^{0}_{d}\to
D^{0}\bar{K}^{0}}}{A\pqty{B^{0}_{d}\to\bar{D}^{0}K^{0}}}$
$\displaystyle=e^{i\omega_{1}},\quad\omega_{1}=\arg{\frac{V_{cb}V^{\star}_{us}}{V^{\star}_{cb}V_{us}}},$
(35) $\displaystyle\frac{A\pqty{B^{0}_{d}\to
D^{0}K^{0}}}{A\pqty{B^{0}_{d}\to\bar{D}^{0}K^{0}}}$
$\displaystyle=r_{B}e^{i\pqty{\delta_{B}+\omega_{2}}},\quad\omega_{2}=\arg{\frac{V_{ub}^{\star}V_{cs}}{V^{\star}_{cb}V_{us}}},$
(36)

where $r_{B}$ and $\delta_{B}$ are the magnitudes and the strong phase of the
ratio, respectively. The parameters for the decay channel $B^{0}_{d}\to
DK^{\star}$ have been determined by the LHCb experiment LHCb:2016aix ;
LHCb:2016bxi , and we take similar values for $B^{0}_{d}\to DK$: $r_{B}=0.366$
and $\delta_{B}=164^{\circ}$. We neglect the direct CP violation in neutral
$K$ meson decay, i.e.,
$\innerproduct{f}{\bar{K}^{0}}=\innerproduct{\bar{f}}{K^{0}}$, as well as the
amplitude of Cabibbo-suppressed process $\bar{D}^{0}\to K^{-}\pi^{+}$ and the
mixing of the neutral $D$ meson. The double-mixing term is

$\displaystyle A_{n,dm}(t_{1},t_{2})=$ $\displaystyle\
\frac{e^{-\Gamma_{B_{d}}t_{1}}}{D(t_{1},t_{2})}S_{n}(t_{2})\sin{\Delta
m_{B_{d}}t_{1}},$ (37) $\displaystyle S_{n}(t_{2})=$ $\displaystyle\
e^{-\Gamma_{K}t_{2}}r_{B}\Big{\\{}\absolutevalue{\frac{q_{K}}{p_{K}}}[\sin{\pqty{\delta_{B}+\delta_{\omega}}}\sinh{\frac{1}{2}\Delta\Gamma_{K}t_{2}}+\cos{\pqty{\delta_{B}+\delta_{\omega}}}\sin{\Delta
m_{K}t_{2}}]$
$\displaystyle-\absolutevalue{\frac{p_{K}}{q_{K}}}[\sin{\pqty{\delta_{B}-\delta_{\omega}}}\sinh{\frac{1}{2}\Delta\Gamma_{K}t_{2}}+\cos{\pqty{\delta_{B}-\delta_{\omega}}}\sin{\Delta
m_{K}t_{2}}]\Big{\\}},$ (38)

where $\delta_{\omega}=\omega_{2}-\omega_{1}+\phi_{B_{d}}-\phi_{K}$,
approximating $2\beta+\gamma$. Applying (11) - (13) yields the result of
$D(t_{1},t_{2})$, with $C^{\prime}_{+}(t_{2})$ has an additional factor of
$r^{2}_{B}$, and $S_{n}^{\prime}(t_{2})$ can be obtained by changing the
negative sign in front of $\absolutevalue{p_{K}/q_{K}}$ in (38) to a positive
sign.

$t_{1}/\tau_{B_{d}}$ | $t_{2}/\tau_{K}$ | $A_{\rm CP}$
---|---|---
0$\sim$3 | 0$\sim$ 2 | 26.36%
0$\sim$2 | 0$\sim$ 2 | 31.28%
0$\sim$3 | 0$\sim$ 1 | 15.68%
0$\sim$2 | 0$\sim$ 1 | 19.79%
0$\sim$3 | 0$\sim$ 0.5 | 8.78%
0$\sim$2 | 0$\sim$ 0.5 | 11.52%
Table 4: Time integrated $A_{\rm CP}$ in $B^{0}_{d}\to
DK\to(K^{-}\pi^{+})(\pi^{+}\ell^{-}\bar{\nu}_{\ell})$.

Our numerical analysis addresses both the total CP asymmetry $A_{\rm CP}$ and
the double-mixing CP asymmetry $A_{dm}$. As illustrated in the left panel of
Figure 7, the maximum value of two-dimensional time-dependent double-mixing
results can surpass 50% when $t_{1}$ is small. From the $t_{1}$ and $t_{2}$
dependence displayed in the middle and right panels, respectively, it can be
seen that the double-mixing effect dominates. This trend resembles that
observed in $B^{0}_{d}\to J/\psi K\to
J/\psi(\pi^{+}\ell^{-}\bar{\nu}_{\ell})$, albeit with differences attributed
to the suppression factor $r_{B}$ and a negative sign in $\delta_{\omega}$.
With integrating $t_{1}$ from 0 to $3\tau_{B_{d}}$, a steady growth of $A_{\rm
CP}$ with respect to $t_{2}$ is observed. This is supported by the time-
integrated $A_{\rm CP}$ presented in Table 4, which shows sensitivity to
changes in the upper limit of $t_{2}$.

Given that the weak phase involved in this decay $\delta_{\omega}\approx
2\beta+\gamma$, its extraction holds significant importance. It should be
emphasized that the weak phase can be directly extracted from experimental
data, as discussed in our previous study Shen:2023nuw .

Figure 7: Time dependence of the double-mixing CP asymmetry $A_{\rm CP}$ in
$B^{0}_{d}(t_{1})\to
DK(t_{2})\to(K^{-}\pi^{+})(\pi^{+}\ell^{-}\bar{\nu}_{\ell})$. The left panel
displays the two-dimensional time dependence. The middle panel and the right
panel display the dependence on $t_{1}$ (with $t_{2}$ integrated from 0 to
$2\tau_{K}$) and $t_{2}$ (with $t_{1}$ integrated from $0$ to
$3\tau_{B_{d}}$), respectively.

####  3.1.4 $B^{0}_{d}\to DK^{0}_{S}\to f_{+}K^{0}_{S}$

Distinct from $B^{0}_{d}\to
DK\to(K^{-}\pi^{+})(\pi^{+}\ell^{-}\bar{\nu}_{\ell})$ discussed previously in
section 3.1.3, when considering the CP-even final state like
$f_{+}=\pi^{+}\pi^{-}$ of $D$ in $B^{0}_{d}\to DK^{0}_{S}\to f_{+}K^{0}_{S}$,
both intermediate $D^{0}$ and $\bar{D}^{0}$ contributions are comparable. With
primary $b\to u$ and $b\to c$ transitions, both $B^{0}_{d}$ and
$\bar{B}^{0}_{d}$ can decay into states involving either $D^{0}$ or
$\bar{D}^{0}$. These make this channel a particularly intricate case,
categorizing it within the third classification.

The terms related to the double-mixing CP violation are expressed as below:

$\displaystyle A_{n,dm,1}(t_{1},t_{2})=$ $\displaystyle\
f(t_{1},t_{2})e^{-\Gamma_{D}t_{2}}\sinh{\frac{1}{2}\Delta\Gamma_{D}t_{2}}\Big{\\{}-\pqty{\absolutevalue{\frac{p_{D}}{q_{D}}}+\absolutevalue{\frac{q_{D}}{p_{D}}}}\sin{(\omega_{1}-\phi_{B_{d}}-\phi_{D})}$
$\displaystyle-
r_{B}\left[\absolutevalue{\frac{p_{D}}{q_{D}}}\sin{\pqty{\delta_{B}+\Phi_{2}}}-\absolutevalue{\frac{q_{D}}{p_{D}}}\sin{\pqty{\delta_{B}-\Phi_{2}}}\right.$
$\displaystyle\left.+\absolutevalue{\frac{p_{D}}{q_{D}}}\sin{\pqty{\delta_{B}+\Phi_{3}}}-\absolutevalue{\frac{q_{D}}{p_{D}}}\sin{\pqty{\delta_{B}-\Phi_{3}}}\right]$
$\displaystyle-
r_{B}^{2}\pqty{\absolutevalue{\frac{p_{D}}{q_{D}}}+\absolutevalue{\frac{q_{D}}{p_{D}}}}\sin{(\Phi_{2}-\omega_{2})}\Big{\\}},$
(39) $\displaystyle A_{n,dm,2}(t_{1},t_{2})=$ $\displaystyle\
f(t_{1},t_{2})e^{-\Gamma_{D}t_{2}}\sin{\Delta
m_{D}t_{2}}\Big{\\{}\pqty{\absolutevalue{\frac{p_{D}}{q_{D}}}-\absolutevalue{\frac{q_{D}}{p_{D}}}}\cos{(\omega_{1}-\phi_{B_{d}}-\phi_{D})}$
$\displaystyle-
r_{B}\left[\absolutevalue{\frac{p_{D}}{q_{D}}}\cos{\pqty{\delta_{B}+\Phi_{2}}}-\absolutevalue{\frac{q_{D}}{p_{D}}}\cos{\pqty{\delta_{B}-\Phi_{2}}}\right.$
$\displaystyle\left.-\absolutevalue{\frac{p_{D}}{q_{D}}}\cos{\pqty{\delta_{B}+\Phi_{3}}}+\absolutevalue{\frac{q_{D}}{p_{D}}}\cos{\pqty{\delta_{B}-\Phi_{3}}}\right]$
$\displaystyle-
r_{B}^{2}\pqty{\absolutevalue{\frac{p_{D}}{q_{D}}}-\absolutevalue{\frac{q_{D}}{p_{D}}}}\cos{(\Phi_{2}-\omega_{2})}\Big{\\}},$
(40)

where $\Phi_{2}=\omega_{1}-\omega_{2}-\phi_{B_{d}}+\phi_{D}$,
$\Phi_{3}=\omega_{1}-\omega_{2}-\phi_{B_{d}}-\phi_{D}$ and
$f(t_{1},t_{2})=e^{-\Gamma_{B_{d}}t_{1}}\sin{\Delta m_{B_{d}}t_{1}}$
$/D(t_{1},t_{2})$, and the double-mixing CP violation is the sum of these two
contributions. Combining (4) and (22) - (25) yields the denominator
$D(t_{1},t_{2})$. The parameters $r_{B}$ and $\delta_{B}$ have been defined in
(35).

Figure 8: Time dependence of the CP asymmetry $A_{\rm CP}$ in
$B^{0}_{d}(t_{1})\to D(t_{2})K^{0}_{S}\to(\pi^{+}\pi^{-})K^{0}_{S}$. The left
panel displays the two-dimensional time dependence. The middle panel and the
right panel display the dependence on $t_{1}$ (with $t_{2}$ integrated from 0
to $5\tau_{D}$) and $t_{2}$ (with $t_{1}$ integrated from $0$ to
$3\tau_{B_{d}}$), respectively.

We perform the numerical analysis assuming the parameter values $r_{B}=0.366$
and $\delta_{B}=164^{\circ}$ as in section 3.1.3. The results are depicted in
Figure 8. On the left panel, the two-dimensional time-dependent CP violation
exhibits a distribution akin to the decay channel $B^{0}_{d}\to J/\psi K\to
J/\psi(\pi^{+}\pi^{-})$, but with peak values significantly lower. Integrating
$t_{2}$ from 0 to $5\tau_{D}$, we present the $t_{1}$-dependent results in the
middle panel, revealing tiny double-mixing CP violation. This is primarily
attributed to the small magnitudes of $x_{D}$ and $y_{D}$, both approximately
$10^{-3}$. These small values reduces both
$\sinh{\frac{1}{2}\Delta\Gamma_{D}t_{2}}$ and $\sin{\Delta m_{D}t_{2}}$. The
right panel of Figure 8 shows the $t_{2}$ dependence of $A_{\rm CP}$ when
integrating $t_{1}$ from 0 to $3\tau_{B_{d}}$. The near-linear dependency of
$A_{dm}$ on $t_{2}$ is primarily due to its denominator being dominated by the
$\cosh{y_{D}t_{2}}$ term and its numerator by the $\sinh{x_{D}t_{2}}$ term. It
is concluded that the CP asymmetry in this channel is dominated by the part
induced by the interference between the $b\to c$ and $b\to u$ amplitudes, and
the part induced by the interference between decays with and without $B$
mixing, while the double-mixing CP violation is negligible owing to tiny
$D$-mixing effect.

####  3.1.5 $B^{0}_{d}\to DK^{0}_{S}\to(K^{-}\pi^{+})K^{0}_{S}$

The last $B^{0}_{d}$ decay channel selected is $B^{0}_{d}\to
DK^{0}_{S}\to(K^{-}\pi^{+})K^{0}_{S}$. The observation of a similar channel
$\bar{B}^{0}_{d}\to D^{0}\bar{K}^{\star 0}$ was reported by Belle:2002jgk ;
BaBar:2005osj , but the branching ratio and relevant parameters were measured
using self-tagging methods and thus the mixing effect was not considered. In
this channel, the intermediate $D$ meson can be either $D^{0}$ or
$\bar{D}^{0}$, although the latter has a suppressed contribution. Therefore,
this channel is categorized as the third type.

To present the result of the CP asymmetry, we introduce the amplitude ratio of
the primary decay as in (35) and the ratio of the secondary decay as

$\displaystyle\frac{A\pqty{\bar{D}^{0}\to K^{-}\pi^{+}}}{A\pqty{D^{0}\to
K^{-}\pi^{+}}}=-r_{D}e^{-i\delta_{D}},$ (41)

where the numerical values for magnitude ratio $r_{D}$ and the strong phase
$\delta_{D}$ are listed in Table 1. The direct CP violation in the $D$ meson
decays is neglected due to the minimal weak phases between the first two
generations of quarks. The double-mixing term in (4) is expressed as

$\displaystyle A_{n,dm}(t_{1},t_{2})=$
$\displaystyle-\frac{e^{-\pqty{\Gamma_{B_{d}}t_{1}+\Gamma_{D}t_{2}}}}{2D(t_{1},t_{2})}\left[\left(\left|\frac{q_{D}}{p_{D}}\right|+\left|\frac{p_{D}}{q_{D}}\right|\right)\sin{(\omega_{1}-\phi_{B_{d}}-\phi_{D})}\sinh{\frac{1}{2}\Delta\Gamma_{D}t_{2}}\right.$
$\displaystyle\left.+\left(\left|\frac{q_{D}}{p_{D}}\right|-\left|\frac{p_{D}}{q_{D}}\right|\right)\cos{(\omega_{1}-\phi_{B_{d}}-\phi_{D})}\sin{\Delta
m_{D}t_{2}}\right]\sin{\Delta m_{B_{d}}t_{1}}\;,$ (42)

where the contributions below $\mathcal{O}(10^{-3})$ have been neglected. It
turns out that the corrections from the doubly Cabibbo-suppressed $D^{0}\to
K^{+}\pi^{-}$ decay are negligible, so practically only the interference term
between $B^{0}_{d}\to\bar{D}^{0}\to D^{0}$ and $B^{0}_{d}\to\bar{B}^{0}_{d}\to
D^{0}$ is retained. With also the contributions below $\mathcal{O}(10^{-3})$
neglected, the non-double-mixing CP asymmetry reads

$\displaystyle A_{non-dm,1}(t_{1},t_{2})=$ $\displaystyle\
r_{B}\Big{\\{}\pqty{\absolutevalue{\frac{p_{D}}{q_{D}}}\cos{\pqty{\delta_{B}+\omega_{2}-\phi_{D}}}-\absolutevalue{\frac{q_{D}}{p_{D}}}\cos{\pqty{\delta_{B}-\omega_{2}+\phi_{D}}}}\sinh{\frac{1}{2}\Delta\Gamma_{D}t_{2}}$
$\displaystyle+\pqty{\absolutevalue{\frac{p_{D}}{q_{D}}}\sin{\pqty{\delta_{B}+\omega_{2}-\phi_{D}}}-\absolutevalue{\frac{q_{D}}{p_{D}}}\sin{\pqty{\delta_{B}-\omega_{2}+\phi_{D}}}}\sin{\Delta
m_{D}t_{2}}$
$\displaystyle+2r_{D}\sin{\pqty{\delta_{B}-\delta_{D}}}\sin{\omega_{2}}\pqty{\cosh{\frac{1}{2}\Delta\Gamma_{D}t_{2}}+\cos{\Delta
m_{D}t_{2}}}\Big{\\}}f(t_{1},t_{2}),$ $\displaystyle A_{non-
dm,2}(t_{1},t_{2})=$
$\displaystyle-2\frac{e^{-\Gamma_{B_{d}}t_{1}}}{D(t_{1},t_{2})}\Big{\\{}r_{B}\cos{\delta_{B}}\sin{(\omega_{1}-\omega_{2}-\phi_{B_{d}})}+r_{D}\cos{\delta_{D}}\sin{\pqty{\omega_{1}-\phi_{B_{d}}}}\Big{\\}}$
$\displaystyle\times\absolutevalue{g_{+,D}(t_{2})}^{2}\sin{\Delta
m_{B_{d}}t_{1}},$ (43)

where $f(t_{1},t_{2})\equiv
e^{-\Gamma_{D}t_{2}}\absolutevalue{g_{+,B_{d}}(t_{1})}^{2}/4D(t_{1},t_{2})$.
The denominator $D(t_{1},t_{2})$ can be derived by combining (4) with (69) -
(74), while maintaining consistency with all the approximations used in the
numerator. The parameters $r_{B}$ and $\delta_{B}$ have been defined in (35).
The non-double-mixing CP asymmetry contains $r_{D}$ in the expression (3.1.5),
indicating that the doubly Cabibbo-suppressed $D^{0}\to K^{+}\pi^{-}$
contribution is considerable here.

$t_{1}/\tau_{B_{d}}$ | $t_{2}/\tau_{D}$ | $A_{\rm CP}$
---|---|---
0$\sim$3 | 0$\sim$ 5 | -60.71%
0$\sim$2 | 0$\sim$ 5 | -69.20%
0$\sim$3 | 0$\sim$ 4 | -60.71%
0$\sim$2 | 0$\sim$ 4 | -69.20%
Table 5: Two-dimensional time integration of $A_{\rm CP}(t_{1},t_{2})$ in
$B^{0}_{d}(t_{1})\to D(t_{2})K^{0}_{S}\to(K^{-}\pi^{+})K^{0}_{S}$.

Figure 9: Time dependence of the CP asymmetry $A_{\rm CP}$ in
$B^{0}_{d}(t_{1})\to D(t_{2})K^{0}_{S}\to(K^{-}\pi^{+})K^{0}_{S}$. The left
panel displays the two-dimensional time dependence. The middle panel and the
right panel display the dependence on $t_{1}$ (with $t_{2}$ integrated from 0
to $5\tau_{D}$) and $t_{2}$ (with $t_{1}$ integrated from $0$ to
$3\tau_{B_{d}}$), respectively.

We perform numerical analysis on this channel. As depicted in the left panel
of Figure 9, the peak of the two-dimensional time-dependent double-mixing
results is on the order of $10^{-2}$, occurring at later $t_{2}$ and earlier
$t_{1}$ times. When integrating $t_{2}$ from 0 to $5\tau_{D}$, the peak value
of the double-mixing effect contributes only 0.8% to the total. The
dependencies of $A_{\rm CP}$ and $A_{dm}$ on $t_{1}$ are similar, differing
only in amplitude magnitude. Upon integrating $t_{1}$ from 0 to
$3\tau_{B_{d}}$, the $t_{2}$ dependence of $A_{\rm CP}$ is shown in the right
panel. The dependence of $A_{dm}$ on $t_{2}$ is nearly linear, attributed to
its nearly hyperbolic sine dependence on $t_{2}$. However, due to the small
value of $y_{D}$, the overall variation of $A_{dm}$ with $t_{2}$ remains
nearly linear. The double-mixing CP violation in the decay channel of this
meson is relatively more significant than in the $\pi^{+}\pi^{-}$ final state,
due to the presence of the $r_{D}$ factor which suppresses the dominant non-
double-mixing CP violation terms. The dominant contribution to the non-double-
mixing CP violation arises from the interference between the
$B^{0}_{d}\to\bar{D}^{0}$ and $B^{0}_{d}\to D^{0}$ path, i.e., the third line
of $A_{non-dm,1}(t_{1},t_{2})$. A slightly smaller contribution comes from the
interference between the $B^{0}_{d}\to D^{0}$ and
$B^{0}_{d}\to\bar{B}^{0}_{d}\to D^{0}$ paths, i.e., the first term of $A_{non-
dm,2}(t_{1},t_{2})$ which involves the factor $r_{B}$. The two-dimensional
time integration of $A_{\rm CP}(t_{1},t_{2})$ for this channel, presented in
Table 5, suggests that the CP asymmetry $A_{\rm CP}$ is relatively insensitive
to the decay of the $D$ meson.

###  3.2 $B_{s}^{0}$ decays

In this subsection, we extend our investigation to the $B^{0}_{s}$ meson, by
considering decays with the same final states as the $B_{d}^{0}$ decays. For
simplicity, we retain the approximation:

$q_{B_{s}}/p_{B_{s}}=e^{-i\phi_{B_{s}}}.$ (44)

Unlike the $B^{0}_{d}$ meson, the $B_{s}^{0}$ meson exhibits a non-negligible
difference in decay widths, as indicated in Table 1. Consequently, the double-
mixing CP violation for the $B_{s}^{0}$ meson comprises both a component
proportional to $\sin{\Delta m_{B_{s}}t_{1}}$, represented by $A_{n,dm}$, and
a component proportional to $\sinh{\frac{1}{2}\Delta\Gamma_{B_{s}}t_{1}}$,
represented by $A_{h,dm}$. Although the underlying calculation process is
similar to that for the $B^{0}_{d}$ meson, the results for the $B^{0}_{s}$
meson are inherently more complicated, implying more fruitful phenomenology.

####  3.2.1 $B^{0}_{s}\to J/\psi K\to J/\psi f_{+}$

We first consider the process $B^{0}_{s}\to J/\psi K\to J/\psi f_{+}$, which
originates from the $\bar{b}\to\bar{c}c\bar{d}$ quark-level transition. The
first observation of such a mode $B_{s}^{0}\to J/\psi K_{S}^{0}$ was reported
by CDF along with the measured branching ratio CDF:2011fhd . In this channel,
both the transitions $B^{0}_{s}\to J/\psi\bar{K}^{0}$ and
$B^{0}_{s}\to\bar{B}^{0}_{s}\to J/\psi K^{0}$ can occur, with final state
$f_{+}$ selected as $\pi^{+}\pi^{-}$ being common to both $K^{0}$ and
$\bar{K}^{0}$, resulting in four possible paths. This makes these channels
fall into the second category of our previously defined classifications.

In the calculation of the CP violation of this channel, direct CP violation in
the primary and secondary decays are neglected. The involved phase difference
between the decay amplitude $\innerproduct{J/\psi K^{0}}{\bar{B}^{0}_{s}}$ and
its charge conjugation $\innerproduct{J/\psi\bar{K}^{0}}{B^{0}_{s}}$ is given
by
$-e^{i\omega_{B_{s}}}=-\pqty{V_{cb}V^{\star}_{cd}}/\pqty{V^{\star}_{cb}V_{cd}}$,
with its numerical value detailed in Table 1. The negative sign arises from
the convention $CP\ket{J/\psi K^{0}}=-\ket{J/\psi\bar{K}^{0}}$. The double-
mixing CP asymmetry is given by

$\begin{split}A_{n,dm}(t_{1},t_{2})=&\
2\frac{e^{-\pqty{\Gamma_{B_{s}}t_{1}+\Gamma_{K}t_{2}}}}{D(t_{1},t_{2})}\Big{\\{}\sinh{\frac{1}{2}\Delta\Gamma_{K}t_{2}}\sin{\Phi_{4}}\pqty{\absolutevalue{\frac{q_{K}}{p_{K}}}+\absolutevalue{\frac{p_{K}}{q_{K}}}}\\\
&+\sin{\Delta
m_{K}t_{2}}\cos{\Phi_{4}}\pqty{\absolutevalue{\frac{q_{K}}{p_{K}}}-\absolutevalue{\frac{p_{K}}{q_{K}}}}\Big{\\}}\sin{\Delta
m_{B_{s}}t_{1}},\end{split}$ (45)

where $\Phi_{4}\equiv\omega_{B_{s}}-\phi_{B_{s}}-\phi_{K}$. It is induced by
the interference between $B^{0}_{s}\to\bar{K}^{0}\to\pi^{+}\pi^{-}$ and
$B^{0}_{s}\to\bar{B}^{0}_{s}\to K^{0}\to\bar{K}^{0}\to\pi^{+}\pi^{-}$, as well
as between the $B^{0}_{s}\to\bar{K}^{0}\to K^{0}\to\pi^{+}\pi^{-}$ and
$B^{0}_{s}\to\bar{B}^{0}_{s}\to K^{0}\to\pi^{+}\pi^{-}$ pathways. The
denominator $D(t_{1},t_{2})$ is obtained from (22) - (25), where the
substitutions
$\absolutevalue{p_{K}/q_{K}}\leftrightarrow\absolutevalue{q_{K}/p_{K}}$ and
$\phi_{K}\to-\phi_{K}$ need to be performed.

We have conducted numerical analyses of this decay channel, obtaining the CP
asymmetry and its time dependence, as illustrated in Figure 10. The left panel
shows that the maximum value of the two-dimensional time-dependent CP
asymmetry can exceed 50%, a substantial value. Integrating the time $t_{2}$
from 0 to $2\tau_{K}$, the middle panel shows the $t_{1}$ dependence of
$A_{\rm CP}$. The rapid oscillation observed is due to the large $x_{s}$ value
of the $B_{s}^{0}$ meson. The total CP asymmetry is dominated by the $S_{n}$
term, which can be further separated into the double-mixing part $A_{n,dm}$
and the non-double-mixing part, both of which have the sine dependence on
$t_{1}$, as defined in (4). The double-mixing part has a magnitude
approximately twice the total CP asymmetry but with an opposite sign.
Integrating $t_{1}$ from 0 to $\tau_{B_{s}}$ reveals the $t_{2}$ dependence of
$A_{\rm CP}$, as shown in the right panel. In this case, $A_{\rm CP}$ is
almost zero because the cancellation between the double-mixing and non-double-
mixing terms in $S_{n}$, while $A_{n,dm}$ varies approximately as
$-\exp(y_{K}t_{2})\sinh{y_{K}t_{2}}$ with respect to $t_{2}$.

Figure 10: Time dependence of the CP asymmetry $A_{\rm CP}$ in $B^{0}_{s}\to
J/\psi K\to J/\psi f_{+}$. The left panel displays the two-dimensional time
dependence. The middle panel and the right panel display the dependence on
$t_{1}$ (with $t_{2}$ integrated from 0 to $2\tau_{K}$) and $t_{2}$ (with
$t_{1}$ integrated from $0$ to $\tau_{B_{s}}$), respectively.

####  3.2.2 $B^{0}_{s}\to J/\psi K\to J/\psi(\pi^{\pm}\ell^{\mp}\nu_{\ell})$

The decay process $B^{0}_{s}\to J/\psi K\to J/\psi(\pi^{-}\ell^{+}\nu_{\ell})$
can occur via the two paths $B^{0}_{s}\to J/\psi\bar{K}^{0}\to J/\psi K^{0}\to
J/\psi(\pi^{-}\ell^{+}\nu_{\ell})$ and $B^{0}_{s}\to\bar{B}^{0}_{s}\to J/\psi
K^{0}\to J/\psi(\pi^{-}\ell^{+}\nu_{\ell})$; while; the process $B^{0}_{s}\to
J/\psi K\to J/\psi(\pi^{+}\ell^{-}\bar{\nu}_{\ell})$ can occur via
$B^{0}_{s}\to J/\psi\bar{K}^{0}\to J/\psi(\pi^{+}\ell^{-}\bar{\nu}_{\ell})$
and $B^{0}_{s}\to\bar{B}^{0}_{s}\to J/\psi K^{0}\to J/\psi\bar{K}^{0}\to
J/\psi(\pi^{+}\ell^{-}\bar{\nu}_{\ell})$. Therefore, they both align with the
first category in our prior discussion.

In the calculation of the CP asymmetry of these channels, we still neglect the
direct CP violation in both the primary and secondary decays. Then, we have
$\innerproduct{f}{K^{0}}=\innerproduct{\bar{f}}{\bar{K}^{0}}$ and the weak
phase difference between the involved primary decay amplitudes is
$\omega_{B_{s}}$ as introduced in previously. For the decay channel with the
final state $\pi^{-}\ell^{+}\nu_{\ell}$, the corresponding double-mixing terms
are given by

$\displaystyle A_{h,dm}(t_{1},t_{2})=$
$\displaystyle-\frac{e^{-\pqty{\Gamma_{B_{s}}t_{1}+\Gamma_{K}t_{2}}}}{2D(t_{1},t_{2})}\Big{\\{}\sinh{\frac{1}{2}\Delta\Gamma_{K}t_{2}}\cos{\Phi_{4}}\pqty{\absolutevalue{\frac{p_{K}}{q_{K}}}-\absolutevalue{\frac{q_{K}}{p_{K}}}}$
$\displaystyle+\sin{\Delta
m_{K}t_{2}}\sin{\Phi_{4}}\pqty{\absolutevalue{\frac{p_{K}}{q_{K}}}+\absolutevalue{\frac{q_{K}}{p_{K}}}}\Big{\\}}\sinh{\frac{1}{2}\Delta\Gamma_{B_{s}}t_{1}},$
(46) $\displaystyle A_{n,dm}(t_{1},t_{2})=$
$\displaystyle-\frac{e^{-\pqty{\Gamma_{B_{s}}t_{1}+\Gamma_{K}t_{2}}}}{2D(t_{1},t_{2})}\Big{\\{}-\sinh{\frac{1}{2}\Delta\Gamma_{K}t_{2}}\pqty{\absolutevalue{\frac{p_{K}}{q_{K}}}+\absolutevalue{\frac{q_{K}}{p_{K}}}}\sin{\Phi_{4}}$
$\displaystyle+\sin{\Delta
m_{K}t_{2}}\cos{\Phi_{4}}\pqty{\absolutevalue{\frac{p_{K}}{q_{K}}}-\absolutevalue{\frac{q_{K}}{p_{K}}}}\Big{\\}}\sin{\Delta
m_{B_{s}}t_{1}}.$ (47)

Combining (4) with (11) - (13) and making the substitutions
$\absolutevalue{p_{K}/q_{K}}\leftrightarrow\absolutevalue{q_{K}/p_{K}}$ and
$\phi_{K}\to-\phi_{K}$ yields the denominator $D(t_{1},t_{2})$.

Figure 11: Time dependence of the CP asymmetry $A_{\rm CP}$ in $B^{0}_{s}\to
J/\psi K\to J/\psi(\pi^{-}\ell^{+}\nu_{\ell})$. The left panel displays the
two-dimensional time dependence. The middle panel and the right panel display
the dependence on $t_{1}$ (with $t_{2}$ integrated from 0 to $\tau_{K}$) and
$t_{2}$ (with $t_{1}$ integrated from $0$ to $2\tau_{B_{s}}$), respectively.

The numerical results for the $\pi^{-}\ell^{+}\nu_{\ell}$ channel are
illustrated in Figure 11. The left panel shows the two-dimensional time
dependence of the CP asymmetry, whose maximal values are of
$\mathcal{O}(10\%)$. Integrating time $t_{2}$ from 0 to $\tau_{K}$, it is
observed from the middle panel that the total asymmetry is dominated by the
$A_{n,dm}$ term. This is attributable to the sine dependence on $t_{1}$ and
hyperbolic sine dependence on $t_{2}$ of $A_{n,dm}$, particularly for the
large $x_{s}$. Conversely, the numerator of $A_{h,dm}$ exhibits a hyperbolic
sine dependence on $t_{1}$ and a sine dependence on $t_{2}$. In the range of
$t_{1}$ from 0 to $2\tau_{B_{s}}$, the numerator of $A_{h,dm}$ increases
almost linearly while the denominator behaves like a sine function, so
$A_{h,dm}$ experiences modulation over time, as reflected in its increasing
peak value. Integrating over $t_{1}$ from 0 to $2\tau_{B_{s}}$, we observe the
$t_{2}$ dependence of $A_{\rm CP}$ in the right panel of Figure 11. Now, the
contributions of the $C$ term and $A_{n,dm}$ term are comparable, slightly
surpassing the $A_{h,dm}$ term.

When the final state is considered as $\pi^{+}\ell^{-}\bar{\nu}_{\ell}$, the
results for the double-mixing terms are similar, but with an extra negative
sign in the term $A_{h,dm}$. The corresponding expressions are

$\displaystyle A_{h,dm}(t_{1},t_{2})=$
$\displaystyle-\frac{e^{-\pqty{\Gamma_{B_{s}}t_{1}+\Gamma_{K}t_{2}}}}{2D(t_{1},t_{2})}\Big{\\{}\sinh{\frac{1}{2}\Delta\Gamma_{K}t_{2}\bqty{\cos{\Phi_{4}}\pqty{\absolutevalue{\frac{q_{K}}{p_{K}}}-\absolutevalue{\frac{p_{K}}{q_{K}}}}}}$
$\displaystyle-\sin{\Delta
m_{K}t_{2}}\bqty{\sin{\Phi_{4}}\pqty{\absolutevalue{\frac{q_{K}}{p_{K}}}+\absolutevalue{\frac{p_{K}}{q_{K}}}}}\Big{\\}}\sinh{\frac{1}{2}\Delta\Gamma_{B_{s}}t_{1}},$
(48) $\displaystyle A_{n,dm}(t_{1},t_{2})=$
$\displaystyle-\frac{e^{-\pqty{\Gamma_{B_{s}}t_{1}+\Gamma_{K}t_{2}}}}{2D(t_{1},t_{2})}\Big{\\{}-\sinh{\frac{1}{2}\Delta\Gamma_{K}t_{2}\bqty{\sin{\Phi_{4}}\pqty{\absolutevalue{\frac{q_{K}}{p_{K}}}+\absolutevalue{\frac{p_{K}}{q_{K}}}}}}$
$\displaystyle-\sin{\Delta
m_{K}t_{2}}\bqty{\cos{\Phi_{4}}\pqty{\absolutevalue{\frac{q_{K}}{p_{K}}}-\absolutevalue{\frac{p_{K}}{q_{K}}}}}\Big{\\}}\sin{\Delta
m_{B_{s}}t_{1}}.$ (49)

The denominator $D(t_{1},t_{2})$ can be obtained by combining (4) with (11) -
(13), while also performing the substitutions
$C^{\prime}_{+}(t_{2})\leftrightarrow C^{\prime}_{-}(t_{2})$,
$\absolutevalue{p_{K}/q_{K}}\leftrightarrow\absolutevalue{q_{K}/p_{K}}$ and
$\phi_{K}\to-\phi_{K}$, and adding an extra negative sign in front of
$S^{\prime}_{n}(t_{2})$.

Figure 12: Time dependence of the CP asymmetry $A_{\rm CP}$ in $B^{0}_{s}\to
J/\psi K\to J/\psi(\pi^{+}\ell^{-}\bar{\nu}_{\ell})$. The left panel displays
the two-dimensional time dependence. The middle panel and the right panel
display the dependence on $t_{1}$ (with $t_{2}$ integrated from 0 to
$\tau_{K}$) and $t_{2}$ (with $t_{1}$ integrated from $0$ to $2\tau_{B_{s}}$),
respectively.

Our numerical analysis for this channel is depicted in Figure 12. The left
panel presents the two-dimensional time dependence. The difference between the
results for this channel and the $\pi^{-}\ell^{+}\nu_{\ell}$ channel lies in
the modulation of the denominator. When $t_{2}$ is fixed, the dominant term in
the denominator has a hyperbolic cosine dependence on $t_{1}$, which is the
same in the results for the two different channels, while the sine term is a
secondary contribution, and it contributes to the two channels with a sign
difference. The middle panel shows the CP asymmetry with $t_{2}$ integrated
from 0 to $\tau_{K}$. While the $A_{n,dm}$ term still dominates, the growth
magnitude of the $A_{h,dm}$ term is larger compared to the
$\pi^{-}\ell^{+}\nu_{\ell}$ channel. Integrating over $t_{1}$ and analyzing
$t_{2}$ dependence in the right panel, the $A_{h,dm}$ term emerges as
dominant. Additionally, both the $A_{\rm CP}$ and $A_{n,dm}$ terms in the
$\pi^{+}\ell^{-}\bar{\nu}_{\ell}$ channel display a sign difference from the
$\pi^{-}\ell^{+}\nu_{\ell}$ channel.

####  3.2.3 $B^{0}_{s}\to\rho^{0}K\to\rho^{0}(\pi^{\pm}\ell^{\mp}\nu)$

The decay processes $B^{0}_{s}\to\rho^{0}K\to\rho^{0}(\pi^{\pm}\ell^{\mp}\nu)$
resemble to $B^{0}_{s}\to J/\psi K^{0}_{L,S}(\to\pi^{-}\ell^{+}\nu_{\ell})$,
differing primarily in the phase difference $\omega_{5}$. This difference
arises from the phase between the decay amplitude
$\innerproduct{\rho^{0}\bar{K}^{0}}{B^{0}_{s}}$ and its charge conjugate
$\innerproduct{\rho^{0}K^{0}}{\bar{B}^{0}_{s}}$, contrasting with the phase
difference $\omega_{B_{s}}$ in the former. Considering
$CP\ket{\rho^{0}K^{0}}=-\ket{\rho^{0}\bar{K}^{0}}$, we define

$\displaystyle\frac{\innerproduct{\rho^{0}K^{0}}{\bar{B}^{0}_{s}}}{\innerproduct{\rho^{0}\bar{K}^{0}}{B^{0}_{s}}}=-e^{i\omega_{5}},\,\omega_{5}=\arg{\frac{V_{ub}V^{\star}_{ud}}{V^{\star}_{ub}V_{ud}}},$
(50)

where again we have neglected the direct CP asymmetry in these processes.

For the final state $\pi^{-}\ell^{+}\nu_{\ell}$, two possible paths exist:
$B^{0}_{s}\to\rho^{0}\bar{K}^{0}\to\rho^{0}K^{0}\to\rho^{0}(\pi^{-}\ell^{+}\nu_{\ell})$
and
$B^{0}_{s}\to\bar{B}^{0}_{s}\to\rho^{0}K^{0}\to\rho^{0}(\pi^{-}\ell^{+}\nu_{\ell})$.
Therefore, it is categorized as the first type in our classification. The
results for the double-mixing terms are

$\displaystyle A_{h,dm}(t_{1},t_{2})=$
$\displaystyle-\frac{e^{-\pqty{\Gamma_{B_{s}}t_{1}+\Gamma_{K}t_{2}}}}{2D(t_{1},t_{2})}\Big{\\{}\pqty{\absolutevalue{\frac{p_{K}}{q_{K}}}-\absolutevalue{\frac{q_{K}}{p_{K}}}}\cos{\Phi_{5}}\sinh{\frac{1}{2}\Delta\Gamma_{K}t_{2}}$
$\displaystyle-\pqty{\absolutevalue{\frac{p_{K}}{q_{K}}}+\absolutevalue{\frac{q_{K}}{p_{K}}}}\sin{\Phi_{5}}\sin{\Delta
m_{K}t_{2}}\Big{\\}}\sinh{\frac{1}{2}\Delta\Gamma_{B_{s}}t_{1}},$ (51)
$\displaystyle A_{n,dm}(t_{1},t_{2})=$
$\displaystyle-\frac{e^{-\pqty{\Gamma_{B_{s}}t_{1}+\Gamma_{K}t_{2}}}}{2D(t_{1},t_{2})}\Big{\\{}\pqty{\absolutevalue{\frac{p_{K}}{q_{K}}}+\absolutevalue{\frac{q_{K}}{p_{K}}}}\sin{\Phi_{5}}\sinh{\frac{1}{2}\Delta\Gamma_{K}t_{2}}$
$\displaystyle+\pqty{\absolutevalue{\frac{p_{K}}{q_{K}}}-\absolutevalue{\frac{q_{K}}{p_{K}}}}\cos{\Phi_{5}}\sin{\Delta
m_{K}t_{2}}\Big{\\}}\sin{\Delta m_{B_{s}}t_{1}},$ (52)

where $\Phi_{5}\equiv\phi_{B_{s}}+\phi_{K}-\omega_{5}$. $D(t_{1},t_{2})$ can
be obtained by replacing $\omega_{B_{s}}$ in the denominator of (46) with
$\omega_{5}$.

Figure 13: Time dependence of the CP asymmetry $A_{\rm CP}$ in
$B^{0}_{s}\to\rho^{0}K\to\rho^{0}(\pi^{-}\ell^{+}\nu_{\ell})$. The left panel
displays the two-dimensional time dependence. The middle panel and the right
panel display the dependence on $t_{1}$ (with $t_{2}$ integrated from 0 to
$\tau_{K}$) and $t_{2}$ (with $t_{1}$ integrated from $\tau_{B_{s}}$ to
$5\tau_{B_{s}}$), respectively.

We have conducted the numerical analysis of both the total CP asymmetry
$A_{\rm CP}$ and the double-mixing CP asymmetry $A_{dm}$, which are depicted
in Figure 13. The two-dimensional time-dependent CP violation is depicted in
the left panel, showing a peak value exceeding $50\%$. By integrating $t_{2}$
from 0 to $\tau_{K}$, we obtain the $t_{1}$-dependent results displayed in the
middle panel, where the double-mixing CP violation is notably significant.
Here, $A_{\rm CP}$ predominantly arises from the double-mixing CP violation,
particularly from the $A_{n,dm}$ term, which sums the contributions of the
decays of $K_{S}^{0}$ and $K_{L}^{0}$. Integrating $t_{1}$ from $\tau_{B_{s}}$
to $5\tau_{B_{s}}$ yields the $t_{2}$-dependent results, showcased in the
right panel. In this scenario, $A_{\rm CP}$ still mainly originates from the
double-mixing CP violation, with the term $A_{h,dm}$ being dominant,
particularly governed by the $K_{S}^{0}-K_{L}^{0}$ interference.

The results for the double-mixing terms of the
$\pi^{+}\ell^{-}\bar{\nu}_{\ell}$ channel are similar to those of the
$\pi^{-}\ell^{+}{\nu}_{\ell}$ channel with an extra overall negative sign in
the term $A_{h,dm}$. They are calculated as

$\displaystyle A_{h,dm}(t_{1},t_{2})=$
$\displaystyle-\frac{e^{-\pqty{\Gamma_{B_{s}}t_{1}+\Gamma_{K}t_{2}}}}{2D(t_{1},t_{2})}\Big{\\{}\sinh{\frac{1}{2}\Delta\Gamma_{K}t_{2}\bqty{\cos{\Phi_{5}}\pqty{\absolutevalue{\frac{q_{K}}{p_{K}}}-\absolutevalue{\frac{p_{K}}{q_{K}}}}}}$
$\displaystyle+\sin{\Delta
m_{K}t_{2}}\bqty{\sin{\Phi_{5}}\pqty{\absolutevalue{\frac{q_{K}}{p_{K}}}+\absolutevalue{\frac{p_{K}}{q_{K}}}}}\Big{\\}}\sinh{\frac{1}{2}\Delta\Gamma_{B_{s}}t_{1}},$
(53) $\displaystyle A_{n,dm}(t_{1},t_{2})=$
$\displaystyle-\frac{e^{-\pqty{\Gamma_{B_{s}}t_{1}+\Gamma_{K}t_{2}}}}{2D(t_{1},t_{2})}\Big{\\{}\sinh{\frac{1}{2}\Delta\Gamma_{K}t_{2}\bqty{\sin{\Phi_{5}}\pqty{\absolutevalue{\frac{q_{K}}{p_{K}}}+\absolutevalue{\frac{p_{K}}{q_{K}}}}}}$
$\displaystyle-\sin{\Delta
m_{K}t_{2}}\bqty{\cos{\Phi_{5}}\pqty{\absolutevalue{\frac{q_{K}}{p_{K}}}-\absolutevalue{\frac{p_{K}}{q_{K}}}}}\Big{\\}}\sin{\Delta
m_{B_{s}}t_{1}},$ (54)

where the denominator $D(t_{1},t_{2})$ can be obtained from the denominator of
(48) by replacing $\omega_{B_{s}}$ with $\omega_{5}$.

Figure 14: Time dependence of the CP asymmetry $A_{\rm CP}$ in
$B^{0}_{s}\to\rho^{0}K\to\rho^{0}(\pi^{+}\ell^{-}\bar{\nu}_{\ell})$. The left
panel displays the two-dimensional time dependence. The middle panel and the
right panel display the dependence on $t_{1}$ (with $t_{2}$ integrated from 0
to $\tau_{K}$) and $t_{2}$ (with $t_{1}$ integrated from $\tau_{B_{s}}$ to
$5\tau_{B_{s}}$), respectively.

Our analysis for this channel reveals a smaller CP violation compared to the
$\pi^{-}\ell^{+}\nu_{\ell}$ final state, as displayed in Figure 14. The left
panel displays the two-dimensional time dependence of the CP asymmetry.
Integrating $t_{2}$ from 0 to $\tau_{K}$, the $t_{1}$-dependent results,
depicted in the middle panel, highlight the dominant role of separate
contributions from $K_{S}^{0}$ and $K_{L}^{0}$ in the double-mixing CP
violation. When integrating $t_{1}$ from $\tau_{B_{s}}$ to $5\tau_{B_{s}}$,
the $t_{2}$-dependent results, as shown in the right panel, are predominantly
influenced by the $K_{S}^{0}-K_{L}^{0}$ interference. Notably, the value of
the $t_{1}$-integrated CP violation is almost entirely positive, contrasting
with the result for the $\pi^{-}\ell^{+}\nu_{\ell}$ final state. This
distinction enables the experimental differentiation of these two processes.

We emphasize that the best approach for the first experimental attempt to
measure the double-mixing CP violation would be to combine the two decay
channels mentioned above. Considering the CP violation defined by the
differences of the summed branching ratios

$\displaystyle{\mathcal{B}[B^{0}_{s}/\bar{B}^{0}_{s}\to\rho^{0}K\to\rho^{0}(\pi^{-}\ell^{+}\nu_{\ell})]-\mathcal{B}[B^{0}_{s}/\bar{B}^{0}_{s}\to\rho^{0}K\to\rho^{0}(\pi^{+}\ell^{-}\bar{\nu}_{\ell})]\over\mathcal{B}[B^{0}_{s}/\bar{B}^{0}_{s}\to\rho^{0}K\to\rho^{0}(\pi^{-}\ell^{+}\nu_{\ell})]+\mathcal{B}[B^{0}_{s}/\bar{B}^{0}_{s}\to\rho^{0}K\to\rho^{0}(\pi^{+}\ell^{-}\bar{\nu}_{\ell})]}\;,$
(55)

initial tagging for $B$ mesons in experiments is not necessary and the
corresponding efficiency loss is prevented. It contains the double-mixing and
non-double-mixing terms, which are calculated to be

$\displaystyle A_{h,dm}(t_{1},t_{2})=$
$\displaystyle-\frac{e^{-\pqty{\Gamma_{B_{s}}t_{1}+\Gamma_{K}t_{2}}}}{D(t_{1},t_{2})}\Big{\\{}\sinh{\frac{1}{2}\Delta\Gamma_{K}t_{2}}\cos{\Phi_{5}}\pqty{\absolutevalue{\frac{p_{K}}{q_{K}}}-\absolutevalue{\frac{q_{K}}{p_{K}}}}$
$\displaystyle-\sin{\Delta
m_{K}t_{2}}\sin{\Phi_{5}}\pqty{\absolutevalue{\frac{p_{K}}{q_{K}}}+\absolutevalue{\frac{q_{K}}{p_{K}}}}\Big{\\}}\sinh{\frac{1}{2}\Delta\Gamma_{B_{s}}t_{1}},$
(56) $\displaystyle A_{non-dm}(t_{1},t_{2})=$ $\displaystyle\
\frac{e^{-\Gamma_{B_{s}}t_{1}}}{D(t_{1},t_{2})}\Big{(}\absolutevalue{\frac{p_{K}}{q_{K}}}^{2}-\absolutevalue{\frac{q_{K}}{p_{K}}}^{2}\Big{)}\absolutevalue{g_{-,K}(t_{2})}^{2}\cosh{\frac{1}{2}\Delta\Gamma_{B_{s}}t_{1}},$
(57) $\displaystyle D(t_{1},t_{2})=$ $\displaystyle\
e^{-\Gamma_{B_{s}}t_{1}}\Big{\\{}\Big{(}\absolutevalue{\frac{p_{K}}{q_{K}}}^{2}+\absolutevalue{\frac{q_{K}}{p_{K}}}^{2}\Big{)}\absolutevalue{g_{-,K}(t_{2})}^{2}+2\absolutevalue{g_{+,K}(t_{2})}^{2}\Big{\\}}\cosh{\frac{1}{2}\Delta\Gamma_{B_{s}}t_{1}}$
$\displaystyle-e^{-\pqty{\Gamma_{B_{s}}t_{1}+\Gamma_{K}t_{2}}}\Big{\\{}\sinh{\frac{1}{2}\Delta\Gamma_{K}t_{2}}\cos{\Phi_{5}}\pqty{\absolutevalue{\frac{p_{K}}{q_{K}}}+\absolutevalue{\frac{q_{K}}{p_{K}}}}$
$\displaystyle-\sin{\Delta
m_{K}t_{2}}\sin{\Phi_{5}}\pqty{\absolutevalue{\frac{p_{K}}{q_{K}}}-\absolutevalue{\frac{q_{K}}{p_{K}}}}\Big{\\}}\sinh{\frac{1}{2}\Delta\Gamma_{B_{s}}t_{1}}.$
(58)

Apparently, the non-double-mixing term is negligible and the total CP
violation is dominated by the double-mixing contribution.

Figure 15: Time dependence of the CP asymmetry $A_{\rm CP}$ in $B^{0}_{s}(t_{1})\to\rho^{0}K(t_{2})\to\rho^{0}(\pi\ell\nu)$ defined by (55). The left panel displays the two-dimensional time dependence. The middle panel and the right panel display the dependence on $t_{1}$ (with $t_{2}$ integrated from 0 to $\tau_{K}$) and $t_{2}$ (with $t_{1}$ integrated from $\tau_{B_{s}}$ to $5\tau_{B_{s}}$), respectively. $t_{1}/\tau_{B_{s}}$ | $t_{2}/\tau_{K}$ | $A_{\rm CP}$
---|---|---
0$\sim$3 | 0$\sim$ 2 | -1.33%
0$\sim$2 | 0$\sim$ 2 | -1.06%
0$\sim$3 | 0$\sim$ 1 | -1.28%
0$\sim$2 | 0$\sim$ 1 | -1.03%
0$\sim$3 | 0$\sim$ 0.5 | -0.83%
0$\sim$2 | 0$\sim$ 0.5 | -0.67%
Table 6: The time integrated $A_{\rm CP}$ in
$B^{0}_{s}\to\rho^{0}K\to\rho^{0}(\pi\ell\nu_{\ell})$.

We conducted a numerical analysis of this process, and the results are
depicted in Figure 15. The left panel exhibits that the total CP asymmetry has
a hyperbolic tangent dependence on $t_{1}$, consequently behaving linearly
when $t_{1}$ is not too large. When integrating $t_{2}$ from 0 to $\tau_{K}$,
the total CP violation is predominantly contributed from the double-mixing CP
violation, as displayed in the middle panel. It exhibits a linear dependence
on $t_{1}$, which is consistent with the two-dimensional time dependence. The
right panel shows the variation of CP violation with respect to $t_{2}$ after
integrating $t_{1}$ from $\tau_{B_{s}}$ to $5\tau_{B_{s}}$. As $t_{2}$
increases, it first reaches a maximum and then rapidly decreases to nearly
zero. This is because the primary contribution of the double-mixing CP
violation in the numerator comes from the interference term between
$K_{S}^{0}$ and $K_{L}^{0}$, i.e., the second term in (58), which has a sine
dependence on $t_{2}$. Meanwhile, the denominator is mainly contributed by the
term that depends on the hyperbolic cosine function of $t_{2}$, which
continues to increase with time. The time-integrated results for the CP
asymmetry (55) for this case are presented in Table 6, for experimental
convenience. With different integrated time ranges, the values are basically
around -1%.

####  3.2.4 $B^{0}_{s}\to DK\to(K^{-}\pi^{+})(\pi^{-}\ell^{+}\nu_{\ell})$

We now focus on the decay channel $B^{0}_{s}\to
DK\to(K^{-}\pi^{+})(\pi^{-}\ell^{+}\nu_{\ell})$. This process involves four
paths, even when we neglect the small mixing of the neutral $D$ meson:
$B^{0}_{s}\to D^{0}\bar{K}^{0}\to D^{0}K^{0}\to f$,
$B^{0}_{s}\to\bar{D}^{0}\bar{K}^{0}\to\bar{D}^{0}K^{0}\to f$,
$B^{0}_{s}\to\bar{B}^{0}_{s}\to D^{0}K^{0}\to f$ and
$B^{0}_{s}\to\bar{B}^{0}_{s}\to\bar{D}^{0}K^{0}\to f$, where
$f\equiv(K^{-}\pi^{+})(\pi\ell\nu)$. This situation is very unique and does
not belong to any of the categories we previously defined, but the expression
for the CP violation can be derived analogously. We define the ratios between
the decay amplitudes as

$\displaystyle\frac{A\pqty{\bar{B}^{0}_{s}\to
D^{0}K^{0}}}{A\pqty{B^{0}_{s}\to\bar{D}^{0}\bar{K}^{0}}}$
$\displaystyle=e^{i\omega_{3}},\quad\omega_{3}=\arg{\frac{V_{cb}V^{\star}_{ud}}{V^{\star}_{cb}V_{ud}}},$
$\displaystyle\frac{A\pqty{B^{0}_{s}\to
D^{0}\bar{K}^{0}}}{A\pqty{B^{0}_{s}\to\bar{D}^{0}\bar{K}^{0}}}$
$\displaystyle=re^{i\pqty{\delta_{4}+\omega_{4}}},\quad\omega_{4}=\arg{\frac{V_{ub}^{\star}V_{cd}}{V^{\star}_{cb}V_{ud}}},$
(59)

where $\omega_{3,4}$ are the weak phases and $\delta_{4}$ is the strong phase.
Then, the double-mixing and non-double-mixing CP violation terms defined in
(4) are calculated as

$\displaystyle A_{h,dm}(t_{1},t_{2})=$
$\displaystyle\frac{N_{1}}{D(t_{1},t_{2})}$ $\displaystyle=$ $\displaystyle\
\frac{e^{-\pqty{\Gamma_{B_{s}}t_{1}+\Gamma_{K}t_{2}}}}{2D(t_{1},t_{2})}\absolutevalue{\frac{p_{K}}{q_{K}}}\Big{\\{}\sinh{\frac{\Delta\Gamma_{K}t_{2}}{2}}\left[r\cos{\pqty{\delta_{4}+\Phi_{6}}}-r_{D}\cos{\pqty{\Phi_{7}-\delta_{D}}}\right]$
$\displaystyle+\sin{\Delta
m_{K}t_{2}}\left[-r\sin{\pqty{\delta_{4}+\Phi_{6}}}+r_{D}\sin{\pqty{\Phi_{7}-\delta_{D}}}\right]\Big{\\}}\sinh{\frac{\Delta\Gamma_{B_{s}}t_{1}}{2}}$
$\displaystyle-\frac{e^{-\pqty{\Gamma_{B_{s}}t_{1}+\Gamma_{K}t_{2}}}}{2D(t_{1},t_{2})}\absolutevalue{\frac{q_{K}}{p_{K}}}\Big{\\{}\sinh{\frac{\Delta\Gamma_{K}t_{2}}{2}}\left[r\cos{\pqty{\delta_{4}-\Phi_{6}}}-r_{D}\cos{\pqty{\Phi_{7}+\delta_{D}}}\right]$
$\displaystyle+\sin{\Delta
m_{K}t_{2}}\left[-r\sin{\pqty{\delta_{4}-\Phi_{6}}}-r_{D}\sin{\pqty{\Phi_{7}+\delta_{D}}}\right]\Big{\\}}\sinh{\frac{\Delta\Gamma_{B_{s}}t_{1}}{2}}$
$\displaystyle-\frac{e^{-\pqty{\Gamma_{B_{s}}t_{1}+\Gamma_{K}t_{2}}}}{2D(t_{1},t_{2})}\absolutevalue{\frac{p_{K}}{q_{K}}}rr_{D}\Big{\\{}\sinh{\frac{\Delta\Gamma_{K}t_{2}}{2}}\pqty{r\cos{\pqty{\Phi_{8}-\delta_{D}}}-r_{D}\cos{\pqty{\delta_{4}-\Phi_{6}}}}$
$\displaystyle+\sin{\Delta
m_{K}t_{2}}\pqty{r\sin{\pqty{\Phi_{8}-\delta_{D}}}-r_{D}\sin{\pqty{\delta_{4}-\Phi_{6}}}}\Big{\\}}\sinh{\frac{\Delta\Gamma_{B_{s}}t_{1}}{2}}$
$\displaystyle+\frac{e^{-\pqty{\Gamma_{B_{s}}t_{1}+\Gamma_{K}t_{2}}}}{2D(t_{1},t_{2})}rr_{D}\absolutevalue{\frac{q_{K}}{p_{K}}}\sinh{\frac{\Delta\Gamma_{B_{s}}t_{1}}{2}}\times$
$\displaystyle\Big{\\{}r\pqty{\sinh{\frac{\Delta\Gamma_{K}t_{2}}{2}}\cos{\pqty{\Phi_{8}+\delta_{D}}}-\sin{\Delta
m_{K}t_{2}}\sin{\pqty{\Phi_{8}+\delta_{D}}}}$ $\displaystyle-
r_{D}\pqty{\sinh{\frac{\Delta\Gamma_{K}t_{2}}{2}}\cos{\pqty{\delta_{4}+\Phi_{6}}}+\sin{\Delta
m_{K}t_{2}}\sin{\pqty{\delta_{4}+\Phi_{6}}}}\Big{\\}},$ (60) $\displaystyle
A_{n,dm}(t_{1},t_{2})=$ $\displaystyle\frac{N_{2}}{D(t_{1},t_{2})}$
$\displaystyle=$ $\displaystyle\
\frac{e^{-\pqty{\Gamma_{B_{s}}t_{1}+\Gamma_{K}t_{2}}}}{2D(t_{1},t_{2})}\absolutevalue{\frac{p_{K}}{q_{K}}}\Big{\\{}\sinh{\frac{\Delta\Gamma_{K}t_{2}}{2}}\left[r\sin{\pqty{\delta_{4}+\Phi_{6}}}-r_{D}\sin{\pqty{\Phi_{7}-\delta_{D}}}\right]$
$\displaystyle+\sin{\Delta
m_{K}t_{2}}\left[r\cos{\pqty{\delta_{4}+\Phi_{6}}}-r_{D}\cos{\pqty{\Phi_{7}-\delta_{D}}}\right]\Big{\\}}\sin{\Delta
m_{B_{s}}t_{1}}$
$\displaystyle-\frac{e^{-\pqty{\Gamma_{B_{s}}t_{1}+\Gamma_{K}t_{2}}}}{2D(t_{1},t_{2})}\absolutevalue{\frac{q_{K}}{p_{K}}}\Big{\\{}\sinh{\frac{\Delta\Gamma_{K}t_{2}}{2}}\left[r\sin{\pqty{\delta_{4}-\Phi_{6}}}+r_{D}\sin{\pqty{\Phi_{7}+\delta_{D}}}\right]$
$\displaystyle+\sin{\Delta
m_{K}t_{2}}\left[r\cos{\pqty{\delta_{4}-\Phi_{6}}}-r_{D}\cos{\pqty{\Phi_{7}+\delta_{D}}}\right]\Big{\\}}\sin{\Delta
m_{B_{s}}t_{1}}$
$\displaystyle-\frac{e^{-\pqty{\Gamma_{B_{s}}t_{1}+\Gamma_{K}t_{2}}}}{2D(t_{1},t_{2})}\absolutevalue{\frac{p_{K}}{q_{K}}}rr_{D}\Big{\\{}\sinh{\frac{\Delta\Gamma_{K}t_{2}}{2}}\left[-r\sin{\pqty{\Phi_{8}-\delta_{D}}}+r_{D}\sin{\pqty{\delta_{4}-\Phi_{6}}}\right]$
$\displaystyle+\sin{\Delta
m_{K}t_{2}}\left[r\cos{\pqty{\Phi_{8}-\delta_{D}}}-r_{D}\cos{\pqty{\delta_{4}-\Phi_{6}}}\right]\Big{\\}}\sin{\Delta
m_{B_{s}}t_{1}}$
$\displaystyle+\frac{e^{-\pqty{\Gamma_{B_{s}}t_{1}+\Gamma_{K}t_{2}}}}{2D(t_{1},t_{2})}rr_{D}\absolutevalue{\frac{q_{K}}{p_{K}}}\sin{\Delta
m_{B_{s}}t_{1}}\times$
$\displaystyle\Big{\\{}r\pqty{\sinh{\frac{\Delta\Gamma_{K}t_{2}}{2}}\sin{\pqty{\Phi_{8}+\delta_{D}}}+\sin{\Delta
m_{K}t_{2}}\cos{\pqty{\Phi_{8}+\delta_{D}}}}$
$\displaystyle+r_{D}\pqty{\sinh{\frac{\Delta\Gamma_{K}t_{2}}{2}}\sin{\pqty{\delta_{4}+\Phi_{6}}}-\sin{\Delta
m_{K}t_{2}}\cos{\pqty{\delta_{4}+\Phi_{6}}}}\Big{\\}},$ (61) $\displaystyle
A_{non-dm}(t_{1},t_{2})=$ $\displaystyle\frac{N_{3}}{D(t_{1},t_{2})}$
$\displaystyle=$
$\displaystyle\Big{\\{}\absolutevalue{g_{+,{B_{s}}}(t_{1})}^{2}\absolutevalue{g_{-,K}(t_{2})}^{2}\Big{\\{}\absolutevalue{\frac{p_{K}}{q_{K}}}^{2}\left[r^{2}+r^{2}_{D}-2r_{D}r\cos{\pqty{\delta_{4}+\delta_{D}+\omega_{4}}}\right]$
$\displaystyle-\absolutevalue{\frac{q_{K}}{p_{K}}}^{2}\left[r^{2}+r^{2}_{D}-2r_{D}r\cos{\pqty{\delta_{4}+\delta_{D}-\omega_{4}}}\right]\Big{\\}}-2rr_{D}\absolutevalue{g_{+,K}(t_{2})}^{2}\times$
$\displaystyle\left[\cos{\pqty{\delta_{4}-\omega_{4}-\delta_{D}}}-\cos{\pqty{\delta_{4}+\omega_{4}-\delta_{D}}}\right]\absolutevalue{g_{-,{B_{s}}}(t_{1})}^{2}\Big{\\}}\frac{1}{D(t_{1},t_{2})},$
(62)

where $\Phi_{6}\equiv\phi_{B_{s}}+\phi_{K}-\omega_{3}+\omega_{4}$,
$\Phi_{7}\equiv\phi_{B_{s}}+\phi_{K}-\omega_{3}$ and
$\Phi_{8}\equiv-\phi_{B_{s}}-\phi_{K}+\omega_{3}-2\omega_{4}$. The denominator
$D(t_{1},t_{2})$ can be divided into three parts
$N_{1}^{\prime}+N_{2}^{\prime}+N_{3}^{\prime}$, which can be obtained by make
some replacements to $N_{1}$, $N_{2}$, and $N_{3}$ in (60) - (62).
$N_{1}^{\prime}$ and $N_{2}^{\prime}$ can be obtained by changing the sign in
front of the exponential in the fourth line and the third-to-last line of (60)
and (61) to an opposite sign. As for $N_{3}^{\prime}$, in addition to changing
the minus sign in front of $\absolutevalue{q_{K}/p_{K}}^{2}$ in the third line
and the minus sign between the two cosines in the fourth line of (62) to a
positive sign in $N_{3}$, we still need to incorporate an additional term
$2\absolutevalue{g_{-,{B_{s}}}(t_{1})}^{2}\absolutevalue{g_{+,K}(t_{2})}^{2}\pqty{r_{D}^{2}r^{2}+1}$
to the modified $N_{3}$.

Figure 16: Time dependence of the CP asymmetry $A_{\rm CP}$ in $B^{0}_{s}\to
DK\to(K^{-}\pi^{+})(\pi^{-}\ell^{+}\nu_{\ell})$. The left panel displays the
two-dimensional time dependence. The middle panel and the right panel display
the dependence on $t_{1}$ (with $t_{2}$ integrated from 0 to $2\tau_{K}$) and
$t_{2}$ (with $t_{1}$ integrated from $0$ to $\tau_{B_{s}}$), respectively.

The numerical result of the CP asymmetry in this process is shown in Figure
16. The two-dimensional time-dependent CP violation is shown in the left
panel. While its peaks surpass 50%, it is only present in the proximity of
specific moments. The result of integrating $t_{2}$ from 0 to $2\tau_{K}$ is
shown in the middle panel. The contribution of $A_{n,dm}$ dominates except at
the positions of positive peaks, as the values of $A_{h,dm}$ and $A_{non-dm}$
are both positive and form upward pulses at each peak position, while
$A_{n,dm}$ rapidly rises at each peak position and then directly becomes a
negative pulse. Additionally, as the sum of $A_{h,dm}$ and $A_{non-dm}$ is
positive when $A_{n,dm}$ is negative, and their contribution is smaller than
$A_{n,dm}$, the overall CP violation $A_{CP}$ has larger positive peaks and
smaller negative peaks compared to $A_{n,dm}$. The result of integrating
$t_{1}$ from 0 to $\tau_{B_{s}}$ is shown in the right panel. $A_{h,dm}$ is
one order of magnitude smaller than $A_{n,dm}$. The interference between
$B^{0}_{s}\to D^{0}\bar{K}^{0}\to D^{0}K^{0}$ and
$B^{0}_{s}\to\bar{B}^{0}_{s}\to D^{0}K^{0}$ is suppressed by the factor $r$.
Considering the contribution from $\bar{D}^{0}$, the interference between
$B^{0}_{s}\to\bar{D}^{0}\bar{K}^{0}\to\bar{D}^{0}K^{0}$ and
$B^{0}_{s}\to\bar{B}^{0}_{s}\to D^{0}K^{0}$ is suppressed by $r_{D}$, while
the remaining interference contributions are all suppressed by factors of
$rr_{D}$ or even more suppressed.

####  3.2.5 $B^{0}_{s}\to DK^{0}_{S}\to(\pi^{+}\pi^{-})K^{0}_{S}$ and
$B^{0}_{s}\to DK^{0}_{S}\to(K^{-}\pi^{+})K^{0}_{S}$

We now analyze the decay modes $B^{0}_{s}\to DK^{0}_{S}\to f_{D}K^{0}_{S}$,
with $f_{D}=\pi^{+}\pi^{-},K^{-}\pi^{+}$. Measurements of similar channels
have been performed by experiments. The branching ratio of
$B^{0}_{s}\to\bar{D}^{0}\bar{K}^{\star 0}$ has been reported by LHCb
LHCb:2011yev ; LHCb:2014ioa , but the final state is self-tagged by the flavor
of the $K^{\star 0}$, which excludes time-dependent analysis. The CP violation
for $A_{\rm CP}(B^{0}_{s}\to f_{D}K^{\star 0})$ has been measured LHCb:2014pae
. Considering the possible decays $B^{0}_{s}\to D^{0}K^{0}_{S}$,
$B^{0}_{s}\to\bar{D}^{0}K^{0}_{S}$, $D^{0}\to f_{D}$, and $\bar{D}^{0}\to
f_{D}$, there are eight pathways for these two process, fitting them into what
is defined as the third type.

Figure 17: Time dependence of the CP asymmetry $A_{\rm CP}$ in $B^{0}_{s}\to
DK^{0}_{S}\to(\pi^{+}\pi^{-})K^{0}_{S}$. The left panel displays the two-
dimensional time dependence. The middle panel and the right panel display the
dependence on $t_{1}$ (with $t_{2}$ integrated from 0 to $5\tau_{D}$) and
$t_{2}$ (with $t_{1}$ integrated from $0$ to $\tau_{B_{s}}$), respectively.

For the channel with $f_{D}=\pi^{+}\pi^{-}$, we derive the results for all the
components of CP violation by setting $r_{1}=r$, $\delta_{1}=\delta_{4}$,
$\theta_{1}=\omega_{4}$, $\theta_{2}=0$, $r_{2}=1$, $\delta_{2}=0$ and
$\theta_{3}=\omega_{3}$ in (26), and then we obtain all the terms, which are
shown in (63) - (74). For $r$ and $\delta_{4}$ defined in (3.2.4), we
numerically use $r=0.02$ and $\delta_{4}=194^{\circ}$ as a benchmark. The
values for $\omega_{3}$ and $\omega_{4}$ are given in Table 1. In this case,
the dominant term in the double-mixing CP violation arises from the
interference between the
$B^{0}_{s}\to\bar{D}^{0}K^{0}_{S}\to(\pi^{+}\pi^{-})K^{0}_{S}$ and
$B^{0}_{s}\to\bar{B}^{0}_{s}\to
D^{0}K^{0}_{S}\to\bar{D}^{0}K^{0}_{S}\to(\pi^{+}\pi^{-})K^{0}_{S}$ amplitudes
as well as that between the $B^{0}_{s}\to\bar{D}^{0}K^{0}_{S}\to
D^{0}K^{0}_{S}\to(\pi^{+}\pi^{-})K^{0}_{S}$ and
$B^{0}_{s}\to\bar{B}^{0}_{s}\to D^{0}K^{0}_{S}\to(\pi^{+}\pi^{-})K^{0}_{S}$
amplitudes. Our numerical results for this channel are presented in Figure 17,
with the two-dimensional time dependence in the left panel. Its peak value is
$\mathcal{O}(10^{-3})$. Integrating $t_{2}$ from 0 to $5\tau_{D}$ yields the
$t_{1}$-dependent result, shown in the middle panel, revealing the double-
mixing CP violation effect is of the order $\mathcal{O}(10^{-4})$. This is due
to the small values $x_{D}$ and $y_{D}$ of the $D$ meson as listed in Table 1.
The $t_{2}$-integrated CP asymmetry is dominated by terms arising from the
interference between the
$B^{0}_{s}\to\bar{D}^{0}K^{0}_{S}\to(\pi^{+}\pi^{-})K^{0}_{S}$ and
$B^{0}_{s}\to\bar{B}^{0}_{s}\to D^{0}K^{0}_{S}\to(\pi^{+}\pi^{-})K^{0}_{S}$
amplitudes, which have a cosine or hyperbolic cosine dependence on $t_{2}$ and
have a sine dependence on $t_{1}$, namely $\sin{\Delta m_{B_{s}}t_{1}}$.
Integrating $t_{1}$ from 0 to $\tau_{B_{s}}$, the magnitude of the double-
mixing term $A_{n,dm}$ climbs slowly as $t_{2}$ increases attributed to small
mixing parameters of $D$ meson, and the total asymmetry $A_{\rm CP}$ does not
show visible changes before $t_{2}$ becomes too large, as depicted in the
right panel.

For the channel with $f_{D}=K^{-}\pi^{+}$, the double-mixing and non-double-
mixing contributions to the CP asymmetry can be obtained by taking (3.1.5) and
(3.1.5) with the substitutions $\phi_{B_{d}}\to\phi_{B_{s}}$, $r_{B}\to r$ and
$\delta_{B}\to\delta_{4}$, and the replacement of the denominator
$D(t_{1},t_{2})$ accordingly. For $r$ and $\delta_{4}$, we still numerically
use $r=0.02$ and $\delta_{4}=194^{\circ}$. The presence of the suppression
factor $r_{D}$ and the strong phase $\delta_{D}$ results in a pattern distinct
from that of the $\pi^{+}\pi^{-}$ channel. The numerical analysis for the
double-mixing CP asymmetry $A_{dm}$ is depicted in Figure 18. The left panel
shows the two-dimensional time dependence, indicating the insignificance of
the CP violation with $t_{2}<3\tau_{D}$, below the threshold of current
experimental detection capacities. This double-mixing effect is also
negligible in the integrated results over $t_{2}$ or $t_{1}$, presented on the
middle and the right panel, respectively.

Figure 18: Time dependence of the CP asymmetry $A_{\rm CP}$ in $B^{0}_{s}\to
DK^{0}_{S}\to(K^{-}\pi^{+})K^{0}_{S}$. The left panel displays the two-
dimensional time dependence. The middle panel and the right panel display the
dependence on $t_{1}$ (with $t_{2}$ integrated from 0 to $5\tau_{D}$) and
$t_{2}$ (with $t_{1}$ integrated from $0$ to $\tau_{B_{s}}$), respectively.

##  4 Conclusions

It was proposed recently by Shen:2023nuw  that a long-overlooked CP violation
effect exists in a decay chain with two neutral mesons involved, namely the
double-mixing CP violation. This violation is induced by the interference
among different oscillation paths of two neutral flavored mesons. In this
work, we have analyzed the double-mixing CP violation in a series of cascade
decay processes $M^{0}_{1}(t_{1})\to M_{2}(t_{2})\to f$, where the primary
neutral meson $M^{0}_{1}$ is either $B^{0}_{d,s}$ mesons and the secondary
neutral meson is $D^{0}$ or $K^{0}$ mesons. The involvement of two time
parameters, $t_{1}$ and $t_{2}$, makes the two-dimensional time analysis of
double-mixing CP violation allowable. We performed numerical analyses for each
decay channel, providing the two-dimensional time-dependent double-mixing CP
violation and the CP violation integrated over $t_{1}$ or $t_{2}$.

Our results show that the double-mixing CP violation can be very significant,
with instances exceeding 50% – notably in the decay channel
$B^{0}_{s}\to\rho^{0}K\to\rho^{0}(\pi\ell\nu)$. In similar cases with kaons
reconstructed by semi-leptonic states, the total CP violation almost entirely
comes from the double-mixing CP violation. Moreover, our findings indicate
that the double-mixing CP violation’s dependency varies distinctly with
$t_{1}$ and $t_{2}$: when integrating over $t_{2}$, the direct decays of
$K^{0}_{S}$ and $K^{0}_{L}$ predominantly contribute to the CP violation,
whereas integrating over $t_{1}$ highlights the dominance of
$K^{0}_{S}-K^{0}_{L}$ interference. We emphasize that one of the first
experimental attempts to measure the double-mixing CP violation would be to
combine the two decay channels
$B^{0}_{s}\to\rho^{0}K\to\rho^{0}(\pi^{-}\ell^{+}\nu_{\ell})$ and
$B^{0}_{s}\to\rho^{0}K\to\rho^{0}(\pi^{+}\ell^{-}\bar{\nu}_{\ell})$. This
method does not require initial tagging of neutral $B$ mesons, thus avoiding
efficiency losses.

In comparison, the significance of double-mixing CP violation is less in decay
channels involving $D^{0}$ as $M^{0}_{2}$, mainly due to the tiny mixing
observed in neutral $D$ mesons, such as in the decay channel $B^{0}_{s}\to
DK^{0}_{S}\to(\pi^{+}\pi^{-})K^{0}_{S}$. Nonetheless, double-mixing CP
violation offers a major advantage by facilitating the extraction of CKM phase
angles without theoretical inputs. For example, the decay channel
$B^{0}_{d}\to DK\to(K^{-}\pi^{+})(\pi\ell\nu)$ involves both strong and weak
phases, with the overall weak phase being approximated by $2\beta+\gamma$.
This allows for the simultaneous extraction of strong and weak phases from
experimental data on double-mixing CP violation, thereby directly testing the
CKM mechanism.

## Acknowledgement

The authors are grateful to Cai-Dian Lü, Wen-Bin Qian, Liang Sun, and Yue-Hong
Xie for useful discussions. This work is supported by the Natural Science
Foundation of China under grant No. 12375086.

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##  Appendix A Formulae for Category 3

Here we present relevant analytical expressions for the terms of the double-
mixing CP violation of Category 3 in Section 2. Based on the definitions of
the parameters provided in (26), we can derive the individual terms in (4) as

$\displaystyle C_{+}(t_{2})=$ $\displaystyle\
e^{-\Gamma_{2}t_{2}}\left\\{r_{2}\sinh{\frac{\Delta\Gamma_{2}t_{2}}{2}}\left[r_{1}^{2}\pqty{\absolutevalue{\frac{q_{2}}{p_{2}}}\cos{\Omega_{3}}-\absolutevalue{\frac{p_{2}}{q_{2}}}\cos{\Omega_{4}}}+\absolutevalue{\frac{p_{2}}{q_{2}}}\cos{\Omega_{3}}-\absolutevalue{\frac{q_{2}}{p_{2}}}\cos{\Omega_{4}}\right]\right.$
$\displaystyle\left.-r_{2}\sin{\Delta
m_{2}t_{2}}\left[r_{1}^{2}\pqty{\absolutevalue{\frac{q_{2}}{p_{2}}}\sin{\Omega_{3}}-\absolutevalue{\frac{p_{2}}{q_{2}}}\sin{\Omega_{4}}}-\absolutevalue{\frac{p_{2}}{q_{2}}}\sin{\Omega_{3}}+\absolutevalue{\frac{q_{2}}{p_{2}}}\sin{\Omega_{4}}\right]\right.$
$\displaystyle\left.+r_{1}\sinh{\frac{\Delta\Gamma_{2}t_{2}}{2}}\left[r_{2}^{2}\pqty{\absolutevalue{\frac{q_{2}}{p_{2}}}\cos{\Omega_{1}}-\absolutevalue{\frac{p_{2}}{q_{2}}}\cos{\Omega_{2}}}+\absolutevalue{\frac{p_{2}}{q_{2}}}\cos{\Omega_{1}}-\absolutevalue{\frac{q_{2}}{p_{2}}}\cos{\Omega_{2}}\right]\right.$
$\displaystyle\left.-r_{1}\sin{\Delta
m_{2}t_{2}}\left[r_{2}^{2}\pqty{\absolutevalue{\frac{q_{2}}{p_{2}}}\sin{\Omega_{1}}-\absolutevalue{\frac{p_{2}}{q_{2}}}\sin{\Omega_{2}}}-\absolutevalue{\frac{p_{2}}{q_{2}}}\sin{\Omega_{1}}+\absolutevalue{\frac{q_{2}}{p_{2}}}\sin{\Omega_{2}}\right]\right\\}$
$\displaystyle+2r_{1}r_{2}\absolutevalue{g_{+,2}(t_{2})}^{2}\Big{[}\cos{\pqty{\Omega_{1}-\Omega_{3}}}-\cos{\pqty{\Omega_{2}-\Omega_{4}}}\Big{]}$
$\displaystyle+2r_{1}r_{2}\absolutevalue{g_{-,2}(t_{2})}^{2}\Big{[}\cos{\pqty{\Omega_{1}+\Omega_{3}}}-\cos{\pqty{\Omega_{2}+\Omega_{4}}}\Big{]}$
$\displaystyle+\pqty{r_{1}^{2}r_{2}^{2}-1}\absolutevalue{g_{-,2}(t_{2})}^{2}\pqty{\absolutevalue{\frac{q_{2}}{p_{2}}}^{2}-\absolutevalue{\frac{p_{2}}{q_{2}}}^{2}},$
(63)

where $\Omega_{1}\equiv\delta_{1}+\theta_{1}-\phi_{2}$,
$\Omega_{2}\equiv\delta_{1}-\theta_{1}+\phi_{2}$,
$\Omega_{3}\equiv\delta_{2}-\phi_{2}$ and
$\Omega_{4}\equiv\delta_{2}+\phi_{2}$,

$\displaystyle C_{-}(t_{2})=$ $\displaystyle\
\absolutevalue{g_{-,2}(t_{2})}^{2}\left\\{r_{1}^{2}\pqty{\absolutevalue{\frac{q_{1}}{p_{1}}}^{2}\absolutevalue{\frac{p_{2}}{q_{2}}}^{2}-\absolutevalue{\frac{p_{1}}{q_{1}}}^{2}\absolutevalue{\frac{q_{2}}{p_{2}}}^{2}}+r_{2}^{2}\pqty{\absolutevalue{\frac{q_{1}}{p_{1}}}^{2}\absolutevalue{\frac{q_{2}}{p_{2}}}^{2}-\absolutevalue{\frac{p_{1}}{q_{1}}}^{2}\absolutevalue{\frac{p_{2}}{q_{2}}}^{2}}\right\\}$
$\displaystyle+2r_{1}r_{2}\absolutevalue{\frac{q_{1}}{p_{1}}}^{2}\left\\{\absolutevalue{g_{+,2}(t_{2})}^{2}\cos{\pqty{\Omega_{2}+\Omega_{3}}}+\absolutevalue{g_{-,2}(t_{2})}^{2}\cos{\pqty{\Omega_{2}-\Omega_{3}}}\right\\}$
$\displaystyle-2r_{1}r_{2}\absolutevalue{\frac{p_{1}}{q_{1}}}^{2}\left\\{\absolutevalue{g_{+,2}(t_{2})}^{2}\cos{\pqty{\Omega_{1}+\Omega_{4}}}+\absolutevalue{g_{-,2}(t_{2})}^{2}\cos{\pqty{\Omega_{1}-\Omega_{4}}}\right\\}$
$\displaystyle+e^{-\Gamma_{2}t_{2}}\left\\{r_{2}\absolutevalue{\frac{q_{1}}{p_{1}}}^{2}\absolutevalue{\frac{q_{2}}{p_{2}}}\pqty{\sinh{\frac{\Delta\Gamma_{2}t_{2}}{2}}\cos{\Omega_{3}}-\sin{\Delta
m_{2}t_{2}}\sin{\Omega_{3}}}\right.$
$\displaystyle\left.+r_{1}^{2}r_{2}\absolutevalue{\frac{q_{1}}{p_{1}}}^{2}\absolutevalue{\frac{p_{2}}{q_{2}}}\pqty{\sinh{\frac{\Delta\Gamma_{2}t_{2}}{2}}\cos{\Omega_{3}}+\sin{\Delta
m_{2}t_{2}}\sin{\Omega_{3}}}\right.$
$\displaystyle\left.-r_{2}\absolutevalue{\frac{p_{1}}{q_{1}}}^{2}\absolutevalue{\frac{p_{2}}{q_{2}}}\pqty{\sinh{\frac{\Delta\Gamma_{2}t_{2}}{2}}\cos{\Omega_{4}}-\sin{\Delta
m_{2}t_{2}}\sin{\Omega_{4}}}\right.$
$\displaystyle\left.-r_{1}^{2}r_{2}\absolutevalue{\frac{p_{1}}{q_{1}}}^{2}\absolutevalue{\frac{q_{2}}{p_{2}}}\pqty{\sinh{\frac{\Delta\Gamma_{2}t_{2}}{2}}\cos{\Omega_{4}}+\sin{\Delta
m_{2}t_{2}}\sin{\Omega_{4}}}\right.$
$\displaystyle\left.+r_{1}\sinh{\frac{\Delta\Gamma_{2}t_{2}}{2}}\left[\absolutevalue{\frac{q_{1}}{p_{1}}}^{2}\cos{\Omega_{2}}\pqty{r_{2}^{2}\absolutevalue{\frac{q_{2}}{p_{2}}}+\absolutevalue{\frac{p_{2}}{q_{2}}}}-\absolutevalue{\frac{p_{1}}{q_{1}}}^{2}\cos{\Omega_{1}}\pqty{r_{2}^{2}\absolutevalue{\frac{p_{2}}{q_{2}}}+\absolutevalue{\frac{q_{2}}{p_{2}}}}\right]\right.$
$\displaystyle\left.+r_{1}\sin{\Delta
m_{2}t_{2}}\left[\absolutevalue{\frac{q_{1}}{p_{1}}}^{2}\sin{\Omega_{2}}\pqty{r_{2}^{2}\absolutevalue{\frac{q_{2}}{p_{2}}}-\absolutevalue{\frac{p_{2}}{q_{2}}}}-\absolutevalue{\frac{p_{1}}{q_{1}}}^{2}\sin{\Omega_{1}}\pqty{r_{2}^{2}\absolutevalue{\frac{p_{2}}{q_{2}}}-\absolutevalue{\frac{q_{2}}{p_{2}}}}\right]\right\\}$
$\displaystyle+\pqty{r_{1}^{2}r_{2}^{2}+1}\pqty{\absolutevalue{\frac{q_{1}}{p_{1}}}^{2}-\absolutevalue{\frac{p_{1}}{q_{1}}}^{2}}\absolutevalue{g_{+,2}(t_{2})}^{2},$
(64) $\displaystyle S_{n,1}(t_{2})=$ $\displaystyle\
\frac{e^{-\Gamma_{2}t_{2}}}{2}\Big{\\{}g(-,-,-,-)\times\sinh{\frac{\Delta\Gamma_{2}t_{2}}{2}}\Big{[}r_{1}^{2}\sin{\pqty{\Omega_{5}-2\Omega_{6}}}+\sin{\Omega_{5}}\Big{]}$
$\displaystyle+g(-,+,-,+)\times\sin{\Delta
m_{2}t_{2}}\Big{[}r_{1}^{2}\cos{\pqty{\Omega_{5}-2\Omega_{6}}}-\cos{\Omega_{5}}\Big{]}\Big{\\}}$
$\displaystyle+r_{1}r_{2}\absolutevalue{\frac{q_{2}}{p_{2}}}\sin{\pqty{\delta_{1}-\Omega_{5}+\Omega_{6}}}e^{-\Gamma_{2}t_{2}}\left\\{\absolutevalue{\frac{q_{1}}{p_{1}}}\left[\sinh{\frac{\Delta\Gamma_{2}t_{2}}{2}}\cos{\Omega_{3}}-\sin{\Delta
m_{2}t_{2}}\sin{\Omega_{3}}\right]\right.$
$\displaystyle\left.+\absolutevalue{\frac{p_{1}}{q_{1}}}\left[\sinh{\frac{\Delta\Gamma_{2}t_{2}}{2}}\cos{\Omega_{4}}+\sin{\Delta
m_{2}t_{2}}\sin{\Omega_{4}}\right]\right\\}$ $\displaystyle-
r_{1}r_{2}\absolutevalue{\frac{p_{2}}{q_{2}}}\sin{\pqty{\delta_{1}+\Omega_{5}-\Omega_{6}}}e^{-\Gamma_{2}t_{2}}\left\\{\absolutevalue{\frac{q_{1}}{p_{1}}}\left[\sinh{\frac{\Delta\Gamma_{2}t_{2}}{2}}\cos{\Omega_{3}}+\sin{\Delta
m_{2}t_{2}}\sin{\Omega_{3}}\right]\right.$
$\displaystyle\left.+\absolutevalue{\frac{p_{1}}{q_{1}}}\left[\sinh{\frac{\Delta\Gamma_{2}t_{2}}{2}}\cos{\Omega_{4}}-\sin{\Delta
m_{2}t_{2}}\sin{\Omega_{4}}\right]\right\\},$ (65)

where $\Omega_{5}\equiv\theta_{3}-\phi_{1}-\phi_{2}$,
$\Omega_{6}\equiv\theta_{1}-\phi_{2}$, and the $g(-,+,-,+)$ is defined as

$\displaystyle
g(-,+,-,+)=-\absolutevalue{\frac{q_{1}}{p_{1}}}\absolutevalue{\frac{p_{2}}{q_{2}}}+r_{2}^{2}\absolutevalue{\frac{q_{1}}{p_{1}}}\absolutevalue{\frac{q_{2}}{p_{2}}}-r_{2}^{2}\absolutevalue{\frac{p_{1}}{q_{1}}}\absolutevalue{\frac{p_{2}}{q_{2}}}+\absolutevalue{\frac{p_{1}}{q_{1}}}\absolutevalue{\frac{q_{2}}{p_{2}}},$

and others $g$ functions with different sign arguments can be obtained by flip
signs in the above expression accordingly,

$\displaystyle S_{n,2}(t_{2})=$ $\displaystyle-
r_{1}^{2}r_{2}\absolutevalue{\frac{q_{1}}{p_{1}}}\left\\{\absolutevalue{g_{+,2}(t_{2})}^{2}\sin{\pqty{\Omega_{3}+\Omega_{5}-2\Omega_{6}}}-\absolutevalue{g_{-,2}(t_{2})}^{2}\sin{\pqty{\Omega_{3}-\Omega_{5}+2\Omega_{6}}}\right\\}$
$\displaystyle+r_{2}\absolutevalue{\frac{q_{1}}{p_{1}}}\left\\{\absolutevalue{g_{+,2}(t_{2})}^{2}\sin{\pqty{\Omega_{3}-\Omega_{5}}}-\absolutevalue{g_{-,2}(t_{2})}^{2}\sin{\pqty{\Omega_{3}+\Omega_{5}}}\right\\}$
$\displaystyle+r_{1}^{2}r_{2}\absolutevalue{\frac{p_{1}}{q_{1}}}\left\\{\absolutevalue{g_{+,2}(t_{2})}^{2}\sin{\pqty{\Omega_{4}-\Omega_{5}+2\Omega_{6}}}-\absolutevalue{g_{-,2}(t_{2})}^{2}\sin{\pqty{\Omega_{4}+\Omega_{5}-2\Omega_{6}}}\right\\}$
$\displaystyle-
r_{2}\absolutevalue{\frac{p_{1}}{q_{1}}}\left\\{\absolutevalue{g_{+,2}(t_{2})}^{2}\sin{\pqty{\Omega_{4}+\Omega_{5}}}-\absolutevalue{g_{-,2}(t_{2})}^{2}\sin{\pqty{\Omega_{4}-\Omega_{5}}}\right\\}$
$\displaystyle+r_{1}\sin{\pqty{\delta_{1}-\Omega_{5}+\Omega_{6}}}f_{1}(q,p)-r_{1}\sin{\pqty{\delta_{1}+\Omega_{5}-\Omega_{6}}}f_{1}(p,q),$
(66)

where

$\displaystyle
f_{1}(q,p)=\absolutevalue{g_{+,2}(t_{2})}^{2}\pqty{\absolutevalue{\frac{q_{1}}{p_{1}}}+r_{2}^{2}\absolutevalue{\frac{p_{1}}{q_{1}}}}+\absolutevalue{\frac{q_{2}}{p_{2}}}^{2}\absolutevalue{g_{-,2}(t_{2})}^{2}\pqty{r_{2}^{2}\absolutevalue{\frac{q_{1}}{p_{1}}}+\absolutevalue{\frac{p_{1}}{q_{1}}}},$

and the function $f_{1}(p,q)$ is obtained by exchanging
$p_{1,2}\leftrightarrow q_{1,2}$ in $f_{1}(q,p)$,

$\displaystyle S_{h,1}(t_{2})=$ $\displaystyle\
\frac{e^{-\Gamma_{2}t_{2}}}{2}\Big{\\{}g(+,+,-,-)\sinh{\frac{\Delta\Gamma_{2}t_{2}}{2}}\Big{[}r_{1}^{2}\cos{\pqty{\Omega_{5}-2\Omega_{6}}}+\cos{\Omega_{5}}\Big{]}$
$\displaystyle-g(+,-,-,+)\sin{\Delta
m_{2}t_{2}}\Big{[}r_{1}^{2}\sin{\pqty{\Omega_{5}-2\Omega_{6}}}-\sin{\Omega_{5}}\Big{]}\Big{\\}}$
$\displaystyle+r_{1}r_{2}\absolutevalue{\frac{q_{2}}{p_{2}}}\cos{\pqty{\delta_{1}-\Omega_{5}+\Omega_{6}}}e^{-\Gamma_{2}t_{2}}\left\\{\absolutevalue{\frac{q_{1}}{p_{1}}}\left[\sinh{\frac{\Delta\Gamma_{2}t_{2}}{2}}\cos{\Omega_{3}}-\sin{\Delta
m_{2}t_{2}}\sin{\Omega_{3}}\right]\right.$
$\displaystyle\left.-\absolutevalue{\frac{p_{1}}{q_{1}}}\left[\sinh{\frac{\Delta\Gamma_{2}t_{2}}{2}}\cos{\Omega_{4}}+\sin{\Delta
m_{2}t_{2}}\sin{\Omega_{4}}\right]\right\\}$
$\displaystyle+r_{1}r_{2}\absolutevalue{\frac{p_{2}}{q_{2}}}\cos{\pqty{\delta_{1}+\Omega_{5}-\Omega_{6}}}e^{-\Gamma_{2}t_{2}}\left\\{\absolutevalue{\frac{q_{1}}{p_{1}}}\left[\sinh{\frac{\Delta\Gamma_{2}t_{2}}{2}}\cos{\Omega_{3}}+\sin{\Delta
m_{2}t_{2}}\sin{\Omega_{3}}\right]\right.$
$\displaystyle\left.-\absolutevalue{\frac{p_{1}}{q_{1}}}\left[\sinh{\frac{\Delta\Gamma_{2}t_{2}}{2}}\cos{\Omega_{4}}-\sin{\Delta
m_{2}t_{2}}\sin{\Omega_{4}}\right]\right\\},$ (67) $\displaystyle
S_{h,2}(t_{2})=$ $\displaystyle\
r_{1}^{2}r_{2}\absolutevalue{\frac{q_{1}}{p_{1}}}\Big{[}\absolutevalue{g_{+,2}(t_{2})}^{2}\cos{\pqty{\Omega_{3}+\Omega_{5}-2\Omega_{6}}}+\absolutevalue{g_{-,2}(t_{2})}^{2}\cos{\pqty{\Omega_{3}-\Omega_{5}+2\Omega_{6}}}\Big{]}$
$\displaystyle+r_{2}\absolutevalue{\frac{q_{1}}{p_{1}}}\Big{[}\absolutevalue{g_{+,2}(t_{2})}^{2}\cos{\pqty{\Omega_{3}-\Omega_{5}}}+\absolutevalue{g_{-,2}(t_{2})}^{2}\cos{\pqty{\Omega_{3}+\Omega_{5}}}\Big{]}$
$\displaystyle-
r_{1}^{2}r_{2}\absolutevalue{\frac{p_{1}}{q_{1}}}\Big{[}\absolutevalue{g_{+,2}(t_{2})}^{2}\cos{\pqty{\Omega_{4}-\Omega_{5}+2\Omega_{6}}}+\absolutevalue{g_{-,2}(t_{2})}^{2}\cos{\pqty{\Omega_{4}+\Omega_{5}-2\Omega_{6}}}\Big{]}$
$\displaystyle-
r_{2}\absolutevalue{\frac{p_{1}}{q_{1}}}\Big{[}\absolutevalue{g_{+,2}(t_{2})}^{2}\cos{\pqty{\Omega_{4}+\Omega_{5}}}+\absolutevalue{g_{-,2}(t_{2})}^{2}\cos{\pqty{\Omega_{4}-\Omega_{5}}}\Big{]}$
$\displaystyle+r_{1}\cos{\pqty{\delta_{1}-\Omega_{5}+\Omega_{6}}}f_{2}(q,p)+r_{1}\cos{\pqty{\delta_{1}+\Omega_{5}-\Omega_{6}}}f_{2}(p,q),$
(68)

where

$\displaystyle
f_{2}(q,p)=\absolutevalue{g_{+,2}(t_{2})}^{2}\pqty{\absolutevalue{\frac{q_{1}}{p_{1}}}-r_{2}^{2}\absolutevalue{\frac{p_{1}}{q_{1}}}}+\absolutevalue{\frac{q_{2}}{p_{2}}}^{2}\absolutevalue{g_{-,2}(t_{2})}^{2}\pqty{r_{2}^{2}\absolutevalue{\frac{q_{1}}{p_{1}}}-\absolutevalue{\frac{p_{1}}{q_{1}}}},$

and the function $f_{2}(p,q)$ is obtained by exchanging
$p_{1,2}\leftrightarrow q_{1,2}$ in $f_{2}(q,p)$,

$\displaystyle C_{+}^{\prime}(t_{2})=$ $\displaystyle\
e^{-\Gamma_{2}t_{2}}\left\\{r_{2}\sinh{\frac{\Delta\Gamma_{2}t_{2}}{2}}\left[r_{1}^{2}\pqty{\absolutevalue{\frac{q_{2}}{p_{2}}}\cos{\Omega_{3}}+\absolutevalue{\frac{p_{2}}{q_{2}}}\cos{\Omega_{4}}}+\absolutevalue{\frac{p_{2}}{q_{2}}}\cos{\Omega_{3}}+\absolutevalue{\frac{q_{2}}{p_{2}}}\cos{\Omega_{4}}\right]\right.$
$\displaystyle\left.-r_{2}\sin{\Delta
m_{2}t_{2}}\left[r_{1}^{2}\pqty{\absolutevalue{\frac{q_{2}}{p_{2}}}\sin{\Omega_{3}}+\absolutevalue{\frac{p_{2}}{q_{2}}}\sin{\Omega_{4}}}-\absolutevalue{\frac{p_{2}}{q_{2}}}\sin{\Omega_{3}}-\absolutevalue{\frac{q_{2}}{p_{2}}}\sin{\Omega_{4}}\right]\right.$
$\displaystyle\left.+r_{1}\sinh{\frac{\Delta\Gamma_{2}t_{2}}{2}}\left[r_{2}^{2}\pqty{\absolutevalue{\frac{q_{2}}{p_{2}}}\cos{\Omega_{1}}+\absolutevalue{\frac{p_{2}}{q_{2}}}\cos{\Omega_{2}}}+\absolutevalue{\frac{p_{2}}{q_{2}}}\cos{\Omega_{1}}+\absolutevalue{\frac{q_{2}}{p_{2}}}\cos{\Omega_{2}}\right]\right.$
$\displaystyle\left.-r_{1}\sin{\Delta
m_{2}t_{2}}\left[r_{2}^{2}\pqty{\absolutevalue{\frac{q_{2}}{p_{2}}}\sin{\Omega_{1}}+\absolutevalue{\frac{p_{2}}{q_{2}}}\sin{\Omega_{2}}}-\absolutevalue{\frac{p_{2}}{q_{2}}}\sin{\Omega_{1}}-\absolutevalue{\frac{q_{2}}{p_{2}}}\sin{\Omega_{2}}\right]\right\\}$
$\displaystyle+2r_{1}r_{2}\absolutevalue{g_{+,2}(t_{2})}^{2}\Big{[}\cos{\pqty{\Omega_{1}-\Omega_{3}}}+\cos{\pqty{\Omega_{2}-\Omega_{4}}}\Big{]}$
$\displaystyle+2r_{1}r_{2}\absolutevalue{g_{-,2}(t_{2})}^{2}\Big{[}\cos{\pqty{\Omega_{1}+\Omega_{3}}}+\cos{\pqty{\Omega_{2}+\Omega_{4}}}\Big{]}$
$\displaystyle+\pqty{r_{1}^{2}r_{2}^{2}+1}\absolutevalue{g_{-,2}(t_{2})}^{2}\pqty{\absolutevalue{\frac{q_{2}}{p_{2}}}^{2}+\absolutevalue{\frac{p_{2}}{q_{2}}}^{2}}+2\pqty{r_{1}^{2}+r_{2}^{2}}\absolutevalue{g_{+,2}(t_{2})}^{2},$
(69) $\displaystyle C_{-}^{\prime}(t_{2})=$ $\displaystyle\
\absolutevalue{g_{-,2}(t_{2})}^{2}\left\\{r_{1}^{2}\pqty{\absolutevalue{\frac{q_{1}}{p_{1}}}^{2}\absolutevalue{\frac{p_{2}}{q_{2}}}^{2}+\absolutevalue{\frac{p_{1}}{q_{1}}}^{2}\absolutevalue{\frac{q_{2}}{p_{2}}}^{2}}+r_{2}^{2}\pqty{\absolutevalue{\frac{q_{1}}{p_{1}}}^{2}\absolutevalue{\frac{q_{2}}{p_{2}}}^{2}+\absolutevalue{\frac{p_{1}}{q_{1}}}^{2}\absolutevalue{\frac{p_{2}}{q_{2}}}^{2}}\right\\}$
$\displaystyle+2r_{1}r_{2}\absolutevalue{\frac{q_{1}}{p_{1}}}^{2}\Big{[}\absolutevalue{g_{+,2}(t_{2})}^{2}\cos{\pqty{\Omega_{2}+\Omega_{3}}}+\absolutevalue{g_{-,2}(t_{2})}^{2}\cos{\pqty{\Omega_{2}-\Omega_{3}}}\Big{]}$
$\displaystyle+2r_{1}r_{2}\absolutevalue{\frac{p_{1}}{q_{1}}}^{2}\Big{[}\absolutevalue{g_{+,2}(t_{2})}^{2}\cos{\pqty{\Omega_{1}+\Omega_{4}}}+\absolutevalue{g_{-,2}(t_{2})}^{2}\cos{\pqty{\Omega_{1}-\Omega_{4}}}\Big{]}$
$\displaystyle+e^{-\Gamma_{2}t_{2}}\left\\{r_{2}\absolutevalue{\frac{q_{1}}{p_{1}}}^{2}\absolutevalue{\frac{q_{2}}{p_{2}}}\pqty{\sinh{\frac{\Delta\Gamma_{2}t_{2}}{2}}\cos{\Omega_{3}}-\sin{\Delta
m_{2}t_{2}}\sin{\Omega_{3}}}\right.$
$\displaystyle\left.+r_{1}^{2}r_{2}\absolutevalue{\frac{q_{1}}{p_{1}}}^{2}\absolutevalue{\frac{p_{2}}{q_{2}}}\pqty{\sinh{\frac{\Delta\Gamma_{2}t_{2}}{2}}\cos{\Omega_{3}}+\sin{\Delta
m_{2}t_{2}}\sin{\Omega_{3}}}\right.$
$\displaystyle\left.+r_{2}\absolutevalue{\frac{p_{1}}{q_{1}}}^{2}\absolutevalue{\frac{p_{2}}{q_{2}}}\pqty{\sinh{\frac{\Delta\Gamma_{2}t_{2}}{2}}\cos{\Omega_{4}}-\sin{\Delta
m_{2}t_{2}}\sin{\Omega_{4}}}\right.$
$\displaystyle\left.+r_{1}^{2}r_{2}\absolutevalue{\frac{p_{1}}{q_{1}}}^{2}\absolutevalue{\frac{q_{2}}{p_{2}}}\pqty{\sinh{\frac{\Delta\Gamma_{2}t_{2}}{2}}\cos{\Omega_{4}}+\sin{\Delta
m_{2}t_{2}}\sin{\Omega_{4}}}\right.$
$\displaystyle+\left.r_{1}\sinh{\frac{\Delta\Gamma_{2}t_{2}}{2}}\left[\absolutevalue{\frac{q_{1}}{p_{1}}}^{2}\cos{\Omega_{2}}\pqty{r_{2}^{2}\absolutevalue{\frac{q_{2}}{p_{2}}}+\absolutevalue{\frac{p_{2}}{q_{2}}}}+\absolutevalue{\frac{p_{1}}{q_{1}}}^{2}\cos{\Omega_{1}}\pqty{r_{2}^{2}\absolutevalue{\frac{p_{2}}{q_{2}}}+\absolutevalue{\frac{q_{2}}{p_{2}}}}\right]\right.$
$\displaystyle\left.+r_{1}\sin{\Delta
m_{2}t_{2}}\left[\absolutevalue{\frac{q_{1}}{p_{1}}}^{2}\sin{\Omega_{2}}\pqty{r_{2}^{2}\absolutevalue{\frac{q_{2}}{p_{2}}}-\absolutevalue{\frac{p_{2}}{q_{2}}}}+\absolutevalue{\frac{p_{1}}{q_{1}}}^{2}\sin{\Omega_{1}}\pqty{r_{2}^{2}\absolutevalue{\frac{p_{2}}{q_{2}}}-\absolutevalue{\frac{q_{2}}{p_{2}}}}\right]\right\\}$
$\displaystyle+\pqty{r_{1}^{2}r_{2}^{2}+1}\pqty{\absolutevalue{\frac{q_{1}}{p_{1}}}^{2}+\absolutevalue{\frac{p_{1}}{q_{1}}}^{2}}\absolutevalue{g_{+,2}(t_{2})}^{2},$
(70) $\displaystyle S_{n,1}^{\prime}(t_{2})=$ $\displaystyle\
\frac{e^{-\Gamma_{2}t_{2}}}{2}\Big{\\{}g(-,-,+,+)\times\sinh{\frac{\Delta\Gamma_{2}t_{2}}{2}}\Big{[}r_{1}^{2}\sin{\pqty{\Omega_{5}-2\Omega_{6}}}+\sin{\Omega_{5}}\Big{]}$
$\displaystyle+g(-,+,+,-)\times\sin{\Delta
m_{2}t_{2}}\Big{[}r_{1}^{2}\cos{\pqty{\Omega_{5}-2\Omega_{6}}}-\cos{\Omega_{5}}\Big{]}\Big{\\}}$
$\displaystyle+r_{1}r_{2}\absolutevalue{\frac{q_{2}}{p_{2}}}\sin{\pqty{\delta_{1}-\Omega_{5}+\Omega_{6}}}e^{-\Gamma_{2}t_{2}}\left\\{\absolutevalue{\frac{q_{1}}{p_{1}}}\left[\sinh{\frac{\Delta\Gamma_{2}t_{2}}{2}}\cos{\Omega_{3}}-\sin{\Delta
m_{2}t_{2}}\sin{\Omega_{3}}\right]\right.$
$\displaystyle\left.-\absolutevalue{\frac{p_{1}}{q_{1}}}\left[\sinh{\frac{\Delta\Gamma_{2}t_{2}}{2}}\cos{\Omega_{4}}+\sin{\Delta
m_{2}t_{2}}\sin{\Omega_{4}}\right]\right\\}$ $\displaystyle-
r_{1}r_{2}\absolutevalue{\frac{p_{2}}{q_{2}}}\sin{\pqty{\delta_{1}+\Omega_{5}-\Omega_{6}}}e^{-\Gamma_{2}t_{2}}\left\\{\absolutevalue{\frac{q_{1}}{p_{1}}}\left[\sinh{\frac{\Delta\Gamma_{2}t_{2}}{2}}\cos{\Omega_{3}}+\sin{\Delta
m_{2}t_{2}}\sin{\Omega_{3}}\right]\right.$
$\displaystyle\left.-\absolutevalue{\frac{p_{1}}{q_{1}}}\left[\sinh{\frac{\Delta\Gamma_{2}t_{2}}{2}}\cos{\Omega_{4}}-\sin{\Delta
m_{2}t_{2}}\sin{\Omega_{4}}\right]\right\\},$ (71) $\displaystyle
S_{n,2}^{\prime}(t_{2})=$ $\displaystyle-
r_{1}^{2}r_{2}\absolutevalue{\frac{q_{1}}{p_{1}}}\left\\{\absolutevalue{g_{+,2}(t_{2})}^{2}\sin{\pqty{\Omega_{3}+\Omega_{5}-2\Omega_{6}}}-\absolutevalue{g_{-,2}(t_{2})}^{2}\sin{\pqty{\Omega_{3}-\Omega_{5}+2\Omega_{6}}}\right\\}$
$\displaystyle+r_{2}\absolutevalue{\frac{q_{1}}{p_{1}}}\left\\{\absolutevalue{g_{+,2}(t_{2})}^{2}\sin{\pqty{\Omega_{3}-\Omega_{5}}}-\absolutevalue{g_{-,2}(t_{2})}^{2}\sin{\pqty{\Omega_{3}+\Omega_{5}}}\right\\}$
$\displaystyle-
r_{1}^{2}r_{2}\absolutevalue{\frac{p_{1}}{q_{1}}}\left\\{\absolutevalue{g_{+,2}(t_{2})}^{2}\sin{\pqty{\Omega_{4}-\Omega_{5}+2\Omega_{6}}}-\absolutevalue{g_{-,2}(t_{2})}^{2}\sin{\pqty{\Omega_{4}+\Omega_{5}-2\Omega_{6}}}\right\\}$
$\displaystyle+r_{2}\absolutevalue{\frac{p_{1}}{q_{1}}}\left\\{\absolutevalue{g_{+,2}(t_{2})}^{2}\sin{\pqty{\Omega_{4}+\Omega_{5}}}+\absolutevalue{g_{-,2}(t_{2})}^{2}\sin{\pqty{\Omega_{4}-\Omega_{5}}}\right\\}$
$\displaystyle+r_{1}\sin{\pqty{\delta_{1}-\Omega_{5}+\Omega_{6}}}f_{2}(q,p)-r_{1}\sin{\pqty{\delta_{1}+\Omega_{5}-\Omega_{6}}}f_{2}(p,q),$
(72) $\displaystyle S_{h,1}^{\prime}(t_{2})=$ $\displaystyle\
\frac{e^{-\Gamma_{2}t_{2}}}{2}\Big{\\{}g(+,+,+,+)\sinh{\frac{\Delta\Gamma_{2}t_{2}}{2}}\Big{[}r_{1}^{2}\cos{\pqty{\Omega_{5}-2\Omega_{6}}}+\cos{\Omega_{5}}\Big{]}$
$\displaystyle-g(+,-,+,-)\sin{\Delta
m_{2}t_{2}}\Big{[}r_{1}^{2}\sin{\pqty{\Omega_{5}-2\Omega_{6}}}-\sin{\Omega_{5}}\Big{]}\Big{\\}}$
$\displaystyle+r_{1}r_{2}\absolutevalue{\frac{q_{2}}{p_{2}}}\cos{\pqty{\delta_{1}-\Omega_{5}+\Omega_{6}}}e^{-\Gamma_{2}t_{2}}\left\\{\absolutevalue{\frac{q_{1}}{p_{1}}}\left[\sinh{\frac{\Delta\Gamma_{2}t_{2}}{2}}\cos{\Omega_{3}}-\sin{\Delta
m_{2}t_{2}}\sin{\Omega_{3}}\right]\right.$
$\displaystyle\left.+\absolutevalue{\frac{p_{1}}{q_{1}}}\left[\sinh{\frac{\Delta\Gamma_{2}t_{2}}{2}}\cos{\Omega_{4}}+\sin{\Delta
m_{2}t_{2}}\sin{\Omega_{4}}\right]\right\\}$
$\displaystyle+r_{1}r_{2}\absolutevalue{\frac{p_{2}}{q_{2}}}\cos{\pqty{\delta_{1}+\Omega_{5}-\Omega_{6}}}e^{-\Gamma_{2}t_{2}}\left\\{\absolutevalue{\frac{q_{1}}{p_{1}}}\left[\sinh{\frac{\Delta\Gamma_{2}t_{2}}{2}}\cos{\Omega_{3}}+\sin{\Delta
m_{2}t_{2}}\sin{\Omega_{3}}\right]\right.$
$\displaystyle\left.+\absolutevalue{\frac{p_{1}}{q_{1}}}\left[\sinh{\frac{\Delta\Gamma_{2}t_{2}}{2}}\cos{\Omega_{4}}-\sin{\Delta
m_{2}t_{2}}\sin{\Omega_{4}}\right]\right\\},$ (73) $\displaystyle
S_{h,2}^{\prime}(t_{2})=$ $\displaystyle\
r_{1}^{2}r_{2}\absolutevalue{\frac{q_{1}}{p_{1}}}\Big{[}\absolutevalue{g_{+,2}(t_{2})}^{2}\cos{\pqty{\Omega_{3}+\Omega_{5}-2\Omega_{6}}}+\absolutevalue{g_{-,2}(t_{2})}^{2}\cos{\pqty{\Omega_{3}-\Omega_{5}+2\Omega_{6}}}\Big{]}$
$\displaystyle+r_{2}\absolutevalue{\frac{q_{1}}{p_{1}}}\left\\{\absolutevalue{g_{+,2}(t_{2})}^{2}\cos{\pqty{\Omega_{3}-\Omega_{5}}}+\absolutevalue{g_{-,2}(t_{2})}^{2}\cos{\pqty{\Omega_{3}+\Omega_{5}}}\right\\}$
$\displaystyle+r_{1}^{2}r_{2}\absolutevalue{\frac{p_{1}}{q_{1}}}\Big{[}\absolutevalue{g_{+,2}(t_{2})}^{2}\cos{\pqty{\Omega_{4}-\Omega_{5}+2\Omega_{6}}}+\absolutevalue{g_{-,2}(t_{2})}^{2}\cos{\pqty{\Omega_{4}+\Omega_{5}-2\Omega_{6}}}\Big{]}$
$\displaystyle+r_{2}\absolutevalue{\frac{p_{1}}{q_{1}}}\left\\{\absolutevalue{g_{+,2}(t_{2})}^{2}\cos{\pqty{\Omega_{4}+\Omega_{5}}}+\absolutevalue{g_{-,2}(t_{2})}^{2}\cos{\pqty{\Omega_{4}-\Omega_{5}}}\right\\}$
$\displaystyle+r_{1}\cos{\pqty{\delta_{1}-\Omega_{5}+\Omega_{6}}}f_{1}(q,p)+r_{1}\cos{\pqty{\delta_{1}+\Omega_{5}-\Omega_{6}}}f_{1}(p,q).$
(74)