Moreover,   Proposition <ref> implies

\begin{align}
\label{20230128 52}
\Big\| S\Big( \sum_{j=1}^{\infty}2^{\boldsymbol{r}(j)}\Delta_j f \Big) \Big\|_{L_p(\bR^d,w\,dx)}
\approx
\Big\| \sum_{j=1}^{\infty}2^{\boldsymbol{r}(j)}\Delta_j f \Big\|_{L_p(\bR^d,w\,dx)},
\end{align}

where the implicit constant depends only on $p$, $d$ and $[w]_{A_p(\bR^d)}$.
Therefore by (<ref>) and (<ref>),

\begin{align*}
\|\pi^{\boldsymbol{r}} f\|_{L_p(\bR^d,w\,dx)} \approx \| S_0 f\|_{L_p(\bR^d,w\,dx)} + \Big\| S\Big( \sum_{j=1}^{\infty}2^{\boldsymbol{r}(j)}\Delta_j f \Big) \Big\|_{L_p(\bR^d,w\,dx)}.
\end{align*}

Next we claim

\begin{equation}\label{ineq 220718 1747}
\begin{aligned}
\Big\| S\Big( \sum_{j=1}^{\infty}2^{\boldsymbol{r}(j)}\Delta_j f \Big) \Big\|_{L_p(\bR^d,w\,dx)}
\leq N\Big\| \Big( \sum_{j=1}^{\infty} | 2^{\boldsymbol{r}(j)}\Delta_j f |^2 \Big)^{1/2} \Big\|_{L_p(\bR^d,w\,dx)}.
\end{aligned}
\end{equation}

Put $T_j:=\Delta_j$ for all $j\in\bZ$ and

\begin{equation*}
\begin{gathered} f_0:=2^{\boldsymbol{r}(1)}\Delta_1f,\quad f_1:=2^{\boldsymbol{r}(1)}\Delta_1f+2^{\boldsymbol{r}(2)}\Delta_2f\\ f_j:=(2^{\boldsymbol{r}(j-1)}\Delta_{j-1}+2^{\boldsymbol{r}(j)}\Delta_{j}+2^{\boldsymbol{r}(j+1)}\Delta_{j+1})f\quad \text{if } j\geq2,\quad f_j:=0 \quad \text{if } j<0.
\end{gathered}
\end{equation*}

Considering the almost orthogonal property of $\Delta_j$ and Remark <ref>, we have

\Big\| S\Big( \sum_{j=1}^{\infty}2^{\boldsymbol{r}(j)}\Delta_j f \Big) \Big\|_{L_p(\bR^d,w\,dx)}
=\Big\| \Big( \sum_{j\in\bZ}|T_jf_j|^2 \Big)^{1/2} \Big\|_{L_p(\bR^d,w\,dx)}\leq N\Big\| \Big( \sum_{j=1}^{\infty} | 2^{\boldsymbol{r}(j)}\Delta_j f |^2 \Big)^{1/2} \Big\|_{L_p(\bR^d,w\,dx)}

For the converse of (<ref>), we make use of the Khintchine inequality (e.g. see [19]),

\begin{align}\label{ineq 220718 1734}
\Big( \sum_{j=1}^{\infty} \big| 2^{\boldsymbol{r}(j)}\Delta_j f \big|^2 \Big)^{1/2}
\approx \Big( \mathbb{E} \Big|\sum_{j=1}^{\infty}  X_j 2^{\boldsymbol{r}(j)}\Delta_j f \Big|^p \Big)^{1/p},\quad p \in (0, \infty)
\end{align}

where the implicit constants depend only on $p$ and $\{X_j\}_{j\in\bN}$ is a sequence of independent and identically distributed random variables with the Rademacher distribution. Let $\Pi_1$ be a Fourier multiplier defined by

\Pi_1(\xi) :=
\frac{ \sum_{j=1}^\infty X_j 2^{\boldsymbol{r}(j)}\cF[\Psi](2^{-j}\xi)}{\cF[\Phi](\xi)+\sum_{j=1}^{\infty} 2^{\boldsymbol{r}(j)}\cF[\Psi](2^{-j}\xi)}.

Recall the notation

\bT_{\Pi_1}f(x):=\cF^{-1}[\Pi_1\cF[f]](x)

\begin{align*}
\pi^{\boldsymbol{r}}:=S_0+\sum_{j=1}^{\infty}2^{\boldsymbol{r}(j)}\Delta_j.
\end{align*}

Then the right-hand side of (<ref>) equals to

\begin{align*}
\left( \bE \left| \bT_{\Pi_1}(\pi^{\boldsymbol{r}}f) \right|^p \right)^{1/p}.
\end{align*}

It is easy to check that $\Pi_1$ satisfies (<ref>), that is, $\Pi_1$ is a weighted Mikhlin multiplier.
Thus it follows that

\begin{equation}\label{ineq 220718 1744}
\begin{aligned}
\Big\| \Big( \mathbb{E} \Big|\sum_{j=1}^{\infty}  X_j 2^{\boldsymbol{r}(j)}\Delta_j f \Big|^p \Big)^{1/p}  \Big\|_{L_p(\bR^d,w\,dx)}
&\leq N\Big( \mathbb{E}  \Big\|\sum_{j=1}^{\infty}  X_j 2^{\boldsymbol{r}(j)}\Delta_j f \Big\|_{L_p(\bR^d,w\,dx)}^p \Big)^{1/p}\\
&= N\Big( \mathbb{E}  \Big\| \bT_{\Pi_1}(\pi^{\boldsymbol{r}}f) \Big\|_{L_p(\bR^d,w\,dx)}^p \Big)^{1/p}
\\
&\leq N [w]_{A_p(\bR^d)}^{\max(1,(p-1)^{-1})}\|\pi^{\boldsymbol{r}} f\|_{L_p(\bR^d,w\,dx)}.
\end{aligned}
\end{equation}

By (<ref>), (<ref>) and (<ref>), we conclude that

\|\pi^{\boldsymbol{r}}f\|_{L_p(\bR^d,w\,dx)}\approx \|f\|_{H_p^{\boldsymbol{r}}(\bR^d,w\,dx)}.

Similarly, we compute $\|\pi^{\boldsymbol{r}}f\|_{B_{p,q}^{\boldsymbol{0}}(\bR^d,w\,dx)}$.
Recall the definition first:

\begin{align*}
\|\pi^{\boldsymbol{r}}f\|_{B_{p,q}^{\boldsymbol{0}}(\bR^d,w\,dx)}
&= \| S_0 \pi^{\boldsymbol{r}}f \|_{L_p(\bR^d,w\,dx)} + \Big( \sum_{j=1}^{\infty} \Big\| \Delta_j \pi^{\boldsymbol{r}}f \Big\|_{L_p(\bR^d,w\,dx)}^q \Big)^{1/q}
\end{align*}

\begin{align*}
\|f\|_{B_{p,q}^{\boldsymbol{r}}(\bR^d,w\,dx)}
&= \| S_0 f \|_{L_p(\bR^d,w\,dx)} + \Big( \sum_{j=1}^{\infty} \Big\| 2^{\boldsymbol{r}(j)}\Delta_j f \Big\|_{L_p(\bR^d,w\,dx)}^q \Big)^{1/q}.
\end{align*}

Due to the almost orthogonal property again, it is obvious that

\begin{equation*}
\begin{gathered}
S_0\pi^{\boldsymbol{r}}f=S_0(S_0+2^{\boldsymbol{r}(1)}\Delta_1)f, \\
\Delta_1\pi^{\boldsymbol{r}}f=\Delta_1(S_0+2^{\boldsymbol{r}(1)}\Delta_1+2^{\boldsymbol{r}(2)}\Delta_2)f, \\
\Delta_j \pi^{\boldsymbol{r}}f = \Delta_{j}(2^{\boldsymbol{r}(j-1)} \Delta_{j-1} +2^{\boldsymbol{r}(j)} \Delta_{j} +2^{\boldsymbol{r}(j+1)} \Delta_{j+1}) f,\quad j\geq 2.
\end{gathered}
\end{equation*}

Moreover, setting

\begin{equation*}
\begin{gathered}
\Pi_2:=\frac{\cF[\Phi]+\cF[\Psi](2^{-1}\cdot)}{\cF[\Phi]+\sum_{j=1}^{\infty}2^{\boldsymbol{r}(j)}\cF[\Psi](2^{-j}\cdot)},\\
\Pi_3:=\frac{2^{\boldsymbol{r}(1)}(\cF[\Phi]+\cF[\Psi](2^{-1}\cdot)+\cF[\Psi](2^{-2}\cdot))}{\cF[\Phi]+\sum_{j=1}^{\infty}2^{\boldsymbol{r}(j)}\cF[\Psi](2^{-j}\cdot)},\\
\Pi_{j+2}:=\frac{2^{\boldsymbol{r}(j)}(\cF[\Psi](2^{-j+1}\cdot)+\cF[\Psi](2^{-j}\cdot)+\cF[\Psi](2^{-j-1}\cdot))}{\cF[\Phi]+\sum_{j=1}^{\infty}2^{\boldsymbol{r}(j)}\cF[\Psi](2^{-j}\cdot)},\quad j\geq 2,
\end{gathered}
\end{equation*}

we have

\begin{equation*}
\begin{gathered}
\quad 2^{\boldsymbol{r}(1)}\Delta_1f=\bT_{\Pi_3}\Delta_1\pi^{\boldsymbol{r}}f,\\
2^{\boldsymbol{r}(j)}\Delta_jf=\bT_{\Pi_{j+2}}\Delta_j \pi^{\boldsymbol{r}}f, \quad j\geq 2.
\end{gathered}
\end{equation*}

Therefore,    by Lemma <ref>,  and Proposition <ref>, we obtain

\|\pi^{\boldsymbol{r}}f\|_{B_{p,q}^{\boldsymbol{0}}(\bR^d,w\,dx)}\approx\|f\|_{B_{p,q}^{\boldsymbol{r}}(\bR^d,w\,dx)}

since all $L_p$-bounds can be chosen uniformly for all $j$ due to the assumption on $\boldsymbol{r}$ (<ref>).
The proposition is proved.

For $s\in\bR$, put $\boldsymbol{r}(j) = sj$.
Then the space $H_p^{\boldsymbol{r}}(\bR^d, w\,dx)$ is equivalent to $H_p^s(\bR^d, w\,dx)$ whose norm is given by

\| f \|_{H_p^s(\bR^d,w\,dx)} := \| (1 - \Delta)^{s/2} f \|_{L_p(\bR^d,w\,dx)}.

Recall the notation

\pi^{\boldsymbol{r}}:=S_0+\sum_{j=1}^{\infty}2^{\boldsymbol{r}(j)}\Delta_j.

By Proposition <ref>, it follows that $\|f\|_{H_p^{\boldsymbol{r}}(\bR^d,w\,dx)} \approx  \|\pi^{\boldsymbol{r}}f\|_{L_p(\bR^d,w\,dx)}
Thus it is sufficient to show

\begin{align*}
\|\pi^{\boldsymbol{r}}f\|_{L_p(\bR^d,w\,dx)} \approx \| (1 - \Delta)^{s/2} f \|_{L_p(\bR^d,w\,dx)}.
\end{align*}

Let $\Psi$ be the Schwartz function appearing in the definition of $\Delta_j$.

\cF[\Phi](\xi) = \sum_{j=-\infty}^{0}\cF[\Psi](2^{-j}\xi).

\Pi(\xi) := \frac{\cF[\Phi](\xi)+\sum_{j=1}^\infty 2^{sj} \cF[\Psi](2^{-j}\xi)}{(1+|\xi|^2)^{s/2}}.

Then $\Pi$ is a weighted Mikhlin multiplier in Proposition <ref>.
Thus we have

\left\|\sum_{j=1}^\infty 2^{\boldsymbol{r}(j)} \Delta_j f\right\|_{L_p(\bR^d,w\,dx)} = \Big\| \bT_{\Pi} \Bigl( (1-\Delta)^{s/2} f \Bigr)\Big\|_{L_p(\bR^d,w\,dx)} \leq N\| (1-\Delta)^{s/2} f\|_{L_p(\bR^d,w\,dx)}.

For the converse direction, it suffices to observe that

\widetilde{\Pi}(\xi) = \frac{(1+|\xi|^2)^{s/2} }{\cF[\Phi](\xi)+\sum_{j=1}^\infty 2^{sj}\cF[\Psi](2^{-j}\xi)}

becomes a weighted Mikhlin multiplier as well. The corollary is proved.

Let $q\in(0,\infty)$, $\boldsymbol{r}:\bZ\to(-\infty,\infty)$, and $B$ be a Banach lattice.

* We denote by $B(\ell_2^{\boldsymbol{r}})$ the set of all $B$-valued sequences $x=(x_0,x_1,\cdots)$ such that

\|x\|_{B(\ell_2^{\boldsymbol{r}})}:=\|x_0\|_B+\left\|\Big(\sum_{j=1}^{\infty}2^{2\boldsymbol{r}(j)}|x_j|^2\Big)^{1/2}\right\|_B<\infty,\quad |x|:

* We denote by $\ell_q^{\boldsymbol{r}}(B)$ the set of all $B$-valued sequences $x=(x_0,x_1,\cdots)$ such that

\|x\|_{\ell_q^{\boldsymbol{r}}(B)}:=\|x_0\|_B+\Big(\sum_{j=1}^{\infty}2^{q\boldsymbol{r}(j)}\|x_j\|_{B}^q\Big)^{1/q}<\infty.

In particular, we use the simpler notation $\ell_q^{\boldsymbol{r}}=\ell_q^{\boldsymbol{r}}(\bR)$.

Let $X$ and $Y$ be quasi-Banach spaces. We say that $X$ is a retract of $Y$ if there exist linear transformation $\frI:X\to Y$ and $\frP:Y\to X$ such that $\frP\frI$ is an identity operator in $X$.

Let $p\in(1,\infty)$, $q\in(0,\infty)$, and $w\in A_p(\bR^d)$. Suppose that

\begin{equation}
\label{locsim}
\sup_{j\in\bZ}|\boldsymbol{r}(j+1)-\boldsymbol{r}(j)|=:C_0<\infty.
\end{equation}

* Then $H_p^{\boldsymbol{r}}(\bR^d,w\,dx)$ is a retract of $L_p(\bR^d,w\,dx)(\ell_2^{\boldsymbol{r}})$.

* Then $B_{p,q}^{\boldsymbol{r}}(\bR^d,w\,dx)$ is a retract of $\ell_q^{\boldsymbol{r}}(L_p(\bR^d,w\,dx))$.

We suggest a unified method to prove both (i) and (ii) simultaneously.

Consider two mappings:

\begin{equation*}
f \mapsto \frI(f):=(S_0f,\Delta_1f,\Delta_2f,\cdots)
\end{equation*}

\begin{equation*}
\boldsymbol{f}=(f_0,f_1,\cdots)\mapsto \frP(\boldsymbol{f}):=S_0(f_0+f_1)+\sum_{j=1}^{\infty}\sum_{l=-1}^1\Delta_{j}f_{j+l}.
\end{equation*}

By using the fact that $f= S_0(f) + \sum_{j=1}^\infty \Delta_j f$,
it is each to check that $\frP\frI$ is an identity operator in $X$. Moreover,
due to the definitions of the function spaces,

\begin{align}
\label{J norm equivalence}
\|f\|_{X}=\|\frI(f)\|_{Y}.
\end{align}

In other words, the target space of the mapping $f \mapsto \frI(f)$ is $Y$
and this implies that $\frI$ is a linear transformation from $X$ to $Y$ since the linearity of the mapping is obvious.

Now, it only remains to show that $\frP$ is a linear transformation from $Y$ to $X$.
Since the linearity is trivial as before, it is sufficient to show that the target space of the mapping
$\boldsymbol{f}=(f_0,f_1,\cdots)\mapsto \frP(\boldsymbol{f})$ is $X$.
By the almost orthogonality of the Littlewood-Paley projections,

\begin{equation}
\label{22.09.13.16.17}
\begin{gathered}
\Delta_1\frP(\boldsymbol{f})=\Delta_1(S_0+\Delta_1)f_0+\Delta_1f_1+\Delta_1(\Delta_1+\Delta_2)f_2+\Delta_1\Delta_2f_3,
\end{gathered}
\end{equation}

\begin{equation}
\begin{aligned}
\Delta_i\frP(\boldsymbol{f})&=\Delta_i\Delta_{i-1}f_{i-2}+\Delta_i(\Delta_{i-1}+\Delta_i)f_{i-1}+\Delta_if_i+\Delta_i(\Delta_i+\Delta_{i+1})f_{i+1}+\Delta_i\Delta_{i+1}f_{i+2}\\
&=:T_i^1f_{i-2}+T_i^2f_{i-1}+T_i^3f_i+T_i^4f_{i+1}+T_i^5f_{i+2},\quad i\geq2.
\end{aligned}
\end{equation}

The remaining part of the proof becomes slightly different depending on $Y=L_p(\bR^d,w\,dx)(\ell_2^{\boldsymbol{r}})$ or $Y=\ell_q^{\boldsymbol{r}}(L_p(\bR^d,w\,dx))$.

* Due to (<ref>),

\sum_{i=2}^{\infty}2^{2\boldsymbol{r}(i)}|\Delta_i\frP(\boldsymbol{f})|^2\leq N\sum_{k=1}^{5}\cI_k,

\cI_k=\sum_{i=2}^{\infty}|T_i^k(2^{\boldsymbol{r}(i)}f_{i+k-3})|^2,\quad k=1,2,3,4,5.

By Lemma <ref>, the sequence of linear operators $\{T_j^k\}_{j=2}^{\infty}$ satisfies (<ref>) for all $k=1,2,3,4,5$. Using Remark <ref> and (<ref>), we have

\Big\|\Big(\sum_{i=2}^{\infty}2^{2\boldsymbol{r}(i)}|\Delta_i\frP(\boldsymbol{f})|^2\Big)^{1/2}\Big\|_{L_p(\bR^d,w\,dx)}\leq N\|\boldsymbol{f}\|_{L_p(\bR^d,w\,dx)(\ell_2^{\boldsymbol{r}})}.

This implies that $\frP(\boldsymbol{f})\in H_p^{\boldsymbol{r}}(\bR^d,w\,dx)$.

* By Lemma <ref>,

\begin{align}
\label{2023012880}
\|S_0\frP(\boldsymbol{f})\|_{L_p(\bR^d,w\,dx)}\leq N \sum_{j=0}^{2}\|f_{j}\|_{L_p(\bR^d,w\,dx)}
\end{align}

\begin{align}
\label{2023012881}
\|\Delta_i\frP(\boldsymbol{f})\|_{L_p(\bR^d,w\,dx)}\leq N \sum_{j=-2}^{2}\|f_{i+j}\|_{L_p(\bR^d,w\,dx)},\quad \forall i\in\bN,
\end{align}

where $N$ is independent of $i$ and $f_i=0$ if $i<0$.
Moreover, due to (<ref>),

\begin{equation}
\label{22.04.11.15.09}
2^{\boldsymbol{r}(i)}\|\Delta_i\frP(\boldsymbol{f})\|_{L_p(\bR^d,w\,dx)}\leq N\sum_{j=-2}^{2}2^{\boldsymbol{r}(i+j)}\|f_{i+j}\|_{L_p(\bR^d,w\,dx)}
\end{equation}

and the constant $N$ is independent of $i$.
Finally, recalling the definition of $B_{p,q}^{\boldsymbol{r}}(\bR^d,w\,dx)$ and combining all (<ref>), (<ref>), and (<ref>), we have

\begin{equation*}
\begin{aligned}
\|\frP(\boldsymbol{f})\|_{B_{p,q}^{\boldsymbol{r}}(\bR^d,w\,dx)} \leq N\|\boldsymbol{f}\|_{\ell_q^{\boldsymbol{r}}(L_p(\bR^d,w\,dx))},\quad q\in(0,\infty),
\end{aligned}
\end{equation*}

where $N$ is independent of $\boldsymbol{f}$. Therefore, $\frP$ is a linear transformation from $Y$ to $X$.

The lemma is proved.

Let $p\in(1,\infty)$, $q\in(0,\infty)$, $w\in A_p(\bR^d)$, and $\boldsymbol{r}:\bZ\to(-\infty,\infty)$ be a sequence satisfying

\sup_{j\in\bZ}|\boldsymbol{r}(j+1)-\boldsymbol{r}(j)|<\infty.

* The space $X$ is a quasi-Banach space

* The closure of $C_c^{\infty}(\bR^d)$ under the quasi-norm $\|\cdot\|_{X}$ is $X$,

where the space $X$ indicates $H_p^{\boldsymbol{r}}(\bR^d,w\,dx)$ or $B_{p,q}^{\boldsymbol{r}}(\bR^d,w\,dx)$.

We put

as in the previous lemma.

* All properties of a quasi-Banach space is obvious except completeness.
Thus we only prove the completeness. Let $\{f_n\}_{n=1}^{\infty}$ be a Cauchy sequence in $X$.
It is obvious that $Y$ is a Banach space due to the completeness of $L_p$-spaces with general measures.
Therefore, using the equivalence of the norms in (<ref>), one can find a $\boldsymbol{f}\in Y$ such that

\begin{align*}
\frI(f_n)=(S_0f_n,\Delta_1f_n,\cdots)\to\boldsymbol{f}\quad \text{ in } Y.
\end{align*}

By Lemma <ref>, $\frP(\boldsymbol{f})\in X$ and

\|f_n-\frP(\boldsymbol{f})\|_{X}\leq N\|\frI(f_n)-\boldsymbol{f}\|_{Y}\to0

as $n\to\infty$. Therefore, $X$ is complete.

* With the help of Proposition <ref>, it is sufficient to consider the case that the order is $\boldsymbol{0}$.
Then $\cS(\bR^d)$ is dense in $X$ due to <cit.>.
Finally, based on the typical truncation and diagonal arguments, the result is obtained.

The proposition is proved.

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aainstitutetext: Department of Physics, University of Oxford,
Denys Wilkinson Building, Keble Road, OX1 3RH, Oxford, United
Kingdombbinstitutetext: Scuola Normale Superiore,
piazza dei Cavalieri 7, 56 126, Pisa, Italyccinstitutetext: INFN, sezione di
Pisa,
largo Pontecorvo 3, 56 127, Pisa, Italy

# Mixing and $C\\!P$ violation in
${{D}^{0}}\\!\rightarrow{{{K}^{-}}{{\pi}^{+}}}$ decays

Tommaso Pajero111Corresponding author. b,c and Michael Joseph Morello
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We review the experimental methods to measure mixing and time-dependent
$C\\!P$ violation in ${D}^{0}$ decays into two hadrons. While these phenomena
are usually neglected for ${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$
decays, this approximation is not always justified. In particular, it produces
a bias on the measurement of the parameter $y_{{C\\!P}}$, when this is
performed by relying on ${{D}^{0}}\\!\rightarrow{{{K}^{-}}{{\pi}^{+}}}$ decays
as a normalisation channel, whose size is around 40% of the precision of the
current world average. Finally, we estimate the sensitivity to the weak mixing
phases achievable by studying ${{D}^{0}}\\!\rightarrow{{{K}^{-}}{{\pi}^{+}}}$
and untagged ${D}\\!\rightarrow{{{K}^{-}}{{\pi}^{+}}}$ decays, where $D$
stands for either of the ${D}^{0}$ and ${\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}$ mesons. Contrary to Cabibbo-
suppressed ${{D}^{0}}\\!\rightarrow{{h}^{+}}{{h}^{-}}$ decays, these decay
channels allow to measure these phases without final-state dependent nuisance
contributions from the decay amplitudes, but their sensitivity is lower by a
factor of six.

###### Keywords:

Charm physics, mixing, $C\\!P$ violation

††arxiv: 2106.02014

##  1 Introduction

Charm hadrons are the only hadrons made up of solely up-type quarks where
mixing and $C\\!P$ violation can be studied. Thus, they provide a unique
opportunity to detect new interactions beyond the Standard Model (SM) that
leave down-type quarks unaffected Grossman:2006jg . Both mixing and $C\\!P$
violation are highly suppressed in charm by the Glashow–Iliopoulos–Maiani
mechanism and by the hierarchical structure of the Cabibbo–Kobayashi–Maskawa
(CKM) matrix, and have been observed only in recent years Aubert:2007wf ;
delAmoSanchez:2010xz ; Lees:2012qh ; Aaltonen:2013pja ; Ko:2014qvu ;
Peng:2014oda ; Staric:2015sta ; LHCb-PAPER-2015-057 ; BaBar:2016kvp ; LHCb-
PAPER-2016-033 ; LHCb-PAPER-2017-046 ; LHCb-PAPER-2018-038 ; LHCb-
PAPER-2019-001 ; LHCb-PAPER-2021-009 ; LHCb-PAPER-2021-033 ; LHCb-
PAPER-2019-006 . The mass and decay-width differences of the ${D}^{0}$ mass
eigenstates are known with relative precision of around 10% LHCb-
PAPER-2021-033 ; while $C\\!P$ violation in the decay amplitudes has been
observed LHCb-PAPER-2019-006 , evidence of $C\\!P$ violation in ${D}^{0}$
mixing is still missing Lees:2012qh ; Aaltonen:2014efa ; Staric:2015sta ;
LHCb-PAPER-2014-069 ; LHCb-PAPER-2016-063 ; LHCb-PAPER-2019-032 ; LHCb-
PAPER-2020-045 ; LHCb-PAPER-2017-046 ; LHCb-PAPER-2019-001 ; LHCb-
PAPER-2021-009 ; LHCb-PAPER-2021-033 . The upgrades of the LHCb experiment and
the Belle II experiment are expected to improve the precision on these
observables by about a factor of 10 within the next two decades
Bediaga:2018lhg ; Kou:2018nap , allowing to attain per-cent precision on the
mixing parameters, and a precision on $C\\!P$ violation in the mixing
comparable to the size of the SM predictions Bigi:2011re ; Bobrowski:2010xg ;
Kagan:2020vri ; Li:2020xrz ; kagan:charm2021 .

The most precise measurements of these phenomena employ multi-body decays such
as ${{D}^{0}}\\!\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{+}}{{\pi}^{-}}$ LHCb-
PAPER-2019-001 ; LHCb-PAPER-2021-009 , or ${D}^{0}$ decays into either of the
Cabibbo-suppressed (CS) final states ${{K}^{+}}{{K}^{-}}$ and
${{\pi}^{+}}{{\pi}^{-}}$ Lees:2012qh ; Staric:2015sta ; LHCb-PAPER-2018-038 ;
LHCb-PAPER-2016-063 ; LHCb-PAPER-2019-032 ; LHCb-PAPER-2020-045 , or into the
wrong-sign (WS) final state ${{K}^{+}}{{\pi}^{-}}$ LHCb-PAPER-2017-046 .222The
${{D}^{0}}\\!\rightarrow{{{K}^{+}}{{\pi}^{-}}}$ decay is named wrong-sign
since, when the ${D}^{0}$ flavour at production is determined by relying on
the ${{D}^{*}(2010)^{+}}\\!\rightarrow{{D}^{0}}{{\pi}^{+}}$ strong decay, the
charges of the pions from the ${D}^{*}(2010)^{+}$ and ${D}^{0}$ decays have
opposite signs. Measurements of two-body decays rely on right-sign (RS)
${{D}^{0}}\\!\rightarrow{{{K}^{-}}{{\pi}^{+}}}$ decays as a normalisation
channel. Since RS decays mainly consist of Cabibbo-favoured (CF) decays
without ${D}^{0}$ flavour oscillation, while the decay amplitude following
flavour oscillation is doubly Cabibbo-suppressed (DCS), the effect of mixing
on their decay rate is usually neglected in the literature. However, it can no
longer be ignored at the current level of experimental precision, and its size
needs to be carefully assessed.

This is the aim of the present article, which extends the discussion in refs.
Xing:1996pn ; Kagan:2020vri  by drawing the experimental consequences of the
results presented therein. The article is structured as follows. Section 2
summarises the theoretical formalism introduced in refs. Grossman:2009mn ;
Kagan:2009gb ; Kagan:2020vri  to parametrise mixing and time-dependent $C\\!P$
violation in ${D}^{0}$ decays, limiting the discussion to decays into two
hadrons. The corresponding time-dependent decay rates are presented in section
3, where the observables ${y_{{C\\!P}}^{{{{K}^{-}}{{\pi}^{+}}}}}$ and ${\Delta
Y_{{{{K}^{-}}{{\pi}^{+}}}}}$, analogues of the observables $y_{{C\\!P}}^{f}$
and $\Delta Y_{f}$ defined for the CS final states $f={{{K}^{+}}{{K}^{-}}}$
and ${{\pi}^{+}}{{\pi}^{-}}$, are introduced to parametrise the first-order
contributions of mixing and time-dependent $C\\!P$ violation to the RS decay
rate. The parameters ${y_{{C\\!P}}^{K\pi\textnormal{,untag}}}$ and $\Delta
Y_{K\pi}^{\textnormal{untag}}$ are similarly defined for untagged
${D}\\!\rightarrow{{{K}^{-}}{{\pi}^{+}}}$ decays, where the flavour at
production of the ${D}^{0}$ meson is unknown. Section 4 reviews the
experimental approaches to measure the time-dependent decay rates, and
pinpoints a bias equal to $-{y_{{C\\!P}}^{{{{K}^{-}}{{\pi}^{+}}}}}$ which
affects most of the measurements of $y_{{C\\!P}}^{f}$ performed to date and
has been missed in the literature. The improvement in precision that might be
achieved by including the $\Delta Y_{{{{K}^{-}}{{\pi}^{+}}}}$ and $\Delta
Y_{K\pi}^{\textnormal{untag}}$ observables in the experimental program of the
LHCb experiment, and by measuring $y_{{C\\!P}}^{f}$ with untagged $D$ decays,
are discussed as well. Section 5 concludes by summarising the main findings.
Finally, appendix A quantifies the contribution of mixing in RS decays to the
parameters measured in the analysis of the time-dependent ratio of the WS to
RS decays rates, and of the terms quadratic in the mixing parameters to the
expansion of the time-dependent asymmetry of the decay rates of ${D}^{0}$ and
${\kern 1.79993pt\overline{\kern-1.79993ptD}}{}^{0}$ mesons into the final
states ${{K}^{+}}{{K}^{-}}$ or ${{\pi}^{+}}{{\pi}^{-}}$. The former
contribution might become relevant at the LHCb upgrades.

##  2 Formalism

This section reviews the theoretical formalism introduced in refs.
Grossman:2009mn ; Kagan:2009gb ; Kagan:2020vri  to parametrise mixing and
time-dependent $C\\!P$ violation in ${D}^{0}$ decays.

###  2.1 ${D}^{0}$ mixing

An arbitrary linear combination of ${D}^{0}$ and ${\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}$ flavour eigenstates,

$a(t)|{{D}^{0}}\rangle+b(t)|{{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}\rangle,$ (1)

evolves in time according to the Schrödinger equation,

$i\frac{\mathrm{d}}{\mathrm{d}t}\begin{pmatrix}a(t)\\\
b(t)\end{pmatrix}=\bm{H}\begin{pmatrix}a(0)\\\ b(0)\end{pmatrix},$ (2)

where $\bm{H}$ is the $2\times 2$ effective Hamiltonian governing the dynamics
of the ${D}^{0}$–${\kern 1.79993pt\overline{\kern-1.79993ptD}}{}^{0}$
system.333Natural units are employed throughout, $\hbar=c=1$. This Hamiltonian
can be expressed in terms of two Hermitian matrices as
$\bm{H}\equiv\bm{M}-\tfrac{i}{2}\bm{\Gamma}$. The $C\\!PT$ symmetry, which is
assumed hereafter, implies $H_{11}=H_{22}=M-\tfrac{i}{2}\Gamma$, with $M$ and
$\Gamma$ the mass and decay width of the ${D}^{0}$ meson. The transition
amplitudes between ${D}^{0}$ and ${\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}$ mesons,

$\langle{{D}^{0}}|H|{{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}\rangle=M_{12}-\tfrac{i}{2}\Gamma_{12},\qquad\langle{{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}|H|{{D}^{0}}\rangle=M^{\ast}_{12}-\tfrac{i}{2}\Gamma^{\ast}_{12},$
(3)

depend on the dispersive and absorptive mixing amplitudes, $M_{12}$ and
$\Gamma_{12}$, which correspond to off-shell and on-shell transitions,
respectively. They are parametrised in terms of two $C\\!P$-even mixing
parameters,

$x_{12}\equiv\frac{2\lvert M_{12}\rvert}{\Gamma},\qquad
y_{12}\equiv\frac{\lvert\Gamma_{12}\rvert}{\Gamma},$ (4)

and one $C\\!P$-odd weak phase,

$\phi_{12}\equiv\arg\left(\frac{M_{12}}{\Gamma_{12}}\right).$ (5)

The additional global phase shared by $M_{12}$ and $\Gamma_{12}$ depends on
the phase conventions for quarks and mesons and is unobservable. The
parameters $x_{12}$ and $y_{12}$ are equal to the absolute value of normalised
mass and decay-width differences of the eigenstates of $\bm{H}$,
$x\equiv\Delta m/\Gamma$ and $y\equiv\Delta\Gamma/(2\Gamma)$, up to
corrections quadratic in $\sin\phi_{12}$.

###  2.2 Time-dependent $C\\!P$ violation

The ${D}^{0}$ and ${\kern 1.79993pt\overline{\kern-1.79993ptD}}{}^{0}$ decay
amplitudes into the $C\\!P$-conjugate final states $f$ and $\bar{f}$ are
denoted as

$\displaystyle{A_{f}}\equiv$ $\displaystyle\langle
f|\mathcal{H}|{{D}^{0}}\rangle,$ $\displaystyle{\bar{A}_{f}}\equiv$
$\displaystyle\langle f|\mathcal{H}|{{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}\rangle,$ (6)
$\displaystyle{A_{\bar{f}}}\equiv$
$\displaystyle\langle\bar{f}|\mathcal{H}|{{D}^{0}}\rangle,$
$\displaystyle{\bar{A}_{\bar{f}}}\equiv$
$\displaystyle\langle\bar{f}|\mathcal{H}|{{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}\rangle,$

where $\mathcal{H}$ is the $\lvert\Delta C\rvert=1$ effective Hamiltonian.

For CS decays into $C\\!P$ eigenstates, $C\\!P$ violation in the interference
of the decay amplitudes with and without flavour oscillation is parametrised
in terms of the following observables Kagan:2020vri ,

$\displaystyle\lambda_{f}^{M}\equiv$ $\displaystyle\frac{M_{12}}{\lvert
M_{12}\rvert}\frac{{A_{f}}}{{\bar{A}_{f}}}\equiv{\eta^{{C\\!P}}_{f}}\left\lvert\frac{{A_{f}}}{{\bar{A}_{f}}}\right\rvert
e^{i\phi_{f}^{M}},$ (7) $\displaystyle\lambda_{f}^{\Gamma}\equiv$
$\displaystyle\frac{\Gamma_{12}}{\lvert\Gamma_{12}\rvert}\frac{{A_{f}}}{{\bar{A}_{f}}}\equiv{\eta^{{C\\!P}}_{f}}\left\lvert\frac{{A_{f}}}{{\bar{A}_{f}}}\right\rvert
e^{i\phi_{f}^{\Gamma}},$

where $\eta^{{C\\!P}}_{f}$ equals $+1$ for $C\\!P$-even final states such as
${{K}^{+}}{{K}^{-}}$ and ${{\pi}^{+}}{{\pi}^{-}}$, and $-1$ for $C\\!P$-odd
final states such as ${{K}^{0}_{\mathrm{S}}}\phi$ and
${{K}^{0}_{\mathrm{S}}}\omega$, and the weak phases $\phi_{f}^{M}$ and
$\phi_{f}^{\Gamma}$ are quark- and meson-rephasing invariant and satisfy
$\phi_{12}=\phi_{f}^{M}-\phi_{f}^{\Gamma}$.

The analogous parameters for the decays into the RS and WS final states
$f={{K}^{-}}{{\pi}^{+}}$ and $\bar{f}={{K}^{+}}{{\pi}^{-}}$ are

$\displaystyle\lambda_{f}^{M}\equiv\frac{M_{12}}{\lvert
M_{12}\rvert}\frac{{A_{f}}}{{\bar{A}_{f}}}\equiv-\left\lvert\frac{{A_{f}}}{{\bar{A}_{f}}}\right\rvert
e^{i(\phi_{f}^{M}-\Delta_{f})},$
$\displaystyle\lambda_{\bar{f}}^{M}\equiv\frac{M_{12}}{\lvert
M_{12}\rvert}\frac{{A_{\bar{f}}}}{{\bar{A}_{\bar{f}}}}\equiv-\left\lvert\frac{{A_{\bar{f}}}}{{\bar{A}_{\bar{f}}}}\right\rvert
e^{i(\phi_{f}^{M}+\Delta_{f})},$ (8)
$\displaystyle\lambda_{f}^{\Gamma}\equiv\frac{\Gamma_{12}}{\lvert\Gamma_{12}\rvert}\frac{{A_{f}}}{{\bar{A}_{f}}}\equiv-\left\lvert\frac{{A_{f}}}{{\bar{A}_{f}}}\right\rvert
e^{i(\phi_{f}^{\Gamma}-\Delta_{f})},$
$\displaystyle\lambda_{\bar{f}}^{\Gamma}\equiv\frac{\Gamma_{12}}{\lvert\Gamma_{12}\rvert}\frac{{A_{\bar{f}}}}{{\bar{A}_{\bar{f}}}}\equiv-\left\lvert\frac{{A_{\bar{f}}}}{{\bar{A}_{\bar{f}}}}\right\rvert
e^{i(\phi_{f}^{\Gamma}+\Delta_{f})},$

where $\Delta_{f}$, the strong-phase difference between the doubly Cabibbo-
suppressed (DCS) and CF decay amplitudes, equals zero in the limit of $U$-spin
symmetry,444Note that this convention for the strong phase differs from those
adopted in refs. HFLAV18  and LHCb-PAPER-2021-033 , respectively:
$\Delta_{{{K}^{-}}{{\pi}^{+}}}=-\delta_{K\pi}=-\delta^{K\pi}_{D}-\pi$. and the
weak phases $\phi_{f}^{M}$ and $\phi_{f}^{\Gamma}$ again satisfy
$\phi_{12}=\phi_{f}^{M}-\phi_{f}^{\Gamma}$, but in general differ from those
of eq. 7. Finally, the minus sign in the right-hand side of the definitions
accounts for the overall minus sign of the CKM matrix elements involved in the
tree-level decay amplitudes, with respect to CS decays.

The chosen conventions imply that all of the phases $\phi_{f}^{M}$ and
$\phi_{f}^{\Gamma}$ in eqs. 7 and 8 equal the intrinsic dispersive and
absorptive mixing phases $\phi_{2}^{M}$ and $\phi_{2}^{\Gamma}$, defined as
the phases of $M_{12}$ and $\Gamma_{12}$ with respect to their dominant
$\Delta U=2$ contributions (hence the subscript “2”) Kagan:2020vri , up to
subleading corrections due to $C\\!P$ violation in the decay. These
corrections depend on the final state, but are shared by the dispersive and
absorptive weak phases:
$\delta\phi_{f}\equiv\phi^{M}_{f}-\phi^{M}_{2}=\phi^{\Gamma}_{f}-\phi^{\Gamma}_{2}$.
In the SM, the phases $\phi_{2}^{M}$ and $\phi_{2}^{\Gamma}$ are naively
predicted to be of the order of $2\text{\,mrad}$, even though enhancements up
to one order of magnitude are not excluded Bigi:2011re ; Bobrowski:2010xg ;
Kagan:2020vri ; Li:2020xrz . An upper bound of $5\text{\,mrad}$ on the size of
$\phi^{\Gamma}_{2}$ has been proposed recently kagan:charm2021 . The final-
state dependent corrections, $\delta\phi_{f}$, are smaller than $10^{-6}$ for
CF and DCS decays, and are suppressed with respect to $\phi_{2}^{M}$ and
$\phi_{2}^{\Gamma}$ by one order in the $U$-spin breaking parameter
$\epsilon\sim 0.4$ for CS decays Kagan:2020vri . Since no evidence of time-
dependent $C\\!P$ violation has been reported to date, they are neglected in
the current combinations of experimental results to maximise the achieved
precision LHCb-PAPER-2021-033 ; Kagan:2020vri ; HFLAV18 . Once an evidence is
found, measurements of CS decays may be dropped from the combinations or
replaced by the arithmetic average of the experimental observables measured in
${{D}^{0}}\\!\rightarrow{{K}^{+}}{{K}^{-}}$ and
${{D}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ decays, which allows to reduce
final-state dependent contributions by a further factor of $\epsilon$
Kagan:2020vri .

##  3 Time-dependent decay rates

The time-dependent decay rates for ${D}^{0}$ and ${\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}$ mesons produced in their flavour
eigenstates at time zero to decay into the final state $f$ at time $t$,
indicated with $\Gamma({{D}^{0}}(t)\\!\rightarrow f)$ and $\Gamma({{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}(t)\\!\rightarrow f)$,
respectively, can be obtained by substituting the definitions above in eq.
(50) of ref. Kagan:2020vri . Since both mixing parameters are smaller than
$1\%$ LHCb-PAPER-2021-033 , the rates are expanded up to quadratic order in
$x_{12}$ and $y_{12}$. This approximation is expected to hold with excellent
precision also at the LHCb and Belle II upgrades as, owing to the exponential
decay of the rates, candidates with $\Gamma t\lesssim 10$ do not contribute
significantly to the experimental precision and are often discarded to ease
the estimation of systematic uncertainties LHCb-PAPER-2018-038 ; LHCb-
PAPER-2020-045 ; LHCb-PAPER-2021-009 . In the following, decay time is
expressed in ${D}^{0}$-lifetime units, $\tau\equiv\Gamma t$, to simplify the
notation. For the same reason, the normalisation factors from phase-space
integration, which are common to ${D}^{0}$ and ${\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}$ decay rates for each final state
$f$ and are additionally equal for the ${{K}^{-}}{{\pi}^{+}}$ and
${{K}^{+}}{{\pi}^{-}}$ final states, are omitted.

###  3.1 Cabibbo-suppressed decays

For CS final states $f$ that are $C\\!P$ eigenstates, the time-dependent decay
rates are parametrised up to second order in the mixing parameters as

$\displaystyle\Gamma({{D}^{0}}(\tau)\\!\rightarrow f)$ $\displaystyle\equiv
e^{-\tau}\lvert{A_{f}}\rvert^{2}\left(1+c^{+}_{f}\tau+c^{\prime+}_{f}\tau^{2}\right),$
(9) $\displaystyle\Gamma({{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}(\tau)\\!\rightarrow f)$
$\displaystyle\equiv
e^{-\tau}\lvert{\bar{A}_{f}}\rvert^{2}\left(1+c^{-}_{f}\tau+c^{\prime-}_{f}\tau^{2}\right),$

where the coefficients $c^{\pm}_{f}$ and $c^{\prime\pm}_{f}$ are equal to

$\displaystyle c^{\pm}_{f}$ $\displaystyle\approx{\eta^{{C\\!P}}_{f}}\left[\mp
x_{12}\sin\phi_{f}^{M}-y_{12}\cos\phi_{f}^{\Gamma}(1\mp{a^{d}_{f}})\right],$
(10a) $\displaystyle c^{\prime\pm}_{f}$
$\displaystyle\approx\frac{1}{2}\left[y_{12}^{2}\pm
x_{12}y_{12}\sin\phi_{12}\mp(x_{12}^{2}+y_{12}^{2}){a^{d}_{f}}\right].$ (10b)

In eq. 10, the parameter

$a_{f}^{d}\equiv\frac{\lvert{A_{f}}\rvert^{2}-\lvert{\bar{A}_{f}}\rvert^{2}}{\lvert{A_{f}}\rvert^{2}+\lvert{\bar{A}_{f}}\rvert^{2}}\approx
1-\left\lvert\frac{{\bar{A}_{f}}}{{A_{f}}}\right\rvert$ (11)

is the $C\\!P$ asymmetry in the decay, and terms proportional to $a^{d}_{f}$
have been expanded to first order in the $C\\!P$-violation parameters
$a^{d}_{f}$, $\sin\phi_{f}^{M}$ and $\sin\phi_{f}^{\Gamma}$.

The following $C\\!P$-even and $C\\!P$-odd combinations of $c^{+}_{f}$ and
$c^{-}_{f}$ are more convenient as experimental observables,

$\displaystyle{y_{{C\\!P}}^{f}}$
$\displaystyle\equiv-\frac{c^{+}_{f}+c^{-}_{f}}{2}\approx{\eta^{{C\\!P}}_{f}}y_{12}\cos\phi_{f}^{\Gamma},$
(12a) $\displaystyle{\Delta Y_{f}}$
$\displaystyle\equiv\frac{c^{+}_{f}-c^{-}_{f}}{2}\approx{\eta^{{C\\!P}}_{f}}(-x_{12}\sin\phi^{M}_{f}+y_{12}a^{d}_{f}).$
(12b)

In the limit of $C\\!P$ symmetry, the parameter $y_{{C\\!P}}^{f}$ equals
$y_{12}$ ($-y_{12}$) for $C\\!P$-even ($C\\!P$-odd) final states, and $\Delta
Y_{f}$ equals zero.

###  3.2 Wrong-sign and right-sign decays

In this section, $f$ ($\bar{f}$) denotes the RS final state
${{K}^{-}}{{\pi}^{+}}$ (the WS final state ${{K}^{+}}{{\pi}^{-}}$), and the
notation follows ref. pajero:2021  rather than ref. Kagan:2020vri . The ratios
of the DCS to CF branching fractions of ${D}^{0}$ and ${\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}$ mesons are denoted as

$R_{f}^{+}\equiv\lvert{A_{\bar{f}}}/{A_{f}}\rvert^{2},\qquad
R_{f}^{-}\equiv\lvert{\bar{A}_{f}}/{\bar{A}_{\bar{f}}}\rvert^{2};$ (13)

their average is denoted as

$R_{f}\equiv\frac{R_{f}^{+}+R_{f}^{-}}{2}$ (14)

and is equal to $(3.43\pm 0.02)\times 10^{-3}$ LHCb-PAPER-2021-033 . Finally,
the $C\\!P$ asymmetry in the decay for CF decays is denoted as

${a^{d}_{f}}\equiv\frac{\lvert{A_{f}}\rvert^{2}-\lvert{\bar{A}_{\bar{f}}}\rvert^{2}}{\lvert{A_{f}}\rvert^{2}+\lvert{\bar{A}_{\bar{f}}}\rvert^{2}}\approx
1-\left\lvert\frac{{\bar{A}_{\bar{f}}}}{{A_{f}}}\right\rvert,$ (15)

and its analogue for DCS decays as

${a^{d}_{\bar{f}}}\equiv\frac{\lvert{A_{\bar{f}}}\rvert^{2}-\lvert{\bar{A}_{f}}\rvert^{2}}{\lvert{A_{\bar{f}}}\rvert^{2}+\lvert{\bar{A}_{f}}\rvert^{2}}\approx
1-\left\lvert\frac{{\bar{A}_{f}}}{{A_{\bar{f}}}}\right\rvert.$ (16)

Both of these asymmetries are expected to be beyond experimental reach in the
SM Grossman:2006jg .555As a consequence, the following equality is expected to
hold, $R^{+}_{f}=R^{-}_{f}=R_{f}$. Allowing for $C\\!P$ violation in the
decay, instead, these parameters are related as $R^{\pm}_{f}\approx
R_{f}[1\pm({a^{d}_{\bar{f}}}-{a^{d}_{f}})]$ up to first order in
$a^{d}_{\bar{f}}$ and $a^{d}_{f}$. The parameter
$A_{D}\equiv(R^{+}_{f}-R^{-}_{f})/(R^{+}_{f}+R^{-}_{f})$ employed in the fits
performed by the HFLAV collaboration HFLAV18  is equal to
${a^{d}_{\bar{f}}}-{a^{d}_{f}}$ up to first order in $a^{d}_{\bar{f}}$ and
$a^{d}_{f}$.

The time-dependent decay rates of RS decays are parametrised as

$\displaystyle\Gamma({{D}^{0}}(\tau)\\!\rightarrow f)$ $\displaystyle\equiv
e^{-\tau}\lvert{A_{f}}\rvert^{2}\left(1+\sqrt{R_{f}}c^{+}_{f}\tau+c^{\prime+}_{f}\tau^{2}\right),$
(17) $\displaystyle\Gamma({{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}(\tau)\\!\rightarrow\bar{f})$
$\displaystyle\equiv
e^{-\tau}\lvert{\bar{A}_{\bar{f}}}\rvert^{2}\left(1+\sqrt{R_{f}}c^{-}_{f}\tau+c^{\prime-}_{f}\tau^{2}\right),$

up to second order in the mixing parameters, where the coefficients
$c^{\pm}_{f}$ and $c^{\prime\pm}_{f}$ are equal to

$\displaystyle c^{\pm}_{f}$
$\displaystyle\approx\Big{[}1\mp\tfrac{1}{2}(a^{d}_{\bar{f}}+a^{d}_{f})\Big{]}(-x_{12}\cos\phi_{f}^{M}\sin\Delta_{f}+y_{12}\cos\phi_{f}^{\Gamma}\cos\Delta_{f})$
(18) $\displaystyle\hskip 156.49014pt\pm
x_{12}\sin\phi_{f}^{M}\cos\Delta_{f}\pm
y_{12}\sin\phi_{f}^{\Gamma}\sin\Delta_{f},$ $\displaystyle c^{\prime\pm}_{f}$
$\displaystyle\approx\tfrac{1}{4}(y_{12}^{2}-x_{12}^{2})+\tfrac{1}{4}R_{f}(1\mp
a^{d}_{\bar{f}}\mp
a^{d}_{f})(x_{12}^{2}+y_{12}^{2})\pm\tfrac{1}{2}R_{f}x_{12}y_{12}\sin\phi_{12}.$

In eq. 18, terms multiplying $a^{d}_{f}$ or $a^{d}_{\bar{f}}$ have been
expanded to first order in the $C\\!P$-violation parameters $a^{d}_{f}$,
$a^{d}_{\bar{f}}$, $\sin\phi_{f}^{M}$ and $\sin\phi_{f}^{\Gamma}$.

Employing the same approximations as above, the time-dependent decay rates of
WS decays are parametrised as

$\displaystyle\Gamma({{D}^{0}}(\tau)\\!\rightarrow\bar{f})$
$\displaystyle\equiv
e^{-\tau}\lvert{A_{f}}\rvert^{2}\left(R_{f}^{+}+\sqrt{R_{f}^{+}}c^{+}_{\bar{f}}\tau+c^{\prime+}_{\bar{f}}\tau^{2}\right),$
(19) $\displaystyle\Gamma({{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}(\tau)\\!\rightarrow f)$
$\displaystyle\equiv
e^{-\tau}\lvert{\bar{A}_{\bar{f}}}\rvert^{2}\left(R_{f}^{-}+\sqrt{R_{f}^{-}}c^{-}_{\bar{f}}\tau+c^{\prime-}_{\bar{f}}\tau^{2}\right),$

where the coefficients $c^{\pm}_{\bar{f}}$ and $c^{\prime\pm}_{\bar{f}}$ are
approximately equal to

$\displaystyle c^{\pm}_{\bar{f}}$ $\displaystyle\approx(1\mp
a_{f}^{d})(x_{12}\cos\phi_{f}^{M}\sin\Delta_{f}+y_{12}\cos\phi_{f}^{\Gamma}\cos\Delta_{f})$
(20) $\displaystyle\hskip 156.49014pt\pm
x_{12}\sin\phi_{f}^{M}\cos\Delta_{f}\mp
y_{12}\sin\phi_{f}^{\Gamma}\sin\Delta_{f},$ $\displaystyle
c^{\prime\pm}_{\bar{f}}$
$\displaystyle\approx\tfrac{1}{4}[(x_{12}^{2}+y_{12}^{2})(1\mp 2a_{f}^{d})\pm
2x_{12}y_{12}\sin\phi_{12}]+\tfrac{1}{4}R_{f}^{\pm}(y_{12}^{2}-x_{12}^{2}).$

The choice of the parametrisation in eqs. 17 and 19 ensures that the
coefficients $c^{\pm}_{f,\bar{f}}$ and $c^{\prime\pm}_{f,\bar{f}}$ are of
similar size to those in eq. 9. The suppression (enhancement) of the terms
linear (and quadratic) in the mixing parameters with respect to the constant
ones for RS (WS) decays, due to the different size of the CF and DCS decay
amplitudes contributing to the decays without and following mixing (or vice
versa), are taken into account by the $\sqrt{R_{f}^{(\pm)}}$ factors. These
factors equal unity up to corrections smaller than $10^{-3}$ for CS decays,
where the same amplitude contributes to both the decays without and following
mixing in the limit of no $C\\!P$ violation.

Contrary to CS decays, where the only strong-phase difference between the
amplitudes without and following mixing is given by the imaginary unit that
multiplies $M_{12}^{(*)}$ in the solution of eq. 2, WS and RS decays receive a
nontrivial contribution also from the strong-phase difference, $\Delta_{f}$,
between the DCS and CF decay amplitudes. Thus, WS and RS decays are sensitive
to the weak phase $\phi^{\Gamma}_{f}$ through sine functions, unlike for CS
decays, which are sensitive only to $\sin\phi^{M}_{f}$ and to
$\cos\phi^{\Gamma}_{f}$. However, this sensitivity is limited by smallness of
$\Delta_{f}$, which is equal to $-0.17\pm 0.07\text{\,rad}$ LHCb-
PAPER-2021-033 .

In analogy with eqs. 12a and 12b, the following parameters are defined for RS
decays,

$\displaystyle{y_{{C\\!P}}^{f}}$ $\displaystyle\begin{aligned}
&\equiv-\sqrt{R_{f}}\,\frac{c^{+}_{f}+c^{-}_{f}}{2}\\\
&\approx\sqrt{R_{f}}\big{(}x_{12}\cos\phi_{f}^{M}\sin\Delta_{f}-y_{12}\cos\phi_{f}^{\Gamma}\cos\Delta_{f}\big{)},\end{aligned}$
(21a) $\displaystyle{\Delta Y_{f}}$ $\displaystyle\begin{aligned}
&\equiv\sqrt{R_{f}}\,\frac{c^{+}_{f}-c^{-}_{f}}{2}\\\
&\approx\sqrt{R_{f}}\big{[}x_{12}\sin\phi_{f}^{M}\cos\Delta_{f}+y_{12}\sin\phi_{f}^{\Gamma}\sin\Delta_{f}+\tfrac{1}{2}(a^{d}_{\bar{f}}+a^{d}_{f})(x_{12}\sin\Delta_{f}-y_{12}\cos\Delta_{f})\big{]}.\end{aligned}$
(21b)

In the limit of no $C\\!P$ violation in the decay and of small strong phase
$\Delta_{f}$, these parameters are equal to the negative of the parameters
$y_{{C\\!P}}^{f}$ and $\Delta Y_{f}$ of CS decays, multiplied by a suppression
factor of $\sqrt{R_{f}}$.

Analogous parameters are defined also for untagged decays, for which the
${D}^{0}$ flavour at production is unknown and the final states $f$ and
$\bar{f}$ are analysed independently of the meson from which they originate,

$\displaystyle{y_{{C\\!P}}^{K\pi\textnormal{,untag}}}$
$\displaystyle\begin{aligned}
&\equiv-\frac{1}{1+R_{f}}\Big{(}\sqrt{R_{f}}\,\frac{c^{+}_{f}+c^{-}_{f}}{2}+\frac{\sqrt{R^{+}_{f}}c^{+}_{\bar{f}}+\sqrt{R^{-}_{f}}c^{-}_{\bar{f}}}{2}\Big{)}\\\
&\approx-\frac{2\sqrt{R_{f}}}{1+R_{f}}\,y_{12}\cos\phi_{f}^{\Gamma}\cos\Delta_{f},\end{aligned}$
(22a) $\displaystyle\Delta Y_{K\pi}^{\textnormal{untag}}$
$\displaystyle\begin{aligned}
&\equiv\frac{\sqrt{R_{f}}c^{+}_{f}+\sqrt{R^{-}_{f}}c^{-}_{\bar{f}}-\sqrt{R_{f}}c^{-}_{f}-\sqrt{R^{+}_{f}}c^{+}_{\bar{f}}}{2(1+R_{f})}\\\
&\approx\frac{\sqrt{R_{f}}}{1+R_{f}}y_{12}[2\sin\phi_{f}^{\Gamma}\sin\Delta_{f}+({a^{d}_{\bar{f}}}-{a^{d}_{f}})\cos\Delta_{f}].\end{aligned}$
(22b)

In the limit of no $C\\!P$ violation in the decay, ${\Delta Y_{f}}$ and
${\Delta Y_{{K}{\pi}}}$ are equal to the time-integrated asymmetries of eqs.
(A1) and (A2) of ref. Kagan:2020vri .666The same formulas had been previously
presented in the phenomenological parametrisation in section V.A of ref.
Xing:1996pn .

##  4 Experimental methods

The following sections discuss how the observables introduced so far can be
measured.

###  4.1 Cabibbo-suppressed decays

The $c^{\pm}_{f}$ coefficients in eq. 10a have been measured at the $B$
factories by modelling the CS decay rates with an exponential function
Lees:2012qh ; Staric:2015sta ,

$\displaystyle\Gamma({{D}^{0}}(\tau)\\!\rightarrow f)$
$\displaystyle=\lvert{A_{f}}\rvert^{2}\exp(-\hat{\Gamma}_{{{D}^{0}}\\!\rightarrow
f}\tau),$ (23) $\displaystyle\Gamma({{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}(\tau)\\!\rightarrow f)$
$\displaystyle=\lvert{\bar{A}_{f}}\rvert^{2}\exp(-\hat{\Gamma}_{{{\kern
1.25995pt\overline{\kern-1.25995ptD}}{}^{0}}\\!\rightarrow f}\tau).$

Neglecting terms quadratic in the mixing parameters, the effective decay
widths defined in eq. 23 satisfy

$\hat{\Gamma}_{{{D}^{0}}/{{\kern
1.25995pt\overline{\kern-1.25995ptD}}{}^{0}}\\!\rightarrow f}\approx
1-c^{\pm}_{f}.$ (24)

Under this approximation, the observables $y_{{C\\!P}}^{f}$ and $\Delta Y_{f}$
defined in eqs. 12a and 12b can be measured as

$\displaystyle{y_{{C\\!P}}^{f}}$
$\displaystyle\approx\frac{\hat{\Gamma}_{{{D}^{0}}\\!\rightarrow
f}+\hat{\Gamma}_{{{\kern
1.25995pt\overline{\kern-1.25995ptD}}{}^{0}}\\!\rightarrow f}}{2}-1,$ (25a)
$\displaystyle{\Delta Y_{f}}$ $\displaystyle\approx\frac{\hat{\Gamma}_{{{\kern
1.25995pt\overline{\kern-1.25995ptD}}{}^{0}}\\!\rightarrow
f}-\hat{\Gamma}_{{{D}^{0}}\\!\rightarrow f}}{2}.$ (25b)

Precision measurements of the effective decay widths require measuring
precisely both the decay time and the decay width $\Gamma$, as they enter
their definition in eq. 23 through $\tau$. Relative errors on the measurement
of decay time or $\Gamma$ provoke relative errors of the same size on the
measurement of the asymmetry parameter $\Delta Y_{f}$ (unless the bias on the
measurement of decay time differs for ${D}^{0}$ and ${\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}$ mesons), and thus affect it only
marginally, as its value is still compatible with zero. On the contrary, they
are added to the value of $y_{{C\\!P}}^{f}$, which quantifies deviations of
the $C\\!P$-averaged effective decay width from unity. Therefore, they can
cause large biases on this small-sized parameter. To reduce the size of
possible biases, determinations of $\Delta Y_{f}$ and $y_{{C\\!P}}^{f}$ based
on the measurement of the effective decay widths are usually normalised to the
measurement of the effective decay width of RS decays, whose deviation from
unity is smaller approximately by a factor of
$\sqrt{R_{{{{K}^{-}}{{\pi}^{+}}}}}=(5.87\pm 0.02)\%$ LHCb-PAPER-2021-033  with
respect to that of CS decays:

$\hat{\Gamma}_{{{D}^{0}}\\!\rightarrow{{{K}^{-}}{{\pi}^{+}}}/{{\kern
1.25995pt\overline{\kern-1.25995ptD}}{}^{0}}\\!\rightarrow{{{K}^{+}}{{\pi}^{-}}}}\approx
1-\sqrt{R_{{{{K}^{-}}{{\pi}^{+}}}}}c^{\pm}_{{{{K}^{-}}{{\pi}^{+}}}}.$ (26)

The eqs. 25a and 25b are modified accordingly, yielding

$\displaystyle{y_{{C\\!P}}^{f}}-{y_{{C\\!P}}^{{{{K}^{-}}{{\pi}^{+}}}}}$
$\displaystyle\approx\frac{\hat{\Gamma}_{{{D}^{0}}\\!\rightarrow
f}+\hat{\Gamma}_{{{\kern
1.25995pt\overline{\kern-1.25995ptD}}{}^{0}}\\!\rightarrow
f}}{\hat{\Gamma}_{{{D}^{0}}\\!\rightarrow{{{K}^{-}}{{\pi}^{+}}}}+\hat{\Gamma}_{{{\kern
1.25995pt\overline{\kern-1.25995ptD}}{}^{0}}\\!\rightarrow{{{K}^{+}}{{\pi}^{-}}}}}-1,$
(27a) $\displaystyle\frac{{\Delta
Y_{f}}}{1+{y_{{C\\!P}}^{{{{K}^{-}}{{\pi}^{+}}}}}}$
$\displaystyle\approx\frac{\hat{\Gamma}_{{{\kern
1.25995pt\overline{\kern-1.25995ptD}}{}^{0}}\\!\rightarrow
f}-\hat{\Gamma}_{{{D}^{0}}\\!\rightarrow
f}}{\hat{\Gamma}_{{{D}^{0}}\\!\rightarrow{{{K}^{-}}{{\pi}^{+}}}}+\hat{\Gamma}_{{{\kern
1.25995pt\overline{\kern-1.25995ptD}}{}^{0}}\\!\rightarrow{{{K}^{+}}{{\pi}^{-}}}}},$
(27b)

where the dependence on $\Gamma$ and on the time scale cancels out in the
ratio of the effective decay widths. A similar cancellation has been exploited
by the LHCb and Belle collaborations, which measured the asymmetry of the
effective lifetimes of ${D}^{0}$ and ${\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}$ mesons LHCb-PAPER-2013-054 ;
Staric:2015sta ,

${A_{\Gamma}^{f}}\equiv\frac{\hat{\Gamma}_{{{D}^{0}}\\!\rightarrow
f}-\hat{\Gamma}_{{{\kern
1.25995pt\overline{\kern-1.25995ptD}}{}^{0}}\\!\rightarrow
f}}{\hat{\Gamma}_{{{D}^{0}}\\!\rightarrow f}+\hat{\Gamma}_{{{\kern
1.25995pt\overline{\kern-1.25995ptD}}{}^{0}}\\!\rightarrow
f}}\approx-\frac{{\Delta Y_{f}}}{1+{y_{{C\\!P}}^{f}}}.$ (28)

This observable differs from that of eq. 27b for the different calibration
channel at denominator as well as for an overall minus sign.

Given the smallness of ${y_{{C\\!P}}^{f}}$ HFLAV18 , the observable
$A_{\Gamma}^{f}$ is equal to the negative of $\Delta Y_{f}$ within 1%
precision. The observable defined in eq. 27b is equal to $\Delta Y_{f}$ with
even better precision, as $y_{{C\\!P}}^{{{{K}^{-}}{{\pi}^{+}}}}$ is smaller
than $y_{{C\\!P}}^{f}$ approximately by a factor of
$\sqrt{R_{{{{K}^{-}}{{\pi}^{+}}}}}$. On the contrary, the approximation that
$\hat{\Gamma}_{{{D}^{0}}\\!\rightarrow{{{K}^{-}}{{\pi}^{+}}}/{{\kern
1.25995pt\overline{\kern-1.25995ptD}}{}^{0}}\\!\rightarrow{{{K}^{+}}{{\pi}^{-}}}}$
is equal to unity cannot be used effectively in the measurement of
$y_{{C\\!P}}^{f}$ through eq. 27a, since the contribution from
$y_{{C\\!P}}^{{{{K}^{-}}{{\pi}^{+}}}}$ in the left-hand side yields relative
corrections to $y_{{C\\!P}}^{f}$ of the order of
$\sqrt{R_{{{{K}^{-}}{{\pi}^{+}}}}}\approx 6\%$ LHCb-PAPER-2021-033 .

In recent years, alternative methods have been preferred to measure $\Delta
Y_{f}$ and $y_{{C\\!P}}^{f}$. Measuring the effective decay widths from the
decay time distributions requires knowing with high precision the
reconstruction efficiency as a function of decay time, even when a
normalisation channel such as ${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ is
available. This task is challenging in case of stringent trigger requirements
on ${D}^{0}$ displacement-related variables, such as those set at hadron
colliders. However, it can be mitigated by measuring the parameter $\Delta
Y_{f}$ from the slope of the time-dependent asymmetry of the decay rates of
${D}^{0}$ and ${\kern 1.79993pt\overline{\kern-1.79993ptD}}{}^{0}$ mesons into
the final state $f$ Aaltonen:2014efa ; LHCb-PAPER-2014-069 ; LHCb-
PAPER-2016-063 ; LHCb-PAPER-2019-032 ; LHCb-PAPER-2020-045 ,

$A_{{C\\!P}}(f,\tau)\equiv\frac{\Gamma({{D}^{0}}(\tau)\\!\rightarrow
f)-\Gamma({{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}(\tau)\\!\rightarrow
f)}{\Gamma({{D}^{0}}(\tau)\\!\rightarrow f)+\Gamma({{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}(\tau)\\!\rightarrow f)}\\\
\approx{a^{d}_{f}}+{\Delta Y_{f}}\tau,$ (29)

where terms quadratic in the mixing parameters or in ${a^{d}_{f}}$ have been
neglected. The average reconstruction efficiency of ${D}^{0}$ and ${\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}$ decays cancels out in the ratio,
and only charge-dependent efficiency effects must be corrected for. In
addition, the $C\\!P$-even terms quadratic in the mixing parameters in eq. 10b
cancel out in the numerator and do not bias the results, while the $C\\!P$-odd
quadratic terms are negligible within the current precision as well as that
attainable at the LHCb upgrades and at the Belle II experiment, as shown in
appendix A. The measurements based on the effective decay widths do not
benefit from this feature: the $C\\!P$-even terms of the $c^{\prime\pm}_{f}$
coefficients in eq. 10b can be of the same order of the $C\\!P$-odd terms of
the $c^{\pm}_{f}$ coefficients in eq. 10a, thus limiting the validity of the
assumption in eq. 24.

The $y_{{C\\!P}}^{f}$ parameter has analogously been measured from the time-
dependent ratio of the decay rates of CS to RS decays LHCb-PAPER-2018-038 ,

$\displaystyle\frac{\Gamma({{D}^{0}}(\tau)\\!\rightarrow f)+\Gamma({{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}(\tau)\\!\rightarrow
f)}{\Gamma({{D}^{0}}(\tau)\\!\rightarrow{{K}^{-}}{{\pi}^{+}})+\Gamma({{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}(\tau)\\!\rightarrow{{K}^{+}}{{\pi}^{-}})}\approx\textnormal{const.}\times\Big{\\{}1-({y_{{C\\!P}}^{f}}-{y_{{C\\!P}}^{{{{K}^{-}}{{\pi}^{+}}}}})\tau$
(30) $\displaystyle\hskip
128.0374pt+\big{[}\tfrac{1}{4}(x_{12}^{2}+y_{12}^{2})(1-R_{{{{K}^{-}}{{\pi}^{+}}}})+{y_{{C\\!P}}^{{{{K}^{-}}{{\pi}^{+}}}}}({y_{{C\\!P}}^{{{{K}^{-}}{{\pi}^{+}}}}}-{y_{{C\\!P}}^{f}})\big{]}\tau^{2}\Big{\\}},$

where terms cubic in the mixing parameters have been neglected and terms
proportional to $a^{d}_{f}$ and $a^{d}_{\bar{f}}$ have been expanded to first
order in the $C\\!P$-violation parameters $a^{d}_{f}$, $a^{d}_{\bar{f}}$,
$\sin\phi^{M}_{f}$ and $\sin\phi^{\Gamma}_{f}$. The negative of the slope
differs from $y_{{C\\!P}}^{f}$ by the same quantity,
$-{y_{{C\\!P}}^{{{{K}^{-}}{{\pi}^{+}}}}}$, as the ratio of the effective decay
widths in eq. 27a, and cannot be neglected. The term quadratic in the mixing
parameters is negligible at the current level of experimental precision, but
will become relevant at large values of $\tau$ in the expected measurements
employing the LHCb Run 2 data sample (2015–2018) and beyond. When this
happens, approximating the time evolution of the ratio with an exponential, as
done in ref. LHCb-PAPER-2018-038 , might provoke biases to the measurement, as
the quadratic term is independent of the linear one, though accidentally its
numerical value is not so different from that expected from the expansion of
the exponential,
$\tfrac{1}{2}({y_{{C\\!P}}^{f}}-{y_{{C\\!P}}^{{{{K}^{-}}{{\pi}^{+}}}}})^{2}$
LHCb-PAPER-2021-033 .

The term $y_{{C\\!P}}^{{{{K}^{-}}{{\pi}^{+}}}}$ in the left-hand side of eq.
27a and in the linear term on the right-hand side of eq. 30 has been neglected
in all combinations of charm mixing and $C\\!P$-violation measurements HFLAV18
; Kagan:2020vri  except for that in ref. LHCb-PAPER-2021-033 , but amounts to
about 40% of the precision of the current world average of
${y_{{C\\!P}}}=(7.19\pm 1.13)\times 10^{-3}$ HFLAV18 , where the convention
${\eta^{{C\\!P}}_{f}}=1$ is employed and measurements of $C\\!P$-odd final
states have been multiplied by $-1$. Since the average is driven by
measurements of $C\\!P$-even final states, it is biased towards larger values.
While its current value contributes marginally to the results of the
combinations, the statistical uncertainty attainable with the data collected
during the Run 2 of the LHCb experiment is smaller by a factor of 5. When this
precision is achieved, correcting the expressions used in the fits according
to eqs. 27a and 30 will be indispensable to avoid large biases, as shown in
fig. 1.

Figure 1:  Impact of neglecting the $y_{{C\\!P}}^{{{{K}^{-}}{{\pi}^{+}}}}$
terms in eqs. 27a and 30 on the world average of the mixing parameters, (left)
as of today and (right) after including a new measurement of
$y_{{C\\!P}}^{f}$, where $f$ equals ${{K}^{+}}{{K}^{-}}$ or
${{\pi}^{+}}{{\pi}^{-}}$, with the precision achievable with the
${D}^{*}(2010)^{+}$-tagged data sample collected by the LHCb experiment during
its Run 2 data taking (2015–2018). In the right plot, the value of
$y_{{C\\!P}}^{f}$ is fixed to its point estimate based on the results of the
fit in ref. HFLAV18 , and its uncertainty is set to $0.3\times 10^{-4}$.
Measurements of the $\gamma$ angle of the CKM unitary triangle relying on two-
body ${D}^{0}$ decays, which improve the precision on the strong-phase
$\Delta_{{{{K}^{-}}{{\pi}^{+}}}}$ and indirectly on the current world average
of $y_{12}$ by around 40% LHCb-PAPER-2021-033 , are not included for
simplicity. The code employed to perform the fits is available on GitHub
pajero:code .

###  4.2 Wrong-sign decays

In the measurements of WS decays into the final state
$\bar{f}={{{K}^{+}}{{\pi}^{-}}}$, modelling the exponential factor $e^{-\tau}$
in the decay rates of eq. 19, as well as the time dependence of the efficiency
due to trigger and reconstruction effects, the $D$-meson production asymmetry
and most of the detection asymmetries, is usually avoided by analysing the
ratio of the WS to RS decay rates Aubert:2007wf ; Ko:2014qvu ;
Aaltonen:2013pja ; LHCb-PAPER-2016-033 ; LHCb-PAPER-2017-046 ,

$\displaystyle\frac{\Gamma({{D}^{0}}(\tau)\\!\rightarrow\bar{f})}{\Gamma({{D}^{0}}(\tau)\\!\rightarrow
f)}$ $\displaystyle\approx
R_{f}^{+}+\sqrt{R_{f}^{+}}c^{+}_{\bar{f}}\tau+c^{\prime+}_{\bar{f}}\tau^{2},$
(31) $\displaystyle\frac{\Gamma({{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}(\tau)\\!\rightarrow
f)}{\Gamma({{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}(\tau)\\!\rightarrow\bar{f})}$
$\displaystyle\approx
R_{f}^{-}+\sqrt{R_{f}^{-}}c^{-}_{\bar{f}}\tau+c^{\prime-}_{\bar{f}}\tau^{2}.$

Here, the contribution of RS decays to the linear and quadratic terms is
suppressed by a relative factor of $R_{f}$ and is neglected. It might become
relevant at the future upgrades of the LHCb experiment, but only for the
$C\\!P$-even observables, as detailed in appendix A.

An alternative approach consists in taking the ratio of the WS decay rates for
a given ${D}^{0}$ initial flavour, over RS decay rates from ${D}^{0}$ mesons
with opposite initial flavour. This new observable,
$\Gamma({{D}^{0}}(\tau)\\!\rightarrow\bar{f})/\Gamma({{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}(\tau)\\!\rightarrow\bar{f})$, is
indistinguishable from that in eq. 31 up to a multiplicative factor of
$(1\pm{a^{d}_{f}})/(1\mp{a^{d}_{f}})\approx 1\pm 2{a^{d}_{f}}$, where the top
(bottom) sign applies to WS decays of ${D}^{0}$ (${\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}$) mesons, but presents different
experimental advantages and challenges. The approximate cancellation in the
ratio of the production asymmetry and of the detection asymmetry of the
accompanying particles employed to infer the ${D}^{0}$ flavour at production
(typically a ${\pi}^{+}$ meson from a
${{D}^{*}(2010)^{+}}\\!\rightarrow{{D}^{0}}{{\pi}^{+}}$ decay or a muon from a
${{\kern
1.79993pt\overline{\kern-1.79993ptB}}}\\!\rightarrow{{D}^{0}}{\mu^{-}}X$
decay, where $X$ stands for an arbitrary number of unreconstructed particles)
does not take place any longer. On the other hand, the detection asymmetry of
the ${{K}^{\pm}}{{\pi}^{\mp}}$ final state approximately cancels out in the
ratio, whereas it needs to be corrected for when using the standard observable
of eq. 31. Depending on the strategy adopted to cancel the asymmetries, one or
other of the two observables could be more convenient.

###  4.3 Right-sign decays

All aforementioned measurements employ RS ${{D}^{0}}\\!\rightarrow f$ decays,
where $f$ stands for ${{{K}^{-}}{{\pi}^{+}}}$, merely as a normalisation
channel, whose decay width is used as a proxy for $\Gamma$. However, RS decays
are sensitive to $C\\!P$ violation in the mixing, although with lower
sensitivity with respect to CS and WS decays. A convenient observable to
measure this effect is the time-dependent asymmetry between the ${D}^{0}$ and
${\kern 1.79993pt\overline{\kern-1.79993ptD}}{}^{0}$ RS decay rates,

$A_{{C\\!P}}(f,\tau)\equiv\frac{\Gamma({{D}^{0}}(\tau)\\!\rightarrow
f)-\Gamma({{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}(\tau)\\!\rightarrow\bar{f})}{\Gamma({{D}^{0}}(\tau)\\!\rightarrow
f)+\Gamma({{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}(\tau)\\!\rightarrow\bar{f})}\approx
a_{f}^{d}+\Delta Y_{f}\tau,$ (32)

where terms quadratic in the mixing parameters and in $a^{d}_{f}$ have been
neglected.

The relative experimental sensitivity to $C\\!P$ violation in the mixing of
measurements of $\Delta Y_{{{{K}^{-}}{{\pi}^{+}}}}$ and $\Delta
Y_{\textnormal{CS}}$ is estimated as follows. The sensitivity of $\Delta
Y_{{{{K}^{-}}{{\pi}^{+}}}}$ to this phenomenon is suppressed roughly by a
factor of $\sqrt{R_{{{{K}^{-}}{{\pi}^{+}}}}}$ with respect to $\Delta
Y_{\textnormal{CS}}$; see eqs. 12b and 21b. On the other hand, the branching
fraction of ${{D}^{0}}\\!\rightarrow{{{K}^{-}}{{\pi}^{+}}}$ decays is larger
by a factor of about 9.7 (27.1) than that of
${{D}^{0}}\\!\rightarrow{{K}^{+}}{{K}^{-}}$
(${{D}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}$) decays PDG2020 . Since the
precision of the measured asymmetries is proportional to $N^{-1/2}$, where $N$
is the number of reconstructed decays, the statistical uncertainty achievable
for $\Delta Y_{{{{K}^{-}}{{\pi}^{+}}}}$ is around 37% of that of the weighted
average of $\Delta Y_{{{{K}^{+}}{{K}^{-}}}}$ and $\Delta
Y_{{{{\pi}^{+}}{{\pi}^{-}}}}$, where the reconstruction efficiency and
background contamination are assumed to be the same for all three decay
channels. This estimate is confirmed by the latest measurements of $\Delta
Y_{{{{K}^{+}}{{K}^{-}}}/{{{\pi}^{+}}{{\pi}^{-}}}}$, where the compatibility of
the measurement of $\Delta Y_{{{{K}^{-}}{{\pi}^{+}}}}$ with zero is used to
cross-check the effectiveness of the analysis procedure in removing nuisance
asymmetries Aaltonen:2014efa ; LHCb-PAPER-2014-069 ; LHCb-PAPER-2016-063 ;
LHCb-PAPER-2019-032 ; LHCb-PAPER-2020-045 . The relative sensitivity to
$C\\!P$ violation in the mixing achievable with
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ with respect to
${{D}^{0}}\\!\rightarrow{{{K}^{+}}{{K}^{-}}}/{{{\pi}^{+}}{{\pi}^{-}}}$ decays
is thus $\sqrt{R_{{{{K}^{-}}{{\pi}^{+}}}}}/37\%\approx 16\%$. While currently
it is not competitive with that of CS decays, it might become of interest once
a nonzero value of $\Delta Y_{{{{K}^{+}}{{K}^{-}}}/{{{\pi}^{+}}{{\pi}^{-}}}}$
is measured, since it provides cleaner access to the mixing phase
$\phi^{M}_{2}$ Kagan:2020vri .

The measurement of $\Delta Y_{{{{K}^{-}}{{\pi}^{+}}}}$ is effectively
uncorrelated with that of the WS-to-RS ratio, in which $C\\!P$-violating
contributions from RS decays are expected to be negligible at all foreseen
upgrades of the LHCb and Belle II experiments; see appendix A. Therefore,
measuring the WS-to-RS ratio and the $\Delta Y_{{{{K}^{-}}{{\pi}^{+}}}}$
parameter is equivalent to measuring the time-integrated asymmetries of WS and
RS decays, as proposed in section V.A of ref. Xing:1996pn  and in appendix A
of ref. Kagan:2020vri . However, it is more convenient experimentally, since
the former observables are by construction less sensitive to time-independent
nuisance asymmetries. Subleading nuisance asymmetries can be corrected for by
relying on control channels for the WS-to-RS ratio LHCb-PAPER-2016-033 ; LHCb-
PAPER-2017-046 , and by building on the methods developed in refs. LHCb-
PAPER-2016-063 ; LHCb-CONF-2019-001 ; LHCb-PAPER-2020-045  for the $\Delta
Y_{{{{K}^{-}}{{\pi}^{+}}}}$ observable. Neither of these approaches degrades
significantly the statistical uncertainty of the measurement, contrary to the
more complex corrections needed to measure time-integrated asymmetries; see
for example ref. LHCb-PAPER-2014-013 .

A direct consequence of the considerations above is that the compatibility of
the measurement of $\Delta Y_{{{{K}^{-}}{{\pi}^{+}}}}$ with zero will not be
utilisable any longer to check the effectiveness of the analysis method in the
measurements of $\Delta Y_{{{{K}^{+}}{{K}^{-}}}}$ and $\Delta
Y_{{{{\pi}^{+}}{{\pi}^{-}}}}$, once these are measured to differ significantly
from zero. This problem might be bypassed by calculating the confidence
interval for $\Delta Y_{{{{K}^{-}}{{\pi}^{+}}}}$ based on the previous
measurements of $\Delta Y_{{{{K}^{+}}{{K}^{-}}}}$, $\Delta
Y_{{{{\pi}^{+}}{{\pi}^{-}}}}$ and other available charm measurements, which
allow to determine the $C\\!P$-violation parameters $\phi^{M}_{2}$ and
$\phi^{\Gamma}_{2}$ with better precision.777This relies on the assumption
that the shifts of $\phi^{M}_{f}$ and $\phi^{\Gamma}_{f}$ with respect to
$\phi^{M}_{2}$ and $\phi^{\Gamma}_{2}$ are small. While this condition is not
necessarily verified for the ${{K}^{+}}{{K}^{-}}$ and ${{\pi}^{+}}{{\pi}^{-}}$
final states, the size of the final-state dependent shifts can be reduced by
utilising the arithmetic average of $\Delta Y_{{{{K}^{+}}{{K}^{-}}}}$ and
$\Delta Y_{{{{\pi}^{+}}{{\pi}^{-}}}}$; see ref. Kagan:2020vri . Issues may
arise if the size of the analysed sample increases considerably from one
measurement to the subsequent, such that the precision achieved with the new
sample of RS decays is comparable to that of the measurements performed with
the previous data samples of CS and WS decays. This would require an increase
of the data sample size by a factor of approximately $(16\%)^{-2}\approx 40$,
where 16% is the relative sensitivity of RS and CS measurements at equal
number of produced ${D}^{0}$ mesons, as estimated above.

###  4.4 Untagged decays

The $C\\!P$ asymmetry of untagged $D$ decays into the final state
$f={{K}^{-}}{{\pi}^{+}}$ is defined, in analogy with eq. 32, as

$\displaystyle A_{{C\\!P}}^{\textnormal{untag}}(f,\tau)$
$\displaystyle\equiv\frac{[\Gamma({{D}^{0}}(\tau)\\!\rightarrow
f)+\Gamma({{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}(\tau)\\!\rightarrow
f)]-[\Gamma({{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}(\tau)\\!\rightarrow\bar{f})+\Gamma({{D}^{0}}(\tau)\\!\rightarrow\bar{f})]}{[\Gamma({{D}^{0}}(\tau)\\!\rightarrow
f)+\Gamma({{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}(\tau)\\!\rightarrow
f)]+[\Gamma({{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}(\tau)\\!\rightarrow\bar{f})+\Gamma({{D}^{0}}(\tau)\\!\rightarrow\bar{f})]}$
(33) $\displaystyle\approx a_{K\pi}^{d\textnormal{,untag}}+\Delta
Y_{K\pi}^{\textnormal{untag}}\,\tau,$

where the $C\\!P$ asymmetry in the decay is equal to

$a_{K\pi}^{d\textnormal{,untag}}\approx{a^{d}_{f}}-R_{f}{a^{d}_{\bar{f}}},$
(34)

$\Delta Y_{K\pi}^{\textnormal{untag}}$ has been defined in eq. 22b, and terms
quadratic in the mixing parameters or in the $C\\!P$ asymmetries $a^{d}_{f}$
and $a^{d}_{\bar{f}}$ have been neglected.

These parameters can be measured with better precision than their analogues
for RS decays. The LHCb experiment currently provides the largest yields of
${D}^{0}$ decays into two charged hadrons, with the largest yields coming from
mesons produced promptly in proton–proton collisions. Since the production
cross-section of ${D}^{0}$ mesons is larger than that of ${D}^{*}(2010)^{+}$
mesons by a factor of 2.6 LHCb-PAPER-2012-041 ; LHCb-PAPER-2015-041 , the
branching fraction of the ${D}^{*}(2010)^{+}$ meson into the
${{D}^{0}}{{\pi}^{+}}$ final state is 68% PDG2020 , and the reconstruction
efficiency of the low-momentum pion from the ${D}^{*}(2010)^{+}$ decay is
relatively low, the ratio of the yields of untagged and tagged decays is close
to 5 LHCb-CONF-2016-005 .

Measuring $a_{K\pi}^{d\textnormal{,untag}}$ is very challenging due to the
need to determine the ${D}^{0}$ production asymmetry and to correct for the
detection asymmetry of the ${{K}^{-}}{{\pi}^{+}}$ final state. On the other
hand, the slope $\Delta Y_{K\pi}^{\textnormal{untag}}$ might be measured
analogously to $\Delta Y_{{{{K}^{-}}{{\pi}^{+}}}}$, by building on the methods
developed in refs. LHCb-PAPER-2016-063 ; LHCb-CONF-2019-001 ; LHCb-
PAPER-2020-045  to correct for the time-dependent component of the nuisance
asymmetries. Since the tagged sample is a subset of the untagged sample, the
measurement of $\Delta Y_{K\pi}^{\textnormal{untag}}$ would be correlated with
those of the WS-to-RS ratio and of $\Delta Y_{{{{K}^{-}}{{\pi}^{+}}}}$.
However, the correlation coefficient can be measured and is expected to be
small due to the very different size of the two samples. While the precision
achievable for $\Delta Y_{K\pi}^{\textnormal{untag}}$ is better by a factor of
$\sqrt{5}\approx 2.2$ with respect to that of $\Delta
Y_{{{{K}^{-}}{{\pi}^{+}}}}$, where 5 is the yields ratio estimated above, the
sensitivity of $\Delta Y_{K\pi}^{\textnormal{untag}}$ to $\phi^{\Gamma}_{f}$
is suppressed by a factor of $2(y_{12}/x_{12})\tan\Delta_{f}\approx 0.54\pm
0.26$ LHCb-PAPER-2021-033  with respect to the sensitivity of $\Delta
Y_{{{{K}^{-}}{{\pi}^{+}}}}$ to $\phi^{M}_{f}$; see eqs. 21b and 22b. The
larger combinatorial background of untagged decays, which is largely rejected
in tagged decays thanks to the low $Q$ value of the ${D}^{*}(2010)^{+}$ decay,
might decrease the relative precision between the two samples, but all in all
RS and untagged decays might provide similar sensitivity to the two weak
mixing phases.

Finally, untagged ${D}\\!\rightarrow{{K}^{+}}{{K}^{-}}/{{\pi}^{+}}{{\pi}^{-}}$
and ${D}\\!\rightarrow K\pi$ decays could be used also to measure
${y_{{C\\!P}}}-{y_{{C\\!P}}^{K\pi\textnormal{,untag}}}$ with the same methods
detailed in section 4.1. Again, this would allow to collect larger yields and
to avoid the reconstruction effects of the low-momentum pion from the
${D}^{*}(2010)^{+}$ decay, at the cost of increased combinatorial background.
The shift of the measured observable from $y_{{C\\!P}}$ due to the
normalisation $D\\!\rightarrow K\pi$ channel would be around twice as large as
that in eq. 30; see eqs. 21a and 22a.

##  5 Summary

The theoretical formalism to describe mixing and $C\\!P$ violation in
${D}^{0}$ decays into two charged hadrons (${{K}^{+}}{{K}^{-}}$,
${{\pi}^{+}}{{\pi}^{-}}$, ${{K}^{+}}{{\pi}^{-}}$ or ${{K}^{-}}{{\pi}^{+}}$)
has been reviewed, along with the experimental methods to measure them. An
analogous discussion employing the phenomenological parametrisation can be
found in appendices A and B of ref. pajero:2021 . The impact of these
phenomena on the RS decay rates, which has usually been neglected in the
literature, has been quantified in sections 3.2 and 4.3.888References
Xing:1996pn  and Kagan:2020vri  have discussed the effect of $C\\!P$ violation
in section V.A and in appendix A, respectively. Here, their discussion has
been extended by including possible contributions from $C\\!P$ violation in
the decay in addition to those from $C\\!P$ violation in the mixing and in the
interference between mixing and decay, and by employing a different
parametrisation for RS decays.

Mixing in RS decays has been shown to affect the measurements of the parameter
$y_{{C\\!P}}$ that rely on RS decays as normalisation channel, shifting the
results by $-{y_{{C\\!P}}^{{{{K}^{-}}{{\pi}^{+}}}}}$, where the latter
observable is defined in eq. 21a; see eqs. 27a and 30. The size of the shift
is about 40% of the uncertainty of the current world average. This effect has
been missed in the literature and is not accounted for in the combinations of
the time-dependent measurements of charm decays in refs. HFLAV18 ;
Kagan:2020vri . Its inclusion in the next iterations of the combinations, as
first done in ref. LHCb-PAPER-2021-033 , will be crucial to avoid
inconsistencies and biases in the fits results.

The $y_{{C\\!P}}$ parameter could be measured also by relying on untagged
decays as in ref. E791:1999bzz , achieving larger yields at the cost of
slightly increased background. The deviation of the observable,
${y_{{C\\!P}}}-{y_{{C\\!P}}^{K\pi\textnormal{,untag}}}$, from $y_{{C\\!P}}$
would be larger than that of RS decays by around a factor of two; see eqs. 21a
and 22a.

The sensitivity of RS and untagged ${D}\\!\rightarrow{{{K}^{-}}{{\pi}^{+}}}$
decays to the weak mixing phases has been estimated in sections 4.3 and 4.4.
While it is worse than that achievable by analysing the ${{K}^{+}}{{K}^{-}}$
and ${{\pi}^{+}}{{\pi}^{-}}$ final states by approximately a factor of 6, it
might become interesting once time-dependent $C\\!P$ violation is observed. In
fact, final-state dependent corrections to the weak mixing phases
$\phi^{M}_{2}$ and $\phi^{\Gamma}_{2}$ are negligible for RS and WS decays,
contrary to what happens for CS decays. The two decay channels would be
complementary, as RS and ${D}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ decays mostly
provide sensitivity to $\phi^{M}_{2}$ and $\phi^{\Gamma}_{2}$, respectively,
with similar experimental precision, and are nearly statistically independent.

Attention should be paid to the usage of ${{K}^{-}}{{\pi}^{+}}$ decays as a
control channel in the measurements of the $\Delta Y_{f}$ parameter for CS
final states $f$, as the compatibility of $\Delta Y_{{{{K}^{-}}{{\pi}^{+}}}}$
with zero will no longer be guaranteed once $\Delta Y_{f}$ is measured to
differ significantly from zero, or if the size of the collected sample of
${D}^{0}$ decays into two charged hadrons increases significantly with respect
to that employed in previous measurements, as shown in section 4.3.

###### Acknowledgements.

We are indebted to Alexander Kagan and Luca Silvestrini, whose recent work
Kagan:2020vri  inspired this article. We are grateful to Roberto Ribatti for
discussions about the new observable proposed in section 4.2. The work of T.P.
was supported by the Science and Technology Facility Council [grant number
ST/S000933/1].

##  Appendix A Subleading corrections

This appendix reports subleading corrections to eqs. 29 and 31, which might
become relevant at the precision foreseen at the end of the LHCb Upgrade II
data taking Bediaga:2018lhg .

The contribution of mixing and $C\\!P$ violation in RS decays has been
neglected in eq. 31 for the WS-to-RS ratios. The complete expressions up to
quadratic order in the mixing parameters and to linear order in the $C\\!P$
asymmetries $a^{d}_{f}$ and $a^{d}_{\bar{f}}$ are

$\displaystyle\frac{\Gamma({{D}^{0}}(\tau)\\!\rightarrow\bar{f})}{\Gamma({{D}^{0}}(\tau)\\!\rightarrow
f)}\Bigg{[}\frac{\Gamma({{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}(\tau)\\!\rightarrow
f)}{\Gamma({{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}(\tau)\\!\rightarrow\bar{f})}\Bigg{]}\approx
R_{f}\Big{[}1\pm({a^{d}_{\bar{f}}}-{a^{d}_{f}})\Big{]}$ (35)
$\displaystyle\hskip
8.5359pt+\sqrt{R_{f}}\bigg{[}x_{12}\cos\phi_{f}^{M}\sin\Delta_{f}\Big{(}1+R_{f}\pm\frac{{a^{d}_{\bar{f}}}-3{a^{d}_{f}}}{2}\Big{)}+y_{12}\cos\phi_{f}^{\Gamma}\cos\Delta_{f}\Big{(}1-R_{f}\pm\frac{{a^{d}_{\bar{f}}}-3{a^{d}_{f}}}{2}\Big{)}$
$\displaystyle\hskip 236.15796pt\pm x_{12}\sin\phi^{M}_{f}\cos\Delta_{f}\mp
y_{12}\sin\phi^{\Gamma}_{f}\sin\Delta_{f}\bigg{]}\tau$ $\displaystyle\hskip
8.5359pt+\bigg{[}\frac{1}{4}(x_{12}^{2}+y_{12}^{2})(1\mp
2{a^{d}_{f}})\pm\frac{1}{2}x_{12}y_{12}(\sin\phi_{f}^{M}-\sin\phi^{\Gamma}_{f})\bigg{]}\tau^{2},$

where the following terms are additionally neglected: terms proportional to
the $C\\!P$ asymmetries $a^{d}_{f}$ and $a^{d}_{\bar{f}}$ and to either of the
parameters $\sin\phi^{M}_{f}$, $\sin\phi^{\Gamma}_{f}$ or $R_{f}$; terms
proportional to $\sin\phi^{M}_{f}$ or $\sin\phi^{\Gamma}_{f}$ and to $R_{f}$;
and terms proportional to the second power of the mixing parameters and to
$R_{f}$. This approximation will have a negligible impact on the theoretical
interpretation of the observables even at the end of the LHCb Upgrade II,
since its target relative precision on the terms linear and quadratic in decay
time is about 2% and 0.5%, respectively Bediaga:2018lhg .999This estimate
assumes that the correlation between the measured parameters will be the same
as in past measurements LHCb-PAPER-2017-046 . Terms of third order in the
mixing parameters will be equally negligible, as they are proportional to
$\sqrt{R_{f}}$. The contribution of RS decays to the time dependence of the
ratios corresponds to the terms proportional to $R_{f}$ in the second line of
eq. 35, and if neglected will cause a bias on the linear coefficient of the
time-dependent ratio of the same order of the statistical precision targeted
by the LHCb Upgrade II.

The analogue of eq. 35 for the new experimental observable proposed in section
4.2 is just the same up to a multiplicative factor of $1+2{a^{d}_{f}}$ (of
$1-2{a^{d}_{f}}$) for
$\Gamma({{D}^{0}}(\tau)\\!\rightarrow\bar{f})/\Gamma({{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}(\tau)\\!\rightarrow\bar{f})$ (for
$\Gamma({{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}(\tau)\\!\rightarrow
f)/\Gamma({{D}^{0}}(\tau)\\!\rightarrow f)$), where the same approximations as
above are employed.

The expansion up to second order in the mixing parameters of the time-
dependent $C\\!P$ asymmetry defined in eq. 29 for ${D}^{0}$ decays into the CS
final state $f$ is

$A_{{C\\!P}}(f,\tau)\approx{a^{d}_{f}}+{\Delta
Y_{f}}\tau+\frac{1}{2}\Big{[}x_{12}y_{12}(\sin\phi^{M}_{f}-\sin\phi^{\Gamma}_{f})-(x_{12}^{2}+y^{2}_{12}){a^{d}_{f}}\Big{]}\tau^{2},$
(36)

where terms multiplying $a^{d}_{f}$ have been expanded to first order in the
$C\\!P$-violation parameters $a^{d}_{f}$, $\sin\phi^{M}_{f}$ and
$\sin\phi^{\Gamma}_{f}$, and terms of order higher than two in
$\sin\phi^{M}_{f}$ and $\sin\phi^{\Gamma}_{f}$ have been neglected. Since both
mixing parameters are of the order of $5\times 10^{-3}$ HFLAV18 ; LHCb-
PAPER-2021-033 , $a^{d}_{f}$ is of the order of $1\times 10^{-3}$ LHCb-
PAPER-2014-013 ; LHCb-PAPER-2019-006  and the effective maximum value of
$\tau$ tested at experiments is of the order of 6 LHCb-PAPER-2020-045 , the
term quadratic in the mixing parameters might become relevant only at large
decay times and at the precision targeted for $\Delta Y_{f}$ by the LHCb
Upgrade II, $1\times 10^{-5}$ Bediaga:2018lhg , and if the magnitude of
$\phi^{M}_{2}$ or $\phi^{\Gamma}_{2}$ is of the same order of the current
experimental limits, $\mathcal{O}(0.05\text{\,rad})$ LHCb-PAPER-2021-033 . The
latter condition is unlikely in the SM Kagan:2020vri ; kagan:charm2021 .

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