# Study of CP violation in $B^{\pm}$ decays to
$\overline{D^{0}}(D^{0})K^{\pm}$ at FCCee

R. Aleksan1, L. Oliver2 and E. Perez3
1 IRFU, CEA, Université Paris-Saclay, 91191 Gif-sur-Yvette cedex, France
2 IJCLab, Pôle Théorie, CNRS/IN2P3 et Université Paris-Saclay, bât. 210, 91405
Orsay, France
3 CERN, EP Department, Geneva, Switzerland

ABSTRACT

The great progress made recently in the sector of Flavor Physics has enabled
to establish CP violation in the B-meson decays. The unitarity triangle
derived from the unitarity relation
$V_{ub}^{*}V_{ud}+V_{cb}^{*}V_{cd}+V_{tb}^{*}V_{td}=0$ has been measured very
precisely. To further asses our understanding of CP violation, it would be
useful to carry out similar measurement of other triangles. In this note, we
investigate the triangle derived from the relation
$V_{ub}^{*}V_{us}+V_{cb}^{*}V_{cs}+V_{tb}^{*}V_{ts}=0$. Two angles of this
triangle ($\alpha_{s}$ and $\beta_{s}$) could be measured very accurately at
FCCee using the decays $B_{s}(\overline{B_{s}})\rightarrow D^{\pm}_{s}K^{\mp}$
and $B_{s}(\overline{B_{s}})\rightarrow J/\psi\phi$ respectively, as discussed
elsewhere by us. This note concentrates on the measurement of the third angle
$\gamma_{s}$ using the modes $B^{\pm}\to\overline{D^{0}}(D^{0})K^{\pm}$. We
show that a direct measurement of the angle $\gamma_{s}$ is possible with some
specific $B^{\pm}$ decays with an estimated resolution of the order of 1∘.

##  1 Introduction

The purpose of this note is to discuss the general features of the reactions
$B^{+}\to\overline{D^{0}}(D^{0})K^{+}$, in particular with respect to the
interference effects which one expects in the Standard Model (SM).

Should there be only 3 families, the CKM matrix elements obey the following
triangular unitarity relations,

$\qquad\qquad\qquad UT_{bd}\equiv
V_{ub}^{*}V_{ud}+V_{cb}^{*}V_{cd}+V_{tb}^{*}V_{td}=0\qquad\qquad\qquad\ \ \ $
$\qquad\qquad UT_{sb}\equiv
V_{ub}^{*}V_{us}+V_{cb}^{*}V_{cs}+V_{tb}^{*}V_{ts}=0\ \ \qquad\qquad$ (1)
$\qquad\qquad\qquad UT_{ds}\equiv
V_{us}^{*}V_{ud}+V_{cs}^{*}V_{cd}+V_{ts}^{*}V_{td}=0\qquad\qquad\qquad\ \ \ $

In the first equation (1), $UT_{bd}$ is known as the Unitarity Triangle, with
the three sides of the same order. The second equation corresponds to a
significantly flatter triangle, while the third one is almost completely flat.
These triangles are visualized in Fig. 1.

Fig. 1. Unitarity Triangle UTbd involving the $3^{rd}$ and $1^{st}$ columns
(top), Unitarity Triangle UTsb involving the $2^{nd}$ and $3^{rd}$ columns
(center) and Unitarity Triangle UTds involving the $1^{nd}$ and $2^{nd}$
columns (bottom) of the CKM matrix (not to scale).

There are also three additional triangles, but they are very similar to those
above. In the SM, the CKM matrix has only four independent parameters.
Therefore the angles of these triangles can be expressed in terms of four
angles. The first relation in (1), with the three sides of the same order, has
been studied extensively. However the other ones would deserve to be studied
in detail as well, in order to investigate further the consistency of the SM.

We define the angles of these triangles as

$\alpha=\arg\left(-\frac{V_{tb}^{*}V_{td}}{V_{ub}^{*}V_{ud}}\right)\ ,\ \ \
\beta=\arg\left(-{V_{cb}^{*}V_{cd}\over V_{tb}^{*}V_{td}}\right)\ ,\ \ \
\gamma=\arg\left(-{V_{ub}^{*}V_{ud}\over V_{cb}^{*}V_{cd}}\right)\qquad$
$\alpha_{s}=\arg\left(-\frac{V_{ub}^{*}V_{us}}{V_{tb}^{*}V_{ts}}\right)\ ,\ \
\ \beta_{s}=\arg\left(-{V_{tb}^{*}V_{ts}\over V_{cb}^{*}V_{cs}}\right)\ ,\ \ \
\gamma_{s}=\arg\left(-{V_{cb}^{*}V_{cs}\over V_{ub}^{*}V_{us}}\right)$ (2)
$\alpha_{ds}=\arg\left(-\frac{V_{us}^{*}V_{ud}}{V_{ts}^{*}V_{td}}\right)\ ,\ \
\ \beta_{ds}=\arg\left(-{V_{ts}^{*}V_{td}\over V_{cs}^{*}V_{cd}}\right)\ ,\ \
\ \gamma_{ds}=\arg\left(-{V_{cs}^{*}V_{cd}\over
V_{us}^{*}V_{ud}}\right)\qquad$

After having studied the direct determination of the angles $\alpha_{s}$ and
$\beta_{s}$ at FCCee in ref. [1], in the present paper we are interested and
we focus on the direct determination of the angle $\gamma_{s}$ of the ”rather
flat” triangle $UT_{sb}$. By direct determination of one angle we mean
measurement of the angle without making use of its relations to other angles
of the triangles in the SM [2].

##  2 Study of interference effect for the reactions
$B^{+}\to\overline{D^{0}}(D^{0})K^{+}$ and their CP conjugates.

The decays $B^{+}\to\overline{D^{0}}K^{+}$ and $B^{+}\to D^{0}K^{+}$, as well
as their CP conjugates, can interfere in some specific cases. These decays are
penguin-free and their interference may lead to clean measurement of the CKM
phase involved and will be discussed in the following.

###  2.1 Direct CP violation in $B^{+}$ decays

We first investigate how CP violation can be generated with the modes
$B^{+}\to\overline{D^{0}}(D^{0})K^{+}$. The decay diagrams for these mode are
shown in Fig. 2 and Fig. 3.

Fig. 2. The leading Feynman diagrams for the $B^{+}$ decay for the final state
$\overline{D^{0}}K^{+}$. The product of CKM elements involved is
$V_{cb}^{*}V_{us}$.

Fig. 3. The leading Feynman diagrams for the $B^{+}$ decay for the final state
$D^{0}K^{+}$. The product of CKM elements involved is $V_{ub}^{*}V_{cs}$.

To generate CP violation in the Standard Model (SM), at least two interfering
amplitudes with different strong phases and different weak phases are
required.

The diagrams in Fig. 2 and in Fig. 3 have distinct CKM weak phases and
different strong phases and thus their interference would generate CP
violation. This is not always possible since, in general, the final states of
$\overline{D^{0}}$ and $D^{0}$ are different, however there are two ways to
get around this by selecting specific $\overline{D^{0}}(D^{0})$ decay final
states.

  1. 1.

Using Doubly Cabibbo Suppressed (DCS) final states for $\overline{D^{0}}$
decays, e.g. $\overline{D^{0}}\rightarrow K^{-}\pi^{+}$, while using the
Cabibbo Allowed (CA) final states for $D^{0}$ decays, i.e. $D^{0}\rightarrow
K^{-}\pi^{+}$. Obviously, similarly, one may use CA states for
$\overline{D^{0}}$ decays and DCS states for $D^{0}$ decays.

  2. 2.

Using $\overline{D^{0}}(D^{0})$ decays to CP eigenstates, e.g.
$\overline{D^{0}}(D^{0})\rightarrow\pi^{+}\pi^{-}$, $K^{+}K^{-}$ or
$K_{s}\pi^{0}$.

One can easily increase the interesting final states by adding $\pi^{0}$ in
the final states.

Let us write for illustration two generic interfering amplitudes and their CP
conjugates, following the diagrams of Figs. 2 and 3.

$A_{1}=A(B^{+}\to\overline{D}^{0}K^{+}\to
f_{i}K^{+})=a_{1}e^{i\alpha_{1}}e^{i\Phi_{1}}$ $A_{2}=A(B^{+}\to D^{0}K^{+}\to
f_{i}K^{+})=a_{2}e^{i\alpha_{2}}e^{i\Phi_{2}}$

$\overline{A_{1}}=A(B^{-}\to
D^{0}K^{-}\to\overline{f}_{i}K^{-})=a_{1}e^{i\alpha_{1}}e^{-i\Phi_{1}}$
$\qquad\qquad\qquad\overline{A_{2}}=A(B^{-}\to\overline{D}^{0}K^{-}\to\overline{f}_{i}K^{-})=a_{2}e^{i\alpha_{2}}e^{-i\Phi_{2}}\
\qquad\qquad$ (3)

where $a_{1}(a_{2})$ are real numbers, $\alpha_{1}(\alpha_{2})$ are the strong
phases and $\Phi_{1}(\Phi_{2})$ are the weak phases. With the notation
$\Delta=\alpha_{1}-\alpha_{2}$ and $\Phi=\Phi_{1}-\Phi_{2}$, one can write the
partial widths

$|A|^{2}=|A_{1}+A_{2}|^{2}=a_{1}^{2}+a_{2}^{2}+2a_{1}a_{2}[\cos\Delta\cos\Phi-\sin\Delta\sin\Phi]\qquad\qquad\
\ \ $
$|\overline{A}|^{2}=|\overline{A_{1}}+\overline{A_{2}}|^{2}=a_{1}^{2}+a_{2}^{2}+2a_{1}a_{2}[\cos\Delta\cos\Phi+\sin\Delta\sin\Phi]\qquad\qquad$
(4)

Should one know $a_{1}^{2}$ and $a_{2}^{2}$, one is left with two equations
with two unknowns, $\Delta$ and $\Phi$. However this is not trivial as
discussed below.

Let us rewrite equations (4) for the final state corresponding to diagrams in
Fig. 2 and Fig. 3, with $D^{0}$ and $\overline{D}^{0}$ decaying into a common
final state $f_{i}$.

The amplitudes $A_{1}$ and $A_{2}$ will respectively correspond e.g. to the
decay chains $B^{+}\to\overline{D}^{0}K^{+}\to f_{i}K^{+}$ and $B^{+}\to
D^{0}K^{+}\to f_{i}K^{+}$. We will have then, factorizing
$\Gamma[B^{+}\to\overline{D^{0}}K^{+}\to f_{i}K^{+}]$ for the decay
$\Gamma\left(B^{+}\to(f_{i})_{D}K^{+}\right)$, and proceeding likewise for its
CP conjugate :

$\mid A\mid^{2}\
\equiv\Gamma\left(B^{+}\to(f_{i})_{D}K^{+}\right)=\Gamma[B^{+}\to\overline{D^{0}}K^{+}\to
f_{i}K^{+}]\qquad\qquad\qquad$ $\times[1+{\cal R}_{B\to f_{i}K}^{2}+2{\cal
R}_{B\to
f_{i}K}(\cos\Delta_{f_{i}}\cos\Phi_{f_{i}}-\sin\Delta_{f_{i}}\sin\Phi_{f_{i}})]$
$\mid\overline{A}\mid^{2}\
\equiv\Gamma\left(B^{-}\to(\overline{f}_{i})_{D}K^{-}\right)=\Gamma[B^{-}\to
D^{0}K^{-}\to\overline{f_{i}}K^{-}]\qquad\qquad\qquad$ $\ \ \
\times[1+\overline{{\cal R}}_{B\to\overline{f_{i}K}}^{2}+2\overline{{\cal
R}}_{B\to\overline{f_{i}K}}(\cos\Delta_{\overline{f_{i}}}\cos\Phi_{\overline{f_{i}}}-\sin\Delta_{\overline{f_{i}}}\sin\Phi_{\overline{f_{i}}})]\qquad\qquad$
(5)

where $f_{i}(\overline{f_{i}})$ is a final state common to $\overline{D^{0}}$
and $D^{0}$, the strong and weak phases satisfy

$\qquad\ \ \ \Delta_{f_{i}}=\Delta_{\overline{f_{i}}}=\alpha_{1}-\alpha_{2}\
,\qquad\Phi_{f_{i}}=-\Phi_{\overline{f_{i}}}=\Phi_{1}-\Phi_{2}\qquad\qquad\qquad$
(6)

and the ratios in (5) are defined by

${\cal R}_{B\to f_{i}K}=\sqrt{{\Gamma[B^{+}\to D^{0}K^{+}\to
f_{i}K^{+}]\over\Gamma[B^{+}\to\overline{D^{0}}K^{+}\to
f_{i}K^{+}]}}\qquad\qquad\qquad$ $=\sqrt{{\Gamma[B^{+}\to
D^{0}K^{+}]\over\Gamma[B^{+}\to\overline{D^{0}}K^{+}]}}\times\sqrt{{\Gamma[D^{0}\to
f_{i}]\over\Gamma[\overline{D^{0}}\to f_{i}]}}\equiv{\cal R}\times{\cal
R}_{D\to f_{i}}\qquad\qquad\qquad$

$\overline{{\cal
R}}_{B\to\overline{f_{i}K}}=\sqrt{{\Gamma[B^{-}\to\overline{D^{0}}K^{-}\to\overline{f_{i}}K^{-}]\over\Gamma[B^{-}\to
D^{0}K^{-}\to\overline{f_{i}}K^{-}]}}\qquad\qquad\qquad\qquad$
$\qquad=\sqrt{{\Gamma[B^{-}\to\overline{D^{0}}K^{-}]\over\Gamma[B^{-}\to
D^{0}K^{-}]}}\times\sqrt{{\Gamma[\overline{D^{0}}\to\overline{f_{i}}]\over\Gamma[D^{0}\to\overline{f_{i}]}}}\equiv\overline{{\cal
R}}\times\overline{{\cal R}}_{D\to\overline{f_{i}}}\qquad\qquad\qquad\ \ \ \ $
(7)

One has the relations

$\qquad\qquad{\cal R}=\overline{{\cal R}}\ ,\qquad\qquad{\cal R}_{D\to
f_{i}}=\overline{{\cal R}}_{D\to\overline{f_{i}}}={1\over{\cal
R}_{D\to\overline{f_{i}}}}\qquad\qquad\qquad\ \ \ $ (8)

that imply

$\qquad\qquad\qquad\qquad\qquad{\cal R}_{B\to f_{i}K}=\overline{{\cal
R}}_{B\to\overline{f_{i}K}}\qquad\qquad\qquad\qquad\qquad\qquad$ (9)

and from eqns. (5)-(9) one gets finally, after some algebra,

$\qquad\qquad\Gamma\left(B^{-}\to(\overline{f}_{i})_{D}K^{-}\right)=\Gamma[B^{+}\to\overline{D^{0}}K^{+}\to
f_{i}K^{+}]\qquad\qquad\qquad\qquad\qquad\qquad\qquad$ $\qquad\times[1+{\cal
R}_{B\to f_{i}K}^{2}+2{\cal R}_{B\to
f_{i}K}(\cos\Delta_{f_{i}}\cos\Phi_{f_{i}}+\sin\Delta_{f_{i}}\sin\Phi_{f_{i}})]\qquad\qquad$
(10)

From an experimental point of view, $|A|^{2}\ (|\overline{A}|^{2})$ is the
number of events $B^{+}\to(f_{i})_{D}K^{+}\
(B^{-}\to(\overline{f_{i}})_{D}K^{-})$ (i.e. the measured signal), while
$\Gamma[B^{+}\to\overline{D^{0}}K^{+}\to f_{i}K^{+}]$ is the number of
$B^{+}\to\overline{D^{0}}K^{+}$ followed by $\overline{D^{0}}\to f_{i}$
decays.

Looking at the first eqn. (5) and eqn. (10), we have 4 unknows,
$\Gamma[B^{+}\to\overline{D^{0}}K^{+}\to f_{i}K^{+}]$, ${\cal R}_{B\to
f_{i}K}$, $\Delta_{f_{i}}$ and $\Phi_{f_{i}}$.

With presently operating and future accelerators like LHC, superKEKB and FCC,
it should be possible to measure ${\cal R}$ and ${\cal R}_{D\to f_{i}}$, and
thus ${\cal R}_{B\to f_{i}K}$, with high accuracy for most of the relevant
final states.

Indeed, one has,

$\qquad{\cal R}^{2}={Br(B^{+}\to D^{0}K^{+})\over
Br(B^{+}\to\overline{D^{0}}K^{+})}={Br(B^{-}\to\overline{D^{0}}K^{-})\over
Br(B^{-}\to D^{0}K^{-})}\qquad\qquad\qquad\qquad$ $\qquad\qquad{\cal R}_{D\to
f_{i}}^{2}={Br(D^{0}\to f_{i})\over Br(\overline{D^{0}}\to
f_{i})}={Br(\overline{D^{0}}\to\overline{f_{i}})\over
Br(D^{0}\to\overline{f_{i}})}\qquad\qquad\qquad\qquad\qquad$ (11)

and these fractions can be determined experimentally, as we argue below

\- ${\cal R}$ is measured by using semileptonics decays of the
$\overline{D^{0}}(D^{0})$. Indeed, the decays $D^{0}\rightarrow\ell^{+}\nu
K^{-}$ and $\overline{D^{0}}\rightarrow\ell^{-}\bar{\nu}K^{+}$, the Branching
Fractions of which are known precisely, tag the flavor of the D mesons.

\- As for the decays $\overline{D^{0}}(D^{0})\to f_{i}$, they can be
determined from the $D^{*+}\to D^{0}\pi^{+}$ and
$D^{*-}\to\overline{D^{0}}\pi^{-}$ decays, where the charge of the pion
identifies the flavor of the $D^{0}$ meson. The $D^{0}-\overline{D^{0}}$
mixing leads to a negligible effect in the extraction of the Branching
Fractions, as expected in the SM.

In summary, we are left with two equations and three unknowns. In principle,
$\Gamma[B^{+}\to\overline{D^{0}}K^{+}\to f_{i}K^{+}]$ could be also obtained
experimentally. The corresponding number of events is

$\qquad N_{B^{+}\to\overline{D^{0}}K^{+}\to f_{i}K^{+}}=N_{B^{+}}\times
Br(B^{+}\to\overline{D^{0}}K^{+})\times Br(\overline{D^{0}}\to f_{i})\qquad$
(12)

Since the individual branching fractions are known, should one know precisely
the initial number of $B^{+}$, one would extract the
$N_{B^{+}\to\overline{D^{0}}K^{+}\to f_{i}K^{+}}$. At the Z-pole $N_{B^{+}}$
is known with a precision of $\sim 2\%$. Although this is not too bad and
might be improved at FCCee, it is preferable to keep
$N_{B^{+}\to\overline{D^{0}}K^{+}\to f_{i}K^{+}}$ as a free parameter, in
which case one would need to use an additional final state $f_{i}$ as is
discussed in [3]. One also derives the CP asymetry :

$A_{CP}=\frac{|A|^{2}-|\overline{A}|^{2}}{|A|^{2}+|\overline{A}|^{2}}=\frac{-2{\cal
R}_{B\to f_{i}K}\sin\Delta_{f_{i}}\sin\Phi_{f_{i}}}{1+{\cal R}_{B\to
f_{i}K}^{2}+2{\cal R}_{B\to
f_{i}K}\cos\Delta_{f_{i}}\cos\Phi_{f_{i}}}\qquad\qquad$ (13)

Let us now investigate further the three types of decays mentioned above.

###  2.2 Using Doubly Cabibbo Suppressed $\overline{D^{0}}$ decays

In the following we consider the decay
$B^{+}\rightarrow\overline{D^{0}}K^{+}\rightarrow(K^{-}\pi^{+})_{\overline{D^{0}}}K^{+}$
and $B^{+}\rightarrow D^{0}K^{+}\rightarrow(K^{-}\pi^{+})_{D^{0}}K^{+}$, where
the former decay channel is Doubly Cabibbo Suppressed for the decay
$\overline{D^{0}}\rightarrow K^{-}\pi^{+}$ and the latter Cabibbo Allowed for
$D^{0}\rightarrow K^{-}\pi^{+}$. Using this type of decays is known as the ADS
method [3]. The complete amplitudes read :

$A_{DCS}=a_{DCS}e^{i\alpha_{1}}e^{i\Phi_{DCS}}\ ,\qquad
A_{CA}=a_{CA}e^{i\alpha_{2}}e^{i\Phi_{CA}}\qquad\qquad$ $\qquad\ \ \
\overline{A}_{DCS}=a_{DCS}e^{i\alpha_{1}}e^{-i\Phi_{DCS}}\
,\qquad\overline{A}_{CA}=a_{CA}e^{i\alpha_{2}}e^{-i\Phi_{CA}}\qquad\qquad\ \ \
$ (14)

where the subscripts DCS and CA stand for Doubly Cabibbo Suppressed and
Cabibbo Allowed, respectively.

The CP violating phases in (14) read,

$\qquad\Phi_{DCS}=\arg(V_{cb}^{*}V_{us}V_{us}^{*}V_{cd})\
,\qquad\Phi_{CA}=\arg(V_{ub}^{*}V_{cs}V_{cs}^{*}V_{ud})\qquad$ (15)

and one gets

$\Phi_{K^{-}\pi^{+}}=\Phi_{DCS}-\Phi_{CA}=\arg\left(V_{cb}^{*}V_{cd}\over
V_{ub}^{*}V_{ud}\right)=-\arg\left(V_{ub}^{*}V_{ud}\over
V_{cb}^{*}V_{cd}\right)$
$\qquad\qquad\qquad\qquad=\pi-\arg\left(-{V_{ub}^{*}V_{ud}\over
V_{cb}^{*}V_{cd}}\right)=\pi-\gamma\qquad\qquad\qquad\qquad\ $ (16)

where $\gamma$ is one of the angles of the usual Unitarity Triangle $UT_{db}$
of Fig. 1. Note also that $\Phi_{DCS}$ and $\Phi_{CA}$ are convention
dependent, while $\Phi_{DCS}-\Phi_{CA}$ is not.

One then writes the decay widths
$\Gamma[B^{+}\rightarrow(K^{-}\pi^{+})_{D}K^{+}]$ and
$\Gamma[B^{-}\rightarrow(K^{+}\pi^{-})_{D}K^{-}]$,

$\Gamma[B^{+}\rightarrow(K^{-}\pi^{+})_{D}K^{+}]=|A_{DCS}+A_{CA}|^{2}\qquad\qquad\qquad\qquad$
$\qquad\qquad\Gamma[B^{-}\rightarrow(K^{+}\pi^{-})_{D}K^{-}]=|\overline{A}_{DCS}+\overline{A}_{CA}|^{2}\qquad\qquad\qquad$
(17)

Thus, as discussed in the section above, with $\Delta=\alpha_{1}-\alpha_{2}$,
one has

$\Gamma[B^{+}\rightarrow(K^{-}\pi^{+})_{D}K^{+}]=a_{DCS}^{2}+a_{CA}^{2}+\
2a_{DCS}a_{CA}[-\cos\Delta\cos\gamma-\sin\Delta\sin\gamma]$
$\Gamma[B^{-}\rightarrow(K^{+}\pi^{-})_{D}K^{-}]=a_{DCS}^{2}+a_{CA}^{2}+\
2a_{DCS}a_{CA}[-\cos\Delta\cos\gamma+\sin\Delta\sin\gamma]$ (18)

where

$a_{DCS}^{2}=\Gamma(B^{+}\to\overline{D^{0}}K^{+})\times\Gamma(\overline{D^{0}}\to
K^{-}\pi^{+})=\Gamma(B^{-}\to D^{0}K^{-})\times\Gamma(D^{0}\to K^{+}\pi^{-})$
$a_{CA}^{2}=\Gamma(B^{+}\to D^{0}K^{+})\times\Gamma(D^{0}\to
K^{-}\pi^{+})=\Gamma(B^{-}\to\overline{D^{0}}K^{-})\times\Gamma(\overline{D^{0}}\to
K^{+}\pi^{-})\ \ \ $ (19)

Thus,

$|A|^{2}=\Gamma[B^{+}\to(f_{i})_{D}K^{+}]=\Gamma[B^{+}\to\overline{D^{0}}K^{+}\to
f_{i}K^{+}]\qquad\qquad$ $\times[1+{\cal R}_{B\to f_{i}K}^{2}+2{\cal R}_{B\to
f_{i}K}(-\cos\Delta_{f_{i}}\cos\gamma-\sin\Delta_{f_{i}}\sin\gamma)]\qquad\qquad$
$|\overline{A}|^{2}=\Gamma[B^{-}\to(f_{i})_{D}K^{-}]=\Gamma[B^{+}\to\overline{D^{0}}K^{+}\to
f_{i}K^{+}]\qquad\qquad$ $\qquad\times[1+{\cal R}_{B\to f_{i}K}^{2}+2{\cal
R}_{B\to
f_{i}K}(-\cos\Delta_{f_{i}}\cos\gamma+\sin\Delta_{f_{i}}\sin\gamma)]\qquad\qquad\
$ (20)

where $f_{i}=K^{-}\pi^{+}$.

We rewrite equations (19) in terms of ${\cal R}$ and ${\cal R}_{D\to f_{i}}$
as defined in equation (7),

$\Gamma[B^{+}\rightarrow(f_{i})_{D}K^{+}]\ =\
\Gamma[B^{+}\to\overline{D^{0}}K^{+}\to f_{i}K^{+}]$ $\times[1+({\cal R}{\cal
R}_{D\to f_{i}})^{2}+2{\cal R}{\cal R}_{D\to
f_{i}}(-\cos\Delta_{f_{i}}\cos\gamma-\sin\Delta_{f_{i}}\sin\gamma)]\qquad\qquad$

$\Gamma[B^{-}\rightarrow(\overline{f_{i}})_{D}K^{-}]\ =\
\Gamma[B^{+}\to\overline{D^{0}}K^{+}\to f_{i}K^{+}]$ $\qquad\times[1+({\cal
R}{\cal R}_{D\to f_{i}})^{2}+2{\cal R}{\cal R}_{D\to
f_{i}}(-\cos\Delta_{f_{i}}\cos\gamma+\sin\Delta_{f_{i}}\sin\gamma)]\qquad$
(21)

As discussed in the previous section, ${\cal R}$ can be determined
experimentally using the $B^{\pm}$ using semileptonics decays of the
$\overline{D^{0}}(D^{0})$ and ${\cal R}_{D\to f_{i}}$ can be determined from
the $D^{*\pm}$ decays, where the charge of the pion identifies the flavor of
the $D^{0}$ meson.

Therefore $\gamma$ can be extracted from equation (20) provided the total
number of produced $B^{\pm}$ is known in order to evaluate
$\Gamma[B^{+}\to\overline{D^{0}}K^{+}\to f_{i}K^{+}]$ following equation (12).

Should one leave $\Gamma[B^{+}\to\overline{D^{0}}K^{+}\to f_{i}K^{+}]$ as a
free parameter, we would be left with three unknowns for two equations and
thus one should use additional modes. For example, one can consider the final
state $\overline{f_{i}}=K^{+}\pi^{-}$. Using equations (21) and the relations
in equation (8), one derives

$\Gamma[B^{+}\rightarrow(\overline{f_{i}})_{D}K^{+}]\ =\
\Gamma[B^{+}\to\overline{D^{0}}K^{+}\to f_{i}K^{+}]\ \qquad\qquad$
$\times[{\cal R}_{D\to f_{i}}^{2}+{\cal R}^{2}+2{\cal R}{\cal R}_{D\to
f_{i}}(-\cos{\overline{\Delta}}_{{\overline{f}}_{i}}\cos\gamma-\sin{\overline{\Delta}}_{{\overline{f}}_{i}}\sin\gamma)]\qquad\qquad$
$\Gamma[B^{-}\rightarrow(f_{i})_{D}K^{-}]\ =\
\Gamma[B^{+}\to\overline{D^{0}}K^{+}\to f_{i}K^{+}]\ \qquad\qquad$
$\qquad\times[{\cal R}_{D\to f_{i}}^{2}+{\cal R}^{2}+2{\cal R}{\cal R}_{D\to
f_{i}}(-\cos{\overline{\Delta}}_{{\overline{f}}_{i}}\cos\gamma+\sin{\overline{\Delta}}_{{\overline{f}}_{i}}\sin\gamma)]\qquad\qquad$
(22)

Compared to equation (20), an additional parameter,
${\overline{\Delta}}_{\overline{f_{i}}}$, has been introduced. It concerns a
strong phase difference as is discussed here below. Altogether, with equations
(21) and (22), we have now four parameters,
$\Gamma[B^{+}\to\overline{D^{0}}K^{+}\to f_{i}K^{+}]$, $\Delta_{f_{i}}$,
${\overline{\Delta}}_{\overline{f_{i}}}$ and $\gamma$, and four equations.
Thus $\gamma$ can be extracted.

For completeness, we write the CP asymetries

${\cal
A_{CP}}(f_{i}=K^{-}\pi^{+})=\frac{\Gamma[B^{+}\rightarrow(K^{-}\pi^{+})_{D}K^{+}]-\Gamma[B^{-}\rightarrow(K^{+}\pi^{-})_{D}K^{-}]}{\Gamma[B^{+}\rightarrow(K^{-}\pi^{+})_{D}K^{+}]+\Gamma[B^{-}\rightarrow(K^{+}\pi^{-})_{D}K^{-}]}$
${\cal A_{CP}}(f_{i}=K^{-}\pi^{+})=\frac{-2{\cal R}{\cal R}_{D\to
f_{i}}\sin\Delta_{f_{i}}\sin\gamma}{1+{\cal R}^{2}{\cal R}_{D\to
f_{i}}^{2}-2{\cal R}{\cal R}_{D\to
f_{i}}\cos\Delta_{f_{i}}\cos\gamma}\qquad\qquad$ (23)

${\cal
A_{CP}}(\overline{f_{i}}=K^{+}\pi^{-})=\frac{\Gamma[B^{+}\rightarrow(K^{+}\pi^{-})_{D}K^{+}]-\Gamma[B^{-}\rightarrow(K^{-}\pi^{+})_{D}K^{-}]}{\Gamma[B^{+}\rightarrow(K^{+}\pi^{-})_{D}K^{+}]+\Gamma[B^{-}\rightarrow(K^{-}\pi^{+})_{D}K^{-}]}$
${\cal A_{CP}}(\overline{f_{i}}=K^{+}\pi^{-})=\frac{-2{\cal R}{\cal R}_{D\to
f_{i}}\sin{\overline{\Delta}}_{\overline{f_{i}}}\sin\gamma}{{\cal R}_{D\to
f_{i}}^{2}+{\cal R}^{2}-2{\cal R}{\cal R}_{D\to
f_{i}}\cos{\overline{\Delta}}_{\overline{f_{i}}}\cos\gamma}\qquad\qquad$ (24)

Comparing equations (23) and (24), the only parameters, which differ are
$\Delta_{f_{i}}$ and ${\overline{\Delta}}_{\overline{f_{i}}}$.

$\qquad\Delta_{f_{i}}=\zeta_{\overline{D^{0}}}-\zeta_{D^{0}}+\delta_{1_{f_{i}}}-\delta_{2_{f_{i}}}=\Delta_{\zeta}+\Delta_{\delta_{f_{i}}}\qquad\qquad\qquad\qquad$
$\qquad\qquad\ \ \
{\overline{\Delta}}_{\overline{f_{i}}}=\zeta_{\overline{D^{0}}}-\zeta_{D^{0}}+\delta_{1_{\overline{f_{i}}}}-\delta_{2_{\overline{f_{i}}}}=\Delta_{\zeta}+\Delta_{\delta_{\overline{f_{i}}}}\qquad\qquad\qquad$
(25)

where $\zeta_{\overline{D^{0}}}$ is related to
$B^{+}\rightarrow\overline{D^{0}}K^{+}$ and $\zeta_{D^{0}}$ to
$B^{+}\rightarrow D^{0}K^{+}$, while $\delta_{1_{f_{i}(\overline{f_{i}})}}$ is
related to $\overline{D^{0}}\to f_{i}(\overline{f_{i}})$ and
$\delta_{2_{f_{i}(\overline{f_{i}})}}$ to $D^{0}\to f_{i}(\overline{f_{i}})$.

It is important to underline that in general neither of these strong phase
differences $\Delta_{\zeta}$ or $\Delta_{\delta}$ are expected to vanish.

As a final comment, the asymmetry ${\cal
A_{CP}}(\overline{f_{i}}=K^{+}\pi^{-})$ is expected to be very small since it
suffers of a very low statistics for the mode $B^{+}\rightarrow
D^{0}K^{+}\rightarrow(K^{+}\pi^{-})_{D^{0}}K^{+}$, as it has multiple
suppression factors compared to
$B^{+}\rightarrow\overline{D^{0}}K^{+}\rightarrow(K^{+}\pi^{-})_{\overline{D^{0}}}K^{+}$,
namely $V_{ub}$ instead of $V_{cb}$, as well as color and double Cabibbo
suppressions.

In the following, we will not pursue the modes
$\overline{D^{0}}(D^{0})\rightarrow K^{\pm}\pi^{\mp}$. The reason is that they
do not measure directly one of the phases of the “flat” triangle UTsb defined
in (1), that is the main interest of the present paper.

###  2.3 Using $\overline{D^{0}}(D^{0})$ decays to CP eigenstates

####  2.3.1 $\overline{D^{0}}(D^{0})$ decays to $\pi^{+}\pi^{-}$

Let us first examine the decay
$\overline{D^{0}}(D^{0})\rightarrow\pi^{+}\pi^{-}$, i.e. $CP=+$. The complete
amplitudes read

$\qquad
A_{1}=a_{\bar{D^{0}}(\pi\pi)}e^{i\alpha_{\bar{D^{0}}(\pi\pi)}}e^{i\Phi_{\bar{D^{0}}(\pi\pi)}}\
,\
A_{2}=a_{D^{0}(\pi\pi)}e^{i\alpha_{D^{0}(\pi\pi)}}e^{i\Phi_{D^{0}(\pi\pi)}}\qquad\qquad$
$\qquad\overline{A_{1}}=a_{\bar{D^{0}}(\pi\pi)}e^{i\alpha_{\bar{D^{0}}(\pi\pi)}}e^{-i\Phi_{\bar{D^{0}}(\pi\pi)}}\
,\
\overline{A_{2}}=a_{D^{0}(\pi\pi)}e^{i\alpha_{D^{0}(\pi\pi)}}e^{-i\Phi_{D^{0}(\pi\pi)}}\
\ \ $ (26)

where

$\qquad\Phi_{\bar{D^{0}}(\pi\pi)}=\arg(V_{cb}^{*}V_{us}V_{ud}^{*}V_{cd})\
,\qquad\Phi_{D^{0}(\pi\pi)}=\arg(V_{ub}^{*}V_{cs}V_{cd}^{*}V_{ud})$ (27)

that gives

$\Phi_{\bar{D^{0}}(\pi\pi)}-\Phi_{D^{0}(\pi\pi)}=\arg\left(\frac{V_{cb}^{*}V_{cd}}{V_{ub}^{*}V_{ud}}\frac{V_{ud}^{*}V_{us}}{V_{cd}^{*}V_{cs}}\right)$
$=\arg\left(-\frac{V_{cb}^{*}V_{cd}}{V_{ub}^{*}V_{ud}}\right)+\arg\left(-\frac{V_{ud}^{*}V_{us}}{V_{cd}^{*}V_{cs}}\right)$
$\qquad\qquad\qquad\qquad\qquad\qquad=-\gamma+\gamma_{ds}\simeq-\gamma\qquad\qquad\qquad\qquad\qquad\qquad\
$ (28)

where $\gamma$ is one of the angles of the usual Unitarity Triangle $UT_{bd}$,
and $\gamma_{ds}$ is a very small angle of the 3rd triangle $UT_{ds}$ in
formulas (1) (2) and Fig. 1.

One obtains the partial widths,

$\Gamma[B^{+}\rightarrow(\pi\pi)_{D}K^{+}]=\Gamma[B^{+}\to\overline{D^{0}}K^{+}\to(\pi\pi)K^{+}]$
$\times[1+{\cal R}^{2}+2{\cal R}(\cos\Delta\cos\gamma+\sin\Delta\sin\gamma)]$

$\Gamma[B^{-}\rightarrow(\pi\pi)_{D}K^{-}]=\Gamma[B^{+}\to\overline{D^{0}}K^{+}\to(\pi\pi)K^{+}]$
$\qquad\qquad\qquad\qquad\times[1+{\cal R}^{2}+2{\cal
R}(\cos\Delta\cos\gamma-\sin\Delta\sin\gamma)]\qquad\qquad\qquad$ (29)

The asymetry reads

${\cal
A}_{CP}^{+}(\pi\pi)=\frac{\Gamma[B^{+}\rightarrow(\pi^{+}\pi^{-})_{D}K^{+}]-\Gamma[B^{-}\rightarrow(\pi^{+}\pi^{-})_{D}K^{-}]}{\Gamma[B^{+}\rightarrow(\pi^{+}\pi^{-})_{D}K^{+}]+\Gamma[B^{-}\rightarrow(\pi^{+}\pi^{-})_{D}K^{-}]}$
$\qquad\qquad\qquad{\cal A}_{CP}^{+}(\pi\pi)\simeq\frac{+2{\cal
R}\sin\Delta\sin\gamma}{1+{\cal R}^{2}+2{\cal
R}\cos\Delta\cos\gamma}\qquad\qquad\qquad\qquad$ (30)

where ${\cal R}$ is defined in equation (5) and
$\Delta=\alpha_{\bar{D^{0}}(\pi\pi)}-\alpha_{D^{0}(\pi\pi)}\simeq\Delta_{\zeta}$.
This is indeed so because $\Delta_{\delta}\simeq 0$ in CP eigenstate decays of
D mesons.

####  2.3.2 $\overline{D^{0}}(D^{0})$ decays to $K^{+}K^{-}$

We consider now the decays $B^{+}\to
D^{0}(\overline{D}^{0})\to(K^{+}K^{-})_{D}K^{+}$, i.e. $CP=+$. The complete
amplitudes read :

$A_{1}=a_{\bar{D^{0}}(KK)}e^{i\alpha_{\bar{D^{0}}(KK)}}e^{i\Phi_{\bar{D^{0}}(KK)}}\
,\ A_{2}=a_{D^{0}(KK)}e^{i\alpha_{D^{0}(KK)}}e^{i\Phi_{D^{0}(KK)}}\qquad$
$\overline{A_{1}}=a_{\bar{D^{0}}(KK)}e^{i\alpha_{\bar{D^{0}}(KK)}}e^{-i\Phi_{\bar{D^{0}}(KK)}}\
,\
\overline{A_{2}}=a_{D^{0}(KK)}e^{i\alpha_{D^{0}(KK)}}e^{-i\Phi_{D^{0}(KK)}}$
(31)

where the CP phases read

$\qquad\qquad\qquad\Phi_{\bar{D^{0}}(KK)}=\arg(V_{cb}^{*}V_{cs})\ ,\
\Phi_{D^{0}(KK)}=\arg(V_{ub}^{*}V_{us})\qquad\qquad$ (32)
$\qquad\qquad\Phi_{\bar{D^{0}}(KK)}-\Phi_{D^{0}(KK)}=\arg\left(\frac{V_{cb}^{*}V_{cs}}{V_{ub}^{*}V_{us}}\right)=\pi+\gamma_{s}\qquad\qquad\qquad$
(33)

The angle $\gamma_{s}$ is one of the angles of the Unitarity Triangle
$UT_{sb}$ in Fig. 1.

One obtains the partial widths from equation (21),

$\Gamma[B^{+}\rightarrow(K^{+}K^{-})_{D}K^{+}]=\Gamma[B^{+}\to\overline{D^{0}}K^{+}\to(K^{+}K^{-})K^{+}]$
$\times[1+{\cal R}^{2}+2{\cal
R}(-\cos\Delta\cos\gamma_{s}+\sin\Delta\sin\gamma_{s})]$
$\Gamma[B^{-}\rightarrow(K^{+}K^{-})_{D}K^{-}]=\Gamma[B^{+}\to\overline{D^{0}}K^{+}\to(K^{+}K^{-})K^{+}]$
$\qquad\qquad\qquad\times[1+{\cal R}^{2}+2{\cal
R}(-\cos\Delta\cos\gamma_{s}-\sin\Delta\sin\gamma_{s})]\qquad\qquad\qquad$
(34)

The asymetry reads

${\cal
A}_{CP}^{+}(K^{+}K^{-})=\frac{\Gamma[B^{+}\rightarrow(K^{+}K^{-})_{D}K^{+}]-\Gamma[B^{-}\rightarrow(K^{+}K^{-})_{D}K^{-}]}{\Gamma[B^{+}\rightarrow(K^{+}K^{-})_{D}K^{+}]+\Gamma[B^{-}\rightarrow(K^{+}K^{-})_{D}K^{-}]}$
$\qquad\qquad\qquad{\cal A}_{CP}^{+}(K^{+}K^{-})=\frac{+2{\cal
R}\sin\Delta\sin\gamma_{s}}{1+{\cal R}^{2}-2{\cal
R}\cos\Delta\cos\gamma_{s}}\qquad\qquad\qquad\ \ $ (35)

where ${\cal R}$ is defined in equation (7) and
$\Delta=\alpha_{\bar{D^{0}}(KK)}-\alpha_{D^{0}(KK)}\simeq\Delta_{\zeta}$.

A similar result is obtained with the decay $\overline{D^{0}}(D^{0})\to
K_{s}\pi^{0}$. However due to the different CP sign ($\eta_{CP}=-1$) for the
eigenstate $K_{s}\pi^{0}$, the CP asymetry reads

$\qquad\qquad\qquad\qquad{\cal A}_{CP}^{-}(K_{s}\pi^{0})=\frac{-2{\cal
R}\sin\Delta\sin\gamma_{s}}{1+{\cal R}^{2}+2{\cal
R}\cos\Delta\cos\gamma_{s}}\qquad\qquad\ \ \ \ \ $ (36)

with
$\Delta_{K_{s}\pi^{0}}=\alpha_{\bar{D^{0}}(K_{s}\pi^{0})}-\alpha_{D^{0}(K_{s}\pi^{0})}\simeq\Delta_{\zeta}$.

Using Eqns. (35) and (36) one can extract $\Delta(=\Delta_{\zeta})$ and
$\gamma_{s}$. This method had been proposed by Gronau, London and Wyler [5]
for the determination of the angle $\gamma$, since $\gamma_{s}\simeq-\gamma$,
up to a very small correction. However, we are now interested in the angle
$\gamma_{s}$ of the $UT_{sb}$ unitarity triangle. Indeed, one gets

$\sin\Delta\sin\gamma_{s}=\frac{{\cal A}_{CP}^{-}(K_{s}\pi^{0})\times{\cal
A}_{CP}^{+}(K^{+}K^{-})}{{\cal A}_{CP}^{-}(K_{s}\pi^{0})-{\cal
A}_{CP}^{+}(K^{+}K^{-})}\times\frac{1+{\cal{R}}^{2}}{\cal{R}}$ $\qquad\ \ \
\cos\Delta\cos\gamma_{s}=-\frac{{\cal A}_{CP}^{-}(K_{s}\pi^{0})+{\cal
A}_{CP}^{+}(K^{+}K^{-})}{{\cal A}_{CP}^{-}(K_{s}\pi^{0})-{\cal
A}_{CP}^{+}(K^{+}K^{-})}\times\frac{1+{\cal{R}}^{2}}{2\cal{R}}\qquad\qquad$
(37)

Thus $\gamma_{s}$ can be measured, though in general with a 8-fold ambiguity.
However depending upon the values of $\cal{A}_{CP}^{+}$ and $\cal{A}_{CP}^{-}$
some solutions may not be physical and thus the ambiguities reduced. Several
other $CP=-$ modes can be used to increase statistics, such as
$\overline{D^{0}}(D^{0})\to K_{s}\eta(\eta^{\prime}$) or $K_{s}\omega$.

###  2.4 Convention of the difference of strong phases

Here we comment on our convention on the difference of the strong phases
$\Delta_{f_{i}}$, that differs from the convention in other works. Let us just
consider the particular case of $B^{\pm}\to D^{0}(\overline{D}^{0})K^{\pm}$,
as made very explicit by Gronau [6].

Let us compare the notation of our amplitudes

$A[B^{+}\to\overline{D}^{0}K^{+}]=a_{1}e^{i\alpha_{1}}e^{i\Phi_{1}}\ ,\qquad
A[B^{+}\to D^{0}K^{+}]=a_{2}e^{i\alpha_{2}}e^{i\Phi_{2}}\ \ \qquad$ (38)

and the CP-conjugated ones,

$A[B^{-}\to D^{0}K^{-}]=a_{1}e^{i\alpha_{1}}e^{-i\Phi_{1}}\ ,\qquad
A[B^{-}\to\overline{D}^{0}K^{-}]=a_{2}e^{i\alpha_{2}}e^{-i\Phi_{2}}\ \ \qquad$
(39)

with Gronau convention, his formula (4),

$\ \ \ \ A[B^{-}\to D^{0}K^{-}]=\ \mid A\mid\ ,\qquad
A[B^{-}\to\overline{D}^{0}K^{-}]=\ \mid\overline{A}\mid e^{i\delta}\
e^{-i\gamma}\ \ \ \ \qquad$ (40)

and its CP-conjugated,

$\ \ \ \ A[B^{+}\to\overline{D}^{0}K^{+}]=\ \mid A\mid\ ,\qquad A[B^{+}\to
D^{0}K^{+}]=\ \mid\overline{A}\mid e^{i\delta}\ e^{i\gamma}\ \ \ \ \qquad$
(41)

One has

$\qquad\qquad\qquad\delta=\alpha_{2}-\alpha_{1}=-\Delta\
,\qquad\qquad\Phi_{2}-\Phi_{1}=-\Phi=\gamma\qquad\qquad$ (42)

and finally

$\qquad\qquad\qquad\qquad\ \ \ \Delta=-\delta\
,\qquad\qquad\Phi=\Phi_{1}-\Phi_{2}=-\gamma\qquad\qquad\qquad$ (43)

So our FSI phase has the opposite sign to Gronau convention, and the CKM phase
is the same in both conventions. The convention of Gronau is the same as the
one used by HFAG [7], where the phase $\delta$ is denoted by $\delta_{B}$. The
opposite sign also holds for our general strong phase difference
$\Delta_{f_{i}}$ and the one of HFAG $\delta_{B}+\delta_{D}$.

##  3 Estimating the size of the modulus of amplitudes

The ratio of amplitudes can be extracted experimentaly from the measured
branching fraction (see Table 1).

Mode | Br | $\eta_{CP}$ | ${\cal R}$ | ${\cal R}_{D\to f_{i}}$
---|---|---|---|---
$B^{+}\to\overline{D^{0}}K^{+}$ | $(3.63\pm 0.12)\cdot 10^{-4}$ |  |  | -
$B^{+}\to D^{0}K^{+}$ | $(3.57\pm 0.35)\cdot 10^{-6}$ |  | 0.099 $\pm$ 0.005 | -
$B^{+}\to\overline{D^{0}}K^{*+}$ | $(5.3\pm 0.4)\cdot 10^{-4}$ |  |  | -
$B^{+}\to D^{0}K^{*+}$ | $(3.1\pm 1.6)\cdot 10^{-6}$ |  | 0.076 $\pm$ 0.020 | -
$\overline{D^{0}}\to K^{+}\pi^{-}$ | $(3.950\pm 0.031)\cdot 10^{-2}$ |  | - | 0.0616$\pm$0.0015
$\overline{D^{0}}\to K^{-}\pi^{+}$ | $(1.50\pm 0.07)\cdot 10^{-4}$ |  | - | 1$6.23\pm 0.38$
$\overline{D^{0}}\to\pi^{+}\pi^{-}$ | $(1.455\pm 0.024)\cdot 10^{-3}$ | $+1$ | - | 1
$\overline{D^{0}}\to K^{+}K^{-}$ | $(4.08\pm 0.06)\cdot 10^{-3}$ | $+1$ | - | 1
$\overline{D^{0}}\to K_{s}\pi^{0}$ | $(1.240\pm 0.022)\cdot 10^{-2}$ | $-1$ | - | 1
$\overline{D^{0}}\to K_{s}\eta$ | $(5.090\pm 0.013)\cdot 10^{-3}$ | $-1$ | - | 1
$\overline{D^{0}}\to K_{s}\eta^{\prime}$ | $(9.49\pm 0.32)\cdot 10^{-3}$ | $-1$ | - | 1
$\overline{D^{0}}\to K_{s}\omega$ | $(1.11\pm 0.06)\cdot 10^{-2}$ | $-1$ | - | 1

Table 1. The branching fraction of $B^{+}$ decays to final states
$\overline{D^{0}}(D^{0})K^{+}(K^{*+})$ and the subsequent
$\overline{D^{0}}(D^{0})$ decays [4].

Using the measured Branching Fractions [4], one gets :

$\qquad\qquad\qquad{\cal R}=\sqrt{\frac{Br(B^{+}\rightarrow
D^{0}K^{+})}{Br(B^{+}\rightarrow\overline{D^{0}}K^{+})}}=0.099\pm
0.005\qquad\qquad\qquad\ \ $ (44)

$\qquad\qquad{\cal R}_{D\to K^{-}\pi^{+}}=\sqrt{\frac{Br(D^{0}\rightarrow
K^{-}\pi^{+})}{Br(\overline{D^{0}}\rightarrow K^{-}\pi^{+})}}=16.23\pm
0.38\qquad\qquad\qquad$ (45) ${\cal R}_{B\to(K^{-}\pi^{+})K}={\cal
R}\times{\cal R}_{D\to K^{-}\pi^{+}}=1.609\pm 0.092\qquad\qquad$

For any $CP$-eigentate, such as $f_{CP}=\pi^{+}\pi^{-}$, $K^{+}K^{-}$ or
$K_{s}\pi^{0}$, ${\cal R}_{B\to(f_{CP})_{D}K}={\cal R}$. Indeed, in the former
equation, $\frac{Br[D^{0}\rightarrow
CP_{eigenstate}]}{Br[\overline{D^{0}}\rightarrow CP_{eigenstate}]}=1$ is
assumed, since
$|D^{0}_{CP\pm}>=\frac{1}{\sqrt{2}}[|D^{0}>\pm|\overline{D^{0}}>]$.

As already mentioned in section 2.1, the ratio $Br(B^{+}\rightarrow
D^{0}K^{+})/Br(B^{+}\rightarrow\overline{D^{0}}K^{+})$ should be measured even
much more precisely at FCCee from the decays $B^{+}\rightarrow
D^{0}K^{+}\rightarrow\ell^{+}\nu K^{-}K^{+}$ and
$B^{+}\rightarrow\overline{D^{0}}K^{+}\rightarrow\ell^{-}\bar{\nu}K^{+}K^{+}$.
The respective branching fractions are about 1.26$\cdot 10^{-7}$ and
1.29$\cdot 10^{-5}$. With the statistics expected at FCCee, one gets $5\cdot
10^{4}$ and $5\cdot 10^{6}$ events, respectively. Despite background levels
orders of magnitude larger, it should be possible to extract the signal rather
cleanly thanks to very accurate vertexing and particle identification for the
kaons. A precision on $\delta({\cal R})/{\cal R}$ at the level of $5\cdot
10^{-3}$ seems attainable.

##  4 Expected Asymetry Sensitivities at FCCee

In this section, we investigate the sensitivity on $\gamma_{s}$, which one may
expect at FCCee. The expected number of events at summarized in Table 2.

|  | $\displaystyle{\mathrm{E_{cm}=91.2\ GeV\ and\ \int L=150ab^{-1}}}$ |
---|---|---|---
$\displaystyle\matrix{\displaystyle{\mathrm{\sigma(e^{+}e^{-}\to Z)}}\\\ \displaystyle{\mathrm{nb}}\\\ }$ | $\displaystyle\matrix{\displaystyle{\mathrm{number}}\\\ \displaystyle{\mathrm{of\ Z}}\\\ }$ | $\displaystyle\matrix{\displaystyle{\mathrm{f(Z\to B^{+})}}\\\ \displaystyle{\mathrm{}}\\\ }$ | $\displaystyle\matrix{\displaystyle{\mathrm{Number\ of}}\\\ \displaystyle{\mathrm{produced\ B^{+}}}\\\ }$
$\displaystyle\sim 42.9$ | $\displaystyle{\mathrm{\sim 6.4\ 10^{12}}}$ | $\displaystyle{\mathrm{\sim 0.061}}$ | $\displaystyle\sim 3.9\ 10^{11}$
$\displaystyle\matrix{\displaystyle{\mathrm{B^{+}\ decay}}\\\ \displaystyle{\mathrm{Mode}}\\\ }$ | $\displaystyle\matrix{\displaystyle{\mathrm{Decay}}\\\ \displaystyle{\mathrm{Mode}}\\\ }$ | $\displaystyle\matrix{\displaystyle{\mathrm{Final}}\\\ \displaystyle{\mathrm{State}}\\\ }$ | $\displaystyle\matrix{\displaystyle{\mathrm{Number\ of}}\\\ \displaystyle{\mathrm{B^{+}\ decays}}\\\ }$
|  | $\displaystyle{\mathrm{\overline{D^{0}}/D^{0}\to nonCP\ eigenstates}}$ |
$\displaystyle\overline{D^{0}}K^{+}$ | $\displaystyle{\mathrm{\overline{D^{0}}\to K^{+}\pi^{-}}}$ | $\displaystyle{\mathrm{K^{+}\pi^{-}K^{+}}}$ | $\displaystyle\sim 5.6\ 10^{6}$
$\displaystyle\overline{D^{0}}K^{+}$ | $\displaystyle{\mathrm{\overline{D^{0}}\to K^{-}\pi^{+}}}$ | $\displaystyle{\mathrm{K^{-}\pi^{+}K^{+}}}$ | $\displaystyle\sim 2.1\ 10^{4}$
$\displaystyle D^{0}K^{+}$ | $\displaystyle{\mathrm{D}^{0}\to K^{-}\pi^{+}}$ | $\displaystyle{\mathrm{K}^{-}\pi^{+}K^{+}}$ | $\displaystyle\sim 5.5\ 10^{4}$
$\displaystyle D^{0}K^{+}$ | $\displaystyle{\mathrm{D}^{0}\to K^{+}\pi^{-}}$ | $\displaystyle{\mathrm{K}^{+}\pi^{-}K^{+}}$ | $\displaystyle\sim 2.1\ 10^{2}$
|  | $\displaystyle{\mathrm{\overline{D^{0}}/D^{0}\to CP\ eigenstates}}$ |
$\displaystyle\overline{D^{0}}K^{+}$ | $\displaystyle{\mathrm{\overline{D^{0}}\to\pi^{+}\pi^{-}}}$ | $\displaystyle{\mathrm{\pi^{+}\pi^{-}K^{+}}}$ | $\displaystyle\sim 2.1\ 10^{5}$
$\displaystyle D^{0}K^{+}$ | $\displaystyle{\mathrm{D^{0}\to\pi^{+}\pi^{-}}}$ | $\displaystyle{\mathrm{\pi^{+}\pi^{-}K^{+}}}$ | $\displaystyle\sim 2.0\ 10^{3}$
$\displaystyle\overline{D^{0}}K^{+}$ | $\displaystyle{\mathrm{\overline{D^{0}}\to K^{+}K^{-}}}$ | $\displaystyle{\mathrm{K^{+}K^{-}K^{+}}}$ | $\displaystyle\sim 5.8\ 10^{5}$
$\displaystyle D^{0}K^{+}$ | $\displaystyle{\mathrm{D^{0}\to K^{+}K^{-}}}$ | $\displaystyle{\mathrm{K^{+}K^{-}K^{+}}}$ | $\displaystyle\sim 5.7\ 10^{3}$
$\displaystyle\overline{D^{0}}K^{+}$ | $\displaystyle{\mathrm{\overline{D^{0}}\to K_{s}\pi^{0}}}$ | $\displaystyle{\mathrm{K_{s}\pi^{0}K^{+}}}$ | $\displaystyle\sim 1.2\ 10^{6}$
$\displaystyle D^{0}K^{+}$ | $\displaystyle{\mathrm{D^{0}\to K_{s}\pi^{0}}}$ | $\displaystyle{\mathrm{K_{s}\pi^{0}K^{+}}}$ | $\displaystyle\sim 1.2\ 10^{4}$
$\displaystyle\overline{D^{0}}K^{+}$ | $\displaystyle{\mathrm{\overline{D^{0}}\to K_{s}\eta}}$ | $\displaystyle{\mathrm{K_{s}\eta K^{+}}}$ | $\displaystyle\sim 5.0\ 10^{5}$
$\displaystyle D^{0}K^{+}$ | $\displaystyle{\mathrm{D^{0}\to K_{s}\eta}}$ | $\displaystyle{\mathrm{K_{s}\eta K^{+}}}$ | $\displaystyle\sim 4.9\ 10^{3}$
$\displaystyle\overline{D^{0}}K^{+}$ | $\displaystyle{\mathrm{\overline{D^{0}}\to K_{s}\eta^{\prime}}}$ | $\displaystyle{\mathrm{K_{s}\eta^{\prime}K^{+}}}$ | $\displaystyle\sim 9.3\ 10^{5}$
$\displaystyle D^{0}K^{+}$ | $\displaystyle{\mathrm{D^{0}\to K_{s}\eta^{\prime}}}$ | $\displaystyle{\mathrm{K_{s}\eta^{\prime}K^{+}}}$ | $\displaystyle\sim 9.1\ 10^{3}$
$\displaystyle\overline{D^{0}}K^{+}$ | $\displaystyle{\mathrm{\overline{D^{0}}\to K_{s}\omega}}$ | $\displaystyle{\mathrm{K_{s}\omega K^{+}}}$ | $\displaystyle\sim 1.1\ 10^{6}$
$\displaystyle D^{0}K^{+}$ | $\displaystyle{\mathrm{D^{0}\to K_{s}\omega}}$ | $\displaystyle{\mathrm{K_{s}\omega K^{+}}}$ | $\displaystyle\sim 1.1\ 10^{4}$

Table 2. The mean expected number of produced $B^{+}$ decays to specific decay
modes at FCCee at a center of mass energy of 91 GeV over 5 years with 2
detectors. These numbers have to be multiplied by 2 when including $B^{-}$
decays. The branching fractions of the PDG [4] have been used (only the
$K_{s}$ decay to $\pi^{+}\pi^{-}$ is included). Note that the number of events
is indicative since the exact numbers depend on the value of the strong phase
differences $(\Delta)$ and the CKM phases involved, as seen in eqn. (4).

###  4.1 Generic Detector acceptance and resolutions

We define below a generic detector, the resolutions of which are parametrized
as follow :

${\rm Acceptance:}|\cos\theta|<0.95$ ${\rm Track\ p_{T}\
resolution:}{\sigma(p_{T})\over p_{T}^{2}}=2.\times 10^{-5}\ \oplus\
{1.2\times 10^{-3}\over p_{T}\sin\theta}$ ${\rm Track\ \phi,\theta\
resolution:}\mathrm{\sigma(\phi,\theta)\ \mu rad}=18\ \oplus\ {1.5\times
10^{3}\over p_{T}\sqrt[3]{\sin\theta}}$ $\qquad\qquad\qquad{\rm Vertex\
resolution:}\mathrm{\sigma(d_{Im})\ \mu m}=1.8\ \oplus\ {5.4\times 10^{1}\over
p_{T}\sqrt{\sin\theta}}\qquad\qquad$ (46) ${\rm Vertex\
resolution:}<\sigma(\mathrm{d_{Im}})>\mathrm{bachelor\ K\ in}\ D_{s}K$
$<\sigma(\rm{d_{Im}})>\simeq 10\ \mu m$ ${\rm Calorimeter\
resolution:}{\sigma(E)\over E}={3\times 10^{-2}\over\sqrt{E}}\ \oplus\ 5\times
10^{-3}$

where $\theta,\phi$ are the track polar and azymutal angle respectively,
$p_{T}$ (in GeV) the track transverse momentum, $E$ the $e^{\pm}\ ,\gamma$
energy and $\mathrm{d_{Im}}$ the impact parameter. Although the tracking
resolutions can be obtained with a combination of silicon vertex detector and
a gazeous central tracking system, the calorimeter resolution requires a
crystal type detector.

The decays
$B^{+}\rightarrow\overline{D^{0}}(D^{0})K^{+}\rightarrow(K^{+}K^{-})_{D}K^{+}$
and
$B^{+}\rightarrow\overline{D^{0}}(D^{0})K^{+}\rightarrow(K_{s}\pi^{0})_{D}K^{+}$
have then been simulated using the resolutions in Equations (46).

Fig. 4. The $B^{\pm}$ resolutions for the $D$ decays $K^{+}K^{-}$ in
$e^{+}e^{-}\to Z\to B^{+}\to D^{0}K^{+}\to K^{+}K^{-}K^{+}$.

Fig. 5. The $B^{\pm}$ resolutions for the $D$ decays $K_{s}\pi^{0}$ in
$e^{+}e^{-}\to Z\to B^{+}\to D^{0}K^{+}\to
K_{s}\pi^{0}K^{+}\to\pi^{+}\pi^{-}\gamma\gamma K^{+}$.

As one can see from Figs. 4 and 5, the $B^{\pm}$ mass resolution is 6.4 MeV
and 12.0 MeV for the $K^{+}K^{-}K^{\pm}$ and $K_{s}\pi^{0}K^{\pm}$ final
states, respectively. For the latter decay, we use the constraint of the
$D^{0}$ mass, which improves significantly the resolution. However, this
resolution should worsen with a complete simulation, which will include
accurately the flight distances of the resonances $B^{\pm}$, $D^{0}$ and
$K_{s}$. It can also be noted that, using the values $\Delta=-130^{\circ}$ and
$\gamma_{s}=108^{\circ}$, one gets about the same number of events for both
modes within $30\%$, once the acceptances are taken into account, in
particular when requiring the $K_{s}$ to decay within a fiducial volume of
$r=1$ meter in the transverse direction and $z=\pm 1$ meter in the
longitudinal direction. Such a fiducial volume for $K_{s}$ reconstruction
implies that the tracking volume of the detector should be large enough, e.g.
$r_{det.}>2$ meter, $z_{det.}>\pm 2$ meter to enable efficient reconstruction
of the $\pi^{+}\pi^{-}$ pair. The tracking system should also consist of a
large number of measurement layers, as shown in the appendix, which presents
an experimental study of the performance of $K_{s}$ reconstruction in a FCC-ee
detector. As a final comment, one should note that in order to keep the
background at a negligible level, an efficient particle identification system
is required enabling to identify the kaons in a wide range from $\sim 1-30$
GeV.

###  4.2 Sensitivity on the CKM angle $\gamma_{s}$

To study the sensitivity, we assume a specific case with $\Delta=-130^{\circ}$
and $\gamma_{s}=108^{\circ}$. With these figures, the asymmetries
$A_{CP}^{+}(K^{+}K^{-})$ and $A_{CP}^{-}(K_{s}\pi^{0})$, as defined in
equations (35) and (36) are -0.1489 and 0.1377, respectively.

Using the expected number of events for the mode
$B^{\pm}\to(K^{+}K^{-})_{D}K^{\pm}$ and $B^{\pm}\to(K_{s}\pi^{0})_{D}K^{\pm}$,
and for a total integrated luminosity of $150\ ab^{-1}$ accumulated by the
addition of all experiments at FCCee, the expected sensitivity for
$\delta(\gamma_{s})$ is about $2.7^{\circ}$ if $\Delta=-130^{\circ}$. Such a
value for $\Delta$ corresponds to the strong phase in the decay $B^{+}\to
D^{0}K^{+}$ as has been measured, and reported in [4],
$\delta_{B}=130^{\circ}$, taking into account our convention
$\Delta=-\delta_{B}$, as exposed in section 2.4.

One should note that the $\gamma_{s}$ sensitivity can vary according to the
actual value of the strong angle difference $\Delta$, as can be seen in Table
3. The most favourable situation occurs when $\Delta=-90^{\circ}$, in which
case $\delta(\gamma_{s})\simeq 0.8^{\circ}$. When $\Delta=\pm\gamma_{s}$, the
first equation in (37) determines the sine-squared of this common (modulo the
sign) angle, and is redundant with the second equation of (37), which
determines the cosine-squared. Consequently, the equations (37) do not allow a
simultaneous extraction of the angles $\Delta$ and $\gamma_{s}$. The same
applies when $\Delta=180^{\circ}\pm\gamma_{s}$. In the vicinity of these
singularities, the experimental uncertainties on
${\cal{A}}^{+}_{CP}(K^{+}K^{-})$ and on ${\cal{A}}^{-}_{CP}(K_{s}\pi^{0})$
cause equation (37) to not always have real solutions – for example, that
would happen with a probability larger than $30\%$ when the angles are within
$8^{\circ}$ of a singularity. These cases are not considered in Table 3.

These results can be improved further should one use additional
$\overline{D^{0}}(D^{0})$ CP eigenstates such as $K_{s}\eta$,
$K_{s}\eta^{\prime}$, $K_{s}\omega\ldots$ and using the mode
$B^{\pm}\rightarrow D_{CP}K^{*\pm}$ such that a sensitivity in the range 1∘-2∘
does not seem out of reach. Combining this with the possible direct
measurements of $\alpha_{s}$ and $\beta_{s}$ with the modes $B_{s}\to
D_{s}^{\pm}K^{\mp}$ and $B_{s}\to J/\psi\phi$, respectively, with the expected
sensitivities [1] $\delta(\alpha_{s})\simeq 0.4^{\circ}$ and
$\delta(\beta_{s})\simeq 0.04^{\circ}$, one would directly measure the three
angles of the Unitarity Triangle UTsb at FCCee, without assuming the unitarity
of the CKM matrix.

$\Delta$ | ${\cal A}_{CP}^{+}(K^{+}K^{-})$ | ${\cal A}_{CP}^{-}(K_{s}\pi^{0})$ | $\delta(\gamma_{s})$
---|---|---|---
$-10^{\circ}$ | $-0.0306$ | $0.0345$ | $\sim 6.5^{\circ}$
$-30^{\circ}$ | $-0.0887$ | $0.0986$ | $\sim 2.8^{\circ}$
$-50^{\circ}$ | $-0.1377$ | $0.1489$ | $\sim 2.8^{\circ}$
$-90^{\circ}$ | $-0.1868$ | $0.1868$ | $\sim 0.8^{\circ}$
$-120^{\circ}$ | $-0.1668$ | $0.1570$ | $\sim 2.8^{\circ}$
$-130^{\circ}$ | $-0.1489$ | $0.1377$ | $\sim 2.8^{\circ}$
$-150^{\circ}$ | $-0.0986$ | $0.0887$ | $\sim 2.8^{\circ}$
$-170^{\circ}$ | $-0.0345$ | $0.0306$ | $\sim 6.5^{\circ}$

Table 3. The experimental sensitivity for measuring the angle $\gamma_{s}$ at
FCCee using the $B^{\pm}\rightarrow(K^{+}K^{-})_{D}K^{\pm}$ and
$B^{\pm}\rightarrow(K_{s}\pi^{0})_{D}K^{\pm}$ decays for different values of
the strong phase difference $\Delta$. The value $\gamma_{s}=108^{\circ}$ has
been used. Finally, one notes that should one use positive values for
$\Delta$, all asymmetries will have opposite signs, but the sensitivities will
remain the same.

##  5 Conclusion

We have investigated in this paper the sensitivity at FCCee of the angle
$\gamma_{s}$ of the Unitarity Triangle UTsb, defined in Fig. 3. This result is
complementary to the ones obtained for the other angles of this triangle
($\alpha_{s}$ and $\beta_{s}$), as discussed in ref. [1]. Altogether, FCCee
should allow one to measure directly all the angles of the rather flat
Unitarity Triangle UTsb, constraining further the CP sector of the Standard
Model.

## Appendix: reconstruction of $K_{s}\rightarrow\pi^{+}\pi^{-}$ decays

The measurement of the decay
$B^{+}\rightarrow\overline{D^{0}}(D^{0})K^{+}\rightarrow(K_{s}(\rightarrow\pi^{+}\pi^{-})\pi^{0})K^{+}$
requires an efficient reconstruction of $K_{s}\rightarrow\pi^{+}\pi^{-}$
decays, up to typically one meter from the interaction point (IP). This
reconstruction, in a FCC-ee detector, has been studied using Monte-Carlo
events that were passed through a fast simulation of the tracking system of
the experiment based on DELPHES [8], and subsequently analysed within the
FCCAnalyses framework [9]. The PYTHIA8 Monte-Carlo generator was used to
simulate the production of $b\bar{b}$ events at $\sqrt{s}=91.2$ GeV, in which
one $b$ quark hadronises into a charged $B$ meson that decays into the mode of
interest, while the other $b-$leg fragments and decays inclusively. The decay
chain of the $B^{\pm}$ mesons was performed with the EVTGEN program. The
resulting charged particles were turned into simulated tracks using a fast
tracking software implemented in DELPHES. It relies on a full description of
the tracker geometry - the vertex detector and the drift chamber of the IDEA
detector [10], which provide resolutions similar to the ones given in Section
4.1, being used here. The software accounts for the finite detector resolution
and for the multiple scattering in each tracker layer and determines the (non
diagonal) covariance matrix of the helix parameters that describe the
trajectory of each charged particle. This matrix is then used to produce a
smeared 5-parameters track, for each charged particle emitted within the
angular acceptance of the tracker.

The reconstruction of $K_{s}\rightarrow\pi^{+}\pi^{-}$ heavily relies on the
reconstruction of displaced vertices. Here, a standalone code is used to fit a
given set of tracks to a common vertex [11], under the assumption that the
trajectories be perfect helices: the track parameters are varied according to
their covariance matrix so that the tracks can meet at a common point, and the
vertex coordinates $(x,y,z)$, together with their covariance matrix, are
obtained from a $\chi^{2}$ minimisation. The vertexing program [11] offers the
possibility to “guide” the first iteration of the fit by providing a guessed
value of the radial position of the vertex, which can be inferred from the
radial positions of the innermost hit of all the fitted tracks. This
functionality is particularly useful when fitting tracks that made no hit in
the ultra-precise silicon layers of the vertex detector surrounding the beam-
pipe.

### Description of the algorithm

The first step in the $K_{s}$ reconstruction consists in finding the primary
vertex and the primary tracks attached to this vertex (and consequently, the
“secondary” tracks which will be used to search for $K_{s}$ candidates). The
identification of the primary vertex and of the primary tracks follows closely
the LCFI+ algorithm [12]. All tracks that have been reconstructed are first
fitted to a common vertex, using the knowledge of the beam-spot as an external
constraint. The track which gives the largest contribution to the $\chi^{2}$
of the fit, if this contribution is larger than a given cut (here set to
$25$), is then removed from the list of tracks, and the vertex fit is run
again. This procedure is iterated until the contribution of all tracks to the
$\chi^{2}$ is below the cut, providing the primary vertex and the “primary”
tracks attached to it. The tracks that are not attached to the primary vertex
are labelled as “secondary tracks”. This selection of secondary tracks has a
typical purity of more than $97\%$, and is very efficient for tracks that come
from a $K_{s}$ decay, in the decay chain considered here.

|
---|---

Fig. 6. Distribution of the mass of reconstructed $K_{s}$ candidates in
$Z\rightarrow b\bar{b}$ events which contain a
$B^{+}\rightarrow(K_{s}\pi^{0})_{D}K^{+}$ decay, for all candidates (red
histogram) and for fake candidates only (black histogram). Right: Distribution
of the $\chi^{2}$ of the vertex fit of reconstructed candidates, that are
matched to a genuine $K_{s}$ that decays at a distance between $50$ cm and
$80$ cm from the interaction point. The overlaid curve represents a $\chi^{2}$
function with one degree of freedom (the number of degrees of freedom of the
two-tracks fit being equal to one.

The identified secondary tracks are used in the second step of the algorithm.
All combinations of two opposite charge tracks are built to form $K_{s}$
candidates. For each combination, the two tracks are fitted to a common
vertex, and the momenta of the tracks at this vertex are used to reconstruct
the mass $M$ of the candidate, assigning the pion mass to each leg. Candidates
are kept if they fulfil the loose mass cut $0.48<M<0.52$ GeV, and if the
$\chi^{2}$ of the vertex fit is smaller than $10$. The latter requirement
selects more than $99.4\%$ of the $K_{s}$ decays in the process of interest.
The association of the reconstructed tracks to the Monte-Carlo particles is
then used to determine whether a candidate is matched to a
$K_{s}\rightarrow\pi^{+}\pi^{-}$ decay (any $K_{s}$ being considered here, not
necessarily the one produced in the $B^{+}\rightarrow DK^{+}$ decay). By
definition, fake candidates are those which are not matched to such a decay.
The mass distribution of all $K_{s}$ candidates reconstructed in the
$B^{+}\rightarrow DK^{+}$ sample, that satisfy the loose cuts listed above, is
shown in Fig. 6 left. A clear peak is observed at the nominal $K_{s}$ mass
used in the Monte-Carlo, on top of a flat background of fake candidates. A fit
of the mass distribution of genuine $K_{s}$ candidates by a sum of two
Gaussian functions leads to a mass resolution of about $400$ keV when the
$K_{s}$ decays within $30$ cm from the interaction point, reaching $1$ to
$1.2$ MeV for $K_{s}$ that decay at more than $1$ m from the IP. The
distribution of the $\chi^{2}$ of the vertex fit for candidates that are
matched to a $K_{s}\rightarrow\pi^{+}\pi^{-}$ decay occuring in the middle of
the drift chamber is shown in Fig. 6 right. It can be seen that the vertex fit
behaves well, even for far detached vertices.

### Kaon reconstruction efficiency

|
---|---

Fig. 7. Reconstruction efficiency of kaons from the
$B^{+}\rightarrow(K_{s}\pi^{0})_{D}K^{+}$ decay, as a function of the distance
$L_{xyz}$ of the decay vertex of the $K_{s}$ to the IP, for (left) several
cuts on the $K_{s}$ mass candidate, and (right) focusing on $L_{xyz}<50$ cm
and defining the efficiency with respect to kaons above a given energy.

The reconstruction efficiency is shown in Fig. 7 left, as a function of the
distance $L_{xyz}$ of the decay vertex of the $K_{s}$ to the interaction
point. The efficiency is defined with respect to the $K_{s}$ that come from
the $B^{+}\rightarrow(K_{s}\pi^{0})_{D}K^{+}$ decay, for which the Monte-Carlo
daughter pions satisfy the acceptance cut $|\cos\theta|<0.95$. The efficiency
obtained with a tighter mass cut of $\pm 5$ MeV or $\pm 2.5$ MeV around the
nominal $K_{s}$ mass, which is in line with the exquisite resolution mentioned
previously, is also shown. The overall efficiency amounts to $90\%$, $88.5\%$
or $85\%$ depending on the mass cut. With a cut of $\pm 5$ MeV, the efficiency
remains above $80\%$ as long as the $K_{s}$ decays within $1.5$ m from the
interaction point. At larger distances, the efficiency drops since the pion
tracks only go through a small distance in the tracker. The drop in efficiency
observed at small flight distances is due to the correlation of the flight
distance with the $K_{s}$ momentum: the pion tracks from $K_{s}$ that decay
close to the IP are softer and are more affected by multiple scattering or may
lead to “loopers”. This correlation is illustrated in Fig. 7 right, which
shows the efficiency to reconstruct kaons above a given energy. The small
remaining inefficiency for energetic kaons that decay close to the interaction
point is due to the occasional mis-assignment of the pion tracks to the
primary vertex.

### Purity of the $K_{s}$ selection

The purity $P$ of the $K_{s}$ selection is studied in an inclusive sample of
$Z$ bosons that decay hadronically. By definition, $1-P$ corresponds to the
fraction of reconstructed $K_{s}$ that are not matched to a Monte-Carlo
$K_{s}\rightarrow\pi^{+}\pi^{-}$. Figure 8 shows this fraction of mis-
identified $K_{s}$, $1-P$, as a function of the distance $L_{xyz}$ of the
decay vertex of the $K_{s}$ candidate to the interaction point. The purity is
shown for two different mass cuts. Since the IDEA detector will provide a
powerful identification of charged hadrons***This identification will be
provided by measurements of the ionisation energy loss along the tracks, in
the drift chamber, and by the measurement of the time-of-flight of particles,
for example in a dedicated layer outside the drift chamber., the purity is
also shown under the assumption that charged pions can be perfectly separated
from kaons and protons. At $L_{xyz}<3$ cm, mis-identified $K_{s}$ usually
correspond to random combinations of tracks coming from the decays of $\rho$,
$\omega$, $D$ or other mesons (in $70\%$ of the cases, the two legs of the
$K_{s}$ candidate do not come from the same parent). As expected, for
candidates that decay within $1$ cm from the interaction point (first bin of
Fig. 8 left), the purity is low, with a contamination of at least $25\%$. This
contamination is drastically reduced as soon as the $K_{s}$ candidates decay
at more than $1$ or $2$ cm from the IP, and this requirement can easily be
applied in a $K_{s}$ selection as it is very efficient†††For example, kaons
from $B^{+}\rightarrow(K_{s}\pi^{0})_{D}K^{+}$ in $Z\rightarrow b\bar{b}$
events decay at more than $1$ cm ($2$ cm) from the IP in more than $97\%$
($94\%$) of the cases.. Above $3$ cm, the fraction of mis-identified $K_{s}$
is at the per-mil level. In $85\%$ of the cases ($98\%$ at $L_{xyz}>50$ cm),
the two legs of the remaining fake candidates come from the same parent, and
they usually correspond to $\Lambda$ baryons that decay into a proton and a
pion. The $\Lambda$ lifetime being larger than that of the $K_{s}$, this
explains the rise of the contamination with $L_{xyz}$, when particle
identification (PID) is not used in the $K_{s}$ identification, as shown in
Fig. 8 right. However, this contamination is easily and efficiently suppressed
by PID.

|
---|---

Fig. 8. Fraction of mis-identified $K_{s}$ in an inclusive
$Z\rightarrow\mbox{hadrons}$ sample, as a function of the distance $L_{xyz}$
of the decay vertex of the candidate $K_{s}$ to the IP, for (left)
$0<L_{xyz}<10$ cm, and (right) for $1~{}\mbox{cm}<L_{xyz}<2$ m. The different
symbols correspond to different cuts used in the $K_{s}$ identification.

### Performance in a Full Silicon tracker

A “continuous tracking”, with a very large number of layers, as is the case in
a drift chamber, is clearly a key for an efficient
$K_{s}\rightarrow\pi^{+}\pi^{-}$ reconstruction. To illustrate this point, the
$B^{+}\rightarrow(K_{s}\pi^{0})_{D}K^{+}$ events used above have been
processed through a simulation of the CLD detector [13]. The vertex detector
of CLD is quite similar to the one of the IDEA detector, but its main tracker
consists of a few layers of silicon sensors (for example six layers in the
central region). Tracks made of at least five or four hits in the full tracker
are used in the $K_{s}$ reconstruction algorithm described above. The
resulting efficiencies are shown in Fig. 9. As expected, the performance
dramatically worsens compared to the efficiency shown in Fig. 7 left. One
clearly sees steps corresponding to the positions of the tracker layers. For
example, as there are only four barrel layers at a radial distance larger than
$40$ cm, the efficiency for selecting 5-hits tracks displaced by more than
$40$ cm vanishes in the central region, explaining the step seen in Fig. 9
left. Some efficiency can be recovered by loosening the requirement on the
minimum number of hits of the tracks to four, as shown in Fig. 9 right, at the
expense of a likely increased rate of fake tracks‡‡‡Fake tracks can not be
studied with the fast simulation tool used here.. Moreover, for $K_{s}$ that
decay within a few tens of centimeters from the IP, the efficiency is also
lower than the one obtained with the IDEA drift chamber. This is because the
material of the full silicon tracker is twice larger than the one of the IDEA
tracker (drift chamber + vertex detector), leading to much larger effects from
multiple scattering and worse resolutions on the $K_{s}$ vertex and on the
$K_{s}$ mass. While some optimisations could be made to the CLD tracker, it is
clear that an efficient reconstruction of $K_{s}\rightarrow\pi^{+}\pi^{-}$
(and of long-lived particles in general) calls for a highly transparent
tracker with a very large number of layers.

|
---|---

Fig. 9. Reconstruction efficiency of $K_{s}$ from the
$B^{+}\rightarrow(K_{s}\pi^{0})_{D}K^{+}$ decay, as a function of the distance
to the IP of the decay vertex of the $K_{s}$, using tracks reconstructed in
the CLD full silicon tracker, demanding at least five hits (left) or at least
four hits (right) on the tracks.

### Acknowledgments

We wish to thank Franco Bedeschi for making his vertexing code available and
for very useful discussions about the reconstruction of far displaced
vertices.

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