# Finite groups with few character values

Sesuai Y. Madanha  Department of Mathematics and Applied Mathematics,
University of Pretoria, Private Bag X20, Hatfield, Pretoria 0028, South Africa
<EMAIL_ADDRESS>
###### Abstract.

A classical theorem on character degrees states that if a finite group has
fewer than four character degrees, then the group is solvable. We prove a
corresponding result on character values by showing that if a finite group has
fewer than eight character values in its character table, then the group is
solvable. This confirms a conjecture of T. Sakurai. We also classify non-
solvable groups with exactly eight character values.

###### Key words and phrases:

character values, character degrees, almost simple groups

###### 2010 Mathematics Subject Classification:

Primary 20C15

##  1\. Introduction

Let $G$ be a finite group and ${\mathrm{Irr}}(G)$ be the set of complex
irreducible characters of $G$. Recall the definition of a character degree
set:

${\mathrm{cd}}(G):=\\{\chi(1)\mid\chi\in{\mathrm{Irr}}(G)\\}$.

The study of the set ${\mathrm{cd}}(G)$ has received much attention from many
authors and has been shown to have strong influence on the structure of $G$.
In particular, we have the famous Ito-Michler theorem which states that $G$
has a normal abelian Sylow $p$-subgroup if and only if $p$ does not divide any
member of ${\mathrm{cd}}(G)$. On the other extreme is Thompson’s theorem,
which states that if $p$ divides every member of
${\mathrm{cd}}(G)\setminus\\{1\\}$, then $G$ has a normal $p$-complement. In
this note we shall study a bigger set found in a character table:

${\mathrm{cv}}(G)=\\{\chi(g)\mid\chi\in{\mathrm{Irr}}(G),g\in G\\}$.

Little has been done in the study of this set. The first article in this
direction is from T. Sakurai [8], who studied groups with very few character
values. We shall compare results in [8] with corresponding character degree
results. It is well known that $|{\mathrm{cd}}(G)|=1$ if and only if $G$ is
abelian. Sakurai showed that if $|{\mathrm{cv}}(G)|\leq 3$, then $G$ is
abelian. The converse is not true since $|{\mathrm{cv}}(\mathrm{C}_{r})|=r$,
where $\mathrm{C}_{r}$ denote the cyclic group of order $r$ for some positive
integer $r$.

If $|{\mathrm{cd}}(G)|=2$ with ${\mathrm{cd}}(G)=\\{1,n\\}$, then either $G$
has an abelian normal subgroup of index $n$, or $n=p^{e}$ for a prime $p$ and
$G$ is direct product of a $p$-group and abelian group. It turns out that a
non-abelian group $G$ is such that $|{\mathrm{cv}}(G)|=4$ if and only if $G$
is the generalized dihedral group of a non-trivial elementary abelian
$3$-group ([8, Theorem]). A generalized dihedral group of a non-trivial
elementary abelian $p$-group is a group
$G_{n}=[\mathrm{C}_{p}\times\mathrm{C}_{p}\times\cdots\times\mathrm{C}_{p}]\rtimes\mathrm{C}_{2}$,
the semidirect product of the direct product of $n$ copies of $\mathrm{C}_{p}$
and $\mathrm{C}_{2}$, where $\mathrm{C}_{2}$ acts on
$[\mathrm{C}_{n}\times\mathrm{C}_{n}\times\cdots\times\mathrm{C}_{n}]$ without
non-trivial fixed points. For groups with more than four character values,
Sakurai remarked on the difficulty to classify these groups. However, a
conjecture was proposed which states that if the group has fewer than eight
character values, then the group is solvable [8, Remark]. Our aim is to settle
this conjecture:

######  Theorem 1.1.

If $|{\mathrm{cv}}(G)|<8$, then $G$ is solvable.

Isaacs [5, Theorem 12] proved a result that is now considered classical which
states that if $|{\mathrm{cd}}(G)|<4$, then $G$ is solvable. Hence Theorem 1.1
is the corresponding result for character values. Our proof uses the
classification of finite simple groups.

Malle and Moretó [7] classified non-solvable groups with
$|{\mathrm{cd}}(G)|=4$ (see Theorem 2.2). We obtain a corresponding result for
character values:

######  Theorem 1.2.

Let $G$ be a finite non-solvable group. Then $|{\mathrm{cv}}(G)|=8$ if and
only if $G\cong{\mathrm{PSL}}_{2}(5)$ or ${\mathrm{PGL}}_{2}(5)$.

##  2\. Preliminary Results

In this section we list some preliminary results that we need to prove our
main results. We shall use freely these well known results in the following
lemma.

######  Lemma 2.1.

Let $G$ be a finite group, $N$ be a normal subgroup of $G$ and $n$ be a
positive integer. Then the following hold:

  * (a)

$|{\mathrm{cv}}(G/N)|\leq|{\mathrm{cv}}(G)|$,

  * (b)

The character table of $\mathrm{C}_{n}$ has a column with pairwise different
values. In particular, $|{\mathrm{cv}}(\mathrm{C}_{n})|=n$,

  * (c)

If $G$ is non-abelian, then $0\in{\mathrm{cv}}(G)$.

The following is a result of Malle and Moretó [7] in which they classified
non-solvable groups with four character degrees.

######  Theorem 2.2.

[7, Theorem A] Let $G$ be a non-solvable group with $|{\mathrm{cd}}(G)|=4$.
Then one of the following holds:

  * (a)

$G\cong{\mathrm{PSL}}_{2}(2^{f})\times A$ for some $f\geq 2$ and some abelian
subgroup $A$;

  * (b)

$G$ has a normal subgroup $U$ such that $U\cong{\mathrm{PSL}}_{2}(q)$ or
${\mathrm{SL}}_{2}(q)$ for some odd $q\geq 5$, and if $C=\textbf{C}_{G}(U)$,
then $C\leq\textbf{Z}(G)$ and $G/C\cong{\mathrm{PGL}}_{2}(q)$; or

  * (c)

the group $G$ has a normal subgroup of index $2$ that is a direct product of
${\mathrm{PSL}}_{2}(9)$ and central subgroup $C$. Furthermore, $G/C\cong
M_{10}$, the Mathieu group which the stabilizer of a point in the Mathieu
group, $M_{11}$, in its natural permutation representation.

Conversely, if one of (a)-(c) holds, then $|{\mathrm{cd}}(G)|=4$.

He and Zhu [4] described non-solvable groups with five and six character
degrees:

######  Theorem 2.3.

[4, Corollary C] Let $G$ be a non-solvable group and $L$ be the solvable
radical of $G$. If $5\leq|{\mathrm{cd}}(G)|\leq 6$, then $G^{\prime}/L$ is
isomorphic to ${\mathrm{PSL}}_{2}(p^{f})$ with $p^{f}\geq 4$,
${\mathrm{PSL}}_{3}(4)$ or ${}^{2}\mathrm{B}_{2}(2^{f})$, where
$2^{f}=2^{2m+1}$ for some integer $m\geq 1$. Moreover, $G/L$ is an almost
simple group.

We classify almost simple groups with five character degrees. We first have
some definitions needed in our result : $q=p^{f}\geqslant 4$,
${\mathrm{Out}}({\mathrm{PSL}}_{2}(q))=\langle\delta\rangle\times\langle\varphi\rangle,|\langle\varphi\rangle|=f,|\langle\delta\rangle|=\gcd(2,q-1)$.

######  Theorem 2.4.

Let $G$ be a finite almost simple group, that is $S\triangleleft G\leq Aut(S)$
for some non-abelian simple group $S$. Then $|{\mathrm{cd}}(G)|=5$ if and only
if one of the following holds:

  * (a)

$G\cong{\mathrm{PSL}}_{2}(q)$, $q$ odd,
${\mathrm{cd}}(G)=\\{1,(q+\epsilon)/2,q\pm 1,q\\}$,

  * (b)

$G\cong{\mathrm{PSL}}_{2}(2^{f})\langle\varphi\rangle$, $f>2$ prime,
${\mathrm{cd}}(G)=\\{1,2^{f}-1,2^{f},(2^{f}\pm 1)f\\}$,

  * (c)

$G\cong{\mathrm{PSL}}_{2}(2^{f})\langle\varphi^{f/2}\rangle$, $f>2$ even,
${\mathrm{cd}}(G)=\\{1,2^{f},2^{f}+1,2(2^{f}\pm 1)\\}$,

  * (d)

$G\cong{\mathrm{PGL}}_{2}(q)\langle\varphi^{f/2}\rangle$, $f>2$ even, $q$ odd,
${\mathrm{cd}}(G)=\\{1,q,q+1,2(q\pm 1)\\}$,

  * (e)

$G\cong{\mathrm{PGL}}_{2}(3^{f})\langle\varphi\rangle$, $f>2$ even,
${\mathrm{cd}}(G)=\\{1,3^{f},3^{f}-1,(3^{f}\pm 1)f\\}$,

  * (f)

$G\cong{\mathrm{PSL}}_{2}(9)\langle\varphi\rangle\cong\mathrm{S}_{6}$, $f=2$,
${\mathrm{cd}}(G)=\\{1,5,9,10,16\\}$,

  * (g)

$G\cong{\mathrm{PSL}}_{2}(5^{f})\langle\varphi\rangle$, $f>2$ prime,
${\mathrm{cd}}(G)=\\{1,5^{f},(5^{f}+1)/2,5^{f}-1,(5^{f}\pm 1)f\\}$,

  * (h)

$G\cong{\mathrm{PSL}}_{2}(q)\langle\delta\varphi^{f/2}\rangle$, $f>2$ even,
$q\neq 9$, $p$ odd, ${\mathrm{cd}}(G)=\\{1,q,q+1,2(q\pm 1)\\}$.

###### Proof.

First, it is well known that every simple group $S$ has a non-trivial
irreducible character $\chi$ that can be extended to ${\mathrm{Aut}}(S)$.
Suppose that $r$ is a prime that divides the character degree of $\chi$. If
$r$ divides every character degree of $G$, then by Thompson’s theorem [5,
Corollary 12.2], $G$ has a normal $r$-complement, a contradiction.

We may thus assume that no prime divides every character degree of $G$. Since
$G$ has five character degrees, there is no prime dividing four character
degrees of $G$. These groups are the groups in [2, Theorem A]. It is
sufficient to consider groups in case (c) since the groups in (a)-(b) and
(d)-(f) can be ruled out using the character tables in the Atlas [1] and [10].

By inspecting [2, Table 1], we have our result. ∎

##  3\. Main Results

######  Theorem 3.1.

Let $G$ be a finite almost simple group, that is $S\triangleleft G\leq Aut(S)$
for some non-abelian simple group $S$. Then either $|{\mathrm{cv}}(G)|\geq 9$
or $|{\mathrm{cv}}(G)|=8$ and
$G\in\\{{\mathrm{PSL}}_{2}(5),{\mathrm{PGL}}_{2}(5)\\}$. In particular, if
$|{\mathrm{cd}}(G)|\geq 5$, then $|{\mathrm{cv}}(G)|\geq 9$.

###### Proof.

Since $G$ is non-solvable, $|{\mathrm{cd}}(G)|\geq 4$ by [5, Theorem 12.15].
Suppose that $|{\mathrm{cd}}(G)|=4$. By Theorem 2.2, it is sufficient to
consider the following groups: ${\mathrm{PSL}}_{2}(2^{f})$ for some $f\geq 2$,
${\mathrm{PGL}}_{2}(q)$ for some odd $q\geq 5$ and $M_{10}$. The character
tables of these groups can be found in the Atlas [1], [3] and [9]. In
particular, $|{\mathrm{cv}}(G)|\geq 9$ except for ${\mathrm{PSL}}_{2}(5)$ and
${\mathrm{PGL}}_{2}(5)$. In the latter cases $|{\mathrm{cv}}(G)|=8$ as
required.

We may assume that $|{\mathrm{cd}}(G)|=5$. It is sufficient to consider groups
in Theorem 2.4. For cases (a) and (f), we have that $|{\mathrm{cv}}(G)|\geq
9$, using character tables in [3] and Atlas [1]. For cases (b), (e) and (g) we
have that $C=G/{\mathrm{PGL}}_{2}(q)$ is cyclic of order greater than $2$.
Then by Lemma 2.1(b), the character table of $C$ has a column with pairwise
different values. In particular, $a,b\in{\mathrm{cv}}(C)\setminus\\{-1,0,1\\}$
with $a\not=b$. Note that the Steinberg character $\psi$ of
${\mathrm{PSL}}_{2}(q)$ is extendible to $G$ and also $\psi(s)=-1$ for some
$s\in{\mathrm{PSL}}_{2}(q)$ by [3, Theorems 4.7 and 4.9]. Hence
$-1\in{\mathrm{cv}}(G)$ and
$|{\mathrm{cv}}(G)|=|{\mathrm{cd}}(G)\cup\\{a,b,-1,0\\}|\geq 9$. Suppose that
$C$ is of order $2$. Groups with this property are the ones in cases (c), (d)
and (h). Consider the characters $\chi_{1},\chi_{2}\in{\mathrm{Irr}}(G)$ such
that $\chi_{1}(1)=2(q+1)$ and $\chi_{2}(1)=2(q-1)$. Then since
${\mathrm{PGL}}_{2}(q)$ has no irreducible characters of degrees $2(q\pm 1)$,
$\chi_{i}=\theta_{i1}+\theta_{i2}$ by [5, Theorem 6.2], where
$\theta_{11},\theta_{12},\theta_{21},\theta_{22}\in{\mathrm{Irr}}(S)$,
$\theta_{11}(1)=\theta_{12}(1)=q+1$ and $\theta_{21}(1)=\theta_{22}(1)=q-1$.
Using [3, Theorems 4.7 and 4.9], we have that
$\theta_{11}(g)=\theta_{12}(g)=1$ and $\theta_{21}(g)=\theta_{22}(g)=-1$ for
some $g\in{\mathrm{PGL}}_{2}(q)$. Hence $\chi_{1}(g)=2$, $\chi_{1}(g)=-2$ and
$|{\mathrm{cd}}(G)\cup\\{-2,-1,0,2\\}|\geq 9$.

Going forward we may assume that $|{\mathrm{cd}}(G)|\geq 6$. Suppose that
$|{\mathrm{cd}}(G)|\geq 6$ and $S\cong{\mathrm{PSL}}_{2}(q)$. We may assume
that $G$ is not isomorphic to ${\mathrm{PSL}}_{2}(2^{f})$ with $f\geq 2$,
${\mathrm{PSL}}_{2}(p^{f})$ with $p$ odd or ${\mathrm{PGL}}_{2}(p^{f})$ with
$p$ odd. Let $H\in\\{{\mathrm{PSL}}_{2}(2^{f}),f\geq
2\\}\cup\\{{\mathrm{PSL}}_{2}(q)$, $q$ odd $\\}\cup\\{{\mathrm{PGL}}_{2}(q),q$
odd$\\}$. If $G\cong H\mathrm{C}_{r}$, where $r\mid f$, $r>2$. Then arguing as
above, we have that $a,b\in{\mathrm{cv}}(\mathrm{C}_{r})\setminus\\{-1,1\\}$
and $|{\mathrm{cv}}(G)|=|{\mathrm{cd}}(G)\cup\\{-1,0,a,b\\}|\geq 10$. If
$G\cong H\mathrm{C}_{2}$, $2\mid f$. Then either $2(q-1)$ or $2(q+1)$ is a
character degree of $G$. Arguing as above, we have that either
$-2\in{\mathrm{cv}}(\mathrm{C})$ or $2\in{\mathrm{cv}}(\mathrm{C})$. Since
$-1,0\in{\mathrm{cv}}(G)$ and $|{\mathrm{cd}}(G)|\geq 6$ we have that
$|{\mathrm{cv}}(G)|\geq 9$.

If $S\cong{\mathrm{PSL}}_{3}(4)$, then the result follows using the character
table in the Atlas [1]. Suppose that $S\cong^{2}\\!\\!\mathrm{B}_{2}(2^{f})$,
where $2^{f}=2^{2m+1}$ for some integer $m\geq 1$. If
$G\cong^{2}\\!\\!\mathrm{B}_{2}(2^{f})$, it follows that
$|{\mathrm{cv}}(G)|\geq 9$ by the character table in [10, Theorem 13]. We may
assume that $S<G<{\mathrm{Aut}}(S)$. Then $C=G/S$ is cyclic of odd order.
Using the argument as in the case of ${\mathrm{PSL}}_{2}(q)$ above, we have
that $a,b\in{\mathrm{cv}}(C)\setminus\\{-1,1\\}$. Hence
$|{\mathrm{cv}}(G)|\geq|{\mathrm{cd}}(G)\cup\\{a,b,0\\}|=9$.

If $|{\mathrm{cd}}(G)|\geq 7$, then by column orthogonality, there is a
character value which is a negative number and so $|{\mathrm{cv}}(G)|\geq 9$.
This concludes our proof. ∎

###### Proof of Theorem 1.1.

We prove our result using induction on $|G|$. Since $|{\mathrm{cv}}(G)|<8$,
every homomorphic image of $G$ has the same property. Let $N$ be a minimal
normal subgroup of $G$. If another such minimal normal subgroup, say $M$,
exists then $G$ is the subdirect product of solvable groups $G/N$ and $G/M$,
and so is solvable. This implies that $N$ is unique and non-abelian. Hence $G$
is almost simple group. The result follows by Theorem 3.1. ∎

Before we prove Theorem 1.2, we recall a definition. If $N\triangleleft G$,
then ${\mathrm{Irr}}(G{\mid}N)=\\{\chi\in{\mathrm{Irr}}(G)\mid
N\not\subseteq\ker\chi\\}$ and
${\mathrm{cd}}(G{\mid}N)=\\{\chi(1)\mid\chi\in{\mathrm{Irr}}(G{\mid}N)\\}$.

###### Proof of Theorem 1.2.

Since $G$ is non-solvable, $|{\mathrm{cd}}(G)|\geq 4$. If
$5\leq|{\mathrm{cd}}(G)|\leq 6$, then by Theorem 2.3, $G^{\prime}/L$ is
isomorphic to ${\mathrm{PSL}}_{2}(p^{f})$, $p^{f}\geq 4$,
${\mathrm{PSL}}_{3}(4)$ or ${}^{2}\mathrm{B}_{2}(2^{f})$, where
$2^{f}=2^{2m+1}$ for some integer $m\geq 1$, where $L$ is the solvable radical
of $G$. Specifically, $G/L$ is an almost simple group. Using Theorem 3.1,
$|{\mathrm{cv}}(G)|\geq|{\mathrm{cv}}(G/L)|\geq 9$.

If $|{\mathrm{cd}}(G)|=4$, then using Theorem 2.2 and Theorem 3.1, it is
sufficient to consider $G\cong{\mathrm{PSL}}_{2}(5)\times A$ for some abelian
subgroup $A$ or $G/C\cong{\mathrm{PGL}}_{2}(5)$, where $C=\textbf{C}_{G}(U)$,
$U={\mathrm{PSL}}_{2}(5)$ or ${\mathrm{SL}}_{2}(5)$ with $C\leq\textbf{Z}(G)$
and $U$ is a normal subgroup of $G$. Suppose that
$G\cong{\mathrm{PSL}}_{2}(5)\times A$ with non-trivial $A$. Consider
$\theta\in{\mathrm{Irr}}({\mathrm{PSL}}_{2}(5))$ such that $\theta(1)=5$ and
$\phi\in{\mathrm{Irr}}(A)$ such that $\phi(g)=k<0$. Then $\chi(1\times
g)=\theta(1)\times\phi(g)=5k<0$ and $|{\mathrm{cv}}(G)|\geq 9$. Suppose that
$G$ is a group in the latter case. We want to show that $C=1$. Suppose that
$C\neq 1$. Let $\textbf{Z}(G)U=N$ and note that ${\mathrm{cd}}(G{\mid}N)\geq
3$ by [6, Theorem B]. Then there is a $\chi\in{\mathrm{Irr}}(G)$ such that
$\chi(1)=m\in{\mathrm{cd}}(G{\mid}N)\setminus\\{1,2\\}$. For $1\neq
z\in\textbf{Z}(G)$, we have that $\chi(z)=-m$. This means that
$|{\mathrm{cv}}(G)|\geq|{\mathrm{PGL}}_{2}(5)\cup\\{-m\\}|=9$. Therefore
$G={\mathrm{PGL}}_{2}(5)$ and that concludes our argument. ∎

## Acknowledgements

The author would like to thank the reviewer for their comments which improved
the presentation of the article.

## References

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p\left({\bm{s}}^{t-1}\right)\right)-\sum\limits_{\bm{\theta}}\delta_{\bm{\theta}^{0}}(\bm{\theta})\mathbb{D}\left(p\left({\bm{s}}^{t-1}\mid{\bm{s}}^{t},\bm{\theta}^{0}\right)\mid\mid
p\left({\bm{s}}^{t-1}\right)\right)\rvert$ (41)
$\displaystyle=\left\lvert\sum\limits_{\bm{\theta}}p\left(\bm{\theta}\mid{\bm{H}}_{t}\right)\mathbb{D}\left(p\left({\bm{s}}^{t-1}\mid{\bm{s}}^{t},\bm{\theta}\right)||p\left({\bm{s}}^{t-1}\right)\right)-\delta_{\bm{\theta}^{0}}(\bm{\theta})\mathbb{D}\left(p\left({\bm{s}}^{t-1}\mid{\bm{s}}^{t},\bm{\theta}^{0}\right)||p\left({\bm{s}}^{t-1}\right)\right)\right\rvert$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{\leq}}\lvert\sum\limits_{\bm{\theta}}\mathbb{D}\left(p\left({\bm{s}}^{t-1}\mid{\bm{s}}^{t},\bm{\theta}\right)||p\left({\bm{s}}^{t-1}\right)\right)\rvert\epsilon=\mathbb{I}_{c}({\bm{s}}^{t-1})\epsilon=\mathcal{O}(\epsilon),$

where $(a)$ follows from the assumption that
$D_{TV}(p(\bm{\theta}\mid{\bm{H}}_{t})\mid\mid\delta_{\bm{\theta}^{0}}(\bm{\theta}))\leq\epsilon$.
(41) shows that as long as the intrinsic information provided by any state
${\bm{s}}^{t}$ is finite, the error intrinsic information between expert and
imitator agents can stay proportional to or below $\mathcal{O}(\epsilon)$.

##  Appendix C Proof of Lemma 2

The error in integrated information can be written as

$\small\mathbb{I}_{\phi,c}^{\eta}-\mathbb{I}_{\phi,c}^{E}=I_{c}({\bm{s}}^{t-1};{\bm{s}}^{t},\bm{\theta})-I_{c}({\bm{s}}^{t-1};{\bm{s}}^{t},\widehat{\bm{\theta}}).\vspace{-3mm}$
(42)

We can rewrite (42) as
$\small\mathbb{I}_{\phi,c}^{\eta}-\mathbb{I}_{\phi,c}^{E}=H\left({\bm{s}}^{t-1}\mid{\bm{s}}^{t},\widehat{\bm{\theta}}({\bm{H}}_{t})\right)-H\left({\bm{s}}^{t-1}\mid{\bm{s}}^{t},\bm{\theta}\right).$
Further, we look at an averaged error measure (averaged across several
estimates of $\widehat{\bm{\theta}}$), which can be written as:

$\small\begin{array}[]{l}\mathbb{I}_{\phi,c}^{\eta}-\mathbb{I}_{\phi,c}^{E}=\sum\limits
p\left(\bm{\theta}=\widehat{\bm{\theta}}\mid{\bm{H}}_{t}\right)p({\bm{s}}^{t-1}\mid{\bm{s}}^{t},\widehat{\bm{\theta}})\log
p\left({\bm{s}}^{t-1}\mid{\bm{s}}^{t},\widehat{\bm{\theta}}({\bm{H}}_{t})\right)-H\left({\bm{s}}^{t-1}\mid{\bm{s}}^{t},\bm{\theta}\right).\end{array}\vspace{-3mm}$
(43)

To simplify this further, we can write
$\frac{p({\bm{s}}^{t-1},{\bm{s}}^{t},\widehat{\bm{\theta}})}{p({\bm{s}}^{t},\widehat{\bm{\theta}})}=p({\bm{s}}^{t}\mid{\bm{s}}^{t-1},\widehat{\bm{\theta}})$.

$\displaystyle\mathbb{I}_{\phi,c}-\mathbb{I}_{\phi,c}^{e}$
$\displaystyle=\sum\limits
p\left(\bm{\theta}=\widehat{\bm{\theta}}\mid{\bm{H}}_{t}\right)p({\bm{s}}^{t-1}\mid{\bm{s}}^{t},\widehat{\bm{\theta}})\log
p\left({\bm{s}}^{t-1}\mid{\bm{s}}^{t},\widehat{\bm{\theta}}({\bm{H}}_{t})\right)-H\left({\bm{s}}^{t-1}\mid{\bm{s}}^{t},\bm{\theta}\right)$
(44) $\displaystyle=\sum\limits
p\left(\bm{\theta}=\widehat{\bm{\theta}}\mid{\bm{H}}_{t}\right)p({\bm{s}}^{t}\mid{\bm{s}}^{t-1},\widehat{\bm{\theta}})\log
p\left({\bm{s}}^{t}\mid{\bm{s}}^{t-1},\widehat{\bm{\theta}}({\bm{H}}_{t})\right)-H\left({\bm{s}}^{t-1}\mid{\bm{s}}^{t},\bm{\theta}\right)$
$\displaystyle=\mathbb{E}_{q}\left(H\left({\bm{s}}^{t-1}\mid{\bm{s}}^{t},\bm{\theta}\right)-H\left({\bm{s}}^{t-1}\mid{\bm{s}}^{t},\bm{\theta})\right)\right).$

##  Appendix D Proof of Theorem 1

We prove the transitive property by a counterexample. Let us suppose that
$\bm{d}_{1}\prec_{w_{1}}\bm{d}_{2}$ and $\bm{d}_{2}\prec_{w_{2}}\bm{d}_{3}$,
where $\phi_{r_{1}}>0$ and $\phi_{r_{2}}>0$. Also, assume that $\bm{d}_{1}$ is
not related to $\bm{d}_{3}$. This implies that there is an overlapping purview
between $\bm{d}_{2}$ and $\bm{d}_{3}$ that is not related to $\bm{d}_{1}$. Let
this partition be denoted by $\bm{d}_{x}$, which is a subset of $\bm{d}_{2}$.
We can then partition $\bm{d}_{2}$ into two sets: $\bm{d}_{x}$ and
$\bm{d}_{y}$, where $\bm{d}_{x}\cap\bm{d}_{y}=\emptyset$. Note that
$\bm{d}_{x}$ is the portion of $\bm{d}_{2}$ that does not have any overlap
with $\bm{d}_{1}$. Since the irreducibility measure of the overlapping purview
between $\bm{d}_{2}$ and $\bm{d}_{3}$ is non-zero, it follows that the
irreducibility measure of $\bm{d}_{x}$ is also non-zero. Now, we can conclude
that $\bm{d}_{2}$ is reducible since it can be partitioned into two subsets
with non-zero irreducibility measures. Furthermore, the integrated information
of a reducible system is zero. Hence, we can conclude that $\bm{d}_{2}$ does
not exist as a causal distinction or a concept. Therefore, we have proven that
if $\bm{d}_{1}\prec_{w_{1}}\bm{d}_{2}$ and
$\bm{d}_{2}\prec_{w_{2}}\bm{d}_{3}$, then $\bm{d}_{1}$ is causally related to
$\bm{d}_{3}$. Otherwise, if $\bm{d}_{1}$ is not causally related to
$\bm{d}_{3}$, then $\bm{d}_{2}$ does not exist as a causal distinction. The
monotonicity property follows directly from the assumption that the number of
causal distinctions in $\bm{d}_{1}$ is less than that in $\bm{d}_{2}$.

##  Appendix E Proof of Lemma 2

We use $V^{\pi}({\bm{s}},\mathcal{E})$ and
$V^{\pi}({\bm{s}},\widehat{\mathcal{E}})$ to refer to the value of policy
$\pi$ when starting from any state ${\bm{s}}$ in the MDP $\mathcal{E}$ and its
variant $\widehat{\mathcal{E}}$, respectively. To simplify the notation, we
introduce the following definition.

$P^{\pi_{\eta}}_{\mathcal{E}}(\cdot\mid{\bm{s}}):=\mathbb{E}_{{\bm{a}}\mid\pi(\cdot\mid{\bm{s}})}P^{\pi_{\eta}}_{\mathcal{E}}(\cdot\mid{\bm{s}},{\bm{a}})\,\,\,\mbox{and}\,\,\,P^{\pi_{\eta}}_{\widehat{\mathcal{E}}}(\cdot\mid{\bm{s}}):=\mathbb{E}_{{\bm{a}}\mid\pi(\cdot\mid{\bm{s}})}P^{\pi_{\eta}}_{\widehat{\mathcal{E}}}(\cdot\mid{\bm{s}},{\bm{a}})\vspace{-3mm}$
(45)

Before the proof, we note the following useful observations.

  * •

Since
$D_{TV}(P_{\mathcal{E}}(\textperiodcentered\mid{\bm{s}},{\bm{a}}),P_{\widehat{\mathcal{E}}}(\textperiodcentered\mid{\bm{s}},{\bm{a}}))\leq\epsilon_{\mathcal{E}}\forall({\bm{s}},{\bm{a}}),$
the inequality also holds for an average over actions, i.e.
$D_{TV}(P_{\mathcal{E}}(\textperiodcentered\mid{\bm{s}}),P_{\widehat{\mathcal{E}}}(\textperiodcentered\mid{\bm{s}}))\leq\epsilon_{\mathcal{E}}\forall{\bm{s}}$.

  * •

Given that the rewards are bounded, it is possible to attain a maximum reward
of $R_{\textrm{max}}$ at each time step. By using a geometric series with a
discount factor of $\gamma$, we can express this as follows:

$\max\limits_{{\bm{s}}\in\mathcal{S}}V^{\pi}({\bm{s}},\mathcal{E})\leq\frac{R_{\textrm{max}}}{1-\gamma}\,\,\,\forall\pi,{\bm{s}}\vspace{-3mm}$
(46)

  * •

Consider a real-valued function $f(x)$ defined over ${\mathcal{X}}$ such that
its output range is limited to $[-f_{\textrm{max}},f_{\textrm{max}}]$. Let
$P_{1}(x)$ and $P_{2}(x)$ be two probability distribution (density) over the
space ${\mathcal{X}}$. Then, we have

$\lvert\mathbb{E}_{x\mid P_{1}(x)}f(x)-\mathbb{E}_{x\mid
P_{2}(x)}f(x)\rvert\leq 2f_{\textrm{max}}D_{TV}(P_{1},P_{2}).\vspace{-3mm}$
(47)

Based on the observations mentioned above, we can establish the following set
of inequalities:

$\displaystyle\lvert
V^{\pi}({\bm{s}},\mathcal{E})-V^{\pi}({\bm{s}},\widehat{\mathcal{E}})\rvert$
$\displaystyle=\lvert\mathcal{R}({\bm{s}})+\gamma{\mathbb{E}}_{{\bm{s}}^{\prime}\sim
P^{\pi}_{\mathcal{E}}(\cdot\mid{\bm{s}})}V^{\pi}({\bm{s}}^{\prime},\mathcal{E})-\mathcal{R}({\bm{s}})-\gamma{\mathbb{E}}_{{\bm{s}}^{\prime}\sim
P^{\pi}_{\widehat{\mathcal{E}}}(\cdot\mid{\bm{s}})}V^{\pi}({\bm{s}}^{\prime},\widehat{\mathcal{E}})\rvert$
(48) $\displaystyle\leq\gamma\lvert{\mathbb{E}}_{{\bm{s}}^{\prime}\sim
P^{\pi}_{\mathcal{E}}(\cdot\mid{\bm{s}})}V^{\pi}({\bm{s}}^{\prime},\mathcal{E})-{\mathbb{E}}_{{\bm{s}}^{\prime}\sim
P^{\pi}_{\widehat{\mathcal{E}}}(\cdot\mid{\bm{s}})}V^{\pi}({\bm{s}}^{\prime},\mathcal{E})\rvert$
$\displaystyle+\gamma\lvert{\mathbb{E}}_{{\bm{s}}^{\prime}\sim
P^{\pi}_{\widehat{\mathcal{E}}}(\cdot\mid{\bm{s}})}\left[V^{\pi}({\bm{s}}^{\prime},\mathcal{E})-V^{\pi}({\bm{s}}^{\prime},\widehat{\mathcal{E}})\right]\rvert$
$\displaystyle\leq
2\gamma\left(\max_{{\bm{s}}^{\prime}\in\mathcal{S}}V^{\pi}({\bm{s}}^{\prime},\mathcal{E})\right)D_{TV}(P^{\pi}_{\mathcal{E}}(\cdot\mid{\bm{s}}),P^{\pi}_{\widehat{\mathcal{E}}}(\cdot\mid{\bm{s}}))$
$\displaystyle+\gamma\left(\max_{{\bm{s}}^{\prime}\in\mathcal{S}}\lvert
V^{\pi}({\bm{s}}^{\prime},\mathcal{E})-V^{\pi}({\bm{s}}^{\prime},\widehat{\mathcal{E}})\rvert\right)$

Since the above bound is applicable to all states, we have $\forall\pi$

$\displaystyle(1-\gamma)\max\limits_{{\bm{s}}^{\prime}\in\mathcal{S}}\lvert
V^{\pi}({\bm{s}}^{\prime},\mathcal{E})-V^{\pi}({\bm{s}}^{\prime},\widehat{\mathcal{E}})\rvert$
$\displaystyle\leq\frac{2\gamma
R_{\textrm{max}}}{1-\gamma}D_{TV}(P^{\pi}_{\mathcal{E}}(\cdot\mid{\bm{s}}),P^{\pi}_{\widehat{\mathcal{E}}}(\cdot\mid{\bm{s}}))$
(49)
$\displaystyle\leq\frac{2\gamma\epsilon_{\mathcal{E}}R_{\textrm{max}}}{1-\gamma}.$

##  Appendix F Proof of Theorem 3

The first step taken is to simplify the performance difference, followed by
bounding the different terms. The proof follows similar steps as in [49]. Let
$\pi_{\widehat{\mathcal{E}}}^{\prime}$ to be an optimal policy in the model,
so that $J(\pi_{\widehat{\mathcal{E}}}^{\prime},\widehat{\mathcal{E}})\geq
J(\pi_{\widehat{\mathcal{E}}},\widehat{\mathcal{E}}),\forall\pi$. We can break
down the performance difference into different contributions as follows:

$\displaystyle
J(\pi^{*},\mathcal{E})-J(\pi,\mathcal{E})=J(\pi^{*},\mathcal{E})-J(\pi^{*},\widehat{\mathcal{E}})+J(\pi^{*},\widehat{\mathcal{E}})-J(\pi,\mathcal{E})$
(50)
$\displaystyle=\underbrace{J(\pi^{*},\mathcal{E})-J(\pi^{*},\widehat{\mathcal{E}})}_{\textrm{Term-I}}+\underbrace{J(\pi^{*},\widehat{\mathcal{E}})-J(\pi,\widehat{\mathcal{E}})}_{\textrm{Term-
II}}+\underbrace{J(\pi,\widehat{\mathcal{E}})-J(\pi,\mathcal{E})}_{\textrm{Term-
III}}$

Let’s focus our attention on Term-II, which pertains to sub-optimality in the
planning problem. Note that
$J(\pi^{*},\widehat{\mathcal{E}})-J(\pi^{*},\widehat{\mathcal{E}})=J(\pi^{*},\mathcal{E})-J(\pi^{*}_{\widehat{\mathcal{E}}},\widehat{\mathcal{E}})+J(\pi^{*}_{\widehat{\mathcal{E}}},\widehat{\mathcal{E}})-J(\pi^{*},\widehat{\mathcal{E}})$.
We have
$J(\pi^{*},\widehat{\mathcal{E}})-J(\pi^{*}_{\widehat{\mathcal{E}}},\widehat{\mathcal{E}})\leq
0$ since $\pi^{*}$ is the optimal policy in the model, and we have
$J(\pi^{*}_{\widehat{\mathcal{E}}},\widehat{\mathcal{E}})-J(\pi^{*},\widehat{\mathcal{E}})\leq\epsilon$
due to the approximate equilibrium condition. For Term-III, we will refer to
the model error performance difference theorem (Theorem 2). It’s worth noting
that the equilibrium condition of low error, along with Pinsker’s inequality
[50], implies that
$\mathbb{E}_{{\bm{s}}\sim\mu{\mathcal{E}}^{\pi}}\left[D_{TV}(P_{\mathcal{E}}(\cdot\mid{\bm{s}},{\bm{a}}),P_{\widehat{\mathcal{E}}}(\cdot\mid{\bm{s}},{\bm{a}}))\right]\leq\sqrt{\epsilon_{\mathcal{E}}}$.
Using this and Theorem 2, we have

$J(\pi,\widehat{\mathcal{E}})-J(\pi,\mathcal{E})\leq\frac{2\gamma\sqrt{\epsilon_{\mathcal{E}}}R_{\textrm{max}}}{(1-\gamma)^{2}\mathbb{I}_{0}t}.\vspace{-1mm}$
(51)

Lastly, Term-I is a transfer learning term that quantifies the error of
$\widehat{\mathcal{E}}$ (which has low error under $\pi$) when operating under
the distribution of $\pi^{*}$. The performance difference can be expressed as:

$\displaystyle J(\pi^{*},\mathcal{E})-J(\pi^{*},\widehat{\mathcal{E}})$
$\displaystyle=\frac{1}{1-\gamma}\mathbb{E}_{({\bm{s}},{\bm{a}})\sim\mu_{\mathcal{E}}^{\pi^{*}}}\left[\mathcal{R}({\bm{s}})\right]-\frac{1}{1-\gamma}\mathbb{E}_{({\bm{s}},{\bm{a}})\sim\mu_{\widehat{\mathcal{E}}}^{\pi^{*}}}\left[\mathcal{R}({\bm{s}})\right]$
(52)
$\displaystyle\leq\frac{2R_{\textrm{max}}}{(1-\gamma)\mathbb{I}_{0}t}D_{TV}(\mu_{\mathcal{E}}^{\pi},\mu_{\widehat{\mathcal{E}}}^{\pi^{*}}).$

Putting all the terms together, we have

$J(\pi^{*},\mathcal{E})-J(\pi,\mathcal{E})\leq\frac{2R_{\textrm{max}}}{(1-\gamma)\mathbb{I}_{0}t}D_{TV}(\mu_{\mathcal{E}}^{\pi^{*}},\mu_{\widehat{\mathcal{E}}}^{\pi^{*}})+\epsilon_{\pi}+\frac{2\gamma\sqrt{\epsilon_{\mathcal{E}}}R_{\textrm{max}}}{(1-\gamma)^{2}\mathbb{I}_{0}t}.\vspace{-3mm}$
(53)

##  Appendix G Proposed Solution for the Bi-level Optimization

The proposed bi-level optimization solution follows a similar procedure to
that of Stackelberg games [51]. To begin, we briefly describe the strategy for
continuous bi-level optimization. Consider a two-level optimization problem
involving problems $A$ and $B$, with their respective parameters denoted by
$\bm{\theta}_{A}$ and $\bm{\theta}_{B}$. The objective function for problem
$A$ is represented by $L_{A}(\bm{\theta}_{A},\bm{\theta}_{B})$, and that for
problem $B$ is represented by $L_{B}(\bm{\theta}_{A},\bm{\theta}_{B})$, and
both aim to minimize their respective losses. The bi-level optimization
problem can be solved usings a nested optimization procedure:

$\displaystyle\min_{\bm{\theta}_{A}}L_{A}(\bm{\theta}_{A},\bm{\theta}_{B}^{*}(\bm{\theta}_{A}))\
$ $\displaystyle\text{subject
to}\quad\bm{\theta}_{B}^{*}(\bm{\theta}_{A})=\min_{\bm{\theta}_{B}}L_{B}(\bm{\theta}_{A},\bm{\theta}_{B}),$
(54)

where the expression $\bm{\theta}_{B}^{*}(\bm{\theta}_{A})$ denotes the best
solution for problem $B$ given the outer level problem $A$’s choice of
$\bm{\theta}_{A}$, and $\bm{\theta}_{A}$ represents the solution to the
optimization problem described earlier. In the inner level, problem $B$
implicitly selects its parameters based on the choice of $\bm{\theta}_{A}$,
and problem $A$ is aware of this relationship and can use this information to
update its own parameters.

To solve the nested optimization, we can focus on optimizing $\bm{\theta}_{A}$
iteratively. Specifically, we can update $\bm{\theta}_{A}$ as:
$\bm{\theta}_{A}\leftarrow\bm{\theta}_{A}\\!-\\!\alpha_{A}\frac{\partial
L_{A}(\bm{\theta}_{A},\bm{\theta}_{B}(\bm{\theta}_{A}^{*}))}{\partial\bm{\theta}_{A}}$,
where $\alpha_{A}$ is the learning rate and the gradient can be written as:

$\begin{array}[]{l}\frac{\partial
L_{A}(\bm{\theta}_{A},\bm{\theta}_{B}(\bm{\theta}_{A}^{*}))}{\partial\bm{\theta}_{A}}=\frac{\partial\bm{\theta}_{B}^{*}(\bm{\theta}_{A})}{\partial\bm{\theta}_{A}}\frac{\partial
L_{A}(\bm{\theta}_{A},\bm{\theta}_{B}(\bm{\theta}_{A}^{*}))}{\partial\bm{\theta}_{B}}\mid_{\bm{\theta}_{B}=\bm{\theta}_{B}^{*}}+\frac{\partial
L_{A}(\bm{\theta}_{A},\bm{\theta}_{B}(\bm{\theta}_{A}^{*}))}{\partial\bm{\theta}_{A}}\mid_{\bm{\theta}_{B}=\bm{\theta}_{B}^{*}}\end{array}\vspace{-2mm}$
(55)

The Jacobian $\frac{\partial\bm{\theta}_{B}^{*}}{(\partial\bm{\theta}_{A})}$
can be computed using the implicit function theorem as in [52]. Thus, in
principle, we can compute the gradient with respect to the $A$’s parameters
and solve the nested optimization, to at least a local minimizer.


# Summary of WG7 at CKM 2018:
“Mixing and violation in the $D$ system: $x_{D}$, $y_{D}$, $|q/p|_{D}$,
$\phi_{D}$, and direct
violation in $D$ decays”

Vishal Bhardwaj  Indian Institute of Science Education and Research Mohali,
India  Mirco Dorigo  CERN and INFN Sezione di Trieste  Fu-Sheng Yu  School of
Nuclear Science and Technology, Lanzhou University, China

###### Abstract

The 10th International Workshop on the CKM Unitary Triangle took place at the
University of Heidelberg on September 17th–21st, 2018. In this write-up, we
summarize the material discussed at the workshop by the Working Group 7, which
focused on latest experimental results and theoretical developments in the
study of mixing and $C\\!P$ violation in the neutral $D$ system, and of
$C\\!P$ violation and decay properties of other charm mesons and baryons.

##  1 Introduction

The study of ${D}^{0}$-${\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$
mixing offers an unique opportunity to probe flavour changing neutral currents
between up-type quarks, which could be affected by physics beyond the standard
model (SM) in very different ways compared to the down-type quarks featuring
the neutral strange and bottom mesons systems. Violation of the $C\\!P$
symmetry in mixing-induced processes is predicted to be very small in the SM,
giving good prospects for the indirect observation of non-SM contributions.
Effects of $C\\!P$ violation in decays of charm mesons and baryons are
difficult to predict; nonetheless, the current level of the most precise
measurements, around $10^{-3}$, is getting closer to that needed for the
observation of the tiny $C\\!P$ asymmetries generally expected in the SM. In
these proceedings, we summarize the most recent efforts in this endeavour
which were discussed by the Working Group 7 at the last International Workshop
on the CKM Unitary Triangle, held at the University of Heidelberg on September
17th–21st, 2018.

We divide the discussion into the following parts. Mixing and indirect $C\\!P$
violation studies are reported in Sect. 2; direct $C\\!P$ violation is
discussed in Sect. 3. A brief report on $C\\!P$-violating triple product
asymmetries is presented in Sect. 4, while Sect. 5 reports results on charmed
baryons, a topic which became very interesting recently. We present some
experimental prospects in Sect. 6, and give a conclusion in Sect. 7. Charge-
conjugation is implied in what follows, and when more than one uncertainty is
assigned to a result, the first refers to the statistical uncertainty, the
second to the systematic. These conventions are applied through the entire
document, unless stated otherwise.

##  2 Mixing and indirect $C\\!P$ violation

The physical states of the time-dependent Hamiltonian describing the neutral
$D$ system are superpositions of the flavor eigenstates,
${|D_{1,2}\rangle=p|{{D}^{0}}\rangle\pm q|{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}\rangle}$, and this generates
flavor mixing. The parameters $p$ and $q$ are complex numbers which satisfy
the relation $|p|^{2}+|q|^{2}=1$. If charge-parity ($C\\!P$) symmetry holds,
$q$ is equal to $p$ and two dimensionless parameters,
$x\equiv(m_{1}-m_{2})/\Gamma$ and $y\equiv(\Gamma_{1}-\Gamma_{2})/2\Gamma$,
fully describe the oscillations. Here, $m_{1(2)}$ and $\Gamma_{1(2)}$ are the
mass and decay width of the $C\\!P$-even (odd) eigenstate $D_{1(2)}$,
respectively, and $\Gamma\equiv(\Gamma_{1}+\Gamma_{2})/2$ is the average decay
width. Violation of the $C\\!P$ symmetry in mixing leads to a non-unity value
of the ratio $|q/p|$. When both ${D}^{0}$ and ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ mesons can decay to a common
final state with amplitudes $\mathcal{A}$ and $\overline{\mathcal{A}}$,
respectively, $C\\!P$ violation can appear in the interference between mixing
and decay amplitudes, and the quantity
$\phi\equiv\arg[q\mathcal{A}/p\overline{\mathcal{A}}]$ differs from zero.
Violation of the $C\\!P$ symmetry in mixing and in the interference between
mixing and decay amplitudes are referred to as manifestations of indirect
$C\\!P$ violation. When $C\\!P$ violation happens in the decay,
$|\mathcal{A}/\overline{\mathcal{A}}|\neq 1$, this is referred to as direct
$C\\!P$ violation.

The importance of studying ${D}^{0}$-${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ mixing is based on its nature
of a flavor-changing neutral-current process, which is potentially sensitive
to physics beyond the SM. The indirect search for new physics in flavor
oscillations needs, however, precise predictions of the SM expectations to be
compared with the measurements. Such predictions are very difficult to obtain
in the charm sector. Theoretical methods like the quark-hadron duality and the
heavy quark expansion work well in $b$-hadron physics, where the short-
distance dynamics is dominant in the expansion of the energy release,
$1/E_{\rm released}\sim 1/m_{b}$, being $m_{b}$ the $b$ quark mass. These
methods can be applied for predicting $b$ hadron lifetimes and
$B^{0}_{(s)}$-$\overline{B}^{0}_{(s)}$ mixing parameters. However, the charm
quark mass ($m_{c}$) is not large enough compared to the energy release in
charm decays, and long-distance dynamics are not negligible. Taking the $KKK$
intermediate state as an example, $E_{\rm released}\sim
m_{D}-3m_{K}\sim\mathcal{O}(\Lambda_{QCD})$. Thus, quark-hadron quality must
be handled carefully in the case of charm hadrons [1].

Except for the inclusive approach using the $1/m_{c}$ expansion just as in $B$
physics [2], the exclusive approach saturates the ${D}^{0}$-${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ mixing correlators by hadronic
states [3, 4, 5, 6]. This is intriguing and intuitively plausible since the
exclusive approach predicted that the values of $x$ and $y$ could reach the
order of $1\%$, which is indeed experimentally observed [3, 4]. Flavor
$SU(3)$-symmetry breaking plays a key role in the understanding of
${D}^{0}$-${\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ mixing, which
would vanish in the limit of perfect $SU(3)$ symmetry, since
$V_{cd}^{*}V_{ud}\approx-V_{cs}^{*}V_{us}$. The advantage of the exclusive
approach is that the experimental uncertainties of hadronic branching
fractions have been significantly improved nowadays. The current most precise
prediction on the width difference is $y_{\rm PP+PV}=(0.21\pm 0.07)\%$ [6],
considering the two-body contributions from the two pseudoscalar modes (PP)
and the pseudoscalar-vector modes (PV), using the Factorization-Assisted
Topological amplitude approach, which includes the dominant $SU(3)$-breaking
effects in the PP and PV modes [7, 8]. The result from two-body decays leads
to expected values still smaller than the experimental data, implying
additional contributions in ${D}^{0}$-${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ mixing. It is proposed to
consider the finite width effect as a new $SU(3)$-breaking effect [1]. The
finite width effect of one-body intermediate states has been manifested in
Ref. [9]. For 2-body contributions like pseudoscalar-vector, pseudoscalar-
scalar, scalar-scalar, etc., the vector, scalar and other excited states have
finite widths and thus contribute to ${D}^{0}$-${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ mixing as a new
$SU(3)$-breaking effect. A Dalitz-plot analysis to understand the finite-width
effects on ${D}^{0}$-${\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$
mixing predictions is ongoing [1].

A measurement of ${D}^{0}$-${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ mixing and $C\\!P$ violation
can be obtained by comparing the ratio of ${{D}^{0}}\to{{K}^{+}}{{\pi}^{-}}$
and ${{D}^{0}}\to{{K}^{-}}{{\pi}^{+}}$ decay rates, as a function of the
${D}^{0}$ decay time, with the corresponding ratio for the charge-conjugate
processes. The flavour of the meson is determined either by the charge of the
soft pion in ${{D}^{*+}}\to{{D}^{0}}\pi^{+}_{s}$ decays or by the charge of
the lepton (muon or electron) in semileptonic $\overline{B}$ decays,
$\overline{B}\to{{D}^{0}}l^{-}\nu_{l}X$. The soft pion or the lepton is then
referred to as the tagging particle. The decays dominated by a Cabibbo-favored
amplitude, ${{D}^{*+}}\to{{D}^{0}}(\to{{K}^{-}}{{\pi}^{+}})\pi_{s}^{+}$ or
$\overline{B}\to{{D}^{0}}(\to{{K}^{-}}{{\pi}^{+}})l^{-}\nu_{l}X$, are defined
as “right-sign” (RS) decays. Wrong-sign (WS) decays,
${{D}^{*+}}\to{{D}^{0}}(\to{{K}^{+}}{{\pi}^{-}})\pi_{s}^{+}$ or
$\overline{B}\to{{D}^{0}}(\to{{K}^{+}}{{\pi}^{-}})l^{-}\nu_{l}X$, are those
due to the doubly Cabibbo-suppressed ${{D}^{0}}\to{{K}^{+}}{{\pi}^{-}}$ decay
and the Cabibbo-favored ${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}\to{{K}^{+}}{{\pi}^{-}}$ decay
that follows a ${D}^{0}$-${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ oscillation. The ratio of WS-
to-RS rates as a function of the neutral $D$-meson decay time $t$ can be
approximated as

$R(t)\approx R_{D}+\sqrt{R_{D}}\,y^{\prime}\,\frac{t}{\tau}+\frac{x^{\prime
2}+y^{\prime 2}}{4}\left(\frac{t}{\tau}\right)^{2}\,,$ (1)

because of the small value of the mixing parameters. Here, $\tau$ is the
average ${D}^{0}$ lifetime and $R_{D}$ is the ratio of suppressed-to-favored
decay rates. The parameters $x^{\prime}$ and $y^{\prime}$ depend on the mixing
parameters, $x^{\prime}\equiv x\cos\delta+y\sin\delta$ and $y^{\prime}\equiv
y\cos\delta-x\sin\delta$, through the strong-phase difference $\delta$ between
the suppressed and favored amplitudes. This phase was measured at the CLEO and
BESIII experiments [10, 11].

A measurement of mixing and $C\\!P$ violation parameters is reported by the
LHCb collaboration using approximately $1.77\times 10^{8}$ RS and $7.22\times
10^{5}$ WS pion-tagged signal decays, reconstructed in a data sample
corresponding to an integrated luminosity of 5.0$\mbox{\,fb}^{-1}$ from
proton-proton ($pp$) collisions at 7, 8, and 13$\mathrm{\,Te\kern-1.00006ptV}$
center-of-mass energies [12]. Assuming $C\\!P$ conservation, the mixing
parameters are measured to be $x^{\prime 2}=(3.9\pm 2.3\pm 1.4)\times
10^{-5}$, $y^{\prime}=(5.28\pm 0.45\pm 0.27)\times 10^{-3}$, and
$R_{D}=(3.454\pm 0.028\pm 0.014)\times 10^{-3}$. Allowing for possible $C\\!P$
violation, the decay-rate ratios of Eq. 1 of mesons produced as ${D}^{0}$ and
${\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ are not the same.
Indirect $C\\!P$ violation generates differences in the value of the
parameters $(x^{\prime})^{2}$ and $y^{\prime}$ for ${D}^{0}$ and ${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}$ mesons, which can constraints
the value of $|q/p|$. If direct $C\\!P$ violation occurs, the asymmetry
$A_{D}\equiv(R_{D}^{+}-R_{D}^{-})/(R_{D}^{+}+R_{D}^{-})$ differs from zero,
being $R_{D}^{+}$ and $R_{D}^{-}$ the ratio of suppressed-to-favored decay
rates for ${D}^{0}$ and ${\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$
mesons. Studying ${D}^{0}$ and ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ decays separately LHCb
determines $1.00<|q/p|<1.35$ at the $68.3\%$ confidence level, and
$A_{D}=(-0.1\pm 8.1\pm 4.2)\times 10^{-3}$. These are the current most
stringent bounds on the parameters $A_{D}$ and $|q/p|$ from a single
measurement, and show no evidence for $C\\!P$ violation.

A sensitive probe of indirect $C\\!P$ violation is given by the study of
${D}^{0}$ mesons decays into $C\\!P$ eigenstates $h^{+}h^{-}$ ($h=K,\pi$).
Being the time-integrated $C\\!P$ asymmetries and the mixing parameters known
to be small, the decay-time-dependent $C\\!P$ asymmetry of these decays can be
approximated as

$A_{{C\\!P}}(t)=\frac{\Lambda({{D}^{0}}(t)\to h^{+}h^{-})-\Lambda({{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}(t)\to
h^{+}h^{-})}{\Lambda({{D}^{0}}(t)\to h^{+}h^{-})+\Lambda({{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}(t)\to h^{+}h^{-})}\approx
a_{\rm dir}^{h^{+}h^{-}}-A_{\Gamma}\frac{t}{\tau}\,,$ (2)

where $\Lambda({{D}^{0}}(t)\to h^{+}h^{-})$ and $\Lambda({{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}(t)\to h^{+}h^{-})$ are the
decay-time-dependent rates of an initial ${D}^{0}$ or ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ decaying into $h^{+}h^{-}$ at
decay time $t$, $a_{\rm dir}^{h^{+}h^{-}}$ is the asymmetry related to direct
$C\\!P$ violation and $A_{\Gamma}$ is the asymmetry between the ${D}^{0}$ and
${\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ effective decay widths,

$A_{\Gamma}\equiv\frac{\hat{\Gamma}_{{{D}^{0}}\to
h^{+}h^{-}}-\hat{\Gamma}_{{{\kern
1.39998pt\overline{\kern-1.39998ptD}{}}{}^{0}}\to
h^{+}h^{-}}}{\hat{\Gamma}_{{{D}^{0}}\to h^{+}h^{-}}+\hat{\Gamma}_{{{\kern
1.39998pt\overline{\kern-1.39998ptD}{}}{}^{0}}\to h^{+}h^{-}}}\,.$ (3)

Here, we have defined the effective decay width $\hat{\Gamma}_{{{D}^{0}}\to
h^{+}h^{-}}$ as the inverse of the effective lifetime
$\int_{0}^{\infty}t\,\Lambda({{D}^{0}}(t)\to h^{+}h^{-}){\rm
d}t/\int_{0}^{\infty}\Lambda({{D}^{0}}(t)\to h^{+}h^{-}){\rm d}t$. In fact,
$A_{\Gamma}$ is the opposite of the indirect $C\\!P$ asymmetry
$a^{h^{+}h^{-}}_{\rm ind}$.

Because of mixing, the effective decay width of decays to $C\\!P$-even final
states $h^{+}h^{-}$ differs from the average decay width, $\Gamma=1/\tau$,
measured in ${{D}^{0}}\to{{K}^{-}}{{\pi}^{+}}$ decays. The quantity

$y_{{C\\!P}}\equiv\frac{\hat{\Gamma}_{{{D}^{0}}\to
h^{+}h^{-}}+\hat{\Gamma}_{{{\kern
1.39998pt\overline{\kern-1.39998ptD}{}}{}^{0}}\to h^{+}h^{-}}}{2\Gamma}-1$ (4)

is equal to $y$ if $C\\!P$ symmetry is conserved. Neglecting the
$\mathcal{O}(10^{-3})$ difference between the phases of the
${{D}^{0}}\to{{K}^{+}}{{K}^{-}}$ and ${{D}^{0}}\to{{\pi}^{+}}{{\pi}^{-}}$
decay amplitudes, $A_{\Gamma}$ and $y_{{C\\!P}}$ are independent of the
$h^{+}h^{-}$ final state, and $\phi\approx\arg(q/p)$. In the limit of small
$C\\!P$ violation in mixing ($|q/p|\approx 1$), $A_{\Gamma}$ can be
approximated as $-x\sin\phi$ and its measurement translates into a constraint
of the mixing phase $\phi$, if the value of $x$ is determined elsewhere. For
$y_{C\\!P}$ instead, $y_{C\\!P}\approx y\cos\phi$.

In a data sample corresponding to an integrated luminosity of
3$\mbox{\,fb}^{-1}$, LHCb selects about 9.6 and 3.0 million of pion-tagged
${{D}^{0}}\to{{K}^{+}}{{K}^{-}}$ and ${{D}^{0}}\to{{\pi}^{+}}{{\pi}^{-}}$
decays, respectively, to measure $A_{\Gamma}({{K}^{+}}{{K}^{-}})=(-0.30\pm
0.32\pm 0.10)\times 10^{-3}$ and $A_{\Gamma}({{\pi}^{+}}{{\pi}^{-}})=(0.46\pm
0.58\pm 0.12)\times 10^{-3}$ [14]. The results are consistent and show no
evidence of $C\\!P$ violation; they are combined to yield a single value of
$A_{\Gamma}=(-0.13\pm 0.28\pm 0.10)\times 10^{-3}$. Combination with a
smaller, independent muon-tagged sample [15], gives $A_{\Gamma}=(-0.29\pm
0.28)\times 10^{-3}$. The measurements of $A_{\Gamma}$ reported by LHCb are
the most precise to date. They are consistent with results from Belle obtained
from the analysis of the full data set collected by the experiment,
corresponding to an integrated luminosity of 976$\mbox{\,fb}^{-1}$, in which
$A_{\Gamma}$ and $y_{{C\\!P}}$ are measured simultaneously with $\tau$, using
$242\times 10^{3}$ ${{D}^{0}}\to{{K}^{+}}{{K}^{-}}$, $114\times 10^{3}$
${{D}^{0}}\to{{\pi}^{+}}{{\pi}^{-}}$ and $2.6\times 10^{6}$
${{D}^{0}}\to{{K}^{-}}{{\pi}^{+}}$ decays, all from
${{D}^{*+}}\to{{D}^{0}}\pi_{s}^{+}$ decays [16]. The values found by Belle are
$y_{{C\\!P}}=(1.11\pm 0.22\pm 0.09)\%$, $A_{\Gamma}=(-0.3\pm 2.0\pm 0.7)\times
10^{-3}$ and $\tau=(408.46\pm 0.54)$ fs.

The precision on $y_{{C\\!P}}$ is greatly improved by a brand new results
presented by LHCb at the workshop for the first time [17]. In a data sample
corresponding to an integrated luminosity of 3$\mbox{\,fb}^{-1}$, $880\times
10^{3}$ ${{D}^{0}}\to{{K}^{+}}{{K}^{-}}$, $310\times 10^{3}$
${{D}^{0}}\to{{\pi}^{+}}{{\pi}^{-}}$ and $4.6\times 10^{6}$
${{D}^{0}}\to{{K}^{-}}{{\pi}^{+}}$ decays, are reconstructed from semimuonic
decays of $\overline{B}$ mesons. The difference between the widths of
${D}^{0}$ decays to $h^{+}h^{-}$ and ${{K}^{-}}{{\pi}^{+}}$ final states,
$\Delta\Gamma\equiv\hat{\Gamma}-\Gamma$, is measured from a fit to the ratio
between ${{D}^{0}}\to h^{+}h^{-}$ and ${{D}^{0}}\to{{K}^{-}}{{\pi}^{+}}$
signal yields, corrected for the selection efficiency, as a function of the
${D}^{0}$ decay time. The parameter $y_{{C\\!P}}$ is then calculated from the
measured value of $\Delta\Gamma$ and the known value of $\Gamma$ as
$y_{CP}=\Delta\Gamma/\Gamma$. The results are
$y_{{C\\!P}}({{K}^{+}}{{K}^{-}})=(0.63\pm 0.15\pm 0.11)\%$, and
$y_{{C\\!P}}=(0.38\pm 0.28\pm 0.15)\%$, consistent between each other. The
value of $y_{{C\\!P}}({{K}^{+}}{{K}^{-}})$ is the most precise to date from a
single experiment. The two measurements are combined and yield
$y_{{C\\!P}}=(0.57\pm 0.13\pm 0.09)\%$, which is as precise as the current
world average value, $(0.84\pm 0.16)\%$ [13]. The result is also consistent
with the known value of the mixing parameter $y=(0.67^{+0.06}_{-0.13})\%$
[13], showing no evidence for $C\\!P$ violation in mixing. A measurement of
$y_{{C\\!P}}$ in correlated $D^{0}\bar{D}^{0}$ is reported also by the BESIII
experiment [18], through a comparison of the combined branching fractions of
several $C\\!P$-even (${{K}^{+}}{{K}^{-}}$, ${{\pi}^{+}}{{\pi}^{-}}$,
${{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{\pi}^{0}}{{\pi}^{0}}$) and
$C\\!P$-odd (${{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{\pi}^{0}}$,
${{K}^{0}_{\mathrm{\scriptscriptstyle S}}}\omega$,
${{K}^{0}_{\mathrm{\scriptscriptstyle S}}}\eta$) final states, although the
sensitivity is low, around $1.5\%$. BESIII is also updating the study by
including $K_{L}^{0}{{\pi}^{0}}({{\pi}^{0}})$.

Mixing and $C\\!P$ violation are also studied through multibody decays. The
golden decay mode in this case is
${{D}^{0}}\to{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{\pi}^{+}}{{\pi}^{-}}$,
which final state is a complex assembly of different resonances including
flavour and $C\\!P$ eigenstates. Sensitivity to mixing and $C\\!P$ violation
can be enhanced by multiple interfering amplitudes with strong phases
continuously varying across phase-space. Two main methods have been proposed
to study these decays: the model-independent and the model-dependent method. A
new model-independent method based on a Fourier analysis was also proposed for
a measurement of the CKM angle $\gamma$ [19], but not yet employed in a
measurement of charm mixing.

Model-dependent analyses measure effective lifetime of individual resonances,
and the choice of the model to describe the Dalitz plot represents irreducible
systematic uncertainties. This method has been used at $B$ factories [20, 21].
Model-independent measurements study the decay-time evolution of ${D}^{0}$
mesons in bins of the Dalitz plot, according to similar strong phase
differences between the interfering amplitudes. This type of analysis is used
by the LHCb experiment [22]. The major limitation comes from the external
input required on the values of the strong phase differences of the
interfering resonances. These are measured at the CLEO-c and BESIII
experiments, exploiting the production of correlated pairs of charm mesons at
threshold. A preliminary (since 2014) result from BESIII [23] provides an
improvement of about 40% on the uncertainty of the measurement by CLEO-c [24],
currently the one exploited in the LHCb analysis; however, the BESIII analysis
is still in progress, and no update (especially on the missing systematic
uncertainties) has been presented at the workshop. Such a measurement would
also be important input in the measurement of the CKM angle $\gamma$ using the
so-called GGSZ method [25]. The results on the mixing parameters $x$ and $y$
obtained with ${{D}^{0}}\to{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{\pi}^{+}}{{\pi}^{-}}$ decays from different experiments are reported in
Table 1. Updates from LHCb are expected soon.

Table 1: Mixing parameters $x$ and $y$ obtained with ${{D}^{0}}\to{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{\pi}^{+}}{{\pi}^{-}}$ decays from different experiments. The result from BaBar is a combination of ${{D}^{0}}\to{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{\pi}^{+}}{{\pi}^{-}}$ and ${{D}^{0}}\to{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{K}^{+}}{{K}^{-}}$ decays. |  $x$ $[\%]$ |  $y$ $[\%]$
---|---|---
LHCb [22] | $-0.86\pm 0.53\pm 0.17$ | $-0.03\pm 0.46\pm 0.13$
Belle [21] | $0.56\pm 0.19^{+0.07}_{-0.13}$ | $0.30\pm 0.15^{+0.05}_{-0.08}$
BaBar [20] | $0.16\pm 0.23\pm 0.14$ | $0.57\pm 0.20\pm 0.15$

These results can be compared with those obtained by BaBar from a model-
dependent analysis of $138\times 10^{3}$
${{D}^{0}}\to{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{0}}$ decays, originating from
${{D}^{*+}}\to{{D}^{0}}\pi^{+}_{s}$ decays, using a sample corresponding to an
integrated luminosity of 468$\mbox{\,fb}^{-1}$, $x=(1.5\pm 1.2\pm 0.6)\%$ and
$y=(0.2\pm 0.9\pm 0.5)\%$ [26]. At LHCb, mixing is also observed in a decay-
time-dependent analysis of the WS-to-RS ratio of
${{D}^{0}}\to{{K}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ decay rates [27],
obtaining $x=(0.41\pm 0.17)\%$ and $y=(0.67\pm 0.80)\%$. This analysis is
sensitive to the phase-space averaged ratio of doubly Cabibbo-suppressed to
Cabibbo-favoured amplitudes $r_{D}^{K3\pi}$, measured to be
$r_{D}^{K3\pi}=(5.67\pm 0.12)\times 10^{-2}$, and the product of the coherence
factor $R_{D}^{K3\pi}$ and a charm mixing parameter $y^{\prime}_{k3\pi}$,
found to be $R_{D}^{K3\pi}y^{\prime}=(0.3\pm 1.8)\times 10^{-3}$. This
provides useful input for determinations of the CKM angle $\gamma$ in
$B^{+}\to{{D}^{0}}(\to{{K}^{-}}{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}){{K}^{+}}$
decays, for which the parameters $r_{D}^{K3\pi}$, $R_{D}^{K3\pi}$ and the
strong phase $\delta_{D}^{K3\pi}$ are required. A combination of the LHCb
results with CLEO-c data significantly improves the precision on these inputs
[28].

##  3 Direct $C\\!P$ violation

Direct $C\\!P$ violation in charm decays is difficult to be predicted
precisely in the SM due to the relatively large non-perturbative effects. It
is nonetheless expected to be small due to the GIM mechanism with
$m_{d,s,b}\ll m_{W}$. Thus, $C\\!P$ violation in charm decays is of good use
as a null test of the SM, as measurements of large $C\\!P$ asymmetries
(generally larger than $\mathcal{O}(1\%)$) would be indication of new physics.
In addition, contrary to the case of the $K$ and $B$ systems, the charm system
is unique for studying $C\\!P$ violation in the up sector of quarks. This is
helpful to distinguish new physics models with different quark structures. The
recent flavor anomalies [29, 30, 31] indicate a possible large non-SM $C\\!P$
phase, which may increase $C\\!P$ violation in charm decays [32]. For example,
the $R(D^{(*)})$ anomaly points to a non-SM structure of the $cb$ quarks
coupling. Concerning $C\\!P$ violation, the $cb$ coupling has no complex phase
in the SM, whereas it might present one in a new physics scenario. This might
manifest in measurements of direct $C\\!P$ violation in charm decays, no
matter what the charm-penguin amplitude considered.

The decay-time-dependent $C\\!P$ asymmetry, $A_{{C\\!P}}(t)$, for ${D}^{0}$
mesons decaying to a $C\\!P$ eigenstate $h^{+}h^{-}$ is given by Eq. 2. In a
sample of reconstructed ${{D}^{0}}\to h^{+}h^{-}$ decays, the decay-time-
integrated asymmetry, $A_{{C\\!P}}$, depends upon the reconstruction
efficiency as a function of the decay time, and can be written as
$A_{{C\\!P}}\approx a_{\rm dir}^{h^{+}h^{-}}(1+\langle
t\rangle/\tau\,y_{{C\\!P}})-\langle t\rangle/\tau\,A_{\Gamma}$, where $\langle
t\rangle$ denotes the mean of the reconstructed decay-time distribution of the
${{D}^{0}}\to h^{+}h^{-}$ decays. What is experimentally accessible is the raw
asymmetry of the number of candidate, $N$, counted as decay of a ${D}^{0}$ or
a ${\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ mesons,

$A_{\rm raw}\equiv\frac{N({{D}^{0}}\to h^{+}h^{-})-N({{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}\to h^{+}h^{-})}{N({{D}^{0}}\to
h^{+}h^{-})+N({{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}\to
h^{+}h^{-})}\,.$ (5)

The raw asymmetry can be written, up to $\mathcal{O}(10^{-6})$, as $A_{\rm
raw}\approx A_{{C\\!P}}+A_{f}+A_{\rm tag}+A_{\rm prod}$, where $A_{f}$ and
$A_{\rm tag}$ are the asymmetries in the reconstruction efficiency of the
${D}^{0}$ final states and of the tagging particle, and $A_{\rm prod}$ is the
production asymmetry for either ${D}^{*+}$ or $\overline{B}$ mesons (relevant
only at hadron colliders). For self-conjugate final states, $A_{f}=0$. To a
good approximation, $A_{\rm tag}$ and $A_{\rm prod}$ are independent of the
final state in any given kinematic region. Thus, the difference of raw
asymmetry between ${{D}^{0}}\to{{K}^{+}}{{K}^{-}}$ and
${{D}^{0}}\to{{\pi}^{+}}{{\pi}^{-}}$ decays give access to

$\Delta A_{{C\\!P}}\equiv
A_{{C\\!P}}^{{{K}^{+}}{{K}^{-}}}-A_{{C\\!P}}^{{{\pi}^{+}}{{\pi}^{-}}}\approx\Delta
a_{\rm dir}\left(1+\frac{\overline{\langle
t\rangle}}{\tau}y_{{C\\!P}}\right)-\frac{\Delta\langle
t\rangle}{\tau}A_{\Gamma}\,,$ (6)

where $\overline{\langle t\rangle}$ is the arithmetic mean of $\langle
t^{{{K}^{+}}{{K}^{-}}}\rangle$ and $\langle
t^{{{\pi}^{+}}{{\pi}^{-}}}\rangle$. This equation holds up to terms of
$\mathcal{O}(10^{-6})$. In the limit of exact U-spin symmetry, $\Delta a_{\rm
dir}\equiv a_{\rm dir}^{{{K}^{+}}{{K}^{-}}}-a_{\rm
dir}^{{{\pi}^{+}}-{{\pi}^{-}}}\approx 2a_{\rm dir}^{{{K}^{+}}{{K}^{-}}}$,
since $a_{\rm dir}^{h^{+}h^{-}}$ is expected to be equal in magnitude and
opposite in sign for ${{K}^{+}}{{K}^{-}}$ and ${{\pi}^{+}}{{\pi}^{-}}$ final
states [33], although U-spin breaking effects could be also present [34].

To take into account an imperfect cancellation of detection and production
asymmetries due to the difference in the kinematic properties of the two decay
modes, a weighting procedure in bins of kinematic distributions is usually
employed to equalize the kinematic of ${{K}^{+}}{{K}^{-}}$ and
${{\pi}^{+}}{{\pi}^{-}}$ final state. With this method, LHCb obtains the most
precise measurement of $\Delta A_{{C\\!P}}$ to date, $\Delta
A_{{C\\!P}}=(-0.10\pm 0.08\pm 0.03)\%$ [35, 36], from a combination of the
results from the pion- and muon-tagged samples. This measurement exploits only
$1/3$ of the currently available data: the analysis of the full data set is
ongoing, and the statistical precision is expected to reach an unprecedented
value, around $0.03\%$.

The direct $C\\!P$ asymmetry in the
${{D}^{0}}\to{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ decay is also interesting to
test the SM prediction of $A_{{C\\!P}}\leq 1.1\%$, or to signal new physics if
data exceeds it [37]. Both $B$-factories and LHCb presented a measurements of
$C\\!P$ asymmetry in this decay. At LHCb, the analysis exploits again the
combination of the raw asymmetries measured in different decays to cancel
spurious asymmetries. The ${{D}^{0}}\to{{K}^{+}}{{K}^{-}}$ decay is used as
the calibration channel to cancel the production and tagging asymmetry, owing
on the precise measurement of $A_{{C\\!P}}^{{{K}^{+}}{{K}^{-}}}$ [38]. Thus,
$A_{{C\\!P}}^{{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}}$ is obtained from a measurement
of $A_{{C\\!P}}^{{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}}-A_{{C\\!P}}^{{{K}^{+}}{{K}^{-}}}$. The result presented at the workshop
exploits the data collected by LHCb in 2015 and 2016, corresponding to an
integrated luminosity of 2$\mbox{\,fb}^{-1}$, and is
$A_{{C\\!P}}^{{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}}=(4.2\pm 3.4\pm 1.0)\%$ [39].
Combining this with a previous result from an independent sample [40], LHCb
obtains $A_{{C\\!P}}^{{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}}=(2.0\pm 2.9\pm 1.0)\%$, which
doesn’t hint to $C\\!P$ violation. This is in agreement with a more precise
result presented by Belle, $A_{{C\\!P}}^{{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}}=(-0.02\pm 1.53\pm 0.17)\%$,
obtained from a sample corresponding to an integrated luminosity of
921$\mbox{\,fb}^{-1}$ [41]. In the same analysis, Belle provides also a
measurement of the branching fraction of this decay,
$\mathcal{B}({{D}^{0}}\to{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}})=(1.321\pm 0.023\pm 0.057)\times
10^{-4}$.

Thanks to the large data samples accumulated, the study of $C\\!P$ violation
of rare charm decays is also possible. A first measurement of $C\\!P$
asymmetries in the rare ${{D}^{0}}\to h^{+}h^{-}{{\mu}^{+}}{{\mu}^{-}}$ decays
is presented by the LHCb collaboration [42], reconstructing about $1000$
${{D}^{0}}\to{{\pi}^{+}}{{\pi}^{-}}{{\mu}^{+}}{{\mu}^{-}}$ and $100$
${{D}^{0}}\to{{K}^{+}}{{K}^{-}}{{\mu}^{+}}{{\mu}^{-}}$ decays in a data set
corresponding to an integrated luminosity of 5$\mbox{\,fb}^{-1}$. In these
decays, $C\\!P$ asymmetry is expected to be below $5\times 10^{-3}$ in the
standard model, but enhanced up to about 1% if new physics is present.
Forward-backward and angular asymmetries are also measured, but they were
discussed at the workshop in more details by the WG3 [43]. Here, we focus on
the $C\\!P$ asymmetry, which is measured in a similar fashion as
$A_{{C\\!P}}^{{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}}$, i.e. by using the
${{D}^{0}}\to{{K}^{+}}{{K}^{-}}$ decays as a calibration channel to cancel the
production and tagging asymmetry, and measuring
$A_{{C\\!P}}^{h^{+}h^{-}{{\mu}^{+}}{{\mu}^{-}}}-A_{{C\\!P}}^{{{K}^{+}}{{K}^{-}}}$.
Using the precise known value of $A_{{C\\!P}}^{{{K}^{+}}{{K}^{-}}}$, LHCb
determines $A_{{C\\!P}}^{{{K}^{+}}{{K}^{-}}{{\mu}^{+}}{{\mu}^{-}}}=(0\pm 11\pm
2)\%$ and $A_{{C\\!P}}^{{{\pi}^{+}}{{\pi}^{-}}{{\mu}^{+}}{{\mu}^{-}}}=(4.9\pm
3.8\pm 0.7)\%$, both in agreement with $C\\!P$ conservation in these decays.

Considering decays of the ${{D}^{+}}$ meson, the BESIII collaboration presents
measurement of $C\\!P$ asymmetries in singly Cabibbo-suppressed decays
${{D}^{+}}\to{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{K}^{+}}$,
${{D}^{+}}\to{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{K}^{+}}{{\pi}^{0}}$,
${{D}^{+}}\to{{K}^{0}_{\mathrm{\scriptscriptstyle L}}}{{K}^{+}}$ and
${{D}^{+}}\to{{K}^{0}_{\mathrm{\scriptscriptstyle L}}}{{K}^{+}}{{\pi}^{0}}$,
obtaining the results: $A_{{C\\!P}}({{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{K}^{+}})=(-1.5\pm 2.8\pm 1.6)\%$,
$A_{{C\\!P}}({{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{K}^{+}}{{\pi}^{0}})=(1.4\pm 4.0\pm 2.4)\%$,
$A_{{C\\!P}}({{K}^{0}_{\mathrm{\scriptscriptstyle L}}}{{K}^{+}})=(-3.0\pm
3.2\pm 1.2)\%$, and $A_{{C\\!P}}({{K}^{0}_{\mathrm{\scriptscriptstyle
L}}}{{K}^{+}}{{\pi}^{0}})=(-0.9\pm 4.1\pm 1.6)\%$ [44]. These do not indicate
$C\\!P$ violation. Note that the last three measurements are provided for the
first time. The decay channel $D^{+}\to\pi^{+}\pi^{0}$ is another excellent
candidates to probe $C\\!P$ violation in charm decays for searching for new
physics [45, 46]. The Belle collaboration reports a measurement which exploits
a sample of about $6.6\times 10^{3}$ pion-tagged and $101\times 10^{3}$
untagged $D\to\pi^{+}\pi^{0}$ decays. They measures
$A_{{C\\!P}}(D^{+}\to\pi^{+}\pi^{0})=(2.31\pm 1.24\pm 0.23)\%$ and
$A_{{C\\!P}}(\Delta I=1/2)=(-2.2\pm 2.7)\%$ [47]. At Belle II, the precision
is expected to improve to reach uncertainties of about $0.18\%$ (statistical)
and $0.20\%$ (systematic).

Results on radiative charm decays $D^{0}\to V\gamma$, where $V$ is a vector
meson, are also reported. Measurements of branching fractions of these decays
are useful test for QCD-based models, since their amplitudes are dominated by
non-perturbative long range dynamics. These decays are also sensitive to new
physics searches via measurement of $C\\!P$ asymmetries: it has been suggested
that those asymmetries can rise to several percent in contrast to
$\mathcal{O}(10^{-3})$ SM expectation in several non-SM scenarios [48, 49,
50]. The Belle collaboration performed first measurements of $C\\!P$ violation
in $D^{0}\to V\gamma$ decays using 943$\mbox{\,fb}^{-1}$ of data [51],
obtaining the results reported in Tab. 2. They are consistent with zero
$C\\!P$ asymmetry in any of the $D^{0}\to V\gamma$ decay mode.

Study of photon polarization in these decays is also a sensitive probe to non-
SM physics. In the inclusive decay of $c\to u\gamma$, the photon polarization
is proportional to $r=C^{\prime}_{7}/C_{7}$, which is almost a null test of
the SM. In the exclusive processes, the photon polarization suffers the
pollution of long-distance contributions, $r^{\rm SM}=\mathcal{O}(\Lambda_{\rm
QCD}/m_{c})$ [52]. It is proposed to use data and $U$-spin symmetry to relate
$D^{0}\to\rho^{0}\gamma$ and $D^{0}\to\overline{K}^{*0}\gamma$ decays [53].
For decays of the type $D\to A(\to P_{1}P_{2}P_{3})\gamma$, where $A$ is an
axial-vector meson, the photon polarization can be determined by the up-down
asymmetry [55]. The charm decays have an advantage compared to $B$ decays: the
phase spaces are suppressed for higher resonances [53]. The angular asymmetry
in charm baryon decays, such as $\Lambda_{c}^{+}\to p\gamma$, is also
sensitive to the chirality-flipped contributions [54].

Table 2: Measurements of branching fractions and $C\\!P$ asymmetries in radiative charm decays presented by the Belle collaboration [51]. Decay |  $\mathcal{B}$ $[10^{-5}]$ |  $A_{{C\\!P}}$ $[\%]$
---|---|---
$D^{0}\to\phi\gamma$ | $2.76\pm 0.19\pm 0.10$ | $-9.4\pm 6.6\pm 0.1$
$D^{0}\to\bar{K}^{*0}\gamma$ | $46.6\pm 2.1\pm 2.1$ | $-0.3\pm 2.0\pm 0.1$
$D^{0}\to\rho^{0}\gamma$ | $1.77\pm 0.30\pm 0.07$ | $5.6\pm 15.2\pm 0.6$

Since $C\\!P$ asymmetries are proportional to the ratio of penguin and tree
amplitudes, $A_{{C\\!P}}\propto|P/T|$, to seek out a process with enhanced
$C\\!P$ violation, it is suggested to look at decays where the penguin
amplitude is enhanced and the tree amplitude is suppressed as much as possible
[32]. The strategy is, i) to avoid $W^{+}\to u\bar{d}$ or $u\bar{s}$ making
charged vector states, such as $\rho^{\pm}$ or $K^{*\pm}$, and ii) to go for
color suppressed or Zweig suppressed final states from tree amplitudes. A
Cabibbo-suppressed decay is automatically forced by the tree-penguin
interference. Possible decay modes are listed in Ref. [56].

All of the above mentioned $C\\!P$ asymmetries arise from the interference
between tree and penguin amplitudes. A basic shortcoming is the unknown
penguin dynamics. We thus lose the power of prediction on $C\\!P$ violation in
charm induced by the tree-penguin interference. On the contrary, the tree
amplitudes are better known with non-perturbative hadronic matrix elements
extracted from measurements of branching fractions [57, 7, 8, 58]. It deserves
to study the tree-tree interference-induced $C\\!P$ violation, i.e. the
interference between Cabibbo-favored (CF) and doubly Cabibbo-suppressed (DCS)
amplitudes, e.g. in $D^{+}\to\pi^{+}K_{S}^{0}$ decay [59]. In literature, it
was always postulated that, subtracting the $C\\!P$ violation in kaon mixing,
data would reveal a direct $C\\!P$ asymmetry in such charm decays. However,
this is not correct when considering the neutral kaon reconstructed by two
charged pions [59]. To consider with the well known $C\\!P$ violation in kaon
mixing and direct $C\\!P$ asymmetry, a new $C\\!P$-violating effect is found
for the first time, arising from the interference between the mother decay and
the daughter mixing. This is more complicated than the ordinary mixing-induced
$C\\!P$ asymmetry in which both decay and mixing occur in the mother particle,
for example the $B^{0}\to\pi^{+}\pi^{-}$ decay. The key point is that we have
to consider all the amplitudes of the decay chain, say, the CF and DCS
amplitudes of the charm decaying into neutral kaons, and the neutral kaons
reconstructed by two charged pions. In the previous studies, either the DCS
amplitude was neglected [60], or the decay chain of neutral kaons decaying
into two charged pions was not considered [61, 62]. Therefore, the total
$C\\!P$ asymmetries in the charm decays into neutral kaons is actually

$\displaystyle
A_{{C\\!P}}(t)=A_{{C\\!P}}^{\overline{K}^{0}}(t)+A_{{C\\!P}}^{\rm
dir}(t)+A_{{C\\!P}}^{\rm int}(t)\,,$ (7)

where the first two terms are the well known $C\\!P$ asymmetries in kaon
mixing and the direct $C\\!P$ violation in charm decays, respectively, and the
third term is the new $C\\!P$-violating effects induced by the interference
between charm decays and kaon mixing. Using the factorization-assisted
topological-amplitude (FAT) approach [7, 63], it is found that the new
$C\\!P$-violating effect can be as large as the order of $10^{-3}$, thus is
accessible at the LHCb and Belle II experiments in the near future. It is
proposed to measure the new $C\\!P$-violating effect via

$\displaystyle\Delta
A_{{C\\!P}}=A_{{C\\!P}}(D^{+}\to\pi^{+}K_{S}^{0})-A_{{C\\!P}}(D_{s}^{+}\to
K^{+}K_{S}^{0})\,,$ (8)

where the dominated $C\\!P$ violation from kaon mixing is cancelled, and the
direct $C\\!P$ asymmetry in charm decays is negligible. Finally, the new
$C\\!P$-violating effect has to be considered in the studies of new physics
using the direct $C\\!P$ asymmetry of charm.

##  4 $C\\!P$-violating triple product asymmetries

Time-reversal asymmetry is sensitive to $C\\!P$ violation via the ${C\\!P}T$
theorem ($T$ violation implies ${C\\!P}$ violation under the assumption of
${C\\!P}T$ invariance.). Measurements of $T$-odd asymmetries are a clean way
to search for $C\\!P$ violation in the charm sector. Analysis of four-body
charm decays allow to probe $C\\!P$ violation in different phase space
regions, where sensitivity can be enhanced due to several interfering
amplitudes with different relative strong phases. In four body $D$ decays,
$D\to abcd$, one can use triple products of final-state particles momenta (in
the $D$ frame), $C_{T}\equiv\vec{p}_{c}\cdot(\vec{p}_{a}\times\vec{p}_{b})$
and
$\overline{C}_{T}\equiv\vec{p}_{\overline{c}}\cdot(\vec{p}_{\overline{a}}\times\vec{p}_{\overline{b}})$
to construct two $T$-odd observables,

$A_{T}\equiv\frac{\Gamma(C_{T}>0)-\Gamma(C_{T}<0)}{\Gamma(C_{T}>0)+\Gamma(C_{T}<0)},\mbox{
and
}\overline{A}_{T}\equiv\frac{\Gamma(-\overline{C}_{T}>0)-\Gamma(-\overline{C}_{T}<0)}{\Gamma(-\overline{C}_{T}>0)+\Gamma(-\overline{C}_{T}<0)}\,,$
(9)

where $\Gamma_{D}$ ($\Gamma_{\overline{D}}$) is the decay width of $D$
($\overline{D}$) decays to $abcd$ ($\overline{abcd}$) in the $C_{T}$
($\overline{C}_{T}$) range. However $A_{T}$ ($\overline{A}_{T}$) can be non-
zero due to final state interaction (FSI) even without $C\\!P$ violation. In
order to cancel FSI, one can create genuine ${C\\!P}$-violating asymmetry
(insensitive also to $D$ production asymmetry and charged-particle
reconstruction asymmetries) as

${a_{{C\\!P}}^{\rm{T}-odd}}=\frac{1}{2}(A_{T}-\overline{A}_{T})\,.$ (10)

A triple product is even (odd) under $C$ ($P$, $T$, and ${C\\!P}$). One can
construct a number of asymmetries that can be computed by integrating over
positive and negative values of the triple product [64].

First measurement of $a_{{C\\!P}}^{\rm{T}-odd}$ in decays $D^{0}\to
K^{+}K^{-}{{\pi}^{+}}{{\pi}^{-}}$ and
$D^{+}\to{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}K^{+}{{\pi}^{+}}{{\pi}^{-}}$
was done by FOCUS [65]. The precision of these measurements was then improved
by BaBar, and the study of the decay $D_{S}^{+}\to
K_{S}^{0}K^{+}{{\pi}^{+}}{{\pi}^{-}}$ was added [67]. Asymmetries consistent
with zero were measured. The LHCb collaboration also measured
${a_{{C\\!P}}^{\rm{T}-odd}}(D^{0}\to K^{+}K^{-}{{\pi}^{+}}{{\pi}^{-}})$,
obtaining the precise result $(0.18\pm 0.29\pm 0.04)\%$, consistent with zero
[68]. For the first time, LHCb also presented ${a_{{C\\!P}}^{\rm{T}-odd}}$ in
different ranges of phase space and in bins of $D^{0}$ decay time. Local
asymmetries up to 30% are seen in bins of phase space. The Belle experiment
reported the first measurement of the
${a_{{C\\!P}}^{\rm{T}-odd}}(D^{0}\to{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{0}})$ to be $(-0.28\pm
1.38^{+0.23}_{-0.76})\times 10^{-3}$, consistent with no ${C\\!P}$ violation
[69], one of the most precise measurement. They also perform
${a_{{C\\!P}}^{\rm{T}-odd}}$ measurements in different region of $D^{0}\to
K_{S}^{0}{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{0}}$ phase space, with no evidence of
${C\\!P}$ violation.

In this workshop, Belle reported their new result on the four body decay of
$D^{0}\to K^{+}K^{-}{{\pi}^{+}}{{\pi}^{-}}$, measuring a set of five
kinematically independent ${C\\!P}$ asymmetries, of which four asymmetries are
measured for the first time [70]. They are reported in Tab. 3, where in the
notation $\theta_{1}$ ($\theta_{2}$) is the helicity angle of $K^{+}K^{-}$
(${{\pi}^{+}}{{\pi}^{-}}$) systems against positive charged particles, and
$\phi$ is the angle between decay planes of these systems. No ${C\\!P}$
violation is observed, however, these new kinematic asymmetries further
constrain new physics models for the first time.

Table 3: Measurement of $C\\!P$-violating kinematic asymmetries measured by Belle [70]. Asymmetry | Result $[\times 10^{-3}]$
---|---
$a_{\cos\phi}$ | $3.4\pm 3.6\pm 0.6$
$a_{\sin\phi}$ | $5.2\pm 3.7\pm 0.7$
$a_{\sin 2\phi}$ | $3.9\pm 3.6\pm 0.7$
$a_{\cos\theta_{1}\cos\theta_{2}\cos\phi}$ | $-0.2\pm 3.6\pm 0.7$
$a_{\cos\theta_{1}\cos\theta_{2}\sin\phi}$ | $0.2\pm 3.7\pm 0.7$

##  5 Results on charm baryons

Violation of the $C\\!P$ symmetry in charm baryons decays is expected to be at
the same level as in charm mesons decays. A first measurement of $C\\!P$
violation in a 3-body decay of the ${\Lambda}^{+}_{c}$ baryons is reported by
the LHCb collaboration, by studying the Cabibbo-suppressed decays
${{\Lambda}^{+}_{c}}\to ph^{+}h^{-}$ originating from semimuonic
${\Lambda}^{0}_{b}$ decays [71]. Similar to the $\Delta A_{{C\\!P}}$
measurement of ${{D}^{0}}\to h^{+}h^{-}$ decays described in Sect. 3, the
analysis measures the difference in the raw asymmetry between the
${{\Lambda}^{+}_{c}}\to p{{K}^{+}}{{K}^{-}}$ and ${{\Lambda}^{+}_{c}}\to
p{{\pi}^{+}}{{\pi}^{-}}$ to cancel spurious asymmetries, and access the
$C\\!P$-asymmetry difference between the two decay modes. The spurious
asymmetries are those due to asymmetries in the reconstruction efficiency of
the proton in the final state, of the muon from the ${\Lambda}^{0}_{b}$
decays, and of the ${\Lambda}^{0}_{b}$ production. To allow for their precise
suppression, kinematic distributions of the ${\Lambda}^{0}_{b}$, the proton
and the muon of the ${{\Lambda}^{+}_{c}}\to p{{\pi}^{+}}{{\pi}^{-}}$ sample
are weighted to match those of the ${{\Lambda}^{+}_{c}}\to
p{{K}^{+}}{{K}^{-}}$ sample. Since ${{\Lambda}^{+}_{c}}\to ph^{+}h^{-}$
dynamics are described by a 5-dimensional phase space, along which $C\\!P$
violation effects may vary, efficiency corrections are considered across the
full phase space. From about $25\times 10^{3}$ ${{\Lambda}^{+}_{c}}\to
p{{K}^{+}}{{K}^{-}}$ and $161\times 10^{3}$ ${{\Lambda}^{+}_{c}}\to
p{{\pi}^{+}}{{\pi}^{-}}$ decays, the LHCb collaboration obtains
$A_{{C\\!P}}^{p{{K}^{+}}{{K}^{-}}}-A_{{C\\!P}}^{p{{\pi}^{+}}{{\pi}^{-}}}=(0.30\pm
0.91\pm 0.61)\%$, finding no signal of $C\\!P$ violation.

Owing on the large cross-section for charm production in $pp$ collisions, the
LHCb experiment can collect large data set of charm baryons decays and open
new avenues in this sector. Indeed, interesting results which go behind
measurements of $C\\!P$ violation were reported at the workshop. A
breakthrough of double-charm baryon physics happened during the past two
years. In 2017 it was firstly pointed out in [72] that
$\Xi_{cc}^{++}\to\Lambda_{c}^{+}K^{-}\pi^{+}\pi^{+}$ and $\Xi_{c}^{+}\pi^{+}$
are the most favorable modes to search for doubly heavy flavor baryons, with
branching fractions of $\mathcal{O}(10\%)$. Soon afterwards, the LHCb
collaboration discovered a first doubly charmed baryon, $\Xi_{cc}^{++}$, via
the final states of $\Lambda_{c}^{+}K^{-}\pi^{+}\pi^{+}$ [73]. The LHCb
collaboration keeps improving the understanding of the newly observed
${\Xi}^{++}_{cc}$ baryon in 2018, by reporting the observation of
${{\Xi}^{++}_{cc}}\to{{\Xi}^{+}_{c}}{{\pi}^{+}}$ [74], with
${{\Xi}^{+}_{c}}\to p{{K}^{-}}{{\pi}^{+}}$, with a significance of $5.9$
standard deviations. A new measurement of the ${\Xi}^{++}_{cc}$ mass is
obtained from this decay mode, $m({{\Xi}^{++}_{cc}})=3620.6\pm 1.5\pm
0.5{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, and the combination of this
result with the measurement in Ref. [73] yields to
$m({{\Xi}^{++}_{cc}})=3621.24\pm 0.65\pm
0.31{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The analysis leads also to the
measurement of the following combination of branching fractions,
$\mathcal{B}({{\Xi}^{++}_{cc}}\to{{\Xi}^{+}_{c}}{{\pi}^{+}})\mathcal{B}({{\Xi}^{+}_{c}}\to
p{{K}^{-}}{{\pi}^{+}})/\mathcal{B}({{\Xi}^{++}_{cc}}\to{{\Lambda}^{+}_{c}}{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}})\mathcal{B}({{\Lambda}^{+}_{c}}\to
p{{K}^{-}}{{\pi}^{+}})=0.035\pm 0.009\pm 0.003$, which is consistent with the
prediction in [72, 75].

Another important result on the ${\Xi}^{++}_{cc}$ baryon is the measurement of
its lifetime [76]. The decay mode employed is the same used for the first
observation, ${{\Xi}^{++}_{cc}}\to{{\Lambda}^{+}_{c}}(\to
p{{K}^{-}}{{\pi}^{+}}){{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$. A correction for the
selection efficiency of the reconstructed decay-time distribution of the
${\Xi}^{++}_{cc}$ decays is extracted from simulation. To suppress biases on
this correction due to potential mismodeling of the simulation, the
efficiency-corrected distribution is normalized to the efficiency-corrected
distribution of ${{\Lambda}^{0}_{b}}\to{{\Lambda}^{+}_{c}}(\to
p{{K}^{-}}{{\pi}^{+}}){{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ decays, reconstructed
in the same data set with a similar selection. From the ratio of the two
decay-time distribution, taking in input the precisely known value of the
${\Lambda}^{0}_{b}$ lifetime, the ${\Xi}^{++}_{cc}$ lifetime is found to be
$256^{+22}_{-20}\pm 14$ fs.

A new lifetime measurement is also reported by LHCb for the ${\Omega}^{0}_{c}$
baryon [77]. This baryon is reconstructed in the decay ${{\Omega}^{0}_{c}}\to
p{{K}^{-}}{{K}^{-}}{{\pi}^{+}}$, where the ${\Omega}^{0}_{c}$ originates from
semileptonic decays,
${{\Omega}^{-}_{b}}\to{{\Omega}^{0}_{c}}{{\mu}^{-}}\nu_{\mu}X$. Collecting a
signal yields of about 980 ${{\Omega}^{0}_{c}}$ decays, i.e. ten times higher
than any previous measurement, the lifetime is measurement using the
${{D}^{+}}\to{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ decay, reconstructed from
$B\to{{D}^{+}}{{\mu}^{-}}\nu_{m}uX$ decays, as a normalization channel. This
allows to suppress potential biases on the modeling of the decay-time-
dependent efficiency (with a similar strategy adopted for the measurement of
the ${\Xi}^{++}_{cc}$ lifetime just reported). The value of the
${\Omega}^{0}_{c}$ lifetime found by LHCb is $268\pm 24\pm 10$ fs, in large
disagreement with the current world average, $69\pm 12$ fs [78]. This is also
in contrast with the hierarchy
$\tau({{\Xi}^{+}_{c}})>\tau({{\Lambda}^{+}_{c}})>\tau({{\Xi}^{0}_{c}})>\tau({{\Omega}^{0}_{c}})$,
as this measurement leads to a value smaller than the ${\Xi}^{+}_{c}$
lifetime, but larger than that of the ${\Lambda}^{+}_{c}$ baryon.

##  6 Belle 2 and LHCb upgrades

In 2018, Belle II successfully completed the “Phase II” commissioning and
accumulated 472$\mbox{\,pb}^{-1}$ of data. All subdetectors except for the
full vertex detector were inserted and operating. Initial plots of
reconstructed charm decays were shown [79, 80], suggesting the readiness of
the experiment to enter the game for charm physics. Belle II, with an ability
to accumulate 50 times more data in comparison to Belle, is expected to play
an important role in the next future. Due to the new vertex detector, which
should be operational in 2019, Belle II can exploit a factor of two of
improvement with respect to Belle on the resolution of the track impact
parameter. This allows to have a decay-time resolution in $D\to hh$ decays of
about $0.14$ ps, which is roughly two times better than that at Belle and
BaBar [81]. It is also reported a good resolution, similar to that of the
Belle experiment, for the $D$ mass in decays into final states with neutral
particles (e.g. $\pi^{0}$, $\eta$). This is important as Belle II is expected
to provide a reach program of measurement in decay channels with neutrals,
which are more difficult to be reconstructed by LHCb. Thank also to the clean
environment, Belle II is expected to measure mixing via interference in modes
such as $D^{0}\to K^{-}\pi^{+}{{\pi}^{0}}$ and achieve sensitivity of about
0.06% (0.05%) for $\sigma_{x"}$ ($\sigma_{y"}$) [80]. Further, the possibility
of reconstructing the entire event allow to explore new avenues in
measurements with semileptonic decays, and search for rare decays with
unreconstructed particles, such as $D^{0}\to\nu\bar{\nu}$, and other searches
with missing energy. New flavour tagging techniques, called rest-of-event
(ROE) tagging, will also enable to tag with advanced statistical methods the
rest of 75% of ${D}^{0}$ mesons not tagged by the pion- and lepton-tagging
techniques [79]. Preliminary studies indicate that combining measurements from
pion-tagged and ROE-tagged samples is equivalent to an effective increase of
luminosity of about 40%. Measurements of ${C\\!P}$ asymmetries on channels
with charged tracks in the final state will also be important to complement
LHCb measurements, even though LHCb is expected to collect larger samples in
these cases.

The LHCb experiments just ended operations at the end of 2018, after
accumulating a data sample of more than 9$\mbox{\,fb}^{-1}$ of $pp$ collisions
since its starting in 2010. This is the largest data set available for
pursuing precision measurements in charm physics. All measurements presented
(and also others) will be updated soon using this full sample reaching an
unprecedented precision. Just to give few examples, the wrong-sign mixing
analysis with ${{D}^{0}}\to{{K}^{\mp}}{{\pi}^{\pm}}$ will reach a precision of
about $0.12$ on $|q/p|$ and $10^{\circ}$ on the mixing phase $\phi$; the
measurement of $A_{\Gamma}$ with ${{D}^{0}}\to{{K}^{+}}{{K}^{-}}$ is expected
to have an uncertainty of $0.013\%$, and that of $\Delta A_{{C\\!P}}$ with
${{D}^{0}}\to h^{+}h^{-}$ decays to have an uncertainty of $0.03\%$. The LHCb
experiment is now undergoing a major upgrade, called Upgrade I, to prepare for
the new LHC runs starting in 2021, where the detector will collect data at an
instantaneous luminosity of $2\times 10^{33}$ cm-2s-1, about four time more
than that of the runs just finished. All the front-end electronics and most of
the subdetectors will be replaced in the next two years to allow a data-
acquisition rate of 40 MHz and integrate about 50$\mbox{\,fb}^{-1}$ of data by
2029. Continuing on the analyses examples given above, this copious data
sample will enable to have precisions of $0.03$ on $|q/p|$ and of $3^{\circ}$
on $\phi$, of $0.0035\%$ on $A_{\Gamma}$, and of $0.01\%$ on $\Delta
A_{{C\\!P}}$ [83]. Consolidation and modest enhancements of the Upgrade I
detector will occur around 2025, when Belle II is expected to accumulate is
full data sample of 50$\mbox{\,ab}^{-1}$ and end operations. It is thus highly
likely that LHCb will be the only large-scale flavour-physics experiment after
that time. To fully profit from the High-Luminosity LHC machine, a second
major upgrade of the detector, Upgrade II, is proposed to be installed in the
long shutdown 4 of the LHC (2030), to build on the strengths of the current
LHCb experiment and the Upgrade I. It will operate at a luminosity up to
$2\times 10^{34}$ cm-2s-1, ten times that of the Upgrade I detector. An
Expression Of Interest proposing Upgrade II was submitted in February 2017 to
the LHCC attention [82], and the physics case for the Upgrade II is presented
in depth in Ref. [83].

##  7 Conclusion

We presented the results on charm physics discussed in the Working Group 7 at
the last CKM Workshop held at the University of Heidelberg. Thank to the
constant effort of both the theoretical and experimental communities, good
progress have been reported, and we could expect to have first evidence of
$C\\!P$ violation in the charm sector soon.

Huge samples collected by the experiments are posing hard challenges in the
analyses of data to control systematic uncertainties; anyhow, refined
techniques and new methods have been presented and an unprecedented precision
have been achieved in measurements of both mixing and $C\\!P$ violation,
further constraining the parameters space for new physics models. Among many
others, we highlight new high-precision measurements reported by LHCb in the
wrong-sign mixing analysis and for the mixing-parameter $y_{{C\\!P}}$. Despite
several years from the end of their operations, $B$ factories are still
producing interesting new results to complement the LHCb fruitful program,
e.g. the study of triple-product $C\\!P$-violating asymmetries and that of
radiative charm decays, which have been suggested as sensitive probes for non-
SM physics. This workshop saw also the first physics plots coming from the
Belle II collaboration, showing the readiness of the experiment to enter the
game. The amount of data collected by LHCb opened new avenues also in the
study of charm baryons and rare $D$ decays for $C\\!P$ violation measurements.

On the theoretical side, advancements to cope with non-perturbative QCD
effects making use of experimental data are on the way, as well as the
understanding of subtle effects, such as those induced by kaon mixing in $D$
decays to final states with neutral kaons. All of these is pivotal to identify
a genuine new physics contribution in measurements at high-precision. New
avenues to search for direct $C\\!P$ violation in the experiments to come have
also been proposed.

Good prospects are ahead of us, with LHCb getting ready for the Upgrade I to
start in 2021, while the Belle II collaboration is completing the detector
with the installation of the full vertex detector. Plans for a second upgrade
of LHCb in 2030 to fully exploit the HL-LHC potential has been also reported.

##  8 Acknowledgements

We would like to thank all the speakers for the excellent presentations in the
WG7. We also thank the organizers of CKM2018 for giving us the opportunity to
convene this fruitful working group. We really enjoyed the hospitality and the
scientific ambience of the workshop. VB would like to thank SERB (India) and
DST (India) for providing financial support from ITS and INSPIRE Faculty
grant.

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