by $\Pi_{\chi}\subseteq\Pi$ the kernel of $\chi$. The profinite group
$\Pi_{\chi}$ admits a natural action of $\Pi$ via the conjugation action. We
put

$N_{\chi}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\\{\pi\in
H_{\chi,p_{\Pi}}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\text{Hom}(\Pi_{\chi},\mathbb{Z}/p_{\Pi}\mathbb{Z})\otimes_{\mathbb{F}_{p_{\Pi}}}\overline{\mathbb{F}}_{p_{\Pi}}\
|\ \tau\cdot\pi=\chi(\tau)\pi\ \text{for all}\ \tau\in\Pi\\},$
$\gamma_{N_{\chi}}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\text{dim}_{\overline{\mathbb{F}}_{p_{\Pi}}}(N_{\chi}),$

where
$(\tau\cdot\pi)(x)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\pi(\tau^{-1}\cdot
x)$ for all $x\in\Pi_{\chi}$. We define a group-theoretical invariant
associated to $\Pi$ as follows:

$\gamma_{\Pi}^{\rm
max}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\text{max}\\{\gamma_{N_{\chi}}\
|\ \chi\in\text{Hom}(\Pi,\overline{\mathbb{F}}_{p_{\Pi}}^{\times})\
\text{and}\ \chi\neq 1\\}.$

Let
$\mu_{m}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\chi(\Pi)\subseteq\overline{\mathbb{F}}_{p_{\Pi}}^{\times}$
be the image of $\chi$ which is the group of $m$th roots of unity for some $m$
prime to $p_{\Pi}$, and $X^{\bullet}_{\chi}=(X_{\chi},D_{X_{\chi}})\rightarrow
X^{\bullet}$ the Galois multi-admissible covering over $k$ with Galois group
$\mu_{m}$ induced by $\chi$. Then we have a natural $\Pi$-equivariant
isomorphism

$H^{1}_{\text{\'{e}t}}(X_{\chi},\mathbb{F}_{p_{\Pi}})\otimes_{\mathbb{F}_{p_{\Pi}}}\overline{\mathbb{F}}_{p_{\Pi}}\cong
H_{\chi,p_{\Pi}}.$

Moreover, since the actions of $\Pi$ on
$H^{1}_{\text{\'{e}t}}(X_{\chi},\mathbb{F}_{p_{\Pi}})\otimes_{\mathbb{F}_{p_{\Pi}}}\overline{\mathbb{F}}_{p_{\Pi}}$
and $H_{\chi,p_{\Pi}}$ factor through $\Pi/\Pi_{\chi}\cong\mu_{m}$, the
isomorphism above is also a $\mu_{m}$-equivariant. Namely, $\gamma_{N_{\chi}}$
is a generalized Hasse-Witt invariant of $\mathbb{Z}/m\mathbb{Z}$-cyclic
admissible coverings of $X^{\bullet}$ (2.2.4). Then we have the following
result.

######  Lemma 6.2.

We maintain the notation introduced above. Then we have $\gamma_{\Pi}^{\rm
max}=\gamma_{X^{\bullet}}^{\rm max}$ (see Definition 3.10 (i) for
$\gamma_{\Pi}^{\rm max}$). In particular, we have

$\gamma_{\Pi}^{\rm max}=g_{X}+n_{X}-2+b_{\Pi}^{2}.$

###### Proof.

The first part of the lemma follows from the above explanation concerning
$\gamma_{N_{\chi}}$. The “in particular” part of the lemma follows immediately
from Theorem 5.4 and Lemma 6.1 (i). ∎

#### 6.1.5.

We have the following anabelian formula for $(g_{X},n_{X})$.

######  Theorem 6.3.

Let $X^{\bullet}$ be an arbitrary pointed stable curve of type $(g_{X},n_{X})$
over an algebraically closed field $k$ of characteristic $p>0$,
$\Pi_{X^{\bullet}}$ the admissible fundamental group of $X^{\bullet}$, and
$\Pi$ an abstract profinite group such that $\Pi\cong\Pi_{X^{\bullet}}$ as
profinite groups. Then we have

$g_{X}=b^{1}_{\Pi}-\gamma_{\Pi}^{\rm max}-1,$ $n_{X}=2\gamma^{\rm
max}_{\Pi}-b_{\Pi}^{1}-b_{\Pi}^{2}+3.$

In particular, $g_{X}$ and $n_{X}$ are group-theoretical invariants associated
to $\Pi$.

###### Proof.

The theorem follows immediately from Lemma 6.1 and Lemma 6.2. ∎

#### 6.1.6.

We maintain the notation introduced above. Moreover, suppose that
$X^{\bullet}$ is smooth over $k$. In the remainder of this subsection, we
discuss a formula for $(g_{X},n_{X})$ which was essentially obtained by
Tamagawa. Let $n\stackrel{{\scriptstyle\mathrm{def}}}{{=}}p^{t}-1$, and let
$K_{n}$ be the kernel of the natural surjection $\Pi\twoheadrightarrow\Pi^{\rm
ab}\otimes\mathbb{Z}/n\mathbb{Z}$. In [T2], Tamagawa introduced

$\text{Avr}_{p}(\Pi)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\lim_{t\rightarrow\infty}\frac{\text{dim}_{\mathbb{F}_{p}}(K_{n}^{\rm
ab}\otimes\mathbb{F}_{p})}{\\#(\Pi^{\rm ab}\otimes\mathbb{Z}/n\mathbb{Z})}$

which is called the limit of $p$-averages associated to $\Pi$. Note that since
$p=p_{\Pi}$ (Lemma 6.1 (ii)), we have that $\text{Avr}_{p}(\Pi)$ is a group-
theoretical invariant associated to $\Pi$. Then the main theorem of [T2]
(i.e., Tamagawa’s $p$-average theorem, see [T2, Theorem 0.5]) is the
following:

$\displaystyle\text{Avr}_{p}(\Pi)=\left\\{\begin{array}[]{ll}g_{X}-1,&\text{if}\
n_{X}\leq 1,\\\ g_{X},&\text{if}\ n_{X}\geq 2.\end{array}\right.$

Let $\ell>>0$ be an arbitrary prime number distinct from $p_{\Pi}$. Write
$\text{Nom}_{\ell}(\Pi)$ for the set of normal subgroups of $\Pi$ of index
$\ell$. Suppose that $b_{\Pi}^{2}=0$ (i.e., $n_{X}\neq 0$). It is well-known
that

$\text{Avr}_{p}(\Pi(\ell))-1=\ell(\text{Avr}_{p}(\Pi))$

for every $\ell\in\mathfrak{P}\setminus\\{p_{\Pi}\\}$ and every
$\Pi(\ell)\in\text{Nom}_{\ell}(\Pi)$ if and only if $n_{X}=1$. We may define a
group-theoretical invariant associated to $\Pi$ as follows:

$\displaystyle
c_{\Pi}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\left\\{\begin{array}[]{ll}1,&\text{if}\
b_{\Pi}^{2}=1,\\\ 1,&\text{if}\ b_{\Pi}^{2}=0,\
\text{Avr}_{p}(\Pi(\ell))-1=\ell(\text{Avr}_{p}(\Pi))\ \text{for}\\\ &\
\text{all}\ \ell\in\mathfrak{P}\setminus\\{p_{\Pi}\\}\ \text{and all}\
\Pi(\ell)\in\text{Nom}_{\ell}(\Pi),\\\ 0,&\text{otherwise}.\end{array}\right.$

Then Tamagawa’s $p$-average theorem implies immediately the following formula:

$g_{X}=\text{Avr}_{p}(\Pi)+c_{\Pi},\
n_{X}=b_{\Pi}^{1}-2\text{Avr}_{p}(\Pi)-2c_{\Pi}-b^{2}_{\Pi}+1.$

In particular, $g_{X}$ and $n_{X}$ are group-theoretical invariants associated
to $\Pi$ when $X^{\bullet}$ is smooth over $k$.

Before Tamagawa proved the above result, he also obtained an étale fundamental
group version formula for $(g_{X},n_{X})$ in a completely different way (by
using wildly ramified coverings) which is much simpler than the case of tame
fundamental groups (see [T1, §1]). Note that, for any smooth pointed stable
curve over an algebraically closed field of positive characteristic, since the
tame fundamental group can be reconstructed group-theoretically from the étale
fundamental group ([T1, Corollary 1.10]), the tame fundamental group version
is stronger than the étale fundamental group version. On the other hand, tame
fundamental groups are much better than étale fundamental groups if we study
anabelian geometry in positive characteristic from the point of view of moduli
spaces (e.g. [T3], [Y2]).

#### 6.1.7.

The approach to finding a group-theoretical formula for $(g_{X},n_{X})$ by
applying the limit of $p$-averages associated to $\Pi$ explained above is
difficult to be generalized to the case where $X^{\bullet}$ is an arbitrary
(possibly singular) pointed stable curve. The reason is as follows. In [Y4],
the author generalized Tamagawa’s $p$-average theorem to the case of pointed
stable curves ([Y4, Theorem 1.3 and Theorem 1.4]). The generalized formula
concerning the limit of $p$-averages associated to $\Pi$ is very complicated
in general when $X^{\bullet}$ is not smooth over $k$, and
$\text{Avr}_{p}(\Pi)$ depends not only on the topological type $(g_{X},n_{X})$
but also on the structure of dual semi-graph $\Gamma_{X^{\bullet}}$.

###  6.2. Field structures associated to inertia subgroups of marked points

In this subsection, we prove that the field structures associated to inertia
subgroups of marked points of arbitrary pointed stable curves can be
reconstructed group-theoretically from admissible fundamental groups and
inertia subgroups of marked points, and that a surjective open continuous
homomorphism of admissible fundamental groups induces a field isomorphism of
the fields associated to inertia subgroups of marked points.

####  6.2.1. Settings

Let $i\in\\{1,2\\}$, and let $X_{i}^{\bullet}$ be a pointed stable curve of
type $(g_{X},n_{X})$ over an algebraically closed field $k_{i}$ of
characteristic $p>0$, $\Gamma_{X_{i}^{\bullet}}$ the dual semi-graph of
$X_{i}^{\bullet}$, and $\Pi_{X_{i}^{\bullet}}$ the admissible fundamental
group of $X_{i}^{\bullet}$. Let
$\widehat{X}_{i}^{\bullet}=(\widehat{X}_{i},D_{\widehat{X}_{i}})$ be the
universal admissible covering of $X_{i}^{\bullet}$ corresponding to
$\Pi_{X_{i}^{\bullet}}$ (2.2.6), $\Gamma_{\widehat{X}_{i}^{\bullet}}$ the dual
semi-graph of $\widehat{X}_{i}^{\bullet}$, $e_{i}\in e^{\rm
op}(\Gamma_{X_{i}^{\bullet}})$ an open edge, and $x_{e_{i}}$ the closed point
of $X_{i}$ corresponding to $e_{i}$. Note that we have
$\text{Aut}(\widehat{X}_{i}^{\bullet}/X_{i}^{\bullet})=\Pi_{X_{i}^{\bullet}}$.

Let $\widehat{e}_{i}\in e^{\rm op}(\Gamma_{\widehat{X}_{i}^{\bullet}})$ be an
open edge over $e_{i}$. We denote by

$I_{\widehat{e}_{i}}\cong\widehat{\mathbb{Z}}(1)^{p^{\prime}}\subseteq\Pi_{X_{i}^{\bullet}}$

the stabilizer subgroup of $\widehat{e}_{i}$.

#### 6.2.2.

Write $\overline{\mathbb{F}}_{p,i}$ for the algebraic closure of
$\mathbb{F}_{p}$ in $k_{i}$. We put

$\mathbb{F}_{\widehat{e}_{i}}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}(I_{\widehat{e}_{i}}\otimes_{\mathbb{Z}}(\mathbb{Q}/\mathbb{Z})_{i}^{p^{\prime}})\sqcup\\{*_{\widehat{e}_{i}}\\},$

where $\\{*_{\widehat{e}_{i}}\\}$ is an one-point set, and
$(\mathbb{Q}/\mathbb{Z})_{i}^{p^{\prime}}$ denotes the prime-to-$p$ part of
$\mathbb{Q}/\mathbb{Z}$ which can be canonically identified with

$\bigcup_{(p,m)=1}\mu_{m}(k_{i}).$

Moreover, $\mathbb{F}_{\widehat{e}_{i}}$ can be identified with
$\overline{\mathbb{F}}_{p,i}$ as sets, hence, admits a structure of field,
whose multiplicative group is
$I_{\widehat{e}_{i}}\otimes_{\mathbb{Z}}(\mathbb{Q}/\mathbb{Z})^{p^{\prime}}_{i}$,
and whose zero element is $*_{\widehat{e}_{i}}$. We have the following
important result.

######  Theorem 6.4.

We maintain the notation introduced above. Let
$\phi:\Pi_{X^{\bullet}_{1}}\twoheadrightarrow\Pi_{X_{2}^{\bullet}}$ be an
arbitrary surjective open continuous homomorphism of admissible fundamental
groups. Suppose that $\phi(I_{\widehat{e}_{1}})=I_{\widehat{e}_{2}}$, and that
$n_{X}=3$. Then there exists a group-theoretical algorithm whose input data
are $\Pi_{X_{i}^{\bullet}}$ and $I_{\widehat{e}_{i}}$, and whose output datum
is $\mathbb{F}_{\widehat{e}_{i}}$ (as a field). Moreover, $\phi$ induces a
field isomorphism

$\phi^{\rm
fd}_{\widehat{e}_{1},\widehat{e}_{2}}:\mathbb{F}_{\widehat{e}_{1}}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathbb{F}_{\widehat{e}_{2}},$

where “fd” means “field”.

###### Proof.

Let $t\in\mathbb{Z}_{>0}$. We denote by $\mathbb{F}_{p^{t},\widehat{e}_{i}}$
the unique subfield of $\mathbb{F}_{\widehat{e}_{i}}$ whose cardinality is
equal to $p^{t}$. On the other hand, we fix a finite field
$\mathbb{F}_{p^{t}}$ of cardinality $p^{t}$ and an algebraic closure
$\overline{\mathbb{F}}_{p}$ of $\mathbb{F}_{p}$ containing
$\mathbb{F}_{p^{t}}$. Note that the field structure of
$\mathbb{F}_{p^{t},\widehat{e}_{i}}$ is equivalent to a subset

$\text{Hom}_{\rm
fd}(\mathbb{F}_{p^{t},\widehat{e}_{i}},\mathbb{F}_{p^{t}})\subseteq\text{Hom}_{\rm
gp}(\mathbb{F}_{p^{t},\widehat{e}_{i}}^{\times},\mathbb{F}_{p^{t}}^{\times}),$

where “gp” means “group”, and “fd” means “field”. Then in order to prove the
first part of the theorem, it is sufficient to prove that there exists a
group-theoretical algorithm whose input data are $\Pi_{X_{i}^{\bullet}}$ and
$I_{\widehat{e}_{i}}$, and whose output datum is the subset $\text{Hom}_{\rm
fd}(\mathbb{F}_{p^{t},\widehat{e}_{i}},\mathbb{F}_{p^{t}})$ for $t$ in a
cofinal subset of $\mathbb{N}$ with respect to division.

Let $n\stackrel{{\scriptstyle\mathrm{def}}}{{=}}p^{t}-1$ and
$\chi_{i}\in\text{Hom}_{\rm gp}(\Pi_{X_{i}^{\bullet}}^{\rm
ab}\otimes\mathbb{Z}/n\mathbb{Z},\mathbb{F}^{\times}_{p^{t}}).$ Write
$H_{\chi_{i}}$ for the kernel of
$\Pi_{X_{i}^{\bullet}}\twoheadrightarrow\Pi_{X_{i}^{\bullet}}^{\rm
ab}\otimes\mathbb{Z}/n\mathbb{Z}\overset{\chi_{i}}{\rightarrow}\mathbb{F}_{p^{t}}^{\times}$,
$M_{\chi_{i}}$ for $H^{\rm ab}_{\chi_{i}}\otimes\mathbb{F}_{p}$, and
$X^{\bullet}_{H_{\chi_{i}}}=(X_{H_{\chi_{i}}},D_{X_{H_{\chi_{i}}}})$ for the
pointed stable curve over $k_{i}$ induced by $H_{\chi_{i}}$. Note that
$M_{\chi_{i}}$ admits a natural action of $\Pi_{X^{\bullet}_{i}}$ via the
conjugation action. Moreover, this action factors through
$\Pi_{X_{i}^{\bullet}}^{\rm ab}\otimes\mathbb{Z}/n\mathbb{Z}$. We put

$M_{\chi_{i}}[\chi_{i}]\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\\{a\in
M_{\chi_{i}}\otimes_{\mathbb{F}_{p}}\overline{\mathbb{F}}_{p}\ |\ \sigma\cdot
a=\chi_{i}(\sigma)a\ \text{for all}\ \sigma\in\Pi_{X_{i}^{\bullet}}^{\rm
ab}\otimes\mathbb{Z}/n\mathbb{Z}\\},$
$\gamma_{\chi_{i}}(M_{\chi_{i}})\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\text{dim}_{\overline{\mathbb{F}}_{p}}(M_{\chi_{i}}[\chi_{i}]).$

Note that $\gamma_{\chi_{i}}(M_{\chi_{i}})$ is a generalized Hasse-Witt
invariant of a cyclic multi-admissible covering of $X_{i}^{\bullet}$ with
Galois group $\mathbb{Z}/n\mathbb{Z}$, and that Lemma 2.5 implies that
$\gamma_{\chi_{i}}(M_{\chi_{i}})\leq g_{X}+1$ if $n_{X}=3$. Moreover, we
define two maps

$\text{Res}_{i,t}:\text{Hom}_{\rm gp}(\Pi_{X_{i}^{\bullet}}^{\rm
ab}\otimes\mathbb{Z}/n\mathbb{Z},\mathbb{F}^{\times}_{p^{t}})\rightarrow\text{Hom}_{\rm
gp}(\mathbb{F}^{\times}_{p^{t},\widehat{e}_{i}},\mathbb{F}_{p^{t}}^{\times}),$
$\Gamma_{i,t}:\text{Hom}_{\rm gp}(\Pi_{X_{i}^{\bullet}}^{\rm
ab}\otimes\mathbb{Z}/n\mathbb{Z},\mathbb{F}^{\times}_{p^{t}})\rightarrow\mathbb{Z}_{\geq
0},\ \chi_{i}\mapsto\gamma_{\chi_{i}}(M_{\chi_{i}}),$

where the map $\text{Res}_{i,t}$ is the restriction with respect to the
natural inclusion

$\mathbb{F}^{\times}_{p^{t},\widehat{e}_{i}}=I_{\widehat{e}_{i}}\otimes\mathbb{Z}/n\mathbb{Z}\hookrightarrow\Pi_{X_{i}^{\bullet}}^{\rm
ab}\otimes\mathbb{Z}/n\mathbb{Z}.$

Let $m_{0}$ be the product of all prime numbers $\leq p-2$ if $p\neq 2,3$ and
$m_{0}=1$ if $p=2,3$. Let $t_{0}$ be the order of $p$ in the multiplicative
group $(\mathbb{Z}/m_{0}\mathbb{Z})^{\times}$. We have the following claim
(see [T2, Claim 5.4] for the case where $X^{\bullet}_{i}$ is smooth over
$k_{i}$):

Claim A: There exists a constant $C(g_{X})$ which depends only on $g_{X}$ such
that, for each natural number
$t>\text{max}\\{\text{log}_{p}(C(g_{X})+1),\log_{p}(\\#(e^{\rm
cl}(\Gamma_{X_{i}^{\bullet}})))\\}$ divisible by $t_{0}$, we have

$\text{Hom}_{\rm
fd}(\mathbb{F}_{p^{t},\widehat{e}_{i}},\mathbb{F}_{p^{t}})=\text{Hom}^{\rm
surj}_{\rm
gp}(\mathbb{F}_{p^{t},\widehat{e}_{i}}^{\times},\mathbb{F}_{p^{t}}^{\times})\setminus\text{Res}_{i,t}(\Gamma_{i,t}^{-1}(\\{g_{X}+1\\})),\
i\in\\{1,2\\},$

where $\text{Hom}^{\rm surj}_{\rm gp}(-,-)$ denotes the set of surjections
whose elements are contained in $\text{Hom}_{\rm gp}(-,-)$.

By applying similar arguments to the arguments given in the proof of [T2,
Claim 5.4], Claim A is equivalent to the following claim:

Claim B: Let $m\in(\mathbb{Z}/n\mathbb{Z})^{\sim}$ (2.2.5). Then the following
statements are equivalent:

(i) We have $m\in\\{p^{r}\ |\ r=0,\dots,t-1\\}$.

(ii) We have that $(m,n)=1$, and that, there does not exist
$D\in(\mathbb{Z}/n\mathbb{Z})^{\sim}[D_{X}]^{0}$ (2.2.5) and
$\alpha\in\text{Rev}_{D}^{\rm adm}(X^{\bullet}_{i})$ (Definition 2.4 (i)) such
that $\text{ord}_{x_{e_{i}}}(D)=m$ and $\gamma_{(\alpha,D)}=g_{X}+1$.

Let us prove Claim B. Firstly, we prove (i) $\Rightarrow$ (ii). If $s(D)=1$
(2.2.5), then Lemma 2.5 implies $\gamma_{(\alpha,D)}\leq g_{X}$. Thus, we may
assume that $s(D)=2$. We put

$D_{X_{i}}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\\{x_{i,1}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}x_{e_{i}},x_{i,2},x_{i,3}\\}.$

By [T2, Proposition 2.21 (iv-a)], either $\text{ord}_{x_{i,1}}(D)=n$ or
$\text{ord}_{x_{i,3}}(D)=n$ holds. This is impossible as
$D\in(\mathbb{Z}/n\mathbb{Z})^{\sim}[D_{X}]^{0}$. Next, we prove (ii)
$\Rightarrow$ (i). Suppose that $m\not\in\\{p^{r}\ |\ r=0,\dots,t-1\\}$, and
that $(m,n)=1$. Since $t$ is divisible by $t_{0}$, $n$ is divisible by all
prime numbers $\leq p-2$. Then the assumption $(m,n)=1$ implies that
$m\not\in\\{ap^{r}\ |\ a=0,\dots,p-2,r=0,\dots,t-1\\}$. Then [T2, Proposition
2.21 (iv-c)] and Theorem 5.7 imply that there exist
$D\in(\mathbb{Z}/n\mathbb{Z})^{\sim}[D_{X}]^{0}$ and
$\alpha\in\text{Rev}_{D}^{\rm adm}(X^{\bullet}_{i})$ such that
$\text{ord}_{x_{e_{i}}}(D)=m$ and $\gamma_{(\alpha,D)}=g_{X}+1$. This
completes the proof of the claim. Then we complete the proof of the first part
of the theorem.

Next, we prove the “moreover” part of the theorem. Let
$\kappa_{2}\in\text{Hom}_{\rm gp}(\Pi_{X^{\bullet}_{2}}^{\rm
ab}\otimes\mathbb{Z}/n\mathbb{Z},\mathbb{F}^{\times}_{p^{t}}).$ Then we obtain
a character

$\kappa_{1}\in\text{Hom}_{\rm gp}(\Pi_{X^{\bullet}_{1}}^{\rm
ab}\otimes\mathbb{Z}/n\mathbb{Z},\mathbb{F}^{\times}_{p^{r}})$

induced by $\phi$. Moreover, the surjection $\phi|_{H_{\kappa_{1}}}$ induces a
surjection $M_{\kappa_{1}}[\kappa_{1}]\twoheadrightarrow
M_{\kappa_{2}}[\kappa_{2}].$ Suppose that
$\kappa_{2}\in\Gamma_{2,r}^{-1}(\\{g_{X}+1\\})$. The surjection
$M_{\kappa_{1}}[\kappa_{1}]\twoheadrightarrow M_{\kappa_{2}}[\kappa_{2}]$
implies that $\gamma_{\kappa_{1}}(M_{\kappa_{1}})=g_{X}+1$. Namely, we have
$\kappa_{1}\in\Gamma_{1,r}^{-1}(\\{g_{X}+1\\})$. On the other hand, the
isomorphism
$\phi|_{I_{\widehat{e}_{1}}}:I_{\widehat{e}_{1}}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}I_{\widehat{e}_{2}}$
induces an injection

$\text{Res}_{2,r}(\Gamma_{2,r}^{-1}(\\{g_{X}+1\\}))\hookrightarrow\text{Res}_{1,r}(\Gamma_{1,r}^{-1}(\\{g_{X}+1\\})).$

Since $\\#(\text{Hom}_{\rm
fd}(\mathbb{F}_{p^{r},\widehat{e}_{1}},\mathbb{F}_{p^{r}}))=\\#(\text{Hom}_{\rm
fd}(\mathbb{F}_{p^{r},\widehat{e}_{2}},\mathbb{F}_{p^{r}}))$, Claim A implies
that $\phi|_{I_{\widetilde{e}_{1}}}$ induces a bijection $\text{Hom}_{\rm
fd}(\mathbb{F}_{p^{r},\widehat{e}_{2}},\mathbb{F}_{p^{r}})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\text{Hom}_{\rm
fd}(\mathbb{F}_{p^{r},\widehat{e}_{1}},\mathbb{F}_{p^{r}}).$ Thus,
$\phi|_{I_{\widehat{e}_{1}}}$ induces a bijection

$\text{Hom}_{\rm
fd}(\mathbb{F}_{\widehat{e}_{2}},\overline{\mathbb{F}}_{p})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\text{Hom}_{\rm
fd}(\mathbb{F}_{\widehat{e}_{1}},\overline{\mathbb{F}}_{p}).$

If we choose $\overline{\mathbb{F}}_{p}=\mathbb{F}_{\widehat{e}_{2}}$, then
the image of $\text{id}_{\mathbb{F}_{\widehat{e}_{2}}}$ via the above
bijection induces a field isomorphism

$\phi^{\rm
fd}_{\widehat{e}_{1},\widehat{e}_{2}}:\mathbb{F}_{\widehat{e}_{1}}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathbb{F}_{\widehat{e}_{2}}.$

This completes the proof of the theorem. ∎

######  Remark 6.4.1.

We maintain the notation introduced above. In fact, by applying Theorem 5.4,
we can prove that $I_{\widehat{e}_{i}}$ can be reconstructed group-
theoretically from $\Pi_{X_{i}^{\bullet}}$, and that
$\phi(I_{\widehat{e}_{1}})$ is always a stabilizer of an open edge of
$\Gamma_{\widehat{X}^{\bullet}_{2}}$ over an open edge of
$\Gamma_{X^{\bullet}_{2}}$ ([Y5, Theorem 4.11]). Moreover, by applying similar
arguments to the arguments given in the proof of [Y2, Proposition 6.1], to
reconstruct the field structures, we may assume that $n_{X}=3$. Thus, Theorem
6.4 implies that the field structures associated to inertia subgroups of
marked points can be reconstructed group-theoretically from surjective open
continuous homomorphisms of admissible fundamental groups ([Y5, Theorem
4.13]).

## References

  * [B] I. Bouw, The $p$-rank of ramified covers of curves, Compositio Math. 126 (2001), 295-–322.
  * [C] R. Crew, Étale $p$-covers in characteristic $p$, Compositio Math. 52 (1984), 31–45.
  * [FJ] M. D. Fried, M. Jarden, Field arithmetic. Third edition. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3\. Folge. A Series of Modern Surveys in Mathematics 11. Springer-Verlag, Berlin, 2008.
  * [H] D. Harbater, Fundamental groups of curves in characteristic $p$, Proceedings of the ICM (Z$\ddot{\text{u}}$rich, 1994), Birkh$\ddot{\text{a}}$user; Basel, 1995, 656–666.
  * [K] F. Knudsen, The projectivity of the moduli space of stable curves, II: The stacks $M_{g,n}$, Math. Scand., 52 (1983), 161–-199.
  * [M] S. Mochizuki, Semi-graphs of anabelioids. Publ. Res. Inst. Math. Sci. 42 (2006), 221–322.
  * [N] S. Nakajima, On generalized Hasse-Witt invariants of an algebraic curve, Galois groups and their representations (Nagoya 1981) (Y. Ihara, ed.), Adv. Stud. Pure Math, 2, North-Holland Publishing Company, Amsterdam, 1983, 69–88.
  * [OP] E. Ozman, R. Pries, Ordinary and almost ordinary Prym varieties. Asian J. Math. 23 (2019), 455–477.
  * [PS] F. Pop, M. Sa$\rm\ddot{\i}$di, On the specialization homomorphism of fundamental groups of curves in positive characteristic. Galois groups and fundamental groups, 107–118, Math. Sci. Res. Inst. Publ., 41, Cambridge Univ. Press, Cambridge, 2003\.
  * [R1] M. Raynaud, Sections des fibrés vectoriels sur une courbe. Bull. Soc. math. France 110 (1982), 103–125.
  * [R2] M. Raynaud, Sur le groupe fondamental d’une courbe complète en caractéristique $p>0$. Arithmetic fundamental groups and noncommutative algebra (Berkeley, CA, 1999), 335–351, Proc. Sympos. Pure Math., 70, Amer. Math. Soc., Providence, RI, 2002\.
  * [S1] J-P. Serre, Sur la topologie des variétés algébriques en caractéristique $p$. Symp. Int. Top. Alg., Mexico (1958), 24–53.
  * [S2] J-P. Serre, Trees. Translated from the French by John Stillwell. Springer-Verlag, Berlin-New York, 1980\.
  * [T1] A. Tamagawa, On the fundamental groups of curves over algebraically closed fields of characteristic $>0$. Internat. Math. Res. Notices (1999), 853–873.
  * [T2] A. Tamagawa, On the tame fundamental groups of curves over algebraically closed fields of characteristic $>0$. Galois groups and fundamental groups, 47–105, Math. Sci. Res. Inst. Publ., 41, Cambridge Univ. Press, Cambridge, 2003\.
  * [T3] A. Tamagawa, Finiteness of isomorphism classes of curves in positive characteristic with prescribed fundamental groups. J. Algebraic Geom. 13 (2004), 675–724.
  * [V] I. Vidal, Contributions à la cohomologie étale des schémas et des log-schémas, Thèse, U. Paris-Sud (2001).
  * [Y1] Y. Yang, On the admissible fundamental groups of curves over algebraically closed fields of characteristic $p>0$, Publ. Res. Inst. Math. Sci. 54 (2018), 649–678.
  * [Y2] Y. Yang, Tame anabelian geometry and moduli spaces of curves over algebraically closed fields of characteristic $p>0$, preprint. See http://www.kurims.kyoto-u.ac.jp/∼yuyang/
  * [Y3] Y. Yang, On topological and combinatorial structures of pointed stable curves over algebraically closed fields of positive characteristic. See http://www.kurims.kyoto-u.ac.jp/∼yuyang/
  * [Y4] Y. Yang, On the averages of generalized Hasse-Witt invariants of pointed stable curves in positive characteristic, Math. Z. 295 (2020), 1–45.
  * [Y5] Y. Yang, Moduli spaces of fundamental groups of curves in positive characteristic I, preprint, arXiv:2010.01806. See also http://www.kurims.kyoto-u.ac.jp/∼yuyang/
  * [Y6] Y. Yang, Moduli spaces of fundamental groups of curves in positive characteristic II, in preparation.
  * [Z] B. Zhang, Revêtements étales abeliens de courbes génériques et ordinarité, Ann. Fac. Sci. Toulouse Math. (5) 6 (1992), 133–138.


* [16] T. M. Aliev, A. Ozpineci, and M. Savci, _$B_{q}\rightarrow l^{+}l^{-}\gamma$ decays in light cone QCD_, Phys. Rev. D55 (1997) 7059, arXiv:hep-ph/9611393
  * [17] C. Q. Geng, C. C. Lih, and W.-M. Zhang, _Study of $B_{s,d}\rightarrow l^{+}l^{-}\gamma$ decays_, Phys. Rev. D62 (2000) 074017, arXiv:hep-ph/0007252
  * [18] D. Melikhov and N. Nikitin, _Rare radiative leptonic decays $B_{d,s}\rightarrow l^{+}l^{-}\gamma$_, Phys. Rev. D70 (2004) 114028, arXiv:hep-ph/0410146
  * [19] A. Kozachuk, D. Melikhov, and N. Nikitin, _Rare FCNC radiative leptonic $B_{s,d}\rightarrow\gamma l^{+}l^{-}$ decays in the standard model_, Phys. Rev. D97 (2018) 053007, arXiv:1712.07926
  * [20] S. Dubnička et al., _Study of $B_{s}\rightarrow\ell^{+}\ell^{-}\gamma$ decays in covariant quark model_, Phys. Rev. D99 (2019) 014042, arXiv:1808.06261
  * [21] M. Beneke, C. Bobeth, and Y.-M. Wang, _$B_{d,s}\rightarrow\gamma\ell\bar{\ell}$ decay with an energetic photon_, JHEP 12 (2020) 148, arXiv:2008.12494
  * [22] F. Dettori, D. Guadagnoli, and M. Reboud, _$B^{0}_{s}\rightarrow\mu^{+}\mu^{-}\gamma$ from $B^{0}_{s}\rightarrow\mu^{+}\mu^{-}$_, Phys. Lett. B768 (2017) 163, arXiv:1610.00629
  * [23] T. M. Aliev, A. Ozpineci, and M. Savci, _Rare $B\rightarrow l^{+}l^{-}\gamma$ decay and new physics effects_, Phys. Lett. B520 (2001) 69, arXiv:hep-ph/0105279
  * [24] D. Guadagnoli, M. Reboud, and R. Zwicky, _$B_{s}^{0}\rightarrow\ell^{+}\ell^{-}\gamma$ as a test of lepton flavor universality_, JHEP 11 (2017) 184, arXiv:1708.02649
  * [25] LHCb collaboration, R. Aaij et al., _Test of lepton universality with ${{B}^{0}}\\!\rightarrow{{K}^{*0}}\ell^{+}\ell^{-}$ decays_, JHEP 08 (2017) 055, arXiv:1705.05802
  * [26] LHCb collaboration, R. Aaij et al., _Search for lepton-universality violation in ${{{B}^{+}}}\\!\rightarrow{{K}^{+}}\ell^{+}\ell^{-}$ decays_, Phys. Rev. Lett. 122 (2019) 191801, arXiv:1903.09252
  * [27] LHCb collaboration, R. Aaij et al., _Measurement of $C\\!P$-averaged observables in the ${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\mu^{+}\mu^{-}}$ decay_, Phys. Rev. Lett. 125 (2020) 011802, arXiv:2003.04831
  * [28] LHCb collaboration, R. Aaij et al., _Angular analysis of the $B^{+}\rightarrow K^{\ast+}\mu^{+}\mu^{-}$ decay_, Phys. Rev. Lett. 126 (2021) 161802, arXiv:2012.13241
  * [29] R. Giles et al., _Two-body decays of $B$ mesons_, Phys. Rev. D30 (1984) 2279
  * [30] LHCb collaboration, R. Aaij et al., _First evidence for the decay ${{B}^{0}_{s}}\\!\rightarrow{\mu^{+}\mu^{-}}$_, Phys. Rev. Lett. 110 (2013) 021801, arXiv:1211.2674
  * [31] CMS and LHCb collaborations, V. Khachatryan et al., _Observation of the rare ${{B}^{0}_{s}}\\!\rightarrow{\mu^{+}\mu^{-}}$ decay from the combined analysis of CMS and LHCb data_, Nature 522 (2015) 68, arXiv:1411.4413
  * [32] LHCb collaboration, R. Aaij et al., _Measurement of the ${{B}^{0}_{s}}\\!\rightarrow{\mu^{+}\mu^{-}}$ branching fraction and effective lifetime and search for ${{B}^{0}}\\!\rightarrow{\mu^{+}\mu^{-}}$ decays_, Phys. Rev. Lett. 118 (2017) 191801, arXiv:1703.05747
  * [33] ATLAS collaboration, M. Aaboud et al., _Study of the rare decays of $B^{0}_{s}$ and $B^{0}$ mesons into muon pairs using data collected during 2015 and 2016 with the ATLAS detector_, JHEP 04 (2019) 098, arXiv:1812.03017
  * [34] CMS collaboration, A. M. Sirunyan et al., _Measurement of properties of B ${}^{0}_{\mathrm{s}}\rightarrow\mu^{+}\mu^{-}$ decays and search for B${}^{0}\rightarrow\mu^{+}\mu^{-}$ with the CMS experiment_, JHEP 04 (2020) 188, arXiv:1910.12127
  * [35] ATLAS, CMS and LHCb collaborations, _Combination of the ATLAS, CMS and LHCb results on the $B_{(s)}^{0}\rightarrow\mu^{+}\mu^{-}$ decays_, LHCb-CONF-2020-002, ATLAS-CONF-2020-049, CMS-PAS-BPH-20-003, 2020
  * [36] BaBar collaboration, B. Aubert et al., _Search for the decays $B^{0}\rightarrow e^{+}e^{-}\gamma$ and $B^{0}\rightarrow\mu^{+}\mu^{-}\gamma$_, Phys. Rev. D77 (2008) 011104, arXiv:0706.2870
  * [37] LHCb collaboration, R. Aaij et al., _Analysis of neutral $B$-meson decays into two muons_, Phys. Rev. Lett. 128 (2022) 041801, arXiv:2108.09284, companion Letter
  * [38] L. Breiman, J. H. Friedman, R. A. Olshen, and C. J. Stone, Classification and regression trees, Wadsworth international group, Belmont, California, USA, 1984
  * [39] LHCb collaboration, A. A. Alves Jr. et al., _The LHCb detector at the LHC_, JINST 3 (2008) S08005
  * [40] LHCb collaboration, R. Aaij et al., _LHCb detector performance_ , Int. J. Mod. Phys. A30 (2015) 1530022, arXiv:1412.6352
  * [41] T. Sjöstrand, S. Mrenna, and P. Skands, _PYTHIA 6.4 physics and manual_ , JHEP 05 (2006) 026, arXiv:hep-ph/0603175
  * [42] T. Sjöstrand, S. Mrenna, and P. Skands, _A brief introduction to PYTHIA 8.1_ , Comput. Phys. Commun. 178 (2008) 852, arXiv:0710.3820
  * [43] I. Belyaev et al., _Handling of the generation of primary events in Gauss, the LHCb simulation framework_ , J. Phys. Conf. Ser. 331 (2011) 032047
  * [44] D. J. Lange, _The EvtGen particle decay simulation package_ , Nucl. Instrum. Meth. A462 (2001) 152
  * [45] C.-H. Chang, C. Driouichi, P. Eerola, and X. G. Wu, _BCVEGPY: An Event generator for hadronic production of the $B_{c}$ meson_, Comput. Phys. Commun. 159 (2004) 192, arXiv:hep-ph/0309120
  * [46] C.-H. Chang, X.-Y. Wang, and X.-G. Wu, _BCVEGPY2.2: A newly upgraded version for hadronic production of the meson Bc and its excited states_ , Comput. Phys. Commun. 197 (2015) 335, arXiv:1507.05176
  * [47] N. Davidson, T. Przedzinski, and Z. Was, _PHOTOS interface in C++: Technical and physics documentation_ , Comput. Phys. Commun. 199 (2016) 86, arXiv:1011.0937
  * [48] M. Bordone, G. Isidori, and A. Pattori, _On the Standard Model predictions for $R_{K}$ and $R_{K^{*}}$_, Eur. Phys. J. C76 (2016) 440, arXiv:1605.07633
  * [49] Geant4 collaboration, J. Allison et al., _Geant4 developments and applications_ , IEEE Trans. Nucl. Sci. 53 (2006) 270
  * [50] Geant4 collaboration, S. Agostinelli et al., _Geant4: A simulation toolkit_ , Nucl. Instrum. Meth. A506 (2003) 250
  * [51] M. Clemencic et al., _The LHCb simulation application, Gauss: Design, evolution and experience_, J. Phys. Conf. Ser. 331 (2011) 032023
  * [52] F. Archilli et al., _Performance of the muon identification at LHCb_ , JINST 8 (2013) P10020, arXiv:1306.0249
  * [53] Particle Data Group, P. A. Zyla et al., _Review of particle physics_ , Prog. Theor. Exp. Phys. 2020 (2020) 083C01
  * [54] Y. Freund and R. E. Schapire, _A decision-theoretic generalization of on-line learning and an application to boosting_ , J. Comput. Syst. Sci. 55 (1997) 119
  * [55] A. Hoecker et al., _TMVA 4 — Toolkit for Multivariate Data Analysis with ROOT. Users Guide._ , arXiv:physics/0703039
  * [56] LHCb collaboration, R. Aaij et al., _Strong constraints on the rare decays ${{B}^{0}_{s}}\\!\rightarrow{\mu^{+}\mu^{-}}$ and ${{B}^{0}}\\!\rightarrow{\mu^{+}\mu^{-}}$_, Phys. Rev. Lett. 108 (2012) 231801, arXiv:1203.4493
  * [57] T. Skwarnicki, A study of the radiative cascade transitions between the Upsilon-prime and Upsilon resonances, PhD thesis, Institute of Nuclear Physics, Krakow, 1986, DESY-F31-86-02
  * [58] LHCb collaboration, R. Aaij et al., _Observation of the decay ${{\kern 1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}}\\!\rightarrow\psi{(2S)}{{K}^{+}}{{\pi}^{-}}$_, Phys. Lett. B747 (2015) 484, arXiv:1503.07112
  * [59] LHCb collaboration, R. Aaij et al., _Measurement of $b$-hadron masses_, Phys. Lett. B708 (2012) 241, arXiv:1112.4896
  * [60] CDF collaboration, D. Acosta et al., _Measurement of $b$ hadron masses in exclusive $J/\psi$ decays with the CDF detector_, Phys. Rev. Lett. 96 (2006) 202001, arXiv:hep-ex/0508022
  * [61] ARGUS collaboration, H. Albrecht et al., _Search for hadronic $b\rightarrow u$ decays_, Phys. Lett. B241 (1990) 278
  * [62] A. Rogozhnikov, _Reweighting with Boosted Decision Trees_ , J. Phys. Conf. Ser. 762 (2016) , arXiv:1608.05806, https://github.com/arogozhnikov/hep_ml
  * [63] LHCb collaboration, R. Aaij et al., _Precise measurement of the $f_{s}/f_{d}$ ratio of fragmentation fractions and of $B^{0}_{s}$ decay branching fractions_, arXiv:2103.06810, to appear in PRD
  * [64] S. Tolk, J. Albrecht, F. Dettori, and A. Pellegrino, _Data driven trigger efficiency determination at LHCb_ , LHCb-PUB-2014-039, 2014
  * [65] L. Anderlini et al., _The PIDCalib package_ , LHCb-PUB-2016-021, 2016
  * [66] BaBar collaboration, B. Aubert et al., _Improved Measurements of the Branching Fractions for $B^{0}\rightarrow\pi^{+}\pi^{-}$ and $B^{0}\rightarrow K^{+}\pi^{-}$, and a Search for $B^{0}\rightarrow K^{+}K^{-}$_, Phys. Rev. D75 (2007) 012008, arXiv:hep-ex/0608003
  * [67] Belle collaboration, Y.-T. Duh et al., _Measurements of branching fractions and direct $CP$ asymmetries for $B\rightarrow K\pi$, $B\rightarrow\pi\pi$ and $B\rightarrow KK$ decays_, Phys. Rev. D87 (2013) 031103, arXiv:1210.1348
  * [68] CLEO collaboration, A. Bornheim et al., _Measurements of charmless hadronic two body B meson decays and the ratio ${\cal B}({B}\rightarrow{D}K)/{\cal B}({B}\rightarrow{D}\pi)$_, Phys. Rev. D68 (2003) 052002, arXiv:hep-ex/0302026, [Erratum: Phys. Rev. D75 (2007) 119907]
  * [69] Belle collaboration, K. Chilikin et al., _Evidence for $B^{+}\rightarrow h_{c}K^{+}$ and observation of $\eta_{c}(2S)\rightarrow p\bar{p}\pi^{+}\pi^{-}$_, Phys. Rev. D100 (2019) 012001, arXiv:1903.06414
  * [70] Belle collaboration, Y. Kato et al., _Measurements of the absolute branching fractions of $B^{+}\rightarrow X_{c\bar{c}}K^{+}$ and $B^{+}\rightarrow\bar{D}^{(\ast)0}\pi^{+}$ at Belle_, Phys. Rev. D97 (2018) 012005, arXiv:1709.06108
  * [71] BaBar collaboration, B. Aubert et al., _Measurements of the absolute branching fractions of $B^{\pm}\rightarrow K^{\pm}$ X($c\bar{c}$)_, Phys. Rev. Lett. 96 (2006) 052002, arXiv:hep-ex/0510070
  * [72] BaBar collaboration, B. Aubert et al., _Measurement of the $B^{+}\rightarrow p\bar{p}K^{+}$ branching fraction and study of the decay dynamics_, Phys. Rev. D72 (2005) 051101, arXiv:hep-ex/0507012
  * [73] BaBar collaboration, B. Aubert et al., _Measurement of branching fractions and charge asymmetries for exclusive $B$ decays to charmonium_, Phys. Rev. Lett. 94 (2005) 141801, arXiv:hep-ex/0412062
  * [74] Belle collaboration, K. Abe et al., _Measurement of branching fractions and charge asymmetries for two-body B meson decays with charmonium_ , Phys. Rev. D67 (2003) 032003, arXiv:hep-ex/0211047
  * [75] CLEO collaboration, C. P. Jessop et al., _Measurement of the decay amplitudes and branching fractions of $B\rightarrow J/\psi K^{*}$ and $B\rightarrow J/\psi K$ decays_, Phys. Rev. Lett. 79 (1997) 4533, arXiv:hep-ex/9702013
  * [76] CLEO collaboration, D. Bortoletto et al., _Inclusive and exclusive decays of $B$ mesons to final states including charm and charmonium mesons_, Phys. Rev. D45 (1992) 21
  * [77] ARGUS collaboration, H. Albrecht et al., _Exclusive Hadronic Decays of $B$ Mesons_, Z. Phys. C48 (1990) 543
  * [78] V. V. Anashin et al., _Final analysis of KEDR data on ${J\mskip-3.0mu/\mskip-2.0mu\psi}$ and $\psi{(2S)}$ masses_, Phys. Lett. B749 (2015) 50
  * [79] E760 collaboration, T. A. Armstrong et al., _Measurement of the ${J\mskip-3.0mu/\mskip-2.0mu\psi}$ and ${\psi}^{{}^{\prime}}$ resonance parameters in $\bar{p}p$ annihilation_, Phys. Rev. D47 (1993) 772
  * [80] OLYA, MD-1 collaborations, A. S. Artamonov et al., _High precision mass measurements in $\psi$ and $\mathchar 28935\relax$ families revisited_, Phys. Lett. B474 (2000) 427, arXiv:hep-ex/0001040
  * [81] C. Baglin et al., _$J/\psi$ Resonance Formation and Mass Measurement in $\overline{p}$-$p$ Annihilations_, Nucl. Phys. B286 (1987) 592
  * [82] D. Martínez Santos and F. Dupertuis, _Mass distributions marginalized over per-event errors_ , Nucl. Instrum. Meth. A764 (2014) 150, arXiv:1312.5000
  * [83] M. van Veghel, Pursuing forbidden beauty : Search for the lepton-flavour violating decays ${{B}^{0}}\rightarrow e^{\pm}\mu^{\mp}$ and ${{B}^{0}_{s}}\rightarrow e^{\pm}\mu^{\mp}$ and study of electron-reconstruction performance at LHCb, PhD thesis, Groningen U., 2020, CERN-THESIS-2020-060
  * [84] LHCb collaboration, R. Aaij et al., _Measurement of $f_{s}/f_{u}$ variation with proton-proton collision energy and $B$-meson kinematics_, Phys. Rev. Lett. 124 (2020) 122002, arXiv:1910.09934
  * [85] LHCb collaboration, R. Aaij et al., _Measurement of $b$ hadron production fractions in 7 TeV $p$ $p$ collisions_, Phys. Rev. D85 (2012) 032008, arXiv:1111.2357
  * [86] LHCb collaboration, R. Aaij et al., _Measurement of $b$-hadron fractions in 13 TeV $p$ $p$ collisions_, Phys. Rev. D100 (2019) 031102(R), arXiv:1902.06794
  * [87] LHCb collaboration, R. Aaij et al., _Measurement of the fragmentation fraction ratio $f_{s}/f_{d}$ and its dependence on $B$ meson kinematics_, JHEP 04 (2013) 001, arXiv:1301.5286
  * [88] LHCb collaboration, R. Aaij et al., _Measurement of the branching fraction of the ${{B}^{0}}\rightarrow{{D}^{+}_{s}}{{\pi}^{-}}$ decay_, Eur. Phys. J. C81 (2021) 314, arXiv:2010.11986
  * [89] LHCb collaboration, R. Aaij et al., _Measurement of the track reconstruction efficiency at LHCb_ , JINST 10 (2015) P02007, arXiv:1408.1251
  * [90] R. Aaij et al., _Selection and processing of calibration samples to measure the particle identification performance of the LHCb experiment in Run 2_ , Eur. Phys. J. Tech. Instr. 6 (2019) 1, arXiv:1803.00824
  * [91] LHCb collaboration, R. Aaij et al., _Measurement of the ${B}_{c}^{-}$ production fraction and asymmetry in 7 and 13 TeV $pp$ collisions_, Phys. Rev. D100 (2019) 112006, arXiv:1910.13404
  * [92] W.-F. Wang and Z.-J. Xiao, _The semileptonic decays $B/B_{s}\rightarrow(\pi,K)(\ell^{+}\ell^{-},\ell\nu,\nu\bar{\nu})$ in the perturbative QCD approach beyond the leading-order_, Phys. Rev. D86 (2012) 114025, arXiv:1207.0265
  * [93] LHCb collaboration, R. Aaij et al., _Determination of the quark coupling strength $|{V_{{u}{b}}^{\phantom{\ast}}}|$ using baryonic decays_, Nature Physics 11 (2015) 743, arXiv:1504.01568
  * [94] LHCb collaboration, R. Aaij et al., _First observation of the decay $B^{0}_{s}\rightarrow K^{-}\mu^{+}\nu_{\mu}$ and measurement of $|V_{ub}|/|V_{cb}|$_, Phys. Rev. Lett. 126 (2021) 081804, arXiv:2012.05143
  * [95] K. S. Cranmer, _Kernel estimation in high-energy physics_ , Comput. Phys. Commun. 136 (2001) 198, arXiv:hep-ex/0011057
  * [96] S. S. Wilks, _The large-sample distribution of the likelihood ratio for testing composite hypotheses_ , Ann. Math. Stat. 9 (1938) 60
  * [97] A. L. Read, _Presentation of search results: The CL S technique_, J. Phys. G28 (2002) 2693
  * [98] G. Cowan, K. Cranmer, E. Gross, and O. Vitells, _Asymptotic formulae for likelihood-based tests of new physics_ , Eur. Phys. J. C71 (2011) 1554, arXiv:1007.1727, [Erratum: Eur. Phys. J. C73 (2013) 2501]
  * [99] M. Kenzie et al., _GammaCombo: A statistical analysis framework for combining measurements, fitting datasets and producing confidence intervals_ ,  doi: 10.5281/zenodo.3371421
  * [100] LHCb collaboration, R. Aaij et al., _Measurement of the CKM angle $\gamma$ from a combination of LHCb results_, JHEP 12 (2016) 087, arXiv:1611.03076
  * [101] S. Bodhisattva, M. Walker, and M. Woodroofe, _On the Unified Method with Nuisance Parameters_ , Statist. Sinica 19 (2009) 301, http://www3.stat.sinica.edu.tw/statistica/oldpdf/A19n116.pdf
  * [102] M. Pivk and F. R. Le Diberder, _sPlot: A statistical tool to unfold data distributions_ , Nucl. Instrum. Meth. A555 (2005) 356, arXiv:physics/0402083
  * [103] LHCb collaboration, R. Aaij et al., _Measurement of ${B}^{0}$, ${B}^{0}_{s}$, ${{B}^{+}}$ and ${\mathchar 28931\relax}^{0}_{b}$ production asymmetries in 7 and 8 TeV proton-proton collisions_, Phys. Lett. B774 (2017) 139, arXiv:1703.08464
  * [104] LHCb collaboration, R. Aaij et al., _Effective lifetime measurements in the ${{B}^{0}_{s}}\\!\rightarrow{{K}^{+}}{{K}^{-}}$, ${{B}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}$ and ${{B}^{0}_{s}}\\!\rightarrow{{\pi}^{+}}{{K}^{-}}$ decays_, Phys. Lett. B736 (2014) 446, arXiv:1406.7204

LHCb collaboration

R. Aaij32, C. Abellán Beteta50, T. Ackernley60, B. Adeva46, M. Adinolfi54, H.
Afsharnia9, C.A. Aidala86, S. Aiola25, Z. Ajaltouni9, S. Akar65, J.
Albrecht15, F. Alessio48, M. Alexander59, A. Alfonso Albero45, Z. Aliouche62,
G. Alkhazov38, P. Alvarez Cartelle55, S. Amato2, Y. Amhis11, L. An48, L.
Anderlini22, A. Andreianov38, M. Andreotti21, F. Archilli17, A. Artamonov44,
M. Artuso68, K. Arzymatov42, E. Aslanides10, M. Atzeni50, B. Audurier12, S.
Bachmann17, M. Bachmayer49, J.J. Back56, P. Baladron Rodriguez46, V.
Balagura12, W. Baldini21, J. Baptista Leite1, R.J. Barlow62, S. Barsuk11, W.
Barter61, M. Bartolini24,h, F. Baryshnikov83, J.M. Basels14, G. Bassi29, B.
Batsukh68, A. Battig15, A. Bay49, M. Becker15, F. Bedeschi29, I. Bediaga1, A.
Beiter68, V. Belavin42, S. Belin27, V. Bellee49, K. Belous44, I. Belov40, I.
Belyaev41, G. Bencivenni23, E. Ben-Haim13, A. Berezhnoy40, R. Bernet50, D.
Berninghoff17, H.C. Bernstein68, C. Bertella48, A. Bertolin28, C.
Betancourt50, F. Betti48, Ia. Bezshyiko50, S. Bhasin54, J. Bhom35, L. Bian73,
M.S. Bieker15, S. Bifani53, P. Billoir13, M. Birch61, F.C.R. Bishop55, A.
Bitadze62, A. Bizzeti22,k, M. Bjørn63, M.P. Blago48, T. Blake56, F. Blanc49,
S. Blusk68, D. Bobulska59, J.A. Boelhauve15, O. Boente Garcia46, T.
Boettcher65, A. Boldyrev82, A. Bondar43, N. Bondar38,48, S. Borghi62, M.
Borisyak42, M. Borsato17, J.T. Borsuk35, S.A. Bouchiba49, T.J.V. Bowcock60, A.
Boyer48, C. Bozzi21, M.J. Bradley61, S. Braun66, A. Brea Rodriguez46, M.
Brodski48, J. Brodzicka35, A. Brossa Gonzalo56, D. Brundu27,48, A. Buonaura50,
C. Burr48, A. Bursche72, A. Butkevich39, J.S. Butter32, J. Buytaert48, W.
Byczynski48, S. Cadeddu27, H. Cai73, R. Calabrese21,f, L. Calefice15,13, L.
Calero Diaz23, S. Cali23, R. Calladine53, M. Calvi26,j, M. Calvo Gomez85, P.
Camargo Magalhaes54, A. Camboni45,85, P. Campana23, A.F. Campoverde Quezada6,
S. Capelli26,j, L. Capriotti20,d, A. Carbone20,d, G. Carboni31, R.
Cardinale24,h, A. Cardini27, I. Carli4, P. Carniti26,j, L. Carus14, K.
Carvalho Akiba32, A. Casais Vidal46, G. Casse60, M. Cattaneo48, G.
Cavallero48, S. Celani49, J. Cerasoli10, A.J. Chadwick60, M.G. Chapman54, M.
Charles13, Ph. Charpentier48, G. Chatzikonstantinidis53, C.A. Chavez
Barajas60, M. Chefdeville8, C. Chen3, S. Chen4, A. Chernov35, V. Chobanova46,
S. Cholak49, M. Chrzaszcz35, A. Chubykin38, V. Chulikov38, P. Ciambrone23,
M.F. Cicala56, X. Cid Vidal46, G. Ciezarek48, P.E.L. Clarke58, M. Clemencic48,
H.V. Cliff55, J. Closier48, J.L. Cobbledick62, V. Coco48, J.A.B. Coelho11, J.
Cogan10, E. Cogneras9, L. Cojocariu37, P. Collins48, T. Colombo48, L.
Congedo19,c, A. Contu27, N. Cooke53, G. Coombs59, G. Corti48, C.M. Costa
Sobral56, B. Couturier48, D.C. Craik64, J. Crkovská67, M. Cruz Torres1, R.
Currie58, C.L. Da Silva67, S. Dadabaev83, E. Dall’Occo15, J. Dalseno46, C.
D’Ambrosio48, A. Danilina41, P. d’Argent48, A. Davis62, O. De Aguiar
Francisco62, K. De Bruyn79, S. De Capua62, M. De Cian49, J.M. De Miranda1, L.
De Paula2, M. De Serio19,c, D. De Simone50, P. De Simone23, F. De Vellis15,
J.A. de Vries80, C.T. Dean67, D. Decamp8, L. Del Buono13, B. Delaney55, H.-P.
Dembinski15, A. Dendek34, V. Denysenko50, D. Derkach82, O. Deschamps9, F.
Desse11, F. Dettori27,e, B. Dey77, A. Di Cicco23, P. Di Nezza23, S. Didenko83,
L. Dieste Maronas46, H. Dijkstra48, V. Dobishuk52, A.M. Donohoe18, F.
Dordei27, A.C. dos Reis1, L. Douglas59, A. Dovbnya51, A.G. Downes8, K.
Dreimanis60, M.W. Dudek35, L. Dufour48, V. Duk78, P. Durante48, J.M. Durham67,
D. Dutta62, A. Dziurda35, A. Dzyuba38, S. Easo57, U. Egede69, V. Egorychev41,
S. Eidelman43,v, S. Eisenhardt58, S. Ek-In49, L. Eklund59,w, S. Ely68, A.
Ene37, E. Epple67, S. Escher14, J. Eschle50, S. Esen13, T. Evans48, A.
Falabella20, J. Fan3, Y. Fan6, B. Fang73, S. Farry60, D. Fazzini26,j, M.
Féo48, A. Fernandez Prieto46, A.D. Fernez66, F. Ferrari20,d, L. Ferreira
Lopes49, F. Ferreira Rodrigues2, S. Ferreres Sole32, M. Ferrillo50, M. Ferro-
Luzzi48, S. Filippov39, R.A. Fini19, M. Fiorini21,f, M. Firlej34, K.M.
Fischer63, D.S. Fitzgerald86, C. Fitzpatrick62, T. Fiutowski34, F. Fleuret12,
M. Fontana13, F. Fontanelli24,h, R. Forty48, V. Franco Lima60, M. Franco
Sevilla66, M. Frank48, E. Franzoso21, G. Frau17, C. Frei48, D.A. Friday59, J.
Fu25, Q. Fuehring15, W. Funk48, E. Gabriel32, T. Gaintseva42, A. Gallas
Torreira46, D. Galli20,d, S. Gambetta58,48, Y. Gan3, M. Gandelman2, P.
Gandini25, Y. Gao5, M. Garau27, L.M. Garcia Martin56, P. Garcia Moreno45, J.
García Pardiñas26,j, B. Garcia Plana46, F.A. Garcia Rosales12, L. Garrido45,
C. Gaspar48, R.E. Geertsema32, D. Gerick17, L.L. Gerken15, E. Gersabeck62, M.
Gersabeck62, T. Gershon56, D. Gerstel10, Ph. Ghez8, V. Gibson55, H.K.
Giemza36, M. Giovannetti23,p, A. Gioventù46, P. Gironella Gironell45, L.
Giubega37, C. Giugliano21,f,48, K. Gizdov58, E.L. Gkougkousis48, V.V.
Gligorov13, C. Göbel70, E. Golobardes85, D. Golubkov41, A. Golutvin61,83, A.
Gomes1,a, S. Gomez Fernandez45, F. Goncalves Abrantes63, M. Goncerz35, G.
Gong3, P. Gorbounov41, I.V. Gorelov40, C. Gotti26, E. Govorkova48, J.P.
Grabowski17, T. Grammatico13, L.A. Granado Cardoso48, E. Graugés45, E.
Graverini49, G. Graziani22, A. Grecu37, L.M. Greeven32, P. Griffith21,f, L.
Grillo62, S. Gromov83, B.R. Gruberg Cazon63, C. Gu3, M. Guarise21, P. A.
Günther17, E. Gushchin39, A. Guth14, Y. Guz44, T. Gys48, T. Hadavizadeh69, G.
Haefeli49, C. Haen48, J. Haimberger48, T. Halewood-leagas60, P.M. Hamilton66,
J.P. Hammerich60, Q. Han7, X. Han17, T.H. Hancock63, S. Hansmann-Menzemer17,
N. Harnew63, T. Harrison60, C. Hasse48, M. Hatch48, J. He6,b, M. Hecker61, K.
Heijhoff32, K. Heinicke15, A.M. Hennequin48, K. Hennessy60, L. Henry48, J.
Heuel14, A. Hicheur2, D. Hill49, M. Hilton62, S.E. Hollitt15, J. Hu17, J.
Hu72, W. Hu7, X. Hu3, W. Huang6, X. Huang73, W. Hulsbergen32, R.J. Hunter56,
M. Hushchyn82, D. Hutchcroft60, D. Hynds32, P. Ibis15, M. Idzik34, D. Ilin38,
P. Ilten65, A. Inglessi38, A. Ishteev83, K. Ivshin38, R. Jacobsson48, S.
Jakobsen48, E. Jans32, B.K. Jashal47, A. Jawahery66, V. Jevtic15, F. Jiang3,
M. John63, D. Johnson48, C.R. Jones55, T.P. Jones56, B. Jost48, N. Jurik48, S.
Kandybei51, Y. Kang3, M. Karacson48, M. Karpov82, F. Keizer48, M. Kenzie56, T.
Ketel33, B. Khanji15, A. Kharisova84, S. Kholodenko44, T. Kirn14, V.S.
Kirsebom49, O. Kitouni64, S. Klaver32, K. Klimaszewski36, S. Koliiev52, A.
Kondybayeva83, A. Konoplyannikov41, P. Kopciewicz34, R. Kopecna17, P.
Koppenburg32, M. Korolev40, I. Kostiuk32,52, O. Kot52, S. Kotriakhova21,38, P.
Kravchenko38, L. Kravchuk39, R.D. Krawczyk48, M. Kreps56, F. Kress61, S.
Kretzschmar14, P. Krokovny43,v, W. Krupa34, W. Krzemien36, W. Kucewicz35,t, M.
Kucharczyk35, V. Kudryavtsev43,v, H.S. Kuindersma32,33, G.J. Kunde67, T.
Kvaratskheliya41, D. Lacarrere48, G. Lafferty62, A. Lai27, A. Lampis27, D.
Lancierini50, J.J. Lane62, R. Lane54, G. Lanfranchi23,48, C. Langenbruch14, J.
Langer15, O. Lantwin50, T. Latham56, F. Lazzari29,q, R. Le Gac10, S.H. Lee86,
R. Lefèvre9, A. Leflat40, S. Legotin83, O. Leroy10, T. Lesiak35, B.
Leverington17, H. Li72, L. Li63, P. Li17, S. Li7, Y. Li4, Y. Li4, Z. Li68, X.
Liang68, T. Lin61, R. Lindner48, V. Lisovskyi15, R. Litvinov27, G. Liu72, H.
Liu6, S. Liu4, A. Loi27, J. Lomba Castro46, I. Longstaff59, J.H. Lopes2, G.H.
Lovell55, Y. Lu4, D. Lucchesi28,l, S. Luchuk39, M. Lucio Martinez32, V.
Lukashenko32,52, Y. Luo3, A. Lupato62, E. Luppi21,f, O. Lupton56, A.
Lusiani29,m, X. Lyu6, L. Ma4, R. Ma6, S. Maccolini20,d, F. Machefert11, F.
Maciuc37, V. Macko49, P. Mackowiak15, S. Maddrell-Mander54, O. Madejczyk34,
L.R. Madhan Mohan54, O. Maev38, A. Maevskiy82, D. Maisuzenko38, M.W.
Majewski34, J.J. Malczewski35, S. Malde63, B. Malecki48, A. Malinin81, T.
Maltsev43,v, H. Malygina17, G. Manca27,e, G. Mancinelli10, D. Manuzzi20,d, D.
Marangotto25,i, J. Maratas9,s, J.F. Marchand8, U. Marconi20, S. Mariani22,g,
C. Marin Benito48, M. Marinangeli49, J. Marks17, A.M. Marshall54, P.J.
Marshall60, G. Martellotti30, L. Martinazzoli48,j, M. Martinelli26,j, D.
Martinez Santos46, F. Martinez Vidal47, A. Massafferri1, M. Materok14, R.
Matev48, A. Mathad50, Z. Mathe48, V. Matiunin41, C. Matteuzzi26, K.R.
Mattioli86, A. Mauri32, E. Maurice12, J. Mauricio45, M. Mazurek48, M.
McCann61, L. Mcconnell18, T.H. Mcgrath62, A. McNab62, R. McNulty18, J.V.
Mead60, B. Meadows65, G. Meier15, N. Meinert76, D. Melnychuk36, S. Meloni26,j,
M. Merk32,80, A. Merli25, L. Meyer Garcia2, M. Mikhasenko48, D.A. Milanes74,
E. Millard56, M. Milovanovic48, M.-N. Minard8, A. Minotti21, L. Minzoni21,f,
S.E. Mitchell58, B. Mitreska62, D.S. Mitzel48, A. Mödden 15, R.A. Mohammed63,
R.D. Moise61, T. Mombächer46, I.A. Monroy74, S. Monteil9, M. Morandin28, G.
Morello23, M.J. Morello29,m, J. Moron34, A.B. Morris75, A.G. Morris56, R.
Mountain68, H. Mu3, F. Muheim58,48, M. Mulder48, D. Müller48, K. Müller50,
C.H. Murphy63, D. Murray62, P. Muzzetto27,48, P. Naik54, T. Nakada49, R.
Nandakumar57, T. Nanut49, I. Nasteva2, M. Needham58, I. Neri21, N. Neri25,i,
S. Neubert75, N. Neufeld48, R. Newcombe61, T.D. Nguyen49, C. Nguyen-Mau49,x,
E.M. Niel11, S. Nieswand14, N. Nikitin40, N.S. Nolte64, C. Normand8, C.
Nunez86, A. Oblakowska-Mucha34, V. Obraztsov44, D.P. O’Hanlon54, R.
Oldeman27,e, M.E. Olivares68, C.J.G. Onderwater79, A. Ossowska35, J.M. Otalora
Goicochea2, T. Ovsiannikova41, P. Owen50, A. Oyanguren47, B. Pagare56, P.R.
Pais48, T. Pajero63, A. Palano19, M. Palutan23, Y. Pan62, G. Panshin84, A.
Papanestis57, M. Pappagallo19,c, L.L. Pappalardo21,f, C. Pappenheimer65, W.
Parker66, C. Parkes62, C.J. Parkinson46, B. Passalacqua21, G. Passaleva22, A.
Pastore19, M. Patel61, C. Patrignani20,d, C.J. Pawley80, A. Pearce48, A.
Pellegrino32, M. Pepe Altarelli48, S. Perazzini20, D. Pereima41, P. Perret9,
M. Petric59,48, K. Petridis54, A. Petrolini24,h, A. Petrov81, S. Petrucci58,
M. Petruzzo25, T.T.H. Pham68, A. Philippov42, L. Pica29,m, M. Piccini78, B.
Pietrzyk8, G. Pietrzyk49, M. Pili63, D. Pinci30, F. Pisani48, Resmi P.K10, V.
Placinta37, J. Plews53, M. Plo Casasus46, F. Polci13, M. Poli Lener23, M.
Poliakova68, A. Poluektov10, N. Polukhina83,u, I. Polyakov68, E. Polycarpo2,
G.J. Pomery54, S. Ponce48, D. Popov6,48, S. Popov42, S. Poslavskii44, K.
Prasanth35, L. Promberger48, C. Prouve46, V. Pugatch52, H. Pullen63, G.
Punzi29,n, H. Qi3, W. Qian6, J. Qin6, N. Qin3, R. Quagliani13, B. Quintana8,
N.V. Raab18, R.I. Rabadan Trejo10, B. Rachwal34, J.H. Rademacker54, M. Rama29,
M. Ramos Pernas56, M.S. Rangel2, F. Ratnikov42,82, G. Raven33, M. Reboud8, F.
Redi49, F. Reiss62, C. Remon Alepuz47, Z. Ren3, V. Renaudin63, R. Ribatti29,
S. Ricciardi57, K. Rinnert60, P. Robbe11, G. Robertson58, A.B. Rodrigues49, E.
Rodrigues60, J.A. Rodriguez Lopez74, A. Rollings63, P. Roloff48, V.
Romanovskiy44, M. Romero Lamas46, A. Romero Vidal46, J.D. Roth86, M.
Rotondo23, M.S. Rudolph68, T. Ruf48, J. Ruiz Vidal47, A. Ryzhikov82, J.
Ryzka34, J.J. Saborido Silva46, N. Sagidova38, N. Sahoo56, B. Saitta27,e, M.
Salomoni48, C. Sanchez Gras32, R. Santacesaria30, C. Santamarina Rios46, M.
Santimaria23, E. Santovetti31,p, D. Saranin83, G. Sarpis14, M. Sarpis75, A.
Sarti30, C. Satriano30,o, A. Satta31, M. Saur15, D. Savrina41,40, H. Sazak9,
L.G. Scantlebury Smead63, A. Scarabotto13, S. Schael14, M. Schellenberg15, M.
Schiller59, H. Schindler48, M. Schmelling16, B. Schmidt48, O. Schneider49, A.
Schopper48, M. Schubiger32, S. Schulte49, M.H. Schune11, R. Schwemmer48, B.
Sciascia23, S. Sellam46, A. Semennikov41, M. Senghi Soares33, A. Sergi24,h, N.
Serra50, L. Sestini28, A. Seuthe15, P. Seyfert48, Y. Shang5, D.M. Shangase86,
M. Shapkin44, I. Shchemerov83, L. Shchutska49, T. Shears60, L. Shekhtman43,v,
Z. Shen5, V. Shevchenko81, E.B. Shields26,j, E. Shmanin83, J.D. Shupperd68,
B.G. Siddi21, R. Silva Coutinho50, G. Simi28, S. Simone19,c, N. Skidmore62, T.
Skwarnicki68, M.W. Slater53, I. Slazyk21,f, J.C. Smallwood63, J.G. Smeaton55,
A. Smetkina41, E. Smith50, M. Smith61, A. Snoch32, M. Soares20, L. Soares
Lavra9, M.D. Sokoloff65, F.J.P. Soler59, A. Solovev38, I. Solovyev38, F.L.
Souza De Almeida2, B. Souza De Paula2, B. Spaan15, E. Spadaro Norella25,i, P.
Spradlin59, F. Stagni48, M. Stahl65, S. Stahl48, P. Stefko49, O.
Steinkamp50,83, O. Stenyakin44, H. Stevens15, S. Stone68, M.E. Stramaglia49,
M. Straticiuc37, D. Strekalina83, F. Suljik63, J. Sun27, L. Sun73, Y. Sun66,
P. Svihra62, P.N. Swallow53, K. Swientek34, A. Szabelski36, T. Szumlak34, M.
Szymanski48, S. Taneja62, A. Terentev83, F. Teubert48, E. Thomas48, K.A.
Thomson60, V. Tisserand9, S. T’Jampens8, M. Tobin4, L. Tomassetti21,f, D.
Torres Machado1, D.Y. Tou13, M.T. Tran49, E. Trifonova83, C. Trippl49, G.
Tuci29,n, A. Tully49, N. Tuning32,48, A. Ukleja36, D.J. Unverzagt17, E.
Ursov83, A. Usachov32, A. Ustyuzhanin42,82, U. Uwer17, A. Vagner84, V.
Vagnoni20, A. Valassi48, G. Valenti20, N. Valls Canudas85, M. van Beuzekom32,
M. Van Dijk49, E. van Herwijnen83, C.B. Van Hulse18, M. van Veghel79, R.
Vazquez Gomez45, P. Vazquez Regueiro46, C. Vázquez Sierra48, S. Vecchi21, J.J.
Velthuis54, M. Veltri22,r, A. Venkateswaran68, M. Veronesi32, M. Vesterinen56,
D. Vieira65, M. Vieites Diaz49, H. Viemann76, X. Vilasis-Cardona85, E. Vilella
Figueras60, A. Villa20, P. Vincent13, D. Vom Bruch10, A. Vorobyev38, V.
Vorobyev43,v, N. Voropaev38, K. Vos80, R. Waldi17, J. Walsh29, C. Wang17, J.
Wang5, J. Wang4, J. Wang3, J. Wang73, M. Wang3, R. Wang54, Y. Wang7, Z.
Wang50, Z. Wang3, H.M. Wark60, N.K. Watson53, S.G. Weber13, D. Websdale61, C.
Weisser64, B.D.C. Westhenry54, D.J. White62, M. Whitehead54, D. Wiedner15, G.
Wilkinson63, M. Wilkinson68, I. Williams55, M. Williams64, M.R.J. Williams58,
F.F. Wilson57, W. Wislicki36, M. Witek35, L. Witola17, G. Wormser11, S.A.
Wotton55, H. Wu68, K. Wyllie48, Z. Xiang6, D. Xiao7, Y. Xie7, A. Xu5, J. Xu6,
L. Xu3, M. Xu7, Q. Xu6, Z. Xu5, Z. Xu6, D. Yang3, S. Yang6, Y. Yang6, Z.
Yang3, Z. Yang66, Y. Yao68, L.E. Yeomans60, H. Yin7, J. Yu71, X. Yuan68, O.
Yushchenko44, E. Zaffaroni49, M. Zavertyaev16,u, M. Zdybal35, O. Zenaiev48, M.
Zeng3, D. Zhang7, L. Zhang3, S. Zhang5, Y. Zhang5, Y. Zhang63, A. Zharkova83,
A. Zhelezov17, Y. Zheng6, X. Zhou6, Y. Zhou6, X. Zhu3, Z. Zhu6, V.
Zhukov14,40, J.B. Zonneveld58, Q. Zou4, S. Zucchelli20,d, D. Zuliani28, G.
Zunica62.

1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4Institute Of High Energy Physics (IHEP), Beijing, China
5School of Physics State Key Laboratory of Nuclear Physics and Technology,
Peking University, Beijing, China
6University of Chinese Academy of Sciences, Beijing, China
7Institute of Particle Physics, Central China Normal University, Wuhan, Hubei,
China
8Univ. Savoie Mont Blanc, CNRS, IN2P3-LAPP, Annecy, France
9Université Clermont Auvergne, CNRS/IN2P3, LPC, Clermont-Ferrand, France
10Aix Marseille Univ, CNRS/IN2P3, CPPM, Marseille, France
11Université Paris-Saclay, CNRS/IN2P3, IJCLab, Orsay, France
12Laboratoire Leprince-Ringuet, CNRS/IN2P3, Ecole Polytechnique, Institut
Polytechnique de Paris, Palaiseau, France
13LPNHE, Sorbonne Université, Paris Diderot Sorbonne Paris Cité, CNRS/IN2P3,
Paris, France
14I. Physikalisches Institut, RWTH Aachen University, Aachen, Germany
15Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
16Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
17Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
18School of Physics, University College Dublin, Dublin, Ireland
19INFN Sezione di Bari, Bari, Italy
20INFN Sezione di Bologna, Bologna, Italy
21INFN Sezione di Ferrara, Ferrara, Italy
22INFN Sezione di Firenze, Firenze, Italy
23INFN Laboratori Nazionali di Frascati, Frascati, Italy
24INFN Sezione di Genova, Genova, Italy
25INFN Sezione di Milano, Milano, Italy
26INFN Sezione di Milano-Bicocca, Milano, Italy
27INFN Sezione di Cagliari, Monserrato, Italy
28Universita degli Studi di Padova, Universita e INFN, Padova, Padova, Italy
29INFN Sezione di Pisa, Pisa, Italy
30INFN Sezione di Roma La Sapienza, Roma, Italy
31INFN Sezione di Roma Tor Vergata, Roma, Italy
32Nikhef National Institute for Subatomic Physics, Amsterdam, Netherlands
33Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, Netherlands
34AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
35Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
36National Center for Nuclear Research (NCBJ), Warsaw, Poland
37Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
38Petersburg Nuclear Physics Institute NRC Kurchatov Institute (PNPI NRC KI),
Gatchina, Russia
39Institute for Nuclear Research of the Russian Academy of Sciences (INR RAS),
Moscow, Russia
40Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
41Institute of Theoretical and Experimental Physics NRC Kurchatov Institute
(ITEP NRC KI), Moscow, Russia
42Yandex School of Data Analysis, Moscow, Russia
43Budker Institute of Nuclear Physics (SB RAS), Novosibirsk, Russia
44Institute for High Energy Physics NRC Kurchatov Institute (IHEP NRC KI),
Protvino, Russia, Protvino, Russia
45ICCUB, Universitat de Barcelona, Barcelona, Spain
46Instituto Galego de Física de Altas Enerxías (IGFAE), Universidade de
Santiago de Compostela, Santiago de Compostela, Spain
47Instituto de Fisica Corpuscular, Centro Mixto Universidad de Valencia -
CSIC, Valencia, Spain
48European Organization for Nuclear Research (CERN), Geneva, Switzerland
49Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL),
Lausanne, Switzerland
50Physik-Institut, Universität Zürich, Zürich, Switzerland
51NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
52Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
53University of Birmingham, Birmingham, United Kingdom
54H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
55Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
56Department of Physics, University of Warwick, Coventry, United Kingdom
57STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
58School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
59School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
60Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
61Imperial College London, London, United Kingdom
62Department of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
63Department of Physics, University of Oxford, Oxford, United Kingdom
64Massachusetts Institute of Technology, Cambridge, MA, United States
65University of Cincinnati, Cincinnati, OH, United States
66University of Maryland, College Park, MD, United States
67Los Alamos National Laboratory (LANL), Los Alamos, United States
68Syracuse University, Syracuse, NY, United States
69School of Physics and Astronomy, Monash University, Melbourne, Australia,
associated to 56
70Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
71Physics and Micro Electronic College, Hunan University, Changsha City,
China, associated to 7
72Guangdong Provincial Key Laboratory of Nuclear Science, Guangdong-Hong Kong
Joint Laboratory of Quantum Matter, Institute of Quantum Matter, South China
Normal University, Guangzhou, China, associated to 3
73School of Physics and Technology, Wuhan University, Wuhan, China, associated
to 3
74Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia,
associated to 13
75Universität Bonn - Helmholtz-Institut für Strahlen und Kernphysik, Bonn,
Germany, associated to 17
76Institut für Physik, Universität Rostock, Rostock, Germany, associated to 17
77Eotvos Lorand University, Budapest, Hungary, associated to 48
78INFN Sezione di Perugia, Perugia, Italy, associated to 21
79Van Swinderen Institute, University of Groningen, Groningen, Netherlands,
associated to 32
80Universiteit Maastricht, Maastricht, Netherlands, associated to 32
81National Research Centre Kurchatov Institute, Moscow, Russia, associated to
41
82National Research University Higher School of Economics, Moscow, Russia,
associated to 42
83National University of Science and Technology “MISIS”, Moscow, Russia,
associated to 41
84National Research Tomsk Polytechnic University, Tomsk, Russia, associated to
41
85DS4DS, La Salle, Universitat Ramon Llull, Barcelona, Spain, associated to 45
86University of Michigan, Ann Arbor, United States, associated to 68

aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil
bHangzhou Institute for Advanced Study, UCAS, Hangzhou, China
cUniversità di Bari, Bari, Italy
dUniversità di Bologna, Bologna, Italy
eUniversità di Cagliari, Cagliari, Italy
fUniversità di Ferrara, Ferrara, Italy
gUniversità di Firenze, Firenze, Italy
hUniversità di Genova, Genova, Italy
iUniversità degli Studi di Milano, Milano, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Modena e Reggio Emilia, Modena, Italy
lUniversità di Padova, Padova, Italy
mScuola Normale Superiore, Pisa, Italy
nUniversità di Pisa, Pisa, Italy
oUniversità della Basilicata, Potenza, Italy
pUniversità di Roma Tor Vergata, Roma, Italy
qUniversità di Siena, Siena, Italy
rUniversità di Urbino, Urbino, Italy
sMSU - Iligan Institute of Technology (MSU-IIT), Iligan, Philippines
tAGH - University of Science and Technology, Faculty of Computer Science,
Electronics and Telecommunications, Kraków, Poland
uP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
vNovosibirsk State University, Novosibirsk, Russia
wDepartment of Physics and Astronomy, Uppsala University, Uppsala, Sweden
xHanoi University of Science, Hanoi, Vietnam