# Lefschetz fixed-object and fixed-morphism theorems for acyclic categories

Samuel Castelo Mourelle  Samuel Castelo Mourelle
Departamento de Matemáticas, Universidade de Santiago de Compostela,
15782-SPAIN<EMAIL_ADDRESS>,  Enrique Macías-Virgós  Enrique
Macías-Virgós
CiTMAga, Departamento de Matemáticas, Universidade de Santiago de Compostela,
15782-SPAIN<EMAIL_ADDRESS>and  David Mosquera-Lois  David Mosquera-
Lois
Departamento de Matemáticas, Universidade de Vigo, SPAIN
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract.

We introduce two novel complementary notions of the Lefschetz number for a
functor from a finite acyclic category to itself and we prove a Lefschetz
fixed-object theorem and a Lefschetz fixed-morphism theorem.

###### Key words and phrases:

acyclic category, Lefschetz number, Lefschetz fixed-point theorem

###### 2020 Mathematics Subject Classification:

Primary: 55M20 Secondary: 55U10

The authors were partially supported by Xunta de Galicia ED431C 2019/10 with
FEDER funds and by MINECO-Spain research project PID2020-MTM-114474GB-100.

## Introduction

Let $X$ be a finite simplicial structure, possibly more general than a
simplicial complex. The Lefschetz number of a simplicial map $f\colon X\to X$
is defined as

(1) $\mathcal{L}(f)=\sum_{i\geq 0}(-1)^{i}\mathrm{trace}(H_{i}(f)),$

where $H_{i}(f):H_{i}(X)\to H_{i}(X)$ denotes the induced morphism in homology
with rational coefficients.

The Lefschetz number stands as a crucial invariant within Algebraic Topology.
Beyond its role as an extension of the Euler-Poincaré characteristic,
$\chi(X)=L(\textup{id}_{X})$, the Lefschetz number serves as a connecting link
between Topology and Dynamics. It ensures the fulfillment of fixed-point
theorems, establishing a significant bridge between these two mathematical
domains.

In 1979 Backlawski and Björner proved a Lefschetz fixed point theorem for
posets, which play an important role as objects of study in Combinatorial
Algebraic Topology:

######  Theorem 1 (Lefschetz fixed point theorem for posets [2, Theorem 1.1]).

Let $X$ be a finite poset and $f\colon X\rightarrow X$ an order-preserving
map. Then

$\mathcal{L}(f)=\chi(X^{f}).$

Moreover, if $\mathcal{L}(f)\neq 0$, then there exists $x\in X$ such that
$f(x)=x$, that is to say there exists at least one fixed point of $f$.

Nowadays, acyclic (or loop-free) categories are gaining much importance in
Combinatorial Algebraic Topology (see for example [1, 5, 6, 3, 4, 7]).

The purpose of this short note is to extend Theorem 1 to finite acyclic
categories. We define two complementary notions of the Lefschetz number for a
functor $F\colon C\rightarrow C$ from a finite acyclic category to itself and
we prove a Lefschetz fixed-object theorem (Theorem 2.2) and a Lefschetz fixed-
morphism theorem (Theorem 3.1) for such a functor.

##  1\. Background

We begin by introducing some notions regarding acyclic categories and fixing
the notation. For a detailed exposition we refer the reader to [5, 6].

A (finite) acyclic category or (finite) loop-free category is a (finite) small
category (the collection of arrows is a (finite) set) such that only the
identity morphisms have inverses and the unique morphism from an object to
itself is the identity. We will represent and denote the acyclic categories by
omitting the identity arrows. The acyclic categories along with the functors
between them form a category, which we call AC. All acyclic categories we work
with will be assumed to have a finite number of morphisms.

Posets are examples of acyclic categories. For a poset $(X,\leq)$, the objects
are the elements of the poset and there is a single morphism between two
objects $x$ and $y$ if and only if $x\leq y$. As a consequence, we can see the
category of posets and order-preserving maps between them, which we denote
$Poset$, as a subcategory of $AC$. Moreover, for an acyclic category $C$,
there is a partial order on the set of objects of $C$ given by $x\leq y$ if
and only if there is an arrow from $x$ to $y$. The resulting poset is denoted
by $R(C)$. Let $C$ and $D$ be acyclic categories and let $F\colon C\rightarrow
D$ be a functor. The map, $R(F)\colon R(C)\rightarrow R(D)$ defined by
$R(F)(x)=F(x)$ is an order-preserving map between posets. Therefore,
$R\colon\textup{ACat}\to\text{Posets}$ is a functor.

Let $C$ be an acyclic category. The nerve of $C$, denoted $\Delta(C)$, is a
regular trisp (polyhedral complex whose cells are simplices, also referred as
generalized simplicial complex or regular $\Delta-$complex depending on the
literature) defined as follows (see [5, Definition 10.4.] and [5, Definition
2.48]):

  * •

The set of vertices is the set of objects of $C$.

  * •

For $k\geq 1$, there is one $k$-simplex $\sigma$ for each chain of $k$ non-
identity composable morphisms
$x_{0}\overset{\alpha_{1}}{\rightarrow}x_{1}\overset{\alpha_{2}}{\rightarrow}\dots\overset{\alpha_{k}}{\rightarrow}x_{k}$.

Intuitively, regular trisps generalize simplicial complexes by allowing
several different simplices to share all its vertices. Let $C$ and $D$ be two
acyclic categories and let $F\colon C\rightarrow D$ be a functor. There is a
regular trisp map $\Delta(F)\colon\Delta(C)\rightarrow\Delta(D)$ given as
follows. For a composable morphism chain
$x_{0}\overset{\alpha_{1}}{\rightarrow}x_{1}\overset{\alpha_{2}}{\rightarrow}\dots\overset{\alpha_{k}}{\rightarrow}x_{k}$
in $\Delta(C)$, $\Delta(F)$ maps it to the composable morphism chain
$F(x_{0})\overset{F(\alpha_{1})}{\rightarrow}F(x_{1})\overset{F(\alpha_{2})}{\rightarrow}\dots\overset{F(\alpha_{k})}{\rightarrow}F(x_{k})$
in $\Delta(D)$. We use RT to denote the category of regular trisps.

We recall the face poset functor $\mathcal{F}\colon RT\to Posets$ for the
particular case of regular trisps and regular trisps maps coming from acyclic
categories by the functor $\Delta$ since they are the only ones we will
consider.

For a regular trisp $\Delta(C)$, $\mathcal{F}(\Delta(C))$ is the poset of
simplices of $\Delta(C)$ ordered by inclusion: $\beta\leq\tau$ if $\beta$ is a
face of $\tau$. For $\Delta(F)\colon\Delta(C)\rightarrow\Delta(D)$,
$\mathcal{F}(\Delta(F))\colon\mathcal{F}(\Delta(C))\rightarrow\mathcal{F}(\Delta(D))$
maps the $k$-simplex (seen as an element of $\mathcal{F}(\Delta(C))$)
$x_{0}\overset{\alpha_{1}}{\rightarrow}x_{1}\overset{\alpha_{2}}{\rightarrow}\dots\overset{\alpha_{k}}{\rightarrow}x_{k}$
in $\Delta(C)$, to
$F(x_{0})\overset{F(\alpha_{1})}{\rightarrow}F(x_{1})\overset{F(\alpha_{2})}{\rightarrow}\dots\overset{F(\alpha_{k})}{\rightarrow}F(x_{k})$
in $\mathcal{F}(\Delta(D))$.

We define the subdivision functor for acyclic categories $\textup{sd}\colon
ACat\to Posets$ as the composition $\textup{sd}=\mathcal{F}\circ\Delta$.

We summarize these functors we will use in this work in Diagram 2.

(2)
ACatPosetsRT$\scriptstyle{\mathrm{sd}}$$\scriptstyle{\Delta}$$\scriptstyle{i}$$\scriptstyle{\mathcal{F}}$

The homology of the acyclic category $C$, which we denote by $H(C)$ is just
the homology $H(\Delta(C))$ of its nerve. Moreover, the face poset functor
(consequently the subdivision functor) induces isomorphisms at the level of
homology.

We introduce some notation. For an acyclic category $C$, we denote by $O$ the
set of objects and by $M$ the set of morphisms. For a functor $F\colon
C\rightarrow C$, we consider the pair of maps between objects and between
arrows: $F_{O}\colon O(C)\rightarrow O(C)$ and $F_{M}\colon M(C)\rightarrow
M(C)$.

  * •

The fixed-object set of $F\colon C\rightarrow C$ is the subset

$O(C)^{F}=\\{x\in O(C)\colon F_{O}(x)=x\\}\subseteq O(C).$

  * •

The fixed-arrow set of $F\colon C\rightarrow C$ is the subset,

$M(C)^{F}=\\{\alpha\in M(C)\colon F_{M}(\alpha)=\alpha\\}\subseteq M(C).$

  * •

The fixed subcategory of $F\colon C\rightarrow C$ is the acyclic category

$C^{F}=(O(C)^{F},M(C)^{F}).$

##  2\. The Lefschetz fixed-object theorem

We introduce two novel complementary notions of Lefschetz number for a functor
$F\colon C\rightarrow C$ between finite acyclic categories and we prove a
Lefschetz fixed-object theorem.

######  Definition 2.1.

Let $C$ be a finite acyclic category and let $F\colon C\rightarrow C$ be a
functor. We define its Lefschetz number as
$\mathcal{L}(F)=\mathcal{L}(\Delta(F))$ and its R-Lefschetz number as
$\mathcal{L}_{R}(F)=\mathcal{L}(R(F))$.

######  Remark 1.

Observe that for a functor between acyclic categories $F\colon C\rightarrow
C$, $\mathcal{L}(F)=\mathcal{L}(\textup{sd}(F))$ and that for an order-
preserving map between posets $F\colon P\rightarrow P$,
$\mathcal{L}(F)=\mathcal{L}_{R}(F)$.

######  Theorem 2.2 (Lefschetz fixed-object theorem).

Let $C$ be a finite acyclic category and $F\colon C\rightarrow C$ be a
functor. Then:

  1. (a)

$\mathcal{L}_{R}(F)=\chi(R(C)^{R(F)}).$

  2. (b)

$\mathcal{L}(F)=\chi(C^{F}).$

Moreover, if $\mathcal{L}_{R}(F)\neq 0$ or $\mathcal{L}(F)\neq 0$, then there
exists at least one object $x\in O(C)$ such that $F(x)=x$, i.e.
$O(C)^{F}\neq\emptyset$.

Theorem 2.2 depends on a crucial observation. Consider Figure 2.1. We
represent a functor $F\colon C\rightarrow C$ between acyclic categories and
the subdivision functor $\textup{sd}(F)\colon\textup{sd}(C)\to\textup{sd}(C)$.
We see that if $\textup{sd}(F)$ fixes $(\gamma,\beta)\in\textup{sd}(C)$, then
$\textup{sd}(F)$ fixes all elements below $(\gamma,\beta)$ in
$\textup{sd}(C)$.

Figure 2.1. A functor $F\colon C\rightarrow C$ between acyclic categories and
the subdivision functor $\textup{sd}(F)\colon\textup{sd}(C)\to\textup{sd}(C)$.

This is not particular to this concrete example but a general fact. Let us
state this as a lemma.

######  Lemma 2.3.

Let $C$ be a finite acyclic category and let $F\colon C\rightarrow C$ be a
functor. If $\textup{sd}(F)\colon\textup{sd}(C)\to\textup{sd}(C)$ fixes an
object $\sigma\in\textup{sd}(C)$ ($\textup{sd}(F)(\sigma)=\sigma$), then
$\textup{sd}(F)(\tau)=\tau$ for all $\tau\leq\sigma$.

###### Proof.

Consider $\sigma\in\textup{sd}(C)$,
$\sigma=x_{0}\overset{\alpha_{1}}{\rightarrow}x_{1}\overset{\alpha_{2}}{\rightarrow}\dots\overset{\alpha_{k}}{\rightarrow}x_{k}$.
If $\textup{sd}(F)\colon\textup{sd}(C)\to\textup{sd}(C)$ fixes
$\sigma\in\textup{sd}(C)$, then it fixes
$x_{1}\overset{\alpha_{2}}{\rightarrow}x_{2}\overset{\alpha_{2}}{\rightarrow}\dots\overset{\alpha_{k}}{\rightarrow}x_{k}$,
$x_{0}\overset{\alpha_{1}}{\rightarrow}x_{1}\dots\overset{\alpha_{k-1}}{\rightarrow}x_{k-1}$
and $x_{0}\overset{\alpha_{1}}{\rightarrow}x_{1}\dots
x_{j-1}\overset{\alpha_{j+1}\circ\alpha_{j}}{\longrightarrow}x_{j+1}\dots\overset{\alpha_{k}}{\rightarrow}x_{k}$
for any $1\leq j\leq k-1$. By applying this argument iteratively a finite
number of times it holds that for all $\tau\leq\sigma$,
$\textup{sd}(F)(\tau)=\tau$. ∎

###### Proof of Theorem 2.2.

First of all, observe that (a) is a direct consequence of Theorem 1. Let us
prove (b). It is enough to check the following chain of equalities:

$\mathcal{L}(F)\overset{(1)}{=}\mathcal{L}(\textup{sd}(F))\overset{(2)}{=}\chi(\textup{sd}(C)^{\textup{sd}(F)})\overset{(3)}{=}\chi(\textup{sd}(C^{F}))\overset{(4)}{=}\chi(C^{F}).$

Equality (1) follows from Remark 1. Equality (2) is consequence of Theorem 1.
Equality (4) follows from the homological invariance of Euler-Poincaré
characteristic. It remains to prove Equality (3). Suppose
$\sigma\in\textup{sd}(C)^{\textup{sd}(F)}$, where
$\sigma=x_{0}\overset{\alpha_{1}}{\rightarrow}x_{1}\overset{\alpha_{2}}{\rightarrow}\dots\overset{\alpha_{k}}{\rightarrow}x_{k}$.
By Lemma 2.3, $\sigma\in\textup{sd}(C^{F})$. Conversely, assume that
$\sigma\in\textup{sd}(C^{F})$ where
$\sigma=x_{0}\overset{\alpha_{1}}{\rightarrow}x_{1}\overset{\alpha_{2}}{\rightarrow}\dots\overset{\alpha_{k}}{\rightarrow}x_{k}$.
By construction,

$\displaystyle\textup{sd}(F)(x_{0}\overset{\alpha_{1}}{\rightarrow}x_{1}\overset{\alpha_{2}}{\rightarrow}\dots\overset{\alpha_{k}}{\rightarrow}x_{k})$
$\displaystyle=F(x_{0})\overset{F(\alpha_{1})}{\rightarrow}F(x_{1})\overset{F(\alpha_{2})}{\rightarrow}\dots\overset{F(\alpha_{k})}{\rightarrow}F(x_{k})$
$\displaystyle=x_{0}\overset{\alpha_{1}}{\rightarrow}x_{1}\overset{\alpha_{2}}{\rightarrow}\dots\overset{\alpha_{k}}{\rightarrow}x_{k}.$

Let us prove now the last part of the theorem. Observe that the elements of
$R(C)$ coincide with the objects of $C$ while $F$ and $R(F)$ fix the same
objects. Therefore, by Theorem 1, if $\mathcal{L}_{R}(F)\neq 0$, then there
exists at least one object $x\in O(C)$ such that $F(x)=x$. Assume now that
$\mathcal{L}(F)\neq 0$. Then, by Theorem 1 there exists
$\sigma\in\textup{sd}(C)^{\textup{sd}(F)}$,
$\sigma=x_{0}\overset{\alpha_{1}}{\rightarrow}x_{1}\overset{\alpha_{2}}{\rightarrow}\dots\overset{\alpha_{k}}{\rightarrow}x_{k}$,
such that $\textup{sd}(F)(\sigma)=\sigma$. By Lemma 2.3,
$\textup{sd}(F)(x_{0})=x_{0}$. Finally, $F(x_{0})=\textup{sd}(F)(x_{0})$ by
definition, which finishes the proof. ∎

We show that for a functor $F\colon C\rightarrow C$ between acyclic
categories, $\mathcal{L}(F)$ and $\mathcal{L}_{R}(F)$ provide complementary
information regarding the fixed objects of $F\colon C\rightarrow C$.

######  Example 2.4.

Consider the acyclic categories $C$ and $D$ depicted in Figure 2.2:

Figure 2.2. Categories $C$ and $D$.

We define functors $F\colon C\to C$ and $G\colon D\to D$ which are the
identity on each category. Therefore, they fix the respective categories (both
arrows and objects). We have $\mathcal{L}_{R}(F)=\chi(R(C))=2-1=1$,
$\mathcal{L}(F)=\chi(C)=2-2=0$, $\mathcal{L}_{R}(G)=\chi(R(D))=4-4=0$ and
$\mathcal{L}(G)=\chi(D)=4-5=-1$.

##  3\. Lefschetz fixed-morphism theorem

We begin by fixing some notation in order to state and prove the Lefschetz
fixed-morphism theorem. Let $P$ be a poset. A morphism is indecomposable if it
cannot be represented as a composition of two nonidentity morphisms. The poset
$P$ is graded if there is a function $\textup{deg}\colon P\to\mathbb{Z}$ such
that whenever $\alpha<\beta$ is an indecomposable morphism, we have
$\textup{deg}(\beta)=\textup{deg}(\alpha)+1$. Moreover, the degree of
$\alpha$, denoted, $\textup{deg}(\alpha)=i$, coincides with the maximum of the
lengths of the chains $\alpha_{0}<\alpha_{1}<\cdots<\alpha$ where $\alpha_{0}$
is a minimal element of $P$.

Let $P$ be a graded poset. We say that the element $\alpha$ belongs to its
i-th layer, denoted $\alpha\in P_{i}$, if $\textup{deg}(\alpha)=i$. We define
$P^{\hskip 2.84526pt\widehat{i}}$ as the deletion of the $i$-th layer from the
poset, that is, the subposet whose elements are $P\setminus P_{i}.$
Analogously, we define $P^{\hskip 2.84526pt\widehat{\leq i}}$ as the deletion
of all the $j$-th layers with $j\leq i$ from the poset $P$.

Let $F\colon P\rightarrow P$ be an order-preserving map. Then $F$ induces
order-preserving maps

$F^{\hskip 2.84526pt\widehat{i}}\colon P^{\hskip
2.84526pt\widehat{i}}\setminus F^{-1}(P_{i})\rightarrow P^{\hskip
2.84526pt\widehat{i}}\setminus F^{-1}(P_{i})$

and

$F^{\hskip 2.84526pt\widehat{\leq i}}\colon P^{\hskip 2.84526pt\widehat{\leq
i}}\setminus F^{-1}\left(\bigcup_{j\leq i}P_{j}\right)\rightarrow P^{\hskip
2.84526pt\widehat{\leq i}}\setminus F^{-1}\left(\bigcup_{j\leq i}P_{j}\right)$

given by $F^{\hskip 2.84526pt\widehat{i}}(\alpha)=F(\alpha)$ and $F^{\hskip
2.84526pt\widehat{\leq i}}(\alpha)=F(\alpha)$, respectively. Finally, let us
denote

${\textup{sd}}_{F}^{\hskip 2.84526pt\widehat{\leq i}}=\textup{sd}(C)^{\hskip
2.84526pt\widehat{\leq i}}\setminus(\textup{sd}({F)^{\hskip
2.84526pt\widehat{\leq i}}})^{-1}\left(\bigcup_{j\leq
i}\textup{sd}(C)_{j})\right)$

so there is an induced order-preserving map

$\textup{(sd}F)^{\hskip 2.84526pt\widehat{\leq
i}}\colon{\textup{sd}}_{F}^{\hskip 2.84526pt\widehat{\leq
i}}\to{\textup{sd}}_{F}^{\hskip 2.84526pt\widehat{\leq i}}.$

######  Theorem 3.1 (Lefschetz fixed-morphism theorem).

Let $C$ be a finite acyclic category and let $F\colon C\rightarrow C$ be a
functor. If $\mathcal{L}(\textup{(sd}F)^{\hskip 2.84526pt\widehat{\leq
i}})\neq 0$, then for all $1\leq k\leq i+1$ there exists at least one chain of
composable morphims
$x_{0}\overset{\alpha_{1}}{\rightarrow}x_{1}\overset{\alpha_{2}}{\rightarrow}\dots\overset{\alpha_{k}}{\rightarrow}x_{k}$
in $C$ such that $F(\alpha_{j})=\alpha_{j}$ for $1\leq j\leq k$ and
$F(x_{0}\overset{\alpha_{1}}{\rightarrow}x_{1}\overset{\alpha_{2}}{\rightarrow}\dots\overset{\alpha_{k}}{\rightarrow}x_{k})=x_{0}\overset{\alpha_{1}}{\rightarrow}x_{1}\overset{\alpha_{2}}{\rightarrow}\dots\overset{\alpha_{k}}{\rightarrow}x_{k}$.
Particularly, if $\mathcal{L}(\textup{(sd}F)^{\hskip
2.84526pt\widehat{0}})\neq 0$, then there exists at least one non-identity
morphism $\alpha$ in $C$ such that $F(\alpha)=\alpha$, i.e,
$M(C)^{F}\neq\emptyset$.

###### Proof.

Consider the order-preserving map $\textup{(sd}F)^{\hskip
2.84526pt\widehat{\leq i}}\colon{\textup{sd}}_{F}^{\hskip
2.84526pt\widehat{\leq i}}\to{\textup{sd}}_{F}^{\hskip 2.84526pt\widehat{\leq
i}}$. By Theorem 1, $\textup{(sd}F)^{\hskip 2.84526pt\widehat{\leq i}}$ has
fixed elements.

The elements of ${\textup{sd}}_{F}^{\hskip 2.84526pt\widehat{\leq i}}$ are of
the form
$x_{0}\overset{\alpha_{1}}{\rightarrow}x_{1}\overset{\alpha_{2}}{\rightarrow}\dots\overset{\alpha_{k}}{\rightarrow}x_{k}$
with $k\geq i+1$ where each $x_{i}$ is an object of $C$ and each
$\alpha_{i}\colon x_{i-1}\to x_{i}$ is a morphism or arrow in $C$. Therefore,
since $(\textup{sd}F)^{\hskip 2.84526pt\widehat{\leq i}}$ fixes an element
$x_{0}\overset{\alpha_{1}}{\rightarrow}x_{1}\overset{\alpha_{2}}{\rightarrow}\dots\overset{\alpha_{k}}{\rightarrow}x_{k}$
in ${\textup{sd}}_{F}^{\hskip 2.84526pt\widehat{\leq i}}$, then $F$ fixes at
least one chain of morphisms of length $k$ in $C$. By Lemma 2.3,
$(\textup{sd}F)^{\hskip 2.84526pt\widehat{\leq i}}$ fixes
$x_{1}\overset{\alpha_{2}}{\rightarrow}x_{2}\dots\overset{\alpha_{k}}{\rightarrow}x_{k}$,
$x_{0}\overset{\alpha_{1}}{\rightarrow}x_{1}\dots\overset{\alpha_{k-1}}{\rightarrow}x_{k-1}$
and $x_{0}\overset{\alpha_{1}}{\rightarrow}x_{1}\dots
x_{j-1}\overset{\alpha_{j+1}\circ\alpha_{j}}{\longrightarrow}x_{j+1}\dots\overset{\alpha_{k}}{\rightarrow}x_{k}$
in ${\textup{sd}}_{F}^{\hskip 2.84526pt\widehat{\leq i}}$ for any $1\leq j\leq
k-1$. Applying Lemma 2.3 repeatedly, it follows that for all $1\leq k\leq i+1$
there exists at least one chain of composable morphims
$x_{0}\overset{\alpha_{1}}{\rightarrow}x_{1}\overset{\alpha_{2}}{\rightarrow}\dots\overset{\alpha_{k}}{\rightarrow}x_{k}$
in $C$ such that $F(\alpha_{j})=\alpha_{j}$ for $1\leq j\leq k$ and
$F(x_{0}\overset{\alpha_{1}}{\rightarrow}x_{1}\overset{\alpha_{2}}{\rightarrow}\dots\overset{\alpha_{k}}{\rightarrow}x_{k})=x_{0}\overset{\alpha_{1}}{\rightarrow}x_{1}\overset{\alpha_{2}}{\rightarrow}\dots\overset{\alpha_{k}}{\rightarrow}x_{k}$.
∎

######  Example 3.2.

Let us consider the category and functor presented in Figure 3.1. Computing
$\mathcal{L}((\textup{sd}F)^{\widehat{0}})=4-3=1$ we determine that $F$ has
fixed arrows (for example $\beta$ and $\gamma\circ\beta$).

Figure 3.1. Example of Theorem 3.1.

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2019 253 28 October

Global baryon number conservation encoded in net-proton fluctuations

ALICE Collaboration††thanks: See Appendix A for the list of collaboration
members ALICE Collaboration

Experimental results are presented on event-by-event net-proton fluctuation
measurements in Pb–Pb collisions at $\sqrt{s_{\mathrm{NN}}}=2.76$ TeV,
recorded by the ALICE detector at the CERN LHC. These measurements have as
their ultimate goal an experimental test of Lattice QCD (LQCD) predictions on
second and higher order cumulants of net-baryon distributions to search for
critical behavior near the QCD phase boundary. Before confronting them with
LQCD predictions, account has to be taken of correlations stemming from baryon
number conservation as well as fluctuations of participating nucleons. Both
effects influence the experimental measurements and are usually not considered
in theoretical calculations. For the first time, it is shown that event-by-
event baryon number conservation leads to subtle long-range correlations
arising from very early interactions in the collisions.

Phase transitions in strongly interacting matter can be addressed by
investigating the response of the system to external perturbations via
measurements of fluctuations of conserved charges such as baryon number or
electric charge [1, 2]. At LHC energies there would be, for vanishing light
quark masses (u and d quarks), a temperature-driven second order phase
transition between a hadron gas and a quark–gluon plasma [3]. For realistic
quark masses this transition becomes a smooth cross over [4, 5]. However,
because of the small masses of the current quarks, one can still probe
critical phenomena at LHC energies (vanishing baryon chemical potential) as
reported in [6]. Indeed, recent LQCD calculations [4, 5] exhibit a rather
strong signal for the existence of a pseudo-critical chiral temperature of
about 156 MeV. Moreover, this pseudo-critical temperature is in agreement with
the chemical freeze-out temperature as extracted by the analysis of hadron
multiplicities [7, 8] measured by the ALICE experiment. This implies that the
strongly interacting matter created in central collisions of Pb nuclei at LHC
energies freezes out very near the chiral phase transition line. Hence, the
singularities arising from the second order phase transition can be captured
by measuring fluctuations of conserved charges such as the net-baryon number.
Evaluated within the framework of the Hadron Resonance Gas (HRG), net-baryon
distributions coincide with the Skellam distribution, which is the probability
distribution of the difference of two random variables, each generated from
statistically independent Poisson distributions [9, 10]. In fact, at the
pseudo-critical temperature of 156 MeV, similar to the HRG framework, LQCD
also predicts a Skellam behavior for the second cumulants of net-baryons,
while the fourth cumulants of net-baryons from LQCD are significantly below
the corresponding Skellam baseline [11, 12].

Conserved quantities of course fluctuate only in sub-regions of the available
total phase space of the reaction. In statistical mechanics they are, hence,
analyzed within the Grand Canonical Ensemble (GCE) [13] formulation, where
only the average number of net-baryons are conserved. In order to compare
theoretical calculations within the GCE, such as the HRG [7] and LQCD [4, 5],
with experimental results, the data are analyzed in different acceptance
windows by imposing selection criteria on rapidity and/or transverse momentum
of the detected particles. Indeed, if the selected acceptance window is too
small, possible dynamical correlations will be washed out and the measured
fluctuations will approach the Poisson limit [14], implying that net-baryons
will be distributed according to the Skellam distribution.

Recently the effects of limited acceptance were studied [15]. There, it was
investigated under which conditions net-baryon fluctuations depend on the size
of the acceptance. An obvious case is fluctuations caused by correlations due
to baryon number conservation. To identify these and other long-range
correlations it is interesting to perform the experimental analysis as a
function of the acceptance size.

The analysis is set up by providing the necessary definitions. Given the
number of baryons $(n_{\mathrm{B}})$ and antibaryons
($n_{\overline{\mathrm{B}}}$), the first and second cumulants of the net-
baryon probability distribution $P(\Delta n_{\mathrm{B}})$, with $\Delta
n_{\mathrm{B}}=n_{\mathrm{B}}-n_{\overline{\mathrm{B}}}$, are defined as

$\displaystyle\kappa_{1}(\Delta n_{\mathrm{B}})=\sum_{\Delta
n_{\mathrm{B}}=-\infty}^{\infty}\Delta n_{\mathrm{B}}P(\Delta
n_{\mathrm{B}})=\left<\Delta n_{\mathrm{B}}\right>,$ (1)
$\displaystyle\kappa_{2}(\Delta n_{\mathrm{B}})=\sum_{\Delta
n_{\mathrm{B}}=-\infty}^{\infty}(\Delta n_{\mathrm{B}}-\left<\Delta
n_{\mathrm{B}}\right>)^{2}P(\Delta n_{\mathrm{B}})=$
$\displaystyle\left<\left(\Delta n_{\mathrm{B}}-\left<\Delta
n_{\mathrm{B}}\right>\right)^{2}\right>.$ (2)

We note that, the first and second cumulants correspond to the expected value
and the variance of net-baryon distribution, respectively. The second cumulant
can be represented as a sum of the corresponding cumulants for single baryons
and antibaryons plus the correlation term for the joint probability
distributions of baryons and antibaryons
$P(n_{\mathrm{B}},n_{\overline{\mathrm{B}}})$

$\displaystyle\kappa_{2}(\Delta
n_{\mathrm{B}})=\kappa_{2}(n_{\mathrm{B}})+\kappa_{2}(n_{\overline{\mathrm{B}}})-2\left(\left<n_{\mathrm{B}}n_{\overline{\mathrm{B}}}\right>-\left<n_{\mathrm{B}}\right>\left<n_{\overline{\mathrm{B}}}\right>\right),$
(3)

where the mixed moment $\left<n_{\mathrm{B}}n_{\overline{\mathrm{B}}}\right>$
is defined as

$\displaystyle\left<n_{\mathrm{B}}n_{\overline{\mathrm{B}}}\right>=\sum_{n_{\mathrm{B}}=0}^{\infty}\sum_{n_{\overline{\mathrm{B}}}=0}^{\infty}n_{\mathrm{B}}n_{\overline{\mathrm{B}}}P(n_{\mathrm{B}},n_{\overline{\mathrm{B}}}).$
(4)

Equation 3 shows that, for vanishing correlations between baryons and
antibaryons
$(\left<n_{\mathrm{B}}n_{\overline{\mathrm{B}}}\right>=\left<n_{\mathrm{B}}\right>\left<n_{\overline{\mathrm{B}}}\right>)$,
the second cumulant of net-baryons is exactly equal to the sum of the
corresponding second cumulants for baryons and antibaryons.

The data presented below were obtained by analyzing about $13\times 10^{6}$
minimum-bias (cf. [16] for definition) Pb–Pb events at a center-of-mass energy
per nucleon–nucleon pair of $\sqrt{s_{\mathrm{NN}}}=2.76$ TeV recorded by the
ALICE detector [17] in the year 2010. Two forward scintillator arrays (V$0$)
are located on either side of the interaction point and cover the
pseudorapidity intervals $2.8<\eta<5.1$ and $-3.7<\eta<-1.7$ [18]. A minimum-
bias trigger condition is defined by requiring a combination of hits in the
two innermost layers of the ITS and coincidence in both V$0$ detectors. The
event centrality is selected based on the signal amplitudes in the V$0$
detectors [18].

The detectors used for track reconstruction are the Time Projection Chamber
(TPC) [19] and the Inner Tracking System (ITS) [20]. In order to keep the
tracking efficiency as high as possible, only the TPC detector was used for
particle identification, while precise vertex determination was performed with
the ITS detector. The track selection criteria used are described in Section
3.1 of [21]. Charged particle tracks with at least 80 out of maximum of 159
specific energy loss (d$E$/d$x$) samples in the TPC were used in this analysis
for the best particle identification. Moreover, in order to suppress
contributions of secondary particles from weak decays, the distance-of-
closest-approach (DCA) of the extrapolated track to the primary collision
vertex was taken to be less than 2 cm along the beam direction. In the
transverse plane, a more restrictive and transverse momentum ($p_{\rm T}$)
dependent DCA cut of less than (0.018 $\rm cm$ \+ 0.035$p_{\rm T}^{-1.01}$)
with $p_{\rm T}$ in GeV/$c$, was imposed [22].

Since the energy loss of charged particles in the gas volume of the TPC
depends explicitly on the particle momentum ($p$), the analysis was performed
in momentum space. Correspondingly, the particle identification (PID)
procedure consists of correlating particle momentum with the specific energy
loss d$E$/d$x$. This allows the event-by-event fluctuation analysis to be
performed with high overall reconstruction efficiency of about 80$\%$ for
protons, almost independent of the collision centrality. The latter is
important because small efficiencies induce Poisson fluctuations. To ensure
the best possible d$E$/d$x$ resolution, the phase space coverage was
restricted to 0.6 $<p<$ 1.5 GeV/$c$ and $|\eta|<0.8$ for the present analysis.
The corresponding $p_{\rm T}$ range at $|\eta|=0.8$ is about 0.45 $<p_{\rm
T}<$ 1.12 GeV/$c$, which gradually approaches the used momentum range towards
midrapidity. Moreover, a differential analysis is provided as function of
$\Delta\eta$ in the range $\Delta\eta$ = 0.2 to 1.6.

The cumulants of net-protons were then reconstructed using the Identity Method
(IM) [23, 24, 25, 26, 27, 21]. This method is designed to deal efficiently
with the overlapping d$E$/d$x$ distributions of protons, kaons, pions and
electrons considered in the present analysis. Their specific probability
distributions were obtained by unfolding the moments of the measured
multiplicity distributions for each particle species. The IM counts proxies of
particle multiplicities $W_{j}$ event-by-event

$W_{j}=\sum_{i=1}^{n}\frac{\rho_{j}(x_{i})}{\rho(x_{i})},\;\>\>\>\;\rho(x_{i})=\sum_{j}\rho_{j}(x_{i}),$
(5)

where $j$ stands for a particle type, $x_{i}$ denotes the measured values of
d$E$/d$x$ for a given track $i$ and $\rho_{j}(x)$ is the inclusive d$E$/d$x$
distribution of particle type $j$ within a specified phase space bin. The
summation in Eq. 5 runs over all selected $n$ tracks in the given event. The
pure and mixed moments of the $W_{j}$ distributions were calculated by
averaging over all events, leading to the moments of the true multiplicity
distributions.

The IM is based on input of moments of $W_{j}$ distributions and inclusive
d$E$/d$x$ fits in bins of momentum and pseudorapidity. The d$E$/d$x$
distributions were fit with generalized Gaussian functions, taking into
account non-Gaussian components of the experimental d$E$/d$x$ distributions.
The fits of inclusive distributions of d$E$/d$x$ were performed separately for
positively and negatively charged particles in 20 MeV/$c$ momentum and 0.1
units of $\eta$ bins.

In the upper panel of Fig. 1 the centrality dependence of the efficiency-
corrected second cumulants of net-protons is compared with the sum of the mean
multiplicities (first cumulants) of protons and antiprotons. Also included are
the first and second cumulants of protons and antiprotons. The efficiency
correction for the cumulants is performed by using proton and anti-proton
efficiencies in analytic formulas derived in [28, 29] assuming binomial
efficiency losses. The characteristics of the ALICE detector response and
applied analysis procedure ensures that this assumption is fulfilled. The
accuracy of the correction procedure was estimated to be on the percent level
and is included in the systematic uncertainties. We note that possible
corrections for volume fluctuations such as discussed in [30, 31] were not
applied to the data since, at LHC energies, second cumulants of net protons,
our main observable, are free from such effects [32].

Figure 1: Measured second cumulants of net-proton distributions (red solid
boxes) compared with the sum of the mean multiplicities (open squares). The
second cumulants of single proton and antiproton distributions are presented
with the filled and open circles, respectively. The first cumulants of protons
and antiprotons are hardly distinguishable because of the nearly equal mean
numbers of protons and antiprotons at LHC energy. In the middle and bottom
panels the normalized cumulants R1and R2 are presented. The band visible in
the bottom panel is the prediction for R2 in the presence of volume
fluctuations [32].

The subsample approach was chosen to estimate the statistical uncertainties of
the reconstructed cumulants [21, 33]. In order to evaluate systematic
uncertainties stemming from track selection criteria, the applied selection
ranges were varied around their nominal values. Other sources of systematic
uncertainties originate from the parameters of the $\rho_{j}(x_{i})$
functions, entering Eq. 5. The latter was estimated by hypothesis testing
using the Kolmogorov-Smirnov (K-S) statistics. For this purpose the parameters
of the $\rho_{j}(x_{i})$ functions were varied by testing the null hypothesis
that measured d$E$/d$x$ distributions and fit functions are similar within a
given significance level of 10% (cf. [21, 33] for details). The final
systematic uncertainties on cumulants were computed by adding in quadrature
the maximum systematic variations from individual contributions.

By their definition, cumulants are extensive quantities, i.e., are
proportional to the system volume. This also explains the centrality
dependence of all cumulants, presented in the upper panel of Fig. 1. To remove
the system size dependence, normalized cumulants R1 and R2 are introduced as

$\mathrm{R}_{1}=\kappa_{2}(n_{\mathrm{p}}-n_{\overline{\mathrm{p}}})/<n_{\mathrm{p}}+n_{\overline{\mathrm{p}}}>,\,\,\
\mathrm{R}_{2}=\kappa_{2}(n_{\mathrm{p}})/<n_{\mathrm{p}}>.$ (6)

In the middle and bottom panels of Fig. 1, deviations from unity are visible
for both R1 and R2. Moreover, the amount of deviation for R2 is about twice as
large compared to that of R1.

In order to shed light on these observations, the results are compared with
predictions from a model constructed recently [32], in which participant
fluctuations are included following the analysis of the ALICE centrality
selection [18]. Within uncertainties, the model predictions are fully
consistent with the measured R2 values, lending support to the interpretation
that volume fluctuations are at the origin of the observed deviation. This is
also supported by the observation of a small structure observed in the 10–20%
centrality class, where, compared to the first two centrality classes, the
centrality bin width is doubled.

On the other hand, by construction, for vanishing net-proton numbers, R1
should not contain any contributions from volume fluctuations, i.e., the
values of R1 obtained from the model should be consistent with unity [32]. In
fact at LHC energies R1 becomes a strongly intensive quantity [34]. The origin
for the deviation of the measured R1 values from unity must therefore be
beyond the volume fluctuations scenario. To further understand these
differences, the acceptance dependence is studied.

Figure 2: Pseudorapidity dependence of the normalized second cumulants of net-
protons R1. Global baryon number conservation is depicted as the pink band.
The dashed lines represent the predictions from the model with local baryon
number conservation [35]. The blue solid line, represents the prediction using
the HIJING generator.

The analysis is performed in eight different pseudorapidity intervals from
$|\eta|<0.1$ up to $|\eta|<0.8$ in steps of 0.1. The obtained normalized
second cumulants R1 of net-protons are presented in Fig. 2. Again the data are
below unity, with the deviation linearly increasing with increasing
acceptance.

Such a behavior was predicted based on the assumption of global baryon number
conservation [36, 37, 32], which induces correlations between protons and
antiprotons leading to the following dependence on the acceptance factor
$\alpha$

$\mathrm{R}_{1}=1-\alpha,$ (7)

where $\alpha=\left<n_{\mathrm{p}}\right>/\left<N_{\mathrm{B}}^{4\pi}\right>$
with $\left<n_{\mathrm{p}}\right>$ and $\left<N_{\mathrm{B}}^{4\pi}\right>$
referring to the mean number of protons inside the acceptance and the mean
number of baryons in full phase space. It should be further noted that, for
non-central collisions, baryon transport to mid-rapidity has to be taken into
account, which is rather model dependent. In order to avoid the model
dependence, the comparison is performed only for the central events and in the
estimate of the alpha parameter only produced baryons are used. In doing so,
the number of baryons are used in the pseudorapidity range of $|\eta|<0.5$ as
reported in [16, 38, 39, 40]. Next, using HIJING and AMPT simulations,
estimates were obtained for the total average number of baryons in full phase
space. The average number of protons $\left<n_{\mathrm{p}}\right>$ entering
into the definition of $\alpha$ (cf. Eq. 7) was taken from the current
analysis for each pseudorapidity interval. Finally, using these values of
$\alpha$, the pink band in Fig. 2 is calculated with Eq. 7. The finite width
of the band reflects the difference between predictions of the two event
generators.

Inspection of Fig. 2 shows that, for small pseudorapidity ranges of
$|\eta|<0.4$ corresponding to $\Delta\eta<0.8$, the experimentally measured
net-proton distributions closely follow a Skellam distribution. This agreement
is expected because of the small acceptance window as discussed above. For
$\Delta\eta>0.8$, deviations from the Skellam distribution are observed. The
amount of deviation is small but significant and in good agreement with the
prediction assuming global baryon number conservation. The observed deviation
is therefore consistent with the assumption of global baryon number
conservation, i.e. conservation within the full phase space.

On the other hand, local baryon number conservation may induce additional
correlations between protons and antiprotons, which would lead to a further
reduction of the measured
$\kappa_{2}(n_{\mathrm{p}}-n_{\overline{\mathrm{p}}})$ [35]. In Fig. 2 the
data are compared to the predictions from an analysis of effects of local
baryon number conservation for different values of correlation width $\Delta
y_{\mathrm{corr}}$ between protons and antiprotons.Within the experimental
uncertainties the data are best described with the assumption of global baryon
number conservation, which corresponds to the correlation width $\Delta
y_{\mathrm{corr}}=2|y_{\mathrm{beam}}|$ but, within one standard deviation
(1.56 for the last point at $\Delta\eta=1.6$), the data are also consistent
with a large correlation width of $\Delta y_{\mathrm{corr}}=5$ [35]. We find
that for $\Delta y_{\mathrm{corr}}=4.5$, with a 5$\%$ significance level, the
last point is not consistent with the experimental data. The results from the
HIJING event generator (cf. blue solid line in Fig. 2), which can be described
with $\Delta y_{\mathrm{corr}}=2$, and from a recent study reported in [41]
would imply much stronger correlations between protons and antiprotons, not
consistent with the experimental data. We note here that correlations arising
from baryon or charge conservation have also been analyzed in the framework of
balance functions [42, 43]. Such an analysis could also shed interesting light
on global vs. local baryon conservation.

From the present results it is concluded that effects due to local baryon
number conservation are not large, if present at all in second cumulants of
net-protons. The large correlation length observed in the data implies that
the normalized second cumulant R1 is determined by collisions in the very
early phase of the Pb–Pb interaction [44]. We note that long range rapidity
correlations were investigated in other contexts in [45, 46]. The search for
critical behavior, as predicted for higher cumulants of net-baryon
distributions [12, 47], will be the topic of future investigations.

## Acknowledgements

The ALICE Collaboration would like to thank all its engineers and technicians for their invaluable contributions to the construction of the experiment and the CERN accelerator teams for the outstanding performance of the LHC complex. The ALICE Collaboration gratefully acknowledges the resources and support provided by all Grid centres and the Worldwide LHC Computing Grid (WLCG) collaboration. The ALICE Collaboration acknowledges the following funding agencies for their support in building and running the ALICE detector: A. I. Alikhanyan National Science Laboratory (Yerevan Physics Institute) Foundation (ANSL), State Committee of Science and World Federation of Scientists (WFS), Armenia; Austrian Academy of Sciences, Austrian Science Fund (FWF): [M 2467-N36] and Nationalstiftung für Forschung, Technologie und Entwicklung, Austria; Ministry of Communications and High Technologies, National Nuclear Research Center, Azerbaijan; Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Financiadora de Estudos e Projetos (Finep), Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) and Universidade Federal do Rio Grande do Sul (UFRGS), Brazil; Ministry of Education of China (MOEC) , Ministry of Science & Technology of China (MSTC) and National Natural Science Foundation of China (NSFC), China; Ministry of Science and Education and Croatian Science Foundation, Croatia; Centro de Aplicaciones Tecnológicas y Desarrollo Nuclear (CEADEN), Cubaenergía, Cuba; Ministry of Education, Youth and Sports of the Czech Republic, Czech Republic; The Danish Council for Independent Research | Natural Sciences, the VILLUM FONDEN and Danish National Research Foundation (DNRF), Denmark; Helsinki Institute of Physics (HIP), Finland; Commissariat à l’Energie Atomique (CEA), Institut National de Physique Nucléaire et de Physique des Particules (IN2P3) and Centre National de la Recherche Scientifique (CNRS) and Région des Pays de la Loire, France; Bundesministerium für Bildung und Forschung (BMBF) and GSI Helmholtzzentrum für Schwerionenforschung GmbH, Germany; General Secretariat for Research and Technology, Ministry of Education, Research and Religions, Greece; National Research, Development and Innovation Office, Hungary; Department of Atomic Energy Government of India (DAE), Department of Science and Technology, Government of India (DST), University Grants Commission, Government of India (UGC) and Council of Scientific and Industrial Research (CSIR), India; Indonesian Institute of Science, Indonesia; Centro Fermi - Museo Storico della Fisica e Centro Studi e Ricerche Enrico Fermi and Istituto Nazionale di Fisica Nucleare (INFN), Italy; Institute for Innovative Science and Technology , Nagasaki Institute of Applied Science (IIST), Japanese Ministry of Education, Culture, Sports, Science and Technology (MEXT) and Japan Society for the Promotion of Science (JSPS) KAKENHI, Japan; Consejo Nacional de Ciencia (CONACYT) y Tecnología, through Fondo de Cooperación Internacional en Ciencia y Tecnología (FONCICYT) and Dirección General de Asuntos del Personal Academico (DGAPA), Mexico; Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO), Netherlands; The Research Council of Norway, Norway; Commission on Science and Technology for Sustainable Development in the South (COMSATS), Pakistan; Pontificia Universidad Católica del Perú, Peru; Ministry of Science and Higher Education and National Science Centre, Poland; Korea Institute of Science and Technology Information and National Research Foundation of Korea (NRF), Republic of Korea; Ministry of Education and Scientific Research, Institute of Atomic Physics and Ministry of Research and Innovation and Institute of Atomic Physics, Romania; Joint Institute for Nuclear Research (JINR), Ministry of Education and Science of the Russian Federation, National Research Centre Kurchatov Institute, Russian Science Foundation and Russian Foundation for Basic Research, Russia; Ministry of Education, Science, Research and Sport of the Slovak Republic, Slovakia; National Research Foundation of South Africa, South Africa; Swedish Research Council (VR) and Knut & Alice Wallenberg Foundation (KAW), Sweden; European Organization for Nuclear Research, Switzerland; Suranaree University of Technology (SUT), National Science and Technology Development Agency (NSDTA) and Office of the Higher Education Commission under NRU project of Thailand, Thailand; Turkish Atomic Energy Agency (TAEK), Turkey; National Academy of Sciences of Ukraine, Ukraine; Science and Technology Facilities Council (STFC), United Kingdom; National Science Foundation of the United States of America (NSF) and United States Department of Energy, Office of Nuclear Physics (DOE NP), United States of America.

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##  Appendix A The ALICE Collaboration

S. Acharyaorg141&D. Adamováorg94&A. Adlerorg74&J. Adolfssonorg80&M.M.
Aggarwalorg99&G. Aglieri Rinellaorg33&M. Agnelloorg30&N. Agrawalorg10,org53&Z.
Ahammedorg141&S. Ahmadorg16&S.U. Ahnorg76&A. Akindinovorg91&M. Al-
Turanyorg106&S.N. Alamorg141&D.S.D. Albuquerqueorg122&D. Aleksandrovorg87&B.
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Trubnikovorg2&W.H. Trzaskaorg126&T.P. Trzcinskiorg142&B.A. Trzeciakorg63&T.
Tsujiorg132&A. Tumkinorg108&R. Turrisiorg56&T.S. Tveterorg20&K.
Ullalandorg21&E.N. Umakaorg125&A. Urasorg135&G.L. Usaiorg23&A.
Utrobicicorg98&M. Valaorg37&N. Valleorg139&S. Valleroorg58&N. van der
Kolkorg63&L.V.R. van Doremalenorg63&M. van Leeuwenorg63&P. Vande Vyvreorg33&D.
Vargaorg145&Z. Vargaorg145&M. Varga-Kofaragoorg145&A. Vargasorg44&M.
Vasileiouorg83&A. Vasilievorg87&O. Vázquez Doceorg104,org117&V.
Vecherninorg112&A.M. Veenorg63&E. Vercellinorg25&S. Vergara Limónorg44&L.
Vermuntorg63&R. Vernetorg7&R. Vértesiorg145&L. Vickovicorg34&Z.
Vilakaziorg131&O. Villalobos Baillieorg110&A. Villatoro Telloorg44&G.
Vinoorg52&A. Vinogradovorg87&T. Virgiliorg29&V. Vislaviciusorg88&A.
Vodopyanovorg75&B. Volkelorg33&M.A. Völklorg102&K. Voloshinorg91&S.A.
Voloshinorg143&G. Volpeorg32&B. von Hallerorg33&I. Vorobyevorg104&D.
Voscekorg116&J. Vrlákováorg37&B. Wagnerorg21&M. Weberorg113&S.G.
Weberorg144&A. Wegrzynekorg33&D.F. Weiserorg103&S.C. Wenzelorg33&J.P.
Wesselsorg144&J. Wiechulaorg68&J. Wikneorg20&G. Wilkorg84&J.
Wilkinsonorg10,org53&G.A. Willemsorg33&E. Willsherorg110&B.
Windelbandorg103&M. Winnorg137&W.E. Wittorg130&Y. Wuorg128&R. Xuorg6&S.
Yalcinorg77&K. Yamakawaorg45&S. Yangorg21&S. Yanoorg137&Z. Yinorg6&H.
Yokoyamaorg63&I.-K. Yooorg17&J.H. Yoonorg60&S. Yuanorg21&A. Yuncuorg103&V.
Yurchenkoorg2&V. Zaccoloorg24&A. Zamanorg14&C. Zampolliorg33&H.J.C.
Zanoliorg63&N. Zardoshtiorg33&A. Zarochentsevorg112&P. Závadaorg66&N.
Zaviyalovorg108&H. Zbroszczykorg142&M. Zhalovorg97&S. Zhangorg39&X.
Zhangorg6&Z. Zhangorg6&V. Zherebchevskiiorg112&D. Zhouorg6&Y. Zhouorg88&Z.
Zhouorg21&J. Zhuorg6,org106&Y. Zhuorg6&A. Zichichiorg10,org26&M.B.
Zimmermannorg33&G. Zinovjevorg2&N. Zurloorg140&

## Affiliation notes

org*Deceased

orgIDipartimento DET del Politecnico di Torino, Turin, Italy

orgIIM.V. Lomonosov Moscow State University, D.V. Skobeltsyn Institute of
Nuclear, Physics, Moscow, Russia

orgIIIDepartment of Applied Physics, Aligarh Muslim University, Aligarh, India

orgIVInstitute of Theoretical Physics, University of Wroclaw, Poland

## Collaboration Institutes

org1A.I. Alikhanyan National Science Laboratory (Yerevan Physics Institute)
Foundation, Yerevan, Armenia

org2Bogolyubov Institute for Theoretical Physics, National Academy of Sciences
of Ukraine, Kiev, Ukraine

org3Bose Institute, Department of Physics and Centre for Astroparticle Physics
and Space Science (CAPSS), Kolkata, India

org4Budker Institute for Nuclear Physics, Novosibirsk, Russia

org5California Polytechnic State University, San Luis Obispo, California,
United States

org6Central China Normal University, Wuhan, China

org7Centre de Calcul de l’IN2P3, Villeurbanne, Lyon, France

org8Centro de Aplicaciones Tecnológicas y Desarrollo Nuclear (CEADEN), Havana,
Cuba

org9Centro de Investigación y de Estudios Avanzados (CINVESTAV), Mexico City
and Mérida, Mexico

org10Centro Fermi - Museo Storico della Fisica e Centro Studi e Ricerche
“Enrico Fermi’, Rome, Italy

org11Chicago State University, Chicago, Illinois, United States

org12China Institute of Atomic Energy, Beijing, China

org13Comenius University Bratislava, Faculty of Mathematics, Physics and
Informatics, Bratislava, Slovakia

org14COMSATS University Islamabad, Islamabad, Pakistan

org15Creighton University, Omaha, Nebraska, United States

org16Department of Physics, Aligarh Muslim University, Aligarh, India

org17Department of Physics, Pusan National University, Pusan, Republic of
Korea

org18Department of Physics, Sejong University, Seoul, Republic of Korea

org19Department of Physics, University of California, Berkeley, California,
United States

org20Department of Physics, University of Oslo, Oslo, Norway

org21Department of Physics and Technology, University of Bergen, Bergen,
Norway

org22Dipartimento di Fisica dell’Università ’La Sapienza’ and Sezione INFN,
Rome, Italy

org23Dipartimento di Fisica dell’Università and Sezione INFN, Cagliari, Italy

org24Dipartimento di Fisica dell’Università and Sezione INFN, Trieste, Italy

org25Dipartimento di Fisica dell’Università and Sezione INFN, Turin, Italy

org26Dipartimento di Fisica e Astronomia dell’Università and Sezione INFN,
Bologna, Italy

org27Dipartimento di Fisica e Astronomia dell’Università and Sezione INFN,
Catania, Italy

org28Dipartimento di Fisica e Astronomia dell’Università and Sezione INFN,
Padova, Italy

org29Dipartimento di Fisica ‘E.R. Caianiello’ dell’Università and Gruppo
Collegato INFN, Salerno, Italy

org30Dipartimento DISAT del Politecnico and Sezione INFN, Turin, Italy

org31Dipartimento di Scienze e Innovazione Tecnologica dell’Università del
Piemonte Orientale and INFN Sezione di Torino, Alessandria, Italy

org32Dipartimento Interateneo di Fisica ‘M. Merlin’ and Sezione INFN, Bari,
Italy

org33European Organization for Nuclear Research (CERN), Geneva, Switzerland

org34Faculty of Electrical Engineering, Mechanical Engineering and Naval
Architecture, University of Split, Split, Croatia

org35Faculty of Engineering and Science, Western Norway University of Applied
Sciences, Bergen, Norway

org36Faculty of Nuclear Sciences and Physical Engineering, Czech Technical
University in Prague, Prague, Czech Republic

org37Faculty of Science, P.J. Šafárik University, Košice, Slovakia

org38Frankfurt Institute for Advanced Studies, Johann Wolfgang Goethe-
Universität Frankfurt, Frankfurt, Germany

org39Fudan University, Shanghai, China

org40Gangneung-Wonju National University, Gangneung, Republic of Korea

org41Gauhati University, Department of Physics, Guwahati, India

org42Helmholtz-Institut für Strahlen- und Kernphysik, Rheinische Friedrich-
Wilhelms-Universität Bonn, Bonn, Germany

org43Helsinki Institute of Physics (HIP), Helsinki, Finland

org44High Energy Physics Group, Universidad Autónoma de Puebla, Puebla, Mexico

org45Hiroshima University, Hiroshima, Japan

org46Hochschule Worms, Zentrum für Technologietransfer und Telekommunikation
(ZTT), Worms, Germany

org47Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest, Romania

org48Indian Institute of Technology Bombay (IIT), Mumbai, India

org49Indian Institute of Technology Indore, Indore, India

org50Indonesian Institute of Sciences, Jakarta, Indonesia

org51INFN, Laboratori Nazionali di Frascati, Frascati, Italy

org52INFN, Sezione di Bari, Bari, Italy

org53INFN, Sezione di Bologna, Bologna, Italy

org54INFN, Sezione di Cagliari, Cagliari, Italy

org55INFN, Sezione di Catania, Catania, Italy

org56INFN, Sezione di Padova, Padova, Italy

org57INFN, Sezione di Roma, Rome, Italy

org58INFN, Sezione di Torino, Turin, Italy

org59INFN, Sezione di Trieste, Trieste, Italy

org60Inha University, Incheon, Republic of Korea

org61Institut de Physique Nucléaire d’Orsay (IPNO), Institut National de
Physique Nucléaire et de Physique des Particules (IN2P3/CNRS), Université de
Paris-Sud, Université Paris-Saclay, Orsay, France

org62Institute for Nuclear Research, Academy of Sciences, Moscow, Russia

org63Institute for Subatomic Physics, Utrecht University/Nikhef, Utrecht,
Netherlands

org64Institute of Experimental Physics, Slovak Academy of Sciences, Košice,
Slovakia

org65Institute of Physics, Homi Bhabha National Institute, Bhubaneswar, India

org66Institute of Physics of the Czech Academy of Sciences, Prague, Czech
Republic

org67Institute of Space Science (ISS), Bucharest, Romania

org68Institut für Kernphysik, Johann Wolfgang Goethe-Universität Frankfurt,
Frankfurt, Germany

org69Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México,
Mexico City, Mexico

org70Instituto de Física, Universidade Federal do Rio Grande do Sul (UFRGS),
Porto Alegre, Brazil

org71Instituto de Física, Universidad Nacional Autónoma de México, Mexico
City, Mexico

org72iThemba LABS, National Research Foundation, Somerset West, South Africa

org73Jeonbuk National University, Jeonju, Republic of Korea

org74Johann-Wolfgang-Goethe Universität Frankfurt Institut für Informatik,
Fachbereich Informatik und Mathematik, Frankfurt, Germany

org75Joint Institute for Nuclear Research (JINR), Dubna, Russia

org76Korea Institute of Science and Technology Information, Daejeon, Republic
of Korea

org77KTO Karatay University, Konya, Turkey

org78Laboratoire de Physique Subatomique et de Cosmologie, Université
Grenoble-Alpes, CNRS-IN2P3, Grenoble, France

org79Lawrence Berkeley National Laboratory, Berkeley, California, United
States

org80Lund University Department of Physics, Division of Particle Physics,
Lund, Sweden

org81Nagasaki Institute of Applied Science, Nagasaki, Japan

org82Nara Women’s University (NWU), Nara, Japan

org83National and Kapodistrian University of Athens, School of Science,
Department of Physics , Athens, Greece

org84National Centre for Nuclear Research, Warsaw, Poland

org85National Institute of Science Education and Research, Homi Bhabha
National Institute, Jatni, India

org86National Nuclear Research Center, Baku, Azerbaijan

org87National Research Centre Kurchatov Institute, Moscow, Russia

org88Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark

org89Nikhef, National institute for subatomic physics, Amsterdam, Netherlands

org90NRC Kurchatov Institute IHEP, Protvino, Russia

org91NRC «Kurchatov Institute» - ITEP, Moscow, Russia

org92NRNU Moscow Engineering Physics Institute, Moscow, Russia

org93Nuclear Physics Group, STFC Daresbury Laboratory, Daresbury, United
Kingdom

org94Nuclear Physics Institute of the Czech Academy of Sciences, Řež u Prahy,
Czech Republic

org95Oak Ridge National Laboratory, Oak Ridge, Tennessee, United States

org96Ohio State University, Columbus, Ohio, United States

org97Petersburg Nuclear Physics Institute, Gatchina, Russia

org98Physics department, Faculty of science, University of Zagreb, Zagreb,
Croatia

org99Physics Department, Panjab University, Chandigarh, India

org100Physics Department, University of Jammu, Jammu, India

org101Physics Department, University of Rajasthan, Jaipur, India

org102Physikalisches Institut, Eberhard-Karls-Universität Tübingen, Tübingen,
Germany

org103Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg,
Heidelberg, Germany

org104Physik Department, Technische Universität München, Munich, Germany

org105Politecnico di Bari, Bari, Italy

org106Research Division and ExtreMe Matter Institute EMMI, GSI
Helmholtzzentrum für Schwerionenforschung GmbH, Darmstadt, Germany

org107Rudjer Bošković Institute, Zagreb, Croatia

org108Russian Federal Nuclear Center (VNIIEF), Sarov, Russia

org109Saha Institute of Nuclear Physics, Homi Bhabha National Institute,
Kolkata, India

org110School of Physics and Astronomy, University of Birmingham, Birmingham,
United Kingdom

org111Sección Física, Departamento de Ciencias, Pontificia Universidad
Católica del Perú, Lima, Peru

org112St. Petersburg State University, St. Petersburg, Russia

org113Stefan Meyer Institut für Subatomare Physik (SMI), Vienna, Austria

org114SUBATECH, IMT Atlantique, Université de Nantes, CNRS-IN2P3, Nantes,
France

org115Suranaree University of Technology, Nakhon Ratchasima, Thailand

org116Technical University of Košice, Košice, Slovakia

org117Technische Universität München, Excellence Cluster ’Universe’, Munich,
Germany

org118The Henryk Niewodniczanski Institute of Nuclear Physics, Polish Academy
of Sciences, Cracow, Poland

org119The University of Texas at Austin, Austin, Texas, United States

org120Universidad Autónoma de Sinaloa, Culiacán, Mexico

org121Universidade de São Paulo (USP), São Paulo, Brazil

org122Universidade Estadual de Campinas (UNICAMP), Campinas, Brazil

org123Universidade Federal do ABC, Santo Andre, Brazil

org124University of Cape Town, Cape Town, South Africa

org125University of Houston, Houston, Texas, United States

org126University of Jyväskylä, Jyväskylä, Finland

org127University of Liverpool, Liverpool, United Kingdom

org128University of Science and Techonology of China, Hefei, China

org129University of South-Eastern Norway, Tonsberg, Norway

org130University of Tennessee, Knoxville, Tennessee, United States

org131University of the Witwatersrand, Johannesburg, South Africa

org132University of Tokyo, Tokyo, Japan

org133University of Tsukuba, Tsukuba, Japan

org134Université Clermont Auvergne, CNRS/IN2P3, LPC, Clermont-Ferrand, France

org135Université de Lyon, Université Lyon 1, CNRS/IN2P3, IPN-Lyon,
Villeurbanne, Lyon, France

org136Université de Strasbourg, CNRS, IPHC UMR 7178, F-67000 Strasbourg,
France, Strasbourg, France

org137Université Paris-Saclay Centre d’Etudes de Saclay (CEA), IRFU,
Départment de Physique Nucléaire (DPhN), Saclay, France

org138Università degli Studi di Foggia, Foggia, Italy

org139Università degli Studi di Pavia, Pavia, Italy

org140Università di Brescia, Brescia, Italy

org141Variable Energy Cyclotron Centre, Homi Bhabha National Institute,
Kolkata, India

org142Warsaw University of Technology, Warsaw, Poland

org143Wayne State University, Detroit, Michigan, United States

org144Westfälische Wilhelms-Universität Münster, Institut für Kernphysik,
Münster, Germany

org145Wigner Research Centre for Physics, Budapest, Hungary

org146Yale University, New Haven, Connecticut, United States

org147Yonsei University, Seoul, Republic of Korea