# Logic and Semiotics in Aristotle’s Topics

Clarence Lewis Protin

###### Abstract

We give a first sketch and outline of our approach to a first-order
formalisation of Aristotle’s Topics.

## Philosophical Preliminary

The medieval scholastics used to distinguish between primary and secondary
intensions. Consider the sentence : the term Man is subordinate to the term
Animal. When using the expression ’term Man’ we are not referencing a concrete
man or even a collection of men. Nor are we referencing an inscription or
physical or perceptual sequence of tokens. Rather we are referencing a complex
consisting both of linguistic and semantic data and their relationship. In
this paper we take the approach that such complexes can be formalised as
linguistic data (in a suitable ideal language) together with a collection of
first-order predicates applying to such data which formalise both grammatical
and semantic information and relations. We are inspired by the work of Richard
Milton Martin and David Parsons’ account of the Inscriptional Meta-Language
with the difference that we do no think of linguistic entities as concrete
’inscriptions’ but in a slightly more abstract and ideal sense which yet
preserves all the crucial formal combinatoric structure of concrete
inscriptions.

##  Introduction

One of the distinguishing features of 20th-century philosophy is the focus on
symbolic logic and its use as a tool for the analysis of language. But was
this really something revolutionary and new ? Philosophy is expressed in
spoken or written language. It is generally believed that such expression must
consist primarily of what are called pursuasive or valid arguments, be these
carried out in the form of debate or as a single text/discourse. Thus it is
quite natural that the preliminary step before engaging in philosophy is to
inquire what a valid or pursuasive argument consists of, as well as what a
specifically philosophical argument is. This question is general enough to
transcend the boundaries between what was later called ’logic’ and
’linguistics’. To put it in another way there is no doubt that to the modern
philosopher the study of this question will involve just as much the
’philosophy of logic’ as the ’philosophy of language’ (which in fact already
overlap).

A remarkable feature of Aristotle’s Topics is it is entirely dedicated to this
radical question and presents us with the first systematic attempt carried out
in ancient times to furnish a detailed answer to this question in which
logical, linguistic and semantic concerns are woven into a systematic whole.
The Topics is focused on the theory of definition. Also the Topics divides all
subjects of discussion into ethics, physics and logic111the kinds of
definition considered by Aristotle would correspond in modern terms to
Mathematics, Ethics, Psychology, Physics, Logic, Ontology and Epistemology. .

Ancient grammar (or ancient linguistics) for all its alleged unsophistication
and naiveté presents us, as we shall see, with a fresh and unprejudiced
approach to language. A paper by Bobzien and
Shogry222https://ora.ox.ac.uk/objects/uuid:b51729b1-6891-4b09-98ed-6d0b6b47bc3c/files/sdf65v7975
documents illuminating connections between quantifier logic and grammar in the
case of Stoic philosophy.

The text of the Topics is difficult and at places corrupt. It is obvious that
different sections where added at different times and that we cannot hope for
complete consistency in the definition of some of the key terms. The Topics
have been traditionally dismissed as an immature and somewhat undigested work
which was superseded by the more profound and rigorous theories of the
Analytics.

The Topics are based on a fairly elaborate system of primitive grammatical,
semantic and logical concepts. These include of course the famous five
predicabilia (genus, species, difference, property and accident) and the ten
categories. Following [16] we consider a topic to be a universally quantified
proposition taken as an axiom which is instantiated in accordance to the needs
of a debate or argument, either for proof or refutation (generally the form of
the topic adequate for refutation follows from applying modern logical rules
to the topic) . All topics must be expressed in terms of the aforementioned
system of primitive concepts. For Aristotle the original meaning of
’syllogism’ was simply a logical rule. Topics are logical-grammatical-semantic
axioms which are converted into logical rules.

The goal of this work is to bring to light these primitive concepts and to
express formally all the topics and also to attempt to reduce the total system
of topics to slighter fewer ones.

Inspired by the work Richard Martin (and also to a certain extent the work of
M. Malink on the First Analytics) we take a first-order approach rather than a
second-order or more complex approach.

Connections can be made with certain formalist or mathematically inspired
schools of modern linguistics as well to structuralist semiotics and cognitive
science.

The rest of this work is dedicated to showing that the theory of the five
predicabilia is very rich philosophically and linguistically and still of
relevance to modern thought333see our previous work [15] for further
discussions..

We propose a detailed comparison with Hegel’s theory of the concept (3rd part
of the Science of Logic), with Husserl’s Logical Investigations and with
modern approaches to the organisation of semantic memory and formal
ontologies.

But the most important and interesting aspect of the Topics is the idea of
formal rule-based (scientific, ethical, philosophical) debate - similar to a
game like Chess - which includes both formal logic and formalised semantic and
syntactic aspects of natural language. It could be hoped that this aspect
might inspire us to elaborate more refined forms of debate based on better
logico-linguistic standards and thereby serve scientific and socio-cultural
progress.

Ancient Greek philosophers loved formal argumentation and debate and there is
evidence that a massive amount of arguments were compiled during the eras of
the late Academia and first Stoic schools, as testified by accounts of the
lives of Chrysippus and Carneades. Ancient philosophy at its best was all
about critical and formal argumentation and debate, not system-building, myth-
making or apologetics in service of the powers that be.

##  Aristotelic Grammar

Unfortunately Aristotle’s view of grammar as well as of many fundamental
questions concerning terms and concepts is only documented in a few scattered
passages. We know more about Stoic grammar and grammar-based logic. For our
purposes the basic linguistic entity is that of sentence or protasis. A
sentence is made up of two linguistic expressions connected by a certain
propositional relation. The two expressions are called subject and predicate
depending on their position in the propositional relation.

Only noun phrases can be a subject but adjectival and verbal constructions can
be predicates. These can becomes subjects after a certain syntactic
transformation.

The most primitive concept of ’verb’ appears to have been that of
predicate444in some Eastern languages verbs and adjectives seem to be less
differentiated than in Western languages.. A ’term’ oros is what can be a
subject. However there is also a sense in which a verb is term to which is
added a temporal aspect. The Stoic concept of kategorema seems to fit more
exactly the concept of predicate. A predicate can be considered in itself, as
unsaturated by any concrete subject.

In Plato’s _Sophist_ there is discussion concerning ’sentences’, ’nouns’ and
’verbs’, sentences being combinations of nouns and verbs, and all sentences
must have a ’subject’. Plato gives the example ’Theaetetus, with whom I am now
speaking, is flying’. This illustrates how Plato’s concept of ’noun’
corresponds to our ’noun phrase’ and very likely to the Aristotelic ’oros’:

__

The formal ’grammar’ required the Topics is rather sophisticated and if we
wish to take a first-order approach then remarkably we are lead in the
direction of modern sententialist formalisations of natural language such as
that of Richard Martin.

From now on we work in the language of classical first-order logic with
equality. The variables of our logic range over a class of linguistic
expressions which are to be take as being constructed from ideal linguistic
tokens. This class is to include nominal, adjectival, adverbial and verbal
constructions555Aristotle probably would accept that these are the class of
expressions which mean something.  but not for instance isolated articles,
prepositions, etc. We assume that there are only a finite number of possible
linguistic expressions and that each one of these is represented by a distinct
constant

$x=c_{1}\vee x=c_{2}\vee...\vee x=c_{n}$

Since we will be dealing with the natural languages ancient Greek and modern
English we will not use constants but cornered brackets enclosing expressions
in these languages. For instance $\ulcorner Socrates\urcorner$ and $\ulcorner
the\,man\,runs\urcorner$.

Our first-order language will contain a finite collection of grammatical
predicates $G_{1},....,G_{g}$ which express purely syntactic (and
morphological) properties of linguistic expressions. Instead of using a number
we will often use a mnemonic tag. For instance $G_{sen}(x)$ for the
grammatical predicate that expresses that $x$ is a sentence and $G_{term}(x)$
the one that expresses that $x$ is a term. $G_{at}(x)$ expresses that $x$ is
an atomic term and $G_{inc}(x,y)$ that the expression $x$ occurs within $y$.
The grammatical predicate $G_{rest}(x,y,z)$ will be important further ahead.
It says that the noun-phrase $z$ is the result of modifying the noun-phrase
$x$ by the adjectival form of the noun $y$. For instance we have
$G_{rest}(\ulcorner animal\urcorner,\ulcorner rationality\urcorner,\ulcorner
rational\,animal\urcorner)$. Aristotle makes frequent use of the
adverbialisation and adjectivisation functions $G_{adv}(x,y)$ and
$G_{adj}(x,y)$ in various topics. It appears that Aristotle conceived the verb
as being a temporalisation of a term.

Besides purely syntactic predicates we have semantic predicates (which overlap
somewhat with the logical, ontological and epistemic domain). We have a finite
collection of semantic relations $S_{1},...,S_{s}$ with distinguished binary
relation $O(x,y)$ called _contrariety_ , $x\multimap y$ the _semantic
dependence_ of $x$ on $y$, $\circledcirc(x)$ representing terms admitting more
or less666to which we add a predicate representing ’having a medium’ $\bot x$
and $y$ being a medium between $x$ and $z$ $M(x,y,z)$. Also $x\gtrdot y$
represents ’better known’. , a quaternary predicate representing _analogy_ (or
homology or similitude) as $x$ is to $y$ so $z$ is to $w$ and written
$\frac{x}{y}=\frac{z}{w}$. One of the most important semantic predicates is
$G_{pol}(x)$ which expresses that $x$ is polysemic or ambiguous. We also have
a predicate for _metaphoric_ predication $\approx$.

The famous ten categories are represented by unary predicates
$C_{1},...,C_{10}$. It seems reasonable that only terms can fall under one of
the categories. However other grammatical categories can easily be brought
within the division of the categories as well. The obvious properties of the
ten categories are

$\neg(C_{i}(x)\&C_{j}(x))\quad i\neq j$ $C_{1}(x)\vee C_{1}(x)\vee...\vee
C_{10}(x)$

$C_{1}$ represents the category of _ousia_. For Aristotle terms in the
category of _ousia_ do not semantically depend on terms in other categories.

$C_{1}(x)\rightarrow(\neg C_{1}(y)\rightarrow\neg x\multimap y)$

Cf. 128a25.

##  The Five Predicabilia

As expected predication will correspond to a binary predicate. The five
predicabilia will correspond to different binary predicates. Aristotle is not
consistent in his definition of ’accident’. We must distinguish between
separable and inseparable accident. In fact the first distinction we make
between different forms of predication is modal. We have necessary predication
$x\in y$ and contingent predication $x\ominus y$ which is to be understood as
$y$ can both hold and not hold of $x$ (bicontingent in the sense of [14]).

$x\in y\leftrightarrow\neg x\ominus y$

Being a genus and species are correlatives. We denote this relation by $x\prec
y$ which can be read $x$ is a species of genus $y$ or $y$ is a genus of
species $x$. The most basic properties which are both explicit and implicit
throughout the Topics are

$x\prec y\&y\prec z\rightarrow x\prec z$ $x\prec y\&x\prec z\rightarrow(y\prec
z\vee z\prec y)$ $x\nprec x$

Because of Axiom 1 there can be no infinite chains $..\prec x_{1}\prec
x_{2}\prec x_{3}\prec...$ extending in either direction and so we have the
ascending and descending chain conditions. It follows that if $x\prec y$ for
some $y$ then there is a $z$ such that $x\prec z$ and there is no $w$ such
that $x\prec w\prec z$. We then write $x\triangleleft z$. Formally

$x\triangleleft z\leftrightarrow\neg\exists y.x\prec y\prec z$

In this situation $x$ is called the immediate species of $z$ and $z$ the
proximate genus of $x$.

Property is denote by $xIy$ ($I$ is for Greek _idion_) and read $y$ is the
property of $x$. Genealogically definition and property seem to share a common
origin.

Inseparable accident is denoted by $x\Sigma y$, $y$ is the inseparable
accident of $x$. We must not confuse separable accident $x\ominus y$ with
inseparable accident $x\Sigma y$.

An attentive reading of Chapter VI reveals that ’difference’ should correspond
to two distinct binary predicates $\Delta_{1},\Delta_{2}$. Indeed Aristotle
speaks of the (immutable, essential) difference of a given species, of
coordinate differences and finally of the various differences that belong to a
given genus. Let the first relation be denoted by $x\Delta_{1}z$ and the third
by $y\Delta_{2}z$. We can define coordinate differences as follows

$z_{1}\Delta_{3}z_{2}\leftrightarrow\exists y.y\Delta_{2}z_{1}\&y\Delta z_{2}$

The following are clear in the Topics

$\neg(x\bigcirc_{1}y\&x\bigcirc_{2}y)\quad\bigcirc_{1},\bigcirc_{2}\in\\{\prec,\Delta_{1},I,\Sigma\\},\bigcirc_{1}\neq\bigcirc_{2}$

$x\bigcirc y\rightarrow x\in y\quad\bigcirc\in\\{\prec,\Delta_{1},I,\Sigma\\}$

There is a problem that Aristotle seems in certain passages to make the five
predicabilia exhaustive for all forms of predication. But this cannot be the
case (unless it is understood in the sense of atomic building blocks) for it
does not include definition. We will discuss $\Delta_{2}$ further ahead. For
now we note that $y\Delta_{2}z\rightarrow z\multimap y\&\neg y\multimap z$.

Now we are equipped to define definition according to the most mature view in
the Topics:

$xDy\leftrightarrow\exists a,b.G_{rest}(a,b,y)\&x\triangleleft
a\&x\Delta_{1}b\&a\Delta_{2}b$

How do we define _infima species_ and individuals ? Note that for Aristotle
$\ulcorner Socrates\urcorner\prec\ulcorner man\urcorner$. We could define
individual

$\iota(x)\leftrightarrow\neg y.y\prec x$

and perhaps add a category condition.

Then we would attempt to define infimate species as

$\wedge(x)\leftrightarrow\forall y.y\prec x\rightarrow\iota(y)$

but this leaves out the important condition that individuals in an infima
species cannot differ from each other specifically. That is, $x$ cannot have
any differences:

$\wedge(x)\leftrightarrow\neg\exists z.x\Delta_{2}z\&\forall y.y\prec
x\rightarrow\iota(y)$

As we shall see an important property of genera is that if they have a
difference then they have more than one

$x\Delta_{2}y\rightarrow\exists z.z\neq y\&x\Delta_{2}z$

which will also imply that a genus with a proximate species has more than one

$x\triangleleft y\rightarrow\exists z.z\neq x\&z\triangleleft y$

Finally we assume that we have distinct Platonic-Parmenidean constants
$\ulcorner being\urcorner$ and $\ulcorner one\urcorner$ such that

$\forall x.x\in\ulcorner being\urcorner\&x\in\ulcorner one\urcorner$

##  Extensions and Finite Set Theory

By Axiom 1 we can easily consider a theory of finite sets or lists of
linguistic expressions. For $x$ we consider the finite set of all constants
$c_{i}$ such that $c_{i}\in x$ and we denote this set by $\\{z:z\in x\\}$ and
we call it the _extension_ of $x$. We can define extensional containment
$x\subset_{e}y$ and equality $x=_{e}y$ in the expected way. Aristotle seems in
various passages to hint at this modern concept of extension with his
expression _pleon legesthai_ , being said of more.

## Topics III, 1

$x\prec\ulcorner good\urcorner$ and $G_{rest}(y,x,z)$ then $x>z$ (more
elligible).

Desirable for something.

## Topics IV, 1

$x\nprec z\&x\prec y\rightarrow y\nprec z$ $(x\ominus y\vee x\Sigma
y)\rightarrow x\nprec y$ $x\prec y\rightarrow C_{i}(x)\leftrightarrow
C_{i}(y),\quad i=1,...,10$ $x\prec y\rightarrow y\notin x$ $x\prec y\&z\in
x\rightarrow z\in y$ $x\prec z\rightarrow\exists y.y\triangleleft z\&x\preceq
y$ $x\prec y\rightarrow x\subsetneq_{e}y$ $x\prec z\&x\prec
y\&\wedge(y)\rightarrow(\forall w.w\prec y\rightarrow w\prec z)$

Vocabulary used in this chapter: _pleasure, good, white, snow, soul, self-
moving, swan, science, beautiful, double, multiple, locomotion, alteration,
movement, origin, principle, opiniable, being, one, line, indivisible_.

## Topics IV, 2

$(\exists z.x\prec z\&z\nprec y\&y\nprec x)\rightarrow x\nprec y$ $(\exists
z.y\prec z\&x\nprec z)\rightarrow x\nprec y$ $x\prec y\rightarrow y\nprec x$
$x\prec z\&y\triangleleft z\&(\forall w.(w\neq y\&w\triangleleft z)\rightarrow
x\nprec w)\rightarrow x\triangleleft y$ $(x\nprec z\&y\prec z)\vee(w\in
x\&x\notin z)\rightarrow x\nprec y$ $yDv\&x\prec y\rightarrow x\in v\&(w\in
x\rightarrow w\in v)$ $x\Delta_{1}y\rightarrow x\nprec y\&y\nprec x$ $x\prec
y\rightarrow x\subsetneq_{e}y$ $x\Delta_{1}y\rightarrow x\subset_{e}y$
$x\Delta_{1}y\rightarrow(x\prec z\rightarrow z\notin y)$ $(\forall
z.x\Delta_{2}z)\rightarrow y\notin z)\rightarrow y\nprec x$

_pleon te gar to genos tês diaphoras dei legesthai_ 123a7

$x\Delta_{2}z\rightarrow z\subsetneq_{e}x$ $x\prec y\rightarrow x\multimap
y\&\neg x\ominus y$

## Topics IV, 3

$x\in y\&O(y,y^{\prime})\rightarrow x\nprec y^{\prime}$

$(x\prec y\rightarrow x\notin z)\&w\in z\rightarrow x\prec y$

$\ulcorner soul\urcorner\in\ulcorner life\urcorner$ Also for all
$N\prec\ulcorner number\urcorner$ we have $N\notin\ulcorner life\urcorner$.
Hence we cannot have $\ulcorner soul\urcorner\prec\ulcorner number\urcorner$ .

$x\prec y\rightarrow\exists z.z\neq x\&z\prec y$ $x\prec y\rightarrow\neg
x\approx y$ $\neg\exists x^{\prime}.O(x,x^{\prime})\&y\prec
x\rightarrow(O(y,y^{\prime})\rightarrow y^{\prime}\prec x)$ $x\prec
y\&O(x,x^{\prime})\&O(y,y^{\prime})\rightarrow x^{\prime}\prec y^{\prime}$
$(\neg\exists y.x\prec y)\&O(x,x^{\prime})\rightarrow\neg\exists
y.x^{\prime}\prec y$ $x\prec y\rightarrow(\bot(x)\leftrightarrow\bot(y))$
$x\prec y\&x^{\prime}\prec y\&M(x,z,x^{\prime})\rightarrow z\prec y$ $x\prec
y\&G_{adv}(x,x^{\prime})\&G_{adv}(y,y^{\prime})\rightarrow x^{\prime}\in
y^{\prime}$ $x\prec y\&G_{adj}(x,x^{\prime})\&G_{adj}(y,y^{\prime})\rightarrow
x^{\prime}\in y^{\prime}$

## Topics IV,4

$\frac{x}{y}=\frac{x^{\prime}}{y^{\prime}}\&y\prec y^{\prime}\rightarrow
x\prec x^{\prime}$ $x\prec
y\&G_{adv}(x,x^{\prime})\&G_{adv}(y,y^{\prime})\rightarrow x^{\prime}\prec
y^{\prime}$

## Topics IV, 5

Here we introduce the semantic relation $S_{loc}$ expressing inherence of one
term in another. Also $x\epsilon y$ expressing that $x$ is a part of $y$. Also
$S_{mod}(x,y,z)$ expresses that $x$ is $z$ only according to aspect $y$. For
example, animals are visible only according to their bodies.

$\ulcorner heksin\urcorner\nprec\ulcorner energeia\urcorner\&\ulcorner
energeia\urcorner\nprec\ulcorner heksin\urcorner$ $\ulcorner
dunamis\urcorner\nprec\ulcorner energeia\urcorner\&\ulcorner
energeia\urcorner\nprec\ulcorner dunamis\urcorner$

Aristotle appears to state that is $a\prec b$ implies that $a$ is the cause of
$b$.

$S_{loc}(x,y)\&y\prec z\rightarrow S_{loc}(x,z)$ $x\prec y\rightarrow x\in
z\rightarrow\neg G_{rel}(y,w,z)$ $x\epsilon y\rightarrow y\nprec x$

What is censurable is not dunamis.

$x\prec y\rightarrow\neg x\Delta_{1}y\&x\Delta_{1}y\rightarrow x\nprec y$
$S_{mod}(x,y,z)\rightarrow\neg x\prec z$

## Topics IV, 6

$xS_{sup}y$ means $x$ is superior to $y$.

$x\biguplus y$ means $x$ is inherent in $y$.

$\wedge(y)\rightarrow x\nprec y$ $\forall x.x\in y\rightarrow z\nprec y\&\neg
zDy$ $x\biguplus y\rightarrow y\nprec x$
$xS_{sup}x^{\prime}\&yS_{sup}y^{\prime}\&O(x,x^{\prime})\&O(y,y^{\prime})\rightarrow(x\prec
y\leftrightarrow x^{\prime}\prec y^{\prime})$ $x\prec
y\rightarrow(\circledcirc x\leftrightarrow\circledcirc y)$ $x\prec
y\&x\biguplus z\rightarrow y\biguplus z$

## Topics V, 2

$xIy\rightarrow(\forall w.G_{inc}(w,y)\rightarrow w\gtrdot x)$
$xIy\rightarrow\neg G_{pol}(y)\&\forall z.G_{inc}(z,y)\rightarrow\neg
G_{pol}(z)$ $xIy\rightarrow\neg G_{pol}(x)$ $xIy\rightarrow\forall
z,w.G_{inc}(z,y)\&G_{inc}(w,y)\rightarrow z=w$ $xIy\rightarrow
G_{1}(w,v,y)\rightarrow\exists u.u\notin v$ $xIy\rightarrow
G_{and}(u,v,y)\rightarrow\neg xIu\&\neg xIv$

## Topics V, 3

$G_{inc}(x,y)\rightarrow\neg xIy$ $G_{inc}(w,y)\&w\prec x\rightarrow\neg wIy$
$G_{inc}(w,y)\&(O(w,x)\vee w\multimap x)\rightarrow\neg xIy$ $x\ominus
y\rightarrow\neg xIy$ $G_{inc}(w,y)\&S_{sens}(w)\rightarrow\neg xIy$
$xDy\rightarrow\neg xIy$

## Topics V, 5

$S_{ess}(x,y)\rightarrow\neg xIy$ $S_{syn}(x,y)\rightarrow\neg xIy$
$S_{hom}(x)\&y\epsilon x\rightarrow(xIz\leftrightarrow yIz)$

## Topics V, 6

$O(x,x^{\prime})\&O(y,y^{\prime})\&xIy\rightarrow x^{\prime}Iy^{\prime}$

## Topics V, 7

$G_{adv}(x,x^{\prime})\&G_{adv}(y,y^{\prime})\&xIy\rightarrow
x^{\prime}Iy^{\prime}$

## Topics V, 8

$S_{more}(a,x,x^{\prime})\&S_{more}(b,y,y^{\prime})\&\neg
x^{\prime}Iy^{\prime}\rightarrow\neg xIy$

## Topics VI, 1

s

$w\in y\&w\notin x\rightarrow\neg xDy$ $(G_{res}(x,y,z)\rightarrow w\nprec
x)\rightarrow\neg wDz$ $x\neq_{e}y\rightarrow\neg xDy$ $\neg
S_{ess}(x,y)\rightarrow\neg xDy$

## Topics VI, 3

$\forall v.w\in v\&G_{inc}(w,x)\rightarrow\neg yDx$ $v\prec
x\&G_{inc}(v,y)\rightarrow\neg xDy$ $G_{inc}(v,w)\&xDv\rightarrow\neg xDw$
$xDy\rightarrow\forall z,w.G_{inc}(z,y)\&G_{inc}(w,y)\rightarrow z=w$

## Topics VI, 4

$xDy\&G_{inc}(w,y)\rightarrow w\gtrdot x$ $xDy\&G_{inc}(z,y)\rightarrow x\neq
z$

## Topics VI, 5

$xDy\rightarrow\exists w,v.G_{res}(w,v,y)\&x\prec w$
$xDy\&G_{res}(w,v,y)\&x\prec w\rightarrow x\triangleleft w$

## Topics VI, 6

A term has a difference iff the difference term belongs to some genus

$\exists y.x\Delta_{1}y\leftrightarrow\exists z.y\Delta_{2}z$

We write $\Delta(x)$ if $x$ satisfies any one of these equivalent conditions.

A difference always has a coordinated difference:

$\Delta(x)\rightarrow\exists y.\Delta_{3}(x,y)$

Coordination is transitive, symmetric and $\neg\Delta_{3}(x,x)$.

If a difference belongs to a genus than so must its coordinates

$y\Delta_{2}z\&\Delta_{3}(z,x)\rightarrow y\Delta_{2}x$

$G_{res}(x,y,z)\&x\Delta_{2}y\&v\ominus y\rightarrow\neg vDz$

$G_{res}(x,y,z)\&x\Delta_{2}y\&(x\in y\vee x\in v)\rightarrow\neg vDz$

$x\Delta_{2}y\rightarrow y\subsetneq_{e}x$

Here is Brunschweig’s view on a difficult topic in this section[2]:

_(…)selon lui, les Platoniciens, partisans de l’existence des Idées. Le lieu
perd son efficacité si 1’on traite le genre comme il faut le traiter à ses
yeux, à savoir précisément non pas comme une chose individuelle, numériquement
une, mais comme un universel : toute longueur particulière est bien soit sans
largeur, soit ayant de la largeur ; mais la longueur en tant que genre est un
universel, qui peut sans contradiction interne regrouper des longueurs
particulières sans largeur et d’autres ayant de la largeur (b28-29). Cette
critique est fondamentale, on le sait, dans l’opposition d’Aristote à Platon._

If a difference is given by negation they there are only two opposite
coordinates.

$x\Delta_{2}y\&z\Delta_{2}y\rightarrow x\prec z\vee z\prec x$

$x\Delta_{2}y\&z\Delta_{1}y\rightarrow z\prec x$

But two subordinate genera of the same genus can have the same difference.
Biped for flying and terrestrial animals, the common genus being animal.

Essence is invariant under spatial and temporal translation. Aristotle
anticipates modern physics.

Difference cannot allow more or less

$x\Delta_{2}y\rightarrow\neg\circledcirc y$

The difference of a relative must be relative:
$x\Delta_{1}y\&S_{rel}(x)\rightarrow S_{rel}(y)$

## Topics VI, 7

Confusing capacity with wanting to so something.

$xDy\rightarrow(\circledcirc x\leftrightarrow\circledcirc y)$

There is also the notion of two things increasing at the same same time

$S_{sim\circledcirc}(x,y)$

$S_{more}(fire,flame,light)$ flame is more fire than light but
$S_{more}(finestparticles,light,flame)$.

Aristotle appears to reject De Morgans law $\neg(A\vee B)\leftrightarrow\neg
A\&\neg B$ in 146a. Beauty = Pleasing to Sight or to the Ear. But what he
means is perhaps $xDy\&xDz$.

## Topics VI, 8

The predicate $G_{rel}(x)$ means that $x$ is a relative. $G_{rel2}(x,y,z)$
means that an expression is a relative saturated by an expression.

## Topics VI, 11

Interesting topics about complex terms and definitions which seems to reflect
at places a much earlier phase of the Topics in which definition and property
were not yer fully differentiated.

## Topic VI, 12

Error of too vague saturation involving $G_{rel2}(x,y,z)$. For instance saying
that medicine is the science of everything that is.

## Topics VI, 13

We need additional semantic primitives $G_{\&}(x,y)$, $S_{prod}(x,y,z)$.

At 150a Aristotle forsakes the requirement that a definition be comprised of
genus and difference.

Aristotle admits that what corresponds to modern ’negation’ is anti-monotonic,
if $A$ implies $B$ then $\neg B$ implies $\neg A$. The semantic relation of
opposition functions in a different way. It is, as we have seen, monotonic for
$\prec$. If we denote the unique $x^{\prime}$ such that $O(x,x^{\prime})$ by
$x^{\circ}$ then $x\prec y$ implies $x^{\circ}\prec y^{\circ}$.

Classical negation intertwines with classical conjunction via the De Morgan
laws $\neg(A\&B)\leftrightarrow\neg A\vee\neg B$. But Aristotle states clearly
that for the operation of ’this and that’ (which we denote by $x+y$) we have
that $(x+y)^{\circ}=x^{\circ}+y^{\circ}$.

The outstanding problem of this section involves how finite sets (or pairs)
are formed and how predicates are (mereologically) applied to them, more
specifically predicates of the form $A+B$.

In the example of justice Aristotle seems to implicitly accept two
definitions:

$\\{x_{1},x_{2}\\}\in y\leftrightarrow x_{1}\in y\vee x_{2}\in y$
$x\in(y+z)\leftrightarrow x\in y\&x\in z\text{ \quad for $x$ not a set}$

Assume as in Aristotle’s example that $x_{1}\in y,x_{1}\in z^{\circ},x_{2}\in
y^{\circ},x_{2}\in z$. Then using the second definition we get that

$\\{x_{1},x_{2}\\}\in(y+z)\leftrightarrow\\{x_{1},x_{2}\\}\in
y\&\\{x_{1},x_{2}\\}\in z$

which by the first definition is equivalent to

$(x_{1}\in y\vee x_{2}\in y)\&(x_{1}\in z\vee x_{2}\in z)$

which obtains. And in the same way we can show that

$\\{x_{1},x_{2}\\}\in(y+z)^{\circ}=y^{\circ}+z^{\circ}$

Thus the pair of men in Aristotle’s example would be both just and unjust.

Ancient mereology apparently distinguished different degrees of ’holism’, from
a mere abstract ’set’ in which they is no difference between a set and the sum
of its part to products in which the elements are changed and incorporated or
integrated into a living whole.

## Topics VI, 14

Need primitives $S_{comp}(x,y,z)$, $S_{comp1}(x,y,z,w)$.

## References

  * [1] Aristote (1967). _Topiques_ , Tome I, Livres I-IV, Texte Établi e Traduit par Jacques Brunschwig, Les Belles Lettres, Paris.
  * [2] Aristote (2007). _Topiques_ , Tome II, Livres V-VIII, Texte Établi e Traduit par Jacques Brunschwig, Les Belles Lettres, Paris.
  * [3] Aristotle, (1853), _The Organon, Or Logical Treatises of Aristotle with the Introduction of Porphyry_ , Literally translated, with notes, syllogistic examples, analysis and introduction by O.F. Owen, Vol. II, Henry G. Bohn, London.
  * [4] Aristotle, (1996). _The Categories, On Interpretation_ , ed. and tr. by H.P. Cooke, _Prior Analytics_ , ed. and tr. by H. Tredennick, Loeb CL 325, Harvard.
  * [5] Bealer, G. (1982). _Quality and Concept_ , Oxford University Press.
  * [6] Bealer, G. (1983). _Completeness in the theory of properties, relations, and propositions_ , Journal of Symbolic Logic 48 (2): 415-426.
  * [7] Bobzien, S. 2019. _Stoic Sequent Logic and Proof Theory_ , History and Philosophy of Logic , Volume 40, Issue 3.
  * [8] Kieffer, J. S. 1964. _Galen’s Institutio Logica: English Translation, Introduction, and Commentary_ , John Hopkins University Press: Baltimore.
  * [9] Hamelin, O. 1920. _Le Système D’Aristote_ , Paris, Librarie Félix Alcan.
  * [10] Kneale, W. and Kneale, M. (1962). _The Development of Logic_ , Oxford University Press, London.
  * [11] Malink, M. (2013). _Aristotle’s Modal Syllogistic_ , Harvard University Press.
  * [12] Porphyry (2003). _Porphyry - Introduction_ , Translated, with a Commentary by Jonathan Barnes, Clarendon Press, Oxford.
  * [13] Primavesi O. 1966. _Die Aristotelische Topik: Ein Interpretationsmodell und sein Erprobung am Beispiel von Topik B_ , C.H. Beck’sche Verlagsbuchhandlung: München.
  * [14] Protin, C. L. (2022). _A Logic for Aristotle’s Modal Syllogistic_ , History and Philosophy of Logic. https://doi.org/10.1080/01445340.2022.2107382.
  * [15] Protin, C. L. (2023). _Modern Light on Ancient Logic_. http://dx.doi.org/10.48550/arXiv.2211.06447.
  * [16] Slomkowski, P. (1997). _Aristotle’s Topics_ , Brill.
  * [17] Zalta, E. (1988)._Intensional Logic and the Metaphysics of Intentionality_ , Cambridge, MA: MIT Press.


EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2016-287 LHCb-PAPER-2016-044 4 April 2017

Search for $CP$ violation in the phase space of
$D^{0}\rightarrow\pi^{+}\pi^{-}\pi^{+}\pi^{-}$ decays

The LHCb collaboration†††Authors are listed at the end of this Letter.

A search for time-integrated $CP$ violation in the Cabibbo-suppressed decay
$D^{0}\rightarrow\pi^{+}\pi^{-}\pi^{+}\pi^{-}$ is performed using an unbinned,
model-independent technique known as the energy test. This is the first
application of the energy test in four-body decays. The search is performed
for $P$-even $CP$ asymmetries and, for the first time, is extended to probe
the $P$-odd case. Using proton-proton collision data corresponding to an
integrated luminosity of 3.0 fb-1 collected by the LHCb detector at centre-of-
mass energies of $\sqrt{s}=$7 TeV and 8 TeV, the world’s best sensitivity to
$CP$ violation in this decay is obtained. The data are found to be consistent
with the hypothesis of $CP$ symmetry with a $p$-value of $(4.6\pm 0.5)\%$ in
the $P$-even case, and marginally consistent with a $p$-value of $(0.6\pm
0.2)\%$ in the $P$-odd case, corresponding to a significance for $CP$ non-
conservation of 2.7 standard deviations.

Published in Phys. Lett. B 769 (2017) 345-356

© CERN on behalf of the LHCb collaboration, licence CC-BY-4.0.

##  1 Introduction

The decay
${{D}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ (charge-
conjugate decays are implied unless stated otherwise) proceeds via a singly
Cabibbo-suppressed ${c}\\!\rightarrow{d}{u}{\overline{d}}$ transition with an
admixture from a ${c}\\!\rightarrow{u}g$ gluonic-penguin transition. Within
the Standard Model (SM), these amplitudes have different weak phases, and the
interference between them may give rise to a violation of the charge-parity
($C\\!P$) symmetry in the decay. Another necessary condition for this direct
$C\\!P$ violation to occur is interference of at least two amplitudes with
different strong phases. The strong phase differences are known to be sizeable
in charm decays and can enter through the resonances that abundantly
contribute to the four-body final states. The sensitivity to $C\\!P$ violation
is usually best for decays where the strong phases between interfering
resonances have large differences. A rich spectrum of resonances contributes
to the decay
${{D}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$, which
according to the amplitude analysis performed by the FOCUS collaboration [1],
is dominated by the amplitudes for ${{D}^{0}}\rightarrow
a_{1}(1260)^{+}\pi^{-}$ with
$a_{1}(1260)^{+}\rightarrow{{\rho}^{0}}(770)\pi^{+}$ and for
${{D}^{0}}\rightarrow{{\rho}^{0}}(770){{\rho}^{0}}(770)$.

In the SM, violation of the $C\\!P$ symmetry in the charm sector is expected
at or below the $\mathcal{O}$($10^{-3}$) level [2]. Contributions from
particles that are proposed to exist in extensions of the SM may participate
in higher-order loop contributions (penguin diagrams) and enhance the level of
$C\\!P$ violation. Multibody decays, such as
${{D}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$, allow
the $C\\!P$ asymmetries to be probed across the phase space of the decay, and
these local $C\\!P$ asymmetries may be larger than global $C\\!P$ asymmetries.

The analysis of
${{D}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ is
primarily sensitive to direct $C\\!P$ violation. In addition to this direct
$C\\!P$ violation, the time-integrated $C\\!P$ asymmetry in
${{D}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ decays
can also receive an indirect contribution arising from either
${D}^{0}$-${\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ mixing or
interference between direct decays and decays following mixing. While direct
$C\\!P$ asymmetry depends on the decay mode, indirect $C\\!P$ violation is
expected to be universal within the SM. The time-dependent measurements of
${{D}^{0}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}$, ${K}^{-}$ ${K}^{+}$ decays
constrain the indirect $C\\!P$ asymmetry below the $\mathcal{O}$($10^{-3}$)
level [3], which is beyond the sensitivity of this analysis.

Previously, the most sensitive search for $C\\!P$ violation in the
${{D}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ decay
was performed by the LHCb collaboration with data collected in 2011 [4]. A
binned $\chi^{2}$ technique ($S_{{C\\!P}}$) was used to exclude
$C\\!P$-violating effects at the 10% level. Four-body decays require five
independent variables to fully represent the phase space. Consequently, binned
analyses will have a trade-off between minimising the number of bins, in order
to maximise the number of events per bin, and retaining sensitivity to the
interference between all contributing resonances. The measurement presented
here includes data collected in 2012, resulting in a signal sample that is
about three times larger than in the previous LHCb analysis. The method
exploited here, known as the energy test, is an unbinned technique, which is
advantageous in the analysis of multibody decays.

The energy test was applied for the first time to search for $C\\!P$ violation
in decays of ${{D}^{0}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{0}}$ [5];
here we present its first application to four-body decays. The energy test is
used to assess the compatibility of the observed data with $C\\!P$ symmetry.
It is sensitive to local $C\\!P$ violation in the phase space and not to
global asymmetries, which may also arise from different production cross-
sections of ${D}^{0}$ and ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ mesons at a proton-proton
collider. The method identifies the phase space regions in which $C\\!P$
violation is observed. Being model-independent, this method neither identifies
which amplitudes contribute to the observed asymmetry nor measures the actual
asymmetry. Consequently, a model-dependent amplitude analysis of ${D}^{0}$ and
${\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ decays would be
required if evidence for a non-zero $C\\!P$ asymmetry is obtained.

The analysis presented here probes separately both $P$-even and $P$-odd
$C\\!P$ asymmetries. The $P$-even test is performed through the comparison of
the distribution of events in the ${D}^{0}$ and ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ phase spaces, characterised by
squared invariant masses. Additionally characterising the events using a
triple product of final-state particle momenta [6, 7, 8] gives sensitivity to
$P$-odd amplitudes, and thus allows the first test for $P$-odd $C\\!P$
asymmetries in an unbinned model-independent technique.

##  2 Detector and reconstruction

The LHCb detector [9, 10] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high-precision tracking
system consisting of a silicon-strip vertex detector surrounding the $pp$
interaction region, a large-area silicon-strip detector located upstream of a
dipole magnet with a bending power of about $4{\mathrm{\,Tm}}$, and three
stations of silicon-strip detectors and straw drift tubes placed downstream of
the magnet. The tracking system provides a measurement of momentum, $p$, of
charged particles with a relative uncertainty that varies from 0.5% at low
momentum to 1.0% at 200${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The minimum
distance of a track to a primary $pp$ interaction vertex (PV), the impact
parameter (IP), is measured with a resolution of
$(15+29/\mbox{$p_{\mathrm{T}}$}){\,\upmu\mathrm{m}}$, where $p_{\mathrm{T}}$
is the component of the momentum transverse to the beam, in
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. Different types of charged hadrons are
distinguished using information from two ring-imaging Cherenkov detectors.
Photons, electrons and hadrons are identified by a calorimeter system
consisting of scintillating-pad and preshower detectors, an electromagnetic
calorimeter and a hadronic calorimeter. Muons are identified by a system
composed of alternating layers of iron and multiwire proportional chambers.

This analysis uses the data from $pp$ collisions collected by the LHCb
experiment in 2011 and 2012 corresponding to integrated luminosities of,
respectively, $1\mbox{\,fb}^{-1}$ and $2\mbox{\,fb}^{-1}$ at centre-of-mass
energies of $7\mathrm{\,Te\kern-1.00006ptV}$ and
$8\mathrm{\,Te\kern-1.00006ptV}$. The polarity of the dipole magnet is
reversed periodically throughout data-taking. The configuration with the
magnetic field pointing upwards (downwards), bends positively (negatively)
charged particles in the horizontal plane towards the centre of the LHC.
Similar amounts of data were recorded with each polarity, which reduces the
effect of charge-dependent detection and reconstruction efficiencies on
results obtained from the full data sample.

In the simulation, $pp$ collisions are generated using Pythia 8 [11] with a
specific LHCb configuration [12]. Decays of hadronic particles are described
by EvtGen [13]. The interaction of the generated particles with the detector
and its response are implemented using the Geant4 toolkit [14,
*Agostinelli:2002hh] as described in Ref. [16].

The online event selection is performed by a trigger, which consists of a
hardware stage, based on high-$p_{\mathrm{T}}$ signatures from the calorimeter
and muon systems, followed by a two-level software stage. Events are required
to pass both hardware and software trigger levels. The software trigger at its
first level applies partial event reconstruction. It requires at least one
good quality track associated with a particle having high $p_{\mathrm{T}}$ and
not originating from a PV.

A second-level software trigger, optimised for four-body hadronic charm
decays, fully reconstructs
${{D}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$
candidates coming from ${{D}^{*+}}\\!\rightarrow{{D}^{0}}\pi_{s}^{+}$ decays.
The charge of the soft pion ($\pi_{s}$) tags the flavour of the $D$ mesons at
production, which is needed as ${\pi}^{+}$ ${\pi}^{-}$ ${\pi}^{+}$ ${\pi}^{-}$
is a self-conjugate final state accessible to both ${D}^{0}$ and ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ decays. The trigger selection
ensures the suppression of combinatorial background while minimising the
distortion of the acceptance in the phase space of the decay. The trigger
requires a four-track secondary vertex with all tracks being of good quality
and passing minimum momentum and transverse momentum requirements. The pions
from the candidate ${D}^{0}$ decay are required to have a large impact
parameter significance ($\chi^{2}_{\rm IP}$) with respect to all PVs, where
$\chi^{2}_{\rm IP}$ is defined as the difference in the vertex-fit $\chi^{2}$
of a PV reconstructed with and without the considered track. A part of the
data collected in 2011 was taken with a different second-level trigger
selection. In this selection only
${{D}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$
candidates are reconstructed while the ${D}^{*+}$ reconstruction is performed
only during the offline selection. For a part of the data collected in 2012,
additional events were selected in the trigger from the partial reconstruction
of ${{D}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$
candidates arising from ${{D}^{*+}}\\!\rightarrow{{D}^{0}}\pi_{s}^{+}$ decays,
using only information from one ${\pi}^{+}$ ${\pi}^{-}$ pair and a soft pion.

##  3 Offline event selection

In the offline selection, signal candidates are required to be associated to
candidates that passed the online selection described in the previous section.
In addition, the offline selection imposes more stringent kinematic criteria
than those applied in the trigger. The ${D}^{*+}$ and ${D}^{0}$ candidates
must have $\mbox{$p_{\mathrm{T}}$}>500{\mathrm{\,Me\kern-1.00006ptV\\!/}c}$.
All the candidate pion tracks, those from the ${D}^{0}$ decay products and the
$\pi_{s}$ mesons, are required to have
$\mbox{$p_{\mathrm{T}}$}>350{\mathrm{\,Me\kern-1.00006ptV\\!/}c}$ and
$\mbox{$p$}>3{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to reduce the combinatorial
background. The candidate ${D}^{0}$ decay products must form a good quality
vertex. As a consequence of the non-negligible ${D}^{0}$ lifetime, the
${D}^{0}$ decay vertex should typically be significantly displaced from the
PV; this is ensured by applying a selection on the significance of the
${D}^{0}$ candidate flight distance. Charm mesons from $b$ hadron decays have
larger IPs due to the comparatively long $b$ hadron lifetimes. This secondary
charm contribution is suppressed by imposing an upper limit on the
$\chi^{2}_{\rm IP}$ of the ${D}^{0}$ candidate. Background from
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}$ decays,
with a kaon misidentified as a pion, is reduced by placing tight requirements
on the ${\pi}^{\pm}$ particle identification based on the ring-imaging
Cherenkov detectors. The contribution of the Cabibbo-favoured
${{D}^{0}}\\!\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{\pi}^{+}}{{\pi}^{-}}$ decay is found to be below the percent level. As
investigated in Ref. [4], partially reconstructed or misreconstructed
multibody ${D}_{(s)}$ decays (e.g., decays with a missing pion or a kaon
misidentified as pion) do not give rise to peaking backgrounds under the
${{D}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ signal.

Constraints on the decay kinematics are applied to improve mass and momentum
resolutions. The four pions from the ${D}^{0}$ decay are constrained to come
from a common vertex and the decay vertex of the ${D}^{*+}$ candidate is
constrained to coincide with its PV [17]. These constraints are applied in the
determination of the mass difference, $\Delta m$, between the ${D}^{0}$ and
the ${D}^{*+}$. The ${D}^{0}$ is constrained to its nominal mass in the
determination of the kinematics in the
${{D}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ decay. A
requirement on fit quality for the ${D}^{*+}$ vertex fits efficiently
suppresses combinatorial background. This requirement also suppresses the
contribution from ${D}^{*+}$ mesons originating from long-lived $b$ hadrons.
The remaining component in the analysis from this source is not sensitive to
$C\\!P$ asymmetries in $b$ hadrons as the flavour tag is obtained from the
charge of the $\pi_{s}$ in the decay of the ${D}^{*+}$ meson.

The $\pi_{s}$ is a low-momentum particle, with the consequence that the large
deflection in the magnetic field leads to different acceptances for the two
charges. Consequently, the soft pion is restricted to the region where the
detection asymmetry is small. This is achieved through the application of
fiducial cuts on the soft pion momentum, following Ref. [18]. As the
kinematics of the slow pion are largely uncorrelated with the ${D}^{0}$ phase
space, the $\pi_{s}$ detection asymmetry would result in a global asymmetry to
which this analysis is not sensitive. There are, however, differences in the
detection efficiencies of the ${D}^{0}$ and ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ daughters that may introduce
additional asymmetries localised in the phase space of ${D}^{0}$ decays, and
which are discussed in detail in Sect. 7.

The signal region in the ${D}^{0}$ invariant mass is defined as
$1852<m({{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}})<1882{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$,
corresponding to a full range of about four times the mass resolution. The
signal yield is estimated from the $\Delta m$ distribution, which is shown in
Fig. 1 for the 2012 data. These $\Delta m$ distributions are modelled by the
sum of three Gaussian functions for signal and a second-order polynomial
multiplied by a threshold function $\sqrt{1-m_{\pi}/{\Delta m}}$, where
$m_{\pi}$ is the pion mass, describing combinatorial and random soft-pion
backgrounds. The selected samples comprise 320,000 and 720,000 signal
candidates in the 2011 and 2012 data with purities of $97\%$ and $96\%$,
respectively. The final signal sample is selected requiring $|{\Delta
m}-145.44|<0.45{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, which corresponds to
selecting a region with a width roughly twice the effective $\Delta m$
resolution.

Figure 1: Distribution of $\Delta m$ with fit overlaid for the selected
${D}^{*+}$ candidates in the 2012 data. The data points and the contributions
from signal, background, and their total obtained from the fit are shown.

##  4 Description of the phase space

Five coordinates are required for a full description of the phase space of
four-body decays [19]. In contrast to three-body decays, there is no standard
or commonly preferred choice of coordinates. Two-body and three-body invariant
mass combinations of the pions are used as coordinates here. The energy test
performed here is a statistical method comparing the distributions of
${D}^{0}$ and ${\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$
candidates in phase space (see Sect. 5). Therefore, it is sensitive to the
position of an event in phase space and to the choice of coordinates spanning
this phase space. The choice of coordinates influences the sensitivity of the
analysis as it will change the distance between events in the phase space.
Furthermore,
${{D}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ decays
contain two ${\pi}^{+}$ and two ${\pi}^{-}$ mesons. The pions of the same
charge can be interchanged; as a result each decay can be placed at four
points in phase space. The energy test is sensitive to such pion interchange.
To obtain both a unique output and optimal sensitivity from the energy test,
an ordering of the input variables of the energy test is defined in the
following.

The order of the charges of the four pions in the ${D}^{0}$ decay
$\pi_{1}\pi_{2}\pi_{3}\pi_{4}$ is fixed to ${\pi}^{+}$ ${\pi}^{-}$ ${\pi}^{+}$
${\pi}^{-}$.111In a ${\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$
decay the order of $\pi_{1}\pi_{2}\pi_{3}\pi_{4}$ is charge-conjugated,
${\pi}^{-}$ ${\pi}^{+}$ ${\pi}^{-}$ ${\pi}^{+}$, with respect to that of the
${D}^{0}$ decay. There are four two-body combinations in which resonances can
be formed and these are the ${\pi}^{+}$ ${\pi}^{-}$ pairs: $\pi_{1}\pi_{2}$,
$\pi_{1}\pi_{4}$, $\pi_{3}\pi_{2}$, and $\pi_{3}\pi_{4}$. Likewise, there are
four three-body combinations: the two of positive charge,
$\pi_{1}\pi_{2}\pi_{3}$ and $\pi_{1}\pi_{3}\pi_{4}$, and the two of negative
charge, $\pi_{1}\pi_{2}\pi_{4}$ and $\pi_{2}\pi_{3}\pi_{4}$. The invariant
masses of all possible ${\pi}^{+}$ ${\pi}^{-}$ pairs are calculated and sorted
for each event. The ${\pi}^{+}$ ${\pi}^{-}$ pair with the largest invariant
mass is fixed to be $\pi_{3}\pi_{4}$, which fully determines the order of all
four pions. As only a small fraction of the $\rho(770)$ resonance, either
produced directly from the ${D}^{0}$ decay or through $a_{1}(1260)$ decays,
contributes to the largest $m({{\pi}^{+}}{{\pi}^{-}})$, the $\pi_{3}\pi_{4}$
combination is excluded from the coordinates used. While any combination of
five variables covers the full phase space, the choice made here is to keep
variables sensitive to the presence of the main resonances. The two-body and
three-body mass combinations that do not contain the $\pi_{3}\pi_{4}$ pair are
used for the energy test coordinates. This results in five invariant masses,
$\pi_{1}\pi_{2}$, $\pi_{1}\pi_{4}$, $\pi_{2}\pi_{3}$, $\pi_{1}\pi_{2}\pi_{3}$
and $\pi_{1}\pi_{2}\pi_{4}$, which are expected to cover most of the
intermediate resonance contributions.

The choice of using only invariant masses has a limitation. Invariant masses
are even under parity transformation, and a comparison of the ${D}^{0}$ and
${\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ samples probes only
$P$-even $C\\!P$ asymmetries. In four-body decays, however, $P$-odd amplitudes
can also be present. In
${{D}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ decays,
there is only one significant $P$-odd amplitude, the perpendicular helicity
($A_{\perp}$) of $D^{0}\rightarrow{{\rho}^{0}}(770){{\rho}^{0}}(770)$ decays.
Alternatively, in the partial-wave basis, it is the amplitude corresponding to
the P-wave $D^{0}\rightarrow{{\rho}^{0}}(770){{\rho}^{0}}(770)$ decays. Its
contribution to the total
${{D}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ width is
about 6% [20]. The default approach is extended to make a complementary test
of the $P$-odd $C\\!P$ asymmetry, which may arise from interference between
$P$-odd and $P$-even amplitudes. This is discussed further in Ref. [8].

Triple products, which are by definition $P$-odd, can be used to probe $P$-odd
$C\\!P$ violation [21]. These asymmetries are proportional to the cosine of
the strong-phase difference between the interfering partial waves [6], and
thus will be enhanced where $P$-even $C\\!P$ asymmetries, proportional to the
sine of the strong-phase difference, lack sensitivity. A triple product
$C_{T}=\vec{p}_{1}\cdot(\vec{p}_{2}\times\vec{p}_{3})$ is constructed for
${D}^{0}$ decays, where pion momenta are measured in the ${D}^{0}$ rest frame.
Here $\vec{p}_{1}$, $\vec{p}_{2}$ and $\vec{p}_{3}$ are the vector momenta of
$\pi_{1}$, $\pi_{2}$ and $\pi_{3}$ sorted as described above (i.e., $\pi_{4}$
is excluded). The corresponding triple product for the ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ decays is obtained by applying
the $C\\!P$ transformation, ${C\\!P}(C_{T})=-C(C_{T})=-\overline{C}_{T}$. The
$\overline{C}_{T}$ observable is constructed by charge conjugating the pions
entering $C_{T}$ (i.e., the excluded pion in $\overline{C}_{T}$ is the
${\pi}^{+}$ in the largest $m({{\pi}^{+}}{{\pi}^{-}})$ combination). The total
sample is divided into four subsamples according to the ${D}^{0}$ flavour and
the sign of the triple product:

$[\rm{I}]\ \textit{{${D}^{0}$}}(\textit{C}_{\textit{T}}>0),\ \ [\rm{II}]\
\textit{{${D}^{0}$}}(\textit{C}_{\textit{T}}<0),\ \ [\rm{III}]\
\textit{{${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$}}(-\overline{\textit{C}}_{\textit{T}}>0),\
\ [\rm{IV}]\ \textit{{${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$}}(-\overline{\textit{C}}_{\textit{T}}<0).$
(1)

Samples $\rm{I}$ and $\rm{III}$ are related by the $C\\!P$ transformation, and
so are $\rm{II}$ and $\rm{IV}$. The yields of these four samples are listed in
Table 1. The test for the presence of $P$-even $C\\!P$ asymmetry is performed
by comparing the combined sample $\rm{I}+\rm{II}$ with the combined sample
$\rm{III}+\rm{IV}$. This corresponds to the integration over $C_{T}$ and is
the default test, in which ${D}^{0}$ and ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ samples are compared in the
phase space spanned with invariant masses only. Similarly, the test for a
$P$-odd $C\\!P$ asymmetry is performed by comparing the combined sample
$\rm{I}+\rm{IV}$ with the combined sample $\rm{II}+\rm{III}$. This comparison
is performed in the same phase space as the default $P$-even approach and
allows the $P$-odd contribution to the $C\\!P$ asymmetry to be probed, since
the $P$-even contribution cancels [8]. No triple-product asymmetry
measurements exist for
${{D}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ decays
and the previous LHCb study [4] was performed in the phase space based on the
invariant masses only. Consequently, this is the first time a $P$-odd $C\\!P$
asymmetry is investigated in this decay mode.

Table 1: The yields of signal events in the four samples that obtained from fits to the $\Delta m$ distribution. Sample | $\rm{I}$ | $\rm{II}$ | $\rm{III}$ | $\rm{IV}$
---|---|---|---|---
Yields | 256 466$\pm$629 | 246 629$\pm$519 | 258 274$\pm$574 | 246 986$\pm$607

##  5 Energy test

Model-independent searches for local $C\\!P$ violation are typically carried
out using a binned $\chi^{2}$ approach to compare the relative density in a
bin of phase space of a decay with that of its $C\\!P$-conjugate. This method
was used in a previous study of
${{D}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ decays
[4]. As discussed in the previous section, five coordinates are required to
describe four-body decays. A model-independent unbinned statistical method
called the energy test was introduced in Refs. [22, 23]. The potential for
increased sensitivity of this method over binned $\chi^{2}$ analyses in Dalitz
plot analyses was shown in Refs. [24, 8] and it was first applied to
experimental data in Ref. [5].

This Letter introduces the first application of the energy test technique to
four-body decays, where it is used to compare two event samples in tests of
both $P$-even and $P$-odd type $C\\!P$ violation. The $P$-even energy test
separates events according to their flavour, and then compares these ${D}^{0}$
and ${\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ samples. The
$P$-odd energy test separates events using both their flavour and sign of the
triple product, as described in the previous section.

A test statistic, $T$, is used to compare the average distances of events in
phase space. The variable $T$ is based on a function
$\psi_{ij}\equiv\psi(d_{ij})$ which depends on the distance $d_{ij}$ between
events $i$ and $j$. It is defined as

$T=\sum_{i,j>i}^{n}\dfrac{\psi_{ij}}{n(n-1)}+\sum_{i,j>i}^{\overline{n}}\dfrac{\psi_{ij}}{\overline{n}(\overline{n}-1)}-\sum_{i,j}^{n,\overline{n}}\dfrac{\psi_{ij}}{n\overline{n}},$
(2)

where the first and second terms correspond to an average weighted distance
between events within the $n$ events of the first sample and between the
$\overline{n}$ events of the second sample, respectively. The third term
measures the average weighted distance between events in the first sample and
events in the second sample. If the distributions of events in both samples
are identical, $T$ will randomly fluctuate around a value close to zero.

The normalisation factors in the denominators of the terms of Eq. 2 remove the
impact of global asymmetries between ${D}^{0}$ and ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ samples. In the $P$-odd test,
subsamples of both ${D}^{0}$ and ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ samples are combined.
Consequently, any global asymmetries in these could result in local
asymmetries in the samples used for the $P$-odd test. Therefore, for the
$P$-odd test the global asymmetry between ${D}^{0}$ and ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ is removed by randomly
rejecting some of the ${\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$
candidates to equalise the size of the samples of the two flavours before
combining the events into the two samples that are compared.

The function $\psi$ should decrease with increasing distance $d_{ij}$ between
events $i$ and $j$, in order to increase the sensitivity to local asymmetries.
A Gaussian function is chosen, $\psi(d_{ij})=e^{-d_{ij}^{2}/2\delta^{2}}$,
with a tuneable parameter $\delta$ (see Sect. 6) that describes the effective
radius in phase space within which a local asymmetry is measured. Thus, this
parameter should be larger than the resolution of $d_{ij}$ but small enough
not to dilute locally varying asymmetries. The distance between two points is
obtained using the five squared invariant masses discussed in the previous
section and calculated as

$d_{ij}^{2}=(m_{12}^{2,j}-m_{12}^{2,i})^{2}+(m_{14}^{2,j}-m_{14}^{2,i})^{2}+(m_{23}^{2,j}-m_{23}^{2,i})^{2}+(m_{123}^{2,j}-m_{123}^{2,i})^{2}+(m_{124}^{2,j}-m_{124}^{2,i})^{2}.$
(3)

In the case of $C\\!P$ violation, the average distances entering in the third
term of Eq. 2 are larger than in the other terms. Due to the characteristics
of the $\psi$ function, this leads to a reduced magnitude of this third term
relative to the other terms. Therefore, larger $C\\!P$ asymmetries lead to
larger values of $T$. This is translated into a $p$-value under the hypothesis
of $C\\!P$ symmetry by comparing the $T$ value observed in data to a
distribution of $T$ values obtained from permutation samples. The permutation
samples are constructed by randomly assigning events to either of the samples,
thus simulating a situation without $C\\!P$ violation. The $p$-value for the
no-$C\\!P$-violation hypothesis is obtained as the fraction of permutation $T$
values greater than the observed $T$ value.

For scenarios where the observed $T$ value lies well within the range of
permutation $T$ values, the $p$-value can be calculated by simply counting how
many permutation $T$ values are larger than the observed one. If large $C\\!P$
violation is observed, the observed $T$ value is likely to lie outside the
range of permutation $T$ values. In this case the permutation $T$ distribution
can be fitted with a generalised-extreme-value (GEV) function, as demonstrated
in Refs. [22, 23] and used in Ref. [5]. The $p$-value from the fitted $T$
distribution can be calculated as the fraction of the integral of the function
above the observed $T$ value. The uncertainty on the $p$-value is obtained by
randomly resampling the fit parameters within their uncertainties, taking into
account their correlations, and by extracting a $p$-value for each of these
generated $T$ distributions. The spread of the resulting $p$-value
distribution is used to set 68% confidence intervals. A 90% confidence-level
upper limit is quoted where no significantly non-zero $p$-value can be
obtained from the fit.

Figure 2:  (a,b) Distribution of permutation $T$-values fitted with a GEV
function for the simulated sample and showing the measured $T$-value as a
vertical line, and (c,d,e,f) local asymmetry significances. Left column plots
are for a $P$-even $C\\!P$-violation test with a $3^{\circ}$ phase $C\\!P$
violation introduced in the $a_{1}(1260)^{+}$ resonance (see text), projected
onto the (c) $m(\pi_{1}\pi_{2}\pi_{3})$ and (e) $m(\pi_{1}\pi_{2})$ axes.
Right column plots are for a $P$-odd $C\\!P$-violation test with $3^{\circ}$
phase $C\\!P$ violation introduced in the P-wave
${{\rho}^{0}}(770){{\rho}^{0}}(770)$ resonance projected onto the same axes.
In plots (c,d,e,f) the grey area corresponds to candidates with a contribution
to the $T$-value of less than one standard deviation. The pink (blue) area
corresponds to candidates with a positive (negative) contribution to the
$T$-value. Light, medium or dark shades of pink and blue correspond to between
one and two, two and three, and more than three standard-deviation
contributions, respectively.

The number of permutations is constrained by the available computing time. The
default $p$-value extraction, defined before obtaining the result from the
data, uses the counting method as long as at least three permutation $T$
values are found to be larger than the observed $T$ value. Otherwise, the
$p$-value is determined by integrating the fitted GEV function. The $p$-values
presented here are based on over 1000 permutations for the default data
results and on 100 permutations for the sensitivity studies (see Sect. 6).

A visualisation of regions of significant asymmetry is obtained by assigning
an asymmetry significance to each event. The contributions of a single event
in one sample, $T_{i}$, and a single event in the other sample,
$\overline{T}_{i}$, to the total $T$ value are given by

$T_{i}=\dfrac{1}{2n\left(n-1\right)}\sum_{j\neq
i}^{n}\psi_{ij}-\dfrac{1}{2n\overline{n}}\sum_{j}^{\overline{n}}\psi_{ij},$
(4)
$\overline{T}_{i}=\dfrac{1}{2\overline{n}\left(\overline{n}-1\right)}\sum_{j\neq
i}^{\overline{n}}\psi_{ij}-\dfrac{1}{2n\overline{n}}\sum_{j}^{n}\psi_{ij}.$
(5)

Having obtained the $T_{i}$ and $\overline{T}_{i}$ values for all events, the
permutation method is also used here to define the significance of each event.
For a given event $i$ the expected $T_{i}$ distribution in the case of $C\\!P$
symmetry is obtained by using the permutation method. The distributions of the
smallest negative and largest positive $T_{i}$ values of each permutation,
$T_{i}^{\rm min}$ and $T_{i}^{\rm max}$, are used to assign significances of
negative and positive asymmetries, respectively. If for real data the $T_{i}$
value falls in the right (left) tail of the distribution of $T_{i}^{\rm max}$
($T_{i}^{\rm min}$) containing 32%, 5% or 0.3% of the values it is assigned a
positive (negative) significance of 1, 2 or 3 $\sigma$, respectively. The same
procedure is applied to the $\overline{T}_{i}$ distribution, leading to a
phase space with an inverted asymmetry pattern. This method is illustrated in
Sect. 6.

The practical limitation of this method is that the number of mathematical
operations scales quadratically with the sample size. Furthermore, a
significant number of permutations is required to get a sufficient precision
on the $p$-value. In this analysis, the method is implemented using parallel
computing on graphics processing units (GPUs) [25].

##  6 Sensitivity studies

To interpret the results, a study of the sensitivity of the present data
sample to different types of $C\\!P$ violation is required. The sensitivity is
examined based on simplified simulation samples generated according to a
preliminary version of the model in Ref. [26] based on CLEO-c measurements.
The generation is performed using MINT, a software package for amplitude
analysis of multibody decays that has also been used by the CLEO collaboration
[27]. Given the high purity obtained in the selection, backgrounds are
neglected in these sensitivity studies.

The variation of the selection efficiency across phase space is taken into
account in these studies. This efficiency is measured using a sample of events
based on the full LHCb detector simulation. The efficiency varies mainly as a
function of the momentum of the lowest momentum pion in the event in the
${D}^{0}$ rest frame, and this efficiency variation is parameterised. However,
the dependence of the efficiency on this parametrisation is relatively weak.
For further studies the efficiency, based on the parametrisation, is then
applied to the simplified simulated data sets.

Table 2: Overview of sensitivities to various $C\\!P$-violation scenarios in simulation. $\Delta A$ and $\Delta\phi$ denote, respectively, the relative change in magnitude and the change in phase of the amplitude of the resonance $R$. The P-wave ${{\rho}^{0}}(770){{\rho}^{0}}(770)$ is a $P$-odd component. The phase change in this resonance is tested with the $P$-odd $C\\!P$-violation test. Results for all other scenarios are given with the standard $P$-even test. $R$ (partial wave) ($\Delta A$, $\Delta\phi$) |  $p$-value (fit)
---|---
$a_{1}\\!\rightarrow{{\rho}^{0}}\pi$ (S) ($5\%,0^{\circ}$) | $2.6^{+3.4}_{-1.7}\times 10^{-4}$
$a_{1}\\!\rightarrow{{\rho}^{0}}\pi$ (S) ($0\%,3^{\circ}$) | $1.2^{+3.6}_{-1.2}\times 10^{-6}$
${\rho}^{0}$ ${\rho}^{0}$ (D) ($5\%,0^{\circ}$) | $3.8^{+2.9}_{-1.9}\times 10^{-3}$
${\rho}^{0}$ ${\rho}^{0}$ (D) ($0\%,4^{\circ}$) | $9.6^{+24}_{-7.2}\times 10^{-6}$
${\rho}^{0}$ ${\rho}^{0}$ (P) ($4\%,0^{\circ}$) | $3.0^{+1.2}_{-0.9}\times 10^{-3}$
${\rho}^{0}$ ${\rho}^{0}$ (P) ($0\%,3^{\circ}$) | $9.8^{+4.4}_{-3.8}\times 10^{-4}$

Various $C\\!P$ asymmetries are introduced by modifying, for a chosen
${D}^{0}$ flavour, either the magnitude or the phase of the dominant amplitude
contributions: $a_{1}(1260)^{+}\\!\rightarrow{{\rho}^{0}}(770)\pi^{+}$ (in
S-wave) and ${{\rho}^{0}}(770){{\rho}^{0}}(770)$ (both P-wave and D-wave). The
resulting sensitivities are shown in Table 2. The $p$-values, including their
statistical uncertainties, are obtained from fits of GEV functions.

The asymmetry significances for each simulated event are shown in Fig. 2 for
$P$-even and $P$-odd $C\\!P$-violation tests and projected onto invariant
masses of the selected three-pion and two-pion subsystems. The $C\\!P$
violation is introduced as a phase difference in either the $a_{1}(1260)^{+}$
or P-wave ${{\rho}^{0}}(770){{\rho}^{0}}(770)$ amplitudes (see Sect. 6). The
shapes of the regions with significant asymmetry that are visible in the one-
dimensional projections in Fig. 2 cannot be easily interpreted in terms of the
amplitudes and phase differences of the contributing resonances; the two-
dimensional spectra are not easy to understand either. However, repeating this
exercise for different scenarios of $C\\!P$ violation, the observed features
are found to be sufficiently distinguishable to help identify the origin of
any $C\\!P$ asymmetry in the data.

The sensitivity of the method also depends on the choice of the effect radius
$\delta$. Studies indicate good stability of the measured sensitivity for
values of $\delta$ from $0.3$ to
$1{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$, which are well above the
resolution of the $d_{ij}$ and small compared to the size of the phase space.
The value $\delta=0.5{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ yields the
best sensitivity to most of the $C\\!P$-violation scenarios studied and was
chosen, prior to the data unblinding, as the default value. The optimal
$\delta$ value may vary with different $C\\!P$-violation scenarios. Hence, the
final results are also quoted for values of
$0.3{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ and
$0.7{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$.

##  7 Systematic effects

There are two main sources of asymmetry that may degrade or bias the energy
test. One is an asymmetry that may arise from background and the other is due
to detection and production asymmetries that could vary across phase space.
The effect of these asymmetries is studied for both the $P$-even and $P$-odd
$C\\!P$-violation measurements.

Background asymmetries are tested by applying the energy test to events in the
sidebands surrounding the signal region in the ${\Delta m}$ vs.
$m({{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}})$ plane. These events are
randomly split into 11 subsamples containing the same number of background
events as expected to contribute under the signal peak. The $p$-values
obtained are compatible with a uniform distribution and no significant
asymmetry is found; $p$-values range between $2\%$ and $96\%$ ($4\%$ and
$91\%$) for $P$-even ($P$-odd). As the background present in the signal region
is found to be $C\\!P$ symmetric, no correction is applied in the $T$ value
calculation discussed in Sect. 5.

Positively and negatively charged pions interact differently with matter; the
differences in the inelastic cross-sections are momentum dependent and
significant for low-momentum particles [20]. The presence of the ${\pi}^{+}$
${\pi}^{-}$ pairs in the final state makes the detection asymmetries cancel to
first order. Residual local asymmetries may remain in certain regions of the
phase space where ${\pi}^{+}$ and ${\pi}^{-}$ have different kinematic
distributions. These effects are tested using the Cabibbo-favoured decay
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}$ as a
control mode. This channel is affected by kaon detection asymmetries, which
are known to be larger than pion detection asymmetries and thus should serve
as a conservative test. The data sample is obtained with the same kinematic
selection criteria as for the signal channel and imposing requirements on the
candidate ${K}^{\pm}$ particles identified using information from the ring-
imaging Cherenkov detectors. The control sample is split into ten subsets,
each of which contains approximately the same amount of data as the signal
sample. The energy test yields results compatible with a uniform distribution
of $p$-values with values between $3\%$ and $87\%$ ($8\%$ and $74\%$) for
$P$-even ($P$-odd), which is consistent with the assumption that this source
of asymmetry is below the current level of sensitivity.

Asymmetries in the soft pion detection, although largely uncorrelated with the
${D}^{0}$ phase space, are reduced using fiducial cuts (see Sect. 3). Charm
mesons produced in the $pp$ interactions exhibit a production asymmetry up to
the percent level, which is slightly dependent on the meson kinematics [28,
29]. More ${D}^{*-}$ particles are observed than ${D}^{*+}$, giving rise to a
global asymmetry, to which the applied method is insensitive by construction.
No significant local $C\\!P$ asymmetry is expected owing to the small observed
correlation of the ${D}^{*}$ momentum and the ${D}^{0}$ phase space.

##  8 Results and conclusions

Figure 3:  (a,b) Distribution of permutation $T$-values fitted with a GEV
function and showing the $T$-value of the data tests as a vertical line, and
(c,d,e,f) local asymmetry significances. Left column plots are for the
$P$-even $C\\!P$-violation test, projected onto the (c)
$m(\pi_{1}\pi_{2}\pi_{3})$ and (e) $m(\pi_{1}\pi_{2})$ axes. Right column
plots are for the $P$-odd $C\\!P$-violation test projected onto the same axes.
In plots (c,d,e,f) the grey area correspond to candidates with a contribution
to the $T$-value of less than one standard deviation. In the $P$-even $C\\!P$
violation test the positive (negative) asymmetry significance is set for the
${D}^{0}$ candidates having positive (negative) contribution to the measured
$T$ value. In the $P$-odd $C\\!P$ violation test the positive (negative)
asymmetry significance is set for sample $I+IV$ having positive (negative)
contribution to the measured $T$ value (see Sect. 5). The pink (blue) area
corresponds to candidates with a positive (negative) contribution to the
$T$-value. Light, medium or dark shades of pink and blue correspond to between
one and two, two and three, and more than three standard deviation
contributions, respectively.

The application of the energy test to all selected
${{D}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$
candidates using an effective radius of
$\delta=0.5{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ yields $T=1.10\times
10^{-6}$ in the $P$-even $C\\!P$-violation test and $T=2.11\times 10^{-6}$ in
the $P$-odd $C\\!P$-violation test. The permutation $T$ value distributions
are shown in Fig. 3 (a,b). By counting the fraction of permutations with a $T$
value above the nominal $T$ value in the data, a $p$-value of $(4.6\pm 0.5)\%$
is obtained in the $P$-even $C\\!P$-violation test and $(0.6\pm 0.2)\%$ in the
$P$-odd $C\\!P$-violation test. The central value of the $P$-odd result would
correspond to being at least 2.7 standard deviations from the mean of a normal
distribution. The significance levels of the $T_{i}$ values are shown in Fig.
3 (c,e) projected onto the $m(\pi_{1}\pi_{2}\pi_{3})$ axis, and Fig. 3 (d,f)
projected onto the $m(\pi_{1}\pi_{2})$ axis. In the $P$-even test, a small
phase space region contains candidates with a local negative asymmetry
exceeding $1\sigma$ significance. Furthermore, in the $P$-odd test, candidates
with a local positive asymmetry exceeding $2\sigma$ significance are seen in a
small phase-space region dominated by the $\rho^{0}$ resonance, which can be
compared with the corresponding plots in Fig. 2. Varying the effective radius
results in the $p$-values listed in Table 3. The central values of the $P$-odd
results for $\delta=0.3{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$
correspond to more than three standard deviations from the mean of a normal
distribution but the significance falls below this level when considering
their uncertainties.

Table 3: Results for the $P$-even and $P$-odd $C\\!P$-violation tests for three different values of the effective radius $\delta$. The $p$-values obtained with both the counting and GEV fitting methods are given (see text). The counting method is the default method. |  $p$-value $P$-even |  $p$-value $P$-odd
---|---|---
$\delta$ [${\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ ] | Counting | GEV fit | Counting | GEV fit
0.3 | $(0.88\pm 0.26)\%$ | $(0.78\pm 0.10)\%$ | $(0.24\pm 0.14)\%$ | $(0.28\pm 0.04)\%$
0.5 | $(4.6\pm 0.5)\%$ | $(4.8\pm 0.3)\%$ | $(0.63\pm 0.20)\%$ | $(0.34\pm 0.05)\%$
0.7 | $(16\pm 2)\%$ | $(17\pm 2)\%$ | $(0.83\pm 0.48)\%$ | $(0.52\pm 0.16)\%$

In summary, a search for time-integrated $C\\!P$ violation in the Cabibbo-
suppressed decay
${{D}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ is
performed using a novel unbinned model-independent technique. This is the
first application of the energy test to four-body decays and extends the
approach to allow $P$-odd $C\\!P$ violation searches. This analysis has the
best sensitivity from a single experiment to $P$-even $C\\!P$ violation and is
the first test for $P$-odd $C\\!P$ violation in this decay. The data are found
to be marginally consistent with the hypothesis of $C\\!P$ symmetry at the
current level of precision.

## Acknowledgements

We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 (France); BMBF, DFG and MPG (Germany); INFN (Italy); FOM
and NWO (The Netherlands); MNiSW and NCN (Poland); MEN/IFA (Romania); MinES
and FASO (Russia); MinECo (Spain); SNSF and SER (Switzerland); NASU (Ukraine);
STFC (United Kingdom); NSF (USA). We acknowledge the computing resources that
are provided by CERN, IN2P3 (France), KIT and DESY (Germany), INFN (Italy),
SURF (The Netherlands), PIC (Spain), GridPP (United Kingdom), RRCKI and Yandex
LLC (Russia), CSCS (Switzerland), IFIN-HH (Romania), CBPF (Brazil), PL-GRID
(Poland) and OSC (USA). This work was supported in part by an allocation of
computing time from the University of Manchester and the Ohio Supercomputer
Center. We are indebted to the communities behind the multiple open source
software packages on which we depend. Individual groups or members have
received support from AvH Foundation (Germany), EPLANET, Marie Skłodowska-
Curie Actions and ERC (European Union), Conseil Général de Haute-Savoie, Labex
ENIGMASS and OCEVU, Région Auvergne (France), RFBR and Yandex LLC (Russia),
GVA, XuntaGal and GENCAT (Spain), Herchel Smith Fund, The Royal Society, Royal
Commission for the Exhibition of 1851 and the Leverhulme Trust (United
Kingdom).

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LHCb collaboration

R. Aaij40, B. Adeva39, M. Adinolfi48, Z. Ajaltouni5, S. Akar6, J. Albrecht10,
F. Alessio40, M. Alexander53, S. Ali43, G. Alkhazov31, P. Alvarez Cartelle55,
A.A. Alves Jr59, S. Amato2, S. Amerio23, Y. Amhis7, L. An41, L. Anderlini18,
G. Andreassi41, M. Andreotti17,g, J.E. Andrews60, R.B. Appleby56, F.
Archilli43, P. d’Argent12, J. Arnau Romeu6, A. Artamonov37, M. Artuso61, E.
Aslanides6, G. Auriemma26, M. Baalouch5, I. Babuschkin56, S. Bachmann12, J.J.
Back50, A. Badalov38, C. Baesso62, S. Baker55, W. Baldini17, R.J. Barlow56, C.
Barschel40, S. Barsuk7, W. Barter40, M. Baszczyk27, V. Batozskaya29, B.
Batsukh61, V. Battista41, A. Bay41, L. Beaucourt4, J. Beddow53, F. Bedeschi24,
I. Bediaga1, L.J. Bel43, V. Bellee41, N. Belloli21,i, K. Belous37, I.
Belyaev32, E. Ben-Haim8, G. Bencivenni19, S. Benson43, J. Benton48, A.
Berezhnoy33, R. Bernet42, A. Bertolin23, C. Betancourt42, F. Betti15, M.-O.
Bettler40, M. van Beuzekom43, Ia. Bezshyiko42, S. Bifani47, P. Billoir8, T.
Bird56, A. Birnkraut10, A. Bitadze56, A. Bizzeti18,u, T. Blake50, F. Blanc41,
J. Blouw11,†, S. Blusk61, V. Bocci26, T. Boettcher58, A. Bondar36,w, N.
Bondar31,40, W. Bonivento16, I. Bordyuzhin32, A. Borgheresi21,i, S. Borghi56,
M. Borisyak35, M. Borsato39, F. Bossu7, M. Boubdir9, T.J.V. Bowcock54, E.
Bowen42, C. Bozzi17,40, S. Braun12, M. Britsch12, T. Britton61, J.
Brodzicka56, E. Buchanan48, C. Burr56, A. Bursche2, J. Buytaert40, S.
Cadeddu16, R. Calabrese17,g, M. Calvi21,i, M. Calvo Gomez38,m, A. Camboni38,
P. Campana19, D.H. Campora Perez40, L. Capriotti56, A. Carbone15,e, G.
Carboni25,j, R. Cardinale20,h, A. Cardini16, P. Carniti21,i, L. Carson52, K.
Carvalho Akiba2, G. Casse54, L. Cassina21,i, L. Castillo Garcia41, M.
Cattaneo40, Ch. Cauet10, G. Cavallero20, R. Cenci24,t, D. Chamont7, M.
Charles8, Ph. Charpentier40, G. Chatzikonstantinidis47, M. Chefdeville4, S.
Chen56, S.-F. Cheung57, V. Chobanova39, M. Chrzaszcz42,27, X. Cid Vidal39, G.
Ciezarek43, P.E.L. Clarke52, M. Clemencic40, H.V. Cliff49, J. Closier40, V.
Coco59, J. Cogan6, E. Cogneras5, V. Cogoni16,40,f, L. Cojocariu30, G.
Collazuol23,o, P. Collins40, A. Comerma-Montells12, A. Contu40, A. Cook48, G.
Coombs40, S. Coquereau38, G. Corti40, M. Corvo17,g, C.M. Costa Sobral50, B.
Couturier40, G.A. Cowan52, D.C. Craik52, A. Crocombe50, M. Cruz Torres62, S.
Cunliffe55, R. Currie55, C. D’Ambrosio40, F. Da Cunha Marinho2, E.
Dall’Occo43, J. Dalseno48, P.N.Y. David43, A. Davis59, O. De Aguiar
Francisco2, K. De Bruyn6, S. De Capua56, M. De Cian12, J.M. De Miranda1, L. De
Paula2, M. De Serio14,d, P. De Simone19, C.-T. Dean53, D. Decamp4, M.
Deckenhoff10, L. Del Buono8, M. Demmer10, A. Dendek28, D. Derkach35, O.
Deschamps5, F. Dettori40, B. Dey22, A. Di Canto40, H. Dijkstra40, F. Dordei40,
M. Dorigo41, A. Dosil Suárez39, A. Dovbnya45, K. Dreimanis54, L. Dufour43, G.
Dujany56, K. Dungs40, P. Durante40, R. Dzhelyadin37, A. Dziurda40, A.
Dzyuba31, N. Déléage4, S. Easo51, M. Ebert52, U. Egede55, V. Egorychev32, S.
Eidelman36,w, S. Eisenhardt52, U. Eitschberger10, R. Ekelhof10, L. Eklund53,
S. Ely61, S. Esen12, H.M. Evans49, T. Evans57, A. Falabella15, N. Farley47, S.
Farry54, R. Fay54, D. Fazzini21,i, D. Ferguson52, A. Fernandez Prieto39, F.
Ferrari15,40, F. Ferreira Rodrigues2, M. Ferro-Luzzi40, S. Filippov34, R.A.
Fini14, M. Fiore17,g, M. Fiorini17,g, M. Firlej28, C. Fitzpatrick41, T.
Fiutowski28, F. Fleuret7,b, K. Fohl40, M. Fontana16,40, F. Fontanelli20,h,
D.C. Forshaw61, R. Forty40, V. Franco Lima54, M. Frank40, C. Frei40, J.
Fu22,q, E. Furfaro25,j, C. Färber40, A. Gallas Torreira39, D. Galli15,e, S.
Gallorini23, S. Gambetta52, M. Gandelman2, P. Gandini57, Y. Gao3, L.M. Garcia
Martin69, J. García Pardiñas39, J. Garra Tico49, L. Garrido38, P.J. Garsed49,
D. Gascon38, C. Gaspar40, L. Gavardi10, G. Gazzoni5, D. Gerick12, E.
Gersabeck12, M. Gersabeck56, T. Gershon50, Ph. Ghez4, S. Gianì41, V. Gibson49,
O.G. Girard41, L. Giubega30, K. Gizdov52, V.V. Gligorov8, D. Golubkov32, A.
Golutvin55,40, A. Gomes1,a, I.V. Gorelov33, C. Gotti21,i, M. Grabalosa
Gándara5, R. Graciani Diaz38, L.A. Granado Cardoso40, E. Graugés38, E.
Graverini42, G. Graziani18, A. Grecu30, P. Griffith47, L. Grillo21,40,i, B.R.
Gruberg Cazon57, O. Grünberg67, E. Gushchin34, Yu. Guz37, T. Gys40, C.
Göbel62, T. Hadavizadeh57, C. Hadjivasiliou5, G. Haefeli41, C. Haen40, S.C.
Haines49, S. Hall55, B. Hamilton60, X. Han12, S. Hansmann-Menzemer12, N.
Harnew57, S.T. Harnew48, J. Harrison56, M. Hatch40, J. He63, T. Head41, A.
Heister9, K. Hennessy54, P. Henrard5, L. Henry8, J.A. Hernando Morata39, E.
van Herwijnen40, M. Heß67, A. Hicheur2, D. Hill57, C. Hombach56, H. Hopchev41,
W. Hulsbergen43, T. Humair55, M. Hushchyn35, N. Hussain57, D. Hutchcroft54, M.
Idzik28, P. Ilten58, R. Jacobsson40, A. Jaeger12, J. Jalocha57, E. Jans43, A.
Jawahery60, F. Jiang3, M. John57, D. Johnson40, C.R. Jones49, C. Joram40, B.
Jost40, N. Jurik61, S. Kandybei45, W. Kanso6, M. Karacson40, J.M. Kariuki48,
S. Karodia53, M. Kecke12, M. Kelsey61, I.R. Kenyon47, M. Kenzie49, T. Ketel44,
E. Khairullin35, B. Khanji12, C. Khurewathanakul41, T. Kirn9, S. Klaver56, K.
Klimaszewski29, S. Koliiev46, M. Kolpin12, I. Komarov41, R.F. Koopman44, P.
Koppenburg43, A. Kosmyntseva32, A. Kozachuk33, M. Kozeiha5, L. Kravchuk34, K.
Kreplin12, M. Kreps50, P. Krokovny36,w, F. Kruse10, W. Krzemien29, W.
Kucewicz27,l, M. Kucharczyk27, V. Kudryavtsev36,w, A.K. Kuonen41, K. Kurek29,
T. Kvaratskheliya32,40, D. Lacarrere40, G. Lafferty56, A. Lai16, G.
Lanfranchi19, C. Langenbruch9, T. Latham50, C. Lazzeroni47, R. Le Gac6, J. van
Leerdam43, J.-P. Lees4, A. Leflat33,40, J. Lefrançois7, R. Lefèvre5, F.
Lemaitre40, E. Lemos Cid39, O. Leroy6, T. Lesiak27, B. Leverington12, Y. Li7,
T. Likhomanenko35,68, R. Lindner40, C. Linn40, F. Lionetto42, B. Liu16, X.
Liu3, D. Loh50, I. Longstaff53, J.H. Lopes2, D. Lucchesi23,o, M. Lucio
Martinez39, H. Luo52, A. Lupato23, E. Luppi17,g, O. Lupton57, A. Lusiani24, X.
Lyu63, F. Machefert7, F. Maciuc30, O. Maev31, K. Maguire56, S. Malde57, A.
Malinin68, T. Maltsev36, G. Manca7, G. Mancinelli6, P. Manning61, J.
Maratas5,v, J.F. Marchand4, U. Marconi15, C. Marin Benito38, P. Marino24,t, J.
Marks12, G. Martellotti26, M. Martin6, M. Martinelli41, D. Martinez Santos39,
F. Martinez Vidal69, D. Martins Tostes2, L.M. Massacrier7, A. Massafferri1, R.
Matev40, A. Mathad50, Z. Mathe40, C. Matteuzzi21, A. Mauri42, B. Maurin41, A.
Mazurov47, M. McCann55, J. McCarthy47, A. McNab56, R. McNulty13, B. Meadows59,
F. Meier10, M. Meissner12, D. Melnychuk29, M. Merk43, A. Merli22,q, E.
Michielin23, D.A. Milanes66, M.-N. Minard4, D.S. Mitzel12, A. Mogini8, J.
Molina Rodriguez1, I.A. Monroy66, S. Monteil5, M. Morandin23, P. Morawski28,
A. Mordà6, M.J. Morello24,t, J. Moron28, A.B. Morris52, R. Mountain61, F.
Muheim52, M. Mulder43, M. Mussini15, D. Müller56, J. Müller10, K. Müller42, V.
Müller10, P. Naik48, T. Nakada41, R. Nandakumar51, A. Nandi57, I. Nasteva2, M.
Needham52, N. Neri22, S. Neubert12, N. Neufeld40, M. Neuner12, A.D. Nguyen41,
T.D. Nguyen41, C. Nguyen-Mau41,n, S. Nieswand9, R. Niet10, N. Nikitin33, T.
Nikodem12, A. Novoselov37, D.P. O’Hanlon50, A. Oblakowska-Mucha28, V.
Obraztsov37, S. Ogilvy19, R. Oldeman49, C.J.G. Onderwater70, J.M. Otalora
Goicochea2, A. Otto40, P. Owen42, A. Oyanguren69,40, P.R. Pais41, A.
Palano14,d, F. Palombo22,q, M. Palutan19, J. Panman40, A. Papanestis51, M.
Pappagallo14,d, L.L. Pappalardo17,g, W. Parker60, C. Parkes56, G. Passaleva18,
A. Pastore14,d, G.D. Patel54, M. Patel55, C. Patrignani15,e, A. Pearce56,51,
A. Pellegrino43, G. Penso26, M. Pepe Altarelli40, S. Perazzini40, P. Perret5,
L. Pescatore47, K. Petridis48, A. Petrolini20,h, A. Petrov68, M. Petruzzo22,q,
E. Picatoste Olloqui38, B. Pietrzyk4, M. Pikies27, D. Pinci26, A. Pistone20,
A. Piucci12, S. Playfer52, M. Plo Casasus39, T. Poikela40, F. Polci8, A.
Poluektov50,36, I. Polyakov61, E. Polycarpo2, G.J. Pomery48, A. Popov37, D.
Popov11,40, B. Popovici30, S. Poslavskii37, C. Potterat2, E. Price48, J.D.
Price54, J. Prisciandaro39, A. Pritchard54, C. Prouve48, V. Pugatch46, A. Puig
Navarro41, G. Punzi24,p, W. Qian57, R. Quagliani7,48, B. Rachwal27, J.H.
Rademacker48, M. Rama24, M. Ramos Pernas39, M.S. Rangel2, I. Raniuk45, F.
Ratnikov35, G. Raven44, F. Redi55, S. Reichert10, A.C. dos Reis1, C. Remon
Alepuz69, V. Renaudin7, S. Ricciardi51, S. Richards48, M. Rihl40, K.
Rinnert54, V. Rives Molina38, P. Robbe7,40, A.B. Rodrigues1, E. Rodrigues59,
J.A. Rodriguez Lopez66, P. Rodriguez Perez56,†, A. Rogozhnikov35, S. Roiser40,
A. Rollings57, V. Romanovskiy37, A. Romero Vidal39, J.W. Ronayne13, M.
Rotondo19, M.S. Rudolph61, T. Ruf40, P. Ruiz Valls69, J.J. Saborido Silva39,
E. Sadykhov32, N. Sagidova31, B. Saitta16,f, V. Salustino Guimaraes2, C.
Sanchez Mayordomo69, B. Sanmartin Sedes39, R. Santacesaria26, C. Santamarina
Rios39, M. Santimaria19, E. Santovetti25,j, A. Sarti19,k, C. Satriano26,s, A.
Satta25, D.M. Saunders48, D. Savrina32,33, S. Schael9, M. Schellenberg10, M.
Schiller40, H. Schindler40, M. Schlupp10, M. Schmelling11, T. Schmelzer10, B.
Schmidt40, O. Schneider41, A. Schopper40, K. Schubert10, M. Schubiger41, M.-H.
Schune7, R. Schwemmer40, B. Sciascia19, A. Sciubba26,k, A. Semennikov32, A.
Sergi47, N. Serra42, J. Serrano6, L. Sestini23, P. Seyfert21, M. Shapkin37, I.
Shapoval45, Y. Shcheglov31, T. Shears54, L. Shekhtman36,w, V. Shevchenko68,
B.G. Siddi17,40, R. Silva Coutinho42, L. Silva de Oliveira2, G. Simi23,o, S.
Simone14,d, M. Sirendi49, N. Skidmore48, T. Skwarnicki61, E. Smith55, I.T.
Smith52, J. Smith49, M. Smith55, H. Snoek43, M.D. Sokoloff59, F.J.P. Soler53,
B. Souza De Paula2, B. Spaan10, P. Spradlin53, S. Sridharan40, F. Stagni40, M.
Stahl12, S. Stahl40, P. Stefko41, S. Stefkova55, O. Steinkamp42, S. Stemmle12,
O. Stenyakin37, S. Stevenson57, S. Stoica30, S. Stone61, B. Storaci42, S.
Stracka24,p, M. Straticiuc30, U. Straumann42, L. Sun64, W. Sutcliffe55, K.
Swientek28, V. Syropoulos44, M. Szczekowski29, T. Szumlak28, S. T’Jampens4, A.
Tayduganov6, T. Tekampe10, M. Teklishyn7, G. Tellarini17,g, F. Teubert40, E.
Thomas40, J. van Tilburg43, M.J. Tilley55, V. Tisserand4, M. Tobin41, S.
Tolk49, L. Tomassetti17,g, D. Tonelli40, S. Topp-Joergensen57, F. Toriello61,
E. Tournefier4, S. Tourneur41, K. Trabelsi41, M. Traill53, M.T. Tran41, M.
Tresch42, A. Trisovic40, A. Tsaregorodtsev6, P. Tsopelas43, A. Tully49, N.
Tuning43, A. Ukleja29, A. Ustyuzhanin35, U. Uwer12, C. Vacca16,f, V.
Vagnoni15,40, A. Valassi40, S. Valat40, G. Valenti15, A. Vallier7, R. Vazquez
Gomez19, P. Vazquez Regueiro39, S. Vecchi17, M. van Veghel43, J.J. Velthuis48,
M. Veltri18,r, G. Veneziano57, A. Venkateswaran61, M. Vernet5, M.
Vesterinen12, B. Viaud7, D. Vieira1, M. Vieites Diaz39, H. Viemann67, X.
Vilasis-Cardona38,m, M. Vitti49, V. Volkov33, A. Vollhardt42, B. Voneki40, A.
Vorobyev31, V. Vorobyev36,w, C. Voß67, J.A. de Vries43, C. Vázquez Sierra39,
R. Waldi67, C. Wallace50, R. Wallace13, J. Walsh24, J. Wang61, D.R. Ward49,
H.M. Wark54, N.K. Watson47, D. Websdale55, A. Weiden42, M. Whitehead40, J.
Wicht50, G. Wilkinson57,40, M. Wilkinson61, M. Williams40, M.P. Williams47, M.
Williams58, T. Williams47, F.F. Wilson51, J. Wimberley60, J. Wishahi10, W.
Wislicki29, M. Witek27, G. Wormser7, S.A. Wotton49, K. Wraight53, K. Wyllie40,
Y. Xie65, Z. Xing61, Z. Xu41, Z. Yang3, Y. Yao61, H. Yin65, J. Yu65, X.
Yuan36,w, O. Yushchenko37, K.A. Zarebski47, M. Zavertyaev11,c, L. Zhang3, Y.
Zhang7, Y. Zhang63, A. Zhelezov12, Y. Zheng63, A. Zhokhov32, X. Zhu3, V.
Zhukov9, S. Zucchelli15.

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
4LAPP, Université Savoie Mont-Blanc, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9I. Physikalisches Institut, RWTH Aachen University, Aachen, Germany
10Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
11Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
12Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
13School of Physics, University College Dublin, Dublin, Ireland
14Sezione INFN di Bari, Bari, Italy
15Sezione INFN di Bologna, Bologna, Italy
16Sezione INFN di Cagliari, Cagliari, Italy
17Sezione INFN di Ferrara, Ferrara, Italy
18Sezione INFN di Firenze, Firenze, Italy
19Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
20Sezione INFN di Genova, Genova, Italy
21Sezione INFN di Milano Bicocca, Milano, Italy
22Sezione INFN di Milano, Milano, Italy
23Sezione INFN di Padova, Padova, Italy
24Sezione INFN di Pisa, Pisa, Italy
25Sezione INFN di Roma Tor Vergata, Roma, Italy
26Sezione INFN di Roma La Sapienza, Roma, Italy
27Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
28AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
29National Center for Nuclear Research (NCBJ), Warsaw, Poland
30Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
31Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
32Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
33Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
34Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
35Yandex School of Data Analysis, Moscow, Russia
36Budker Institute of Nuclear Physics (SB RAS), Novosibirsk, Russia
37Institute for High Energy Physics (IHEP), Protvino, Russia
38ICCUB, Universitat de Barcelona, Barcelona, Spain
39Universidad de Santiago de Compostela, Santiago de Compostela, Spain
40European Organization for Nuclear Research (CERN), Geneva, Switzerland
41Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL),
Lausanne, Switzerland
42Physik-Institut, Universität Zürich, Zürich, Switzerland
43Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
44Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
45NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
46Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
47University of Birmingham, Birmingham, United Kingdom
48H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
49Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
50Department of Physics, University of Warwick, Coventry, United Kingdom
51STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
52School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
53School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
54Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
55Imperial College London, London, United Kingdom
56School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
57Department of Physics, University of Oxford, Oxford, United Kingdom
58Massachusetts Institute of Technology, Cambridge, MA, United States
59University of Cincinnati, Cincinnati, OH, United States
60University of Maryland, College Park, MD, United States
61Syracuse University, Syracuse, NY, United States
62Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
63University of Chinese Academy of Sciences, Beijing, China, associated to 3
64School of Physics and Technology, Wuhan University, Wuhan, China, associated
to 3
65Institute of Particle Physics, Central China Normal University, Wuhan,
Hubei, China, associated to 3
66Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia,
associated to 8
67Institut für Physik, Universität Rostock, Rostock, Germany, associated to 12
68National Research Centre Kurchatov Institute, Moscow, Russia, associated to
32
69Instituto de Fisica Corpuscular (IFIC), Universitat de Valencia-CSIC,
Valencia, Spain, associated to 38
70Van Swinderen Institute, University of Groningen, Groningen, The
Netherlands, associated to 43

aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil
bLaboratoire Leprince-Ringuet, Palaiseau, France
cP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
dUniversità di Bari, Bari, Italy
eUniversità di Bologna, Bologna, Italy
fUniversità di Cagliari, Cagliari, Italy
gUniversità di Ferrara, Ferrara, Italy
hUniversità di Genova, Genova, Italy
iUniversità di Milano Bicocca, Milano, Italy
jUniversità di Roma Tor Vergata, Roma, Italy
kUniversità di Roma La Sapienza, Roma, Italy
lAGH - University of Science and Technology, Faculty of Computer Science,
Electronics and Telecommunications, Kraków, Poland
mLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
nHanoi University of Science, Hanoi, Viet Nam
oUniversità di Padova, Padova, Italy
pUniversità di Pisa, Pisa, Italy
qUniversità degli Studi di Milano, Milano, Italy
rUniversità di Urbino, Urbino, Italy
sUniversità della Basilicata, Potenza, Italy
tScuola Normale Superiore, Pisa, Italy
uUniversità di Modena e Reggio Emilia, Modena, Italy
vIligan Institute of Technology (IIT), Iligan, Philippines
wNovosibirsk State University, Novosibirsk, Russia
†Deceased