.

# Configurations of eigenpoints

Valentina Beorchia  Dipartimento di Matematica e Geoscienze, Università di
Trieste, Via Valerio 12/1, 34127 Trieste, Italy<EMAIL_ADDRESS>ORCID
0000-0003-3681-9045 and  Rosa M. Miró-Roig  Facultat de Matemàtiques i
Informàtica, Universitat de Barcelona, Gran Via des les Corts Catalanes 585,
08007 Barcelona, Spain<EMAIL_ADDRESS>ORCID 0000-0003-1375-6547

###### Abstract.

This note is motivated by the Question 16 of http://cubics.wikidot.com (see
also [16]): Which configurations of 15 points in $\mathbb{P}^{3}$ arise as
eigenpoints of a cubic surface? We prove that a general eigenscheme in
$\mathbb{P}^{n}$ is the complete intersection of two suitable smooth
determinantal curves on a smooth determinantal surface. Moreover, we prove
that the converse result holds if $n=3$, providing an answer in any degree to
the cited question. Finally, we show that any general set of points in
$\mathbb{P}^{3}$ can be enlarged to an eigenscheme of a partially symmetric
tensor.

###### Key words and phrases:

Tensor eigenvector, determinantal variety, complete intersection, surface
containing a line.

###### 2020 Mathematics Subject Classification:

Primary 15A18, 14M12; Secondary 15A69

The first author is a member of GNSAGA of INdAM and is supported by the fund
Università degli Studi di Trieste - FRA 2022, by the MIUR Excellence
Department Project awarded to the DMG of Trieste, 2018-2023, and by the
Italian MIUR funds, PRIN project Moduli Theory and Birational Classification
(2017).

The second author has been partially supported by the grant
PID2019-104844GB-I00

##  1\. Introduction

Tensor spectral theory fits in a natural way in the context of Algebraic
Geometry, and it has recently received a lot of attention, also because of the
extended range of applications, like hypergraphs theory, quantum physics,
dynamical systems, polynomial optimization, signal processing and medical
imaging among others (see, for instance, [15]). In this article we focus on
eigenvectors of tensors, which were introduced independently in [11] and [14].
The definition relies on the choice of a nondegenerate quadratic form $Q$ on
$\mathbb{C}^{n+1}$, or equivalently on the choice of an isomorphism
$\mathbb{C}^{n+1}\cong{\mathbb{C}^{n+1}}^{\vee}$. With such a choice, given
$T\in({\mathbb{C}^{n+1}}^{\vee})^{\otimes d}$, a non zero vector
$v\in\mathbb{C}^{n+1}$ is said to be an eigenvector for $T$ if the contraction
$T(v^{\otimes d-1})\in{\mathbb{C}^{n+1}}^{\vee}$ is equal to $\lambda v$ for
some $\lambda\in\mathbb{C}$ under the isomorphism
${\mathbb{C}^{n+1}}^{\vee}\cong\mathbb{C}^{n+1}$. Since the property of being
an eigenvector is preserved under scalar multiplication, it is natural to
regard eigenvectors as points in $\mathbb{P}^{n}$, in which case we shall call
them eigenpoints. By choosing an orthonomal basis with respect to $Q$ and
coordinates $T=(t_{i_{1}i_{2}\cdots i_{d}})$, the eigenpoint condition can be
expressed with the algebraic equations

(1.1) $\sum_{i_{2}\dots i_{d}=0}^{n}t_{ji_{2}\dots i_{d}}x_{i_{2}}\dots
x_{i_{d}}=\lambda x_{j},\ {\text{f}or}\ j=0\dots n.$

The complete set of eigenpoints with the natural scheme structure arising from
its equations will be called an eigenscheme.

There are several natural problems related to eigenschemes: their dimension,
their degree and their configurations. Concerning their dimension, Abo proved
in [1] that the discriminant locus of tensors in
$\mathbb{P}(({\mathbb{C}^{n+1}}^{\vee})^{\otimes d})$ with positive-
dimensional or non-reduced eigenscheme is a hypersurface, and he computed its
degree. So the general tensor has a $0$-dimensional eigenscheme, and its
degree has been determined in [6], see also Lemma 2.3.

Regarding the geometry of their configurations, the planar case has been
settled in [2] for $d=3$ and in [3] for $d\geq 4$. It turns out that in the
planar case the eigenschemes of general tensors are characterized
geometrically by the property of being the base locus of a net of degree $d$
curves, and of having no aligned subschemes of length $d+1$, nor length $kd$
subschemes on a curve of degree $k$ for any $2\leq k\leq d-1$.

Finally, in the case $n=3=d$, some partial results are given in [5], and for
$n\geq 3$, a characterization of the sets of determinantal equations is given
in [4]. As far as we know, a general description of the geometry of
eigenpoints in the projective space is widely open.

This note is motivated by Question 16 of the website http://cubics.wikidot.com
(see also [16]): Which configurations of 15 points in $\mathbb{P}^{3}$ arise
as eigenpoints of a cubic surface? We give a geometric descriptions of
eigenpoints configurations in $\mathbb{P}^{n}$. Our main results are the
following: a general eigenscheme in $\mathbb{P}^{n}$ is the complete
intersection of two smooth determinantal curves on a smooth determinantal
surface, see Theorem 3.2. We prove that the converse result holds if $n=3$
(see Theorem 3.4), providing an answer to the cited question. Finally, in
Proposition 3.6, we show that any general set of points in $\mathbb{P}^{3}$
can be enlarged to an eigenscheme of a partially symmetric tensor.

Our approach relies on the fact that eigenschemes of general tensors turn out
to be standard determinantal subschemes, see Definition 2.2. By deleting any
column of the defining matrix, the maximal minors of the resulting matrix
define a smooth irreducible standard determinantal curve, and deleting two
columns we similarly find a smooth standard determinantal surface. Such a
surface is not general, since it contains a line. By using the results of [10]
and [8] it is possible to determine the class of the curves on the surface,
and to obtain the first result.

The proof of the converse in $\mathbb{P}^{3}$ relies on results of [12] on the
Picard group of a general surface containing a line and on the use of the
Mapping cone process.

Finally, some similar ideas allow to proof Proposition 3.6.

Acknowledgement. Most of this work was done while the second author was a
guest of the University of Trieste, and she would like to thank the people of
the Dipartimento di Matematica e Geoscienze for their warm hospitality.

Notation. Throughout this note, $R=\mathbb{C}[x_{0},x_{1},\cdots,x_{n}]$ and
$\mathbb{P}^{n}=\operatorname{Proj}(R)$. Given a homogeneous matrix $M$, i.e.
a matrix representing a degree 0 morphism $\Phi$ of free graded $R$-modules,
we denote by $I_{t}(M)$ the ideal of $R$ generated by the $t\times t$ minors
of $M$.

##  2\. Preliminaries

This section contains the basic definitions and results on eigenschemes and it
lays the groundwork for the results in the later sections.

As observed in [2, Section 1], the eigenscheme equations (1.1) involve $n+1$
homogeneous polynomials of degree $d-1$, so that it is not restrictive to
assume that $T$ is a partially symmetric tensors, that is for any fixed
$j\in\\{0,\dots,n\\}$, the slice $(t_{ji_{2}\dots i_{d}})$ is a symmetric
tensor of order $d-1$. Indeed, it is clear from (1.1) that any set of
eigenvectors can be realized as the set of eigenvectors of a partially
symmetric tensor $T\in(Sym^{d-1}\mathbb{C}^{n+1})\otimes\mathbb{C}^{n+1}$.
Finally, we can identify $T$ with a $(n+1)$-uple $(g_{0},\cdots,g_{n})$ of
homogeneous polynomials of degree $d-1$. With such identifications, we
immediately see that the equations (1.1) are of determinantal type, and we are
in position to give the explicit definition of eigenscheme. We first recall
the following notion.

######  Definition 2.1.

A codimension $c$ subscheme $X\subset\mathbb{P}^{n}$ is called a standard
determinantal scheme if the saturated homogeneous ideal $I(X)=I_{t}(A)$ for
some $t\times(t+c-1)$ homogeneous matrix $A$.

######  Definition 2.2.

Let $T=(g_{0},\cdots,g_{n})\in(Sym^{d-1}\mathbb{C}^{n+1})^{\oplus(n+1)}$ be a
partially symmetric tensor. The eigenscheme of $T$ is the closed subscheme
$E(T)\subset\mathbb{P}^{n}$ defined by the $2\times 2$ minors of

(2.1) $M=\begin{pmatrix}x_{0}&x_{1}&\cdots&x_{n}\\\
g_{0}&g_{1}&\cdots&g_{n}\end{pmatrix};$

i.e. $I(E(T))=I_{2}(M)$. If $T$ is general, then $E(T)$ is $0$-dimensional
(see, for instance, [1]). Moreover, by the Hochster-Eagon Theorem [9], the
quotient $R/I(E(T))$ is a Cohen-Macaulay ring, and as a consequence, $I(E(T))$
is saturated. Hence $E(T)$ is a standard determinantal scheme. When $T$ is
symmetric, i.e., there is a homogeneous polynomial $f$ and
$g_{i}=\partial_{i}f$ we denote its eigenscheme by $E(f)$.

The eigenschemes arise also in the following framework: if we consider the
rational map

$\mu:\mathbb{P}^{n}\dashrightarrow\mathbb{P}^{n},\quad
p\mapsto\mu(p)=(g_{0}(p),g_{1}(p),\cdots,g_{n}(p)),$

then $p\in\mathbb{P}^{n}$ is an eigenpoint if and only if either $p$ is a
fixed point of $\mu$ or $\mu$ is not defined at $p$. Therefore, in the
symmetric case, the eigenscheme $E(f)$ consists of the Jacobian scheme and the
fixed points of the polar map $\nabla f=(\partial_{0}f,\cdots,\partial_{n}f)$.
Basic questions concerning the degree, the dimension and the minimal free
resolution of eigenschemes $E(T)$ of a general symmetric tensor
$T=(g_{0},\cdots,g_{n})\in(Sym^{d-1}\mathbb{C}^{n+1})^{\oplus(n+1)}$ are well
known and we recall them in next proposition for seek of completeness. In next
section, we will use the geometry of arithmetically Cohen-Macaulay curves and
surfaces naturally associated with eigenschemes to gain some knowledge about
the configuration of eigenschemes.

######  Lemma 2.3.

Fix integers $n\geq 2$ and $d\geq 2$. Let
$T=(g_{0},\cdots,g_{n})\in(Sym^{d-1}\mathbb{C}^{n+1})^{\oplus(n+1)}$ be a
general partially symmetric tensor. It holds:

  * (1)

$E(T)$ is a reduced 0-dimensional scheme of length

$\frac{(d-1)^{n+1}-1}{d-2}.$

  * (2)

The homogeneous ideal $I(E(T))\subset R$ has a minimal free $R$-resolution

$0\longrightarrow\oplus_{i=0}^{n-1}R(-(i+1)d-n+2i+1)\longrightarrow\cdots\longrightarrow$
$R(-1-d)^{\binom{n+1}{3}}\oplus R(1-2d)^{\binom{n+1}{3}}\longrightarrow
R(-d)^{\binom{n+1}{2}}\longrightarrow I(E(T))\longrightarrow 0.$

###### Proof.

(1) See [6].

(2) Since $E(T)$ is a standard determinantal scheme of codimension $n$, its
minimal free resolution is given by the Eagon-Northcott complex (see [7,
Theorem A2.10]). ∎

It is worthwhile to point out that for any partially symmetric tensor
$T=(g_{0},\cdots,g_{n})\in(Sym^{d-1}\mathbb{C}^{n+1})^{\oplus(n+1)}$ the
associated eigenscheme $E(T)$ contains no $d+1$ points on a line and, more
general, no $sd+1$ points on a curve of degree $s$. Indeed, $E(T)$ is a
0-dimensional scheme whose homogeneous ideal is generated by forms of degree
$d$. If $E(T)$ contains $sd+1$ points on a curve $C$ of degree $s$ then, by
Bezout’s theorem, the whole curve $C$ will be contained in the base locus
contradicting the fact that $E(T)$ is 0-dimensional.

We will end this preliminary section with a concrete example.

######  Example 2.4.

We consider the Fermat cubic
$f=x_{0}^{3}+x_{1}^{3}+x_{2}^{3}+x_{3}^{3}\in\mathbb{C}[x_{0},x_{1},x_{2},x_{2}]$.
The eigenscheme $E(f)$ is the 0-dimensional subscheme of $\mathbb{P}^{3}$ of
length 15 defined by the maximal minors of

$M=\begin{pmatrix}x_{0}&x_{1}&x_{2}&x_{3}\\\
x^{2}_{0}&x^{2}_{1}&x_{2}^{2}&x_{3}^{2}\end{pmatrix}.$

Therefore,
$E(f)=\\{(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1),(1,1,0,0),(1,0,1,0),(1,0,0,1),\\\
(0,1,1,0),(0,1,0,1),(0,0,1,1),(1,1,1,0),(1,1,0,1),(1,0,1,1),(0,1,1,1),(1,1,1,1)\\}$

##  3\. Eigenschemes of partially symmetric tensors

In this section, we deal with partially symmetric tensors and we address the
open problem of determining all possible subschemes of $\mathbb{P}^{n}$
arising as their eigenschemes. We will be able to give for any $n\geq 3$ a
necessary condition which turns out to be sufficient when $n=3$.

######  Remark 3.1.

Let $Z\subset\mathbb{P}^{n}$ with $n\geq 3$ be an eigenscheme of a general
partially symmetric tensor
$T=(g_{0},\cdots,g_{n})\in(Sym^{d-1}\mathbb{C}^{n+1})^{\oplus(n+1)}$, with
defining matrix $M$ as in (2.1).

Moreover, denote by

$M_{i}:=\begin{pmatrix}x_{0}&x_{1}&\cdots&\widehat{x_{i}}&\cdots&x_{n}\\\
g_{0}&g_{1}&\cdots&\widehat{g_{i}}&\cdots&g_{n}\end{pmatrix}$

the matrix obtained from $M$ by deleting the $i$-th column.

Then the $2\times 2$ minors of $M_{i}$ define a standard aCM determinantal
curve $C_{i}\subset\mathbb{P}^{n}$, which is smooth and irreducible if the
forms $g_{j}$ are general.

We claim that this holds also in the symmetric case, where
$g_{i}=\partial_{i}f$, if $f$ is a sufficiently general homogeneous
polynomial.

Indeed, first observe that for a general $f$, the determinantal subscheme
$C_{i}$ defined by the $2\times 2$ minors of

$\begin{pmatrix}x_{0}&x_{1}&\cdots&\widehat{x_{i}}&\cdots&x_{n}\\\
\partial_{0}f&\partial_{1}f&\cdots&\widehat{\partial_{i}f}&\cdots&\partial_{n}f\end{pmatrix}$

has pure dimension $1$, hence it is of the maximal codimension $n-1$. To
verify this statement, note that such a property is satisfied by an open
subscheme $U$ of the space of all partially symmetric tensors. The symmetric
tensors form an irreducible closed subvariety, so in order to prove the
nonemptyness of the intersection of $U$ with the symmetric locus, it suffices
to exhibit one such example. This is given, for instance, by the Fermat
polynomial

$f=x_{0}^{d}+\dots+x_{n}^{d},$

where the defining matrix of the eigenscheme is of the type of the Example
2.4. It is not difficult to see that by deleting a column, the locus $C_{i}$ a
set of lines, all passing through one of the fundamental points.

Next we observe that, if $f$ is general, we have that $C_{i}$ is smooth.
Indeed, the singular locus is the common zero locus of the matrix entries, and
this is, in general, empty.

As a consequence, the ideal sheaf of such a subscheme is resolved by the
Eagon-Northcott complex and, hence, $C_{i}$ is an aCM subscheme. In
particular, we have $H^{1}(\mathcal{I}_{C_{i}})=0$. Therefore
$H^{0}(\mathcal{O}_{C_{i}})=\mathbb{C}$. This implies that $C_{i}$ is
connected and being smooth we get that $C_{i}$ is irreducible.

Now we turn again to the general case of partially symmetric tensors, and we
determine the degree of $C_{i}$. Since the section of $C_{i}$ with the
hyperplane $x_{i}=0$ is, by construction, a general 0-dimensional eigenscheme
in $\mathbb{P}^{n-1}$, its degree is (Lemma 2.3(1))

${\rm deg}\ C_{i}=\frac{(d-1)^{n}-1}{d-2}.$

Similarly, for any pair of indices $i\neq j$, we can consider the matrix

$M_{ij}:=\begin{pmatrix}x_{0}&x_{1}&\cdots&\widehat{x_{i}}&\cdots&\widehat{x_{j}}&\cdots&x_{n}\\\
g_{0}&g_{1}&\cdots&\widehat{g_{i}}&\cdots&\widehat{g_{j}}&\cdots&g_{n}\\\
\end{pmatrix}.$

Then the $2\times 2$ minors of $M_{ij}$ define a standard determinantal
surface $S_{ij}\subset\mathbb{P}^{n}$, which is smooth and irreducible if the
forms $g_{k}$ are general.

Again, by similar arguments, we see that this holds also in the symmetric
case, if $f$ is a sufficiently general homogeneous polynomial.

Since the section of $S_{ij}$ with the codimension $2$ linear subspace
$x_{i}=x_{j}=0$ is by construction a general eigenscheme in
$\mathbb{P}^{n-2}$, its degree is (Lemma 2.3(1))

${\rm deg}\ S_{ij}=\frac{(d-1)^{n-1}-1}{d-2}.$

We are now in the position to state and prove our first result.

######  Theorem 3.2.

Let $Z\subset\mathbb{P}^{n}$ with $n\geq 3$ be an eigenscheme of a general
partially symmetric, or general symmetric, tensor of order $d\geq 3$, and let
$i\neq j$, $i,j\in\\{0,\dots,n\\}$ be arbitrary.

Then $Z$ is the complete intersection of the two aCM curves

$Z=C_{i}\cap C_{j}$

on the smooth surface $S_{ij}$.

###### Proof.

Without loss of generality we can assume, for simplicity, that $i=0$ and
$j=1$.

We assume first that

$n\geq 4\text{ or }d\geq 4.$

By denoting with $H$ a hyperplane divisor of $S_{01}$ and with $L$ the line
given by the equations

$L:\quad x_{2}=x_{3}=\cdots=x_{n}=0,$

by [10, Theorem 4.1] we have

${\rm Pic}(S_{01})\cong\mathbb{Z}^{2}\cong\langle H,L\rangle.$

To determine the class of $C_{i}$ as a divisor, we recall the well known fact
that ${\mathcal{I}}_{C_{i}/S_{01}}\cong{\mathcal{I}}_{L/S_{01}}(\lambda)$ for
some $\lambda\in\mathbb{Z}$, i.e. $C_{i}\sim L+\lambda H$ for some
$\lambda\in\mathbb{Z}$ (see [8]). By considering the relation

$C_{i}\cdot H=\frac{(d-1)^{n}-1}{d-2},$

we get that

$\lambda=d-1.$

Next we determine the self-intersection $L^{2}$ of the line $L$. Observe that
by [10, Proposition 2.2 (iii)] we have that the canonical class

$K_{S_{01}}\sim(n-3)C+(d-n-1)H\sim(n-3)L+((n-3)(d-1)+d-n-1))H.$

Then by the adjunction formula we have

$2g(L)-2=-2=(K_{S_{01}}+L)\cdot L=((nd-2d-2n+2)H+(n-2)L))\cdot$

This gives

$(n-2)L^{2}=(n-2)(d-2),\ i.e.$ $L^{2}=2-d.$

As a consequence we have

$C_{0}\cdot C_{1}=((d-1)H+L)^{2}=\frac{(d-1)^{n+1}-1}{d-2}={\rm deg}\ (Z).$

Therefore $Z$ is the complete intersection of $C_{0}$ and $C_{1}$.

Finally, let us treat the case $n=d=3$. In this case, $S_{01}$ is the blow up
of $\mathbb{P}^{2}$ at 6 points and ${\rm
Pic}(S_{01})\cong\mathbb{Z}^{7}\cong\langle\ell,e_{1},e_{2},e_{3},e_{4},e_{5},e_{6}\rangle.$
where $\ell$ is the pullback of a line in $\mathbb{P}^{2}$ and $e_{i}$ are the
exceptional divisors. The line $L:\quad x_{2}=x_{3}=0$ of $S_{01}$ can be
identified with one of the following divisors: $e_{j}$, $\ell-
e_{j_{1}}-e_{j_{2}}$, $2\ell-
e_{j_{1}}-e_{j_{2}}-e_{j_{3}}-e_{j_{4}}-e_{j_{5}}$ and $H\sim 3\ell-
e_{1}-e_{2}-e_{3}-e_{4}-e_{5}-e_{6}$. Applying again [8] we have
${\mathcal{I}}_{C_{i}/S_{01}}\cong{\mathcal{I}}_{L/S_{01}}(\lambda)$ for some
$\lambda\in\mathbb{Z}$, i.e. $C_{i}\sim L+\lambda H$ for some
$\lambda\in\mathbb{Z}$ and using the fact that $C_{i}\cdot H=7$ we get that
$\lambda=2$. The remaining part of the proof works as in the case $n\geq 4$ or
$d\geq 4$. ∎

In what follows we shall see that the converse holds if $n=3$ which for the
particular case $d=3$ will provide an answer to Question 16 of the website
http://cubics.wikidot.com: Which configurations of 15 points in
$\mathbb{P}^{3}$ arise as eigenpoints of a cubic surface?

The following result clarifies the necessity of the assumptions of Theorem
3.4.

######  Proposition 3.3.

Let $Z=E(T)\subset\mathbb{P}^{n}$ be a $0$ \- dimensional general reduced
eigenscheme. Then no $(d-1)\frac{(d-1)^{n}-1}{d-2}$ points of $Z$ lie on a
hypersurface of degree $d-1$.

###### Proof.

Let (2.1) be the matrix associated with $Z$. Assume by contradiction that $Z$
admits a subscheme $Z^{\prime}\subset Z$ of $(d-1)\frac{(d-1)^{n}-1}{d-2}$
points lying on a hypersurface

$\Sigma=V(h)\subseteq\mathbb{P}^{n}$

of degree $d-1$.

By using the notations of Remark 3.1 and as in the proof of Theorem 3.2 we
denote by $H$ a hyperplane divisor of $S_{01}$ and by $L$ the line given by
the equations

$L:\quad x_{2}=x_{3}=\cdots=x_{n}=0.$

We set $D^{\prime}=S_{01}\cap\Sigma\sim(d-1)H$. As $Z^{\prime}\subset Z$ and
$Z$ is a complete intersection of two irreducible divisors
$C_{0},C_{1}\in|(d-1)H+L|$ by Theorem 3.2, we have that

$Z^{\prime}\subseteq D^{\prime}\cap C_{i}.$

Since we have

$D^{\prime}\cdot C_{i}=(d-1)H\cdot C_{i}=(d-1)\frac{(d-1)^{n}-1}{d-2}={\rm
deg}\ Z^{\prime},$

we see that $Z^{\prime}=D^{\prime}\cap C_{i}$. Moreover, observe that
residually to $Z^{\prime}$ in $Z$ we get one point $P$.

Now we claim that $P\in L$. Indeed, since both $Z$ and $Z^{\prime}$ are
complete intersection schemes in $S_{01}$, the minimal resolutions of their
$\mathcal{I}_{Z,S_{01}}$ and $\mathcal{I}_{Z^{\prime},S_{01}}$ in $S_{01}$ are
given by the Koszul complex. More precisely, we have

$0\to\mathcal{O}_{S_{01}}(-2(d-1)H-2L)\to
2\mathcal{O}_{S_{01}}(-(d-1)H-L)\to\mathcal{I}_{Z,S_{01}}\to 0,\text{ and }$
$0\to\mathcal{O}_{S_{01}}(-2(d-1)H-L)\to\mathcal{O}_{S_{01}}(-(d-1)H-L)\oplus\mathcal{O}_{S_{01}}(-(d-1)H)\to\mathcal{I}_{Z^{\prime},S_{01}}\to
0.$

Since $Z^{\prime}$ and $P$ are directly linked by $Z$, by applying the Mapping
cone process, we get a resolution of the ideal sheaf $\mathcal{I}_{P,S_{01}}$:

$0\to\mathcal{O}_{S_{01}}(-(d-1)H-L)\oplus\mathcal{O}_{S_{01}}(-(d-1)H-2L)\to\qquad\qquad\qquad\qquad$
$\qquad\qquad\qquad\qquad\to
2\mathcal{O}_{S_{01}}(-(d-1)H-L)\oplus\mathcal{O}_{S_{01}}(-L)\to\mathcal{I}_{P,S_{01}}\to
0,$

which allows us to conclude that $P\in L$.

As a consequence, the divisor $D^{\prime}+L$ contains $Z$. So, it belongs to
the pencil

$D^{\prime}+L\in\langle C_{0},C_{1}\rangle$

whose base locus is $Z$. Therefore $Z$ is a complete intersection also of
$C_{1}$ and the reducible curve $D^{\prime}\cup L$.

By construction, the equations in $S_{01}$ of $D^{\prime}\cup L$ are

$x_{i}h=0,\ {\rm for}\ i=2,\dots,n+1$

hence the saturated homogeneous ideal of $D^{\prime}\cup L$ in
$\mathbb{P}^{n}$ is generated by the $2\times 2$ minors of the matrix

$\begin{pmatrix}x_{2}&x_{3}&\cdots&x_{n}&0\\\
g_{2}&g_{3}&\cdots&g_{n}&h\end{pmatrix}.$

Finally, we set

$N:=\begin{pmatrix}x_{1}&x_{2}&x_{3}&\cdots&x_{n}&0\\\
g_{1}&g_{2}&g_{3}&\cdots&g_{n}&h\end{pmatrix},$

and we consider $W\subset\mathbb{P}^{n}$ the determinantal scheme defined by
the $2\times 2$ minors of $N$:

$I(W)=I_{2}(N).$

By construction we have that $W$ is $0$-dimensional and $Z\subseteq W$. As
${\rm deg}\ (Z)={\rm deg}\ (W)$, the equality $Z=W$ holds.

So $I(Z)=I_{2}(M)=I_{2}(N)$, but this is a contradiction, since any defining
matrix for an eigenscheme has the property that the linear entries of the
first row are linearly independent (see, for instance, [4, Corollary 2.9]).

∎

Taking into account all the necessary conditions that we have proved so far,
we can state the converse result for points in $\mathbb{P}^{3}$.

######  Theorem 3.4.

Let $Z\subset\mathbb{P}^{3}$ be a reduced $0$ \- dimensional subscheme of
degree

${\rm deg}\ (Z)=\frac{(d-1)^{4}-1}{d-2}=d(d^{2}-2d+2)$

for some $d\geq 3$.

If $Z$ is a complete intersection of two irreducible aCM curves $C_{0}$ and
$C_{1}$ on a smooth general surface $S$ of degree $d$ containing a line $L$,
and the following conditions are satisfied:

  1. (1)

no $(d-1)(d^{2}-d+1)$ points of $Z$ lie on a surface of degree $d-1$;

  2. (2)

$C_{0}\sim C_{1}$;

  3. (3)

${\rm deg}\ C_{0}={\rm deg}\ C_{1}=\frac{(d-1)^{3}-1}{d-2}=d^{2}-d+1$;

  4. (4)

$g(C_{0})=g(C_{1})=d^{3}-7\frac{d(d-1)}{2}-1$,

then $Z$ is the eigenscheme of some partially symmetric tensor of order $d$.

###### Proof.

With no loss of generality we may assume that $L$ has equations

$L:\quad x_{0}=x_{1}=0.$

Since $L\subset S$, the equation of $S$ can be written in the form

$S:\quad x_{0}g_{1}-x_{1}g_{0}=0,$

for some suitable degree $d-1$ forms $g_{0}$ and $g_{1}$.

Our next goal is to show that both $C_{0}$ and $C_{1}$ are determinantal
curves, whose equations are the $2\times 2$ minors of a matrix given by adding
a suitable column to

$\begin{pmatrix}x_{0}&x_{1}\\\ g_{0}&g_{1}\\\ \end{pmatrix}.$

We start by determining the divisor class of $C_{i}$ in $S$.

If $d\geq 4$, by [12] we have that $Pic(S)\cong\mathbb{Z}^{2}\cong\langle
H,L\rangle$, where $H$ is a hyperplane divisor on $S$. So $C_{i}\sim\alpha
H+\beta L$ for some integers $\alpha$ and $\beta$. By recalling
$K_{S}\sim(d-4)H$, the relations

$C_{i}\cdot H=d^{2}-d+1,\quad 2g(C_{i})-2=((d-4)H+C_{i})\cdot C_{i}$

yield the equations

$d\alpha+\beta=d^{2}-d+1,$
$2d^{3}-7d(d-1)-4=d\alpha^{2}+(2-d)\beta^{2}+2\alpha\beta+(d-4)(\alpha
d+\beta),$

where we used the fact that $L^{2}=2-d$ Indeed, $-2=2g(L)-2=(K_{s}+L)\cdot
L=((d-4)H+L)\cdot L=d-4+L^{2}$). The two possible solutions are

(3.1) $\alpha=d-1,\quad\beta=1,$

and

(3.2) $\alpha=\frac{(d^{2}-d+2)}{d},\quad\beta=-1.$

As for $d\geq 3$ the solution (3.2) is not integer, it holds (3.1). Therefore,
we have

$C_{i}\sim(d-1)H+L.$

If $d=3$, by [13, Theorem 4.7], any aCM curve $C$ of degree $7$ and genus $5$
on a smooth cubic surface $S$ has class either

$C\sim e_{i}+2H,\quad{\rm or}\ C\sim(l-e_{j}-e_{k})+2H,\quad{\rm or}\ C\sim
2l-e_{i_{1}}-e_{i_{2}}-e_{i_{3}}-e_{i_{4}}-e_{i_{5}}+2H,$

where the $e_{i}$’s are the exceptional divisors of $S$, when we identify it
with the blow up of $\mathbb{P}^{2}$ in $6$ points, and $l$ is the pull back
of a line of $\mathbb{P}^{2}$. We observe that the divisors

$e_{i},\ l-e_{j}-e_{k},\ 2l-e_{i_{1}}-e_{i_{2}}-e_{i_{3}}-e_{i_{4}}-e_{i_{5}}$

give rise to the $27$ lines of a smooth cubic surface, and by the assumption
$C_{0}\sim C_{1}$ we finally get that

$C_{0}\sim C_{1}\sim L+2H,$

for some line $L\subset S$.

In particular, in all cases we get

(3.3)
$\mathcal{I}_{C_{i},S}=\mathcal{O}_{S}(-C_{i})=\mathcal{O}_{S}(-(d-1)H-L)=\mathcal{I}_{L,S}(-(d-1)H),$

so we are led to determine a resolution of $\mathcal{I}_{L,S}(-(d-1)H)$.

By applying the Mapping Cone process (see, for instance, [17, Section 1.5]) to
the two exact sequences

$\begin{array}[]{r}0\to\mathcal{O}_{\mathbb{P}^{3}}(-d)\to\mathcal{I}_{S,\mathbb{P}^{3}}\to
0\\\ \\\ 0\to\mathcal{O}_{\mathbb{P}^{3}}(-2)\to
2\mathcal{O}_{\mathbb{P}^{3}}(-1)\to\mathcal{I}_{L,\mathbb{P}^{3}}\to 0\\\
\end{array}$

we get the resolution

$0\to\mathcal{O}_{\mathbb{P}^{3}}(-2)\oplus\mathcal{O}_{\mathbb{P}^{3}}(-d)\xrightarrow{\left(\begin{array}[]{cc}x_{0}&g_{0}\\\
x_{1}&g_{1}\\\
\end{array}\right)}2\mathcal{O}_{\mathbb{P}^{3}}(-1)\to\mathcal{I}_{L,S}\to
0.$

By (3.3) we have $\mathcal{I}_{C_{i},S}\cong\mathcal{I}_{L,S}(-(d-1)H)$ and we
get, as a consequence, the resolution of $\mathcal{I}_{C_{i},S}$:

$0\to\mathcal{O}_{\mathbb{P}^{3}}(-d-1)\oplus\mathcal{O}_{\mathbb{P}^{3}}(-2d+1)\to
2\mathcal{O}_{\mathbb{P}^{3}}(-d)\to\mathcal{I}_{C_{i},S}\to 0.$

Finally, we apply the Horseshoe Lemma to the diagram

$\begin{array}[]{rcl}&0&\\\ &\downarrow&\\\
0\to\mathcal{O}_{\mathbb{P}^{3}}(-d)\to&\mathcal{I}_{S,\mathbb{P}^{3}}&\to
0\\\ &\downarrow&\\\ &\mathcal{I}_{C_{i},\mathbb{P}3}&\\\ &\downarrow&\\\
0\to\mathcal{O}_{\mathbb{P}^{3}}(-d-1)\oplus\mathcal{O}_{\mathbb{P}^{3}}(-2d+1)\to
2\mathcal{O}_{\mathbb{P}^{3}}(-d)\to&\mathcal{I}_{C_{i},S}&\to 0\\\
&\downarrow&\\\ &0&\\\ \end{array}$

and we obtain

$0\to\mathcal{O}_{\mathbb{P}^{3}}(-d-1)\oplus\mathcal{O}_{\mathbb{P}^{3}}(-2d+1)\xrightarrow{M_{i}:=\left(\begin{array}[]{cc}x_{0}&g_{0}\\\
x_{1}&g_{1}\\\ l_{i}&f_{i}\\\
\end{array}\right)}3\mathcal{O}_{\mathbb{P}^{3}}(-d)\to\mathcal{I}_{C_{i},\mathbb{P}^{3}}\to
0,$

for suitable linear forms $l_{i}$ and suitable degree $d-1$ forms $f_{i}$. In
particular, the saturated homogeneous ideal $I_{C_{i},\mathbb{P}^{3}}$ is
generated in degree $d$ by the three $2\times 2$ minors of the matrix above.

Now we claim that the linear forms $x_{0},x_{1},l_{i}$ are linearly
independent in both cases $i=0$ and $i=1$. Indeed, if this were not the case,
by a base change we could transform the matrix $M_{i}$ in the matrix

$\left(\begin{array}[]{cc}x_{0}&g_{0}\\\ x_{1}&g_{1}\\\ 0&h_{i}\\\
\end{array}\right),$

so the curve $C_{i}$ would be contained in the zero locus

$C_{i}\subset V(x_{0}h_{i},x_{1}h_{i}).$

Since $C_{i}$ is irreducible by assumption, we necessarily have

$C_{i}\subset V(h_{i}).$

As $C_{i}$ is also contained in the degree $d$ surface $S$, by Bézout Theorem
we would have

${\rm deg}\ C_{i}\leq(d-1)d,$

and this contradicts the assumption (iii).

Finally, we set

(3.4) $M:=\left(\begin{array}[]{cc}l_{0}&f_{0}\\\ x_{0}&g_{0}\\\
x_{1}&g_{1}\\\ l_{1}&f_{1}\\\ \end{array}\right),$

and set $W\subset\mathbb{P}^{3}$ to be the determinantal scheme defined by the
$2\times 2$ minors of $M$:

$I(W)=I_{2}(M).$

Since $Z$ is the complete intersection of the two determinantal curves $C_{0}$
and $C_{1}$, we have

$Z\subseteq W.$

As ${\rm deg}\ (Z)={\rm deg}\ (W)$, the equality $Z=W$ holds.

Finally, we claim that the four linear forms

$x_{0},x_{1},l_{0},l_{1}$

are linearly independent.

Assume by contradiction that this is not the case; then possibly performing a
base change, we have that $Z$ is the degeneracy locus also of the matrix

(3.5) $M^{\prime}:=\left(\begin{array}[]{cc}l_{0}&f_{0}\\\ x_{0}&g_{0}\\\
x_{1}&g_{1}\\\ 0&h_{1}\\\ \end{array}\right),$

so we would have $Z\supseteq V(l_{0}h_{1},x_{0}h_{1},x_{1}h_{1})$, and $Z$
would contain $(d-1)(d^{2}-d+1)$ points on the degree $d-1$ surface
$V(h_{1})$, contradicting assumption (1).

To conclude, we observe that possibly performing a base change we may assume
that $M$ is of the type

$\left(\begin{array}[]{cc}x_{0}&g_{0}\\\ x_{1}&g_{1}\\\ x_{2}&g_{2}\\\
x_{3}&g_{3}\end{array}\right),$

so $Z$ is indeed an eigenscheme. ∎

######  Remark 3.5.

Theorem 3.4 gives a geometric criterion for a set of points
$Z\subset\mathbb{P}^{3}$ to be an eigenscheme of some partially symmetric
tensor $T$, and our construction furnishes explicitly a set of generators for
its ideal. Then by applying the criterion given in [4, Theorem 3.4, and
Algorithm 2] it is possible to determine whether $Z$ is the eigenscheme of a
symmetric tensor.

We conclude this note by proving that any general set of points in
$\mathbb{P}^{3}$ can be enlarged to an eigenscheme of a partially symmetric
tensor.

######  Proposition 3.6.

Let $W\subset\mathbb{P}^{3}$ be a set of $k$ general points and set $d\geq 3$
to be an integer such that

(3.6) ${\rm deg}\ W=k\leq\binom{d-1}{3}+3\binom{d}{2}+1.$

Then $W$ can be embedded into an eigenscheme $Z=E(T)$ of a partially symmetric
tensor of order $d$.

###### Proof.

We observe that if $W$ is general enough, there always exists a smooth surface
$S\subset\mathbb{P}^{3}$ of degree $d$, containing a line $L$ and $W$. Indeed,
we have $h^{0}(\mathcal{O}_{\mathbb{P}^{3}}(d))=\binom{d+3}{3}$, and the
condition of containing a fixed line $L$ imposes $d+1$ linear conditions. By
(3.6) we have

$\binom{d+3}{3}-(d+1)-{\rm deg}\ W\geq\frac{d(d+1)}{2},$

so the claim follows.

Without loss of generality we may assume that $L$ has equations
$x_{0}=x_{1}=0$, so that $S$ is given by

$S:\ \ x_{0}g_{1}-x_{1}g_{0}=0,$

for some degree $d-1$ polynomials $g_{0}$ and $g_{1}$.

By [12] the Picard group of such a surface is

${\rm Pic}(S)\cong\mathbb{Z}^{2}\cong\langle H,L\rangle,$

where $H$ is a hyperplane divisor. Consider now the linear system
$|(d-1)H+L|$. By Riemann Roch we have

$\chi(\mathcal{O}_{S}((d-1)H+L)=\frac{1}{2}((d-1)H+L)\cdot((d-1)H+L-K_{S}))+1+p_{a}(S);$

recalling that $K_{S}\sim(d-4)H$ and $p_{a}(S)=\binom{d-1}{3}$, we obtain

$\chi(\mathcal{O}_{S}((d-1)H+L)=\frac{1}{2}((d-1)H+L)\cdot(3H+L)+1+\binom{d-1}{3}=3\binom{d}{2}+3+\binom{d-1}{3}.$

Moreover, since

$(d-1)H+L\sim K_{S}+3H+L$

and $3H+L$ is an ample divisor, by the Kodaira Vanishing Theorem we have
$h^{1}(\mathcal{O}_{S}((d-1)H+L)=h^{2}(\mathcal{O}_{S}((d-1)H+L)=0$, so that

$\chi(\mathcal{O}_{S}((d-1)H+L)=h^{0}(\mathcal{O}_{S}((d-1)H+L).$

By the assumption (3.6) there exists a pencil $\Lambda=\langle
C_{0},C_{1}\rangle$ of curves in $|(d-1)H+L|$ containing $W$. Furthermore, by
the generality of $W$ we may assume that no curve in $\Lambda$ has $L$ as an
irreducible component, and that the base locus $Z$ of $\Lambda$ is zero -
dimensional.

Then we have the isomorphisms given in (3.3) and we may apply the same
argument of the Theorem 3.4 to show that the complete intersection subscheme
$C_{0}\cap C_{1}$ is the degeneracy locus of a matrix of the type
$\eqref{matrix_oftwocurves}$, where the triples of linear forms
$x_{0},x_{1},l_{0}$ and $x_{0},x_{1},l_{1}$ are linearly independent. We claim
that also $x_{0},x_{1},l_{0},l_{1}$ are linearly independent.

Assume by contradiction that this is not the case; then possibly performing a
base change, we have that $Z$ is the degeneracy locus also of the matrix
(3.5), and we see that there will be a curve $C$ in $\Lambda$ with equations
$V(x_{0}g_{1}-x_{1}g_{0},x_{0}h_{1},x_{1}h_{1})$. Since $C\supset L$, this
contradicts our choice of $\Lambda$.

Hence we finally get that $Z$ is the degeneracy locus of a matrix

$\left(\begin{array}[]{cc}x_{2}&g_{2}\\\ x_{0}&g_{0}\\\ x_{1}&g_{1}\\\
x_{3}&g_{3}\\\ \end{array}\right),$

so it is an eigenscheme and $Z\supset W$ by construction.

∎

## References

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  * [17] Ch. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics 38 (1994) Cambridge University Press.


EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

CERN-EP-2016-028 LHCb-PAPER-2015-053 April 25, 2016

Measurement of the ${{B}^{0}_{s}}\rightarrow{D}^{(*)+}_{s}{D}^{(*)-}_{s}$
branching fractions

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

The branching fraction of the decay
${{B}^{0}_{s}}\rightarrow{D}^{(*)+}_{s}{D}^{(*)-}_{s}$ is measured using $pp$
collision data corresponding to an integrated luminosity of
$1.0$$\mbox{\,fb}^{-1}$, collected using the LHCb detector at a centre-of-mass
energy of $7$$\mathrm{\,Te\kern-1.00006ptV}$. It is found to be

$\displaystyle{\mathcal{B}}(B_{s}^{0}\rightarrow~{}D_{s}^{(\ast)+}D_{s}^{(\ast)-})=(3.05\pm
0.10\pm 0.20\pm 0.34)\%,$

where the uncertainties are statistical, systematic, and due to the
normalisation channel, respectively. The branching fractions of the individual
decays corresponding to the presence of one or two ${D}^{*\pm}_{s}$ are also
measured. The individual branching fractions are found to be

$\displaystyle{\mathcal{B}}({{B}^{0}_{s}}\rightarrow{{D}^{*\pm}_{s}}{{D}^{\mp}_{s}})=(1.35\pm
0.06\pm 0.09\pm 0.15)\%,$
$\displaystyle{\mathcal{B}}({{B}^{0}_{s}}\rightarrow{{D}^{*+}_{s}}{{D}^{*-}_{s}})=(1.27\pm
0.08\pm 0.10\pm 0.14)\%.$

All three results are the most precise determinations to date.

Published in Phys. Rev. D 92, 092008 on 20th May 2016

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

##  1 Introduction

Because of ${{B}^{0}_{s}}-{{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}}$ oscillations, the mass and
flavour eigenstates of the ${B}^{0}_{s}$ system do not coincide. The
${B}^{0}_{s}$ meson mass eigenstates have a relative decay width difference
$\Delta\Gamma_{{s}}/\Gamma_{{s}}$, where $\Delta\Gamma_{{s}}$ ($\Gamma_{{s}}$)
is the difference (average) of the decay widths between the heavy and light
states. The relative decay width difference is one of the key parameters of
the ${B}^{0}_{s}$ system, and its precise determination allows stringent tests
of the flavour sector of the Standard Model.

Under certain theoretical assumptions,
${{B}^{0}_{s}}\rightarrow{D}^{(*)+}_{s}{D}^{(*)-}_{s}$ decays were thought to
saturate the $C\\!P$-even contribution to $\Delta\Gamma_{{s}}$, and therefore
the branching fraction of
${{B}^{0}_{s}}\rightarrow{D}^{(*)+}_{s}{D}^{(*)-}_{s}$ was used as a means of
approximating $\Delta\Gamma_{{s}}/\Gamma_{{s}}$ [1]. This approximation is now
considered to be a poor one [2], as the decay modes containing at least one
${D}^{*\pm}_{s}$ have a non-negligible $C\\!P$-odd component, and other three-
body ${B}^{0}_{s}$ decays can contribute to the value of $\Delta\Gamma_{{s}}$
at a similar level as ${{B}^{0}_{s}}\rightarrow{D}^{(*)+}_{s}{D}^{(*)-}_{s}$
decays. A detailed discussion of theoretical predictions of the
${{B}^{0}_{s}}\rightarrow{D}^{(*)+}_{s}{D}^{(*)-}_{s}$ branching fractions,
and the predicted contribution of other modes to the value of
$\Delta\Gamma_{{s}}/\Gamma_{{s}}$, is given in Ref. [3].

In a more general context, since the branching fraction of
${{B}^{0}_{s}}\rightarrow{D}^{(*)+}_{s}{D}^{(*)-}_{s}$ decays is one of the
dominant contributions to the total inclusive
${b}\\!\rightarrow{c}{\overline{c}}{s}$ branching fraction, its precise
measurement is an important ingredient in model-independent searches for
physics beyond the Standard Model in $B$ meson decays [4]. The most recent
measurements are provided by the Belle [5], CDF [6], and D0 [7] collaborations
who obtain, respectively,

$\displaystyle{\mathcal{B}}({{B}^{0}_{s}}\rightarrow{D}^{(*)+}_{s}{D}^{(*)-}_{s})$
$\displaystyle=(4.32_{-0.39-1.03}^{+0.42+1.04})\%,$
$\displaystyle{\mathcal{B}}({{B}^{0}_{s}}\rightarrow{D}^{(*)+}_{s}{D}^{(*)-}_{s})$
$\displaystyle=(3.38\pm 0.25\pm 0.30\pm 0.56)\%,$
$\displaystyle{\mathcal{B}}({{B}^{0}_{s}}\rightarrow{D}^{(*)+}_{s}{D}^{(*)-}_{s})$
$\displaystyle=(3.5\pm 1.0\pm 1.1)\%.$

The data used in the analysis presented in this paper correspond to an
integrated luminosity of $1.0\mbox{\,fb}^{-1}$, collected by the LHCb
experiment during the 2011 run period. The branching fraction of the full
${{B}^{0}_{s}}\rightarrow{D}^{(*)+}_{s}{D}^{(*)-}_{s}$ decay is determined
relative to the ${{B}^{0}}\rightarrow{{D}^{+}_{s}}{{D}^{-}}$ decay, which has
a similar final state and a precisely measured branching fraction. The charm
daughters are reconstructed using the
${{D}^{+}_{s}}\rightarrow{{K}^{+}}{{K}^{-}}{{\pi}^{+}}$ and
${{D}^{-}}\rightarrow{{K}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ final states. Throughout
the paper, unless stated otherwise, charge-conjugate modes are implied and
summed over. The branching fraction ratio is determined as

$\frac{{\mathcal{B}}({{B}^{0}_{s}}\rightarrow{D}^{(*)+}_{s}{D}^{(*)-}_{s})}{{\mathcal{B}}(B^{0}\rightarrow{{D}^{+}_{s}}{{D}^{-}})}=\frac{f_{d}}{f_{s}}\cdot\frac{\epsilon^{B^{0}}}{\epsilon^{B_{s}^{0}}}\cdot\frac{{\mathcal{B}}(D^{-}\rightarrow~{}K^{+}\pi^{-}\pi^{-})}{{\mathcal{B}}(D_{s}^{+}\rightarrow~{}K^{-}K^{+}\pi^{+})}\cdot\frac{N_{B_{s}^{0}}}{N_{B^{0}}},$
(1)

where $f_{d}$ ($f_{s}$) is the fraction of ${\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}$ (${\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}$) mesons produced in the
fragmentation of a $b$ quark, $\epsilon^{B^{0}}/\epsilon^{B_{s}^{0}}$ is the
relative efficiency of the ${B}^{0}$ to the ${B}^{0}_{s}$ selections,
${\mathcal{B}}(D^{-}\rightarrow~{}K^{+}\pi^{-}\pi^{-})$ and
${\mathcal{B}}(D_{s}^{+}\rightarrow~{}K^{-}K^{+}\pi^{+})$ are the branching
fractions of the charm daughter decays, and $N_{B_{s}^{0}}/N_{B^{0}}$ is the
relative yield of ${B}^{0}_{s}$ and ${B}^{0}$ candidates.

The branching fraction of the exclusive
${{B}^{0}_{s}}\rightarrow{{D}^{*+}_{s}}{{D}^{*-}_{s}}$ decay is determined in
the same way, along with the branching fraction of
${{B}^{0}_{s}}\rightarrow{{D}^{*\pm}_{s}}{{D}^{\mp}_{s}}$. The branching
fraction of the ${{B}^{0}_{s}}\rightarrow{{D}^{+}_{s}}{{D}^{-}_{s}}$ decay has
been previously measured by LHCb using the same data as this analysis [8], and
is therefore not determined in this study. However, the selection efficiency
and yield in the ${{B}^{0}_{s}}\rightarrow{{D}^{+}_{s}}{{D}^{-}_{s}}$ channel
are determined in this analysis, as both are needed for the calculation of the
total ${{B}^{0}_{s}}\rightarrow{D}^{(*)+}_{s}{D}^{(*)-}_{s}$ selection
efficiency.

##  2 Detector and simulation

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 vertex, the impact parameter, 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. The event selection is performed in two stages, with an initial
online selection followed by a tighter offline selection. The online event
selection is performed by a trigger [11], which consists of a hardware stage,
based on information from the calorimeter and muon systems, followed by a
software stage, which performs a full event reconstruction.

In the simulation, $pp$ collisions are generated using Pythia 6 [12] with a
specific LHCb configuration [13]. Decays of hadronic particles are described
by EvtGen [14], in which final-state radiation is generated using Photos [15].
The interaction of the generated particles with the detector, and its
response, are implemented using the Geant4 toolkit [16, *Agostinelli:2002hh]
as described in Ref. [18].

##  3 Signal selection

The ${D}^{*+}_{s}$ meson decays to a ${D}^{+}_{s}$ meson and either a photon
or a neutral pion $(93.5\pm 0.7)\%$ and $(5.8\pm 0.7)\%$ of the time,
respectively, nearly saturating the total branching fraction. The remainder of
the decays are ignored in this analysis. Neither of the neutral particles is
reconstructed in the decay chain, and the individual
${{B}^{0}_{s}}\rightarrow{{D}^{*\pm}_{s}}{{D}^{\mp}_{s}}$ and
${{B}^{0}_{s}}\rightarrow{{D}^{*+}_{s}}{{D}^{*-}_{s}}$ decays are identified
through the reconstructed invariant mass of the ${{D}^{+}_{s}}{{D}^{-}_{s}}$
system. The individual peaks from
${{B}^{0}_{s}}\rightarrow{{D}^{*\pm}_{s}}(\rightarrow{{D}^{\pm}_{s}}{\gamma}){{D}^{\mp}_{s}}$
and
${{B}^{0}_{s}}\rightarrow{{D}^{*\pm}_{s}}(\rightarrow{{D}^{\pm}_{s}}{{\pi}^{0}}){{D}^{\mp}_{s}}$
are not resolved. Therefore the reconstructed ${{D}^{+}_{s}}{{D}^{-}_{s}}$
mass distribution has three separate peaks, corresponding to decays containing
zero, one, or two ${D}^{*\pm}_{s}$ particles.

At the hardware trigger stage, events are required to have a muon with high
$p_{\mathrm{T}}$ or a hadron, photon or electron with high transverse energy
in the calorimeters. For hadrons, the transverse energy threshold is
$3.5\mathrm{\,Ge\kern-1.00006ptV}$. Candidate ${B}^{0}_{s}$ and ${B}^{0}$
mesons are used in the analysis if at least one of the associated tracks is
selected by the hardware trigger, or if the event is triggered independently
of the particles in the signal decay. The software trigger considers all
charged particles with
$\mbox{$p_{\mathrm{T}}$}>500{\mathrm{\,Me\kern-1.00006ptV\\!/}c}$ and
constructs two-, three-, or four-track secondary vertices which require a
significant displacement from the primary $pp$ interaction vertices. At least
one charged particle must have a transverse momentum
$\mbox{$p_{\mathrm{T}}$}>1.7{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and be
inconsistent with originating from a primary vertex. A multivariate algorithm
[19] is used for the identification of secondary vertices consistent with the
decay of a $b$ hadron. The selection to this point is hereafter referred to as
the initial selection.

Signal ${B}^{0}_{s}$ and normalisation
${{B}^{0}}\rightarrow{{D}^{+}_{s}}{{D}^{-}}$ candidates are required to
satisfy a number of additional conditions in order to be included in the final
samples. Kaons and pions are required to be identified by the particle
identification (PID) system. All ${D}^{+}_{s}$ and ${D}^{-}$ candidates must
have an invariant mass within $\pm 30{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
of their known values [20]. Signal ${B}^{0}_{s}$ candidates are required to
have a reconstructed mass in the range
$4750-5800{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, whereas ${B}^{0}$
candidates must have a mass in the range
$5050-5500{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. After these requirements
are applied there are still contributions from other $b$-hadron decays into
final states with two charm particles. The decays ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}(\rightarrow{p}{{K}^{-}}{{\pi}^{+}}){{D}^{-}_{s}}$, where
the $p$ is misidentified as a ${K}^{+}$, and
${{B}^{0}}\rightarrow{{D}^{+}_{s}}{{D}^{-}}(\rightarrow{{K}^{+}}{{\pi}^{-}}{{\pi}^{-}})$,
where a ${\pi}^{-}$ is misidentified as a ${K}^{-}$, result in background
contamination in the signal channel, while the decay
${{B}^{0}_{s}}\rightarrow{{D}^{+}_{s}}{{D}^{-}_{s}}$ contributes to the
background in the normalisation channel if the ${K}^{+}$ in
${{D}^{+}_{s}}\rightarrow{{K}^{+}}{{K}^{-}}{{\pi}^{+}}$ is misidentified as a
${\pi}^{+}$. As these backgrounds accumulate in reconstructed mass close to
the signal peaks, candidates consistent with any one of these background decay
hypotheses are rejected in the selection by applying a veto based on the
invariant mass of the candidate under the alternative particle type
hypotheses. Candidate ${D}^{+}_{s}$ mesons are vetoed if they have a
reconstructed mass in the range
$2271-2301{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ when the ${K}^{+}$
candidate is assumed to be a proton, or a mass in the range
$1835-1905{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ when the ${K}^{+}$
candidate is assigned the ${\pi}^{+}$ mass. Candidate ${D}^{-}$ mesons are
vetoed if they have a reconstructed mass in the range
$1950-1990{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ when a ${\pi}^{-}$
candidate is assigned the kaon mass. In a simulated sample of
${{B}^{0}}\rightarrow{{D}^{+}_{s}}{{D}^{-}}$ decays, $17.7\%$ of the events
meet all of the ${{B}^{0}_{s}}\rightarrow{D}^{(*)+}_{s}{D}^{(*)-}_{s}$
selection criteria before the ${D}^{\pm}$ veto is applied. After the veto,
only $0.05\%$ of the simulated ${{B}^{0}}\rightarrow{{D}^{+}_{s}}{{D}^{-}}$
sample still pass the full
${{B}^{0}_{s}}\rightarrow{D}^{(*)+}_{s}{D}^{(*)-}_{s}$ selection. The decay
${{B}^{0}}\rightarrow{{D}^{+}_{s}}{{D}^{-}_{s}}$ and three-body
${{B}^{\pm}}\rightarrow D_{(s)}^{+}D_{(s)}^{-}h^{\pm}$ decays, where $h$ is
either a kaon or pion, are examined as other potential background sources, but
are all disregarded because of either a small selection efficiency or small
branching fraction relative to the signal channels.

In order to further improve the purity of the signal and normalisation
samples, a boosted decision tree (BDT) classifier is used to distinguish real
$B^{0}_{(s)}$ decays from combinatorial background [21]. The BDT is trained
using the AdaBoost algorithm [22] to distinguish simulated ${B}^{0}_{s}$
signal decays from background candidates obtained from mass sidebands in the
data. Background candidates must contain a ${B}^{0}_{s}$ candidate with a mass
greater than $5600{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and two
${D}^{\pm}_{s}$ candidates with masses less than
$1930{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ or greater than
$2010{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The set of 14 variables used
as input to the BDT exploits the topology of the ${B}^{0}_{s}$ decay chain and
includes the transverse momentum of the ${B}^{0}_{s}$ candidate and of the two
${D}^{\pm}_{s}$ daughters, as well as the product of the absolute transverse
momenta of the pions and kaons produced in the decay of each ${D}^{\pm}_{s}$.
The decay times of the two ${D}^{\pm}_{s}$ candidates with respect to the
primary vertex and variables related to the consistency of the ${B}^{0}_{s}$
and of the two ${D}^{\pm}_{s}$ to come from the primary vertex are also used.
The optimal BDT requirement is chosen to maximise the value of
$N_{s}/\sqrt{N_{s}+N_{b}}$, where $N_{s}$ is the total number of signal
candidates matching any of the three exclusive decays in
${{B}^{0}_{s}}\rightarrow{D}^{(*)+}_{s}{D}^{(*)-}_{s}$ and $N_{b}$ is the
total number of combinatorial background events as taken from the fit. The
same BDT classifier and selection criteria are also applied to the
normalisation sample.

Table 1:  Efficiencies of the various selection criteria for the three individual channels of ${{B}^{0}_{s}}\rightarrow{D}^{(*)+}_{s}{D}^{(*)-}_{s}$, and for ${{B}^{0}}\rightarrow{{D}^{+}_{s}}{{D}^{-}}$. Each efficiency is presented relative to the previous cut and measured using simulated events, except for the PID efficiency which is obtained from data. The ${D}^{+}_{s}$ veto was only applied to the normalisation mode, ${{B}^{0}}\rightarrow{{D}^{+}_{s}}{{D}^{-}}$. |  Selection efficiency (%)
---|---
Selection | ${{B}^{0}_{s}}\rightarrow{{D}^{+}_{s}}{{D}^{-}_{s}}$ | ${{B}^{0}_{s}}\rightarrow{{D}^{*\pm}_{s}}{{D}^{\mp}_{s}}$ | ${{B}^{0}_{s}}\rightarrow{{D}^{*+}_{s}}{{D}^{*-}_{s}}$ |  ${{B}^{0}}\rightarrow{{D}^{+}_{s}}{{D}^{-}}$
Reconstruction |  0.1184 $\pm$ | 0.0003 |  0.1127 $\pm$ | 0.0005 |  0.1061 $\pm$ | 0.0005 |  0.1071 $\pm$ |  0.0002
Initial selection |  1.362 $\pm$ | 0.008 |  1.250 $\pm$ | 0.010 |  1.100 $\pm$ | 0.010 |  1.416 $\pm$ | 0.009
Mass requirements |  89.4 $\pm$ | 0.6 |  87.8 $\pm$ | 1.0 |  88.3 $\pm$ | 1.0 |  88.5 $\pm$ | 0.6
BDT |  97.9 $\pm$ | 0.7 |  96.6 $\pm$ | 1.1 |  96.7 $\pm$ | 1.1 |  97.6 $\pm$ | 0.7
${D}^{+}$ veto |  48.7 $\pm$ | 0.5 |  50.3 $\pm$ | 0.8 |  48.9 $\pm$ | 0.8 |  68.7 $\pm$ | 0.6
${D}^{+}_{s}$ veto | $-$ | $-$ | $-$ |  64.8 $\pm$ | 0.7
${\mathchar 28931\relax}^{+}_{c}$ veto |  96.3 $\pm$ | 1.0 |  96.3 $\pm$ | 1.6 |  95.9 $\pm$ | 1.6 |  98.2 $\pm$ | 0.8
Trig. requirement |  96.6 $\pm$ | 0.7 |  96.7 $\pm$ | 1.1 |  96.6 $\pm$ | 1.1 |  96.8 $\pm$ | 0.7
PID requirements |  82.4 $\pm$ | 0.2 |  82.4 $\pm$ | 0.2 |  82.4 $\pm$ | 0.2 |  84.2 $\pm$ | 0.1
Total |  0.0527 $\pm$ | 0.0067 |  0.0460 $\pm$ | 0.0095 |  0.0372 $\pm$ | 0.0081 |  0.0467 $\pm$ |  0.0060

The efficiencies of the selection criteria in both the signal and
normalisation channels are listed in Table 1. The efficiencies of the
background vetoes, trigger, reconstruction, and BDT selection are determined
using simulated signal samples. The efficiencies of identifying ${K}^{+}$ and
${\pi}^{+}$ mesons are determined using a calibration data sample of
${{D}^{*+}}\rightarrow{{D}^{0}}(\rightarrow{{K}^{-}}{{\pi}^{+}}){{\pi}^{+}}$
decays, with kinematic quantities reweighted to match those of the signal
candidates. The efficiency of the PID selection is found to be $(82.4\pm
0.2)\%$ for signal ${B}^{0}_{s}$ decays and $(84.2\pm 0.1)\%$ for ${B}^{0}$
decays. The efficiency of the full
${{B}^{0}_{s}}\rightarrow{D}^{(*)+}_{s}{D}^{(*)-}_{s}$ decay is determined by
calculating a weighted average of the individual signal channels, with weights
given by the relative yields in data. The relative efficiencies of the
${B}^{0}$ decay to the three individual channels and the full decay are given
in Table 2.

Table 2:  Efficiency of the normalisation channel ${{B}^{0}}\rightarrow{{D}^{+}_{s}}{{D}^{-}}$ relative to the signal decays. Channel |  $\epsilon^{{{B}^{0}}}/\epsilon^{{{B}^{0}_{s}}}$
---|---
${{B}^{0}_{s}}\rightarrow{{D}^{+}_{s}}{{D}^{-}_{s}}$ |  $0.89\pm 0.02$
${{B}^{0}_{s}}\rightarrow{{D}^{*\pm}_{s}}{{D}^{\mp}_{s}}$ | $1.02\pm 0.03$
${{B}^{0}_{s}}\rightarrow{{D}^{*+}_{s}}{{D}^{*-}_{s}}$ | $1.26\pm 0.03$
${{B}^{0}_{s}}\rightarrow{D}^{(*)+}_{s}{D}^{(*)-}_{s}$ | $1.06\pm 0.02$

##  4 Signal and background shapes

The ${B}^{0}_{s}$ and ${B}^{0}$ yields in the signal channels and the
normalisation mode are extracted by performing a three-dimensional extended
unbinned maximum likelihood fit to the mass distributions of the $B^{0}_{(s)}$
meson and the two charm daughters.

In order to determine the yields for the individual signal peaks, the
${B}^{0}_{s}$ candidate mass distribution in each channel is modelled using
simulated signal events. The
${{B}^{0}_{s}}\rightarrow{{D}^{+}_{s}}{{D}^{-}_{s}}$ peak is parameterised as
the sum of a Crystal Ball function [23] and a Gaussian function. The tail
parameters of the Crystal Ball function, the ratio of the width of its
Gaussian core to the width of the Gaussian function, and the relative weight
of each function in the full distribution, are taken from simulation. The mean
and width of the Gaussian core are allowed to float. The two ${D}^{\pm}_{s}$
distributions are also parameterised using this model, with all shape
parameters fixed to the values found in simulation.

Because of the kinematic differences between the
${{D}^{*+}_{s}}\rightarrow{{D}^{+}_{s}}{\gamma}$ and
${{D}^{*+}_{s}}\rightarrow{{D}^{+}_{s}}{{\pi}^{0}}$ decays, the peak of the
${{B}^{0}_{s}}\rightarrow{{D}^{*\pm}_{s}}{{D}^{\mp}_{s}}$ mass distribution is
parameterised by a superposition of two Gaussian functions. The individual
mean values, the ratio of the widths, and the fraction of each Gaussian
function in the full distribution are fixed to values taken from simulation.
The peak corresponding to
${{B}^{0}_{s}}\rightarrow{{D}^{*+}_{s}}{{D}^{*-}_{s}}$ decays is modelled
using a single Gaussian function, with the mean fixed to the value found from
simulated events.

There is also a component in the fit to describe the presence of background
decays of the form ${{B}^{0}_{s}}\rightarrow D_{{s}J}^{+}{{D}^{-}_{s}}$, where
the $D_{{s}J}^{+}$ can be either a $D_{{s}1}(2460)^{+}$ or a
$D_{{s}0}(2317)^{+}$ meson that decays to a ${D}^{+}_{s}$ along with some
combination of photons and neutral or charged pions. As some decay products
are missed, this background is present only at the low mass region of the
signal distribution. The shape of the distribution is determined by fitting to
${{B}^{0}_{s}}\rightarrow D_{{s}1}(2460)^{+}{{D}^{-}_{s}}$ simulated events,
as the contribution from $D_{{s}1}(2460)^{+}$ is currently the best understood
among the $D_{{s}J}^{+}$ decays. It is found to be well modelled by an Argus
function [24], all shape parameters for which are fixed to the values found in
simulation.

The combinatorial background shape in the ${B}^{0}_{s}$ candidate mass
distribution is parameterised by a second-order polynomial, and the model is
validated with candidates passing a wrong-sign version of the selection. The
wrong-sign selection is identical to the signal selection but instead looks
for events containing two ${{D}^{+}_{s}}$ mesons. The parameters of the
combinatorial background distribution are allowed to float in the full fit to
data, and are found to be compatible with the values obtained from the fit to
the wrong-sign sample. The combinatorial background shape in the
${D}^{\pm}_{s}$ distribution is determined using events taken from the high-
mass sideband region of the ${B}^{0}_{s}$ distribution, and is found to be
consistent with a first-order polynomial. The impact of adding a small
Gaussian contribution to account for the presence of real ${D}^{\pm}_{s}$
mesons in the combinatorial background was found to be minimal, with the
observed deviations from the nominal signal yields being smaller than the
statistical uncertainty in each case.

The ${B}^{0}$ distribution is modelled using the same parameterisation as for
the full ${B}^{0}_{s}$ distribution, with one exception. The peak where either
the ${D}^{+}_{s}$ or ${D}^{-}$ comes from the decay of an excited state is
modelled by a superposition of three Gaussian functions, rather than the two-
Gaussian model used in the ${B}^{0}_{s}$ case, to account for the difference
in distributions from ${D}^{*+}_{s}$ and ${D}^{*-}$ decays, as the ${D}^{*-}$
decay contains a ${\pi}^{0}$ in the final state more frequently than
${D}^{*+}_{s}$ decays. There is also a small contribution from the decay
${{B}^{0}_{s}}\rightarrow{{D}^{-}_{s}}{{D}^{+}}$, which is modelled with the
same distribution as for the signal ${B}^{0}_{s}$ candidates.

##  5 Fit results

The fit to the signal data samples is shown in Fig. 1, where the triple peaked
structure of the full decay is clearly visible. The yields for the individual
signal channels and the two backgrounds are given in Table 3. The total
${{B}^{0}_{s}}\rightarrow{D}^{(*)+}_{s}{D}^{(*)-}_{s}$ yield is the sum of the
individual signal channel yields, with the uncertainty calculated using the
correlation coefficients between the individual yields, and is found to be
$2230\pm 63$. The full fit to the data sample for the normalisation mode is
shown in Fig. 2, and the yields are given in Table 4. Almost all
${{B}^{0}}\rightarrow{{D}^{*\pm}_{s}}{{D}^{*\mp}}$ decays are reconstructed
with a mass lower than the 5050${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ mass
cut imposed on the ${B}^{0}$ candidates. There is thus a relatively small
yield from this channel. Only the main
${{B}^{0}}\rightarrow{{D}^{+}_{s}}{{D}^{-}}$ peak is used for normalisation
purposes.

Figure 1:  Invariant mass distribution of the ${{B}^{0}_{s}}\rightarrow{D}^{(*)+}_{s}{D}^{(*)-}_{s}$ candidates. Also shown is the fit function and the individual components of the fit model. Table 3:  The yields extracted from the fit to the ${{B}^{0}_{s}}\rightarrow{D}^{(*)+}_{s}{D}^{(*)-}_{s}$ candidate sample. Decay Mode | Yield
---|---
${{B}^{0}_{s}}\rightarrow{{D}^{+}_{s}}{{D}^{-}_{s}}$ | 412 $\pm$ | 23
${{B}^{0}_{s}}\rightarrow{{D}^{*\pm}_{s}}{{D}^{\mp}_{s}}$ | 1032 $\pm$ | 39
${{B}^{0}_{s}}\rightarrow{{D}^{*\pm}_{s}}{{D}^{*\mp}_{s}}$ | 786 $\pm$ | 48
Combinatorial background | 1342 $\pm$ | 47
${{B}^{0}_{s}}\rightarrow D_{s}(2460)^{\pm}{{D}^{\mp}_{s}}$ | 432 $\pm$ | 42
Figure 2:  Invariant mass distribution of the ${{{B}^{0}}\rightarrow D^{(*)\pm}_{s}D^{(*)\mp}}$ candidates. Also shown is the fit function and the individual components of the fit model. Table 4:  The yields extracted from the fit to the ${{{B}^{0}}\rightarrow D^{(*)\pm}_{s}D^{(*)\mp}}$ candidate sample. Decay Mode | Yield
---|---
${{B}^{0}}\rightarrow{{D}^{+}_{s}}{{D}^{-}}$ | 3636 $\pm$ | 64
${{B}^{0}}\rightarrow{{D}^{*+}_{s}}{{D}^{-}}/{{D}^{*-}}{{D}^{+}_{s}}$ | 3579 $\pm$ | 110
${{B}^{0}}\rightarrow{{D}^{*+}_{s}}{{D}^{*-}}$ | 166 $\pm$ | 86
${{B}^{0}_{s}}\rightarrow D_{s}^{\pm}D^{\mp}$ | 85 $\pm$ | 13
Combinatorial background | 1542 $\pm$ | 56

##  6 Systematic uncertainties

A number of systematic uncertainties affect the measurements of the ratios of
branching fractions; the sources and magnitudes of these uncertainties are
summarised in Table 5. The dominant source of uncertainty for two of the three
branching fractions comes from the ${b}$ fragmentation fraction ratio,
$f_{s}/f_{d}=0.259\pm 0.015$ [25]. Part of the uncertainty on this ratio is
due to the ratio of the charm branching fractions
${B}({{D}^{+}_{s}}\rightarrow{{K}^{+}}{{K}^{-}}{{\pi}^{+}})/{B}({{D}^{-}}\rightarrow{{K}^{+}}{{\pi}^{-}}{{\pi}^{-}})=0.594\pm
0.020$ [25], the inverse of which is used in the measurements presented in
this paper, as shown in Eq. 1. With the two values from Ref. [25], the part of
the uncertainty on $f_{s}/f_{d}$ due to the charm branching fractions cancels,
leading to a total uncertainty for the product
$f_{d}/f_{s}\times{B}({{D}^{-}}\rightarrow{{K}^{+}}{{\pi}^{-}}{{\pi}^{-}})/{B}({{D}^{+}_{s}}\rightarrow{{K}^{+}}{{K}^{-}}{{\pi}^{+}})$
of $4.7\%$.

The fit model used for the yield extraction is validated using
pseudoexperiments and is found to be unbiased. The uncertainty due to the
imperfect knowledge of the shape of the full mass distribution is investigated
by measuring the yields using alternative models for each of the peaks. The
${{D}^{+}_{s}}{{D}^{-}_{s}}$ peak is modelled with an Apollonios function [26]
or a Cruijff function [27], the ${{D}^{*\pm}_{s}}{{D}^{\mp}_{s}}$ peak is
modelled using a single Gaussian function, and the
${{D}^{*+}_{s}}{{D}^{*-}_{s}}$ peak is modelled using a combination of two
Gaussian functions. The ${{{B}^{0}}\rightarrow D^{(*)\pm}_{s}D^{(*)\mp}}$ fit
model uncertainty is assessed by modeling the
${{B}^{0}}\rightarrow{{D}^{-}_{s}}{{D}^{+}}$ peak with both an Apollonios
function and a Cruijff function. In all cases, the systematic uncertainty is
taken to be the RMS deviation of the sets of yields with respect to the
nominal yields found using the standard fits. The
${{B}^{0}}\rightarrow{{D}^{-}_{s}}{{D}^{+}}$ uncertainty is added in
quadrature to the signal channel uncertainties, leading to a systematic
uncertainty of $3.4\%$ for the ${{D}^{*\pm}_{s}}{{D}^{\mp}_{s}}$ branching
fraction ratio, $2.2\%$ for the ${{D}^{*+}_{s}}{{D}^{*-}_{s}}$ branching
fraction ratio, and $2.2\%$ for the total
${{B}^{0}_{s}}\rightarrow{D}^{(*)+}_{s}{D}^{(*)-}_{s}$ branching fraction
ratio.

The uncertainty on the combinatorial background yield is determined by
considering the differences when instead fitting this background with an
exponential function, and is of the order of $1.5\%$ for all of the branching
fraction ratios.

The dominant uncertainty for the
${{B}^{0}_{s}}\rightarrow{{D}^{*+}_{s}}{{D}^{*-}_{s}}$ decay channel results
from the lack of knowledge of the ${{B}^{0}_{s}}\rightarrow
D_{{s}J}^{+}{{D}^{-}_{s}}$ background decays. The shape of this background
overlaps mostly with the
${{B}^{0}_{s}}\rightarrow{{D}^{*+}_{s}}{{D}^{*-}_{s}}$ signal decays, and
therefore the systematic uncertainty due to this background shape is much
larger for this channel ($5.0\%$) than for the other two exclusive branching
fractions ($0.2\%-0.4\%$). The uncertainty is measured by repeating the fit
with the cut-off point of the Argus function varied from
$5050{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ to
$5200{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, where the upper limit is
chosen in order to account for the presence of decays containing
$D_{{s}0}(2317)^{+}$ mesons. The changes to the yields from the values found
in the nominal fit are calculated in each case. The systematic uncertainty in
each channel is then assigned as the RMS of the full set of deviations. The
uncertainty on the overall branching fraction ratio is also determined in this
way, and is found to be $1.9\%$.

The uncertainties on the overall efficiencies due to the limited size of the
simulated samples are calculated individually for each channel. For the total
measurement,
${\mathcal{B}}({{B}^{0}_{s}}\rightarrow{D}^{(*)+}_{s}{D}^{(*)-}_{s})$, a
weighted average of the individual uncertainties is used, with weights
proportional to the final yield values obtained from data. These uncertainties
on the efficiencies are then propagated to the branching fraction ratios.

There is a systematic uncertainty arising from the calculation of the
efficiencies of the PID cuts. The calibration of the data samples is performed
in bins of momentum and pseudorapidity, which results in an uncertainty on the
calculated efficiencies owing to the finite size of the
${{D}^{*+}}\rightarrow{{D}^{0}}{{\pi}^{+}}$ calibration samples and the
binning scheme used. The uncertainty resulting from the calibration sample
size and binning scheme is determined by redoing the calibration using
different binning schemes. Another systematic uncertainty is due to the
presence of a small combinatorial background component in the samples that are
used to determine the PID efficiencies. The systematic uncertainty due to this
contamination is estimated by comparing the efficiencies found in data to
those found when calibrating simulated signal events. The total uncertainties
due to the PID efficiency calculation for the three branching fraction ratios
presented in this paper are shown in Table 5. The value for
${{B}^{0}_{s}}\rightarrow{D}^{(*)+}_{s}{D}^{(*)-}_{s}$ is again the weighted
average of the contributing channels, with the uncertainty for the
${{D}^{+}_{s}}{{D}^{-}_{s}}$ contribution being $1.1\%$.

The uncertainty of $1.5\%$ from the trigger response is assessed by
considering variations in the response between data and simulation. The
individual uncertainties are combined in quadrature to give the total relative
systematic uncertainties for each measurement given in Table 5.

Table 5:  Systematic uncertainties, in % of the relevant branching fraction ratio, for the ${{B}^{0}_{s}}\rightarrow{D}^{(*)+}_{s}{D}^{(*)-}_{s}$ branching fraction ratios. Source | ${{B}^{0}_{s}}\rightarrow{{D}^{*\pm}_{s}}{{D}^{\mp}_{s}}$ | ${{B}^{0}_{s}}\rightarrow{{D}^{*+}_{s}}{{D}^{*-}_{s}}$ |  ${{B}^{0}_{s}}\rightarrow{D}^{(*)+}_{s}{D}^{(*)-}_{s}$
---|---|---|---
$f_{d}/f_{s}\times\frac{{B}({{D}^{-}}\rightarrow{{K}^{+}}{{\pi}^{-}}{{\pi}^{-}})}{{B}({{D}^{+}_{s}}\rightarrow{{K}^{+}}{{K}^{-}}{{\pi}^{+}})}$ | $4.7$ | $4.7$ |  $4.7$
Fit model | $3.4$ | $2.2$ | $2.2$
Comb. background | $1.2$ | $1.9$ | $1.5$
$D_{{s}J}^{+}$ background | $0.4$ | $5.0$ | $1.9$
Simulation statistics | $1.9$ | $2.1$ | $1.9$
PID efficiency | $1.4$ | $1.8$ | $1.5$
Trigger efficiency | $1.5$ | $1.5$ | $1.5$
Total | $6.6$ | $8.1$ |  $6.4$

##  7 Summary and discussion

Inserting the measured yields and relative efficiencies into Eq. 1, along with
the $f_{s}/f_{d}$ and
${B}({{D}^{-}}\rightarrow{{K}^{+}}{{\pi}^{-}}{{\pi}^{-}})/{B}({{D}^{+}_{s}}\rightarrow{{K}^{+}}{{K}^{-}}{{\pi}^{+}})$
values taken from [25], gives

$\displaystyle\frac{{\mathcal{B}}(B_{s}^{0}\rightarrow~{}D_{s}^{(\ast)+}D_{s}^{(\ast)-})}{{\mathcal{B}}({{B}^{0}}\rightarrow{{D}^{+}_{s}}{{D}^{-}})}$
$\displaystyle=4.24\pm 0.14\mathrm{\,(stat)}\pm 0.27\mathrm{\,(syst)},$
$\displaystyle\frac{{\mathcal{B}}({{B}^{0}_{s}}\rightarrow{{D}^{*\pm}_{s}}{{D}^{\mp}_{s}})}{{\mathcal{B}}({{B}^{0}}\rightarrow{{D}^{+}_{s}}{{D}^{-}})}$
$\displaystyle=1.88\pm 0.08\mathrm{\,(stat)}\pm 0.12\mathrm{\,(syst)},$
$\displaystyle\frac{{\mathcal{B}}({{B}^{0}_{s}}\rightarrow{{D}^{*+}_{s}}{{D}^{*-}_{s}})}{{\mathcal{B}}({{B}^{0}}\rightarrow{{D}^{+}_{s}}{{D}^{-}})}$
$\displaystyle=1.76\pm 0.11\mathrm{\,(stat)}\pm 0.14\mathrm{\,(syst)}.$

Using the current world average measurement of the
${{B}^{0}}\rightarrow{{D}^{+}_{s}}{{D}^{-}}$ branching fraction of $(7.2\pm
0.8)\times 10^{-3}$ [20], gives

$\displaystyle{\mathcal{B}}(B_{s}^{0}\rightarrow~{}D_{s}^{(\ast)+}D_{s}^{(\ast)-})$
$\displaystyle=(3.05\pm 0.10\pm 0.20\pm 0.34)\%,$
$\displaystyle{\mathcal{B}}({{B}^{0}_{s}}\rightarrow{{D}^{*\pm}_{s}}{{D}^{\mp}_{s}})$
$\displaystyle=(1.35\pm 0.06\pm 0.09\pm 0.15)\%,$
$\displaystyle{\mathcal{B}}({{B}^{0}_{s}}\rightarrow{{D}^{*+}_{s}}{{D}^{*-}_{s}})$
$\displaystyle=(1.27\pm 0.08\pm 0.10\pm 0.14)\%,$

where the uncertainties are statistical, systematic, and due to the branching
fraction of the normalisation channel, respectively.

Figure 3 shows the LHCb measurement of the total
${{B}^{0}_{s}}\rightarrow{D}^{(*)+}_{s}{D}^{(*)-}_{s}$ branching fraction,
along with the previous measurements by Belle [5], CDF [6], and D0 [7], the
average of these previous measurements as calculated by HFAG [28], and the
theoretical value from Ref. [3]. The theoretical prediction is for a decay
time $t=0$, while the measurements integrate over the ${B}^{0}_{s}$ meson
lifetime; the correction factor for mixing is known [29], but has not been
applied. The LHCb result is consistent with all previous measurements and
calculations, and is the most precise determination to date. In addition, the
LHCb measurements of the individual
${{B}^{0}_{s}}\rightarrow{{D}^{*\pm}_{s}}{{D}^{\mp}_{s}}$ and
${{B}^{0}_{s}}\rightarrow{{D}^{*+}_{s}}{{D}^{*-}_{s}}$ branching fractions are
consistent with, and more precise than, all previous measurements.

Figure 3:  The ${{B}^{0}_{s}}\rightarrow{D}^{(*)+}_{s}{D}^{(*)-}_{s}$
branching fraction measurements by Belle, CDF, and D0, the current world
average, the theoretical prediction from Ref. [3], and the new LHCb result.

Using this measurement of the branching fraction of
${{B}^{0}_{s}}\rightarrow{D}^{(*)+}_{s}{D}^{(*)-}_{s}$ decays to calculate
$\Delta\Gamma_{{s}}/\Gamma_{{s}}$ as detailed in Ref. [1] gives a value
approximately half as large as the most recent HFAG determination [28],
suggesting that indeed ${{B}^{0}_{s}}\rightarrow{D}^{(*)+}_{s}{D}^{(*)-}_{s}$
decays do not saturate the $C\\!P$-even modes [2]. The measurements presented
in this analysis will be useful in improving the understanding of
hadronisation effects in ${B}^{0}_{s}$ decays via the
${b}\\!\rightarrow{c}{\overline{c}}{s}$ quark transition, and in determining a
precise value of the inclusive branching fraction of these
${b}\\!\rightarrow{c}{\overline{c}}{s}$ decays.

## 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 FANO (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). 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), The Royal Society, Royal
Commission for the Exhibition of 1851 and the Leverhulme Trust (United
Kingdom).

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LHCb collaboration

R. Aaij39, C. Abellán Beteta41, B. Adeva38, M. Adinolfi47, A. Affolder53, Z.
Ajaltouni5, S. Akar6, J. Albrecht10, F. Alessio39, M. Alexander52, S. Ali42,
G. Alkhazov31, P. Alvarez Cartelle54, A.A. Alves Jr58, S. Amato2, S. Amerio23,
Y. Amhis7, L. An3,40, L. Anderlini18, G. Andreassi40, M. Andreotti17,g, J.E.
Andrews59, R.B. Appleby55, O. Aquines Gutierrez11, F. Archilli39, P.
d’Argent12, A. Artamonov36, M. Artuso60, E. Aslanides6, G. Auriemma26,n, M.
Baalouch5, S. Bachmann12, J.J. Back49, A. Badalov37, C. Baesso61, W.
Baldini17,39, R.J. Barlow55, C. Barschel39, S. Barsuk7, W. Barter39, V.
Batozskaya29, V. Battista40, A. Bay40, L. Beaucourt4, J. Beddow52, F.
Bedeschi24, I. Bediaga1, L.J. Bel42, V. Bellee40, N. Belloli21,k, I.
Belyaev32, E. Ben-Haim8, G. Bencivenni19, S. Benson39, J. Benton47, A.
Berezhnoy33, R. Bernet41, A. Bertolin23, M.-O. Bettler39, M. van Beuzekom42,
S. Bifani46, P. Billoir8, T. Bird55, A. Birnkraut10, A. Bizzeti18,i, T.
Blake49, F. Blanc40, J. Blouw11, S. Blusk60, V. Bocci26, A. Bondar35, N.
Bondar31,39, W. Bonivento16, S. Borghi55, M. Borisyak66, M. Borsato38, T.J.V.
Bowcock53, E. Bowen41, C. Bozzi17,39, S. Braun12, M. Britsch12, T. Britton60,
J. Brodzicka55, N.H. Brook47, E. Buchanan47, C. Burr55, A. Bursche41, J.
Buytaert39, S. Cadeddu16, R. Calabrese17,g, M. Calvi21,k, M. Calvo Gomez37,p,
P. Campana19, D. Campora Perez39, L. Capriotti55, A. Carbone15,e, G.
Carboni25,l, R. Cardinale20,j, A. Cardini16, P. Carniti21,k, L. Carson51, K.
Carvalho Akiba2, G. Casse53, L. Cassina21,k, L. Castillo Garcia40, M.
Cattaneo39, Ch. Cauet10, G. Cavallero20, R. Cenci24,t, M. Charles8, Ph.
Charpentier39, M. Chefdeville4, S. Chen55, S.-F. Cheung56, N. Chiapolini41, M.
Chrzaszcz41,27, X. Cid Vidal39, G. Ciezarek42, P.E.L. Clarke51, M.
Clemencic39, H.V. Cliff48, J. Closier39, V. Coco39, J. Cogan6, E. Cogneras5,
V. Cogoni16,f, L. Cojocariu30, G. Collazuol23,r, P. Collins39, A. Comerma-
Montells12, A. Contu39, A. Cook47, M. Coombes47, S. Coquereau8, G. Corti39, M.
Corvo17,g, B. Couturier39, G.A. Cowan51, D.C. Craik51, A. Crocombe49, M. Cruz
Torres61, S. Cunliffe54, R. Currie54, C. D’Ambrosio39, E. Dall’Occo42, J.
Dalseno47, P.N.Y. David42, A. Davis58, O. De Aguiar Francisco2, K. De Bruyn6,
S. De Capua55, M. De Cian12, J.M. De Miranda1, L. De Paula2, P. De Simone19,
C.-T. Dean52, D. Decamp4, M. Deckenhoff10, L. Del Buono8, N. Déléage4, M.
Demmer10, D. Derkach66, O. Deschamps5, F. Dettori39, B. Dey22, A. Di Canto39,
F. Di Ruscio25, H. Dijkstra39, S. Donleavy53, F. Dordei39, M. Dorigo40, A.
Dosil Suárez38, A. Dovbnya44, K. Dreimanis53, L. Dufour42, G. Dujany55, K.
Dungs39, P. Durante39, R. Dzhelyadin36, A. Dziurda27, A. Dzyuba31, S.
Easo50,39, U. Egede54, V. Egorychev32, S. Eidelman35, S. Eisenhardt51, U.
Eitschberger10, R. Ekelhof10, L. Eklund52, I. El Rifai5, Ch. Elsasser41, S.
Ely60, S. Esen12, H.M. Evans48, T. Evans56, A. Falabella15, C. Färber39, N.
Farley46, S. Farry53, R. Fay53, D. Ferguson51, V. Fernandez Albor38, F.
Ferrari15, F. Ferreira Rodrigues1, M. Ferro-Luzzi39, S. Filippov34, M.
Fiore17,39,g, M. Fiorini17,g, M. Firlej28, C. Fitzpatrick40, T. Fiutowski28,
F. Fleuret7,b, K. Fohl39, P. Fol54, M. Fontana16, F. Fontanelli20,j, D. C.
Forshaw60, R. Forty39, M. Frank39, C. Frei39, M. Frosini18, J. Fu22, E.
Furfaro25,l, A. Gallas Torreira38, D. Galli15,e, S. Gallorini23, S.
Gambetta51, M. Gandelman2, P. Gandini56, Y. Gao3, J. García Pardiñas38, J.
Garra Tico48, L. Garrido37, D. Gascon37, C. Gaspar39, R. Gauld56, L.
Gavardi10, G. Gazzoni5, D. Gerick12, E. Gersabeck12, M. Gersabeck55, T.
Gershon49, Ph. Ghez4, S. Gianì40, V. Gibson48, O.G. Girard40, L. Giubega30,
V.V. Gligorov39, C. Göbel61, D. Golubkov32, A. Golutvin54,39, A. Gomes1,a, C.
Gotti21,k, M. Grabalosa Gándara5, R. Graciani Diaz37, L.A. Granado Cardoso39,
E. Graugés37, E. Graverini41, G. Graziani18, A. Grecu30, E. Greening56, P.
Griffith46, L. Grillo12, O. Grünberg64, B. Gui60, E. Gushchin34, Yu. Guz36,39,
T. Gys39, T. Hadavizadeh56, C. Hadjivasiliou60, G. Haefeli40, C. Haen39, S.C.
Haines48, S. Hall54, B. Hamilton59, X. Han12, S. Hansmann-Menzemer12, N.
Harnew56, S.T. Harnew47, J. Harrison55, J. He39, T. Head40, V. Heijne42, A.
Heister9, K. Hennessy53, P. Henrard5, L. Henry8, J.A. Hernando Morata38, E.
van Herwijnen39, M. Heß64, A. Hicheur2, D. Hill56, M. Hoballah5, C. Hombach55,
W. Hulsbergen42, T. Humair54, M. Hushchyn66, N. Hussain56, D. Hutchcroft53, D.
Hynds52, M. Idzik28, P. Ilten57, R. Jacobsson39, A. Jaeger12, J. Jalocha56, E.
Jans42, A. Jawahery59, M. John56, D. Johnson39, C.R. Jones48, C. Joram39, B.
Jost39, N. Jurik60, S. Kandybei44, W. Kanso6, M. Karacson39, T.M. Karbach39,†,
S. Karodia52, M. Kecke12, M. Kelsey60, I.R. Kenyon46, M. Kenzie39, T. Ketel43,
E. Khairullin66, B. Khanji21,39,k, C. Khurewathanakul40, T. Kirn9, S.
Klaver55, K. Klimaszewski29, O. Kochebina7, M. Kolpin12, I. Komarov40, R.F.
Koopman43, P. Koppenburg42,39, M. Kozeiha5, L. Kravchuk34, K. Kreplin12, M.
Kreps49, P. Krokovny35, F. Kruse10, W. Krzemien29, W. Kucewicz27,o, M.
Kucharczyk27, V. Kudryavtsev35, A. K. Kuonen40, K. Kurek29, T.
Kvaratskheliya32, D. Lacarrere39, G. Lafferty55,39, A. Lai16, D. Lambert51, G.
Lanfranchi19, C. Langenbruch49, B. Langhans39, T. Latham49, C. Lazzeroni46, R.
Le Gac6, J. van Leerdam42, J.-P. Lees4, R. Lefèvre5, A. Leflat33,39, J.
Lefrançois7, E. Lemos Cid38, O. Leroy6, T. Lesiak27, B. Leverington12, Y. Li7,
T. Likhomanenko66,65, M. Liles53, R. Lindner39, C. Linn39, F. Lionetto41, B.
Liu16, X. Liu3, D. Loh49, I. Longstaff52, J.H. Lopes2, D. Lucchesi23,r, M.
Lucio Martinez38, H. Luo51, A. Lupato23, E. Luppi17,g, O. Lupton56, N.
Lusardi22, A. Lusiani24, F. Machefert7, F. Maciuc30, O. Maev31, K. Maguire55,
S. Malde56, A. Malinin65, G. Manca7, G. Mancinelli6, P. Manning60, A.
Mapelli39, J. Maratas5, J.F. Marchand4, U. Marconi15, C. Marin Benito37, P.
Marino24,39,t, J. Marks12, G. Martellotti26, M. Martin6, M. Martinelli40, D.
Martinez Santos38, F. Martinez Vidal67, D. Martins Tostes2, L.M. Massacrier7,
A. Massafferri1, R. Matev39, A. Mathad49, Z. Mathe39, C. Matteuzzi21, A.
Mauri41, B. Maurin40, A. Mazurov46, M. McCann54, J. McCarthy46, A. McNab55, R.
McNulty13, B. Meadows58, F. Meier10, M. Meissner12, D. Melnychuk29, M. Merk42,
E Michielin23, D.A. Milanes63, M.-N. Minard4, D.S. Mitzel12, J. Molina
Rodriguez61, I.A. Monroy63, S. Monteil5, M. Morandin23, P. Morawski28, A.
Mordà6, M.J. Morello24,t, J. Moron28, A.B. Morris51, R. Mountain60, F.
Muheim51, D. Müller55, J. Müller10, K. Müller41, V. Müller10, M. Mussini15, B.
Muster40, P. Naik47, T. Nakada40, R. Nandakumar50, A. Nandi56, I. Nasteva2, M.
Needham51, N. Neri22, S. Neubert12, N. Neufeld39, M. Neuner12, A.D. Nguyen40,
T.D. Nguyen40, C. Nguyen-Mau40,q, V. Niess5, R. Niet10, N. Nikitin33, T.
Nikodem12, A. Novoselov36, D.P. O’Hanlon49, A. Oblakowska-Mucha28, V.
Obraztsov36, S. Ogilvy52, O. Okhrimenko45, R. Oldeman16,48,f, C.J.G.
Onderwater68, B. Osorio Rodrigues1, J.M. Otalora Goicochea2, A. Otto39, P.
Owen54, A. Oyanguren67, A. Palano14,d, F. Palombo22,u, M. Palutan19, J.
Panman39, A. Papanestis50, M. Pappagallo52, L.L. Pappalardo17,g, C.
Pappenheimer58, W. Parker59, C. Parkes55, G. Passaleva18, G.D. Patel53, M.
Patel54, C. Patrignani20,j, A. Pearce55,50, A. Pellegrino42, G. Penso26,m, M.
Pepe Altarelli39, S. Perazzini15,e, P. Perret5, L. Pescatore46, K. Petridis47,
A. Petrolini20,j, M. Petruzzo22, E. Picatoste Olloqui37, B. Pietrzyk4, M.
Pikies27, D. Pinci26, A. Pistone20, A. Piucci12, S. Playfer51, M. Plo
Casasus38, T. Poikela39, F. Polci8, A. Poluektov49,35, I. Polyakov32, E.
Polycarpo2, A. Popov36, D. Popov11,39, B. Popovici30, C. Potterat2, E.
Price47, J.D. Price53, J. Prisciandaro38, A. Pritchard53, C. Prouve47, V.
Pugatch45, A. Puig Navarro40, G. Punzi24,s, W. Qian4, R. Quagliani7,47, B.
Rachwal27, J.H. Rademacker47, M. Rama24, M. Ramos Pernas38, M.S. Rangel2, I.
Raniuk44, N. Rauschmayr39, G. Raven43, F. Redi54, S. Reichert55, A.C. dos
Reis1, V. Renaudin7, S. Ricciardi50, S. Richards47, M. Rihl39, K.
Rinnert53,39, V. Rives Molina37, P. Robbe7,39, A.B. Rodrigues1, E.
Rodrigues55, J.A. Rodriguez Lopez63, P. Rodriguez Perez55, S. Roiser39, V.
Romanovsky36, A. Romero Vidal38, J. W. Ronayne13, M. Rotondo23, T. Ruf39, P.
Ruiz Valls67, J.J. Saborido Silva38, N. Sagidova31, B. Saitta16,f, V.
Salustino Guimaraes2, C. Sanchez Mayordomo67, B. Sanmartin Sedes38, R.
Santacesaria26, C. Santamarina Rios38, M. Santimaria19, E. Santovetti25,l, A.
Sarti19,m, C. Satriano26,n, A. Satta25, D.M. Saunders47, D. Savrina32,33, S.
Schael9, M. Schiller39, H. Schindler39, M. Schlupp10, M. Schmelling11, T.
Schmelzer10, B. Schmidt39, O. Schneider40, A. Schopper39, M. Schubiger40,
M.-H. Schune7, R. Schwemmer39, B. Sciascia19, A. Sciubba26,m, A. Semennikov32,
A. Sergi46, N. Serra41, J. Serrano6, L. Sestini23, P. Seyfert21, M. Shapkin36,
I. Shapoval17,44,g, Y. Shcheglov31, T. Shears53, L. Shekhtman35, V.
Shevchenko65, A. Shires10, B.G. Siddi17, R. Silva Coutinho41, L. Silva de
Oliveira2, G. Simi23,s, M. Sirendi48, N. Skidmore47, T. Skwarnicki60, E.
Smith56,50, E. Smith54, I.T. Smith51, J. Smith48, M. Smith55, H. Snoek42, M.D.
Sokoloff58,39, F.J.P. Soler52, F. Soomro40, D. Souza47, B. Souza De Paula2, B.
Spaan10, P. Spradlin52, S. Sridharan39, F. Stagni39, M. Stahl12, S. Stahl39,
S. Stefkova54, O. Steinkamp41, O. Stenyakin36, S. Stevenson56, S. Stoica30, S.
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S. Topp-Joergensen56, N. Torr56, E. Tournefier4, S. Tourneur40, K. Trabelsi40,
M. Traill52, M.T. Tran40, M. Tresch41, A. Trisovic39, A. Tsaregorodtsev6, P.
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Zhokhov32, L. Zhong3, 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
35Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
36Institute for High Energy Physics (IHEP), Protvino, Russia
37Universitat de Barcelona, Barcelona, Spain
38Universidad de Santiago de Compostela, Santiago de Compostela, Spain
39European Organization for Nuclear Research (CERN), Geneva, Switzerland
40Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
41Physik-Institut, Universität Zürich, Zürich, Switzerland
42Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
43Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
44NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
45Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
46University of Birmingham, Birmingham, United Kingdom
47H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
48Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
49Department of Physics, University of Warwick, Coventry, United Kingdom
50STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
51School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
52School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
53Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
54Imperial College London, London, United Kingdom
55School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
56Department of Physics, University of Oxford, Oxford, United Kingdom
57Massachusetts Institute of Technology, Cambridge, MA, United States
58University of Cincinnati, Cincinnati, OH, United States
59University of Maryland, College Park, MD, United States
60Syracuse University, Syracuse, NY, United States
61Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
62Institute of Particle Physics, Central China Normal University, Wuhan,
Hubei, China, associated to 3
63Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia,
associated to 8
64Institut für Physik, Universität Rostock, Rostock, Germany, associated to 12
65National Research Centre Kurchatov Institute, Moscow, Russia, associated to
32
66Yandex School of Data Analysis, Moscow, Russia, associated to 32
67Instituto de Fisica Corpuscular (IFIC), Universitat de Valencia-CSIC,
Valencia, Spain, associated to 37
68Van Swinderen Institute, University of Groningen, Groningen, The
Netherlands, associated to 42

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 Urbino, Urbino, Italy
iUniversità di Modena e Reggio Emilia, Modena, Italy
jUniversità di Genova, Genova, Italy
kUniversità di Milano Bicocca, Milano, Italy
lUniversità di Roma Tor Vergata, Roma, Italy
mUniversità di Roma La Sapienza, Roma, Italy
nUniversità della Basilicata, Potenza, Italy
oAGH - University of Science and Technology, Faculty of Computer Science,
Electronics and Telecommunications, Kraków, Poland
pLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
qHanoi University of Science, Hanoi, Viet Nam
rUniversità di Padova, Padova, Italy
sUniversità di Pisa, Pisa, Italy
tScuola Normale Superiore, Pisa, Italy
uUniversità degli Studi di Milano, Milano, Italy
†Deceased