# Cosmetic crossing conjecture for genus one knots with non-trivial Alexander
polynomial

Tetsuya Ito

###### Abstract.

We prove the cosmetic crossing conjecture for genus one knots with non-trivial
Alexander polynomial. We also prove the conjecture for genus one knots with
trivial Alexander polynomial, under some additional assumptions.

##  1\. Introduction

A _cosmetic crossing_ is a non-nugatory crossing such that the crossing change
at the crossing preserves the knot. A cosmetic crossing conjecture [Kir,
Problem 1.58] asserts there are no such crossings.

######  Conjecture 1 (Cosmetic crossing conjecture).

An oriented knot $K$ in $S^{3}$ does not have cosmetic crossings.

Here a crossing $c$ of a knot diagram $D$ is _nugatory_ if there is a circle
$C$ on the projection plane that transverse to the diagram $D$ only at $c$.
Obviously the crossing change at a nugatory crossing always preserves the
knot, so the cosmetic crossing conjecture can be rephrased that when a
crossing change at a crossing $c$ preserves the knot, then $c$ is nugatory.

In [BFKP] Balm-Friedl-Kalfagianni-Powell proved the following constraints for
genus one knots to admit a cosmetic crossing.

######  Theorem 1.1.

[BFKP, Theorem 1.1, Theorem 5.1] Let $K$ be a genus one knot that admits a
cosmetic crossing. Then $K$ has the following properties.

  * •

$K$ is algebraically slice.

  * •

For the double branched covering $\Sigma_{2}(K)$ of $K$,
$H_{1}(\Sigma_{2}(K);\mathbb{Z})$ is finite cyclic.

  * •

If $K$ has a unique genus one Seifert surface, $\Delta_{K}(t)=1$.

In this paper, by using the 2-loop part of the Kontsevich invariant, we prove
the cosmetic crossing conjecture for genus one knot with non-trivial Alexander
polynomial.

######  Theorem 1.2.

Let $K$ be a genus one knot. If $\Delta_{K}(t)\neq 1$, then $K$ satisfies the
cosmetic crossing conjecture.

For genus one knot $K$ with $\Delta_{K}(t)=1$ we get an additional constraint
for $K$ to admit a cosmetic crossing. Let $\lambda$ be the Casson invariant of
integral homology spheres and let
$w_{3}(K)=\frac{1}{36}V^{\prime\prime\prime}_{K}(1)+\frac{1}{12}V^{\prime\prime}_{K}(1)$
be the primitive integer-valued degree $3$ finite type invariant of $K$. Here
$V_{K}(t)$ is the Jones polynomial of $K$.

######  Theorem 1.3.

Let $K$ be a genus one knot with $\Delta_{K}(t)=1$. If
$\lambda(\Sigma_{2}(K))-2w_{3}(K)\not\equiv 0\pmod{16}$, then $K$ satisfies
the cosmetic crossing conjecture.

The cosmetic crossing conjecture has been confirmed for several cases;
2-bridge knots [Tor], fibered knots [Kal], knots whose double branched
coverings are L-spaces with square-free 1st homology [LiMo], and some
satellite knots [BaKa]. Except the last satellite cases and the unknot, all
the knots mentioned so far, including knots treated in Theorem 1.1, has non-
trivial Alexander polynomial.

Theorem 1.3 gives examples of non-satellite knots with trivial Alexander
polynomial satisfying the cosmetic crossing conjecture. Let $K=P(p,q,r)$ be
the pretzel knot for odd $p,q,r$. Obviously, as long as $K$ is non-trivial,
$g(K)=1$. The Alexander polynomial of $K$ is

$\Delta_{K}(t)=\frac{pq+qr+rp+1}{4}t+\frac{-2pq-2qr-2rp+1}{2}+\frac{pq+qr+rp+1}{4}t^{-1}.$

Hence, for example, the pretzel knot $P(4k+1,4k+3,-(2k+1))$ has the trivial
Alexander polynomial.

######  Corollary 1.4.

If $k\equiv 1,2\pmod{4}$, the pretzel knot $P(4k+1,4k+3,-(2k+1))$ satisfies
the cosmetic crossing conjecture.

##  2\. Cosmetic crossing of genus one knot and Seifert surface

We review an argument of [BFKP, Section 2, Section 3] that relates a cosmetic
crossing change and Seifert matrix.

A _crossing disk_ $D$ of an oriented knot $K$ is an embedded disk having
exactly one positive and one negative crossing with $K$. A crossing change can
be seen as $\varepsilon=\pm 1$ Dehn surgery on $\partial D$ for an appropriate
crossing disk $D$, and the crossing is nugatory if and only if $\partial D$
bounds an embedded disk in $S^{3}\setminus K$.

Assume that $K$ admits a cosmetic crossing with the crossing disk $D$. Then as
is discussed in [BFKP, Section 2], there is a minimum genus Seifert surface
$S$ of $K$ such that $\alpha:=D\cap S$ is a properly embedded, essential arc
in $S$.

If $g(S)=1$, such an arc $\alpha$ is non-separating. We take simple closed
curves $a_{x},a_{y}$ of $S$ so that

  * •

$a_{x}$ intersects $\alpha$ exactly once.

  * •

$a_{x}$ and $a_{y}$ form a symplectic basis of $H_{1}(S;\mathbb{Z})$.

Then we view $K=\partial S$ as a neighborhood of $a_{x}\cup a_{y}$ and express
$K$ by a framed 2-tangle $T$ as depicted in Figure 1.

$D$$\alpha$$a_{x}$$a_{y}$$x$$y$2-parallel ofa 2-tangle$T=$

Figure 1. A spine tangle $T$ adapted to the cosmetic crossing

We call the framed tangle $T$ a _spine tangle_ of $K$ adapted to the cosmetic
crossing of a genus one knot $K$. Let $M=\begin{pmatrix}n&\ell\\\
\ell&m\end{pmatrix}$ be the linking matrix of $T$, where $n$ (resp. $m$) is
the framing of the strand $x$ (resp. $y$) and $\ell$ is the linking number of
two strands of $T$.

Let $K^{\prime}$ be a knot obtained from $K$ by crossing change along the
crossing disk $D$. Then $S$ gives rise to a Seifert surface $S^{\prime}$ of
$K^{\prime}$ ([BFKP, Proposition 2.1]). $K^{\prime}$ has a spine tangle
presentation $T^{\prime}$, so that $T^{\prime}$ and $T$ are the same as
unframed tangles, and that the linking matrix of $T^{\prime}$ is
$M^{\prime}=\begin{pmatrix}n\pm 1&\ell\\\ \ell&m\end{pmatrix}$

With respect to the basis $\\{a_{x},a_{y}\\}$, the Seifert matrix $V$ of $K$
and the Seifert matrix $V^{\prime}$ of $K^{\prime}$ are given by

$V=\begin{pmatrix}n&\ell\\\ \ell\pm 1&m\end{pmatrix},\quad
V^{\prime}=\begin{pmatrix}n\pm 1&\ell\\\ \ell\pm 1&m\end{pmatrix}$

respectively. Since $K$ and $K^{\prime}$ are the same knot,

$\Delta_{K}(t)\doteq\det(V-tV^{T})\doteq\det(V^{\prime}-tV^{\prime}{}^{T})\doteq\Delta_{K^{\prime}}(t).$

By direct computation, this implies that

(2.1) $m=0.$

In particular, $K$ is algebraically slice.

##  3\. 2-loop polynomial of genus one knot

Here we quickly review the 2-loop polynomial. For details, see [Oht]. Let
$\mathcal{B}$ be the space of open Jacobi diagram. For a knot $K$ in $S^{3}$,
let $Z^{\sigma}(K)\in\mathcal{B}$ be the Kontsevich invariant of $K$, viewed
so that it takes value in $\mathcal{B}$ by composing the inverse of the
Poincaré-Birkoff-Witt isomorphism
$\sigma:\mathcal{A}(S^{1})\rightarrow\mathcal{B}$.

A Jacobi diagram whose edge is labeled by a power series
$f(\hbar)=c_{0}+c_{1}\hbar+c_{2}\hbar^{2}+c_{3}\hbar^{3}+\cdots$ represents
the Jacobi diagram

$\raisebox{-11.38109pt}{
\begin{picture}(20.0,30.0)\put(2.0,0.0){\line(0,1){30.0}}
\put(5.0,22.0){\scriptsize$f(\hbar)$} \end{picture}
}=c_{0}\raisebox{-11.38109pt}{
\begin{picture}(10.0,30.0)\put(2.0,0.0){\line(0,1){30.0}} \end{picture}
}+c_{1}\raisebox{-11.38109pt}{
\begin{picture}(10.0,30.0)\put(2.0,0.0){\line(0,1){30.0}}
\put(2.0,15.0){\line(1,0){10.0}} \end{picture} }+c_{2}\raisebox{-11.38109pt}{
\begin{picture}(10.0,30.0)\put(2.0,0.0){\line(0,1){30.0}}
\put(2.0,12.0){\line(1,0){10.0}} \put(2.0,18.0){\line(1,0){10.0}}
\end{picture} }+c_{3}\raisebox{-11.38109pt}{
\begin{picture}(10.0,30.0)\put(2.0,0.0){\line(0,1){30.0}}
\put(2.0,9.0){\line(1,0){10.0}} \put(2.0,15.0){\line(1,0){10.0}}
\put(2.0,21.0){\line(1,0){10.0}} \end{picture} }+\cdots$

It is known that (the logarithm of) the Kontsevich invariant $Z^{\sigma}(K)$
is written in the following form [GaKr, Kri].

$\displaystyle\log_{\sqcup}Z^{\sigma}(K)$
$\displaystyle=\raisebox{-19.91692pt}{
\begin{picture}(140.0,60.0)\put(70.0,22.0){\oval(140.0,20.0)}
\put(5.0,40.0){\footnotesize$\frac{1}{2}\log\left(\frac{\sinh(\hbar/2)}{\hbar/2}\right)-\frac{1}{2}\log(\Delta_{K}(e^{\hbar}))$}
\end{picture} }+\sum_{i:\textrm{finite}}\raisebox{-22.76219pt}{
\begin{picture}(80.0,60.0)\put(40.0,25.0){\oval(80.0,50.0)}
\put(0.0,25.0){\line(1,0){80.0}}
\put(10.0,54.0){\footnotesize$p_{i,1}(e^{\hbar})/\Delta_{K}(e^{\hbar})$}
\put(10.0,29.0){\footnotesize$p_{i,2}(e^{\hbar})/\Delta_{K}(e^{\hbar})$}
\put(10.0,6.0){\footnotesize$p_{i,3}(e^{\hbar})/\Delta_{K}(e^{\hbar})$}
\end{picture} }$ $\displaystyle\qquad+((\ell>2)\mbox{-loop parts}).$

Here

  * •

$\Delta_{K}(t)$ is the Alexander polynomial of $K$, normalized so that
$\Delta_{K}(1)=1$ and $\Delta_{K}(t)=\Delta_{K}(t^{-1})$ hold.

  * •

$\log_{\sqcup}$ is the logarithm with respect to the disjoint union product
$\sqcup$ of $\mathcal{B}$, given by

$\log_{\sqcup}(1+D)=D-\frac{1}{2}D\sqcup D+\frac{1}{3}D\sqcup D\sqcup
D+\cdots.$

  * •

$p_{i,j}(e^{\hbar})$ is a polynomial of $e^{\hbar}$.

Let

$\Theta(t_{1},t_{2},t_{3};K)=\sum_{\varepsilon\in\\{\pm 1\\}}\sum_{\sigma\in
S_{3}}p_{i,1}(t^{\varepsilon}_{\sigma(1)})p_{i,2}(t^{\varepsilon}_{\sigma(2)})p_{i,3}(t^{\varepsilon}_{\sigma(3)}).$

Here $S_{3}$ is the symmetric group of degree $3$. The _2-loop polynomial_
$\Theta_{K}(t_{1},t_{2})\in\mathbb{Q}[t_{1}^{\pm 1},t_{2}^{\pm 1}]$ of a knot
$K$ is defined by

$\Theta_{K}(t_{1},t_{2})=\Theta(t_{1},t_{2},t_{3};K)|_{t_{3}=t_{1}^{-1}t_{2}^{-1}}.$

The _reduced 2-loop polynomial_ is a reduction of the $2$-loop polynomial
defined by

$\widehat{\Theta}_{K}(t)=\frac{\Theta_{K}(t,1)}{(t^{\frac{1}{2}}-t^{-\frac{1}{2}})^{2}}\in\mathbb{Q}[t^{\pm
1}].$

In general, although Ohtsuki developed fundamental techniques and machineries
that enable us to compute $\Theta_{K}(t_{1},t_{2})$, the computation of the
2-loop polynomial is much more complicated than the computation of the 1-loop
part (i.e., the Alexander polynomial). Fortunately, when the knot has genus
one, Ohtsuki proved a direct formula of $\Theta_{K}(t_{1},t_{2})$ [Oht,
Theorem 3.1]. Consequently he gave the following formula of the reduced 2-loop
polynomial of genus one knots.

######  Theorem 3.1.

[Oht, Corollary 3.5] Let $K$ be a genus one knot expressed by using a framed
2-tangle $T$ as in Figure 1, and let $M=\begin{pmatrix}n&\ell\\\
\ell&m\end{pmatrix}$ be the linking matrix of $T$. Then

$\displaystyle\widehat{\Theta}_{K}(t)$
$\displaystyle=\Bigl{(}(n+m)(d-\frac{nm}{2})-\ell(\ell+\frac{1}{2})(\ell+1)+12v_{3}\Bigr{)}\Bigl{(}-2-\frac{2d+1}{3}(t+t^{-1}-2)\Bigr{)}$
$\displaystyle\quad-4\left(mv_{2}^{xx}+nv_{2}^{yy}-(\ell+\frac{1}{2})v_{2}^{xy}+3v_{3}\right)\Delta_{K}(t)$

Here

  * •

$d=nm-\ell^{2}-\ell$. In particular, $\Delta_{K}(t)=dt+(1-2d)+dt^{-1}$.

  * •

$v_{2}^{xx},v_{2}^{yy},v_{2}^{xy}$ (resp. $v_{3}$) are some integer-valued
finite type invariant of $T$ whose degree is $2$ (resp. $3$), which do not
depend on the framing.

##  4\. Constraint for cosmetic crossings

We prove the Theorem 1.2 and Theorem 1.3 at the same time.

######  Theorem 4.1.

Let $K$ be a genus one knot. If $K$ admits a cosmetic crossing, then
$\Delta_{K}(t)=1$ and $\lambda(\Sigma_{2}(K))-2w_{3}(K)\equiv 0\pmod{16}$.

###### Proof.

Assume that $K$ is a genus one knot admitting a cosmetic crossing. We express
$K$ using a spine tangle $T$ adapted to the cosmetic crossing. Then as we have
seen (2.1), the linking matrix of $T$ is $M=\begin{pmatrix}n&\ell\\\
\ell&0\end{pmatrix}$. Moreover, for the knot $K^{\prime}$ obtained by the
crossing change, $K^{\prime}$ has a spine tangle $T^{\prime}$ which is
identical with $T$ as an unframed tangle with linking matrix is
$M^{\prime}=\begin{pmatrix}n\pm 1&\ell\\\ \ell&0\end{pmatrix}$.

Since the finite type invariants $v_{2}^{xx},v_{2}^{yy},v_{2}^{xy}$ and
$v_{3}$ do not depend on the framing, by Theorem 3.1,

$\displaystyle 0$
$\displaystyle=\widehat{\Theta}_{K}(t)-\widehat{\Theta}_{K^{\prime}}(t)$
$\displaystyle=d(-2-\frac{2d+1}{3}(t+t^{-1}-2))-4v_{2}^{yy}(dt+(1-2d)+dt^{-1})$
$\displaystyle=d\left(-\frac{2d+1}{3}-4v_{2}^{yy}\right)t+\frac{d(4d-4)}{3}+4v_{2}^{yy}(2d-1)+d\left(-\frac{2d+1}{3}-4v_{2}^{yy}\right)t^{-1}.$

Therefore

(4.1)
$d\left(-\frac{2d+1}{3}-4v_{2}^{yy}\right)=\frac{d(4d-4)}{3}+4v_{2}^{yy}(2d-1)=0.$

If $d\neq 0$, by (4.1) $d=\frac{1}{4}$. Since $d\in\mathbb{Z}$, this is a
contradiction so we conclude $d=0$ and $\Delta_{K}(t)=1$.

Then by (4.1), $d=0$ implies $v_{2}^{yy}=0$. Moreover, since
$d=nm-\ell^{2}-\ell=-\ell(\ell+1)$, we get $\ell=0,-1$. Thus by Theorem 3.1,
the reduced 2-loop polynomial is

$\widehat{\Theta}_{K}(t)=12v_{3}\left(-2-\frac{1}{3}(t+t^{-1}-2)\right)-4\left(-\left(\ell+\frac{1}{2}\right)v_{2}^{xy}-3v_{3}\right)$

hence

$\widehat{\Theta}_{K}(1)=-12v_{3}+4\left(\ell+\frac{1}{2}\right)v_{2}^{xy},\
\widehat{\Theta}_{K}(-1)=4v_{3}+4\left(\ell+\frac{1}{2}\right)v_{2}^{xy}.$

On the other hand, by [Oht, Proposition 1.1]

$\widehat{\Theta}_{K}(1)=2w_{3}(K),\quad\widehat{\Theta}_{K}(-1)=-\frac{1}{12}V^{\prime}_{K}(-1)V_{K}(-1).$

Since $\Delta_{K}(-1)=V_{K}(-1)=1$, by Mullins’ formula of the Casson-Walker
invariant $\lambda_{w}$ of the double branched coverings [Mul],

$\lambda_{w}(\Sigma_{2}(K))=-\frac{V^{\prime}_{K}(-1)}{6V_{K}(-1)}+\frac{\sigma(K)}{4}$

we get

$\widehat{\Theta}_{K}(-1)=\frac{1}{2}\lambda_{w}(\Sigma_{2}(K)).$

For an integral homology sphere, the Casson invariant $\lambda$ is twice of
the Casson-Walker invariant $\lambda_{w}$ hence we conclude

$\displaystyle\lambda(\Sigma_{2}(K))-2w_{3}(K)$
$\displaystyle=\widehat{\Theta}_{K}(-1)-\widehat{\Theta}_{K}(1)=16v_{3}.$

∎

###### Proof of Corollary 1.4.

The reduced $2$-loop polynomial of genus pretzel knots $P(p,q,r)$ was given in
[Oht, Example 3.6]. In particular, for $K=P(4k+1,4k+3,-(2k+1))$,
$\widehat{\Theta}_{K}(1)$ and $\widehat{\Theta}_{K}(-1)$ are given by

$\widehat{\Theta}_{K}(1)=-\frac{1}{8}(4k+2)(4k+4)(-2k),\widehat{\Theta}_{K}(-1)=-\frac{1}{24}(4k+2)(4k+4)(-2k)$

hence

$\displaystyle\lambda(\Sigma_{2}(K))-2w_{3}(K)$
$\displaystyle=\widehat{\Theta}_{K}(-1)-\widehat{\Theta}_{K}(1)=\frac{1}{12}(4k+2)(4k+4)(-2k)$
$\displaystyle=-16\frac{(2k+1)(k+1)k}{12}.$

When $k\equiv 1,2\pmod{4}$, $\frac{(2k+1)(k+1)k}{12}\not\in\mathbb{Z}$ hence
$K$ does not admit cosmetic crossing by Theorem 1.3. ∎

## Acknowledgement

The author has been partially supported by JSPS KAKENHI Grant Number
19K03490,16H02145.

## References

  * [BFKP] C. Balm, S. Friedl, E. Kalfagianni, and M. Powell, Cosmetic crossings and Seifert matrices, Comm. Anal. Geom. 20 (2012), 235–253.
  * [BaKa] C. Balm and E. Kalfagianni, Knots without cosmetic crossings, Topology Appl. 207 (2016), 33–42.
  * [GaKr] S. Garoufalidis and A. Kricker, A rational noncommutative invariant of boundary links, Geom. Topol. 8 (2004), 115–204.
  * [Kal] E. Kalfagianni, Cosmetic crossing changes of fibered knots, J. reine angew. Math. 669 (2012), 151–164.
  * [Kir] R. Kirby (ed). Problems in low-dimensional topology, Edited by Rob Kirby. AMS/IP Stud. Adv. Math., 2.2, Geometric topology (Athens, GA, 1993), 3–473, Amer. Math. Soc., Providence, RI, 1997.
  * [Kri] A. Kricker, The lines of the Kontsevich integral and Rozansky’s rationality conjecture, arXiv:math/0005284.
  * [LiMo] T. Lidman and A. Moore, Cosmetic surgery in L-spaces and nugatory crossings, Trans. Amer, Math. Soc. 369 (2017), 3639–3654.
  * [Mul] D. Mullins, The generalized Casson invariant for 2-fold branched covers of $S^{3}$ and the Jones polynomial, Topology 32 (1993), 419–438.
  * [Oht] T. Ohtsuki, On the 2-loop polynomial of knots, Geom. Topol. 11 (2007), 1357–1475.
  * [Tor] I. Torisu, On nugatory crossings for knots, Topology Appl. 92 (1999), 119–129.


EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2016-315 LHCb-PAPER-2016-041 2 April 2017

Measurement of $\boldmath{C\\!P}$ asymmetries in
$\boldmath{D^{\pm}\rightarrow\eta^{\prime}\pi^{\pm}}$ and
$\boldmath{D_{s}^{\pm}\rightarrow\eta^{\prime}\pi^{\pm}}$ decays

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

A search for $C\\!P$ violation in $D^{\pm}\rightarrow\eta^{\prime}\pi^{\pm}$
and $D^{\pm}_{s}\rightarrow\eta^{\prime}\pi^{\pm}$ decays is performed using
proton-proton collision data, corresponding to an integrated luminosity of
3$\mbox{\,fb}^{-1}$, recorded by the LHCb experiment at centre-of-mass
energies of $7$ and $8\,\mathrm{\,Te\kern-1.00006ptV}$. The measured
$C\\!P$-violating charge asymmetries are
$\mathcal{A}_{{C\\!P}}({{D}^{\pm}}\rightarrow\eta^{\prime}{{\pi}^{\pm}})=(-0.61\pm
0.72\pm 0.53\pm 0.12)\%$ and
$\mathcal{A}_{{C\\!P}}({{D}^{\pm}_{s}}\rightarrow\eta^{\prime}{{\pi}^{\pm}})=(-0.82\pm
0.36\pm 0.22\pm 0.27)\%$, where the first uncertainties are statistical, the
second systematic, and the third are the uncertainties on the
$\mathcal{A}_{{C\\!P}}({{D}^{\pm}}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{\pi}^{\pm}})$ and
$\mathcal{A}_{{C\\!P}}({{D}^{\pm}_{s}}\rightarrow\phi{{\pi}^{\pm}})$
measurements used for calibration. The results represent the most precise
measurements of these asymmetries to date.

Published in Phys. Lett. B 771 (2017) 21-30

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

##  1 Introduction

The decays of charmed mesons offer a unique opportunity for the experimental
investigation of hitherto unobserved $C\\!P$ violation in the up-type quark
sector. The Standard Model (SM) predicts $C\\!P$ violation to occur in the
charm sector, albeit at a level of $\mathcal{O}\left(10^{-3}\right)$ at
leading order in $1/m_{c}$, compatible with the lack of evidence in current
measurements. Larger values are possible if new sources of $C\\!P$ violation
beyond the SM exist. The study of charm systems is a unique tool to probe
sources of $C\\!P$ violation that affect only the dynamics of up-type quarks
[1].

In order for non-zero $C\\!P$ asymmetries to be observable in a process, two
or more interfering amplitudes with different $C\\!P$-odd and $C\\!P$-even
phases are needed. In the SM, no direct $C\\!P$ violation can therefore emerge
at leading order in Cabibbo-favoured charm decays, which are mediated by a
single weak amplitude, while small $C\\!P$ asymmetries are expected in singly-
Cabibbo-suppressed decays [2] due to the interference of colour-allowed tree-
level amplitudes with loop- (penguin) and colour-suppressed tree-level
amplitudes. Since these asymmetries may be enhanced by nonperturbative effects
[3], theoretical interpretations of experimental results require the analysis
of several channels with similar sensitivity. In particular, the study of
charm decays to pseudoscalar mesons tests flavour topology [4] and SU(3)
predictions, and may constrain amplitudes through triangle relations or shed
light on sources of SU(3) flavour symmetry breaking [5, 6, 7]. To date, the
most precise measurements of $C\\!P$ asymmetries in singly-Cabibbo-suppressed
two-body charm decays have been performed in
${{D}^{0}}\rightarrow{{K}^{+}}{{K}^{-}}$ and
${{D}^{0}}\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ decays by the LHCb collaboration
[8, 9], and have shown no evidence for $C\\!P$ violation. Among the other
charm decays to two pseudoscalar mesons with significant branching fractions,
thus far ${{D}^{\pm}}\rightarrow\eta^{\prime}{{\pi}^{\pm}}$ and
${{D}^{\pm}_{s}}\rightarrow\eta^{\prime}{{\pi}^{\pm}}$ have been studied only
in $e^{+}e^{-}$ collisions [10, 11] due to the experimental difficulty of
reconstructing $\eta^{(\prime)}$ mesons in hadron collisions. The most recent
studies of these decays at the Belle and CLEO experiments yielded a $C\\!P$
asymmetry of $(-0.12\pm 1.12\pm 0.17)\%$ [11] for the singly-Cabibbo-
suppressed ${{D}^{\pm}}\rightarrow\eta^{\prime}{{\pi}^{\pm}}$ decay and
$(-2.2\pm 2.2\pm 0.6)\%$ [10] for the Cabibbo-favoured
${{D}^{\pm}_{s}}\rightarrow\eta^{\prime}{{\pi}^{\pm}}$ decay, respectively.

In this Letter the first analysis of
$D^{\pm}_{(s)}\rightarrow\eta^{\prime}{{\pi}^{\pm}}$ decays at a hadron
collider is presented, using proton-proton ($pp$) collision data corresponding
to an integrated luminosity of approximately 3$\mbox{\,fb}^{-1}$, collected by
the LHCb experiment. This allows for the large charm yields available at the
LHC to be exploited, resulting in the most precise measurement of $C\\!P$
asymmetries in these decays to date.

##  2 Method

The $C\\!P$ asymmetries $\mathcal{A}_{{C\\!P}}$ are determined from the
measured (raw) asymmetries

$\displaystyle\mathcal{A}_{\mathrm{raw}}(D^{\pm}_{(s)}\rightarrow
f^{\pm})=\frac{N(D^{+}_{(s)}\rightarrow f^{+})-N(D^{-}_{(s)}\rightarrow
f^{-})}{N(D^{+}_{(s)}\rightarrow f^{+})+N(D^{-}_{(s)}\rightarrow f^{-})},$ (1)

where $N$ denotes the observed yield for the decay to a given charged final
state $f^{\pm}$. The measured asymmetries include additional contributions
other than $\mathcal{A}_{{C\\!P}}(D^{\pm}_{(s)}\rightarrow f^{\pm})$. For
small asymmetries, it is possible to approximate to first order

$\displaystyle\mathcal{A}_{\mathrm{raw}}\approx\mathcal{A}_{{C\\!P}}+\mathcal{A}_{\mathrm{P}}+\mathcal{A}_{\mathrm{D}},$
(2)

where $\mathcal{A}_{\mathrm{P}}$ is the asymmetry in the production of
${D}^{\pm}_{(s)}$ mesons in high-energy $pp$ collisions in the LHCb
acceptance, and $\mathcal{A}_{\mathrm{D}}$ arises from the difference in
detection efficiencies between positively and negatively charged hadrons.

These effects are studied using control decay modes for which
$\mathcal{A}_{{C\\!P}}$ is known precisely. The control decays, which have
similar decay topologies as the signal decays, are the Cabibbo-favoured
${{D}^{\pm}}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{\pi}^{\pm}}$
and ${{D}^{\pm}_{s}}\rightarrow\phi{{\pi}^{\pm}}$ decays for
${{D}^{\pm}}\rightarrow\eta^{\prime}{{\pi}^{\pm}}$ and
${{D}^{\pm}_{s}}\rightarrow\eta^{\prime}{{\pi}^{\pm}}$, respectively. The
$C\\!P$ asymmetries in these control decays have been measured at the
$10^{-3}$ level by the Belle and D0 collaborations [12, 13].

The differences between the $C\\!P$ asymmetries measured in the
$D^{\pm}_{(s)}\rightarrow\eta^{\prime}{{\pi}^{\pm}}$ decays and in the
corresponding control channels are defined as

$\displaystyle\Delta\mathcal{A}_{{C\\!P}}({{D}^{\pm}}\rightarrow\eta^{\prime}{{\pi}^{\pm}})$
$\displaystyle\equiv$
$\displaystyle\mathcal{A}_{{C\\!P}}({{D}^{\pm}}\rightarrow\eta^{\prime}{{\pi}^{\pm}})-\mathcal{A}_{{C\\!P}}({{D}^{\pm}}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{\pi}^{\pm}})$ $\displaystyle=$
$\displaystyle\mathcal{A}_{\mathrm{raw}}({{D}^{\pm}}\rightarrow\eta^{\prime}{{\pi}^{\pm}})-\mathcal{A}_{\mathrm{raw}}({{D}^{\pm}}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{\pi}^{\pm}})+\mathcal{A}({{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{0}}-{{K}^{0}}}),$
$\displaystyle\Delta\mathcal{A}_{{C\\!P}}({{D}^{\pm}_{s}}\rightarrow\eta^{\prime}{{\pi}^{\pm}})$
$\displaystyle\equiv$
$\displaystyle\mathcal{A}_{{C\\!P}}({{D}^{\pm}_{s}}\rightarrow\eta^{\prime}{{\pi}^{\pm}})-\mathcal{A}_{{C\\!P}}({{D}^{\pm}_{s}}\rightarrow\phi{{\pi}^{\pm}})$
$\displaystyle=$
$\displaystyle\mathcal{A}_{\mathrm{raw}}({{D}^{\pm}_{s}}\rightarrow\eta^{\prime}{{\pi}^{\pm}})-\mathcal{A}_{\mathrm{raw}}({{D}^{\pm}_{s}}\rightarrow\phi{{\pi}^{\pm}}).$

These equations assume that the kinematic distributions of the pion and of the
$D_{(s)}$ meson are similar in the signal and control channels, so that
detection and production asymmetries largely cancel in the difference. The
uncertainty associated to this assumption is discussed in Sec. 5. The
$\mathcal{A}({{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{0}}-{{K}^{0}}})$ term in Eq. 2
represents the kaon asymmetry in
${{D}^{\pm}}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{\pi}^{\pm}}$
decays, which arises from regeneration and from mixing and $C\\!P$ violation
in the ${{\kern 1.99997pt\overline{\kern-1.99997ptK}{}}{}^{0}}-{{K}^{0}}$
system. This contribution is estimated using simulations, as described in Ref.
[9], to be $(-0.08\pm 0.01)\%$. The $C\\!P$ asymmetry in the singly-Cabibbo-
suppressed ${{D}^{\pm}}\rightarrow\eta^{\prime}{{\pi}^{\pm}}$ decay is
therefore given by

$\mathcal{A}_{{C\\!P}}({{D}^{\pm}}\rightarrow\eta^{\prime}{{\pi}^{\pm}})\approx\Delta\mathcal{A}_{{C\\!P}}({{D}^{\pm}}\rightarrow\eta^{\prime}{{\pi}^{\pm}})+\mathcal{A}_{{C\\!P}}({{D}^{\pm}}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{\pi}^{\pm}}).$ (4)

Similarly, the $C\\!P$ asymmetry for the Cabibbo-favoured
${{D}^{\pm}_{s}}\rightarrow\eta^{\prime}{{\pi}^{\pm}}$ decay is approximated
as

$\mathcal{A}_{{C\\!P}}({{D}^{\pm}_{s}}\rightarrow\eta^{\prime}{{\pi}^{\pm}})\approx\Delta\mathcal{A}_{{C\\!P}}({{D}^{\pm}_{s}}\rightarrow\eta^{\prime}{{\pi}^{\pm}})+\mathcal{A}_{{C\\!P}}({{D}^{\pm}_{s}}\rightarrow\phi{{\pi}^{\pm}}).$
(5)

##  3 Detector

The LHCb detector [14, 15] 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 polarity of the dipole magnet is reversed periodically
throughout data taking. The configuration with the magnetic field vertically
upwards (downwards) bends positively (negatively) charged particles in the
horizontal plane towards the centre of the LHC. 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 (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.
The online event selection is performed by a trigger, which consists of a
hardware stage, based on information from the calorimeter and muon systems,
followed by a software stage, which applies a full event reconstruction. 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
deposit in the calorimeters.

In the simulation, $pp$ collisions are generated using Pythia 6.4 [16] with a
specific LHCb configuration [17]. Decays of hadronic particles are described
by EvtGen [18], in which final-state radiation is generated using Photos [19].
The interaction of the generated particles with the detector, and its
response, are implemented using the Geant4 toolkit [20, *Agostinelli:2002hh]
as described in Ref. [22].

##  4 Reconstruction and sample composition

The data correspond to an integrated luminosity of approximately
3$\mbox{\,fb}^{-1}$ recorded in $pp$ collisions at centre-of-mass energies of
$\sqrt{s}=7$$\mathrm{\,Te\kern-1.00006ptV}$ ($1$$\mbox{\,fb}^{-1}$) and
8$\mathrm{\,Te\kern-1.00006ptV}$ ($2$$\mbox{\,fb}^{-1}$). Approximately 50% of
the data were collected in each configuration of magnet polarity. The
$\mathcal{A}_{\mathrm{raw}}$ measurements are performed separately for the two
field polarities and the two centre-of-mass energies.

The signal $D^{\pm}_{(s)}\rightarrow\eta^{\prime}{{\pi}^{\pm}}$ candidates, as
well as control ${{D}^{\pm}}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{\pi}^{\pm}}$ and ${{D}^{\pm}_{s}}\rightarrow\phi{{\pi}^{\pm}}$
candidates, are reconstructed through the intermediate resonance decays
$\eta^{\prime}\rightarrow{{\pi}^{+}}{{\pi}^{-}}\gamma$,
${{K}^{0}_{\mathrm{\scriptscriptstyle S}}}\rightarrow{{\pi}^{+}}{{\pi}^{-}}$,
and $\phi\rightarrow{{K}^{+}}{{K}^{-}}$. The sample is divided into three
mutually exclusive subsamples according to the fulfilled hardware trigger
requirements. The first subsample, T1, consists of events for which the
trigger decision is based on the transverse energy deposited in the hadronic
calorimeter by a charged particle from the decay of the $\eta^{\prime}$,
${K}^{0}_{\mathrm{\scriptscriptstyle S}}$, or $\phi$ meson. The second
subsample, T2, consists of the subset of the remaining events for which a
particle other than the decay products of the $D^{\pm}_{(s)}$ candidate is
associated with a high transverse-energy deposit in the hadronic calorimeter.
The third subsample, T3, consists of the events accepted because of a high
transverse-energy deposit in the electromagnetic calorimeter or a
high-$p_{\mathrm{T}}$ muon, not associated with the $D^{\pm}_{(s)}$ decay and
not included in the other subsamples. The hardware trigger selections do not
rely on information associated with the same-charge pion from the
$D^{\pm}_{(s)}$ decay.

One or more of the charged decay products from the $\eta^{\prime}$,
${K}^{0}_{\mathrm{\scriptscriptstyle S}}$, or $\phi$ meson is required to
activate the first stage of the software trigger, which selects a sample with
enhanced heavy-flavour content by requiring the presence of a large-IP charged
particle with
$\mbox{$p_{\mathrm{T}}$}>1.6\,{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$
($\mbox{$p_{\mathrm{T}}$}>1.7\,{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$) in the
$8\mathrm{\,Te\kern-1.00006ptV}$ ($7\mathrm{\,Te\kern-1.00006ptV}$) data. In
the second stage of the software trigger, each selected event is required to
contain at least one combination of three tracks that meet loose requirements
on the IP of the final-state particles and on the invariant mass of the
charged decay products.

For the $D^{\pm}_{(s)}\rightarrow\eta^{\prime}{{\pi}^{\pm}}$ channels, the
$\eta^{\prime}$ candidates are reconstructed by combining pairs of oppositely
charged particles with a photon of
$\mbox{$p_{\mathrm{T}}$}>1\,{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The
$\eta^{\prime}$ charged decay products must not be identified as kaons by the
particle identification system [15], and must be displaced from the PV. Photon
candidates are reconstructed from clusters of energy deposits in the
electromagnetic calorimeter. The absence of tracks pointing to the energy-
cluster barycentre is used to distinguish neutral from charged particles. For
high-$p_{\mathrm{T}}$ photons a multivariate algorithm based on the shape
parameters of the cluster is used to reject $\pi^{0}\rightarrow\gamma\gamma$
background in which the two photons are reconstructed as a single cluster
[15]. To maximize sensitivity to
$\mathcal{A}_{\mathrm{raw}}$($D^{\pm}_{(s)}\rightarrow\eta^{\prime}{{\pi}^{\pm}}$),
the three-particle mass is required to satisfy
$0.934<m({{\pi}^{+}}{{\pi}^{-}}\gamma)<0.982\,{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$,
as shown in Fig. 1(a) by the light-shaded region.

(a)(b)

Figure 1:  Distribution of (a) $m({{\pi}^{+}}{{\pi}^{-}}\gamma)$ for
$D^{\pm}_{(s)}\rightarrow\eta^{\prime}{{\pi}^{\pm}}$ candidates. The grey
solid area represents the signal region, while the black hatched area
represents the $m({{\pi}^{+}}{{\pi}^{-}}\gamma)$ sideband. Distribution of (b)
$m(\eta^{\prime}{{\pi}^{\pm}})$ for
$D^{\pm}_{(s)}\rightarrow\eta^{\prime}{{\pi}^{\pm}}$ candidates in the
$m({{\pi}^{+}}{{\pi}^{-}}\gamma)$ sideband.

The ${K}^{0}_{\mathrm{\scriptscriptstyle S}}$ candidates are formed from a
pair of non-prompt, oppositely charged high-momentum particles reconstructed
in the vertex detector. A good-quality vertex fit and sufficient separation
from the PV are required for the decay vertex of the
${K}^{0}_{\mathrm{\scriptscriptstyle S}}$ candidate. The
${{\pi}^{+}}{{\pi}^{-}}$ mass is required to lie in the range
$0.4626<m({{\pi}^{+}}{{\pi}^{-}})<0.5326\,{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$.

To reconstruct $\phi$ candidates, two oppositely charged, large-IP particles,
classified as kaons by the particle identification system [15], are combined.
The $K^{+}K^{-}$ mass is required to be within $\pm
20\,{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the known $\phi$ mass [23].

Selected $\eta^{\prime}$, ${K}^{0}_{\mathrm{\scriptscriptstyle S}}$, and
$\phi$ candidates are combined with a third non-prompt charged pion (bachelor
particle) to form a $D^{\pm}_{(s)}$ candidate. The selection criteria for the
bachelor pion are chosen to be as similar as possible between signal and
control samples. To suppress background contributions from
$D^{\pm}_{(s)}\rightarrow X\ell^{\pm}\nu$ and $D^{\pm}_{(s)}\rightarrow
X{{K}^{\pm}}$ decays, with $X=\eta^{\prime}$, $\phi$, or
${K}^{0}_{\mathrm{\scriptscriptstyle S}}$, the bachelor particle must be
identified as a pion rather than as an electron, muon or kaon. The lepton veto
removes more than $95\%$ of the electrons and muons and $9\%$ of the pions,
and the kaon veto rejects about $95\%$ of the kaons while retaining $90\%$ of
all pions [15]. Fiducial requirements are imposed to exclude kinematic regions
where reconstruction and particle identification of the bachelor pion suffer
from large charge-dependent asymmetries [24].

Candidate $D^{\pm}_{(s)}$ mesons are required to have
$\mbox{$p_{\mathrm{T}}$}>2\,{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ in all decay
modes, and mass in the range
$1.82<m(\eta^{\prime}{{\pi}^{\pm}})<2.03\,{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$
for the signal $D^{\pm}_{(s)}\rightarrow\eta^{\prime}{{\pi}^{\pm}}$ decays and
$1.80<m({{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{\pi}^{\pm}})<2.03\,{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$
($1.80<m(\phi{{\pi}^{\pm}})<2.03\,{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$)
for the ${{D}^{\pm}}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{\pi}^{\pm}}$ ($D^{\pm}_{(s)}\rightarrow\phi{{\pi}^{\pm}}$) control mode.
To calculate the $D^{\pm}_{(s)}$ mass [25], the $\eta^{\prime}$ candidate mass
is constrained to its known value [23], without placing constraints on the
origin of the $D^{\pm}_{(s)}$ meson. The charged decay products of the
reconstructed $D^{\pm}_{(s)}$ candidates are required to match one of the
three-track combinations that activated the second stage of the software
trigger. The scalar sum of the transverse momenta of charged decay products
must exceed $2.8\,{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ for all decay modes.
In events with multiple $D^{\pm}_{(s)}$ candidates only one randomly selected
candidate is kept. This procedure removes less than $2\%$ of the original
candidates.

A combinatorial background contribution is present in all decay modes.
Background from partially reconstructed
${{D}^{\pm}_{s}}\rightarrow\eta^{\prime}\rho^{\pm}$ decays is suppressed by
requiring
$m(\eta^{\prime}{{\pi}^{\pm}})>1.82\,{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$.
Background from
$D^{\pm}_{(s)}\rightarrow{{\pi}^{\mp}}{{\pi}^{\pm}}{{\pi}^{\pm}}$ decays,
paired with a random photon, is suppressed by requiring the invariant mass of
the three charged hadrons to be less than
$1.80\,{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. A contribution from
$D^{\pm}_{(s)}\rightarrow\phi{{\pi}^{\pm}}$ decays, with
$\phi\rightarrow{{\pi}^{+}}{{\pi}^{-}}\pi^{0}$ (denoted below as
$D^{\pm}_{(s)}\rightarrow\phi_{3\pi}{{\pi}^{\pm}}$), is also present, where
one of the photons in the $\pi^{0}\rightarrow\gamma\gamma$ decay is not
reconstructed or the two photons are reconstructed as a single cluster.

Background from
${{D}^{\pm}_{s}}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{K}^{\pm}}$ and
${{D}^{\pm}_{s}}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{\pi}^{\pm}}\pi^{0}$ decays
(${{D}^{\pm}_{s}}\rightarrow\phi{{\pi}^{\pm}}\pi^{0}$ and non-resonant
${{D}^{\pm}_{s}}\rightarrow{{K}^{+}}{{K}^{-}}{{\pi}^{\pm}}$ decays), where the
bachelor kaon is misidentified as a pion or the $\pi^{0}$ is not
reconstructed, contributes negligibly to the
${{D}^{\pm}}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{\pi}^{\pm}}$
(${{D}^{\pm}_{s}}\rightarrow\phi{{\pi}^{\pm}}$) candidate mass spectrum.

The $D^{\pm}_{(s)}\rightarrow\eta^{\prime}{{\pi}^{\pm}}$ candidates
originating from the decays of $b$ hadrons are suppressed by requiring a good
quality of the $D^{\pm}_{(s)}$ vertex fit, performed with the origin of the
$D^{\pm}_{(s)}$ constrained to the associated PV but without a constraint on
the $\eta^{\prime}$ candidate mass. Non-prompt
${{D}^{\pm}_{s}}\rightarrow\phi{{\pi}^{\pm}}$ and
${{D}^{\pm}}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{\pi}^{\pm}}$
candidates are rejected by requiring a small difference between the quality of
the fit of the PV formed with and without the tracks assigned to the
reconstructed $D^{\pm}_{(s)}$ candidate.

##  5 Determination of the asymmetries

For each final state, the data are divided into twelve mutually exclusive
subsamples, according to the two $pp$ centre-of-mass energies, two magnet
polarities, and three hardware trigger selections. Since detection asymmetries
depend on the kinematic properties of the process under study, $D^{\pm}_{(s)}$
candidates in each subsample are divided into $3\times 3$ bins of transverse
momentum and pseudorapidity of the bachelor pion. The bin edges in
$p_{\mathrm{T}}$ are defined as $0.5$, $1.5$, $3.0$, and
$20.0\,{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and the bin edges in $\eta$ are
defined as $2.0$, $2.8$, $3.2$, and $5.0$. While the kinematic distributions
of the bachelor pion for the signal and
${{D}^{\pm}_{s}}\rightarrow\phi{{\pi}^{\pm}}$ control decays are in good
agreement, the average bachelor-pion $p_{\mathrm{T}}$ ($\eta$) is $30\%$ lower
($5\%$ higher) in the
${{D}^{\pm}}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{\pi}^{\pm}}$
control channel. The binning reduces the effect of the discrepancies between
the bachelor-pion kinematic distributions for signal and control decays, thus
improving the suppression of $\mathcal{A}_{\mathrm{D}}$ in the differences of
raw asymmetries. For each of the twelve subsamples, the raw $C\\!P$
asymmetries of the $D^{\pm}_{(s)}\rightarrow\eta^{\prime}{{\pi}^{\pm}}$ signal
channels are determined with a maximum likelihood fit to the unbinned
$\eta^{\prime}\pi$ invariant mass distribution, performed simultaneously for
positively and negatively charged $D^{\pm}_{(s)}$ candidates, and for the nine
$\mbox{$p_{\mathrm{T}}$}-\eta$ bins.

The fit model comprises two signal components for the ${D}^{\pm}$ and
${D}^{\pm}_{s}$ resonances, a combinatorial background component, and two
peaking components accounting for background from
$D^{\pm}_{(s)}\rightarrow\phi_{3\pi}{{\pi}^{\pm}}$ decays. The signal
components are modelled by Johnson $S_{U}$ distributions [26]:

$f(x;\mu,\sigma,\delta,\gamma)\propto\left[1+\left(\frac{x-\mu}{\sigma}\right)^{2}\right]^{-\frac{1}{2}}\exp\left\\{-\frac{1}{2}\left[\gamma+\delta\sinh^{-1}\left(\frac{x-\mu}{\sigma}\right)\right]^{2}\right\\}.$
(6)

The parameters $\mu$ and $\sigma$, which govern the mean and width of each
distribution, are fitted independently for ${D}^{\pm}$ and ${D}^{\pm}_{s}$,
and can vary with the charge and pseudorapidity of the bachelor pion. The
remaining two parameters, $\delta$ and $\gamma$, characterising the tails of
the Johnson $S_{U}$ distributions, are common between the two signal
components and are required to be the same across all
$\mbox{$p_{\mathrm{T}}$}-\eta$ bins, but can vary with the charge of the
bachelor pion. The combinatorial background is parametrised by a fourth-order
polynomial, whose parameters can vary independently for positive and negative
charges and for different bins in the bachelor-pion pseudorapidity. The
parameters of the background model, for each charge and each bin in
pseudorapidity of the bachelor pion, are Gaussian-constrained to the results
of fits of the same functional form to the corresponding
$m(\eta^{\prime}{{\pi}^{\pm}})$ distributions from the
$m({{\pi}^{+}}{{\pi}^{-}}\gamma)$ sideband (Fig. 1(b)). The
$D^{\pm}_{(s)}\rightarrow\phi_{3\pi}{{\pi}^{\pm}}$ background components are
described by empirical functions [27] derived from simulated events. The
yields and charge asymmetries of signal and combinatorial components in each
$\mbox{$p_{\mathrm{T}}$}-\eta$ bin, and the total yields of the
$D^{\pm}_{(s)}\rightarrow\phi_{3\pi}{{\pi}^{\pm}}$ contributions are free
parameters in the fit. For the
$D^{\pm}_{(s)}\rightarrow\phi_{3\pi}{{\pi}^{\pm}}$ components, the raw $C\\!P$
asymmetries and the fraction of the total yields in each
$\mbox{$p_{\mathrm{T}}$}-\eta$ bin are determined from
$D^{\pm}_{(s)}\rightarrow\phi{{\pi}^{\pm}}$ control decays, with
$\phi\rightarrow K^{+}K^{-}$. The model well reproduces the charge-integrated
$m(\eta^{\prime}{{\pi}^{\pm}})$ distributions in all $p_{\mathrm{T}}$ bins. To
estimate the goodness of fit, in each of the twelve subsamples the $\chi^{2}$
of the fitted model is calculated for the binned
$m(\eta^{\prime}{{\pi}^{\pm}})$ distribution in all
$\mbox{$p_{\mathrm{T}}$}-\eta$ bins. The $p$-value is greater than $5\%$ in
all cases. The results of the fit to the $\eta^{\prime}\pi$ mass distribution
for $D^{\pm}_{(s)}\rightarrow\eta^{\prime}{{\pi}^{\pm}}$ candidates are shown
in Fig. 2. The signal yields, combined over all kinematic bins, $pp$ centre-
of-mass energies, and hardware trigger selections, are
$N({{D}^{\pm}}\rightarrow\eta^{\prime}{{\pi}^{\pm}})=(62.7\pm 0.4)\times
10^{3}$ and $N({{D}^{\pm}_{s}}\rightarrow\eta^{\prime}{{\pi}^{\pm}})=(152.2\pm
0.5)\times 10^{3}$, respectively.

Due to the high purity of the control samples, the raw $C\\!P$ asymmetries for
the ${{D}^{\pm}_{s}}\rightarrow\phi{{\pi}^{\pm}}$ and
${{D}^{\pm}}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{\pi}^{\pm}}$
decay modes are extracted by counting the numbers of positively and negatively
charged candidates in the signal mass range and subtracting the corresponding
numbers in the sidebands, shown in Fig. 3. For the
${{D}^{\pm}}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{\pi}^{\pm}}$
decay, the sidebands are defined as 1.800 –
1.835${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ and 1.905 –
1.940${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, and the signal range as 1.835
– 1.905${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. For the
${{D}^{\pm}_{s}}\rightarrow\phi{{\pi}^{\pm}}$ channel the sidebands are
defined as 1.910 – 1.935${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ and 2.005 –
2.030${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, and the signal range as 1.935
– 2.005${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. The event yields determined
in the ${{D}^{\pm}_{s}}\rightarrow\phi{{\pi}^{\pm}}$ sidebands are scaled by a
factor $1.4$ to account for the different widths of the sideband and signal
ranges. Background from
${{D}^{\pm}_{s}}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{K}^{\pm}}$,
${{D}^{\pm}_{s}}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{\pi}^{\pm}}\pi^{0}$,
${{D}^{\pm}_{s}}\rightarrow\phi{{\pi}^{\pm}}\pi^{0}$, and non-resonant
${{D}^{\pm}_{s}}\rightarrow{{K}^{+}}{{K}^{-}}{{\pi}^{\pm}}$ decays is
neglected. The effect of the small fraction of $D^{\pm}_{(s)}$ signal leaking
into the sidebands, which may depend on the charge, $p_{T}$ and pseudorapidity
of the bachelor pion, is considered as a source of systematic uncertainty.

For each subsample, the differences of raw asymmetries for signal and
associated control channels are calculated in each
$\mbox{$p_{\mathrm{T}}$}-\eta$ bin. The weighted averages of the results
obtained in the nine bins are then evaluated, taking into account the
covariance matrix $V$, calculated as the sum of the covariance matrices for
the results of the $D^{\pm}_{(s)}\rightarrow\eta^{\prime}{{\pi}^{\pm}}$ fit
and of the sideband subtraction for control decays. The weights are
$w_{i}=\sum_{k}V_{ik}^{-1}/\left(\sum_{j}\sum_{k}V_{jk}^{-1}\right)$, where
$i$, $j$, and $k$ run over the $\mbox{$p_{\mathrm{T}}$}-\eta$ bins. The
resulting $\Delta\mathcal{A}_{{C\\!P}}$ values are averaged with equal weights
over the two magnet polarities. Detection asymmetries that differ between the
signal and control decays are suppressed in this average. The results for the
signal channels are shown in Fig. 4. Finally, the inverse-variance weighted
average of the $\Delta\mathcal{A}_{{C\\!P}}$ values obtained for the two $pp$
centre-of-mass energies and the three hardware trigger selections is
calculated. No significant charge asymmetry is observed for the combinatorial
background component in any of the subsamples. The inverse-variance weighted
average of $\mathcal{A}_{\mathrm{raw}}$ for the combinatorial background is
$(0.92\pm 0.72)\%$, where the error is statistical only.

(a)(b)

Figure 2:  Mass distribution of $\eta^{\prime}\pi^{\pm}$ candidates, combined
over all kinematic bins, $pp$ centre-of-mass energies, and hardware trigger
selections, for (a) positively and (b) negatively charged $D^{\pm}_{(s)}$
candidates. Points with errors represent data, while the curves represent the
fitted model (solid), the ${{D}^{\pm}_{s}}\rightarrow\phi_{3\pi}{{\pi}^{\pm}}$
(dashed) and ${{D}^{\pm}}\rightarrow\phi_{3\pi}{{\pi}^{\pm}}$ (long-dashed)
components, and the sum of all background contributions (dotted), including
combinatorial background. Residuals divided by the corresponding uncertainty
are shown under each plot.

(a)(b)
(c)(d)

Figure 3:  Top: ${{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{\pi}^{\pm}}$ mass
distribution for (a) positively and (b) negatively charged ${D}^{\pm}$
candidates. Bottom: $\phi{{\pi}^{\pm}}$ mass distribution for (c) positively
and (d) negatively charged ${D}^{\pm}_{s}$ candidates. The signal regions are
enclosed within the vertical dashed lines. The mass distributions are combined
over all kinematic bins, $pp$ centre-of-mass energies, and hardware trigger
selections.

(a)
(b)

Figure 4: $\Delta\mathcal{A}_{{C\\!P}}$ results for (a)
${{D}^{\pm}}\rightarrow\eta^{\prime}{{\pi}^{\pm}}$ and (b)
${{D}^{\pm}_{s}}\rightarrow\eta^{\prime}{{\pi}^{\pm}}$ decays, as a function
of $pp$ centre-of-mass energy and trigger selection. Uncertainties are
statistical only. A shaded band representing the $68.3\%$ confidence intervals
obtained from the weighted average over all the samples is shown to guide the
eye.

##  6 Systematic uncertainties

The contributions to the systematic uncertainty on the inverse-variance
weighted $\Delta\mathcal{A}_{{C\\!P}}$ average are described below and
summarised in Table 1. The overall systematic uncertainties are obtained by
adding the individual contributions in quadrature.

Table 1: Systematic uncertainties (absolute values in %) on $\Delta\mathcal{A}_{{C\\!P}}$. The total systematic uncertainty is the sum in quadrature of the individual contributions.  Source | $\delta[\Delta\mathcal{A}_{{C\\!P}}({{D}^{\pm}})]$ | $\delta[\Delta\mathcal{A}_{{C\\!P}}({{D}^{\pm}_{s}})]$
---|---|---
Non-prompt charm | $0.03$ | $0.03$
Trigger | $0.09$ | $0.09$
Background model | $0.50$ | $0.19$
Fit procedure | $0.08$ | $0.04$
Sideband subtraction | $0.03$ | $0.02$
${{K}^{0}}$ asymmetry | $0.08$ | $-$
$\pi^{\pm}$ detection asymmetry | $0.06$ | $0.01$
$D^{\pm}_{(s)}$ production asymmetry | $0.07$ | $0.02$
Total | $0.53$ | $0.22$

The selection of signal and control sample candidates removes the majority of
background from non-prompt $D^{\pm}_{(s)}$ mesons, originating from the decay
of a $b$ hadron. The remaining secondary $D^{\pm}_{(s)}$ mesons may introduce
a bias in the measured $C\\!P$ asymmetries due to a difference in the
production asymmetries for $b$ hadrons and $D^{\pm}_{(s)}$ mesons. This bias
might not cancel in the difference of measured asymmetries for signal and
control channels, due to differences in the final-state reconstruction. In
order to investigate this bias, the $D^{\pm}_{(s)}$ production asymmetries in
$D^{\pm}_{(s)}\rightarrow\eta^{\prime}{{\pi}^{\pm}}$ decays are modified using
$\mathcal{A}_{\mathrm{P}}^{\prime}=(\mathcal{A}_{\mathrm{P}}+f\mathcal{A}_{\mathrm{P}}^{b})/(1+f)$,
where $f$ is the fraction of secondary $D^{\pm}_{(s)}$ candidates in a
particular decay channel and $\mathcal{A}_{\mathrm{P}}^{b}$ is the
corresponding $b$-hadron production asymmetry. The fraction $f$ is estimated
from the measured cross-sections for inclusive production of ${{D}^{\pm}}$,
${D}_{s}^{\pm}$, and $b$ hadrons [28, 29], the inclusive branching fractions
${\mathcal{B}}(b\rightarrow D^{\pm}X)$ and ${\mathcal{B}}(b\rightarrow
D^{\pm}_{s}X)$ [23], and the efficiencies calculated from simulation. The
resulting values of $f$ are below $6\%$. The $b$-hadron production asymmetry
$\mathcal{A}_{\mathrm{P}}^{b}$ is taken from existing measurements for
${B}^{\pm}$, $B_{(s)}^{0}$, and ${\mathchar 28931\relax}^{0}_{b}$ hadrons [30,
31, 32, 33, 34]. Under the assumption that the bias due to
$\mathcal{A}_{\mathrm{P}}^{b}$ does not cancel in the difference of measured
asymmetries for signal and control channels, the systematic uncertainty on
$\Delta\mathcal{A}_{{C\\!P}}$ is evaluated by recalculating the $C\\!P$
asymmetries using $\mathcal{A}_{\mathrm{P}}^{\prime}$ for the signal decay
modes and $\mathcal{A}_{\mathrm{P}}$ for the control samples.

Potential trigger biases are studied using
$D^{\pm}_{(s)}\rightarrow\phi{{\pi}^{\pm}}$ decays, with $\phi\rightarrow
K^{+}K^{-}$. The $C\\!P$ asymmetries measured in the subsamples defined by the
T2 and T3 trigger selections are compared to the asymmetries from the T1
subsample, which is based on charge-symmetric combinations of tracks. No
statistically significant discrepancy is observed, and the statistical
uncertainty of the difference is assigned as a systematic uncertainty. This
systematic uncertainty accounts for residual trigger-induced biases in the
difference of measured asymmetries for signal and control channels.

Different background parametrizations can change the ratio of signal and
background and affect the observed asymmetry. The nominal model is modified by
replacing, for all subsamples, the fourth-order polynomial with other
empirically chosen functions, a second-order polynomial or an ARGUS function
[35]. Different fit configurations are tested, in which the background
parameters are fixed according to the results of a fit to the
$m({{\pi}^{+}}{{\pi}^{-}}\gamma)$ sideband, or in which the
$D^{\pm}_{(s)}\rightarrow\phi_{3\pi}{{\pi}^{\pm}}$ background fractions are
varied. The maximum deviations from the results of the nominal fit, observed
with any of the alternative models providing a reasonable fit to the data, are
assigned as systematic uncertainties. This represents the largest contribution
to the systematic uncertainties in both channels. This estimate is in
agreement with an independent assessment, based on the increased statistical
uncertainties on $\mathcal{A}_{\mathrm{raw}}$ when the constraints on the
background parameters are removed from the nominal model.

The fitting procedure is validated with several pseudoexperiments using events
simulated according to the fit model, varying the $\mathcal{A}_{\mathrm{raw}}$
value used in generation. The sum in quadrature of the bias and of its
statistical uncertainty is taken as a systematic uncertainty.

A systematic uncertainty is introduced for the background contributions
neglected in the measurement of the raw asymmetries for the
${{D}^{\pm}}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{\pi}^{\pm}}$
and ${{D}^{\pm}_{s}}\rightarrow\phi{{\pi}^{\pm}}$ control decays, and for the
neglected fraction of $D^{\pm}_{(s)}$ signal leaking into the sidebands. The
effect of non-resonant
${{D}^{\pm}_{s}}\rightarrow{{K}^{+}}{{K}^{-}}{{\pi}^{\pm}}$ contributions to
the ${{D}^{\pm}_{s}}\rightarrow\phi{{\pi}^{\pm}}$ control sample is evaluated
by observing the variation of
$\Delta\mathcal{A}_{{C\\!P}}({{D}^{\pm}_{s}}\rightarrow\eta^{\prime}{{\pi}^{\pm}})$
when the $K^{+}K^{-}$ mass is required to be within $\pm
10\,{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ (instead of $\pm
20\,{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$) of the known $\phi$ mass. The
systematic uncertainty due to
${{D}^{\pm}_{s}}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{K}^{\pm}}$,
${{D}^{\pm}_{s}}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{\pi}^{\pm}}\pi^{0}$, and
${{D}^{\pm}_{s}}\rightarrow\phi{{\pi}^{\pm}}\pi^{0}$ is calculated from the
estimated fraction of background events, assuming a negligible $C\\!P$
violation and using the production asymmetries in LHCb acceptance as an input.
The difference of raw asymmetries in
$\Delta\mathcal{A}_{{C\\!P}}({{D}^{\pm}}\rightarrow\eta^{\prime}{{\pi}^{\pm}})$
is corrected for the ${{K}^{0}}$ asymmetry [9] and an associated systematic
uncertainty equal to the applied correction is included.

The potential discrepancy in the bachelor pion kinematic distribution within
each $\mbox{$p_{\mathrm{T}}$}-\eta$ bin between signal and control samples,
associated to the finite number of bins, might result in an incomplete
cancellation of detection asymmetries. The discrepancy in
$\Delta\mathcal{A}_{{C\\!P}}$ with respect to the nominal binning, resulting
from using no kinematic binning, is assigned as a systematic uncertainty.

The $D^{\pm}_{(s)}$ production asymmetry may show a dependence on
$p_{\mathrm{T}}$ and $\eta$ of the charm meson. Therefore, the cancellation of
production effects in $\Delta\mathcal{A}_{{C\\!P}}$ may be partial, since
$D^{\pm}_{(s)}$ kinematic distributions are different for signal and control
channels. To estimate this effect, in each bin of the bachelor-pion kinematic
distribution, the ${{D}^{\pm}}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{\pi}^{\pm}}$ and ${{D}^{\pm}_{s}}\rightarrow\phi{{\pi}^{\pm}}$
candidates are given a weight depending on either the $p_{\mathrm{T}}$ or the
$\eta$ value of the $D^{\pm}_{(s)}$ meson, to reproduce the $D^{\pm}_{(s)}$
kinematic distribution of signal candidates. The effect on
$\Delta\mathcal{A}_{{C\\!P}}$ is assigned as a systematic uncertainty.

The $\Delta\mathcal{A}_{{C\\!P}}$ results are stable when the requirements on
the bachelor-pion particle identification and track quality are tightened, and
when the constraints on the parameters of the combinatorial background
component are removed from the fit to
$D^{\pm}_{(s)}\rightarrow\eta^{\prime}{{\pi}^{\pm}}$ candidates. The stability
of $\Delta\mathcal{A}_{{C\\!P}}$ is also investigated as a function of beam
energy and hardware trigger decision. No significant dependence is observed,
as shown in Fig. 4.

##  7 Results and summary

Using $pp$ collision data collected by the LHCb experiment at centre-of-mass
energies of $7$ and $8\,\mathrm{\,Te\kern-1.00006ptV}$, the differences in
$C\\!P$ asymmetries between ${{D}^{\pm}}\rightarrow\eta^{\prime}{{\pi}^{\pm}}$
and ${{D}^{\pm}}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{\pi}^{\pm}}$ decays, and between
${{D}^{\pm}_{s}}\rightarrow\eta^{\prime}{{\pi}^{\pm}}$ and
${{D}^{\pm}_{s}}\rightarrow\phi{{\pi}^{\pm}}$ decays, are measured to be

$\displaystyle\Delta\mathcal{A}_{{C\\!P}}({{D}^{\pm}}\rightarrow\eta^{\prime}{{\pi}^{\pm}})$
$\displaystyle=$ $\displaystyle(-0.58\pm 0.72\pm 0.53)\%,$
$\displaystyle\Delta\mathcal{A}_{{C\\!P}}({{D}^{\pm}_{s}}\rightarrow\eta^{\prime}{{\pi}^{\pm}})$
$\displaystyle=$ $\displaystyle(-0.44\pm 0.36\pm 0.22)\%.$

In all cases, the first uncertainties are statistical and the second are
systematic.

Using the previously measured values of the $C\\!P$ asymmetries in control
decays,
$\mathcal{A}_{{C\\!P}}({{D}^{\pm}}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{\pi}^{\pm}})=(-0.024\pm 0.094\pm 0.067)\%$ [12] and
$\mathcal{A}_{{C\\!P}}({{D}^{\pm}_{s}}\rightarrow\phi{{\pi}^{\pm}})=(-0.38\pm
0.26\pm 0.08)\%$ [13], the individual $C\\!P$ asymmetries are found to be

$\displaystyle\mathcal{A}_{{C\\!P}}({{D}^{\pm}}\rightarrow\eta^{\prime}{{\pi}^{\pm}})$
$\displaystyle=$ $\displaystyle(-0.61\pm 0.72\pm 0.53\pm 0.12)\%,$
$\displaystyle\mathcal{A}_{{C\\!P}}({{D}^{\pm}_{s}}\rightarrow\eta^{\prime}{{\pi}^{\pm}})$
$\displaystyle=$ $\displaystyle(-0.82\pm 0.36\pm 0.22\pm 0.27)\%,$

where the last contribution to the uncertainty comes from the
$\mathcal{A}_{{C\\!P}}({{D}^{\pm}}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{\pi}^{\pm}})$ and
$\mathcal{A}_{{C\\!P}}({{D}^{\pm}_{s}}\rightarrow\phi{{\pi}^{\pm}})$
measurements.

The measured values show no evidence of $C\\!P$ violation, and are consistent
with SM expectations [36, 37, 38] and with previous results obtained in
$e^{+}e^{-}$ collisions [10, 11]. The results represent the most precise
measurements of these quantities to date.

## 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). 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).

## References

  * [1] Y. Grossman, A. L. Kagan, and Y. Nir, _New physics and CP violation in singly Cabibbo suppressed D decays_ , Phys. Rev. D75 (2007) 036008, arXiv:hep-ph/0609178
  * [2] F. Buccella, M. Lusignoli, and A. Pugliese, _Charm nonleptonic decays and final state interactions_ , Phys. Lett. B379 (1996) 249, arXiv:hep-ph/9601343
  * [3] M. Golden and B. Grinstein, _Enhanced CP violations in hadronic charm decays_ , Phys. Lett. B222 (1989) 501
  * [4] L.-L. Chau, _Quark mixing in weak interactions_ , Phys. Rept. 95 (1983) 1
  * [5] I. Hinchliffe and T. A. Kaeding, _Nonleptonic two-body decays of D mesons in broken SU(3)_ , Phys. Rev. D54 (1996) 914, arXiv:hep-ph/9502275
  * [6] B. Bhattacharya and J. L. Rosner, _Charmed meson decays to two pseudoscalars_ , Phys. Rev. D81 (2010) 014026, arXiv:0911.2812
  * [7] H.-Y. Cheng and C.-W. Chiang, _SU(3) symmetry breaking and CP violation in $D\rightarrow PP$ decays_, Phys. Rev. D86 (2012) 014014, arXiv:1205.0580
  * [8] LHCb collaboration, R. Aaij et al., _Measurement of the difference of time-integrated ${C\\!P}$ asymmetries in ${{D}^{0}}\rightarrow{{K}^{-}}{{K}^{+}}$ and ${{D}^{0}}\rightarrow{{\pi}^{-}}{{\pi}^{+}}$ decays_, Phys. Rev. Lett. 116 (2016) 191601, arXiv:1602.03160
  * [9] LHCb collaboration, R. Aaij et al., _Measurement of ${C\\!P}$ asymmetry in ${{D}^{0}}\rightarrow{{K}^{-}}{{K}^{+}}$ and ${{D}^{0}}\rightarrow{{\pi}^{-}}{{\pi}^{+}}$ decays_, JHEP 07 (2014) 041, arXiv:1405.2797
  * [10] CLEO collaboration, P. U. E. Onyisi et al., _Improved measurement of absolute hadronic branching fractions of the $D_{s}^{+}$ meson_, Phys. Rev. D88 (2013) 032009, arXiv:1306.5363
  * [11] Belle collaboration, E. Won et al., _Observation of $D^{+}\rightarrow K^{+}\eta^{(\prime)}$ and search for CP violation in $D^{+}\rightarrow\pi^{+}\eta^{(\prime)}$ decays_, Phys. Rev. Lett. 107 (2011) 221801, arXiv:1107.0553
  * [12] Belle collaboration, B. R. Ko et al., _Evidence for CP violation in the decay $D^{+}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}\pi^{+}$_, Phys. Rev. Lett. 109 (2012) 021601, Erratum ibid. 109 (2012) 119903, arXiv:1203.6409
  * [13] D0 collaboration, V. M. Abazov et al., _Measurement of the direct CP-violating charge asymmetry in ${D_{s}^{\pm}\rightarrow\phi\pi^{\pm}}$ decays_, Phys. Rev. Lett. 112 (2014) 111804, arXiv:1312.0741
  * [14] LHCb collaboration, A. A. Alves Jr. et al., _The LHCb detector at the LHC_, JINST 3 (2008) S08005
  * [15] LHCb collaboration, R. Aaij et al., _LHCb detector performance_ , Int. J. Mod. Phys. A30 (2015) 1530022, arXiv:1412.6352
  * [16] T. Sjöstrand, S. Mrenna, and P. Skands, _PYTHIA 6.4 physics and manual_ , JHEP 05 (2006) 026, arXiv:hep-ph/0603175
  * [17] I. Belyaev et al., _Handling of the generation of primary events in Gauss, the LHCb simulation framework_ , J. Phys. Conf. Ser. 331 (2011) 032047
  * [18] D. J. Lange, _The EvtGen particle decay simulation package_ , Nucl. Instrum. Meth. A462 (2001) 152
  * [19] P. Golonka and Z. Was, _PHOTOS Monte Carlo: A precision tool for QED corrections in $Z$ and $W$ decays_, Eur. Phys. J. C45 (2006) 97, arXiv:hep-ph/0506026
  * [20] Geant4 collaboration, J. Allison et al., _Geant4 developments and applications_ , IEEE Trans. Nucl. Sci. 53 (2006) 270
  * [21] Geant4 collaboration, S. Agostinelli et al., _Geant4: A simulation toolkit_ , Nucl. Instrum. Meth. A506 (2003) 250
  * [22] M. Clemencic et al., _The LHCb simulation application, Gauss: Design, evolution and experience_, J. Phys. Conf. Ser. 331 (2011) 032023
  * [23] Particle Data Group, C. Patrignani et al., _Review of particle physics_ , Chin. Phys. C40 (2016) 100001
  * [24] LHCb collaboration, R. Aaij et al., _Evidence for CP violation in time-integrated $D^{0}\rightarrow h^{-}h^{+}$ decay rates_, Phys. Rev. Lett. 108 (2012) 111602, arXiv:1112.0938
  * [25] W. D. Hulsbergen, _Decay chain fitting with a Kalman filter_ , Nucl. Instrum. Meth. A552 (2005) 566, arXiv:physics/0503191
  * [26] N. L. Johnson, _Systems of frequency curves generated by methods of translation_ , Biometrika 36 (1949) 149
  * [27] BaBar collaboration, P. del Amo Sanchez et al., _Study of $B\rightarrow X\gamma$ decays and determination of $|V_{td}/V_{ts}|$_, Phys. Rev. D82 (2010) 051101, arXiv:1005.4087
  * [28] LHCb collaboration, R. Aaij et al., _Prompt charm production in ${p}{p}$ collisions at $\sqrt{s}=7$$~{}T\kern-0.50003pteV$_, Nucl. Phys. B871 (2013) 1, arXiv:1302.2864
  * [29] LHCb collaboration, R. Aaij et al., _Measurement of $\sigma(pp\rightarrow{{b}{\overline{b}}}X)$ at $\sqrt{s}=7$ TeV in the forward region_, Phys. Lett. B694 (2010) 209, arXiv:1009.2731
  * [30] LHCb collaboration, R. Aaij et al., _Measurements of the branching fractions and ${C\\!P}$ asymmetries of ${{B}^{\pm}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{\pi}^{\pm}}$ and ${{B}^{\pm}}\rightarrow{\psi{(2S)}}{{\pi}^{\pm}}$ decays_, Phys. Rev. D85 (2012) 091105(R), arXiv:1203.3592
  * [31] LHCb collaboration, R. Aaij et al., _Measurement of ${C\\!P}$ asymmetries in the decays ${{B}^{0}}\rightarrow{{K}^{*0}}{\mu^{+}}{\mu^{-}}$ and ${{{B}^{+}}}\rightarrow{{K}^{+}}{\mu^{+}}{\mu^{-}}$_, JHEP 09 (2014) 177, arXiv:1408.0978
  * [32] LHCb collaboration, R. Aaij et al., _Measurement of the ${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}$–${{B}^{0}}$ and ${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}}$–${{B}^{0}_{s}}$ production asymmetries in ${p}{p}$ collisions at $\sqrt{s}=7$$~{}T\kern-0.50003pteV$_, Phys. Lett. B739 (2014) 218, arXiv:1408.0275
  * [33] LHCb collaboration, R. Aaij et al., _Measurement of the semileptonic ${C\\!P}$ asymmetry in ${{B}^{0}}$–${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}$ mixing_, Phys. Rev. Lett. 114 (2015) 041601, arXiv:1409.8586
  * [34] LHCb collaboration, R. Aaij et al., _Observation of the ${{\mathchar 28931\relax}^{0}_{b}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{p}{{\pi}^{-}}$ decay_, JHEP 07 (2014) 103, arXiv:1406.0755
  * [35] ARGUS collaboration, H. Albrecht et al., _Search for hadronic $b\rightarrow u$ decays_, Phys. Lett. B241 (1990) 278
  * [36] H.-Y. Cheng and C.-W. Chiang, _Two-body hadronic charmed meson decays_ , Phys. Rev. D81 (2010) 074021, arXiv:1001.0987
  * [37] H.-Y. Cheng and C.-W. Chiang, _Direct CP violation in two-body hadronic charmed meson decays_ , Phys. Rev. D85 (2012) 034036, Erratum ibid. D85 (2012) 079903, arXiv:1201.0785
  * [38] H.-n. Li, C.-D. Lu, and F.-S. Yu, _Branching ratios and direct CP asymmetries in $D\rightarrow PP$ decays_, Phys. Rev. D86 (2012) 036012, arXiv:1203.3120

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.
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Coco59, J. Cogan6, E. Cogneras5, V. Cogoni16,40,f, L. Cojocariu30, G.
Collazuol23,o, P. Collins40, A. Comerma-Montells12, A. Contu40, A. Cook48, G.
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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.
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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
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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,
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Morawski28, A. Mordà6, M.J. Morello24,t, J. Moron28, A.B. Morris52, R.
Mountain61, F. Muheim52, M. Mulder43, M. Mussini15, D. Müller56, J. Müller10,
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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, Centro Mixto Universidad 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