EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2016-203 LHCb-PAPER-2016-037 January 12, 2017

Measurement of $C\\!P$ violation in $B^{0}\\!\rightarrow D^{+}D^{-}$ decays

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

The $C\\!P$ violation observables $S$ and $C$ in the decay channel
$B^{0}\\!\rightarrow D^{+}D^{-}$ are determined from a sample of proton-proton
collisions at center-of-mass energies of
$78\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$, collected by the LHCb experiment
and corresponding to an integrated luminosity of
$3\text{\,}\mathrm{f}\mathrm{b}^{-1}$. The observable $S$ describes $C\\!P$
violation in the interference between mixing and the decay amplitude, and $C$
parametrizes direct $C\\!P$ violation in the decay. The following values are
obtained from a flavor-tagged, decay-time-dependent analysis:

$\displaystyle S$ $\displaystyle=-0.54\,^{+0.17}_{-0.16}\,\text{(stat)}\pm
0.05\,\text{(syst)}\,,$ $\displaystyle C$
$\displaystyle=\phantom{-}0.26\,^{+0.18}_{-0.17}\,\text{(stat)}\pm
0.02\,\text{(syst)}\,.$

These values provide evidence for $C\\!P$ violation at a significance level of
$4.0$ standard deviations. The phase shift due to higher-order Standard Model
corrections is constrained to a small value of

$\displaystyle\Delta\phi=-0.16\,^{+0.19}_{-0.21}\,$\mathrm{rad}$\,.$

Published in Phys. Rev. Lett. 117, 261801 (2016)

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

Studies of beauty hadron decays into pairs of charm hadrons give access to a
multitude of observables that probe the Cabibbo-Kobayashi-Maskawa (CKM) quark
mixing matrix [1, 2] of the Standard Model (SM). Comparisons of these
observables with each other and with similar observables from beauty hadron
decays to charmonia allow higher-order SM contributions, like loop diagrams,
to be separated from effects caused by physics beyond the SM [3, 4, 5, 6, 7].
For example, under the assumption that flavor symmetries hold to a good
approximation, higher-order corrections in the measurement of $\phi_{{s}}$ in
$B^{0}_{{s}}\\!\rightarrow D^{+}_{{s}}D^{-}_{{s}}$ [8] can be
constrained.111The inclusion of charge-conjugate processes is implied
throughout the Letter, unless otherwise noted.

In the $B^{0}$ meson system, $C\\!P$ violation in the mixing is negligible, as
is the decay width difference $\mathrm{\Delta}\Gamma$ of the mass eigenstates
[9]. In contrast, sizable $C\\!P$ violation from the interference between the
direct (unmixed) decay into the $C\\!P$-even final state $D^{+}D^{-}$ and the
decay to the same final state after $B^{0}$–$\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ mixing, or from the interference
of different decay processes, leads to a decay-time-dependent decay rate of

$\displaystyle\frac{\mathrm{d}\Gamma(t,d)}{\mathrm{d}t}$
$\displaystyle\propto\mathrm{e}^{-t/\tau}\Big{(}1-d\,S\sin{\left(\mathrm{\Delta}mt\right)}+d\,C\cos{\left(\mathrm{\Delta}mt\right)}\Big{)}\,,$
(1)

where $t$ is the proper decay time, $d$ represents the $B^{0}$ flavor at
production and takes a value of $+1$ for mesons whose initial flavor is
$B^{0}$ and $-1$ for ${\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}}$,
$\tau$ is the mean lifetime and $\mathrm{\Delta}m$ is the mass difference
between the physical $B^{0}$ meson eigenstates. The $C\\!P$ observables $S$
and $C$ are related to the $B^{0}$ mixing phase $\phi_{d}$ and a phase shift
$\Delta{\phi}$ from the decay amplitudes via
${S}/{\sqrt{1-C^{2}}}=-\sin(\phi_{d}+\Delta{\phi})$ [10]. In the SM,
$\phi_{d}=2\beta$, where
$\beta\equiv\arg[-(V_{cd}V_{cb}^{\ast})/(V_{td}V_{tb}^{\ast})]$ is an angle of
one of the CKM unitary triangles and $V_{qq^{\prime}}$ are elements of the CKM
matrix. If the $B^{0}\\!\rightarrow D^{+}D^{-}$ decay amplitude can be
described by a dominant tree-level $b\\!\rightarrow c\overline{c}d$
transition, the phase shift $\Delta{\phi}$ vanishes and the $C\\!P$
observables are given by $C=0$ and $S=-\sin\phi_{d}$. The value of the latter
has been measured to be $\sin\phi_{d}=+0.679\pm 0.020$ [9] in $b\\!\rightarrow
c\overline{c}s$ decays such as
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}{K^{0}_{\rm\scriptscriptstyle S}}$, in which the contribution from loop
processes in the decay can be constrained to high precision [11]. In contrast,
previous measurements of the $C\\!P$ observables in the decay
$B^{0}\\!\rightarrow D^{+}D^{-}$ by the BaBar and Belle collaborations [12,
13] give world average values of $S=$-0.98\,\pm 0.17$$ and $C=$-0.31\,\pm
0.14$$ [9]. The values are at the edge of the physically allowed region of
$S^{2}+C^{2}\leq 1$, which leaves room for a large value of $\Delta\phi$.

This Letter reports a measurement of $C\\!P$ violation in $B^{0}\\!\rightarrow
D^{+}D^{-}$ decays with the LHCb experiment. The measurement is based on
samples of $pp$ collision data corresponding to integrated luminosities of $1$
and $2\text{\,}\mathrm{f}\mathrm{b}^{-1}$ at center-of-mass energies of $7$
and $8\,\mathrm{\,Te\kern-1.00006ptV}$, respectively, recorded by the LHCb
experiment. The LHCb detector is a single-arm forward spectrometer covering
the pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks, and is described in detail in Refs. [14, 15].
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.
Simulated events are produced with the software described in Refs. [16,
*Sjostrand:2006za, 18, 19, 20, 21, *Agostinelli:2002hh, 23].

Candidate $B^{0}\\!\rightarrow D^{+}D^{-}$ decays are reconstructed through
the subsequent decays $D^{+}\\!\rightarrow K^{-}\pi^{+}\pi^{+}$ and
$D^{+}\\!\rightarrow K^{-}K^{+}\pi^{+}$, with combinations of two
$D\\!\rightarrow KK\pi$ candidates omitted due to the low branching fraction.
The kaon and pion candidates, which must fulfill loose particle identification
(PID) criteria, are required to have transverse momentum
$p_{\text{T}}>$100\text{\,}\mathrm{M}\mathrm{eV}\\!\mathrm{/}\mathrm{\mathit{c}}$$,
to have a good track quality and to be inconsistent with originating from a
primary vertex (PV). The three hadron tracks must form a good common vertex
and their combined invariant mass has to be in the range $\pm
25\text{\,}\mathrm{M}\mathrm{eV}\\!\mathrm{/}\mathrm{{}^{2}}\mathrm{\mathit{c}}$
around the known $D^{+}$ mass [24]. The scalar sum of the $p_{\text{T}}$ of
the three hadrons has to exceed
$1800\text{\,}\mathrm{M}\mathrm{eV}\\!\mathrm{/}\mathrm{\mathit{c}}$ and the
$D^{+}$ vertex has to be significantly displaced from all PVs. Defining
$\theta_{X}$ as the angle between the momentum vector of a particle $X$ and
the displacement vector from the best-matched PV to the $X$ decay vertex,
$\cos\theta_{D^{+}}$ is required to be positive.

To suppress contributions from misreconstructed $D^{+}_{s}\\!\rightarrow
K^{-}K^{+}\pi^{+}$ decays, which proceed predominantly through
$D^{+}_{s}\\!\rightarrow\phi\pi^{+}$, $D^{+}\\!\rightarrow
K^{-}\pi^{+}\pi^{+}$ candidates are rejected if, after assigning the kaon mass
hypothesis to the $\pi^{+}$ with the higher $p_{\text{T}}$, the invariant mass
$m(K^{-}K^{+})$ is within
$10\text{\,}\mathrm{M}\mathrm{eV}\\!\mathrm{/}\mathrm{{}^{2}}\mathrm{\mathit{c}}$
of the known $\phi$ meson mass. Furthermore, if the invariant mass
$m(K^{-}K^{+}\pi^{+})$ is within
$25\text{\,}\mathrm{M}\mathrm{eV}\\!\mathrm{/}\mathrm{{}^{2}}\mathrm{\mathit{c}}$
of the known $D^{+}_{s}$ meson mass, the requirement on the PID information of
the higher-$p_{\text{T}}$ pion to be consistent with the pion hypothesis is
tightened. Similarly, protons can be misidentified as pions, resulting in
background contributions from $\Lambda^{+}_{c}\\!\rightarrow K^{-}p\pi^{+}$.
To suppress these processes, the pion candidate with the higher $p_{\text{T}}$
of $D^{+}\\!\rightarrow K^{-}\pi^{+}\pi^{+}$ is required to be well identified
as a pion if
$|m(K^{-}p\pi^{+})−-m_{\Lambda^{+}_{c}}|<$25\text{\,}\mathrm{M}\mathrm{eV}\\!\mathrm{/}\mathrm{{}^{2}}\mathrm{\mathit{c}}$$.

Candidate $B^{0}$ mesons are reconstructed from pairs of oppositely charged
$D^{\pm}$ candidates that form a common vertex. The scalar sum of the
$p_{\text{T}}$ of the $D^{\pm}$ mesons must exceed
$5\text{\,}\mathrm{G}\mathrm{eV}\\!\mathrm{/}\mathrm{\mathit{c}}$. The decay
time significance of each $D^{\pm}$ meson, defined as its decay time divided
by its estimated uncertainty, is required to be greater than $0$, or greater
than $3$ if one of the $D^{\pm}$ mesons is reconstructed in the
$K^{-}K^{+}\pi^{+}$ final state. This reduces the contamination of
$B^{0}\\!\rightarrow D^{-}K^{-}K^{+}\pi^{+}$ decays. The $B^{0}$ candidate is
required to have momentum
$p>$10\text{\,}\mathrm{G}\mathrm{eV}\\!\mathrm{/}\mathrm{\mathit{c}}$$,
$\cos\theta_{B^{0}}>$0.999$$, and to not originate from the associated PV. A
fit to the full decay chain, in which the $B^{0}$ production vertex is
constrained to the position of the associated PV, is performed to determine
the reconstructed decay time $t^{\prime}$ of the $B^{0}$ candidate, which
differs from the true time $t$. Only candidates with decay times in the range
$0.2510.25\text{\,}\mathrm{ps}$ are kept. The invariant mass $m_{D^{+}D^{-}}$
of the $B^{0}$ candidate is calculated from a similar fit to the full decay
chain, while additionally constraining the invariant masses of
$K^{-}\pi^{+}\pi^{+}$ and $K^{-}K^{+}\pi^{+}$ to the known $D^{+}$ mass, and
is required to be in the range
$51505500\text{\,}\mathrm{M}\mathrm{eV}\\!\mathrm{/}\mathrm{{}^{2}}\mathrm{\mathit{c}}$.

Two boosted decision trees (BDTs) [25, 26], for $B^{0}$ final states with two
and three kaons, are used to suppress the combinatorial background. Both are
trained on simulated signal samples and on background samples formed from
$B^{0}$ candidates at high invariant masses
($>$5500\text{\,}\mathrm{M}\mathrm{eV}\\!\mathrm{/}\mathrm{{}^{2}}\mathrm{\mathit{c}}$$),
and exploit observables related to the kinematics of the decay, PID
information, and track and vertex quality. The requirements on the BDT
classifier outputs are chosen to optimize the precision of both $C\\!P$
observables, $S$ and $C$.

To separate the remaining background from the signal a fit to the $D^{+}D^{-}$
invariant mass distribution is performed to calculate signal candidate weights
via the sPlot technique [27]. The mass fit is performed simultaneously in four
categories, split by the data-taking period ($7\text{\,}\mathrm{TeV}$,
$8\text{\,}\mathrm{TeV}$) and the number of kaons in the final state. The
probability density function (PDF) used to parametrize the mass distribution
consists of four contributions: signal, $B^{0}_{s}\\!\rightarrow D^{+}D^{-}$,
combinatorial background, and a component that includes both
$B^{0}\\!\rightarrow D^{+}_{s}D^{-}$ and $B^{0}_{s}\\!\rightarrow
D^{-}_{s}D^{+}$ decays. The signal is modeled by the sum of three Crystal Ball
functions [28] with a common mean. The parameters of the tails (two towards
lower and one towards higher mass) and the three widths are determined from
simulated samples. To account for differences in the mass resolution in
simulation and data, the width parameters are multiplied by a common scale
factor, which is free to vary in the fit to data. The $B^{0}_{s}\\!\rightarrow
D^{+}D^{-}$ component shares all shape parameters with the signal PDF except
for the peak position, which is constrained by the known value of the
difference between the $B^{0}$ and the $B^{0}_{s}$ masses [24]. Each peak in
the $B^{0}\\!\rightarrow D^{+}_{s}D^{-}$ and $B^{0}_{s}\\!\rightarrow
D^{-}_{s}D^{+}$ component is described by the sum of two Crystal Ball
functions (one with a tail towards lower and one with a tail towards higher
masses) whose parameters are taken from simulation. The widths and the $B^{0}$
peak position are free to vary in the fit while the $B^{0}_{s}$ peak offset is
constrained in the same way as that of the $B^{0}_{s}\\!\rightarrow
D^{+}D^{-}$ component. The combinatorial background is parametrized with an
exponential function, with separate exponents used for the final states with
two or three kaons. Partially reconstructed $B^{0}\\!\rightarrow D^{*+}D^{-}$
decays with $D^{*+}\\!\rightarrow D^{+}\pi^{0}$, where the neutral pion is
missed, lie outside the mass range used for the fit. The equivalent
$B^{0}_{s}\\!\rightarrow D^{*+}D^{-}$ decays and decay modes with only one or
no charm meson, such as $B^{0}\\!\rightarrow D^{-}K^{-}K^{+}\pi^{+}$, are also
neglected in the mass fit. The influence of their omission on the $C\\!P$
measurement is treated as a systematic uncertainty. The mass distribution is
shown in Fig. 1 (a). The combined $B^{0}\\!\rightarrow D^{+}D^{-}$ signal
yield is $1610\,\pm 50$, of which $1347\,\pm 45$ are in the Cabibbo-favored
final state with two $D^{+}\\!\rightarrow K^{-}\pi^{+}\pi^{+}$ decays.

Figure 1: Distribution of the reconstructed mass of all $B^{0}\\!\rightarrow
D^{+}D^{-}$ candidates (a) and background-subtracted decay time distribution
for tagged candidates (b). In plot (a) besides the data points and the
projection of the full PDF (solid black) the projections of the $B^{0}$ signal
(dashed blue), the $B^{0}_{s}\\!\rightarrow D^{+}D^{-}$ background (short-
dash-dotted turquoise), the $B^{0}\\!\rightarrow D^{+}_{s}D^{-}$ background
(dotted green), the $B^{0}_{s}\\!\rightarrow D^{-}_{s}D^{+}$ background (long-
dash-three-dotted red) and the combinatorial background (long-dash-dotted
purple) are shown.

The measurement of decay-time-dependent $C\\!P$ violation requires knowledge
of the initial flavor of each reconstructed $B^{0}$ meson. Flavor-tagging
algorithms deliver a measured tag decision $d^{\prime}$ for the flavor of the
$B^{0}$ meson, which takes the value $+1$ for a $B^{0}$, $-1$ for a ${\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}}$ initial state, and $0$ if no
decision is possible, and an estimate $\eta$ of the probability for the tag
decision to be incorrect. The latter is referred to as the mistag probability.
Two classes of flavor-tagging algorithms are used: opposite-side (OS) and
same-side (SS) taggers [29, 30, 31]. In $b\overline{b}$ pair production, the
dominant source of $b$ hadrons at LHCb, the signal $B^{0}$ meson is
accompanied by a second $b$ hadron. The OS taggers determine the flavor of the
signal by examining the decay products of this second $b$ hadron. The
information from the decay products consists of the charge of muons or
electrons produced in semileptonic decays, the charge of kaons from
$b\rightarrow c\rightarrow s$ transitions, the charge of charm hadrons from
$b\rightarrow c$ transitions, and the net charge of all decay products. The SS
taggers analyze pions and protons related to the hadronization process of the
$B^{0}$ meson. This is the first analysis to use the LHCb SS proton and OS
charm taggers, and the first to use the new SS pion tagger.

The outputs of all OS algorithms are combined into an overall OS tagging
decision and mistag estimate, and the same is done for the SS algorithms. The
mistag estimates $\eta\in\\{\eta_{\text{OS}},\eta_{\text{SS}}\\}$ are
calibrated using linear functions $\omega(\eta|d)$, so that $\eta$ on average
matches the true mistag probability $\omega$, which depends on the true
production flavor $d$ of the $B^{0}$ meson. The calibration studies are
performed with a sample of $B^{0}\\!\rightarrow D^{+}_{s}D^{-}$ decays, for
which the final state determines the flavor of the $B^{0}$ at decay. Since the
calibration and signal channels are kinematically very similar, the
calibration can be applied to the signal channel without further corrections.
To ensure that the same calibration is valid for both, the same selection is
used as for the signal decay with one $D^{+}\\!\rightarrow K^{-}K^{+}\pi^{+}$,
apart from requiring that the $K^{-}K^{+}\pi^{+}$ invariant mass lie within
$25\text{\,}\mathrm{M}\mathrm{eV}\\!\mathrm{/}\mathrm{{}^{2}}\mathrm{\mathit{c}}$
of the known $D^{+}_{s}$ mass [24] and dropping the vetoes against
misidentified backgrounds. Background is subtracted from the calibration
sample via the sPlot technique [27]. The tagging calibration parameters are
determined from a fit to the decay time and tag distributions of
$B^{0}\\!\rightarrow D^{+}_{s}D^{-}$ candidates, in which the detection
asymmetry, the production asymmetry of the $B^{0}$ mesons, and the flavor-
specific semileptonic asymmetry $a_{\text{sl}}^{d}$ are taken into account.
Here, the detection asymmetry describes the difference in reconstruction
efficiency between the $D^{+}_{s}D^{-}$ and $D^{-}_{s}D^{+}$ final states, and
${A_{\mathrm{P}}}\equiv{[\sigma(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0})-\sigma(B^{0})]}/{[\sigma(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0})+\sigma(B^{0})]}$, where $\sigma$
denotes the production cross-section inside the LHCb acceptance. The values of
all these parameters are fixed according to the latest LHCb measurements [32,
33], and their uncertainties are treated as sources of systematic uncertainty
on the calibration parameters. Further systematic uncertainties are assigned
due to the calibration method, the dependence of the efficiency on decay time,
the decay time resolution, and the background subtraction. More details on the
calibration studies are given in the supplemental material.

In the $B^{0}\\!\rightarrow D^{+}D^{-}$ signal data sample, the correlation
between the OS and the SS mistag estimates is found to be negligible. A small
correlation of the mistag probability with decay time is seen; this is
neglected in the main fit but considered as a source of systematic
uncertainty.

The effective tagging efficiency is the product of the probability for
reaching a tagging decision, $\varepsilon_{\mathrm{tag}}$ = $87.6\,\pm
0.8\text{\,}\mathrm{\char 37\relax}$, and the square of the effective dilution
$D=1-2\omega=$30.3\,\pm 1.1\text{\,}\mathrm{\char 37\relax}$$. Its value is
$\varepsilon_{\mathrm{tag}}D^{2}=$8.1\,\pm 0.6\text{\,}\mathrm{\char
37\relax}$$, the highest effective tagging efficiency to date in tagged
$C\\!P$ violation measurements at LHCb thanks to the improved flavor-tagging
algorithms and the kinematic properties of the selected $B^{0}\\!\rightarrow
D^{+}D^{-}$ decays.

The $C\\!P$ violation observables $S$ and $C$ are determined from a
multidimensional fit to the background-subtracted tag and decay time
distributions of the tagged $B^{0}\\!\rightarrow D^{+}D^{-}$ candidates; a
projection of the decay time distribution summed over the non-zero tag
decisions is shown in Fig. 1 (b). The conditional PDF describing the
reconstructed decay time $t^{\prime}$ and tag decisions
$\vec{d^{\prime}}=(d^{\prime}_{\text{OS}},d^{\prime}_{\text{SS}})$, given a
per-event decay time resolution $\sigma_{t^{\prime}}$ and per-event mistag
probability estimates $\vec{\eta}=(\eta_{\text{OS}},\eta_{\text{SS}})$, is

$P\left(t^{\prime},\vec{d^{\prime}}\>|\>\sigma_{t^{\prime}},\vec{\eta}\right)\propto\epsilon(t^{\prime})\left(\mathcal{P}(t,\vec{d^{\prime}}\>|\>\vec{\eta})\otimes\mathcal{R}(t^{\prime}-t\>|\>\sigma_{t^{\prime}})\right)\,,$
(2)

where

$\mathcal{P}(t,\vec{d^{\prime}}\>|\>\vec{\eta})\\\
\propto\sum_{d}\mathcal{P}(\vec{d^{\prime}}\>|\>d,\vec{\eta})[1-d\,A_{\text{P}}]\,e^{-t/\tau}\left\\{1-d\,S\sin({\Delta
m}t)+d\,C\cos({\Delta m}t)\right\\}\,,$ (3)

and where $t$ is the true decay time, $d$ is the true production flavor,
$A_{\text{P}}$ is the production asymmetry, and
$\mathcal{P}(\vec{d^{\prime}}\>|\>d,\vec{\eta})$ is a two-dimensional binomial
PDF describing the distribution of tagging decisions given $\vec{\eta}$ and
$d$. Normalization factors are omitted for brevity. In the fit, the mass
difference $\mathrm{\Delta}m$ and the lifetime $\tau$ are constrained to their
known values within uncertainties [24]. The production asymmetry
$A_{\text{P}}$ is constrained separately for the $78\text{\,}\mathrm{TeV}$
samples to the values obtained from weighting the results from the
measurements in Ref. [33] according to the kinematic distribution of the
$B^{0}$ signal candidates. The decay time resolution model $\mathcal{R}$ is
the sum of three Gaussian functions, two of which have event-dependent widths
proportional to $\sigma_{t^{\prime}}$, and one which has a global width that
describes the effect of candidates matched to a wrong PV; all three share a
common mean. All parameters of the resolution model are determined from
simulation. The average decay time resolution in data is
$49\text{\,}\mathrm{fs}$. The function $\epsilon(t^{\prime})$ describes the
efficiency for all reconstruction and selection steps as a function of the
reconstructed decay time. It is represented by cubic splines [34], with the
spline coefficients left unconstrained in the fit.

The statistical uncertainties are estimated using the bootstrap method [35].
Individual bootstrap samples are drawn from the candidates in data that pass
the full selection; the analysis procedure described above, consisting of the
mass fit, background subtraction, and decay time fit, is then applied to
obtain the values of the $C\\!P$ observables for each such sample. Two-sided
$68\text{\,}\mathrm{\char 37\relax}$ confidence intervals, with equal tail
probabilities on either side, are obtained from the distributions of fitted
parameters in the bootstrapped samples. To account for the uncertainties of
the flavor-tagging calibration parameters, which are fixed in the likelihood
fit, further pseudoexperiments are generated in which these flavor-tagging
calibration parameters are varied within their combined statistical and
systematic uncertainties. The results are then used to correct the
uncertainties from the bootstrapping procedure. The $C\\!P$ observables are
measured to be $S=$-0.54$\,^{+0.17}_{-0.16}$ and $C=$0.26$\,^{+0.18}_{-0.17}$
with a correlation coefficient of $\rho=$0.48$$. The decay-time-dependent
signal yield asymmetry $(N_{{\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}}}-N_{B^{0}})/(N_{{\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}}}+N_{B^{0}})$, where $N_{B^{0}}$ is
the number of $B^{0}\\!\rightarrow D^{+}D^{-}$ decays with a $B^{0}$ flavor
tag, and $N_{{\kern 1.25995pt\overline{\kern-1.25995ptB}{}^{0}}}$ the number
with a ${\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}}$ tag, is shown in
Fig. 2.

Figure 2: Decay-time-dependent signal yield asymmetry. The solid curve is the
projection of the signal PDF given in Eq. 2.

Several sources of systematic uncertainties on the $C\\!P$ observables are
studied with pseudoexperiments. The largest systematic uncertainty arises from
neglecting backgrounds in which the final state contains only one charm meson,
such as $B^{0}\\!\rightarrow D^{-}K^{-}K^{+}\pi^{+}$. The yield of these
backgrounds is estimated to be about $2\text{\,}\mathrm{\char 37\relax}$ of
the signal yield and their impact is assessed by assuming that they maximally
violate $C\\!P$ symmetry and have the eigenvalue opposite to the signal mode.
This leads to a systematic uncertainty of $\pm 0.05$ on $S$ and $\pm 0.013$ on
$C$. Further systematic uncertainties on $S$ are related to the assumption
$\mathrm{\Delta}\Gamma=0$ ($\pm 0.014$), and to the modeling of the dependence
of the efficiency on decay time ($\pm 0.007$). For $C$ the second largest
systematic uncertainty of $\pm 0.007$ is due to neglecting the correlation
between the invariant mass and the decay time. Additional systematic
uncertainties arise from the decay time resolution, the uncertainty on the
knowledge of the length scale, the parametrization of the mass model, and from
uncertainties on the $B^{0}$ production asymmetry and mass difference
$\mathrm{\Delta}m$. The total systematic uncertainty, calculated as the sum in
quadrature of all contributions, is $\pm 0.05$ for $S$ and $\pm 0.02$ for $C$,
with a correlation coefficient of $\rho=$-0.69$$.

In conclusion, a measurement of the $C\\!P$ observables $S$ and $C$ in the
decay channel $B^{0}\\!\rightarrow D^{+}D^{-}$ is performed. Using the full
data sample collected by the LHCb experiment during Run 1, which corresponds
to a total integrated luminosity of $3\text{\,}{\mathrm{fb}}^{-1}$, they are
determined to be

$\displaystyle S$ $\displaystyle=-0.54\,^{+0.17}_{-0.16}\,\text{(stat)}\pm
0.05\,\text{(syst)}\,,$ $\displaystyle C$
$\displaystyle=\phantom{-}0.26\,^{+0.18}_{-0.17}\,\text{(stat)}\pm
0.02\,\text{(syst)}\,,$

with a statistical correlation coefficient of $\rho=$0.48$$. This result
excludes the conservation of $C\\!P$ symmetry by $4.0$ standard deviations. It
is compatible with the previous measurement by the BaBar experiment of
$S=$-0.63\pm 0.36\pm 0.05$$ and $C=$-0.07\pm 0.23\pm 0.03$$ [12] while being
significantly more precise. A proper evaluation of the compatibility with the
result from the Belle experiment [13] could not be performed due to its non-
Gaussian uncertainties. The result presented here corresponds to
$\sin(\phi_{d}+\Delta\phi)=0.56\,^{+0.16}_{-0.17}$, which constrains the phase
shift to the world’s most precise value of
$\Delta\phi=-0.16\,^{+0.19}_{-0.21}$ $\mathrm{rad}$, and thus implies only a
small contribution from higher-order Standard Model corrections.

## 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).

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## Supplemental material

### Tagging calibration

For the flavor tagging calibration, a fit to the background-subtracted decay
time distribution of $B^{0}\\!\rightarrow D^{+}_{s}D^{-}$ candidates is used.
In the first step, the mass fit, the $B^{0}\\!\rightarrow D^{+}_{s}D^{-}$
signal is modeled as two Crystal Ball functions that share a common mean, but
have different widths and tail parameters that are obtained from simulations.
The $B^{0}_{s}\\!\rightarrow D^{-}_{s}D^{+}$ background component is modeled
similarly and shares all shape parameters with $B^{0}\\!\rightarrow
D^{+}_{s}D^{-}$ except for the peak position, which is constrained to be
$87.35\text{\,}\mathrm{M}\mathrm{eV}\\!\mathrm{/}\mathrm{{}^{2}}\mathrm{\mathit{c}}$
larger. The combinatorial background component is modeled as an exponential
function. The $B^{0}\\!\rightarrow D^{+}_{s}D^{-}$ yield is found to be
$16\,736\,\pm 134$. The invariant mass distribution of $B^{0}\\!\rightarrow
D^{+}_{s}D^{-}$ candidates is shown in Fig. 3 with the PDF projection
overlaid.

Figure 3: Masses of $B^{0}\\!\rightarrow D^{+}_{s}D^{-}$ candidates and
projected PDFs, shown with a linear scale on the vertical axis (a) and a
logarithmic scale (b). The solid line is the PDF projection, the blue dashed
line represents the $B^{0}\\!\rightarrow D^{+}_{s}D^{-}$ component, while the
dash-dotted cyan (dotted green) line represents the combinatorial
($B^{0}_{s}\\!\rightarrow D^{-}_{s}D^{+}$) background.

The decay time fit to the $B^{0}\\!\rightarrow D^{+}_{s}D^{-}$ candidates uses
a modified version of the PDF from the $B^{0}\\!\rightarrow D^{+}D^{-}$ fit
where $S$ and $C$ are fixed to zero and unity, respectively, and the
production flavor variables $d^{\prime}$ and $d$ are replaced by the mixing
state ($+1$ if the production flavor and the reconstructed decay flavor are
the same, and $-1$ otherwise). Additional modifications are implemented to
treat the production asymmetry correctly after replacing $d^{\prime}$ by the
mixing state, to allow for a flavor-specific asymmetry $a_{\mathrm{sl}}^{d}$
(fixed to the latest LHCb measurement [32]) and to include an asymmetry in the
detection efficiency for $D^{+}_{s}D^{-}$ and $D^{-}_{s}D^{+}$ (only for the
evaluation of systematic uncertainties). The $B^{0}$ oscillation frequency,
$\mathrm{\Delta}m$, and the mean lifetime, $\tau$, are fixed. The associated
systematic uncertainties are taken to be the changes in the calibration
parameters when the quantities that were fixed are varied within their
uncertainties.

The calibration function for initial $B^{0}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ mesons is given by

$\omega(\eta|d)=p_{0}+d\frac{\Delta p_{0}}{2}+\left(p_{1}+d\frac{\Delta
p_{1}}{2}\right)(\eta-\langle\eta\rangle)\,.$ (4)

Here, $d$ is $+1$ for mesons whose initial flavor is $B^{0}$ and $-1$ for
${\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}}$. The predicted, per-
candidate mistag rate is $\eta$ and $\omega$ the calibrated, per-candidate
mistag rate. The value of $\eta$ when averaged over all tagged candidates is
$\langle\eta\rangle$. The calibration parameters are $p_{0}$, $\Delta p_{0}$,
$p_{1}$, and $\Delta p_{1}$, and are to be understood as follows. Averaging
the calibrated mistag rate over all candidates, $p_{0}$ is the value obtained
when further averaging over $B^{0}$ and ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}$, and $\Delta p_{0}$ is the
difference between the average mistag rates for $B^{0}$ and ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}$. The coefficients $p_{1}$ and
$\Delta p_{1}$ describe the linear relationship between $\eta$ and $\omega$;
again, $p_{1}$ is averaged over $B^{0}$ and ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}$ and $\Delta p_{1}$ is the
difference between $B^{0}$ and ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}$. For perfectly calibrated
taggers the following two relations hold

$\displaystyle p_{0}$ $\displaystyle=\langle\eta\rangle$ (5) $\displaystyle
p_{1}$ $\displaystyle=1\,,$

and the tagging asymmetries vanish: $\Delta p_{1}=\Delta p_{0}=0$. The OS
calibration parameters are determined to be

$\displaystyle p_{1,\mathrm{OS}}$ $\displaystyle=1.07\pm
0.07\,(\mathrm{stat})\pm 0.01\,(\mathrm{syst})\,,$ (6) $\displaystyle
p_{0,\mathrm{OS}}$ $\displaystyle=0.369\pm 0.008\,(\mathrm{stat})\pm
0.010\,(\mathrm{syst})\,,$ $\displaystyle\langle\eta_{\mathrm{OS}}\rangle$
$\displaystyle=0.3627\,,$ $\displaystyle\Delta p_{1,\mathrm{OS}}$
$\displaystyle=0.03\pm 0.11\,(\mathrm{stat})\pm 0.03\,(\mathrm{syst})\,,$
$\displaystyle\Delta p_{0,\mathrm{OS}}$ $\displaystyle=-0.009\pm
0.012\,(\mathrm{stat})\pm 0.001\,(\mathrm{syst})\,.$

The SS calibration parameters are determined to be

$\displaystyle p_{1,\mathrm{SS}}$ $\displaystyle=0.84\pm
0.09\,(\mathrm{stat})\pm 0.01\,(\mathrm{syst})\,,$ (7) $\displaystyle
p_{0,\mathrm{SS}}$ $\displaystyle=0.430\pm 0.006\,(\mathrm{stat})\pm
0.009\,(\mathrm{syst})\,,$ $\displaystyle\langle\eta_{\mathrm{SS}}\rangle$
$\displaystyle=0.4282\,,$ $\displaystyle\Delta p_{1,\mathrm{SS}}$
$\displaystyle=0.07\pm 0.13\,(\mathrm{stat})\pm 0.05\,(\mathrm{syst})\,,$
$\displaystyle\Delta p_{0,\mathrm{SS}}$ $\displaystyle=-0.007\pm
0.009\,(\mathrm{stat})\pm 0.001\,(\mathrm{syst})\,.$

The time-dependent raw mixing asymmetry
$(N_{\text{unmixed}}-N_{\text{mixed}})/(N_{\text{unmixed}}+N_{\text{mixed}})$,
where $N_{\text{unmixed}}$ is the number of $B^{0}\\!\rightarrow
D^{+}_{s}D^{-}$ decays with a final state that does correspond to the flavor
tag, and $N_{\text{mixed}}$ the number with a final state that does not, as
measured using OS or SS taggers, is shown in Fig. 4. In the
$B^{0}\\!\rightarrow D^{+}D^{-}$ dataset, the tagging power for events which
are tagged only by OS taggers is $1.02\,\pm 0.09\text{\,}\mathrm{\char
37\relax}$, and for events tagged only by SS taggers $1.36\,\pm
0.19\text{\,}\mathrm{\char 37\relax}$; for events tagged by both OS and SS
taggers, the combined tagging power is $5.7\,\pm 0.5\text{\,}\mathrm{\char
37\relax}$. These sum to a tagging power of $8.1\,\pm
0.5\text{\,}\mathrm{\char 37\relax}$ for events tagged by OS, SS, or OS and SS
taggers.

Figure 4: Raw mixing asymmetry as a function of the $B^{0}$ decay time for
events tagged by (a) the OS tagger and (b) the SS tagger. The solid line
represents the PDF projection.

LHCb collaboration

R. Aaij40, B. Adeva39, M. Adinolfi48, Z. Ajaltouni5, S. Akar6, J. Albrecht10,
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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, G. Raven44, F. Redi55,
S. Reichert10, A.C. dos Reis1, C. Remon Alepuz68, V. Renaudin7, S.
Ricciardi51, S. Richards48, M. Rihl40, K. Rinnert54, V. Rives Molina38, P.
Robbe7,40, A.B. Rodrigues1, E. Rodrigues59, J.A. Rodriguez Lopez65, 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 Valls68, J.J. Saborido Silva39, E. Sadykhov32, N.
Sagidova31, B. Saitta16,f, V. Salustino Guimaraes2, C. Sanchez Mayordomo68, 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, V. Shevchenko67, A. Shires10, 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.
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Yushchenko37, K.A. Zarebski47, M. Zavertyaev11,c, L. Zhang3, Y. Zhang7, Y.
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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) and Novosibirsk State
University, 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
41Ecole 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
64Institute of Particle Physics, Central China Normal University, Wuhan,
Hubei, China, associated to 3
65Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia,
associated to 8
66Institut für Physik, Universität Rostock, Rostock, Germany, associated to 12
67National Research Centre Kurchatov Institute, Moscow, Russia, associated to
32
68Instituto de Fisica Corpuscular (IFIC), Universitat de Valencia-CSIC,
Valencia, Spain, associated to 38
69Van 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