EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2016-325 LHCb-PAPER-2016-054 19 January 2017

Measurement of the ${{B}^{\pm}}$ production asymmetry and the $C\\!P$
asymmetry in ${{B}^{\pm}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{\pm}}$ decays

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

The ${B}^{\pm}$ meson production asymmetry in $pp$ collisions is measured
using ${{{B}^{+}}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{+}}$ decays. The data
were recorded by the LHCb experiment during Run 1 of the LHC at centre-of-mass
energies of $\sqrt{s}=$ 7 and 8$\mathrm{\,Te\kern-1.00006ptV}$. The production
asymmetries, integrated over transverse momenta in the range
$2<\mbox{$p_{\mathrm{T}}$}<30$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and
rapidities in the range $2.1<y<4.5$, are measured to be

$\displaystyle\mathcal{A}_{\rm
prod}({{{B}^{+}}},\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV})$
$\displaystyle=(-0.41\pm 0.49\pm 0.10)\times 10^{-2},$
$\displaystyle\mathcal{A}_{\rm
prod}({{{B}^{+}}},\sqrt{s}=8\mathrm{\,Te\kern-1.00006ptV})$
$\displaystyle=(-0.53\pm 0.31\pm 0.10)\times 10^{-2},$

where the first uncertainties are statistical and the second are systematic.
These production asymmetries are used to correct the raw asymmetries of
${{{B}^{+}}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}$ decays, thus allowing a measurement of the $C\\!P$
asymmetry,

$\mathcal{A}_{{C\\!P}}({{{B}^{+}}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}})=\left(0.09\pm 0.27\pm 0.07\right)\times 10^{-2}.$

Submitted to Phys. Rev. D

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

##  1 Introduction

One of the primary goals of the LHCb experiment is to search for effects of
physics beyond the Standard Model through measurements of $C\\!P$-violating
asymmetries in beauty- and charm-hadron decays. A challenge for such
measurements in $pp$ collisions is that the heavy flavour production rates
differ between particles and antiparticles. These production asymmetries
cannot be precisely predicted since they arise in the non perturbative $b$ or
$c$ quark hadronisation process [1, 2, 3]. The effects of production
asymmetries cancel in measurements of the difference between $C\\!P$
asymmetries of two different decays of the same hadron species.

The $C\\!P$ asymmetries of ${B}^{+}$ meson decay rates111The inclusion of
charge-conjugate processes is implied throughout, except in the discussion of
asymmetries. are often measured relative to that of the decay
${{{B}^{+}}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}$. The leading tree-level diagram for this decay, shown in
Fig. 1 (left), is colour-suppressed and the total decay amplitude may receive
a sizeable contribution from the gluonic loop diagram shown in Fig. 1 (right).
Therefore, the ${{{B}^{+}}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}$ decay can in principle exhibit a $C\\!P$ asymmetry due to
the interference between these amplitudes. The current world average value of
the $C\\!P$ asymmetry is
$\mathcal{A}_{{C\\!P}}({{{B}^{+}}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}})=(0.3\pm 0.6)\%$ [4] and the uncertainty represents a
limitation in many ${{B}^{+}}$ meson ${C\\!P}$ asymmetry measurements that use
this channel as a reference.

Figure 1: Tree and loop (penguin) diagrams for the
${{{B}^{+}}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}$ decay.

This analysis exploits the decay ${{{B}^{+}}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{+}}$, which is dominated
by a Cabibbo- and colour-favoured tree-level amplitude and is therefore
expected to have a $C\\!P$ asymmetry with a smaller value and uncertainty than
for the ${{{B}^{+}}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}$ mode. The ${{{B}^{+}}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{+}}$ decay mode is used
to measure the production asymmetry between the cross-sections for
${{{B}^{-}}}$ and ${{{B}^{+}}}$ mesons, defined as

$\mathcal{A}_{\rm
prod}({{{B}^{+}}})\equiv\frac{\sigma({{{B}^{-}}})-\sigma({{{B}^{+}}})}{\sigma({{{B}^{-}}})+\sigma({{{B}^{+}}})}.$
(1)

Since the production asymmetry is expected to be a function of the kinematics,
the measurement is performed in nine bins of ${{B}^{+}}$ transverse momentum,
$p_{\mathrm{T}}$, and rapidity, $y$, within the fiducial region
$2<\mbox{$p_{\mathrm{T}}$}<30$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and
$2.1<y<4.5$. Measurements are performed on two data sets corresponding to
integrated luminosities of 1$\mbox{\,fb}^{-1}$ and 2$\mbox{\,fb}^{-1}$,
recorded at centre-of-mass energies of 7 and 8 TeV in 2011 and 2012,
respectively. These measurements complement the existing LHCb studies of heavy
flavour production asymmetries [5, 6, 7, 8]. A combined analysis of
${{{B}^{+}}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{+}}$ and
${{{B}^{+}}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}$ decays allows a measurement of the $C\\!P$ asymmetry in the
latter mode. The raw charge asymmetry for a flavour-specific decay to the
final state $f$ ($\bar{f}$) accessible in decays of $B^{-}$ ($B^{+}$) mesons
is defined as

$\displaystyle\mathcal{A}_{\rm raw}({{{B}^{+}}}\rightarrow\bar{f})$
$\displaystyle\equiv\frac{N({{{B}^{-}}}\rightarrow
f)-N({{{B}^{+}}}\rightarrow\bar{f})}{N({{{B}^{-}}}\rightarrow
f)+N({{{B}^{+}}}\rightarrow\bar{f})}.$ (2)

For the two decay modes under study, the asymmetries are well approximated by

$\displaystyle\mathcal{A}_{\rm raw}({{{B}^{+}}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{+}})$
$\displaystyle=\mathcal{A}_{\rm prod}({{{B}^{+}}})+\mathcal{A}_{\rm
det}({{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{+}})+\mathcal{A}_{{C\\!P}}({{{B}^{+}}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{+}}),$ (3)
$\displaystyle\mathcal{A}_{\rm
raw}({{{B}^{+}}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}})$ $\displaystyle=\mathcal{A}_{\rm
prod}({{{B}^{+}}})+\mathcal{A}_{\rm
det}({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}})+\mathcal{A}_{{C\\!P}}({{{B}^{+}}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}),$

where $\mathcal{A}_{\rm det}$ is the detector-induced asymmetry resulting from
differences in the detection efficiencies between particles and antiparticles.
All contributions to $\mathcal{A}_{\rm det}$ are measured on independent
control samples from the same data set. The high correlation of
$\mathcal{A}_{\rm det}$ between the two decay modes implies a partial
cancellation in their difference. This cancellation and the low level of
$C\\!P$ violation in the ${{{B}^{+}}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{+}}$ decay mode enable a
precise measurement of
$\mathcal{A}_{{C\\!P}}({{{B}^{+}}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}})$.

##  2 The LHCb detector

The LHCb detector [9, 10] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high-precision tracking
system consisting of a silicon-strip vertex detector surrounding the $pp$
interaction region, a large-area silicon-strip detector located upstream of a
dipole magnet with a bending power of about $4{\mathrm{\,Tm}}$, and three
stations of silicon-strip detectors and straw drift tubes placed downstream of
the magnet. Data samples corresponding to roughly equal integrated
luminosities were recorded with configurations in which the magnetic field was
pointing vertically upwards and downwards. This largely cancels any charge
asymmetries in the reconstruction efficiency for charged particles. 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.
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 [11], which consists of a hardware stage, based on information from
the calorimeter and muon systems, followed by a two-stage software trigger,
which applies a full event reconstruction. This analysis makes use of
inclusive dimuon and beauty selections at the software trigger stages.

##  3 Selection of ${{{B}^{+}}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{+}}$ decays

The selection of signal candidate ${{{B}^{+}}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{+}}$ decays closely
follows a recent LHCb analysis involving the same decay channel [12]. Events
are considered for the analysis if they contain a track with large enough
$p_{\mathrm{T}}$ and IP to satisfy the requirements of the first stage of the
software trigger. An inclusive beauty selection is applied at the second stage
of the software trigger. Candidate ${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}\rightarrow{{K}^{+}}{{\pi}^{-}}({{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}\rightarrow{{K}^{+}}{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{-}})$
decays are constructed from the intersection of two (four) tracks that satisfy
appropriate kaon or pion particle identification (PID) criteria, and that have
a large $p_{\mathrm{T}}$ and significant IP with respect to all primary
vertices. These candidates must have a mass within $\pm
25$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the ${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}$ mass [4]. Each ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ candidate is combined with a
high $p_{\mathrm{T}}$ track that is identified as a pion to create a displaced
vertex that is consistent with a decay of a ${{{B}^{+}}}$ meson. The
${{{B}^{+}}}$ candidates are required to have a mass within the range
$5079$–$5899$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. To reduce to a
negligible level the uncertainty related to L0 trigger asymmetries, it is
explicitly required that a positive L0 trigger decision was caused by a
particle that is distinct from any of the final-state particles that compose
the signal candidate. This requirement is independent of whether or not the
signal candidate itself also caused a positive L0 trigger decision and is
therefore referred to as triggering independently of signal (TIS) [11].

For both the two- and four-body ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$-mode selections, a pair of
boosted decision tree (BDT) discriminators [13], implementing the gradient
boost algorithm [14], is used to achieve further background suppression. The
first of these BDTs is trained to reject candidates with fake ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ decays, and the second to
reject backgrounds with real ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ decays. The BDTs are trained
using simulated ${{{B}^{+}}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{+}}$ signal decays and a
sample of decays from data with masses in the range
$5900$–$7200$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ to model the
combinatorial background in the nominal mass range. For the training of the
first BDT, a background sample is provided by candidates with ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ masses that differ by more than
$\pm 30$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ from the known ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ mass. The second BDT is trained
using a background sample of candidates with ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ masses within $\pm
25$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the known ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ mass. A loose cut on the
classifier response of the first BDT is applied prior to training the second
one. The inputs to the BDTs include properties of each particle ($p$,
$p_{\mathrm{T}}$, and the IP significance) and additional properties of the
$B$ and ${D}^{0}$ composite particles (decay time, flight distance, decay
vertex quality, radial distance between the decay vertex and the PV, and the
angle between the reconstructed momentum vector and the line connecting the
production and decay vertex). A further input to the BDTs is an isolation
variable

$I_{\mbox{$p_{\mathrm{T}}$}}=\frac{\mbox{$p_{\mathrm{T}}$}({{B}^{\pm}})-\Sigma\mbox{$p_{\mathrm{T}}$}}{\mbox{$p_{\mathrm{T}}$}({{B}^{\pm}})+\Sigma\mbox{$p_{\mathrm{T}}$}},$
(4)

for which the sum is taken over tracks that are not part of the signal
candidate but fall within a cone of half-angle $\Delta R<1.5$ radians, where
$(\Delta R)^{2}=(\Delta\theta)^{2}+(\Delta\phi)^{2}$, and $\Delta\theta$ and
$\Delta\phi$ are the differences in polar and azimuthal angle of each track
with respect to the ${{B}^{+}}$ candidate direction. Tracks are only
considered in the isolation cone if they are associated, by smallest IP, to
the same primary vertex as the signal candidate. Signal decays are expected to
have larger values of $I_{\mbox{$p_{\mathrm{T}}$}}$ than background.

The cut on the second BDT response is optimised by minimising the expected
uncertainty on the asymmetry between the yields of
${{{B}^{-}}}\rightarrow{{D}^{0}}{{\pi}^{-}}$ and
${{{B}^{+}}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{+}}$. No PID information
is used in the BDT training, but the purity of the sample is further improved
by requiring all kaon and pion candidates to satisfy PID criteria. Events
containing more than one ${{{B}^{+}}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{+}}$ candidate amount to
less than 1 %, and in these cases the candidate with the highest quality
${{B}^{+}}$ decay vertex is selected.

The raw asymmetries between the yields of
${{{B}^{-}}}\rightarrow{{D}^{0}}{{\pi}^{-}}$ and
${{{B}^{+}}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{+}}$ decays are
determined by binned maximum likelihood fits to the mass distributions of
selected ${{B}^{-}}$ and ${{B}^{+}}$ candidates, treating the two- and four-
body ${\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ modes separately.
The fit function is built from a signal component and three background
components. A sum of two Gaussian functions with asymmetric power-law tails
and an additional Gaussian function are combined to model
${{{B}^{+}}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{+}}$ decays [12].
Misidentified ${{{B}^{+}}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{K}^{+}}$ decays have a
distribution that is below the signal peak with a tail that extends to lower
masses. They are modelled by the sum of two Gaussian functions with asymmetric
power-law tail components. Partially reconstructed decays with an additional
particle from a ${D}^{*}$ or $\rho$ meson decay form a background at masses
lower than that of the signal peak. This component is described by a
combination of analytical functions with shapes that depend on the spin-parity
of the missing particle, following the method described in Ref. [12]. A linear
function is adequate to describe the combinatorial background distribution.
The yield of misidentified ${{{B}^{+}}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{K}^{+}}$ decays is constrained
with an independent control sample of these decays, combined with the
calibrated particle identification efficiencies and misidentification rates
[15]. With the exception of the tail parameters, which are fixed to values
obtained from simulation, all parameters are allowed to vary in the fit.

Figure 2 shows the fits to the mass distributions in the bin with
$4.5<\mbox{$p_{\mathrm{T}}$}<9.5$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and
$2.10<y<2.85$. The subsequent analysis is based on separate fits for the nine
kinematic bins and two centre-of-mass energies. The signal yields for each of
the nine kinematic bins are listed in Table 1. The $p_{\mathrm{T}}$ and $y$
intervals of each bin are defined in the second and third columns. The yields
sum over ${B}^{\pm}$ meson charges and centre-of-mass energies. Integrated
over the fiducial acceptance,
$2<\mbox{$p_{\mathrm{T}}$}<30$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and
$2.1<y<4.5$, the fits return signal yields of around $2.3\times 10^{5}$ decays
for the ${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}\rightarrow{{K}^{+}}{{\pi}^{-}}$
mode and around $1.3\times 10^{5}$ decays for the ${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}\rightarrow{{K}^{+}}{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{-}}$
mode.

##  4 Selection of
${{{B}^{+}}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}$ decays

The selection of ${{{B}^{+}}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}$ decays with ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\rightarrow{\mu^{+}}{\mu^{-}}$ is based on events in which a muon or a
generic track, with large $p_{\mathrm{T}}$ and IP, satisfies the requirements
of the first-stage software trigger. Events must be selected based on a dimuon
signature by the second-level software trigger. Candidate
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\rightarrow{\mu^{+}}{\mu^{-}}$
decays are reconstructed from high-$p_{\mathrm{T}}$ muon candidates with large
IP with respect to all PVs. A mass interval of $3057$–$3127$
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ is imposed on the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidates. These candidates
are combined with a high-$p_{\mathrm{T}}$ identified kaon with a significant
IP with respect to all PVs, where the $J/\psi$ candidate invariant mass is
constrained to its known value in the combination. The L0 trigger TIS
requirement is applied in the same way as for the ${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{+}}$ selection. A single
BDT classifier is used to improve the purity of the
${{{B}^{+}}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}$ sample. This classifier is trained on a similar set of
variables as that for the ${{{B}^{+}}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{+}}$ selection, and
exhibits very similar performance in terms of signal efficiency and background
rejection. Events containing more than one
${{{B}^{+}}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}$ candidate amount to less than 1 %, and in these cases the
candidate with the highest quality ${{B}^{+}}$ decay vertex is selected.

A simultaneous fit of the mass distributions across the kinematic bins is
performed, where the same value of
$\mathcal{A}_{{C\\!P}}({{{B}^{+}}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}})$ is assumed for all bins. The signal peak is described using
a Gaussian function with an additional asymmetric power-law tail component.
The mean of the Gaussian is constrained to be the same in all kinematic bins,
while its width and the tail parameters are allowed to vary between bins. A
small background from misidentified
${{B}^{+}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}}$ decays is described by a similar function, with fixed
shape parameters taken from simulation. The yield of this contribution is
allowed to vary in each kinematic bin, but a single raw asymmetry is shared
between all bins. The contribution from random particle combinations is
described by a linear function. The yield of this component and the slope
parameter are allowed to vary in each kinematic bin. The yield is also fitted
separately for each ${B}^{\pm}$ charge.

Integrated over the full fiducial acceptance, a signal yield of about
$2.3\times 10^{5}$ events is measured. Table 1 lists the yields of each signal
decay mode in each of the kinematic bins summing over the two centre-of-mass
energies. An example of the fit in the bin with
$4.5<\mbox{$p_{\mathrm{T}}$}<9.5$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and
$2.10<y<2.85$ is displayed in Fig. 3.

Figure 2: Mass distributions of selected (top) ${{B}^{\pm}}\rightarrow[{{K}^{\pm}}{{\pi}^{\mp}}]_{D}\pi^{\pm}$ and (bottom) ${{B}^{\pm}}\rightarrow[{{K}^{\pm}}{{\pi}^{\pm}}{{\pi}^{\mp}}{{\pi}^{\mp}}]_{D}\pi^{\pm}$ candidates in the bin with $4.5<\mbox{$p_{\mathrm{T}}$}<9.5$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $2.10<y<2.85$. These distributions sum over the two centre-of-mass energies. $B^{-}$ candidates are displayed on the left, and $B^{+}$ candidates on the right. The red dotted lines indicate the contribution from ${{B}^{\pm}}\rightarrow D\pi^{\pm}$ decays. The purple dash-dotted lines indicate the contribution from misidentified ${{B}^{\pm}}\rightarrow DK^{\pm}$ decays. The grey shaded regions at low values of reconstructed mass indicate the contribution from various partially reconstructed $B$ decays, and the green dashed lines indicate the combinatorial background. The total fit function is shown by the blue solid lines. The fit in other kinematic bins is similar, aside from the specific signal and background component yields.  Figure 3: Mass distribution of selected $B^{\pm}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}K^{\pm}$ candidates in the bin with $4.5<\mbox{$p_{\mathrm{T}}$}<9.5$ ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $2.10<y<2.85$. These distributions sum over the two centre-of-mass energies. $B^{-}$ candidates are displayed on the left, and $B^{+}$ candidates on the right. The signal components are displayed as red dotted lines while the background from combinatorial events is shown by the green dashed lines. The fit in other kinematic bins is similar, aside from the specific signal and background component yields.  Table 1: The $p_{\mathrm{T}}$ and $y$ intervals for each kinematic bin, and the corresponding signal yields in each of the $B^{+}$ decay modes, summing over the two centre-of-mass energies. Bin | $p_{\mathrm{T}}$ | $y$ | $B^{+}\rightarrow{{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}\pi^{+}$ | $B^{+}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}K^{+}$
---|---|---|---|---
| [${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ ] |  | ${{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}\rightarrow{{K}^{+}}{{\pi}^{-}}$ | ${{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}\rightarrow{{K}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ | ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\rightarrow\mu^{+}\mu^{-}$
1 | 2.0– | 4.5 | 2.10– | 2.85 | 13604 $\pm$ | 118 | 1549 $\pm$ | 42 | 17319 $\pm$ | 194
2 | 2.0– | 4.5 | 2.85– | 3.3 | 18587 $\pm$ | 145 | 4022 $\pm$ | 66 | 26038 $\pm$ | 229
3 | 2.0– | 4.5 | 3.3– | 4.5 | 19946 $\pm$ | 151 | 6347 $\pm$ | 87 | 31110 $\pm$ | 260
4 | 4.5– | 9.5 | 2.10– | 2.85 | 44470 $\pm$ | 219 | 14209 $\pm$ | 131 | 34939 $\pm$ | 231
5 | 4.5– | 9.5 | 2.85– | 3.3 | 47597 $\pm$ | 240 | 23895 $\pm$ | 163 | 36682 $\pm$ | 230
6 | 4.5– | 9.5 | 3.3– | 4.5 | 31137 $\pm$ | 200 | 24014 $\pm$ | 170 | 31345 $\pm$ | 212
7 | 9.5– | 30 | 2.10– | 2.85 | 33516 $\pm$ | 195 | 23378 $\pm$ | 167 | 25174 $\pm$ | 189
8 | 9.5– | 30 | 2.85– | 3.3 | 20176 $\pm$ | 159 | 20332 $\pm$ | 151 | 15110 $\pm$ | 136
9 | 9.5– | 30 | 3.3– | 4.5 | 4767 $\pm$ | 73 | 8832 $\pm$ | 97 | 8602 $\pm$ | 191
Integrated | 233390 $\pm$ | 537 | 126350 $\pm$ | 393 | 226319 $\pm$ | 632

##  5 Measurement of the ${B}^{+}$ production asymmetry

Table 2:  A summary of the terms that enter the production asymmetry
determination (Eq. 5) in the 7 TeV data set. The $p_{\mathrm{T}}$ and $y$
intervals of each bin are provided in Tab. 1. The L0 trigger asymmetry
$\mathcal{A}_{\rm det}^{\rm TIS}$ is omitted from this table since it is
assumed to be independent of the ${{B}^{+}}$ kinematics. All uncertainties are
statistical.

Bin |  $\mathcal{A}_{\rm raw}^{D\pi}$ ($\times 10^{-2}$) |  $\mathcal{A}_{{C\\!P}}^{D\pi}$ ($\times 10^{-2}$) |  $\mathcal{A}_{\rm det}^{K\pi}$ ($\times 10^{-2}$) |  $\mathcal{A}_{\rm det}^{\pi}$ ($\times 10^{-2}$) |  $\mathcal{A}_{\rm det}^{\rm PID}$ ($\times 10^{-2}$)
---|---|---|---|---|---
1 |  $-1.1$ $\pm$ | $1.5$ |  $+0.08$ $\pm$ | $0.05$ |  $-1.39$ $\pm$ | $0.22$ |  $-0.04$ $\pm$ | $0.13$ |  $-0.066$ $\pm$ | $0.006$
2 |  $-1.5$ $\pm$ | $1.3$ |  $+0.08$ $\pm$ | $0.05$ |  $-1.18$ $\pm$ | $0.15$ |  $-0.05$ $\pm$ | $0.08$ |  $+0.017$ $\pm$ | $0.017$
3 |  $-1.7$ $\pm$ | $1.1$ |  $+0.07$ $\pm$ | $0.05$ |  $-1.19$ $\pm$ | $0.16$ |  $-0.04$ $\pm$ | $0.09$ |  $+0.077$ $\pm$ | $0.007$
4 |  $-1.1$ $\pm$ | $0.8$ |  $+0.07$ $\pm$ | $0.05$ |  $-1.23$ $\pm$ | $0.21$ |  $+0.03$ $\pm$ | $0.11$ |  $-0.0875$ $\pm$ | $0.0021$
5 |  $-1.6$ $\pm$ | $0.7$ |  $+0.07$ $\pm$ | $0.04$ |  $-1.03$ $\pm$ | $0.13$ |  $+0.03$ $\pm$ | $0.08$ |  $-0.049$ $\pm$ | $0.004$
6 |  $-1.5$ $\pm$ | $0.8$ |  $+0.06$ $\pm$ | $0.04$ |  $-1.10$ $\pm$ | $0.13$ |  $-0.02$ $\pm$ | $0.08$ |  $+0.2092$ $\pm$ | $0.0033$
7 |  $-0.7$ $\pm$ | $0.8$ |  $+0.06$ $\pm$ | $0.04$ |  $-0.84$ $\pm$ | $0.20$ |  $+0.04$ $\pm$ | $0.13$ |  $-0.0606$ $\pm$ | $0.0026$
8 |  $-2.6$ $\pm$ | $0.9$ |  $+0.05$ $\pm$ | $0.04$ |  $-0.65$ $\pm$ | $0.12$ |  $+0.05$ $\pm$ | $0.12$ |  $+0.0645$ $\pm$ | $0.0022$
9 |  $-0.2$ $\pm$ | $1.6$ |  $+0.04$ $\pm$ | $0.04$ |  $-1.07$ $\pm$ | $0.11$ |  $+0.06$ $\pm$ | $0.12$ |  $+0.3951$ $\pm$ | $0.0032$

Table 3:  A summary of the terms that enter the production asymmetry
determination (Eq. 5) in the 8 TeV data set. The L0 trigger asymmetry
$\mathcal{A}_{\rm det}^{\rm TIS}$ is omitted from this table since it is
assumed to be independent of the ${{B}^{+}}$ kinematics. All uncertainties are
statistical.

Bin |  $\mathcal{A}_{\rm raw}^{D\pi}$ ($\times 10^{-2}$) |  $\mathcal{A}_{{C\\!P}}^{D\pi}$ ($\times 10^{-2}$) |  $\mathcal{A}_{\rm det}^{K\pi}$ ($\times 10^{-2}$) |  $\mathcal{A}_{\rm det}^{\pi}$ ($\times 10^{-2}$) |  $\mathcal{A}_{\rm det}^{\rm PID}$ ($\times 10^{-2}$)
---|---|---|---|---|---
1 |  $-0.7$ $\pm$ | $1.0$ |  $+0.08$ $\pm$ | $0.05$ |  $-1.16$ $\pm$ | $0.13$ |  $-0.17$ $\pm$ | $0.09$ |  $+0.059$ $\pm$ | $0.004$
2 |  $-1.2$ $\pm$ | $0.9$ |  $+0.07$ $\pm$ | $0.05$ |  $-1.08$ $\pm$ | $0.09$ |  $-0.10$ $\pm$ | $0.06$ |  $+0.0855$ $\pm$ | $0.0029$
3 |  $-2.8$ $\pm$ | $0.8$ |  $+0.07$ $\pm$ | $0.05$ |  $-0.93$ $\pm$ | $0.10$ |  $-0.07$ $\pm$ | $0.06$ |  $+0.0659$ $\pm$ | $0.0026$
4 |  $-1.3$ $\pm$ | $0.5$ |  $+0.07$ $\pm$ | $0.05$ |  $-1.07$ $\pm$ | $0.12$ |  $-0.10$ $\pm$ | $0.07$ |  $-0.0144$ $\pm$ | $0.0008$
5 |  $-1.7$ $\pm$ | $0.4$ |  $+0.07$ $\pm$ | $0.04$ |  $-0.99$ $\pm$ | $0.08$ |  $-0.11$ $\pm$ | $0.05$ |  $+0.0963$ $\pm$ | $0.0013$
6 |  $-1.2$ $\pm$ | $0.5$ |  $+0.06$ $\pm$ | $0.04$ |  $-0.79$ $\pm$ | $0.08$ |  $-0.06$ $\pm$ | $0.06$ |  $+0.1323$ $\pm$ | $0.0024$
7 |  $-1.0$ $\pm$ | $0.5$ |  $+0.06$ $\pm$ | $0.04$ |  $-0.93$ $\pm$ | $0.11$ |  $-0.02$ $\pm$ | $0.08$ |  $+0.0120$ $\pm$ | $0.0012$
8 |  $-1.0$ $\pm$ | $0.6$ |  $+0.05$ $\pm$ | $0.04$ |  $-0.78$ $\pm$ | $0.07$ |  $-0.14$ $\pm$ | $0.08$ |  $+0.0581$ $\pm$ | $0.0029$
9 |  $-1.8$ $\pm$ | $1.0$ |  $+0.04$ $\pm$ | $0.04$ |  $-0.56$ $\pm$ | $0.07$ |  $-0.10$ $\pm$ | $0.08$ |  $+0.0914$ $\pm$ | $0.0017$

The ${{{B}^{+}}}$ production asymmetry is determined in the nine bins of
$p_{\mathrm{T}}$ and $y$ according to

$\mathcal{A}_{\rm prod}({{{B}^{+}}})=\mathcal{A}_{\rm
raw}^{D\pi}-\mathcal{A}_{{C\\!P}}^{D\pi}-\mathcal{A}_{\rm
det}^{K\pi}-\mathcal{A}_{\rm det}^{\pi}-\mathcal{A}_{\rm det}^{\rm
PID}-\mathcal{A}_{\rm det}^{\rm TIS},$ (5)

where $\mathcal{A}_{\rm raw}^{D\pi}$ and $\mathcal{A}_{{C\\!P}}^{D\pi}$ are
the raw charge asymmetry and $C\\!P$ asymmetry in the
${{{B}^{+}}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{+}}$ decay,
respectively. The four $\mathcal{A}_{\rm det}$ terms correct for detector-
induced asymmetries and will be described in the following. All terms other
than $\mathcal{A}_{{C\\!P}}^{D\pi}$ are evaluated separately for the four
disjoint data sets corresponding to the two centre-of-mass energies and the
two magnet polarities. An average of the $\mathcal{A}_{\rm raw}^{D\pi}$ values
for the two ${{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}$ decay
modes is computed with weights that are chosen to minimise the uncertainty.
The same weights are used to compute averages over the two ${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}$ decay modes for all other
terms in Eq. 5 apart from $\mathcal{A}_{\rm det}^{\rm TIS}$, which is
independent of the ${{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}$
decay. Tables 2 and 3 list the values of the first five terms in Eq. 5 for the
7 and 8$\mathrm{\,Te\kern-1.00006ptV}$ data sets, respectively. The overall
detection asymmetry has two main contributions. The first arises because
${{K}^{-}}$ mesons have a larger nuclear interaction cross-section than
${{K}^{+}}$ mesons. This means that more ${{K}^{-}}$ mesons than ${{K}^{+}}$
mesons interact inelastically with the detector material before they leave
enough hits to be reconstructed in the tracking stations. The resulting
${{K}^{-}}$–${{K}^{+}}$ detection asymmetry is around $10^{-2}$. The second
cause of asymmetry is the different trajectories of positively and negatively
charged particles, which therefore have different sensitivities to
misalignments and inhomogeneities of the detector. This source contributes to
all detection asymmetry terms. It is partially cancelled when averaging
measurements over data recorded with the dipole magnet in the two polarities.

The ${\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ detection
asymmetry, $\mathcal{A}_{\rm det}^{K\pi}$, is measured using samples of
${{D}^{-}}$ mesons that are produced in the primary $pp$ interactions and
decay to the ${{K}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ and
${{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{\pi}^{-}}$ final states. The
${{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ mesons are reconstructed in their
decay to $\pi^{+}\pi^{-}$. Within a small phase-space region in terms of the
${D}^{-}$ decay products, it is assumed that the detection asymmetry for a
$K^{+}\pi^{-}$ pair can be determined using

$\mathcal{A}_{\rm det}^{K\pi}=\mathcal{A}_{\rm
raw}({{D}^{-}}\rightarrow{{K}^{+}}{{\pi}^{-}}{{\pi}^{-}})-\mathcal{A}_{\rm
raw}({{D}^{-}}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{\pi}^{-}}),$ (6)

with a small correction for the effects of $C\\!P$ violation in
${{K}^{0}}-{{\kern 1.99997pt\overline{\kern-1.99997ptK}{}}{}^{0}}$ mixing and
the different material interactions of ${{K}^{0}}$ and ${{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{0}}$. For each of the
${{D}^{-}}\rightarrow{{K}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ candidates one of the
two ${{\pi}^{-}}$ mesons is randomly labelled as being matched to the
${{{B}^{+}}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{+}}$ signal. A weight is
assigned to each ${{D}^{-}}\rightarrow{{K}^{+}}{{\pi}^{-}}{{\pi}^{-}}$
candidate such that the kinematic distributions of the ${{K}^{+}}$ and the
matched ${{\pi}^{-}}$ agree with those from the signal ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ decays. For the ${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}\rightarrow{{K}^{+}}{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{-}}$
sample, the procedure is repeated for each of the two possible pions with
opposite charge to the kaon, averaging over the two. Each
${{D}^{-}}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{\pi}^{-}}$
candidate is assigned a weight, such that the ${{\pi}^{-}}$ kinematic
distributions agree with those of the unmatched ${{\pi}^{-}}$ in the weighted
${{D}^{-}}\rightarrow{{K}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ sample, and the
${D}^{-}$ kinematic distributions are equalised between the two ${D}^{-}$
decay modes. This ensures cancellation of the ${D}^{-}$ production asymmetry,
and means that any detection asymmetry associated with the unmatched
${{\pi}^{-}}$ is cancelled with a corresponding asymmetry affecting the
${{D}^{-}}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{\pi}^{-}}$
sample. This weighting procedure is performed for each of the nine
${{{B}^{+}}}$ kinematic bins. The raw asymmetries that enter Eq. 6 are
determined by fitting the weighted mass spectra for the four combinations of
${{D}^{\pm}}$ decay modes and charges.

Using a detailed description of the LHCb detector and cross-section
measurements from fixed target experiments [4] the nuclear interaction
contribution to the pion asymmetry is estimated to be negligibly small. The
tracking asymmetry can therefore be assumed to be the same for pions and
muons. The ${\pi}^{+}$ tracking asymmetry, $\mathcal{A}_{\rm det}^{\pi}$, is
therefore inferred from that of muons measured using a sample of
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\rightarrow{\mu^{+}}{\mu^{-}}$
decays in which one of the muons is reconstructed without requiring hits in
all tracking stations [16]. Weights are assigned to the
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ candidates such that the
kinematic distributions of this muon match those of the ${{\pi}^{-}}$ in the
${{{B}^{+}}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{+}}$ sample.

The PID requirements on the ${{{B}^{+}}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{+}}$ decays can
introduce asymmetries. Corrections are determined using a control sample of
${{D}^{*+}}\rightarrow{{D}^{0}}{{\pi}^{+}}$ decays, with
${{D}^{0}}\rightarrow{{K}^{-}}{{\pi}^{+}}$, in which no PID requirements are
imposed on the ${K}^{-}$ or ${\pi}^{+}$ from the ${D}^{0}$ decay. The
asymmetry associated with PID requirements on the ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ decays is partially accounted
for in the $\mathcal{A}_{\rm det}^{K\pi}$ correction, since PID requirements
are imposed on the final state kaons and pions in the ${D}^{-}$ control
samples. The requirements are tighter in these control samples, and so a
residual correction must still be applied. The sum of this correction, and a
corresponding correction for the PID requirement on the ${{\pi}^{+}}$ from the
${{D}^{*+}}\rightarrow{{D}^{0}}{{\pi}^{+}}$ decays, is denoted
$\mathcal{A}_{\rm det}^{\rm PID}$.

The asymmetry associated with the TIS trigger efficiency, $\mathcal{A}_{\rm
det}^{\rm TIS}$, is determined using a sample of $b$-hadron decays to the
final state ${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}\mu^{+}\nu_{\mu}X$ with ${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}\rightarrow{{K}^{+}}{{\pi}^{-}}$.
An unbiased probe of the TIS trigger efficiency is provided by the subset of
these in which the muon prompted a positive decision by the L0 muon trigger.
The corresponding asymmetries do not exhibit any kinematic dependence, and so
a single correction is determined for each centre-of-mass energy, and is
applied to all kinematic bins. The measured $\mathcal{A}_{\rm det}^{\rm TIS}$
values are $(+0.16\pm 0.16)\times 10^{-2}$ and $(+0.02\pm 0.10)\times 10^{-2}$
for the 7 TeV and 8 TeV data sets, respectively.

The $C\\!P$ asymmetry, $\mathcal{A}_{{C\\!P}}^{D\pi}$, is estimated from
measurements of the CKM angle $\gamma$ and the hadronic parameters of
${{{B}^{+}}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{+}}$ decays [17].
Different values are obtained for the ${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}\rightarrow{{K}^{+}}{{\pi}^{-}}$
and ${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}\rightarrow{{K}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$
decay modes due to the smaller coherence factor from the competing hadronic
resonances in the four-body mode. The asymmetries are

$\displaystyle\mathcal{A}_{{C\\!P}}^{D\pi}(K^{-}\pi^{+})$
$\displaystyle=\left(0.09^{+0.05}_{-0.04}\right)\times 10^{-2},$
$\displaystyle\mathcal{A}_{{C\\!P}}^{D\pi}(K^{-}\pi^{+}\pi^{-}\pi^{+})$
$\displaystyle=\left(0.00^{+0.05}_{-0.02}\right)\times 10^{-2},$

with a 55% correlation between the uncertainties on these two quantities. The
$\mathcal{A}_{{C\\!P}}^{D\pi}$ values reported in Tables 2 and 3 are averaged
over the two- and four-body modes. These values vary between the kinematic
bins due to the different weights of the two- and four-body modes.

Several sources of systematic uncertainty arise in the determination of the
production asymmetries. Their contributions are listed in Table 4. Variations
in the weighting procedures that are used to determine $\mathcal{A}_{\rm
det}^{K\pi}$ and $\mathcal{A}_{\rm det}^{\pi}$ yield uncertainties of
$0.07\times 10^{-2}$ and $0.04\times 10^{-2}$, respectively. An uncertainty of
$0.04\times 10^{-2}$ is assigned to a possible pion nuclear interaction
asymmetry that is not accounted for in the tracking efficiency measurements
with muons from ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ decays.
Finally, the $\mathcal{A}_{{C\\!P}}^{D\pi}$ uncertainties are included in the
total systematic uncertainty, which is taken to be correlated between the
kinematic bins.

Table 4: Systematic uncertainties on the $\mathcal{A}_{\rm prod}({{{B}^{+}}})$ measurement. The $\mathcal{A}_{{C\\!P}}^{D\pi}$ uncertainty varies between the kinematic bins and the range is indicated. All systematic uncertainties are considered to be correlated between kinematic bins. Source | Size ($\times 10^{-2}$)
---|---
$\mathcal{A}_{\rm det}^{K\pi}$ method | $\pm 0.07$
$\mathcal{A}_{\rm det}^{\pi}$ method | $\pm 0.04$
Pion nuclear interactions | $\pm 0.04$
$\mathcal{A}_{{C\\!P}}^{D\pi}$ | $\pm(0.04-0.05)$
Table 5: The measured $\mathcal{A}_{\rm prod}$ values for each kinematic bin
and integrated over the full kinematic acceptance,
$2<\mbox{$p_{\mathrm{T}}$}<30$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and
$2.1<y<4.5$. The integrated values sum over the asymmetries in each bin,
weighted by the values, $w$, in the second and fourth columns for the two
centre-of-mass energies. The first uncertainty is the statistical uncertainty
on $\mathcal{A}_{\rm raw}^{D\pi}$ and is uncorrelated between the kinematic
bins. The second uncertainty is the statistical uncertainty on the detection
asymmetry corrections and is taken to be correlated between the kinematic
bins. The third uncertainty is purely systematic and is assumed to be
correlated between bins.

Bin | $w(7\mathrm{\,Te\kern-1.00006ptV})$ |  $\mathcal{A}_{\rm prod}({{{B}^{+}}},7\mathrm{\,Te\kern-1.00006ptV})$ ($\times 10^{-2}$) | $w(8\mathrm{\,Te\kern-1.00006ptV})$ |  $\mathcal{A}_{\rm prod}({{{B}^{+}}},8\mathrm{\,Te\kern-1.00006ptV})$ ($\times 10^{-2}$)
---|---|---|---|---
1 | 0.182 | $+0.12\pm 1.54\pm 0.30\pm 0.10$ | 0.174 | $+0.42\pm 0.96\pm 0.19\pm 0.10$
2 | 0.092 | $-0.54\pm 1.25\pm 0.24\pm 0.10$ | 0.088 | $-0.15\pm 0.89\pm 0.14\pm 0.10$
3 | 0.156 | $-0.78\pm 1.13\pm 0.24\pm 0.10$ | 0.156 | $-1.95\pm 0.75\pm 0.16\pm 0.10$
4 | 0.208 | $-0.04\pm 0.78\pm 0.29\pm 0.10$ | 0.202 | $-0.22\pm 0.50\pm 0.17\pm 0.10$
5 | 0.094 | $-0.78\pm 0.70\pm 0.22\pm 0.10$ | 0.095 | $-0.83\pm 0.45\pm 0.14\pm 0.10$
6 | 0.144 | $-0.82\pm 0.80\pm 0.22\pm 0.10$ | 0.151 | $-0.61\pm 0.52\pm 0.14\pm 0.10$
7 | 0.064 | $-0.04\pm 0.79\pm 0.28\pm 0.10$ | 0.068 | $-0.17\pm 0.51\pm 0.17\pm 0.10$
8 | 0.028 | $-2.24\pm 0.92\pm 0.23\pm 0.10$ | 0.030 | $-0.19\pm 0.60\pm 0.15\pm 0.10$
9 | 0.032 | $+0.23\pm 1.59\pm 0.23\pm 0.10$ | 0.038 | $-1.33\pm 1.05\pm 0.14\pm 0.10$
Integrated | $-0.41\pm 0.42\pm 0.26\pm 0.10$ |  | $-0.53\pm 0.26\pm 0.16\pm 0.10$

The measured $\mathcal{A}_{\rm prod}({{{B}^{+}}})$ values for each kinematic
bin are listed in Table 5 for both centre-of-mass energies. They are shown as
a function of rapidity for the three $p_{\mathrm{T}}$ ranges in Fig. 4.
Samples of simulated ${B}^{\pm}$ decays are produced using Pythia 8 [18, 19]
with a specific LHCb configuration [20], and are used to determine the weights
that are assigned to each of the nine bins, such that the sum corresponds to
the asymmetry integrated over the full fiducial region covering
$2<\mbox{$p_{\mathrm{T}}$}<30$ ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and
$2.1<y<4.5$. These weights are listed in Table 5. The integrated asymmetries,
which are also reported in Table 5, are

$\displaystyle\mathcal{A}_{\rm
prod}({{{B}^{+}}},\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV})$
$\displaystyle=(-0.41\pm 0.49\pm 0.10)\times 10^{-2},$
$\displaystyle\mathcal{A}_{\rm
prod}({{{B}^{+}}},\sqrt{s}=8\mathrm{\,Te\kern-1.00006ptV})$
$\displaystyle=(-0.53\pm 0.31\pm 0.10)\times 10^{-2},$

where the first uncertainty is statistical and includes contributions from
$\mathcal{A}_{\rm raw}^{D\pi}$ and the detection asymmetry corrections which
are inherently statistical in nature. The second uncertainty is systematic.
Several cross-checks are performed. The measured value of $\mathcal{A}_{\rm
prod}({{{B}^{+}}})$ is found to have no statistically significant dependence
on the ${{B}^{+}}$ decay time or kaon momentum. Statistically compatible
results are obtained for the two magnet polarities.

Figure 4: The measured $\mathcal{A}_{\rm prod}({{{B}^{+}}})$ as a function of
rapidity of the $B$ meson in three bins of $p_{\mathrm{T}}$. The ranges of
$p_{\mathrm{T}}$ are indicated in the legends. The left- and right-hand
figures correspond to 7 and 8 TeV centre-of-mass energies, respectively.

##  6 Measurement of
$\mathcal{A}_{{C\\!P}}({{{B}^{+}}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}})$

The value of
$\mathcal{A}_{{C\\!P}}({{{B}^{+}}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}})$ is determined according to

$\mathcal{A}_{{C\\!P}}({{{B}^{+}}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}})=\mathcal{A}_{\rm raw}^{\psi K}-\delta\mathcal{A}_{\rm
det}^{K\pi}-\mathcal{A}_{\rm raw}^{D\pi}+\mathcal{A}_{{C\\!P}}^{D\pi},$ (7)

where $\mathcal{A}_{\rm raw}^{\psi K}$ is the raw asymmetry of
${{B}^{\pm}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{\pm}}$ decays and $\delta\mathcal{A}_{\rm det}^{K\pi}$ corrects
for the different detection asymmetries of the two decay modes. The two final
states differ by the transformation of a $\pi^{+}\pi^{-}$ pair to a
$\mu^{+}\mu^{-}$ pair, where the only significant contribution to the
difference between the overall detection asymmetries arises from the charged
kaon asymmetry. The method used to determine $\mathcal{A}_{\rm det}^{K\pi}$,
as described in the previous section, is applied to the
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}K^{+}$ final state by
considering the muon with opposite charge to the kaon as a pion. The
difference between this and the corresponding asymmetry for the
${{B}^{+}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{+}}$ mode is defined as
$\delta\mathcal{A}_{\rm det}^{K\pi}=\mathcal{A}_{\rm
det}^{K\pi}(B\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}K)-\mathcal{A}_{\rm det}^{K\pi}(B\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}\pi)$. The uncertainties are
cancelled to a large degree in this difference. Table 6 lists the values of
$\delta\mathcal{A}_{\rm det}^{K\pi}$ for each kinematic bin. The values of
$\delta\mathcal{A}_{\rm det}^{K\pi}$ are positive, since the kaons in the
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}K^{+}$ decays tend to have
higher momenta than those in the ${{B}^{+}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{+}}$ decays. A further
asymmetry could result from differences between the kinematic distributions of
the pion in the ${{B}^{+}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{+}}$ decay compared to
the $\mu^{+}$ in the ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}K^{+}$
decay, but this is estimated to be negligibly small.

Table 6: Residual differences $\delta\mathcal{A}_{\rm det}^{K\pi}$, measured in each bin of $B$ kinematics. These are the effective values after summing over centre-of-mass energies and averaging over the two ${\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ decay modes. Bin |  $\delta\mathcal{A}_{\rm det}^{K\pi}$ ($\times 10^{-2}$)
---|---
1 | 0.15 $\pm$ 0.04
2 | 0.22 $\pm$ 0.03
3 | 0.24 $\pm$ 0.05
4 | 0.26 $\pm$ 0.02
5 | 0.29 $\pm$ 0.02
6 | 0.21 $\pm$ 0.02
7 | 0.27 $\pm$ 0.02
8 | 0.23 $\pm$ 0.01
9 | 0.05 $\pm$ 0.02

The values of $\mathcal{A}_{\rm raw}^{\psi K}$ in each bin are corrected
according to Eq. 7 using measurements of $\mathcal{A}_{\rm raw}^{D\pi}$,
$\delta\mathcal{A}_{\rm det}^{K\pi}$ and $\mathcal{A}_{{C\\!P}}^{D\pi}$ in
order to extract
$\mathcal{A}_{{C\\!P}}({{{B}^{+}}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}})$. Gaussian constraints are applied to the values of
$\mathcal{A}_{\rm raw}^{D\pi}$ and $\delta\mathcal{A}_{\rm det}^{K\pi}$, such
that the statistical uncertainty on these parameters is included in the
overall statistical uncertainty for
$\mathcal{A}_{{C\\!P}}({{{B}^{+}}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}})$. A systematic uncertainty of $0.02\times 10^{-2}$ is
assigned for the use of fixed parameters in the mass fits, while a systematic
uncertainty of $0.05\times 10^{-2}$ is assigned for the method used to measure
$\delta\mathcal{A}_{\rm det}^{K\pi}$. The $\mathcal{A}_{{C\\!P}}^{D\pi}$
values contribute a systematic uncertainty of $0.04\times 10^{-2}$. The final
result is

$\displaystyle\mathcal{A}_{{C\\!P}}({{{B}^{+}}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}})=\left(0.09\pm 0.27\pm 0.07\right)\times 10^{-2},$

where the first uncertainty is statistical and the second is systematic. By
fixing all Gaussian constrained parameters to have zero uncertainty, the
contribution from the finite
${{{B}^{+}}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}$ statistics is found to be $\pm 0.20\times 10^{-2}$. This
result is consistent with, and improves upon, the current world average value
of
$\mathcal{A}_{{C\\!P}}({{{B}^{+}}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}})=(0.3\pm 0.6)\%$ [4].

##  7 Summary and conclusions

The ${B}^{+}$ meson production asymmetry is a crucial input in the measurement
of $C\\!P$ asymmetries in ${B}^{+}$ decays. A sample of
${{{B}^{+}}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{+}}$ decays is used to
measure the production asymmetry. The analysed data set corresponds to
integrated luminosities of 1 and 2$\mbox{\,fb}^{-1}$ recorded during 2011 and
2012 at proton-proton centre-of-mass energies of 7 and
8$\mathrm{\,Te\kern-1.00006ptV}$, respectively. The production asymmetries are
measured in nine bins of transverse momenta and rapidity, covering the region
$2<\mbox{$p_{\mathrm{T}}$}<30$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and
$2.1<y<4.5$, and separately for the two centre-of-mass energies. The
measurements are generally consistent with zero asymmetry within typical
uncertainties of roughly $10^{-2}$, which is in agreement with $b$-quark
hadronisation models [1, 2, 3]. Integrated over the full $p_{\mathrm{T}}$ and
$y$ ranges, the production asymmetries are measured to be

$\displaystyle\mathcal{A}_{\rm
prod}({{{B}^{+}}},\sqrt{s}=7~{}\mathrm{\,Te\kern-1.00006ptV})$
$\displaystyle=(-0.41\pm 0.49\pm 0.10)\times 10^{-2},$
$\displaystyle\mathcal{A}_{\rm
prod}({{{B}^{+}}},\sqrt{s}=8~{}\mathrm{\,Te\kern-1.00006ptV})$
$\displaystyle=(-0.53\pm 0.31\pm 0.10)\times 10^{-2},$

where the first uncertainty accounts for all statistical sources, and the
second accounts for all systematic sources. A simultaneous study of the
${{B}^{+}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}K^{+}$
decay allows a measurement of its $C\\!P$ asymmetry,

$\mathcal{A}_{{C\\!P}}({{{B}^{+}}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}})=\left(0.09\pm 0.27\pm 0.07\right)\times 10^{-2}.$

## 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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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