11institutetext: 1MBZUAI, UAE, 2Aalto University, Finland
3Weizmann Institute of Science, Israel
4University of Central Florida, USA
11email<EMAIL_ADDRESS>
# 3D Mitochondria Instance Segmentation with Spatio-Temporal Transformers

Omkar Thawakar 11 Rao Muhammad Anwer 1122 Jorma Laaksonen 22
Orly Reiner 33 Mubarak Shah 44 Fahad Shahbaz Khan 11

###### Abstract

Accurate 3D mitochondria instance segmentation in electron microscopy (EM) is
a challenging problem and serves as a prerequisite to empirically analyze
their distributions and morphology. Most existing approaches employ 3D
convolutions to obtain representative features. However, these convolution-
based approaches struggle to effectively capture long-range dependencies in
the volume mitochondria data, due to their limited local receptive field. To
address this, we propose a hybrid encoder-decoder framework based on a split
spatio-temporal attention module that efficiently computes spatial and
temporal self-attentions in parallel, which are later fused through a
deformable convolution. Further, we introduce a semantic foreground-background
adversarial loss during training that aids in delineating the region of
mitochondria instances from the background clutter. Our extensive experiments
on three benchmarks, Lucchi, MitoEM-R and MitoEM-H, reveal the benefits of the
proposed contributions achieving state-of-the-art results on all three
datasets. Our code and models are available at
https://github.com/OmkarThawakar/STT-UNET.

###### Keywords:

Mitochondria segmentation Spatio-temporal Transformer

##  1 Introduction

Mitochondria are membrane-bound organelles that generate the primary energy
required to power the cell activities, thereby crucial for metabolism.
Mitochondrial dysfunction, which occurs when mitochondria are not functioning
properly has been witnessed as a major factor in numerous diseases, including
noncommunicable chronic diseases (e.g, cardiovascular and cancer), metabolic
(e.g, obesity) and neurodegenerative (e.g, Alzheimer and Parkinson) disorders
[19, 21]. Electron microscopy (EM) images are typically utilized to reveal the
corresponding 3D geometry and size of mitochondria at a nanometer scale,
thereby facilitating basic biological research at finer scales. Therefore,
automatic instance segmentation of mitochondria is desired, since manually
segmenting from a large amount of data is particularly laborious and
demanding. However, automatic 3D mitochondria instance segmentation is a
challenging task, since complete shape of mitochondria can be sophisticated
and multiple instances can also experience entanglement with each other
resulting in unclear boundaries. Here, we look into the problem of accurate 3D
mitochondria instance segmentation.

Earlier works on mitochondria segmentation employ standard image processing
and machine learning methods [28, 16, 17]. Recent approaches address [22, 3,
12] this problem by leveraging either 2D or 3D deep convolutional neural
network (CNNs) architectures. These existing CNN-based approaches can be
roughly categorized [30] into bottom-up [12, 3, 24, 11, 2] and top-down [9].
In case of bottom-up mitochondria instance segmentation approaches, a binary
segmentation mask, an affinity map or a binary mask with boundary instances is
computed typically using a 3D U-Net [4], followed by a post-processing step to
distinguish the different instances. On the other hand, top-down methods
typically rely on techniques such as Mask R-CNN [6] for segmentation. However,
Mask R-CNN based approaches struggle due to undefined bounding-box scale in EM
data volume.

As discussed above, most recent approaches for 3D mitochondria instance
segmentation utilize convolution-based designs within the U-shaped 3D encoder-
decoder architecture. In such an architecture, the encoder aims to generate a
low-dimensional representation of the 3D data by gradually performing the
downsampling of the extracted features. On the other hand, the decoder
performs upsampling of these extracted feature representations to the input
resolution for segmentation prediction. Although such a CNN-based design has
achieved promising segmentation results compared to traditional methods, they
struggle to effectively capture long-range dependencies due to their limited
local receptive field. Inspired from success in natural language processing
[27], recently vision transformers (ViTs) [5, 15, 26, 10, 25] have been
successfully utilized in different computer vision problems due to their
capabilities at modelling long-range dependencies and enabling the model to
attend to all the elements in the input sequence. The core component in ViTs
is the self-attention mechanism that that learns the relationships between
sequence elements by performing relevance estimation of one item to other
items. Inspired by ViTs and based on the observation that attention-based
vision transformers architectures are an intuitive design choice for modelling
long-range global contextual relationships in volume data, we investigate
designing a CNN-transformers based framework for the task of 3D mitochondria
instance segmentation.

Figure 1: Qualitative 3D instance segmentation comparison between the recent
Res-UNET [13] and our proposed STT-UNET approach on the example input regions
from MitoEM-H and MitoEM-R validation sets. Here, we present the corresponding
segmentation predictions of the baseline and our approach along with the
ground truth. Our STT-UNET approach achieves superior segmentation performance
by accurately segmenting 16% more cell instances in these examples, compared
to Res-UNET-R.

When designing a attention-based framework for 3D mitochondria instance
segmentation, a straightforward way is to compute joint spatio-temporal self-
attention where all pairwise interactions are modelled between all spatio-
temporal tokens. However, such a joint spatio-temporal attention computation
is computation and memory intensive as the number of tokens increases linearly
with the number of input slices in the volume. In this work, we look into an
alternative way to compute spatio-temporal attention that captures long-range
global contextual relationships without significantly increasing the
computational complexity. Our contributions are as follows:

  * •

We propose a hybrid CNN-transformers based encoder-decoder framework, named
STT-UNET. The focus of our design is the introduction of a split spatio-
temporal attention (SST) module that captures long-range dependencies within
the cubic volume of human and rat mitochondria samples. The SST module
independently computes spatial and temporal self-attentions in parallel, which
are then later fused through a deformable convolution.

  * •

To accurately delineate the region of mitochondria instances from the
cluttered background, we further introduce a semantic foreground-background
(FG-BG) adversarial loss during the training that aids in learning improved
instance-level features.

  * •

We conduct experiments on three commonly used benchmarks: Lucchi [16],
MitoEM-R [30] and MitoEM-H [30]. Our STT-UNET achieves state-of-the-art
segmentation performance on all three datasets. On Lucchi test set, our STT-
UNET outperforms the recent [3] with an absolute gain of 3.0% in terms of
Jaccard-index coefficient. On MitoEM-H val. set, STT-UNET achieves AP-75 score
of 0.842 and outperforms the recent 3D Res-UNET [13] by 3.0%. Fig. 1 shows a
qualitative comparison between our STT-UNET and 3D Res-UNET [13] on examples
from MitoEM-R and MitoEM-H datasets.

##  2 Method

###  2.1 Baseline Framework

We base our approach on the recent Res-UNET [13], which utilizes encoder-
decoder structure of 3D UNET [29] with skip-connections between encoder and
decoder. Here, 3D input patch of mitochondria volume ($32\times 320\times
320$) is taken from the entire volume of ($400\times 4096\times 4096$). The
input volume is denoised using an interpolation network adapted for medical
images [7]. The denoised volume is then processed utilizing an encoder-decoder
structure containing residual anisotropic convolution blocks (ACB). The ACB
contains three layers of 3D convolutions with kernels ($1\times 3\times 3$),
($3\times 3\times 3$), ($3\times 3\times 3$) having skip connections between
first and third layers. The decoder outputs semantic mask and instance
boundary, which are then post-processed using connected component labelling to
generate final instance masks. We refer to [13] for more details.

Limitations: As discussed above, the recent Res-UNET approach utilizes 3D
convolutions to handle the volumetric input data. However, 3D convolutions are
designed to encode short-range spatio-temporal feature information and
struggle to model global contextual dependencies that extend beyond the
designated receptive field. In contrast, the self-attention mechanism within
the vision transformers possesses the capabilities to effectively encode both
local and global long-range dependencies by directly performing a comparison
of feature activations at all the space-time locations. In this way, self-
attention mechanism goes much beyond the receptive field of the conventional
convolutional filters. While self-attention has been shown to be beneficial
when combined with convolutional layers for different medical imaging tasks,
to the best of our knowledge, no previous attempt to design spatio-temporal
self-attention as an exclusive building block for the problem of 3D
mitochondria instance segmentation exists in literature. Next, we present our
approach that effectively utilizes an efficient spatio-temporal attention
mechanism for 3D mitochondria instance segmentation.

Figure 2: (a) Overall architecture of our STT-UNET framework for 3D
mitochondria instance segmentation. A 3D volume patch of mitochondria is first
pre-processed using the interpolation network. The resulting reconstructed
volume is then fed to our split spatio-temporal attention based encoder-
decoder to generate the semantic-level mitochondria segmentation masks. The
focus of our design is the introduction of split spatio-temporal attention
(SST) module within the encoder-decoder. (b) The SST module first computes
spatial and temporal attentions independently, which are later combined
through a deformable convolution. Consequently, the semantic masks from the
decoder are then input to the instance segmentation module to generate the
final instance masks. The entire framework is trained using the standard BCE
loss ($L_{BCE}$) and our semantic foreground-background (FG-BG) adversarial
loss ($L_{fg-bg}$). (c) The $L_{fg-bg}$ loss improves the instance-level
features, thereby aiding in the better separability of the region of
mitochondria instances from the cluttered background.

###  2.2 Spatio-Temporal Transformer Res-UNET (STT-UNET)

Fig. 2(a) presents the overall architecture of the proposed hybrid
transformers-CNN based 3D mitochondria instance segmentation approach, named
STT-UNET. It comprises a denoising module, transformer based encoder-decoder
with split spatio-temporal attention and an instance segmentation block. The
denoising module alleviates the segmentation faults caused by anomalies in the
EM images, as in the baseline. The denoising is performed by convolving the
current frame with two adjacent frames using predicted kernels, thereby
generating the resultant frame by adding the convolution outputs. The
resulting denoised output is then processed by our transformer based encoder-
decoder with split spatio-temporal attention to generate the semantic masks.
Consequently, these semantic masks are post-processed by an instance
segmentation module using a connected component labelling scheme, thereby
generating the final instance-level segmentation output prediction. To further
enhance the semantic segmentation quality with cluttered background we
introduced semantic adversarial loss which leads to improved semantic
segmentation in noisy background.

Split Spatio-Temporal Attention based Encoder-Decoder: Our STT-UNET framework
comprises four encoder and three decoder layers. Within each layer, we
introduce a split spatio-temporal attention-based (SST) module, Fig. 2(b),
that strives to capture long-range dependencies within the cubic volume of
human and rat samples. Instead of the memory expensive joint spatio-temporal
representation, our SST module splits the attention computation into a spatial
and a temporal parallel stream. The spatial attention refines the instance
level features from input features along the spatial dimensions, whereas the
temporal attention effectively learns the inter-dependencies between the input
volume. The resulting spatial and temporal attention representations are
combined through a deformable convolution, thereby generating spatio-temporal
features. As shown in Fig 2 (b), the normalized 3D input volume of denoised
features $X$ of size $(T\times H\times W\times C)$ where $T$ is volume size,
$(H\times W)$ is spatial dimension of volume and $C$ is number of channels.
The spatial and temporal attention blocks project $X$ through linear layer to
generate $Q_{s}$, $K_{s}$, $V_{s}$ and $Q_{t}$, $K_{t}$, $V_{t}$. In temporal
attention $Q_{t}$, $K_{t}$, $V_{t}$ is permuted to generate $Q_{tp}$,
$K_{tp}$, $V_{tp}$ for temporal dot product. The spatial and temporal
attention is defined as,

$X_{s}=softmax(\frac{Q_{s}K_{s}^{T}}{\sqrt{d_{k}}})V_{s}$ (1)

$X_{t}=softmax(\frac{Q_{tp}K_{tp}^{T}}{\sqrt{d_{k}}})V_{tp}$ (2)

Where, $X_{s}$ is spatial attention map, $X_{t}$ is temporal attention map and
$d_{k}$ is dimension of $Q_{s}$ and $K_{s}$. To fuse spatial and temporal
attention maps, $X_{s}$ and $X_{t}$, we employ deformable convolution. The
deformable convolution generates offsets according to temporal attention map
$X_{t}$ and by using these offsets the spatial attention map $X_{s}$ is
aligned. The deformable fusion is given as,

$X=\int_{c=1}^{C}\sum_{k_{n}\in R}W(k_{n})\cdot X_{s}(k_{0}+k_{n}+\Delta
K_{n})$ (3)

Where, $C$ is no of channels, $X$ is spatially aligned attention map with
respect to $X_{t}$. $W$ is the weight matrix of kernels, $X_{s}$ is spatial
attention map, $k_{0}$ is starting position of kernel, $k_{n}$ is enumerating
along all the positions in kernel size of $R$ and $\Delta K_{n}$ is the offset
sampled from temporal attention map $X_{t}$. We empirically observe that
fusing spatial and temporal features through a deformable convolution, instead
of concatenation through a conv. layer or addition, leads to better
performance. The resulting spatio-temporal features of decoder are then input
to instance segmentation block to generate final instance masks, as in
baseline.

Semantic FG-BG Adversarial Loss: As discussed earlier, a common challenge in
mitochondria instance segmentation is to accurately delineate the region of
mitochondria instances from the cluttered background. To address this, we
introduce a semantic foreground-background (FG-BG) adversarial loss during the
training to enhance the FG-BG separability. Here, we introduce the auxiliary
discriminator network $D$ with two layers of 3D convolutions with stride $2$
during the training as shown in Fig. 2(c). The discriminator takes the input
volume $I$ along with the corresponding mask as an input. Here, the mask $M$
is obtained either from the ground truth or predictions, such that all
mitochondria instances within a frame are marked as foreground. While the
discriminator $D$ attempts to distinguish between ground truth and predicted
masks ($M_{gt}$ and $M_{pred}$, respectively), the model $\Psi$ learns to
output semantic mask such that the predicted masks $M_{pred}$ are close to
ground truth $M_{gt}$. Let
$\mathbf{F}_{gt}=\mathrm{CONCAT}(\mathbf{I},\mathbf{M}_{gt})$ and
$\mathbf{F}_{pr}=\mathrm{CONCAT}(\mathbf{I},\mathbf{M}_{pred})$ denote the
real and fake input, respectively, to the discriminator $D$. Similar to [8],
the adversarial loss is then given by,

$\small L_{fg-bg}=\min_{\Psi}\max_{D}\Psi[\log
D(F_{gt})]+\Psi[\log(1-D(F_{pr}))]+\lambda_{1}\Psi[D(F_{gt})-D(F_{pr})]$ (4)

Consequently, the overall loss for training is: $L=L_{BCE}+\lambda\cdot L_{fg-
bg}$, Where, $L_{BCE}$ is BCE loss, $\lambda=0.5$ and $L_{fg-bg}$ is semantic
adversarial loss.

##  3 Experiments

Dataset: We evaluate our approach on three datasets: MitoEM-R [30], MitoEM-H
[30] and Lucchi [18]. The MitoEM [30] is a dense mitochondria instance
segmentation dataset from ISBI 2021 challenge. The dataset consists of 2 EM
image volumes (30 $\mu m^{3}$) of resolution of $8\times 8\times 30$ nm, from
rat tissues (MitoEM-R) and human tissue (MitoEM-H) samples, respectively. Each
volume has 1000 grayscale images of resolution $(4096\times 4096)$ of
mitochondria, out of which train set has 400, validation set contains 100 and
test set has 500 images. Lucchi [18] is a sparse mitochondria semantic
segmentation dataset with training and test volume size of $165\times
1024\times 768$.

Implementation Details: We implement our approach using Pytorch1.9 [23] (rcom
env) and models are trained using 2 AMD MI250X GPUs. During training of
MitoEM, for the fair comparison, we adopt same data augmentation technique
from [30]. The 3D patch of size $(32\times 320\times 320)$ is input to the
model and trained using batch size of 2. The model is optimized by Adam
optimizer with learning rate of $1e^{-4}$. Unlike baseline [13], we do not
follow multi-scale training and perform single stage training for 200k
iterations. For Lucchi, we follow training details of [13, 30] for semantic
segmentation. For fair comparison with previous works, we use the same
evaluation metrics as in the literature for both datasets. We use 3D AP-75
metric [30] for MitoEM-R and MitoEM-H datasets. For Lucchi, we use jaccard-
index coefficient (Jaccard) and dice similarity coefficient (DSC).

Table 1: State-of-the-art comparison in terms of AP on Mit-EM-R and MitoEM-H
validation sets. Best results are in bold.

Methods | MitoEM-R | MitoEM-H
---|---|---
Wei [30] | 0.521 | 0.605
Nightingale [20] | 0.715 | 0.625
Li [14] | 0.890 | 0.787
Chen [13] | 0.917 | 0.82
STT-UNET (Ours) | 0.958 | 0.849

Table 2: State-of-the-art comparison in terms of Jaccard and DSC on Lucchi
test set. Best results are in bold.

Methods | Jaccard | DSC
---|---|---
Yuan [31] | 0.865 | 0.927
Casser [1] | 0.890 | 0.942
Res-UNET-R [13] | 0.895 | 0.945
Res-UNET-R + MRDA [3] | 0.897 | 0.946
STT-UNET (Ours) | 0.913 | 0.962

###  3.1 Results

State-of-the-Art Comparison: Tab. 2 shows the comparison on MitoEm-R and
MitoEM-H validation sets. Our STT-UNET achieves state-of-the-art performance
on both sets. Compared to the recent [13], our STT-UNET achieves an absolute
gains of 4.1% and 2.9% on MitoEM-R and MitoEM-H sets, respectively. Note that
[13] employs two decoders for MitoEM-H. In contrast, we utilize only a single
decoder for both MitoEM-H and MitoEM-R sets, while still achieving improved
segmentation performance. Fig 3 presents the segmentation predictions of our
approach on example input regions from the validation set. Our approach
achieves promising segmentation results despite the noise in the input
samples. Tab. 2 presents the comparison on Lucchi test set. Our STT-UNET sets
a new state-of-the-art on this dataset in terms of both Jaccard and DSC.

Figure 3: Qualitative 3D instance segmentation results of our STT-UNET on the
example input regions from MitoEM-H and MitoEM-R val sets. Our STT-UNET
achieves promising results on these input examples containing noise.

Table 3:  Baseline performance comparison.

Methods | MitoEM-R | MitoEM-H
---|---|---
Baseline | 0.921 | 0.823
\+ SST | 0.948 | 0.839
\+ $L_{fg-bg}$ | 0.958 | 0.849

Table 4: Ablation study on the impact of feature fusion.

Feature Fusion | MitoEM-R | MitoEM-H
---|---|---
addition | 0.950 | 0.841
concat | 0.952 | 0.842
def-conv | 0.958 | 0.849

Table 5:  Ablation study on the impact of design choice.

Deisgn choice | MitoEM-R | MitoEM-H
---|---|---
spatial-temporal | 0.922 | 0.817
temporal-spatial | 0.937 | 0.832
spatial|temporal | 0.958 | 0.849

Ablation study: Tab. 5 shows a baseline comparison when progressively
integrating our contributions: SST module and semantic foreground-background
adversarial loss. The introduction of SST module improves performance from
0.921 to 0.941 with a gain of 2.7%. The performance is further improved by 1%,
when introducing our semantic foreground-background adversarial loss. Our
final approach achieves absolute gains of 3.7% and 2.6% over the baseline on
MitoEM-R and MitoEM-H, respectively. Tab. 5 shows ablation study with feature
fusion strategies in our SST module: addition, concat and deformable-conv. The
best results are obtained with deformable-conv on both datasets. For encoding
spatial and temporal information, we analyze two design choices with SST
module: cascaded and split, as shown in Tab. 5. The best results are obtained
using our split design choice (row 3) with spatial and temporal information
encoded in parallel and later combined. We also evaluate with different input
volumes: 4,8,16,32. We observe best results are obtained when using 32 input
volume.

##  4 Conclusion

We propose a hybrid CNN-transformers based encoder-decoder approach for 3D
mitochorndia instance segmentation. We introduce a split spatio-temporal
attention (SST) module to capture long-range dependencies within the cubic
volume of human and rat mitochondria samples. The SST module computes spatial
and temporal attention in parallel, which are later fused. Further, we
introduce a semantic adversarial loss for better delineation of mitochondria
instances from background. Experiments on three datasets demonstrate the
effectiveness of our approach, leading to state-of-the-art segmentation
performance.

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EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-PH-EP-2013-106 LHCb-PAPER-2013-021 June 26 2013

First observation of the decay
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$

The LHCb collaboration†††Authors are listed on the following pages.

The decay $B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}$ is observed for the first time using a data sample, corresponding
to an integrated luminosity of 1.0 $\mbox{\,fb}^{-1}$, collected by the LHCb
experiment in $pp$ collisions at a centre-of-mass energy of
7$\mathrm{\,Te\kern-1.00006ptV}$. A yield of $46\pm 12$ events is reported,
with a significance of 5.0 standard deviations. The ratio of the branching
fraction of $B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}$ to that of
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}$ is
measured to be $0.069\pm 0.019\pm 0.005$, where the first uncertainty is
statistical and the second is systematic.

Submitted to JHEP

© CERN on behalf of the LHCb collaboration, license CC-BY-3.0.

LHCb collaboration

R. Aaij40, C. Abellan Beteta35,n, B. Adeva36, M. Adinolfi45, C. Adrover6, A.
Affolder51, Z. Ajaltouni5, J. Albrecht9, F. Alessio37, M. Alexander50, S.
Ali40, G. Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr24,37, S. Amato2, S.
Amerio21, Y. Amhis7, L. Anderlini17,f, J. Anderson39, R. Andreassen56, R.B.
Appleby53, O. Aquines Gutierrez10, F. Archilli18, A. Artamonov34, M. Artuso58,
E. Aslanides6, G. Auriemma24,m, S. Bachmann11, J.J. Back47, C. Baesso59, V.
Balagura30, W. Baldini16, R.J. Barlow53, C. Barschel37, S. Barsuk7, W.
Barter46, Th. Bauer40, A. Bay38, J. Beddow50, F. Bedeschi22, I. Bediaga1, S.
Belogurov30, K. Belous34, I. Belyaev30, E. Ben-Haim8, G. Bencivenni18, S.
Benson49, J. Benton45, A. Berezhnoy31, R. Bernet39, M.-O. Bettler46, M. van
Beuzekom40, A. Bien11, S. Bifani44, T. Bird53, A. Bizzeti17,h, P.M.
Bjørnstad53, T. Blake37, F. Blanc38, J. Blouw11, S. Blusk58, V. Bocci24, A.
Bondar33, N. Bondar29, W. Bonivento15, S. Borghi53, A. Borgia58, T.J.V.
Bowcock51, E. Bowen39, C. Bozzi16, T. Brambach9, J. van den Brand41, J.
Bressieux38, D. Brett53, M. Britsch10, T. Britton58, N.H. Brook45, H. Brown51,
I. Burducea28, A. Bursche39, G. Busetto21,p, J. Buytaert37, S. Cadeddu15, O.
Callot7, M. Calvi20,j, M. Calvo Gomez35,n, A. Camboni35, P. Campana18,37, D.
Campora Perez37, A. Carbone14,c, G. Carboni23,k, R. Cardinale19,i, A.
Cardini15, H. Carranza-Mejia49, L. Carson52, K. Carvalho Akiba2, G. Casse51,
L. Castillo Garcia37, M. Cattaneo37, Ch. Cauet9, M. Charles54, Ph.
Charpentier37, P. Chen3,38, N. Chiapolini39, M. Chrzaszcz25, K. Ciba37, X. Cid
Vidal37, G. Ciezarek52, P.E.L. Clarke49, M. Clemencic37, H.V. Cliff46, J.
Closier37, C. Coca28, V. Coco40, J. Cogan6, E. Cogneras5, P. Collins37, A.
Comerma-Montells35, A. Contu15, A. Cook45, M. Coombes45, S. Coquereau8, G.
Corti37, B. Couturier37, G.A. Cowan49, D.C. Craik47, S. Cunliffe52, R.
Currie49, C. D’Ambrosio37, P. David8, P.N.Y. David40, A. Davis56, I. De
Bonis4, K. De Bruyn40, S. De Capua53, M. De Cian39, J.M. De Miranda1, L. De
Paula2, W. De Silva56, P. De Simone18, D. Decamp4, M. Deckenhoff9, L. Del
Buono8, N. Déléage4, D. Derkach14, O. Deschamps5, F. Dettori41, A. Di Canto11,
H. Dijkstra37, M. Dogaru28, S. Donleavy51, F. Dordei11, A. Dosil Suárez36, D.
Dossett47, A. Dovbnya42, F. Dupertuis38, R. Dzhelyadin34, A. Dziurda25, A.
Dzyuba29, S. Easo48,37, U. Egede52, V. Egorychev30, S. Eidelman33, D. van
Eijk40, S. Eisenhardt49, U. Eitschberger9, R. Ekelhof9, L. Eklund50,37, I. El
Rifai5, Ch. Elsasser39, D. Elsby44, A. Falabella14,e, C. Färber11, G.
Fardell49, C. Farinelli40, S. Farry51, V. Fave38, D. Ferguson49, V. Fernandez
Albor36, F. Ferreira Rodrigues1, M. Ferro-Luzzi37, S. Filippov32, M. Fiore16,
C. Fitzpatrick37, M. Fontana10, F. Fontanelli19,i, R. Forty37, O. Francisco2,
M. Frank37, C. Frei37, M. Frosini17,f, S. Furcas20, E. Furfaro23,k, A. Gallas
Torreira36, D. Galli14,c, M. Gandelman2, P. Gandini58, Y. Gao3, J. Garofoli58,
P. Garosi53, J. Garra Tico46, L. Garrido35, C. Gaspar37, R. Gauld54, E.
Gersabeck11, M. Gersabeck53, T. Gershon47,37, Ph. Ghez4, V. Gibson46, V.V.
Gligorov37, C. Göbel59, D. Golubkov30, A. Golutvin52,30,37, A. Gomes2, H.
Gordon54, M. Grabalosa Gándara5, R. Graciani Diaz35, L.A. Granado Cardoso37,
E. Graugés35, G. Graziani17, A. Grecu28, E. Greening54, S. Gregson46, P.
Griffith44, O. Grünberg60, B. Gui58, E. Gushchin32, Yu. Guz34,37, T. Gys37, C.
Hadjivasiliou58, G. Haefeli38, C. Haen37, S.C. Haines46, S. Hall52, T.
Hampson45, S. Hansmann-Menzemer11, N. Harnew54, S.T. Harnew45, J. Harrison53,
T. Hartmann60, J. He37, V. Heijne40, K. Hennessy51, P. Henrard5, J.A. Hernando
Morata36, E. van Herwijnen37, A. Hicheur1, E. Hicks51, D. Hill54, M.
Hoballah5, C. Hombach53, P. Hopchev4, W. Hulsbergen40, P. Hunt54, T. Huse51,
N. Hussain54, D. Hutchcroft51, D. Hynds50, V. Iakovenko43, M. Idzik26, P.
Ilten12, R. Jacobsson37, A. Jaeger11, E. Jans40, P. Jaton38, A. Jawahery57, F.
Jing3, M. John54, D. Johnson54, C.R. Jones46, C. Joram37, B. Jost37, M.
Kaballo9, S. Kandybei42, M. Karacson37, T.M. Karbach37, I.R. Kenyon44, U.
Kerzel37, T. Ketel41, A. Keune38, B. Khanji20, O. Kochebina7, I. Komarov38,
R.F. Koopman41, P. Koppenburg40, M. Korolev31, A. Kozlinskiy40, L. Kravchuk32,
K. Kreplin11, M. Kreps47, G. Krocker11, P. Krokovny33, F. Kruse9, M.
Kucharczyk20,25,j, V. Kudryavtsev33, T. Kvaratskheliya30,37, V.N. La Thi38, D.
Lacarrere37, G. Lafferty53, A. Lai15, D. Lambert49, R.W. Lambert41, E.
Lanciotti37, G. Lanfranchi18,37, C. Langenbruch37, T. Latham47, C.
Lazzeroni44, R. Le Gac6, J. van Leerdam40, J.-P. Lees4, R. Lefèvre5, A.
Leflat31, J. Lefrançois7, S. Leo22, O. Leroy6, T. Lesiak25, B. Leverington11,
Y. Li3, L. Li Gioi5, M. Liles51, R. Lindner37, C. Linn11, B. Liu3, G. Liu37,
S. Lohn37, I. Longstaff50, J.H. Lopes2, E. Lopez Asamar35, N. Lopez-March38,
H. Lu3, D. Lucchesi21,p, J. Luisier38, H. Luo49, F. Machefert7, I.V.
Machikhiliyan4,30, F. Maciuc28, O. Maev29,37, S. Malde54, G. Manca15,d, G.
Mancinelli6, U. Marconi14, R. Märki38, J. Marks11, G. Martellotti24, A.
Martens8, A. Martín Sánchez7, M. Martinelli40, D. Martinez Santos41, D.
Martins Tostes2, A. Massafferri1, R. Matev37, Z. Mathe37, C. Matteuzzi20, E.
Maurice6, A. Mazurov16,32,37,e, B. Mc Skelly51, J. McCarthy44, A. McNab53, R.
McNulty12, B. Meadows56,54, F. Meier9, M. Meissner11, M. Merk40, D.A.
Milanes8, M.-N. Minard4, J. Molina Rodriguez59, S. Monteil5, D. Moran53, P.
Morawski25, M.J. Morello22,r, R. Mountain58, I. Mous40, F. Muheim49, K.
Müller39, R. Muresan28, B. Muryn26, B. Muster38, P. Naik45, T. Nakada38, R.
Nandakumar48, I. Nasteva1, M. Needham49, N. Neufeld37, A.D. Nguyen38, T.D.
Nguyen38, C. Nguyen-Mau38,o, M. Nicol7, V. Niess5, R. Niet9, N. Nikitin31, T.
Nikodem11, A. Nomerotski54, A. Novoselov34, A. Oblakowska-Mucha26, V.
Obraztsov34, S. Oggero40, S. Ogilvy50, O. Okhrimenko43, R. Oldeman15,d, M.
Orlandea28, J.M. Otalora Goicochea2, P. Owen52, A. Oyanguren35, B.K. Pal58, A.
Palano13,b, M. Palutan18, J. Panman37, A. Papanestis48, M. Pappagallo50, C.
Parkes53, C.J. Parkinson52, G. Passaleva17, G.D. Patel51, M. Patel52, G.N.
Patrick48, C. Patrignani19,i, C. Pavel-Nicorescu28, A. Pazos Alvarez36, A.
Pellegrino40, G. Penso24,l, M. Pepe Altarelli37, S. Perazzini14,c, D.L.
Perego20,j, E. Perez Trigo36, A. Pérez-Calero Yzquierdo35, P. Perret5, M.
Perrin-Terrin6, G. Pessina20, K. Petridis52, A. Petrolini19,i, A. Phan58, E.
Picatoste Olloqui35, B. Pietrzyk4, T. Pilař47, D. Pinci24, S. Playfer49, M.
Plo Casasus36, F. Polci8, G. Polok25, A. Poluektov47,33, E. Polycarpo2, A.
Popov34, D. Popov10, B. Popovici28, C. Potterat35, A. Powell54, J.
Prisciandaro38, A. Pritchard51, C. Prouve7, V. Pugatch43, A. Puig Navarro38,
G. Punzi22,q, W. Qian4, J.H. Rademacker45, B. Rakotomiaramanana38, M.S.
Rangel2, I. Raniuk42, N. Rauschmayr37, G. Raven41, S. Redford54, M.M. Reid47,
A.C. dos Reis1, S. Ricciardi48, A. Richards52, K. Rinnert51, V. Rives
Molina35, D.A. Roa Romero5, P. Robbe7, E. Rodrigues53, P. Rodriguez Perez36,
S. Roiser37, V. Romanovsky34, A. Romero Vidal36, J. Rouvinet38, T. Ruf37, F.
Ruffini22, H. Ruiz35, P. Ruiz Valls35, G. Sabatino24,k, J.J. Saborido Silva36,
N. Sagidova29, P. Sail50, B. Saitta15,d, V. Salustino Guimaraes2, C.
Salzmann39, B. Sanmartin Sedes36, M. Sannino19,i, R. Santacesaria24, C.
Santamarina Rios36, E. Santovetti23,k, M. Sapunov6, A. Sarti18,l, C.
Satriano24,m, A. Satta23, M. Savrie16,e, D. Savrina30,31, P. Schaack52, M.
Schiller41, H. Schindler37, M. Schlupp9, M. Schmelling10, B. Schmidt37, O.
Schneider38, A. Schopper37, M.-H. Schune7, R. Schwemmer37, B. Sciascia18, A.
Sciubba24, M. Seco36, A. Semennikov30, K. Senderowska26, I. Sepp52, N.
Serra39, J. Serrano6, P. Seyfert11, M. Shapkin34, I. Shapoval16,42, P.
Shatalov30, Y. Shcheglov29, T. Shears51,37, L. Shekhtman33, O. Shevchenko42,
V. Shevchenko30, A. Shires52, R. Silva Coutinho47, T. Skwarnicki58, N.A.
Smith51, E. Smith54,48, M. Smith53, M.D. Sokoloff56, F.J.P. Soler50, F.
Soomro18, D. Souza45, B. Souza De Paula2, B. Spaan9, A. Sparkes49, P.
Spradlin50, F. Stagni37, S. Stahl11, O. Steinkamp39, S. Stoica28, S. Stone58,
B. Storaci39, M. Straticiuc28, U. Straumann39, V.K. Subbiah37, L. Sun56, S.
Swientek9, V. Syropoulos41, M. Szczekowski27, P. Szczypka38,37, T. Szumlak26,
S. T’Jampens4, M. Teklishyn7, E. Teodorescu28, F. Teubert37, C. Thomas54, E.
Thomas37, J. van Tilburg11, V. Tisserand4, M. Tobin38, S. Tolk41, D.
Tonelli37, S. Topp-Joergensen54, N. Torr54, E. Tournefier4,52, S. Tourneur38,
M.T. Tran38, M. Tresch39, A. Tsaregorodtsev6, P. Tsopelas40, N. Tuning40, M.
Ubeda Garcia37, A. Ukleja27, D. Urner53, U. Uwer11, V. Vagnoni14, G.
Valenti14, R. Vazquez Gomez35, P. Vazquez Regueiro36, S. Vecchi16, J.J.
Velthuis45, M. Veltri17,g, G. Veneziano38, M. Vesterinen37, B. Viaud7, D.
Vieira2, X. Vilasis-Cardona35,n, A. Vollhardt39, D. Volyanskyy10, D. Voong45,
A. Vorobyev29, V. Vorobyev33, C. Voß60, H. Voss10, R. Waldi60, R. Wallace12,
S. Wandernoth11, J. Wang58, D.R. Ward46, N.K. Watson44, A.D. Webber53, D.
Websdale52, M. Whitehead47, J. Wicht37, J. Wiechczynski25, D. Wiedner11, L.
Wiggers40, G. Wilkinson54, M.P. Williams47,48, M. Williams55, F.F. Wilson48,
J. Wishahi9, M. Witek25, S.A. Wotton46, S. Wright46, S. Wu3, K. Wyllie37, Y.
Xie49,37, Z. Xing58, Z. Yang3, R. Young49, X. Yuan3, O. Yushchenko34, M.
Zangoli14, M. Zavertyaev10,a, F. Zhang3, L. Zhang58, W.C. Zhang12, Y. Zhang3,
A. Zhelezov11, A. Zhokhov30, L. Zhong3, A. Zvyagin37.

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 de Savoie, 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
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Padova, Padova, Italy
22Sezione INFN di Pisa, Pisa, Italy
23Sezione INFN di Roma Tor Vergata, Roma, Italy
24Sezione INFN di Roma La Sapienza, Roma, Italy
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
26AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
27National Center for Nuclear Research (NCBJ), Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
41Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
42NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
43Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
44University of Birmingham, Birmingham, United Kingdom
45H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
46Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
47Department of Physics, University of Warwick, Coventry, United Kingdom
48STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
49School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
50School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
51Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
52Imperial College London, London, United Kingdom
53School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
54Department of Physics, University of Oxford, Oxford, United Kingdom
55Massachusetts Institute of Technology, Cambridge, MA, United States
56University of Cincinnati, Cincinnati, OH, United States
57University of Maryland, College Park, MD, United States
58Syracuse University, Syracuse, NY, United States
59Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
60Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11

aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oHanoi University of Science, Hanoi, Viet Nam
pUniversità di Padova, Padova, Italy
qUniversità di Pisa, Pisa, Italy
rScuola Normale Superiore, Pisa, Italy

The $B_{c}^{+}$ meson is composed of two heavy valence quarks, and has a wide
range of expected decay modes[1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. Prior to LHCb
taking data, only a few decay channels, such as
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}$ and
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\mu^{+}\nu$
had been observed[11, 12]. For $pp$ collisions at a centre-of-mass energy of
7$\mathrm{\,Te\kern-1.00006ptV}$, the total $B_{c}^{+}$ production cross-
section is predicted to be about 0.4$\rm\,\upmu b$, one order of magnitude
higher than that at the Tevatron [13, 14]. LHCb has thus been able to observe
new decay modes, such as
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}\pi^{+}$[15], $B_{c}^{+}\rightarrow\psi{(2S)}\pi^{+}$[16]
and $B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}D_{s}^{(*)+}$[17], and to measure precisely the mass of the $B_{c}^{+}$
meson[18].

In this paper, we report the first observation of the decay channel
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$
(inclusion of charge conjugate modes is implied throughout the paper). The
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ meson is reconstructed in the
dimuon final state. The branching fraction is measured relative to that of the
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}$
decay mode, which has identical topology and similar kinematic properties, as
shown in Fig. 1. No absolute branching fraction of the $B_{c}^{+}$ meson is
known to date. The predicted ratio of branching fractions ${\cal
B}(B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+})/{\cal
B}(B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+})$
is dominated by the ratio of the relevant Cabibbo-Kobayashi-Maskawa (CKM)
matrix elements $\left|V_{ud}/V_{us}\right|^{2}\approx 0.05$[19]. However,
after including the decay constants, $f_{K^{+}(\pi^{+})}$, the ratio is
enhanced,

$\displaystyle{{\cal
B}(B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+})\over{\cal
B}(B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+})}\approx\left|{V_{us}f_{K^{+}}\over
V_{ud}f_{\pi^{+}}}\right|^{2}=0.077\,,$ (1)

where the values of $f_{K^{+}(\pi^{+})}$ are given in Ref. [19]. Taking into
account the contributions of the $B_{c}^{+}$ form factor and the kinematics,
the theoretical predictions for the ratio of branching fractions lie in the
range from 0.054 to 0.088 [2, 3, 5, 6, 7, 9, 10]. The large span of these
predictions is due to the various models and the uncertainties on the
phenomenological parameters. The measurement of ${\cal
B}(B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+})/{\cal
B}(B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+})$
therefore provides a test of the theoretical predictions of hadronisation.

Figure 1:  Diagram for a
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}(K^{+})$ decay.

The analysis is based on a data sample, corresponding to an integrated
luminosity of 1.0$\mbox{\,fb}^{-1}$ of $pp$ collisions, collected by the LHCb
experiment at a centre-of-mass energy of 7$\mathrm{\,Te\kern-1.00006ptV}$. The
LHCb detector[20] is a single-arm, forward spectrometer covering the
pseudorapidity range $2<\eta<5$ and is designed for precise measurements in
the $b$ and $c$ quark sectors. 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$\rm\,Tm$, and three stations of
silicon-strip detectors and straw drift tubes placed downstream. The combined
tracking system has momentum resolution $\Delta p/p$ that varies from 0.4% at
5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to 0.6% at
100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and impact parameter (IP)
resolution of 20$\,\upmu\rm m$ for tracks with high transverse momentum
($p_{\rm T}$). Charged hadrons are identified using two ring-imaging Cherenkov
(RICH) detectors and good kaon-pion separation is achieved for tracks with
momentum between 5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and
100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ [21]. Photon, electron and hadron
candidates are identified by a calorimeter system consisting of scintillating-
pad and preshower detectors, an electromagnetic calorimeter and a hadronic
calorimeter. Muons are identified by a system composed of alternating layers
of iron and multiwire proportional chambers. The trigger system[22] consists
of a hardware stage, based on information from the calorimeter and muon
systems, followed by a two-stage software trigger that applies event
reconstruction and reduces the event rate from 1${\rm\,MHz}$ to around
3${\rm\,kHz}$.

In the hardware trigger, events are selected by requiring a single muon or
dimuon candidate with high $p_{\rm T}$. In the software trigger, events are
selected by requiring dimuon candidates with invariant mass close to the known
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass[19] and with decay length
significance greater than 3 with respect to the associated primary vertex
(PV). For events with several PVs, the one with the smallest $\chi^{2}_{\rm
IP}$ is chosen, where $\chi^{2}_{\rm IP}$ is defined as the difference in
$\chi^{2}$ of a given PV reconstructed with and without the considered track.
The bachelor hadrons ($K^{+}$ for
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ and
$\pi^{+}$ for $B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}$ decays) are required to be separated from the $B_{c}^{+}$ PV
and have $\mbox{$p_{\rm T}$}>0.5{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The
$B_{c}^{+}$ candidates are required to have good vertex quality with vertex
fit $\chi^{2}_{\rm vtx}$ per degree of freedom less than 5, and mass within
$500{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the world average value of
the $B_{c}^{+}$ mass [19].

A boosted decision tree (BDT)[23] is used for the final event selection. The
BDT is trained using a simulated
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}$
sample as a proxy for signal and the high-mass sideband
($m_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}}>6650{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$) in data for
background. The BDT cut value is optimised to maximise the expected
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ signal
significance. In the simulation, $pp$ collisions are generated using Pythia
6.4 [24] with a specific LHCb configuration [25]. The $B_{c}^{+}$ meson
production is simulated with the dedicated generator Bcvegpy [26]. Decays of
hadronic particles are described by EvtGen [27], in which final state
radiation is generated using Photos [28]. The interaction of the generated
particles with the detector and its response are implemented using the Geant4
toolkit [29, *Agostinelli:2002hh] as described in Ref. [31]. The BDT takes the
following variables into account: the $\chi^{2}_{\rm IP}$ of the bachelor
hadron and $B_{c}^{+}$ mesons with respect to the PV; the $B_{c}^{+}$ vertex
quality; the distance between the $B_{c}^{+}$ decay vertex and the PV; the
$p_{\rm T}$ of the $B_{c}^{+}$ candidate; the $\chi^{2}$ from the $B_{c}^{+}$
decay vertex refit[32], obtained with a constraint on the PV and the
reconstructed ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass; and the
cosine of the angle between the momentum of the $B_{c}^{+}$ meson and the
direction vector from the PV to the $B_{c}^{+}$ decay vertex. These variables
are chosen as they discriminate the signal from the background, and have
similar distributions for
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ and
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}$
decays, ensuring that the systematic uncertainty due to the relative selection
efficiency is minimal. After the BDT selection, no event with multiple
candidates remains.

The branching fraction ratio is computed as

$\displaystyle{{\cal
B}(B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+})\over{\cal
B}(B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+})}={N(B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+})\over N(B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+})}\cdot{\epsilon(B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+})\over\epsilon(B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+})}\,,$ (2)

where $N$ is the signal yield of
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ or
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}$
decays and $\epsilon$ is the total efficiency, which takes into account the
geometrical acceptance, detection, reconstruction, selection and trigger
effects.

An unbinned maximum likelihood fit is used to determine the yields from the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ mass distribution of the
$B_{c}^{+}$ candidates, under the kaon mass hypothesis. The total probability
density function for the fit has four components: signals for
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ and
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}$
decays; the combinatorial background; and the partially reconstructed
background.

To discriminate between pion and kaon bachelor tracks, the quantity

$\displaystyle\mathrm{DLL}_{K\pi}=\ln\mathcal{L}(K)-\ln\mathcal{L}(\pi)$ (3)

is used, where $\mathcal{L}(K)$ and $\mathcal{L}(\pi)$ are the likelihood
values provided by the RICH system under the kaon and pion hypotheses,
respectively. Since the momentum spectra of the bachelor pions and kaons are
correlated with the $\mathrm{DLL}_{K\pi}$, the shapes of the mass distribution
used in the fit vary as a function of $\mathrm{DLL}_{K\pi}$. To reduce this
dependence and separate the two signals, the $\mathrm{DLL}_{K\pi}$ range is
divided into four bins, $\mathrm{DLL}_{K\pi}<-5$, $-5<\mathrm{DLL}_{K\pi}<0$,
$0<\mathrm{DLL}_{K\pi}<5$ and $\mathrm{DLL}_{K\pi}>5$. The ratio of the total
signal yields is defined as
$\mathcal{R}_{K^{+}/\pi^{+}}=\sum^{4}_{i=1}N^{i}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}}/\sum^{4}_{i=1}N^{i}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}}$, where $N_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}(\pi^{+})}^{i}$ is the signal yield in each $\mathrm{DLL}_{K\pi}$
bin $i$. Due to the limited sample size of the
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ signal
in the bins with $\mathrm{DLL}_{K\pi}<-5$ and $-5<\mathrm{DLL}_{K\pi}<0$,
their signal yields are fixed, respectively, to be zero and
$P\times\sum^{4}_{i=1}N^{i}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}}$ where the $P$ is the probability that the kaon from the
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ decay
has $-5<\mathrm{DLL}_{K\pi}<0$, as estimated from simulation.

Figure 2:  Mass distributions of $B_{c}^{+}$ candidates in four
$\mathrm{DLL}_{K\pi}$ bins and the superimposed fit results. The solid shaded
area (red) represents the
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ signal
and the hatched area (blue) the
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}$
signal. The dot-dashed line (blue) indicates the partially reconstructed
background and the dotted (red) the combinatorial background. The solid line
(black) represents the sum of the above components and the points with error
bars (black) show the data. The labels $(a)$, $(b)$, $(c)$ and $(d)$
correspond to $\mathrm{DLL}_{K\pi}<-5$, $-5<\mathrm{DLL}_{K\pi}<0$,
$0<\mathrm{DLL}_{K\pi}<5$ and $\mathrm{DLL}_{K\pi}>5$ for the bachelor track,
respectively.

Figure 2 shows the invariant mass distributions of the $B_{c}^{+}$ candidates,
calculated with the kaon mass hypothesis in the four $\mathrm{DLL}_{K\pi}$
bins. In the fit to the $B_{c}^{+}$ mass spectrum, the shape of the
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ signal
is modelled by a double-sided Crystal Ball (DSCB) function[33] as

$\displaystyle f(x;M,\sigma,a_{l},n_{l},a_{r},n_{r})=\left\\{\begin{aligned}
&e^{-a^{2}_{l}\over 2}\left({n_{l}\over a_{l}}\right)^{n_{l}}\left({n_{l}\over
a_{l}}-a_{l}-{x-M\over\sigma}\right)^{-n_{l}}&{x-M\over\sigma}<-a_{l}\\\
&\exp\left[-{1\over
2}\left({x-M\over\sigma}\right)^{2}\right]&-a_{l}\leq{x-M\over\sigma}\leq-
a_{r}\\\ &e^{-a_{r}^{2}\over 2}\left({n_{r}\over
a_{r}}\right)^{n_{r}}\left({n_{r}\over
a_{r}}-a_{r}+{x-M\over\sigma}\right)^{-n_{r}}&{x-M\over\sigma}>-a_{r}\\\
\end{aligned}\right.$ (4)

where the peak position is fixed to that from an independent fit to the
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}$ mass
distribution, and the tail parameters $a_{l,r}$ and $n_{l,r}$ on both sides
are taken from simulation.

As the decay $B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}$ is reconstructed with the kaon mass replacing the pion mass,
the signal is shifted to higher mass values and is modelled by another DSCB
function whose shape and the relative position to the
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ signal
are also derived from simulation. Two corrections are applied to the
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}$
simulation sample. Firstly, since the resolution of the detector is
overestimated, the momenta of charged particles are smeared to make the
resolution on the $B_{c}^{+}$ mass in the
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}$
simulation sample the same as that of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}$ mass distribution of the
$B_{c}^{+}$ candidates in the data sample. Secondly, the shapes of the
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}$ mass
distribution in the four $\mathrm{DLL}_{K\pi}$ bins depend on the
$\mathrm{DLL}_{K\pi}$ distribution, which is different in data and simulation.
To reduce the effect of this difference, each simulated event is reweighted by
a $\mathrm{DLL}_{K\pi}$ dependent correction factor, which is derived from a
linear fit to the ratio of the $\mathrm{DLL}_{K\pi}$ distribution in
background-subtracted data, to that of the simulation sample. The background
subtraction [34] is performed with the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}$ mass distribution of the $B_{c}^{+}$ candidates in the data
sample with the pion mass hypothesis.

The combinatorial background is modelled as an exponential function with a
different freely varying parameter in each $\mathrm{DLL}_{K\pi}$ bin. The
contribution of the partially reconstructed background is modelled by an ARGUS
function[35]. The contribution of the partially reconstructed background is
dominated by events with bachelor pions, which are suppressed in the high-
value $\mathrm{DLL}_{K\pi}$ bins, therefore the number of the partially
reconstructed events in the $\mathrm{DLL}_{K\pi}>5$ bin is assumed to be zero.
All parameters of the partially reconstructed background are allowed to vary.
The observed $B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}$ signal yield is $46\pm 12$ and the ratio of yields is

$\displaystyle\mathcal{R}_{K^{+}/\pi^{+}}={N(B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+})\over N(B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+})}=0.071\pm 0.020\,\mathrm{\,(stat)}\,.$

The ratio of the total efficiencies computed over the full
$\mathrm{DLL}_{K\pi}$ range is

$\displaystyle{\epsilon(B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+})\over\epsilon(B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+})}=1.029\pm 0.007\,,$

which is determined from simulation and the uncertainty is due to the finite
size of the simulation samples.

The $B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}$
signal has a long tail that may extend into the high mass region. A systematic
uncertainty is assigned due to the choice of fit range, and is determined to
be 0.9% by changing the mass window from 6000-6600
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ to
6200-6700${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and comparing the results.
To estimate the systematic uncertainty due to the potentially different
performance of the BDT on data and simulation, several BDT cut values were
tested. A 5.7% spread in the final result is obtained and is propagated to the
quoted systematic uncertainty.

To estimate the uncertainty due to the shapes of the
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ and
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}$
signals, the fit is repeated many times by varying the parameters of the tails
of these DSCB functions that were kept constant in the fit within one standard
deviation of their values in simulation. A spread of 0.7% is observed. For the
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}$
signal the assigned systematic uncertainty is 0.5%.

To estimate the systematic uncertainty due to the choice of signal shape, an
alternative $B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}$ mass shape is used, which is determined from the data sample by
subtracting the background in the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}$ mass distribution of the $B_{c}^{+}$ candidates with the pion
hypothesis. A $2.7\%$ difference with the ratio obtained with the nominal
signal shape is observed.

For the systematic uncertainty due to the choice of the partially
reconstructed background shape in each $\mathrm{DLL}_{K\pi}$ bin, the shape is
modelled with the ARGUS function convolved with a Gaussian function. The
observed 2.3% deviation from the default fit is assigned as the systematic
uncertainty.

For the $B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}$ yields in the two bins with $\mathrm{DLL}_{K\pi}<0$, half of the
probability estimated from the simulation, namely 1.8%, is taken as systematic
uncertainty.

To estimate the uncertainty due to the choice of the $\mathrm{DLL}_{K\pi}$
binning, two other binning choices are tried: $\mathrm{DLL}_{K\pi}<-6$,
$-6<\mathrm{DLL}_{K\pi}<-1$, $-1<\mathrm{DLL}_{K\pi}<4$,
$\mathrm{DLL}_{K\pi}>4$ and $\mathrm{DLL}_{K\pi}<-4$,
$-4<\mathrm{DLL}_{K\pi}<1$, $1<\mathrm{DLL}_{K\pi}<6$,
$\mathrm{DLL}_{K\pi}>6$. The average value of the results with these two
binning choices has a 1.2% deviation from the default value, which is taken as
the systematic uncertainty.

There is a systematic uncertainty due to the different track reconstruction
efficiencies for kaons and pions. Since the simulation does not describe
hadronic interactions with detector material perfectly, a 2% uncertainty is
assumed, as in Ref.[36].

An uncertainty of 0.7% arises from the statistical uncertainty of the ratio of
the total efficiencies, which is due to the finite size of the simulation
sample.

Table 1: Relative systematic uncertainties on the ratio of branching fractions. Source | Uncertainty (%)
---|---
Mass window | 0.9
BDT selection | 5.7
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ signal model | 0.7
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}$ signal model | 0.5
Choice of signal shape | 2.7
Partially reconstructed background shape | 2.3
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ signals in $\mathrm{DLL}_{K\pi}<0$ bins | 1.8
$\mathrm{DLL}_{K\pi}$ binning choice | 1.2
$K^{+}$ and $\pi^{+}$ interaction length | 2.0
Simulation sample size | 0.7
Total | 7.5

The systematic uncertainties are summarised in Table 1. The total systematic
uncertainty, obtained as the quadratic sum of the individual uncertainties, is
7.5%.

The asymptotic formula for a likelihood-based test
$\sqrt{-2\ln(\mathcal{L}_{\mathrm{B}}/\mathcal{L}_{\mathrm{S+B}})}$ is used to
estimate the $B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}$ signal significance, where $\mathcal{L}_{\mathrm{B}}$ and
$\mathcal{L}_{\mathrm{S+B}}$ stand for the likelihood of the background-only
hypothesis and the signal and background hypothesis respectively. A deviation
from the background-only hypothesis with 5.2 standard deviations is found when
only the statistical uncertainty is considered. When taking the systematic
uncertainty into account, the total significance of the
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ signal
is 5.0 $\sigma$.

In summary, a search for the
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ decay
is performed using a data sample, corresponding to an integrated luminosity of
1.0 $\mbox{\,fb}^{-1}$ of $pp$ collisions, collected by the LHCb experiment.
The signal yield is $46\pm 12$ candidates, and represents the first
observation of this decay channel. The branching fraction of
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ with
respect to that of $B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}$ is measured as

$\displaystyle{{\cal
B}(B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+})\over{\cal
B}(B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+})}=0.069\pm 0.019\pm 0.005\,,$

where the first uncertainty is the statistical and the second is systematic.
The measurement is in agreement with the theoretical predictions[2, 3, 5, 6,
7, 9, 10].

Assuming factorization holds, the naïve prediction of the ratio ${\cal
B}(B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+})/{\cal
B}(B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+})$
can be compared to other $B$ meson decays with a similar topology

$\displaystyle{{\cal B}(B\rightarrow DK^{+})\over{\cal B}(B\rightarrow
D\pi^{+})}=\left\\{\begin{aligned} &0.0646\pm 0.0043\pm
0.0025&\mathrm{for}\,&B^{0}_{s}\rightarrow D^{-}_{s}K^{+}(\pi^{+})\\\
&0.0774\pm 0.0012\pm 0.0019&\mathrm{for}\,&B^{+}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{+}(\pi^{+})\\\ &0.074\pm
0.009&\mathrm{for}\,&B^{0}\rightarrow D^{-}K^{+}(\pi^{+})\\\
\end{aligned}\right.$ (5)

taken from Ref.[37, 38, 19]. Hence, this measurement of ${\cal
B}(B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+})/{\cal
B}(B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+})$
is consistent with naïve factorisation in $B$ decays.

## Acknowledgements

We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG
(Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR
(Poland); MEN/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER
(Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We
also acknowledge the support received from the ERC under FP7. The Tier1
computing centres are supported by IN2P3 (France), KIT and BMBF (Germany),
INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United
Kingdom). We are thankful for the computing resources put at our disposal by
Yandex LLC (Russia), as well as to the communities behind the multiple open
source software packages that we depend on.

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