EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2017-306 LHCb-PAPER-2017-042 February 12, 2018

Search for excited $B_{c}^{+}$ states

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

A search is performed in the invariant mass spectrum of the
${{B}_{c}^{+}}{{\pi}^{+}}{{\pi}^{-}}$ system for the excited ${{B}_{c}^{+}}$
states ${B}_{c}(2^{1}S_{0})^{+}$ and ${B}_{c}(2^{3}S_{1})^{+}$ using a data
sample of $pp$ collisions collected by the LHCb experiment at the centre-of-
mass energy of $\sqrt{s}=8\mathrm{\,Te\kern-1.00006ptV}$, corresponding to an
integrated luminosity of $2\mbox{\,fb}^{-1}$. No evidence is seen for either
state. Upper limits on the ratios of the production cross-sections of the
${{B}_{c}(2^{1}S_{0})^{+}}$ and ${{B}_{c}(2^{3}S_{1})^{+}}$ states times the
branching fractions of
${{B}_{c}(2^{1}S_{0})^{+}}\\!\rightarrow{{B}_{c}^{+}}{{\pi}^{+}}{{\pi}^{-}}$
and
${{B}_{c}(2^{3}S_{1})^{+}}\\!\rightarrow{{B}_{c}^{*+}}{{\pi}^{+}}{{\pi}^{-}}$
over the production cross-section of the ${{B}_{c}^{+}}$ state are given as a
function of their masses. They are found to be between 0.02 and 0.14 at $95\%$
confidence level for ${{B}_{c}(2^{1}S_{0})^{+}}$ and
${{B}_{c}(2^{3}S_{1})^{+}}$ in the mass ranges
$[6830,6890]{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and
$[6795,6890]{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, respectively.

Published in JHEP 01 (2018) 138

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

##  1 Introduction

The $B_{c}$ meson family is unique in the Standard Model, as its states
contain two different heavy-flavour valence quarks. It has a rich
spectroscopy, predicted by various models [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,
12, 13, 14] and lattice QCD [15]. The ground state of the $B_{c}$ meson
family, the ${{B}_{c}^{+}}$ meson, was first observed by the CDF experiment
[16, 17] at the Tevatron collider in 1998.111Sums over charge-conjugated modes
are implied throughout this paper. Recently, the ATLAS collaboration reported
observation of an excited $B_{c}$ state with a mass of $6842\pm
4\mathrm{\,(stat)}\pm
5\mathrm{\,(syst)}{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ [18]. Since the
production cross-section of the ${{B}_{c}(2^{3}S_{1})^{+}}$ state is predicted
to be more than twice that of the ${{B}_{c}(2^{1}S_{0})^{+}}$ state [19, 13,
8, 20], the most probable interpretation of the single peak is either a signal
for
${{B}_{c}(2^{3}S_{1})^{+}}\\!\rightarrow{{B}_{c}^{*+}}{{\pi}^{+}}{{\pi}^{-}}$,
followed by ${{B}_{c}^{*+}}\rightarrow{{B}_{c}^{+}}\gamma$ with a missing low-
energy photon, or an unresolved pair of peaks from the decays
${{B}_{c}(2^{1}S_{0})^{+}}\\!\rightarrow{{B}_{c}^{+}}{{\pi}^{+}}{{\pi}^{-}}$
and
${{B}_{c}(2^{3}S_{1})^{+}}\\!\rightarrow{{B}_{c}^{*+}}{{\pi}^{+}}{{\pi}^{-}}$.222The
spectroscopic notation $n^{2s+1}L_{J}$ is used, where $n$ is the radial
quantum number, $s$ the total spin of the two valence quarks, $L$ their
relative angular momentum ($S$ implies $L=0$), and $J$ the total angular
momentum of the system, i.e. spin of the excited state. ${{B}_{c}^{*+}}$
denotes the ${{B}_{c}(1^{3}S_{1})^{+}}$ state. The ${{B}_{c}(2^{1}S_{0})^{+}}$
and ${{B}_{c}(2^{3}S_{1})^{+}}$ states are denoted as $B_{c}(2S)^{+}$ and
$B_{c}^{*}(2S)^{+}$ hereafter, and ${{B}_{c}^{(*)}(2S)^{+}}$ denotes either
state.

In the present paper, the ${{B}_{c}(2S)^{+}}$ and ${{B}_{c}^{*}(2S)^{+}}$
mesons are searched for using $pp$ collision data collected by the LHCb
experiment at $\sqrt{s}=8\mathrm{\,Te\kern-1.00006ptV}$, corresponding to an
integrated luminosity of $2\mbox{\,fb}^{-1}$. The ${{B}_{c}(2S)^{+}}$ and
${{B}_{c}^{*}(2S)^{+}}$ mesons are reconstructed through the decays
${{B}_{c}(2S)^{+}}\\!\rightarrow{{B}_{c}^{+}}{{\pi}^{+}}{{\pi}^{-}}$ and
${{B}_{c}^{*}(2S)^{+}}\\!\rightarrow{{B}_{c}^{*+}}{{\pi}^{+}}{{\pi}^{-}}$ with
${{B}_{c}^{*+}}\rightarrow{{B}_{c}^{+}}\gamma$,
${{B}_{c}^{+}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}}$ and ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\rightarrow{\mu^{+}}{\mu^{-}}$. The branching fraction of the
$B_{c}^{(*)}(2S)^{+}\rightarrow B_{c}^{(*)+}{{\pi}^{+}}{{\pi}^{-}}$ decay,
${\mathcal{B}}(B_{c}^{(*)}(2S)^{+}\rightarrow
B_{c}^{(*)+}{{\pi}^{+}}{{\pi}^{-}})$, is predicted to be between 39% and 59%
[13, 8]. The low-energy photon in the ${{B}_{c}^{*}(2S)^{+}}$ decay chain is
not reconstructed. The ${{B}_{c}^{*}(2S)^{+}}$ state still appears in the
invariant mass $M({{B}_{c}^{+}}{{\pi}^{+}}{{\pi}^{-}})$ spectrum as a narrow
mass peak [20, 21], which is centered at $M({{B}_{c}(2S)^{+}})-\Delta M$,
where

$\Delta
M\equiv\left[M({{B}_{c}^{*+}})-M({{B}_{c}^{+}})\right]-\left[M({{B}_{c}^{*}(2S)^{+}})-M({{B}_{c}(2S)^{+}})\right],$
(1)

and $M({{B}_{c}^{+}})$ is the known mass of ${{B}_{c}^{+}}$. According to
theoretical predictions [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], the mass of the
${{B}_{c}(2S)^{+}}$ state, $M({{B}_{c}(2S)^{+}})$, is expected to be in the
range $[6830,6890]{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, and $\Delta M$ in
the range $[0,35]{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, such that the peak
position of the ${{B}_{c}^{*}(2S)^{+}}$ state in
$M({{B}_{c}^{+}}{{\pi}^{+}}{{\pi}^{-}})$ is expected to be in the range
$[6795,6890]{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.

##  2 Detector and simulation

The LHCb detector [22, 23] 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 (TT) located upstream
of a dipole magnet with a bending power of about $4{\mathrm{\,Tm}}$, and three
stations of silicon-strip detectors and straw drift tubes placed downstream of
the magnet. The tracking system provides a measurement of momentum, $p$, of
charged particles with a relative uncertainty that varies from 0.5% at low
momentum to 1.0% at 200${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The minimum
distance of a track to a primary vertex (PV), the impact parameter (IP), is
measured with a resolution of
$(15+29/\mbox{$p_{\mathrm{T}}$}){\,\upmu\mathrm{m}}$, where $p_{\mathrm{T}}$
is the component of the momentum transverse to the beam, in
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. Different types of charged hadrons are
distinguished using information from two ring-imaging Cherenkov detectors.
Photons, electrons and hadrons are identified by a calorimeter system
consisting of scintillating-pad and preshower detectors, an electromagnetic
calorimeter and a hadronic calorimeter. Muons are identified by a system
composed of alternating layers of iron and multiwire proportional chambers.
The online event selection is performed by a trigger, which consists of a
hardware stage, based on information from the calorimeter and muon systems,
followed by a software stage, which applies a full event reconstruction. At
the hardware stage, events are required to have at least one muon with high
$p_{\mathrm{T}}$ or a hadron with high transverse energy. At the software
stage, two muon tracks or three charged tracks are required to have high
$p_{\mathrm{T}}$ and to form a secondary vertex with a significant
displacement from the interaction point.

In the simulation, $pp$ collisions are generated using Pythia 6 [24] with a
specific LHCb configuration [25]. The generator Bcvegpy [19] is used to
simulate the production of $B_{c}$ mesons. Decays of hadronic particles are
described by EvtGen [26], in which final-state radiation is generated using
Photos [27]. The interaction of the generated particles with the detector, and
its response, are implemented using the Geant4 toolkit [28] as described in
Ref. [29]. In the default simulation, the masses of the excited $B_{c}$ states
are set as
$M({{B}_{c}(2S)^{+}})=6858{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$,
$M({{B}_{c}^{*}(2S)^{+}})=6890{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and
$M({{B}_{c}^{*+}})=6342{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$,
corresponding to $\Delta M=35{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, and
the ${{B}_{c}^{*}(2S)^{+}}$ state is assumed to be produced unpolarised.
Simulated samples with different mass settings, which cover the expected mass
range of the ${{B}_{c}^{(*)}(2S)^{+}}$ states, are generated to study
variations in the reconstruction efficiency.

##  3 Event selection

To select ${{B}_{c}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}}$ decays, ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$
candidates are formed from pairs of opposite-charge tracks. The tracks are
required to have $p_{\mathrm{T}}$ larger than
$0.55{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and good track-fit quality, to be
identified as muons, and to originate from a common vertex. Each
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ candidate with an invariant
mass between $3.04{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ and
$3.14{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ is combined with a charged pion
to form a ${{B}_{c}^{+}}$ candidate. The pion is required to have
$\mbox{$p_{\mathrm{T}}$}>1.0\,{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and good
track-fit quality. The ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
candidate and the charged pion are required to originate from a common vertex,
and the ${{B}_{c}^{+}}$ candidates must have a decay time larger than
$0.2\,{\rm ps}$. Each of the particles is associated to the PV that has the
smallest $\chi^{2}_{\text{IP}}$, where $\chi^{2}_{\text{IP}}$ is defined as
the difference in the vertex-fit $\chi^{2}$ of a given PV reconstructed with
and without the particle under consideration. The $\chi^{2}_{\text{IP}}$ of
the ${{B}_{c}^{+}}$ (${{\pi}^{+}}$) candidate is required to be $<25$ ($>9$)
with respect to the associated PV of the ${{B}_{c}^{+}}$ candidate. To further
suppress background, a requirement on a boosted decision tree (BDT) [30, 31]
classifier is applied. The BDT classifier uses information from the
$\chi^{2}_{\text{IP}}$ of the two muons, the pion, the
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$, and the ${{B}_{c}^{+}}$
mesons with respect to the associated PV; the $p_{\mathrm{T}}$ of both muons,
the $J/\psi$ and ${{\pi}^{+}}$ mesons; and the decay length, decay time, and
the vertex-fit $\chi^{2}$ of the ${{B}_{c}^{+}}$ meson. The BDT is trained
with signal events taken from simulation and background events from the upper
sideband containing ${{B}_{c}^{+}}$ candidates with masses in the range
$[6370,6600]{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The distributions of
the BDT response for the simulation and the background subtracted data are in
agreement. The criterion on the BDT output is chosen to maximise the figure of
merit $S/\sqrt{S+B}$, where $S$ and $B$ are the expected numbers of signal and
background in the range $M({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}})\in[6251,6301]{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.
The mass of the ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ candidates is
constrained to the known value [32] to improve the ${{B}_{c}^{+}}$ mass
resolution.333The ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ mass is
taken to be $3096.916{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ according to
the 2014 edition of the Review of Particle Physics [32], rather than
$3096.900{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ in the 2016 edition [33].
The effect of this choice on the final result is negligible. The
${{B}_{c}^{+}}$ signal yield is obtained by performing an unbinned extended
maximum likelihood fit to the $M({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}})$ mass distribution, as shown in Fig. 1. The signal
component is modelled by a Gaussian function with asymmetric power-law tails
as determined from simulation. The mean and resolution of the Gaussian
function are free parameters in the fit. The combinatorial background is
described with an exponential function. The contamination from the Cabibbo-
suppressed channel
${{B}_{c}^{+}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}K^{+}$,
with the kaon misidentified as a pion, is described by a Gaussian function
with asymmetric power-law tails. The parameters are also fixed from
simulation, with only the Gaussian mean related to the
${{B}_{c}^{+}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}}$ signal as a free parameter to account for the possible
small mass difference in data and simulation. The signal yield of
${{B}_{c}^{+}}$ decays is determined to be $3325\pm 73$.

Figure 1:  Invariant mass distribution of the selected
${{B}_{c}^{+}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}}$ candidates. The points with error bars represent the data.
The blue solid line is the fit to data. The red cross-hatched area shows the
signal. The green shaded area represents the
${{B}_{c}^{+}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}K^{+}$
background. The violet dash-dotted line is the combinatorial background.

To reconstruct the ${{B}_{c}^{(*)}(2S)^{+}}$ states, the ${{B}_{c}^{+}}$
candidates with $M({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}})\in[6200,6340]{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ are
combined with two opposite-charge tracks. The tracks are required to have
$\mbox{$p_{\mathrm{T}}$}>0.25{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, momenta
larger than $2{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and good track-fit
quality, and to be identified as pions. The ${{B}_{c}^{(*)}(2S)^{+}}$
candidates are required to have good ${{B}_{c}^{+}}{{\pi}^{+}}{{\pi}^{-}}$
vertex-fit quality. To improve the ${{B}_{c}^{(*)}(2S)^{+}}$ mass resolution,
the mass of ${{B}_{c}^{+}}$ candidates is constrained to the known
${{B}_{c}^{+}}$ mass [34], and the reconstructed ${{B}_{c}^{(*)}(2S)^{+}}$
mesons are constrained to originate from the associated PV. To optimise the
sensitivity of the analysis, a selection based on a multilayer perceptron
(MLP) [35] classifier is applied. To distinguish the signal candidates from
combinatorial background, the MLP classifier uses information on the angles
between the ${{B}_{c}^{+}}$ and ${{\pi}^{+}}$, ${{B}_{c}^{+}}$ and
${{\pi}^{-}}$, and ${{\pi}^{+}}$ and ${{\pi}^{-}}$ candidate momenta projected
in the plane transverse to the beam axis; the angles between the
${{B}_{c}^{(*)}(2S)^{+}}$ momentum and the ${{B}_{c}^{+}}$, ${{\pi}^{+}}$, and
${{\pi}^{-}}$ momenta in the ${{B}_{c}^{(*)}(2S)^{+}}$ centre-of-mass frame;
the minimum cosine value of the angles between the momentum of the
${{B}_{c}^{+}}$ meson or of one of the pions from ${{B}_{c}^{(*)}(2S)^{+}}$
and the momentum of the muons or pion from the ${{B}_{c}^{+}}$ meson; and the
vertex-fit $\chi^{2}$ of the ${{B}_{c}^{(*)}(2S)^{+}}$ meson. In simulation,
these variables have similar distributions for the
${{B}_{c}(2S)^{+}}\\!\rightarrow{{B}_{c}^{+}}{{\pi}^{+}}{{\pi}^{-}}$ and
${{B}_{c}^{*}(2S)^{+}}\\!\rightarrow{{B}_{c}^{*+}}(\rightarrow{{B}_{c}^{+}}\gamma){{\pi}^{+}}{{\pi}^{-}}$
decays. Therefore, the combination of the simulated candidates for the decays
${{B}_{c}(2S)^{+}}\\!\rightarrow{{B}_{c}^{+}}{{\pi}^{+}}{{\pi}^{-}}$ and
${{B}_{c}^{*}(2S)^{+}}\\!\rightarrow{{B}_{c}^{*+}}(\rightarrow{{B}_{c}^{+}}\gamma){{\pi}^{+}}{{\pi}^{-}}$
is used as signal for the MLP training, and the background sample consists of
the candidates in the lower and upper sidebands of the
$M({{B}_{c}^{+}}{{\pi}^{+}}{{\pi}^{-}})$ mass spectrum in data, with
$M({{B}_{c}^{+}}{{\pi}^{+}}{{\pi}^{-}})\in[6555,6785]{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
and $[6900,7500]{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, respectively. The
MLP response is transformed to make the signal candidates distributed evenly
between zero and unity, and the background candidates cluster near zero. Only
the candidates with transformed output values smaller than 0.02 are rejected,
retaining 98% of the signal. The remaining candidates are divided into four
categories with the MLP response falling in $(0.02,0.2)$, $[0.2,0.4)$,
$[0.4,0.6)$ and $[0.6,1.0]$, respectively. The
$M({{B}_{c}^{+}}{{\pi}^{+}}{{\pi}^{-}})$ distributions in the expected signal
region for the four MLP categories are shown in Fig. 2. The mass resolutions
on $M({{B}_{c}^{+}}{{\pi}^{+}}{{\pi}^{-}})$ for the ${{B}_{c}^{(*)}(2S)^{+}}$
state, $\sigma_{\rm w}({{B}_{c}^{(*)}(2S)^{+}})$, can be determined from the
simulated samples of the
${{B}_{c}(2S)^{+}}\\!\rightarrow{{B}_{c}^{+}}{{\pi}^{+}}{{\pi}^{-}}$ and
${{B}_{c}^{*}(2S)^{+}}\\!\rightarrow{{B}_{c}^{*+}}(\rightarrow{{B}_{c}^{+}}\gamma){{\pi}^{+}}{{\pi}^{-}}$
decays. The differences between the mass resolutions in data and simulation
are evaluated with the control decay mode
${{B}_{c}^{+}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}$, which has the same final state as
the signal and a large yield, and are corrected by applying a scale factor.
The obtained mass resolutions are $\sigma_{\rm w}({{B}_{c}(2S)^{+}})=2.05\pm
0.05{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and $\sigma_{\rm
w}({{B}_{c}^{*}(2S)^{+}})=3.17\pm
0.03{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The
$M({{B}_{c}^{+}}{{\pi}^{+}}{{\pi}^{-}})$ distributions are consistent with the
background-only hypothesis, as determined by the scan described below.

(a) MLP category: (0.02,0.2)  (b) MLP category: [0.2,0.4)  (c) MLP category:
[0.4,0.6)  (d) MLP category: [0.6,1.0]

Figure 2: Mass distributions of the selected
${{B}_{c}^{+}}{{\pi}^{+}}{{\pi}^{-}}$ candidates in the range
$[6795,6890]{\mathrm{\,Me\kern-0.90005ptV\\!/}c^{2}}$ for the four MLP
categories.

##  4 Upper limits

As no significant ${{B}_{c}^{(*)}(2S)^{+}}$ signal is found, upper limits are
set, for each ${{B}_{c}^{(*)}(2S)^{+}}$ mass hypothesis, on the ratio
$\mathcal{R}$ of the ${{B}_{c}^{(*)}(2S)^{+}}$ production cross-section times
the branching fraction of
$B_{c}^{(*)}(2S)^{+}\rightarrow{B}_{c}^{(*)+}{{\pi}^{+}}{{\pi}^{-}}$ to the
production cross-section of the ${{B}_{c}^{+}}$ state. The ratio $\mathcal{R}$
is determined for ${{B}_{c}^{(*)}(2S)^{+}}$ and ${{B}_{c}^{+}}$ candidates in
the kinematic ranges
$\mbox{$p_{\mathrm{T}}$}\in[0,20]{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and
rapidity $y\in[2.0,4.5]$, and is expressed as

$\displaystyle\begin{aligned}
\mathcal{R}&=\frac{\sigma_{B_{c}^{(*)}(2S)^{+}}}{\sigma_{{{B}_{c}^{+}}}}\cdot{\mathcal{B}}(B_{c}^{(*)}(2S)^{+}\rightarrow{B}_{c}^{(*)+}{{\pi}^{+}}{{\pi}^{-}})\\\
&=\frac{N_{B_{c}^{(*)}(2S)^{+}}}{N_{{{B}_{c}^{+}}}}\cdot\frac{\varepsilon_{{{B}_{c}^{+}}}}{\varepsilon_{B_{c}^{(*)}(2S)^{+}}},\end{aligned}$
(2)

where $\sigma$ is the production cross-section, $N$ the yield, and
$\varepsilon$ the efficiency of reconstructing and selecting the
${{B}_{c}^{+}}$ or ${{B}_{c}^{(*)}(2S)^{+}}$ candidates in the required
$p_{\mathrm{T}}$ and $y$ regions. In the case $\Delta M=0$, the reconstructed
${{B}_{c}(2S)^{+}}$ and ${{B}_{c}^{*}(2S)^{+}}$ states fully overlap, and the
ratio $\mathcal{R}$ corresponds to the sum of the $\mathcal{R}$ values of the
${{B}_{c}(2S)^{+}}$ and ${{B}_{c}^{*}(2S)^{+}}$ states. The upper limits are
calculated using the ${\rm CL_{s}}$ method [36], in which the upper limit for
each mass hypothesis is obtained from the ${\rm CL_{s}}$ value calculated as a
function of the ratio $\mathcal{R}$. The test statistic is the ratio of the
likelihoods of the signal-plus-background hypothesis and the background-only
hypothesis, defined as

$\mathcal{Q}(N_{\rm obs};N_{S},N_{B})=\frac{\mathcal{L}(N_{\rm
obs};N_{S}+N_{B})}{\mathcal{L}(N_{\rm obs};N_{B})},$ (3)

where $N_{\rm obs}$ is the number of observed candidates, $N_{B}$ is the
expected background yield, and $N_{S}$ is the expected signal yield. For a
given value of the ratio $\mathcal{R}$, $N_{S}$ is determined as

$\displaystyle\begin{aligned} N_{S}=\mathcal{R}\cdot
N_{{{B}_{c}^{+}}}\cdot\frac{\varepsilon_{B_{c}^{(*)}(2S)^{+}}}{\varepsilon_{{{B}_{c}^{+}}}}.\end{aligned}$
(4)

The likelihood $\mathcal{L}$ is defined as

$\mathcal{L}(n;x)=\frac{e^{-x}}{n!}x^{n}.$ (5)

The total statistical test value $\mathcal{Q}_{\rm tot}$ is the product of
that for each of the four MLP categories. The ${\rm CL_{s}}$ value is the
ratio of ${\rm CL_{s+b}}$ to ${\rm CL_{b}}$, where ${\rm CL_{s+b}}$ is the
probability to find a $\mathcal{Q}_{\rm tot}$ value smaller than the
$\mathcal{Q}_{\rm tot}$ value found in the data sample under the signal-plus-
background hypothesis, and ${\rm CL_{b}}$ is equivalent probability under the
background-only hypothesis. The ${\rm CL_{s+b}}$ and ${\rm CL_{b}}$ values are
obtained from pseudoexperiments, in which the input variables are varied
within their statistical and systematic uncertainties. The ${{B}_{c}(2S)^{+}}$
state is searched for by scanning the mass region
$M({{B}_{c}^{+}}{{\pi}^{+}}{{\pi}^{-}})\in[6830,6890]{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$,
which is motivated by theoretical predictions [1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
11]. The value of $\Delta M$ is successively fixed to $0$, $15$, $25$ and
$35{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The search windows are within
$\pm 1.4\sigma_{\rm w}({{B}_{c}^{(*)}(2S)^{+}})$ of the
${{B}_{c}^{(*)}(2S)^{+}}$ mass hypotheses. This choice of the search window
gives the best sensitivity according to Ref. [37].

The selection efficiencies $\varepsilon_{{{B}_{c}^{+}}}$ and
$\varepsilon_{B_{c}^{(*)}(2S)^{+}}$ are estimated using simulation. The track
reconstruction efficiency is studied in a data control sample of
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\rightarrow{\mu^{+}}{\mu^{-}}$
decays using a tag-and-probe technique [38], in which one of the muons is
fully reconstructed as the tag track, and the other muon, the probe track, is
reconstructed using only information from the TT detector and the muon
stations. The track reconstruction efficiency is the fraction of
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ candidates whose probe tracks
match fully reconstructed tracks. The particle-identification (PID) efficiency
of the two opposite-charge pions is determined with a data-driven method,
using a ${{\pi}^{+}}$ sample from $D^{*}$-tagged
$D^{0}\rightarrow{{K}^{-}}{{\pi}^{+}}$ decays. The total efficiency
$\varepsilon_{{{B}_{c}^{+}}}$ is determined to be $0.0931\pm 0.0005$, where
the uncertainty is the statistical uncertainty of the simulated sample. The
${{B}_{c}^{(*)}(2S)^{+}}$ efficiencies obtained from the default simulation,
where $M({{B}_{c}(2S)^{+}})=6858{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and
$M({{B}_{c}^{*}(2S)^{+}})=6890{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, are
summarised in Table 1. The variation of the efficiencies with respect to
$M({{B}_{c}(2S)^{+}})$ and $M({{B}_{c}^{*}(2S)^{+}})$, assumed to be linear,
is studied using the data simulated with different mass settings. This
variation is considered when searching for the ${{B}_{c}^{(*)}(2S)^{+}}$
states at other masses. The expected background yield in each of the
${{B}_{c}^{(*)}(2S)^{+}}$ signal regions, $N_{B}$, is estimated via
extrapolation from the $M({{B}_{c}^{+}}{{\pi}^{+}}{{\pi}^{-}})$ sidebands for
each MLP category. The background is modelled by an empirical threshold
function as shown in Fig. 3, where the threshold is taken to be
$M({{B}_{c}^{+}})+M({{\pi}^{+}})+M({{\pi}^{-}})=6555{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.
The other parameters are fixed according to the
$M({{B}_{c}^{+}}{{\pi}^{+}}{{\pi}^{-}})$ distribution of the same-sign sample,
which is constructed with ${{B}_{c}^{+}}{{\pi}^{+}}{{\pi}^{+}}$ or
${{B}_{c}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ combinations.

Table 1:  Efficiencies for the ${{B}_{c}^{(*)}(2S)^{+}}$ states in the regions
$\mbox{$p_{\mathrm{T}}$}\in[0,20]{\mathrm{\,Ge\kern-0.90005ptV\\!/}c}$ and
$y\in[2.0,4.5]$ for each MLP category. The efficiencies obtained before
applying the MLP classifier are $0.0091\pm 0.0002$ and $0.0086\pm 0.0001$ for
${{B}_{c}(2S)^{+}}$ and ${{B}_{c}^{*}(2S)^{+}}$, respectively. The
uncertainties are statistical only, and are due to the limited size of the
simulated sample.

MLP category | $(0.02,0.2)$ | $[0.2,0.4)$ | $[0.4,0.6)$ | $[0.6,1.0]$
---|---|---|---|---
| Efficiencies in %
${{B}_{c}(2S)^{+}}$ | $0.148\pm 0.006$ | $0.140\pm 0.006$ | $0.130\pm 0.006$ | $0.256\pm 0.008$
${{B}_{c}^{*}(2S)^{+}}$ | $0.118\pm 0.003$ | $0.140\pm 0.004$ | $0.144\pm 0.004$ | $0.288\pm 0.005$

(a) MLP category: (0.02,0.2)  (b) MLP category: [0.2,0.4)  (c) MLP category:
[0.4,0.6)  (d) MLP category: [0.6,1.0]

Figure 3: The $M({{B}_{c}^{+}}{{\pi}^{+}}{{\pi}^{-}})$ distributions in the
same-sign (darkgreen shaded areas) and data (points with error bars) samples
in the range $[6600,7300]{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ with the
background model (blue solid line) overlaid, for the four MLP categories. The
areas between the two vertical red lines are the signal regions.

The sources of systematic uncertainties that affect the upper limit
calculation are studied and summarised in Table 2. The systematic uncertainty
on $N_{{{B}_{c}^{+}}}$ comes from the potentially imperfect modelling of the
signal, and has been studied using pseudoexperiments. The uncertainty on
$\varepsilon_{{{B}_{c}^{+}}}$ is due to the limited size of the simulated
sample. The uncertainty on $N_{B}$ comes both from differences between the
combinatorial backgrounds in the opposite-sign and the same-sign data samples
and from the potential mismodelling of the background. The former is studied
by performing a large set of pseudoexperiments, in which the samples are
generated by randomly taking candidates from the data sample, while the
candidates in
$M({{B}_{c}^{+}}{{\pi}^{+}}{{\pi}^{-}})\in[6785,6900]{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
are taken from the same-sign sample. The
$M({{B}_{c}^{+}}{{\pi}^{+}}{{\pi}^{-}})$ distributions of the pseudosamples
are fit using the same function as in the nominal background modelling. The
difference between the mean value of $N_{B}$ obtained from the
pseudoexperiments and the nominal value is taken as the systematic
uncertainty. The potential mismodelling of the background is estimated by
using the Bukin function [39] as an alternative model and the differences to
the nominal results are taken as systematic uncertainties. The uncertainties
on $\varepsilon_{{{B}_{c}^{(*)}(2S)^{+}}}$ are dominated by the uncertainty
due to the finite size of the simulated samples, but also include the
systematic uncertainties on the PID and track reconstruction efficiency
calibration, which come from the limited size and the binning scheme of the
calibration samples. The variations of efficiency with respect to
$M({{B}_{c}(2S)^{+}})$ and $M({{B}_{c}^{*}(2S)^{+}})$ are fitted with linear
functions, and the uncertainties of such fits are taken as systematic
uncertainties.

Table 2:  Summary of the systematic uncertainties entering the upper limit
calculation for the four MLP categories.

MLP category | $(0.02,0.2)$ | $[0.2,0.4)$ | $[0.4,0.6)$ | $[0.6,1.0]$
---|---|---|---|---
$N_{{{B}_{c}^{+}}}$ | 1.0%
$\varepsilon_{{{B}_{c}^{+}}}$ | 0.5%
$N_{B}$ | $4.2\%$ | $9.0\%$ | $15.0\%$ | $6.9\%$
${{B}_{c}(2S)^{+}}\\!\rightarrow{{B}_{c}^{+}}{{\pi}^{+}}{{\pi}^{-}}$
$\varepsilon_{{{B}_{c}(2S)^{+}}}$ | $4.6\%$ | $4.7\%$ | $4.9\%$ | $3.6\%$
Efficiency variation vs. $M({{B}_{c}(2S)^{+}})$ | $0.6\%$ | $1.3\%$ | $1.8\%$ | $2.7\%$
${{B}_{c}^{*}(2S)^{+}}\\!\rightarrow{{B}_{c}^{*+}}{{\pi}^{+}}{{\pi}^{-}}$
$\varepsilon_{{{B}_{c}^{*}(2S)^{+}}}$ | $3.5\%$ | $3.3\%$ | $3.3\%$ | $2.7\%$
Efficiency variation vs. $M({{B}_{c}^{*}(2S)^{+}})$ | $1.0\%$ | $1.8\%$ | $2.5\%$ | $4.3\%$

No evidence of the ${{B}_{c}^{(*)}(2S)^{+}}$ signal is observed. The
measurement is consistent with the background-only hypothesis for all mass
assumptions. The upper limits at $90\%$ and $95\%$ confidence levels (CL) on
the ratio $\mathcal{R}$, as functions of the ${{B}_{c}^{(*)}(2S)^{+}}$ mass
states, are shown in Fig. 4. All the upper limits at 95% CL on the ratio
$\mathcal{R}$ are contained between $0.02$ and $0.14$. Theoretical models
predict that the ratio $\mathcal{R}$ has no significant dependence on $y$ and
$p_{\mathrm{T}}$ of the ${{B}_{c}^{+}}$ mesons [19], allowing comparison with
the ATLAS result [18]. The most probable interpretation of the ATLAS
measurement is that it is either the ${{B}_{c}^{*}(2S)^{+}}$ state or a sum of
${{B}_{c}(2S)^{+}}$ and ${{B}_{c}^{*}(2S)^{+}}$ signals under the $\Delta
M\sim 0$ scenario. For both interpretations of the ATLAS measurement, the
comparison of the ratio $\mathcal{R}$ between the LHCb upper limits in the
vicinity of the peak claimed by ATLAS at
$M(B_{c}^{(*)}(2S)^{+})=6842~{}{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and
the ratios determined by ATLAS are given in Table 3. The LHCb and ATLAS
results are compatible only in case of very large (unpublished) relative
efficiency of reconstructing the ${{B}_{c}^{(*)}(2S)^{+}}$ candidates with
respect to the ${{B}_{c}^{+}}$ signals for the ATLAS measurement.

(a) $\Delta M=0{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ (b) $\Delta
M=15{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ (c) $\Delta
M=25{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ (d) $\Delta
M=35{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$

Figure 4: The upper limits on the ratio $\mathcal{R}({{B}_{c}^{(*)}(2S)^{+}})$
at $95\%$ and $90\%$ confidence levels under different mass splitting $\Delta
M$ hypotheses.  Table 3:  Comparison of the $\mathcal{R}$ value between the
LHCb upper limits at 95% CL and the ATLAS measurement [18], where
$0<\varepsilon_{7,8}\leq 1$ are the relative efficiencies of reconstructing
the ${{B}_{c}^{(*)}(2S)^{+}}$ candidates with respect to the ${{B}_{c}^{+}}$
signals for the 7 and $8{\mathrm{\,Te\kern-0.90005ptV}}$ data, respectively.

| $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$ | $\sqrt{s}=8\mathrm{\,Te\kern-1.00006ptV}$
---|---|---
ATLAS | $(0.22\pm 0.08\mathrm{\,(stat)})/\varepsilon_{7}$ | $(0.15\pm 0.06\mathrm{\,(stat)})/\varepsilon_{8}$
LHCb | – | $<[0.04,0.09]$

##  5 Summary

In summary, a search for the ${B}_{c}(2S)^{+}$ and ${B}_{c}^{*}(2S)^{+}$
states is performed at LHCb with a data sample of $pp$ collisions,
corresponding to an integrated luminosity of $2\mbox{\,fb}^{-1}$, recorded at
a centre-of-mass energy of 8 TeV. No significant signal is found. Upper limits
on the ${B}_{c}(2S)^{+}$ and ${B}_{c}^{*}(2S)^{+}$ production cross-sections
times the branching fraction of
${B}_{c}^{(*)}(2S)^{+}\rightarrow{B}_{c}^{(*)+}\pi^{+}\pi^{-}$ relative to the
${{B}_{c}^{+}}$ cross-section, are given as a function of the
${{B}_{c}(2S)^{+}}$ and ${{B}_{c}^{*}(2S)^{+}}$ masses.

## Acknowledgements

We thank Chao-Hsi Chang and Xing-Gang Wu for frequent and interesting
discussions on the production of the $B_{c}$ mesons. 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); MOST and NSFC (China);
CNRS/IN2P3 (France); BMBF, DFG and MPG (Germany); INFN (Italy); 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), ANR, Labex P2IO, ENIGMASS
and OCEVU, and Région Auvergne-Rhône-Alpes (France), RFBR and Yandex LLC
(Russia), GVA, XuntaGal and GENCAT (Spain), Herchel Smith Fund, the Royal
Society, the English-Speaking Union and the Leverhulme Trust (United Kingdom).

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LHCb collaboration

R. Aaij40, B. Adeva39, M. Adinolfi48, Z. Ajaltouni5, S. Akar59, J. Albrecht10,
F. Alessio40, M. Alexander53, A. Alfonso Albero38, S. Ali43, G. Alkhazov31, P.
Alvarez Cartelle55, A.A. Alves Jr59, S. Amato2, S. Amerio23, Y. Amhis7, L.
An3, L. Anderlini18, G. Andreassi41, M. Andreotti17,g, J.E. Andrews60, R.B.
Appleby56, F. Archilli43, P. d’Argent12, J. Arnau Romeu6, A. Artamonov37, M.
Artuso61, E. Aslanides6, M. Atzeni42, G. Auriemma26, M. Baalouch5, I.
Babuschkin56, S. Bachmann12, J.J. Back50, A. Badalov38,m, C. Baesso62, S.
Baker55, V. Balagura7,b, W. Baldini17, A. Baranov35, R.J. Barlow56, C.
Barschel40, S. Barsuk7, W. Barter56, F. Baryshnikov32, V. Batozskaya29, V.
Battista41, A. Bay41, L. Beaucourt4, J. Beddow53, F. Bedeschi24, I. Bediaga1,
A. Beiter61, L.J. Bel43, N. Beliy63, V. Bellee41, N. Belloli21,i, K. Belous37,
I. Belyaev32,40, E. Ben-Haim8, G. Bencivenni19, S. Benson43, S. Beranek9, A.
Berezhnoy33, R. Bernet42, D. Berninghoff12, E. Bertholet8, A. Bertolin23, C.
Betancourt42, F. Betti15, M.O. Bettler40, M. van Beuzekom43, Ia. Bezshyiko42,
S. Bifani47, P. Billoir8, A. Birnkraut10, A. Bizzeti18,u, M. Bjørn57, T.
Blake50, F. Blanc41, S. Blusk61, V. Bocci26, T. Boettcher58, A. Bondar36,w, N.
Bondar31, I. Bordyuzhin32, S. Borghi56,40, M. Borisyak35, M. Borsato39, F.
Bossu7, M. Boubdir9, T.J.V. Bowcock54, E. Bowen42, C. Bozzi17,40, S. Braun12,
J. Brodzicka27, D. Brundu16, E. Buchanan48, C. Burr56, A. Bursche16,f, J.
Buytaert40, W. Byczynski40, S. Cadeddu16, H. Cai64, R. Calabrese17,g, R.
Calladine47, M. Calvi21,i, M. Calvo Gomez38,m, A. Camboni38,m, P. Campana19,
D.H. Campora Perez40, L. Capriotti56, A. Carbone15,e, G. Carboni25,j, R.
Cardinale20,h, A. Cardini16, P. Carniti21,i, L. Carson52, K. Carvalho Akiba2,
G. Casse54, L. Cassina21, M. Cattaneo40, G. Cavallero20,40,h, R. Cenci24,t, D.
Chamont7, M.G. Chapman48, M. Charles8, Ph. Charpentier40, G.
Chatzikonstantinidis47, M. Chefdeville4, S. Chen16, S.F. Cheung57, S.-G.
Chitic40, V. Chobanova39, M. Chrzaszcz42, A. Chubykin31, P. Ciambrone19, X.
Cid Vidal39, G. Ciezarek40, P.E.L. Clarke52, M. Clemencic40, H.V. Cliff49, J.
Closier40, V. Coco40, J. Cogan6, E. Cogneras5, V. Cogoni16,f, L. Cojocariu30,
P. Collins40, T. Colombo40, A. Comerma-Montells12, A. Contu16, G. Coombs40, S.
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G.A. Cowan52, D.C. Craik58, A. Crocombe50, M. Cruz Torres1, R. Currie52, C.
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Massafferri1, R. Matev40, A. Mathad50, Z. Mathe40, C. Matteuzzi21, A. Mauri42,
E. Maurice7,b, B. Maurin41, A. Mazurov47, M. McCann55,40, A. McNab56, R.
McNulty13, J.V. Mead54, B. Meadows59, C. Meaux6, F. Meier10, N. Meinert67, D.
Melnychuk29, M. Merk43, A. Merli22,40,q, E. Michielin23, D.A. Milanes66, E.
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Rodriguez1, T. Mombächer10, I.A. Monroy66, S. Monteil5, M. Morandin23, M.J.
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F. Ratnikov35, G. Raven44, M. Ravonel Salzgeber40, M. Reboud4, F. Redi41, S.
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S. Richards48, M. Rihl40, K. Rinnert54, P. Robbe7, A. Robert8, A.B.
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M.S. Rudolph61, T. Ruf40, P. Ruiz Valls70, J. Ruiz Vidal70, J.J. Saborido
Silva39, E. Sadykhov32, N. Sagidova31, B. Saitta16,f, V. Salustino
Guimaraes62, C. Sanchez Mayordomo70, B. Sanmartin Sedes39, R. Santacesaria26,
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Schubiger41, M.H. Schune7, R. Schwemmer40, B. Sciascia19, A. Sciubba26,k, A.
Semennikov32, E.S. Sepulveda8, A. Sergi47, N. Serra42, J. Serrano6, L.
Sestini23, P. Seyfert40, M. Shapkin37, I. Shapoval45, Y. Shcheglov31, T.
Shears54, L. Shekhtman36,w, V. Shevchenko68, B.G. Siddi17, R. Silva
Coutinho42, L. Silva de Oliveira2, G. Simi23,o, S. Simone14,d, M. Sirendi49,
N. Skidmore48, T. Skwarnicki61, I.T. Smith52, J. Smith49, M. Smith55, l.
Soares Lavra1, M.D. Sokoloff59, F.J.P. Soler53, B. Souza De Paula2, B.
Spaan10, P. Spradlin53, S. Sridharan40, F. Stagni40, M. Stahl12, S. Stahl40,
P. Stefko41, S. Stefkova55, O. Steinkamp42, S. Stemmle12, O. Stenyakin37, M.
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Tilburg43, M.J. Tilley55, V. Tisserand5, M. Tobin41, S. Tolk49, L.
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Zhang3, Y. Zhang7, A. Zhelezov12, Y. Zheng63, X. Zhu3, V. Zhukov9,33, J.B.
Zonneveld52, S. Zucchelli15.

1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, IN2P3-LAPP, Annecy,
France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6Aix Marseille Univ, CNRS/IN2P3, CPPM, Marseille, France
7LAL, Univ. Paris-Sud, CNRS/IN2P3, Université Paris-Saclay, 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
17Universita e INFN, 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 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
39Instituto Galego de Física de Altas Enerxías (IGFAE), Universidade 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
69National Research Tomsk Polytechnic University, Tomsk, Russia, associated to
32
70Instituto de Fisica Corpuscular, Centro Mixto Universidad de Valencia -
CSIC, Valencia, Spain, associated to 38
71Van Swinderen Institute, University of Groningen, Groningen, The
Netherlands, associated to 43
72Los Alamos National Laboratory (LANL), Los Alamos, United States, associated
to 61

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, Vietnam
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


Dark Sector searches with jets

J. von Ahnen1$\star$, on behalf of the ATLAS and CMS Collaborations

1 DESY, Hamburg, Germany

*<EMAIL_ADDRESS>
50th International Symposium on Multiparticle Dynamics
(ISMD2021)
12-16 July 2021 10.21468/SciPostPhysProc.?

## Abstract

The presence of a non-baryonic Dark Matter (DM) component in the Universe is
inferred from the observation of its gravitational interaction. The ATLAS and
CMS experiments located at the LHC have developed a broad search program for
DM candidates, including resonance searches for the mediator and searches with
large missing transverse momentum. Additionally, searches have been conducted
in models where the Higgs Sector and the Dark Sector are connected leading for
example to invisible Higgs boson decays. The results of recent searches on 13
TeV $pp$ data, their interplay and interpretation are presented.

##  1 Introduction

Several astrophysical observations exist that can not be explained with the
particle contents of the standard model (SM) of particle physics [1]. One
approach to explain these observations is the introduction of non-baryonic
Dark Matter (DM), which could explain the phenomenons with its gravitational
impact. This DM does not interact via the strong or the electromagnetic force.

There are three different categories of experiments that try to detect this
DM. One of these is the category of indirect detection experiments in which
cosmic particles are investigated to look for hints of DM annihilation. The
second category is direct detection. Here one looks for DM by measuring the
recoil of SM particles coming from potential scattering with DM. The last type
of search looks for the production of DM using particle colliders.

The LHC is a proton-proton collider located at CERN in Geneva. With its high
centre-of-mass energy of 13 TeV, it could produce DM in its collisions. ATLAS
[2] and CMS [3] are two general-purpose detectors at the LHC [4] and have a
broad program of searches for DM. In high energy collisions of protons, the
presence of jets is very likely. Therefore, it is crucial to explore and
understand final states with jets.

To interpret and guide analyses, models that extend the SM can be a good tool.
Simplified models are minimal extensions of the SM that generally have a rich
phenomenology and small number of free parameters.

##  2 $E_{\text{T}}^{\text{miss}}$ +X searches

Detectors like ATLAS or CMS can not directly detect DM. Therefore, to identify
events with DM, other SM particles have to be present. These events lead to
event topologies in which there is an imbalance in the transverse momentum
($E_{\text{T}}^{\text{miss}}$). Analyses that target these events are called
$E_{\text{T}}^{\text{miss}}$ +X searches. The $E_{\text{T}}^{\text{miss}}$
+jet search is one of these [5, 6].

###  2.1 $E_{\text{T}}^{\text{miss}}$ +jet strategy

The signal signature has large $E_{\text{T}}^{\text{miss}}$ ($\gtrapprox
200\text{Ge\kern-1.00006ptV}$) recoiling against a jet with high transverse
momentum. No leptons are expected in the event.

The major backgrounds for this signature are Z+jets, W+jets and
$t\bar{t}$-production. To minimize the uncertainties on the prediction of
these backgrounds, control regions (CR) are constructed. In these CRs the
lepton requirements are changed in order to increase the contribution from
backgrounds and discard the majority of signal events.

###  2.2 $E_{\text{T}}^{\text{miss}}$ +jet results

Figure 1(a) shows the post fit distribution of the total transverse momentum
of all visible objects ($p_{\text{T}}^{\text{recoil}}$). This analysis can be
interpreted in the context of different beyond the SM theories, e.g. invisible
Higgs boson decays, large extra spatial dimensions or supersymmetric
particles. Figure 1(b) shows the exclusion contour on a simplified model with
an Axial-vector mediator.

(a)

(b)

Figure 1: (1(a)) Measured distributions of $p_{\text{T}}^{\text{recoil}}$
compared with the SM prediction in the SR. (1(b)) 95% CL exclusion contours in
the $m_{Z_{A}}-m_{\chi}$ parameter plane for the axial-vector mediator model
[5].

##  3 Resonance searches

A new mediator that couples both to SM and DM particles, and that can be
produced in collisions at the LHC would also have a resonant signature. This
would lead to an increased number of events where the collision energy is
close to the mediator mass.

###  3.1 Dijet resonance search strategy

The dijet resonance search analysis looks for the enhanced number of events in
the dijet mass spectrum [7, 8]. In the SM this is a smoothly falling spectrum
and can therefore be described by an analytical function. A new resonance
would create a bump on this spectrum.

###  3.2 Results

Because of the high event rate for dijet events in the low dijet mass region
($\lessapprox 1.5\,\mathrm{TeV}$), not all collisions can be recorded. This
leads to a limited sensitivity for the dijet resonance analysis relying on two
separated jets. By for example requiring additional objects in the events
(e.g. photons) also the lower dijet mass region can be explored. Figure 2
shows a summary plot for dijet resonance searches produced by the CMS
collaboration.

Figure 2: Limits on the universal coupling $g^{\prime}_{q}$ between a
leptophobic $Z^{\prime}$ boson and quarks [9] from various CMS dijet analyses
[10].

The dijet analyses and the $E_{\text{T}}^{\text{miss}}$ +X searches provide a
complementary way of exploring certain simplified models. Figure 3(a) shows
the 95% exclusion contours for a simplified model with a spin-1 mediator. The
exclusion contours depend on the choice of parameters.

The limits on the mediator mass can also be translated into DM-nucleon
scattering cross-section limits which allows the results to be compared with
those from direct detection experiments as shown in figure 3(b). Again the
exclusion contours depend on the assumptions and choice of parameters.

(a)

(b)

Figure 3: (3(a)) 95% CL exclusion contours for dijet, dilepton and
$E_{\text{T}}^{\text{miss}}$ +X searches for leptophilic vector mediator
simplified models. (3(b)) Comparison of spin-independent WIMP-nucleon
scattering cross-section for various DM searches performed by ATLAS and limits
from direct detection experiments [11].

##  4 Search for invisible Higgs boson decays

The Higgs boson couples to all known particles with mass and could therefore
also couple to DM. Models in which the Higgs boson connects the SM particles
with the Dark Sector are called Higgs-portal models. In this case, the
branching ratio of the Higgs boson to invisible particles
($\mathcal{BR}_{\text{inv}}$) would be larger than predicted by the SM. In
general, analyses focus on one of the four Higgs boson production modes
(gluon-gluon fusion, vector-boson fusion (VBF), $t\bar{t}$-fusion,
Higgsstrahlung) and measure $\mathcal{BR}_{\text{inv}}$. The analyses focusing
on the VBF production are the most sensitive [12, 13].

###  4.1 VBF-H(inv) strategy

The signal signature consists of two jets with a large separation in
pseudorapidity ($\eta\gtrapprox 3.8$) and a high invariant dijet mass
($m_{jj}\gtrapprox 800\,\mathrm{GeV}$). No isolated leptons are expected and
the invisible Higgs boson decay results in large $E_{\text{T}}^{\text{miss}}$
($\gtrapprox 200\,\text{Ge\kern-1.00006ptV}$). The major backgrounds are
W+jets and Z+jets production. These are constraints by CRs, which are
constructed by inverting the lepton veto of the SR.

###  4.2 Results

No excess from the SM is observed, and upper limits on
$\mathcal{BR}_{\text{inv}}$ are set. The best upper limit
($\mathcal{BR}_{\text{inv}}>11\%$) comes from the preliminary combination of
Run 1 and Run 2 ATLAS results (see figure 4(a)) [14]. With certain
assumptions, the upper limit on $\mathcal{BR}_{\text{inv}}$ can be translated
to a WIMP-nucleon scattering cross-section and then compared to the limits
achieved by direct detection experiments (see figure 4(b)).

(a)

(b)

Figure 4: (4(a)) Preliminary combination of Run 1 and Run 2 ATLAS results for
invisible Higgs boson decays [14]. (4(b)) Comparison of upper limits on the
WIMP-nucleon scattering cross-section for direct detection experiments and a
Higgs portal interpretation [12].

##  5 Models with extended Higgs sector

Several extensions of the SM involve changes to the Higgs sector. The Two-
Higgs-Doublet model (2HDM) extends the SM with a second Higgs doublet [15].
One of the motivations for the 2HDM is that it could solve the problem of the
observed baryon asymmetry in the universe. The analysis focusing on DM
produced in association with a SM Higgs boson decaying to $b$-quarks is
sensitive to these models [16, 17].

###  5.1 Mono-H($bb$)+$E_{\text{T}}^{\text{miss}}$ strategy

The signal signature is two $b$-jets with an invariant mass close to the SM
Higgs boson mass ($\approx 125\,\mathrm{GeV}$). Furthermore, this di-$b$-jet
system is recoiling against large $E_{\text{T}}^{\text{miss}}$ ($\gtrapprox
150\,\mathrm{GeV}$) and no leptons are present in the event.

The major backgrounds are Z+jets, W+jets and $t\bar{t}$-production. By
inverting the lepton requirements of the SR, two CRs are defined and included
in the fit to constrain the normalisation of the major backgrounds.

###  5.2 Mono-H($bb$)+$E_{\text{T}}^{\text{miss}}$ results

The data show no significant deviation from the SM prediction. Figure 5 shows
the 95% confidence level exclusion contour for the 2HDM+a model [18].

Figure 5: 95% CL exclusion contours for the 2HDM+a signal [16].

##  6 Conclusion

A wide range of analyses focusing on final states with jets to explore the
Dark Sector is presented. The data used in these analyses was collected by the
ATLAS and CMS detectors at the LHC. Both experiments have collected $\approx
140\,\mathrm{fb}^{-1}$ of proton-proton collision data at a centre-of-mass
energy of 13 TeV during Run 2.

The data show no evidence for DM. Besides preparing for the upcoming Run 3 to
collect more data the two collaborations are also working on improving their
analysis techniques as well as exploring new final states.

## Copyright

Copyright 2021 CERN for the benefit of the ATLAS and CMS Collaborations.
Reproduction of this article or parts of it is allowed as specified in the CC-
BY-4.0 license.

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