EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2019-258 LHCb-PAPER-2019-037 February 11, 2020

Precision measurement of the ${{\mathchar 28932\relax}^{++}_{{c}{c}}}$ mass

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

A measurement of the ${\mathchar 28932\relax}^{++}_{{c}{c}}$ mass is performed
using data collected by the LHCb experiment between 2016 and 2018 in $p$ $p$
collisions at a centre-of-mass energy of 13 TeV, corresponding to an
integrated luminosity of $5.6\text{\,fb}^{-1}$. The ${{\mathchar
28932\relax}^{++}_{{c}{c}}}$ candidates are reconstructed via the decay modes
${{\mathchar 28932\relax}^{++}_{{c}{c}}}\\!\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ and ${{\mathchar
28932\relax}^{++}_{{c}{c}}}\rightarrow{{\mathchar
28932\relax}^{+}_{c}}{{\pi}^{+}}$. The result, $3621.55\,\pm
0.23\,{\rm(stat)\,}\pm 0.30\,{\rm(syst)}$
$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$, is the most precise measurement of the
${{\mathchar 28932\relax}^{++}_{{c}{c}}}$ mass to date.

Published in JHEP 02 (2020) 049

© 2024 CERN for the benefit of the LHCb collaboration. CC-BY-4.0 licence.

##  1 Introduction

Baryons containing two charm quarks and a lighter quark are predicted by the
quark model [1, 2, *Zweig:570209] and provide an ideal system to study the
dynamics of bound states of quarks. Observation of the ${{\mathchar
28932\relax}^{++}_{{c}{c}}}$ ($ccu$) baryon via decays to ${{\mathchar
28931\relax}^{+}_{c}}{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ and ${{\mathchar
28932\relax}^{+}_{c}}{{\pi}^{+}}$ final states has been reported by the LHCb
collaboration [4, 5].111The inclusion of charge conjugate modes is implied
throughout this paper. The ${{\mathchar 28932\relax}^{++}_{{c}{c}}}$ mass was
measured by the LHCb collaboration to be $3621.24\pm 0.65\text{\,(stat)}\pm
0.31\text{\,(syst)}\text{\,Me\kern-1.00006ptV\\!/}c^{2}$. Before the LHCb
observation, theoretical calculations using quark models[6, 7, 8], bag models
[9], the Faddeev method [10], quantum chromodynamics (QCD) sum rules[11, 12,
13, 14], potential models [15] and lattice QCD [16, 17, 18] predicted the mass
of this state in the range 3450–3750$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$.
Most of the predictions using quark models are around
3600$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ while other methods have a larger
spread. Theoretical calculations of the ${{\mathchar
28932\relax}^{++}_{{c}{c}}}$ mass [19, 20, 21, 22] after the LHCb observation
fall into a $\pm 20\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ window around the
experimental value measured by LHCb.

At present, the experimental uncertainty on the ${\mathchar
28932\relax}^{++}_{{c}{c}}$ mass is still large compared to that of the singly
charmed baryons. This paper presents an updated measurement of the ${\mathchar
28932\relax}^{++}_{{c}{c}}$ mass using the decay modes ${{\mathchar
28932\relax}^{++}_{{c}{c}}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}(\rightarrow{p}{{K}^{-}}{{\pi}^{+}}){{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$
and ${{\mathchar 28932\relax}^{++}_{{c}{c}}}\rightarrow{{\mathchar
28932\relax}^{+}_{c}}(\rightarrow{p}{{K}^{-}}{{\pi}^{+}}){{\pi}^{+}}$. The
analysis uses a data sample corresponding to an integrated luminosity of
5.6$\text{\,fb}^{-1}$, collected by the LHCb experiment during 2016–2018 in
$p$ $p$ collisions at a centre-of-mass energy of 13 TeV. This measurement
supersedes the results reported on the ${\mathchar 28932\relax}^{++}_{{c}{c}}$
mass in Refs. [4, 5], which only use $pp$ collision data at 13 TeV taken in
2016, corresponding to an integrated luminosity of 1.7$\text{\,fb}^{-1}$.

##  2 Detector and simulation

The LHCb detector [23, 24] 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 [25], a large-area silicon-strip detector located upstream
of a dipole magnet with a bending power of about $4{\mathrm{\,Tm}}$, and three
stations of silicon-strip detectors and straw drift tubes [26, 27] placed
downstream of the magnet. The tracking system provides a measurement of the
momentum of charged particles with a relative uncertainty that varies from
0.5% at low momentum to 1.0% at 200$\text{\,Ge\kern-1.00006ptV\\!/}c$. The
momentum scale is calibrated using samples of
${{B}^{+}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}K^{+}$ and
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\rightarrow\mu^{+}\mu^{-}$
decays collected concurrently with the data sample used for this analysis [28,
29]. The relative accuracy of this procedure is estimated to be $3\times
10^{-4}$ using samples of other fully reconstructed ${b}$-hadron, $\mathchar
28935\relax$ and ${{K}^{0}_{\mathrm{S}}}$ decays. The minimum distance of a
track to a primary $pp$ collision vertex (PV), the impact parameter (IP), is
measured with a resolution of
$(15+29/(p_{\mathrm{T}}/\text{\,Ge\kern-1.00006ptV\\!/}c))\,\upmu\text{m}$,
where $p_{\mathrm{T}}$ is the momentum component transverse to the beam axis.
Different types of charged hadrons are distinguished using information from
two ring-imaging Cherenkov detectors [30]. Photons, electrons and hadrons are
identified by a calorimeter system consisting of scintillating-pad and
preshower detectors, an electromagnetic and a hadronic calorimeter. Muons are
identified by a system composed of alternating layers of iron and multiwire
proportional chambers [31].

The online event selection is performed by a trigger [32], 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. In
between the two software stages, an alignment and calibration of the detector
is performed in near real-time [33]. This process allows the reconstruction of
the ${\mathchar 28932\relax}^{++}_{{c}{c}}$ decays to be performed entirely in
the software trigger, whose output is used as input to the present analysis.

Simulated samples are used to model the effects of the detector acceptance,
optimise selections and verify the validity of the methods used in the
measurement. In the simulation, $p$ $p$ collisions are generated using Pythia
8 [34] with a LHCb specific configuration [35]. The production of doubly
charmed ${{\mathchar 28932\relax}^{++}_{{c}{c}}}$ baryons is simulated using
the dedicated generator GenXicc2.0 [36, *Wang:2012vj]. Decays of hadrons are
described by EvtGen [38], in which final-state radiation is generated using
Photos [39]. The interaction of the generated particles with the detector, and
its response, are implemented using the Geant4 toolkit [40,
*Agostinelli:2002hh] as described in Ref. [42]. Sources of background, such as
those from ${{\mathchar
28932\relax}^{++}_{{c}{c}}}\rightarrow{\it\Xi^{{}^{\prime}+}_{c}}(\rightarrow{{\mathchar
28932\relax}^{+}_{c}}\gamma){{\pi}^{+}}$ and ${{\mathchar
28932\relax}^{++}_{{c}{c}}}\rightarrow{{\mathchar
28932\relax}^{+}_{c}}{{\rho}^{+}}(\rightarrow{{\pi}^{+}}{{\pi}^{0}})$, are
studied using the fast simulation package RapidSim [43].

##  3 Event selection

The reconstruction of ${{\mathchar
28932\relax}^{++}_{{c}{c}}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}(\rightarrow{p}{{K}^{-}}{{\pi}^{+}}){{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$
and ${{\mathchar 28932\relax}^{++}_{{c}{c}}}\rightarrow{{\mathchar
28932\relax}^{+}_{c}}(\rightarrow{p}{{K}^{-}}{{\pi}^{+}}){{\pi}^{+}}$ decays
is similar to that used in previous LHCb analyses [4, 5]. Candidate
${{\mathchar 28932\relax}^{+}_{c}}({{\mathchar
28931\relax}^{+}_{c}})\rightarrow pK^{-}{{\pi}^{+}}$ decays are reconstructed
from three charged particles identified as a $p$, $K^{-}$ and ${{\pi}^{+}}$
using information from the RICH detectors [30]. The charged particles are
required to form a good-quality vertex and be inconsistent with originating
from any PV. The ${{\mathchar 28932\relax}^{+}_{c}}({{\mathchar
28931\relax}^{+}_{c}})$ candidate is then combined with one (three) additional
charged particle(s) to form a ${{\mathchar
28932\relax}^{++}_{{c}{c}}}\rightarrow{{\mathchar
28932\relax}^{+}_{c}}{{\pi}^{+}}$ (${{\mathchar
28932\relax}^{++}_{{c}{c}}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$) decay candidate. These
additional particles must form a good-quality vertex with the ${{\mathchar
28932\relax}^{+}_{c}}({{\mathchar 28931\relax}^{+}_{c}})$ candidate, which is
required to be upstream of the ${{\mathchar 28932\relax}^{+}_{c}}({{\mathchar
28931\relax}^{+}_{c}})$ decay vertex. Each ${\mathchar
28932\relax}^{++}_{{c}{c}}$ candidate is required to have
$p_{\mathrm{T}}>2\text{\,Ge\kern-1.00006ptV\\!/}c$ and to be consistent with
originating from its associated PV. The associated PV is that with respect to
which the ${{\mathchar 28932\relax}^{++}_{{c}{c}}}$ candidate has the smallest
$\chi^{2}_{\text{IP}}$. The $\chi^{2}_{\text{IP}}$ is defined as the
difference in $\chi^{2}$ of the PV fit with and without the particle in
question. To avoid candidates including duplicated tracks, each track pair is
required to have an opening angle larger than 0.5 mrad or a momentum
difference larger than 5% of the minimum momentum of the track pair.

In order to improve the signal purity, multivariate classifiers are trained to
separate signal from background. The choice of classifier algorithms is based
on their performance for each decay mode. A classifier based on the Boosted
Decision Tree (BDT) algorithm [44, 45] implemented in the TMVA toolkit [46,
*TMVA4] is used for the the ${{\mathchar
28932\relax}^{++}_{{c}{c}}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ mode, while a Multilayer
Perceptron (MLP) algorithm [46, *TMVA4] is used for the ${{\mathchar
28932\relax}^{++}_{{c}{c}}}\rightarrow{{\mathchar
28932\relax}^{+}_{c}}{{\pi}^{+}}$ mode. The BDT for the ${{\mathchar
28932\relax}^{++}_{{c}{c}}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ decay is trained with
simulated signal events as a signal proxy and wrong-sign ${\mathchar
28931\relax}^{+}_{c}$ ${K}^{-}$ ${\pi}^{+}$ ${\pi}^{-}$ combinations
(3525–3725$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$) in data, where the two
final-state pions have opposite charges, as a background proxy. Both the
signal and background proxies are required to pass the selection described
above. Variables associated with the ${\mathchar 28932\relax}^{++}_{{c}{c}}$
candidates used in the training include the vertex-fit quality, the
$\chi^{2}_{\text{IP}}$, the angle between the momentum and the displacement
vector, the flight-distance $\chi^{2}$ between the PV and the decay vertex.
The flight-distance $\chi^{2}$ is defined as the $\chi^{2}$ of the hypothesis
that the decay vertex of the candidate coincides with its associated PV.
Variables associated with the decay products of the ${\mathchar
28932\relax}^{++}_{{c}{c}}$ candidates (the ${{\mathchar
28931\relax}^{+}_{c}}$, ${{K}^{-}}$ and ${{\pi}^{+}}$) used in the training
include their $p_{\mathrm{T}}$ and $\chi^{2}_{\text{IP}}$, the vertex-fit
quality of the ${\mathchar 28931\relax}^{+}_{c}$ candidates and the smallest
$p_{\mathrm{T}}$ among the ${\mathchar 28931\relax}^{+}_{c}$ decay products
(${p}$, ${{K}^{-}}$ and ${{\pi}^{+}}$). Particle-identification information
for the final-state particles is also used.

The threshold applied to the classifier response is determined by maximising
the signal significance $S/\sqrt{S+B}$, where $S$ is the expected signal yield
estimated using simulation, and $B$ is the background yield evaluated in the
upper sideband of data (3800–3900$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$)
extrapolated to the signal region
(3607–3635$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$). The multivariate classifier
for the ${{\mathchar 28932\relax}^{++}_{{c}{c}}}\rightarrow{{\mathchar
28932\relax}^{+}_{c}}{{\pi}^{+}}$ decay is developed following the same
strategy as that for the ${{\mathchar
28932\relax}^{++}_{{c}{c}}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ decay.

After the full selection, an event may still contain more than one
${{\mathchar 28932\relax}^{++}_{{c}{c}}}$ candidate. According to studies on
simulated decays and the wrong-sign control sample, multiple candidates in the
same event may form a peaking structure in the mass distribution of the
${{\mathchar 28932\relax}^{++}_{{c}{c}}}$ candidates if they are obtained from
the same final-state tracks, but via swapping two final state tracks (e.g. the
${K}^{-}$ from the ${\mathchar 28931\relax}^{+}_{c}$ decay and the ${K}^{-}$
from the ${\mathchar 28932\relax}^{++}_{{c}{c}}$ decay). In this case, one
candidate is chosen randomly. Other kinds of multiple candidates, which
account for 8% ($<1$%) of the ${{\mathchar
28932\relax}^{++}_{{c}{c}}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ (${{\mathchar
28932\relax}^{++}_{{c}{c}}}\rightarrow{{\mathchar
28932\relax}^{+}_{c}}{{\pi}^{+}}$) signal events, are not removed since they
do not form a peaking background.

##  4 Mass measurement

To improve the mass resolution, the invariant mass of a ${{\mathchar
28932\relax}^{++}_{{c}{c}}}$ candidate is computed as

$\displaystyle m_{\rm cand}({{\mathchar 28932\relax}^{++}_{{c}{c}}})$
$\displaystyle=m({{\mathchar
28931\relax}^{+}_{c}}{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}})-m({{\mathchar
28931\relax}^{+}_{c}})+M_{\rm PDG}({{\mathchar 28931\relax}^{+}_{c}}),$ (1)
$\displaystyle m_{\rm cand}({{\mathchar 28932\relax}^{++}_{{c}{c}}})$
$\displaystyle=m({{\mathchar 28932\relax}^{+}_{c}}{{\pi}^{+}})-m({{\mathchar
28932\relax}^{+}_{c}})+M_{\rm LHCb}({{\mathchar 28932\relax}^{+}_{c}}),$

where $m({{\mathchar 28931\relax}^{+}_{c}}{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}})$
and $m({{\mathchar 28932\relax}^{+}_{c}}{{\pi}^{+}})$ are the reconstructed
${{\mathchar 28932\relax}^{++}_{{c}{c}}}$ masses; $m({{\mathchar
28931\relax}^{+}_{c}})$ and $m({{\mathchar 28932\relax}^{+}_{c}})$ are the
reconstructed ${{\mathchar 28931\relax}^{+}_{c}}$ and ${{\mathchar
28932\relax}^{+}_{c}}$ masses; $M_{\rm PDG}({{\mathchar
28931\relax}^{+}_{c}})$ is the known value of the ${\mathchar
28931\relax}^{+}_{c}$ mass; $M_{\rm LHCb}({{\mathchar 28932\relax}^{+}_{c}})$
is the known value of the ${\mathchar 28932\relax}^{+}_{c}$ mass. The known
value of the ${{\mathchar 28931\relax}^{+}_{c}}$ mass is $2286.46\pm
0.14\text{\,Me\kern-1.00006ptV\\!/}c^{2}$[48, 49], and that of the
${{\mathchar 28932\relax}^{+}_{c}}$ mass is determined to be $2467.97\pm
0.22\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ using $M_{\rm PDG}({{\mathchar
28931\relax}^{+}_{c}})$ and the difference between $m({{\mathchar
28932\relax}^{+}_{c}})$ and $m({{\mathchar 28931\relax}^{+}_{c}})$ of
$181.51\pm 0.14\pm 0.10\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ measured by the
LHCb collaboration [50].

The $m_{\rm cand}({{\mathchar 28932\relax}^{++}_{{c}{c}}})$ mass distributions
of the selected ${{\mathchar 28932\relax}^{++}_{{c}{c}}}$ candidates are shown
in Fig. 1 for the ${{\mathchar
28932\relax}^{++}_{{c}{c}}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ and ${{\mathchar
28932\relax}^{++}_{{c}{c}}}\rightarrow{{\mathchar
28932\relax}^{+}_{c}}{{\pi}^{+}}$ decay modes. The ${{\mathchar
28932\relax}^{++}_{{c}{c}}}$ mass is determined by performing unbinned
extended maximum-likelihood fits to the two $m_{\rm cand}({{\mathchar
28932\relax}^{++}_{{c}{c}}})$ mass distributions. The signal components are
described by a modified Gaussian function with a power-law tail on the left-
hand side of the distribution [51], parameterised as

$f(x;\alpha,N,\bar{x},\sigma)=\begin{cases}e^{-\frac{1}{2}(\frac{x-\bar{x}}{\sigma})^{2}}&{\rm
for\,}\frac{x-\bar{x}}{\sigma}>-\alpha\\\
\left(\frac{N}{\lvert\alpha\rvert}\right)^{N}e^{-\frac{\lvert\alpha\rvert^{2}}{2}}\left(\frac{N}{\lvert\alpha\rvert}-\lvert\alpha\rvert-\frac{x-\bar{x}}{\sigma}\right)^{-N}&{\rm
for\,}\frac{x-\bar{x}}{\sigma}\leq-\alpha.\end{cases}$ (2)

The peak position, $\overline{x}$, and width, $\sigma$, of the function are
allowed to vary in the fit. The power-law tail parameters, $N$ and $\alpha$,
are fixed from simulation. The background from randomly associated tracks is
modelled using an exponential function. Background contributions from the
partially reconstructed decays ${{\mathchar
28932\relax}^{++}_{{c}{c}}}\rightarrow{\it\Xi^{{}^{\prime}+}_{c}}(\rightarrow{{\mathchar
28932\relax}^{+}_{c}}\gamma){{\pi}^{+}}$ and ${{\mathchar
28932\relax}^{++}_{{c}{c}}}\rightarrow{{\mathchar
28932\relax}^{+}_{c}}{{\rho}^{+}}(\rightarrow{{\pi}^{+}}{{\pi}^{0}})$, where
photons and neutral ${{\pi}^{0}}$ mesons are not reconstructed, can contribute
to the ${{\mathchar 28932\relax}^{++}_{{c}{c}}}\rightarrow{{\mathchar
28932\relax}^{+}_{c}}{{\pi}^{+}}$ decay mode. The mass line shapes of these
partially reconstructed backgrounds are determined from simulation.

The fits return signal yields of $1598\pm 64$ and $616\pm 47$ for the
${{{\mathchar 28932\relax}^{++}_{{c}{c}}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}}$ and ${{{\mathchar
28932\relax}^{++}_{{c}{c}}}\rightarrow{{\mathchar
28932\relax}^{+}_{c}}{{\pi}^{+}}}$ decay modes, respectively. The peak
positions are determined with the ${{{\mathchar
28932\relax}^{++}_{{c}{c}}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}}$ and ${{{\mathchar
28932\relax}^{++}_{{c}{c}}}\rightarrow{{\mathchar
28932\relax}^{+}_{c}}{{\pi}^{+}}}$ decay modes to be $3622.08\pm
0.24\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ and $3622.37\pm
0.60\text{\,Me\kern-1.00006ptV\\!/}c^{2}$, respectively, where the uncertainty
is statistical only.

Due to multiple scattering, the opening angle between the ${{\mathchar
28932\relax}^{++}_{{c}{c}}}$ decay products can be increased or decreased.
This can bias both the resulting ${\mathchar 28932\relax}^{++}_{{c}{c}}$ mass
and the measured decay length. Since the selection is more efficient for
candidates with larger reconstructed decay lengths, and the decay length is
correlated with the mass by the effect of the multiple scattering, this can
bias the ${{\mathchar 28932\relax}^{++}_{{c}{c}}}$ mass measurement. This
effect was studied with charmed hadrons,
${{D}^{+}},{{D}^{0}},{{D}^{+}_{s}},{{\mathchar 28931\relax}^{+}_{c}}$, and was
found to be well reproduced by simulation [4]. Corresponding corrections of
$-0.61\pm 0.09\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ for ${{\mathchar
28932\relax}^{++}_{{c}{c}}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ and $-0.45\pm
0.09\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ for ${{\mathchar
28932\relax}^{++}_{{c}{c}}}\rightarrow{{\mathchar
28932\relax}^{+}_{c}}{{\pi}^{+}}$ are determined using simulated candidates by
comparing the fitted mass with signal candidates before and after applying the
event selection. These corrections are applied to the fitted mass values. The
uncertainties are due to the limited size of simulated samples, and are taken
as the systematic uncertainties from the selection-induced bias on the
${{\mathchar 28932\relax}^{++}_{{c}{c}}}$ mass. The difference of the
kinematic distributions in simulation and data is considered as a systematic
uncertainty and is discussed in Sec. 5.

Figure 1: Invariant-mass distributions of (left) ${{\mathchar
28932\relax}^{++}_{{c}{c}}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ and (right) ${{\mathchar
28932\relax}^{++}_{{c}{c}}}\rightarrow{{\mathchar
28932\relax}^{+}_{c}}{{\pi}^{+}}$ decay candidates, in the mass range of
3470–3770$\text{\,Me\kern-0.90005ptV\\!/}c^{2}$ and
3350–3800$\text{\,Me\kern-0.90005ptV\\!/}c^{2}$, respectively. The results of
unbinned extended maximum-likelihood fits to the mass distributions are
indicated by the blue solid lines.

Low-momentum photons emitted by the final-state particles are not
reconstructed. This distorts the reconstructed mass distribution and can bias
the fitted mass value. This effect is studied with the simulation. To
disentangle this effect from those due to reconstruction, the mass of the
${\mathchar 28932\relax}^{++}_{{c}{c}}$ candidates calculated with the true
momenta of the final-state particles is smeared with different resolution
values. The difference between the fitted and input mass values is studied as
a function of the mass resolution, and the difference corresponding to the
mass resolution in data is taken as a correction. Alternative signal models
are also considered. The largest difference of the fitted mass with final-
state radiation corrections between the nominal and the alternative is quoted
as the uncertainty. Following the procedure described above, the corrections
due to the final-state radiation are determined as $0.06\pm
0.05\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ and $0.03\pm
0.16\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ for the ${{\mathchar
28932\relax}^{++}_{{c}{c}}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ and ${{\mathchar
28932\relax}^{++}_{{c}{c}}}\rightarrow{{\mathchar
28932\relax}^{+}_{c}}{{\pi}^{+}}$ decay modes, respectively. The uncertainties
on the corrections are due to the limited size of the simulated samples, and
the difference between the corrections with different signal models.

##  5 Systematic uncertainties

The dominant source of systematic uncertainty on the mass measurement is due
to the momentum-scale calibration [28, 29]. It amounts to
0.21$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ for the ${{\mathchar
28932\relax}^{++}_{{c}{c}}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ decay, and
0.34$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ for the ${{\mathchar
28932\relax}^{++}_{{c}{c}}}\rightarrow{{\mathchar
28932\relax}^{+}_{c}}{{\pi}^{+}}$ decay due to larger $Q$-value. A further
uncertainty arises from the correction for energy loss in the spectrometer,
which is known with 10% accuracy [24]. This uncertainty was studied in Ref.
[29], and amounts to 0.03$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ for ${D}^{0}$
$\rightarrow$ ${K}^{+}$ ${K}^{-}$ ${\pi}^{+}$ ${\pi}^{-}$ decays. The
uncertainties on the ${\mathchar 28932\relax}^{++}_{{c}{c}}$ mass are scaled
from that of the ${D}^{0}$ decay by the number of final-state particles, and
are determined to be 0.05$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ and
0.03$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ for the ${{\mathchar
28932\relax}^{++}_{{c}{c}}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ and ${{\mathchar
28932\relax}^{++}_{{c}{c}}}\rightarrow{{\mathchar
28932\relax}^{+}_{c}}{{\pi}^{+}}$ decays, respectively.

Differences between kinematic distributions in simulation and data are treated
as sources of systematic uncertainties on the corrections due to the selection
procedure. The kinematic variables used in the event selection that are found
to affect the corrections are listed below: $p_{\mathrm{T}}$ of ${\mathchar
28932\relax}^{++}_{{c}{c}}$ candidates; the angle between the momentum and the
displacement vector from the PV to the decay vertex of the ${\mathchar
28932\relax}^{++}_{{c}{c}}$ and ${\mathchar 28931\relax}^{+}_{c}$ (${\mathchar
28932\relax}^{+}_{c}$) candidates; the $\chi^{2}_{\text{IP}}$ of ${\mathchar
28932\relax}^{++}_{{c}{c}}$ and ${\mathchar 28931\relax}^{+}_{c}$ (${\mathchar
28932\relax}^{+}_{c}$) candidates and their decay products; the BDT (MLP)
response; and the particle identification information. The distributions of
these variables in simulation are weighted to match those in data where the
background is subtracted by means of the sPlot technique [52]. Then, the
corrections obtained with the weights are compared to those without weights,
and largest variations of the corrections are taken as systematic
uncertainties, which are 0.09$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ and
0.05$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ for the ${{\mathchar
28932\relax}^{++}_{{c}{c}}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ and ${{\mathchar
28932\relax}^{++}_{{c}{c}}}\rightarrow{{\mathchar
28932\relax}^{+}_{c}}{{\pi}^{+}}$ decays, respectively.

The uncertainty related to the background description is estimated by
repeating the fits with alternative models which include first and second-
order polynomial functions. For the ${{\mathchar
28932\relax}^{++}_{{c}{c}}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ decay, the fit with a
second-order polynomial function has better fit quality, but returns identical
fitted mass. The largest changes on the fitted mass value are found to be
0.01$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ and
0.04$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ for the ${{\mathchar
28932\relax}^{++}_{{c}{c}}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ and ${{\mathchar
28932\relax}^{++}_{{c}{c}}}\rightarrow{{\mathchar
28932\relax}^{+}_{c}}{{\pi}^{+}}$ decays, respectively, which are assigned as
systematic uncertainties.

The mass of ${\mathchar 28932\relax}^{++}_{{c}{c}}$ candidates also depends on
the value of the known ${\mathchar 28931\relax}^{+}_{c}$ and ${\mathchar
28932\relax}^{+}_{c}$ masses. The uncertainties on the ${\mathchar
28931\relax}^{+}_{c}$ mass and on the mass difference between the ${\mathchar
28932\relax}^{+}_{c}$ and ${\mathchar 28931\relax}^{+}_{c}$ are propagated to
the ${\mathchar 28932\relax}^{++}_{{c}{c}}$ mass measurement. The
corresponding uncertainties on the ${\mathchar 28932\relax}^{++}_{{c}{c}}$
mass are $0.14\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ and
$0.22\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ for the ${{\mathchar
28932\relax}^{++}_{{c}{c}}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ decay and the
${{\mathchar 28932\relax}^{++}_{{c}{c}}}\rightarrow{{\mathchar
28932\relax}^{+}_{c}}{{\pi}^{+}}$ decay, respectively.

The sources of systematic uncertainty considered in this analysis are
summarised in Table 1. When computing the total uncertainty, the uncertainty
on the momentum-scale calibration of the ${{\mathchar 28932\relax}^{+}_{c}}$
mass from Ref. [50] is assumed to be fully correlated to that of the
${\mathchar 28932\relax}^{++}_{{c}{c}}$ mass. The total systematic uncertainty
is calculated by summing the individual sources of uncertainty in quadrature.

Table 1: Systematic uncertainties on the ${\mathchar 28932\relax}^{++}_{{c}{c}}$ mass measurements using ${{\mathchar 28932\relax}^{++}_{{c}{c}}}\rightarrow{{\mathchar 28931\relax}^{+}_{c}}{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ and ${{\mathchar 28932\relax}^{++}_{{c}{c}}}\rightarrow{{\mathchar 28932\relax}^{+}_{c}}{{\pi}^{+}}$ decays. The total systematic uncertainty on each mode is calculated by summing the individual sources of uncertainty in quadrature, except for the uncertainty on the momentum-scale calibration of the ${{\mathchar 28932\relax}^{+}_{c}}$ mass [50], that is added linearly to that of the ${\mathchar 28932\relax}^{++}_{{c}{c}}$ mass. | Uncertainty $[\text{\,Me\kern-1.00006ptV\\!/}c^{2}]$
---|---
Source | ${{\mathchar 28932\relax}^{++}_{{c}{c}}}\rightarrow{{\mathchar 28931\relax}^{+}_{c}}{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ | ${{\mathchar 28932\relax}^{++}_{{c}{c}}}\rightarrow{{\mathchar 28932\relax}^{+}_{c}}{{\pi}^{+}}$
Momentum-scale calibration | 0.21 | 0.34
Energy-loss correction | 0.05 | 0.03
Simulation/data agreement | 0.09 | 0.05
Selection-induced bias on the ${\mathchar 28932\relax}^{++}_{{c}{c}}$ mass | 0.09 | 0.09
Final-state radiation | 0.05 | 0.16
Background model | 0.01 | 0.04
${\mathchar 28931\relax}^{+}_{c}$, ${\mathchar 28932\relax}^{+}_{c}$ mass | 0.14 | 0.22
Total | 0.29 | 0.49

##  6 Results and summary

The resulting values of the ${{\mathchar 28932\relax}^{++}_{{c}{c}}}$ mass
using the ${{\mathchar 28932\relax}^{++}_{{c}{c}}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ and ${{\mathchar
28932\relax}^{++}_{{c}{c}}}\rightarrow{{\mathchar
28932\relax}^{+}_{c}}{{\pi}^{+}}$ decay modes are $3621.53\pm 0.24\pm
0.29\text{\,Me\kern-1.00006ptV\\!/}c^{2}$, and $3621.95\pm 0.60\pm
0.49\text{\,Me\kern-1.00006ptV\\!/}c^{2}$, respectively, including corrections
and systematic uncertainties. The combination of the two measurements is
performed using the Best Linear Unbiased Estimator (BLUE) [53, 54]. The
combined ${\mathchar 28932\relax}^{++}_{{c}{c}}$ mass is determined to be

$\displaystyle\mbox{$3621.55\,\pm 0.23\,{\rm(stat)\,}\pm 0.30\,{\rm(syst)}$
$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$}.$

In the combination, the correlation between the ${{\mathchar
28931\relax}^{+}_{c}}$ and ${{\mathchar 28932\relax}^{+}_{c}}$ masses [50, 49]
is taken into account. Uncertainties arising from the momentum-scale
calibration, energy-loss corrections, and final-state radiation are assumed to
be 100% correlated while other sources of systematic uncertainty are assumed
to be uncorrelated. The individual mass measurements and the resulting
combination are illustrated in Fig. 2.

Figure 2: Measured ${{\mathchar 28932\relax}^{++}_{{c}{c}}}$ mass values and
uncertainties obtained with the decay modes ${{\mathchar
28932\relax}^{++}_{{c}{c}}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ and ${{\mathchar
28932\relax}^{++}_{{c}{c}}}\rightarrow{{\mathchar
28932\relax}^{+}_{c}}{{\pi}^{+}}$. The combination is performed using the best
linear unbiased estimator [53, 54]. The inner error bars represent statistical
uncertainties and the outer error bars represent the quadratic sum of
statistical and systematic uncertainties. The inner and outer green bands
correspond to the uncertainties on the averaged value.

The averaged mass is dominated by the result for the ${{\mathchar
28932\relax}^{++}_{{c}{c}}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ mode, due to its larger
yield and smaller momentum-scale uncertainty relative to that of the
${{\mathchar 28932\relax}^{++}_{{c}{c}}}\rightarrow{{\mathchar
28932\relax}^{+}_{c}}{{\pi}^{+}}$ decay. This is the most precise measurement
of the ${{\mathchar 28932\relax}^{++}_{{c}{c}}}$ mass to date, improving upon
the previous weighted average mass value of $3621.24\,\pm
0.65\,{\rm(stat)\,}\pm
0.31\,{\rm(syst)}$$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ from Ref. [5].

## Acknowledgements

We thank Chao-Hsi Chang, Cai-Dian Lü, Xing-Gang Wu, and Fu-Sheng Yu for the
discussions on the production and decays of double-heavy-flavor baryons. 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 (Netherlands); MNiSW and NCN (Poland); MEN/IFA (Romania); MSHE
(Russia); MinECo (Spain); SNSF and SER (Switzerland); NASU (Ukraine); STFC
(United Kingdom); DOE NP and NSF (USA). We acknowledge the computing resources
that are provided by CERN, IN2P3 (France), KIT and DESY (Germany), INFN
(Italy), SURF (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 and OCEVU,
and Région Auvergne-Rhône-Alpes (France); Key Research Program of Frontier
Sciences of CAS, CAS PIFI, and the Thousand Talents Program (China); RFBR, RSF
and Yandex LLC (Russia); GVA, XuntaGal and GENCAT (Spain); the Royal Society
and the Leverhulme Trust (United Kingdom).

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LHCb collaboration

R. Aaij31, C. Abellán Beteta49, T. Ackernley59, B. Adeva45, M. Adinolfi53, H.
Afsharnia9, C.A. Aidala80, S. Aiola25, Z. Ajaltouni9, S. Akar66, P.
Albicocco22, J. Albrecht14, F. Alessio47, M. Alexander58, A. Alfonso Albero44,
G. Alkhazov37, P. Alvarez Cartelle60, A.A. Alves Jr45, S. Amato2, Y. Amhis11,
L. An21, L. Anderlini21, G. Andreassi48, M. Andreotti20, F. Archilli16, J.
Arnau Romeu10, A. Artamonov43, M. Artuso67, K. Arzymatov41, E. Aslanides10, M.
Atzeni49, B. Audurier26, S. Bachmann16, J.J. Back55, S. Baker60, V.
Balagura11,b, W. Baldini20,47, A. Baranov41, R.J. Barlow61, S. Barsuk11, W.
Barter60, M. Bartolini23,47,h, F. Baryshnikov77, G. Bassi28, V. Batozskaya35,
B. Batsukh67, A. Battig14, A. Bay48, M. Becker14, F. Bedeschi28, I. Bediaga1,
A. Beiter67, L.J. Bel31, V. Belavin41, S. Belin26, N. Beliy5, V. Bellee48, K.
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Bifani52, P. Billoir12, A. Bizzeti21,u, M. Bjørn62, M.P. Blago47, T. Blake55,
F. Blanc48, S. Blusk67, D. Bobulska58, V. Bocci30, O. Boente Garcia45, T.
Boettcher63, A. Boldyrev78, A. Bondar42,x, N. Bondar37, S. Borghi61,47, M.
Borisyak41, M. Borsato16, J.T. Borsuk33, T.J.V. Bowcock59, C. Bozzi20, M.J.
Bradley60, S. Braun16, A. Brea Rodriguez45, M. Brodski47, J. Brodzicka33, A.
Brossa Gonzalo55, D. Brundu26, E. Buchanan53, A. Buonaura49, C. Burr47, A.
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Cai72, R. Calabrese20,g, L. Calero Diaz22, S. Cali22, R. Calladine52, M.
Calvi24,i, M. Calvo Gomez44,m, A. Camboni44, P. Campana22, D.H. Campora
Perez31, L. Capriotti19,e, A. Carbone19,e, G. Carboni29, R. Cardinale23,h, A.
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J.J. Lane61, G. Lanfranchi22, C. Langenbruch13, T. Latham55, F. Lazzari28,v,
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Oliveira2, G. Simi27,o, S. Simone18,d, I. Skiba20, N. Skidmore16, T.
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J. Yu7,aa, X. Yuan67, O. Yushchenko43, K.A. Zarebski52, M. Zavertyaev15,c, M.
Zdybal33, M. Zeng3, D. Zhang7, L. Zhang3, S. Zhang3, W.C. Zhang3,z, Y.
Zhang47, A. Zhelezov16, Y. Zheng5, X. Zhou5, Y. Zhou5, X. Zhu3, V.
Zhukov13,39, J.B. Zonneveld57, S. Zucchelli19,e.

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
4School of Physics State Key Laboratory of Nuclear Physics and Technology,
Peking University, Beijing, China
5University of Chinese Academy of Sciences, Beijing, China
6Institute Of High Energy Physics (IHEP), Beijing, China
7Institute of Particle Physics, Central China Normal University, Wuhan, Hubei,
China
8Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, IN2P3-LAPP, Annecy,
France
9Université Clermont Auvergne, CNRS/IN2P3, LPC, Clermont-Ferrand, France
10Aix Marseille Univ, CNRS/IN2P3, CPPM, Marseille, France
11LAL, Univ. Paris-Sud, CNRS/IN2P3, Université Paris-Saclay, Orsay, France
12LPNHE, Sorbonne Université, Paris Diderot Sorbonne Paris Cité, CNRS/IN2P3,
Paris, France
13I. Physikalisches Institut, RWTH Aachen University, Aachen, Germany
14Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
15Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
16Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
17School of Physics, University College Dublin, Dublin, Ireland
18INFN Sezione di Bari, Bari, Italy
19INFN Sezione di Bologna, Bologna, Italy
20INFN Sezione di Ferrara, Ferrara, Italy
21INFN Sezione di Firenze, Firenze, Italy
22INFN Laboratori Nazionali di Frascati, Frascati, Italy
23INFN Sezione di Genova, Genova, Italy
24INFN Sezione di Milano-Bicocca, Milano, Italy
25INFN Sezione di Milano, Milano, Italy
26INFN Sezione di Cagliari, Monserrato, Italy
27INFN Sezione di Padova, Padova, Italy
28INFN Sezione di Pisa, Pisa, Italy
29INFN Sezione di Roma Tor Vergata, Roma, Italy
30INFN Sezione di Roma La Sapienza, Roma, Italy
31Nikhef National Institute for Subatomic Physics, Amsterdam, Netherlands
32Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, Netherlands
33Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
34AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
35National Center for Nuclear Research (NCBJ), Warsaw, Poland
36Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
37Petersburg Nuclear Physics Institute NRC Kurchatov Institute (PNPI NRC KI),
Gatchina, Russia
38Institute of Theoretical and Experimental Physics NRC Kurchatov Institute
(ITEP NRC KI), Moscow, Russia, Moscow, Russia
39Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
40Institute for Nuclear Research of the Russian Academy of Sciences (INR RAS),
Moscow, Russia
41Yandex School of Data Analysis, Moscow, Russia
42Budker Institute of Nuclear Physics (SB RAS), Novosibirsk, Russia
43Institute for High Energy Physics NRC Kurchatov Institute (IHEP NRC KI),
Protvino, Russia, Protvino, Russia
44ICCUB, Universitat de Barcelona, Barcelona, Spain
45Instituto Galego de Física de Altas Enerxías (IGFAE), Universidade de
Santiago de Compostela, Santiago de Compostela, Spain
46Instituto de Fisica Corpuscular, Centro Mixto Universidad de Valencia -
CSIC, Valencia, Spain
47European Organization for Nuclear Research (CERN), Geneva, Switzerland
48Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL),
Lausanne, Switzerland
49Physik-Institut, Universität Zürich, Zürich, Switzerland
50NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
51Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
52University of Birmingham, Birmingham, United Kingdom
53H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
54Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
55Department of Physics, University of Warwick, Coventry, United Kingdom
56STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
57School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
58School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
59Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
60Imperial College London, London, United Kingdom
61Department of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
62Department of Physics, University of Oxford, Oxford, United Kingdom
63Massachusetts Institute of Technology, Cambridge, MA, United States
64University of Cincinnati, Cincinnati, OH, United States
65University of Maryland, College Park, MD, United States
66Los Alamos National Laboratory (LANL), Los Alamos, United States
67Syracuse University, Syracuse, NY, United States
68Laboratory of Mathematical and Subatomic Physics , Constantine, Algeria,
associated to 2
69School of Physics and Astronomy, Monash University, Melbourne, Australia,
associated to 55
70Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
71South China Normal University, Guangzhou, China, associated to 3
72School of Physics and Technology, Wuhan University, Wuhan, China, associated
to 3
73Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia,
associated to 12
74Institut für Physik, Universität Rostock, Rostock, Germany, associated to 16
75Van Swinderen Institute, University of Groningen, Groningen, Netherlands,
associated to 31
76National Research Centre Kurchatov Institute, Moscow, Russia, associated to
38
77National University of Science and Technology “MISIS”, Moscow, Russia,
associated to 38
78National Research University Higher School of Economics, Moscow, Russia,
associated to 41
79National Research Tomsk Polytechnic University, Tomsk, Russia, associated to
38
80University of Michigan, Ann Arbor, United States, associated to 67

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
vUniversità di Siena, Siena, Italy
wMSU - Iligan Institute of Technology (MSU-IIT), Iligan, Philippines
xNovosibirsk State University, Novosibirsk, Russia
yINFN Sezione di Trieste, Trieste, Italy
zSchool of Physics and Information Technology, Shaanxi Normal University
(SNNU), Xi’an, China
aaPhysics and Micro Electronic College, Hunan University, Changsha City, China
†Deceased


# Simultaneous Translation for Unsegmented Input:
A Sliding Window Approach

Sukanta Sen
University of Edinburgh
<EMAIL_ADDRESS>
&Ondřej Bojar
Charles University
MFF ÚFAL
<EMAIL_ADDRESS>
&Barry Haddow
University of Edinburgh
<EMAIL_ADDRESS>Work performed while at the University of Edinburgh

###### Abstract

In the cascaded approach to spoken language translation (SLT), the ASR output
is typically punctuated and segmented into sentences before being passed to
MT, since the latter is typically trained on written text. However, erroneous
segmentation, due to poor sentence-final punctuation by the ASR system, leads
to degradation in translation quality, especially in the simultaneous (online)
setting where the input is continuously updated. SSAlso, in the inference
time, we are using online text-flow which means we are not handling unstable
input. I am not sure how do I rephrase the last sentence without a risk. For
now I am keeping as it is for the submission purpose. To reduce the influence
of automatic segmentation, we present a sliding window approach to translate
raw ASR outputs (online or offline) without needing to rely on an automatic
segmenter. We train translation models using parallel windows (instead of
parallel sentences) extracted from the original training data. At test time,
we translate at the window level and join the translated windows using a
simple approach to generate the final translation. Experiments on English-to-
German and English-to-Czech show that our approach improves 1.3–2.0 BLEU
points over the usual ASR-segmenter pipeline, and the fixed-length window
considerably reduces flicker compared to a baseline retranslation-based online
SLT system.

Simultaneous Translation for Unsegmented Input:
A Sliding Window Approach

Sukanta Sen††thanks:  Work performed while at the University of Edinburgh
University of Edinburgh<EMAIL_ADDRESS>Ondřej Bojar Charles University
MFF ÚFAL<EMAIL_ADDRESS>Barry Haddow University of Edinburgh
<EMAIL_ADDRESS>

##  1 Introduction

For machine translation (MT) with textual input, it is usual to segment the
text into sentences before translation, with the boundaries of sentences in
most text types indicated by punctuation. For spoken language translation
(SLT), in contrast, the input is audio so there is no punctuation provided to
assist segmentation. Segmentation thus has to be guessed by the ASR system or
a separate component. Perhaps more importantly, for many speech genres the
input cannot easily be segmented into well-formed sentences as found in MT
training data, giving a mismatch between training and test.

In order to address the segmentation problem in SLT, systems often include a
segmentation component in their pipeline, e.g. Cho et al. (2017). In other
words, a typical _cascaded_ SLT system consists of automatic speech
recognition (ASR – which outputs lowercased, unpunctuated text) a
punctuator/segmenter (which adds punctuation and so defines segments) and an
MT system. The segmenter can be a sequence-sequence model, and training data
is easily synthesised from punctuated text. However adding segmentation as an
extra step has the disadvantage of introducing an extra component to be
managed and deployed. Furthermore, errors in segmentation have been shown to
contribute significantly to overall errors in SLT (Li et al., 2021), since
neural MT is known to be susceptible to degradation from noisy input
Khayrallah and Koehn (2018).

These issues with segmentation can be exacerbated in the _online_ or
_simultaneous_ setting. This is an important use case for SLT where we want to
produce the translations from live speech, as the speaker is talking. To
minimise the latency of the translation, we would like to start translating
before speaker has finished their sentence. Some online low-latency ASR
approaches will also revise their output after it has been produced, creating
additional difficulties for the downstream components. In this scenario, the
segmentation into sentences will be more uncertain and we are faced with the
choice of waiting for the input to stabilise (so increasing latency) or
translating early (potentially introducing more errors, or having to correct
the output when the ASR is extended and updated).

To address the segmentation issue in SLT, Li et al. (2021) has proposed to a
data augmentation technique which simulates the bad segmentation in the
training data. They concatenate two adjacent source sentences (and also the
corresponding targets) and then start and end of the concatenated sentences
are truncated proportionally.

We use a _sliding window_ approach to translate unsegmented input. In this
approach, we translate the ASR output as a series of overlapping windows,
using a merging algorithm to turn the translated windows into a single
continuous (but still sometimes updated) stream. The process is illustrated in
Figure 1. To generate the training data, we convert the sentence-aligned
training data into window-window pairs, and remove punctuation and casing from
the source. We explain our algorithms in detail in Section 2.

For online SLT, we use a _retranslation_ approach (Niehues et al., 2016;
Arivazhagan et al., 2020a), where the MT system retranslates a recent portion
of the input each time there is an update from ASR. This approach has the
advantage that it can use standard MT inference, including beam search, and
does not require a modified inference engine as in streaming approaches (e.g.
Ma et al. (2019)). Retranslation may introduce flicker, i.e. potentially
disruptive changes of displayed text, when outputs are updated. Flicker can be
traded off with latency by masking the last $k$ words of the output
(Arivazhagan et al., 2020a).111This paper also introduced the idea of _biased
beam search_ , where the translation of an extended prefix is soft-constrained
to stay close to the translation of the prefix. Biased beam search
significantly reduces flicker, but it requires that ASR output has a fixed
segmentation, and uses a modified MT inference engine. Our sliding window
approach is easily combined with retranslation to create an online SLT system
which can operate on unsegmented ASR. Each time there is an update from ASR,
we retranslate the last $n$ tokens and merge the latest translation into the
output stream. Using the fixed size window has the advantage of reducing
flicker, since we control how much of the output stream can change on each
retranslation.

Experiments on English$\rightarrow$Czech and English$\rightarrow$German show
that our sliding window approach improves BLEU scores for both online and
offline SLT. For the online case, our approach improves the tradeoff between
latency and flicker.

Figure 1: Example of how our proposed window-based translation works at test
time in case of a match and no-match of translations of two subsequent
windows. The text inside the rectangular box is the source window at time $t$,
which is translated into output window (Tt) by the MT system. The text in blue
(dark) shade shows the common segment between the output window (Tt) and the
output stream (Ot) at time $t$. The text in red shade shows the segment newly
added from the output window Tt into the output stream Ot+1. With no common
segment between Tt and Ot (“No match”), we extend the input window into the
history and translate again. $\bullet\bullet\bullet$ indicates there are more
tokens. Note that we used characters here (instead of tokens) just for
explanation.

##  2 Window-Based Translation

###  2.1 Preprocessing

To make the parallel corpus resemble ASR output, we remove all punctuation
(and other special characters) from the source sentences and replace it with
spaces. We then remove repeated spaces, and lowercase the source.

###  2.2 Generating the Window Pairs for Training

To convert the parallel corpus into a set of parallel windows, we use a word-
alignment based approach. We first word-align the pre-processed parallel
corpus using fast_align (Dyer et al., 2013), then we concatenate each side of
the corpus to give two long lines. Note that the word alignments will however
never cross sentence boundaries. We randomly select windows of length 15–25
from the target side, and use the word alignment to get the corresponding
source window. The algorithms are described in Appendix B.

A subtle detail is whether the original corpus was or was not shuffled at the
level of sentences. An original, non-shuffled corpus provides the MT system
with useful examples of cross-sentence conditioning, a very useful feature
especially for spontaneous speech translation. A minority of our cs-en data is
shuffled, which adds some noise to the process, but our method works despite
this.

###  2.3 Translating Input Windows

In our simultaneous MT setting, we assume that the ASR system is transcribing
the incoming speech signal into a continuous stream of text. To obtain a new
input to the MT system, a fixed-side window is shifted by one token to right
every time the stream is extended. For every input window, the MT system
translates it and sends it to the module that joins the output windows to the
output stream as described in the next section.

###  2.4 Joining the Output Windows

Since two consecutive source windows overlap, the corresponding output windows
normally have an overlap. We use this overlap to join an output window to the
output stream.

We show the pseudo code for merging an output window to the output stream in
Algorithm 1. We assume that ASR produces an input stream $I$ which is
continuously growing by one token at a time. Our algorithm requires a window
length $w_{l}$, a threshold $r$ and the current output stream $O_{t}$. For
every new token in $I$, our merge module in Algorithm 1 is triggered. The MT
system translates the last $w_{l}$ tokens of $I$ to a target window $T_{t}$.
For any translated window $T_{t}$ and output stream $O_{t}$ at that time step,
we find the longest common substring $s$. The threshold $r$ gives the required
minimum length of the common substring. If the match is sufficiently long
(“significant” in the following), we merge the current target window $T_{t}$,
otherwise we extend input window by 1 token to the left and translate again.

In our experiments, we extend the history to maximum of 5 tokens until we have
found a significant match. A higher $r$ assures that the translation of the
current window will not accidentally match a random segment in the stream, and
as the successive windows are just 1 token apart, we find a match almost
always (see Appendix C for details). Once we have found a significant match,
we merge $T_{t}$ with $O_{t}$ around the match, chopping the part of $T_{t}$
before match. This approach of joining windows is able to handle both the
online and offline situations.

Algorithm 1 Pseudo code for merging newly translated window into existing
output.

0: The current output stream $O_{t}$, input stream $I$, an $MT$ system, window
length $w_{l}$, threshold $r\in(0,1)$.

1: $k=0$ {extra history considered}

2: while true do

3: $T_{t}\leftarrow MT(I[\>\lvert I\rvert\>-(w_{l}+k):\>\lvert I\rvert\>])$

4: $O_{t}^{{}^{\prime}}\leftarrow O_{t}[\>\lvert O_{t}\rvert\>-\>\lvert
T_{t}\rvert\>:\>\lvert O_{t}\rvert\>]$

5: $s,i,j\leftarrow T_{t}\Psi O_{t}^{{}^{\prime}}$ {$s$ is longest common
substring. $i$ and $j$ are the start indices of match in $O_{t}^{{}^{\prime}}$
and $T_{t}$}

6: $k\leftarrow k+1$

7: if $\>\lvert s\rvert\>\geq\>\lvert T_{t}\rvert\>*r$ or $k>5$ then

8: break

9: end if

10: end while

11: if $\>\lvert s\rvert\>=0$ then

12: $i\leftarrow\>\lvert T_{t}\rvert\>$

13: $j\leftarrow 0$

14: end if

15: $O_{t+1}\leftarrow O_{t}[0:\lvert O_{t}\rvert-\lvert
T_{t}\rvert+i]+T_{t}[j:\lvert T_{t}\rvert]$

16: return $O_{t+1}$

##  3 Datasets and Experimental Settings

For training, we use parallel datasets from WMT 2020 (Barrault et al., 2020)
for English-German and from WMT 2021 (Akhbardeh et al., 2021) for English-
Czech (see Appendix A for details). For the validation set, we use the
concatenation of IWSLT 2014,15 test sets for English-German, and newstest2019
for English-Czech. We use the ESIC test set for evaluation. ESIC (Macháček et
al., 2021) is a corpus derived from the European parliament proceedings which
has transcripts of source English speech and interpreted German and Czech
transcripts. This test set is aligned at document level.

We use the SentencePiece Kudo and Richardson (2018) tokenizer for
preprocessing the windows with a shared subword (Sennrich et al., 2016)
vocabulary size of 32k. We train transformer-based222with 60 millions
parameters. One model using 4 GPUs took on an average 2 days. (Vaswani et al.,
2017) NMT models using the Marian toolkit (Junczys-Dowmunt et al., 2018). MT
models are trained to convergence (using early stopping of 10) with a learning
rate of $0.0003$, and translate using a beam of 6. We train the following two
types of models: i) Baseline: trained on gold-segmented data and evaluated on
segmented data generated by the ASR system; ii) Window: trained on windows of
10-25 tokens and evaluated on fixed length windows of ASR output.

SSI think Barry is talking about the test set which we obtain after doing ASR
and applying online text-flow (OTF). But I am not aware much about the ASR
part. We are using OTF for getting the stable part.

##  4 Results

| Baseline | Window
---|---|---
Pair | SF | SO | 8 | 10 | 12 | 14 | 16 | 18 | 20
en-de | 11.2 | 11.4 | 12.5 | 12.8 | 13.0 | 13.0 | 13.1 | 13.2 | 13.2
en-cs | 9.4 | 9.4 | 10.0 | 10.3 | 10.4 | 10.5 | 10.6 | 10.6 | 10.7
Table 1: Sacrebleu scores of segmented and window based approaches. SF:
Offline segment level. SO: Online segment level.

We evaluate both the offline and online SLT. For offline SLT, the baseline
system is trained using parallel sentences, and for the online version, the
baseline system is a prefix-prefix retranslation system (Niehues et al., 2016;
Arivazhagan et al., 2020a). For our proposed window-based system, the offline
and online are the same system. We evaluate our proposed approach on ESIC
using Sacrebleu333nrefs:1—case:mixed—eff:no—tok:13a—smooth:exp—version:2.0.0
Papineni et al. (2002); Post (2018) score. As the test set is not sentence
aligned, we translate each document and then align the output sentences
(hypothesis) to corresponding reference document using mwerSegmenter (Matusov
et al., 2005), before calculating BLEU.

For the baseline, we translate the test set using the segmentations produced
by ASR. For our proposed window-based method, we evaluate using different
fixed-size windows of length 8, 10, $\ldots$, 20 tokens. The results are shown
in Table 1where we observe that the proposed method outperforms the baseline
with margins of 1.3 and 2.0 BLEU. These BLEU scores in Table 1 across
different window length are the best scores obtained after exploring different
threshold ($r$) of match (refer to line 7 of Algorithm 1). We show the BLEU
scores for each threshold in Appendix Table 3.

(a) English-German

(b) English-Czech

Figure 2: Mask vs Flicker plots for different window lengths at threshold $r$
= 0.4, and the baseline.

For online SLT, since our system uses retranslation, we evaluate quality using
BLEU, and flicker using normalised erasure (NE; Arivazhagan et al. 2020a). We
first note that flicker is affected by both window length and thresholds –
shorter windows force commitment earlier which gives lower flicker. Low
thresholds promote spurious matches, make translation flicker more, whilst
high thresholds force too many retranslations, and will cause extra flicker
when the maximum backoff is exceeded. After exploration (Appendix C), we set
the threshold to 0.4 for the rest of our experiments.

Figure 2 shows the flicker-latency tradeoff of our sliding-window approach to
online SLT, as we vary the fixed mask. We can see that the tradeoff is
improved at all window sizes. This improvement is because the window approach
only allows updates that are within the window length. The quality of the
online SLT (as measured on full sentences) is the same as the offline SLT. The
flexible mask allows further improvements in flicker, for matched latency

##  5 Conclusion

We proposed window-based approach which works at window (of fixed length of
tokens) level, and removes the need of automatic sentence-segmentation of ASR
output in cascaded SLT. We experimented with English-German and English-Czech
language pairs and found that our proposed approach performs better than the
segmentation based translation obtaining an improvement of 1.3-2 BLEU points.
We also observed that masking the output reduced the flicker by a considerable
margin as compared to the baseline.

## Acknowledgements

eu-logo.pngThis work has received funding from the European Union’s Horizon
2020 Research and Innovation Programme under Grant Agreement No 825460
(ELITR).

Ondřej Bojar would also like to acknowledge the support from the 19-26934X
(NEUREM3) grant of the Czech Science Foundation.

This work used the Cirrus UK National Tier-2 HPC Service at EPCC
(http://www.cirrus.ac.uk) funded by the University of Edinburgh and EPSRC
(EP/P020267/1)

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##  Appendix A Training data statistics

In Table 2 we show the breakdown of our training data.

Corpus | Sentence pairs
---|---
English-German
Europarl | 1.79 M
Rapid | 1.45 M
News Commentary | 0.35 M
OpenSubtitle | 22.51 M
TED corpus | 206 K
MuST-C.v2 | 248 K
English-Czech
Europarl | 645 K
ParaCrawl | 14 M
CommonCrawl | 161 K
News Commentary | 260 K
CzEng2.0 | 36 M444deduped; original has 75 M
Wikititles | 410 K
Rapid | 452 K
Table 2: Corpora used in training the systems

##  Appendix B Creation of Windowed Parallel Corpus

First, we word-align the pre-processed parallel corpus $D$ to obtain alignment
$A$ using fast_align (Dyer et al., 2013). Then we concatenate all the source-
target sentence pairs $(s_{k},t_{k})$ into a single, very long, pair $(s,t)$
and subsequently, revise the alignment using the Algorithm 2, so that the
indexes are still correct in the concatenated corpus.

Algorithm 2 Pseudo code for collapsing a word-aligned parallel corpus into a
single pair of sentences preserving the word alignments.

0: Parallel corpus $D=\\{(s_{1},t_{1}),$ $(s_{2},t_{2}),$ $\dots,$
$(s_{n},t_{n})\\}$, alignment
$A=\\{a_{1},a_{2},...,a_{n}\\},s=\epsilon,t=\epsilon$, revised alignment
$A^{{}^{\prime}}=\\{\\}$

1: for $k\leftarrow 1$ to $|D|$ do

2: for each $i,j\in a_{k}$ do

3: $i\leftarrow i+|s|$

4: $j\leftarrow j+|t|$

5: $A^{{}^{\prime}}\leftarrow A^{{}^{\prime}}\cup(i,j)$

6: end for

7: $s\leftarrow s+s_{k}$ {concatenation}

8: $t\leftarrow t+t_{k}$ {concatenation}

9: end for

10: return $s,\quad t,\quad A^{{}^{\prime}}$

Once we have combined the parallel corpus into a pair of sentences $(s,t)$, we
use the revised alignment $A^{{}^{\prime}}$ to generate parallel windows of
length 15-25 tokens using Algorithm 3.

Algorithm 3 Pseudo-code for extracting windows from the concatenated corpus

0: Unsegmented source $s$, target $t$, and word alignment $A^{{}^{\prime}}$

1: Initialize: $idx\leftarrow 0$

2: while $idx<|t|$ do

3: $l\leftarrow random(10,25)$

4: $W_{t}\leftarrow t[idx:idx+l]$ {target window}

5: $p=\min_{i}\\{(i,j)\in A^{{}^{\prime}},idx\leq j<idx+l\\}$

6: $q=\max_{i}\\{(i,j)\in A^{{}^{\prime}},idx\leq j<idx+l\\}$

7: $W_{s}\leftarrow s[p:q]$ {source window}

8: $idx\leftarrow idx+l$

9: end while

##  Appendix C Exploration of Match Threshold

We have two hyperparameters to consider: window length and threshold, when
generating the output. We explore their combination to find the best threshold
value. Table 3 shows BLEU scores with different window length and threshold.
We plot the flicker against the threshold for each window in Figure 3 and we
found 0.4 to be the best choice for threshold. Shorter windows force
commitment earlier producing lower flicker. Low thresholds promote spurious
matches making translation flicker more, whilst high thresholds force too many
retranslations. We have shown the number of retranslation in Table 4 for
different combination of window length and threshold. The reason why higher
threshold forces too many retranslations is that even if we set higher
threshold, it matches only with the match ratio between 0.5 to 0.6 on average.
We have shown the average match ratio after joining every combination of
window length and threshold in Table 5. We observe in Figure 3 that higher
threshold increases the flicker. The reason is that: as mentioned before, in
one hand, it never reaches a match of $>$ 0.6 on average thus it retranslates
more and generates longer output window, on the other hand, flicker depends on
actual number of token mismatch - longer window will have more mismatch for
the same threshold. In addition to that, these extra retranslations incurs an
increase in computation requirement. However, this increase in complexity can
be easily ignored, as in real life settings, largest source of latency is
waiting for new source content from the speaker (Arivazhagan et al., 2020b).

| Match Threshold ($r$) |
---|---|---
Window($w_{l}$) | 0.1 | 0.2 | 0.4 | 0.5 | 0.6 | 0.8
en$\rightarrow$de
8 | 10.8 | 11.3 | 12.3 | 12.5 | 12.4 | 12.5
10 | 12.0 | 12.3 | 12.7 | 12.8 | 12.8 | 12.7
12 | 12.5 | 12.7 | 12.9 | 13.0 | 13.0 | 12.9
14 | 12.7 | 12.8 | 13.0 | 13.0 | 12.9 | 12.8
16 | 12.9 | 12.9 | 13.1 | 13.1 | 13.1 | 13.0
18 | 13.0 | 13.0 | 13.2 | 13.2 | 13.2 | 13.1
20 | 13.1 | 13.0 | 13.2 | 13.2 | 13.2 | 13.2
en$\rightarrow$cs
8 | 8.3 | 9.1 | 9.8 | 10.0 | 10.0 | 9.9
10 | 9.5 | 9.7 | 10.2 | 10.3 | 10.2 | 10.2
12 | 10.0 | 10.2 | 10.4 | 10.4 | 10.4 | 10.4
14 | 10.2 | 10.4 | 10.5 | 10.4 | 10.5 | 10.4
16 | 10.5 | 10.6 | 10.6 | 10.6 | 10.6 | 10.5
18 | 10.5 | 10.5 | 10.6 | 10.6 | 10.6 | 10.5
20 | 10.5 | 10.6 | 10.7 | 10.7 | 10.5 | 10.5
Table 3: Results with different window length and threshold. Sacrebleu computed after sentence aligning each document using mwerSegmenter. Bleu scores in green have the lowest flickers. | Match Threshold ($r$) |  |
---|---|---|---
$w_{l}$ | 0.1 | 0.2 | 0.4 | 0.5 | 0.6 | 0.8 | #windows
en$\rightarrow$de |
8 | 1724 | 10513 | 66471 | 103034 | 140889 | 200441 | 45879
10 | 1303 | 7352 | 50345 | 82775 | 118991 | 185398 | 45497
12 | 956 | 6394 | 46528 | 74698 | 110669 | 178805 | 45115
14 | 702 | 4809 | 42886 | 69847 | 105207 | 173017 | 44733
16 | 432 | 4098 | 40447 | 66391 | 100591 | 167585 | 44351
18 | 308 | 3809 | 38774 | 65132 | 99410 | 163935 | 43969
20 | 215 | 3407 | 37358 | 64266 | 96701 | 162025 | 43587
en$\rightarrow$cs |
8 | 2388 | 14757 | 74465 | 111900 | 148605 | 206238 | 45879
10 | 1257 | 8906 | 53651 | 84838 | 120135 | 188964 | 45497
12 | 1374 | 7170 | 44905 | 71432 | 105294 | 176580 | 45115
14 | 1094 | 5825 | 40480 | 64564 | 97436 | 169418 | 44733
16 | 806 | 4762 | 37067 | 60457 | 92346 | 163384 | 44351
18 | 489 | 4118 | 34710 | 58321 | 89392 | 158114 | 43969
20 | 292 | 3807 | 33440 | 57187 | 87418 | 154827 | 43587
Table 4: Number of extra retranslations due to history extension. $w_{l}$ is
window length.

Figure 3: Threshold ($r$) vs Flicker plots.  | Match Threshold ($r$) |  |
---|---|---|---
$w_{l}$ | 0.1 | 0.2 | 0.4 | 0.5 | 0.6 | 0.8 | #windows
en$\rightarrow$de |  |
8 | 0.40 | 0.42 | 0.52 | 0.56 | 0.57 | 0.55 | 45879
10 | 0.47 | 0.49 | 0.57 | 0.59 | 0.60 | 0.58 | 45497
12 | 0.51 | 0.52 | 0.59 | 0.62 | 0.62 | 0.59 | 45115
14 | 0.52 | 0.54 | 0.60 | 0.63 | 0.64 | 0.60 | 44733
16 | 0.53 | 0.55 | 0.61 | 0.64 | 0.65 | 0.62 | 44351
18 | 0.54 | 0.55 | 0.61 | 0.64 | 0.65 | 0.62 | 43969
20 | 0.55 | 0.56 | 0.62 | 0.64 | 0.65 | 0.62 | 43587
en$\rightarrow$cs |  |
8 | 0.37 | 0.41 | 0.51 | 0.54 | 0.56 | 0.54 | 45879
10 | 0.45 | 0.47 | 0.56 | 0.59 | 0.60 | 0.57 | 45497
12 | 0.50 | 0.52 | 0.59 | 0.61 | 0.63 | 0.60 | 45115
14 | 0.53 | 0.54 | 0.61 | 0.63 | 0.65 | 0.61 | 44733
16 | 0.54 | 0.55 | 0.62 | 0.64 | 0.65 | 0.63 | 44351
18 | 0.55 | 0.56 | 0.62 | 0.65 | 0.66 | 0.63 | 43969
20 | 0.56 | 0.57 | 0.63 | 0.65 | 0.66 | 0.64 | 43587
Table 5: Average match ratio after joining all the windows across different
window length and threshold. We define average match ratio as
$\frac{1}{\\#window}\sum^{\\#window}\frac{\mathrm{match\\_length}}{\mathrm{output\\_window\\_length}}$