EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-LHCb-DP-2020-002

Muon identification for LHCb Run 3

L. Anderlini1, F. Archilli2, A. Cardini3, V. Cogoni3, M. Fontana4, G.
Graziani1, N. Kazeev5,6, H. Kuindersma7, R. Oldeman3, M. Palutan8, M.
Santimaria8, B. Sciascia8, P. De Simone8, R. Vazquez Gomez8,9

1Sezione di Firenze, INFN, Italy
2Physikalisches Institut, Ruprecht-Karls-Universitat Heidelberg, Germany
3Sezione di Cagliari, INFN, Italy
4CERN, Switzerland
5National Research University Higher School of Economics, Russia
6The Sapienza University of Rome, Italy
7Nikhef, National Institute for Subatomic Physics, Netherlands
8Laboratori Nazionali di Frascati, INFN, Italy
9IGFAE, Universidade de Santiago de Compostela, Spain

Muon identification is of paramount importance for the physics programme of
LHCb. In the upgrade phase, starting from Run 3 of the LHC, the trigger of the
experiment will be solely based on software. The luminosity increase to
$2\times 10^{33}$ cm-2s-1 will require an improvement of the muon
identification criteria, aiming at performances equal or better than those of
Run 2, but in a much more challenging environment. In this paper, two new muon
identification algorithms developed in view of the LHCb upgrade are presented,
and their performance in terms of signal efficiency versus background
reduction is presented.

Submitted to JINST

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

##  1 Introduction

The LHCb experiment [1] at the LHC is a single-arm forward spectrometer
specialised in studying particles containing $b$ or $c$ quarks. Thanks to a
versatile reconstruction and trigger system, the LHCb physics programme has
been extended to electroweak, soft QCD and even heavy-ion physics. Many of the
physics channels are identified by their very clean muon signatures, therefore
muon identification and trigger are crucial for the success of the experiment.

A brief description of the Run 2 muon detector and reconstruction techniques
follows, which sets the basis for the improvements later discussed in view of
Run 3. During Run 1 and Run 2, the tracking system of LHCb provided a
measurement of the momentum $(p)$ of charged particles with a relative
uncertainty that varied 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 $pp$ collision vertex (PV), the impact parameter (IP), was 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 ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. Muons were identified and
triggered by a system composed of five stations, M1-M5, of rectangular shape,
placed along the beam axis as shown in Fig. 1.

Figure 1: Side view of the muon system in the $y-z$ plane.

Stations M2 to M5 were placed downstream the calorimeters, and were
interleaved with $80$ cm-thick iron absorbers to select penetrating muons. The
M1 station was placed in front of the calorimeters and used to improve the
$p_{\mathrm{T}}$ measurement in the trigger.

Each muon station is subdivided into four regions, as shown in Fig. 2, with
different read-out schemes defining the $x,y$ resolutions. The dimensions of
the logical pads were chosen such that their contribution to the
$p_{\mathrm{T}}$ resolution was approximately equal to the multiple scattering
contribution [2].

Figure 2: Front view of one quadrant of M2 showing the four regions. The
intersection of a horizontal and a vertical strip defines a logical pad. The
region and channel dimensions scale by a factor of two from one region to the
following.

These logical pads were obtained from the crossing of horizontal and vertical
strips (either cathodic pads or group of wires), with the exception of the
full M1 station and the innermost regions of stations M4 and M5, where the
logical pads corresponded to physical channels on the detector and were
readout directly.

A schematic diagram showing the trigger data flow in Run 2 is depicted in Fig.
3.

Figure 3: The LHCb trigger scheme in Run 2.

The trigger and reconstruction scheme followed three basic steps:

  * •

A hardware trigger (L0), based on selected calorimeter and muon information,
to reduce the interaction rate of 20${\mathrm{\,MHz}}$ 111Out of the total LHC
bunch crossing rate of 40${\mathrm{\,MHz}}$, there are about
30${\mathrm{\,MHz}}$ of inelastic collisions, of which around 2/3 are visible
in the LHCb detector. to 1${\mathrm{\,MHz}}$, which corresponds to the readout
bandwidth of the detector. The L0 muon trigger was based on the coincidence of
one hit in each of the five stations, selected in a projective Field Of
Interest (FOI) defined in the $x-y$ plane, from which a muon standalone
$p_{\mathrm{T}}$ reconstruction was performed with $\sim 20\%$ resolution [2].
Candidate tracks above a $p_{\mathrm{T}}$ threshold of about
$1.5{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ were then used to build single and
dimuon topologies.

  * •

A first software stage (HLT1) based on partial reconstruction of tracks from
the spectrometer, which allowed to put more strict constraints on the
candidate $p_{\mathrm{T}}$ and IP. Concerning muons, a loose and efficient
selection was performed, called IsMuon, based on the coincidence of hits in M2
to M5 stations, and combined with the information of the spectrometer. The
muon hits were selected in a FOI centered around the track extrapolation
position on the muon stations: the number of hits required was two, three or
four in the momentum ranges $3-6$, $6-10$ and above
$10{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, respectively, as expected from the
muon penetration power in the iron absorbers [3].

  * •

A more refined software trigger (HLT2), exploiting the full reconstruction of
the detector information to reconstruct more complex signal topologies.
Concerning muons, a better discrimination than IsMuon was achieved by using a
likelihood variable (MuonDLL), built upon the uncorrelated sum of the spatial
residuals of the muon hits with respect to the track extrapolation position in
each station [3], defined as:

$D^{2}=\frac{1}{N}\sum_{i=1}^{N}\left[\left(\frac{x^{i}_{\text{closest}}-x^{i}_{\text{track}}}{\text{pad}^{i}_{x}}\right)^{2}+\left(\frac{y^{i}_{\text{closest}}-y^{i}_{\text{track}}}{\text{pad}^{i}_{y}}\right)^{2}\right],$
(1)

where the index $i$ runs over the $N$ stations containing hits inside the FOI,
and the closest coordinates represent the position of the hit which is closest
to the track extrapolation point. The hit residuals were normalised to the
logical pad size in the $x$ and $y$ directions, padx and pady respectively.
The $D^{2}$ distribution for muons exhibits a narrow peak at 0, while hadrons
satisfying the IsMuon criterion have a broader distribution. Using the $D^{2}$
spectra of muons as a signal proxy and of protons as a background proxy (pions
are instead contaminated by real muons from decays in flight), the MuonDLL
likelihood was defined, which measured the difference in probability for a
candidate track to match the signal or background hypotheses. Using the above
variable, on top of IsMuon, the misidentification probability for protons was
kept at the 2-3 per mille level on the full momentum spectrum, while keeping
the muon efficiency above 90%. For pions, similar performances were obtained
only for momenta higher than 50${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, with
decays in flight contributing for another few per mille at low momenta [4].

The LHCb detector will be upgraded for Run 3 to sustain a factor of five
increase in the instantaneous luminosity, up to $2\times 10^{33}$ cm-2s-1. The
1${\mathrm{\,MHz}}$ readout limitation of the current detector will be
removed, allowing for the full event rate to be processed in software without
the need for a hardware stage [5]. For this reason, a full software trigger
has been implemented, which will allow to select signal events with higher
efficiency, and with the goal of achieving an order of magnitude increase in
the physics bandwidth with respect to Run 2.

In preparation of Run 3, the M1 station has been removed due to the much
higher occupancy which will be reached in front of the calorimeter, where the
station is located (Fig. 1). In addition, its main contribution, consisting in
the improvement of the standalone muon $p_{\mathrm{T}}$ determination in the
hardware L0 trigger is no longer relevant. When working on the implementation
of the future software muon trigger lines, two aspects have to be taken into
account:

  * •

The need to keep a high efficiency at HLT1, with a smooth dependence on the
running conditions and on the phase space, and with a fast execution time.
Concerning the bandwidth, a high rejection power must be guaranteed against
the combinatorial background, especially important at low momentum. This
background mainly originates from pion tracks extrapolated to the muon
detector and paired to accidental hits in the muon stations, and is expected
to increase almost linearly with the luminosity.

  * •

The possibility to tune highly selective cuts in order to achieve very low
mis-identification levels at HLT2, especially useful for example in rare decay
searches. The full information from the LHCb PID detectors may conveniently be
used in this case as the constraints on the execution time are less stringent.

These functionalities can be implemented following different approaches. In
this paper we discuss a baseline strategy for the HLT1 which is an evolution
of the present scheme. This assumes that tracks in the spectrometer are
reconstructed upfront, and that the muon identification is applied in two
steps: a first step based on IsMuon as it is now, plus a second step based on
a correlated $\chi^{2}$ variable (Section 2), which represents an improvement
of the MuonDLL. At the HLT2 stage, the muon identification performances are
further refined by means of a multivariate classifier (Section 3).

##  2 Correlated $\chi^{2}$

The misidentification of charged pions and kaons to muons has an almost
irreducible component due to decays in flight, together with a combinatorial
component that is relevant especially at
$p<10{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, where the hit coincidence is less
stringent. The present muon identification algorithm was optimised for a low
occupancy scenario, without prioritising the rejection of the combinatorial
background. The higher luminosity of Run 3 will require instead to suppress
this background more effectively, especially in the central detector regions
where the occupancies are higher.

An obvious limitation of the present approach based on $D^{2}$ variable is
that it does not include the information from the multiple scattering
experienced by charged particles while traversing the calorimeter and the iron
absorbers, as well as the correlation between the hit positions across the
muon system. The importance of taking into account these correlations is
evident in Fig. 4, where two very different hit combinations are shown, yet
giving a similar MuonDLL value. On the left, a random combination of hits
scattered around the track extrapolation is shown, receiving contributions
from uncrossed logical pads, indicated by the larger error bars. Such events
are more affected by electronic noise and spillover hits. On the right-hand
side, a clear pattern of hits is visible, which are displaced with respect to
the extrapolated track due to multiple scattering.

Figure 4: Two different combination of muon hits having a similar value of
MuonDLL: a combinatorial background event (left) and a clear muon pattern
(right).

These two topologies can be discriminated by using a $\chi^{2}$ variable,
expressed as

$\chi^{2}_{\mathrm{CORR}}=\delta\overrightarrow{x}^{T}\text{V}_{x}^{-1}\delta\overrightarrow{x}+\delta\overrightarrow{y}^{T}\text{V}_{y}^{-1}\delta\overrightarrow{y},$
(2)

where $\delta\overrightarrow{x}$ and $\delta\overrightarrow{y}$ are the
distances, in the $x$ and $y$ directions, between the track extrapolation
points and the closest hit positions, with indices running over the stations
M2 to M5. The covariance matrices $\text{V}_{x}$ and $\text{V}_{y}$ both have
a diagonal contribution from the detector resolution and an off-diagonal
contribution from the multiple scattering (MS). The diagonal terms are of the
form

$\text{V}^{\text{RES}}_{jj}=\left(\text{pad}^{j}_{x,y}/\sqrt{12}\right)^{2},$
(3)

where the pad size along $x$ and $y$ corresponding to the muon hit in the
given station are used. The off-diagonal terms, accounting for MS, are
modelled as

$\text{V}^{\text{MS}}_{jk}=\sum_{z_{i}<z_{j},z_{k}}(z_{j}-z_{i})(z_{k}-z_{i})\sigma_{\text{MS},i}^{2},$
(4)

where $z_{j,k}$ represent the coordinates of stations M2 to M5 along the beam
axis, $z_{i}$ represents the coordinates of the main absorbers, namely the
calorimeters and the muon iron filters, as listed in Tab. 1, and
$\sigma_{\text{MS},i}$ represents the MS deviation. This term takes the usual
expression [6]

$\sigma_{\text{MS},i}=\frac{13.6\text{MeV}}{\beta cp}\sqrt{\Delta
z_{i}/X_{0}},$ (5)

where $p$ and $\beta c$ are the momentum and the velocity of the incident
particle, respectively, and $\Delta z_{i}/X_{0}$ is the thickness of the
absorber at the given position $z_{i}$ in units of radiation length, also
listed in Tab. 1.

Absorber |  $z$ position (m) |  $\Delta z_{i}$/X0
---|---|---
ECAL | 12.8 | 25
HCAL | 14.3 | 53
Muon filter 1 | 15.8 | 47.5
Muon filter 2 | 17.1 | 47.5
Muon filter 3 | 18.3 | 47.5
Table 1: Position along the beam axis and thickness in units of radiation
length for the main scattering media contributing to the multiple scattering
experienced by particles reaching the muon detector.

The probability for a muon to penetrate the iron absorbers and reach a given
muon station depends on its momentum. In particular, below
$6\,{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ the probability to reach M4 and M5
stations can be substantially smaller than one, so that hits falling in the
FOI of the track are in this case often due to accidental background. For this
reason, in that momentum interval only the hits on M2 and M3 stations are
included in the $\chi^{2}_{\mathrm{CORR}}$ computation.

The performances of the $\chi^{2}_{\mathrm{CORR}}$ variable are evaluated on
muons and protons from data control samples collected in 2016. An abundant
source of muons is provided by $J/\psi\rightarrow\mu^{+}\mu^{-}$ decays: by
requiring the reconstructed $J/\psi$ to have a large flight distance
significance and good decay vertex quality, most of the combinatorial
background from the tracks originating from the primary vertex is removed, and
the sample gets enriched by $B\rightarrow J/\psi X$ candidates. To further
reduce the background, one of the decay tracks, the $tag$ muon, is required to
be positively identified in the muon detector; the other track, the $probe$
muon, is unbiased with respect to particle identification and trigger
requirements and it is therefore used to measure the algorithm performances.
Protons are selected from $\Lambda\rightarrow p\pi^{-}$ decays, selected with
vertex quality criteria and detachment of the decay vertex from the primary
one. In addition, the invariant mass obtained by assigning the $\pi$ mass to
the two daughters is required to be outside the nominal $K^{0}_{S}$ mass
window. Examples of mass spectra for muon and proton calibration samples are
shown in Fig. 5.

Figure 5: Typical invariant mass distributions for
$J/\psi\rightarrow\mu^{+}\mu^{-}$ (left) and $\Lambda\rightarrow p\pi^{-}$
(right) calibration samples. The superimposed fit (red line) is composed of a
signal (dashed blue) and background (dotted dash green) component [7].

For both samples, the residual background contribution is subtracted by using
the sPlot method [8]. To perform unbiased studies, the muon and proton samples
have been weighted in order to equalize their momentum, transverse momentum,
and track multiplicity spectra. In addition, since the main challenge for Run
3 is the fivefold luminosity increase with respect to Run 2, a weighting
procedure that adds more emphasis on high multiplicity events is applied to
each calibration sample. Since there is not enough data to accurately emulate
the upgrade conditions, the samples have been weighted in such a way to
reproduce an occupancy spectrum which is in-between the two actual running
conditions.

As a result, in Fig. 6 the $\chi^{2}_{\mathrm{CORR}}$ spectrum for muons and
protons satisfying the IsMuon requirement is shown, demonstrating a good
separation between signal and background.

Figure 6: Spectrum of the $\chi^{2}_{\mathrm{CORR}}$, normalised by the
degrees of freedom, for muons and protons as evaluated on 2016 calibration
samples

A quantitative comparison between the performance of the
$\chi^{2}_{\mathrm{CORR}}$ and MuonDLL variables is shown in Fig. 7, where the
proton rejection as a function of the muon efficiency, ROC curve in the
following, is displayed for tracks satisfying the IsMuon requirement. The ROCs
are shown in different momentum and transverse momentum intervals, which allow
to probe the response of the muon identification algorithms in different
regions of the detector and in different momentum regimes.

Figure 7: Proton rejection as a function of muon efficiency for tracks
satisfying IsMuon obtained with the $\chi^{2}_{\mathrm{CORR}}$ (blue) and
MuonDLL (black) variables on 2016 calibration data. Low momentum bins, which
are not covered by the calibration samples, are not shown.

The performance of the $\chi^{2}_{\mathrm{CORR}}$ variable is definitely
better than the MuonDLL in all regions of the phase space, and especially at
low momenta. In particular, at muon efficiency of $\sim 98\%$, which is a good
working point for efficient trigger selections, the gain in background
rejection is a factor $\sim 1.4$ in the region
$p>10\,{\mathrm{\,Ge\kern-1.00006ptV\\!/}c},~{}\mbox{$p_{\mathrm{T}}$}<2\,{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$,
and exceeds a factor of $2$ in the rest of the phase space.

###  2.1 Performance in HLT1

As discussed in Sec. 1, for the HLT1 in Run 3 it will be crucial to guarantee
a high efficiency for muons and a fast execution time of the algorithms.
Moreover, a tighter rejection against combinatorial background with respect to
the present IsMuon selection will be certainly needed.

Given the good performances of the $\chi^{2}_{\mathrm{CORR}}$ variable in
rejecting protons, which constitute pure combinatorial background to the muon
detector, we consider as interesting to provide the rejection estimates on
trigger unbiased events, which are mostly populated by pions, selected from a
Run 2 data sample without any trigger requirement. As a preliminary selection
for this benchmark, the events are filtered by requiring at least one track to
satisfy IsMuon and the cuts $p_{T}>800{\mathrm{\,Me\kern-1.00006ptV\\!/}c}$
and $\text{IP}\chi^{2}>35$ 222The impact parameter $\chi^{2}$,
$\text{IP}\chi^{2}$, is defined as the difference in the primary vertex fit
$\chi^{2}$ with and without the given track., which represent the main
requirements of the Run 2 HLT1 single muon line. The rejection is therefore
computed relatively to the above selection, and thus represents the
improvement with respect to the present HLT1, and by removing the L0 trigger.
To select high multiplicity events, only those having at least $3$ primary
vertices (nPVs) are used, whereas average Run 2 events have one primary
vertex.

This study is done in three momentum intervals, $3-6$, $6-10$ and
$p>10{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, since the number of hits selected
by IsMuon is different in each one, as described in Sec. 1. In each interval,
a $\chi^{2}_{\mathrm{CORR}}$ cut with $\sim 98\%$ muon efficiency, as
evaluated on muon calibration data, is chosen. The results are shown in Tab. 2
and demonstrate the effectiveness of this variable in rejecting about half of
the trigger unbiased events, with a very small efficiency loss, on top of the
Run 2 HLT1 muon selection. In particular, the highest rejection is achieved
for $6<p<10{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, where the fraction of pion
decays in flight is lower with respect to
$3<p<6{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and the MS correlations provide
sensible discrimination as the momentum is not too high.

Table 2: Rejection factors of the $\chi^{2}_{\mathrm{CORR}}$ variable on trigger unbiased events, for a muon efficiency of $\sim 98\%$. The rejection is evaluated on top of the $p_{T}>800{\mathrm{\,Me\kern-0.90005ptV\\!/}c}$, $\text{IP}\chi^{2}>35$, IsMuon and nPVs $\geq 3$ requirements. Momentum range | Rejection factor
---|---
$3<p<6\,{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ | $1.8$
$6<p<10\,{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ | $3.2$
$p>10\,{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ | $2.2$

Finally, the $\chi^{2}_{\mathrm{CORR}}$ execution time is tested within the
HLT1 upgrade sequence. Throughput tests333On nodes mounting two Intel Xeon
E5-2630 v4 CPUs at $2.20$ GHz (40 threads/node). are performed on simulated
Run 3 data and show a $\chi^{2}_{\mathrm{CORR}}$ resource usage of about
$0.4\%$ out of a total HLT1 throughput rate of $\sim 36$ MHz, and in view of a
data taking rate of $30$ MHz. This result makes the $\chi^{2}_{\mathrm{CORR}}$
well suited for a usage in the upgraded HLT1 trigger of the experiment.

##  3 Multivariate algorithm

At the second stage of the trigger, HLT2, and offline, the timing budget
allows to use more complex algorithms. Besides the spatial information, each
muon hit also carries two different time counters, one for each view, $x$ and
$y$. The number of views (i.e. the fact that the hit is crossed or uncrossed)
also provides valuable information, as noise or spillover hits typically have
one view only. This information, along with its correlations, can be exploited
in a multivariate operator. To this purpose, a recent variant of gradient tree
boosting available in the CatBoost library from Yandex [9] has been
implemented [10]. It uses oblivious decision trees as weak learners, as
explained in the following.

A regular decision tree selects each split independently, while an oblivious
decision tree has the same split on each level. The difference is illustrated
in Fig. 8. An oblivious decision tree is less expressive but is much faster to
evaluate, as it makes possible to unwrap the tree into a table and look up the
correct leaf in one operation, instead of the multiple conditional jumps of a
regular tree. According to a benchmark study by the CatBoost authors, this
provides 30–100 faster prediction compared to the competing state-of-the-art
gradient boosting libraries [11].

Figure 8: Classic versus oblivious decision trees. Reproduced from [12].

For the muon identification, five variables for each muon station M2 to M5 are
used as input to the CatBoost algorithm:

  * •

$x_{\text{res}}$: the difference between the closest hit $x$ position and the
track extrapolation, normalised to the total uncertainty;

  * •

$y_{\text{res}}$: the difference between the closest hit $y$ position and the
track extrapolation, normalised to the total uncertainty;

  * •

$t_{x}$: the time of the $x$ view;

  * •

$dt=t_{x}-t_{y}$: the temporal difference between the $x$ and $y$ views;

  * •

$N_{\text{views}}$: the number of views.

The uncertainty in the residuals $x_{\text{res}}$ and $y_{\text{res}}$
contains the pad size and the contribution from the multiple scattering (Eq.
5), summed in quadrature. In addition to the hit information, for each event
the track extrapolation $x$ and $y$ coordinates on M2 are used, to allow the
algorithm to discriminate between different detector regions. Finally, the
aforementioned $\chi^{2}_{\mathrm{CORR}}$ variable of the track is added. This
set of up to 23 variables per event has been found to be the smallest one
containing the maximum associated information, without introducing excessive
correlations. This feature is very important in order to decrease the
complexity and hence the computation time of the operator in the trigger.

The classifier is trained using samples from 2016 data, to which the IsMuon
requirement is applied. Since the IsMuon algorithm is very fast to execute and
already provides rejections of $\mathcal{O}(1\%)$, evaluating the classifier
only for events that pass the IsMuon requirement allows to significantly
reduce the computational cost and to focus on reducing the remaining
background. The data samples used in the training are:

  * •

muons from $J/\psi\rightarrow\mu^{+}\mu^{-}$ decays,

  * •

pions from $D^{*}\rightarrow D^{0+}(\rightarrow K^{-}\pi^{+})\pi^{-}$ decays,

  * •

protons from $\Lambda\rightarrow p\pi^{-}$ decays.

While protons represent pure combinatorial events, the pion sample is included
to boost the training statistics and accounts for another source of
classification error due to particles that decay in flight to muons before
reaching the muon stations. These samples have been treated as described in
Sec. 2, including kinematic reweighting, background subtraction and
multiplicity weights.

To deal with negative sWeights, the solution proposed in Ref. [10] is used,
which consists of first using a machine learning regression to estimate the
expected sWeight in each point of the training variable phase space and,
second, using the expected sWeight as event weight during classification.

Finally, since the classifier is trained on the same 2016 calibration data
which are used to evaluate its performance, a cross-validation method is used
to obtain unbiased predictions. The dataset is split into $5$ subsets of equal
size, and the model is independently trained on all subsets but the $i$-th,
for which predictions are made. The ROC curves of the CatBoost algorithm are
shown in Fig. 9 for muon efficiencies above $90\%$. For comparison, the ROC
curves for the $\chi^{2}_{\mathrm{CORR}}$ variable are superimposed.

Figure 9: Proton rejection as a function of muon efficiency for tracks
satisfying IsMuon obtained with the CatBoost algorithm (magenta) and
$\chi^{2}_{\mathrm{CORR}}$ (blue) on 2016 calibration data. Low momentum bins,
which are not covered by the calibration samples, are not shown.

As a result, for high muon efficiency the CatBoost algorithm has better
discriminating power than $\chi^{2}_{\mathrm{CORR}}$ in all the momentum bins.
The difference in background rejection has a break point around 98% muon
efficiency, where for $p<10\,{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ it lies in
20 – 40 % range and 4 – 10 % for $p\geq
10\,{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$.

With a similar setup as the one for HLT1 (Sec. 2.1), throughput tests are
performed on simulated Run 3 data and show resource usage of about 0.4% out of
a total HLT2 throughput rate of $\sim 129$ Hz. Therefore, this CatBoost
operator is fast enough to be employed in the upgraded HLT2 trigger of the
experiment.

##  4 Conclusions

Two new muon identification algorithms have been developed in view of the LHCb
Run 3 upgrade. The first one, $\chi^{2}_{\mathrm{CORR}}$, expands on the muon
likelihood variable developed in Run 1 by including the correlation among the
muon hits. The second one features a multivariate algorithm based on the
CatBoost machine learning toolkit. The performances of both algorithms in
terms of background rejection versus signal efficiency are characterised on
2016 proton calibration data, and in both cases are found to improve
considerably those of the muon likelihood used during Run 1 and Run 2, with
the CatBoost classifier offering a slightly better performance. As far as the
computational times are concerned, the $\chi^{2}_{\mathrm{CORR}}$ has been
proven to be fast enough to be included in the upgrade muon trigger lines. The
CatBoost algorithm, while improving the competing state-of-art gradient
boosting libraries, can be computed in the HLT2, where the time constraints
are less stringent.

## Acknowledgements

We would like to thank the LHCb RTA team for supporting this publication, and
in particular Vladimir Gligorov for the review and Manuel Schiller for the
code optimisation. 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, Yandex LLC, HPC facilities at NRU HSE (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); Basic Research
Program of the NRU HSE and Yandex LLC (Russia); GVA, XuntaGal and GENCAT
(Spain); the Royal Society and the Leverhulme Trust (United Kingdom).

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