# Genetic Algorithm for determination of the event collision time and particle
identification by time-of-flight at NICA SPD

Semyon Yurchenko1, Mikhail Zhabitsky2 Correspondence:
<EMAIL_ADDRESS>to MDPI Physics journal

(1Saint Petersburg State University, Laboratory of ultra-high energy physics
2Joint Institute for Nuclear Research
)

###### Abstract

Particle identiﬁcation is an important feature of the future SPD experiment at
the NICA collider. In particular, identification of particles with momenta up
to a few $\text{GeV}/c$ by their time-of-flight will facilitate reconstruction
of events of interest. High time-resolution of modern TOF detectors dictates
the need to obtain the event collision time $t_{0}$ with comparable accuracy.
While determination of the collision time is feasible through the use of TOF
signals supplemented by track reconstruction, it proves to be computationally
expensive. In this work we have developed a dedicated Genetic Algorithm as a
fast and accurate method to determine the pp-collision time by the
measurements of the TOF detector at the SPD experiment. By using this reliable
method for $t_{0}$ determination we compare different approaches for the
particle identification procedure based on TOF-signals.

Keywords: Genetic Algorithm; Time-Of-Flight; Particle identification

##  1 Introduction

The Spin Physics Detector (SPD) is a future experiment that will be placed in
one of the two interaction points of the NICA collider in the Joint Institute
for Nuclear Research. By studying collisions of polarized proton and deuteron
beams, the SPD collaboration will perform a comprehensive study of the
unpolarized and polarized gluon content of nucleons and other spin related
phenomena [1]. With polarized proton-proton collision energies $\sqrt{s}$ up
to $27~{}\text{GeV}$, SPD will cover a kinematic range between the low-energy
measurements at ANKE-COSY and SATURNE and the high-energy measurements at RHIC
and LHC.

The SPD experimental setup is planned as a general-purpose $4\pi$ detector
with advanced tracking and particle identification capabilities. The particle
identification will be performed by means of $dE/dx$, Time-Of-Flight (TOF),
Electromagnetic calorimetry and Muon-filtering techniques. The experiment will
use a system of Multigap Resistive Plate Chambers (MRPC) [2] as the TOF
detector. Its main aim is to provide $\pi/K/$p-identification of charged
particles with momenta up to a few $\text{GeV}/c$.

Identification of particles’ types by their time-of-flight is a well
established technique in high energy physics collider experiments [3, 4, 5, 6,
7]. It requires just three ingredients: $p$ – momentum of the particle, $L$ –
arc length of its trajectory between the primary collision point and the TOF
detector, $\tau$ — the corresponding time-of-flight. The latter is calculated
as a difference between the stop and start ($t_{0}$) signals. While the stop
signal is measured with high precision by the TOF detector, the collision time
$t_{0}$ cannot be obtained directly with the same accuracy. The collision time
can be estimated from timing of the accelerator or deduced from signals of a
corresponding T0-detector, which will detect appearance of secondary particles
scattered on small angles with respect to the beams axis, but in this case
uncertainty of the collision time will dominate uncertainty of the difference
between stop and start signals. Fortunately, the event collision time $t_{0}$
can be determined with sufficient precision from the TOF measurements by means
of $\chi^{2}$ minimization procedure [3, 5].

As a significant number of secondary particles, originated from the proton-
proton collision, subsequently enters the acceptance and is detected by the
TOF detector, one can reconstruct $t_{0}$ as a common value for all detected
particles through minimization of the sum of the squares of residuals. A
residual is defined as a difference between the measured TOF signal and its
expected arrival time, assuming a particle’s mass hypothesis. Thus the
$\chi^{2}$ is minimised with respect not only to $t_{0}$ but to all particles’
masses, which proved to be a difficult computational problem [4].

As the bulk of secondary particles are pions, kaons or protons, it is natural
to try different combinations of their masses in order to minimize $\chi^{2}$,
thus the minimization is performed over a discrete set of particles’ types.
The global minimum can be found by the Brute Force Algorithm (BFA), which is
characterized by a very long run-time. Authors have developed an Asynchronous
Differential Evolution-inspired [9] Genetic Algorithm (ADE-GA), which solves
the $\chi^{2}$ minimisation problem within significantly reduced computational
time.

The reliable method for $t_{0}$ determination facilitates the identification
of particles by their time-of-flight. Several approaches can be used for the
PID procedure [10]. In this work we compare performance of the Bayesian
approach, ”n-sigma” criteria and the direct solution of the
$\chi^{2}$-minimisation problem.

##  2 Time-Of-Flight detector and event selection

Figure 1: General layout of the SPD detector [1]

The TOF system will consist of a barrel and two end-cap parts with a radius of
about 105 cm and a length of 370 cm. It will have an overall active area of
$27~{}\text{m}^{2}$. The short distance from the TOF detector to the
interaction point dictates the TOF resolution to be within $50\div 60$ ps,
which can be achieved with the MRPC technologies [2]. In our study we use a
conservative estimation $\sigma_{t}=70$ ps. The TOF detector will be located
within the solenoidal magnetic field ($B=1$ T, parallel to the beams axis)
outside the inner tracker, which will measure momenta with relative precision
$\frac{\sigma_{p}}{p}=2\%$ [1].

PYTHIA8 Monte Carlo generator [8] has been used for simulation of proton-
proton collisions at the center-of-mass energy $\sqrt{s}=27$ GeV, SoftQCD:all
settings have been selected to simulate minimum-bias events.

All charged particles have been propagated through the uniform magnetic field.
Intersection points of helix trajectories with the TOF detector have been
calculated through the closed-form expressions.

Only charged tracks with momenta greater than $0.5~{}\text{GeV}/c$ have been
used in the analysis, as relativistic particles are characterized by
negligible track distortions due to multiple scattering and $dE/dx$-losses in
the walls of the beam pipe and material of the inner tracker. In the following
the generated momentum value $p_{0}$ and the arrival time $t_{i,0}$ to the TOF
detector are smeared according to Normal distributions $p=N(p_{0},\sigma_{p})$
and $t_{i}=N(t_{i,0},\sigma_{t})$.

##  3 Event collision time measurement performed by the TOF detector

###  3.1 Method to reconstruct the event collision time

A large fraction of particles produced in proton-proton collisions at
$\sqrt{s}=27~{}\text{GeV}$ has momenta below $2~{}\text{GeV}/c$, which
suggests a time-of-flight as a powerful technique for particles
identification. Statistically more than half of secondary charged particles
detected by the TOF detector are pions. Other major contributions are protons
and charged kaons. Admixtures of electrons and muons do not exceed a few
percent and will be identified by the Electromagnetic Calorimeter and the Muon
Range System [1]. One can calculate all possible times-of-flight

$\tau_{ij}=\frac{L_{ij}}{c}\sqrt{1+\frac{m_{j}^{2}}{p_{i}^{2}}},$ (1)

by assigning independently for each track $i$ a certain particle type $j$.
Formula (1) works well only for relativistic particles, for low momenta it has
to be exchanged to piecewise summation along the particle’s trajectory
intersecting coordinate detectors. Arc length $L_{ij}$ should take into
account details of a type $j$ particle interaction with matter. In this
analysis we selected only tracks with momentum above $0.5~{}\text{GeV}/c$
where matter effects can be neglected. For the event with $N$ reconstructed
tracks the event collision time can be found by a $\chi^{2}$ minimization
procedure:

$\chi^{2}=\sum_{i}^{N}\frac{(t_{0}+\tau_{ij}-t_{i})^{2}}{\sigma_{t}^{2}+\sigma_{\tau(j,p_{i})}^{2}}=\sum_{i}^{N}\frac{(t_{0}+\tau_{ij}-t_{i})^{2}}{\sigma_{ij}^{2}}.$
(2)

Where time-of-flight uncertainty $\sigma_{\tau(j,p_{i})}$ due to uncertainty
in momentum is given by

$\sigma_{\tau(j,p_{i})}=\frac{L_{ij}}{c}\cdot\frac{m^{2}_{j}}{p^{2}_{i}}\left(\sqrt{1+\frac{m^{2}_{j}}{p^{2}_{i}}}\right)^{-1}\cdot\frac{\sigma_{p}}{p}=0.02\cdot\frac{L_{ij}}{c}\cdot\frac{m^{2}_{j}}{p^{2}_{i}}\left(\sqrt{1+\frac{m^{2}_{j}}{p^{2}_{i}}}\right)^{-1}.$
(3)

For a certain mass hypothesis an analytic solution for $t_{0}$ reads:

$t_{0}=\sigma_{0}^{2}\sum_{i}^{N}\frac{t_{i}-\tau_{ij}}{\sigma_{ij}^{2}},\quad\text{where}\quad\frac{1}{\sigma_{0}^{2}}=\sum_{i}^{N}\frac{1}{\sigma_{ij}^{2}}.$
(4)

So task is reduced to minimisation of $\chi^{2}$ (Eqs. 2–4) by finding the
proper mass hypothesis — vector of masses $(m_{1},m_{2},\ldots,m_{N})$ for
tracks in the event-by-event way. The emphasis is paid to deduce an accurate
and unbiased estimation of the collision time $t_{0}$. Sections 3.2 and 3.3
are dedicated for developing algorithms to perform the minimisation step.

###  3.2 Brute Force Algorithm

Most straightforward solution is to check all mass hypotheses and thus locate
the global minimum — a combination of masses which has minimal $\chi^{2}$ (so-
called exhaustive search or Brute Force Algorithm — BFA). If $N_{m}$ — number
of possible masses (possible particles types), then the total number of
combinations is $N_{m}^{N}$ and time complexity of this algorithm will be
$O(N\cdot N_{m}^{N})$. Exponential running time means that this algorithm is
computationally expensive if the number of reconstructed tracks exceeds 10. To
keep BFA execution time reasonable, possible particle types are restricted to
$\pi^{\pm}$, $K^{\pm}$ and protons ($N_{m}=3$).

###  3.3 Genetic Algorithm

The minimisation of $\chi^{2}$ (Eqs. 2–4) is performed over discrete set of
particle species, thus represents a typical problem in the domain of the
discrete optimization. To solve the problem we have developed an Asynchronous
Differential Evolution-inspired [9] Genetic Algorithm (ADE-GA).

All possible types of particles are represented as an ordered by mass set
(genetic representation), e.g. $[m_{\pi},m_{K},m_{p}]\rightarrow[0,1,2]$. The
algorithm maintains a set of candidate solutions called a population. It
optimizes a problem by iteratively improving the population through generation
of new candidate solutions, which can replace inferior population members by
means of natural (Darwinian) selection. Opposite to the exhaustive search, the
algorithm does not check all possible mass combinations, but identifies better
solutions and performs further searches around them.

Algorithm’s workflow for an event with $N$ tracks is as follows:

  1. 1.

Create an initial population of $N_{pop}$ random candidates solutions. Each
candidate solution is a random set of $N$ masses associated with corresponding
tracks, each species has equal probability $\frac{1}{N_{m}}$ to be assigned to
a given track. Initialization procedure enforces that all population members
are unique and for each track there are at least two different masses within
the population.
Example of a population in event with 5 tracks and size of population
$N_{pop}=5$:
$v_{1}$ $(0,1,1,2,0)\leftrightarrow(m_{\pi},m_{K},m_{K},m_{p},m_{\pi})$
$v_{2}$ $(2,2,0,1,0)\leftrightarrow(m_{p},m_{p},m_{\pi},m_{K},m_{\pi})$
$v_{3}$ $(1,1,1,0,0)\leftrightarrow(m_{K},m_{K},m_{K},m_{\pi},m_{\pi})$
$v_{4}$ $(0,0,0,2,1)\leftrightarrow(m_{\pi},m_{\pi},m_{\pi},m_{p},m_{K})$
$v_{5}$ $(2,0,1,1,2)\leftrightarrow(m_{p},m_{\pi},m_{K},m_{K},m_{p})$

  2. 2.

Create a new candidate solution (offspring generation):

    1. (a)

Choose three distinct random solution vectors from the current population and
create a mutant vector:

${}v_{mut}=v_{p}+(v_{r}-v_{q}).$ (5)

Vector $v_{p}$ is called a parent vector. Two other vectors form a difference
vector. If any coordinate falls outside the range $[0,N_{m}-1]$, it is
projected back to the corresponding boundary. The mutant vector has to be
different from any population member, otherwise the generation is repeated.
Example: $v_{p}=(0,1,1,2,0)$, $v_{r}=(2,2,0,1,0)$, $v_{q}=(1,1,1,0,0)$,

$v_{mut}=(0,1,1,2,0)+(2,2,0,1,0)-(1,1,1,0,0)=(1,2,0,2,0).$ (6)

    2. (b)

Calculate fitness of the offspring: $t_{0}^{mut}$ and $\chi^{2}_{mut}$ (see
Eqs. 2–4),

    3. (c)

Compare $\chi^{2}_{p}$ and $\chi^{2}_{mut}$,

    4. (d)

If $\chi^{2}_{mut}<\chi^{2}_{p}$ — the new mutant vector is better than the
parent, then the offspring supersedes the parent vector in the population.
Otherwise the population remains unchanged. This step is called natural
(Darwinian) selection.

  3. 3.

Steps 2a–2d are repeated until a stop criteria is met. After a predefined
number of iteration $N_{steps}$, the solution with the smallest $\chi^{2}$ is
chosen as the best combination.

This algorithm has only one control parameter — size of the population
$N_{pop}$. Numerical simulations proved that $N_{pop}=15$ is sufficient to
solve the problem. Time complexity of Genetic Algorithm is $O(N\cdot
N_{pop}\cdot N_{steps})$, where $800<N_{steps}<1000$. Time complexity
increases only linearly as a function of track number, which makes the
algorithm suitable for high-multiplicity events. Also the Genetic algorithm is
not limited to $N_{m}=3$, but can perform the global search for a wider range
of possible particle types without loss of performance.

Besides the population size $N_{pop}$ the canonical Differential Evolution has
two other control parameters: crossover rate $C_{r}$ and scale factor $F$. In
this study $C_{r}=1$ because minimization variables are correlated, $F=1$ is
chosen due to granularity of the mass spectra. The developed ADE-GA algorithm
represents the asynchronous Evolutionary Algorithm: it updates randomly
selected population members by the ADE/rand/rand/1 strategy [9]. In this
approach fitness of many candidate solutions can be calculated in parallel,
which will further speed-up calculations.

##  4 Results

###  4.1 Comparison of the Genetic Algorithm with the Brute Force Algorithm

Brute Force Algorithm finds the global minimum of $\chi^{2}$ minimisation and
is used as a reference to check performance of the Genetic Algorithm. Due to
high time complexity we can use Brute Force Algorithm as a reference only in
events with low multiplicities ($5\leq N\leq 14$). Distributions of errors
$\Delta t_{0}=t_{0}-t^{true}_{0}$ for such events are presented in Fig. 2.
Only $\pi^{\pm},K^{\pm},p^{\pm}$ are used as allowed particle types. Both
algorithms provide unbiased estimation of the reconstructed event collision
time with resolutions of $29$ ps for the Brute Force Algorithm and $30$ ps for
the Genetic Algorithm.

Figure 2: Distributions of errors $\Delta t_{0}=t_{0}-t^{true}_{0}$ for the
event collision time reconstructed by the Brute-Force algorithm (left) and by
the DE-inspired Genetic Algorithm (right)

Another important metric is the overall percentage of tracks that were
identified correctly: $97.2\%$ for the Brute Force Algorithm, $96.8\%$ for the
Genetic Algorithm. Non-zero PID inefficiency by BFA looks counter-intuitive,
but it appears due to finite resolution of the TOF detector when uncertainty
of its measurement exceeds typical time-of-flight difference between two
different particle types at a given momentum. In this case a particle will be
misidentified if the global minimum of $\chi^{2}$ minimization is deeper than
the $\chi^{2}$ of the actual particle configuration.

The Genetic Algorithm performance to solve $\chi^{2}$ minimization problem is
on a par with the exhaustive search, but it demonstrates a different time
complexity for high-multiplicity events (see Fig. 3). While for events with
less than 8 tracks Brute Force Algorithm has shorter run time, it
exponentially slows down as multiplicity grows. Average run time of Brute
Force Algorithm on events with $5\leq N\leq 14$ is $5$ ms, while Genetic
Algorithm is much faster — $160\;\mu$s. Both BFA and ADE-GA are intrinsically
parallel algorithms. Run-times, cited in this article, have been measured in a
single-thread calculation mode to simplify the comparison. Faster execution
time is achieved with multithreading.

Analysis by ADE-GA of events with any number of reconstructed tracks confirms
that the uncertainty in the collision time $\sigma_{0}$ decreases from about
$32$ ps for 5-track events down to about $20$ ps for high-multiplicity events,
it scales as $1/\sqrt{N}$. The achieved uncertainty in $t_{0}$ is much better
than the resolution of the TOF detector $\sigma_{t}=70$ ps, thus the latter
will dominate uncertainty in the time-of-flight between the collision point
and the TOF detector. The efficiency of the ADE-GA to accurately reconstruct
the collision time is estimated to be about 97%.

Figure 3: Time complexity comparison of the Brute-Force and the DE-inspired
Genetic algorithms: the median run-time $\langle t\rangle$ as a function of
event multiplicity

Thanks to the swiftness of the Genetic algorithm, a wider than $N_{m}=3$ range
of possible mass types can be taken into account. If electrons/positrons are
added then the reconstructed event collision time becomes biased (Fig. 4). Due
to short flight paths, the expected arrival time of pions with momenta above
$1~{}\text{GeV}/c$ to the TOF detector is delayed to electrons less than the
TOF time-resolution. In this case some pions will be misidentified as
electrons whenever such a mass hypothesis provides a deeper minimum for
$\chi^{2}$. As pions are much more abundant than electrons/positrons such
misidentification will result in a biased estimation of the collision time.

Figure 4: Distributions of errors $\Delta t_{0}=t_{0}-t^{true}_{0}$ for the
event collision time reconstructed by ADE-GA with
$\mathrm{e}^{\pm},\pi^{\pm},K^{\pm},p^{\pm}$ species

To remedy the probability of misidentifications one can remove from
consideration any track which type is not identified in a unique way for sure
(for example by $n$-sigma criteria, see below). This approach was derived in
work [11]: using a priori knowledge about dominant prevalence of pions in the
sample of registered particles, one can consider all low momenta tracks as
pions, for each track calculate estimation of the collision time, identify the
most probable one and reject all heavier-than-pion particles. In this way the
collision time can be found with uncertainty about $32\;$ps for events with a
fairly high number of tracks, but not in events with less than 3 pions below
$1.5~{}\text{GeV}/c$.

As the accurate and unbiased estimation of the event collision time is the
main goal of this study, the correctness of the obtained $t_{0}$ value is
further verified by iteratively removing major addends from the $\chi^{2}$-sum
(Eq. 2) followed by the $\chi^{2}$ minimization over the rest of tracks in the
event. Statistically significant shift of the $t_{0}$ value indicates a
possible outlier due to noise or misidentification.

###  4.2 Particle identification by time-of-flight

As the reliable method to reconstruct the event collision time $t_{0}$ is
developed one can perform particle identification through comparison of track
timing by the TOF detector to the expected time of particle’s arrival to the
detector. There are several strategies for PID by time-of-flight:

  1. 1.

One can assign particle type for each track from the result of
$\chi^{2}$-minimization: the track type is accepted as the most likely species
(maximal probability).

  2. 2.

Or for every track $i$ in event one can exclude it from determination of the
collision time $t_{0}$ to avoid correlations. Let’s denote as $t_{i0}$ the
event collision time calculated over the rest of tracks in the event. Then
there are two common strategies to perform PID by time-of-flight [10]:

    1. (a)

$n$-sigma selection — the most simple threshold discriminator:

$n_{ij}=\frac{t_{i}-(t_{i0}+\tau_{ij})}{\sigma_{ij}}=\frac{S_{i}-\hat{S_{i}}(m_{j})}{\sigma_{ij}}.$
(7)

Here $S_{i}$ is a signal obtained for track $i$, $\hat{S}_{i}(m_{j})$ is the
expected signal for a particle of species $j$ with momenta $p_{i}$. If the
signal belongs to the range $\pm 2\sigma$ or $\pm 3\sigma$ of a certain
species this track is accepted as the particle of this species. Track can be
accepted as multiple species.

    2. (b)

Bayesian method: takes into account yield of particle species. The conditional
probability for track $i$ to be a particle of species $j$ reads:

$P(H_{j}|S_{i})=\frac{P(S_{i}|H_{j})C(H_{j})}{\sum_{\alpha=\pi,K,p}P(S_{i}|H_{\alpha})C(H_{\alpha})}.$
(8)

Here $C(H_{j})$ is a prior probability that is calculated iteratively. It
takes into account relative abundance of species $j$, which depends on
particle momenta and emission angle. The likelihood function $P(S_{i}|H_{j})$
is given by:

$P(S_{i}|H_{j})=\frac{1}{\sqrt{2\pi}\sigma_{ij}}\exp\left(-\frac{1}{2}n^{2}_{ij}\right).$
(9)

Separation power $n\sigma_{\pi K}=(\tau_{iK}-\tau_{i\pi})/\sigma_{iK}$ can be
used as a measure of the PID performance [5]. In the SPD, identification of
particles by their time-of-flight can be performed up to $1.7~{}\text{GeV}/c$
for $\pi/K$ separation and up to $3~{}\text{GeV}/c$ for $K/$p at $3\sigma$
level (Fig. 5).

Figure 5: $\pi/K$ and $K/$p separation powers as functions of momenta

###  4.3 PID benchmarks for two-prong decay channels

Different PID methods have been compared by reconstructing different decay
channels with two oppositely charged particles in the final state:
$\phi\rightarrow K^{+}K^{-}$, $\Lambda\rightarrow p^{+}\pi^{-}$ and
$K^{0}_{s}\rightarrow\pi^{+}\pi^{-}$. In Figs. 6–8 invariant mass of all pairs
of oppositely charged tracks is shown. The same combination of masses has been
assigned to each pair as for the channel of interest (”no pid” in figures). In
the $n$-sigma approach only tracks which have TOF signals within $\pm
3\sigma_{ij}$ of a certain species $j$ are selected (”$3$ sigma”). In case of
the weighted Bayesian PID all combinations are included with the pair’s weight
as a product of conditional probabilities (Eq. 8) for each prong (”bayesian”).
The possible pair combinations corresponding to the global minimum of the
$\chi^{2}$-minimization are marked as ”chi2_min”. Finally, ”ideal” corresponds
to Monte-Carlo combinations with known particle types.

Figure 6: Invariant mass of pairs of oppositely charged tracks assumed to be
$K^{+}K^{-}$ (left) and corresponding signal-to-background ratios (right) with
different PID strategies

Figure 7: Invariant mass of pairs of oppositely charged tracks assumed to be
$\pi^{+}\pi^{-}$ (left) and corresponding signal-to-background ratios (right)
with different PID strategies

Figure 8: Invariant mass of pairs of oppositely charged tracks assumed to be
p$\pi^{-}$ (left) and corresponding signal-to-background ratios (right) with
different PID strategies

The weighted Bayesian approach, which exploits both PID-by-TOF capabilities
and abundance of particle species, provides the best suppression of background
while preserving particles of interest. Kaons are less abundant then pions and
protons, therefore the advantage of the Bayesian approach is more pronounced
if kaons are the decay products (Fig. 6). By applying the $n$-sigma approach
one can preserve more signal events, but at the same time the suppression of
the combinatorial background is reduced. Benchmarks analyses, shown in Figs.
6–8, demonstrate the power of the identification of particles by the time-of-
flight method in the SPD experimental conditions.

Study of open-charm $D$-mesons production is one of the main goals of the SPD
experiment. $D$-mesons preferentially decay by weak interaction into kaons and
pions. In the case of $D^{0}\rightarrow\pi^{+}K^{-}$ (and charge conjugate)
decays, the Bayesian PID by time-of-flight allows to suppress combinatorial
two-prong background by a factor 15, thus enhancing the analysis of open-charm
decays.

##  5 Conclusions

In the SPD experiment the accurate determination of the event collision time
is required to perform $\pi/K/$p identification in low momenta range
($0.5{-}3~{}\text{GeV}/c$) by the time-of-flight method. The collision time
can be reconstructed on an event-by-event basis by a minimization procedure,
which uses measurements of the TOF detector combined with track
reconstruction, but the corresponding solution is computationally expensive.
In this work we present a dedicated Asynchronous Differential Evolution-
inspired Genetic Algorithm which solves the optimization problem in the direct
way without any simplifications, thus providing the fast and reliable
measurement of the event collision time throughout the range of track
multiplicities. Finally, different strategies of particle identification by
time-of-flight are tested to prove the power of the PID-by-TOF method in the
SPD experimental conditions.

## Author contributions

Conceptualization, supervision: Mikhail Zhabitsky; formal analysis,
methodology, software, investigations: Semyon Yurchenko. Both authors have
read and agreed to the published version of the manuscript.

## Funding

This work was in part supported by the JINR START 2022 program.

## Acknowledgments

The authors would like to express their deepest appreciation to members of the
SPD Collaboration for their valuable feedback.

## Conflict of interest

The authors declare that the research was conducted in the absence of any
commercial or financial relationships that could be construed as a potential
conflict of interest.

## Abbreviations

The following abbreviations are used in this manuscript:
ADE-GA Asynchronous Differential Evolution-inspired Genetic Algorithm BFA
Brute Force Algorithm DE Differential Evolution GA Genetic Algorithm MRPC
Multigap Resistive Plate Chamber NICA Nuclotron-based Ion Collider fAcility
PID Particle IDentification SPD Spin Physics Detector TOF Time-Of-Flight

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