Response simulation and primary vertex reconstruction in the SPD NICA straw tracker

Full Text

Introduction
At the present time, the spin structure of the nucleon is one of the challenging topics of highenergy
physics. Nowadays, a new accelerator complex, the NICA (Nuclotron based Ion Collider
fAcility) collider, is being built at the Joint Institute for Nuclear Research (Dubna, Russia). The Spin
Physics Detector (SPD) is proposed to be placed in one of the two interaction points of the NICA
to study the nucleon spin structure and other spin-related phenomena with polarized proton and
deuteron beams [1]. The SPD physics program is described in detail in [2–4] and mainly aimed to
extract information on the gluon Transverse-Momentum-Dependent Parton Distribution Functions
(TMD PDFs) in the proton and deuteron, as well as the gluon transversity distribution and tensor
PDFs in the deuteron, via the measurement of specific single- and double-spin asymmetries using
such complementary probes as charmonia, open charm, and prompt photon production processes.
These phenomena are planned to study at the center-of-mass collision energy
√
s up to 27 GeV and
a total luminosity up to 1032 cm−2 s−1. Since the
√
s exceeds the typical hard scale of the studied
processes moderately, the 4π geometry resolution of the SPD is planned, together with triggerless
data acquisition system (DAQ), purposed to minimize systematic uncertainties of the measurements.
For the effective data recognition under specified conditions, the related software should provide fast
data processing, event selection, and primary vertex reconstruction at the online data filtering stage.
Particularly, for the rational use of memory and high-performance the simplest procedure of track
fitting and further primary vertex recovery is anticipated [5–7].
1. Straw tracker simulation
The straw tracker (ST) is the inner part of the SPD detector aimed at reconstructing tracks of
primary and secondary particles with high efficiency for the precise measurements of their momenta
and to identify particles via their energy deposition. Its construction includes the barrel part and
two end-caps [1], constituted of two different kinds of the low-mass straw tubes, similar to those
used in such modern experiments as NA64 [8], COMPASS [9; 10] and others [1]. The most of the
produced particles will be registered in the barrel part consisting of the 8 azimuthal modules, each
containing the 31 double layers of straw tubes, which schematic view is presented at the Fig. 1 [1].
Morozova S.D., Shipilova A.V. Response simulation and primary vertex reconstruction...
Морозова С.Д., Шипилова А.В. Моделирование отклика и восстановление первичных вершин... 76 из 85
The geometrical ST model used in this study was developed by A. Allakhverdieva using the
GeoModel toolkit [11; 12], which is a class library for the description of detector geometry. The
designed ST geometry is stored in the local GeoModel SQLite file (.db) being the input of the Geant4
package [13–15], generating the detector model automatically. Such an approach for the detector
simulation provides the flexibility of the model, simplifying the purely construction changes. To
describe the ST geometry, we use a global coordinate system, where the z-axis is oriented along
the nominal beam direction, the y-axis is vertical, and the x-axis is perpendicular to them and is
directed toward the center of the collider ring. The origin of the coordinate system is the nominal
center of the setup being the ideal interaction point. The ST model of the present study is specified
by the setup with inner and outer radii of 270 and 867 mm, respectively, consisting of 8 octants
filled with layers of cylindrical tubes, to be shown in the XOY plane in the Fig. 2. The base of the
detector, the ST tubes, are constructed from a thick polyethylene shell and an inner cylindrical
volume of radius R = 4.78 mm filled by the gas mixture of Ar(70%) + CO2(30%). The internal
volume of each tube is represented by the GEANT4 sensitive detector object ConstructSDandField
and the HitsCollection array, which elements (Hits) are recorded every time a particle looses energy
in the sensitive volume, and contain information about the particle type, energy loss, coordinates,
track data, and the unique number of the tube where the hit is arised.
Fig. 1. Schematic representation of the Straw Tracker (ST) [1]
Рис. 1. Схематичное представление строу-трекера (SТ) [1]
2. Experiment simulation
We consider the p-p mode of the experiment where the proton beams collide at the energy
of
√
s = 27 GeV in the center-of-mass system. We suppose the proton bunch crossing every 76 ns,
while the temporal interval of the data storing (timeslice), defined by the DAQ system, to be
10 μs. The heavy charged primary particles are simulated by positive muons with a given energy
of E = 1 GeV and momentum ⃗p, which direction has a uniform spatial distribution, using the
G4PrimaryVertex object of the Geant4. The such choice of primary particles has been made due to
their small cross section of interaction with ST material resulting the small number of secondary
tracks while the major energy loss of a muon is due to a gas ionization, which is favourable to initiate
the electron avalanche into a straw drift chamber.
The probability of a hard interaction between two protons in a given beam crossing is simulated
by the Poisson distribution f (k) =
λk
k!
e−λ with the expected value λ = 0.3. We impose the
coordinates of the interaction point and the corresponding primary vertex to be the same and equal
to (0, 0, z), where z is normally distributed around the central value z0 = 0 by a Gaussian distribution
f (z) =
1
σ
√
2π
e−1
2 ( z−z0
σ )
2
with σ = 30 cm. The number of muons produced in the one pp hard
interaction is determined by a Poisson distribution with the expected value λ = 7. The propagation
of a charged particle through a gas leads to energy loss due to ionization. The snapshot of the
Vestnik of Samara University. Natural Science Series 2025, vol. 31, no. 1, pp. 75–85
Вестник Самарского университета. Естественнонаучная серия 2025. Том 31, № 1. С. 75–85 77 из 85
Fig. 2. The ST model used in this work in XOY plane
Рис. 2. Модель строу-трекера, используемая в представленной работе, в плоскости XOY
particle state at the moment of energy loss within the sensitive volume was recorded as a hit. The
visualization of launch of one event into the ST is shown in Fig. 3, where the hits and tracks are
illustrated by green dots and blue lines, respectively.
Fig. 3. Visualization of launch of one event into the ST: XOY plane (left), ZOY plane (right). Hits and
tracks are illustrated by green dots and blue lines, respectively (see the color online on the wersite)
Рис. 3. Визуализация запуска одного события в строу-трекере: в плоскости XOY (слева), в плоскости
ZOY (справа). Зеленые точки — точки потери энергии в чувствительном объеме, голубые линии —
треки (цвет см. в онлайн-варианте на сайте)
We collected the hits into hit arrays applying the following selection criteria: only the energy loss
of primary particles were considered with the threshold for the total energy loss of Etot > 100 MeV,
and all the secondary particles are neglected. The set of coordinates of each muon energy loss points
were used to determine the shortest distance to the tube axis, anode wire, to calculate the response
time of the straw tube. This time depends on the electron avalanche drift time from the ionization
point to the anode wire t(r), where r is the shortest distance from the track to the anode. Using
Morozova S.D., Shipilova A.V. Response simulation and primary vertex reconstruction...
Морозова С.Д., Шипилова А.В. Моделирование отклика и восстановление первичных вершин... 78 из 85
the complete hit collection array for a given muon we can reconstruct its track and apply this
procedure for all the muons in the simulation. If there are several points of muon energy loss in one
logical volume, we approximate the muon track by a straight line connecting the first and the last hits,
then r is calculated as the shortest distance between this line and tube axis. The hit reconstruction
step was omitted for the transparency of the simulation results, that means the 3D coordinates of
particle hits are assumed to be known. The distance from the hit to the anode wire is calculated using
the local coordinates, relatively to the axis of a particular tube, and the tracks are approximated
in global coordinates, relative to the entire detector. Since each particle is tracked individually in
GEANT4, it is not suitable to simulate interactions for a large number of particles, as the electron
avalanche is. Thus, we used the t(r) dependence [1] simulated using the GARFIELD software [16; 17]),
for which we found the analytical approximation: t(r) = 2.7101+1.2156r +6.8287r2. We present the
resulting temporal distribution averaged by 100 time slices in the Fig. 4, where grey area corresponds
to the times of the sensitive volume intersections by the sample particle, and colored area is straw
tubes response time distribution. Here and after all the histograms were generated using the CERN
ROOT [18] tools. We obtain the significant overlap between the straw tubes response times from the
different bunch crossings appearing in the same time slice. This fact indicates the problem of signal
decoding for track and primary vertex reconstruction during data collection in a real experiment.
Fig. 4. Temporal distribution averaged by 100 time slices, grey area – times of the sensitive volume
intersections by the sample particle, colored area – straw tubes response time distribution
Рис. 4. Временное распределение, усредненное по 100 временным срезам. Серая область – время
пересечения чувствительного объема частицей из выборки, цветная область – распределение времен
отклика строу-трубок
3. Primary vertex reconstruction
To reconstruct primary vertices assigned with bunch crossings points, we develop an algorithm
based on the simplest approximation of particle tracks, making it suitable in the online evaluations
for the triggerless regime of ST. For the pure examination of the algorithm, we omit the step of
hit reconstruction, and reconstruct the tracks using the direct coordinates of the hits collected in
Sec. 1. In our simulation we suppose a magnetic field oriented along z-axis to be equal to B = 1 T
and uniform everywhere inside the ST, providing a spiral trajectory of a charged particle. We
approximate the muon motion in the XOY plane by the parabola and a straight line in the ZOX or
YOZ planes, see Fig. 3. The set of hits {xi, yi}N in the XOY plane for the each track numbered N
is approximated by the parabolic function ˜ y = a2,N ˜ x2 + a1,N ˜ x + a0,N = f ( ˜ x) where the coefficients
ai,N are determined using the least-squares method. Since the system has a rotational symmetry in
the XOY plane, but the coordinate system is fixed, we should choose the desired approximation
Vestnik of Samara University. Natural Science Series 2025, vol. 31, no. 1, pp. 75–85
Вестник Самарского университета. Естественнонаучная серия 2025. Том 31, № 1. С. 75–85 79 из 85
function to be Y(X) or X(Y) for the each track. It is done by the following procedure: if any of the
sets {xi}N, {yi}N is not ordered by ascending or descending, its elements correspond to function
values ˜ y( ˜ x) while the elements of another to argument values ˜ x. In the case whether both sets are
ordered, the adjusted R square values are calculated using the following formulas:
R2
Adj = 1 −
(1 − R2)(k − 1)
k − n − 1
R2 = 1 −
kΣ
i=1
(yi − f (xi))2
kΣ
i=1
(yi − ¯ y)2
. (1)
Then we compare values of R2
Adj for {xi, yi}N corresponding to choices ( ˜ x, ˜ y) and ( ˜ y, ˜ x) to adopt
one providing the higher R2
Adj.
The coefficients ai for each primary particle determine the arc length of the parabolic segment
LN which is calculated using a simple formula
LN( ˜ xi,N) =
Z ˜ xi,N
0
q
1 + ( f ′( ˜ x))2d ˜ x = (2)
=
1
4a2,N

ln
    
q
( f ′( ˜ xi,N))2 + 1 + f ′( ˜ xi,N)
    
+ f ′( ˜ xi,N)
q
( f ′( ˜ xi,N))2 + 1

.
Then we can use it to obtain the dependence z(L)N = b1,NLN + b0,N, where z(L)N can be
approximated by a linear function, which must be extrapolated to intersect with the Z-axis in order
to determine the position of the primary vertex. The simulated tracks and hits in the ZOY plane
from the primary particles in one time slice are shown in Fig. 3.
The coordinates of the hit are approximated assuming that the particle is produced at the point
{0, 0, Z0}. The approximated tracks are passed then through the two selection criteria. The first
one is based on the polar angle θ of the track, which is determined as the angle between the line
connecting the start point of the track with the first hit and the Z axis.
Analysing the distribution of the vertex reconstruction error by polar angle, we can conclude
that particle tracks close to Z-axis do not intersect a sufficient number of detector layers, resulting a
decrease in the reconstruction accuracy, since the extrapolation error multiplies the approximation
error. Therefore, we decided to exclude all tracks with polar angle θ < 0.5 and θ > π −0.5, and this
reduces the number of tracks to 80 % from the initial one. In the Fig. 5 we present the distance
distribution between the reconstructed vertex and the true vertex depending on the polar angle.
The second track selection criterion is derived by the RSS (regression residuals squares) parameter
value stated by the formula:
RSSz =
nΣ i=1
(zi,N − (b0,N + b1,NLN( ˜ xi,N)))2. (3)
Tracks with an RSS below the value of 1.0 are discarded. This critical value was chosen as
corresponding to the optimal ratio of discarded to actually poorly reconstructed tracks, leading to
the 56% of the remaining from the initial number of tracks.
The obtained distortion of the distribution relative to θ = π, see Fig. 5, is due to the asymmetry
of the trajectories of positive muons propagating in and against the direction of the magnetic
induction lines. Assuming the Z0 − Znach data are normally distributed within a certain range of θ
values, we defined the parameters of Gaussian distribution corresponding to the each θ interval:
f (Z0, θ) =
1
σ(θ)
√
2π
e−12

Z0−μ(θ)
σ(θ)
2
. (4)
The obtained parameters μ(θ) and σ(θ) will be used for further track processing. The obtained
dependence of the standard deviation and the mean value on the polar angle is shown in Fig. 6.
Using the mean value, the reconstructed Z0s were corrected. The σ(θ) is the corresponding
reconstruction error of a particular vertex, which was used for vertex clustering and for
Morozova S.D., Shipilova A.V. Response simulation and primary vertex reconstruction...
Морозова С.Д., Шипилова А.В. Моделирование отклика и восстановление первичных вершин... 80 из 85
Fig. 5. Distance distribution between the reconstructed vertex and the true vertex versus the polar angle θ
Рис. 5. Распределение расстояния между реконструированной вершиной и истинной вершиной
в зависимости от полярного угла θ
Fig. 6. Dependence of the standard deviation (σ) and the mean value (μ) on the polar angle
Рис. 6. Зависимость стандартного отклонения (σ) и среднего значения (μ) от полярного угла (θ)
correcting the reconstructed vertices. The adjusted distance distribution between the reconstructed
vertex and the true vertex are shown in Fig. 7.
The next step is to group the tracks into clusters with common vertices expected as primary
vertices. The set of initial coordinate values Z0 was sorted in ascending order. The reconstructed Z0
values were taken into account with a corresponding error interval {Z0,k ± σk}. In case error intervals
overlap, all Z0s in a given region are considered as one cluster treated as reconstructed primary vertex
with the coordinate Z0,C =
1
K ΣK
k=1 Z0,k. After all the clusters has been found by this procedure we
compare their coordinates with true primary vertices. In case of two or more vertices are present
within a single cluster they are combined into one primary vertex only if the distance between them
does not exceed 3σ0, where σ0 = 1 mm. If several vertices within a cluster are at a greater distance,
they are all considered to be reconstructed incorrectly. A correctly reconstructed vertex is one that
can be distinguished in the cluster, and all assigned tracks actually belong to it. To determine
this, we used the ID of the track (TrackID), which is matched between the reconstructed and true
vertex in this algorithm. Among all the incorrectly reconstructed vertices Nwr the fraction related
to indistinguishability
Nwr.vert
Nwr
= 96 %. The vertex reconstruction efficiency was calculated as the
Vestnik of Samara University. Natural Science Series 2025, vol. 31, no. 1, pp. 75–85
Вестник Самарского университета. Естественнонаучная серия 2025. Том 31, № 1. С. 75–85 81 из 85
Fig. 7. Adjusted distribution of the distance between the reconstructed vertex and the true vertex versus
polar angle θ
Рис. 7. Скорректированное распределение расстояния между реконструированной вершиной
и истинной вершиной в зависимости от полярного угла θ
ratio of the number of correctly reconstructed true vertices to the total number of true vertices in
the dataset. The efficiency obtained by this method is
Nreco
Nall
= 97.1 %.
As a result, if the initial vertices are far enough apart each other, they are easily distinguishable
and can be correctly reconstructed. However, if the primary vertices are closely located, the error
regions in the reconstructed Z0 can overlap, reducing the efficiency of the reconstruction process.
In the Fig. 8, left, we present the dependence of primary vertex reconstruction efficiency which is
defined as the ratio of all the reconstructed vertices to the all initial true vertices in the simulation,
against the critical value of the RSSz parameter. While on the right panel we demonstrate the
percentage of the vertices lost by use the RSSz-related criterion. One can find the maximal efficiency
is achieved at the RSSz = 1 being equal to 77.1 %.
Fig. 8. Efficiency dependence (left) and vertices information loss (right) on RSSz of z(l) approximation
Рис. 8. Зависимость эффективности реконструкции вершин (слева) и потери информации о вершинах
(справа) от порогового значения RSSz аппроксимации z(l)
Morozova S.D., Shipilova A.V. Response simulation and primary vertex reconstruction...
Морозова С.Д., Шипилова А.В. Моделирование отклика и восстановление первичных вершин... 82 из 85
Conclusions
We present the algorithm for the primary vertex reconstruction in the SPD NICA straw tracker
which gives an opportunity to resolve a complicated temporal structure of the ST response within
the timeslice. Due to the simplicity of the algorithm it is fast and can be used for the online data
processing in the triggerless regime of the detector operation. We found the optimal parameters
providing the maximal efficiency with minimal information loss, and a good quality of the vertex
reconstruction on the conditionally-reduced sample of the true vertices.
The further study implies the verification of the algorithm performance using the hit data with
less purity and using more complicated functions for track approximations.

About the authors

S. D. Morozova

Samara National Research University

Author for correspondence.
Email: svtmorozova09@gmail.com
ORCID iD: 0009-0009-1168-8529

Master’s degree student of the Department of General and Theoretical Physics

Russian Federation, 34, Moskovskoye shosse, Samara, 443086, Russian Federation

A. V. Shipilova

Samara National Research University; Joint
Institute for Nuclear Research

Email: shipilova.av@ssau.ru
ORCID iD: 0000-0003-3965-3757

senior research fellow

Russian Federation, 34, Moskovskoye shosse, Samara, 443086, Russian Federation; 6, Joliot-Curie Street, Dubna, 141980, Russian Federation

References

Supplementary files