# Charged particle discrimination via time-of-flight capabilities (ToF) of plastic scintillators: A paragon over $\mu^+ + \mu^-$ separation through GEANT4 simulations

Ahmet Ilker Topuz

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Ahmet Ilker Topuz. Charged particle discrimination via time-of-flight capabilities (ToF) of plastic scintillators: A paragon over $\mu^+ + \mu^-$ separation through GEANT4 simulations. 2025. hal-05211863

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# Charged particle discrimination via time-of-flight capabilities (ToF) of plastic scintillators: A paragon over μ⁺ - proton separation through GEANT4 simulations

Ahmet Ilker Topuzª

ªDepartment of Physics, Erciyes University, Kayseri, Türkiye, 38030 ahmetilkertopuz@erciyes.edu.tr, aitopuz@zohomail.eu

## Abstract

This study investigates the discrimination capabilities of the time-of-flight (ToF) measurements with the plastic scintillators for the charged particles by specifically focusing on the separation of the positive muons (μ⁺) from the protons. The Monte Carlo simulations are conducted by using the GEANT4 framework in order to evaluate the temporal difference across the different momenta lying on the interval between 1.7 and 6.87 GeV/c. The simulation setup comprised a planar source geometry with a vertically directed beam and the plastic scintillating hodoscopes that are positioned to measure the ToF over a fixed distance of 1 m. The simulation outcomes are contrasted with those of Gruppen and Shwartz, 2008 where an analytical formulation to determine the ToF of the relativistic charged particles exists. The results in the present study in accordance with Gruppen and Shwartz, 2008 demonstrate that the ToF discrimination becomes increasingly challenging at higher momenta, i.e after 4.85 GeV/c, due to the converging relativistic velocities with the time differences decreasing from 0.478 ns at 1.7 GeV/c to 0.042 ns at 6.87 GeV/c.

**Keywords:** Relativistic charged particles; Time-of-flight; Particle discrimination; Plastic scintillators; Monte Carlo simulations; GEANT4

## 1 Introduction

The particle identification represents a fundamental challenge in the modern high-energy physics experiments where the accurate discrimination between the different charged particle species is crucial for understanding the concepts such as collision dynamics, decay processes, and fundamental interactions [1]. The time-of-flight (ToF) measurements have emerged as a powerful technique for the particle identification by exploiting the mass-dependent velocity differences of particles traveling at the relativistic speeds over the known distances [2, 3].

The discrimination of the positive muons (μ⁺) from the protons presents a particular significance in various experimental contexts, including cosmic ray studies, neutrino oscillation experiments, and hadron collider physics [4]. The positive muons (μ⁺) as well as the protons, despite having substantially different masses (105.66 MeV/c² for muons versus 938.27 MeV/c² for protons), exhibit converging velocities at high momenta due to the relativistic effects, making their separation increasingly challenging as energy increases [4].

The plastic scintillators have become the detector technology of choice for the ToF measurements due to their fast response times (typically sub-nanosecond), high light output, and cost-effectiveness for large-area coverage. These organic scintillators provide excellent timing resolution while maintaining reasonable spatial resolution, which makes them ideal for the hodoscope configurations in the particle physics detectors [5, 6].
A robust framework for evaluating detector performance and optimizing experimental designs is provided by the GEANT4 code. The comprehensive physics models of GEANT4 account for the electromagnetic interactions, the hadronic processes, and the detector response effects that the analytical calculations cannot fully capture, particularly in the complex geometries and the material configurations [7, 8].

The previous studies have established the theoretical foundations for the ToF-based particle discrimination, but detailed simulations comparing experimental reality with the analytical predictions remain limited, especially for the specific particle pairs across the extended momentum ranges [2, 3]. This work addresses this gap by providing the comprehensive GEANT4 simulations of µ⁺-proton discrimination via using the plastic scintillator plates.

This study is organized as follows. Section 2 describes the methodology that is conducted in this study. Section 3 states the simulation properties in this study, while section 4 exhibits the simulation outcomes by comparing with those of Gruppen and Shwartz, 2008 [11]. Finally, the conclusions are drawn in section 5.

## 2 Methodology

A direct way to compute the particle velocity is to measure its ToF between two points separated by a distance $L$, which is 1 m in the current study. These two points on the plastic scintillators can be defined by two counters that provide "star" and "stop" signals or by the moment of particle production and a stop counter. In the latter case, the "start" signal that is synchronized with the beam-beam or beam-target collision can be produced by the accelerator system. A more detailed review of ToF detectors in high-energy particle experiments can be referred to [9, 10]. Two particles of mass $m_1$ and $m_2$, which are respectively 938.27 MeV/c² for the protons and 105.66 MeV/c² for the µ⁺ in the present study, are considered to approximately have the same momentum and a flight distance of $L$. Then, the ToF difference denoted by $\Delta t_{\text{Reference}}$ is expressed as

$$ \Delta t_{\text{Reference}} = L \left( \frac{1}{v_1} - \frac{1}{v_2} \right) = \frac{L}{c} \left( \frac{1}{\beta_1} - \frac{1}{\beta_2} \right) \quad (1) $$

Since this study is devoted to the µ⁺ - proton separation, then Eq. (1) becomes

$$ \Delta t_{\text{Reference}} = L \left( \frac{1}{v_{\text{Proton}}} - \frac{1}{v_{\text{µ}^+}} \right) = \frac{L}{c} \left( \frac{1}{\beta_{\text{Proton}}} - \frac{1}{\beta_{\text{µ}^+}} \right) \quad (2) $$

By using $pc = \beta E$ where the velocity $\beta$ is $v/c$ with $c = 3 \times 10^8$ m/s, one obtains

$$ \Delta t_{\text{Reference}} = \frac{L}{pc^2} (E_{\text{Proton}} - E_{\text{µ}^+}) = \frac{L}{pc^2} \left( \sqrt{p^2 c^2 + m_{\text{Proton}}^2 c^4} - \sqrt{p^2 c^2 + m_{\text{µ}^+}^2 c^4} \right) \quad (3) $$

where $p$ is the momentum of the corresponding particle. Since $pc^2 \gg m_{\text{Proton}}^2, \mu^+ c^4$, the expansion of the square roots in Eq. (3) finally leads to

$$ \Delta t_{\text{Reference}} = \frac{L}{2pc^2} \left( m_{\text{Proton}}^2 - m_{\text{µ}^+}^2 \right) \quad (4) $$
<figure><img src="image_2.png" /><figcaption>Figure 1: Simulation setup for the computation of the time-of-flight (ToF) by using the GEANT4 simulations.</figcaption></figure>

In the GEANT4 simulations, the time difference between separate hodoscope sections depicted in Fig. 1 is determined by using the following expression:

$$ \Delta T = T_{\text{Stop}} - T_{\text{Start}} \qquad (5) $$

where $T_{\text{Start}}$ is the start point indicated by the yellow disc, while $T_{\text{Stop}}$ is the stop point symbolized by the red disc. Since a substantial number of particles reach the detector system, the average time difference at a certain energy is determined by averaging the previously determined time differences over N number of the muons and protons as written in

$$ \overline{\Delta T} = \frac{1}{N} \sum_{i=1}^{N} \Delta T_{i} \qquad (6) $$

Please note that the standard deviations in these calculations are tolerable under the condition of < 5%.

### 3 Simulation scheme

To perform the computation of the ToF, the geometrical scheme is illustrated in Fig. 1, and it is shown that the plastic scintillators are separated by a distance of 1 m. Furthermore, the dimensions of the detector layers are 100×0.4×100 cm³. By satisfying the geometrical properties of the detection system, the Monte Carlo simulations are conducted via the GEANT4 code in order to record the temporal values at the plastic scintillators. The simulation parameters are listed in Table 1, and the dimension of the simulation box is 100 × 170 × 100 cm³ where the Cartesian components are situated symmetrically in the interval of (-50 cm, 50 cm), (-85 cm, 85 cm), and (-50 cm, 50 cm), respectively as indicated in Fig. 1. A narrow planar multi-energetic mono-directional beam that is generated at ([-0.5, 0.5] cm, 85 cm, [-0.5, 0.5] cm) via G4ParticleGun is used, and the generated positive muons as well the protons are propagating
in the vertically downward direction as shown by the black arrow in Fig. 1, i.e. from the top edge of the simulation box through the bottom edge.

<table><caption>Table 1: Simulation properties.</caption><tr><td>Particle</td><td>μ<sup>+</sup>, proton</td></tr><tr><td>Beam direction</td><td>Vertical</td></tr><tr><td>Momentum direction</td><td>(0, -1, 0)</td></tr><tr><td>Source geometry</td><td>Planar</td></tr><tr><td>Initial position (cm)</td><td>([-0.5, 0.5], 85, [-0.5, 0.5])</td></tr><tr><td>Number of particles</td><td>10<sup>5</sup></td></tr><tr><td>Muon energy (GeV)</td><td>1.6, 2.7, 3.7, 4.75, 5.75, and 6.75</td></tr><tr><td>Proton energy (GeV)</td><td>1, 2, 3, 4, 5, and 6</td></tr><tr><td>Energy distribution</td><td>Constant</td></tr><tr><td>Material database</td><td>G4/NIST</td></tr><tr><td>Reference physics list</td><td>FTFP_BERTHP</td></tr></table>

$10^5$ is the total number of the generated $\mu^+$ and protons in each simulation. As stated in section 2, in order to compare the simulation results with Eq. (4), the initial momenta of the associated particles are supposed to be fixed. By recalling this fact, the initial kinetic energies of the $\mu^+$ are different from those of the protons in the present study. While the initial kinetic energies of $\mu^+$ are 1.6, 2.7, 3.7, 4.75, 5.75, and 6.75 GeV, these values become 1, 2, 3, 4, 5, and 6 GeV. The reference physics list utilized in this work is FTFP_BERTHP, and all of the materials in the simulation geometry are defined in accordance with the GEANT4/NIST material database. G4Step maintains the particle tracking, and a Python script is used to post-process the registered times. This initially involves the determination of the temporal difference for each corresponding particle, and then, the obtained time difference is averaged over the number of particles.

## 4 Simulation outcomes

The GEANT4 simulations are conducted to evaluate the ToF discrimination capabilities between the $\mu^+$ and protons across the momentum ranges from 1.7 to 6.87 GeV/c. The simulation results are presented in Tables 2 and 3, which show the average ToF measurements for the $\mu^+$ and protons respectively, over the fixed distance of 1 m.

<table><caption>Table 2: ToF of $\mu^+$ according to given kinetic energies as well as momenta.</caption><thead><tr><th></th><th>μ<sup>+</sup></th><th></th></tr><tr><th>E<sub>Kinetic</sub> [GeV]</th><th>p [GeV/c]</th><th>ΔT<sub>μ+</sub> [ns]</th></tr></thead><tbody><tr><td>1.60</td><td>1.70</td><td>3.355</td></tr><tr><td>2.70</td><td>2.80</td><td>3.351</td></tr><tr><td>3.70</td><td>3.80</td><td>3.350</td></tr><tr><td>4.75</td><td>4.85</td><td>3.349</td></tr><tr><td>5.75</td><td>5.85</td><td>3.349</td></tr><tr><td>6.75</td><td>6.87</td><td>3.349</td></tr></tbody></table>
The simulation results reveal distinct temporal characteristics for both particle species across the investigated momentum range. For the $\mu^+$ particles as listed in Table 2, the ToF values demonstrate the minimal variation with the momentum, ranging from 3.355 ns at 1.7 GeV/c to 3.349 ns at 6.87 GeV/c. In contrast, the proton ToF measurements as tabulated in Table 3 exhibit a more pronounced momentum dependence, decreasing from 3.834 ns at 1.7 GeV/c to 3.391 ns at 6.87 GeV/c.

<table><caption>Table 3: ToF of proton according to given kinetic energies as well as momenta.</caption><thead><tr><th colspan="3">Proton</th></tr><tr><th>E<sub>Kinetic</sub> [GeV]</th><th>p [GeV/c]</th><th>ΔT<sub>Proton</sub> [ns]</th></tr></thead><tbody><tr><td>1.0</td><td>1.70</td><td>3.834</td></tr><tr><td>2.0</td><td>2.80</td><td>3.539</td></tr><tr><td>3.0</td><td>3.80</td><td>3.453</td></tr><tr><td>4.0</td><td>4.85</td><td>3.418</td></tr><tr><td>5.0</td><td>5.85</td><td>3.402</td></tr><tr><td>6.0</td><td>6.87</td><td>3.391</td></tr></tbody></table>

Fig. 2 illustrates the momentum-dependent behavior of both particle species, showing the convergence of their ToF values at higher momenta.

<figure><img src="image_3.png" /><figcaption>Figure 2: Average ToF of $\mu^+$ and protons with respect to the initial momentum in the GEANT4 simulations.</figcaption></figure>

The discrimination capability, quantified by the time difference $\Delta t = \overline{\Delta T}_{Proton} - \overline{\Delta T}_{\mu^+}$, demonstrates a systematic decrease with increasing momentum as detailed in Table 4.
<table><caption>Table 4: Difference between the average ToFs of protons and μ⁺ with respect to the initial momenta including the GEANT4 simulations as well as the reference from Gruppen and Shwartz, 2008 [11].</caption><thead><tr><th>p [GeV/c]</th><th>Δt<sub>GEANT4</sub> = ΔT<sub>Proton</sub> - ΔT<sub>μ+</sub> [ns]</th><th>Δt<sub>Reference</sub> = L/2pc<sup>2</sup> (m<sup>2</sup><sub>Proton</sub> - m<sup>2</sup><sub>μ+</sub>) [ns]</th><th>δΔt [%]</th></tr></thead><tbody><tr><td>1.70</td><td>0.478</td><td>0.500</td><td>4.400</td></tr><tr><td>2.80</td><td>0.187</td><td>0.185</td><td>1.081</td></tr><tr><td>3.80</td><td>0.103</td><td>0.100</td><td>3.000</td></tr><tr><td>4.85</td><td>0.069</td><td>0.062</td><td>11.290</td></tr><tr><td>5.85</td><td>0.053</td><td>0.042</td><td>26.190</td></tr><tr><td>6.87</td><td>0.042</td><td>0.030</td><td>40.000</td></tr></tbody></table>

At the lowest momentum (i.e. 1.7 GeV/c), the GEANT4 simulation yields a time difference of 0.478 ns, providing substantial discrimination power. This value represents approximately 14% relative difference between the two species, well above typical detector resolution limits. The discrimination remains robust through intermediate momenta, with values of 0.187 ns at 2.8 GeV/c and 0.103 ns at 3.8 GeV/c. The critical transition occurs beyond 4.85 GeV/c, where the time difference drops to 0.069 ns. At this momentum, the discrimination begins to approach the practical limits of the conventional plastic scintillator timing systems, which typically achieve resolutions of 50-100 ps for optimal configurations.

<figure><img src="image_4.png" /><figcaption>Figure 3: Difference between the average ToFs of protons and μ⁺ with respect to the initial momenta.</figcaption></figure>

The comparison between the GEANT4 simulation results and the analytical reference values from Gruppen and Shwartz, 2008 provides a crucial validation of both the simulation methodology and the theoretical framework. A good match is observed at the intermediate momenta with the deviations of only 1.081% at 2.8 GeV/c and 3.000% at 3.8 GeV/c. These minimal discrepancies validate the accuracy of both the GEANT4 physics models and the analytical
approximations within this momentum regime. However, systematic deviations emerge at the
momentum extremes with the substantial deviations occurring at the high momenta, reaching
11.290% at 4.85 GeV/c, 26.190% at 5.85 GeV/c, and 40.000% at 6.87 GeV/c.

The increasing discrepancies at the high momenta as illustrated in Fig. 3 warrant detailed
examination as they reveal the limitations of simplified analytical treatments. Several factors
contribute to these deviations. First, the analytical formula assumes $pc^2 \gg m^2c^4$ for the
square root expansion in Eq. (3). While this approximation holds reasonably well for muons
across all studied momenta, it becomes less accurate for protons at lower momenta, where the
mass term contributes more significantly to the total energy. Second, GEANT4 simulations
incorporate realistic detector response phenomena that analytical calculations cannot capture,
including multiple scattering effects as particles traverse the plastic scintillator material, energy
loss fluctuations described by Landau-Vavilov distributions, finite detector thickness effects on
timing measurements, and secondary particle production.

The simulation results provide crucial insights for experimental design considerations. Based
on the results, the plastic scintillator ToF systems provide a satisfactory $\mu^+$-proton discrimina-
tion for the momenta up to approximately 3.8 GeV/c, where the time differences exceed 0.1 ns.
This threshold provides comfortable margins above the typical detector resolution limits. The
momentum range between 3.8 and 4.85 GeV/c represents a transitional region where discrim-
ination becomes increasingly difficult but may still be feasible with high-performance timing
systems achieving sub-50 ps resolution. Beyond 4.85 GeV/c, the diminishing time differences
fall below practical detection limits for conventional plastic scintillator systems, necessitating
alternative identification strategies.

Table 4 presents the critical comparison between GEANT4 simulation results and the ana-
lytical reference values from Gruppen and Shwartz, 2008. The difference between the proton and
muon ToFs decreases dramatically with the increasing momentum, from 0.478 ns at 1.7 GeV/c
to 0.042 ns at 6.87 GeV/c according to the GEANT4 simulations, and this trend closely follows
the analytical predictions although notable deviations emerge at higher momenta. The simu-
lation results confirm that time-of-flight discrimination by using plastic scintillators provides
the robust particle identification capabilities at lower momenta but faces significant challenges
in the high-momentum regime. The converging relativistic velocities fundamentally limit the
discrimination power, requiring either longer flight paths or improved timing resolution for
effective separation at momenta exceeding 3.8 GeV/c.

# 5 Conclusion

This study has comprehensively evaluated the ToF discrimination capabilities of plastic scintil-
lators for separating the positive muons from the protons by using the GEANT4 simulations.
The investigation covers the momentum ranges from 1.7 to 6.87 GeV/c by providing the de-
tailed comparisons with the analytical predictions from the established relativistic mechanics
formulations. The simulation results demonstrate that the ToF-based particle discrimination
exhibits a strong momentum dependence with discrimination power decreasing exponentially
as the particle momenta increase. At the low momenta (e.g. 1.7 GeV/c), the time difference
between the protons and the muons reaches 0.478 ns by providing excellent separation capa-
bilities well within the resolution limits of modern plastic scintillator systems. However, this
discrimination advantage rapidly diminishes, falling to 0.042 ns at 6.87 GeV/c. The compar-
ison with the analytical reference calculations reveals a good agreement at the intermediate
momenta (2.8-3.8 GeV/c) with the deviations below 3%, by validating both the simulation
methodology and the theoretical framework. The increasing discrepancies observed at higher
momenta (reaching 40% at 6.87 GeV/c) highlight the importance of Monte Carlo simulations
in capturing the realistic detector response effects that the analytical calculations cannot fully account for, including the multiple scattering, the energy loss fluctuations, and the finite detector resolution. The practical threshold for the reliable µ⁺-proton discrimination using the 1-m flight path plastic scintillator hodoscopes appears to be approximately 4.85 GeV/c beyond which alternative or complementary identification techniques become necessary. For the higher momentum applications, the potential solutions include extending flight paths, implementing higher resolution timing systems, or combining the ToF measurements with the other discrimination methods such as energy loss (dE/dx) measurements or the Cherenkov radiation detection. The findings of this study provide a valuable guidance for the experimental design in the high-energy physics applications that require the µ⁺-proton discrimination, particularly in the cosmic ray experiments, the neutrino physics investigations, and the hadron collider detector systems. The detailed simulation framework established here can be readily adapted to evaluate the other particle discrimination scenarios and optimize the detector configurations for the specific experimental requirements. A future work might focus on investigating the impact of the detector geometry optimization by exploring the advanced timing techniques as well as evaluating the performance of hybrid discrimination approaches that combine the multiple identification methods to extend the effective momentum range for the reliable particle separation.

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