# Forecasting performance in particle identification with potential TOF detectors at HL-LHC

Antonin Maire

► To cite this version:

Antonin Maire. Forecasting performance in particle identification with potential TOF detectors at HL-LHC. 2022. [arXiv:2206.03619]

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# Forecasting performance in particle identification with potential TOF detectors at HL-LHC

Antonin Maire *

1. Université de Strasbourg, IPHC, 23 rue du Loess 67037 Strasbourg, France.

2. CNRS/In2p3, UMR7178, 67037 Strasbourg, France.

Analysis note

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**Abstract**

[ANA-1073 - Version 2.0 - (git rev. dummy) - Tuesday 9th March, 2021, 04:57]

The document addresses comparative performances expected for various configurations of Time-Of-Flight detectors, for several CERN experiments (ALICE-1, ALICE-3, ATLAS, CMS). The figures are based on analytical formulae derived for charged particles traveling in a solenoidal magnetic field. While these performances correspond to ideal limits (with respect to real data or full simulations including particle transport), the main point is to assess the theoretical separation among various identified species (e$^{±}$, μ$^{±}$, π$^{±}$, K$^{±}$, p$^{±}$, d$^{±}$, t$^{±}$, $^{3}$He2$^{±}$, $^{4}$He2$^{±}$) at the horizon of runs IV (≈ 2027-2029) or V (> 2030) of the High-Luminosity LHC, HL-LHC. The intent is thus to appreciate the experimental realm of possible physics cases.

**Keywords:**

- TOF separation, PID, central barrel, forward endcap / ALICE-1, ALICE-3, ATLAS, CMS / HL-LHC.

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<footnote>* contact mail : mail at CERN</footnote>
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<table><tbody><tr><td>I</td><td>Introduction : context of the HL-LHC run V</td><td>4</td></tr><tr><td>I-A</td><td>Experimental landscape at HL-LHC</td><td>4</td></tr><tr><td>I-B</td><td>ALICE-3 proposal</td><td>4</td></tr><tr><td>II</td><td>Math of time of flight</td><td>5</td></tr><tr><td>II-A</td><td>Prime equations related to TOF quantities</td><td>5</td></tr><tr><td>II-B</td><td>Core working hypotheses : purely helical trajectory</td><td>7</td></tr><tr><td>II-C</td><td>Parametric equations of the relativistic helical motion</td><td>8</td></tr><tr><td>II-C.i</td><td>Differential equation of relativistic motion</td><td>8</td></tr><tr><td>II-C.ii</td><td>Sketch of the transverse projection (frames and notations)</td><td>9</td></tr><tr><td>II-C.iii</td><td>Acceleration vector as a function of time</td><td>10</td></tr><tr><td>II-C.iv</td><td>Velocity vector as a function of time</td><td>12</td></tr><tr><td>II-C.v</td><td>Spatial coordinates as a function of time</td><td>13</td></tr><tr><td>II-D</td><td>Deriving path lengths and times</td><td>14</td></tr><tr><td>II-D.i</td><td>Path length formula for a regular helix</td><td>14</td></tr><tr><td>II-D.ii</td><td>Date to reach the TOF barrel layer</td><td>15</td></tr><tr><td>II-D.iii</td><td>Date to reach the TOF endcap plane</td><td>16</td></tr><tr><td>II-E</td><td>Visualising concrete helical trajectories calls for comments</td><td>16</td></tr><tr><td>II-E.i</td><td>Introduction of the trajectory illustrations</td><td>16</td></tr><tr><td>II-E.ii</td><td>Generic comment 1 : what to be seen actually behind the wording “η acceptance”</td><td>22</td></tr><tr><td>II-E.iii</td><td>Generic comment 2 : Helix versus straight-line approximations</td><td>22</td></tr><tr><td>II-E.iv</td><td>Generic comment 3 : looper particles spinning towards endcap</td><td>23</td></tr><tr><td>II-F</td><td>Acceptance thresholds : minimal p<sub>T</sub> and inclination angles</td><td>24</td></tr><tr><td>II-F.i</td><td>Minimal transverse momentum to reach TOF (barrel and endcap)</td><td>24</td></tr><tr><td>II-F.ii</td><td>Inclination angle of the track with respect to the barrel detector surface</td><td>26</td></tr><tr><td>II-F.iii</td><td>Inclination angle of the track with respect to the endcap detector surface</td><td>27</td></tr><tr><td>III</td><td>TOF separation power</td><td>29</td></tr><tr><td>III-A</td><td>General intent and different configurations</td><td>29</td></tr><tr><td>III-B</td><td>Typical layout of the figures to come</td><td>30</td></tr><tr><td>IV</td><td>Using ALICE-1 in LHC runs I and II as a benchmark exercise</td><td>30</td></tr><tr><td>IV-A</td><td>Analytics-based performances of the ALICE-1 TOF</td><td>34</td></tr><tr><td>IV-B</td><td>Measured performances of the ALICE-1 TOF sub-detector</td><td>34</td></tr><tr><td>IV-C</td><td>Comparing and discussing analytics and real performances</td><td>37</td></tr><tr><td>IV-C.i</td><td>Acceptance × efficiency to reach TOF</td><td>37</td></tr><tr><td>IV-C.ii</td><td>Validation with the measured separation among (π<sup>±</sup>, K<sup>±</sup>, p<sup>±</sup>) species</td><td>37</td></tr></tbody></table>
<table><tbody><tr><td>62</td><td>IV-C.iii</td><td>Impact of multiplicity on the overall TOF performance, the start time</td><td>38</td></tr><tr><td></td><td>IV-C.iv</td><td>Light nuclei, illustration with the deuteron identification</td><td>39</td></tr><tr><td>64</td><td>V</td><td>Synopsis of the figures and corresponding TOF configurations tested</td><td>41</td></tr><tr><td>66</td><td>VI</td><td>Specifications and technological choices, ALICE-3 compared to ATLAS and CMS</td><td>48</td></tr><tr><td></td><td>VII</td><td>TOF measurements in ALICE-3 (HL-LHC run V)</td><td>49</td></tr><tr><td>68</td><td>VII-A</td><td>Configuration with a TOF as barrel outer layer at R = 1 m</td><td>49</td></tr><tr><td></td><td>VII-B</td><td>Configuration with a TOF as barrel intermediate layer at R = 0.2 m</td><td>50</td></tr><tr><td>70</td><td>VII-C</td><td>Possibility to have endcap TOF planes</td><td>51</td></tr><tr><td></td><td>VIII</td><td>TOF measurement in CMS (HL-LHC run IV and V)</td><td>54</td></tr><tr><td>72</td><td>IX</td><td>TOF measurements in ATLAS (HL-LHC run IV and V)</td><td>58</td></tr><tr><td></td><td>X</td><td>Conclusion</td><td>92</td></tr><tr><td>74</td><td>A</td><td>ROOT code snippet</td><td>94</td></tr><tr><td></td><td>References</td><td></td><td>102</td></tr></tbody></table>
# I Introduction : context of the HL-LHC run V

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## I-A Experimental landscape at HL-LHC

At the horizon 2030, the LHC will have already entered its High-Luminosity mode, HL-LHC, normally since one achieved run (Run IV, 2027-2030). By then, the ATLAS and CMS collaborations will have already carried out their major experiment upgrades (during Long Shutdown 3, 2025-2027); the LHCb and ALICE collaborations will be in similar positions, i.e. planning to renew their instrumentations in order to pursue/extend their respective physics cases while coping with/making the best of the enhanced instantaneous luminosity available at the time.

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## I-B ALICE-3 proposal

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In the context of the European Strategy for Particle Physics (ESPP) initiated in 2018, a proposal for a new experiment has been submitted with the intent to take over the scientific programme from the ALICE-2 apparatus and to be located at Point 2 of the LHC. The timeline foreseen for such a novel instrument would develop along an installation during LHC Long Shutdown 4 (≈ 2030-2032) and a start of operations at HL-LHC run V (> 2032).

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Such an experiment has been outlined and roughly sketched in an expression of interest entitled *A Next-Generation LHC Heavy-Ion Experiment* [1]. While I have been using the acronym ANGHIE as a temporary nickname in the past months including in official presentations<fnref n="1" />, as of today, an official acronym, ALICE-3, is proposed by the ALICE people interested in such a potential experiment at CERN. If such a name could be subject to yet another change in the coming years, it is for now a name officially approved internally. The intent is not to square the ALICE territory in order to forbid any outsider participation (experimentalists, theoreticians) but is essentially meant to keep a name immediately recognisable by authorities. Such a nickname will be used in the following.

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The main physics focuses motivating this original experiment are :

(i) **quark flavour physics** : a goal is to cover equivalently any measurement in the light-flavour sector ($g + u,d,s$) as in the heavy-flavour one ($c,b$). This is of particular importance in order to be in position to correlate such observables.

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(ii) **electrons and soft photons** : the QCD physics cases must be enlarged decisively to electromagnetic probes, such probes being at first order insensitive to interactions with the medium, once they are produced by the latter. On that front, studies of chiral symmetry restoration [2] and soft theorems stay [3, 4] the prime targets. Those physics cases have been promoted since several decades but have never been fully at reach : they are known to be especially delicate to address but remain of unique prominence.

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(iii) **hermiticity** : being after an advanced exhaustiveness in the foreseen measurements, the phase space coverage should be widened to the greatest extent, this to get the picture of the particle production as complete as possible, on an event-by-event basis. This is of particular relevance for fluctuation measurements : i) the fluctuations stemming from the initial state, propagated in an almost unaltered way, through barely viscous hydrodynamics, up to the final state and ii) the fluctuations in net quantum numbers (electric charge, baryon number, strangeness, ...).

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<footnote n="1">¹ Note that this was a temporary shortcut just resorted to for the sake of commodity, i.e. to avoid repeated periphrases and to simplify the uniqueness of labelling versus other experiments. It should have been considered as a notation and by no mean as an official acronym already approved by the proto-collaboration or by CERN authorities.</footnote>
(iv) low momentum : the focus must be given to the phase space areas where the bulk production of any identified particle arises, i.e. at intermediate ($2 \le p_T < 10 \text{ GeV/c}$) and, even more significantly, at low ($0 < p_T \le 2 \text{ GeV/c}$) momenta, with special abilities of detecting properly also in an ultra-low momentum domain ($0.03 \le p_T < 0.15 \text{ GeV/c}$).

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(v) high integrated luminosities : the statistical reach will be a key value of the experiment. It is not only about the ability of the apparatus to inspect steadily high instantaneous luminosity, then potentially trigger on it or, most of the time, not trigger on it; the point is rather to record a very large genuine collection of inelastic ($\approx$ Minimum Bias) collisions (e.g. $\mathcal{O}(1-10\text{fb}^{-1})$) of pp collisions at $\sqrt{s} = 14$ TeV, having typically a total inelastic cross-section of 80 mb, thus corresponding to an overall sample of about 80 000 – 800 000.10⁹ events).

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By correspondence, such key physics incentives clearly drive the key specifications of the instrument :

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(a) hyper-granularity with space resolution per layer as low as $\mathcal{O}(3-5)$ µm,

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(b) large pseudo-rapidity coverage with sub-detectors equipped on $|\eta| < 4$ units,

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(c) moderate magnetic field with $B \approx 0.2-1.0$ T,

(d) ultra-low material budget ranging from 0.05% of $x/X_0$ per layer for innermost layers to $\mathcal{O}(1)\%$ per layer for outermost ones,

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(e) extended particle identification making use of TOF measurements with timing resolution of $\mathcal{O}(10-30)$ ps,

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(f) swift readout with full-event record of $\mathcal{O}(1)$ MHz.

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To remain on a realistic path for the 10 years to come, the feasibility of such an experiment has to be regarded within the scope of concrete detector R&D, of actual technologies that should become both, i) mature and ii) financially affordable at the horizon 2030. Under those conditions, extra constraints must be brought to the reader's considerations.

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The full experiment will have to be installed within a time span of one LHC Long Shutdown (1.5-2.5 years or so). To that end, the full apparatus is compelled to be rather “compact” and “simple”, i.e. the active surface to be equipped needs to remain moderate and the technology should be essentially ready to be produced along an almost-industrial model. This militates against a vast diversification of technologies brought into operations but rather advocates for narrowing the choices to a limited set of detector technologies, ideally to a single type, to be applied and declined as needed everywhere in the experiment. For those reasons, the candidate technology to that purpose could only be at the time being silicon based, more specifically CMOS based.

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Figure 1 illustrates the first thoughts given to a such an experiment and gives a schematic view of the possible anticipated layout.

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# II Math of time of flight

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## II-A Prime equations related to TOF quantities

Generic considerations for equations at stakes for TOF are given in section 4 of [5] and can start from the Lorentz ratio $\beta$.

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$$ \beta_{\text{tot}} = \frac{v_{\text{tot}}}{c} \qquad (1) $$

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$$ \beta_{\text{tot}} = \frac{\mathcal{L}}{t.c} \qquad (2) $$

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<figure><img src="image_2.png" /><figcaption>Fig. 1: First detector layout proposed in the early 2019 for the ALICE-3 experiment [1], with barrel covering a pseudo-rapidity range of $|\eta| < 1.4$ and endcaps extending the acceptance to $1.4 < |\eta| < 4.0$. The detector hinges on 3 types of sub-detectors: a pixellated tracker, a pixellated time-of-flight layer and a silicon-sampled electromagnetic preshower.</figcaption></figure>

giving :

$$t_{\text{TOF}} = \frac{\mathcal{L}_{\text{flight}}}{c \cdot \beta_{\text{tot}}} \qquad (3)$$

with :

■ c the speed of light in vacuum,

■ $v_{\text{tot}}$ the norm in the 3 spatial dimensions of the velocity $\vec{v}_{\text{tot}}$, associated to Lorentz factors, $\beta_{\text{tot}} = \frac{v_{\text{tot}}}{c}$ (Eq. 1) and $\gamma_{\text{tot}} = \frac{1}{\sqrt{1 - \beta_{\text{tot}}^2}}$, (4)

■ t the elapsed time for the particle to cover its flight between two considered points, typically the production and the arrival points,

■ $\mathcal{L}$ the integrated path length followed by the particle along such a flight, → all these quantities being considered in the *laboratory frame*.

The TOF separation power $n\sigma_{\text{TOF}}$ between two particle species A and B can then be written as a multiple of the overall timing resolution of the TOF detector, $\sigma_{\text{TOF}}$:

$$n\sigma_{\text{TOF}} = \frac{|t_A - t_B|}{\sigma_{\text{TOF}}} \qquad (5)$$

$$n\sigma_{\text{TOF}} = \frac{1}{\sigma_{\text{TOF}}} \cdot \frac{1}{c} \cdot \left| \frac{\mathcal{L}_A}{\beta_{\text{tot},A}} - \frac{\mathcal{L}_B}{\beta_{\text{tot},B}} \right| \qquad (5a)$$

with the 2 key components :

$$\frac{1}{\beta_{\text{tot},\mathbb{H}}} = \sqrt{1 + \left(\frac{m_{0,\mathbb{H}} \cdot c}{p_{\text{tot},\mathbb{H}}}\right)^2} \qquad (6)$$

$$\frac{1}{\beta_{\text{tot},\mathbb{H}}} = \sqrt{1 + \left(\frac{m_{0,\mathbb{H}} \cdot c}{p_{\text{T,\mathbb{H}}} \cdot \cosh(\eta_{\mathbb{H}})}\right)^2} \qquad (6a)$$

$$[\text{because : } p_{\text{tot},\mathbb{H}} = p_{\text{T,\mathbb{H}}} \cdot \cosh(\eta_{\mathbb{H}})] \qquad (7)$$
$$ \circ \mathcal{L}_{\mathbb{H}} = f(m_{0,\mathbb{H}}, q_{\mathbb{H}}, \eta_{\mathbb{H}}, p_{\mathrm{T},\mathbb{H}}, B) \qquad (8) $$

depending on quantities related to the particle of interest, labelled generically $\mathbb{H}$ here :

$$ \bullet m_{0,\mathbb{H}} \quad \text{the invariant mass at rest,} \\ \bullet q_{\mathbb{H}} = n_{\mathbb{H}} \cdot e \quad \text{the electric charge,} \qquad (9) $$

$$ \text{with : } n \quad \text{multiple of } e \text{ units and } n \in \mathbb{Z}, \text{ set of integers,} \\ e \quad \text{the primary electric charge, } e = 1.602 176 634 \cdot 10^{-19} \text{ C,} $$

$$ \bullet p_{\mathrm{tot},\mathbb{H}} \quad \text{the norm of the total momentum,} $$

$$ \bullet \eta_{\mathbb{H}} \quad \text{the initial pseudo-rapidity,} $$

$$ \bullet p_{\mathrm{T},\mathbb{H}} \quad \text{the norm of the transverse momentum,} $$

together with :

$$ \bullet B \quad \text{the intensity of the magnetic field in the experiment,} $$

In HEP experiments, a solenoidal magnetic field is frequently in place, parallel to the beam axis ($\pm \vec{e}_z$). This is in particular true for LHC experiments such as ALICE, ATLAS and CMS.

## II-B Core working hypotheses : purely helical trajectory

In the above set of equations, it is not obvious a priori that $\beta_{\mathrm{tot},\mathbb{H}}$ stays constant over the whole flight, from the primary production point up to the arrival point into the TOF detector, nor that the path itself is smooth or devoid of any sharp point. Indeed, along the way, interactions of various types can happen in the instruments : the trajectory can suddenly change due to multiple Coulomb scattering (direction of $\vec{p}_{\mathrm{tot}}$), to some lineic energy loss happening while crossing material (direction and norm of $\vec{p}_{\mathrm{tot}}$), as well as to quasi-continuous energy losses along the path with processes like bremsstrahlung (direction and norm of $\vec{p}_{\mathrm{tot}}$).

In practice, the integrated time covering the full flight can rather be seen as the sum of segments taken each between two consecutive (particle-to-detector) interactions of such types, the distance $\mathcal{L}_{\mathrm{flight},i}$ and the norm $\beta_{\mathrm{tot},i}$ staying clearly defined in between such dates.

$$ t_{\mathrm{TOF}} = \sum_{\mathrm{segment} i} \frac{\mathcal{L}_{\mathrm{flight},i}}{c \cdot \beta_{\mathrm{tot},i}} \qquad (10) $$

In order to predict results including such phenomena, it usually entails Monte-Carlo simulations with full particle transport into the numeric detector geometry, addressed with software toolkits such as FLUKA [6] or GEANT4 [7].

**Hypothesis 1** In the current work, the intent is to derive analytically the performance of various TOF detectors. To that end, the ideal case is considered, meaning that along the particle flight, there is **no extra interaction other than the magnetic component of the Lorentz force**, $\vec{f}_L(t) = q \cdot \vec{v}_{\mathrm{tot}} \times \vec{B}$, i.e. no energy loss, no scattering, no absorption, no bremsstrahlung, no $\vec{E} \times \vec{B}$ effects, ... are taken into account.

**Hypothesis 2** The magnetic field is assumed to be **perfectly static and homogeneous, exactly aligned along ($\pm \vec{e}_z$)**.
212 After such hypotheses, the exercise can be seen as an assessment of the ultimate performances, the horizons beyond which the experiments could simply not go. Under various detector configurations for ALICE, ATLAS and CMS, such limits of accessibilities will be posed. The consequences of such a strategic bias imply that, along the trajectory, the norm $||\vec{\beta}_{tot}||$ stays constant, i.e. the direction of $\vec{p}_{tot}$ evolves along with the influence of the sole external magnetic field but $||\vec{p}_{T}||$ and $||\vec{p}_{Z}||$ stay constant along the particle propagation.

218 We thus fall back to the equation previously listed, Eq. 11.

$$t_{\text{TOF}} = \frac{\mathcal{L}_{\text{flight}}}{c \cdot \beta_{\text{tot}}} \qquad (11)$$

## II-C Parametric equations of the relativistic helical motion

220 If there is not much ambiguity about the literal expression to be used for $\beta_{tot}$ (see Eq. 6a), there are on the other hand various options followed in the literature to approximate the path length, $\mathcal{L}$.

224 Under the aforementioned longitudinal B field configuration, the trajectory of any relativistic charged particle subjected to the Lorentz force can be represented following at least four approaches:

228 a) a straight line extending the initial tangent: derive the path and time as if there were no B field, the track thus becomes simply the straight flight continuation of the tangent to the trajectory at $t_0$ ($p_T(t_0)$, $\eta(t_0)$), prolonged up to the intersection with the TOF layer or plane. This option is likely the spontaneous idea that one may have as a first approximation.

232 b) a straight line connecting the primary vertex and the TOF hit location: such an option requires significant mathematical derivations, since the actual location ($x(t_f)$, $y(t_f)$, $z(t_f)$) of the final TOF hit needs to be obtained beforehand. This option is certainly not the most natural a priori but has the advantage to provide a distance which will remain closely related to the actual trajectory; for the user, it avoids the integration of the curvilinear path length $\mathcal{L}$.

238 c) a circular arc in the transverse plane: the bending of the trajectory is now explicitly taken into account but the trajectory is only thought through its projection onto the transverse plane (O; $\vec{e}_X$, $\vec{e}_Y$), ignoring the z component $\vec{p}_Z$; this provides a fair approximation of the situation in the barrel case near mid-rapidity $y \approx 0$.

242 d) a 3D helix segment: the most generic case meant to work without further approximation, for both barrel- and endcap-oriented particles.

244 In this document, the avenue pursued under any experimental configuration is the last one, option (d).

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### II-C.i Differential equation of relativistic motion

248 In order to develop such a line, the parametric equations of motion have been derived, applying the fundamental principle of dynamics (FPD) in the relativistic case to a charged particle immersed into a static and homogeneous magnetic field.

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Expressed with the four-force and four-acceleration, the FPD is written like :

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$$ \vec{F} = m_0 \cdot \vec{A} \qquad (12) $$

$$ \vec{F} = \left( \gamma_{\text{tot}} \frac{\vec{f} \cdot \vec{v}}{c}, \gamma_{\text{tot}} \vec{f} \right) \qquad (12a) $$

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with :

$$ \bullet \vec{f} \cdot \vec{v} = \frac{d}{dt} [\gamma_{\text{tot}}(t) m_0 c^2] = \frac{dE}{dt} \qquad (13) $$

$$ \bullet \vec{f} = \frac{d}{dt} [\gamma_{\text{tot}}(t) m_0 \vec{v}_{\text{tot}}(t)] = \frac{d\vec{p}_{\text{tot}}(t)}{dt} \qquad (14) $$

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In the present case, with the sole force to be considered being the Lorentz force :

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$$ \vec{f}_{\text{L}}(t) = q \cdot \vec{v}_{\text{tot}}(t) \times \vec{B} \qquad (15) $$

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$$ \text{It implies that :} \qquad \forall t \in \mathbb{R}^{+}, \quad \vec{f}_{\text{L}}(t) \perp \vec{v}_{\text{tot}}(t) \qquad (16) $$

$$ \text{so :} \qquad \forall t \in \mathbb{R}^{+}, \quad \vec{f}_{\text{L}}(t) \cdot \vec{v}_{\text{tot}}(t) = 0 \qquad (17) $$

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$$ \text{implying from Eq. 13 :} \qquad \forall t \in \mathbb{R}^{+}, \frac{dE}{dt} = 0 \qquad (18) $$

$$ \text{and then :} \qquad \forall t \in \mathbb{R}^{+}, \frac{d\gamma_{\text{tot}}}{dt} = 0 \qquad (19) $$

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Under such circumstances, the energy is conserved along the motion, so do in turn $\gamma_{\text{tot}}$ and $||\vec{v}_{\text{tot}}||$. The equation 14 becomes :

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$$ \vec{f}_{\text{L}}(t) = \frac{d\vec{p}_{\text{tot}}(t)}{dt} \qquad (20) $$

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$$ q \cdot \vec{v}_{\text{tot}}(t) \times \vec{B} = \frac{d\vec{p}_{\text{tot}}(t)}{dt} \qquad (20a) $$

$$ q \cdot \vec{v}_{\text{tot}}(t) \times \vec{B} = \frac{d}{dt} [\gamma_{\text{tot}} m_0 \vec{v}_{\text{tot}}(t)] \qquad (20b) $$

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**Highlight II.1. Differential equation of motion for a relativistic charged particle in a static and homogeneous magnetic field, $\vec{B}$**.

$$ q \cdot \vec{v}_{\text{tot}}(t) \times \vec{B} = \gamma_{\text{tot}} m_0 \frac{d\vec{v}_{\text{tot}}(t)}{dt} \qquad (20c) $$

## II-C.ii Sketch of the transverse projection (frames and notations)

272

Before moving to the resolution of the differential equation, the notations need to be defined. The sketch depicted in Fig. 2 illustrates a projection of the trajectory onto the transverse plane;

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it highlights the angle $\alpha(t)$ to be considered for the motion of rotation around the curvature centre C, for a particle initiated at the primary vertex O in the laboratory frame.
276 **Hypothesis 3** The particles considered here are generically supposed to be **emitted at the exact centre** of the experiment, conveniently coinciding with the origin $O(0,0,0)$ of the laboratory frame. This essentially relates to three assumptions:

i) the derivations being performed here will only hold for **primary-particle** population (or "primary-like" as are decay products of short-lived resonances directly stemming the primary vertex);

ii) the accelerator beams are perfectly aligned on the z axis, i.e. there is no $(x,y)$ displacement of the beam (typically by a few mm) due to the accelerator optics as well as no further spread of the primary vertices in $(x,y)$ [$O(mm)$];

iii) there is no shallow spread of the primary vertices in $z$ [$O(10 \text{ cm})$] due to any actual length of crossing bunches.

278 **Hypothesis 4** Geometry of both TOF barrel layer and endcap plane are considered fully **invariant under rotation** around the axis ($O; \vec{e}_Z$). Let's assume they are indeed plain cylinder and plain ring, respectively, without finite elements that would break the invariance in practice. With such a simplification, starting from the origin $O(0,0,0)$, one can conveniently make a choice for the initial orientation of $\vec{p}_T$. In the following, whatever will be considered for ($m_0, p_T, \eta$), we will always define trajectories for which $\vec{p}_T(t_0) = + \|\vec{p}_T\| \cdot \vec{e}_X$. Any other case with different initial orientation in azimuth, i.e. with an angle, $\varphi(t_0)$, between $\vec{e}_X$ and $\vec{e}_r(t_0)$, can be derived from the previous case by a rotation of each point of the trajectory by $\varphi(t_0)$.

280 Figure 3 shows the frames exploited to derive the solution for Eq. 20c. The math for the moving particle $M$ is primarily performed in the cylindrical frame centered around the curvature centre ($C; \vec{e}_r, \vec{e}_\varphi, \vec{e}_Z$), before moving back ultimately to the Cartesian coordinate system of the laboratory, ($O; \vec{e}_X, \vec{e}_Y, \vec{e}_Z$).

284 **II-C.iii** *Acceleration vector as a function of time*

286 It follows the expression of the centripetal acceleration, derived in (C; $\vec{e}_r, \vec{e}_\varphi, \vec{e}_Z$).

$$ \forall t[s] \in \mathbb{R}^+, \quad \vec{a}_M(t) = -|\rho| \cdot \left( \frac{qB_Z}{\gamma_{\text{tot}}m_0} \right)^2 \vec{e}_r \quad (21) $$

$$ \vec{a}_M(t) = -|\rho| \cdot (\omega_T)^2 \vec{e}_r, \quad (21a) $$

with the 2 notations:

**Notation II.1.** *Curvature radius which is constant with time.*

(Beware that this is taken here as a signed quantity, depending on $q$ and $B_Z$.)

$$ r(t) \cdot \vec{e}_r(t) \stackrel{\text{not}}{=} \rho \cdot \vec{e}_r(t), \\ \text{with : } \rho = \frac{\|\vec{p}_T\|}{\left(\frac{c \cdot 10^{-9}}{1V}\right) n B_Z} \quad (22) $$

290 The units in the present formula may need to be further explicated here, since they are a bit
<figure><img src="image_3.png" /><figcaption>**Fig. 2:** Sketch introducing the notations for the angles at stakes in the component of motion that is transverse to *B* field.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>**Fig. 3:** Sketch laying the frame notations for ($C$; $\vec{e}_r, \vec{e}_\varphi, \vec{e}_Z$) anchored to the centre of rotation $C$ of the motion and for ($O$; $\vec{e}_X, \vec{e}_Y, \vec{e}_Z$) anchored to the lab frame.</figcaption></figure>

The illustration here is outlined for:

i) $\vec{B} = B_Z \cdot \vec{e}_Z$ with $B_Z < 0$

ii) $q > 0$.

**Note:**

* the initial angular position,
  $\varphi(t_0) = -\pi/2$.

* the curvature radius,
  $\rho = \overline{CM} \propto (+\vec{e}_r)$ but $\propto (-\vec{e}_Y)$.

* the pulsation,
  $-\omega_T > 0$.
specific for concrete numerical applications that will be performed later.

292

$$ \underset{[\text{m}]}{\rho} = \frac{\underset{[\text{m.s}^{-1}, \text{V}^{-1}]}{\overbrace{\|\vec{p}_T\|}} \underset{[\text{T}]}{B_Z} \underset{[\emptyset, \text{multiple of e units}]}{n}}{0.299\ 792\ 458} \quad (22a) $$

**Notation II.2.** Larmor angular frequency (or gyromagnetic pulsation) in the relativistic case.

$$ \frac{q B_Z}{\gamma_{\text{tot}} m_0} \stackrel{\text{not}^o}{=} \omega_T, \quad (23) $$

294

Contrary to the situation considered for $\rho$ in Eq. 22, the formula of $\omega_T$ has to be considered with standard units, meaning especially that $m_0$ is expressed in kg.

$$ \underset{[\text{rad.s}^{-1}]}{\omega_T} = \frac{\underset{[\emptyset]}{\overbrace{(q = n.e)}^{[\text{C}]}} \underset{[\text{kg}]}{B_Z}}{\underset{[\emptyset]}{\overbrace{\gamma_{\text{tot}}}} \underset{[\text{kg}]}{m_0}} \quad (23a) $$

296

**II-C.iv** Velocity vector as a function of time

298

Looking for a primitive defined with the initial conditions on $\vec{v}_{\text{tot}}(t_0)$, one obtains the velocity expression as a function of time :

$$ \forall t[s] \in \mathbb{R}^+, \quad \vec{v}_M(t) \stackrel{\text{not}^o}{=} \vec{v}_{\text{tot}}(t) \quad (24) $$

300

$$ \vec{v}_M(t) = \frac{d\vec{CM}}{dt} \quad (25) $$

$$ \vec{v}_M(t) = \underset{\|\vec{v}_T\|(t)}{\underbrace{v_\phi(t) \cdot \vec{e}_\phi}} + v_Z(t) \cdot \vec{e}_Z \quad (25a) $$

302

$$ \vec{v}_M(t) = v_T(t_0) \cdot \vec{e}_\phi(t) + v_Z(t_0) \cdot \vec{e}_Z \quad (25b) $$

304

$$ \vec{v}_M(t) = [-|\rho| \omega_T] \cdot \vec{e}_\phi(t) + [\beta_Z(t_0) c] \cdot \vec{e}_Z \quad (25c) $$

$$ \vec{v}_M(t) = [-|\rho| \omega_T] \cdot \vec{e}_\phi(t) + \underbrace{\left[ \frac{p_Z(t_0)}{E_{\text{tot}}(t_0)} c \right]}_{\text{beta Z expressed here in HEP units ("natural units"),}} \cdot \vec{e}_Z \quad (25d) $$

306

$\beta_Z$ expressed here in HEP units ("natural units"), i.e. ratio of $p_Z[\text{GeV/c}]$ and $E_{\text{tot}}[\text{GeV}]$ ("c=1" HEP logic)<fnref n="2" />

**Highlight II.2.** Velocity vector as a function of time

$$ \vec{v}_M(t) = \left[ -|\rho| \omega_T \right] \cdot \vec{e}_\phi(t) + \left[ \frac{\|\vec{p}_T\| \sinh(\eta)}{\gamma_{\text{tot}} m_0} c \right] \cdot \vec{e}_Z \quad (25e) $$

308

because :

<footnote n="2">² In standard units, one would have $\beta_Z = \frac{c p_Z [\text{kg m.s}^{-1}]}{E_{\text{tot}}[\text{J}]}$, i.e. with an extra c appearing at the numerator.</footnote>
$$p_z = \underbrace{\|\vec{p}_T\| \sinh(\eta)}_{[\text{GeV/c}]} \qquad (26)$$

310

$$E_{\text{tot}} = \underbrace{\gamma_{\text{tot}} m_0}_{[\text{GeV}/c^2]} \qquad (27)$$

As previously done, Eq. 25e can be revised with the perspective of units, revealing on the way what has to be taken for the component $v_Z$.

312

$$\rightarrow \vec{v}_M(t) = - \underbrace{|\rho| \omega_T}_{[\text{m}] [\text{rad.s}^{-1}]} \cdot \underbrace{\vec{e}_\phi(t)}_{[\emptyset]} + \underbrace{\frac{\underbrace{\|\vec{p}_T\|}_{[\text{GeV/c}]} \sinh(\eta)}{\gamma_{\text{tot}} \underbrace{m_0}_{[\text{GeV}/c^2]}}}_{[\text{m.s}^{-1}]} \cdot \underbrace{299\,792\,458 \cdot \vec{e}_Z}_{[\text{GeV/c}]} \qquad (27a)$$

314

It should be noted that, if the direction of the velocity vector evolves with time, notably due to $\vec{e}_\phi(t)$, the individual components of the velocity, $\|\vec{v}_T\|$ and $\|\vec{v}_Z\|$, are constant with time and, in turn, $\|\vec{v}_{\text{tot}}\|$ is constant as well, as we anticipated earlier. So instead of taking the norm of the aforementioned Eq. 25e, the norm of the velocity can be rewritten into a simpler form (with $m_0$ in $[\text{GeV}/c^2]$ and $\|\vec{p}_T\|$ in $[\text{GeV}/c]$):

316

318

$$\forall t[s] \in \mathbb{R}^+, \qquad \|\vec{v}_M(t)\| = \|\vec{v}_{\text{tot}}\| \qquad (28)$$

320

$$\|\vec{v}_M(t)\| = \beta_{\text{tot}} \cdot c \qquad (28a)$$

$$(\text{see Eq. 6a}) \qquad \|\vec{v}_M(t)\| = \left( \sqrt{1 + \frac{m_0^2}{\|\vec{p}_T\|^2 \cosh^2(\eta)}} \right)^{-1} \cdot c \qquad (28b)$$

322

### II-C.v Spatial coordinates as a function of time

The last integration step leads to the spatial coordinates of the relativistic motion, taking into account initial positions, first for $\vec{CM}$ and finally for $\vec{OM}$ in $(O; \vec{e}_X, \vec{e}_Y, \vec{e}_Z)$. The set of parametric equations is meant to be generic to accommodate the possible change of sign for the $B_Z$ configuration ($\pm \vec{e}_Z$) together with the sign of the electric charge of the particle of interest ($q = n.e < 0$ or $q > 0$), leading to 4 configurations of $[B_Z, q]$ to consider.

324

326
**Highlight II.3.** Parametric equations of the spatial coordinates as a function of time, for a charged particle propagating into a static and homogeneous magnetic field, $\vec{B} = B_z \vec{e}_z$, expressed in the frame at rest in the laboratory (O; $\vec{e}_x$, $\vec{e}_y$, $\vec{e}_z$).

$$ \forall t[s] \in \mathbb{R}^+, \quad \vec{OM}(t) = \overbrace{\frac{\|\vec{p}_T\|}{0.299\,792\,458 \ nB_z}}^{-\rho} \sin\left(\underbrace{\frac{neB_z}{\gamma_{tot}m_0} t}_{\omega_T}\right) \cdot \vec{e}_x \\ + \frac{\|\vec{p}_T\|}{0.299\,792\,458 \ nB_z} \left[\cos\left(\underbrace{\frac{neB_z}{\gamma_{tot}m_0} t}_{\gamma_{tot}m_0} - 1\right) - 1\right] \cdot \vec{e}_y \\ + \underbrace{\left[\frac{\|\vec{p}_T\| \sinh(\eta)}{\gamma_{tot} m_0} c t\right]}_{\beta_z} \cdot \vec{e}_z \quad (29) $$

$$ \text{with : } \gamma_{tot} = \sqrt{1 + \frac{\|\vec{p}_{tot}\|^2}{m_0^2}} = \sqrt{1 + \frac{\|\vec{p}_T\|^2 \cosh^2(\eta)}{m_0^2}} \quad (30) $$

328

Note in particular that $n$ and $B_z$ are considered as signed quantities and that the equations encompass a two-fold logic as far as the units are concerned, depending on the considered term: $\rho$ (Eq. 22a), $\beta_z$ (Eq. 26 combined with Eq. 27) and $\gamma_{tot}$ (Eq. 30) abide by the HEP units hinged on GeV; $\omega_T$ follows the standard physics units, as exposed in Eq. 23a. All in all, $t$ has to be in [s] and the 3 space components of $\vec{OM}(t)$ will be in [m].

330

332

The vectorial equation 29 is meant to cover both trajectory cases, should it hit a barrel layer or an endcap plane, and to work for particle rotations for which the absolute angle around the curvature centre C stays within $[0; +\pi]$ (barrel and endcap) but also rotations which cover more than a half-turn $([+\pi; \infty[$, “looper tracks” met in the endcap case).

334

336

## II-D Deriving path lengths and times

338

### II-D.i Path length formula for a regular helix

After the derivation of the parametric equations, one can calculate the actual quantities of interest, the path length $L$ and the time needed to cover such a curvilinear distance, $t_{TOF}$, introduced in Eq. 5a. Starting from the curvilinear abscissa $s$ along the curve, the path length is the integral from the starting time $t_0$ to the final one $t_f$.

340

342

$$ \mathcal{L} = \int_{t_0}^{t_f} ds \qquad (31) $$

344

$$ \mathcal{L} = \int_{t_0}^{t_f} \|d\vec{C}\vec{M}\| = \int_{t_0}^{t_f} \|d\vec{O}\vec{M}\| \qquad (31a) $$

$$ \mathcal{L} = \int_{t_0}^{t_f} \left\| \frac{d\vec{O}\vec{M}}{dt} \right\| dt \qquad (31b) $$

346

$$ \mathcal{L} = \int_{t_0}^{t_f} \| \vec{v}_{tot} \| dt \qquad (31c) $$
Since $||\vec{v}_{tot}||$ is constant with time (see Eq. 28b), the path length can finally be expressed as :

$$ \mathcal{L} = \| \vec{v}_{tot} \| \int_{t_0}^{t_f} dt \qquad (31d) $$

$$ \mathcal{L} = \| \vec{v}_{tot} \| [t]_{t_0}^{t_f} \qquad (31e) $$

**Highlight II.4. Path length along the regular helical trajectory covered by a (relativistic) charged particle in a static and homogeneous magnetic field.**

$$ \mathcal{L} = \| \vec{v}_{tot} \| t_f \qquad (31f) $$

From the above equation, $||\vec{v}_{tot}||$ being known from the initial conditions (Eq. 28b), it is now understood that the whole stress to assess the path length is in fact reported onto the determination of the date $t_f$, for either the flight to reach the barrel layer or the endcap plane.

### II-D.ii Date to reach the TOF barrel layer

In the barrel case, the TOF layer is reached when the transverse location is equal for the first time to the TOF radius. We look for :

$$ t_f \in \mathbb{R}^+ \quad / \quad \|\overrightarrow{OM}(t_f) \cdot \vec{e}_r\| = R_{\text{TOF}} \qquad (32) $$

$$ \text{i.e.} \quad t_f \quad / \quad \sqrt{x^2(t_f) + y^2(t_f)} = R_{\text{TOF}} \qquad (33) $$

Using the abscissa and ordinate components defined in Eq. 29 to solve this equation, we obtain:

**Highlight II.5. TOF date in the barrel case.**

$$ t_f^{\text{barrel}} = \underbrace{\frac{1}{|\omega_T|}}_{\text{barrel}} \arccos \left( 1 - \frac{R_{\text{TOF}}^2}{2\rho^2} \right) \qquad (34) $$

$$ = \frac{\gamma_{\text{tot}} m_0}{|n e B_Z|} \arccos \left( 1 - \frac{R_{\text{TOF}}^2}{2} \frac{[0.299\,792\,458 |n B_Z|]^2}{\|\vec{p}_T\|^2} \right) (34a) $$

$$ \text{alternatively : } t_f^{\text{barrel}} = \frac{1}{|\omega_T|} \quad 2 \arcsin \left( \frac{R_{\text{TOF}}}{2|\rho|} \right) \qquad (35) $$

see Eq. 30

$$ = \underbrace{\frac{\gamma_{\text{tot}} m_0}{|n e B_Z|}}_{\text{barrel}} \arcsin \left( \frac{R_{\text{TOF}}}{2} \frac{0.299\,792\,458 |n B_Z|}{\|\vec{p}_T\|} \right) \qquad (35a) $$

$$ \text{leading to : } \mathcal{L} = \underbrace{\|\vec{v}_{tot}\|}_{\text{barrel}} t_f^{\text{barrel}} \qquad (36) $$

see Eq. 28b
### II-D.iii Date to reach the TOF endcap plane

362 In the endcap case, the TOF plane is reached when the longitudinal position is equal to the TOF location in z. We look for :

$$ t_f \in \mathbb{R}^+ \quad / \quad \|\vec{OM}(t_f) \cdot \vec{e}_z\| = Z_{\text{TOF}} \qquad (38) $$

$$ \text{i.e.} \quad t_f \quad / \quad z(t_f) = Z_{\text{TOF}} \qquad (39) $$

366 Using the longitudinal component defined in Eq. 29 to solve this equation, we obtain :

**Highlight II.6. TOF date in the endcap case.**

$$ t_f^{\text{endcap}} = \frac{|Z_{\text{TOF}}|}{|v_z|} = \frac{|Z_{\text{TOF}}|}{|\beta_z c|} \qquad (40) $$

$$ = |Z_{\text{TOF}}| \frac{\gamma_{\text{tot}} m_0}{\|\vec{p}_T\| |\sinh(\eta)| c} \quad (\eta \neq 0) \qquad (40a) $$

$$ \text{leading to : } \mathcal{L} = \underbrace{\|\vec{v}_{\text{tot}}\|}_{\text{see Eq. 28b}} t_f^{\text{endcap}} \qquad (41) $$

$$ \text{in fact : } \mathcal{L} = |Z_{\text{TOF}}| \frac{1}{|\tanh(\eta)|} = |Z_{\text{TOF}}| \frac{\|\vec{p}_{\text{tot}}\|}{|p_z|} \qquad (41a) $$

368 In the present situation, the path length turns out to be only function of $Z_{\text{TOF}}$ and $\eta$, and this way, independent of $B_z$. Should $B_z$ be zero, be of small value or be set to a large one, once $p_T$ and $p_z$ are decided at $t_0$, the Larmor radius will admittedly get smaller and smaller for increasing magnetic field but the corresponding path length will be in any case the same.

372

## II-E Visualising concrete helical trajectories calls for comments...

### II-E.i Introduction of the trajectory illustrations

374 Before moving into the actual map in $(p_T, y)$ of the TOF separation for various species, it could be profitable to illustrate some aspects of helical trajectories, based on a few selected cases. There are indeed several subtleties that the reader should be made aware of beforehand.

378 For such a purpose, we will use the configuration foreseen for the CMS detector in run IV (2027-2029) at HL-LHC, it is the first experiment planning to be concretely instrumented with both a barrel- and an endcap TOF [8].

380 On the one hand, Figs. 4 and 5 are examples of trajectories for a proton and an antiproton, respectively, both reaching the TOF barrel, placed at a radial position of $R_{\text{TOF}} = 1.16$ m. Figure 6 is also derived for a proton reaching the TOF barrel layer, but for a higher $p_T$. On the other hand, Figs. 7, 8 and 9 embody three cases of trajectories towards the endcap plane, located at $Z_{\text{TOF}} = 3.04$ m.

386 For each figure, the initial kinematic hypotheses are given ($m_0, q, p_T, \eta$ and $\beta_{\text{tot}}$), together with the detector key characteristics ($B_z, \sigma_{\text{TOF}}, R_{\text{TOF}}, Z_{\text{TOF}}$). Beyond the simple visualisation of the trajectories, the outcome in the figures also includes quantitative pieces of information. It concerns :
(i) the TOF hit itself such as its location $x(t_f), y(t_f), z(t_f)$,

(ii) the overall rotation and the Larmor radius, both imprinted by the magnetic field,

(iii) three path lengths and associated times $t_f$ to cover such flight distances

* the exact one, $\mathcal{L}$, along the curvilinear path (solid green curve) standing for option (d) among the possibilities described p.8

* the straight segment, $a$, along the tangent to the helix at the origin (O; 0,0,0) prolonged up to the TOF detector, displayed as a diagonal dotted azure line in figures, standing for approximation (a) given on the same page 8.

* the straight segment, $b$, connecting the origin (O; 0,0,0) to the TOF hit, displayed as the diagonal dashed olive line in figures, standing for approximation (b) given on p.8.

Both distances $a$ and $b$ are further related to $\mathcal{L}$ via their respective difference and ratio to the curvilinear distance, the same comparisons are also performed for the pair of corresponding time $t_f$. The intent with such tests is to assess down to which point a straight line approximation holds for a given $\sigma_{\text{TOF}}$ anticipated, i.e. is the straight-line approximation transparent for our performance study so that any difference is absorbed under the blurring of $\sigma_{\text{TOF}}$ ? or is it like the approximation too coarse so that the TOF results become clearly altered by it ?

Out of such illustrations of trajectories, several generic comments emerge.
<figure><img src="image_5.png" /><figcaption>**Fig. 4:** Helical trajectory for a proton of $p_T = 1.0$ GeV/c with $\eta = 1.2$, emitted with $\vec{p}_T(t_0)$ along $+\vec{e}_X$ direction, into the standard CMS configuration foreseen for the barrel pile-up tagger (MTD-BTL) at the HL-LHC run IV (2027-2029).</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>**Fig. 5:** Helical trajectory for an antiproton of $p_T = 1.0$ GeV/c with $\eta = 1.2$, emitted with $\vec{p}_T(t_0)$ along $+\vec{e}_X$ direction, into the standard CMS configuration foreseen for the barrel pile-up tagger (MTD-BTL) at the HL-LHC run IV (2027-2029).</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>**Fig. 6:** Helical trajectory for an antiproton of $p_T = 4.20$ GeV/c with $\eta = 1.2$, emitted with $\bar{p}_T(t_0)$ along $+\bar{e}_X$ direction, into the standard CMS configuration foreseen for the barrel pile-up tagger (MTD-BTL) at the HL-LHC run IV (2027-2029).</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>**Fig. 7:** Helical trajectory for a proton of $p_T = 0.5$ GeV/c with $\eta = 0.6$, emitted with $\vec{p}_T(t_0)$ along $+\vec{e}_X$ direction, into the standard CMS configuration foreseen for the endcap pile-up tagger (MTD-ETL) at the HL-LHC run IV (2027-2029).</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>**Fig. 8:** Helical trajectory for a proton of $p_T = 0.5$ GeV/c with $\eta = 0.9$, emitted with $\vec{p}_T(t_0)$ along $+\vec{e}_X$ direction, into the standard CMS configuration foreseen for the endcap pile-up tagger (MTD-ETL) at the HL-LHC run IV (2027-2029).</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>**Fig. 9:** Helical trajectory for a proton of $p_T = 0.65$ GeV/c with $\eta = 2.0$, emitted with $\vec{p}_T(t_0)$ along $+\vec{e}_X$ direction, into the standard CMS configuration foreseen for the endcap pile-up tagger (MTD-ETL) at the HL-LHC run IV (2027-2029).</figcaption></figure>
II-E.ii Generic comment 1 : what to be seen actually behind the wording “$\eta$ acceptance”

It is quite casual to quote the zenithal acceptance of a given sub-detector in terms of pseudo-rapidity. For instance, the detector MTD-BTL for CMS is meant to cover “up to $\pm 1.48$ units of $\eta$” (sec. 2.1 p.29 in [8]). However such a sentence has to be read as a shortcut to the actual situations met in concrete detection. Behind the scene, it is silently assuming the case of straight tracks, so particle of infinite $p_{\text{tot}}$ on which the solenoidal magnetic field has barely an influence. That means that the track is very near the straight prolongation of the direction taken originally, defined by the tangent to the curve at $t_0$ time. In practice, seen from the TOF point of view, the particle will always have an “apparent” $\eta$ which is larger... Such a sentence may look odd to the reader : the pseudorapidity is a kinematic property of the particle; in the ideal case pursued in the present document, there is no reason for it to change along the trajectory, isn’t there ? - It is tried to be more concrete in the following.

In the barrel, the final location $z(t_f)$ will always be larger than what is anticipated as $z(t_f)$ for a straight track being the continuation of the initial tangent at $t_0$. In Fig. 4, the proton is defined with $\eta = 1.20$. The final $z(t_f)$ is equal to 1.9148 m, This corresponds to the final $z$ position of a straight track of $\eta = 1.27$. In the figure, the dotted red arc circles drawn along the TOF cylinder stand for the apparent $\eta$ of +0.2$k$ with $k \in [0; 7]$; the curved trajectory arrives beyond such a dotted line at $\eta = 1.2$.

In the endcap, the final transverse radius of the TOF hit will always occur at a lower radius than what is anticipated for the straight track characterised by the initial $\eta$. In Fig. 9, the final hit is located below the ring representing the apparent $\eta$ of 2.0, at a final radius $r_T = 0.7648$ m, corresponding to an apparent $\eta = 2.09$.

It is clear that such differences vanish as $p_{\text{tot}}$ grows. The concrete consequence of this is that the actual acceptance in $\eta$ is in fact B-field and/or momentum-dependent<fnref n="3" />: such an acceptance is in fact narrower than the announced number, the latter being only the limit case for high-momentum track. For instance, in the CMS MTD case, for a track of $p_T = 0.661$ GeV/c, the maximum acceptance of the barrel BTL is in fact reduced from ($\eta = \pm 1.48$) down to ($\eta \approx \pm 1.12$), that is a difference of $\pm 0.36$ units in $\eta$. For a track of $p_T > 3.5$ GeV/c, the barrel acceptance of the CMS MTD becomes indeed close to $\pm 1.48$ in $\eta$. For the endcap, we can gain a bit of acceptance on the low $\eta$ side, since particle a priori expected outside the endcap $\eta$ may finally sneak in nearby the outer radius of the instrumented disk. At the inner ring, towards the maximum $\eta$, we lose again some $\eta$ acceptance. Note that, in the endcap case, the gains and losses in $\eta$ will be more moderate than in the central barrel region, and the $p_T$ needed to converge towards the limit acceptance is lowered: with growing $\eta$, $p_Z$ and, in turn, $p_{\text{tot}}$, are growing faster than at mid-rapidity. For instance, a track of $p_T = 0.661$ GeV/c and $\eta = 1.61$ will hit the endcap at $r_T = 1.0356$ m thus giving an apparent $\eta$ of 1.80, i.e. +0.2 $\eta$ units, a $p_T > 2.5$ GeV/c only (i.e. not 3.5 GeV/c as in the barrel case above) is enough to look like a straight track. For $p_T = 0.661$ GeV/c but $\eta = 3.00$, the particle escapes the endcap acceptance below the ETL inner radius by less than 3 mm, corresponding already to less than 0.01 unit of $\eta$.

II-E.iii Generic comment 2 : Helix versus straight-line approximations

The various figures discussed in the previous sub-section give a glimpse of the straight line approximations with respect to the helical computation. In the casual cases where particle per-

<footnote n="3">³ The $\eta$ acceptance is not species-dependent on the other hand : once $p_T$, $\eta$ and thus $p_Z$ are fixed, any particle species will follow the same helical path, but with a velocity which will then depend on $m_0$ so with a different timing.</footnote>
form less than half a turn before reaching the TOF detector, barrel or endcap, the difference in terms of covered distances often reduces to several centimeters over a flight distance necessarily ranging above the $R_{\text{TOF}}$ or $Z_{\text{TOF}}$ values, so above $O(1 \text{ m})$ usually. As a length difference, this can easily be perceived as something relatively negligible. In the TOF perspective, one can be all the more conforted into such an intuition that particles typically travel at highly relativistic velocities (e.g. for a proton of $p_T = 0.661 \text{ GeV/c}$, $\beta_{\text{tot}}$ ranges from 0.57 at $\eta = 0$ to 0.99 at $\eta = 3.0$), so a few cm more covered at a velocity close to speed of light should not make much difference in terms of time ($t = l/v$) at the end of the flight... However, TOF detectors that we are considering here, be they from ALICE, ATLAS or CMS, typically have a time resolution $\sigma_{\text{TOF}} O(10 - 10^2 \text{ ps})$. And an extra 5-cm distance covered at 1.0 c already leads to a time supplement of 166 ps, meaning already several $\sigma_{\text{TOF}}$ apart. The straight line approximations can thus easily falsify the TOF results and should be considered with caution, such approximations being most of the time everything but neutral in view of the timing resolution at stakes.

“Most of the time” ... Among the cases proposed by the illustrative figures, the reader may need to stop on the endcap cases (Figs. 7, 8, 9). There is indeed a peculiarity that must be underlined : in any endcap case, the approximation (a) is in fact not an approximation anymore: moving with $\beta_{\text{tot}}$ along the tangent at $t_0$ to the trajectory, further prolonged up to the TOF endcap, yields in fact the exact same length and time as for the associated helix path, as announced already in subsection II-D.iii. In the endcap figures, to be concrete, it means that the length of the straight azur dotted line and of the curled solid green line are the same. The fact that the path length towards the endcap is independent of $B_z$ (Eq. 41a) implies that “approximation” (a) is an alternative but equivalent way to determine the value of the curvilinear length $L$. The determination of $L$ boils down to the determination of the length covered by the trajectory followed by the particle under all the same initial conditions ($\eta, p_T, m_0, ...$) except the dummy condition of $B_z = 0$, so when the trajectory becomes naturally straight. The reason for this counter-intuitive fact is due to the magnetic force $\vec{f}_L(t)$ which does not apply any work along $\vec{e}_z$ direction ($\forall t, \vec{f}_L(t) \perp B_z \vec{e}_z$, so $\vec{f}_L(t) \cdot \vec{v}_z = 0$), so that the time to cover a certain $z$ distance should remain constant once the ratio $||\vec{p}_{\text{tot}}||/||\vec{p}_z||$, i.e. the inclination angle between the velocity components, is set.

### II-E.iv Generic comment 3 : looper particles spinning towards endcap

Looking at Figs. 7 and 8, the reader is driven to consider that the TOF endcap can become a low-$p_T$ TOF detector for mid-rapidity particles. In the case of small $p_T$, the particle may be unable to reach the TOF barrel layer, the Larmor radius being too small for that; it will instead start to loop towards the encap plane, driven by the $p_z$ momentum. Exploring this singular possibility requires the tracking algorithms of the experiment to be tailored to quest looper tracks. This tracking feature may come with 2 limits. i) In practice, low momentum particle are of course more vulnerable to multiple scattering, absorption, ... On the looping path towards endcap, the particle will multiply the chances to be scattered, especially by crossing many layers which are usually of increasing radiation length $x/X_0$ as one goes away from mid-rapidity towards forward or backward directions. ii) Provided that multiple scattering is not dooming the full idea, the detection is not granted under any circumstances a priori. Figure 8 illustrates a point of phase space ($p_T(t_0), \eta(t_0), \phi(t_0)$) for which the particle escapes finally the TOF endcap acceptance, arriving at the end of the flight into the inner part of the endcap plane which is not instrumented, next to the beam pipe, below the inner radius of the detector. Even in the ideal case exposed in this document, due to simple geometric considerations and the way the helix evolves, there will be some deadzones in phase space for detecting looper particles.
## II-F Acceptance thresholds : minimal $p_T$ and minimal inclination angle to detector surface

Two extra aspects of the problem that can still be explored analytically are related to acceptance matters, momentum thresholds and incidental angle to detector surface.

### II-F.i Minimal transverse momentum to reach TOF (barrel and endcap)

For Eq. 35a to be defined, the (arcsin) function needs to be defined, i.e. $R_{\text{TOF}}^{\text{barrel}}/2|\rho|$ needs to be in [-1;1], that means in the current situation, smaller than 1. This is obtained for $R_{\text{TOF}}^{\text{barrel}}/2 < |\rho|$. $R_{\text{TOF}}^{\text{barrel}}/2$ is a minimal radius of curvature which is necessary to touch the TOF barrel layer. Translated in terms of $p_T$, and using thus 22a, one gets :

$$ \left( \frac{R_{\text{TOF}}^{\text{barrel}}}{2} 0.299\,792\,458 |B_Z n| = p_{\text{Tmin}}^{\text{barrel}} \right) < p_T \quad (42) $$

(Note that the above formula is a sort of bare minimum to be required for the barrel case : when $p_T \to p_{Tmin}^{barrel}$, the particle performs almost a half-turn in the transverse plane and approaches tangentially the surface of the barrel cylinder.)

Similar considerations have to be taken into account for the endcap. There exists a lower radius for the endcap ring, near the beampipe, $R_{\text{TOF,min}}^{\text{endcap}}$. Such a radius can be obtained via the alleged $\eta_{max}$ acceptance of the endcap sub-detector (see II-E.ii).

$$ R_{\text{TOF,min}}^{\text{endcap}} = R(\eta_{\text{max}}^{\text{endcap}}) \quad (43) $$

$$ R_{\text{TOF,min}}^{\text{endcap}} = \frac{|Z_{\text{TOF}}^{\text{endcap}}|}{\sinh(\eta_{\text{TOF,max}}^{\text{endcap}})} \quad (43a) $$

On the same model as Eq. 42, one can then define for the endcap a minimal $p_T$ requested for regular particles flying towards the endcap as well as for looper particles :

$$ \left( \frac{R_{\text{TOF,min}}^{\text{endcap}}}{2} 0.299\,792\,458 |B_Z n| = p_{\text{Tmin}}^{\text{endcap}} \right) < p_T \quad (44) $$

**Highlight II.7. Minimum $p_T$ threshold necessary in endcap to reach TOF plane above the inner ring radius, $R_{\text{TOF,min}}^{\text{endcap}}$**

$$ \left( \frac{|Z_{\text{TOF}}^{\text{endcap}}|}{2 \sinh(\eta_{\text{TOF,max}}^{\text{endcap}})} 0.299\,792\,458 |B_Z n| = p_{\text{Tmin}}^{\text{endcap}} \right) < p_T \quad (44a) $$

In such an endcap case, there is no specific angle of approach to be anticipated towards the vertical plane of the TOF endcap. To encounter a tangential arrival, one needs to consider an almost infinite looper, with $\eta \approx 0^+$ or $0^-$ but $\neq 0$.

In order to anticipate the future discussion regarding the various experimental configurations that will be studied later in this document, Tab. 1 lists a few examples of $p_T$ threshold for different setups, depending on $B_Z$, $R_{\text{TOF}}^{\text{barrel}}$ for barrel TOF and $B_Z$, $\eta_{\text{TOF,max}}^{\text{endcap}}$ and $Z_{\text{TOF}}^{\text{endcap}}$ for endcap TOF.
<table><caption>**Table 1:** Minimal $p_T$ thresholds based on Eq. 42 and 44a obtained for particles of electric charge $q = n.e = \pm 1.e$ Coulomb, like $\pi^\pm$. Several TOF detector configurations are considered, among the ones found or possibly expected at the (HL-)LHC. The detector cases which are conjectured here without being endorsed officially by the concerned collaboration are notified with the label (hyp.), the sub-detector acronym is mentioned otherwise.</caption><thead><tr><th></th><th>Configuration label</th><th>|B<sub>Z</sub>|</th><th>sub-det.</th><th>⟨R<sup>barrel</sup><sub>TOF</sub>⟩</th><th>p<sup>barrel</sup><sub>Tmin</sub></th></tr></thead><tbody><tr><td rowspan="6">BARREL</td><td>ALICE-1 run II</td><td>0.5 T</td><td>(TOF)</td><td>3.80 m</td><td>0.285 GeV/c</td></tr><tr><td>ATLAS run IV+V</td><td>2.0 T</td><td>(hyp.)</td><td>1.00 m</td><td>0.300 GeV/c</td></tr><tr><td>ATLAS run IV+V</td><td>2.0 T</td><td>(hyp.)</td><td>0.29 m</td><td>0.087 GeV/c</td></tr><tr><td>CMS run IV</td><td>3.8 T</td><td>(BTL)</td><td>1.16 m</td><td>0.661 GeV/c</td></tr><tr><td>ALICE-3 run V</td><td>0.5 T</td><td>(hyp.)</td><td>1.00 m</td><td>0.075 GeV/c</td></tr><tr><td>ALICE-3 run V</td><td>0.5 T</td><td>(hyp.)</td><td>0.20 m</td><td>0.015 GeV/c</td></tr><tr><td rowspan="6">ENDCAP</td><td>ALICE-1 run II</td><td>0.5 T</td><td>-</td><td>Ø</td><td>-</td></tr><tr><td>ATLAS run IV+V</td><td>2.0 T</td><td>(HGTD)</td><td>4.0</td><td>3.45 m</td></tr><tr><td>ATLAS run IV+V</td><td>2.0 T</td><td>(hyp.)</td><td>2.1</td><td>1.50 m</td></tr><tr><td>CMS run IV</td><td>3.8 T</td><td>(ETL)</td><td>3.0</td><td>3.04 m</td></tr><tr><td>ALICE-3 run V</td><td>0.5 T</td><td>(hyp.)</td><td>4.0</td><td>2.00 m</td></tr></tbody></table>
### II-F.ii Inclination angle of the track with respect to the barrel detector surface

The track will arrive onto the detector surface with a certain inclination angle. It can be anticipated that a tangential approach will not give a decent detection. To cross properly the active thickness of the detector and avoid malfunctioning of the TOF response, it is likely that a minimal angle is required in practice.

In the case of the TOF barrel, the angle to focus on is sketched in Fig. 10. The goal is to determine the expression of the angle $\delta$ between $\vec{p}_{tot}(t_f)$ and the local plane of the TOF barrel nearby the TOF hit. Using the notations defined in the figure, we have:

$$ \sin(\delta) = \frac{AB}{BI} \qquad (45) $$

...

$$ \sin(\delta) = \frac{1}{\cosh(\eta)} \sqrt{1 - \left(\frac{R_{\text{TOF}}^{\text{barrel}}}{2 \rho}\right)^2} \qquad (45a) $$

$$ \sin(\delta) = \frac{1}{\cosh(\eta)} \sqrt{1 - \left(\frac{R_{\text{TOF}}^{\text{barrel}}}{2}\right)^2 \left(\frac{0.299\ 792\ 458 |B_z\ n|}{\|\vec{p}_T\|}\right)^2} \qquad (45b) $$

$\delta$, being then a function of $\eta$, $R_{\text{TOF}}^{\text{barrel}}$, $n$, $B_z$ and $p_T$

From Eq. 45b, we revert the logic and now express $p_T$ as a function of $(\delta, \eta, ...)$.

From Eq. 45b, we revert the logic and now express $p_T$ as a function of $(\delta, \eta, ...)$. Doing so, we arrive to the final constraint:

$$ \|\vec{p}_T\| > \left( p_{T\min}^{\text{barrel}}(\delta) = \underbrace{\frac{R_{\text{TOF}}^{\text{barrel}}}{2} 0.299\ 792\ 458 |B_z\ n|}_{\text{not } \eta} \frac{1}{\sqrt{1 - \sin^2(\delta) \cosh^2(\eta)}} \right) (46) \\ = p_{T\min}^{\text{barrel}}(\text{Eq. 42}) \qquad \underbrace{\eta} \qquad \underbrace{(k_\delta)^{-1}} \qquad (47) $$

So, imposing a minimal angle of approach to the TOF surface implies, in the barrel, a modulation of the bare minimum case of $p_{T\min}^{\text{barrel}}$ (Eq. 42, i.e. tangential approach) by the extra factor $(k_\delta)^{-1}$. Note that, for the factor $(k_\delta)^{-1}$ to be defined, there is a preliminary condition to be met, meaning that :

$$ [k_{\delta}^{2} = 1 - \sin^{2}(\delta) \cosh^{2}(\eta)] > 0 \qquad (47) $$

As an underlying constraint, this leads to define a maximum pseudorapidity value $\eta_\delta$ allowed for the particle to tackle the TOF surface with a large enough inclination angle.

$$ |\eta| < \left( \operatorname{arccosh} \left[ \frac{1}{\sin(\delta)} \right] \stackrel{\text{not}^o}{=} \eta_\delta \right) \qquad (47a) $$

Table 2 gives a set of example values to bear in mind for $\eta_\delta$, once the inclination angle to the TOF surface is limited to a range $[\delta; \pi/2]$.
<table><caption>**Table 2:** Values of $\eta_{\delta}$, the maximum allowed pseudo-rapidity value affordable to cross the TOF barrel surface with a tilt angle bigger than $\delta$.</caption><thead><tr><th>Min. δ angle</th><th>Max. η<sub>δ</sub></th></tr></thead><tbody><tr><td>45°</td><td>0.88</td></tr><tr><td>30°</td><td>1.32</td></tr><tr><td>25°</td><td>1.51</td></tr><tr><td>20°</td><td>1.74</td></tr><tr><td>15°</td><td>2.03</td></tr><tr><td>10°</td><td>2.44</td></tr></tbody></table>

558 So, provided that one has $|\eta| < \eta_{\delta}$,

$$ \text{then :} \quad \forall \delta \in [0; \frac{\pi}{2}], \quad k_{\delta} \in ]0; 1] $$

$$ \text{thus :} \quad (k_{\delta})^{-1} > 1 $$

562 meaning that $p_{T\min}^{\text{barrel}}(\delta)$ will always be lifted to higher $p_T$ values with respect to $p_{T\min}^{\text{barrel}}$, and that such an increase depends on $\eta$.

564 Table 3 further illustrates by how much the values of $p_{T\min}^{\text{barrel}}$ listed in Tab. 1 are raised depending on $\eta$ of the particle of interest. The non-linear evolution of the factor $(k_{\delta})^{-1}$ can be noted : it rises monotonously with $\eta$ slowly at first but then starts to increase exponentially as $\eta$ gets closer to $\eta_{\delta}$.

<table><caption>**Table 3:** Values of $(k_{\delta})^{-1}$, the extra factor that shift upwards the value of $p_{T\min}^{\text{barrel}}$ as one request a minimal inclination angle $\delta$ for the approach of the particle to the TOF barrel surface. Note that, for a retained value of $\delta$, this in turn restricts the range allowed for $|\eta|$ to $[0;\eta_{\delta}]$.</caption><thead><tr><th colspan="2">&eta; (&lt; &eta;<sub>&delta;</sub>)</th><th></th><th></th><th></th><th></th><th></th><th></th><th></th><th></th></tr><tr><th>&delta;</th><th>&eta;<sub>&delta;</sub></th><th>0.0</th><th>0.2</th><th>0.5</th><th>0.8</th><th>1.0</th><th>1.2</th><th>1.3</th><th>1.5</th><th>1.7</th></tr></thead><tbody><tr><td>30&deg;</td><td>1.32</td><td>1.155</td><td>1.162</td><td>1.211</td><td>1.345</td><td>1.572</td><td>2.355</td><td>5.885</td><td>&emptyset;</td><td>&emptyset;</td></tr><tr><td>25&deg;</td><td>1.51</td><td>1.103</td><td>1.108</td><td>1.137</td><td>1.212</td><td>1.319</td><td>1.553</td><td>1.807</td><td>9.275</td><td>&emptyset;</td></tr><tr><td>20&deg;</td><td>1.74</td><td>1.064</td><td>1.067</td><td>1.084</td><td>1.125</td><td>1.177</td><td>1.274</td><td>1.476</td><td>1.684</td><td>3.945</td></tr></tbody><tfoot><tr><td colspan="2"></td><td colspan="8" style="text-align: center;">(k<sub>&delta;</sub>)<sup>-1</sup></td></tr></tfoot></table>

## II-F.iii Inclination angle of the track with respect to the endcap detector surface

568 In the case of the TOF endcap, the angle to focus on is sketched in Fig. 12. Under such circumstances, the inclination angle is in fact defined from the very beginning, by the initial conditions. The inclination angle with respect to the TOF vertical plane corresponds directly to $\lambda$, the angle between $\vec{p}_{tot}$ and $\vec{p}_{T}^{4}$. Using the notations defined in the figure, we have:

$$ \tan(\lambda) = \frac{\|\vec{p}_z\|}{\|\vec{p}_T\|} = \frac{\|\vec{p}_T\| \sinh(\eta)}{\|\vec{p}_T\|} \quad (48) \\ \tan(\lambda) = \sinh(\eta) \quad (48a) $$

<footnote n="4">⁴ Note that such an angle is the $\pi/2$ complement to $\theta$, the angle usually bound to the pseudo-rapidity definition, the angle between $\vec{p}_z$ and $\vec{p}_{tot}$.</footnote>
<figure><img src="image_11.png" /></figure>

**Fig. 10:** Sketch of the various angles at stakes when considering the inclination angle of the trajectory to the detector surface of the TOF barrel. The angle that is necessary to be assessed is the angle $\delta$ in the figure, to be related to $\alpha(t_f)$, the transverse rotation and/or $\lambda$, the inclination angle between $\vec{p}_{tot}$ and $\vec{p}_T$.

<figure><img src="image_12.png" /></figure>

**Fig. 11:** Sketch of the unfolded hexix of the trajectory, letting appearing the angle $\lambda$, the inclination angle between $\vec{p}_{tot}$ and $\vec{p}_T$. The horizontal axis is the flattened length of the transverse projection to the B field [arc circle onto $(O;\vec{e}_X,\vec{e}_Y)$]; the vertical axis stands for the $z(t_f)$ component along $\vec{e}_Z$.

<figure><img src="image_13.png" /></figure>

**Fig. 12:** Sketch of the various angles at stakes when considering the inclination angle of the trajectory to the detector surface of the TOF endcap. The angle that is necessary to be assessed is simply $\lambda$, the inclination angle between $\vec{p}_{tot}$ and $\vec{p}_T$.
574 Meaning that if we want to set a minimal value for $\lambda$, one needs to require :

$$ \tan(\lambda) > \tan(\lambda_{\min}) \qquad (49) $$

576 giving :

**Highlight II.9.** Minimum $\eta$ threshold, $\eta_{min}^{\lambda}$, necessary to reach the TOF endcap plane under a certain inclination angle $\lambda$.

$$ \eta > \left( \arcsinh[\tan(\lambda_{min})] \stackrel{\text{not}^o}{=} \eta_{min}^{\lambda} \right) \qquad (49a) $$

578 There is a minimal pseudorapidity value to satisfy in order to reach the TOF end-plane under a sizeable angle. This is clearly not an issue a priori for particles expected directly in the endcap, because the incidental angle is large anyhow and converging towards $\pi/2$ with increasing $\eta$. For instance, for $\eta = +1.5$, $\lambda = 64.8^{\circ}$; for $\eta = +4.0$, $\lambda = 87.9^{\circ}$ (using Eq. 48a, $\lambda = \arctan[\sinh(\eta)]$). However, Eq. 49a has to be borne in mind for low-$p_T$ looper particles : tracks that do not reach the TOF barrel will not necessarily be properly detected in the endcap, even if they manage to survive and actually reach the vertical instrumented plane.

586 Table 4 gives a glimpse of what can be the actual minimum $\eta_{min}^{\lambda}$ induced by an explicit requirement on the authorised range for the tilt angle to endcap [$\lambda_{min}; \pi/2$].

<table><caption>**Table 4:** Values of $\eta_{min}^{\lambda}$ taken for various minimal angles requested when a particle is about to hit the TOF endcap plane.</caption><thead><tr><th>λ<sub>min</sub> angle</th><th>η<sub>min</sub><sup>λ</sup></th></tr></thead><tbody><tr><td>45°</td><td>0.881</td></tr><tr><td>30°</td><td>0.549</td></tr><tr><td>25°</td><td>0.451</td></tr><tr><td>20°</td><td>0.356</td></tr><tr><td>15°</td><td>0.265</td></tr><tr><td>10°</td><td>0.175</td></tr></tbody></table>

## III TOF separation power

### III-A General intent and different configurations

590 The goal is to determine the ability of TOF detectors, present or foreseen at the (HL-)LHC, to discriminate between particle species in terms of $n\sigma_{TOF}$ (Eq. 5a p.6), based on the analytical formulae derived in earlier sections. Such a separation power will be derived as 2D maps in phase space ($p_T$, $y$).

594 The code performing such an assessment of the TOF separation power under various experimental configurations is written in C++ within the ROOT framework. A code snippet is provided as an appendix (see App. A); it gives the core of the code, computing the actual times of flight along $p_T$ and $y$. The complete source code is further publicly released under github.com under the project name TOFseparationPowerAsFuncPtY [9].
### III-B Typical layout of the figures to come

In Figure 13, the content of the figures to come is detailed, this allows us to settle the type of information to expect. For a given experimental TOF setup ($B_Z$, $\sigma_{TOF}$, $R_{TOF}$, $Z_{TOF}$), corresponding typically to one of the LHC experiment configurations, (ALICE-1, ALICE-3, ATLAS, CMS), 8 sub-figures forecast the formula-based separation among identified species : $e^{\pm}$, $\mu^{\pm}$, $\pi^{\pm}$, $K^{\pm}$, $p^{\pm}$, $d^{\pm}$, $t^{\pm}$, $^{3}\text{He}^{2\pm}$, $^{4}\text{He}^{2\pm}$. The separation power is addressed with 2-by-2 strategy among “neighbouring” species, i.e. one given species and the closest particle(s) in mass at lower and/or higher $m_0$ in the previous list.

<figure><img src="image_14.png" /><figcaption>**Fig. 13:** Template layout of the typical sub-figures to be shown for each configuration tested per experiment, based on the set of working hypotheses drawn : detector timing resolution ($\sigma_{TOF}$), magnetic solenoidal field ($B_Z$), radial position of central barrel TOF ($R_{TOF}$), longitudinal position of the endcap TOF ($Z_{TOF}$) and the respective $\eta$ coverages in barrel and endcap.</figcaption></figure>

## IV Using ALICE-1 in LHC runs I and II as a benchmark exercise

At the LHC, there is already an operational TOF detector in a central rapidity region, the one from the ALICE experiment. It is a sub-detector based on Multi-Resistive Plate Chamber (MRPC), located at $\langle R_{TOF} \rangle = 3.80$ m, covering up to $|\eta| = 0.88$, and it is a sub-detector for which real performances are already known, measured and monitored. In the following, we focus on the ALICE-1 configuration<fnref n="5" />, i.e. the setup running during LHC runs I (2009-2013) and II (2015-2018). Such a concrete example can be used to assess the realm of the proposed exercise : by letting the reader know to what extent the analytical formulae at use here do match the performances met under real conditions, one can better grasp the limits of the forecast provided in later sections; there, the point will be to provide analytically-derived projections for future detector configurations at the LHC that are currently developed or simply considered (...) but in any case not yet installed and operated.

<footnote n="5">⁵ The same detector will still be in place in ALICE-2, after the LHC Long Shutdown 2 [2019-2021], for the LHC runs III and IV (≥ 2022).</footnote>
<figure><img src="image_15.png" /><figcaption>**Fig. 14:** Expected TOF performances in terms of particle separations as a function of $p_T$ and $y$ for the configuration already foreseen or possibly anticipated for the ALICE-1 experiment, with hypotheses : i) $\langle r_{\text{barrel}}^{\text{barrel}} \rangle = 3.80$ m, ii) $\langle z_{\text{TOF}}^{\text{endcap}} \rangle = -0.5$ m, iii) TOF timing resolution = 80 ps, iv) $B_{\text{field}} = -0.5$ T.</figcaption></figure>
<figure><img src="image_16.png" /><figcaption>**Fig. 15:** Expected TOF performances in terms of particle separations as a function of $p_T$ and $y$ for the configuration already foreseen or possibly anticipated for the ALICE-1 experiment, with hypotheses : i) $\langle r_{\text{barrel}}^{\text{barrel}} \rangle = 3.80$ m, ii) $\langle z_{\text{TOF}}^{\text{endcap}} \rangle = -0.5$ m, iii) TOF timing resolution = 56 ps, iv) $B_{\text{field}} = -0.5$ T.</figcaption></figure>
<figure><img src="image_17.png" /><figcaption>**Fig. 16:** Expected TOF performances in terms of particle separations as a function of $p_T$ and $y$ for the configuration already foreseen or possibly anticipated for the ALICE-1 experiment, with hypotheses : i) $\langle r_{\text{barrel}}^{\text{barrel}} \rangle = 3.80$ m, ii) $\langle z_{\text{TOF}}^{\text{endcap}} \rangle = -0.2$ m, iii) TOF timing resolution = 56 ps, iv) $B_{\text{field}} = -0.2$ T.</figcaption></figure>
## IV-A Analytical-formula-based performances of the ALICE-1 TOF

622 To that end, three sets of figures, making use of the aforementioned code and formulae, have been derived in the ALICE-1 case, corresponding to:

624 Figure 14: ALICE run I configuration, with timing resolution set to 80 ps, with a L3 solenoidal field of 0.5 T, valid in Pb-Pb or pp,

626 Figure 15: ALICE run II configuration, with improved timing resolution of the order of 56 ps, with a L3 solenoidal field of 0.5 T, achieved in run II Pb-Pb at least,

628 Figure 16: ALICE run II “low field” configuration, with timing resolution of the order of 56 ps, with a L3 solenoidal field of 0.2 T, achieved in run II Pb-Pb and Xe-Xe at least.

## IV-B Measured performances of the ALICE-1 TOF sub-detector

632 The previous figures have to be compared with the concrete performances of the TOF detector in real conditions.

634 For ALICE TOF in Run I, generic performances can be found in the reference [10] and in the dedicated section 7.3 in the publication about the ALICE Run-I general performances [11]. For the same detector in Run II, the improved performances (in Pb-Pb notably) are summarised in the proceedings [12].

636 To illustrate the detector achievements, several ALICE figures are brought to the reader's attention in the following.

640 Figure 17: the figure illustrates the $\beta$ distributions a function of total momentum $p$. The individual sub-figures are obtained with LHC run II data, for 4 different collision systems (pp, p-Pb, Xe-Xe, Pb-Pb) at TeV-scale collisions energies and for 2 magnetic field configurations of the L3 solenoid (0.5 T and 0.2 T). There on any graph, several bands densely populated with counts emerge : $e^\pm$ appear at very low momenta at $\beta_{tot} \approx 1$, followed with clear constrained bands of $\pi^\pm$, $K^\pm$, $p^\pm$ and to a lesser extent, $d^\pm$.

644 Figure 18: TPC-TOF matching efficiency (here in A-A) i.e. ability of matching a track prolongation from TPC as detector placed at lower radius for (a) charged particles $h^\pm$ in $B = 0.5$ T and (b) positively-charged identified hadrons in $B = 0.2$ T.

648 Figure 19 and Figure 20: the summary plots indicate the mid-rapidity separation power in terms of $n.\sigma_{PID}$ for 3 stable hadrons ($\pi^\pm/K^\pm$ and $K^\pm/p^\pm$). (Note especially in the lower panels the $p_T$ range covered by the ALICE TOF for a > 2. $\sigma_{PID}$ separation). The first set of graphs is taken from [11] (Fig. 46) and is obtained considering Pb-Pb detector conditions in LHC run I. The second one from [13] (Fig. 1), for Pb-Pb and pp detector conditions in LHC run II.

654 Figure 21: Resolution on the event starting time, $t_0$, assessed with the TOF itself (a) for low TOF-hit multiplicities (pp, p-Pb, peripheral A-A) and (b) for high TOF-hit multiplicities (A-A).
<figure><img src="image_18.png" /><figcaption>**Fig. 17:** Measured $\beta$ distributions as a function of total momentum $p$, obtained with the ALICE TOF detector in data recorded in run II for various collision systems :</figcaption></figure>

(a) pp, $\sqrt{s} = 13$ TeV [$B = 0.5$ T], (b) p-Pb, $\sqrt{s_{NN}} = 8.16$ TeV [$B = 0.5$ T] and

(c) Xe-Xe, $\sqrt{s_{NN}} = 5.44$ TeV [$B = 0.2$ T], (d) Pb-Pb, $\sqrt{s_{NN}} = 5.02$ TeV [$B = 0.5$ T].

<figure><img src="image_19.png" /><figcaption>**Fig. 18:** Efficiency in TPC-TOF track matching (here for A–A) in ALICE-1, i.e. ability of matching a track prolongation out of TPC towards TOF, the TPC being the detector placed at lower radius for (a) charged particles $h^{\pm}$ in $B = 0.5$ T and (b) positively-charged identified particles in $B = 0.2$ T.</figcaption></figure>
<figure><img src="image_20.png" /><figcaption>**Fig. 19:** Separation power for particle identification averaged over $|η| < 0.5$ for $π^{±}/K^{±}$ and $K^{±}/p^{±}$ with the dedicated ALICE sub-detectors under run-I conditions for Pb-Pb with the collision energy $\sqrt{s_{NN}} = 2.76$ TeV. The lower panels represent the range over which the different sub-detectors have a separation power of more than $2σ$ (Fig. 46 in [11]).</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>**Fig. 20:** Separation power for particle identification averaged over $|η| < 0.5$ for $π^{±}/K^{±}$ and $K^{±}/p^{±}$ with the dedicated ALICE sub-detectors under run-II conditions for pp and Pb-Pb with $\sqrt{s_{NN}} = 5.02$ TeV. The lower panels represent the range over which the different sub-detectors have a separation power of more than $2σ$ (Fig. 1 in [13]).</figcaption></figure>
<figure><img src="image_22.png" /><figcaption>**Fig. 21:** Resolution on the event starting time, $t_0$, determined with the TOF sub-detector itself (a) for low TOF-hit multiplicities (pp, p-Pb, peripheral A–A), updated by the ALICE collaboration from [14] and (b) for high TOF-hit multiplicities (A–A).</figcaption></figure>

## IV-C Comparing and discussing analytics and real performances

Such figures call for dedicated comments, that would pinpoint several specific aspects of the confrontation between analytical formulae and real performances.

### IV-C.i Acceptance × efficiency to reach TOF

As pointed by Tab. 1, the $p_{T\min}^{\text{barrel}}$ of mid-rapidity acceptance in case of ALICE-1 is on paper equal to 0.285 GeV/c. With a tangential approach to the TOF layer, tracks with this $p_T$ are essentially at the edge of the tracking efficiency ($A.\epsilon O(1%)$), illustrated by Fig. 18 (b). It stands for the bare minimum minimorum, that is moreover usually achieved with $\pi^\pm$ only. With such a value of $p_T$, one is not yet in a sort of comfort zone around a desirable functioning point ($A.\epsilon \approx 50\%$).

Still in the same figure, the reader may notice, for $\pi^\pm$, $K^\pm$ and $p^\pm$ cases, that the tracking of a particle up to the TOF sub-detector is a particle-species dependent feature, that, in addition, changes from one experiment to the next. For a given experiment, $A.\epsilon$ becomes independent of the particle species only above a certain momentum; for instance, here in ALICE-1, $p_T$ has to be above 2.5 GeV/c.

### IV-C.ii Validation with the measured separation among ($\pi^\pm$, $K^\pm$, $p^\pm$) species

Figures 19 and 20 give a summary of the separation among the most abundantly produced hadrons. The separation is derived from concrete analyses of primary-like particles and concerns inclusive production of identified particles. The $p_T$ ranges accessible with the help of TOF are further summarised for different LHC runs, systems and collision energies in Tab. 5. Note that the measurements are:

(i) all performed under a field $|B_z| = 0.5$ T,

(ii) averaged over a certain extent in $|\eta| (< 0.5)$,

(iii) dependent on the signal extraction methods that can subtly differ from one analysis to the next, improving as experience grows.

Considering the values listed in the table, it should be noted that the lower identification threshold values as well as the upper cut-off values are not universal but vary a little with LHC periods, collision systems, $\sqrt{s_{NN}}$ and thus, with running conditions and detector stress. All in all, comparing the table values to 4-$\sigma_{TOF}$ and, more especially, to 3-$\sigma_{TOF}$ of the “analytical” figures
<table><caption>**Table 5:** ALICE-1 TOF measurements of identified hadrons, $\pi^{\pm}$, $K^{\pm}$, $p^{\pm}$, in several analyses of LHC run-I and -II data.</caption><thead><tr><th rowspan="2">LHC run</th><th rowspan="2">System</th><th rowspan="2">√s<sub>NN</sub></th><th colspan="3">p<sub>T</sub> range of measurements (GeV/c)</th><th rowspan="2">Ref.</th></tr><tr><th>π<sup>±</sup></th><th>K<sup>±</sup></th><th>p<sup>±</sup></th></tr></thead><tbody><tr><td>run I</td><td>Pb-Pb</td><td>2.76 TeV</td><td>0.50-3.00</td><td>0.45-3.00</td><td>0.50-4.60</td><td>[15]</td></tr><tr><td>run I</td><td>pp</td><td>7.00 TeV</td><td>0.50-2.50</td><td>0.50-2.40</td><td>0.80-4.00</td><td>[16]</td></tr><tr><td>run II</td><td>Pb-Pb</td><td>5.02 TeV</td><td>0.50-3.50</td><td>0.60-3.50</td><td>0.80-4.50</td><td>[13]</td></tr><tr><td>run II</td><td>pp</td><td>5.02 TeV</td><td>0.50-3.50</td><td>0.65-3.50</td><td>0.80-4.50</td><td>[13]</td></tr><tr><td>run II</td><td>pp</td><td>13.00 TeV</td><td>0.70-4.00</td><td>0.60-3.00</td><td>0.90-4.00</td><td>[17]</td></tr></tbody></table>

(Figs. 14 and 15, sub-figures (c) and (d) related to $\pi^{\pm}$-$K^{\pm}$ and $K^{\pm}$-$p^{\pm}$ separations), one can state that:

(1) the lower threshold is, in the analytic projections, always 0.33 GeV/c, independently of the species. Such a value is never met in real data, the TOF measurements systematically start at higher values, $\approx +0.2$ GeV/c being at least necessary for charged pions, rather +0.3 GeV/c required for charged kaons, and up to +0.5 GeV/c for protons.

(2) the upper momentum bounds of separation are, for each particle species, in a similar ballpark (i.e. within 0.5 GeV/c, as is the spread among the various data analyses), showing a convergence of the analytic computation with measured performances. In turn, it reveals that, on the high momentum front, the location of the TOF layer $R_{\text{TOF}}^{\text{barrel}}$ and its intrinsic time resolution $\sigma_{\text{TOF}}$ become the two key parameters driving the TOF performances.

### IV-C.iii Impact of multiplicity on the overall TOF performance, the start time

As detailed in [14] and mentioned in [12], the intrinsic resolution of the TOF sub-detector, which has been considered up to here, $\sigma_{\text{TOF}}$, is not the only component entering the final timing resolution at stakes for particle identification. The total timing resolution $\sigma_{\text{timing}}^{\text{tot}}$ depends on:

$$ (\sigma_{\text{timing}}^{\text{tot}})^2 = \sigma_{\text{TOF}}^2 + \sigma_{\text{tracking}}^2 + \sigma_{\text{evt-t}_0}^2 \quad (50) $$

* anything related to the assessment of actual length $\mathcal{L}$ of the track can be wrapped in the uncertainty associated to tracking, $\sigma_{\text{tracking}}$

* The event timing, $t_0$, can have a substantial impact on the total timing uncertainty. In ALICE-1, the event collision time can be provided, in a self-consistent way, by the TOF sub-detector itself and/or by another sub-detector T0. In both options, the quality of the information is weighed event-by-event based on the corresponding charged particle acceptance: on the one hand, the ALICE-1 TOF is covering $\approx 2 \times 0.88$ units of pseudo-rapidity, but is 3.80 m away from the interaction point, on the other hand, T0 is a two-side set of Cerenkov counters (T0A and T0C) at forward and backward rapidities, one being rather far away from the interaction point ($z_{T0A} = +374$ cm), the other rather close ($z_{T0C} = -70$ cm), but in both sides with a rather limited $\eta$ coverage, ($+4.61 \le \eta_{T0A} \le +4.92$) and ($-3.28 \le \eta_{T0C} \le -2.97$). Focussing on the sole TOF detector and the standalone timing strategy, one can gauge (Fig. 21) the accuracy of the TOF-based $\sigma_{\text{evt-t}_0}$ as a function of the TOF hit multiplicity. While the contribution in semi-central and central Pb-Pb is contained with respect to $\sigma_{\text{TOF}}$ [$\sigma_{\text{evt-t}_0} < 5-10$ ps, sub-Fig. (b)], the situation at low multiplicity [pp, p-Pb and peripheral Pb-Pb in sub-Fig. (a)] indicates an effect that is of the same order as $\sigma_{\text{TOF}}$: for a multiplicity below 30 charged particles reaching the TOF layers, $\sigma_{\text{evt-t}_0}$ varies between [30;90] ps. Such an extra uncertainty can only degrade in a
sizeable manner the timing accuracy of the measurements via a larger $\sigma_{\text{timing}}^{\text{tot}}$, thus reducing ineluctably the upper $p_T$ reach of any TOF measurement. In details, if it is certainly well-founded that such an uncertainty can be essentially neglected for the most of A–A cross-section, it will be on the contrary unavoidable for small systems like pp or p–Pb Minimum Bias.

IV-C.iv *Light nuclei, illustration with the deuteron identification*

In Fig. 17, the reader can guess there the appearance of the deuteron band, typically above 1 GeV/c, for any collision system presented here (pp, p–Pb, Xe–Xe and Pb–Pb). That effective threshold value looks substantially larger than the value $p_{\text{min}}^{\text{barrel}} = 0.285$ GeV/c, the value holding a priori for any $\pm 1e$ charged particle, as pointed by Tab. 1. What can be noticed here on the graph is in line with various ALICE publications, where the TOF parts of the deuteron measurements indeed starts only for $p_T$ above 1 GeV/c in pp ([18], [19] and [20]), and 1.4 GeV/c in Pb–Pb [21].

The reason for this is that the propagation of a light nucleus up to the TOF layers may encounter various issues.

First, the path itself to TOF detector is modified by the increased probability of interactions with detector material (tracking layers and their services), meaning that, compared to lower-mass/lower-charge particles, the nucleus itself can be more easily lost on the way to TOF. Secondly, even if the heavy particle manages to hit the TOF layer, the nucleus signal can be drowned by the existing background. The contaminants located within the same expected zone in $(\beta_{\text{tot}}, p_{\text{tot}})$ phase space can be made of any species : stable hadrons such as $\pi^{\pm}$, $K^{\pm}$ or $p^{\pm}$, as well as light nuclei, deuteron themselves (!) and other light nuclei.

Such a signal pollution can be due to :

(1) TOF hit mismatches where the tracking fails, among the combinatorics of hits, to connect TOF to tracking sub-detectors of lower radii (TPC and/or, TRD in the ALICE-1 case);

(2) secondary particles emitted within the experimental volume, as secondary products from weak decays or as genuine products of inelastic interactions with detector material<fnref n="6" />. For a fixed momentum, secondaries will have in turn a shorter $\mathcal{L}$ and duration of flight, thus luring the TOF detector with wrong time assignment.

(3) the tails of the $(\beta_{\text{tot}}, p_{\text{tot}})$ distributions of other “regular” primary particles.

The background problem is essentially a question of relative frequency of occurrence between the background cases and the natural production rate of the light nucleus of interest in the given momentum range. The production of primary deuteron, for instance, is a rather rare process ($dN(d^{\pm})/dy|_{y \approx 0} \approx 2-4.10^{-4}$ in pp [18]), such a signal can easily be overwhelmed by the misidentification or ill-identification, admittedly rare, of particles that are far more copiously produced like $\pi^{\pm}$, $K^{\pm}$ or $p^{\pm}$ (e.g. at $y \approx 0$, in pp [16], the integrated yields $dN(\pi^{\pm})/dy \approx 18 dN(p^{\pm})/dy$, $dN(K^{\pm})/dy \approx 2.3 dN(p^{\pm})/dy$ and $dN(p^{\pm})/dy = 0.25$). For low-momentum nuclei below 1 GeV/c, in the ALICE-1 case, it is in fact the unique capabilities of lineic energy loss $dE/dx$ in the TPC that allow clean identification (up to 159 $dE/dx$ sampling). The case will have to be carefully considered in the future LHC configuration where no TPC is foreseen and, if $dE/dx$ measurements may exist, they will be limited along each track to a few hits ($\mathcal{O}(5)$) in the silicon layers instrumented with analogous readout (typically micro-strips).

<footnote n="6">⁶ Considering knock-out nuclei, such a phenomenon is essentially absent for anti-nucleus production but is important for nucleus one at low momenta, e.g., in pp 7 TeV or 13 TeV, knock-out deuterons typically stand for 40% of the raw deuteron signal at $p_T = 0.6$ GeV/c and decreases exponentially with momentum, down to less than 5% above 1.4 GeV/c [19, 20].</footnote>
At the other side of the momentum spectrum, towards high $p_{\text{tot}}$, the TOF identification proves to be very efficient and clean. The slow motion of the heavy objects that the light nuclei are at last ($m_0 \approx A.m(p) > 2.m(p) \approx 1.87 \text{ GeV/c}^2$) allows for an extended momentum reach at high momentum : for a given $p_{\text{tot}}$, the light nuclei will have a clearly separated and lower $\beta_{\text{tot}}$ compared to primary stable hadrons like $\pi^{\pm}$, K$^{\pm}$ or p$^{\pm}$.

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In Fig. 14(e) (80-ps), the maximum $p_T$ possible for deuteron identification is anticipated to be between 7.17 GeV/c ($\eta=0$) and 6.05 GeV/c ($\eta=0.88$) for a $4-\sigma_{\text{TOF}}$ separation, it slightly overshoot the casual upper bound met in the measurements with the corresponding ALICE-1 set-up : 6 GeV/c achieved recently in a Pb-Pb analysis (Fig. 2 in [21]). However, in pp, the situation is dimmed, with an upper limit reduced typically to 3 GeV/c [18, 19] or 3.8 GeV/c [20]. – Why is that ? The impact of the start time resolution $\sigma_{\text{evt}-t_0}$ (see previous sub-section IV-C.iii) clearly becomes relevant but cannot explain solely the situation : with a time resolution increasing from 80 ps to 120 ps, the analytical formulae developed here predicts a reduction of the $p_T$ reach further down, but only to 5.83 GeV/c ($\eta=0$) and 4.91 GeV/c ($\eta=0.88$) for a $4-\sigma_{\text{TOF}}$ separation. Very likely, the concrete situation is also a consequence of the impact of contaminants in view of the natural production rate of deuteron. At such $p_T$ above 3 GeV/c, despite the already high integrated luminosities recorded with Minimum Bias triggers$^7$, the deuteron production is certainly still scarce, a penalty factor being in place in small systems compared to Pb-Pb.

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For nuclei, it should be finally noted that using time-of-flight measurements as identification strategy comes with one clear drawback, that is, the separation among nuclei with the same mass number A. Time-of-flight principles make it indeed difficult to distinguish between species very close in $m_0$. This is especially the case for A = 3, between t and $^3$He. There essentially, the TOF difference can only be an indirect consequence of the curvature due to different electric charges : the helium nucleus will travel at very similar $\beta_{\text{tot}}$ as triton, but due to $Z(t) = 1$ and $Z(^3\text{He}) = 2$, the helium will fly on a trajectory that is more curved [$\rho(^3\text{He})$, half of $\rho(t)$] and so, of longer path length for a same [$p_T(t_0)$, $\eta(t_0)$]. Beyond the separation of primary t and $^3$He, such light nuclei enter frequently the decay chain of hypernuclei. For instance, $^3$He plays a pivotal role in the reconstruction of a hypernucleus like hypertriton $^{3}_{\Lambda}\text{H}$ ($^{3}_{\Lambda}\text{H}^+ \rightarrow {}^3\text{He}^{2+}\pi^-$ with B.R. $\approx 25\%$) [22, 23]; a blunted separation power among t and $^3$He can clearly endanger the hyperphysics case in practice.

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<footnote>$^7$950.10$^{-6}$ MB pp events have been recorded from years 2016 and 2017 for [20], corresponding to about 12 pb$^{-1}$ of integrated luminosity for inelastic pp cross-section.</footnote>
794 **V Synopsis of the figures and corresponding TOF configurations**
**tested**

796 In the following, several TOF configurations for LHC detectors will be exercised as announced earlier. Each exercise takes into account most of the problem aspects that have been pointed out and discussed in previous sections. The list of what has been retained or left aside is given in the overview Tab. 6.
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800 The table 7 further gives the complete list of the experimental configurations tested in view of TOF performances, together with key parameters that make the differences from one configuration to the next.
802

804 To smooth the peregrinations of the reader through the various figures still to come (about 31 experimental configurations × 8 two-by-two TOF separations = 248 sub-figures), two figures have been isolated and annotated. This is meant to explicit the generic comments applicable to any of the results and, this way, to provide a reading template in order to tackle each figure with key items in mind. Both figures give the outcome of the K$^{±}$/p$^{±}$ separation in a CMS-like TOF configuration with a standard (-3.8 T) magnetic field, having barrel (red and pink shades) and endcap (orange and ochre shades) 2D maps based upon the timing resolution of 30 ps anticipated for HL-LHC run IV. The |η| acceptance in barrel is set to ± 1.50 and the range for endcap is defined between [1.60; 3.00].
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Figure 22 is an illustration based on major hypotheses that are mostly artificial, i.e. in order to
let the reader become aware of the manifold corners of potential phase space, the figure has
been derived with :

(i) very loose requests on the inclination angles to barrel and endcap surfaces, $\delta > 10°$ and $\lambda > 5°$ respectively;

(ii) the assumption that any detector layer remains sensitive while fully permeable as well to charged particles. In particular, the map is obtained with the idea that electromagnetic calorimeter in the barrel (ECal), usually present next to the last tracking/TOF layer, neither stop nor disturb the particles but simply let them pass, unaltered, beyond the calorimeter lower radius.

824 Figure 23 corresponds to a more realistic case, where both $\delta$ and $\lambda$ must exceed 30° and where the barrel ECal location signs definitively the end of any track hitting it; even at the beginning of its $\mathcal{A} \cdot \epsilon$, the material budget (radiation length and nuclear interaction length) of the calorimeter is enough to destroy the ideal trajectory and the set of hypotheses discussed so far in the current study.

830 **Labels** $P_b$ and $P_e$ : due to the curvature of trajectory implied by the magnetic field, any particle must have a minimal $p_T$ to be able to touch the detector surface (the curvature radius needs to be large enough), and this is required for both barrel and endcap TOF ($p_{T\min}^{\text{barrel}}$ from Eq. 42, $p_{T\min}^{\text{endcap}}$ from Eq. 44a). Such values just depend on the magnetic field $B_Z$ and the electric charge $q$ and is not dependent on $m_0$, i.e. the numerical $p_T$ values are shared among species of same charge.
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**Labels** $A_e^+$, $A_e^-$ and $A_b^-$: the zenith angle acceptance of a given detector is not an area that can be framed regularly by fixed edges given in $\eta$. In other words, in both figures, the pink solid line at 1.50 does not limit properly the barrel acceptance, the two orange solid lines at $\eta$ at 1.60 and 3.00 do not circumscribe correctly the endcap acceptance.
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<figure><img src="image_23.png" /><figcaption>**Fig. 22:** Typical figure content with key elements highlighted, illustrated here with the K$^{±}$/p$^{±}$ TOF separation in a CMS-like case. In the present illustration, the requested constraints on the inclination angles in barrel (δ) and endcap (λ) are relaxed, to 10° and 5° respectively. The phase space is artificially mapped further as if any detector layer remains active but fully transparent, i.e. it means in particular that the map is derived as if ECal did not disturb or stop the particles but let them fly unaltered beyond the calorimeter lower radius. See text for explanation of the different labels.</figcaption></figure>
<figure><img src="image_24.png" /><figcaption>**Fig. 23:** Typical figure content with key elements highlighted, illustrated here with the K$^{±}$/p$^{±}$ TOF separation in a CMS-like case. In the present illustration, the requested constraints on the inclination angles in barrel (δ) and endcap (λ) are set in both cases to 30°, a more realistic value. If the tracking and TOF layer are still considered as without effect on particles, the ECal is here envisaged as the term of the helical trajectory. See text for explanation of the different labels.</figcaption></figure>
As explained in sub-section II-E.ii, the actual acceptances do depend on ($p_T$, $y$, $B_Z$). Figure 22 gives a concrete illustration of the effect, where one can see the losses (red dashed broken lines) and gains (red solid broken line) in phase space, for barrel ($A_b^-$) and endcap ($A_e^+$ and $A_e^-$). Given the experimental layout chosen here, it is even up to the level where there are particles naively foreseen in barrel that in fact end up first and foremost into the endcap (e.g. primary protons of $p_T = 0.7$ GeV/c, $\eta = 1.45$).

**Label L**i: a peculiar case that to scrutinize concerns looper particles that finish their trajectory into the endcap ring. They can be defined as particles that perform at least one completed half-turn before reaching the TOF vertical plane. They are represented by the area framed in violet on Fig. 22 and Fig. 23. Their possible phase space is delimited by:

**south**: the TOF endcap comes necessarily with a lowest radius, below which the plane is not equipped anymore (beam pipe, services, ...). Such a minimal radius implies a minimum transverse momentum $p_{T\min}^{\text{endcap}}$ (Eq. 44a) for particles to have any chance to reach the active zone the TOF endcap ring.

**west**: an imposed minimal inclination angle to the endcap surface, translated in terms of minimal $\eta_{\min}^{\lambda}$, as is seen with the blue exclusion line labelled $I_e$

**north**: on the upper edge of the domain, two $p_T$ quantities can border the looper acceptance:

* $p_{T\min}^{\text{barrel}}$ that favours the first detection into the TOF barrel,

* $p_{T\min}^{\text{ECal}}$, the $p_T$ acceptance to enter the calorimeter.

Among the two thresholds, if looping up to $p_{T\min}^{\text{barrel}}$ only would be the most spontaneous option, it turns out that loops can decently happen up to the higher value of $p_{T\min}^{\text{ECal}}$. Such a quantity is defined by the $B_Z$ field and the radial position of the barrel calorimeter, in the spirit of Eq. 42 defining $p_{T\min}^{\text{barrel}}$. In practice, setting the northern limit to $p_{T\min}^{\text{ECal}}$ means that one allows charged particles to cross the TOF barrel layer and continue their courses as long as they can, potentially up to the TOF endcap, implying a double acceptance, in both barrel and endcap, for some regions of phase space ($p_T$, $y$). Such a double detection is apparently forbidden concretely in the specific setup studied in Fig. 23, but can be there potentially, behind the scene, as exposed in Fig. 22 where we see overlap regions between the red (barrel) and orange (endcap) areas. We will see later in this document experimental configurations where such barrel+endcap detections are present and may be more than the phoney case shown here for illustration. In a word, this will be important for cases where the barrel TOF layer is possibly at a much lower radius than ECal, stationed among the concentric layers of the tracker...

In terms of time separation, looper trajectories are generally from low-$p_T$ particles shaping long track, this makes the identification clean ($> 5\sigma_{\text{TOF}}$ separation). The decisive issues are certainly not on the TOF timing itself but much more on the acceptance and efficiency aspects.

The tracking efficiency up to TOF plane is a prime issue, because with more and more half-turns, it is a longer and longer path to be carried out in the middle of the tracker material and thus, with more and more chances to loose energy, being scattered, etc. This makes the TOF detection less and less probable as the number of complete rotations grows (i.e. as the particle starts back with $y \to 0$). In real experiment, it is likely that the western limit of the looper domain will, rather than be defined by $\eta_{\min}^{\lambda}$ usually taking values in a mid-rapidity scope, be set by a progressive degradation in efficiency, becoming irretrievable already with some still forward rapidities.
**east**: the looper area is in fact the set of sub-areas, made by populations of tracks having accomplished different numbers of half-turns. This is marked with violet numbers in the figures (0,1,2,3,4,5, ... half-turns). The limit between two consecutive half-turns is displayed with white solid lines. The first curled line at the largest rapidities stands for the first frontier, between 0 and 1 half-turn. In the “0” region, tracks stay in the same quadrant as the one they have been originating from (the azimuth difference [$\varphi(t_f) - \varphi(t_0)$] $\in [0; \pi/2]$), in the region “1”, they now rotate by more than 180°. For instance, the trajectory displayed in Fig. 9 with a rotation of 84° populates the region “0” (floor of the division 84./180.), while Fig. 7 with a rotation of 623° enters the region “3” (floor of 623/180), having done more than 1.5 full turns.

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**Labels $D_e(R_{min})$ and $D_e(R_{max})$**: coming back to acceptance considerations for loopers into endcap, one should bear in mind that the looper area is not a continuous region but is in fact split into several pieces, each being associated to a certain number of half-turns and further separated by acceptance deadzones. To that respect, the fact that the endcap is not an infinite plane but a finite ring, with $R_{min}$ and $R_{max}$, is the reason for such a twisted acceptance.

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On the low $p_T$ side, the $R_{min}$ value is responsible of the acceptance losses, the large white gaps in the shape of an orca dorsal fin, appearing between region “1” and “2”, “3” and “4”, visible on both Figs. 22 and 23. The trajectory visualised in Fig. 8, escaping surprisingly the TOF endcap acceptance by the low radius, is an example of trajectory that counts among the ones populated the first large white deadzone, between the orange regions “1” and “2”.

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On the high $p_T$ front, conversely, the acceptance irregularities stem from the outmost limit of the endcap ring with $R_{max}$; depending on its value, the dip (only visible in Fig. 22) separating region “2” and “3”, “4” and “5”, etc can be more or less pronounced: with a larger radius, the acceptance losses are reduced, the accessible phase space being pushed to higher $p_T$ values.

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**Labels $I_{b1}$ and $I_{b2}$**: Requiring a minimal inclination angle $\delta$ to tackle the barrel TOF surface translates into 2 conditions to satisfy, on $\eta$ and $p_T$. The upper bound $\eta_\delta$ in pseudo-rapidity defines the vertical asymptote to the curve that uplifts the $p_{T\min}^{\text{barrel}}$ as a function of $\eta$ via the factor $(k_\delta)^{-1}$ (Eq. 46). The trend previously exposed with numbers in Tab. 3 can be visualised in the figures, the slow increase at first followed by an exponential rise near the asymptote. Note that, for inclination angles $\delta = 30^\circ$ or $25^\circ$, such a modulation of $p_{T\min}^{\text{barrel}}$ becomes the most stringent constraint, superseding the acceptance considerations of $A_b^-$. At more moderate angles ($20^\circ, 15^\circ$), the effects remain sensible but only up to a certain $p_T$ after which the acceptance losses $A_b^-$ become dominant, if any. Lower angles ($< 15^\circ$) are almost of no consequence in terms of acceptance losses, as can be seen from Fig. 22 with $\delta = 10^\circ$, $\eta_\delta$ being rejected to high pseudo-rapidity values that are more typical of endcap detectors rather than barrel ones.

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**Label $I_c$**: The requirement of a minimal inclination angle $\lambda$ for the TOF endcap results in a minimal $\eta_{\min}^\lambda$, typically repelled back to mid-rapidity values. It is of no effect for casual track directed to endcap, but is the ultimate limit for looper tracks.

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<table><caption>**Table 6:** List of analytical aspects that are considered in the plotted TOF separation performances.</caption><thead><tr><th>Physics aspect</th><th>Considered here ?</th><th>Label in Fig. 22</th><th>Section</th><th>Eq.</th></tr></thead><tbody><tr><td colspan="5"><strong>Core hypotheses</strong></td></tr><tr><td>. Hyp. 1, no energy loss of any kind</td><td>✓ -</td><td>-</td><td>II-B</td><td>-</td></tr><tr><td>. Hyp. 2, static and homogeneous B<sub>Z</sub> field</td><td>✓ -</td><td>-</td><td>II-B</td><td>-</td></tr><tr><td>. Hyp. 3, (primary) particle emitted at (0,0,0)</td><td>✓ -</td><td>-</td><td>II-C.ii</td><td>-</td></tr><tr><td>. Hyp. 4, simplified TOF geometry (rotation inv.)</td><td>✓ -</td><td>-</td><td>II-C.ii</td><td>-</td></tr><tr><td colspan="5"><strong>Trajectory approximations</strong></td></tr><tr><td>. Ideal helix derivation (L, t<sub>f</sub>)</td><td>✓ -</td><td>-</td><td>II-E.iii</td><td>Eq. 31f</td></tr><tr><td>. Straight line approx. (a) [tangent(t<sub>0</sub>)]</td><td>- X</td><td>-</td><td>II-E.iii</td><td>-</td></tr><tr><td>. Straight line approx. (b) [PV to TOF hit]</td><td>- X</td><td>-</td><td>II-E.iii</td><td>-</td></tr><tr><td colspan="5"><strong>Trajectory details</strong></td></tr><tr><td>. TOF η acceptance = f(p<sub>T</sub>, η)</td><td>✓ -</td><td>A<sub>x</sub><sup>±</sup></td><td>II-E.ii</td><td>-</td></tr><tr><td>. Looper particles towards endcap</td><td>✓ -</td><td>L<sub>t</sub></td><td>II-E.iv</td><td>-</td></tr><tr><td colspan="5"><strong>Minimum p<sub>T</sub></strong></td></tr><tr><td>. Barrel minimal p<sub>T</sub></td><td>✓ -</td><td>P<sub>b</sub></td><td>II-F.i</td><td>Eq. 42</td></tr><tr><td>. Endcap minimal p<sub>T</sub></td><td>✓ -</td><td>P<sub>e</sub></td><td>II-F.i</td><td>Eq. 44a</td></tr><tr><td colspan="5"><strong>Inclination angle to TOF surface</strong></td></tr><tr><td>. max. η<sub>δ</sub> for barrel tracks</td><td>✓ -</td><td>I<sub>b1</sub></td><td>II-F.ii</td><td>Eq. 47a</td></tr><tr><td>. (k<sub>δ</sub>)<sup>-1</sup> factor for barrel tracks</td><td>✓ -</td><td>I<sub>b2</sub></td><td>II-F.ii</td><td>Eq. 46</td></tr><tr><td>. η<sub>min</sub><sup>λ</sup> for looper in endcap</td><td>✓ -</td><td>I<sub>e</sub></td><td>II-F.iii</td><td>Eq. 49a</td></tr><tr><td colspan="5"><strong>Extra concrete considerations</strong></td></tr><tr><td>. Endcap deadzones for loopers</td><td>✓ -</td><td>D<sub>e</sub>(R)</td><td>II-E.iv</td><td>-</td></tr><tr><td>. (A·ε)<sub>TOF</sub> = f(species)</td><td>- X</td><td></td><td>IV-C.i</td><td>-</td></tr><tr><td>. p<sub>T</sub> resolution = f(species, p<sub>T</sub>, η, x/X<sub>0</sub>, B<sub>Z</sub>)</td><td>- X</td><td></td><td></td><td>-</td></tr><tr><td>. Electric charge determination</td><td>- X</td><td></td><td></td><td>-</td></tr><tr><td>. σ<sub>evt-t<sub>0</sub></sub> = f(h<sup>±</sup> multiplicity)</td><td>- X</td><td></td><td>IV-C.iii</td><td>-</td></tr><tr><td>. TOF hit background effects</td><td>- X</td><td></td><td>IV-C.iv</td><td>-</td></tr></tbody></table>
<table><caption>**Table 7:** List of the various TOF configurations tested and the corresponding figures. Values marked with • are the numerical values essentially established, either because they are in fact actual (ALICE-1), because they are quoted in the Technical Design Report of the corresponding sub-detectors anticipated for HL-LHC run IV (CMS MTD, ATLAS HGTD) or because they are simply the most plausible (B field).</caption><thead><tr><th>Experiment</th><th>⟨R<sup>barrel</sup>⟩</th><th>Δη<sub>barrel</sub></th><th>⟨Z<sup>endcap</sup>⟩</th><th>Δη<sub>endcap</sub></th><th>Timing resolution</th><th>B<sub>Z</sub> field</th><th>Figure</th></tr></thead><tbody><tr><td>ALICE-1</td><td>• 3.80 m</td><td>• < 0.88</td><td>-</td><td>-</td><td>• 80 ps</td><td>• 0.5 T</td><td>Fig. 14</td></tr><tr><td>ALICE-1</td><td>• 3.80 m</td><td>• < 0.88</td><td>-</td><td>-</td><td>• 56 ps</td><td>• 0.5 T</td><td>Fig. 15</td></tr><tr><td>ALICE-1</td><td>• 3.80 m</td><td>• < 0.88</td><td>-</td><td>-</td><td>• 56 ps</td><td>• 0.2 T</td><td>Fig. 16</td></tr><tr><td>ALICE-3</td><td>1.00 m</td><td>< 1.40</td><td>2.00 m</td><td>1.5; 4.0</td><td>10 ps</td><td>• 0.2 T</td><td>Fig. 30</td></tr><tr><td>ALICE-3</td><td>1.00 m</td><td>< 1.40</td><td>2.00 m</td><td>1.5; 4.0</td><td>10 ps</td><td>• 0.5 T</td><td>Fig. 31</td></tr><tr><td>ALICE-3</td><td>1.00 m</td><td>< 1.40</td><td>2.00 m</td><td>1.5; 4.0</td><td>10 ps</td><td>1.0 T</td><td>Fig. 32</td></tr><tr><td>ALICE-3</td><td>1.00 m</td><td>< 1.40</td><td>2.00 m</td><td>1.5; 4.0</td><td>20 ps</td><td>• 0.2 T</td><td>Fig. 33</td></tr><tr><td>ALICE-3</td><td>1.00 m</td><td>< 1.40</td><td>2.00 m</td><td>1.5; 4.0</td><td>20 ps</td><td>• 0.5 T</td><td>Fig. 34</td></tr><tr><td>ALICE-3</td><td>1.00 m</td><td>< 1.40</td><td>2.00 m</td><td>1.5; 4.0</td><td>20 ps</td><td>1.0 T</td><td>Fig. 35</td></tr><tr><td>ALICE-3</td><td>1.00 m</td><td>< 1.40</td><td>2.00 m</td><td>1.5; 4.0</td><td>30 ps</td><td>• 0.2 T</td><td>Fig. 36</td></tr><tr><td>ALICE-3</td><td>1.00 m</td><td>< 1.40</td><td>2.00 m</td><td>1.5; 4.0</td><td>30 ps</td><td>• 0.5 T</td><td>Fig. 37</td></tr><tr><td>ALICE-3</td><td>1.00 m</td><td>< 1.40</td><td>2.00 m</td><td>1.5; 4.0</td><td>30 ps</td><td>1.0 T</td><td>Fig. 38</td></tr><tr><td>ALICE-3</td><td>0.20 m</td><td>< 1.40</td><td>2.00 m</td><td>1.5; 4.0</td><td>10 ps</td><td>• 0.2 T</td><td>Fig. 39</td></tr><tr><td>ALICE-3</td><td>0.20 m</td><td>< 1.40</td><td>2.00 m</td><td>1.5; 4.0</td><td>10 ps</td><td>• 0.5 T</td><td>Fig. 40</td></tr><tr><td>ALICE-3</td><td>0.20 m</td><td>< 1.40</td><td>2.00 m</td><td>1.5; 4.0</td><td>10 ps</td><td>1.0 T</td><td>Fig. 41</td></tr><tr><td>ALICE-3</td><td>0.20 m</td><td>< 1.40</td><td>2.00 m</td><td>1.5; 4.0</td><td>20 ps</td><td>• 0.2 T</td><td>Fig. 42</td></tr><tr><td>ALICE-3</td><td>0.20 m</td><td>< 1.40</td><td>2.00 m</td><td>1.5; 4.0</td><td>20 ps</td><td>• 0.5 T</td><td>Fig. 43</td></tr><tr><td>ALICE-3</td><td>0.20 m</td><td>< 1.40</td><td>2.00 m</td><td>1.5; 4.0</td><td>20 ps</td><td>1.0 T</td><td>Fig. 44</td></tr><tr><td>ALICE-3</td><td>0.20 m</td><td>< 1.40</td><td>2.00 m</td><td>1.5; 4.0</td><td>30 ps</td><td>• 0.2 T</td><td>Fig. 45</td></tr><tr><td>ALICE-3</td><td>0.20 m</td><td>< 1.40</td><td>2.00 m</td><td>1.5; 4.0</td><td>30 ps</td><td>• 0.5 T</td><td>Fig. 46</td></tr><tr><td>ALICE-3</td><td>0.20 m</td><td>< 1.40</td><td>2.00 m</td><td>1.5; 4.0</td><td>30 ps</td><td>1.0 T</td><td>Fig. 47</td></tr><tr><td>CMS</td><td>• 1.16 m</td><td>• < 1.48</td><td>• 3.04 m</td><td>• 1.6; 3.0</td><td>• 30 ps</td><td>• 3.8 T</td><td>Fig. 48</td></tr><tr><td>CMS</td><td>• 1.16 m</td><td>• < 1.48</td><td>• 3.04 m</td><td>• 1.6; 3.0</td><td>• 30 ps</td><td>1.9 T</td><td>Fig. 49</td></tr><tr><td>ATLAS</td><td>0.29 m</td><td>< 1.00</td><td>• 3.45 m</td><td>• 2.4; 4.0</td><td>• 32 ps</td><td>• 2.0 T</td><td>Fig. 50</td></tr><tr><td>ATLAS</td><td>0.29 m</td><td>< 1.00</td><td>• 3.45 m</td><td>• 2.4; 4.0</td><td>• 32 ps</td><td>1.0 T</td><td>Fig. 51</td></tr><tr><td>ATLAS</td><td>1.00 m</td><td>< 1.00</td><td>• 3.45 m</td><td>• 2.4; 4.0</td><td>• 32 ps</td><td>• 2.0 T</td><td>Fig. 52</td></tr><tr><td>ATLAS</td><td>1.00 m</td><td>< 1.00</td><td>• 3.45 m</td><td>• 2.4; 4.0</td><td>• 32 ps</td><td>1.0 T</td><td>Fig. 53</td></tr><tr><td>ATLAS</td><td>0.29 m</td><td>< 1.00</td><td>1.50 m</td><td>1.22; 2.1</td><td>20 ps</td><td>• 2.0 T</td><td>Fig. 54</td></tr><tr><td>ATLAS</td><td>0.29 m</td><td>< 1.00</td><td>1.50 m</td><td>1.22; 2.1</td><td>20 ps</td><td>1.0 T</td><td>Fig. 55</td></tr><tr><td>ATLAS</td><td>1.00 m</td><td>< 1.00</td><td>1.50 m</td><td>1.22; 2.1</td><td>20 ps</td><td>• 2.0 T</td><td>Fig. 56</td></tr><tr><td>ATLAS</td><td>1.00 m</td><td>< 1.00</td><td>1.50 m</td><td>1.22; 2.1</td><td>20 ps</td><td>1.0 T</td><td>Fig. 57</td></tr></tbody></table>
## VI Specifications and technological choices, ALICE-3 compared to ATLAS and CMS

The initial proposal for the ALICE-3 experiment [1] includes one TOF sub-detector, in the central barrel, towards the outer radius of the experiment, at about 1 m radial distance from the beam interaction region.

On the hardware side, such a TOF detector would consist of CMOS Monolithic Active Pixel Sensors (MAPS) combining both the sensitive part and the readout part, likely either based on Low-Gain Avalanche Detectors (LGAD) architecture (with moderate reverse bias voltage of the photodiode to work under a linear gain) or ideally on Single-Photon Avalanche Diode (SPAD) architecture (with reverse bias voltage above the breakdown value letting the diode operating in Geiger mode). The exact architecture among these two families is not yet decided and calls for dedicated R&D in the coming years.

This poses the question of what will be the desired specifications for such a TOF detector in ALICE-3.

Before answering to that question, let us first present the associated situation of instrumentation research, to my current knowledge, dominated by LGAD technologies. At CERN, there is an ongoing R&D programme, RD50, dedicated to "Radiation hard semiconductor devices for very high luminosity colliders" [24]. LGAD technologies count among the sub-projects explored. The goal is to provide tracker (i.e. not TOF as prime intention) technologies that are time aware and time accurate, to enable tracking in 3+1D in the context of experiments exposed to very high instantaneous flux of particles. With the ever increasing luminosity of HL-LHC, some instrumented regions of ATLAS and CMS in runs IV and beyond will be more and more severely irradiated; the radiation tolerance is typically the most stringent factor to satisfy in this framework and is kept as the principal headline. In the context of the R&D programme, the targeted level of radiation hardness are typically up to $O(10^{16} \text{ 1-MeV } n_{\text{eq}}/\text{cm}^{-2})$.

For ALICE-3, the radiation tolerance will be a much less critical parameter: levels below $O(10^{12} - 10^{14}) \text{ 1-MeV } n_{\text{eq}}/\text{cm}^{-2}$ depending on the radial position within the experiment would suffice, whereas such levels are nowadays already in the ballpark of the achieved radiation tolerances $O(10^{15}) \text{ 1-MeV } n_{\text{eq}}/\text{cm}^{-2}$. In the case of ALICE-3, the stress for the TOF detector will be focused elsewhere, on other critical design parameters:

* the timing resolution must be as low as $O(10-20 \text{ ps})$;

* the spatial resolution must be precise and as much as possible in line with the tracker precision, $O(5-10 \mu\text{m})$, usually obtained with pixels $O(25 \times 25 \mu\text{m}^2)$;

* last but not least, the material budget should absolutely be drastically limited [$O(0.1\% \text{ x/X}_0)$ per silicon layer (sensor + read-out), considering that adjunct mechanics and services to MAPS will typically contribute dominantly to an extra 1% x/X₀ per layer or so].

The LGAD chosen by CMS and ATLAS for their respective endcap pile-up tagger in HL-LHC run IV will not be properly suited for ALICE-3:

* the timing resolution is a bit higher ($O(30-40 \text{ ps})$) than what would be desired for ALICE-3.

* the spatial resolution is degraded by rather large pixel/pads size. It will be $1.3 \times 1.3 \text{ mm}^2$ for both CMS and ATLAS LGAD in endcap, giving a typical spatial resolution of $1.3 \text{ mm}/\sqrt{12} = 375 \mu\text{m}$.

* ATLAS and CMS LGAD are hybrid pads, that is the front-end electronics is not embedded in the chip itself but is a secondary and independent application-specific integrated circuit (ASIC), separated from the sensitive volume; it needs to be bump-bonded thus coming with substantial added costs for the detector production (3.5 MCHF for electronics out of
a total 10.9 MCHF (Tab. 6.5 in [8]), to equip the 2 × 31.6 m² – 2 sides, forward and backward, × (2+2) disks per side, with 4.0 m² active per disk – for the CMS timing endcaps; 0.73 MCHF for front-end ASIC and 0.90 MCHF for bump-bonding in the ATLAS timing endcap out of a total cost at 8.5 MCHF (Tab. 6.3 in [25]), to instrument 2 × 4.2 m² – 2 sides, forward and backward, × (2+2) disks per side, with 1.0 m² active per disk – for the ATLAS timing endcaps.

* the material budget ranges from moderate to high (e.g. ≈ 2 cm-thick per disk, Fig. 4.11 and Tab. 4.3 in [25]), out of which silicon constitutes only a 50-µm-thick active part overlaid on a 300 µm substrate. If the active LGAD sensor limits its power density to about 30 mW/cm², the associated ASIC consumes and dissipates O(200 mW/cm²) (section 5.1 in [25]), needing for substantial cooling (liquid CO₂), and thus extra material, to maintain operations at -30° C.

* the so-called *fill factor*, defined by the ratio between the active area of the sensor to the total surface of the latter, is restricted, e.g. 85% in the case of CMS LGAD (section 3.2.4.1 in [8]) and would in fact further decrease for smaller pad sizes due to incompressible no-gain areas and inter-pad guard rings (≈ 40-80 µm currently). To recover a complete hermiticity, a stack of several contiguous layers is required (2 double-sided disks, i.e. 4 planes, for both CMS (Fig. 3.2 in [8]) and ATLAS (section 4.1 in [25]) and thus, coming with extra material budget.

Note that on the hybrid LGAD front, there are recent developments promising substantial improvements in terms of spatial resolution O(50 µm), thickness of the sensitive part O(50 µm)<fnref n="8" /> and fill factor pushed at 100% while keeping similar timing resolution O(40 ps). This progress are connected to Trench-Isolated Low Gain Avalanche Diodes, TI-LGADs [26] and more especially to Resistive AC-coupled LGAD referred to RSD and equivalently to AC-LGAD in the literature [27, 28].

# VII TOF measurements in ALICE-3 (HL-LHC run V)

## VII-A Configuration with a TOF as barrel outer layer at R = 1 m

In the original proposal for the ALICE-3 experiment [1], there is only one TOF sub-detector, covering a η range of ±1.4, located in the barrel region at a radius of about 1 m, immediately between the last tracking layer and the electromagnetic pre-shower.

Such a configuration is part and parcel of the first set of exercises in place in the current work. As the exact magnetic field is not decided, 3 options are tested, two corresponding to ALICE-3 continuing to use the L3 magnet already available at the experimental point, offering the possibility of 2 distinct field intensities, $B_Z = \pm 0.2$ T and $\pm 0.5$ T, one introducing the possibility to build and use a new magnet delivering higher field, here considered at $\pm 1.0$ T.

As far as the timing resolution is concerned, given the uncertainty on the retained technology, 3 values are tested, 10 ps, 20 ps and 30 ps. All in all, conjugating the $B_Z$ fields with the various timing resolutions, 9 figures are needed to present the TOF performances for the 1.0-m central TOF configurations in ALICE-3, ranging from Fig. 30 to Fig. 38.

For any figure, the minimal inclination angle in barrel has been set to a value $\delta > 30°$. Note that, under such a request, if it turned out that particle detection or tracking indeed failed at such a low inclination angle, the graphs argue further that it would not make any sense to instrument the central barrel to values above $\eta_\delta = 1.32$: particles beyond that threshold will

<footnote n="8">⁸ i.e. read-out thickness excluded. The sensitive silicon part that is thinned further from about 300 µm to the mentioned 50 µm.</footnote>
never be properly reconstructed and identified; the barrel detector in that more forward range would then be worth for nothing except producing secondary interactions.

1020

On the domain where particles cross the TOF layer with a large enough angle, one can now discuss the accessible momentum span. A general observation is that, depending on $B_Z$ but independently of the timing resolution, the minimal $p_T$ reach goes from 40 MeV/c to 180 MeV/c via 90 MeV/c for ($q = \pm 1e$) particles and with twice larger values for ($q = \pm 2e$).

1022

1024

At the other end of the momentum range, at high $p_T$, the magnetic field is of almost no consequence. For a given $R_{\text{TOF}}^{\text{barrel}}$, the timing resolution $\sigma_{\text{TOF}}$ essentially drives the particle separation and sets the upper $p_T$ limit, the latter being species dependent ($m_0$ and $y$). The larger the mass, the further upwards the last $p_T$ is theoretically repelled. On that upper front, to maintain more or less the same separation power as in ALICE-1 ($R_{\text{TOF}}^{\text{barrel}} = 3.8$ m, $\sigma_{\text{TOF}} = 80$ ps), a timing resolution of 20 ps or below is necessary for the ALICE-3 TOF. (The TOF radial location being divided by a factor 4, the timing resolution needs correspondingly to be improved by a factor 4 or so...) This can be observed by comparing for instance Fig. 14 for ALICE-1 with Fig. 34 for ALICE-3, both having $B_Z = 0.5$ T.

1026

1028

1030

1032

One may give dedicated considerations to specific separations:

1034

*separation* ($e^{\pm}/\pi^{\pm}$): the separation covers very low $p_T$, in the region where we can reconstruct low-mass di-electrons (study of the thermal radiations of virtual photons in the low energy side – that is, late in the medium history, when it is getting colder –, chiral symmetry restoration effects). The separation is such that, in a default configuration [$B_Z = 0.5$ T; $\sigma_{\text{TOF}} = 20$ ps, Fig. 34(a)], the distinction $e^{\pm}/\pi^{\pm}$ would be effective up to $2 \times p_T(e^{\pm}) \approx m[\phi(1020)] = 1.020 \text{ GeV/c}^2$. On the opposite side, at high $p_T$, even with a 10-ps resolution, the identification will not permit di-electron identification in view of the reconstruction of charmonia $c\bar{c}$ (J/ψ, ψ(2S), $\chi_{cJ} \to J/\psi$, ...) with $m[J/\psi] = 3.097 \text{ GeV/c}^2$, and even less for bottomonia $b\bar{b}$ (Υ(nS), ...) with $m[Υ(1S)] = 9.460 \text{ GeV/c}^2$. The pre-shower detector of ALICE-3 needs to relay TOF on such a higher-momentum front.

1036

1038

1040

1042

1044

*separation* (t/$^3$He): the case appears rather delicate, as already suggested earlier in subsection IV-C.iv. The difference of path length $\mathcal{L}$ between ($q = \pm 1e$)- and ($q = \pm 2e$) particles of similar masses remains a small quantity. In the most favourable case – high $B_Z$ field (1.0 T), lowest $\sigma_{\text{TOF}}$ (10 ps), at the foreseen outmost radius for a TOF layer (1.0 m), that is Fig. 32 – the accessible phase space where a separation is possible is restricted to a momentum range going from $p_T = 0.35 \text{ GeV/c}$ to $1.49 \text{ GeV/c}$ at $y = 0$ and in fact, unlike any other separation, (mildly) grows while moving to the edge of the central barrel (1.61 $\text{GeV/c}$). In contrast, for instance, the analytical separation $d^{\pm}/t$ covers 13 $\text{GeV/c}$ under the same experimental configuration. The endcap plays here no role: t and $^3$He heading into the endcap acceptance, loopers or regulars, are simply never separated. Broadly speaking, the case is very sensitive to $\sigma_{\text{TOF}}$ and $B_Z$, it can vanish rapidly with reduced $B_Z$ and degraded $\sigma_{\text{TOF}}$.

1046

1048

1050

1052

1054

1056

## VII-B Configuration with a TOF as barrel intermediate layer at R = 0.2 m

With a default field at $|B_Z| = 0.5$ T, the transverse momentum entry of the TOF barrel acceptance at 1.0 m is evaluated analytically at 90 MeV/c. In the real experiment, the threshold will certainly be uplifted, especially for species of increasing mass $m_0$ for which the efficiency curve is usually shifted towards higher momenta (see sub-section IV-C.i).

1058

1060

In order to bridge the gap and still, uniquely at the LHC, enable particle identification at the lowest momenta, i.e. $p_T < 0.15 \text{ GeV/c}$, one is led to consider a second TOF layer at lower radial

1062
1064 position. This would be a prominent feature to have, in order to fully explore the “ultra soft sector”. The physics cases go from the identification of soft electrons coming from thermal radiations or from photon conversion of the soft $\gamma$ issued by a $\chi_{cJ}$ decay that one is after for years, to Fourier coefficients $v_n$ for identified hadrons down to mild-relativistic momenta to dispute the validity limit of hydrodynamics, via ultra-soft charged pions to test the pions correlations at the chiral phase transition (Bose-Einstein pion condensate, Disoriented Chiral Condensate, ...) [29] or measurement of net quantum number fluctuations (charge, baryon number, strangeness) to challenge almost directly Lattice QCD sheer predictions [30].

1072 Here a configuration with $R = 0.2$ m is tested, again under the same various $B_Z$ and $\sigma_{TOF}$, with $\delta > 30^\circ$. The results can be seen from Fig. 39 to Fig. 47

1074 Note that the point would not be to choose among the TOF layer location at $R = 1.0$ m and at a lower radius, but to have both layers, thus providing frequently 2 TOF measurements per track. This would help in terms of PID (for primary particles but also for identification of secondaries), it would help also in terms of tracking and pile-up management (primarily out-of-bunch but also to a certain extent, in-bunch pile-up) by introducing a time component as extra information.

1080 For elegant the idea may appear, it needs nonetheless to be precisely evaluated technically. It goes without saying that placing a hybrid LGAD packed in a 7.5 cm-thick integration environment in the middle of one of the most lightweight tracker ever considered is not a viable option. For such a TOF idea to happen, it necessarily requests the TOF detection based on a MAPS architecture<fnref n="9" /> to become a fact in the next 8 years. Furthermore, with a fill factor close to 100% in order to limit the number of sensor layers (2-3) to be stacked for the sake of hermeticity. This way only, the extra material budget due to the integration of an extra intermediate TOF layer coming to complement a tracker one will be bearable, +O(1% $x/X_0$).

1088 In terms of area to instrument, a TOF at $R_{\text{TOF}}^{\text{barrel}} = 0.2$ m covering $\pm 1.4$ $\eta$ units is embodied by a cylinder extended to $\pm Z_{\text{TOF,max}}^{\text{barrel}} = \pm 0.38$ m, thus covering an area of $2\pi R_{\text{TOF}}^{\text{barrel}} \cdot 2Z_{\text{TOF,max}}^{\text{barrel}} \approx 1$ m² per sensitive layer entering the TOF stack (typically 1-, 2- or 3-cylinder stack). Such a surface can be compared to the surface of the basic barrel option with $R_{\text{TOF}}^{\text{barrel}} = 1.0$ m; the surface per sensitive cylinder is there $2\pi R_{\text{TOF}}^{\text{barrel}} \cdot 2Z_{\text{TOF,max}}^{\text{barrel}} \approx 2\pi(1\ \text{m}) \cdot 2(1.9\ \text{m}) \approx 24$ m².

## VII-C Possibility to have endcap TOF planes

1094 In the first thoughts given to ALICE-3 [1], no TOF endcap has been anticipated. The benefit of such an option has been studied in this document and the case is brought to the limelight for the sake of the discussion. Note that, as the first attempt here, the TOF endcap has been chosen to be located at the extreme position of the forward tracker, at $Z_{\text{TOF}}^{\text{endcap}} = 2.0$ m, just before the ultimate plane that the pre-shower defines. It is further considered here to cover a pseudorapidity scope ranging from 1.5 to 4.0. If it makes sense physics-wise to push the instrument as forward as possible (and ignore for a moment the issues of radiation tolerance, integration, etc encountered there...), one must be at least reasonable for the starting upper edge of the endcap. A gap of at least 0.1 unit in $\eta$ is likely necessary to fit in the mechanical structure, to let the wires and services pass between the barrel and the endcap, not only for TOF layers and planes, but also for tracker layers and planes. Still, at $R = 1$ m, 0.1 unit in $\eta$ corresponds to a gap of about 10 cm only between the outmost $z$ value of the TOF barrel $Z_{\text{TOF,max}}^{\text{barrel}}$ and the location $Z_{\text{TOF}}^{\text{endcap}}$.

For any figure, the minimal inclination angle in endcap has been requested to a value $\lambda > 30^\circ$.

<footnote n="9">⁹ Given the prime role led by SPAD architectures in industry, SPAD would be the most promising candidates a priori to achieve this.</footnote>
1108 On a general note, one can draw the reader's attention upon the effects of the magnetic field. These are manifold. As for the barrel region, $B_Z$ does not change (this time, strictly does not change) the upper limits of TOF separations. $B_Z$ modifies however the acceptance in a significant way. Stronger $B_Z$ implies that the “north” and “south” borders, $p_{T\min}^{ECal}$ and $p_{T\min}^{endcap}$, are uplifted. Stronger and stronger magnetic also dig further the deadzones for the looper particles (the aforementioned orca’s dorsal fin appearing regularly), making the detection phase space more and more splitted. Behind the scene, and on a positive note, one should however acknowledge that stronger $B_Z$ certainly also means that the $p_T$ resolution of particles in endcap acceptance may be characterised with a more precise transverse momentum resolution (stronger curvature would be beneficial for tracks which total momentum can quickly become quite high ($p_{tot} = p_T \cosh(\eta)$) and thus complicating the $p_T$ determination: the particle escapes the experiment promptly before being bent timely).

1120 In terms of PID performances, looking at the endcap maps in ALICE-3 figures, the reader may object that the $4-\sigma_{TOF}$ separation remains limited, in terms of $p_T$ reach. Even sometimes very limited in $p_T$ as we move to more and more forward $\eta$... “The endcap case for ALICE-3 looks feeble”, one may think. However, after second thoughts given along the present lines, several reasons could in fact argue decisively in favour of having a TOF plane in the endcap. In the following, we list such rationale, from the most immediate to maybe the least obvious argument.

1. **Forward PID**: The possibility to have PID information at forward rapidity would improve the detector hermiticity on the identification front. It means that PID could be further exploited over a complete $\eta$ coverage and fair $p_T$ range for complex cascade decays hinged on 3 to 6 final state particles spread over large $\eta$ domains [like multi-charm baryons: $\Lambda_c^+(udc)$, $\Xi_c^+(usc)$, $\Xi_c^0(dsc)$, $\Omega_c^0(ssc)$, $\Xi_{cc}^{2+}(ucc)$ $\Omega_{ccc}^{2+}(ccc)$]. It means that studies of $n$-particle correlations could be performed at (very) low $p_T$ with PID over the whole $\eta$ coverage of the experiment. It means that multiplicity-dependent studies like the ones performed with V0M in ALICE-1 [31] could be further tailored from $p_T \approx 0$ but now with supplemental PID information exploited for the multiplicity estimator.

2. **Possible acceptance redundancy**: for a TOF barrel layer located among the tracker layers (e.g. $R=0.2$ m), a TOF endcap would have in common some regions of phase space with such an inner TOF layer, offering possibly double TOF information with a first hit in barrel followed by its complement in endcap. This would be beneficial at least for mutual calibration purposes. Figures 39 to Fig. 47 pin point the ($p_T$, $y$, $m_0$) locations where such redundancies would happen.

In concrete terms, the exact frequency of such opportunities per event would be to be assessed with fast or full simulations, the exact experimental layout and the material budget distribution will be of strong influence.

3. **Pile-up tagger**: if CMS and ATLAS may use their pile-up tagger as TOF detector, it could be conversely convenient for ALICE-3 to use its TOF detectors as pile-up tagger. On the one hand, ALICE intends to harvest Minimum Bias events in their individual entirety¹⁰ in astronomic quantities. On the other hand, such a collection has to be performed with, ideally native “zero” pile-up contamination per event, at the very least, with thorough and fully mastered pile-up tagging. Such a mitigation is a crest line to find in the data taking strategy. It inescapably implies that the full-event record take its (necessary) time, while we always wish/are wished to be faster. To that end, comprehensive timing information will be much appreciated when LHC conditions will become more and more stringent for the experiment / sub-detectors with the slowest read-out. (To give a leading-order idea
1156

of the problem, bear in mind the 25-ns bunch spacing in LHC compared to the typical $\mathcal{O}(1-10 \ \mu\text{s})$ resolution of MAPS read-out in tracker.)

1158

4. Reducing $\sigma_{\text{evt}-t_0}$: by extending the TOF acceptance and thus the additional particles at hand, one can reduce the uncertainty on $\sigma_{\text{evt}-t_0}$ (see sub-section IV-C.iii). This will be especially beneficial in low-multiplicity collisions like pp p–A and peripheral A–A.

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1166

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5. Escaping $y \to 0$: the reader should note in the figures how the barrel coverage narrows down to $y$ almost zero as one moves towards lower and lower $p_T$. For $p_T < 0.1 \ \text{GeV/c}$, the full rapidity coverage of the central barrel is for most of the hadronic species reduced to $|y| < 0.3$; at $p_T < 0.05 \ \text{GeV/c}$, to $|y| < 0.1$. And it is essentially $|y| = 0$ for light nuclei below $0.1 \ \text{GeV/c}$. Provided that a low material budget preserves some decent or high $\mathcal{A} \cdot \varepsilon$ to the forward directions, the opportunity of an endcap TOF information put the ALICE experiment in capacity to extend its low-$p_T$ studies to more forward rapidities. For protons and light nuclei, by profiting from the extra Lorentz boost given along $p_z$, TOF forward endcaps could become perhaps the only effective way to identify mid-rapidity populations at $p_T \approx 0$.

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To follow-up on such a proposal, and foster any subsequent discussion to have in term of cost and integration issues, one can mention the corresponding detector surface at stakes. It would be a matter of instrumenting 2 sides of the experiments, forward and backward regions, to cover 1.5 to 4.0 units of $|\eta|$, at $Z_{\text{TOF}}^{\text{endcap}} \approx \pm 2.0 \ \text{m}$. It translates to disks extending from $R_{\text{TOF,min}}^{\text{endcap}} = 0.073 \ \text{m}$ to $R_{\text{TOF,max}}^{\text{endcap}} = 0.939 \ \text{m}$. The corresponding ring surface to equip become then $\pi(R_{\text{TOF,max}}^{\text{endcap}} - R_{\text{TOF,min}}^{\text{endcap}})^2 = 2.75 \ \text{m}^2$ per disk. Each TOF endcap would then consist of a stack of disks of that kind, 1 to 4 disks typically depending on the desired redundancy and the fill factor of the chosen MAPS architecture. Such an endcap surface of $2.75 \ \text{m}^2$ per active disk can be related to the aforementioned $24 \ \text{m}^2$ per active cylinder of the basic option for barrel TOF ($R_{\text{TOF}}^{\text{barrel}} = 1 \ \text{m}$).

1180

The integration and the radiation hardness specifications could potentially be eased by reducing the $\eta$ most forward edge from 4.0 to a lower value (3.5, 3.0, ...).

1182

The cost could be reduced essentially by reducing the equipped area. There are two pragmatic lever arms to do that:

1184

* at a given $Z_{\text{TOF}}^{\text{endcap}}$, the upper radius of the endcap could be shrunk, which is likely not so desirable if we want to assure as much as possible the continuity of measurement ($p_T$ upper reach for separations) with the barrel TOF at $R_{\text{TOF}}^{\text{barrel}} = 1.0 \ \text{m}$.

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1190

1192

1194

1196

* for a given $\eta$ coverage, we could move the TOF endcap plane closer to the interaction point. This will come with a mild reduction of the $p_T$ upper reach for separation but a clearly reduced area. Located at $Z_{\text{TOF}}^{\text{endcap}} = 1.5 \ \text{m}$, still covering 1.5 to 4.0 units of pseudorapidity, the surface per disk would be reduced by 45% to $\pi(0.70^2 - 0.05^2) = 1.5 \ \text{m}^2$. This implies that the chosen architecture and its integration appendices are of limited material budget, because under such circumstances, the TOF endcap will be located in the middle of the forward tracker. The forward direction coming intrinsically with the longitudinal boost $p_z$ makes the consequences of the material budget less dramatic compared to mid-rapidity TOF at $R_{\text{TOF}}^{\text{barrel}} = 0.2 \ \text{m}$ but still, given the targeted low momenta, the issue at forward rapidity may well remain a bottleneck.

<footnote n="10">¹⁰ We are not focusing on individual particles, QGP physics is a high-energy physics in which the event context does matter, being on a same footing with the particle species of interest. It entails that the collision “event” cannot be amputated, easily and harmlessly, without a potential loss of scientific material...</footnote>
# VIII TOF measurement in CMS (HL-LHC run IV and V)

1198 Among the choices already made by the CMS collaboration for its strategical upgrades (Long Shutdown 3, 2025-27) in view of HL-LHC run IV, a detector meant to tag and sort pile-up events has been casted, MIP Timing Detector (MTD) (see Technical Design Report [8]). The task for such a detector is to stamp particle detection with a time resolution below the HL-LHC bunch spacing (typically $O(25 \text{ ns})$) and further help to parse in-bunch pile-up. The detector consists of two components :

1204 (i) one cylindrical layer at central rapidity, the Barrel Timing Layer, BTL, covering $|\eta| < 1.48$, based on LYSO/Ce crystals and SiPM readout, positioned at an average radius $r = 1.16$ m.

1206 (ii) two endcap planes at forward and backward pseudo-rapidities, the Endcap Timing Layer, ETL, covering $1.6 < |\eta| < 3.0$, based on Low-Gain Avalanche Detectors (LGAD), positioned at an average distance $|z| = \pm 3.04$ m.

1210 On both parts, the expected timing resolution should be of 30 ps, degrading a bit to 50 ps with the accumulated luminosity (Fig. B.10 in [8]).

1212 A side-product of such a detector is its usage as a TOF detector, as reported in the same TDR [8] in Fig. 1.5 p.8 (separation power for $\pi^{\pm}/K^{\pm}$ and $K^{\pm}/p^{\pm}$) and Fig. 5.23 p.222 (simulation of $\beta_{tot}$ distributions as a function of total momentum $p_{tot}$, with Hydjet in A-A for BTL and for ETL).

1216 In the present document, the default results exposed in the TDR (Fig. 1.5) for a magnetic field of 3.8 T have been extended with further cases of particle separations (Fig. 48). As an extra scenario, an example configuration with twice lower magnetic field (1.9 T) has been tested (Fig. 49), in order to assess a possible gain towards low $p_T$ at both mid- and forward rapidities due to a reduced magnetic field. In any case, the minimal inclination angle in the barrel has been set to a value $\delta > 30^{\circ}$, in the endcap, to $\lambda > 30^{\circ}$.

1222 The TOF PID competitiveness of CMS with respect to ALICE-3 will be driven essentially by aspects escaping the analytical formulae presented here. These aspects encompass considerations of scientific strategy, of detector as well as of native physics :

1. *One run ahead*: the simple fact that MTD will exist and will be a reality almost for certain : the project is already approved by CERN authorities and collaboration, the R&D has been carried out, the production is financed. And furthermore it should become a fact with one HL-LHC run in advance compared to ALICE-3, that is, HL-LHC run IV by 2027.

1230 **2. $B_Z$ field"s**: the possibility to collect data under various $B_Z$ field intensities (3.8 T only ? 3.8 T and possibly lower intensities ?)$^{11}$.

1232 **3. Spatial resolution**: the spatial granularity driven by the TOF pad sizes, in barrel ($\Delta r\phi = 57 \text{ mm} \times \Delta z = 3 \text{ mm}^2$ for LYSO bar crystals, Fig. 2.60 in [8]) and endcap ($1.3 \times 1.3 \text{ mm}^2$ for LGAD, Sec. 3.2 in [8]).

1236 **4. Occupancy**: the sub-detector occupancy depending on the collision system and pile-up conditions.

1238 **5. A.ε**: the impact of the tracker material budget on A.ε to reach the TOF sub-detector for the individual identifiable species.
6. Relative abundancies: the differential production rate per event for each particle species with respect to all others, making more or less favourable balance between signal and background. The issue may not be on $\pi^{\pm}$, K$^{\pm}$ and p$^{\pm}$, but rather on light nuclei, naturally scarce and so, being more vulnerable in that respect.

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Among the undecided points above, mostly all of them could be addressed with dedicated full simulations, in order to judge better of the underlying difficulties and relevance. Such Monte Carlo studies will naturally be as many long-standing and detailed work that would need to be carried out collectively, not necessarily by relying on CMS members' shoulders only but potentially led by the “heavy-ion” LHC community as a whole.

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In the meantime, as extra considerations, it will be valuable to keep in mind the characteristics of the Phase-2 CMS tracker [32] in place by the time of the HL-LHC run IV and later, occupying the volume between the interaction point and the MTD. Figure 24 gives the experiment layout, how the barrel and endcap components are organised. Figure 25 specifies the expected material budget distribution in $X/X_0$ as function of $|\eta|$. Figures Fig. 26(a) and (b) provides respectively an estimate of the fluence (in 1-MeV $n_{eq}/\text{cm}^{-2}$) and the total ionising dose TID (in Gray, Gy) distributed over the tracker geometry.

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<footnote n="11">¹¹ Running the CMS experiment under various $B_Z$ fields recovers several orthogonal or complementary aspects: the collaboration-wide interest and approval for such magnetic configurations, ability of the 3.8-T supraconducting magnet to sustain additional hysteresis cycles, the existence of accurate field maps per $B_Z$ configuration throughout the geometry of the experiment, steering of the LHC beams through the experiment under unusual $B_Z$, tracking recommissioning adapted to new $B_Z$, recommissioning of the calorimeter responses as well, overall data taking time allotted (hours, days, weeks ?) per $B_Z$ configuration and per collision system (pp, p–A, A–A).</footnote>
<figure><img src="image_25.png" /><figcaption>**Fig. 24:** CMS detector layout for the Phase-2 tracker, showing the barrel and endcap elements corresponding to Fig. 2.3 in [32]. Orange and green lines stand for the Inner Tracker (IT) consisting of pixel sensors, blue dashes and red segment define together the Outer Tracker (OT), with modules combining macro-pixels with micro-strips in blue (PS), and double-sided micro-strips in red (2S).</figcaption></figure>

<figure><img src="image_26.png" /><figcaption>**Fig. 25:** CMS repartition of the radiation length as a function of $|\eta|$ corresponding to Fig. 6.2 in [32].</figcaption></figure>
<figure><img src="image_27.png" /><figcaption>**Fig. 26:** CMS 2D maps for fluence (a) and total ionising dose (b) in the Phase-2 tracker detector, corresponding respectively to Figs. 2.2 and 8.3 in [32].</figcaption></figure>
# IX TOF measurements in ATLAS (HL-LHC run IV and V)

In a similar path followed by CMS, the ATLAS collaboration has also made the choice to set up a detector dedicated to pile-up tagging for HL-LHC run IV, the High Granularity Timing detector (HGTD). See the corresponding Technical Design Report [25].

The peculiarity of this sub-detector is that it is only foreseen at forward and backward rapidities, as two endcap disks, covering $2.4 < |η| < 4.0$, based on LGAD architecture (as in the CMS case, with the same pad size of $1.3 \times 1.3 \text{ mm}^2$), occupying about 5-cm thickness, at distances $|z|$ between 3.435 m and 3.485 m, with a time resolution of 32 ps. The physics cases associated to TOF capabilities have not been discussed in the TDR but will certainly be a fact, as in the CMS case.

In the work done here, performances of a HGTD configuration in the ATLAS default magnetic solenoidal field of 2.0 T is provided (Fig. 50, Fig. 52, Fig. 54 and Fig. 56). As proposed with the CMS configuration, a second configuration has been tested with a twice lower magnetic field, i.e. at 1.0 T here. (Fig. 51, Fig. 53, Fig. 55 and Fig. 57).

The fact that, at the run IV horizon, the central rapidities will not be equipped with any timing detector opens the hypothesis of a future instrumentation at such a location, for instance, at the next milestone of Long Shutdown 4 ($\approx 2030$) before HL-LHC run V for a timeline similar to ALICE-3. To that end, two mid-rapidity configurations have been further added to the plots, considering a timing layer positioned either at a radius $R = 1.0$ m or at $R = 0.29$ m.

The motivations of such chosen radii rely on a few prime considerations.

(i) ITk layout illustrated in Fig. 27, taken from recent [33],

(ii) ITk material budget estimated in Fig. 28, taken from the same note [33],

(iii) ITk fluence and dose maps illustrated in Fig. 29, available from [34].

If SPAD architecture were developed and ready for ALICE-3, with $1\% X/X_0$ per layer and radiation tolerance up to $\mathcal{O}(10^{15})$ 1-MeV $n_{eq}/\text{cm}^{-2}$, why not imagining ATLAS using it as well in the middle of the tracking layers ? – This calls for specific considerations about the radiation levels as a function of space and, looking at Fig. 29, the domain around $R = 0.29$ m or a fortiori around $R = 1.0$ m would match such an instrumentation idea.

If the location would be defined, the next question to come concerns the zenith angle coverage or the $\eta$ coverage. Let's take a second look at the Fig. 27. At both radii, $R = 1.0$ m or $R = 0.29$ m, it would be first accessible to cover $|η| < 1.0$. The corresponding foreseen layers at both radii in ITk must be constituted by simple cylindric geometries. Replacing such a given layer with one technology by another may be more manageable in terms of mechanics and integration.

Back to Fig. 27, what if one would now decide to enlarge the $\eta$ coverage to $\pm 2.0$ units ? ...

* at $R \approx 1.0$ m, this would call for a revision of several endcap planes in addition to the previous layer, complexifying the enterprise and raising the associated costs;

* at $R = 0.29$ m, this would go with a modification in azimuth of all the tilted double-segments at the same radius (illustrated by the 9 red segments on the figure) in addition of the regular cylinder at mid-rapidity. The pros and cons would need to be discussed in details with experts but it is a priori not the most immediate sensor change to be integrated.

A way out solving potentially the situation for both radii could be in fact to change only the outer endcap disk, appearing as blue vertical segment at $z = 1.5$ m in Fig. 27. Doing so, the pseudorapidity coverage from $\approx 1.0$ to $\approx 2.2$ would be assured by a single plane to be modified; this implies that this domain that is yet still at mid-rapidity would then be covered not by a cylindric *barrel* layer but by a vertical *endcap* disk.
In a similar spirit to what has been discussed for ALICE-3, this would augur that the ultimate ATLAS TOF configuration could be hinged on the combination of four different TOF locations :

(i) an outer barrel layer at $R_{\text{TOF}}^{\text{barrel}} = 1.0$ m, covering $|\eta| < 1.0$, based on MAPS,

(ii) an intermediate barrel layer at $R_{\text{TOF}}^{\text{barrel}} = 0.29$ m, covering $|\eta| < 1.0$, based on MAPS,

(iii) an endcap disk at $Z_{\text{TOF}}^{\text{endcap}} = 1.50$ m, covering the range $1.1 < |\eta| < 2.2$, based on MAPS,

(iv) a distant endcap disk embodied by the HGTD, located at $Z_{\text{TOF}}^{\text{endcap}} = 3.45$ m, spanning $2.4 < |\eta| < 4.0$, based on LGAD architecture.

In any of the aforementioned ATLAS cases, the minimal inclination angle in barrel has been set to a value $\delta > 30^{\circ}$, for any vertical endcap planes, to $\lambda > 30^{\circ}$.

As for The TOF PID competitiveness of ATLAS with respect to ALICE-3 and CMS, the same comments stressed in previous CMS section IX can be stressed again here.

<figure><img src="image_28.png" /><figcaption>Fig. 27: Layout of the ATLAS tracker ITk corresponding to Fig. 1 in [33].</figcaption></figure>

<figure><img src="image_29.png" /><figcaption>Fig. 28: Repartition of the material budget (in $x/X_0$) for the ATLAS tracker ITk corresponding to Fig. 3 in [33].</figcaption></figure>
<figure><img src="image_30.png" /><figcaption>Fig. 29: ATLAS 2D maps for fluence (a) and total ionising dose (b) in the ITk detector taken from [34].</figcaption></figure>
1314

ALICE-3 configurations
<figure><img src="image_31.png" /><figcaption>**Fig. 30:** Expected TOF performances in terms of particle separations as a function of $p_T$ and $y$ for the configuration already foreseen or possibly anticipated for the ALICE-3 experiment, with hypotheses : i) $\langle r_{\text{barrel}} \rangle = 1.00$ m, ii) $\langle z_{\text{TOF}}^{\text{endcap}} \rangle = 2.00$ m, iii) TOF timing resolution = 10 ps, iv) $B_{\text{field}} = -0.2$ T.</figcaption></figure>
<figure><img src="image_32.png" /><figcaption>**Fig. 31:** Expected TOF performances in terms of particle separations as a function of $p_T$ and $y$ for the configuration already foreseen or possibly anticipated for the ALICE-3 experiment, with hypotheses : i) $\langle r_{\text{barrel}} \rangle = 1.00$ m, ii) $\langle z_{\text{TOF}}^{\text{endcap}} \rangle = 2.00$ m, iii) TOF timing resolution = 10 ps, iv) $B_{\text{field}} = -0.5$ T.</figcaption></figure>
<figure><img src="image_33.png" /><figcaption>**Fig. 32:** Expected TOF performances in terms of particle separations as a function of $p_T$ and $y$ for the configuration already foreseen or possibly anticipated for the ALICE-3 experiment, with hypotheses : i) $\langle r_{\text{barrel}} \rangle = 1.00$ m, ii) $\langle z_{\text{TOF}}^{\text{endcap}} \rangle = 2.00$ m, iii) TOF timing resolution = 10 ps, iv) $B_{\text{field}} = -1.0$ T.</figcaption></figure>
<figure><img src="image_34.png" /><figcaption>**Fig. 33:** Expected TOF performances in terms of particle separations as a function of $p_T$ and $y$ for the configuration already foreseen or possibly anticipated for the ALICE-3 experiment, with hypotheses : i) $\langle r_{\text{barrel}} \rangle = 1.00$ m, ii) $\langle z_{\text{TOF}}^{\text{endcap}} \rangle = 2.00$ m, iii) TOF timing resolution = 20 ps, iv) $B_{\text{field}} = -0.2$ T.</figcaption></figure>
<figure><img src="image_35.png" /><figcaption>**Fig. 34:** Expected TOF performances in terms of particle separations as a function of $p_T$ and $y$ for the configuration already foreseen or possibly anticipated for the ALICE-3 experiment, with hypotheses : i) $\langle r_{\text{barrel}} \rangle = 1.00$ m, ii) $\langle z_{\text{TOF}}^{\text{endcap}} \rangle = 2.00$ m, iii) TOF timing resolution = 20 ps, iv) $B_{\text{field}} = -0.5$ T.</figcaption></figure>
<figure><img src="image_36.png" /><figcaption>**Fig. 35:** Expected TOF performances in terms of particle separations as a function of $p_T$ and $y$ for the configuration already foreseen or possibly anticipated for the ALICE-3 experiment, with hypotheses : i) $\langle r_{\text{barrel}} \rangle = 1.00$ m, ii) $\langle z_{\text{TOF}}^{\text{endcap}} \rangle = 2.00$ m, iii) TOF timing resolution = 20 ps, iv) $B_{\text{field}} = -1.0$ T.</figcaption></figure>
<figure><img src="image_37.png" /><figcaption>**Fig. 36:** Expected TOF performances in terms of particle separations as a function of $p_T$ and $y$ for the configuration already foreseen or possibly anticipated for the ALICE-3 experiment, with hypotheses : i) $\langle r_{\text{barrel}} \rangle = 1.00$ m, ii) $\langle z_{\text{TOF}}^{\text{endcap}} \rangle = 2.00$ m, iii) TOF timing resolution = 30 ps, iv) $B_{\text{field}} = -0.2$ T.</figcaption></figure>
<figure><img src="image_38.png" /><figcaption>**Fig. 37:** Expected TOF performances in terms of particle separations as a function of $p_T$ and $y$ for the configuration already foreseen or possibly anticipated for the ALICE-3 experiment, with hypotheses : i) $\langle r_{\text{barrel}} \rangle = 1.00$ m, ii) $\langle z_{\text{TOF}}^{\text{endcap}} \rangle = 2.00$ m, iii) TOF timing resolution = 30 ps, iv) $B_{\text{field}} = -0.5$ T.</figcaption></figure>
<figure><img src="image_39.png" /><figcaption>**Fig. 38:** Expected TOF performances in terms of particle separations as a function of $p_T$ and $y$ for the configuration already foreseen or possibly anticipated for the ALICE-3 experiment, with hypotheses : i) $\langle r_{\text{barrel}} \rangle = 1.00$ m, ii) $\langle z_{\text{TOF}}^{\text{endcap}} \rangle = 2.00$ m, iii) TOF timing resolution = 30 ps, iv) $B_{\text{field}} = -1.0$ T.</figcaption></figure>
<figure><img src="image_40.png" /><figcaption>**Fig. 39:** Expected TOF performances in terms of particle separations as a function of $p_T$ and $y$ for the configuration already foreseen or possibly anticipated for the ALICE-3 experiment, with hypotheses : i) $\langle r_{\text{barrel}} \rangle = 0.20$ m, ii) $\langle z_{\text{TOF}}^{\text{endcap}} \rangle = 2.00$ m, iii) TOF timing resolution = 10 ps, iv) $B_{\text{field}} = -0.2$ T.</figcaption></figure>
<figure><img src="image_41.png" /><figcaption>**Fig. 40:** Expected TOF performances in terms of particle separations as a function of $p_T$ and $y$ for the configuration already foreseen or possibly anticipated for the ALICE-3 experiment, with hypotheses : i) $\langle r_{\text{barrel}} \rangle = 0.20$ m, ii) $\langle z_{\text{TOF}}^{\text{endcap}} \rangle = 2.00$ m, iii) TOF timing resolution = 10 ps, iv) $B_{\text{field}} = -0.5$ T.</figcaption></figure>
<figure><img src="image_42.png" /><figcaption>**Fig. 41:** Expected TOF performances in terms of particle separations as a function of $p_T$ and $y$ for the configuration already foreseen or possibly anticipated for the ALICE-3 experiment, with hypotheses : i) $\langle r_{\text{barrel}} \rangle = 0.20$ m, ii) $\langle z_{\text{TOF}}^{\text{endcap}} \rangle = 2.00$ m, iii) TOF timing resolution = 10 ps, iv) $B_{\text{field}} = -1.0$ T.</figcaption></figure>
<figure><img src="image_43.png" /><figcaption>**Fig. 42:** Expected TOF performances in terms of particle separations as a function of $p_T$ and $y$ for the configuration already foreseen or possibly anticipated for the ALICE-3 experiment, with hypotheses : i) $\langle r_{\text{barrel}} \rangle = 0.20$ m, ii) $\langle z_{\text{TOF}}^{\text{endcap}} \rangle = 2.00$ m, iii) TOF timing resolution = 20 ps, iv) $B_{\text{field}} = -0.2$ T.</figcaption></figure>
<figure><img src="image_44.png" /><figcaption>**Fig. 43:** Expected TOF performances in terms of particle separations as a function of $p_T$ and $y$ for the configuration already foreseen or possibly anticipated for the ALICE-3 experiment, with hypotheses : i) $\langle r_{\text{barrel}} \rangle = 0.20$ m, ii) $\langle z_{\text{endcap}}^{\text{endcap}} \rangle = 2.00$ m, iii) TOF timing resolution = 20 ps, iv) $B_{\text{field}} = -0.5$ T.</figcaption></figure>
<figure><img src="image_45.png" /><figcaption>**Fig. 44:** Expected TOF performances in terms of particle separations as a function of $p_T$ and $y$ for the configuration already foreseen or possibly anticipated for the ALICE-3 experiment, with hypotheses : i) $\langle r_{\text{barrel}} \rangle = 0.20$ m, ii) $\langle z_{\text{TOF}}^{\text{endcap}} \rangle = 2.00$ m, iii) TOF timing resolution = 20 ps, iv) $B_{\text{field}} = -1.0$ T.</figcaption></figure>
<figure><img src="image_46.png" /><figcaption>**Fig. 45:** Expected TOF performances in terms of particle separations as a function of $p_T$ and $y$ for the configuration already foreseen or possibly anticipated for the ALICE-3 experiment, with hypotheses : i) $\langle r_{\text{barrel}} \rangle = 0.20$ m, ii) $\langle z_{\text{TOF}}^{\text{endcap}} \rangle = 2.00$ m, iii) TOF timing resolution = 30 ps, iv) $B_{\text{field}} = -0.2$ T.</figcaption></figure>
<figure><img src="image_47.png" /><figcaption>**Fig. 46:** Expected TOF performances in terms of particle separations as a function of $p_T$ and $y$ for the configuration already foreseen or possibly anticipated for the ALICE-3 experiment, with hypotheses : i) $\langle r_{\text{barrel}} \rangle = 0.20$ m, ii) $\langle z_{\text{TOF}}^{\text{endcap}} \rangle = 2.00$ m, iii) TOF timing resolution = 30 ps, iv) $B_{\text{field}} = -0.5$ T.</figcaption></figure>
<figure><img src="image_48.png" /><figcaption>**Fig. 47:** Expected TOF performances in terms of particle separations as a function of $p_T$ and $y$ for the configuration already foreseen or possibly anticipated for the ALICE-3 experiment, with hypotheses : i) $\langle r_{\text{barrel}} \rangle = 0.20$ m, ii) $\langle z_{\text{TOF}}^{\text{endcap}} \rangle = 2.00$ m, iii) TOF timing resolution = 30 ps, iv) $B_{\text{field}} = -1.0$ T.</figcaption></figure>
CMS configurations
<caption>**Fig. 48:** Expected TOF performances in terms of particle separations as a function of $p_T$ and $y$ for the configuration already foreseen or possibly anticipated for the CMS experiment, with hypotheses :</caption>

i) $\langle r_{\text{barrel}} \rangle = 1.16$ m, ii) $\langle z_{\text{TOF}}^{\text{endcap}} \rangle = 3.04$ m, iii) TOF timing resolution = 30 ps, iv) $B_{\text{field}} = -3.8$ T.

<figure><img src="image_49.png" /></figure>
<figure><img src="image_50.png" /><figcaption>**Fig. 49:** Expected TOF performances in terms of particle separations as a function of $p_T$ and $y$ for the configuration already foreseen or possibly anticipated for the CMS experiment, with hypotheses : i) $\langle r_{\text{barrel}} \rangle = 1.16$ m, ii) $\langle z_{\text{TOF}}^{\text{endcap}} \rangle = 3.04$ m, iii) TOF timing resolution = 30 ps, iv) $B_{\text{field}} = -1.9$ T.</figcaption></figure>
1316 ATLAS configurations

[ANA-1073 - Version 2.0 - (git rev. dummy) - Tuesday 9th March, 2021, 04:57]
<figure><img src="image_51.png" /><figcaption>**Fig. 50:** Expected TOF performances in terms of particle separations as a function of $p_T$ and $y$ for the configuration already foreseen or possibly anticipated for the ATLAS experiment, with hypotheses : i) $\langle r_{\text{barrel}} \rangle = 0.29$ m, ii) $\langle z_{\text{TOF}}^{\text{endcap}} \rangle = 3.45$ m, iii) TOF timing resolution = 32 ps, iv) $B_{\text{field}} = -2.0$ T.</figcaption></figure>
<figure><img src="image_52.png" /><figcaption>**Fig. 51:** Expected TOF performances in terms of particle separations as a function of $p_T$ and $y$ for the configuration already foreseen or possibly anticipated for the **ATLAS** experiment, with hypotheses : i) $\langle r_{\text{barrel}} \rangle = 0.29$ m, ii) $\langle z_{\text{TOF}}^{\text{endcap}} \rangle = 3.45$ m, iii) TOF timing resolution = 32 ps, iv) $B_{\text{field}} = -1.0$ T.</figcaption></figure>
<figure><img src="image_53.png" /><figcaption>**Fig. 52:** Expected TOF performances in terms of particle separations as a function of $p_T$ and $y$ for the configuration already foreseen or possibly anticipated for the ATLAS experiment, with hypotheses : i) $\langle r_{\text{barrel}} \rangle = 1.00$ m, ii) $\langle z_{\text{TOF}}^{\text{endcap}} \rangle = 3.45$ m, iii) TOF timing resolution = 32 ps, iv) $B_{\text{field}} = -2.0$ T.</figcaption></figure>
<figure><img src="image_54.png" /><figcaption>**Fig. 53:** Expected TOF performances in terms of particle separations as a function of $p_T$ and $y$ for the configuration already foreseen or possibly anticipated for the ATLAS experiment, with hypotheses : i) $\langle r_{\text{barrel}} \rangle = 1.00$ m, ii) $\langle z_{\text{TOF}}^{\text{endcap}} \rangle = 3.45$ m, iii) TOF timing resolution = 32 ps, iv) $B_{\text{field}} = -1.0$ T.</figcaption></figure>
<figure><img src="image_55.png" /><figcaption>**Fig. 54:** Expected TOF performances in terms of particle separations as a function of $p_T$ and $y$ for the configuration already foreseen or possibly anticipated for the **ATLAS** experiment, with hypotheses : i) $\langle r_{\text{barrel}} \rangle = 0.29$ m, ii) $\langle z_{\text{TOF}}^{\text{endcap}} \rangle = 1.50$ m, iii) TOF timing resolution = 20 ps, iv) $B_{\text{field}} = -2.0$ T.</figcaption></figure>
<figure><img src="image_56.png" /><figcaption>**Fig. 55:** Expected TOF performances in terms of particle separations as a function of $p_T$ and $y$ for the configuration already foreseen or possibly anticipated for the ATLAS experiment, with hypotheses : i) $\langle r_{\text{barrel}} \rangle = 0.29$ m, ii) $\langle z_{\text{TOF}}^{\text{endcap}} \rangle = 1.50$ m, iii) TOF timing resolution = 20 ps, iv) $B_{\text{field}} = -1.0$ T.</figcaption></figure>
<figure><img src="image_57.png" /><figcaption>**Fig. 56:** Expected TOF performances in terms of particle separations as a function of $p_T$ and $y$ for the configuration already foreseen or possibly anticipated for the **ATLAS** experiment, with hypotheses : i) $\langle r_{\text{barrel}} \rangle = 1.00$ m, ii) $\langle z_{\text{TOF}}^{\text{endcap}} \rangle = 1.50$ m, iii) TOF timing resolution = 20 ps, iv) $B_{\text{field}} = -2.0$ T.</figcaption></figure>
<figure><img src="image_58.png" /><figcaption>**Fig. 57:** Expected TOF performances in terms of particle separations as a function of $p_T$ and $y$ for the configuration already foreseen or possibly anticipated for the ATLAS experiment, with hypotheses : i) $\langle r_{\text{barrel}} \rangle = 1.00$ m, ii) $\langle z_{\text{TOF}}^{\text{endcap}} \rangle = 1.50$ m, iii) TOF timing resolution = 20 ps, iv) $B_{\text{field}} = -1.0$ T.</figcaption></figure>
# X Conclusion

The document has presented the “analytical” figures of merit of various Time-Of-Flight configurations either at a proposal stage or already planned to take place at the HL-LHC, in the experiments ALICE-3, ATLAS and CMS, at the horizon of HL-LHC runs IV (≈ 2027-2029) or V (> 2030). The calculated performances have been first discussed in view of the known performances of the existing TOF sub-detector in the ALICE-1 experiment at the LHC; the case can be considered as a litmus test to comprehend where the limits of the analytical exercise lay. The figures have been derived from analytical formulae describing purely helical trajectories in a solenoidal magnetic field of relativistic charged primary particles. The study is developed under the potentially crude hypothesis that there will be no alteration of any kind of the particle path and energy. This is carried out for different species (e$^{±}$, μ$^{±}$, π$^{±}$, K$^{±}$, p$^{±}$, d$^{±}$, t$^{±}$, $^{3}$He2$^{±}$, $^{4}$He2$^{±}$) allowing to assess the separation power 2-by-2 among the species, depending on the TOF locations, magnetic field intensity and time resolution.

Among the tested configurations, ALICE-3 shows the most extended potential in terms of $p_T$ coverage, towards the low as well as the high transverse momenta.

On the high $p_T$ side, the 3 experiments exhibit more or less the same reach. For instance, for the casual $π^±$, K$^±$, p$^±$, the three experiments and their various configurations tested stay in a similar ballpark of 0.5 GeV/c. At high momentum, the magnitude of the magnetic field is of second-order effects; the impact of the material budget and $A \cdot \varepsilon$ is getting less and less significant for $p_T > 2$ GeV/c. The crucial parameters are the position of the TOF sub-detector (the further away from the collision point, the better) and first and foremost the intrinsic timing resolution $\sigma_{TOF}$. The possible configurations of the various experiments are based more or less on the same distances to TOF, $O(1~\text{m})$ at most for the barrels, $O(2-3~\text{m})$ for the endcaps, meaning that the stress is put, from one instrument to the next, on the $\sigma_{TOF}$ difference.

On the low-momentum side ($p_T < 1$ GeV/c), there, the access to phase space is quite different from one experiment to the other; the intensity of magnetic field is the prime parameter on paper, the material budget and $A \cdot \varepsilon$ will be as well of decisive influences in practice. These are certainly the keys that will make measurements possible or simply forbid them irretrievably. Along the same line but with an impact even more exacerbated, on the ultra-low $p_T$ ($p_T < 0.1$ GeV/c), only ALICE-3 may have the potential to explore such uncharted territories.

All in all, ALICE-3 shows theoretically the most extended potential. But if the potential is the deepest, this is for the moment still only on paper... This raises the question of the competitiveness of ATLAS and CMS on the real ground. ATLAS and CMS already have a starting basis on track. Incomplete, one may say, with some caveats, admittedly, but still in position to contribute on the identification front. It is clear that, in the move from analytics presented here to concrete detector realisations in 5 to 10 years or so, ALICE-3 should be the experiment with the least PID chasm between the theoretical design and the real behaviour : with specifications on total material budget restricted to $O(10\% \times/X_0)$ only at mid-rapidity and similarly at forward rapidities, the distortions and surprises must be minimalised. But this will come with a certain price (after all, it is essentially a brand new experiment...) and with its lot of challenges : the detector prototype still needs to be demonstrated in great details, funded and, ultimately, produced.

ALICE-3 appears as the ideal instrument to perform particle identification measurements in HL-LHC run V. On several aspects, it even seems that measurements will happen with ALICE-3 or nowhere else. The ultimate question can thus become how far the ALICE collaboration and experiment could really go, how far the measurements with the real instrument can be pushed.
1364 Acknowledgements

I would like here to warmly thank Wei Li from Rice University (link), member of the CMS collaboration. The reader may notice similarities between the figures plotted in the present work and the Fig. 1.5 p.8 in the Technical Design Report of MTD detector [8]. And indeed, Wei have kindly shared the original source code producing such a CMS figure, it has been the first inspiration of the current code, adapted and enriched for the present work.

1370 I would also like to thank Iouri Belikov, Daniel Husson and Romain Schotter from the IPHC for their time and contradiction on the mathematical derivation of the analytical formulae; my gratitude also goes to Boris Hippolyte and Yves Schutz as well as Federico Antinori, Jérôme Baudot, David d'Enterria, Tanguy Pierog, Patrick Robbe and Isabelle Wingerter-Seez for having taken the long time of a careful and complete reading, in the frame of the "Habilitation" (HDR) process.
1376

A ROOT code snippet

[ANA-1073 - Version 2.0 - (git rev. dummy) - Tuesday 9th March, 2021, 04:57]
1432 55 lAbsNbHalfTurns_Barrell1 = 0;
1434 56 lAbsNbHalfTurns_Endcap1 = 0;
1436 58 if(pt < lPtMinEndcap) continue; // below the endcap minimum pT, so no detection at all allowed
1438 60 else kAbovePtMinEndcap = kTRUE;
1440 62
1442 64 if(pt > lPtMinBarrel) kAbovePtMinBarrel = kTRUE;
1444 66 // above the endcap minimum pT (loopers, regular endcap, regular barrel)
1446 68 if(pt > lPtMinCalo) kAbovePtMinCalo = kTRUE;
1448 70 // above the reach of barrel calorimeter -> forget about double acceptance (barrel +endcap) loopers
1450 72 Double_t eta = iEta*etastep;
1452 74 Double_t ptot = TMath::Sqrt(p1*pl + pt*pt);
1454 77 Double_t energy1 = TMath::Sqrt(pt*pt + pl*pl + m1*m1);
1456 79 Double_t energy2 = TMath::Sqrt(pt*pt + pl*pl + m2*m2);
1458 81 Double_t rapidity1 = 0.5*log((energy1+pl)/(energy1-pl));
1458 82 Double_t rapidity2 = 0.5*log((energy2+pl)/(energy2-pl));
1458 83 Double_t gamma1 = TMath::Sqrt(1. + ptot*ptot/m1/m1);
1460 85 if( eta > EtaMinLambda) kAboveLambdaMinAngle = kTRUE;
1462 86 if( eta < EtaMaxDelta-etastep && pt > fnPtMinBarrelDelta->Eval(eta) ) kAboveDeltaMinAngle = kTRUE;
1464 87 // NOTE : due to discretisation of eta, one needs to remove one last etastep to avoid rounding effects between EtaMaxDelta and eta = i*etastep
1466 88
1466 90 if( kAboveLambdaMinAngle
1468 91 && kAbovePtMinEndcap
1470 92 && kAbovePtMinCalo == kFALSE
1472 93 && eta < lEtaMinEndcap
1472 94 && kActivateEndcap) kLoop = kTRUE;
1474 95
1476 99 // Wei's Lee code
1478 101 // Wei's Lee code
1478 102 Bool_t kWeiCodeSwitch = kFALSE;
1480 103
1482 105 if(!kWeiCodeSwitch) // Antonin Maire's math core of the code
1482 106
1484 107 Double_t AngFreq1 = (q1*qElem * 1Bz) / (gamma1 * m1InKg); // in rad.s-1, from var with international system unit q*e (C), m (kg), etc
1484 108 Double_t AngFreq2 = (q2*qElem * 1Bz) / (gamma2 * m2InKg);
1486 109
1488 111 Double_t rho1 = pt / (clight * 1Bz * q1);
1488 112 Double_t rho2 = pt / (clight * 1Bz * q2);
Double_t lPhit0 = 0.; // Initial azimuthal angle, by default = 0, could be anything in [0, 2pi] : TMath::Pi()/2. , ... // NOTE : ok for the trajectory // but still to be cured for the other displays : centre of curvature location, tan(t0)... // The physics quantities : path length, TOF time, straight lengths, Rt, rotation angle ... // should be left unchanged by this rotation // NOTE the helix will be rotated in block around the z axis, like a door on its hinges, by the side // so that in terms of acceptance the initial phi(t0) has NO effect, if the detector is invariant by rotation // the final TOF hit will end up always in the same circle, same radius, same z, just at a different location on it. // -> it has been proved here with phi(t0) = 90, 60, 45 deg, all the output looks the same

Double_t lTempX = 0;
Double_t lTempY = 0;
Double_t ltf1 = 0;
Double_t ltf2 = 0;

// Math for potential barrel case

Double_t ltf_Barrel1 = TMath::Abs(1./AngFreq1) * 2 * TMath::ASin( 1RbarrelTOF/ (2. * TMath::Abs(rho1)) );
Double_t ltf_Barrel2 = TMath::Abs(1./AngFreq2) * 2 * TMath::ASin( 1RbarrelTOF/ (2. * TMath::Abs(rho2)) );

Double_t lXtf_Barrel1 = -rho1 * TMath::Sin( AngFreq1 * ltf_Barrel1 ) ;
Double_t lYtf_Barrel1 = -rho1 * ( TMath::Cos( AngFreq1 * ltf_Barrel1 ) - 1);
Double_t lZtf_Barrel1 = betaZ1 * TMath::C() * ltf_Barrel1 ;

1TempX = lXtf_Barrel1;
1TempY = lYtf_Barrel1;

lXtf_Barrel1 = 1TempX * TMath::Cos( 1Phit0 ) - 1TempY * TMath::Sin( 1Phit0 ); // trigo formula cos(a+b) : x = r. cos (phit) -> x' = r. cos( phit + phi0 ) -> x'
= x. cosphi0) - y. sin(

lYtf_Barrel1 = 1TempY * TMath::Cos( 1Phit0 ) + 1TempX * TMath::Sin( 1Phit0 ); // trigo formula sin(a+b) : y = r. sin (phit) -> x' = r. sin( phit + phi0 ) -> y'
= y. cos(

Double_t lXtf_Barrel2 = -rho2 * TMath::Sin( AngFreq2 * ltf_Barrel2 ) ;
Double_t lYtf_Barrel2 = -rho2 * ( TMath::Cos( AngFreq2 * ltf_Barrel2 ) - 1);
Double_t lZtf_Barrel2 = betaZ2 * TMath::C() * ltf_Barrel2 ;

1TempX = lXtf_Barrel2;
1TempY = lYtf_Barrel2;

lXtf_Barrel2 = 1TempX * TMath::Cos( 1Phit0 ) - 1TempY * TMath::Sin( 1Phit0 ); // trigo formula cos(a+b)
lYtf_Barrel2 = 1TempY * TMath::Cos( 1Phit0 ) + 1TempX * TMath::Sin( 1Phit0 ); // trigo formula sin(a+b)

Double_t lRt_tfBarrel1 = TMath::Sqrt( lXtf_Barrel1 * lXtf_Barrel1 + lYtf_Barrel1 * lYtf_Barrel1 );
Double_t lRt_tfBarrel2 = TMath::Sqrt( lXtf_Barrel2 * lXtf_Barrel2 + lYtf_Barrel2 * lYtf_Barrel2 );

Double_t lTransvRotAngle_Barrel1 = 2 * TMath::ASin( 1Rt_tfBarrel1 /2. *1/(-rho1) );
```cpp
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```
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// acceptance check : track ends up below 1RminEndcap, so undo the flag if necessary.
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// if(kLooper && lAbsNbHalfTurns_Endcap1 != 1) kLooper = kFALSE;
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// 2nd thought needed : Reevaluate the assignment in terms of acceptance : only in barrel, only in endcap or in both, finally
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}

if( kWithinRZbarrelAcceptce && kWithinRZendcapAcceptce && kActivateEndcap) {
++iDoubleAcceptance;
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}

if( kWithinRZbarrelAcceptce < 1RminEndcap1 || 1Rt TF Endcap1 > 1RmaxEndcap) {
&& (1Rt TF Endcap1 < 1RminEndcap || 1Rt TF Endcap1 > 1RmaxEndcap)
) kLooper = kFALSE;

// NOTE : For investigations on rotation for loopers : isolate a given number of half turn + apply 1.1427 (~15 lines below)

if( TMath::Abs(1Ztf Barrel1) < 1ZmaxBarrel
&& TMath::Abs(1Rt TF Barrel1 - 1RbarrelTOF) < 2e-3
//&& lAbsNbHalfTurns Barrel1 == 0
) {

kWithinRZbarrelAcceptce = kTRUE; }

if( 1Rt TF Endcap1 < 1RmaxEndcap
&& 1Rt TF Endcap1 > 1RminEndcap
&& TMath::Abs(1Ztf Endcap1 - 1ZendcapTOF) < 2e-3
//&& lAbsNbHalfTurns Endcap1 == 1
) {

kWithinRZendcapAcceptce = kTRUE; }

if( kWithinRZbarrelAcceptce && kWithinRZendcapAcceptce && kActivateEndcap) {
++iDoubleAcceptance;
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}

if( kWithinRZbarrelAcceptce && kAbovePtMinCalo )

kWithinRZendcapAcceptce = kFALSE;

// NOTE : the track, first and foremost a barrel track, will stop in calorimeter,
// so for sure will not manage to further loop back towards the endcap plane
// In practice, it can only be a barrel TOF measurement

if( kWithinRZbarrelAcceptce && kAboveLambdaMinAngle == kFALSE )

kWithinRZendcapAcceptce = kFALSE;

// NOTE : track, first and foremost a barrel track, is here a priori allowed to loop
// - above the TOF barrel radius
// - but below the calorimeter one
// and thus loop towards the endcap !
// But then , the requirement about the minimal inclination angle is not met : lambda too small
// In practice, it will remain a clean measurement only in barrel TOF.

if( kRelaxAngConstraints)

kWithinRZendcapAcceptce = kTRUE;

// NOTE : to visualise the full phase "without constraints (calo, lambda, delta angles)"
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if( cT_diff > n5sigma*cDeltaT ) h5sigm_barrel[ iCan ] ->SetBinContent( h5sigm_barrel[ iCan ] ->FindBin(rapidity1,pt), 1);
if( cT_diff > n4sigma*cDeltaT ) h4sigm_barrel[ iCan ] ->SetBinContent( h4sigm_barrel[ iCan ] ->FindBin(rapidity1,pt), 1);
if( cT_diff > n3sigma*cDeltaT ) h3sigm_barrel[ iCan ] ->SetBinContent( h3sigm_barrel[ iCan ] ->FindBin(rapidity1,pt), 1);
if( cT_diff > n2sigma*cDeltaT ) h2sigm_barrel[ iCan ] ->SetBinContent( h2sigm_barrel[ iCan ] ->FindBin(rapidity1,pt), 1);
if( cT_diff > n1sigma*cDeltaT ) h1sigm_barrel[ iCan ] ->SetBinContent( h1sigm_barrel[ iCan ] ->FindBin(rapidity1,pt), 1);
}

if( IsPartForeseenInEndcap && kAboveLambdaMinAngle) {
    if( cT_diff > n5sigma*cDeltaT ) h5sigm_ecap[ iCan ] ->SetBinContent( h5sigm_ecap[ iCan ] ->FindBin(rapidity1,pt), 10+1AbsNbHalfTurns_Endcap1);
    if( cT_diff > n4sigma*cDeltaT ) h4sigm_ecap[ iCan ] ->SetBinContent( h4sigm_ecap[ iCan ] ->FindBin(rapidity1,pt), 10+1AbsNbHalfTurns_Endcap1);
    if( cT_diff > n3sigma*cDeltaT ) h3sigm_ecap[ iCan ] ->SetBinContent( h3sigm_ecap[ iCan ] ->FindBin(rapidity1,pt), 10+1AbsNbHalfTurns_Endcap1);
    if( cT_diff > n2sigma*cDeltaT ) h2sigm_ecap[ iCan ] ->SetBinContent( h2sigm_ecap[ iCan ] ->FindBin(rapidity1,pt), 10+1AbsNbHalfTurns_Endcap1);
    if( cT_diff > n1sigma*cDeltaT ) h1sigm_ecap[ iCan ] ->SetBinContent( h1sigm_ecap[ iCan ] ->FindBin(rapidity1,pt), 10+1AbsNbHalfTurns_Endcap1);
}

if( kLooper) {
    if( cT_diff > n5sigma*cDeltaT ) h5sigm_loop[ iCan ] ->SetBinContent( h5sigm_loop[ iCan ] ->FindBin(rapidity1,pt), 10+1AbsNbHalfTurns_Endcap1);
    if( cT_diff > n4sigma*cDeltaT ) h4sigm_loop[ iCan ] ->SetBinContent( h4sigm_loop[ iCan ] ->FindBin(rapidity1,pt), 10+1AbsNbHalfTurns_Endcap1);
    if( cT_diff > n3sigma*cDeltaT ) h3sigm_loop[ iCan ] ->SetBinContent( h3sigm_loop[ iCan ] ->FindBin(rapidity1,pt), 10+1AbsNbHalfTurns_Endcap1);
    if( cT_diff > n2sigma*cDeltaT ) h2sigm_loop[ iCan ] ->SetBinContent( h2sigm_loop[ iCan ] ->FindBin(rapidity1,pt), 10+1AbsNbHalfTurns_Endcap1);
    if( cT_diff > n1sigma*cDeltaT ) h1sigm_loop[ iCan ] ->SetBinContent( h1sigm_loop[ iCan ] ->FindBin(rapidity1,pt), 10+1AbsNbHalfTurns_Endcap1);
}

// end loop iEta
// end loop pt
References

[1] F. Antinori, A. Morsch, L. Musa, et al., A next-generation LHC heavy-ion experiment, arXiv, Input for the 2019 CERN council Open Symposium in Granada (May 2019), in view of the 2020 update of the European Strategy for Particle Physics, Feb. 2019. arXiv:1902.01211.

[2] J. W. Holt, M. Rho, and W. Weise, Chiral symmetry and effective field theories for hadronic, nuclear and stellar matter, Phys. Rept. 621 (2016) 2–75, [arXiv:1411.6681].

[3] F. Low, Bremsstrahlung of very low-energy quanta in elementary particle collisions, Phys. Rev. Lett. 110 (1958) 974–977. DOI Link.

[4] A. Strominger, Lectures on the Infrared Structure of Gravity and Gauge Theory, arXiv:1703.05448.

[5] C. Lippmann, Particle identification, Nucl. Instrum. and Methods A 666 (2011) 148–172, [arXiv:1101.3276]. DOI Link.

[6] Fluka collab., Fluka home page, Web, 2020. www.fluka.org/.

[7] Geant4 collab., Geant4 home page, Web, 2020. /geant4.web.cern.ch/.

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