Jet timing

The measurement of the arrival time of a particle, such as a lepton, a photon, or a pion, reaching the detector provides valuable information. A similar measurement for a hadronic final state, however, is much more challenging as one has to extract the relevant information from a collection of particles. In this paper, we explore various possibilities in defining the time of a jet through the measurable arrival times of the jet constituents. We find that a definition of jet time based on a transverse momentum weighted sum of the times of the constituents has the best performance. For prompt jets, the performance depends on the jet trajectory. For delayed jets, the performance depends on the trajectory of the jet, the trajectory of the mother particle, and the location of the displaced vertex. Compared to the next-best-performing jet time definition, the transverse momentum weighted sum has roughly a factor of ten times better jet time resolution. We give a detailed discussion of the relevant effects and characterize the full geometrical dependence of the performance. These results highlight the critical importance of using a proper definition of jet time with its corresponding detector-dependent calibration and the exciting possibility of deepening our understanding of jets in the time domain.


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detector as a collection of particles, with a corresponding collection of arrival times. A jet is the standard object that combines these particles into a single object that can be used in analyses and searches. In momentum-space variables, summing the four-vectors of the constituents provides a natural definition of the four-vector of the jet. Unfortunately, there is not an obvious choice for the definition of the arrival time of a jet.
The selection of a proper jet time definition is pivotal. A proper definition will enable efficient separation of the Standard Model (SM) prompt background and BSM long-lived signatures. A poor definition, on the other hand, will not allow us to take full advantage of the precision timing capabilities at the level of 30 − 40 picoseconds, that will be part of upgrades to ATLAS [39], CMS [40], and LHCb [41]. Already, CMS has demonstrated sensitivity to delayed jets in their search for displaced gluinos [42].
Beyond just performance, a proper definition of jet time may help identify exciting properties of quantum chromodynamics (QCD), enable new jet tagging possibilities, and provide additional inputs for machine learning applications. Even pileup suppression may benefit substantially from an effective usage of jet time because generically pileup vertices have a spread both in space and in time. At the high-luminosity LHC any improvements to pileup suppression are indispensable.
The purpose of this paper is to explore a variety of definitions of jet time and characterize their performances. As with any measurement tool, there are two aspects: accuracy and precision. For jet time, as we will discuss in detail later, the "correct" time is somewhat ambiguous. The precision, or resolution, is well-defined and will be the main criterion in comparing different approaches.
The structure of the paper is as follows. In section 2 we provide a brief overview of various possible definitions of jet time. The general behavior, both for prompt jets and delayed jets, is discussed in section 3. In section 4 we perform an in-depth numerical study of the behavior of each jet time definition, paying special attention to the dependence on the event geometry. Finally, our conclusions are in section 5 along with outlook for future studies. Several appendices are included for cross-checks and studies of additional effects. We discuss the behavior of jet time when endcaps are also used to measure arrival times in appendix A, the impact of pileup and jet grooming in appendix B, the effects of detector resolution in appendix C, and the impact of hadronization in appendix D.

Definitions for jet time
In this section we briefly describe the definitions of jet time that we study. More detailed descriptions will follow in section 3 and simulation results will be shown in section 4. We first define our notation. A single particle i has a four-momentum (E i , p i ) and a particle time t i , which is the time that it crosses a particular layer of the detector. A jet J is a set of particles which we write as J = {i}. While a particle has an unambiguous time, the jet has a set of times {t i } associated to it. In the same way that it is often useful to treat the jet as a single four-vector, e.g. in new physics searches, it is also useful to be able to assign a jet a single time t J , that we call the jet time.

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There are a number of possibilities that can be used. One can choose a single constituent i in the jet and use its particle time t i to represent the jet time. Jet time definitions of this type include: • median time: take t J to be the median value of the particle times {t i }, • hardest time: take t J to be the time t i h , that corresponds to the time of the constituent i h with the largest transverse momentum, • random time: take t J to be a randomly-drawn value of the particle times {t i }.
The median time has been used by CMS in their search for gluinos with displaced decays [42]. The hardest particle in a jet is likely to be very close to the jet axis so it may be a good proxy for the time of the jet. We do not expect choosing a random particle time as the jet time to perform well, but it is useful as a baseline comparison. Another option is to calculate t J from a weighted sum of {t i }, similar to a jet shape. Generically, this would take the form 2 where ∆R i is the ηφ-distance between particle i and the jet axis. The two simplest versions of a weighted sum are the average time where (α, β, γ) = (0, 0, 1) and the p T -weighted time where (α, β, γ) = (1, 0, 1): 3 • average time: take t J to be where there are N particles in the jet, • p T -weighted time: take t J to be where H T,J is a normalization factor.
Finally, one could simply disregard the particles times, treat the jet as a single particle, and calculate its time based on the jet kinematics. There are two variations depending on whether the jet is treated as a massless particle or massive particle: • null time: treat the jet J as a massless particle and calculate its crossing time using the three-momentum p J of the jet (assuming knowledge of the production vertex), 2 A number of alternatives are possible, such as particle energy E in place of pT or angle θ in place of ∆R. In a brief survey, we did not find these to outperform the variables used in eq. (2.1). 3 We briefly studied a few additional cases such as p 2 T -weighted but did not see an improvement over the pT -weighted time or average time. An optimization of α, β, and γ is beyond the scope of this work.

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• kinematic time: treat the jet J as a massive particle and calculate its crossing time using the four-momentum (E J , p J ) of the jet (assuming knowledge of the production vertex).
Since these definitions do not utilize the information available from timing measurements, they do not correspond to experimentally measurable arrival times. They are useful, however, to determine what constitutes good performance from a jet time definition. These times are constant for fixed parent particle trajectory and jet kinematics. In addition, the null time represents the crossing time if the parton did not undergo showering and hadronization and will serve as a useful reference time. 4,5 After having chosen a definition for t J we also need to choose a metric to evaluate which definition is the most useful. For a jet time definition, we will compare the relative time difference ∆t/t ref defined as When the particle i points along the same direction as the jet axis, the relative time difference is always zero. When there is a fixed angular distance ∆η between the particle i and the jet axis, however, the relative time difference changes with η J . Due to the detector geometry, as the jet becomes more forward, the relative time difference will grow. The furthest that a particle i can be from the axis of the jet is approximately given by the jet radius R jet . Therefore, for a given η J there is a maximum relative time difference given by ∆t where the + applies for positive η J and the − applies for negative η J . Figure 1. Illustration of the bounds for a prompt jet (left). The slowest particle time, eq. (3.7), follows the high |η| boundary of the jet (blue). The fastest particle time, eq. (3.8), follows the low |η| boundary of the jet (green) or the η = 0 line (red). Plot of the boundaries for the relative time difference for a jet with R jet = 0.5 (right) where the colored lines correspond to the boundaries in the left image.

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The minimum is similar except that for |η J | < R jet there is a stronger bound that comes from the fact that η i = 0 for a massless particle corresponds to the fastest time possible since it is the shortest path from the origin. The bound for |η J | < R jet consequently only depends on η J . We find Eqs. (3.7) and (3.8) taken together specify boundaries in the space of pseudorapidity vs. relative time difference. The different jet time definitions will have different distributions within these boundaries. Figure 1 illustrates these boundaries graphically. At η J = 0 the minimum of ∆t/t ref is 0 while the maximum is cosh(R jet ) − 1 > 0 which means that the relative time difference cannot be negative. Therefore, for very central jets we expect the relative time difference distributions to skew towards positive values since there is more available phase space. For less central jets we do not expect a strong preference for positive or negative values based only on phase space.
For the kinematic time rather a bound, one can relate the relative time difference to a kinematic quantity. From eqs. (3.4) and (3.5) we find that for the kinematic time where β J is the velocity. Since β J ≤ 1 the relative time difference for the kinematic time is always non-negative. This is expected since the kinematic time points in the same direction as the null time, but adjusts for the mass of the jet. Next, we move to the relative time difference for the p T -weighted time. This has a simple form given our cylindrical detector model. The p T -weighted time is Written in the form after the second equality we recognize the factor E J /| p J | ≥ 1 from eq. (3.9). The other factor p T,J /H T,J is the ratio of the jet p T to the scalar sum of the constituent p T values. Since p T,J is a vector sum we have p T,J /H T,J ≤ 1. The distribution at a given η J is determined by the η-dependence of each of these two terms. For small mass jets E J /| p J | ≈ 1 + m 2 J / p J 2 and schematically for QCD jets the mass is [43]. Consequently, E J /| p J | ∼ 1 + R 2 sech 2 η J which peaks at η J = 0 and reduces as |η J | grows. The other quantity p T,J /H T,J depends on the energy distribution in the jet and is not strongly correlated with η J . Therefore, we expect that the relative time distribution for the p T -weighted time to be positively-skewed for central jets and switch over to negatively-skewed as the jets become more forward.

Delayed particles
Next, we study the jet time behavior for delayed particles. Our benchmark scenario involves a mother particle M that travels a macroscopic distance, then decays into two daughter particles D and D. We assume that M and D are unobserved while D is colored and results in a jet due to showering and hadronization. 8 In our numerical study we take M as a gluino, D as a gravitino, and D as a gluon.
Let the mother M have four-momentum (E M , p M ) and decay at the displaced vertex x M at a time t M . The daughter D showers and hadronizes into a delayed jet. A particle i in the delayed jet has a four-momentum (E i , p i ) and originates from x M at t M . Let the vector pointing from x M to where i crosses the detector be x i . 9 Since | x i | only makes up a fraction of the total traversed distance, the impact of the per-particle time spread is generically lessened for delayed jets. We will refer to this effect as the daughter time fraction.
If particle i is measured, but the displaced vertex is not identified, then i will be assumed to have come from the origin, having traveled along x i = x M + x i . This is illustrated in figure 2. We call the kinematics computed using x i the observed kinematics and those using x i the truth kinematics.
four-vector assumed trajectory For transverse displacements 10 cm the tracking efficiency is 40% in CMS and drops off further above 50 cm [44]. Conservatively, we assume that the displaced vertex is not 8 If M or D (or both) are colored, they will propagate as color-neutral R-hadrons. They will be unobserved if the resulting R-hadrons are electrically-neutral. 9 For simplicity we neglect the effects of curvature in the magnetic field of the detector. For a magnetic field of 3.8 T the effect on the measurement of time or momentum is less than 1% for particles with pT > 2.5 GeV. In appendix B where we study pileup, we do include curvature induced by the magnetic field. Figure 2. The mother particle M travels along x M and decays to a delayed jet (shaded brown). The daughter particle i travels along x i until it crosses the detector. If the displaced vertex is not identified then i is assumed to have traveled along x i .

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identified and work with the observed kinematics. The most direct impact of this is that the transverse momentum of any given constituent can be incorrectly assigned. This occasionally leads to noticeable broadening of time definitions that directly depend on the constituents' p T . While those that do not, e.g. the median time, are mostly insensitive to whether the truth or observed kinematics are used. A more subtle consequence of using observed kinematics is that the observed opening angle does not correspond to the true opening angle, leading to the inclusion/exclusion of particles with extreme arrival times. We will refer to this discrepancy in opening angles as the effective radius.
In the following, we study in detail these three primary effects which control the performance of the timing of a delayed jet: the daughter time fraction, observed kinematics, and the effective radius.
Observed kinematics. For a massless particle i, if it is prompt its time is fully specified by its pseudorapidity η i (see eq. (3.2)). When i is delayed, its time depends now on its pseudorapidity η i , the pseudorapidity of the mother η M , the azimuthal angle difference ∆φ = φ i −φ M , the speed of the mother β M , and the transverse decay location of the mother x T,M : (3.13) The transverse distance x T,i traveled by i is calculated to be The observed kinematics, (p T,i , η i , φ i ), can be computed in terms of the variables in eq. (3.13) and x T,i . The observed pseudorapidity η i is found from solving In terms of the true transverse momentum p T,i , the observed transverse momentum p T,i is Finally, the observed azimuthal angle φ i is Jets are clustered using the observed kinematics. The time t i of a particle i is not impacted by using observed kinematics because the arrival time of a particle is an independent measurement. Since x i and x i cross the detector at the same location, the effect on clustering using observed kinematics is nearly negligible (comparable to the difference between different jet algorithms). The primary impact of using observed kinematics is on jet time definitions that utilize p T information. We expect the p T -weighted time to be impacted at a noticeable level (the size of this effect will be studied in section 4). The hardest time could be affected if which jet constituent is the hardest changes under the observed kinematics. In practice, this is rare due to the hierarchical nature of the parton shower. The median time, likewise, is minimally affected.
Effective radius. The radius of a jet, R jet , is a parameter in the jet finding algorithm that determines which particles are included in the same jet. It determines the catchment area of a jet in ηφ-space which is approximately a circle with radius R jet for cone-like algorithms used on isolated jets [45].
When choosing a jet radius there are trade-offs. If the radius is too small, then particles coming from the showering of a hard particle could fall outside a particular jet and the jet will not be a useful proxy for the underlying hard quark or gluon. If the radius is too large, the jet is more susceptible to contamination like underlying event and pileup [43].
One consequence for prompt jets of using a fixed R jet for jet finding is that an optimal jet radius may be different for central jets as compared to forward jets. This is because for a fixed R jet in ηφ-space, the corresponding angular distance, ∆θ, between a pair of particles is smaller for forward jets than for central jets. Physically, if a set of central particles within a jet with radius R jet were shifted to larger |η| values, then they may not all fit within a radius R jet anymore.
In the prompt case, we consider the effective radius of the jet to be the angular distance ∆θ that is required to keep ∆R fixed. This means that forward jets have a smaller effective JHEP01(2022)014 radius than central jets because for fixed ∆R the required ∆θ distance shrinks. See figure 3 for an illustration. Variable R jets were proposed to account for this by letting the jet radius grow at larger |η| by scaling the radius inversely with transverse momentum [46].
In contrast, for delayed jets we consider the effective radius of a jet to be the true ∆R distance that is required to keep observed ∆R fixed. There are two factors that alter the effective radius for delayed jets. The first is that a non-zero value of x T,M means the jet originates closer to the detector radius r T . The same way that the image from a projector is smaller as you move the projector closer to the screen, a fixed observed ∆R value corresponds to a larger true ∆R value as x T,M grows. See figure 4 for an illustration.
The second effect is that both η J and η M can vary. Changing η J tilts the direction of the jet and generally causes the effective radius to shrink with η J similar to the prompt case. Changing η M is not a tilt, but rather a shift of the origin point of the particles. Due to the geometry of ηφ-space the effective radius generally increases as |η M | grows.
The effective radius can be estimated numerically. As shorthand we write the observed pseudorapidity of a jet as η J = f (η M , x T,M , η J ) where the function is found in eq. (3.15). We define the effective jet radius as R eff and find it by solving The definition is not rigorous but rather is meant to provide intuition for the general behavior. We also set ∆φ zero in the above for simplicity. In figure 5 we plot R eff as a function of η J for several sample points of x T,M and η M with fixed jet radius R jet = 0.5.
For isolated QCD jets, we expect the jet properties to change slowly with respect to increasing R eff . In a typical parton shower there are both more and higher momentum particles near the center of the jet. Including additional soft particles further from the jet JHEP01(2022)014  Figure 6. The time of flight of the mother particle is t M and the time of flight of a daughter particle is t D .
axis will not perturb the jet four-vector by much. When R eff decreases the jet properties should change faster as more and higher momentum particles are excluded.
Daughter time fraction. The third effect is the fraction of time that comes from time of flight of the daughter i as compared to the time of flight of the mother, as shown in figure 6. Intuitively, when the mother travels more of the distance from the origin to the detector, there is less variation among the times of the particles in a jet. Consequently, the distribution of jet times becomes narrower.
The time of a delayed particle i is The null time of a jet, which we continue to use as the reference time, is t null Let us first consider the relative time difference for the median time so that i = i m . We have representing the "null" time from the daughter segment. If we approximate x T,i = x T,J (i.e. the jet is narrow) and particle i as massless, then we find The first factor is the fraction of the particle's time that is traveled by the daughter and the second factor we recognize from eq. (3.6) as the prompt distribution evaluated at particle i's true pseudorapidity. As the distance the daughter travels, x T,J , shrinks, so does the spread in ∆t/t ref . In figure 7, we plot the suppression factor, t D /(t M + t D ), as a function of η J for several sample points of x T,M and η M using β M = 0.4 Delayed relative time difference. Here we briefly review our expectations for the relative time difference in delayed jets. Recall that for single particle measures, like the median or the hardest, the prompt relative time is given by The three effects that cause the delayed distribution to differ from eq. (3.23) are: • the daughter time fraction, • the effective radius of the jet, • the difference between observed and truth kinematics.

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Let us now contrast a few jet time definitions to assess the impact of each delayed effect on the relative time distribution. We start with the hardest time. The difference in observed kinematics should have a negligible effect except in rare instances when the hardest particle in a jet changes between observed and truth kinematics. The effective radius should also have a minimal effect because the hardest particle in a jet tends to be near the jet axis. The daughter time fraction, however, is an irreducible effect.
From eq. (3.22) we see that the delayed distribution inherits the prompt dependence on the daughter's true pseudorapidity, but with an additional suppression from the fact that spread between particles occurs over a smaller distance. The suppression comes from the prefactor .
The times scale with their respective pseudorapidities, t M ∝ cosh(η M ) and t D ∝ cosh(η i ), so that the prefactor is closest to 1 when η D ≈ 0 and |η i | is large, and closest to 0 when η i ≈ 0 and |η D | is large. The prefactor can range from 0 to 1 and it plays a large role in the relative time difference distribution.
Next, we consider the median time. Again, we expect the observed kinematics to have a negligible effect on the relative time difference. The effective radius, however, can now have an impact because each particle has an equal effect on the median time of a jet. As R eff grows, particles further from the jet axis are included in the jet and in the calculation of the jet time. Being far from the jet axis, these particles act like noise for the particle time distribution leading to more variation in the relative time difference distribution. Conversely, a shrinking R eff will tend to narrow the distribution somewhat. The daughter time fraction is irreducible and has an O(1) effect on the median time.
Finally, we consider the p T -weighted time. The daughter time fraction is again a driving effect. The impact of the effective radius should be smaller than in the median case, because particles far from the jet axis are typically soft so their contribution to the p T -weighted time is suppressed by their p T . The observed kinematics, however, can now have a large effect. The p T -weighted time for delayed jets is From eq. (3.16) we see that the ratio p T,i /p T,i is independent of momentum. This means in the infinitely-narrow jet limit the p T -weighted time is not affected by the observed kinematics. Beyond this limit, the effect of using the observed p T can be large if the variation of p T,i /p T,i is large over the area of the jet. The ability to accurately identify displaced vertices can eliminate the impact of the observed kinematics. Such an upgrade would be expected to improve the performance of the p T -weighted time, but have a small effect on the hardest time and the median time.

JHEP01(2022)014 4 Numerical results
In this section, we compute the relative time differences for several jet time definitions in simulated data. Results will be compared with the derived behaviors from section 3 and are found to follow the predicted trends.

Simulation details
For prompt jets, we generate pp → Z → qq events, where q = u, d, using Pythia v8.240 [47] at a center of mass energy of √ s = 14 TeV and with a Z mass of m Z = 1 TeV. Initial state radiation (ISR) and multiparton interactions (MPI) are turned off. Particles with p T < 0.5 GeV or with |η| > 4 are discarded. Particles are clustered into anti-k T jets [48] with R jet = 0.5 using FastJet v3.3.2 [49]. The results are presented at particle-level without any detector resolution or time resolution included. The impact of these effects is shown in appendix C to be small.
For delayed jets, we generate pp →gg → (gG)(gG) at parton-level using MadGraph5 v2.7.3 [50] at a center of mass energy of √ s = 14 TeV and with particle masses of mg = 2 TeV and mG = 10 −16 TeV. 10 Events are showered with Pythia with ISR and MPI turned off. Particles with an observed transverse momentum below p T < 0.5 GeV or with an observed pseudorapidity |η | > 4 are discarded, where a cylindrical detector with a radius of r T = 1 m is used.
In both the prompt and delayed samples, hadronization is turned off. In Pythia, when there are both prompt and displaced particles, due to the hadronization procedure, some particles that descend from the displaced gluon can be assigned to the prompt vertex. Using unhadronized events avoids the issue of determining to which vertex a hadron should be assigned. In appendix D we compare the relative time distributions, in a prompt sample, with and without hadronization and find that the impact is at most a few percent.

Prompt jets
We first look at prompt jets because the prompt distributions are inputs to understanding the delayed distributions. In each event we only consider the hardest jet and require that it has p T > 250 GeV.
In figure 8 (left) we show the distribution of ∆t/t ref for jets with |η J | < 0.5 for the jet time definitions of p T -weighted, median, hardest, average, and random. As expected, selecting a random particle in the jet to represent the jet time yields the widest distribution. of the resolution of a method and a useful figure of merit. Since these distributions are non-Gaussian, the 1σ standard deviation does not fully characterize the shapes, and in particular does not provide useful information about the tails. For that reason, we use the 3σ width (i.e. the bounds of the integral containing 99.7% of events) for comparison. 11 With this as the resolution, the p T -weighted time performs 5 times better than the hardest time and 6 times better than the median time.
In figure 8 (right) we look at the relative time distribution for jets with 1.0 < |η J | < 1.5. The same pattern is present here where the random time is the widest distribution, followed by the median, hardest, and average times with similar widths, and the p T -weighted time with the narrowest distribution. Again, comparing the resolutions, we find that the p Tweighted time is 16 times better than the hardest time and 17 times better than the median time. The distributions of the median and hardest times widen noticeably in this 1.0 < |η J | < 1.5 range, as compared to |η J | < 0.5, as predicted by eqs. (3.7) and (3.8). The p T -weighted time instead depends on the interplay between E J /| p J | and the ratio of transverse momentum to the scalar sum of the constituents' transverse momenta.
To better understand the difference between η J regions, we look at the two-dimensional distribution of ∆t/t ref vs. η J . Figure 9 shows this distribution for random time with the bounds from eqs. (3.7) and (3.8) overlaid. The majority of events fill out the region between the bounds with a few events above the maximum, due to mass effects, and a few events below the minimum due to the fact that particles can be slightly farther than R jet from the jet axis. Figure 9 (right) shows ∆t/t ref vs. η J for the median time. The red solid lines are the boundaries from eqs. (3.7) and (3.8). This distribution clusters closer around ∆t/t ref values near zero. In fact, the dashed lines are boundaries for a jet with radius (2/3)R jet which corresponds to the empirical observation that the behavior of the median time is similar to choosing a random particle from a narrower jet.
In figure 10 we show the same distribution for the p T -weighted time. Here, we see that the behavior predicted by eq.  time differences near η J = 0 result from the E J /| p J | factor. The shape in that region even follows sech 2 (η J ) as discussed in section 3.1. As |η J | grows past ≈ 2 the E J /| p J | factor approaches unity and the p T,J /H T,J factor determines the shape. Both of these factors have a narrow distribution leading to an overall narrow distribution for the relative time difference for the p T -weighted time. and decays at a transverse distance x T,M to a gluon that points along η J at parton-level. This same event is then re-showered 100,000 times to produce a sample of jets with fixed parton-level kinematics. We consider only the hardest jet (that originates from inside the detector) in the event and require it to have an observed p T > 50 GeV. In order to identify effects that are dependent on the event topology, we discard events that differ by more than 0.25 in pseudorapidity or 0.25 in azimuthal angle before and after showering.

Delayed jets
We first look at distributions of ∆t/t ref in figure   slight suppression of ≈ 0.6 from the daughter time fraction. This is due to a sizable variation between the observed and truth kinematics. The median and hardest distributions do not differ much from their prompt counterparts. Figure 11 (bottom right) shows the distributions for (η J , η M ) = (−2, 2). In this case, each distribution is very narrow. This is primarily a consequence of the jet having R eff = 0.27. 12 While the p T -weighted distribution is still narrow, the difference is not as large as for other jet times because of the discrepancy between observed and truth kinematics for this configuration. . In these two plots, we see very different behavior as x T,M increases. The left plot shows the distribution narrowing with x T,M , the expected behavior from the decrease in the daughter time fraction. By contrast, the right plot shows the distribution broadening and skewing positive with x T,M . This broadening is the expected behavior from both the larger effective cone size, and larger difference between the true and observed p T . Furthermore, the relative increase in observed p T is larger forward particles, resulting in the positive skew.
The parameter scans in figures 11 and 12 are useful for emphasizing a few physics points. Firstly, the p T -weighted time is consistently better than other jet time definitions across the variation of key kinematics, namely η M and η J . Secondly, the p T -weighted time has a different spread at different kinematical points which means one should compute calibrations and efficiencies at each point rather than using a single value over all of parameter space. 13 12 Note that because the particles in a jet are not uniformly distributed, excluding particles that are farther from the jet axis does result in a narrower relative time difference distribution. 13 A parameter scan of (ηJ , ηM ) with finer steps can be found in appendix E. The trends shown there agree with the analysis provided here.

Conclusions
The time of a jet is a theoretically ambiguous and yet experimentally highly relevant quantity. The time profile of a jet provides a new independent probe of jet properties, potentially deepening our understanding of QCD. Experimentally, the jet time is an observable with strong discrimination power in searches for long-lived particles. Like how the jet clustering algorithm itself defines a jet using a collection of particles, the choice of jet time definition determines its properties and performance. A useful definition should have predictable behavior, give the closest representation to the parton-level information, and, more importantly, minimize the spread in arrival time.
In this work, we primarily studied five definitions of jet time. The first was the p Tweighted time where the jet time is a p T -weighted sum of the jet constituent arrival times. The second was the median time which uses the median constituent time as the jet time. The third was the hardest time where the time of the highest p T constituent is used as the jet time. The fourth was the average time where the jet time is taken as the average of the constituent times. The fifth was the random time where the time of a constituent was randomly chosen to be used as the jet time.
To evaluate the various definitions, we both predicted and computed in simulation the relative time difference of a definition compared to the time it would take a massless parton to travel along the jet's trajectory. The width of the relative time difference distribution tells us how precisely the jet time can be measured. For prompt jets, we showed that the performance depends on the pseudorapidity of the jet. Due to the geometry of the detector barrel, all jet time definitions have wider distributions as the jets become more forward. We found that the p T -weighted jet time consistently has the best performance. For instance, for central jets with 1.0 < |η| < 1.5, the p T -weighted time has a 16-fold improvement over the (next-best-performing) hardest time. For central jets with |η| < 0.5, the p T -weighted time has a 5-fold improvement over hardest time.
For delayed jets, the full kinematics of the event affects the performance. Specifically, the direction of the mother particle, the direction of the jet, and the transverse decay location of the mother particle determine the behavior of the jet times. We show that delayed jet timing behavior can be understood through three effects. The first is the daughter time fraction which is the fact that as the displaced vertex gets closer to the detector there is less distance for the constituents to travel and consequently less spread in their times. The second is the effective radius of the jet that is an effect of the displaced vertex. The third is the that observed p T differs from the true p T , which occurs when the displaced vertex is not identified. Just as for prompt jets, the p T -weighted time has the best performance over the full parameter space. Furthermore, the strong dependence on the event kinematics emphasizes the importance of having an efficiency map that depends on the long-lived particle's direction, its decay location, and the direction of the daughter jet.
This work is the first study that looks at the impact of different definitions of jet time. There are many related directions that can be explored. For instance, finding the jet time definition that is most amenable to direct calculation may help reduce theory uncertainties. More practically, given the trigger computation complexity budget, it would be useful to JHEP01(2022)014 understand the best alternative jet time definition for a low-level delayed jet trigger. On the analysis side, studies could be done on the interplay between jet time and pileup and grooming. Other new physics models with different event topologies would be interesting to study. More detailed signal-specific studies are needed to evaluate the direct impact of using the jet time in new physics searches. We are optimistic that the jet time has the potential to be a standard tool in long-lived particle searches in the near future.

A Finite length detectors
In the main text, we consider a detector with only a barrel capable of timing measurements. If one includes endcaps, then the timing distributions are different for jets with times that the endcaps would measure.
If the pseudorapidity at which the barrel connects to the endcap is η EC , then the arrival time of a particle i at the endcap is where z EC = r T cosh η EC . If all of the jet's constituents lie solely in the endcap, the trajectory that yields the shortest (largest) arrival time is now the most forward (central) constituent.
For jets with constituents in the intermediate region, the trajectory that yields the largest arrival time is always the trajectory intersecting the barrel-endcap corner. Depending on the jet axis, the shortest arrival time can be a constituent that intersects the barrel or the endcap.
In figure 13 we show the maximum and minimum relative time differences (for a singleparticle measure) for r T = 1 m and a total barrel length of L = 6 m which corresponds to η EC = 1.76. 14 Once all jet constituents lie within the endcap, the allowed spread in relative time difference sharply drops for prompt jets. pileup vertices follows a Gaussian spread in both z and t, with σ z = cσ t = 60 mm [53].

B Pileup and grooming
The events with pileup include all of the detector effects discussed in appendix C. The relative time differences without any form of pileup mitigation are shown in figure 14 for |η J | < 0.5 (left) and 1.0 < |η J | < 1.5 (right). The average time distribution gets distorted for the |η J | < 0.5 bin, and the peak shifts away from zero considerably. Like those from pileup, low-energy particles have a smaller curvature radius from the magnetic field and are therefore delayed more than higher-energy particles. Since a sizable fraction of a jet's constituents can come from pileup, this causes the average time to shift considerably. The same reasoning is responsible for the broadening of the distributions of the median and p T -weighted times. The hardest time is affected very little. The 1.0 < |η J | < 1.5 bin shows less impact from pileup as can be seen, for example, by the peak of the average time distribution remaining close to zero. Similarly, the distributions of the other times broaden slightly, but their peaks do not shift. This is the result of the p T cut restricting to more energetic particles at larger pseudorapidities.
For pileup mitigation, we use an idealized version of charged hadron subtraction [54] where we assume all charged pileup can be removed. The remaining particles were then clustered into R jet = 0.5 anti-k T jets and trimmed [55] with R sub = 0.2 and f cut = 0.03. The choice of keeping R jet the same is to ensure that the jet times with and without pileup mitigation are directly comparable. The distributions for the |η J | < 0.5 bin are shown in figure 15. The improvement is predominantly due to the removal of soft charged pileup particles by charged hadron subtraction. These constituents are the ones that are mainly delayed by mass effects and the magnetic field. Trimming plays a minor role because R sub is not significantly smaller than R jet and the number of pileup vertices is large. More aggressive trimming may improve results slightly. 15

C Detector effects
In this section we show the effects of implementing a simple detector model. We first implement time resolution, followed by time and spatial resolution. The effects are shown in a prompt sample for |η J | < 0.5 in figure 16 and for 1.5 < |η J | < 2.0 in figure 17.
For time resolution, we round each particle's time to the nearest multiple of 30 ps. This is the expected resolution of LHC upgrades [39][40][41]. 16 The effect on the ∆t/t ref distribution can be seen by comparing the left plots (no timing resolution) to the center plots (30 ps timing resolution) in figures 16 and 17. In both cases, the time resolution has a negligible effect on the shape of the distribution.
For spatial resolution, we consider an η × φ grid of 0.05 × 0.05 cells. The four-momenta are replaced with a massless four-vector with the same energy as the particle, and the direction shifted pointing to the center of the corresponding ηφ-cell. If multiple particles fall into the same cell and the same time window, their energies are added, and they are combined into a single cell. The effect on the ∆t/t ref distribution can be seen in figure 16 (right) and figure 17 (right).
In the |η J | < 0.5 bin, we see that spatial resolution does not significantly affect the timing distributions. By contrast, the 1.0 < |η J | < 1.5 bin has a noticeable broadening in 15 One could also study the performance using pileup mitigation techniques that are better suited to large values of n PU such as jet cleansing [56], constituent subtraction [57], PUPPI [58], soft killer [59], or PUMML [60]. This is beyond the scope of this work. 16 This time resolution does not include non-Gaussian tail effects (e.g. hot cells) which may affect perparticle arrival time measurements. These effects are very detector-dependent and is beyond the scope of this work. the p T -weighted distribution and moderate broadening in the median and hardest distributions. This is because the fractional momentum resolution induced by the spatial resolution increases with |η|. This impacts both the momentum of the jets and their constituents.
Lastly, we considered the impact of including a 4T magnetic field. In this case, the particles were hadronized (in order to get the correct electric charge of the hadrons), and at the same time and spatial resolution was applied. The effect on the relative time distribution for both bins are shown in figure 18. There is a very slight positive pull in the |η J | < 0.5 bin (left) while the 1.0 < |η J | < 1.5 bin (right) has no noticeable change. This difference is due to the p T > 0.5 GeV cut imposed on the constituents. As η increases, the energy required to pass the p T cut also increases. The shift in arrival time due to the change in path length is inversely proportional to the energy. Hardest time Figure 19. The relative time difference distributions with and without hadronization for the p Tweighted time (left), median time (center), and hardest time (right) with 1.0 < |η J | < 1.5.

D Hadronization
In this study the events are not hadronized to ensure that Pythia assigns the correct vertex to each delayed particle. Figure 19 compares the relative time difference for prompt jets with and without hadronization for the p T -weighted time (left), median time (center), and hardest time (right) in the range 1.0 < |η J | < 1.5. Of the three, only the median time shows a slight observable change.

E Parameter scans
In figure 20 we show a scan over η J in the x-direction and η M in the y-direction for the p T -weighted, median, and hardest times. Here, we observe the general trend primarily follows eq. (3.22). Changing η J we see that distribution width tends to track with the corresponding pseudorapidity for the prompt distribution. The slight narrowing at large |η M | is due to the changing daughter time fraction, as in eq. (3.24). The few deviations from this pattern are caused by larger changes in the effective radius and additionally from the observed kinematics for the p T -weighted time.