Probing the time structure of the quark-gluon plasma with top quarks

The tiny droplets of Quark Gluon Plasma (QGP) created in high-energy nuclear collisions experience fast expansion and cooling with a lifetime of a few $\text{fm}/c$. Despite the information provided by probes such as jet quenching and quarkonium suppression, and the excellent description by hydrodynamical models, direct access to the time evolution of the system remains elusive. We point out that the study of hadronically-decaying $W$ bosons, notably in events with a top-antitop quark pair, can provide key novel insight, into the time structure of the QGP. This is because of a unique feature, namely a time delay between the moment of the collision and that when the $W$-boson decay products start interacting with the medium. Furthermore, the length of the time delay can be constrained by selecting specific reconstructed top-quark momenta.

The quark-gluon plasma (QGP), a state that characterised the first microseconds of the universe, is regularly produced and studied in ultrarelativistic heavy-ion collisions at both RHIC and the LHC.A range of complementary probes is used to study the QGP.These include properties that can be ascribed to hydrodynamic flow patterns, suppression of heavy-quark bound states, hadrochemistry of the final state, and modifications of the fragmentation of energetic partons that traverse the medium (see e.g.[1]).A property common to all these probes is that they are sensitive to the properties of the QGP integrated over its lifetime.Hydrodynamic simulation codes [2] predict a strong time-dependence of the QGP's properties associated with its expansion and cooldown, which last about 10 fm/c at the LHC.It would be invaluable to develop a way of probing this time-dependence.The recent discovery (see e.g.[3,4] and references therein) that high-multiplicity proton-proton (pp) and proton-nucleus (pA) collisions show signatures suggestive of collective effects, in systems with significantly smaller lifetimes than typical PbPb or AuAu collisions, is an additional motivation for devising a way of probing the time-structure of the QCD medium.
One powerful probe of the QGP is "jet quenching", i.e. the study of modifications of jets that pass through the QGP (see e.g.Ref. [5]).In all hard processes used so far for this purpose, dijet, γ+jet or Z+jet production, the jets are produced simultaneously with the collision of the ions.
In this Letter, we point out that top-antitop (t t) production offers a unique novel opportunity to study the quark-gluon plasma, in particular its time structure.This is because, at variance with all other jet measurements considered so far in the literature, the jets that come from the decay products of the W-boson start interacting with the medium only at later times, due to a series of time delays. 1 At rest, top quarks decay with a lifetime of about τ top 0.15 fm/c and the W that is produced in the top-quark decay has a lifetime of about τ W 0.09 fm/c.When the W boson decays hadronically, the resulting colour-singlet quark-antiquark (q q) pair is not immediately resolved by the medium [7].Only after the q and q have propagated and separated a certain distance do they start interacting independently with the medium.We call this delay a decoherence time, τ d .Thus the jets that are produced in the t → b + W → q q decay chain do not see the full QGP, but only the part of the QGP that remains after the sum of decay and decoherence times.That sum of times is correlated to the momentum of the top quark, a feature that may be exploited given a sufficient number of events.
To carry out a first investigation of the potential of using top quarks for probing the time structure of the QGP, we proceed as follows.We take the average total delay time before the W decay products start interacting with the medium to be For the decay times, we use a transverse boost factor, , defined in terms of the mass m X , and transverse momentum p t,X of particle X.The transverse component is the natural choice, because the frame in which the top-quark has no longitudinal momentum is also the one in which it is most natural to describe its interaction with the QGP, which is approximately longitudinally-invariant.We take the average de- 1 In light of Ref. [6], similar measurements of the time structure of the QGP could be accessible with W +jet events.However, we will focus here on the t t avenue which we consider more promising (cf. the supplemental material).
coherence time to be [7] in natural units, = c = 1, and with θ q q the opening angle between the two decay products of the W , again in a longitudinal frame where the z component of W momentum is zero.The quantity q is the transport coefficient of the medium (squared transverse momentum broadening per unit length, see e.g.[5]).While in practice it is expected to be a function of time, for our proof of principle illustration here, we take it to be constant, q = 4 GeV 2 / fm (conservatively taken larger than found in Refs.[8,9]).To get an event-by-event estimate of the interaction start time, we will associate each component with a randomly distributed exponential distribution.With these choices, for inclusive top-quark production at the LHC with centre-of-mass energy (per nucleon pair) √ s N N = 5.5 TeV, the average times are, γ t,top τ top 0.18 fm/c , γ t,W τ W 0.14 fm/c , and τ d 0.34 fm/c , with dispersions that are comparable.The 1/3 power in Eq. ( 2), means that τ d is only weakly dependent on the value of q.
To probe jet quenching and its time dependence in t t production, we here suggest measuring the invariant mass m jj of the dijet system that is produced from hadronic W decays.In pp events, m jj is closely related to the W mass, modulo final-state-radiation (FSR) effects.The difference in reconstructed m jj in central ion-ion (AA) collisions as compared to pp will be our measure of jet quenching.
To evaluate the potential of such a study we examine semi-muonic t t events, i.e.where one top decays to bW (W → µν), while the other decays hadronically to bW (W → jj).In pA collisions it has been demonstrated that it is possible to tag this class of events with essentially no background [10] as long as two b-tags are required, and so we only consider signal events.
For a quantitative analysis, we use events from the "hvq" (heavy-quark) process [11] in revision 3180 of the POWHEGbox [12] generator, which simulates topquark production to next-to-leading (NLO) accuracy in the strong coupling constant.We use it with the PDF4LHC15 nlo 30 PDF set [13], and shower events with Pythia 8.223 [14,15], tune 4C [16].Our final results will be based on events at hadron level, without underlying event.The number of events that we can expect for an integrated luminosity pp is the pp cross section for t t production and A is the atomic mass of the ions being collided (see also Ref. [17]).The c(f ) factor accounts for the centrality range f .We will concentrate on f = 0−10%, and so use c(0−10%) 0.42 [18].
To keep the analysis and simulation relatively simple, we choose not to embed events in a heavy-ion medium.
Instead we introduce a single factor to mimic the combination of all sources of fluctuations: those from the embedding and medium-subtraction procedure, from finite detector resolution and also from jet quenching dynamics.Specifically, we rescale the momentum of each particle i by a factor (1 + rσ pt / p t,i + 1 GeV) where r is a Gaussian-distributed random number (different for each particle) with a standard deviation of 1; σ pt is taken to be 1.5 GeV 1/2 .This leads to an effective relative jet energy resolution of about 1.5 GeV 1/2 / √ p t for high-p t jets, or about 15% for p t = 100 GeV, consistent with Ref. [19].
To simulate baseline full quenching in 0−10% central PbPb systems, we apply a constant energy loss rescaling factor Q 0 = 0.85 to all particle momenta, which is consistent with observations in γ/Z+jets measurements performed by ATLAS and CMS [20,21].Recall that the fluctuations associated with quenching are included in our single global fluctuation factor.A more sophisticated analysis would be possible, but is perhaps best carried out in the context of a full experimental study.
To account for dependence of the quenching on the time τ tot at which the W decay products start to interact with the medium, all particles from the W decay are scaled by a factor Q(τ tot ) rather than Q 0 .We will return to the exact form of Q(τ tot ) below.
To tag potential t t events, we require the presence of a muon, two b-tagged jets and at least two non-b-tagged jets.The muon should have p t > 25 GeV and rapidity |y| < 2.5.Jets are obtained using the anti-k t jet algorithm [22] with radius R = 0.3 and subsequent partial declustering [23,24] with the k t algorithm [25,26], all performed within FastJet v3.2.1 [27].A selection requirement of p t > 30 GeV and |y| < 2.5 is applied to the antik t jets.We assume a b-tagging efficiency of b = 70% per b, as obtained in pPb events in [10] and anticipating the expected improvements in b-tagging in high-multiplicity environments from HL-LHC detector upgrades [28,29].We also assume that fake b-tags do not introduce any substantial background.
Our W and top-quark reconstruction procedure is inspired by the pseudo-top definition of Refs.[30,31], adapted to be more resilient to the presence of additional jets from initial-state radiation (ISR) and at the same time robust with respect to effects of quenching on the energy scales of W and top candidates.The use of R = 0.3 anti-k t clustering and then k t declustering to obtain the input jets helps ensure adequate performance across a broad range of top-quark transverse boosts (similar in spirit though different in its details to Ref. [32]).The full procedure is detailed in the supplemental material.
For each event that satisfies the reconstruction requirements, we consider two observables: m reco W , the mass of the reconstructed hadronic W -boson candidate and p reco t,top , the p t of the corresponding top candidate.The  former will provide our measure of quenching (and was once before studied for this purpose [33]).The latter can be translated to an average τ tot and for 200 GeV p reco t,top 1 TeV the relation reads (see figure 6 in the supplemental material) τ tot (p reco t,top ) (0.37 + 0.0022 The distribution of τ tot values is given in Fig. 1 for the LHC √ s N N = 5.5 TeV, inclusively over p reco t,top , and for a future-circular-collider (FCC) with √ s N N = 39 TeV, considering events with p reco t,top > 400 GeV.Note the long tails in both cases, which will contribute sensitivity to times substantially beyond τ tot .
Fig. 2 shows the distribution of m reco W , again for the LHC and FCC, with a p reco t,top cut in the latter case.Results are shown with baseline full quenching for all particles and without quenching (the latter being equivalent to pp events embedded in heavy-ion events to account for the effect of the underlying event).One sees clear W -mass peaks, superposed on a continuum associated with events where the W decay jets have not been correctly identified.The continuum is significantly reduced The average (points) and standard deviation (width of band) for m reco W across many pseudo-experiments, as a function of luminosity for an inclusive sample of t t events, as a function of the integrated PbPb luminosity at the LHC (left) and the HE-LHC (right).
at high p reco t,top .The W peaks in the quenched case are shifted to the left, and the extent of the shift provides an experimental measure of the quenching.The peaks are also lower in the quenched case, reflecting the smaller fractions of events that pass the reconstruction (and, for FCC, p reco t,top ) cuts.To estimate the sensitivity of top-quark measurements to the time-dependence of quenching in the medium, we consider a toy model in which the quenching is proportional to the time between the moment when the W decay products decohere, τ tot , and a moment when the medium quenching effect stops being active, τ m .This gives a τ tot -dependent quenching factor Q(τ tot ) for the W decay products of Recall that all other hadronic particles undergo quenching with the factor Q 0 .
For each choice of τ m we obtain a m reco W histogram as in Fig. 2. We carry out a binned likelihood fit for the histogram and the background of incorrectly reconstructed W 's using the functional form which yields good fits.The free parameters a, b, c, σ and m fit W are constrained to sensible ranges so as to increase the stability of the fit in low statistics samples.Fig. 3  obtain when we carry out fits for a large number of replica pseudo-experiments.Two of the bands are independent of the PbPb luminosity: the top, unquenched band, corresponds to the result that would be obtained by embedding 2 fb −1 of pp (unquenched) data into minimum-bias PbPb events.The bottom band is obtained by a similar procedure, but with the pp jets' particles simply scaled down by the quenching factor Q 0 , i.e. by the quenching factor that would be expected if the W decay products were present and started interacting from time 0. In a real experiment, the corresponding scaling factor could be obtained by measuring quenching in another quarkjet dominated process (e.g. with γ+jet or Z+jet balance), as a function of the jet p t .For short values of the effective medium lifetime, τ m , the m fit W result is close to the unquenched result.This reflects the fact that the W decay products start interacting only towards the end of the medium lifetime.For larger values of τ m they instead still see most of the medium duration, and most of the quenching.A very short-lived medium, τ m = 1 fm/c, could be distinguished from the full quenching baseline at the LHC with its currently approved L PbPb = 10 nb −1 .However, to distinguish larger values of τ m would require either higher luminosities or higher energies.This is illustrated in the right-hand plot of Fig. 3 for a future HE-LHC ( √ s N N = 11 TeV), where the t t cross section is 6 times larger.At higher-energies it becomes advantageous to explore the p reco t,top dependence of m fit W , illustrated in Fig. 4 for the HE-LHC and the FCC ( √ s N N = 39 TeV).For each bin of p reco t,top , the upper axis shows the corresponding average τ tot .For a given band of τ m , when p reco t,top is large  enough so that τ tot τ m , the band merges with the unquenched expectation.Thus the shape of the p reco t,top dependence gives powerful information on the medium time-structure. 2 Fig. 5 shows our estimate of the maximum τ m that can be distinguished at two standard deviations from the baseline full quenched result, for different colliders as a function of L PbPb .The number of standard deviations takes into account the statistical uncertainty of m fit W , for both the actual heavy-ion data and a reference sample as well as an additional 1% systematic uncertainty (see supplemental material and Refs.[20,34]).The reference sample is obtained using the same procedure as for the bottom bands in Figs. 3 and 4, i.e. using 2 fb −1 of pp events with a rescaling of particle momenta by a factor Q 0 and inclusion of underlying-event fluctuations.
For each collider luminosity and energy the results are obtained by choosing a p reco t,top cut so as to maximise the significance.We have verified that if we increase the fluctuations, σ pt , the required luminosity scales as σ 2 pt , in line with expectations.
Lighter ions such as Kr are potentially promising, despite their smaller quenching effects [35], because of the potential for order-of-magnitude higher effective integrated nucleon-nucleon luminosities [36,37].They are discussed further in the supplemental material.
To conclude, in this work we have shown that the study of top quarks and their decays has a unique potential to 2 The unquenched and baseline-quenched bands also have a p reco t,top dependence, induced by the underlying jet and muon pt cuts, as well as different amounts of final-state radiation outside the R = 0.3 jet as a function of p reco t,top .
resolve the time dimension in jet-quenching studies of the QGP.To benefit from this potential requires a sufficiently large sample of top quarks, in particular to enhance event rates on the high-p t tail, which gives the sensitivity to the longer timescales.At the LHC, with currently planned luminosity, such a programme could begin.With higher energy colliders or a significantly increased luminosity at the LHC (whether from longer running or lighter ion species), there would be substantial prospects for using jet quenching to study the evolution of the QGP over the first few fm/c.Overall, our results provide a strong motivation for a programme of experimental studies of top-quark production in heavy-ion collisions.

SUPPLEMENTAL MATERIAL Using W +jet events
One question that one might ask is whether one could use W +jet events rather than t t events given that such events have two of the sources of time delay: the W decay and decoherence times.It is especially natural to ask this question in light of the clear hadronic W peak observed recently for high-p t W +jet events by the CMS collaboration [6].However, we argue here that measuring t t looks more promising.
For √ s = 5.5 TeV, the cross section for the W +jet process with a p t cut of ∼ 100 GeV is comparable to the total t t cross section.In the case of semi-leptonic t t events, the 2 b quarks, lepton and (potentially) missing transverse momentum provide powerful scope for tagging, albeit with a substantial price to pay in terms of efficiency for the identification of all the decay products.
In the case of hadronic W +jet events there is a huge hadronic background.So far it has been possible to reduce this background sufficiently to clearly see a W (and Z) mass peak only at p t 's of several hundred GeV and with a W tagging procedure that has moderate efficiency, of the order of 10%.This tagging procedure relies strongly on the pattern of radiation from the W decay products.Any attempt to use similar methods for W tagging in heavy-ion collisions would have the drawback that they could bias the reconstructed mass, for example by selecting W events which do not have additional medium-induced radiation.The availability of tagging handles that are independent of the hadronic W in t t events thus makes the t t process more attractive.

Top reconstruction procedure
Our reconstruction procedure requires as its starting point a muon, two b-tagged jets and at least two non b-tagged jets.Experimentally there would also be muonisolation and possibly missing-energy requirements, however these are not straightforward to simulate correctly in heavy-ion collisions and we believe that neglecting them here should not critically change the results.
One particularity of the reconstruction procedure is how we obtain the input jets.Firstly, we cluster all particles except the muon (and any neutrinos) with the anti-k t jet finder [22] (from FastJet v3.2.1 [27]) with a radius of R = 0.3.Only jets with p t > 30 GeV and |y| < 2.5 are accepted at this stage.For each anti-k t jet, its constituents are then reclustered with the exclusive longitudinally-invariant k t algorithm [25,26] with R = 1 and d cut = (20 GeV) 2 .For low p t anti-k t jets this procedure usually yields just a jet that is identical in particle content to the original anti-k t jet.However for high p t anti-k t jets it can yield multiple exclusive k t jets, notably associated with the substructure of boosted top quarks and W bosons.This procedure is similar to the declustering approach of Ref. [23] and to the Y -splitter algorithm of Ref. [24], but applied to small-R jets rather than large-R jets.It is intended to provide the inputs for a top-reconstruction approach that works at both low and high top-quark transverse boosts.It is thus similar in its aims to the (technically different) procedure of Ref. [32].
Given the muon, two b-tagged jets and at least two non b-tagged jets obtained in this way, we use a procedure to identify the W and top-quark candidates that is in part inspired by the pseudo-top definition of Refs.[30,31].The b-jet that has the smallest distance to the muon is taken to come from the leptonically decaying top (b , as opposed to b h ).The distance that is used is ∆R 2 , where are y i and φ i are the rapidity and azimuthal angle of particle i.
To reconstruct the hadronically decaying W the standard pseudo-top approach is to consider the two highestp t non b-tagged jets.However we found that this yielded poor reconstruction for high p t top quarks, because quite often one of the two highest-p t non b-tagged jets comes from initial state radiation.In full PbPb events this problem might be further exacerbated by jets from other nucleon-nucleon interactions.
A widespread alternative approach (e.g.Ref. [31]) is to select and assign jets so as to minimise a suitable kinematic fit variable.This corresponds to choosing the subset of jets that is most consistent with the known W and top masses.However, there is tendency for such a procedure to induce a peak around m W and m top in the mass distributions of incorrectly reconstructed W bosons and top quarks.This complicates fits for the mass distribution of correctly reconstructed (potentially quenched) W candidates.
Ultimately the procedure that we adopted to identify the hadronic W -decay jets was to consider all pairs (j, k) of non b-tagged jets that satisfy m jk < 130 GeV and m jkb h < 250 GeV and select the pair with the largest p t,j +p t,k .This pair is considered to be the reconstructed hadronically decaying W boson candidate, with mass m reco W ≡ m jk .The reconstructed top-quark candidate is the combination of the W candidate with the b h jet and its transverse momentum is denoted as p reco t,top ≡ p t,jkbj .If no pair is found satisfying the above m jk and m jkb h mass conditions, the reconstruction is deemed to have failed and the event is discarded.From Fig. 2, one sees that the mass distribution of incorrectly reconstructed W 's is free of substantial shaping.
Our procedure, while adequate for the study of scenarios ranging from the LHC to FCC, has not been the subject of extensive optimisation.We believe that such an optimisation is better performed in the context of a more detailed study that includes also full consideration of all heavy-ion effects at a given specific collider.1), for q = 4GeV 2 /fm.The average contribution of each component is shown as coloured stacked bands (see legend).For comparison, the total delay time for q = 1 GeV 2 /fm is shown as a dashed line.
The result of Eq. ( 1) is shown as a function of the reconstructed top jet transverse momentum in Fig. 6, broken into its three components, represented as stacked bands.The range of p t 's shown is guided by expectations as to what will be accessible at widely discussed scenarios of potential future colliders [38,39].The dispersion σ τtot of the sum of the three components is also represented in Fig. 6, as vertical black lines.To illustrate the weak dependence of τ tot on the value of q, the average total delay time assuming a q = 1 GeV 2 / fm (rather than q = 4 GeV 2 / fm) is shown as a dashed line.The larger result for τ tot would translate to a larger reach in τ m values for a given collider setup.

Control of the jet energy scale
To be able to identify the time-induced difference between quenching of W jets in t t events from full quenching, it is crucial to have a reliable estimate of the expected reconstructed W mass were quenching of the W jets to be unaffected by coherence delays and the W lifetime.
The procedure that we envisage for this purpose is to use measurements of the Z-jet and γ-jet balance in events with cleanly identified (leptonic) Z bosons and photons to determine the expectations for full quenching and to then apply that determination to embedded t t events.
To estimate the potential precision of such an approach, we examined how well the average x jZ = p tj /p tZ ratio could be determined at the HL-LHC.Ref. [34] from CMS gives a projection for the uncertainties on the x jZ distribution with L PbPb = 10 nb −1 .We took that distribution and created replica distributions by fluctuating each bin with a Gaussian uncertainty set by the projection.We then evaluated the standard deviation of the x jZ values across many replicas.The result for the standard deviation was 1.2%.This guides our choice of 1% for the systematic uncertainty on the impact of standard quenching for the purpose of producing Fig. 5.
We also note that Ref. [20] from ATLAS, shows a 1% uncertainty (blue lines, bottom panel of Fig. 3) for the cross-calibration uncertainty between PbPb and pp collisions.One should keep in mind that other jet-energy scale uncertainties that are common to the pp and PbPb cases should largely cancel when considering the difference between embedded pp results and PbPb data (and it is precisely this difference that interests us).

Lighter ions
Following the recent successful XeXe machinedevelopment run at the LHC, the prospect has been raised [36] that with ions lighter than Pb it might be possible to achieve effective nucleon-nucleon luminosities (i.e. total number of hard collisions) that are up to an order of magnitude larger than for PbPb, in part because of the reduction of effects such as bound-free pair production [37].Generically, higher luminosities would bring substantially increased sensitivity to the longer time structure of the QGP medium.
Aside from luminosity considerations, smaller ion species have both an advantage and a disadvantage.The advantage is that the intrinsic time scales associated with the smaller, cooler QGP might be shorter than for PbPb and so more accessible with top-quark probes.However a smaller, cooler QGP is also likely to result in less quenching.It is for the purpose of illustrating the tradeoffs associated with lighter species that in Fig. 5 we show a curve labelled KrKr.It uses a quenching of 10% rather than 15%, in line with observations in CuCu [35] that are consistent with quenching that goes as A 1/3 , where A is the nuclear mass.The reduced quenching means that the equivalent of Fig. 3 for KrKr would have the bands more closely spaced.Accordingly one needs to go to higher luminosities in order to distinguish any two given time scenarios.At low luminosities the extra factor is relatively limited, about 1.5, while at higher luminosities it increases to about 3. Note that at higher luminosities the systematic and pp statistical uncertainties on the expected standard quenching results start to dominate, since we have taken them to be independent of the PbPb equivalent luminosity.

FIG. 1 .
FIG.1.Distribution of τtot for events that pass all reconstruction cuts and have a top-quark candidate (independently of the reconstructed top-quark and W -boson masses).

FIG. 2 .
FIG. 2. Differential fiducial proton-proton t t reconstruction cross section as a function of m recoW at the LHC and FCC.
FIG.5.The maximum medium quenching end-time, τm, that can be distinguished from full quenching with two standard deviations, as a function of luminosity for different collider energies and species.For the KrKr points, the LKrKr value that is used is equal to L PbPb • (A Pb /AKr) 2 , i.e. maintaining an equal number of nucleon-nucleon collisions.

FIG. 6 .
FIG.6.Total delay time and its standard deviation (markers and corresponding error bars), as given by Eq. (1), for q = 4GeV 2 /fm.The average contribution of each component is shown as coloured stacked bands (see legend).For comparison, the total delay time for q = 1 GeV 2 /fm is shown as a dashed line.
shows the results for m fit W .They are plotted as bands for different τ m values, as a function of the PbPb integrated luminosity, L PbPb .The width of each band represents the standard deviation of m fit W values that we Dependence of the reconstructed W mass on the reconstructed top pt for HE-LHC (left) and FCC (right) collisions.The quenched result corresponds to baseline full modification of the pp results, which would in practice be obtained using knowledge of quenching from other measurements.
maximum medium quenching end-time, τm, that can be distinguished from full quenching with two standard deviations, as a function of luminosity for different collider energies and species.For the KrKr points, the LKrKr value that is used is equal to L PbPb • (A Pb /AKr) 2 , i.e. maintaining an equal number of nucleon-nucleon collisions.