Jet substructure at the Tevatron and LHC: new results, new tools, new benchmarks^*

In the context of measurements of the mass, angularity and planar flow of high-p_T jets [9], a new method of correcting the jet mass for pile-up was developed. The use of a complementary cone at right angles to the jet in azimuthal angle ϕ and at approximately the same pseudorapidity allows the energy density from both underlying event (UE) and multiple interactions (MI) to be measured. The UE refers to the interactions and hadronization of partons in the colliding protons other than the partons in the hard process. MI refer to both 'in-time' and 'out-of-time' pile-up: the former describing multiple proton–proton collisions in a single bunch crossing, and the latter describing the delayed instrumental effects of previous crossings. The incoherent contributions to the shift in jet mass from MI were isolated from the partially coherent contributions due to UE by examining the mass shift as a function of the number of good vertices in the event. The average shift in the jet mass when adding the towers from the complementary cone into the jet is shown in figure 1 along with the MI+UE corrections measured for angularity and planar flow. The high mass selections made as part of the angularity and planar flow measurements resulted in too few events to separate the UE (single-vertex) and MI (single- and multiple-vertex) components.

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The results of a search for pair production of a supersymmetric particle that decays strongly and violates R-parity were described in [10]. The final state of interest was at least six quarks, most observed as separate jets, and no missing transverse energy in the event. The challenge for this search was to reduce the large backgrounds from QCD multijet production, which was done by placing specific kinematic requirements on three-jet triplets in the final state.

The analysis sought events in a CDF sample that satisfied a trigger requiring at least four jets with p_T > 15 GeV and the sum of the calorimeter transverse energy greater than 175 GeV. Events were furthermore required to have at least six jets with p_T > 15 GeV and |η| < 2.5, and the scalar sum of the six highest-energy jets was required to be greater than 250 GeV. Any jets beyond the six highest-energy jets were not used in this analysis. The missing transverse energy, ${\unicode{x246}}_{T}$, was required to be less than 50 GeV in order to reduce contributions from W boson final states and mis-measured QCD events.

All 20 combinations of two three-jet triplets that could be produced from six jets were considered and the invariant mass of each combination, M_jjj, and the sum of the magnitudes of the transverse momenta of the three jets, ∑_jjj|p_T|, were formed. Monte Carlo studies have shown that requiring

$$\begin{eqnarray} &&\sum _{jjj} |p_{T} | - M_{jjj} > 115 \;{\rm GeV} \end{eqnarray} \tag{ 1 }$$

is an efficient way of separating potential signal combinations (where M_jjj would be a constant reflecting the mass of the parent supersymmetric parent) from the QCD and combinatorial backgrounds. This is in effect a boosted three-jet final state. The shape of the backgrounds in M_jjj were estimated by using a five-jet final state, showing that the background is expected to peak around 100 GeV.

The M_jjj distribution was then fit to a combination of signal and background terms, where the signal was defined by a PYTHIA Monte Carlo calculation for R-parity violating (RPV) gluino pair production. Although the acceptance for the gluino final state is quite low (roughly 5 × 10⁻⁵), CDF was able to set significant limits on the RPV gluino cross section, which were then converted into lower limits on the gluino mass. Lower mass limits ranging from 144 to 154 GeV at 95% C.L. were set on the gluino mass, depending on the assumptions about the spectrum of intermediate supersymmetric final states. What is perhaps as interesting is that CDF observed evidence for a boosted top quark signature. In the top quark mass region of M_jjj ∼ 175 GeV, CDF observed 11 ± 5 jet triplets. Although one expects on average only one top quark event in this kinematic region, the shape of the mass distribution is consistent with what one expects from MC simulations.

CDF presented updated results on the measurements of jet mass, angularity and planar flow for jets with p_T > 400 GeV from a sample of 5.95 fb⁻¹ [11]. The measured distributions were compared with analytical expressions from NLO QCD calculations, as well as PYTHIA 6.1.4 predictions incorporating full detector simulation. The theory predictions for jet mass were in good agreement with the data, whereas the angularity and planar flow predictions by PYTHIA showed disagreement in detail (primarily at low angularity and low planar flow).

CDF used these data to also search for a signal of boosted top quark production. Candidate events were selected in two channels: the fully hadronic decays where both top quarks produce massive high-p_T jets (the '1+1' channel), and the decays where one top quark decays semi-leptonically resulting in one massive high-p_T jet recoiling against a second lower-mass jet and significant missing transverse energy (the 'SL' final state). A signal region in the 1+1 mode was defined by requiring both leading jets to have a mass between 130 and 210 GeV (see figure 2), while the signal region in the SL mode requires the leading jet to have 130 < m^jet1 < 210 GeV and the event to have missing transverse energy significance

$$\begin{eqnarray} &&S_{{\rm MET}} \equiv {\unicode{x246}}_{T} / \sqrt{\sum E_T} \in (4,10). \end{eqnarray} \tag{ 2 }$$

This cut rejects primarily QCD dijet events. The remaining QCD backgrounds are estimated by looking at the event rates in sideband regions of jet mass and S_MET. The analysis observes 31 and 26 candidate events in the 1+1 and SL channels, respectively, and the backgrounds are estimated to be 14.6 ± 2.7 and 31.3 ± 8.1 events, respectively. The total number of top quark events expected in the two channels is 4.9 ± 2.1 candidates.

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Although the data are consistent with a boosted top quark signature, they are not statistically strong enough to claim observation. Rather, they are used to set an upper limit of 38 fb on top quarks produced with p_T > 400 GeV. The SM expectation for this cross section is 4.5 fb.

There are several theories beyond the standard model that predict the production of cascades of particles that appear in the final state as a 'lepton jet'. The CDF collaboration reported on a search for such objects using 5.1 fb⁻¹ of $\sqrt{s} = 1.96$ TeV proton–antiproton collisions at the Tevatron. The search looked for events with a large number of low-energy leptons produced in association with a W or Z boson.

Events were selected by performing the standard selection for W or Z boson candidates, requiring at least one well-identified electron or muon candidate and then either significant ${\unicode{x246}}_{T}$ or a second well-identified charged lepton of the same flavour but opposite charge. A 'soft lepton' algorithm was then employed to identify additional electron or muon candidates down to a transverse momentum of 1 GeV for electrons and 3 GeV for muons. The numbers of events in the zero or one additional lepton bins were used to scale the expected backgrounds for the signal region defined by two or more soft lepton candidates. The analysis found that the dominant backgrounds came from inclusive W+jets production where one or more of the jets were mis-identified or the Drell–Yan production where the additional leptons were mis-identified jets. Other sources of background, such as W+c-quark, W+b-quark, $t\bar{t}$ and di-vector boson production, were also evaluated.

The potential signal was modelled on a benchmark process defined by a neutralino model with a 'hidden' Higgs boson coupled to a dark sector, where the dark sector particles decay to pairs of charged leptons [12]. The channels with the best signal-to-background for this model were those with a W or Z boson with at least three additional muons, either with none or one additional electron; for example, in the channel with a W + 3 additional muons, one expected 1.5 ± 1.2 background events and nine signal events (only two events were observed). There was no signal observed above background, and a 95% C.L. upper limit on the production cross section of a W or Z boson produced in association with a Higgs boson with the expected couplings of 27 fb was set. This allowed the collaboration to rule out the benchmark model.

This analysis showed the effectiveness of using the lepton jet signatures to search for evidence of new physics.

The D0 experiment presented recent results showing how the colour flow that is expected to arise between two jets can be used as a discriminant to identify W boson hadronic decays in a 5.3 fb⁻¹ sample of $t\bar{t}$ events [13]. Since the W decay products form a colour singlet, they produce an antenna radiation pattern, with most soft particles emitted between the two jet directions. The 'pull' of the jets' radiation towards each other can be used to more effectively identify dijet systems arising from a specific colour state.

Events were selected by requiring the traditional lepton+jets final state, with a charged lepton, missing transverse energy and four or more jets. At least two of the jets had to be tagged as b-quark candidates, resulting in 728 candidate events with an estimated background rate of 82 ± 9 events in the sample not arising from $t\bar{t}$ production. The jet pairs that had an invariant mass within 30 GeV of the W boson mass were then selected and the shape of the energy depositions studied to look for a signal consistent with the colour singlet. The minimum relative pull angle for jet pairs satisfying |m_W − m_j1j2| < 30 GeV, ΔR_{j1, j2} < 2 and |η| < 1 is shown in figure 3(a). The expected colour effect in the relative orientations of the jet pulls was observed when comparing the daughter jets from W boson decays and the b-quark jets, though the statistical power of the measurement was modest and the kinematics of the pairs of jets were not identical. A more direct comparison was made by producing Monte Carlo $t\bar{t}$ decays with a W boson produced as a colour octet compared with the expected colour singlet state, and a series of detailed studies were performed to evaluate the systematic uncertainties on this measurement.

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The actual quantity measured was the fraction of singlet W boson decays compared with the total number, giving

$$\begin{eqnarray} &&f_{\rm singlet} = 0.65^{+0.37}_{-0.38}\, {\rm (stat.)} \pm 0.22\, {\rm (syst.)}. \end{eqnarray} \tag{ 3 }$$

This could be turned into a lower limit on the fraction of singlet decays of f_singlet > 0.277 at 95% C.L. Although this is somewhat higher than the limit that might have been expected given the data sample, it is consistent with the expectation within statistical uncertainties. The expected C.L. bands for this limit are shown in figure 3(b).

This means of measuring the colour flow provides an additional discriminant that can be used to search for decays of other boosted objects (such as $H \rightarrow b\bar{b}$) and gives us confidence that the modelling of colour flow is being done well in current Monte Carlo calculations.

In the recent ATLAS analysis of 35 pb⁻¹ of data [4], the sensitivity of the individual jet mass to pile-up is directly tested. The increase in the mean jet mass as a function of the number of reconstructed good vertices (good vertices being defined as those with at least 5 tracks with p_T > 150 MeV), N_PV, is measured for jets with a p_T of at least 300 GeV and rapidity |y| < 2. The results are shown in figure 4 for jets of various radii and built using two different jet algorithms. The mean jet mass is observed to increase linearly with N_PV. For events with only a single primary vertex, the mean jet mass increases linearly with R. The slope of the mean jet mass as a function of the number of primary vertices, though, exhibits a dependence on R that is consistent with the ratio of the third power of the jet resolution parameter. This follows the prediction of pQCD calculations [14]: the extra mass from an additional particle k scales like δm² ≈ 2p_T · k ∼ p_Tp^k_TΔR²_k, so the net change from a uniform pile-up p_T density ρ is δm² ∼ p_Tρ∫dAΔR² ∼ p_TρR⁴, and hence δm ∼ R⁴ρp_T/m. Since m ∼ R, the derivative dm/dρ then scales like R³ for fixed p_T.

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Figure 4 also shows the impact of pile-up on each of the three jet algorithms used in the ATLAS study. Specifically, figure 4(b) shows the dependence on N_PV for the mean jet mass before and after the splitting and filtering procedure. The filtering procedure significantly reduces the effect of pile-up on jet mass. In fact, the slope of a straight line fit is consistent with zero. This result demonstrates that certain well-designed jet grooming procedures can significantly reduce the impact of pile-up and that the efficacy of the method can be tested in situ. Methods to correct for these effects and to provide more general pile-up mitigation procedures using jet grooming are underway in ATLAS.

The study of jet shapes in proton–proton collisions provides information about the details of the parton-to-jet fragmentation process [15]. The internal structure of sufficiently energetic jets is primarily dictated by the emission of multiple gluons from the primary parton, calculable in perturbative QCD (pQCD). The shape of the jet depends on the type of parton (quark or gluon) and is also sensitive to non-perturbative fragmentation effects and UE contributions from the interaction between proton remnants. A proper modelling of the soft contributions is crucial for the understanding of jet production in hadron–hadron collisions and for comparison of the jet cross section measurements with pQCD theoretical predictions.

The ATLAS collaboration recently published results on differential and integrated jet shapes for jets constructed with the anti-k_T algorithm [16] with distance parameter R = 0.6 [2]. The jet shapes were calculated using the constituents associated with the jets, which were 3D topological calorimeter clusters [17].

The analysis follows similar measurements undertaken by CDF for $p\bar{p}$ collisions [18–20] by ZEUS for e^±p [21–24] and OPAL for e⁺e⁻ [25, 26] collisions.

The ATLAS analysis uses data corresponding to 3 pb⁻¹ of total integrated luminosity, collected during 2010. This period of data collection can safely be referred to as ATLAS's early days, when pile-up was not the burning issue that it quickly became in 2011, after about 35 pb⁻¹ had been collected. In 2010, the average number of good vertices in a single hard scatter was approximately 2. In 2011 data, this average is closer to 10, and increasing with luminosity.

The measurements are carried out for jets with p_T > 30 GeV and |y| < 2.8.

The measurements of integrated and differential jet shapes, Ψ (equation (5)) and ρ (equation (4)), are corrected to hadron level and compared to several Monte Carlo predictions. Different phenomenological models are employed to describe the fragmentation processes and UE contributions.

The comparisons presented in [2] were complemented with additional results in a second publication [27] including the predictions from new generators and up-to-date MC tunes, and are reported in this contribution to the proceedings.

The differential jet shape ρ(r) as a function of the distance $r=\sqrt{\Delta y^2 + \Delta \phi ^2}$ to the jet axis is defined as the average fraction of the jet p_T that lies inside an annulus of inner radius r − Δr/2 and outer radius r + Δr/2 around the jet axis:

$$\begin{equation} \rho (r) = \frac{1}{\Delta r}\frac{1}{N^{\rm jet}}\sum _{\rm jets}\frac{p_{T}(r-\Delta r/2,r+\Delta r/2)}{p_{T} (0,R)},\ \Delta r/2 \le r \le R-\Delta r/2, \end{equation} \tag{ 4 }$$

where p_T(r₁, r₂) denotes the summed p_T of the clusters in the annulus between radius r₁ and r₂, N^jet is the number of jets, and R = 0.6 and Δr = 0.1 are used.

The points from the differential jet shape at different r values are correlated since, by definition, ∑^R₀ρ(r) Δr = 1. Alternatively, the integrated jet shape Ψ(r) is defined as the average fraction of the jet p_T that lies inside a cone of radius r concentric with the jet cone:

$$\begin{equation} \Psi (r) = \frac{1}{N^{\rm jet}}\sum _{\rm jets}\frac{p_{T}(0,r)}{p_{T}(0,R)},\ 0 \le r \le R, \end{equation} \tag{ 5 }$$

where, by definition, Ψ(r = R) = 1, and the points at different r values are correlated.

A summary of the results presented in [2] are given in figure 5 for the differential jet shape in the bin 30 GeV < p_T < 40 GeV and in figure 6 for the integrated jet shape at r = 0.3 in jets with 30 GeV < p_T < 600 GeV.

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The measured jets become narrower as the jet transverse momentum and rapidity increase, although with a rather mild rapidity dependence. The results indicate that the predicted jet shapes are mainly dictated by the details of the implementation of the parton showers and the modelling of the UE in the Monte Carlo generators. The presence of contributions from higher-order pQCD matrix elements in the final state does not affect the predicted jet shapes significantly.

The PYTHIA, HERWIG++ and HERWIG/JIMMY Monte Carlo generators, using the latest tunes to describe minimum bias and UE-related observables in data, provide a good description of the measured jet shapes, although the latest version of HERWIG++ tends to produce jets narrower than the data. As expected, PYTHIA 6 AMBT1 tends to produce jets slightly narrower than the data as it underestimates the UE activity in dijet events [28] and lacks a tuned final-state parton shower from the initial-state radiation (ISR). The effect of the wrong setting in 'HERWIG++ 2.4.2 bug'⁴³ is only visible for jets with p_T < 40 GeV .

The different SHERPA predictions are similar and provide a reasonable description of the data. This indicates that the presence of additional partons from higher-order matrix element contributions does not affect the predicted jet shapes, mainly dictated by the soft radiation in the parton shower. The comparison between SHERPA 1.2.3 and SHERPA 1.3.0 shows that the NLL-inspired corrections included in the latter for the parton shower do not impact significantly the predicted jet shapes. ALPGEN interfaced with PYTHIA predicts too-narrow jets and does not describe the data. This was already the case for ALPGEN interfaced with HERWIG/JIMMY [2] and requires further investigation to determine whether the disagreement observed with the data can be completely attributed to the UE modelling in the MC samples or is also related to the prescription followed by ALPGEN in merging the partons from the matrix elements with the parton showers in the final state.

Finally, POWHEG interfaced with HERWIG/JIMMY provides a reasonable description of the data while the interface with PYTHIA predicts too-narrow jets, which is mainly attributed to the details of the UE modelling.

These results confirm the sensitivity jet shape measurements, and their potential to constrain different Monte Carlo models.

First steps have been made this year by ATLAS in the quest to reconstruct the decay of a boosted top quark in a single jet. The first publication has been delivered in the context of the search for massive exotic resonances decaying to $t\bar{t}$ [29]. This analysis was able to isolate a clean sample of boosted top quarks. These events are the first candidates for boosted top quarks reconstructed as single jets at ATLAS.

Although the statistics with the 2010 dataset were too low to provide a testing ground for the many substructure techniques and variables we are interested in testing on such a clean sample of boosted top quarks, the groundwork was laid down to make the prospects for boosted top analyses on 2011 data very exciting.

One of the most important tasks for the early ATLAS boosted object analyses was to begin subjecting the various grooming methods and variables, initially implemented in Monte Carlo, to the tests that come with the conditions of a real detector. The recent ATLAS analysis on jet mass and substructure has done just that [4]. The analysis was based on 35 pb⁻¹ of data taken in 2010 and presented the mass and splitting scale of high p_T (>300 GeV) jets reconstructed with the anti-k_T (R = 1.0) algorithm and with the Cambridge–Aachen (R = 1.2) algorithm at different stages of grooming. The grooming method employed was a mass-drop filtering method which looks for hard subjets within the very large parent jet, and discards any radiation not falling within the subjets designated to the hard splitting. As already discussed in section 2.3.1, the behaviour of the mass of large-area jets as a function of the number of good vertices in an event was studied and the process of filtering away the soft radiation in such jets was found to effectively remove this dependence. This is perhaps unsurprising, as the radiation present in an event due to pile-up or UE is expected to be a fairly soft continuum. The analysis discarded all events with more than one good vertex, focusing on the 28% of the 2010 dataset that is regarded as being free from pile-up⁴⁴.

The final distributions resulting from the analysis were unfolded to hadron level. The mass distributions of Cambridge–Aachen jets with R = 1.2 are shown in figure 7 before and after employing mass-drop filtering with the requirement that the heavier subjet carry less than μ = 67% of the mass of the parent jet and the splitting between subjets is fairly symmetric, $y_2 \equiv \frac{\min \left(p^2_{T,a}, \,\, p^2_{T,b}\right)}{m^2_{\rm jet}} \Delta R^2_{ab} < y_2^{\rm cut}=0.09$. Figure 8 shows the mass and splitting scale ($\sqrt{d_{12}}= \min (p_{T,a}, p_{T,b})\times \Delta R_{ab}$) of anti-k_T jets with R = 1.0.

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In all cases, the distributions are found to be well modelled by a range of leading-order parton shower Monte Carlos to within the systematic uncertainties.

One advantage of the CMS detector and reconstruction pipeline is the so-called particle flow algorithm for a more holistic approach to jet reconstruction. Because of this approach, it is possible to remove around 60% of the pile-up from jets directly as it corresponds to charged tracks which can be associated with subleading primary vertices. This aids in the reduction of pile-up in jets with substructure at CMS, and studies show that there is only moderate dependence of the substructure variables on pile-up for the luminosities considered [5].

The CMS collaboration plans to retain a data-driven approach to measuring mis-tag rates, ameliorating pile-up issues there. Further studies are ongoing to utilize other advanced jet reconstruction techniques to further reduce pile-up dependence.

Several jet shape measurements have been performed at CMS similar to those at ATLAS described above in section 2.3.2. The first is the examination of the classic jet shapes, and the second is the examination of the jet charged component structure, both in [3].

Several differences exist between the ATLAS and CMS results. While both collaborations use the anti-k_T jet algorithm, CMS uses different values of the radius parameter R. For the inclusive shape measurements, two types of jets were used: calorimeter-only jets and jets where the calorimeter response was corrected using associated tracks ('jet-plus-tracks' jets) [30]. To reduce out-of-cone radiation, a large R of 0.7 was used. The jets were required to have (track-corrected) transverse momenta greater than 15 GeV. For the charged multiplicity measurement, only the jet-plus-tracks jets were used, with R = 0.5. The (track-corrected) transverse momenta for the first two jets were required to be larger than 20 and 10 GeV, respectively.

The distribution of Ψ(r) (equation (5)) is shown in figure 9. The Monte Carlo simulation is doing a very reasonable job at predicting the data, and the agreement improves with larger transverse momentum.

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CMS has developed algorithms to identify the hadronic decays of boosted top quarks and W bosons. The top-tagging algorithm is based on the Johns Hopkins tagger [31], with small modifications as described in [32, 1]. The W-tagging algorithm is based on jet pruning [33, 34], adding a requirement on the mass drop of the subjet splitting, which was motivated by the BDRS subjet/filter algorithm [35].

The algorithms were characterized as of the BOOST 2011 conference [5], and subsequent studies have advanced this characterization [36]. The strategy for the studies here is to have data-driven mis-tag predictions which can accurately represent non-top and non-W backgrounds. Sidebands in the data are selected that have little to no true top or W contributions, and the tagging rate is derived as a function of the jet transverse momentum. These per-jet parameterizations are then applied to each jet in a signal selection, which is used to form a data-driven background estimate. This has several advantages, primarily that modelling effects are not relevant. Moreover, pile-up effects are handled directly, because the same data sample is used in the sidebands and the signal region, so the pile-up contribution is the same.

The efficiency for the tagging algorithms is derived from Monte Carlo, and corrected for a data-to-Monte-Carlo scale factor using standard model $t\overline{t}$ events in the semi-leptonic channel in [36]. In that public result, limits are set on the possible cross section for narrow resonances decaying to $t\overline{t}$ pairs. Figure 10 shows the mis-tag rates and efficiencies for the top and W tagging algorithms from CMS.

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The study of the performance of jet substructure techniques in generic QCD samples is critically important to understand these algorithms, and to derive appropriate sideband regions for data-driven background estimates at CMS. As such, CMS has performed detailed comparisons of jet substructure observables in [5].

For example, figure 11 shows the jet mass for the top-tagging and W-tagging algorithms, compared to several different predictions from Monte Carlo. The simulation seems to be predicting the data quite nicely for all of the generators examined, with slight variations based on the shower model and UE tunes. This seems to have a much larger effect on the substructure than the pile-up contributions.

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We consider LHC proton–proton collisions at a centre-of-mass energy of 7 TeV. We have simulated the standard model processes $t\bar{t} +X$, W/Z + X and pure QCD jets. All samples were produced with version 1.3.1 of the SHERPA event generator [67, 68]. For the electroweak gauge bosons as well as the top quarks, we consider fully hadronic decays only. For each process we generated samples of 100 000 unweighted events for p_T slices of 100 GeV between 0, ..., 1.6 TeV, resulting in an almost flat p_T distribution when combining all samples. As a slicing parameter, we used the maximum of the Monte Carlo top/anti-top p_T, the gauge-boson p_T for the W/Z channels and the leading-jet p_T for pure jet production, where parton-level jets were reconstructed according to the anti-k_T algorithm with R = 1.5. For all processes, the slicing cut is applied at the matrix-element level before parton showering is invoked.

To accurately simulate high-p_T events and to account for configurations with more than one hard parton inside a large jet, we employ SHERPA's matrix-element–parton-shower matching algorithm [69, 70]. We consider the complete sets of tree-level matrix elements with up to two (four) additional final-state partons for $t\bar{t}$ and dijet (W/Z) production from SHERPA's matrix-element generator COMIX [71]. The matrix elements of varying multiplicity are consistently combined to form inclusive samples of fully exclusive events through subsequent parton shower evolution [72] before being subjected to hadronization. The renormalization and factorization scales are dynamically determined on an event-by-event basis according to the matching algorithm [69]. For the parton separation parameter of the matching procedure, we use Q_cut = 20 GeV. The default tune of SHERPA 1.3.1 for the hadronization and UE model parameters was used.

For comparison with last year's HERWIG events, we have also produced samples using HERWIG++ 2.5.0. The initial state consists of two beams of protons at a centre-of-mass energy of 7 TeV. The UE has been simulated using the UE7-2 tune, whose parameters can be found at http://projects.hepforge.org/herwig/MB_UE_tunes/input_cards/LHC-UE7-2.in. The samples have been produced using standard model matrix elements for QCD 2 → 2 scattering and $t\bar{t}$ production. Only fully hadronic final states are considered. For each process 10 000 events were produced for each p_T bin of a width of 100 GeV, from 200 GeV to 800 GeV for $t\bar{t}$ production and from 200 GeV to 1.6 TeV for QCD 2 → 2 scattering. The resulting p_T distribution when combining all samples is approximately flat. The p_T cut is applied at the level of the hard process. B hadrons have been set stable through a lifetime cut of 1 ps for consistency with the HERWIG samples used for BOOST 2010⁴⁹. All other settings have been left as default.

We first discuss the perturbative contributions to the cross section, in particular, the resummation. We focus on jets in the anti-k_T algorithm, since jets defined in other algorithms are complicated by the non-trivial role of clustering between soft gluons. Soft gluon clustering in these algorithms means that the soft phase space does not factorize into single particle phase space for each soft gluon, affecting the subleading logarithms in the perturbative series starting at $\mathcal {O}(\alpha _s^2 \ln ^2)$ [87, 88]. In the anti-k_T algorithm, the phase space for a single soft particle is not affected by the presence of other soft particles.

The leading logs in the cross section are given by the exponentiation of single soft gluon emission in QCD, and single-log and running-coupling corrections can be included. Equivalently in SCET, the leading logs are specified by the cusp anomalous dimensions of the one-loop jet and soft functions, with the single-log terms coming from the non-cusp anomalous dimensions. Running-coupling effects arise in renormalization group evolution. In the limit of small jet radius (R ≪ 1), the leading logs have the form

$$\begin{equation} \int _0^m \frac{\mathrm{d}\sigma }{\mathrm{d}m} \sim \sigma _0 \exp \left( - \frac{\alpha _s C_F}{2\pi } \ln ^2 \frac{m}{p_T R^2} \right)\!. \end{equation} \tag{ 11 }$$

In principle, subleading logarithms can be straightforwardly computed and resummed. The extension of resummation techniques beyond next-to-leading-log (NLL) is natural using the SCET framework. However, general jet shapes suffer from non-global logarithms that contribute at NLL [89]. Nearly every jet measurement made at the LHC will have non-global observables, and their resummation is more challenging and can only be performed in the leading-colour approximation [89–92]. The physics of the leading non-global logs at fixed order (α²_s) has been well understood in QCD, and their resummation calculated numerically in the large-N_C limit. Recent explorations in SCET have provided additional insights into their structure and origin, providing a possible new path to renormalization group-based resummation [93–95]. Non-global logs are reduced by soft gluon clustering, and so are reduced for algorithms other than anti-k_T [90, 95].

The small-R approximation is advantageous because a simple picture of non-global logs emerges. Each jet has its own non-global correction factor, and cross talk between jets is relegated to a power-suppressed contribution in R. The non-global factor for a jet was calculated in the large-N_c limit in [89] and used for jet mass in [92]. The large-N_c approximation will receive corrections from subleading colour terms which will modify the non-global log contribution at around the 10% level. Since non-global logs contribute at NLL and start at $\mathcal {O}(\alpha _s^2 \ln ^2)$, subleading terms in colour ought not to matter phenomenologically, as was the case for deep inelastic scattering event shapes [96]. Thus, in the small-R approximation, the cross section is described by a resummed factor for each jet dressed by a calculated non-global factor, and one can include jet mass measurements for each jet [92].

Beyond the small-R approximation, one can account for the corrections by including terms that vanish as powers of R. In QCD, one computes soft gluon emission from all of the colour dipoles in the event, including the incoming partons. Expanding the result as a power series in R, one finds that the terms of relative order R² make a numerical impact (at around the 10% level even for R = 0.4), but terms scaling as R⁴ do not matter even for R = 1. Similarly, the non-global factor must be corrected beyond the small-R limit. In SCET, the sum over soft gluon emission from colour dipoles is given in the soft function, where one expands in

$$\begin{equation} t_{ij}^{-1} \equiv \frac{\tan ^2 R/2}{\tan ^2 \theta _{ij}/2} \,, \end{equation} \tag{ 12 }$$

with θ_ij the angle between jets i and j [97, 54]. The requirement 1/t_ij ≪ 1 forces the jets to be well separated, which is needed to factorize the cross section. When jets become close together, additional modes in SCET are required to separate the scale dependence on the small dijet invariant mass [98]. These modes parameterize soft radiation between the nearby jets, and SCET can be formally extended to SCET₊, which includes these modes.

Therefore, for a wide range of jet mass measurements, it should be possible to obtain a resummed result that accounts for non-global logarithms and goes beyond the small-R approximation to sufficient accuracy so as to be phenomenologically useful.

A resummed calculation describes the dominant shape of the distribution. However, many other effects must be taken into account. The shape of the large-mass tail of the distribution is described well by fixed-order QCD. The resummed and fixed-order calculations can be combined in a simple way:

$$\begin{equation} \frac{\mathrm{d}\sigma }{\mathrm{d}m} = \frac{\mathrm{d}\sigma }{\mathrm{d}m}\bigg |_{{\rm resum}} - \frac{\mathrm{d}\sigma }{\mathrm{d}m}\bigg |_{{\rm resum}}^{{\rm FO exp}} + \frac{\mathrm{d}\sigma }{\mathrm{d}m}\bigg |_{{\rm FO}} . \end{equation} \tag{ 13 }$$

The resummed and fixed order distributions are added, and the fixed order expansion of the resummed distribution is subtracted. In the small-mass limit, the two fixed order functions cancel up to power corrections in m/p_T, and the resummed distribution correctly describes the distribution. In the large-mass tail, the resummed distributions (all orders and fixed-order expanded) cancel up to the fixed order in α_s of the expansion, so the fixed-order QCD distribution describes the shape up to higher-order corrections.

Hadronization, pile-up and UE contributions must also be included in a calculation of the jet mass. These non-perturbative contributions affect the small-mass regime of the distribution, and are needed for the calculation to be compared to data. While important, these effects are challenging to include, as no framework exists to systemically include them and they require a more detailed understanding of the data.

Basic models have been formulated to quantify contributions from the UE and pile-up [99, 14, 6, 100]. As jet shapes are measured and calculated more extensively, the comparison between the two is a good testing ground to study these models and fold them into calculations. In the case of 2-jet event shapes in e⁺e⁻ collisions, the correction to the first moment of the event shape from non-perturbative effects has been shown to be related to a universal, measurable non-perturbative matrix element for a wide class of event shapes [101, 102]. In the SCET framework, model soft functions that parameterize non-perturbative effects have also been introduced [103]. Further study is warranted to understand how non-perturbative effects in a hadron collider environment can be introduced, and whether similar universality relations that exist in e⁺e⁻ collisions apply in the hadron collider case.

There are several possible channels in which a jet mass distribution can be measured. These include pure QCD events with inclusive and exclusive jet cross sections, jets produced in association with electroweak bosons and light quark jets produced in association with top quarks. Because the calculation of a jet mass distribution depends on the precise event topologies studied, to compare theory and experiment it is important to carefully choose the measurements. Several key topologies will provide important jet mass measurements. The three we focus on are γ/W/Z + jet, dijet and multijet events. In each topology, there are particular jet mass observables of interest, specialized to the final state.

• γ/W/Z + jet events. These have only a single jet in the final state (assuming the W or Z decays leptonically) and provide the cleanest jet measurements. This class of events is an important reference sample for jet energy scale calibration, and can be used to measure fake rates for certain substructure tagging algorithms. They are a source of jet + lepton and jet + MET final states, and are a background to many new physics searches. Theoretically, they can be used as precision tests of QCD and provide a more calculable environment due to the single jet multiplicity.The observable of interest is the differential jet mass distribution, dσ/dm_J, in bins of jet p_T. A jet veto must be imposed to obtain an exclusive sample of single jet events. A hard veto on additional jets, such as $p_{T_J} < 0.1\, p_T^{\gamma }$, can be useful for jet energy scale calibration. The jet mass distribution in this case is of interest because of the restriction on radiation outside the singe jet. For any value of the jet veto, if the jet mass distribution is finely binned in p_T (several bins instead of one or two), the impact of various components of the calculation can be compared across p_T.
• Dijet events. Dijet events with high-p_T jets have a large cross section and make up a significant fraction of the pure QCD events containing high-p_T jets at the LHC. These events are backgrounds to many new physics searches, and their large cross section means that they must be precisely understood.Because there are multiple jets in the event, different jet mass measurements can be made. The doubly differential distribution,
  $$\begin{equation} \frac{\mathrm{d}^2 \sigma }{\mathrm{d}m_1 \mathrm{d}m_2} , \end{equation} \tag{ 14 }$$
  with p_T1 > p_T2, is interesting because correlations between the jet masses can be studied. This was explored in the CDF analysis of jet mass [104], where the ratio of event numbers in different bins in the double differential distribution was measured. Define two single-jet mass bins b₁ and b₂ (in the CDF study b₁ = (30, 50) GeV and b₂ = (130, 210) GeV). These bins define four regions:
  $$\begin{eqnarray} A&: m_1 \in b_1 , \quad m_2 \in b_1 , \nonumber \\ B&: m_1 \in b_2 , \quad m_2 \in b_1 , \nonumber \\ C&: m_1 \in b_1 , \quad m_2 \in b_2 , \\ D&: m_1 \in b_2 , \quad m_2 \in b_2 . \nonumber \end{eqnarray} \tag{ 15 }$$
  Then, define the ratio R_mass to be
  $$\begin{equation} R_{{\rm mass}} \equiv \frac{N_B N_C}{N_A N_D} . \end{equation} \tag{ 16 }$$
  If the jet mass distributions were independent, then R_mass = 1. Correlations between these distributions exist because of soft gluon exchange, although the expected value of R is not known. This observable is of interest due to the discrepancy between Monte Carlo simulation and data, and a theory calculation could help resolve the differences.In addition to the doubly differential mass distribution, the projection onto a single mass is theoretically accessible. Common choices are the heavy jet mass and the sum of jet masses, which have been studied at e⁺e⁻ colliders. The hardest jet's mass is harder to predict theoretically because the relative p_T in dijet events is affected primarily by soft physics.
• Multijet events. Events with more than two jets provide a more complex background. The colour correlations between jets are enhanced as the detector becomes more crowded, and these events typically have a hierarchy of jet p_T's. Multijet events are more important at the LHC due to the increased phase space for high-p_T parton production, and significant attention has been paid to understanding them better theoretically and experimentally. Multijet events not only present a theoretical challenge to describe but also offer the opportunity to learn about effects such as colour correlations that figure less prominently in simpler events.In multijet events, placing a hard p_T cut on the leading three jets will give a multijet event sample that allows one to study the structure of the hardest jet as the kinematics of the events change. In particular, the jet mass distribution of the hardest jet can be measured in bins of the ΔR separation to the nearest jet. As jets become closer together, inter-jet soft radiation can have a large effect on the mass distribution. This can be predicted theoretically, and the result is compared with the Monte Carlo.

Because the jet mass is sensitive to the environment of the event, it is important to examine the effect of event-wide characteristics on the mass. As pile-up increases, the jet mass can serve as a probe of the size of the pile-up and the effectiveness of subtraction methods (see section 2.3.1).

The rapidities of the hard jets will affect the jet mass distribution for a number of reasons. First, large-rapidity jets imply a large momentum asymmetry between the initial partons, meaning that one carries a large fraction of the proton momentum (x ∼ 0.1). Since the quark parton distribution functions dominate in this regime, the fraction of quark jets will be enhanced relative to central events. Second, the shape of the parton distribution functions implies that there is likely to be more ISR, since they are less steep in the large x region. Coupled with the fact that the large-rapidity jets are closer to the beams, it is interesting to see the dependence of jet mass on the jet rapidity.

Finally, the total transverse momentum in the event, H_T, is a measure of the activity of the event. Soft jets that are vetoed contribute to H_T and will affect the jet mass, but still fit into the same event selection criteria. The contribution of UE and pile-up will similarly affect the jet mass distribution, and H_T is a simple proxy for the size of these observables.

Jet shapes measure the radiation pattern within a jet by calculating a function of the momenta in the jet. The jet mass is a special jet shape, but there are other important jet shapes whose measurement would be very informative. We discuss several. Both N-subjettiness and planar flow are calculable, as they take moments of the momenta in the jet.

• N-subjettiness is a useful variable for jet substructure [38, 56]. N-subjettiness is defined with respect to a set of subjet axes q_i:
  $$\begin{equation} \tau _N^{(\beta )} = \frac{\sum _k p_{T,k} (\min \lbrace \Delta R_{1,k},\Delta R_{2,k},\ldots , \Delta R_{N,k} \rbrace )^{\beta }}{\sum _k p_{T,k} (R_0)^{\beta }}, \end{equation} \tag{ 17 }$$
  where β > 0 is a real parameter, R₀ is the radius of the jet and ΔR_{i, k} is the angular separation between particle k and subjet axis q_i. The shape τ^(β)_N probes how 'N-subjet-like' a jet is, and provides an effective veto on additional subjets. Because adding more axes only lowers the value of τ^(β)_N, N-subjettiness cannot provide a veto against fewer subjets.The case N = 1, β = 1 is also known as girth, broadening, or width; β = 2 is related to the jet mass for small masses. These are the most common choices for β, and it would be interesting to compare them. Additionally, other choices of β can be informative, as they will probe the structure of the jet as a function of the angle to the 1-subjettiness axis. τ^(β)₁ is similar to jet angularities, a jet shaped defined for e⁺e⁻ events [53, 105, 97, 54]. The distribution in jet angularity τ_a was calculated to next-to-leading log accuracy in τ_a [54], and the theoretical tools exist to extend this calculation to the distribution of τ^(β)₁ at the LHC.2- and 3-subjettiness are also of interest. These can probe jets with multiple subjets, and the substructure of these jets will more closely resemble a boosted heavy object decay. (Using τ₃/τ₂ to identify top jets was studied in section 5.)
• Planar flow is an interesting shape variable that was proposed to distinguish top and QCD jets [106, 74]. Planar flow is defined in terms of the eigenvalues of a 2-by-2 matrix, I_w, constructed from the components of momenta perpendicular to the jet axis:
  $$\begin{equation} I_w^{kl} = \frac{1}{m_J} \sum _{i\in J} \frac{p_i^{\perp k} p_i^{\perp l}}{E_i} \,, \end{equation} \tag{ 18 }$$
  where p^⊥k is the kth component of p perpendicular to the jet axes (with respect to a pair of basis axes in the perpendicular plane). From I_w, planar flow is defined as
  $$\begin{equation} Pf = \frac{4\, {\rm det}\, {I_w}}{{\rm tr}(I_w)^2} . \end{equation} \tag{ 19 }$$
  If the radiation in the jet is clustered along a line in the perpendicular plane going through the jet axis, then Pf will be close to 0. This will be the case for QCD jets and jets with 2-body kinematics, such as Higgs or W jets. For jets with 3-body (or more) kinematics, such as the top, Pf will be $\mathcal {O}(1)$.
• Subjet and track multiplicities have been shown to be effective discriminators of quark and gluon jets [45, 46]. Subjets can be found by reclustering the jet with a small radius, and finer resolution of the subjet size can improve discrimination between jets from different sources. The study of subjet multiplicity offers the chance to study the sensitivity to p_T or energy cutoffs placed on jet constituents or subjets. Does the energy resolution for subjets significantly improve with a higher energy cutoff on jet constituents? Does the multiplicity become sensitive to pile-up and the UE if the cutoff is taken too low? Similarly, the charged track multiplicity is a useful observable whose power depends on the angular resolution for track identification as well as the performance over the range of charged particle momenta in the jet.

Many new physics searches can utilize jet grooming algorithms. Filtering, pruning and trimming are jet grooming algorithms that modify the substructure of the jet to remove contributions from the UE and pile-up [35, 33, 34, 64]. These algorithms can be applied to a wide range of channels, and comparing the performance of the three algorithms can be instructive (cf section 6 of [1]). Because each algorithm operates on the substructure differently, it is useful to compare all of them. Experimental studies can quantify the performance of each algorithm and determine which is best for a given application.

Each grooming algorithm operates on a single jet and produces a groomed jet. Therefore, any observable can be measured on the original jet and the groomed jet. The distributions before and after grooming are always of interest to understand the effects of grooming, and grooming algorithms can be used to study a variety of jet properties. The most interesting observables and objects to study with jet grooming are as follows.

• Jet mass.
• Pile-up and the UE are a large motivation for jet grooming methods. The performance with respect to pile-up for some jet grooming algorithms is currently being studied, and is quantified by studying tracks from secondary vertices. Quantifying the performance with respect to the UE is more challenging, but measures exist [100].
• If N-subjettiness is measured, then the effect of the grooming algorithms will be interesting to study. Grooming algorithms are designed to remove isolated soft radiation that can come from the UE or pile-up and contribute to poor mass resolution in the jet. This radiation will also have a large effect on N-subjettiness. Studying the N-subjettiness distribution (for N = 1, 2, 3) will give insight to grooming methods using an observable where it is simple to visualize jets with multiple energetic subjets.
• Given the good tracking and angular resolution at the LHC, the experiments can study the performance of grooming methods using only charged tracks.

Finally, event shapes are also of interest. While these are not direct jet substructure observables, they have a strong connection to jet substructure. Initial studies of jet substructure were driven by Monte Carlo studies, but more recently the theoretical foundations of jet substructure have been explored. Event shapes have traditionally been a testing ground for theoretical tools, and jet shapes are a by-product of these studies. N-subjettiness was built in analogy to N-jettiness, an event shape used to veto against additional jets in an event [50, 107, 108]. Like N-subjettiness, N-jettiness measures how ⩽N-jet-like an event is. Measurements of event shapes can be compared directly to calculations, and the comparison is useful in building theoretical tools to calculate jet substructure properties. These observables are especially amenable to calculation, and in fact distributions of these observables have already been calculated in a variety of applications.

Observables of interest include the following.

• 0-jettiness, also known as beam thrust, is a measure of total hadronic activity away from the beams. In events without energetic jets, this can be used to study the UE and assist in tuning the Monte Carlo. In new physics searches with a veto on any jets (e.g. h → WW), the beam thrust distribution can be used to understand the theory uncertainties on the jet veto.
• 1-jettiness can be similarly implemented on events with a single jet, and can be used to veto against additional jets for an exclusive 1-jet sample. An ample background of γ/W/Z + jet exists in which we study the 1-jettiness distribution.
• A number of event shapes have been proposed for study in dijet-type events in [109] and examined in detail in [110], including observables related to thrust, broadening and jet rates (Y₂₃). Thrust-like observables show enhanced sensitivity to the UE, possibly allowing for powerful constraints on its properties, while Y₂₃ provides a more direct sensitivity to perturbative effects. The distribution of Y₂₃ has also been studied in [73]. The above observables have all been defined with particles as inputs. One can also study event shapes with jets as inputs and in one such study [111], Monte Carlos with matched matrix elements covering a range of jet multiplicities gave a worse description of event shape data than parton shower Monte Carlos (which only include 2 → 2 matrix elements matched directly onto the parton shower). This is troubling, and comparison to more event shapes can provide a way to differentiate different effects in the Monte Carlo simulations that may be contributing to this issue.