Fractal based observables to probe jet substructure of quarks and gluons
EEPJ manuscript No. (will be inserted by the editor)
Fractal based observables to probe jet substructure of quarksand gluons
Joe Davighi and Philip Harris Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge, UK CERN, European Organization for Nuclear Research, Geneva, Switzerlandthe date of receipt and acceptance should be inserted later
Abstract.
New jet observables are defined which characterize both fractal and scale-dependent contribu-tions to the distribution of hadrons in a jet. These infrared safe observables, named Extended FractalObservables (EFOs), have been applied to quark-gluon discrimination to demonstrate their potential util-ity. The EFOs are found to be individually discriminating and only weakly correlated to variables used inexisting discriminators. Consequently, their inclusion improves discriminator performance, as here demon-strated with particle level simulation from the parton shower.
A hadronic jet is produced from an initial parton via a se-quence of perturbative QCD branching interactions (theparton shower), followed by the non-perturbative conver-sion of partons to the hadrons we observe in experiments(hadronization). A Markov chain description of the partonshower suggests the spatial distribution of partons will ex-hibit some fractal character [1,2,3,4,5,6], and this will beinherited by the final hadron distribution (invoking localparton-hadron duality [7]). However, true scale invari-ance of the hadron distribution within a jet is broken bythe running of the branching probability, termination ofthe shower due to hadronization, and finite detector res-olution. Here we define new observables to characterizejet branching structure, named Extended Fractal Observ-ables (EFOs), which accommodate deviations from fractalstructure through simple parametrizations. The idea is toapply box-counting techniques, used widely in the studyof dynamical systems and scale invariant objects, to thesubstructure of QCD jets. Box counting has previouslybeen employed in particle physics to calculate the fractaldimension of electromagnetic showers [8] for highly gran-ular calorimetric reconstruction. Here, we extend the gen-erality and information content of this technique in ourcharacterization of QCD jets.The motivation for this study is two-fold. Firstly, wewould like to characterize the spatial substructure of jetsinto a set of new observables. Secondly, we would like todemonstrate the use of such observables in the discrimi-nation of quark and gluon jets. Quark and gluon discrim-ination has long been used as a tool to enhance the sensi-tivity of signatures with additional quarks [9,10,11,12]. Inparticular, weak boson fusion induced Higgs-production isenhanced due to the distinct signature of two additional hard quark jets in the gluon-dominated forward region ofthe detector [13,14,15,16,17,18,19,20,9,21]. Quark andgluon tagging are also expected to be useful for physicssearches beyond the Standard Model, including the detec-tion of supersymmetric particles [22,23]. Additionally, ifwell designed, these taggers can be further extended tothe subjets of boosted boson signatures [24]. We demon-strate that modest improvements can be made to existingquark-gluon taggers by incorporating the new jet observ-ables defined in this paper.Finally, our construction of pixel-based jet observablesresonates with the recent development of the jet imageparadigm [25,26], in which the energy measured in eachdetector cell is interpreted as the intensity of a pixel in a2D image. Within this approach, powerful machine-learningalgorithms for classifying images have been brought tobear on a range of jet classification problems. This has in-cluded tagging boosted weak bosons [27,26], boosted topquarks [28], and heavy-flavors [29,30].We define EFOs in the following section. In section 3we analyze the performance of these observables in quark-gluon discrimination, before concluding.
The computation of the EFOs is performed on a jet by jetbasis using a variation of the Minkowski-Bouligand (box-counting) dimension, as follows.
To define our variables we implement a two-stage recipe:firstly, the jet cone is divided in the familiar ( η, φ ) angu- a r X i v : . [ h e p - ph ] M a y Joe Davighi, Philip Harris: Fractal based observables to probe jet substructure of quarks and gluons
Fig. 1.
An illustration of the iterated box-counting procedure used to calculate fractal-based quantities on a set of points.The filled blue circles are the ( η, φ ) angular coordinates of the hadrons within a particular sample jet (in particular, this jethas total p T = 157 GeV, and 30 constituent hadrons). The box-counting is illustrated for four sample scales, corresponding tosuccessively finer (cid:15) values of 0 .
2, 0 .
1, 0 .
067 and 0 .
05. The cells registering particle hits are highlighted with red shading. lar coordinates into a square grid of cells, each cell havingside-length (cid:15) . For a given scale (cid:15) , we count the numberof cells N hits ( (cid:15) ) which register particle hits with a totaltransverse momentum greater than some pixel-level softcutoff, in this study chosen to be p T > . y = log N hits ( (cid:15) )with x = log (1 /(cid:15) ), and to extract the parameters of thefit as a set of (correlated) jet observables, which we callExtended Fractal Observables (EFOs). This is a general-ization of the traditional box-counting method, in whichonly linear functions y = mx + c are fitted, with the gra-dient m identified as the fractal dimension [8]. Indeed, in Figure 2 there is no distinct region of linearscaling, as would be needed to extract a fractal dimension.Rather, log N hits ( (cid:15) ) levels off smoothly from large to smallscales as saturation is approached, motivating a non-linearfit to extract whatever information this curve might en-code about the jet. In particular, the hadronization region(i.e. at small (cid:15) ) obviously carries non-perturbative infor-mation sensitive to the flavor of the jet. The observedcurves are distinct between quarks, gluons and b-quarks,as summarized in Figures 2 and 3. This scaling is a funda-mental property of QCD resulting from the differences inthe splitting of quarks and gluons. Further measurementsof this scaling allows for an alternative approach to extractQCD properties such as the strong coupling constant [32,33].The generic plateauing curves in Figure 2 can be fit-ted by almost any non-linear function (given a suitablyrestricted range in x ), so we studied fit functions with atmost three parameters, for speed and robustness of fitting. oe Davighi, Philip Harris: Fractal based observables to probe jet substructure of quarks and gluons 3 (1/scale) log0.6 0.8 1 1.2 1.4 ) h i t s ( N l og b-quarkgluonquark (1/scale) log0.6 0.8 1 1.2 1.4 ) h i t s ( N l og b-quarkgluonquark Fig. 2.
Left: logarithmic fits to log N hits ( (cid:15) ) against log (1 /(cid:15) ) for light quarks, bottom quarks, and gluons, of the form y = p + p x + p log x . The values of the fitted parameters { p i } define one possible set of Extended Fractal Observables. Right: fitsto log( N hits ) against log (1 /(cid:15) ) using an asymptotically saturating fitting function, specifically y = p + p tanh( x − p ). (1/scale) log0.6 0.8 1 1.2 1.4 ) / qua r k h i t s ( N l og b-quark/quarkgluon/quark (1/scale) log0.6 0.8 1 1.2 1.4 )( X - qua r k ) h i t s ( N l og D - - b-quark/quarkgluon/quark Fig. 3.
Left: the ratio of log( N hits ) with respect to the quark values, for b-quarks and gluons, as a function of log (1 /(cid:15) ). A linearfit is added for comparison. Right: the difference of log( N hits ) with respect to the quark values, for b-quarks and gluons. In theModified Leading Logarithmic Approximation (MLLA), the differences in hadron multiplicity between quarks, b-quarks andgluons are predicted to be energy independent [31]. The small but non-zero slopes in this plot reflect the fact that box-countingat a given angular scale probes spatial information in addition to the rate of splitting at the corresponding energy scale. Fits were carried out simply by a binned χ minimizationof the chosen function. Example fit functions included thefollowing:1. logarithmic fits of the form y = p + p x + p log x .2. quadratic fits: y = p + p x + p x .3. hyperbolic tangent fits: y = p + p tanh( x − p ).The values of the best fit parameters { p i } for each fit-ting function constitute three possible sets of EFOs. Fora polynomial in x = log(1 /(cid:15) ), like the quadratic fit func-tion, the fit reduces to a matrix inversion and thus has awell-defined convergence. The other two parametrizationsare not polynomials, hence we perform a χ minimization.Functions which actually saturate, such as the hyper-bolic tangent parametrization above, are more physicallymotivated because they can model the saturation itself(asymptoting to the jet multiplicity). However, for therange of box scales used in our study (of width (cid:15) ≥ .
05, -see 2.2 below), and for all but the lowest p T jets, the non - saturating fit functions also provide adequate models forthe observed scaling. For the purpose of quark-gluon dis-crimination (see section 3), the logarithmic fitting func-tion was found to give the best discrimination performanceof the three functions above (see Figure 6 to compare theperformance between the logarithmic and hyperbolic tan-gent fitting functions). The range of angular scales (cid:15) has been chosen by pavingthe jet cone with a square grid of N × N cells, wherethe splitting scale N ranges in integer steps from 3 to 16.For each N , the angular scale is (cid:15) = 2 R/N , where R isthe jet radius, in this study R = 0 .
4. The coarsest (cid:15) scalechosen, corresponding to N = 3, is essentially the coarsestscale carrying potentially discriminating information (for N = 2 the jet cone would be divided into four quarters, Joe Davighi, Philip Harris: Fractal based observables to probe jet substructure of quarks and gluons all of which will register a hit for realistic jet shapes). Thefinest (cid:15) scale chosen is (cid:15) min = 0 . /
16 = 0 .
05, becausethis is approximately the angular detector resolution inboth LHC experiments, CMS and ATLAS [34,35]. For the p T ≥
100 GeV jets studied here, the number of hits is justbeginning to saturate at this scale (see Figure 2), so weare probing into the hadronization region prior to the flatplateau.Finally, we would like to highlight that these fractal-based observables are similar in spirit to calculating subjetrates of jets [15,36], given subjets clustered using the p T -independent Cambridge-Aachen algorithm [37]. Both ob-servables compute p T -independent branching informationon a succession of angular scales down to some thresh-old. And both observables perform what is essentially afurther clustering on the substructure of the jet to extractthis information pertaining to the branching history of thejet. In light of this, the EFO approach could be extendedto utilize subjet counts (instead of hit grid cell counts) toassign scale-dependent multiplicities N ( (cid:15) ). Preserving infrared and collinear safety ensure calculabil-ity in perturbative QCD. An observable is infrared (collinear)safe if its value is unchanged by the emission of soft (co-moving) particles. The EFOs, as defined in 2.1 with apixel-level soft cutoff, are fully IRC safe.Firstly, the box counting procedure is intrinsically collinearsafe: if one particle splits into two particles with the same( η, φ ) coordinates, we still count just one cell hit by bothdaughter particles, at any finite scale of probing. Hencecollinear splittings will not affect the number of cells N hits ( (cid:15) )to register particle hits at any choice of scale. On theother hand, infrared safety of the EFOs can only be engi-neered by imposing some low momentum cutoff to cleansethe jet of its soft constituents. However, this soft cutoffmust be implemented consistently with collinear safety.If we simply discarded all soft hadrons with, say p T < p T = 1 . p T = 0 . p T = 0 . N hits ( (cid:15) ) wouldnot be invariant under this collinear splitting.This is remedied by defining a pixel-level (rather thanparticle-level) sort cutoff. That is, we only consider a cellto register a hit if it measures a total p T greater thanour soft cutoff of 1 GeV. This way, if the troublesome1 . p T of 1 . p T cut (over valuesbetween 0 . . p T cut of 1 GeV isused throughout. Finally, we acknowledge that pixel-levelcutoffs have been used previously in the context of jet im-ages analyses (for example in [25]) to ensure IRC safetyin the same context. We now investigate whether these observables might be auseful new tool in the important and challenging problemof distinguishing light quarks from gluon jets.
In this study, we use QCD dijet samples at a center-of-mass energy of 13 TeV. Because previous quark-gluonstudies have revealed that discrimination performance variesa lot between the different generators [38,9,10,14,11] , wehere produce and shower events (at leading order) usingboth Herwig++ (version 2.7.0 with tune UE-EE-5C ) [39,40] and Pythia 8 (version 8.185 with tune CUETP8M1)[41],with order 150k events in each. Jets are clustered with theanti- k T algorithm using the final state particles followingshowering and hadronization; a cone size of R = 0 . p T jet ineach event. We define the flavor of that jet by matchingto the highest- p T parton within R < . .As a baseline for comparison, we shall consider thevariables currently used by the Compact Muon Solenoid(CMS) quark-gluon tagger, which are [10]: i) the totalnumber of reconstructed particles in the jet (the multi-plicity) [43]; ii) the p T D variable ( C β =01 )[44], p T D = (cid:113) Σ i p T,i Σ i p T,i , (1)where i sums over the constituents of the jet, which de-scribes the distribution of transverse momentum betweenthe particles in the jet; and iii) σ , the ( p T -weighted) semi-minor axis of the jet in the ( η, φ ) plane [10], defined by σ = ( λ /Σ i p T,i ) / , (2) Herwig has been consistently seen to give the more conser-vative estimates of discrimination power, both with respect toPythia and real LHC data. Note that b(bottom)-jets may be efficiently identified usinga secondary vertex tagger, and separately vetoed.oe Davighi, Philip Harris: Fractal based observables to probe jet substructure of quarks and gluons 5 gluon ˛ qua r k ˛ Multiplicity s DT p ,p ,p p - - - - - s D T p pa r t N F r a c t a l D i m . p p p s DT p part NFractal Dim. p p p
100 -36 -3 33 41 -39 41-36 100 -62 -46 -19 -15-3 -62 100 60 28 -3 2333 -46 60 100 79 -65 7941 -19 28 79 100 -95 99-39 -3 -65 -95 100 -9741 -15 23 79 99 -97 100
Fig. 4.
Left: single variable performance ROC curves. The EFOs, minor axis, and p T D are significantly more discriminatingthan multiplicity. The EFOs are most discriminating for high signal efficiency ( (cid:38) where λ is the smaller eigenvalue of the 2 × M = Σ i p T,i ∆η i , M = Σ i p T,i ∆φ i , and M = − Σ i p T,i ∆η i ∆φ i . Throughout thisstudy, we build multi-variable quark-gluon discriminantsusing a boosted decision tree (BDT), implemented usingthe Toolkit for Multivariate Analysis (TMVA) via adap-tive boosting. The p T of the quark and gluon samples arereweighted to match the exact same kinematics in bothcases, so as to avoid selection biases induced by kinematicdifferences in the simulation. We first compare the discriminator performance of singlevariables and the correlations between them, before go-ing on to compare multi-variable taggers built with andwithout inclusion of the new EFO observables.We can measure discriminator performance by receiveroperator characteristic (ROC) curves, which plot back-ground rejection against signal efficiency. Roughly speak-ing, the more convex the curve, the better the perfor-mance. The left plot of Figure 4, made using the Her-wig samples, shows that the EFOs are individually well-discriminating, particularly if we seek high signal efficiency.Their performance is significantly better than that of thejet multiplicity variable. We use a BDT discriminator built from the combinationof the three EFOs, p , p and p . While the combination of allthree EFOs adds little discrimination beyond that of a singleEFO due to their near-perfect correlation, the selection of anysingle p i would be arbitrary for the sake of this comparison. The right plot of Figure 4 presents the linear correla-tion coefficients (calculated using the TMVA toolkit) be-tween the EFOs and the existing CMS quark-gluon tag-ger variables: multiplicity, p T D and σ . We also include acomputation of the fractal dimension, which has been cal-culated from a linear fit over a small range of box scales.Strong correlations are present amongst the EFOs, as isnatural given they are parameters derived from the samefit. However, their correlations with the other variablesare no greater than 43% (for either quarks or gluons) .Interestingly, the EFOs are most highly correlated with σ , not multiplicity as might have been expected. Thisevidence suggests the discrimination power of the EFOsis not simply a result of higher multiplicities in gluon jets,and therefore that the addition of these parameters to aquark-gluon discriminator might improve performance.We find that replacing the multiplicity variable in theexisting CMS quark-gluon tagger with the EFO variableyields a gain in discriminator performance, albeit onlya modest one. This gain is seen using both Herwig andPythia event generators (with the setup described above)in the ROCs presented in Figure 5, which are for jets with p T ≥
100 GeV. We see the performance in Pythia is signif-icantly better than Herwig for each combination of vari-ables, consistent with previous studies [9,10,14,11].Moreover, the incremental gain upon replacing multi-plicity with the EFOs is larger in Pythia than Herwig, soHerwig gives the more conservative estimate of the im-pact of including the EFOs. We see the gain in perfor-mance (relative to a baseline tagger using just p T D and Note that the traditional fractal dimension is more stronglycorrelated to existing QGD variables, particularly multiplicity. Joe Davighi, Philip Harris: Fractal based observables to probe jet substructure of quarks and gluons
Fig. 5.
ROC curves for BDT discriminators constructed from various combinations of observables, as indicated by the legend,for events showered using both Herwig and Pythia with jet p T ≥
100 GeV. The discrimination is superior in Pythia. We see inboth event generators that including the EFOs rather than multiplicity (which is used in the CMS tagger) yields a marginallybetter performance. gluon ˛ D T , p s G a i n w r t Herwig DT ,p s ,Multiplicity DT ,p s ,EFOs DT ,p s ,EFOs tanh DT ,p s gluon ˛ D T , p s G a i n w r t Pythia DT ,p s ,Multiplicity DT ,p s ,EFOs DT ,p s ,EFOs tanh DT ,p s Fig. 6.
Left: the relative gain for the three-variable taggers with respect to a baseline tagger using just p T D and σ , for theHerwig events (which yield more conservative discrimination estimates). The gain is also plotted for EFOs computed with thehyperbolic tangent fitting function specified in subsection 2.1, for which the performance is worse. Right: for Pythia events.Note the wider range of the y-axis, to accommodate the larger gains found in Pythia. σ ) more clearly in Figure 6, with the left panel for Herwigand the right for Pythia. The gain is at the level of 1 − p T cut. Finally, weinvestigated how the performance varies with energy scale,by performing the analysis in p T bins of 50 −
100 GeV, 100 −
200 GeV, and 200 −
500 GeV. Discrimination wasfound to increase with p T in both Herwig and Pythia (seeFigure 7 for the Herwig results).Combining all four variables (multiplicity, p T D , σ andthe EFOs) was seen to give no further improvement. Thissuggests all the information from multiplicity is captured oe Davighi, Philip Harris: Fractal based observables to probe jet substructure of quarks and gluons 7 Fig. 7.
Performance of a possible new quark-gluon tagger (us-ing p T D , σ , and the EFOs), in three p T bins, for Herwig-generated dijet events. Quarks and gluons are found to be eas-ier to distinguish using this tagger at higher p T . by the EFOs , while the converse is not true. In summary,we have presented evidence in this study that the Ex-tended Fractal Observables provide an additional handlethat captures the salient features of jet multiplicity, incor-porates new information from showering and hadroniza-tion, and which is also better behaved under IRC emission(see 2.3). In this study we defined new jet observables, the Ex-tended Fractal Observables, by a generalization of thebox-counting method used in the study of fractal systems.Defined with a pixel-level low momentum cutoff, these ob-servables are infrared and collinear safe. We have thensought to apply the EFOs to improve quark-gluon dis-crimination. At the generator level, we find some modestimprovement in discrimination by gluon rejection when wereplace multiplicity with the EFOs in the existing CMStagger, across both Herwig++ and Pythia 8. Extendingthe performance of these new variables to include detec-tor effects can naturally be performed in the LHC en-vironment with the CMS Particle Flow algorithm [45] inconjunction with the PUPPI algorithm [46] to reconstructparticle candidates in the presence of high pile-up.
This method of studying jet substructure is a new ap-proach. As such, there are many directions in which wewould like to proceed, including: This is unsurprising, because jet multiplicity is simply theasymptotic number of hits as we approach the saturation re-gion.
1. Exploring particle hits in a 3-dimensional coordinatespace spanned by η , φ and z − , where z is the frac-tional transverse momentum of the jet constituent.2. Applying the EFOs beyond Quark-Gluon discrimina-tion, for example to the identification of pile-up jets,or initial state radiation.3. These box-counting methods extend very naturally fromthe substructure of a single jet to a whole-event anal-ysis. Such a novel approach may provide new insightinto searches for new physics topologies such as thosein supersymmetry or top quark pair production [47].4. Furthermore, box-counting analyses could provide auseful characterization of event shapes in heavy ioncollisions, where studies of jet properties beyond jetreconstruction are traditionally difficult, but well mo-tivated [48,49,50].5. Finally, we would like to emphasize that the calcula-tion of EFOs on quark and gluon jets probes partonshower scaling that results from the QCD color fac-tor ratio. Calculating EFOs on cosmic ray air showerprofiles [51] could therefore help discriminate QCD-induced air showers from more interesting signals; ofparticular interest, showers induced by electroweak sphalerons.Experimentally, the calculation of EFOs in this airshower context is conceptually appealing: the 1660 in-dividual Cerenkov detectors (spread over 3000 km ) ofthe Pierre Auger Observatory in Argentina [52] wouldnaturally function as the finest-scale cells in our box-counting algorithm. These techniques could thereforebe useful in probing physics at energies far beyond thatof the LHC. JD’s work has been supported by The Cambridge Trust,and by the STFC consolidated grant ST/L000385/1. Wethank the CERN summer student program where thiswork was initiated. We also thank Andrew Larkoski forhis insightful comments when performing these studies,and Bryan Webber for helpful discussions. Finally, wethank Eric Metodiev for helpful comments.
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