Boosting the standard model Higgs signal with the Template Overlap Method

Journal of High Energy Physics, Jul 2013

We show that the Template Overlap Method can improve the signal to background ratio of boosted h → \( b\overline{b} \) events produced in association with a leptonically decaying W. We introduce several improvements on the previous formulations of the template method. Varying three-particle template subcones increases the rejection power against the backgrounds, while the longitudinally invariant, sequential template generation ensures an efficient coverage in template phase space. We integrate b-tagging information into the template overlap framework and introduce a new template based observable, the template stretch. We further improve on the previous formulation of Template Planar Flow by including the effects of finite size subcones centered at each template momentum. Our analysis takes into account the contamination from the charm daughters of top decays in \( t\overline{t} \) events, and includes nearly-realistic effects of pileup and underlying events. We show that the Template Overlap Method displays very low sensitivity to pileup, hence providing a self-contained alternative to other methods of pile up subtraction. The developments described in this work are quite general, and may apply to other searches for massive boosted objects.

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Boosting the standard model Higgs signal with the Template Overlap Method

JHE Boosting the standard model Higgs signal with the Template Overlap Method Mihailo Backovic 1 Jose Juknevich 1 Gilad Perez 0 1 0 CERN, Theory Division , CH1211 Geneva 23 , Switzerland 1 Department of Particle Physics and Astrophysics, Weizmann Institute of Science , Rehovot 76100 , Israel We show that the Template Overlap Method can improve the signal to background ratio of boosted h ! bb events produced in association with a leptonically decaying W . We introduce several improvements on the previous formulations of the template method. Varying three-particle template subcones increases the rejection power against the backgrounds, while the longitudinally invariant, sequential template generation ensures an e cient coverage in template phase space. We integrate b-tagging information into the template overlap framework and introduce a new template based observable, the template stretch. We further improve on the previous formulation of Template Planar Flow by including the e ects of nite size subcones centered at each template momentum. Our analysis takes into account the contamination from the charm daughters of top decays in tt events, and includes nearly-realistic e ects of pileup and underlying events. We show that the Template Overlap Method displays very low sensitivity to pileup, hence providing a self-contained alternative to other methods of pile up subtraction. The developments described in this work are quite general, and may apply to other searches for massive boosted objects. Jets; Hadronic Colliders - 4 1 Introduction 2 3 Template Overlap Method 2.1 Other peak template observables 2.1.1 Angular correlations 2.1.2 Template Planar Flow 2.1.3 Template b identi cation 2.1.4 Template Stretch 2.2 Varying subcone templates and showering correction 2.3 Stability of the Template Overlap Method Data simulation and analysis 3.1 Higgs tagging with Template Overlap - no pileup 3.1.1 Background rejection power at ps = 8 TeV 3.1.2 Background rejection power at ps = 13 TeV 3.2 Higgs tagging with Template Overlap - e ects of pileup 3.2.1 Rejection power 4 Conclusions A More on template Planar Flow B E cient generation of template libraries B.1 The case of 2-body templates B.2 The case of 3-body templates B.3 Extension to arbitrary N B.4 Numerical simulations Introduction Recent results from ATLAS [1] and CMS [2] have con rmed the discovery of a new boson of mass roughly 125 GeV, decaying to , ZZ and likely W W (for the Tevatron combination of the Higgs searches see ref. [3]). The results are so far consistent with the interpretation of the new particle as the Standard Model (SM) Higgs boson. While the data agrees with the SM Higgs boson, our understanding of the new particle's properties remains incomplete. A strong test of the SM Higgs theory consists of a detailed experimental study of its sharply predicted characteristics. This includes, among others, observing all the major decay modes of the SM Higgs, as well as establishing no deviations from the predicted SM Higgs production and decay rates. Global ts to the Higgs boson { 1 { production and decay rates allow for extraction of its couplings to various other SM elds, as well as possible invisible channels (see refs. [4{8] for recent analyses). A SM Higgs boson of 125 GeV predicts a dominant decay mode to a bb pair, which calls for a direct veri cation. An enormous QCD background, however, makes it rather di cult to observe this channel at the LHC. The hadronic Higgs decay mode has so far only been reported by the Tevatron experiments, at 2:8 in the CDF/D0 combinations. Despite an impressive progress by both CMS and ATLAS, the extraction of a statistically signi cant measurement of h ! bb rate from the LHC data at 8 TeV remains challenging. One way to reduce the QCD background in h ! bb is by focusing on associated Higgs production with W and Z bosons. However, the cross section of the background pp ! V bb process is still much higher than the cross section of pp ! V h. Even after a cut on the mass of the bb system around the known Higgs mass, the signal is swamped by the background. Authors of ref. [ 9 ] showed that when considering moderately boosted Higgs events, traditional jet clustering algorithms with large cone sizes (R 1) can be used to increase the signal to background ratio (S/B). The method of ref. [ 9 ] is based on the fact that decay products of a boosted Higgs are collimated and can be captured within a single \fat" jet. This, in principle, reduces the combinatorial background, contamination from soft and incoherent components and allows to better characterize the structure of the energy ow within the fat jet. The last few years have seen a proliferation of new theoretical and experimental techniques to identify high-pT jets at the LHC (see refs. [10{17] for recent reviews and references therein). Two main classes of approaches have emerged: Filtering [ 9 ] (see also refs. [18, 19]) and Template Overlap Method [20]. Filtering algorithms act on the list of jet constituents by removing the soft components based on some measure which de nes the \hard" part of the jet. The remaining constituents are then reclustered into the \ ltered" jet. The Template Overlap Method, discussed in detail below, does not manipulate the measured jet's list of constituents, nor does it require any special clustering algorithm. Instead the method compares the jet to a set of parton level states built according to a xed-order distribution of signal jets called templates. The comparison makes use of an \overlap function" which evaluates the level of agreement between each measured jet and a set of templates. Ref. [20] focused on building the templates according to the leading order decay modes, namely two body (N = 2) for the boosted Higgs and three body (N = 3) for the boosted top. In ref. [21], the ATLAS collaboration used the Template Overlap Method together with the the HEPTopTagger [22] to search for heavy tt resonances. Authors of ref. [ 23 ] showed how to extend the Template Overlap Method beyond leading order as well as how to construct templates which describe the energy ow of say h ! bbg in an infrared (IR) safe manner. The method allows one to gain access to \partonic-like" observables which correspond to the template con gurations with the maximal overlap score. The resulting information can further improve the ability to distinguish the signal from various background channels. Furthermore, ref. [ 23 ] introduced the concept of template jet shapes, such as Template Planar Flow (tPf) [23{26] and template-angularity [ 24, 27 ]). Template jet shapes can further boost the background rejection power of the overlap method. The results of ref. [23] { 2 { H E P 0 7 ( 2 0 1 3 ) 1 1 4 showed that template overlap is capable of delivering background rejection factors of O(100) against the W j's background in the idealistic ultra-boosted Higgs regime (i.e. pT 1 TeV), when combined with other jet substructure observables. In this paper, we examine the decay and radiation patterns of a boosted SM Higgs boson, with focus on a realistic pT kinematic regime (i.e, 300 400 GeV). We argue that a boosted Higgs search using the Template Overlap Method is viable in the future LHC run. We achieve the best signal sensitivity by combining templates in the full phase space for N = 3 and N = 2 overlaps in addition to other template based observables. Our treatment of Template Overlap Method improves on the previous formulations in several ways. First, we de ne templates in terms of longitudinally boost-invariant variables. Second, and more importantly, we entirely revamp the method of template generation. In ref. [ 23 ], the minimum number of templates required to adequately describe the jet energy ow in the medium pT range, was roughly two orders of magnitude larger than in this paper. The reason is that in ref. [ 23 ], templates were generated in the Higgs rest frame (with a MonteCarlo-like method) and then boosted to the lab frame on an event by event basis. Generating templates in the lab frame and \tiling" them according to the event kinematics leads to better coverage of phase space at lower pT and to a great improvement in the overall performance of the analysis. Third, we introduce b-tagging into the Template Overlap framework. Information about b-jets combined with peak templates serves to improve the rejection power, de ned as the signal e ciency divided by the e ciency for the background. Fourth, the optimal radius of the template subcones is not necessarily the same for every parton in a template, as low momentum subjets tend to have wider angular pro les. We allow the three-body template subcones to vary with pT providing a more adequate description of the showering patterns within a fat jet. Varying cones improve the tagging performance of three-body overlap as well as most of the other template-derived observables. Fifth, we introduce a new variable, Template Stretch, which exploits the di erence in plain distance of the two leading b-tagged subjets relative to the signal expectations. Template stretch is correlated with the jet mass, but with an important advantage of robustness against pileup. Finally, we include the template subcones in the formulation of tPf. Inclusion of subcones serves to simulate the e ects of showering on the numerical value of tPf. Our analysis includes nearly-realistic e ects of pileup and underlying event (UE). In high luminosity environments, the large jet cone radius allows for severe e ects of pileup on the spectrum of both jet and substructure observables. We show that the template jet shapes give us an additional handle on pileup, as they are based on best matched templates and not jet constituents. The \spikiness" of the jet energy distribution naturally avoids the complication of soft un-correlated backgrounds. The Template Overlap Method is thus less susceptible to pileup compared to other kinematic observables such as jet invariant mass and pT . This feature invites us to re-consider several jet-shape observables in terms of the \partonic" distribution of the peak templates. For instance, jet Planar Flow (Pf) is known to exhibit high susceptibility to pileup [ 28, 29 ], however it is a useful background discriminant when the hard and coherent part of the massive jet is considered [ 24, 25, 30 ]. We thus introduce a pileup-insensitive alternative to Planar Flow constructed from the template states alone. { 3 { H E P 0 7 ( 2 0 1 3 ) 1 1 4 We use parton-shower simulations to illustrate the non-susceptibility of the Template Overlap Method to a high-pileup environment. Our results agree with the 7 TeV ATLAS data analysis in ref. [21], which showed that the overlap method is indeed fairly robust to presence of moderate pileup contamination. In section 2, we give an overview of the Template Overlap Method and introduce several template based observables sensitive to the QCD radiation patterns. Section 3 describes our Monte Carlo (MC) data generation, and shows the results for boosted Higgs searches with a mass of 125 GeV at ps = 8 TeV and ps = 13 TeV. Section 3 also contains a detailed discussion of pileup e ects. We give a detailed review of template Planar Flow in appendix A, while appendix B describes the new method of template states generation. 2 Template Overlap Method Template Overlap Method is based on the quantitative comparison between the energy ow inside physical jets and the energy carried by partons modeled after the boosted signal events (templates). We de ne libraries of templates as sets of N four-momenta jf i = jp1; p2; ; pN i representing the decay products of a SM Higgs boson at a xed momenta P and Higgs mass mh: N X pa = P; a=1 P 2 = m2h: (2.1) We require that the N quanta of energy be captured within an anti-kT jet of varying size, scaled according to the pT of the Higgs (or according to the pT of the associated vector boson). The number of partons in the templates is calculated at xed order in perturbation theory,1 but is not limited to the leading order con guration, and \next-to-leading-order" templates with more than the minimum number of partons are possible. Below we focus on combining the information from templates in the full phase space for N = 3 and N = 2 partonic con gurations, and show that they provide some additional rejection power against the dominant W bb and tt backgrounds. Next, a functional measure quanti es the agreement in energy ow between a given Higgs decay hypothesis (a template) f and an observed jet j. A scan over a large set of templates that cover the N -body phase space of a Higgs decay results in f [j], the template which maximizes the functional measure. Our primary jet substructure observable is the N -body overlap OvN , the value of the functional measure for the best matched template. For each jet candidate, we de ne the (maximum) overlap as 2 2 OvN (j; f [j]) = max 4exp 4 ffg 0 XN 1 2 @ pT;a a=1 a 1233 X pT;i F (n^i; n^a)A 55 ; i2j (2.2) where ff g collectively denotes a template library for the given jet pT , pT;a is the transverse momentum of the ath template parton and pT;i is the transverse momentum of the ith jet 1In principle, one can re-sum the soft radiation from each of the partons. Here we simply stick to a xed order perturbation theory description. { 4 { E P 0 7 ( 2 0 1 3 ) 1 1 4 constituent (or calorimeter tower, topocluster, etc.). The rst sum is over the N partons in the template and the sum inside the parentheses is over jet constituents. The kernel functions F (n^; n^a) restrict the angular sums to (nonintersecting) regions surrounding each of the template momenta. We refer to the template state which maximizes the functional measure as the \peak template" f [j] of the jet j. The parameter allows us to correct for the energy not captured by the template overlap. In the case of N = 3, we use = 0:8. For more details see section 2.2. The de nition of OvN in eq. (2.2) di ers from the previous de nitions of overlap in refs. [ 20, 23 ]. We nd that the formulation of OvN in terms of longitudinally boost-invariant quantities is more appropriate for a detailed hadron collider study. Notice, however, that since we will be focusing on Higgs events with high transverse momentum, the di erences between the results obtained with previous de nitions of OvN and eq. (2.2) should not be signi cant. In this analysis, we take the kernel function to be a normalized step function that is nonzero only in non-overlapping angular regions around the directions of the template momenta pa: F (n^i; n^a) = ( 1 if R < ra ; 0 otherwise where R is the plain distance between the template parton and a jet constituent in the ( , ) plane. The parameters ra determine the angular scale of the template subjet. Together with the energy resolutions a, these are the only tunable parameters of the model. The kernel de nition of eq. (2.3) assumes that the angular regions around the template momenta are non-intersecting, meaning that any pair of template momenta i; j satis es the condition Rij > ri + rj ; where ri;j are the template subcones. Eq. 2 serves to ensure that a single particle in an event can not be assigned to more than one template momentum in the overlap calculation. If the condition for non-intersecting templates is not satis ed, the template is discarded as non-valid. A few ideas for possible strategies to determine the values of ra and a are listed below: Choose the single best parameter according to some optimization criterion (e.g., optimize the tagging e ciency and background rejection), and use the same values for all the partons within a set of templates. Choose the parameters separately for each template, e.g. using a pT -dependent scale for template matching. Based on a combination of these two criteria, we x a (for the ath parton) by that parton's transverse momentum, a = pT;a=3: { 5 { (2.3) (2.4) H E P 0 7 ( 2 0 1 3 ) 1 1 4 We use a subcone of radius r2 = 0:3 for the two-body template analysis, while the threebody subcone is dynamically determined on a template-by-template basis. Section 2.2 contains a detailed discussion on the optimal scaling of subcone radii. We should nevertheless emphasize that the overall performance of the method can be maximized for a wide range of r by rescaling other parameters. Next, we generate libraries of O(104) 2-body templates and O(106) 3-body templates in steps of Higgs pT of 30 GeV starting from 315 GeV. For each event, we achieve the best signal sensitivity by dynamically selecting a set of templates based on the transverse momentum of the W boson, pTW 2 hpbTinmin; pbTinmax ; (2.5) where pbTinmin; pbTinmax are the limits of the bin corresponding to the template set with ptTemp = (pbTinmax pbTinmin)=2. For instance, an event with pTW = 320 GeV would be analyzed by a template set with pT = 315 GeV etc. We give more details on template properties and generation in the appendix. Replacing the criteria for the template set selection from jet pT to pTW has a enormous advantage when considering e ects of pileup, however it is not the only choice. As an alternative, one could analyze each jet with all template sets, but at a huge expense in computation time. We use the TemplateTagger [31] numerical package for the template matching analysis.2 The package is a C++ code which provides basic implementation of the Template Overlap Method for jet substructure, as well as several other jet analysis tools. 2.1 Other peak template observables Template overlap provides a mapping of nal states j to partonic con gurations f [j] at any given order in perturbation theory. Once the best matched template f [j] is found, it can be used to characterize the energy ow of the state, giving information on the likelihood that the event is signal or background. The scope of template overlap does not stop with Ov2 and Ov3. Peak templates contain additional information about correlations within the fat jet, some of which we explore in the following sections. 2.1.1 Angular correlations Of particular value are angular correlations between template momenta which can otherwise be concealed in the numerical values of the peak overlap. For instance, the angular distribution of jet radiation can be measured with the variable [ 23 ], de ned as = X sin i RiJ ; (2.6) where RiJ is the distance in the ( ; ) plane between the ith template partonic momentum and the jet axis. When measured using three-body templates, the variable exploits the fact that signal events tend to have smaller emission angles. Notice that for highly boosted jets, the 2-body version of simply reduces to the angle between the two partons [ 24 ]. 2Publicly available at: { 6 { J H E P 0 7 ( 2 0 1 3 ) 1 1 4 Another useful background discriminant is Planar Flow [ 24, 25 ] (see ref. [26] for a recent study of the Pf distribution of QCD massive boosted jets). Previous de nitions of Pf were de ned in terms of the two momenta in the plane perpendicular to the jet axis. Here, we choose to de ne the Pf variable in terms of longitudinally boost-invariant quantities. We begin by de ning the two-momenta pik as with i; i measured relative to the jet axis. Next, we de ne the \jet inertia tensor" Ikl as: pi1 piT i; pi2 piT i; Ikl 1 mJ N X pkpl iT i ; i pi Pf 4 det(I) tr(I)2 : where mJ is the jet mass, i runs over the jet constituents and pk;l are de ned in i eq. (2.7). The Pf jet shape is then given by Notice that because of the fact that the trace of Ikl is proportional to the jet mass [32], Planar Flow is only well de ned for massive jets [ 29 ]. Planar ow is particularly helpful in distinguishing energy ow distributions which lie on a line (Pf ! 0) from uniformly distributed energy ow (Pf ! 1). For instance, Planar Flow of a boosted Higgs will tend to be smaller than that of a massive QCD jet that in turn will be smaller than that of a boosted top or gluino [33]. Planar Flow of a jet is useful when considering only the hard and coherent part of the jet. Because of high sensitivity of jet Planar Flow to pileup and UE, here we consider Template Planar Flow (tPf) [ 23 ], as an alternative. We de ne tPf using peak template momenta as well as the template subcones to include physical e ects of energy smearing. To compute tPf, we use the de nition of eq. (2.9) with two major di erences. First, in the de nition of tPf, the index i runs over the three template momenta of a three-body template con guration. Second, the inertia tensor takes the form Ikl ! Iskl = Ikl + 1 X3 piT ri kl; mJ i=1 (2.10) where ri is the template subcone radius and Ikl is de ned in eq. (2.8). Note that other de nitions of Iskl are possible. Appendix A gives a more detailed discussion. The advantage of tPf is that it is constructed purely out of peak template states and thus less susceptible to pileup. We demonstrate this point using a Monte Carlo simulation in section 3.2. In addition, the non zero mass of the peak template states guarantees infrared safety of tPf, whereas jet Pf is infra-red safe only after a cut on the jet invariant mass is applied. { 7 { (2.7) (2.8) (2.9) J H E P 0 7 ( 2 0 1 3 ) 1 1 4 A simple but useful way to reject backgrounds and accept signal events is to incorporate the information related to b-tagging into the template overlap framework. The identi cation of b-tagged jets relies on information beyond what is provided by the calorimeters (say from the presence of a displaced vertex and/or a hard lepton). This information (e.g. direction of the b-tagged jets in and ) is fairly uncorrelated with the information about the direction of the peak template partons. We integrate the b-tagging information into the template overlap framework by assigning a b-quark tag, t(bf) to each peak template f [j]. A two body template parton is assigned a b-tag if an anti-kT (r = 0:4) b-jet lies within a template cone of radius r2 around the template parton axis. For simplicity, we take a jet to be b-tagged if it has pT > ptag (default: ptag = 20 GeV) and contains a b or b quark. T T Information about template b-tags can be of particular use in discriminating the large tt background. For instance, consider a typical doubly-b-tagged jet coming from a tt event. A fragment of another light (or c) jet is likely to fall in the cone of R 1. If only the criterion of a doubly b-tagged jet is used, there is no guarantee that the peak two body template will select the two b subjets, making the tt event more likely to pass the kinematic constraints of the template states. On the contrary, if we require that the b-tagged jets coincide with the template momenta, we discriminate against jets in which only one b-jet is tagged by the template. Notice that the e ect of b-tagging on the templates should not have a large e ect on the signal. 2.1.4 Template Stretch In order to increase the signal e ciency of the Tempate Overlap Method we chose a working point where the energy resolution of each of the template partons is rather loose (i.e. a = pT;a=3). This implies that even after both Ov2 and Ov3 cuts the mass distribution for the background events is still broad and certainly more spread than that of the signal. This, as well as the fact that for template b-tagging we require a large r = 0:4 anti-kT jet parameter (motivated by the current experimental defaults), implies that the angular distance between the partonic, \b" candidates with a high Ov2 score would still have some smearing with respect to the actual distance between the two anti-kT b-tagged jets. To capture this e ect, we de ne a new observable, Template Stretch, as Sb(bt) = Rbb : Rt (2.11) where Rt is the distance between the peak two-body template momenta and Rbb is the distance between the two b-tagged subjets. We expect that the background events will have a broader distribution of Sb(bt) compared to the signal events. The functionality of the template stretch is correlated with the mass of the jet, but with an important advantage. The mass of a fat jet is subject to a large jet cone radius R 1, making it highly susceptible to e ects of pileup and underlying event. Since Sb(bt) is constructed out of subjets with r = 0:4 and template states, it is bound to be less sensitive to pileup. We will illustrate this point further in section 3.2 (See gure 13). { 8 { J H E P 0 7 ( 2 0 1 3 ) 1 1 4 Varying subcone templates and showering correction Fixed template cones are limited by the fact that di erent pT subjets yield a di erent energy pro le in ; . This fact is important for three body template analysis, where we expect the pT of the three peak template partons to be non-uniform. For instance, one would nd that a template subcone of radius r = 0:05 is adequate to capture the radiation pattern of a 500 GeV quark. Yet, the same subcone would completely fail to adequately describe the radiation pattern of a quark with energy of 100 GeV, resulting in a poor overlap score. We thus introduce the concept of scaled three body subcones into the template overlap framework. Varying template subcones allow us to correct for energy deposition outside the template subcone radii. This in turn leads to an an improved template-level energy resolution while keeping systematic uncertanties well under control.3 Jets become narrower as pT increases, meaning that a smaller jet area is needed to collect some xed fraction of the jet energy at higher transverse momentum. The corresponding distribution of jet areas (at a xed energy fraction) is generically non calculable but is measured by experiments, and is commonly denoted as the \jet shape" variable. Refs. [34] and [35] present ATLAS and CMS studies of the jet shape variable respectively. We used the ATLAS jet shape study in ref. [34] to establish a scaling rule for the template subcones. The di erential jet shape drops rapidly as r increases: at low pT , more than 80% of the transverse momentum is contained within a cone of radius r = 0:3 around the jet direction. This fraction increases up to 95% at very high pT . We t the numerical values of the integrated jet shape in di erent pT regions to obtain the minimum cone radius required to capture 80% of the jet transverse momentum. In the overlap analysis, we correct for the 80% e ciency by scaling the template pT accordingly. Figure 1 shows the result of our dynamical scaling. The points represent the minimum radius necessary to capture 80% of jet's energy as a function of jet pT . The resulting curve gives a shape to the optimal scaling rule for template subcones. The error bars on the data points are small enough that they can be omitted for the purpose of our analysis. To obtain the subcone values in the pT < 30 GeV region, we extrapolate the data. An overall shift in the optimal jet-shape curve remains a free parameter. To calibrate it we choose a benchmark point of r3(100 GeV). We demonstrate in the following section that other calibrations are possible and may perform in a similar manner, within a reasonable range. The dotted curve shows that a shift of r = 0:063 units provides an excellent t. Notice that the naive scaling r3(ptT ) = r3(100 GeV) 100 GeV ptT ; (2.12) is in excellent agreement in the pT > 60 GeV region, while the discrepancy with data becomes large at lower jet pT . 3At low pT , the leakage of QCD radiation outside the template subcones can be especially large when using xed subcones. To account for this, one has to include energy correction factors into the pT of the templates, which are largely a ected by systematic uncertanties. { 9 { J H E P 0 7 ( 2 0 1 3 ) 1 1 4 r 0.5 Note that the extrapolation procedure in gure 1, while somewhat arbitrary, also does not a ect the results of our analysis. We impose the condition that for all template momenta. The lower limit on r3 takes into account the detector granularity, while the upper limit provides consistency with the two body template analysis. In addition, we nd that the shape of the optimal curve crosses the maximum value of r3 = 0:3 a bit below pT = 30 GeV, keeping the method largely insensitive to the extrapolation procedure. In the ultra high energy regime, the extrapolation also becomes irrelevant, as the template cone size becomes limited by the detector resolution. Figure 2 shows an example of the e ect of varying subcones on peak templates compared to xed ones. The blue, dotted circles represent the peak two body template with radius r2 = 0:3. The red, dashed circles are peak three body templates with a xed r3 = 0:12 (left panel) and varying r3 (right panel). Notice that a higher percentage of the lowest pT subject is \encompassed" by the varying cone resulting in an overall increase in the Ov3 score. Varying subcones improve the performance of templates on the distribution level as well. Figure 3 shows a comparative example for Ov3. The panel on the left was obtained using a xed r3 = 0:12 while r3 was allowed to vary according to the scaling rule of gure 1 in the right panel. Notice that the varying subcones result in background overlap distributions which are signi cantly more peaked in the region of low overlap values. On the other hand the signal remains mildly a ected, resulting in improved performance. E P 0 7 ( 2 0 1 3 ) 1 1 4 η 0.5 -0.5 -1 0.5 φ 1.5 -0.5 1 0 J H E P 0 7 ( 2 0 1 3 ) 1 1 4 -0.5 -1 0 04. Ov 3 ( p T > 30 GVe , Ov 06. 2 > 0 . 8 ) 04. Ov 3 ( p T > 30 GVe , Ov 06. 2 > 0 . 8 ) How sensitive is the Template Overlap Method rejection power to the choice of subcone radius rshape r3(100 GeV ) and the overall normalization of the working curve of gure 1? Signi cantly modifying the overall scale for subcone radius rshape does alter the distribution of our kinematic variables; in particular, reducing the size of rshape tends to shift Ov2 and Ov3 to lower values and vice versa. The sensitivity of the template method to the speci c choice of rshape calls for a dedicated experimental study on a control sample to x the corresponding value. One can use either boosted top analysis in the signal area oPwer .04 6 8 tSabily fo voerlap .06 .08 .01 .014 .016 .018 .02 .02 blue line shows rejection power with a rigid Ov3 > 0:6 cut. The green line is the rejection power with a cut on Ov3 varied with rshape, while keeping a xed e ciency. The rejection power is relative to cross sections with no Basic Cuts (see next section). (in fact the Ov3 distribution of boosted tops and their backgrounds was already studied experimentally by ATLAS in ref. [21]) or look at hadronic W inside a boosted top jet as a way to experimentally analyze the above dependence. The sensitivity to rshape is of course not unique to our proposal, as even if xed cones are used, all jet substructure methods will depend on the choice of the corresponding subcone parameters. Varying the subcone size to keep the enclosed energy fraction xed is in fact a more covariant way to proceed with the substructure analysis. We further wish to point out that it is in general quite possible to choose values of cuts on Ov2 and Ov3 such that the overall signal and background e ciencies are essentially the same for di erent subcone radii. To illustrate this point, in gure 4 we show curves of rejection power for several choices of the subcone parameter rshape. The meaning of rshape in gure 4 is de ned as the choice of r3(100 GeV), while the actual value of r3 is allowed to vary according to the scaling rule in gure 1. The blue squares represent rejection power against the W bb background as a function of the threebody subcone radius r3(100GeV ), and a xed Ov3 > 0:6 cut, while the Ov3 cut is allowed to scale with rshape to provide a ve percent e ciency for each rshape . Signal and background e ciency of a xed Ov3 cut shift with the choice of rshape, but disproportionately. Rigid Ov3 cuts thus show high sensitivity to the choice of rshape as shown by the blue squares in gure 4. Alternatively, xing signal e ciency also xes the background e ciency for a wide range of rshape, thus preserving the rejection power over a wide range of rshape (i.e. rshape < 0:12). 0 7 ( 2 0 1 3 ) 1 1 4 Data simulation and analysis In this section, we investigate the tagging e ciencies for Higgs jets and the mistag rates for QCD backgrounds using template overlap. We consider events in which a boosted h ! bb jet is produced in association with a leptonically decaying W boson (only rst two generations of leptons). The most dominant backgrounds for this process come from tt and W bb, while other channels do not signi cantly contribute after b-tagging requirement. The scope of our analysis includes data at both ps = 8 TeV and ps = 13 TeV with and without pileup. We use MG/ME v5.1.33 [36] interfaced to Pythia v6 [37] (with MLM matching [38]) as well as Sherpa v1.4.0 [39] (with CKKW matching [40]) for data generation and the CTEQ6L1 [ 41 ] parton distribution functions. We perform jet clustering with a Fastjet [ 42 ] implementation of the anti-kT jet clustering algorithm. To simplify the notation from here on we solely refer to Pythia and Sherpa results and leave the fact that all our data samples are matched implicit. We simulate W h, W bb and tt samples using Pythia, while Sherpa simulations of W h and W bb data serve to illustrate the e ects of various showering algorithms by comparison. We only include the W h and W bb data in the comparison, as the tt distributions are characterized by hard scales and therefore less sensitive to detail of the showering. We scale the fat anti-kT jet radius according to the W momenta where pTW is the transverse momentum of pl + p=T : Note that the momentum of the W is highly correlated with the momentum of the Higgs as they recoil against each other. However, scaling the cone according to pTW has an advantage in that it is not susceptible to pileup. Continuing, we limit the value of R from below to be higher than 0.8 as to be able to accommodate two 0.4 anti-kT jets used for b-tagging. The scaled fat-jet cone ful lls three tasks; it is designed to capture the bb at a xed e ciency rate of 80% ; it reduces the amount of contamination from soft radiation and pileup for events with high pT Higgs jets; and it lowers the overall tt background. Note that the scaling rule of eq. (3.1) has only a minor e ect on the correlation between the Higgs fat jet momenta and that of the W as is shown in gure 5. Next,we normalize our data to next-to-leading-order (NLO) cross sections obtained from MCFM 6.3 [ 43 ]. The cross sections assume a xed renormalization/factorization scale of = pTmin = 300 GeV at 8 TeV and = pTmin = 350 GeV at 13 TeV and CTEQ6.6M [ 44 ] parton distribution functions. For each event, we nd the jet with the highest transverse momentum j and impose the following Basic Cuts: pjT > pTmin ; j ; l < 2:5 ; Nb = 2 ; pTW > pTmin ; pT > 40 GeV ; = Rbb 0:4 ; Nk(pT > 20 GeV) < 2 ; Nl(pT > 20 GeV) = 1 ; (3.2) J H E P 0 7 ( 2 0 1 3 ) 1 1 4 where Nk;l is the number of jets (anti-kT , r = 0:4) and leptons outside the highest pT fat jet (of radius R, determined according to eq. (3.1)) and Nb is the required number of b-tagged (anti-kT r = 0:4) subjets. For a fat anti-kT jet j (of radius R, determined according to eq. (3.1)), and an anti-kT , r = 0:4 jet k, a jet is considered to be outside the fat jet if the plain distance R(j; k) > R + r. Similarly, a lepton l is considered outside if R(j; l) > R. Table 1 summarizes the cross section results with and without Basic Cuts. We consider pTmin = 300; 350 GeV respectively at 8 TeV and 13 TeV. For b-tagging we assume an e ciency of 75% and fake rate of 1% for light jets [ 45 ]. The current studies suggest a charm fake rate of 18% [ 45 ], which is likely a conservative estimate. Charms are extremely important when considering boosted Higgs decays, as the largest part of the tt background comes from events in which one top decays leptonically, while the hadronic W from the other top decays to a charm. We emphasize that omitting the charms as a source of background (as is done in some of the boosted Higgs analyses) will result in an improved performance for our tagger. Yet, at present, it is not clear whether this is possible, and the burden of proof is thus placed on the experimental collaborations. We analyze the cases with and without pileup separately in order to illustrate the sensitivity of the Template Overlap Method to a pileup environment. This allows us E P 0 7 ( 2 0 1 3 ) 1 1 4 ✂ R 1 p/ TW 0.6 d md 1 0.4 80 m ( Ov 2 10 > 0 . 9 , 2 120 t b , Ov 3 140 > 0 . 8 ) 160 to determine the range of background rejection power as a function of the e ciency of pileup subtraction. In addition, it also allows for a comparative study of the various jet substructure observables in a pileup environment. Higgs tagging with Template Overlap - no pileup We proceed to discuss the ability of the Template Overlap Method to discriminate between di erent sources of coherent QCD radiation. In terms of pileup ltering, analysis without pileup is equivalent to stating that the e ciency of pileup subtraction is 100%. In this section we present only the results on jets with pT > 300 GeV, simulated at ps = 8 TeV, while we postpone the discussion of the future 13 TeV LHC run until section 3.1.2. An important feature of the Template Overlap Method is that it is designed to identify a particular kinematic jet substructure con guration, including the jet pT and mass. High peak overlap score implies that the kinematics of a fat jet matches the kinematics of the peak template state. In gure 6 we plot the jet mass distribution without (left panel) and with (right panel) template overlap cuts of Ov2 > 0:9 and Ov3 > 0:8, after the Basic Cuts of eq. (3.2) have been applied. It is evident from gure 6 that sizable chunk of the background is removed as a result of the overlap cuts though the resolution of the fat jet Higgs mass is only moderately improved. The mass resolution of the peak templates depends largely on the chosen parameters of the method, namely the template cone radii ra and their energy resolution a which we choose rather loosely as to keep the signal e ciency at a reasonable level. Recently, authors of ref. [21] presented a jet substructure analysis of boosted tt pairs at ATLAS. Their results showed that even with a high peak overlap cut, an additional mass window improved the background rejection power by a factor of two. Our result in gure 6 agrees with the ATLAS result. It appears that even after the overlap cuts, a mass window of (say) 110 GeV < mj < 130 GeV would improve the background rejection power (this would, however, require an additional procedure of pileup removal). In the following sections we remain agnostic about this issue and show results with an without a mass cut. H E P 0 7 ( 2 0 1 3 ) 1 1 4 We proceed to discuss the rejection power of the method for jets with pT > 300 GeV at ps = 8 TeV. We de ne the signal e ciency s to be the ratio of the signal cross section after various kinematic cuts are applied, to the cross section after Basic Cuts, i.e. where BC denotes Basic Cuts of eq. 3.2 and cuts are any kinematic cuts in addition to the Basic Cuts (see eq. (3.7)). Each cross section in eq. (3.3) assumes b-tagging e ciency of 75%, c mistag rate of 18% and light jet mistag rate of 1%. Similarly, we de ne the total background fake rate to be and the fake rate for individual background channels as Finally, we de ne the background rejection power RP as W bb f cuts tt : BC tt s cuts W h BC W h Wcutbsb + tt ; cuts WBCbb + tt BC cuts WBCbb ; W bb RP tt s : f (3.3) (3.4) (3.5) (3.6) (3.7) E P 0 7 ( 2 0 1 3 ) 1 1 4 For the purpose of illustration, we consider several combinations of cuts on both template and jet observables in addition to Basic Cuts of eq. (3.2), while we leave Ov3min a free parameter. We label the cuts as following: Cuts 1 : Cuts 2 : Cuts 3 : Cuts 4 : Cuts 5 : Ov2 > 0:9: Ov2 > 0:9; 2tb: Ov2 > 0:9; 2tb; Rbb= Rt < 1:0: Ov2 > 0:9; 2tb; tPf < 0:3: Ov2 > 0:9; 2tb; Rbb= Rt < 1:0: 110 GeV < m < 130 GeV; where 2tb denotes that both two-particle peak template momenta are b-tagged. afke %() 10 arte 0 5 0 feicny %() 5 e ciencies are relative to Basic Cuts of eq. (3.2). our analysis, with and without a mass window cut. The signal e ciency and fake rates are measured relative to the cross sections with Basic Cuts from table 1. The curves also include b-tagging e ciencies we discussed the previous section. Each curve represents fake rate as a function of signal e ciency with a set of xed cuts on template observables, while a cut on Ov3 in the range of (0; 1) runs along the curve. Our results show that template observables can signi cantly improve the background rejection power relative to Basic Cuts of eq. (3.2). Figure 9 illustrates the rejection power over individual background channels. Template Overlap method alone performs signi cantly better in rejecting W bb events for most signal e ciencies, as shows in the left panel of gure 9. This is reasonable since tt events typically consist of two b-tagged jets and an additional fragment of a hadronically decaying W boson. Such a con guration is more likely to be tagged with a higher Ov3 score than the typical two body substructure of a light QCD jet. However, cuts on additional kinematic observables such as the Template Stretch or tP f , as well heavy avor tagging requirements, can result in tt events being rejected at a rate higher than W bb. Table 2 shows an example of benchmark signal e ciency points. At 14% e ciency, a rejection factor of 4 is achievable with template based observables only. Additional mass window boosts the rejection power by 25%, leading to a S=B 0:3, with roughly 20% e ciency. Template Overlap Method can achieve enough rejection power at 8 TeV to overcome the backgrounds at the severe cost of signal e ciency. Yet, the 20 fb 1 of integrated luminosity collected during the LHC 8 TeV run is insu cient to yield a statistically signi cant Higgs signal in the boosted regime, primarily due to the tiny boosted signal cross section. For instance, take our results from table 2 with Cuts 5. The signal e ciency of 23% and 20 fb 1 of data would yield 1 Higgs event with S= B 0:45. A 3 signal would p E 0 7 ( 2 0 1 3 ) 1 1 4 01 51 0 5 0 01 bW „b t „t Cuts 3 arte arte afke afke 01 51 0 5 0 5 stuC ,3 stuC ,3 01 bW „b t „t 51 E P 0 7 ( 2 0 1 3 ) 1 1 4 0.3 0.8 0.3 0.8 0.8 0.3 0.8 0.3 0.8 0.8 The left panel shows the overall e ciency and fake rate for tt and W bb separately with Cuts 1 of eq. (3.7). A cut on Ov3min runs along the curves. The right panel shows the signal e ciency and fake rate for Cuts 3 of eq. (3.7) . All e ciencies are relative to Basic Cuts of eq. (3.2). Cut Set Ov3min s (% ) s (fb) Wbb Wbb (fb) tt tt (fb) overall rejection power S=B S=pB (20 fb 1) s = 8 TeV. The values in the table show the signal e ciencies ( s) and fake rates ( W bb; tt) relative to the cross sections with Basic Cuts of eq. (3.2) as well as the signal and background cross sections after the cuts speci ed in the rst and second column. The overlap rejection power includes both the W bb and tt. start appearing only with the un-realistic for the future 13 TeV run. 3.1.2 Background rejection power at p s = 13 TeV The composition of background channels at p 1000 fb 1 of data. We thus turn to projections s = 13 TeV changes relative to 8 TeV, with tt amounting to 60% of the total. Higher center of mass energy also allows us to push the minimum pT to higher values; we opt for pT jet cone scaling rule of eq. 3.1 with pT 350 GeV. Figure 10 shows the result. The 350 GeV will always yield R = 0:8. We will thus use a xed value of R = 0:8 in this section and neglect the scaling rule. We again nd that template overlap can signi cantly improve the rejection rate over p s = 8 TeV and pT 300 GeV; as expected. traditional jet observables. The overall performance of templates improves relative to 51 arte afke 0 5 0 E 0 7 ( 2 0 1 3 ) 1 1 4 gure shows the overall e ciency and fake rate with xed cuts of eq. (3.7). A cut on Ov3min runs along the curves. All e ciencies are relative to Basic Cuts of eq. (3.2). Cut Set Ov3min s (% ) s (fb) Wbb s = 13 TeV. The values in the table show the signal e ciencies ( s) and fake rates ( W bb; tt) relative to the cross sections with Basic Cuts of eq. (3.2) as well as the signal and background cross sections after the cuts speci ed in the rst and second column. The Overall Rejection Power includes both the W bb and tt. Figure 10 shows that an overall rejection power of 10 (E : 20%) is achievable at p p s = 13 TeV, leading to an overall S=B 0:5. This constitutes an improvement over the 8 TeV result where the maximum rejection power was 5. Our results show that Template Overlap is capable of delivering a statistically signi cant boosted Higgs signal at s = 13 TeV. At 22% signal e ciency (Cuts 5), we nd that S=B 0:5 is possible. The 3 signal signi cance can be achieved with 250 fb 1, the amount of data possible with the future LHC run. 3.2 Higgs tagging with Template Overlap - e ects of pileup A foe to most jet substructure observables, pileup has become an LHC fact of life. The strive for high luminosity resulted in pileup levels of whopping 20 average interactions per bunch crossing during the current 8 TeV run. Pileup events contribute both to the fat jet constituent multiplicity and the energy distribution within a jet, resulting in possibly dramatic e ects on any jet substructure observable constructed out of jet constituents. feicny %() 0.3 0.2 .0 σd dM 1 σ 1 σ R ∼ 1 p/ TW .0 M ( p T > 30 GV)e 05 01 02 520 05 01 M ( p T > 30 GVe 510 , Ov 3 > 0 . 8) 02 The left panels show distributions after the Basic Cuts of eq. (3.2), while the right panel shows the same distributions with Ov2 > 0:9 Ov3 > 0:8. Jet mass is perhaps the best illustration of this point. Figure 11 shows an example. The left panel shows mass distributions in the presence of average 20 interactions per bunch crossing. In addition to shifting the mass peaks by as much as 30 GeV (relative to the plots in gure 6) pileup reduces the mass resolution. In a recent study of ref. [21], the ATLAS collabration tested this feature with the LHC 7 TeV data. The mass resolution does not improve signi cantly even after cuts on the overlap are applied, again due to a low energy scale resolution. Note that reducing the value of a and thus increasing the mass resolution of the Template Overlap Method could be used to study e ects of pileup on jet observables. Many \post processing" pileup subtraction techniques exist in the literature, such as trimming [18], pruning [19] and jet area techniques [ 29, 46 ]. Alternatively, ref. [47] presents a data driven method of correcting for pileup e ects for jet shape variable of massive narrow jets. Finally, particle tracking information can be used to subtract pileup events, a method already used by the CMS collaboration [48]. In this section we do not consider any pileup subtraction. Instead, we show that the Template Overlap Method is largely una ected by pileup. Robustness of the Template Overlap Method against pileup comes from the de nition of template overlap. Consider for instance a single template momentum pt. The core of the overlap measure is the di erence pT = ptT X p j T j (r3 Rtj ); (3.8) where pj are momenta of jet constituents and selects the ones which fall into a cone of radius r3 around pt. The size of the template subcone r3 thus limits the e ects of pileup. Matching pT of a template state to the jet pT with pileup is problematic. Jet pT is shifted to higher values by pileup and as such is inappropriate as a criterion for template selection. Furthermore, a lower cut on the jet pT with pileup will include jets which would not pass the cut if pileup was not present. We instead use the pT of the leptonically t „t bW „b 520 E P 0 7 ( 2 0 1 3 ) 1 1 4 η 0.5 0 -0.5 -1 (Ov2 = 0.980532 , Ov3 = 0.801652) η -0.5 0 1 decaying W boson, as an infra-red safe, pileup independent observable (recall that since the Higgs recoils agains a W boson, pW pH , as we showed in section 3). We simulate the e ects of pileup by adding a random number of minimum bias events (MBE) to every event we analyze. The number of added MBEs is distributed according to a Poisson distribution with the mean Nvtx = 20, consistent with the LHC conditions at ps = 8 TeV. Figure 12 shows an example of the pileup e ects on the overlap analysis of a single event. The analysis of a jet with no pileup (left panel) yields nearly identical peak template state as the jet with pileup (right panel). Notice that the overlap scores remained within 10% of each other. On distribution level, the situation is similar. Figure 13 shows examples of several template based observables with and without pileup. Even at 20 interactions per bunch crossing, we nd no signi cant e ects of pileup on the distribution shapes of Ov3; tPf and template stretch. In fact, the di erence between susceptibility of tPf and Pf to a pileup environment is striking. While jet Planar Flow is signi cantly shifted to higher values by pileup, template Planar Flow remains mostly una ected. The overlap method used in this study e ectively reduces the area of fat jets and is therefore less sensitive to pileup. As a simple application of this idea, in gure 14 we show fake rate vs. e ciency with the cuts of eq. (3.7), obtained from Pythia data at ps = 8 TeV and Nvtx = 20. For a given cut, the e ciency is controlled by the lower cut on Ov3. The results depend on the choice of cuts, but it is clear that the overall performance of overlap approach remains largely unchanged when compared to the case of events without pileup. Our nal results can be summarized in table 4 for a few benchmark e ciency points. We choose to omit the result of adding a mass cut since it requires an additional mechanism for pile up subtraction, and is thus beyond the scope of this paper. In each case, we nd background rejections comparable to our results for events wihout pileup. E P 0 7 ( 2 0 1 3 ) 1 1 4 Cut Set Ov3min W h e ciency (%) W bb fake rate (%) tt fake rate (%) overall rejection power Pythia 51 afke %() 01 arte 0 5 0 s = 8 TeV with Nxvt = 20. The values in the table show the signal e ciencies and fake rates relative to the cross sections with Basic Cuts. The overall rejection power includes both the W bb and tt. Nvtx = 20. The gure shows the overall e ciency and fake rate with xed cuts of eq. (3.7). A cut on Ov3min runs along the curves. All e ciencies are relative to Basic Cuts of eq. (3.2). 4 20 fb 1 a viable boosted Higgs search at p Hadronic decay channel of the Standard Model Higgs boson is one of the most challenging measurements in Higgs physics at the LHC. Traditional jet observables such as jet mass and pT are inadequate to combat the large QCD background as well as high luminosity environments characteristic of the LHC. Jet substructure techniques can be used to overcome the large backgrounds in the boosted Higgs regime. Our results show that with a statistically signi cant signal in the realistic LHC conditions at p the template overlap method is not able to deliver su cient background rejection power s = 8 TeV with . We also showed that one can achieve a good signal/background discrimination for s = 13 TeV, starting with 250 fb 1 of data. We note that the above statement is based upon the assumption that the detector response would not result in a reduction of the e ciencies or in an increase of the fake rates estimated above. Detector e ects (like calorimeter resolution or the smearing of the energy) were determined not to change qualitatively the phenomenological estimations in ref. [21], and we have neglected them in our analysis. A more complete study which makes use of the full detector simulation and reconstruction chain, including more realistic b-tagging, is be0 7 ( 2 0 1 3 ) 1 1 4 yond of the scope of this work, and we leave it to the experimental collaborations to study precisely how template-based variables are a ected by detector response and systematic uncertainties. In general the template-based variables presented here are robust against contamination from pileup at an average of 20 interactions per bunch crossing. We introduced several improvements into the template overlap framework. Varying the three-body subcones according to the transverse momentum of the template allows for more e cient signal tagging, while improving the background rejection power. Sequential template generation over longitudinally boost invariant variables results in a better phase space coverage at lower pT ( 200; 300 GeV), while keeping the number of templates at a reasonably low number of O(106). For the rst time, we integrated b-tagging identi cation into the peak templates. We further combined the b-tagging information with boosted event kinematics into template stretch, a new, useful observable. Template stretch is highly correlated with the jet mass, but with an important distinction that it is much more weakly a ected by pileup. Finally, we improved on the existing de nitions of Template Planar Flow, by including the simulated e ects of parton showering via nite template subcones. Acknowledgments We thank Raz Alon, Ehud Duchovni, Ohad Silbert and Pekka Sinervo for useful comments and encouragement and especially Jan Winter for an early collaboration on this project. We also thank Gavin Salam for useful comments on the TemplateTagger code, and Ciaran Williams for discussions about MCFM. MB and JJ would like to thank the CERN theory group for their hospitality, where a portion of this research was conducted. The authors also express their gratitude to Pierre Choukroun and Lorne Levinson for their enormous help regarding the computing needs of this work. GP holds the Shlomo and Michla Tomarin development chair and is supported by the Gruber foundation, IRG, ISF and Minerva. A More on template Planar Flow To illustrate how energy smearing a ects the parton level tPf consider a peak three-body template state with an inertial tensor It. Energy smearing a ects only the diagonal elements of It while o -diagonals remain unchanged due to symmetry, leading to H E P 0 7 ( 2 0 1 3 ) 1 1 4 (A.1) (A.2) where PT = Pi3=1 piT and piT is the transverse momentum of the ith template momentum. The e ects of energy smearing are summarized in the symbol where g(p^i) is the energy smearing distribution. It ! Is = It + PMT I; = Z d g(p^i); For simplicity, we consider a uniform distribution over a disc of radius r3, giving Alternatively, one could also consider a Gaussian distribution centered around the template momentum, with a width , resulting in Continuing, the determinant of Is becomes det(Is) = det It + PMT I Similarly, the trace becomes For a jet cone radius R 1 it is reasonable to assume that 1. Furthermore, PT =M is typically of O(1), allowing us to expand det(Is) in . Keeping only the leading term in both the trace and the determinant we get = det It I + It 1 PMT I = det(It)det I + It 1 PMT I : tr(Is) = tr It + PMT I = T r(It) + 2 PT M tPft ! tPfs = tPf 1 + tr(It 1) PMT : 1 + 4 MPtTr(It) tr(It 1) = tr(It) = det(It) tPf 4 tr(It) : tPfs = tPf 1 + =tPf ; 1 + 4 PT : tr(It) M Since It is by de nition a 2 2 matrix, we can write Combining eqs. (A.7) and (A.8) we nally obtain where As r3 ! 0, eq. (A.11) correctly reduces to the expression for tPf. In the limit of tPf ! 1, the showering e ects become irrelevant as well, keeping eq. (A.7) bounded in accordance with the de nition of tPf. Notice that eq. (A.11) is not valid in the tPf ! 0 limit, as we made an assumption that det(It) 6= 0 during its derivation. (A.3) (A.4) (A.5) (A.6) (A.7) (A.8) (A.9) (A.10) J H E P 0 7 ( 2 0 1 3 ) 1 1 4 6 5 2 1 0 ( 2 0 1 3 ) 1 1 4 04. Pf ( p T > 30 GVe Continuing, in the narrow angle approximation tr(It) = M=ptT , where ptT is the total template transverse momentum. We have veri ed numerically that this result is satisfactory even at R=1.4 We can then rewrite eq. (A.7) as tPfs = tPf 1 + 0=tPf ; 1 + 0 0 4 PT ptT M 2 : (A.11) (A.12) where eq. (A.11) provides a useful qualitative insight into the e ects of showering on Planar Flow. Template momentum con gurations with low Planar Flow are more sensitive to showering e ects. This is important when considering Planar Flow of Higgs jets which are expected to peak at low values. Figure 15 shows the results of a data analysis with par tonic tPf, and template Planar Flow calculated a in eq. (A.1), for both varying (tPf) and xed r3. Addition of subcones to the Planar Flow calculation produces results which match the physical jet distribution much better than the parton level tPf(r = 0). Notice that the match between tPf and jet Planar Flow is excellent in the region of Pf > 0:4, where perturbative expansions are expected to hold. E cient generation of template libraries Template Overlap Method is a systematic framework aimed to identify kinematic characteristics of an boosted jet. A typical template con guration consists of a model template, f , calculated in perturbation theory, which describes a \prong-like" shape of the underlying hard subprocess of a jet. Template construction typically employs prior, theoretical knowledge of the signal kinematics and dynamics, as well as possible experimental input. The simplest template con gurations are the ones describing the kinematics of twobody processes such as the decay of SM Higgs or W=Z bosons into quark-antiquark pairs. These are easily dealt with by assuming the rest frame of the parent particle and producing two decay products with equal and opposite, isotropically-selected momenta and magnitude, subject to energy conservation. The problem of a N -body decay subtracts four constraints from the decay products' 3N degrees of freedom: three for overall conservation of momentum and one for energy.4 The nal states can therefore be found on a (3N 4)dimensional manifold in the multi-particle phase space. Note that the dimensionality of the template space increases rapidly with additional patrons. For instance, the two-body templates require only two degrees of freedom, while a corresponding four-body template space is already eight dimensional. The question of which kinematic frame the templates should be generated in requires careful consideration. Authors of ref. [ 23 ] argued that a search for the global maximum of OvN could be too computationally intensive. To improve the computation time, the template states were generated in the Higgs rest frame using a Monte Carlo routine, and then boosted into the lab frame. While this method worked su ciently well for tagging a highly-boosted object(i.e. a 1 TeV Higgs jet), it introduced residual algorithmic dependence and a certain sense of arbitrariness in the jet shape. At lower pT the Monte Carlo approach samples mainly the templates within the soft-collinear region, leaving other regions of phase space unpopulated. An enormous number of templates is required to adequately cover the phase space at pT O(100 GeV), thus fully diminishing the motivation for a Monte-Carlo approach. The simplest and most robust choice is then to generate templates directly in the lab frame and then rotate them into the frame of the jet axis. The result is a well covered template phase space in all relevant boosted frames. In addition, the lab frame templates result in a signi cant decrease in computation time as a much smaller number of templates are needed. We proceed to show how to generate the phase space for 2- and 3-parton nal states as well as how to generalize the results to arbitrary N . J H E P 0 7 ( 2 0 1 3 ) 1 1 4 B.1 The case of 2-body templates First, we summarize our notation and conventions. The model template consists of a set of four vectors, p1; ; pN , on the hyperplane determined by the energy-momentum conservation, (B.1) 4For our purpose, a template is an object with no other properties other than its four-momenta. X pi = P; P 2 = M 2; i where M; P are the mass and four momentum of a heavy boosted particle,i.e. the Higgs. For simplicity, we treat all template particles to be massless. We work in an ( ; ; pT ) space, where is pseudorapidity, azimuthal angle and pT transverse momentum. Without loss of generality, we can assume that the template points in the x direction ( = = 0). The templates are distributed according to pi = pT;i(cos i; sin i; sinh i; cosh i); i = 1; 2; 3 subject to the constraint N X pi = P = (pT ; 0; 0; EJ ) i=1 with EJ = qM 2 + p2T . We nd it useful to de ne unit vectors by p^i = (cos i; sin i; sinh i; cosh i); i = 1; 2; (B.2) (B.3) (B.4) (B.5) (B.7) (B.8) (B.9) J H 0 1 3 ) 1 1 4 so that pi = pT;i p^i. Phase space for the 2-body decay processes is characterized by particularly simple kinematic parameters. To illustrate, rst note that the 2-particle templates are uniquely determined by one single four momentum, p1 subject to the condition Writing p1 = pT;1p^1, we can solve for pT;1 in terms of the angles of the rst parton pT;1 = 2(P p^1) : (B.6) We see that a 2-particle template is therefore completely determined in terms of the unit vector p^1 as follows: (P p1)2 = 0: M 2 M 2 p1 = 2(P p^1) p^1 p2 = P p1: Note that we can represent such a template as a point (^; ^) in plane. These are the two degrees of freedom, in accordance with the general result that the dimensionality of the N template space is 3N 4. B.2 The case of 3-body templates A space of ve degrees of freedom allows for 3-particle templates to di er from one another in more than one way. The 3-particle templates are determined by two four momenta, p1 and p2, subject to the constraint, (P p1 p2)2 = 0: Using p1 = pT;1p^1 and p2 = pT;2p^2, we can solve for pT;2 in terms of the angles of rst two partons and pT;1, pT;2 = 2(PM 2p^2 2Pp1 p1p^2) : (B.10) A general 3-particle template is then completely speci ed by pT;1 and two unit vectors (or, equivalently, four angles) p^1 and p^2. A generalization to an arbitrary number of particles is straight-forward. Proceeding as above, the N -particle templates are determined by p1; ; pN 1 subject to the constraint, P N 1 X pi i=1 !2 = 0: Using pi = pT;ip^i, we can now solve for pT;N 1 in terms of the p1; ; pN 2 and p^N 1, (B.11) (B.12) (B.13) J H 0 7 ( 2 0 1 3 ) 1 1 4 pT;N 1 = M 2 + 2 PiN<j 2 pi pj 2 (p^N 1 P ) 2 p^N 1 PN 2 i PiN 2 pi pi For the special cases of N = 2 and N = 3, this formula reduces to the above results . B.4 Numerical simulations We choose to cover the phase space uniformly in the (3N 4) variables p^1; ; p^N 1 and pT;1; pT;N 2. For a nite number of points this method covers regions of phase space with large Planar Flow much more uniformly than a Monte Carlo based approach. We survey the kinematically-allowed templates by xing the total four momentum of each of the analyzed con gurations and scanning over the possible values of (pT;i) and the angles (p^i) within the bounded interval. The number of variables depends on the number of degrees of freedom of the template states. The value of P = (pT ; 0; 0; EJ ) can be imposed by an additional equation: pN = P N 1 X pi: i=1 We generate template libraries in the plane with a sequential scan in steps of = = 0:05. Three-particle templates we rehire and additional scan over the transverse momentum with which we perform in steps of pT = 5 GeV. The resulting four momenta are a requirement that they \ t" into an anti-kT jet of xed radius R. 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Mihailo Backović, José Juknevich, Gilad Perez. Boosting the standard model Higgs signal with the Template Overlap Method, Journal of High Energy Physics, 2013, 114, DOI: 10.1007/JHEP07(2013)114