CLASSIFICATION OF HIGGS JETS AS DECAY PRODUCTS OF A RANDALL-SUNDRUM GRAVITON AT THE ATLAS EXPERIMENT

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1 CLASSIFICATION OF HIGGS JETS AS DECAY PRODUCTS OF A RANDALL-SUNDRUM GRAVITON AT THE ATLAS EXPERIMENT AVIV CUKIERMAN, ZIHAO JIANG. Introducton In 202, physcsts at the Large Hadron Collder announced the dscovery of the Hggs boson, whch earned the monker God Partcle n the popular meda due to ts far-reachng theoretcal mplcatons. The Standard Model, whch s by far the most successful model we have descrbng matter and ts nteractons, predcts that the Hggs boson exsts and gves mass to quarks. The Hggs boson was the fnal partcle predcted by the Standard Model that had not yet been dscovered and so the dscovery was a major trumph for the Standard Model. However, there are problems wth the Standard Model. For nstance, there s a queston of why the mass of the Hggs tself s so small, when nave theoretcal extensons of the Standard Model predct t should be much hgher. Among all the more sophstcated models whch account for ths problem, one model that yelds detectable quanttes s the Randall-Sundrum Model, whch predcts there s a new partcle, a Gravton, whch can decay to two Hggs bosons. A search for such a partcle requres dstngushng the Hggs bosons from background processes, mostly quark and gluon showerng from quantum chromodynamc (QCD) effects. In ths project, we use machne learnng technques to dstngush the sgnal of a Randall-Sundrum Gravton (RSG) decay to two Hggs from the QCD background, usng knematc varables that would be observed n the ATLAS detector at CERN. In partcular, we look at the case where each of the Hggs bosons decay to a par of bottom-antbottom (bb) quarks. 2. Data & Features The study uses Monte Carlo smulated data wth full reconstructon of ATLAS detector response. For the sgnal, we assume the mass of the RSM Gravton to be 000 GeV/c 2, consstng of events. For the background, we use ATLAS tuned QCD smulatons consstng of unweghted events. In a sgnal event we have two Hggs bosons, each of whch decay to a bb par of quarks. These quarks are are observed as collmated energy deposts n the detector, whch are called jets. We know from physcs that n fact the quark-antquark par wll be detected wth small angular separaton n the detector, and so the quark jets form the sub-structure of a larger Hggs jet. From these energy deposts we can reconstruct the space-tme four momenta of the two quarks and the Hggs bosons ( 3 4 = 2 features). In addton, we have b-taggng varables constructed from other observables that descrbe the probablty that the quarks are bottoms ( 2 features). So n total we have (2+2=4) features for each potental Hggs jet. 3. Models 3.. Nave Bayes. The Nave Bayes model holds the assumpton that the probablty of a set of features {F n } condtoned on a varable X s p(f,..., F n X) = n p(f X) In our case, X = 0, denotng whether the a jet s or s not a Hggs boson. The features are the contnuous knematc varables of the jets and the sub-jets. We model the condtonal probabltes as a Gaussan p(f X) = exp (x µ)2, where the parameters µ and σ are calculated drectly from the mean and RMS of the 2πσ 2 2σ 2 knematc varables. Heurstcally, the features that are most relevant to ths classfcaton problem are the nvarant mass and transverse momenta (pt) of the jets. Hggs jets have nvarant mass around 25 GeV, whle the QCD jets have smaller mass because the showerng of gluon and quarks usually produces lght mesons. The mass and energy of sub-jets (decay products) are predctve. The Hggs tend to decay to heavy-flavor quarks lke Bottom and Charm, whle the sub-jets n QCD come from radaton of lght partcles. For the Nave Bayes algorthm, we consder the parameters n the followng order: mass of the jet, pt of the jet, mass of the two sub-jets, pt of the two sub-jets, and b-taggng score of the two sub-jets. We add these varables to our feature set one by one, and each tme we re-run the learnng algorthm and testng for 00 teratons. For each teraton, we use 70% of the sgnal and background to tran the Gaussan parameters and test the predcton on the remanng 30% of the data set. We take the tranng error and testng error as the mean of these 00 teratons and RMS as the uncertantes. The results of ths algorthm can be seen n Fgure. The tagger tranng and testng errors quckly get down to percent level as we ncrease the sze of the feature set. For the sgnal, the errors are gettng sgnfcantly smaller as we consder

2 2 AVIV CUKIERMAN, ZIHAO JIANG more features. However ncludng the sub-jets nformaton doesn t mprove the performance of taggng background. We also note that that after ncludng the frst two features (mass and pt of the Hggs jet), almost all of the dscrmnatve power s acheved. Fgure. Nave Bayes Drven Hggs Tagger. Wth 8 features, the sgnal error was 5% and the background error was 8% Logstc Regresson. In logstc regresson we have a hypothess of the form h θ (x) = g(θ T x) = + exp( θ T x) Where x s a (D + )-dmensonal vector wth x 0 = and x,...,d the features, and θ R D+ are the parameters of the model. The predcton for y {0, } s {h θ (x) > 0.5}. We splt the data nto 70% tranng set and 30% testng set. We optmze the parameters of the regresson va batch gradent ascent on the log-lkelhood functon on the tranng set. I.e. we run the followng algorthm: Repeat for N teratons: θ j+ := θ j + α = (y() h θ (x () ))x () j Where n the algorthm we set α =, and x (),..., x (m) s the tranng set. We then calculate the error n the testng set (and tranng set) by calculatng ɛ Test/Tran Sgnal,N ɛ Test/Tran Test/Tran = = {y () = }{h θ (x () ) < 0.5}} m Test/Tran Sgnal Test/Tran,N = ɛ Test/Tran N = {y () = 0}{h θ (x () ) > 0.5}} = ɛ Test/Tran Sgnal,N m Test/Tran + ɛtest/tran,n For each N. Sometmes ɛ Tran M > ɛtran N for M > N as the algorthm moves out of local mnma. So we look at M = argmn {,...,N} (ɛ Tran ) Snce n practce we can choose whatever θ j mnmzes the tranng error over the course of the algorthm, whch sn t necessarly θ N. Runnng the algorthm wth all the features, we get convergence of ɛ Tran M,mn at around N = 000 teratons, whch can be seen n Fg. 2. In order to compare across algorthms, we are concerned wth the mnmum total error when ɛ Sgnal So we defne smlarly The results after N = 5000 teratons are shown n Table. ɛ Tran Sgnal,M M 5% = argmn S(ɛ Tran ) S = { : ɛ Tran Sgnal, 0.05 < 0.0} ɛtran,m ɛtran M ɛ Test Sgnal,M ɛtest,m ɛ Test M M = M M = M5%

3 CLASSIFICATION OF HIGGS JETS AS DECAY PRODUCTS OF A RANDALL-SUNDRUM GRAVITON AT THE ATLAS EXPERIMENT 3 Fgure 2. ɛ Tran/Test N,mn as a functon of N usng all the features. By about N = 000, the algorthm has converged to wthn a few percent level of the N lmt. Table. Results of logstc regresson algorthm k-clusterng. We examne a sgnfcantly modfed k-means clusterng algorthm desgned by us. Frst, we run an ordnary unsupervsed clusterng algorthm on the tranng data:. Choose k centrods µ,..., µ k randomly from the data 2. For every, set c() = argmn j x () µ j 2 3. For every j, set µ j = = {c() = j} x() 4. Repeat the last two steps untl convergence 5. Repeat steps -4 a few tmes to globally mnmze the cost functon J = = x() µ c() 2. Then we supervse the unsupervsed algorthm. For each cluster j we look at what fracton of the data classfed n that cluster s sgnal: = f j = {y() = }{c() = j} = {c() = j} We then set a sgnal threshold s and background threshold b to label sgnal (f j > s) and background (f j < b) centrods, respectvely. Then usng only the centrods that passed the sgnal or background threshold cuts (.e. the number of clusters used n the testng phase k Test k), we cluster and label the testng data to get sgnal and background dentfcaton effcences: e Test/Tran Sgnal (s) = e Test/Tran (b) = m Test/Tran m Test/Tran Sgnal m Test/Tran m Test/Tran Sgnal And the error ɛ = e. By varyng s and b, we get a ROC curve, whch s shown n Fgure 3. = = y () {f c () > s} ( y () ){f c () < b} 3.4. Multvarate Analyss. There are correlatons between dfferent knematcs varables n the dataset we consder. For nstance, both the Hggs jets and QCD background jets wll splt to two sub-jets of the same amount of energy. Therefore, the pt of the sub-jets are roughly the same. To account for such correlatons we mplement more advanced multvarate analyss tools BDT (Boosted Decson Tree) and k-nn (k th nearest neghborhood) whch are wdely used n ATLAS Collaboraton Boosted Decson Tree. The Boosted Decson Tree (BDT) s an ensemble of weak classfers called decson trees. The algorthm nvolves two steps: () buldng a forest of decson trees, and (2) adaptve boostng. A decson tree s a mult-layer bnary classfer. The tranng starts wth the root node and splts to two subsets, each of whch searches for the best varable and ts correspondng cut based on calculaton of the Gn Index, defned as p( p), where p s the purty of the sample. If n a sample, sgnal and background are mxed equally then p = 0.5 and the Gn Index s maxmzed, whle t falls to zero f the sample s entrely sgnal or background. In ths search step, we partton each varable nto 20 equal segments and decde the cut based on the maxmum change between the parent node Gn Index

4 4 AVIV CUKIERMAN, ZIHAO JIANG Fgure 3. The ROC curves for the the modfed k-clusterng algorthm. The model mproved wth ncreasng numbers of clusters, but low data statstcs prevented makng more clusters. and the sum of the daughter nodes Gn Indces. The tree keeps splttng untl the resultng splt subsets have fewer than 200 data ponts. Once we obtan 200 such weak classfers h, we pass them to the boostng step. The goal of the boostng s to construct a strong classfer as a lnear combnaton of the 200 weak classfers we obtaned, f(x) = T =200 t= α t h t (x). For b {, }, we come up wth a predcton model H b (x) = {f(x) > b}. Gven m tranng samples, the α s are chosen such that ɛ b = m {H b(x ) y } s mnmzed. The mnmzaton scheme we use s adaptve boostng, whch estmates and mnmzes an upper bound of ɛ. By varyng b, we get a ROC curve, whch can be seen n Fgure 4. The BDT algorthm s also useful n determnng whch varables are most useful for separatng sgnal and background. We calculated the dscrmnatng power of each feature as the number of splts occurred for that varable weghted by the gan-squared and number of events n t. The gan s calculated as the change n nformaton entropy H from a prevous stage to a new stage wth more nformaton. As seen n Table 2, we fnd that the Hggs jet transverse momentum and mass have the most dscrmnatng power. Varable Importance Varable Importance jet pt 0.9 leadng sub-jet b-taggng score 0.3 jet mass 0.5 sub-leadng sub-jet b-taggng score 0.2 sub-leadng sub-jet mass 0.4 leadng sub-jet mass 0.09 sub-leadng sub-jet pt 0.3 leadng sub-jet pt 0.07 Table 2. Importance calculated by BDT algorthm. Most ndcatve varables are the jet pt and jet mass k-nn. The phlosophy of k-nn s neghborhood majorty vote. The label of a pont s decded by the labels of the majorty of ts neghborhood ponts. To formalze the problem, we need a metrc to decde for two ponts to be close or neghbors. For ths problem, we use the conventonal Eucldean metrc. Gven a test pont p and a set of features ( n {f, f 2,..., f n }, we calculate the metrc R = a w 2 b j, where a and b j are the th feature value for p and j th tranng pont. Then we select the closest k tranng ponts (n our case, we use k = 20) and assgn a probablty of p beng sgnal P s = ks k, where k s s the number of sgnal ponts of the k-nearest ponts. The weght factor w n the metrc s the wdth of feature for all the data. By varyng the cut on the sgnal probablty, we come up wth a ROC curve, whch can be seen n Fgure 4. ) 2 4. Results The results are detaled n the Models secton. But we wsh to compare the models used. Table 3 shows the mnmum test background error for each model wth test sgnal error 5%. Model Error Nave Bayes 0.08 Logstc Regresson Modfed k-clusterng 0.29 BDT k-nn Table 3. Results of the models used here for test sgnal error 5%.

5 CLASSIFICATION OF HIGGS JETS AS DECAY PRODUCTS OF A RANDALL-SUNDRUM GRAVITON AT THE ATLAS EXPERIMENT 5 Fgure 4. ROC curves for BDT and k-nn algorthms. The BDT algorthm outperforms k-nn at all sgnal effcences. As we can see, the BDT algorthm s the best performng. 5. Dscusson We examne both lnear and non-lnear classfcaton algorthms. In general we fnd the non-lnear classfers to perform better than the lnear ones. The best performng lnear classfer was Nave Bayes, and the best performng classfer overall was BDT. We note that the ATLAS collaboraton has generally agreed on usng BDT to classfy sgnal versus background n a wde varety of problems, and our results support that decson. We also desgned a modfed k-clusterng algorthm. Our new algorthm was the worst performng of all the ones we examned. However t was nterestng to see that an algorthm wth unsupervsed components could stll do a decent job of classfcaton. We also analyzed and measured the mportance of each feature used n our algorthms. Both our Nave Bayes algorthm and BDT fnd that the two most mportant features are the Hggs jet mass and transverse momentum. 6. Future Work Most of the algorthms dscussed here have seen years of research and development. The k-clusterng algorthm developed here, on the other hand, shows some promse, even though t was the worst performng of all the algorthms. Wth more tme t would be nterestng to look at ths algorthm from a theoretcal standpont to understand ts lmtatons and under what model t performs best. If we were to work on ths project for a few more months, we would also probably look at other algorthms lke SVM. 7. Concluson We fnd that machne learnng algorthms can be very powerful n dstngushng Hggs sgnal from background n smulated data from ATLAS. Its partcularly nterestng that the algorthms themselves had no knowledge of the physcs, only of the data and tranng classfcatons. The prospect of dscoverng or rulng out a Randall-Sundrum gravton va decay to two Hggs bosons at ATLAS seems feasble gven the power of the classfcaton algorthms dscovered here. From a wder perspectve, machne learnng algorthms can be used n many physcs analyses other than the specfc one studed here, such as searches for supersymmetrc partcles, dark matter, or exotc partcles lke axons. We beleve that the future of physcs done at experments lke ATLAS nevtably les wth the use of machne learnng technques lke the ones presented here. 8. References [] ATLAS Collaboraton (202). Observaton of a new partcle n the search for the Standard Model Hggs boson wth the ATLAS detector at the LHC. Phys. Lett. B 76 (39) [2] M. Caccar, G. Salam, G. Soyez (2008), The Ant-Kt jet clusterng algorthm, JHEP 0804:063 [3] A. Hoecker et al. (2007). TMVA - Toolkt for Multvarate Data Analyss, PoS ACAT 040 [4] A. Ng. CS229 Lecture Notes, Retreved from CS 229 Autumn 204 Course Webste, [5] L. Randall, R. Sundrum (999). Large Mass Herarchy from a Small Extra Dmenson. Physcal Revew Letters 83 (7) [6] T. Sjostrand, S. Mrenna and P. Skands (2007). A Bref ntroducton to PYTHIA 8., arxv: