Jet-finding for LHC. Gavin Salam work with M. Cacciari and (in progress) G. Soyez. LPTHE, Universities of Paris VI and VII and CNRS

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1 Jet-finding for LHC Gavin Salam work with M. Cacciari and (in progress) G. Soyez LPTHE, Universities of Paris VI and VII and CNRS Robi is 60, SPhT Saclay 24 November 2006

2 Jet clustering in the LHC era (p. 2) Introduction Introduction Jet finding at LEP/HERA: At LHC it will be more complex: Clean environment Moderate number of particles ( 50) Up to 25 simultaneous pp collisions (10 in above event) Thousands of particles

3 Jet clustering in the LHC era (p. 2) Introduction Introduction Jet finding at LEP/HERA: At LHC it will be more complex: Clean environment Moderate number of particles ( 50) Up to 25 simultaneous pp collisions (10 in above event) Thousands of particles

4 Jet clustering in the LHC era (p. 3) Introduction New issues at LHC Two characteristically new issues appear for jet finding in the complex environment of of LHC: How, computationally, to deal with the clustering of thousands of particles Solutions exploit interesting links between jet-finding and computational geometry How to reconstruct correct jet kinematics despite the significant additional energy from the pileup events Useful to introduce the concept of a jet area

5 Jet clustering in the LHC era (p. 4) Introduction Jet finder defs. Clustering jet finders 1. Calculate distances dij between all particles i and j dib between i and beam 2. Find smallest of d ij and d ib If dij is smallest, recombine i and j if dib is smallest call i a jet 3. Goto step 1 if anything s left Two variants (& one parameter, R) k t jet finder [1991] d ij = min(k 2 ti, k 2 tj) R 2 ij, d ib = k 2 tir 2 Cambridge/Aachen [1998] d ij = R 2 ij, d ib = R 2 [ R 2 ij = y 2 ij+ φ 2 ij]

6 Jet clustering in the LHC era (p. 4) Introduction Jet finder defs. Clustering jet finders 1. Calculate distances dij between all particles i and j dib between i and beam 2. Find smallest of d ij and d ib If dij is smallest, recombine i and j if dib is smallest call i a jet 3. Goto step 1 if anything s left Two variants (& one parameter, R) k t jet finder [1991] d ij = min(k 2 ti, k 2 tj) R 2 ij, d ib = k 2 tir 2 Cambridge/Aachen [1998] d ij = R 2 ij, d ib = R 2 [ R 2 ij = y 2 ij+ φ 2 ij]

7 Jet clustering in the LHC era (p. 4) Introduction Clustering jet finders 1. Calculate distances dij between all particles i and j dib between i and beam Cone jet finders e.g. Jet finder defs. 2. Find smallest of d ij and d ib If dij is smallest, recombine i and j if dib is smallest call i a jet 3. Goto step 1 if anything s left Two variants (& one parameter, R) k t jet finder [1991] d ij = min(k 2 ti, k 2 tj) R 2 ij, d ib = k 2 tir 2 Cambridge/Aachen [1998] d ij = R 2 ij, d ib = R 2 [ R 2 ij = y 2 ij+ φ 2 ij] [from W. Plano]

8 Jet clustering in the LHC era (p. 5) Speed Time to cluster N particles t (s) KtJet OJF Tevatron MidPoint N JetClu (IR unsafe cone) LHC (single LHC (c. 20 LHC interaction) interactions) Heavy Ion Standard C++ (and fortran) k t -clustering takes time N 3. a Pb-Pb event takes 1 day! JetClu (cone) is fast, but IR unsafe at NLO. being phased out at Tevatron IR-safer cone (Midpoint) is as slow as k t IR unsafe at NNLO Jet-clustering speed is an issue for high-luminosity pp ( 10 8 events) and Pb-Pb ( 10 7 events) collisions at LHC.

9 Jet clustering in the LHC era (p. 5) Speed Time to cluster N particles t (s) KtJet OJF Tevatron MidPoint N JetClu (IR unsafe cone) LHC (single LHC (c. 20 LHC interaction) interactions) Heavy Ion Standard C++ (and fortran) k t -clustering takes time N 3. a Pb-Pb event takes 1 day! JetClu (cone) is fast, but IR unsafe at NLO. being phased out at Tevatron IR-safer cone (Midpoint) is as slow as k t IR unsafe at NNLO Jet-clustering speed is an issue for high-luminosity pp ( 10 8 events) and Pb-Pb ( 10 7 events) collisions at LHC.

10 Jet clustering in the LHC era (p. 6) Speed Why is k t an N 3 algorithm? 1. Given the initial set of particles, construct a table of all the d ij, d ib. [O ( N 2) operations, done once] 2. Scan the table to find the minimal value d min of the d ij, d ib. [O ( N 2) operations, done N times] 3. Merge or remove the particles corresponding to d min as appropriate. [O (1) operations, done N times] 4. Update the table of d ij, d ib to take into account the merging or removal, and if any particles are left go to step 2. [O (N) operations, done N times] This is the brute-force or naive method

11 Jet clustering in the LHC era (p. 7) Speed Can one do better than N 2? There are N(N 1)/2 distances d ij surely we have to calculate them all in order to find smallest? k t distance measure is partly geometrical: Consider smallest d ij = min(k 2 ti,k2 tj )R2 ij Suppose k ti < k tj Then: R ij <= R il for any l j. [If l s.t. R il < R ij then d il < d ij ] In words: if i,j form smallest d ij then j is geometrical nearest neighbour (GNN) of i. k t distance need only be calculated between GNNs Each point has 1 GNN need only calculate N d ij s

12 Jet clustering in the LHC era (p. 7) Speed Can one do better than N 2? There are N(N 1)/2 distances d ij surely we have to calculate them all in order to find smallest? k t distance measure is partly geometrical: Consider smallest d ij = min(k 2 ti,k2 tj )R2 ij Suppose k ti < k tj Then: R ij <= R il for any l j. [If l s.t. R il < R ij then d il < d ij ] In words: if i,j form smallest d ij then j is geometrical nearest neighbour (GNN) of i. k t distance need only be calculated between GNNs Each point has 1 GNN need only calculate N d ij s

13 Jet clustering in the LHC era (p. 7) Speed Can one do better than N 2? There are N(N 1)/2 distances d ij surely we have to calculate them all in order to find smallest? k t distance measure is partly geometrical: Consider smallest d ij = min(k 2 ti,k2 tj )R2 ij Suppose k ti < k tj Then: R ij <= R il for any l j. [If l s.t. R il < R ij then d il < d ij ] In words: if i,j form smallest d ij then j is geometrical nearest neighbour (GNN) of i. k t distance need only be calculated between GNNs Each point has 1 GNN need only calculate N d ij s

14 Jet clustering in the LHC era (p. 8) Speed Finding Geom Nearest Neighbours Given a set of vertices on plane (1...10) a Voronoi diagram partitions plane into cells containing all points closest to each vertex Dirichlet 1850, Voronoi 1908 A vertex s nearest other vertex is always in an adjacent cell. E.g. GNN of point 7 will be found among 1,4,2,8,3 (it turns out to be 3) Construction of Voronoi diagram for N points: N ln N time Fortune 88 Update of 1 point in Voronoi diagram: ln N time Devillers 99 [+ related work by other authors] Convenient C++ package available: CGAL

15 Jet clustering in the LHC era (p. 8) Speed Finding Geom Nearest Neighbours Given a set of vertices on plane (1...10) a Voronoi diagram partitions plane into cells containing all points closest to each vertex Dirichlet 1850, Voronoi 1908 A vertex s nearest other vertex is always in an adjacent cell. E.g. GNN of point 7 will be found among 1,4,2,8,3 (it turns out to be 3) Construction of Voronoi diagram for N points: N ln N time Fortune 88 Update of 1 point in Voronoi diagram: ln N time Devillers 99 [+ related work by other authors] Convenient C++ package available: CGAL

16 Jet clustering in the LHC era (p. 8) Speed Finding Geom Nearest Neighbours Given a set of vertices on plane (1...10) a Voronoi diagram partitions plane into cells containing all points closest to each vertex Dirichlet 1850, Voronoi 1908 A vertex s nearest other vertex is always in an adjacent cell. E.g. GNN of point 7 will be found among 1,4,2,8,3 (it turns out to be 3) Construction of Voronoi diagram for N points: N ln N time Fortune 88 Update of 1 point in Voronoi diagram: ln N time Devillers 99 [+ related work by other authors] Convenient C++ package available: CGAL

17 Jet clustering in the LHC era (p. 9) Speed Assembling fast k t clustering The FastJet algorithm: Construct the Voronoi diagram of the N particles with CGAL O (N ln N) Find the GNN of each of the N particles, calculate d ij store result in a priority queue (C++ map) O (N ln N) Repeat following steps N times: Find smallest d ij, merge/eliminate i,j N O (1) Update Voronoi diagram and distance map N O (ln N) Overall an O (N ln N) algorithm Cacciari & GPS, hep-ph/ salam/fastjet/ Results identical to standard N 3 implementations

18 Jet clustering in the LHC era (p. 9) Speed Assembling fast k t clustering The FastJet algorithm: Construct the Voronoi diagram of the N particles with CGAL O (N ln N) Find the GNN of each of the N particles, calculate d ij store result in a priority queue (C++ map) O (N ln N) Repeat following steps N times: Find smallest d ij, merge/eliminate i,j N O (1) Update Voronoi diagram and distance map N O (ln N) Overall an O (N ln N) algorithm Cacciari & GPS, hep-ph/ salam/fastjet/ Results identical to standard N 3 implementations

19 Jet clustering in the LHC era (p. 10) Speed FastJet performance (kt) OJF MidPoint JetClu (almost IR unsafe) t (s) KtJet FastJet Tevatron LHC (single LHC (c. 20 LHC interaction) interactions) Heavy Ion N For N 10 4, FastJet algorithm scales as N lnn For N 10 4, FastJet switches to a related geometrical N 2 alg.

20 Jet clustering in the LHC era (p. 11) Speed Jets Comp. Geom. Brute force Geom. jet.alg. scaling Geometrical concepts scaling k t N 3 dynamic nearest-neighbour graph N lnn Dynamic voronoi diagram Devillers 99 (and others) Cam / N 3 dynamic closest pairs N lnn Aachen Shuffles, quad-trees T. Chan 02 Seedless N 2 N All circular partitions of a 2D set N 5/2 Cone of points (+ range-searching) not clear if studied Cam/Aachen: Cacciari & GPS 06 Seedless cone (replaces IR unsafe midpoint cone): GPS & Soyez, in progress

21 Jet clustering in the LHC era (p. 12) Areas What is speed good for? Standard hard event Two well isolated jets 200 particles Easy even with old methods

22 Jet clustering in the LHC era (p. 12) Areas What is speed good for? Add 10 min-bias events (moderately high lumi) 2000 particles Clustering takes O (10s) with old methods. 20ms with FastJet.

23 Jet clustering in the LHC era (p. 12) Areas What is speed good for? Add dense coverage of infinitely soft ghosts See how many end up in jet to measure jet area particles Clustering takes with old methods. 0.6s with FastJet. 20 minutes

24 Jet clustering in the LHC era (p. 13) Areas Jet areas P t,jet dijet event + 10 minbias (Kt-alg, R=1) median (pt/area) jet area Jet areas in k t algorithm are quite varied Because k t -alg adapts to the jet structure Contamination from min-bias area Complicates corrections: minbias subtraction is different for each jet. Cone supposedly simpler Area = πr 2? (Not quite...) But: area can be measured for each jet, as can typical median p t /area.

25 Jet clustering in the LHC era (p. 14) Areas Application: semi-leptonic t LHC 0.02 R=0.4, LHC semileptonic ttbar events k t, no pileup k t, pileup k t, pileup, corrected 1/N dn/dmass 0.01 W mass top mass Each jet corrected by area median (P t /area) reconstructed W / top mass [GeV] Naive analysis: no cuts; assume both b s tagged Take two hardest non-b jets call them a W Take correct sign b, combine with W top

26 Jet clustering in the LHC era (p. 15) Areas 1/N dn/da Distribution of jet areas purely soft jets kt algorithm A / πr 2 Distribution of jet areas Since areas are a crucial quantity in reconstructing jet kinematics, study them further... Two simple cases: 1. run clustering on many soft particles & look at areas of jets that come out 2. Add one hard particle, and examine area of its jet Conclusion: jet area expands when it is anchored by structure. Can one obtain analytical insight into this? To some extent in 1D Hierarchical clustering is used in many fields (bio, computing,...) are similar features of relevance there?

27 Jet clustering in the LHC era (p. 15) Areas 1/N dn/da Distribution of jet areas purely soft jets jets with 1 hard parton kt algorithm A / πr 2 Distribution of jet areas Since areas are a crucial quantity in reconstructing jet kinematics, study them further... Two simple cases: 1. run clustering on many soft particles & look at areas of jets that come out 2. Add one hard particle, and examine area of its jet Conclusion: jet area expands when it is anchored by structure. Can one obtain analytical insight into this? To some extent in 1D Hierarchical clustering is used in many fields (bio, computing,...) are similar features of relevance there?

28 Jet clustering in the LHC era (p. 15) Areas 1/N dn/da Distribution of jet areas purely soft jets jets with 1 hard parton kt algorithm A / πr 2 Distribution of jet areas Since areas are a crucial quantity in reconstructing jet kinematics, study them further... Two simple cases: 1. run clustering on many soft particles & look at areas of jets that come out 2. Add one hard particle, and examine area of its jet Conclusion: jet area expands when it is anchored by structure. Can one obtain analytical insight into this? To some extent in 1D Hierarchical clustering is used in many fields (bio, computing,...) are similar features of relevance there?

29 Jet clustering in the LHC era (p. 15) Areas 1/N dn/da Distribution of jet areas purely soft jets jets with 1 hard parton kt algorithm A / πr 2 Distribution of jet areas Since areas are a crucial quantity in reconstructing jet kinematics, study them further... Two simple cases: 1. run clustering on many soft particles & look at areas of jets that come out 2. Add one hard particle, and examine area of its jet Conclusion: jet area expands when it is anchored by structure. Can one obtain analytical insight into this? To some extent in 1D Hierarchical clustering is used in many fields (bio, computing,...) are similar features of relevance there?

30 Jet clustering in the LHC era (p. 16) Areas Conclusions Jet finding in the context of LHC is a rich subject! Speed (a major issue) improves enormously when one exploits the geometrical structures that underly jet algorithms especially together with recent developments by computational geometers. Jet areas are a new concept of particular relevance in high-noise environments much is still to be understood about them. NB: more is known than could be shown here!

31 Jet clustering in the LHC era (p. 17) Extra material EXTRA MATERIAL

32 Jet clustering in the LHC era (p. 18) Extra material Inclusive jets in LHC 1/n coll d n jets / d P t k t alg, R=0.4 scaled pp y < 5 raw Pb-Pb FastJet Pb-Pb with subtraction PRELIMINARY Hydjet v 1.1 [Pythia P t,min = 10 GeV, unquenched] P t [GeV] Most HI studies use just particles with p t > a few GeV IR unsafe affected by quenching We use all particles and area-based subtraction. Good results despite the huge subtraction being performed.