A Global Jet Finding Algorithm Yang Bai University of Wisconsin-Madison MC4BSM Workshop@LPC May 20, 2015
In collaboration with: Zhenyu Han Ran Lu University of Oregon Univ. of Wisconsin-Madison arxiv:1411.3705 and work in progress 2
Outline • Motivation: physics beyond the Standard Model • Review of jet finding algorithms • Introduction of a global definition • Application to QCD-like jets • Extension to boosted two-prong jets • Conclusion 3
What is a jet? An ensemble of particles in detectors can be called a jet Jet-finding algorithm: how to group particles together? 4
Jets in BSM ¯ q • Mono-jet plus MET events as the ¯ χ dark matter signature q χ • Multi-jets plus MET for RPC SUSY or without MET for RPV SUSY j P 2 ˜ t j j ˜ t ∗ P 1 j 5
Fat-jet object Search for a few TeV resonance decaying into t, W, Z, h … • Boosted top quark, W/Z, Higgs …… j j j j pp → Z 0 → WW Z ′ W W Z ′ p T ( W ) ∼ M W p T ( W ) � M W 6
Jet substructure A jet may not be just a parton and it could have an internal structure Many new objects: (incomplete list) • …; Butterworth, Cox, Forshaw, WW scattering, hep-ph/0201098 • Butterworth, Davison, Rubin, Salam, boosted Higgs, 0802.2470 • Thaler and Wang, boosted top, 0806.0023 • Kaplan, Rehermann, Schwartz, Tweedie, boosted top, 0806.0848 • Almeida, Lee, Perez, Sterman, Sung, boosted top, 0807.0234;… Many new variables or procedures: • mass drop, N-subjettiness, pull, dipolarity, without trees, … • Jet grooming: filtering, trimming, pruning … 7
Jet substructure: an example Boosted Higgs for measuring the decay h → b ¯ b Two steps: Butterworth et.al., 0802.2470 (1) start from a jet-finding algorithm (C/A) to cover a wider area (2) mass-drop: (the QCD quark is massless) some subset of particles inside a Higgs-jet can have a much smaller mass. Filter: (reduce underlying events) introduce a finer angular scale R R filt b b R b b R bb g mass drop filter 8
Our motivation Can we combine this two-step procedure into a single one? • Hope: keep more hard process information and less underlying event contamination • Method: define a new jet-finding algorithm suitable for a boosted heavy object To proceed, let’s start with traditional jet-finding algorithms for QCD jets 9
A brief review of jet-finding algorithms ★ Cone algorithm • Started by Sterman and Weinberg in 70’s • CDF SearchCone, Mid point, SISCone … • Used at UA1, Tevatron ★ Sequential recombination algorithm • Started by the JADE collaboration in 80’s • , Cambridge/Aachen, anti- k t k t • Extensively used at the LHC 10
Cone algorithm Iterative process: • choose particle with highest transverse momentum as the seed particle • draw a cone of radius R around the seed particle • sum the momenta of all particles in the cone as the jet axis • if the jet axis does not agree with the original one, continue; otherwise find a stable cone and stop c) Colinear safety? SISCone (split-merge) jet 1 jet 1 jet 2 11 n α x (+ ∞ )
algorithm Anti- k t tj ) ∆ R 2 ij d ij = min( p − 2 ti , p − 2 ij = ( y i − y j ) 2 + ( φ i − φ j ) 2 d iB = p − 2 ∆ R 2 R 2 ti Iterative process: • Find the minimum of the and d ij d iB • If it is a , recombine i and j into a single new particle, d ij and repeat • otherwise, if it is a , declare i to be a jet, and remove d iB it from the list of particles • stop when no particles remain Infrared and collinear safe ! 12
Behaviors of different algorithms SISCone, R=1, f=0.75 anti-k , R=1 p [GeV] p p [GeV] t t t 25 25 20 20 15 15 10 10 5 5 6 0 6 0 5 5 4 4 φ φ 3 3 2 2 1 1 6 6 4 4 2 2 0 0 0 -2 0 -2 -4 -4 -6 -6 y y Salam, 0906.1833 13
Quantify the goodness of algorithms Back-reaction: how much adding soft background particles changes the original particles in a jet 1 SISCone (f=0.75) Pythia 6.4 Cam/Aachen LHC (high lumi) k t 2 hardest jets anti-k t 1/N dN/dp t (GeV -1 ) p t,jet > 1 TeV 0.1 |y|<2 R=1 0.01 0.001 -20 -15 -10 -5 0 5 10 15 (B) (GeV) ∆ p t Salam, 0906.1833 14
Can one has a more intuitive way to define a jet-finding algorithm? events with a jet with N particles subset particles 15
Can one has a more intuitive way to define a jet-finding algorithm? events with a jet with N particles subset particles A jet function 15
Can one has a more intuitive way to define a jet-finding algorithm? events with a jet with N particles subset particles A jet function Look for a simple jet definition function 15
Start with a QCD jet ★ QCD partons are massless ★ The jet function should • prefer increasing jet energy • disfavor increasing jet mass 16
Start with a QCD jet ★ QCD partons are massless ★ The jet function should • prefer increasing jet energy • disfavor increasing jet mass ★ The simple option at a lepton collider: J ( P µ ) = E − β m 2 [H. Georgi, 1408.1161] E 16
Start with a QCD jet ★ QCD partons are massless ★ The jet function should • prefer increasing jet energy • disfavor increasing jet mass ★ The simple option at a lepton collider: J ( P µ ) = E − β m 2 [H. Georgi, 1408.1161] E For N particles and possibilities, find the one 2 N maximizing this jet function. One does this iteratively to find all jets in one event. 16
Special cases J ( P µ ) = E − β m 2 E E • : β = 0 J = E include all particles in one jet 17
Special cases J ( P µ ) = E − β m 2 E E • : β = 0 J = E include all particles in one jet J = | ~ • : β = 1 P | hemisphere way for two jets 17
General cases ★ A group of particles will have a boost factor from its rest frame and has a jet function bigger than a soft particle 2 −β m 2 ) 2 J =( E 2 −β) m =(γ E ≥ 0 E γ≥ √ β ★ Relativistic beaming effect sin θ≤ √ 1 β ★ The particles are inside a jet cone A larger value of means a smaller cone size β 18
Extension to hadron colliders ★ The center-of-mass frame is likely to be highly boosted in the beam direction ★ The simplest way to extend the jet definition is J ) ≡ E T − β m 2 J E T ( P µ E T ★ One could also try other powers T (1 − β m 2 J E T ( P µ J ) ≡ E α ) E 2 T ★ Does this new function has a similar cone geometry? 19
Try an “easier” function ★ For , α = 2 T = E 2 T − β m 2 = E 2 − P 2 z − β m 2 J E 2 ★ Requiring , the boundary satisfies T ( P µ T ( P µ J − p µ J ) > J E 2 j ) J E 2 | p || P | ( P x p x + P y p y + ( 1 − 1 β ) P z p z ) = 1 v ( 1 − 1 β ) 1 20
Try an “easier” function ★ For , α = 2 T = E 2 T − β m 2 = E 2 − P 2 z − β m 2 J E 2 ★ Requiring , the boundary satisfies T ( P µ T ( P µ J − p µ J ) > J E 2 j ) J E 2 | p || P | ( P x p x + P y p y + ( 1 − 1 β ) P z p z ) = 1 v ( 1 − 1 β ) 1 { 2 + p y 2 + p z 2 = C 1 ( P ) p x 2 + ( p z − ( 1 − 1 β ) P z ) 2 2 +( p y − P y ) ( p x − P x ) = C 2 ( P ) ★ Can be interpreted as intersection of two spheres 20
Still a cone jet ★ For a general , the boundary is α 2 2 β + α− 2 m 1 κ= 1 − α | p || P | ( P x p x + P y p y +κ P z p z ) = κ 2 2 v E T the center is shifted from the jet momentum towards the central region 1 ~ ˆ ( ˆ J , ˆ J , ˆ P y r < 1 P x P z P c = J ) q 1 � (1 � 2 ) ˆ P z 2 J particles belong to the jet is within a cone from the center z c � 1 � (1 � 2 ) cos 2 ✓ J p v J 21
Still a cone jet ★ For a general , the boundary is α 2 2 β + α− 2 m 1 κ= 1 − α | p || P | ( P x p x + P y p y +κ P z p z ) = κ 2 2 v E T the center is shifted from the jet momentum towards the central region 1 ~ ˆ ( ˆ J , ˆ J , ˆ P y r < 1 P x P z P c = J ) q 1 � (1 � 2 ) ˆ P z 2 J particles belong to the jet is within a cone from the center z c � 1 � (1 � 2 ) cos 2 ✓ J p v J ★ The beam direction always stays away from the jet and does not need any special treatment 21
Cone identification R x x x P P P Q Q (I) (II) (III) theoretical physical boundary boundary 22
Cone identification R x x x P P P Q Q (I) (II) (III) theoretical physical boundary boundary one can use three particles to identify a cone 22
Alternative boundaries P P P Q Q R' (a) (b) (c) 23
Numerical implementation ★ In general, we need to check all possible subsets of 2 N particles for a general function, which is not possible ★ Knowing the geometrical shape of jets, one only need to check all possible cones and choose the one maximizing the jet function — “global” ★ For each particle, one can also determine its fiducial region such that one only needs to check “n << N” nearby particles as a neighbor ★ For each particle, the physically distinct cones is O ( n 3 ) , the total operation time is O ( N n 3 ) https://github.com/LHCJet/JET 24
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