A Global Jet Finding Algorithm Yang Bai University of - - PowerPoint PPT Presentation

A Global Jet Finding Algorithm

University of Wisconsin-Madison

Yang Bai

MC4BSM Workshop@LPC May 20, 2015

2

In collaboration with:

Zhenyu Han University of Oregon Ran Lu

• Univ. of Wisconsin-Madison

arxiv:1411.3705 and work in progress

3

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

4

What is a jet?

An ensemble of particles in detectors can be called a jet Jet-finding algorithm: how to group particles together?

5

Jets in BSM

• Mono-jet plus MET events as the

dark matter signature

q ¯ q χ ¯ χ

• Multi-jets plus MET for RPC SUSY or without MET

for RPV SUSY

P1 P2 ˜ t∗ ˜ t

j j j j

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Fat-jet object

• Boosted top quark, W/Z, Higgs ……

pp → Z0 → WW

j Z′ j j Z′ j W W

pT (W) ∼ MW

pT (W) MW

Search for a few TeV resonance decaying into t, W, Z, h …

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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
• Kaplan, Rehermann, Schwartz, Tweedie, boosted top, 0806.0848
• Thaler and Wang, boosted top, 0806.0023

Many new variables or procedures:

• mass drop, N-subjettiness, pull, dipolarity, without trees, …
• Jet grooming: filtering, trimming, pruning …
• Almeida, Lee, Perez, Sterman, Sung, boosted top, 0807.0234;…

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Jet substructure: an example

Boosted Higgs for measuring the decay

h → b¯ b

(1) start from a jet-finding algorithm (C/A) to cover a wider area

Two steps:

(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

Butterworth et.al., 0802.2470

b R

b b

Rfilt Rbb g b R mass drop filter

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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

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A brief review of jet-finding algorithms

★ Cone algorithm ★ Sequential recombination algorithm

• Started by Sterman and Weinberg in 70’s
• CDF SearchCone, Mid point, SISCone …
• Started by the JADE collaboration in 80’s
• Used at UA1, Tevatron
• , Cambridge/Aachen,

kt anti-kt

• Extensively used at the LHC

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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

jet 1 c) jet 2 jet 1

α x (+ ) ∞

n

Colinear safety? SISCone (split-merge)

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algorithm

Iterative process:

• Find the minimum of the and
• If it is a , recombine i and j into a single new particle,

and repeat

Anti-kt

dij = min(p−2

ti , p−2 tj )∆R2 ij

R2 diB = p−2

ti

∆R2

ij = (yi − yj)2 + (φi − φj)2

dij

diB

dij

• otherwise, if it is a , declare i to be a jet, and remove

it from the list of particles

diB

• stop when no particles remain

Infrared and collinear safe !

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Behaviors of different algorithms

• 6
• 4
• 2

2 4 6 1 2 3 4 5 6 0 5 10 15 20 25 SISCone, R=1, f=0.75 y [GeV]

t

p φ p

• 6
• 4
• 2

2 4 6 1 2 3 4 5 6 0 5 10 15 20 25

, R=1

t

anti-k

y [GeV]

t

p φ

Salam, 0906.1833

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Quantify the goodness of algorithms

Salam, 0906.1833

0.001 0.01 0.1 1

• 20
• 15
• 10
• 5

5 10 15 1/N dN/dpt (GeV-1) ∆pt

(B) (GeV)

R=1

Pythia 6.4 LHC (high lumi) 2 hardest jets pt,jet> 1 TeV |y|<2

SISCone (f=0.75) Cam/Aachen kt anti-kt

Back-reaction: how much adding soft background particles changes the original particles in a jet

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Can one has a more intuitive way to define a jet-finding algorithm?

events with N particles a jet with subset particles

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Can one has a more intuitive way to define a jet-finding algorithm?

events with N particles a jet with subset particles

A jet function

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Can one has a more intuitive way to define a jet-finding algorithm?

events with N particles a jet with subset particles

A jet function

Look for a simple jet definition function

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Start with a QCD jet

★ QCD partons are massless ★ The jet function should

• prefer increasing jet energy
• disfavor increasing jet mass

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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 m2 E

[H. Georgi, 1408.1161]

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Start with a QCD jet

★ QCD partons are massless ★ The jet function should

• prefer increasing jet energy
• disfavor increasing jet mass

For N particles and possibilities, find the one maximizing this jet function. One does this iteratively to find all jets in one event.

2N

★ The simple option at a lepton collider:

J(P µ) = E − β m2 E

[H. Georgi, 1408.1161]

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Special cases

J(P µ) = E − β m2 E

• :

β = 0

J = E

E

include all particles in one jet

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Special cases

J(P µ) = E − β m2 E

• :

β = 0

J = E

E

include all particles in one jet

• :

β = 1

J = |~ P|

hemisphere way for two jets

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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 J =(E

2−βm 2)

E =(γ

2−β) m 2

E ≥0

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

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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

JET (P µ

J ) ≡ ET − β m2

ET

★ One could also try other powers

JET (P µ

J ) ≡ Eα T (1 − β m2

E2

T

)

★ Does this new function has a similar cone geometry?

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Try an “easier” function

★ For ,

α = 2

JE2

T = E2

T − βm2 = E2 − P 2 z − βm2

★ Requiring , the boundary satisfies JE2

T (P µ

J ) > JE2

T (P µ

J − pµ j )

1 |p||P|(P x p x+P y p y+(1−1 β) P z pz)=1 v (1− 1 β)

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Try an “easier” function

★ For ,

α = 2

JE2

T = E2

T − βm2 = E2 − P 2 z − βm2

★ Requiring , the boundary satisfies JE2

T (P µ

J ) > JE2

T (P µ

J − pµ j )

1 |p||P|(P x p x+P y p y+(1−1 β) P z pz)=1 v (1− 1 β)

{

p x

2+ p y 2+ pz 2=C1(P)

( p x−P x)

2+( p y−P y) 2+( p z−(1−1

β )P z)

2

=C 2(P)

★ Can be interpreted as intersection of two spheres

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Still a cone jet

★ For a general , the boundary is

α

1 |p||P|(P x p x+P y p y+κ P z pz )= κ v

κ=1− α 2β + α−2 2 m

2

ET

2

the center is shifted from the jet momentum towards the central region

~ ˆ Pc = 1 q 1 (1 2) ˆ P z 2

J

( ˆ P x

J , ˆ

P y

J ,  ˆ

P z

J )

r  < 1 zc  vJ p 1 (1 2) cos2 ✓J

particles belong to the jet is within a cone from the center

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Still a cone jet

★ For a general , the boundary is

α

1 |p||P|(P x p x+P y p y+κ P z pz )= κ v

κ=1− α 2β + α−2 2 m

2

ET

2

the center is shifted from the jet momentum towards the central region

~ ˆ Pc = 1 q 1 (1 2) ˆ P z 2

J

( ˆ P x

J , ˆ

P y

J ,  ˆ

P z

J )

r  < 1 zc  vJ p 1 (1 2) cos2 ✓J

particles belong to the jet is within a cone from the center

★ The beam direction always stays away from the jet and

does not need any special treatment

22

Cone identification

(I) P (II) P Q (III) P Q R

theoretical boundary

x x

physical boundary

x

22

Cone identification

(I) P (II) P Q (III) P Q R

theoretical boundary

x x

physical boundary

x

• ne can use three particles to identify a cone

23

Alternative boundaries

(a) P Q R' (b) P Q (c) P

24

Numerical implementation

★ In general, we need to check all possible subsets of

particles for a general function, which is not possible

2N

★ 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

, the total operation time is

O(n3)

O(N n3)

https://github.com/LHCJet/JET

25

Comparison: shape

• 10
• 8
• 6
• 4
• 2

2 4 6 8 10 1 2 3 4 5 6 50 100 150 200 250 300 350

y [GeV]

t

p φ

• 10
• 8
• 6
• 4
• 2

2 4 6 8 10 1 2 3 4 5 6 50 100 150 200 250 300 350

y [GeV]

t

p φ

JET with β = 1.4

anti-kT

with R = 1.0

26

Comparison: size

100 200 300 400 500 600 pT (GeV) 200 400 600 800 1000 1200 number of jets JET(β = 6) anti-kt (R = 0.88) anti-kt (R = 0.43) anti-kt (R = 0.33)

match anti-kt results very well for a QCD jet

27

Comparison: back-reaction

again, similar to the anti-kt results

−20 −15 −10 −5 5 10 15 20 ∆p(b)

t

(GeV) 10−4 10−3 10−2 10−1 100 1/N dN/dpt (GeV−1) anti-kt (R = 0.43) kt (R = 0.43) JET (β = 6.0) JE2

t (β = 12.0)

28

Comparison: dijet Z’ mass

again, similar to the anti-kt results

500 1000 1500 2000 2500 3000 Mj1,j2 (GeV) 200 400 600 800 1000 1200 1400 number of events MZ = 2 TeV JET (β = 6) anti-kt (R = 0.43)

29

A naive comparison for W-jet

20 40 60 80 100 Jet Mass (GeV) 500 1000 1500 2000 2500 3000 MW JET(β = 6) anti-kt (R = 0.43) JET(β = 6) with PU anti-kt (R = 0.43) with PU

pT (W) > 250 GeV

• ur jet-finding algorithm is designed for QCD jets so far

30

Design a W-jet-finding function

• A boosted W-jet contains

a two-prong structure

• Need to incorporate a jet

shape in the function

• The existing part of

may be kept

JET

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Design a W-jet-finding function

• The new function need to prefer two-prong

JW

ET (P µ J ) = Eα T

 1 − β m2 E2

T

+ γH2,J

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Design a W-jet-finding function

• The new function need to prefer two-prong
• try the jet energy correlation functions:

≡

• i̸=k

|⃗ pi||⃗ pk| E2

J

| sin ϕik|a(1 − | cos ϕik|)1−a .

Banfi, Salam, Zanderighi, hep-ph/0407286

ECF(N, β) = X

i1<i2<...<iN∈J

N Y

a=1

pT ia ! N−1 Y

b=1 N

Y

c=b+1

Ribic !

Larkoski, Salam, Thaler, 1305.0007

JW

ET (P µ J ) = Eα T

 1 − β m2 E2

T

+ γH2,J

32

A working function

• It is Lorentz invariant except the overall factor
• It becomes transparent in the jet rest frame
• One can easily show that this function reaches

its maximum for a two-prong structure

H2,J = @X

i,k

|~ pi||~ pk| E2

T

cos2 'ik 1 A

rest

= @X

i,k

(~ pi · ~ pk)2 E2

T |~

pi||~ pk| 1 A

rest

H2,J ≡ H2,J E2

T

= 1 E2

T

X

i,k

⇥ m2

J pi · pk − (PJ · pi)(PJ · pk)

⇤2 m2

J(PJ · pi)(PJ · pk)

32

A working function

• It is Lorentz invariant except the overall factor
• It becomes transparent in the jet rest frame
• One can easily show that this function reaches

its maximum for a two-prong structure

• The function in rest frame is the Fox-Wolfram

moment, introduced as an event shape at lepton colliders

H2,J = @X

i,k

|~ pi||~ pk| E2

T

cos2 'ik 1 A

rest

= @X

i,k

(~ pi · ~ pk)2 E2

T |~

pi||~ pk| 1 A

rest

H2,J ≡ H2,J E2

T

= 1 E2

T

X

i,k

⇥ m2

J pi · pk − (PJ · pi)(PJ · pk)

⇤2 m2

J(PJ · pi)(PJ · pk)

33

Double-cone shape

in the rest frame in the lab frame W W

• a double-cone structure with the subjet size

determined dynamically

JW

ET (P µ J ) = Eα T

 1 − β m2 E2

T

+ γH2,J

33

Double-cone shape

in the rest frame in the lab frame W W

• a double-cone structure with the subjet size

determined dynamically

JW

ET (P µ J ) = Eα T

 1 − β m2 E2

T

+ γH2,J

• controls the subjet size and

controls the fat jet size

1/ p β

1/(β − γ)

34

results

JW

E2

T

no pile-up included yet

pruning jet: S. Ellis, Vermilion, Walsh; 0912.0033

14 TeV LHC WW

pT (W) > 200 GeV

35

Variables used in CMS

sig

ε

0.2 0.4 0.6 0.8 1

bkg

ε 1 -

0.2 0.4 0.6 0.8 1

CA R = 0.8 < 350 GeV

T

250 < p | < 2.4 η | < 100 GeV

jet

m 60 <

W+jet

MLP neural network Naive Bayes classifier

1

τ /

2

τ

Qjet

Γ pruned

1

τ /

2

τ no axes optimization

1

τ /

2

τ =1.7) β (

2

C Mass drop

+

= 1.0) W κ Jet charge (

8 TeV

CMS

Simulation

CMS; 1410.4227 N-subjettiness: Thaler and Tilburg; 1011.2268 Q-jets: Ellis, Hornig, Roy, Krohn, Schwartz; 1201.1914

36

Performance w. Jet-sub. Variables

preliminary A better jet-finding algorithm makes some improvement

37

Byproduct: A New Event Shape Variable

preliminary

38

Conclusions

★ A global jet-finding algorithm for maximizing a jet

function works for a QCD jet

★ Our preliminary results show that our W-jet

function can tag a W-jet very well

★ We are finalizing the numerical code with a trade-off

between finding a global maximum and running speed

★ Other jet functions to tag top quark, black-hole multi-

jets and new conformal gauge sector signatures are also interesting to explore

39

Thanks

40

Real proof for a cone jet

★ Check the angular distance of a soft particle from the

jet momentum

, z=cosθ= p x P x+ p y P y+ pz P z |p||P|

★ For a soft particle belongs to the jet:

j

J (P)>J (P− p j)

1−β(1−vα

2 )>1−r j−β 1−vα 2−2r j(1−z vα)

1−r j

z> β(1+vα

2 )−(1−r j)

2βvα >β(1+vα

2 )−1

2βvα = 1 vα(1− 1 2β(1+β m

2

E

2))

41

Real proof for a cone jet

★ For a soft particle belongs to the jet:

j

z> β(1+vα

2 )−(1−r j)

2βvα >β(1+vα

2 )−1

2βvα = 1 vα(1− 1 2β(1+β m

2

E

2))

★ For a soft particle not belongs to the jet:

k

z< β(1+vα

2 )−(1+r k)

2βvα <β(1+vα

2 )−1

2βvα = 1 vα(1− 1 2β(1+β m

2

E

2))

★ Soft particles are on the boundary; very IR safe ★ So, a cone-like boundary for individual jets