Spectral Analysis of Jet
Substructure with Neural Networks:
Boosted Higgs and Sgluon Case
Sung Hak Lim

Beyond the BSM, Ikaho Onsen, Gunma, Japan
Oct. 2018
based on S. H. Lim, M. M. Nojiri, arXiv:1807.03312, submitted to JHEP
A. Chakraborty, S. H. Lim, M. M. Nojiri, work in progress

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Introduction A spectral function of jet substructure Spectral Analysis of Jet Substructure: Higgs Spectral Analysis of Jet Substructure: Sgluon Conclusion

Jets and Boosted Particles
At a hadron-hadron collider, such as LHC, a collimated particle cluster
called a jet often appears when a colored parton is produced.
As LHC stacking up multi TeV center-of-mass energy events, boosted
heavy particles is produced and form
√ a single collimated cluster of particles
similar to the QCD jets. (mEW / ŝ = O(0.1))
QCD jetqpartonparton shower

hadronized particles

Higgs jethpartonparton shower

hadronized particles

We have to differentiate these non-QCD jets from QCD jets to maximize
sensitivity of channels involving boosted particles.
Jet Substructure !

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Introduction A spectral function of jet substructure Spectral Analysis of Jet Substructure: Higgs Spectral Analysis of Jet Substructure: Sgluon Conclusion

A Catalog of Observables of Jet Substructure
There are lots of observables of jet substructure, which are focussing on
particular substructure. A few examples are:mass of the mother particle:
mjet , trimmed mjet , · · ·n-prong jet:n-subjettiness ratio, D2 , · · ·hvs Z /Wvs h/Z /Wjvssubjet pT asymmetry:mass drop tagger, · · ·hcolor charge:
jet girth, · · ·qvscolor flow:
jet pull between subjets, · · ·

h

vs g

vs

t

j
g

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Introduction A spectral function of jet substructure Spectral Analysis of Jet Substructure: Higgs Spectral Analysis of Jet Substructure: Sgluon Conclusion

A Catalog of Observables of Jet Substructure
There are lots of observables of jet substructure, which are focussing on
particular substructure. A few examples are:mass of the mother particle:
mjet , trimmed mjet , · · ·n-prong jet:n-subjettiness ratio, D2 , · · ·hvs Z /Wvs h/Z /Wjvssubjet pT asymmetry:mass drop tagger, · · ·hcolor charge:
jet girth, · · ·qvscolor flow:
jet pull between subjets, · · ·

h

vs g

vs

t

j
g

Q: is there any generic framework which is highly visual and unifies theseanalyses?
I want to introduce an analogy from
1
H-NMR Analysis in Organic Chemistry.

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Introduction A spectral function of jet substructure Spectral Analysis of Jet Substructure: Higgs Spectral Analysis of Jet Substructure: Sgluon Conclusion

A Spectral Function from
Analogy of Jet Substructures and 1 H-NMR Analysis in Organic Chemistry

H

H

CH2

C

OH

H

OH

Intensity[arb. unit]

The 1 H-NMR spectrum can be used for identifying the topologicalskeleton of an organic molecule, the carbon skeleton, from interaction
between hydrogen nuclei and the rest of molecular structure.
CH3

C

4
H

H

3
2
1
chemical shift [ppm]0Chemical shift (δ): a resonant frequency of precession reletaive to areference in a magnetic field.
Bs : from molecular structureγ-electron cloudν=Beff , Beff = B0 + Bs
2π-nearby hydrogen spin
External: B0
νsample − νrefBeff : nucleus seesδ=νref

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Introduction A spectral function of jet substructure Spectral Analysis of Jet Substructure: Higgs Spectral Analysis of Jet Substructure: Sgluon Conclusion

A Spectral Function from
Analogy of Jet Substructures and 1 H-NMR Analysis in Organic Chemistry

H

H

CH2

C

OH

H

OH

Intensity[arb. unit]

The 1 H-NMR spectrum can be used for identifying the topologicalskeleton of an organic molecule, the carbon skeleton, from interaction
between hydrogen nuclei and the rest of molecular structure.
CH3

C

4
H

H

3
2
1
chemical shift [ppm]0Jets also have a topological skeleton, the originating partons.

bb̄h

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Introduction A spectral function of jet substructure Spectral Analysis of Jet Substructure: Higgs Spectral Analysis of Jet Substructure: Sgluon Conclusion

A Spectral Function from
Analogy of Jet Substructures and 1 H-NMR Analysis in Organic Chemistry

H

H

CH2

C

OH

H

OH

Intensity[arb. unit]

The 1 H-NMR spectrum can be used for identifying the topologicalskeleton of an organic molecule, the carbon skeleton, from interaction
between hydrogen nuclei and the rest of molecular structure.
CH3

C

4
H

H

3
2
1
chemical shift [ppm]0Jets also have a topological skeleton, the originating partons.

bb̄

Can we build a similar analysis framework of jet
substructure to identify the originating partons of
the jet?

h

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Introduction A spectral function of jet substructure Spectral Analysis of Jet Substructure: Higgs Spectral Analysis of Jet Substructure: Sgluon Conclusion

A Spectral Function from
Analogy of Jet Substructures and 1 H-NMR Analysis in Organic Chemistry

H

H

CH2

C
H

CH3

OH

OH

Intensity[arb. unit]

The 1 H-NMR spectrum can be used for identifying the topologicalskeleton of an organic molecule, the carbon skeleton, from interaction
between hydrogen nuclei and the rest of molecular structure.

C

4
H

H

3
2
1
chemical shift [ppm]0Jets also have a topological skeleton, the originating partons.
To measure correlation between jet constituents,
we define an IRC safe binned spectral function ofangular distance R,bb̄h

S2 (R; ∆R)

=

1
∆R

X

pT ,i pT ,j ,i,j∈jetRij ∈[R,R+∆R)

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Introduction A spectral function of jet substructure Spectral Analysis of Jet Substructure: Higgs Spectral Analysis of Jet Substructure: Sgluon Conclusion

A typical Higgs jet
The spectrum contains information on non-local correlation in the jet.

S2 (R)bb̄
h0
S2 (R; ∆R)

=

1
∆R

Rjet
X

2Rjet RpT ,i pT ,ji,j∈jetRij ∈[R,R+∆R)

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Introduction A spectral function of jet substructure Spectral Analysis of Jet Substructure: Higgs Spectral Analysis of Jet Substructure: Sgluon Conclusion

A typical Higgs jet
The spectrum contains information on non-local correlation in the jet.
(pT2 ,b + pT2 ,b̄ )S2 (R)bb̄
h0
S2 (R; ∆R)

=

1
∆R

Rjet
X

2Rjet RpT ,i pT ,ji,j∈jetRij ∈[R,R+∆R)

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Introduction A spectral function of jet substructure Spectral Analysis of Jet Substructure: Higgs Spectral Analysis of Jet Substructure: Sgluon Conclusion

A typical Higgs jet
The spectrum contains information on non-local correlation in the jet.
(pT2 ,b + pT2 ,b̄ )S2 (R)(2pT ,b pT ,b̄ )bsubjet pT :symmetricb̄h0
S2 (R; ∆R)

=

1
∆R

Rbb̄ Rjet
X

2Rjet RpT ,i pT ,ji,j∈jetRij ∈[R,R+∆R)

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Introduction A spectral function of jet substructure Spectral Analysis of Jet Substructure: Higgs Spectral Analysis of Jet Substructure: Sgluon Conclusion

A typical Higgs jet
The spectrum contains information on non-local correlation in the jet.
(pT2 ,b + pT2 ,b̄ )S2 (R)(2pT ,b pT ,b̄ )bsubjet pT :symmetric

b b̄

b̄
h

0
S2 (R; ∆R)

=

1
∆R

Rbb̄ Rjet
X

2Rjet RpT ,i pT ,ji,j∈jetRij ∈[R,R+∆R)

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Introduction A spectral function of jet substructure Spectral Analysis of Jet Substructure: Higgs Spectral Analysis of Jet Substructure: Sgluon Conclusion

A typical Higgs jet
The spectrum contains information on non-local correlation in the jet.
(pT2 ,b + pT2 ,b̄ )S2 (R)

subjet pT :
(2pT ,b pT ,b̄ ) symmetricempty region beyond Rbb̄h: color singlet

b
b b̄

b̄
h

0
S2 (R; ∆R)

=

1
∆R

Rbb̄ Rjet
X

2Rjet RpT ,i pT ,ji,j∈jetRij ∈[R,R+∆R)

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Introduction A spectral function of jet substructure Spectral Analysis of Jet Substructure: Higgs Spectral Analysis of Jet Substructure: Sgluon Conclusion

A typical Higgs jet
The spectrum contains information on non-local correlation in the jet.
(pT2 ,b + pT2 ,b̄ )S2 (R)

subjet pT :
(2pT ,b pT ,b̄ ) symmetricempty region beyond Rbb̄h: color singlet

b
b b̄

b̄
h

0

Rbb̄ Rjet

2Rjet Rp T,jet = 305 GeV
m jet = 120 GeV
m jet, tr = 117 GeV

0

−1

b
−1

normalized S2(R ;0.1)1

Σ i ∈ pixel pT,i [arb. unit]

φ - φ jetThis is a typical Higgs jet from MG5 aMC@NLO+Pythia8+Delphes.
0.6

D2 = 0.24 ph(YD 2) = 99.2%
D2,tr = 0.17 ph(YD 2+ tr ) = 99.6%
ph(Y S2) = 99.9%
ph(YS2+ tr ) = 99.8%

0.4

0.2

g
0

1

η - η jet

00

0.5

1

1.5

2

R

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Introduction A spectral function of jet substructure Spectral Analysis of Jet Substructure: Higgs Spectral Analysis of Jet Substructure: Sgluon Conclusion

A typical QCD jet
Meanwhile, a typical QCD jet has a smoothly falling behavior according to
the parton splitting kernels.

S2 (R)qg
q

0
S2 (R; ∆R)

=

1
∆R

Rjet
X

2Rjet RpT ,i pT ,ji,j∈jetRij ∈[R,R+∆R)

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Introduction A spectral function of jet substructure Spectral Analysis of Jet Substructure: Higgs Spectral Analysis of Jet Substructure: Sgluon Conclusion

A typical QCD jet
Meanwhile, a typical QCD jet has a smoothly falling behavior according to
the parton splitting kernels.
(1 − 2z(1 − z))pT2 ,q/g = O[1]S2 (R)q

2z(1 − z)pT2 ,q/g = O[z]gq0
S2 (R; ∆R)

=

1
∆R

Rjet
X

2Rjet RpT ,i pT ,ji,j∈jetRij ∈[R,R+∆R)

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Introduction A spectral function of jet substructure Spectral Analysis of Jet Substructure: Higgs Spectral Analysis of Jet Substructure: Sgluon Conclusion

A typical QCD jet
Meanwhile, a typical QCD jet has a smoothly falling behavior according to
the parton splitting kernels.
(1 − 2z(1 − z))pT2 ,q/g = O[1]S2 (R)q

2z(1 − z)pT2 ,q/g = O[z]large angle radiation:
making mjet largegq0
S2 (R; ∆R)

=

1
∆R

Rjet
X

2Rjet RpT ,i pT ,ji,j∈jetRij ∈[R,R+∆R)

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Introduction A spectral function of jet substructure Spectral Analysis of Jet Substructure: Higgs Spectral Analysis of Jet Substructure: Sgluon Conclusion

A typical QCD jet
Meanwhile, a typical QCD jet has a smoothly falling behavior according to
the parton splitting kernels.
(1 − 2z(1 − z))pT2 ,q/g = O[1]S2 (R)q

2z(1 − z)pT2 ,q/g = O[z]large angle radiation:
making mjet largegq0
S2 (R; ∆R)

=

1
∆R

Rjet
X

2Rjet RpT ,i pT ,ji,j∈jetRij ∈[R,R+∆R)

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Introduction A spectral function of jet substructure Spectral Analysis of Jet Substructure: Higgs Spectral Analysis of Jet Substructure: Sgluon Conclusion

A typical QCD jet
Meanwhile, a typical QCD jet has a smoothly falling behavior according to
the parton splitting kernels.

S2 (R)

q

hard

soft

g
q
0
S2 (R; ∆R)

=

1
∆R

Rjet
X

2Rjet RpT ,i pT ,ji,j∈jetRij ∈[R,R+∆R)

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Introduction A spectral function of jet substructure Spectral Analysis of Jet Substructure: Higgs Spectral Analysis of Jet Substructure: Sgluon Conclusion

A typical QCD jet
Meanwhile, a typical QCD jet has a smoothly falling behavior according to
the parton splitting kernels.

S2 (R)

q

hard

soft

g
q
0

2Rjet R

Rjet

0

−1

q
−1

0.6

normalized S2,tr (R ;0.1)

p T,jet = 348 GeVm jet = 100 GeV
m jet, tr = 53 GeV

normalized S2(R ;0.1)1

Σ i ∈ pixel pT,i [arb. unit]

φ - φ jetThis is a typical quark jet from MG5 aMC@NLO+Pythia8+Delphes.
D2 = 3.20 ph(YD 2) = 1.6%
D2,tr = 2.45 ph(YD 2+ tr ) = 1.4%
ph(Y S2) = 0.4%
ph(YS2+ tr ) = 0.2%

0.4

0.2

0.6

0.4

0.2

g
0

1

η - η jet

00

0.5

1

1.5

2

R

00

0.5

1

1.5

2

R

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Introduction A spectral function of jet substructure Spectral Analysis of Jet Substructure: Higgs Spectral Analysis of Jet Substructure: Sgluon Conclusion

Intermediate Check: a quick classifier between the Higgs jets and QCD jets
The typical Higgs jets and QCD jets have a different spectrum. Theenergy deposit difference could be used as a quick classifier.
R min[a0 R̂bb̄ ,Rjet ]dR S2 (R)aR̂bb̄
2 · 125 GeV
RS2 = R, R̂bb̄ =RaR̂bb̄
∞pT ,jetdR S2 (R) +dR S2 (R)min[a0 R̂bb̄ ,Rjet ]0

S2 (R)

S2 (R)

R

R
0

Rjet
Higgs jet

2Rjet

0

Rjet
QCD jet2Rjet

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Introduction A spectral function of jet substructure Spectral Analysis of Jet Substructure: Higgs Spectral Analysis of Jet Substructure: Sgluon Conclusion

Intermediate Check: a quick classifier between the Higgs jets and QCD jets
The typical Higgs jets and QCD jets have a different spectrum. Theenergy deposit difference could be used as a quick classifier.
R min[a0 R̂bb̄ ,Rjet ]dR S2 (R)aR̂bb̄
2 · 125 GeV, R̂bb̄ =RS2 = RR∞
aR̂bb̄
pT ,jetdR S2 (R)dR S2 (R) +0min[a0 R̂bb̄ ,Rjet ]Event preselection: mjet ∈ [100, 150] GeV, and pT ,jet ∈ [300, 400] GeV.

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Introduction A spectral function of jet substructure Spectral Analysis of Jet Substructure: Higgs Spectral Analysis of Jet Substructure: Sgluon Conclusion

An Artificial Neural Network?
Nevertheless, the energy deposit ratio of the spectrum is not the only
information available in S2 (R).
We will introduce a neural network to use all the information available in
S2 (R).
An artificial neural network is a mathematical model of a function.
F̂ (xi ) = vj · f (wij · xi + bj )where f is a nonlinear function, for example, we may consider ReLU,(
x x >0fReLU (x) =0 x <= 0
A graphical representation is follows.
Input
Hidden1{xi }(1)

OutputF̂ (xi )1“Universal approximation theorem” states that if we have an enoughnumber of the hidden neurons, which is the hidden dimension of j, thenwe can approximate any bounded continuous functions.

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Introduction A spectral function of jet substructure Spectral Analysis of Jet Substructure: Higgs Spectral Analysis of Jet Substructure: Sgluon Conclusion

Our ANN setup
Hence, we trained a neural network for the Higgs jet and QCD jet
classification, and used the network as a classifier.
(0)

=

xi

(n)

=

ŷi

=

fReLU (wij z (n−1) + bi )
i
i X
h
h
(5) (4)(5) (4)/
exp wij zj
exp wij zj

zi
zi

(n)

(n)

i

This network is trained for predicting a class vector((1, 0) Higgs jet~
y=(0, 1) QCD jet
by minimizing the categorical cross entropyXXL=−yi log ŷi
event

i

The predicted class vector ŷ , which is the output of the neural network,
can be used as a classifier.

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Introduction A spectral function of jet substructure Spectral Analysis of Jet Substructure: Higgs Spectral Analysis of Jet Substructure: Sgluon Conclusion

Comparing S2 (R) and D2

To compare the information
contained in S2 (R) and D2 , wecompare the ROC curves of neuralnetworks with folowing inputs.
ND2 : {pT ,jet , mjet , D2β=2 }NS2 : {pT ,jet , mjet ,S2 (0; 0.1), · · · , S2 (1.9; 0.1)}+tr : trimmed variables areincluded.
Q: what’s the source of thedifference?
Reduce mistagging
Going beyond two-prong

Event preselection: mjet ∈ [100, 150] GeV, and pT ,jet ∈ [300, 400] GeV.

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Introduction A spectral function of jet substructure Spectral Analysis of Jet Substructure: Higgs Spectral Analysis of Jet Substructure: Sgluon Conclusion

A jet tagged in D2 analysis only
S2 (R) subjet pT : asymmetricqgq

p T,jet = 364 GeV
m jet = 112 GeVm jet, tr = 106 GeV

0

−1

q
−1

Rjet
normalized S2(R ;0.1)1

Σ i ∈ pixel pT,i [arb. unit]

φ - φ jet

00.6

2Rjet R

D2 = 0.69 ph(YD 2) = 76.7%
D2,tr = 0.15 ph(YD 2+ tr ) = 85.2%
ph(Y S2) = 24.2%
ph(YS2+ tr ) = 33.3%

0.4

0.2

g
0

1

η - η jet

00

0.5

1

1.5

2

R

ND2 : {pT ,jet , mjet , D2β=2 }Even though this jet is two-prong, the likelihood for QCD jet is not that
small.

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Introduction A spectral function of jet substructure Spectral Analysis of Jet Substructure: Higgs Spectral Analysis of Jet Substructure: Sgluon Conclusion

A jet tagged in S2 analysis with trimming only
subjet pT : symmetric

S2 (R)

moderate contaminationsbRbb̄

large R FSRb̄

p T,jet = 329 GeV
m jet = 147 GeV
m jet, tr = 130 GeV

0

−1

b
−1

normalized S2(R ;0.1)1

Σ i ∈ pixel pT,i [arb. unit]

φ - φ jet

0
0.6

Rjet

D2 = 2.28 ph(YD 2) = 7.7%
D2,tr = 2.38 ph(YD 2+ tr ) = 17.0%
ph(Y S2) = 18.6%
ph(YS2+ tr ) = 82.2%

0.4

0.2

2Rjet R
normalized S2,tr (R ;0.1)

h

0.6

0.4

0.2

g
0

1

η - η jet

00

0.5

1

1.5

2

R

00

0.5

1

1.5

2

R

This jet has two-prong substructure, but the subjets are wide andcontaminated by other QCD activities.
S2 (R) together with jet trimming and a neural network can identify hardand soft substructure.

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Introduction A spectral function of jet substructure Spectral Analysis of Jet Substructure: Higgs Spectral Analysis of Jet Substructure: Sgluon Conclusion

Color charge of the boosted Higgs jets

Preliminary

S2 (R) shows a good performance on identifying boosted Higgs jets fromQCD jets.
Is it possible to identify color charge of the Higgs jet from S2 (R)?
We considered a color octet scalar, sgluon, decaying to b b̄ and did a
multi-class classification with a neural network ŷ predicting(1, 0, 0) Higgs jet~
y = (0, 1, 0) Sgluon jet
(0, 0, 1) QCD jet

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Introduction A spectral function of jet substructure Spectral Analysis of Jet Substructure: Higgs Spectral Analysis of Jet Substructure: Sgluon Conclusion

A typical Sgluon jet

Preliminary

The sgluon jet has a different radiation pattern compared to the Higgs jet
because of its nontrivial color charge.
(pT2 ,b + pT2 ,b̄ )S2 (R)(2pT ,b pT ,b̄ )bsubjet pT :symmetricb̄
sgluon0
S2 (R; ∆R)

=

1
∆R

X

Rbb̄ Rjet

2Rjet RpT ,i pT ,ji,j∈jetRij ∈[R,R+∆R)

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Introduction A spectral function of jet substructure Spectral Analysis of Jet Substructure: Higgs Spectral Analysis of Jet Substructure: Sgluon Conclusion

A typical Sgluon jet

Preliminary

The sgluon jet has a different radiation pattern compared to the Higgs jet
because of its nontrivial color charge.
(pT2 ,b + pT2 ,b̄ )S2 (R)(2pT ,b pT ,b̄ )bsubjet pT :symmetriccolor flowb̄
sgluon0
S2 (R; ∆R)

=

1
∆R

X

Rbb̄ Rjet

2Rjet RpT ,i pT ,ji,j∈jetRij ∈[R,R+∆R)

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Introduction A spectral function of jet substructure Spectral Analysis of Jet Substructure: Higgs Spectral Analysis of Jet Substructure: Sgluon Conclusion

A typical Sgluon jet

Preliminary

The sgluon jet has a different radiation pattern compared to the Higgs jet
because of its nontrivial color charge.
(pT2 ,b + pT2 ,b̄ )S2 (R)(2pT ,b pT ,b̄ )bsubjet pT :symmetriccolor flowb̄
sgluon0

Rbb̄ Rjet

2Rjet R

Preliminary
p T,jet = 336 GeV
m jet = 145 GeV
m jet, tr = 111 GeV

0

−1

b
−1

normalized S2(R ;0.1)1

Σ i ∈ pixel pT,i [arb. unit]

φ - φ jetThis is a typical sgluon jet from MG5 aMC@NLO+Pythia8+Delphes.
Preliminary ph(YD 2+ tr ) = 16.4%
D2 = 1.31 likelihoods from S2
D2,tr = 0.63 ph(h vs j ) = 44.0%
ph(h vs sg) = 13.9%
psg(sg vs j ) = 82.4%
0.4
0.6

0.2

g
0

1

00

0.5

1

1.5

2

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Introduction A spectral function of jet substructure Spectral Analysis of Jet Substructure: Higgs Spectral Analysis of Jet Substructure: Sgluon Conclusion

A typical Sgluon jet

Preliminary

The sgluon jet has a different radiation pattern compared to the Higgs jet
because of its nontrivial color charge.
(pT2 ,b + pT2 ,b̄ )subjet pT :
(2pT ,b pT ,b̄ ) symmetriclarge angle radiations
color octet

S2 (R)bb̄
sgluon0

Rbb̄ Rjet

2Rjet R

Preliminary
p T,jet = 336 GeV
m jet = 145 GeV
m jet, tr = 111 GeV

0

−1

b
−1

normalized S2(R ;0.1)1

Σ i ∈ pixel pT,i [arb. unit]

φ - φ jetThis is a typical sgluon jet from MG5 aMC@NLO+Pythia8+Delphes.
Preliminary ph(YD 2+ tr ) = 16.4%
D2 = 1.31 likelihoods from S2
D2,tr = 0.63 ph(h vs j ) = 44.0%
ph(h vs sg) = 13.9%
psg(sg vs j ) = 82.4%
0.4
0.6

0.2

g
0

1

00

0.5

1

1.5

2

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Introduction A spectral function of jet substructure Spectral Analysis of Jet Substructure: Higgs Spectral Analysis of Jet Substructure: Sgluon Conclusion

A typical Sgluon jet

Preliminary

The sgluon jet has a different radiation pattern compared to the Higgs jet
because of its nontrivial color charge.
(pT2 ,b + pT2 ,b̄ )subjet pT :
(2pT ,b pT ,b̄ ) symmetriclarge angle radiations
color octet

S2 (R)bb̄
sgluon0

Rbb̄ Rjet

2Rjet R

Preliminary
p T,jet = 384 GeV
m jet = 138 GeVm jet, tr = 108 GeV

0

−1

b
−1

normalized S2(R ;0.1)1

Σ i ∈ pixel pT,i [arb. unit]

φ - φ jetThis is a typical sgluon jet from MG5 aMC@NLO+Pythia8+Delphes.
Preliminary ph(YD 2+ tr ) = 19.9%
D2 = 1.97 likelihoods from S2
D2,tr = 0.66 ph(h vs j ) = 29.3%
ph(h vs sg) = 3.6%
psg(sg vs j ) = 73.4%
0.4
0.6

0.2

g
0

1

00

0.5

1

1.5

2

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Introduction A spectral function of jet substructure Spectral Analysis of Jet Substructure: Higgs Spectral Analysis of Jet Substructure: Sgluon Conclusion

Conclusion
We have introduced a spectral analysis of jet substructure for classifying
Higgs jets and QCD jets.
The spetrum S2 (R) is highly visual and useful in describing jet
substructure with large angular scale.
The improvement of performance of Higgs jet vs QCD jet classification is
not large, but the spectral function S2 (R) provide us a systematicalframework for investigating jet substructure over various angular scale.
For example, S2 (R) can be used for understanding a color charge ofboosted jets.
(pT2 ,b + pT2 ,b̄ )S2 (R)b b̄
0(pT2 ,b + pT2 ,b̄ )subjet pT :S2 (R)(2pT ,b pT ,b̄ ) symmetricempty region beyond Rbb̄h: color singlet

Rbb̄ Rjet

2Rjet R

subjet pT :
(2pT ,b pT ,b̄ ) symmetriclarge angle radiations
color octet
color flow0

Rbb̄ Rjet

2Rjet R

There are more wider applications will be come out soon, so please staytuned!

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Backup

Backups

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Backup

IRC safety of S2 (R)The S2 (R) spectrum is better to be infrared and collinear (IRC) safe,
namely invariant under soft and collinear radiations.
Soft radiation: the parton radiates a pT = 0 parton.
pT
pT
pT = 0~Collinear radiation: the parton splits in same direction R.
zpT
pT(1 − z)pT

Otherwise, the virtual and real corrections in higher order perturbation
theory do not sum up, and such a IRC unsafe observables are hard to be
estimated from perturbative QCD calculations. (KNL theorem)++= finite(2)

S2 (R) is IRC safe because soft or collinear radiation does not introduce
any new angular scale R.

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Backup

IRC safety of S2 (R)The S2 (R) spectrum is IRC safe.
S2 (R; ∆R)

=

1
∆R

X

pT ,i pT ,j ,(3)i,j∈jetRij ∈[R,R+∆R)

Soft radiation is ignored safely because pT ,i pT ,j = 0 if i or j is a softradiation.
S2 (R)
0
b
h
b̄

Rbb̄
R

~ does not change and the radiation
Collinear radation is okay because R
products always sum up.
b

0

b̄

Rbb̄

S2 (R)

hR

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BackupComparison to another variable

For comparison, we prepare an ANN anlysis with D2 variable (1409.6298)
e2β

=

e3β

=

D2β

=

1

X

pT2 ,jet i,j∈jeti<j1

pT ,i pT ,j Rijβ ,(4)

pT ,i pT ,j pT ,k Rijβ Rjkβ Rkiβ ,(5)X

pT3 ,jet i,j,k∈jet
i<j<k

e3β
4
∼
,
(−)3
(e2β )3(6)

For Higgs jets, D2 is small because three-point energy correlation have tocount soft or collinear radiation.
b
≈
1
D2β ≈
h
(−)3
b̄

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Backup

Preliminary
p T,jet = 328 GeVm jet = 126 GeV
m jet, tr = 123 GeV

0

−1

b
−1

normalized S2(R ;0.1)1PreliminaryΣ i ∈ pixel pT,i [arb. unit]

φ - φ jet

Higgs jet and color octet classifier

Preliminary ph(YD 2+ tr ) = 97.6%
D2 = 0.31 likelihoods from S2
D2,tr = 0.28 ph(h vs j ) = 99.1%
ph(h vs sg) = 98.3%
psg(sg vs j ) = 95.8%
0.4
0.6

0.2

g
0

1

η - η jet

00

0.5

1

1.5

2

R

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Backup

Preliminary
p T,jet = 336 GeV
m jet = 149 GeV
m jet, tr = 129 GeV

0

−1

b
−1

normalized S2(R ;0.1)1

Σ i ∈ pixel pT,i [arb. unit]

Preliminary

φ - φ jetsgluon!

Preliminary ph(YD 2+ tr ) = 21.4%
D2 = 0.97 likelihoods from S2
D2,tr = 0.62 ph(h vs j ) = 24.7%
ph(h vs sg) = 0.6%
psg(sg vs j ) = 77.2%
0.4
0.6

0.2

g
0

1

η - η jet

00

0.5

1

1.5

2

R

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