a theory of quark vs. gluon...

A Theory of Quark vs.

Gluon DiscriminationAndrew Larkoski

Reed College

with Eric Metodiev 1906.01639

BOOST 2019, July 24, 2019

2

So, what is a jet?

Experimental Representation

{pi}i2JJ

N particles 3N - 4 unique real numbers

3

So, what is a jet?

Experimental Representation

{pi}i2JJ


Abstract Representation


~f({pi})

x

y

z

xiyi

zi

additional dimensions suppressed

4

So, what is a jet?

Examples of mappings

{pi}i2JJ

pixels

{pi}i2JJ

Energy Flow Polynomials

{pi}i2JJ

clustering history

de Oliveira, Kagan, Mackey, Nachman, Schwartzman 2015

Komiske, Metodiev, Thaler 2017

Louppe, Cho, Becot, Cranmer 2017

5

So, what is a jet?

Universal Approximation TheoremAll mappings are equal, but some are more equal than others.

Choose the representation that works for your problem

Theory of Quark vs. Gluon Jet Discrimination

eliminates four-vectors, pixels, clustering history, etc.

Want infrared and collinear safe representation

6

So, what is a jet?

Universal Approximation TheoremAll mappings are equal, but some are more equal than others.

Choose the representation that works for your problem

Theory of Quark vs. Gluon Jet Discrimination

Want infrared and collinear safe representation

Want simple, additive all-orders properties

eN eN+1 eN + es

= +

7

So, what is a jet?

N-subjettinesses and related observables accomplish this

{pi}i2JJ

N-subjettiness (also EFPs, ECFs,…)

Datta, AJL 2017

history:Thaler, van Tilburg, 2010, 2011Stewart, Tackmann, Waalewijn 2010Brandt, Dahmen 1979Wu, Zobernig 1979Nachtmann, Reiter 1982

⌧1

⌧2

⌧3

⌧ (�)N =1

pTJ

X

i2J

pTi minnR�

1i, R�2i, . . . , R

�Ni

o

8

Where do jets live?

For visualization simplicity, just consider (𝜏1,𝜏2)

⌧1

⌧2

9

Where do jets live?


⌧1

⌧2

IRC safety + additivity = exponential suppression

Exponentially small probability in regions where ⌧N ! 0

Particle production as Poisson process

10

Where do jets live?


⌧1

⌧2

/ CA

/ CF < CA

gluon jet⌧gluonN > ⌧quarkN

Additivity then implies

Exponentially more likely to be quark than gluon here

quark jet

11

Where do jets live?


⌧1

⌧2

Arbitrarily pure sample of quark jets for ⌧N ! 0

Quark “reducibility factor” q = 0


12

Where do jets live?


⌧1

⌧2

Can also determine best gluon purity

Argument is more subtle; details in our paper

More likely to be gluon than quark here

13

Where do gluon jets live?

Simplified argument:

For all 𝜏N ~ 1, probabilities described at fixed-order:

pg({⌧N ⇠ 1}) ' (↵sCA)Nfg({⌧N})

pq({⌧N ⇠ 1}) ' (↵sCF )Nfq({⌧N})

No non-analytic structure where 𝜏N ~ 1

Quark likelihood = Gluon “reducibility factor” = g ⇠✓CF

CA

◆N

14

Where do jets live?


⌧1

⌧2

Gluon jets are always contaminated by some quark jets

In practice small because CF/CA ~ 0.44

Resolving only 6 emissions:✓CF

CA

◆�6

< 1%



15

One final observation

What is the optimal quark versus gluon discriminant?

⌧1

⌧2

Ans: Jerzy Neyman and Egon Pearson say likelihood ratioNeyman, Pearson 1933

L =pg({⌧N})pq({⌧N})



16


What are the properties of the likelihood ratio?

⌧1

⌧2

Ans: No clue. Some nasty function of {𝜏N}AJL BOOST 2019

L =pg({⌧N})pq({⌧N})




17


What are the properties of the likelihood ratio?

⌧1

⌧2 More likely to be gluon than quark here

Ans: As any 𝜏N → 0, likelihood vanishes

𝜏N → 0 is the fixed-order divergent limit

Quark vs. Gluon likelihood ratio is IRC safe!

L =pg({⌧N})pq({⌧N}) ! 0

L =pg({⌧N})pq({⌧N}) !

✓CA

CF

◆N

18

IRC Safety of the Likelihood

Consequences

L(q, g)

GeneralIndependent of number of

resolved emissions N

quar

k

gluon

Non-vacuousPronginess discriminators

(D2, 𝜏2,1, 𝜏3,2, …) all IRC unsafe

PracticalIRC safe observables are good q/g discriminants out of the box

good 𝜏N discrimination well-knownGallicchio, Schwartz 2012

Caveat EmptorDoes not mean that likelihood

can be calculated at fixed-order

19

Conclusions

Three results from simple considerations:

Can always purify quark jet sample

⌧1

⌧2

L =pg({⌧N})pq({⌧N}) ! 0

L =pg({⌧N})pq({⌧N}) !

✓CA

CF

◆N

Gluons contaminated by (CF/CA)N quarks

Quark/gluon likelihood ratio is IRC safe

Moral: Choice of jet representation matters for understanding!

20

Other results

Many other results presented in our paper:

Explicit calculations up through 𝛼s3

Derivation of quantitative performance bounds

Relationship between multiplicity and 𝜏N

0 1 2 3 4 5N

0.0

0.1

0.2

0.3

0.4

0.5

Quar

kvs

.G

luon

AU

C

Multiplicity

N-subjettiness: ⌧ (�)N

Pythia 8.226,p

s = 14 TeV

R = 0.4, pT 2 [1000, 1100] GeV

� = 0.5

� = 1.0

� = 2.0

0.90 0.92 0.94 0.96 0.98 1.00Quark Jet Signal E�ciency

0.86

0.88

0.90

0.92

0.94

0.96

0.98

1.00

Glu

onJe

tB

ackg

round

Rej

ection

N-subjettiness DNN, � = 2.0

Pythia 8.226,p

s = 14 TeV

R = 0.4, pT 2 [1000, 1100] GeV

DNN(⌧1)

DNN(⌧1, ⌧2)

DNN(⌧1, ⌧2, ⌧3)

DNN(⌧1, ⌧2, ⌧3, ⌧4)

1

B

1

1

S

Signal Fraction

Backg

roundFraction

Validation in simulation

1

�0

d3�C3

Fq

d⌧1 d⌧2 d⌧3=

⇣2↵s

⇡

⌘3C3

F

Z 1

0

dz1z1

Z 1

0

d✓1✓1

Z 1

0

dz2z2

Z ✓1

0

d✓2✓2

Z 1

0

dz3z3

Z ✓2

0

d✓3✓3

⇥ �(⌧1 � z1✓1)�(⌧2 � z2✓2)�(⌧3 � z3✓3)

=⇣2↵s

⇡

⌘3 1

2

C3F

⌧1⌧2⌧3

✓1

3log3

⌧3⌧1

� 1

3log3 ⌧3 + log ⌧1 log

2 ⌧2⌧3

◆

�1.0 �0.5 0.0 0.5 1.0↵

0.16

0.18

0.20

0.22

0.24

0.26

0.28

0.30

0.32

0.34

0.36

Quar

kvs

.G

luon

AU

C

N-subjettiness: ⌧ (�)↵1 ⌧ (�)

2

Pythia 8.226,p

s = 14 TeV

R = 0.4, pT 2 [1000, 1100] GeV

� = 0.5

� = 1.0

� = 2.0

21

Now Available!

Bonus Slides

22

Reducibility Factors for QCD vs. Z jets

23

d�Z(⌧(2)1 )

d⌧ (2)2

= 2↵s

⇡

CF

⌧ (2)2

log⌧ (2)1

⌧ (2)2

d�q(⌧(2)1 )

d⌧ (2)2

= �↵s

⇡

1

⌧ (2)2

"CF

2log ⌧ (2)1 + (CF + CA) log

⌧ (2)2

⌧ (2)1

#

q = �2CF log ⌧ (2)

1

⌧ (2)2

CF2 log ⌧ (2)1 + (CF + CA) log

⌧ (2)2

⌧ (2)1

��⌧ (2)2 !⌧ (2)

1

= 0

Form of Calculation

log1

z

log1

✓log

1

⌧1log

1

⌧2

log1

⌧1

log1

⌧2

log1

z

log1

✓log

1

⌧1log

1

⌧2

log1

⌧1

log1

⌧2

p(⌧1, ⌧2) =

Zdz1 p(⌧1) p(z1|⌧1) p(⌧2|⌧1, z1)

24

AreaCF =↵s

⇡CF

✓log2

⌧2⌧1

+ 2 log z1 log⌧2⌧1

◆AreaCA =

↵s

⇡CA log2

⌧2⌧1

Robust bound on AUC

1

B

1

1

S

Signal Fraction

Backg

roundFraction

AUC � S + B � 2SB

2� 2SB

25

26

Gluon Reducibility Factor in Simulation

0.90 0.92 0.94 0.96 0.98 1.00Quark Jet Signal E�ciency

0.86

0.88

0.90

0.92

0.94

0.96

0.98

1.00

Glu

onJe

tB

ackg

round

Rej

ection

N-subjettiness DNN, � = 2.0

Pythia 8.226,p

s = 14 TeV

R = 0.4, pT 2 [1000, 1100] GeV

DNN(⌧1)

DNN(⌧1, ⌧2)

DNN(⌧1, ⌧2, ⌧3)

DNN(⌧1, ⌧2, ⌧3, ⌧4)

Slope of ROC compared to predicted (CA/CF)N

Relationship to Multiplicity

0 10 20 30 40 50 60 70 80 90 100N

0.0

0.1

0.2

0.3

0.4

0.5

Quar

kvs

.G

luon

AU

C

Multiplicity


Pythia 8.226,p

s = 14 TeV

R = 0.4, pT 2 [1000, 1100] GeV

� = 0.5

� = 1.0

� = 2.0

0 1 2 3 4 5N

0.0

0.1

0.2

0.3

0.4

0.5

Quar

kvs

.G

luon

AU

C

Multiplicity


Pythia 8.226,p

s = 14 TeV

R = 0.4, pT 2 [1000, 1100] GeV

� = 0.5

� = 1.0

� = 2.0

27

0 25 50 75 100 125 150Constituent Multiplicity

0.000

0.005

0.010

0.015

0.020

0.025

Cro

ssSec

tion

(Nor

mal

ized

)

Constituent Multiplicity

Pythia 8.226,p

s = 14 TeV

R = 0.4, pT 2 [1000, 1100] GeV

Quarks

Gluons

a theory of quark vs. gluon...

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