Towards an understanding of jet substructure · nose experimentally, ... on signal jets and how to optimize the parameters of a method k nown as ﬁltering [1]. Ref. [13] discussed

Towards an understanding of

jet substructure

1

Gavin Salam (CERN)with Mrinal Dasgupta, Alessandro Fregoso & Simone Marzani

Boost 2013Flagstaff, Arizona

August 2013

Gavin Salam (CERN) Towards an understanding of jet substructure Boost 2013, Flagstaff, August 2013

As a field we’ve devised O(10-20) powerful methods to tag jet substructure.

Many of the methods have been tried out in searches and work; these kinds of methods will be crucial for

searches in the years to come.

2


As a field we’ve devised O(10-20) powerful methods to tag jet substructure.

Many of the methods have been tried out in searches and work; these kinds of methods will be crucial for

searches in the years to come.

2

But from outside, the many methods make the field look pretty confusing.

And from inside, I get the impression we don’t always know why or how the methods work – which is bad if we’re looking for

robustness.

Is it time to get back to basics?


What do we know currently?

At the time:• No clear picture of why the taggers might be similar or

different• No clear picture of how the parameter choices affect

the taggers

3

Boost 2010 proceedings:

1. Introduction

The Large Hadron Collider (LHC) at CERN is increasingly exploring phenomena at ener-

gies far above the electroweak scale. One of the features of this exploration is that analysis

techniques developed for earlier colliders, in which electroweak-scale particles could be con-

sidered “heavy”, have to be fundamentally reconsidered at the LHC. In particular, in the

context of jet-related studies, the large boost of electroweak bosons and top quarks causes

their hadronic decays to become collimated inside a single jet. Consequently a vibrant

research field has emerged in recent years, investigating how best to tag the characteristic

substructure that appears inside the single “fat” jets from electroweak scale objects, as

reviewed in Refs. [?,?,26]. In parallel, the methods that have been developed have started

to be tested and applied in numerous experimental analyses (e.g. [23–25] for studies on

QCD jets and [some searches]).

The taggers’ action is twofold: they aim to suppress or reshape backgrounds, while re-

taining signal jets and enhancing their characteristic jet-mass peak at the W/Z/H/top/etc.

mass. Nearly all the discussion of these aspects has taken place in the context of Monte

Carlo simulation studies [Some list], with tools such as Herwig [?, ?], Pythia [?, ?] and

Sherpa [?]. While Monte Carlo simulation is an extremely powerful tool, its intrinsic nu-

merical nature can make it di!cult to extract the key characteristics of individual taggers

and the relations between taggers (examining appropriate variables, as in [4], can be helpful

in this respect). As an example of the kind of statements that exist about them in the

literature, we quote from the Boost 2010 proceedings:

The [Monte Carlo] findings discussed above indicate that while [pruning,

trimming and filtering] have qualitatively similar e"ects, there are important

di"erences. For our choice of parameters, pruning acts most aggressively on the

signal and background followed by trimming and filtering.

While true, this brings no insight about whether the di"erences are due to intrinsic proper-

ties of the taggers or instead due to the particular parameters that were chosen; nor does it

allow one to understand whether any di"erences are generic, or restricted to some specific

kinematic range, e.g. in jet transverse momentum. Furthermore there can be significant

di"erences between Monte Carlo simulation tools (see e.g. [22]), which may be hard to diag-

nose experimentally, because of the many kinds of physics e"ect that contribute to the jet

structure (final-state showering, initial-state showering, underlying event, hadronisation,

etc.). Overall, this points to a need to carry out analytical calculations to understand the

interplay between the taggers and the quantum chromodynamical (QCD) showering that

occurs in both signal and background jets.

So far there have been three investigations into the analytical features that emerge from

substructure taggers. Ref. [19, 20] investigated the mass resolution that can be obtained

on signal jets and how to optimize the parameters of a method known as filtering [1].

Ref. [13] discussed constraints that might arise if one is to apply Soft Collinear E"ective

Theory (SCET) to jet substructure calculations. Ref. [14] observed that for narrow jets the

distribution of the N -subjettiness shape variable for 2-body signal decays can be resummed

– 2 –


This talk → analytical understanding. Why?

4

Better Insight

Can guide taggers’ use in experimental

analyses

It may help us design better taggers


This talk → analytical understanding. Why?

5

Better Insight

Can guide taggers’ use in experimental

analyses

It may help us design better taggers

Robustness

You know what you predict, what you don’t

Unlike MC, you have powerful handles for

cross-checks & accuracy estimates

There is a “right” answer

Gavin Salam (CERN) Towards an understanding of jet substructure Boost 2013, Flagstaff, August 2013 6

Scope of our study


What kind of events?

7

To fully understand “Boost” you want to study all possible signal (W/Z/H/top/…) and QCD jets.

But you need to start somewhere.We chose the QCD jets because:

(a) they have the richest structure.

(b) once you know understand the QCD jets,the route for understanding signal jets becomes clear too.


study 3 taggers/groomers

8

Mass-drop tagger (MDT, aka BDRS)

Trimming

Pruning

Cannot possibly study all toolsThese 3 are widely used

Recluster

on scale Rsub

discard subjetswith < zcut pt

decluster &

discard soft junkrepeat until

find hard struct

jet mass/ptsets Rprune discard large-angle

soft clusteringsRecluster


First use MC to get a panorama


The key variables

10

For phenomenology

Jet mass: m

[as compared to W/Z/Hor top mass]

For QCD calculations

[R is jet opening angle– or radius]

Because ρ is invariant underboosts along jet direction

⇢ =m2

p2tR2⇢ =

m2

p2tR2⇢ =

m2

p2tR2


!/"

d"

/ d

!

! = m2/(pt2 R2)

quark jets (Pythia 6 MC)

m [GeV], for pt = 3 TeV, R=1

Jets: C

/A w

ith R

=1. M

C: P

ythia

6.4

, DW

tune, p

arto

n-le

vel (n

o M

PI), q

q#

qq, p

t > 3

TeV

plain jet mass

0

0.1

0.2

0.3

10-6 10-4 0.01 0.1 1

10 100 1000

11

Start with “plain” jet mass


!/"

d"

/ d

!

! = m2/(pt2 R2)



Jets: C

/A w

ith R

=1. M

C: P

ythia

6.4

, DW

tune, p

arto

n-le

vel (n

o M

PI), q

q#

qq, p

t > 3

TeV

plain jet mass

0

0.1

0.2

0.3

10-6 10-4 0.01 0.1 1

10 100 1000

11

Physical mass for 3 TeV, R=1 jets

ρ ~ Rescaled mass2

(i.e. the QCD variable)



mEW

!/"

d"

/ d

!

! = m2/(pt2 R2)



Jets: C

/A w

ith R

=1. M

C: P

ythia

6.4

, DW

tune, p

arto

n-le

vel (n

o M

PI), q

q#

qq, p

t > 3

TeV

plain jet mass

0

0.1

0.2

0.3

10-6 10-4 0.01 0.1 1

10 100 1000

11

Physical mass for 3 TeV, R=1 jets

ρ ~ Rescaled mass2

(i.e. the QCD variable)



mEW

!/"

d"

/ d

!

! = m2/(pt2 R2)



Jets: C

/A w

ith R

=1. M

C: P

ythia

6.4

, DW

tune, p

arto

n-le

vel (n

o M

PI), q

q#

qq, p

t > 3

TeV

plain jet mass

0

0.1

0.2

0.3

10-6 10-4 0.01 0.1 1

10 100 1000

12

Quark and gluonjets are different,so we treat themseparately



mEW

13

!/"

d"

/ d

!

! = m2/(pt2 R2)

gluon jets (Pythia 6 MC)


Jets: C

/A w

ith R

=1. M

C: P

ythia

6.4

, DW

tune, p

arto

n-le

vel (n

o M

PI), q

q#

qq, p

t > 3

TeV

plain jet mass

0

0.1

0.2

0.3

10-6 10-4 0.01 0.1 1

10 100 1000

Quark and gluonjets are different,so we treat themseparately



!/"

d"

/ d

!

! = m2/(pt2 R2)



Jets: C

/A w

ith R

=1. M

C: P

ythia

6.4

, DW

tune, p

arto

n-le

vel (n

o M

PI), q

q#

qq, p

t > 3

TeV

plain jet mass

Trimmer (zcut=0.05, Rsub=0.3)

Pruner (zcut=0.1)

MDT (ycut=0.09, µ=0.67)

0

0.1

0.2

0.3

10-6 10-4 0.01 0.1 1

10 100 1000

Different taggerscan be

quite similar

Now examine “taggers/groomers”


!/"

d"

/ d

!

! = m2/(pt2 R2)



Jets: C

/A w

ith R

=1. M

C: P

ythia

6.4

, DW

tune, p

arto

n-le

vel (n

o M

PI), q

q#

qq, p

t > 3

TeV

plain jet mass


Pruner (zcut=0.1)

MDT (ycut=0.09, µ=0.67)

0

0.1

0.2

0.3

10-6 10-4 0.01 0.1 1

10 100 1000

But only for a limited range

of masses

Now examine “taggers/groomers”


!/"

d"

/ d

!

! = m2/(pt2 R2)



Jets: C

/A w

ith R

=1. M

C: P

ythia

6.4

, DW

tune, p

arto

n-le

vel (n

o M

PI), q

q#

qq, p

t > 3

TeV


0

0.1

0.2

0.3

10-6 10-4 0.01 0.1 1

10 100 1000

What do we want to find out?

Where exactly are the kinks? How do their locations depend

on zcut, Rsub?Kinks are especially dangerous for data-driver backgrounds

What physics is relevant in the different regions?

Because then you have an idea of how well you

control itAnd maybe you can make better taggers


The parameters & approximations

17

TrimmingTake all particles in a

jet of radius R and recluster them into subjets with a jet

definition with radius Rsub < R

The subjets that satisfy the condition

pt(subjet) > zcut pt(jet) are kept and merged to

form the trimmed jet.

Krohn, Thaler & Wang ’09

Our approximations

• ρ ≪ 1logs of ρ get resummed

• pretend R ≪ 1• zcut ≪ 1,

but (log zcut) not large

These approximations are not as “wild”as they might sound.

They can also be relaxed. But our aim for now is to

understand the taggers — we leave highest precision

calculations till later.


Leading Order — 2-body kinematic plane

18

At O(αs), a quark jet emits a gluon. We study thisas a function of the gluon momentum fraction zand the quark-gluon opening angle θ

z

sub

Rsub

(1!z)

R

!

R

0.01

0.1

1 0.1 1

both particles in 1 subjet 2 subjets

z

�qgRsub


Leading Order — 2-body kinematic plane

19

z

sub

Rsub

(1!z)

R

!

R

0.01

0.1

1 0.1 1


z

�qgRsub

zcut

trimmed away

At O(αs), a quark jet emits a gluon. We study thisas a function of the gluon momentum fraction zand the quark-gluon opening angle θ


0.01

0.1

1 0.1 1


z

�qgRsub

zcut

trimmed away

↵sCF

⇡

d✓2

✓2dz

z

matrix element

emission probability ~ constantin log θ – log z plane


0.01

0.1

1 0.1 1


z

�qgRsub

zcut

� = 0.200

trimmed away

jet mass↵sCF

⇡

d✓2

✓2dz

z

matrix element


length of fixed-ρ contour givesLO differential cross section

⇢ = z(1� z)✓2


0.01

0.1

1 0.1 1


z

�qgRsub

zcut

� = 0.200

trimmed away

jet mass↵sCF

⇡

d✓2

✓2dz

z

matrix element


length of fixed-ρ contour givesLO differential cross section

⇢ = z(1� z)✓2

!/"

d"

/ d

!

! = m2/(pt2 R2)

trimmed quark jets: LO


0

0.1

0.2

10-6 10-4 0.01 0.1 1

10 100 1000


0.01

0.1

1 0.1 1


z

�qgRsub

zcut

� = 0.150

trimmed away

jet mass↵sCF

⇡

d✓2

✓2dz

z

matrix element


length of fixed-ρ contour ~LO differential cross section

⇢ = z(1� z)✓2

!/"

d"

/ d

!

! = m2/(pt2 R2)



0

0.1

0.2

10-6 10-4 0.01 0.1 1

10 100 1000


0.01

0.1

1 0.1 1


z

�qgRsub

zcut

� = 0.100

trimmed away

jet mass↵sCF

⇡

d✓2

✓2dz

z

matrix element



⇢ = z(1� z)✓2

!/"

d"

/ d

!

! = m2/(pt2 R2)



0

0.1

0.2

10-6 10-4 0.01 0.1 1

10 100 1000


0.01

0.1

1 0.1 1


z

�qgRsub

zcut

� = 0.030

trimmed away

jet mass↵sCF

⇡

d✓2

✓2dz

z

matrix element



⇢ = z(1� z)✓2

!/"

d"

/ d

!

! = m2/(pt2 R2)



0

0.1

0.2

10-6 10-4 0.01 0.1 1

10 100 1000


0.01

0.1

1 0.1 1


z

�qgRsub

zcut

� = 0.009

trimmed away

jet mass↵sCF

⇡

d✓2

✓2dz

z

matrix element



⇢ = z(1� z)✓2

!/"

d"

/ d

!

! = m2/(pt2 R2)



0

0.1

0.2

10-6 10-4 0.01 0.1 1

10 100 1000


0.01

0.1

1 0.1 1


z

�qgRsub

zcut

� = 0.003

trimmed away

jet mass↵sCF

⇡

d✓2

✓2dz

z

matrix element



⇢ = z(1� z)✓2

!/"

d"

/ d

!

! = m2/(pt2 R2)



0

0.1

0.2

10-6 10-4 0.01 0.1 1

10 100 1000


0.01

0.1

1 0.1 1


z

�qgRsub

zcut

� = 0.001

trimmed away

jet mass↵sCF

⇡

d✓2

✓2dz

z

matrix element



⇢ = z(1� z)✓2

!/"

d"

/ d

!

! = m2/(pt2 R2)



0

0.1

0.2

10-6 10-4 0.01 0.1 1

10 100 1000


↵sCF

⇡

✓ln

r2

⇢� 3

4

◆

↵sCF

⇡

✓ln

1

zcut� 3

4

◆

↵sCF

⇡

✓ln

1

⇢� 3

4

◆

!/"

d"

/ d

!

! = m2/(pt2 R2)



0

0.1

0.2

10-6 10-4 0.01 0.1 1

10 100 1000

zcut

r2zcut

r =Rsub

R

Trimming at LO in αs

⇢

�

d�(trim,LO)

d⇢=


continue with all-orderresummation of terms

↵ns lnm ⇢


→ all orders in αs

31

QCD pattern of multiple

soft/collinear emission Establish which

simplifying approximations

to use for tagger & matrix

elements

Inputs

Analysis of taggers’

behaviour for 1, 2, 3, … n,emissions

Output

approx. formula for

tagger’s mass distribution for

ρ ≪ 1

keeping only terms with largest powers of ln ρ,

e.g. m = 2n, 2n-1

⇢

�

d�

d⇢=

1X

n=1

cnm ↵ns lnm ⇢


Trimming at all orders

32

Trimming

⇢trim(k1, k2, . . . kn) 'nX

i

⇢trim(ki)

⇠ max

i{⇢trim(ki)}

Trimmed jet reduces (~) to sum of trimmed emissions

Matrix element

X

n

1

n!

nY

i

d✓2i✓2i

dzizi

↵s(✓izipjett )CF

⇡

can use QED-like independent emissions, as if gluons don’t split



33

!/"

d"

/ d

!

! = m2/(pt2 R2)



0

0.1

0.2

10-6 10-4 0.01 0.1 1

10 100 1000

LO

d�trim,resum

d⇢=

d�trim,LO

d⇢



34

d�trim,resum

d⇢=

d�trim,LO

d⇢exp

�Z

⇢d⇢0

1

�

d�trim,LO

d⇢0

�

!/"

d"

/ d

!

! = m2/(pt2 R2)

trimmed quark jets


0

0.1

0.2

10-6 10-4 0.01 0.1 1

10 100 1000

LO

Resummed(fixed coupling)



35

d�trim,resum

d⇢=

d�trim,LO

d⇢exp

�Z

⇢d⇢0

1

�

d�trim,LO

d⇢0

�

!/"

d"

/ d

!

! = m2/(pt2 R2)

trimmed quark jets


0

0.1

0.2

10-6 10-4 0.01 0.1 1

10 100 1000

LO


0.01

0.1

1 0.1 1


z

�qgRsub

zcut

� = 0.030

trimmed away

VETO

Sudakov



36

d�trim,resum

d⇢=

d�trim,LO

d⇢exp

�Z

⇢d⇢0

1

�

d�trim,LO

d⇢0

�

!/"

d"

/ d

!

! = m2/(pt2 R2)

trimmed quark jets


0

0.1

0.2

10-6 10-4 0.01 0.1 1

10 100 1000

LO


0.01

0.1

1 0.1 1


z

�qgRsub

zcut

� = 0.009

trimmed away

VETO

Sudakov



37

d�trim,resum

d⇢=

d�trim,LO

d⇢exp

�Z

⇢d⇢0

1

�

d�trim,LO

d⇢0

�

!/"

d"

/ d

!

! = m2/(pt2 R2)

trimmed quark jets


0

0.1

0.2

10-6 10-4 0.01 0.1 1

10 100 1000

LO


0.01

0.1

1 0.1 1


z

�qgRsub

zcut

� = 0.003

trimmed away

VETO

Sudakov



38

d�trim,resum

d⇢=

d�trim,LO

d⇢exp

�Z

⇢d⇢0

1

�

d�trim,LO

d⇢0

�

!/"

d"

/ d

!

! = m2/(pt2 R2)

trimmed quark jets


0

0.1

0.2

10-6 10-4 0.01 0.1 1

10 100 1000

LO


0.01

0.1

1 0.1 1


z

�qgRsub

zcut

� = 0.001

trimmed away

VETO

Sudakov



39

d�trim,resum

d⇢=

d�trim,LO

d⇢exp

�Z

⇢d⇢0

1

�

d�trim,LO

d⇢0

�


!/"

d"

/ d

!

! = m2/(pt2 R2)

trimmed quark jets


0

0.1

0.2

10-6 10-4 0.01 0.1 1

10 100 1000

Resummed(running coupling) Full resummation also

needs treatment of running coupling


Trimming: MC v. analytics

40

!/"

d"

/ d

!

! = m2/(pt2 R2)

Analytic Calculation: quark jets

m [GeV], for pt = 3 TeV, R = 1

Trimming

Rsub=0.3, zcut=0.05

Rsub=0.3, zcut=0.1

0

0.05

0.1

0.15

0.2

0.25

0.3

10-6 10-4 0.01 0.1 1

10 100 1000

!/"

d"

/ d

!

! = m2/(pt2 R2)

Pythia 6 MC: quark jets


Trimming

Rsub = 0.3, zcut = 0.05

Rsub = 0.3, zcut = 0.1

0

0.05

0.1

0.15

0.2

0.25

0.3

10-6 10-4 0.01 0.1 1

10 100 1000

Monte Carlo Analytic

quark jets

Non-trivial agreement!(also for dependence on parameters)


Trimming: MC v. analytics

41

!/"

d"

/ d

!

! = m2/(pt2 R2)

Analytic Calculation: gluon jets


Trimming

Rsub=0.3, zcut=0.05

Rsub=0.3, zcut=0.1

0

0.05

0.1

0.15

0.2

0.25

0.3

10-6 10-4 0.01 0.1 1

10 100 1000

!/"

d"

/ d

!

! = m2/(pt2 R2)

Pythia 6 MC: gluon jets


Trimming

Rsub = 0.3, zcut = 0.05

Rsub = 0.3, zcut = 0.1

0

0.05

0.1

0.15

0.2

0.25

0.3

10-6 10-4 0.01 0.1 1

10 100 1000


gluon jets

Non-trivial agreement!(also for dependence on parameters)


What logs, what accuracy?

42

⌃(⇢) =

Z ⇢

0d⇢0

1

�

d�

d⇢0Theorists state accuracy for the “cumulant” Σ(ρ):

Trimming’s leading logs (LL, in Σ) are:

We also have next-to-leading logs (NLL):

↵sL2, ↵2

sL4, .... I.e. ↵n

sL2n↵n

sL2n↵n

sL2n Just like the

jet mass

↵nsL

2n�1↵nsL

2n�1↵nsL

2n�1

Use shorthand L = log 1/ρ


What logs, what accuracy?

42

⌃(⇢) =

Z ⇢

0d⇢0

1

�

d�

d⇢0Theorists state accuracy for the “cumulant” Σ(ρ):

Trimming’s leading logs (LL, in Σ) are:

We also have next-to-leading logs (NLL):

↵sL2, ↵2

sL4, .... I.e. ↵n

sL2n↵n

sL2n↵n

sL2n Just like the

jet mass

↵nsL

2n�1↵nsL

2n�1↵nsL

2n�1

Use shorthand L = log 1/ρ

Could we do better? Yes: NLL in ln Σ:

Trimmed mass is like plain jet mass (with R → Rsub), and this accuracy involves non-global logs, clustering logs

ln⌃: ↵nsL

n+1 and ↵nsL

n


Everything so far was at parton levelAre partons sufficient?


mEW

44

!/"

d"

/ d

!

! = m2/(pt2 R2)

trimming: hadronisation (quark jets)


Pyth

ia 6

DW

, C/A

R =

1, R

sub =

0.3

, zcu

t = 0

.05

parton level

hadron level (no UE)

hadron level (with UE)

0

0.1

0.2

0.3

0.4

10-6 10-4 0.01 0.1 1

10 100 1000

hadronisation adds roughly

to the trimmed-jet squared mass

�m2 ⇠ µNP pjett Rsub

⇠ (30GeV)2

non-perturbative QCD important for EW-scale

phenomenology...

It’s even worse for plain jet massbut better for other taggers...


Comments Past years → a vast trove of ideas for jet substructure tagging.

But maybe it’s time to try to go back to “basics” → detailed understanding about how our methods work.

Trimming was a particularly illustrative case:

• has non-trivial structure, relevant for phenomenology

• can mostly be understood from LO calculation & standard resummation techniques — quite similar to jet mass

• non-perturbative effects are relevant

Now over to Simone, who will discuss pruning and MDT

45

46

EXTRAS


Trimming v. jet pt

47

!/"

d"

/ d

!

! = m2/(pt2 R2)


m [GeV], for pt = 3.0 TeV, R = 1

Pyth

ia 6

DW

, C/A

R =

1, R

sub =

0.3

, zcu

t = 0

.05

parton level



0

0.1

0.2

0.3

0.4

10-6 10-4 0.01 0.1 1

10 100 1000

EW

3000 GeV


Trimming v. jet pt

48

!/"

d"

/ d

!

! = m2/(pt2 R2)



Pyth

ia 6

DW

, C/A

R =

1, R

sub =

0.3

, zcu

t = 0

.05

parton level



0

0.1

0.2

0.3

0.4

10-6 10-4 0.01 0.1 1

10 100 1000

EW

2000 GeV


Trimming v. jet pt

49

!/"

d"

/ d

!

! = m2/(pt2 R2)



Pyth

ia 6

DW

, C/A

R =

1, R

sub =

0.3

, zcu

t = 0

.05

parton level



0

0.1

0.2

0.3

0.4

10-6 10-4 0.01 0.1 1

10 100 1000

EW

1300 GeV


Trimming v. jet pt

50

!/"

d"

/ d

!

! = m2/(pt2 R2)



Pyth

ia 6

DW

, C/A

R =

1, R

sub =

0.3

, zcu

t = 0

.05

parton level



0

0.1

0.2

0.3

0.4

10-6 10-4 0.01 0.1 1

1 10 100

EW

800 GeV


Trimming v. jet pt

51

!/"

d"

/ d

!

! = m2/(pt2 R2)



Pyth

ia 6

DW

, C/A

R =

1, R

sub =

0.3

, zcu

t = 0

.05

parton level



0

0.1

0.2

0.3

0.4

10-6 10-4 0.01 0.1 1

1 10 100

EW

500 GeV


Trimming v. jet pt

52

!/"

d"

/ d

!

! = m2/(pt2 R2)



Pyth

ia 6

DW

, C/A

R =

1, R

sub =

0.3

, zcu

t = 0

.05

parton level



0

0.1

0.2

0.3

0.4

10-6 10-4 0.01 0.1 1

1 10 100

EW

300 GeV


cuts on N-subjettiness, etc.?

• These cuts are nearly always for a jet whose mass is somehow groomed. All the structure from the grooming persists.

• So tagging & shape must probably be calculated together

53

!/"

d"

/ d

!

! = m2/(pt2 R2)

trimming & #21 cut (MC quark jets)


Pyth

ia 6

DW

, C/A

R =

1, R

sub =

0.2

, zcu

t = 0

.1

Trimmed

Trimmed + #21$=1, kt axes < 0.3

0

0.1

0.2

10-6 10-4 0.01 0.1 1

10 100 1000


kinematic plane and jet masses

54

Figure 2. Lund diagrams [45] representemission kinematics in terms of two vari-ables: vertically, the logarithm of an emis-sion’s transverse momentum kt with respectto the jet axis, and horizontally, the loga-rithm of the inverse of the emission’s angle !with respect to the jet axis, i.e. its rapiditywith respect to the jet axis. Here the dia-gram shows a line of constant jet mass, to-gether with a shaded region corresponding tothe part of the kinematic plane where emis-sions are vetoed, leading to a Sudakov formfactor.

jet m

ass

θln zln kt

ln 1/θ

regionsoft collinear

smallangles

largeangles

Lundkinematicdiagram

~

y ~

hard collinear region

line o

f con

stant

soft large−angle region

Let us define

D(") =

! 1

!

d"!

"!

! 1

!!dz pgq(z)

#s(z"!R2p2t )CF

$, (3.1a)

! #sCF

$

"

1

2ln2

1

"" 3

4ln

1

"+O (1)

#

, (fixed coupling approx.) , (3.1b)

where pgq =1+(1"z)2

2z is the quark-gluon splitting function, stripped of its colour factor, and

the fixed-coupling approximation in the second line helps visualise the double-logarithmic

structure of D(").

To NLL accuracy,3 i.e. control of terms #nsL

n+1 and #nsL

n in ln!("), where L # ln 1! ,

the integrated jet mass distribution is given by

!(") = e"D(!) · e""ED!(!)

"(1 +D!("))· N (") . (3.2)

The first factor, which is double logarithmic, accounts for the Sudakov suppression of

emissions that would induce a (squared, normalised) jet mass greater than ". In terms of

the “Lund” representation of the kinematic plane [45], Fig. 2, it accounts for the probability

of there being no emissions in the shaded region, with the 12 ln

2 1/" term in Eq. (3.1b) for

D(") coming from the bulk of the area (soft divergence of pgq), while the "34 ln 1/" term

comes from the hard collinear region (finite z). The second factor in Eq. (3.2), defined in

terms of D!(") # %LD, encodes the single-logarithmic corrections associated with the fact

that the e#ects of multiple emissions add together to give the jet’s overall mass. These

emissions tend to be close to the constant-jet-mass boundary in Fig. 2. The third factor,

3Which requires the coupling in Eq. (3.1) to run with a two-loop !-function, and to be evaluated in the

CMW scheme [46], or equivalently taking into account the two-loop cusp anomalous dimension.

– 6 –

for QCD emissions

log of 1/(emission angle)

log of transverse

momentumof emission wrt jet axis



55


jet m

ass

θln zln kt

ln 1/θ


smallangles

largeangles


~

y ~


line o

f con

stant


Let us define

D(") =

! 1

!

d"!

"!

! 1

!!dz pgq(z)

#s(z"!R2p2t )CF

$, (3.1a)

! #sCF

$

"

1

2ln2

1

"" 3

4ln

1

"+O (1)

#


where pgq =1+(1"z)2



structure of D(").


n+1 and #nsL



!(") = e"D(!) · e""ED!(!)

"(1 +D!("))· N (") . (3.2)













– 6 –

!/"

d"

/ d

!

! = m2/(pt2 R2)



Jets: C

/A w

ith R

=1. M

C: P

ythia

6.4

, DW

tune, p

arto

n-le

vel (n

o M

PI), q

q#

qq, p

t > 3

TeV

plain jet mass

0

0.1

0.2

0.3

10-6 10-4 0.01 0.1 1

10 100 1000



56


jet m

ass

θln zln kt

ln 1/θ


smallangles

largeangles


~

y ~


line o

f con

stant


Let us define

D(") =

! 1

!

d"!

"!

! 1

!!dz pgq(z)

#s(z"!R2p2t )CF

$, (3.1a)

! #sCF

$

"

1

2ln2

1

"" 3

4ln

1

"+O (1)

#


where pgq =1+(1"z)2



structure of D(").


n+1 and #nsL



!(") = e"D(!) · e""ED!(!)

"(1 +D!("))· N (") . (3.2)













– 6 –

!/"

d"

/ d

!

! = m2/(pt2 R2)



Jets: C

/A w

ith R

=1. M

C: P

ythia

6.4

, DW

tune, p

arto

n-le

vel (n

o M

PI), q

q#

qq, p

t > 3

TeV

plain jet mass

0

0.1

0.2

0.3

10-6 10-4 0.01 0.1 1

10 100 1000

increase inphasespace



57


jet m

ass

θln zln kt

ln 1/θ


smallangles

largeangles


~

y ~


line o

f con

stant


Let us define

D(") =

! 1

!

d"!

"!

! 1

!!dz pgq(z)

#s(z"!R2p2t )CF

$, (3.1a)

! #sCF

$

"

1

2ln2

1

"" 3

4ln

1

"+O (1)

#


where pgq =1+(1"z)2



structure of D(").


n+1 and #nsL



!(") = e"D(!) · e""ED!(!)

"(1 +D!("))· N (") . (3.2)













– 6 –

!/"

d"

/ d

!

! = m2/(pt2 R2)



Jets: C

/A w

ith R

=1. M

C: P

ythia

6.4

, DW

tune, p

arto

n-le

vel (n

o M

PI), q

q#

qq, p

t > 3

TeV

plain jet mass

0

0.1

0.2

0.3

10-6 10-4 0.01 0.1 1

10 100 1000

increase inphasespace

Sudakovsuppression


Some key calculations related to jet mass

• Catani, Turnock, Trentadue & Webber, ’91: heavy-jet mass in e+e–

• Dasgupta & GPS, ‘01: hemisphere jet mass in e+e– (and DIS)

• Appleby & Seymour, ’02Delenda, Appleby, Dasgupta & Banfi ’06: impact of jet boundary

• Gehrmann, Gehrmann de Ridder, Glover ’08; Weinzierl ’08Chien & Schwartz ’10: heavy-jet mass in e+e– to higher accuracy

• Dasgupta, Khelifa-Kerfa, Marzani & Spannowsky ’12,Chien & Schwartz ’12,Jouttenus, Stewart, Tackmann, Waalewijn ’13:jet masses at hadron colliders

58


Trimming

59

Take all particles in ajet of radius R and recluster them into subjets with a jet




form the trimmed jet.Krohn, Thaler & Wang ’09


Trimming

59

Take all particles in ajet of radius R and recluster them into subjets with a jet




form the trimmed jet.Krohn, Thaler & Wang ’09

l < z cut r 2

l = z cut

r 2

= R

sub

e

Egluon = z

cut Ejet

l = z cut

ln 1/ e

Trimming

KEPT

REJ

ECTE

D


Trimming

60

l < z cut r 2

l = z cut

r 2

= R

sub

e

Egluon = z

cut Ejet

l = z cut

ln 1/ e

Trimming

KEPT

REJ

ECTE

D!/"

d"

/ d

!

! = m2/(pt2 R2)

trimmed quark jets (Pythia 6 MC)


Jets: C

/A w

ith R

=1. M

C: P

ythia

6.4

, DW

tune, p

arto

n-le

vel (n

o M

PI), g

g#

gg, p

t > 3

TeV 0

0.1

0.2

10-6 10-4 0.01 0.1 1

10 100 1000


Trimming

61

l < z cut r 2

l = z cut

r 2

= R

sub

e

Egluon = z

cut Ejet

l = z cut

ln 1/ e

Trimming

KEPT

REJ

ECTE

D!/"

d"

/ d

!

! = m2/(pt2 R2)



Jets: C

/A w

ith R

=1. M

C: P

ythia

6.4

, DW

tune, p

arto

n-le

vel (n

o M

PI), g

g#

gg, p

t > 3

TeV 0

0.1

0.2

10-6 10-4 0.01 0.1 1

10 100 1000


Trimming

62

l < z cut r 2

l = z cut

r 2

= R

sub

e

Egluon = z

cut Ejet

l = z cut

ln 1/ e

Trimming

KEPT

REJ

ECTE

D!/"

d"

/ d

!

! = m2/(pt2 R2)



Jets: C

/A w

ith R

=1. M

C: P

ythia

6.4

, DW

tune, p

arto

n-le

vel (n

o M

PI), g

g#

gg, p

t > 3

TeV 0

0.1

0.2

10-6 10-4 0.01 0.1 1

10 100 1000


Trimming

63

l < z cut r 2

l = z cut

r 2

= R

sub

e

Egluon = z

cut Ejet

l = z cut

ln 1/ e

Trimming

KEPT

REJ

ECTE

D!/"

d"

/ d

!

! = m2/(pt2 R2)



Jets: C

/A w

ith R

=1. M

C: P

ythia

6.4

, DW

tune, p

arto

n-le

vel (n

o M

PI), g

g#

gg, p

t > 3

TeV 0

0.1

0.2

10-6 10-4 0.01 0.1 1

10 100 1000


What about actual calculations of the taggers?

Simpler More complex

mMDT > plain mass ~ trimming > Y-pruning > full pruning ≫ MDT

Dasgupta et al (full NLL, including non-global part)Chien et al (partial NNLL)Jouttenus et al (they say full NNLL; I remain to be convinced!)

LL in all cases, plus some subleading logs [NB: LL doesn’t mean the same thing in all cases!)

64


Kinematic regions for different taggers

65

l < z cut r 2

l = z cut

r 2

= R

sub

e

Egluon = z

cut Ejet

l = z cut

ln 1/ e

Trimming

KEPT

REJ

ECTE

D

fatl < z cut l

fat

= zl

cut l

gluon

jet

E

E

= zcut

= zl

cut

= R

epr

une

eln 1/

Pruning

KEPTR

EJEC

TED

l cut < z

Ejet

cut

Egluon = z

cut

l = z

ln 1/e

eln

z mMDT

b

a

KEPT

REJ

ECTE

D


Kinematic regions for different taggers

65

l < z cut r 2

l = z cut

r 2

= R

sub

e

Egluon = z

cut Ejet

l = z cut

ln 1/ e

Trimming

KEPT

REJ

ECTE

D

fatl < z cut l

fat

= zl

cut l

gluon

jet

E

E

= zcut

= zl

cut

= R

epr

une

eln 1/

Pruning

KEPTR

EJEC

TED

l cut < z

Ejet

cut

Egluon = z

cut

l = z

ln 1/e

eln

z mMDT

b

a

KEPT

REJ

ECTE

D

Signals live here


Summary table

66

highest logs transition(s) Sudakov peak NGLs NP: m2 !

plain mass !nsL

2n — L ! 1/"!̄s yes µNP ptR

trimming !nsL

2n zcut, r2zcut L ! 1/"!̄s # 2 ln r yes µNP ptRsub

pruning !nsL

2n zcut, z2cut L ! 2.3/"!̄s yes µNP ptR

MDT !nsL

2n!1 ycut,14y

2cut, y

3cut — yes µNP ptR

Y-pruning !nsL

2n!1 zcut (Sudakov tail) yes µNP ptR

mMDT !nsL

n ycut — no µ2NP/ycut

Table 1. Table summarising the main features for the plain jet mass, the three original taggers of

our study and the two variants introduced here. In all cases, L = ln 1

!= ln R2p2

t

m2 , r = Rsub/R andthe log counting applies to the region below the smallest transition point. The transition pointsthemselves are given as " values. Sudakov peak positions are quoted for d#/dL; they are expressedin terms of !̄s $ !sCF /$ for quark jets and !̄s $ !sCA/$ for gluon jets and neglect corrections ofO (1). “NGLs” stands for non-global logarithms. The last column indicates the mass-squared belowwhich the non-perturbative (NP) region starts, with µNP parametrising the scale where perturbationtheory is deemed to break down.

9 Conclusions

In this paper we have developed an extensive analytical understanding of the action of

widely used boosted-object taggers and groomers on quark and gluon jets.

We initially intended to study three methods: trimming, pruning and the mass-drop

tagger (MDT). The lessons that we learnt there led us to introduce new variants, Y-pruning

and the modified mass-drop tagger (mMDT). The key features of the di!erent taggers are

summarised in table 1. We found, analytically, that the taggers are similar in certain

phase-space regions and di!erent in others, identified the transition points between these

regions and carried out resummations of the dominant logarithms of pt/m to all orders.

One tagger has emerged as special, mMDT, in that it eliminates all sensitivity to

the soft divergences of QCD. As a result its dominant logarithms are !nsL

n, entirely of

collinear origin. It is the first time, to our knowledge, that such a feature is observed,

and indeed all the other taggers involve terms with more logarithms than powers of !s.

One consequence of having just single, collinear logarithms is that the complex non-global

(and super-leading [57]) logarithms are absent. Another is that fixed-order calculations

have an enhanced range of validity, up to L % 1/!s rather than L % 1/"!s. The

modified mass-drop tagger is also the least a!ected by non-perturbative corrections. Finally

the parameters of the tagger can be chosen so as to ensure a mass distribution that is

nearly flat, which can facilitate the reliable identification of small signals. Also of interest

19In this context it may be beneficial to study a range of variables, such as N-subjettiness [26] and energy

correlations [32], or even combinations of observables as done in Refs [81, 82]. It is also of interest to examine

observables specifically designed to show sensitivity to colour flows, such as pull [83] and dipolarity [84],

though it is not immediately apparent that these exploit di!erences in the double logarithmic structure.

It would also, of course, be interesting to extend our analysis to other types of method such as template

tagging [85].

– 45 –

NEW

Special: only single logarithms (L = ln ρ)→ more accurately calculable

Special: better exploits signal/bkgd

differences


!/"

d"

/ d

!

! = m2/(pt2 R2)



mMDT ycut=0.03

ycut=0.13

ycut=0.35 (some finite ycut)

0

0.1

0.2

10-6 10-4 0.01 0.1 1

10 100 1000

Modified Mass Drop Tagger

67

[mMDT is closest we have to a scale-invariant tagger,though exact behaviour depends on q/g fractions]

!/"

d"

/ d

!

! = m2/(pt2 R2)



mMDT ycut=0.03

ycut=0.13

ycut=0.35

0

0.1

0.2

10-6 10-4 0.01 0.1 1

10 100 1000


quark jets


Pruning

68

!/"

d"

/ d

!

! = m2/(pt2 R2)



Pruning, zcut=0.1

Y-pruning, zcut=0.1

I-pruning, zcut=0.1

0

0.1

0.2

10-6 10-4 0.01 0.1 1

10 100 1000

!/"

d"

/ d

!

! = m2/(pt2 R2)



Pruning, zcut=0.1

Y-pruning, zcut=0.1

I-pruning, zcut=0.1

0

0.1

0.2

10-6 10-4 0.01 0.1 1

10 100 1000


quark jets


Comparing MC & other tools

69

!/"

d"

/ d

!

! = m2/(pt2 R2)

LO v. Pythia showers (quark jets)


mMDT (ycut = 0.13)

pt,jet > 3 TeV

v6.425 (DW) virtuality orderedv6.425 (P11) pt ordered

v6.428pre (P11) pt ordered

v8.165 (4C) pt ordered

Leading Order

0

0.1

10-6 10-4 0.01 0.1 1

10 100 1000

!/"

d"

/ d

!

! = m2/(pt2 R2)

LO v. NLO v. resummation (quark jets)


mMDT (ycut = 0.13)

pt,jet > 3 TeV

Leading Order

Next-to-Leading Order

Resummed

0

0.1

10-6 10-4 0.01 0.1 1

10 100 1000


Performance for finding signals

70

0

0.2

0.4

0.6

0.8

1

300 500 1000 3000

tagg

ing

effic

ienc

y ε

pt,min [GeV]

W tagging efficiencies

hadron level with UE

mMDT (ycut = 0.11)

pruning (zcut=0.1)

Y-pruning (zcut=0.1)

trimming (Rsub=0.3, zcut=0.05)

Figure 17. E!ciencies for tagging hadronically-decaying W ’s, for a range of taggers/groomers,shown as a function of the W transverse momen-tum generation cut in the Monte Carlo samples(Pythia 6, DW tune). Further details are givenin the text.

It receives O (!s) corrections from gluon radiation o" the W ! qq̄! system. Monte Carlo

simulation suggests these e"ects are responsible, roughly, for a 10% reduction in the tagging

e!ciencies. Secondly, Eq. (8.9) was for unpolarized decays. By studying leptonic decays of

the W in the pp ! WZ process, one finds that the degree of polarization is pt dependent,

and the expected tree-level tagging-e!ciency ranges from about 76% at low pt to 84%

at high pt. These two e"ects explain the bulk of the modest di"erences between Fig. 17

and the result of Eq. (8.9). However, the main conclusion that one draws from Fig. 17

is that the ultimate performance of the di"erent taggers will be driven by their e"ect on

the background rather than by the fine details of their interplay with signal events. This

provides an a posteriori justification of our choice to concentrate our study on background

jets.

2

3

4

5

300 500 1000 3000

ε S / √ε

B

pt,min [GeV]

signal significance with quark bkgds


(Rsub=0.3, zcut=0.05)

mMDT (ycut = 0.11)

pruning (zcut=0.1)


trimming

2

3

4

5

300 500 1000 3000ε S

/ √ε

B

pt,min [GeV]

signal significance with gluon bkgds


(Rsub=0.3, zcut=0.05)

mMDT (ycut = 0.11)

pruning (zcut=0.1)


trimming

Figure 18. The significance obtained for tagging signal (W ’s) versus background, defined as"S/

""B, for a range of taggers/groomers, shown as a function of the transverse momentum gen-

eration cut in the Monte Carlo samples (Pythia 6, DW tune) Further details are given in thetext.

– 43 –

Documents

Towards an understanding of jet substructure · nose experimentally, ... on signal jets and how to optimize the parameters of a method k nown as ﬁltering [1]. Ref. [13] discussed