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Simone Marzani Institute for Particle Physics Phenomenology Durham University RADCOR 2013 11 th International Symposium on Radiative Corrections (Applications of Quantum Field Theory to Phenomenology) 22 nd -27 th September 2013 with M. Dasgupta, A. Fregoso, G. P . Salam and A. Powling arXiv:1307.0007 and 1307.0013 QCD CALCULATIONS FOR JET SUBSTRUCTURE
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Page 1: QCD CALCULATIONS FOR JET SUBSTRUCTURE

Simone MarzaniInstitute for Particle Physics Phenomenology

Durham University

RADCOR 201311th International Symposium on Radiative Corrections

(Applications of Quantum Field Theory to Phenomenology) 22nd-27th September 2013

withM. Dasgupta, A. Fregoso, G. P. Salam and A. Powling

arXiv:1307.0007 and 1307.0013

QCD CALCULATIONS FOR

JET SUBSTRUCTURE

Page 2: QCD CALCULATIONS FOR JET SUBSTRUCTURE

• The LHC is exploring phenomena at energies above the EW scale

• Z/W/H/top can no longer be considered heavy particles

•These particles are abundantly produced with a large boost

• Their hadronic decays are collimated and can be reconstructed within a single jet. Need to distinguish:

The boosted regime

pt � mt

Wb

q

R R

from QCD

Page 3: QCD CALCULATIONS FOR JET SUBSTRUCTURE

• The last few years have seen a rapid development in substructure techniques: O(10-20) powerful methods to tag jet substructure

• Many of the methods have been tried out in searches and work; they will be crucial for searches in the years to come

• Many methods can lead to some confusion• Do we understand how / why they work ?• Only analytic understanding can give this field robustness

Grooming and tagging

Page 4: QCD CALCULATIONS FOR JET SUBSTRUCTURE

Where to start ?

Mass-drop tagger (MDT, aka BDRS)

Trimming

Pruning

Cannot possibly study all toolsThese 3 are widely used

We concentrate on background (QCD jets)

recluster

on scale Rsub

discard subjetswith < zcut pt

decluster &

discard soft junkrepeat until

find hard struct

Rprune ~ mj/pt

discard large-anglesoft clusterings

recluster

Krohn, Thaler and Wang (2010)

Ellis, Vermillion and Walsh (2009)

Butterworth, Davison, Rubin and Salam (2008)

Page 5: QCD CALCULATIONS FOR JET SUBSTRUCTURE

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 difficult 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 effects, there are important

differences. 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 differences 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 differences are generic, or restricted to some specific

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

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

nose experimentally, because of the many kinds of physics effect 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 Effective

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 –

Our understanding so far

• To what extent are the taggers above similar ?• How does the statement of aggressive behaviour depend on the taggers’ parameters and on the jet’s kinematics ?

•The “right” MC study can be instructive

Page 6: QCD CALCULATIONS FOR JET SUBSTRUCTURE

3

Comparison of taggers

Plain jet mass: characteristic Sudakov peak

0

0.1

0.2

0.3

10-6 10-4 0.01 0.1 1

10 100 1000

l/m

dm

/ dl

l = m2/(pt2 R2)

gluon jets: m [GeV], for pt = 3 TeV

Jets: C/A w

ith R=1. M

C: Pythia 6.4, D

W tune, parton-level (no M

PI), ggAgg, pt > 3 TeV

plain jet mass

0

0.1

0.2

0.3

10-6 10-4 0.01 0.1 1

10 100 1000

l/m

dm

/ dl

l = m2/(pt2 R2)

quark jets: m [GeV], for pt = 3 TeV

Jets: C/A w

ith R=1. M

C: Pythia 6.4, D

W tune, parton-level (no M

PI), qqAqq, pt > 3 TeV

plain jet mass

Page 7: QCD CALCULATIONS FOR JET SUBSTRUCTURE

3

Comparison of taggers

Different taggers appear to behave quite similarly

l/m

dm

/ dl

l = m2/(pt2 R2)

gluon jets: m [GeV], for pt = 3 TeV

Jets: C/A w

ith R=1. M

C: Pythia 6.4, D

W tune, parton-level (no M

PI), ggAgg, pt > 3 TeV

plain jet massTrimmerPrunerMDT

0

0.1

0.2

0.3

10-6 10-4 0.01 0.1 1

10 100 1000

l/m

dm

/ dl

l = m2/(pt2 R2)

quark jets: m [GeV], for pt = 3 TeV

Jets: C/A w

ith R=1. M

C: Pythia 6.4, D

W tune, parton-level (no M

PI), qqAqq, pt > 3 TeV

plain jet massTrimmer (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

Page 8: QCD CALCULATIONS FOR JET SUBSTRUCTURE

3

Comparison of taggers

But only for a limited kinematic region !

l/m

dm

/ dl

l = m2/(pt2 R2)

gluon jets: m [GeV], for pt = 3 TeV

Jets: C/A w

ith R=1. M

C: Pythia 6.4, D

W tune, parton-level (no M

PI), ggAgg, pt > 3 TeV

plain jet massTrimmerPrunerMDT

0

0.1

0.2

0.3

10-6 10-4 0.01 0.1 1

10 100 1000

l/m

dm

/ dl

l = m2/(pt2 R2)

quark jets: m [GeV], for pt = 3 TeV

Jets: C/A w

ith R=1. M

C: Pythia 6.4, D

W tune, parton-level (no M

PI), qqAqq, pt > 3 TeV

plain jet massTrimmer (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

Page 9: QCD CALCULATIONS FOR JET SUBSTRUCTURE

3

Comparison of taggers

Let’s translate from QCD variables to ``search’’ variables: ρ ➞ m, for pt = 3 TeV, R =1.0

l/m

dm

/ dl

l = m2/(pt2 R2)

gluon jets: m [GeV], for pt = 3 TeV

Jets: C/A w

ith R=1. M

C: Pythia 6.4, D

W tune, parton-level (no M

PI), ggAgg, pt > 3 TeV

plain jet massTrimmerPrunerMDT

0

0.1

0.2

0.3

10-6 10-4 0.01 0.1 1

10 100 1000

l/m

dm

/ dl

l = m2/(pt2 R2)

quark jets: m [GeV], for pt = 3 TeV

Jets: C/A w

ith R=1. M

C: Pythia 6.4, D

W tune, parton-level (no M

PI), qqAqq, pt > 3 TeV

plain jet massTrimmer (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

Page 10: QCD CALCULATIONS FOR JET SUBSTRUCTURE

Questions that arise

• Can we understand the different shapes (flatness vs peaks) ?• What’s the origin of the transition points ?• How do they depend on the taggers’ parameters ?

• What’s the perturbative structure of tagged mass distributions ? • Cumulative distribution for plain jet mass contains (soft & collinear) double logs

• Do the taggers ameliorate this behaviour ? • If so, what’s the applicability of FO calculations ?

⌃(⇢) ⌘ 1�

Z ⇢ d�

d⇢0 d⇢0 ⇠X

n

↵ns ln2n 1

⇢+ . . .

Page 11: QCD CALCULATIONS FOR JET SUBSTRUCTURE

11

↵sCF

✓ln

r2

⇢� 3

4

↵sCF

✓ln

1

zcut� 3

4

↵sCF

✓ln

1

⇢� 3

4

ρ/σ

/ d

ρ

ρ = m2/(pt2 R2)

trimmed quark jets: LO

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

0

0.1

0.2

10-6 10-4 0.01 0.1 1

10 100 1000

zcut

r2zcut

r =Rsub

R

d�(trim,LO)

d⇢=

Trimming at LO

Emissions within Rsub are never tested for zcut: double logs

Intermediate region in which zcut is

effective: single logs

Page 12: QCD CALCULATIONS FOR JET SUBSTRUCTURE

3

Trimming: all ordersOne gets exponentiation of LO (+ running coupling)

0

0.1

0.2

0.3

10-6 10-4 0.01 0.1 1

10 100 1000

l/m

dm

/ dl

l = m2/(pt2 R2)

Analytic Calculation: quark jets

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

Trimming

Rsub=0.2, zcut=0.05Rsub=0.2, zcut=0.1

0

0.1

0.2

0.3

10-6 10-4 0.01 0.1 1

10 100 1000

l/m

dm

/ dl

l = m2/(pt2 R2)

Pythia 6 MC: quark jets

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

Trimming

Rsub = 0.2, zcut = 0.05Rsub = 0.2, zcut = 0.1

All-order calculation done in the small-zcut limit

d�trim,resum

d⇢=

d�trim,LO

d⇢exp

�Z

⇢d⇢0

1

d�trim,LO

d⇢0

Page 13: QCD CALCULATIONS FOR JET SUBSTRUCTURE

3

Pruning & MDT at LO• The pruning radius is set dynamically: Rprune < dij• The 2 prongs are always tested for zcut: single logs

ρ/σ

/ d

ρ

ρ = m2/(pt2 R2)

pruned quark jets: LO

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

0

0.1

0.2

10-6 10-4 0.01 0.1 1

10 100 1000

single-logregion zcut

MDT result is identical to LO pruning (in the small-ycut limit)

Page 14: QCD CALCULATIONS FOR JET SUBSTRUCTURE

Structures beyond LOAll-order MDT and pruning distributions are NOT given by exponentiation of LO

(a)

1 p2

p3

p1

p3p2

(b)

p

Figure 2: Two characteristic partonic configurations that arise at in the tree-level O(

α2s

)

contri-bution. The dashed cone provides a schematic representation of the boundary of the jet.

whole is tagged. If E3/E12 < ycut, then the MDT recurses, into the heavier of the two

subjets, i.e. j12, which can be analysed as in the previous, LO section. The key point

here is that in the limit in which E3 ! Ejet, the presence of gluon 3 has no effect on

whether the j12 system gets tagged. This is true even if mjet is dominated by emission

3, such that mjet " m12. This was part of the intended design of the MDT: if the jet

contains hard substructure, the tagger should find it, even if there is other soft structure

(including underlying event and pileup) that strongly affects the original jet mass. One

of the consequences of this design is that when evaluated, the NLO contribution that

comes from configuration (a) and the corresponding virtual graphs, one finds a logarithmic

structure for the integrated cross section of C2Fα

2s ln

2 ρ [5]. This is suggestive of an all-orders

logarithmic structure of the form (αs ln ρ)n. We will return to this shortly.

Configuration (b) in Fig. 2 reveals an unintended behaviour of the tagger. Here we

have θ23 ! θ12 # θ13, so the first unclustering leads to j1 and j23 subjets. It may happen

that the parent gluon of the j23 subjet was soft, so that E23 < ycutEjet. The jet therefore

fails the symmetry at this stage, and so recurses one step down. The formulation of the

MDT is such that one recurses into the more massive of the two prongs, i.e. only follows the

j23 prong, even though this is soft. This was not what was intended in the original design,

and is to be considered a flaw — in essence one follows the wrong branch. It is interesting

to determine the logarithmic structure that results from it, which can be straightforwardly

evaluated as follows:

ρ

σ

(MDT,NLOflaw)

= −CFρ(αs

π

)2∫

dxpgq(x)dθ2

θ2Θ(

R2 − θ2)

Θ (ycut − x)×

×∫

dz

(

1

2CApgg(z) + nfTRpqg(z)

)

dθ223θ223

δ

(

ρ− z(1− z)x2θ223R2

)

×

×Θ (z − ycut)Θ (1− z − ycut)Θ(

θ2 − θ223)

=CF

4

(αs

π

)2[

CA

(

ln1

ycut− 11

12

)

+nf

6

]

ln21

ρ+O

(

α2s ln

1

ρ

)

(4.5)

where θ is the angle between j1 and the j23 system, while x = E23/Ejet and z = E2/E23,

and pgg(z) = (1 − z)/z + z/(1 − z) + z(1 − z), pqg(z) =12(z

2 + (1 − z)2). Considering the

integrated distribution, this corresponds to a logarithmic structure α2s ln

3 ρ, i.e. enhanced

– 9 –

What MDT does wrongIf the energy condition fails, MDT iterates on the more massive subjet. It can follow a soft branch (p2+p3 < ycut ptjet), when the “right” answer was that the (massless) hard branch

had no substructure

p3

p1Rprune p2

R

Figure 5: Configuration that illustrates generation of double logs in pruning at O(

α2s

)

. Soft gluonp3 dominates the jet mass, thus determining the pruning radius. However, because of p3’s softness,it is then pruned away, leaving only the central core of the jet, which has a usual double-logarithmictype mass distribution.

ycut → zcut):

ρ

σ

(pruned, LO)

=αsCF

π

[

Θ(zcut − ρ) ln1

zcut+Θ(ρ− zcut) ln

1

ρ− 3

4

]

. (6.1)

6.1 3-particle configurations and “sane” and “anomalous” pruning

As was the case for the original mass-drop tagger, once we consider 3-particle configurations

the behaviour of pruning develops a certain degree of complexity. Fig. 5 illustrates the type

of configuration that is responsible: there is a soft parton that dominates the total jet mass

and so sets the pruning radius (p3), but does not pass the pruning zcut, meaning that it

does not contribute to the pruned mass; meanwhile there is another parton (p2), within

the pruning radius, that contributes to the pruned jet mass independently of how soft it

is. We call this anomalous pruning, because the emission that dominates the final pruned

jet mass never gets tested for the pruning zcut condition.

Let us work through this quantitatively. For gluon 3 to be discarded by pruning it must

have x3 < zcut # 1, i.e. it must be soft. Then the pruning radius is given by R2prune = x3θ23

and for p2 to be within the pruning core we have θ2 < Rprune. This implies θ2 # θ3, which

allows us to treat p2 and p3 as being emitted independently (i.e. due to angular ordering)

and also means that the C/A algorithm will first cluster 1 + 2 and then (1 + 2) + 3. The

leading-logarithmic contribution that one then obtains at O(

α2s

)

is then

ρ

σ

dσanom-pruned

dρ$(

CFαs

π

)2 ∫ zcut

0

dx3x3

∫ R2dθ23θ23

∫ 1

0

dx2x2

∫ x3θ23

0

dθ22θ22

ρ δ

(

ρ− x2θ22R2

)

(6.2a)

=

(

CFαs

π

)2 1

6ln3

zcutρ

+O(

α2s ln

2 1

ρ

)

, (valid for ρ < zcut). (6.2b)

where we have directly taken the soft limits of the relevant splitting functions.

The ln3 ρ contribution that one observes here in the differential distribution corre-

sponds to a double logarithmic (α2s ln

4 ρ) behaviour of the integrated cross-section, i.e. it

has as many logs as the raw jet mass, with both soft and collinear origins. This term is

– 14 –

What pruning sometimes doesChooses Rprune based on a soft p3

(dominates total jet mass), and leads to a single narrow subjet whose mass is also

dominated by a soft emission (p2, within Rprune of p1, so not pruned away).

Page 15: QCD CALCULATIONS FOR JET SUBSTRUCTURE

The modified Mass Drop Tagger

• In practice the soft-branch contribution is very small

• However, this modification makes the all-order structure

particularly interestingl/m

dm

/ dl

l = m2/(pt2 R2)

Pythia 6 MC: quark jets

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

1-4 ycut2

MDT, ycut=0.09wrong branch

mMDT

0

0.05

0.1

0.15

10-6 10-4 0.01 0.1 1

10 100 1000

• The soft-branch issue can be considered a flaw of the tagger• It worsens the logarithmic structure ~αs2 L3

• It makes all-order treatment difficult• It calls for a modification: always follow be the subjet with highest transverse mass

Page 16: QCD CALCULATIONS FOR JET SUBSTRUCTURE

All-order structure of mMDT• In the small ycut limit, it is just the exponentiation of LO• The mMDT has single logs to all orders (i.e. αsn Ln)

0

0.1

0.2

10-6 10-4 0.01 0.1 1

10 100 1000

l/m

dm

/ dl

l = m2/(pt2 R2)

Analytic Calculation: quark jets

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

mMDT ycut=0.03ycut=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

l/m

dm

/ dl

l = m2/(pt2 R2)

Pythia 6 MC: quark jets

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

mMDT ycut=0.03ycut=0.13ycut=0.35

Remarkable agreement !Interesting feature: flat mass distribution (more in backup slides)

Page 17: QCD CALCULATIONS FOR JET SUBSTRUCTURE

All-order structure of mMDT• In the small ycut limit, it is just the exponentiation of LO• The mMDT has single logs to all orders (i.e. αsn Ln)

• Single logs: extended validity of FO calculations

• Single logs of collinear origi• Remarkable consequence: mMDT is free of non-global logs!

l/m

dm

/ dl

l = m2/(pt2 R2)

LO v. NLO v. resummation (quark jets)

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

mMDT (ycut = 0.13)

pt,jet > 3 TeV

Leading OrderNext-to-Leading Order

Resummed

0

0.1

10-6 10-4 0.01 0.1 1

10 100 1000

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

10-2 10-1 100

1/m

dm

/ dc

c = mJ/pTJ

Z+jet, R=1.0, pTJ > 200 GeV

LONLL

NLL+LO

only a qualitative comparison: different process/kinematics!

plain mass

Page 18: QCD CALCULATIONS FOR JET SUBSTRUCTURE

All-order structure of mMDT• In the small ycut limit, it is just the exponentiation of LO• The mMDT has single logs to all orders (i.e. αsn Ln)

• Single logs: extended validity of FO calculations

• Single logs of collinear origin• Remarkable consequence: mMDT is free of non-global logs!

l/m

dm

/ dl

l = m2/(pt2 R2)

LO v. NLO v. resummation (quark jets)

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

mMDT (ycut = 0.13)

pt,jet > 3 TeV

Leading OrderNext-to-Leading Order

Resummed

0

0.1

10-6 10-4 0.01 0.1 1

10 100 1000

Page 19: QCD CALCULATIONS FOR JET SUBSTRUCTURE

Pruning: I & Y components• Pruning @ NLO ~αs2 L4 (like plain jet mass)• Single-pronged component (I-pruning) is active for ρ < zcut2

• A simple modification: require at least one successful merging with ΔR > Rprune and z > zcut (Y-pruning)

• It is convenient to resum the two components separately• Y-pruning: essentially Sudakov suppression of LO ~ αsn L2n-1

• I-pruning: convolution between the pruned and the original mass ~ αsn L2n

Page 20: QCD CALCULATIONS FOR JET SUBSTRUCTURE

All-order results

All-order calculation done in the small-zcut limit

l/m

dm

/ dl

l = m2/(pt2 R2)

Pythia 6 MC: quark jets

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

Pruning, zcut=0.1Y-pruning, zcut=0.1I-pruning, zcut=0.1

0

0.1

0.2

10-6 10-4 0.01 0.1 1

10 100 1000

l/m

dm

/ dl

l = m2/(pt2 R2)

Analytic Calculation: quark jets

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

Pruning, zcut=0.1Y-pruning, zcut=0.1I-pruning, zcut=0.1

0

0.1

0.2

10-6 10-4 0.01 0.1 1

10 100 1000

• Full Pruning: single-log region for zcut2 <ρ<zcut

• We control αsn L2n and αsn L2n-1 in the expansion • NG logs present but deferred to NNLO

Page 21: QCD CALCULATIONS FOR JET SUBSTRUCTURE

Non-perturbative effects

hadr

on (w

ith U

E) /

hadr

on (n

o U

E)l = m2/(pt

2 R2)

UE summary (quark jets)

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

Plain massTrimmerpruning

Y-pruningmMDT

0

0.5

1

1.5

2

2.5

10-6 10-4 0.01 0.1 1

10 100 1000

hadr

on /

parto

n

l = m2/(pt2 R2)

hadronisation summary (quark jets)

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

plain masstrimmerpruning

Y-pruningmMDT (zcut)

0

0.5

1

1.5

2

2.5

10-6 10-4 0.01 0.1 1

10 100 1000

• Most taggers have reduced sensitivity to NP physics• mMDT particularly so (it’s the most calculable)• Y-pruning sensitive to UE because of the role played by the fat jet mass

hadron / parton wUE / noUE

Page 22: QCD CALCULATIONS FOR JET SUBSTRUCTURE

Performances for finding signals (Ws)

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. Efficiencies 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 (αs) corrections from gluon radiation off the W → qq′ system. Monte Carlo

simulation suggests these effects are responsible, roughly, for a 10% reduction in the tagging

efficiencies. 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-efficiency ranges from about 76% at low pt to 84%

at high pt. These two effects explain the bulk of the modest differences 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 different taggers will be driven by their effect 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

hadron level with UE

(Rsub=0.3, zcut=0.05)

mMDT (ycut = 0.11)

pruning (zcut=0.1)

Y-pruning (zcut=0.1)

trimming

2

3

4

5

300 500 1000 3000ε S

/ √ε

Bpt,min [GeV]

signal significance with gluon bkgds

hadron level with UE

(Rsub=0.3, zcut=0.05)

mMDT (ycut = 0.11)

pruning (zcut=0.1)

Y-pruning (zcut=0.1)

trimming

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

√εB, 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 –

Y-pruning gives a visible improvement

Page 23: QCD CALCULATIONS FOR JET SUBSTRUCTURE

In summary ...• Analytic studies of the taggers reveal their properties• Particularly useful if MCs don’t agree

Page 24: QCD CALCULATIONS FOR JET SUBSTRUCTURE

In summary ...• Analytic studies of the taggers reveal their properties• Particularly useful if MCs don’t agree

l/m

dm

/ dl

l = m2/(pt2 R2)

Different showers

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

mMDT (ycut = 0.13)

pt,jet > 3 TeV

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

v6.428pre (P11) pt orderedv8.165 (4C) pt ordered

Analytics

0

0.1

10-6 10-4 0.01 0.1 1

10 100 1000

Page 25: QCD CALCULATIONS FOR JET SUBSTRUCTURE

In summary ...• Analytic studies of the taggers reveal their properties• Particularly useful if MCs don’t agree

• They also indicate how to develop better taggers

• Y-pruning:• improved log behaviour wrt pruning (αsn L2n-1 vs αsn L2n)• better rejection of QCD background

• mMDT:• exceptionally simple structure (single logs, no non-global)• reduced sensitivity to non-perturbative physics

Page 26: QCD CALCULATIONS FOR JET SUBSTRUCTURE

BACKUP SLIDES

Page 27: QCD CALCULATIONS FOR JET SUBSTRUCTURE

Summary table

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

plain mass αns L2n — L ! 1/

√αs yes µNP pt R

trimming αns L2n zcut, r2zcut L ! 1/

√αs − 2 ln r yes µNP pt Rsub

pruning αns L2n zcut, z2

cut L ! 2.3/√

αs yes µNP pt R

MDT αns L2n−1 ycut,

14y2

cut, y3cut — yes µNP pt R

Y-pruning αns L2n−1 zcut (Sudakov tail) yes µNP pt R

mMDT αns Ln ycut — no µ2

NP/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

ρ= 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 ρ values. Sudakov peak positions are quoted for dσ/dL; they are expressedin terms of αs ≡ αsCF /π for quark jets and αs ≡ αsCA/π 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.

performance of pruning relative to mMDT is mitigated. Most interesting, perhaps, is

Y-pruning. Its background enjoys a double-logarithmic Sudakov suppression for small

m/pt, due to the factor e−D(ρ) in Eq. (5.10a). The analogous effect for the signal is, we

believe, single-logarithmic, hence the modest reduction in signal yields in Fig. 17. Overall

the background suppression dominates, leading to improved tagging significance at high

pt. This is most striking in the gluon case, because of the CA colour factor in the e−D(ρ)

Sudakov suppression. Despite this apparent advantage, one should be aware of a defect

of Y-pruning, namely that at high pt the Y/I classification can be significantly affected

by underlying event and pileup, because of the way in which they modify the original jet

mass and the resulting pruning radius. It remains of interest to develop a tagger that

exploits the same double-logarithmic background suppression while not suffering from this

drawback.21

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 different taggers are

21In 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 differences 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 –

Page 28: QCD CALCULATIONS FOR JET SUBSTRUCTURE

Lund diagrams for mMDT

l cut < z

Ejet

cut

Egluon = z

cut

l = z

ln 1/e

eln

z mMDT

b

a

Page 29: QCD CALCULATIONS FOR JET SUBSTRUCTURE

Lund diagrams for pruning

gluon

jet

E

E

= zcut

= zl

cut

cut > zl

eln 1/

ln ze

fat

= zl

cut l

< z cut < lzcut l fat

gluon

jet

E

E

= zcut

I

Y

= zl

cut

= R

epr

une

eln 1/

andRfat

prune

sets emission that

l

fatl < z cut l

fat

= zl

cut l

gluon

jet

E

E

= zcut

= zl

cut

= R

epr

une

eln 1/

Pruning

Page 30: QCD CALCULATIONS FOR JET SUBSTRUCTURE

Hadronisation effects for mMDTha

dron

/ pa

rton

l = m2/(pt2 R2)

hadronisation v. analytics (quark jets)

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

mMDT (zcut = 0.10)0-mass mMDT (zcut = 0.10)0-mass mMDT (ycut = 0.11)

mMDT (ycut = 0.11)analytics

analytics (×2.4)

0.5

1

1.5

10-6 10-4 0.01 0.1 1

10 100 1000Hadronisation produces:1. a shift in the squared

jet mass 2. a shift in the jet’s (or

prong’s) momentum

Same power behaviour but with competing signs:

d�NP

dm' d�PT

dm

1 + a

⇤NP

m

Page 31: QCD CALCULATIONS FOR JET SUBSTRUCTURE

Examples of NLO checks

-20

0

20

40

60

80

100

120

140

-12 -10 -8 -6 -4 -2

d m

/ d

ln v

ln v

Coefficient of (CF _s//)2 for pruning R=0.8, zcut=0.4

Event2Event2 - Analytic

-30

-25

-20

-15

-10

-5

0

5

10

15

-12 -10 -8 -6 -4 -2

d m

/ d

ln v

ln v

Coefficient of (CF_s//)2 for modified mass-drop R=0.8, ycut=0.1

Event2Event2 - Analytic

0

2

4

6

8

10

12

14

16

18

20

-12 -10 -8 -6 -4 -2

d m

/ d

ln v

ln v

Coefficient of CF CA (_s//)2 for modified mass-drop R=0.8, ycut=0.2

Event2Event2 - Analytic

-250

-200

-150

-100

-50

0

50

100

-12 -10 -8 -6 -4 -2

d m

/ d

ln v

ln v

Coefficient of (CF _s//)2 for trimming R=0.8, Rsub=0.2, zcut=0.15

Event2Event2 - Analytic

Page 32: QCD CALCULATIONS FOR JET SUBSTRUCTURE

Other properties of mMDT• Flatness of the background is a desirable property (data-driven analysis, side bands)• ycut can be adjusted to obtain it (analytic relation)• Role of μ, not mentioned so far• It contributes to subleading logs and has small impact if not too small (μ>0.4)• Filtering only affects subleading (NnfiltLL) terms

l/m

dm

/ dl

l = m2/(pt2 R2)

Effect of filtering: quark jets

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

mMDT (ycut=0.13)mMDT + filtering

0

0.1

10-6 10-4 0.01 0.1 1

10 100 1000

l/m

dm

/ dl

l = m2/(pt2 R2)

Effect of µ parameter: quark jets

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

µ = 1.00µ = 0.67µ = 0.40µ = 0.30µ = 0.20

0

0.1

10-6 10-4 0.01 0.1 1

10 100 1000

Page 33: QCD CALCULATIONS FOR JET SUBSTRUCTURE

ATLAS MDT

0 50 100 150 200 250 300

GeV1

dmm

d m1

00.002

0.0040.0060.008

0.01

0.0120.014

0.016

0.018 -1 L = 35 pb02010 Data,

Statistical Unc.

Total Unc.

Pythia

Herwig++

Cambridge-Aachen R=1.2 > 0.3qqSplit/Filtered with R

< 600 GeVT500 < p = 1, |y| < 2PVN

ATLAS

Jet Mass [GeV]0 50 100 150 200 250 300

MC

/ D

ata

0.20.40.60.8

11.21.41.61.8

MC

/ D

ata

0.20.40.60.8

11.21.41.61.8

•ATLAS measured the jet mass with MDT• Different version of the tagger with Rmin=0.3 between the prongs

Page 34: QCD CALCULATIONS FOR JET SUBSTRUCTURE

ATLAS MDT

0 50 100 150 200 250 300

GeV1

dmm

d m1

00.002

0.0040.0060.008

0.01

0.0120.014

0.016

0.018 -1 L = 35 pb02010 Data,

Statistical Unc.

Total Unc.

Pythia

Herwig++

Cambridge-Aachen R=1.2 > 0.3qqSplit/Filtered with R

< 600 GeVT500 < p = 1, |y| < 2PVN

ATLAS

Jet Mass [GeV]0 50 100 150 200 250 300

MC

/ D

ata

0.20.40.60.8

11.21.41.61.8

MC

/ D

ata

0.20.40.60.8

11.21.41.61.8

l/m

dm

/ dl

l = m2/(pt2 R2)

ATLAS MDT (Pythia 6 MC, quarks)

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

MDT (ycut=0.09)

ATLAS MDT (Rmin=0.3, ycut=0.09)

0

0.1

10-6 10-4 0.01 0.1 1

10 100 1000

•ATLAS measured the jet mass with MDT• Different version of the tagger with Rmin=0.3 between the prongs

•This cut significantly changes the tagger’s behaviour: mass minimum• The single-log region is reduced (and can even disappear)• We hope that future studies will be able to avoid this


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