matteo cacciari work in collaboration with - lpthe · journées de la frif jets 31 janvier 2008...

Journées de la FRIF31 Janvier 2008

JetsMatteo Cacciari

LPTHEWork in collaboration with

Gavin Salam, Juan Rojo and Gregory Soyez

A jet is something that happens in high energy events:

a collimated bunch of hadrons flying roughly in the same direction

Why?

Often you don’t need a fancy algorithm to ‘see’ it

Why?

Jets are usually related to an underlying perturbative dynamics

(i.e. quarks and gluons)

The purpose of a ‘jet clustering’ algorithm is then toreduce the complexity of the final state, simplifying many hadrons to

simpler objects that one can hope to calculate

Why?

Jets (pretty much like phenomenologists...) lie therefore at the interface between data and theory

Jet Algorithm

{pi} {jk}jet algorithm

particles,4-momenta,

calorimeter towers, ....

jets

A jet algorithm maps the momenta of the final state particles into the momenta of a certain number of jets:

Most algorithms contain a resolution parameter, R, which controls the extension of the jet

In order for them to work properly jet algorithms should be

Infrared safe (for the calculation to converge)

fast (if many particles have to be clustered)

These (and other) requirements where widely acknowledged as early as 1990 (Snowmass accord)

Perhaps quite surprisingly, they had not yet been met in 2005

soft emission shouldn’t change jetscollinear splitting shouldn’t change jets

In order to be realistically applicable at detector level

Jet Algorithm

sequential recombination algorithmsbottom-up approach: combine particles starting from closest one in some distance measure. Repeat until few left: call them jets

cone algorithmstop-down approach: find coarse regions of energy flow. How? Find stable cones (i.e. their axis coincides with sum of momenta of particles in it)

Infrared safe but slow with many particles (O(N3))

Not Infrared safe in approximate implementations

Impossibly slow (O(N2N)) if done exactly

[One day to cluster 30000 particles, a single heavy ions collision at LHC]

[1017 years to cluster just 100 particles]

Two main classes of jet algorithms

Work because of mapping closeness ⇔QCD divergenceLoved by e+e-, ep and theorists

Work because QCD only modifies energy flow on small scalesLoved by pp and (fewer) theorists

Until some time ago cone was infrared unsafe and kt was slow

What happened next?

- kt made fast (MC, Salam, hep-ph/0512210)

- cone made safe (Salam, Soyez, arXiv: 0704.0292)

Both implementations (and a lot more) available via FastJetwww.lpthe.jussieu.fr/~salam/fastjet

(presently used by all the LHC Collaborations)

Recombination algorithms

kt algorithm

The definition of a sequential clustering algorithm is extremely simple.

For instance, take the longitudinally invariant kt:

Calculate the distances between the particles:

Calculate the beam distances:

Combine particles with smallest distance or, if diB is smallest, call it a jet

Find again smallest distance and repeat procedure until no particles are left

diB = k2

ti

S. Catani, Y. Dokshitzer, M. Seymour and B. Webber, Nucl. Phys. B406 (1993) 187S.D. Ellis and D.E. Soper, Phys. Rev. D48 (1993) 3160

di j =min(k2ti,k2

t j)!"2+!#2

R2

The kt algorithm has, however, an apparent drawback:

finding all the distances is an N2 operation, to be repeated N times

⇒ naively, the kt jet algorithm scales like N3

FastJetTo improve the speed of the algorithm we must find more efficiently which

particle is “close” to another and therefore gets combined with it

Observation (MC, G.P. Salam, hep-ph/0512210):

If i and j form the smallestand

kti < ktj ⇒ ΔRij ≤ ΔRik ∀ k ≠ j

Translation from mathematics:

When a particle gets combined with another, and has the smallest kt, its partner will be its geometrical nearest neighbour on the cylinder

spanned by y and ϕ

This means that we need to look for partners only among the O(N) nearest neighbours of all particles

(a few neighbours each × N particles)

i.e. j is the geometrical nearest neighbour of i

di j =min(k2ti,k2

t j)!"2+!#2

R2

FastJet

Our problem has now become a geometrical one:how to find efficiently the (nearest) neighbour(s) of a point

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Widely studied problem in computational geometryTool: Voronoi diagram

Definition: each cell contains the locations which have the given point as nearest neighbour

Key feature: once the Voronoi diagram is constructed, the nearest neighbour of a point will be in one of the O(1) cells sharing an edge with its own cell

Example : the G(eometrical) N(earest) N(eighbour) of point 7 will be found among 1,4,2,8 and 3 (it turns out to be 3)

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The dual of a Voronoi diagram is a Delaunay triangulation

FastJet

The FastJet algorithm:

Construct the Voronoi diagram of the N particles using the CGAL library

O(N lnN)

Find the GNN of each of the N particles. Construct the dij distances, store the results in a priority queue (C++ map) O(N lnN)

Merge/eliminate particles appropriately

Update Voronoi diagram and distances’ map O(lnN)repeat N times

MC and G.P. Salam, hep-ph/0512210

Overall, an O(N ln N) algorithm

NB. Results identical to standard kt algorithm. This is NOT a new jet-finder.

FastJet performance

Time taken to cluster N particles:

1 ms

10 s

Almost two orders of magnitude gain at small N (related O(N2) implementation)

Large-N region now reachable

10-5

10-4

10-3

10-2

10-1

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t (s

)

N

KtJet FastJet

OJFMidPoint

TevatronLHC (single LHC (c. 20 LHCinteraction) interactions) Heavy Ion

Cone algorithms

!"#$%& !"#$%&

'() '*)

+

,+ - . /!-

0++

/++

.++

-++

+

,+ - . /!-

0++

/++

.++

-++

MidPoint Cone Infrared Unsafety

Three hard particles clustered into two cones

Addition of a soft particles changes the hard jets

configuration: three stable cones are found

The problem is that the specific stable-cone search procedure used by MidPoint cannot find all stable cones

Text

Calculations cost real money:~ 100 theorists ×15 years ≈100 M€

[Salam, Soyez, arXiv: 0704.0292]

Infrared (un)safety

Q: How often are the hard jets changed by the addition of a soft particle?

A:

10-5

10-4

10-3

10-2

10-1

1

Fraction of hard events failing IR safety test

JetClu

SearchCone

MidPoint

Midpoint-3

PxCone

Seedless (bad split-merge)

Seedless (good split-merge)

50.1%

48.2%

16.4%

15.6%

16.6%

0.05%

< 10-9SISCone

badgood

Sala

m &

Soy

ez

SISCone speed

SISCone

MidPoint

kt (FastJet)

SISCone as fast as MidPoint → no penalty for infrared safety!

Jet areas and subtraction

So far, old jet clustering, just better and/or faster

High speed and infrared safety allow for a qualitatively new use of jet clustering, through new features:

Jet areas

pT (parton)

A simple event

pT (jet) ~ pT (parton)

A simple event

The parton radiates, but we can usually collect most of its momentum into a jet

pT (jet) ~ pT (parton)

A messier (i.e. more realistic) event

+

Average underlyingmomentum density

×‘size’ of the jet

In a realistic set-up underlying event (UE) and pile-up (PU) from multiple collisions produce many soft particles

0

0.01

0.02

0.03

0.04

0 50 100 150 200 250

1/N

dN

/dm

ass

reconstructed Z mass [GeV]

R=0.7, LHC

kt, no UE

+ UE

+ high-lumi (100 fb-1

/yr)

Challenge at high-luminosity machines:reconstruct objects from jets when a lot of spurious activity is present

You’d like to be able to subtract this extra stuff

from the jets and get back to the correct Z mass

Consequences of UE and PU

Can we get to know the momentum density of the UE/PU?Can we subtract it from the jet to find the ‘true’ momentum?

But...wait...what is the ‘size’ of a jet??

What is the ‘size’ of a jet?

rapidity-azimuth plane

φ

y

Consider an event made up of a number of particles

{pi} {jk}jet-finder algorithmparticles, 4-momenta,

calorimeter towers, .... jets


φ

y

But... where exactly does a jet end, and another begins?

What is the ‘size’ of a jet?

The clustering procedure assigns each particle to a jet:

Jet Area

One idea: tile the plane, count the cells of a jet, sum the areas


φ

y

But what do I do when different jets share a cell?

Jet Area

Obviously, make the cells smaller to improve accuracy


φ

y

Unfortunately, particles being pointlike, the area tends to zero!

Jet Area

Next try, use the convex hull


φ

yBut what do I do if they overlap?

Moreover, what about the ‘no man’s land’ ?

The Active Jet AreaWe propose the following definition:


φ

y

The ‘active area’ of a jet is (proportional to) the number of uniformly distributed infinitely soft

particles that get clustered in it

The Active Jet Area

After the clustering, a given set of ghosts belong to each jet


φ

y

Their number (times the average area of a single ghost) defines the catchment area of the jet

The Jet AreaThe definition of active area mimics the behaviour of the jet-clustering algorithms in the presence of a large number of

randomly distributed soft particles

Tools needed to implement it:

1. An infrared safe jet algorithm (the ghosts should not change the jets)

2. A reasonably fast implementation (we are adding thousands of ghosts)

This is the link to physics!

NB. A passive area can also be defined. Determined by clustering a single ghost particles, it mimics the behaviour of point-like radiation.

Both are available

Jet areas

kt Cam/Aa

SISCone anti-kt

[MC, Salam, Soyez]

Active and Passive areas: analytical approach

A jet of ‘radius’ R will surely have area πR2, right?

Well, it depends.....

Passive areas of a single hard particle are indeed πR2

However, active areas are not: { kt 0.81 πR2

Cam/Aa 0.81 πR2

SISCone 0.25 πR2

Recall that ‘area’ is how much rubbish a jet can pick up.Its knowledge is essential in order to subtract it from measurements

[MC, Salam, Soyez]


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A(!

12)

/ "R

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!12/R

thin lines: passive area

thick lines: active area

(a)

kt

cam

SISCone

Real events have more than a single hard particle. Add a second (soft) one at a distance Δ12

Δ12

hard soft

The area depends on the distance

between the particles

1 2


Finally, weigh the probability of emission of the soft particle with the leading QCD matrix element:

Δ12

hard soft

1 2

The result is an anomalous dimension: areas change with transverse momentum of the jet in a predictable way:

〈!area〉 =ZC1"s(pt2!12)

#

dpt2

pt2

[d!12

!12

]+

C1

!b0ln

"s(Q0)"s(Rpt1)

〈!area〉 = coeff

kt Cam/Aa SISCone

passive active0.56 0.520.08 0.09

-0.06 0.12

( (

!AJA

,R"/#R

2

Herwig, R=1

kt, qq

theory (Q0=0.62 GeV)

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2.55 10 100 1000

!AJA

,R"/#R

2

Cam, qq


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!AJA

,R"/#R

2

pt (GeV)

SISCone(f=0.75), qq


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!AJA

,R"/#R

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Herwig, R=1

kt, gg


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!AJA

,R"/#R

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Cam, gg


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2pt (GeV)

SISCone(f=0.75), gg


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Areas anomalous dimensions at (simulated) LHC

Averages and dispersions evolution

from Monte Carlo simulations in good

agreement with simple LL calculations

A practical application of areas: subtraction

When a hard event is superimposed on a roughly uniformly distributed background, study of transverse momentum/area of each jet allows one to determine the noise density ρ (and its fluctuation) on an event-by-event basis

Once measured, the background density can be used to correct the transverse momentum of the hard jets:

phard jet, correctedT = p

hard jet, rawT −!×Areahard jet

[MC, Salam, arXiv:0707.1378]

Reconstructed Z’ mass

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0.005

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0.015

1900 2000 2100

1/N

dN

/dm

[G

eV

-1]

m [GeV]

kt, R=0.7

LHC, high lumi

Z! at 2 TeV

no pileup

no pileup, sub

pileup

pileup, sub

Let’s discover a leptophobic Z’ and measure its mass:

MC simulation:m = 2000 GeV, width ~ 10 GeV

Naive measurement with PU: m ~ 2050 GeV, width ~ 60 GeV

Measurement after subtraction:m ~ 2000 GeV, width ~ 25 GeV

Conclusions

Both cone- and recombination-type jets can be properly defined and implemented

Their infrared safety allows one to analyse them as legitimate observables in QCD, including more exotic (and previously unexplored) characteristics like their area

The area itself can be used for background subtraction, opening the way to a more accurate, and theoretically motivated, use of jet clustering in high luminosity and even heavy ions collisions environments

(Hopefully) more to come....

matteo cacciari work in collaboration with - lpthe · journées de la frif jets 31 janvier 2008...

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