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?
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)
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
!"#$%& !"#$%&
'() '*)
+
,+ - . /!-
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
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 (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
בsize’ 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
rapidity-azimuth plane
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
rapidity-azimuth plane
φ
y
But what do I do when different jets share a cell?
Jet Area
Obviously, make the cells smaller to improve accuracy
rapidity-azimuth plane
φ
y
Unfortunately, particles being pointlike, the area tends to zero!
Jet Area
Next try, use the convex hull
rapidity-azimuth plane
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:
rapidity-azimuth plane
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
rapidity-azimuth plane
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
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]
Active and Passive areas: analytical approach
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A(!
12)
/ "R
2
!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
Active and Passive areas: analytical approach
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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1.5
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2.55 10 100 1000
!AJA
,R"/#R
2
Cam, qq
theory (Q0=0.27 GeV)
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!AJA
,R"/#R
2
pt (GeV)
SISCone(f=0.75), qq
theory (Q0=0.21 GeV)
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5 10 100 1000
!AJA
,R"/#R
2
Herwig, R=1
kt, gg
theory (Q0=2.24 GeV)
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!AJA
,R"/#R
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Cam, gg
theory (Q0=0.62 GeV)
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!AJA
,R"/#R
2pt (GeV)
SISCone(f=0.75), gg
theory (Q0=0.34 GeV)
5 10 100 1000 0
0.5
1
1.5
2
2.5
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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0.015
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....
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