OPTIMAL JET FINDER

1
OPTIMAL JET FINDER

• Ernest Jankowski
• University of Alberta

in collaboration with D. Grigoriev F. Tkachov
2
Acknowledgements

• Alberta Ingenuity Fund
• J Gordin Kaplan Graduate Student Award

3
Plan of the talk

• introduction
• Optimal Jet Definition and its implementation
• comparison with cone and kT algorithms
• benchmark physics application test based on
• motivations behind OJD

4
Introduction

• final states of particle collisions in HEP
  experiments often consist of sprays of hadrons
• simple interpretation in the collision quarks
  with high kinetic energy are produced or released
  from the colliding particles
• but quarks interact very strongly at large
  distances and never appear as free particles
  (confinement)
• any attempt to separate free quarks results in
  production of extra pairs of quarks and
  antiquarks
• which recombine into colorless states hadrons
• if the collision energy is high enough the
  hadrons appear in sprays, called jets

5
Hadronic jets an example
6
Jet algorithms

• roughly jets correspond to quarks and gluons
  produced in hard scattering process
• jet algorithm takes the final state hadrons and
  assigns them to jets
• a jets momentum is then computed from the
  momenta of the particles that belong to that jet
  (so called recombination scheme)
• the jets momentum corresponds approximately to
  the momentum of quarks and gluons in hard
  scattering process (partons in perturbative
  calculations)
• jet algorithms used cone algorithm and
  successive recombination algorithms, such as
  Durham (kT)

7
Optimal Jet Definition

• I will present so called Optimal Jet Definition
  (OJD)
• proposed by Fyodor Tkachov
• a short introduction to the subject is
• Phys. Rev. Lett. 91, 061801 (2003)
• FORTRAN 77 implementation of OJD
• called Optimal Jet Finder (OJF) is described in
• hep-ph/0301226 (Comp. Phys. Commun., in print)

8
Recombination matrix zaj
HEP event list of particles
(partons hadrons
calorimeter cells towers preclusters)
recombination matrix
the 4-momentum qj of the j-th jet expressed by
4-momenta pa of the particles
result list of jets

9
Recombination matrix zaj

• zaj describes the fraction of the a-th particle
  that belongs to the j-th jet (i.e. we split up
  particles between jets)
• conventional jet algorithms have zaj equal to 0
  or 1, i.e. a particle either entirely belongs to
  some jet or does not belong to that jet at all
• fragmentation and hadronization is always effect
  of interaction of (at least) two hard partons
  evolving into two jets, so some hadrons that
  emerge in this process can belong partially to
  both jets
• but this is also very convenient
  algorithmically, because a jet configuration is
  described by a set of continuous numbers zaj

10
Recombination matrix zaj
the 4-momentum qj of the j-th jet expressed by
4-momenta pa of the particles (a1,2,...,nparts)
the fraction of the energy of the a-th particle
can be positive only
the fraction of the energy of the a-th particle
that does not go into any jet
i.e. no more than 100 of each particle is
assigned to jets
11
Optimal Jet Definition

• any allowed value of the recombination matrix
  zaj describes some jet configuration
• the desired optimal jet configuration is the one
  that minimizes some function ?(zaj)
• details of ? are different for CM lepton-lepton
  collisions (spherical kinematics) and collisions
  involving hadrons (cylindrical kinematics), where
  boost invariance along the beam axis should be
  maintained

12
Optimal Jet Definition spherical kinematics
width of the j-th jet
energy outside jets
?aj is the angle between the a-th particle and
j-th jet Ea is the energy of the a-th
particle Rgt0 is a parameter with a similar
meaning as the cone radius
13
(No Transcript)
14
(No Transcript)
15
(No Transcript)
16
(No Transcript)
17
(No Transcript)
18
(No Transcript)
19
(No Transcript)
20
(No Transcript)
21
(No Transcript)
22
Optimal Jet Finder fixed number of jets

• desired optimal jet configuration corresponds to
  minimum of ?(zaj)
• the program finds a minimum iteratively using a
  simple gradient-based method
• facilitated by analytical formulas for gradient
• start with some candidate minimum (the initial
  jet configuration) and descend into a local
  minimum in subsequent iterations
• the initial jet configuration may be completely
  random

23
Minimization 2 jets
a point on the boundary of the simplex
a point inside the simplex
24
iteration 0
25
iteration 1
26
iteration 2
27
iteration 3
28
iteration 4
29
iteration 5
30
iteration 6
31
Local minima of ?

• each time the programs starts with a random
  configuration, it finds a local minimum which
  does not need to be the global minimum of
  ?(zaj)
• this is similar to how the result of cone
  algorithm depends on the initial position for the
  cone iterations
• but here in contrast with the cone algorithm, we
  know which local minimum we should choose the
  one that gives the smallest value of ?

32
Optimal Jet Finder number of tries

• in order to find the global minimum or increase
  the probability of finding it, the program tries
  a couple different random initial configurations
• OJF parameter number of tries ntries
• it was sufficient to take ntries ? 10
• (even ntries ? 3) in the cases we studied
• compromise between quality of jets found and
  computing time used

33
OJF number of jets to be determined

• assume some small positive parameter ?cut, which
  is analogous to the jet resolution parameter ycut
  in binary recombination algorithms
• start with (for example) njets1
• find the best configuration with the number of
  jets equal to njets
• as described previously (starting from ntries
  different initial configurations and choosing the
  best configuration)
• check if
• if so, this is the final jet configuration
• and the final number of jets is njets
• if not, increase njets by 1 and go to point 3

34
Cone algorithm

• cone algorithm defines a jet as all particles
  within a cone of certain radius in ??? space
• ?j, ?j are coordinates of the geometrical center
  of the cone ?a, ?a are coordinates of the
  particles
• ?j, ?j are found from the requirement that
  E?-weighted centroid computed from all particles
  in the cone coincides with the geometrical center
  of the cone

35
Cone algorithm

• ?j, ?j are found in an iterative procedure
• start with some trial position ?j, ?j for a cone
• for all particles within the cone compute
  E?-weighted centroid ?j, ?j (usually different
  from ?j, ?j)
• use the centroid as a new position of the center
  of the cone
• the content of the cone will change accordingly,
• update it and go to ?
• procedure ends when the cone center and
  E?_weighted centroid for all particles in the
  cone coincide a stable cone

36
Cone algorithm seeds

• the iterative procedure described above needs
  some initial cone position
• look for stable cones starting everywhere
• (i.e. at every tower or cell) very expensive
  computationally
• start only at energetive towers, so called seeds

37
Problems with seeds

• problems arise when we compare theory and
  experimental data
• when we apply the cone algorithm involving seeds
  to partons beyond the LO, soft radiation or
  collinear splitting of partons may change the jet
  configuration significantly
• whereas the experimental jet configuration
  remains unchanged

38
Soft radiation at NLO
with a soft gluon as a seed the event is
reconstructed into 1 jet
R lt (angle between two partons) lt 2R the event is
reconstructed to 2 jets
39
Collinear splitting at NNLO
the red parton splits into 2 collinear pieces
now the blue parton is the most energetic and
the event is reconstructed into 2 jets
R lt (angle between two partons) lt 2R the event is
reconstructed to 1 jet using the most energetic
parton (red) as the seed
40
Collinear splitting in the experiment
the particle in the center hits a single
calorimeter cell the cell has sufficient energy
to be a seed
the particle in the center hits two separate
cells none of the cells has enough energy to be
a seed
41
Cone algorithm overlapping cones

• after all stable cones are found, deal with
  overlapping cones
• if the fraction f of overlapping energy (with
  respect to the smaller energy jet) exceeds some
  threshold (e.g. fgt50) merge the two jets
• otherwise split two jets, assigning particles to
  the closest jet
• if there are more than two jets overlapping, the
  final result may depend on the ordering of this
  procedure (so some standard order has to be
  specified)
• it is difficult to take account of the merging \
  splitting procedure in theoretical calculations

42
Cone algorithm overlapping cones
experiment
theory
43
Solution to seed problems

• cone algorithm involving seeds becomes unstable
  at higher order perturbative calculations
• seedless algorithm look for jets everywhere
• Improved Legacy Cone Algorithm (ILCA)
• midpoints pabpapb, pabcpapbpc, ... are used
  as seeds
• still problem with overlapping cones
• successive recombination algorithms (binary
  algorithms), such as JADE, Durham (kT)

44
is an infrared and collinear safe
45
Successive recombination algorithms

• for each pair of particles compute the distance
  dab between the particles in the pair, for
  example
• choose the pair with the smallest distance and
• if min(dab) lt ycut (ycut is the jet resolution
  parameter) combine the two particles into one
  particle using pnewpapb and go back to the
  beginning (now having the number of particles
  decreased by one)
• if min(dab) ? ycut then stop (we achieved the
  final jet configuration)

46
OJF and kT

• kT merges only 2 particles at a time (binary
  recombination 2?1)
• other recombination schemes also considered 3?2,
  m?n
• OJF takes into account the global structure of
  the energy flow in the event, i.e.
• jet configuration is found from the momenta of
  all particles in the event (corresponds to m?n)
• OJF finds more regular jets that kT (still a jet
  is not a cone)

47
OJF and kT

• OJF is much faster that kT if a large number of
  particles has to be analyzed
• average time per event
• ? nparts for OJF
• ? n3parts for kT
• this does not matter when we apply the algorithm
  to theoretical calculations, but in experiments
  we have to analyze large number of calorimeter
  cells
• D0 ? 45000 cells, ATLAS ? 200 000 cells

48
OJF and kT

• the cubic dependence determines the way kT has to
  be used in experiments
• it cannot be applied directly at the level of
  cells
• a preclustering step is needed to reduce the
  input data to ? 200 preclusters (D0 and CDF
  practice)
• how does the preclustering affect measurements?
• it is difficult to account for the preclustering
  in theoretical calculations
• the preclustering is a completely independent
  procedure from the kT algorithm itself

49
Benchmark test W-boson mass extraction

• benchmark test based on the W-boson mass
  extraction from the process
• modeled on the OPAL analysis (CERN-EP-2000-099)
• we compared OJF with JADE and Durham (kT)
  algorithm (the best algorithm used by the OPAL
  collaboration)
• we obtained the same accuracy as Durham (still we
  did not explore all possibilities)
• we studied the speed of OJF and we found that it
  is much faster then Durham when a large number of
  calorimeter cells need to be analyzed

50
Quality of jets
ALGORITHM statistical error of W-boson mass (corresponding to 1000 experimental events) based on Fishers information MeV (3)
Durham (kT) 105
JADE 118
OJF 106
51
average time per event
time seconds
kT
OJF ntries10
OJF ntries5
number of particles \ cells
52
average time per event
time seconds
kT
OJF ntries10, ntries5
number of particles \ cells
53
Method of moments

• parameter estimation, a fundamental problem of
  mathematical statistic
• for an event P theory gives the probability
  density ?M(P) that depends on some parameter M,
  i.e. MW or ?s
• given an experimental sample of events, we want
  to estimate the best value of the parameter M
  with possibly small statistical error
• method of maximal likelihood
• reinterpreted as method of moments

54
Method of moments
depends on M
computed from the experimental sample
55
Method of moments
informativeness of the observable f, based on the
statistical error that f gives
Fishers information
Rao-Cramer inequality
56
Event representation basic shape observables
event particles
event as energy flow collinearly invariant
basic shape observables
f function of a direction only
57
Factorial estimate

• event ? values of all basic shape observables
  f(P)
• the optimal observable for the measurement of
  some parameter can be expressed as a combination
  of basic shape observables
• applying a jet algorithm we reduce the available
  information about the event P to make the
  analysis computationally manageable
• we use values of all basic shape observables
  f(Q), taken on the jet configuration

58
Hadronic event approximated by jets

• good approximations for f(P) could exist among
  functions that depend only on Q, which is a
  parameterization of P in terms of a few jets,
  found from the condition
• modeled after
• (q is partonic structure of the event) this
  expresses the fact that most of the information
  about the event is inherited from its
  gluon-and-quark structure

59
Factorial estimate
for any f Cf,R depends only on f, it does not
depend on the event or jet configuration
configuration
? depends only on the event P and the jet
configuration Q we choose the jet configuration
so that the loss of information about the event
is minimal generically for all f (we minimize
?) this is essentially the Optimal Jet Definition
60
Summary

• I presented the Optimal Jet Finder
• based on the global energy flow in the event
• infra-red and collinear safe no seed-related
  problems
• no overlapping jets
• returns additional numerical characteristics of
  the jet configuration found (so called dynamical
  width and soft energy) which may be helpful in
  construction of (quasi-) optimal observables in
  statistical problems
• much faster than kT for a large number of input
  cells

61
(No Transcript)
62
Optimal Jet Definition cylindrical kinematics