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 k T 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 jet’s momentum is then computed from the momenta of the particles that belong to that jet (so called recombination scheme) the jet’s 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 (k T )

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 z aj HEP event: list of particles (partons hadrons calorimeter cells towers preclusters) result: list of jets recombination matrix the 4-momentum q j of the j-th jet expressed by 4-momenta p a of the particles

9. Recombination matrix z aj z aj 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 z aj 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 z aj

10. Recombination matrix z aj i.e. no more than 100% of each particle is assigned to jets the fraction of the energy of the a-th particle that does not go into any jet the 4-momentum q j of the j-th jet expressed by 4-momenta p a of the particles (a=1,2,...,n parts ) the fraction of the energy of the a-th particle can be positive only

11. Optimal Jet Definition any allowed value of the recombination matrix {z aj } describes some jet configuration the desired optimal jet configuration is the one that minimizes some function  ({z aj }) 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  aj is the angle between the a-th particle and j-th jet E a is the energy of the a-th particle R>0 is a parameter with a similar meaning as the cone radius width of the j-th jetenergy outside jets

22. Optimal Jet Finder: fixed number of jets desired optimal jet configuration corresponds to minimum of  ({z aj }) 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 1 1 1 1 a point inside the simplex a point on the boundary of 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  ({z aj }) 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 n tries it was sufficient to take n tries  10 (even n tries  3) in the cases we studied compromise between quality of jets found and computing time used

33. OJF: number of jets to be determined 1.assume some small positive parameter  cut, which is analogous to the jet resolution parameter y cut in binary recombination algorithms 2.start with (for example) n jets =1 3.find the best configuration with the number of jets equal to n jets as described previously (starting from n tries different initial configurations and choosing the best configuration) 4.check if 5.if so, this is the final jet configuration and the final number of jets is n jets 6.if not, increase n jets 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 R < (angle between two partons) < 2R the event is reconstructed to 2 jets with a soft gluon as a seed the event is reconstructed into 1 jet

39. Collinear splitting at NNLO R < (angle between two partons) < 2R the event is reconstructed to 1 jet using the most energetic parton (red) as the seed the “red” parton splits into 2 collinear pieces; now the “blue” parton is the most energetic and the event is reconstructed into 2 jets

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. f>50%) 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 p ab =p a +p b, p abc =p a +p b +p c,... are used as seeds still problem with overlapping cones successive recombination algorithms (binary algorithms), such as JADE, Durham (k T )

44.  is an infrared and collinear safe

45. Successive recombination algorithms for each pair of particles compute the “distance” d ab between the particles in the pair, for example choose the pair with the smallest distance and if min(d ab ) < y cut (y cut is the jet resolution parameter) combine the two particles into one particle using p new =p a +p b and go back to the beginning (now having the number of particles decreased by one) if min(d ab )  y cut then stop (we achieved the final jet configuration)

46. OJF and k T k T 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 k T (still a jet is not a cone)

47. OJF and k T OJF is much faster that k T if a large number of particles has to be analyzed average time per event  n parts for OJF  n 3 parts for k T 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 k T the cubic dependence determines the way k T 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 k T 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 (k T ) 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 ALGORITHMstatistical error of W-boson mass (corresponding to 1000 experimental events) based on Fisher’s information [MeV] (±3) Durham (k T )105 JADE118 OJF106

51. time [seconds] number of particles \ cells average time per event kTkT OJF n tries =10 OJF n tries =5

52. time [seconds] number of particles \ cells average time per event kTkT OJF n tries =10, n tries =5

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. M W 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 Fisher’s 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; C f,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 k T for a large number of input cells

62. Optimal Jet Definition: cylindrical kinematics