# Presentation: 919709

## Jet finding Algorithms at Tevatron

- B.Andrieu (LPNHE, Paris)
-  On behalf of the             collaboration

- Outline:

- Introduction
- The Ideal Jet Algorithm
- Cone Jet Algorithms: RunII/RunI, D0/CDF
*k* Jet Algorithm
- Summary

- 

- 

- 

- 

- 

## Jets: from parton to detector level

- Non-perturbative processes not predictable  QCD inspired phenomenology

## Jets: from parton to detector level

- Infrared unsafety

- Collinear unsafety

- Figures from hep-ex/0005012

## Jet definition

- Associate “close” to each other “particles” ****** Clustering (Jet Algorithm)**
- “particles” can be:
- “close” ?  Distance ****** ***R *=    or  *Y * (preferred in RunII) for Cone Algorithm****** **relative *p**T*  for *k* algorithm
- Calculate jet 4-momentum from “particles” 4-momenta ****** Recombination scheme**
- invariant under longitudinal boosts
-  Snowmass scheme (RunI): *E**T *-weighted recombination scheme in (,)
-  covariant or E-scheme (preferred for RunII): 4-momenta addition
- used at the end of clustering but also during clustering process(not necessarily the same, still preferrable)

**Two things need to be done to define a jet:**

- partons (analytical calculations or parton showers MC)
- “hadrons” = final state particles (MC particles or charged particles in trackers)
- towers (or cells or preclusters or any local energy deposits)

- independent of the distance from interaction point
- invariant under longitudinal boosts

## The ideal jet algorithm for pp

- infrared and collinear safety
- invariance under longitudinal boosts
- fully specified and straightforward to implement
- same algorithm at the parton, hadron and detector level
- boundary stability (kinematic limit of inclusive jet cross section at *E**T *=** *s*/2)
- factorisation (universal parton densities)
- independence of detector detailed geometry and granularity
- minimal sensitivity to non-perturbative processesand multiple scatterings at high luminosity
- minimization of resolution smearing/angle bias
- reliable calibration
- maximal reconstruction efficiency (find all jets) vs minimal CPU time
- replicate RunI cross sections while avoiding theoretical problems

**Theory**

**Experiment**

- -

**General**

- Compare jets at the parton, hadron and detector level
- ****** ****Jet algorithms should ensure**

## Run I Cone Algorithm

- Based on Snowmass algorithm: *E**T *-weighted recombination scheme in ()
**Preclustering** (D0, similar algorithm for CDF)Note: Tower segmentation in () space: **D0 ******** 0.1 X 0.1****, ****CDF ******** 0.11 X 0.26**
- start from seeds (= hadronic towers with *p**T* >1 GeV ordered in decreasing *p**T*)
- cluster (and remove) all contiguous calorimeter towers around seed in a *R*= 0.3 cone
**Clustering**
- start from preclusters (ordered in decreasing *E**T*)
- proto-jet candidate = all particles within *R**cone* of the precluster axis in () space**CDF: keep towers of the original precluster through all iterations (ratcheting)**
- proto-jet direction compared before/after recombination  iterate until it is stable

**Merging/Splitting** (treat overlapping proto-jets)
*E*12 > *f* . Min(*E*1*,E*2)  Merge jets
*E*12 < *f* . Min(*E*1*,E*2)  Split jets = assign each particle to its closest jet
**D0: *****f***** = 50 %, use only clusters with *****E******T *****> 8 GeV****    -    ****CDF: *****f***** = 75 %**
** ****Final calculation of jet variables** (modified Snowmass scheme)
- scalar addition of *E**T * (D0) or *E**  *(CDF) of particles to determine jet *E**T * or *E*
- addition of 3-momenta of particles to determine jet direction, then ()Note: this procedure is not Lorentz invariant for boosts along beam axis**CDF: *****E******T *****= *****E *****sin(********)**

## Why new algorithms for Run II?

- Different in D0 and CDF
- Not infrared and collinear safe due to the use of seeds(collinear safety ensured at sufficiently large *E**T* : *E**T *>20 GeV with *p**T**min *(seed) = 1 GeV in D0)
- Preclustering difficult to match at parton or hadron level
- CDF ratcheting not modelled in theory
- Need to introduce a new parameter (*R**sep*) in jet algorithm at parton level to match theory predictions to measurements (S.D. Ellis *et al.*, PRL69, 3615 (1992))
- Not invariant under boosts along beam axis
****** ****2 new Cone Algorithms proposed for RunII**  (G.C. Blazey *et al.*, “RunII Jet Physics”, hep-ex/0005012)
- Seedless Cone Algorithm
- RunII (= Improved Legacy or Midpoint) Cone Algorithm
****** ****Use *****k********* algorithm (already used in RunI)**

**Run I Cone algorithms have many drawbacks**

## Seedless Cone Algorithm

**Not really “seedless” **
****** ****Use enough seeds (all towers) to find all stable cones**
- First step:
- form cone around seed, recalculate cone direction (Snowmass recombination)
- stop processing seed if the cone centroid is outside of the seed tower**CDF: use tower size X 1.1 to avoid boundary problems**
- Secund step similar to Run I Cone algorithm:
- use the cones formed in first step (pre-protojets) as seeds
- form cone around seed and recalculate cone direction (*E*-scheme = 4-momentum addition)
- iterate until cone direction after/before recombination is stable
- Streamlined (faster) option
- Stop iteration in second step if the cone centroid is outside of the seed tower Only miss low *E**T * protojets or stable directions within the same tower

-  Infrared and collinear safe
-  Probably close to Ideal for a Cone algorithm
-  Even the streamlined version is very computational intensive
****** ****Use an approximation of Seedless Algorithm ******** RunII Cone**

## RunII Cone Algorithm (hep-ex/0005012)

- QCD calculation at fixed order *N* only 2*N* –1 possible positions for stable cones (*p**i *, *p**i**+p**j*, *p**i**+p**j**+p**k *,...)
- Data: consider seeds used in RunI Cone algorithms as partons in addition to seeds, use ‘midpoints’ i.e. *p**i**+p**j*, *p**i**+p**j**+p**k *,...
- only need to consider seeds all within a distance *R* < 2*R**cone*
- only use midpoints between proto-jets (reduce computing time)
- otherwise algorithm similar to RunI

**How to build a valid approximation of the seedless algorithm?**

*E*-scheme recombination = 4-momenta addition
- use true rapidity *Y*  instead of pseudo-rapidity in *R*
- use all towers as seeds (*p**T *> 1 GeV)
- splitting/merging: *p**T*  ordered, *f* = 50 %

**Other specifications of the suggested RunII cone Algorithm**

## D0 Run II Cone Algorithm: Preclustering

- Simple Cone Algorithm
- Start from particles withhighest *p**T * and *p**T* >500 MeV
- Precluster formed from all particles within a cone of*r *=0.3 (*r *=0.2) for Cone jets with *R **** *0.5 (*R *= 0.3) (RunI: only neighbouring cells)
- Remove particles as soon as they belong to a precluster
- No cone drifting
- Precluster 4-momentum calculated using theE-scheme

## D0 Run II Cone Algorithm: Clustering

*p**T* < 0.5 Jet *p**T**min*
- # Iterations = 50(avoids infinite cycles)

- D0 Run II Cone Algorithm: Clustering

- Use all preclusters as seeds (*p**T* ordered), except those close to already found protojets(*R* (precluster,protojet)< 0.5 *R**cone *)

- Repeat same clustering for midpoints* except:
- No condition on close protojet
- No removal of duplicates

- Cone drifting until cone axiscoincides with jet direction

- Remove duplicates

- * - only pairs are considered
-    - calculated using *p**T* –weighted mean

## D0 Run II Cone Algorithm: Merge/Split

- D0 Run II Cone Algorithm: Merge/Split

- Use *p**T** * ordered list of proto-jets (from seeds and midpoints)

- If some energy is shared between two proto-jets, decide to split/merge depending on shared fraction

- Re-order list of merged/splitted jets

- Recalculate 4-momenta of merged/splitted jets

## The Smaller Search Cone Algorithm

- Jets might be missed by RunII Cone Algorithm (S.D. Ellis *et al.*, hep-ph/0111434) low *p**T * jets
- too close to high *p**T * jet to form a stable cone (cone will drift towards high *p**T * jet)
- too far away from high *p**T * jet to be part of the high *p**T * jet stable cone
- proposed solution
- remove stability requirement of cone
- run cone algorithm with smaller cone radius to limit cone drifting(*R**search * = *R**cone* / ** 2)
- form cone jets of radius *R**cone* around protojets found with radius *R**search*

- Problem of lost jets seen by CDF, not seen by D0 A physics or an experimental problem?
- Proposed solution not satisfactory in terms of elegance and simplicity
-  D0 prefers using RunII Cone without Smaller Search Cone

**Remarks**

## k Algorithm

- originally proposed for *e*+*e*- colliders, then adapted to hadron colliders (S. Catani *et al.*, NPB406,187 (1993))
- universal factorisation of initial-state collinear singularities
- infrared safe: soft partons are combined first with harder partons  result stable when energy of soft partons -> 0
- collinear safe: two collinear partons are combined first in the original parton
- no issue with merging/splitting

*p**T*  ordered list of particles  form the list of *d**i* = (*p**T*i)2
- calculate for all pairs of particles, *d**i* *j *= Min((*p**T*i)2, (*p**T*j)2) *R/D*
- find the minimum of all *d**i*  and *d**i* *j*
- if it is a *d**i* , form a jet candidate with particle *i*  and remove *i*  from the list
- if not, combine *i *and *j  *according to the *E*-scheme
- use combined particle *i *+ *j * as a new particle in next iteration
- need to reorder list at each iteration  computing time  O(N3) (N particles)
- proceed until the list of preclusters is exhausted

**Description of inclusive**** *****k********* ****algorithm** (Ellis&Soper, PRD48, 3160, (1993))

**Remarks**

## D0 Run II k Algorithm

- Use *E *-scheme for recombination
- Use *p**T** * ordered list of preclusters (geometrical 2x2 preclustering)
- Remove preclusters with *E* < 0

- Either merge pairs of preclusters which are closest to each other in relative *p**T* or form a jet with each isolated low *p**T* precluster
- When all preclusters have been associated  to a jet, calculate 4-momenta of all jets
- Apply a *p**T**min * cut on  jets (*p**T* > 8 GeV)

## Summary

- RunII (Midpoint) Cone Algorithm clear improvement over RunI Algorithm
- Many problems or questions still remain open (not exhaustive list):
- D0 uses only RunII Cone (Midpoint) Algorithm (no smaller search cone)
- CDF still uses JetClu (RunI) Cone Algorithm + Smaller Search Cone Algorithm
- D0 implementation does not fully follow RunII Cone recommendations
*p**T**min* / 2 cut on proto-jets candidates
- preclustering
- seeds too close to already found protojets not used
- influence of parameters for precluster formation?
- usefulness of a *p**T* cut on proto-jets before merging/splitting at high luminosity?
- procedure chosen for merging/splitting optimal?
- origin of the difference D0 vs CDF for lost jets problem?
- In contrast*, k* algorithm is conceptually simpler, theoretically well-behaved, although less intuitive. It also needs studies, as for the RunII Cone Algorithm (jet masses, sensitivity to experimental effects, ...).   However, shouldn’t we put more effort on using *k* algorithm and less         on reproducing results obtained with RunI algorithms? (personal statement)