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### Building Better JetsA Work in Progress

(Largely with Joey Huston, Matthias Tönnesmann, Dave Soper and Walter Giele)

S. D. Ellis CDF/D0/Theory Jet Workshop 12/16/02

The Goal is 1% strong Interaction Physics (where Run I was ~ 10%)

Want to precisely connect

- What we can measure, e.g., E(y,) in the detector

To

- What we can calculate, e.g., arising from small numbers of partons as functions of E, y,

Issues: Uncertainties in pdf’s

Higher orders in perturbation theory

Non-perturbative hadronization (& showering)

Details (especially differences between groups) of algorithms & kinematics

S.D. Ellis: CDF/DO/Theory Jet Workshop 12/16/02

Why Jet Algorithms?

- We “understand” what happens at the level of partons and leptons, i.e., LO theory is simple.
- We want to map the observed (hadronic) final states onto a representation that mimics the kinematics of the energetic partons; ideally on a event-by-event basis.
- But we know that the partons shower (perturbatively) and hadronize (nonperturbatively), i.e., spread out.

S.D. Ellis: CDF/DO/Theory Jet Workshop 12/16/02

Thus we want to associate “nearby” hadrons or partons into JETS

- Nearby in angle – Cone Algorithms - issue is “splashout”
- Nearby in momentum space – kT Algorithm - issue is “splashin”
- But mapping of hadrons to partons can never be 1 to 1, event-by-event!

S.D. Ellis: CDF/DO/Theory Jet Workshop 12/16/02

Think of the algorithm as a “microscope” for seeing the (colorful) underlying structure -

S.D. Ellis: CDF/DO/Theory Jet Workshop 12/16/02

Note – 2 logically distinct phases

- Identify contents of jet – particles, calorimeter towers or partons – jet IDscheme
- Combine kinematic properties of jet contents (e.g., 4-vectors) to find jet kinematic properties – recombination scheme
- May not want to do both steps with the same parameters!?

S.D. Ellis: CDF/DO/Theory Jet Workshop 12/16/02

History – Starting in Snowmass

Start over 10 years ago with the “Snowmass Accord” (or the Snowmass Cone Algorithm).

Idea was to have an agreed upon algorithm (hence accord) that everyone would use. But, in practice, it was flawed

Was not efficient – experimenters used seeds to limit where one looked for jets – this introduces IR sensitivity at NNLO

Did not treat issue of overlapping cones – split/merge question

S.D. Ellis: CDF/DO/Theory Jet Workshop 12/16/02

Snowmass Cone Algorithm

- Cone Algorithm – particles, calorimeter towers, partons in cone of size R, defined in angular space, e.g., Snowmass (,)
- CONE center - (C,C)
- CONE i C iff
- Energy
- Centroid

S.D. Ellis: CDF/DO/Theory Jet Workshop 12/16/02

“Flow vector”

- Jet is defined by “stable” cone:
- Stable cones found by iteration: start with cone anywhere (and, in principle, everywhere), calculate the centroid of this cone, put new cone at centroid, iterate until cone stops “flowing”, i.e., stable Proto-jets (prior to split/merge) unique, discrete jets event-by-event (at least in principle)

S.D. Ellis: CDF/DO/Theory Jet Workshop 12/16/02

Consider the Snowmass “Potential”

- In terms of 2-D vector ordefine a potential
- Extrema are the positions of the stable cones; gradient is “force” that pushes trial cone to the stable cone, i.e., the flow vector

S.D. Ellis: CDF/DO/Theory Jet Workshop 12/16/02

But note:

- Theoretically can look “everywhere” and find all stable cones
- Experimentally reduce size of analysis by putting initial cones only at seeds – energetic towers or clusters of towers – thus introducing undesirable IR sensitivity and missing certain possible 2-jets-in-1 configurations
- May NOT find 3rd(middle) cone

S.D. Ellis: CDF/DO/Theory Jet Workshop 12/16/02

For example, consider 2 partons: yields potential with 3 minima – trial cones will migrate to minimum

S.D. Ellis: CDF/DO/Theory Jet Workshop 12/16/02

One of a list of HIDDEN issues, all of which influence the result

- Energy Cut on towers kept in analysis (e.g., to avoid noise)
- (Pre)Clustering to find seeds (and distribute “negative energy”
- Energy Cut on precluster towers
- Energy cut on clusters
- Energy cut on seeds kept

Starting with seeds find stable cones by iteration

In JETCLU, “once in a seed cone, always in a cone”, the “ratchet” effect

S.D. Ellis: CDF/DO/Theory Jet Workshop 12/16/02

Overlapping stable cones must be split/merged

Depends on overlap parameter fmerge

Order of operations matters

All of these issues impact the content of the “found” jets

- Shape may not be a cone
- Number of towers can differ, i.e., different energy
- Corrections for underlying event must be tower by tower

S.D. Ellis: CDF/DO/Theory Jet Workshop 12/16/02

To address these issues, the Run II Study group Recommended

Both experiments use

- (legacy) Midpoint Algorithm – always look for stable cone at midpoint between found cones
- Seedless Algorithm
- kT Algorithms
- Use identical versions except for issues required by physical differences – all of this in preclustering??
- Use (4-vector) E-scheme variables for jet ID and recombination

S.D. Ellis: CDF/DO/Theory Jet Workshop 12/16/02

E-scheme (4-vector)

- CONE i C iff
- 4-vector
- ”Centroid”
- Stable (Arithmetically more complex than Snowmass)

S.D. Ellis: CDF/DO/Theory Jet Workshop 12/16/02

Actually used by CDF and D in run I for cone finding, and approximately equivalent to Snowmass. For jet ET used -

- Snowmass (D) –
- CDF -
- E-Scheme (Run II study proposal) –

The differences matter! (in a 1% game)

S.D. Ellis: CDF/DO/Theory Jet Workshop 12/16/02

For example, consider 2 partons: p1=zp2

S.D. Ellis: CDF/DO/Theory Jet Workshop 12/16/02

Thus ET,4D, CDF may be larger or smaller than ET,scalar, depending on the kinematics

S.D. Ellis: CDF/DO/Theory Jet Workshop 12/16/02

5% Differences (at NLO) !!

S.D. Ellis: CDF/DO/Theory Jet Workshop 12/16/02

A different (and not completely consistent) view comes from Matthias – EKS style NLO calculation with CTEQ4m pdfs

S.D. Ellis: CDF/DO/Theory Jet Workshop 12/16/02

Note that the PDFs are also still different on this scale

S.D. Ellis: CDF/DO/Theory Jet Workshop 12/16/02

Streamlined Seedless Algorithm

- Data in form of 4 vectors in(,)
- Lay down grid of cells (~ calorimeter cells) and put trial cone at center of each cell
- Calculate the centroid of each trial cone
- If centroid is outside cell, remove that trial cone from analysis, otherwise iterate as before
- Approximates looking everywhere; converges rapidly
- Split/Merge as before

S.D. Ellis: CDF/DO/Theory Jet Workshop 12/16/02

A NEW issue for Midpoint & Seedless Cone Algorithms

- Compare jets found by JETCLU (with ratcheting) to those found by MidPoint and Seedless Algorithms
- “Missed Energy” – when energy is smeared by showering/hadronization do not always find 2 partons in 1 cone solutions that are found in perturbation theory, underestimate ET– new kind of Splashout
- See Ellis, Huston & Tönnesmann, hep-ph/0111434

S.D. Ellis: CDF/DO/Theory Jet Workshop 12/16/02

Lost Energy!? (ET/ET~1%, /~5%)

S.D. Ellis: CDF/DO/Theory Jet Workshop 12/16/02

Missed Towers – How can that happen?

S.D. Ellis: CDF/DO/Theory Jet Workshop 12/16/02

Consider a simple model with 2 partons, ET in ratio z and separated in angle by r

Look at energy in cone of radius R Energy Distribution

S.D. Ellis: CDF/DO/Theory Jet Workshop 12/16/02

seed

NLO Perturbation Theory – r = parton separation, z = E2/E1Rsep simulates the cones missed due to no middle seedNaïve Snowmass

With Rsep

r

r

S.D. Ellis: CDF/DO/Theory Jet Workshop 12/16/02

Consider the corresponding “potential” with 3 minima, expect via MidPoint or Seedless to find middle stable cone

S.D. Ellis: CDF/DO/Theory Jet Workshop 12/16/02

But in “real” life the parton’s energy is smeared by hadronization, etc. Simulate with gaussian smearing in angle of width s. Smooths the energy in the cone distribution, larger s, larger effect. First s = 0.1 -

Smeared parton energy

Energy in cone

S.D. Ellis: CDF/DO/Theory Jet Workshop 12/16/02

Next s = 0.25 -larger effect, but the desired cones are still “obvious”!?

Smeared parton energy

Energy in cone

S.D. Ellis: CDF/DO/Theory Jet Workshop 12/16/02

But it matters for the potential: as we increase wewash out middle minimum and lose middle cone

S.D. Ellis: CDF/DO/Theory Jet Workshop 12/16/02

Then washout out second minima, find only 1 stable cone

S.D. Ellis: CDF/DO/Theory Jet Workshop 12/16/02

“Fix”

- Use R<R, e.g.,R/2, during stable cone discovery, less sensitivity to energy at periphery
- Use R during jet construction

restores right cone, but not middle cone

- Helps some with Midpoint algorithm
- Does not help with Seedless (need even smaller R ?)

still no stable middle cone

S.D. Ellis: CDF/DO/Theory Jet Workshop 12/16/02

The Fixed potential (in red)

S.D. Ellis: CDF/DO/Theory Jet Workshop 12/16/02

With Fix

S.D. Ellis: CDF/DO/Theory Jet Workshop 12/16/02

Consider the number of events versus the jet ET difference for various R' values, distribution ~ symmetric for 1/2 reduction

S.D. Ellis: CDF/DO/Theory Jet Workshop 12/16/02

Make a second pass to find jets in the “leftovers”, R2nd = R/2, most have previously found “ jet neighbors”

Irreducible (JetClu)level at aboutR‘ ~ R/2= R0.25

S.D. Ellis: CDF/DO/Theory Jet Workshop 12/16/02

The -z plane, from Matthiasblack = 1jet, green = 2 jets, red = 3 jets (merged to 1)

= 0

= 0.1

= 0.25

= 0.25, fix50

= 0.25, fix25

S.D. Ellis: CDF/DO/Theory Jet Workshop 12/16/02

But Note – we are “fixing” to match JETCLU which is NOT the same as perturbation theory

S.D. Ellis: CDF/DO/Theory Jet Workshop 12/16/02

Racheting – Why did it work?Must consider seeds and subsequent migration history of trial cones – yields separate potential for each seed

INDEPENDENT of smearing, first potential finds stable cone near 0, while second finds stable cone in middle (even when right cone is washed out)! ~ NLO Perturbation Theory!!

S.D. Ellis: CDF/DO/Theory Jet Workshop 12/16/02

The “ratcheted” potential function looks like:Note the missing functions, those terms can be positive far from the seed, hence the “cutoffs”

S.D. Ellis: CDF/DO/Theory Jet Workshop 12/16/02

With the kT algorithm we can avoid seeds, Rsep, merging etc., but “splash-in” can be an issue

- In this algorithm we deal with a list of 4-vectors (preclusters and/or protojets) – in terms of a “size” parameter D define
- If the smallest object is dii, remove i from the list and define it to be a jet, if the smallest object is dij, remove i and j from the list and replace them with themerged object. For the new list (with one fewer item), repeat the calculation as above, until the list is empty.

S.D. Ellis: CDF/DO/Theory Jet Workshop 12/16/02

Apply to a the simple 2 parton configuration we used earlier, find 2 jets for > D even for = 0.25 unless z is small

D = 0.7, = 0.71, z = 1.0

D = 0.7, =0.8, z = 0.1

2 jets

1 jet

S.D. Ellis: CDF/DO/Theory Jet Workshop 12/16/02

So little splash-out problem but splash-in is real – vacuums up extra energy that happens to be around

A more realistic (Pythia) DØ event with D = 1.0, and “preclustering”; last view shows R = 1 circles around jets

S.D. Ellis: CDF/DO/Theory Jet Workshop 12/16/02

To test the robustness of the kT jets found consider the results of various analyses applied to the event we just looked at – ET’s of leading 2 jets; only the leading jet is nearly invariant (but ET still varies)

S.D. Ellis: CDF/DO/Theory Jet Workshop 12/16/02

At NLO the kT algorithm is just the cone algorithm with Rsep = 1 and D = R. The original study suggested that R = 0.7 (Rsep = 2) was comparable to D = 1. For the better (phenomenological) value Rsep = 1.3, D = 0.83 is a better match to R = 0.7.

Snowmass Kinematics

4-D kinematics

S.D. Ellis: CDF/DO/Theory Jet Workshop 12/16/02

Better yet, DØ has data for D = 1.0 (4-D)

Assume 2 Gev Splash-in

S.D. Ellis: CDF/DO/Theory Jet Workshop 12/16/02

To Test for splash-in try measuring the D dependence of the cross section; assume splash-in D2 (~area)At ET = 100 GeV

S.D. Ellis: CDF/DO/Theory Jet Workshop 12/16/02

BUT .. Want to get rid of seeds, ratcheting and all that!Time for a new idea!! (?)Forget jets event-by-eventUse JEF – Jet Energy Flow

- See Tkachov, et al. (circa 1995); Giele & Glover (1997); Sterman, et al. (2001), Berger, et al. hep-ph/0202207 (Snowmass 2001)

S.D. Ellis: CDF/DO/Theory Jet Workshop 12/16/02

Each event produces a JEF distribution,not discrete jets

- Each event = list of 4-vectors
- Define 4-vector distribution where the unit vector is a function of a 2-dimensional angular variable
- With a “smearing” functione.g.,

S.D. Ellis: CDF/DO/Theory Jet Workshop 12/16/02

We can define JEFs

or

Corresponding to

The Cone jets are the same function evaluated at the discrete solutions of (stable cones)

S.D. Ellis: CDF/DO/Theory Jet Workshop 12/16/02

Simulated calorimeter data & JEF

S.D. Ellis: CDF/DO/Theory Jet Workshop 12/16/02

“Typical” CDF event in y,

Found cone jets

JEF distribution

S.D. Ellis: CDF/DO/Theory Jet Workshop 12/16/02

Here is the JEF version of the event we saw earlier

S.D. Ellis: CDF/DO/Theory Jet Workshop 12/16/02

Since JEF yields a “smooth” distribution for each event (compared to “non-analytic algorithms), we expect that

- The JEF analysis is more amenable to resummation techniques and power corrections analysis in perturbative calculations.
- The required multi-particle phase space integrations are largely unconstrained, i.e.,more analytic, and easier (and faster) to implement.
- The analysis of the experimental data from an individual event should proceed more quickly (no need to identify jets event-by-event).
- Signal to background optimization can now include the JEF parameters (and distributions).

S.D. Ellis: CDF/DO/Theory Jet Workshop 12/16/02

The trick with JEF is defining observables, e.g.

- The probability distribution (for a CDF type rapidity acceptance and CDF ET = E sin definition) is i.e., probabilities area/R2
- The corresponding number of jets (JEFs) above ET,min, per event, is

S.D. Ellis: CDF/DO/Theory Jet Workshop 12/16/02

Apply to the “CDF” event and find, where the data points are the CDF found jets

Jet ET

Jet ET

Jet ET

S.D. Ellis: CDF/DO/Theory Jet Workshop 12/16/02

Apply to Pythia event, see cone (R = 0.7) analysis jets as bumps in the distribution

S.D. Ellis: CDF/DO/Theory Jet Workshop 12/16/02

The JEF definition in NLO yields a cross section much like the usual cone algorithm:

S.D. Ellis: CDF/DO/Theory Jet Workshop 12/16/02

The mass of a single JEF (jet) is

- With probability density
- And event occupancy probability

S.D. Ellis: CDF/DO/Theory Jet Workshop 12/16/02

Applied to a W1 jet in (simulated events)

From

J.M. Butterworth

S.D. Ellis: CDF/DO/Theory Jet Workshop 12/16/02

Summary

- There are many challenges before we get to 1% precision QCD! The details now matter!
- At the same time we have many possible avenues to study! Need to “optimize” Cone & kT algorithms Study the JEF idea
- It is essential that we share the details during Run II! (which often did not happen in Run I)

S.D. Ellis: CDF/DO/Theory Jet Workshop 12/16/02

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