1	Building Better Jets A Work in Progress (Largely with Joey Huston, Matthias Tönnesmann, Dave Soper and Walter Giele)
2	The Goal is 1% strong Interaction Physics (where Run I was ~ 10%) Want to precisely connect What we can measure, e.g., E(y,f) in the detector To What we can calculate, e.g., arising from small numbers of partons as functions of E, y,f Issues: Uncertainties in pdf’s Higher orders in perturbation theory Non-perturbative hadronization (& showering) Details (especially differences between groups) of algorithms & kinematics
3	"Warning:" Warning: We must all use the same algorithm!!
4	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.
5	Thus we want to associate “nearby” hadrons or partons into JETS Nearby in angle – Cone Algorithms - issue is “splashout” Nearby in momentum space – k_T Algorithm - issue is “splashin” But mapping of hadrons to partons can never be 1 to 1, event-by-event!
6	Think of the algorithm as a “microscope” for seeing the (colorful) underlying structure -
7	Note – 2 logically distinct phases Identify contents of jet – particles, calorimeter towers or partons – jet ID scheme 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!?
8	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
9	Snowmass Cone Algorithm Cone Algorithm – particles, calorimeter towers, partons in cone of size R, defined in angular space, e.g., Snowmass (h,j) CONE center - (h^C,j^C) CONE i Î C iff Energy Centroid
10	"“Flow vector”" “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 Þ Proto-jets (prior to split/merge) Þ unique, discrete jets event-by-event (at least in principle)
11	Consider the Snowmass “Potential” In terms of 2-D vector or define 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
12	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 3^rd (middle) cone
13	For example, consider 2 partons: yields potential with 3 minima – trial cones will migrate to minimum
14	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
15	"Overlapping stable cones must be..." Overlapping stable cones must be split/merged Depends on overlap parameter f_merge 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
16	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 k_T 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
17	E-scheme (4-vector) CONE i Ì C iff 4-vector ”Centroid” Stable (Arithmetically more complex than Snowmass)
18	Actually used by CDF and DÆ in run I for cone finding, and approximately equivalent to Snowmass. For jet E_T used - Snowmass (DÆ) – CDF - E-Scheme (Run II study proposal) – The differences matter! (in a 1% game)
19	For example, consider 2 partons: p₁=zp₂
20	Thus E_{T,4D, CDF} may be larger or smaller than E_T,scalar, depending on the kinematics
21	5% Differences (at NLO) !!
22	A different (and not completely consistent) view comes from Matthias – EKS style NLO calculation with CTEQ4m pdfs
23	Note that the PDFs are also still different on this scale
24	Streamlined Seedless Algorithm Data in form of 4 vectors in (h,j) 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
25	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 E_T– new kind of Splashout See Ellis, Huston & Tönnesmann, hep-ph/0111434
26	Lost Energy!? (DE_T/E_T~1%, Ds/s~5%)
27	Missed Towers – How can that happen?
28	Consider a simple model with 2 partons, E_T in ratio z and separated in angle by r
29	NLO Perturbation Theory – r = parton separation, z = E₂/E₁R_sep simulates the cones missed due to no middle seed
30	Consider the corresponding “potential” with 3 minima, expect via MidPoint or Seedless to find middle stable cone
31	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 -
32	Next s = 0.25 - larger effect, but the desired cones are still “obvious”!?
33	But it matters for the potential: as we increase s we wash out middle minimum and lose middle cone
34	Then washout out second minima, find only 1 stable cone
35	“Fix” Use R¢<R, e.g.,R/Ö2, during stable cone discovery, less sensitivity to energy at periphery Use R during jet construction Þ restores right cone, but not middle cone Helps some with Midpoint algorithm Does not help with Seedless (need even smaller R¢ ?) Þ still no stable middle cone
36	The Fixed potential (in red)
37	With Fix
38	Consider the number of events versus the jet ET difference for various R' values, distribution ~ symmetric for 1/Ö2 reduction
39	Make a second pass to find jets in the “leftovers”, R_2nd = R/Ö2, most have previously found “ jet neighbors”
40	The r -z plane, from Matthias black = 1 jet, green = 2 jets, red = 3 jets (merged to 1)
41	But Note – we are “fixing” to match JETCLU which is NOT the same as perturbation theory
42	Racheting – Why did it work? Must consider seeds and subsequent migration history of trial cones – yields separate potential for each seed
43	The “ratcheted” potential function looks like: Note the missing Q functions, those terms can be positive far from the seed, hence the “cutoffs”
44	With the k_T algorithm we can avoid seeds, R_sep, 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 d_ii, remove i from the list and define it to be a jet, if the smallest object is d_ij, remove i and j from the list and replace them with the merged object. For the new list (with one fewer item), repeat the calculation as above, until the list is empty.
45	r -z plane from Matthias
46	Apply to a the simple 2 parton configuration we used earlier, find 2 jets for r > D even for s = 0.25 unless z is small
47	So little splash-out problem but splash-in is real – vacuums up extra energy that happens to be around
48	To test the robustness of the k_T jets found consider the results of various analyses applied to the event we just looked at – E_T’s of leading 2 jets; only the leading jet is nearly invariant (but E_T still varies)
49	At NLO the k_T algorithm is just the cone algorithm with R_sep = 1 and D = R. The original study suggested that R = 0.7 (R_sep = 2) was comparable to D = 1. For the better (phenomenological) value R_sep = 1.3, D = 0.83 is a better match to R = 0.7.
50	Better yet, DØ has data for D = 1.0 (4-D)
51	With more “Modern” pdf’s
52	To Test for splash-in try measuring the D dependence of the cross section; assume splash-in a D²(~area) At E_T = 100 GeV
53	BUT .. Want to get rid of seeds, ratcheting and all that! Time for a new idea!! (?) Forget jets event-by-event Use 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)
54	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” function e.g.,
55	We can define JEFs or Corresponding to The Cone jets are the same function evaluated at the discrete solutions of (stable cones)
56	Simulated calorimeter data & JEF
57	“Typical” CDF event in y, f
58	Here is the JEF version of the event we saw earlier
59	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).
60	The trick with JEF is defining observables, e.g. The probability distribution (for a CDF type rapidity acceptance and CDF E_T = E sinq definition) is i.e., probabilities µ area/pR² The corresponding number of jets (JEFs) above E_T,min, per event, is
61	Apply to the “CDF” event and find, where the data points are the CDF found jets Jet E_T
62	Apply to Pythia event, see cone (R = 0.7) analysis jets as bumps in the distribution
63	The JEF definition in NLO yields a cross section much like the usual cone algorithm:
64	"The mass of a single..." The mass of a single JEF (jet) is With probability density And event occupancy probability
65	Applied to a W®1 jet in (simulated events)
66	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 & k_T algorithms Study the JEF idea It is essential that we share the details during Run II! (which often did not happen in Run I)