DLM and Its Applications at CDF March 8, Tsukuba K.Kondo 1. New Formulation of DLM Single path as a quantum process ★ Parton level cross section and likelihood ★ Transfer function Full dynamical likelihood ２. Applications 3 ． Mtop at CDF

Ｄｙｎａｍｉｃａｌ Ｌｉｋｅｌｉｈｏｏ ｄ Ｍｅｔｈｏｄ 量子力学的尤度法 Ｍｏｔｉｖａｔｉｏｎ Uncertainties in collider events quarks, gluons → jets momentum/energy measurements missing particles →MET parton-jet identification unknown initial state signal/background identification Spectroscopy: H 0 Collider ： L int d  Reconstruction of parton process based on theory and observation ＜ use of differential cross section ＞ Traditional d  /dx many events ＤＬＭ d  /d   single event

Ｃｏｎｃｅｐｔ of DLM DLM : Reconstruction/Inference of Parton Process beam observed theoreticalexperimental parton process

Differential cross section for a single event When (c 1,…,c n ) is given, integrate by (a,b): I (a*,b*) : parton flux factor d  /d  n (f) : Probability to have final phase space d  ｎ  f  for a unit luminosity of the incident beam. flux(particles/cm 2 /sec) particles/sec luminosity(particles/cm 2 ) particles

Cross section and probability density function (p.d.f.) Take an example of  = M. f(x;  ) = p.d.f. (Axiom No.1 for p.d.f.) ( Dalitz-Goldstein,... ) This is quite reasonable for a general p.d.f., but Nd  /dx depends on the choice of x, while d  /d  n does not. d  /d  n is a probability per unit beam luminosity. This concept is missing in the general p.d.f. f(x:M).

Normalization of likelihood at parton level We assume Bayes’ Postulate ( Principle of Equidistribution of Ignorance ). Quantities parameters:   variables: x, L int and N tot ( depend on . P(  ), l 0 : Prior probabilities. [ l 0 ] = [L] - 2 true value of 

A single path as a quantum process QuantizedDLM For a single path, the phase volume and Jacobian- scaled transfer variables are both quantized.

Phase space and observables  n (f)  3n p  3n x*  3n y* d  /d  n (f) d  /dy* d  /dx* w(y|x) JxJx

Resonance “s ｒ ” for reconstruction Phase space : If parton kinematics is determined by inferring s r with propagator factor  (s r ｒ ), resonance r occupies a unit p.s. The cross section per unit p.s. is given by whereif s ｒ is inferred if all c i are given n – j+1 j =1

Transfer Function defined by M.C. events Prior TF: M.C. events with the same cuts as applied to data Variable transformation from to : efficiency included ydx dy x 

Posterior TF x, y : indicates1-d quantity in this page TF includes efficiency.

Summary of QDLM We assume a single path occupies a unit p.s.  n *=1. The cross section for the path is given by The variable space in a single path is The likelihood for the k-th path in the i-th event is defined by Prior TF is obtained by the MC events satisfying the event selection criteria. Posterior TF is given by The expectation value of the event likelihood is given by The likelihood of dynamical constant  is =1 const.

Applications

Correction factor (mapping function) Why? Parton-jet matching is not guaranteed. 1 parton→parton shower→fragmentation→detector→jet(s) Even if a parton and a jet are matched, topology assignment is not unique. Many solutions for unmeasured momenta

Solving problems by likelihood The values of likelihood can be used for process (S vs. B) determination, right topology identification, right solution for missing momentum. By iteration, one gets better estimation of  measurement of dynamical parameter study of kinematics

戸谷・蛯名 TTbar 6 jet channel ： parton-jet idetification by Likelihood Quark level Observation level

久保 Parton ID by Moments in TTbar to l4j(1b tagged) Charged particle momentum distribution in a jet: M-. E-moments An example of  (I T )

TC-mass M(   ） 既知として事象再構成 How much does d  /d  help ?

Mtop

Top Detection  Events are energetic –Ht,SumEt  Events are central and spherical –|  |< 2.0, aplanarity  Two high E T b-jets –Displaced secondary vertex –Soft lepton inside jet  High energy jets and isolated leptons –missing Et from neutrino in leptonic modes –High Et jets  Possible additional jets from gluon radiation (isr,fsr)

CDF approaches to the top mass measurement History: DLM (1988) Top discovery (1994) Template, Multi-variant template DLM DZero ( Dalitz-Goldstein ) type MEAT, FLAME,MADCOW,... QDLM

CDF Takes Three Paths for The Top Mass (Fermilab today 05-May-25)

Why Top Mass Mt  175 GeV  Yukawa coupling  1Special role in EWSB? Dominant parameter in radiative corrections: quadratic in m t, logarithmic in m H Mt from precision EW measurements

Mtop Measurements Combined RunI mass: m t =178.0 ± 4.3 GeV/c 2 was: ± 5.1 GeV/c 2 –Higgs mass –Best-fit M H  113 GeV/c 2 –95% C.L. : M H < 237 GeV/c 2 Run II measurements –Systematic uncertainty largely dominated by jet energy correction: will be reduced –RunII goal is  m~2-3 GeV/c 2

Mtop: templates Mass templete method –Reconstruct one top mass for each event –Compare Mtop distribution to simulated top templates at various masses –Minimize -ln L vs Mtop to extract mass Dileptons: b-tagged l+jets: b-tagged l+jets w/ multivar templates: Probability for Mt from likelihood based on MC multidimesional templates loop over j-p assignments loop over P Z for impose kinematic constrains choose m that best fits event

Signal/Background separation by Likelihood Distributions We expect the background makes likelihood peak down when it is multiplied to signal events. CDF simulationParton Level x = Mtop y = –2ln(likelihood) Signal background Signal:175GeV background Signal : input mass Backgrounds: lower mass Peak : QDLM

More Checks on MC vs Data More likely to be background QDLM

Extracted top mass from RunII Observed :Total 22 events; electrons 12, Muons 10 Correct background-pulling 4.2 events expected. Fit : Two 2 nd order polynomials for positive/negative errors. QDLM

ILC での精密測定の物理 Top クォークの閾値近辺でのエネルギースキャン W/ トップクォークの質量 ワインバーグアングル の精密測定 → 高次補正に現れる新しい物理の精密検証 topmass

Conclusion DLM is easy to use for any process. It should be the standard method in the near future. The parton-jet, S/B and solution identifications can be improved by the likelihood itself. This is not fully demonstrated yet. Mapping function saves you at any stage of your analysis. It is ammusing to notice that Lagrangian Likelihood

Backup slides

DLM and DGM

Likelihood in DLM A single path likelihood ： A single event likelihood: expectation value of path likelihood Joint likelihood from multiple events ：

Maximum Likelihood Method (MLM) MLM in general:  i : data in i-th event,  : parameter f(  ;  ) : probability density function. Define L n (  ) =  i=1 n f(  i ;  ), then (  0 = true value of  ) MLM in DLM : c i0 : common to all paths in an event.   : common to all events For c i0  ik = c* ik, f(  ik ;  ) = l 0 d  /dc* ik  k  L 1 ik in an event For    i = y* i, f(  i ;  ) = l 0  i L 1 i for multi-events parameters

Top Production at Tevatron 15% Tevatron: 85% top-antitop pairs: single top: 0.88 pb 1.98 pb ~ one top event every 10 BILLION inelastic collisions

Jet Energy Corrections Determine true “particle”, “parton” jet E from measured jet E Non-linear response Uninstrumented regions Response to different particles Out of cone E loss Spectator interactions Underlying event Jet Energy Scale (MC derived) Total Systematics