57
Ian Ross - University of Wisconsin - Preliminary Exam 1 Study of ZZ→4 Leptons with CMS at the LHC Ian Ross Preliminary Exam

Study of ZZ→4 Leptons with CMS at the LHC

  • Upload
    steffi

  • View
    40

  • Download
    3

Embed Size (px)

DESCRIPTION

Study of ZZ→4 Leptons with CMS at the LHC. Ian Ross Preliminary Exam. Outline. ZZ→4ℓ The LHC and CMS Analysis Simulation and reconstruction Comparison to Tevatron data Future Next steps of analysis Conclusions. Standard Model. Describes fundamental particles and interactions - PowerPoint PPT Presentation

Citation preview

Page 1: Study of ZZ→4 Leptons with CMS at the LHC

Ian Ross - University of Wisconsin - Preliminary Exam1

Study of ZZ→4 Leptons with CMS at the LHC

Ian RossPreliminary Exam

Page 2: Study of ZZ→4 Leptons with CMS at the LHC

Ian Ross - University of Wisconsin - Preliminary Exam2

Outline

• ZZ→4ℓ

• The LHC and CMS

• Analysis

• Simulation and reconstruction

• Comparison to Tevatron data

• Future

• Next steps of analysis

• Conclusions

Page 3: Study of ZZ→4 Leptons with CMS at the LHC

Ian Ross - University of Wisconsin - Preliminary Exam3

Standard Model

• Describes fundamental particles and interactions• Fermions compose matter• Bosons mediate forces

• Higgs boson• Infuses other particles with

mass• H→ZZ→4ℓ: “Golden Channel”

for discovery at the LHC

Page 4: Study of ZZ→4 Leptons with CMS at the LHC

Ian Ross - University of Wisconsin - Preliminary Exam4

Importance of ZZ→4ℓ signals

• Higgs discovery• H→ZZ*→4ℓ• e/μ modes give clean signature

across range of mH

• BR becomes prominent for MH > 2 MZ

• WW branch has higher BR, but more difficult reconstruction

• Beyond the Standard Model• ZZ decays of new particles• Anomalous triple gauge

couplings

Higgs Decay Modes

Page 5: Study of ZZ→4 Leptons with CMS at the LHC

Ian Ross - University of Wisconsin - Preliminary Exam5

ZZ Production at the LHC• LHC collisions have √s=7 TeV

• MC analysis done at √s=10 TeV

• ZZ decays• ZZ→4ℓ decays rare, but provide

cleanest signalDecay Mode BR

ZZ→4 Leptons 1%ZZ→Hadrons 49%

ZZ→Hadrons+Neutrinos 28%ZZ→Hadrons+Leptons 14%

ZZ→Neutrinos 4%ZZ→Neutrinos+Leptons 4%

Process σ at √s=7 TeV σ at √s=10 TeVpp→ZZ ~5 pb 7.3 pb

gg→H→ZZ(400 GeV MH)

~1.4 pb 2 pb

Page 6: Study of ZZ→4 Leptons with CMS at the LHC

Ian Ross - University of Wisconsin - Preliminary Exam6

Large Hadron Collider (LHC)

• pp collider• 27 km circumference• Four primary detectors:

• ATLAS, CMS – General purpose detectors• ALICE – Heavy ion experiments• LHCb – Forward detector for b-physics LHC Dipole

Page 7: Study of ZZ→4 Leptons with CMS at the LHC

Ian Ross - University of Wisconsin - Preliminary Exam7

LHC Proton-Proton Collisions

• Designed for:• 14 TeV center of mass energy• 2808 bunches per beam• 1011 protons per bunch• Luminosity up to 1034 cm-2 s-1

• Collect 100 fb-1 per year• 2010-2011 plan:

• 7 TeV center of mass energy• 720 bunches per beam• 7∙1010 protons per bunch• Luminosity up to ~1032 cm-2 s-1

• Collect 1 fb-1 of data

Page 8: Study of ZZ→4 Leptons with CMS at the LHC

Ian Ross - University of Wisconsin - Preliminary Exam8

Compact Muon Solenoid (CMS)

Hadronic Calorimeter (HCAL)

TrackerElectromagnetic Calorimeter

(ECAL)

SolenoidMuon System

Page 9: Study of ZZ→4 Leptons with CMS at the LHC

Ian Ross - University of Wisconsin - Preliminary Exam9

Detecting Particles in CMS

Page 10: Study of ZZ→4 Leptons with CMS at the LHC

Ian Ross - University of Wisconsin - Preliminary Exam10

CMS: Silicon Tracker• Identifies tracks, measures

charge and transverse momentum (pT)

• Pixel detectors nearest interaction point, then layers of strip detectors

• 66M pixel readout channels• 9.6M strip readout channels• 1.2 m in radius, 5.6 m in length• 205 m2 of Si coverage• Extends to |η| ≤ 2.5• Resolution:

Page 11: Study of ZZ→4 Leptons with CMS at the LHC

Ian Ross - University of Wisconsin - Preliminary Exam11

Tracker Performance – LHC Collisions @ 7 TeV

D+→K-π+π+ andD-→K+π-π-

MD=1.869GeV• Momentum requirements:

• p > 1.5 GeV• pT > 0.1 GeV

• Require secondary vertex with total charge ±1

• Shows excellent tracker performance

• Momentum resolution • Secondary vertex finding

Page 12: Study of ZZ→4 Leptons with CMS at the LHC

Ian Ross - University of Wisconsin - Preliminary Exam12

CMS: ECAL

σE

=2.7%E

+ 0.5% +150MeVE

Noise

Page 13: Study of ZZ→4 Leptons with CMS at the LHC

Ian Ross - University of Wisconsin - Preliminary Exam13

ECAL Performance -- LHC Collisions @ 7 TeV

• π0 and η observations indicate excellent early ECAL performance

Page 14: Study of ZZ→4 Leptons with CMS at the LHC

Ian Ross - University of Wisconsin - Preliminary Exam14

CMS: HCAL• Measures energy of hadronic

showers and missing ET

• 9500 readout channels• Central and barrel -- brass

and scintillator calorimeter coverage: |η| ≤ 3• Resolution:

• Forward -- steel and quartz calorimeter coverage: 3 ≤ |η| ≤ 5• Resolution:

σE ⎛ ⎝ ⎜

⎞ ⎠ ⎟2

=115%E

⎛ ⎝ ⎜

⎞ ⎠ ⎟2

+ 5.5%( )2

σE ⎛ ⎝ ⎜

⎞ ⎠ ⎟2

=280%E

⎛ ⎝ ⎜

⎞ ⎠ ⎟2

+ 11%( )2

Page 15: Study of ZZ→4 Leptons with CMS at the LHC

Ian Ross - University of Wisconsin - Preliminary Exam15

HCAL Performance -- LHC Collisions @ 7 TeV

• Reasonable agreement between MC and data

Page 16: Study of ZZ→4 Leptons with CMS at the LHC

Ian Ross - University of Wisconsin - Preliminary Exam

Dijet event @ 7 TeV

• Nice dijet event showing ECAL and HCAL deposits.

16

Page 17: Study of ZZ→4 Leptons with CMS at the LHC

Ian Ross - University of Wisconsin - Preliminary Exam17

CMS: Muon System• Provides muon detection and

momentum measurements for high-pT muons

• Drift Tube Chambers (DTs) and Cathode Strip Chambers (CSCs) give precise position measurements• DT coverage -- 0 < |η| < 1.2• CSC coverage -- 0.9 < |η| < 2.4

• Resistive Plate Chambers (RPCs) provide precise timing• 0 < |η| < 1.6

• Muon chamber hits matched to tracker hits for low-pT momentum resolution

Page 18: Study of ZZ→4 Leptons with CMS at the LHC

Ian Ross - University of Wisconsin - Preliminary Exam18

Muon System Performance – LHC Collisions @ 7 TeV

• Good muon system performance so far

Muons built in muon system matched to

tracker hits

MJ/Ψ=3.096 GeV

Page 19: Study of ZZ→4 Leptons with CMS at the LHC

Ian Ross - University of Wisconsin - Preliminary Exam19

CMS Trigger

• 25 ns bunch crossings yield up to 1 GHz event rate at 1034cm-2sec-1 (40 MHz ∙ 25 interactions per crossing )

• Level 1 Trigger• Hardware selection reduces

rate to 100 kHz• High Level Trigger (HLT)

• Software selection utilizing computing farm• Reconstructs events, reduces

rate to 300Hz

Page 20: Study of ZZ→4 Leptons with CMS at the LHC

Ian Ross - University of Wisconsin - Preliminary Exam20

CMS: Trigger (Level 1)• Calorimeter Trigger

• Regional Calorimeter Trigger (RCT) -- Find e/γ, hadronic deposits

• Global Calorimeter Trigger (GCT)• Finds jets, calculates missing

transverse energy• Sorts objects from RCT

• Muon Trigger• Finds segments, tracks

regionally• Global Muon Trigger

• Sorts muons, checks isolation

Muon TriggerMuon TriggerHFHF HCALHCAL ECALECAL RPCRPC CSCCSC DTDT

PatternPatternComparatorComparator

Trigger Trigger

RegionalRegionalCalorimeterCalorimeter

TriggerTrigger

4 4 μμ

e, J, Ee, J, ETT, H, HTT, E, ETTmissmiss

Calorimeter TriggerCalorimeter Trigger

max. 100 kHz L1 Accept

Global TriggerGlobal Trigger

Global Muon TriggerGlobal Muon Trigger

GlobalGlobalCalorimeterCalorimeter

TriggerTrigger

Local Local DT TriggerDT Trigger

Local Local CSC TriggerCSC Trigger

DT TrackDT TrackFinderFinder

CSC TrackCSC TrackFinderFinder

40 M

Hz

pipe

line,

lat

ency

< 3

.2 μ

s

Muon TriggerMuon Trigger

• Global trigger makes acceptance decision, passes to HLT

Page 21: Study of ZZ→4 Leptons with CMS at the LHC

Ian Ross - University of Wisconsin - Preliminary Exam

L1 ECAL Trigger Performance @ 7 TeV

21

• For 2 GeV trigger threshold, clusters over 4 GeV trigger 100% of the time in both barrel (left) and endcap (right)

Barrel Endcap

Page 22: Study of ZZ→4 Leptons with CMS at the LHC

Ian Ross - University of Wisconsin - Preliminary Exam22

CMS: High Level Trigger• e/γ HLT

• Start with ECAL objects, match to tracks and pixel hits from tracker• Reconstruction includes

bremsstrahlung losses• Apply ET threshold, isolation,

other criteria• μ HLT

• Find tracks in muon systems• Find consistent tracks in

tracker system• Match muon tracks to tracker

candidates

Page 23: Study of ZZ→4 Leptons with CMS at the LHC

Ian Ross - University of Wisconsin - Preliminary Exam23

HLT Performance (e/γ) @ 7 TeV

• Reasonable agreement between data and MC

ΔR = Δϕ 2 + Δη 2

Page 24: Study of ZZ→4 Leptons with CMS at the LHC

Ian Ross - University of Wisconsin - Preliminary Exam

Z candidate @ 7 TeV

24

Page 25: Study of ZZ→4 Leptons with CMS at the LHC

Ian Ross - University of Wisconsin - Preliminary Exam25

Event Simulation

Generation Detector Simulation

ReconstructionCMSSW

GEANT4PYTHIA,

MADGRAPH•Simulate physical processes

•Simulate interaction with matter•Creates detectors hits like real data

•Rebuild physics objects•Same software is used on real data

Page 26: Study of ZZ→4 Leptons with CMS at the LHC

Ian Ross - University of Wisconsin - Preliminary Exam26

Muon Reconstruction

•Combined reconstruction• Muon tracks matched to

tracker hits• Provides vastly improved low

pt resolution

•Require < 0.15 to match with generated muons

Muon pt (GeV)

Effi

cien

cy

Reconstruction Efficiency

Muon Reconstruction Efficiency (MC)

Efficiency = (Generated muons matched to reconstructed muons)(All generated muons)

ΔR = Δϕ 2 + Δη 2

Page 27: Study of ZZ→4 Leptons with CMS at the LHC

Ian Ross - University of Wisconsin - Preliminary Exam27

Electron Reconstruction

•Calorimeter Reconstruction•ECAL superclusters generated

to include bremmstrahlung photons

•Find tracker hits consistent with ECAL deposits

• Require < 0.15 to match with generated electrons

ΔR = Δϕ 2 + Δη 2

Electron pt (GeV)E

ffici

ency

Reconstruction Efficiency

e Reconstruction Efficiency (MC)

Efficiency = (Generated electrons matched to reconstructed electrons)(All generated electrons)

Page 28: Study of ZZ→4 Leptons with CMS at the LHC

Ian Ross - University of Wisconsin - Preliminary Exam

Important Backgrounds

• ZZ→4ℓ (Signal)• σ = 7.3 pb @ 10 TeV

• qq/gg→ttbar (Background)• σ = 242.8 pb @ 10 TeV• ttbar→WbWb→ℓνℓνbb

• b’s hadronize, can produce (or fake) leptons

• Z+jets (Background)• σ = 3600 pb @10 TeV• Z decays leptonically• Jets can produce (or fake) leptons

28

Page 29: Study of ZZ→4 Leptons with CMS at the LHC

Ian Ross - University of Wisconsin - Preliminary Exam29

MC Analysis @ 10 TeV• pp→ZZ→4ℓ Monte Carlo Dataset

• σ = 0.1033 pb• 100,000 events generated with PYTHIA6• Event weight = 10-3

• Most Important Backgrounds:• Z+jets

• σ = 3600 pb• 1,000,000 events generated with MADGRAPH• Event weight = 3.6

• ttbar• σ = 242.8 pb• 500,000 events generated with PYTHIA6• Event weight = 0.5

All datasets normalized to 1fb-1

Page 30: Study of ZZ→4 Leptons with CMS at the LHC

Ian Ross - University of Wisconsin - Preliminary Exam30

Preselection• Require 4 leptons (e/μ with pT > 5 GeV) combined into two Z

candidates with: • | Mℓℓ1 - 91.2 | < 15 GeV

• Z candidate with higher pt lepton within 15 GeV of MZ

• | Mℓℓ2 - 91.2 | < 25 GeV• Second Z candidate within 25 GeV of MZ to remove ZZ*

events• Require leptons to be within tracker acceptance of |η| < 2.5

Process Events expected in 1 fb-1

ZZ→4ℓ 8.0

Z+jets 112

ttbar 47

Signal

Background

Need to apply criteria to

distinguish signal from background

Page 31: Study of ZZ→4 Leptons with CMS at the LHC

Ian Ross - University of Wisconsin - Preliminary Exam31

Starting M4ℓ Spectrum

• Only preselection applied (previous slide) • Stacked Plot

Events

ZZ→4ℓ 8.0

Z+jets 112

ttbar 47

Page 32: Study of ZZ→4 Leptons with CMS at the LHC

Ian Ross - University of Wisconsin - Preliminary Exam

Z mass peaks after preselection

• Signal is indistinguishable• Applying discrimination cuts to electrons will allow us to

distinguish signal from background

32

Page 33: Study of ZZ→4 Leptons with CMS at the LHC

Ian Ross - University of Wisconsin - Preliminary Exam•33

Electron Isolation

•Veto cone

•Outer Cone

•Track Isolation•Outer cone radius = 0.3; veto cone radius = 0.015

•TrackIso=∑pT>0.7 GeV tracks in outer cone with veto cone removed

•Electrons•Brehmsstrahlung photons•Soft underlying event photon

•ECAL Isolation•Center cone on electron deposit

•Outer cone radius = 0.4

•Veto cone radius = 0.045 (barrel) 0.07 (endcap)

•Strip half-width Δη= 0.02

•ECALIso=∑energy deposits in outer cone with veto cone, strip contributions removed

• Reconstruction process may misidentify particles as electrons• True electrons will be well isolated, with little energy nearby

Page 34: Study of ZZ→4 Leptons with CMS at the LHC

Ian Ross - University of Wisconsin - Preliminary Exam34

Electron Isolation (cont.)HCAL Isolation

•Sum in outer cone, centered on ECAL supercluster

•Use all available HCAL depths

•Outer cone radius R=0.4

•Veto cone radius R=0.15

•HCAL Iso=∑energy deposits in outer cone with veto cone removed

•Electron isolation expected to provide discrimination power against jets.•Combined Iso = ECAL Iso + HCAL Iso + Tracker Iso

•Measured in GeV

ECAL deposit

HCAL deposit

Outer cone Veto cone

Page 35: Study of ZZ→4 Leptons with CMS at the LHC

Ian Ross - University of Wisconsin - Preliminary Exam35

Electron IsolationCut above 8 GeV Cut above 8 GeV

Cut above 8 GeV Cut above 8 GeV

• Cuts remove 90% of background and keep 90% of signal

∑e Iso, Leading electron (GeV) ∑e Iso, second electron (GeV)

∑e Iso, third electron (GeV) ∑e Iso, fourth electron (GeV)

Leptons ordered by pT

Page 36: Study of ZZ→4 Leptons with CMS at the LHC

Ian Ross - University of Wisconsin - Preliminary Exam36

M4l Spectrum After Isolation

• With electron isolation applied (stacked plot)

Process Events

ZZ→4ℓ (e/μ) 7.2

Z+jets 7.5

ttbar 8.6

Page 37: Study of ZZ→4 Leptons with CMS at the LHC

Ian Ross - University of Wisconsin - Preliminary Exam37

Electron ID• Additional electron

discrimination applied via ID variables• Low fraction of energy reaches

HCAL (H/E)• φ/η separations between

ECAL deposit and tracks (Δφin, Δηin)

• Cluster shape covariance (σiηiη)

• Expect loose electron ID cuts to further eliminate jets ‘faking’ electrons, while keeping most signal electrons

Variable Barrel Value Endcap ValueH/E 0.075 0.083

|Δηin| 0.0077 0.0100

|Δφin| 0.058 0.042

σiηiη (Not used-- see next slide)

e Reconstruction Efficiency (Signal only)

• Default e Reconstruction• e Reconstruction with relaxed loose ID applied

Page 38: Study of ZZ→4 Leptons with CMS at the LHC

Ian Ross - University of Wisconsin - Preliminary Exam

Electon ID (cont.)

• σiηiη cuts too many signal electrons in lower pt regions• Apply only H/E, |Δηin|, and |Δφin| cuts

38

Efficiency for Electron ID Cuts

Electron pt (GeV)

Effi

cien

cy

Page 39: Study of ZZ→4 Leptons with CMS at the LHC

Ian Ross - University of Wisconsin - Preliminary Exam39

M4l Spectrum After Isolation

Process Events

ZZ→4ℓ (e/μ) 6.2

Z+jets 0

ttbar 1.0

• With electron isolation and ID applied (stacked plot)

Page 40: Study of ZZ→4 Leptons with CMS at the LHC

Ian Ross - University of Wisconsin - Preliminary Exam

Final Z mass peaks

40

Process Events

ZZ→4ℓ (e/μ) 6.2

Z+jets 0

ttbar 1.0

• With electron isolation and ID applied (stacked plot)

Page 41: Study of ZZ→4 Leptons with CMS at the LHC

Ian Ross - University of Wisconsin - Preliminary Exam41

ZZ→4ℓ Signal

• With only a few cuts, the ZZ→4ℓ (e/μ) signal can be distinguished from relevant backgrounds

• 78% of the signal is kept, while 0.6% of total background remains

• ZZ→4ℓ signal:background is 6

• Backgrounds will be established using data as it becomes available

Process After Preselection After Isolation After IDZZ→4ℓ (e/μ) 8.0 7.2 6.2

Z+jets 112 7.5 0

ttbar 47 8.6 1

Page 42: Study of ZZ→4 Leptons with CMS at the LHC

Ian Ross - University of Wisconsin - Preliminary Exam42

Tevatron ZZ→4ℓ (e/μ) Observations•SM predicts σppbar→ZZ = 1.60 pb @ 2 TeV

•CDF -- 5 signal events in 4.8fb-1 with 5.7σ significance in observation•σppbar→ZZ = 1.56+0.80

-0.63(stat)±0.25(syst.) pb

•D0 -- 3 signal events in 1.7fb-1 with 5.3σ significance in observation•σppbar→ZZ = 1.75+1.27

-0.86(stat)±0.08 (syst.) ± 0.10 (lumi.) pb

LHC yield is expected to be competitive with Fermilab results

Page 43: Study of ZZ→4 Leptons with CMS at the LHC

Ian Ross - University of Wisconsin - Preliminary Exam43

ZZ→4ℓ (e/μ) Signal

Experiment Energy Data Signal Events

Background Events

CMS (MC Analysis)

10 TeV 1 fb-1 6.2 1

D0 1.96 TeV 1.7 fb-1 3 0.2

CDF 1.96 TeV 4.8 fb-1 5 0.04

CMS (Estimate)

7 TeV 1 fb-1 4.4 0.7

• LHC yield is expected to be competitive with Fermilab results• Signal/Background needs improvement

Page 44: Study of ZZ→4 Leptons with CMS at the LHC

Ian Ross - University of Wisconsin - Preliminary Exam44

Conclusions and Next Steps

• With the first fb-1 of data at CMS, a handful of ZZ→4ℓ (e/μ) events are expected

• Comparable to Tevatron results, but measured at 7 TeV

• Next steps:

• Improve analysis by further optimizing cuts and extend to other decay modes

• Full uncertainty studies.

• Data will provide further statistics on backgrounds

• Develop data-driven reconstruction efficiencies

• Prepare for Higgs search

Page 45: Study of ZZ→4 Leptons with CMS at the LHC

Ian Ross - University of Wisconsin - Preliminary Exam

Backup Slides

45

Page 46: Study of ZZ→4 Leptons with CMS at the LHC

Ian Ross - University of Wisconsin - Preliminary Exam

Higgs Production BR @ the LHC

46

Page 47: Study of ZZ→4 Leptons with CMS at the LHC

Ian Ross - University of Wisconsin - Preliminary Exam

Higgs Sensitivity @ CMS

47

• No Higgs masses can be excluded with 1 fb-1 @ 7 TeV using the H→ZZ

Page 48: Study of ZZ→4 Leptons with CMS at the LHC

Ian Ross - University of Wisconsin - Preliminary Exam

Higgs Sensitivity @ CMS

48

• Combined Hγγ, HWW, and HZZ channels provide a mH

exclusion range of 145-190 GeV

Page 49: Study of ZZ→4 Leptons with CMS at the LHC

Ian Ross - University of Wisconsin - Preliminary Exam49

Tracker Performance – LHC Collisions @ 900 GeV

• Λ0 kinematic distributions• First data closely matches Monte Carlo

Page 50: Study of ZZ→4 Leptons with CMS at the LHC

Ian Ross - University of Wisconsin - Preliminary Exam50

Tracker Performance – LHC Collisions @ 7 TeV

Page 51: Study of ZZ→4 Leptons with CMS at the LHC

Ian Ross - University of Wisconsin - Preliminary Exam51

ECAL Performance -- LHC Collisions @ 900 GeV

• Mass and width compatible with MC

• η/π0 ratio agrees with MC

• N(η) / N(π0) = 0.020 ± 0.003 DATA (left)

• N(η) / N(π0) = 0.021 ± 0.003 MC (right)

η→γγ

Page 52: Study of ZZ→4 Leptons with CMS at the LHC

Ian Ross - University of Wisconsin - Preliminary Exam52

HCAL Performance -- LHC Collisions @ 900 GeV

• Di-jet events in the calorimeter

• Spectra shapes match Monte Carlo

Page 53: Study of ZZ→4 Leptons with CMS at the LHC

Ian Ross - University of Wisconsin - Preliminary Exam53

Dimuon Event at 2.36 TeV

pt(μ1)=3.6 GeVpt(μ2)=2.6 GeV

Mμ1μ2 = 3.03 GeV

Page 54: Study of ZZ→4 Leptons with CMS at the LHC

Ian Ross - University of Wisconsin - Preliminary Exam

L1 Performance (e/γ) @ 900 GeV

54

• Clusters above 1 GeV trigger at almost 100% efficiency

Page 55: Study of ZZ→4 Leptons with CMS at the LHC

Ian Ross - University of Wisconsin - Preliminary Exam55

ECAL Performance -- LHC Collisions @ 7 TeV

• Electrons – Data/Monte Carlo comparisonSC – Supercluster– Collection of ECAL deposits associated with a candidate

φSC – φ coordinate of the supercluster

ηSC – η coordinate of the supercluster

σiηiη – shower shape covariance – measures spread of ECAL deposits

Page 56: Study of ZZ→4 Leptons with CMS at the LHC

Ian Ross - University of Wisconsin - Preliminary Exam

L1 Performance (μ) with 2008 Cosmic Data

56

Page 57: Study of ZZ→4 Leptons with CMS at the LHC

Ian Ross - University of Wisconsin - Preliminary Exam57

Tracker Performance – LHC Collisions @ 7 TeV

• Data shows agreement with Monte Carlo