Top mass error predictions with variable JES for projected luminosities

Joshua QuallsCentre College

josh.qualls@centre.eduMentor: Michael Wang

Contents

Motivation- Top mass

Theory- Top decay- Matrix Element method- Ensemble tests

Analysis- ROOT Macros/plots

Conclusion

Precise knowledge of MW and mt constrain mh, the Higgs mass

Why measure top mass?

Consider W boson mass(1+r)

1

Radiative corrections

Radiative corrections to Feynman diagram

W

FW

GM

22

sin

2

W

tFtop

mGr

22

2

tan

1

28

3)(

2

2

2

22

ln224

cos11)(

Z

hWZFHiggs M

mMGr

- tt produced from pp collisions in Tevatron

- From dozens to thousands annually

- Three decay modes of interest:

1) All jets

2) Dilepton

3) Lepton + Jets

Top Quark Production

- -

Decay channel 1: All Jets

p pt

t

All jets = 44%

Pros- Large branching fraction- Jet energy calibration using hadronic W

Cons- High background levels

-W

qq

b

b W+q

q

Decay channel 2: Dilepton

p p

t

t

b W

W

+

-

l

lv

vAll jets = 44%Dilepton = 5%

Pros- Low background levels

Cons- Low branching fraction- No hadronic W

b

b

Decay channel 3: Lepton + Jets

p p

t

t

W -

All jets = 44%Dilepton = 5%Lepton + Jets = 29%Other = 22%

W -

l

v

Pros

- Reasonable branching fraction- Jet energy calibration from hadronic W- Medium background levels- Traditionally yielded best results

b W+q

q

b

The General Method

Event specified by xi in volume dxi

Probability for configuration of N observed events within infinitesimal phase space dxi containing empty finite elements Δxi is:

P(x1,…xN)dx1…dxN = Prob(0 events in Δx1) x Prob(1 event in dx1) x …

By Poisson statistics, total probability is

N

iii

dxxPN

NN dxxPNedxdxxxP V

1

)(

11 )(...),...(

However,

actual events occur in more than

one dimension

- Extend method to k-dimensional space V

- Probability density depends on parameter(s) α

- Likelihood function given by

- Maximize L(α), OR (due to rapid variations in L(α) ) minimize -ln L(α)

Likelihood Function

N

ii

dxxPNxPeL i

1

);();()(

- Event with four jets, electron, and missing ET might not be tt

- W + 4 jets

- five jets, with one improperly reconstructed

- W + 3 jets, with one jet splitting

- Correct for this by calculating the Acceptance:

- Includes all conditions for accepting or rejecting an event

- geometric acceptance- trigger efficiencies- reconstruction efficiencies- selection criteria

Detector Complications

);()();( xPxAccxP i

tt ?

Jet Complications

1) Detector sees 4 jets, a lepton, missing ET, interaction vertex

- Can’t definitively match jets to quarks

- Must try all 12(ish) permutations

2) Determining jet energy scale (JES)

- Jet energy determined by scintillator sheets

- Numerous effects spoil the accuracy of conversion of light into jet energy

- Consider two situations: 1) fixed JES 2) variable JES

b? u?

- Event probabilities calculated directly

- Have signal and probability component

- For good detector, Psig is proportional to the cross section:

Where the cross section is given by

And |M| corresponds to the matrix element

Matrix Element Method

)()1();();( 11 xPcmxPcmxP bkgtsigtevt

)(

);();(

topobs

toptopsig m

mxdmxP

),()()()()(

12121 xyWqfqfdqdqyd

mtopobs

6

2121

24

(4

)2(

d

mmqqd

M

minm

Event nEvent n-1Event 3Event 2Event 1

To extract mtop from a sample of n events, probabilities are calculated for each individual event as a function of mtop :

From these we build

the likelihood function

The best estimate of the top mass is then determined

by minimizing:

);...(ln 1 topn mxxL

0.5

And the statistical error can be

estimated from:

5.0)(ln)(ln minmin mm LL

N

ii

dxxPNxPeL i

1

);();()(

Matrix Element Method (2)

From a large pool of M monte carlo events, we perform ensemble tests by randomly drawing n

events N number of times to form N pseudo-experiments:

Expt NExpt 1 Expt N-1Expt 3Expt 2

min min min min min

σ σ σ σ σ

mm

pull

min

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 . . . . . . . . . . . . . . . . . . . . . M

A. The error is estimated for each experiment and entered into the “Mass Error” histogram

B. The mass at the minimum for each experiment is entered into the “Top Mass” histogram

C. The pulls are calculated for each experiment by dividing the deviation of the mass at the minimum from the mean of this mass for all experiments by the estimated error

Ensemble tests

1) Writing numerous scripts to streamline the process of performing probability calculations

- Creating job submission template files - Writing out generated events- Scanning output files for errors, and creating new submission files

2) Modifying ROOT macros to create ensemble test histograms and compare mass error vs. beam luminosity

My Projects

- Theory predicts scaling of mass error with increased luminosity for fixed JES ≈ 1

- Beam luminosity corresponds to the number of simulated events

- - Observed events should scale with luminosity- - 150 events corresponds to 0.4 fb-1

- Theory had only predicted (accurately) the situation with fixed JES

- Mass error scales as 1/x2

- My mentor (among others) made theoretical calculations saying that the mass error for variable JES should scale as a constant times the fixed JES

Variable JES

1)Modified code to perform n pseudo experiments

- - n ranges from 150 to 2400

2) Generated likelihood histograms for fixed and variable JES

3) Generate the three histograms mentioned previously:

-mass error-top mass-mass pull

4) Debug this endlessly

ROOT Macros

1-D Likelihood

2-D Likelihood

Sample Histogram

Mass Error Plot

Top Mass Plot

- Evidence that variable JES mass errors will scale at approximately 1.5 times fixed JES mass errors

- Calculations for the exact theoretical value are still being performed

- Tentative value of 1.5 does not account for W mass

- This will lower the value, hopefully to the experimentally determined

~1.4

- Scripts for job submission are still being used, both to submit jobs locally and to the Open Science Grid

Results

Acknowledgements:

Michael Wang, Gaston Gutierrez, FNAL, D0 Collaboration, DOE, etc.

Thank You

Questions?