Using Maximum Likelihood Analysisto Determine the π+ → e+νe Branching Ratio

Anthony Palladinofor the PEN Collaboration

University of Virginiaand

Paul Scherrer Institute

The American Physical Society’s April MeetingWashington, DC14 February 2010

Anthony Palladino (UVa&PSI) ML Analysis for the π+ → e+νe BR 14 Feb ’10 1 / 13

Outline

IntroductionThe PEN Experiment

Maximum Likelihood AnalysisModel: Probability Density FunctionsMeasurement: DataTechniquesPreliminary Results

Conclusions

Anthony Palladino (UVa&PSI) ML Analysis for the π+ → e+νe BR 14 Feb ’10 2 / 13

Introduction The PEN Experiment

The PEN Experiment

• Precision Measurement of the π+ → e+ν branching ratio.

B = Γ(π+→e+νe(γ))Γ(π+→µ+νµ(γ)) =

(ge

gµ

)2 (memµ

)2 (1−m2e/m2

µ)2

(1−m2µ/m2

π)2 (1 + δR)

Bcalc = (1.2352± 0.0005)× 10−4Marciano & Sirlin, [PRL 71, 3629 (1993)]

Bcalc = (1.2354± 0.0002)× 10−4Finkemeier, [Phys. Lett. B 387, 391 (1996)]

Bcalc = (1.2352± 0.0001)× 10−4Cirigliano & Rosel, [PRL 99, 231801 (2007)]

Bexp = (1.230± 0.004)× 10−4Experiment World Average (Current PDG)

Lepton Universality: W. Loinaz, et. al., Phys. Rev. D 65, 113004 (2004) [hep-ph/0403306](ge

gµ

)π

= 1.0021± 0.0016

Our Goal:∆Bexp

Bexp≤ 5× 10−4

Anthony Palladino (UVa&PSI) ML Analysis for the π+ → e+νe BR 14 Feb ’10 3 / 13

Introduction The PEN Experiment

Mass Limits on Leptoquark and Supersymmetric Particles

We will be able to give lower bounds on the masses of some hypotheticalparticles in theories beyond the standard model.Following the calculations in Shanker, NP B204 (82) 375:

Particle Projected Lower Bound Current Bounds

Charged Higgs Boson: mH > 6.9 TeV mH > 2 TeV

Pseudoscalar Leptoquark: mp > 3.8 TeV mp > 1.3 TeV

Vector Leptoquark: MG > 630 TeV MG > 220 TeV

Anthony Palladino (UVa&PSI) ML Analysis for the π+ → e+νe BR 14 Feb ’10 4 / 13

Introduction The PEN Experiment

PEN Detector

Anthony Palladino (UVa&PSI) ML Analysis for the π+ → e+νe BR 14 Feb ’10 5 / 13

Introduction The PEN Experiment

Experimental Method

Detector cross-sections2008 RunWedged Degrader

Anthony Palladino (UVa&PSI) ML Analysis for the π+ → e+νe BR 14 Feb ’10 6 / 13

Maximum Likelihood Analysis Model: Probability Density Functions

Maximum Likelihood Analysis

Model: Probability Density FunctionsOne combined p.d.f. encompassing many observables and processes.

L (x;N) =n∏

i=1

Fi (xi ;N) where N = Nj=1,...,m

Fi (xi ;N) =m∑

j=1

fj (xi ;Nj)

n Observables (xi )

• Time between π+ and e+

• Total Positron Energy

• “Probability” of Pile-up

“P“pile-up = ln

[∑k=1

e−|dtk |/τµ

]• Pion Decay Vertex

m Processes with normalization (Nj)

• Np2e, π+ → e+ (p2e)

• Nmich, π+ → µ+ → e+

(Michel)

• Nacc, Accidentals / Pile-up

• Ndif, Pion Decays-in-flight

Anthony Palladino (UVa&PSI) ML Analysis for the π+ → e+νe BR 14 Feb ’10 7 / 13

Maximum Likelihood Analysis Model: Probability Density Functions

Model: Probability Density Functions

L (E, t;Np2e ,Nmich,Nacc ,Ndif ) = f1 (E;N) f2 (t;N)

(MeV)+eE0 10 20 30 40 50 60 70 80

Eve

nts

310

410

510

610

(MeV)+eE0 10 20 30 40 50 60 70 80

Eve

nts

310

410

510

610

Time between pion and positron (ns)0 50 100 150 200

(M

eV)

+ eE

0

10

20

30

40

50

60

70

80

0

0.05

0.1

0.15

0.2

0.25

0.3

-310×

Time between pion and positron (ns)0 50 100 150 200

Eve

nts

210

310

410

510

610

Time between pion and positron (ns)0 50 100 150 200

Eve

nts

210

310

410

510

610

Measurement DataTotal p.d.f.p2e ComponentAccidental ComponentMichel ComponentDIF Component

Maximization Techniques:• Binned vs. Unbinned• Maximum Likelihood Fit• χ2 Fit• Negative Log Likelihood,

` = −lnL

.

Anthony Palladino (UVa&PSI) ML Analysis for the π+ → e+νe BR 14 Feb ’10 8 / 13

Maximum Likelihood Analysis Measurement: Data

Measurement: Data

Time between pion and positron0 50 100 150 200

Eve

nts

310

410

510

610

Time between pion and positron0 50 100 150 200

Eve

nts

310

410

510

610

+eE0 10 20 30 40 50 60 70 80

Eve

nts

310

410

510

610

+eE0 10 20 30 40 50 60 70 80

Eve

nts

310

410

510

610Measurement Data

Sum p.d.f.

p2e Component

Accidental / Pile-up

Michel Component

DIF Component

Variable Binning → Improvement in calculation speed.Anthony Palladino (UVa&PSI) ML Analysis for the π+ → e+νe BR 14 Feb ’10 9 / 13

Maximum Likelihood Analysis Techniques

Techniques: Negative Log Likelihood

` = −lnL L = e−` Np2e vs. Nmichel

Number of Michel events

19.5619.58 19.619.6219.6419.6619.68 19.7

610×

events

ν e→ π

Number of

3600036500

3700037500

3800038500

39000

-ln(

L)

0

10

20

30

40

50

Number of Michel events

19.5619.58 19.619.6219.6419.6619.68 19.7

610×

events

ν e→ π

Number of

3600036500

3700037500

3800038500

39000

L

0

0.2

0.4

0.6

0.8

1

Number of Michel events19.56 19.58 19.6 19.62 19.64 19.66 19.68 19.7

610×

eve

nts

ν e

→ πN

umbe

r of

36000

36500

37000

37500

38000

38500

39000

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8-310×

Use of standard software libraries MINUIT, MIGRAD, HESSE, and MINOSon `. Use of ROOFIT and ROOSTATS to set up model p.d.f.

Anthony Palladino (UVa&PSI) ML Analysis for the π+ → e+νe BR 14 Feb ’10 10 / 13

Maximum Likelihood Analysis Preliminary Results

Preliminary Results

Symmetric Confidence Intervals:

(This is only 8.1% of 2008 data → 1√Np2e

' 6× 10−3 )

68.3% (1σ): B = (1.228± 0.142)× 10−4

∆BB = 0.116

Dominated by systematic uncertainties... no well defined DIF p.d.f. yet.

Proper confidence levels are calculatedusing the Likelihood Ratio,

R = L(x;N)

L(x;N)

where N maximizes L (x;N). Calculate

R for all N and rank the Nj values

according to their R values. Nj are

added to the confidence region first until∫L (x;N) dx over the confidence region

reaches our desired confidence level.Number of Michel events

19.56 19.58 19.6 19.62 19.64 19.66 19.68 19.7

610×

eve

nts

ν e

→ πN

umbe

r of

36000

36500

37000

37500

38000

38500

39000

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8-310×

Anthony Palladino (UVa&PSI) ML Analysis for the π+ → e+νe BR 14 Feb ’10 11 / 13

Conclusions

Conclusions

Maximum Likelihood benefits:

• Provides a unique, unbiased, minimum variance estimate (for a largeenough sample).

• Practical, tractable approach via product p.d.f.’s

• Use all the information in the data to determine the parameters;minimal cuts.

Drawbacks:

• Critical dependence on p.d.f.

Anthony Palladino (UVa&PSI) ML Analysis for the π+ → e+νe BR 14 Feb ’10 12 / 13

Conclusions

PEN Experiment collaboration members:

L.P. Alonzi,a V. A. Baranov,c W. Bertl,b M. Bychkov,a Yu.M. Bystritsky,c

E. Frlez,a V. Kalinnikov,c N.V. Khomutov,c A.S. Korenchenko,c

S.M. Korenchenko,c M. Korolija,f T. Kozlowski,d N.P. Kravchuk,c

N.A. Kuchinsky,c M.C. Lehman,a D. Mekterovic,f D. Mzhavia,c,e

A. Palladino,a,b D. Pocanic,a? P. Robmann,g O.A. Rondon-Aramayo,a

A.M. Rozhdestvensky,c T. Sakhelashvili,b V.V. Sidorkin,c U. Straumann,g

I. Supek,f Z. Tsamalaidze,e A. van der Schaaf,g? E.P. Velicheva,c

V.V. Volnykh,c

aDept of Physics, Univ of Virginia, Charlottesville, VA 22904-4714, USAbPaul Scherrer Institut, CH-5232 Villigen PSI, SwitzerlandcJoint Institute for Nuclear Research, RU-141980 Dubna, RussiadInstitute for Nuclear Studies, PL-05-400 Swierk, PolandeIHEP, Tbilisi, State University, GUS-380086 Tbilisi, GeorgiafRudjer Boskovic Institute, HR-10000 Zagreb, CroatiagPhysik Institut der Universitat Zurich, CH-8057 Zurich, Switzerland

blue = Graduate Student? = Spokesperson

Anthony Palladino (UVa&PSI) ML Analysis for the π+ → e+νe BR 14 Feb ’10 13 / 13