Statistical Methods used in Reactor Neutrino Experiments Xin Qian - - PowerPoint PPT Presentation

Statistical Methods used in Reactor Neutrino Experiments

Xin Qian BNL

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Reactor Neutrinos

2

• Coincidence signal from

Inverse Beta Decay:

– Prompt: e+ & annihilation  – Delayed: n + Gd  8 MeV with 30 us capture time

• ~ 200 MeV per fission
• ~ 6 anti-νe per fission from

daughters decay

• ~ 2 x 1020 anti-νe/GWth/sec

n e p

e

  





0.78

prompt n

E E E MeV



  

Anti-νe Disappearance

Daya Bay (China) RENO (Korea) Double Chooz (France) Gd Target 80 ton 16.1 ton 10 ton Reactor Thermal Power 17.4 GW 16.4 GW 8.7 GW Baseline ~1.7 km ~1.4 km ~1.0 km

3

2 2 2 4 2 2 2 13 13 12 21

( ) 1 sin 2 sin cos sin 2 sin

e e ee

L L P m m E E                            

Far/Near Ratio

2 2 3 2 3 2 5 2 21

| |~| | 2.4 10 | | 7.6 10

ee x

m m eV m eV

 

       

Near/far ratio to cancel uncertainty in reactor flux, firstly proposed by Mikaelyan&Sinev Phys. Atmo.

• Nucl. 63, (2000)

Near/Far Ratio

• 100% cancellation of flux uncertainty with one

reactor, one near and one far detector

4

Double Chooz ~88% suppression of systematic uncertainties RENO ~77% Daya Bay ~95% Statement (~80% suppression) in arXiv:1501.00356 regarding DYB is incorrect

Current Status of sin22Θ13 and ∆m2

32

After Neutrino 2016

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In the following, I will focus on the statistical methods used in Daya Bay in fitting these parameters

Log-likelihood profiling

• Also Pearson chi-square with pull terms in PRL, 108,

171803 (2012)

• According to Wilks’ theorem, assuming

∆T=T – Tmin following a chi-square distribution

• Advantages: simple to program and easy to examine
• Disadvantages: When number of nuisance parameters is

large, can be slow to minimize

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 

 

   

, Re 2 2

2Log 2Log 2 Log Example format with N

stat sys

• bs

ADs bin j pred

• bs
• bs

stat j j j pred j j sys Detector Background actor Oscillation pred sys j

T L L C N T N N N N T T T T T T                                  



AD: Antineutrino Detector

Covariance matrix in PRL, 115, 11802 (2015)

• Approximating impacts of all systematics on the event counts

as normal distributions

• Advantages: Since “V” can be pre-calculated, the minimization

process to obtain Tmin can be very fast

• Disadvantages: “V” may have dependences on the

parameters of interest (i.e. θ13 and ∆m2), additional cares are needed

– Also Gaussian-Hermite technique to calculate integration in-flight

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     

 

 

 

     

1 , 2 2 , , 1

1 ( ) exp ( ) ( ) d 2 2 1 ( ) ( )

• bs

pred stat sys

• bs

pred i i j j ij i j

• bs

pred

• bs

pred ij i i j j N psuedo k pred psuedo k pred i i j j k

T F F V V F F V F F F F F F F F N



           

   

                       

  

 

 

2 2 1

1 (y ) 1 ( ) exp (y)dy 2 2 2

n i y i i y y

y E h y h w h x y     

  

               

 

“F” is a function of

• bserved events

“i” is a energy bin label for a detector

Hybrid Approach in PRL,112, 061801 (2014)

• Sometimes, the number of nuisance parameters can be too

many  numerical instability in finding the minimum

• For example, for reactor-related systematics (26 energy bins),

we have

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 

 

 

   

 

1 ,

2Log 2Log

stat

• ther sys

i reactor j ij i j

T L L V C   



      



Given three sites, the number of event bins is about 26x3=78 Given the nature of these systematics, expect many degeneracies  potential difficulties in finding the minimum Use Covariance Matrix (rank 78) to reduce 151 uncertainties  78 nuisance parameters (one on each event bin 4 isotopes 6 reactors 26 bins Also NDF difference can be used to check the covariance matrix

Combining nH + nGd (I)

• n + Gd (nGd)  ~ 8 MeV gammas
• n + p (nH)  2.2 MeV gamma

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n e p

e

  





Combing nH + nGd (II)

• Approximately, one can estimate the combination through the

Best Linear Unbiased Estimate (BLUE)

• A. C. Aitken, Proc. Ry. Soc. Edinburgh 55, 42 (1935)

Lyons&Gibaut&Clifford, NIMA 270, 110 (1988)

• Alternatively, a single fitter can be written to take into account

all correlations in systematics

• Both methods reach

similar results

• Combined result reported in

PRD 90, 071101(R) (2014) PRD 93, 072011 (2016).

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Combining Daya Bay, RENO, and Double Chooz? Expect <10% improvement

• We reported 0.943 +- 0.008 (exp.)
• Many literatures reported 0.928 (~ 1.5% lower)
• A tricky statistical mistake, they used

the measured values to build the theoretical covariance matrix

• See G. D’Agostini NIMA 346, 306 (1994),
• V. Blobel, SLAC-R-0703, p101,
• B. Roe arXiv:1506.09077

One Note About Global Average

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PRD 83, 073006 (2011)

 

2 1 exp 2 2

(R ) (R ) V R ( ) should be ( )

past past past g g i ij g j theory th theory the eory

• bs
• bs

theory i j theory th

• r

e r y j y i

• R

R V V V V R R V R R   



       

PRL, 116, 061801 (2016) and arXiv:1607.05378

The 5-MeV “Bump”

• Unambiguous observations of discrepancies between data and

spectrum calculation at ~ 5 MeV from all three experiments

• Uncertainties in flux calculation is underestimated (> 5% from

Hayes et al. PRL 112, 202501, 2014)

• Also saw in NEOS. Which isotopes? arXiv:1609.03910 (Huber)

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Daya Bay RENO Double Chooz

Absolute Neutrino Spectrum

• Compare to Huber+Mueller

model

• 3σ discrepancy at the full energy

range

• 4.4σ local significance at 4~6

MeV

13

arXiv:1607.05378

2

48.1 24 NDF  

2

37.4 8 NDF    

2.6σ and 4.0σ in PRL 116, 061801 Nested-hypothesis test: eight nuisance parameters controlling the shape in 2 MeV window are allowed to freely move

Neutrino Spectrum Extraction (Unfolding)

• Unfolding “original” neutrino spectrum with reduced information

from the measured prompt energy spectrum is desired for simpler usage

14

Stat+Sys

• One challenge of the unfolding is the smearing

due to finite energy resolution and statistical fluctuations

• Therefore, regularization is needed

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 

2 2 2 2 2 2 2

''

i ij j regularization i j regularization i ij j i i j

M R S c S F S                      

    

2 2 1 1 2

S (1 )

k

F R M S R 

 

     

• Basically, smearing due to detector response “R” (typically irregular) is

replaced by a regular response (1+F2/R2)-1

• With existence of uncertainties, smearing represents an information

loss, and cannot be fully recovered

• The optimal regularization depends on the existing smearing and

statistics

An independent Check

Possible light sterile neutrino oscillation

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P(e e) 1cos414 sin2 213sin2 mee

2 L

4E      sin2 214 sin2 m41

2 E

4E      

• A minimum extension of the 3-ν model: 3(active) + 1(sterile)-ν model
• Search for a higher frequency oscillation pattern besides |∆m2

ee|

16

[km/MeV]



/ E

eff

L 0.2 0.4 0.6 0.8 )

e

 

e

 P( 0.9 0.95 1

EH1 EH2 EH3 best fit  3 + sterile (illustration)  3

Daya Bay: Full 6 AD data

Search for a Light Sterile Neutrino

• Confidence Intervals are
• btained from Covariance

matrix method (fast) with the Feldman-Cousins (FC)

– PRD 57, 3873 (1998)

• Due to FC’s computing

demands, CLs method (A.L. Read, J. Phys. G28, 2693

• T. Junk, NIMA 434,435) is

chosen for “likelihood + pull”

– Gaussian CLs method is used

• G. Cowan et al. Eur. Phys. J. C71,

1544 (2011)

• XQ, A. Tan et al. NIMA 827, 63

(2016)

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arXiv:1607.01174 (to be published in PRL), factor of 2 improvement to the previous result (PRL 113, 141802, 2014) See A. Tan’s talk

Combined Sterile Search

• CLs method is easy to combine results

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arXiv:1607.01177 (DYB+MINOS) to be published in PRL

• See past Wine&Cheese seminars

MINOS  θ24 with νμ disappearance Daya Bay/Bugey-3  θ14 with (anti)νe disappearance

Further Prospect of Current Reactor Neutrino Experiments

• Daya Bay:

– Expect to reach < 3% uncertainty for both sin22θ13 and ∆m2

ee by 2020

– Another factor of two improvement in the limit of sterile neutrino search at low ∆m2

41

• Complimentary to the expected results from short-baseline

reactor experiments (i.e, PROSPECT) at high ∆m2

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• Combination among Daya Bay, RENO, and

Double Chooz is under discussion

– Below 3% precision of sin22θ13 by 2017

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JUNO

• Reactor Power: 36 GW
• Baseline: 53 km
• Detector: 20 kton LS
• E: 3% (2% at 2.5 MeV)
•  rate: ~60/day
• Background:
• Accidentals (10%)
• 9Li (<1%)
• Fast neutrons (<1%) Yangjiang NPP

Taishan NPP Daya Bay NPP

53 km 53 km

Hong Kong Macau Guangzhou Shenzhen

700 m underground

Acrylic Tank + SS structure m2

21

m2

32

JUNO DUNE

sin2212 0.7% m2

21

0.6% |m2

32|

0.5% 0.3% MH 3–4σ >5σ sin2213 14% 3% sin22 3% CP 10°

• J. Phys. G: Nucl. Part. Phys. 43, 030401 (2016)

MH Sensitivity (Non-nested Hypothesis Test)

• What’s the meaning of MH sensitivity?

– XQ, A. Tan et al. PRD86, 113011 (2012) – M. Blennow et al. JHEP 03, 028 (2014) among others

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Summary

• Reactor neutrinos have been and will continue

to play an important role in understanding the neutrino properties

– Previous: KamLAND – Current: Daya Bay, RENO, Double Chooz – Future: JUNO, PROSPECT …

• Data analysis of reactor neutrinos involves a

wide range of statistical techniques

– Parameter fit, (nested/non-nested) hypothesis tests, unfolding …

22

23

Rate-only vs. Shape-only

• Rate-only:
• Shape-only:

24

i , , , ,

2 Log 2 Log 2 Log 2 Log

• bs

ADs j pred

• bs
• bs

j j j pred

• bs

ADs j j ji pred

• bs
• bs

stat ji ji ji pred

• bs
• bs

ADs j ji ji j

• bs
• bin

bin bs ji j pred pred j ji j i

N N N N N N T N N N N N N N N N N                                                          

  

ADs j

                  



2 Log

• bs

ADs j pred

• bs
• bs

j j j pred j j

N N N N N                   



, ,i

2 Log

• bs

ADs ji

• bs

ji pred j ji pred

• bs

ji i i bin j i

N N N with N N          

  

Multinomial distribution first discussed in Baker&Cousins, NIMA, 221, 437 (1984) PRL,112, 061801 (2014) AD: Antineutrino Detector

Absolute Reactor Anti-Neutrino Flux

• 621 days data
• Effective fission fraction

235U 238U 239Pu 241Pu

56.1% 7.6% 30.7% 5.6%

Daya Bay’s absolute reactor flux measurement is consistent with previous short baseline experiments

Rglobe = 0.943 ± 0.008

• The World Average:
• Daya Bay result:

Rdyb = 0.946 ± 0.020

PRL, 116, 061801 (2016) and arXiv:1607.05378

Energy Nonlinearity Calibration

Sources of detector energy nonlinearity

• Scintillator quenching (Birks Law)
• Cherenkov light
• PMT readout electronics
• Modeled with MC and single channel

FADC measurement

Energy model is constrained with gamma (Improved fitting upon Crystal Ball in arXiv:1603.04433) and electron sources ~1% uncertainty (correlated among detectors)

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An Independent Check

• When treating the (still smeared) unfolded spectrum

as the real (unsmeared) spectrum, additional uncertainties (bias) by are needed, which represents an additional information loss

– The price that we have to pay for the simpler usage – Otherwise, same amount of information generally

• Bias be estimated through pseudo experiments

– In Daya Bay, we use various predictions of neutrino spectrum  pseudo measurements  unfolded spectrum to be compared with MC truth  determine the size of bias and additional uncertainties needed

27

2 2 4 2 4 2 2 3 2 3

1 8 1 8

data data

Erf CLs Erf

   

                                                

Statistical tests: 3-ν or 4-ν ?

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• Data is consistent with 3-ν

hypothesis with FC test No evidence for sterile neutrino

• ∆χ2

data = 5.6; p-value is 0.41

p0 p1 1-p0 1-p1 ∆χ2 = χ2

3ν – χ2 4ν

CLs  1 p1 1 p0

A.L. Read J. Phys. G28, 2693

• T. Junk NIMA434, 435

NIMA 827, 63 (2016)