Statistical Methods used in Reactor Neutrino Experiments

Xin QianBNL

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Reactor Neutrinos

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• 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

nepe

0.78prompt nE E E MeV

Anti-νe Disappearance

DayaBay (China)

RENO (Korea)

Double Chooz(France)

GdTarget

80 ton 16.1 ton 10 ton

Reactor ThermalPower

17.4 GW

16.4 GW

8.7 GW

Baseline ~1.7 km ~1.4 km ~1.0 km

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2 2 2 4 2 2 213 13 12 21( ) 1 sin 2 sin cos sin 2 sine e ee

L LP m mE E

Far/NearRatio

2 2 3 23

2 5 221

| |~| | 2.4 10

| | 7.6 10ee xm 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

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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 ∆m232

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 6

,

Re

2

2

2Log 2Log

2 Log

Example format with N

stat sys

obsADs binjpred obs obs

stat j j j predj j

sys Detector Background actor Oscillation

predsys j

T L L C

NT 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 ( ) ( ) d22

1 ( ) ( )

obs pred stat sys obs predi i j jij

i j

obs pred obs predij i i j j

Npsuedo k pred psuedo k pred

i i j jk

T F F V V F F

V F F F F

F F F FN

2

21

1 (y ) 1( ) exp (y)dy 222

n

i y iiyy

yE h y h w h x y

“F” is a function of observed 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 2Logstat other sys i reactor jiji 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 isotopes6 reactors26 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 9

nepe

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 pastg g i ij g j

theory

th

theory the

eory obs obs theoryi j

theory thor e ryj

yi

o

R R

V V VV 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)12

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

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arXiv:1607.05378

2 48.124NDF

2 37.48NDF

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

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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

222 2 ''

i ij j regularizationi j

regularization i ij ji i j

M R S

c S F S

2 2

1 120 S (1 )

k

F R MS 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

16P(e e ) 1 cos414 sin2 213 sin2 mee

2 L4E

sin2 214 sin2 m41

2 E4E

• 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]

/ Eeff

L0 0.2 0.4 0.6 0.8

) e

e

P(

0.9

0.95

1EH1EH2EH3

best fit3 + sterile (illustration)3

Daya Bay: Full 6 AD data

Search for a Light Sterile Neutrino• Confidence Intervals are

obtained 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 disappearanceDaya Bay/Bugey-3 θ14 with (anti)νedisappearance

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 ∆m241

• 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 km53 km

Hong Kong

Macau

GuangzhouShenzhen

700 m underground

Acrylic Tank + SS structure

m221

m232

JUNO DUNEsin2212 0.7%m2

21 0.6%|m2

32| 0.5% 0.3%MH 3–4σ >5σsin2213 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 …

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Rate-only vs. Shape-only

• Rate-only:

• Shape-only:

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i

,

,,

,

2 Log

2 Log

2 Log 2 Log

obsADsjpred obs obs

j j j predobsADs j jjipred obs obsstat ji ji ji pred obs obsADsj ji ji jobs o

bin

binbs

ji jpred predj ji j

i

NN N N

NNT N N N

N N NN N

N N

ADs

j

2 LogobsADsjpred obs obs

j j j predj j

NN N N

N

,

,i2 Log

obsADsjiobs

ji predj ji

pred obsji i

i

bin

ji

NN

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 241Pu56.1% 7.6% 30.7% 5.6%

Daya Bay’s absolute reactor flux measurement is consistentwith 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) 26

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

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2 24

24

2 23

23

18

18

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

• ∆χ2data = 5.6; p-value is 0.41

p0

p11-p01-p1

∆χ2 = χ23ν – χ2

4ν

CLs 1 p1

1 p0

A.L. Read J. Phys. G28, 2693T. Junk NIMA434, 435

NIMA 827, 63 (2016)