statistical methods used in reactor neutrino experiments
TRANSCRIPT
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)