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

Statistical Methods used in Reactor Neutrino Experiments Xin Qian BNL 1

Reactor Neutrinos • ~ 200 MeV per fission • ~ 6 anti- ν e per fission from daughters decay • ~ 2 x 10 20 anti- ν e /GW th /sec      p e n e • Coincidence signal from Inverse Beta Decay: – Prompt: e + & annihilation     0.78 E E E MeV  prompt n – Delayed: n + Gd  8 MeV with 30 us capture time 2

Anti- ν e Disappearance     L L                 2 2 2 4 2 2 2 ( ) 1 sin 2 sin   cos sin 2 sin   P m m 13 13 12 21 e e ee     E E Daya RENO Double Bay (Korea) Chooz (China) (France) Gd 80 ton 16.1 ton 10 ton Far/Near Target Ratio Reactor 17.4 16.4 8.7 GW Thermal GW GW Power Baseline ~1.7 km ~1.4 km ~1.0 km      Near/far ratio to cancel uncertainty 2 2 3 2 | |~| | 2.4 10 m m eV 3 ee x in reactor flux, firstly proposed by      2 5 2 Mikaelyan&Sinev Phys. Atmo. | | 7.6 10 m eV 21 Nucl. 63, (2000) 3

Near/Far Ratio • 100% cancellation of flux uncertainty with one reactor, one near and one far detector Double Chooz RENO Daya Bay ~88% suppression of ~77% ~95% systematic uncertainties Statement (~80% suppression) in arXiv:1501.00356 regarding DYB is incorrect 4

Current Status of sin 2 2 Θ 13 and ∆ m 2 32 After Neutrino 2016 In the following, I will focus on the statistical methods used in Daya Bay in fitting these parameters 5

Log-likelihood profiling • Also Pearson chi-square with pull terms in PRL, 108, 171803 (2012)         2Log 2Log T L L C stat sys     obs , ADs bin N         j  pred obs obs 2 Log T N N N     stat j j j pred N     j j     T T T T T Re sys Detector Background actor Oscillation   2      T    pred AD: Antineutrino Detector Example format with N  sys j 2 • According to Wilks’ theorem, assuming ∆ T=T – T min 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

Covariance matrix in PRL, 115, 11802 (2015)         1       “F” is a function of obs pred stat sys obs pred T F F V V F F i i j j ij observed events , i j          2  1 ( )                    obs pred obs pred exp ( ) ( ) d V F F F F     ij 2 i i j j  2 2   “i” is a energy bin     N 1        label for a detector , , psuedo k pred psuedo k pred ( ) ( ) F F F F i i j j N  1 k • Approximating impacts of all systematics on the event counts as normal distributions • Advantages: Since “V” can be pre-calculated, the minimization process to obtain T min can be very fast • Disadvantages: “V” may have dependences on the parameters of interest (i.e. θ 13 and ∆ m 2 ), additional cares are needed – Also Gaussian-Hermite technique to calculate integration in-flight      2 n 1 (y ) 1  y              ( ) exp (y)dy 2 E h y h w h x y        i y i 2  2 2    7 i 1 y y

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 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 4 isotopes nuisance parameters (one on 6 reactors each event bin 26 bins               1           2Log 2Log T L L V C stat other sys i reactor j ij i j , 8 Also NDF difference can be used to check the covariance matrix

Combining nH + nGd (I) • n + Gd (nGd)  ~ 8 MeV gammas      p e n e • n + p (nH)  2.2 MeV gamma 9

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

One Note About Global Average PRL, 116, 061801 (2016) • We reported 0.943 +- 0.008 (exp.) and arXiv:1607.05378 • Many literatures reported 0.928 (~ 1.5% lower) PRD 83, 073006 (2011) • A tricky statistical mistake, they used         2 1 past past past (R ) (R ) V R R R the measured values to build the g g i ij g j theoretical covariance matrix   exp theory V V V   th eory obs obs theory 2 ( ) V R R i j • See G. D’Agostini NIMA 346, 306 (1994),   2 theory theory the or y th e r o y should be ( ) V R R V. Blobel, SLAC-R-0703, p101, i j B. Roe arXiv:1506.09077 11

The 5-MeV “Bump” Daya Bay RENO Double Chooz • 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

Absolute Neutrino Spectrum • Compare to Huber+Mueller model • 3 σ discrepancy at the full energy range  2 48.1  24 NDF • 4.4 σ local significance at 4~6 MeV   2 37.4   8 NDF Nested-hypothesis test: eight nuisance parameters controlling the shape in 2 MeV window are allowed to freely move arXiv:1607.05378 2.6 σ and 4.0 σ in PRL 116, 061801 13

Neutrino Spectrum Extraction (Unfolding) • Unfolding “original” neutrino spectrum with reduced information from the measured prompt energy spectrum is desired for simpler usage Stat+Sys 14

An independent Check • One challenge of the unfolding is the smearing due to finite energy resolution and statistical fluctuations • Therefore, regularization is needed 2           2 2   M R S i ij j regularization   i j 2        2     2 2   '' c S F S regularization i ij j   i i j   2 2 F       1 1 0 S (1 ) R M  2 S R k • Basically, smearing due to detector response “R” (typically irregular) is replaced by a regular response (1+F 2 /R 2 ) -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 15

Possible light sterile neutrino oscillation 1 EH1 EH2 EH3 )  3 best fit e   3 + sterile (illustration)  0.95 e  P( 0.9 Daya Bay: Full 6 AD data 0 0.2 0.4 0.6 0.8 L / E [km/MeV] eff      2 L 2 E P (  e   e )  1  cos 4  14 sin 2 2  13 sin 2  m ee  sin 2 2  14 sin 2  m 41        4 E  4 E  16 • A minimum extension of the 3- ν model: 3(active) + 1(sterile)- ν model • Search for a higher frequency oscillation pattern besides | ∆ m 2 ee | 16

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) arXiv:1607.01174 (to be published in PRL), factor of 2 improvement to the See A. Tan’s talk previous result (PRL 113, 141802, 2014) 17

Combined Sterile Search • CLs method is easy to combine results arXiv:1607.01177 (DYB+MINOS) to MINOS  θ 24 with ν μ disappearance be published in PRL Daya Bay/Bugey-3  θ 14 with (anti) ν e disappearance 18 • See past Wine&Cheese seminars