THE SUPER-KAMIOKANDE SOLAR ANALYSIS χ 2 To determine what oscillation parameters the Super-Kamiokande solar neutrino data set favors, a χ 2 minimization is performed on the observed recoil electron spectrum and the expected Monte Carlo simulated spectrum with oscillations. The total BP2004 SSM flux [1] is assumed Φ8 B,SSM = 5.79 10 6 cm 2 s 1, Φ hep,ssm = 7.88 10 3 cm 2 s 1 ) and the χ 2 is minimized with respect to 8 B and hep flux scaling factors to achieve the best possible match between spectra. The general χ 2 is given as N di χ 2 SK β, η) = βb i ηh i ) f E i, δ B, δ S, δ R ) δb σ B δs δr 2 logl), σ S σ R where d i is the data spectrum, N is the number of energy bins, and β and η are the unit-less flux scaling factors for the 8 B b i ) and hep h i ) simulated spectra respectively. The term f E i, δ B, δ S, δ R ) and related constraints are also chosen to minimized the χ 2 to take into account any effects from energy correlated systematic uncertainties. The last term is the unbinned time-variation with an extended maximum likelihood fit and is described in [2]. 1 Flux Scaling Factor Uncertainties Dropping the time-variation term, the spectrum contribution to the χ 2 can be written as ) T ) β χ 2 β, η) = χ 2 βm β βm mβ m, η m ) C, η η m η η m for any given combination of δ B, δ S, and δ R. The χ 2 m term is the minimized χ 2 value with factors β m and η m. The curvature matrix C, written explicitly as C = α N b 2 i b i h i b i h i h 2 i, 1
includes the modifier α to reflect the energy uncorrelated systematic error on the total flux σ sys ). It is defined as σ 2 N 0 α = σ 2 0, where σ 2 σ2 0 = sys 1 σ 2 stat,i The χ 2 is minimized in its matrix form and the uncertainties of the determined values of β and η are recognized to be the inverse elements of C: σ 2 β = α N b 2 i, σ 2 η = α where σ 2 c.t. is the cross-term correlation of the fluxes. N h 2 i ) 1., σ 2 c.t. = α N b i h i, 2 Energy Correlated Systematic Uncertainties To account for uncertainties in the 8 B spectrum shape and the detector s energy scale and resolution, the simulated spectrum βb i ηh i is shifted by the energy shape factor f E i, δ B, δ S, δ R ) which depends on the assigned energy correlated systematic errors. The shape factor can be written as f E i, δ B, δ S, δ R ) = f B E i, δ B ) f S E i, δ S ) f R E i, δ R ), where δ B describes the 8 B spectrum shape uncertainty, δ S describes the energy scale uncertainty, and δ R describes the energy resolution uncertainty. Each δ is varied until the χ 2 reaches a minimum. This is done by a simplex search with the constraints δ B /σ B, δ S /σ S, and δ R /σ R assuring sensible one sigma region values. The final simulated spectra is βb i ηh i ) f B E i, δ B ) f S E i, δ S ) f R E i, δ R ), when β, η, δ B, δ S, and δ R are at their minimizing values. The 8 B neutrino spectrum is taken from [3] and its uncertainties from [4]. 3 SK-I and SK-II Combined Analysis Both SK-I and SK-II data sets employ the same χ 2 as explained above but with their respective binning 21 bins in SK-I and 17 bins in SK-II), timevariation thresholds 5.0 MeV and 7.5 MeV), systematic errors and response functions. The combined SK-I and SK-II χ 2 is a sum of the two separate χ 2 s with common factors β and η. Care must be taken though when considering the 8 B spectrum shape uncertainty since both SK-I and SK-II analyses utilize 2
the same distribution. All but δ B are repeated in a combined χ 2 making a total of five constraints on the energy correlated systematic uncertainties. The total SK χ 2 is χ 2 SK β, η) = χ2 SK-I β, η) χ2 SK-II β, η) = 21 di βb i ηh i ) f E i, δ B, δ S,SK-I, δ R,SK-I ) 17 di βb i ηh i ) f E i, δ B, δ S,SK-II, δ R,SK-II ) δb σ B δs σ S SK-I δr σ R SK-I δs σ S SK-II δr σ R SK-II 2 log L SK-I ) 2 log L SK-II ) Additional references are [5] and [6]. 4 The SK χ 2 Map A χ 2 map of selected quantities from the SK-I and SK- II combined analysis as explained above can be found at http://www-sk.icrr.u-tokyo.ac.jp/sk/lowe/sk2 data/. The following 3
list relates the column labels in the map with those described in this text. tanˆ2th) tan 2 θ deltaˆm2 m 2 chiˆ2 ˆ8B hep ˆ8B un hep un χ 2 SK β Φ8 B,SSM η Φ hep,ssm σ β Φ8 B,SSM σ η Φ hep,ssm cross σ c.t. Φ8 B,SSMΦ hep,ssm ) 1/2 ˆ8B sh δ B /σ B ) es-i er-i es-ii er-ii δ S /σ S ) SK-I δ R /σ R ) SK-I δ S /σ S ) SK-II δ R /σ R ) SK-II When using these quantities in an oscillation χ 2 analysis, it is recommended that free fitting parameters describing the 8 B and hep fluxes be constrained to SK values and that the SK χ 2 minimum for each combination of oscillation parameters be added to the total χ 2. For example, χ 2 = χ 2 xβ, η) χ 2 SK β β SK σ 2 β,sk η η SK σ 2 η,sk 2β β SK)η η SK ), σ 2 c.t.,sk where the SK-subscripted terms represent those values in the SK χ 2 map and χ 2 x is a separate analysis sensitive to the 8 B and hep fluxes. The total χ 2 is then minimized with respect to β and η. When one wants to do a global solar analysis, i.e. combine SK results with SNO and other experiments, the correlation of the 8 B neutrino spectrum shape may need to be considered. To incorporate this uncertainty, one may perform a spectrum fit with the SK-I and SK-II recoil total) electron energy spectra. The SK-II spectrum and errors can also be found on http://www-sk.icrr.u-tokyo.ac.jp/sk/lowe/sk2 data/ with other information in the pre-print arxiv:0803.4312. SK-I data can be found in [6]. Lastly, when including the SK χ 2 map information in an analysis, please cite as The Super-Kamiokande Collaboration J.P. Cravens et al.), arxiv:0803.4312. 4
References [1] J.N.Bahcall and M.H.Pinsonneault, Phys. Rev. Lett. 92 2004) 121301. [2] S. Fukuda et al., Phys. Rev. Lett. 86 2001) 5651-5655. [3] C.E. Ortiz et al., Phys. Rev. Lett. 85 2000909. [4] J.N.Bahcall et al., Phys. Rev. C54 1996) 411. [5] M.B.Smy, arxiv:hep-ex/0202020. [6] J. Hosaka et al., Phys. Rev. D73 2006) 112001. 5