Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://nuclphys.sinp.msu.ru/news/kamland/PRD53010.pdf Äàòà èçìåíåíèÿ: Wed Sep 17 10:09:40 2003 Äàòà èíäåêñèðîâàíèÿ: Tue Oct 2 00:35:01 2012 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: guide 8.0

PHYSICAL REVIEW D 66, 053010 2002

Getting the most from the statistical analysis of solar neutrino oscillations
G. L. Fogli,1 E. Lisi,1 A. Marrone,1 D. Montanino,2 and A. Palazzo1
2

Dipartimento di Fisica and Sezione INFN di Bari, Via Amendola 173, 70126 Bari, Italy Dipartimento di Scienza dei Materiali and Sezione INFN di Lecce, Via Arnesano, 73100 Lecce, Italy Received 25 June 2002; published 30 September 2002

1

We present a thorough analysis of the current solar neutrino data, in the context of two-flavor active neutrino oscillations. We aim at performing an accurate and exhaustive statistical treatment of both the input and the output information. Concerning the input information, we analyze 81 observables, including the total event rate from the chlorine experiment, the total gallium event rate and its winter-summer difference, the 44 bins of the Super-Kamiokande SK energy-nadir electron spectrum, and the 34 day-night energy spectrum bins from the Sudbury Neutrino Observatory SNO experiment. We carefully evaluate and propagate the effects of 31 sources of correlated systematic uncertainties, including 12 standard solar model SSM input errors, the 8 B neutrino energy spectrum uncertainty, as well as 11 and 7 systematics in SK and SNO, respectively. Concerning the output information, we express the 2 analysis results in terms of ``pulls,'' embedding the single contributions to the total 2 coming from both the observables and the systematics. It is shown that the pull method, as compared to the numerically equivalent covariance matrix approach, is not only simpler and more advantageous, but also includes useful indications about the preferred variations of the neutrino fluxes with respect to their SSM predictions. Our final results confirm the current best-fit solution at large mixing angle LMA , but also allow, with acceptable statistical significance, other solutions in the low-mass LOW or in the quasivacuum oscillation QVO regime. Concerning the LMA solution, our analysis provides conservative bounds on the oscillation parameters, and shows that the contribution of correlated systematics to the total 2 is rather modest. In addition, within the LMA solution, the allowed variations from SSM neutrino fluxes are presented in detail. Concerning the LOW and QVO solutions, the analysis of the pull distributions clearly shows that they are still statistically acceptable, while the small mixing angle SMA solution could be recovered only by ad hoc ``recalibrations'' of several SSM and experimental systematics. A series of Appendixes elucidate various topics related to the 2 statistics, the winter-summer difference in GALLEX-GNO, the treatment of the SK and SNO spectra, and a quasi-model-independent comparison of the SK and SNO total rates. DOI: 10.1103/PhysRevD.66.053010 PACS number s : 26.65. t, 13.15. g, 14.60.Pq, 91.35. x

I. INTRODUCTION

The data from the Homestake 1 , SAGE 2 , GALLEXGNO 3,4 , Kamiokande 5 , Super-Kamiokande SK 6 , and Sudbury Neutrino Observatory SNO 7­9 experiments have consistently established that electron neutrinos emitted from the Sun 10 undergo flavor transitions to the or ) . Neutrino oscillations 11 , other active states ( possibly affected by matter effects in the Sun or in the Earth 12 , represent a beautiful explanation of such transitions. Assuming the simplest scenario of two-family oscillations among active neutrinos, an important task for the next future is to refine the current constraints on the neutrino squared mass difference m 2 m 2 m 2 0 and on the mixing angle 2 1 0, /2 . In order to accomplish this task, one needs: a 12 new or more precise measurements; b accurate calculations of the e survival probability P ee ( m 2 , 12) and of related observable quantities; and c powerful statistical analyses to compare the increasingly large solar data set with theoretical expectations. The point a , not discussed in this work, will soon be addressed by the decisive reactor ¯ e experiment KamLAND 13 , as well as by the solar experiments which are currently running 2,4,9 , being restored 6 , or in construction 14 . Concerning the point b , since the current numerical
0556-2821/2002/66 5 /053010 22 /$20.00

and analytical understanding of the oscillation probability and related observables is quite mature in the whole ( m 2 , 12) plane, we will only make a few remarks when needed. In this paper, we rather focus on point c , aiming at an exhaustive statistical analysis including all known observables and uncertainties in input, and providing very detailed information in output, in order to better appreciate the current status of the solutions to the solar problem in terms of active flavor oscillations. Although some details will be specific of solar data, the analysis method that we discuss is quite general, and can be easily extended to any kind of global fit. The structure of our paper is the following. In Sec. II we discuss the equivalence between the 2 approaches in terms of the covariance matrix and of ``pulls'' of observables and systematics. In Sec. III we describe the input and output of the pull approach, as applied to the analysis of 81 solar neutrino observables and of 31 input systematics, in the context of 2 active oscillations. In Sec. IV we discuss the 2 analysis results in terms of multiple allowed regions in the massmixing parameter space ( m 2 , tan2 12) and in terms of the associated pull distributions, which provide additional information about the relative likelihood of the various solutions and about the allowed deviations from standard solar model SSM fluxes. We draw our conclusions in Sec. V. More technical but sometimes substantial issues are discussed in
©2002 The American Physical Society

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a series of Appendixes, which deal with the 2 statistics Appendix A , the winter-summer asymmetry in GALLEXGNO Appendix B , the treatment of the SK spectrum uncertainties Appendix C , the analysis of the SNO data Appendix D , and a quasi-model-independent comparison of SK and SNO total rates Appendix E . As a conclusion to this Introduction, we would like to stress that deepening the statistical analysis and improving the evaluation of the uncertainties is an important task in neutrino oscillation physics, just as it happens or happened in other areas of ``precision'' physics. Indeed, after the observation of two large oscillation effects the disappearance and of solar e , and their upcoming tests of atmospheric at long-baseline accelerator and reactor experiments , we are likely to face an era of delicate searches for smaller effects related, e.g., to the angle 13 , to leptonic CP violation, or to subleading contributions induced by nonstandard states or interactions. Moreover, one should not forget that, so far, there is no direct evidence for a vacuum oscillation pattern disappearance and reappearance of a specific flavor or for matter effects in the Sun or the Earth. Such effects might well generate only small signals in present or planned experiments, and any effort should be made in order to quantify them if any with accurate analyses. From this viewpoint, we think that our thorough analysis can add valuable inforfits mation and useful technical tools to other solar 6,9,15 ­ 20 that appeared soon after the release of the SNO neutral current data 8 .
II. TWO EQUIVALENT WAYS OF DEFINING THE FUNCTION
2

Given the input numbers in Eq. 1 , two 2 definitions have been basically used in global analyses with some variants or combinations . They will be referred to as the ``covariance'' approach Sec. II A and the ``pull'' approach Sec. II B . Although seemingly different, the two approaches are strictly equivalent Sec. II C . The pull approach, however, proves to be much more advantageous, and will be used throughout this paper.
A. The covariance approach

In the ``covariance approach,'' one builds the covariance matrix of squared errors as
K 2 nm nm u n u m k1 k ckcm , n

4

then inverts it, and evaluates the quadratic form
N 2 covar

R
n,m 1

expt n

R

theor n

2 nm

1

R

expt m

R

theor m

.

5

This approach, proposed in 21 for the data available at that time, has been later used in the majority of solar analyses of total events rates, with some variants related to the treatment of the 8 B flux free or SSM and to the separation of spectral and total rate information in the SK data.
B. The pull approach

Let us consider a set of N observables R n n 1,. . . , N with their associated sets of experimental observations R expt and n theoretical predictions R theor . In general, one wants to build n a 2 function which measures the differences R expt R theor in n n units of the total experimental and theoretical uncertainties. This task is completely determined if, for any difference R expt R theor , one can estimate an uncorrelated error u n , and n n a set of K correlated systematic errors c k induced by K inn dependent sources, namely R with un ,u c ,c where
1

The alternative ``pull approach'' embeds the effect of each independent k-th source of systematics through a shift of the n difference R expt R theor by an amount k c k , where k is a n n 2 univariate Gaussian random variable
K

R

expt n

R

theor n

R

expt n

R

theor n

k1

k kc n

.

6

expt n

R

theor n

u

n

c

1 n

c 2 ... n

c

K n

n 1, . . ., N ,

1

The normalization condition for the k 's is implemented through quadratic penalties in the 2 , which is then minimized with respect to all k 's,
K 2 k kc n K k1 2 k expt n theor n

m h m

nm

, n,m ,

2 3

N 2 pull

R

R

min
k

k1

k n

n1

u

. 7

n

kh

represents the correlation index.1

The error c k represents the shift of the n-th observable induced n by a 1 variation in the k-th systematic error source. Linear propagation of errors is assumed, namely, possible 1 asymmetries and second-order systematic effects c k c h are consistently nn neglected in computing the uncertainties of the difference R expt n R theor . n

Denoting as ¯ k ``pulls'' of the systematics the values of the k 's at the minimum, and defining the ``pulls'' x n of the observables as
The minus sign preceding the terms k c k is conventional. It n amounts to attribute all the shifts to the theoretical estimate: R theor n K k R theor n k 1 kc n .
2

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K

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R ¯ x
n

expt n

R

theor n

k1

¯k c

k n

u

,

8

n

2 the value of pull is then split into two diagonalized pieces, embedding the contribution to the 2 from the residuals of the observables and of the systematics, 2 pull 2 obs N 2 sys K

9 ¯2 . k 10

¯ x
n1

2 n

k1

The pull approach has often been used by the SK Collaboration in their day-night spectral analysis, in combination with the covariance method for non-SK data 22 . More recently, the SK energy-nadir spectrum has been analyzed through a mixed pull-covariance approach 23 . The link between the covariance and pull method, discussed in the next section, is also mentioned in passing in Ref. 24 . To our knowledge, however, a complete analysis of solar neutrino 2 data in terms of pull has not been performed, prior to the present work.
C. Comparison and equivalence of the covariance and pull approaches

It is perhaps not generally known that, although seemingly different, the covariance and pull approaches are strictly equivalent,
2 covar 2 pull.

matrices. The situation might become even more problematic in future high-statistics experiments, such as the neutrino factories from muon storage rings or superbeams, where the oscillation parameters will be inferred from the analysis and comparison of densely binned and correlated anti neutrino event spectra.4 The pull approach is clearly more practical than the covariance one when K N . In fact, the minimization in Eq. 7 leads to a set of K linear equations in the ¯ k 's, and to an associated K K matrix inversion, rather than the N N covariance matrix inversion see Appendix A . Moreover, the final decomposition in terms of pulls of observables and systematics Eqs. 9 and 10 allows to trace the individual contributions to the 2 , and to easily detect anomalously large residuals. Indeed, the pull distribution has been recognized as a useful diagnostic tool in many areas of physics, including electroweak precision physics 30 and, more recently, solar physics, as discussed in 32 see also 19 .5 In general, this method is useful to gauge the mutual agreement of data in a global fit, or to diagnose tension among data if any , for any given point in the model parameter 2 space ( m 2 , tan2 12) in our case . The analysis of the obs 2 and sys components Eq. 9 can also be useful to trace possible sources of good or bad fits. 2 Given the advantages of pull in cases where K N , we have redesigned the statistical analysis of solar neutrino oscillations in terms of pulls, and applied it to the current data set where N 81 and K 31) , as discussed in the next section.
III. INPUT AND OUTPUT FOR THE
2 pull

11

ANALYSIS

Our proof of the above identity is given in Appendix A.3 Given the equivalence in Eq. 11 , the choice between the covariance and the pull approach must be dictated by their relative merits. In particle physics, the covariance approach is typically used either when the experimental collaborations provide detailed information about the correlation matrix as in the case of the CERN e e collider LEP Electroweak Working Group 30 , or when N K as in the case of solar neutrino fits to total rates only 21 . However, for increasing N the approach becomes increasingly complicated. The inversion of large N N covariance matrices, in addition to being numerically tricky, can make it difficult to fully understand the results of global analyses. Indeed, the current solar or atmospheric neutrino data fits, involving N O (102 ) , are getting close to their manageability limits in terms of covariance
3

In this section we describe the main input and output 2 quantities, related to the pull analysis of solar neutrino data. In input we consider a set of N 81 observables R n with associated uncorrelated errors u n ) , and a set of K 31 sources of correlated systematic errors c k , in part related to n the SSM and in part to the experiments.6 In output we con2 sider the total pull , its decomposition in individual pulls, and the shifts of the neutrino fluxes from their SSM value.
A. Input observables and uncorrelated errors

The first two observables in our list are the chlorine total rate 1 , R
expt Cl

2.56 0.23 SNU,

12

We have recently realized that Eq. 11 and its implications have been also discussed 25 and are routinely used 26,27 in the context of parton density distribution fitting 28 . Closely related results have also been recently found in the context of cosmic microwave background data fitting 29 . Although the connection between covariance matrix and pulls appears thus to be an ubiquitous result in physics data analysis, we have been unable to trace explicit references to Eq. 11 prior to Ref. 25 .

The pull approach has been recently applied to prospective studies in this context 31 . 5 Notice that in Refs. 19,30,32 the pulls are defined in a somewhat different way, namely, without the shifts in Eq. 8 . 6 Systematic error sources are defined as ``correlated'' if they act upon two or more observables at the same time. Systematics which act upon one observable only e.g., the -Cl absorption cross section error simply contribute quadratically to the uncorrelated error for that observable the Cl total rate, in the example .

4

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and the average Gallium total rate SAGE 2 GNO 4 , R
expt Ga

GALLEX-

scale hep from the value in 41 ( 9.3 103 cm our default value
hep

2

s

1

) to 15

70.8 4.4 SNU,

13

8.3 103 cm

2

s

1

,

where the errors include the statistical and experimental systematic contributions to the uncorrelated errors u Cl and u Ga , respectively. In the analysis, the corresponding cross section uncertainties must also added in quadrature. Following the suggestion in 33 , the cross section error components R X , i for X ( Cl, Ga) are first added linearly and then quadratically into low L and ``high'' H energy parts L pp, pep, Be, N, O and H B, hep , respectively 7
2 2 u X cross section 2

R
iL

theor X,i

R
iH

theor X,i

. 14

For X Cl, the cross section error components are evaluated as R X , i R X , i ln CX,i , where R X , i are the oscillated rate components, and the fractional 1 cross section uncertainties lnCX,i can be taken from the compilation in 37 . For X Ga, the value of R X , i is computed by taking 1/3 of the variations induced by the 3 perturbed cross sections 38 on the i-th Ga rate component for each point of the oscillation parameter space, as suggested in 33 .8 Our third observable is the winter-summer ( W S ) rate difference 39 measured in GALLEX-GNO 4 , here introduced for the first time in the oscillation analysis. This datum is described in detail in Appendix B. The SK experiments provides 44 observables, in terms of binned absolute event rates for the energy-nadir differential spectrum of electrons 6 . Our treatment of the SK spectral information is described in detail in Sec. III C. observables is completed by the 34 The set of solar day-night energy spectrum bins from the SNO experiment 8,9 , which include contributions from elastic scattering ES , charged current CC and neutral current NC interactions, and backgrounds 40 . Our treatment of the SNO spectral information is described in Appendix D.
B. Input correlated systematics

according to the recent evaluation of the associated S hep factor in 42 . The SSM also embeds a set of eleven sources X k of correlated uncertainties the cross section factors S 11 , S 13 , S 34 , S 1,14 , S 17 , the Be capture cross section C Be , the Sun luminosity, metallicity Z / X , age, opacity, and element diffusion , with fractional uncertainties ln Xk , as listed in 37 . With respect to the compilation in 37 , we update lnZ/X 0.061 from 41 , and we add X 12 S hep , with lnShep 0.3. 9 The effects of such sources of uncertainties on the neutrino fluxes i are characterized by log-derivatives ln i / ln Xk , as compiled in 37 . Concern10,21 , ik ing X 12 , the only nonzero log-derivative is hep,12 1. The collective effect of the SSM sources of systematics X k amounts to shift the neutrino fluxes as
12 i



i

1
k1

k

ik

lnX

k

,

16

The 31 sources of correlated systematics include 12 uncertainties related to SSM input, the 8 B energy shape uncertainty, 11 SK error sources and 7 SNO error sources. Concerning the SSM input, we take from 41 the central values for the fluxes ( pp , pep , Be , B , N , O) , but re7

The prescription in Eq. 14 is intermediate between the extreme possibilities of quadratic sum 21 and linear sum 34 over all flux components. We observe that this prescription is not only justified by the physics of the e absorption processes in Cl and Ga 33 , but also by the effective separation of the Cl and the Ga solar response functions into two ``L '' and ``H '' clusters in the energy domain 35,36 . 8 We conservatively assume the largest between the 1 and 1 asymmetric Ga cross section errors.

12 where the k 's, penalized by the quadratic term k 1 2 in k 2 the expression of pull , are minimized away in the fit. Notice that the above equation is the linearized form of the power laws connecting each flux to the X k 's 10,43 . Such linear form satisfies the luminosity constraint for the fluxes 44 by construction, due to the sum rule discussed in Refs. 21,37 see also 45 . Within the pull approach, the shifts in Eq. 16 are easily propagated to all theoretical predictions R theor . In particular, n theor if R n , i is the i-th flux contribution to R theor , then the asson k ciated correlated shift from the k-th source is k c n , i theor kR n,i ik ln Xk . The net effect of the shifts in Eq. 16 is thus the generation of correlated errors on the R n 's, which is strictly equivalent to the construction of the astrophysical error matrix defined in the earlier covariance approach 21,37 , as also noticed in 24 . The 13th source of correlated systematics in our list is the 8 B spectrum shape uncertainty 46 around the currently adopted ``central'' spectrum 47 , which affects all the 81 observables10 at the same time. In the absence of oscillations, we estimate that a 1 perturbation of the 8 B spectrum in the direction of higher energies generates a 2.2% and a 1.7% increase of the B component of the R Ga and R Cl theoretical rates, respectively the evaluation is repeated in each point of the mass-mixing plane . The corresponding shifts for the SK and SNO spectra are evaluated in Appendix C and D, respectively. Notice that we conservatively use the

9 The authors of 42 quote an uncertainty of 15% for S hep , that we conservatively double to 30% . 10 With the possible exception of the Ga winter-summer difference, where its effects cancel to a large extent, and can be safely neglected as compared with the rather large statistical error see Appendix B .

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TABLE I. Positions and values of the absolute minimum LMA and of three relevant local minima 2 2 LOW, QVO, SMA of pull , together with the separate contributions from pulls of observables ( obs) and of 2 2 variations are also given. correlated systematics ( sys) . The corresponding Solution LMA LOW QVO SMA m 2 (eV2 ) 5.5 7.3 6.5 5.2 10 10 10 10
5 8 10 6

tan2

12

2 obs

2 obs

2 sys

2 sys

2 pull

2 pull

0.42 0.67 1.33 1.1 10

3

71.3 79.7 74.9 83.1

-- 8.4 3.6 11.8

2.1 4.1 6.3 13.8

-- 2.0 4.2 11.7

73.4 83.8 81.2 96.9

-- 10.4 7.8 23.5

B spectrum shape uncertainties estimated in 46 rather than the smaller ones estimated in 47 since, in our opinion, the issue of significant O ( 100) keV energy calibration differences among existing 8 B decay spectrum measurements 46,47 is not completely clarified, and warrants further experimental investigations.11 There are also eleven sources of correlated systematics, which affect only the SK spectrum. They include the SK energy scale and resolution uncertainties, an overall SK rate offset, and eight sources of systematics separately affecting the eight energy bins, with full correlation in nadir 6 see Appendix C . Finally, there are seven sources of correlated systematics, which affect only the SNO spectrum. They include: the uncertainties affecting the SNO energy scale and resolution, the event vertex reconstruction, the neutron capture efficiency, the neutron and low-energy LE background estimates and the interaction cross sections. See Appendix D for more details.
C. Output

8

Finally, it is useful to isolate the twelve SSM systematic pulls ¯ k k 1, . . . ,12 which, on the basis of Eq. 16 , allow us to derive the induced neutrino flux shifts from the SSM central values, namely
12 i i k1

¯

k

ik

ln X k .

17

These shifts provide valuable and luminosity-constrained information about the preferred departures from the SSM within the various oscillation solutions to the solar neutrino problem. Summarizing, we will show and discuss results about pulls, ¯ x ¯
n n 1 , . . . ,8 1

pulls of the observables, pulls of the correlated systematics,

18 19

k k 1 , . . . ,3 1 2

about

values,
81 2 obs n1 31 2 sys k1 2 obs

As output of the pull analysis, we get the function m 2 , tan2 12) essential to identify absolute and local minima and to draw confidence level contours , as well as other useful statistical indicators. 2 Concerning pull , for any fixed point in the parameter 2 2 2 space ( m , tan 12) , the goodness-of-fit test requires pull N ( N 81) for an acceptable fit. Further information can 2 be gained by splitting pull into the separate contributions 2 2 obs and sys Eqs. 9 and 10 , obtained by summing up the squared pulls of the N 81 observables ¯ n , see Eq. 8 x and of the K 31 systematics ¯ k , see Eq. A6 . The larger 2 the value of sys , the more the fit tends to ``stretch'' one or more correlated systematics to get a better agreement between data and expectations. Apart from global features, the x analysis of the pull sets ¯ n and ¯ k allows to quantify individual contributions to the 2 , which, if anomalously large, might be indicative of problems either in the theoretical predictions or in the experimental measurements. Therefore, we think it useful to present, besides the global values 2 2 2 of pull obs sys , also some selected lists of pulls.
2 pull(

¯2 , xn

20

¯2 , k
2 sys,

21 22

2 pull

and about fractional shifts from the SSM predictions,
i

/

i

flux shifts.
2 pull

23

IV. RESULTS OF THE

ANALYSIS

In this section we start by describing the global results of 2 the pull analysis, and then we break down such results at increasing levels of detail.
A. Global results

We have been informed that a new 8 B( ) 8 Be( 2 ) decay spectral measurement is in progress at Argonne W. T. Winter private communication .

11

The global results of our solar marized in Table I and in Fig. 1. ( m 2 , tan2 12) coordinates of the large mixing angle LMA solution

oscillation fit are sumIn Table I we report the best-fit point so-called and of the three deepest

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the data and of the systematics offsets from zero. Concerning the parameter estimation test13 based on 2 pull variations around the minimum Fig. 1 shows the results of our global analysis in the usual mass-mixing plane. 2 4.61, 5.99, The confidence level isolines are drawn at 9.21, and 11.83, corresponding to 90%, 95%, 99%, and 99.73% joint probability regions for the two ( m 2 , tan2 12) parameters. The QVO and LOW parameters are still acceptable at the 99% and 99.73% C.L., respectively, while the SMA parameters are basically ruled out. Our LMA bounds appear to be sometimes significantly more conservative than in other recent analyses 6,9,15 ­ 20 .14 In particular, at the 99.73% C.L. we derive from Fig. 1 that: i maximal mixing is marginally allowed in the LMA region, and ii the highest m 2 allowed values hit the region independently disfavored by CHOOZ data 53 .15 We think that the detailed treatment of all known uncertainties and of their propagation to all relevant experimental observables plays a role in such different results, also for non-LMA solutions. Concerning the LOW solution, we note that the inclusion of the winter-summer datum from GALLEX-GNO contributes to decrease its likelihood in our analysis.
B. Separating experimental bounds

FIG. 1. Global results of the solar neutrino data analysis, including 81 observables and 31 sources of correlated systematics. The parameter space ( m 2 , tan2 12) refers to the scenario of 2 oscillations among active states. The relevant 2 minima in the LMA, LOW, and QVO regions are given in Table I.
2 local pull minima in the regions of low m 2 LOW , quasivacuum oscillations12 QVO , and small mixing angle SMA . Concerning the goodness-of-fit test, we recall that, at the absolute minimum, one expects the total 2 to be in the 2 N DF 50 . In our case ( N DF 81 2), 1 range N DF 2 it is pull 73.4 at the LMA best-fit point, well within the expected range 79 12.6. Also the LOW and especially the 2 QVO solutions have acceptable values of pull , while the SMA value appears to be significantly larger than expected. The pull analysis will confirm that the LOW and QVO solutions are still viable, while the SMA solution is no longer statistically acceptable. Notice that at the LMA point, most 2 of the contribution to pull comes from pulls of observables 2 2 ( obs) rather than systematics ( sys) . All the other solutions 2 2 in Table I show an increase of both pull and sys , implying an increasing departure of the theoretical predictions from

Figure 2 show the decomposition of the global results into contributions from the Cl experiment total rate , from the Gallium experiments total rate and winter-summer difference , from the SK energy-nadir spectrum 44 bins , and from the SNO day-night spectrum 34 bins . In each panel, the results are shown in terms of allowed regions, for the same confidence levels as in Fig. 1 referred to the absolute minimum in each panel . 2 Concerning the pull minima in Fig. 2, their positions are not particularly interesting for the Cl and Ga cases, where they are essentially degenerate. More interesting is the case 2 of the SK experiment alone, where the best fit ( pull 38.4) is reached at maximal mixing and for m 2 6.5 10 2 eV in agreement with the results in 54 , very 10 close to the QVO coordinates in Table I. Concerning the fit to SNO data only, we find the best fit at m 2 3.7 10 5 eV2 and tan2 12 0.47, close to the LMA coordi2 nates in Table I, with pull 25.7. The latter value appears to 2 be on the lower side of the 1 expected range for the pull 16 in SNO ( 32 64) .
13

12 We do not find acceptable solutions in the octant-symmetric vacuum oscillation VO regime. For the QVO solution in Table I, only the highest energy flux components ( B and hep) have reached the VO regime, while the lowest energy ones are still affected by octant-asymmetric quasivacuum effects 48,49 in the Sun.

Useful discussions of the applications and differences between the goodness-of-fit test and the parameter estimation test can be found in 50­52 . 14 The closest agreement is reached with the global allowed LMA region in Ref. 18 . 15 CHOOZ data are not included in the present analysis, in order to show more clearly the strength of the upper bound on m 2 placed by solar neutrino data alone. 16 We think that this feature might be partly due to nonoptimal binning of the SNO spectrum. Although, at low energy, relatively dense binning is required to enhance the effects of the neutral current component, at high energies it is preferable to enlarge the bin

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FIG. 2. Results of the solar neutrino data analysis, as obtained by separating four classes of observables: i the chlorine rate; ii the average SAGE GALLEX-GNO gallium rate plus the GALLEX-GNO winter-summer difference; iii the SK energy-nadir spectrum; and iv the SNO day-night spectrum.

Concerning the shapes of the allowed regions in Fig. 2, we note the following facts. None of the experiments excludes maximal mixing and m 2 at 99% C.L. This feature is rather well known for Ga and SK data, but does not appear in all recent analyses for the Cl and SNO data. For instance, the Cl contours in Ref. 6 appear to be more restrictive than ours, which might be due to different estimates of the Cl errors. We also note some differences between our SNO bounds in Fig. 2 and the SNO official analysis in Fig. 4 a of 9 : a in the QVO region, our contours are smooth as they should ; b we do not find Q VO solutions at maximal mixing for m 2 10 10 eV2 ; c our bounds in the LMA region allow maximal mixing and m 2 well within the 99.73% C.L. Concerning the points a and b , we think that the differences might depend in 9 on the numerically delicate averaging of the oscillating terms in the energy or time domain. Concerning the points b and c , some differences might also be due to the fact that the analysis in Fig. 4 a of 9 is done without SSM input. Concerning the point c , we

width so as to match the SNO energy resolution width analogously to the current SK energy spectrum binning .

note that the exclusion of m 2 and of tan2 12 1 at the 99.73% level by the SNO data alone as found in Fig. 4 a of 9 would be equivalent to the exclusion of the constant P ee 1/2 case or of equal e and , fluxes, e , ) at the same confidence level. Although the SNO data clearly prefer P ee 1/3 8 see also Appendix E , a 3 rejection of P ee 1/2 might be premature. Indeed, from Fig. 3 of 8 it appears that the e , line touches the 95% error ellipse determined by the total SNO rates. Given that the exclusion of relatively large values of m 2 and tan2 has profound implications in lepton physics both phenomenologically and theoretically , we think that the impact of the SNO data on such values warrants further investigations. From Fig. 2 it also appears that all experiments largely agree in the LMA region, and that a few QVO ``islands'' below 10 9 eV2 also happen to be consistent with all experiments. Moreover, all the experiments appear to be generically consistent with some ``LOW'' or ``SMA'' regions at least at 99% C.L. However, such regions are somewhat different for the different panels in Fig. 2. Concerning the LOW case, there is a reasonable overlap of the Cl, Ga, and SK bounds at m 2 10 7 eV2 , while SNO prefers m 2

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FIG. 3. Results of the solar neutrino data analysis, as obtained by excluding each of the four data sets in Fig. 2 from the global set used in Fig. 1.

10 7 eV2 , where the LOW parameters more easily adapt to the preferred value P ee 1/3. This tension generates an overall decrease of the LOW likelihood. Concerning the SMA case, the low-tan2 12 regions separately allowed by Cl, Ga, SK, and SNO at the 99.73% C.L. do not overlap. Figure 3 allows to appreciate the impact of each experiment in the global fit, by removing one experiment at a time. In a sense, Fig. 3 is the ``difference'' between Fig. 1 and Fig. 2. It can be seen that the removal of either the Cl or the Ga experiments upper panels in Fig. 3 weakens the bounds on large mixing, on large values of m 2 , and on the LOW parameters, but does not alter the situation for vacuum oscillations, which are excluded in both cases.17 The lower left panel in Fig. 3 shows that the removal of the SK experiment
17 In Ref. 20 , the Ga impact on the LOW solution has been studied by lowering the total rate from 70.8 4.4 to 66.1 5.3 SNU. 2 The authors of 20 find a corresponding reduction of the LOW from 6.9 to 3.0, with respect to the LMA minimum. By repeating 2 the same exercise, we find a smaller effect: our LOW decreases from 10.4 see Table I to 9.0.

would diminish the likelihood of the solutions characterized by no or mild energy spectrum distortion e.g., m 2 or QVO cases , and, conversely, would make some strongly energy-dependent solutions marginally reappear at small mixing and in the vacuum regime . Finally, the lower right panel in Fig. 3 shows the pre-SNO situation but with updated SK, Cl, and Ga data , with all the well-known multiple solutions to the solar neutrino problem. The comparison of Figs. 1 ­ 3 shows the dramatic impact of SK and SNO in determining the preference for large mixing and especially for the LMA solutions and the rejection of the SMA solution. However, the Cl and Ga data still play an important role in determining the shape of the LMA contours, as well as the likelihood of the less favored solutions LOW and QVO , which cannot be rejected on the basis of the present global information.
C. Separating and grouping pulls
2 We discuss the decomposition of pull into separate and x grouped pulls of observables ¯ n and of correlated systematics ¯ k see Eqs. 18 ­ 22 for the various solutions.

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Table II shows the pulls of each of the 81 observables used in our analysis, corresponding to the four solutions in Table I.18 For the best-fit LMA solution, we also give the range spanned by each pull within the 99.73% C.L. LMA region shown in Fig. 1. It can be seen that, in any case, there are no anomalously large say, 3 standard deviation pulls--not even in the SMA solution or at the borders of the LMA region. This fact confirms the results of the previous sections, namely, that the allowance or rejection of the various solutions comes from a ``collective'' effect of several experimental observables rather than by a small subset of them. Figure 4 provides a graphical version of the first three columns in Table II, together with an histogram of the pull distribution, compared with a Gaussian distribution conventionally normalized to N 81) to guide the eye. It appears that the pulls of the observables have a reasonably symmetrical and Gaussian distribution, confirming the goodness of the LMA fit at a deeper level than the global 2 values. The relatively large pull for the Cl datum ( 1.74) may be regarded as a statistical fluctuation among the others, at the LMA best fit. Notice that the Cl pull in the SMA solution is instead very small 0.14 . Retrospectively, this small pull, together with theoretical prejudices against large mixing, appears to be at the origin of a very long detour towards smallmixing oscillations in matter. The distributions of pulls of the observables for the QVO and LOW cases not graphically shown are also reasonably Gaussian as for the LMA case, although with a slightly larger area given by the correspond2 ing obs values in Table I . The distribution of pulls for the SMA case not shown , besides having an even larger area 2 ( obs 83.1) , appears also to be slightly skewed, with an excess of positive pulls. This adds to the statistical problems of this solution. It is interesting to group the separate experimental contri2 butions to the global obs in Table I, by summing up the corresponding squared pulls from Table II. The results of this exercise, as reported in Table III, show which experiment ``wins'' or ``loses'' in the various global solutions. In particular, the radiochemical experiments clearly win in the SMA and loose in the QVO and LOW solutions, while the fit to the SK spectrum observables appears to be rather stable, with 2 3 variations in the various solutions. The only obs SMA solution tries to make a compromise between SK and SNO data, in which SK dominates having smaller spectral 2 errors , leaving SNO with a worse fit ( obs 38.5) as com2 pared with the LMA case ( obs 26.2) . Let us now consider the contributions of correlated systematics shifts to the global fit. Table IV shows the contributions of the pulls of systematics in the various solutions. As in Table II, for the LMA solution we also show the range spanned by the pulls within the 99.73% C.L. LMA region in Fig. 1. It appears that such pulls are generally rather small,
18

and typically assume minimal values in the LMA solution. In particular, Fig. 5 shows the LMA pull diagram for the correlated systematics, none of which presents a significant offset. The smallness of all SSM input offsets with the possible exception of S 17) is particularly impressive. In general, at the LMA best-fit point there is no need to stretch any correlated systematics to fit the data. In the LOW and QVO solutions the situation is less ideal several offsets in Table IV are at the level of 0.5 ) but certainly still acceptable. In the SMA case, however, several SNO systematic offsets in Table IV are at the level of 1 , and the 8 B shape uncertainty is stretched by 1.5 . Such offsets, which act in the direction of reducing the spectral differences between theory and 2 data, produce a significant contribution to the SMA fit ( sys 13.8) . It is quite unlikely that future, possible recalibrations of systematics may just happen to cancel out all these offsets, thus giving more chances to the SMA solution. It is interesting to group some 2 contributions of systematics, according to their origin. This exercise is done in Table 2 V. The SK contribution to sys is rather stable and small , with a minimum at the QVO solution. This situation parallels 2 the SK contribution to obs see Table III and related comments , and show the pivoting role of the SK spectrum in determining the likelihood of all solutions. Conversely, the 2 contributions to sys from SNO and from the standard neutrino flux input SSM and 8 B shape systematics increase significantly when passing from the LMA to the SMA solution. 2 The grand total of the various contributions to the pull from both observables and systematics is shown in Table VI. This table shows quantitatively that the LMA solution is in very good agreement with both the experimental data and with the SSM. The LOW and QVO solutions provide a slightly less good agreement with the SSM and with SK SNO, and are somehow ``borderline'' from the point of view of radiochemical experiments. However, they cannot be really excluded by any data at present. The SMA solution is instead safely ruled out, mainly as a consequence of the SNO 2 data fit ( SNO 45.1) .
D. Implications for the SSM neutrino fluxes

The SK and SNO spectrum bins are identified by their energy range and by their nadir interval day and night bins . See also Appendixes C and D for notation.

In the previous section, we have seen that the LMA solution does not require any significant offset ¯ k in the SSM input systematics ( k 1,... ,12) , and that such offsets are relatively small also in the other solutions. Equation 17 allows to translate such offsets into preferred shift of the neutrino fluxes i from their SSM central values. The results are shown in Table VII for the various solutions. In the LMA best-fit point, all solar neutrino fluxes are basically confirmed: just for illustration, an approximate translation of the LMA flux shifts into variations of the ``effective solar core temperature'' ( t c / t c 37,43,45 would formally provide t c / t c 0.2% . The QVO and SMA solutions would formally require t C / t C 0.5 ­ 1 % a slightly cooler Sun , the LOW case being intermediate between the latter and the LMA one. In all solutions, the preferred pp values are within a percent from the SSM. The preferred negative

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TABLE II. Pulls of the observables in the various solutions. n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 Observable Cl rate Ga rate Ga W S SK 5.0,5.5 SK 5.5,6.5 day SK 5.5,6.5 M1 SK 5.5,6.5 M2 SK 5.5,6.5 M3 SK 5.5,6.5 M4 SK 5.5,6.5 M5 SK 5.5,6.5 core SK 6.5,8.0 day SK 6.5,8.0 M1 SK 6.5,8.0 M2 SK 6.5,8.0 M3 SK 6.5,8.0 M4 SK 6.5,8.0 M5 SK 6.5,8.0 core SK 8.0,9.5 day SK 8.0,9.5 M1 SK 8.0,9.5 M2 SK 8.0,9.5 M3 SK 8.0,9.5 M4 SK 8.0,9.5 M5 SK 8.0,9.5 core SK 9.5,11.5 day SK 9.5,11.5 M1 SK 9.5,11.5 M2 SK 9.5,11.5 M3 SK 9.5,11.5 M4 SK 9.5,11.5 M5 SK 9.5,11.5 core SK 11.5,13.5 day SK 11.5,13.5 M1 SK 11.5,13.5 M2 SK 11.5,13.5 M3 SK 11.5,13.5 M4 SK 11.5,13.5 M5 SK 11.5,13.5 core SK 13.5,16.0 day SK 13.5,16.0 M1 SK 13.5,16.0 M2 SK 13.5,16.0 M3 SK 13.5,16.0 M4 SK 13.5,16.0 M5 SK 13.5,16.0 core SK 16.0,20.0 SNO 5.0,5.5 day SNO 5.0,5.5 night SNO 5.5,6.0 day SNO 5.5,6.0 night SNO 6.0,6.5 day SNO 6.0,6.5 night LMA 1.74 0.39 1.23 0.09 0.36 0.41 1.77 0.03 1.11 0.62 0.90 1.20 1.97 1.31 1.29 0.19 0.81 1.12 0.40 0.17 0.24 0.17 1.40 0.26 0.51 0.67 0.20 1.01 0.55 0.08 0.17 0.33 0.72 0.67 2.21 1.76 0.36 0.80 0.67 0.65 0.59 2.03 1.51 1.31 0.48 0.22 0.34 0.99 0.23 0.94 0.31 0.09 0.72 LMA min, max at 3 2.56, 1.49, 1.23, 0.46, 1.06, 0.63, 2.00, 0.20, 0.93, 0.47, 1.04, 0.48, 1.80, 0.82, 1.80, 0.23, 1.17, 1.41, 0.93, 0.05, 0.32, 0.73, 0.92, 0.65, 0.83, 1.19, 0.36, 0.45, 1.11, 0.43, 0.21, 0.01, 0.36, 0.57, 2.64, 2.11, 0.02, 1.02, 0.46, 0.36, 0.52, 1.88, 1.39, 1.18, 0.57, 0.30, 0.06, 1.32, 0.17, 0.63, 0.65, 0.53, 1.55, 1.17 1.90 1.22 0.10 0.40 0.29 1.64 0.26 1.36 0.86 0.67 2.35 2.04 1.38 1.07 0.46 0.54 0.87 0.56 0.19 0.42 0.15 1.76 0.09 0.21 0.19 0.13 1.30 0.10 0.57 0.63 0.69 1.31 0.71 1.98 1.43 0.69 0.50 0.92 1.02 0.61 2.06 1.57 1.39 0.41 0.15 0.47 0.28 1.01 1.42 0.12 0.64 0.02 LOW 1.86 1.27 1.75 0.14 0.06 0.27 1.84 0.05 1.46 0.71 0.75 1.15 1.99 1.08 1.39 0.62 0.78 0.99 0.67 0.13 0.06 0.22 1.75 0.20 0.39 1.07 0.29 0.88 0.56 0.40 0.26 0.43 0.42 0.61 2.25 1.72 0.57 0.72 0.76 0.50 0.56 2.03 1.54 1.42 0.42 0.15 0.31 0.49 0.78 1.23 0.36 0.26 1.27 QVO 2.68 0.84 1.25 0.01 0.52 0.40 1.72 0.19 1.30 0.80 0.73 0.56 1.84 1.29 1.13 0.41 0.60 0.94 0.91 0.09 0.31 0.08 1.70 0.02 0.28 0.83 0.17 1.22 0.13 0.55 0.57 0.62 0.66 0.71 2.00 1.39 0.70 0.59 0.76 0.37 0.25 1.79 1.26 1.05 0.85 0.60 0.19 0.66 0.65 1.06 0.64 0.60 1.52 SMA 0.14 0.19 1.22 0.58 0.17 0.14 1.43 0.52 1.64 1.14 0.41 0.72 1.90 1.37 1.02 0.55 0.40 0.73 1.29 0.05 0.17 0.05 1.61 0.03 0.23 1.70 0.49 0.90 0.49 0.26 0.45 0.52 0.11 0.52 2.23 1.65 0.55 0.54 0.82 0.48 0.57 2.08 1.62 1.47 0.26 0.08 0.35 0.08 1.19 1.33 0.60 0.74 1.88

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GETTING THE MOST FROM THE STATISTICAL. . . TABLE II. Continued . n 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 Observable SNO 6.5,7.0 day SNO 6.5,7.0 night SNO 7.0,7.5 day SNO 7.0,7.5 night SNO 7.5,8.0 day SNO 7.5,8.0 night SNO 8.0,8.5 day SNO 8.0,8.5 night SNO 8.5,9.0 day SNO 8.5,9.0 night SNO 9.0,9.5 day SNO 9.0,9.5 night SNO 9.5,10.0 day SNO 9.5,10.0 night SNO 10.0,10.5 day SNO 10.0,10.5 night SNO 10.5,11.0 day SNO 10.5,11.0 night SNO 11.0,11.5 day SNO 11.0,11.5 night SNO 11.5,12.0 day SNO 11.5,12.0 night SNO 12.0,12.5 day SNO 12.0,12.5 night SNO 12.5,13.0 day SNO 12.5,13.0 night SNO 13.0,20.0 day SNO 13.0,20.0 night LMA 0.64 1.34 0.66 0.27 0.41 0.39 0.20 0.69 1.31 0.43 0.72 0.54 0.95 0.66 0.76 2.19 0.60 1.76 0.78 0.69 0.27 0.27 1.28 0.89 0.45 0.64 1.37 1.03 LMA min, max at 3 1.46, 0.70, 1.24, 0.13, 0.15, 0.25, 0.26, 0.73, 1.39, 0.35, 0.96, 0.07, 1.59, 1.13, 0.44, 1.70, 0.09, 2.54, 0.15, 1.25, 0.17, 0.64, 1.01, 0.66, 0.69, 0.44, 1.21, 1.41, 0.10 1.95 0.05 0.74 0.79 0.68 0.04 0.47 1.07 0.60 0.39 1.35 0.07 0.07 0.85 2.43 1.02 1.22 1.24 0.21 0.69 0.10 1.54 1.14 0.13 0.92 1.60 0.57

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LOW 1.24 0.84 1.14 0.09 0.16 0.24 0.27 0.68 1.25 0.51 0.51 1.03 0.29 0.27 0.80 1.96 0.20 2.24 0.41 0.98 0.10 0.33 1.31 1.00 0.26 0.82 1.54 0.64

QVO 1.20 1.17 0.57 0.38 0.37 0.13 0.65 1.20 1.78 0.27 0.20 1.09 0.19 0.21 0.75 1.84 0.18 1.93 0.99 0.21 0.69 0.08 1.17 0.59 0.84 0.40 1.33 0.19

SMA 1.82 0.40 1.51 0.31 0.06 0.22 0.21 0.58 1.10 0.60 0.44 1.71 0.50 0.17 0.84 1.75 0.14 2.60 0.17 1.13 0.08 0.25 1.45 1.19 0.03 1.02 1.68 0.51

variations of the B values from its SSM value are instead significant in non-LMA solutions, and appear to be consistent with those derived in some B -free analyses 18,19,54 . This might seem surprising, since we do not exclude any SSM input in our analysis. The reason for such agreement is that the current SNO data implicitly provide an experimental determination of B which is already significantly more accurate than the SSM estimate; therefore, there is little difference in making the analysis with or without SSM input for B . Notice that, in addition, we can also quantify the preferred variations of Be , N , and O , which appear to be all negative in non-LMA solutions.19 Concerning hep , we do not find any significant preferred variation with respect to Eq. 15 . This fact is mainly due to the poor sensitivity of the data to this flux confined to the last few bins in the SK and SNO spectrum and possibly to our careful treatment of SK and SNO spectral uncertainties see Appendixes C and D . Finally, Fig. 6 shows isolines of the preferred neutrino flux shifts within the LMA solution superposed at 99% C.L. . The variations of pp and of pep are limited to about a percent, but can be an order of magnitude larger for the
19 This information is relevant for prospective studies in the BOREXINO experiment 14 .

other neutrino fluxes. In the upper or right part of the LMA region, the current global fits seem to prefer variations of the fluxes roughly corresponding to a slightly ``cooler '' Sun, while in the lower left corner of the LMA region the trend is opposite slightly ``hotter '' Sun . Should the best-fit LMA value be confirmed by the KamLand experiment 13 , the current SSM input would be just ``perfect.''
V. SUMMARY AND CONCLUSIONS

We have performed a global analysis of solar neutrino oscillations within the 2 active scenario,20 including all the current solar neutrino data and all relevant sources of uncertainties. The statistical analysis has been performed in a way which clearly traces the residual contribution of each observable and of each source of correlated systematics in the 2 fit ``pull'' method . It turns out that there is still a multiplicity of acceptable solutions at large mixing in the so-called LMA, LOW, and QVO regions Fig. 1 and Table I , none of which contradict any data Figs. 2 and 3 , as also confirmed by a detailed pull analysis Tables II ­ VI . In particular, the bestfit LMA solution appears to be in very good agreement with
20

Three-flavor results will be shown elsewhere.

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FIG. 4. Diagram of pulls ¯ n n 1, . . . ,81 for obx servables at the LMA best-fit point. See the text for details.

all the data Fig. 4 , without requiring any stretching of the correlated systematics Fig. 5 , contrary to the strongly disfavored solution at small mixing angle. The striking LMA agreement with all standard solar model fluxes Table VII is only slightly worsened when moving away from the best fit Fig. 6 . The statistical techniques and the treatment of the data underlying these results have been discussed in detail in the Appendixes.
TABLE III. Separate experimental contributions to the global for the various solutions reported in Table I, as obtained by grouping squared pulls from Table II.
2 obs

Note added. After completion of this work, the SAGE Collaboration has added the winter-summer difference datum in Ref. 2 . We thank B. Cleveland for related communications.
ACKNOWLEDGMENTS

n 1 2­3 4 ­ 48 49 ­ 81 1­ 81

Experiment Cl rate Ga rate and W S ) SK 44 bins SNO 34 bins 2 All global obs)

LMA 3.0 1.7 40.4 26.2 71.3

LOW 3.4 4.7 42.9 28.7 79.7

QVO 7.2 2.3 37.0 28.4 74.9

SMA 0.0 1.5 43.1 38.5 83.1

We thank the organizers of the 20th International Conference on Neutrino Physics and Astrophysics in Munich-- where preliminary results of this work were presented--for kind hospitality. We also thank the many participants to this Conference who have contributed useful comments, remarks, and suggestions. This work was supported in part by INFN and in part by the Italian Ministero dell'Istruzione, Univer` sita e Ricerca through the ``Astroparticle Physics'' research project.
APPENDIX A: PROOF OF
2 covar

ä

2 pull

Let us recall that basic ingredients of any 2 statistics are: the experimental and theoretical values ( R expt and R theor) of n n

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TABLE IV. Pulls of the systematics in the various solutions. k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 Systematic S 11 S 33 S 34 S 1,14 S 17 Luminosity Z/X Age Opacity Diffusion C Be S hep 8 B shape SK scale SK resolution SK offset SK 5.0,5.5 SK 5.5,6.5 SK 6.5,8.0 SK 8.0,9.5 SK 9.5,11.5 SK 11.5,13.5 SK 13.5,16.0 SK 16.0,20.0 SNO scale SNO resolution SNO vertex SNO n capture SNO n background SNO LE background SNO cross section LMA 0.05 0.00 0.01 0.15 0.38 0.04 0.03 0.00 0.05 0.02 0.07 0.03 0.17 0.78 0.61 0.44 0.03 0.26 0.54 0.01 0.14 0.21 0.26 0.01 0.15 0.32 0.13 0.10 0.06 0.16 0.04 LMA min, max at 3 0.31, 0.13, 0.98, 0.49, 0.71, 0.31, 0.89, 0.06, 0.36, 0.26, 0.22, 0.04, 0.71, 0.51, 0.54, 0.33, 0.18, 0.61, 0.32, 0.06, 0.25, 0.31, 0.23, 0.00, 0.90, 0.41, 0.60, 0.46, 0.27, 0.49, 0.16, 0.42 0.32 0.42 0.02 1.15 0.17 0.48 0.04 0.52 0.43 0.13 0.00 1.24 1.82 0.87 0.70 0.05 0.13 0.67 0.19 0.21 0.06 0.34 0.02 0.58 0.05 0.65 0.60 0.35 0.53 0.21 LOW 0.19 0.10 0.32 0.08 0.35 0.08 0.35 0.02 0.22 0.18 0.07 0.02 0.66 0.49 0.61 0.57 0.06 0.10 0.70 0.08 0.30 0.29 0.32 0.02 0.86 0.16 0.52 0.42 0.25 0.33 0.16 QVO 0.35 0.24 0.72 0.21 0.47 0.20 0.68 0.05 0.42 0.34 0.09 0.10 0.80 0.49 0.06 0.68 0.00 0.28 0.52 0.03 0.14 0.10 0.11 0.01 1.49 0.52 0.20 0.34 0.20 0.28 0.02 SMA 0.34 0.18 0.59 0.01 0.83 0.22 0.60 0.04 0.41 0.31 0.16 0.02 1.56 0.31 0.73 0.34 0.27 0.34 0.89 0.42 0.76 0.45 0.44 0.02 1.48 0.47 1.42 0.94 0.55 0.87 0.52

the N observables to be fitted; the associated uncorrelated errors u n ; and the associated set of fully correlated errors c k , due to K independent sources of systematics ( k n 1, . . . , K ). In order to simplify the notation, we normalize both the differences R expt R theor and the correlated errors c k to the n n n u n 's, by defining R
n expt n

while Eq. 7 reads
N 2 pull 2 n K k1 2 k

min
k

q
k

n1

k n

k

,

A4

R u
n

theor n

,

A1

where the k are Gaussian random variables with k 0 and 2 1. k The minimization in Eq. A4 leads to a set of K linear equations in the unknowns ¯ k ,
K N kh n

and q Equation 5 reads then
N 2 covar n,m 1 n nm k 1 k n

c u

k n n

.

A2

h1

qkqh ¯ nn

h

n1

k nq n

,

A5

whose solution is
K m N

qkq n

k m

,

A3
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¯

k

S
h1

kh

n1

h nq n

,

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2 TABLE V. Separate contributions to the global sys for the various solutions reported in Table I, as obtained by grouping squared pulls from Table IV.

k 1 ­ 13 14 ­ 24 25 ­ 31 1­31

Systematic sources SSM and B shape SK systematics SNO systematics 2 All global sys)
8

LMA 0.2 1.7 0.2 2.1

LOW 1.0 1.7 1.4 4.1

QVO 2.4 1.1 2.8 6.3

SMA 4.3 2.9 6.6 13.8

and to substitute Eq. A6 in Eq. A9 . Expanding the righthand side RHS of Eq. A9 , and making use of Eq. A8 , one recovers the RHS of Eq. A3 , namely,
2 covar 2 pull.

A10

Finally, we observe that setting
K

¯ x

n

n

k1

q k¯ k , n

A11

as in Eq. 8 , one gets from Eq. A9 a ``diagonal'' form for 2 pull ,
N 2 pull K

¯ x
n1

2 n

k1

¯2 , k

A12

FIG. 5. Diagram of pulls ¯ k k 1,...,31 for correlated systematics at the LMA best-fit point. See the text for details.

as anticipated A different cussed in the where the pull

in Eq. 10 . proof of the previous relations have been discontext of parton distribution fitting 25 , method is now routinely used 26,27 .

where S is the inverse matrix21
1

APPENDIX B: WINTER-SUMMER DIFFERENCE IN GALLEX-GNO

S

kh

kh

n

qkq n

h n

.

A7

It turns out that the matrix S is also related to the inversion of the covariance matrix,
1 nm K nm

Earth matter effects can generate an observable wintersummer difference ( R W R S ) in the event rates measured in gallium experiments. Such a difference can be as large as 6 SNU around the LOW solution 39 . The GALLEX-GNO Collaboration has recently reported the measurement 4 see also 55 R
W

q
k

kk nq m

S
k,h 1

kh

h qm qk . n

A8

R

S

11 9 SNU GALLEX-GNO ,

B1

Indeed, the product of the above two matrices gives the unit matrix. The above equation reduces the inversion of the N 2 N covariance matrix in covar to the inversion of the generally much smaller K K matrix in Eq. A7 . 2 The last step is to write the pull in terms of the ¯ k 's,
N 2 pull n1 n k 2 K k1

whose uncertainty is almost entirely statistical 55 , all systematics being largely cancelled in the difference 56 . In our
TABLE VI. Separate contributions to the global various solutions reported in Table I. n k Contributions
8 2 pull

for the

LMA LOW QVO SMA 1.0 3.4 4.7 44.6 30.1 83.8 2.4 7.2 2.3 38.1 31.2 81.2 4.3 0.0 1.5 46.0 45.1 96.9

qk¯ n

k

¯2 , k

A9

Here we deal only with symmetric matrices. Therefore, given a matrix equation such as A B 1 , we can conventionally write it as A nm B nm 1 without index ambiguity.

21

-- 1 ­ 13 SSM and B shape 0.2 1 -- Cl experiment 3.0 2­3 -- Ga experiments 1.7 4 ­ 48 14 ­ 24 SK experiment 42.1 49 ­ 81 25 ­ 31 SNO experiment 26.4 2 1­ 81 1­31 73.4 All global )

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GETTING THE MOST FROM THE STATISTICAL . . . TABLE VII. Fractional neutrino flux shifts from the SSM central values ( i / i 100) , for the various solutions. pp LMA LOW QVO SMA 0.0 0.5 1.1 0.9 pep 0.0 0.8 1.6 1.3 Be 0.5 5.5 11.5 9.9 B 5.2 12.2 22.2 24.2 hep 0.8 0.0 1.8 1.8 N 1.0 8.3 15.1 11.4 O 1.2 8.3 17.0 12.8

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``pull'' approach, lated error u WS The definition 4,55 is slightly corresponds to

we simply attach to R W R S an uncorre9 SNU, with no systematics. of ``Winter '' and ``Summer '' adopted in different from the astronomical one,22 and 3 months, 3 months. B3 B2

``Winter '' perihelion 5 Jan. ``Summer '' aphelion 5 July

Therefore, the solar exposure functions for the above periods, as shown in Fig. 7 in terms of the nadir angle ( ) , are slightly different from the ones given in 39 . In particular, the comparison of Fig. 7 in this paper with Fig. 5 in 39 shows the the above definitions lead to a lower higher exposure of the innermost trajectories in the mantle during winter summer . The total annual exposure is, of course, unaltered. The definitions in Eqs. B2 and B3 are particularly useful to smoothly extend the theoretical calculation of R W R S from the matter-enhanced regime 39 down to the vacuum oscillation regime, where seasonal variations are instead induced by the eccentricity variations of the orbital distance L purely geometrical ( 1/L 2 ) effects being factored out . Indeed, with the definitions in Eqs. B2 and B3 , the winter-summer rate difference in vacuum happens to coincide with the near-far rate difference ( R N R F ) previously defined in 57 . We have then matched the results found in 39 and in 57 , by considering both matter-induced and eccentricity-induced contributions to R W R S , so as to calculate this quantity in the whole oscillation parameter space. The match between the matter and vacuum regimes occurs in the quasivacuum range, and is made easier by the lucky circumstance that Earth matter effects vanish just when the oscillating terms in the e survival probability start to be important 48 . The datum in Eq. B1 is compatible with no seasonal asymmetry, adds a slight penalty to the region roughly corresponding to the LOW solution where 0 R W R S 6 SNU) , and modulates the likelihood of the solutions in the Q VO regime, where both positive and negative values 26 SNU) , of the asymmetry can occur ( 26 R W R S the negative ones being slightly favored by the GALLEX-

FIG. 6. Preferred deviations of the solar fluxes from their SSM central values 41 , in the region of the LMA solution superposed at 99% C.L. . Dotted solid isolines roughly correspond to a slightly ``cooler '' hotter Sun. See the text for details.

GNO datum. We remark that positive or negative seasonal effects in gallium experiments, being largely driven by lowenergy solar neutrino components, can be compatible with the nonobservation of ( 8 B-driven seasonal effects in SK 54 . Finally, we mention that the SAGE experiment has recently reported event rates in gallium, grouped in bi monthly intervals 2 , which appear to be consistent with no seasonal variations although no explicit R W R S estimate is given in 2 . Taking the data in 2 at face value, we argue a slightly positive value for R W R S in SAGE, with a total uncertainty comparable to that of GALLEX-GNO. The combination of a slightly positive SAGE and a slightly negative GALLEX-GNO winter-summer difference would then provide a central value closer to zero for R W R S , very consistent with the LMA solution 39 . An official evaluation of R W R S from SAGE to be combined with the one in Eq. B1 from GALLEX-GNO appears thus a desirable input for future analyses.
APPENDIX C: THE SK ENERGY-NADIR DIFFERENTIAL SPECTRUM

Winter and summer were defined in 39 as six-months intervals separated by equinoxes and centered, respectively, at the winter solstice 21 Dec and at the summer solstice 21 Jun .

22

The SK energy-nadir spectrum of electrons induced by neutrino elastic scattering is fundamental to constrain the solar neutrino parameter space. Therefore, we think it useful

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FIG. 7. Solar exposure functions at the Gran Sasso latitude for the ``winter '' and ``summer '' periods defined in Eqs. B2 and B3 .

to describe in some detail our improved error estimates and theoretical calculations. Concerning the ``error format,'' the pull method used in this paper does not leave any freedom in the 2 treatment of the spectrum, which is uniquely defined by providing, for each bin rate R n , the uncorrelated error component u n , and the fully correlated error components c k due to independent n k-th sources of systematic errors.23 The task is thus reduced to a careful evaluation of such components. To reach this goal, we combine information from SK 6 and from our own evaluation of systematics. Table VIII shows the main characteristics of the SK binned energy spectrum. The energy bins 2 ­ 7 are further divided into seven nadir angle bins 6 , reported in Table IX. In each energy-nadir bin, the statistical error represents the only SK uncorrelated error component ( u n ) . All other error sources in SK are correlated among bins, at least in nadir. Indeed, apart from the obvious SSM systematics, the SK bin rates are further affected by 8 3 1 12 correlated systematics, as discussed in the following. The first eight systematic errors, listed in the fifth column of Table VIII, are uncorrelated among energy bins. In each energy bin, however, they are fully correlated in nadir 6 . Such errors represent those nadir-independent data reduction uncertainties which are specific of each energy bin, independently of the others.
Therefore, we abandon our previous 2 approach in terms of separated SK total rate and spectral shape information 58 which, although correct, cannot be exactly cast in a ``pull'' form.
23

Three further systematics, fully correlated among all energy-nadir bins, are induced by 8 B spectrum shape uncertainties 46 and by the SK energy scale and resolution uncertainties 6 . We note two facts. First, these sources of systematics act differently on the B and hep components in SK: for instance, the hep component is obviously unaffected by the 8 B shape uncertainty.24 Secondly, the relative error signs for each source of systematics are relevant: for instance, an increase in the energy resolution width flattens the--steeply falling--SK energy spectrum, the high-energy low-energy part being thus enhanced suppressed , as evident from the eleventh and thirteenth column in Table VIII. Since the information about the relative error signs and about the separate B and hep components is not given in 6 , we perform our own evaluation as follows. The 8 B spectrum shape error is evaluated by attaching to the default neutrino spectrum 47 the 1 shape perturbations evaluated in 46 , and calculating the corresponding fractional variations in the absolute rates.25 The energy scale uncertainty is evaluated by shifting the centroid of the Gaussian energy resolution function by 0.64% 6 , namely, by taking T T (1 0.0064) , where T is the true electron kinetic energy the quantity calibrated in SK . The resolution uncertainty is evaluated by perturbing the energy resolution width T see Eq. E3 by 2.5% 6 , T T (1 0.025) . In all cases, the same uncertainties are calculated for opposite perturbations, and small error asymmetries are averaged. The results are given in the last five columns of Table VIII, where the contributions from B and hep are separated. As far as the B flux component is concerned, there is reasonable agreement in size with the corresponding SK error evaluation in 6 , except for the 8 B spectrum shape. We are unable to explain such difference. Our error assignment is completed by an overall SK systematic offset ( 2.75% , symmetrized from 6 , which is attached to all energy-nadir theoretical rates with full correlation. This error mainly represents the overall uncertainty of the data reduction efficiency 54 , affecting the whole spectrum. Notice that, in Table VIII, the last eight columns refer to the no oscillation case. In the presence of oscillations, one theor usually updates only the theoretical rates R m , and assumes that the fractional errors in the last five columns are approximately unchanged. The latter assumption is a very good approximation for spectra with no or mild distortions, but is not strictly applicable in the whole oscillation parameter space. Therefore, for the sake of accuracy, in our oscillation analysis we recalculate the fractional errors in the last five col24 Analyses where the hep / B ratio is taken as a free parameter should separately rescale the different B and hep error components in the evaluation of the total error. In fact, one cannot calculate first the total ( B hep) systematic error assuming the SSM ratio for hep / B , and then use the same error when such ratio is significantly varied say, by factors 10) . We think that this remark should be taken into account, when placing upper bounds on hep from the analysis of the high-energy tail of the SK spectrum. 25 Conventionally, we denote as 1 perturbation of the 8 B shape the one which moves the energy spectrum to higher energies.

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TABLE VIII. Characteristics of the SK energy spectrum. The first two columns define the bins in terms of observed total electron energy E. The third column is the ratio of the SK observed event rates to the SSM 41 predictions, as taken from 6 , with a slight correction to account for our default hep flux we take hep 8.3 rather than 9.3 41 in units of 103 cm 2 s 1 ; see Eq. 15 . The boron flux is 6 2 s 1 41 , as in 6 . The fourth column gives the fractional uncorrelated error in each bin, whose only component is the B 5.05 10 cm statistical error taken from 6 . The fifth column represents those fractional systematic errors which are uncorrelated among energy bins, but correlated in nadir bins from 6 . The sixth, seventh and eighth columns represent our evaluation of the total, boron, and hep event rate in SK in the absence of oscillation , for 1 kton year kty exposure. The ninth, tenth, and eleventh column represent our evaluation of theor the three main systematic effects given as 1 fractional contributions to R m , B ) which are fully correlated in energy and in nadir, and are 8 generated by uncertainties in the B energy shape, SK energy scale, and SK resolution width. The last two uncertainties also affect the hep theor contribution to the total rate, as reported in the last two columns our evaluation, given as 1 fractional contributions to R m , hep) . Finally, we theor off 2.75% 6 to all binned rates R m , with full correlation in energy and nadir. In the presence of attach an overall systematic offset c oscillations, the last eight columns properly separated into nadir bins are recalculated for each ( m 2 , tan2 12) point. Bin m 1 2 3 4 5 6 7 8
a

E range MeV 5.0,5.5 5.5,6.5 6.5,8.0 8.0,9.5 9.5,11.5 11.5,13.5 13.5,16.0 16.5,20.0

R

expt m

/R

theor m

um /R

expt m

bin cm /R

theor m

B hep 0.4672 0.4581 0.4730 0.4601 0.4630 0.4626 0.5683 0.5637

100 8.65 3.08 1.78 2.02 2.23 3.64 6.88 26.3

100 3.24 1.43 1.37 1.37 1.37 1.37 1.37 1.37

theor Rm 1/kty

a

R

theor m,B

R

theor m , hep 8

100 c B shape 0.52 0.69 1.05 1.61 2.48 3.87 5.82 8.09

m,B

/R

theor m,B

100 c Scale 0.38 0.29 0.09 0.24 0.78 1.67 3.14 6.09

m , hep

/R

theor m , hep

1/kty 81.624 140.654 155.014 95.239 59.688 18.004 3.660 0.212

1/kty 0.171 0.320 0.419 0.338 0.323 0.193 0.108 0.033

Scale 0.07 0.14 0.58 1.32 2.50 4.50 7.47 12.32

Resol. 0.19 0.20 0.20 0.09 0.00 1.59 4.51 11.26

Resol. 0.09 0.11 0.15 0.18 0.19 0.07 0.42 2.41

81.795 140.974 155.433 95.577 60.011 18.197 3.768 0.245

The total event rate R theor is here normalized to the efficiency-corrected, no-oscillation value of 556 events/kty, as graphically derived from 59 , with a correction for the updated SSM B , hep values. This specific value for R theor is unimportant in practice, as far as the ratios in the 3rd column are used in the analysis.

umns of Table VIII, for any ( m 2 , tan2 12) grid point and for any nadir bin . This improvement leads to minimal dif2 ferences in the local pull minima corresponding to almost undistorted spectra , but is not totally negligible in deriving the borders of, say, the 99.73% C.L. allowed regions, where 2 it can lead to 1 differences. Concerning the calculations of the e survival probability, we compute Earth matter effects 60 through eight relevant shells of the preliminary reference earth model PREM 61 , namely in the language of 61 : 1 ocean; 2,3 crust layers; 4 LID low velocity zone; 5 transition zone; 6 lower mantle D-zone; 7 outer core; and 8 inner core. In
TABLE IX. Fractional contributions to the total event rate ( R / R tot , our evaluation , in the case of no oscillation, from each of the seven SK nadir intervals, identified as ``day'' bin, ``mantle'' bins M1­M5 , and ``core'' bin. The observed SK event rates in each energy-nadir bin, together with their statistical errors, are reported in 6 . Nadir bin Day M1 M2 M3 M4 M5 Core Day Night cos range R/R
tot

each of them, we approximate the radial density profile through a biquadratic parametrization, which allows a fast and accurate analytical calculation of the relevant transition probabilities in the Earth 60 . In the quasi vacuum oscillation regime, the oscillating term in the e survival probability P ee depends implicitly upon the daily time through the orbital distance L ( d ), where d 2 · day/365, with d 0 at winter solstice.26 As noticed in 62 , the different exposure of the ``day'' and ``night'' bins in terms of d induces a slight day-night vacuum difference in the time-averaged P ee . Concerning the SK energy-nadir spectrum, we take into account the different exposures E ( 1 , 2 ) for each nadir bin range 1 , 2 in Table IX as follows. In a given day of the year ( d ) , the range spanned by a detector at latitude equal to 36.48° for SK is min( d), max( d) , where min( d) S( d), and max( d) S( d), having defined the solar declination sin i cos d , with sini 0.3978 incliS ( d ) through sin S nation of Earth axis . The daily exposure function E for a generic 1 , 2 nadir bin is then E
d

1,0 0,0.16 0.16,0.33 0.33,0.50 0.50,0.67 0.67,0.84 0.84,1 1,1

0.5000 0.0685 0.0777 0.0984 0.1025 0.0839 0.0690 1.0000

,

1

,

2

h h

d

, min
d

2

,
1

max

, max

,

min

/,

C1 and

where

h

(

d

, ) is the hourly time normalized to 2

26 The purely geometrical 1/L 2 ( d ) variation is assumed to be already corrected for in the SK data.

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as reported in Fig. 2 of 6 we also obtain very good agreement for the normalization-free case, not shown . A final remark is in order. All the SK systematic error sources discussed in 6 and adopted in this work are fully correlated in nadir. This means that no allowance is given in 6 for uncorrelated nadir shape variations, apart from the obvious statistical fluctuations. However, it is known from previous SK publications 22,63 and presentations 59 that some systematics do not cancel in day-night differences, and may thus, in general, affect the nadir bins in uncorrelated ways. In addition, Earth matter density and chemical composition uncertainties may also contribute to small and different errors 64,65 in the various nadir bins 66 via e regeneration effects. In particular, one should not forget that the widely used PREM model is spherically symmetric by construction, and that local departures from the worldaveraged density are to be expected. For instance, the first half of the SK mantle bin M1 see Table IX is sensitive to the PREM ``ocean crust'' density, whose local characteristics at the SK site may well be different from the world average. The inclusion of such additional uncertainties might give a little more freedom to fit possible distortions in the nadir distributions. Therefore, if the currently weak hints for an excess of night events in SK 54 and in SNO 9 will be corroborated by future data, an improved discussion of the nadir spectrum uncertainties will be useful to precisely assess their statistical significance.
APPENDIX D: TREATMENT OF THE SNO DAY-NIGHT ENERGY SPECTRUM

FIG. 8. Daytime solar exposures of each SK nadir bin and of the SNO day and night bins. These exposure functions are used for accurate time averages of the oscillating terms in the quasi vacuum regime.

starting at midnight corresponding to the angle day d , 27
h d

during the

,

arccos

cos cos cos

tan tan
S

S

.

C2

Figure 8 shows the exposure functions E ( d ) for the seven SK nadir bins listed in Table IX, as well as for the SNO day and night bins the functions sum up to unity in both cases . It can be seen, as intuitively expected, that the SK core bin is sensitive only to extreme values of L ( d ) close to the orbital perihelion and aphelion , while the two outermost mantle bins M1 and M2 are almost equally sensitive to all values of L ( d ) , being crossed by solar neutrinos during the whole year. We take into account the different bin exposures in Fig. 8 when time-averaging the oscillating terms in the quasi vacuum regime, for both SK and SNO. The results of our oscillation analysis of the SK spectrum are given in the SK panel of the previously discussed Fig. 2. The C.L. contours compare well with the official SK ones,

Any angle min , plains the appearance of of Eq. C1 .

27

max

is spanned twice each day. This exrather than of 2 ) in the denominator

The solar neutrino events observed in the SNO spectrum 8 cannot be currently identified as being of ES, CC, or NC type on an event-by-event basis. A statistical separation is possible, however, by exploiting their different distribution in terms of suitable variables, the most important being the observed effective electron kinetic energy T. Since the ES and CC distribution shapes in T depend upon the oscillation parameters, also the inferred CC and NC rates depend on such parameters, as stressed in 40 . In the oscillation analysis, however, it is not necessary to perform a separate fit to the ES, CC, and NC components as done in 18 . Given the oscillated predictions for these components, one can simply add them up together with the known background rates in each energy bin, calculate the corresponding systematics, and fit the observed SNO daynight energy spectrum. This method, which has been dubbed ``forward fitting'' by the SNO Collaboration 40 , allows us to take into account the full spectral information central values and errors of each bin . In the following, we describe our 2 implementation of such method in the pull evaluation. Table X reports some relevant characteristics of the SNO spectrum, including our evaluation of the SNO neutrino signal components and their errors small error asymmetries being averaged out . The effects of the 8 B spectrum shape error not included in the official SNO analysis 8,9 are estimated as in SK see Appendix C . Notice that a 1 shift of this systematic uncertainty increases the number of signal events, especially at high energies. The fractional in-

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TABLE X. Characteristics of the SNO spectrum, divided in 17 day and 17 night bins first two columns . The experimental rates 8 are given in the third and fourth columns, and include signal CC ES NC and background events. Our evaluation for the corresponding CC, ES, and NC components and of their 8 B shape, energy scale and resolution errors in the absence of oscillation, day night are given in the remaining 12 columns. The NC signal is negligible in the last 8 bins. All the quoted numbers refer to the total B hep contributions. In the analysis, however, the B and hep contributions and their errors are separately evaluated, analogously to the SK spectrum. In the presence of oscillations, the last twelve columns are split into day and night contributions and updated in each point of the parameter space. Backgrounds are oscillation-independent and are treated according to 40 . Bin m 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 T range MeV 5,5.5 5.5,6 6,6.5 6.5,7 7,7.5 7.5,8 8,8.5 8.5,9 9,9.5 9.5,10 10,10.5 10.5,11 11,11.5 11.5,12 12,12.5 12.5,13 13,20 R expt day 1/kty 351.1 330.9 299.6 222.4 191.2 148.9 128.7 139.7 90.1 82.7 66.2 49.6 31.3 18.4 9.2 11.0 9.2
expt R night 1/kty theor R CC 1/kty

fractional errors 100 Shape Scale Resol. 0.42 0.63 0.86 1.12 1.41 1.73 2.08 2.47 2.89 3.36 3.86 4.40 4.97 5.56 6.17 6.79 7.68 2.30 1.89 1.40 0.85 0.22 0.51 1.33 2.25 3.29 4.45 5.74 7.16 8.70 10.37 12.15 14.01 17.83 0.10 0.21 0.53 0.83 1.09 1.30 1.42 1.41 1.24 0.85 0.18 0.82 2.21 4.04 6.35 9.13 16.34

theor R ES 1/kty

fractional errors 100 Shape Scale Resol. 0.64 0.77 0.91 1.07 1.24 1.44 1.66 1.90 2.16 2.45 2.77 3.12 3.51 3.92 4.36 4.83 5.97 0.26 0.57 0.93 1.32 1.77 2.28 2.85 3.48 4.19 4.99 5.87 6.86 7.94 9.14 10.45 11.87 15.71 0.26 0.35 0.46 0.54 0.58 0.58 0.54 0.43 0.23 0.07 0.51 1.13 1.97 3.07 4.47 6.23 12.48

theor R NC 1/kty

fractional errors 100 Shape Scale Resol. 1.55 1.55 1.55 1.55 1.55 1.55 1.55 1.55 1.55 -- -- -- -- -- -- -- -- 0.88 3.73 7.05 10.85 15.16 19.96 25.26 31.19 37.69 -- -- -- -- -- -- -- -- 4.33 2.84 0.43 5.43 12.18 20.68 31.02 43.38 58.11 -- -- -- -- -- -- -- --

399.7 313.4 272.2 249.6 235.0 176.6 122.1 134.1 95.6 86.3 62.4 59.8 41.2 21.2 18.6 15.9 9.3

822.2 875.6 903.0 902.0 872.8 817.7 740.9 648.5 547.4 444.9 347.5 260.4 186.8 128.2 84.0 52.6 71.1

135.8 121.7 107.8 94.1 81.0 68.7 57.2 46.9 37.7 29.6 22.8 17.1 12.5 8.9 6.1 4.1 6.5

293.8 248.5 172.1 97.7 45.4 17.3 5.4 1.4 0.3 0 0 0 0 0 0 0 0

crease is obviously constant for the NC events, which have no memory of the original energy. From Table X it appears that, in the bulk of the SNO spectrum, the 8 B shape uncertainties are not negligible as compared with the purely instrumental SNO energy calibration and resolution uncertainties. The latter two errors are evaluated by shifting the centroid of the energy resolution function and varying its width in the way described in Ref. 40 , and calculating the corresponding variations in each bin of the CC, NC, and ES spectrum. Actually we separate the B and hep components not shown in Table X , as for the SK spectrum. In the presence of oscillations, we not only update the day and night rates in each bin, but also their fractional systematic errors, for each point in the mass-mixing plane. As for SK, the effect of this improved estimate of fractional systematic errors is rather small in the local 2 minima, but increases towards the borders of the 99.73% C.L. regions, where the spectral distortions can be more sizeable. In the fit to SNO 2 data only, such improvement can lead to pull variations as 2 large as 4 at the 3 borders, and is thus not totally negligible in deriving precise C.L. contours for the less favored solutions. Finally, notice that the three systematic error sources in Table X can induce, in the various bins, event rate variations with different relative signs, which are taken into account in the analysis. Besides the previous three systematic error sources, we include a vertex reconstruction uncertainty, whose 1 effect is to increase the CC and ES rates by 3 % and the NC rates by 1.45% 40 . The CC and NC binned rates are also

affected by a systematic cross section uncertainty, whose 1 effect is to increase them by 1.8% and 1.3% , respectively 8 .28 The CC and NC cross section errors embed differences between independent theoretical calculations 67,68 and uncertainties related to additional radiative corrections 69,70 , as discussed in 8 . Finally, the various SNO background components 40 are affected by the socalled neutron capture, neutron background, and low-energy LE background correlated systematics, whose oscillationindependent effects included in our analysis are reported in 40 and not repeated here. When all such inputs are included, our fit to SNO data only provides the allowed regions given in the lower right panel of Fig. 2. Such regions appear to be somewhat different from those found in Ref. 9 , as discussed in Sec. IV B.
APPENDIX E: QUASI-MODEL-INDEPENDENT ANALYSIS OF SK AND SNO

In this last appendix we present a new version of the model-independent comparison of SK ES and SNO CC and NC total rates, first proposed in 71 and then applied in 58,72 to earlier SNO CC data. For the current SNO threshold T SNO 5 MeV 8 , the SK threshold equalizing the SNO CC and SK ES response func28 The correlation between the NC and CC cross section errors can be safely taken 1 in the SNO analysis.

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tions 71 turns out to be T SK 6.74 MeV, corresponding to a total electron energy E SK 7.25 MeV, SNO CC response T
SNO

5 MeV
SK

SK ES response E

7.25 MeV .

E1

In calculating the response functions, we have taken into account the latest evaluation of the energy resolution width T in SNO 8 ,
T

SNO

0.0684 0.331 T

0.0425 T ,

E2

where T is the true electron kinetic energy, and both T and T are given in MeV. We have cast, in the same form, the latest evaluation of T for SK graphically reported in 23 ,
T

SK

0.25 0.20 T

0.06 T .

E3

From the spectral information discussed in Appendix C we estimate derivation omitted
SK ES

2.35 0.029 0.080 0.045 E 7.25 MeV , E4
FIG. 9. Quasi-model-independent comparison of SK ES and SNO CC and NC total event rates in the plane charted by the boron flux B and by the average e survival probability P ee , for equalized SK and SNO response functions. The evaluation of the SNO rates includes possible linear distortions in the energy spectrum. The 2 bands for each datum ( 2 4 ) appear to be in very good agreement with each other for P ee 1/3 and for B close to its SSM prediction 41 . The combination of the 2 bands is also shown slanted elliptical region .
SNO CC SNO NC

in units of 106 cm 2 s 1 . In the above equation, the first error is statistical, while the second represents the properly propagated sum of systematic errors, with the exception of the 8 B shape uncertainty, given separately as a third error component. While the above SK ES flux estimate is safe for modelindependent analyses, the current SNO CC flux 8 is not, since its value does depend on model-dependent assumptions, governing the CC spectrum distortions and thus the CC-NC separation. 40 However, we can resort to a quasimodel-independent analysis, by assuming that the only observable effect of new neutrino physics can be embedded in a shift of the first spectral moment 73 , namely, in a linear distortion tilt of the CC energy spectrum.29 This assumption is reasonably general, since the current SNO statistics is not high enough to really constrain higher moments, and since we know from SK that only the scenarios with mild distortions can survive. In this case, the normalizationpreserving form of a generic linear distortion for the CC differential energy spectrum reads dN
CC

1.78 0.10 stat , 4.90 0.66 stat ,

E6 E7

T

dT



dN

CC

T

dT

1

T T
2

T T
2

T

,

E5

where T and T 2 are the undistorted spectrum, and ment. Assuming T free, our gives the following CC and

first and second moment of the T is the shift in the first mofit to the SNO energy spectrum NC fluxes,

The numerical results for a linear distortion of the CC component of the SNO spectrum are very similar to those obtained for a linear distortion of the CC ES components not shown , since the ES contribution is relatively small in SNO.

29

with correlation 0.8. By setting T 0 , we would SNO SNO 1.73 0.06( stat.) and 5.29 instead obtain CC NC 0.62, in good agreement 0.43( stat.) with correlation with the results of 8,9 for undistorted spectrum. The larger statistical errors in Eqs. E6 and E7 are the price to pay to allow possible linear distortions in the fit. Systematic errors are attached as follows. From the positively and negatively correlated SNO error components reported in Table II of 8 , we estimate the CC,NC experimental systematics as ( 5.2% ,8.8%), with correlation 0.5. The corresponding theoretical cross section uncertainties 1.8%,1.3% from Table II of 8 are assumed to have 1 . Finally, if we repeat to the fit leading to Eqs. E6 and E7 with a 1 perturbed 8 B neutrino energy spectrum, we obtain the variations ( 1.1% , 3.3% ) , which are fully correlated among themselves and with the third SK error component in Eq. E4 . In conclusion, we get in units of 106 cm 2 s 1 ),
SK ES

2.35 0.10,

E8

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1.78 0.14, 4.90 0.80,

E9 E10

SNO CC SNO NC

B

P ,

ee

,

E13 E14

B

with correlation matrix 1 0.07 1 0.09 0.65 . 1 E11

The above SK and SNO equalized fluxes can be useful to constrain generic models of new physics alternative to--or coexisting with--usual mass-mixing oscillations , as far as their main distortion effect on the SNO CC spectrum is approximately linear in T. Within this quasi-model-independent assumption, and in the hypothesis of active oscillations, the fluxes in Eqs. E8 ­ E10 are still linked by the exact relations 71 .
SK ES B

P

ee

r1

P

ee

,

E12

where P ee ( E ) is not necessarily constant in energy, and r 0.154 is the ratio of , and e CC cross sections averaged over the current SNO equalized response function . In the above equations, B is the true 8 B flux from the Sun generally different from the SSM value , and P ee is the energy average of P ee ( E ) over the response function equal in SK and SNO . Equations E8 ­ E11 and E12 ­ E14 overconstrain the P ee . Figure 9 shows, in two parameters B and ( B , P ee ) coordinates, both separate and combined bounds at the 2 level for each datum ( 2 4 ) . There is very good agreement between any two out of the three data in Eqs. E8 ­ E10 . Their combination strengthens previous model-independent indications 58,72 for a consistency of 1/3 in the B with the SSM prediction 41 and for P ee SK-SNO energy range.

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