Abstract
The existing measurements of the solar neutrino flux are compared with the predictions of all models capable of reproducing the other solar observables. These predictions are supplemented by the hypothesis of neutrino oscillations with mass differences large enough to render energyindependent the depletion of the solar $\nu_{e}$ flux. It is concluded that the data are consistent with this hypothesis and that an energydependence of the solar neutrino deficit must be regarded as an attractive possibility but not as a compelling reality.
UB/FI 981
April 1998
SOLAR MODELS AND NEUTRINO DEFICIT ^{*}^{*}Contributed paper to the XVIII International Conference on Neutrino Physics and Astrophysics, Takayama, Japan, 4  9 June 1998
G. Conforto, C. Grimani, F. Martelli and F. Vetrano
UniversitΓ degli Studi, I61029 Urbino, Italy
Istituto Nazionale di Fisica Nucleare, I50125 Firenze, Italy
1 Introduction
In a previous paper [1]we have addressed the question of a possible energydependence of the solarneutrino deficit coming to the conclusion that, at least for the time being, its existence must be regarded only as an attractive possibility and by no means as an established reality.
Our analysis was based on the comparison between the existing experimental data [2, 3, 4, 5]and the theoretical predictions, assumed to be well represented by those of the βreference modelβ of ref. [6].
The latter assumption is actually rather questionable. There exist in fact several solar models capable of reproducing all experimentally known facts but giving rise to somewhat different neutrino flux predictions [7]. As different models of the same sun cannot be all simultaneously right, it follows that theoretical predictions can only be said to be known within indeterminations which are actually larger than those associated with the results of any single model.
In our previous analysis [1]we mentioned this effect but did not take it into account. In this paper, we try to quantify this additional theoretical uncertainty by looking at the spread of the various predictions. The procedure adopted is described in section 2. With modified theoretical predictions and their errors, the statistical analysis of ref. [1]is then repeated in section 3. Section 4 summarises our conclusions.
2 Solar models
An extensive review of solar models is presented in ref. [7]. We have restricted ourselves to βstandardβ models, i.e. to those capable of reproducing all solar observational results, including the helioseismology data.
The solar neutrino event rates for the Gallium (Ga), Clorine (Cl) and Kamiokande (Ka) experiments predicted by the four considered models [6, 8, 9, 10]are reported in table 1. Within each model, the quoted errors reflect the uncertainties on the input parameters. These errors are highly correlated. However, it is only for the model of ref. [6]that the error correlation matrix is readily available [11, 1]. Consequently, we have taken the errors of this model to be typical of and to apply to all this type of calculations.
Experiment (Units)  Ga(SNU)  Cl(SNU)  Ka(10${}^{6}$ cm${}^{2}$ s${}^{1}$) 

Ref.[6]  136.8 $\pm^{8}_{7}$  9.5 $\pm^{1.2}_{1.4}$  6.62 $\pm^{0.93}_{1.12}$ 
Ref.[8]  136 $\pm^{7}_{6}$  8.9 $\pm 1.1$  6.4 $\pm 0.9$ 
Ref.[9]  132.8 $\pm 6.9$  8.5 $\pm 1.1$  6.3 $\pm 0.9$ 
Ref.[10]  128 $\pm 6$  7.4 $\pm 0.8$  5.16 $\pm 0.75$ 
Although rather similar, the central values of the rates are not identical. This is due to different assumptions entering in the various models. To take this effect into account, we have treated these results as independent determinations of the same quantity with a common variance. Accordingly, we have calculated the average values of the predicted rates and the components of their errors due to their spread. This procedure yields the results
$(133.4\pm 2.1)\,{\textrm{SNU}},\>(8.57\pm 0.45)\,{\textrm{SNU and}}\>(6.12\pm 0% .33)\times 10^{6}{\textrm{cm}}^{2}{\textrm{s}}^{1}$ 
for the Ga, Cl and Ka experiments respectively. The final covariance matrix on the average rate values is then obtained by adding in quadrature the errors above to the diagonal elements of the covariance matrix of the model of ref. [6]. These results are reported in table 2.
$P_{1}$ = 133.4 SNU  $P_{2}$ = 8.57 SNU  $P_{3}$ = $6.12\times 10^{6}$cm${}^{2}$s${}^{1}$  

6.30  4.81  
68.5  5.51  5.52  
53.5  7.35  5.79  
6.43  5.06  
6.30  1.09  
5.51  1.64  1.27  
V =  7.35  2.16  1.31 
6.43  1.53  
4.81  1.09  
5.52  1.27  0.97  
5.79  1.31  1.36  
5.06  1.53 
Fig. 1 illustrates the results of the models. The predictions from the four solar models and their average values are shown in the GaClKa space. The ellipses centered on the stars are the $1\sigma$ contour for the model of ref. [6], calculated by averaging the asymmetric errors. The black dot represents the average rate values and the smaller and larger ellipses centered on it are respectively the estimated uncertanties on the average rate values due to their differences and the $1\sigma$ contour for the average rate values (see table 2), calculated for averaged asymmetric errors.
3 Statistical analysis
The experimental input data used in the analysis are shown in table 3 [12]. For each result, statistical and systematic errors have been combined in quadrature. For the Ga and Ka experiments the weighted averages of the two available results have been used.
Experiment (Units)  Result 

Gallex (SNU)  69.7 $\pm 6.7\pm^{3.9}_{4.5}$ 
Sage (SNU)  73 $\pm^{10}_{11}$ 
Chlorine (SNU)  $2.54\pm 0.14\pm 0.14$ 
Kamiokande ($10^{6}{\textrm{cm}}^{2}{\textrm{s}}^{1}$)  $2.80\pm 0.19\pm 0.33$ 
Superkamiokande($10^{6}{\textrm{cm}}^{2}{\textrm{s}}^{1}$)  $2.44\pm 0.06\pm^{0.25}_{0.09}$ 
Following ref. [1], the hypothesis of an energyindependent depletion of the solar $\nu_{e}$ flux due to oscillations is tested by studying the function
$\chi^{2}(F)=\Sigma_{i}\Sigma_{j}\;(e_{i}p_{i})\;(e_{j}p_{j})\;(S^{1})_{ij}$ 
where:

β’
F is the common factor by which all calculated $\nu_{e}$ fluxes are reduced;

β’
the indices $i$ and $j$ run over the three (Ga, Cl and Ka) experiments;

β’
$e_{i}$ are the experimental results;

β’
$p_{i}$ are the theoretical predictions. They are obtained from the $P_{i}$ of table 2 through the relations
$\begin{array}[]{rcl}p_{Ga,Cl}&=&FP_{Ga,Cl}\\ p_{Ka}&=&F(1f)P_{Ka}+fP_{Ka}\end{array}$ where $f=0.155$ is the fraction of the Kamiokande detection efficiency due to flavourblind Neutral Currents;

β’
S is the covariance matrix obtained from that of table 2 by multiplying the elements $V_{ij}$ with $i,j\neq 3$ by $F^{2}$ , those with either $i$ or $j=3$ by $[F^{2}(1f)+Ff]$ and $V_{33}$ by $[F(1f)+f]^{2}$ and by adding in quadrature the experimental errors to the diagonal elements.
We have at first replaced all asymmetric errors by their average values. With this procedure, the $\chi^{2}$ analysis yields
$F=0.504\pm 0.064$ 
with $\chi^{2}_{{min}}=8.13$ corresponding to a Confidence Level (C.L.) of 1.7 $\%$ .
For comparison, if the theoretical predictions and their covariance matrix are taken from the model of ref. [6] alone, one obtains
$F=0.437\pm 0.056$ 
with $\chi^{2}_{{min}}=14.46$ corresponding to a C.L. of 0.072 %.
To test the sensitivity of these results to the procedure adopted in treating the errors, we have introduced a modification. In all operations involving two quantities (weighted averages of the two Ga and Ka results and evaluations of the $\chi^{2}$ terms) we have used the positive (negative) error on the first and the negative (positive) error on the second if the first quantity was smaller (larger) than the second. In this case the results are
$F=0.508\pm 0.066$ 
with $\chi^{2}_{{min}}=7.61$ corresponding to a C.L. of 2.2 %.
The results obtained by using the second procedure are shown in fig. 2. In each plane the two ellipses are the projections of the volumes in which the points representing respectively the experimental and calculated rates lie with 68.27 % probability.
4 Conclusions
Although not unacceptably low, the confidence level for an energyindependent depletion of the solar $\nu_{e}$ flux due to oscillations obtained using the results of the model of ref. [6] is admittedly rather marginal.
In the analysis presented in this paper we have used the average values of the predictions of four models introducing also additional fewpercent uncertainties on them due to their spread. The confidence levels obtained are remarkably good.
In conclusion, for the time being, oscillation solutions of the solar neutrino problem based on an energy modulation of the solar neutrino deficit must be considered as perhaps suggested but certainly not compelled by the data.
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