# Constraints on Vacuum Oscillations

from Recent Solar Neutrino Data

###### Abstract

A detailed study of the solar neutrino vacuum oscillation is made taking into account three neutrino flavours and seasonal effect. A set of x regions is calculated for a range of the parameter from 0 to 1, with and without the inclusion of the recoil-electron spectrum in the rates. The averaged survival probabilities for as a function of the energy are obtained, what reveal that solutions with values of and the maximum exclusion of neutrinos and minimum exclusion of neutrinos give a better explanation for the suppression rates of all detectors.

Centro Brasileiro de Pesquisas Físicas, Rio de Janeiro, RJ, 22290-180, Brazil

Universidade Federal Fluminense, Niterói, RJ, 24210-340, Brazil

## 1 Introduction

For three decades the deficit on the observed flux of solar neutrinos compared to that one expected from Standard Solar Models (SSM) has puzzled the scientists and originated several explanations for this problem. These solutions are divided into astrophysical and beyond the standard model [1]. Nowadays, the astrophysical solutions have been highly disfavoured by the helioseismological data which are well explained by standard solar models, however, solutions with new physics, in special the neutrino oscillations, that can be in vacuum [2] or in matter (by MSW effect) [3], gives a good fit to the solar neutrino data.

In the present work we will focus attention only on vacuum oscillations, which occur in the neutrino way from the Sun to Earth. The evidence of the existence of vacuum oscillations come from the observation of the seasonal effect. Because the Earth orbit is not circular, the variations in the Earth-Sun distance produce a difference in the oscillation probability what can be very important for neutrino line sources, in special, the (0.861 MeV) and is to be confirmed by the Borexino detector [4]. In our study we will deal with two and three neutrino flavours to try to explain the solar neutrino problem. We will vary the parameter that is concerned with the oscillations in order to discover which scenario (two or three generations) gives a better description for the experimental data. This analysis will be made by the calculation of the x regions considering first only the rates of the detectors without the spectral distortion, caused by the recoil-electron spectrum in Kamiokande and Super-Kamiokande, and after including these spectrum within the calculation. An investigation in the survival probability will be made, what will reveal the behavior of the neutrino as a function of the energy.

In section 2 we give the relations needed to calculate the mass regions and mixing angles allowed by the experiments. In section 3 we present the survival probability as well the and conversion probabilities. In section 4 we analise the obtained data to describe the probabilities as a function of the energy for a set of values of . In section 5 we present the conclusions of this work.

## 2 Suppression Rates

The first procedure is to calculate the suppression rates of the experiments using neutrino oscillations and compare with the observed ones to search for the values of the parameters and that fit the five experiments (Homestake [5], GALLEX [6], SAGE [7], Kamiokande [8] and Super-Kamiokande [9]) together. The suppression rate, to the experimental case, is obtained by dividing the capture rate of the detector to that one expected from the SSM and theoretically, dividing the capture rate calculated using neutrino oscillations to the SSM one. We will make this calculation using the data from the BP98 Standard Solar Model [10] with 99% C.L.

The Solar Neutrino Problem is evidenced when we compare the theoretical capture rates with those obtained experimentally. When this comparison is made we discover that these rates never match, giving values varying from 33% for Homestake to 60% for GALLEX. See table 1.

Experiment | Result | BP98 SSM | Result/SSM |
---|---|---|---|

Homestake [5] | |||

GALLEX [6] | |||

SAGE [7] | |||

Kamiokande [8] | |||

Super-Kamiokande [9] |

The solar neutrino suppression rate using neutrino oscillation is given by

(1) |

where

(2) |

and is the same expression with

The equation (2) is available for Homestake, GALLEX and SAGE experiments, where runs for each neutrino source, is the total neutrino flux from the source , is its normalized neutrino energy spectrum, is the cross section for the experiment considered and is the electron neutrino survival probability. The neutrino sources that we consider for these experiments are , , , , and , except for Homestake that is not sensitive to neutrinos.

For Super-Kamiokande we consider the neutrino source only and we have to take into account the fact that this experiment is also sensitive to the and scattering, so becomes

(3) |

since and the equation (3) is simplified to

(4) |

For the Kamiokande experiment only the source is considered again but the sensitivity to is neglected.

As Super-Kamiokande and Kamiokande are neutrino-electron scattering detectors, different from the other three which use neutrino absorption, we can also take into account the energy resolution in order to observe the influence of the spectral distortion caused by the recoil-electrons in the suppression rates, giving the following expressions for and

(5) |

(6) |

(7) |

The energy resolution is described by

(8) |

where and are the measured and true electron kinetic energy, respectively, and the limit in the third integral of the eq.(7) is the maximum kinetic energy that an electron can achieve given the neutrino energy , 11, 1]. . For more details see [

The limits of energy used to calculate these rates are not the same the neutrinos have in the Sun, so we have to calculate the normalized detector sensitivity to know these limits, as follows

(9) |

We can notice in the figure 1 the existence of regions where the spectral sensitivity is zero.

## 3 Electron Neutrino Survival Probability

For three neutrino flavours the survival probability is written as [12]

(10) |

where the terms , , , and are short forms for , , , and , respectively.

In this work we will consider the following mass hierarchy [13]

(11) |

what leads us to and , since , thus we can simplify the equation (10) to the form

(12) |

Now we have two choices

i) if then the term tends to be very small and we fall in the case of two generations with as the mixing angle and as the mass parameter;

ii) if , for the range of energy considered the term can be averaged to , giving the final equation for the survival probability, which applies for two and three neutrino oscillations

(13) |

The introduction of the seasonal effect causes a modification in the probability since the distance varies with the time on the following way [14]

(14) |

where is the mean Earth-Sun distance ( . ), is the elipticity of the orbit and = 365 days. Once we are taking a temporal average over the probability the eq.(13) turns.

(15) |

where we integrate in a time interval , which in this work we consider as one year.

For and we have

(16) |

and

(17) |

In figure 2 it is shown the expected maximal and minimal variation of these probabilities as a function of .

## 4 Analysis of the Results

The figure 3 illustrates the plots of x considering the seasonal effect. We can notice an increase in the regions that explain the five experiments together as runs from 0 to 1. The first column shows the regions without taking into account the recoil-electron spectrum of Kamiokande and Super-Kamiokande and the second column gives the regions with the inclusion of that spectrum, what causes a decrease in the mass regions in the order of 33% for to 23% for . The analysis was made following the references [14, 15] and is shown in table 2 and we can observe that the inclusion of the recoil-electron spectrum gives a better fit.

With the and points from figure 3 we can obtain the averaged electron neutrino survival probabilities as a function of the energy, what are ploted on figure 4. In a first sight we observe a considerable decrease of the probability on the energy region dominated by the neutrino and a slight decrease in the line source, because it lies in a region where there are CNO neutrinos also. As the parameter grows, the probability range decrease with the energy. For , , . and for

To make a better analysis it is useful to adopt the procedure given by Barger et al., [13], where the suppression rates of the detectors are given by means of the averaged probabilities. So we have for the detectors the following expressions,

Column 1 | |||
---|---|---|---|

0 | . | ||

0.25 | . | ||

0.50 | . | ||

0.75 | . | ||

1 | . | ||

Column 2 | |||

0 | . | ||

0.25 | . | ||

0.50 | . | ||

0.75 | . | ||

1 | . |

(18) |

where for simplification on the calculus we take into account the fact that in Kamiokande and Super-Kamiokande. In table 3 there are the values (99% C.L.) of the suppression rates using the probabilities from figure 4 considering the following range of energies: for and for . For we used for the radiochemical detectors and and for Super-Kamiokande and Kamiokande, respectively. For neutrinos the range is for Homestake and for the gallium detectors.

Homestake | Gallium | Kamiokande | Super-Kam | |

rates only | ||||

0 | ||||

0.25 | ||||

0.50 | ||||

0.75 | ||||

1 | ||||

rates + recoil-electron spectrum | ||||

0 | ||||

0.25 | ||||

0.50 | ||||

0.75 | ||||

1 |

Homestake | Gallium | |

rates only | ||

0 | ||

0.25 | ||

0.50 | ||

0.75 | ||

1 | ||

rates + recoil-electron spectrum | ||

0 | ||

0.25 | ||

0.50 | ||

0.75 | ||

1 |

A comparison between table 3 and the experimental data from table 1 shows us that the suppression rates for Homestake are over the experimental ones for all values of , but as this parameter tends to 1 the fit of the values turns better and the rates of Homestake can be well explained within for , considering the rates with or without recoil-electron spectrum^{1}^{1}1Note that Homestake, GALLEX and SAGE are not affected individualy by the recoil spectrum as Kamiokande and Super-Kamiokande are, but we are dealing with the data from the plots that fits the five experiments together.. This is also verified in Super-Kamiokande for . For the gallium experiments all rates are checked for any with , but small values of this parameter give a better agreement, so we can check GALLEX and SAGE within an experimental error for . For Kamiokande the data are explained for all values of within , but for it is checked with .

A better fit for Homestake is obtained considering the maximum possible suppression of neutrino^{2}^{2}2The total suppression of the neutrino only occurs for (see figure 2). (this suppression is limited by eq.15); in special for this agreement is found whitin error. Unfortunatelly this fact leads to a decrease in the GALLEX and SAGE rates. To explain these rates with the maximum suppression of neutrinos, we need to assume a
. In this case the fits are obtained, within error, for any values of . See table 4 for more details.

So a better agreement for the five experiments is found, with 99% C.L., for . The allowed regions that fit the observed rates of the five experiments with 99% C.L. are shown in the figure 5. In this figure are plotted the calculated rates using the maximum suppression of neutrinos and . From the figure we see that the allowed regions of x grows when rises. For , the range of the parameters are,

Column 1 | |||
---|---|---|---|

0 | . | ||

0.25 | . | ||

0.50 | . | ||

0.75 | . | ||

1 | . | ||

Column 2 | |||

0 | . | ||

0.25 | . | ||

0.50 | . | ||

0.75 | . | ||

1 | . |

## 5 Conclusions

The recent Super-Kamiokande data on solar neutrinos brought new constraints on the previously allowed solutions and have restricted them. In this work we show that the vacuum oscillations still give a good explanation to the solar neutrino problem and that three neutrino solutions gives better results. In three neutrino scenario the allowed regions for and parameters that fits all experimental data increase as grows.

The calculated suppression rates including the recoil-electron spectrum leads to a reduction of approximately 30% in the mass regions, but it does not affect the fitting of the data because the averaged probabilities are almost the same as those obtained considering the rates only. An analysis with and without the recoil-electron spectrum shows that the inclusion of this spectrum checks better the experimental data.

A better fit for the experiments are obtained, with 99% C.L., if we consider the maximum suppression of neutrinos and a suppression of neutrinos from 0 to 40% as well a suppression of about 60% of neutrinos. In this case the best agreement with the experimental data is obtained for . The inclusion of the recoil-electron spectrum gives a better fit than the calculated with rates only, for any value of .

## Acknowledgments

We would like to thank the CNPq (Brazilian Council of Scientific and Technologic Developments) for the financial support and the CBPF (Centro Brasileiro de Pesquisas Físicas) for the facilities.

## References

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Figure 1 - Normalized spectral sensitivity as a function of the neutrino energy, for the five experiments.

Figure 2 - Limits of variation for , and as a function of .

Figure 3 - Mass regions versus mixing angles for the BP98 SSM with 99% C.L. in vacuum, with seasonal effect, for different values of . The first column is for the rates only and the second column is for the rates with recoil-electron spectrum. The black dot in each plot represents the best fit.

Figure 4 - Electron neutrino survival probability as a function of the energy for the BP98 SSM with 99% C.L. in vacuum, for different values of . The full lines represent the best fit for calculated with the rates only (column 1 on figure 3) and the dashed lines the same fits with rates and recoil-electron spectrum (column 2 on figure 3).

Figure 5 - Mass regions versus mixing angles for the BP98 SSM with 99% C.L. in vacuum, with seasonal effect, for different values of considering maximal suppression of neutrinos. The first column is for the rates only and the second column is for the rates with recoil-electron spectrum. The black dot in each plot represents the best fit.