# The Solar Neutrino Problem

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2 served neutrino oscillations, playing an important part in the solution of the solar neutrino problem, require a nonzero mass. 2 The solar neutrino problem 2.1 The Standard Solar Model The Standard Solar Model (SSM) desribes the structure and the thermodynamical properties of the Sun. According to the SSM the Sun generally consists of hydrogen plasma which is designed by the equilibrium of the gravitational force and the energy density. At the core temperatures of about K [21] and pressures of 150 g/cm 3[8] enable nuclear fusion where mainly a proton is transformed into a neutron. To verify the SSM different radiation or fluxes might be observed: photons in the sense of light, plasma namely the solar wind, and neutrinos. In contrast to all other particles including photons, neutrinos are coming right from the core and therefore can give particular insights into the Sun. Table 2.1: Neutrino production from the protonproton chain reaction (upper) and the CNO cycle (bottom) based on the SSM. The total solar flux at the Earth is cm 2 s 1. The majority of the solar neutrinos come from the pp chain (more than 91%); while the 7 Be, pep, and 8 B chains correspond to about 7%, 0.2%, and 0.008% of the total flux, respectively. The hep contribution is minuscule and mostly neglected. [9] This is table 7 of [6]. Reaction Label Flux cm 1 s 1 p + p 2 H + e + + ν e pp p + e + p 2 H + ν e pep He + p 4 He + e + + ν e hep Be + e 7 Li + ν e 7 Be B 8 Be + e + + ν e 8 B N 13 C + e + + ν e 13 N O 15 N + e + + ν e 15 O F 17 O + e + + ν e 17 F Neutrinos from the Sun Whenever a proton is converted in a neutron a quark has to change its flavor. The only interaction being responsible for this is the weak interaction and therefore the nuclear beta decay. Thus, each time a neutron is formed, an electron neutrino is produced, too. [9] p + X n + e + + ν e + X (1) These neutrinos have a continuous energy spectrum with a maximal energy depending on the reaction, see Tab. 2.1 as well as Fig However, the inverse beta decay of 7 Be produces a discret energy spectrum with lines at 0.9 MeV and 0.4 MeV. The flux of the 8 B neutrinos is of particular importance since it is Φ( 8 B) T 25, so highly sensitive. [5] This is only one example how neutrinos might give applicable information about a property of a system. 2.3 Homestake The first investigation of solar neutrions were made by R. Davis in the mid-1960 s. [4] The so-called Homestake experiment was based on the neutrino reaction on chlorine making an isotope of argon Figure 2.1: The solar neutrino spectra predicted by the SSM. Note, that the continuum neutrino fluxes at one astronomical unit are given in units of cm 2 s 1 MeV 1, while the line ones are given in cm 2 s 1. This is figure 2 of [7]. ν e + 37 Cl e + 37 Ar (2) 2

4 solar neutrinos, the number of electron neutrinos at the night side is larger than on the day side which is why the Sun occurs brighter at night. Finally, the SuperKamiokande was looking for antineutrinos and the conversion probability from a solar electron neutrino to an electron anti-neutrino, respectively. They found an upper limit of 0.8% of the total flux between 8 MeV and 20 MeV predicted by the SSM [17]. 2.6 Sudbury Neutrino Observatory The Sudbury Neutrino Observatory (SNO) was the first detector explicitly searching for electron neutrinos as well as myon or tau neutrinos from the 8 B reaction. The detector was quite similar to Kamiokande but worked with heavy water D 2 O instead of light water. This allowed analysing three reaction at once. [16] First, the elastic scattering (ES) of electrons by neutrinos similar to Eq. (4) but this time explicit sensitive for all flavors of neutrinos, ν x + e ν x + e (x = e, µ, τ). (5) This reaction guarantees the Sun as source of the neutrinos. Second, a charged current (CC) reaction d + ν e p + p + e, (6) which is sensitive for electron neutrinos only. Third, a neutral-current (NC) reaction d + ν x ν x + p + n (x = e, µ, τ), (7) where a deuteron is fragmented into a proton and a neutron. This is sensitive for all types equally. This leads to the following relations between specific neutrino flux and to be measured rates, which are Φ CC = Φ e Φ ES = Φ e Φ µτ (8) Φ NC = Φ e + Φ µτ Finally, the resulting neutrino fluxes determined by SNO were [13][14] Φ total = ( ) cm 2 s 1 Φ e = (1.76 ± 0.05 ± 0.09) 10 6 cm 2 s 1 Φ µτ = ( 3.41 ± ) 10 6 cm 2 s 1. Not only the predicted total flux of 8 B neutrinos by the SSM is Φ SSM = ( ) 10 6 cm 2 s 1, and therewith very close, but also the ascertained muon and tau neutrion rate is 5.3σ above zero [13], i.e. it has to be assumed that flavor transformation exists with a probability bordering on certainty. In addition, the amount of electron neutrinos of the total flux coincides with the measurements of the former experiments, with respect to influences of the other neutrino flavors, very well. The solar neutrino problem seems to be solved. 3 Neutrino oscillations The SNO experiment has shown that neutrinos can change their flavor. One possible explanation might be neutrino oscillations. Like every particle neutrinos can be described as a wave function or a superposition of wavefunctions which is summarized by ν α = i U αi ν i. (9) We can only measure the resulting eigenstate on the left-hand side of Eq. (9), which we call flavor. It is generally accepted that there are three different flavors, namently electron, muon and tau neutrinos corresponding to the three charged leptons, i.e. α = e, µ, τ. On the right-hand side we find the mass eigenstates ν i with i = 1, 2, 3 and the mixing matrix U αi which is called the Pontecorvo-Maki-Nakagawa-Sakata (MNSP) matrix and is basically a rotation matrix with three Euler angles. It describes a charged-current interaction, i.e. a weak interaction mediated by the W boson, in the leptonic sector. [9] The assumption that only electron neutrinos are produced in the Sun is right. Now, we describe the propagation of these neutrinos in vacuum, which is given in interstellar space in good approximation, as a plane wave. Since these neutrinos are ultrahighenergetic, i.e. p m and t L, we can approximate E = p 2 + m 2 p + m2 2p E + m2 2E and it is ν i (t) = e im2 i L/2E ν i (0). (10) The eigenstates ν i are of course the mass eigenstates. But this means, that they are travel with different velocities depending on m i. The electron neutrino, as a superposition of different ν i, flows apart. Constructive interference then leads to the transition probability from flavor α to β P αβ = ν β (0) ν α (t) 2 2 = U αi Uβie im2 i L/2E. i (11) 4

5 Table 3.1: Fit values for the three-flavor neutrino oscillation parameters based on global data including solar, atmospheric, reactor and accelerator experiments. Last update Sept This is table D1 of [23]. parameter best fit 2σ 3σ m 2 21 [10 5 ev 2 ] m 2 31 [10 3 ev 2 ] sin 2 θ sin 2 θ sin 2 θ Assuming two types of neutrinos the mixing matrix U only depends on one mixing angle θ. Now, the transition probability can be written as ( m P αβ = sin 2 2θ sin 2 2 ) L, (12) 4E where m 2 = m 2 α m 2 β. According to the Standard Model of particles physics this difference would always give zero since all masses are zero. However, we can observe neutrino oscillations, so the model has to be extended. In the case of three neutrino types there are not only two parameters but at least five. Depending on possible symmetry violations there could be more. Current measurements, not only of solar neutrinos but also in particle accelerators and of atmospheric as well as reactor neutrinos, yield the values given in Tab. 3.1 by [23]. In matter scattering becomes important. As used in the SNO experiment, the cross section for elastic scattering and charged current reactions are different for electron and non-electron neutrinos. The Hamiltonian depends on the electron density and resonances occurs now. This is called the Mikheev-Smirnov- Wolfenstein (MSW) effect and is responsible for the day/night asymmetry and the energy dependence of the transistion probability. [3] 4 Current research and future questions The Solar Neutrino Problem seems to be solved. However, some questions are still of interest such as the total mass of the neutrinos or the energy dependence of neutrino oscillations. Yet, ultra-high energetic neutrinos above 5 M ev were investigated. Nevertheless, low energy neutrinos up to 2 M ev include many information. Unfortunately, a new generation of detectors is needed since these energies are below threshold for production of Čerenkov light and radiochemical reactions have too small cross sections. Also background radiation becomes important again. The 7 Be neutrinos are monoenergetic so that a time dependence can be measured well. The KamLAND 2 and Borexino 3 are supposed to research at this energy scale. And finally, pp neutrinos represent about 91% of the total neutrino flux but have a maximum energy of 0.42 MeV. Since the observation of neutrino oscillations only leads to the mass difference between two types, just a lower limit can be estimated assuming the lightest neutrino is indeed massless. An upper limit can be estimated by measuring the electron energy spectrum at the beta decay while this ends at one neutrino mass before the total energy of the system is reached. The mass of an electron antineutrino could be indicated to < 2 ev [2]. The KATRIN experiment 4 was planed to be started in 2009 and is designed to measure the mass of the electron antineutrino using the tritium beta decay to an upper limit of 0.2 ev [15]. If a neutrinoless double beta decay is possible, neutrinos would be their own antiparticles. The evidence for this decay mode is 97% (2.2σ) [22] but still in discussion since an evidence of 5σ is necessary to prove a theory. References [1] MINOS Collaboration: P. Adamson. Measurement of neutrino velocity with the minos detectors and numi neutrino beam. Physical Review D, 76:072005, [2] C. Amsler and Particle Data Group. Review of particle physics, [3] J. N. Bahcall, P. I. Krastev, and A. Yu Smirnov. Is large mixing angle msw the solution of the solar neutrino problems? Physical Review D, 60:093001, [4] John N. Bahcall. Solar neutrinos. Phys. Rev. Lett., 17(7): , [5] John N. Bahcall. How many σs is the solar neutrino effect? Phys. Rev. C, 65(1):015802, [6] John N. Bahcall, M. H. Pinsonneault, and Sarbani Basu. Solar models: current epoch and time dependences, neutrinos, and helioseismological properties, Kamioka Liquid Scintillator Antineutrino Detector 3 Boron Experiment 4 Karlsruhe Tritium Neutrino Experiment 5

6 [7] John N. Bahcall, Aldo M. Serenelli, and Sarbani Basu. New solar opacities, abundances, helioseismology, and neutrino fluxes, [8] Sarbani Basu, William J. Chaplin, Yvonne Elsworth, Roger New, and Aldo M. Serenelli. Fresh insights on the structure of the solar core, [9] Alain Bellerive. Review of solar neutrino experiments. International Journal of Modern Physics A, 19:1167, [10] B. T. Cleveland et al. Measurement of the solar electron neutrino flux with the Homestake chlorine detector. Astrophys. J., 496: , [11] The Super-Kamiokande Collabora. Measurements of the solar neutrino flux from super-kamiokande s first 300 days. ERRATUM-IBID., 81:4279, [12] DONUT Collaboration. Observation of tau neutrino interactions. Physics Letters B, 504:218, [13] SNO Collaboration. Direct evidence for neutrino flavor transformation from neutral-current interactions in the sudbury neutrino observatory. Physical Review Letters, 89:011301, [14] SNO Collaboration. Independent Measurement of the Total Active 8 B Solar Neutrino Flux Using an Array of 3 He Proportional Counters at the Sudbury Neutrino Observatory. Physical Review Letters, 101:111301, [20] S. Eidelman. Review of Particle Physics. Physics Letters B, 592:1+, [21] G. Fiorentini and B. Ricci. What have we learnt about the Sun from the measurement of the 8 B neutrino flux? Physics Letters B, 526:186, [22] H. V. Klapdor-Kleingrothaus, A. Dietz, H. L. Harney, and I. V. Krivosheina. Evidence for neutrinoless double beta decay. MOD.PHYS.LETT.A, 16:2409, [23] M. Maltoni, T. Schwetz, M. A. Tortola, and J. W. F. Valle. Status of global fits to neutrino oscillations, [24] W. Pauli. Offener Brief an die Gruppe der Radioaktiven bei der Gauvereins-Tagung zu Tübingen, [25] M. L. Perl, G. S. Abrams, A. M. Boyarski, M. Breidenbach, D. D. Briggs, F. Bulos, W. Chinowsky, J. T. Dakin, G. J. Feldman, C. E. Friedberg, D. Fryberger, G. Goldhaber, G. Hanson, F. B. Heile, B. Jean-Marie, J. A. Kadyk, R. R. Larsen, A. M. Litke, D. Lüke, B. A. Lulu, V. Lüth, D. Lyon, C. C. Morehouse, J. M. Paterson, F. M. Pierre, T. P. Pun, and P. A. Rapidis. Evidence for Anomalous Lepton Production in e + e Annihilation. Phys. Rev. Lett., 35(22): , [26] B. Pontecorvo. Lepton mixing and neutrino oscillations. J.Exptl. Theoret. Phys., 33:549, [15] The KATRIN Collaboration. Katrin design report 2004, [16] The SNO Collaboration. The sudbury neutrino observatory. NUCL.INSTRUM.METH.A, 449:172, [17] Super-Kamiokande collaboration Gando. Search for anti-electron-neutrinos from the sun at superkamiokande-i. Physical Review Letters, 90:171302, [18] C. L. Cowan, F. Reines, F. B. Harrison, H. W. Kruse, and A. D. McGuire. Detection of the free neutrino: A Confirmation. Science, 124: , [19] G. Danby. Observation of High-Energy Neutrino Reactions and the Existence of Two Kinds of Neutrinos. Phys. Rev. Lett., 9:36 44,

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