1 LMA parameters and non-zero Ue3 effects on atmospheric ν data? O. L. G. Peresa MCSD]Instituto de Fisica Gleb Wataghin, Universidade Estadual de Campinas, UNICAMP 13083-970 Campinas SP, Brazil∗ , A. Yu. Smirnovb c a [ b The Abdus Salam International Centre for Theoretical Physics, I-34100 Trieste, Italy c Institute for Nuclear Research of Russian Academy of Sciences, Moscow 117312, Russia Recent results on atmospheric neutrinos  as well as results from the long base-line experiment K2K  further confirmed the interpretation of the atmospheric neutrino anomaly in terms of νµ ↔ ντ oscillations with maximal or close to maximal mixing and mass squared difference in the interval, ∆m2atm = (1.5 − 4) × 10−3 eV2 , sin2 2θatm > 0.88, at 90 % C.L. . A sub-dominant oscillation of electron neutrinos is not excluded yet. It seems that there is an excess of the e−like events in the low energy part of the sub-GeV sample (p < 0.4 GeV, where p is the momentum of lepton). In comparison with predictions based on the atmospheric neutrino flux from Ref. the excess is about (12 15)%. For higher energies, the excess is much smaller. Can these results be related to the νe −oscillations? What could be the implications of the positive answer? We have some preliminary results that we will discuss in next sections. 1.4 1.3 1.2 1.1 1 0.9 0.8 0.7 0.6 Ne/Ne0 1. Introduction Ne/Ne0 We study the possible manifestation of the interference between the effects produced in the atmospheric neutrinos due to oscillation driven by the solar parameters parameters ∆m221 , sin2 2θ21 and due to oscillation driven by Ue3 . 1.4 1.3 1.2 1.1 1 0.9 0.8 0.7 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 cos(ΘL) ∗ O.L.G.P. thanks the hospitability of ICTP when this work began. O.L.G.P. was supported by Funda¸ca ˜o de Amparo ` a Pesquisa do Estado de S˜ ao Paulo (FAPESP) and by Conselho Nacional de Ciˆ encia e Tecnologia (CNPq). Figure 1. Zenith distribution for sub-GeV events with p< 0.4 GeV and for p > 0.4 GeV. We assume the parameters showed in the plot. 2 2. Formalism 3. Ue3 and induced interference In the three neutrino schemes which explain the atmospheric and solar neutrino data, there are two possible channels of the νe − oscillations: 1. νe −oscillations driven by ∆m2atm responsible for dominant mode of the atmospheric neutrino oscillations . These oscillations require non-zero value of Ue3 . The effects are restricted by the CHOOZ result . 2. νe −oscillations driven by the solar mass splitting ∆m2 . The detailed study of the effect have been done in our previous paper  where we have shown that neutrino oscillations with parameters in the LMA MSW allowed region  ∆m2 = (2 − 30) · 10−5 eV2 , sin2 2θ > 0.65, favored by analyzes of solar neutrino data from SNO  and SuperKamiokande  data, can lead to an observable excess of the e-like events in the sub-GeV atmospheric neutrino sample. It was shown that the excess is determined by the two neutrino transition probability P2 and the “screening” factor: We consider the three-flavor neutrino system with hierarchical mass squared differences: ∆m221 = ∆m2 << ∆m231 = ∆m2atm . The evolution of the neutrino vector of state νf ≡ (νe , νµ , ντ )T is described by the equation U M 2U † dνf = + V νf , (2) i dt 2E Fe − 1 = P2 (rc223 − 1) , Fe0 (1) where Fe and Fe0 are the electron neutrino fluxes with and without oscillations and r is the ratio of the original muon and electron neutrino fluxes. In the sub-GeV region r ≈ 2, so that the screening factor is zero when the νµ −ντ mixing is maximal. We show in Fig. 1 our previous results compared with the latest data on Super-Kamiokande . In previous studies the effects of oscillations driven by the solar and atmospheric ∆m2 have been considered separately: The studies of the ∆m2atm −driven oscillations where performed in the framework of the so called “one level dominating scheme” when the effect of solar mass splitting between two lightest states, ∆m221 , is neglected. In studies of the solar ∆m221 driven oscillations it was assumed that Ue3 is negligible. In this paper we study the effects of the interplay of oscillations with the LMA parameters and non-zero Ue3 . where E is the neutrino energy and M 2 = diag(0, ∆m221 , ∆m231 ) is the diagonal matrix of neutrino mass squared eigenvalues. V = neudiag(Ve , 0, 0) is the matrix of matter-induced √ trino potentials with Ve = 2GF Ne , GF and Ne being the Fermi constant and the electron number density, respectively. The mixing matrix U is defined through νf = U νmass , where νmass ≡ (ν1 , ν2 , ν3 )T is the vector of neutrino mass eigenstates. It can be parameterized as U = U23 U13 U12 . The matrix Uij = Uij (θij ) performs the rotation in the ij- plane by the angle θij . Here we have neglected possible CP-violation effects in the lepton sector. 3.1. Propagation basis The dynamics of oscillations is simplified in the “propagation” basis ν˜ = (˜ νe , ν˜2 , ν˜3 )T , which is ˜ ν˜. We related with the flavor basis by νf = U define the propagation basis in such a way that ˜ equals: U ˜ = U23 U13 . The projection matrix U propagation basis can be introduced in the following way. First, let us perform the rotation νf = U23 U13 ν 0 . Using Eq. (2) we find that in the basis ν 0 the Hamiltonian takes the form, H2 0 0 H ≈ , (3) 0 ∆m231 /2E + Ve s213 † /2E + Ve c213 , and M2 = where H2 = U12 M2 U12 2 diag(0, ∆m21 ). We neglect off-diagonal terms in the evolution equation, Eq. (2). The evolution matrix S in the propagation basis (˜ νe , ν˜µ , ν˜τ ) has the following form: 0 A˜ee A˜eµ S˜ ≈ A˜µe A˜µµ (4) 0 , ˜ 0 0 Aτ τ where Aτ τ ≈ exp(−i∆m231 L/2E) , and L is the total distance traveled by the neutrinos. 3 i h −s213 2W23 + P˜µe (r − 2) + s413 W23 (2 − P˜µe ) Figure 2. Zenith distribution for sub-GeV events with p< 0.4 GeV. We assume the parameters showed in the plot and ∆m221 = 5 10−5 eV2 . 3.2. Flavor transitions and interference Let us find the probabilities of (νµ ↔ νe ) oscillations, Pµe , and (νe ↔ νe ) oscillations, Pee , relevant for our problem. The S−matrix in the ˜ S˜U ˜ † , and we find flavor basis equals: S = U 2 Pµe = −s13 c13 s23 A˜ee + c13 c23 A˜µe + s213 c213 s223 , and Pee = c413 (1− P˜µe )+s413 . For sub-GeV sample oscillations driven by ∆m231 are averaged out, so that there is no interference effect due to state ν˜τ . At the same time, according to (5) the amplitudes A˜ee and A˜µe interfere. It this interference which produces effect we are interested in this paper. Notice that amplitudes A˜ee and A˜µe are both due to solar oscillation parameters. However their interference appears due to presence of the third neutrino (non-zero s13 ). In what follows we will call the interference of the amplitudes (with solar oscillation parameters) due to non-zero Ue3 ∼ s13 as induced interference. Combining Pµe and Pee , the excess of the νe −flux equals: Fe − 1 = (rc223 − 1)P˜µe − rs13 c213 sin 2θ23 Q Fe0 and Q ≡ Re(A˜∗ee A˜µe ) and W23 ≡ (1 − rs223 ). The first term on the left hand side (zero order in s213 ) corresponds to the contribution we have discussed in . The second term is the effect of the induced interference. Let us stress its properties: 1). The interference term depends on s13 linearly. So its effect may not be strongly suppressed even for small s13 . 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