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 [1] as well as results from the long base-line experiment K2K [2] 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.[3] 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 [4]. These oscillations require non-zero value of Ue3 . The effects are restricted by the CHOOZ result [5]. 2. νe −oscillations driven by the solar mass splitting ∆m2 [6]. The detailed study of the effect have been done in our previous paper [6] where we have shown that neutrino oscillations with parameters in the LMA MSW allowed region [9] ∆m2 = (2 − 30) · 10−5 eV2 , sin2 2θ > 0.65, favored by analyzes of solar neutrino data from SNO [7] and SuperKamiokande [8] 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 [1]. 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 [6]. 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 . The interference depends on the sign of s13 , also does not have screening factor, and its smallness is mainly due to smallness of s13 . 3). Beside this term has opposite signs for neutrinos and anti-neutrinos. We have calculated dependences of the excess of the e−like events on the zenith angle of electron, Θe . The procedure was described before in Ref. [10]. In Fig. 2 we show the zenith angle dependences of the excess of the e-like events for different values of oscillation parameter. Concluding, we show that if the LMA solution is the correct one and for θ23 = 45◦ ( in this case the effects due the oscillations driven by ∆m221 only are suppressed) we can have a direct way to determine Ue3 , from the electron neutrino zenith distribution as is shown in Figure 2. REFERENCES 1. Y. Totsuka, these proceedings. 2. T. Hasegawa, these proceedings. 3. M. Honda et al., Phys. Rev. D 54, 4985 (1995). 4. E. Kh. Akhmedov, A. Dighe, P. Lipari, A. Yu. Smirnov, Nucl. Phys. B 542 (1999) 3. 5. M. Apollonio et al. (CHOOZ Collaboration), Phys.Lett. B466, 415 (1999). 6. O. L. G. Peres and A. Yu. Smirnov, Phys. Lett. B456 (1999) 204. 7. A. McDonald, these proceedings. 8. Y. Fukuda et al. (Super-Kamiokande Collaboration), Phys. Rev. Lett. 82, 1810 (1999); idem 86, 5651 (2001). 9. A.M. Gago et al., hep-ph/0112060 accepted for publication in Phys. Rev. D. 10. M. C. Gonzalez-Garcia et al., Phys. Rev. D58, 033004 (1998).

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