LMA parameters and non-zero Ue3 effects on atmospheric ν data?

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
The Abdus Salam International Centre for Theoretical Physics, I-34100 Trieste, Italy
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
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. Introduction
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 .
∗ 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. 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 †
+ V νf ,
− 1 = P2 (rc223 − 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,
H ≈
0 ∆m231 /2E + Ve s213
/2E + Ve c213 , and M2 =
where H2 = U12 M2 U12
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:
A˜ee A˜eµ
S˜ ≈  A˜µe A˜µµ
0  ,
Aτ τ
where Aτ τ ≈ exp(−i∆m231 L/2E) , and L is the
total distance traveled by the neutrinos.
−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
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:
− 1 = (rc223 − 1)P˜µe − rs13 c213 sin 2θ23 Q
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.
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Peres and A. Yu.
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