# New Enhancement Mechanism of the Transitions in the Earth of the Solar and Atmospheric Neutrinos Crossing the Earth Core

###### Abstract

It is shown that the and () transitions respectively of the solar and atmospheric neutrinos in the Earth in the case of mixing in vacuum, are strongly enhanced by a new type of resonance when the neutrinos cross the Earth core. The resonance is operative at small mixing angles but differs from the MSW one. It is in many respects similar to the electron paramagnetic resonance taking place in a specific configuration of two magnetic fields. The conditions for existence of the new resonance include, in particular, specific constraints on the neutrino oscillation lengths in the Earth mantle and in the Earth core, thus the resonance is a “neutrino oscillation length resonance”. It leads also to enhancement of the and transitions in the case of mixing and of the (or ) transitions at small mixing angles. The presence of the neutrino oscillation length resonance in the transitions of solar and atmospheric neutrinos traversing the Earth core has important implications for current and future solar and atmospheric neutrino experiments, and more specifically, for the interpretation of the results of the Super-Kamiokande experiment.

## 1 Introduction

When the solar and atmospheric neutrinos traverse the Earth,
the and
() transitions/oscillations
they undergo due to small mixing in vacuum
^{1}^{1}1As is well-known, the
transition probability accounts for the Earth effect
in the solar neutrino survival probability
in the case of the MSW two-neutrino
and
transition solutions of the
solar neutrino problem, being a sterile neutrino.
can be dramatically
enhanced by a new type of resonance
which differs from the MSW one and takes place
when the neutrinos cross the
Earth core [1].
The resonance is present
in the and
() transition
probabilities, and
,
if the neutrino oscillation length (and mixing angles) in the
Earth mantle and in the Earth core
obey specific conditions [1].
When satisfied, these conditions ensure that the
relevant oscillating factors in the
probabilities and
are maximal
^{2}^{2}2Note that, in contrast, the MSW effect is
a resonance amplifying the neutrino mixing.
and that this
produces a resonance maximum in and
.
Accordingly, the term “neutrino oscillation
length resonance” or simply “oscillation length resonance” was used
in [1] to denote the resonance of interest.
There exists a beautiful analogy between the neutrino
oscillation length resonance and the electron
spin-flip resonance realized
in a specific configuration of
magnetic fields ^{3}^{3}3This analogy was brought
to the attention of the author by L. Wolfenstein.
(see [1] for further details).

At small mixing angles () the maxima due to the neutrino oscillation length resonance in and ) are absolute maxima and dominate in and : the values of the probabilities at these maxima in the simplest case of two-neutrino mixing are considerably larger - by a factor of (), than the values of and at the local maxima associated with the MSW effect taking place in the Earth core (mantle). The magnitude of the enhancement due to the oscillation length resonance depends on the neutrino trajectory through the Earth core: the enhancement is maximal for the center-crossing neutrinos [1, 2]. Even at small mixing angles the resonance is relatively wide both in the neutrino energy (or resonance density) [1] - it is somewhat wider than the MSW resonance, and in the Nadir angle [2], , specifying the neutrino trajectory in the Earth. It also exhibits strong energy dependence.

The presence of the oscillation length resonance in the transitions of solar and atmospheric neutrinos traversing the Earth has important implications [1, 2, 3, 4, 5] for the interpretation of the results, e.g., of the Super-Kamiokande experiment [6, 7].

The Earth enhancement of the two-neutrino transitions of interest has been discussed rather extensively, see, e.g., refs. [5, 8]. Some of the articles contain plots of the probabilities and/or or on which one can recognize now the dominating maximum due to the neutrino oscillation length resonance (see, e.g., [8]). However, this maximum was invariably interpreted to be due to the MSW effect in the Earth core before the appearance of [1].

## 2 The Neutrino Oscillation Length Resonance (NOLR)

All the interesting features of the
solar and atmospheric neutrino transitions
in the Earth, including those related to
the neutrino oscillation length resonance,
can be understood quantitatively
in the framework of the two-layer model of the Earth density
distribution [1, 2, 9, 10].
The density profile of the Earth in the two-layer model
is assumed to consist of two structures -
the mantle and the core,
having different constant densities, and ,
and different constant electron fraction numbers,
and
^{4}^{4}4The densities
should be considered as mean
densities along the neutrino trajectories.
In the Earth model [11]
one has: and
.
For one can use the standard
values [11, 12]
(see also [3]) and ..
The transitions of interest of the neutrinos
traversing the Earth are essentially
caused by two-neutrino oscillations
taking place i) first in the mantle over a distance
with a mixing angle and oscillation length
, ii) then in the core over a
distance with different mixing angle
and oscillation length , and iii) again
in the mantle over a distance
with and .
Due to the matter effect
and , being the
oscillation length in vacuum
(see, e.g., [13]).
For fixed and
the neutrino oscillation length
resonance occurs [1]
if i) the relative phases acquired by the
energy eigenstate neutrinos
in the mantle and in the core,
and ,
are correlated, being odd multiples of ,
so that

where , and if ii) the inequality

is fulfilled. Condition (2) is valid for the probability (). When equalities (1) hold, (2) ensures that () has a maximum. In the region of the NOLR maximum where, e.g., , is given in the case of mixing by [1]:

At the NOLR maximum takes the form [1]

The analogs of eqs. (3) - (4) for the probability can be obtained by formally setting while keeping and in (3) - (4). Note that one of the two NOLR requirements is equivalent at to the physical condition [1]

Remarkably enough, for the and transitions in the Earth, the NOLR conditions (1) with are approximately fulfilled at small mixing angles () in the regions where (2) holds [1]. The associated NOLR maxima in and are absolute maxima (Figs. 1 - 2).

Let us note that the study performed in [1]
and discussed briefly above
^{5}^{5}5For analysis of the NOLR effects
in the
and
transitions ( mixing)
and in the
(or )
transitions at small mixing angles see [1, 2].
differs substantially from
the studies [14]. The authors of
[14] considered the
possibility of resonance enhancement of the
transitions of
neutrinos propagating in matter with density, varying
periodically along the neutrino path
(parametric resonance). It was found, in particular,
that at small mixing angles strong enhancement
is possible only if the neutrinos
traverse at least 2 - 3 periods (in length)
of the density oscillations.
The density distribution in the
Earth is not periodic ^{6}^{6}6The density change
along the path of a neutrino crossing the Earth core is not
periodic even in the two-layer model: it falls short of making
even one and a half periods.;
and in order for the oscillation length resonance
^{7}^{7}7Although this term may not be perfect,
it underlines the physical essence of the new resonance.
The objection to it raised in [15]
is not convincing.
The term “parametric resonance” used in [15], e.g.,
suggests incorrect analogies.
to occur periodic variation of the density is not required.

In [16] the transitions in the Earth were considered for . It was noticed that in the region where , being the neutron number density, a new maximum in appears when , which was found to hold at . The height of the maximum is comparable to the heights of the other “ordinary” maxima present in for . It is stated in [16] that the effect does not take place in the transitions, which is incorrect both for and [1, 2].

## 3 Implications of the Neutrino Oscillation Length Resonance

The implications of the oscillation length resonance enhancement of the probability for the Earth core crossing solar neutrinos, for the tests of the MSW and solutions of the solar neutrino problem are discussed in refs. [1, 2, 3, 4]. It is remarkable that for values of from the small mixing angle (SMA) MSW solution region and the geographical latitudes at which the Super-Kamiokande, SNO and ICARUS detectors are located, the enhancement takes place in the case for values of the B neutrino energy

lying in the interval MeV to which these detectors are sensitive. The resonance maximum in at for the trajectory with , for instance, is located at MeV if . Accordingly, at small mixing angles the NOLR is predicted [3] to produce a much bigger - by a factor of , day-night (D-N) asymmetry in the Super-Kamiokande sample of solar neutrino events, whose night fraction is due to the core-crossing neutrinos, in comparison with the asymmetry determined by using the whole night event sample. On the basis of these results it was concluded in [3] that it can be possible to test a substantial part of the MSW SMA solution region in the plane by

performing core D-N asymmetry measurements. The Super-Kamiokande collaboration has already successfully applied this approach to the analysis of their solar neutrino data [6]: the limit the collaboration has obtained on the D-N asymmetry utilizing only the core event sample permitted to exclude a part of the MSW SMA solution region located in the area , , which is allowed by the mean event rate data from all solar neutrino experiments (Homestake, GALLEX, SAGE, Kamiokande and Super-Kamiokande). In contrast, the current Super-Kamiokande upper limit on the whole night D-N asymmetry [6] does not permit to probe the SMA solution region: the predicted asymmetry is too small [3].

The strong NOLR enhancement of the and transitions of atmospheric neutrinos crossing the Earth core can take place at small mixing angles practically for all neutrino trajectories through the core [1], e.g., for the trajectories with (Fig. 2). This is particularly relevant for the interpretation of the results of the atmospheric neutrino experiments and for the future studies of the oscillations/transitions of atmospheric neutrinos crossing the Earth. The Super-Kamiokande collaboration has reported at this Conference strong evidences for oscillations of the atmospheric () [7]. Assuming two-neutrino mixing, the data is best described in terms of vacuum oscillations with parameters and . The possibility of two-neutrino large mixing oscillations is disfavored by the data [7]; at it is ruled out [17].

It is a remarkable coincidence that for
and small mixing, ,
the oscillation length resonance
in
occurs [1] for values of the
energy of the atmospheric
and
which contribute either to the
sub-GeV or to the multi-GeV
like and like Super-Kamiokande event samples [7].
For ,
,
and (Earth center crossing),
for instance, the absolute maximum in
due to the NOLR
takes place at GeV.
Thus, for values of
from the region of the
oscillation solution of the atmospheric neutrino problem,
the NOLR
strongly enhances the (and
)
transitions of the atmospheric
neutrinos crossing the Earth core, making
the transitions
detectable even at small mixing angles.
It was suggested in [1] that the
excess of e-like events
in the region ,
being the Zenith angle, either
in the sub-GeV or in the
multi-GeV sample,
observed (in both samples) in the
Super-Kamiokande experiment [7], is due to
small mixing angle transitions, ,
with or
respectively ,
strongly enhanced by the NOLR
^{8}^{8}8A more detailed investigation [2, 19]
performed within the indicated three-neutrino mixing scheme
reveals, in particular, that the excess
of e-like events in the Super-Kamiokande sub-GeV data
at
seems unlikely to be due to small mixing angle
transitions amplified by the oscillation length resonance..
The same resonantly enhanced transitions with
()
should produce
at least part of the strong zenith angle dependence,
exhibited by the like multi-GeV
(sub-GeV) Super-Kamiokande data [2].

The transitions of interest arise in a three-neutrino mixing scheme, in which the small mixing angle MSW transitions with , or large mixing angle oscillations with , provide the solution of the solar neutrino problem and the atmospheric neutrino anomaly is due to oscillations with [1]. For the three-neutrino and transition probabilities reduce [18] to the two-neutrino transition probability (Fig. 2) with and playing the role of the two-neutrino oscillation parameters, where is the element of the lepton mixing matrix, being the heaviest massive neutrino. The data [7, 17] implies: . Thus, searching for the and transitions of atmospheric neutrinos, amplified by the oscillation length resonance, can provide also unique information about the magnitude of [19].

## 4 Conclusions

The neutrino oscillation length resonance should be present in the transitions taking place when the solar neutrinos cross the Earth core on the way to the detector, if the solar neutrino problem is due to small mixing angle MSW transitions in the Sun. The same resonance should be operative also in the () small mixing angle transitions of the atmospheric neutrinos crossing the Earth core if the atmospheric and indeed take part in large mixing vacuum , oscillations with , as is strongly suggested by the Super-Kamiokande data [7], and if all three flavour neutrinos are mixed in vacuum. The existence of three-flavour-neutrino mixing in vacuum is a very natural possibility in view of the present experimental evidences for oscillations/transitions of the flavour neutrinos. In both cases the oscillation length resonance produces a strong enhancement of the corresponding transitions probabilities, making the effects of the transitions observable even at rather small mixing angles. Actually, the resonance may have already manifested itself in the excess of e-like events at observed in the Super-Kamiokande multi-GeV atmospheric neutrino data [1, 2, 19]. And it can be responsible for at least part of the strong zenith angle dependence present in the Super-Kamiokande multi-GeV and sub-GeV like data [2, 19].

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