On the evolution of the neutrino state inside the sun Page: 4 of 34
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" What physical criteria determine whether the neutrino evolution is adi-
abatic or not?
" At what radius in the Sun does the nonadiabatic "jumping" between
the eigenstates of the instantaneous Hamiltonian take place?
The traditional wisdom is that one should analyze the density profile around
the so-called resonance point, i.e., the point where the difference of the eigen-
values of the instantaneous Hamiltonian is minimal and the local value of the
mixing angle is Om = fr/4 (see, e.g., [20, 21, 22, 23, 24, 25, 26]). This, how-
ever, clearly needs to be modified for large mixing angles. In particular, for
O > fr/4 the resonance, defined in this way, simply does not exist. We will
show how this contradiction is resolved in Section 2.2. In Section 2.3 we for-
mulate the adiabaticity condition that, unlike the standard result, remains
valid for 0 ? 7r/4.
In Section 3 we present the results of numerical calculations of the jump-
ing probability P, for the neutrino propagating in the realistic solar profile.
The calculations are carried out for a wide range of Am2 and tan2 0, from
the VO region to the region where the exponential density approximation is
valid. We show how the adiabaticity prescription of Section 2.3 applies to
this case. We also give a simple empirical prescription on how to compute P,
in this range of the parameters in terms of only elementary functions. Such
an empirical parametrization of the numerical results can be helpful if one
would like to be able to quickly estimate the value of P, anywhere in the
range in question without having to solve the differential equation each time.
In Section 3.2, we discuss what happens in the transitional region between
the adiabatic and nonadiabatic regimes (QVO). In particular, we determine
what part of the solar electron density profile is primarily responsible for the
matter effects in this region.
Finally, in Section 4 we comment on the four known exact analytical
solutions for the neutrino jumping probability P,. Such solutions have been
found for the linear, exponential, 1/r, and the hyperbolic tangent matter
density profiles. A natural question to ask is whether these profiles have
something in common that makes finding exact solutions possible. Using
the formulation of the evolution equations introduced in Section 2, we show
that all four results are not independent and that, given the formula for
the hyperbolic tangent profile, one can very simply obtain the other three
solutions. As an added benefit, we obtain an exact expression for the density
distribution Ne o (coth(x/l) + 1).
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Friedland, Alexander. On the evolution of the neutrino state inside the sun, article, January 26, 2001; Berkeley, California. (https://digital.library.unt.edu/ark:/67531/metadc778694/m1/4/: accessed April 22, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.