# MSW Effects in Vacuum Oscillations Page: 2 of 5

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MSW Effects in Vacuum Oscillations

Alexander Friedland

Department of Physics, University of California, Berkeley, CA 94720, USA;

Theory Group, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA

(February 1, 2008)

We point out that for solar neutrino oscillations with the mass squared difference of Am2 t

1010 - 10-9 eV2, traditionally known as "vacuum oscillation" range, the solar matter effects are

non-negligible, particularly for the low energy pp neutrinos. One consequence of this is that the

values of the mixing angle a and </2 - a are not equivalent, leading to the need to consider theentire physical range of the mixing angle 0 < a

neutrino oscillation parameters.

1. The field of solar neutrino physics is currently un-

dergoing a remarkable change. For 30 years the goal

was simply to confirm the deficit of solar neutrinos. The

latest experiments, however, such as Super-Kamiokande,

SNO, Borexino, KamLAND, etc, aim to accomplish more

than that. By collecting high statistics real time data

sets on different components of the solar neutrino spec-

trum, they hope to obtain unequivocal proof of neu-

trino oscillations and measure the oscillation parameters.

With the physics of solar neutrinos quickly becoming a

precision science, it is more important then ever to ensure

that all relevant physical effects are taken into account

and the right parameter set is used.

It has been a long standing tradition in solar neutrino

physics to present experimental results in the Am2 -

sin2 20 space and to treat separately the "vacuum os-

cillation" (Am2 ~ 10- - 10-9 eV2) and the MSW

(Am2 i-8 - 10-s eV2) regions. In the vacuum oscil-

lation region the neutrino survival probability (i.e. the

probability to be detected as ve) was always computed

according to the canonical formula,

P 1 -sin2 20 sin2 (1.27A 2L) , (1)

where the neutrino energy E is in GeV, the distance L in

km, and the mass squared splitting Am2 in eV2. Eq. (1)

makes sin2 20 seem like a natural parameter choice. As

sin2 20 runs from 0 to 1, the corresponding range of the

mixing angle is 0 < 0 < r/4. There is no need to

treat separately the case of Am2 < 0 (or equivalently

r/4 < 0 < r/2), since Eq. (1) is invariant with respect

to Am2 -> -Am2 (0 -> /2 -0).

The situation is different in the MSW region, since

neutrino interactions with matter are manifestly flavor-

dependent. It is well known that for Am2 > 10-8 eV2

matter effects in the Sun and Earth can be quite large.

In this case, if one still chooses to limit the range of

the mixing angle to 0 < 0 < </4, one must consider

both signs of Am2 to describe all physically inequivalent

situations. As was argued in [1], to exhibit the continuity

of physics around the maximal mixing, it is more natural

to keep the same sign of Am2 and to vary the mixing

angle in the range 0 < 0 < r/2.< </2 when determining the allowed values of the

Historically, a possible argument in favor of not consid-

ering 0 > </4 in the MSW region might have been that

this half of the parameter space is "uninteresting", since

for 0 > r/4 there is no level-crossing in the Sun and the

neutrino survival probability is always greater than 1/2.

However, a detailed analysis reveals that allowed MSW

regions can extend to maximal mixing and beyond, as

was explored in [2] (see also [3] and [4] for a treatment of

3- and 4- neutrino mixing schemes).

In this letter we point out that for solar neutrinos with

low energies, particularly the pp neutrinos, the solar mat-

ter effects can be relevant even for neutrino oscillations

with Am2 ~ 10--1 - 10-9 eV2. These effects break the

symmetry between 0 and /2 - 0 making it necessary

to consider the full physical range of the mixing angle

0 < 0 r/2 even in the "vacuum oscillation" case.

2. For simplicity, we will only consider here the two-

generation mixing. If neutrino masses are nonzero then,

in general, the mass eigenstates vi,2) are different from

the flavor eigenstates ve, ). The relationship between

the two bases is given in terms of the mixing angle 0:

vi) cos0 ve) - sin06v ),v22) sin6ve) + cos06v ).

(2)

In our convention v2) is always the heavier of the two

eigenstates, i.e. Am2 m - m ;> 0. Then, as already

mentioned, 0 < 0 < r/2 encompasses all physically dif-

ferent situations.

Neutrinos are created in the Sun's core and exit the

Sun in the superposition of vi) and v2). For Am2 in

the vacuum oscillation region, the neutrino is produced

almost completely in the heavy Hamiltonian eigenstate

v+). In this case, if the evolution inside the Sun is adia-

batic, the exit state is purely v2). In the case of a nona-

diabatic transition there is also a nonzero probability Pc

to find the neutrino in the vi) state (a "level crossing"

probability). For a given value of Pc, the survival prob-

ability for neutrinos arriving at the Earth is determined

by simple 2-state quantum mechanics [5,7,8]:

P = zPccos20+ (1 -Pc)sin20

. ( ~Am 2L (3

+ 2 Pc(1 -P) sin B cos B cos 2.54 E. + , . (3)1

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Friedland, Alexander. MSW Effects in Vacuum Oscillations, article, February 6, 2000; Berkeley, California. (https://digital.library.unt.edu/ark:/67531/metadc927089/m1/2/: accessed June 17, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.