Stars as Laboratories for Fundamental Physics - MPP Theory Group

Neutrino Oscillations 301
8.3.5 The Triangle and the Bathtub
The most important application of resonant neutrino oscillations is the
possible reduction of the solar ν e flux, an issue to be discussed more
fully in Chapter 10. Here, I will use the above simple analytic results to
discuss schematically the survival probability prob(ν e →ν e ) of neutrinos
produced in the Sun.
To this end I use an exponential profile for the electron density,
n e = n c e −r/R 0
, which is a reasonable first approximation with R 0 =
R ⊙ /10.54 (Bahcall 1989). Then |∇ ln n e | = R0 −1 is a quantity independent
of location so that there is no need to evaluate it specifically on
resonance. Thus |∇ ln n e | res = 3×10 −15 eV is independent of neutrino
parameters. Therefore, the adiabaticity parameter is
γ = 1 6 × 103 sin 2θ 0 tan 2θ 0
∆m 2
meV 2 MeV
E ν
. (8.45)
An electron density at the center of n c = 1.6×10 26 cm −3 yields
ξ c = 40 meV2
∆m 2
E ν
MeV . (8.46)
The quantity ξ was defined in Eq. (8.31).
It is further assumed that all neutrinos are produced with a fixed
energy by a point-like source at the solar center. They will encounter
a resonance on their way out if ξ c > cos 2θ 0 . In this case one may
calculate the “jump probability” p according to Eq. (8.42) with F from
Eq. (8.43) for the exponential profile. The survival probability is then
given by Eq. (8.40) with the mixing angle at the solar center
cos 2θ =
cos 2θ 0 − ξ c
[(cos 2θ 0 − ξ c ) 2 + sin 2 . (8.47)
2θ 0 ]
1/2
If ξ c < cos 2θ 0 no resonance is encountered; the vacuum parameters
dominate throughout the Sun. In this case one may use p = 0.
In the framework of these approximations it is straightforward to
evaluate prob(ν e →ν e ) as a function of ∆m 2 and sin 2 2θ 0 . In Fig. 8.9
contours for the survival probability are shown for E ν = 1 MeV. Because
the energy always appears in the combination ∆m 2 /2E ν one may
obtain an analogous plot for other energies by an appropriate vertical
shift. This kind of plot represents the well-known MSW triangle. The
dashed line marks the condition γ = 1 and thus divides the parameter
plane into a region where the oscillations are adiabatic and one