357
21.4. Neutrino mixing and oscillations
Suppose now that L/|λ 31 (E)| ∼ 1, while L/|λ 21 (E)| ≪ 1. Then expression
(21.119) reduces to (problem 21.7)
P (ν α → ν β ; L, E) ≈ δ αβ − 4|U α3 |
2 [δ αβ − |U β3 |
2 ] sin
2 Δm 31
4E
L
= P (¯ ν α → ¯
ν β ; L, E).
(21.126)
In particular,
P (¯ ν e → ¯
ν e ; L, E) = 1 − 4|U e3 |
2 (1 − |U e3 |
2 ) sin
2 Δm
2
31
4E
L,
(21.127)
2
which can describe the survival probability of reactor ¯
ν e s, for example.
Adopting a parametrization of the form (20.166), with rows labelled by
2
e, μ and τ , and columns by 1, 2, and 3, |U e3 | is sin
2 θ e3 , which is found
experimentally to be small (see the following section). It is often a good
2
approximation to set |U e3 | to zero, in which case |U μ3 | = sin
2 θ μ3 . Then
(21.126) gives the ν μ survival probability
P (ν μ → ν μ ; L, E) = P (¯ ν μ → ν ¯ μ ; L, E) ≈ 1 − sin
2 2θ μ3 sin
2 (L/2λ 31 (E))
(21.128)
and the flavour-change probability
P (ν μ → ν τ ; L, E) = P (¯ ν μ → ν ¯ τ ; L, E) ≈ sin
2 2θ μ3 sin
2 (L/2λ 31 (E)). (21.129)
In this approximation, P (ν μ → ν e ) = P (¯ ν μ → ν ¯ e ) = 0. Formulae (21.128)
and (21.129) can be used to describe the dominant atmospheric ν μ and ¯
ν μ
2
oscillations (see the following section), and the parameters θ μ3 and Δm 31 (or
2
Δm ) are referred to as the atmospheric mixing angle and mass squared
32
2
difference. The smaller mass squared difference Δm , and the angle θ e2 , are
21
associated with solar ν e oscillations.
The formulae (21.128) and (21.129) are, in fact, exactly what a simple
2-state mixing model would give. Suppose that the effective mixing matrix
for the 2-state system has the form (see problem 1.6)
(
)
−a cos 2θ asin 2θ ,
(21.130)
a sin 2θ a cos 2θ
where rows are labelled by e, μ and columns by 1, 3; then the survival probability is just
1 − sin
2 2θ sin
2 (La),
(21.131)
where we have taken L ≈ T as before. We can therefore identify the mixing
parameter as
2
31
a = [2λ 31 (E)]
−1 =
Δm .
(21.132)
4E
Note that the energies are here measured relative to a common average energy;
if |e› is the lighter eigenstate and |h› the heavier, then
|ν e › = cosθ|e› + sin θ|h›
|ν μ › = − sin θ|e› + cos θ|h›.
(21.133)
21.4. Neutrino mixing and oscillations
Suppose now that L/|λ 31 (E)| ∼ 1, while L/|λ 21 (E)| ≪ 1. Then expression
(21.119) reduces to (problem 21.7)
P (ν α → ν β ; L, E) ≈ δ αβ − 4|U α3 |
2 [δ αβ − |U β3 |
2 ] sin
2 Δm 31
4E
L
= P (¯ ν α → ¯
ν β ; L, E).
(21.126)
In particular,
P (¯ ν e → ¯
ν e ; L, E) = 1 − 4|U e3 |
2 (1 − |U e3 |
2 ) sin
2 Δm
2
31
4E
L,
(21.127)
2
which can describe the survival probability of reactor ¯
ν e s, for example.
Adopting a parametrization of the form (20.166), with rows labelled by
2
e, μ and τ , and columns by 1, 2, and 3, |U e3 | is sin
2 θ e3 , which is found
experimentally to be small (see the following section). It is often a good
2
approximation to set |U e3 | to zero, in which case |U μ3 | = sin
2 θ μ3 . Then
(21.126) gives the ν μ survival probability
P (ν μ → ν μ ; L, E) = P (¯ ν μ → ν ¯ μ ; L, E) ≈ 1 − sin
2 2θ μ3 sin
2 (L/2λ 31 (E))
(21.128)
and the flavour-change probability
P (ν μ → ν τ ; L, E) = P (¯ ν μ → ν ¯ τ ; L, E) ≈ sin
2 2θ μ3 sin
2 (L/2λ 31 (E)). (21.129)
In this approximation, P (ν μ → ν e ) = P (¯ ν μ → ν ¯ e ) = 0. Formulae (21.128)
and (21.129) can be used to describe the dominant atmospheric ν μ and ¯
ν μ
2
oscillations (see the following section), and the parameters θ μ3 and Δm 31 (or
2
Δm ) are referred to as the atmospheric mixing angle and mass squared
32
2
difference. The smaller mass squared difference Δm , and the angle θ e2 , are
21
associated with solar ν e oscillations.
The formulae (21.128) and (21.129) are, in fact, exactly what a simple
2-state mixing model would give. Suppose that the effective mixing matrix
for the 2-state system has the form (see problem 1.6)
(
)
−a cos 2θ asin 2θ ,
(21.130)
a sin 2θ a cos 2θ
where rows are labelled by e, μ and columns by 1, 3; then the survival probability is just
1 − sin
2 2θ sin
2 (La),
(21.131)
where we have taken L ≈ T as before. We can therefore identify the mixing
parameter as
2
31
a = [2λ 31 (E)]
−1 =
Δm .
(21.132)
4E
Note that the energies are here measured relative to a common average energy;
if |e› is the lighter eigenstate and |h› the heavier, then
|ν e › = cosθ|e› + sin θ|h›
|ν μ › = − sin θ|e› + cos θ|h›.
(21.133)