354
21. CP Violation and Oscillation Phenomena
from the ν α source the probability P (ν α → ν β ; E, L) of detecting a neutrino of
a different flavour ν β is non-zero.
2 Such a flavour change will of course imply
that the ν α survival probability, P (ν α → ν α ; E, L), is less than 1. We shall
give a simplified version of the derivation of such probabilities, following the
approach of the review by Nakamura and Petcov in section 13 of Nakamura
et al. (2010). This review includes a large list of references to the timedependent formalism; we mention here the contributions of Kayser (1981),
Nauenberg (1999) and Cohen et al. (2009). We shall treat all the neutrinos
as stable particles.
We consider the evolution of the state |ν α › in the frame in which the
detector which measures its flavour is at rest (the lab frame). As in the meson
case, the states with simple space-time evolution in a vacuum are the mass
eigenstates |ν i › (i = 1, 2, 3), a superposition of which is equal to |ν α ›:
∑
|ν α › =
U
∗
αi
j
|ν i , p i ›,
(21.111)
the complex conjugation arising from taking the dagger of the relation (21.106)
for the field operators. Here U stands for either the Dirac or the Majorana
matrix, and p i is the 4-momentum of ν i . Similarly,
∑
|ν ¯ α › =
U αi |ν ¯ i , p i
i
›.
(21.112)
We will consider highly relativistic neutrinos, as is the case for the experiments
under discussion. We will assume that there are no degeneracies among the
masses m j . The states in the superpositions (21.111) and (21.112) will all
have, in general, different energies and momenta E i , p i . We shall also treat
the evolution as occurring in one spatial dimension, taking all the momenta
to lie in the direction from the source to the detector. Note that the fractional
deviation of E i and p i from the massless case E = p is of order m
2 /E
2 which
will be extremely small, of order one part in 10
16 , say.
Suppose now that the neutrinos of flavour ν α started in the state (21.111)
at time t = 0 in the detector frame are detected at time T after production,
having travelled a distance L. Then the amplitude for finding a neutrino of
flavour ν β at (L, T ) is
∑
A(ν α → ν β ; L, E) =
U αi
∗ e
−iEiT +ipiL
‹ν β |ν i , p i ›
∑
i
=
U
∗
αi U βi e
−iEiT +ipiL .
(21.113)
i
We make two immediate comments on (21.113). First, the Majorana phases
in (21.54) cancel in A(ν α → ν β ; L, E) = δ αβ , since the same phase appears
2 We shall not indicate the chirality explicitly from now on, it being assumed that we are
referring to the L (R) component for neutrinos (antineutrinos).
21. CP Violation and Oscillation Phenomena
from the ν α source the probability P (ν α → ν β ; E, L) of detecting a neutrino of
a different flavour ν β is non-zero.
2 Such a flavour change will of course imply
that the ν α survival probability, P (ν α → ν α ; E, L), is less than 1. We shall
give a simplified version of the derivation of such probabilities, following the
approach of the review by Nakamura and Petcov in section 13 of Nakamura
et al. (2010). This review includes a large list of references to the timedependent formalism; we mention here the contributions of Kayser (1981),
Nauenberg (1999) and Cohen et al. (2009). We shall treat all the neutrinos
as stable particles.
We consider the evolution of the state |ν α › in the frame in which the
detector which measures its flavour is at rest (the lab frame). As in the meson
case, the states with simple space-time evolution in a vacuum are the mass
eigenstates |ν i › (i = 1, 2, 3), a superposition of which is equal to |ν α ›:
∑
|ν α › =
U
∗
αi
j
|ν i , p i ›,
(21.111)
the complex conjugation arising from taking the dagger of the relation (21.106)
for the field operators. Here U stands for either the Dirac or the Majorana
matrix, and p i is the 4-momentum of ν i . Similarly,
∑
|ν ¯ α › =
U αi |ν ¯ i , p i
i
›.
(21.112)
We will consider highly relativistic neutrinos, as is the case for the experiments
under discussion. We will assume that there are no degeneracies among the
masses m j . The states in the superpositions (21.111) and (21.112) will all
have, in general, different energies and momenta E i , p i . We shall also treat
the evolution as occurring in one spatial dimension, taking all the momenta
to lie in the direction from the source to the detector. Note that the fractional
deviation of E i and p i from the massless case E = p is of order m
2 /E
2 which
will be extremely small, of order one part in 10
16 , say.
Suppose now that the neutrinos of flavour ν α started in the state (21.111)
at time t = 0 in the detector frame are detected at time T after production,
having travelled a distance L. Then the amplitude for finding a neutrino of
flavour ν β at (L, T ) is
∑
A(ν α → ν β ; L, E) =
U αi
∗ e
−iEiT +ipiL
‹ν β |ν i , p i ›
∑
i
=
U
∗
αi U βi e
−iEiT +ipiL .
(21.113)
i
We make two immediate comments on (21.113). First, the Majorana phases
in (21.54) cancel in A(ν α → ν β ; L, E) = δ αβ , since the same phase appears
2 We shall not indicate the chirality explicitly from now on, it being assumed that we are
referring to the L (R) component for neutrinos (antineutrinos).