352
21. CP Violation and Oscillation Phenomena
for the charged leptons, and also for Dirac neutrinos, since evidently the mass
′
term (21.104) is invariant under a global U(1) transformation ψ ˆ = exp(iθ)ψ ˆ .
Hence the matrix U will, in this Dirac case, have a parametrization of the
CKM form, with one CP-violating phase.
The mixing described by (21.106) implies that the individual lepton flavour
numbers L e , L μ , L τ are no longer conserved. However, since we are here taking
the neutrinos to be Dirac particles, there will be a quantum number carried by
ν e , ν μ and ν τ which is conserved by the interactions. This could, for example,
be the total lepton number L e + L μ + L τ , assigning L(ν α ) = 1 for α = e, μ, τ,
′
which would follow from invariance under the global U(1) transformation e ˆ =
α
′
exp(iδ)e ˆ α , ν ˆ = exp(iδ)ˆ ν α , where δ is independent of the flavour α.
α
This ‘Dirac’ option, though simple, may be thought uneconomical, however. As noted, the R components of neutrino fields have no interactions of
Standard Model type. The charged leptons do have electromagnetic interactions, of course, as do the quarks, which also have strong interactions. But
the neutral neutrinos only have weak interactions, which involve only their
L-components. Why, then, enlarge the field content to include hypothetical
ν ˆ R fields, which don’t have any SM interactions? It seems more economical to
make do with only the ˆ
ν L fields. In this case, the Dirac mass term (21.104) is
not possible, but a Majorana mass term (21.105) can still exist. Clearly, such
a mass term is not invariant under U(1) global phase transformations, and it
breaks lepton number conservation explicitly. As in the Dirac case, the chiral
L component will include a ‘wrong’ (i.e. positive) helicity component with an
amplitude proportional to m/E.
The fact that global phase changes on the neutrino fields are now no longer
freely available, because that symmetry is lost if they are Majorana fields, has
implications for the mixing matrix, call it U M , in this case. Since the three
Majorana neutrino fields can no longer absorb phases, we have only the three
phases from the charged leptons at our disposal, which leaves three phase
parameters in U M , after rephasing. The PMNS matrix in the Majorana case
therefore has two more irreducible phase parameters than the CKM matrix,
and is conventionally parametrized as
iα21/2 iα31/2 ).
U M = U(CKM − type) × diag.(1, e
, e
(21.107)
There are three CP-violating phases in the Majorana neutrino case.
The only information at present (2012) concerning the entries in U comes
from neutrino oscillation experiments, which we shall discuss in the next sections. We shall see that the Majorana phases α 21 and α 31 cancel in the
probabilities calculated for neutrino transitions, and no experiment so far is
sensitive to CP-violating effects in the neutrino sector. We shall discuss how
the values of the parameters θ 12 , θ 13 and θ 23 can be inferred from the observed
oscillations, and also the differences in the squared masses of the neutrinos.
Anticipating these results, we state here that the two independent squared
2
2
2
2
mass differences, m 2 − m 1 and m 3 − m 1 , turn out to be very small indeed, and
rather different from each other: namely approximately 7.6 × 10
−5 eV
2 and
21. CP Violation and Oscillation Phenomena
for the charged leptons, and also for Dirac neutrinos, since evidently the mass
′
term (21.104) is invariant under a global U(1) transformation ψ ˆ = exp(iθ)ψ ˆ .
Hence the matrix U will, in this Dirac case, have a parametrization of the
CKM form, with one CP-violating phase.
The mixing described by (21.106) implies that the individual lepton flavour
numbers L e , L μ , L τ are no longer conserved. However, since we are here taking
the neutrinos to be Dirac particles, there will be a quantum number carried by
ν e , ν μ and ν τ which is conserved by the interactions. This could, for example,
be the total lepton number L e + L μ + L τ , assigning L(ν α ) = 1 for α = e, μ, τ,
′
which would follow from invariance under the global U(1) transformation e ˆ =
α
′
exp(iδ)e ˆ α , ν ˆ = exp(iδ)ˆ ν α , where δ is independent of the flavour α.
α
This ‘Dirac’ option, though simple, may be thought uneconomical, however. As noted, the R components of neutrino fields have no interactions of
Standard Model type. The charged leptons do have electromagnetic interactions, of course, as do the quarks, which also have strong interactions. But
the neutral neutrinos only have weak interactions, which involve only their
L-components. Why, then, enlarge the field content to include hypothetical
ν ˆ R fields, which don’t have any SM interactions? It seems more economical to
make do with only the ˆ
ν L fields. In this case, the Dirac mass term (21.104) is
not possible, but a Majorana mass term (21.105) can still exist. Clearly, such
a mass term is not invariant under U(1) global phase transformations, and it
breaks lepton number conservation explicitly. As in the Dirac case, the chiral
L component will include a ‘wrong’ (i.e. positive) helicity component with an
amplitude proportional to m/E.
The fact that global phase changes on the neutrino fields are now no longer
freely available, because that symmetry is lost if they are Majorana fields, has
implications for the mixing matrix, call it U M , in this case. Since the three
Majorana neutrino fields can no longer absorb phases, we have only the three
phases from the charged leptons at our disposal, which leaves three phase
parameters in U M , after rephasing. The PMNS matrix in the Majorana case
therefore has two more irreducible phase parameters than the CKM matrix,
and is conventionally parametrized as
iα21/2 iα31/2 ).
U M = U(CKM − type) × diag.(1, e
, e
(21.107)
There are three CP-violating phases in the Majorana neutrino case.
The only information at present (2012) concerning the entries in U comes
from neutrino oscillation experiments, which we shall discuss in the next sections. We shall see that the Majorana phases α 21 and α 31 cancel in the
probabilities calculated for neutrino transitions, and no experiment so far is
sensitive to CP-violating effects in the neutrino sector. We shall discuss how
the values of the parameters θ 12 , θ 13 and θ 23 can be inferred from the observed
oscillations, and also the differences in the squared masses of the neutrinos.
Anticipating these results, we state here that the two independent squared
2
2
2
2
mass differences, m 2 − m 1 and m 3 − m 1 , turn out to be very small indeed, and
rather different from each other: namely approximately 7.6 × 10
−5 eV
2 and