21.4. Neutrino mixing and oscillations
351
topic, which is a highly active field of research in particle physics; there are
analogies with the meson oscillations we have been considering.
It is fair to say that in the original Standard Model the neutrinos were
taken to be massless, but there was no compelling theoretical reason for this,
and the framework of the Standard Model can easily be extended to include
massive neutrinos. However, one question immediately arises: are neutrinos
Dirac or Majorana fermions? As explained in section 20.3, we do not yet know
the answer, and it may be some time before we do. The way the mass terms
enter the Lagrangian is, in fact, different in the two cases. We are familiar
with the Dirac mass term
¯
¯
¯
ˆ ˆ
ˆ
mψψ = m(ψ ˆ
ˆ ˆ
R ψ L + ψ L ψ R ),
(21.104)
ˆ
where ψ is a four-component Dirac field, and R and L refer to the chirality
components. We learned in section 7.5.2 that a Majorana mass term can be
written in the form
mχ ˆ
T
L iσ 2 χ ˆ L + h.c.
(21.105)
where χ ˆ L is a two-component field of L chirality. A similar expression could
be written using a two-component R-chirality field. The difference in form
between the Dirac and Majorana mass terms leads to a difference in the
parametrization of neutrino mixing, as we shall see.
Suppose, first, that the neutrinos are Dirac particles, with both L and R
chiralities (or equivalently either helicity) for a given mass. We remind the
reader that this is not ruled out experimentally, since the non-observation of
the ‘wrong’ helicity component may be accounted for by the appearance of a
suppression factor (m/E), where m is a neutrino mass and E is an average
neutrino energy (see section 20.2.2). We also assume that their interactions
have the V-A structure indicated by the phenomenology of the previous chapter. Then only the L (R) chirality component of a neutrino (antineutrino) field
feels the weak force; the R (L) component of a neutrino (antineutrino) field
has no interactions of Standard Model type. But, just as in the quark case, it
will in general be necessary to allow for the possibility that the L-components
of the fields which have definite neutrino mass, call them ν ˆ 1L , ν ˆ 2L , ν ˆ 3L , are not
the same as the fields ν ˆ eL , ν ˆ μL , ν ˆ τ L which enter into the charged current V-A
interaction. For Dirac neutrinos, we therefore write
(
)
(
) (
)
(
)
ν ˆ e
U e1 U e2 U e3
ν ˆ 1
ν ˆ 1
( ν ˆ μ ) = ( U μ1 U μ2 U μ3 ) ( ν ˆ 2 ) ≡ U ( ν ˆ 2 ) ,
(21.106)
ν ˆ τ
U τ 1 U τ 2 U τ 3
ν ˆ 3
ν ˆ 3
L
L
L
where the unitary matrix U is the PMNS matrix, named after Pontecorvo
(1957, 1958, 1967), and Maki, Nakagawa and Sakata (1962).
Now we showed in section 20.7.3 that the general 3 × 3 unitary matrix has
three real (rotation angle) parameters, and 6 phase parameters, five of which
we could get rid of by rephasing the quark fields by global U(1) transformations
of the form q ˆ
′ = exp(iθ)ˆ q. Such rephasing transformations are equally allowed
351
topic, which is a highly active field of research in particle physics; there are
analogies with the meson oscillations we have been considering.
It is fair to say that in the original Standard Model the neutrinos were
taken to be massless, but there was no compelling theoretical reason for this,
and the framework of the Standard Model can easily be extended to include
massive neutrinos. However, one question immediately arises: are neutrinos
Dirac or Majorana fermions? As explained in section 20.3, we do not yet know
the answer, and it may be some time before we do. The way the mass terms
enter the Lagrangian is, in fact, different in the two cases. We are familiar
with the Dirac mass term
¯
¯
¯
ˆ ˆ
ˆ
mψψ = m(ψ ˆ
ˆ ˆ
R ψ L + ψ L ψ R ),
(21.104)
ˆ
where ψ is a four-component Dirac field, and R and L refer to the chirality
components. We learned in section 7.5.2 that a Majorana mass term can be
written in the form
mχ ˆ
T
L iσ 2 χ ˆ L + h.c.
(21.105)
where χ ˆ L is a two-component field of L chirality. A similar expression could
be written using a two-component R-chirality field. The difference in form
between the Dirac and Majorana mass terms leads to a difference in the
parametrization of neutrino mixing, as we shall see.
Suppose, first, that the neutrinos are Dirac particles, with both L and R
chiralities (or equivalently either helicity) for a given mass. We remind the
reader that this is not ruled out experimentally, since the non-observation of
the ‘wrong’ helicity component may be accounted for by the appearance of a
suppression factor (m/E), where m is a neutrino mass and E is an average
neutrino energy (see section 20.2.2). We also assume that their interactions
have the V-A structure indicated by the phenomenology of the previous chapter. Then only the L (R) chirality component of a neutrino (antineutrino) field
feels the weak force; the R (L) component of a neutrino (antineutrino) field
has no interactions of Standard Model type. But, just as in the quark case, it
will in general be necessary to allow for the possibility that the L-components
of the fields which have definite neutrino mass, call them ν ˆ 1L , ν ˆ 2L , ν ˆ 3L , are not
the same as the fields ν ˆ eL , ν ˆ μL , ν ˆ τ L which enter into the charged current V-A
interaction. For Dirac neutrinos, we therefore write
(
)
(
) (
)
(
)
ν ˆ e
U e1 U e2 U e3
ν ˆ 1
ν ˆ 1
( ν ˆ μ ) = ( U μ1 U μ2 U μ3 ) ( ν ˆ 2 ) ≡ U ( ν ˆ 2 ) ,
(21.106)
ν ˆ τ
U τ 1 U τ 2 U τ 3
ν ˆ 3
ν ˆ 3
L
L
L
where the unitary matrix U is the PMNS matrix, named after Pontecorvo
(1957, 1958, 1967), and Maki, Nakagawa and Sakata (1962).
Now we showed in section 20.7.3 that the general 3 × 3 unitary matrix has
three real (rotation angle) parameters, and 6 phase parameters, five of which
we could get rid of by rephasing the quark fields by global U(1) transformations
of the form q ˆ
′ = exp(iθ)ˆ q. Such rephasing transformations are equally allowed