22.5. Fermion masses
407
to the Higgs field, of the form (22.133). Then the Yukawa and the mass terms
ν ˆ R are
¯
1
L Y,R = −
ˆ ˆ
¯ ˆ
g (e eL φ C ν ˆ R + ν ˆ R φ
† ˆ
R
C e eL ) − m R [(ˆ ν R ) C ν ˆ R + h.c.].
(22.144)
2
Then, in the Higgs vacuum the first term in (22.144) becomes
−
¯
¯
m D (ν ˆ eL ν ˆ R + ν ˆ R ν ˆ eL )
(22.145)
where m D = g R v/
√
2. The term (22.145) couples the fields ν ˆ R and ν ˆ eL , so
that we need to do a diagonalization to find the true mass eigenvalues and
eigenstates. The combined mass terms from (22.144) and (22.145) can be
written as
1
−
ˆ
ˆ
(N L ) C M N L + h.c.
(22.146)
2
where
(
)
(
)
ν ˆ
0
m
ˆ
N L ≡
eL
, M =
D
.
(22.147)
(ˆ ν R ) C
m D m R
CP invariance would imply that the parameters m R and m D are real, as we
will assume, for simplicity.
Suppose now that m D ≪ m R . Then the eigenvalues of M are approximately
m 1 ≈ m R , m 2 ≈ −m
2
D /m R .
(22.148)
The apparently troubling minus sign can be absorbed into the mixing parameters. Thus one eigenvalue is (by assumption) very large compared to m D ,
and one is very much smaller. The vanishing of the first element in M ensures
that the lepton number violating term (22.137) is characterized by a large
mass scale m R . It may be natural to assume that m D is a ‘typical’ quark or
lepton mass term, which would then imply that m 2 of (22.148) is very much
lighter than that – as appears to be true for the neutrinos. This is the famous
‘see-saw’ mechanism of Minkowski (1977), Gell-Mann et al. (1979), Yanagida
(1979) and Mohapatra and Senjanovic (1980, 1981). If in fact m R ∼ 10
16 Gev,
we recover an estimate for m 2 which is similar to that in (22.143). It is worth
emphasizing that the Majorana nature of the massive neutrinos is an essential
part of the see-saw mechanism.
These considerations are tending to take us ‘beyond the Standard Model’,
so we shall not pursue them at any greater length. Instead, we must now
generalize the discussion of fermion masses to the three-generation case.
22.5.2 Three-generation mixing
We introduce three doublets of left-handed quark fields
(
)
(
)
(
)
u ˆ L1
u ˆ L2
u ˆ L3
q ˆ L1 =
,
q ˆ L2 =
,
q ˆ
ˆ
L3 =
(22.149)
ˆ
ˆ
d L1
d L2
d L3
407
to the Higgs field, of the form (22.133). Then the Yukawa and the mass terms
ν ˆ R are
¯
1
L Y,R = −
ˆ ˆ
¯ ˆ
g (e eL φ C ν ˆ R + ν ˆ R φ
† ˆ
R
C e eL ) − m R [(ˆ ν R ) C ν ˆ R + h.c.].
(22.144)
2
Then, in the Higgs vacuum the first term in (22.144) becomes
−
¯
¯
m D (ν ˆ eL ν ˆ R + ν ˆ R ν ˆ eL )
(22.145)
where m D = g R v/
√
2. The term (22.145) couples the fields ν ˆ R and ν ˆ eL , so
that we need to do a diagonalization to find the true mass eigenvalues and
eigenstates. The combined mass terms from (22.144) and (22.145) can be
written as
1
−
ˆ
ˆ
(N L ) C M N L + h.c.
(22.146)
2
where
(
)
(
)
ν ˆ
0
m
ˆ
N L ≡
eL
, M =
D
.
(22.147)
(ˆ ν R ) C
m D m R
CP invariance would imply that the parameters m R and m D are real, as we
will assume, for simplicity.
Suppose now that m D ≪ m R . Then the eigenvalues of M are approximately
m 1 ≈ m R , m 2 ≈ −m
2
D /m R .
(22.148)
The apparently troubling minus sign can be absorbed into the mixing parameters. Thus one eigenvalue is (by assumption) very large compared to m D ,
and one is very much smaller. The vanishing of the first element in M ensures
that the lepton number violating term (22.137) is characterized by a large
mass scale m R . It may be natural to assume that m D is a ‘typical’ quark or
lepton mass term, which would then imply that m 2 of (22.148) is very much
lighter than that – as appears to be true for the neutrinos. This is the famous
‘see-saw’ mechanism of Minkowski (1977), Gell-Mann et al. (1979), Yanagida
(1979) and Mohapatra and Senjanovic (1980, 1981). If in fact m R ∼ 10
16 Gev,
we recover an estimate for m 2 which is similar to that in (22.143). It is worth
emphasizing that the Majorana nature of the massive neutrinos is an essential
part of the see-saw mechanism.
These considerations are tending to take us ‘beyond the Standard Model’,
so we shall not pursue them at any greater length. Instead, we must now
generalize the discussion of fermion masses to the three-generation case.
22.5.2 Three-generation mixing
We introduce three doublets of left-handed quark fields
(
)
(
)
(
)
u ˆ L1
u ˆ L2
u ˆ L3
q ˆ L1 =
,
q ˆ L2 =
,
q ˆ
ˆ
L3 =
(22.149)
ˆ
ˆ
d L1
d L2
d L3