3 The Standard Model of Electroweak Interactions
55
where m D and M are the Dirac and Majorana mass matrices (M is the matrix M ij
in Eq. (3.80)). The corresponding eigenvalues are three very heavy neutrinos with
masses of order M and three light neutrinos with masses
m ν = −m
T
D M
−1 m D ,
(3.82)
which are possibly very small if M is large enough. This is the see-saw mechanism
for neutrino masses [24]. Note that if no ν iR exist a Majorana mass term could
still be built out of ν jL . But ν jL have weak isospin 1/2, being part of the left
handed lepton doublet l. Thus, the left handed Majorana mass term has total weak
isospin equal to one and needs two Higgs fields to make a gauge invariant term. The
resulting mass term:
O 5 = λl
T
i λ ij l j H H /M ,
(3.83)
with M a large scale (apriori comparable to the scale of M RR ) and λ a dimensionless
coupling generically of o(1), is a non renormalizable operator of dimension 5. The
corresponding mass terms are of the order m ν ∼ λv 2 /M, hence of the same generic
order of the light neutrino masses from Eq. (3.82).
In conclusion, neutrino masses are believed to be small because neutrinos are
Majorana particles with masses inversely proportional to the large scale M of energy
where L non conservation is induced. It is interesting that the observed magnitudes
of the mass squared splittings of neutrinos are well compatible with a scale M
remarkably close to the Grand Unification scale, where in fact L non conservation
is naturally expected.
In the previous Section we have discussed flavour mixing for quarks. But, clearly,
given that non vanishing neutrino masses have been established, a similar mixing
matrix is also introduced in the leptonic sector, but will not be discussed here (see
Chapter 11).
3.8 Renormalization of the Electroweak Theory
The Higgs mechanism gives masses to the Z, the W ± and to fermions while the
lagrangian density is still symmetric. In particular the gauge Ward identities and the
symmetric form of the gauge currents are preserved. The validity of these relations
is an essential ingredient for renormalizability. In the previous Sections we have
specified the Feynman vertices in the “unitary” gauge where only physical particles
appear. However, as discussed in Chap. 2, in this gauge the massive gauge boson
propagator would have a bad ultraviolet behaviour:
W μν =
−g μν +
q μ q ν
m 2
W
q 2 − m 2
W
.
(3.84)
55
where m D and M are the Dirac and Majorana mass matrices (M is the matrix M ij
in Eq. (3.80)). The corresponding eigenvalues are three very heavy neutrinos with
masses of order M and three light neutrinos with masses
m ν = −m
T
D M
−1 m D ,
(3.82)
which are possibly very small if M is large enough. This is the see-saw mechanism
for neutrino masses [24]. Note that if no ν iR exist a Majorana mass term could
still be built out of ν jL . But ν jL have weak isospin 1/2, being part of the left
handed lepton doublet l. Thus, the left handed Majorana mass term has total weak
isospin equal to one and needs two Higgs fields to make a gauge invariant term. The
resulting mass term:
O 5 = λl
T
i λ ij l j H H /M ,
(3.83)
with M a large scale (apriori comparable to the scale of M RR ) and λ a dimensionless
coupling generically of o(1), is a non renormalizable operator of dimension 5. The
corresponding mass terms are of the order m ν ∼ λv 2 /M, hence of the same generic
order of the light neutrino masses from Eq. (3.82).
In conclusion, neutrino masses are believed to be small because neutrinos are
Majorana particles with masses inversely proportional to the large scale M of energy
where L non conservation is induced. It is interesting that the observed magnitudes
of the mass squared splittings of neutrinos are well compatible with a scale M
remarkably close to the Grand Unification scale, where in fact L non conservation
is naturally expected.
In the previous Section we have discussed flavour mixing for quarks. But, clearly,
given that non vanishing neutrino masses have been established, a similar mixing
matrix is also introduced in the leptonic sector, but will not be discussed here (see
Chapter 11).
3.8 Renormalization of the Electroweak Theory
The Higgs mechanism gives masses to the Z, the W ± and to fermions while the
lagrangian density is still symmetric. In particular the gauge Ward identities and the
symmetric form of the gauge currents are preserved. The validity of these relations
is an essential ingredient for renormalizability. In the previous Sections we have
specified the Feynman vertices in the “unitary” gauge where only physical particles
appear. However, as discussed in Chap. 2, in this gauge the massive gauge boson
propagator would have a bad ultraviolet behaviour:
W μν =
−g μν +
q μ q ν
m 2
W
q 2 − m 2
W
.
(3.84)