54
G. Altarelli and S. Forte
3.7 Neutrino Masses
In the minimal version of the SM the right handed neutrinos ν iR , which have no
gauge interactions, are not present at all. With no ν R no Dirac mass is possible
for neutrinos. If lepton number conservation is also imposed, then no Majorana
mass is allowed either and, as a consequence, all neutrinos are massless. But, at
present, from neutrino oscillation experiments (see Chapter 11 of the present work),
we know that at least 2 out of the 3 known neutrinos have non vanishing masses:
the two mass squared differences measured from solar (m 2
12 ) and atmospheric
oscillations (m 2
23 ) are given by m 2
12 ∼ 8 10 −5 eV 2 and m 2
23 ∼ 2.5 10 −3
[21]. The absolute values of the masses are very small, with an upper limit of a
fraction of eV , obtained from laboratory experiments (tritium β decay near the end
point: m ν 2 eV [5], absence of visible neutrinoless double β decay : |m ee |
0.3−0.7 eV (m ee is a combination of neutrino masses; for a review, see, for example
[22]) and from cosmological observations: m ν 0.1 − 0.7 eV (depending on the
cosmological model assumptions) [23]. If ν iR are added to the minimal model and
lepton number is imposed by hand, then neutrino masses would in general appear as
Dirac masses, generated by the Higgs mechanism, like for any other fermion. But,
for Dirac neutrinos, to explain the extreme smallness of neutrino masses, one should
allow for very small Yukawa couplings. However, we stress that, in the SM, baryon
B and lepton L number conservation, which are not guaranteed by gauge symmetries
(as is the case for the electric charge Q), are understood as “accidental” symmetries,
due to the fact that, out of the SM fields, it is not possible to construct gauge invariant
operators which are renormalizable (i.e. of operator dimension d ≤ 4) and violate
B and/or L. In fact the SM lagrangian should contain all terms allowed by gauge
symmetry and renormalizability. The most general renormalizable lagrangian, built
from the SM fields, compatible with the SM gauge symmetry, in absence of ν iR , is
automatically B and L conserving. But in presence of ν iR , this is no more true and
the right handed Majorana mass term is allowed:
M RR = ¯
ν
c
iR M ij ν jR = ν
T
iR CM ij ν jR ,
(3.80)
where ν
c
iR = C ¯
ν T
iR is the charge conjugated neutrino field and C is the charge
conjugation matrix in Dirac spinor space. The Majorana mass term is an operator
of dimension d = 3 with L = 2. Since the ν iR are gauge singlets the Majorana
mass M RR is fully allowed by the gauge symmetry and a coupling with the Higgs is
not needed to generate this type of mass. As a consequence, the entries of the mass
matrix M ij do not need to be of the order of the EW symmetry breaking scale v and
could be much larger. If one starts from the Dirac and RR Majorana mass terms for
neutrinos, the resulting mass matrix, in the L, R space, has the form:
m ν =
0 m D
m D M
(3.81)
G. Altarelli and S. Forte
3.7 Neutrino Masses
In the minimal version of the SM the right handed neutrinos ν iR , which have no
gauge interactions, are not present at all. With no ν R no Dirac mass is possible
for neutrinos. If lepton number conservation is also imposed, then no Majorana
mass is allowed either and, as a consequence, all neutrinos are massless. But, at
present, from neutrino oscillation experiments (see Chapter 11 of the present work),
we know that at least 2 out of the 3 known neutrinos have non vanishing masses:
the two mass squared differences measured from solar (m 2
12 ) and atmospheric
oscillations (m 2
23 ) are given by m 2
12 ∼ 8 10 −5 eV 2 and m 2
23 ∼ 2.5 10 −3
[21]. The absolute values of the masses are very small, with an upper limit of a
fraction of eV , obtained from laboratory experiments (tritium β decay near the end
point: m ν 2 eV [5], absence of visible neutrinoless double β decay : |m ee |
0.3−0.7 eV (m ee is a combination of neutrino masses; for a review, see, for example
[22]) and from cosmological observations: m ν 0.1 − 0.7 eV (depending on the
cosmological model assumptions) [23]. If ν iR are added to the minimal model and
lepton number is imposed by hand, then neutrino masses would in general appear as
Dirac masses, generated by the Higgs mechanism, like for any other fermion. But,
for Dirac neutrinos, to explain the extreme smallness of neutrino masses, one should
allow for very small Yukawa couplings. However, we stress that, in the SM, baryon
B and lepton L number conservation, which are not guaranteed by gauge symmetries
(as is the case for the electric charge Q), are understood as “accidental” symmetries,
due to the fact that, out of the SM fields, it is not possible to construct gauge invariant
operators which are renormalizable (i.e. of operator dimension d ≤ 4) and violate
B and/or L. In fact the SM lagrangian should contain all terms allowed by gauge
symmetry and renormalizability. The most general renormalizable lagrangian, built
from the SM fields, compatible with the SM gauge symmetry, in absence of ν iR , is
automatically B and L conserving. But in presence of ν iR , this is no more true and
the right handed Majorana mass term is allowed:
M RR = ¯
ν
c
iR M ij ν jR = ν
T
iR CM ij ν jR ,
(3.80)
where ν
c
iR = C ¯
ν T
iR is the charge conjugated neutrino field and C is the charge
conjugation matrix in Dirac spinor space. The Majorana mass term is an operator
of dimension d = 3 with L = 2. Since the ν iR are gauge singlets the Majorana
mass M RR is fully allowed by the gauge symmetry and a coupling with the Higgs is
not needed to generate this type of mass. As a consequence, the entries of the mass
matrix M ij do not need to be of the order of the EW symmetry breaking scale v and
could be much larger. If one starts from the Dirac and RR Majorana mass terms for
neutrinos, the resulting mass matrix, in the L, R space, has the form:
m ν =
0 m D
m D M
(3.81)