196 Dark matter and particle physics
of the electron-neutrino) requires an exceedingly small neutrino Yukawa coupling
ofO( 10 −^11 )or so. Although quite economical, we do not consider this option
particularly satisfactory.
(4) The other possibility is to link the presence of neutrino masses to the
violation ofL. In this case one introduces a new mass scale, in addition to the
electroweak Fermi scale, into the problem. Indeed, lepton number can be violated
at a very high or a very low mass scale. The former choice represents, in our view,
the most satisfactory way to have massive neutrinos with a very small mass. The
idea (see-saw mechanism [19, 20]) is to introduce a right-handed neutrino into the
fermion mass spectrum with a Majorana massMmuch larger thanMW. Indeed,
being the right-handed neutrino, a singlet under the electroweak symmetry group,
its mass is not chirally protected. The simultaneous presence of a very large
chirally unprotected Majorana mass for the right-handed component together with
a ‘regular’ Dirac mass term (which can be at most ofO(100 GeV)gives rise to
two Majorana eigenstates with masses very far apart.
The Lagrangian for neutrino masses is given by
Lmass=−1
2
(νLN
c
L)(
0 mD
mD M)(
νcR
NR)
+h.c. (5.28)whereνcRis theCP-conjugated ofνLandNcLofNR. It holds thatmDM.
Diagonalizing the mass matrix we find two Majorana eigenstatesn 1 andn 2 with
masses very far apart:
m 1 m^2 D
M, m 2 M.The light eigenstaten 1 is mainly in theνLdirection and is the neutrino that we
‘observe’ experimentally while the heavy onen 2 is in theNRone. The key point
is that the smallness of its mass (in comparison with all the other fermion masses
in the SM) finds a ‘natural’ explanation in the appearance of a new, large mass
scale whereLis violated explicitly (by two units) in the right-handed neutrino
mass term.
5.4.3 Thermal history of neutrinos
Let us consider a stable massive neutrino (of mass less than 1 MeV) (see for
example [5]). If its mass is less than 10−^4 eV it is still relativistic today and
its contribution toMis negligible. In the opposite case it is non-relativistic
and its contribution to the energy density of the universe is simply given by its
number density multiplied by its mass. The number density is determined by the
temperature at which the neutrino decouples and, hence, by the strength of the
weak interactions. Neutrinos decouple when their mean free path exceeds the
horizon size or equivalently<H. Using natural units (c=h ̄=1), we have
that
∼σνne±∼G^2 FT^5 (5.29)