Physical Foundations of Cosmology

214 The very early universe

is consistent with the gauge symmetries of the theory; here theνRcare the charge-
conjugated wave functions of the right-handed neutrinos. After symmetry breaking
the fieldχacquires the expectation valueχ 0 and the Yukawa term induces the Dirac
masses described by the mass matrix(mD)ij=fij(ν)χ 0 .Considering for simplicity
the case of one generation, we can write the total mass term as

L(ν)=−

1

2

(

ν ̄Lν ̄cR

)( 0 mD

mD M

)(

νcL

νR

)

+h.c., (4.226)

taking into account that ̄νLνR=ν ̄cRνcL.The mass matrix in (4.226) is not diagonal.
WhenmDM,the mass eigenvalues

mν−

m^2 D

M

, mNM (4.227)

correspond to the eigenstates

ννL+νLc, NνR+νcR, (4.228)

which describe the Majorana fermions (ν=νc,N=Nc)−light and heavy neutri-
nos respectively. Choosing formDthe largest known fermion mass of order the top
quark mass,mD∼mt∼170 GeV,and takingM 3 × 1015 GeV,we find from
(4.227) thatmν 10 −^2 eV, which is favored by neutrino oscillation measurements.
IfmD∼me∼ 0 .5 MeV,then the mass of the heavy neutrino should be 2 × 106
GeV.It is important to note that the Majorana mass terms are not generated via
the Higgs mechanism, and therefore can be much larger than the masses of ordi-
nary quarks and leptons. This leads to light neutrino masses that are very small,
according to (4.227 ). Such a method of obtaining very small masses is known as
theseesaw mechanism. If one were restricted to having only Dirac mass terms, the
Yukawa couplings would have to be unnaturally small.
The Majorana mass terms violate lepton number by two units. The heavy Ma-
jorana neutrinosN=Ncare coupled to the Higgs particles via (4.224) and they
can decay into a lepton–Higgs pair,N→lφ,or into theCPconjugated state,
N→ ̄lφ, ̄ thus violating lepton number (Figure 4.22). Returning to the case of
three generations, we see that the neutrino mass eigenstates do not necessarily

NN

l

φ

l

φ

Fig. 4.22.