4.6 Beyond the Standard Model 213theory and the resulting baryon asymmetry is many orders of magnitude smaller
than the required 10−^10 .Another unhappy feature of theSU( 5 )model is thatB−L
is conserved in this theory. In this caseB−L=0evenifB=0 and any baryon
asymmetry generated will be washed out in subsequent topological transitions. For
successful baryogenesis we actually need to generate nonvanishingB−L.In more
complex models both these obstacles can, in principle, be overcome. For example,
in theSO( 10 )model, neitherBnorB−Lis conserved, and the necessaryεcan
be arranged.
In reality the situation with Grand Unified Theory baryogenesis is more com-
plicated than described above. We will see later that inflation most likely ends at
energy scales below the Grand Unified Theory scale and hence that relativisticX
bosons were never in thermal equilibrium. However, one can produce them (out of
equilibrium) during the preheating phase (see Section 5.5).
Baryogenesis via leptogenesis Baryon asymmetry can also be generated via lep-
togenesis. What is required is a nonvanishing initial value of(B−L)i.Even if
Bi=0 andLi= 0 ,then lepton number will be partially converted to baryon num-
ber in subsequent topological transitions. SinceB+aLvanishes in these transitions
whileB−Lis conserved, the final baryon number is
Bf=−
a
1 +aLi, (4.223)wherea= 28 /51 in the Standard Model with one Higgs doublet. In turn, the initial
nonvanishing lepton numberLican be generated in out-of-equilibrium decay of
heavy neutrinos.
Let us briefly discuss the motivation for the existence of such heavy neutrinos.
The neutrino oscillations measured can be explained only if the neutrinos have non-
vanishing masses. To generate the neutrino masses we need right-handed neutrinos
νR.Then the Yukawa coupling term generating the Dirac masses can be written as
in (4.82):
L(Yν)=−fij(ν)χL ̄iLφ 1 νRj+h.c.=−fij(ν)χν ̄iLνRj+h.c., (4.224)wherei= 1 , 2 ,3 is the lepton generation index andνL are theSU( 2 )gauge-
invariant left-handed neutrinos defined in Section 4.3.4. Under theU( 1 )group,
νLtransforms according to (4.72) and, becauseY(Lν)= 1 / 2 ,it remains invariant;
hence the Yukawa term (4.224) is gauge-invariant only if the hypercharges of the
right-handed neutrinos are equal to zero. The right-handed neutrinos areSU( 2 )
singlets, have no color and do not carry hypercharge. Therefore a Majorana mass
term,
L(Mν)=−^12 Mij(
ν ̄cR)i
νRj, (4.225)