162 The very early universe
respectively,wherei= 1 , 2 ,3 is the generation index. The lower components are
linear superpositions of the appropriate flavors,
d′i=Vjidj, (4.81)whereVij is the unitary 3× 3 Kobayashi–Maskawa matrix. The general quark
Yukawa term can then be written as
LqY=−fijdχQ ̄iLφ 0 dR′j−fijuχQ ̄iLφ 1 uRj+h.c., (4.82)where
QiL=(
ui
d′i)
, φ 1 =(
1
0
)
. (4.83)
The second term on the right hand side in (4.82) is also gauge-invariant and generates
the masses of the upper components of the doublets. Expression (4.82), rewritten
in terms of the original flavors, gives the mass term
Lmq=−(
Vmi∗fijdVkj)
χd ̄LmdRk−fijuχu ̄iLujR+h.c. (4.84)Taking the matricesfijdandfijusuch that
(
Vmi∗fijdVkj
)
χ 0 =mdkδmk (4.85)and
fijuχ 0 =muiδij, (4.86)we get the usual quark mass terms. The Yukawa coupling constant is largest for the
top quark,ft 0 .7(mt170 GeV).For the other quarks it is more than 10 times
smaller.
Neutrino masses, which – according to measurements – are different from zero,
can be generated in an almost identical manner. In this case, the neutrino flavors
naturally mix in a similar way to the quark generations and this leads to neutrino
oscillations.
4.3.6 CP violation
The parity operationPcorresponds to reflection(t,x,y,z)→(t,−x,−y,−z)and
converts left-handed particles into right-handed particles without changing their
other properties. Charge conjugationCreplaces particles by antiparticles without
changing handedness. For instance, theCoperation converts a left-handed electron
to a left-handed positron. Any chiral gauge theory is not invariant with respect to