152 The very early universe
described by four component spinors with the first two components equal to zero.
To make concrete calculations of processes, the reader should be familiar with the
standard algebra of Dirac matrices, which can be found in any book on field theory.
We will not need it here.
Other leptons also come in doublets and singlets:
(
νμ
μ
)
L,μR;(
ντ
τ)
L,τR. (4.38)The three differentgenerationsof leptons have very similar properties. Because
weak interactions convert particles only within a particular generation, the lepton
numbers are conserved separately.
The six quark flavors also form three generations under weak interaction:
(
u
d′
)
L,uR,dR′;(
c
s′)
L,cR,s′R;(
t
b′)
L,tR,b′R. (4.39)We have skipped the color indices which are irrelevant for electroweak interactions.
The flavorsd′,s′,b′entering the doublets are linear superpositions of the flavors
d,s,b,conserved in strong interactions.As a consequence, the weak interactions
violate all flavor conservation laws.
Since the individual Lagrangians for each generation have the same form, the
fermionic part of electroweak Lagrangian is obtained essentially by replication
of the Lagrangian for one particular lepton generation. Therefore we consider, for
example, only the electron and its corresponding neutrino. However, the importance
of the quark generations should not be underestimated. The anomalies which would
spoil renormalizability are canceled only if the number of quark generations is equal
to the number of lepton generations.
TheSU( 2 )group has three gauge bosons. As we have already mentioned, two of
them are responsible for the charged weak interaction. The third boson is electrically
neutral since only then is it its own antiparticle. However, it cannot be identified
with the photon, because the photon should be an AbelianU( 1 )gauge boson.
Because one of the partners in the doublet (4.37) is electrically charged, it makes
sense to try to incorporate both the electromagnetic and weak interactions into the
SU( 2 )×U( 1 )group. The corresponding Lagrangian
Lf=iψ ̄Lγμ(
∂μ+igAμ+ig′YLBμ)
ψL+iψ ̄Rγμ(
∂μ+ig′YRBμ)
ψR (4.40)is invariant under bothSU( 2 )transformations,
ψL→UψL,ψR→ψR,