Physical Foundations of Cosmology

4.3 Electroweak theory 159

Alternatively, using definitions (4.56), (4.57) and (4.71), we obtain

Lf= i

(

e ̄γμ∂μe+ν ̄Lγμ∂μνL

)

−e(e ̄γμe)Aμ+

g

√

2

(

(ν ̄LγμeL)Wμ++(e ̄LγμνL)Wμ−

)

+

[

sin^2 θw

cosθw

g(e ̄RγμeR)−

cos 2θw

2 cosθw

g( ̄eLγμeL)+

g

2 cosθw

(ν ̄LγμνL)

]

Zμ,

(4.74)

where the well known properties of the Dirac matrices have been used to write

e ̄LγμeL+e ̄RγμeR= ̄eγμe.

The first cubic term is the familiar electromagnetic interaction and the next terms
describe the charged and neutral weak interactions due to the exchange ofW±
andZbosons respectively. Note that the right-handed electrons participate only in
electromagnetic and neutral weak interactions, and not in the charged interactions.
Replacinge,νein (4.74) byμ,νμ/τ,ντ,we obtain the Lagrangian for the sec-
ond/third generation of leptons. Let us consider muon decay. The appropriate tree
diagram is shown in Figure 4.7 (the reader must take care to correctly identify the no-
tation used for wave functions and particles in diagrams; for example, the conjugated
wave function ̄νcan describe a neutrino as well as an antineutrino depending on the
orientation of the diagram). Because the muon mass is much smaller than the mass of
theWboson, the boson propagator can be replaced byigμν/M^2 Wand the diagram in
Figure 4.7 reduces to the four-fermion diagram corresponding to the coupling term

2

√

2 GF( ̄νμγαμL)( ̄eLγανe),

μL μ

L

νe

GF

νe

νμ

W−

−

e−L

ν−μ

e−L

Fig. 4.7.