Physical Foundations of Cosmology

158 The very early universe

fine structure constant”

αw≡

g^2

4 π

=

e^2

4 πsin^2 θw

(4.69)

is in fact larger thanα≡e^2 / 4 π 1 / 137.

4.3.4 Fermion interactions

Combining the left-handed doubletψeLwith the scalar field, we can easily build
the corresponding fermionicSU( 2 )gauge-invariant variables:

ΨeL=ζ−^1 ψeL. (4.70)

The right-handed electron,ψeR≡eR,is a singlet with respect to theSU( 2 )group.
Because underU( 1 )transformationsψeL→e−ig
′YLλ
ψeLandζ→ζE,we obtain

ΨeL→e−ig

′YLλ

E−^1 ΨeL.

Defining theSU( 2 )gauge-invariant left-handed electron and neutrino as

ΨeL≡

(

νL

eL

)

, (4.71)

we have

νL→eig

′( (^12) −YL)λ
νL, eL→e−ig
′( (^12) +YL)λ
eL, eR→e−ig
′YRλ
eL. (4.72)
The neutrino has no electrical charge and therefore should not transform. Hence the
hypercharge of the left-handed doublet should be taken to beYL= 1 / 2 .In this case
the left-handed electron transforms aseL→e−ig
′λ
eL.To ensure the same value for
the electric charges of the right- and left-handed electrons we have to putYR=1.
Taking into account the transformation law for the vector potentialAμ(see (4.67)),
we conclude that the electric charge of the electron is equal toegiven in (4.68).
SubstitutingψeL=ζΨLin (4.40 ) and using definition (4.49), we can rewrite
the Lagrangian for fermions in terms of the gauge-invariant variables:
Lf=iΨ ̄eLγμ

(

∂μ+igGμ+

i

2

g′Bμ

)

ΨeL+iψ ̄Rγμ

(

∂μ+ig′Bμ

)

ψR. (4.73)