Physical Foundations of Cosmology

4.3 Electroweak theory 153

andU( 1 )transformations,

ψL→e−ig

′YLλ(x)

ψL,ψR→e−ig

′YRλ(x)

ψR,

if the gauge fieldsAμandBμtransform according to (4.11) and (4.5) respectively.
TheU( 1 )hyperchargesYLandYRcan be different for the right- and left-handed
electrons. The only requirement is that they should be able to reproduce the correct
values of the observed electric charges.
In electroweak theory, three out of four gauge bosons should acquire masses and
one boson should remain massless. Additionally, the fermions should become mas-
sive. These masses can be generated in a soft way via interaction with the classical
scalar field. In this case the theory remains gauge-invariant and renormalizable.
To demonstrate how this mechanism works, let us first consider the simplestU( 1 )
Abelian gauge field which interacts with a complex scalar field.

4.3.2 “Spontaneous breaking” ofU(1) symmetry

The Lagrangian

L=^12 ((∂μ+ieAμ)φ)∗((∂μ+ieAμ)φ)−V(φ∗φ)−^14 F^2 (A), (4.41)

whereF^2 ≡FμνFμν,is invariant under the gauge transformations

φ→e−ieλφ, Aμ→Aμ+∂μλ.

Forφ= 0 ,we can parameterize the complex scalar fieldφby two real scalar fields
χandζ,defined via

φ=χexp(ieζ). (4.42)

The fieldχis gauge-invariant and the fieldζtransforms asζ→ζ−λ.We can
combine the fieldAμandζto form the gauge-invariant variable

Gμ≡Aμ+∂μζ. (4.43)

Lagrangian (4.41) can then be rewritten entirely in terms of the gauge-invariant
fieldsχandGμ:

L=

1

2

∂μχ∂μχ−V(χ^2 )−

1

4

F^2 (G)+

e^2

2

χ^2 GμGμ. (4.44)

If the potentialVhas a minimum atχ 0 =const= 0 ,we can consider small pertur-
bations around this minimum,χ=χ 0 +φ,and expand the Lagrangian in powers
ofφ.It then describes the real scalar fieldφof massmH=

√

V,χχ(χ 0 ),which inter-
acts with the massive vector fieldGμ.The mass of the vector field isMG=eχ 0 .If
Lagrangian (4.41 ) is renormalizable, one expects that after rewriting it in explicitly