Physical Foundations of Cosmology

4.3 Electroweak theory 157

where

Fμν(A)≡∂μAν−∂νAμ+igsinθw

(

Wμ−Wν+−Wν−Wμ+

)

,

Fμν(Z)≡∂μZν−∂νZμ+igcosθw

(

Wμ−Wν+−Wν−Wμ+

)

,

(4.62)

and

Fμν(W±)≡D±μWν±−D±νWμ±,

D±μ ≡∂μ∓igsinθwAμ∓igcosθwZμ.

(4.63)

The terms which are third and fourth order in the field strength describe the inter-
actions of the gauge fields. Draw the corresponding vertices.

We now turn to the renormalizable scalar field potential

V

(

χ^2

)

=

λ

4

(

χ^2 −χ 02

) 2

. (4.64)

In this case, the fieldχacquires a vacuum expectation valueχ 0 ,corresponding
to the minimum of this potential. Let us consider small perturbations around this
minimum, so thatχ=χ 0 +φ.It is obvious that the LagrangianLφ+LFthen
describes the Higgs scalar fieldφof mass

mH=

√

V,χχ(χ 0 )=

√

2 λχ 0 , (4.65)

massive vector fieldsZμandWν±,with masses

MZ=

√

g^2 +g′^2

χ 0

2

, MW=

gχ 0

2

=MZcosθw, (4.66)

and the massless fieldAμ.This massless field is responsible for long-range inter-
actions and should be identified with the electromagnetic field.

Problem 4.13Using (4.55), verify that underU( 1 )transformations

Wμ±→e±ig

′λ

Wμ±, Aμ→Aμ+

1

cosθw

∂μλ, Zμ→Zμ. (4.67)

Thus we see thatWμ±transform as electrically charged fields.Comparing these
transformation laws with those of electrodynamics, we can identify the electric
charge of theWμ±bosons (see also ( 4.63)) as

e=g′cosθw=gsinθw. (4.68)

The bosonZμis electrically neutral. As we will see,WandZbosons are responsible
for the charged and neutral weak interactions respectively, and the “weakness” of
these interactions is due to the large masses of the intermediate bosons rather than
the smallness of the weak coupling constantg.It follows from (4.68) that the “weak