Physical Foundations of Cosmology

156 The very early universe

The fieldsBμandGμchange underU( 1 )transformations. FieldBμtransforms
as

Bμ→Bμ+∂μλ. (4.54)

Taking into account thatAμ→Aμandζ→ζE,where matrixEis defined in
(4.52), we find that

Gμ→G ̃μ=E−^1 GμE−

i

g

E−^1 ∂μE (4.55)

under theU( 1 )transformations.
MatrixGμis the Hermitian traceless matrix

Gμ≡

(

−G^3 μ/ 2 −Wμ+/

√

2

−Wμ−/

√

2 G^3 μ/ 2

)

, (4.56)

whereWμ±are a conjugate pair of complex vector fields andG^3 μis a real vector
field. In parameterizing matrixGμwe have used the standard sign convention and
normalization adopted in the literature. Substituting this expression in (4.48) and
replacing fieldsG^3 μandBμwith the “orthogonal” linear combinationsZμandAμ,

(

Aμ

Zμ

)

≡

(

cosθw sinθw

−sinθw cosθw

)(

Bμ

G^3 μ

)

, (4.57)

whereθwis the Weinberg angle and

cosθw=

g

√

g^2 +g′^2

, (4.58)

we can rewrite (4.48) in the following form:

Lφ=

1

2

∂μχ∂μχ−V(χ^2 )+

(g^2 +g′^2 )χ^2

8

ZμZμ+

g^2 χ^2

4

Wμ+W−μ. (4.59)

Because

trF^2 (A)=trF^2 (G),

whereF^2 ≡FμνFμν,the Lagrangian for the gauge fields is

LF=−^14 F^2 (B)−^12 trF^2 (G). (4.60)

Problem 4.12Substituting (4.56) in (4.60) and using the definitions (4.13) and
(4.57), verify that (4.60) can be rewritten as

LF=−^14 F^2 (A)−^14 F^2 (Z)−^12 Fμν(W+)Fμν(W−), (4.61)