MODERN COSMOLOGY

190 Dark matter and particle physics

LHf=−

gener∑.

b,c

(λbcdQLbHDRc+λbcuQLbHU ̃ Rc+λebcELbHERc)+h.c. (5.3)

where the parametersμandλare real constants,λdbc,λubcandλebcare 3× 3
matrices on the generation space. H ̃indicates the charge conjugate of H:
H ̃a=abHb†.
While the Lagrangian is invariant under gauge symmetry the vacuum is not
and the neutral component of the doubletHdevelops a vacuum expectation value
(vev):

〈H^0 〉=

(

0

v

)

. (5.4)

This breaks the symmetrySU( 2 )L⊗U( 1 )Ydown toU( 1 )EM. We recall that
when a global symmetry is spontaneously broken, a massless Goldstone boson
appears in the theory; if the symmetry is local (gauge) these Goldstone bosons
become the longitudinal components of the vector bosons (it is said that they are
eaten up by the gauge bosons). The gauge bosons relative to the broken symmetry
acquire a mass as shown inLM gauge:

LM gauge=−

1

2

v^2

4

[g^2 (Wμ^1 )^2 +g^2 (Wμ^2 )^2 +(−gWμ^3 +g′Bμ)^2 ]. (5.5)

Therefore there are three massive vectorsWμ±andZ^0 μ:

Wμ±=

1

√

2

(Wμ^1 ∓iWμ^2 ), (5.6)

Z^0 μ=

1

√

g^2 +g′^2

(gWμ^3 −g′Bμ), (5.7)

whose masses are given by

mW=g

v

2

, (5.8)

mZ=

√

(g^2 +g′^2 )

v

2

, (5.9)

while the gauge boson

Aμ≡

1

√

g^2 +g′^2

(gWμ^3 +g′Bμ),

relative toU( 1 )EM, remains massless as imposed by the gauge symmetry. Such
a mechanism is called the Higgs mechanism and preserves renormalizability.