154 The very early universe
gauge-invariant form, it will remain renormalizable. This is what really happens in
spite of the fact that the vector field acquires mass. Ifχ 0 = 0 ,the physical fields cor-
responding to observable particles are the gauge-invariant real scalar field (Higgs
field) and the massive vector field. Of course, after we have rewritten the Lagrangian
in terms of the new variables, the total number of physical degrees of freedom does
not change. In fact, the system described by (4.41) has four degrees of freedom per
point in space: namely, two for the complex scalar field and two corresponding to
the transverse components of the vector field. The Lagrangian expressed in terms
of gauge-invariant variables describes a real scalar field with one degree of freedom
and a massive vector field with three degrees of freedom.
This method of generating the mass term is known as the Higgs mechanism and its
main advantage is that it does not spoil renormalizability. The vector field acquires
mass and its longitudinal degree of freedom becomes physical at the expense of
a classical scalar field. Of course, (4.44) is invariant with respect to the original
gauge transformations which become trivial:χ→χandGμ→Gμ. However, if
one tries to interpret the gauge-invariant fieldGμas a gauge field likeAμ,then one
erroneously concludes that gauge invariance is gone. This is why one often says that
the symmetry is spontaneously broken. Such a statement is somewhat misleading
but we will nevertheless use this wide accepted, standard terminology.
The gauge-invariant variables can be introduced and interpreted as physical
degrees of freedom only ifχ 0 =0 and only when the perturbations aroundχ 0 are
small,so thatχ=χ 0 +φ=0 everywhere in the space. Otherwise (4.42) becomes
singular atχ=0 and the fieldsχ andζused to construct the gauge-invariant
variables become ill-defined. In the caseχ 0 = 0 ,one has to work directly with the
Lagrangian in its original form (4.41).
4.3.3 Gauge bosons
In electroweak theory, the masses of the gauge bosons can also be generated
using the Higgs mechanism. Let us consider theSU( 2 )×U( 1 )gauge-invariant
Lagrangian
Lφ=^12 (Dμφ)†(Dμφ)−V(φ†φ), (4.45)whereφis anSU( 2 )doublet of complex scalar fields,†denotes the Hermitian
conjugation and
Dμ≡∂μ+igAμ−i
2g′Bμ. (4.46)We have assumed here that the hypercharge of the scalar doublet isYφ=− 1 / 2 ,
so that under theU( 1 )group it transforms asφ→e
2 ig′λ
φ.The scalar field can be