Subatomic Physics

12.4. The Higgs Mechanism; Spontaneous Symmetry Breaking 393
The coupling constant g in Eqs. (12.23) and
(12.25) is similar to the charge e in quantum
electrodynamics and is thus sometimes referred
to as the “charge”. The non-Abelian theory
thus describes a “charged” field, in contrast
to the “uncharged” or neutral electromagnetic
field Aµ. To examine the consequences of this
“charge”, which is related to isospin, we look
at the energy of the field, corresponding to the
Hamiltonian. The energy density u is given by
u = 1
2 (� E 2
+ � B 2
) (12.31)
If we substitute Eqs. (12.26) into (12.31), we
observe that the extra term in Eq. (12.26) leads
to cubic and quartic self-interactions of the
non-Abelian “free” field; examples are
cubic terms ∝ g
�
1
c
quartic terms ∝ g 2 ( � V0 × � V ) 2 .
g 2
g
Figure 12.2: Feynman diagrams for
self-interactions of a “charged” field.
∂ � V
∂t + ∇� �
V0 ·( � V0 × � V )
(12.32)
Thereisnofreefield! ThegaugefieldV (a) and its quanta are “charged”, thus the
quanta interact directly with each other, unlike photons. The self-interactions are
the cubic terms, proportional to the “charge” g, and the quartic ones proportional
to g 2 in Eq. (12.32). Feynman diagrams for these interactions are shown in Fig. 12.2.
The strengths of these interactions are given in terms of the unique coupling g. If
g is the “charge” of the matter field, as q was in the electromagnetic case, then the
gauge vector fields are seen to carry this “charge”; they are not “neutral.”
Quantum chromodynamics (QCD) is a theory quite analogous to, but somewhat
more general than what we have developed in this section. In QCD, the
charge is called “color charge” and the massless vector gauge bosons are the colored
gluons. The gluons, however, come in eight colors, not just three charges. The
self-interactions are present and there is no free gluon field. Since the gluons are
color-charged, they always interact with each other. Our model can be generalized
to this situation.
12.4 The Higgs Mechanism; Spontaneous Symmetry Breaking
We saw in Section 12.1 that gauge theories require massless vector bosons. Any
connection to a theory of weak interactions, where the vector bosons are very massive,
therefore appears to be lost. A non gauge-invariant theory, however, leads to a