4.1 The Standard Model: A Typical Gauge Theory 161
This makes the following trick possible. All particle masses M
i
are replaced
by
M
i
→
Φ(x)
v
M
i
. (4.12)
For physical states one has to set φ(x) = v +χ(x). This yields just M
i
plus an ad-
ditional interaction term, whose meaning will be explained later. So long as this
term does not cause significant physical effects, the original theory has practi-
cally not been changed. In this case the theory is then exactly renormalizable if
it is also renormalizable for a quantization around φ(x) =0, i.e., for vanishing
mass terms. As it is possible to prove that certain classes of massless gauge the-
ories are renormalizable, this is also true for the resulting theories with masses
generated by (4.12), i.e., by spontaneous symmetry breaking. The crucial point
is that spontaneous symmetry breaking preserves gauge invariance, which is
essential for the proof of renormalizability.
In general, φ(x) is not a scalar with respect to the current gauge symmetry
(i.e. not a singlet) but rather a multiplet. The vacuum expectation value v then
defines a particular direction within this multiplet. Because of this, one speaks
of spontaneous symmetry breaking. This expression is quite misleading since
the symmetry is really preserved. The choice of v should instead be compared
with the choice of a specific direction of quantization in a spherically symmet-
ric problem. Now we want to discuss spontaneous symmetry breaking for the
special case of the standard model.
The standard model describes the electromagnetic and weak interactions.
The latter was first understood as a four-fermion interaction between, for ex-
ample, a neutron, proton, electron, and antineutrino. But it has been clear for
a long time that this can be an effective description only for a microscopic pro-
cess, e.g., the exchange of charged so-called W bosons (see Fig. 4.2). Since the
W fields are vectors and mediate an interaction (just like the photon), it is natural
to regard them as gauge fields of a gauge group. Choosing possible candidates
for this gauge group does not cause any difficulties, if the following points are
considered:
(1) The leptons exclusively couple to the corresponding neutrinos, for ex-
ample the electron only to the electron neutrino. Therefore we have to choose
a group with an irreducible two-dimensional representation if we want to avoid
additional artificial assumptions.
Fig. 4.2. The Fermi interac-
tion and its interpretation as
the exchange of charged W
bosons