267
19.3. Spontaneously broken local U(1) symmetry
alone via (19.48), we have evidently chosen a gauge function (cf (19.42))
α ˆ(x) = −θ ˆ (x)/v.
(19.53)
Recalling then the form of the associated local phase change on φ ˆ (x)
φ ˆ (x) → φ ˆ ′ (x) = e
−i ˆ
α(x) φ ˆ (x)
(19.54)
we see that the phase of φ ˆ in (19.45) has been reduced to zero, in this choice
of gauge. Thus it is indeed possible to ‘gauge θ ˆ away’ in (19.45), but then
the vector field we must use is A ˆ ′μ , satisfying the massive equation (19.50)
(ignoring other interactions). In superconductivity, the choice of gauge which
takes the macroscopic wavefunction to be real (i.e. φ = 0 in (19.26)) is called
the ‘London gauge’. In the next section we shall discuss a subtlety in the
argument which applies in the case of real superconductors, and which leads
to the phenomenon of flux quantization.
The fact that this ‘Higgs mechanism’ leads to a massive vector field can
be seen very economically by working in the particular gauge for which φ ˆ is
real, and inserting the parametrization (cf (19.45))
1
ˆ
+ ˆ
φ = √ (v h)
(19.55)
2
into the Lagrangian L ˆ H . Retaining only the terms quadratic in the fields one
finds (problem 19.5)
L
quad
1
1 2 2 ˆ A ˆ μ
ˆ
H
= − (∂ μ A ˆ ν − ∂ ν A ˆ μ )(∂
μ A ˆ ν − ∂
ν A ˆ μ ) + q v A μ
4
2
+
1 ∂ μ ˆ h∂
μ h ˆ − μ
2 h ˆ 2 .
(19.56)
2
The first line of (19.56) is exactly the Lagrangian for a spin-1 field of mass
vq – i.e. the Maxwell part with the addition of a mass term (note that the
sign of the mass term is correct for the spatial (physical) degrees of freedom);
√
the second line is the Lagrangian of a scalar particle of mass 2μ. The latter
is the mass of excitations of the Higgs field ˆ h away from its vacuum value
(compare the global U(1) case discussed in section 17.5). The necessity for
the existence of one or more massive scalar particles (‘Higgs bosons’), when
a gauge symmetry is spontaneously broken in this way, was pointed out by
Higgs (1964).
We may now ask: what happens if we start with a certain phase θ ˆ for φ ˆ
but do not make use of the gauge freedom in A ˆ ν to reduce θ ˆ to zero? We shall
see in section 19.5 that the equation of motion, and hence the propagator,
for the vector particle depends on the choice of gauge; furthermore, Feynman
graphs involving quanta corresponding to the degree of freedom associated
with the phase field θ ˆ will have to be included for a consistent theory, even
though this must be an unphysical degree of freedom, as follows from the fact
19.3. Spontaneously broken local U(1) symmetry
alone via (19.48), we have evidently chosen a gauge function (cf (19.42))
α ˆ(x) = −θ ˆ (x)/v.
(19.53)
Recalling then the form of the associated local phase change on φ ˆ (x)
φ ˆ (x) → φ ˆ ′ (x) = e
−i ˆ
α(x) φ ˆ (x)
(19.54)
we see that the phase of φ ˆ in (19.45) has been reduced to zero, in this choice
of gauge. Thus it is indeed possible to ‘gauge θ ˆ away’ in (19.45), but then
the vector field we must use is A ˆ ′μ , satisfying the massive equation (19.50)
(ignoring other interactions). In superconductivity, the choice of gauge which
takes the macroscopic wavefunction to be real (i.e. φ = 0 in (19.26)) is called
the ‘London gauge’. In the next section we shall discuss a subtlety in the
argument which applies in the case of real superconductors, and which leads
to the phenomenon of flux quantization.
The fact that this ‘Higgs mechanism’ leads to a massive vector field can
be seen very economically by working in the particular gauge for which φ ˆ is
real, and inserting the parametrization (cf (19.45))
1
ˆ
+ ˆ
φ = √ (v h)
(19.55)
2
into the Lagrangian L ˆ H . Retaining only the terms quadratic in the fields one
finds (problem 19.5)
L
quad
1
1 2 2 ˆ A ˆ μ
ˆ
H
= − (∂ μ A ˆ ν − ∂ ν A ˆ μ )(∂
μ A ˆ ν − ∂
ν A ˆ μ ) + q v A μ
4
2
+
1 ∂ μ ˆ h∂
μ h ˆ − μ
2 h ˆ 2 .
(19.56)
2
The first line of (19.56) is exactly the Lagrangian for a spin-1 field of mass
vq – i.e. the Maxwell part with the addition of a mass term (note that the
sign of the mass term is correct for the spatial (physical) degrees of freedom);
√
the second line is the Lagrangian of a scalar particle of mass 2μ. The latter
is the mass of excitations of the Higgs field ˆ h away from its vacuum value
(compare the global U(1) case discussed in section 17.5). The necessity for
the existence of one or more massive scalar particles (‘Higgs bosons’), when
a gauge symmetry is spontaneously broken in this way, was pointed out by
Higgs (1964).
We may now ask: what happens if we start with a certain phase θ ˆ for φ ˆ
but do not make use of the gauge freedom in A ˆ ν to reduce θ ˆ to zero? We shall
see in section 19.5 that the equation of motion, and hence the propagator,
for the vector particle depends on the choice of gauge; furthermore, Feynman
graphs involving quanta corresponding to the degree of freedom associated
with the phase field θ ˆ will have to be included for a consistent theory, even
though this must be an unphysical degree of freedom, as follows from the fact