268
19. Spontaneously Broken Local Symmetry
that a gauge can be chosen in which this field vanishes. That the propagator
is gauge dependent should, on reflection, come as a relief. After all, if the
massive vector boson generated in this way were simply described by the
wave equation (19.50), all the troubles with massive vector particles outlined
in section 19.1 would be completely unresolved. As we shall see, a different
choice of gauge from that which renders φ ˆ real has precisely the effect of
ameliorating the bad high-energy behaviour associated with (19.50). This is
ultimately the reason for the following wonderful fact: massive vector theories,
in which the vector particles acquire mass through the spontaneous symmetry
breaking mechanism, are renormalizable (’t Hooft 1971b).
However, before discussing other gauges than the one in which φ ˆ is given
by (19.55), we first explore another interesting aspect of superconductivity.
19.4 Flux quantization in a superconductor
Though a slight diversion, it is convenient to include a discussion of flux quantization at this point, while we have a number of relevant results assembled.
Apart from its intrinsic interest, the phenomenon may also provide a useful
physical model for the ‘confining’ property of QCD, as already discussed in
sections 1.3.6 and 16.5.3.
Our discussion of superconductivity so far has dealt, in fact, with only
one class of superconductors, called type I; these remain superconducting
throughout the bulk of the material (exhibiting a complete Meissner effect),
when an external magnetic field of less than a certain critical value is applied.
There is a quite separate class – type II superconductors – which allow partial
entry of the external field, in the form of thin filaments of flux. Within each
filament the field is high, and the material is not superconducting. Outside the
core of the filaments, the material is superconducting and the field dies off over
the characteristic penetration length λ. Around each filament of magnetic flux
there circulates a vortex of screening current; the filaments are often called
vortex lines. It is as if numerous thin cylinders, each enclosing flux, had been
drilled in a block of type I material, thereby producing a non-simply connected
geometry.
In real superconductors, screening currents are associated with the macroscopic pair wavefunction (field) ψ. For type II behaviour to be possible, |ψ|
must vanish at the centre of a flux filament, and rise to the constant value
appropriate to the superconducting state over a distance ξ < λ, where ξ is the
‘coherence length’ of section 17.7. According to the Ginzburg–Landau (GL)
theory, a more precise criterion is that type II behaviour holds if ξ < 2
1/2 λ;
both ξ and λ are, of course, temperature-dependent. The behaviour of |ψ| and
B in the vicinity of a flux filament is shown in figure 19.2. Thus, whereas for
simple type I superconductivity, |ψ| is simply set equal to a constant, in the
19. Spontaneously Broken Local Symmetry
that a gauge can be chosen in which this field vanishes. That the propagator
is gauge dependent should, on reflection, come as a relief. After all, if the
massive vector boson generated in this way were simply described by the
wave equation (19.50), all the troubles with massive vector particles outlined
in section 19.1 would be completely unresolved. As we shall see, a different
choice of gauge from that which renders φ ˆ real has precisely the effect of
ameliorating the bad high-energy behaviour associated with (19.50). This is
ultimately the reason for the following wonderful fact: massive vector theories,
in which the vector particles acquire mass through the spontaneous symmetry
breaking mechanism, are renormalizable (’t Hooft 1971b).
However, before discussing other gauges than the one in which φ ˆ is given
by (19.55), we first explore another interesting aspect of superconductivity.
19.4 Flux quantization in a superconductor
Though a slight diversion, it is convenient to include a discussion of flux quantization at this point, while we have a number of relevant results assembled.
Apart from its intrinsic interest, the phenomenon may also provide a useful
physical model for the ‘confining’ property of QCD, as already discussed in
sections 1.3.6 and 16.5.3.
Our discussion of superconductivity so far has dealt, in fact, with only
one class of superconductors, called type I; these remain superconducting
throughout the bulk of the material (exhibiting a complete Meissner effect),
when an external magnetic field of less than a certain critical value is applied.
There is a quite separate class – type II superconductors – which allow partial
entry of the external field, in the form of thin filaments of flux. Within each
filament the field is high, and the material is not superconducting. Outside the
core of the filaments, the material is superconducting and the field dies off over
the characteristic penetration length λ. Around each filament of magnetic flux
there circulates a vortex of screening current; the filaments are often called
vortex lines. It is as if numerous thin cylinders, each enclosing flux, had been
drilled in a block of type I material, thereby producing a non-simply connected
geometry.
In real superconductors, screening currents are associated with the macroscopic pair wavefunction (field) ψ. For type II behaviour to be possible, |ψ|
must vanish at the centre of a flux filament, and rise to the constant value
appropriate to the superconducting state over a distance ξ < λ, where ξ is the
‘coherence length’ of section 17.7. According to the Ginzburg–Landau (GL)
theory, a more precise criterion is that type II behaviour holds if ξ < 2
1/2 λ;
both ξ and λ are, of course, temperature-dependent. The behaviour of |ψ| and
B in the vicinity of a flux filament is shown in figure 19.2. Thus, whereas for
simple type I superconductivity, |ψ| is simply set equal to a constant, in the