271
19.5. ’t Hooft’s gauges
however, solutions of the magnetic monopole type do occur in the case of
non-Abelian gauge field theories, whose symmetry is spontaneously broken
to an electromagnetic U(1) em gauge group. Just this circumstance can arise
in a grand unified theory which contains SU(3) c and a residual U(1) em . Incidentally, these monopole solutions provide an illuminating way of thinking
about charge quantization: as Dirac (1931) pointed out, the existence of just
one monopole implies, from his quantization condition (19.62), that charge is
quantized.
When these ideas are applied to QCD, E and B must be understood as
the appropriate colour fields (i.e. they carry an SU(3) c index). The group
structure of SU(3) is also quite different from that of U(1) models, and we do
not want to be restricted just to static solutions (as in the GL theory, here
used as an analogue). Whether in fact the real QCD vacuum (ground state)
is formed as some such coherent plasma of monopoles, with confinement of
electric charges and flux, is a subject of continuing research; other schemes are
also possible. As so often stressed, the difficulty lies in the non-perturbative
nature of the confinement problem.
19.5 ’t Hooft’s gauges
We must now at last grasp the nettle and consider what happens if, in the
parametrization
φ ˆ = |φ ˆ | exp(iθ ˆ (x)/v)
(19.63)
we do not choose the gauge (cf (19.52))
∂ μ A ˆ μ = ❗ ˆ
(19.64)
θ/M.
This was the gauge that enabled us to transform away the phase degree of
freedom and reduce the equation of motion for the electromagnetic field to
that of a massive vector boson. Instead of using the modulus and phase as
the two independent degrees of freedom for the complex Higgs field φ ˆ , we now
choose to parametrize φ ˆ , quite generally, by the decomposition
φ ˆ = 2
−1/2 [v + ˆ
χ 1 (x) + iˆ χ 2 (x)],
(19.65)
where the vacuum values of ˆ
χ 1 and ˆ
χ 2 are zero. Substituting this form for φ ˆ
into the master equation for A ˆ ν (obtained from (19.43) and (19.44))
ˆ
φ
† ∂
ν ˆ
2 A ˆ ν φ ˆ † ˆ
❗A ˆ ν − ∂
ν (∂ μ A
μ ) = iq[ ˆ φ − (∂
ν φ ˆ )
† φ ˆ ] − 2q
φ,
(19.66)
leads to the equation of motion
(❗ + M
2 )A ˆ ν − ∂
ν (∂ μ A ˆ μ ) = −M ∂
ν χ ˆ 2 + q( ˆ
χ 2 ∂
ν χ ˆ 1 − χ ˆ 1 ∂
ν χ ˆ 2 )
2 ˆ χ
2
χ
2
− q A
ν ( ˆ 1 + 2vχ ˆ 1 + ˆ 2 )
(19.67)
19.5. ’t Hooft’s gauges
however, solutions of the magnetic monopole type do occur in the case of
non-Abelian gauge field theories, whose symmetry is spontaneously broken
to an electromagnetic U(1) em gauge group. Just this circumstance can arise
in a grand unified theory which contains SU(3) c and a residual U(1) em . Incidentally, these monopole solutions provide an illuminating way of thinking
about charge quantization: as Dirac (1931) pointed out, the existence of just
one monopole implies, from his quantization condition (19.62), that charge is
quantized.
When these ideas are applied to QCD, E and B must be understood as
the appropriate colour fields (i.e. they carry an SU(3) c index). The group
structure of SU(3) is also quite different from that of U(1) models, and we do
not want to be restricted just to static solutions (as in the GL theory, here
used as an analogue). Whether in fact the real QCD vacuum (ground state)
is formed as some such coherent plasma of monopoles, with confinement of
electric charges and flux, is a subject of continuing research; other schemes are
also possible. As so often stressed, the difficulty lies in the non-perturbative
nature of the confinement problem.
19.5 ’t Hooft’s gauges
We must now at last grasp the nettle and consider what happens if, in the
parametrization
φ ˆ = |φ ˆ | exp(iθ ˆ (x)/v)
(19.63)
we do not choose the gauge (cf (19.52))
∂ μ A ˆ μ = ❗ ˆ
(19.64)
θ/M.
This was the gauge that enabled us to transform away the phase degree of
freedom and reduce the equation of motion for the electromagnetic field to
that of a massive vector boson. Instead of using the modulus and phase as
the two independent degrees of freedom for the complex Higgs field φ ˆ , we now
choose to parametrize φ ˆ , quite generally, by the decomposition
φ ˆ = 2
−1/2 [v + ˆ
χ 1 (x) + iˆ χ 2 (x)],
(19.65)
where the vacuum values of ˆ
χ 1 and ˆ
χ 2 are zero. Substituting this form for φ ˆ
into the master equation for A ˆ ν (obtained from (19.43) and (19.44))
ˆ
φ
† ∂
ν ˆ
2 A ˆ ν φ ˆ † ˆ
❗A ˆ ν − ∂
ν (∂ μ A
μ ) = iq[ ˆ φ − (∂
ν φ ˆ )
† φ ˆ ] − 2q
φ,
(19.66)
leads to the equation of motion
(❗ + M
2 )A ˆ ν − ∂
ν (∂ μ A ˆ μ ) = −M ∂
ν χ ˆ 2 + q( ˆ
χ 2 ∂
ν χ ˆ 1 − χ ˆ 1 ∂
ν χ ˆ 2 )
2 ˆ χ
2
χ
2
− q A
ν ( ˆ 1 + 2vχ ˆ 1 + ˆ 2 )
(19.67)