54
13. Local Non-Abelian (Gauge) Symmetries
13.3.2
Gauge field self-interactions
We start by pointing out an interesting ambiguity in the prescription for
‘covariantizing’ wave equations which we have followed, namely ‘replace ∂
μ
by D
μ ’. Suppose we wished to consider the electromagnetic interactions of
charged massless spin-1 particles, call them X’s, carrying charge e. The standard wave equation for such free massless vector particles would be the same
as for A
μ , namely
❗X
μ
− ∂
μ ∂
ν X ν = 0.
(13.74)
To ‘covariantize’ this (i.e. introduce the electromagnetic coupling) we would
replace ∂
μ by D
μ = ∂
μ + ieA
μ so as to obtain
D
2 X
μ
− D
μ D
ν X ν = 0.
(13.75)
But this procedure is not unique: if we had started from the perfectly equivalent wave equation
❗X
μ
− ∂
ν ∂
μ X ν = 0
(13.76)
we would have arrived at
D
2 X
μ
− D
ν D
μ X ν = 0
(13.77)
which is not the same as (13.75), since (cf (13.45))
[D
μ , D
ν ] = ieF
μν .
(13.78)
The simple prescription ∂
μ
→ D
μ has, in this case, failed to produce a
unique wave equation. We can allow for this ambiguity by introducing an
arbitrary parameter δ in the wave equation, which we write as
D
2 X
μ
− D
ν D
μ X ν + ieδF
μν X ν = 0.
(13.79)
The δ term in (13.79) contributes to the magnetic moment coupling of the
X-particle to the electromagnetic field, and is called the ‘ambiguous magnetic
moment’. Just such an ambiguity would seem to arise in the case of the
charged weak interaction quanta W
± (their masses do not affect this argument). For the photon itself, of course, e = 0 and there is no such ambiguity.
It is important to be clear that (13.79) is fully U(1) gauge-covariant, so that
δ cannot be fixed by further appeal to the local U(1) symmetry. Moreover, it
turns out that the theory for arbitrary δ is not renormalizable (though we shall
not show this here): thus the quantum electrodynamics of charged massless
vector bosons is in general non-renormalizable.
However, the theory is renormalizable if – to continue with the present
terminology –
X are the
the photon, the X-particle, and its antiparticle the ¯
members of an SU(2) gauge triplet (like the W’s), with gauge coupling constant e. This is, indeed, very much how the photon and the W
± are ‘unified’,
but there is a complication (as always!) in that case, having to do with the
13. Local Non-Abelian (Gauge) Symmetries
13.3.2
Gauge field self-interactions
We start by pointing out an interesting ambiguity in the prescription for
‘covariantizing’ wave equations which we have followed, namely ‘replace ∂
μ
by D
μ ’. Suppose we wished to consider the electromagnetic interactions of
charged massless spin-1 particles, call them X’s, carrying charge e. The standard wave equation for such free massless vector particles would be the same
as for A
μ , namely
❗X
μ
− ∂
μ ∂
ν X ν = 0.
(13.74)
To ‘covariantize’ this (i.e. introduce the electromagnetic coupling) we would
replace ∂
μ by D
μ = ∂
μ + ieA
μ so as to obtain
D
2 X
μ
− D
μ D
ν X ν = 0.
(13.75)
But this procedure is not unique: if we had started from the perfectly equivalent wave equation
❗X
μ
− ∂
ν ∂
μ X ν = 0
(13.76)
we would have arrived at
D
2 X
μ
− D
ν D
μ X ν = 0
(13.77)
which is not the same as (13.75), since (cf (13.45))
[D
μ , D
ν ] = ieF
μν .
(13.78)
The simple prescription ∂
μ
→ D
μ has, in this case, failed to produce a
unique wave equation. We can allow for this ambiguity by introducing an
arbitrary parameter δ in the wave equation, which we write as
D
2 X
μ
− D
ν D
μ X ν + ieδF
μν X ν = 0.
(13.79)
The δ term in (13.79) contributes to the magnetic moment coupling of the
X-particle to the electromagnetic field, and is called the ‘ambiguous magnetic
moment’. Just such an ambiguity would seem to arise in the case of the
charged weak interaction quanta W
± (their masses do not affect this argument). For the photon itself, of course, e = 0 and there is no such ambiguity.
It is important to be clear that (13.79) is fully U(1) gauge-covariant, so that
δ cannot be fixed by further appeal to the local U(1) symmetry. Moreover, it
turns out that the theory for arbitrary δ is not renormalizable (though we shall
not show this here): thus the quantum electrodynamics of charged massless
vector bosons is in general non-renormalizable.
However, the theory is renormalizable if – to continue with the present
terminology –
X are the
the photon, the X-particle, and its antiparticle the ¯
members of an SU(2) gauge triplet (like the W’s), with gauge coupling constant e. This is, indeed, very much how the photon and the W
± are ‘unified’,
but there is a complication (as always!) in that case, having to do with the