264
19. Spontaneously Broken Local Symmetry
what about Lorentz invariance? Can we provide a Lagrangian description of
the phenomenon? The answers to these questions are mostly contained in the
model to which we now turn, which is due to Higgs (1964) and is essentially
the local version of the U(1) Goldstone model of section 17.5.
19.3 Spontaneously broken local U(1) symmetry:
the Abelian Higgs model
This model is just L ˆ G of (17.69) and (17.77), extended so as to be locally,
rather than merely globally, U(1) invariant. Due originally to Higgs (1964), it
provides a deservedly famous and beautifully simple model for investigating
what happens when a gauge symmetry is spontaneously broken.
To make (17.69) locally U(1) invariant, we need only replace the ∂’s by
ˆ
D’s according to the rule (7.123), and add the Maxwell piece. This produces
F
μν
−
L ˆ H = [(∂
μ +iqA ˆ μ )φ ˆ ]
† [(∂ μ +iqA ˆ μ )φ ˆ ]−
1 F ˆ μν ˆ
1 λ(φ ˆ † φ ˆ )
2 +μ
2 (φ ˆ † φ ˆ ). (19.40)
4
4
(19.40) is invariant under the local version of (17.72), namely
α(x)φ ˆ (x)
φ ˆ (x) → φ ˆ ′ (x) = e
−i ˆ
(19.41)
when accompanied by the gauge transformation on the potentials
ˆ
′
A
μ (x) → A ˆ μ (x) = A ˆ μ (x) +
1 ∂
μ α ˆ(x).
(19.42)
q
Before proceeding any further, we note at once that this model contains four
field degrees of freedom – two in the complex scalar Higgs field φ ˆ , and two in
the massless gauge field A ˆ μ .
We learned in section 17.5 that the form of the potential terms in (19.40)
(specifically the μ
2 one) does not lend itself to a natural particle interpretation, which only appears after making a ‘shift to the classical minimum’, as
in (17.84). But there is a remarkable difference between the global and local
cases. In the present (local) case, the phase of φ ˆ is completely arbitrary, since
any change in ˆ
α of (19.41) can be compensated by an appropriate transformation (19.42) on A ˆ μ , leaving L ˆ H the same as before. Thus the field θ ˆ in
(17.84) can be ‘gauged away’ altogether, if we choose! But θ ˆ was precisely the
Goldstone field, in the global case. This must mean that there is somehow
no longer any physical manifestation of the massless mode. This is the first
unexpected result in the local case. We may also be reminded of our desire to
‘gauge away’ the longitudinal polarization states for a ‘massive gauge’ boson:
we shall return to this later.
19. Spontaneously Broken Local Symmetry
what about Lorentz invariance? Can we provide a Lagrangian description of
the phenomenon? The answers to these questions are mostly contained in the
model to which we now turn, which is due to Higgs (1964) and is essentially
the local version of the U(1) Goldstone model of section 17.5.
19.3 Spontaneously broken local U(1) symmetry:
the Abelian Higgs model
This model is just L ˆ G of (17.69) and (17.77), extended so as to be locally,
rather than merely globally, U(1) invariant. Due originally to Higgs (1964), it
provides a deservedly famous and beautifully simple model for investigating
what happens when a gauge symmetry is spontaneously broken.
To make (17.69) locally U(1) invariant, we need only replace the ∂’s by
ˆ
D’s according to the rule (7.123), and add the Maxwell piece. This produces
F
μν
−
L ˆ H = [(∂
μ +iqA ˆ μ )φ ˆ ]
† [(∂ μ +iqA ˆ μ )φ ˆ ]−
1 F ˆ μν ˆ
1 λ(φ ˆ † φ ˆ )
2 +μ
2 (φ ˆ † φ ˆ ). (19.40)
4
4
(19.40) is invariant under the local version of (17.72), namely
α(x)φ ˆ (x)
φ ˆ (x) → φ ˆ ′ (x) = e
−i ˆ
(19.41)
when accompanied by the gauge transformation on the potentials
ˆ
′
A
μ (x) → A ˆ μ (x) = A ˆ μ (x) +
1 ∂
μ α ˆ(x).
(19.42)
q
Before proceeding any further, we note at once that this model contains four
field degrees of freedom – two in the complex scalar Higgs field φ ˆ , and two in
the massless gauge field A ˆ μ .
We learned in section 17.5 that the form of the potential terms in (19.40)
(specifically the μ
2 one) does not lend itself to a natural particle interpretation, which only appears after making a ‘shift to the classical minimum’, as
in (17.84). But there is a remarkable difference between the global and local
cases. In the present (local) case, the phase of φ ˆ is completely arbitrary, since
any change in ˆ
α of (19.41) can be compensated by an appropriate transformation (19.42) on A ˆ μ , leaving L ˆ H the same as before. Thus the field θ ˆ in
(17.84) can be ‘gauged away’ altogether, if we choose! But θ ˆ was precisely the
Goldstone field, in the global case. This must mean that there is somehow
no longer any physical manifestation of the massless mode. This is the first
unexpected result in the local case. We may also be reminded of our desire to
‘gauge away’ the longitudinal polarization states for a ‘massive gauge’ boson:
we shall return to this later.