216
17. Spontaneously Broken Global Symmetry
So (17.88) becomes
φ ˆ U ˆ −1
B ‹0, α|φ ˆ |0, α› B = B ‹0|U ˆ α
|0› B
(17.91)
α
U
−1
and we may interpret ˆ |0› B as the ‘alternative vacuum’ |0, α› B (this arα
gument is, as usual, not valid in the infinite volume limit where N ˆ φ fails to
exist).
It is interesting to find out what happens to the symmetry current corresponding to the invariance (17.72), in the ‘broken symmetry’ case. This
current is given in (7.23) which we write again here in slightly different notation:
μ
ˆ j = i(φ ˆ † ∂
μ φ ˆ − (∂
μ φ ˆ )
† φ ˆ ),
(17.92)
φ
normal ordering being understood. Written in terms of the h ˆ and θ ˆ of (17.84),
μ
ˆ j becomes
φ
μ
ˆ
h
2 ∂
μ ˆ
j = θ + 2 ˆ θ + ˆ
θ/v.
(17.93)
v∂
μ ˆ h∂
μ ˆ
φ
The term involving just the single field θ ˆ is very remarkable: it tells us that
there is a non-zero matrix element of the form
μ
μ −ip·x
B ‹0| ˆ j (x)|θ, p› = −ip ve
(17.94)
φ
where |θ, p› stands for the state with one θ-quantum (Goldstone boson), with
momentum p
μ . This is easily seen by writing the usual normal mode expansion
†
for θ ˆ , and using the standard bosonic commutation relations for ˆ
a θ (k), a ˆ (k
′ ).
θ
In words, (17.94) asserts that, when the symmetry is spontaneously broken,
the symmetry current connects the vacuum to a state with one Goldstone
quantum, with an amplitude which is proportional to the symmetry breaking
vacuum expectation value v, and which vanishes as the 4-momentum goes to
zero. The matrix element (17.94), with x = 0, is precisely of the type that was
shown to be non-zero in the proof of the Goldstone theorem, after (17.67).
μ
ˆ
2
Note also that (17.94) is consistent with ∂ μ j = 0 only if p = 0, as is required
φ
for the massless θ.
We are now ready to generalize the Abelian U(1) model to the (global)
non-Abelian case.
17.6 Spontaneously broken global non-Abelian
symmetry
We can illustrate the essential features by considering a particular example,
which in fact forms part of the Higgs sector of the Standard Model. We
consider an SU(2) doublet, but this time not of fermions as in section 12.3,
17. Spontaneously Broken Global Symmetry
So (17.88) becomes
φ ˆ U ˆ −1
B ‹0, α|φ ˆ |0, α› B = B ‹0|U ˆ α
|0› B
(17.91)
α
U
−1
and we may interpret ˆ |0› B as the ‘alternative vacuum’ |0, α› B (this arα
gument is, as usual, not valid in the infinite volume limit where N ˆ φ fails to
exist).
It is interesting to find out what happens to the symmetry current corresponding to the invariance (17.72), in the ‘broken symmetry’ case. This
current is given in (7.23) which we write again here in slightly different notation:
μ
ˆ j = i(φ ˆ † ∂
μ φ ˆ − (∂
μ φ ˆ )
† φ ˆ ),
(17.92)
φ
normal ordering being understood. Written in terms of the h ˆ and θ ˆ of (17.84),
μ
ˆ j becomes
φ
μ
ˆ
h
2 ∂
μ ˆ
j = θ + 2 ˆ θ + ˆ
θ/v.
(17.93)
v∂
μ ˆ h∂
μ ˆ
φ
The term involving just the single field θ ˆ is very remarkable: it tells us that
there is a non-zero matrix element of the form
μ
μ −ip·x
B ‹0| ˆ j (x)|θ, p› = −ip ve
(17.94)
φ
where |θ, p› stands for the state with one θ-quantum (Goldstone boson), with
momentum p
μ . This is easily seen by writing the usual normal mode expansion
†
for θ ˆ , and using the standard bosonic commutation relations for ˆ
a θ (k), a ˆ (k
′ ).
θ
In words, (17.94) asserts that, when the symmetry is spontaneously broken,
the symmetry current connects the vacuum to a state with one Goldstone
quantum, with an amplitude which is proportional to the symmetry breaking
vacuum expectation value v, and which vanishes as the 4-momentum goes to
zero. The matrix element (17.94), with x = 0, is precisely of the type that was
shown to be non-zero in the proof of the Goldstone theorem, after (17.67).
μ
ˆ
2
Note also that (17.94) is consistent with ∂ μ j = 0 only if p = 0, as is required
φ
for the massless θ.
We are now ready to generalize the Abelian U(1) model to the (global)
non-Abelian case.
17.6 Spontaneously broken global non-Abelian
symmetry
We can illustrate the essential features by considering a particular example,
which in fact forms part of the Higgs sector of the Standard Model. We
consider an SU(2) doublet, but this time not of fermions as in section 12.3,