19.6. Spontaneously broken local SU(2) × U(1) symmetry
275
19.6 Spontaneously broken local SU(2)×U(1) symmetry
We shall limit our discussion of the spontaneous breaking of a local nonAbelian symmetry to the particular case needed for the electroweak part of the
Standard Model. This is, in fact, just the local version of the model studied
ˆ
in section 17.6. As noted there, the Lagrangian L Φ of (17.97) is invariant
under global SU(2) transformations of the form (17.100), and also global U(1)
transformations (17.101). Thus in the local version we shall have to introduce
ˆ μ
three SU(2) gauge fields (as in section 13.1), which we call W i (x)(i = 1, 2, 3),
ˆ
ˆ
and one U(1) gauge field B
μ (x). We recall that the scalar field φ is an SU(2)doublet
(
)
ˆ
φ
+
ˆ
φ =
,
(19.76)
ˆ
φ
0
ˆ
so that the SU(2) covariant derivative acting on φ is as given in (13.10), namely
D ˆ μ
ˆ
μ
= ∂
μ + igτ · W /2.
(19.77)
To this must be added the U(1) piece, which we write as ig
′ ˆ
B
μ /2, the
1 being
2
for later convenience. The Lagrangian (without gauge-fixing and ghost terms)
is therefore
λ
1
1
L ˆ
ˆ ˆ † ˆ μ ˆ
2 ˆ † ˆ
μν
−
ˆ † ˆ 2 − ˆ
ˆ
ˆ ˆ
GΦ = (D μ φ) (D φ) + μ φ φ
(φ φ)
F
μν
μν F
4
·
− G μν G (19.78)
4
4
where
ˆ ˆ
ˆ
μ
D
μ φ = (∂
μ + igτ · W /2 + ig
′ ˆ
B
μ
ˆ
/2)φ,
(19.79)
ˆ
μν
ˆ
ν
ˆ
μ
ˆ
μ
ˆ
ν
F
= ∂
μ
W − ∂
ν
W − gW × W ,
(19.80)
and
ˆ μν
μ ˆ ν −
ν ˆ
G = ∂ B
∂ B
μ .
(19.81)
We must now decide how to choose the non-zero vacuum expectation value
that breaks this symmetry. The essential point for the electroweak application is that, after symmetry breaking, we should be left with three massive
boson gauge bosons (which will be the W
± and Z
0 ) and one massless gauge
boson, the photon. We may reasonably guess that the massless boson will
be associated with a symmetry that is unbroken by the vacuum expectation
value. Put differently, we certainly do not want a ‘superconducting’ massive
photon to emerge from the theory in this case, as the physical vacuum is not
an electromagnetic superconductor. This means that we do not want to give a
vacuum value to a charged field (as is done in the BCS ground state). On the
other hand, we do want it to behave as a ‘weak’ superconductor, generating
mass for W
± and Z
0 . The choice suggested by Weinberg (1967) was
(
)
0
‹ | ˆ
0 φ|0› =
v/
√
(19.82)
2
275
19.6 Spontaneously broken local SU(2)×U(1) symmetry
We shall limit our discussion of the spontaneous breaking of a local nonAbelian symmetry to the particular case needed for the electroweak part of the
Standard Model. This is, in fact, just the local version of the model studied
ˆ
in section 17.6. As noted there, the Lagrangian L Φ of (17.97) is invariant
under global SU(2) transformations of the form (17.100), and also global U(1)
transformations (17.101). Thus in the local version we shall have to introduce
ˆ μ
three SU(2) gauge fields (as in section 13.1), which we call W i (x)(i = 1, 2, 3),
ˆ
ˆ
and one U(1) gauge field B
μ (x). We recall that the scalar field φ is an SU(2)doublet
(
)
ˆ
φ
+
ˆ
φ =
,
(19.76)
ˆ
φ
0
ˆ
so that the SU(2) covariant derivative acting on φ is as given in (13.10), namely
D ˆ μ
ˆ
μ
= ∂
μ + igτ · W /2.
(19.77)
To this must be added the U(1) piece, which we write as ig
′ ˆ
B
μ /2, the
1 being
2
for later convenience. The Lagrangian (without gauge-fixing and ghost terms)
is therefore
λ
1
1
L ˆ
ˆ ˆ † ˆ μ ˆ
2 ˆ † ˆ
μν
−
ˆ † ˆ 2 − ˆ
ˆ
ˆ ˆ
GΦ = (D μ φ) (D φ) + μ φ φ
(φ φ)
F
μν
μν F
4
·
− G μν G (19.78)
4
4
where
ˆ ˆ
ˆ
μ
D
μ φ = (∂
μ + igτ · W /2 + ig
′ ˆ
B
μ
ˆ
/2)φ,
(19.79)
ˆ
μν
ˆ
ν
ˆ
μ
ˆ
μ
ˆ
ν
F
= ∂
μ
W − ∂
ν
W − gW × W ,
(19.80)
and
ˆ μν
μ ˆ ν −
ν ˆ
G = ∂ B
∂ B
μ .
(19.81)
We must now decide how to choose the non-zero vacuum expectation value
that breaks this symmetry. The essential point for the electroweak application is that, after symmetry breaking, we should be left with three massive
boson gauge bosons (which will be the W
± and Z
0 ) and one massless gauge
boson, the photon. We may reasonably guess that the massless boson will
be associated with a symmetry that is unbroken by the vacuum expectation
value. Put differently, we certainly do not want a ‘superconducting’ massive
photon to emerge from the theory in this case, as the physical vacuum is not
an electromagnetic superconductor. This means that we do not want to give a
vacuum value to a charged field (as is done in the BCS ground state). On the
other hand, we do want it to behave as a ‘weak’ superconductor, generating
mass for W
± and Z
0 . The choice suggested by Weinberg (1967) was
(
)
0
‹ | ˆ
0 φ|0› =
v/
√
(19.82)
2