380
22. The GSW Gauge Theory of Electroweak Interactions
A key contribution was made by Glashow (1961); similar ideas were also
advanced by Salam and Ward (1964). Glashow suggested enlarging the
Schwinger–Bludman SU(2) schemes by inclusion of an additional U(1) gauge
group, resulting in an ‘SU(2) L × U(1)’ group structure. The new Abelian U(1)
group is associated with a weak analogue of hypercharge – ‘weak hypercharge’
– just as SU(2) L was associated with ‘weak isospin’. Indeed, Glashow proposed that the Gell-Mann–Nishijima relation for charges should also hold for
these weak analogues, giving
eQ = e(t 3 + y/2)
(22.26)
for the electric charge Q (in units of e) of the t 3 member of a weak isomultiplet, assigned a weak hypercharge y. Clearly, therefore, the lepton doublets,
(ν e
′ , e
− ), etc, then have y = −1, while the quark doublets (u, d
′ ), etc, have
y = +
1 . Now, when this group is gauged, everything falls marvellously into
3
place: the charged vector bosons appear as before, but there are now two
neutral vector bosons, which between them will be responsible for the weak
neutral current processes, and for electromagnetism. This is exactly the piece
of mathematics we went through in section 19.6, which we now appropriate
as an important part of the Standard Model.
For convenience, we reproduce here the main results of section 19.6. The
ˆ
Higgs field φ is an SU(2) doublet
(
)
ˆ
φ
+
ˆ
φ =
(22.27)
ˆ
φ
0
with an assumed vacuum expectation value (in unitary gauge) given by
(
)
0
‹ | ˆ
0 φ|0› =
v/
√
.
(22.28)
2
Fluctuations about this value are parametrized in this gauge by
(
)
0
ˆ
φ =
√ 1
(22.29)
ˆ
(v + H)
2
where H ˆ is the (physical) Higgs field. The Lagrangian for the sector consisting
of the gauge fields and the Higgs fields is
λ
1
1
L
ˆ ˆ
GΦ = (D μ φ)
† ˆ
(D
μ ˆ
2 ˆ ˆ
ˆ ˆ
ˆ
ˆ
μν
ˆ ˆ
φ) + μ φ
† φ − (φ
† φ)
2
− F μν · F − G G
μν
μν
, (22.30)
4
4
4
ˆ
ˆ
μ
where F μν is the SU(2) field strength tensor (19.80) for the gauge fields W
ˆ
and G μν is the U(1) field strength tensor (19.81) for the gauge field B
μ , and
ˆ
D
μ ˆ
φ is given by (19.79). After symmetry breaking (i.e. the insertion of (22.29)
22. The GSW Gauge Theory of Electroweak Interactions
A key contribution was made by Glashow (1961); similar ideas were also
advanced by Salam and Ward (1964). Glashow suggested enlarging the
Schwinger–Bludman SU(2) schemes by inclusion of an additional U(1) gauge
group, resulting in an ‘SU(2) L × U(1)’ group structure. The new Abelian U(1)
group is associated with a weak analogue of hypercharge – ‘weak hypercharge’
– just as SU(2) L was associated with ‘weak isospin’. Indeed, Glashow proposed that the Gell-Mann–Nishijima relation for charges should also hold for
these weak analogues, giving
eQ = e(t 3 + y/2)
(22.26)
for the electric charge Q (in units of e) of the t 3 member of a weak isomultiplet, assigned a weak hypercharge y. Clearly, therefore, the lepton doublets,
(ν e
′ , e
− ), etc, then have y = −1, while the quark doublets (u, d
′ ), etc, have
y = +
1 . Now, when this group is gauged, everything falls marvellously into
3
place: the charged vector bosons appear as before, but there are now two
neutral vector bosons, which between them will be responsible for the weak
neutral current processes, and for electromagnetism. This is exactly the piece
of mathematics we went through in section 19.6, which we now appropriate
as an important part of the Standard Model.
For convenience, we reproduce here the main results of section 19.6. The
ˆ
Higgs field φ is an SU(2) doublet
(
)
ˆ
φ
+
ˆ
φ =
(22.27)
ˆ
φ
0
with an assumed vacuum expectation value (in unitary gauge) given by
(
)
0
‹ | ˆ
0 φ|0› =
v/
√
.
(22.28)
2
Fluctuations about this value are parametrized in this gauge by
(
)
0
ˆ
φ =
√ 1
(22.29)
ˆ
(v + H)
2
where H ˆ is the (physical) Higgs field. The Lagrangian for the sector consisting
of the gauge fields and the Higgs fields is
λ
1
1
L
ˆ ˆ
GΦ = (D μ φ)
† ˆ
(D
μ ˆ
2 ˆ ˆ
ˆ ˆ
ˆ
ˆ
μν
ˆ ˆ
φ) + μ φ
† φ − (φ
† φ)
2
− F μν · F − G G
μν
μν
, (22.30)
4
4
4
ˆ
ˆ
μ
where F μν is the SU(2) field strength tensor (19.80) for the gauge fields W
ˆ
and G μν is the U(1) field strength tensor (19.81) for the gauge field B
μ , and
ˆ
D
μ ˆ
φ is given by (19.79). After symmetry breaking (i.e. the insertion of (22.29)