PHY431 Lecture 11
26
5/1/2000
14.
Symmetries of the Standard Model:
U
(1),
SU
(2),
SU
(3)
We’ve now constructed a powerful toolset: we know how to describe free elementary particles: bosons
using the KleinGordon Lagrangian; and fermions using the Dirac Lagrangian. We introduce interac
tions by the use of the gauge principle: requiring invariance of the Lagrangian under local phase/gauge
transformations of the particle fields, we are led to introduce ‘compensating gauge fields’ of the proper
form. The gauge fields have to be massless in order to preserve gauge invariance, which is clearly a
problem when considering weak interactions where the gauge bosons are massive. However, we can
overcome this problem by invoking the Higgs mechanism: postulating the existence of a boson field
with an odd shape, i.e. a nonzero expectation value, and by reexpressing the Higgs field with respect
to a true minimum (vacuum), we break the manifest symmetry of the Lagrangian, but reap great bene
fits as well: the gauge boson acquires mass, just what we need for the weak gauge bosons. A final
benefit from the use of a theory with local gauge invariance is that such theories are inherently self
consistent when higherorder diagrams are considered. Very elegant cancellations between diagrams
occur, which cure divergencies that would otherwise make the theory meaningless.
We label the gauge symmetry by their group structure. The simplest gauge transformation is the multi
plication of the field by a phase factor function: exp{
i
α
(
x
)}
≡
1 +
i
α
+ (
i
α
)
2
/2! + (
i
α
)
3
/3! + ... , which is
the (infinite) group of complex functions that have modulus 1. The group operation is multiplication
(addition of the phase angles), and the unit element is exp{0} = 1. The group’s elements are clearly
unitary: [exp{
i
α
(
x
)}]
†
= exp{
−
i
α
(
x
)} = [exp{
i
α
(
x
)}]
−
1
. The official name of the group is “unitary
group of dimension one”:
U
(1) for short.
More complicated gauge groups are formed by bringing in matrices: e.g. the 2
×
2 Pauli matrices
exp{
i
½
σ⋅α
(
x
)}
≡
1
2
+
i
½
σ⋅α
+ (
i
½
σ⋅α
)
2
/2! + (
i
½
σ⋅α
)
3
/3! + ... . This group is again an infinite group of
complex operator functions that have determinant 1 and are unitary. They are unitary because the Pauli
matrices are Hermitian (and so is
i
½
σ⋅α
); because the Pauli matrices are traceless we also find that
det[exp{
i
½
σ⋅α
}] = exp{Tr[
i
½
σ⋅α
]} = exp{0} = 1. This group’s name is special unitary group of di
mension 2,
SU
(2). Its generators are the Pauli Matrices, because any member of the group can be
formed by appropriate combinations of the three Pauli matrices.
14.1
The
SU
(2)
L
of Weak Isospin
The basic particle for the weak interactions is the doublet consisting of the (electron) neutrino and the
electron. The weak interaction processes that we have seen – muon decay, or neutron decay – proceed
by coupling of the
W
vector boson to the lefthanded neutrinoelectron or updown quark doublets:
*
*
*
*
*
*
*
*
,
,
,
,
e
e
e
e
W
W
e
n
p
W
W
e
d
u
W
W
e
W
W
e
µ
µ
ν
ν
ν
ν
π
ν
−
−
−
−
−
−
−
−
−
−
−
−
−
−
→
+
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 Spring '01
 Rijssenbeek
 Physics, Power, Higgs, weak isospin

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