17.6. Spontaneously broken global non-Abelian symmetry
217
but of bosons:
(
) (
)
ˆ
√ 1
φ
+
(φ
ˆ
φ =
ˆ
φ
≡
1 + iφ 2 )
2
0
√ 1
(17.95)
(φ 3 + iφ 4 )
2
ˆ
where the complex scalar field φ
+ destroys positively charged particles and
ˆ
creates negatively charged ones, and the complex scalar field φ
0 destroys neutral particles and creates neutral antiparticles. As we shall see in a moment,
the Lagrangian we shall use has an additional U(1) symmetry, so that the
full symmetry is SU(2) × U(1). This U(1) symmetry leads to a conserved
quantum number which we call y. We associate the physical charge Q with
the eigenvalue t 3 of the SU(2) generator t ˆ 3 , and with y, via
Q = e(t 3 + y/2)
(17.96)
so that y(φ
+ ) = 1 = y(φ
0 ). Thus φ
+ and φ
0 can be thought of as analogous
to the hadronic iso-doublet (K
+ , K
0 ).
The Lagrangian we choose is a simple generalization of (17.69) and (17.77):
λ
L ˆ
ˆ
ˆ
ˆ ˆ
ˆ ˆ
Φ = (∂ μ φ
† )(∂
μ φ) + μ
2 φ
† φ − (φ
† φ)
2
(17.97)
4
which has the ‘spontaneous symmetry breaking’ choice of sign for the param2
2
ˆ ˆ
eter μ . Plainly, for the ‘normal’ sign of μ , in which ‘+μ
2 φ
† φ’ is replaced by
‘−μ
2 ˆ
φ
† ˆ
φ’, with μ
2 positive in both cases, the free (λ = 0) part would describe
a complex doublet, with four degrees of freedom, each with the same mass μ.
Let us see what happens in the broken symmetry case.
For the Lagrangian (17.97) with μ
2 > 0, the minimum of the classical
potential is at the point
(φ
† φ)
2
min = 2μ /λ ≡ v
2 /2.
(17.98)
ˆ ˆ
As in the U(1) case, we interpret (17.98) as a condition on the vev of φ
† φ,
‹0| ˆ
φ
† ˆ
φ|0› = v
2 /2.
(17.99)
Before proceeding we note that (17.97) is invariant under global SU(2) transformations
ˆ
φ → ˆ
φ
′
ˆ
= exp(−iα · τ /2)φ
(17.100)
but also under a separate global U(1) transformation
ˆ
φ → ˆ
φ
′ = exp(− ˆ
iα)φ
(17.101)
where α is to be distinguished from α ≡ (α 1 , α 2 , α 3 ). The symmetry is then
referred to as SU(2) × U(1), which is the symmetry of the electroweak sector
of the Standard Model, except that in that case it is a local symmetry.
As before, in order to get a sensible particle spectrum we must expand the
ˆ
ˆ
fields φ not about φ = 0 but about a point satisfying the stable ground state
217
but of bosons:
(
) (
)
ˆ
√ 1
φ
+
(φ
ˆ
φ =
ˆ
φ
≡
1 + iφ 2 )
2
0
√ 1
(17.95)
(φ 3 + iφ 4 )
2
ˆ
where the complex scalar field φ
+ destroys positively charged particles and
ˆ
creates negatively charged ones, and the complex scalar field φ
0 destroys neutral particles and creates neutral antiparticles. As we shall see in a moment,
the Lagrangian we shall use has an additional U(1) symmetry, so that the
full symmetry is SU(2) × U(1). This U(1) symmetry leads to a conserved
quantum number which we call y. We associate the physical charge Q with
the eigenvalue t 3 of the SU(2) generator t ˆ 3 , and with y, via
Q = e(t 3 + y/2)
(17.96)
so that y(φ
+ ) = 1 = y(φ
0 ). Thus φ
+ and φ
0 can be thought of as analogous
to the hadronic iso-doublet (K
+ , K
0 ).
The Lagrangian we choose is a simple generalization of (17.69) and (17.77):
λ
L ˆ
ˆ
ˆ
ˆ ˆ
ˆ ˆ
Φ = (∂ μ φ
† )(∂
μ φ) + μ
2 φ
† φ − (φ
† φ)
2
(17.97)
4
which has the ‘spontaneous symmetry breaking’ choice of sign for the param2
2
ˆ ˆ
eter μ . Plainly, for the ‘normal’ sign of μ , in which ‘+μ
2 φ
† φ’ is replaced by
‘−μ
2 ˆ
φ
† ˆ
φ’, with μ
2 positive in both cases, the free (λ = 0) part would describe
a complex doublet, with four degrees of freedom, each with the same mass μ.
Let us see what happens in the broken symmetry case.
For the Lagrangian (17.97) with μ
2 > 0, the minimum of the classical
potential is at the point
(φ
† φ)
2
min = 2μ /λ ≡ v
2 /2.
(17.98)
ˆ ˆ
As in the U(1) case, we interpret (17.98) as a condition on the vev of φ
† φ,
‹0| ˆ
φ
† ˆ
φ|0› = v
2 /2.
(17.99)
Before proceeding we note that (17.97) is invariant under global SU(2) transformations
ˆ
φ → ˆ
φ
′
ˆ
= exp(−iα · τ /2)φ
(17.100)
but also under a separate global U(1) transformation
ˆ
φ → ˆ
φ
′ = exp(− ˆ
iα)φ
(17.101)
where α is to be distinguished from α ≡ (α 1 , α 2 , α 3 ). The symmetry is then
referred to as SU(2) × U(1), which is the symmetry of the electroweak sector
of the Standard Model, except that in that case it is a local symmetry.
As before, in order to get a sensible particle spectrum we must expand the
ˆ
ˆ
fields φ not about φ = 0 but about a point satisfying the stable ground state