13.3. Local non-Abelian symmetries in Lagrangian quantum field theory 55
necessity for finding room in the scheme for the neutral weak boson Z
0 as
well. We shall see how this works in chapter 19; meanwhile we continue with
this X − γ model. We shall show that when the X − γ interaction contained
in (13.79) is regarded as a 3 − X vertex in a local SU(2) gauge theory, the
value of δ has to equal 1; for this value the theory is renormalizable. In this
μ
μ
interpretation, the X
μ wave function is identified with ‘ √ 1 (X 1 + iX 2 )’ and
2
¯
μ
μ
X
μ with ‘ √ 1 (X 1 −
μ
iX 2 )’ in terms of components of the SU(2) triplet X i ,
2
while A
μ
μ
is identified with X 3 .
Consider then equation (13.79) written in the form
2
❗X
μ
− ∂
ν ∂
μ
ˆ
X ν = V X
μ
(13.80)
where
ˆ
V X
μ = −ie {[∂
ν (A ν X
μ ) + A
ν ∂ ν X
μ ]
− (1 + δ) [∂
ν (A
μ X ) + A
ν ∂
μ
ν
X ν ]
+ δ [∂
μ (A
ν X ν ) + A
μ ∂
ν X ν ] } ,
(13.81)
and we have dropped terms of O(e
2 ) which appear in the ‘D
2 ’ term; we shall
come back to them later. The terms inside the { } brackets have been written
in such a way that each [ ] bracket has the structure
∂(AX) + A(∂X)
(13.82)
which will be convenient for the following evaluation.
The lowest-order (O(e)) perturbation theory amplitude for ‘X → X’ under
ˆ
the potential V is then
∫
−i X μ
∗
ˆ
(f)V X
μ (i)d
4 x.
(13.83)
Inserting (13.81) into (13.83) clearly gives something involving two ‘X’-wavefunctions and one ‘A’ one, i.e. a triple-X vertex (with A
μ
≡ X
μ
3 ), shown in
figure 13.4. To obtain the rule for this vertex from (13.83), consider the first
[ ] bracket in (13.81). It contributes
∫
−i(−ie) X
∗ (2){∂
ν (X (3)X
μ (1)) + X
ν (3)∂ X
μ (1)}d
4
μ
3ν
3
ν
x
(13.84)
where the (1), (2), (3) refer to the momenta as shown in figure 13.4, and for
reasons of symmetry are all taken to be ingoing; thus
μ
μ
X 3 (3) = ∈ 3 exp(−ik 3 · x)
(13.85)
2 The sign chosen for V ˆ here apparently differs from that in the KG case (3.101), but
it does agree when allowance is made, in the amplitude (13.83), for the fact that the dot
product of the polarization vectors is negative (cf (7.87)).
necessity for finding room in the scheme for the neutral weak boson Z
0 as
well. We shall see how this works in chapter 19; meanwhile we continue with
this X − γ model. We shall show that when the X − γ interaction contained
in (13.79) is regarded as a 3 − X vertex in a local SU(2) gauge theory, the
value of δ has to equal 1; for this value the theory is renormalizable. In this
μ
μ
interpretation, the X
μ wave function is identified with ‘ √ 1 (X 1 + iX 2 )’ and
2
¯
μ
μ
X
μ with ‘ √ 1 (X 1 −
μ
iX 2 )’ in terms of components of the SU(2) triplet X i ,
2
while A
μ
μ
is identified with X 3 .
Consider then equation (13.79) written in the form
2
❗X
μ
− ∂
ν ∂
μ
ˆ
X ν = V X
μ
(13.80)
where
ˆ
V X
μ = −ie {[∂
ν (A ν X
μ ) + A
ν ∂ ν X
μ ]
− (1 + δ) [∂
ν (A
μ X ) + A
ν ∂
μ
ν
X ν ]
+ δ [∂
μ (A
ν X ν ) + A
μ ∂
ν X ν ] } ,
(13.81)
and we have dropped terms of O(e
2 ) which appear in the ‘D
2 ’ term; we shall
come back to them later. The terms inside the { } brackets have been written
in such a way that each [ ] bracket has the structure
∂(AX) + A(∂X)
(13.82)
which will be convenient for the following evaluation.
The lowest-order (O(e)) perturbation theory amplitude for ‘X → X’ under
ˆ
the potential V is then
∫
−i X μ
∗
ˆ
(f)V X
μ (i)d
4 x.
(13.83)
Inserting (13.81) into (13.83) clearly gives something involving two ‘X’-wavefunctions and one ‘A’ one, i.e. a triple-X vertex (with A
μ
≡ X
μ
3 ), shown in
figure 13.4. To obtain the rule for this vertex from (13.83), consider the first
[ ] bracket in (13.81). It contributes
∫
−i(−ie) X
∗ (2){∂
ν (X (3)X
μ (1)) + X
ν (3)∂ X
μ (1)}d
4
μ
3ν
3
ν
x
(13.84)
where the (1), (2), (3) refer to the momenta as shown in figure 13.4, and for
reasons of symmetry are all taken to be ingoing; thus
μ
μ
X 3 (3) = ∈ 3 exp(−ik 3 · x)
(13.85)
2 The sign chosen for V ˆ here apparently differs from that in the KG case (3.101), but
it does agree when allowance is made, in the amplitude (13.83), for the fact that the dot
product of the polarization vectors is negative (cf (7.87)).