15.2. The renormalization group and related ideas in QED
117
where, from (11.23) and (11.24),
∫ 1
∫
4
Π
[2]
γ (q
2
d k
′
x(1 x)
) = 8e
2 i
dx
0
(2 4
−
(15.9)
π) (k ′2 − Δ γ + i∈) 2
and Δ γ = m
2
e − x(1 − x)q
2 with q
2 < 0. We regularize the k
′ integral by a
cut-off Λ as explained in sections 10.3.1 and 10.3.2, obtaining (problem 15.1)
( (
√
)
(
e
2 ∫ 1
Π
[2]
Λ + Λ 2 + Δ
Λ
γ (q
2
−
dx x(1
π 2
0
−
γ
) =
x) ln
1
.
2
−
(Λ 2 + Δ γ ) 1/2
Δ γ
(15.10)
Setting q
2 = 0 and retaining the dominant ln Λ term, we find that
(
) 1
( )
[2] 2
α
Z 3
= 1 −
ln(Λ/m e ).
(15.11)
3π
It is not a coincidence that the coefficient α/3π of the ultraviolet divergence
is also the coefficient of the ln(|q
2
|/m
2
e ) term in (11.55)–(11.57); we need to
understand why.
We first recall how (11.55) was arrived at. It refers to the renormalized
self-energy part, which is defined by the ‘subtracted’ form
¯
Π
[2]
γ (q
2 ) = Π
[2]
γ (q
2 ) − Π
[2]
γ (0).
(15.12)
In the process of subtraction, the dependence on the cut-off Λ disappears and
we are left with
∫
]
¯
Π
[2]
γ (q
2
2α
1
[
m
2
) = −
dx x(1 − x) ln
e
(15.13)
π 0
m 2
e − q 2 x(1 − x)
as in equation (11.34). For large values of
2
2
|q
2
| this leads to the ‘large log’
term (α/3π) ln(|q |/m e ). Now, in order to form such a term, it is obviously
not possible to have just ‘ln |q
2
|’ appearing: the argument of the logarithm
must be dimensionless, so that some mass scale must be present, to which
|q
2
| can be compared. In the present case, that mass scale is evidently m e ,
[2]
which entered via the quantity Π γ (0), or equivalently via the renormalization
[2]
constant Z 3 (cf (15.11)). This is the beginning of the answer to our questions.
[2]
Why is it m e that enters into Π γ (0) or Z 3 ? Part of the answer – once
again – is of course that a ‘ln Λ’ cannot appear in that form, but must be
‘ln(Λ/some mass)’. So we must enquire: what determines the ‘some mass’ ?
With this question we have reached the heart of the problem (for the moment).
The answer is, in fact, not immediately obvious: it lies in the prescription used
to define the renormalized coupling constant; this prescription, whatever it is,
determines Z 3 .
The value (15.8) (or (11.31)) was determined from the requirement that
the O(e
2 ) corrected photon propagator (in ξ = 1 gauge) had the simple form
117
where, from (11.23) and (11.24),
∫ 1
∫
4
Π
[2]
γ (q
2
d k
′
x(1 x)
) = 8e
2 i
dx
0
(2 4
−
(15.9)
π) (k ′2 − Δ γ + i∈) 2
and Δ γ = m
2
e − x(1 − x)q
2 with q
2 < 0. We regularize the k
′ integral by a
cut-off Λ as explained in sections 10.3.1 and 10.3.2, obtaining (problem 15.1)
( (
√
)
(
e
2 ∫ 1
Π
[2]
Λ + Λ 2 + Δ
Λ
γ (q
2
−
dx x(1
π 2
0
−
γ
) =
x) ln
1
.
2
−
(Λ 2 + Δ γ ) 1/2
Δ γ
(15.10)
Setting q
2 = 0 and retaining the dominant ln Λ term, we find that
(
) 1
( )
[2] 2
α
Z 3
= 1 −
ln(Λ/m e ).
(15.11)
3π
It is not a coincidence that the coefficient α/3π of the ultraviolet divergence
is also the coefficient of the ln(|q
2
|/m
2
e ) term in (11.55)–(11.57); we need to
understand why.
We first recall how (11.55) was arrived at. It refers to the renormalized
self-energy part, which is defined by the ‘subtracted’ form
¯
Π
[2]
γ (q
2 ) = Π
[2]
γ (q
2 ) − Π
[2]
γ (0).
(15.12)
In the process of subtraction, the dependence on the cut-off Λ disappears and
we are left with
∫
]
¯
Π
[2]
γ (q
2
2α
1
[
m
2
) = −
dx x(1 − x) ln
e
(15.13)
π 0
m 2
e − q 2 x(1 − x)
as in equation (11.34). For large values of
2
2
|q
2
| this leads to the ‘large log’
term (α/3π) ln(|q |/m e ). Now, in order to form such a term, it is obviously
not possible to have just ‘ln |q
2
|’ appearing: the argument of the logarithm
must be dimensionless, so that some mass scale must be present, to which
|q
2
| can be compared. In the present case, that mass scale is evidently m e ,
[2]
which entered via the quantity Π γ (0), or equivalently via the renormalization
[2]
constant Z 3 (cf (15.11)). This is the beginning of the answer to our questions.
[2]
Why is it m e that enters into Π γ (0) or Z 3 ? Part of the answer – once
again – is of course that a ‘ln Λ’ cannot appear in that form, but must be
‘ln(Λ/some mass)’. So we must enquire: what determines the ‘some mass’ ?
With this question we have reached the heart of the problem (for the moment).
The answer is, in fact, not immediately obvious: it lies in the prescription used
to define the renormalized coupling constant; this prescription, whatever it is,
determines Z 3 .
The value (15.8) (or (11.31)) was determined from the requirement that
the O(e
2 ) corrected photon propagator (in ξ = 1 gauge) had the simple form