121
15.2. The renormalization group and related ideas in QED
2
is to understand how this can help us with the large −q behaviour of our
cross section, the problem we originally started from.
15.2.3 The RGE and large –q 2 behaviour in QED
To see the connection we need to implement the fundamental requirement,
stated at the end of section 15.2.2, that predictions for physically measurable
quantities must not depend on the renormalization scale μ. Consider, for
−
example, our annihilation cross section σ for e
+ e → hadrons, pretending
that the one-loop corrections we are interested in are those due to QED rather
than QCD. We need to work in the spacelike region, so as to be consistent
with all the foregoing discussion. To make this clear, we shall now denote
2
the 4-momentum of the virtual photon by q rather than Q, and take q <
0 as in sections 15.2.1 and 15.2.2. Bearing in mind the way we used the
‘dimensionless-ness’ of the e’s in (15.20), let us focus on the dimensionless
ratio σ/σ pt ≡ S. Neglecting all masses, S can only be a function of the
dimensionless ratio |q
2
|/μ
2 and of e μ :
S = S(|q
2
|/μ
2 , e μ ).
(15.31)
But S must ultimately have no μ dependence. It follows that the μ
2 dependence arising via the |q
2
|/μ
2 argument must cancel that associated with e μ .
This is why the μ
2 -dependence of e μ controls the |q
2
| dependence of S, and
hence of σ. In symbols, this condition is represented by the equation
(
|
|
)
∂ |
|
| +
de μ |
|
|
∂
S
(
|q
2
|/μ
2 , e μ
)
= 0,
(15.32)
∂μ
dμ
∂e μ
eμ
e0
or
(
)
∂
∂
(
)
μ ∂μ
|
|
|
| + β em (e μ ) ∂e μ
S |q|
2 /μ
2 , e μ = 0.
(15.33)
eμ
Equation (15.33) is referred to as ‘the renormalization group equation
(RGE) for S’. The terminology goes back to Stueckelberg and Peterman
(1953), who were the first to discuss the freedom associated with the choice
of renormalization scale. The ‘group’ connotation is a trifle obscure – but all
it really amounts to is the idea that if we do one infinitesimal shift in μ
2 , and
then another, the result will be a third such shift; in other words, it is a kind of
‘translation group’. It was, however, Gell-Mann and Low (1954) who realized
how equation (15.33) could be used to calculate the large |q
2
| behaviour of S,
as we now explain.
It is convenient to work in terms of μ
2 and α rather than μ and e. Equation
(15.33) is then
(
)
μ
2
∂μ
∂
2
|
|
|
| + β em (α μ ) ∂α
∂
μ
S
( |q
2
|/μ
2 , α μ
)
= 0,
(15.34)
αμ
15.2. The renormalization group and related ideas in QED
2
is to understand how this can help us with the large −q behaviour of our
cross section, the problem we originally started from.
15.2.3 The RGE and large –q 2 behaviour in QED
To see the connection we need to implement the fundamental requirement,
stated at the end of section 15.2.2, that predictions for physically measurable
quantities must not depend on the renormalization scale μ. Consider, for
−
example, our annihilation cross section σ for e
+ e → hadrons, pretending
that the one-loop corrections we are interested in are those due to QED rather
than QCD. We need to work in the spacelike region, so as to be consistent
with all the foregoing discussion. To make this clear, we shall now denote
2
the 4-momentum of the virtual photon by q rather than Q, and take q <
0 as in sections 15.2.1 and 15.2.2. Bearing in mind the way we used the
‘dimensionless-ness’ of the e’s in (15.20), let us focus on the dimensionless
ratio σ/σ pt ≡ S. Neglecting all masses, S can only be a function of the
dimensionless ratio |q
2
|/μ
2 and of e μ :
S = S(|q
2
|/μ
2 , e μ ).
(15.31)
But S must ultimately have no μ dependence. It follows that the μ
2 dependence arising via the |q
2
|/μ
2 argument must cancel that associated with e μ .
This is why the μ
2 -dependence of e μ controls the |q
2
| dependence of S, and
hence of σ. In symbols, this condition is represented by the equation
(
|
|
)
∂ |
|
| +
de μ |
|
|
∂
S
(
|q
2
|/μ
2 , e μ
)
= 0,
(15.32)
∂μ
dμ
∂e μ
eμ
e0
or
(
)
∂
∂
(
)
μ ∂μ
|
|
|
| + β em (e μ ) ∂e μ
S |q|
2 /μ
2 , e μ = 0.
(15.33)
eμ
Equation (15.33) is referred to as ‘the renormalization group equation
(RGE) for S’. The terminology goes back to Stueckelberg and Peterman
(1953), who were the first to discuss the freedom associated with the choice
of renormalization scale. The ‘group’ connotation is a trifle obscure – but all
it really amounts to is the idea that if we do one infinitesimal shift in μ
2 , and
then another, the result will be a third such shift; in other words, it is a kind of
‘translation group’. It was, however, Gell-Mann and Low (1954) who realized
how equation (15.33) could be used to calculate the large |q
2
| behaviour of S,
as we now explain.
It is convenient to work in terms of μ
2 and α rather than μ and e. Equation
(15.33) is then
(
)
μ
2
∂μ
∂
2
|
|
|
| + β em (α μ ) ∂α
∂
μ
S
( |q
2
|/μ
2 , α μ
)
= 0,
(15.34)
αμ