123
15.2. The renormalization group and related ideas in QED
This is a remarkable result. It shows that all the dependence of S on
the (momentum)
2 variable |q
2
| enters through that of the running coupling
2
α(|q
2
|). Of course, this result is only valid in a regime of −q which is much
greater than all quantities with dimension (mass)
2 – for example the squares of
all particle masses, which do not appear in (15.31). This is why the technique
applies only at ‘high’ −q
2 . The result implies that if we can calculate S(1, α μ )
2
(i.e. S at the point q = −μ
2 ) at some definite order in perturbation theory,
then replacing α μ by α(|q
2
|) will allow us to predict the q
2 -dependence (at
large −q
2 ). All we need to do is solve (15.39). Indeed, for QED with one
+ −
[2]
e e loop we have seen that β em (α) = α
2 /3π. Hence integrating (15.39) we
obtain
α μ
α μ
α(|q
2
|) =
αμ =
.
(15.44)
αμ
1 − t
1 −
ln(|q 2 |/μ 2 )
3π
3π
This is almost exactly the formula we proposed in (11.57), on plausibility
grounds.
1
Suppose now that the leading QED perturbative contribution to S(1, α μ )
is S 1 α μ . Then the terms contained in S(1, α(|q
2
|)) in this approximation can
be found by expanding in powers of α μ :
[
] −1
α μ
S(1, α(|q
2
|)) ≈ 1 + S 1 α(|q
2
|) = 1 + S 1 α μ 1 −
t
3π
[
(
) 2
]
α μ t
α μ t
= 1 + S 1 α μ 1 +
+
+ . . . ,
(15.45)
3π
3π
where t = ln(|q
2
|/μ
2 ). The next-higher-order calculation of S(1, α μ ) would be
S 2 α
2 , say, which generates the terms
μ
[
]
2α μ t
S 2 α
2 (|q
2
|) = S 2 α
2 1 +
+ . . . .
(15.46)
μ
3π
Comparing (15.45) and (15.46) we see that each power of the large log factor
appearing in (15.46) comes with one more power of α μ than in (15.45). Provided α μ is small, then, the leading terms in t, t
2 , . . . are contained in (15.45).
It is in this sense that replacing S(1, α μ ) by S(1, α(|q
2
|)) sums all ‘leading log
terms’.
In fact, of course, the one-loop (and higher) corrections to S in which we
are really interested are those due to QCD, rather than QED, corrections. But
the logic is exactly the same. The leading (O(α s )) perturbative contribution
2
to S = σ/σ pt at q = −μ
2 is given in (15.1) as α s (μ
2 )/π. It follows that
2
the ‘leading log corrections’ at high −q are summed up by replacing this
expression by α s (|q
2
|)/π, where the running α s (|q
2
|) is determined by solving
(15.39) with the QCD analogue of (15.36) – to which we now turn.
1 The difference has to do, of course, with the different renormalization prescriptions. Eq
(11.57) is written in terms of an ‘α’ defined at q 2 = 0, and without neglect of me.
15.2. The renormalization group and related ideas in QED
This is a remarkable result. It shows that all the dependence of S on
the (momentum)
2 variable |q
2
| enters through that of the running coupling
2
α(|q
2
|). Of course, this result is only valid in a regime of −q which is much
greater than all quantities with dimension (mass)
2 – for example the squares of
all particle masses, which do not appear in (15.31). This is why the technique
applies only at ‘high’ −q
2 . The result implies that if we can calculate S(1, α μ )
2
(i.e. S at the point q = −μ
2 ) at some definite order in perturbation theory,
then replacing α μ by α(|q
2
|) will allow us to predict the q
2 -dependence (at
large −q
2 ). All we need to do is solve (15.39). Indeed, for QED with one
+ −
[2]
e e loop we have seen that β em (α) = α
2 /3π. Hence integrating (15.39) we
obtain
α μ
α μ
α(|q
2
|) =
αμ =
.
(15.44)
αμ
1 − t
1 −
ln(|q 2 |/μ 2 )
3π
3π
This is almost exactly the formula we proposed in (11.57), on plausibility
grounds.
1
Suppose now that the leading QED perturbative contribution to S(1, α μ )
is S 1 α μ . Then the terms contained in S(1, α(|q
2
|)) in this approximation can
be found by expanding in powers of α μ :
[
] −1
α μ
S(1, α(|q
2
|)) ≈ 1 + S 1 α(|q
2
|) = 1 + S 1 α μ 1 −
t
3π
[
(
) 2
]
α μ t
α μ t
= 1 + S 1 α μ 1 +
+
+ . . . ,
(15.45)
3π
3π
where t = ln(|q
2
|/μ
2 ). The next-higher-order calculation of S(1, α μ ) would be
S 2 α
2 , say, which generates the terms
μ
[
]
2α μ t
S 2 α
2 (|q
2
|) = S 2 α
2 1 +
+ . . . .
(15.46)
μ
3π
Comparing (15.45) and (15.46) we see that each power of the large log factor
appearing in (15.46) comes with one more power of α μ than in (15.45). Provided α μ is small, then, the leading terms in t, t
2 , . . . are contained in (15.45).
It is in this sense that replacing S(1, α μ ) by S(1, α(|q
2
|)) sums all ‘leading log
terms’.
In fact, of course, the one-loop (and higher) corrections to S in which we
are really interested are those due to QCD, rather than QED, corrections. But
the logic is exactly the same. The leading (O(α s )) perturbative contribution
2
to S = σ/σ pt at q = −μ
2 is given in (15.1) as α s (μ
2 )/π. It follows that
2
the ‘leading log corrections’ at high −q are summed up by replacing this
expression by α s (|q
2
|)/π, where the running α s (|q
2
|) is determined by solving
(15.39) with the QCD analogue of (15.36) – to which we now turn.
1 The difference has to do, of course, with the different renormalization prescriptions. Eq
(11.57) is written in terms of an ‘α’ defined at q 2 = 0, and without neglect of me.