132
15. QCD II: Asymptotic Freedom, the Renormalization Group
the latter was dimensionless whereas (recalling that ψ ˆ has mass dimension
3 )
2
S ˜ ′ (q
2 ) has dimension M
−1 . This dimensionality is, of course, just what a
F
propagator of the free-field form i/(/ q − m) would provide.
Accordingly, we extract this (/ q)
−1 factor (compare σ/σ pt ) and consider
the dimensionless ratio R ˜ ′ (|q
2
|/μ
2 , α μ ) = / qS ˜ ′ (q
2 ). We might guess that, just
F
F
as for S(|q
2
|/μ
2 , α μ ), to get the leading large |q
2
| behaviour we will need to
calculate R ˜ ′ to some order in α μ , and then replace α μ by α(|q
2
|/μ
2 ). But this
F
is not quite all. The factor Z 2 in (15.71) will – as noted above – depend on
the renormalization scale μ, just as Z 3 of (15.15) did. Thus when we change
2
μ, the normalization of the ψ ˆ ’s will change via the Z
1
factors – of course by
2
a finite amount here – and we must include this change when writing down
the analogue of (15.33) for this case (i.e. the condition that the ‘total change,
on changing μ, is zero’). The required equation is
[
]
∂
∂
μ
2
R ˜ ′
∂μ 2
|
|
|
| + β(α μ ) ∂α μ
+ γ 2 (α μ )
F (|q
2
|/μ
2 , α μ ) = 0.
(15.73)
αμ
The solution of (15.73) is somewhat more complicated than that of (15.33).
We can gain insight into the essential difference caused by the presence of γ 2
by considering the special case β(α μ ) = 0. In this case, we easily find
˜
R F
′ (|q
2
|/μ
2 , α μ ) ∝ (μ
2 )
−γ2 (αμ) .
(15.74)
But since R ˜ ′ can only depend on μ via |q
2
|/μ
2 , we learn that if β = 0 then
F
the large |q
2
| behaviour of R ˜ ′ is given by (|q
2
|/μ
2 )
γ2 – or, in other words,
F
that at large |q
2
|
S ˜
F
′ (|q
2
|/μ
2 , α μ ) ∝
1
( |q
2
|
) γ2(αμ )
.
(15.75)
/ q μ 2
Thus, at a zero of the β-function, S ˜ ′ has an ‘anomalous’power law dependence
F
on |q
2
| (i.e. in addition to the obvious / q
−1 factor), which is controlled by the
parameter γ 2 . The latter is called the ‘anomalous dimension’ of the fermion
field, since its presence effectively means that the |q
2
| behaviour of S ˜ ′ is not
F
determined by its ‘normal’ dimensionality M
−1 . The behaviour (15.75) is
often referred to as ‘scaling with anomalous dimension’, meaning that if we
multiply |q
2
| by a scale factor λ, then S ˜ ′ is multiplied by λ
γ2(αμ )−1 rather than
F
just λ
−1 . Anomalous dimensions turn out to play a vital role in the theory of
critical phenomena – they are, in fact, closely related to ‘critical exponents’
(see section 16.4.3, and Peskin and Schroeder 1995, chapter 13). Scaling with
anomalous dimensions is also exactly what occurs in deep inelastic scattering
of leptons from nucleons, as we shall see in section 15.6.
The full solution of (15.73) for β / 0 is elegantly discussed in Coleman
=
(1985), chapter 3; see also Peskin and Schroeder (1995) section 12.3. We quote
15. QCD II: Asymptotic Freedom, the Renormalization Group
the latter was dimensionless whereas (recalling that ψ ˆ has mass dimension
3 )
2
S ˜ ′ (q
2 ) has dimension M
−1 . This dimensionality is, of course, just what a
F
propagator of the free-field form i/(/ q − m) would provide.
Accordingly, we extract this (/ q)
−1 factor (compare σ/σ pt ) and consider
the dimensionless ratio R ˜ ′ (|q
2
|/μ
2 , α μ ) = / qS ˜ ′ (q
2 ). We might guess that, just
F
F
as for S(|q
2
|/μ
2 , α μ ), to get the leading large |q
2
| behaviour we will need to
calculate R ˜ ′ to some order in α μ , and then replace α μ by α(|q
2
|/μ
2 ). But this
F
is not quite all. The factor Z 2 in (15.71) will – as noted above – depend on
the renormalization scale μ, just as Z 3 of (15.15) did. Thus when we change
2
μ, the normalization of the ψ ˆ ’s will change via the Z
1
factors – of course by
2
a finite amount here – and we must include this change when writing down
the analogue of (15.33) for this case (i.e. the condition that the ‘total change,
on changing μ, is zero’). The required equation is
[
]
∂
∂
μ
2
R ˜ ′
∂μ 2
|
|
|
| + β(α μ ) ∂α μ
+ γ 2 (α μ )
F (|q
2
|/μ
2 , α μ ) = 0.
(15.73)
αμ
The solution of (15.73) is somewhat more complicated than that of (15.33).
We can gain insight into the essential difference caused by the presence of γ 2
by considering the special case β(α μ ) = 0. In this case, we easily find
˜
R F
′ (|q
2
|/μ
2 , α μ ) ∝ (μ
2 )
−γ2 (αμ) .
(15.74)
But since R ˜ ′ can only depend on μ via |q
2
|/μ
2 , we learn that if β = 0 then
F
the large |q
2
| behaviour of R ˜ ′ is given by (|q
2
|/μ
2 )
γ2 – or, in other words,
F
that at large |q
2
|
S ˜
F
′ (|q
2
|/μ
2 , α μ ) ∝
1
( |q
2
|
) γ2(αμ )
.
(15.75)
/ q μ 2
Thus, at a zero of the β-function, S ˜ ′ has an ‘anomalous’power law dependence
F
on |q
2
| (i.e. in addition to the obvious / q
−1 factor), which is controlled by the
parameter γ 2 . The latter is called the ‘anomalous dimension’ of the fermion
field, since its presence effectively means that the |q
2
| behaviour of S ˜ ′ is not
F
determined by its ‘normal’ dimensionality M
−1 . The behaviour (15.75) is
often referred to as ‘scaling with anomalous dimension’, meaning that if we
multiply |q
2
| by a scale factor λ, then S ˜ ′ is multiplied by λ
γ2(αμ )−1 rather than
F
just λ
−1 . Anomalous dimensions turn out to play a vital role in the theory of
critical phenomena – they are, in fact, closely related to ‘critical exponents’
(see section 16.4.3, and Peskin and Schroeder 1995, chapter 13). Scaling with
anomalous dimensions is also exactly what occurs in deep inelastic scattering
of leptons from nucleons, as we shall see in section 15.6.
The full solution of (15.73) for β / 0 is elegantly discussed in Coleman
=
(1985), chapter 3; see also Peskin and Schroeder (1995) section 12.3. We quote