4 QCD: The Theory of Strong Interactions
95
or
Z G [
∂
∂ log μ 2 +
∂α
∂ log μ 2
∂
∂α
+
1
Z G
∂Z G
∂ log μ 2 ]G ren = 0
(4.20)
Finally the renormalisation group equation (RGE) can be written as:
[
∂
∂ log μ 2 + β(α)
∂
∂α
+ γ G (α)]G ren = 0
(4.21)
where
β(α) =
∂α
∂ log μ 2
(4.22)
and
γ G (α) =
∂ log Z G
∂ log μ 2
(4.23)
Note that β(α) does not depend on which Green function G we are considering, but
it is a property of the theory and the renormalisation scheme adopted, while γ G (α)
also depends on G. Strictly speaking the RGE as written above is only valid in the
Landau gauge (λ = 0). In other gauges an additional term that takes the variation
of the gauge fixing parameter λ should also be included. We omit this term, for
simplicity, as it is not relevant at the 1-loop level.
Assume that we want to apply the RGE to some hard process at a large scale
Q, related to a Green function G that we can always take as dimensionless (by
multiplication by a suitable power of Q). Since the interesting dependence on Q
will be logarithmic we introduce the variable t as :
t = log
Q 2
μ 2
(4.24)
Then we can write G ren ≡ F (t, α, x i ) where x i are scaling variables (we often omit
to write them in the following). In the naive scaling limit F should be independent
of t. To find the actual dependence on t, we want to solve the RGE
[−
∂
∂t
+ β(α)
∂
∂α
+ γ G (α)]G ren = 0
(4.25)
with a given boundary condition at t = 0 (or Q 2 = μ 2 ): F (0, α).
We first solve the RGE in the simplest case that γ G (α) = 0. This is not an
unphysical case: for example, it applies to R e + e − where the vanishing of γ is related
to the non renormalisation of the electric charge in QCD (otherwise the proton and
95
or
Z G [
∂
∂ log μ 2 +
∂α
∂ log μ 2
∂
∂α
+
1
Z G
∂Z G
∂ log μ 2 ]G ren = 0
(4.20)
Finally the renormalisation group equation (RGE) can be written as:
[
∂
∂ log μ 2 + β(α)
∂
∂α
+ γ G (α)]G ren = 0
(4.21)
where
β(α) =
∂α
∂ log μ 2
(4.22)
and
γ G (α) =
∂ log Z G
∂ log μ 2
(4.23)
Note that β(α) does not depend on which Green function G we are considering, but
it is a property of the theory and the renormalisation scheme adopted, while γ G (α)
also depends on G. Strictly speaking the RGE as written above is only valid in the
Landau gauge (λ = 0). In other gauges an additional term that takes the variation
of the gauge fixing parameter λ should also be included. We omit this term, for
simplicity, as it is not relevant at the 1-loop level.
Assume that we want to apply the RGE to some hard process at a large scale
Q, related to a Green function G that we can always take as dimensionless (by
multiplication by a suitable power of Q). Since the interesting dependence on Q
will be logarithmic we introduce the variable t as :
t = log
Q 2
μ 2
(4.24)
Then we can write G ren ≡ F (t, α, x i ) where x i are scaling variables (we often omit
to write them in the following). In the naive scaling limit F should be independent
of t. To find the actual dependence on t, we want to solve the RGE
[−
∂
∂t
+ β(α)
∂
∂α
+ γ G (α)]G ren = 0
(4.25)
with a given boundary condition at t = 0 (or Q 2 = μ 2 ): F (0, α).
We first solve the RGE in the simplest case that γ G (α) = 0. This is not an
unphysical case: for example, it applies to R e + e − where the vanishing of γ is related
to the non renormalisation of the electric charge in QCD (otherwise the proton and