4 QCD: The Theory of Strong Interactions
97
In fact the sum of the two derivatives acting on the factor F [0, α(t)] vanishes and the
exponential is by itself a solution of the complete equation. Note that the boundary
condition is also satisfied.
The important point is the appearance of the running coupling that determines the
asymptotic departures from scaling. The next step is to study the functional form of
the running coupling. From Eq. (4.31) we see that the rate of change with t of the
running coupling is determined by the β function. In turn β(α) is determined by
the μ dependence of the renormalised coupling through Eq. (4.22). Clearly there
is no dependence on μ of the basic 3-gluon vertex in lowest order (order e). The
dependence starts at 1-loop, that is at order e 3 (one extra gluon has to be emitted
and reabsorbed). Thus we obtain that in perturbation theory:
∂e
∂ log μ 2 ∝ e
3
(4.34)
Recalling that α = e 2 /4π, we have:
∂α
∂ log μ 2 ∝ 2e
∂e
∂ log μ 2 ∝ e
4
∝ α
2
(4.35)
Thus the behaviour of β(α) in perturbation theory is as follows:
β(α) = ± bα
2
[1 + b
α + . . .]
(4.36)
Since the sign of the leading term is crucial in the following discussion, we stipulate
that always b > 0 and we make the sign explicit in front.
Let us make the procedure for computing the 1-loop beta function in QCD (or,
similarly, in QED) more precise. The result of the 1loop 1PI diagrams for V ren can
be written down as (we denote e s and α s by e and α, for shorthand):
V ren = e[1 + αB 3g log
μ 2
−p 2 + . . . ]
(4.37)
V ren satisfies the RGE:
[
∂
∂ log μ 2 + β(α)
∂e
∂α
∂
∂e
−
3
2
γ g (α)]V ren = 0
(4.38)
With respect to Eq. (4.21) the beta function term has been rewritten taking into
account that V ren starts with e and the anomalous dimension term arises from
a factor Z
−1/2
g
for each gluon leg. In general for a n-leg 1PI Green function
V n,bare = Z
−n/2
g
V n,ren , if all external legs are gluons. Note that in the particular case
of V = V 3 that is used to define e other Z factors are absorbed in the replacement
97
In fact the sum of the two derivatives acting on the factor F [0, α(t)] vanishes and the
exponential is by itself a solution of the complete equation. Note that the boundary
condition is also satisfied.
The important point is the appearance of the running coupling that determines the
asymptotic departures from scaling. The next step is to study the functional form of
the running coupling. From Eq. (4.31) we see that the rate of change with t of the
running coupling is determined by the β function. In turn β(α) is determined by
the μ dependence of the renormalised coupling through Eq. (4.22). Clearly there
is no dependence on μ of the basic 3-gluon vertex in lowest order (order e). The
dependence starts at 1-loop, that is at order e 3 (one extra gluon has to be emitted
and reabsorbed). Thus we obtain that in perturbation theory:
∂e
∂ log μ 2 ∝ e
3
(4.34)
Recalling that α = e 2 /4π, we have:
∂α
∂ log μ 2 ∝ 2e
∂e
∂ log μ 2 ∝ e
4
∝ α
2
(4.35)
Thus the behaviour of β(α) in perturbation theory is as follows:
β(α) = ± bα
2
[1 + b
α + . . .]
(4.36)
Since the sign of the leading term is crucial in the following discussion, we stipulate
that always b > 0 and we make the sign explicit in front.
Let us make the procedure for computing the 1-loop beta function in QCD (or,
similarly, in QED) more precise. The result of the 1loop 1PI diagrams for V ren can
be written down as (we denote e s and α s by e and α, for shorthand):
V ren = e[1 + αB 3g log
μ 2
−p 2 + . . . ]
(4.37)
V ren satisfies the RGE:
[
∂
∂ log μ 2 + β(α)
∂e
∂α
∂
∂e
−
3
2
γ g (α)]V ren = 0
(4.38)
With respect to Eq. (4.21) the beta function term has been rewritten taking into
account that V ren starts with e and the anomalous dimension term arises from
a factor Z
−1/2
g
for each gluon leg. In general for a n-leg 1PI Green function
V n,bare = Z
−n/2
g
V n,ren , if all external legs are gluons. Note that in the particular case
of V = V 3 that is used to define e other Z factors are absorbed in the replacement