15
QCD II: Asymptotic Freedom, the
Renormalization Group, and Scaling
Violations
In the previous chapter we learned that QCD amplitudes contributing to
+ −
e e → jets generally have IRC singularities, but that finite physical cross
sections can be obtained by including together kinematically indistinguishable
−
final states. The partial cross sections (for example σ(e
+ e → 2 jets)) will
depend on the IRC cut-off parameter(s). What about the fully inclusive pro+
cess e e
−
→ hadrons, where all final states are summed over? At order αα s ,
the parton-level diagrams contributing to this process are the same ones we
considered in section 14.5, namely figures 14.16, 14.21 and 14.22. If we denote
the amplitudes for these contributions by F rg (for real gluon emission), F vg
(for virtual gluon emission) and F γ for the Born graph, then the partial cross
−
∗
section σ(e
+ e → 2 jets) includes |F γ |
2 , the interference term 2Re(F γ F ),
vg
and the integral of |F rg |
2 over strips near the boundaries of figure 14.17. At
this order, the partial cross section σ(e
+ e
−
→ 3 jets) is given by the integral
of |F rg |
2 over the remaining (interior) region of figure 14.17. The correspond∗
ing total cross section is thus simply the sum of |F γ |
2 , 2Re(F γ F ), and the
vg
integral of |F rg |
2 over the whole of the x 1 − x 2 phase space. Clearly the IRC
singularities will cancel, as in the 2-jet cross section, and the result will not
depend on any IRC cut-off parameter. Indeed, the result is (see for example
Muta 2010, section 5.1.2)
σ(e
+ e
−
→ hadrons) = σ pt (Q
2 )(1 + α s /π).
(15.1)
This fully inclusive cross section is finite and free of IRC cut-offs.
At first sight, this result might appear satisfactory. It predicts a cross
section somewhat greater than σ pt , as is observed in figure 14.1 – from which
we might infer that α s ∼ 0.5 or less. Assuming the expansion parameter is
α s /π, the implied perturbation series in powers of α s would seem to be rapidly
convergent. However, this is an illusion, which is dispelled as soon as we go
to the next order in α s (i.e. to the order α
2 α
2 in the cross section).
s
113
QCD II: Asymptotic Freedom, the
Renormalization Group, and Scaling
Violations
In the previous chapter we learned that QCD amplitudes contributing to
+ −
e e → jets generally have IRC singularities, but that finite physical cross
sections can be obtained by including together kinematically indistinguishable
−
final states. The partial cross sections (for example σ(e
+ e → 2 jets)) will
depend on the IRC cut-off parameter(s). What about the fully inclusive pro+
cess e e
−
→ hadrons, where all final states are summed over? At order αα s ,
the parton-level diagrams contributing to this process are the same ones we
considered in section 14.5, namely figures 14.16, 14.21 and 14.22. If we denote
the amplitudes for these contributions by F rg (for real gluon emission), F vg
(for virtual gluon emission) and F γ for the Born graph, then the partial cross
−
∗
section σ(e
+ e → 2 jets) includes |F γ |
2 , the interference term 2Re(F γ F ),
vg
and the integral of |F rg |
2 over strips near the boundaries of figure 14.17. At
this order, the partial cross section σ(e
+ e
−
→ 3 jets) is given by the integral
of |F rg |
2 over the remaining (interior) region of figure 14.17. The correspond∗
ing total cross section is thus simply the sum of |F γ |
2 , 2Re(F γ F ), and the
vg
integral of |F rg |
2 over the whole of the x 1 − x 2 phase space. Clearly the IRC
singularities will cancel, as in the 2-jet cross section, and the result will not
depend on any IRC cut-off parameter. Indeed, the result is (see for example
Muta 2010, section 5.1.2)
σ(e
+ e
−
→ hadrons) = σ pt (Q
2 )(1 + α s /π).
(15.1)
This fully inclusive cross section is finite and free of IRC cut-offs.
At first sight, this result might appear satisfactory. It predicts a cross
section somewhat greater than σ pt , as is observed in figure 14.1 – from which
we might infer that α s ∼ 0.5 or less. Assuming the expansion parameter is
α s /π, the implied perturbation series in powers of α s would seem to be rapidly
convergent. However, this is an illusion, which is dispelled as soon as we go
to the next order in α s (i.e. to the order α
2 α
2 in the cross section).
s
113