4 QCD: The Theory of Strong Interactions
103
of diagrams with factorial growth at large order n is made up by dressing gluon
propagators by any number of quark bubbles together with their gauge completions
(renormalons).The problem of the precise relation between the ambiguities of the
perturbative expansion and the higher twist corrections has been discussed in recent
years [14].
4.5 Application to Hard Processes
4.5.1 R e + e − and Related Processes
The simplest hard process is R e + e − that we have already started to discuss. R is
dimensionless and in perturbation theory is given by R = N C
i Q 2
i F (t, α s ),
where F = 1 + o(α S ). We have already mentioned that for this process the
“anomalous dimension” function vanishes: γ (α s ) = 0 because of electric charge
non renormalisation by strong interactions. Let us review how this happens in detail.
The diagrams that are relevant for charge renormalisation in QED at 1-loop are
shown in Fig. 4.8. The Ward identity that follows from gauge invariance in QED
imposes that the vertex (Z V ) and the self-energy (Z f ) renormalisation factors cancel
and the only divergence remains in Z γ , the vacuum polarization of the photon. So
the charge is only renormalised by the photon blob, hence it is universal (the same
factor for all fermions, independent of their charge) and is not affected by QCD
at 1-loop. It is true that at higher orders the photon vacuum polarization diagram
is affected by QCD (for example, at 2-loops we can exchange a gluon between
the quarks in the photon loop) but the renormalisation induced by the vacuum
polarisation diagram remains independent of the nature of the fermion to which
the photon line is attached. The gluon contributions to the vertex (Z V ) and to the
Z J
Z v
Z f
Z f
J
J
J
J
f
f
+
+
+
+
+
J
f
f
f
f
f
f
J
f
f
Fig. 4.8 Diagrams for charge renormalisation in QED at 1-loop
103
of diagrams with factorial growth at large order n is made up by dressing gluon
propagators by any number of quark bubbles together with their gauge completions
(renormalons).The problem of the precise relation between the ambiguities of the
perturbative expansion and the higher twist corrections has been discussed in recent
years [14].
4.5 Application to Hard Processes
4.5.1 R e + e − and Related Processes
The simplest hard process is R e + e − that we have already started to discuss. R is
dimensionless and in perturbation theory is given by R = N C
i Q 2
i F (t, α s ),
where F = 1 + o(α S ). We have already mentioned that for this process the
“anomalous dimension” function vanishes: γ (α s ) = 0 because of electric charge
non renormalisation by strong interactions. Let us review how this happens in detail.
The diagrams that are relevant for charge renormalisation in QED at 1-loop are
shown in Fig. 4.8. The Ward identity that follows from gauge invariance in QED
imposes that the vertex (Z V ) and the self-energy (Z f ) renormalisation factors cancel
and the only divergence remains in Z γ , the vacuum polarization of the photon. So
the charge is only renormalised by the photon blob, hence it is universal (the same
factor for all fermions, independent of their charge) and is not affected by QCD
at 1-loop. It is true that at higher orders the photon vacuum polarization diagram
is affected by QCD (for example, at 2-loops we can exchange a gluon between
the quarks in the photon loop) but the renormalisation induced by the vacuum
polarisation diagram remains independent of the nature of the fermion to which
the photon line is attached. The gluon contributions to the vertex (Z V ) and to the
Z J
Z v
Z f
Z f
J
J
J
J
f
f
+
+
+
+
+
J
f
f
f
f
f
f
J
f
f
Fig. 4.8 Diagrams for charge renormalisation in QED at 1-loop