92
G. Altarelli and S. Forte
Fig. 4.6 Deep inelastic
lepto-production
N
T
hadrons does not affect the rate (it happens with probability 1). We have already
mentioned that in order for this to be true within a given accuracy an averaging over
a sufficiently large bin of Q must be understood. The binning width is larger in the
vicinity of thresholds: for example when one goes across the charm c ¯
c threshold
the physical cross-section shows resonance bumps which are absent in the smooth
partonic counterpart which however gives an average of the cross-section.
A very important class of hard processes is Deep Inelastic Scattering (DIS)
l + N → l
+ X
l= e
± , μ
± , ν, ¯
ν
(4.13)
which has played and still plays a very important role for our understanding of QCD
and of nucleon structure. For the processes in Eq. (4.13), shown in Fig. 4.6, we have,
in the lab system where the nucleon of mass m is at rest:
Q
2
= − q
2
= − (k − k
)
2
= 4EE
sin
2 θ/2;
mν = (p.q);
x =
Q 2
2mν
(4.14)
In this case the virtual momentum q of the gauge boson is spacelike. x is the
familiar Bjorken variable. The DIS processes in QCD will be extensively discussed
in Sect. 4.5
4.3 The Renormalisation Group and Asymptotic Freedom
In this section we aim at providing a reasonably detailed introduction to the
renormalisation group formalism and the concept of running coupling which leads
to the result that QCD has the property of asymptotic freedom. We start with a
summary on how renormalisation works.
In the simplest conceptual situation imagine that we implement regularisation of
divergent integrals by introducing a dimensional cut-off K that respects gauge and
Lorentz invariance. The dependence of renormalised quantities on K is eliminated
by absorbing it into a redefinition of m (the quark mass: for simplicity we assume
a single flavour here), the gauge coupling e (can be e in QED or e s in QCD)
G. Altarelli and S. Forte
Fig. 4.6 Deep inelastic
lepto-production
N
T
hadrons does not affect the rate (it happens with probability 1). We have already
mentioned that in order for this to be true within a given accuracy an averaging over
a sufficiently large bin of Q must be understood. The binning width is larger in the
vicinity of thresholds: for example when one goes across the charm c ¯
c threshold
the physical cross-section shows resonance bumps which are absent in the smooth
partonic counterpart which however gives an average of the cross-section.
A very important class of hard processes is Deep Inelastic Scattering (DIS)
l + N → l
+ X
l= e
± , μ
± , ν, ¯
ν
(4.13)
which has played and still plays a very important role for our understanding of QCD
and of nucleon structure. For the processes in Eq. (4.13), shown in Fig. 4.6, we have,
in the lab system where the nucleon of mass m is at rest:
Q
2
= − q
2
= − (k − k
)
2
= 4EE
sin
2 θ/2;
mν = (p.q);
x =
Q 2
2mν
(4.14)
In this case the virtual momentum q of the gauge boson is spacelike. x is the
familiar Bjorken variable. The DIS processes in QCD will be extensively discussed
in Sect. 4.5
4.3 The Renormalisation Group and Asymptotic Freedom
In this section we aim at providing a reasonably detailed introduction to the
renormalisation group formalism and the concept of running coupling which leads
to the result that QCD has the property of asymptotic freedom. We start with a
summary on how renormalisation works.
In the simplest conceptual situation imagine that we implement regularisation of
divergent integrals by introducing a dimensional cut-off K that respects gauge and
Lorentz invariance. The dependence of renormalised quantities on K is eliminated
by absorbing it into a redefinition of m (the quark mass: for simplicity we assume
a single flavour here), the gauge coupling e (can be e in QED or e s in QCD)