4 QCD: The Theory of Strong Interactions
119
g can be measured indirectly by scaling violations and directly from asymmetries,
e.g. in c ¯
c production. Existing measurements by Hermes, Compass, and at RHIC are
still crude but show no hint of a large g at accessible value of x and Q 2 . Present
data are consistent with g large enough to sizeably contribute to the spin sum rule
but there is no indication that α s g can explain the difference between constituents
and parton quarks. The perspectives of better measurements are good at Compass
and RHIC in the near future.
4.5.4 Factorisation and the QCD Improved Parton Model
The parton densities defined and measured in DIS are instrumental to compute
hard processes initiated by hadronic collisions via the Factorisation Theorem (FT).
Suppose you have a hadronic process of the form h 1 + h 2 → X + all where h i are
hadrons and X is some triggering particle or pair of particles which specify the large
scale Q 2 relevant for the process, in general somewhat, but not much, smaller than
s, the total c.o.m. squared mass. For example, in pp or p ¯
p collisions, X can be a W
or a Z or a virtual photon with large Q 2 , or a jet at large transverse momentum p T ,
or a pair of heavy quark-antiquark of mass M. By “all” we mean a totally inclusive
collection of gluons and light quark pairs. The FT states that for the total crosssection or some other sufficiently inclusive distribution we can write, apart from
power suppressed corrections, the expression:
σ (s, τ ) =
AB
dx 1 dx 2 p 1A (x 1 , Q
2 )p 2B (x 2 , Q
2 )σ AB (x 1 x 2 s, τ )
(4.97)
Here τ = Q 2 /s is a scaling variable, p iC are the densities for a parton of type
C inside the hadron h i , σ AB is the partonic cross-section for parton-A + partonB→ X + all . This result is based on the fact that the mass singularities that are
associated with the initial legs are of universal nature, so that one can reproduce
the same modified parton densities, by absorbing these singularities into the bare
parton densities, as in deep inelastic scattering. Once the parton densities and α s are
known from other measurements, the prediction of the rate for a given hard process
is obtained with not much ambiguity (e.g from scale dependence or hadronisation
effects). The NLO calculation of the reduced partonic cross-section is needed in
order to correctly specify the scale and in general the definition of the parton
densities and of the running coupling in the leading term. The residual scale and
scheme dependence is often the most important source of theoretical error. In the
following we consider a few examples.
A comparison of data and predictions on the production of jets at large
√
s and
p T in pp or p ¯
p collisions is shown in Fig. 4.15 [9, 35].
This is a particularly significant test because the rates at different c.o.m. energies
and, for each energy, at different values of p T span over many orders of magnitude.
119
g can be measured indirectly by scaling violations and directly from asymmetries,
e.g. in c ¯
c production. Existing measurements by Hermes, Compass, and at RHIC are
still crude but show no hint of a large g at accessible value of x and Q 2 . Present
data are consistent with g large enough to sizeably contribute to the spin sum rule
but there is no indication that α s g can explain the difference between constituents
and parton quarks. The perspectives of better measurements are good at Compass
and RHIC in the near future.
4.5.4 Factorisation and the QCD Improved Parton Model
The parton densities defined and measured in DIS are instrumental to compute
hard processes initiated by hadronic collisions via the Factorisation Theorem (FT).
Suppose you have a hadronic process of the form h 1 + h 2 → X + all where h i are
hadrons and X is some triggering particle or pair of particles which specify the large
scale Q 2 relevant for the process, in general somewhat, but not much, smaller than
s, the total c.o.m. squared mass. For example, in pp or p ¯
p collisions, X can be a W
or a Z or a virtual photon with large Q 2 , or a jet at large transverse momentum p T ,
or a pair of heavy quark-antiquark of mass M. By “all” we mean a totally inclusive
collection of gluons and light quark pairs. The FT states that for the total crosssection or some other sufficiently inclusive distribution we can write, apart from
power suppressed corrections, the expression:
σ (s, τ ) =
AB
dx 1 dx 2 p 1A (x 1 , Q
2 )p 2B (x 2 , Q
2 )σ AB (x 1 x 2 s, τ )
(4.97)
Here τ = Q 2 /s is a scaling variable, p iC are the densities for a parton of type
C inside the hadron h i , σ AB is the partonic cross-section for parton-A + partonB→ X + all . This result is based on the fact that the mass singularities that are
associated with the initial legs are of universal nature, so that one can reproduce
the same modified parton densities, by absorbing these singularities into the bare
parton densities, as in deep inelastic scattering. Once the parton densities and α s are
known from other measurements, the prediction of the rate for a given hard process
is obtained with not much ambiguity (e.g from scale dependence or hadronisation
effects). The NLO calculation of the reduced partonic cross-section is needed in
order to correctly specify the scale and in general the definition of the parton
densities and of the running coupling in the leading term. The residual scale and
scheme dependence is often the most important source of theoretical error. In the
following we consider a few examples.
A comparison of data and predictions on the production of jets at large
√
s and
p T in pp or p ¯
p collisions is shown in Fig. 4.15 [9, 35].
This is a particularly significant test because the rates at different c.o.m. energies
and, for each energy, at different values of p T span over many orders of magnitude.