The different observables

We have discussed in the previous two parts of this book, deep inelastic scatterings in hadron colliders and different hard processes in e+e– annihilations. These hard processes have been used for testing the underlying ideas of perturbative QCD at short distances. Among others, one has studied and measured:

• Scaling violations in different parton model sum rules.

• Structure functions.

• Spin content of the proton.

• Fragmentation functions.

• Spin of the photon.

• One hadron inclusive production.

• Jets.

• Total inclusive e+e– cross-sections.

• Hadronic τ and Z0 decays.

In all these hard processes, most of the perturbative QCD predictions based on the SU(3)c colour group and on asymptotic freedom properties have been confirmed by the data.

Different tests of QCD

The main outcomes of these analysis in the previous parts of the book are given in the following sections.

Deep inelastic scatterings

• A measurement of the scaling violations to parton model predictions in deep inelastic processes using different moments of the structure functions as predicted by QCD. In the unpolarized case, one has used these processes to extract the value of the QCD running coupling. In the polarized case, one has been able to emphasize the important universal rôle of the QCD anomaly for explaining the relative suppression of the first moment of the structure function compared to the OZI prediction (so-called proton spin) and a proposal for testing its effect from the measurement of the photon spin in γ-γ scattering processes, and of some semi-inclusive processes.

• […]

Pseudoscalar gluonia

We shall be concerned with the correlator:

where QR(x) is the renormalized gluon topological density which mixes under renormalization with the divergence of the flavour singlet axial current mix as follows [129] (see Section 10.3.3 in Part III):

where:

and we have quoted the formulae for nf flavours. The correlation function X(k2) obeys the inhomogeneous RGE [260]:

The anomalous dimension is:

The extra RG function β(L) (so called because it appears in the longitudinal part of the Green function of two axial currents) is given by

The RGE is solved in the standard way, giving

where αs(t) is the running coupling. The different QCD diagrams contributing to the correlator are shown in Fig. 37.1.

The perturbative expression for the two-point correlation function in the scheme is [455]:

The non-perturbative contribution from the gluon condensates (coming from the next lowest dimension operators in the OPE) is [382]:

Pseudoscalar meson-gluonium mixing

Let us consider the off-diagonal two-point correlator:

shown in Fig. 37.2.

Its QCD expression reads [458]:

where one can notice that the mixing from the OPE vanishes in the chiral limit. However, one should notice that this mixing acts on the gluonium propagator, that is, it affects the mass splitting but not its decay width which is governed by a three-point function. Unfortunately, several authors mix these two features in the literature.