Quantum Chromodynamics (QCD) is the field theory governing the
strong interactions of hadrons. At high energies, due to asymptotic
freedom, perturbation theory is applicable, whereas at low energies
relevant for hadronic bound states (strong QCD), non-perturbative
techniques are required. One of these techniques, the field-theoretical approach of the Dyson-Schwinger Equations (DSEs),
is utilized in the present study.
Mesons are the simplest hadrons, and thus are an excellent
“laboratory” to investigate strong QCD. In particular the properties
of the pion, the lightest pseudoscalar meson, is determined by
non-perturbative effects such as dynamical chiral symmetry breaking
(DCSB). In order to gain a deeper understanding of strong QCD, we
investigate two rather different aspects of non-perturbative dynamics for
meson physics in this work.
The first aspect deals with the transition between the perturbative
and non-perturbative regimes of QCD, in particular the determination
of the distance scale for the onset of non-perturbative dynamics.
Correlation functions (correlators) with meson quantum numbers,
which are vacuum expectation
values of products of gauge-invariant local operators, are ideally
suited for this type of investigation.
We consider the vector and axial-vector
correlators built from vector and axial-vector currents respectively.
We investigate the difference (V-A correlator), sum (V+A correlator),
and ratio of the difference and
sum of these correlators. In the chiral (massless) limit,
to any finite order of perturbation theory, the vector and
are identical. Thus the way the difference (V-A) correlator increases
as momentum decreases is a measure of the onset of non-perturbative
dynamics. It can provide information on the associated distance scale
and the four quark condensate. The V+A correlator remains close to
free-field behavior for distances as large as 1 fm. We therefore use
the ratio of the V-A and V+A correlators as a probe. The requisite
non-perturbative inputs to the calculation are DSE solution for the
dressed quark propagator and an Ansatz for the vector and axial vector
The extracted four-quark
condensate is compared to results from other models and to the
prediction of the vacuum saturation Ansatz.
Using Fourier transforms, we
calculate the distance scale relevant to the onset of dynamical chiral
symmetry breaking and, by implication, of non-perturbative dynamics.
Our results are compared to results from QCD sum rule calculations,
lattice QCD, and instanton physics.
The second aspect involves the evaluation of the valence quark
distributions in the light pseudoscalar mesons: pions and kaons.
Quark distributions in hadrons are intrinsically non-perturbative
and thus are currently determined from
structure functions measured experimentally in processes like the
deep inelastic hadron-lepton scattering and the Drell-Yan lepton-pair
production. These distributions give the probability densities of
finding a quark carrying a fraction x of the parent hadron's momentum,
at a resolving scale Q. We work in the Bjorken limit (very large Q)
and concentrate on the valence quark for which the so-called
“handbag” diagram mechanism is considered sufficient.
Non-perturbative inputs such as the dressed quark propagators and the
bound state wave function are taken from DSE solution and
Bethe-Salpeter solution respectively. The valence quark distributions
in the pion and kaon are compared to available data. This is the first
time that bound state descriptions of the quality provided by the
Bethe-Salpeter solutions have been compared to the quark distributions
measured in deep inelastic scattering.
Using the leading order DGLAP evolution
equation for the nonsinglet structure function to evolve to relevant
experimental scales, we compare our results with existing FermiLab data on
the pion at Q = 4.05 GeV, the recent reanalysis of data at Q =
5.2 GeV, and an earlier theoretical model which is a primitive
version of the current model.
We also compare the ratio of the kaon to pion distributions
with the Drell-Yan experimental data that produced such
information. Approximations used in the formulation are critically
evaluated and discussed.