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

axial-vector correlators

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

vertices.

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.