We explore possible consequences of the dynamic chiral symmetry breaking arising due to the Nambu and Jona-Lasinio (NJL) mechanism. The local four-quark interactions of NJL type are known to be a useful ground to construct the effective Lagrangians describing the dynamics of collective quark-antiquark bound states formed in the strong coupling regime. These Lagrangians suffer from the mixing between the Goldstone and axial-vector modes. A standard diagonalization creates the problem in the description of anomalous processes which includes the electromagnetic interactions of soft pions and leads to the violation of a number of famous low-energy theorems of quantum chromodynamics. Here we present a solution to this problem. It is based on the careful treatment of surface terms arising in the calculation of anomalous triangle diagrams. The proposed formalism does not support the vector meson dominance hypotheses.
The other aspect of the NJL mechanism is the Nambu sum rules, which relates the spectrum of collective modes with the energy gap in the spectrum of quasi-particle excitations. We calculate the spectrum of spinless modes in the NJL type model with the $SU(2)_L \times U(1)_R$ symmetrical four-quark-interaction proposed by Miransky, Tanabashi, and Yamawaki to explain the huge mass of the top-quark in the standard model of electroweak interactions. We show that the $U(1)_A$ symmetry breaking, for which the ’t Hooft four-quark interaction is responsible, causes deviation from the standard Nambu sum rule. Nonetheless, we demonstrate that the Nambu sum rule is not violated to the leading $1/N_c$ order.
Taking into account the principle of causality the relativistic equations for the multi-channel hadron scattering amplitudes are derived within the $S$-matrix method of Bogolyubov. The resulting equations have the form of three-dimensional time-ordered relativistic equations. The form and structure of these three-dimensional relativistic equations does not change if the quark degrees of freedom are taken into account based on the Huang-Weldon formulation, where hadrons are considered as bound states of quarks.
The equivalence of the field-theoretical $S$-matrix methods of Bogolyubov and Lehman-Simanczyk-Zimmerman is considered.
It is shown that take into account the causality principle in the suggested field-theoretical equations reduces to the special shifts on magnitudes of the momentums in ${\overline s}$, $u$, ${\overline u}$, $t$, ${\overline t}$ channels. At the same time, unitarity in the $s$-channel is preserved for equations with and without quark degrees of freedom.
Numerical solutions of the obtained equations do not require additional efforts in comparison with similar Lippmann-Schwinger equations in non-relativistic collision theory. A good description was obtained of the experimental phases of elastic $\pi N$ and $NN$ scattering in the low energy region Quark-parton model for inclusive generation of particles with large transverse momentum is reproduced using a separable approximation. This made it possible to describe experimental data of the inclusive production of a $\rho$-meson with a large transverse momentum ($< 1.5~GeV/s$) in a proton-proton collision in the energy region $2.9 < \sqrt{s} < 64~GeV$.
The processes $\tau \to K^{-} [\pi^{0}, \eta, K^{0}] \nu_{\tau}$ are considered with the use of the Nambu--Jona-Lasinio model. The final state interaction is taken into account as an additional corrections which are outside of the approximation of the model. The additional parameter arising from this corrections is fixed by experimental data.