Chiral symmetry restoration in excited hadrons, quantum fluctuations, and quasiclassics

In this paper, we discuss the transition to the semiclassical regime in excited hadrons, and consequently, the restoration of chiral symmetry for these states. We use a generalized Nambu–Jona-Lasinio model with the interaction between quarks in the form of the instantaneous Lorentz-vector confining potential. This model is known to provide spontaneous breaking of chiral symmetry in the vacuum via the standard self-energy loops for valence quarks. It has been shown recently that the effective single-quark potential is of the Lorentz-scalar nature, for the low-lying hadrons, while, for the high-lying states, it becomes a pure Lorentz vector and hence the model exhibits the restoration of chiral symmetry. We demonstrate explicitly the quantum nature of chiral symmetry breaking, the absence of chiral symmetry breaking in the classical limit, as well as the transition to the semiclassical regime for excited states, where the effect of chiral symmetry breaking becomes only a small correction to the classical contributions.

Figure 1

Graphical representation of the equations for the dressed-quark propagator, Eq. (5), and for the quark mass operator, Eq. (7).

Figure 2

Planar [figure (a)] and nonplanar, suppressed by $N_C$ [figure (b)], diagrams.

Figure 3

Nontrivial solution to the mass-gap equation (17) with $m=0$ and for the linear confinement (see, for example, Refs. 41, 42 for the details). The momentum $p$ is given in the units of $\sqrt{\sigma }$.

Figure 4

Graphical representation of the Bethe-Salpeter equation (27) for the quark-antiquark bound state, in the ladder approximation.