Spectral density in the resonance region and analytic confinement

We study the role of finite widths of resonances in a nonlocal version of the Wick-Cutkosky model. The spectrum of bound states is known analytically in this model and forms linear Regge tragectories. We compute the widths of resonances, calculate the spectral density in an extension of the Breit-Wigner ansatz and discuss a mechanism for the damping of unphysical exponential growth of observables at high energy due to finite widths of resonances.

Figure 1

Graphical representation of various terms in the effective action.

Figure 2

The two-particle decay width ${\Gamma }_n(M_n^2)$ as a function of radial quantum number. Deviation from strictly exponential behavior is due to the polynomial $P_n(s)$ in ${\Gamma }_n(s)\propto P_n(s)\exp (\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}s)$.

Figure 3

Graphical representation of the polarization function.

Figure 4

Spectral density in the Breit-Wigner approximation as defined by Eq. (22)

Figure 5

Spectral density $\rho (s)$ as a function of the energy variable $s$ (upper plot) and the contributions ${\rho }_n(s)$ of few lowest radial numbers to the total spectral density (lower plot)

Figure 6

Two-particle decay widths ${\Gamma }_n(s)$ as a function of energy variable $s$ for the few lowest resonances.