Dynamical generation of extended objects in a ( 1 + 1 ) -dimensional chiral field theory: Nonperturbative Dirac operator resolvent analysis

We analyze the $(1+1)$-dimensional Nambu\char21{}Jona-Lasinio (NJL) model nonperturbatively. In addition to its simple ground-state saddle points, the effective action of this model has a rich collection of nontrivial saddle points in which the composite fields $\ensuremath{\sigma}(x)=〈\ensuremath{\psi}\phantom{\rule{0ex}{0ex}}\ifmmode\bar\else\textasciimacron\fi{}\ensuremath{\psi}〉$ and $\ensuremath{\pi}(x)=〈\ensuremath{\psi}\phantom{\rule{0ex}{0ex}}\ifmmode\bar\else\textasciimacron\fi{}i{\ensuremath{\gamma}}_{5}\ensuremath{\psi}〉$ form static space-dependent configurations because of nontrivial dynamics. These configurations may be viewed as one-dimensional chiral ``bags.'' We start our analysis of such configurations by asking what kind of initially static ${\ensuremath{\sigma}(x),\ensuremath{\pi}(x)}$ background configurations will remain so under fermionic back reaction. By simply looking at the asymptotic spatial behavior of the expectation value of the fermion number current we show, independently of the large-$N$ limit, that a necessary condition for this situation to occur is that ${\ensuremath{\sigma}(x),\ensuremath{\pi}(x)}$ give rise to a reflectionless Dirac operator. We provide an explicit formula for the diagonal resolvent of the Dirac operator in a reflectionless ${\ensuremath{\sigma}(x),\ensuremath{\pi}(x)}$ background which produces a prescribed number of bound states. We analyze in detail the cases of a single as well as two bound states. We explicitly check that these reflectionless backgrounds may be tuned such that the large-$N$ saddle-point condition is satisfied. Thus, in the case of the NJL model, reflectionlessness is also sufficient to assure the time independence of the background. In our view, these facts make our work conceptually simpler than the previous work of Shei and of Dashen, Hasslacher, and Neveu which were based on the inverse scattering formalism. Our method of finding such nontrivial static configurations may be applied to other $(1+1)$-dimensional field theories.