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Dynamical Generation of Extended Objects in a $1+1$ Dimensional Chiral Field Theory: Non-Perturbative Dirac Operator Resolvent Analysis

We analyze the $1+1$ dimensional Nambu-Jona-Lasinio model non-perturbatively. In addition to its simple ground state saddle points, the effective action of this model has a rich collection of non-trivial saddle points in which the composite fields $\sigx=\lag\barψψ\rag$ and $\pix=\lag\barψi\gam_5ψ\rag$ form static space dependent configurations because of non-trivial dynamics. These configurations may be viewed as one dimensional chiral bags that trap the original fermions (``quarks") into stable extended entities (``hadrons"). We provide explicit expressions for the profiles of these objects and calculate their masses. Our analysis of these saddle points is based on an explicit representation we find for the diagonal resolvent of the Dirac operator in a $\{\sigx, \pix\}$ background which produces a prescribed number of bound states. We analyse in detail the cases of a single as well as two bound states. We find that bags that trap $N$ fermions are the most stable ones, because they release all the fermion rest mass as binding energy and become massless. Our explicit construction of the diagonal resolvent is based on elementary Sturm-Liouville theory and simple dimensional analysis and does not depend on the large $N$ approximation. These facts make it, in our view, simpler and more direct than the calculations previously done by Shei, using the inverse scattering method following Dashen, Hasslacher, and Neveu. Our method of finding such non-trivial static configurations may be applied to other $1+1$ dimensional field theories.