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

Abstract

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.

References in corpus (2)

• On Kinks and Bound States in the Gross-Neveu Model
• Non Trivial Saddle Points and Band Structure of Bound States of the Two Dimensional O(N) Vector Model

Cited by in corpus (21)

• Inhomogeneous Condensates in the Thermodynamics of the Chiral NJL_2 model
• Effective Tachyon Dynamics in Superstring Theory
• Self-consistent crystalline condensate in chiral Gross-Neveu and Bogoliubov-de Gennes systems
• A Twisted Kink Crystal in the Chiral Gross-Neveu model
• All about the Static Fermion Bags in the Gross-Neveu Model
• Exotic supersymmetry of the kink-antikink crystal, and the infinite period limit
• Crystals of superconducting Baryonic tubes in the low energy limit of QCD at finite density
• Supersymmetric twisting of carbon nanotubes
• Pion crystals hosting topologically stable baryons
• Analytic SU(N) Skyrmions at finite Baryon density
• Self-Consistent Large-$N$ Analytical Solutions of Inhomogneous Condensates in Quantum $\mathbb{C}{P}^{N-1}$ Model
• Extended supersymmetry of the self-isospectral crystalline and soliton chains
• Divergence of the axial current and fermion density in Gross-Neveu models
• Covariant boost and structure functions of baryons in Gross-Neveu models
• The periodic table of static fermion bags in the Gross-Neveu Model
• Confining solitons in the Higgs phase of $\mathbb{C}{P}^{N-1}$ model: Self-consistent exact solutions in large-$N$ limit
• Analytic non-homogeneous condensates in the $(2+1)$-dimensional Yang-Mills-Higgs-Chern-Simons theory at finite density
• Fermion Bag Solitons in the Massive Gross-Neveu and Massive Nambu-Jona-Lasinio Models in 1+1 Dimensions: Inverse Scattering Analysis
• Analytic Baby Skyrmions at Finite Density
• On instability of ground states in 2D CP(N-1) and O(N) models at large N
• The Dirac Operator in a Fermion Bag Background in 1+1 Dimensions and Generalized Supersymmetric Quantum Mechanics