A variational study of some hadron bag models

TL;DR

This paper rigorously analyzes hadron bag models from quantum chromodynamics using variational methods, proving existence of solutions, Gamma-convergence, and deriving the M.I.T. bag equations through mathematical techniques.

Contribution

It introduces a variational framework for hadron bag models, proving existence of solutions, Gamma-convergence of energy functionals, and rigorously deriving the M.I.T. bag equations.

Findings

Existence of excited and ground state solutions for bag models.

Gamma-convergence of bag approximation energy functionals to soliton bag models.

Rigorous derivation of the M.I.T. bag equations from the models.

Abstract

Quantum chromodynamics (QCD) is the theory of strong interaction and accounts for the internal structure of hadrons. Physicists introduced phe- nomenological models such as the M.I.T. bag model, the bag approximation and the soliton bag model to study the hadronic properties. We prove, in this paper, the existence of excited state solutions in the symmetric case and of a ground state solution in the non-symmetric case for the soliton bag and the bag approximation models thanks to the concentration compactness method. We show that the energy functionals of the bag approximation model are Gamma -limits of sequences of soliton bag model energy functionals for the ground and excited state problems. The pre- compactness, up to translation, of the sequence of ground state solutions associated with the soliton bag energy functionals in the non-symmetric case is obtained combining the Gamma -convergence theory and the concentration-compactness method. Finally, we give a rigorous proof of the original derivation of the M.I.T. bag equations done by Chodos, Jaffe, Johnson, Thorn and Weisskopf via a limit of bag approximation ground state solutions in the spherical case. The supersymmetry property of the Dirac operator is the key point in many of our arguments.