The MIT Bag Model as an infinite mass limit

Naiara Arrizabalaga,Albert Mas,Loïc Le Treust,Nicolas Raymond

doi:10.5802/jep.95

Abstract

The Dirac operator, acting in three dimensions, is considered. Assuming that a large mass $m>0$ lies outside a smooth and bounded open set $\Omega\subset\R^3$, it is proved that its spectrum is approximated by the one of the Dirac operator on $\Omega$ with the MIT bag boundary condition. The approximation, which is developed up to and error of order $o(1/\sqrt m)$, is carried out by introducing tubular coordinates in a neighborhood of $\partial\Omega$ and analyzing the corresponding one dimensional optimization problems in the normal direction.

Highlights

Context. — This paper is devoted to the spectral analysis of the Dirac operator with high scalar potential barrier in three dimensions
See [8, 10], it is expected that, when m becomes large, the eigenfunctions of low energy do not visit R3 Ω and tend to satisfy the so-called MIT bag condition on ∂Ω. This boundary condition, that we will define is usually chosen by the physicists [13, 10, 11], in order to get a vanishing normal flux at the bag surface. It was originally introduced by Bogolioubov in the late 60 s [8] to describe the confinement of the quarks in the hadrons with the help of an infinite scalar potential barrier outside a fixed set Ω
In the mid 70 s, this model has been revisited into a shape optimization problem named MIT bag model [13, 10, 11] in which the optimized energy takes the form

Summary

Introduction

Regarding the first eigenvalue of the MIT bag Dirac operator, we find the first order term in the asymptotic expansion of the eigenvalues given by the high scalar potential barrier, showing its dependence on geometric quantities related to ∂Ω This is a novel result with respect to the ones in [16]. We consider particles with large effective mass m m0 outside Ω Their kinetic energy is associated with the self-adjoint operator (Hm, Dom(Hm)) defined by. It is the MIT bag Dirac operator with reversed boundary condition (see Definition 1.2). In [14] the authors study interactions of the free Dirac operator in R3 with potentials that shrink towards ∂Ω, proving convergence in the strong resolvent sense to δ-shell interactions with precise coupling constants. In order to ease the reading, we provide here a list of notation regarding the spaces and the quadratic forms, as well as the equation number where they are introduced, that we will use in the sequel: Key Space domain Variational space Quadratic form Infimum

About the exterior optimization problem

Lower bounds

Proof of the main theorem