Critical Charges on Strange Quark Nuggets and Other Extended Objects

TL;DR

This paper studies the critical charge for spontaneous pair production in extended charged objects like strange quark nuggets, confirming a linear relation between critical charge and radius for large sizes using Dirac equation solutions.

Contribution

It provides a detailed analysis of the critical charge behavior for various charge distributions, extending previous models with new numerical results and potential modifications.

Findings

Confirmed linear relation Z_C=0.71 R for large R in spherical shell models.

Found similar Z_C~R relation for uniformly charged spheres.

Explored effects of potential modifications on Z_C behavior.

Abstract

We investigate the behavior of the critical charge for spontaneous pair production, $Z_C$, defined as the charge at which the total energy of a $K$-shell electron is $E=-m_e$, as a function of the radius $R$ of the charge distribution. Our approach is to solve the Dirac equation for a potential $V(r)$ consisting of a spherically symmetrical charge distribution of radius $R$ and a Coulomb tail. For a spherical shell distribution of the type usually associated with color-flavor locked strange quark nuggets, we confirm the relation $Z_C=0.71 R({\rm fm})$ for sufficiently large $R$ obtained by Madsen, who used an approach based on the Thomas-Fermi model. We also present results for a uniformly charged sphere and again find that $Z_C\sim R$ for large enough $R$. Also discussed is the behavior of $Z_C$ when simple {\it ad hoc} modifications are made to the potential for $0\leq r < R$.