New class of quantum bound states: Diprotons in extreme magnetic fields

Abstract

This paper considers the possibility that two charged particles with an attractive short-ranged potential between them which is not strong enough to form bound states in free space, may bind in uniform magnetic fields. It is shown that in the formal limit where Coulomb repulsion is negligible ($q\ensuremath{\rightarrow}0\phantom{\rule{0.3em}{0ex}}\text{and}\phantom{\rule{0.3em}{0ex}}{B}_{0}\ensuremath{\rightarrow}\ensuremath{\infty}\phantom{\rule{0.3em}{0ex}}\text{with}\phantom{\rule{0.3em}{0ex}}q{B}_{0}\phantom{\rule{0.3em}{0ex}}\text{fixed}$, where $q$ is the charge and ${B}_{0}$ the field strength) there always exists a bound state for a system of two identical charged particles in a constant magnetic field, provided that there is a short-range uniformly attractive potential between them. Moreover, it is shown that in this limit any potential with an attractive s-wave scattering length will possess bound states provided that the range of the potential is much smaller than the characteristic magnetic length, ${r}_{0}\ensuremath{\equiv}(\frac{q{B}_{0}}{4}){}^{\ensuremath{-}1/2}$. For this case, the binding is computed numerically. We estimate the size of the magnetic field needed to approximately reach a regime where the formal limit considered becomes a good approximation to the dynamics. These numerical estimates indicate that two protons in an extremely strong magnetic field such as might be found in a magnetar will bind to form a diproton.