Collapse of Coulomb Bound States of Vector Bosons

Abstract

Charged spin 1 (vector) particles behave very differently from electrons or scalars in a Coulomb field. For an infinitely heavy point-like nucleus their bound state wave functions fall to the centre, and embedding the system in a renormalisable electroweak-type theory does not remedy this short-distance pathology. We therefore solve the pure Coulomb problem for a finite nuclear radius $R$ and recover the point nucleus limit by letting $R\to 0$. This approach allows us to include the crucial Upsilon term in the wave equations, which for the point-like nucleus is proportional to delta(r) and was ignored in the previous calculations of the energy spectrum. Several unusual effects emerge: (i) The Upsilon term supports a tower of states located mainly inside the nucleus. As R -> 0 their number diverges, most lying in the negative energy continuum (energy epsilon < - m c^2). They trigger vacuum breakdown - particle-antiparticle pair creation that ultimately screens the nuclear charge. (ii) Ordinary Sommerfeld-like states (with binding energy smaller m c^2) persist, but a finite fraction of each wave function leaks into the nucleus, even as R -> 0. (iii) Charge density of a negatively charged vector particle changes sign in a vicinity of the nucleus and becomes positive charge density, whereas the Upsilon term ensures its density inside the nucleus remains negative. (iv) For weak coupling, Z alpha << 1, yet with mR <Z alpha, the non-relativistic solution differs qualitatively from Schrodinger theory despite binding energies are well below m c^2; agreement is recovered only when Z alpha << mR. These phenomena highlight the distinctive and subtle behaviour of spin-1 particles in the Coulomb field.