Collapse of Coulomb Bound States of Vector Bosons

TL;DR

This work analyzes a charged vector (spin-1) particle in a Coulomb field, showing that the short-distance collapse present for a point-like nucleus is regularized by introducing a finite nuclear radius $R$ and by including the $\Upsilon$ term in the wave equation. The problem is solved for $mR \ll Z\alpha \ll 1$, focusing on the longitudinal $j=0$ sector, and reveals two spectral sectors: a tower of intranuclear states that accumulate as $R\to 0$ and lie in the negative-energy continuum, causing vacuum breakdown and charge screening, and ordinary Sommerfeld-like states that leak into the nucleus. Vacuum polarization modifies the short-distance behavior, producing different barriers depending on the theory’s ultraviolet behavior, while the charge density exhibits sign changes and a finite intranuclear fraction due to the $\Upsilon$ term. The results highlight distinctive spin-1 Coulomb dynamics, showing that short-distance regularization, vacuum effects, and finite-size corrections fundamentally alter the bound-state spectrum compared to scalar or spinor cases.

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.