No, it is not a portrait of one of the authors! This is merely a Feynman’s diagram, which represents propagation of a W-boson (head and hands) in the Coulomb field of a heavy particle (feet), which is screened by the vacuum polarization (body). Main result: the body (vacuum polarization) keeps the head (W-boson) apart from the feet, thus resolving the Corben-Schwinger paradox.

We have resolved a challenging theoretical puzzle, which manifested itself in the Coulomb problem for W-bosons (massive charged vector particles with spin S=1). It has been found previously that in an attractive Coulomb field the W-bosons fall to the Coulomb centre (provided the latter has small size). This collapse of W-bosons was discovered in the pioneering works of several research groups that included a number of brilliant scientists (Oppenheimer, as well as two Nobel Prize laureates Tamm and Schwinger, were among them) and presented a serious nuisance for the theory known as the Standard Model.

The point was that the fall of the boson to the centre came into contradiction with a very general property of the theory known as the renormalizability. Speaking plainly, renormalizability means that no weird phenomena should take place at small distances. Thus, direct calculations unambiguously demonstrated that the boson falls to the center, while the renormalizability states that this event should not take place. This controversy persisted for 66 long years, in spite of a number of attempts to resolve it.

Our work provides an elegant way out of the problem by including the vacuum polarization into calculations, which modifies the Coulomb field at small distances. This solution looks so simple that at first glance it is unclear why it took so long to suggest it. This simplicity is deceptive though. The vacuum polarization is known to make the attractive potential stronger. One could have anticipated, therefore, that the polarization should stimulate the collapse of the W-bosons, making the mess in the Coulomb problem only worse.

A careful analysis shows that vector bosons exhibit a very specific behaviour (different from scalars and spinors). The vacuum polarization for vector particles results in an effective, very strong repulsion at small distances, which prevents the collapse, making the Coulomb problem for vector bosons well defined for the first time. It also ensures that it complies with the renormalizability of the Standard Model. Along with its scientific significance, the Coulomb problem for vector bosons can serve as a new interesting example of a solvable problem in quantum field theory for teaching and education purposes.

Michael Kuchiev and Victor Flambaum