Solvable Relativistic Hydrogenlike System in Supersymmetric Yang-Mills Theory

TL;DR

This paper demonstrates that the dual conformal symmetry in planar  super Yang-Mills theory extends the hidden symmetries of the Hydrogen atom to a relativistic quantum field theory context, enabling new spectrum computations.

Contribution

It establishes a connection between dual conformal symmetry and superintegrability in a relativistic quantum field theory, providing a novel approach to bound state spectrum calculations.

Findings

Identifies dual conformal symmetry as a relativistic extension of Hydrogen atom symmetries.

Provides tests of the symmetry's implications at weak and strong coupling.

Suggests potential for extending the approach to arbitrary coupling values.

Abstract

The classical Kepler problem, as well as its quantum mechanical version, the Hydrogen atom, enjoy a well-known hidden symmetry, the conservation of the Laplace-Runge-Lenz vector, which makes these problems superintegrable. Is there a relativistic quantum field theory extension that preserves this symmetry? In this Letter we show that the answer is positive: in the non-relativistic limit, we identify the dual conformal symmetry of planar $\mathcal{N}=4$ super Yang-Mills with the well-known symmetries of the Hydrogen atom. We point out that the dual conformal symmetry offers a novel way to compute the spectrum of bound states of massive $W$ bosons in the theory. We perform nontrivial tests of this setup at weak and strong coupling, and comment on the possible extension to arbitrary values of the coupling.