Solvable Relativistic Hydrogenlike System in Supersymmetric Yang-Mills Theory

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.

4 pages, 3 figures. Clarifications added; published version

References in corpus (6)

• Gluon scattering amplitudes at strong coupling
• Iterative structure of finite loop integrals
• Higgs-regularized three-loop four-gluon amplitude in N=4 SYM: exponentiation and Regge limits
• Spinor Helicity and Dual Conformal Symmetry in Ten Dimensions
• Dual Conformal Properties of Six-Dimensional Maximal Super Yang-Mills Amplitudes
• The static potential in {\cal N}=4 supersymmetric Yang-Mills at weak coupling

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• On hydrogen-like bound states in $\mathcal{N}=4$ super Yang-Mills
• Extremal Black Holes as Relativistic Systems with Kepler Dynamics
• Leading anomalies, the drift Hamiltonian and the relativistic two-body system