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.

arXiv comment4 pages, 3 figures. Clarifications added; published version

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• Quantum Spectral Curve for a Cusped Wilson Line in N=4 SYM
• Quark--anti-quark potential in N=4 SYM
• Four-graviton scattering to three loops in $\mathcal{N}=8$ supergravity
• Massive Conformal Symmetry and Integrability for Feynman Integrals
• Constructing $\mathcal{N}=4$ Coulomb Branch Superamplitudes
• Subleading Regge limit from a soft anomalous dimension
• Yangian Bootstrap for Massive Feynman Integrals
• What can we learn about QCD and collider physics from N=4 super Yang-Mills?
• Conformal Four-Point Integrals: Recursive Structure, Toda Equations and Double Copy
• Relativistic Runge-Lenz vector: from $\mathcal{N}=4$ SYM to SO(4) scalar field theory
• On hydrogen-like bound states in $\mathcal{N}=4$ super Yang-Mills
• Cusp Anomalous dimension and rotating open strings in AdS/CFT
• Leading anomalies, the drift Hamiltonian and the relativistic two-body system
• Extremal Black Holes as Relativistic Systems with Kepler Dynamics
• A note on scattering amplitudes on the moduli space of ABJM