Solvable Relativistic Hydrogenlike System in Supersymmetric Yang-Mills Theory

The classical Kepler problem, as well as its quantum mechanical version, the hydrogen atom, enjoys 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 nonrelativistic limit, we identify the dual conformal symmetry of planar $\mathcal{N}=4$ super Yang-Mills theory 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.

Figure 1

Four-point amplitude in $\mathcal{N}=4$ SYM with nontrivial scalar vacuum expectation values. Thick lines correspond to massive $W$ bosons, while dashed lines correspond to massless particles.

Figure 2

Different limits of the four-point amplitude that are equivalent thanks to dual conformal symmetry. The double lines denote Wilson lines.

Figure 3

Regge trajectories of hydrogenlike states in $\mathcal{N}=4$ SYM theory for $\lambda =5$, 10, 10, 30, 100 (bottom to top). The solid-blue lines use the weak-coupling formulas while the dashed-red lines use the large-$\lambda$ formulas (see text). The bound states (crosses) are obtained by equating $j$ to an integer. The inset shows the same curves with the total energy in units of mass on the horizontal axis.