Solvable Relativistic Hydrogenlike System in Supersymmetric Yang-Mills Theory

Abstract

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 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.

On hydrogen-like bound states in N $$ \mathcal{N} $$ = 4 super Yang-Mills

“…They have found that the split of the degeneracy due to the relativistic correction is milder with more supersymmetry. The result of [4] and ours can be thought of as a confirmation of the continuation of this trend to N = 4 supersymmetry. 1 The rest of the note is organized as follows.…”

“…shows an n 2 degeneracy for each principal quantum number n. This can be understood from the existence of an accidental conserved charge [1] which extends the rotational symmetry so(3) to so (4). One way to see this is as follows.…”

“…[3]. Recently, it was shown by Caron-Huot and Henn [4] that in N = 4 super Yang-Mills theory, there is a two-particle bound state where relativistic corrections still preserve the extended so(4) symmetry. The analysis there was carried out using sophisticated techniques of integrability of the N = 4 theory, developed during the last decade and a half.…”

“…More specifically, he mapped the momentum space R 3 by an inverse stereographic projection to S 3 , and showed that the Coulomb interaction written in this manner is invariant under the natural so(4) action on this S 3 . Therefore it is indeed natural to attribute this so(4) symmetry to be the unbroken part of the dual conformal symmetry so (4,2) in the presence of the rest mass of the bound state.…”

Quark-anti-quark potential in N $$ \mathcal{N} $$ = 4 SYM

“…Remarkably, we also observed the appearance of several distinct scales in the QSC which may be thought of as a counterpart to this effective field theory description. In the future it will be also interesting to apply the QSC to study the energies of hydrogen-like bound states in N = 4 SYM [30] which are also related to a φ → π limit. Moreover, our weak coupling results may be useful to establish connections with QCD, similarly to e.g.…”