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

## TL;DR

This paper investigates the spectrum of a two-particle bound state in massive $ ext{N}=4$ super Yang-Mills using relativistic quantum mechanics, revealing an exact degeneracy pattern consistent with non-relativistic limits and integrability insights.

## Contribution

It provides a relativistic quantum mechanical analysis of bound states in $ ext{N}=4$ super Yang-Mills, confirming degeneracy patterns previously observed in integrability studies.

## Key findings

- Spectrum exhibits exact $n^2$ degeneracy for each principal quantum number $n$.
- Results align with integrability and dual conformal symmetry findings.
- Supports non-relativistic limit behavior in a relativistic framework.

## Abstract

Using relativistic quantum mechanics, we study the spectrum of a non-BPS two-particle bound state in the massive phase of $\mathcal{N}=4$ super Yang-Mills, in the limit when one of the particles is infinitely heavier than the other. We find that the spectrum shows the exact $n^2$ degeneracy for each principal quantum number $n$, just as in the strict non-relativistic limit. This is in line with the findings of Caron-Huot and Henn, who studied the same system in the large $N$ limit with the technique of integrability and the dual conformal symmetry.