  This search, a type not previously performed at ATLAS, uses a comparison of
the production cross sections for $e^+ \mu^-$ and $e^- \mu^+$ pairs to
constrain physics processes beyond the Standard Model. It uses $139
\text{fb}^{-1}$ of proton$-$proton collision data recorded at $\sqrt{s} = 13$
TeV at the LHC. Targeting sources of new physics which prefer final states
containing $e^{+}\mu^{-}$ to $e^{-}\mu^{+}$, the search contains two broad
signal regions which are used to provide model-independent constraints on the
ratio of cross sections at the 2% level. The search also has two special
selections targeting supersymmetric models and leptoquark signatures.
Observations using one of these selections are able to exclude, at 95%
confidence level, singly produced smuons with masses up to 640 GeV in a model
in which the only other light sparticle is a neutralino when the
$R$-parity-violating coupling $\lambda'_{231}$ is close to unity. Observations
using the other selection exclude scalar leptoquarks with masses below 1880 GeV
when $g_{\text{1R}}^{eu}=g_{\text{1R}}^{\mu c}=1$, at 95% confidence level. The
limit on the coupling reduces to $g_{\text{1R}}^{eu}=g_{\text{1R}}^{\mu
c}=0.46$ for a mass of 1420 GeV.
