Searches for the Higgs boson decays $H \to ee$ and $H \to e \mu$ are performed using data corresponding to an integrated luminosity of 139 fb$^{-1}$ collected with the ATLAS detector in $pp$ collisions at $\sqrt{s} = 13$ TeV at the LHC. No significant signals are observed, in agreement with the Standard Model expectation. For a Higgs boson mass of 125 GeV, the observed (expected) upper limit at the 95% confidence level on the branching fraction $\mathcal{B}(H \to ee)$ is $3.6 \times 10^{-4}$ ($3.5 \times 10^{-4}$) and on $\mathcal{B}(H \to e \mu)$ is $6.1\times 10^{-5}$ ($5.8\times 10^{-5}$). These results represent improvements by factors of about five and six on the previous best limits on $\mathcal{B}(H \to ee)$ and $\mathcal{B}(H \to e \mu)$ respectively.

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