Formation of H${\epsilon}$ in the solar atmosphere

Aims. We aim to understand how Hepsilon is formed in the quiet Sun. In particular, we consider the particular physical mechanism that sets its source function and extinction, how it is formed in different solar structures, and why it is sometimes observed in emission. Methods. We used a 3D radiative magnetohydrodynamic (MHD) simulation that accounts for non-equilibrium hydrogen ionization, run with the Bifrost code. To synthesize Hepsilon and Ca II H spectra, we made use of the RH code, which was modified to take into account the non-equilibrium hydrogen ionization. To determine the dominant terms in the H${\epsilon}$ source function, we adopted a multi-level description of the source function. Using synthetic spectra and simulation, we studied the contribution function to the relative line absorption or emission and compared it with atmospheric quantities at different locations. Results. Our multi-level source function description suggests that the H${\epsilon}$ source function is dominated by interlocking, with the dominant interlocking transition being through the ground level, populating the upper level of H${\epsilon}$ via the Lyman series. This makes the H${\epsilon}$ source function partly sensitive to temperature. The H${\epsilon}$ extinction is set by Lyman-${\alpha}$. In some cases, this temperature dependence gives rise to H${\epsilon}$ emission, indicating heating. High-resolution observations reveal that H${\epsilon}$ is not just a weak absorption line. Regions with H${\epsilon}$ in emission are especially interesting to detect small-scale heating events in the lower solar atmosphere, such as Ellerman bombs. Thus, H${\epsilon}$ can be an important new diagnostic tool for studies of heating in the solar atmosphere, augmenting the diagnostic potential of Ca II H when observed simultaneously