  Hydrogen Balmer lines are commonly used as spectroscopic effective
temperature diagnostics of late-type stars. However, the absolute accuracy of
classical methods that are based on one-dimensional (1D) hydrostatic model
atmospheres and local thermodynamic equilibrium (LTE) is still unclear. To
investigate this, we carry out 3D non-LTE calculations for the Balmer lines,
performed, for the first time, over an extensive grid of 3D hydrodynamic
STAGGER model atmospheres. For H$\alpha$, H$\beta$, and H$\gamma$, we find
significant 1D non-LTE versus 3D non-LTE differences (3D effects): the outer
wings tend to be stronger in 3D models, particularly for H$\gamma$, while the
inner wings can be weaker in 3D models, particularly for H$\alpha$. For
H$\alpha$, we also find significant 3D LTE versus 3D non-LTE differences
(non-LTE effects): in warmer stars ($T_{\text{eff}}\approx6500$K) the inner
wings tend to be weaker in non-LTE models, while at lower effective
temperatures ($T_{\text{eff}}\approx4500$K) the inner wings can be stronger in
non-LTE models; the non-LTE effects are more severe at lower metallicities. We
test our 3D non-LTE models against observations of well-studied benchmark
stars. For the Sun, we infer concordant effective temperatures from H$\alpha$,
H$\beta$, and H$\gamma$; however the value is too low by around 50K which could
signal residual modelling shortcomings. For other benchmark stars, our 3D
non-LTE models generally reproduce the effective temperatures to within
$1\sigma$ uncertainties. For H$\alpha$, the absolute 3D effects and non-LTE
effects can separately reach around 100K, in terms of inferred effective
temperatures. For metal-poor turn-off stars, 1D LTE models of H$\alpha$ can
underestimate effective temperatures by around 150K. Our 3D non-LTE model
spectra are publicly available, and can be used for more reliable spectroscopic
effective temperature determinations.
