Supernova production of axion-like particles coupling to electrons, reloaded

We revisit the production of axion-like particles (ALPs) coupled to electrons at tree-level in a relativistic plasma. We explicitly demonstrate the equivalence between pseudoscalar and derivative couplings, incorporate previously neglected processes for the first time-namely, semi-Compton production ($γe^-\rightarrow a e^-$) and pair annihilation ($e^+e^-\rightarrow aγ$)-and derive analytical expressions for the bremsstrahlung ($e^- N\to e^- N a$) production rate, enabling a more computationally efficient evaluation of the ALP flux. Additionally, we assess uncertainties in the production rate arising from electron thermal mass corrections, electron-electron Coulomb interactions, and the Landau-Pomeranchuk-Migdal effect. The ALP emissivity is made available in a public repository as a function of the ALP mass, the temperature, and the electron chemical potential of the plasma. Finally, we examine the impact of ALP production and subsequent decays on astrophysical observables, deriving the leading bounds on ALPs coupling to electrons. At small couplings, the dominant constraints come from the previously neglected decay $a\to e^+ e^-γ$, except for a region of fireball formation where SN 1987A X-ray observations offer the best probe. At large couplings, bounds are dominated by the energy deposition argument, with a recently developed new prescription for the trapping regime.