Title: Coma cluster $γ$-ray and radio emission is consistent with a secondary electron origin for the radio halo

Abstract: Observations of diffuse, non-thermal radio emission spanning several megaparsecs have been documented in over 100 galaxy clusters. This emission, classified as giant radio halos (GHs), mini halos, and radio relics based mainly on their location and morphology, is interpreted as synchrotron radiation and implies the presence of relativistic electrons and magnetic fields in the intra-cluster medium (ICM). GHs were initially thought to be generated by secondary electrons resulting from inelastic $p+p\rightarrow X+\pi^{\pm}$ collisions. However, recent literature has leaned towards primary-electron turbulent (re)acceleration models, partly due to claimed upper limits on the $\gamma$-ray emission from $\pi^0$ decay. We demonstrate that the observed GH and $\gamma$-ray flux in the Coma cluster are consistent with a secondary origin for the GH across a broad range of magnetic field values. Although the constraints on magnetic field configuration are not stringent, they align well with previous estimates for Coma. Within this magnetic field range, the energy density of cosmic-ray protons (CRp) constitutes a few percent to tens of percent of the ICM energy density, as predicted and observed for a sample of radio-emitting galaxy clusters. Notably, we detect a rise in the ratio of CRp to ICM energy densities towards the outer regions of the cluster. This phenomenon was anticipated to arise from either adiabatic compression of CRp accelerated by accretion shocks or, more likely, from strong CRp diffusion.