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A fraction of merging galaxy clusters host diffuse radio emission in theircentral region, termed as a giant radio halo (GRH). The most promisingmechanism of GRHs is the re-acceleration of non-thermal electrons and positronsby merger-induced turbulence. However, the origin of these seed leptons hasbeen under debate, and either protons or electrons can be primarily-acceleratedparticles. In this work, we demonstrate that neutrinos can be used as a probeof physical processes in galaxy clusters, and discuss possible constraints onthe amount of relativistic protons in the intra-cluster medium with theexisting upper limits by IceCube. We calculate radio and neutrino emission frommassive ($>10^{14}M_\odot$) galaxy clusters, using the cluster population modelof Nishiwaki & Asano (2022). This model is compatible with the observedstatistics of GRHs, and we find that the contribution of GRHs to the isotropicradio background observed with the ARCADE-2 experiment should be subdominant.Our fiducial model predicts the all-sky neutrino flux that is consistent withIceCube's upper limit from the stacking analysis. We also show that theneutrino upper limit gives meaningful constraints on the parameter space of there-acceleration model, such as the electron-to-proton ratio of primarycosmic-rays and the magnetic field, and in particular the secondary scenario,where the seed electrons mostly originate from inelastic $pp$ collisions, canbe constrained even in the presence of re-acceleration.

High-energy neutrino constraints on cosmic-ray re-acceleration in radio halos of massive galaxy clusters