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One of the most promising mechanisms for producing radio halos (RHs) ingalaxy clusters is the reacceleration of cosmic-ray electrons by turbulences.However, the origin of the seed electrons for the reacceleration is stillpoorly constrained. In the secondary scenario, most of the seed electrons areinjected via collision of proton cosmic-rays, while non-thermal electrons aredirectly injected in the primary scenario. In this paper, we examine the twoscenarios for the seed electrons with the observed statistical properties ofRHs, combining two methods: following the temporal evolutions of the electronenergy and radial distributions in a cluster, and the merger history ofclusters. We find that the RH lifetime largely depends on the seed origin, asit could be longer than the cosmological timescale in the secondary scenario.We study the condition for the onset of RHs with the observed RH fraction andthe RH lifetime we obtained, and find that long-lived RHs in the secondaryscenario should be originated from major mergers with a mass ratio of$\xi\sim0.1$, while the short lifetime in the primary scenario requires morefrequent onsets by minor mergers with $\xi\sim0.01$. Our simple model of theturbulence acceleration can reproduce the observed radio luminosity-massrelation. The RH luminosity functions we obtained suggest that the expected RHnumber count with the ASKAP survey will detect $\approx10^3$ RHs in both thescenarios.

Statistical Properties of Radio Halos in Galaxy Clusters and Origin of Seed Electrons for Reacceleration