We discuss a new technique to constrain models for the origin of radio relics in galaxy clusters using the correlation between the shock Mach number and the radio power of relics. This analysis is carried out using a sample of relics with information on both the Mach numbers derived from X-ray observation, $mathcal{M}_X$, and using spectral information from radio observations of the peak and the average values of the spectral index along the relic, $mathcal{M}_R$. We find that there is a lack of correlation between $mathcal{M}_X$ and $mathcal{M}_R$; this result is an indication that the spectral index of the relic is likely not due to the acceleration of particles operated by the shock but it is related to the properties of a fossil electrons population. We also find that the available data on the correlation between the radio power $P_{1.4}$ and Mach numbers ($mathcal{M}_R$ and $mathcal{M}_X$) in relics indicate that neither the DSA nor the adiabatic compression can simply reproduce the observed $P_{1.4}-mathcal{M}$ correlations. Furthermore, we find that the radio power is not correlated with $mathcal{M}_X$, whereas it is not possible to exclude a correlation with $mathcal{M}_R$. This also indicates that the relic power is mainly determined by the properties of a fossil electron population rather than by the properties of the shock. Our results require either to consider models of shock (re)acceleration that go beyond the proposed scenarios of DSA and adiabatic compression at shocks, or to reconsider the origin of radio relics in terms of other physical scenarios.
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