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Radio relics in galaxy clusters are associated with powerful shocks that(re)accelerate relativistic electrons. It is widely believed that theacceleration proceeds via diffusive shock acceleration. In the framework ofthermal leakage, the ratio of the energy in relativistic electrons to theenergy in relativistic protons should should be smaller than $K_{\rm e/p} \sim10^{-2}$. The relativistic protons interact with the thermal gas to produce$\gamma$-rays in hadronic interactions. Combining observations of radio relicswith upper limits from $\gamma$-ray observatories can constrain the ratio$K_{\rm e/p}$. In this work we selected 10 galaxy clusters that contain doubleradio relics, and derive new upper limits from the stacking of $\gamma$-rayobservations by FERMI. We modelled the propagation of shocks using asemi-analytical model, where we assumed a simple geometry for shocks and thatcosmic ray protons are trapped in the intracluster medium. Our analysis showsthat diffusive shock acceleration has difficulties in matching simultaneouslythe observed radio emission and the constraints imposed by FERMI, unless themagnetic field in relics is unrealistically large ($\gg 10 ~\rm \mu G$). In allinvestigated cases (also including realistic variations of our basic model andthe effect of re-acceleration) the mean emission of the sample is of the orderof the stacking limit by FERMI, or larger. These findings put tension on thecommonly adopted model for the powering of radio relics, and imply that therelative acceleration efficiency of electrons and protons is at odds withpredictions of diffusive shock acceleration, requiring $K_{\rm e/p} \geq10-10^{-2}$.

Constraining the electron and proton acceleration efficiencies in merger shocks in galaxy clusters