  Radio relics in galaxy clusters are giant diffuse synchrotron sources powered
in cluster outskirts by merger shocks. Although the relic-shock connection has
been consolidated in recent years by a number of observations, the details of
the mechanisms leading to the formation of relativistic particles in this
environment are still not well understood. The diffusive shock acceleration
(DSA) theory is a commonly adopted scenario to explain the origin of cosmic
rays at astrophysical shocks, including those in radio relics in galaxy
clusters. However, in a few specific cases it has been shown that the energy
dissipated by cluster shocks is not enough to reproduce the luminosity of the
relics via DSA of thermal particles. Studies based on samples of radio relics
are required to further address this limitation of the mechanism. In this
paper, we focus on ten well-studied radio relics with underlying shocks
observed in the X-rays and calculate the electron acceleration efficiency of
these shocks that is necessary to reproduce the observed radio luminosity of
the relics. We find that in general the standard DSA cannot explain the origin
of the relics if electrons are accelerated from the thermal pool with an
efficiency significantly smaller than 10%. Our results show that other
mechanisms, such as shock re-acceleration of supra-thermal seed electrons or a
modification of standard DSA, are required to explain the formation of radio
relics.
