  In this paper we derive new expected scaling relations for clusters with
giant radio halos in the framework of the re-acceleration scenario in a
simplified, but physically motivated, form, namely: radio power (P_R) vs size
of the radio emitting region (R_H), and P_R vs total cluster mass (M_H)
contained in the emitting region and cluster velocity dispersion (sigma_H) in
this region.
  We search for these correlations by analyzing the most recent radio and X-ray
data available in the literature for a well known sample of clusters with giant
radio halos. In particular we find a good correlation between P_R and R_H and a
very tight ``geometrical'' scaling between M_H and R_H. From these correlations
P_R is also expected to scale with M_H and sigma_H and this is confirmed by our
analysis. We show that all the observed trends can be well reconciled with
expectations in the case of a slight variation of the mean magnetic field
strength in the radio halo volume with M_H. A byproduct correlation between R_H
and sigma_H is also found, and can be further tested by optical studies. In
addition, we find that observationally R_H scales non-linearly with the virial
radius of the host cluster, and this immediately means that the fraction of the
cluster volume which is radio emitting increases with cluster mass and thus
that the non-thermal component in clusters is not self-similar.
