  The evolution of protoplanetary disks is regulated by an interplay of several
processes, either internal to the system or related to the environment. As most
of the stars and planets have formed in massive stellar clusters, studying the
effects of UV radiation on disk evolution is of paramount importance. Here we
test the impact of external photoevaporation on the evolution of disks in the
$\sigma$ Orionis cluster by conducting the first combined large-scale UV to IR
spectroscopic and mm-continuum survey of this region. We study a sample of 50
targets located at increasing distances from the central, OB system $\sigma$
Ori. We combine new VLT/X-Shooter spectra with new and previously published
ALMA measurements of disk dust and gas fluxes and masses. We confirm the
previously found decrease of $M_{\rm dust}$ in the inner $\sim$0.5 pc of the
cluster. This is particularly evident when considering the disks around the
more massive stars ($\ge$ 0.4 $M_{\odot}$), where those located in the inner
part ($<$ 0.5 pc) have $M_{\rm dust}$ about an order of magnitude lower than
the more distant ones. About half of the sample is located in the region of the
$\dot{M}_{\rm acc}$ vs $M_{\rm disk}$ expected by models of external
photoevaporation, namely showing shorter disk lifetimes. These are observed for
all targets with projected separation from $\sigma$ Ori $<$ 0.5 pc, proving
that the presence of a massive stellar system affects disk evolution. External
photoevaporation is a viable mechanism to explain the observed shorter disk
lifetimes and lower $M_{\rm dust}$ in the inner $\sim$0.5 pc of the cluster.
Follow-up observations of the low stellar mass targets are crucial to confirm
the dependence of the external photoevaporation process with stellar host mass.
This work confirms that the effects of external photoevaporation are
significant down to impinging radiation as low as $\sim 10^{4}$ G$_0$.
