Update on the GRB universal scaling E$_{\rm{X,iso}}$-E$_{\rm{γ,iso}}$-E$_{\rm{pk}}$ with ten years of $Swift$ data

Abstract: From a comprehensive statistical analysis of $Swift$ X-ray light-curves of gamma-ray bursts (GRBs) collected from December 2004 to the end of 2010, we found a three-parameter correlation between the isotropic energy emitted in the rest frame 1-10$^4$ keV energy band during the prompt emission (E${\rm{\gamma,iso}}$), the rest frame peak of the prompt emission energy spectrum (E${\rm{pk}}$), and the X-ray energy emitted in the rest frame 0.3-30 keV observed energy band (E$_{\rm{X,iso}}$), computed excluding the contribution of the flares. In this paper, we update this correlation with the data collected until June 2014, expanding the sample size with $\sim$35% more objects, where the number of short GRBs doubled. With this larger sample we confirm the existence of a universal correlation that connects the prompt and afterglow properties of long and short GRBs. We show that this correlation does not depend on the X-ray light-curve morphology and that further analysis is necessary to firmly exclude possible biases derived by redshift measurements. In addition we discuss about the behavior of the peculiar objects as ultra-long GRBs and we propose the existence of an intermediate group between long and short GRBs. Interestingly, two GRBs with uncertain classification fall into this category. Finally, we discuss the physics underlying this correlation, in the contest of the efficiency of conversion of the prompt $\gamma$-ray emission energy into the kinetic energy of the afterglow, the photosferic model, and the cannonball model.