Distributions of Gamma-Ray Bursts and Blazars in the $L_{\mathrm{p}}-E_{\mathrm{p}}$ Plane and Possible Implications for their Radiation Physics

We present a spectral analysis for a sample of redshift known GRBs observed with {\em Fermi}/GBM. Together with the results derived from our systematical spectral energy distribution modeling with the leptonic models for a {\em Fermi}/LAT blazar sample, we compare the distributions of the GRBs and the blazars by plotting the synchrotron peak luminosity ($L_{\mathrm{s}}$) and the corresponding peak photon energy $E_{\mathrm{s}}$ of blazars in the $L_{\rm p}-E_{\rm p}$planeofGRBs,where$L_{\rm p}$and$E_{\rm p}$ are the peak luminosity and peak photon energy of the GRB time-integrated $\nu f_{\nu }$ spectrum, respectively. The GRBs are in the high-$L_{\mathrm{p}}$, high-$E_{\mathrm{p}}$ corner of the plane and a tight $L_{\mathrm{p}}-E_{\mathrm{p}}$ relation is found, i.e., $L_{\mathrm{p}}\propto E_{\mathrm{p}}^{2.13_{-0.46}^{+0.54}}$. Both FSRQs and LBLs are clustered in the low-$E_{\mathrm{p}}$, low-$L_{\mathrm{p}}$ corner. IBLs and HBLs have $E_{\mathrm{s}}\sim 2\times {10}^{-3}-{10}^2$ keV and $L_{\rm s} \sim 10^{44} - 10^{47}$ergs$^{-1}$,butnodependenceof$L_{\rm s}$on$E_{\rm s}$ is found. We show that the tight $L_p-E_p$ relation of GRBs is potentially explained with the synchrotron radiation of fast-cooling electrons in a highly magnetized ejecta, and the weak anti-correlation of $L_{\mathrm{s}}-E_{\mathrm{s}}$ for FSRQs and LBLs may be attributed to synchrotron radiation of slow-cooling electrons in a moderately magnetized ejecta. The distributions of IBLs and HBLs in the $L_{\mathrm{p}}-E_{\mathrm{p}}$ plane may be interpreted with synchrotron radiation of fast-cooling electrons in a matter-dominated ejecta. These results may present a unified picture for the radiation physics of relativistic jets in GRBs and blazars within the framework of the leptonic synchrotron radiation models.