Scale invariance in X-ray flares of gamma-ray bursts

X-ray flares are generally believed to be produced by the reactivation of the central engine, and may have the same energy dissipation mechanism as the prompt emission of gamma-ray bursts (GRBs). X-ray flares can therefore provide important clues to understanding the nature of the central engines of GRBs. In this work, we study for the first time the physical connection between differential size and return distributions of X-ray flares of GRBs with known redshifts. We find that the differential distributions of duration, energy, and waiting time can be well fitted by a power-law function. In particular, the distributions for the differences of durations, energies, and waiting times at different times (i.e., the return distributions) well follow a $q$-Gaussian form. The $q$ values in the $q$-Gaussian distributions remain nearly steady for different temporal interval scales, implying a scale-invariant structure of GRB X-ray flares. Moreover, we verify that the $q$ parameters are related to the power-law indices $\alpha$ of the differential size distributions, characterized as $q=(\alpha+2)/\alpha$. These statistical features can be well explained within the physical framework of a self-organizing criticality system.