Statistical analysis of multi-band plateaus in gamma-ray burst afterglows

Plateau features are frequently observed in the afterglows of gamma-ray bursts (GRBs), yet their physical origins remain under debate. In this work, we compile a sample of 124 GRBs with known redshifts and simultaneous X-ray and optical afterglow observations. We categorize them into four subsets based on the existence of plateaus and the bands in which they appear. Namely, Dataset 1: plateaus are detected simultaneously in both X-ray and optical bands (75 bursts); Dataset 2: plateaus are only in X-rays (15 bursts); Dataset 3: plateaus appear only in the optical (17 bursts); Dataset 4: no plateaus in either band (17 bursts). We employ these datasets to test the applicability of the energy-injection model by examining whether the temporal decay index $\alpha$ and the spectral index $\beta$ of GRB afterglows simultaneously satisfy the closure relations in X-ray and optical bands. We find that 47 bursts of Dataset 1 simultaneously obey the closure relations in both bands under the conditions of the electron spectral index $p>2$ and the injection parameter $q\in (0, 0.5)$, and 69 of the dataset for $p>1$ and $q\in (0, 0.8)$, providing a strong support for the energy-injection interpretation. However, for Datasets 2 and 3, although $\alpha$ and $\beta$ of the plateaus mostly satisfy the closure relations, those in the other band show significant deviations, which implies that bursts with a single-band plateau are inconsistent with the interpretation of energy injection. Furthermore, we also compare the isotropic X-ray energy of plateaus with the rotational energy budget of millisecond magnetars.