The Northern Cross Fast Radio Burst project IV. Multi-wavelength study of the actively repeating FRB 20220912A

Fast radio bursts (FRBs) are energetic, millisecond-duration radio pulses observed at extragalactic distances and whose origins are still a subject of heated debate. A fraction of the FRB population have shown repeating bursts, however it's still unclear whether these represent a distinct class of sources. We investigated the bursting behaviour of FRB 20220912A, one of the most active repeating FRBs known thus far. In particular, we focused on its burst energy distribution, linked to the source energetics, and its emission spectrum, with the latter directly related to the underlying emission mechanism. We monitored FRB 20220912A at $408$ MHz with the Northern Cross radio telescope and at $1.4$ GHz using the $32$-m Medicina Grueff radio telescope. Additionally, we conducted $1.2$ GHz observations taken with the upgraded Giant Meter Wave Radio Telescope (uGMRT) searching for a persistent radio source coincident with FRB 20220912A, which included high energy observations in the $0.3-10$ keV, $0.4-100$ MeV and $0.03-30$ GeV energy range. We report $16$ new bursts from FRB 20220912A at $408$ MHz during the period between October 16$^{\rm th}$ 2022 and December 31$^{\rm st}$ 2023. Their cumulative spectral energy distribution follows a power law with slope $\alpha_E = -1.3 \pm 0.2$ and we measured a repetition rate of $0.19 \pm 0.03$ hr$^{-1}$ for bursts having a fluence of $\mathcal{F} \geq 17$ Jy ms. Furthermore, we report no detections at 1.4 GHz for $\mathcal{F} \geq 20$ Jy ms. These non-detections imply an upper limit of $\beta < -2.3$, with $\beta$ being the $408$ MHz $-$ $1.4$ GHz spectral index of FRB 20220912A. This is inconsistent with positive $\beta$ values found for the only two known cases in which an FRB has been detected in separate spectral bands. We find that FRB 20220912A shows a decline of four orders of magnitude in its bursting activity at $1.4$ GHz over a .. (abridged)