Shadow and greybody bounding of a regular scale-dependent black hole solution

In this manuscript, we explore the shadow and the greybody bounding characteristics of a regular black hole within 4-dimensional space-time, employing the context of gravity that is scale-dependent. Our focus lies in determining limitations on the parameter denoted as $\tilde{\epsilon}$, which serves as a descriptor for the scale-dependent solution with respect to the classically observed shadow radius $R_\text{sh}$ as indicated in the documented data from the Event Horizon Telescope (EHT). Our result indicates that there is a unique value for $\tilde{\epsilon}$ occurring at the reported mean of $R_\text{sh}$ and the uncertainties could be the result of the fluctuating value of the scale-dependent parameter. We found that $\tilde{\epsilon} > 0$ is positive for Sgr. A*, but $\tilde{\epsilon}<0$ for M87*. Utilizing M87* as a reference model, we scrutinized the shadow radius and weak deflection angle within the specified constraints. Discrepancies were observed not only in the shadow but also in the deflection angle of photons, particularly when the photon's impact parameter closely approached the critical impact parameter.