So only Airbus and Safety department doing the investigation as far as I know
You would think the Authority would be involved, will look into it
Depends on the severity of the incident against standards.
Nobody seriously injured and the aircraft landed safely, might only require the incident be investigated by the certificate holder under "Continued Airworthiness" and reported to the regulator. The decision of standards of reporting lie with the regulator and/or a countries investigating agency.