ARS 13-3112 Paragraph G:
G. On receipt of a concealed weapon permit application, the department of public safety shall conduct a check of the applicant's criminal history record pursuant to section 41-1750. The department of public safety may exchange fingerprint card information with the federal bureau of investigation for federal criminal history record checks.
Note the word "may". Don't know if the FBI retains them or not; I'm guessing probably everything submitted to AFIS is being kept as part of the Partiot Act.
DPS does diitize and keep them in a dedicated database. When you renew your permit, they do a new state and federal background check including checking for afingerprint match against several criminal databases:
Arizona Criminal Justice Information System (ACJIS)
National Crime Information Center (NCIC)
Automatic Fingerprint Information System (AFIS) - run by the FBI