Scalar-tensor-vector gravity theory is tested by black hole photon rings

This paper investigates the photon ring and shadow structure of the Reissner-Nordstr\"om black hole in the scalar-tensor-vector gravitational framework. The black hole is characterized by the ( MOG) parameter (\alpha) and the charge (Q). The study finds that as (\alpha) increases, the event horizon radius (r_h), photon sphere radius (r_{ph}), and critical impact parameter (b_{ph}) all increase, while these decrease as (Q) increases. The innermost stable circular orbit radius (r_{isco}) exhibits similar monotonic behavior. Ray-tracing shows that as (Q) increases, the impact parameter (b) interval between the lensing ring and photon ring widens; (b_{\text{ph}}) is non-degenerate, and the photon ring radius is uniquely determined by (\alpha) and (Q). Using $EHT$ constraints on (SgrA^*) and (M87^*), the bounds on (\alpha) and (Q) are derived. For (Q = 0), (0.5), and (1), the allowed ranges are (\alpha \in [0, 0.06]), ([0, 0.11]), and ([0.19, 0.36]), respectively. Radiative simulations show that for fixed (Q), larger (\alpha) leads to a larger, non-degenerate photon ring. The Schwarzschild case is approached only when both (\alpha) and (Q) are small. This provides a computational basis for testing modified black holes and offers a non-degenerate observational criterion for distinguishing quantum gravity models, consistent with current $EHT$ data. Future observations with $ngEHT$ and multi-band polarization can further test this. The results suggest that studying the photon ring structure of a Reissner-Nordstr\"om black hole in scalar-tensor-vector gravity provides a unique optical diagnostic for potential quantum-gravity tests and black hole properties.