Correction to black hole radiation due to pair annihilation

We consider the emission of charged scalar particles from a Schwarzschild black hole. It is shown that these can interact with each other through pair annihilation, and as result, produce photons. These photons make correction to spectrum are directly emitted By solving field equations using WKB approximation, annihilation rate taken into account for most probable case, i.e., first order, therefore will be order $e^2$. Considering this scenario, we show interactions take place near event horizon hole, yet number not large enough remarkable modification in radiation particles. results refute claims cause immense changes Hawking