Photopion threshold on a photon background ($E_{\mathrm{th}}=\frac{m_{\pi }{c}^2(2m_p{c}^2+m_{\pi }{c}^2)}{4ϵ}$)

For an ultrarelativistic proton of energy $E$ colliding head-on with a photon of energy $ϵ$, the invariant squared collision energy is approximately $(m_p{c}^2{)}^2+4Eϵ$. Producing a final proton and neutral pion requires at least $(m_p{c}^2+m_{\pi }{c}^2{)}^2$. Thus $E_{\mathrm{th}}=m_{\pi }{c}^2(2m_p{c}^2+m_{\pi }{c}^2)/(4ϵ)$. A broad isotropic photon background turns this threshold into an energy-loss feature depending on the photon distribution and source distance, not an absolute ceiling for all arriving protons.