Next-to-leading-order relativistic and QCD corrections to prompt $\boldsymbol{J/\psi}$ pair photoproduction at future $\boldsymbol{e^+e^-}$ colliders

Within the framework of nonrelativistic-QCD factorization, we calculate both the next-to-leading-order relativistic and QCD corrections to prompt $J/\psi$ pair production, with feeddown from $\psi(2S)$ mesons, via photon-photon collisions at future $e^+e^-$ colliders including the Future Circular Lepton Collider (FCC-ee), the Circular Electron Positron Collider (CEPC), and the Compact Linear Collider (CLIC). We present total cross sections and distributions in single $J/\psi$ transverse momentum and rapidity, and in $J/\psi$ pair invariant mass. The relativistic and QCD corrections both turn out to be large and negative. Yet, the production rates are large enough for useful experimental studies.