Differential luminosity under multiphoton beamstrahlung

For the next generation of $e^{+}{e}^{-}$ linear colliders in the TeV range, the energy loss due to beamstrahlung during the collision of the $e^{+}{e}^{-}$ beams is expected to be substantial. One consequence is that the center-of-mass energy between the colliding particles can be largely degraded from the designed value. The knowledge on the differential luminosity as a function of the center-of-mass energy is essential for particle physics analysis on the interesting events. On the other hand, the beamstrahlung photon spectrum provides useful information on the low-energy backgrounds and high-energy $\gamma \gamma$ luminosity. In this paper, we derive analytic formulas for the $e^{+}{e}^{-}$ and $\gamma$ energy spectra under multiple beamstrahlung process, and the $e^{+}{e}^{-}$ and $\gamma \gamma$ differential luminosities. Major characteristics of these formulas are discussed.