The role of beamstrahlung in high-energy e+e- storage-ring colliders (SRCs) is examined. Particle loss due to the emission of single energetic beamstrahlung photons is shown to impose a fundamental limit on SRC luminosities at energies 2E_0 >~ 140 GeV for head-on collisions and 2E_0 >~ 40 GeV for crab-waist collisions. With beamstrahlung taken into account, we explore the viability of SRCs in the E_0=240-500 GeV range, which is of interest in the precision study of the Higgs boson. At 2E_0=240 GeV, SRCs are found to be competitive with linear colliders; however, at 2E_0=400-500 GeV, the attainable SRC luminosity would be a factor 15-25 smaller than desired.
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