Lineshape of the Higgs boson in future lepton colliders

The effect of the photon emission (initial-state radiation) in the cross section of the process of direct production of the Higgs boson in future high luminosity electron and muon colliders is calculated. It was found that the cross section at the top of the Higgs boson resonance peak is reduced by a factor 0.348 for the electron collider and 0.548 for the muon collider. A centre-of-mass energy spread of the centre-of-mass energy of 4.2 MeV (equal to the Higgs width) would reduce peak cross section further, by a factor 0.170 and 0.256 (QED and energy spread) for electron and muon beams respectively. Possible uncertainties in the resummed QED calculations are discussed. Numerical results for the lineshape cross section including QED and many values of the centre-of-mass energy spread are provided.