Our beam-beam parameter study using beam-beam simulations and PWFA (particle-driven plasma acceleration) beam parameters indicates that at 3 TeV, for examined electron beam lengths ${2~mumathrm{m}leqsigma_zleq 10~mumathrm{m}}$, the total luminosity, as well as the sharpness of the luminosity spectrum for a $gammagamma$ collider are independent of the beam length of the electron beams used to scatter the photons, given that the hourglass effect is avoided. The total luminosity can consequently be maximised by minimising the horizontal and vertical beta functions $beta_{x,y}^*$ at the interaction point. Furthermore, we performed background studies in GUINEA-PIG where we considered the smallest currently achievable $beta_{x,y}^*$ combined with PWFA beam parameters. Simulations results show that our proposed parameter set for a 3 TeV PWFA $gammagamma$ collider is able to deliver a total luminosity significantly higher than a $gammagamma$ collider based on CLIC parameters, but gives rise to more background particles.
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