Center-of-mass energy determination using $e^+e^-$ to $\mu^+\mu^-$ $(\gamma)$ events at future $e^+e^-$ colliders

Methods for measuring the absolute center-of-mass energy, $\sqrt{s}$, and its distribution, are investigated for future $e^+e^-$ Higgs-factory colliders using in situ $e^+e^-$ collisions. We emphasize the potential of an estimator based on the measurement of muon momenta that we denote $\sqrt{s}_p$. It can be determined with high precision in $e^+e^-$ to $\mu^+\mu^-$($\gamma$) events while being sensitive to effects from beam energy spread, beamstrahlung, initial-state radiation (ISR), final-state radiation (FSR), crossing angle, and detector resolution. The measurement precision is enabled by a high-precision low-mass tracker; the reported performance estimates are based on full simulation of the tracker response of the ILD detector concept for the ILC operating at $\sqrt{s} = 250$ GeV. The underlying statistical precision is 1.9 ppm for a 2.0 ab$^{-1}$ dataset at ILC. The ultimate utility will depend largely on how well one can calibrate and maintain the tracker momentum scale.