CERN Accelerating science

The Higgs lineshapes with various effects as a function of $E_{\rm com}-m_H$, the actual hard scattering center-of-mass energy difference with respect to the Higgs pole mass. Here $E$ is $x E_{\rm com}$, $\delta(x-1)x\overline E_{\rm com}$, $x \overline E_{\rm com}$, and $\overline E_{\rm com}$ for the Breit-Wigner, Breit-Wigner plus BES, Breit-Wigner plus ISR, and Breit-Wigner with both ISR and BES, respectively.\footnote{Note here the less dependence on convolution parameters, $x$, $E_{\rm com}$, the closer to the final, post-convolution distribution.} The detailed meanings of these quantities are described \autoref{sec:widthlinshape}. We show the theoretical predictions in linear ({\em left panel}) and logarithmic ({\em right panel}) scales. To obtain physical observable, one needs to decide on the scan range, separation, and luminosity assignment of each scan step. These will form a discrete set of event counts that are further subject to statistical fluctuations, forming the pseudo experimental data set to feed into the Higgs fit.

The Higgs lineshapes with various effects as a function of $E_{\rm com}-m_H$, the actual hard scattering center-of-mass energy difference with respect to the Higgs pole mass. Here $E$ is $x E_{\rm com}$, $\delta(x-1)x\overline E_{\rm com}$, $x \overline E_{\rm com}$, and $\overline E_{\rm com}$ for the Breit-Wigner, Breit-Wigner plus BES, Breit-Wigner plus ISR, and Breit-Wigner with both ISR and BES, respectively.\footnote{Note here the less dependence on convolution parameters, $x$, $E_{\rm com}$, the closer to the final, post-convolution distribution.} The detailed meanings of these quantities are described \autoref{sec:widthlinshape}. We show the theoretical predictions in linear ({\em left panel}) and logarithmic ({\em right panel}) scales. To obtain physical observable, one needs to decide on the scan range, separation, and luminosity assignment of each scan step. These will form a discrete set of event counts that are further subject to statistical fluctuations, forming the pseudo experimental data set to feed into the Higgs fit.

The projected width sensitivity as a function of total luminosity for the lineshape scan. We also highlight two benchmarks of integrated luminosity considered in this study in green and cyan asterisk symbol.\footnote{Note that this scanning range of $m_H\pm 12.5~\mev$ is sub-optimal compared to our final scanning range of $m_H\pm 8~\mev$. However, this does not affect our discussion here, which is the precision scaling with luminosity. The precision dependence on the scan range is discussed next.}

The projected width sensitivity as a function of scanning steps for various choices of scanning ranges around the Higgs pole mass $m_H$, with a fixed total luminosity of $20~\fbi$.

The differential cross-section for some of the non-Higgs backgrounds in terms of the production polar angle $\theta$. On the left panel, we assume the two final state particles can be distinguished and the sign of $\cos\theta$ is measured. On the right panel, only the folded distribution in terms of $|\!\cos\theta|$ is measured.

The differential cross-section for some of the non-Higgs backgrounds in terms of the production polar angle $\theta$. On the left panel, we assume the two final state particles can be distinguished and the sign of $\cos\theta$ is measured. On the right panel, only the folded distribution in terms of $|\!\cos\theta|$ is measured.

The Higgs coupling precision from a global fit of Higgs measurements in the $\kappa$-framework. The four columns represent the HL-LHC S2 scenario, a circular $e^+e^-$ collider at $240\,$GeV, a muon collider at 125\,GeV with a total integrated luminosity of $20\ifb$, and the combination of the $e^+e^-$ and the muon collider, respectively. The measurements are combined with the HL-LHC S2 for all the lepton collider scenarios. The column shows results with $\Gamma_H$ treated as a free parameter; the horizontal marks show the ones assuming that the Higgs has no exotic decay.

The same as \autoref{fig:kapparesult} but using the luminosity benchmark of 5 fb$^{-1}$ for the muon collider.

The one-sigma precision reach on the effective Higgs couplings from a global fit of the Higgs and electroweak measurements in the SMEFT framework. The four columns represent the HL-LHC S2 scenario with electroweak measurements at LEP and SLD, a circular $e^+e^-$ collider with center-of-mass energy up to $240\,$GeV, a muon collider at 125\,GeV with a total integrated luminosity of $20\ifb$, and the combination of the $e^+e^-$ and the muon collider, respectively. The measurements are combined with the HL-LHC S2 and LEP/SLD measurements for all the lepton collider scenarios. For the last two scenarios, the diboson $hZ$ and $WW$ measurements at a 3\,TeV muon collider ($1\iab$) are also considered to show their impact on different operators.\footnote{Note that we do not include the Higgs precision input from a 3 TeV muon collider here. One can research on the complementarity between 125 GeV muon collider with high energy muon collider in a future study.} Results with (without) the 3\,TeV $hZ/WW$ measurements are shown with solid (light-shaded) columns.

Same as \autoref{fig:eft1} but assuming $5\ifb$ at the 125\,GeV muon collider.