$g_{\star}$, respectively). In Figure 7 we translate the results of the fit in
Figure 6 in terms of the 95% probability constraints in the
$(g_{\star},m_{\star})$ plane under such assumptions, and setting all $O(1)$
coefficients exactly to 1, i.e.

$\begin{split}\frac{c_{\phi,6,y_{f}}}{\Lambda^{2}}&=\frac{g_{\star}^{2}}{m_{\star}^{2}},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
\frac{c_{W,B}}{\Lambda^{2}}=\frac{1}{m_{\star}^{2}},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\frac{c_{2W,2B,2G}}{\Lambda^{2}}=\frac{1}{g_{\star}^{2}}\frac{1}{m_{\star}^{2}},\\\
\frac{c_{T}}{\Lambda^{2}}&=\frac{y_{t}^{4}}{16\pi^{2}}\frac{1}{m_{\star}^{2}},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\frac{c_{\gamma,g}}{\Lambda^{2}}=\frac{y_{t}^{2}}{16\pi^{2}}\frac{1}{m_{\star}^{2}},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \frac{c_{\phi W,\phi
B}}{\Lambda^{2}}=\frac{g_{\star}^{2}}{16\pi^{2}}\frac{1}{m_{\star}^{2}},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\frac{c_{3W,3G}}{\Lambda^{2}}=\frac{1}{16\pi^{2}}\frac{1}{m_{\star}^{2}}.\end{split}$
(23)

We focus the comparison, again, on the full physics program at each future
collider project (solid regions), but also show the region delimited by the
low energy runs, or the FCC-ee for the case of the FCC project (the boundaries
are indicated by the dashed lines). In the right panel of that figure we also
show, for illustration purposes, the individual constraints set by several of
the operators in (19) for the FCC fit. The modifications of the on-shell Higgs
properties discussed in this report are mainly controlled, within the SILH
assumptions, by the contributions to the operators ${\cal O}_{\phi}$ and
${\cal O}_{y_{f}}$, both of which set similar constraints in the global fit
for this collider. These give the leading constraints in strongly coupled
scenarios. Electroweak precision measurements, on the other hand, are more
affected by a combination of ${\cal O}_{W,B}$ and set bounds independently of
the new physics coupling. Finally, some of the high-energy probes included in
the analysis provide the most efficient way of testing weakly coupled
scenarios.

|
---|---
Figure 7:  (Left) 2-$\sigma$ exclusion regions in the $(g_{\star},m_{\star})$
plane from the fit presented in Figure 6, using the SILH power-counting
described in Eq. (20) and below (solid regions). Dashed lines indicate the
regions constrained by the corresponding low-energy runs (or FCC-ee only for
the case of the FCC project). (Right) The same comparing the results from the
global fit with the constraints set by some of the operators individually, for
the illustrative case of the HL-LHC+FCC-ee/eh/hh. In this case, the
constraints from the on-shell Higgs measurements mainly affect ${\cal
O}_{\phi}$ and ${\cal O}_{y_{f}}$.

###  3.5 Impact of Standard Model theory uncertainties in Higgs calculations

As important as it is to have very precise experimental measurements of the
different Higgs processes, it is also fundamental from the point of view of
their physical interpretation to have theoretical calculations for the
predictions of such processes with comparable or better precision. In this
sense, to quantify to what extent an experimental measurement with uncertainty
$\delta_{\mathrm{exp}}$ can be translated into a constraint on new
physics,171717Or, equivalently, to what extent a measurement agrees with the
SM. one needs to know the corresponding uncertainty $\delta_{\mathrm{SM}}$ for
the SM prediction. In order to extract the maximum experimental information,
ideally, $\delta_{\mathrm{SM}}\ll\delta_{\mathrm{exp}}$. The sources of the SM
uncertainty are typically separated in two types of contributions:

  * •

Parametric theory uncertainties (ThPar). For a given observable $O$, this is
the error associated to the propagation of the experimental error of the SM
input parameters to the SM prediction $O_{\mathrm{SM}}$.

  * •

The second source of uncertainty is due to the fact that, in practice,
$O_{\mathrm{SM}}$ is only known to a finite order in perturbation theory. The
estimate of the net size associated with the contribution to $O_{\mathrm{SM}}$
from missing higher-order corrections is usually referred to as intrinsic
theory uncertainty (ThIntr).

Of course, in the interpretation of any measurement in a particular extension
of the SM, there are also errors associated with the missing corrections in
the expansion(s) including the new physics parameters. In the particular case
of the EFT framework, these would come from NLO corrections in the
perturbative expansion including dimension-6 interactions or, from the point
of view of the EFT expansion, from $q^{4}/\Lambda^{4}$ effects coming from
either the square of the dimension-6 contributions to the amplitudes, or the
SM interference with amplitudes involving dimension-8 operators or double
insertions of the dimension-6 ones. Note that all these corrections affect the
interpretation of a measurement in terms of pinpointing what is the source of
the deformation from the SM, i.e. which particular operator and how large its
coefficient can be, but not on the size of the overall deformation per se. The
latter is only controlled by the SM theoretical uncertainty. Because of that,
and in the absence of a fully developed program including such contributions
in the SMEFT framework, we restrict the discussion in this section to SM
uncertainties only.

In the previous sections the results for future colliders after the HL/HE-LHC
era were presented taking into account parametric uncertainties only. This was
done to illustrate the final sensitivity to BSM deformations in Higgs
couplings, as given directly by the experimental measurements of the different
inputs (i.e. Higgs rates, diBoson measurements, EWPO or the processes used to
determine the values of the SM input parameters). On the other hand, for this
scenario to be meaningful, it is crucial to also study the effect in such
results of the projections for the future intrinsic errors. This is needed to
be able to quantify how far we will be from the assumption that such intrinsic
errors become subdominant and, therefore, which aspects of theory calculations
should the theory community focus on to make sure we reach the maximum
experimental sensitivity at future colliders.

In this section we discuss more in detail the impact of the two types of SM
theory errors described above, from the point of view of the calculations of
the predictions for Higgs observables. This will be done both within the
$\kappa$ framework and also in the context of the EFT results. For the results
from the $\kappa$-framework we will use the most general scenario considered
in Section 3.1, i.e. kappa-3, which allows non-SM decays. On the EFT side, we
will use the scenario SMEFTPEW, where the uncertainty associated with the
precision of EWPO has already been “factorized”. In this scenario each fermion
coupling is also treated separately, thus being sensitive to the uncertainties
in the different $H\to f\bar{f}$ decay widths. Finally, we will also restrict
the study in this subsection to the case of future lepton colliders only (we
always consider them in combination with the HL-LHC projections. For the
latter we keep the theory uncertainties as reported by the WG2 studies [13]).

In Table 10 we show the results of the $\kappa$ fit for the benchmark scenario
kappa-3, indicating the results obtained including/excluding the different
sources of SM theory uncertainties. Similarly, Table 11 shows the results of
the EFT fit for the benchmark scenario SMEFTPEW. For the EFT results the
impact of the different theory uncertainties is also illustrated in Figure 8.
As can be seen, if the SM errors were reduced to a level where they become
sub-dominant, the experimental precision would allow to test deviations in
some of the couplings at the one per-mille level, e.g. the coupling to vector
bosons at CLIC in the SMEFT framework (the presence of extra decays would
however reduce the precision to the $0.4\%$ level, as shown in the kappa-3
results). The assumed precision of the SM theory calculations and inputs,
however, prevents reaching this level of sensitivity. The most notable
obstacle to achieve this close to per-mille level of precision are the
intrinsic uncertainties for the $e^{+}e^{-}\to ZH$ and, especially, in
$e^{+}e^{-}\to H\bar{\nu}\nu$, estimated to be $\sim$0.5%. In reaching this
level of theoretical precision it was assumed that predictions at NNLO in the
EW coupling for both processes will be available. This is within reach for
$ZH$ production, but it may be more challenging for $e^{+}e^{-}\to
H\bar{\nu}\nu$ (and $H\to VV^{*}\to 4f$). However, with enough effort on the
theory side [55, 56, 57], this type of uncertainties can be reduced. If the
necessary resources are dedicated to develop these types of calculations, it
should be possible to achieve, or even surpass, the required level of
precision. This is not the case for the SM parametric errors, which depend on
the experimental measurements of the corresponding input parameters. From the
results of the fits, the largest effect of this type of uncertainty on the
determination of the fermion couplings affects the effective coupling of the
bottom to the Higgs. The corresponding SM error in $H\to b\bar{b}$ depends on
the precision of the bottom quark mass, whose projected future determination
was assumed to be $\sim 13$ MeV. Taking into account the projected
improvements from Lattice QCD calculations, this should be a conservative
estimate [55]. Other parametric uncertainties, e.g. in $H\to c\bar{c},gg$ and
associated with $m_{c}$ and $\alpha_{S}$, are larger than the one for $H\to
b\bar{b}$ but have a smaller effect in the results due to the also larger
experimental errors expected in the corresponding channels. From the point of
view of the Higgs decays into vector bosons, the predictions of $H\to
ZZ^{*},WW^{*}$ have a strong dependence on the value of the Higgs mass. It it
therefore important to accompany the precise measurements of the Higgs
couplings with equally precise measurements of the Higgs mass, to the level of
10 MeV. This would be possible at 240/250 GeV lepton colliders but more
challenging at CLIC, where the final precision on $M_{H}$ is expected at the
level of 20-30 MeV (see Section 7). In the kappa-framework, the fact that the
dependence of the production $e^{+}e^{-}$ Higgs cross sections on $M_{H}$ is
less severe helps to reduce the impact of the $M_{H}$ uncertainty in the CLIC
results. This is no longer the case once we move to the more general
description of the SMEFT. In that case, non-SM like interactions contribute to
the effective $HZZ$ and $HWW$ couplings, and the information on $H\to WW^{*}$
becomes relevant to determine $g_{HZZ}^{\rm eff}$. The measurement of $M_{H}$
at the HL-LHC at the 10-20 MeV level prevents this from becoming an issue at
the lower energy stages at CLIC. But there is still a factor $\sim 2$
deterioration in the precision of the $g_{HZZ}^{\rm eff}$ coupling in the
final CLIC results, emphasising again the necessity of a precise determination
of $M_{H}$.

Table 10:  Comparison of the sensitivity at 68% probability to deviations in the different Higgs couplings modifiers in the kappa-3 fit, under different assumptions for the SM theory uncertainties. We compare the results obtained neglecting both intrinsic and parametric uncertainties, including each of them separately, and adding the full SM uncertainty.  | Benchmark | HL-LHC +
---|---|---
| kappa-3 | ILC | CLIC | CEPC | FCC-ee
|  | 250 | 500 | 1000 | 380 | 1500 | 3000 |  | 240 | 365
$\kappa_{W}[\%]$ |  ExpStat | $1.$ | $0.28$ | $0.24$ | $0.73$ | $0.4$ | $0.38$ | $0.87$ | $0.87$ | $0.4$
|  ExpStat \+ ThPar | $1.$ | $0.29$ | $0.24$ | $0.73$ | $0.4$ | $0.38$ | $0.88$ | $0.88$ | $0.41$
|  ExpStat \+ ThIntr | $1.$ | $0.51$ | $0.47$ | $0.82$ | $0.53$ | $0.49$ | $0.89$ | $0.89$ | $0.56$
|  ExpStat \+ Th | $1.$ | $0.51$ | $0.47$ | $0.81$ | $0.53$ | $0.63$ | $0.89$ | $0.89$ | $0.56$
$\kappa_{Z}[\%]$ | ExpStat | $0.28$ | $0.22$ | $0.22$ | $0.44$ | $0.4$ | $0.39$ | $0.17$ | $0.19$ | $0.16$
| ExpStat \+ ThPar | $0.29$ | $0.23$ | $0.22$ | $0.44$ | $0.4$ | $0.39$ | $0.18$ | $0.2$ | $0.17$
| ExpStat \+ ThIntr | $0.32$ | $0.27$ | $0.26$ | $0.46$ | $0.42$ | $0.41$ | $0.23$ | $0.24$ | $0.22$
| ExpStat \+ Th | $0.32$ | $0.27$ | $0.27$ | $0.46$ | $0.42$ | $1.2$ | $0.23$ | $0.24$ | $0.23$
$\kappa_{g}[\%]$ |  ExpStat | $1.3$ | $0.83$ | $0.58$ | $1.5$ | $1.1$ | $0.83$ | $1.$ | $1.1$ | $0.87$
|  ExpStat \+ ThPar | $1.4$ | $0.85$ | $0.63$ | $1.5$ | $1.1$ | $0.86$ | $1.$ | $1.2$ | $0.9$
|  ExpStat \+ ThIntr | $1.4$ | $0.97$ | $0.8$ | $1.6$ | $1.1$ | $0.95$ | $1.1$ | $1.2$ | $1.$
|  ExpStat \+ Th | $1.4$ | $0.99$ | $0.82$ | $1.6$ | $1.1$ | $2.1$ | $1.1$ | $1.2$ | $1.$
$\kappa_{\gamma}[\%]$ | ExpStat | $1.4$ | $1.2$ | $1.1$ | $1.4$ | $1.3$ | $1.2$ | $1.3$ | $1.3$ | $1.3$
| ExpStat \+ ThPar | $1.4$ | $1.2$ | $1.1$ | $1.4$ | $1.3$ | $1.2$ | $1.3$ | $1.3$ | $1.3$
| ExpStat \+ ThIntr | $1.4$ | $1.3$ | $1.1$ | $1.5$ | $1.3$ | $1.2$ | $1.3$ | $1.3$ | $1.3$
| ExpStat \+ Th | $1.4$ | $1.3$ | $1.1$ | $1.4$ | $1.3$ | $5.9$ | $1.3$ | $1.4$ | $1.3$
$\kappa_{Z\gamma}[\%]$ |  ExpStat | $10.$ | $10.$ | $10.$ | $10.$ | $8.2$ | $5.7$ | $6.3$ | $10.$ | $10.$
|  ExpStat \+ ThPar | $10.$ | $10.$ | $10.$ | $10.$ | $8.2$ | $5.7$ | $6.3$ | $10.$ | $10.$
|  ExpStat \+ ThIntr | $10.$ | $10.$ | $10.$ | $10.$ | $8.2$ | $5.7$ | $6.3$ | $10.$ | $10.$
|  ExpStat \+ Th | $10.$ | $10.$ | $10.$ | $10.$ | $8.2$ | $17.$ | $6.3$ | $10.$ | $10.$
$\kappa_{c}[\%]$ | ExpStat | $1.9$ | $1.1$ | $0.74$ | $4.$ | $1.8$ | $1.3$ | $1.9$ | $1.4$ | $1.2$
| ExpStat \+ ThPar | $2.$ | $1.2$ | $0.9$ | $4.1$ | $1.9$ | $1.4$ | $2.$ | $1.5$ | $1.3$
| ExpStat \+ ThIntr | $1.9$ | $1.2$ | $0.84$ | $4.$ | $1.8$ | $1.4$ | $1.9$ | $1.5$ | $1.2$
| ExpStat \+ Th | $2.$ | $1.3$ | $0.99$ | $4.1$ | $1.9$ | $3.6$ | $2.$ | $1.5$ | $1.3$
$\kappa_{t}[\%]$ |  ExpStat | $3.1$ | $2.8$ | $1.4$ | $3.2$ | $2.1$ | $2.1$ | $3.1$ | $3.1$ | $3.1$
|  ExpStat \+ ThPar | $3.1$ | $2.8$ | $1.4$ | $3.2$ | $2.1$ | $2.1$ | $3.1$ | $3.1$ | $3.1$
|  ExpStat \+ ThIntr | $3.2$ | $2.9$ | $1.4$ | $3.2$ | $2.1$ | $2.1$ | $3.1$ | $3.1$ | $3.1$
|  ExpStat \+ Th | $3.1$ | $2.8$ | $1.4$ | $3.2$ | $2.1$ | $7.$ | $3.1$ | $3.1$ | $3.1$
$\kappa_{b}[\%]$ | ExpStat | $1.1$ | $0.47$ | $0.36$ | $1.2$ | $0.5$ | $0.41$ | $0.82$ | $0.91$ | $0.56$
| ExpStat \+ ThPar | $1.1$ | $0.56$ | $0.47$ | $1.2$ | $0.59$ | $0.52$ | $0.9$ | $0.98$ | $0.64$
| ExpStat \+ ThIntr | $1.1$ | $0.64$ | $0.54$ | $1.2$ | $0.64$ | $0.54$ | $0.86$ | $0.94$ | $0.68$
| ExpStat \+ Th | $1.2$ | $0.71$ | $0.62$ | $1.3$ | $0.71$ | $0.87$ | $0.93$ | $1.$ | $0.76$
$\kappa_{\mu}[\%]$ |  ExpStat | $4.2$ | $3.9$ | $3.6$ | $4.4$ | $4.1$ | $3.5$ | $3.9$ | $4.$ | $3.9$
|  ExpStat \+ ThPar | $4.2$ | $3.9$ | $3.6$ | $4.4$ | $4.1$ | $3.5$ | $3.9$ | $4.$ | $3.9$
|  ExpStat \+ ThIntr | $4.2$ | $3.9$ | $3.6$ | $4.4$ | $4.1$ | $3.5$ | $3.9$ | $4.$ | $3.9$
|  ExpStat \+ Th | $4.2$ | $4.$ | $3.6$ | $4.4$ | $4.1$ | $4.4$ | $3.9$ | $4.$ | $3.9$
$\kappa_{\tau}[\%]$ | ExpStat | $1.1$ | $0.64$ | $0.53$ | $1.4$ | $0.99$ | $0.81$ | $0.91$ | $0.94$ | $0.65$
| ExpStat \+ ThPar | $1.1$ | $0.64$ | $0.54$ | $1.4$ | $1.$ | $0.82$ | $0.91$ | $0.94$ | $0.66$
| ExpStat \+ ThIntr | $1.1$ | $0.74$ | $0.64$ | $1.4$ | $1.$ | $0.85$ | $0.93$ | $0.96$ | $0.74$
| ExpStat \+ Th | $1.1$ | $0.75$ | $0.65$ | $1.4$ | $1.$ | $3.3$ | $0.94$ | $0.96$ | $0.75$
$\mathrm{BR}_{\mathrm{inv}}^{95\%}<$ |  ExpStat | $0.26$ | $0.22$ | $0.23$ | $0.63$ | $0.62$ | $0.62$ | $0.27$ | $0.22$ | $0.19$
|  ExpStat \+ ThPar | $0.26$ | $0.23$ | $0.22$ | $0.63$ | $0.62$ | $0.62$ | $0.27$ | $0.22$ | $0.19$
|  ExpStat \+ ThIntr | $0.26$ | $0.23$ | $0.23$ | $0.63$ | $0.62$ | $0.62$ | $0.27$ | $0.22$ | $0.19$
|  ExpStat \+ Th | $0.26$ | $0.23$ | $0.23$ | $0.63$ | $0.62$ | $0.69$ | $0.27$ | $0.22$ | $0.19$
$\mathrm{BR}_{\mathrm{unt}}^{95\%}<$ | ExpStat | $1.8$ | $1.4$ | $1.4$ | $2.8$ | $2.4$ | $2.4$ | $1.1$ | $1.2$ | $1.$
| ExpStat \+ ThPar | $1.8$ | $1.4$ | $1.4$ | $2.7$ | $2.4$ | $2.4$ | $1.1$ | $1.2$ | $1.$
| ExpStat \+ ThIntr | $1.8$ | $1.4$ | $1.4$ | $2.8$ | $2.4$ | $2.4$ | $1.1$ | $1.2$ | $1.$
| ExpStat \+ Th | $1.8$ | $1.4$ | $1.4$ | $2.7$ | $2.4$ | $2.4$ | $1.1$ | $1.2$ | $1.1$
Figure 8:  Impact of the different sources of SM theory uncertainties in the coupling reach at the different lepton-collider projects based on the SMEFTPEW fit. Using dark to light shades we show the results without SM theory uncertainties (darkest shade), only with the intrinsic uncertainty (medium), and the full SM error (lightest shade). The solid line indicates the result with SM parametric uncertainties only. The most significant differences are found for the effective coupling to vector bosons (dominated by intrinsic uncertainties) and to bottom quarks (controlled by the parametric error associated with $m_{b}$). See Table 11 and text for details. Table 11:  Comparison, within the SMEFTPEW fit, of the sensitivity at 68% probability to deviations in the different effective Higgs couplings and aTGC under different assumptions for the SM theory uncertainties. We compare the results obtained neglecting both intrinsic and parametric uncertainties, including each of them separately, and finally adding the full SM uncertainty.  | Benchmark | HL-LHC +
---|---|---
|  SMEFTPEW | ILC | CLIC | CEPC | FCC-ee
|  | 250 | 500 | 1000 | 380 | 1500 | 3000 |  | 240 | 365
$g_{HZZ}^{\mathrm{eff}}[\%]$ |  ExpStat | $0.26$ | $0.16$ | $0.12$ | $0.4$ | $0.14$ | $0.089$ | $0.43$ | $0.46$ | $0.25$
|  ExpStat \+ ThPar | $0.27$ | $0.17$ | $0.13$ | $0.42$ | $0.19$ | $0.16$ | $0.44$ | $0.46$ | $0.26$
|  ExpStat \+ ThIntr | $0.33$ | $0.29$ | $0.28$ | $0.45$ | $0.33$ | $0.32$ | $0.47$ | $0.5$ | $0.34$
|  ExpStat \+ Th | $0.34$ | $0.3$ | $0.29$ | $0.47$ | $0.36$ | $0.34$ | $0.48$ | $0.5$ | $0.35$
$g_{HWW}^{\mathrm{eff}}[\%]$ | ExpStat | $0.28$ | $0.17$ | $0.13$ | $0.41$ | $0.14$ | $0.091$ | $0.41$ | $0.45$ | $0.26$
| ExpStat \+ ThPar | $0.29$ | $0.18$ | $0.14$ | $0.43$ | $0.18$ | $0.15$ | $0.41$ | $0.45$ | $0.27$
| ExpStat \+ ThIntr | $0.37$ | $0.32$ | $0.31$ | $0.48$ | $0.36$ | $0.34$ | $0.47$ | $0.5$ | $0.37$
| ExpStat \+ Th | $0.37$ | $0.33$ | $0.32$ | $0.5$ | $0.38$ | $0.36$ | $0.47$ | $0.51$ | $0.38$
$g_{H\gamma\gamma}^{\mathrm{eff}}[\%]$ |  ExpStat | $1.3$ | $1.2$ | $1.1$ | $1.4$ | $1.3$ | $1.1$ | $1.3$ | $1.3$ | $1.2$
|  ExpStat \+ ThPar | $1.3$ | $1.2$ | $1.1$ | $1.4$ | $1.3$ | $1.1$ | $1.3$ | $1.3$ | $1.3$
|  ExpStat \+ ThIntr | $1.4$ | $1.3$ | $1.1$ | $1.5$ | $1.4$ | $1.2$ | $1.4$ | $1.4$ | $1.3$
|  ExpStat \+ Th | $1.4$ | $1.3$ | $1.2$ | $1.5$ | $1.4$ | $1.2$ | $1.4$ | $1.4$ | $1.3$
$g_{HZ\gamma}^{\mathrm{eff}}[\%]$ | ExpStat | $8.1$ | $5.9$ | $5.4$ | $8.9$ | $4.3$ | $3.5$ | $6.2$ | $9.9$ | $9.3$
| ExpStat \+ ThPar | $8.1$ | $6.$ | $5.5$ | $9.$ | $4.4$ | $3.5$ | $6.2$ | $9.9$ | $9.3$
| ExpStat \+ ThIntr | $8.1$ | $6.6$ | $6.1$ | $9.$ | $5.3$ | $4.2$ | $6.2$ | $9.9$ | $9.6$
| ExpStat \+ Th | $8.1$ | $6.5$ | $6.1$ | $9.1$ | $5.3$ | $4.2$ | $6.2$ | $9.9$ | $9.5$
$g_{Hgg}^{\mathrm{eff}}[\%]$ |  ExpStat | $1.1$ | $0.78$ | $0.54$ | $1.3$ | $0.95$ | $0.74$ | $0.75$ | $0.94$ | $0.81$
|  ExpStat \+ ThPar | $1.1$ | $0.79$ | $0.54$ | $1.3$ | $0.96$ | $0.75$ | $0.75$ | $0.95$ | $0.82$
|  ExpStat \+ ThIntr | $1.2$ | $0.82$ | $0.6$ | $1.4$ | $0.99$ | $0.78$ | $0.78$ | $0.98$ | $0.85$
|  ExpStat \+ Th | $1.2$ | $0.82$ | $0.6$ | $1.4$ | $1.$ | $0.79$ | $0.78$ | $0.98$ | $0.85$
$g_{Htt}^{\mathrm{eff}}[\%]$ | ExpStat | $3.1$ | $2.8$ | $1.5$ | $3.2$ | $2.1$ | $2.1$ | $3.1$ | $3.1$ | $3.1$
| ExpStat \+ ThPar | $3.2$ | $2.8$ | $1.5$ | $3.2$ | $2.1$ | $2.1$ | $3.1$ | $3.1$ | $3.1$
| ExpStat \+ ThIntr | $3.1$ | $2.8$ | $1.5$ | $3.2$ | $2.1$ | $2.1$ | $3.1$ | $3.1$ | $3.1$
| ExpStat \+ Th | $3.2$ | $2.8$ | $1.5$ | $3.2$ | $2.1$ | $2.1$ | $3.1$ | $3.1$ | $3.1$
$g_{Hcc}^{\mathrm{eff}}[\%]$ |  ExpStat | $1.7$ | $1.1$ | $0.72$ | $3.9$ | $1.8$ | $1.3$ | $1.8$ | $1.3$ | $1.1$
|  ExpStat \+ ThPar | $1.8$ | $1.2$ | $0.88$ | $4.$ | $1.9$ | $1.4$ | $1.8$ | $1.4$ | $1.3$
|  ExpStat \+ ThIntr | $1.7$ | $1.2$ | $0.77$ | $4.$ | $1.8$ | $1.3$ | $1.8$ | $1.3$ | $1.2$
|  ExpStat \+ Th | $1.8$ | $1.3$ | $0.92$ | $4.$ | $1.9$ | $1.4$ | $1.9$ | $1.4$ | $1.3$
$g_{Hbb}^{\mathrm{eff}}[\%]$ | ExpStat | $0.66$ | $0.4$ | $0.29$ | $0.92$ | $0.3$ | $0.17$ | $0.52$ | $0.61$ | $0.46$
| ExpStat \+ ThPar | $0.73$ | $0.5$ | $0.42$ | $0.96$ | $0.44$ | $0.37$ | $0.62$ | $0.7$ | $0.56$
| ExpStat \+ ThIntr | $0.7$ | $0.49$ | $0.41$ | $0.97$ | $0.45$ | $0.38$ | $0.57$ | $0.66$ | $0.53$
| ExpStat \+ Th | $0.76$ | $0.58$ | $0.52$ | $1.$ | $0.56$ | $0.5$ | $0.67$ | $0.74$ | $0.62$
$g_{H\tau\tau}^{\mathrm{eff}}[\%]$ |  ExpStat | $0.77$ | $0.57$ | $0.48$ | $1.3$ | $0.92$ | $0.73$ | $0.64$ | $0.69$ | $0.57$
|  ExpStat \+ ThPar | $0.77$ | $0.57$ | $0.48$ | $1.3$ | $0.93$ | $0.74$ | $0.65$ | $0.69$ | $0.57$
|  ExpStat \+ ThIntr | $0.79$ | $0.61$ | $0.53$ | $1.3$ | $0.95$ | $0.77$ | $0.67$ | $0.72$ | $0.61$
|  ExpStat \+ Th | $0.79$ | $0.61$ | $0.53$ | $1.3$ | $0.95$ | $0.78$ | $0.68$ | $0.72$ | $0.62$
$g_{H\mu\mu}^{\mathrm{eff}}[\%]$ | ExpStat | $4.1$ | $3.9$ | $3.5$ | $4.3$ | $4.1$ | $3.5$ | $3.9$ | $3.9$ | $3.8$
| ExpStat \+ ThPar | $4.1$ | $3.9$ | $3.5$ | $4.3$ | $4.1$ | $3.4$ | $3.9$ | $3.9$ | $3.9$
| ExpStat \+ ThIntr | $4.1$ | $3.9$ | $3.5$ | $4.3$ | $4.1$ | $3.5$ | $3.9$ | $3.9$ | $3.9$
| ExpStat \+ Th | $4.1$ | $3.9$ | $3.5$ | $4.3$ | $4.1$ | $3.5$ | $3.9$ | $3.9$ | $3.9$
$\delta g_{1Z}[\times 10^{2}]$ |  ExpStat | $0.039$ | $0.015$ | $0.013$ | $0.03$ | $0.0034$ | $0.0012$ | $0.087$ | $0.085$ | $0.036$
|  ExpStat \+ ThPar | $0.039$ | $0.015$ | $0.013$ | $0.03$ | $0.0034$ | $0.0012$ | $0.087$ | $0.085$ | $0.036$
|  ExpStat \+ ThIntr | $0.039$ | $0.015$ | $0.013$ | $0.031$ | $0.0034$ | $0.0012$ | $0.087$ | $0.086$ | $0.037$
|  ExpStat \+ Th | $0.039$ | $0.015$ | $0.013$ | $0.031$ | $0.0034$ | $0.0012$ | $0.088$ | $0.086$ | $0.037$
$\delta\kappa_{\gamma}[\times 10^{2}]$ | ExpStat | $0.056$ | $0.019$ | $0.015$ | $0.043$ | $0.0073$ | $0.0026$ | $0.089$ | $0.086$ | $0.049$
| ExpStat \+ ThPar | $0.056$ | $0.019$ | $0.015$ | $0.043$ | $0.0074$ | $0.0026$ | $0.089$ | $0.086$ | $0.049$
| ExpStat \+ ThIntr | $0.056$ | $0.02$ | $0.016$ | $0.044$ | $0.0074$ | $0.0026$ | $0.09$ | $0.086$ | $0.05$
| ExpStat \+ Th | $0.056$ | $0.02$ | $0.016$ | $0.044$ | $0.0074$ | $0.0026$ | $0.09$ | $0.086$ | $0.05$
$\lambda_{Z}[\times 10^{2}]$ |  ExpStat | $0.041$ | $0.019$ | $0.014$ | $0.042$ | $0.0053$ | $0.0018$ | $0.11$ | $0.1$ | $0.05$
|  ExpStat \+ ThPar | $0.041$ | $0.019$ | $0.014$ | $0.042$ | $0.0053$ | $0.0018$ | $0.11$ | $0.1$ | $0.05$
|  ExpStat \+ ThIntr | $0.041$ | $0.019$ | $0.014$ | $0.043$ | $0.0053$ | $0.0018$ | $0.11$ | $0.1$ | $0.05$
|  ExpStat \+ Th | $0.041$ | $0.019$ | $0.014$ | $0.042$ | $0.0053$ | $0.0018$ | $0.11$ | $0.1$ | $0.05$

##  4 The Higgs boson self-coupling

The Higgs field is responsible for the spontaneous breaking of the electroweak
symmetry, and for the generation of all the SM particle masses, because its
potential features a global minimum away from the origin. Within the SM, this
potential is fully characterised by two parameters, the Higgs mass $m_{h}$,
and $v$, which can be experimentally inferred from the measurements of the
Fermi constant ($v=1/\sqrt{\sqrt{2}G_{F}}\approx 246$ GeV).

$V(h)=\frac{1}{2}m_{H}^{2}h^{2}+\lambda_{3}vh^{3}+\frac{1}{4}\lambda_{4}h^{4},\qquad\textrm{with}\quad\lambda_{3}^{\rm
SM}=\lambda_{4}^{\rm SM}=\frac{m_{H}^{2}}{2v^{2}}.$ (24)

However, the Higgs potential could show sizeable departures from the SM form,
described in eq. (24). The understanding of EW symmetry breaking will remain
hypothetical until experimental measurements reconstruct the shape of the
Higgs potential. The measurement of the Higgs potential is therefore a high
priority goal on the physics programme of all future colliders.

Unfortunately, the Higgs self-interactions, apart from the simple kinematical
2-point interaction that corresponds to the Higgs boson mass, are not physical
observables. Therefore, a theoretical framework is needed to infer their
values from experimental measurements. One needs a general parametrisation of
the departures from the SM that allows the various Higgs couplings to vary
continuously. Within this framework, one makes accurate predictions of various
observables as a function of the modified Higgs couplings and a global fit
then leads to a determination of all these couplings. Effective Field Theory
offers us such a theoretically sound framework in which higher order
calculations can be performed to provide solid and improvable predictions able
to cope with systematic and statistic experimental uncertainties. As in
Section 3.3, we will focus our attention on EFT where the EW symmetry is
linearly realised, i.e. under the assumption that no new heavy degree of
freedom acquires its mass from the Higgs expectation value. In that case,
there are only two dimension-6 operators that induce a deviation of the Higgs
self-couplings

$\displaystyle\mathcal{L}=\mathcal{L}^{\rm
SM}+\frac{c_{\phi}}{2\Lambda^{2}}\partial_{\mu}|\phi|^{2}\partial^{\mu}|\phi|^{2}-\frac{c_{6}\,\lambda_{3}^{\rm
SM}}{\Lambda^{2}}|\phi|^{6}$
$\displaystyle\qquad\qquad\Rightarrow\quad\kappa_{3}\equiv\frac{\lambda_{3}}{\lambda_{3}^{\rm
SM}}=1+\left(c_{6}-\frac{3}{2}c_{\phi}\right)\frac{v^{2}}{\Lambda^{2}},\quad\kappa_{4}\equiv\frac{\lambda_{4}}{\lambda_{4}^{\rm
SM}}=1+\left(6c_{6}-\frac{25}{3}c_{\phi}\right)\frac{v^{2}}{\Lambda^{2}}.$
(25)

In particular, the operator proportional to $c_{\phi}$ requires a non-linear
field definition to keep the Higgs boson kinetic term canonically normalised.
The modifications of the cubic and quartic self-interactions are related in
this model. Independent modifications are only obtained when operators of
dimension 8 are considered.

The most direct way to assess the Higgs cubic self-interaction is through the
measurement of double Higgs production either at hadron colliders, where the
production is dominated by gluon fusion, $gg\to HH$, or at lepton colliders
via double Higgs-strahlung, $e^{+}e^{-}\to ZHH$, particularly relevant at low
energies, or via vector boson fusion (VBF), $e^{+}e^{-}\to
HH\nu_{e}\bar{\nu}_{e}$, more important at centre-of-mass energies of 1 TeV
and above. At leading order, double Higgs production receives a contribution
proportional to the cubic coupling, for both $pp$ and $e^{+}e^{-}$ collisions,
as shown in Fig. 9. Figure 10 shows the dependence of the inclusive double
Higgs production cross section when the value of the Higgs cubic coupling is
varied, assuming no other deviation from the SM. Gluon fusion production at a
hadron collider has been computed within the SM at NNLO accuracy in the
infinite top mass limit[58, 59, 60, 61] and at NLO with the full top mass
dependence[62, 63, 64], leading to a prediction whose theoretical and
parametric uncertainties are of the order of a few percent.

For the LHC at 14 TeV, the cross section is predicted to be
$36.69^{+2.1\%}_{-4.9\%}$ fb, about three orders of magnitude smaller than the
single Higgs production, which makes the double Higgs channel a challenging
process to observe. The most up-to-date analysis relies on the combination of
the $b\bar{b}\gamma\gamma$ and $b\bar{b}\tau\tau$ decay channels to reach
almost 5 standard deviation evidence for double Higgs production at HL-LHC
(see Table 55 and Fig. 65 of Ref [13]), which can be translated into a 68% CL
bound of order 50% on the deviation of the Higgs cubic coupling relative to
the SM prediction. Note that the mapping of the inclusive $gg\to HH$ cross
section onto a value of the Higgs cubic self-coupling is not unique: for
instance, at 14 TeV LHC, a value of the cross section equal to the SM
prediction corresponds either to $\kappa_{3}=1$ or to $\kappa_{3}\approx 6.2$.
This ambiguity can however be resolved by analysing the shape of the invariant
mass distribution of the reconstructed two Higgs boson system: the larger the
value of $\kappa_{3}$, the closer to threshold the $m_{HH}$ distribution is
peaked. This kinematic information is a crucial element of Boosted Decision
Trees (BDT) based analysis performed at HL-LHC. However the BDT and the final
selection cuts are often devised to optimise the significance of the SM cross
section for double Higgs production and therefore it is not necessarily
optimised for the determination of the Higgs self-coupling directly, leaving
room for possible improvement towards an even higher sensitivity. At lepton
colliders, double Higgs-strahlung, $e^{+}e^{-}\to ZHH$, gives stronger
constraints on positive deviations ($\kappa_{3}>1$), while VBF is better in
constraining negative deviations, ($\kappa_{3}<1$). While at HL-LHC, values of
$\kappa_{3}>1$, as expected in models of strong first order phase transition,
result in a smaller double-Higgs production cross section due to the
destructive interference, at lepton colliders for the $ZHH$ process they
actually result in a larger cross section, and hence into an increased
precision. For instance at ILC500, the sensitivity around the SM value is 27%
but it would reach 18% around $\kappa_{3}=1.5$.

Figure 9:  Representative Feynman diagrams for the leading contribution to
double Higgs production at hadron (left) and lepton (right) colliders.
Extracting the value of the Higgs self-coupling, in red, requires a knowledge
of the other Higgs couplings that also contribute to the same process. See
Table 18 for the SM rates. At lepton colliders, double Higgs production can
also occur via vector boson fusion with neutral currents but the rate is about
ten times smaller. The contribution proportional to the cubic Higgs self-
coupling involves an extra Higgs propagator that dies off at high energy.
Therefore, the kinematic region close to threshold is more sensitive to the
Higgs self-coupling.  Figure 10:  Double Higgs production at hadron (left)
[65] and lepton (right) [66] colliders as a function of the modified Higgs
cubic self-coupling. See Table 18 for the SM rates. At lepton colliders, the
production cross sections do depend on the polarisation but this dependence
drops out in the ratios to the SM rates (beam spectrum and QED ISR effects
have been included).

Modified Higgs self-interactions can also affect, at higher orders, the single
Higgs processes [67, 68, 69] and even the electroweak precision observables
[70, 71, 72]. Since the experimental sensitivities for these observables are
better than for double Higgs production, one can devise alternative ways to
assess the value of the Higgs self-interactions. For a 240 GeV lepton
collider, the change of the $ZH$ production cross section at NLO induced by a
deviation of the Higgs cubic coupling amounts to

$\sigma^{\rm NLO}_{ZH}\approx\sigma^{\rm
NLO,SM}_{ZH}(1+0.014\,\delta\kappa_{3}).$ (26)

Thus, to be competitive with the HL-LHC constraint, the $ZH$ cross section
needs to be measured with an accuracy below 1%, but this is expected to be
achieved by $e^{+}e^{-}$ Higgs factories at 240/250 GeV. However, one needs to
be able to disentangle a variation due to a modified Higgs self-interaction
from variations due to another deformation of the SM. This cannot always be
done relying only on inclusive measurements [73, 74] and it calls for detailed
studies of kinematical distributions with an accurate estimate of the relevant
uncertainties [75]. Inclusive rate measurements performed at two different
energies also help lifting the degeneracy among the different Higgs coupling
deviations (see for instance the $\kappa_{3}$ sensitivities reported in Table
12 for FCC-ee240 vs FCC-ee365; it is the combination of the two runs at
different energies that improve the global fit, a single run at 365 GeV alone
would not improve much compared to a single run at 240 GeV).

In principle, large deformations of $\kappa_{3}$ could also alter the fit of
single Higgs processes often performed at leading order, i.e. neglecting the
contribution of $\kappa_{3}$ at next-to-leading order. The results presented
in Section 3.4 were obtained along that line. It was shown in [73] that a 200%
uncertainty on $\kappa_{3}$ could for instance increase the uncertainty in
$g_{Htt}$ or $g_{Hgg}^{\rm eff}$ by around 30–40%. The fact that HL-LHC from
the double Higgs channel analysis will limit the deviations of $\kappa_{3}$ to
50% prevents such a large deterioration of the global fits to single Higgs
couplings when also allowing $\kappa_{3}$ to float. In the effective coupling
basis we are considering in this report, the effect of $\kappa_{3}$ would be
mostly in the correlations among the single Higgs couplings. In other bases,
like the Warsaw basis, there would be a deterioration up to 15-20% in the
sensitivity of the operator $\mathcal{O}_{\phi\Box}$. Anyway, one should keep
in mind that such a deterioration only concerns specific models where the
deviations of the Higgs self-coupling is parametrically larger than the
deviations of the single Higgs couplings and in generic situations, the
results of Section 3.4 hold.

In order to set quantitative goals in the determination of the Higgs self-
interactions, it is useful to understand how large the deviations from the SM
could be while remaining compatible with the existing constraints on the
different single Higgs couplings. From an agnostic point of view, the Higgs
cubic coupling can always be linked to the independent higher dimensional
operator $|H|^{6}$ that does not alter any other Higgs couplings. Still,
theoretical considerations set an upper bound on the deviation of the
trilinear Higgs couplings. Within the plausible linear EFT assumption
discussed above, perturbativity imposes a maximum deviation of the Higgs cubic
self-interaction, relative to the SM value, of the order of [73, 27]

$|\kappa_{3}|\mathrel{\hbox to0.0pt{\lower 3.0pt\hbox{\hskip
0.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}{\rm Min}(600\,\xi,4\pi)\,,$ (27)

where $\xi$ is the typical size of the deviation of the single Higgs couplings
to other SM particles [30]. However, the stability condition of the EW vacuum,
i.e. the requirement that no other deeper minimum results from the inclusion
of higher dimensional operators in the Higgs potential, gives the bound [27,
76]

$|\kappa_{3}|\mathrel{\hbox to0.0pt{\lower 3.0pt\hbox{\hskip
0.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}70\,\xi\,.$ (28)

At HL-LHC, $\xi$ can be determined with a precision of 1.5% at best,
corresponding to a sensitivity on the Higgs self-coupling of about 100%, and
thus somewhat inferior but roughly comparable to the direct sensitivity of 50%
[13]. Parametric enhancements of the deviations of Higgs cubic self-coupling
relative to the single Higgs couplings require a particular dynamics for the
new physics. An example is encountered in Higgs portal models where the Higgs
boson mixes with a SM neutral scalar field, possibly contributing to the dark
matter relic abundance [41, 73]. In more traditional scenarios addressing the
hierarchy problem, such as supersymmetric or composite models, the deviation
of $\kappa_{3}$ is expected to be of the order $\xi$ and is likely to remain
below the experimental sensitivity.

The sensitivity of the various future colliders to the Higgs cubic coupling
can be obtained using five different methods (1, 2(a), 2(b), 3, and 4):

  1. 1.

an exclusive analysis of HH production, i.e., a fit of the double Higgs cross
section considering only deformation of the Higgs cubic coupling;

  2. 2.

a global analysis of HH production, i.e., a fit of of the double Higgs cross
section considering also all possible deformations of the single Higgs
couplings that are constrained by single Higgs processes;

    1. (a)

the global fit does not consider the effects at higher order of the modified
Higgs cubic coupling to single Higgs production and to Higgs decays;

    2. (b)

these higher order effects are included;

  3. 3.

an exclusive analysis of single Higgs processes at higher order, i.e.,
considering only deformation of the Higgs cubic coupling; technically, this
will be a one-dimensional EFT fit where only the linear combination of the two
operators of Eq. (25) corresponding to the $\kappa_{3}$ deformation is turned
on;

  4. 4.

a global analysis of single Higgs processes at higher order, i.e., considering
also all possible deformations of the single Higgs couplings. Technically,
this will be a 30-parameter EFT fit done within the scenario SMEFTND scenario
of Eq. (16). The contribution of $\kappa_{3}$ to EWPO at 2-loop could also be
included but for the range of $\kappa_{3}$ values discussed here, the size of
effects would be totally negligible.

Most of the studies of the Higgs self-couplings at Future Colliders were done
following Method (1). In order to maximize the sensitivity to $\lambda_{3}$,
the analyses rely on sophisticated BDTs, and a simple recasting within an EFT
framework is not an easy task. A pragmatic approach was followed along the
line of what was proposed in [73]: different bins in $m_{HH}$ are considered
and the experimental uncertainty on the total rate is distributed in the
different bins according to their number of expected events. This certainly
ignores the bin-to-bin correlations and it does not take into account either
that the background itself has a non-trivial shape as a function of $m_{HH}$.
Nevertheless, the results obtained that way are in good agreement with those
quoted by the different collaborations. This approach has the advantage that
it can be easily generalised to a global EFT analysis that considers all the
operators modifying also the single Higgs couplings, Methods (2). One should
keep in mind that the bounds derived that way represent a crude estimate that
waits for a proper experimental study181818A detailed $m_{HH}$ binned analysis
was not available for HE-LHC, hence we could not estimate the $\kappa_{3}$
sensitivity along Method (2) for that collider. Similarly, for CLIC3000, the
granularity of the available information was not sufficient to match the
announced sensitivity, and therefore we did not venture into a complete study
along Method (2) either. In both cases, our checks led to the conclusion that
there will not be any noticeable difference between the sensitivity obtained
in Methods (1) and (2)..

For most colliders, the single Higgs constraints are strong enough that they
give a contribution to the double Higgs production below its experimental
sensitivity. And Method (1) and Method (2) lead to rather similar bound on
$\kappa_{3}$. A notable exception is at FCC-hh where the 1% uncertainty on the
top Yukawa coupling results in a deviation of the double Higgs production rate
at a level comparable to the one induced by a shift of $\kappa_{3}$ by 5%.
While a parametric enhancement of the deviation in $\kappa_{3}$ compared to
the other Higgs couplings deviations could make its higher order contributions
to single Higgs processes as important as the leading order ones and thus
could in principle modify the global fit, in practice, the constraints set by
the double Higgs production are strong enough that there is hardly any
difference in the results obtained using Methods (2a) and (2b). Methods (3)
and (4) are particularly relevant for low-energy colliders below the double
Higgs production threshold. Above this threshold, these methods can still be
relevant to complement results from the double Higgs analysis, for instance by
helping to resolve the degeneracy between the SM and a second minimum of the
likelihood. While this does not modify the 1$\sigma$ bound on $\kappa_{3}$, it
can impact the bound starting at the 2$\sigma$ level due to the non-Gaussian
profile of the likelihood. It should be remembered that the single Higgs data
used in Methods (3) and (4) have not been optimised for the extraction of the
Higgs self-coupling that would benefit from further differential information.
Therefore, the bounds on $\kappa_{3}$ should be considered as conservative and
are certainly improvable.

Table 12 reports the sensitivity at the various colliders for the Higgs cubic
coupling determination. For the global EFT fits, we limit ourselves to the
SMEFTND scenario, see Eq. (16), extended with Eq. (7). For all results a
simple combination with the HL-LHC results is done, i.e. by using a 50%
uncertainty on $\kappa_{3}$. It is seen that the results for Methods (1) and
(2) are very similar, showing that the determination of $\kappa_{3}$ is
dominated by the di-Higgs measurements when these are included. When comparing
Methods (3) and (4) one observes that the exclusive results appear to be more
constraining that the global results. But they overestimate the sensitivity as
a priori it is not known which operator coefficients to fix and the same
single-Higgs data should be used to constrain all operators. Method (4) is
significantly more robust than Method (3). In the following we focus on
Methods (1) and (4).

Table 12:  Sensitivity at 68% probability on the Higgs cubic self-coupling at the various future colliders. All the numbers reported correspond to a simplified combination of the considered collider with HL-LHC, which is approximated by a 50% constraint on $\kappa_{3}$. The numbers in the first column (i.e. "di-H excl." or Method (1)) correspond to the results given by the future collider collaborations and in parenthesis, we report our derived estimate obtained in the binned analysis described in the text. In the three last columns, i.e. Methods (2a), (3) and (4), we report the results computed by the Higgs@FC working group. For the leptonic colliders, the runs are considered in sequence. For the colliders with $\sqrt{s}\lesssim 400$ GeV, Methods (1) and (2.a) cannot be used, hence the dash signs in the corresponding cells. No sensitivity was computed along Method (2.a) for HE-LHC and CLIC3000 but our initial checks do not show any difference with the sensitivity obtained for Method (1). In the global analyses, Methods (2.a) and (4), we consider the flavour scenario of Neutral Diagonality (the results show little difference compared to the ones reported in the first version of this report within the Neutral Diagonality scenario). Due to the lack of results available for the $ep$ cross section in SMEFT, we do not present any result for LHeC nor HE-LHeC, and only results with Method (1) for FCC-eh. For Method (3) results are shown with and without combination with HL-LHC for many of the colliders (in several cases, the fit for Method (4) does not converge for the standalone collider without HL-LHC input).  collider | (1) di-H excl. | (2.a) di-H glob. | (3) single-H excl. | (4) single-H glob.
---|---|---|---|---
|  |  | with HL-LHC | w/o HL-LHC |
HL-LHC |  ${}_{-50}^{+60}$% (50%) | 52% | 47% | 125% | 50%
HE-LHC | 10-20% (n.a.) | n.a. | 40% | 90% | 50%
ILC250 | $-$ | $-$ | 29% | 126% | 49%
ILC350 | $-$ | $-$ | 28% | 37% | 46%
ILC500 | 27% (27%) | 27% | 27% | 32% | 38%
ILC1000 | 10% (n.a.) | 10% | 25% | n.a. | 36%
CLIC380 | $-$ | $-$ | 46% | 120% | 50%
CLIC1500 | 36% (36%) | 36% | 41% | 80% | 49%
CLIC3000 |  ${}_{-7}^{+11}$% (n.a.) | n.a. | 35% | 65% | 49%
FCC-ee240 | $-$ | $-$ | 19% | 21% | 49%
FCC-ee365 | $-$ | $-$ | 19% | 21% | 33%
FCC-ee${}_{365}^{\mathrm{4IP}}$ | $-$ | $-$ | 14% | n.a. | 24%
FCC-eh | 17-24% (n.a.) | n.a. | n.a. | n.a. | n.a.
FCC-ee/eh/hh | 5% (5%) | 6% | 18% | 19% | 25%
LE-FCC | 15% (n.a) | n.a | n.a. | n.a. | n.a.
CEPC | $-$ | $-$ | 17% | n.a. | 49%

The results are also summarised in Fig. 11 for these two methods. Even though
the likelihood is not a symmetric function of $\kappa_{3}$, the current level
of precision in this EFT analysis is not good enough to robustly assess an
asymmetric error and we report only symmetrised bounds in the Figure.

Based on di-Higgs measurements, with a 50% sensitivity on $\kappa_{3}$, HL-LHC
will exclude the absence of the Higgs self-interaction ($\kappa_{3}$ = 0) at
95%CL. Several of the proposed FCs (HE-LHC, LE-FCC and LHeC) will reach a
sensitivity of order 20% based on di-Higgs production, thus establishing the
existence of the self-interaction at 5$\sigma$. Even more remarkable, CLIC3000
and ILC1000 are expected to reach a sensitivity of order 10% and FCChh of the
order of 5%, where one could start probing the size of the quantum corrections
to the Higgs potential directly.

With single-Higgs production at FCC-ee and ILC500 and ILC1000, in combination
with di-Higgs results from HL-LHC, a sensitivity of $\sim 30\%$ can be
reached. For FCC-ee with 4 interaction points (IPs) this is reduced to 24%.
For the other collider options with $\sqrt{s}<400$ GeV no improvement w.r.t.
the HL-LHC result is seen.

Even though we do not report any sensitivity on $\kappa_{3}$ at muon-collider,
we note that preliminary studies [77] indicate that a 10 TeV (resp. 30 TeV)
machine could reach a 3% (resp. 1%) sensitivity.

Figure 11:  Sensitivity at 68% probability on the Higgs cubic self-coupling at
the various FCs. All values reported correspond to a simplified combination of
the considered collider with HL-LHC. Only numbers for Method (1), i.e. "di-H
excl.", corresponding to the results given by the future collider
collaborations, and for Method (4), i.e. "single-H glob." are shown (the
results for Method (3) are reported in parenthesis). For Method (4) we report
the results computed by the Higgs@FC working group. For the leptonic
colliders, the runs are considered in sequence. For the colliders with
$\sqrt{s}\lesssim 400$ GeV, Method (1) cannot be used, hence the dash signs.
Due to the lack of results available for the $ep$ cross section in SMEFT, we
do not present any result for LHeC nor HE-LHeC, and only results with Method
(1) for FCC-eh.

##  5 Rare Higgs boson decays

There are many reasons for the interest in rare Higgs boson decays. First,
they provide access to Higgs couplings which are expected to be small in the
SM and have not yet been directly probed. A leading example is the coupling to
second and first generation fermions, whose determination would test the
hypothesis that the same Higgs doublet is responsible for the mass generation
of the lighter states of the SM. More specifically, the measurement of several
Yukawa couplings will allow the comparison of ratios of couplings with ratios
of masses on the one hand, and test constants of proportionality on the other.
The second motivation is that processes which are predicted to be rare in the
SM, offer enhanced sensitivity to new physics residing at high scales. A
leading example is the search for flavour-changing neutral interactions, which
are extremely suppressed in the SM and if detected would reliably point to the
existence of new physics. Third, peculiar and rare final state signatures can
have a special connection with BSM scenarios. One example is $H$ decaying to
invisible particles, which is used to constrain scenarios featuring DM
candidates. In the SM, the Higgs boson can decay invisibly via $H\to 4\nu$
with a branching ratio of $0.11\%$. Finally, Yukawa interactions with first
generation fermions are the cornerstone of the low-energy constraints on $CP$
violation of the couplings on the third generation. The typical example here
are limits obtained by the EDM’s on the CP-odd interaction of the third
generation fermions (Section 6).

The reach of various colliders for rare decays, depends in the first place on
the available statistics of the Higgs bosons being produced. The expected
rates are presented in the Appendix B, Table 18.

In the following, we restrict ourselves to a summary of the prospects to bound
or determine the size of the interactions of the Higgs to the other SM
particles through decays. These can occur either directly, through a process
which is proportional to a tree-level coupling squared, i.e. all decays
$H\to\bar{f}f$, where $f$ is any SM fermion of the first or second generation,
or indirectly, i.e. through interfering amplitudes or loops, such as
$H\to\gamma\gamma$ and $H\to\gamma Z$. We will also briefly present results on
very rare exclusive decays, which could provide indirect information on the
light-quark Yukawa couplings. We follow the notation introduced in the
$\kappa$-framework and consider the rescaling factors
$\kappa_{i}=y_{i}/y^{SM}_{i}$ introduced previously for the couplings to
quarks $\kappa_{u},\kappa_{d},\kappa_{c},\kappa_{s}$ and for $\kappa_{\mu}$,
and for the loop induced processes, $\kappa_{\gamma}$ and $\kappa_{Z\gamma}$.
The values of $\kappa_{\mu},\kappa_{\gamma},\kappa_{Z\gamma},\kappa_{c}$ have
been obtained from the kappa-3,-4 fits presented in Section 3.2 and we do not
reproduce them here, while the upper bounds on
$\kappa_{u},\kappa_{d},\kappa_{s}$ ($\kappa_{c}$ for hadron colliders) are
obtained from the upper limits on ${\rm BR}_{\rm unt}$. Constraints on
flavour-changing Higgs boson interactions are not reported here.

The constraints of the couplings to first and second generation quarks are
given in Table 13 and displayed in Fig. 12, based on the results on
BR${}_{\textrm{unt}}$. For $\kappa_{c}$ the hadron colliders reach values of
${\cal{O}}(1)$, and lepton colliders and LHeC are expected to improve the
precision by about two orders of magnitude, to a 1-2%. For the strange quarks
the constraints are about 5-10$\times$ the SM value while for the first
generation it ranges between 100-600$\times$ the SM value. For the latter,
future colliders could improve the limits obtained at the HL-LHC by about a
factor of two. For HL-LHC, HE-LHC and LHeC, the determination of
BR${}_{\textrm{unt}}$ relies on assuming $\kappa_{V}\leq 1$. For
$\kappa_{\gamma}$, $\kappa_{Z\gamma}$ and $\kappa_{\mu}$ the lepton colliders
do not significantly improve the precision compared to HL-LHC but the higher
energy hadron colliders, HE-LHC and FCChh, achieve improvements of factor of
2-3 and 5-10, respectively, in these couplings.

For the electron Yukawa coupling, the current limit $\kappa_{e}<611$ [78] is
based on the direct search for $H\to e^{+}e^{-}$. A preliminary study at the
FCC-ee [79] has assessed the reach of a dedicated run at $\sqrt{s}=m_{H}$. At
this energy the cross section for $e^{+}e^{-}\to H$ is 1.64 fb, which reduces
to 0.3 with an energy spread equal to the SM Higgs width. According to the
study, with 2 ab-1 per year achievable with an energy spread of 6 MeV, a
significance of 0.4 standard deviations could be achieved, equivalent to an
upper limit of $2.5$ times the SM value, while the SM sensitivity would be
reached in a five year run.

Table 13:  Upper bounds on the $\kappa_{i}$ for $u$, $d$, $s$ and $c$ (at hadron colliders) at 95% CL, obtained from the upper bounds on BR${}_{\textrm{unt}}$ in the kappa-3 scenario. | HL-LHC | +LHeC | +HE-LHC | +ILC500 | +CLIC3000 | +CEPC | +FCC-ee240 | +FCC-ee/eh/hh
---|---|---|---|---|---|---|---|---
$\kappa_{u}$ | 560. | 320. | 430. | 330. | 430. | 290. | 310. | 280.
$\kappa_{d}$ | 260. | 150. | 200. | 160. | 200. | 140. | 140. | 130.
$\kappa_{s}$ | 13. | 7.3 | 9.9 | 7.5 | 9.9 | 6.7 | 7. | 6.4
$\kappa_{c}$ | 1.2 |  | 0.87 | measured directly
Figure 12:  Summary plot illustrating the limits that can be obtained from rare Higgs decays on the couplings.  Table 14:  Expected upper limits on the invisible and untagged BRs of the Higgs boson. The SM decay, $H\to 4\nu$, has been subtracted as a background. Given are the values of the direct searches using missing (transverse) momentum searches, the constraint derived from the coupling fit (see Table 5) in the kappa-3 scenario, and the result from a fit in the $\kappa$ framework where only modifications of ${\rm BR}_{\rm inv}$ are allowed. The last two columns show the corresponding information for untagged BR of the Higgs, ${\rm BR}_{\rm unt}$. For all fits the direct search for invisible decays is included.  Collider | 95% CL upper bound on ${\rm BR}_{\rm inv}$ [%] | 95% CL upper bound on ${\rm BR}_{\rm unt}$ [%]
---|---|---
| Direct searches | kappa-3 fit | Fit to ${\rm BR}_{\rm inv}$ only | kappa-3 fit | Fit to ${\rm BR}_{\rm unt}$ only
HL-LHC | 2.6 | 1.9 | 1.9 | 4.0 | 3.6
HL-LHC + HE-LHC($S_{2}^{\prime}$) |  | 1.5 | 1.5 | 2.4 | 1.9
FCC-hh | 0.025 | 0.024 | 0.024 | 1.0 | 0.36
HL-LHC + LHeC | 2.3 | 1.1 | 1.1 | 1.3 | 1.3
HL-LHC + CEPC | 0.3 | 0.27 | 0.26 | 1.1 | 0.49
HL-LHC + FCC-ee240 | 0.3 | 0.22 | 0.22 | 1.2 | 0.62
HL-LHC + FCC-ee365 |  | 0.19 | 0.19 | 1.0 | 0.54
HL-LHC + ILC250 | 0.3 | 0.26 | 0.25 | 1.8 | 0.85
HL-LHC + ILC500 |  | 0.23 | 0.22 | 1.4 | 0.55
HL-LHC + ILC1000 |  | 0.22 | 0.20 | 1.4 | 0.43
HL-LHC + CLIC380 | 0.69 | 0.63 | 0.56 | 2.7 | 1.0
HL-LHC + CLIC1500 |  | 0.62 | 0.40 | 2.4 | 0.51
HL-LHC + CLIC3000 |  | 0.62 | 0.30 | 2.4 | 0.33

While the limits quoted on $\kappa_{c}$ from hadron colliders (see Table 13)
have been obtained indirectly, we mention that progress in inclusive direct
searches for $H\rightarrow c\bar{c}$ at the LHC has been reported from ATLAS
together with a projection for the HL-LHC. Currently the upper bound on the
charm coupling is $\kappa_{c}\mathrel{\hbox to0.0pt{\lower 3.0pt\hbox{\hskip
0.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}10$ [80]. With HL-LHC, it is expected
to improve the sensitivity to values of $\kappa_{c}<2.1$ (based on Ref. [81]),
while LHCb, with the foreseen detector improvement, could reach a sensitivity
on $\kappa_{c}$ of 2-3 [13].

Exclusive Higgs decays to a vector meson ($V$) and a photon, $H\to V\gamma$,
$V=\rho,\omega,\phi,J/\psi,\Upsilon$ directly probe the Higgs bottom, charm
strange, down and up quark Yukawas [82, 83, 84]. Within the LHC, the Higgs
exclusive decays are the only direct probe of the $u$ and $d$ Yukawa
couplings, while if $s$-tagging could be implemented at the LHC [84], then the
strange Yukawa could be probed both inclusively and exclusively. On the
experimental side, both ATLAS and CMS have reported upper bounds on $H\to
J/\psi\gamma$ [85, 86], $H\to\phi\gamma$ and $h\to\rho\gamma$ [87, 88]. These
processes receive contributions from two amplitudes, only one of which is
proportional to the Yukawa coupling. Since the contribution proportional to
the Yukawa is smaller, the largest sensitivity to the Higgs $q$-quark coupling
is via the interference between the two diagrams. The prospects for probing
light quark Yukawas within future LHC runs employing the direct probe from
exclusive decays are not competitive with indirect limits that can be set from
production or global fit or inclusive search for $c$-Yukawa [81, 13]. However,
the information coming from exclusive decays will be relevant regardless of
the global fit sensitivity. For example, a limit of $|y_{s}/y_{b}|\lesssim 50$
could be set HL-LHC [13] and $y_{s}/y_{b}\lesssim 25$ at FCChh [1].

The constraints on invisible and untagged BRs to new particles are reported in
Table 14. For the invisible decays the SM $H\to 4\nu$ process (${\rm BR}^{\rm
SM}_{\rm inv}={\rm BR}(H\to 4\nu)=0.11\%$) is treated as background. Shown are
the estimated projections for direct searches for invisible decays using
signatures of missing transverse or total energy, and the results from the
kappa-3 fit presented earlier in Table 5. Also shown is a kappa-fit where all
SM BR values are fixed and only BR${}_{\textrm{inv}}$ is free in the fit. It
is seen that the $e^{+}e^{-}$ colliders generally improve the sensitivity by
about a factor 10 compared to HL-LHC. FCC-hh improves it by another order of
magnitude and will probe values below that of the SM. Comparing the three
determinations of the BR${}_{\textrm{inv}}$ for the various colliders, it is
seen that in most cases the difference is small, indicating that the
BR${}_{\textrm{inv}}$ is mostly constrained by the direct search. An exception
is LHeC where the kappa-fits improve the direct search result by a factor two.

Finally, comparing the bounds on the invisible and untagged BR one notices the
latter are always weaker as the untagged BR is not constrained by any direct
search here. For the untagged BR, the kappa-3 fit sensitivity is significantly
worse than that obtained by fitting only BR${}_{\textrm{unt}}$ as the kappa-3
fit implicitly takes into account the experimental uncertainties on all other
BR values.

##  6 Sensitivity to Higgs CP

Barring the strong-CP problem, in the SM the only source of CP violation stems
from fermion mixing in the charged currents, while the Higgs boson is
predicted to have CP-even, flavour-diagonal interactions. Detecting non-zero
CP-odd components in the Higgs interactions with the SM particles, would
therefore clearly point to physics beyond the Standard Model. Departures from
the SM can be efficiently parametrised in terms of a limited set of (flavour
conserving) dimension-6 operators. Employing the Higgs basis, the
(P-violating/C-conserving) CP-violating (CPV) $HVV$ couplings are given by

$\displaystyle\delta{\cal L}^{hVV}_{\rm
CPV}=\frac{h}{v}\Big{[}{\tilde{c}_{gg}}\frac{g_{s}^{2}}{4}G^{a}_{\mu\nu}\tilde{G}^{a}_{\mu\nu}+{\tilde{c}_{aa}}\frac{e^{2}}{4}A_{\mu\nu}\tilde{A}_{\mu\nu}+{\tilde{c}_{za}}\frac{e\sqrt{g^{2}+g^{\prime
2}}}{2}Z_{\mu\nu}\tilde{A}_{\mu\nu}+{\tilde{c}_{zz}}\frac{g^{2}+g^{\prime
2}}{4}Z_{\mu\nu}\tilde{Z}_{\mu\nu}+{\tilde{c}_{ww}}\frac{g^{2}}{2}W^{+}_{\mu\nu}\tilde{W}^{-}_{\mu\nu}\Big{]}\,,$
(29)

where, $g_{s}$, $g$ and $g^{\prime}$ are the $SU(3)$, $SU(2)_{L}$ and
$U(1)_{Y}$ gauge coupling constants and
$\tilde{V}_{\mu\nu}=\frac{1}{2}\epsilon^{\mu\nu\rho\sigma}V_{\rho\sigma}$. Out
of the four electroweak parameters, only three are independent at this order
in the EFT expansion. In particular,

${\tilde{c}_{ww}}={\tilde{c}_{zz}}+2\sin^{2}{\theta_{w}}\leavevmode\nobreak\
\\!{\tilde{c}_{za}}+\sin^{4}{\theta_{w}}\leavevmode\nobreak\
\\!{\tilde{c}_{aa}}.$ (30)

The (P-violating/C-violating) CP-violating (yet flavour-diagonal) interactions
of the Higgs boson with fermions can be parametrised as

${\cal L}^{hff}_{\rm
CPV}=-\bar{\kappa}_{f}m_{f}\frac{h}{v}\bar{\psi}_{f}(\cos\alpha+i\gamma_{5}\sin\alpha)\psi_{f}\,,$
(31)

where the angle $\alpha$ parametrizes the departure from the CP-even case.
Another, equivalent parametrization employs
$\kappa_{f}=\bar{\kappa}_{f}\cos\alpha$ and
$\tilde{\kappa}_{f}=\bar{\kappa}_{f}\sin\alpha$, where $\kappa_{f}=1+\delta
y_{f}$ in the notation used for the CP conserving cases in the
$\kappa$-framework (with $\kappa>0$). The pure scalar coupling corresponds to
$\alpha=0$ ($\tilde{\kappa}_{f}=0$), a pure pseudoscalar coupling to
$\alpha=90^{\circ}$ ($\kappa_{f}=0$), while CP violation occurs in all other
intermediate cases.

Sensitivity to the CP-odd operators can arise from two distinct classes of
observables. The first class includes CP-even observables, such as total cross
sections or single particle inclusive distributions. In this case, CP-odd
operators contribute in a way that is analogous to CP-even operators, i.e.
affecting rates and shapes. The second class includes observables that are
built to be directly sensitive to CP violation, i.e. they are zero (at the
lowest order) if CP is conserved. Limits obtained from this second class are
therefore automatically insensitive to the presence of higher-dimensional
$CP$-conserving operators and deviations from zero would uniquely point to CP
violation.

Sensitivity to the CP-odd $hgg$ interaction comes from gluon fusion processes
at the inclusive level, while direct sensitivity to CP violation can arise
only starting from final states featuring at least two jets in the final
state. Studies performed at the LHC exist, yet no dedicated investigation for
future colliders has been documented. Sensitivity to the CP-odd $hVV$ weak
operators comes from Higgs-strahlung processes ($WH$ and $ZH$), the vector
boson fusion and the Higgs decay into four charged leptons ($H\to 4\ell$).
Studies have been performed both at the level of rates/distributions and via
CP-sensitive observables [13].

CP-violation effects in the couplings to fermions have been considered for the
top quark and the tau lepton. Proposals to access information on CP violation
in top quark interactions exist for both classes of observables, yet studies
at future colliders have been mostly based on rates and distributions. These
focus on $ttH$ at hadron colliders and on $ttH$ and $tH$ final states at
$e^{+}e^{-}$ colliders and $ep$ colliders, respectively, which are also
sensitive to the absolute signs of CP-even and CP-odd interactions through
interference effects. For example, by studying distributions in $ttH$, the HL-
LHC will be able to exclude a CP-odd Higgs at 95%CL with about 200 fb-1 of
integrated luminosity. CLIC 1.5 TeV foresees to measure the mixing angle for
the top quark, $\alpha_{t}$ in $t\bar{t}H$ to better than 15∘. At LHeC, a
Higgs interacting with the top quarks with a CP-odd coupling can be excluded
at $3\sigma$ with 3 ab-1. At FCCeh a precision of 1.9% could be achieved on
$\alpha_{t}$.

The most promising direct probe of CP violation in fermionic Higgs decays is
the $\tau^{+}\tau^{-}$ decay channel, which benefits from a relatively large
branching fraction ($6.3\%$). Accessing the CP violating phase requires a
measurement of the linear polarisations of both $\tau$ leptons and the
azimuthal angle between them. This can be done by analysing the angular
distribution of the various components of the tau decay products and by
building suitable CP sensitive quantities (such as triple products of three-
vectors or acoplanarities). The estimated sensitivities for the CP-violating
phases, $\alpha_{\tau}$ of the $\tau$ Yukawa coupling and $\tilde{c}_{zz}$
extracted from CP-sensitive variables are collected in Table 15.

Table 15:  Upper bounds on the CP phase $\alpha$ of the Yukawa coupling for $\tau$ leptons and the CP-violating coefficient $\tilde{c}_{zz}$ entering the HZZ coupling. The result in parenthesis for the HL-LHC is obtained with the same method used for the CEPC study. Name | $\alpha_{\tau}$ | $\tilde{c}_{zz}$ | Ref.
---|---|---|---
HL-LHC | 8∘ | 0.45 (0.13) | [13]
HE-LHC | – | 0.18 | [13]
CEPC | – | 0.11 | [2]
FCC-ee240 | 10∘ | – | [1]
ILC250 | 4∘ | 0.014 | [3]

Before concluding this section, we recall that CP-violating Yukawa couplings
are well constrained from bounds on the electric dipole moments (EDMs)[89, 90,
91, 92, 93, 94] under the assumptions of i) no cancellation with other
contributions to EDMs, ii) SM values for the CP-even part of the Yukawa
couplings.

CP violation in the top quark sector can be constrained by the EDM of the
electron, giving $\tilde{\kappa}_{t}<0.001$ once the latest limits of the ACME
collaboration are considered [95]. For the bottom and charm Yukawas the
strongest limits come from the neutron EDM, $\tilde{\kappa}_{b}<5$ and
$\tilde{\kappa}_{c}<21$ when theory errors are taken into account. For the
light quark CPV Yukawas, measurements of the neutron EDM give a rather weak
constraint on the strange quark Yukawa of $\tilde{\kappa}_{s}<7.2$, while the
bound on the mercury EDM translates into strong bounds on the up and down
Yukawas of $\tilde{\kappa}_{u}<0.11$ and $\tilde{\kappa}_{d}<0.05$ (no theory
errors, $90\%$ CL). For the $\tau$ Yukawa coupling, using the latest ACME
measurement gives $\tilde{\kappa}_{\tau}<0.3$, while for the electron Yukawa,
provides an upper bound of $\tilde{\kappa}_{e}<1.9\times 10^{-3}$.

Assuming a SM Yukawa coupling of the Higgs to the electron, one can easily
compare the indirect limits from EDMs with the prospects for direct ones.
Using the relations between $(\bar{\kappa},\alpha)$ and
$(\kappa,\tilde{\kappa})$ one can convert the results for both the top quark
(given above) and for the $\tau$ lepton (collected in Table 15). One finds
that the direct top quark limits are not competitive with the indirect ones,
while those on the $\tau$ lepton are comparable with the current indirect
ones.

##  7 The Higgs boson mass and full width

The current best measurement of the Higgs boson mass, based on the ATLAS and
CMS analyses of $H\rightarrow ZZ^{*}$ and $H\rightarrow\gamma\gamma$ events in
the LHC Run-2 data is $125.18\pm 0.16$ GeV[96]. Future accelerators are
expected to substantially improve the precision of this mass measurement.

The mass measurements at lepton colliders in the centre-of-mass energy range
240-350 GeV analyse the recoil mass of the Higgs boson in
$e^{+}e^{-}\rightarrow ZH$ events, with $Z\rightarrow
e^{+}e^{-},\mu^{+}\mu^{-}$. Only the statistical uncertainties on the mass
measurements are shown, as systematic uncertainties in this recoil mass
analysis are expected to be negligible. The CLIC mass measurements at higher
centre-of-mass energies analyse the $H\rightarrow b\bar{b}$ invariant mass
distribution in $e^{+}e^{-}\rightarrow H(\rightarrow b\bar{b})\nu\nu$ events.
The quoted mass resolutions based on $m_{bb}$ measurements account only for
statistical uncertainties, but are sensitive to $b$-jet energy scale
uncertainties. This systematic uncertainty can be constrained with a
$e^{+}e^{-}\rightarrow Z(\rightarrow b\bar{b})\nu\nu$ calibration sample which
is expected to yield comparable statistics to the Higgs sample. The mass
measurement at HL-LHC is based on the analysis of $H\rightarrow
Z(l^{+}l^{-})Z(l^{+}l^{-})$ events. While the calibration of lepton momentum
scales has not been studied in detail, a resolution of 10-20 MeV is projected
to be plausibly in reach with the assumption that the higher statistics can
help to significantly improve muon $p_{T}$ systematic uncertainties.

Table 16 summarizes the expected precision of Higgs boson mass measurements of
future accelerators. Also shown is the impact of the $m_{H}$ uncertainty on
the $H\to ZZ^{*}$ partial decay width. Already with HL-LHC, it will be
possible to reduce this impact to the level of about 0.2%. At this value, the
parametric uncertainty on Higgs partial widths, (primarily on $ZZ^{*},WW^{*}$)
is much smaller than the expected precision at any hadron collider. For the
$e^{+}e^{-}$ colliders the precision on the $W$ and $Z$ couplings is of that
order, so that the $m_{H}$ precision needs to be further improved to about 10
MeV to avoid any limitations on the Higgs coupling extraction precision
(assuming the uncertainty due to higher order processes gets improved in the
future, see Table 19).

Table 16:  Overview of expected precision of Higgs boson mass measurements for future accelerators scenarios. For the lepton colliders (ILC, CLIC, CEPC) the projected uncertainties listed are statistical only. The impact of $\delta m_{H}$ on $\delta\Gamma_{ZZ^{*}}$ reported in this table is calculated as $1.2\%\cdot(\delta m_{H}/100\leavevmode\nobreak\ \rm{MeV})$, following Ref. [97].  Collider Scenario | Strategy |  $\delta m_{H}$ (MeV) | Ref. |  $\delta(\Gamma_{ZZ^{*}})$ [%]
---|---|---|---|---
LHC Run-2 | $m(ZZ),m(\gamma\gamma)$ | 160 | [96] | 1.9
HL-LHC | $m(ZZ)$ | 10-20 | [13] | 0.12-0.24
ILC250 |  $ZH$ recoil | 14 | [3] | 0.17
CLIC380 |  $ZH$ recoil | 78 | [98] | 0.94
CLIC1500 |  $m(bb)$ in $H\nu\nu$ | 30191919In Ref. [98] the values are 36 MeV (for $\sqrt{s}=1.5$ TeV) and 28 MeV (for $\sqrt{s}=3$ TeV) are based on unpolarized beams. The values quoted here are for the default scenario of 80% electron polarisation assumed throughout. | [98] | 0.36
CLIC3000 |  $m(bb)$ in $H\nu\nu$ | 23 | [98] | 0.28
FCC-ee |  $ZH$ recoil | 11 | [99] | 0.13
CEPC |  $ZH$ recoil | 5.9 | [2] | 0.07

In the SM, the width of a 125 GeV $H$ boson is predicted to be around $4$ MeV,
i.e. three orders of magnitude smaller than that of the weak bosons and of the
top quark. It is therefore very challenging to measure it directly. All
methods considered so far at colliders are in fact indirect and model
dependent to various degrees. Three methods have been proposed at the LHC, and
are considered for future hadron colliders.

The most direct method involves the diphoton decay mode and it is based on the
measurement of the shape of the invariant mass of the diphoton close to the
Higgs boson mass. This observable has a dependence on the width from signal-
background interference effects. The foreseen sensitivity, however, will not
allow to probe values close to the SM predictions, and can provide constraints
of about $8-22\times\Gamma_{\textrm{SM}}$ [13].

A second method extracts the width indirectly from a global fit of the Higgs
boson couplings by employing specific assumptions. For example, in the
$\kappa$-framework, assuming $\kappa_{Z}\leq 1$ and ${\rm BR}_{\rm unt}=0$ one
can determine the width from the fit. 202020In fact, the width and the
branching ratio to undetected final states are not independent observables. In
the analysis presented in Section 3.2 we opted to fit ${\rm BR}_{\rm unt}$ and
calculate $\Gamma_{H}$ from Eq. (4).

A third method is based on the combination of two independent measurements in
gluon fusion production of a $H$ boson with subsequent decay into a $ZZ$ final
state: $gg\to H\to ZZ^{*}$, where the $H$ boson is on shell (and at least one
of the final state $Z$ off shell) and $gg\to ZZ$ with two on-shell $Z$ bosons,
where the $H$ boson contribution is off shell [100]. The ratio of the off-
shell over the on-shell rate is directly proportional to the total width
[101]. Even though in generic BSM scenarios including the EFT, the
interpretation of the off-shell/on-shell ratio as an extraction of the width
is model dependent, this ratio can provide useful information on other key
aspects of the Higgs couplings, e.g. their energy dependence [102]. It is
foreseen that, with the HL-LHC and improvements in the theoretical
calculations, $\Gamma_{H}$ can be measured with a precision of up to 20% using
this method [13].

At lepton colliders, the mass recoil method allows to measure the inclusive
cross section of the $ZH$ process directly, without making any assumption
about the Higgs BR’s. This possibility is unique to lepton colliders as it
relies on the precise knowledge of the total initial energy of the event. In
combination with measurements of exclusive Higgs decay cross sections, it
allows to extract the total width $\Gamma_{H}$ with a mild model dependence.
The simplest way is to consider the ratio of the $ZH$ cross section (from the
recoil method) with the $H\to ZZ$ branching ratio (extracted from the $ZH,H\to
ZZ^{*}$ rate)

$\frac{\sigma(e^{+}e^{-}\to ZH)}{{\rm BR}(H\to
ZZ^{*})}=\frac{\sigma(e^{+}e^{-}\to ZH)}{\Gamma(H\to
ZZ^{*})/\Gamma_{H}}\simeq\left[\frac{\sigma(e^{+}e^{-}\to ZH)}{\Gamma(H\to
ZZ^{*})}\right]_{\rm SM}\times\Gamma_{H}\,,$ (32)

where the last approximate equality assumes a cancellation of new physics
effects, which holds, for instance, in the $\kappa$-framework. This method is
limited by the relatively poor statistical precision of the $H\to ZZ$ BR
measurement. More in general, even in scenarios where such a cancellation does
not hold, e.g. in an EFT, a global fit can be performed to extract information
on the width, using other decays (particularly the $bb$ and $WW$ decays) and
channels ($e^{+}e^{-}\to H\nu\bar{\nu}$). This method is used for CEPC. For
FCC-ee and CLIC the $\kappa$-formalism is used to extract the width, similar
to what is done in this report for Table 5. For ILC, the width reported here
was extracted using an EFT formalism that does not assume that there is only
one operator that governs the interactions between the Higgs boson and the Z
boson (as is done implicitly in the $\kappa$-framework). In this determination
of $\Gamma_{H}$, angular distributions and polarisation asymmetries are used
to constrain the free parameters that result from relaxing this assumption
[18], in addition to the parameters used by the $\kappa$-formalism for the
other lepton colliders. This fit is different from the EFT fits performed in
Section 3.4.

Table 17 summarizes the expected relative precision that can be reached on the
Higgs width at future lepton colliders, comparing the estimates of the
standalone estimates of the future lepton colliders to the results of the
kappa-3 scenario fits performed in this article (with HL-LHC data included).
It is seen that the result obtained in the kappa-3 fit is generally more
constraining than the results quoted in the references, primarily as this
result also includes the constraint from the HL-LHC data, and, in some cases,
uses a different approach to modelling changes to the total width. In both
cases, the best precision is obtained for the ILC500 and FCC-ee365 scenarios.

Table 17:  Overview of expected precision of Higgs boson width measurements for future accelerator scenarios. The result given in the second column refers to the width extraction as performed by the future lepton colliders using the stated technique, and as provided in the references given. The last column of the table lists the width extracted from the kappa-3 scenario fit. It also includes the HL-LHC measurements (but excludes the constraint $\kappa_{V}<1$ that is used in HL-LHC-only fits). Collider |  $\delta\Gamma_{H}$ [%] | Extraction technique standalone result |  $\delta\Gamma_{H}$ [%]
---|---|---|---
| from Ref. |  | kappa-3 fit
ILC250 | 2.3 | EFT fit [3, 4] | 2.2
ILC500 | 1.6 | EFT fit [14, 3, 4] | 1.1
ILC1000 | 1.4 | EFT fit [4] | 1.0
CLIC380 | 4.7 |  $\kappa$-framework [98] | 2.5
CLIC1500 | 2.6 |  $\kappa$-framework [98] | 1.7
CLIC3000 | 2.5 |  $\kappa$-framework [98] | 1.6
CEPC | 2.8 |  $\kappa$-framework [103, 104] | 1.7
FCC-ee240 | 2.7 |  $\kappa$-framework [1] | 1.8
FCC-ee365 | 1.3 |  $\kappa$-framework [1] | 1.1

##  8 Future studies of the Higgs sector, post-European Strategy

###  8.1 Higgs prospects at the muon collider

Electron-positron colliders offer a well-defined value of the collision energy
of the hard-scattering process and a relatively clean event, as opposed to
hadron collisions where the underlying event and the high-level of event
pileup challenge the reconstruction of the hard scattering event and its
measurement.

The main limitation to the collision energy of circular electron-positron
colliders is due to the low mass of the electrons/positrons which leads to
large fraction of their energy emitted as synchrotron radiation. The solutions
pursued so far to reach high lepton collision energies are based on limiting
the energy loss by synchrotron radiation by reducing the curvature either by
increasing the radius of the circular colliders or by employing linear
colliders. However, the beam acceleration does require a number of RF cavities
imposing a machine of large dimensions.

With a mass of about two hundred times that of electrons, muons do not suffer
significant energy losses due to synchrotron radiation (the loss goes as the
inverse of the fourth power of the mass) and therefore could be accelerated up
to multi-TeV collision energies. For example, if the LHC ring were used, with
the proposed HE-LHC dipoles (Nb3Sn, 16 T), muons would collide at an energy
close to $\sqrt{s}$= 14 TeV, compared to the 0.2 to 0.4 TeV of an electron-
positron collider.

Alternatively, a collider with $\sqrt{s}=125$ GeV could be a very compact
(diameter $\sim 60$ m) Higgs factory using $s$-channel production of Higgs
bosons [105]. However, it should be noted that the expected rate of produced
Higgs bosons by $s$-channel is small, given the instantaneous luminosity
possible at this machine [106], and the limited production cross section
(taking into account both the beam energy spread and the initial state
radiation effects) [107, 108, 109]. Estimates of the achievable precision on
Higgs couplings for such a machine are given in [110].

Muon production, cooling, lifetime and physics background [111] pose severe
challenges to the accelerator and detector technologies. Although the study of
a Muon Collider (machine and physics prospects) is not as mature as those of
other future proposed colliders, its physics potential certainly merits
consideration.

Currently, two different configurations have been proposed for the muon
collider. In the first configuration, muons are produced by the decay of
hadronically produced charged pions or kaons, and cooled before they undergo
the acceleration [106]. In the second configuration, muons are produced at
threshold (in the centre of mass frame) by high energy positron collisions
with atomic electrons [112]. The first configuration has been originally
proposed for $\mu^{+}\mu^{-}$ collision at the Higgs boson pole
($\sqrt{s}\sim$125 GeV), while the second is mainly considered for very high
energy collisions, in the range of $O(10)$ TeV.

At muon collision c.m. energies $\sqrt{s}\gtrsim 10$ TeV, assuming the point
cross section $\sigma\simeq 4\pi\alpha^{2}/(3s)\simeq 1\leavevmode\nobreak\
$fb$\cdot(10\,$TeV$/\sqrt{s})^{2}$, the requirement of a percent statistical
precision in the measurement of heavy particle pair production would imply an
integrated luminosity of the order $L\sim 10$
ab${}^{-1}(\sqrt{s}/10\,$TeV$)^{2})$. This could correspond to a 10-year
physics run with an instantaneous luminosity of the order
10${}^{35}(\sqrt{s}/10$TeV$)^{2}$cm-2s-1 [113]. At such large values of
$\sqrt{s}$, both the single-Higgs and the multi-Higgs production mechanisms
are dominated by vector-boson fusion (VBF) processers, which provides very
large statistical Higgs samples [77]. As an example, at $\sqrt{s}\sim 14$ TeV,
with 20 ab-1, one would produce about 20 million single Higgs, 90,000 Higgs
pairs, and 140 triple Higgs final states, with presumably quite moderate
background. Although there is currently only preliminary analysis of the Higgs
production in such an environment this would be a robust basis to considerably
advance on the Higgs couplings determination. The Higgs self-coupling sector
might be explored with unprecedented precision. In particular, with the above
Higgs production statistics, and no unexpectedly difficult background, an
accuracy of few percent for the trilinear Higgs coupling, and a few tens of
percent for the quadrilinear Higgs coupling might be reached at $\sqrt{s}\sim
14$ TeV, with 20 ab-1, assuming all the remaining Higgs and EW parameters at
their SM value. Many other investigations of the Higgs properties might
significantly benefit from such collider configuration [113, 77].

###  8.2 Higgs physics at multi-TeV electron-positron colliders

The length of linear accelerators proposed today, is largely determined by the
electric field gradients that can be achieved with RF cavities. For the
superconducting RF technology used by ILC the limit is about 35 MV/m while for
the drive-beam technology, envisaged for CLIC, it is about 100 MV/m.

Much higher gradients (up to 1000 times more acceleration compared to RF) can
be achieved using plasma-wakefield acceleration, where laser pulses [114, 115,
116, 117], electron [118, 119] or proton [120] bunches (called drivers) can
excite ultra-high fields in plasma devices. Thus this is a very promising
technique for future high energy $e^{+}e^{-}$ and $\gamma\gamma$ colliders
212121For $\gamma\gamma$ colliders it is sufficient to accelerate two $e^{-}$
beams which is technically less demanding than accelerating positrons. The
ALEGRO Collaboration [9] has been formed with the goal of designing an
Advanced Linear Collider (ALIC) based on this technology. A summary of the
facilities operating today and planned for the future, as well as the R&D
needed, are given in [9]. The physics opportunities of an $e^{+}e^{-}$
collider with $\sqrt{s}$ up to $100$ TeV are also discussed there.

The minimum instantaneous luminosity that needs to be achieved for probing
cross sections of new particles interacting weakly at energies in the $10-100$
TeV is found to be $10^{36}$ cm-2s-1. With such a collider, an integrated
luminosity of 30 ab-1 could be collected within a few years. With this
dataset, the Higgs physics programme is similar to that of a Multi-TeV muon
collider outlined above. It is also being considered to have such a collider
at lower collision energies, in the range between $m_{Z}$ and 3 TeV. Here, it
would have the same physics programme as the other proposed colliders,
assuming that comparable luminosity values can be achieved and background
conditions are similar.

The proposed ALIC collider [9] would achieve $\sqrt{s}=30$ TeV with a peak
luminosity of $10^{36}$ cm-2s-1 in a tunnel of 9 km length. While the
principle of acceleration has been proven, there are many issues that need to
be resolved before a collider based on plasma-technology can be achieved, but
none are considered to be show-stoppers at present. The primary focus of the
R&D programme is the beam quality which is addressed at lower-energy
applications (e.g. free-electron lasers, fixed target experiments) and will
benefit the development of a collider based on this technology.

###  8.3 What and Why: Higgs prospect studies beyond this report

The purpose of this subsection is to place the Higgs coupling measurements in
perspective with other new physics studies performed at future colliders with
the aim of providing answers to the following two questions: What are we going
to learn?, What can we possibly discover?. The unknown territory of energy and
precision to be explored may have different discoveries in store, including
unexpected ones. Given the scope of this document, a discussion of the various
options would hardly be self-contained, and would miss, by definition, the
most exciting case of unexpected discoveries. On the other hand, by focusing
on some open problems in particle physics, it is possible to structure a self-
contained discussion at least around the first question. The hierarchy problem
(HP), dark matter (DM) and the electroweak phase transition (EW$\Phi$T) are
issues on which we shall definitely make progress. Flavour could also be added
to this list, but mostly in the measure in which it is connected to the HP.

In view of its centrality, and of the controversial regard in which it is
sometimes held, a succinct but modern appraisal of the HP is needed. The HP is
a paradox challenging the modern effective field theory (EFT) view of particle
physics. The challenge is presented by the clash between infrared (IR)
Simplicity and Naturalness. IR Simplicity is an unavoidable feature of any EFT
when making observations at energies much below its fundamental scale
$\Lambda_{UV}$. In practice that is due to the decoupling of the infinite
(complicated) set of non-renormalizable couplings in favor of the finite
(simple) set of renormalizable ones. Naturalness instead arises by viewing EFT
parameters as functions of more fundamental ones: in this point of view it is
expected that any specific structure, like the presence of a very small
parameter, should be accounted for by symmetries and selection rules rather
than by accidents. Now, the structure of the SM is such that several crucial
experimental facts like approximate baryon and lepton numbers, lightness of
neutrinos, GIM suppression of FCNC, custodial symmetry all  remarkably and
beautifully follow from IR Simplification. That is by assuming
$\Lambda_{UV}\gg m_{weak}$. However when considering the Higgs mass parameter,
one famously finds that $\Lambda_{UV}\gg m_{H}$ is inconsistent with the
predicate of Naturalness. In the SM, IR Simplicity can thus only be obtained
at the price of un-Naturalness. But this is only half of the problem. The
other half is that models realizing Naturalness, like supersymmetry (SUSY) or
Composite Higgs (CH), invariably sacrifice Simplicity. Indeed all these
natural extensions have concrete structural difficulties in reproducing the
observed simplicity in flavour, CP violating and electroweak observables. In
order to meet the corresponding experimental constraints, these scenarios must
rely on clever constructions mostly associated with ad hoc symmetries, like
flavour symmetries or custodial symmetry, which in the SM are either not
needed or automatic. The paradoxical tension between Simplicity and
Naturalness is what defines the hierarchy problem: no win-win scenario seems
to be available.

The paradox could already be formulated before LEP, and gained in importance
with more and more precise flavour and electroweak data that demands a more
elaborate structure in natural models. Futhermore, the ever stronger bounds
from direct searches for ‘Natural’ agents at Tevatron and LHC imply the need
for some amount of un-Naturalness, or fine tuning, even in models like SUSY or
CH that aimed at full Naturalness. Depending on the scenario, the finesse of
the cancellation in the Higgs mass parameter needed to lift new physics out of
LHC reach can be quantified to roughly range from $1/10$ to $1/10^{3}$.

The test of Naturalness vs. Simplicity offers one concrete criterion to
compare future machines across their reach in three different sets of
measurements: direct searches, Higgs couplings, EW precision tests (EWPT).

  * •

Direct searches: Natural models all possess computational control over the
Higgs squared mass. The result varies in a finite range, and a small or
vanishing result can only be achieved by tuning different contributions
against one another. Indicating by $\Delta m_{H}^{2}$ the most sizeable
contribution, the tuning is simply measured by

$\epsilon\equiv m_{H}^{2}/\Delta m_{H}^{2}\,,$ (33)

with $m_{H}$ the observed Higgs mass. Because of its large Yukawa coupling,
the most sizeable effects come from coloured states associated with the top,
the so-called top-partners. Models can be broadly classified into three
classes according to the dependence of $\Delta m_{H}^{2}$ on the top partner
mass $m_{T}$:

    * –

Soft: $\Delta m_{H}^{2}\sim m_{T}^{2}$. This situation is realized in SUSY
with soft terms generated at a high scale. In the absence of any tuning
$m_{T}\sim m_{H}\sim 100$ GeV, within the energy range of LEP and Tevatron.

    * –

SuperSoft: $\Delta m_{H}^{2}\sim(3y_{t}^{2})/(4\pi^{2})\,m_{T}^{2}$. This
situation is realized in SUSY with low scale mediation and in CH. Without any
tuning one expects $m_{T}\sim m_{H}/\sqrt{3y_{t}^{2}/4\pi^{2}}\sim 450$ GeV,
within the reach of the LHC.

    * –

HyperSoft: $\Delta m_{H}^{2}\sim(3\lambda_{h})/(16\pi^{2})\,m_{T}^{2}$. The
mechanism of Neutral Naturalness is a prime example. The top partner mass is
naturally pushed around 1.5 TeV.

A $\sim 10$ TeV reach on $m_{T}$ like offered by FCC-hh or muon-collider (the
top partners have often EW quantum numbers) will thus probe Naturalness down
to $\epsilon=10^{-4},10^{-3},10^{-2}$ in respectively Soft, SuperSoft and
HyperSoft scenarios.

  * •

Higgs couplings: The deviations $\delta g_{h}$ from the SM in single and
multi-Higgs couplings satisfy

${\delta g_{H}}/{g_{H}}^{\rm SM}\sim c\,\epsilon\,,$ (34)

with $c$ a coupling-dependent coefficient, and $\epsilon$ the Higgs mass
correction defined in (33). In basically all models, there always exists a set
of couplings where $c\sim O(1)$. The only exception is strictly supersoft
SUSY, where one can cleverly go down to $c\sim 0.1$. Not surprisingly full
Naturalness basically mandates $O(1)$ deviations in Higgs couplings.

The best measurements that will be carried out at future machines aim at
$10^{-3}$ precision on some of the Higgs couplings, in particular $g_{HWW}$
and $g_{HZZ}$. This should be compared to the reach in $\epsilon$ in direct
searches. In particular, Higgs couplings probe less than direct searches in
the simplest high scale SUSY models. But one must not forget that these models
admit countless variants, with additional states, in particular SM singlets,
and with a spread spectra. So one cannot completely discount the relevance of
Higgs couplings to probe these models. In any case, one should not
underestimate the value of Higgs precision programme that can measure the
Higgs couplings with a $10^{-3}$ precision. The equal relevance of Higgs
studies and direct searches for CH models seems robust.

In view of parametric uncertainties, $10^{-3}$ seems like a limiting (or at
least a critical) sensitivity to BSM deviations in single Higgs production
near threshold. However these deviations are all associated with operators of
dimension $\geq 6$, whose effects grow with energy when considering processes
with a sufficient number of legs, like $gg\to HH,hV_{L}$ or $VV\to
HH,V_{L}V_{L}$. And one must then consider the possibility of obtaining a
better sensitivity by measuring such processes. For instance, FCC-hh can probe
$gg\to HV_{L}$ and $VV\to VV$ up to $\epsilon\sim 1\%$ [79]. Lepton machines
might compete better: while CLIC can reach a sensitivity to $\epsilon\sim
10^{-2}$ in $VV\to HH$ [121], still one order of magnitude poorer than single
Higgs measurements, a recent analysis [77] of a muon collider shows the
$\epsilon=10^{-3}$ wall is beaten for a CM energy of 10 TeV. A $\mu$-dream
machine running at 30 TeV could go down to $\epsilon=2\times 10^{-4}$ [77],
which competes well even with the reach on top partners ($\sim 14$ TeV) for
such a machine.

  * •

EWPT: While Higgs couplings are prime sensors of Naturalness, EWPT sense the
dynamics of EWSB indirectly, via loops. To make this concrete, consider the
$\hat{S}$ parameter, defined with the normalization of [48]. For all models,
encompassing supersymmetry, CH or technicolor one can write a parametric
formula

$\widehat{S}\sim\frac{\alpha_{W}}{4\pi}\frac{g_{*}^{2}v^{2}}{m_{*}^{2}}N\mathrel{\hbox
to0.0pt{\lower 3.0pt\hbox{\hskip 0.0pt$\sim$}\hss}\raise
1.0pt\hbox{$<$}}\frac{m_{W}^{2}}{m_{*}^{2}},$ (35)

where $m_{*},g_{*}$ indicate overall mass and coupling of the new dynamics
(the most obvious expectation being $m_{*}\sim m_{T}$), while $N$ measures the
number of new degrees of freedom. Theoretical considerations set the upper
bound $g_{*}\sqrt{N}\mathrel{\hbox to0.0pt{\lower 3.0pt\hbox{\hskip
0.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}4\pi$, which is saturated in CH and
technicolor where $\widehat{S}\sim m_{W}^{2}/m_{*}^{2}$. In these models a
measurement of $\widehat{S}$ translates into an indirect measurement of the
scale $m_{*}$. In the case of CH, one obtains $\widehat{S}\equiv 3\times
10^{-2}\epsilon$, indicating that a sensitivity to $\widehat{S}\sim\,{\rm
few}\,\times 10^{-5}$ corresponds to $10^{-3}$ sensitivity in Higgs
couplings/fine tuning. Supersymmetric models are instead well below the
saturation of the upper bound, as in that case the $g_{*}$ is of the order of
SM couplings, principally $g_{W}$ and $y_{t}$, while the multiplicity $N$ is
$O(1)$ [122]. One can then very roughly write
$\widehat{S}\sim(\alpha_{W}/4\pi)(m_{weak}^{2}/m_{*}^{2})$ implying
$m_{*}\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise
1.0pt\hbox{$>$}}1$ TeV is enough to make $\widehat{S}\mathrel{\hbox
to0.0pt{\lower 3.0pt\hbox{\hskip 0.0pt$\sim$}\hss}\raise
1.0pt\hbox{$<$}}10^{-5}$, below the wildest dreams of an FCC-ee.

Very much like for Higgs couplings, we can consider the sensitivity to the
same class of dim-6 operators contributing to $\widehat{S}$ in processes with
more legs, where the growth with energy can be exploited A crucial comparison
here is that between the reach of a $Z$-pole machine like the FCC-ee and CLIC
which can study processes such as $e^{+}e^{-}\to hZ,h\gamma,WW$ at higher
energies222222The latter processes are sensitive to slightly different
combinations of operator coefficients than $\widehat{S}$ at low energy, but in
well motivated models like CH, this difference is often subdominant, and at
worse they represent equally interesting but different combinations.. The
available CLIC studies estimate its reach as $\widehat{S}\simeq 0.5\times
10^{-5}$. This should be compared to the estimated reach of $5\times 10^{-5}$
at FCC-ee. Again the systematics of the two measures would be drastically
different, with CLIC dominated by statistics and with FCC-ee dominated by
parametric and intrinsic systematics.

The above analysis also offers the starting point for the consideration of
other motivations and other viewpoints. As we mentioned at the beginning the
EW phase transition and DM offer alternative motivations. We will discuss them
briefly in what follows.

The interest in the order of the EW phase transition is largely related to
baryogenesis. A strongly first order transition with sizeable sources of CP
violation from BSM dynamics could generate the observed cosmological baryon
asymmetry. The corresponding new physics would impact both future colliders
and precision low energy studies. In particular a first order phase transition
implies a $O(1)$ change in the Higgs potential at finite temperature,
indicating the possibility for important effects also at zero temperature, the
regime we can test at colliders. The connection between $T\not=0$ and $T=0$ is
however model dependent and one can broadly distinguish two scenarios. In the
absence of new symmetries $T\not=0$ and $T=0$ are directly connected and the
Higgs trilinear is expected to be $O(1)$ off its SM prediction (see e.g.
[123]). On the other hand in the presence of extra symmetries [124] there
could be a further phase separation between $T\not=0$ and $T=0$ in which case
deviations can be smaller but hardly smaller than a few $\%$ [125]. The low
energy implications of EW baryogenesis concern electric dipole moments (edms)
from new sources of CP violation. Here it must be noticed that the present
bound $d_{e}<10^{-29}$ e$\cdot$cm on the electron edm is already very strongly
constraining many scenarios. Moreover there are serious plans to improve the
sensitivity on $d_{e}$ by a few orders of magnitude in the future. A thorough
analysis is not available to our knowledge, but it would be interesting in
order to better appreciate the impact of such improved measurements on the
space of possibilities, as that conditions the importance of Higgs trilinear
studies. It should however be kept in mind that the EW$\Phi$T could be viewed
as interesting per se, regardless of baryogenesis, as it is an essential part
of the history of our Universe. It also offers a new bridge between Higgs
physics and gravitational physics: in case of a strong first order phase
transition, the EW$\Phi$T would be the source of a stochastic gravitational
wave (GW) background and future GW experiments like LISA could proffer
complementary probes of the Higgs potential [126].

The search for Dark Matter (DM) at future colliders is a broad field whose
implications cannot be even partially described here. One crucial strength of
a machine with a reach in the multi-TeV range is that it should be able to
discover the minimal incarnation of DM, the one which arises from the purely
radiatively split EW multiplets. Amazingly, this simple and well motivated
class of models is hardly directly detectable in view of its loop suppressed
spin independent cross section (see Table 1 in [127]). Focussing on Higgs
studies the basic question is: to what extent can an invisible Higgs width be
associated with DM? The answer is given in Fig. 4 of ref. [128]: considering
all present constraints there remains a very small region around
$m_{DM}=m_{H}/2$, and part of this region will be explored, by direct DM
detection experiments a long time before the future precision Higgs studies.
The chance that DM sits in this region seems slim but a discovery in direct
detection would certainly add to the Higgs programme. One should also keep in
mind that the Higgs boson can decay invisibly to new particles that are not by
themselves stable with the right relic abundance, but that would be part of a
more complex DM sector whose abundance would not set by its interactions with
the Higgs but rather by its own internal dynamics. Twin sectors of Mirror Twin
Higgs models [129] and strongly coupled dark sectors [130] are examples of
this type and they offer a motivation to search for invisible decays of the
Higgs.

##  9 Summary

The precision study of the Higgs boson will be the primary guaranteed
deliverable of any future collider facility. The apparent uniqueness of the
Higgs boson, as the only fundamental scalar boson thus far discovered,
justifies the importance which it is accorded in planning for future
facilities.

Several future colliders have been proposed in the context of the update of
the European Strategy for Particle Physics. The potential of these machines
for Higgs boson physics has been intensely studied by the proponents of these
machines. The purpose of the Higgs@FutureCollider Working Group (Higgs@FC WG)
and this report is to provide a coherent comparison of the reach of these
machines in the exploration of the Higgs sector. We have assumed the baseline
provided by the approved programme of the HL-LHC and quantified the additional
information that would be provided by each of the future facilities.

Taking into account the inputs submitted to the strategy process and our
dedicated discussions with proponents of future colliders, we provide this
report on the comparisons achieved, using both the simplified kappa framework
and an alternative EFT framework. The comparisons are made in reasonable
frameworks developed based on current knowledge, with the prime objective to
allow a clear and coherent comparison. Where relevant we note the potential
caveats in the approaches taken. We have also reported on the rare decays of
the Higgs boson, on measurements of its mass and width, and on the
expectations for CP violation studies.

This report updates a preliminary version posted on the (arXiv:1905.03764v1).
This version of the document has been used to provide input for Physics
Briefing Book of the Physics Preparatory Group, which will, in turn, serve as
input for strategic discussions in the European Strategy Group.

## Acknowledgements

We would like to express our appreciation to the numerous colleagues who have
presented and discussed material to our working group, and in addition for the
valuable comments made on a draft of this report by representatives of the
future collider projects.

We wish to express special thanks to J. Gu for discussions, for providing some
of the inputs for the EFT studies in this report, and for his help in
producing some of the results presented in Section 4. We are also very
grateful to J. Tian for his cross-checks of the EFT fit results and to J.
Reuter for his results on the double Higgs cross-sections at lepton colliders.

We thank S. Heinemeyer and collaborators for kindly sharing an updated version
of their study of the theory uncertainties in electroweak and Higgs precision
observables, whose results were also include in this report. We also benefited
from helpful discussions with R. Contino, G. Durieux, G.F. Giudice, M. Kumar,
M. Mangano, M. McCullough, G. Panico, A. Paul, M.E. Peskin, A. Pomarol, J.
Reuter, F. Riva, A. Shivaji, Y. Soreq, E. Vryonidou, A. Wulzer and X. Zhao.

The work of CG and BH was in part funded by the Deutsche
Forschungsgemeinschaft under Germany‘s Excellence Strategy – EXC 2121 “Quantum
Universe" – 390833306. The work of JDH and FM was partly supported by
F.R.S.-FNRS under the “Excellence of Science – EOS” – be.h project n.
30820817.

## References

  * [1] FCC Collaboration, “FCC Physics Opportunities”,  Eur. Phys. J. C79 (2019), no. 6, 474, 10.1140/epjc/s10052-019-6904-3.
  * [2] CEPC Study Group Collaboration, “CEPC Conceptual Design Report: Volume 2 - Physics & Detector”, arXiv:1811.10545.
  * [3] P. Bambade et al., “The International Linear Collider: A Global Project”, arXiv:1903.01629.
  * [4] K. Fujii et al., “Tests of the Standard Model at the International Linear Collider”, arXiv:1908.11299.
  * [5] CLICdp, CLIC Collaboration, “The Compact Linear Collider (CLIC) - 2018 Summary Report”,  CERN Yellow Rep. Monogr. 1802 (2018) 1–98, 10.23731/CYRM-2018-002, arXiv:1812.06018.
  * [6] LHeC Study Group Collaboration, “A Large Hadron Electron Collider at CERN: Report on the Physics and Design Concepts for Machine and Detector”,  J. Phys. G39 (2012) 075001, 10.1088/0954-3899/39/7/075001, arXiv:1206.2913.
  * [7] D. M. Asner, J. B. Gronberg, and J. F. Gunion, “Detecting and studying Higgs bosons at a photon-photon collider”,  Phys. Rev. D67 (2003) 035009, 10.1103/PhysRevD.67.035009, arXiv:hep-ph/0110320.
  * [8] S. A. Bogacz et al., “SAPPHiRE: a Small Gamma-Gamma Higgs Factory”, arXiv:1208.2827.
  * [9] ALEGRO Collaboration, “Towards an Advanced Linear International Collider”, arXiv:1901.10370.
  * [10] M. Mangano, “Physics potential of a low-energy FCC-hh”. https://cds.cern.ch/record/2681366.
  * [11] J.-J. Blaising and P. Roloff, “Electroweak couplings of the $Z$ boson at CLIC”. private communication.
  * [12] F. Bordry et al., “Machine Parameters and Projected Luminosity Performance of Proposed Future Colliders at CERN”, arXiv:1810.13022.
  * [13] Physics of the HL-LHC Working Group Collaboration, “Higgs Physics at the HL-LHC and HE-LHC”, arXiv:1902.00134.
  * [14] K. Fujii et al., “Physics Case for the 250 GeV Stage of the International Linear Collider”, arXiv:1710.07621.
  * [15] CLIC, CLICdp Collaboration, “The Compact Linear e+e- Collider (CLIC): Physics Potential”, arXiv:1812.07986.
  * [16] LHC Higgs Cross Section Working Group Collaboration, “LHC HXSWG interim recommendations to explore the coupling structure of a Higgs-like particle”, arXiv:1209.0040.
  * [17] LHC Higgs Cross Section Working Group Collaboration, “Handbook of LHC Higgs Cross Sections: 3. Higgs Properties”, 10.5170/CERN-2013-004, arXiv:1307.1347.
  * [18] T. Barklow et al., “Improved Formalism for Precision Higgs Coupling Fits”,  Phys. Rev. D97 (2018), no. 5, 053003, 10.1103/PhysRevD.97.053003, arXiv:1708.08912.
  * [19] M. Klein et al., 2019.  LHeC Higgs studies (in preparation).
  * [20] HEPFit Collaboration, “HEPfit: a code for the combination of indirect and direct constrains on high energy physics models”. http://hepfit.roma1.infn.it/.
  * [21] J. de Blas et al., “Electroweak precision observables and Higgs-boson signal strengths in the Standard Model and beyond: present and future”,  JHEP 12 (2016) 135, 10.1007/JHEP12(2016)135, arXiv:1608.01509.
  * [22] J. de Blas et al., “Electroweak precision constraints at present and future colliders”,  PoS ICHEP2016 (2017) 690, 10.22323/1.282.0690, arXiv:1611.05354.
  * [23] J. de Blas et al., “The Global Electroweak and Higgs Fits in the LHC era”,  PoS EPS-HEP2017 (2017) 467, 10.22323/1.314.0467, arXiv:1710.05402.
  * [24] J. de Blas, O. Eberhardt, and C. Krause, “Current and Future Constraints on Higgs Couplings in the Nonlinear Effective Theory”,  JHEP 07 (2018) 048, 10.1007/JHEP07(2018)048, arXiv:1803.00939.
  * [25] F. Beaujean et al., “Bayesian Analysis Toolkit: 1.0 and beyond”,  J. Phys. Conf. Ser. 664 (2015), no. 7, 072003, 10.1088/1742-6596/664/7/072003.
  * [26] A. Falkowski, S. Rychkov, and A. Urbano, “What if the Higgs couplings to W and Z bosons are larger than in the Standard Model?”,  JHEP 04 (2012) 073, 10.1007/JHEP04(2012)073, arXiv:1202.1532.
  * [27] A. Falkowski and R. Rattazzi, “Which EFT”, arXiv:1902.05936.
  * [28] S. Chang and M. A. Luty, “The Higgs Trilinear Coupling and the Scale of New Physics”, arXiv:1902.05556.
  * [29] J. de Blas, J. C. Criado, M. Perez-Victoria, and J. Santiago, “Effective description of general extensions of the Standard Model: the complete tree-level dictionary”,  JHEP 03 (2018) 109, 10.1007/JHEP03(2018)109, arXiv:1711.10391.
  * [30] G. F. Giudice, C. Grojean, A. Pomarol, and R. Rattazzi, “The Strongly-Interacting Light Higgs”,  JHEP 06 (2007) 045, 10.1088/1126-6708/2007/06/045, arXiv:hep-ph/0703164.
  * [31] D. Liu, A. Pomarol, R. Rattazzi, and F. Riva, “Patterns of Strong Coupling for LHC Searches”,  JHEP 11 (2016) 141, 10.1007/JHEP11(2016)141, arXiv:1603.03064.
  * [32] B. Grzadkowski, M. Iskrzynski, M. Misiak, and J. Rosiek, “Dimension-Six Terms in the Standard Model Lagrangian”,  JHEP 10 (2010) 085, 10.1007/JHEP10(2010)085, arXiv:1008.4884.
  * [33] R. Alonso, E. E. Jenkins, A. V. Manohar, and M. Trott, “Renormalization Group Evolution of the Standard Model Dimension Six Operators III: Gauge Coupling Dependence and Phenomenology”,  JHEP 04 (2014) 159, 10.1007/JHEP04(2014)159, arXiv:1312.2014.
  * [34] G. D’Ambrosio, G. F. Giudice, G. Isidori, and A. Strumia, “Minimal flavor violation: An Effective field theory approach”,  Nucl. Phys. B645 (2002) 155–187, 10.1016/S0550-3213(02)00836-2, arXiv:hep-ph/0207036.
  * [35] A. Falkowski, “Higgs Basis: Proposal for an EFT basis choice for LHC HXSWG”,  LHCHXSWG-INT-2015-001 (2015).
  * [36] A. Falkowski, “Effective field theory approach to LHC Higgs data”,  Pramana 87 (2016), no. 3, 39, 10.1007/s12043-016-1251-5, arXiv:1505.00046.
  * [37] R. Contino, “The Higgs as a Composite Nambu-Goldstone Boson”, in  Physics of the large and the small, TASI 09, proceedings of the Theoretical Advanced Study Institute in Elementary Particle Physics, Boulder, Colorado, USA, 1-26 June 2009, pp. 235–306.  2011\.  arXiv:1005.4269.  10.1142/9789814327183_0005.
  * [38] G. Panico and A. Wulzer, “The Composite Nambu-Goldstone Higgs”,  Lect. Notes Phys. 913 (2016) pp.1–316, 10.1007/978-3-319-22617-0, arXiv:1506.01961.
  * [39] G. Ferretti and D. Karateev, “Fermionic UV completions of Composite Higgs models”,  JHEP 03 (2014) 077, 10.1007/JHEP03(2014)077, arXiv:1312.5330.
  * [40] G. Cacciapaglia and F. Sannino, “Fundamental Composite (Goldstone) Higgs Dynamics”,  JHEP 04 (2014) 111, 10.1007/JHEP04(2014)111, arXiv:1402.0233.
  * [41] A. Azatov, R. Contino, G. Panico, and M. Son, “Effective field theory analysis of double Higgs boson production via gluon fusion”,  Phys. Rev. D92 (2015), no. 3, 035001, 10.1103/PhysRevD.92.035001, arXiv:1502.00539.
  * [42] M. S. Amjad et al., “A precise characterisation of the top quark electro-weak vertices at the ILC”,  Eur. Phys. J. C75 (2015), no. 10, 512, 10.1140/epjc/s10052-015-3746-5, arXiv:1505.06020.
  * [43] CLICdp Collaboration, “Top-Quark Physics at the CLIC Electron-Positron Linear Collider”, arXiv:1807.02441.
  * [44] P. Janot, “Top-quark electroweak couplings at the FCC-ee”,  JHEP 04 (2015) 182, 10.1007/JHEP04(2015)182, arXiv:1503.01325.
  * [45] J. de Blas et al., “On the future of Higgs, electroweak and diboson measurements at lepton colliders”, arXiv:1907.04311.
  * [46] J. de Blas et al., “The CLIC Potential for New Physics”, 10.23731/CYRM-2018-003, arXiv:1812.02093.
  * [47] T. Barklow et al., “Model-Independent Determination of the Triple Higgs Coupling at e+e- Colliders”,  Phys. Rev. D97 (2018), no. 5, 053004, 10.1103/PhysRevD.97.053004, arXiv:1708.09079.
  * [48] R. Barbieri, A. Pomarol, R. Rattazzi, and A. Strumia, “Electroweak symmetry breaking after LEP-1 and LEP-2”,  Nucl. Phys. B703 (2004) 127–146, 10.1016/j.nuclphysb.2004.10.014, arXiv:hep-ph/0405040.
  * [49] M. Farina et al., “Energy helps accuracy: electroweak precision tests at hadron colliders”,  Phys. Lett. B772 (2017) 210–215, 10.1016/j.physletb.2017.06.043, arXiv:1609.08157.
  * [50] S. Banerjee, C. Englert, R. S. Gupta, and M. Spannowsky, “Probing Electroweak Precision Physics via boosted Higgs-strahlung at the LHC”,  Phys. Rev. D98 (2018), no. 9, 095012, 10.1103/PhysRevD.98.095012, arXiv:1807.01796.
  * [51] K. Fujii, J. List, M. Peskin and J. Tiang, private communication.
  * [52] R. Franceschini et al., “Electroweak Precision Tests in High-Energy Diboson Processes”,  JHEP 02 (2018) 111, 10.1007/JHEP02(2018)111, arXiv:1712.01310.
  * [53] B. Henning, D. Lombardo, M. Riembau, and F. Riva, “Higgs Couplings without the Higgs”, arXiv:1812.09299.
  * [54] F. Maltoni, L. Mantani, and K. Mimasu, “Top-quark electroweak interactions at high energy”, arXiv:1904.05637.
  * [55] G. P. Lepage, P. B. Mackenzie, and M. E. Peskin, “Expected Precision of Higgs Boson Partial Widths within the Standard Model”, arXiv:1404.0319.
  * [56] A. Blondel et al., “Theory Requirements and Possibilities for the FCC-ee and other Future High Energy and Precision Frontier Lepton Colliders”, arXiv:1901.02648.
  * [57] A. Blondel et al., “Standard Model Theory for the FCC-ee: The Tera-Z”, in  Mini Workshop on Precision EW and QCD Calculations for the FCC Studies : Methods and Techniques CERN, Geneva, Switzerland, January 12-13, 2018.  2018\.  arXiv:1809.01830.
  * [58] D. de Florian and J. Mazzitelli, “Two-loop virtual corrections to Higgs pair production”,  Phys. Lett. B724 (2013) 306–309, 10.1016/j.physletb.2013.06.046, arXiv:1305.5206.
  * [59] D. de Florian and J. Mazzitelli, “Higgs Boson Pair Production at Next-to-Next-to-Leading Order in QCD”,  Phys. Rev. Lett. 111 (2013) 201801, 10.1103/PhysRevLett.111.201801, arXiv:1309.6594.
  * [60] J. Grigo, K. Melnikov, and M. Steinhauser, “Virtual corrections to Higgs boson pair production in the large top quark mass limit”,  Nucl. Phys. B888 (2014) 17–29, 10.1016/j.nuclphysb.2014.09.003, arXiv:1408.2422.
  * [61] D. de Florian et al., “Differential Higgs Boson Pair Production at Next-to-Next-to-Leading Order in QCD”,  JHEP 09 (2016) 151, 10.1007/JHEP09(2016)151, arXiv:1606.09519.
  * [62] S. Borowka et al., “Higgs Boson Pair Production in Gluon Fusion at Next-to-Leading Order with Full Top-Quark Mass Dependence”,  Phys. Rev. Lett. 117 (2016), no. 1, 012001, 10.1103/PhysRevLett.117.079901, 10.1103/PhysRevLett.117.012001, arXiv:1604.06447. [Erratum: Phys. Rev. Lett.117,no.7,079901(2016)].
  * [63] S. Borowka et al., “Full top quark mass dependence in Higgs boson pair production at NLO”,  JHEP 10 (2016) 107, 10.1007/JHEP10(2016)107, arXiv:1608.04798.
  * [64] J. Baglio et al., “Gluon fusion into Higgs pairs at NLO QCD and the top mass scheme”,  Eur. Phys. J. C79 (2019), no. 6, 459, 10.1140/epjc/s10052-019-6973-3, arXiv:1811.05692.
  * [65] R. Frederix et al., “Higgs pair production at the LHC with NLO and parton-shower effects”,  Phys. Lett. B732 (2014) 142–149, 10.1016/j.physletb.2014.03.026, arXiv:1401.7340.
  * [66] J. Reuter (DESY), WHIZARD team, private communication.
  * [67] M. McCullough, “An Indirect Model-Dependent Probe of the Higgs Self-Coupling”,  Phys. Rev. D90 (2014), no. 1, 015001, 10.1103/PhysRevD.90.015001, 10.1103/PhysRevD.92.039903, arXiv:1312.3322. [Erratum: Phys. Rev.D92,no.3,039903(2015)].
  * [68] G. Degrassi, P. P. Giardino, F. Maltoni, and D. Pagani, “Probing the Higgs self coupling via single Higgs production at the LHC”,  JHEP 12 (2016) 080, 10.1007/JHEP12(2016)080, arXiv:1607.04251.
  * [69] W. Bizon, M. Gorbahn, U. Haisch, and G. Zanderighi, “Constraints on the trilinear Higgs coupling from vector boson fusion and associated Higgs production at the LHC”,  JHEP 07 (2017) 083, 10.1007/JHEP07(2017)083, arXiv:1610.05771.
  * [70] J. J. van der Bij, “Does Low-energy Physics Depend on the Potential of a Heavy Higgs Particle?”,  Nucl. Phys. B267 (1986) 557–565, 10.1016/0550-3213(86)90131-8.
  * [71] G. Degrassi, M. Fedele, and P. P. Giardino, “Constraints on the trilinear Higgs self coupling from precision observables”,  JHEP 04 (2017) 155, 10.1007/JHEP04(2017)155, arXiv:1702.01737.
  * [72] G. D. Kribs et al., “Electroweak oblique parameters as a probe of the trilinear Higgs boson self-interaction”,  Phys. Rev. D95 (2017), no. 9, 093004, 10.1103/PhysRevD.95.093004, arXiv:1702.07678.
  * [73] S. Di Vita et al., “A global view on the Higgs self-coupling”,  JHEP 09 (2017) 069, 10.1007/JHEP09(2017)069, arXiv:1704.01953.
  * [74] S. Di Vita et al., “A global view on the Higgs self-coupling at lepton colliders”,  JHEP 02 (2018) 178, 10.1007/JHEP02(2018)178, arXiv:1711.03978.
  * [75] F. Maltoni, D. Pagani, A. Shivaji, and X. Zhao, “Trilinear Higgs coupling determination via single-Higgs differential measurements at the LHC”,  Eur. Phys. J. C77 (2017), no. 12, 887, 10.1140/epjc/s10052-017-5410-8, arXiv:1709.08649.
  * [76] L. Di Luzio, R. Gröber, and M. Spannowsky, “Maxi-sizing the trilinear Higgs self-coupling: how large could it be?”,  Eur. Phys. J. C77 (2017), no. 11, 788, 10.1140/epjc/s10052-017-5361-0, arXiv:1704.02311.
  * [77] “Muon Collider - Preparatory Meeting”, 10-11 April 2019, CERN. https://indico.cern.ch/event/801616.
  * [78] CMS Collaboration, “Search for a standard model-like Higgs boson in the $\mu^{+}\mu^{-}$ and $e^{+}e^{-}$ decay channels at the LHC”,  Phys. Lett. B744 (2015) 184–207, 10.1016/j.physletb.2015.03.048, arXiv:1410.6679.
  * [79] FCC Collaboration, “FCC-ee: The Lepton Collider”,  Eur. Phys. J. ST 228 (2019), no. 2, 261–623, 10.1140/epjst/e2019-900045-4.
  * [80] ATLAS Collaboration, “Search for the Decay of the Higgs Boson to Charm Quarks with the ATLAS Experiment”,  Phys. Rev. Lett. 120 (2018), no. 21, 211802, 10.1103/PhysRevLett.120.211802, arXiv:1802.04329.
  * [81] G. Perez, Y. Soreq, E. Stamou, and K. Tobioka, “Prospects for measuring the Higgs boson coupling to light quarks”,  Phys. Rev. D93 (2016), no. 1, 013001, 10.1103/PhysRevD.93.013001, arXiv:1505.06689.
  * [82] G. T. Bodwin, F. Petriello, S. Stoynev, and M. Velasco, “Higgs boson decays to quarkonia and the $H\bar{c}c$ coupling”,  Phys. Rev. D88 (2013), no. 5, 053003, 10.1103/PhysRevD.88.053003, arXiv:1306.5770.
  * [83] G. T. Bodwin et al., “Relativistic corrections to Higgs boson decays to quarkonia”,  Phys. Rev. D90 (2014), no. 11, 113010, 10.1103/PhysRevD.90.113010, arXiv:1407.6695.
  * [84] A. L. Kagan et al., “Exclusive Window onto Higgs Yukawa Couplings”,  Phys. Rev. Lett. 114 (2015), no. 10, 101802, 10.1103/PhysRevLett.114.101802, arXiv:1406.1722.
  * [85] ATLAS Collaboration, “Searches for exclusive Higgs and $Z$ boson decays into $J/\psi\gamma$, $\psi(2S)\gamma$, and $\Upsilon(nS)\gamma$ at $\sqrt{s}=13$ TeV with the ATLAS detector”,  Phys. Lett. B786 (2018) 134–155, 10.1016/j.physletb.2018.09.024, arXiv:1807.00802.
  * [86] CMS Collaboration, “Search for rare decays of Z and Higgs bosons to J$/\psi$ and a photon in proton-proton collisions at $\sqrt{s}=$ 13 TeV”,  Eur. Phys. J. C79 (2019), no. 2, 94, 10.1140/epjc/s10052-019-6562-5, arXiv:1810.10056.
  * [87] ATLAS Collaboration, “Search for exclusive Higgs and $Z$ boson decays to $\phi\gamma$ and $\rho\gamma$ with the ATLAS detector”,  JHEP 07 (2018) 127, 10.1007/JHEP07(2018)127, arXiv:1712.02758.
  * [88] CMS Collaboration, “Search for a Higgs boson decaying into $\gamma^{*}\gamma\to\ell\ell\gamma$ with low dilepton mass in pp collisions at $\sqrt{s}=$ 8 TeV”,  Phys. Lett. B753 (2016) 341–362, 10.1016/j.physletb.2015.12.039, arXiv:1507.03031.
  * [89] J. Brod, U. Haisch, and J. Zupan, “Constraints on CP-violating Higgs couplings to the third generation”,  JHEP 11 (2013) 180, 10.1007/JHEP11(2013)180, arXiv:1310.1385.
  * [90] Y. T. Chien et al., “Direct and indirect constraints on CP-violating Higgs-quark and Higgs-gluon interactions”,  JHEP 02 (2016) 011, 10.1007/JHEP02(2016)011, arXiv:1510.00725. [JHEP02,011(2016)].
  * [91] W. Altmannshofer, J. Brod, and M. Schmaltz, “Experimental constraints on the coupling of the Higgs boson to electrons”,  JHEP 05 (2015) 125, 10.1007/JHEP05(2015)125, arXiv:1503.04830.
  * [92] D. Egana-Ugrinovic and S. Thomas, “Higgs Boson Contributions to the Electron Electric Dipole Moment”, arXiv:1810.08631.
  * [93] J. Brod and E. Stamou, “Electric dipole moment constraints on CP-violating heavy-quark Yukawas at next-to-leading order”, arXiv:1810.12303.
  * [94] J. Brod and D. Skodras, “Electric dipole moment constraints on CP-violating light-quark Yukawas”,  JHEP 01 (2019) 233, 10.1007/JHEP01(2019)233, arXiv:1811.05480.
  * [95] ACME Collaboration, “Improved limit on the electric dipole moment of the electron”,  Nature 562 (2018), no. 7727, 355–360, 10.1038/s41586-018-0599-8.
  * [96] Particle Data Group Collaboration, “Review of Particle Physics”,  Phys. Rev. D 98 (Aug, 2018) 030001, 10.1103/PhysRevD.98.030001.
  * [97] LHC Higgs Cross Section Working Group Collaboration, “Handbook of LHC Higgs Cross Sections: 4. Deciphering the Nature of the Higgs Sector”, 10.23731/CYRM-2017-002, arXiv:1610.07922.
  * [98] A. Robson and P. Roloff, “Updated CLIC luminosity staging baseline and Higgs coupling prospects”, arXiv:1812.01644.
  * [99] P. Azzi et al., “Prospective Studies for LEP3 with the CMS Detector”, arXiv:1208.1662.
  * [100] N. Kauer and G. Passarino, “Inadequacy of zero-width approximation for a light Higgs boson signal”,  JHEP 08 (2012) 116, 10.1007/JHEP08(2012)116, arXiv:1206.4803.
  * [101] F. Caola and K. Melnikov, “Constraining the Higgs boson width with ZZ production at the LHC”,  Phys. Rev. D88 (2013) 054024, 10.1103/PhysRevD.88.054024, arXiv:1307.4935.
  * [102] A. Azatov, C. Grojean, A. Paul, and E. Salvioni, “Taming the off-shell Higgs boson”,  Zh. Eksp. Teor. Fiz. 147 (2015) 410–425, 10.1134/S1063776115030140, 10.7868/S0044451015030039, arXiv:1406.6338. [J. Exp. Theor. Phys.120,354(2015)].
  * [103] CEPC Physics-Detector Study Group Collaboration, “The CEPC input for the European Strategy for Particle Physics - Physics and Detector”, arXiv:1901.03170.
  * [104] F. An et al., “Precision Higgs physics at the CEPC”,  Chin. Phys. C43 (2019), no. 4, 043002, 10.1088/1674-1137/43/4/043002, arXiv:1810.09037.
  * [105] V. D. Barger, M. Berger, J. F. Gunion, and T. Han, “Physics of Higgs factories”,  eConf C010630 (2001) E110, arXiv:hep-ph/0110340.
  * [106] C. Rubbia, “Further searches of the Higgs scalar sector, Input to the European Particle Physics Strategy Update 2018-2020”. https://indico.cern.ch/event/765096/contributions/3295605/, 2018.
  * [107] M. Greco, T. Han, and Z. Liu, “ISR effects for resonant Higgs production at future lepton colliders”,  Phys. Lett. B763 (2016) 409–415, 10.1016/j.physletb.2016.10.078, arXiv:1607.03210.
  * [108] T. Han and Z. Liu, “Potential precision of a direct measurement of the Higgs boson total width at a muon collider”,  Phys. Rev. D87 (2013), no. 3, 033007, 10.1103/PhysRevD.87.033007, arXiv:1210.7803.
  * [109] Y. Alexahin et al., “Muon Collider Higgs Factory for Snowmass 2013”, in  Proceedings, 2013 Community Summer Study on the Future of U.S. Particle Physics: Snowmass on the Mississippi (CSS2013): Minneapolis, MN, USA, July 29-August 6, 2013.  2013\.  arXiv:1308.2143.
  * [110] P. Janot, “Circular lepton colliders (FCC-ee + CEPC + Muon collider)”, (2018). https://indico.cern.ch/event/716380/.
  * [111] N. Bartosik et al., “Preliminary Report on the Study of Beam-Induced Background Effects at a Muon Collider”, arXiv:1905.03725.
  * [112] N. Pastrone, “Muon Colliders, Input to the European Particle Physics Strategy Update 2018-2020”. https://indico.cern.ch/event/765096/contributions/3295784/, 2018.
  * [113] J. P. Delahaye et al., “Muon Colliders”, arXiv:1901.06150.
  * [114] S. P. D. Mangles et al., “Monoenergetic beams of relativistic electrons from intense laser-plasma interactions”,  Nature 431 (2004) 535–538, 10.1038/nature02939.
  * [115] S. Lee, T. C. Katsouleas, R. Hemker, and W. B. Mori, “Simulations of a meter-long plasma wake field accelerator”,  Phys. Rev. E61 (2000) 7014–7021, 10.1103/PhysRevE.61.7014.
  * [116] J. Krall and G. Joyce, “Transverse equilibrium and stability of the primary beam in the plasma wake-field accelerator”,  Phys. Plasmas 2 (1995), no. 4, 1326–1331, 10.1063/1.871344.
  * [117] D. H. Whittum, “Transverse two stream instability of a beam with a Bennett profile”,  Phys. Plasmas 4 (1997) 1154–1159, 10.1063/1.872202.
  * [118] I. Blumenfeld et al., “Energy doubling of 42 GeV electrons in a metre-scale plasma wakefield accelerator”,  Nature 445 (2007) 741–744, 10.1038/nature05538.
  * [119] M. Litos et al., “High-efficiency acceleration of an electron beam in a plasma wakefield accelerator”,  Nature 515 (2014), no. 7525, 92–95, 10.1038/nature13882.
  * [120] AWAKE Collaboration, “Acceleration of electrons in the plasma wakefield of a proton bunch”,  Nature 561 (2018), no. 7723, 363–367, 10.1038/s41586-018-0485-4, arXiv:1808.09759.
  * [121] R. Contino et al., “Strong Higgs Interactions at a Linear Collider”,  JHEP 02 (2014) 006, 10.1007/JHEP02(2014)006, arXiv:1309.7038.
  * [122] G. Marandella, C. Schappacher, and A. Strumia, “Supersymmetry and precision data after LEP2”,  Nucl. Phys. B715 (2005) 173–189, 10.1016/j.nuclphysb.2005.03.001, arXiv:hep-ph/0502095.
  * [123] C. Grojean, G. Servant, and J. D. Wells, “First-order electroweak phase transition in the standard model with a low cutoff”,  Phys. Rev. D71 (2005) 036001, 10.1103/PhysRevD.71.036001, arXiv:hep-ph/0407019.
  * [124] J. R. Espinosa, T. Konstandin, and F. Riva, “Strong Electroweak Phase Transitions in the Standard Model with a Singlet”,  Nucl. Phys. B854 (2012) 592–630, 10.1016/j.nuclphysb.2011.09.010, arXiv:1107.5441.
  * [125] D. Curtin, P. Meade, and C.-T. Yu, “Testing Electroweak Baryogenesis with Future Colliders”,  JHEP 11 (2014) 127, 10.1007/JHEP11(2014)127, arXiv:1409.0005.
  * [126] C. Grojean and G. Servant, “Gravitational Waves from Phase Transitions at the Electroweak Scale and Beyond”,  Phys. Rev. D75 (2007) 043507, 10.1103/PhysRevD.75.043507, arXiv:hep-ph/0607107.
  * [127] M. Farina, D. Pappadopulo, and A. Strumia, “A modified naturalness principle and its experimental tests”,  JHEP 08 (2013) 022, 10.1007/JHEP08(2013)022, arXiv:1303.7244.
  * [128] A. De Simone, G. F. Giudice, and A. Strumia, “Benchmarks for Dark Matter Searches at the LHC”,  JHEP 06 (2014) 081, 10.1007/JHEP06(2014)081, arXiv:1402.6287.
  * [129] Z. Chacko, D. Curtin, M. Geller, and Y. Tsai, “Cosmological Signatures of a Mirror Twin Higgs”,  JHEP 09 (2018) 163, 10.1007/JHEP09(2018)163, arXiv:1803.03263.
  * [130] R. Contino, A. Mitridate, A. Podo, and M. Redi, “Gluequark Dark Matter”,  JHEP 02 (2019) 187, 10.1007/JHEP02(2019)187, arXiv:1811.06975.
  * [131] S. Heinemeyer et al., “Theoretical uncertainties for electroweak and Higgs-boson precision measurements at FCC-ee”, (2019). IFT-UAM/CSIC-18-021, TUM-HEP-1185/19.
  * [132] F. Simon, “Impact of Theory Uncertainties on the Precision of the Top Quark Mass in a Threshold Scan at Future e+e- Colliders”,  PoS ICHEP2016 (2017) 872, 10.22323/1.282.0872, arXiv:1611.03399.
  * [133] The CEPC Study Group Collaboration, “CEPC Electroweak Precision Measurements - CEPC-PHY-EW-2019-008”. private communication.
  * [134] A. Abada et al., “Future Circular Collider”,  CERN-ACC-2018-0056 (2018).
  * [135] N. Alipour Tehrani et al., “FCC-ee: Your Questions Answered”, in  CERN Council Open Symposium on the Update of European Strategy for Particle Physics (EPPSU) Granada, Spain, May 13-16, 2019, A. Blondel and P. Janot, eds.  2019\.  arXiv:1906.02693.
  * [136] R. Tenchini, “Electroweak Precision Measurements at the FCC-ee”. https://indico.cern.ch/event/789349/contributions/3298720/attachments/1805901/2947167/FCC-EWK_CDR-presentation.pdf.
  * [137] G. W. Wilson, “Updated Study of a Precision Measurement of the W Mass from a Threshold Scan Using Polarized $\rm{e}^{-}$ and $\rm{e}^{+}$ at ILC”, in  Proceedings, International Workshop on Future Linear Colliders (LCWS15): Whistler, B.C., Canada, November 02-06, 2015.  2016\.  arXiv:1603.06016.
  * [138] ALEPH, DELPHI, L3, OPAL, SLD, LEP Electroweak Working Group, SLD Electroweak Group, SLD Heavy Flavour Group Collaboration, “Precision electroweak measurements on the $Z$ resonance”,  Phys. Rept. 427 (2006) 257–454, 10.1016/j.physrep.2005.12.006, arXiv:hep-ex/0509008.
  * [139] M. Mangano, “Higgs Physics Potential of FCC-hh Standalone”. https://cds.cern.ch/record/2681378.
  * [140] G. Altarelli and R. Barbieri, “Vacuum polarization effects of new physics on electroweak processes”,  Phys. Lett. B253 (1991) 161–167, 10.1016/0370-2693(91)91378-9.
  * [141] M. E. Peskin and T. Takeuchi, “A New constraint on a strongly interacting Higgs sector”,  Phys. Rev. Lett. 65 (1990) 964–967, 10.1103/PhysRevLett.65.964.
  * [142] B. Holdom and J. Terning, “Large corrections to electroweak parameters in technicolor theories”,  Phys. Lett. B247 (1990) 88–92, 10.1016/0370-2693(90)91054-F.
  * [143] M. Golden and L. Randall, “Radiative Corrections to Electroweak Parameters in Technicolor Theories”,  Nucl. Phys. B361 (1991) 3–23, 10.1016/0550-3213(91)90614-4.
  * [144] M. E. Peskin and T. Takeuchi, “Estimation of oblique electroweak corrections”,  Phys. Rev. D46 (1992) 381–409, 10.1103/PhysRevD.46.381.
  * [145] B. Grinstein and M. B. Wise, “Operator analysis for precision electroweak physics”,  Phys. Lett. B265 (1991) 326–334, 10.1016/0370-2693(91)90061-T.

##  Appendix A Mandate agreed by RECFA in consultation with the PPG “Higgs
physics with future colliders in parallel and beyond the HL-LHC”

In the context of exploring the Higgs sector, provide a coherent comparison of
the reach with all future collider programmes proposed for the European
Strategy update, and to project the information on a timeline.

  * •

For the benefit of the comparison, motivate the choice for an adequate
interpretation framework (e.g. EFT, $\kappa$, …) and apply it, and map the
potential prerequisites related to the validity and use of such framework(s).

  * •

For at least the following aspects, where achievable, comparisons should be
aim for:

    * –

Precision on couplings and self-couplings (through direct and indirect
methods);

    * –

Sensitivities to anomalous and rare Higgs decays (SM and BSM), and precision
on the total width;

    * –

Sensitivity to new high-scale physics through loop corrections;

    * –

Sensitivities to flavour violation and CP violating effects.

  * •

In all cases the future collider information is to be combined with the
expected HL-LHC reach, and the combined extended reach is to be compared with
the baseline reach of the HL-LHC.

  * •

In April 2019, provide a comprehensive and public report to inform the
community.

  * •

ECFA helps in the creation of a working group relevant for the Strategy
process, especially for the Physics Preparatory Group (PPG).

  * •

Towards the Open Symposium the working group will work together with the PPG
to provide a comprehensive and public report to inform the community, i.e.
this is not an ECFA report.

  * •

The working group has a scientific nature, i.e. not a strategic nature; it