$t\bar{t}H$, multilepton ($3\ell$) † | $35.9\oplus 41.5$ | $0.82\begin{subarray}{c}+0.77\\\ -0.71\end{subarray}\oplus 0.29\begin{subarray}{c}+0.82\\\ -0.62\end{subarray}$ | [92, 93]
$t\bar{t}H$, multilepton ($4\ell$) † | $35.9\oplus 41.5$ | $0.57\begin{subarray}{c}+2.29\\\ -1.57\end{subarray}\oplus 0.99\begin{subarray}{c}+3.31\\\ -1.69\end{subarray}$ | [92, 93]
|  |  $\sigma_{\text{obs}}$ [pb] |  $\sigma_{\text{SM}}$ [pb] |
$gg\to H$, $H\to W^{+}W^{-}$ ($0j$) | $137.0$ | $0.0423\begin{subarray}{c}+0.0063\\\ -0.0059\end{subarray}$ | $0.0457\begin{subarray}{c}+0.0029\\\ -0.0018\end{subarray}$ | [94]
$gg\to H$, $H\to W^{+}W^{-}$ ($1j$) | $137.0$ | $0.0240\begin{subarray}{c}+0.0057\\\ -0.0051\end{subarray}$ | $0.0217\begin{subarray}{c}+0.0023\\\ -0.0022\end{subarray}$ | [94]
$gg\to H$, $H\to W^{+}W^{-}$ ($2j$) | $137.0$ | $0.0151\begin{subarray}{c}+0.0051\\\ -0.0046\end{subarray}$ | $0.0100\begin{subarray}{c}+0.0020\\\ -0.0011\end{subarray}$ | [94]
$gg\to H$, $H\to W^{+}W^{-}$ ($3j$) | $137.0$ | $0.0050\begin{subarray}{c}+0.0045\\\ -0.0042\end{subarray}$ | $0.0033\begin{subarray}{c}+0.0002\\\ -0.0004\end{subarray}$ | [94]
$gg\to H$, $H\to W^{+}W^{-}$ ($4j$) | $137.0$ | $0.0064\begin{subarray}{c}+0.0039\\\ -0.0034\end{subarray}$ | $0.0018\begin{subarray}{c}+0.0001\\\ -0.0002\end{subarray}$ | [94]
VBF, $H\to ZZ$ | $137.1$ | $0.279\begin{subarray}{c}+0.211\\\ -0.162\end{subarray}$ | $0.450\pm 0.010$ | [95]
$gg/b\bar{b}\to H$, $H\to ZZ$ | $137.1$ | $5.328\pm 0.611$ | $5.550\begin{subarray}{c}+0.600\\\ -0.650\end{subarray}$ | [95]
$VH$, $H\to ZZ$ | $137.1$ | $0.305\begin{subarray}{c}+0.243\\\ -0.194\end{subarray}$ | $0.270\pm 0.010$ | [95]
$t\bar{t}H,tH$, $H\to ZZ$ † | $137.1$ | $0.0078\pm 0.0552$ | $0.060\begin{subarray}{c}+0.011\\\ -0.012\end{subarray}$ | [95]
$gg\to H$, $H\to\gamma\gamma$ ($0j$) | $77.4$ | $0.072\pm 0.0122$ | $0.0610\begin{subarray}{c}+0.0037\\\ -0.0031\end{subarray}$ | [96]
$gg\to H$, $H\to\gamma\gamma$ ($1j,~{}p_{T,H}$ high) | $77.4$ | $0.0029\begin{subarray}{c}+0.0017\\\ -0.0012\end{subarray}$ | $0.0017\pm 0.0002$ | [96]
$gg\to H$, $H\to\gamma\gamma$ ($1j,~{}p_{T,H}$ low) | $77.4$ | $0.021\begin{subarray}{c}+0.0090\\\ -0.0075\end{subarray}$ | $0.015\pm 0.0015$ | [96]
$gg\to H$, $H\to\gamma\gamma$ ($1j,~{}p_{T,H}$ med.) | $77.4$ | $0.0076\pm 0.0040$ | $0.010\pm 0.001$ | [96]
$gg\to H$, $H\to\gamma\gamma$ ($2j$) | $77.4$ | $0.0084\begin{subarray}{c}+0.0066\\\ -0.0055\end{subarray}$ | $0.011\pm 0.002$ | [96]
$gg\to H$, $H\to\gamma\gamma$ (BSM) | $77.4$ | $0.0029\pm 0.00104$ | $0.0013\pm 0.0003$ | [96]
VBF, $H\to\gamma\gamma$ | $77.4$ | $0.0091\begin{subarray}{c}+0.0044\\\ -0.0033\end{subarray}$ | $0.0011\pm 0.002$ | [96]
$t\bar{t}H$, $H\to\gamma\gamma$ † | $137.0$ | $0.00155\begin{subarray}{c}+0.00034\\\ -0.00032\end{subarray}$ | $0.00113\begin{subarray}{c}+0.00008\\\ -0.00011\end{subarray}$ | [25]
$V(\text{had})H$, $H\to\tau^{+}\tau^{-}$ | $77.4$ | $-0.0433\begin{subarray}{c}+0.057\\\ -0.054\end{subarray}$ | $0.037\pm 0.001$ | [97]
VBF, $H\to\tau^{+}\tau^{-}$ | $77.4$ | $0.114\begin{subarray}{c}+0.034\\\ -0.033\end{subarray}$ | $0.114\pm 0.009$ | [97]
$gg\to H$, $H\to\tau^{+}\tau^{-}$ ($0j$) | $77.4$ | $-0.680\begin{subarray}{c}+1.292\\\ -1.275\end{subarray}$ | $1.70\pm 0.10$ | [97]
$gg\to H$, $H\to\tau^{+}\tau^{-}$ ($1j,~{}p_{T,H}$ high) | $77.4$ | $0.108\begin{subarray}{c}+0.071\\\ -0.061\end{subarray}$ | $0.060\pm 0.010$ | [97]
$gg\to H$, $H\to\tau^{+}\tau^{-}$ ($1j,~{}p_{T,H}$ low) | $77.4$ | $-0.139\begin{subarray}{c}+0.562\\\ -0.570\end{subarray}$ | $0.410\pm 0.060$ | [97]
$gg\to H$, $H\to\tau^{+}\tau^{-}$ ($1j,~{}p_{T,H}$ med.) | $77.4$ | $0.353\begin{subarray}{c}+0.437\\\ -0.420\end{subarray}$ | $0.280\pm 0.040$ | [97]
$gg\to H$, $H\to\tau^{+}\tau^{-}$ ($2j$) | $77.4$ | $0.0987\begin{subarray}{c}+0.1911\\\ -0.1806\end{subarray}$ | $0.210\pm 0.050$ | [97]
$gg\to H$, $H\to\tau^{+}\tau^{-}$ ($1j,~{}p_{T}^{j_{1}}>200$ GeV) | $77.4$ | $0.0199\begin{subarray}{c}+0.0145\\\ -0.0148\end{subarray}$ | $0.0141\pm 0.0004$ | [97]
$gg\to H$, $H\to\tau^{+}\tau^{-}$ (Rest) | $77.4$ | $-0.195\begin{subarray}{c}+0.506\\\ -0.491\end{subarray}$ | $0.184\pm 0.005$ | [97]
Table 5: CMS Higgs rate measurements from LHC Run-2 included in the fit via
HiggsSignals. Measurements from dedicated $t\bar{t}H+tH$ analyses are marked
with † and are not included in a specific fit that we performed for comparison
(see Sec. 5.4).

These measurements are implemented in HiggsSignals along with correlation
matrices and detailed information about the composition of the signal in terms
of the various relevant Higgs boson production and decay processes. If this
information is not provided by the experiments, the signal is assumed to be
composed of the relevant Higgs processes with equal acceptances, and only
correlations of the luminosity uncertainty (within one experiment) and
theoretical rate uncertainties are taken into account. For details, see Ref.
[75].

An important point for our study arises, however, in the implementation of
measurements targeting Higgs production in association with top quarks. While
current experimental studies mainly focus on $t\bar{t}H$ production, which is
the dominant top associated production mode in the SM, these searches are
often also sensitive to $tH$ and $tWH$ production. These processes are, due to
their rather small rate in the SM, either neglected or assumed to be equal to
the SM prediction when performing the $t\bar{t}H$ measurements, and no
detailed information about their potential contribution to the signal is
quoted.141414Currently, the only measurement quoting the $tH$ contribution to
the signal is the $t\bar{t}H,H\to\gamma\gamma$ analysis by CMS, see Tab. 1 of
Ref. [25].

In the current HiggsSignals implementation, we treat $t\bar{t}H$ measurements
without public information about the $tH$ and $tWH$ contribution as follows:
(_i_) the signal acceptances of $t\bar{t}H$ and $tWH$ are assumed to be
identical; (_ii_) the $tH$ contribution is neglected in the $2$-lepton
category of $t\bar{t}H$, $H\to b\bar{b}$ analyses, and in all other
observables the $tH$ signal acceptance is assumed to be identical to the
$t\bar{t}H$ acceptance. The first point (_i_) is certainly a reasonable
approximation as the final states of $tWH$ and $t\bar{t}H$ are experimentally
very similar. In contrast, $tH$ production can only lead to one lepton (at
parton level), which motivates our choice in the second point (_ii_). However,
the approximation of equal acceptances of $tH$ and $t\bar{t}H$ in the
remaining measurements is clearly a simplification. Given the multivariate
nature of most of the recent data analyses, it is difficult to improve upon
this.

Figure 21: Fit results in the ($c_{t}$, $\tilde{c}_{t}$) parameter plane of
the 5D parametrization for two variations of the assumed signal acceptance of
$tH$ production in all included Higgs boson signal rate measurements. In the
_left panel_ we completely neglected the $tH$ production, in the _right panel_
we doubled the $tH$ signal acceptance. The corresponding result for the
default $tH$ signal acceptances is shown in Fig. 8 (_bottom right panel_).

In order to quantify the impact of this assumption on our fit and the
associated uncertainty, we repeat the fit in the 5D parametrization –
($c_{t}$, $\tilde{c}_{t}$, $c_{V}$, $\kappa_{g}$, $\kappa_{\gamma}$) free –
for two variations of the $tH$ signal acceptance in _all_ implemented
measurements. In the first variation, we set the $tH$ acceptance to zero, i.e.
we neglect entirely the contribution from $tH$ production, and in the second
variation we double the $tH$ acceptance. One should keep in mind, though, that
this variation is clearly very drastic, as in reality the $tH$ acceptance will
certainly not be under- or over-estimated in the same way in all observables.
The results in the ($c_{t}$, $\tilde{c}_{t}$) parameter plane for these two
variations are shown in Fig. 21. These results should be compared with Fig. 8
(bottom, right). The impact of the variation is largest in the parameter
region with an enhanced $tH$ cross section, i.e. the negative $c_{t}$ range
and/or regions with large $\tilde{c}_{t}$.

##  Appendix C Additional information about the best-fit points

The $F$-test is a possibility to quantify how much a model B with more free
parameters improves over the description of the data compared to model A with
less free parameters. If B contains A, as in extensions of the SM, B will
always provide a better or equal fit to the data than A. The $F$-test weighs
the higher complexity of B over A against the improvement in the fit. Here,
model A is a more restricted model with number of parameters,
$n_{\text{par,{A}}}$, smaller than the number of parameters of model B,
$n_{\text{par,{B}}}$. Model A is furthermore nested within model B, i.e., it
can be obtained for a specific parameter choice of model B. The test statistic
$f$ is then calculated as

$\displaystyle
f=\frac{\chi^{2}_{\textbf{A}}}{\nu_{\textbf{A}}}\cdot\left(\frac{\chi^{2}_{\textbf{B}}}{\nu_{\textbf{B}}}\right)^{-1},$
(32)

with the number of degrees of freedom
$\nu_{\textbf{A,B}}=n_{\text{obs}}-n_{\text{par,{A,B}}}$, and $n_{\text{obs}}$
the number of measurements. The $\chi^{2}$ values refer to the minimal
$\chi^{2}$ value found in the parameter space (i.e. the best-fit value). In
our case, we consider the SM as the restricted model A. As the SM is contained
within all models we consider, the minimal $\chi^{2}$ in the models is smaller
(or equal) to the $\chi^{2}$ in the SM.

The cumulative probability $F$ quantifies the significance of the $\chi^{2}$
improvement found in the more general model, while accounting for the larger
number of parameters. It is found by integrating the test statistic $f$ from
zero to the $f$-value determined by the data via the fit. The null hypothesis,
which is that model B does not provide a significantly better fit to the data
than model A (i.e. the SM), can be rejected, for instance, at the $68\%$
($95\%$) confidence level if $F>0.68~{}(0.95)$. In the last column of Tab. 6
we give the calculated cumulative probability of this $F$-test for all
considered model fits. We see that none of the considered models is favored
over the SM even at the $68\%$ C.L. In fact, as all considered models feature
$F<0.50$, for the current data the relative simplicity of the SM can be
considered as preferable over the minimally better fit of the SM extensions
fitted here.

Fit observables | Best-fit parameter point | $\chi_{\text{BF}}^{2}/n_{\text{obs}}$ |  $F$-test
---|---|---|---
$c_{t}$ | $\tilde{c}_{t}$ | $c_{V}$ | $\kappa_{g}$ | $\kappa_{\gamma}$ | $\kappa_{ggZH}$ | (w.r.t. SM)
all | $0.991$ | $-0.009$ | – | – | – | – | $84.32/106$ | $46\%$
all | $0.996$ | $0.004$ | $0.997$ | – | – | – | $84.30/106$ | $45\%$
all | $0.997$ | $-0.015$ | $0.989$ | – | $1.001$ | – | $84.28/106$ | $43\%$
all | $1.032$ | $-0.048$ | $0.991$ | $0.993$ | $0.994$ | – | $84.08/106$ | $41\%$
all, no shape mod. | $1.038$ | $0.065$ | $0.999$ | $0.989$ | $1.001$ | – | $84.06/106$ | $41\%$
no $t\bar{t}H$ obs., | $1.285$ | $0.667$ | $0.989$ | $0.993$ | $0.979$ | – | $61.55/77$ | $46\%$
no shape mod.
all, no shape mod. | $1.032$ | $0.075$ | $0.996$ | $0.991$ | $1.004$ | $0.948$ | $84.01/106$ | $39\%$
no $t\bar{t}H$ obs., | $0.323$ | $-1.629$ | $0.998$ | $0.990$ | $0.978$ | $0.988$ | $61.49/77$ | $43\%$
no shape mod.
Table 6: Results for the best-fit (BF) points for all performed fits: The
first column indicates the experimental input for the fit, the middle columns
give the parameter values for the free fit parameters (– signifies
fixed/derived parameters), and the two last columns give the total $\chi^{2}$
over the number of measurements, $n_{\text{obs}}$, and the $F$-test value (see
text for definition), using the SM as the reference model. The SM $\chi^{2}$
values are $84.45$ ($64.10$) for the full observable set (observable set
without dedicated $t\bar{t}H$ analyses).

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