In the SM likeness test of the current \texttt{HiggsBounds} version we treat production and decay modes of the Higgs search channels separately. For this, we defined
\begin{align}
\bar s = \frac{1}{N_s}\sum_{i=1}^{N_s} s_i, \quad\quad&\quad\quad\bar b = \frac{1}{N_b}\sum_{k=1}^{N_b} b_k,\\
where we have chosen $\epsilon=2\%$. The normalized signal rate (or signal strength modifier) is then approximated by $\mu=\bar s \bar b$ and then tested against the experimental results.
Although the first and second term in the parenthesis of Eq.~\eqref{Eq:oldDelta} may compensate each other (both can be positive or negative), the third term will always contribute irrespective of the signs of $\delta s_i$ and $\delta b_k$.
The short-comings of this method are (see also Oliver's notes):
\begin{enumerate}
\item (\textit{no weighting}) The criteria does not take into account the relative contribution of a search channel to the total expected Higgs boson event rate. Thus, even for search channels with tiny contributions to the signal rate, the parameter point may fail if its contribution differs by more than $\epsilon$.
\item (\textit{testing more channels than needed}) Since we specify production and decay modes seperately (and not their combinations) the model likeness criteria considers all possible combinations of production and decay modes. Thus, we include channels in the model likeness test which may not be included in the Higgs search. This has two disadvantages: (\textit{i}) The test may fail due to a large deviation in the irrelevant channel, and (\textit{ii}) the signal strength modifier is influenced by the irrelevant channel\footnote{In the current implementation, (\textit{ii}) is not a problem because if the signal strength modifier is changed notably by the irrelevant channel the test will fail. However, with the inclusion of weights the picture may change (\textit{see below}).}.
\end{enumerate}
\subsection{The new implementation}
In the new SM likeness test proposed here we use only the channels composed of one production and one decay process. We define
\begin{align}
\bar c = \frac{1}{N_c}\sum_{j=1}^{N_c} c_i, \label{Eq:newsignalstrength}\\
In this way of implementation, production and decay mode of one particular channel can fully compensate each other. In contrast to the old method, this allows us to consider only the relevant search channels, such that we do not have to include all possible combinations of production and decay modes (\textit{solving the 2nd short-coming}). $\bar c$ is used as the signal strength modifier.
\subsection{Introduction of a weighting factor}
Usually, only a few production and decay modes are relevant for the Higgs search and for certain Higgs mass regions. However, in many analyses all production modes are included in order to maximize the significance of SM Higgs searches. The SM likeness criteria presented in Eq.~\eqref{Eq:oldDelta} and \eqref{Eq:newDelta} would then put very strong constraints on the applicability of search, i.e. the relative contributions of \textit{every} search channel - irrespective of how much it actually contributes to the signal rate - has to be very close to the SM expectation (\textit{see also 1st short-coming}).
We now want to allow for a slightly larger deviation of the subdominant channels. For this, we introduce a weighting factor $\omega$ in Eq.~\eqref{Eq:oldDelta} and \eqref{Eq:newDelta}, such that
Furthermore, in the new implementation the signal strength modifier is now a \textit{weighted} average of the channel ratios,
\begin{align}
\bar c = \sum_{j=1}^{N_c}\omega_j c_j. \label{Eq:muweighted}
\end{align}
Note that this value of $\bar c$ is also used in the definition of $\delta c_j$, Eq.~\eqref{Eq:deltacj}.
\begin{figure}
\centering
\subfigure[LHC7 production weights.]{
\scalebox{0.72}{
\input{lhc7_weights}
}}
\hspace{-2cm}
\subfigure[Tevatron production weights.]{
\scalebox{0.72}{
\input{tev_weights}
}}
\caption{Channel weights for different production modes (assuming everywhere the same decay mode which thus factors out) for LHC7 and Tevatron.}
\label{Fig:weights}
\end{figure}
As an illustration, we show in Fig.~\ref{Fig:weights} the channel weights for LHC7 and Tevatron for a Higgs search which includes all production modes but only one decay mode. In this case, the branching ratio $\mathcal{B}(H\to F)$ in Eq.~\eqref{Eq:weightik} and \eqref{Eq:weightj} factors out and the weights correspond to the relative SM contributions of the production modes. At both LHC7 and Tevatron, the single Higgs production mode gains the largest weight.
\subsection{Performance of the different implementations}
We now test the weighting procedure with a recently published SM combined Higgs analysis from CMS (CMS-HIG-12-008). The search channels considered here are all possible combinations of the five production processes (single $H$, VBF, $HZ$, $HW$, $Ht\bar t$) and the five decay modes $H\to WW,~ZZ,~\gamma\gamma,~\tau\tau,~b\bar b$. We consider one SM-like Higgs and use the effective coupling input of \texttt{HiggsBounds}. All squared effective couplings are set to $1$ unless stated differently. All results are obtained with the new implementation (using the \textit{channels} directly) either with or without the weighting procedure.
In Fig.~\ref{Fig:mh_g2hgg} we show the applicability of the search for the relevant Higgs mass range under variation of either the squared normalized effective Higgs coupling to gluons, $g^2_{hgg}$, or to gauge bosons, $g^2_{hWW} = g^2_{hZZ}$. We show the results using the new implementation without weights in Fig.~\ref{Fig:mh_g2hgg}(a,c) and with weights in Fig~\ref{Fig:mh_g2hgg}(b,d).% Without weights, the results using the old implementation are the same.
Without weights, the maximal allowed variation of $g^2_{hgg}$ is $2\%$ for $m_h \le305\gev$ and $\lesssim4\%$ else. The reason for this jump at $m_h =305\gev$ is the following: The SM cross section functions for $WH$, $ZH$ and $Ht\bar t$ are only fitted for $m_h \le305$ and set to zero for larger Higgs masses. In the latter case, they are not considered in the SM likeness test, which reduces the number of search channels, $N_c$. This however affects the signal strength modifier $\bar c$, cf. Eq.~\eqref{Eq:newsignalstrength}, in such a way, that every remaining channel gains more weight in the averaged sum. (The weight is $1/N_c$ and thus turns from $20\%$ to $50\%$ for the remaining channels). Since the modification of the coupling $g^2_{hgg}$ tears $\bar c$ into the same direction, a larger deviation from $g^2_{hgg}=1$ is allowed if the corresponding $gg\to h$ production channel has more impact on $\bar c$. Note also, that the allowed modified range of $g^2_{hgg}$ is slightly asymmetric, cf. Fig.~\ref{Fig:mh_g2hgg}.
%(This behavior is expected from the limiting case, where only one channel is dominant and remains in Eq.~\eqref{Eq:newsignalstrength}. Then, the search should always be applied.)
If we introduce weights to the SM likeness criterion, cf. Fig~\ref{Fig:mh_g2hgg}(b), larger deviations from $g^2_{hgg} =1$ are allowed. Comparing with Fig.~\ref{Fig:weights}(a), the largest allowed deviation appears where the weight is the largest. Again, this is due to the fact that the signal strength modifier is influenced by the varied coupling. Recall that we introduced the weight also in the calculation of the signal strength modifier, Eq.~\eqref{Eq:muweighted}. \ttext{Do we expect this shape of the ``applicablility curve''? What happens at $m_H \approx(130-140)\gev$?}
The applicability of the SM combined Higgs search under variations of the $g^2_{hWW} = g^2_{hZZ}$ coupling is shown in Fig.~\ref{Fig:mh_g2hgg}(c,d). \ttext{These couplings influence approximately in the same way the VBF, HZ and HW channel. Thus, the influence of these modified couplings in the calculation of the signal strength modifier has a weight of $3/5$. Beyond $m_H =305\gev$, only the VBF channel remains with a weight of $1/2$. This jump ($60\%$ to 50\%) is not as drastic as the one observed above, thus we do not observe a change in Fig.~\ref{Fig:mh_g2hgg}(c). (\textit{Tim: I corrected this bit.})} Including weights changes the applicability in a similar way as seen in Fig.~\ref{Fig:mh_g2hgg}(b).
We also evaluate the applicability of the Higgs search under the variation of two effective couplings at the same time, holding the Higgs mass fixed at $m_H=250\gev$. This is shown in Fig.~\ref{Fig:SMcombined_variation_couplings} where we investigate the performance by modifying $g^2_{hgg}$, $g^2_{hWW} = g^2_{hZZ}$ and $g^2_{htt}$. In Fig.~\ref{Fig:SMcombined_variation_couplings}(a) we show the allowed range for the modified couplings $g^2_{hgg}$ and $g^2_{hWW} = g^2_{hZZ}$ for the implementation without weights. The maximally allowed modification is around $2\%$. Since all channels are treated equally, the same range is expected for the other effective squared couplings that are responsible for the Higgs production.
Fig.~\ref{Fig:SMcombined_variation_couplings}(b,c,d) show the results with the inclusion of weights. In (b) we vary the two couplings with the highest weight, i.e. $g^2_{hgg}$ and $g^2_{hWW} = g^2_{hZZ}$. The allowed parameter region has a cone shape opening towards large effective couplings (and continuing forever). Since the signal strength modifier is dominantly influenced by these couplings, it also increases when both couplings are increased at the same time. Therefore, larger coupling deviations are allowed for larger couplings.
If the signal strength wanders to far from 1 we might have expected other channels based on the unmodified couplings like $g^2_{htt}$ to fail the likeness test at some point. However, this is sometimes not the case because the weight of these channels can be simply too small. Here, the weight of the $Ht\bar t$ channels is around $\omega\approx2.5\cdot10^{-3}$ at $m_H=250\gev$ and in general we have $\delta c /\bar c \le1$ (if $\delta c < \bar c$). Since both quantities multiplied have to be larger than $2\%$ to fail the test, this does not happen here.
In contrast, if the subdominant channel is modified by increasing its effective coupling, we may have $\delta c /\bar c \ge1$, also because its impact on the signal strength modifier $\bar c$ is rather negligible (due to the weighted sum, Eq.~\eqref{Eq:muweighted}). Then, these channels can lead to a failure of the likeness test even if their weight is less than $0.02$. This is illustrated in Fig.~\ref{Fig:SMcombined_variation_couplings}. (Note that \texttt{obsratio} (and thus $\bar c$) does not vary notably with $g_{Htt}^2$.)
We perform the same test for the SM $H\to\gamma\gamma$ ATLAS search (arXiv:1202.1414). In this search, all five production modes are included but only the $H\to\gamma\gamma$ decay mode. It is illustrated in Fig~\ref{Fig:1414mh_g2} and Fig.~\ref{Fig:1414g2hgg_h2hWW}. The behavior is essentially the same as above.
In order to better understand which channel is responsible for a test failure in each case of Fig~\ref{Fig:1414mh_g2}, we set the Higgs mass to $m_H=125\gev$ and study the individual channel ratios in Fig.~\ref{Fig:1414_channels}. Remember that the test fails if $\omega_j (c_j-\bar c)/\bar c \ge0.02$, where $\omega_j =1$ in the unweighted case and otherwise given by Eq.~\eqref{Eq:weightj}.
Under variation of $g^2_{Hgg}$ in the unweighted case, Fig.~\ref{Fig:1414mh_g2}(a), the failure is caused by the single Higgs channel. In contrast to $c(pp\to H)$ the signal strength modifier $\bar c$ does not change drastically with $g^2_{Hgg}$ and thus, at some point, the difference between these two becomes too large.
Including the weights, Fig.~\ref{Fig:1414mh_g2}(b), however changes the picture: The signal strength modifier $\bar c$ rather follows $c(pp\to H)$ due to its large weight, cf. Eq.~\ref{Eq:muweighted}. Since the other channel ratios are not influenced by $g^2_{Hgg}$, they are equal to 1. The dominant single Higgs channel [subdominant VBF channel] has a weight of $\omega_{pp\to H} \approx0.877$ [$\omega_\mathrm{VBF} \approx0.068$]. At the lower edge of the applicable region, $g^2_{Hgg}=0.835$, we have
and thus, the single H channel is still responsible for the test failure. The same happens at the upper edge at $g^2_{Hgg}=1.225$. However, if the weights were even more unbalanced, i.e. the single Higgs channel even more dominant, $\bar c$ would be closer to $c(pp\to H)$ and the test failure would be caused by the subdominant VBF channel.
In Fig.~\ref{Fig:1414mh_g2}(c) and ~\ref{Fig:1414mh_g2}(d) we show the channel ratios under variation of the couplings $g^2_{HWW}=g^2_{HZZ}$ for the unweighted and the weighted case, respectively. These couplings influence the VBF, HW and HZ channel. The VBF and the HW lines lie on top of each other. Since 3 of 5 channels are affected by the varied effective couplings, the signal strength modifier $\bar c$ is moderately influenced in the unweighted case, Fig~\ref{Fig:1414mh_g2}(c). Here, the failure of the SM likeness test is caused by the deviation of the single Higgs and $Ht\bar t$ channel, which stay constant at 1.
In the weighted case, the signal strength modifier $\bar c$ stays fairly close to 1 due to the small weights of the modified channels. Again, we take a closer look at the numbers at the lower edge, $g^2_{HWW}=g^2_{HZZ}=0.81$:
Thus, also here the test failure is due to the single Higgs channel.
\begin{figure}
\centering
\subfigure[not weighted.]{
\scalebox{0.75}{
\input{SMtest_mh_g2hjgg_CnoW}
}}
\hspace{-3cm}
\subfigure[weighted.]{
\scalebox{0.75}{
\input{SMtest_mh_g2hjgg_CW_ver2}
}}
\subfigure[not weighted.]{
\scalebox{0.75}{
\input{SMtest_mh_g2hjWW_CnoW}
}}
\hspace{-3cm}
\subfigure[weighted.]{
\scalebox{0.75}{
\input{SMtest_mh_g2hjWW_CW_ver2}
}}
\caption{Applicability of the combined Higgs search from CMS-HIG-12-008 under variation of effective Higgs couplings for different implementations of the SM likeness test. In (\textit{a,b}) [(\textit{c,d})], we vary the normalized squared effective Higgs-gluon-gluon, $g^2_{hgg}$, [Higgs-vector boson, $g^2_{hWW} = g^2_{hZZ}$] coupling over the searches Higgs mass range. The implementation in (\textit{a,c}) does not employ weights while they are used in (\textit{b,d}). The color corresponds to \texttt{obsratio}.}
\label{Fig:mh_g2hgg}
\end{figure}
\begin{figure}
\centering
\subfigure[not weighted.]{
\scalebox{0.75}{
\input{SMtest_g2hjWW_g2hjgg_CnoW}
}}
\hspace{-3cm}
\subfigure[weighted.]{
\scalebox{0.75}{
\input{SMtest_g2hjWW_g2hjgg_CW_ver2}
}}
\subfigure[weighted.]{
\scalebox{0.75}{
\input{SMtest_g2hjgg_g2hjtt_CW_ver2}
}}
\hspace{-3cm}
\subfigure[weighted.]{
\scalebox{0.75}{
\input{SMtest_g2hjWW_g2hjtt_CW_ver2}
}}
\caption{Applicability of the combined Higgs search from CMS-HIG-12-008 under variation of the squared effective Higgs couplings $g^2_{hgg}$, $g^2_{hWW} = g^2_{hZZ}$ and $g^2_{htt}$: (\textit{a}) without weights, (\textit{b, c, d}) with weights. We set the Higgs mass to $m_H=250\gev$. The color corresponds to \texttt{obsratio}.}
\label{Fig:SMcombined_variation_couplings}
\end{figure}
\begin{figure}
\centering
\subfigure[not weighted.]{
\scalebox{0.75}{
\input{SMtest_1414_mh_g2hjgg_CnoW}
}}
\hspace{-3cm}
\subfigure[weighted.]{
\scalebox{0.75}{
\input{SMtest_1414_mh_g2hjgg_CW_ver2}
}}
\subfigure[not weighted.]{
\scalebox{0.75}{
\input{SMtest_1414_mh_g2hjWW_CnoW}
}}
\hspace{-3cm}
\subfigure[weighted.]{
\scalebox{0.75}{
\input{SMtest_1414_mh_g2hjWW_CW_ver2}
}}
\caption{Applicability of the ATLAS Higgs search arXiv:1202.1414 ($H\to\gamma\gamma$) under variation of effective Higgs couplings for different implementations of the SM likeness test. In (\textit{a,b}) [(\textit{c,d})], we vary the normalized squared effective Higgs-gluon-gluon, $g^2_{hgg}$, [Higgs-vector boson, $g^2_{hWW} = g^2_{hZZ}$] coupling over the relevant Higgs mass range. The implementation in (\textit{a,c}) does not employ weights while they are used in (\textit{b,d}). The color corresponds to \texttt{obsratio}.}
\label{Fig:1414mh_g2}
\end{figure}
\begin{figure}
\centering
\subfigure[not weighted.]{
\scalebox{0.75}{
\input{1414_SMtest_g2hjWW_g2hjgg_CnoW}
}}
\hspace{-3cm}
\subfigure[weighted.]{
\scalebox{0.75}{
\input{1414_SMtest_g2hjWW_g2hjgg_CW_ver2}
}}
\subfigure[weighted.]{
\scalebox{0.75}{
\input{1414_SMtest_g2hjgg_g2hjtt_CW_ver2}
}}
\hspace{-3cm}
\subfigure[weighted.]{
\scalebox{0.75}{
\input{1414_SMtest_g2hjWW_g2hjtt_CW_ver2}
}}
\caption{Applicability of the ATLAS Higgs search arXiv:1202.1414 under variation of the squared effective Higgs couplings $g^2_{hgg}$, $g^2_{hWW} = g^2_{hZZ}$ and $g^2_{htt}$: (\textit{a}) without weights, (\textit{b, c, d}) with weights. We set the Higgs mass to $m_H=125\gev$. The color corresponds to \texttt{obsratio}.}
\label{Fig:1414g2hgg_h2hWW}
\end{figure}
\begin{figure}
\centering
\subfigure[not weighted.]{
\scalebox{0.75}{
\input{1414_mh125_g2hgg}
}}
\hspace{-3cm}
\subfigure[weighted.]{
\scalebox{0.75}{
\input{1414_mh125_g2hgg_weights}
}}
\subfigure[not weighted.]{
\scalebox{0.75}{
\input{1414_mh125_g2hWW_ZZ}
}}
\hspace{-3cm}
\subfigure[weighted.]{
\scalebox{0.75}{
\input{1414_mh125_g2hWW_ZZ_weights}
}}
\caption{Channel ratio dependence on the modified effective squared couplings for the implementation with and without weights for the channels of the $H\to\gamma\gamma$ search \texttt{arXiv:1202.1414}. We set the Higgs mass to $m_H=125\gev$. In the gray regions the SM likeness test fails. Other channel ratios not plotted are equal to 1.}