\texttt{HiggsSignals} is a \texttt{Fortran90} computer code that
allows to test the compatibility of Higgs sector predictions against
Higgs rates and masses measured at the LHC or the Tevatron. Arbitrary
models with any number of Higgs bosons can be \htbd{used with}\htb{investigated using} a
model-independent input scheme based on \texttt{HiggsBounds}. The test
is based on the calculation of a $\chi^2$ derived from the predictions
and the measured Higgs rates and masses, with the ability of \htr{fully
taking into account} systematics and correlations for the signal rate
predictions, luminosity and Higgs mass predictions. It features two
complementary methods for the test. First, the peak-centered method,
in which each observable is defined by a Higgs signal rate measured at
a specific hypothetical Higgs mass, corresponding to a tentative Higgs
signal. Second, the mass-centered method, where the test is evaluated by
comparing the signal rate measurement to the theory prediction at the
Higgs mass predicted by the model. The program allows for the
simultaneous use of both methods, which is useful in testing models
with multiple Higgs bosons. The code automatically combines the
signal rates of multiple Higgs bosons if their signals cannot be
resolved by the experimental analysis. In that case the signal rates
of the corresponding Higgs bosons are added incoherently.
\htbd{We validate \texttt{HiggsSignals} with a number of comparisons of Higgs property
determinations, where the results obtained using \texttt{HiggsSignals}
are confronted with the official results from ATLAS and CMS, and find
very good consistency.}
\htb{We compare results obtained with \HS\ with official ATLAS and CMS results in various scenarios of Higgs property determinations. We find very good agreement, supporting the validity of the \HS\ methodology and its underlying assumptions and approximations.}
A few examples of \texttt{HiggsSignals} applications \htr{are provided,
going beyond the scenarios that have already been investigated by the
LHC collaborations.} It is straightforward to use \texttt{HiggsSignals} and
\texttt{HiggsBounds} in parallel. For models with more than one Higgs boson this is recommended to exploit the full
constraining power of Higgs search exclusion limits and the
measurements of the signal seen at $m_H\approx125.5$~GeV.
% OLD ABSTRACT:
%
%\HS\
%is a computer code that performs a $\chi^2$ test of the Higgs sector
%predictions in arbitrary models with any number of Higgs
% bosons against the signal rates and masses measured in Higgs searches
%by the Tevatron and LHC experiments. It features two complementary
%methods for the $\chi^2$ evaluation: the \textit{peak-centered $\chi^2$
% method}, in which each observable is defined by a Higgs signal
% rate measured at a specific hypothetical Higgs mass, corresponding to
%a tentative Higgs signal. In the second method, the
%\textit{mass-centered $\chi^2$ method}, the $\chi^2$ is evaluated by
%comparing the signal rate measurement to the theory prediction at the
%Higgs mass predicted by the model. The program allows for the
%simultaneous use of both methods, which is useful in testing models with
%multiple Higgs bosons. In the statistical procedure, systematic
%uncertainties of the signal rate prediction, luminosity and Higgs mass predictions are fully taken into
% account, including their correlations.
%The code automatically combines the signal rates of Higgs bosons if their signals cannot be resolved by the experimental analysis. In that case
%the signal rates of the corresponding Higgs bosons are added incoherently.
% The model input framework of \HS~is based on the the \HB~input format. It is therefore straightforward to use both
%codes in parallel, \htod{something} which is highly recommended to exploit
% the full constraining power of Higgs search exclusion limits and the measurements
% of the signal seen at $m_H \simeq \mass \gev$. \htr{TODO: Rewrite to something less technical!}
\end{abstract}
\noindent{\it Keywords\/}: Higgs bosons, Higgs searches, Higgs signal, Tevatron, LHC, Beyond the Standard Model, Supersymmetry, Higgs coupling determination, Global fits
% PACS codes here, in the form: \PACS code \sep code
Searches for a Higgs boson~\cite{Englert:1964et,*Higgs:1964ia,*Higgs:1964pj,*Guralnik:1964eu,*Higgs:1966ev,*Kibble:1967sv} have been one of the driving factors behind
experimental particle physics over many years. Until recently, results
from these searches have always been in the form of exclusion limits,
where different Higgs mass hypotheses are rejected at a certain
confidence level (usually $95\%$) by the non-observation of any
signal. This has been the case for Standard Model (SM) Higgs searches at
LEP~\cite{Barate:2003sz}, the Tevatron \cite{TEVNPH:2012ab}, and (until
July 2012) also for the LHC experiments
\cite{Aad:2012an,*Chatrchyan:2012tx}. Limits have also been presented on
extended Higgs sectors in theories beyond the SM, where one prominent
example are the combined limits on the Higgs sector of the minimal
supersymmetric standard model (MSSM) from the LEP
experiments~\cite{Schael:2006cr,Abbiendi:2013hk}. To test the
predictions of models with arbitrary
Higgs sectors consistently against all the available experimental data
on Higgs exclusion \htb{limits}, we \htr{have presented}%in spring 2009
the public tool \HB\ \cite{Bechtle:2011sb,*Bechtle:2008jh}, which
recently appeared in version \vers{4}{0}{0}\cite{HB4,Bechtle:2013gu}.
With the recent discovery of a new state---compatible with a SM Higgs
boson---by \htb{the LHC experiments}\ATLAS\ \cite{ATLASDiscovery} and \CMS\ \cite{CMSDiscovery},
models with extended Higgs sectors are facing new constraints. It is no
longer sufficient to test for non-exclusion, but the model predictions
must be tested against the measured mass and
rates of the observed state, which contains more information. To test
the predictions of an arbitrary Higgs sector against this Higgs
signal\footnote{Here, and in the following, the phrase \emph{Higgs signal} refers
to any hint or observation of a signal in the data of the Tevatron/LHC Higgs searches,
regardless of whether in reality this is due to the presence of a
Higgs boson. In fact, the user can directly define the Higgs signals,
\htr{i.e. the signal strength at a given mass peak or as a function of Higgs
masses, } which should be considered as observables in \HS, see Section~\ref{Sect:expdata} for more details.} (and potentially against other signals of
additional Higgs states discovered in the future) is the purpose of a new public computer
program, \HS, which we present here.
\HS\ is a \texttt{Fortran90/2003} code, which evaluates a $\chi^2$ measure to provide a quantitative
answer to the statistical question of how compatible the Higgs
search data (measured signal strengths and masses) is with the model
predictions. This $\chi^2$\htb{value} can be evaluated with two distinct
methods, namely the \textit{peak-centered} and the
\textit{mass-centered}$\chi^2$ method. In the \textit{peak-centered}
$\chi^2$ method, the (neutral) Higgs signal rates and masses predicted
by the model are tested against the various signal rate measurements
published by the experimental collaborations for a fixed
Higgs mass hypothesis. This hypothetical Higgs mass is typically motivated by the
signal ``peak'' observed in the channels with high mass resolution, \ie the
searches for $H\to\gamma\gamma$ and $H\to ZZ^{(*)}\to4\ell$. In this
way, the model is tested \textit{at the mass
position of the observed peak}. In the \textit{mass-centered}$\chi^2$ method on the other hand, \HS~tries
to find for every neutral Higgs boson in the model the corresponding
signal rate measurements, which are performed under the assumption of a
Higgs boson mass equal to the predicted Higgs mass. Thus, the $\chi^2$
is evaluated at the \textit{model-predicted mass position}. For this
method to be applicable, the experimental measurements therefore have to be given for a
certain mass range.
%\begin{itemize}
%\item \htod{How likely is it that one or several observed Higgs signal(s) are due to the Higgs boson(s) of this model?}
%\item \htod{How compatible is the Higgs search data (measured signal strengths) with the model predictions?}
%\end{itemize}
The input from the user is given in the form of Higgs masses, production cross
sections, and decay rates in a format similar to that used in \HB.
The experimental data from Tevatron and
LHC Higgs searches is provided with the program, so there is no need for the user to include these values
manually. However, it is possible for the user to modify or add to the data
at will.
Like \HB, the aim is to always keep \HS~updated with the latest experimental results.
The usefulness of a generic code such as \HS~has become apparent in the
last year, given the intense work by theorists to use the new Higgs
measurements as constraints on the SM and theories for new
The experimental data used in \HS~is collected at hadron colliders, mainly the LHC, but there are also some complementary measurements from the
Tevatron collider. This will remain the case for the foreseeable future,
but the \HS\ methods can be easily extended to include data from, for instance, a
future $e^+e^-$ linear collider. In this section we give a very brief
review of Higgs searches at hadron colliders, focussing the description
on the experimental data that provides the basic input for \HS. For a more complete review see, e.g., Ref.~\cite{Djouadi:2005gi,*Djouadi:2005gj,*Dittmaier:2012nh}.
Most searches for Higgs bosons at the LHC are performed under the
assumption of the SM. This fixes completely the couplings of the Higgs
state to fermions and vector bosons, and both the cross sections and
branching ratios are fully specified as a function of the Higgs boson
mass, $\mH$. Most up-to-date predictions, including an extensive
list of references, can be found
in~\cite{Dittmaier:2011ti,Dittmaier:2012vm}.
This allows experiments to measure one-parameter scalings
of the total SM rate \hto{of a certain (ensemble of) signal channel(s)}, so-called \emph{signal strength modifiers},
corresponding to the best fit to the data. These measurements are the
basic experimental input used by \HS. Two examples of this (from \ATLAS) are shown in
Fig.~\ref{fig:muplots}. The left plot (taken from
\cite{ATLAS-CONF-2013-013}) shows the measured value of the signal
strength modifier, which we denote by $\muhat$, in the inclusive \htb{$pp\to~H\to ZZ^{(*)}\to4\ell$}
process as a function of $\mH$ (black line). The cyan band gives a
$\pm1\,\sigma$ uncertainty on the measured rate. Since the signal
strength modifier is measured relative to its SM value ($\muhat=1$,
displayed in Fig.~\ref{fig:muplots} by a \htb{dotted} line), this contains also
the theory uncertainties on the SM Higgs cross section and branching
ratios~\hto{\cite{Dittmaier:2011ti,Dittmaier:2012vm,Denner:2011mq}}. As can be seen from Fig.~\ref{fig:muplots}, the measured value
of $\muhat$ is allowed to take on negative values. In the absence of
sizable signal-background interference---as is the case for the
SM---the signal model would not give $\muhat<0$. This must therefore be
understood as statistical downward fluctuations of the data w.r.t. the background expectation (the
average background-only expectation is $\muhat=0$). To keep $\muhat$ as
an unbiased estimator of the true signal strength, it is however
essential that the full range of values is retained. As we shall see in
more detail below, the applicability of \HS\ is limited to the mass
range for which measurements of $\muhat$ are reported. It is therefore
highly desirable that experiments publish this information even for
mass regions where a SM Higgs signal has been excluded.
A second example of \HS\ input is shown in the right plot of
Fig.~\ref{fig:muplots} (from~\cite{ATLAS-CONF-2013-034}). This figure
summarizes the measured signal strength modifiers for \emph{all}
\htb{relevant Higgs decay} channels at an interesting value of the Higgs mass, here
$\mH=125.5\gev$. This particular value is typically
selected to correspond to the maximal significance for a signal seen in
the data. It is important to note that, once a value for
$\mH$ has been selected, this plot shows a compilation of information
for the separate channels that is also available directly from the
mass-dependent plots (as shown in Fig.~\ref{fig:muplots}(a)). Again, the
error bars on the measured $\muhat$ values correspond to $1\sigma$
uncertainties that include both experimental (systematic and
statistical) uncertainties, as well as SM theory uncertainties.
%\begin{figure}
%\subfigure[The best-fit signal strength $\muobs$ for the LHC process
%$(pp)\to H\to\gamma\gamma$ is given as a function of the assumed Higgs
The SM weights contain the relative experimental \emph{efficiencies},
$\epsilon_i$, for the different channels. Unfortunately, these are
rarely quoted in experimental publications.If they are available, these numbers can be used by \HS, which leads to a
more reliable comparison between theory predictions and the
experimental data for these channels. In the case of unknown efficiencies, all channels considered by the analysis are treated equally, \ie we set all $\epsilon_i \equiv1$.
where $N_H$ is the number of (neutral) Higgs bosons of the model. The calculation of the individual contributions from the signal strength modifiers, $\chi^2_\mu$, and the Higgs masses, $\chi^2_{m_i}$, will be discussed below.
The input data used in this method is based on the prejudice that a
Higgs signal has been observed at a particular Higgs mass value, which
does not necessarily have to be the exact same value for all observables. Technically, each observable is defined by a single text file, which contains all relevant information needed by \HS. An experimental dataset\footnote{The most up-to-date experimental data is contained in the folder \texttt{Expt\_tables/latestresults}. A summary of these observables, as included in the \HSv{1.0.0} release, is given in \refse{sect:Examples}, \reffi{Fig:peakobservables}.} is then a collection of observables, whose text files are stored in a certain subdirectory of the \HS\ distribution. The user may add, modify or remove the experimental data for \hto{her} own purpose, see Section~\ref{Sect:expdata} for more details.
Currently, an obvious and prominent application of the peak-centered $\chi^2$ method would be the test of a single Higgs boson against the rate and mass measurements performed at around $125$--$126$~GeV in all channels reported by \htbd{ATLAS and CMS}\htb{the experimental collaborations at the LHC and Tevatron}. \htb{This scenario will be discussed in detail in Section~\ref{sect:Examples}.} However, \HS\ is implemented in a way that
is much more general: Firstly, contributions from other Higgs bosons in the model to the Higgs signals will be considered, and if relevant, included in the test automatically. Secondly, the extension of this test to more Higgs signals (in other mass regions) can simply be achieved by the inclusion of the proper experimental data, or for a phenomenological study, the desired pseudo-data.
\subsubsection{Signal strength modifiers}
\label{Sect:chisq_mu}
For $N$ defined signal observables, the total $\chi^2$ contribution is given by
where the observed and predicted signal strength modifiers are contained in the $N$-dimensional vectors $\muobsvec$ and $\muvec$, respectively. $\mathbf{C_\mu}$ is the signal strength covariance matrix.
%
%\htb{(Tim: the following is the old way of writing it. Which way do you prefer?)}
%\htbd{and the signal strength covariance matrix, $\mathbf{C_\mu}$. As introduced above, the measured quantities (associated with the signal) are denoted by a hat, \emph{i.e.}~$\hat\mu_\alpha$ is the best-fit signal strength modifier for signal $\alpha$, and $\mu_\alpha$ is the corresponding model prediction for the particular analysis in which the signal $\alpha$ was observed.}
The signal strength covariance matrix $\covmu$ is constructed in the
following way. The diagonal elements $(\covmu)_{\alpha\alpha}$
(corresponding to signal \htb{observable}$\alpha$) should first of all contain the
intrinsic experimental (statistical and systematic) $1\,\sigma$
uncertainties on the signal strengths squared, denoted by $(\Delta
\hat\mu_\alpha^*)^2$. These will be treated as uncorrelated
uncertainties, since there is no information publicly available on the\htb{ir}
correlations. We define these uncorrelated uncertainties by subtracting
from the total uncertainty $\Delta\hat\mu_\alpha$ (which is given
directly from the $1\,\sigma$ error band in the experimental data,
cf.~Fig.~\ref{fig:muplots}) the luminosity uncertainty as well as
the theory uncertainties on the predicted signal rate (which we
shall include later as correlated uncertainties). \htb{Hereby, we assume that these uncertainties can be treated as Gaussian errors.} This gives
Here, $\Delta\mathcal{L}$ is the relative uncertainty on the luminosity, and $\Delta c_a^\mathrm{SM}$ is the SM channel rate uncertainty (for a total of $k$ channels contributing to the analysis with signal $\alpha$) given by
The SM channel weights, $\omega_a$, have been defined in \refeq{Eq:omega}.
%In the \textit{rare cases} where the experimental collaborations quote the efficiencies for each channel, they are included in the calculation of the weights.
%Recall, that the predicted signal strength modifier is calculated by
Here, $k_\alpha$ and $k_\beta$ are the respective numbers of Higgs
(production~$\times$~decay) channels considered in the experimental
analyses where the signals $\alpha$ and $\beta$ are observed. We use the
index notation $p(a)$ and $d(a)$, to map the channel $a$ onto its
production and decay processes, respectively. In other words, analyses
where the signals share a common production and/or decay mode have
correlated systematic uncertainties. \htb{These channel rate uncertainties are inserted in the covariance matrix according to their relative contributions to the total signal rate \textit{in the model}, i.e. via the channel weight evaluated from the model predictions,
where the $\alpha$-th entry of the predicted mass vector $\mvec_i$ is
given by $m_i$, if the of Higgs boson $h_i$ is assigned to the signal $\alpha$, or $\mobs_\alpha$ otherwise (thus leading to a zero $\chi^2$ contribution from \textit{this} observable and \textit{this} Higgs boson).
As can be seen from Eq.~\eqref{Eq:chisq_mh}, we construct a mass covariance matrix
$\mathbf{C_{m_{i}}}$ for each Higgs boson $h_i$ in the model. The diagonal
elements $(\mathbf{C_{m_{i}}})_{\alpha\alpha}$ contain the experimental
mass resolution squared, $(\dmexp_\alpha)^2$, of the analysis in which
the signal $\alpha$ is observed. The squared theory mass uncertainty,
$(\dmth_i )^2$, enters all matrix elements
$(\mathbf{C_{m_{i}}})_{\alpha\beta}$ (including the diagonal)
where the Higgs boson $h_i$ is assigned to both signal \htb{observables}$\alpha$ and
$\beta$. \htb{Thus, the theoretical mass uncertainty is treated as fully correlated.}
\htb{However, the sign of this correlation depends on the relative position of the predicted Higgs boson mass, $m_i$, with respect to the two (different) observed mass values, $\hat{m}_{\alpha,\beta}$ (where we assume $\hat{m}_\alpha < \hat{m}_\beta$ for the following discussion): If the predicted mass lies outside the two measurements, i.e. $m_i < \hat{m}_\alpha, \hat{m}_\beta$ or $m_i > \hat{m}_\alpha, \hat{m}_\beta$, then the
correlation is assumed to be positive. If it lies in between the two mass measurements, $\hat{m}_\alpha < m_i < \hat{m}_\beta$, the correlation is negative (i.e. we have anti-correlated observables). The necessity of this sign dependence can be illustrated as follows: Let us assume the predicted Higgs mass is varied within its theoretical uncertainty. In the first case, the deviations of $m_i$ from the mass measurements $\hat{m}_{\alpha,\beta}$ both either increase or decrease (depending on the direction of the mass variation). Thus, the mass measurements are positively correlated. However, in the latter case, a variation of $m_i$ towards one mass measurement always corresponds to a larger deviation of $m_i$ from the other mass measurements. Therefore, the theoretical mass uncertainties for these observables have to be anti-correlated.}
\htbd{Finally, the total mass $\chi^2$ contribution is obtained by
summing over the individual contributions of all the Higgs bosons. }
\subsubsection{Assignment of multiple Higgs bosons}
\label{Sect:peakassignment}
\htb{If a model contains an extended (neutral) Higgs sector, it is a priori not clear which neutral Higgs boson(s) of the model give the best explanation of the experimental observations. Moreover, possible superpositions of the signal strengths of the Higgs bosons have to be taken into account. Another (yet hypothetical) complication arises if \emph{more than one} Higgs signal has been discovered in the \emph{same} Higgs search, indicating the discovery of \emph{another} Higgs boson. In this case, care has to be taken that a Higgs boson of the model is only considered as an explanation of one of these signals.
In the peak-centered $\chi^2$ method, these complications are taken into account by the automatic \emph{assignment} of the Higgs bosons in the model to the signal observables.}
\htbd{An essential feature of the peak-centered $\chi^2$ method to go beyond the case of a single observed signal, and to test models with multiple Higgs bosons, is the automatic \emph{assignment} of the Higgs bosons in the model to the observed signals.} In this procedure, \HS~tests whether the combined signal strength of several Higgs bosons might yield a better fit than the assignment of a single Higgs boson to one signal in an analysis. Moreover, based on the predicted and observed Higgs mass values, as well as their uncertainties, the program decides whether a comparison of the predicted and observed signal rates is valid for the considered Higgs boson. A priori, all possible Higgs combinations which can be assigned to the observed signal(s) of an analysis are considered. If more than one signal exists in \textit{one} analysis, it is taken care of that each Higgs boson is assigned to at most one signal to avoid double-counting. A signal to which no Higgs boson is assigned contributes a $\chi^2$ penalty given by Eq.~\eqref{Eq:chimu} with the corresponding model prediction ${\mu_\alpha=0}$. This corresponds to the case where an observed signal cannot be explained by any of the Higgs bosons in the model.
For each Higgs search analysis the best Higgs boson assignment is found in the following way: For every possible assignment $\eta$ of a Higgs boson combination to the signal $\alpha$ observed in the analysis, its corresponding tentative $\chi^2$ contribution, $\chi_{\alpha,\eta}^2$, based on both the signal strength and potentially the Higgs mass measurement, is evaluated. There are two requirements the Higgs combination has to fulfill in order to be considered for the assignment:
\begin{itemize}
\item Higgs bosons which have a mass $m_i$ close enough to the signal mass $\hat{m}_\alpha$, \ie
are required to be assigned to the signal $\alpha$. \htb{Here, $\Lambda$ denotes the \textit{assignment range}, which can be modified by the user, see Section~\ref{Sect:subroutines} (the default setting is $\Lambda=1$).}
\item If the $\chi^2$ contribution from the measured Higgs mass is deactivated for this signal, combinations with a Higgs boson that does not fulfill Eq.~\eqref{Eq:massoverlap} are not taken into account for a possible assignment.
\end{itemize}
In the case where multiple Higgs bosons are assigned to the same signal, the combined signal strength modifier $\mu$ is taken as the sum over their predicted signal strength modifiers (corresponding to incoherently adding their rates). The best Higgs-to-signals assignment $\eta_0$ in an analysis is that which minimizes the lowest overall $\chi^2$ contribution, i.e.
Here, the sum runs over all signals observed within \emph{this particular}
analysis.\footnote{\htbd{Note that the case of having multiple signals
observed in one analysis is hypothetical, since there are currently no
multiple signals seen in any Higgs search by ATLAS or CMS. However,
\HS~is general enough to cope with this potential future situation.}}
In this procedure, \HS~only considers assignments $\eta$ where each
Higgs boson is not assigned to more than one signal within the same
analysis in order to avoid double counting.
\htb{Finally, there is also the possibility to enforce that a collection of peak observables is either
assigned or not assigned in parallel. This can be useful if certain peak observables stem from the same Higgs analysis but correspond to the measurements performed for specific tags or categories. See Section~\ref{Sect:expdata} for a description of these \textit{assignment groups}.}
%A special case which can follow from this procedure is the possibility that no Higgs boson is assigned to a signal (\eg if no Higgs boson in the \htb{model} is sufficiently close in mass). In that case, there is no $\chi^2$ contribution from the Higgs mass measurement, and the $\chi^2$ contribution from the signal strength is evaluated for a predicted %\htb{signal strength} $\mu=0$. In the covariance matrix for the signal strength then enters the systematic rate uncertainties using the SM weights evaluated at the mass position of the signal. (\htb{Tim: This last sentence is more meant as a technical note, maybe don't need to mention it here?})
In the $\muobs$ plot the experimental mass uncertainty is already taken into account in the experimental analysis. However,
%One complication of comparing the model predictions $\mu(\mpred_i)$, directly to the experimental data, $\muhat(\mpred_i)$, is that the experimental mass uncertainty is already folded into the $\muhat$ measurement by the experimental collaboration \htb{(Tim: "folded into" sounds as if it was introduced by hand, although it is simply (in most cases?) a physical effect (distribution shapes etc.).)}.
we also want to take into account a possible theoretical
uncertainty on the predicted Higgs mass, $\dmth_i$. \HS~provides two
different methods to include theoretical Higgs mass uncertainties in the
mass-centered $\chi^2$ evaluation:
\begin{itemize}
\item[\textit{(i)}] (\textit{default setting}) In the first method the
predicted Higgs mass is varied around $\mpred_i$ within its
uncertainties. We denote this varied mass by $m'$ in the
following. For a uniform (box) parametrization of the theoretical mass
\item If either $m_i$ or $m_j$ is known exactly, for instance $\dmth_i =0$,
the mass of the new Higgs cluster is chosen equal to this mass,
$m_k = m_i$, with zero combined theory mass uncertainty,
$\dmth_k =\dmth_i =0.$
\item If both $m_i$ and $m_j$ are known exactly, $\dmth_i =\dmth_j =0$,
the Higgs cluster is assigned an averaged mass $m_k =(m_i + m_j)/2$,
with $\dmth_k =0$.
\end{itemize}
\item[3.] The procedure is repeated from step 1. The entities considered
for further clustering include both the unclustered (initial) Higgs
bosons, as well as the already combined Higgs clusters. The single
Higgs bosons which form part of a cluster are no longer present.
\item[4.] Each single Higgs boson or Higgs cluster $h_k$ that remains
after the clustering according to steps 1-3 enters the mass-centered
$\chi^2$ test. Their predicted signal strength modifiers are formed
from the incoherent sum (again, neglecting interference effects)
of the individual signal strength modifiers for the combined Higgs
bosons,
\begin{equation}
%\mu_k(m_k) = \sum_i^\Ncomb \mu_i(\mpred_i).
\mu_k(m_k) = \sum_i \mu_i(\mpred_i).
\label{Eq:mc_cluster_mu}
\end{equation}
\end{enumerate}
In this way, the predictions that are compared to \textit{one}
implemented analysis are determined. \HS~repeats this procedure for all
implemented experimental analyses. Since the experimental mass
resolution can vary significantly between different analyses, the
resulting clustering in each case may differ.
The two different treatments of the theoretical mass uncertainties, as
discussed above, have to be slightly extended for the case of Higgs
clusters:
\begin{itemize}
\item[\textit{(i)}] If the Higgs boson $h_i$ is contained within a Higgs
cluster $h_k$ for one analysis $a$, the considered mass region for the
variation of $m'$ in \refeq{Eq:chisq_dmth_variation_box} is now the
overlap region $M_i \cap M_k$, with $M_i =\left[m_i -\dmth_i , m_i +
\dmth_i \right]$ in the case of a uniform (box) Higgs mass
pdf.\footnote{If $M_i \cap M_k =\varnothing$, we increase $M_k$ until
there is a (minimal) overlap. This will effectively lead to an
evaluation of the tentative $\chi^2$ at the boundary of $M_i$ which
is closest to the mass $m_k$ of the Higgs cluster.} We denote the
resulting tentative total $\chi^2$ from the variation of the mass of
Higgs boson $h_i$ by $\chi^2_i$. The variation is done for every Higgs
boson contained in the cluster $h_k$. When the cluster $h_k$ is evaluated
against the observed results for analysis $a$, the observed
values $\muobs_a$ and $\dmuobs_a$ are defined at the value of $m'$
where the global $\chi^2$, composed of all $\chi_i^2$ distributions,
is minimal.\footnote{The global $\chi^2$ is defined in the mass region
$(M_i \cap M_k)\cup(M_j \cap M_k)\cup\dots$, when the Higgs
bosons $h_i,~h_j,\dots$ are combined in the cluster $h_k$.}
\item[\textit{(ii)}] In the second approach, the convolution of
the experimental $\muobs$ values with theory uncertainties is
performed separately for each Higgs boson, or Higgs cluster $k$, with
the combined Higgs mass pdf
\begin{equation}
%g_k(m',m) = \frac{1}{N}\sum_i^\Ncomb g_i(m',m).
g_k(m',m) = \frac{1}{N}\sum_i g_i(m',m).
\end{equation}
The normalization factor $N =\int_{M_k}\mathrm{d}m' g_k(m',m)$ to
preserve probability. The sum runs over all Higgs bosons which have been
combined for this cluster.
\end{itemize}
%Note again, that in this procedure only those Higgs bosons are considered whose mass falls in the relevant mass range.
Once all model predictions \htb{and \emph{mass-centered observables}} have been defined, when necessary using
Stockholm clustering as discussed above, the total mass-centered
$\chi^2$ is evaluated with a signal strength vector\footnote{The length
of this vector depends in this case on the Higgs masses and the result
of the clustering. Each analysis may contribute any number of entries
$\alpha$, where $0\leq\alpha\leq N_{\mathrm{Higgs}}$.} and
covariance matrix constructed analogously as in the peak-centered
$\chi^2$ method, \cf\refeq{Eq:chimu}. The uncertainties of production
cross sections, decay rates, and the luminosity are again treated as
fully correlated Gaussian errors. Note that, in this method, there is no
contribution from Higgs mass measurements to the total $\chi^2$, since
the evaluation is done directly against the experimental data at the
predicted Higgs mass values \htb{(within their uncertainties)}.
As a final remark, we would like to point out that the $\muobs$ plots necessary for this method are so far only published for a few selected analyses.\footnote{Currently, the full data ($\muobs$ plot) is published only for the $H\to\gamma\gamma$, $H\to ZZ^{(*)}$ and $H\to WW^{(*)}$ searches.} Thus, there is not (yet) a full coverage of the various Higgs signal topologies with the mass-centered $\chi^2$ method. Furthermore, the published results cover only a limited range in the Higgs mass, which is a further limit to its applicability.
Since the two methods presented here are complementary---they test
inherently different statistical hypotheses---\HS~allows for the
possibility to apply the peak-centered and mass-centered $\chi^2$
methods simultaneously. We present here one approach, which attempts to
make maximal use of the available experimental information when testing
models with multiple Higgs bosons. The user of \HS\ is of course
free to use other combinations of the two results, which can be %reported
derived completely independently.
In the provided combined approach, \HS~first runs the peak-centered
$\chi^2$ method and assigns the Higgs bosons to the observed signals,
tracing the assigned combination for each analysis. In the second step,
all remaining Higgs bosons (which have not been assigned) are
considered with the mass-centered $\chi^2$ method; their respective
(mass-centered) $\chi^2$ contributions are constructed. In this way, a possible double-counting, where a Higgs boson is tested with \textit{both} the peak- and mass-centered $\chi^2$ method against the same data, is avoided. In the last
step, the total $\chi^2$ is evaluated. Here, the Higgs mass $\chi^2$
from the (relevant) signals, as well as the $\chi^2$ from combined
signal strength vectors from both the peak-centered \textit{and}
the mass-centered approach, are evaluated with a full covariance
matrix. This method thus tests the model predictions against the data in
the maximal possible way, while ensuring that no Higgs boson is tested
more than once against the same experimental data.
As a final recommendation, it should be noted that the mass ranges for the measured $\muhat$ values are still much smaller than the mass ranges for (SM) Higgs exclusion limits. To constrain theories with Higgs bosons outside this smaller range (or below the lower limit of the range currently considered by LHC searches), it is still highly recommended to run \HB\ \cite{Bechtle:2011sb,*Bechtle:2008jh,HB4} in parallel to \HS.
%\ttext{Tim: (Note for future implementation:) In the Stockholm clustering we should also include the Higgs bosons which have been already used in the peak-centered method, and if they are combined with another Higgs boson, their combination should be rejected. This avoids the problem of comparing a Higgs boson against a mu value, which might be already explained by a nearby Higgs through the peak-centered method. \textit{Maybe actually not necessary because in channels with low mass resolution these Higgs bosons are forced to be combined in the peak-centered method anyways.}}
which is also the home of \HB. Since \HS\ depends on the \HB\ libraries,
this code (version 4.0.0 or newer) should be downloaded and installed as
well. For further detail on how to do this, we refer to the \HB\ manual
\cite{Bechtle:2011sb,*Bechtle:2008jh,HB4}. Like \HB, \HS\ is written in
{\tt Fortran 90/2003}. Both codes can be compiled, for example,
using {\tt gfortran} (version 4.2 or higher). After unpacking the
downloaded source files, which should create a new directory for \HS,
the user possibly needs to set the correct path to the \HB\ installation in the \texttt{configure} file. Optionally, the path to a \FH\ installation (version 2.9.4 or higher recommended)~\cite{Heinemeyer:1998yj,*Heinemeyer:1998np,*Degrassi:2002fi,*Frank:2006yh,*Hahn:2009zz} can be set in order to use some of the example programs which use \FH\ subroutines (see below).
Furthermore,
compiler flags necessary for specific platforms can be placed here.
Configuration and installation starts with running
\cbox{./configure}
which will generate a \texttt{makefile} from the initial file
\texttt{makefile.in}. Once this is done, run
\cbox{make}
to produce the \HS~Fortran library (called {\tt libHS.a}) and the command line executable.
In addition, the user may conveniently use a bash script,
\cbox{./run\_tests.bat}
to build the \HS\ library and executable as well as the provided example programs (described in Section~\ref{Sect:examples}). The script will then perform a few test runs.
\HS~is designed to require mostly the same input as \HB, so that users
already familiar with this code should be able to transfer their
existing analyses to also use \HS\ with a minimal amount of extra
work. There are two ways to run \HS: either from the command line, or via the subroutines contained in the \HS~library {\tt libHS.a}. For the command line version, the model predictions (Higgs masses,
their theory uncertainties, total widths,
production and decay rates) have to be specified in data files using the same format as \HB-4, see Ref.~\cite{HB4}. The command line version of \HS~is presented in more detail in Section~\ref{Sect:HScmdline}.
In the subroutine version, the model predictions (which can be given as effective couplings, or as cross sections either at partonic or hadronic level) have to be provided via subroutines. Most of these subroutines are shared with the \HB~library (for details we refer again to \cite{HB4}). In addition to the \HB\ input, \HS\ requires input of the theoretical uncertainties on both the Higgs masses and the
rate predictions. Therefore, \HS~contains two additional input subroutines to set these quantities, see Section~\ref{Sect:subroutines} for more details. An accessible demonstration of how to use the \HS\ subroutines is provided by the example programs, discussed further in Section~\ref{Sect:examples}.
As already mentioned, the required input of Higgs production and decay
rates can be given either as effective couplings, or as cross sections
at partonic or hadronic level. For supersymmetric models there is an
option of using the SUSY Les Houches Accord
(SLHA)~\cite{Skands:2003cj,SLHA2} for input (either using data files
or subroutines). In this case, the production rates are always
approximated using the effective couplings specified in the two
\HB~specific input SLHA blocks (as specified in Ref.~\cite{HB4}),
whereas the Higgs branching ratios are taken directly from the
corresponding decay blocks. If present, the theoretical mass
uncertainties are read in from the SLHA block \texttt{DMASS} (as
available e.g.\ from \texttt{FeynHiggs}). Since there is no
consensus yet on how to encode the theoretical rate uncertainties in
the SLHA format, these have to be given to \HS\ explicitly by
hand.\footnote{This can be done by either calling the
subroutine \texttt{setup\_rate\_uncertainties} (see below) or by
including the rate uncertainties directly in the file
\texttt{usefulbits\_HS.f90} in case the subroutine cannot be
used (i.e. if \HS\ is run on the command line). \htb{If the user does not specify the rate uncertainties (in either case), they are assumed to be identical to the SM rate uncertainties, Eq.~\eqref{Eq:SMrateuncertainties}.}}
%\htr{\bf What about new \HS\ input of uncertainties??}
%\subsubsection{\HS\ output}
The main results from \HS~are reported in the form of a $\chi^2$ value
and the number of considered observables. For reference, the code also
calculates the $p$-value associated to the total $\chi^2$ and the
number of degrees of freedom $N$. The user may specify the number of
free model parameters $N_p$ (see below). Then, the number of degrees of
freedom is given by $N = N_\mathrm{obs} - N_p$, where $N_\mathrm{obs}$
is the total number of the included observables. Note that if the user does not specify $N_p$, the $p$-value is evaluated assuming $N_p =0$.
%\footnote{It should be noted that this value should not be considered part of the main result, since in most realistic cases the number of degrees of freedom is \emph{not} equal to the number of observables due to free model parameters. An appropriate $p$-value should then be (externally) evaluated by the user from the total $\chi^2$ value given by \HS.}
%respectively. \htb{(TS: Do we want to keep these formulas?)}
%\htr{SH: I think this is too misleading}
In the case of running with input data files, the \HS\ output is written into new files as described in Section~\ref{Sect:HScmdline}. There also exist subroutines, see Section~\ref{Sect:subroutines}, to specify the extent of screen output and to retrieve many quantities of interest for further analysis.
If \HS~is run in the SLHA mode, the results can be appended to the SLHA file in
the form of new \htb{SLHA-inspired\footnote{\htb{These blocks deviate from the SLHA conventions~\cite{Skands:2003cj,SLHA2} in the way that they contain string values (without whitespaces), which are parenthesized by the symbols `\texttt{||}'.}}} blocks. The main results are then collected in
\cbox{BLOCK HiggsSignalsResults,}
as shown for a specific example in Tab.~\ref{Tab:Block_HSoutput}. The first entries of this \texttt{BLOCK} contain general
information on the global settings of the \HS~run, \ie the version
number, the experimental data set, the $\chi^2$ method and the Higgs
mass parametrization used. Moreover, it lists the number of analyzed
observables of the different types (\texttt{BLOCK} entries \texttt{4}--\texttt{6}), as well as the total number (\texttt{BLOCK} entry \texttt{7}). \htg{Next, it} gives the corresponding
$\chi^2$ values separately from the signal strength peak observables (\texttt{BLOCK} entry \texttt{8}), the Higgs mass peak observables (\texttt{BLOCK} entry \texttt{9}), and the mass-centered observables (\texttt{BLOCK} entry \texttt{10}). The total signal strength $\chi^2$ for both methods (the sum of \texttt{BLOCK} entries \texttt{8} and \texttt{10}) is provided (\texttt{BLOCK} entry \texttt{11}), as is the total $\chi^2$ sum (\texttt{BLOCK} entry \texttt{12}). The final element (\texttt{BLOCK} entry \texttt{13}) gives the reference $p$-value, as discussed above.
%, $\chi^2_{\mathrm{total},\mu}$,
%(\texttt{BLOCK} entry \texttt{11}) is given by the sum of the signal
%strength $\chi^2$ part of the peak-centered $\chi^2$ method
%(\texttt{BLOCK} entry \texttt{8}) and the $\chi^2$ from the
%whereas the total $\chi^2$ value, $\chi^2_\mathrm{total}$,
%(\texttt{BLOCK} entry \texttt{12}) also contains the $\chi^2$
%contribution from the Higgs mass (from the peak-centered $\chi^2$
%method) (\texttt{BLOCK} entry \texttt{9}). The probability
%$p(N,\chi^2)$ quoted in \texttt{BLOCK} entry \texttt{13} corresponds
%to the probability that the observed $\chi^2$ will exceed
%$\chi^2_\mathrm{total}$ by chance \textit{even} if the investigated
%model is correct, assuming that the total number of observables $N$
%(\texttt{BLOCK} entry \texttt{7}) \textit{equals} the number of
%degrees of freedom .
% (\textit{Tim: Philip, could you please verify this statement about the probability carefully? Most of it is taken from the Numerical Recipes book (http://www.haoli.org/nr/bookfpdf/f6-2.pdf). Why do we divide by 2 in Eq.~\eqref{Eq:Pvalue}?})
\caption{Example for the SLHA block \texttt{HiggsSignalsPeakObservables}. The first column enumerates through all considered peak observables, as indicated by the dots at the bottom.}
\label{Tab:Block_HSpeakobs}
\end{table}
We show an excerpt from this extensive \texttt{BLOCK} for an example
(MSSM) parameter point in Tab.~\ref{Tab:Block_HSpeakobs}. The first
identifier, \texttt{OBS}, in the \texttt{BLOCK} enumerates the peak
observables, whereas the second number, \texttt{FLAG}, labels the specific quantity (for this peak observable). For every peak observable, the first entries (\texttt{FLAG=1-11}) give general information about the experimental data defining the observable. This is followed
by model-specific information and the results from the
\HS~run. \texttt{FLAG=12} displays a binary code representing the Higgs boson combination
which has been assigned to the signal. It has the same length as the number of Higgs bosons\footnote{For technical reasons, \HS\ is currently limited to models with $n_H \leq9$ neutral Higgs bosons, but this could easily be extended if there is a demand for more.}, such that an assigned Higgs boson with index $k$ corresponds to the binary value $2^{k-1}$. A code of only zeroes means that no Higgs boson has been assigned to this peak observable.
%Internally, the neutral Higgs
%bosons are labeled with an index $i$, for instance, in the MSSM we have
%$i=1,2,3$ for the $h, H, A$, respectively. If the Higgs boson index $i$
%appears \htb{anywhere} in the assignment code this Higgs boson has been assigned to the
%peak observable. \htr{SH: unclear; do we say that only 9 Higgs bosons
%are allowed?}\htb{(TS: Yes, so far the whole HB/HS is setup for $n_H \le 9$. It could be changed, though, but then we %%also have to change the coding here.)} The index $i=0$ \htr{SH: how can this happen? Or do we
%mean $00 \ldots 00$ corresponding to the number of Higgses?}
%can usually be ignored, however, it is
%also possible that the assignment code only contains zeros, which means
%that no Higgs boson was assigned.
In the specific example shown in Tab.~\ref{Tab:Block_HSpeakobs}, the lightest of the three neutral
Higgs bosons in the MSSM (with $k=1$) has been assigned.
%\htb{Note, that although the length of the code is equal to the number of the neutral Higgs bosons of the model, the ordering has no meaning. For instance,
%\begin{itemize}
%\item the code \texttt{100} (which is equivalent to \texttt{010} and \texttt{001}) means, that the Higgs boson $h_1$ was %assigned (out of three neutral Higgs bosons of the model),
%\item the code \texttt{3100} means, that the two Higgs bosons $h_1$ and $h_3$ were assigned (out of four neutral Higgs %bosons) and thus their signal strengths are combined.
%\end{itemize}
%(TS: Another possible way of coding this information would be binary code of length $n_H$, such that $1$ ($0$) means (not) assigned. Would you prefer that?)
%}
This \texttt{BLOCK} also contains additional information
(index $i$, Particle data group (PDG) number, mass, and signal strength contribution under \texttt{FLAG=13-16})
about the assigned Higgs boson that gives the largest contribution to
the total predicted signal strength. The total
predicted signal strength is given by \texttt{FLAG=17}. The \HS\ results (\texttt{FLAG=18-20}) contain
the $\chi^2$ contribution from the signal strength and
Higgs mass test from this observable, as well as the total $\chi^2$
contribution obtained for the assigned Higgs boson combination. Finally, the $\chi^2$ obtained for the case with no predicted signal, $\mu=0$, is given for \texttt{FLAG=21}. It should be noted that the
quoted $\chi^2$ values correspond to intermediate results in the total $\chi^2$
evaluation, where correlated uncertainties are taken into account by the
covariance matrix. For instance, the signal strength $\chi^2$ (\texttt{FLAG=18})
corresponds to $\cmua^2$ in Eq.~\eqref{Eq:chimu}, where $\alpha$ is the
index of the peak observable given in the first column of the
\texttt{BLOCK}. Thus, this quantity differs from the na\"ively
calculated $\chi^2=(\mu-\muobs)^2/(\dmuobs)^2$, and might in the extreme case even be
negative due to the impact of correlated uncertainties.
The results from the mass-centered $\chi^2$ method are summarized in
\cbox{BLOCK HiggsSignalsMassCenteredObservables}
in a similar way as in \texttt{BLOCK HiggsSignalsPeakObservables}. An example is given in Tab.~\ref{Tab:Block_HSmassobs}. The model-independent information about the observable (\texttt{FLAG=1-7}) is identical to the corresponding information in \texttt{BLOCK HiggsSignalsPeakObservables}. However, since the evaluated experimental quantities of the mass-centered observable depend on the model prediction, \cf~Section~\ref{Sect:mc_chisq}, we give the information of the tested Higgs boson (cluster) at first (\texttt{FLAG=8-10}), corresponding to Eqs.~\eqref{Eq:mc_cluster_m}--\eqref{Eq:mc_cluster_mu}. The number and binary code of the combined Higgs bosons, which form a Stockholm Higgs cluster, is given by \texttt{FLAG=11} and \texttt{12}, respectively.
From the experimental data is given the mass position (\texttt{FLAG=13}), and the measured signal strength with its lower and upper uncertainties (\texttt{FLAG=14-16}). Finally, the resulting $\chi^2$ contribution from this mass-centered observable is given at \texttt{FLAG=17}.
\htb{Note that there is also the possibility to create a new SLHA file with the \HS\ output blocks even if the input was not provided in SLHA format. Moreover,}\HS~can give an extensive screen output with similar information as encoded in the three SLHA output blocks. The level of information that is desired should then be specified before the \HS~run via the subroutine \texttt{setup\_output\_level}. See Section~\ref{Sect:subroutines} for more details.
\caption{Example for the SLHA output block \texttt{HiggsSignalsMassCenteredObservables} containing information about the observables and results from the mass-centered $\chi^2$ method.}
This command line call is very similar to the one of \HB~and the last four arguments have been directly taken over from \HB. The user may consult the \HB~manual~\cite{HB4} for more details on these arguments. The number of neutral and charged Higgs bosons of the model are specified by \texttt{nHzero} and \texttt{nHplus}, respectively. As in \HB, the model predictions are read in from the data files specified by \texttt{prefix}. Which data files are required as input depends on the argument \texttt{whichinput}, which can take the string values \texttt{effC}, \texttt{part}, \texttt{hadr} and \texttt{SLHA} for the various input formats. The theory mass uncertainties are read in from the data file \texttt{<prefix>MHall\_uncertainties.dat} for both the neutral and charged Higgs bosons. If this file is absent these uncertainties are set to zero. For more information of the data file structure we refer to the \HB-4 manual~\cite{HB4}. Note that for \texttt{whichinput=SLHA}, all the input is read in from the SLHA input file which, like the ordinary data files, should be specified by \texttt{<prefix>}.
The first three arguments are intrinsic \HS~options. The string \texttt{<expdata>} specifies which experimental data set should be used. \HS~will read in the observables found in the directory \texttt{Expt\_tables/<expdata>}. The second argument, \texttt{<mode>}, specifies which $\chi^2$ method should be used; it can take the string values \texttt{peak} (for the peak-centered $\chi^2$ method, described in Section~\ref{Sect:pc_chisq}), \texttt{mass} (for the mass-centered $\chi^2$ method, see Section~\ref{Sect:mc_chisq}), or \texttt{both} (for the simultaneous use of both methods, as described in Section~\ref{Sect:bothmethods}).
Finally, the \texttt{<pdf>} argument takes an integer selecting the parametrization for the Higgs mass uncertainty as either \texttt{1} (box), \texttt{2} (Gaussian), or \texttt{3} (box+Gaussian) pdf.
As an example, the user may run \cbox{./HiggsSignals latestresults
peak 2 effC 3 1 example\_data/mhmax/mhmax\_} which runs the
peak-centered $\chi^2$ method on the provided parameter points in the
$(M_A,~\tb)$ plane of the $m_h^\mathrm{max}$ benchmark
scenario~\cite{Carena:2002qg} of the
MSSM, using the most recent Higgs data contained in the directory
\texttt{Expt\_tables/latestresults/}.
The \HS~output from a successful command line run is collected in the data file \texttt{<prefix>HiggsSignals\_results.dat}, except for the case \texttt{whichinput=SLHA}, where the results are attached as SLHA output blocks to the SLHA file, \cf Section~\ref{Sect:HS_io}.
% of the generalized NMSSM~\cite{Ross:2011xv, SchmidtHoberg:2012yy} provided with the \HS~package. \htr{(Tim: TODO!)}
Note, that the SUSY spectrum generator \texttt{SPheno}~\cite{Porod:2003um,*Porod:2011nf}, used in conjunction with the model building tool \texttt{SARAH}~\cite{Staub:2008uz,*Staub:2009bi,*Staub:2010jh}, can write directly the \HB~(and thus \HS) data files for input in the effective couplings format.
\subsection{\HS~subroutines}
\label{Sect:subroutines}
In this section we present the subroutines needed for the use of \HS. First, we go step-by-step through the user subroutines encountered during a normal run of \HS. Then, we list additional (optional) subroutines for specific applications of \HS, and for a convenient handling of the output.
which sets up the \HS~framework: It allocates internal arrays according to the number of neutral (\texttt{nHzero}) and charged\footnote{At this point, there are no measurements available of signal strength quantities for charged Higgs bosons, which are therefore not considered in any way by \HS.} (\texttt{nHplus}) Higgs bosons in the model and reads in the tables for the SM branching ratios in the same way as done in \HB. Furthermore, it calls the subroutine \texttt{setup\_observables}, which reads in the experimental data contained in the directory \texttt{Expt\_tables/}(\texttt{expdata}). The user may create a new directory in \texttt{Expt\_tables/} containing the relevant observables for his study, see Section~\ref{Sect:expdata} for more details. For convenience, we also provide a wrapper subroutine
which does not require the third argument but uses the experimental data from the folder \texttt{Expt\_tables/}\texttt{latestresults/}.
\subroutine{setup\_pdf(}{\textit{int} pdf)}
The next step is to specify the probability density function (pdf) for the Higgs masses, which is done using {\tt setup\_pdf}. Available settings are $\texttt{pdf}=1$ for a uniform (box-shaped) distribution, $\texttt{pdf}=2$ for a Gaussian, and $\texttt{pdf}=3$ for a box-shaped pdf with Gaussian tails. The impact of this choice has been discussed in detail in Section~\ref{sect:Statistics} and will furthermore be demonstrated in Section~\ref{sect:Examples}. With the subroutine
\htbd{Once the Higgs mass pdf has been chosen with {\tt setup\_pdf},} values for the theory mass uncertainties $\dmth_i$ can be specified.
This subroutine sets the theoretical uncertainties of the neutral Higgs boson masses (in GeV) of the model via the array \texttt{dMh}. The default values (in case this subroutine is not invoked) is for all uncertainties to be zero. Note that \HB-4~also contains a similar subroutine (\texttt{set\_mass\_uncertainties}) to set theoretical mass uncertainties of the neutral and charged Higgs bosons. These uncertainties are taken into account via mass variation in the \HB~run. Since the treatment of these uncertainties is intrinsically different between the two codes, we allow the user to set the theoretical mass uncertainties for \HS\ independently using this subroutine.\footnote{The use of different theoretical mass uncertainties in \HB~and \HS~is restricted to the subroutine version. In the command line version of both programs, the theoretical uncertainties will be read in from the same data file, namely \texttt{<prefix>MHall\_uncertainties.dat}.}
For models with different uncertainties on the Higgs production cross sections and branching ratios than those for a SM Higgs boson, these should be specified using this subroutine, which sets the theoretical uncertainties of the production and decay rates (in \%) in the considered model. In the current implementation, LHC and Tevatron channels are considered to have the same relative rate uncertainties, and the rate uncertainties are assumed to be the same for all neutral Higgs bosons, independent of their masses. The input arrays should follow the structure of Table~\ref{tab:uncorder}.
The remaining required input (Higgs boson masses, total widths, branching ratios, cross sections) is identical to the \HB~input and should be set via the \HB~input subroutines, \cf Ref.~\cite{HB4}.
\begin{table}[b]
\centering
\footnotesize
\begin{tabular}{c | c c c c c}
\br
Array &\multicolumn{5}{c}{Element}\\
& 1 & 2 & 3 & 4 & 5 \\
\mr
%\hline
\texttt{dCS}& singleH & VBF &$HW$&$HZ$&$t\bar t H$\\
%\texttt{dCS\_SM}& SM & singleH & VBF & $HW$ & $HZ$ & $t\bar t H$ \\
\caption{Ordering of the elements of the input arrays \texttt{dCS} and \texttt{dBR} for the relative uncertainties of the hadronic production cross sections and branching ratios, respectively. Recall that the hadronic production mode ``singleH'' usually contains both the partonic processes $gg\to H$ and $b\bar{b}\to H$, currently assuming equal experimental efficiencies. The latter can change in the future once search categories with $b$-tags are included. \htb{Also, this table will possibly be extended once measurements in new channels (e.g. $H\to Z\gamma$) are performed.}}
\label{tab:uncorder}
\end{table}
%\line(1,0){470}
\subroutine{setup\_nparam(}{\textit{int} Np)}
In order to evaluate a meaningful $p$-value during the \HS\ run, the program has to know the number of free model parameters, $N_p$, \cf Section~\ref{Sect:HS_io}. This number is specified by the subroutine \texttt{setup\_nparam}. If this subroutine is not called before the main \HS\ run, the code assumes no free model parameters, $N_p =0$.
Once all the input has been specified, the main \HS\ evaluation can be run by calling the {\tt run\_HiggsSignals} subroutine to start the $\chi^2$ evaluation. The \texttt{mode} flag specifies the $\chi^2$ method which is used in the following evaluation process. Possible values are $\texttt{mode}=\texttt{1}$ (peak-centered method, \cf Section~\ref{Sect:pc_chisq}), $\texttt{mode}=\texttt{2}$ (mass-centered method, \cf Section~\ref{Sect:mc_chisq}), or $\texttt{mode}=\texttt{3}$ (simultaneous use of both methods, \cf Section~\ref{Sect:bothmethods}).
%Internally, \texttt{run\_HiggsSignals} will call the subroutine \texttt{evaluate\_model}.
After a successful run, this subroutine returns the $\chi^2$ contribution from the signal strength measurements (\texttt{csqmu}),\footnote{If $\texttt{mode}=\texttt{3}$, \texttt{csqmu} contains the contributions from peak and mass-centered observables.} the $\chi^2$ contribution from the Higgs mass measurements (\texttt{csqmh}), and the total $\chi^2$ value (\texttt{csqtot}). It also returns the number of observables involved in the $\chi^2$ evaluation (\texttt{nobs}). If the mass-centered $\chi^2$ method is employed, it is important to realize that \texttt{nobs} can depend on many parameters, such as the Higgs boson masses of the model (which may be inside or outside the range of an analysis). The Stockholm clustering can also affect the number of observables that are evaluated in the final $\chi^2$ calculation. Finally, the associated $p$-value (\texttt{Pvalue}) for the total $\chi^2$ with \texttt{nobs}$-N_p$ degrees of freedom is calculated.
%In the case where all model parameters are fixed, \texttt{ndf} corresponds to the number of degrees of freedom (neglecting the effect of correlations). We therefore calculate the associated $p$-value. The returned \texttt{Pvalue} gives the probability corresponding to the quoted $\chi^2$ value \texttt{csqtot}, for \texttt{nobs} degrees of freedom, as given by Eq.~\eqref{Eq:Pvalue}.
%\line(1,0){470}
\subroutine{finish\_HiggsSignals}{()}
At the end of a \HS\ run, the user should call this routine to deallocate all internal arrays.
%\line(1,0){470}
\subsubsection*{\bf Specific user subroutines}
This section provides a list (alphabetically ordered) of subroutines handling more special features of \HS.
If the user wants to perform a dedicated statistical study using pseudo-measurements (also called toy-measurements) for the Higgs signal rates and mass measurements, they can be set via this subroutine for the peak observable with the identification number \texttt{obsID}. This \textit{observable ID} is unique to the peak observable and is encoded in the experimental data, see Section~\ref{Sect:expdata} for more details. Note, that after a (dummy) run of \HS~the observable ID can also be read out with the subroutine \texttt{get\_ID\_of\_peakobservable} (see below).
%In order to set pseudo-measurements to the peak observables, the user has to know how the peak observables are structured within the code. This can be simply found by running \HS~once on the wanted set of experimental data. The first three arguments tell \HS~the peak observable for which the pseudo-measurements should be set. The argument \texttt{ii} gives the internal analysis number, \texttt{peakindex} gives the peak observable number and \texttt{npeaks} tells the code how many peak observables to expect for analysis \texttt{ii}. In the special case where we have only one peak observable for each analysis, \texttt{peakindex~$=1$} and \texttt{npeaks~$=1$}, and pseudo-measurements can be easiliy set to all peak observables by looping \texttt{ii} over all analyses. \htb{(Tim: A simpler way of implementing would be to ask for the peak observable ID.)
The arguments \texttt{mu\_obs} and \texttt{mh\_obs} are the pseudo-measured values for the signal strength modifier $\muobs$ and the Higgs mass $\mobs$. Note that the uncertainties are kept at their original values.
If the user wants to scale the uncertainties of the Higgs signal rate and mass measurements, this can be done via this subroutine in an analogous way as setting the toy measurements (using \texttt{assign\_toyvalues\_to\_peak}). Here, \texttt{scale\_mu} is the scale factor for the experimental uncertainty on the signal strength of the peak with identification number \texttt{obsID}. The theoretical rate uncertainties, which can be set independently via the subroutine \texttt{setup\_rate\_uncertainties} (see above), are unaffected by this scale factor. In this way, \HS~allows the user to scale the experimental and theoretical rate uncertainties independently. This is useful if the user is interested in a future projection of the compatibility between the model and the experimental data, assuming that a certain improvement in the precision of the measurements and/or theoretical predictions can be achieved.
%\line(1,0){470}
After the \HS~run the user can employ the following ``\texttt{get\_}'' subroutines to obtain useful information from the \HS~output. \htb{The following three subroutines are contained in the Fortran module \texttt{io}.}
If the peak-centered $\chi^2$ method is used, the peak observables are internally enumerated in \HS~based on their alphabetical appearance in the directory \texttt{Expt\_tables/(expdata)} of the used experimental dataset. This ordering is reflected \eg in the screen output and the SLHA output. However, a safer way to access the peak observables (for instance to set toy observables) is the usage of the unique observable ID of the peak observable. For this, the user may call this subroutine
which returns the observable ID \texttt{obsID} of the peak observable, which is internally structured at the position $i$.
This subroutine returns the total number of various observables: \texttt{ntotal} is the total number of observables, \texttt{npeakmu} and \texttt{npeakmh} are the number of signal strength and Higgs mass observables entering the peak-centered $\chi^2$ method, respectively, \texttt{nmpred} is the number of observables considered in the mass-centered $\chi^2$ method, and \texttt{nanalyses} gives the number of implemented analyses. Note that several mass-centered and peak observables can in general exist for each experimental analysis.
More information about the \HS\ result can be obtained by calling this subroutine. It returns the total predicted signal strength modifier, the index of the dominantly contributing Higgs boson and the number of combined Higgs bosons for the peak observable with observable identifier \texttt{obsID} as \texttt{mupred}, \texttt{npeak} and \texttt{nHcomb}, respectively.% These arrays are allocated in the subroutines call and their size is of the total number of peak observables.
This subroutine allows the user to read out the predicted signal rate for an arbitrary channel combination. This channel combination is specified by the number of combined channels, \texttt{Nchannels}, and the array \texttt{IDchannels}, which contains the two-digit IDs of these channels as specified in \cf~Tab.~\ref{tab:codes}. The output (\texttt{rate}) is the combined rate. It is more general than {\tt get\_Rvalues} (see below).
This returns the model-predicted signal rates (normalized to the SM signal rates) of Higgs boson \texttt{i} for the six different processes listed in Tab.~\ref{Tab:Rvalues}. These signal rates are calculated via Eq.~\eqref{Eq:mu}, assuming that all channels have the same relative efficiency, $\epsilon_i=1$. These quantities are evaluated either for the Tevatron or LHC with $\sqrt{s} =7\tev$ or $8\tev$, as specified by the argument \texttt{collider}, taking the values \texttt{1}, \texttt{2} or \texttt{3} for Tevatron, LHC7 or LHC8, respectively.
\caption{Production and decay modes considered in the signal rate ratio quantities which are returned by the subroutine \texttt{get\_Rvalues}.}
\label{Tab:Rvalues}
\end{table}
%\line(1,0){470}
\htb{In order to write the \HS\ SLHA output blocks, we provide three different SLHA output subroutines, contained in the Fortran module \texttt{io}. For more information about these output blocks, see Section~\ref{Sect:HS_io}.
If the user does not use the SLHA input format of \HS, or rather wants to write the output into a different file, this subroutine can be used to create a new file as specified by the argument \texttt{filename}. Note, that if this file already exists, \HS\ will \textit{not} overwrite this file but give a warning. The integer argument \texttt{detailed} takes values of \texttt{0} or \texttt{1}, determining whether only the block \texttt{HiggsSignalsResults} or all possible output blocks (i.e. also the block \texttt{HiggsSignalsPeakObservables} and/or \texttt{HiggsSignalsMassCenteredObservables}), respectively, are written to the file. The wrapper subroutine
If \HS~is run on an SLHA input file, the subroutine {\tt HiggsSignals\_SLHA\_output} appends the \HS~results as blocks to the SLHA input file.
%\line(1,0){470}
The following ``\texttt{setup\_}'' subroutines can be used to change the default settings of the \HS\ run. Thus, they should be called before the subroutine \texttt{run\_HiggsSignals}.
\htb{This subroutine can be used to change the mass range, in which a Higgs boson is forced to be assigned to a peak observable, see Section~\ref{Sect:peakassignment}. The value \texttt{Lambda} corresponds to $\Lambda$ in Eq.~\eqref{Eq:massoverlap}.}
\htb{The subroutine can be used to switch off (on) the correlations among the systematic uncertainties in the $\chi^2$ evaluation of the signal strength [Higgs mass] part by setting \texttt{corr\_mu} [\texttt{corr\_mh}]\texttt{ = 0}~(\texttt{1}). If this subroutine is not called, the default is to evaluate the $\chi^2$\textit{with} correlated uncertainties (\texttt{corr\_mu = corr\_mh = 1}).}
%can be used to control the evaluation of the total $\chi^2$ value. By setting \texttt{corr} to 0 (1), the inclusion of correlations among the systematic uncertainties can be switched off (on). The arguments \texttt{maxcsqcut} and \texttt{mincsqcut} can be set to 0 (1) in order to deactivate (activate) a maximal and minimal $\chi^2$ cutoff on the individual $\chi^2$ contributions, respectively. In case these cutoffs are activated the $\chi_\alpha^2$ value from signal observable $\alpha$ is cut off at the maximal value, corresponding to the $\chi_\alpha^2$ obtained by a predicted zero signal strength, $\mu=0$, and is restricted to positive values\footnote{Note, that in the $\chi^2$ evaluation including correlations, some of the individual $\chi^2_\alpha$ values may happen to be negative.}. The default values are \texttt{corr=1}, \texttt{maxcsqcut=0} and \texttt{mincsqcut=0}.
%}
%If the peak-centered $\chi^2$ method is used, the subroutine
%can be used to set the number of iterations in the assignment procedure of (multiple) Higgs bosons to the signals. For a detailed description of this iterative method we refer to Appendix~\ref{Append:HtoPiterations}. By setting \texttt{iter~$=0$}, correlations among systematic uncertainties are neglected in the assignment of the Higgs bosons to the peak observables. If \texttt{iter~$=1$} the assignment is carried out assuming \textit{maximally} correlated Higgs mass uncertainties. Thus, in the first iteration step, the method tries to assign as many Higgs bosons as possible to the signals. If \texttt{iter~$>1$} the $\chi^2$ value influencing the Higgs boson signal assignment is evaluated using the covariance matrices which are constructed assuming the Higgs boson signal assignment determined in the previous iteration step. Note, that the runtime of \HS~increases significantly with the number of iterations, especially for a larger number of signal observables. If the user prefers speed over accuracy (e.g. in a large parameter scan or a fit), we recommend to set \texttt{iter~$=0$}. The default value is \texttt{iter~$=1$}.
If the mass-centered $\chi^2$ method is used, the treatment of the Higgs mass theory uncertainty can be set by calling this subroutine with \texttt{mode}=\texttt{1} to use the mass variation (default), or \texttt{mode}=\texttt{2} for convolving the theory mass uncertainty with the $\muobs$ plot. See Section~\ref{Sect:mc_chisq} for more details of these methods.
%\HS~also provides the possibility to perform the Standard Model (SM) likeness test of \HB, see also Ref.~\cite{HB:PoSchargedH2012} for a detailed discussion of this test. By calling the subroutine
%the SM likeness test can be switched on (off) by setting \texttt{SMtest~$=1~(0)$}. The parameter \texttt{epsilon} sets the maximally allowed relative deviation $\epsilon$ of the individual channel signal strength from the averaged signal strength, \cf Ref.~\cite{HB:PoSchargedH2012}. In \HB~it is set to $\epsilon=0.02$. If the SM likeness test is used in \HS, the predicted signal strength of an analysis where the model fails the SM likeness test is set to zero. Thus, we still get a $\chi^2$ contribution from observables where the SM likeness test fails. The default (and recommended) use of \HS~does \textit{not} employ the SM likeness test. \ttext{Tim: We could also remove the SM likeness test completely?}
The user may control the screen output from the \HS~run with the subroutine,
where \texttt{level} takes values from $0$ to $3$, corresponding to the following output:
\begin{itemize}
\item[0] Silent mode (suitable for model parameter scans, etc.) \htb{(\textit{default})},
\item[1] Screen output for each analysis with its peak and/or mass-centered observables. The channel signal strength modifiers and SM channel weights, \cf~Eq.~\eqref{Eq:ci} and \eqref{Eq:omega}, respectively, are given for all channels considered by the analysis.
\item[2] Screen output of the \htb{essential experimental data of the peak-observables and/or implemented $\muobs$ plots (as used for the mass-centered $\chi^2$ method). For each observable, the signal channels are listed with the implemented efficiencies.}
\item[3]\htb{Creates text files holding essential information about the experimental data and the model predictions for each observable. In the peak-centered $\chi^2$ run mode}, the files \texttt{peak\_information.txt} and \texttt{peak\_massesandrates.txt} are created. The first file lists all peak observables, including a description and references to the publications, whereas the second file gives the observed and model-predicted values for the Higgs mass\footnote{If multiple Higgs bosons are assigned to the peak, we give the mass of the Higgs boson contributing dominantly to the signal rate.} and signal rates and their corresponding pull values, which we define as:
\htb{Note that in this expression the effect of correlated uncertainties is not taken into account. In the mass-centered $\chi^2$ run mode, the files \texttt{mctables\_information.txt} and \texttt{mcobservables\_information.txt} are created. The first file gives general information about the analyses with an implemented $\muobs$-plot. The second file lists all mass-centered observables, which have been constructed during the \HS\ run, including the mass position, the observed and predicted signal strength values as well as their pull values.}
\end{itemize}
For any of the options \texttt{level}~$=1-3$, the main \HS~results are printed to the screen at the end of the run.
\HS~provides the \htb{seven} example programs \texttt{HSeffC}, \htb{\texttt{HShadr}}, \texttt{HSwithSLHA}, \texttt{HBandHSwithSLHA}, \texttt{HSwithToys}, \texttt{HS\_scale\_uncertainties}, and \texttt{HBandHSwithFH}. They are contained in the subfolder
\cbox{./example\_programs/}
of the main \HS\ distribution and can be compiled \htb{all together (except \texttt{HBandHSwithFH})} by running
\cbox{make HSexamples}
or separately by calling:
\cbox{make <name of example program>}
The first program, \texttt{HSeffC}, considers a model with one neutral Higgs boson and uses the effective couplings input subroutines of \HB~to set the input. It demonstrates how to scan over a certain Higgs mass range and/or over various effective couplings while calculating the total $\chi^2$ for every scan point. The code furthermore contains two functions: \texttt{get\_g2hgaga}, which calculates the loop-induced $H\gamma\gamma$ effective coupling from the effective (tree-level) Higgs couplings to third generation fermions and gauge bosons~\cite{LHCHiggsCrossSectionWorkingGroup:2012nn} (assuming a Higgs boson mass of $126\gev$), and a second function which interpolates the cross section uncertainty of the composed single Higgs production from the uncertainties of the gluon fusion and $b\bar{b}\to H$ processes using the effective $Hgg$ and $Hb\bar{b}$ couplings. This can be relevant if the Higgs coupling to bottom quark is strongly enhanced.
%Some results of running this example program with different choices for the Higgs mass pdf are shown in Fig.~\ref{Fig:SM} below.
The second example program, \texttt{HShadr}, performs a two dimensional scan over common scale factors of the hadronic production cross sections of $p\accentset{(-)}{p} \to H$ and $p\accentset{(-)}{p} \to t\bar{t}H$ on the one side, denoted by $\mu_{ggf+ttH}$, and of $p\accentset{(-)}{p} \to q\bar{q}H$, $p\accentset{(-)}{p} \to WH$ and $p\accentset{(-)}{p} \to ZH$ on the other side, denoted by $\mu_{\mathrm{VBF}+VH}$. The Higgs branching ratios are kept at their SM values.
The third example program, \texttt{HSwithSLHA}, uses the SLHA input of \HB, \ie~an SLHA file which contains the two special input blocks for \HB. It can be executed with
\cbox{./HSwithSLHA <number of SLHA files> <SLHA filename>}
The program can test several SLHA files in one call. The total number of SLHA files must therefore be given as the first argument. The SLHA files must all have the same name, and should be enumerated by \texttt{SLHA\_filename.x}, where \texttt{x} is a number. Running, for example,
\htb{\cbox{./HSwithSLHA 2 SLHAexample.fh}
would require the two SLHA files \texttt{SLHAexample.fh.1} and \texttt{SLHAexample.fh.2} to be present}. The output is written as SLHA blocks, \cf Section~\ref{Sect:HS_io}, which are appended to each input SLHA file. The example program \texttt{HBandHSwithSLHA} can be run in an analogous way. It employs both \HB\ and \HS\ on the provided SLHA file(s), demonstrating how these two codes can be run together efficiently.
The example program \texttt{HSwithToys} demonstrates how to set new values (corresponding to pseudo-measurements) for $\hat\mu$ and $\hat{m}$ for each signal. In the code, \HS~is first run on the SM with a Higgs mass \htb{around $126\gev$} using the effective couplings input format. Then, the predicted signal strengths are read out from the \HS~output and set as pseudo-measurements. A second \HS~run on these modified observables then results in a total $\chi^2$ of zero.
The example program \texttt{HS\_scale\_uncertainties} also runs on the SM with a Higgs mass \htb{around $126\gev$}. It scans over a universal scale factor for \textit{(i)} the experimental uncertainty of the signal strength $\muobs$ only, \textit{(ii)} the theoretical uncertainties of the production cross sections and branching ratios only, and finally \textit{(iii)} both experimental and theoretical uncertainties. The output of each scan is saved in text files. In this way, rough projections of the model compatibility to a more accurate measurement in the future (with the same central values) can be made.
The last example, \texttt{HBandHSwithFH}, demonstrates how to run \HB\ and \HS\ simultaneously on a realistic model, in this case the MSSM. Here, \FH~\cite{Heinemeyer:1998yj,*Heinemeyer:1998np,*Degrassi:2002fi,*Frank:2006yh} is used to calculated the MSSM predictions needed as input for \HB\ and \HS.
%For this, the user has to know already the different signals and their ordering. In the example program, the new values are set by hand, but the user might find it more convenient to input these with an interface. The new values are passed to \HS~through the subroutine \texttt{assign\_toyvalues\_to\_mutable}.
%}
\subsection{Input of new experimental data into \HS}
\label{Sect:expdata}
The ambition with \HS\ is to always keep the code updated with the latest experimental results. Nevertheless, there are several situations when a user may want to manually add new data (or pseudo-data) to the program, for example to assess the impact of a hypothetical future measurement. For advanced users, we therefore provide a full description of the data file format used by \HS.
For each observable that should be considered by \HS, there must exist a textfile (file suffix: \texttt{.txt}). This file should be placed in a directory
\cbox{Expt\_tables/(expdata)/}
where \texttt{(expdata)} is the name identifying the new (or existing) experimental dataset.\footnote{The identifier \texttt{(expdata)} is the argument which has to be passed to \texttt{initialize\_HiggsSignals} at initialization, \cf~Section~\ref{Sect:subroutines}.} All analysis files in this directory will then be read in automatically by \HS~during the initialization.
%After editing the experimental data we recommend to recompile the code by typing
%\cbox{make hyperclean}
%\cbox{./configure}
%\cbox{make}
%Note that, in order to speed up this process, two temporary files are created. Firstly, the analyses are listed in the textfile \texttt{Expt\_tables/analyses.txt}. Secondly, the main data of these analyses is written into a binary file
%\cbox{Expt\_tables/mutables.binary}
%Both files remains until the subroutine \texttt{finalize\_HiggsSignals} is called.
%Therefore, if any of the analyses text files in \texttt{Expt\_tables/newtables/} are modified, added, or removed, this binary file also has to be removed so that a new file, containing the new information, is created in the next \HS run. Similarly, in the \HS~main directory, \HS~creates a file called \texttt{list}, which also has to be removed in this case. Both these files can be simply removed by issuing
%\cbox{make newdata}
%in the \HS~main directory.
As an example we show in Tab.~\ref{Tab:datafile} and~\ref{Tab:datafile2} the two data files for the \texttt{ATLAS} search for SM $H\to\gamma\gamma$~\cite{ATLAS-CONF-2012-091}, which define a peak observable and provide the full $\muobs$ plot as needed by the mass-centered $\chi^2$ method, respectively. The first $11$ rows of these files encode general information about the analysis and the observable (each row is required), as described in Tab.~\ref{tab:input}. Comments can be included in the top rows if they are starting with a \texttt{\#} symbol. Note, that the \textit{observable ID} must be unique, whereas the \textit{analysis ID} must be the same for (peak- or mass-centered) observables, which correspond to the same analysis \htb{and where a multiple assignment of the same Higgs boson to the corresponding observables shall be avoided. In the (yet hypothetical) case that two distinct signals have been observed within the same analysis, their peak observables thus need to have the same analysis ID, otherwise a Higgs boson might be assigned to both signals}. All integers should not have more than 10 digits.
\caption{Example file for an implemented peak observable. This file is located in \texttt{Expt\_tables/ICHEP2012/} (with name \texttt{ATL\_H-gaga\_incl\_8TeV\_5.9fb-1\_2012091102.txt}) and contains the information from the \texttt{ATLAS} search for the SM Higgs boson in the channel $H\to\gamma\gamma$~\cite{ATLAS-CONF-2012-091}. For a detailed description of each line in the file, see Tab.~\ref{tab:input}.}
\caption{Example file for an analysis with a full $\muobs$ plot as needed for the mass-centered $\chi^2$ method. This file is located in \texttt{Expt\_tables/ICHEP2012/ATL\_H-gaga\_incl\_8TeV\_5.9fb-1\_2012091202.txt}. It is the same analysis for which we already defined a peak observable in Tab.~\ref{Tab:datafile}. For a detailed description of each line in the file, see Tab.~\ref{tab:input}.}
\label{Tab:datafile2}
\end{table}
%\begin{figure}
%\centering
%\includegraphics[width=0.7\textwidth]{datatable}
%\caption{Example file for an implemented analysis. This file is located in \texttt{Expt\_tables/newtables/2012091.txt} and contains the information from the \texttt{ATLAS} search \cite{} for the SM Higgs boson in the channel $H\to\gamma\gamma$. For a detailed description of each line in the file, see the text.}
5 & CM energy (TeV), Integrated luminosity (fb$^{-1}$), Relative luminosity uncertainty\\
6 & Higgs boson type (1: neutral, 2: charged), Enable $\chi^2$ from $m_H$ (0: no, 1: yes)\\
7 & Mass resolution of analysis (GeV), \htb{assignment group (optional string without whitespaces)}\\
8 & Lowest Higgs mass, highest Higgs mass, Higgs mass interval (of the following datatable)\\
9 & Number of search channels, reference mass for efficiencies (-1: no efficiencies given)\\
10 & Search channel codes (see Tab.~\ref{tab:codes}) (\# entries must equal \# channels))\\
11 & Channel efficiencies (\# entries must equal \# channels)\\
\br
\end{tabular}
\caption{Input format for general analysis information encoded in the
first 11 rows of the experimental data file.}
\label{tab:input}
\end{table}
The channel codes in the $10^\mathrm{th}$ row are given as two-digit integers, where the first digit encodes the production mode, and the second digit the decay mode. The corresponding numbers are given in Tab.~\ref{tab:codes}. For example, the channel code of $(pp)\to HW \to(b\bar b) W$ is 35. In the example of Tab.~\ref{Tab:datafile}, we thus consider all five production modes, but only a single decay mode, \ie$H\to\gamma\gamma$.
The channel efficiencies in the $11^\mathrm{th}$ row correspond to the channels as defined by the channel codes on the previous row, and thus have to be given in the same order. If the experimental channel efficiencies are unknown, the reference mass in the $9^\mathrm{th}$ row should be set equal to $-1$, in which case the $11^\mathrm{th}$ row will be ignored. Since it must still be present, it could be left blank for the sake of clarity. Note that the channel efficiencies are defined as the fraction of events passing the analysis cuts, and \textit{not} the relative contribution of this channel to total signal yield. The latter would use information about the channel cross section, which in our case is already taken care of by the channel weights $\omega$, cf.~Eq.~\eqref{Eq:omega}. Furthermore, it is only the relative efficiencies among the channels that are important, and not their overall normalization (for the same reason).
From the $12^\mathrm{th}$ row \htr{onwards}, the signal strength data is listed. Each row contains four values: the Higgs mass, the measured signal strength modifier at the lower edge of the $1\sigma$ uncertainty (``cyan'') band, $\hat\mu-\Delta\hat{\mu}$, the central value (best-fit) $\hat\mu$, and finally the signal strength modifier at the upper edge of the $1\,\sigma$ uncertainty band, $\hat\mu+\Delta\hat{\mu}$. In the case of a peak observable definition, as in Tab.~\ref{Tab:datafile}, the data file ends after the $12^\mathrm{th}$ row, since the signal strength is only measured at a single Higgs mass value (corresponding to the signal). \htb{In contrast, for the construction of mass-centered observables, the data is listed here for the full investigated mass range, which is typically extracted from the corresponding $\muobs$ plot using \texttt{EasyNData}~\cite{Uwer:2007rs}.}
As a \htr{further} remark, we point out a general limitation in the implementation of experimental data: some results from the LHC experiments are given for the combination of data collected at different center-of-mass energies, \eg at $7\tev$ and $8\tev$. These results cannot be disentangled by \HS. Therefore, these observables are implemented as if the data was collected at the center-of-mass energy, which can be assumed to be dominating the experimental data. This approximation is valid, since both the observed and the predicted signal strengths are treated as SM normalized quantities. The only remaining inaccuracy lies in the SM channel weights, Eq.~\eqref{Eq:omega}, which depend on the center-of-mass energy.
\htb{A complication arises in the assignment of Higgs observables, if an
analysis with one measured mass peak value is split up in several
categories, each containing an individual signal rate measurement, see e.g.~\cite{ATLAS-CONF-2013-012,CMS-PAS-HIG-12-045}. In this case, each category result defines a peak observable, however only one of these observables can be associated with the mass
measurement from the analysis, which is going to contribute to the $\chi^2$. In all other categories this contribution has to be switched off. Nevertheless, this difference in the implementation can lead to inconsistent assignments of the Higgs boson(s) to the category observables. In order to enforce a consistent
assignment, peak observables can build an \textit{assignment group}. This enforces that the Higgs boson(s) are assigned to either all or none of the observables in this group, judged by the assignment status of the observable containing the mass measurement. For each peak observable, the assignment group can be specified in the experimental table, cf.
Tab.~\ref{tab:input}. Note, that the analysis IDs of the category peak observables have to be different from each other.}
In this section we discuss a few example applications which demonstrate
the performance of \HS. Most of the examples are chosen such that their
results can be validated with official results from ATLAS and CMS. The
quality of agreement of the reproduced \HS~results with the official
results justifies the gaussian limit approximation in
the statistical approach of \HS. Note, that to a certain extent (which
is difficult to estimate), the accuracy of the reproduced results
suffers from the lack of publicly available information of the
analysis efficiencies on the various production modes.
\htb{At the end of this section, we briefly discuss a few \HS\ example applications, where \htr{the results incorporate }all presently available Higgs data \htr{from the LHC and the Tevatron}. Another}
example application of \HS~within the context of the MSSM \htbd{can be found}\htb{was presented} in Ref.~\cite{Bechtle:2013gu}.
and the Tevatron experiments CDF~\cite{Aaltonen:2013ipa} and D\O~\cite{Abazov:2013zea}, as they are implemented in
\HSv{1.0.0}~as \textit{peak observables}. The left panel shows the Higgs
mass value for which the signal strength was measured. A value with
error bars indicates that the mass value is treated as a Higgs mass
observable in the peak-centered $\chi^2$ method, whereas a gray asterisk
only serves as an indication of the Higgs mass value, which was assumed in the rate measurement. \htbd{mass peak measured for this group of
observables, and} This value does not enter directly the total $\chi^2$. For some
LHC analyses, measurements for both the $7\tev$ and $8\tev$ data exist,
shown in blue and red, respectively. If the measurement is based on the
combined $7/8\tev$ dataset, we treat it as an $8\tev$ measurement only. \htb{For the $H\to\gamma\gamma$ analyses from ATLAS and CMS, the special tagged categories were implemented as separate peak observables, including their efficiencies, but collected together in assignment groups.} In total \htr{there are} 4 Higgs mass observables and 45 Higgs signal rate observables. This data is used for the performance scans in Fig.~\ref{Fig:SM} and the example applications in Section~\ref{sect:Combinedfits}.}
\label{Fig:peakobservables}
\end{figure}
As a first application we discuss the performance of the peak-centered
$\chi^2$ method on a SM-like Higgs boson. As already shown in
Fig.~\ref{fig:muplots}(b), a simple one parameter fit can be performed
to the signal strength modifier $\mu$, which scales the predicted signal
rates of all investigated Higgs channels uniformly. In this fit the
Higgs mass is held fixed at \eg$\htb{m_H=125.5\gev}$. Using the signal strength
measurements of the individual search channels \htr{obtained by the
ATLAS collaboration as given} in Fig.~\ref{fig:muplots}(b), the best-fit signal strength reconstructed with \HS~is $\muobs_\mathrm{comb} =\htb{1.17\pm0.23}$. This agrees with the official ATLAS result~\cite{ATLAS-CONF-2013-034}\htb{$\muobs=1.30\pm0.20$} within $68\%~\mathrm{C.L.}$, however, it is lower by \htb{$\sim0.5\sigma$}.
\htg{(TS: Without correlations, the fit gives $1.24\pm0.18$. In my opinion, it is very unlucky that we start the discussion with an discrepancy.)}
A possible reason for this discrepancy might be the correlations among
the systematic uncertainties of the jet energy scale (JES), which are
not taken into account in \HS\ (by default). The JES uncertainty is
most relevant for the \htb{two} channels \htr{$VH$$(V=W,Z)$, $H\to b\bar{b}$} and
$H\to\tau\tau$. A larger correlation of the systematic uncertainties
among these search channels than implemented in \HS~leads to a smaller
weight of these channels in the combined fit. Thus, these effects
enhance the impact of the $H\to\gamma\gamma$ search, which \htr{moves} the combined best-fit value $\muobs_\mathrm{comb}$ to a larger value in the official ATLAS result.
%\htb{Insert here as first example a simple fit on a combined $\muobs$ from the individual observables, using the same observables as in ATLAS/CMS as well as all of Fig.~\ref{Fig:peakobservables}. A first attempt to reconstruct the result of ATLAS ~\cite{ATLAS-CONF-2012-170} for $m_H=125\gev$ yields $\muobs = 1.24\pm 0.26$, whereas the official ATLAS result is $\muobs = 1.35\pm 0.24$. From a na\"ively calculated $\chi^2$ (neglecting any correlations), we obtain $\muobs = 1.28 \pm 0.23$.}
Now, we collect as peak observables the measured signal rates from the
and CMS~\cite{CMS-PAS-HIG-12-045,CMS-PAS-HIG-13-002,CMS-PAS-HIG-13-001,*CMS-PAS-HIG-13-003,*CMS-PAS-HIG-13-004,CMS-PAS-HIG-12-039,CMS-PAS-HIG-12-015,CMS-PAS-HIG-13-005}, as
well as the
Tevatron \htbd{combined analysis (TCB) of}\htb{experiments} CDF~\cite{Aaltonen:2013ipa} and D\O~\cite{Abazov:2013zea}, as}
summarized in Fig.~\ref{Fig:peakobservables}. If possible, we implement
results from the $7\tev$ and $8\tev$ LHC run as separate observables.
\htr{However,} if the only quoted result is a combination of both
center-of-mass energies we implement it as an $8\tev$ result. The $H\to\gamma\gamma$ and
$H\to ZZ^{(*)}\to4\ell$ analyses of ATLAS and CMS have a
\htr{rather} precise mass
resolution, thus we treat the implemented mass value of their signal as
a measurement which enters the Higgs mass part of the total $\chi^2$,
\cf Section~\ref{Sect:pc_chisq}. Note however that the implemented mass
value is not necessarily the most precise measurement of the Higgs mass
but rather the mass value for which the signal strength was published
by the experimental analysis. The Higgs mass can be determined more
accurately from a simultaneous fit to the mass and the signal
strength. This can be done with the mass-centered $\chi^2$ method, as
discussed in the next subsection. \htb{Note also, that the Higgs mass values assumed in the signal strength measurements can differ by up to $\sim2.5\gev$. It would be desirable if all experiments would present their best-fit signal strengths for all available Higgs channels (including specially tagged categories) for a common Higgs mass (equal or close to the Higgs mass value preferred by the combined data) in the future. In the present case, global fits combining the signal strength measurements performed at different Higgs masses rely on the assumption that these measurements do not vary too much within these mass differences.}
Nevertheless, we want to discuss the total $\chi^2$ distribution,
obtained by the peak-centered $\chi^2$ method, as a function of the
Higgs mass, $m_H$, which is the only free parameter of the SM. This
serves as a demonstration of the three different Higgs mass
uncertainty parametrizations (box, Gaussian, box+Gaussian pdfs) as
%\marginpar{\htr{\tiny Put comment\\ (also in the text)\\ about applying\\ in this example\\ theory\\ uncertainties\\ for the SM!}}
well as the implications of taking into account the correlations among
the systematic uncertainties in the $\chi^2$ calculation. Furthermore,
features of the automatic assignment of the Higgs boson to the peak
observables can be studied. \htb{In the following example, we set the predicted signal strength for all Higgs channels to the SM value ($\mu_i \equiv1$) and set the production and decay rate uncertainties to the values given in Eq.~\eqref{Eq:SMrateuncertainties}, as recommended by the LHC Higgs Cross Section Working Group for the SM Higgs boson around $m_H \simeq125\gev$. We then evaluate the total peak-centered $\chi^2$ for each Higgs boson mass $m_H \in[110,~140]\gev$ using the peak observables presented in Fig.~\ref{Fig:peakobservables}.}
\caption{\htb{Total $\chi^2$ distribution obtained by the peak-centered
$\chi^2$ method for a SM Higgs boson with mass $m_H$ obtained \htr{from} the 45 peak observables (status: April 2013) shown in~Fig.~\ref{Fig:peakobservables}. In (a, b), the total $\chi^2$ is evaluated without taking into account the correlations among the systematic uncertainties, whereas they are fully included in (c, d). In (a, c) no theoretical mass uncertainty $\Delta m$ is assumed (like in the SM) whereas in (b, d) we set $\Delta m=2\gev$. For each setting, we show the total $\chi^2$ obtained for all three parametrizations of the \htr{theoretical Higgs mass uncertainty}: box (solid red), Gaussian (dashed green) and box+Gaussian (dotted blue) pdf. For each case, we also give the total number of peak observables, which have been assigned with the Higgs boson, depicted by the corresponding faint lines.}}
%\caption{\htb{Total $\chi^2$ distribution (\textit{green, dashed}) for a SM-like Higgs boson, where the Higgs mass uncertainty of $\Delta m = 2\gev$ is parametrized by a Gaussian pdf. The predicted total signal strength modifier $\mu$ is profiled and also displayed (\textit{blue, dotted}). We show the results obtained both (\textit{a}) without and (\textit{b}) with the inclusion of correlations of systematic uncertainties. \htr{(TS: For comparison, will not (?) be included in final version of draft.)}}}
%\label{Fig:SMprofile}
%\end{figure}
%
%\htr{(TS: Further tests on-going. Strange features around the minimum seem to originate from different mass measurement implementations of the $H\to\gamma\gamma$ category observables.)}
The total $\chi^2$ mass distribution is shown in Fig.~\ref{Fig:SM} for
four different cases: In Fig.~\ref{Fig:SM}(a,b) the correlations among
the systematic uncertainties of \htb{the signal rates, luminosity and Higgs mass predictions} are neglected, whereas they are taken into
account in Fig.~\ref{Fig:SM}(c,d). In order to demonstrate
the difference between the three parametrizations of the Higgs mass
\htr{uncertainty} we
show the $\chi^2$ distribution assuming a theoretical Higgs mass
uncertainty of $\dmth=0\gev$ in Fig.~\ref{Fig:SM}(a,c) and
$\dmth=2\gev$ in Fig.~\ref{Fig:SM}(b,d), respectively. \htb{Furthermore, Fig.~\ref{Fig:SM} includes the number of peak observables, which have been assigned with the Higgs boson, as a function of the Higgs mass. These are depicted by the faint graphs for each Higgs mass uncertainty parametrization.}
The discontinuous shape of the $\chi^2$ distribution is caused by
changes in the Higgs boson assignment to the individual
observables. \htbd{For instance,}\htb{Recall that,} if the Higgs mass $m_H$ is so far away from
the implemented mass position of \htbd{a signal}\htb{the peak observable} that
Eq.~\eqref{Eq:massoverlap} is not fulfilled, the Higgs boson is not
assigned to the signal, thus yielding a $\chi^2$ contribution
corresponding to no predicted signal, $\mu=0$,
\cf Section~\ref{Sect:pc_chisq}. Most of the peak observables have
different mass resolutions, therefore the $\chi^2$ distribution has a
staircase-like shape. \htb{At each step, the total number of peak observable assignments changes.}
As can be seen in Fig.~\ref{Fig:SM} all three
\htr{parametrizations of the theoretical Higgs mass uncertainty}
yield the same total $\chi^2$ values if the Higgs
mass $m_H$ is far away from the implemented signal mass position,
because typically observables which enter the Higgs mass part of the
$\chi^2$ in the Gaussian parametrization exhibit a decent mass
resolution, and the Higgs boson is only assigned if this $\chi^2$ is
low, \ie$m_H \approx\mobs$. \htb{Conversely, at the $\chi^2$ minimum at a Higgs mass $m_H\sim125-126\gev$, we obtain slightly different $\chi^2$ values for the three parametrizations: Firstly, assuming that every observable is assigned with the Higgs boson, the minimal $\chi^2$ is in general slightly higher in the Gaussian case than in the box and box+Gaussian case if the Higgs mass measurements do not have the same central values for all (mass sensitive) peak observables. In that case, there will always be a non-zero $\chi^2$ contribution from the Higgs mass measurements for any predicted value of the Higgs mass. Secondly, in the case of no theoretical mass uncertainty, the box parametrization does not exhibit a full assignment of all currently implemented peak observables at any Higgs mass value. This is because the mass measurements of the ATLAS $H\to\gamma\gamma$~\cite{ATLAS-CONF-2013-012} and $H\to ZZ^{(*)}\to4\ell$~\cite{ATLAS-CONF-2013-013} observables have a mass difference of $2.5\gev$, which corresponds to a discrepancy of around $2.5~\sigma$~\cite{ATLAS-CONF-2013-014}. Thus, the Higgs boson is only assigned to either of these (group of) observables, receiving a maximal $\chi^2$ penalty from the other observable (group). In fact, we observe a double minimum structure in Fig.\ref{Fig:SM}(a,c), because for a Higgs mass $m_H \in[125.4,~125.8]\gev$, neither the ATLAS $H\to\gamma\gamma$ nor the $H\to ZZ^{(*)}\to4\ell$ observables are assigned with the Higgs boson, leading to a large total $\chi^2$.}
A difference between the Gaussian and
the theory box with experimental Gaussian (box+Gaussian)
parametrization appears only for non-zero $\dmth$. For $\dmth=2\gev$
the minimal $\chi^2$ is obtained for a plateau \htb{$m_H \approx(124.8-
126.5)\gev$} in the box+Gaussian case, whereas in the Gaussian case we
have a non-degenerate minimum at \htb{$m_H=125.7\gev$}. However, outside this plateau the
$\chi^2$ shape of the box+Gaussian increases faster than in the
Gaussian case, since the uncertainty governing this Gaussian slope is
smaller.
For the \htb{Gaussian}\htr{parametrization of the theoretical Higgs mass uncertainty}\htb{and no theoretical mass uncertainty}
the minimal $\chi^2$ at $m_H=125.7\gev$ changes from $40.2$ to $37.5$
if we include the correlations among the systematic
uncertainties in the $\chi^2$ evaluation.
%\htbd{This effect is even more visible at larger $\chi^2$ values. For instance, at $m_H=110\gev$, we have $\chi^2=47.5~(74.8)$ with (without) correlations. (FIXME: Update chi2 values to values from latest plots) Furthermore, the inclusion of correlations has also an impact on the $\chi^2$ shape in case of the Gaussian and box-Gaussian Higgs mass parametrizations. This impact is rather complicated, since the Higgs boson assignment to the signal (following a total $\chi^2$ minimization procedure) and the total $\chi^2$ evaluation (in particular the construction of the covariance matrices) depend on each other.}
%, see Appendix~\ref{Append:HtoPiterations} for more details. In this example we do not iterate the Higgs boson to signal assignment, thus the minimization is based on a $\chi^2$ calculation neglecting correlations. (\htb{Tim: Have to correct this discussion, take out the iteration discussion...})
\htb{In the case of a non-zero theoretical mass uncertainty, also the shape of the total $\chi^2$ distribution can be affected when the correlations are taken into account. Recall, that only in the Gaussian parametrization the correlations of the theoretical mass uncertainties enter the $\chi^2$ evaluation, featuring a sign dependence on the relative position of the predicted Higgs mass value with respect to the two observed Higgs mass values, cf. Section~\ref{Sec:HiggsMassObs}. This results in a shallower slope of the $\chi^2$ distribution at Higgs masses larger than all mass measurements, $m_H \gtrsim126.8\gev$, since all mass observables are positively correlated in this case.}
\htb{In conclusion we would like to emphasize that, although the direct $\chi^2$ contribution from (the few) mass measurements to the total $\chi^2$ might appear small in comparison to the $\chi^2$ contribution from (many) signal strength measurements, the automatic assignment of Higgs boson(s) to the peak observables introduces a strong mass dependence, even for peak observables without an implemented mass measurement. Hereby, the procedure tries to ensure that a comparison of the predicted and observed signal strength is valid for each observable (depending on the mass resolution of the corresponding Higgs analysis), or otherwise considers the signal as not explainable by the model.}
%\htbd{In this analysis, we make use of the assignment group for
% the ATLAS~\cite{ATLAS-CONF-2013-012} and
% CMS~\cite{CMS-PAS-HIG-12-045} $h\to\gamma\gamma$ analyses. Also note
% that the (anti-)correlation of the theoretical Higgs mass
% uncertainty with all mass measurements, introduced in
% Section~\ref{Sec:HiggsMassObs} and used in the Gaussian case, leads
% to a narrower $\chi^2$ shape for the Gaussian case than for the
%box-Gaussian
% and box case in all instances where the Higgs boson mass
% prediction falls in between several mass peaks. Finally, note that the
% discrepancy between the measured mass peak value of the ATLAS
% $h\to\gamma\gamma$ and $h\to
% ZZ^{(*)}$ analyses leads to a non-trivial
% minimal $\chi^2$ contribution for the box case without
%theoretical Higgs mass uncertainty.}
%\subsubsection{Simultaneous evaluation of different search channels with the mass-centered $\chi^2$ method}
\subsubsection{Combining search channels with the mass-centered $\chi^2$ method}
\begin{figure}[h]
\centering
\subfigure[Simultaneous evaluation of $7$ and $8\tev$ results from the ATLAS SM $H\to \gamma\gamma$ search~\cite{ATLAS-CONF-2012-091}.]{\includegraphics[width=0.47\textwidth]{ATL_gaga_combination7and8TeV}}\hfill
\subfigure[Simultaneous evaluation of ATLAS searches for $H\to \gamma\gamma,~ZZ$ and $WW$~\cite{ATLAS-CONF-2012-091,ATLAS-CONF-2012-098,ATLAS-CONF-2012-092}.]{\includegraphics[width=0.47\textwidth]{ATL_mpred_combination}}
\subfigure[Official ATLAS combination of $7$ and $8\tev$ results from the ATLAS SM $H\to \gamma\gamma$ search~\cite{ATLAS-CONF-2012-091}.]{\includegraphics[width=0.47\textwidth]{fig_13}}\hfill
\subfigure[Official ATLAS combination of the SM $H\to \gamma\gamma,~ZZ,~WW,~b\bar{b}$ and $\tau^+\tau^-$ searches~\cite{ATLAS:2012gk}.]{\includegraphics[width=0.47\textwidth]{figaux_013}}
\caption{Reconstruction of the combined best-fit signal strength from the results of the individual dataset / channels with the mass-centered $\chi^2$ method (a, b). For comparison, we give the official ATLAS results in (c, d).
%\subfigure[\HS~result using the mass-centered $\chi^2$ method. The results of the combined fit are shown in gray.]{\includegraphics[width=0.47\textwidth]{mhmu_comb}}
%\subfigure[Official ATLAS result from~\cite{ATLAS:2012gk}.]{\includegraphics[width=0.47\textwidth]{ATLAS_mhmu}}\hfill
%\caption{Simultaneous fit to the Higgs mass and signal strength using the experimental data from the ATLAS searches $H\to\gamma\gamma$~\cite{ATLAS-CONF-2012-091}, $H\to WW^{(*)}\to \ell\nu\ell\nu$~\cite{ATLAS-CONF-2012-098} and $H\to ZZ^{(*)}\to 4\ell$~\cite{ATLAS-CONF-2012-092}.}
%\label{Fig:mhmufit}
%\end{figure}
\begin{figure}[h]
\centering
\subfigure[\HS~result using the mass-centered $\chi^2$ method. The results of the combined fit are shown in gray.]{\includegraphics[width=0.47\textwidth]{mhmu_ATLAS_Moriond2013}}
\subfigure[Official ATLAS result from~\cite{ATLAS-CONF-2013-030}.]{\includegraphics[width=0.47\textwidth]{mhmu_ATLAS_Moriond2013_official}}\hfill
\caption{Simultaneous fit to the Higgs mass and signal strength using the experimental data from the ATLAS searches $H\to\gamma\gamma$~\cite{ATLAS-CONF-2012-168}, $H\to WW^{(*)}\to\ell\nu\ell\nu$~\cite{ATLAS-CONF-2013-030} and $H\to ZZ^{(*)}\to4\ell$~\cite{ATLAS-CONF-2013-013}. For comparison, we show a corresponding result from ATLAS in (b), which includes also the updated $H\to\gamma\gamma$ search~\cite{ATLAS-CONF-2013-012}.}
\label{Fig:mhmufit_Moriond2013}
\end{figure}
As a first demonstration of the mass-centered $\chi^2$ method we evaluate simultaneously the $7\tev$ and $8\tev$ results from ATLAS for the Higgs searches $H\to\gamma\gamma$~\cite{ATLAS-CONF-2012-091}, as well as its evaluation together with the $H\to WW^{(*)}\to\ell\nu\ell\nu$~\cite{ATLAS-CONF-2012-098} and $H\to ZZ^{(*)}\to4\ell$~\cite{ATLAS-CONF-2012-092} searches. This is possible because the full $\muobs$ plot was published for these analyses for $7\tev$ and $8\tev$, except for the $H\to ZZ^{(*)}\to4\ell$ search where only the combined $7/8\tev$ result is available.\footnote{Since it is not possible to disentangle this result into $7\tev$ and $8\tev$, we implemented this observable as $8\tev$ only data in \HS.}
We scan the relevant Higgs mass range $m_H=(110-150)\gev$, as well as
the signal strength $\mu$, and at each point ($m_H,~\mu$) evaluate the
mass-centered $\chi^2$ using the corresponding $\muobs$ plots as
\textit{mass-centered observables}. We then find the best-fit $\mu$
value (and the corresponding $1\sigma$ and $2\sigma$ regions) by
minimizing the $\chi^2$ (finding $\Delta\chi^2=1$ and $\Delta\chi^2
=4$, respectively) for a fixed Higgs mass $m_H$. This is shown in
Fig.~\ref{Fig:mc_comb}(a) and~\ref{Fig:mc_comb}(b) for the
$H\to\gamma\gamma$\htr{channel}
and the combination of $H\to\gamma\gamma$, $H\to WW^{(*)}\to\ell\nu\ell\nu$ and $H\to ZZ^{(*)}\to4\ell$, respectively. These results nicely agree with the corresponding official ATLAS results~\cite{ATLAS-CONF-2012-091, ATLAS:2012gk}, which are shown in Fig.~\ref{Fig:mc_comb}(c,d) for comparison. Especially at the signal around $\simeq126\gev$ the Gaussian limit approximation works very well due to a good event sampling (in the $H\to\gamma\gamma$ analysis). Note that in Fig.~\ref{Fig:mc_comb}(d) also the channels $H\to\tau\tau$ and $VH \to b\bar{b}$ are included, however, these observables are rather insignificant for this result due to large uncertainties on the signal strength measurement as well as a poor mass resolution.
Instead of minimizing the $\chi^2$ for a fixed Higgs mass $m_H$, we now perform a two parameter fit to $m_H$ and $\mu$, using the latest currently available $\muobs$ plots from the ATLAS searches\footnote{ATLAS did not include a new $\muobs$ plot in their $H\to\gamma\gamma$ search update at the Moriond 2013 conference~\cite{ATLAS-CONF-2013-012}. Therefore, we have to use an older result here. \htb{We use the $\muobs$ plot from~\cite{ATLAS-CONF-2012-168} which includes the mass scale systematic (MSS) uncertainty.}}$H\to\gamma\gamma$~\cite{ATLAS-CONF-2012-168}, $H\to WW^{(*)}\to\ell\nu\ell\nu$~\cite{ATLAS-CONF-2013-030} and $H\to ZZ^{(*)}\to4\ell$~\cite{ATLAS-CONF-2013-013}. For a given signal hypothesis, ($m_H,~\mu)$, we scan the full mass range, $m_H'\in[120,~150]\gev$ with a reasonably small step size of $\mathcal{O}(0.1-1)\gev$. For each scanning point we evaluate the mass-centered $\chi^2$ value\htb{, $\chi^2_\mathrm{MC}$,} for the hypothesis ($m_H',~\mu'$), where
\begin{equation}
\mu' = \left\{\begin{array}{ll}
\mu&\mbox{if}\quad m_H' = m_H, \\
0 &\mbox{if}\quad m_H' \ne m_H. \\
\end{array}
\right.
\end{equation}
The obtained $\chi^2$ values from this scan are summed and associated
with the point ($m_H,~\mu)$. \htb{Mathematically, this corresponds approximately to evaluating
where we used $\mu' =\mu~\delta(m_H - m_H')$ in the last step.}
Thus we test the \htb{(combined)} hypothesis of having a
Higgs boson at $m_H$ with signal strength $\mu$ while at any other
mass position, $m_H' \ne m_H$, we have no signal ($\mu'=0$). The
procedure is then repeated for all points in the two-dimensional
($m_H, \mu$) plane to obtain the 2D $\chi^2$ likelihood map. The results are
shown in Fig.~\ref{Fig:mhmufit_Moriond2013}(a), where the Higgs
searches are investigated separately as well as in combination. For
comparison, we also show the official ATLAS
result~\cite{ATLAS-CONF-2013-030} in
Fig.~\ref{Fig:mhmufit_Moriond2013}(b). It also contains the
confidence regions of the updated $H\to\gamma\gamma$
search~\cite{ATLAS-CONF-2013-012} which we could not include in our
fit \htb{due to the absence of the $\muobs$ plot in the public data}. Qualitatively, the obtained
$68\%$ and $95\%$ C.L. regions agree fairly well for all investigated
channels. Note that the spiky structures of the contour ellipses
in Fig.~\ref{Fig:mhmufit_Moriond2013}(a) are rather an artifact of
our data extraction with \texttt{EasyNData}~\cite{Uwer:2007rs} than a physical
effect.\footnote{It would therefore be desirable if the experimental
collaborations published the data of the $\muobs$ plots also in
tabular form in accurate precision.}
%\htbd{ for $H\to\gamma\gamma$ and $H\to WW^{(*)}\to \ell \nu \ell \nu$ agree fairly well with the ATLAS results, whereas they appear to be larger for $H\to ZZ^{(*)}\to4\ell$ in the reconstructed \HS~result. The reason for this is the limited event sample in the $H\to ZZ^{(*)}\to4\ell$ analysis, resulting in a visible difference between the log-likelihood method used by ATLAS and the $\chi^2$ method of \HS, which is based on the Gaussian limit assumption. This difference however can be expected to vanish soon with more experimental data being analyzed. }
%\hto{Do this plot again with the ATLAS December results.} \htb{(Tim: Done, was sent around. Should we include it here instead of Fig. 5?)}
The best-fit point of our simultaneous fit to the ATLAS Higgs channels $H\to\gamma\gamma$, $H\to ZZ^{(*)}\to4\ell$ and $H\to WW^{(*)}\to\ell\nu\ell\nu$ is found at
\begin{equation}
% THIS IS FOR THE COMBINATION OF ATLAS AND CMS ANALYSES:
where the uncertainties \htr{given refer} to the 1D profiled $68\%$ confidence interval. Note that the scan was performed with step sizes of $\delta m_H =0.1\gev$ and $\delta\mu=0.05$.
%Profiling either $\mu$ or $m_H$, we obtain the parameter ranges
%(\htb{Tim: Note, these intervals were obtained by simply holding one parameter at the best-fit point value and then reading off the 68\% and 95\% C.L. values for the other parameter. This is certainly not strictly the meaning of "profiling".})
%\subsection{Determining the coupling structure of a Higgs boson at $126\gev$}
\subsection[Validation with official fit results for Higgs coupling scaling factors]{Validation with official \htr{fit results for Higgs coupling scaling factors}\sectionmark{Validation with official fit results\dots}}\label{Sec:ValidationOffEff}
\sectionmark{Validation with official fit results\dots}
A major task after the discovery of \htr{a Higgs-like state is the}
determination of its coupling properties and thus a thorough test of its
\htr{compatibility with the SM}. Both ATLAS~\cite{ATLAS-CONF-2013-034,ATLAS-CONF-2012-127} and
CMS~\cite{CMS-PAS-HIG-12-045,CMS-PAS-HIG-13-005} have \htr{obtained results for Higgs
coupling scaling factors} in the
framework of restricted benchmark models proposed by the LHC Higgs Cross
Section Working Group~\cite{LHCHiggsCrossSectionWorkingGroup:2012nn}.
\htr{Numerous other studies have been performed, both for Higgs coupling
scaling factors \htb{in a model-independent or effective coupling approach}}~\citeHiggsEffC\ \htr{as well as for particular models\htb{, including composite Higgs scenarios~\citeCompHiggsAndEffC, Two Higgs Doublet Models (2HDMs)~\citeTHDM, supersymmetric models~\citeHiggsSUSY\ as well as other, more exotic extensions of the SM~\citeSMextensions.}}
Here, we want to focus on the reproduction of the official
ATLAS and CMS results using the \htr{Higgs coupling scaling factors as
defined in the benchmark models} of Ref.~\cite{LHCHiggsCrossSectionWorkingGroup:2012nn} in order to validate the \HS\ implementation.
%\caption{Signal strength measurements from various ATLAS Higgs searches implemented in \HS as peak observables at the mass position $\mobs =126.5\gev$. Results from combined $7/8\tev$ data are implemented as $8\tev$-only in \HS.}
\caption{Signal strength measurements from various ATLAS Higgs searches implemented in \HS\ as peak observables. Results from combined $7/8\tev$ data are implemented as $8\tev$-only in \HS.}
\caption{Signal strength measurements from various CMS Higgs searches implemented in \HS~as peak observables at the mass position $\mobs=125.8\gev$.
%\ttext{(Tim: Maybe include also reference to original analysis. Note that these channels are mostly implemented without efficiencies, thus we interpret the VBF tagged channels as purely VBF produced Higgs events. As long as CMS doesn't give the signal efficiencies, we must live with it.)}
%\subfigure[Result obtained using \HS. The $\Delta \chi^2 = \chi^2 - \chi^2_\mathrm{best-fit}$ distribution is given in colour.]{\includegraphics[ width=8.6cm, height=6.6cm]{ATLanalyses_7and8TeV}}
%\subfigure[Official ATLAS result from Ref.~\cite{ATLAS-CONF-2012-127}.]{\includegraphics[trim = 0cm -1.4cm 0cm 0cm, clip,width=7.7cm]{ATLAS_FV_fit}}\hfill
%\caption{Comparison of the 2-parameter fits probing different coupling strength scale factors for fermions, $\kappa_F=g_{Hff}$, and vector bosons, $\kappa_V = g_{HVV}$, derived by ATLAS~\cite{ATLAS-CONF-2012-127} (a) and \HS~(b). The signal rates of the Higgs searches for $VH\to b\bar{b}$, $H\to \tau\bar{\tau}$, $H\to WW^{(*)}\to \ell \nu\ell\nu$, $H\to\gamma\gamma$ and $H\to ZZ^{(*)}\to 4\ell$ are measured assuming a Higgs mass of $m_H=126\gev$.}
\htb{We validate with the ATLAS results, as presented at the Moriond 2013 conference~\cite{ATLAS-CONF-2013-034}, as well as earlier CMS results published for the HCP 2012 conference~\cite{CMS-PAS-HIG-12-045}. The measurements from ATLAS and CMS, which are used as observables for our reproduced fits, are summarized in Tab.~\ref{Tab:ATLAS_peakobs} and~\ref{Tab:CMS_peakobs}, respectively.}
%First, we investigate two two-dimensional \htr{Higgs coupling scaling factor} fits \htb{presented by ATLAS at the Moriond 2013 conference}~\cite{ATLAS-CONF-2013-034}.
\htb{In the ATLAS coupling fits} the Higgs mass is assumed to be $m_H =125.5\gev$.
% The observables used for our reproduced fits are summarized in Tab.~ref{Tab:ATLAS_peakobs}.
\htb{However, for} a Higgs mass of $125.5\gev$ there
are no signal strengths measurements for the $H\to\gamma\gamma$
categories available in the literature, thus we use the $\muobs$
measurements performed at $m_H =126.8\gev$ in~\cite{ATLAS-CONF-2013-012}, keeping in mind that this might lead to
some inaccuracies. The two \htb{ATLAS}$H\to WW^{(*)}$ signal strength measurements
are extracted from the corresponding $\muobs$ plots in~\cite{ATLAS-CONF-2013-030}. Note that for the remaining channels,
$H\to ZZ^{(*)}$, $H\to\tau\tau$ and $VH \to Vb\bar{b}$, only the inclusive $\muobs$ measurements are available in the literature, whereas the ATLAS fit also includes information of their sub-channels~\cite{ATLAS-CONF-2013-034}. \htb{In the CMS coupling fits a Higgs mass of $m_H=125.8\gev$ is assumed. All signal strength measurements, as listed in Tab.~\ref{Tab:CMS_peakobs}, have been performed for this assumed Higgs mass value.}
\htb{The first benchmark model we want to investigate is a}
%The \HS\ result for the
two-dimensional fit to universal scale factors for the Higgs coupling to the massive SM vector bosons, $\kappa_V$, and to SM
fermions, $\kappa_F$. In this fit it is assumed that no other modifications to the total width than those induced by the
\htr{coupling scale factors}$\kappa_F$ and $\kappa_V$ are present, \hto{allowing for a fit to the coupling strength modifiers individually \htr{rather than to ratios of the scale factors}~\cite{LHCHiggsCrossSectionWorkingGroup:2012nn}.}\htb{Note that the loop-induced effective $H\gamma\gamma$ coupling is derived from the (scaled) tree-level couplings and thus exhibits a non-trivial scaling behavior. In particular the $tW$ interference term introduces a non-negligible dependence on the relative sign of the scale factors $\kappa_F$ and $\kappa_V$. In the case of a relative minus sign this interference term gives a positive contribution to the $H\gamma\gamma$ coupling.}
\subfigure[Result from \HS. The $\Delta \chi^2$ distribution is given in color.]{\includegraphics[ width=8.0cm, height=6.2cm]{ATLAS_KVKF_Moriond2013}}
\subfigure[Official ATLAS result from Ref.~\cite{ATLAS-CONF-2013-034}.]{\includegraphics[trim = 0cm -1.4cm 0cm 0cm, clip,width=7.2cm]{ATLAS_KVKF_Moriond2013_official}}\hfill
\caption{Comparison of the two-parameter fits probing different coupling strength scale factors for fermions, $\kappa_F$, and vector bosons, $\kappa_V$, derived by \HS\ (a) and ATLAS~\cite{ATLAS-CONF-2013-034}~(b). The signal strength measurements used for the \HS\ fit are listed in Tab.~\ref{Tab:ATLAS_peakobs}. The Higgs mass is chosen to be $m_H=125.5\gev$.
%The signal rates of the Higgs searches for $VH\to b\bar{b}$, $H\to \tau\bar{\tau}$, $H\to WW^{(*)}\to \ell \nu\ell\nu$, $H\to\gamma\gamma$ and $H\to ZZ^{(*)}\to 4\ell$ are measured assuming a Higgs mass of $m_H=126\gev$.
\subfigure[Official CMS result from Ref.~\cite{CMS-PAS-HIG-12-045}.]{\includegraphics[trim=0cm -1.3cm 0cm 0cm, clip, height=7cm, width=0.47\textwidth]{CMS-12-045-KV-KF-color_original}}\hfill
\caption{Comparison of the two-parameter fits probing different coupling strength scale factors for fermions, $\kappa_F$, and vector bosons, $\kappa_V$, obtained using \HS~(a), and by CMS~\cite{CMS-PAS-HIG-12-045} (b). The signal strength measurements used for the \HS\ fit are listed in Tab.~\ref{Tab:CMS_peakobs}. The Higgs mass is chosen to be $m_H=125.8\gev$.}
\htb{The reconstructed ATLAS and CMS fits obtained with \HS\ are shown in Fig.~\ref{Fig:ATLAS_FV-fit}(a) and~\ref{Fig:CMS_FV-fit}(a), respectively. For comparison, we show the official fit results
from ATLAS~\cite{ATLAS-CONF-2013-034} and CMS~\cite{CMS-PAS-HIG-12-045}\htr{in
We find overall very good agreement. In the reconstructed ATLAS fit we observe a weak
tendency to slightly lower values of $\kappa_V$\htr{than} in the official ATLAS result.
\marginpar{\htr{\small comment\\ on this\\ feature?}}
Our best-fit point lies at $(\kappa_V,\kappa_F)=(1.11, 0.85)$ with $\chi^2/\mathrm{ndf}=10.2/17$.}
%\htbd{Due to the sign difference in the couplings, the $tW$ interference term in the loop-induced $H\gamma\gamma$ coupling gives a positive contribution and thus enhances the $H\to\gamma\gamma$ decay mode. Nevertheless, the preference for the ($+$,$-$) sector of the $(\kappa_V,~\kappa_F)$ plane is very weak. Restricting the fit to positive couplings, we find the best-fit point at $(\kappa_V,~\kappa_F) = (1.24,~0.75)$ with $\chi^2/\mathrm{ndf}=3.45/7$.}
The (2D) compatibility of the SM hypothesis with this best-fit point is \htb{$17.8\%$}.
%We perform the same fit of the scaling factors $\kappa_V$ and $\kappa_F$, as previously discussed for the ATLAS measurements, now for the CMS observables. The result and the corresponding official CMS result are shown in Fig.~\ref{Fig:CMS_FV-fit}. \htr{[GW: Shouldn't we order the results by topic rather than by collaboration? I would find it better to discuss Fig.~9 together with Fig.~6.]}
%Overall the agreement is fairly good.
\htb{In the reconstructed CMS fit, we find the} best-fit point in the sector with a
relative sign difference between the couplings, \htb{in agreement with the CMS result}. The preference for this
sector is slightly more pronounced in the \HS~reconstruction,
\marginpar{\htr{\small comment\\ on this\\ feature?}}
such that the $68\%~\mathrm{C.L.}$ region lies solely in this sector. We find the best-fit point at $(\kappa_V,~\kappa_F)=(0.86,~-0.85)$ with $\chi^2/\mathrm{ndf} =3.6/9$ and the (2D) compatibility of the SM hypothesis with this point is $16.5\%$.
\htr{Some care is necessary regarding the interpretation of the fit
result in the parameter region where the relative sign of $\kappa_V$ and
$\kappa_F$ differs from the SM case. First of all, it should be kept in
mind that in this fit only the two parameters
$\kappa_V$ and $\kappa_F$ are allowed to
deviate from their SM values, while all other Higgs couplings and
partial decay widths have been fixed to their SM values. The way an
observed deviation from the SM manifests itself in the parameter space
of coupling strength modifiers $\kappa_i$ will sensitively depend on how
general the basis of the $\kappa_i$ is that one has chosen. Furthermore
the framework of the coupling strength modifiers $\kappa_i$ as defined
in Ref.~\cite{LHCHiggsCrossSectionWorkingGroup:2012nn} is designed for
the analysis of relatively small deviations from the SM. In case a firm
preference should be established in a parameter region that is very
different from the SM case (like a different relative sign of Higgs
couplings), the framework of the coupling strength modifiers $\kappa_i$
would have to be \htbd{a} replaced by a more general parametrisation.}
\subfigure[Result from \HS. The $\Delta \chi^2$ distribution is given in color.]{\includegraphics[ width=8.0cm, height=6.2cm]{ATLAS_KgaKg_Moriond2013}}
\subfigure[Official ATLAS result from Ref.~\cite{ATLAS-CONF-2013-034}.]{\includegraphics[trim = 0cm -1.4cm 0cm 0cm, clip,width=7.2cm]{ATLAS_KgaKg_Moriond2013_official}}\hfill
\caption{Comparison of the two-parameter fits probing different coupling
strength scale factors to gluons, $\kappa_g$, and photons,
$\kappa_\gamma$, obtained by \HS~(a), and
ATLAS~\cite{ATLAS-CONF-2013-034} (b). It is assumed that no new Higgs
boson decay modes are open, $\Gamma_\mathrm{BSM} =0\gev$, \htr{and that
no other modifications of the couplings occur with respect to their SM values}. The signal strength measurements used for the \HS\ fit are listed in Tab.~\ref{Tab:ATLAS_peakobs}. The Higgs mass is chosen to be $m_H=125.5\gev$.}
In order to probe the presence of BSM physics in the Higgs boson
phenomenology a fit to the loop-induced Higgs couplings to gluons,
$\kappa_g$, and photons, $\kappa_\gamma$, can be performed. In this fit
\htr{it is assumed that all other (tree-level) Higgs couplings} are as in the SM and no new Higgs boson decay modes exist. Fig.~\ref{Fig:ATLAS_kgkga-fit} shows the 2D likelihood map in the $(\kappa_\gamma,~\kappa_g)$ parameter plane for both the \HS~result [Fig.~\ref{Fig:ATLAS_kgkga-fit}(a)] and the official ATLAS~\cite{ATLAS-CONF-2013-034} result [Fig.~\ref{Fig:ATLAS_kgkga-fit}(b)].
Again, we observe good agreement between the two results. \htb{The
$\chi^2$ distribution of the \HS\ result is slightly shallower \htr{in
the $\kappa_g$ direction} than in \htr{the ATLAS result},
\marginpar{\htr{\small comment\\ on this\\ feature?}}
leading to slightly larger confidence regions. The best-fit point is found at $(\kappa_\gamma,~\kappa_g)=(1.27,~1.00)$, which is (2D) compatible with the SM at the level of $18\%$, based on the $\chi^2$~${\cal P}$-value.}
\htb{The corresponding fit result for the CMS observables is shown in Fig.~\ref{Fig:CMS_kgkga-fit}(a), together with the official CMS result, Fig.~\ref{Fig:CMS_kgkga-fit}(b).} The best-fit point at $(\kappa_\gamma,~\kappa_g)=(1.40,~0.84)$ as well as the $68\%$ and $95\%$ C.L. regions of the \HS~result are in excellent agreement with the CMS result. \htb{The (2D) compatibility of the SM hypothesis with the best-fit point is at the $15.5\%$ level.}
%We now turn to the reconstruction of some coupling fits from CMS presented \htb{at the HCP2012 conference}~\cite{CMS-PAS-HIG-12-045}. In the following we assume a Higgs mass of $m_H=125.8\gev$. The signal rate measurements used as peak observables in \HS~are listed in Tab.~\ref{Tab:CMS_peakobs}.
\begin{figure}[th]
\centering
\subfigure[\HS~result for different channels (as indicated by the legend). We furthermore show the $68\%~\mathrm{C.L.}$ region for the combination of the individual channels.]{\includegraphics[trim=2cm 0.25cm 2.1cm 0cm, clip, width=0.47\textwidth]{CMS-12-045_CS-scaling-comb}}
\subfigure[Official CMS result from Ref.~\cite{CMS-PAS-HIG-12-045}.]{\includegraphics[trim=0.32cm -0.28cm 0.32cm 0cm, clip,width=0.47\textwidth]{CMS-12-045_CSscaling_original}}\hfill
\caption{The $68\%~\mathrm{C.L.}$ regions for the universal scale factors for the production cross sections of gluon-gluon fusion (ggf) and top quark pair associated Higgs production (ttH), $\mu_\mathrm{ggf+ttH}$, and of vector boson fusion (qqH) and vector boson associated Higgs production (VH), $\mu_\mathrm{qqH+VH}$, as obtained from the individual Higgs search channels by CMS. \htb{The signal strength measurements used for the \HS\ fit are listed in Tab.~\ref{Tab:CMS_peakobs}. The Higgs mass is chosen to be $m_H=125.8\gev$.}}
\label{Fig:CSscaling}
\end{figure}
\htb{Furthermore,} CMS has combined the channels with a
particular decay mode to explicitly target the different production
modes. A two parameter fit was performed for each of these decay modes
to a signal strength modifier associated with the ggf and $t\bar{t}H$ production
mechanisms, $\mu_\mathrm{ggf+ttH}$, and a signal strength modifier for
the VBF and $VH$ production modes, $\mu_\mathrm{VBF+VH}$. The result of
the same fit performed with \HS~is shown in Fig.~\ref{Fig:CSscaling}(a)
in direct comparison with the CMS result, displayed in
Fig.~\ref{Fig:CSscaling}(b). All $68\%~\mathrm{C.L.}$ regions are in
good agreement with the CMS results. Small differences are observed for
the $H\to WW$ and $H\to\gamma\gamma$ contour ellipses. This is most
likely due to the absence\footnote{CMS states in
Ref.~\cite{CMS-PAS-HIG-12-045} that the signal purities of the
production mode tagged subchannels \htr{vary} substantially. For
instance, the ggf fraction in the dijet VBF tagged
channels typically amounts to roughly $20-50\%$. Unfortunately, CMS
\htr{has not published} these important estimates on the signal contamination. Therefore, we can only treat the channels as \textit{pure} tags.} of public information of the signal efficiencies of the various production modes for the subchannels. In Fig.~\ref{Fig:CSscaling}(a) we also show the $68\%~\mathrm{C.L.}$ region obtained for a combination of all measurements\htb{, assuming SM Higgs branching fractions}. The SM \htb{is well compatible with the combined fit result in this simplified production mode scaling scenario}.
\htr{[GW: In their paper CMS writes about this kind of plot: ``A
combination of the different decay modes is not performed since that
would require introducing hypotheses on the relative branching
fractions.'' Do we assume the SM in the combination? I think we should
comment on that point.]}
\subsection{Example applications of~\HS}
\label{sect:Combinedfits}
We now go beyond validation and repeat the two discussed \htr{Higgs coupling scaling factor} fits including the full presently available data from the LHC and Tevatron experiments, as they were presented at the Moriond 2013 conference and listed in Fig.~\ref{Fig:peakobservables}. The \htb{fit} results are shown in Fig.~\ref{Fig:Moriond2013_couplingfit}, \htb{where we assume a Higgs boson mass of $126\gev$}.
%The corresponding SM best-fit point exhibits $\chi^2 (\mu~\mbox{part}) = 33.5$. \htr{[GW: it's not clear to me what is meant here.]}
\htr{[GW: Should we explicitly refer to the Tables? \htb{TS: Which tables?}]}
%For $\kappa_V$, agreement on the
%$1\sigma$ level with the SM is found, while $\kappa_F$ is only
%compatible at the $2\sigma$ level.
\htr{For the ($\kappa_V, \kappa_F$) fit the SM point is found to be
located just outside the 68\% C.L.\ contour.}
Compared to the individual results from ATLAS and CMS\footnote{\htb{However, note that the CMS results used here are newer then those used for the validation in Fig.~\ref{Fig:CMS_FV-fit} and~\ref{Fig:CMS_kgkga-fit}. We refer to~\cite{CMS-PAS-HIG-13-005} for a comparison with the corresponding official results.}} presented in Fig.~\ref{Fig:ATLAS_FV-fit} and \ref{Fig:CMS_FV-fit}, a significant degradation of
the fit quality of the non-SM minimum \htg{(i.e. for negative $\kappa_F$)} is observed, which highlights
the power of such simultaneous global analyses.
A similar improvement is seen for the ($\kappa_\gamma, \kappa_g$)
which \htbd{ought to}\htb{can} be compared \htbd{to}\htb{with} Fig.~\ref{Fig:ATLAS_kgkga-fit} and
\ref{Fig:CMS_kgkga-fit}. \htr{Here the SM \htbd{point} is compatible with the
fit result within its 95\% C.L.\ contour, where the observed slight
deviation is mainly associated with $\kappa_g$.}\htb{Note, that the discrimination power on this scale factor will increase only slowly with more data, since the large uncertainty of the rate prediction for single Higgs production is already the dominant limitation of the precision of the combined fit~\citeTheoUncertainties.}
%Also here, a compatibility on the
% $1\sigma$ ($2\sigma$) level with the SM is observed for
\caption{\htb{Two-dimensional fit results for the two different
\htr{benchmark scenarios of Higgs coupling scaling factors discussed
above:}
(a) Common
scale factors for the vector boson and fermion couplings, $\kappa_V$
and $\kappa_F$, respectively; (b) \htr{Scale} factors for the
loop-induced Higgs couplings to photons, $\kappa_\gamma$, and
gluons, $\kappa_g$. In these fits, the Higgs boson mass is assumed
to be $126\gev$. The full available data from Tevatron and LHC
experiments, as presented at the Moriond 2013 conference and
summarized in Fig.~\ref{Fig:peakobservables}, is used.}}
\label{Fig:Moriond2013_couplingfit}
\end{figure}
\htr{As a further \htb{example application} we \htbd{have} performed fits}
in three of the \htb{MSSM} benchmark
scenarios recently proposed for the interpretation of the SUSY Higgs
search results at the LHC in~\cite{Carena:2013qia}. These scenarios
are defined in terms of two free parameters, \htbd{either $\tb$ and $\MA$,
or $\tb$ and $\mu$}\htb{$\tb$ and either $\MA$ or $\mu$}. The other parameters are fixed to exhibit
certain features of the MSSM Higgs phenomenology. \htb{For each parameter point in these two-dimensional planes we calculated the model predictions with \texttt{FeynHiggs-2.9.4} and} evaluated the
total $\chi^2$\htb{, comprised of the LEP Higgs exclusion $\chi^2$ value\cite{Barate:2003sz,Schael:2006cr} obtained}\htr{from \texttt{HiggsBounds\htb{-4}}~\cite{HB4,Bechtle:2013gu}\htb{as well as the total $\chi^2$ from}\texttt{HiggsSignals}}\htb{using the \emph{peak-centered}$\chi^2$ method. The
other MSSM parameters are set} to their default values as specified in~\cite{Carena:2013qia}. \htb{In all three cases, the theoretical mass uncertainty of the lightest Higgs boson is either set to $2\gev$ in the case of a Gaussian uncertainty treatment (i.e. in the LEP exclusion $\chi^2$ from \HB\ and in \HS), or set to $3\gev$ for the $95\%~\mathrm{C.L.}$ LHC exclusions obtained with \HB.}
\begin{figure}[t]
\centering
\includegraphics[width=0.8\textwidth]{mhmax_HSHB}
\caption{\htb{$\Delta\chi^2$ distribution (\HS\ and \HB\ LEP exclusion $\chi^2$ added) in the (updated) \mhmax\ benchmark scenario of the MSSM~\cite{Carena:2013qia}. The patterned areas indicate $95\%~\mathrm{C.L.}$ excluded parameter regions from the following LHC Higgs searches, as obtained by \HB: CMS $h/H/A \to\tau\tau$ search~\cite{Chatrchyan:2012vp} (orange, checkered), ATLAS search for a light charged Higgs boson in top decays, $t\to H^+ b \to\tau^+\nu_\tau b$~\cite{Aad:2012tj} (green, coarsely striped), CMS combination of SM Higgs searches~\cite{CMS-PAS-HIG-12-045} (wine-red, striped). The $95\%~\mathrm{C.L.}$ LEP excluded region~\cite{Barate:2003sz,Schael:2006cr}, corresponding to $\chi^2_\mathrm{LEP,HB} =4.0$, is below the black solid line. The best-fit point (indicated by a green star) is found at $(\MA, \tan\beta)=(390\gev, 5.3)$ with $\chi^2/\mathrm{ndf} =33.0/48$. We furthermore include the $68\%$ and $95\%$ C.L. preferred regions ((based on the 2D $\Delta\chi^2$ probability w.r.t. the best fit point) as solid and dashed gray lines, respectively.}}
\label{Fig:mhmax}
\end{figure}
The first scenario is an updated version of the \htr{well-known
\mhmax\ benchmark} scenario~\cite{Carena:2013qia,Carena:1999xa}, where
the masses of the gluino and the squarks of the first and second
generation were set to higher values \htr{in view of the latest} bounds from
\htb{SUSY searches at} the LHC, see~\cite{Carena:2013qia} for details. The results are shown in
Fig.~\ref{Fig:mhmax} in the ($\MA$, $\tb$) plane. Besides the colors
indicating the \htb{$\De\chi^2=\chi^2-\chi^2_\mathrm{best-fit}$}\htr{distribution relative to the best-fit
point (shown as a green star)} we also show \htr{the parameter regions
that are excluded \htb{at $95\%~\mathrm{C.L.}$} by LHC searches \htb{for a light charged Higgs boson (dark-green, coarsely striped)~\cite{Aad:2012tj}, neutral Higgs boson(s) in the $\tau\tau$ final state (orange, checkered)~\cite{Chatrchyan:2012vp}} and the combination of SM search channels (wine-red, finely striped)~\cite{CMS-PAS-HIG-12-045},
as obtained using \HB. \htb{As an indication for the parameter regions that are $95\%~\mathrm{C.L.}$ excluded by neutral Higgs searches at LEP~\cite{Barate:2003sz,Schael:2006cr} we include a corresponding contour (black, dotted) for the value $\chi^2_\mathrm{LEP,HB} =4.0$. Conversely, the parameter regions favored by the fit are shown as $68\%$ and $95\%$ C.L. regions (based on the 2D $\Delta\chi^2$ probability w.r.t. the best fit point) by the solid and dashed gray lines, respectively.}
%
%{\bf [GW: Which HB excluded regions are actually shown? Mention colour coding of the HB excluded regions in the text and in the figure caption! We need to discuss / explain the treatment of the LEP exclusion bounds! In particular, we need to explain why no LEP exclusion is shown in the low $\tan\beta$ region (see also below). Discuss ``double-branch structure of the 68\% C.L.\ region!]}
%
}
As can be seen in the figure, the best fit regions are obtained in a
strip at relatively small values of \htb{$\tb\approx4.5-7$}, where in this scenario
$\Mh\sim125.5\gev$ is found. At larger $\tb$ values the light Higgs
mass \htr{in this benchmark scenario (which was designed to
maximise $\Mh$ for a given $\tb$ in the region of large $\MA$)
turns out to be {\em higher\/} than the measured mass of the observed
signal, resulting in} a corresponding $\chi^2$ penalty. At
very low $\tb$ values the light Higgs mass is found to be \htr{below the
preferred mass region, again
resulting in} a $\chi^2$ penalty. \htb{Here, the $\chi^2$ steeply rises (for $m_h \lesssim122\gev$), because the mass-sensitive observables ($H\to\gamma\gamma, ZZ^{(*)}$) cannot be explained by the light Higgs boson anymore, \cf Section~\ref{Sect:pc_performance}.}
The values of $\MA$ that are allowed in
this scenario, in particular where $\Mh\sim125.5\gev$, are larger than
$200\gev$,
%
\htr{
{\bf [GW: This statement is not strictly correct, since we do not
show a LEP-excluded region on the plot. Smaller values of $\MA$ are
``allowed'', although they have a high $\Delta\chi^2$. One possibility
would be to show the region that is excluded by LEP at the 95\% C.L.\
level although we use the $\chi^2$ information from LEP in the fit.
Otherwise we should rephrase the above statement.]}}
%
and thus the light Higgs has mainly SM-like
couplings. Consequently, the $\chi^2$ contribution from the rate
measurements is similar to the one for a SM Higgs boson. \htb{In this regime, the Higgs mass dependence of the total $\chi^2$ (from \HS) is comparable to the results shown in Fig.~\ref{Fig:SM}(d).}
\htbd{Overall} We
find the best fit point
\htr{at $(M_A, \tanb)=(390\gev, 5.3)$ with $\chi^2/\mathrm{ndf} =33.0/48$}.
\htbd{yielding $p =0.ZZ$.}\htb{The (naively counted) number of degrees of freedom (ndf) are comprised of the $45$ signal strength and $4$ mass measurements, presented in Fig.~\ref{Fig:peakobservables}, as well as one LEP exclusion observable from \HB. Note, that due to the correlations among the measurements, the actual ndf is (slightly) reduced.}
\caption{\htb{$\Delta\chi^2$ distribution (\HS\ and \HB\ LEP exclusion $\chi^2$ added) in the $\mhmodp$ benchmark scenario of the MSSM~\cite{Carena:2013qia}. The contour lines correspond to the same as in Fig.~\ref{Fig:mhmax}, in particular, the same Higgs analyses are responsible for the $95\%~\mathrm{C.L.}$ excluded regions. The best-fit point (indicated by a green star) is found at $(\MA, \tan\beta)=(372\gev, 10.3)$ with $\chi^2/\mathrm{ndf} =33.4/48$.}}
\label{Fig:mhmod}
\end{figure}
The second scenario \htr{that we discuss here}
is a modification of the \mhmax\ scenario with a
lower value of $\Xt$, leading to $\Mh\sim125.5\gev$ over nearly the
Fig.~\ref{Fig:mhmod}\htb{(same colors and contours as for the \mhmax\ scenario, Fig.~\ref{Fig:mhmax})}.
%
%\htr{
%{\bf [GW: Change Fig.~13: the x-axis should be $\MA$ rather than $\mu$!
%See comments about HB excluded regions from above.]}
%}
%
The best fit point is
found \htb{at $(\MA,\tb)=(372\gev, 10.3)$ with $\chi^2=33.4/48$},
\marginpar{\htr{\small put value!}\\\htb{TS: I think\\ we should\\ not give\\ p-values\\ here.}}
\htbd{yielding
$p =0.ZZ$.} Only slightly larger $\chi^2$ values are found over the rest
of the plane, except for the lowest $\tb$ values, where $\Mh$
\htr{is found to be below the preferred mass region.}
%
\htr{
{\bf [GW: A separate discussion is required if the region of
small $\MA$ is regarded as unexcluded, see above.]}
}
%
As in the \mhmax\ scenario the light Higgs behaves mostly
SM-like, and the $\chi^2$ from the rates is close to the one found in the
\mhmax\ scenario.
%
\htr{
{\bf [GW: also this statement is only correct if the region of
small $\MA$ is regarded as excluded.]}
}
%
\begin{figure}[t]
\centering
%\includegraphics[width=0.8\textwidth]{lowmH}
\includegraphics[width=0.9\textwidth]{mHlow_HSHB}
\caption{\htb{$\Delta\chi^2$ distribution (\HS\ and \HB\ LEP exclusion $\chi^2$ added) in the $\mHlow$ benchmark scenario of the MSSM~\cite{Carena:2013qia}. The contour lines correspond to the same as in Fig.~\ref{Fig:mhmax}, except the wine-red, finely striped region, which gives the $95\%~\mathrm{C.L.}$ exclusion from the CMS Higgs search $H\to ZZ^{(*)}\to4\ell$~\cite{CMS-PAS-HIG-13-002}, applied to the SM-like heavy $\cp$ even Higgs boson. The best-fit point (indicated by a green star) is found at $(\mu, \tan\beta)=(3070\gev, 6.0)$ with $\chi^2/\mathrm{ndf} =47.4/48$.}}
\label{Fig:lowmH}
\end{figure}
\htr{As a final example, we \htbd{have} performed a fit in the $\mHlow$
benchmark scenario of the MSSM~\cite{Carena:2013qia}.
This} scenario is based on the assumption that the Higgs observed at
$\sim125.5\gev$ is the heavy $\cp$-even Higgs boson of the MSSM.
In this case the light $\cp$-even Higgs has a mass below the LEP limit
\htb{for a SM Higgs boson} of $114.4\gev$~\cite{Barate:2003sz}, but is effectively decoupled from
the SM gauge bosons. The \htr{other states of the Higgs spectrum are
also rather light, with masses around $\sim130\gev$, so that this
scenario offers good prospects for the searches for additional Higgs
bosons~\citeheavyH.}
Since $\MA$ must be relatively small in this case the ($\mu$, $\tb$)
plane is scanned~\cite{Carena:2013qia}, where only $\tb\lesssim10$ is
considered. The $\cp$-odd Higgs boson mass is fixed to $\MA=110\gev$.
Our results are shown in Fig.~\ref{Fig:lowmH}. \htb{The $95\%~\mathrm{C.L.}$ excluded regions are obtained from the same Higgs searches as in Fig.~\ref{Fig:mhmax}, except for the wine-red, finely patterned region, which results from applying the limit from the CMS SM Higgs search $H\to ZZ^{(*)}\to4\ell$~\cite{CMS-PAS-HIG-13-002} to the SM-like, heavy $\cp$-even Higgs boson.}
%
%\htr{
%{\bf [GW: Mention colour coding of the HB excluded regions in the text and in the figure caption!]}
%}
%
\htb{Two distinct best-fit regions are found~\cite{Carena:2013qia}: The parameter space with $\mu\sim1.8-2.0\tev$ and $\tb\sim4-5$ predicts a heavy $\cp$-even Higgs boson with a well compatible mass value $m_H \approx126\gev$ and SM-like couplings. However, large parts (at low $\tanb\lesssim4.9$) of this region favored by the rate and mass measurements are severely constrained by charged Higgs searches~\cite{Aad:2012tj}. The second region favored by the fit is located at large values of $\mu\sim2.6-3.2\tev$ and $\tb\sim6-7$. Here, the masses of the $\cp$-even Higgs bosons are generally lower. For instance, at the best-fit point at $(\mu,~\tb)\sim(3070\gev, 6.0)$, we have $m_h\approx76.1\gev$ and $m_H\approx122.8\gev$. For slightly larger (lower) values of $\mu$ ($\tb$) we find a steep edge in the \HS\ $\chi^2$ distribution, because the Higgs mass becomes too low to allow for an assignment of the SM-like heavy $\cp$-even Higgs boson to the mass-sensitive peak observables, cf.~Section~\ref{Sect:pc_performance}. Due to the low mass of the light $\cp$-even Higgs boson in this region, the LEP channel $e^+e^-\to h A$~\cite{Schael:2006cr} is kinematically accessible and contributes a non-negligible $\chi^2$ which increases with $\mu$.}
%The best fit regions are found at $(\mu,~\tb)\sim (2000 \gev, 4)$
%
%\htr{\bf [GW: Discuss impact of the HB excluded regions!]}
%
%and also, with a slightly higher $\chi^2$, in the region of the largest $\mu$ values shown in the plot, $\mu \gtrsim 3000 \gev$.The best fit point is found at $(\mu, \tanb) = (1.82\tev, 4.5)$ with $\chi^2 = 49.8$, yielding} \marginpar{\htr{\small put value!}} $p = 0.ZZ$.
%\htbd{It can be seen, however, that the best-fit regions only partially
%overlap with the regions still allowed by LEP and LHC Higgs searches,
%\marginpar{\htr{\small mention\\ LEP here?}}
%see \cite{Carena:2013qia} for details. The lowest $\chi^2$ found in the
%\marginpar{\htr{\small put values!}}
%allowed region is $\chi^2 = XY$ with $p = 0.ZZ$.}
\htr{
The \htb{parameter space} between the two preferred regions is excluded \htb{at $95\%~\mathrm{C.L.}$}\htbd{in particular} because it \htbd{would} give\htb{s} rise to too high rates \htb{in particular} for \htb{the}$H \to ZZ^{(*)}, WW^*$\htb{channels} as compared to the LHC measurements, thus leading
to a rather large $\chi^2$ penalty. }
\htb{At the best-fit point we find a $\chi^2/\mathrm{ndf} =47.4/48$. Compared with the light $\cp$-even Higgs interpretation of the observed signal, as discussed in the \mhmax\ and $\mhmodp$ scenarios, this fit is slightly worse, however, overall it is still very acceptable.}
\htbd{However, the $p$ values found for the
allowed points corresponding to the
interpretation of the heavy $\cp$-even Higgs as the new state
discovered at $\sim125.5\gev$ are only slightly worse than the ones
for the best fit region in the case where the observed signal is
interpreted as the light $\cp$-even Higgs boson of the MSSM.
}
%\begin{figure}[h]
%\centering
%\subfigure[SM $H\to \gamma\gamma$ searches (inclusive analyses and 2jet/VBF-tagged analyses) from ATLAS and CMS (4 total).]{
%\caption{$\chi^2$ distribution for a Higgs boson with mass $m_H = 125.7\gev$ and modified squared SM normalized effective Higgs couplings to electroweak vector bosons, $g^2_{HVV} \equiv g^2_{HWW}=g^2_{HZZ}$, and fermions, $g^2_{Hff}\equiv g^2_{Ht\bar{t}} = g^2_{Hb\bar{b}} =g^2_{H\tau\tau}$. In this parametrization, the gluon vertex loop is effectively a fermion loop and thus $g^2_{Hgg} = g^2_{Hff}$. Only the photon vertex loop has non-trivial scaling, including the contributions from top and bottom quarks, the $\tau$ lepton, the $W$ boson, as well as their (destructive) interference. The green asterisk is indicating the point with minimal $\chi^2$ in the two-dimensional plane. We show contour lines for $\Delta \chi^2 = 2.297$ ($1\sigma$ with $2$ ndf) and $\Delta \chi^2 = 5.99$ ($2\sigma$ with $2$ ndf) as gray dashed and finely dotted, respectively. The SM is shown as white triangle at $(1,1)$. The subfigures (a-d) show the results for different sets of Higgs searches. \ttext{Tim: This scenario corresponds to the benchmark scenario described in Section~4.2 in Ref.~\cite{LHCHiggsCrossSectionWorkingGroup:2012at}} (assuming no invisible or undetectable widths).}
The input that has to be provided by the user (\htb{and which is similar to the \HB\ input}) consists of the Higgs boson masses, \htr{preferably}
the corresponding theory uncertainties, the Higgs production cross sections and decay
branching ratios, where several levels of approximation are possible. In
case of the MSSM also the SLHA~\htr{\cite{Skands:2003cj,SLHA2}}
can be used as input/output format.
We presented in detail the two statistical methods provided by \HS: the
\textit{peak-centered $\chi^2$ method}, in which each observable is
defined by a Higgs signal rate measured at a specific hypothetical
Higgs mass, corresponding to a tentative Higgs signal. In the second,
the \textit{mass-centered $\chi^2$ method}, the $\chi^2$
is evaluated by comparing the signal rate measurement to the theory
prediction at the Higgs mass predicted by the model. It was described
how these two methods can be combined, as it is an option of \HS, to
yield the most reliable consistency test. \hto{In this combination, the
mass-centered $\chi^2$ method is applied only to those Higgs bosons which have not yet been tested with the peak-centered $\chi^2$ method against the same data.} Similarly, \htb{in order to include a more complete set of constraints on the Higgs sector,} it is recommended
to use \HS\ together with \HB\ to test the model under consideration
also against the existing Higgs exclusion bounds.
The installation, usage and subroutines of \HS\ were explained in
detail, together with the various input and output formats. It was
explained how the user can add new (hypothetical) experimental
data. Several pre-defined example codes were presented that permit the
user to get familiar with \HS\ and, by modifying the example
codes, analyze own models of interest. As an example, by linking
\HS\ to {\tt FeynHiggs}, the consistency of any MSSM parameter point with the
observed LHC signal can be analyzed in a simple way. \htb{Furthermore, some example codes demonstrate how to use \HB\ and \HS\ simultaneously in an efficient way.}
We have presented several examples of the use of \HS. As a first example
\htr{the combined best-fit signal strength has been determined. \htb{For the peak-centered $\chi^2$ method the mass dependence,} the impact of correlations
between the systematic uncertainties and the treatment of theoretical
uncertainties has been discussed \htb{in detail}. For the case of \htb{a} SM\htb{-like Higgs boson, we demonstrated how} the mass \htbd{of the Higgs boson has been}\htb{can be} determined from a fit to the \htbd{available Higgs sector observables}\htb{signal rate measurements as a function of the mass using the mass-centered $\chi^2$ method}. \htb{Moreover,} we employed this method for a combination of different search channels \htb{over the full investigated mass range}. \htb{Various} fits for coupling strength
modifiers have been carried out using the peak-centered $\chi^2$ method. \htb{Their} results have been compared for validation purposes with official results from the ATLAS and CMS
collaborations, and very good agreement has been found.
%the SM Higgs boson mass is determined in a fit to all available
%data. Next, various combined $\muobs$ plots (of the ATLAS collaboration) were
%reproduced, showing remarkable agreement with the officially presented
%results. Two-dimensional results (of the ATLAS collaboration) in the
%plane of the SM Higgs mass and the respective signal strengths in the
%$H \to \ga\ga, ZZ^{(*)}, WW^{(*)}$ channels, together with our
%\HS\ average were presented. While the $\ga\ga$ and the $WW^{(*)}$ channels are very
%well reproduced, in the $ZZ^{(*)}$ channel somewhat larger uncertainties
%are obtained in the Gaussian approximation due to the limited statistics. Various examples as presented by
%ATLAS and CMS on the fit to coupling strenth modifiers, \eg\ in the
%$\kappa_V$-$\kappa_F$ plane, were reproduced, again showing remarkable
%agreement with the official results.
It is expected that the agreement with the official results published by
ATLAS and CMS could be improved even further}
%The \htgd{impressive} \htg{very good} agreement with the official
%results could probably even be improved further
if relative signal
efficiencies of different production modes in all search channels
would be publicly provided by the experimental collaborations. The same
applies to a more complete description of the impact of individual
experimental systematic uncertainties and their correlations amongst
search channels. In particular, \htr{it would be useful if}
systematic uncertainties \htr{were}
given as a relative error on the quoted signal strength.
We would furthermore welcome the publication of the full $\muobs$ plot for every analysis to allow a $\chi^2$ test at various Higgs masses.
\htr{Going beyond just a validation of \HS\ results, we have also given
a few examples of \HS\ applications. In particular, we have performed
fits of Higgs coupling scaling factors including the full presently
available data from both the LHC and the Tevatron.
Furthermore we have investigated \htbd{recently proposed}
benchmark scenarios \htb{recently proposed} for the SUSY Higgs search at the LHC, where we have
taken into account both the limits obtained from the searches at LEP,
the Tevatron and the LHC, as well as the information about the observed
signal at about $126\gev$. The provided examples give only a first
glimpse of the capabilities of \HS. The applicability of \HS\ goes far
beyond those examples, and in particular it should be a useful tool for
taking into account Higgs sector information in global fits.}
%The provided examples give only a first glimpse of \HS, demonstrating
%how the fundamental search
%information can be used for all kinds of models. The applicability of
%\HS\ is far more general than these examples show. It \hto{could}, for
%example, be directly useful to global analyses and fits rather than just
%analyzing the
%two-dimensional plots of couplings versus masses (or similarly).
%\hto{
%\begin{itemize}
%\item Summary of the nice agreements in Section 5. Examples show that
% \HS~allows to apply the fundamental search information to all kinds of
% models. This is far more general and directly applicable to global
% analyses/fits than just looking at 2D plots of couplings vs masses,
% etc.
%\item Make case for collaborations to continue to publish the $\muobs$
% plots. This is the most general information, everything else can be
% constructed from it.
%\item Impressive agreement could probably be improved if relative signal
% efficiencies of different production modes in all search channels
% would be disclosed as well as a more complete description of the
% impact of individual experimental systematic uncertainties and their
% correlations amongst search channels.Systematic errors should
% preferably be given as a relative error on the quoted signal
\htb{We thank Oliver Brein and Karina Williams for their great contributions to the \HB\ project, which was the basis for the development of \HS. We thank the Fittino collaboration, in particular Sebastian Heer, Xavier Prudent, Bj\"orn Sarrazin and Mathias Uhlenbrock, for comments and suggestions on the code development. We are grateful for helpful discussions with Michael D\"uhrssen, Michael Kr\"amer, Stefan Liebler, Jana Schaarschmidt, Florian Staub and Lisa Zeune. T.S. would like to thank the Bonn-Cologne-Graduate-School for financial support and is grateful for the hospitality of the Stockholm University, where part of the concepts of \HS\ were developed. This work is supported by the Helmholtz Alliance ``Physics at the Terascale'' and the Collaborative Research Center SFB676 of the DFG, ``Particles, Strings, and the Early Universe''. \htg{The work of S.H.\ was supported in part by CICYT (grant FPA 2010--22163-C02-01) and by the Spanish MICINN's Consolider-Ingenio 2010 Program under grant MultiDark CSD2009-00064.}
%\section{Iterative assignment of Higgs bosons to signals}
%\label{Append:HtoPiterations}
%
%\htb{(Tim: I will take this appendix out after Sven's iteration.)}
%
%\htb{In the peak-centered $\chi^2$ method, the assignment of Higgs boson combinations to the signals follows a $\chi^2$ minimization procedure, as discussed in Section~\ref{Sect:pc_chisq}. The \textit{Higgs bosons-to-signals assignment} which yields the least total $\chi^2$ (and fulfills certain requirements, \cf~Section~\ref{Sect:pc_chisq}) should be considered for the final $\chi^2$ evaluation.
%
%For this tentative $\chi^2$ calculation, which has to be performed for every possible Higgs bosons-to-signals assignment, we provide different options for the treatment of correlated uncertainties among the signals. In the exact evaluation, every possible Higgs bosons-to-signals assignment leads to new covariance matrices, which have to be constructed.} However, due to the \htb{large number of possible assignments we encounter for an increased number of neutral Higgs bosons in the model and observables\footnote{To illustrate this problem, assume we had 2 peak observables and 2 Higgs bosons. Then, to every peak observable there are 4 possible Higgs combinations (none, $h_1$, $h_2$, $h_1h_2$) to be assigned. Thus, we would have to calculate the total $\chi^2$ for 16 possible Higgs-to-peaks assignments, each requiring the construction of its own covariance matrices. It is obvious that this number grows very fast with increasing number of Higgs bosons and/or peak observables.}}, we chose to follow an iterative method \htb{instead of constructing the correct covariance matrix for each Higgs bosons-to-signals assignment}. The user can control this method by setting the number of iterations via the subroutine
%\noindent before \HS~is run. In the first \htbd{iteration} step, the best \htb{Higgs bosons-to-signals assignment} is determined without the use of covariance matrices, and thus, correlations among the systematic uncertainties of the peak observables do not influence the choice of assigned Higgs bosons. After this iteration step, the covariance matrices are constructed \htb{as usual} on the basis of the determined \htb{Higgs bosons-to-signals assignment}. If no more iterations are required by the user, the total $\chi^2$ value is evaluated employing the constructed covariance matrices.
%
%In the \htb{first iterated} step \htb{(Tim: Strictly speaking, this is not really an iteration on the first evaluation.)}, the \htb{Higgs bosons-to-signals assignment} subroutine is run again, but now the tentative total $\chi^2$ of the assignments are evaluated in a different way: The \textit{mass} covariance matrices are constructed such that they contain \textit{maximal correlations}, \htb{which assumes} that every Higgs boson \htb{of the model} is assigned to every signal. Thus, every matrix element of the covariance matrix $\mathbf{C_{m_{i}}}$ contains the squared theoretical mass uncertainty \htb{of Higgs boson $h_i$}. The tentative total $\chi^2$ of the possible \htb{Higgs bosons-to-signals assignments} are now evaluated with these covariance matrices.
%\caption{Comparison of different iteration steps in the Higgs-to-peak assignment routine: We show the total $\chi^2$ value for different Higgs masses $m_H$ using the dataset "latestresults" assuming SM signal strengths.}
%\label{Fig:mh_cutoff}
%\end{figure}
%
%
%
%Due to this construction, this iteration step tries to assign as many Higgs bosons as possible to the peak observables, \htb{since the mass covariance matrices, containing maximally correlated uncertainties, decrease the influence of Higgs masses on the total $\chi^2$}.
%
%In the second (and all successive) iteration step(s), the Higgs mass covariance matrices used \htb{for the tentative total $\chi^2$ evaluation} are based on the \htb{Higgs bosons-to-signals assignment} from the previous iteration step.
%
%Due to possibly changing \htb{Higgs bosons-to-signals assignments} and thus different covariance matrices for different model parameter values (\eg the SM Higgs boson mass $\MH$), there might be discontinuities in the $\chi^2$ distribution\htb{, as already discussed in Section~\ref{sect:Examples}.} In rare cases, a (non-trivial) Higgs boson combination is assigned to a signal $\alpha$ even if \htb{it} is allowed \htb{to have \textit{no} Higgs boson assigned \textit{and} this would} yield a lower $\chi^2$ value, \htb{denoted by} $\chi_0^2$. In this \htb{(and only this)} case, we cut the total $\chi^2$ contribution, \htb{$\chi^2_{\mu,\alpha} + \sum_i \chi^2_{m_i,\alpha}$, from this observable $\alpha$} at the value $\chi_0^2$, which is determined solely from the signal strength $\chi^2$ contribution by setting the predicted signal strength to zero ($\mu=0$). Note that this cut is \textit{not} applied, if the \textit{no-Higgs bosons-combination} is not allowed \htb{to be assigned to the signal}, \ie if Eq.~\eqref{Eq:massoverlap} is fulfilled.
%
%\htb{We now discuss the impact of the iterations in the Higgs bosons-to-signals assignment on the example discussed in Section~\ref{sect:Examples} in Fig.~\ref{Fig:SM}. We show the $\chi^2$ distribution of the for a SM Higgs boson with mass $m_H$ and predicted signal strength $\mu=1$ in Fig.~\ref{Fig:mh_cutoff} for no, one, two and three iterations. The Higgs mass theoretical uncertainty is set to $2\gev$.} We use the peak observables shown in Fig.~\ref{Fig:peakobservables}. Note, that Fig.~\ref{Fig:mh_cutoff}(a) is identical to Fig.~\ref{Fig:SM}(d).
%
%\htb{(Tim: The interpretation of these figures is a bit unclear to me. We would expect that with more iterations, the $\chi^2$ distribution for the Gaussian can only become shallower, but this seems not necessarily to be the case. After the first iteration, every other iteration can only decrease the number of assigned Higgs bosons. This does not necessarily lead to a better $\chi^2$.)}
%
%\htbd{In Fig.}~\ref{Fig:mh_cutoff} \htbd{we observe some spikes in the gaussian and box-gaussian distributions at Higgs mass values $m_H \approx 118-121\gev$ and $m_H \approx 132-135\gev$. Here, the total $\chi^2$ value even exceeds the asymptotic $\chi^2$ value obtained if no Higgs bosons are assigned to any of the peak observables. In fact, these spikes are artifacts from the Higgs-to-peaks-assignment method because (\textit{i}) the Higgs boson mass $m_H$ lies outside the overlap region defined by Eq.}~\eqref{Eq:massoverlap} \htbd{and (\textit{ii}) the $\chi^2$ contribution from the respective peak observable and its assigned, non-trivial Higgs combination is in fact \textit{not} the least $\chi^2$ value.}
%
%Thus, if we wanted to take into account correlations to find the best (global) Higgs-to-peaks-assignment, we would have to go through \textit{every possible Higgs-to-peaks assignment (for all peaks)} and then \textit{find the least total $\chi^2$ (from all peaks)}. This would be computationally expensive (but maybe necessary, as you can see below).
%Now, the following may happen: Two different Higgs combinations may have very similar $\chi^2$ (\textit{without correlations}) contributions. Thus, slight changes of the input parameters may lead to a change of the Higgs-to-peaks assignment. However, the true $\chi^2$ contribution (\textit{with correlations}) might be quite different. Thus, in total $\chi^2$ returned by \HS~might have a jump at this transition of Higgs-to-peak assignments.
%An example is shown in Fig.~\ref{Fig:heavyH}. Here, all three neutral Higgs bosons of the model are possibly assigned to peaks, so there are many transitions in this plane. Comparing the plot without correlations (a) with the one with correlations (b), we see that there two structures (around $(m_A,\tan\beta)\approx (115,3)~\mbox{and}~(140,3)$) in (b) where the total-$\chi^2$ makes a jump to higher values. The total $\chi^2$ without correlations shown in (a) has a much smoother shape. Note, that the Higgs-to-peak assignment for every parameter point is identical for both (a) and (b).
%\caption{Comparison of different implementations and settings of the Higgs-to-peak assignment routine: We show the total $\chi^2$ value for different Higgs masses $m_H$ obtained from the four $H\to\gamma\gamma$ peak observables for a predicted signal strength $\mu=1$. On the left (a, c, e, g), no cut-off at $\chi_0^2$ is employed (see text), whereas it is used on the right-hand side plots (b, d, f, h). Furthermore, different iteration steps ($0,~1,~2,~3$) are used. }
%\label{Fig:mh_cutoff}
%\end{figure}
%\section{A toy example with two neutral Higgs bosons}
%To illustrate the procedure of evaluating the peak-centered $\chi^2$ and in particular the construction of the covariance matrices we go through a toy example. Assume \HS~has identified the following peak observables:
%\begin{center}
%\begin{tabular}{c|c|c}
%peak number & Collaboration & Included Higgs channels \\
%Each peak has a measured value for the signal strength modifier, $\hat\mu^i$, and the Higgs mass, $\hat m^i$, where $i=1,\dots,4$ runs over the peak numbers. The uncertainties of these measurements are denoted by $\Delta\hat\mu^i$ and $\Delta \hat m^i$ ($\hat{=}$ mass resolution of the analysis), respectively.
%
%Peak 1 and peak 3 have correlated uncertainties in the production (VBF) and decay rate ($H\to\gamma\gamma$) and peak 2 and peak 4 are correlated in the $ZH$ production rate uncertainty. Furthermore, peak 1 and 2 (peak 3 and 4) have a correlated luminosity uncertainty since they are both from the ATLAS (CMS) experiment.
%
%Now, lets have a model with two neutral Higgs bosons with predicted masses $m_1$ and $m_2$ in the relevant mass region. Each Higgs boson has a predicted signal strength modifier for each peak, denoted by $\mu_a^i$, where $a=1,2$ enumerates the two Higgses and $i=1,\dots,4$ the peaks.
%
%After running the Higgs-to-peaks assignment described in Section~\ref{Sect:Higgstopeaks}, we have the following assignment:
%\begin{center}
%\begin{tabular}{c|cccc}
%peak number ($i$): & 1 & 2 & 3 & 4 \\
%\hline
%Higgs bosons ($a$):& 1 & 1+2 & 1 & 2 \\
%\end{tabular}
%\end{center}
%
%Now we can construct the covariance matrices and thus evaluate the $\chi^2$. Lets assume the production and decay rate uncertainties of the model to be equal to those in the SM. Then, for the contribution from the signal strength, we have
%Here, the notation of the weights is $\omega_\alpha^i$, with $\alpha$ indexing the channels investigated in the analysis of peak $i$. For example, $\omega_2^1$ means the weight of the ($\mathrm{VBF} \times H\to \gamma\gamma$) channel of peak 1. Note, that $\omega_1^2=\omega_1^3=1$, since these analyses investigate only one channel. Furthermore note, that the weight is the only model-dependent quantity in the covariance matrix $\mathbf{C}_\mu$ because it depends on the Higgs mass. If more than one Higgs boson is assigned to the peak, the weight is evaluated at the mass of the Higgs boson which contributes dominantly to the signal strength modifier. (\ttext{Tim: Alternatively, we could evaluate the weight at the peak mass, this is however programming-wise not so nice...)}
%
%If the production and decay rate uncertainties of the model and the SM are different, the diagonal elements are modified by first subtracting the signal rate uncertainties of the SM according to Eq.~\eqref{Eq:intrinsic_dmusq} and then adding the model's signal rate uncertainties as described in Section~\ref{Sect:chisq_mu_cov}.
%
%Now, let's turn to the $\chi^2$ contribution from the Higgs masses. We demonstrate here the use of the gaussian Higgs mass pdf and denote the theoretical uncertainties on the Higgs masses by $\Delta m_1^\mathrm{th}$ and $\Delta m_2^\mathrm{th}$, respectively. For the first Higgs boson we get
%The total $\chi^2$ can now be obtained from Eq.~\eqref{Eq:chisqtot}, where the maximal $\chi^2$ contributions of each peak, $(\chi_{\mu,\alpha}^\mathrm{max})^2$, are obtained by evaluating the $\chi^2$ contribution from the signal strength according to Eq.~\eqref{Eq:example_vmu} and Eq.~\eqref{Eq:example_Cmu}, however, setting all predicted signal strengths modifiers to zero.
\section{Theory mass uncertainties in the mass-centered $\chi^2$ method}
\label{App:MCuncertainties}
In order to illustrate the two possible treatments of theoretical mass uncertainties in the mass-centered $\chi^2$ method we first discuss a constructed toy example (Example 1). Then we show how a typical $\muobs$ plot changes if it is convolved with a Higgs mass pdf, which parametrizes the theoretical mass uncertainty (Example 2).
\subsection*{Example 1: Variation of the predicted Higgs mass}
We look at a simple toy model with three neutral Higgs bosons $h_i$ ($i=1,2,3$) with masses $m_1=125\gev$, $m_2=135\gev$, $m_3=140\gev$. For every Higgs boson the theoretical mass uncertainty is set to $2\gev$. We test this model \htb{using the experimental data from}\htbd{against} the four $\muobs$ plots of the ATLAS searches for $H\to\gamma\gamma$~\cite{ATLAS-CONF-2012-091} ($7$ and $8\tev$ separately), $H\to ZZ^{(*)}\to4\ell$~\cite{ATLAS-CONF-2012-092} and $H\to WW^{(*)}\to\ell\nu\ell\nu$~\cite{ATLAS-CONF-2012-098} (both $7+8\tev$ combination). The predicted signal strength modifiers are set for every analysis to $\mu_1=1.0$, $\mu_2=0.5$ and $\mu_3=0.2$ for the three neutral Higgs bosons, respectively. Note that the experimental mass resolution of the $H\to WW$ search is estimated to $8\gev$, while the $H\to ZZ$ and $H\to\gamma\gamma$ searches have a lower experimental mass uncertainty of $\lesssim2\gev$. All $\muobs$ plots include the mass region between $120\gev$ and $150\gev$, thus all three Higgs bosons can be tested with all four analyses.
In the first step of the mass-centered $\chi^2$ method, \HS~constructs possible Higgs boson combinations following the \textit{Stockholm clustering scheme}. In our example, $h_2$ and $h_3$ are combined in a Higgs cluster, denoted by $h_{23}$, for the $H\to WW$ analysis since their mass difference is lower than the experimental mass resolution. In all other cases, the Higgs bosons are tested singly, thus we have in total 11 observables. The mass and its uncertainty associated with the Higgs cluster $h_{23}$ are derived from Eq.~\eqref{Eq:mc_cluster_m} and~\eqref{Eq:mc_cluster_dm} to $m_{23} =137.5\gev$ and $\Delta m_{23} =1.4\gev$. Its predicted signal strength is $\mu_{23} =0.7$.
\begin{figure}
\subfigure[Box-shaped parametrization of the theory mass uncertainties. The light gray striped regions show the scanned mass regions $M_i$ of the three Higgs bosons, whereas the darker gray striped region corresponds to $M_k$ of the Higgs boson cluster $k$.\label{Fig:mc_box}]{\includegraphics[width=0.48\textwidth]{dmth_variation_box}}\hfill
\subfigure[Gaussian parametrization of the theory mass uncertainties. The light gray striped regions now indicate the $\chi^2$ contribution to the tentative total $\chi_i^2$ from the Higgs mass, \cf \refeq{Eq:chisq_dmth_variation_gaussian}.\label{Fig:mc_gauss}]{\includegraphics[width=0.48\textwidth]{dmth_variation_gaussian}}
\caption{Illustration of the treatment of the theoretical mass uncertainties by variation of the predicted Higgs boson masses (\textit{first option}) for the toy model and observables discussed (see text).
%\htbd{We use a toy model with 3 neutral Higgs bosons $h_i$ ($i=1,2,3$) with masses $m_1 = 125\gev$, $m_2 = 135\gev$, $m_3 = 140\gev$ and theoretical mass uncertainties of 2 GeV. The signal strength modifier for all analyses are set to $\mu_1 = 1.0$, $\mu_2 = 0.5$ and $\mu_3 = 0.2$ for the three neutral Higgs bosons, respectively. We consider the following four ATLAS analyses ($\muobs$-plots): $H\to WW$, $H\to ZZ$ (both $7+8\tev$ combination) and $H\to \gamma \gamma$ ($7$ and $8\tev$ separately).}
For the $H\to WW$ analysis, $h_2$ and $h_3$ are combined
%\htbd{due to the low experimental mass resolution, $\dmexp = 8\gev$, and thus define}
in a Higgs cluster $h_{23}$ with $m_{23}=137.5\gev$ and $\dmth_{23} =1.4\gev$. We show the tentative total $\chi_i^2(m')$ distributions for each Higgs boson $h_i$ for the box-shaped [Fig.~A\ref{Fig:mc_box}] and gaussian parametrization [Fig.~A\ref{Fig:mc_gauss}].}
\label{Fig:mcmethod_dmth_variation}
\end{figure}
In the second step, the observed quantities $\muobs_\alpha$ and $\dmuobs_\alpha$ have to be determined from the $\muobs$ plots for each observable $\alpha$. In order to take into account the theoretical mass uncertainties, the relevant mass region is scanned \htb{to construct}\htbd{in order to evaluate} the tentative total $\chi_i^2(m')$ distribution for each Higgs boson $h_i$, as described in Section~\ref{Sect:mc_chisq}. For this example, the $\chi_i^2(m')$ distributions for the box-shaped and Gaussian parametrization of the theoretical mass uncertainty are shown in Fig.~A\ref{Fig:mc_box} and Fig.~A\ref{Fig:mc_gauss}, respectively. At the mass position $\mobs_i$, where $\chi^2_i (m')$ is minimal, the observed quantities $\muobs_\alpha$ and $\dmuobs_\alpha$ are extracted from the $\muobs$ plots for those observables $\alpha$, which test the Higgs boson $i$.
In the box-shaped parametrization, the measured signal strengths of all \textit{mass-centered} observables which test $h_1$ are defined at $\mobs_1=124.7\gev$, where $\chi^2_1$ is minimal. In contrast, the Higgs bosons $h_2$ and $h_3$ form the Higgs cluster $h_{23}$ in the $H\to WW$ analysis, therefore their allowed mass variations are restricted to the overlap regions $M_2\cap M_{23}$ and $M_3\cap M_{23}$, \cf Eq.~\eqref{Eq:massrange}, respectively. In those observables, where $h_2$ ($h_3$) is tested singly, the measured quantities are defined at $\mobs=136.1\gev~(138.9\gev)$. For the observable testing the Higgs cluster $h_{23}$ the observable is defined by the minimum of the joint $\chi^2$ distribution, which is located at $\mobs=138.9\gev$.
In the Gaussian parametrization the mass variation is less restricted. In contrast to the box-shaped parametrization, each mass variation is allowed over the full available mass range of the analyses, however, the additional contribution of the Higgs mass to the tentative $\chi^2$, \cf~Eq.~\eqref{Eq:chisq_dmth_variation_gaussian}, tries to keep the varied mass close the its original predicted value. From the minimum of each tentative $\chi^2$ distribution, the observed quantities of analyses, which test either $h_1$, $h_2$ or $h_3$ singly, are defined at $\mobs=124.8\gev,~133.2\gev$ and $140.3\gev$, respectively. For the Higgs cluster $h_{23}$ the position $\mobs=140.3\gev$ is chosen.
%In the case of the observable for the $H\to WW$ analysis, which tests the combination of $h_2$ and $h_3$, the position of the minimum of the joint $\chi^2$ distribution of $\chi_2^2$ and $\chi_3^2$ is chosen, \ie $\mobs = ~138.0~(140.2)\gev$ is selected in the box-shaped (gaussian) parametrization.
\subsection*{Example 2: Smearing of the $\muobs$-plot with $\dmth$}
We want to illustrate how the experimental data changes, if we choose to fold the theoretical Higgs mass uncertainty, $\dmth$, into the original $\muobs$ plot, as discussed in Section~\ref{Sect:mc_chisq}. For this, we look at the $\muobs$ plot published by ATLAS for the $H\to ZZ^{(*)}\to4\ell$ search~\cite{ATLAS-CONF-2012-092} and convolve it with a uniform (box) or Gaussian Higgs mass pdf, centered at $m_H$, for various theoretical mass uncertainties $\dmth=(0,~2,~5)\gev$, following Eq.~\eqref{Eq:muobssm} and \eqref{Eq:dmuobssm}. This is done over the full mass range, $m_H\in[112,~160]\gev$, to obtain the results shown in Fig.~\ref{Fig:mcmethod_smearing}. For $\dmth=0\gev$, the $\muobs$ plot is unchanged, whereas for increasing $\dmth$ it becomes smoother and fluctuations tend to vanish. This happens faster for the Gaussian pdf.
\begin{figure}
\centering
\subfigure[Original $\mu$-plot (from \cite{ATLAS-CONF-2012-092}) after the convolution with zero mass theory uncertainty.]{\includegraphics[width=5.5cm, angle=270]{mu_smeared_dm0}}\hfill
%\subfigure[$\chi^2$ map for zero mass theory uncertainty.]{\includegraphics[width=5.5cm, angle=270]{2D_dm0}}
\subfigure[$\mu$-plot after the convolution with a box-shaped mass pdf with $\dmth= 2\gev$.]{\includegraphics[width=5.5cm, angle=270]{mu_smeared_dm2_box}}\hfill
%\subfigure[$\chi^2$ map for a box-shaped mass pdf with $\dmth=2\gev$.]{\includegraphics[width=5.5cm, angle=270]{2D_dm2_box}}
\subfigure[$\mu$-plot after the convolution with a Gaussian mass pdf with $\dmth = 2\gev$.]{\includegraphics[width=5.5cm, angle=270]{mu_smeared_dm2}}\hfill
%\subfigure[$\chi^2$ map for a gaussian mass pdf with $\dmth = 2\gev$.]{\includegraphics[width=5.5cm, angle=270]{2D_dm2}}
\subfigure[$\mu$-plot after the convolution with a box-shaped mass pdf with $\dmth= 5\gev$.]{\includegraphics[width=5.5cm, angle=270]{mu_smeared_dm5_box}}\hfill
\subfigure[$\mu$-plot after the convolution with a Gaussian mass pdf with $\dmth = 5\gev$.]{\includegraphics[width=5.5cm, angle=270]{mu_smeared_dm5}}\hfill
\caption{Plots for the ATLAS $H\to ZZ$ analysis~\cite{ATLAS-CONF-2012-092} after convolution with the Higgs mass pdf for $\dmth=0\gev$ (a), $\dmth=2\gev$ (b),(c), and $\dmth=5\gev$ (d),(e), respectively. In (b) and (d) a uniform (box) pdf is used for the theoretical Higgs mass uncertainty, whereas a Gaussian parametrization was used in (c) and (e).% In the left panel we show the smeared $\muobs$-plot, $\muobssm(m)$, obtained for a mass pdf centered at $m$. In the right panel, we show the $\chi^2$ likelihood map in the ($m,~\mu$) plane, including contour lines for $1\sigma$ and $2\sigma$ regions.
}
\label{Fig:mcmethod_smearing}
\end{figure}
%\ttext{TS: Oscar and I had some difficulties to understand whether this parametrization of the theoretical mass uncertainty makes sense. There could be cases where the $\chi^2$ becomes larger for a higher theoretical mass uncertainty. For instance, if the model predicts a Higgs which is located at a minimum or maximum of the $\muobs$-plot and fits nicely with the observed signal strength, $\mu \approx \muobs$. Now, smearing the $\muobs$-plot with the theory uncertainty will decrease $|\muobs|$ since we look at the extremum. Thus, the discrepancy between $\mu$ and $\muobssm$ becomes larger, leading to a larger $\chi^2$. This is the opposite of what you would expect for an increasing uncertainty.
%On the other hand, this method is not particularly designed for describing peaks. For this, we have the peak-centered $\chi^2$ method.}