claudioc/OSNote2010/limit.tex

\section{Limit on new physics}
\label{sec:limit}

%{\bf \color{red} The numbers in this Section need to be double checked.}

\subsection{Limit on number of events}
\label{sec:limnumevents}
As discussed in Section~\ref{sec:results}, we see one event 
in the signal region, defined as SumJetPt$>$300 GeV and 
\met/$\sqrt{\rm SumJetPt}>8.5$ GeV$^{\frac{1}{2}}$.

The background prediction from the SM Monte Carlo is 1.3 events.
%, where the uncertainty comes from 
%the jet energy scale (30\%, see Section~\ref{sec:systematics}),
%the luminosity (10\%), and the lepton/trigger 
%efficiency (10\%)\footnote{Other uncertainties associated with 
%the modeling of $t\bar{t}$ in MadGraph have not been evaluated.
%The uncertainty on $pp \to \sigma(t\bar{t})$ is also not included.}.
The data driven background predictions from the ABCD method 
and the $P_T(\ell\ell)$ method are $1.3 \pm 0.8({\rm stat}) \pm 0.3({\rm syst})$ 
and $2.1 \pm 2.1({\rm stat}) \pm 0.6({\rm syst})$, respectively.

These three predictions are in good agreement with each other
and with the observation of one event in the signal region.
We calculate a Bayesian 95\% CL upper limit\cite{ref:bayes.f} 
on the number of non SM events in the signal region to be 4.1.
We have also calculated this limit using 
% a profile likelihood method
% as implemented in 
the cl95cms software\cite{ref:cl95cms}, 
and we also find 4.1.  (This is not surprising, since cl95cms
also gives baysean upper limits with a flat prior).
These limits were calculated using a background prediction of $N_{BG} = 1.4 \pm 0.8$
events, the error-weighted average of the ABCD and $P_T(\ell\ell)$ background 
predictions.  The upper limit is not very sensitive to the choice of
$N_{BG}$ and its uncertainty.

To get a feeling for the sensitivity of this search to some
popular SUSY models, we remind the reader of the number of expected
LM0 and LM1 events from Table~\ref{tab:sigcont}: $8.6 \pm 1.6$ 
events and $3.6 \pm 0.5$ events respectively, where the uncertainties
are from energy scale (Section~\ref{sec:systematics}), luminosity,
and lepton efficiency.


\subsection{Outreach}
\label{sec:outreach}
Conveying additional useful information about the results of
a generic ``signature-based'' search such as the one described
in this note is a difficult issue.  
Here we attempt to present our result in the most general 
way.

Models of new physics in the dilepton final state 
can be confronted in an approximate way by simple 
generator-level studies that 
compare the expected number of events in 34.0~pb$^{-1}$
with our upper limit of 4.1 events.  The key ingredients
of such studies are the kinematical cuts described 
in this note, the lepton efficiencies, and the detector
responses for SumJetPt and \met/$\sqrt{\rm SumJetPt}$.
The muon identification efficiency is $\approx 95\%$;
the electron identification efficiency varies from $\approx$ 63\% at 
$P_T = 10$ GeV to 91\% for $P_T > 30$ GeV.  The isolation
efficiency in top events varies from $\approx 83\%$ (muons)
and $\approx 89\%$ (electrons) at $P_T=10$ GeV to 
$\approx 95\%$ for $P_T>60$ GeV. 
%{\bf \color{red} The following numbers were derived from Fall 10 samples. } 
The average detector 
responses for SumJetPt and $\met/\sqrt{\rm SumJetPt}$ are 
$1.02 \pm 0.05$ and $0.94 \pm 0.05$ respectively, where
the uncertainties are from the jet energy scale uncertainty.
The experimental resolutions on these quantities are 11\% and 
16\% respectively.

To justify the statements in the previous paragraph 
about the detector responses, we plot 
in Figure~\ref{fig:response} the average response for 
SumJetPt and \met/$\sqrt{\rm SumJetPt}$ in MC, as well as the
efficiency for the cuts on these quantities used in defining the
signal region.
% (SumJetPt $>$ 300 GeV and \met/$\sqrt{\rm SumJetPt} > 8.5$
% Gev$^{\frac{1}{2}}$).  
%{\bf \color{red} The following numbers were derived from Fall10 samples }
We find that the average SumJetPt response 
in the Monte Carlo is about 1.02, with an RMS of order 11\% while
the response of \met/$\sqrt{\rm SumJetPt}$ is approximately 0.94 with an
RMS of 16\%.

%Using this information as well as the kinematical
%cuts described in Section~\ref{sec:eventSel} and the lepton efficiencies
%of Figures~\ref{fig:effttbar}, one should be able to confront
%any existing or future model via a relatively simple generator 
%level study by comparing the expected number of events in 35 pb$^{-1}$
%with our upper limit of 4.1 events.

\begin{figure}[tbh]
\begin{center}
\includegraphics[width=\linewidth]{selectionEffDec10.png}
\caption{\label{fig:response} Left plots: the efficiencies
as a function of the true quantities for the SumJetPt (top) and
tcMET/$\sqrt{\rm SumJetPt}$ (bottom) requirements for the signal 
region as a function of their true values.  The value of the 
cuts is indicated by the vertical line.
Right plots: The average response and its RMS for the SumJetPt
(top) and tcMET/$\sqrt{\rm SumJetPt}$ (bottom) measurements.
The response is defined as the ratio of the reconstructed quantity
to the true quantity in MC.  These plots are done using the LM0
Monte Carlo, but they are not expected to depend strongly on 
the underlying physics.
%{\bf \color{red} These plots were made with Fall10 samples. } 
}
\end{center}
\end{figure}


%%%  Nominal
% ----------------------------------------- 
% observed events                         1
% relative error on acceptance        0.000
% expected background                 1.400
% absolute error on background        0.770
% desired confidence level             0.95
% integration upper limit             30.00
% integration step size              0.0100
% ----------------------------------------- 
% Are the above correct? y
%    1  16.685     0.29375E-06
%
% limit: less than     4.112 signal events 


%%%  Add 20% acceptance uncertainty based on LM0
% ----------------------------------------- 
% observed events                         1
% relative error on acceptance        0.200
% expected background                 1.400
% absolute error on background        0.770
% desired confidence level             0.95
% integration upper limit             30.00
% integration step size              0.0100
% ----------------------------------------- 
% Are the above correct? y
%    1  29.995     0.50457E-06
%
% limit: less than     4.689 signal events 


\subsection{mSUGRA scan}
\label{sec:mSUGRA}
We also perform a scan of the mSUGRA parameter space, as recomended
by the SUSY group convenors\cite{ref:scan}.
The goal of the scan is to determine an exclusion region in the 
$m_0$ vs. $m_{1/2}$ plane for
$\tan\beta=3$, 
sign of $\mu = +$, and $A_{0}=0$~GeV.  This scan is based on events
generated with FastSim.

The first order of business is to verify that results using
Fastsim and Fullsim are compatible.  To this end we compare the
expected yield for the LM1 point in FullSim (3.56 $\pm$ 0.06) and
FastSim (3.29 $\pm$ 0.27), where the uncertainties are statistical only.
These two numbers are in agreement, which gives us confidence in
using FastSim for this study.

The FastSim events are generated with different values of $m_0$
and $m_{1/2}$ in steps of 10 GeV.  For each point in the 
$m_0$ vs. $m_{1/2}$ plane, we compute the expected number of
events at NLO.  We then also calculate an upper limit $N_{UL}$
using cl95cms at each point using the following inputs:
\begin{itemize} 
\item Number of BG events = 1.40 $\pm$ 0.77
\item Luminosity uncertainty = 11\%
\item The acceptance uncertainty is calculated at each point
as the quadrature sum of
\begin{itemize}
\item The uncertainty due to JES for that point, as calculated 
using the method described in Section~\ref{sec:systematics}
\item A 5\% uncertainty due to lepton efficiencies
\item An uncertaity on the NLO cross-section obtained by varying the 
factorization and renormalization scale by a factor of two\cite{ref:sanjay}. 
\item The PDF uncertainty on the product of cross-section and acceptance
calculated using the method of Reference~\cite{ref:pdf}.
\end{itemize}
\item We use the ``log-normal'' model for the nuisance parameters
in cl95cms
\end{itemize}

An mSUGRA point is excluded if the resulting $N_{UL}$ is smaller
than the expected number of events.  Because of the quantization 
of the available MC points in the $m_0$ vs $m_{1/2}$ plane, the 
boundaries of the excluded region are also quantized.  We smooth
the boundaries using the method recommended by the SUSY 
group\cite{ref:smooth}.  In addition, we show a limit
curve based on the LO cross-section, as well as the 
``expected'' limit curve.  The expected limit curve is 
calculated using the CLA function also available in cl95cms.
Cross-section uncertainties due to variations of the factorization 
and renormalization scale are not included for the LO curve.
The results are shown in Figure~\ref{fig:msugra}


\begin{figure}[tbh]
\begin{center}
\includegraphics[width=\linewidth]{exclusion_noPDF.pdf}
\caption{\label{fig:msugra}\protect Exclusion curves in the mSUGRA parameter space, 
assuming $\tan\beta=3$, sign of $\mu = +$ and $A_{0}=0$~GeVs.  THIS IS STILL MISSING
THE PDF UNCERTAINTIES.  WE ALSO WANT TO IMPROVE THE SMOOTHING PROCEDURE.}
\end{center}
\end{figure}


\subsubsection{Check of the nuisance parameter models}
We repeat the procedure outlined above but changing the 
lognormal nuisance parameter model to a gaussian or 
gamma-function model.  The results are shown in 
Figure~\ref{fig:nuisance}.  (In this case,
to avoid smoothing artifacts, we 
show the raw results, without smoothing).

\begin{figure}[tbh]
\begin{center}
\includegraphics[width=0.5\linewidth]{nuissance.png}
\caption{\label{fig:nuisance}\protect Exclusion curves in the 
mSUGRA parameter space, 
assuming $\tan\beta=3$, sign of $\mu = +$ and $A_{0}=0$~GeVs
using different models for the nuisance parameters.
PDF UNCERTAINTIES ARE NOT INCLUDED.}
\end{center}
\end{figure}

We find that different assumptions on the PDFs for the nuisance
parameters make very small differences to the set of excluded
points.
Following the recommendation of Reference~\cite{ref:cousins},
we use the lognormal nuisance parameter model as the default.


\clearpage


\subsubsection{Effect of signal contamination}
\label{sec:contlimit}

Signal contamination could affect the limit by inflating the 
background expectation.  In our case we see no evidence of signal
contamination, within statistics.
The yields in the control regions  
$A$, $B$, and $C$ (Table~\ref{tab:datayield}) are just 
as expected in the SM, and the check 
of the $P_T(\ell \ell)$ method in the control region is
also consistent with expectations (Table~\ref{tab:victory}).
Since we have two data driven methods, with different 
signal contamination issues, giving consistent
results that are in agreement with the SM, we 
argue for not making any correction to our procedure
because of signal contamination.  In some sense this would 
be equivalent to using the SM background prediction, and using 
the data driven methods as confirmations of that prediction.

Nevertheless, here we explore the possible effect of 
signal contamination.  The procedure suggested to us
for the ABCD method is to modify the 
ABCD background prediction from $A_D \cdot C_D/B_D$ to
$(A_D-A_S) \cdot (C_D-C_S) / (B_D - B_S)$, where the 
subscripts $D$ and $S$ refer to the number of observed data
events and expected SUSY events, respectively, in a given region. 
We then recalculate $N_{UL}$ at each point using this modified
ABCD background estimation.  For simplicity we ignore
information from the $P_T(\ell \ell)$
background estimation.  This is conservative, since
the $P_T(\ell\ell)$ background estimation happens to
be numerically larger than the one from ABCD. 

Note, however, that in some cases this procedure is 
nonsensical.  For example, take LM0 as a SUSY 
point.  In region $C$ we have a SM prediction of 5.1
events and $C_D = 4$ in agreement with the Standard Model,
see Table~\ref{tab:datayield}.  From the LM0 Monte Carlo,
we find $C_S = 8.6$ events.   Thus, including information
on $C_D$ and $C_S$ should {\bf strengthen} the limit, since there
is clearly a deficit of events in the $C$ region in the 
LM0 hypothesis.  Instead, we now get a negative ABCD 
BG prediction (which is nonsense, so we set it to zero),
and therefore a weaker limit.


\begin{figure}[tbh]
\begin{center}
\includegraphics[width=0.5\linewidth]{sigcont.png}
\caption{\label{fig:sigcont}\protect Exclusion curves in the 
mSUGRA parameter space, 
assuming $\tan\beta=3$, sign of $\mu = +$ and $A_{0}=0$~GeVs
with and without the effects of signal contamination.
PDF UNCERTAINTIES ARE NOT INCLUDED.}  
\end{center}
\end{figure}

A comparison of the exclusion region with and without
signal contamination is shown in Figure~\ref{fig:sigcont}
(with no smoothing).  The effect of signal contamination is 
small, of the same order as the quantization of the scan.


\subsubsection{mSUGRA scans with different values of tan$\beta$}
\label{sec:tanbetascan}

For completeness, we also show the exclusion regions calculated
using $\tan\beta = 10$ (Figure~\ref{fig:msugratb10}).

\begin{figure}[tbh]
\begin{center}
\includegraphics[width=\linewidth]{exclusion_tanbeta10.pdf}
\caption{\label{fig:msugratb10}\protect Exclusion curves in the mSUGRA parameter space, 
assuming $\tan\beta=10$, sign of $\mu = +$ and $A_{0}=0$~GeVs.  THIS IS STILL MISSING
THE PDF UNCERTAINTIES.  WE ALSO WANT TO IMPROVE THE SMOOTHING PROCEDURE.}
\end{center}
\end{figure}


Revision:	1.34
Committed:	Fri Jan 14 00:02:59 2011 UTC (14 years, 3 months ago) by claudioc
Content type:	application/x-tex
Branch:	MAIN
Changes since 1.33:	+22 -24 lines
Log Message:	small fixes
#	User	Rev	Content
1	claudioc	1.1	\section{Limit on new physics}
2			\label{sec:limit}
3	claudioc	1.2
4	claudioc	1.10	%{\bf \color{red} The numbers in this Section need to be double checked.}
5	claudioc	1.2
6	claudioc	1.32	\subsection{Limit on number of events}
7			\label{sec:limnumevents}
8	claudioc	1.2	As discussed in Section~\ref{sec:results}, we see one event
9			in the signal region, defined as SumJetPt$>$300 GeV and
10			\met/$\sqrt{\rm SumJetPt}>8.5$ GeV$^{\frac{1}{2}}$.
11
12	benhoob	1.22	The background prediction from the SM Monte Carlo is 1.3 events.
13	benhoob	1.15	%, where the uncertainty comes from
14			%the jet energy scale (30\%, see Section~\ref{sec:systematics}),
15			%the luminosity (10\%), and the lepton/trigger
16			%efficiency (10\%)\footnote{Other uncertainties associated with
17			%the modeling of $t\bar{t}$ in MadGraph have not been evaluated.
18			%The uncertainty on $pp \to \sigma(t\bar{t})$ is also not included.}.
19	claudioc	1.2	The data driven background predictions from the ABCD method
20	benhoob	1.22	and the $P_T(\ell\ell)$ method are $1.3 \pm 0.8({\rm stat}) \pm 0.3({\rm syst})$
21			and $2.1 \pm 2.1({\rm stat}) \pm 0.6({\rm syst})$, respectively.
22	claudioc	1.2
23			These three predictions are in good agreement with each other
24			and with the observation of one event in the signal region.
25	benhoob	1.5	We calculate a Bayesian 95\% CL upper limit\cite{ref:bayes.f}
26	benhoob	1.17	on the number of non SM events in the signal region to be 4.1.
27	claudioc	1.32	We have also calculated this limit using
28			% a profile likelihood method
29			% as implemented in
30			the cl95cms software\cite{ref:cl95cms},
31			and we also find 4.1. (This is not surprising, since cl95cms
32			also gives baysean upper limits with a flat prior).
33	benhoob	1.22	These limits were calculated using a background prediction of $N_{BG} = 1.4 \pm 0.8$
34			events, the error-weighted average of the ABCD and $P_T(\ell\ell)$ background
35			predictions. The upper limit is not very sensitive to the choice of
36	claudioc	1.2	$N_{BG}$ and its uncertainty.
37
38			To get a feeling for the sensitivity of this search to some
39			popular SUSY models, we remind the reader of the number of expected
40	benhoob	1.22	LM0 and LM1 events from Table~\ref{tab:sigcont}: $8.6 \pm 1.6$
41			events and $3.6 \pm 0.5$ events respectively, where the uncertainties
42	claudioc	1.2	are from energy scale (Section~\ref{sec:systematics}), luminosity,
43	benhoob	1.23	and lepton efficiency.
44
45	claudioc	1.2
46	claudioc	1.32	\subsection{Outreach}
47			\label{sec:outreach}
48	claudioc	1.10	Conveying additional useful information about the results of
49			a generic ``signature-based'' search such as the one described
50	claudioc	1.32	in this note is a difficult issue.
51			Here we attempt to present our result in the most general
52			way.
53	claudioc	1.10
54	claudioc	1.32	Models of new physics in the dilepton final state
55	claudioc	1.10	can be confronted in an approximate way by simple
56			generator-level studies that
57	benhoob	1.23	compare the expected number of events in 34.0~pb$^{-1}$
58	claudioc	1.10	with our upper limit of 4.1 events. The key ingredients
59			of such studies are the kinematical cuts described
60			in this note, the lepton efficiencies, and the detector
61	benhoob	1.21	responses for SumJetPt and \met/$\sqrt{\rm SumJetPt}$.
62	claudioc	1.10	The muon identification efficiency is $\approx 95\%$;
63			the electron identification efficiency varies from $\approx$ 63\% at
64			$P_T = 10$ GeV to 91\% for $P_T > 30$ GeV. The isolation
65			efficiency in top events varies from $\approx 83\%$ (muons)
66			and $\approx 89\%$ (electrons) at $P_T=10$ GeV to
67	benhoob	1.30	$\approx 95\%$ for $P_T>60$ GeV.
68			%{\bf \color{red} The following numbers were derived from Fall 10 samples. }
69	dbarge	1.29	The average detector
70	claudioc	1.10	responses for SumJetPt and $\met/\sqrt{\rm SumJetPt}$ are
71	claudioc	1.31	$1.02 \pm 0.05$ and $0.94 \pm 0.05$ respectively, where
72	claudioc	1.10	the uncertainties are from the jet energy scale uncertainty.
73	dbarge	1.28	The experimental resolutions on these quantities are 11\% and
74			16\% respectively.
75	claudioc	1.10
76			To justify the statements in the previous paragraph
77			about the detector responses, we plot
78			in Figure~\ref{fig:response} the average response for
79	claudioc	1.8	SumJetPt and \met/$\sqrt{\rm SumJetPt}$ in MC, as well as the
80			efficiency for the cuts on these quantities used in defining the
81	claudioc	1.9	signal region.
82			% (SumJetPt $>$ 300 GeV and \met/$\sqrt{\rm SumJetPt} > 8.5$
83			% Gev$^{\frac{1}{2}}$).
84	benhoob	1.30	%{\bf \color{red} The following numbers were derived from Fall10 samples }
85	claudioc	1.9	We find that the average SumJetPt response
86	claudioc	1.31	in the Monte Carlo is about 1.02, with an RMS of order 11\% while
87			the response of \met/$\sqrt{\rm SumJetPt}$ is approximately 0.94 with an
88	dbarge	1.28	RMS of 16\%.
89	claudioc	1.8
90	claudioc	1.10	%Using this information as well as the kinematical
91			%cuts described in Section~\ref{sec:eventSel} and the lepton efficiencies
92			%of Figures~\ref{fig:effttbar}, one should be able to confront
93			%any existing or future model via a relatively simple generator
94			%level study by comparing the expected number of events in 35 pb$^{-1}$
95			%with our upper limit of 4.1 events.
96	claudioc	1.8
97			\begin{figure}[tbh]
98			\begin{center}
99	dbarge	1.25	\includegraphics[width=\linewidth]{selectionEffDec10.png}
100	claudioc	1.8	\caption{\label{fig:response} Left plots: the efficiencies
101			as a function of the true quantities for the SumJetPt (top) and
102			tcMET/$\sqrt{\rm SumJetPt}$ (bottom) requirements for the signal
103			region as a function of their true values. The value of the
104			cuts is indicated by the vertical line.
105			Right plots: The average response and its RMS for the SumJetPt
106			(top) and tcMET/$\sqrt{\rm SumJetPt}$ (bottom) measurements.
107			The response is defined as the ratio of the reconstructed quantity
108			to the true quantity in MC. These plots are done using the LM0
109			Monte Carlo, but they are not expected to depend strongly on
110	benhoob	1.20	the underlying physics.
111	benhoob	1.30	%{\bf \color{red} These plots were made with Fall10 samples. }
112			}
113	claudioc	1.8	\end{center}
114			\end{figure}
115	benhoob	1.22
116
117
118			%%% Nominal
119			% -----------------------------------------
120			% observed events 1
121			% relative error on acceptance 0.000
122			% expected background 1.400
123			% absolute error on background 0.770
124			% desired confidence level 0.95
125			% integration upper limit 30.00
126			% integration step size 0.0100
127			% -----------------------------------------
128			% Are the above correct? y
129			% 1 16.685 0.29375E-06
130			%
131			% limit: less than 4.112 signal events
132
133
134
135			%%% Add 20% acceptance uncertainty based on LM0
136			% -----------------------------------------
137			% observed events 1
138			% relative error on acceptance 0.200
139			% expected background 1.400
140			% absolute error on background 0.770
141			% desired confidence level 0.95
142			% integration upper limit 30.00
143			% integration step size 0.0100
144			% -----------------------------------------
145			% Are the above correct? y
146			% 1 29.995 0.50457E-06
147			%
148	dbarge	1.25	% limit: less than 4.689 signal events
149	claudioc	1.32
150
151			\subsection{mSUGRA scan}
152			\label{sec:mSUGRA}
153			We also perform a scan of the mSUGRA parameter space, as recomended
154			by the SUSY group convenors\cite{ref:scan}.
155			The goal of the scan is to determine an exclusion region in the
156			$m_0$ vs. $m_{1/2}$ plane for
157			$\tan\beta=3$,
158			sign of $\mu = +$, and $A_{0}=0$~GeV. This scan is based on events
159			generated with FastSim.
160
161			The first order of business is to verify that results using
162			Fastsim and Fullsim are compatible. To this end we compare the
163			expected yield for the LM1 point in FullSim (3.56 $\pm$ 0.06) and
164			FastSim (3.29 $\pm$ 0.27), where the uncertainties are statistical only.
165			These two numbers are in agreement, which gives us confidence in
166	claudioc	1.33	using FastSim for this study.
167	claudioc	1.32
168			The FastSim events are generated with different values of $m_0$
169			and $m_{1/2}$ in steps of 10 GeV. For each point in the
170			$m_0$ vs. $m_{1/2}$ plane, we compute the expected number of
171			events at NLO. We then also calculate an upper limit $N_{UL}$
172			using cl95cms at each point using the following inputs:
173			\begin{itemize}
174			\item Number of BG events = 1.40 $\pm$ 0.77
175			\item Luminosity uncertainty = 11\%
176			\item The acceptance uncertainty is calculated at each point
177			as the quadrature sum of
178			\begin{itemize}
179			\item The uncertainty due to JES for that point, as calculated
180			using the method described in Section~\ref{sec:systematics}
181			\item A 5\% uncertainty due to lepton efficiencies
182			\item An uncertaity on the NLO cross-section obtained by varying the
183			factorization and renormalization scale by a factor of two\cite{ref:sanjay}.
184			\item The PDF uncertainty on the product of cross-section and acceptance
185			calculated using the method of Reference~\cite{ref:pdf}.
186			\end{itemize}
187			\item We use the ``log-normal'' model for the nuisance parameters
188			in cl95cms
189			\end{itemize}
190
191			An mSUGRA point is excluded if the resulting $N_{UL}$ is smaller
192			than the expected number of events. Because of the quantization
193			of the available MC points in the $m_0$ vs $m_{1/2}$ plane, the
194			boundaries of the excluded region are also quantized. We smooth
195			the boundaries using the method recommended by the SUSY
196			group\cite{ref:smooth}. In addition, we show a limit
197			curve based on the LO cross-section, as well as the
198			``expected'' limit curve. The expected limit curve is
199			calculated using the CLA function also available in cl95cms.
200	claudioc	1.33	Cross-section uncertainties due to variations of the factorization
201			and renormalization scale are not included for the LO curve.
202	claudioc	1.32	The results are shown in Figure~\ref{fig:msugra}
203
204
205			\begin{figure}[tbh]
206			\begin{center}
207	claudioc	1.33	\includegraphics[width=\linewidth]{exclusion_noPDF.pdf}
208	claudioc	1.32	\caption{\label{fig:msugra}\protect Exclusion curves in the mSUGRA parameter space,
209			assuming $\tan\beta=3$, sign of $\mu = +$ and $A_{0}=0$~GeVs. THIS IS STILL MISSING
210	claudioc	1.33	THE PDF UNCERTAINTIES. WE ALSO WANT TO IMPROVE THE SMOOTHING PROCEDURE.}
211	claudioc	1.32	\end{center}
212			\end{figure}
213
214
215			\subsubsection{Check of the nuisance parameter models}
216			We repeat the procedure outlined above but changing the
217			lognormal nuisance parameter model to a gaussian or
218			gamma-function model. The results are shown in
219			Figure~\ref{fig:nuisance}. (In this case,
220			to avoid smoothing artifacts, we
221			show the raw results, without smoothing).
222
223			\begin{figure}[tbh]
224			\begin{center}
225			\includegraphics[width=0.5\linewidth]{nuissance.png}
226			\caption{\label{fig:nuisance}\protect Exclusion curves in the
227			mSUGRA parameter space,
228			assuming $\tan\beta=3$, sign of $\mu = +$ and $A_{0}=0$~GeVs
229			using different models for the nuisance parameters.
230	claudioc	1.34	PDF UNCERTAINTIES ARE NOT INCLUDED.}
231	claudioc	1.32	\end{center}
232			\end{figure}
233
234	claudioc	1.34	We find that different assumptions on the PDFs for the nuisance
235			parameters make very small differences to the set of excluded
236			points.
237			Following the recommendation of Reference~\cite{ref:cousins},
238			we use the lognormal nuisance parameter model as the default.
239	claudioc	1.32
240
241			\clearpage
242
243
244			\subsubsection{Effect of signal contamination}
245			\label{sec:contlimit}
246	claudioc	1.34
247	claudioc	1.32	Signal contamination could affect the limit by inflating the
248			background expectation. In our case we see no evidence of signal
249			contamination, within statistics.
250			The yields in the control regions
251			$A$, $B$, and $C$ (Table~\ref{tab:datayield}) are just
252			as expected in the SM, and the check
253			of the $P_T(\ell \ell)$ method in the control region is
254			also consistent with expectations (Table~\ref{tab:victory}).
255			Since we have two data driven methods, with different
256			signal contamination issues, giving consistent
257			results that are in agreement with the SM, we
258			argue for not making any correction to our procedure
259			because of signal contamination. In some sense this would
260			be equivalent to using the SM background prediction, and using
261			the data driven methods as confirmations of that prediction.
262
263			Nevertheless, here we explore the possible effect of
264			signal contamination. The procedure suggested to us
265	claudioc	1.34	for the ABCD method is to modify the
266	claudioc	1.32	ABCD background prediction from $A_D \cdot C_D/B_D$ to
267			$(A_D-A_S) \cdot (C_D-C_S) / (B_D - B_S)$, where the
268	claudioc	1.34	subscripts $D$ and $S$ refer to the number of observed data
269	claudioc	1.32	events and expected SUSY events, respectively, in a given region.
270	claudioc	1.34	We then recalculate $N_{UL}$ at each point using this modified
271			ABCD background estimation. For simplicity we ignore
272			information from the $P_T(\ell \ell)$
273			background estimation. This is conservative, since
274			the $P_T(\ell\ell)$ background estimation happens to
275			be numerically larger than the one from ABCD.
276	claudioc	1.32
277			Note, however, that in some cases this procedure is
278			nonsensical. For example, take LM0 as a SUSY
279			point. In region $C$ we have a SM prediction of 5.1
280			events and $C_D = 4$ in agreement with the Standard Model,
281			see Table~\ref{tab:datayield}. From the LM0 Monte Carlo,
282			we find $C_S = 8.6$ events. Thus, including information
283			on $C_D$ and $C_S$ should {\bf strengthen} the limit, since there
284			is clearly a deficit of events in the $C$ region in the
285			LM0 hypothesis. Instead, we now get a negative ABCD
286			BG prediction (which is nonsense, so we set it to zero),
287			and therefore a weaker limit.
288
289	claudioc	1.34
290
291
292	claudioc	1.32	\begin{figure}[tbh]
293			\begin{center}
294			\includegraphics[width=0.5\linewidth]{sigcont.png}
295			\caption{\label{fig:sigcont}\protect Exclusion curves in the
296			mSUGRA parameter space,
297			assuming $\tan\beta=3$, sign of $\mu = +$ and $A_{0}=0$~GeVs
298	claudioc	1.33	with and without the effects of signal contamination.
299			PDF UNCERTAINTIES ARE NOT INCLUDED.}
300	claudioc	1.32	\end{center}
301			\end{figure}
302
303	claudioc	1.34	A comparison of the exclusion region with and without
304			signal contamination is shown in Figure~\ref{fig:sigcont}
305	claudioc	1.32	(with no smoothing). The effect of signal contamination is
306	claudioc	1.34	small, of the same order as the quantization of the scan.
307
308	claudioc	1.32
309			\subsubsection{mSUGRA scans with different values of tan$\beta$}
310			\label{sec:tanbetascan}
311
312			For completeness, we also show the exclusion regions calculated
313	claudioc	1.33	using $\tan\beta = 10$ (Figure~\ref{fig:msugratb10}).
314
315			\begin{figure}[tbh]
316			\begin{center}
317			\includegraphics[width=\linewidth]{exclusion_tanbeta10.pdf}
318			\caption{\label{fig:msugratb10}\protect Exclusion curves in the mSUGRA parameter space,
319			assuming $\tan\beta=10$, sign of $\mu = +$ and $A_{0}=0$~GeVs. THIS IS STILL MISSING
320			THE PDF UNCERTAINTIES. WE ALSO WANT TO IMPROVE THE SMOOTHING PROCEDURE.}
321			\end{center}
322			\end{figure}
323	claudioc	1.32
324
325
326
327
328