\documentstyle[times,pramana,epsf,floats,feynmp]{ias}
\def\abinv{ab$\rm ^{-1}$}\def\W{\mbox{$\hbox{\rm W}$}}\def\SM{standard model} \def\h{\mbox{$ \hbox{\rm h}$}}\def\mh{\mbox{$m_{\mathrm h}$}}\def\bbbar{\mbox{${\mathrm {b\bar{b}}}$}}\def\epem{\mbox{$\mathrm{e^+ e^-}$}}\def\goes{\mbox{$\rightarrow$}}\def\WpWm{\mbox{$\rm \W^+\W^-$}}\def\Z{\mbox{$\hbox{\rm Z}$}}\def\nnbar{\mbox{$\nu\bar\nu$}}\def\ttbar{\mbox{${\mathrm {t\bar{t}}}$}}\def\infb{\mbox{$\hbox{fb}^{-1}$}}\def\e{\mbox{$ \hbox{\rm e}$}}\def\trilin{$\lambda_{\h\h\h}$}\def\Gcs{\hbox{GeV}/\mbox{$\mathrm{c^2}$}}\def\invfb{\mbox{$\hbox{fb}^{-1}$}}\def\sqrtofs{\mbox{$\rm \sqrt s $}}\def\invab{$\rm ab^{-1}$} \def\Evis{$E_{vis}$}\def\eflow{{\it eflow}}\def\hhh{hhh}\def\fom{$s/\sqrt{s+b}$}\def\effhhh{$\rm\epsilon_{\hhh}$}\def\signalhhh{$S_{\hhh}$}\def\Btot{$B_{tot}$}\def\dhhh{$\Delta\lambda/\lambda$}\def\lumisim{${\cal L}_{\rm sim} $} \def\lumi{$\cal L$}
 
\begin{document}
\mark{{Higgs self-coupling}{P. Gay}}
\title{Higgs self coupling measurement in e$^+$e$^-$ collisions at center-of-mass energy of 800~GeV}
\author{P.~Gay}
\address{Laboratoire de Physique Corpusculaire, Univ. B. Pascal/$ \rm IN^2P^3$-CNRS,24 Av. des Landais, F-63177 Aubi\`ere Cedex, France}
\keywords{Higgs, trilinear coupling, Linear Collider}
\pacs{2.0}
\abstract{Feasibility of the measurement of the trilinear self-coupling of 
the Higgs boson is studied at centre of mass energy of 800 GeV. Double-Higgs strahlung as well as \W\W\ fusion processes has been investigated with a realistic simulation of a typical detector. With the combinaison of the two processes, 
the self Higgs coupling may be measured with a 14~\% precision when an integrated luminosity of 2~\abinv\ would be collected at the Linear Collider. }

\maketitle
\section{Introduction}
\label{intro} 

In the framework of the standard model, the generation of mass occurs
through the Higgs mechanism. This mechanism assumes a Higgs potential which behaves as 
\mbox{{$\boldmath{\lambda}$}($\Phi^2$-$\frac{1}{2}${\it v}$^2$)$^2$}, 
where $\phi$ is an isodoublet scalar field, and {\it v}$\sim$246 GeV is the
vacuum expectation value of its neutral component.  Measurement
of the trilinear self-coupling $\lambda_{\h\h\h}$=$\frac{6}{\sqrt{2}}$
$\lambda${\it v} offers an independent determination of the Higgs
potential shape and a direct experimental test of the Higgs mechanism.  
A feasibility of the measurement of self Higgs coupling has already be performed 
at center-of-mass energy of 500~GeV. A new analysis has been performed at a higher 
centre-of-mass energy of 800~GeV, in order to include the \W\W-fusion process.  

\section{Simulation and analysis} 
\label{signal} 

The trilinear Higgs self-coupling could be
extracted from the measurement of the cross-section of each of the
following processes~: double Higgs-strahlung ($e^+e^-\to Z\h\h$) or
$WW$ double-Higgs fusion ($e^+e^-\to \bar\nu_e\nu_e \h\h$)~\cite{djouadi}.
Feynman diagrams involved are indicated on Fig.~\ref{double} and \ref{fusion}.
\begin{fmffile}{fd}
\begin{figure}[htb]
\begin{center}
%\vspace{.7cm}
{\footnotesize
\unitlength1mm
\hspace{10mm}
\begin{fmfshrink}{0.7}
\begin{fmfgraph*}(24,12)
  \fmfstraight
  \fmfleftn{i}{3} \fmfrightn{o}{3}
  \fmf{fermion}{i1,v1,i3}
  \fmflabel{$e^-$}{i1} \fmflabel{$e^+$}{i3}
  \fmf{boson,lab=$Z$,lab.s=left,tens=3/2}{v1,v2}
  \fmf{boson}{v2,o3} \fmflabel{$Z$}{o3}
  \fmf{phantom}{v2,o1}
  \fmffreeze
  \fmf{dashes,lab=$H$,lab.s=right}{v2,v3} \fmf{dashes}{v3,o1}
  \fmffreeze
  \fmf{dashes}{v3,o2} 
  \fmflabel{$H$}{o2} \fmflabel{$H$}{o1}
  \fmfdot{v3}
\end{fmfgraph*}
\hspace{15mm}
\begin{fmfgraph*}(24,12)
  \fmfstraight
  \fmfleftn{i}{3} \fmfrightn{o}{3}
  \fmf{fermion}{i1,v1,i3}
  \fmf{boson,lab=$Z$,lab.s=left,tens=3/2}{v1,v2}
  \fmf{dashes}{v2,o1} \fmflabel{$H$}{o1}
  \fmf{phantom}{v2,o3}
  \fmffreeze
  \fmf{boson}{v2,v3,o3} \fmflabel{$Z$}{o3}
  \fmffreeze
  \fmf{dashes}{v3,o2} 
  \fmflabel{$H$}{o2} \fmflabel{$H$}{o1}
\end{fmfgraph*}
\hspace{15mm}
\begin{fmfgraph*}(24,12)
  \fmfstraight
  \fmfleftn{i}{3} \fmfrightn{o}{3}
  \fmf{fermion}{i1,v1,i3}
  \fmf{boson,lab=$Z$,lab.s=left,tens=3/2}{v1,v2}
  \fmf{dashes}{v2,o1} \fmflabel{$H$}{o1}
  \fmf{dashes}{v2,o2} \fmflabel{$H$}{o2}
  \fmf{boson}{v2,o3} \fmflabel{$Z$}{o3}
\end{fmfgraph*}
\end{fmfshrink}
} \\
\end{center} 
\caption{ Main Feynman diagrams involved in the signal cross-section via the double Higgs-strahlung.\label{double}}
\end{figure}
\begin{figure}[htb]
\begin{center}
%\vspace{.3cm}
{\footnotesize
\unitlength1mm
\hspace{10mm}
\begin{fmfshrink}{0.7}
\begin{fmfgraph*}(24,20)
  \fmfstraight
  \fmfleftn{i}{8} \fmfrightn{o}{8}
  \fmf{fermion,tens=3/2}{i2,v1} \fmf{phantom}{v1,o2}
  \fmflabel{$e^-$}{i2}
  \fmf{phantom}{o7,v2} \fmf{fermion,tens=3/2}{v2,i7}
  \fmflabel{$e^+$}{i7}
  \fmffreeze
  \fmf{fermion}{v1,o1} \fmflabel{$\nu_e$}{o1}
  \fmf{fermion}{o8,v2} \fmflabel{$\bar\nu_e$}{o8}
  \fmf{boson}{v1,v3} 
  \fmf{boson}{v3,v2}
  \fmf{dashes,lab=$H$}{v3,v4}
  \fmf{dashes}{v4,o3} \fmf{dashes}{v4,o6}
  \fmflabel{$H$}{o3} \fmflabel{$H$}{o6}
  \fmffreeze
  \fmf{phantom,lab=$W$,lab.s=left}{v1,x1} \fmf{phantom}{x1,v3} 
  \fmf{phantom,lab=$W$,lab.s=left}{x2,v2} \fmf{phantom}{v3,x2}
  \fmfdot{v4}
\end{fmfgraph*}
\hspace{15mm}
\begin{fmfgraph*}(24,20)
  \fmfstraight
  \fmfleftn{i}{8} \fmfrightn{o}{8}
  \fmf{fermion,tens=3/2}{i2,v1} \fmf{phantom}{v1,o2}
  \fmf{phantom}{o7,v2} \fmf{fermion,tens=3/2}{v2,i7}
  \fmffreeze
  \fmf{fermion}{v1,o1}
  \fmf{fermion}{o8,v2}
  \fmf{boson}{v1,v3} 
  \fmf{boson}{v4,v2}
  \fmf{boson,lab=$W$,lab.s=left}{v3,v4}
  \fmf{dashes}{v3,o3} \fmf{dashes}{v4,o6}
  \fmflabel{$H$}{o3} \fmflabel{$H$}{o6}
\end{fmfgraph*}
\hspace{15mm}
\begin{fmfgraph*}(24,20)
  \fmfstraight
  \fmfleftn{i}{8} \fmfrightn{o}{8}
  \fmf{fermion,tens=3/2}{i2,v1} \fmf{phantom}{v1,o2}
  \fmf{phantom}{o7,v2} \fmf{fermion,tens=3/2}{v2,i7}
  \fmffreeze
  \fmf{fermion}{v1,o1}
  \fmf{fermion}{o8,v2}
  \fmf{boson}{v1,v3} 
  \fmf{boson}{v3,v2}
  \fmf{dashes}{v3,o3} \fmf{dashes}{v3,o6}
  \fmflabel{$H$}{o3} \fmflabel{$H$}{o6}
\end{fmfgraph*}
\end{fmfshrink}
}
\end{center} 
\caption{ Main Feynman diagrams involved in the signal cross-section via the \W\W\ fusion.\label{fusion}}
\end{figure}
%\end{figure}
\end{fmffile}

At centre-of-mass energy around 500 GeV the double Higgs-strahlung is dominant with 
respect to \W\W\ fusion one but at centre-of-mass energy close to 800 GeV, 
the \W\W\ fusion is more important.
The study presented here will be performed for \mh=120~\Gcs{} and \sqrtofs=800~GeV in the \SM\ framework. 
With an expected integrated luminosity of 500~\invfb, the number of events 
expected from the signal is only of 90 events, while the background processes 
yield to an sample of events which is greater by many order of magnitude.  
Only part of the diagrams involve self Higgs coupling and  sample of events corresponding to those diagrams  have been simulated in order to evaluate the efficiencies of the analysis to that particular contribution from double-Higgssthralung and \W\W-fusion processes.
For light Higgs boson masses, the Higgs boson decays predominantly in \bbbar{} pair and the signature of those final states is caracterised by multijet environment with a high b-like content with or without missing energy. 
The signal and background event samples have been simulated with the {\tt WHIZARD}
generator~\cite{whizard} and {\tt PYTHIA}~\cite{pythia} has been used to perform the 
hadronisation of the primary parton. The {\tt CIRCE } package~\cite{circe} has been use to introduce the initial energy dispersion. 
The background sources are either four and six fermions final state processes. 
The following processes \epem\goes\WpWm, \Z\Z, \h\Z\ and \epem\goes\W\W\Z{}, \Z\Z\Z{} have been simulated. In order to take into account the diffusion at low angle dedicated samples have been generated for \epem\goes\W e$\nu$, \Z ee, \W\W\nnbar, \Z\Z\epem, \Z\Z\nnbar\ and \Z\W$e\nu$ processes. The top pair production, which leads to six fermions final state has been simulated as well as \epem\goes\ttbar\h, \ttbar\Z. A dedicated  sample of \epem\goes\ttbar\nnbar\ has been generated. Table~\ref{suptab} summaries the size of the samples used and the cross-section of each process. 
\begin{table}[h]
\caption{Cross-sections of the processes, Monte Carlo statistics 
and simulated luminosity (\lumisim). Numbers of events expected from signal (\signalhhh), background processes (\Btot) and contribution for selection A and B ( resp. $N_{sel A}$ and $N_{sel B}$).  Relative error (\dhhh) on $\lambda$ for selection (A) and (B)  and an 2~\invab\ integrated luminosity. The signal number of event (\signalhhh) and signal efficiency (\effhhh) corresponds to the sub-sample of events involving \hhh\ coupling. }\label{suptab} 

\begin{tabular}{cr|c|c|c||c|c|c}
& { process} 		&  { $N_{gen}$ }& {$ \sigma$ (fb) }& { \lumisim (\infb)} & \multicolumn{2}{c|}{$N_{sel A}$ } & $N_{sel B}$\\ 
&								&								&									 &										 & loose cut        & tight cut &           \\ \hline
& signal        & 10k 	& 0.1844 & 54227 &		& 	  	&	\\   
&\W\W           & 9.2M	& 668.8 	& 13890	&  -    &				 	&				\\ 
&\Z\Z           & 1.5M  & 222.4		& 6744 	& 32.6	& 			 	&1.48		\\
&\h\Z           & 160k	& 22.65		& 7064.	& 64.3	& 			 	&5.64		\\
&\ttbar   			& 1M 	  & 260.			& 3840.	& 391.	& 	  		& 				\\
& \ttbar\h      & 20k	  & 2.498    & 8006 	& 69.3  & 10.5	 	& 				\\ 
& \ttbar\Z      & 25k   & 4.528    & 5521 	& 40.6	& 7.24 	  & 				\\  
& \ttbar\nnbar\ & 20k		& 0.787		& 25400.& 0.16	&				 	&				\\
&\W\W\Z\     	  & 100k	& 56.96		& 1755.	& 1.12	& 			 	&				\\ 
&\Z\Z\Z\     	  & 25k 	& 0.729		& 34280.& 10.9 	& 4.12	 	& 				\\ 
&\W\W\nnbar     & 60k 	& 12.43 	& 4827.	&		-		&					&				\\	
&\Z\Z\epem  		& 3k 		& 0.287		& 10400.& 1.72 	& 			 	& 				\\
&\Z\Z\nnbar    	& 25k		& 3.477	  & 7190.	& 0.28	& 			 	&  1.96\\
&\Z\W\e$\nu$ 	  & 12k 	& 10.09		& 1176.	&		-		&					&    \\ \hline 
\multicolumn{4}{c}{ }			&\Btot	& 612.	&21.3			&9.1 \\
\multicolumn{4}{c}{ }				&	\signalhhh & 137.6 &88. 			&34.\\
\multicolumn{4}{c}{ }					&	\effhhh & 42.6\%& 30.2\% 	&37.3\% \\
\multicolumn{4}{c}{ }					&\fom 	& 5.    &8.2 			&5.2 \\ 
\multicolumn{4}{c}{ }			&	\dhhh&       & 21.7\%& 17.6\%\\ 
\end{tabular} 
\end{table} 
\noindent The detector simulation was performed with a Parametric Monte Carlo~\cite{simdet}. The tracking system consists on a vertex detector (VDET) surrounding the beam tube followed by a time projection chamber (TPC). The tracking system is complemented with a forward tracker and muon chambers. The hermeticity of the detector is reenforced at low angle with a luminometer (LCAL). Energy and direction of the photons and neutral hadrons are measured thanks to the electromagnetic (ECAL) and hadronic (HCAL) calorimeters with an energy resolutions of $\Delta E/E$=10.2\%/$\sqrt{E(GeV)}$ and  $\Delta E/E$=40.5\%/$\sqrt{E(GeV)}$  respectively for ECAL/LCAL and HCAL. Smeared clusters in calorimeters and smeared tracks are combined in order to form the best reconstructed objects so-called here after \eflow\ objects. 
The selection is split in two branches according the value of total visible energy (\Evis) measured in the detector. This energy is defined from the \eflow\ reconstructed objets. Selection (A) corresponds to the events where \Evis\ is greater than 500~GeV. The event is therefore forced in six jets with the {\tt DURHAM} scheme. 
Selection (B) collected the events with \Evis\ less 500~GeV, and forced into four jets. 
In a second part, combination of the di-jets reconstructed masses are
used to reject the three-bosons and di-bosons final-state contribution taking advantage of the reconstruction of two Higgs bosons. The flavour tagging will play an important r\^o le. The b-tagging and c-jet contamination is performed through a parametrisation derived from full reconstruction~\cite{rh}. 
Finally, relevant informations are combined with a multivariable method (neural network) as illustrated on Fig.~\ref{nn_hhz} for the six jets topology.

\begin{figure}[htbp]
\epsfxsize=8cm
\centerline{\epsfbox{PG_hhh_coupling_fig_1.eps}}
\caption{Neural Network output distribution ($ NN_{output}$) for $\rm H$$\rm H$$\rm Z$\ signal (full histogram) 
and background (empty histogram) with 500~{\mbox{$\hbox{fb}^{-1}$}}\ and 
{\mbox{$m_{\mathrm h}$}}=120~{\hbox{GeV}/\mbox{$\mathrm{c^2}$}}. 
\label{nn_hhz}}
\end{figure}

The contribution from the various background sources are reported in Table~\ref{suptab} for the selection (A) with the loose and tightened cuts and for the selection (B). The signal efficiencies are  30.2\% and 37.3\% respectively for the selection (A) and (B). Those efficiencies are derived from a sample of signal events generated with only diagramma involving the trilinear \hhh\ coupling in the cross-section. Assuming an integrated luminosty of 2000\invfb\, the combination of the two selections leads to 30.4 events expected from  background processes and 122 events from the signal sensible to the \hhh\ coupling.   


\section{Conclusion}
To establish the Higgs mechanism in an unambiguous way, the
self-energy potential of the Higgs field must be reconstructed. This
implies the determination of the trilinear self-coupling.  
The experimental feasibility of the measurement has
been explored trough a detailed analysis of the reconstruction of the
double Higgs-strahlung and \W-\W fusion events. 
To derive the relative error on \trilin\  the
relation between \trilin\ and the cross-section has to be taken into
account and the relative error on the trilinear Higgs coupling ($\Delta\lambda/\lambda$) is displayed on Table~\ref{suptab} 
for an integrated luminosity of 2~\invab.
With the same integrated luminosity, the combination of the two analysis indicated that an relative error on on $\lambda_{\rm hhh}$ of 14\%
may be achivied. 

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\end{thebibliography}

\end{document}