\documentclass{elsart3}
%\documentstyle[epsf]{article}
\usepackage{natbib}
\input{epsf}

\begin{document}
%\runauthor{Andrei Nomerotski}
\begin{frontmatter}
\title{Optimization of Vertex Detector for International Linear  Collider}
\author[Paestum]{Andrei Nomerotski}
\address[Paestum]{University of Oxford, Denys Wilkinson Building, Keble Road, Oxford OX1 3RH, United Kingdom}

\begin{abstract}
The vertex detector for the future Linear Collider 
needs to be as close as possible to the beampipe and to be as thin as possible. Importance of these 
and other parameters for the optimization is discussed together with mechanical 
options for thin ladders based on rigid foams.
\end{abstract}

\begin{keyword}
vertex detector; ILC; pixel silicon detector 
\end{keyword}

\end{frontmatter}

\section{Comparison of vertex detectors}

The last vertex detector of SLD, VDX3, was arguably the best vertex detector for particle 
physics experiments ever built \cite{SLD}.
The detector had three layers of CCD sensors with pixels 20~$\mu$m~x~20~$\mu$m 
around a small diameter beampipe and was very thin, 
0.4\% radiation length, X$_0$ per layer. It is instructive to compare the typical impact parameter 
(IP) resolution of VDX3 with that of the LEP \cite{LEP} and Tevatron \cite{tevatron} 
vertex detectors, all shown in Table 1. 
In the Table R is the radius of the first layer,
$\sigma_h$ is the single hit resolution,  
X$_0$ is the radiation length per layer,
$\sigma_{IP}^{res}$ is the single hit resolution term and 
$\sigma_{IP}^{ms}$ is the multiple scattering term.
The resulting IP resolution for particles with momentum $p$ passing perpendicular to the sensor 
can be determined using the formula:
\begin{equation}
\sigma_{IP} = \sqrt{ (\sigma_{IP}^{res})^2 + (\sigma_{IP}^{ms} / \mathrm{p})^2 }.
\end{equation}
\begin{table}
%\caption{Impact parameter resolution for various vertex detectors}
%\label{defparagcl} 
\begin{center}
\begin{tabular}{c c c c c c}
\hline 
detector & R [cm] & ~$\sigma_h$~[$\mu$m] & $X_0$ [\%] & $\sigma_{IP}^{res}$ [$\mu$m]& $\sigma_{IP}^{ms}$ %%@
[$\mu$m]\\ 
\hline 
SLD VDX3 & 2.8    & 4          & 0.4         & 8                   & 30 \\  
SLD VDX2 & 2.8    & 5          & 1.2         & 11                  & 70 \\  
LEP      & 6.3    & 8          & 1           & 20                  & 65 \\  
Tevatron & 2.0    & 8          & 2-3         & 10                  & 40 \\  
\hline 
ILC      & 1.5    & 4          & 0.1         & 4                   & 8 \\  
\hline 
\end{tabular}
\end{center}
\end{table}
A larger beampipe for the LEP experiments is explained by the synchrotron radiation 
in the circular $e^+e^-$ machine and a worse single hit resolution for the Tevatron 
and LEP follows from a larger pitch of the stip silicon sensors used by the experiments. 
The table also shows the resolution targeted for the ILC vertex detector.   

The SLD IP resolution was a factor of two better than that of the LEP vertex detectors 
providing excellent vertexing and tracking capabilities. 
Comparison of the b-tagging performance for SLD and LEP experiments is 
shown in Figure \ref{btagging}. One can see that the performance of VDX3 is considerably 
better than for the other detectors including VDX2, the predecessor of VDX3 at SLD. 
\begin{figure}
\epsfysize=7cm
\epsfxsize=7cm
\epsffile{SLDbtagging_Chris.eps}
\caption{Comparison of the b-tagging performance for SLD and LEP experiments.}
\label{btagging}
\end{figure}
 
\section{Algorithms Based on Precise Vertexing}

The superior resolution and pattern recognition of VDX3 allowed 
for innovative algorithms based on correct assignment of tracks to secondary and tertiary 
vertices. The tertiary vertices originate from the decay chains of B hadrons through D hadrons 
as both b- and c-quarks have considerable lifetime. 

The vertex mass algorithm \cite{vertexmass}, for example, calculates the invariant mass 
of all tracks assigned to the secondary vertex assuming that they all are pions. This algorithm 
provides excellent separation between various species of quarks. This includes a 
good separation even between b- and c-quarks enabled by the difference in their masses. The vertex mass
is the most important single contributor to the combined b-tagging procedure 
represented in Figure \ref{btagging}.   

Two other important algorithms aimed to determine the sign of the quark charge are 
the vertex charge and vertex dipole \cite{vertexcharge}. In the first one the 
total charge of tracks associated with a secondary vertex is calculated. Tracks from 
a possible tertiary vertex are included as well as they carry information about the charge 
of the parent. Figure \ref{vertexcharge} shows the reconstructed vertex charge for charged B mesons
in Monte-Carlo simulations for ILC \cite{snowmass}. 
\begin{figure}
\epsfysize=7cm
\epsfxsize=7cm
\epsffile{VtxChargeRecoComparisonCharged.eps}
\caption{Reconstructed vertex charge for B$^+$ and B$^-$ mesons.}
\label{vertexcharge}
\end{figure}
The vertex charge has a binary behaviour - a lost or wrongly assigned track changes 
the charge of the vertex which makes every track important for the correct charge reconstruction. 
This particular feature makes the vertex charge algorithm sensitive to low p$_T$ tracks hence
sensitive to the amount of material as this enters the multiple scattering term in the IP 
resolution formula.

One of the goals of the vertex detector optimisation is comparison of different detector layouts 
to determine trade-offs between various design parameters. From the discussion above
it is clear that the vertex charge algorithm should have a better discrimination for the designs
than the b-tagging as the b-tagging relies on all tracks and contribution of a single track is 
averaged out. 

The vertex charge algorithm is a crucial tool in a number of physics analyses. One class of such 
analyses includes pair production of quarks. The forward-backward asymmetry of this production 
in the Standard Model can be modified by various new phenomena processes 
(e.g. extra dimensions \cite{extradimensions}). The full sensitivity is obtained when the asymmetry 
is measured as a function of the polar angle with respect to the direction of colliding particles and
 the flavour of the b and anti-b quarks is determined by the vertex charge. 
Other applications include angle correlations between heavy flavour 
jets and other objects in the event. This can identify polarization of decay products of the top
quarks which is also sensitive to various new physics scenarios \cite{topquark}. 
The angle correlations 
help to distinguish SUSY from the Little Higgs models \cite{susy} which both result in 
similar signatures different only in the spin of exotic neutral particles which are 
not detected directly. However angle correlations measured through the vertex charge of heavy 
quark jets are sensitive to this difference.

Another important algorithm of flavour tagging 
is the vertex dipole where the b or anti-b quark fragments into 
neutral B hadrons so it cannot be identified by the vertex charge as it is equal to zero in 
both cases. The vertex dipole is defined as a signed distance between secondary and tertiary
vertices where the sign is defined by the difference of vertex charges in two vertices. 

Other algorithms useful for optimization of the vertex detector are the bottom, charm and $\tau$ tagging. 
They can be especially sensitive to the detector parameters for soft b/c/$\tau$ jets predicted by
the SUSY scenarios which are constrained by Cold Dark Matter results \cite{susy}. Also a smaller 
lifetime of the charm quark and $\tau$ lepton makes their identification through displaced 
vertices more demanding.

\section{Dependence on Parameters}

The LCFI (Linear Collider Flavour Identification) collaboration 
used the vertex charge as optimisation tool 
for initial studies of the ILC vertex detector \cite{snowmass}. One of the most important result 
of the study is the sensitivity to the beampipe radius in the interaction point.
Figure \ref{lambda} shows the mistag rate (probability 
to reconstruct B$^0$ as B$^+$ or B$^-$) as function of the jet energy for three geometries different in
 location of the first layer of the detector. 
\begin{figure}
\epsfysize=7cm
\epsfxsize=7cm
%\epsffile{lambdavsE.eps}
\epsffile{LeakageRateVsSqrts_Rbp.eps}
\caption{Probability 
to reconstruct B$^0$ as B$^+$ or B$^-$ as function of the jet energy 
for three geometries different in
location of the first layer of the detector.}
\label{lambda}
\end{figure}
As expected the mistag is a strong function of the minimal radius which needs to be as small as possible
to obtain the best performance. To quantify the gain the corresponding ‘luminosity factors’ based on 
the efficiency deterioration at a constant mistag have been calculated.
For example, 70\% more luminosity is needed for 50 GeV jets if the radius is increased from 15 mm to 25 mm
assuming double-tagging of b-jets. Note that the increase will also require the increase 
of the thickness for the Be beampipe from 0.4 mm to 1 mm \cite{snowmass}.

Figure \ref{degrad} shows the effect of various parameters on the mistag rate. The default configuration 
(minimal radius 15 mm, P$_T$ cutoff in tracking 0.1 GeV and thickness per layer 0.1\% X$_0$) is compared to
configurations where one of these three parameters has been changed (minimal radius 25 mm, 
P$_T$ cutoff in tracking 0.2 GeV and thickness per layer 0.4\% X$_0$). Strong sensitivity to material and 
p$_T$ cutoff has been observed. Note that in these studies the ideal tracking (no fake tracks) has been used 
and no secondaries from interactions with material have been assumed so the real performance is expected to %%@
be somewhat worse.
\begin{figure}
\epsfysize=7cm
\epsfxsize=7cm
%\epsffile{QvtxDegrad.eps}
\epsffile{LeakageVsCosTheta_MaterialPtRbp.eps}
\caption{Degradation in performance for minimal radius 25 mm, 
P$_T$ cutoff in tracking 0.2 GeV and thickness per layer 0.4\% X$_0$.}
\label{degrad}
\end{figure}
%It should be noted that the deterioration is caused ultimately by the impact parameter resolution. 
%The resolution is a function of both radius and material so the small radius of the first layer 
%helps to reduce multiple scattering effects from the material downstream. For example Figure \ref{L00} 
%shows the effect of the CDF Layer00 on the IP resolution in the hybrid region \cite{L00}. 
%The improvement of the resolution is especially dramatic in the regions corresponding to the hybrid in 
%the second layer of the vertex detector. 
%\begin{figure}
%\epsfysize=7cm
%\epsfxsize=7cm
%\epsffile{L00hybrid.eps}
%\caption{Effect of the CDF Layer00 on the IP resolution in the hybrid region}
%\label{L00}
%\end{figure}
Apart from the multiple scattering term in IP resolution 
there are some additional material effects that are worth mentioning - production of 
the secondaries and also tails in the resolution function. The gamma conversions and generation of 
energetic delta-electrons in the material of the detector will increase the overall 
occupancy and will create
displaced vertices in the vertex detector volume. Their effect will depend on the sophistication 
of the reconstruction 
software so its evaluation will require full simulation of the detector
and a realistic tracking and vertexing code. 
%To illustrate the effect see Figure \ref{conv} which shows the reconstructed vertices of gamma conversions 
%in the two inner barrels of DZero SMT \cite{D0}. 
%Studies done at CDF indicated that secondary particles account for a half of all tracks in the silicon 
%tracker \cite{stuart}.

%\begin{figure}
%\epsfysize=7cm
%\epsfxsize=7cm
%\epsffile{gammaconversions.eps}
%\caption{SLD}
%\label{conv}
%\end{figure}

%The typical thickness of the SMT ladder was factor of ten larger than the 0.1\% X$_0$ expected for the ILC 
%vertex detector and the pixel 
%sensors are indeed more robust for the pattern recognition than the SMT strip detectors but on the other %%@
%hand the 
%goal of sub 10 $\mu$m impact parameter resolution is very ambitious and requires to take 
%into consideration all factors.
  
Another important topic in the discussion of material balance in the vertex detector is the inhomogeneity of 
the material distribution which will result in the tails of the resolution function. 
This can be accounted for by accurate description of material in the reconstruction software. 
The tails of the resolution can also originate from several other reasons
like the angle and p$_T$ dependence of the resolution 
or the clustering algorithm. 
%From the analysis prospective the poor 
%knowledge of which events have worse resolution (tails in the 'pull' distributions) 
%is a bigger problem (tails in the 'pull' distributions) than a somewhat 
%worse resolution for a fraction of events by itself.
Accurate description of material and sophistication of the reconstruction software are important 
to take advantage of a good detector.

To fully explore the sensitivity of various designs to the mentioned above parameters one needs a suite of 
software tools which should include realistic simulations of interactions with the detector material and
advanced tracking and vertexing packages. One of them, the vertexing package ZVTOP developed by 
the LCFI collaboration, includes the topological vertex finder ZVTOP, additional ‘ghost track’ 
enhancement algorithm ZVKIN and is fully integrated into the ILC MonteCarlo, tracking and software %%@
frameworks. 
The package has been inspired by the vertexing code used by the SLD detector \cite{zvtop}. 
The package also has a flavour tagging part based on a neural net algorithm.

Considerations above show that the thinnest possible detector is needed but, of course, the final design
will be a trade-off between the ideal and reality. The TESLA TDR released in 2001 had 
the thickness of the vertex detector at
0.06\% X$_0$ per layer. This has been changed in 2003 to a more realistic number 
0.1\% and stayed at this number till now. All effort should be made to
maintain this goal as this is a crucial parameter.

The above considerations were to demonstrate that the Linear Collider conditions allow to do much 
more sophisticated vertexing algorithms than at other machines through the correct particle assignment 
to secondary and tertiary vertices. The two big items in the vertex detector optimization 
are the small diameter beampipe 
and the thin detector. Next section discusses how to achieve the 0.1\% radiation length per layer.

\section{Mechanical Options}

The target of 0.1\% X$_0$ per layer which is equivalent to 100 $\mu$m of silicon is a huge challenge. 
Studies carried out during the last years by the LCFI collaboration already investigated several methods.
Unsupported thin silicon where the longitudinal tensioning provides stiffness has no lateral stability and is
not believed to be promising. The other technique involved thin substrates of various materials. In this case 
a silicon sensor is thinned to the epitaxial layer of about 25 $\mu$m and then glued to a low mass substrate 
which provides lateral stability. The longitudinal stiffness in this case is still coming from 
a moderate tension. Beryllium has the best specific stiffness and has been used before at SLD where 
the ladder was composed of 0.15 mm of Si and 0.37 mm of Be.

Ladders based on Be and carbon fibre (CF) substrates have been constructed with the total 
thickness of 0.15\% X$_0$ achieved for Be and 
0.09\% for CF. The CTE mismatch was causing a rippling effect for Be while for the CF substrate the mismatch %%@
is 
much smaller and the rippling does not occur. However in both cases the lateral stability was determined to %%@
be 
insufficient.

The most promising technique investigated so far is based on rigid foam structures made of 
RVC (Reticulated Vitreous Carbon) or SiC (Silicon Carbide) 
%Figure \ref{foam} shows a microphotograph of SiC foam 
which form macroscopically uniform open cells.
%\begin{figure}
%\begin{center}
%\epsfysize=5cm
%\epsfxsize=5cm
%\epsffile{foam.eps}
%\caption{Microphotograph of SiC foam.}
%\label{foam}
%\end{center}
%\end{figure}
Several prototypes have been constructed of a 3\% fill factor RVC foam sandwiched between two pieces of thin
silicon and of a 8\% SiC foam glued to one piece of silicon with the total thickness of 0.09\% and 
0.14\% X$_0$ correspondingly. The first structure has been determined to be mechanically unsatisfactory while
the second one performed well. Figure \ref{bend} shows deviation from the original position as function of %%@
the 
location and temperature for the SiC ladder. The laser displacement meter from Keyence Corp. 
has been used for the measurement. 
The survey system equipped with 2D motorized stages provided the 3D precision of about 1 $\mu$m and was able %%@
to perform 
measurements for a ladder placed in a cryostat. Work is in progress to build and test further prototypes %%@
using 
the both types of foam with smaller fill factors. 
\begin{figure}
\begin{center}
\epsfysize=7.2cm
\epsfxsize=7.2cm
\epsffile{SIC_Diff.eps}
\caption{Deviation from the original position as function of the 
location and temperature for the SiC ladder.}
\label{bend}
\end{center}
\end{figure}

\section{Summary}

A number of refined vertex algorithms are possible with a precise and thin detector at ILC.
These algorithms are sensitive to the vertex detector design and can be used for its optimization,
the most compelling being the tagging of the heavy quark flavour based on vertex charge. The beampipe %%@
diameter, 
spatial resolution and the thickness of the detector are three most essential parameters for the
optimization. Mechanically, ladders  with thickness of 0.1\% $X_0$ seem achievable exploiting rigid foams %%@
based on
silicon carbide or RVC.

\begin{thebibliography}{999}

\bibitem{SLD}K. Abe {\it et al}, {\em Nucl. Inst. Meth.} {\bf A386} (1997) 46.
\bibitem{LEP}V. Chabaud {\it et al.}, {\em Nucl. Inst. Meth.} {\bf A368} (1996) 314;
%\bibitem{ALEPH}
D. Creanza {\it et al.}, {\em Nucl. Inst. Meth.} {\bf A409} (1998) 157;
%\bibitem{OPAL}
P. P. Allport {\it et al.}, {\em Nucl. Inst. and Meth.} {\bf A346 (1994)} 476.
\bibitem{tevatron}A. Sill for CDF collaboration, {\em Nucl. Inst. Meth.} {\bf A447} (2000) 1;
%\bibitem{D0}
E. Kajfasz for D0 collaboration, {\em Nucl. Inst. Meth.} {\bf A511} (2003) 16. 
\bibitem{vertexmass}N. de Groot, {\em Nucl. Inst. Meth.} {\bf A501} (2003) 229. 
\bibitem{vertexcharge}SLD collaboration, {\em Int J Mod Phys A} {\bf 16S1A} (2001) 274.
\bibitem{snowmass} S. Hillert, C.J.S. Damerell on behalf of the LCFI Collaboration, Proceedings of the 2005 %%@
International Linear Collider Physics 
and Detector Workshop and 2nd ILC Accelerator Workshop (Snowmass 2005), SLAC-R-798 eConf: %%@
C0508141:ALCPG0101,(2005). 
%S. Hillert, The Vertex Detector in the LDC Concept, contribution to the ILC Vertex Detector Workshop at %%@
%Ringberg, May 2006, available at http://www.hll.mpg.de/~lca/ringberg/Talks/1-Monday/2-4-hillert.ppt.

%\bibitem{L00}C. S. Hill for CDF collaboration, {\em Nucl. Inst. Meth.} {\bf A530} (2004) 1.
\bibitem{zvtop} D. J. Jackson,{\em Nucl. Intr. Meth.} {\bf A388} (1997) 247. 
\bibitem{extradimensions} M. Battaglia, S. De Curtis, D. Dominici, S. Riemann, 
e-Print Archive: hep-ph/0112270.
\bibitem{topquark} G. A. Moortgat-Pick, 
Proceedings of the 2005 International Linear Collider Physics and Detector Workshop and 2nd ILC Accelerator %%@
Workshop (Snowmass 2005),
eConf C0508141:ALCPG0306,2005, eConf C0508141:ALCPG0401,2005.
\bibitem{susy} D. Auto, H. Baer, A. Belyaev, T. Krupovnickas, {\em Jour. High Ener. Phys.} 0410:066 (2004) %%@
066. 
%\bibitem{keyence} 

\end{thebibliography}
\end{document}
