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\begin{titlepage}

\noindent
\begin{center}
%{\it {\large version of \today}} \\[.3em] 
\begin{small}
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%Submitted to & \multicolumn{3}{r}{\footnotesize Electronic Access: {\it http://www-h1.desy.de/h1/www/publications/conf/conf\_list.html}} \\[.2em] \hline 
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\multicolumn{4}{l}{{\bf
                International Europhysics
                Conference on High Energy Physics},
                July~12,~2001,~Budapest} \\
 {\bf EPS 2001:} 
                 & Abstract:        & {\bf 877}    &\\
                 & Parallel Session & {\bf 13}   &\\ \hline
%                 & Plenary Session  & {\bf 4, 7}   &\\[.7em]
%\multicolumn{4}{l}{{\bf
%               XX International Symposium on Lepton and Photon Interactions}, 
%               July~23,~2001,~Rome} \\ 
%{\bf LP 2001:}  
%                 & Abstract:        & {\bf 1301} &\\
%                 & Parallel Session & {\bf 23}   &\\
%                 & Plenary Session  & {\bf 34}   &\\ \hline
 & \multicolumn{3}{r}{\footnotesize {\it
    www-h1.desy.de/h1/www/publications/conf/conf\_list.html}} \\[.2em]
\end{tabular}
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\end{center}
\vspace*{2cm}

\begin{center}
%  \Large
  {\LARGE\bf Electron polarization measurement using a\\
  Fabry-P\'erot cavity at HERA}

  \vspace*{1cm}
{\large E.~Barrelet$^1$,V.~Brisson$^2$, M.~Jacquet-Lemire$^2$, A.~Reboux$^2$, 
\\
C.~Pascaud$^2$, Z.~Zhang$^2$, F.~Zomer$^2$\\
and the POL2000 group$^3$\\
\small{$^1$LPNHE Jussieu, IN2P3-CNRS}\\
\small{$^2$Laboratoire de l'Acc\'el\'erateur Lin\'eaire, IN2P3-CNRS
 et Universit\'e de Paris-Sud}\\
 \small{B.P. 34,  F-91898 Orsay Cedex,  France}\\
 \small{$^3$members from DESY, HERMES, H1 and ZEUS\footnote{See DESY
 PRC-99-01 for a detailed list of members of this group. Contact
Ties.Behnke{@}desy.de}}} 
\end{center}

\begin{abstract}
\noindent
A new Compton
 longitudinal polarimeter for HERA is under construction. The key component
of the calorimeter is 
 a Fabry-P\'erot cavity located around the electron beam pipe. With such an
optical cavity,
 a continuous laser power of 5\,kW can be achieved leading to backscattered
 photon rate around one per bunch crossing. This `few-photon mode' will allow
 a very precise determination of the calorimeter response with little
 systematic uncertainty.
% function together to a very large statistics. A determination of 
The electron polarization measurement at the per mill level is expected.
\end{abstract}

\end{titlepage}

\pagestyle{plain}

\section{Introduction}
%
The study of polarized electron-proton scattering at very high
energies will be one of the major new physics topics after the 
upgrade of the HERA storage ring. The colliding-beam experiments 
H1 and ZEUS will for the first time have access to longitudinally 
polarized leptons. This will allow detailed studies of 
the helicity structure of the weak currents. 

To fully exploit the potential of the machine and the luminosity to be 
delivered to the experiments, a significant 
upgrade of the existing longitudinal polarimeter (LPOL) has been
proposed~\cite{loi,proposal}. The main components of the LPOL are a
Fabry-P\'erot cavity, a continuous wave laser of 0.7\,W, 
a series of optical components to circularly polarize the laser light, 
a system of monitoring devices to measure and control the properties of the
laser beam, a feedback system for control the frequency of the
laser and to keep the cavity in resonance, and a fast data acquisition system
to store all histograms needed for a polarization measurement on a 
bunch-by-bunch basis. The heart of the polarimeter is the high finesse 
monolithic
Fabry-P\'erot cavity, which allows to reach an effective power of the laser
close to 5\,kW, sufficient to obtain one backscattered photon per bunch
crossing. With this system the statistical uncertainty of the polarization
can be reduced to 0.1\% per minute and per group of bunches in HERA, or 1\%
per minute and per bunch.

The structure of this paper is as follows. We will start in Sec.\ref{sec:nlpol}
with a short description of the new LPOL and point out the difference with 
the existing ones. The necessities of having a fast and precise polarimeter 
is discussed in Sec.\ref{sec:why},
%The status and the schedule of the project is presented in Sec.\ref{sec:when},
followed by a conclusion in Sec.\ref{sec:sum}.

\section{The New Longitudinal Polarimeter} \label{sec:nlpol}
%
At HERA, the electron or positron beam of 27.5\,GeV in collision with the
proton beam of 820-920\,GeV is naturally polarized transversely through the
emission of synchrotron radiation which is known as the Sokolov-Ternov
effect~\cite{steffect}. Longitudinal polarization at East area near the
HERMES experiment was achieved since 1994 by installing a pair of spin
rotators.
During the upgrade, the H1 and ZEUS interaction regions have also been
equipped with spin rotators.
Currently the measurement of the electron\footnote{Unless stated otherwise,
the word ``electron'' will refer to both electron and positron in the
following.} polarization at HERA is done
with two different devices. The transverse polarimeter (TPOL), which is 
installed in the West area of HERA, measures the polarization of the
transversely polarized beam (Fig.\ref{fig:hera_pol}).
The LPOL in contrast measures the longitudinal
polarization of the beam near the HERMES interaction point.
\begin{figure}[htbp]
\begin{center}
\setlength{\unitlength}{1mm}
\begin{picture}(160,80)(0,-80)
\put(15,-85){
\epsfig{file=hera.eps,
bbllx=0pt,bblly=0pt,bburx=594pt,bbury=842pt,width=15cm}}
\end{picture}
\end{center}
\caption{A schematic view of the HERA $ep$ collider showing the locations of 
the interaction points, the spin rotators, and the longitudinal and 
transverse polarimeters (LPOL, TPOL).}
\label{fig:hera_pol}
\end{figure}
Since the absolute value of the degree of polarization of the electron beam
is the same everywhere in the ring, the location of the polarization
measurement is not confined to the experiment's interaction region.

The principle of the measurement relies on the non-destructive method of
detecting backscattered photons produced in Compton scattering of an 
intense circularly polarized laser (photon) beam off the electron beam.
The transverse polarization is measured from the vertical asymmetry, whereas
the longitudinal polarization is determined from the energy asymmetry.
%For longitudinally polarized electrons, the Compton scattering cross-section 
%can be written as
%\begin{equation}
%\frac{d\sigma}{dE_\gamma}=\frac{d\sigma_0}{dE_\gamma}
%\left[1-P_eS_\gamma A_z(E_\gamma)\right]\,
%\end{equation}
%where $d\sigma_0/dE_\gamma$ is the unpolarized cross-section, $E_\gamma$
%is the energy of the backscattered Compton photons, $S_\gamma$ is the circular
%polarization of the incident photons for two helicity states $\pm 1$,
%$P_e$ is the longitudinal polarization of the electron beam, and
%$A_z(E_\gamma)$ is the longitudinal asymmetry function, which is known.
The current statistical precision of the LPOL is around 1\% for 2 minutes of
data taking averaged over all bunches in the ring (up to 190). The systematic
uncertainty is about 2\% which is dominated by the energy scale uncertainty 
of the calorimeter measuring the backscattered photons in multi-photon
mode\footnote{Three different modes of operation can be distinguished for a
polarimeter detecting the backscattered photons:
\begin{itemize}
\item Single-photon mode: The probability to scatter a photon is very low
(0.001 for the existing TPOL), so that at most one photon is detected in
the calorimeter).
\item Few-photon mode: The probability is high enough that on average one
photon per bunch crossing is observed. In the calorimeter a distribution
ranging from one to a few photon is detected. The shape of the Compton
spectra is still visible.
\item Multi-photon mode: Per bunch crossing a large number of photons is
scattered into the calorimeter (e.g.\ around 1000 for the LPOL).
\end{itemize}}.
To reach a same statistical precision
for a single bunch, more than 30 minutes is required due to the low pulsed 
laser frequency (up to 100\,Hz). The statistical precision of the
TPOL is 1\% per minute again averaged over all bunches (the current
TPOL data acquisition (DAQ) system does not allow the differentiation of
bunch to be made). The systematic uncertainty is about twice that of the LPOL 
due mostly to the limited statistics of the dedicated rise-time runs taken with
a flat machine and needed to define the absolute polarization scale.

The new longitudinal polarimeter is under construction and will be installed
next to the existing LPOL during the shutdown in the Spring of 2002. 
The new LPOL
should provide the most precise measurement of the polarization with 
a statistical error of 0.1\% (0.3\%) for the colliding (non-colliding) bunches 
and 1\% per minute for every single bunch, and a systematic error at per mill 
level.

\subsection{A High Gain Fabry-P\'erot Cavity}
%
The new laser system will operate in the single- or few-photon mode.
To have an averaged probability of 0.1 photon per interaction, the power
of the laser system at the intersection with the lepton beam has to be
around 5\,kW, if a beam current of 1\,mA and a crossing angle of 58\,mrad
are assumed. Such power levels cannot be reached with commercially available
lasers. A Fabry-P\'erot cavity located around the interaction point is
therefore proposed~\cite{proposal} to boost the equivalent laser power to
the required level. The expected gain of the device is $7.3\times 10^3$,
and the bandwidth is 7\,kHz when pumped with an infrared laser of
$\lambda=1064$\,nm.
%A somewhat smaller cavity has been installed and
%operated successfully at the CEBAF laboratory since serval years~\cite{cebaf}.

The conceptual layout is shown schematically in Fig.\ref{fig:cavity} 
and a detailed view is shown in Fig.\ref{fig:cavity2}.
The cavity surrounds the interaction point for a length of 2\,m. The main
part of the cavity is a tube made from stainless steel, with a wall
thickness of around 0.3\,cm. The tube is mounted on an optical table. 
The cavity mirrors, mounted on specially constructed mirror
holders, are rigidly fixed to the table allowing the initial alignment of 
the mirrors. The cavity is thus a monolithic cavity.
Tuning of the system is done by changing the frequency of the
laser in a feedback loop, and not by varying the mechanical dimensions
of the cavity. Such a design has been successfully employed at
CEBAF~\cite{cebaf}.
\begin{figure}[htbp]
\begin{center}
\setlength{\unitlength}{1mm}
\begin{picture}(160,70)(0,-70)
\put(5,-142.5){
\epsfig{file=cavity_shema.eps,
bbllx=0pt,bblly=0pt,bburx=594pt,bbury=842pt,width=15cm}}
\end{picture}
\end{center}
\caption{A schematic view of a Fabry-P\'erot cavity and the laser system
located around the electron beam. The power indicated in the cavity is the
equivalent power a continuous wave laser must deliver to obtain the same
Compton scattering rate.}
\label{fig:cavity}
\end{figure}
\begin{figure}[htbp]
\begin{center}
\setlength{\unitlength}{1mm}
\begin{picture}(160,85)(0,-85)
\put(32.5,-102.5){
\epsfig{file=cavity_en.eps,
bbllx=0pt,bblly=0pt,bburx=594pt,bbury=842pt,width=9cm}}
\end{picture}
\end{center}
\caption{Layout of the cavity}
\label{fig:cavity2}
\end{figure}

The feedback system is able to correct for mechanical deformation of
the cavity if they stay below $70\mu m$. Therefore it is essential
that the cavity is stable at this level. The biggest problem is that
of thermal expansion and contraction of the system, if the ambient
temperature changes. A temperature change of
$2^\circ$C corresponds to the limit of an overall length change of
$70\,\mu$m.

Temperature changes are coming from changes of the temperature 
in the tunnel, and from local heating effects of the beam pipe inside the
cavity. The first problem
can be solved by placing the whole system in a temperature controlled
hut, maintaining a stable temperature within $1^\circ$C in the hut.
The temperature of the beam pipe at the present LPOL location has been
measured~\cite{schuler} for different intensities
of the e beam 
(between 0 and 48\,mA), and during different times within a fill. 
The maximum temperature variation found is about $\sim 2.5^\circ$C. 
In addition, some heat will be produced by higher order modes induced by the
beam due to the holes in the beam pipe needed for the laser tube. The power
lost has been estimated to be 18\,W during injection, and less than one W
during normal running~\cite{wipf}. This should not present a problem, since
in the current design, without considering heat conduction, purely by
radiative cooling the beam pipe heats by less than one degree for 25\,W
of heating power.

Another important source of concern is vibrations. One source 
originates from ground motion. A careful study of these has been made
in the HERA tunnel~\cite{Montag}. Other sources of vibrations under
investigation 
come from the machine itself, through cooling water circulation,
magnet power supplies, etc. 

The design shown in Fig.\ref{fig:cavity2} takes these effects into account.
Indeed, the optical elements of the cavity are mechanically decoupled
from the beam pipe and are supported from a stable optical platform.
The latter is isolated from the HERA tunnel through pneumatic isolators. 
The central part of the cavity is connected to two end pieces by means of
two bellows. Only the end pieces are in contact with the beam pipe. 
The central part of the cavity therefore is effectively thermally
isolated from the rest of HERA. Heat produced in the beam pipe
surrounding the cavity can only be transmitted by radiation, not by
thermal heat conduction. In addition the bellows act as a
vibration filter, blocking vibrations in the machine from reaching the
mirrors. Preliminary calculations suggest that this system will be
insensitive to the level of temperature changes and vibrations
expected in the tunnel. 

\subsection{Other Optical Components} \label{sec:optics}
%
In addition to the Fabry-P\'erot cavity, there are a few other optical
components mounted on the same optical table, in particular
the laser source (Nd:YAG, a small power solid state laser). 
The role of the these external optical elements is twofold: 
the laser beam has to
be transported into the cavity, and the properties of the laser beam have to
be measured and monitored. The laser beam has to be highly circularly
polarized (better than 99.5\%), and should be focused in the center of the
cavity with a waist of $\sim 0.6$\,mm.

A layout of the optical system is shown in Fig.\ref{fig:optics}. The main
parameters of the laser (spatial mode TEM00, linewidth $<5$\,kHZ, coarse
(fine) frequency tuning range 30\,GHz (30\,MHz)) are well adapted to the
system for both pumping the cavity and locking the system into resonance.
\begin{figure}[htb]
\begin{center}
\setlength{\unitlength}{1mm}
\begin{picture}(160,100)(0,-100)
\put(-8,-122.5){
\epsfig{file=optics.ps,
bbllx=0pt,bblly=0pt,bburx=594pt,bbury=842pt,width=14.5cm}}
\end{picture}
\end{center}
\caption{Layout of the optical components needed to operate the cavity,
where FI is the Faraday insulator, Mc1 and Mc2 are two mirrors located
inside the cavity, PDR, HBS, and WP stand respectively for photodetectors, 
holographic beam sampling, and Wollaston prism.}
\label{fig:optics}
\end{figure}

The laser is protected from light reflected from the entrance mirror through
a Faraday insulator. A filter is used to linearly polarize the laser light.
A system of $\lambda/2$ plate and beam splitter splits off a part of the beam
and reflects it into an integrating sphere and photodetector. These are
used to derive a signal for the mode-locking system.

A remote controlled $\lambda/4$ plate transforms the linearly polarized
laser light into circularly polarized light. Switching between left and
right polarized light is done by rotating the plate.

An optical telescope then focuses the laser beam into the cavity, and
matches the focal point of the laser with that of the cavity. One of these
lenses (L3 in the drawing) is remote-controllable, so that the focus can be
fine-tuned during operation.

It is important that the mirrors used to reflect the light into the
cavity do not disturb the circular polarization. Mirrors Mr1-Mr3
and Mr2-Mr4 are thus matched to have the same properties.
In order to give the system enough degrees of freedom for matching
the laser beam into the cavity, the mirrors Mr1 and Mr3 are furthermore
remote-controllable.

A small fraction of the laser power exits the cavity and is
reflected into a system of optical measurement devices. They are designed to
determine the exact polarization of the laser light and to measure the power
of the laser beam.


\subsection{Radiation Effects on the Cavity Mirrors}
%
A particular concern is the level of radiation exposure of the 
optical elements of the cavity, throughout its lifetime. 
During the month of July and August 2000, tests were done in 
HERA to investigate the effects of radiation on the 
proposed mirrors. 
The Laboratoire des materiaux avanc\'es (IPN LYON) lent us two mirrors
from the same batch used
for the CEBAF cavity. 
These mirrors were installed next to the HERA beam pipe, 
at the same place where the final cavity will locate. 
One mirror was exposed directly to the radiation levels, 
the other one was shielded by 2\,mm of lead. The dose absorbed 
by the mirrors was in both cases monitored with 
dosimeters, and was found to be $\approx 0.1$\,Gy/month inside 
the lead shielding, and around 10 times more without 
any shielding. 

After the exposure both mirrors were tested and the 
quality of the mirror coating was determined. 
Within the measurement accuracy no change in the 
properties of the mirrors were detected. 

\subsection{Data Acquisition System}
%
The significantly higher rates with which data are taken with the new LPOL
make an addition to the existing DAQ necessary. It is intended to continue
to use the existing slow controls and monitoring system, and only to add a
fast DAQ system to record and analyze the data from the few-photon operation
of the LPOL. 
%Compared to the system currently under construction for the 
%TPOL, significantly higher rates need to be handled for the new LPOL. 
In its basic option the new LPOL DAQ consists of two 
almost independent subsystems: 
\begin{itemize}
  \item a laser and cavity control system and 
its auxiliary electronics for laser alignment, feedback control, 
switching and monitoring of the cavity power and polarization. This 
system is similar as that developed 
for the CEBAF polarimeter;
  \item a VME based fast DAQ system for the few-photon mode LPOL, 
which is adapted from the upgraded TPOL DAQ. 
\end{itemize}
For a transition period the slow control will be handled through the DAQ 
of the existing LPOL system. However, a new system specific to this device
will have to be added when the new LPOL will eventually take over the
existing one.

\subsubsection{Fast LPOL DAQ}
%
The fast part of the DAQ for the new LPOL in essence consists of 
custom-made fast ADC's and TDC's, which are read out 
by a dedicated Power-PC based Multifunction Computing Core (MFCC), 
while a second Power-PC takes care of the 
analysis and control of the data. 
The bandwidth of the ADC allows for the transfer of 
two data words per HERA bunch, which is enough for the 
new system (calorimeter total energy and pedestal). 
The readout and storage of the data however will have to 
be handled differently from the TPOL to allow for the larger rates. 

\subsubsection{Laser and Cavity Control}
%
The laser control system is taken from the similar system developed for the
CEBAF polarimeter. The system has two operation modes: manual and automatic.
Under manual control the operator has direct control of the moving parts
with corresponding data displays. Under automatic mode the following
procedures are implemented:
\begin{itemize}
\item Laser feed back startup: The feedback electronics is started and the
finding of the resonance is monitored.
\item Laser beam alignment: 4 mirror motors, 8 monitoring diodes from 2 four
cell quadrants, and beam transmission monitors are used to control the laser
beam. A CCD camera permits a visual control of the resonant modes present in
the cavity.
\item Run control: Based on instructions received from the main run control,
parameters like the light polarization, the power etc are varied and monitored.
\end{itemize}

All components in the system are commercial ones, except for the laser
feedback electronics~\cite{loi}.

\subsection{Calibration}
%
One of the main advantages of operating the LPOL in
the single-photon mode is the simplicity in the calibration of the
detector. The energy spectra of the photons scattered into the
detector have a simple physical meaning, and provide a
number of unambiguous calibration points for the
energy calibration of the detector.
The main idea is to use the Compton edges for 1, 2, and maybe
three photons, and the bremsstrahlung edge in the energy spectra,
to define the energy scale of the calorimeter. In Fig.\ref{fig:spectrum}
the spectrum (left) and the differential spectrum (right) for a simulated
operation of the device with $n_\gamma=0.1$ and $n_\gamma=2$ is shown.
Clearly visible are the locations of the single and double Compton peaks, and
the position of the bremsstrahlung edge. Preliminary studies 
showed that this procedure allows 
a calibration of the device on a few per mill level. 
\begin{figure}[ht]
  \begin{center}
\setlength{\unitlength}{1mm}
\begin{picture}(160,75)(0,-75)
\put(-2.5,-82.5){
\epsfig{file=spectrum.eps,
bbllx=0pt,bblly=0pt,bburx=594pt,bbury=842pt,width=9cm}}
\put(77.5,-82.5){
\epsfig{file=nonlin.eps,
bbllx=0pt,bblly=0pt,bburx=594pt,bbury=842pt,width=9cm}}
\end{picture}
  \end{center}
  \caption{\label{fig:spectrum} Compton spectra (left) including background, 
  and
differential spectra (right), for $n_\gamma=0.1$ and 2. The dotted curves on
the right are obtained by including a 1\% nonlinear contribution in the
energy calibration of the calorimeter. The lowest curve is an estimate of
the background, obtained from the simulation when the laser is off.}
\end{figure}

To estimate the precision with which the polarization can be measured with
this device, a number of systematic effects have been simulated and their
impact on the polarization has been studied:
\begin{itemize}
\item The knowledge of the energy scale of the calorimeter 
  translates into an error on the polarization. Simulations have 
  shown that a $0.5\%$ uncertainty of the energy scale corresponds
  to an error of the polarization of $0.1\%$ for a $50\%$ polarization.
\item If a nonlinear component exists in the response of the 
  calorimeter this changes the analyzing power and therefore 
  the measured polarization. However, a 1\% effect at 10\,GeV leads to a
very large non-linearity in the position of the peaks (see right plot of
Fig.\ref{fig:spectrum}). Therefore, such a nonlinear effect can be controlled 
at the few per mill level.  Assuming a possible residual non-linearity 
of 0.2\% at 10\,GeV  we find a contribution to the uncertainty of the 
  polarization of a few per mill. 
  \item The polarization of the laser light inside the cavity is the 
  same as the polarization of the light entering the cavity 
  within $0.1\%$, as measured with the CEBAF cavity. 
  \item Fluctuations of the laser power inside the cavity 
  would change the form of the spectra, and in turn would influence 
  the way in which calibration constants are derived from the
  spectra. From CEBAF such fluctuations are known to be 
  controllable to better than $0.5\%$, which translates into 
  an error of the polarization of $0.05\%$.
\item The shape of the spectra is modified by additional material 
  in front of the polarimeter. As an example in Fig.\ref{nopb} the 
  effect of an additional 10\% of a radiation length of material in 
  front of the polarimeter is shown. At 50\% beam polarization, this
  translates into an error of the polarization of around 0.3\%.
  Therefore it is important that the amount of the material in front of the
polarimeter is carefully controlled at the 1\% of a radiation length level.
\item The properties of the electron beam directly influence the 
  distribution of photons on the front face of the calorimeter. 
  With a well calibrated device, with a homogeneous response across the 
  calorimeter, the influence of this can be reduced to the level of a few 
  pre mill. 
  In addition it is planned to install a fiber detector in front of the 
  calorimeter. This detector will be able to measure the position of the
  Compton cone. Therefore corrections can be applied as a function of the 
  Compton cone position. This scheme only works if enough data can be
accumulated fast enough by the system, on a time scale which is small
compared to the typical time scale of a precise determination of the
polarization in the machine. This is clearly the case when a precise 
determination of the polarization is done within a few minutes, so it
should not present any problem to the new system under construction.
\end{itemize}
\begin{figure}[htbp]
  \begin{center}
\setlength{\unitlength}{1mm}
\begin{picture}(160,100)(0,-100)
\put(20,-105){
\epsfig{file=sys_material.eps,
bbllx=0pt,bblly=0pt,bburx=594pt,bbury=842pt,width=12.5cm}}
\end{picture}
    \caption{Change in the polarization, if the photon beam has to traverse
an additional 10\% of a radiation length of material in front of the
calorimeter, for different levels of beam polarization and for different
Compton photon rates.}
    \label{nopb}
  \end{center}
\end{figure} 

In conclusion, it seems likely that the systematic errors of the polarization
can be controlled very well with the new LPOL. An essential condition is
that the LPOL can be operated in the single-/few-photon mode, and that the
polarization is precisely determined within a few minutes, on a time scale
short in comparison to the intrinsic variations of the polarization
in the machine.

\section{The Necessities of Having a Fast and Precise Polarization
Measurement} \label{sec:why}
%
Physics with polarized lepton beams will be a central part of the 
physics program of HERA after the luminosity upgrade. 
The basic program has been described in a series of workshops 
\cite{proce1}, \cite{proce2} and \cite{proce3}. 
We will discuss first in Sec.\ref{sec:phyics} the role the 
polarization will play in this and in Sec.\ref{sec:machine} its roles
from the machine's point of view.
 
\subsection{The Physics Case} \label{sec:phyics}
%
Three topics are of particular relevance: 
 the charged current (CC) cross-section measurement,
 the right-handed CC
 and the measurement of the light quark electroweak couplings.
In the following each of these topics will be discussed in detail.

\subsubsection{Charged Current Cross-Section Measurements}
%
A precise determination of CC cross-section 
at HERA can serve as a sensitive test ground of 
the Standard Model. One test is the helicity structure of 
the CC. Another such test is the precise 
measurement of the total cross-section, and its 
interpretation in the Standard Model. 

One particular 
interesting quantity is the mass of the W boson. Although 
no large data sample of directly reconstructed W bosons is available at
HERA, a precise indirect determination of the W mass is 
possible. As an example, with 
1000\,pb$^{-1}$ CC and NC data and a polarization of 
70$\%$, a total systematic error of 1\%
leads to a determination of $M_W$ with an accuracy of 55\,MeV.
This determination would be competitive to the LEP and Tevatron results and, 
because the W is accessed via the t channel at HERA, it would
provide a test of the universality of the Standard Model.

\subsubsection{Search for Right Handed Charged Currents} 
%
In the Standard Model of the electroweak and strong interactions,
CCs are carried by left-handed W bosons. Right-handed 
CCs should not exist. So far no experimental evidence 
for right-handed CCs has been found. The 
present limit, expressed in terms of the mass of a right-handed 
W boson, is  $750$\,GeV at 95\% confidence level (CL)~\cite{MWR_D0}.
In studying collisions between polarized leptons and unpolarized 
protons, as will be the case at HERA after the upgrade, 
sensitive and direct searches for 
right-handed CCs can be done. 

To estimate the reach of HERA we shall assume the following 
parameter: HERA operates with two helicities of the 
lepton (electron or positron) beam. The level of 
polarization for each helicity state is taken to be $50\%$. 
For each helicity an integrated luminosity of $250$\,pb$^{-1}$ 
is considered. This corresponds to about one year 
running per helicity state at the nominal luminosity of the 
machine after the upgrade. 

The observed CC cross-sections 
as a function of the lepton polarization $P$ is given by
\begin{equation}\label{sig2}
  \sigma_{\rm obs,CC}^{e^-}(P)=\frac{1-P}{2}\sigma^{e^-}_{L}+
                           \frac{1+P}{2}\sigma^{e^-}_{R}
\end{equation}
where 
$\sigma^{e^-}_{R}$ is the right-handed CC cross-section, 
$\sigma^{e^-}_{L}$ the left-handed CC cross-section,
and $P$ is the luminosity weighted polarization valid for the
whole data sample:
\begin{equation}\label{pweight}
   P =\frac{\sum_{i}^{N_{\rm run}}
      \int_{t_{{\rm min},i}}^{t_{{\rm max},i}}P_{i}(t){\cal L}_{i}(t)dt}
      {\sum_{i}^{N_{\rm run}}\int_{t_{{\rm min},i}}^{t_{{\rm max},i}}{\cal L}_{i}(t)dt}\;.
\end{equation}
In this equation the number of runs taken is denoted with $N_{\rm run}$, 
and  $(t_{{\rm max},i}-t_{{\rm min},i})$ is the duration of the $i$-th run, 
${\cal L}_{i}(t)$ and $P_{i}(t)$ are
the instantaneous luminosity and polarization respectively.

Using Eq.(\ref{sig2}) a measurement can be performed of the
right-handed CC cross-section, if the two data
samples taken with positive and negative helicities are combined.
Such a measurement with an assumed data sample of $250\,$pb$^{-1}$
is shown in Fig.\ref{fig-CC} for data at large $Q^2>1000\,{\rm GeV}^2$. 
\begin{figure}[ht]
  \begin{center}
\setlength{\unitlength}{1mm}
\begin{picture}(160,100)(0,-100)
\put(20,-110){
\epsfig{file=cc_sigma.eps,
bbllx=0pt,bblly=0pt,bburx=594pt,bbury=842pt,width=12.5cm}}
\end{picture}
  \end{center}
  \caption{\label{fig-CC} Measured charged current cross-sections 
  at positive and negative helicities, for $Q^2>1000$~GeV, and extrapolation 
  to the point of $P=+100\%$. The hashed band indicates the 
  extrapolation error assuming a data sample of $250$\,pb$^{-1}$ per 
  point (see text for more details). }
\end{figure}

To estimate the uncertainty of the right-handed CC cross-section, 
it is convenient to define the ratio 
$ r=\sigma^{e^-}_{R}/\sigma^{e^-}_{L}$. This ratio should be zero 
in the Standard Model. 
From the observed CC cross-sections, 
$\sigma_{\rm obs,CC}$, at 
$+|P|$ and at $-|P|$, the ratio of the measured quantities
$R=\sigma_{\rm obs,CC}^{e^-}(|P|)/\sigma_{\rm obs,CC}^{e^-}(-|P|)$
is defined, which can be expressed as a function of the theoretical 
ratio of cross-sections, $r$, as
\begin{equation}\label{asymm}
  R(|P|)=\frac{\alpha+r}{1+\alpha r}  \;,\;{\rm with}
\;\;\;\;\alpha=\frac{1-|P|}{1+|P|}\;.
\end{equation}
From this one obtains: 
\begin{equation}\label{rinvert}
  r=\frac{R-\alpha}{1-\alpha R}\;.
\end{equation}
Fig.\ref{pprime} shows how
$(\delta r)_{\rm stat}$ changes with the uncertainty on the polarization
measurement $(\delta P)_{\rm stat}$. 
\begin{figure}[htb]
  \begin{center}
\setlength{\unitlength}{1mm}
\begin{picture}(160,100)(0,-100)
\put(20,-110){
\epsfig{file=CC_error.eps,
bbllx=0pt,bblly=0pt,bburx=594pt,bbury=842pt,width=12.5cm}}
\end{picture}
  \end{center}
  \caption{\label{pprime} Relative uncertainty of the right-handed charged
  current cross-section, as a function of the error of the 
  polarization measurement. The solid curve is for $50\%$
  polarization, the dashed one for $70\%$ polarization. Also indicated 
  is the contribution to the uncertainty which does not depend on the 
  polarization. }
\end{figure}
 The two solid curves
shown correspond to different levels of polarization of the lepton
beam, of $50\%$  and $70\%$. It is clearly visible that a 
higher polarization minimizes the error on $r$. 

In this measurement, one must carefully distinguish systematic uncertainties
which are proportional to the CC cross-section (scaling systematics:
luminosity uncertainty, hadronic energy scale uncertainty, etc) from those
which are independent of the cross-section (additive systematics:
background). A measurement of the right-handed CC cross-section is only
sensitive to the second part, the additive systematic uncertainties. The
luminosity uncertainty, at least in first order, does not enter the
extrapolation to $P=+1$. From the current CC analysis in the H1 experiment
for the high $Q^2$ analysis ($Q^2>1000\,{\rm GeV}^2$), the first type of
uncertainties is estimated to be around 4\%. Therefore, to a good
approximation, the right-handed cross-section uncertainty depends only on
the statistical accuracy of the cross-section measurements and on the
statistical and systematic uncertainties of the polarization measurement.

From these studies, one concludes that in order that the errors from the
polarization do not limit the final accuracy of the measurement, the total
error of the polarization should be kept around $2\times 10^{-3}$.

This simple picture changes slightly if the polarization measurement
introduces correlation between the measured cross-section. If e.g.\
systematic uncertainties are fully correlated between the cross-sections at
different beam helicities, it can be easily shown, that
\begin{equation}
(\delta r)_{\rm sys}\approx \frac{(\delta P)_{\rm sys}}{2}\,,
\end{equation}
where $(\delta P)_{\rm sys}$ is the correlated part (scaling or additive) of
the systematic uncertainty of the polarization measurement. Such effects
therefore should be controlled at the 0.5\% level in order not to limit the
precision on the final result.

The reach of HERA compared to other experiments is 
best expressed in terms of a mass of a hypothetical 
right-handed W boson. Given the current estimates of 
systematics, and assuming a high precision 
determination of the polarization to $0.2\%$, the results 
are summarized in Fig.\ref{fig-HERAlimit}.
It shows clearly that for competing with the Tevatron we need a very large
data sample. If the uncertainties can be 
tightly controlled, the reach of HERA becomes comparable to 
that of the Tevatron with the advantage of a pure and direct measurement. 
\begin{figure}[htb]
  \begin{center}
\setlength{\unitlength}{1mm}
\begin{picture}(160,100)(0,-100)
\put(20,-110){
\epsfig{file=M_WR_2.eps,
bbllx=0pt,bblly=0pt,bburx=594pt,bbury=842pt,width=12.5cm}}
\end{picture}
  \end{center}
  \caption{\label{fig-HERAlimit}Possible exclusion limit ($95\%$ CL)
  obtainable at HERA, for a right-handed W boson, as a function of
  the precision of the polarization measurement. Full curves are for
$P=0.7$, dotted ones for $P=0.5$. For each polarization value, the top curve
corresponds to 1000\,pb$^{-1}$, the middle one to 500\,pb$^{-1}$, and the
lower one to 250\,pb$^{-1}$.}
\end{figure}

In conclusion the investigation of the right-handed CC requires the highest
possible precision from all parts of the experiment and the machine:
luminosity, polarization and detector systematics all should be controlled
very tightly to maximize the reach of HERA.

\subsubsection{Neutral Current Cross-Section Measurements} 
%
The cross-section for the neutral current (NC) events in 
polarized lepton-proton scattering can be written as : 
\begin{eqnarray}
\label{eq-NC}
  \sigma_{\rm NC}^{e^{\pm} p}\ &=&\ \frac{2 \pi \alpha^2}{x Q^4} 
  \left(H_0^{\pm} + P H_P^{\pm} \right) \ \ \ \ {\rm with}\nonumber \\
  H_{0,P}^{\pm}\ &=&\ \left(1+(1-y)^2\right)\ 
  F_2^{0,P}\ \mp\ \left(1-(1-y)^2\right)\ xF_3^{0,P}\;. 
\end{eqnarray}
\noindent
For large $Q^2$, where the 
longitudinal function $F_L$ can be neglected,
$F_2^{0,{\cal P}}$ and $xF_3^{0,{\cal P}}$ are the deep-inelastic-scattering 
structure functions and contain linear combination of the quark densities and
quadratic functions of the vector and
axial couplings of the fermions to the $Z^0$.
By measuring the NC cross-section for different 
lepton charges and different helicity states the vector and 
axial coupling constants  $v_u$,
$v_d$, $a_u$ and $a_d$ can be measured.

The analysis consists of a $\chi^2$ fit to four
NC differential cross-section measurements: 
$e^-_{+|P|}$, $e^-_{-|P|}$, 
$e^+_{+|P|}$ and $e^+_{-|P|}$. 
Unknown parameters of this fit are the light quark electroweak
couplings $a_u$, $v_u$, $a_d$ and $v_d$. 

In Fig.\ref{upsys} the $1 \sigma$ contour plot in the 
$a_u-v_u$ plane is shown. 
It was assumed that at each of 
the four data points an integrated luminosity of 
$250\,{\rm pb}^{-1}$ is available. 
\begin{figure}[ht]
  \begin{center}
\setlength{\unitlength}{1mm}
\begin{picture}(160,100)(0,-100)
\put(25,-107.5){
\epsfig{file=up_lep.eps,
bbllx=0pt,bblly=0pt,bburx=594pt,bbury=842pt,width=11cm}}
\end{picture}
  \end{center}
  \caption{\label{upsys} Couplings constants for up-type 
  quarks, as measured at LEP, and expected measurement at HERA.}
\end{figure}
The ellipse corresponds to a 50\% lepton beam 
polarization. Also shown is a second shifted ellipse which corresponds to
the result of a measurement if a systematic shift of 1\% is present in the
polarization measurement. From this curve it is clear that such systematic
effects on the polarization should be controlled to be better than one percent.

In conclusion the determination of the NC coupling constants for light
quarks at HERA will provide a stringent test of the consistency of the
Standard Model. To achieve a precision comparable to that of LEP a
determination of both luminosity and polarization to around 1\% is needed.

\subsection{Other Considerations} \label{sec:machine}
%
In this section, we discuss a number of other considerations which also
point to the necessity of having a
fast and precise polarization measurement.

\subsubsection{Luminosity - Polarization Correlations}
%
The effective polarization needed for the different 
measurements discussed is a luminosity weighted sum, 
as defined in Eq.(\ref{pweight}). Based on 
typical HERA run durations of around 12h, and a typical 
turn-on time of the experiments after the declaration of 
physics, the effective polarization in the 
machine is reduced from its asymptotic value by $6-10\%$
(e.g., assuming an asymptotic polarization of $50\%$, 
averaged over the run the luminosity weighted polarization 
is may be $47\%$). 

In Fig.\ref{fig:tpol}, the measured polarization from a typical HERA fill is
shown, as a function of the bunch number. Two things are visible: there are
clear differences between colliding and non-colliding bunches, and there
appears to be a clear structure in the polarization of the colliding bunches. 
\begin{figure}[htb]
  \begin{center}
\setlength{\unitlength}{1mm}
\begin{picture}(160,52.5)(0,-52.5)
\put(2.5,-131){
\epsfig{file=tpolmoncol.ps,
bbllx=0pt,bblly=0pt,bburx=594pt,bbury=842pt,width=15cm}}
\end{picture}
  \end{center}
  \caption{\label{fig:tpol} Transverse polarization measured during the test
of the new TPOL DAQ system, as a function of the bunch number. Data points
with polarization $P\approx 0.55$ are the non-colliding bunches.}
\end{figure}
In Fig.\ref{fig:pol_time}, the variation of the polarization is shown as a
function of time throughout a fill. The observed variations are seen by both
polarimeters. Typical time-scales for these variations are a few
10 minutes. Taking both plots together it is important that the
polarization of individual bunch can be tracked with sufficient accuracy
($<1$\%) on a time-scale of a few minutes per point. In this way, a
correction to the overall polarization can be done, based on the actual
polarization of the bunches and on their individual behavior.
\begin{figure}[htb]
  \begin{center}
\setlength{\unitlength}{1mm}
\begin{picture}(160,70)(0,-70)
\put(12,-155){
\epsfig{file=lpol_tpol_time.ps,
bbllx=0pt,bblly=0pt,bburx=594pt,bbury=842pt,width=14cm}}
\end{picture}
  \end{center}
  \caption{\label{fig:pol_time} Transverse and longitudinal polarizations
averaged over all bunches, for a given fill, as a function of time. Time
variations are observed simultaneously by both calorimeters.}
\end{figure}

Since the luminosity is highest early in the fill, it would be good that
these data are also used in the analysis. For this, one has to take possible
correlations between the polarization and the luminosity into account. There
are two possible sources of correlation: bunch-to-bunch correlation and time
evolution. The latter source has already been described above. To control it
one must measure accurately the polarization during the early phase of the
polarization during its buildup. This implies a very fast polarimeter.

The time evolution of the bunch-to-bunch correlation coefficient between
$P$ and a variable $X$ is defined as:
\begin{equation}
C^t(P,X)=\frac{\left[\sum^N_{i=1}(P_i-\overline{P})\right]
\left[\sum^N_{i=1}(X_i-\overline{X})\right]}{\sqrt{\sigma^2_P+\sigma^2_X}}
\end{equation}
with $\overline{P}$ and $\overline{X}$ being respectively the average of
$P_i$ and $X_i$. The summation is over all colliding bunches $N$, and $X$
can be e.g.\ the luminosity ${\cal L}$, the lepton beam intensity $I_e$ or
the proton beam intensity $I_P$. In Fig.\ref{fig:correlation}(a) the
polarization as a function of time is shown for a typical HERA fill. For
this fill, the correlations to luminosity, the electron and proton currents
are shown in Fig.\ref{fig:correlation}(b), (c), and (d). The luminosity and
the current intensity data are taken every 10\,s and the polarization is
calculated for 10\,min intervals.
\begin{figure}[htb]
  \begin{center}
\setlength{\unitlength}{1mm}
\begin{picture}(160,75)(0,-75)
\put(0,-175){
\epsfig{file=correl_vs_time_proposal.ps,
bbllx=0pt,bblly=0pt,bburx=594pt,bbury=842pt,width=16cm}}
\end{picture}
  \end{center}
  \caption{\label{fig:correlation} Polarization (a) and its correlation with
luminosity $C^t(P,L)$ (b), lepton beam intensity $C^t(P,I_e)$, and proton
beam intensity $C^t(P,I_P)$ (d) as a function of time for one typical fill.}
\end{figure}

With the present accuracy it is, however, difficult to determine if a
significant level of correlation exists. To do so a precise bunch-by-bunch
polarization measurement is desirable. 

\subsubsection{Optimization of the Polarization}
%
From the above discussions it is clear that the most important input to the
physics reach of HERA for polarized physics is, aside from the luminosity,
the degree of polarization in the machine. The higher the polarization, the
smaller the errors are. The optimization of the polarization therefore will
play a crucial role in the running of the machine after the upgrade.

One way to reach higher polarization is to take advantage of the
time-to-time and bunch-to-bunch fluctuations (see Figs.\ref{fig:tpol} and
\ref{fig:pol_time}). To use this possibility, a fast polarimeter, allowing a
precise bunch-by-bunch polarization measurement, is required.

At the moment the polarization is optimized in an empirical procedure, 
by using so-called harmonic bumps. These are ensembles of 
magnets, arranged at different locations symmetrically around the
ring. They are used to spin-match the machine, and also 
to compensate for the effect from magnet misalignments. 

An important input in the procedure is the effect on the 
polarization. A fast and precise measurement of the 
polarization can help to speed up the procedure, to 
detect changes in the polarization faster, and to 
start corrective actions earlier before a significant loss 
of polarization occurs. 
The result of an adjustment of the harmonic bumps is shown in 
Fig.\ref{fig_polar}.
The steps observed in the polarization are the result of an optimization. As can
be seen the time between adjustments is of the order of hours, before the
full effect of an optimization can be judged.
\begin{figure}[htb]
  \begin{center}
\setlength{\unitlength}{1mm}
\begin{picture}(160,40)(0,-40)
\put(-2.5,100){
\epsfig{file=polarisation.eps,
bbllx=0pt,bblly=0pt,bburx=594pt,bbury=842pt,width=16cm,angle=-90}}
\end{picture}
  \end{center}
  \caption{\label{fig_polar}Development of the polarization 
  throughout a typical HERA fill. Visible are a number of 
  increases of the polarization, which are the result of
  harmonic bump optimizations.}
\end{figure}

A fast and precise measurement of the polarization can 
help in two ways: 
\begin{itemize}
  \item a change of the polarization after a change in the 
  machine parameters can be quickly detected. A feedback 
  therefore is available to the machine on whether or 
  not a particular operation was successful;
  \item if the measurements is precise enough a determination of 
  the speed with which the polarization changes after an 
  operation can be made. This information can be used to 
  determine the theoretically possible maximum polarization 
  in the machine, if no further changes are done. 
  Thus at least in theory it is possible to perform a 
  number of optimization steps within a relatively short time, 
  optimizing the asymtotic polarization without 
  having to wait each time for the full effect of the 
  preceding optimization step.
\end{itemize}
Note that in order to avoid oscillations the measurement does 
not only need to be fast, but also precise, to avoid taking
decisions based on statistical fluctuations.

To estimate what one could gain with a fast and precise polarization 
 measurement, a simple model is considered. The rise-up curve is
 modeled by
\begin{equation}\label{polar-formula}
P(t)=P_\infty-(P_\infty-P_0)e^{-\frac{t-t_0}{\tau}}
\end{equation}
where $P_0$ is the polarization at $t=t_0$ and
 $P_\infty=P_{\rm ST}/\tau_{\rm ST}\times \tau$ with $P_{\rm ST}=0.916$, 
 $\tau=43.2$\,s.
$\tau_{\rm ST}$ is the intrinsic rise-time of the polarization in the
machine, and $\tau$ the time of the measurement.

It is conceivable to install a simple feedback in the machine, which would
help keeping the polarization at a maximum. This procedure would rely on a
fast and precise determination of the polarization in the following ways:
over a time interval $\Delta T$ the polarization is measured each a number
of times, e.g.\ every 6\,s, and the data points are fitted with a simple
model, as e.g.\ given in Eq.(\ref{polar-formula}). If the fit is based on
data point of sufficient precision, a prediction can be made whether the
polarization in the machine is stable, or is changing in a significant
fashion. Among a large number of decision criteria, we have investigated the
determination of the asymtotic value $P_\infty$: after a period $\Delta T$
one decides if a significant polarization change is observed if the
uncertainty $\delta P_\infty$, determined by the fit, is smaller than a
given value. However, such criteria are reliable only for a short period of
time because of machine changes and would be, in this case, more efficient
if applied on the non-colliding bunches polarization measurement.

Fig.\ref{fig:deltat} shows $\Delta T$ as a function of $P_0$ (see
Eq.(\ref{polar-formula})) for $P_\infty=0.5$ and a required $\delta
P_\infty=1$\%. The three curves correspond to different polarization
measurement accuracies $\delta P=0.1$\% (colliding bunches), 0.3\%
(non-colliding bunches) and 1\% (individual bunch).
\begin{figure}[htb]
  \begin{center}
\setlength{\unitlength}{1mm}
\begin{picture}(160,100)(0,-100)
\put(20,-110){
\epsfig{file=bump_optimisation.eps,
bbllx=0pt,bblly=0pt,bburx=594pt,bbury=842pt,width=12.5cm}}
\end{picture}
  \end{center}
  \caption{\label{fig:deltat}$\Delta T$ as a function of $P_0$ (see text) for
$P_\infty=0.5$ and $\delta P=1$\%. The three curves correspond to different
polarization measurement accuracies $\delta P=0.1$\%, 0.3\% and 1\%.}
\end{figure}

From this simple simulations and from comparisons to typical HERA fills, it
is found that reliable statements can be made if the polarization is
measured over periods of a few minutes with an accuracy of a few per mill
over a few seconds.

\section{Conclusion} \label{sec:sum}
%
The new longitudinal polarimeter under construction uses a Fabry-P\'erot
cavity to significantly gain the effective power of a laser light.
The increased power will enable us
to operate the LPOL in the single-/few-photon mode, and still reaching a
high statistical accuracy of 0.2\%. The new polarimeter has a number of 
important advantages over the existing ones:
\begin{itemize}
\item The polarimeter can be calibrated using the same data as for the
polarization measurement;
\item Systematic errors can be tracked and controlled based on information
obtained from the data;
\item Fast and reliable feedback can be given to the machine, helping to
optimize the polarization.
\end{itemize}

The precision which this device should be able to reach will be around
$0.2-0.4$\% on the polarization. This will guarantee that the physics
potential of HERA is not limited by the polarimetry, and that high
precision measurements for the study of neutral and charged currents are
possible.

Based on all investigations made so far, we are confident that the
new polarimeter will meet the goals both in terms of performance and in
terms of reliability.


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\end{document}