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\title{GlueX Requirements for 12~GeV \\
Electron Beam Properties}
\author{Richard Jones, University of Connecticut}
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\begin{center}
Draft 1
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\begin{abstract}
The electron beam requirements for the GlueX experiment were first
published in the GlueX Design Report.  The latest version of the design
report that was released in November 2002 presents the same list of
requirements that were shown at the GlueX Detector Review in October 2004
and at the Lehman Review in July 2005.  Considering the time that has
passed since these requirements were set and the progress that is now
being made in optimizing the design of CEBAF for 12~GeV operation,
the GlueX collaboration has been asked to conduct a careful review of
its requirements.  The purpose of this note is to examine key requirements
in a quantitative way and show how the requirements are connected to
the ability of the GlueX experiment to reach its ultimate physics goals.
\end{abstract}

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This note is intended to be a self-contained description of the GlueX
requirements for the 12~GeV electron beam for Hall D at Jefferson Lab.
It has been written with two purposes in mind.  The first is to serve
as a record for the GlueX collaboration of how these requirements were
defined and what implications each requirement has for the physics of
GlueX.  The second purpose is to explain these requirements to 12~GeV
Upgrade Project leaders and designers, providing quantitative input 
that will enable them to make design decisions that will maximize the
physics potential of GlueX.  The reader who is only interested in the
final requirements can skip the bulk of this paper and to directly to
the final section entitled ``Graduated Requirements.''  The reader who
wants to know how these requirements were obtained and to see how 
sensitive the experimental results of GlueX are to a given beam
parameter should read all sections below in order.

Beam requirements (performance that is essential to the success
of the experiment), as opposed to design goals (performance that 
would be helpful to achieve if possible), are the focus of this note.
This does not mean, however, that every requirement can be
considered a Boolean variable.  For example, missing the 12~GeV
energy goal by 50~MeV would not disable the GlueX experiment.  This
paper not only explains the basis for the 12~GeV energy requirement
but uses metrics to show what the consequences would be if the energy
were less than 12~GeV by a given amount.  A similar argument is made
for the emittance requirement.  The following sections
show that electron beam energy and emittance are of similar importance
in terms of their impact on the physics potential of GlueX.
The GlueX experiment is apparently unique among Jefferson Lab
experiments in that its required emittance is close to the ideal
limits of the machine.  For this reason, special attention is given
in this paper to the implications of electron beam emittance for the
GlueX experiment.

The GlueX collaboration is aware that some conceptual design work has
already been carried out for the 12~GeV upgrade and we are encouraged
that results so far appear to be compatible with the GlueX requirements.
However for the purposes of this document, the only design parameter
that is being taken as given is the maximum energy of 12~GeV.  All other
intrinsic beam parameters have been studied as free variables without
factoring in cost or technical difficulty.  The information contained
in this document is intended to provide the 12~GeV upgrade team with
the quantitative information that is needed to make decisions based
upon all factors.

\section{\label{sec:metrics}Metrics}

Making the connection between beam properties and experimental results
quantitative requires finding an appropriate set of metrics to quantify
the effects of variations in electron beam parameters on experimental
results.  Electron beam properties translate into photon beam properties,
and photon beam properties determine the rate at which the GlueX
experiment advances towards its physics goals.

The GlueX experiment requires a photon beam of approximately 9~GeV.
This energy is high enough to access mesons in the mass region of 2.0
to 2.5~GeV/c$^2$ where exotic hybrid mesons are expected to appear,
yet low enough to allow measurement of charged particle momenta in
a solenoidal detector with good resolution.  Linear polarization is
needed to identify the spin-parity of produced meson states.

The polarized photon beam is generated from the electron beam by the
process of coherent bremsstrahlung in a thin diamond crystal.
Of course, photons of 9~GeV require electrons of at
least 9~GeV, but photon energy is not the consideration that drives
the choice of electron energy.  The two properties of the photon beam
that drive the electron beam requirements are photon beam polarization
and tagged photon intensity.  Associated with these two properties are
the two metrics that will be the focus of this note, the {\em
polarization figure of merit} and the {\em tagging efficiency}. 

The polarization figure of merit is the product of the intensity of
the tagged photon beam at the entrance to the GlueX target multiplied
by its mean-square linear polarization.  More precisely,
\begin{equation}
\mbox{\em fom} = \int_{E_0}^{E_1}{\frac{dN}{dE}P^2(E)\,dE}
\label{eq:fom}
\end{equation}
where $dN/dE$ is the spectral intensity of the collimated photon
beam, $P(E)$ is its linear polarization, and $E_0$ and $E_1$ are the
limits of the tagged region of the spectrum.  This expression is
derived by considering a random variable $x$ that has a frequency
spectrum $N_+(x)$ when the beam is polarized $+1$ and $N_-(x)$ when
the beam is polarized $-1$.  The polarization asymmetry of $x$ is
defined as the ratio $A = (N_+ - N_-)/(N_+ + N_-)$ in the limit
of 100\% polarization, which must be multiplied by the factor $1/P$
for the more general case of polarization $P$.  Here one must
keep in mind that the polarization of a coherent bremsstrahlung
beam $P(E)$ depends on the photon energy $E$.  Standard error
analysis shows that the statistical error on the extracted value
for $A$ is inversely proportional to the square root of the
experimental run time.  If this relationship is expressed in
terms of the variance $V(A)$ then one finds that the quantity
$1/V(A)$ is proportional to the run time of the measurement with
a proportionality constant given by Eq.~\ref{eq:fom}.  Thus the
figure of merit is inversely proportional to the run time required
for an experiment to achieve a given level of statistical precision
on a measured asymmetry, or stated another way, the figure of
merit measures the rate at which the experiment is progressing
towards some goal of statistical precision on a polarization
observable.  

The tagging efficiency of a tagged photon experiment is defined as
\begin{equation}
\varepsilon_i = \frac{R_i}{S_i}
\label{eq:te}
\end{equation}
where the subscript $i$ refers to a given energy bin in the tagged
beam energy spectrum, $S_i$ is the counting rate for that bin in
the tagging spectrometer focal plane, and $R_i$ is the rate of
tagged photons entering the experimental target within that same
energy window.  While it is generally valid to assume that the rate
$R_i$ includes all beam gammas entering the target within a given energy
range, this definition includes only {\em tagged} gammas, i.e.\ those
that are in coincidence with an electron that was actually detected
in the given tagging focal plane channel.  Defined in this way, the
variable $\varepsilon_i$ is always in the range [0,1) and depends on
photon energy.  Tagged photon experiments are typically designed to
operate with tagging efficiencies of order 90\% or greater within the
desired energy window. The GlueX experiment, however, plans to run
with tagging efficiencies of order 50\% in the region of interest
because of tight collimation that is necessary
for reasons that require a brief diversion to explain.

Bremsstrahlung photons with linear polarization are produced
when the electron scatters from the oriented planes of atoms in the
diamond crystal. For a fixed photon energy the degree of linear
polarization increases approximately as the square of the
difference between the photon and electron energies.
This polarization is significantly enhanced
by strictly collimating the bremsstrahlung photons downstream of the
radiator, and the ability to effectively collimate is dependent on
the low emittance of the electron beam at the diamond radiator.
The thickness of the diamond radiator has been chosen to be small
enough that multiple-scattering makes a relatively small contribution
to the angular spread of the electron beam in the radiator.
It should be noted that the collimator affects both factors inside
the integral of Eq.~\ref{eq:fom}, by increasing the polarization $P$
and by reducing the tagging efficiency and hence the collimated
intensity $dN/dE$.

The photon beam produced through the bremsstrahlung process contains
photons of all energies up to the energy of the electron beam.  The
GlueX experiment can tolerate the presence of all of these photons in
the beam, provided that it has some means of distinguishing them
from those whose energies lie in the desired coherent peak.
GlueX plans to do this using the tagging
technique in which a coincidence is formed between the products of
photon interactions in the GlueX detector and an electron in the tagging
spectrometer.  An electron which enters the diamond radiator with 12~GeV
and radiates a 9~GeV photon as it passes through the crystal exits with
only 3~GeV.  Instead of being deflected with the 12~GeV electron beam
into the dump it is bent into an array of detectors called tagging
counters.  A hit in the tagging counter at 3~GeV amounts to a a
prediction that a 9~GeV photon is on its way toward the GlueX target.
Of course, not all 9~GeV photons that exit the radiator actually arrive
at the target because the collimator removes those photons which are
produced at large bremsstrahlung angles.  The tagging efficiency measures
the fraction that pass through the collimator and hit the target,
provided that background rates in the tagging counters are negligible.

Tagging efficiency is a critical parameter for the GlueX experiment
because the tagging counters must operate at a very high aggregate rate
in order not to limit the rate capabilities of the GlueX detector and 
trigger.  At such high rates there is a significant number of accidental
tags in which a false association between a beam photon event and a tagger
electron hit occurs in the same beam
bucket.  The rate of experimental triggers scales linearly with photon
beam intensity, whereas the accidentals rate scales with the product of
the photon beam intensity and the rate of electrons in the tagger, or
in other words the photon beam intensity squared divided by the tagging
efficiency.  An upper bound on the operating beam intensity that is
compatible with tagging is obtained by requiring that the rate of
accidental tags be some small fraction of the total tagged event rate.
For example, the maximum rate of $10^8$ tagged photons on target per
second at which GlueX is designed to run was obtained by limiting the
accidental tagging fraction to 30\% under nominal beam conditions.

The accidental tagging fraction itself is not suitable as a beam
performance metric because its value is a choice, not a given, and
the choice of an appropriate value depends on a balance of
event rate and background considerations which depend in turn on the
physics emphasis during a given run period.  What one would like to do
is to fix the tagged photon event rate and then obtain as low an
accidental fraction as possible.  This the equivalent to maximizing
the tagging efficiency.

Both of these two metrics have a similar straight-forward interpretation
as the rate at which the experiment is making progress towards its
scientific goals.  In general it is best to treat them separately
because each is sensitive to different combinations of electron beam
parameters and they affect the experimental results in different ways.
However the two metrics are coupled in that the polarization figure of
merit is proportional to the tagged beam intensity, and the tagging
efficiency is one of the factors that determines the upper bound on
usable beam intensity.
%In cases where variations in a single beam
%parameter significantly degrade both the tagging efficiency and the
%polarization, the metrics are combined by taking the tagging efficiency
%to limit the rate that goes into the polarization figure of merit.
In cases where the two metrics are considered individually, care must
taken to avoid double-counting by using a criterion other than
tagging efficiency to define the rate that goes into the
polarization figure of merit.

\section{\label{sec:parameters}Electron Beam Parameters}

Two different categories of electron beam properties were considered
in this study, intrinsic and extrinsic.  Intrinsic properties like
beam energy and emittance are determined by the accelerator, while
extrinsic properties like beam dispersion and transverse size can
be selected by the choice of optics in the beam delivery system.
GlueX photon beam properties depend on both.
The focus of this note is on the intrinsic properties.
In the following sections five intrinsic beam properties of
relevance to GlueX are examined and their implications for the
experiment quantified in terms of the metrics described above.
A discussion of extrinsic properties follows, showing how the results
for intrinsic properties can be used to derive the sensitivity to
optics parameters.

Each of the GlueX beam metrics is a multi-dimensional function
of many parameters.  The exploration of their dependence on electron
beam parameters begins with a reference configuration that is
internally consistent and that preliminary studies have shown to
be close to optimum for GlueX.  Individual beam parameters are
then allowed to vary in either direction relative to the reference
values and the resulting changes in the metrics displayed.
Consistency sometimes requires that more than one parameter be
varied at once. For example, the focal spot size must grow when
the beam emittance increases to prevent the beam spot size at
the radiator from expanding outside the dimensions of the crystal.
In cases like this, all relevant constraints have been applied
when the independent parameter is varied so as to always maintain
a configuration that is physically self-consistent.

The following conditions constitute the reference configuration.
The electron beam energy is 12~GeV.  The crystal is mounted
so that the primary peak corresponds to reciprocal lattice
vector $(0,2,\overline{2})$ and the non-defining crystal angle
is 50~mr.  The orientation of the crystal is adjusted to keep
the primary coherent edge at 9~GeV.

The emittance of the electron beam is $10^{-8}$~m$\cdot$r in the
horizontal plane and $2.5\cdot 10^{-9}$ ~m$\cdot$r in the vertical.
The beam spot at the crystal is an elliptical Gaussian with a
horizontal r.m.s.\ width of 1.6~mm and a vertical r.m.s.\ of
0.5~mm.  The electron beam at the radiator is converging in
the horizontal direction so that it forms a virtual focus at
the entrance face of the collimator whose r.m.s.\ radius is
given by the beam emittance and the spot size at the radiator.
In the vertical plane the beam is weakly converging at the
radiator and forms a weak focus mid-way between the radiator
and the collimator, so that the vertical r.m.s.\ beam spot size
at the radiator and collimator are the same.

The collimator is treated in these studies as a perfectly
absorbing disk with a perfectly transparent circular hole
in the middle.  The virtual focal spot of the
electron beam is centered on the collimator hole.  The
tagging efficiency is computed under the simple assumption
that every electron that exits the radiator with a given
energy is detected in the corresponding tagging counter.
The quoted tagging efficiency is the unweighted average
value over the tagging range from 8.4 to 9.0~GeV.
Tagging efficiency improvements coming from the vertical
segmentation of the tagger hodoscope are not taken into
account in this study, but are expected to hold independent
of variations in electron beam parameters over most of the
range covered in this study.  There is no electron beam
halo in the reference configuration.

The beam profile at the radiator has been kept fixed
throughout all studies because it minimizes the size of the
virtual focal spot on the collimator consistent with the
combined constraints of the size of the crystal, the ratio
of approximately 4 between horizontal and vertical emittances,
and the fact that the virtual focal spot should be circular.
Variations in the emittance in this study were applied by
proportionally varying the focal spot diameter while leaving
the spot at the radiator unchanged.

\subsection{\label{sec:parameters:energy}electron energy}

Electron beam energy is an important beam property for achieving
the physics goals of GlueX.  The tagging efficiency for 9~GeV photons
is essentially independent of electron beam energy, but the polarized
beam intensity and polarization are strongly affected.  Beam intensity
is controlled by electron beam current, so the reader may wonder how
this is coupled to the beam energy.  The reason for this is tied to the
presence of all of the low-energy photons that are present in the photon
beam together with the tagged photons are 9~GeV.  At the low-energy end
of the photon beam spectrum the spectral intensity scales like $1/k$
where $k$ is the photon energy, so in terms of photon count the beam is
dominated by low-energy photons.  These low-energy photons undergo both
electromagnetic and hadronic interactions in the target which are the
primary source of background in the detector.  The GlueX detector and
trigger are designed to operate in the presence of these backgrounds
and ignore them, but there are limits on how high a background
is acceptable.

Independent of what those exact limits will be, it is possible to compute
a relative figure of merit as a function of electron beam energy.
The results are shown in Fig.~\ref{fom_energy} for a variety of
collimator diameters.  These curves were generated by computing
the square of the average polarization in the photon beam multiplied
by the flux of tagged photons between 8.4 and 9~GeV divided by the
total hadronic interaction rate in the target.  Each curve
is a smooth interpolation between points at several discrete energies
over the range 9 - 13~GeV.  Normalizing to the total electromagnetic rate
instead of the hadronic rate would produce essentially the same result.
The lowest curve in the figure corresponds to the photon beam
without any collimator in place.  It is included to show that the
improvement in the beam polarization figure of merit in going from
9~GeV to 12~GeV electrons is mainly achieved through collimation. 
This fact is relevant in considering the role played by
electron beam emittance, to be discussed next.

\begin{figure}[t]
\begin{center}
\includegraphics[width=5in]{fom_E.eps}
\caption{\label{fom_energy}
Polarization figure of merit (arb. units) as a function of
electron beam energy for the reference design with (a) a 1.6~mm diameter
collimator, (b) a 3.2~mm diameter collimator, (c) a 6.4~mm diameter
collimator, and (d) without a collimator.  The vertical axis has been
normalized to unity for an uncollimated beam at 12~GeV.  The reference
configuration is represented by the intersection of curve (b) with the
vertical grid line at 12~GeV.}
\end{center}
\end{figure}

\subsection{\label{sec:parameters:emittance}transverse emittance}

The definition taken here for the transverse emittance of the electron
beam is
\begin{equation}
\epsilon_x = <x^2 x'\,^2> - <xx'>^2
\label{eq:emit}
\end{equation}
where $x$ represents one of the two transverse coordinates of a
particle in the beam with respect to an origin at the center of
the beam, and $x'$ is the corresponding slope of the particle's
trajectory with respect to the nominal beam axis.  This corresponds
to the product of the half-axes of the 1-sigma beam ellipse in
the transverse coordinate.  In the following discussion, $\epsilon_x$
refers to the horizontal and $\epsilon_y$ to the vertical emittance
in a reference system where $z$ represents the nominal beam direction.
This is the same definition of beam emittance as is used by the
Jefferson Lab beam physics group.

The electron beam emittance at the entrance to the crystal radiator
is very important to the GlueX experiment.
Having a small-emittance electron beam, along with a thin (20 micron)
diamond radiator, is what makes it possible to
use strict collimation to significantly enhance the photon beam 9~GeV
flux and polarization.  Both the tagging efficiency and the polarization
figure of merit are sensitive to emittance.  The tagging efficiency is
plotted versus horizontal beam emittance in Fig.~\ref{te_emittance}.
The polarization figure of merit is shown in Fig.~\ref{fom_emittance}.
Both figures show results obtained for three different collimator
apertures.

Increasing the emittance increases the photon spot size on the
collimator.  Because of this, the photon intensity at the GlueX
target tends to decrease with increasing emittance for fixed
electron beam current.  This decrease can be compensated by
increasing the electron beam current as long as the current 
remains within the operating range of the beam line.
If the rate-limiting factor is background rate in the detector
and trigger then it makes sense to recompute the beam current at each
value of the emittance so as to keep the background rate constant in 
the detector, similar to the way Fig.~\ref{fom_energy} was produced.
The curves in Fig.~\ref{fom_emittance} were computed under these
conditions.

As can be seen in Figs.~\ref{te_emittance}-\ref{fom_emittance},
our two metrics provide competing criteria for choosing the optimum
collimator aperture.  There is no unique way to combine the two
metrics, but doing so should demonstrate how the value best choice
for the collimator diameter changes with the beam emittance.  One
way to form a combined figure of merit is simply to take the product
of the tagging efficiency with the polarization figure of merit.
The results are shown in Fig.~\ref{cfom_emittance}.  From this
plot it is apparent that the choice of 3.2~mm for the reference
configuration is closely tied to the reference emittance of
$10^{-8}$~m$\cdot$r.

%In the discussion above it was pointed out that background rates are
%a key consideration that places an upper bound on the photon beam
%intensity that is acceptable to GlueX.  A second consideration that
%must be taking into account in setting the operating intensity is
%the tagging accidentals rate, which depends upon the tagging efficiency.
%It is the more restrictive condition which takes precedence over the other. 
%If one takes a fixed accidentals rate to set the beam current instead of
%background rates then the intensity factor that appears in the definition
%of the polarization figure of merit must decrease with increasing emittance
%instead of increasing.  Quantitatively, this intensity scaling factor is
%just the tagging efficiency shown in Fig.~\ref{te_emittance}.  Hence the
%solid curve in Fig.~\ref{fom_emittance} is simply the product of the
%tagging efficiency with the polarization figure of merit at constant
%electron beam current.

\begin{figure}[t]
\begin{center}
\includegraphics[width=5in]{te_e.eps}
\caption{\label{te_emittance}
Tagging efficiency as a function of horizontal electron beam
emittance for (a) a 1.6~mm diameter collimator, (b) a 3.2~mm diameter
collimator, and (c) a 6.4~mm diameter collimator.  The electron beam
energy is 12~GeV.  The reference configuration is represented by the
intersection of curve (b) with the vertical grid line at 
$10^{-8}$~m$\cdot$r.}
\end{center}
\end{figure}

\begin{figure}[t]
\begin{center}
\includegraphics[width=5in]{fom_e.eps}
\caption{\label{fom_emittance}
Polarization figure of merit as a function of horizontal
electron beam emittance for (a) a 1.6~mm diameter collimator, (b)
a 3.2~mm diameter collimator, and (c) a 6.4~mm diameter collimator.
The electron beam energy is 12~GeV.  The reference configuration is
represented by the intersection of curve (b) with the vertical
grid line at $10^{-8}$~m$\cdot$r.}
\end{center}
\end{figure}

\begin{figure}[t]
\begin{center}
\includegraphics[width=5in]{cfom_e.eps}
\caption{\label{cfom_emittance}
Combined figure of merit as a function of horizontal
electron beam emittance for (a) a 1.6~mm diameter collimator, (b)
a 3.2~mm diameter collimator, and (c) a 6.4~mm diameter collimator.
The tagging efficiency and polarization figure of merit were combined
by simply taking their product.  The reference configuration is
represented by the intersection of curve (b) with the vertical
grid line at $10^{-8}$~m$\cdot$r.}
\end{center}
\end{figure}

\subsection{\label{sec:parameters:spread}energy spread}

The energy resolution of the photon beam depends on the energy spread
of the electron beam folded with the resolution of the tagging 
spectrometer, which depends in turn on the granularity of the
focal plane counters and the uniformity of the tagger magnet field.
Knowing the energy of the photon that caused an event
provides an important constraint that GlueX needs in order
to optimize its momentum and mass resolution in reconstructing
exclusive final states.  The energy resolution determines
to what degree it will be possible to eliminate reconstructed events with
missing particles in the final state, and to reconstruct final states with
neutrons or other neutral particles that are not seen in the detector by
exploiting energy and momentum conservation from the initial state.

The following general argument can be made to estimate the required
photon beam energy resolution for GlueX.  The way that initial
state energy is applied to final state reconstruction is through 
momentum conservation, requiring the sum of the momenta of all final state
particles, seen and unseen, to equal the momentum of the initial state.
The momentum of at least one final-state particle must be measured in the
detector and, even in cases where missing particles are allowed, typically
at least 50\% of the total energy in the final state comes from particles
whose momentum is measured.  If one takes a final state with several
charged tracks, each with an optimum momentum resolution of 2\% in the
GlueX tracking detectors, and let their energies sum up to 5~GeV then the
uncertainty on the sum of their energies is roughly $100/\sqrt{n}$~MeV,
where $n$ is the number of tracks.  These energies are compared with
the initial state energy when the constraint of momentum conservation
is applied, so improving the resolution on the initial-state photon
below something of order 50~MeV produces diminishing returns.  This 
argument leads to an upper bound on the energy spread of the electron
beam of roughly 50~MeV.

%This estimate was checked using reconstructed Monte Carlo events for
%the reaction $\gamma p\rightarrow K^+K^-\pi^+\pi^-\pi^0p$ where the
%final $\pi^0$ was missing from the reconstructed event.  One way to
%detect the presence of a missing particle in a reconstructed event is
%to plot the missing mass, a measure of the mismatch between the total
%momentum of the initial and final states.  The missing-mass resolution
%of the detector + beam combination provides a useful metric for gauging
%the relative effect of varying the tagged photon energy resolution on
%the physics capabilities of GlueX.  In Fig.~\ref{mm_deltaE} is shown
%the missing-mass resolution for the above reaction with a missing $\pi^0$
%as a function of the r.m.s.\ error on the energy of the initial photon.
%This simulated reaction contains five reconstructed charged particles
%and one neutral.  Not all charged particles are measured with the
%optimum 2\% precision and not all events in this reaction have 50\%
%missing energy, but at the level of a factor-of-two estimate the above
%analytic formula agrees with the data.  Optimum missing-mass resolution
%in the GlueX detector can be obtained with a tagger energy resolution
%of 0.5\% or better.

Clearly the energy spread from CEBAF at 12~GeV will be better than
this by about an order of magnitude, as will the resolution of the
tagger.  The intrinsic resolution of the tagging spectrometer is on
the order of a few MeV, and the full width of individual tagging
channels in the tagger microscope array is 8~MeV, with segmentation
driven by rate considerations.  All of this means that the intrinsic
energy resolution of the Hall D photon beam will be much better than
what is required for the GlueX physics program, as currently envisioned.
It is never a bad thing to exceed physics requirements in a beam
parameter, provided that it does not require significant special
expenditures to obtain it.

%\begin{figure}[t]
%\begin{center}
%\includegraphics[width=5in]{ch6_eresl.eps}
%\caption{\label{mm_deltaE}
%Missing mass resolution as a function of the uncertainty on
%the initial state photon energy measured by the tagger.  The reaction
%used for this study was $\gamma p \rightarrow K^+K^-\pi^+\pi^-\pi^0p$
%where the $\pi^0$ is undetected and must be reconstructed using missing
%momentum.  The incident photon energy energy is 9~GeV.}
%\end{center}
%\end{figure}

\subsection{\label{sec:parameters:halo}beam tails}

Transverse emittance reflects the r.m.s.\ widths of the transverse
position and angle distributions of electrons in the beam.  These 
distributions follow a Gaussian profile, typically over several orders
of magnitude.  Beyond some radius, however, the distribution begins to
fall more slowly than the Gaussian profile or even becomes relatively
flat.  These tails (or halo) contain relatively few particles relative
to the central core of the beam, but can be important because they can
interact with the dense materials surrounding the beam line and target
and produce background.  In the case of GlueX, the distance of order
100~m from the electron beam to the target (with the photon collimator
in between) prevents such off-axis electrons from producing background
in the GlueX detector.  The same is not true of the tagging counters,
however. 

Photons produced by halo electrons have essentially zero chance of 
getting through the photon collimator.  The reason for this is that
they are produced by bremsstrahlung in materials that are orders of
magnitude thicker than the crystal radiator, otherwise their photon
yield would be negligible by reason of the low intensity of halo
electrons relative to the core of the beam.  Electrons passing through
such a thick radiator undergo so much multiple scattering that the
photon spot that they produce projected out to the distance of the
collimator plane is orders of magnitude larger than the collimator
aperture.  By contrast, the probabilities are
somewhat higher that the degraded halo electron might find its way into
one of the tagging counters.  Beam particles which create hits in the
tagging counters but no corresponding photon in the photon beam inflate
the tagging efficiency.  To make a quantitative estimate of this effect
a model is needed for the distribution of materials in the vicinity of
the beam axis upstream of the tagger and a spatial and momentum
distribution for the halo particles.

A zeroth-order estimate for a safe upper limit for beam halo intensity
is obtained as follows. Fit the transverse beam position distribution to
a Gaussian function and subtract this distribution from the full beam
population.  The remaining particles, described as the halo population,
are spread out over a relatively large spot compared to the original
Gaussian radius. Assume that all of the halo beam particles end up
striking some vacuum, support or magnetic element on their way through
the tagger and that they all either scatter into a tagging counter
themselves or produce secondaries that do.  Under these assumptions, one
may ask what fraction of the original beam may belong to the halo
population without degrading the tagging efficiency by more than 1\%.
Under these somewhat perverse assumptions, the upper limit on the halo
integral is $10^{-6}$. 

If this halo level were easy to guarantee coming from CEBAF at 12~GeV
then our work would be done.  However we have been assured that it is not.
To make progress in refining this estimate, more information regarding the
distribution of materials around the beam and the halo phase space
distribution is needed.  Experience with CEBAF at 6~GeV has shown that
the latter can be difficult to predict and sometimes to control.  On the
other hand, we can configure the beamline elements to minimize the
amount of material in the region upstream of the tagger which may cause
the electron beam tails to scrape and produce background in the tagging
counters.
The following preliminary description of the region surrounding the beam in
the region of the radiator has been studied using a GlueX tagger Monte
Carlo simulation.
\begin{itemize}
\item A square aluminum frame with an inner cutout region of dimensions
$1.5\times 1.5$~cm$^2$, outer dimensions $3\times 3$~cm$^2$ and
thickness 3~mm.
\item A stainless steel beam pipe leading through the tagger quadrupole
from the radiator housing to the tagger vacuum box of outer diameter
3.8~cm and thickness 1.5~mm.
\item A pressure of $10^{-4}$ Torr in the tagger vacuum.
\item A 3~cm gap between the poles of the tagger dipoles.
\end{itemize}
In order to estimate the background in the tagging spectrometer,
particles were generated uniformly in a disk of radius 2.5~cm at
the position of the Hall D radiator.  This is a very pessimistic model
for what might emerge from a 1.5 inch beam pipe coming from the tunnel,
so the results can be taken as a conservative estimate for the expected
halo rates in the tagger.

Using this model, $10^6$ halo events were tracked through the tagger
geometry with magnetic field simulation and full shower generation.
Fig.~\ref{halo_image} shows the transverse profile at the position of
the radiator of all electrons in this sample which produced hits in
the tagging counters.  The second panel in the figure is provided to
help with interpreting the hit pattern.  The central red square in
the second panel is the diamond radiator.  The crystal mounting frame
is shown by a light blue rectangular outline around the crystal.
Next in order of increasing radius is the entrance flange to the
tagging spectrometer vacuum box (purple ring) whose inner diameter
is equal to the gap between the poles of 3~cm, somewhat smaller than
the 3.5~cm ID pipe leading into it (solid blue ring).  The green
horizontal structures at the top and bottom of the figure are the
pole shoes of the first spectrometer dipole.  Surrounding the beam
pipe with a gap of a few mm is the spectrometer quadrupole (solid
yellow) which is represented in the model as a large block of iron
with a circular hole cut out of the middle.  Clearly visible in the
hit pattern are the outlines of the radiator crystal holder and the
vacuum chamber entrance flange, while the beam pipe material and the
quadrupole and dipole magnets appear as partial voids because they
act as absorbers rather than sources of background events.

Fig.~\ref{halo_image} shows that the entrance flange to the vacuum
box is the largest single source of tagger backgrounds in the
present simulation geometry.  This conclusion may change, however,
when the halo distribution is improved beyond the crude model of a
uniform disk extending out to 2.5~cm in radius.  For the purposes
of an initial estimate, the current model is sufficient.
of 
expected to reduce the background rates by a factor of 3-5 in 

\begin{figure}[t]
\begin{center}
\includegraphics[width=4in]{halo_image.eps}
\includegraphics[width=4in]{hi.eps}
\caption{\label{halo_image}
Transverse profile (top frame) at the radiator of particles in the
beam halo that went on to create hits in the tagging counters.
Each point is the intersection of an electron track with the plane
containing the radiator.  The hit pattern reflects the material
distribution (bottom frame) as seen by the incoming electron beam.
The two panels have matching dimensions.  See text for details of
the material map.}
\end{center}
\end{figure}

In this sample a total of 9480 hits were observed leaving energy
more than 200~keV in the tagging microscope counters.
One million beam particles represents 50~ns of
real time under full-intensity running conditions of $10^{8}$ tagged
$\gamma/s$ on the GlueX target.  Hence if $10^{-5}$ of the total electron
beam population were in the halo then the halo rate in the tagging
counters would be 10k per 5~ms or 2~MHz, which corresponds to
a inflation factor of 1\% in the tagging efficiency.  From this it
follows that an upper bound of $10^{-5}$ on the halo fraction is
sufficient to insure that its effect on the tagging efficiency is
negligible.  The only part of the halo that contributes significantly
to the background in the tagging counters is what lies between the radii
of 1.0 and 1.5~cm.

In addition to the tagger microscope counters, there is also an second
array of tagging counters which covers a much broader range in photon
beam energy is more coarsely segmented.  This broad-band array is not
used for tagging during polarized photon running, but it is extremely
important for monitoring the quality and stability of the photon beam.
It is also needed to align the crystal at the beginning of each run
period.  The simulation showed that a $10^{-5}$ halo creates a hit
rate below 1\% of the tagged photon rate across most of the counters,
with the exception of the very low-energy electron end of the spectrum.
The broad-band array extends up to 95\% of the bremsstrahlung end-point,
which means that it sees electrons of only 600~MeV energy at the
extreme end.  This end of the array is also the closest to the radiator,
which means that the solid angle of these counters relative to sources in the
radiator region is the greatest in the same place where the bremsstrahlung
spectral intensity is at its minimum.  Fig.~\ref{halo_frac} shows the
fraction of the total hits in each of the counters in the broad-band array
that would be generated by a halo with a beam fraction of $10^{-5}$.
Maintaining a 1\% upper bound on the halo contribution to the count rate
in the tagger at the high-energy photon end of the broad-band array would
push the requirement on the total halo fraction down to $10^{-6}$.

\begin{figure}[t]
\begin{center}
\includegraphics[width=5in]{halo_frac.eps}
\caption{\label{halo_frac}
Percentage of hits in the broad-band tagging counters that 
are generated from electron beam halo particles, assuming a beam halo
fraction of $10^{-5}$.  The electron beam energy scale is roughly
linear in the $z$ coordinate, varying from 600~MeV at the left end
of the plot to 9~GeV at the right.  The shaded region indicates the
coverage of the microscope.}
\end{center}
\end{figure}

\subsection{\label{sec:parameters:polarization}electron beam polarization}

Up to this point, the beam polarization has referred to the linear
polarization of the photon beam that is produced through the coherent
bremsstrahlung process and enhanced by collimation.  In this section
we consider the degree of freedom of electron beam polarization and its
implications for GlueX physics.  Any non-zero polarization in the electron
beam is transferred by the bremsstrahlung process onto the circular
polarization of the photon beam.  More precisely, it is the projection
of the electron polarization onto the electron momentum axis that is
transferred to the circular polarization of the radiated photon.  This
transfer is quite efficient, more than 90\% for a 12~GeV electron
radiating a 9~GeV photon.  Circular polarization is not required for
the GlueX physics program but it does have an effect on the results,
so if it is present then that fact must be known so that it can be
taken into account in the analysis.

The simplest scenario for GlueX is if the electron beam is unpolarized.
The beam in the LEP ring was
naturally polarized to a high degree by the process of synchrotron
radiation, but the relaxation time for that process was on the order
of hours, whereas the CEBAF beam spends only microseconds circulating
in the machine.  On that basis it appears unlikely that the beam can
acquire a significant spontaneous polarization during acceleration, 
so the polarization of the beam delivered to Hall D should be controlled
at the source.  The GlueX requirement for the precision with which we
know the average degree of linear polarization in the photon beam is
1-2\% absolute.  Knowing that the polarization of the electron beam
is zero at a similar level of precision should be sufficient.

It is important to emphasize that it is only the time-averaged electron
beam polarization that is of interest to GlueX.  A beam with a high
degree of electron polarization would be acceptable to GlueX if it
were reversed frequently and had a zero time-averaged value over a time
scale of several hours.

\subsection{\label{sec:parameters:optics}optics}

In the preceding sections, the metrics of tagging
efficiency and polarization figure of merit were shown to be sensitive
to the electron beam energy and transverse emittance.
The validity of these results relies
upon two assumptions, that the beam line optics that were assumed in
the design of the GlueX photon beam can actually be achieved, and that
no way can be found to produce the same photon beam properties under
relaxed electron beam conditions.  Examining these assumptions requires
a closer look at the optics for Hall D.

In what follows, the beam energy is considered to be fixed at 12~GeV
and the beam emittance is a variable that is within an order of magnitude
of 10~mm$\cdot\mu$r.  The angle between the electron direction and
the direction of the radiated photon is random, but its scale is set by
the characteristic angle $m/E$ which is about 40~$\mu$r.  A typical photon
trajectory intersects the front surface of the collimator about 3~mm from
the linear projection onto the same plane of its electron's trajectory at
the point of radiation.  The collimator is located 75~m downstream of the
radiator.  While incoherently radiated photons have an angular
spread of order m/E0 independent of photon energy, the angle of
coherent bremsstrahlung photons is strongly correlated with photon
energy so that photons closest to the coherent edge have emission
angles much smaller than the characteristic angle. 
This statement is correct regardless of the emittance
of the electron beam.  The electrons are dumped far upstream of the
collimator, however, so the only way these low-angle photons can be
distinguished from the others at the collimator is by their proximity
to the center of the photon spot.  The center of the spot is only as
well defined as the electron beam emittance allows it to be.

The optics that optimizes this angular collimation are those which create
a virtual electron beam focus at the collimator position, while keeping
the size of the beam at the radiator within the limits of the crystal
dimensions.  A solution for the Hall D beam line that achieves these
goals in the horizontal direction and keeps the virtual spot circular
at the position of the collimator has been found and been subjected
to one or two optimizing iterations.  So far no scheme has been found
which is able to exploit a given electron beam emittance to produce
a collimated photon beam with significantly higher values for
the metrics in Figs.~\ref{fom_energy}-\ref{cfom_emittance}.

\section{\label{sec:grading}Graduated Requirements}

The physics goals of GlueX will be achieved over a period of time.
Likewise it is foreseen that the ultimate performance of CEBAF at 12~GeV
will be achieved in a series of steps.  Without attempting to predict
what the limiting factors in 12~GeV accelerator operations will be, a
graded schedule for achieving certain performance figures for critical
beam properties is presented that would insure that GlueX can make optimal
progress toward achieving its scientific objectives.

The GlueX run plan for the first 2.5 years is broken down into
three phases.  During the first 6 months we will be doing low-level
detector commissioning that does not require 12~GeV electrons or the
ultimate emittance from the machine.  During the following 12 months
we will systematically go through physics commissioning by the study
of well-known reactions, e.g.\ measure the density matrix elements of
diffractively produced rho, omega, and phi mesons to check out our
event reconstruction and particle ID capabilities.  During this
phase we want to also look at the photon beam linear polarization
and measure how it tracks with emittance.  Finally in the next 12
months we want to map out some known higher mass states and
start the first exploratory physics.

Table~\ref{tab:reqs} lists a graduated set of electron beam
requirements that are consistent with the initial GlueX run plan
outlined above.  The minimum beam current stated in the table
refers to the minimum current at which position controls in the
Hall D beam transport line are capable of measuring and stabilizing
the beam, which is a minimum requirement for stable operation of
the collimated photon beam.  The current design for beam controls
and monitoring entails three modes of operation with varying 
capabilities for stabilization.
\begin{enumerate}
\item Full visibility of the Hall D beam in its path through the
accelerator and transport line all the way to the electron beam
dump, including feedback mechanisms that keep the beam orbit and
intensity within defined limits of the desired values.
\item Visibility of the Hall D beam only in special low-current
monitors installed toward the downstream end of the Hall D
transport line and beam dump that measure the beam position,
direction, and current at the entrance to the radiator, with
feedback mechanisms that keep these variables within defined
limits of the desired values.
\item Visibility of the Hall D beam only at the beam dump, with
limited or no information regarding beam position or direction
at locations upstream of the dump.
\end{enumerate}
The GlueX operating current during initial running will be
300~nA, increasing eventually to a maximum current of 3~$\mu$A.
These currents are well above the minimum of 50~nA required for
mode 1 operation.  The minimum current requirement in
Table~\ref{tab:reqs} is tied to the ability to measure the
photon beam spectrum directly with a total absorption counter
in the beam.  Such measurements require running in mode 2.
Experience with controls on the Hall B transport line indicates
that 1~nA is the minimum current at which the sensors can 
detect the beam.  There are circumstances under which currents
as low as 100~pA will be required for short periods of time,
but it is understood that during these periods all locks
will be disabled and mode 3 will apply.  Of all of the above
running modes, stable operation at 1~nA is deemed to be the
most challenging in terms of beam line design, so it has been
highlighted in the list of requirements.

Listed in Table~\ref{tab:pars} are the operating parameters for the
GlueX photon beam, based on our best estimates for the 12~GeV electron
beam properties at Hall D.  This is a copy of Table 4.3 from the GlueX
Design Report and have been presented a number of times over the last
two-year period.  The values in this table are best estimates and are
expected to change over time, as more is learned about the electron beam
properties and the design for the upgrade moves forward.  They are provided
here for reference only, not to be confused with requirements, which are
listed in Table~\ref{tab:reqs}.

\begin{table}[b]
\begin{center}
\caption{\label{tab:reqs}
Values for 12~GeV electron beam requirements at various
stages in the commissioning and execution of the GlueX experimental program.}
\begin{tabular}{lccc}
\hline\hline
& first 6 months & months 6-12 & year 2 and following \\
\hline
min energy & 10~GeV & 11~GeV & 12~GeV \\
max current & 3~$\mu$A & 3~$\mu$A & 3~$\mu$A \\
min current\footnotemark[1]
 & 1~nA & 1~nA & 1~nA \\
max emittance 
 & 50~mm$\cdot\mu$r & 20~mm$\cdot\mu$r & 10~mm$\cdot\mu$r \\
max energy spread & $<0.5$\% & $<0.5$\% & $<0.5$\% \\
max halo fraction & $10^{-4}$ & $10^{-5}$ & $10^{-5}$ \\
max $e^-$ polarization\footnotemark[2]
 & {\em unspec.} & {\em unspec.} & 1\% \\
\hline\hline
\end{tabular}
\end{center}
\footnotemark[1]{operating mode 2, see text}\\
\footnotemark[2]{refers to time-averaged polarization}\\
\end{table}

\begin{table}
\begin{center}
\caption{\label{tab:pars}
Electron beam parameters that are actually used by GlueX
in the design and simulation of the photon beam.  The table is
reproduced from Table 4.3 in the GlueX design report.}
\vspace{3mm}
\begin{tabular}{lc}
\hline\hline
parameter & design value \\
\hline
energy & 12~GeV \\
electron polarization & 0 \\
minimum useful current & 100~pA \\
maximum useful current & 3~$\mu$A \\
r.m.s.\ energy spread & 7~MeV \\
transverse $x$ emittance & 10~mm$\cdot\mu$r \\
transverse $y$ emittance & 2.3~mm$\cdot\mu$r \\
x-dispersion at radiator & 0 \\
y-dispersion at radiator & 0 \\
$x$ spot size at radiator & 1.55~mm r.m.s. \\
$y$ spot size at radiator & 0.55~mm r.m.s. \\
$x$ image size at collimator & 0.54~mm r.m.s.\\
$y$ image size at collimator & 0.52~mm r.m.s.\\
distance radiator to collimator & 75~m \\
position stability & $\pm$200~$\mu$ \\
\hline\hline
\end{tabular}
\end{center}
\end{table}

%The figure of merit metric has an additional meaning that is not
%immediately clear from the above discussion, that is very important
%to GlueX physics.  Optimizing the experimental design to achieve
%as high a polarization figure of merit as possible is not merely a
%matter of reducing the number of hours required to run the experiment,
%as important as that is.  Even if the experiment had sufficient resources
%to run indefinitely, there would be a point at which it would make no more
%sense to continue because intrinsic uncertainties that enter into the
%way the data is analyzed and interpreted ultimately limit the precision
%of the results.  These systematic errors do not depend upon the beam
%polarization.  But because they add in quadrature with statistical errors
%they determine the number of events that can be analyzed for a given
%reaction, beyond which taking more data produces diminishing returns.
%In a sense the GlueX experiment cannot exhaust its ability to press forward
%in its search for new states because it can investigate reactions with
%successively smaller cross sections and decay branching ratios, but for
%a given channel the systematic errors will impose a practical limit on 
%the duration...

\end{document}
