<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\section{RADIATION AND BACKGROUNDS}

\subsection{Conventional radiation}
%
The proton source generates a 4~MW proton beam, which is comparable to the
proposed spallation source \cite{sns}. This is a very high power and will, as
in the spallation source, require great care in reducing unwanted particle
losses, as well as  careful machine shielding,  and target and beam dump
design. Initial studies of the target and capture solenoid region have been
performed with the MARS code, and preliminary specifications for shielding
determined, but more work is needed.

The cooling and accelerator chain is rather clean, since a relatively small
fraction of the muons decay, and their energies are low. Power deposited in the
accelerators is typically 10-30~W/m (see table~\ref{acceleration1} and table~\ref{acceleration3}).

If no muons are lost, then the only sources of radiation are the muon decays
yielding electrons and neutrinos. The neutrino radiation we discuss below. The
electrons shower in the collider beam pipe shields, depositing most of their
energy there and a relatively small amount in the magnet coils and yoke. Radioactivation levels, as calculated by MARS \cite{snowrad}, after five years of 4~TeV collider operation are given in table~\ref{beamradiation} for the cases immediately after turn off and 1 day after turn off. It is seen that the areas in the tunnel that are
outside the magnets are relatively free of radioactivation. Special procedures
will be needed when the shield pipe has to be opened, as for instance when a
magnet is changed. For the lower energy colliders, the  radioactivation levels
are proportionally less.

   %
\begin{table}[htb!]
\caption[4~TeV (CoM) collider ring radioactivation levels ]{4~TeV (CoM)
collider ring radioactivation levels (mrem/hour) after turn off, for parameters
in table~\ref{sum}}
\label{beamradiation}
\begin{tabular}{lcc}
             &amp; immediate   &amp; after 1 day    \\
\hline
Inside face of shield     &amp; 9000 &amp; 4000  \\
Outside face of shield    &amp;  200 &amp; 170   \\
Outside of coils          &amp; 30   &amp;  14   \\
Outside of yoke           &amp; 3    &amp;  1.4  \\
\end{tabular}
\end{table}
If muons are lost either accidentally, by scraping, or deliberately after some
 number of turns, then the muons penetrate to considerable distances in the 
soil/rock (3.5~km at 2~TeV, 800~m at 250~GeV) and deposit their energy directly
 or through their interaction products. Figure~\ref{snowmass10.6} 
and Fig.~\ref{snowmass10.7} show the distribution of radiation levels, assuming 25\% of all muons (4 bunches of $2 \times 10^{12}$ at 15~Hz) are dumped into 
soil/rock with density 2.24~g/cm$^3.$ 
 The outer contours correspond to the federal limits, reached at maximum radii  of 18~m (2~TeV) and 14.5~m (250~GeV). To confine this radiation beneath the ground
one can deflect the extracted beams down by 4.5~mrad at 2~TeV and about 10~mrad
at $\leq$250~GeV.
If any water were present in the soil/rock, then the first two meters around
the tunnel and around the
aborted beam axis  would require insulation or drainage up to a distance of
2.5~km at 2~TeV or 550~m at 250~GeV.

\begin{figure*}[tbh!]
%\centerline{\epsfig{file=10fig6.eps,height=3.5in,width=3.5in}}
\centerline{\epsfig{file=dose2.eps,height=3.5in,width=4.5in}}
\caption[MARS isodose contours in the soil/rock for 2~TeV muons extracted at 
$3\times 10^{13}$ per second]
{Isodose contours in the soil/rock ($\rho$=2.24\,g/cm$^3$) for 2~TeV muons 
extracted at $3\times 10^{13}$ per second. Right scale is dose
rate in rem/s.} 
\label{snowmass10.6} 
\end{figure*}
\begin{figure*}[tbh!]
%\centerline{\epsfig{file=10fig7.eps,height=3.5in,width=3.5in}}
\centerline{\epsfig{file=dose250.eps,height=3.5in,width=4.5in}}
\caption[MARS isodose contours in the soil/rock for 250~GeV muons extracted at 
$3 \times 10^{13}$ per second]
{Isodose contours in the soil/rock ($\rho$=2.24\,g/cm$^3$) for 250~GeV muons extracted at $3 \times 10^{13}$ per second. Right scale is dose
rate in rem/s.} 
\label{snowmass10.7} 
\end{figure*}
\subsection{Neutrino induced radiation}
It has been shown \cite{ref8b,king_phd,ref41a,ref41b,ref41c} that the neutrinos created
in muon beam decays can generate excessive secondary radiation at large
distances from a muon collider (see Fig.~\ref{nurad}). The surface radiation
dose $D_B(Sv)$ in units of equivalent \cite{data_book} doses ($Sv$) over a time
$t(s)$, in the plane of a bending magnet of field B(T), in a circular collider
with beam energy $E(TeV)$, average bending field $&lt;B(T)&gt;$, at a depth $d(m)$
(assuming a spherical earth), with muon current (of each sign) of
$I_{\mu}\textrm{(muons/s/sign)}$ is given by:
\begin{equation}
D_B\ \approx\ 4.4\times 10^{-24}\ {I_\mu \ E^3\ t \over d}\ {&lt;B&gt; \over B}
\end{equation}
and the dose $D_S$ at a location on the surface, in line with a high
beta straight section of length $\ell(m),$ is:
\begin{equation}
D_S\ \approx\ 6.7\times 10^{-24}\ {I_\mu \ E^3\ t \over d}\ {\ell  &lt;B&gt;}.
\end{equation}
The equation for $D_S$ assumes that the average divergence angles satisfy the
condition: $\sigma_\theta &lt;&lt; {1\over \gamma}$. This condition is not satisfied
in the straight sections approaching the IP, and these regions, despite their
length, do not contribute a significant dose. 

\begin{figure*}[tbh!]
\centerline{\epsfig{file=nurad.eps,height=2.5in,width=3.5in}}
\caption[Neutrino radiation disk    ]{Neutrino radiation disk. For a 3~TeV CoM
collider the neutrino radiation width is $\approx $~4 m at a distance of 30~km.
A \textit{hot spot} produced by 0.1 m straight section in the ring contains
roughly twice the number of neutrinos on the disk on average, depending on the
details of the collider lattice. } 
\label{nurad} 
\end{figure*}
For the 3~TeV parameters given in table~\ref{sum} and muon currents 
$I_{\mu}=6\times 10^{20}\,\mu^-/yr,$ $&lt;B&gt;= 6~T,$ $B= 10~T$ and
$\textrm{depth}=500$~m, and taking the Federal limit on off-site radiation
dose/year, $D_{\textrm{Fed}},$ to be 1~mSv/year (100~mrem/year), the annual dose
$D_B$ (1 year is defined as $10^7$ s), in the plane of a bending dipole is,
\begin{equation}
D_B = 1.07\times 10^{-5}\  \textrm{Sv}\ \approx 1\% \ D_{\textrm{Fed}},
\end{equation}
and for a straight section of length 0.6 m is:
\begin{equation} 
D_S = 9.7\times 10^{-5}\ \textrm{Sv}\ \approx 10\% \ D_{\textrm{Fed}},
\end{equation}
which may be taken to be within a reasonable limit. The general trend of these  expressions has been verified by Monte Carlo simulations \cite{mokhovrada} using MARS. In particular, for the 3~TeV case the needed depth to stay within 
$1\%\ D_{\textrm{Fed}},$ is 300~m instead of 500~m. 

Special care will be required in the lattice design to assure that no field-
free region longer than 0.6~m  is present. This may sound difficult, but it may
be noted that the presence of a field of even 1~T, is enough
to reduce the dose to a level below the Federal limit. The application of such
a field over all rf and other components seems
possible \cite{mokhovrada}. 

For lower energy machines, the requirements rapidly get easier: a 0.5~TeV
machine at 100~m depth could have 25~m long sections, for the same surface
dose. For a 100~GeV machine the doses are negligible. 

For machines above 3~TeV, various strategies can be employed: 
\begin{itemize}
\item  The machines could be built at greater depths (mines many km deep are
common). 
\item  The vertical beam orbits in the machine could be varied so as
to spread the plane of radiation and thus reduce the peak doses. \item  The
specific locations in line with straight sections could be purchased and
restricted. 
\item  Straight sections could be shortened further by using
continuous combined function magnets. 
\item The machines could be built on an
island, but this could have difficulties associated with access to power and
other utilities. 
\end{itemize}

But for any large increase in energy, to 10~TeV for instance, some reduction
in muon beam flux probably will  be required. The resultant loss of luminosity
might be made up in a number of ways \cite{skrinskiultimate}:

\begin{itemize} 
\item  The beam-beam tune shift constraint could be avoided by
introducing a conducting medium (e.g. liquid lithium) at the interaction
point \cite{skrinskibeambeam}. 
\item  The focusing strength could be increased
by the use of plasma or other exotic focusing method. 
\item  Better cooling
could be developed. Optical stochastic cooling \cite{ref21}, for instance,
might reduce the emittances by many orders of magnitude, thus greatly reducing
the required beam currents. Indeed, such cooling would require lower currents
to function appropriately. 
\end{itemize}

Such options will need future study.
\subsection{Muon decay background}
 
With $4\times 10^{12}$ muons per bunch in a $2\: +\: 2$~TeV  collider ring there
are approximately $4\times 10^5$ muon decays per meter giving rise to high
energy electrons. These off-energy, off-axis electrons undergo bremsstrahlung
when they traverse magnetic fields. When they exit the beam pipe they interact and
produce electromagnetic showers and, to a lesser extent, hadrons and muons. Much
of this debris can be locally shielded, so the primary concern is muon decays
near the interaction point \cite{ref6a}. This is the background we discuss
in some detail below.

\begin{figure*}[tbh!]
\centerline{\epsfig{file=ip_draw.eps,height=4.in,width=6.0in}}
%\vspace{0.5cm}
\caption[Region up to 130~m from the IP, of the 2 + 2~TeV interaction
region modeled in GEANT]{Region up to 130~m from the IP, of the 2 + 2~TeV interaction region modeled in GEANT. The triangular blue regions represent
tungsten shielding. On the right hand of the figure, the red areas represent
quadrupoles in the beam line. The areas around the IP represent the various
detector volumes used in the calculations of particle fluences. The detector
(white and green areas) is 10~m in diameter and 20~m long.}
\label{ip_fig}
\end{figure*}

Detailed Monte Carlo simulations of electromagnetic, hadronic and muon
components of 
the background \cite{ref6a,carnik96,ref42,shield96,mokhovrada,ref39,nikolai96}
have been performed using the MARS \cite{mars} and GEANT \cite{ref40} codes.
The most recent study \cite{ref39} has been done with GEANT. 
Figure~\ref{ip_fig} shows the final 130~meters of the 2 + 2~TeV detector
region in this study.
It includes the final four quadrupoles, dipoles and a solenoidal field
surrounding the detector.
This study:
   \begin{itemize}
\item
followed shower neutrons and photons down to 
$40\,$keV and electrons to $25\,$keV.
\item 
used a tungsten shield over the beam, extending outward to an angle of
20 degrees from the axis.
%extending in to within 14 cm of the interaction point.
\item
Inside this shield, the clear radius has a minimum, in the high energy
cases, at a distance from the IP of 1.1~m (80~cm for 50+50~GeV). At this
point, and in an expanding cone beyond it, the clear radius is maintained at
approximately 4~sigma of the beam size. 
  \item Between this minimum aperture point and the IP, the clear radius
follows an inverse cone, increasing as it approaches the IP, with an angle
a little greater than the 4~sigma of the beam divergence. These cones are 
designed so that the detector could not `see' any surface directly
illuminated by the initial decay electrons, whether in the 
forward or backward (albedo) direction (see figure~\ref{shield}).          
  \item
The resulting open space between the IP and the tip of the cone is
approximately 3~cm in the 4~TeV and 500~GeV CoM cases, and approximately 6~cm in the 100~GeV CoM case.
   \item
The inner surface of each shield is shaped into a series of collimating
steps and slopes to maximize the absorption of electron showers from
electrons at very small angles to the cone surface, thus reducing the
funnelling of low energy electrons down the pipes.
     \item
Further upstream, prior to the first quadrupole (from 2.5 to 4~m in the
Higgs case), an 8~T dipole, with collimators inside, is used to sweep decay
electrons before the final collimation.
   \end{itemize}

Note that there is currently an inconsistency, in the very low $\Delta p/p$ Higgs
Factory Case, between the short open space between shields (+/- 6 cm) and
the rms source length ($\sigma_{\textrm{source}}=1/\sqrt(2) \sigma_z$) of 10~cm. Some modifications to the parameters and shielding design will be
required for this case.

\begin{figure*}[tbh!]
\centerline{\epsfig{file=shield511.eps,height=5.in,width=5.0in,clip=}}
\vspace{0.25cm}
\caption[Detail of the tungsten shielding designed for the 50 + 50 GeV
case.   ]{ Detail of the tungsten shielding designed for the 50 + 50 GeV
case. It is designed so that the detector is not connected by a straight
line with any surface hit by decay electrons in forward or backward
directions. The picture extends out to a radius of 6~cm and, on the right, to a
distance 4~m from the IP. The dipole from 2.5-4.0~m is not shown. }
\label{shield}
\end{figure*}

Every modern detector will have to be able to identify and reconstruct
secondary vertices such as those associated with b-quark decays. In order to estimate the viability of a vertex
detector we have to show that the occupancy of its elements is not higher than
about $1\%.$ Figure~\ref{occ} shows the occupancy as a function of radial
distance from the interaction point for the three CoM energies studied : 0.1,
 0.5 and 4~TeV. The occupancy was calculated for silicon pads of $300~\mu m \times 300~\mu m$, and assuming interaction probabilities of 0.003 and 0.0003 for low
energy photons and neutrons respectively. One can observe that the
total occupancy (left figure) is above one percent for small radii. Most of the hits is due  to conversions of photons.  The 
occupancy due to hits resulting from charged particles is below $1\%$ 
(right hand figure). One can lower the
occupancy at small radii by using smaller pixel sizes, as indicated in table~\ref{backgnd} below, as well as by using innovative detector
ideas as described in the next section.

Table~\ref{backgnd} gives the hit density for the Higgs factory from the
various sources and the occupancy of pixels of the given sizes; in each case
the number is given per bunch crossing. The hit density for the higher energy
machines is found to be somewhat lower due to the smaller decay
angles of the electrons. %
\begin{table*}[htb!]
\caption{Detector backgrounds from $\mu$ decay}
\label{backgnd}
%\centering \protect
\renewcommand{\arraystretch}{1,1}
\begin{tabular}{llcccc}
%\tableline
Radius             &amp;$cm$   &amp;      5   &amp; 10  &amp;20 &amp; 100   \\
\hline
Photons hits     &amp;$cm^{-2}$&amp; 26 &amp; 6.6 &amp; 1.6 &amp; 0.06   \\
Neutrons hits    &amp;$cm^{-2}$ &amp; 0.06 &amp; 0.08 &amp; 0.2 &amp; 0.04     \\
Charged hits     &amp;$cm^{-2}$ &amp;8 &amp; 1.2 &amp; 0.2 &amp; 0.01    \\
Total hits       &amp;$cm^{-2}$ &amp; 34 &amp; 8 &amp; 2 &amp; 0.12   \\
\hline
Pixel size     &amp; $\mu m \times \mu m $&amp; $60\times 150$&amp;$60\times 150$&amp;$300\times
300$&amp;$300\times 300$ \\
Total occupancy  &amp; \%  &amp; 0.6 &amp; 0.14 &amp; 0.4 &amp; 0.02 \\
Occupancy charged &amp; \% &amp; 0.14&amp; 0.02 &amp; 0.04 &amp; 0.002 \\
\end{tabular}

\end{table*}

\begin{figure*}[tbh!]
%\centerline{\hbox to 4.in{\hfil      \hfil}\hfill
%   \hbox to 4.in{\hfil       \hfil}}
%\dofigs{3.5in}{compoccup1.eps}{3.5in}{compoccup1_charged.eps}
\includegraphics[width=5.in,angle=90,bb=0 -25 450 623]{occup.eps}
\caption[Occupancy as a function of the radius]{Occupancy for $300~\mu\,m\times 300~\mu\,m$ silicon pads, as a function of the radius for the three energies studied.
Left figure shows the total occupancy and the right figure shows the occupancy
from hits resulting from charged particles.}
\label{occ}
\end{figure*}
The radiation damage by the neutrons on a silicon detector has also been
estimated. In the Higgs case, at 5~cm from the vertex, the number of hits from
neutrons above 100~keV is found to be $1.8\times 10^{13}/\textrm{cm}^2$ per year.
This is significantly less than that  expected at the LHC which is now ordering
silicon detectors claimed to survive $5\times 10^{14}$ hits, 
approximately three times that assumed here. The damage for silicon detectors
in the higher energy machines is of the same order (see table~\ref{neutron}).

\begin{table*}[htb!]
\caption[Radiation damage by neutrons on silicon detectors   ]{Radiation damage
by neutrons on silicon detectors. The working assumptions are: 1000 turns,
15~Hz and 1~year=$10^7$~s. An acceptable number of hits per year is $1.5\times
10^{14}$. }
\label{neutron}
%\centering \protect
\renewcommand{\arraystretch}{1,1}
\begin{tabular}{ccccc}
%\tableline
CoM &amp;$\mu$'s/bunch &amp;neutrons/cm$^2$/crossing &amp; Hits/cm$^2$/year&amp;Lifetime   \\
(TeV) &amp; $(10^{12})$ &amp; (above 100~KeV) &amp; $(10^{13})$ &amp; (years)\\
4 &amp; 2 &amp; 100 &amp; 3 &amp; 5 \\
0.5 &amp; 4 &amp; 50 &amp; 3 &amp; 5    \\
0.1 &amp; 4 &amp; 30 &amp; 1.8 &amp; 8   \\
\end{tabular}
\end{table*}
   


\subsection{Halo background}

Muon halo refers to those muons which are lost from the beam bunch as it
circulates around the collider ring. In conventional electron or proton
accelerators, beam particles which are lost away from the IP are of little
concern as they can be locally shielded. However, muons can traverse long
distances and therefore have the potential to generate background in a
detector. The magnitude of this background depends on a detailed knowledge of
the injected beam profile and a credible model for beam halo and beam losses.
More work is needed before these are well enough understood. Nevertheless, it
is clear that the beam will need careful preparation  before  injection into
the collider, and  the injection system will have to be precise and free of
ripple. 
\begin{figure*}[bht!]
%\dofigs{3.5in}{pair_2t.ps}{3.5in}{pair_4t.ps}
\dofigs{3.5in}{track1.ps}{3.5in}{track2.ps}
\caption[Radius vs. length of electron pair tracks ]
{Radius vs. length of electron pair tracks for
initial momenta from 3.8 to 3000 MeV 
in geometric steps of $\sqrt{2};$ (a) for
a solenoid field of 2 T, (b) for 4 T.
\label{pair}}
\end{figure*}

The collimation system described in the previous subsection was designed to
scrape the beam both initially and during the 1000 turns, to assure that all
loss occurs at the scraper and not near the IP. That study indicated
suppressions  better than 10$^3$ of background in the detector \cite{scraping}. 

Beam loss must be limited as far as possible. Gas scattering has been
studied \cite{snowmass8.5.3} and shown to give a negligible contribution. The
effects of beam-beam scattering are under study and need
further work.  Momentum spread tails from uncorrected wakefield effects must
be controlled. Assuming that the total loss from all causes, after injection
and the first few turns is less than $10^{-4}$ in 1000~turns, (i.e. $10^{-7}$
per turn), then the number of background muons passing through the detector
should be less than 800 ($2\times 4\times 10^{12} \times 10^{-7}\times 10^{-3}$) per turn. This is a low density of tracks per cm$^2$ and should be acceptable, but lower losses or better scraping would be desirable.
\subsection{Pair production}
Coherent beam-beam electron pair production (beamstrahlung) has been shown \cite{chen,ginzburg} to be negligible, but the
incoherent pair production (i.e. \mumu  $\rightarrow$ \mumu \ee) in
the 4~TeV collider case is significant.
     
The cross section is estimated to be  $10\,$mb \cite{ginzburg}, which would give
rise to a  background of  $\approx 3\times 10^4$  electron pairs per bunch
crossing.  The electrons at production do not have significant  transverse
momentum but the fields of the on-coming $3 \mu$m bunch can deflect them
towards the detector. A simple program was written to track electrons from
close to the axis  (the worst case) as they are deflected away from the bunch
center.  Once clear of the opposing bunch the tracks spiral under the 
influence of the experimental solenoid field. Figures~\ref{pair} shows the
radii vs. length of these electron tracks for initial momenta from 3.8 to
3000~MeV in geometric steps of $\sqrt{2}.$ Fig.~\ref{pair}(a) is for a solenoidal
field of 2~T and Fig.~\ref{pair}(b) for 4~T. In the  2~T case tracks with
initial energy below 30~MeV do not make it out  to a detector at 10~cm, while
those above 100~MeV have too small an  initial angle and remain within the
shield. Approximately 10\%~(3000 tracks) of  these are in this energy range and
pass through a detector at 10~cm. The track fluences at the ends of the detector
are less  than 10~tracks per cm$^2$ which should not present a serious problem.
At 5~cm, there are 4500~tracks giving a fluence of 30~per cm$^2$, which is also
probably acceptable. If  the detector solenoid field is raised to 4~T, then no
electrons reach 10~cm and the flux at 5~cm is reduced by a factor of 2. 

\subsection{Bethe-Heitler muons}


\begin{table*}[htb!]
\caption{Bethe-Heitler Muons }
\label{tab-bheitler}
%\centering \protect
\renewcommand{\arraystretch}{1,1}
\begin{tabular}{lccc}
%\tableline
CoM Collider Energy (TeV)  &amp; 4 &amp; 0.5  &amp; 0.1  \\
Assumed source length (m) &amp; 130 &amp; 33 &amp; 20 \\
\hline
$\mu\ (p_{muon }&gt;1\ \textrm{GeV}/c)$ per electron &amp; $5.4\times 10^{-4}$ &amp; $8.3\times
10^{-5}$ &amp; $9.6\times 10^{-6}$ \\
Beam $\mu$'s per bunch         &amp;   $2\times 10^{12}$ &amp;    $2\times 10^{12}$ &amp;   
$4\times 10^{12}$ \\
Bethe-Heitler $\mu$'s per bunch crossing ($\times 10^3$)   &amp; 28  &amp;  17.5  &amp;
6.1  \\
$&lt;p_{muon}&gt;$ initial (GeV) &amp;     22      &amp;   9.5    &amp;
4.4  \\
\hline
$\mu$'s entering calorimeter     &amp;     220      &amp;     160      &amp; 25    \\
$&lt;p_{muon}&gt;$  (GeV)  &amp;    15.4    &amp;   6.3   &amp;
1.8  \\
$&lt;E_{dep}&gt;$   (GeV)                &amp; 2.9 &amp;    1.3 &amp;   0.4 \\
Total $E_{dep}$  (GeV)          &amp;  640    &amp;    210 &amp;    10    \\
$E_{dep}$ pedestal subtracted  (GeV)&amp; 50    &amp;  25  &amp;   1    \\
Fluctuation in $E_{dep}$ (GeV)  &amp;    55    &amp;   15    &amp;   1   \\
$E_{trans}$ pedestal subtracted  (GeV)  &amp;  15    &amp;   15          &amp;   .5  \\
Fluctuation in $E_{trans}$  (GeV)  &amp;   40      &amp;  8     &amp;  0.5  \\
\end{tabular}
\end{table*}
  
The GEANT/MARS studies \cite{ref6a,shield96,snowrad} also found a significant
flux of muons with quite high energies, from $\mu$ pair production in
electromagnetic showers (Bethe-Heitler).  Figures~\ref{fig-bheitler1}
and~\ref{fig-bheitler2} show the trajectories of typical muons from their
sources in the shielding around the beam pipe to the detector.
Figure~\ref{fig-bheitler1} is for a 4~TeV CoM collider, where the muons have
high energy  and long path lengths. A relatively long  (130~m) section of beam
pipe prior to the detector is shown. Figure~\ref{fig-bheitler2} is for the
100~GeV CoM collider for which, since the muons have rather short path lengths,
only a limited length of beam pipe is shown. Note that the scales are extremely
distorted: the 20$^o$ shielding cones on the right hand  of the figures
appear at steeper angles.
\begin{figure*}[bht!]
\centerline{\epsfig{file=disp_dep_new_new_20.eps,height=7.0in,width=4.0in,angle=-90}}
\vskip 0.5cm
\caption[Trajectories of Bethe-Heitler muons for a 4~TeV collider]{Trajectories of typical Bethe-Heitler muons from their source in the
shielding around the beam pipe to the detector for a 4~TeV CoM collider. As indicated in the text the scales are extremely distorted, the total horizontal length is $\approx 130$~m and the outer edge of the calorimeter is $\approx 4$~m.  Notice that $&lt;1\%$ of the tracks end in the calorimeter (see
table~\ref{tab-bheitler}).}
\label{fig-bheitler1}
\end{figure*}

\begin{figure*}[hbt!]
\centerline{\epsfig{file=dep_new_new_50.eps,height=7.0in,width=4.0in,angle=-90}}
\vskip 0.5cm
\caption[Trajectories of Bethe-Heitler muons for a 100~GeV collider]{Trajectories of typical Bethe-Heitler muons from their source in
the shielding around the beam pipe to the detector for a 100~GeV CoM collider. As indicated in the text the scale is extremely distorted, the total horizontal length is $\approx 20$~m and the outer edge of the calorimeter is $\approx 4$~m. Notice that $&lt;0.5\%$ of the tracks end in the calorimeter (see
table~\ref{tab-bheitler}).}
\label{fig-bheitler2}
\end{figure*}

The most serious effect appears to arise when these muons make deeply inelastic
interactions and deposit spikes of energy in the electromagnetic and
hadronic calorimeters. This is not serious in the Higgs case, for which the
fluxes
and cross sections are low, but at the higher collider energies they generate
significant fluctuations in global parameters, such as transverse energy and
missing transverse energy.

Table~\ref{tab-bheitler} gives some parameters of the muons for three different
machine energies. In the 4~TeV and 500~GeV CoM cases, massive lead
shielding outside the focus quadrupoles has been included. 

Figures~\ref{spikes1} and~\ref{spikes2} show energy deposition from 
Bethe-Heitler muons in a typical bunch crossing. These depositions are 
 plotted against the cosine
of the polar angle and azimuthal angle in the calorimeter for 4~TeV and for
500~GeV CoM, respectively. The massive lead shielding referred to above was not
included in this study. Right hand plots in  Figs.~\ref{spikes1}
and~\ref{spikes2} show the same distributions with a 1~ns timing cut. It is
seen that the timing cut, if it is possible, is effective in removing energy
spikes at small rapidity, but has little effect in the forward and backward
directions. The overall reduction in energy deposition is about a factor of
two.  

The energy spikes can cause at least three problems: 1) they affect the triggers and
event selections based on overall or transverse energy balance; 2) they can
generate false jets and 3) they can give errors in the energies of real jets.
After a pedestal subtraction, the effects on energy balances do not seem
serious. The generation of false jets can be eliminated by a longitudinal
energy distribution cut without introducing significant inefficiency. Energy
errors in real jets appear to be the most serious problem. They can be reduced
by the application of radial energy distribution cuts, but such cuts introduce
significant inefficiencies for lower energy jets. More study is needed.

Earlier studies \cite{nikolai96} with MARS, using less sophisticated
shielding, gave results qualitatively in agreement with those from GEANT.

\begin{figure*}[hbt!]
\centerline{\epsfig{file=mumu_dep_new_2000.eps,height=5.0in,width=3.33in,
angle=-90}}
\vskip 1cm
\caption[Energy deposition from Bethe-Heitler muons for a 4~TeV CoM collider]{Left hand plot shows the energy deposition from Bethe-Heitler muons
\textit{vs.}  the cosine of the polar angle and azimuthal angle in the
calorimeter for a 4~TeV CoM collider. Right hand plot shows the same
distributions with a 1~ns timing cut.}
\label{spikes1}
\end{figure*}

\begin{figure*}[hbt!]
\centerline{\epsfig{file=mumu_dep_new_250.eps,height=5.0in,width=3.33in,
angle=-90}}
\vskip 1cm
\caption[Energy deposition from Bethe-Heitler muons for a 0.5~TeV CoM collider  ]{Left
hand plot shows the energy deposition from Bethe-Heitler muons
\textit{vs.}  the cosine of the polar angle and azimuthal angle in the
calorimeter for a 0.5~TeV CoM collider. Right hand plot shows the same
distributions with a 1~ns timing cut.}
\label{spikes2}
\end{figure*}
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