The equation of motion of a same-charge particle during a beam-beam collision is expressed by[5]
(similarly for y), where N is the beam intensity, 2L the bunch
length, 
the classical electron radius, 
the Lorentz
factor of the beam energy and 
the energy fraction of the
particle. For a uniform charge distribution inside an elliptic
cylinder of 2L long with radii a and b in the horizontal(x)
and vertical(y) directions, respectively, the Coulomb potential is
exactly given by
where q(x,y) is the positive solution to the equation 
. Assuming that a particle is created
uniformly at a position 
with zero scattering
angle 
, its trajectory inside the
ellipsoid is
where 
is the aspect ratio of the transverse beam
profile, which is equivalent to 
for a Gaussian
beam.
We first consider the azimuthal angular distribution of
same-charge particles while ignoring the Coulomb field outside of the
cylinder. The azimuthal angle (
) is defined by
where
under the condition
The distribution function (
), which is normalized to unity,
is expressed by
where 
is a step function.
For 
and 
(or 
), where we are
interested in measuring the aspect ratio for future linear
colliders, can be well approximated as
As can be clearly seen in the above
equations(8,9), the
distribution in the angular region near to the horizontal plane is
depleted for a large aspect ratio as 1/R. On the contrary, the
distribution in the vertical direction 
has
apparently no dependence on R. Therefore, we hereafter concentrate
on calculations near to the horizontal plane in this section. We then
estimate the effect of the Coulomb force outside of the elliptic
cylinder, where the equation of motion is approximated for 
and 
, as follows:
and
Since the vertical force in Eq.(11) is very small
in this region, except for 
, we neglect it for the moment.
Integrating the x component of Eq.(10) once, for
large 
we obtain
where 
and 
, which is the x-velocity at the
position (x, y) = (a, 0) . The resultant distribution function at
is give for large 
by
We further estimated the effect of the vertical force neglected in
the above calculation by numerically integrating
Eqs.(10) and (11). The major
effect was found to be an overall change in the magnitude by a
constant factor of 
. Therefore, the R dependence of
F(0) becomes 
instead of 1 / R (
Eq.(9)) mainly due to a horizontal force
outside of the elliptic cylinder. Thus, for the case of a flat beam (
), the same-charge particles are strongly
deflected in the vertical direction; their azimuthal angular
distribution gives us direct information about R. It should also be
noticed that the particles tend to be deflected more in a direction
where their creation points are displaced by 
and
, since the forces are proportional to 
and 
, respectively, provided by
differentiating Eq.(2). This gives us the sensitivity
to measure the alignment between two beams, which is discussed in the
next section.
The maximum deflection angle for low-energy particles can be
expressed by[5]
where 
, 
, 
and 
. Using these
relations, Gaussian beams 
can be
well described in terms of variables (a,b,L) of an elliptic
cylinder. Assuming that the beam intensity and the bunch length are
known, the horizontal beam size can be obtained from a measurement
of the maximum deflection angle, especially for a large aspect ratio,
as can be clearly seen in Eq.(14).