  Beam dump experiments have been used to search for new particles with null
results interpreted in terms of limits on masses $m_\phi$ and coupling
constants $\epsilon$. However these limits have been obtained by using
approximations [including the Weizs\"{a}cker-Williams (WW) approximation] or
Monte-Carlo simulations. We display methods, using a new scalar boson as an
example, to obtain the cross section and the resulting particle production
numbers without using approximations or Monte-Carlo simulations. We show that
the approximations cannot be used to obtain accurate values of cross sections.
The corresponding exclusion plots differ by substantial amounts when seen on a
linear scale. In the event of a discovery, we generate pseudodata (assuming
given values of $m_\phi$ and $\epsilon$) in the currently allowed regions of
parameter space. The use of approximations to analyze the pseudodata for the
future experiments is shown to lead to considerable errors in determining the
parameters. Furthermore, a new region of parameter space can be explored
without using one of the common approximations, $m_\phi\gg m_e$. Our method can
be used as a consistency check for Monte-Carlo simulations.
