Validity of the Weizs\"acker-Williams Approximation and the Analysis of Beam Dump Experiments: Production of an axion, a dark photon, or a new axial-vector boson

Beam dump experiments have been used to search for new particles, $\phi$, 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 to obtain the cross section and the resulting particle production rates without using approximations on the phase space integral or Monte-Carlo simulations. In our previous work we examined the case of the new scalar boson production; in this paper we explore all possible new spin-0 and spin-1 particles. 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. Furthermore, a new region ($m_\phi<2m_e$) of parameter space can be explored without using one of the common approximations, $m_\phi\gg m_e$. We derive new expressions for the three photon decays of dark photon and four photon decays of new axial-vector bosons. As a result, the production cross section and exclusion region of different low mass ($m_\phi<2m_e$) bosons are very different. Moreover, our method can be used as a consistency check for Monte-Carlo simulations.