The analytic properties of the box diagram amplitude: I

A double spectral representation of Mandelstam type is obtained for the invariant amplitude T of the binary collision process AB → CD involving scalar particles, using unitarity to evaluate the contribution to Im T from a two-particle intermediate state (EF) and making one-pole approximations to the amplitudes for the processes AB → EF, CD → EF. For the case where the masses satisfy the inequalities (m_A+m_B) ≤ (m_E+m_F), m_C+m_D) ≤ (m_E+m_F) the above approximations are shown to give the same result as that obtained from the Feynman parametrization of the amplitude arising from the box diagram. The proof proceeds by directly evaluating the box diagram amplitude and in this way we obtain a spectral representation of the amplitude for all allowed mass configurations, including cases for which there are anomalous thresholds and cases for which a double spectral representation no longer exists.