Boundary Dispersion Relations

We derive a dispersion relation for direct-channel reactions of the type $a+b\to c+d$ with $m_b=m_d$, e.g., $\gamma N\to \pi N$, which involves contributions from both forward and backward scattering of the direct channel and backward scattering of the crossed channel but no contribution from unphysical regions of the direct channel. In particular, the integral involving the direct-channel contribution is evaluated along the boundary of the physical region. Our result, in the case of elastic scattering, i.e., $m_a=m_c$, reduces to the familiar backward dispersion relation.