  We solve exactly the "boundary sine-Gordon" system of a massless scalar field
\phi with a \cos[\beta\phi/2] potential at a boundary. This model has appeared
in several contexts, including tunneling between quantum-Hall edge states and
in dissipative quantum mechanics. For \beta^2 < 8\pi, this system exhibits a
boundary renormalization-group flow from Neumann to Dirichlet boundary
conditions. By taking the massless limit of the sine-Gordon model with boundary
potential, we find the exact S matrix for particles scattering off the
boundary. Using the thermodynamic Bethe ansatz, we calculate the boundary
entropy along the entire flow. We show how these particles correspond to wave
packets in the classical Klein-Gordon equation, thus giving a more precise
explanation of scattering in a massless theory.
