Algebraic Factorization of Scattering Amplitudes at Physical Landau Singularities

The behavior of the scattering amplitude in the vicinity of a physical Landau singularity is considered. It is shown that its singular part may be written as an algebraic product of the scattering amplitudes for each vertex of the corresponding Landau graph times a certain explicitly determined singularity factor which depends only on the type of singularity (triangle graph, square graph, etc.) and on the masses and spins of the internal particles. Thus the well-known result for single-particle-exchange poles is generalized to arbitrary physical Landau singularities. Also, it is shown that for any Landau singularity there exists a finite polynomial in the scalar products of the external four-momenta whose vanishing gives the Landau singularity curve. A general, purely algebraic, method is given for constructing this polynomial.