Anomalous Threshold as the Pivot of Feynman Amplitudes

where D is a partition of {1 . . . n} into 4 nonempty sets, P i is the sum of momenta in i ∈ D and D0 a scalar box. In other words, scalar one-loop integrals (up to boxes) form a basis. Thus, coefficients in the expansion (BD etc.) are uniquely determined, although some reduction method can be more efficient than others. However, troublesome points where the numerical stability of the result is at stake will always be there. What to do in these cases? We can change (adapt) bases, or avoid bases (expansion). We explain our idea via examples; first, we consider factorization of Feynman amplitudes, the Kershaw theorem of Ref. [ 2]: any Feynman diagram is particularly simple when evaluated around its anomalous threshold. The singular part of a scattering amplitude around its leading Landau singularity 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, box graph, etc.) and on