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The Landau equations give a physically useful criterion for how singularitiesarise in Feynman amplitudes. Furthermore, they are fundamental to the uses ofperturbative QCD, by determining the important regions of momentum space inasymptotic problems. Generalizations are also useful. We will show that inexisting treatments there are significant gaps in derivations, and in somecases implicit assumptions that will be shown here to be false in importantcases like the massless Feynman graphs ubiquitous in QCD applications. In thispaper is given a new proof that the Landau condition is both necessary andsufficient for physical-region pinches in the kinds of integral typified byFeynman graphs. The proof's range is broad enough to include the modifiedFeynman graphs that are used in QCD applications. Unlike many existingderivations, there is no need to use the Feynman parameter method. Somepossible further applications of the new proof and its subsidiary results areproposed.

A new and complete proof of the Landau condition for pinch singularities of Feynman graphs and other integrals