Analytic Structure of Collision Amplitudes in Perturbation Theory

Abstract

Some methods are developed for studying the singularities of collision amplitudes in perturbation theory as functions of two of the invariant energies, $s$, $t$, and $u$. It is shown that: (i) There are no singularities other than normal thresholds in the physical regions of the physical sheet.(ii) For the equal-mass case there are no singularities in the Euclidean region of the physical sheet.(iii) The only straight lines of singularities on the real boundary of the physical sheet are normal singularities in the equal-mass case, and in the general-mass case are either normal singularities or they intersect the Euclidean region.(iv) The curves of singularities on the real $s$, $t$ plane in the physical sheet do not connect to surfaces extending into the region $s$ real, $t$ complex except at turning points of the curves.(v) Turning points of curves of singularities in the physical sheet may occur either when sufficient coincident singularities become also end-point singularities, or when there is an accidental relation between the Feynman variables at coincident singularities. The former correspond to anomalous thresholds; the latter are called spurious turning points.(vi) For the equal-mass case there are no anomalous thresholds and no anomalous turning points in the curves of singularities.(vii) Spurious turning points do occur in negative spectral regions, but here it appears that they may not lead to complex singularities on the physical sheet. There are no spurious turning points in positive spectral regions in low orders in perturbation theory and to all orders for some types of diagram. It is plausible that there are none for any diagram, but this is not proved.The relation of this work to the Mandelstam representation is discussed. All the proven results in this paper are consistent with this representation. Some points are noted which require further investigation before the validity of the representation can be established to all orders in perturbation theory.