Lehmann Ellipses

Abstract

This chapter gives a detailed treatment of both the small as well as the large Lehmann Ellipses. These are indispensable tools in enlarging the domains of analyticity. They play central roles in analytical continuations of many kinds including the ones in particle masses. The discussion starts with the demonstration that continuing scattering angles to complex values is equivalent to extending the domain of analyticity in $\cos {\theta }$ to ellipses. Lehmann ellipses can be understood this way. Lehmann’s integral representation is then derived based on the well-known Jost-Lehmann-Dyson theorem. This is applied to the earlier integral representation of Lehmann to obtain yet another integral representation, based on which Lehmann gave an integral representation in which the entire dependence on scattering angle is isolated in a simple denominator. This enabled analytical continuations to complex scattering angles straightforwardly. Consequently, Lehmann obtains an explicit expression for the imaginary part of the scattering angle in terms of masses and energies. This determines the sizes of Lehmann ellipses as a function of these physical quantities. These are central to the derivation of the fixed-t dispersion relations.