Forward Compton Scattering Amplitude as a Simultaneous Analytic Function of Complex Photon Mass and Energy

The analytic structure of the virtual forward Compton scattering amplitude as a function of one and two complex variables is investigated for various combinations of variables which involve the virtual photon mass. This is done both by using the Deser-Gilbert-Sudarshan representation and by using the Feynman perturbation theory. The role and significance of complex Landau singularities is discussed. In general these are branch points, but it is found that at $t=0\mathrm{}$ some of these become poles. The effect of these complex singularities and of overlapping cuts on the ordinary single-variable dispersion relations and the mass extrapolations in the vector-meson-dominance model is explained. Using the Cutkosky discontinuity formula for Feynman graphs, the analytic structure of the discontinuities across various normal threshold cuts of the non-Born-term part of this amplitude is deduced. One finds that the two-variable analyticity of the amplitude implies a single-variable analyticity for these discontinuities. It is shown that in general the inelastic structure functions $W$ and $\overline{W}$ cannot be expected to have a simply determinable analytic structure. But under certain conditions $W$ and $\overline{W}$ can be identified with boundary values of the discontinuities across the $s$-and $u$-channel normal threshold cuts in the virtual forward Compton scattering amplitude, respectively. This is used to show that the contribution to $W$ and $\overline{W}$ from certain types of Feynman graphs under certain conditions are analytic functions of one complex variable, and only for such cases can one use crossing symmetry to relate the inelastic-electron-scattering structure functions and the annihilation structure functions. The motion of the Landau singularities of $\nu W_2$ is shown to provide a possible explanation for its observed rapid approach to "scaling" for finite but large final-state masses.