Analyticity in the Forward Tube, Complex Scaling, and New Sum Rules for Virtual Compton Amplitudes

The forward virtual Compton amplitude is studied as an analytic function of two complex variables. A method is developed for deriving sum rules that follow from analyticity in two variables where the domain of analyticity is the forward tube. Using this method, new sum rules are derived. These sum rules follow from analyticity in the tube and "complex" scaling. They relate a double integral over the inelastic form factor W₂(ν,q²) to an integral over the scaling function F₂(ω). The sum rules provide a tool for directly testing whether scaling occurs when |q²|→∞ in complex directions inside the tube domain, and whether this scaling starts at relatively low values of |q²| as in the real case. The possibility of finding more restrictive sum rules by enlarging the domain of analyticity is also briefly explored and a mathematical but unphysical example is given.