Some Consequences of the Wu-Yang Asymptotic Behavior of Form Factors

We investigate the properties of a class $\mathcal{a}$ of form factors $F\left(t\right)\mathrm{}$ which include some with the asymptotic behavior suggested by Wu and Yang: $F\left(t\right)\mathrm{}\sim A\exp [-a{\mathrm{}(-t)\mathrm{}}^{\frac{1}{2}}]$, $a>\mathrm{}0\mathrm{}$, for large negative $t$. A simple representation theorem for such form factors is proved. This theorem is then used as the main tool to obtain the results which follow. "Asymptotic lower bounds" are derived for any $F\left(t\right)\mathrm{}$ of class $\mathcal{a}$, both at large positive and at large negative $t$. It is shown that any form factor belonging to class $\mathcal{a}$ satisfies a generalized unsubtracted dispersion relation. A whole set of sum rules for $\mathrm{Im}F$ and $\mathrm{Re}F$ on the cut are obtained, generalizing those which follow from a superconvergent dispersion relation. An asymptotic lower bound on the number of changes of sign of $\mathrm{Im}F$ and of $\mathrm{Re}F$ along the cut at large positive $t$ is derived. As an illustration, we first construct according to a simple procedure $F\left(t\right)\text{'}\mathrm{s}$ which have the given asymptotic behavior and whose only singularities are simple poles. We then derive an asymptotic lower bound on the number of these poles at large positive $t$ for any $F\left(t\right)\mathrm{}$ of this type.