New Type of Dispersion-Theory Sum Rule

We show that a new class of sum rules can be obtained by comparing, for example, fixed-$t$ and fixed-$u$ dispersion relations. In particular, if an amplitude obeys both fixed-$t$ and fixed-$u$ dispersion relations at $t^{*}$ and $u^{*}$, respectively, we obtain a sum rule by equating the two dispersion relations for the amplitude evaluated at $t^{*}$ and $u^{*}$. Under special circumstances the no-subtraction requirement can be lifted. We apply our procedure to the $A^{\mathrm{}(-1\mathrm{})\mathrm{}}$ and $B^{\mathrm{}(-)\mathrm{}}\pi N$ amplitudes to derive new sum rules, and we show that these sum rules are reasonably well satisfied with only $\rho$, $N$, and $N^{*}$ contributions for the choice $t^{*}=u^{*}=0\mathrm{}$.