Dispersion Relations and Asymptotic Behavior of the Veneziano Partial-Wave Amplitude in the ComplexsPlane

The asymptotic behavior of the Veneziano partial-wave $I=1\mathrm{}$ amplitude $V_l\mathrm{}\left(s\right)\mathrm{}$ for $\pi \pi$ scattering is studied in the complex $s$ plane for physical $l$ values. The $\rho -f^0$ exchange-degenerate trajectory is of the form $\alpha \mathrm{}\left(s\right)=as+b$. For $b<\mathrm{}1\mathrm{}$ and $3b+4a{m_{\pi }}^2\ge 1$, it is shown that, asymptotically, $V_l\mathrm{}\left(s\right)\mathrm{}\sim o\left(s^{b-1\mathrm{}}\right)$. Under the same conditions, the resonance partial widths for fixed $l$ have the property $\Gamma s_R\sim o\left({s_R}^{b-\frac{3}{2}}\right)$. The discontinuity of $V_l\mathrm{}\left(s\right)\mathrm{}$ across the left-hand cut oscillates, and if $b<\mathrm{}1\mathrm{}$, then, asymptotically, disc $V_l\mathrm{}\left(s\right)\mathrm{}\sim o\left(s^{-2b-4a{m_{\pi }}^2}\right)$. In the case $-2a{m_{\pi }}^2<b<\mathrm{}1\mathrm{}$, disc $V_l\mathrm{}\left(s\right)\mathrm{}\to 0$ as $s\to -\infty$ and $V_l\mathrm{}\left(s\right)\mathrm{}\to 0$ as $\mathrm{}\left|s\right|\mathrm{}\to \infty$ and $V_l\mathrm{}\left(s\right)\mathrm{}$ can be written in the form of unsubtracted partial-wave dispersion relations, i.e., as an integral along the left-hand cut plus the sum of an infinite number of poles along the right-hand real axis. Thus for the particular case of the $\rho$-trajectory ($b\approx \frac{1}{2},a\approx 1$ Be${\mathrm{V}}^{-2\mathrm{}}$), an unsubtracted dispersion relation can be written.