Unitarization of the Veneziano Model, Pomeranchuk Singularity, and the Pion Mass

We suggest a general framework for unitarizing the Veneziano model for $\pi \pi$ scattering, and deduce some immediate consequences. By duality the unitarity $s$ cut generates a moving Regge cut with branch-point trajectory $1+{\alpha }^{\prime }t$ in the approximation $m_{\pi }=0\mathrm{}$, with ${\alpha }^{\prime }$ the slope of the $\rho -f^0$ trajectory. In principle $m_{\pi }$ may be calculated from the shift of the Adler zero. A crude estimate yields $m_{\pi }\approx {\left({\alpha }^{\prime }\right)}^{\frac{1}{2}}{m}_{\rho }{\Gamma }_{\rho }=95\mathrm{}$ MeV.