Approximate Formula for a Regge Trajectory with Right-Hand Cut and Its Application

We derive a general ansatz for the imaginary part of a Regge trajectory, with only a right-hand cut, using conformal mapping and threshold behavior. Next we use the dispersion relation and a three-parameter approximation of this ansatz to obtain an analytical formula for the trajectory. If the Pomeranchuk trajectory has no left-hand cut, then application of this formula shows that $f^0$ does not lie on this trajectory. Further, we find ${\alpha }_P(\pm \infty )>-1\mathrm{} \mathrm{and} \gtrsim 0$. Applying the formula to $f^0$ and $\rho$, we conclude that, if our approximation is good, then the Regge trajectories corresponding to these resonances must have left-hand cuts.