Following up on recent work by Caron-Huot et al. we consider a generalization of the old Lovelace-Shapiro model as a toy model for Pi-Pi scattering satisfying (most of) the properties expected to hold in (t Hoofts) large-N limit of massless QCD. In particular, the model has asymptotically linear and parallel Regge trajectories at positive t, a positive leading Regge intercept $alpha_0 < 1$, and an effective bending of the trajectories in the negative-t region producing a fixed branch point at J=0 for $t < t_0 < 0$. Fixed (physical) angle scattering can be tuned to match the power-like (including logarithmic corrections) behavior predicted by perturbative QCD: $A(s,t) ~ s^{-beta} log(s)^{-gamma} F(theta)$. Tree-level unitarity (i.e. positivity of residues for all values of s and J) imposes strong constraints on the allowed region in the alpha_0-beta-gamma parameter space, which nicely includes a physically interesting region around $alpha_0 = 0.5$, $beta = 2$ and $gamma = 3$. The full consistency of the model would require an extension to multi-pion processes, a program we do not undertake in this paper.
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