  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 Hooft's) 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.
