  The sum over planar multi-loop diagrams in the NS+ sector of type 0 open
strings in flat spacetime has been proposed by Thorn as a candidate to resolve
non-perturbative issues of gauge theories in the large $N$ limit. With $SU (N)$
Chan-Paton factors, the sum over planar open string multi-loop diagrams
describes the 't Hooft limit $N\to \infty$ with $Ng_s^2$ held fixed. By
including only planar diagrams in the sum the usual mechanism for the
cancellation of loop divergences (which occurs, for example, among the planar
and M\"obius strip diagrams by choosing a specific gauge group) is not
available and a renormalization procedure is needed. In this article the
renormalization is achieved by suspending total momentum conservation by an
amount $p\equiv \sum_i^n k_i\neq 0$ at the level of the integrands in the
integrals over the moduli and analytically continuing them to $p=0$ at the very
end. This procedure has been successfully tested for the 2 and 3 gluon planar
loop amplitudes by Thorn. Gauge invariance is respected and the correct running
of the coupling in the limiting gauge field theory was also correctly obtained.
In this article we extend those results in two directions. First, we generalize
the renormalization method to an arbitrary $n$-gluon planar loop amplitude
giving full details for the 4-point case. One of our main results is to provide
a fully renormalized amplitude which is free of both UV and the usual spurious
divergences leaving only the physical singularities in it. Second, using the
complete renormalized amplitude, we extract the high-energy scattering regime
at fixed angle (tensionless limit). Apart from obtaining the usual exponential
falloff at high energies, we compute the full dependence on the scattering
angle which shows the existence of a smooth connection between the Regge and
hard scattering regimes.
