Soft interaction at high energy and N=4 SYM

In this paper we show that the N=4 SYM total cross section violates the Froissart theorem, and in the huge range of energy this cross section is proportional to $s^{1/3}$. The graviton reggeization will change this increase to the normal logarithmic behaviour $\sigma \propto \ln^2 s$. However, we demonstrated that this happens at ultra high energy, much higher than the LHC energy. In the region of accessible energy we need to assume that there is a different source for the total cross section, with the value of the cross section about 40 mb. With this assumption we successfully describe $\sigma_{tot}, \sigma_{el}$ and $\sigma_{diff}$ for the accessible range of energy from the fixed target Fermilab to the Tevatron energies. It turns out that the N=4 SYM mechanism can be responsible only for a small part of the inelastic cross section for this energy region (about $2 mb$). However, at the LHC energy the N=4 SYM theory can describe the multipartticle production with $\sigma_{in} \approx 28 mb$. The second surprise is the fact that the total cross section and the diffraction cross section can increase considerably from the Tevatron to the LHC energy. The bad description of $B_{el}$ gives the strong argument that the non N=4 SYM background should depend on energy. We believe that we have a dilemma: to find a new mechanism for the inelastic production in the framework of N=4 SYM other than the reggeized graviton interaction, or to accept that N=4 SYM is irrelevant to any experimental data that has been measured before the LHC era.