# How do soft Pomerons become hard?

The exchange of soft Pomerons (and Reggeons) ($$\alpha_R(0)=0.55$$ and $$\alpha_P(0)=1.08$$) seem to describe total hadron-hadron cross sections pretty well in the Regge limit. See, for example:

https://arxiv.org/abs/hep-ph/9209205

In this limit, QCD is of very little use since the exchanged momentum scale is rather low and, therefore, the coupling constant is very large. Perturbative computations seem to be completely out of the question.

In the Regge limit of Deep Inelastic Scattering experiments (small x-physics),

https://arxiv.org/abs/hep-ph/9507320

where the exchanged momentum scale is larger, p-QCD and Regge theory seem to find a common ground. However, at large $$Q^2$$, or for that matter, very small $$x$$, the soft Pomeron does not correctly describe the proton structure function $$F_2(x)$$. A hard Pomeron with a $$\alpha_P(0)=1.3$$ value seems to be required.

How and why does this happen? It may seem that Reggeon Field Theory (in the form of the BFKL equation) is needed, a theory that takes into account Pomeron-Pomeron interactions:

https://arxiv.org/abs/1304.8022

It is, however, tempting to give an alternative interpretation:

Reggeon Field Theory explains the critical exponents of the Directed Percolation Models:

https://arxiv.org/abs/cond-mat/0001070

The fundamental critical exponents of the DP model in three space plus one time dimensions are:

$$\beta=0.81\\ \nu_\bot=0.581\\ \nu_\|=1.105$$

while in two space plus one time dimesions are:

$$\beta=0.584\\ \nu_\bot=0.734\\ \nu_\|=1.295$$

Could the hard Pomeron be explained in terms of a change of the number of relevant spatial dimensions from three to two at higher exchanged momenta? Does this make any sense?