How do soft Pomerons become hard?

[Anashim]

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 dimensions 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?

 

[AndyW]

I find the connection between Reggeon Field Theory and Directed Percolation Models to be intriguing. It is possible that the change in the number of relevant spatial dimensions could explain the need for a hard Pomeron in the Regge limit of Deep Inelastic Scattering experiments.

One possible explanation for this connection could be the concept of universality, which states that different physical systems can exhibit similar behavior at critical points. In the case of the Directed Percolation Models, the critical exponents describe the behavior of the system near a phase transition point, and it is possible that the same critical exponents can also describe the behavior of the soft and hard Pomerons in the Regge limit.

Another possibility could be the role of Pomeron-Pomeron interactions in the BFKL equation. It is possible that these interactions could play a similar role to the interactions between particles in the Directed Percolation Models, leading to a change in the critical exponents at higher exchanged momenta.

Further research and experimentation would be needed to fully understand the connection between Reggeon Field Theory and Directed Percolation Models and its implications for the behavior of the soft and hard Pomerons in the Regge limit. However, the potential for a new understanding of these phenomena is certainly an exciting prospect for future studies in this field.