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https://arxiv.org/pdf/1711.03288

https://arxiv.org/abs/1808.08580

Three different Regge-trajectories are needed: one Reggeon, one (soft) Pomeron and one Odderon. The Pomeron carries the same quantum numbers as the vacuum and the Odderon has odd symmetry under crossing. This "particle" or trajectory explains why ##pp## and ##p\bar p## total cross-sections are different even at high energies (Pomeranchuck's theorem states that they should be equal and experiments prove otherwise). An additional "hard" Pomeron might be needed to explain ##ep## deep inelastic scattering results.

Regge-theory allows the construction of different efective field theories that are used in QCD in different asymptotic cases (BFKL field equation, for example).

These three trajectories are taken from experiments. Intensive research efforts are being made to explain these three "particles" from first principles (QCD) and the consensus is that they are some sort of collective gluonic behavior involving ladder gluons or "reggeized-gluons", although the problem is far from being solved.

The physical bases of the multiperipheral model are explained in (paywalled):

https://link.springer.com/content/pdf/10.1007/BF02781901.pdf

and in:

"High-Energy Particle Diffraction" 2002, by V. Barone and E. Pedrazzi.

**Section 5.9**.

The emerging picture is that Regge-trajectories are

**essentially non loca**l objects, involving the addition of multiple gluon ladders, as described in the multiperipheral model:

See, for example, "High-Energy Particle Diffraction" 2002,

**Chapter 8**.

I thought that, perhaps, this collective gluon behavior could be related to some (unknown) critical dynamic phenomenon belonging to one of the known Universality Classes. If you look at the experimental results of total cross-sections of different hadrons and their best fit curves, you cannot help wondering if there is some kind of

__universal behavior__. See for example slides (35) and (36) in:

https://indico2.riken.jp/event/2729/attachments/7480/8729/PomeronRIKEN.pdf

So, the question was if the three trajectories, defined by:

$$\alpha_{\mathbb R}(t)=\alpha_{\mathbb R}(0)+\alpha_{\mathbb R}^{'}(0)t$$

$$\alpha_{\mathbb P}(t)=\alpha_{\mathbb P}(0)+\alpha_{\mathbb P}^{'}(0)t$$

$$\alpha_{\mathbb O}(t)=\alpha_{\mathbb O}(0)+\alpha_{\mathbb O}^{'}(0)t$$

where:

- ##\alpha_{\mathbb R}(0)\simeq 0.55##
- ##\alpha_{\mathbb P}(0)\simeq 1.1##
- ##\alpha_{\mathbb O}(0)\simeq 1##
- ##\alpha_{\mathbb R}^{'}(0)\simeq 0.86 GeV^{-2}##
- ##\alpha_{\mathbb P}^{'}(0)\simeq 0.25 GeV^{-2}##
- ##\alpha_{\mathbb O}^{'}(0)\sim 0.2 GeV^{-2}##

could belong to some Universality Class.

I had a quick look and realized that the three first non-dimesional numbers are close to some of the critical exponents belonging to the 3D-directed percolation universality class.

Someone attracted my attention to the fact that the directed percolation universality classes had been originaly claculated in:

https://sunclipse.org/wp-content/downloads/2013/04/cardy-etal1980.pdf

using a "Reggeon-Field Theory". That could mean that the Regge-trajectories had been inserted by hand in the theory from the very beginning.

This kept me thinking for a long while. However I do not think that it is an issue, because you cannot introduce anything by hand because of renormalization. If this F.T. represents a critical phenomenon, it MUST have a fixed point (possibly in the IR) whose running coupling constants stop changing (possibly below a momentum scale). This FP defines the critical exponents of the theory.

Another important concern is that critical exponents

**must**be non-dimensional. Can the non-dimensional ##\alpha_{\mathbb R}^{'}(0)##, ##\alpha_{\mathbb P}^{'}(0)## and ##\alpha_{\mathbb O}^{'}(0)## be obtained? I think that the answer might be yes (using the impact parameter representation), which is briefly explained in:

http://school-diff2013.physi.uni-heidelberg.de/Talks/Poghosyan.pdf

In slide (22) it can seen that the diffraction peak shrinks with the C.O.M. energy ##s## and, therefore, so does the diffraction dip energy ##t_{min}##.

In:

https://arxiv.org/pdf/1609.08847.pdf

expression (11) gives the diffraction dip energy, ##t_{min}##, and, if I am not wrong, the Pomeron non-dimensional slope is denoted by ##a_2\sim 0.01##.

Could any of the DP critical models be consistent with these results? IMO yes, the 3-DP model

**be consistent with them (although I am well aware that it is a bit of a leap to consider that the 3DP model**

*could**critical dynamic phenomenon explaining Regge-trajectories). Its critical exponents are shown in page 56 of this reference:*

**could be the**https://lanl.arxiv.org/pdf/cond-mat/0001070v2

Moreover, I realized a few weeks ago, that the scattering amplitude ##A_{pp}(s,t)## depends only on

**two squared momenta**and the model expected to be consistent with the data was the 2-DP model which, definitely

**is not**. However,

**now**I think that since the collective phenomenon involves

**gluons**, which are not the relevant interacting "particles" in Regge Theory, this is no longer a problem. They just "live" in different configuration spaces.

Well, I guess I am going to be laughed at for suggesting this idea. However, I have tried to explain and document every single step that I have followed before making this question. If I am totally wrong it will not be difficult to find out where it is that I am wrong.