# Could the 3D percolation model explain Regge-Trajectories?

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Regge-theory succesfully explains the latest LHC ##pp## elastic scattering experimental results and total cross-sections:

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):

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 local 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:

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 could be consistent with them (although I am well aware that it is a bit of a leap to consider that the 3DP model could be the critical dynamic phenomenon explaining Regge-trajectories). Its critical exponents are shown in page 56 of this reference:

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.

There is something relevant that I did not mention.

In (paywalled, the last two papers):

https://www.nature.com/collections/rxsztdqblr/

it is experimentally checked that the directed percolation models describe well the critical transition from laminar to turbulent flow. However, the model only works in a narrow neighbourhood of the critical condition.

The problem is that the incompressible Navier Stokes equations are Galileo-invariant, not Lorentz-invariant. This disease may be somewhat "cured" doing a "pseudo-relativistic" correction. Please, bear in mind that the incompressible Navier-Stokes equations approximation breaks down as Mach's number approaches 1 from (well) below.

Instead of considering the critical opertator:

$$p_c:=\frac{|\mathbb{Re}-\mathbb{Re}_c|}{\mathbb{Re}_c}\propto \frac{|U-U_c|}{U_c}$$

where ##Re## denotes Reynold's number, ##c## critical condition and ##U## the reference velocity, you can use, instead:

$$p_c:=\frac{|\mathbb{Re}-\mathbb{Re}_c|}{\mathbb{Re}_c\Big [1+\frac{\mathbb{Re}\mathbb{Re}_c}{\mathbb{Re}_m^2}\Big ]}\propto \frac{|U-U_c|}{U_c\Big [1+\frac{UU_c}{U_m^2}\Big ]}$$

where ##m## denotes the speed of sound condition (Mach's number=1). In this way, the validity of the percolation model may be extended deep into the turbulence region. The speed of sound may play here a similar roled to the speed of light in relativity.

Poincare's invariance must be somehow restored if incompressible Navier-Stokes equations and hadron interactions are to be mutually related by the same Universality Class model.

Now, I'm finally done with this issue.

If everything I said in the previous posts turns out to be true (I am going to the the calculations to see if the hypotheses hold true), a trivial consequence would be that:

$$p_c:=\frac{U_c}{U_m}$$

is a universal number. It would only depend on the spacial dimension ##d## number of the laminar to turbulent critical transition. ##U_c## is the critical velocity and ##U_m## the speed of sound in the incompressible fluid. These numbers should match those of the different ##p_c## of the Directed Percolation Models. If this is not true then some of my hypotheses do not hold true (I would be wrong).

Could anyone with access to the data, please, do the calculations? I'm going to try to collect the data with a collegue who is specialized in fluid dynamics, but I'm not sure if we'll be able to get them.

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There is something relevant that I did not mention.

In (paywalled, the last two papers):

https://www.nature.com/collections/rxsztdqblr/

it is experimentally checked that the directed percolation models describe well the critical transition from laminar to turbulent flow. However, the model only works in a narrow neighbourhood of the critical condition.

The problem is that the incompressible Navier Stokes equations are Galileo-invariant, not Lorentz-invariant. This disease may be somewhat "cured" doing a "pseudo-relativistic" correction. Please, bear in mind that the incompressible Navier-Stokes equations approximation breaks down as Mach's number approaches 1 from (well) below.

Instead of considering the critical opertator:

$$p_c:=\frac{|\mathbb{Re}-\mathbb{Re}_c|}{\mathbb{Re}_c}\propto \frac{|U-U_c|}{U_c}$$

where ##Re## denotes Reynold's number, ##c## critical condition and ##U## the reference velocity, you can use, instead:

$$p_c:=\frac{|\mathbb{Re}-\mathbb{Re}_c|}{\mathbb{Re}_c\Big [1+\frac{\mathbb{Re}\mathbb{Re}_c}{\mathbb{Re}_m^2}\Big ]}\propto \frac{|U-U_c|}{U_c\Big [1+\frac{UU_c}{U_m^2}\Big ]}$$

where ##m## denotes the speed of sound condition (Mach's number=1). In this way, the validity of the percolation model may be extended deep into the turbulence region. The speed of sound may play here a similar roled to the speed of light in relativity.

Poincare's invariance must be somehow restored if incompressible Navier-Stokes equations and hadron interactions are to be mutually related by the same Universality Class model.
.

Typical critical Reynolds ##\mathbb{Re}_c## are so low compared to sound Reynolds ##\mathbb{ Re}_m## in usual incompressible fluids that it is pretty sure that the "pseudo-relativistic" corrections do not play any role in the deviations from critical behaviour deep inside the turbulent phase (which are, of course, expected to happen).

I was totally wrong.

Even though the critical condition can be established for Non-Directed Percolation Models, I have not been able to find any sotochastic model that clearly established a threshold probability (or critical probability) for continuous d-Dimensional Directed Percolation Model.

The hypothesis if the universality nature of ##\frac{U_c}{U_m}## is probably wrong too.

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Typical critical Reynolds ##\mathbb{Re}_c## are so low compared to sound Reynolds ##\mathbb{ Re}_m## in usual incompressible fluids that it is pretty sure that the "pseudo-relativistic" corrections do not play any role in the deviations from critical behaviour deep inside the turbulent phase (which are, of course, expected to happen).

I was totally wrong.

Even though the critical condition can be established for Non-Directed Percolation Models, I have not been able to find any sotochastic model that clearly established a threshold probability (or critical probability) for continuous d-Dimensional Directed Percolation Model.

The hypothesis if the universality nature of ##\frac{U_c}{U_m}## is probably wrong too.
For reference:

Nature's monograph on transition to turbulence and Directed Percolation models (paywalled):

https://www.nature.com/collections/rxsztdqblr/

I am very confused. Letter number (5) of Nature's monograph on turbulence shows NO sign of turbulence density saturation and the agreement between experimental resuts, simulations and the 1+1 Directed Percolation Model is impressive even in the fully developed turbulent phase, where the agreement is unexpected.

However, Letter number (4) shows a good degree of agreement between experiments and the 2+1 Directed Percolation Model only in a narrow interval of Re numbers. I find two issues of this paper a bit strange:

1. Turbulent conditions are enforced at the inlet (so that transients are quicker, so I unerstand why they've done it but I'm not sure what its implications are).
2. Figure (2.c) shows that the non-turbulent regions seem to be correlated with the edges of the channel. Could this at the origin of the turbulent-phase saturation reported in this letter?

I will try to talk again to a collegue who is specialyzed in Fluid Dynamics and ask him what he thinks about this issue, because the authors of Letter number (5) are very assertive about their results (saturation of the turbulent phase density).

What I wrote about critical velocity being much smaller than the speed of sound is, of course, true. I was completely wrong about pseudo-relativistic corrections being able to explain the above mentioned saturation effect.