$pp$ and $p\bar p$ scattering can be approximately described (in the Regge limit, that is, when $s \gg m \gt |t|$) by the exchange of Reggeons defined by the following Regge trajectory (low $s$):

$$\alpha_R(t)=\alpha_R(0)+\frac{\mathrm d\alpha_R}{\mathrm dt}t$$

and the exchange of one (or several) Pomerons (whose quantum number must be equal to those of the vacuum) that are defined by the following Regge trajectory (high $s$):

$$\alpha_P(t)=\alpha_P(0)+\frac{\mathrm d\alpha_P}{\mathrm dt}t$$

where $s$ is the COM frame energy, $t$ is the 4-momentum transfer, $\alpha_R(0)\simeq 0.55$, $\mathrm d\alpha_R/\mathrm dt\simeq 0.86$, $\alpha_P(0)\simeq 1.08$, $\mathrm d\alpha_P/\mathrm dt\simeq 0.25$.

Surprisingly, the critical exponents of the 3d directed percolation theory, that happens to be described by a Reggeon Field Theory, have the following critical exponents: $\eta_\bot\simeq 0.581$, $\beta\simeq 0.81$, $\eta_\|\simeq 1.105$.

Hadron interaction $s$ and $t$ exponents seem to be related to the critical exponents of the 3d directed percolation model. Their numerical values have a good degree of agreement and both problems are described by a Regge Field Theory, so it makes sense think that they might be the same numbers.

The specific question is:

Why are these critical exponents values so closed?

This question is relevant because the nature of Reggeons and Pomerons (and the Odderon, that is not included here) is unknown.

They need to be explained in terms of the Standar Model of particle physics (this is one of the unsolved problemas in physics, see, for example https://en.wikipedia.org/wiki/Regge_theory) but, although there are some speculations (glueballs?), nothing is known for sure.

Any link between this problem (Reggeon and Pomeron as interaction mediators between hadrons) and a well known problem (the directed percolation model of Critical Phenomena) may provide useful information about their nature.

This is a path that I am willing to explore, but my knowledge about Regge Theory in its many variants is very limited and it will take me some time. I thought that, maybe, more experienced people than myself will be able to see a connection that is crystal clear but that I am not able to see due to my lack of knowledge.

To be able to understand better the question, you need to be familiar with hadron interactions in the Regge limit as well as with the theory of Critical Phenomena. But the questions is, IMO, very clear.

The following reference provides information on the Regge limit of hadron interactions. Slide 5 plots $pp$ and $p\bar p$ total cross-sections where two of the critical exponents are used to fit the experimental results. Since total cross-sections are related to forward scattering, $\frac{\mathrm d\alpha_R}{\mathrm dt}$ and $\frac{\mathrm d\alpha_P}{\mathrm dt}$ are not needed to fit the experimental results.

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

The following reference provides information about the directed percolation model critical exponents (page 56, Table 2.) and its relation to Reggeon Field Theory (page 58, Section Relation to Reggeon field theory).

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

There's an interesting reference in the coments bellow:

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

However, I think it hardly explains anything. I suspect that the authors of this letter have implicity used the meson Chew-Frautschi plot, that is:

$$\alpha_R(t)=\alpha_R(0)+\frac{\mathrm d\alpha_R}{\mathrm dt}t$$ in their $x$ definition.

I have to check this carefully. However I tend to think that it is much more important the fact that the 3d directed percolation model seems to describe the transition from a steady incompressible laminar flow, to a turbulente one (where vorticity is created at all scales) as suggested in:

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

which is completely unrelated to the hadron scattering problem.

If the model used in c) to derive the 3d directed percolation model also includes the Pomeron trajectory it is reasonable that its three independent scaling exponents can be similar to $\alpha_R(0)\simeq 0.55$, $\mathrm d\alpha_R/\mathrm dt\simeq 0.86$, $\alpha_P(0)\simeq 1.08$.

The question now is what's the relationship between hadron scattering and the transition to an almost non-viscous incompressible fluid (almost) described by the Euler equations of fluid dynamics because d) strongly suggests this.

I've been thinking hard about this question for a while and I think I can explain things in a much more straightforward and clear way (the question is not solved yet, only better posed):

1.- There is considerable evidence to assert that the imcompressible fluid flow transition from laminar to turbulent may well be described by the 3d directional percolation critical model. Please read the papers included in this reference:


IMO the papers that present empirical data are much more important than those that only express opinions, but, of course, that's up to you to decide.

2.- In this model there is a single variable (a probability $P$) that when exactly equal to a given value, $P_c$, the transition happens.

In the neighbourhood of this critical point all physica properties can be expressed as $(P-P_c)^{C_e}$, where $C_e$ are critical exponents (there are quite a few of them but only THREE of them are independent).Critical exponents are very usually non-rational numbers.

3.- The calculation of the critical exponents of the 3d directed percolation model was carried out in:


and Regge Theory was used. This may suggest that the relationship between the 3d percolation model and hadron interactions has been enforced from the very begining and is, therefore, useless.

4.- However, the 3d percolation model, also describes a very important physical problem: the flow transition on an incompressible fluid from laminar to turbulent.

IMO it can be concluded that there's a strong relationship between hadron interaction and transition to turbulence, since all the statistical information of both problems can be extracted from the same model.

5.- For the sake of concreteness I will concentrate on $pp$ elastic scattering.

We know that all the information is encoded in the complex $\mathbb{T}$ matrix elements $A_{pp}\big(s,t\big)$. For example the total $pp$ cross-section is directly related to $Im\Big[A_{pp}\big(s,0)\Big]$. The differential elastic scattering is directly related to $|A_{pp}\big(s,t\big)|^2$.

6.- Here, there seems to be a problem, since $A_{pp}$ depends both on $s$ and $t$ and the laminar to turbulent flow transition depends only on Reynolds number. However, we have to take into account that $t\ll s$, so as a first order approximation will be tolerated (for a short while).

Please, notice that $$\sigma^{tot}_{pp}=\frac{Im\Big[A_{pp}\big(s,t=0\big)\Big]}{2|p_1|\sqrt{s}}$$

7.- Let's analyze the expression of Reynolds number:

$$Re=\frac{\rho u L}{\mu\big(T\big)}$$


$\rho$: liquid density.

$u$: liquid average velocity.

$L$: characteristic linear dimension.

$\mu$: dynamical viscosity (depends slightly on $T$).

$T$: Temperature.

We, therefore, have:

$$Re=\frac{\rho u L}{\mu\big(T_0\big)}-\frac{\rho u L\mu'\big(T_0\big) \big(T-T_0\big)}{\mu^2\big(T_0\big)}$$

Where $T_0$ is a reference temperature. So, we see that, in fact, there is a second independent variable (operator) that takes much smaller values.

However there's a slight problem here. $t$ or $-\frac{\rho u L\mu'\big(T_0\big) \big(T-T_0\big)}{\mu^2\big(T_0\big)}$ is a relevant variable (it's essential if we want to calculate the total elastic cross-section), but the directed percolation model has one single relevant variable (or operator) in phase-space.

8.- The directed percolation model is the simplest example of dynamical critical phenomena.

9.- My objective now is to find out if there exists a dynamical critical model, strongly related to the directed percolation model, that has two relevant operators.

As mentioned in a comment below, there are, at least, two things that I got wrong:

  1. If any of them is, the 3D percolation model cannot be relevant in $pp$ and $p\bar p$ scattering since their amplitude depends on only two squared momenta, $s$ and $t$: $A_{pp}(s,t)$: Only the 2D model can be relevant.
  2. The critical exponents MUST be non dimensional and $\alpha_{\mathbb R}^{'}(0)$, $\alpha_{\mathbb P}^{'}(0)$ and $\alpha_{\mathbb 0}^{'}(0)$ do have dimesions ($GeV^{-2}$) (and I do not know of any natural energy scale in QCD to make them non dimensional).



three critical exponents are given for each model (1D, 2D and 3D). IMO, the Montecarlo-simulation results must be more accurate, so I will calculate the rest of the CE using scaling and hyperscaling relations (for the 2D model) and check if they fit or not. If they do not, then this idea MUST be ruled out.

From the above reference:

  • $\nu=0.734$
  • $\beta=0.584$
  • $z=1.763$

$$\nu d=2-\alpha=2\beta+\gamma=\beta(1+\delta) \Rightarrow \alpha=2-2\nu=0.532; \gamma=2\nu-2\beta=0.3$$

$$ \delta=\frac{2\nu}{\beta}-1=1.514;2-\eta =\frac{\gamma}{\nu}\Rightarrow \eta=2-\frac{\gamma}{\nu}=1.591$$

Taking into account that:

$$\sigma_{pp}^{tot}\sim B_{\mathbb R}s^{\alpha_{\mathbb R}(0)-1}+ B_{\mathbb P}s^{\alpha_{\mathbb p}(0)-1}$$

$$\frac{d\sigma_{pp}^{el}}{dt}\simeq F_{\mathbb R}(t)s^{2\alpha_{\mathbb R}(t)-2}+F_{\mathbb P}(t)s^{2\alpha_{\mathbb P}(t)-2}$$

Either $\alpha_{\mathbb R}(0)-1\simeq -0.45$ and $\alpha_{\mathbb P}(0)-1\simeq 0.1$ or $2\alpha_{\mathbb R}(0)-2\simeq -0.9$ and $2\alpha_{\mathbb P}(0)-2\simeq 0.2$ must show up in the critical exponent list. (I guess it could also be $\alpha_{\mathbb R}(0)\simeq 0.55$ and $\alpha_{\mathbb P}(0)\simeq 1.1$).

Since I do not see them there, I guess my idea was wrong.

  • $\begingroup$ I think this address the question : sunclipse.org/wp-content/downloads/2013/04/cardy-etal1980.pdf $\endgroup$
    – Adam
    Jul 31, 2018 at 13:47
  • $\begingroup$ D=3 results are missing in the letter. Maybe the perturbative expansion is good enough for this case too. I'll check and thanks again $\endgroup$ Jul 31, 2018 at 14:15
  • $\begingroup$ The 3d directed percolation model seems to describe the transition from a steady incompressible laminar flow, to a turbulente one (where vorticity is created at all scales). I wonder if this idea may help. $\endgroup$ Jul 31, 2018 at 14:34
  • $\begingroup$ @Adam: This cannot be an issue. The critical exponents MUST be associated with a fix point of the renormalization group and, therefore, cannot be inserted by hand in the model. If they were, renormalization would change their value. $\endgroup$ Oct 3, 2018 at 13:22
  • $\begingroup$ Not sure that helps, but this paper computes the critical exponents of directed percolation using some RG : arxiv.org/pdf/cond-mat/0309504.pdf $\endgroup$
    – Adam
    Oct 3, 2018 at 13:32

1 Answer 1


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



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.

These three trajectories are taken from experiments. Intensive research efforts are being made to explain these three "particles" from QCD and the consensus is that they are some sort of collective phenomena 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 anf the best fits you cannot help wondering if there is some kind of universal behavior. See for example slides 35 and 36 in:


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

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


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

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.

It was quickly pointed out 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 issue has kept me distracted for a long while. However I do not think now 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? The answer is yes (using the impact parameter representation), which is briefly explained in:


In slide 22 we can see that the diffraction peak shrinks with the C.O.M. energy $s$ and, therefore, so does the diffraction dip energy $t_{min}$.



expression (11) gives the diffraction dip energy, $t_{min}$, and 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. Its critical exponents are shown in page 56 of this reference:


However, 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.

Well, now I think that this is no problem either. The collective phenomenon where gluons produce Regge-trajectories may be happen in a 3d space, and the elastic scattering effective theory is 2d because of rotational invariance.

In (paywalled, last two letters):


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 Reynolds 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 (Machs 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.

If everything I said in the previous paragraphs 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:


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(s) 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.

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):


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 developped turbulent phase.

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 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 be the cause of the turbulent-phase saturation reported in this letter?

I will try to talk again with a collegue who is specialyzed in Fluid Dynamics and ask him what he thinks about this issue.

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.

  • $\begingroup$ Minor comment: it's better to link to abstracts of papers than to PDFs (e.g. http://arxiv.org/abs/whatever) $\endgroup$
    – rob
    Oct 5, 2018 at 18:56
  • $\begingroup$ @rob: I changed one of them (in arXiv), but I do not how to fix the Pomeron slides and the paywalled references. $\endgroup$ Oct 5, 2018 at 19:48
  • $\begingroup$ I am very confused. Critical models either provide exact results or are completely wrong. It is a hit or miss situation but, in this case, there seems to be no response from anybody at all. $\endgroup$ Oct 5, 2018 at 19:57
  • $\begingroup$ I am starting to think that, like someone said in another comment I read, Regge-theory is lost knowledge. $\endgroup$ Oct 5, 2018 at 20:06
  • $\begingroup$ I'm done with this issue. $\endgroup$ Oct 8, 2018 at 19:39

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