$pp$ 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:
https://www.nature.com/collections/rxsztdqblr/
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:
https://sunclipse.org/wp-content/downloads/2013/04/
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:
$$Re=\frac{\rho u L}{\mu\big(T\big)}$$
where:
$\rho$: 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:
- 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.
- 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).
In:
https://arxiv.org/pdf/cond-mat/0309504.pdf
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.
