# Is field theory needed in Regge-theory?

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I have been reading some papers from G.F. Chew and S. C. Frautschi and they do not even bother to introduce the concept of "Field" when they describe hadron interactions. My impression is that they do not need to because interactions seem to be described by single Regge-trajectories. However Gribov introduced by the end of the 1960s the "Reggeon Field Theory".

When is it mandatory to introduce fields in the theory of complex angular momenta-Regge theory?

Is it when Reggeons (Pomerons) interact among themselves?

Do these fields verify "locality" and "causality"?

I apologyze for asking, again, about this subject. I'm trying to make sense of the papers and books that I have to figure out what I need to study.

From what I have read, it seems that you do not need any "Reggeon Field Theory" if you're just studying either the differential elastic ##pp## and ##p\bar p## cross-sections or their total cross-sections. However, if you are interested in diffractive processes, I have the impression that you do need to take into account multipomeron interactions and, therefore, a Field Theory is needed. However, I have the impression that these field theories are neither local nor causal.

Could anyone, please, help me to either validate or dismiss these ideas?

Yes, I am well aware that he was. However, you also seem to be a smart guy. You gave me a good hint about the ##\gamma \rho## mixing effect. Are you sure that you cannot help me out here?

I am less than an amateur with respect to this topic. You're helping me out. However, I did have one thought. Chew hoped to create an alternative to field theory, the bootstrap theory. Something called bootstrap has been revived and applied to Ising model. Meanwhile, you pointed out that the reggeon description of pp and ppbar scattering resembles 3d directed percolation. So maybe the answers are in that area.

Perhaps slide 31-32 of this will help, unfortunately I'm still trying to understand Landau's introduction of regge theory in a non-relativistic context in his elastic scattering chapter

I am less than an amateur with respect to this topic. You're helping me out. However, I did have one thought. Chew hoped to create an alternative to field theory, the bootstrap theory. Something called bootstrap has been revived and applied to Ising model. Meanwhile, you pointed out that the reggeon description of pp and ppbar scattering resembles 3d directed percolation. So maybe the answers are in that area.

What I am about to write is purely speculative. They are just a set of ideas that some researchers are working with and some ideas that I have in mind. They are just phenomenological models that try to fit the experimental hadronic interactions and speculations of my own. If any of these models turns out to be fully succesful (my own ideas will probably be wrong), it should later be derived from first principles (QCD). I am not trying to confuse anybody.

OK, yes there might be a connection between hadron scattering and the 2+1 directed percolation model. However, that connection has been stablished through a "Reggeon Field Theory" used in hadronic interactions, as you can see in (and references therein) (I do not know if it is paywalled):

http://iopscience.iop.org/article/10.1088/0305-4470/13/12/002/pdf

That, IMO, might mean that the relationship has been enforced from the very beginning and might not be relevant (I have to think a lot more about it because there are other issues to be considered that I am not mentioning for the moment, see point 4).

Let us assume that there might (speculation) be a relationship between the 2+1 DP model and elastic hadron interactions:

1.- In the directed percolation model you have to fix a single relevant operator to be at the critical point (there is a single reduced relevant variable). Once you are there (critical point) (or, at least not too far from it), every relevant physical quantity is related to the reduced operator by a non-trivial critical exponent. There are only three independent critical exponents. Any other relevant operator MUST be null at the critical point. Other critical exponents are related by scaling relations to the three independent ones.

2.- Is all this related to elastic hadron interactions? To be honest I do not know. I have a few "crazy" hypothesis in mind: for example, elastic ##pp## and ##p\bar p## elastic scattering are clearly ##s##-channel interactions and the average diffraction angles are tiny. Perhaps, only perhaps, it may not be unreasonable to think that ##t=0## (zero momentum transfer) could be a critical condition and ##s\gt 4m^2## the relevant operator, maybe, there might be a conection between the 1+2 DP and elastic hadron interactions.

3.- The simplest (and, to the best of my knowledge, only) model that fully explains the experimental results (so far) of elastic ##pp## and ##p\bar p## scattering is published here:

https://arxiv.org/pdf/1711.03288.pdf. Here ##t=0## (total cross-sections).
https://arxiv.org/abs/1808.08580. Here ##t\lt0## (elastic differential cross-sections).

Where three different "particles" (I do not really know how to call them) are used: several "Reggeons" (collectively secondary Reggeons), several "Pomerons" (collectively called Froissaron) and a several "Odderons" (collectively called maximal Odderon). That means 3 different "critical exponents" at ##t=0## and 6 at ##t\leq 0##. We know of a Regge trajectory for the "secondary Reggeons". If there happens to be Regge trajectories for the "Pomeron" and "Odderon" it could be argued that there are only THREE independent "critical exponents" like in the 1+2 DP even at ##t\leq 0##. Or, maybe, the scaling relationships of the 1+2 DP model could be used as a guide to look for the missing Regge trajectories. These ideas are purely speculative but they can be checked. If ##\alpha_R\big(t=0\big)##, ## \frac{d\alpha_R\big ( t\big )}{dt}\Big|_{t=0}##, ##\alpha_P\big(t=0\big)##, ## \frac{d\alpha_P\big ( t\big )}{dt}\Big|_{t=0}##,##\alpha_O\big(t=0\big)## and ## \frac{d\alpha_O\big ( t\big )}{dt}\Big|_{t=0}## are all contained in the list of 1+2 DP critical exponents, then there must be a deep relationship between this model and, at least,the ##pp## and ##p\bar p## elastic scattering amplitude functions ##T(s,t)_{pp}##, ##T(t,s)_{p\bar p}## and ##T(u,t)_{p\bar p}##. If not the relationship has been artificially enforced. At the time when:

http://iopscience.iop.org/article/10.1088/0305-4470/13/12/002/pdf

was written, to the best of my knowledge, the Odderon was not included as an interacting field.

4.- The quark-gluon plasma seems to behave like a superfluid liquid (almost free of viscosity).

https://arxiv.org/pdf/1802.04801.pdf

The directed percolation models seems to describe the critical transition from laminar flow to turbulent flow in incompressible fluids (where viscosity just limits the minimum size of the eddies and otherwise is negligeable outside the tiny boudary layer)(paywalled):

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

The 3+1 Ising Model is a very "nice" Field theory (it is a conformal field theory) "too nice for my taste" but, of course, that's just a prejudice and I will take a close look at it. Thank you for the information.

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