# At what stage is it necessary to introduce a field theory in the regeon-pomeron-odderon model of hadron interactions?

I've 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?

Is it when Reggeons 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 I have to figure out what I need to study.

I will try to rephrase the question in terms of what I have been reading the last few days:

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? Thank you very much in advance for your help.

OK, now I have found a partial answer to my own question. I will post it here just in case someone is interested in this topic:

1. If you are only interested in elastic scattering you do not need any field at all. Regge-theory on its own works just fine as long as $$\frac{s}{|t|}$$ is very large (and this is always the case for elastic scattering in particle accelerators). If you knew the singularity structure of $$A_{pp} (s,t)$$ in the $$s$$ and $$t$$ complex planes and the relevant Regge trajectories, you would be able to calculate everything.The problem is that you do not know a priori the Regge trajectories that you need to calculate the elastic amplitude function $$A(s,t)$$. (Or, at least, nobody has been able to deduce them from first principles). You also ignore the weights of each Regge-trajectory. All that information is taken from experiments (total, $$\sigma_{tot}$$ and elastic, $$\sigma_{el}$$ cross-sections measurements and Chew-Frautschi plots). For the sake of concreteness, I will give the final expressions for the one-particle intermediate-state approximation of the Optical Theorem:

$$A(s,t)=\sum_{\xi=\pm}\sum_{i=Reg-trajs}\eta_{\xi}\big [ \alpha _{i}^{\xi} (t)\big ]\gamma _{i}^{\xi}(t)\Big (\frac{s}{s_0}\Big )^{\alpha_{i}^{\xi} (t)}$$

• $$\xi=+1$$ for Regge trajectories associated to mesons (or whatever other hadrons) with even angular momentum and $$\xi=-1$$ for Regge trajectories associated to mesons (or whatever other hadrons) with odd angular momentum.
• $$\eta(t)$$ is called the signature factor.

$$\eta(t)=-\frac{1+\xi e^{-i\pi\alpha(t)}}{\sin\big [\pi\alpha (t)\big ]}$$

• $$\alpha^{\pm}_{i}(t)$$ is called Regge trajectory and, as far as we know, can be (very well) approximated by a straight line (Chew-Frautschi plots). $$\alpha ^{\pm}_{i}(t) \simeq \alpha ^{\pm}_{i} (0)+\frac{d} {dt}\alpha ^{\pm}_{i}(t)|_{t=0}t$$.
• $$\gamma _{i}^{\xi}(t)$$ is the residue of each Regge trajectory. Regge trajectories represent, in the $$t$$ channel, resonances among mesons (or whatever other hadrons), for each $$t$$ value there is a complex residue in the complex $$t$$ plane. The $$t$$ channel is the analytic continuation of $$A_{pp}(s,t)$$ to the region where $$t>4m^2$$ and $$s<0$$ and where we have $$A_{p\bar p}(t,s)$$ instead of $$A_{pp}(s,t)$$ because of crossing symmetry.
2. The latest experiments ( $$\sigma_{tot}^{pp}$$, $$\sigma_{tot}^{p\bar p}$$ and $$\sigma_{el}^{pp}$$) at the LHC are correctly described only by a Regge theory that contains THREE Regge-trajectories named the Reggeon(s), the Pomeron(s) and the Odderon(s), theoretically predicted a long time ago.

3. If you want to consider diffractive events you do need an effective field theory and there are several models depending on what what you are interested in. In these theories the Regge-trajectories (or Regge particles) are considered as interacting fields.
4. If you think that Regge-theory is incorrect you are wrong. It belongs to S-matrix theory where axioms are clear and proofs are mathematically correct. If you think that Regge-theory is useless then you are very probably wrong.
5. I do not know anything about the different Regge field theories so I cannot say anything about the locality and causality issues.
6. This is just an opinion, but I think I am entitled to give one. IMO Regge-theory has been dismissed as irrelevant or wrong since the discovery of quarks and the success of the Standard Model. However, there has been very little progress in QCD beyond the perturbative limit (where Regge-theory works well in some asymptotic cases). This attitude is, IMO, just an unjustifiable prejudice. Regge-theory should be derived from first principles (QCD), but this cannot be done if the theory remains widely ignored.
• If you found an answer to your own question, you can just post it as an answer and accept it! – knzhou Sep 26 '18 at 13:11
• Would I not be cheating? It's a partial answer. – Carlos L. Janer Sep 26 '18 at 13:36
• The bounty period is over, anyway. – Carlos L. Janer Sep 26 '18 at 13:40

OK, now I have found a partial answer to my own question. I will post it here just in case someone is interested in this topic:

1. If you are only interested in elastic scattering you do not need any field at all. Regge-theory on its own works just fine as long as $$\frac{s}{|t|}$$ is very large (and this is always the case for elastic scattering in particle accelerators). If you knew the singularity structure of $$A_{pp} (s,t)$$ in the $$s$$ and $$t$$ complex planes and the relevant Regge trajectories, you would be able to calculate everything.The problem is that you do not know a priori the Regge trajectories that you need to calculate the elastic amplitude function $$A(s,t)$$. (Or, at least, nobody has been able to deduce them from first principles). You also ignore the weights of each Regge-trajectory. All that information is taken from experiments (total, $$\sigma_{tot}$$ and elastic, $$\sigma_{el}$$ cross-sections measurements and Chew-Frautschi plots). For the sake of concreteness, I will give the final expressions for the one-particle intermediate-state approximation of the Optical Theorem:

$$A(s,t)=\sum_{\xi=\pm}\sum_{i=Reg-trajs}\eta_{\xi}\big [ \alpha _{i}^{\xi} (t)\big ]\gamma _{i}^{\xi}(t)\Big (\frac{s}{s_0}\Big )^{\alpha_{i}^{\xi} (t)}$$

• $$\xi=+1$$ for Regge trajectories associated to mesons (or whatever other hadrons) with even angular momentum and $$\xi=-1$$ for Regge trajectories associated to mesons (or whatever other hadrons) with odd angular momentum.
• $$\eta(t)$$ is called the signature factor.

$$\eta(t)=-\frac{1+\xi e^{-i\pi\alpha(t)}}{\sin\big [\pi\alpha (t)\big ]}$$

• $$\alpha^{\pm}_{i}(t)$$ is called Regge trajectory and, as far as we know, can be (very well) approximated by a straight line (Chew-Frautschi plots). $$\alpha ^{\pm}_{i}(t) \simeq \alpha ^{\pm}_{i} (0)+\frac{d} {dt}\alpha ^{\pm}_{i}(t)|_{t=0}t$$.
• $$\gamma _{i}^{\xi}(t)$$ is the residue of each Regge trajectory. Regge trajectories represent, in the $$t$$ channel, resonances among mesons (or whatever other hadrons), for each $$t$$ value there is a complex residue in the complex $$t$$ plane. The $$t$$ channel is the analytic continuation of $$A_{pp}(s,t)$$ to the region where $$t>4m^2$$ and $$s<0$$ and where we have $$A_{p\bar p}(t,s)$$ instead of $$A_{pp}(s,t)$$ because of crossing symmetry.
2. The latest experiments ( $$\sigma_{tot}^{pp}$$, $$\sigma_{tot}^{p\bar p}$$ and $$\sigma_{el}^{pp}$$) at the LHC are correctly described only by a Regge theory that contains THREE Regge-trajectories named the Reggeon(s), the Pomeron(s) and the Odderon(s), theoretically predicted a long time ago.

3. If you want to consider diffractive events you do need an effective field theory and there are several models depending on what what you are interested in. In these theories the Regge-trajectories (or Regge particles) are considered as interacting fields.
4. If you think that Regge-theory is incorrect you are wrong. It belongs to S-matrix theory where axioms are clear and proofs are mathematically correct. If you think that Regge-theory is useless then you are very probably wrong.
5. 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): https://link.springer.com/content/pdf/10.1007%2FBF02781901.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 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.

6. This is just an opinion, but I think I am entitled to give one. IMO Regge-theory has been dismissed as irrelevant or wrong since the discovery of quarks and the success of the Standard Model. However, there has been very little progress in QCD beyond the perturbative limit (where Regge-theory works well in some asymptotic cases). This attitude is, IMO, just an unjustifiable prejudice. Regge-theory should be derived from first principles (QCD), but this cannot be done if the theory remains widely ignored.