Use of the Optical Theorem and Regge trajectories

[Anashim]

Cutkosky rule states that:

$$2Im \big(A_{ab}\big)=(2\pi)^4\sum_c \delta\Big(\sum_c p^{\mu}_{c}-\sum_a p^{\mu}_{a}\Big)|A_{cb}|^2\hspace{0.5cm} (1)$$

putting $a=b=p$ in Cutkosky rule we deduce the Optical Theorem for $pp$ scattering:

$$2Im \big(A_{pp}\big)=(2\pi)^4\sum_c \delta\Big(\sum_c p^{\mu}_{c}-\sum_p p^{\mu}_{p}\Big)|A_{cp}|^2\hspace{0.5cm} (2)$$

From which the following relations can be deduced:

$$\sigma_{total}^{pp}=\frac{Im\big[A_{pp}(s,t=0)\big]}{2|p_1|\sqrt s}\hspace{4.5cm}(3)$$

$$\frac{d\sigma_{el}^{pp}}{dt}=\frac{|A_{pp}(s,t)|^2}{64\pi |p_1|^2s}\hspace{5.85cm}(4)$$

where $s,t,u,$ are Mandelstam variables and $A_{pp}(s,t)$ are the $\mathbb T$ matrix $(i\mathbb T=\mathbb S- \mathbb I)$ elements of elastic $pp$ scatterings. $\sigma_{total}^{pp}$ total cross-section, includes both elastic and non-elastic collisions.

My question is:

Do we know the singularity structure of $A_{pp}(s)$ in the $s$ plane so that $Re\big[A_{pp}(s)\big]$ can be calculated from $Im\big[A_{pp}(s)\big]$?

Please, notice that $\sigma_{total}^{pp}$ has been measured up to $s=13 TeV$ and, therefore, $Im\big[A_{pp}(s,t=0)\big]$ is known.

I am very confused because in $(1)$ and $(2)$ it is very clear that all intermediate states, $c$, are described by "on-shell" particles. However in Regge theory, Reggeons, Pomerons and even Odderons (they all seem to be one particle intermediate states) seem to be identified with Regge trajectories of the form $\alpha(t)=\alpha(0)+\frac{d\alpha}{dt}t$ (straight lines).

This seems reasonable, since the transmited momentum $(t)$ in a $pp$ scattering process should be a continuous variable and not a discrete one, but this clearly contradicts $(2)$. So, there's a second question directly related to the first one:

Are really Regge trajectories used instead of the particles belonging to them?, if so, why?, since most points of a Regge trajectory do not represent real particles that can be "on-shell".

Moreover, Regge trajectories completely change the singularity structure of $A_{pp}(s,t=0)$. (Cut branches seem to be replacing singular points).

What is it that I'm getting so wrong??

Could the answer be that all real particles belonging to a given Regge trajectory can resonate among themselves giving rise to the whole trajectory?? Or is this pure madness??

Any help will be much appreciated.

Since the subject "calculation of total cross-sections in hadron interaction" is rather specialized, the following links are given for reference:

https://link.springer.com/content/pdf/10.1140/epjc/s10052-016-4585-8.pdf

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

http://school-diff2013.physi.uni-heidelberg.de/Talks/Poghosyan.pdf

Please, notice that Regge theory seems to be working pretty well as you can check in:

https://arxiv.org/pdf/1711.03288.pdf

https://arxiv.org/pdf/1807.06471.pdf

In fact, it is the only phenomenological model that seems to be providing the right answers.

 

[Anashim]

P.S.

Equation $(3)$ is very difficult for me to interpret. It relates the forward elastic scattering of protons $A_{pp}\big(s,t=0\big)$ with the total cross-section, that is to say, the cross-section of $p+p\rightarrow anything$. I know that $(3)$ is mathematically correct but I completely miss its physical interpretation. Forward scattering is a purely EM interaction, while outside a very small cone, the interaction is predominantly strong so why are these two quantities related? and, I repeat, I understand how $(3)$ is derived from $(2)$.

I feel so frustrated...