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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:

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:

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

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.

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

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.

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