# Use of the Optical Theorem and Regge trajectories

• A
• Anashim
In summary, the Cutkosky rule states that the imaginary part of the amplitude for a process involving on-shell particles is related to the total cross-section for the same process. This can be seen in equations (1) and (2), which also lead to equations (3) and (4) that relate the total and elastic cross-sections to the amplitude. However, there is confusion about the singularity structure of the amplitude in the complex plane, particularly when considering Regge trajectories. These trajectories do not correspond to physical particles, but rather represent resonances among them. This can be seen in the successful application of Regge theory in calculating total cross-sections in hadron interactions. However, the physical interpretation of equation (3), which
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://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.

Last edited:
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...

## 1. What is the Optical Theorem?

The Optical Theorem is a fundamental principle in theoretical physics that relates the scattering of particles to the total cross section of the interaction. It states that the total cross section is equal to four times the imaginary part of the forward scattering amplitude.

## 2. How is the Optical Theorem used in research?

The Optical Theorem is used in research to calculate the total cross section of particle interactions and to validate theoretical models. It is also used in the study of quantum chromodynamics and high-energy physics, where it is applied to calculate scattering amplitudes.

## 3. What are Regge trajectories?

Regge trajectories are a series of curves that describe the behavior of particles with different spins and masses as a function of their angular momentum. They were first proposed by physicist Tullio Regge in the 1950s and have been used to study the properties of high-energy particles.

## 4. How are Regge trajectories related to the Optical Theorem?

Regge trajectories and the Optical Theorem are closely related as they both involve the study of particle scattering and interactions. Regge trajectories are used to predict the behavior of particles at high energies, while the Optical Theorem is used to calculate the total cross section of these interactions.

## 5. What are some practical applications of the use of the Optical Theorem and Regge trajectories?

The use of the Optical Theorem and Regge trajectories has practical applications in fields such as high-energy physics, nuclear physics, and cosmology. They are used to study the properties of particles, validate theoretical models, and understand the fundamental forces of nature. They also have potential applications in medical imaging and radiation therapy.

• High Energy, Nuclear, Particle Physics
Replies
4
Views
2K
• High Energy, Nuclear, Particle Physics
Replies
9
Views
2K
• High Energy, Nuclear, Particle Physics
Replies
1
Views
1K
• Quantum Physics
Replies
8
Views
1K
Replies
1
Views
3K
• Quantum Physics
Replies
1
Views
1K
Replies
4
Views
2K
• Quantum Physics
Replies
1
Views
1K
• Quantum Physics
Replies
1
Views
2K