From matrix element to hadronic cross section

In summary, the conversation discusses the process of calculating the hadronic cross section from a matrix element for a 2->3 scattering process containing quarks. The phase space factor and kinematical factors are important components in determining the collision of two particles, and the integration of the four δ-functions can be complex. References to further information on this topic are also provided.
  • #1
ayseo
1
0
Hello,

currently I work on 2->3 scattering process. So there exist five external momenta and in my case 5 different Feynman diagrams, for which I have already calculated the full matrix element.
The matrix element is a function of various scalar products of the four-momenta. This scattering contains quarks. My next aim is to calculate the hadronic cross section from the matrix element.

Can someone explain me how can I do this?
 
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  • #2
ayseo, I think this topic is mentioned in almost every treatment of scattering theory, and 99 percent of those treatments are way too brief to be of any use. The most complete discussion I've found is in a book on QFT by S.J. Chang, which devotes ten pages to it.

You need to put a factor in front due to the initial flux, and a factor in back due to the final phase space. The general result for a collision of two particles a and b is

dσ = (2ωa2ωb vab)-1|ℳ|2 dΦ

where dΦ is the phase space factor,

dΦ = (2π)4 δ4(∑ki - P) Π (d3ki/((2π)3 2ωi)

ℳ of course is the relativistically invariant amplitude calculated from the Feynman diagram. 1/2ωa is a boson kinematical factor ("wavefunction normalization"), one for each particle both incoming and outgoing. For fermions use M/E instead.

vab = |va - vb| is the relative velocity of a and b. ωi, ki are the energy/momenta of the individual outgoing particles, while E, P are the total energy/momentum.

Bad enough already, but the real ugly part comes when you go to integrate out the four δ-functions and express the (redundant) ki's in terms of the desired experimental parameters. Quoting Chang, for two outgoing particles,

dΦ = k13 dΩ1/(16π2 (E k12 - ω1 P·k1))

For three outgoing particles, (take a deep breath!)

dΦ = k12k22 dk1 dΩ1 dΩ2/((2π)5 8ω1 [k22(E - ω1) - ω2 k2·(P - k1)])
 
  • #3
Theres an older paper that sets up a lot of these phase space integrals, and I believe a production process with 2->3 is done in the appendix of :

http://prola.aps.org/abstract/PR/v185/i5/p1865_1
and some more in
http://prd.aps.org/abstract/PRD/v2/i9/p1902_1
 

1. What is a matrix element?

A matrix element is a mathematical quantity that describes the transition amplitude between two quantum states in a physical system. In particle physics, it is used to calculate the probability of a specific particle interaction occurring.

2. How is a matrix element calculated?

The calculation of a matrix element involves using the fundamental principles of quantum mechanics and the laws of conservation of energy and momentum. It also requires a thorough understanding of the specific physical system being studied.

3. What is a hadronic cross section?

A hadronic cross section is a measure of the probability of a hadronic process (a process involving hadrons, such as protons and neutrons) occurring. It is used to describe the interaction between particles and is an important quantity in particle physics experiments.

4. How is the hadronic cross section related to the matrix element?

The hadronic cross section is related to the matrix element through the Feynman-Hellmann theorem, which states that the derivative of the total cross section with respect to a coupling constant is equal to the imaginary part of the matrix element. In other words, the hadronic cross section is directly proportional to the probability of a particle interaction occurring, which is described by the matrix element.

5. What are some applications of studying the matrix element and hadronic cross section?

Studying the matrix element and hadronic cross section is crucial in understanding the fundamental interactions of particles in the universe. It is also important in the development of new theories and models in particle physics, as well as the design and analysis of experiments to test these theories. Additionally, knowledge of these quantities can have practical applications, such as in medical imaging and nuclear energy production.

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