From matrix element to hadronic cross section

[ayseo]

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?

 

[Bill_K]

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 = (2ω_a2ω_b v_ab)^-1|ℳ|² dΦ

where dΦ is the phase space factor,

dΦ = (2π)⁴ δ⁴(∑k_i - P) Π (d³k_i/((2π)³ 2ω_i)

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

v_ab = |v_a - v_b| is the relative velocity of a and b. ω_i, k_i 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) k_i'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)])

 

[Hepth]

there's 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