How Do You Calculate Electron-Muon Scattering Amplitude in Particle Physics?

[gn0m0n]

Homework Statement

Griffiths Particle Physics, problem 7.24.

Evaluate the amplitude for electron-muon scattering in the center-of-momentum system, assuming the e and mu approach one another along the z-axis, repel, and return back along the z-axis. Assume initial and final particles to have helicity +1. [Answer is given: M=-2(g^2)]

Homework Equations

The solutions to the Dirac equation. Also, the form of the amplitude derived for two-particle scattering from Feynman rules.

I'll have to scan the equation because it's complicated and I don't know Latex.

Sorry. I will upload a scan of the eq's and my work later tonight. I just am hoping someone will be be familiar enough with the subject to help anyway.

The Attempt at a Solution

The equation for this process obtained from the lowest-level Feynman diagram contains, u's, v's, p's (4-vectors of the 4 particles, or 2 before and after), and gamma matricex (gamma_mu). The u's and v's are the spinors that satisfy the Dirac equation, and u-bar is (u*)(gamma_0) where the * denotes the adjoint or Hermitian conjugate.

The particular u's and v's to use depend on the spins of the particle, so taking the assumption given regarding their helicity, I assume they all spin counter-clockwise around their direction of motion - i.e., the right-hand rule. Therefore, taking the direction of motion to be along the z-axis, have particles 1 and 4 with spin up, 2 and 3 spin down. Is this properly applied?

Then I simply plug in the values for the u,v spinors and their transpose conjugates from Griffiths section on the Dirac equation (section 7.2). The gamma_0 's just switch the sign of the last two entries in the transposed spinors, and the "N_i" coefficients of the spinors can just come out of the expression and group together.

So basically I am not sure now how to multiply something of the form (u*)(gamma_mu)(u) Do I just add (u*)gamma_0 + (u*)gamma_1 + ... (u*)gamma_3 ?

Actually here it will be (u3*)(gamma_mu)(u1)(u4*)(gamma_mu)(u2) where the numbers just indicate the particle's label.

I'd really appreciate any help!

 

[gn0m0n]

Oops, probably shouldn't have called those spinors... I guess they're "bispinors" or "Dirac spinors".

Also, the gamma_mu's are using the "Bjorken and Drell convention".

Oh, and it's not just that I'm unsure how to do those gamma_mu multiplications, it also doesn't look like what I have so far will give the right answer...

Finally, if anyone knows any sites specifically for discussing Griffiths problems, please let me know! :)

 

[nrqed]

Homework Statement

Griffiths Particle Physics, problem 7.24.

Evaluate the amplitude for electron-muon scattering in the center-of-momentum system, assuming the e and mu approach one another along the z-axis, repel, and return back along the z-axis. Assume initial and final particles to have helicity +1. [Answer is given: M=-2(g^2)]

Homework Equations

I'll have to scan the equation because it's complicated and I don't know Latex.

Sorry. I will upload a scan of the eq's and my work later tonight. I just am hoping someone will be be familiar enough with the subject to help anyway.

The Attempt at a Solution

The equation for this process obtained from the lowest-level Feynman diagram contains, u's, v's, p's (4-vectors of the 4 particles, or 2 before and after), and gamma matricex (gamma_mu). The u's and v's are the spinors that satisfy the Dirac equation, and u-bar is (u*)(gamma_0) where the * denotes the adjoint or Hermitian conjugate.

The particular u's and v's to use depend on the spins of the particle, so taking the assumption given regarding their helicity, I assume they all spin counter-clockwise around their direction of motion - i.e., the right-hand rule. Therefore, taking the direction of motion to be along the z-axis, have particles 1 and 4 with spin up, 2 and 3 spin down. Is this properly applied?

Then I simply plug in the values for the u,v spinors and their transpose conjugates from Griffiths section on the Dirac equation (section 7.2). The gamma_0 's just switch the sign of the last two entries in the transposed spinors, and the "N_i" coefficients of the spinors can just come out of the expression and group together.

So basically I am not sure now how to multiply something of the form (u*)(gamma_mu)(u)

Actually here it will be (u3*)(gamma_mu)(u1)(u4*)(gamma_mu)(u2) where the numbers just indicate the particle's label.

I'd really appreciate any help!

Well, there are some tricks to simplify the calculation using projectors but you can do it the long way (it's just ordinary matrix multiplications) :

\bar{u_3} \gamma_0 u_1 ~~\bar{u_4} \gamma_0 u_2 ~~ - \bar{u_3} \gamma_1 u_1 ~~\bar{u_4} \gamma_1 u_2 - \bar{u_3} \gamma_2 u_1 ~~\bar{u_4} \gamma_2 u_2 - \bar{u_3} \gamma_3 u_1 ~~\bar{u_4} \gamma_3 u_2

 

[gn0m0n]

Cool, thanks. I just want to understand ANY way to get through it right now, I'm not too concerned if it's not pretty ;)

So are we sort of looking at the two gamma_mu's and acting the same as if they were right next to each other, i.e., summing over like indices (subtracting where appropriate)? Or would we do as you wrote even if it was just, say, (u*)(gamma_mu)(u)? In that second case, I wasn't sure if we would add or subtract for mu=1,2,3.

 

[gn0m0n]

I'm still having some trouble getting this to come out right so if anyone has any other thoughts, please share! I ended up with -(1/2)g^2 instead of -2g^2 ! So close! I am wondering if I made proper assumptions, e.g., in this case, p1=-p2=-p3=p4, right? And E1=E3, E2=E4? Also the helicity hint needs to have been applied correctly to assigning their spins...