How to incorporate the neutral current into the Dirac equation

[BobbyV]

Hi Everyone,

I'm a math grad student working on numerical procedures for the Dirac equation, and I'd like to be able to incorporate the neutral current interaction

neutrino + fermion -> Z bozon -> neutrino + fermion <- poorly impersonated Feynman diagram

into the Dirac equation as a potential term to allow me to calculate more physically meaningful results. The problem is, I'm familiar enough with the notation to expand the neutral current interaction into full matrix form so I can start programming it into the computer.

Here's the neutral current equation from Wikipedia's entry on Neutral Current

and here's the neutral current component of the Electroweak Lagrangian

Which of these is most appropriate to integrate into the Dirac equation, and how so?

How can I find the formal definition of each of the terms so I can expand them? (the gamma matrices are pretty straightforward, of course, but I have trouble with other terms like the matrix definition of weak isospin and the vector and axial vector couplings)

Any help would be appreciated, including links to papers on this topic.

Thanks!

 

[Bill_K]

BobbyV, The vertex that couples to the Z is -iψγ^μ(g_V - g_Aγ⁵)ψ Z_μ. The relationship you need to use is g_V - g_Aγ⁵ ≡ (g/cos θ_W) (c_V - c_Aγ⁵) = (g/cos θ_W) (½(1 - γ⁵)T³ - sin²θ_WQ) where g is the weak coupling constant, T³ is the third component of isospin, Q is the charge, and θ_W is the weak mixing angle: M_W/M_Z = cos θ_W.

 

[BobbyV]

Hi Bill_K! Thank you for your prompt reply. Because I'm unfamiliar with particle physics, I have a couple of questions for you from your equation:

-iψγ^μ(g_V - g_Aγ⁵)ψ Z_μ

1. How does one find the components of the Z spinor, Z_μ above? I'm told this should be available, but I haven't been able to find an accessible paper on this.

2. Is the wave function on the left ψ the wave function conjugate, or is the wave function for the second particle involved in the scattering?

3. If the answer to 2. is that it's the conjugate, how do we take the second particle into account in this interaction? I'm assuming that we may multiply the second particle's wavefunction on the right and it's conjugate on the left, but I could easily be mistaken.

4. How does one finally integrate this with the Dirac equation? My guess would be to do something like what I'm talking about in question 3, but simply remove the two conjugate wave functions from the left side of the equation to get something like this:

γ^μ(g_V - g_Aγ⁵)ψ Z_μ + (-iγ^μ∂_μ + m)ψ = 0

5. Instead of considering two particles, could we impersonate the entire neutral current with potential field? I remember from the example of the hydrogen atom, the proton is represented as a coulomb potential instead of a particle, and was wondering if something similar could be done here? I'm trying to model neutrino - matter interactions, so the momentum of the neutrinos (pretty high) might make this approach invalid...

I'm really sorry if this is too many questions. I really appreciate the opportunity to ask questions with someone who knows about physics :)

Thanks!