Calculating Eq.(23) from Eq.(15): Seeking Help!

[junhui.liao]

Hi, guys,

I'm reading a theoretical paper on dark matter, http://arxiv.org/pdf/hep-ph/0307185.pdf .

Above the Eq.(23) of section 3, it says Eq.(23) could be calculated easily from Eq.(15).
I've asked a theorist(senior PhD student), but no satisfied calculation obtained.

Can any people here please tell me how the Eq.(23) has been deprived ?
Any useful hit / tip is also welcome !

Thank !Jun

 

[Hepth]

Step 1: Derive the Feynman Rules from the Lagrangian

This should be fairly easy. Just set some in/out states and reduce it. The first term should give a vertex where 1 nucleon and 1 DM go in, and they go back out (like a big X), and the coefficient is $$ -i 4 f_N$$, then the second term will give something similar but with the spin structure still there.

Step 2: Find the Amplitude of the sum of diagrams for what process you want

So write down the diagrams, there are two, one from each of those terms.
Keep intact the spinors, making their indices explicit $\vec{S} \rightarrow S^a$
Then square it to get the amplitude squared. $|M|^2 = M^{\dagger} M$ (amplitude times the hermitian conjugate of amplitude). These are spin-dependent amplitudes still.

Then you have to choose. For "SI = Spin-Independent" you sum over the final spins, average over the initial. There should be some identities for weyl spinors for this, and it reduces.

Also, look into if there are low-momentum versions (q^2-> 0) for the explicit spinors. Not sure if that matters.

For the SD one I'm not sure right away, there should be some reduction allowed from the spin states, and then you can relate the dot products to the total angular momentums (thats why there's J's).

Then go find the 2-> 2 cross section formula, either the PDG kinematics pdf or peskin and schroder should have it. You then just have to integrate over the kinematic variables. I think in the low-moment approximation maybe they'll have done a taylor expansion before integration.

Hope that's a start?

 

[junhui.liao]

Hi, Hepth,

Thanks for your comments !
Sorry if I forgot to tell you I'm an experimentalist.

What you mentioned is more detailed than the PhD student told me before.
However, I have to say, I still don't know how to finish it quantitatively(derive it step by step).

If possible, could you please give me more detailed explanation ?
For simple, let's forget the Spin-depedent part, and only focus on the spin-independent, saying, the first term of Eq.(15).
I have Peskin on my hands, but it's too much work for me to find the corresponding chapter/section.

Thanks again !Jun