Wick contraction and derivatives

[earth2]

Hey guys,

one quick question about Wick contractions and derivatives:

Suppose I want to write down all (non-vacuum) Wick contractions of two fields a, and one field A into a cubic QCD-like interaction term of the form

$\partial^\nu A^{\mu, a}(p_1) a_\mu^b(p_2) a_\nu^c(p_3) f^{abc}$

to get the corresponding Feynman rule (where each field carries also a color index).
How do I cope with the derivative? Normally I'd do a Fourier transform, but how do I know to which field the momentum belongs to? In other words if I Fourier trafo the derivative will it become a $p_1$, $p_2$, or $p_3$?

Cheers,
earth2

 

[AndyW]

sky

Hello earth2sky,

Thank you for your question about Wick contractions and derivatives. In order to determine the Feynman rule for this interaction term, we need to consider the momentum conservation at each vertex. In this case, the derivative term \partial^\nu A^{\mu, a}(p_1) will contribute a momentum factor p_1^\nu at the vertex where it is attached. This means that the momentum conservation at this vertex will be p_1 + p_2 + p_3 = 0.

In terms of Wick contractions, this means that the derivative will be contracted with one of the fields, and the remaining two fields will be contracted among themselves. This gives us a total of three possible Wick contractions, each corresponding to a different field being contracted with the derivative.

To determine which field the momentum p_1 belongs to, we can use the momentum conservation at each vertex. For example, if p_1 is contracted with A^{\mu, a}(p_1), then the momentum conservation at that vertex would be p_1 + p_2 = p_1 + p_2 + p_3 = 0. This means that p_1 must belong to the field A^{\mu, a}(p_1).

I hope this helps clarify how to determine the Feynman rule for this interaction term. If you have any further questions, please don't hesitate to ask.