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I wanted to evaluate the Wilson Coefficients for the process $c\rightarrow su\bar{d}$ since wherever I look they seem to be always given, for the exception of Schwart's book Quantum Field theory and the Standard Model where the calculation is done in a slightly different manner to what I'm used to.

For reference, I'm following G. Buchalla et al. review. In equation III.42 they give the result of the calculation for the one loop corrections to the cited amplitude. I want to reproduce this result and the results of III.45-III.46 which are the amplitudes for the effective theory.

To evaluate the relevant Feynman diagrams (in fig. 2 and 3 of the review) they use the Feynman gauge and take all the external quark lines massless and carrying the off-shell momentum $p$. What I understand then is that all external lines carry momentum $p$ which in turn means that the $W$ propagator should carry zero momentum. So, if this is what they are intending, for diagram in figure $2$ (a) (which I put here for reference)

$\hspace{100pt}$Feynman amplitude

I should get the following amplitude

$$\int\frac{\mathrm{d}^4k}{(2\pi)^4}\bar{u}_u^i(p)\left(-ig_sT^a_{ij}\gamma_\mu\right)\frac{-i}{\not p-\not k}\left(-ig\gamma_\alpha P_L\right)\frac{-i}{\not p-\not k}\left(-ig_sT^b_{jl}\gamma_\nu\right)u^l_d(p)\frac{-ig^{\mu\nu}\delta^{ab}}{k^2+i\epsilon}\frac{-ig^{\alpha\beta}}{-M_W^2+i\epsilon}\bar{u}_s(p)(-ig\gamma_\beta P_L)u_c(p)$$

I made the indices evident where needed, $P_L = \dfrac{1-\gamma_5}{2}$ is the left spinor projector and $T^a$ are $SU(3)$ generators.

Now the question: Is this amplitude correct? Is the fact that the $W$ carries zero momentum a sensible choice? I understand that is a choice that we can make since the $W$ is virtual and so off-shell, but it's not clear to me if this is what the authors imply in their text. Because it seems to me that this amplitude is nowhere different from the amplitude I would get in the effective theory.

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They probably want to do this in the low energy limit which means taking $p_{1},p_{2},k_{1},k_{2} \approx p \ll M_{w}$ where $p$ signals incoming momentum and $k$ outgoing so that they can get the low energy Wilson coefficient.

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  • $\begingroup$ And this makes sense, but in this limit what momentum should I use for the $W$ propagator? $\endgroup$ Mar 31, 2021 at 15:39

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