# How does the introduction of the charm quark suppress FCNC?

I did some reading on the GIM mechanism today, and simply fail to understand how it works. I understand how the CKM-matrix can be used to do the basic calculation of the probability of, say, observing an up quark after a strange quark's decay over the weak interaction. However, I don't understand how the CKM matrix is applied to these Feynman diagrams:

I see that the vertices for the weak interactions of the quarks are labeled with the corresponding matrix elements, but how do I take these two Feynman diagrams and infer that they cancel each other out?

• Consider to spell out acronyms. – Qmechanic Aug 20 '19 at 8:40

To the extent the masses of the two quarks in the internal lines differ, the effect of them on the respective propagators differ, and so the respective results of the loops differ. In fact, good SM books compute the nonvanishing, but vastly suppressed amplitude. It is a function of $$m_c/m_u$$ which goes to 0 as that ratio goes to 1. Something like $$\propto g^4 \frac{m_c^2}{M_W^2} ( 1-m_u^2/m_c^2)$$.
So you might sensibly object that the term in the parenthesis is much closer to 1 than it is to 0. But, look at the factor multiplying it, $$\alpha^2 m_c^2/M_W^2$$, and how small it is: do this. (Still, if that were a part of your puzzlement, perversely, the introduction of c actually increases the $$\Delta S=1$$ rate instead of suppressing it! Historically, the rate was used to bound the mass of the then hypothetical c from above!)