This is my idea of the Feynman diagram of the $B^0$ to a $K^0$ decay:enter image description here

The photon is radiated off by one of the particles, and by $up$ quarks I just mean ($u$,$c$ and $t$) and their antiquarks.

how would I go on to calculate the amplitude for this decay, and how would I know if it is allowed in the Standard Model?

  • $\begingroup$ In general (particle) interactions have certain conservation laws as compatibility conditions. So if these do not hold the interaction is not compatible with the standard model. Some of them are: baryon number, lepton number, color, charge, momentum etc... i would have to look a bit more throroughly to given a better answer.. $\endgroup$
    – Nikos M.
    Oct 22 '14 at 10:26
  • $\begingroup$ Any two-W process is going to be weak-surpressed relative the dominate singly weak decays. Looking is the copy of the particle physics booklet on my desk I see some other double weak decay channels listed with branching ration limits in the $10^{-4}$--$10^{-6}$ range. The listed limit for decay to $K^0 l^+ l^-$ is $3 \times 10^{-7}$. $\endgroup$ Oct 22 '14 at 14:46
  • $\begingroup$ "suppressed". How do you quantify this? $\endgroup$
    – SuperCiocia
    Oct 22 '14 at 18:02
  • $\begingroup$ The branching ratio quantifies it. Singly weak decays are much more common than doubly weak decays (by ratios of 10000:1 or more). $\endgroup$ Oct 23 '14 at 1:22

Yes, it is allowed. This is a typical flavor changing neutral process. In the Standard Model, this kind of process is highly suppressed, because the neutral bosons ($\gamma$, $Z$) always couple to quarks of the same flavor at tree-level. So, a vertex connecting two quarks of different flavors with a neutral gauge boson only appears at higher orders in perturbation theory.

For the process $B\to K \gamma$, there are two types of diagrams that may contribute: (i) box diagrams and (ii) penguin diagrams. The first type of diagram is already drawn in you question and you just have to be careful if you are considering $K^0$ or $\bar{K}^0$ in the final state, since this diagram changes flavor two times ($\Delta F =2$).

The second type of diagram is given by

$\hskip{5cm}$ penguin diagram

where the second quark is just a spectator and the $W$ boson and the $t$ quark are running in the loop. So, in this case you change flavor only 1 time ($\Delta F =1$). There are other diagrams similar to this one that are also called "penguin diagrams", because you can also exchange $W$ and $t$, or draw a self-energy like diagram.

Now, how do we compute this kind of diagram? Usually, we write an effective Hamiltonian with all allowed dimension 6 operators multiplying Wilson Coefficients. Then, we compute theses diagrams in the full theory (with propagating $W$ bosons) and we make the matching of the effective theory with the full theory, extracting the values of the Wilson coefficients. You can find more details in the review of Buras, for example.


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