Why the rare kaon decay K+ -> pi+ nu nubar req. penguins?

[rjseen]

Hello,

my question is, why does the decay of

K⁺ -> π⁺ ν ν-bar

require a loop to be allowed. See images below.*

-What is it that makes the decay forbidden in figure 1.4 and not forbidden in figure 1.5?
-What does being a first order weak decay and a second order weak decay mean?
-I suspect the decay is allowed in the standard model for the three modes of decay as seen in the figure 1.5. How are these modes determined?

Thanks in advance,
rjseen
*With acknowledgments to Bipul Bhuyan thesis:
https://www.bnl.gov/userscenter/Thesis/2004/BB-Thesis.pdf

 

[dukwon]

There are no flavour-changing neutral currents at tree level in the Standard Model.

The W boson changes quark flavour, but the Z boson cannot (neither can photons or gluons for that matter)

 

[rjseen]

Thanks for the quick answer.

I have come across that explanation in various theses, so that's apparently the real thing I have to understand in this.

So, I have understood that the K⁺ -> π⁺ ν ν-bar is called an FCNC, is the flavor part because of the strange anti-quark ending up as a down anti-quark? And the neutral part because the charge of the strange anti-quark equals the down anti-quark?

What is the tree level? Does it refer to first order weak decays? What is the meaning of first order and second order weak decays?rjseen

 

[mfb]

Tree level is without loops. It can be second order (count the number of vertices where W or Z participate) but I don't think that happens.
FCNC: right

 

[vanhees71]

One should also stress that the absence of flavor-changing neutral currents (here an anti-s quark changes to an anti-u quark) in the 2nd diagram in the posting #1, i.e., it's due to a Z-boson exchange and thus "neutral") is built in into quantum flavor dynamics by construction. It's one of the basic observable facts going into the model building. It's historically important, because it lead to the socalled GIM mechanism (named after Glashow Iliopolous, and Maiani), i.e., the prediction of the existence of a fourth quark flavor, the charm quark. It has been discovered in terms of the $J/\psi$ meson in November 1974 ("November Revolution"), which is a bound state of an anti-c and a c quark ("Charmonium").

https://en.wikipedia.org/wiki/GIM_mechanism
http://www.scholarpedia.org/article/Glashow-Iliopoulos-Maiani_mechanism
https://en.wikipedia.org/wiki/J/psi_meson

 

[rjseen]

Thank you for your replies! And especial thank you to Vanhees for those links. The one at scholarpedia is really well written for rudimentary understanding.

For future browsers I can try and conclude (or be corrected!):

-There are no Flavor-Changing Neutral Currents (FCNC) in the Tree level (diagram without loops (1)). It has been experimentally determined that flavour changing processes involve charged currents at tree level (2), i.e. stemming from the involvement of a W⁺-boson, which gives a net difference in charge of the in and out products. For example in the first diagram in figure 1.4, the strange anti-quark of charge -1/3 decays into an up anti-quark of charge 2/3. This observation and one in which the, outdated quantum number, strangeness changes by at most one unit in the tree level lead to the formulation of the GIM mechanism (2). Decays with changes in strangeness of two units, as well as decays containing FCNCs, are restricted to second order weak processes which decays by two weak vertices (3). As for the second diagram in figure 1.5, there are three vector bosons but the second W⁺ isn't counted because it connects back to the u,c,t anti-quarks (speculative).

(1): https://en.wikipedia.org/wiki/Feynman_diagram#Loop_order
(2): http://www.scholarpedia.org/article/Glashow-Iliopoulos-Maiani_mechanism#GIM_mechanism
(3)http://www.physnet.org/modules/pdf_modules/m281.pdf

 

[vanhees71]

In the scholarpedia is however a typo in Eq. (8), concerning the axial U(1) anomaly. It should read
$$\partial_{\mu} j_{5}^{\mu}=-\frac{e^2}{8 \pi^2} {^{\dagger} F}{}^{\mu \nu} F_{\mu \nu}=-\frac{e^2}{16 \pi^2} \epsilon^{\mu \nu \rho \sigma} F_{\mu \nu} F_{\rho \sigma}.$$
See my lecture notes on QFT (the chapter on anomalies in the gauge-theory chapter):

http://fias.uni-frankfurt.de/~hees/publ/lect.pdf

Modulo the sign, which may be due to different conventions for the $\epsilon$-tensor (a pain when comparing results in different textbooks and/or papers) there's a factor 2.