Penguin diagrams and CP violation

1. Apr 10, 2014

Einj

Hi everyone. I have been studying CP violation in kaon systems. I would like to know is someone can explain why, to leading order, strong penguin diagrams (i.e. involving a gluon) only contribute to the $K\to (\pi\pi)_{I=0}$ amplitude, while the isospin 2 amplitude is given by the electro-weak penguins (i.e. involving a photon or a Z boson).

Thank you very much

Edit: I can add more information to the question. Take the gluon penguin diagram. It contains a $(\bar ds)$ current which is an SU(3) octet and another current which is a flavor singlet. Therefore, it transform as an $(8_L,1_R)$ representation of $SU(3)_L\times SU(3)_R$. In many books I've found that this implies that it can only carry a change in isospin $\Delta I=1/2$. Can anyone explain why?

Last edited: Apr 10, 2014
2. Apr 10, 2014

andrien

Gluonic penguin operators do not differentiate between a final state uu- and final state dd-,so they only contribute to final states of zero isospin.Those electroweak contribute for both.And that ΔI=1/2 selection rule has it's origin in nonperturbative hadronic physics,and I don't think there is any good explanation for it.It is rather an experimental data to fit the small value of I=2 to I=0.

3. Apr 10, 2014

Einj

Thank you very much for the answer! Could you explain a little bit further why they don't differential between the two final states and why this means that they only contribute to I=0 final states? Thank you very much

4. Apr 11, 2014

andrien

The gluonic penguin operators contain factor like ∑(q-q)V+/-A,Which contribute equally to uu- and dd-,while the electroweak part brings in factors like ∑eq(q-q)V+/-A.Eletroweak penguins with different final states acquire different factors like eq=2/3 for q=u and eq=-1/3 for q=d.You can show now that gluonic ones only contributes to I=0 case.

5. Apr 11, 2014

Einj

I'm sorry to bother you but that's exactly what I can't show.

Edit: I found the following properties on Donoghue's book. Could you tell me if I'm doing correctly? The action of the isospin rising and lowering operators on the quark fields are:
$$I_+d=u,\;\; I_+\bar u=-\bar d,\;\;I_-u=d,\;\;I_-\bar d=-\bar u,$$
all the others are zero. Therefore, we have, in the case of QCD penguins:
$$I_+(\bar u u+\bar d d)=(I_+\bar u)u+\bar u(I_+u)+(I_+\bar d)d+\bar d(I_+d)=\bar d u-\bar d u=0$$
and
$$I_-(\bar u u+\bar d d)=(I_-\bar u)u+\bar u(I_-u)+(I_-\bar d)d+\bar d(I_-d)=-\bar u d+\bar u d=0.$$

Therefore, since both the lowering and rising operators give zero it must be an I=0 operator. Is that correct?

Last edited: Apr 11, 2014
6. Apr 11, 2014

andrien

You can use ladder operators,but see simply that quark and antiquark pair can have isospin of 0 or 1,and the 1 case is already ruled out as per symmetry requirement.The state symmetric in interchange of u and d is indeed isospin zero state.

7. Apr 11, 2014

Einj

Got it! Thanks