Chiral Perturbation Theory : pi0 pi0 Z vertex?

[Hepth]

Not sure if anyone has any experience with chiral perturbation theory, but I'm trying to see what all of the vertices are for interactions with a single Z boson. I've looked at the lagrangian up to order p^4 so far, and it seems that the Z only interacts with charged pions/kaons.
I'm using the external field method. Is there a reason why these don't appear?
I imagine that I have the generic Z current:
$$\sum_{q = u,d,s} \bar{q} \left( c_L \gamma^{\mu} P_L + c_R \gamma^{\mu} P_R\right) q \cdot Z_{\mu}$$
where the constants are calculable, but it should be noted that due to weak isospin and charge, the down and strange currents are the same constant, with the up being different.

I imagine this then correlates to chiral perturbation theory as an external field matrix where these constants lie along the diagonal for the left and right handed external fields, which then go into the covariant derivative.

So just naively I would think that if I'm looking for say, K0 in, Z and K0 out:
$$\langle K0(p_i - p_Z) | \bar{s} \left( c_L \gamma^{\mu} P_L + c_R \gamma^{\mu} P_R\right) s | K0(p_i) \rangle \eta_{p_Z}^{*}$$

As the Vector parts of the current are -1 parity, the axial +1, and the Kaons -1, We have
(-1)(c_v (-1)) (-1) = -1 c_v
so a vector, such as the momenta of the kaons or the Z.

The axial parts
(-1) c_a (+1) (-1) = +1 c_a, we have no way to construct a +1 axial vector (polarization is already factorized out)

so only the vector component remains, which is a combination of the left and right components. (I guess I could have just started with that form).

So its momenta dependent, and we have two degrees of freedom, the pZ and the pK.

Now, due to indices we can have things of the form
$$p_Z \cdot \eta_{p_Z} \rightarrow 0\\ p_K \cdot \eta_{p_Z}$$

So I expect my vertex to be of the form of the second product.

Now when looking for where this contribution comes from in the chiral lagrangian, at O(p^2) we would expect the kinetic term:
$$L_{kin} = -\frac{f^2}{4} tr\left(D^{\mu} U D_{\mu} U^{\dagger}\right)$$
to generate these, and it does for the K+-, pi+- vertices, but if you work it out (and I think I'm doing it right) the neutral contributions all disappear.

Perhaps its due to the d/s being the same mass in this case,(though the pions/eta also disappear) and I need to break that symmetry to get this term, but then when I do the lagrangians at p^4 with mass included, I still don't get a term with it...

Any ideas or suggestions? http://arxiv.org/abs/hep-ph/9406283 , section 21( page 44) has both the p^2 and p^4 lagrangians.

I'm just trying to see which order I have to go to in order to get this vertex, as it MUST exist right? I mean, there's a definite cross section for e+ e- -> pi0 pi0, and I believe it occurs through the Z -> (ssbar)-> glu/photon off leg -> (ddbar) and the opposite diagram.

Any suggestions, even if only reading materials would be greatly appreciated!

-Hepth

ps Its possible I'm doing the matrix algebra incorrectly, but I'm fairly sure I'm doing it right. (95% sure).

 

[AndyW]

Hello Hepth,

Thank you for your post. I have experience with chiral perturbation theory and may be able to help answer your questions. First, let me clarify that chiral perturbation theory is a low-energy effective theory of QCD, which describes the interactions of pions and other low-energy hadrons. It is based on the concept of spontaneous chiral symmetry breaking, which is responsible for the low mass of the pions and their interactions with the Z boson.

Now, to address your specific questions. From what you have described, it seems that you are looking for the vertex that describes the interaction of the Z boson with a single pion or kaon. This vertex can indeed be found in the chiral lagrangian at order p^4, as you have correctly noted. However, there are a few things to keep in mind when looking for this vertex.

First, the lagrangian you have mentioned, L_{kin}, only describes the kinetic terms for the pions and kaons. In order to find the vertex you are looking for, you will need to include additional terms in the lagrangian that describe the interactions of the Z boson with the pions and kaons. These terms can be found in the paper you have referenced, in the sections on "Electroweak Interactions of Pseudoscalar Mesons" and "Electroweak Interactions of Vector Mesons".

Second, the Z boson only couples to charged pions and kaons, not neutral ones. This is due to the fact that the Z boson is a neutral particle and therefore does not interact with neutral particles like the pion or eta. In order to see the Z boson coupling to the pion or kaon, one must look at processes like e+ e- -> pi+ pi- or e+ e- -> K+ K-.

Finally, I would suggest checking your matrix algebra to make sure you are calculating the vertex correctly. It is possible that there may be a mistake in your calculation, which is why you are not seeing the vertex you are expecting. If you are still having trouble, I would recommend consulting with a colleague or professor who has experience with chiral perturbation theory.

I hope this helps. Good luck with your research!