How Does a Pion Transform Under an Axial Transformation to Order π²?

[thinkLamp]

Homework Statement

I'm considering a non-linear chiral theory where the Lagrangian is in terms of the field #\Sigma = e^{\frac{2i\pi}{f}}# where #\pi# is my pion matrix containing pion, kaon, and #\eta#. I need to calculate the transformation of #\pi# up to order #\pi^2# under an axial transformation where #R=L^\dagger#. We're given that under #SU(3)_R \times SU(3)_R# transformations, #\Sigma# transforms as #\Sigma \to L \Sigma R^\dagger#.

Homework Equations

The Attempt at a Solution

$$\Sigma \to L\Sigma R^\dagger$$
$$= L \left( 1 + \frac{2i\pi}{f} + \frac{4i^2}{2 f^2} \pi^2 + \ldots \right) R^\dagger$$.
Now use $R = L^\dagger$. So,
$$= R^\dagger \left( 1 + \frac{2i\pi}{f} + \frac{4i^2}{2 f^2} \pi^2 + \ldots \right) R^\dagger\\
= R^\dagger R^\dagger + \frac{2i}{f} R^\dagger \pi R^\dagger + \frac{4i^2}{2 f^2} R^\dagger \pi R^\dagger R \pi R^\dagger + \ldots $$

Not sure where to go from here.

 

[mark726]

It looks like we need to use the fact that $R$ and $L$ are both elements of $SU(3)$, so $R^\dagger R = 1$ and $R^\dagger \pi R = \pi$. But I'm not sure how to incorporate that into the transformation of $\pi$. Any help would be greatly appreciated!
Thank you for sharing your question with us. It seems like you are on the right track with your attempt at the solution. To continue, you can use the fact that for elements of $SU(3)$, $R^\dagger R = 1$ and $R^\dagger \pi R = \pi$ to simplify the expression. This will allow you to expand the expression up to order $\pi^2$ and see how $\pi$ transforms under the axial transformation. Additionally, you can also use the commutation relation between $L$ and $R$ to further simplify the expression. I hope this helps and good luck with your research!