# Homework Help: Axial Transformation of a pion

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1. Nov 10, 2018

### thinkLamp

1. The problem statement, all variables and given/known data
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#.

2. Relevant equations

3. 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.

2. Nov 15, 2018 at 6:00 PM

### PF_Help_Bot

Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.