# Axial Transformation of a pion

#### 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.

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#### PF_Help_Bot

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"Axial Transformation of a pion"

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