# 180deg Rotation in Isospin space

1. Feb 15, 2009

### malawi_glenn

The G-partiy of the pion is -1, which is rotation in isospin space around y-axis by angle $\pi$ followed by C-conjugation.

The rotation matrix around y-axis , with angle $\pi$, is: (e.g. Sakurai, Halzen ..)

$$\begin{pmatrix}0 & 0 & 1 \\ 0 & -1 & 0 \\ 1 & 0 & 0 \end{pmatrix}$$

Thus on $\pi^+$: T = 1, Tz = +1:
$$\begin{pmatrix}0 & 0 & 1 \\ 0 & -1 & 0 \\ 1 & 0 & 0 \end{pmatrix} \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} = \begin{pmatrix} 0\\ 0 \\ 1 \end{pmatrix} = \pi^-$$

But this is not $-\pi^-$, which is required (see e.g. http://arxiv.org/PS_cache/hep-ph/pdf/9903/9903256v1.pdf page 14)

I feel really stupid now, I don't get the required minus sign for pi- and pi+, only for pi0....

The rotation matrix I use is:
$$\begin{pmatrix} \tfrac{1}{2}(1+\cos \beta ) & - \tfrac{1}{\sqrt{2}}\sin \beta & \tfrac{1}{2}(1 - \cos \beta ) \\ -\tfrac{1}{\sqrt{2}}\sin \beta& \cos \beta & - \tfrac{1}{\sqrt{2}}\sin \beta \\ \tfrac{1}{2}(1-\cos \beta ) & \tfrac{1}{\sqrt{2}}\sin \beta & \tfrac{1}{2}(1 + \cos \beta ) \end{pmatrix}$$

2. Feb 19, 2009

### malawi_glenn

I found that the author has made an error, one should rotate around x-axis with that convention of C-parity.