180deg Rotation in Isospin space

[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:
$$\begin{pmatrix} \tfrac{1}{2}(1+\cos \beta ) & - \tfrac{1}{\sqrt{2}}\sin \beta & \tfrac{1}{2}(1 - \cos \beta ) \\ -\tfrac{1}{\sqrt{2}}\sin \beta& \cos \beta & - \tfrac{1}{\sqrt{2}}\sin \beta \\ \tfrac{1}{2}(1-\cos \beta ) & \tfrac{1}{\sqrt{2}}\sin \beta & \tfrac{1}{2}(1 + \cos \beta ) \end{pmatrix}$$

 

[malawi_glenn]

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

 

[sir-pinski]

First of all, don't feel stupid! Understanding concepts in physics can be challenging, and it's important to ask questions and seek clarification.

In this case, the issue is with the definition of the rotation matrix. The matrix you are using is for a rotation around the y-axis in a 3-dimensional space, but isospin space is a 2-dimensional space. Therefore, the rotation matrix needs to be adjusted accordingly.

The correct rotation matrix for a rotation around the y-axis in isospin space is:
\begin{pmatrix} \cos \beta & \sin \beta \\ -\sin \beta & \cos \beta \end{pmatrix}

Using this matrix, we can see that for a rotation of \pi around the y-axis, the pion with isospin T=1 and Tz=+1 will become a pion with T=-1 and Tz=-1, as required.

I hope this helps clarify the issue and allows you to continue your understanding of isospin rotations. Keep asking questions and seeking understanding, it's the best way to learn!