- #1

- 6,735

- 2,456

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

[tex]\begin{pmatrix}0 & 0 & 1 \\ 0 & -1 & 0 \\ 1 & 0 & 0 \end{pmatrix} [/tex]

Thus on [itex] \pi^+ [/itex]: T = 1, Tz = +1:

[tex]\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^-[/tex]

But this is not [itex] -\pi^- [/itex], 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:

[tex]\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}[/tex]